Research Article Parametrically Excited...
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Research Article Parametrically Excited Oscillations of Second-Order Functional Differential Equations and Application to Duffing Equations with Time Delay Feedback Mervan PašiT Department of Applied Mathematics, Faculty of Electrical Engineering and Computing, University of Zagreb, 10000 Zagreb, Croatia Correspondence should be addressed to Mervan Paˇ si´ c; [email protected] Received 8 December 2013; Accepted 12 February 2014; Published 16 April 2014 Academic Editor: Zhengqiu Zhang Copyright © 2014 Mervan Paˇ si´ c. is is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. We study oscillatory behaviour of a large class of second-order functional differential equations with three freedom real nonnegative parameters. According to a new oscillation criterion, we show that if at least one of these three parameters is large enough, then the main equation must be oscillatory. As an application, we study a class of Duffing type quasilinear equations with nonlinear time delayed feedback and their oscillations excited by the control gain parameter or amplitude of forcing term. Finally, some open questions and comments are given for the purpose of further study on this topic. 1. Introduction Let , , be three nonnegative parameters and let (V), (, , V), (, ), (, ), and () be continuous functions in all their variables satisfying some conditions determined in Section 2. We consider the following large class of second- order functional differential equations: ( () ( ())) + (, () , ()) + (, ( ())) + (, ( ())) = () , ≥ 0 , (1) where as usual the functional terms () and () satisfy () ≤ , lim →∞ () = ∞, and () ≥ . ree cases are studied simultaneously: delay (() ̸ ≡, () = ), advanced (() = , () ̸ ≡), and delay-advanced (() ̸ ≡, () ̸ ≡), and both () and () are increasing functions. A continuous function () is called nonoscillatory if there is a point ≥ 0 such that () ̸ =0 for all ≥ . Otherwise, () is an oscillatory function. A function = (), ∈ 2 (( 0 , ∞), R), is called the (extendable) solution of (1) if it satisfies equality in (1) for all > 0 . Equation (1) is oscillatory if all its solutions are oscillatory. In the paper, we investigate conditions on (, ), (, ), and () under which (1) is oscillatory provided at least one of parameters , , and is large enough. It is shortly called the parametrically excited oscillations of (1). In Section 4 we discuss some known oscillation criteria published in [1–9] which allow the parametrically excited oscillations but only in the form of examples. On various problems concerning the functional differential equations we refer the reader to [10–15] and the references therein. In Section 2 we state a fundamental lemma proposing a new oscillation criterion that plays a crucial role in the formulation of the main results illustrated on some suitable chosen examples. In Section 3 we consider an application of the main results to the Duffing type quasilinear equations with time delayed feedback, taking into account the known results in applied sciences concerning such kind of nonlinear oscillators without time delay, see [16–37], and with time delay, see [38–48] and the references therein. In Section 5 we present some open questions and comments for further study that can follow our main results. And in Section 6, we describe the method for proving the main results of the paper. Hindawi Publishing Corporation Discrete Dynamics in Nature and Society Volume 2014, Article ID 875020, 17 pages http://dx.doi.org/10.1155/2014/875020
Transcript of Research Article Parametrically Excited...
Research ArticleParametrically Excited Oscillations of Second-OrderFunctional Differential Equations and Application toDuffing Equations with Time Delay Feedback
Mervan PašiT
Department of Applied Mathematics Faculty of Electrical Engineering and ComputingUniversity of Zagreb 10000 Zagreb Croatia
Correspondence should be addressed to Mervan Pasic mervanpasicgmailcom
Received 8 December 2013 Accepted 12 February 2014 Published 16 April 2014
Academic Editor Zhengqiu Zhang
Copyright copy 2014 Mervan PasicThis is an open access article distributed under the Creative Commons Attribution License whichpermits unrestricted use distribution and reproduction in any medium provided the original work is properly cited
We study oscillatory behaviour of a large class of second-order functional differential equationswith three freedom real nonnegativeparameters According to a new oscillation criterion we show that if at least one of these three parameters is large enough thenthe main equation must be oscillatory As an application we study a class of Duffing type quasilinear equations with nonlineartime delayed feedback and their oscillations excited by the control gain parameter or amplitude of forcing term Finally some openquestions and comments are given for the purpose of further study on this topic
1 Introduction
Let 120582 120583 120588 be three nonnegative parameters and let 119860(V)119861(119905 119906 V) 119865(119905 119906) 119866(119905 119906) and 119890(119905) be continuous functions inall their variables satisfying some conditions determined inSection 2 We consider the following large class of second-order functional differential equations
(119903 (119905) 119860 (1199091015840
(119905)))1015840
+ 119861 (119905 119909 (119905) 1199091015840
(119905)) + 120582119865 (119905 119909 (120591 (119905)))
+ 120583119866 (119905 119909 (120590 (119905))) = 120588119890 (119905) 119905 ge 1199050
(1)
where as usual the functional terms 120591(119905) and120590(119905) satisfy 120591(119905) le119905 lim119905rarrinfin
120591(119905) = infin and 120590(119905) ge 119905 Three cases are studiedsimultaneously delay (120591(119905) equiv 119905120590(119905) = 119905) advanced (120591(119905) = 119905120590(119905) equiv 119905) and delay-advanced (120591(119905) equiv 119905 120590(119905) equiv 119905) and both120591(119905) and 120590(119905) are increasing functions
A continuous function 119909(119905) is called nonoscillatory ifthere is a point 119879 ge 119905
0such that 119909(119905) = 0 for all 119905 ge
119879 Otherwise 119909(119905) is an oscillatory function A function119909 = 119909(119905) 119909 isin 119862
2((1199050infin)R) is called the (extendable)
solution of (1) if it satisfies equality in (1) for all 119905 gt 1199050
Equation (1) is oscillatory if all its solutions are oscillatory
In the paper we investigate conditions on 119865(119905 119906) 119866(119905 119906)and 119890(119905) under which (1) is oscillatory provided at least oneof parameters 120582 120583 and 120588 is large enough It is shortly calledthe parametrically excited oscillations of (1) In Section 4 wediscuss some known oscillation criteria published in [1ndash9]which allow the parametrically excited oscillations but onlyin the form of examples On various problems concerning thefunctional differential equationswe refer the reader to [10ndash15]and the references therein
In Section 2 we state a fundamental lemma proposinga new oscillation criterion that plays a crucial role in theformulation of the main results illustrated on some suitablechosen examples In Section 3 we consider an application ofthe main results to the Duffing type quasilinear equationswith time delayed feedback taking into account the knownresults in applied sciences concerning such kind of nonlinearoscillators without time delay see [16ndash37] and with timedelay see [38ndash48] and the references therein In Section 5we present some open questions and comments for furtherstudy that can follow our main results And in Section 6 wedescribe themethod for proving themain results of the paper
Hindawi Publishing CorporationDiscrete Dynamics in Nature and SocietyVolume 2014 Article ID 875020 17 pageshttpdxdoiorg1011552014875020
2 Discrete Dynamics in Nature and Society
2 Main Assumptions and Results
Let R+= (0infin) For the functions 119903 = 119903(119905) and 119860 = 119860(V)
both appearing in the second-order differential operator of(1) we suppose the following
0 lt 119903 (119905) le 1199030on [1199050infin) 119903 isin 119862
1([1199050infin) R
+) (2)
and 119860(V) is odd increasing and
|119906|minus119901minus1
119860 (V) V ge 120572 (|119860 (V)| |119906|minus119901)
for some 119901 gt 0 and all 119906 = 0 V isin R
120572 = 120572 (119904) 120572 isin 1198621
([0infin) [0infin))
1
1 + 120572isin 1198711(R+R+) 120587
lowast= int
infin
0
2119889119904
1 + 120572 (119904)
120572 (119888119904) ge 119888120574120572 (119904) for some 120574 ge 1 and all 119888 gt 0 119904 ge 0
(3)
For instance it is simple to check that for 120572(119904) = 119904120574
hypothesis (3) is fulfilled in the next three most importantcases of 119860(V) the linear operator 119860(V) = V if 119901 = 1 and120574 = 2 the quasilinear 119901-Laplacian operator 119860(V) = |V|119901minus1V if119901 ge 1 and 120574 = (119901 + 1)119901 and the quasilinear mean curvatureoperator 119860(V) = V(1 + V2)minus12 if 119901 = 1 and 120574 = 2 Somedetails about the number 120587
lowastwhich depends on the function
120572(119904) given in (3) are presented in Section 6The damped term 119861 = 119861(119905 119906 V) satisfies the strong
condition
119861 (119905 119906 V) sgn (119906) ge 0 forall119905 ge 1199050 119906 = 0 V isin R (4)
We hope that (4) can be relaxed with some weaker conditionwhich is commented as an open problem in Section 5 below
In the following fundamental lemma which plays acrucial role in the proof of the main results we are workingwith such solutions 119909(119905) of (1) that satisfies the inequality
[120582119865 (119905 119909 (120591 (119905))) + 120583119866 (119905 119909 (120590 (119905))) minus 120588119890 (119905)]
times (|119909 (119905) |119901minus1
119909 (119905))minus1
ge 119896 (120582 120583 120588)119876 (119905)
(5)
for all 119905 isin 119869 and some interval 119869 where the functions 119896(120582 120583 120588)and 119876(119905) do not depend on 119909(119905) but only on 119865 119866 119890 and theyare determined in the process below The functions 119896(120582 120583 120588)and 119876(119905) present the key point in the parametrically excitedoscillations
Lemma 1 Let assumptions (2) (3) and (4) hold Let (119886 119887) and(119888 119889) be two disjoint open intervals such that 120590(119887) le 120591(119888) Letthe functions 119896 = 119896(120582 120583 120588) 119896 isin 119862([0infin)
3R+) and 119876 =
119876(119905) 119876 isin 119862( 119869 [0infin)) 119876(119905) equiv 0 on 119869 be such that
1199011120574
1199031minus1120574
0
[119896 (120582 120583 120588)]1minus1120574
120587lowast
int119869
119876 (119905) 119889119905 ge (max119905isin 119869
119876 (119905))
1120574
(6)
for both 119869 = (119886 119887) and 119869 = (119888 119889) for all 120582 ge 1205820 120583 ge 120583
0 and
120588 ge 1205880 and for some (120582
0 1205830 1205880) isin R3+ where 119903
0 119901 120574 and 120587
lowast
are constants defined respectively in (2) and (3) Let 119909(119905) be asolution of (1) satisfying the next two statements
119894119891 119909 (119905) gt 0 119909 (120591 (119905)) gt 0 119909 (120590 (119905)) gt 0
119900119899 (120591 (119886) 120590 (119887))
119905ℎ119890119899 119909 (119905) 119904119886119905119894119904119891119894119890119904 (5) 119900119899 (119886 119887)
(7)
119894119891 119909 (119905) lt 0 119909 (120591 (119905)) lt 0 119909 (120590 (119905)) lt 0
119900119899 (120591 (119888) 120590 (119889))
119905ℎ119890119899 (119905) 119904119886119905119894119904119891119894119890119904 (5) 119900119899 (119888 119889)
(8)
Then 119909(119905) has at least one zero point in (120591(120591(119886)) 120590(120590(119889)))
This lemma simultaneously holds for all three types offunctional arguments delay advanced and delay-advancedIt will be proved in Section 6 In Corollaries 9 10 and 11below we give some simple conditions on two functions119891(119905) 119892(119905) and two numbers 119901
1 1199012such that all solutions
of (1) with the functions 119865(119905 119906) = 119891(119905 119906)|119906|1199011 sgn(119906) and
119866(119905 119906) = 119892(119905 119906)|119906|1199012 sgn(119906) satisfy required statements
(7) and (8) with respect to some intervals (119886119899 119887119899) and the
explicitly given functions 119896(120582 120583 120588) and 119876119899(119905) satisfying (6)
where 119890(119905) satisfies a basic assumptionIn what follows (119886
119899 119887119899) 119899 isin N denotes a sequence of
disjoint open intervals such that 1199050
le 120591(119886119899) lt 120590(119887
119899) le
120591(119886119899+1
) lt 120590(119887119899+1
) le sdot sdot sdot and 119886119899rarr infin as 119899 rarr infin Now
we present several variations of Lemma 1 in which essentialinequality (6) is relaxed with some asymptotic assumptionsthat are simpler to be verified in several applications
Lemma 2 Let assumptions (2) (3) and (4) hold Let thecontinuous function 119896 = 119896(120582 120583 120588) gt 0 and the sequence offunctions 119876
119899= 119876119899(119905) 119876
119899isin 119862([119886
119899 119887119899] [0infin)) 119876
119899(119905) equiv 0
on (119886119899 119887119899) satisfy the next two inequalities there are constants
1198880gt 0 119899
0isin N and (120582
0 1205830 1205880) isin R3+such that
119896 (120582 120583 120588) ge 1199030119901minus1(120574minus1)
(2120587lowast
1198880
)
120574(120574minus1)
forall120582 ge 1205820 120583 ge 120583
0 120588 ge 120588
0
(9)
where the numbers 1199030 119901 120574 and 120587
lowastare respectively from (2)
and (3) and
lim119899rarrinfin
(1
(max119905isin[119886119899119887119899]119876119899(119905))1120574
int
119887119899
119886119899
119876119899(119905) 119889119905) ge 119888
0gt 0
(10)
If 119909(119905) is a solution of (1) that satisfies (7) and (8) with 119886 =
1198862119899minus1
119887 = 1198872119899minus1
119888 = 1198862119899 119889 = 119887
2119899 and 119876(119905) = 119876
119899(119905) then 119909(119905)
has at least one zero point in (120591(120591(1198862119899minus1
)) 120590(120590(1198872119899))) forall119899 ge 119899
0
In the following slightly simpler version of Lemma 2inequality (9) is replaced with an asymptotic condition andat the same time the limit in (10) is relaxed with the limitinferiorThus conditions (9) and (10) are replaced with morepractical ones
Discrete Dynamics in Nature and Society 3
Lemma 3 Let assumptions (2) (3) and (4) hold Let thecontinuous function 119896 = 119896(120582 120583 120588) gt 0 and the sequence offunctions 119876
119899= 119876119899(119905) 119876119899isin 119862([119886
119899 119887119899] [0infin)) 119876
119899(119905) equiv 0 on
(119886119899 119887119899) satisfy respectively
119896 (120582 120583 120588) 997888rarr infin 119886119904 120582 997888rarr infin
119900119903 120583 997888rarr infin 119900119903 120588 997888rarr infin
(11)
lim inf119899rarrinfin
(1
(max119905isin[119886119899119887119899]119876119899(119905))1120574
int
119887119899
119886119899
119876119899(119905) 119889119905) gt 0 (12)
If 119909(119905) is a solution of (1) that satisfies (7) and (8) with 119886 =
1198862119899minus1
119887 = 1198872119899minus1
119888 = 1198862119899 119889 = 119887
2119899 and 119876(119905) = 119876
119899(119905) then 119909(119905)
has at least one zero point in (120591(120591(1198862119899minus1
)) 120590(120590(1198872119899))) forall119899 ge 119899
0
and for some 1198990isin N
In some concrete cases we use the next version ofLemmas 2 and 3 where condition (10) or (12) is replaced withappropriate one that appears in (1) with periodic coefficients
Lemma 4 Let assumptions (2) (3) and (4) hold Let thecontinuous function 119896 = 119896(120582 120583 120588) gt 0 and the sequence offunctions 119876
119899= 119876119899(119905) 119876119899isin 119862([119886
119899 119887119899] [0infin)) 119876
119899(119905) equiv 0 on
(119886119899 119887119899) satisfy respectively (11) and for some 119862
0 1198881isin R
0 lt max119905isin[119886119899 119887119899]
119876119899(119905) le 119862
0 int
119887119899
119886119899
119876119899(119905) 119889119905 ge 119888
1gt 0
forall119899 ge 1198990
(13)
and some 1198990isin N If 119909(119905) is a solution of (1) that satisfies
(7) and (8) with 119886 = 1198862119899minus1
119887 = 1198872119899minus1
119888 = 1198862119899 119889 = 119887
2119899
and 119876(119905) = 119876119899(119905) then 119909(119905) has at least one zero point in
(120591(120591(1198862119899minus1
)) 120590(120590(1198872119899))) forall119899 ge 119899
0
Next we suppose that the coefficient 119903(119905) additionallysatisfies
119903 (119904) le 119903 (119905) in delay
119903 (119904) ge 119903 (119905) in advanced case forall119904 le 119905
119903 (119905) is a constant in delay-advanced case
(14)
and the forcing term 119890(119905) satisfies
119890 (119905) le 0 on [120591 (1198862119899minus1
) 120590 (1198872119899minus1
)]
119890 (119905) ge 0 on [120591 (1198862119899) 120590 (119887
2119899)] 119899 isin N
(15)
We remark that 119890(119905) remains arbitrary function outside theset⋃119899(120591(119886119899) 120590(119887119899))
The first result of the paper deals with delay equation (1)
Theorem 5 Let assumptions (2) (3) (4) (14) and (15) hold120591(119905) = 119905 minus 120591 120591 ge 0 120590(119905) equiv 119905 119866(119905 119906) equiv 0 and let 119865(119905 119906) satisfy
119865 (119905 119906) sgn (119906) ge 119891 (119905) |119906|119902
forall119906 = 0 119905 ge 1199050
119891 (119905) ge 0 119891 (119905) equiv 0 119900119899 [120591 (119886119899) 119887119899]
(16)
where 119902 ge 119901 number 119901 is from (3) sequence (119886119899 119887119899) is from
(15) and 119891 isin 119862([1199050infin)R) is a periodic function with period
119879lowastgt 0 such that
[1198862119899minus1
+ 119879lowast 1198872119899minus1
+ 119879lowast] sube [119886
2119899+1 1198872119899+1
]
[1198862119899+ 119879lowast 1198872119899+ 119879lowast] sube [119886
2119899+2 1198872119899+2
] 119899 isin N(17)
Then (1) is oscillatory in the next two cases 119902 = 119901 andparameter 120582 is large enough 119902 gt 119901 120582 gt 0 120588 gt 0 and at leastone of parameters 120582 and 120588 is large enough
The proof ofTheorem 5 is presented in Section 6 and it isbased on Lemma 4 where
119896 (120582 120583 120588) = 120582119901119902
1205881minus(119901119902)
119902 ge 119901 (18)
119876119899(119905)
=
119891 (119905) (119905 minus 119886119899
119905 minus 119886119899+ 120591
)
119901
if 119902 = 119901
119902
119901(
119901
119902 minus 119901)
(119902119901)minus1
|119890 (119905)|1minus(119901119902)
(119891 (119905))119901119902
times(119905 minus 119886119899
119905 minus 119886119899+ 120591
)
119901
if 119902 gt 119901
(19)
The second result deals with advanced equation (1)
Theorem 6 Let assumptions (2) (3) (4) (14) and (15) hold120591(119905) equiv 119905 120590(119905) = 119905 + 120590 120590 ge 0 119865(119905 119906) equiv 0 and let 119866(119905 119906) satisfy
119866 (119905 119906) sgn (119906) ge 119892 (119905) |119906|119902
forall119906 = 0 119905 ge 1199050
119892 (119905) ge 0 119892 (119905) equiv 0 119900119899 [119886119899 120590 (119887119899)]
(20)
where 119902 ge 119901 number 119901 is from (3) sequence (119886119899 119887119899) is from
(15) and 119892 isin 119862([1199050infin)R) is a periodic function with period
119879lowastgt 0 such that (17) is fulfilled Then (1) is oscillatory in the
next two cases 119902 = 119901 and parameter 120583 is large enough 119902 gt 119901120583 gt 0 120588 gt 0 and at least one of parameters 120583 and 120588 is largeenough
The proof of Theorem 6 is based on Lemma 4 (seeSection 6) where
119896 (120582 120583 120588) = 120583119901119902
1205881minus(119901119902)
119902 ge 119901 (21)
119876119899(119905)
=
119892 (119905) (119887119899minus 119905
119887119899minus 119905 + 120590
)
119901
if 119902 = 119901
119902
119901(
119901
119902 minus 119901)
(119902119901)minus1
|119890 (119905)|1minus(119901119902)
(119892 (119905))119901119902
times(119887119899minus 119905
119887119899minus 119905 + 120590
)
119901
if 119902 gt 119901
(22)
The third result deals with delay-advanced equation (1)
4 Discrete Dynamics in Nature and Society
Theorem 7 Let assumptions (2) (3) (4) (14) and (15) hold120591(119905) = 119905 minus 120591 120591 ge 0 120590(119905) = 119905 + 120590 120590 ge 0 and 119865(119905 119906) and 119866(119905 119906)satisfy
119865 (119905 119906) sgn (119906) ge 119891 (119905) |119906|1199011 119866 (119905 119906) sgn (119906) ge 119892 (119905) |119906|
1199012
forall119906 = 0 119905 ge 1199050
119891 (119905) ge 0 119891 (119905) equiv 0 119892 (119905) ge 0
119892 (119905) equiv 0 119900119899 [120591 (119886119899) 120590 (119887
119899)]
(23)
where additionally 119891 119892 119890 isin 119862([1199050infin)R) are three periodic
functions having a common period 119879lowast
gt 0 such that (17)is fulfilled where 119890(119905) is the forcing term in (1) Then (1) isoscillatory provided one of the next two cases is fulfilled wherethe number 119901 is from (3) (1) (in superlinear delay-advancedcase) 119901
1gt 119901 119901
2gt 119901 120588 gt 0 and either parameter 120588 is
large enough or at least one of 120582 and 120583 is large enough (2) (insupersublinear delay-advanced case) 119901
1gt 119901 gt 119901
2gt 0 120582 gt 0
120583 gt 0 120588 gt 0 and at least one of parameters 120582 120583 and 120588 is largeenough
The proof of Theorem 7 is based on Lemma 4 (seeSection 6) where
119896 (120582 120583 120588) =
min 12058211990111990111205881minus(1199011199011) 12058311990111990121205881minus(1199011199012) superlinear case
120582120578112058312057811205881205780
supersublinear case
(24)
119876119899(119905)
=
1199011
119901(
119901
2 (1199011minus 119901)
)
(1199011119901)minus1
times |119890 (119905)|1minus(119901119901
1)(119891 (119905))
1199011199011
120591119899(119905)
+1199012
119901(
119901
2 (1199012minus 119901)
)
(1199012119901)minus1
|119890 (119905)|1minus(119901119901
2)
times (119892 (119905))1199011199012
120590119899(119905)
superlinear case|119890 (119905)|1205780(119891 (119905))
1205781
(119892 (119905))1205782
(120591119899(119905))1205781(1199011119901)
times (120590119899(119905))1205782(1199012119901)
2
prod
119894=0
120578minus120578119894
119894
supersublinear case
(25)
where we denote
120591119899(119905) = (
119905 minus 119886119899
119905 minus 119886119899+ 120591
)
119901
120590119899(119905) = (
119887119899minus 119905
119887119899minus 119905 + 120590
)
119901
(26)
Here the numbers 1205780 1205781 1205782isin (0 1) are chosen such that 120578
0+
1205781+ 1205782= 1 and 119901
11205781+ 11990121205782= 119901 Let us mention that if
1199011= 52 119901 = 1 and 119901
2= 12 and 120578
0= 1205781= 1205782= 13 then
(1199011 1199012) and (120578
0 1205781 1205782) satisfy previous two equalities About
the existence of such (119873+1)-tuple (1205780 1205781 120578
119873) in a general
case we refer to [49]
Remark 8 A difference between assumptions ofTheorems 56 and 7 is that 119890(119905) in Theorems 5 and 6 is not necessarilyperiodic or bounded function as it is supposed inTheorem 7
Now we study an important class of second-order func-tional differential equations as a particular case of (1)
(119903 (119905) 119860 (1199091015840
(119905)))1015840
+ 119861 (119905 119909 (119905) 1199091015840
(119905))
+ 120582119891 (119905) |119909 (120591 (119905))|1199011 sgn (119909 (120591 (119905)))
+ 120583119892 (119905) |119909 (120590 (119905))|1199012 sgn (119909 (120590 (119905))) = 120588119890 (119905) 119905 ge 119905
0
(27)
where 120591(119905) = 119905 minus 120591 120590(119905) = 119905 + 120590 and 120582 120583 120588 120591 120590 ge 0 and1199011 1199012gt 0 Using previous theorems we are able to state
the main consequences showing the parametrically excitedoscillations in (27)
Corollary 9 (delay equation) Let assumptions (2) (3) (4)(14) and (15) hold Let 119892(119905) equiv 0 119891(119905) ge 0 and 119891(119905) equiv 0 on[119886119899minus120591 119887119899] where119891 isin 119862([119905
0infin)R) is a periodic function with
period 119879lowast satisfying (16) Equation (27) is oscillatory in the
following two cases 1199011= 119901 and parameter 120582 is large enough
1199011gt 119901 120582 gt 0 120588 gt 0 and at least one of parameters 120582 and 120588 is
large enough
Corollary 10 (advanced equation) Let assumptions (2) (3)(4) (14) and (15) hold Let 119891(119905) equiv 0 119892(119905) ge 0 and 119892(119905) equiv 0
on [119886119899 119887119899+ 120590] where 119892 isin 119862([119905
0infin)R) is a periodic function
with period 119879lowast satisfying (17) Then (27) is oscillatory in the
following two cases 1199012= 119901 and parameter 120583 is large enough
1199012gt 119901 120583 gt 0 120588 gt 0 and at least one of parameters 120583 and 120588 is
large enough
Corollary 11 (delay-advanced equation) Let assumptions (2)(3) (4) (14) and (15) hold and 119891(119905) and 119892(119905) satisfy
119891 (119905) ge 0 119891 (119905) equiv 0
119892 (119905) ge 0 119892 (119905) equiv 0
119900119899 [119886119899minus 120591 119887119899+ 120590]
(28)
where additionally 119891 119892 119890 isin 119862([1199050infin)R) are three periodic
functions having a common period 119879lowast
gt 0 such that (17)is fulfilled where 119890(119905) is the forcing term in (1) Then (27) isoscillatory in the next two cases where the number 119901 is from(3) (1) (superlinear case) 119901
1gt 119901 119901
2gt 119901 120588 gt 0 and either
parameter 120588 is large enough or at least one of 120582 and 120583 is largeenough (2) (supersublinear case) 119901
1gt 119901 gt 119901
2gt 0 120582 gt 0
120583 gt 0 120588 gt 0 and at least one of parameters 120582 120583 and 120588 is largeenough
According to previous corollaries we can derive thefollowing examples
Example 12 (delay case) Let 119901 ge 1 120576 ge 0 and 119898 isin N
be fixed and 120591119898
isin R 0 le 120591119898
lt 120587(4119898) With the help of
Discrete Dynamics in Nature and Society 5
Corollary 9 the following two different classes of quasilineardelay differential equations
(100381610038161003816100381610038161199091015840
(119905)10038161003816100381610038161003816
119901minus1
1199091015840
(119905))1015840
+ 120582 sin (119898119905) 1003816100381610038161003816119909 (119905 minus 120591119898)1003816100381610038161003816119901+120576
times sgn (119909 (119905 minus 120591119898)) = minus120588 cos (2119898119905)
(1199091015840(119905)
radic1 + 11990910158402
(119905)
)
1015840
+ 120582 cos (119898119905) 1003816100381610038161003816119909 (119905 minus 120591119898)10038161003816100381610038161+120576
times sgn (119909 (119905 minus 120591119898)) = minus120588 cos (2119898119905)
(29)
are oscillatory provided at least one of 120582 gt 0 and 120588 gt 0 is largeenough (the case 120588 = 0 is possible if 120576 = 0) It is because forall 119899 isin N we have
minus cos (2119898119905)
le 0 on [2119899120587
1198982119899120587
119898+
120587
4119898]
ge 0 on [2119899120587
119898+
120587
41198982119899120587
119898+
120587
2119898]
sin (119898119905) ge 0 cos (119898119905) ge 0 on [2119899120587
1198982119899120587
119898+
120587
2119898]
[2119899120587
119898+ 120591119898+ 119879lowast2119899120587
119898+
120587
4119898+ 119879lowast]
= [(2119899 + 2) 120587
119898+ 120591119898(2119899 + 2) 120587
119898+
120587
4119898]
[2119899120587
119898+
120587
4119898+ 120591119898+ 119879lowast2119899120587
119898+
120587
2119898+ 119879lowast]
= [(2119899 + 2) 120587
119898+
120587
4119898+ 120591119898(2119899 + 2) 120587
119898+
120587
2119898]
(30)
where 119879lowast= 2120587119898 is the common period of the functions
sin(119898119905) and cos(119898119905) Thus in order to apply Corollary 9 wecan choose 119886
2119899minus1= 2119899120587119898 + 120591
119898 1198872119899minus1
= 2119899120587119898 + 120587(4119898)1198862119899= 2119899120587119898 + 120587(4119898) + 120591
119898 and 119887
2119899= 2119899120587119898 + 120587(2119898)
Example 13 (advanced case) Let 119901 ge 1 120576 ge 0 and 119898 isin
N be fixed and 120590119898
isin R 0 le 120590119898
lt 120587(4119898) With thehelp of Corollary 10 the following two classes of quasilinearadvanced differential equations
(100381610038161003816100381610038161199091015840
(119905)10038161003816100381610038161003816
119901minus1
1199091015840
(119905))1015840
+ 120583 sin (119898119905) 1003816100381610038161003816119909 (119905 + 120590119898)1003816100381610038161003816119901+120576
times sgn (119909 (119905 + 120590119898)) = minus120588 cos (2119898119905)
(1199091015840(119905)
radic1 + 11990910158402
(119905)
)
1015840
+ 120583 cos (119898119905) 1003816100381610038161003816119909 (119905 + 120590119898)10038161003816100381610038161+120576
times sgn (119909 (119905 + 120590119898)) = minus120588 cos (2119898119905)
(31)
are oscillatory provided at least one of 120583 gt 0 and 120588 gt 0 islarge enough (the case 120588 = 0 is possible if 120576 = 0) In order toapply Corollary 10 we can choose 119886
2119899minus1= 2119899120587119898 119887
2119899minus1=
2119899120587119898 + 120587(4119898) minus 120590119898 1198862119899
= 2119899120587119898 + 120587(4119898) and 1198872119899
=
2119899120587119898 + 120587(2119898) minus 120590119898
Example 14 (delay-advanced case) Let 119901 ge 1 1205761gt 0 1205762gt 0
and 119898 isin N be fixed and 120591119898ge 0 and 120590
119898ge 0 0 le 120591
119898+ 120590119898lt
120587(4119898) With the help of Corollary 11 the following class ofquasilinear delay-advanced differential equations
(100381610038161003816100381610038161199091015840
(119905)10038161003816100381610038161003816
119901minus1
1199091015840
(119905))1015840
+ 120582 sin (119898119905)
times1003816100381610038161003816119909 (119905 minus 120591119898)
1003816100381610038161003816119901+1205761 sgn (119909 (119905 minus 120591
119898))
+ 120583 cos (119898119905) 1003816100381610038161003816119909 (119905 + 120590119898)1003816100381610038161003816119901+1205762
times sgn (119909 (119905 + 120590119898)) = minus120588 cos (2119898119905)
(32)
is oscillatory provided either 120588 gt 0 is large enough or at leastone of 120582 gt 0 and 120583 gt 0 is large enough In order to applyCorollary 11 we can choose 119886
2119899minus1= 2119899120587119898 + 120591
119898 1198872119899minus1
=
2119899120587119898 + 120587(4119898) minus 120590119898 1198862119899
= 2119899120587119898 + 120587(4119898) + 120591119898 and
1198872119899= 2119899120587119898 + 120587(2119898) minus 120590
119898
3 Application to Duffing Equations withTime Delay Feedback
Let 120582 ge 0 denote the control gain parameter (often calledldquodisplacement feedback coefficientrdquo) 120591 gt 0 the time delayand 120588 ge 0 and 120596 gt 0 the amplitude and frequency of theexternal force respectively Let the function Φ = Φ(119905 119906) thatwill appear in the delay feedback term Φ(119905 119909(119905 minus 120591)) satisfythe general condition
Φ (119905 119906) sgn (119906) ge 1206010|119906|119902
forall119905 ge 1199050
119906 = 0 and some 119902 ge 1 1206010gt 0
(33)
For instance Φ(119905 119906) = 1199062119898minus1 119898 isin N or more general
Φ(119905 119906) = sum119898
119896=11206011198961199062119896minus1 120601
119896gt 0119898 isin N
In this section we consider the following large class ofundamped possible nonautonomous and nonconservativeDuffing equations without or with the general time delayfeedback Φ(119905 119909(119905 minus 120591))
(10038161003816100381610038161003816119909101584010038161003816100381610038161003816
119901minus1
1199091015840)1015840
+ 1205962
0119909 +
1205831|119909|1199031 sgn (119909)
(1205832+ 12058331199092)1199032
+
119898
sum
119894=1
120573119894(119905) |119909|
120572119894minus1119909 + 120582Φ (119905 119909 (119905 minus 120591)) = 120588 cos (120596119905)
(34)
where 1205960is the natural frequency 120583
1ge 0 is the density of the
nonlinear potential (or rigidity coefficient) and 1205832 1205833 1199031 1199032
are nonnegative constants 120573119894(119905) ge 0 and 120572
119894ge 1
When 119901 = 1 120582 = 0 and 120573119894(119905) equiv 120573
119894= const
(34) contains many most important classes of undampedautonomous Duffing oscillators such as the following
(i) the strongly nonlinear Duffing oscillator with smoothodd nonlinearity is given in (34) provided 120583
1= 0 and
120572119894= 2119894 + 1 let us recall some of its known particular
cases
(a) the classic Duffing oscillator 11990910158401015840 +12059620119909+120573119909
3= 0
has been recently studied in the searching of
6 Discrete Dynamics in Nature and Society
solitarywave solutions of classic and generalizedZakharov equations of plasma physics (see [16])and of nonlinear Schrodinger equation (see[17]) also it is strongly connected with theJacobi elliptic equation (see [18])
(b) the cubic-quintic oscillator 11990910158401015840 + 1205962
0119909 + 120573
11199093+
12057321199095
= 0 is used as a model for the non-linear dynamics of a slender elastica (see [19])in nonlinear wave systems (see [20]) for thepropagation of a short electromagnetic pulsein a nonlinear medium (see [21]) and in theunimodal Duffing temporal problem (see [22])
(c) the cubic truly nonlinear oscillator 11990910158401015840 + 1205731199093=
0 models the motion of a ball bearing thatoscillates in a glass tube that is bent into acurve (see [23]) as well as the motion of a massattached to identical stretched elastic wires (see[24])
(d) the nonhomogeneous Duffing oscillator 11990910158401015840 +1205962
0119909 + 120573119909
3= 120588 cos(120596119905) describes various forced
vibrations of beams springs with nonlinearstiffness cables plates shells and optical fibresin electrical circuits in nonlinear isolators andso forth (see for instance [25 26])
(ii) the general Duffing-harmonic oscillator (with rationalor irrational nonlinear restoring-force) is given in(30) if 120583
1= 0 120573119894= 0 and 120588 = 0 the most known
subclasses of these oscillators are
(a) the classic Duffing-harmonic oscillator 11990910158401015840+
(12058311199093(1205832+ 12058331199092)) = 0 which models many
conservative nonlinear oscillatory systems see[27]
(b) the relativistic harmonic oscillator11990910158401015840+ (1205831119909radic1 + 1199092) = 0 see [28]
(c) the nonlinear oscillator11990910158401015840+119909minus(1205831119909radic1 + 1199092) =
0 1205831
isin [0 1] which is typified as a massattached to a stretched elastic wire see [29 30]
(d) the nonlinear oscillator 11990910158401015840
+
(1205831119909(radic(1 + 1199092 )
3
) = 0 which presentsnonlinear oscillations of a punctual charge inthe electric field of charged ring see [31]
Finding several explicit forms of periodic approximate solu-tions for these oscillators has been intensively studied lastyears by many authors see for instance [28 30 32ndash37] andalso the references therein
When 120582 = 0 and linear time delay feedbackΦ(119905 119909(119905minus120591)) =119909(119905 minus 120591) the following topics have been studied for varioustypes of Duffing oscillators with time delayed feedback in[38] authors constructed a low-order approximate solutionunder weak feedback gain parameter about the low- andhigh-order approximations see also [39] in [40] with 120588 = 0the Hopf bifurcation diagrams have been explored for theapproximate periodic solutions (amplitude versus time delay120591 and feedback gain 120582 versus time delay 120591) moreover in [41]
authors made an analysis on the effect of the control gainand time delay parameters on the amplitude of approximateperiod solution from the theoretical and numerical pointsof view see also [42] in [43] authors studied the chaoticbehaviour with respect to gains and time delay parameterssee also [44]
Equations under time delay control such as (34) (espe-cially with damped term) are used as a model for variouscontrolled physical mechanical and engineering systemswith time delays see for instance [39 45ndash48] and thereferences therein
Here (34) contains very general nonlinear time delayfeedback Φ(119905 119909(119905 minus 120591)) with Φ satisfying (33) and the lineartime delay feedback 119909(119905 minus 120591) is only a particular case ofit and to the best of our knowledge the previous topicsare not considered for (34) as yet Moreover with suchan Φ the oscillations of (34) can be taken under a doubteven with the linear time delay feedback (see the nature ofthe approximations given in [38 39]) Hence we can posethe following question under what conditions on equationrsquosparameters (34) is a nonlinear oscillator that is possessesonly oscillatory solutions An answer is given in the nextresult as an easy consequence of the parametrically excitedoscillations by Theorem 5
Theorem 15 Let 120591 isin (0 120587120596) and (33) hold Equation (34) isoscillatory in the next two cases
(i) 119902 = 119901 and 120582 is large enough(ii) 119902 gt 119901 120582 gt 0 120588 gt 0 and at least one of 120582 and 120588 is large
enough
Proof Let 119903(119905) equiv 1 119860(V) = |V|119901minus1 119865(119905 119906) = Φ(119905 119906) 119866(119905 119906) equiv0 119890(119905) = cos(120596119905) and
119861 (119905 119906 V) = 1205962
0119906 +
1205831|119906|1199031s119892119899 (119906)
(1205832+ 12058331199062)1199032
+
119898
sum
119894=1
120573119894(119905) |119906|
120572119894minus1119906 (35)
It is easy to check that all assumptions of Theorem 5 arefulfilled with respect to the sequence 119886
119899= minus1205872120596 + 119899120587120596 + 120591
and 119887119899= 1205872120596 + 119899120587120596 + 120591 where 119886
119899lt 119887119899since it is supposed
that 120591 lt 120587120596 Hence Theorem 5 proves this theorem
Remark 16 Even in the linear forced case (119890(119905) equiv 0) it isnot easy to establish the oscillations of all solutions since theoscillation and nonoscillation can occur simultaneously Themost simple and important example for the coincidence ofoscillation and nonoscillation is the following linear forceddifferential equation 11990910158401015840 + (2119905)119909
1015840+ 119909 = 2119905 119905 gt 0 that
allows an oscillatory solution 1199091(119905) = (3 sin 119905)119905 + 2119905 and a
nonoscillatory solution 1199092(119905) = 2119905 This is not possible in
the linear case with 119890(119905) equiv 0 because of Sturmrsquos separationtheorem
4 Parametrically Excited Oscillations andWell-Known Oscillation Criteria
In this section we would like to draw the readerrsquos attentionto the fact that the parametrically excited oscillations have
Discrete Dynamics in Nature and Society 7
been already appearing in some published papers on theoscillation of functional differential equations but only insome examples illustrating certain main oscillation criteriaHowever with the help of our main results in which theparametrically excited oscillations are studied in a generalsetting the equations from these examples are replaced withgeneral ones also having parameters 120582 and 120583
In [1] (see also [2 Example 31] with 120591 = 0 [3 Example31] and [4 Section 3]) the author considers the oscillationof the second-order delay differential equation
11990910158401015840
(119905) + 119891 (119905) |119909 (120591 (119905))|120574 sgn119909 (120591 (119905)) = 119890 (119905) (36)
in the linear case (120574 = 1) and the superlinear (120574 gt 1)In the linear case (analogously for the superlinear case see[1 Theorem 2]) the author proved the following oscillationcriterion In what follows we denote
119863 (119886 119887) = 119906 isin 1198621
([119886 119887] R) 119906 (119905) equiv 0 119906 (119886) = 119906 (119887) = 0
(37)
Theorem 17 ([1 Theorem 1]) Suppose that for any 119879 ge 0there exist constants 119886
1 1198871 1198862 1198872such that 119879 le 119886
1lt 1198871 119879 le
1198862lt 1198872 and 119891(119905) ge 0 on [120591(119886
1) 1198871] cup [120591(119886
2) 1198872] 119890(119905) le 0
on [120591(1198861) 1198871] and 119890(119905) ge 0 on [120591(119886
2) 1198872] If there exists 119906 isin
119863(119886119894 119887119894) 119894 = 1 2 such that
int
119887119894
119886119894
[1199062
(119905) 119891 (119905)120591 (119905) minus 120591 (119886
119894)
119905 minus 120591 (119886119894)
minus (1199061015840
(119905))2
]119889119905 ge 0 (38)
then (36) with 120574 = 1 is oscillatory
Previous criterion has been applied on the followingparticular equation
11990910158401015840
(119905) + 120582 sin (119905)1003816100381610038161003816100381610038161003816119909 (119905 minus
120587
4)
1003816100381610038161003816100381610038161003816
120574
times sgn 119909(119905 minus120587
4) = cos (119905) 119905 ge 0
(39)
where 120582 ge 0 and 120574 = 1 Applying Theorem 17 to (39) theauthor proved that (39) is oscillatory provided the followinginequality
120582int
119887119894
119886119894
sin2 (2119905) cos2 (2119905) sin (119905)119905 minus 119886119894
119905 minus 119886119894+ 1205874
119889119905 ge120587
2 (40)
holds for sufficiently large 120582 Thus the oscillation of (39) isexcited by the large enough parameter 120582 However accordingto Theorems 5 and 6 we are able to show that the nextparametric equation that corresponds to general equation(36)
11990910158401015840
(119905) + 120582119891 (119905) |119909 (120591 (119905))|120574 sgn119909 (120591 (119905)) = 119890 (119905) (41)
is oscillatory provided 120582 is large enough where 1199011= 1199012= 120574
120583 = 0 and 120588 = 1Next in [5] (see also [6ndash8]) the authors consider the
oscillation of the following class of second-order differentialequations with delay and advanced arguments
(119903 (119905) 1199091015840
(119905))1015840
+ 119891 (119905) |119909 (120591 (119905))|1199011 sgn119909 (120591 (119905))
+ 119892 (119905) |119909 (120590 (119905))|1199012 sgn119909 (120590 (119905)) = 119890 (119905) 119905 ge 0
(42)
where 1199011 1199012ge 1 When 119901
1= 1199012= 1 the authors prove the
following result (for other cases see [5Theorems 32 33 and34]
Theorem 18 ([5 Theorem 31]) Suppose that for any 119879 ge
0 there exist intervals [120591(1198861) 1198871] [120591(119886
2) 1198872] [1198881 120590(1198891)] and
[1198882 120590(1198892)] contained in [119879infin) such that 119886
1lt 1198871 1198862lt 1198872
1198881lt 1198891 1198882lt 1198892 and
119891 (119905) ge 0 119900119899 [120591 (1198861) 1198871] cup [120591 (119886
2) 1198872]
119892 (119905) ge 0 119900119899 [1198881 120590 (1198891)] cup [119888
2 120590 (1198892)]
119890 (119905) le 0 119900119899 [120591 (1198861) 1198871] cup [1198881 120590 (1198891)]
119890 (119905) ge 0 119900119899 [120591 (1198862) 1198872] cup [1198882 120590 (1198892)]
(43)
and 119888119894= 120591(119886
119894) 119889119894= 119886119894 and 119887
119894= 120590(119889
119894) 119894 = 1 2 If there exist
1199061isin 119863(119886
119894 119887119894) and 119906
2isin 119863(119888119894 119889119894) such that either
int
119887119894
119886119894
[1199062
1(119905) 119891 (119905)
120591 (119905) minus 120591 (119886119894)
119905 minus 120591 (119886119894)
minus (1199061015840
1(119905))2
119903 (119905)] 119889119905 ge 0 (44)
or
int
119889119894
119888119894
[1199062
2(119905) 119891 (119905)
120590 (119889119894) minus 120590 (119905)
120590 (119889119894) minus 119905
minus (1199061015840
2(119905))2
119903 (119905)] 119889119905 ge 0 (45)
for 119894 = 1 2 then (42) with 1199011= 1199012= 1 is oscillatory
As a consequence of this result it has been concluded thatthe particular equation
(119903 (119905) 1199091015840
(119905))1015840
+ 120582 sin (119905) 119909 (119905 minus 120587
12)
+ 120583 cos (119905) 119909 (119905 + 120587
6) = cos (2119905) 119905 ge 0
(46)
is oscillatory provided either 120582 or 120583 is large enough Howeverby following Theorems 5 and 6 one can obtain the sameconclusion for the following general equation associated with(42)
(119903 (119905) 1199091015840
(119905))1015840
+ 120582119891 (119905) |119909 (120591 (119905))|1199011 sgn119909 (120591 (119905))
+ 120583119892 (119905) |119909 (120590 (119905))|1199012 sgn119909 (120590 (119905)) = 119890 (119905)
(47)
Related observation can be done with [8 Example 33]and [9 Example 21] where the quasilinear second-orderfunctional differential equations have been considered It isleft to the reader
5 Some Open Questions and Comments
In this section we discuss some problems related to ourmainresults that are not studied here
(1) Quasiperiodic Case In the theory of nonlinear oscillatorsa particularly important case occurs when the periodiccoefficients in the oscillator do not have any common periodIt is called the quasiperiodic (or two-frequency) nonlinear
8 Discrete Dynamics in Nature and Society
oscillator and studied for instance in [50ndash52] Since inTheorems 5 6 and 7 we assume that the correspondingperiodic functions have a commonperiod it is natural to posethe next question
Open Question 1 Is it possible to derive sufficient conditionsfor the oscillation of (27) in the casewhen119891(119905) and119892(119905) (resp119891(119905) 119892(119905) and ℎ(119905)) are two (resp three) periodic functionsnot having a common period
(2) Equation with More Functional Arguments Next regard-ing some second-order functional differential equationsconsidered in the references of this paper more than twononlinear functional terms are appearing and thereforeinstead of main equation (1) and corresponding particularequation (27) considered inTheorems 5 6 and 7 we suggestthe following classes of equations
(119903 (119905) 119860 (1199091015840
(119905)))1015840
+ 119861 (119905 119909 (119905) 1199091015840
(119905))
+
1198981
sum
119896=1
120582119896119865119896(119905 119909 (120591
119896(119905)))
+
1198982
sum
119896=1
120583119896119866119896(119905 119909 (120590
119896(119905))) = 120588119890 (119905)
(48)
where 0 le 120591119896(119905) le 119905 lim
119905rarrinfin120591119896(119905) = infin 120590
119896(119905) ge 119905 119898
1 1198982isin
N and
(119903 (119905) 119860 (1199091015840
(119905)))1015840
+ 119861 (119905 119909 (119905) 1199091015840
(119905))
+
1198981
sum
119896=1
120582119896119891119896(119905)
1003816100381610038161003816119909 (119905 minus 120591119896)1003816100381610038161003816119901119896 sgn (119909 (119905 minus 120591
119896))
+
1198982
sum
119896=1
120583119896119892119896(119905)
1003816100381610038161003816119909 (119905 + 120590119896)1003816100381610038161003816119902119896 sgn (119909 (119905 + 120590
119896)) = 120588119890 (119905)
(49)
where 120582119896 120583119896 120588 120591119896 120590119896ge 0 and 119901
119896 119902119896gt 0
Comment We suggest the reader to enlarge the main resultsof this paper to (48) and (49)
(3) Damped Duffing Equation In the application the Duffingequation (34) is often appearing with the linear damped term1199091015840(119905) that is
11990910158401015840+ 11988901199091015840+ 1205962
0119909 + 120573119909
3+ 120582Φ (119909 (119905 minus 120591)) = 120588 cos (120596119905) (50)
where 1198890
is the damped coefficient which can in anactive way influence various behaviours of (50) Since119861(119905 119909(119905) 119909
1015840(119905)) = 119889
01199091015840(119905) does not satisfy the required
assumption (4) we are not able to apply our main results to(50) Hence we pose the following questionOpen Question 2 Is it possible to obtain the parametricallyexcited oscillation for (1) in the case when the damped term119861(119905 119906 V) satisfies a larger condition than (4) in which thelinear damped term 120573119909
1015840(119905) is especially included
(4) Functional Argument in Damped Term In a class of Duff-ing equations we have two time delayed feedback and hence
besides the control gain parameter 1205821another parameter 120582
2
appears the so-called velocity gain parameter Hence insteadof (34) one can consider
11990910158401015840+ 11988901199091015840+ 1205962
0119909 + 120573119909
3+ 1205821119909 (119905 minus 120591)
+ 12058221199091015840
(119905 minus 120591) = 120588 cos (120596119905) (51)
Therefore we suggest the following problem for further studyOpen Question 3 Is it possible to obtain the parametricallyexcited oscillation for the following more general functionaldifferential equation than (1) in which the functional argu-ment appears in the damped term too as follows
(119903 (119905) 119860 (1199091015840
(119905)))1015840
+ 119861 (119905 119909 (119905) 1199091015840
(119905)) + 1205821119865 (119905 119909 (120591 (119905)))
+ 1205822119867(119905 119909
1015840
(120591 (119905))) = 120588119890 (119905) 119905 ge 1199050
(52)or
(119903 (119905) 119860 (1199091015840
(119905)))1015840
+ 119861 (119905 119909 (119905) 1199091015840
(119905)) + 1205831119866 (119905 119909 (120590 (119905)))
+ 1205832119867(119905 119909
1015840
(120590 (119905))) = 120588119890 (119905) 119905 ge 1199050
(53)
About known oscillation criteria for the second-order func-tional differential equations having the functional argumentin the damped term we refer the reader to for instance [53]and the references therein
6 Proofs of Main Results
The proof of Lemma 1 is based on the following three stepstwo working forms of condition (6) (see Lemmas 19 and 20)the existence of an explosive solution of a suitable Riccatidifferential inequality (see Proposition 22) and a comparisonprinciple (see Proposition 24)
Lemma 19 (a necessary condition to (6)) Let 0 lt 119903(119905) le 1199030
on [1199050infin) If assumption (6) is fulfilled then there is a positive
real number 120576 such that1
120587lowast
int119869
min119901
(120576119903 (119905))120574minus1
120576119896 (120582 120583 120588)119876 (119905) 119889119905 ge 1 (54)
for all 120582 ge 1205820 120583 ge 120583
0 and 120588 ge 120588
0and some (120582
0 1205830 1205880) isin R3+
Proof Since 0 lt 119903(119905) le 1199030for 119905 ge 119905
0 we conclude that for
120576 = (119901
119903120574minus1
0119896 (120582 120583 120588)max
119905isin 119869119876 (119905)
)
1120574
(120582 120583 120588) isin R3
+
(55)
it holds that 119901(120576119903(119905))120574minus1
ge 119901(1205761199030)120574minus1
= 120576119896(120582 120583 120588)
max119905isin 119869119876(119905) ge 120576119896(120582 120583 120588)119876(119905) 119905 isin 119869 and hence
int119869
min119901
(120576119903 (119905))120574minus1
120576119896 (120582 120583 120588)119876 (119905) 119889119905
= 120576119896 (120582 120583 120588) int119869
119876 (119905) 119889119905
(56)
Discrete Dynamics in Nature and Society 9
On the other hand from (6) we observe
1
120587lowast
int119869
119876 (119905) 119889119905 ge1199031minus(1120574)
0
1199011120574[119896 (120582 120583 120588)]1minus(1120574)
(max119905isin 119869
119876 (119905))
1120574
(57)
which together with (55) and (56) gives
1
120587lowast
int119869
min119901
(120576119903 (119905))120574minus1
120576119896 (120582 120583 120588)119876 (119905) 119889119905
= 120576119896 (120582 120583 120588)1
120587lowast
int119869
119876 (119905) 119889119905
ge 1205761199031minus(1120574)
0
1199011120574[119896 (120582 120583 120588)]
1120574
(max119905isin 119869
119876 (119905))
1120574
= 1
(58)
for all 119899 ge 1198990 120582 ge 120582
0 120583 ge 120583
0 and 120588 ge 120588
0 It proves this
lemma
Lemma 20 (an equivalent condition to (54)) Assumption(54) is fulfilled if and only if there is a real number 120576 gt 0 and acontinuous function 119870(119905) ge 0 119905 isin 119869 such that
1198880= int119869
119870 (119905) 119889119905 gt 0119870 (119905)
1198880
le1
120587lowast
timesmin119901
(120576119903 (119905))120574minus1
120576119896 (120582 120583 120588)119876 (119905)
(59)
for all 119905 isin 119869 120582 ge 1205820 120583 ge 120583
0 and 120588 ge 120588
0and some (120582
0 1205830 1205880) isin
R3+
Proof This proof is very elementary Indeed if (54) holdsthen the function119870(119905) and number 119888
0 defined by
119870 (119905) =1
120587lowast
min119901
(120576119903 (119905))120574minus1
120576119896 (120582 120583 120588)119876 (119905)
1198880= int119869
119870 (119905) 119889119905
(60)
obviously satisfy 1198880
ge 1 and 119870(119905)1198880
le 119870(119905) = (1120587lowast)
min119901(120576119903(119905))120574minus1 120576119896(120582 120583 120588)119876(119905) which shows (59) Con-versely if (59) holds then integrating both sides of thesecond inequality in (59) we obtain
int119869
1
120587lowast
min119901
(120576119903 (119905))120574minus1
120576119896 (120582 120583 120588)119876 (119905) ge int119869
119870 (119905)
1198880
119889119905 = 1
(61)
which shows (54)
In conclusion according to previous two lemmas wesee that supposed condition (6) implies (59) which plays animportant role in the proof of the main results
The second step in the proof of Lemma 1 is to prove theexistence of a function 120596(119905) which blows up in the finitetime and satisfies a generalized Riccati differential lowerinequality we briefly present the existence and properties
of the so-called generalized tangent type function In whatfollows let 120587
lowastbe a positive real number defined in (3) Let us
remark that 120572(119904) = 119904120574 120574 gt 1 implies 120587
lowast= (2120587)(120574 sin(120587120574))
see for instance [54] and obviously for 120574 = 2wehave120587lowast= 120587
Lemma 21 Let 120572 [0infin) rarr [0infin) be a continuousfunction such that
int
infin
0
119889120591
1 + 120572 (120591)lt infin (62)
Then there is a real number 120587lowastgt 0 and a function 119911 = 119911(119904)
119911 isin 1198621((minus120587lowast2 120587lowast2)R) such that
119889119911
119889119904= 1 + 120572 (|119911 (119904)|) 119904 isin (minus
120587lowast
2120587lowast
2)
119911 (0) = 0
(63)
Moreover 119911(119904) is increasing and odd
lim119904rarr120587lowast2
119911 (119904) = infin 120587lowast=
2120587
120574 sin (120587120574)for 120572 (119904) = 119904
120574
120574 gt 1
(64)
In particular for 120572(119904) = 1199042 one can take 119911(119904) = tan(119904) and
120587lowast= 120587
Proof Let 119885 = 119885(119905) 119905 isin R be a function defined by
119885 (119905) = int
119905
0
1
1 + 120572 (|120591|)119889120591 119905 isin R (65)
The function 119885(119905) is well defined since 120572(119904) is positive andcontinuous on [0infin) 119885(119905) is increasing and odd functionand
119889119885
119889119905=
1
1 + 120572 (|119905|) 119905 isin R
119885 (0) = 0 119885 isin 1198621
(RR)
(66)
Moreover because of (62) there is a real number 120587lowastgt 0 such
that120587lowast
2= int
infin
0
119889120591
1 + 120572 (120591) (67)
Thus 119885 R rarr (minus120587lowast2 120587lowast2) and there exists an inverse
function 119885minus1 = 119885minus1(119904) of the original function 119885 = 119885(119905) and
119885minus1
(minus120587lowast2 120587lowast2) rarr R Also from 119885(119885
minus1(119904)) = 119904 and
119889119885119889119905 = 0 onR we also derive that119889119885minus1119889119904 = 0 on its domain(minus120587lowast2 120587lowast2) and
119889119885
119889119905(119885minus1
(119904)) =1
(119889119885minus1119889119904) 119904 isin (minus
120587lowast
2120587lowast
2) (68)
Putting 119905 = 119885minus1(119904) for 119904 isin (minus120587
lowast2 120587lowast2) into (66) and using
(68) we easily obtain
119889119885minus1
119889119904= 1 + 120572 (
10038161003816100381610038161003816119885minus1
(119904)10038161003816100381610038161003816) 119904 isin (minus
120587lowast
2120587lowast
2)
119885minus1
(0) = 0 119885minus1isin 1198621((minus
120587lowast
2120587lowast
2) R)
(69)
10 Discrete Dynamics in Nature and Society
Moreover from (67) we have lim119904rarr120587lowast2119885minus1(119904) = 119885
minus1
(lim119905rarrinfin
119885(119905)) = lim119905rarrinfin
119885minus1119885(119905) = lim
119905rarrinfin119905 = infin Thus
if we set 119911(119904) = 119885minus1(119904) then previous two statements and
(67) prove this lemma
Next we prove the main result of this section
Proposition 22 Let (2) and (6) hold where 119869 = (119886 119887) Let 120576 gt0 be a real number and let119870(119905) ge 0 119905 isin [119886 119887] be a continuousfunction both obtained in Lemma 20 Let 120587
lowastbe from (3) and
1198880from (59) and let 119877
119886isin R be an arbitrary real number If
119911 = 119911(119904) is the generalized tangens function defined in (63)and 119881(119905) is a function defined by
119881 (119905) =120587lowast
1198880
int
119905
119886
119870 (120591) 119889120591 + 119911minus1(119877119886) 119905 isin [119886 119887] (70)
then there is a 119879lowast119886isin [119886 119887) such that
119881 (119879lowast
119886) =
120587lowast
2 119881 ([119886 119879
lowast
119886)) sub (minus
120587lowast
2120587lowast
2) (71)
Moreover for a function 120596(119905) defined by120596 (119905) = 119911 (119881 (119905)) 119905 isin [119886 119879
lowast
119886) (72)
one has 120596(119886) = 119877119886 lim119905rarr119879
lowast
119886
120596(119905) = infin and
119889120596
119889119905le
119901
(120576119903 (119905))120574minus1
120572 (1003816100381610038161003816120596 (119905)
1003816100381610038161003816)
+ 120576119896 (120582 120583 120588)119876 (119905) 119905 isin (119886 119879lowast
119886)
(73)
where the numbers 119901 and 120574 are from (3) and the functions119896(120582 120583 120588) and 119876(119905) are from (6)
Proof Under assumptions (2) and (6) and because of Lem-mas 19 and 20 we obtain 120576 gt 0 and 119870(119905) gt 0 119905 isin [119886 119887]satisfying inequality (59)
Next since 119911minus1(119877119886) isin (minus120587
lowast2 120587lowast2) (see Lemma 21)
from (70) we directly obtain
119881 (119886) = 119911minus1(119877119886) lt
120587lowast
2 119881 (119887) = 120587
lowast+ 119911minus1(119877119886) gt
120587lowast
2
(74)Since 119870 isin 119862([119886 119887] [0infin)) we obtain 119881 isin 119862([119886 119887]R) cap
1198621((119886 119887)R) and from (74) we observe that there exist
numbers 119879lowast119886isin (119886 119887) such that119881(119879lowast
119886) = 120587lowast2 Also119870(119905)119888
0ge
0 gives 119881([119886 119879lowast119886)) sub (minus120587
lowast2 120587lowast2) which proves statement
(71) Moreover it together with Lemma 21 and (72) provesthat
lim119905rarr119879
lowast
119886
120596 (119905) = lim119905rarr119879
lowast
119886
119911 (119881 (119905)) = 119911 (120587lowast
2) = infin (75)
Next according to (59) (63) and (72) we make thefollowing calculation on the interval [119886 119879lowast
119886)
1205961015840
(119905) = 1199111015840
(119881 (119905)) 1198811015840
(119905) = [1 + 120572 (|119911 (119881 (119905))|)]120587lowast
1198880
119870 (119905)
= [1 + 120572 (1003816100381610038161003816120596 (119905)
1003816100381610038161003816)]120587lowast
1198880
119870 (119905)
le119901
(120576119903 (119905))120574minus1
120572 (1003816100381610038161003816120596 (119905)
1003816100381610038161003816) + 120576119896 (120582 120583 120588)119876 (119905)
(76)
Thus all assertions of this proposition are proved
Remark 23 In the proof of the main result the number 119877119886
is determined by 119877119886= 120596(119886) where 120596(119905) denotes a function
associated with a nonoscillatory solution and it is given by(84) below
The third step in the proof of Lemma 1 is to show thefollowing pointwise comparison principle for the functions120596and120596 satisfying respectively the lower and upper differentialinequalities (73) and
119889120596
119889119905ge
119901
(120576119903 (119905))120574minus1
120572 (|120596 (119905)|) + 120576119896 (120582 120583 120588)119876 (119905) 119905 isin (119886 119887)
(77)
Proposition 24 Let [119886 119887) sub [1199050infin) be an arbitrary inter-
val One supposes that all coefficients of Riccati differentialinequalities (73) and (77) are continuous and strictly positivefunctions Let 120596 120596 isin 119862
1([119886 119887)R) be two functions satisfying
respectively (73) and (77) on the interval [119886 119887) Then
120596 (119886) le 120596 (119886) 119894119898119901119897119894119890119904 120596 (119905) le 120596 (119905) forall119905 isin [119886 119887) (78)
Proof Let119867(119905 119906) be a function defined by
119867(119905 119906) =119901
(120576119903 (119905))120574minus1
120572 (|119906|) + 120576119896 (120582 120583 120588)119876 (119905)
119905 isin [119886 119887) 119906 isin R
(79)
Let 119868 sub [119886 119887) and 119872 gt 0 be arbitrary For any two 1199061
1199062 minus119872 le 119906
1lt 1199062le 119872 let 119868
12be an interval defined
by 11986812
= (min|1199061| |1199062|max|119906
1| |1199062|) Since 120572(119904) is a 1198621-
function on [0infin) we know by the Lagrange mean valuetheorem applied on 119868
12that there is a 120585 isin 119868
12such that
120572 (10038161003816100381610038161199062
1003816100381610038161003816) minus 120572 (10038161003816100381610038161199061
1003816100381610038161003816)
1199062minus 1199061
le
1003816100381610038161003816120572 (10038161003816100381610038161199062
1003816100381610038161003816) minus 120572 (10038161003816100381610038161199061
1003816100381610038161003816)1003816100381610038161003816
1199062minus 1199061
=100381610038161003816100381610038161205721015840
(120585)10038161003816100381610038161003816
100381610038161003816100381610038161003816100381610038161199062
1003816100381610038161003816 minus10038161003816100381610038161199061
10038161003816100381610038161003816100381610038161003816
1199062minus 1199061
le100381610038161003816100381610038161205721015840
(120585)10038161003816100381610038161003816
le max119904isin11986812
100381610038161003816100381610038161205721015840
(119904)10038161003816100381610038161003816
(80)
since ||1199062| minus |1199061|| le 119906
2minus 1199061 Hence for any 119905 isin 119868 and 119906
1 1199062
minus119872 le 1199061lt 1199062le 119872 we have
119867(119905 1199062) minus 119867 (119905 119906
1)
1199062minus 1199061
= 1205880(119905)
120572 (10038161003816100381610038161199062
1003816100381610038161003816) minus 120572 (10038161003816100381610038161199061
1003816100381610038161003816)
1199062minus 1199061
le 1205880(119905)max119904isin11986812
100381610038161003816100381610038161205721015840
(119904)10038161003816100381610038161003816= 1198710(119905)
(81)
Thus the function119867(119905 119906) from (79) satisfies required condi-tion of [55 Lemma 19] and applying it to (73) and (77) weprove this proposition
Proof of Lemma 1 On the contrary let 119909(119905) be a solution of(1) such that
119909 (119905) = 0 on (120591 (120591 (119886)) 120590 (120590 (119889))) (82)
Discrete Dynamics in Nature and Society 11
that is 119909(119905) gt 0 on (120591(120591(119886)) 120590(120590(119889))) or 119909(119905) lt 0 on(120591(120591(119886)) 120590(120590(119889))) since 119909(119905) is a continuous function on[1199050infin) Let for instance
119909 (119905) gt 0 on (120591 (120591 (119886)) 120590 (120590 (119889))) (83)
Another case can be analogously treated let us see thecomment at the end of this proof In particular from (83)we have 119909(119905) gt 0 on (120591(120591(119886)) 120590(120590(119887))) which implies (since120591(119905) and 120590(119905) are increasing functions) 119909(119904) gt 0 for all 119904 isin
(120591(119886) 120590(119887)) cup (120591(120591(119886)) 120591(120590(119887))) cup (120590(120591(119886)) 120590(120590(119887))) whichyields 119909(119905) gt 0 119909(120591(119905)) gt 0 and 119909(120590(119905)) gt 0 on (120591(119886) 120590(119887))Hence by assumption (7) we may use inequality (5) on theinterval (119886 119887)
Firstly we show that the following classic Riccati transfor-mation of 119909(119905)
120596 (119905) = minus120576119903 (119905) 119860 (119909
1015840(119905))
|119909 (119905)|119901minus1
119909 (119905) 119905 isin (119886 119887) 120576 gt 0 (84)
satisfies upper Riccati differential inequality (77) Let usremark that from (1) we have in particular
minus(119903 (119905) 119860 (1199091015840
(119905)))1015840
= 119861 (119905 119909 (119905) 1199091015840
(119905)) + 120582119865 (119905 119909 (120591 (119905)))
+ 120583119866 (119905 119909 (120590 (119905))) minus 120588119890 (119905) 119905 ge 1199050
(85)
Taking the first derivative on both sides of (84) and usingassumptions (3) (4) and (5) as well as equality (85) and(|119909(119905)|
119901minus1119909(119905))1015840
= 119901|119909(119905)|119901minus1
1199091015840(119905) we obtain
119889120596
119889119905= 120576119901 119903 (119905)
119860 (1199091015840(119905)) 1199091015840(119905)
|119909 (119905)|119901+1
minus 1205761
|119909 (119905)|119901minus1
119909 (119905)(119903 (119905) 119860 (119909
1015840
(119905)))1015840
= 120576119901119903 (119905)119860 (1199091015840(119905)) 1199091015840(119905)
|119909 (119905)|119901+1
+120576
|119909 (119905)|119901minus1
119909 (119905)
times [120582119861 (119905 119909 (119905) 1199091015840
(119905)) + 119865 (119905 119909 (120591 (119905)))
+120583119866 (119905 119909 (120590 (119905))) minus 120588119890 (119905) ]
ge 120576119901119903 (119905)119860 (1199091015840(119905)) 1199091015840(119905)
|119909 (119905)|119901+1
+120576
|119909 (119905)|119901minus1
119909 (119905)
times [120582119865 (119905 119909 (120591 (119905))) + 120583119866 (119905 119909 (120590 (119905))) minus 120588119890 (119905)]
ge 120576119901119903 (119905) 120572(
10038161003816100381610038161003816119860 (1199091015840(119905))
10038161003816100381610038161003816
|119909 (119905)|119901
) + 120576119896 (120582 120583 120588)119876 (119905)
= 120576119901119903 (119905) 120572 (|120596 (119905)|
120576119903 (119905)) + 120576119896 (120582 120583 120588)119876 (119905)
ge119901
(120576119903 (119905))120574minus1
120572 (|120596 (119905)|) + 120576119896 (120582 120583 120588)119876 (119905) 119905 isin (119886 119887)
(86)
Thus according to inequality (5) it is shown that if 119909(119905) isa solution of (1) which satisfies (83) then the function 120596(119905)
defined by (84) satisfies the Riccati differential inequality(77) and 120596 isin 119862((119886 119887)R) On the other hand let 119877
119886be a
real number defined by 119877119886= 120596(119886) According to (6) and
Lemma 19 we obtain (54) which together with Lemma 20ensures that we may use Proposition 22 for such chosen realnumber 119877
119886 Hence we obtain a function 120596(119905) defined by (72)
which satisfies the lower Riccati differential inequality (73) on[119886 119879lowast
119886) 119879lowast119886isin (119886 119887) such that 120596(a) = 119877
119886and lim
119905rarr119879lowast
119886
120596(119905) =
infin Therefore by 120596(119886) = 119877119886= 120596(119886) and Proposition 24 we
conclude that lim119905rarr119879
lowast
119886
120596(119905) = infin too which is a contradictionwith the above conclusion saying that 120596 isin 119862((119886 119887)R) Thushypothesis (82) is not true and consequently Lemma 1 isshown
For the analogous case 119909(119905) lt 0 on (120591(120591(119886)) 120590(120590(119889))) wealso have 119909(119905) lt 0 on (120591(120591(119888)) 120590(120590(119889))) which implies (since120591(119905) and 120590(119905) are increasing functions)
119909 (119904) lt 0 forall119904 isin (120591 (119888) 120590 (119889)) cup (120591 (120591 (119888)) 120591 (120590 (119889)))
cup (120590 (120591 (119888)) 120590 (120590 (119889)))
(87)
which yields 119909(119905) lt 0 119909(120591(119905)) lt 0 and 119909(120590(119905)) lt 0 on(120591(119888) 120590(119889)) Now we can repeat the preceding procedure buton interval (119888 119889) and using (8) instead of (119886 119887) and (7)
Proof of Lemma 2 From assumption (10) we obtain the exis-tence of an 119899
0isin N such that
int
119887119899
119886119899
119876119899(119905) 119889119905 ge
1198880
2( max119905isin[119886119899 119887119899]
119876119899(119905))
1120574
119899 ge 1198990 (88)
that is
2
1198880
int
119887119899
119886119899
119876119899(119905) 119889119905 ge ( max
119905isin[119886119899 119887119899]119876119899(119905))
1120574
119899 ge 1198990 (89)
Now from (9) and previous inequality we deduce that forlarge enough 120582 120583 120588 and 119899
1199011120574
1199031minus1120574
0
[119896 (120582 120583 120588)]1minus1120574
120587lowast
int
119887119899
119886119899
119876119899(119905) 119889119905
ge2
1198880
int
119887119899
119886119899
119876119899(119905) 119889119905 ge ( max
119905isin[119886119899 119887119899]119876119899(119905))
1120574
(90)
which shows (6) Thus all assumptions of Lemma 1 arefulfilled and hence Lemma 2 immediately follows fromLemma 1
Proof of Lemma 3 Obviously assumption (11) is a particularcase of assumption (9) Hence this proof is very similar tothe proof of Lemma 2 and so it is left to the reader
Proof of Lemma 4 It is clear that from assumption (13) weobtain
1
(max119905isin[119886119899119887119899]119876119899(119905))1120574
int
119887119899
119886119899
119876119899(119905) 119889119905 ge
1198881
1198621120574
0
gt 0 forall119899 ge 1198990
(91)
12 Discrete Dynamics in Nature and Society
Thus hypothesis (12) is fulfilled and therefore Lemma 3proves this lemma
Proof of Theorems 5 6 and 7 This proof is based onLemma 4 In order to simplify notation in many placesin this proof we set 120591(119905) = 119905 minus 120591 and 120590(119905) = 119905 + 120590 Sinceassumptions (2) (3) and (4) have been already supposed inTheorems 5 6 and 7 in order to prove these theorems byLemma 4 we are going to show that the functions 119896(120582 120583 120588)and 119876
119899(119905) explicitly given respectively in (18) (21) or (24)
and (19) (22) or (25) satisfy required conditions (11) and(13) respectively and that every solution 119909(119905) of (27) satisfiesconditions (7) and (8) with respect to functions 119896(120582 120583 120588)and 119876
119899(119905) where 119886 = 119886
2119899minus1 119887 = 119887
2119899minus1 119888 = 119886
2119899 and 119889 = 119887
2119899
The proof that the function 119896(120582 120583 120588) given in (18) (21) or(24) satisfies (11) Passing to the limit in (18) (21) or (24) it isvery simple to show (11)
The proof that the function 119876119899(119905) given in (19) (22) or
(25) satisfies the first claim in (13) From (25) we immediatelyobtain
1003816100381610038161003816120591119899 (119905)1003816100381610038161003816 =
100381610038161003816100381610038161003816100381610038161003816
(119905 minus 119886119899
119905 minus 119886119899+ 120591
)
119901100381610038161003816100381610038161003816100381610038161003816
le 1
1003816100381610038161003816120590119899 (119905)1003816100381610038161003816 =
100381610038161003816100381610038161003816100381610038161003816
(119887119899minus 119905
119887119899minus 119905 + 120590
)
119901100381610038161003816100381610038161003816100381610038161003816
le 1 forall119899 isin N
(92)
Next by assumptions of this corollary we can conclude thatthere are three positive constants 119891
0 1198920 1198900such that |119891(119905)| le
1198910and |119892(119905)| le 119892
0on [1199050infin) in cases (i) and (ii) and
|119890(119905)| le 1198900on [1199050infin) in cases (iii) and (iv) Putting previous
inequalities into (19) (22) or (25) for all 119899 isin N and 119905 isin
[1199050infin) it holds that
1003816100381610038161003816119876119899 (119905)1003816100381610038161003816 le
119902
119901(
119901
119902 minus 119901)
(119902119901)minus1
1198901minus(119901119902)
0119891119901119902
0
delay case with 119902 gt 119901
119902
119901(
119901
119902 minus 119901)
(119902119901)minus1
1198901minus(119901119902)
0119892119901119902
0
advanced case with 119902 gt 119901
1199011
119901(
119901
2 (1199011minus 119901)
)
(1199011119901)minus1
1198901minus(119901119901
1)
0119891119901119902
0+1199012
119901
times(119901
2 (1199012minus 119901)
)
(1199012119901)minus1
1198901minus(119901119901
2)
0119892119901119902
0
delay-advanced case (i)
1198901205780
01198911205781
01198921205782
0
2
prod
119894=0
120578minus120578119894
119894
delay-advanced case (ii) (93)
which shows the first claim in (13)
The proof that the function119876119899(119905) given in (19) (22) or (25)
satisfies the second claim in (13)Without loss of generality weprove this claim only in case (i) since for other cases the prooffollows analogously In this sense let119876
119899(119905) = 119891(119905)120591
119899(119905) Since
1198862119899+1
minus 1198862119899minus1
le 119879lowast 1198872119899+1
minus 1198872119899minus1
ge 119879lowast 1198862119899+2
minus 1198862119899le 119879lowast and
1198872119899+2
minus 1198872119899
ge 119879lowast where 119879
lowastgt 0 is the period of the function
119891(119905) we have 1198862119899minus1
le 1198861+(119899minus1)119879
lowastand 1198872119899minus1
ge 1198871+(119899minus1)119879
lowast
119899 isin N Hence
int
1198872119899minus1
1198862119899minus1
119876119899(119905) 119889119905
= int
1198872119899minus1
1198862119899minus1
119891 (119905) (119905 minus 1198862119899minus1
119905 minus 1198862119899minus1
+ 120591)
119901
119889119905
ge int
1198871+(119899minus1)119879
lowast
1198861+(119899minus1)119879lowast
119891 (119905) (119905 minus 1198861minus (119899 minus 1) 119879
lowast
119905 minus 1198861minus (119899 minus 1) 119879
lowast+ 120591
)
119901
119889119905
= int
1198871
1198861
119891 (119904 + (119899 minus 1) 119879lowast) (
119904 minus 1198861
119904 minus 1198861+ 120591
)
119901
119889119904
= int
1198871
1198861
119891 (119904) (119904 minus 1198861
119904 minus 1198861+ 120591
)
119901
119889119904
(94)
which proves that the integral on the left hand side does notdepend on 119899 isin N that is the second claim in (13) is shown on[1198862119899minus1
1198872119899minus1
] This claim follows in the same way on [1198862119899 1198872119899]
Thus the second claim in (13) is proved on [119886119899 119887119899]
Next to the end of this proof let 119909(119905) be a solu-tion of (1) In particular it implies that (119903(119905)119860(1199091015840(119905)))1015840 =
minus119861(119905 119909(119905) 1199091015840(119905)) minus 120582119865(119905 119909(120591(119905))) minus 120583119866(119905 119909(120590(119905))) + 120588119890(119905) It
together with assumptions (15) (16) (20) and (23) easilygives the next two statements
if 119909 (119905) gt 0 119909 (120591 (119905)) gt 0 119909 (120590 (119905)) gt 0
on (120591 (1198862119899minus1
) 120590 (1198872119899minus1
))
then 119909 (119905) satisfies 119903 (119905) 119860 (1199091015840
(119905)) le 0
on (120591 (1198862119899minus1
) 120590 (1198872119899minus1
)) 119899 ge 1198990
(95)
if 119909 (119905) lt 0 119909 (120591 (119905)) lt 0 119909 (120590 (119905)) lt 0
on (120591 (1198862119899) 120590 (119887
2119899))
then 119909 (119905) satisfies 119903 (119905) 119860 (1199091015840
(119905)) ge 0
on (120591 (1198862119899) 120590 (119887
2119899)) 119899 ge 119899
0
(96)
Now we need the following lemma
Discrete Dynamics in Nature and Society 13
Lemma 25 Let 120591119886119887(119905) and 120590
119886119887(119905) be defined by
120591119886119887(119905) = (
120591 (119905) minus 120591 (119886)
119905 minus 120591 (119886))
119901
120590119886119887(119905) = (
120590 (119887) minus 120590 (119905)
120590 (119887) minus 119905)
119901
119905 isin (119886 119887)
(97)
and let 119909 isin 1198622([1198790infin)R) be an arbitrary function If
(119903(119905)119860(1199091015840(119905)) le 0 for all 119905 isin (120591(119886) 120590(119887)) or (119903(119905)119860(1199091015840(119905)) ge 0
for all 119905 isin (120591(119886) 120590(119887)) then
119909 (120591 (119905))
119909 (119905)ge (120591119886119887(119905))1119901
119909 (120590 (119905))
119909 (119905)ge (120590119886119887(119905))1119901
119905 isin (119886 119887)
(98)
Since119860(V) is supposed to be odd and increasing functionjust before (3) and 119903(119905) satisfies (14) the proof of Lemma 25in the first case that is 119903(119905)119860(1199091015840(119905)) le 0 for all 119905 isin (120591(119886) 120590(119887))is the same as the proof of [9 Corollaries 17 and 18] But in thesecond case that is 119903(119905)119860(1199091015840(119905)) ge 0 for all 119905 isin (120591(119886) 120590(119887))the proof is as follows if previous inequality holds then119903(119905)119860(minus119909
1015840(119905)) le 0 for all 119905 isin (120591(119886) 120590(119887)) and therefore to
the function minus119909(119905) one can apply the first case of this lemmaand consequently one obtains
119909 (120591 (119905))
119909 (119905)=minus119909 (120591 (119905))
minus119909 (119905)ge (120591119886119887(119905))1119901
119909 (120590 (119905))
119909 (119905)=minus119909 (120590 (119905))
minus119909 (119905)ge (120590119886119887(119905))1119901
119905 isin (119886 119887)
(99)
which proves this lemma in the second caseNow combining statements (95) (96) and (98) one
easily obtains
if 119909 (119905) gt 0 119909 (120591 (119905)) gt 0 119909 (120590 (119905)) gt 0
on (120591 (1198862119899minus1
) 120590 (1198872119899minus1
))
then 119909 (119905) satisfies 119909 (120591 (119905))
119909 (119905)ge (120591119899(119905))1119901
on (1198862119899minus1
1198872119899minus1
) 119899 ge 1198990
(100)
if 119909 (119905) lt 0 119909 (120591 (119905)) lt 0 119909 (120590 (119905)) lt 0
on (120591 (1198862119899) 120590 (119887
2119899))
then 119909 (119905) satisfies 119909 (120590 (119905))
119909 (119905)ge (120590119899(119905))1119901
on (1198862119899 1198872119899) 119899 ge 119899
0
(101)
where 120591119899(119905) and 120590
119899(119905) are defined in (26)
The proof that 119909(119905) satisfies (7) and (8) In this proofwe frequently use assumptions (16) (20) and (23) andstatements (100) and (101) Also because of (15) and 119865(119905 119906) =
119891(119905)|119906|1199011 sgn(119906) 119866(119905 119906) = 119892(119905)|119906|
1199012 sgn(119906) in both cases
(100) and (101) we can simultaneously use
minus119890 (119905) (|119909 (119905)|119901minus1
119909 (119905))minus1
= |119890 (119905)| |119909 (119905)|minus119901
ge 0 on 119869119899
119865 (119905 119909 (120591 (119905))) (|119909 (119905)|119901minus1
119909 (119905))minus1
ge 119891 (119905) |119909 (120591 (119905))|1199011 |119909 (119905)|
minus119901ge 0 on 119869
119899
119866 (119905 119909 (120590 (119905))) (|119909 (119905)|119901minus1
119909 (119905))minus1
ge 119892 (119905) |119909 (120590 (119905))|1199012 |119909 (119905)|
minus119901ge 0 on 119869
119899
|119909 (120591 (119905))| |119909 (119905)|minus1=119909 (120591 (119905))
119909 (119905)
|119909 (120590 (119905))| |119909 (119905)|minus1=119909 (120590 (119905))
119909 (119905)on 119869119899
(102)
where 119869119899= (1198862119899minus1
1198872119899minus1
) in the case of (100) and 119869119899= (1198862119899 1198872119899)
in the case of (101)
(i) Delay or Advanced Case with 119902 = 119901 Since 119902 = 119901 we obtain
[120582119865 (119905 119909 (120591 (119905))) + 120583119866 (119905 119909 (120590 (119905))) minus 120588119890 (119905)]
times (|119909 (119905)|119901minus1
119909 (119905))minus1
ge [120582119891 (119905) |119909 (120591 (119905))|119901+ 120583119892 (119905) |119909 (120590 (119905))|
119901
+120588 |119890 (119905)| ] |119909 (119905)|minus119901
ge [120582119891 (119905) |119909 (120591 (119905))|119901+ 120583119892 (119905) |119909 (120590 (119905))|
119901] |119909 (119905)|
minus119901
= 120582119891 (119905) (119909 (120591 (119905))
119909 (119905))
119901
+ 120583119892 (119905) (119909 (120590 (119905))
119909 (119905))
119901
ge 120582119891 (119905) 120591119899(119905) + 120583119892 (119905) 120590
119899(119905) 119905 isin 119869
119899
(103)
where the functions 120591119899(119905) and 120590
119899(119905) are defined in (26)
(ii) Delay Case with 119902 gt 119901 In this part we use the nextelementary inequality
119883120574+ (120574 minus 1) 119884
120574ge 120574119883119884
120574minus1 120574 gt 1 119883 119884 ge 0 (104)
Since 119902 gt 119901 and using (104) especially for
120574 =119902
119901gt 1 119883 = (120582119891 (119905))
119901119902
|119909 (120591 (119905))|119901
119884 = (119901
119902 minus 119901120588 |119890 (119905)|)
119901119902
(105)
14 Discrete Dynamics in Nature and Society
for all 119905 isin 119869119899we obtain
[120582119865 (119905 119909 (120591 (119905))) + 120583119866 (119905 119909 (120590 (119905))) minus 120588119890 (119905)]
times (|119909 (119905)|119901minus1
119909 (119905))minus1
= [120582119865 (119905 119909 (120591 (119905))) minus 120588119890 (119905)] (|119909 (119905)|119901minus1
119909 (119905))minus1
ge [120582119891 (119905) |119909 (120591 (119905))|1199011 + 120588 |119890 (119905)|] |119909 (119905)|
minus119901
= [119883120574+ (120574 minus 1) 119884
120574] |119909 (119905)|
minus119901
ge119902
119901(120582119891 (119905))
119901119902
|119909 (120591 (119905))|119901
times (119901
119902 minus 119901120588 |119890 (119905)|)
(119901119902)((119902119901)minus1)
|119909 (119905)|minus119901
= 120582119901119902
1205881minus(119901119902)
119902
119901(
119901
119902 minus 119901)
1minus(119901119902)
|119890 (119905)|1minus(119901119902)
times (119891 (119905))119901119902
(119909 (120591 (119905))
119909 (119905))
119901
ge 119896 (120582 120583 120588)119902
119901(
119901
119902 minus 119901)
1minus(119901119902)
|119890 (119905)|1minus(119901119902)
times (119891 (119905))119901119902
120591119899(119905)
(106)
where the function 119896(120582 120583 120588) is from (18)
(iii) Advanced Case with 119902 gt 119901 Using the same line ofarguments as in the proof of the previous case for all 119905 isin 119869
119899
we obtain
[120582119865 (119905 119909 (120591 (119905))) + 120583119866 (119905 119909 (120590 (119905))) minus 120588119890 (119905)]
times (|119909 (119905)|119901minus1
119909 (119905))minus1
= [120583119866 (119905 119909 (120590 (119905))) minus 120588119890 (119905)] (|119909 (119905)|119901minus1
119909 (119905))minus1
ge 119896 (120582 120583 120588)119902
119901(
119901
119902 minus 119901)
1minus(119901119902)
|119890 (119905)|1minus(119901119902)
times (119892 (119905))119901119902
120590119899(119905)
(107)
where the function 119896(120582 120583 120588) is from (21)
(iv) Superlinear Delay-Advanced Case Since 1199011 1199012gt 119901 for
all 119905 isin 119869119899we obtain
[120582119865 (119905 119909 (120591 (119905))) + 120583119866 (119905 119909 (120590 (119905))) minus 120588119890 (119905)]
times (|119909 (119905)|119901minus1
119909 (119905))minus1
= [120582119865 (119905 119909 (120591 (119905))) minus120588
2119890 (119905)] (|119909 (119905)|
119901minus1119909 (119905))minus1
+ [120583119866 (119905 119909 (120590 (119905))) minus120588
2119890 (119905)] (|119909 (119905)|
119901minus1119909 (119905))minus1
ge [120582119891 (119905) |119909 (120591 (119905))|1199011 +
120588
2|119890 (119905)|] |119909 (119905)|
minus119901
+ [120583119892 (119905) |119909 (120590 (119905))|1199012 +
120588
2|119890 (119905)|] |119909 (119905)|
minus119901
(108)
Now just the same as in the proofs of previous delay andadvanced cases with 119902 gt 119901 and with the help of (104) inparticular for
120574 =1199011
119901gt 1 119883 = (120582119891 (119905))
1199011199011
|119909 (120591 (119905))|119901
119884 = (119901
1199011minus 119901
120588
2|119890 (119905)|)
1199011199011
(109)
we have
[120582119891 (119905) |119909 (120591 (119905))|1199011 +
120588
2 |119890 (119905)|] |119909 (119905)|
minus119901
= [119883120574+ (120574 minus 1) 119884
120574] |119909 (119905)|
minus119901
ge1199011
119901(120582119891 (119905))
1199011199011
|119909 (120591 (119905))|119901
times (119901
1199011minus 119901
120588
2|119890 (119905)|)
(1199011199011)((1199011119901)minus1)
|119909 (119905)|minus119901
= 12058211990111990111205881minus(119901119901
1)1199011
119901(
119901
2 (1199011minus 119901)
)
1minus(1199011199011)
times |119890 (119905)|1minus(119901119901
1)(119891 (119905))
1199011199011
(119909 (120591 (119905))
119909 (119905))
119901
ge 119896 (120582 120583 120588)1199011
119901(
119901
2 (1199011minus 119901)
)
1minus(1199011199011)
times |119890 (119905)|1minus(119901119901
1)(119891 (119905))
1199011199011
120591119899(119905)
(110)
where the function 119896(120582 120583 120588) is from (24) Analogously weshow that
[120583119892 (119905) |119909 (120590 (119905))|1199012 +
120588
2|119890 (119905)|] |119909 (119905)|
minus119901
ge 119896 (120582 120583 120588)1199012
119901(
119901
2 (1199012minus 119901)
)
1minus(1199011199012)
times |119890 (119905)|1minus(119901119901
2)(119891 (119905))
1199011199012
120590119899(119905)
(111)
Discrete Dynamics in Nature and Society 15
Summarizing previous calculation we obtain
[120582119865 (119905 119909 (120591 (119905))) + 120583119866 (119905 119909 (120590 (119905))) minus 120588119890 (119905)]
times (|119909 (119905)|119901minus1
119909 (119905))minus1
ge 119896 (120582 120583 120588)119876119899(119905) 119905 isin 119869
119899
(112)
where the function 119896(120582 120583 120588) is from (24)
(v) Supersublinear Delay-Advanced Case Since 1199011gt 119901 gt 119901
2
and the following well-known elementary inequality holds
12057801199060+ 12057811199061+ 12057821199062ge 1199061205780
01199061205781
11199061205782
2 120578119894ge 0 119906
119894ge 0 (113)
from 1205780 1205781 1205782isin (0 1) 120578
0+ 1205781+ 1205782= 1 and 119901
11205781+ 11990121205782= 119901
we obtain for all 119905 isin 119869119899 for all 119905 isin 119869
119899we obtain
[120582119865 (119905 119909 (120591 (119905))) + 120583119866 (119905 119909 (120590 (119905))) minus 120588119890 (119905)]
times (|119909 (119905)|119901minus1
119909 (119905))minus1
ge [120582119891 (119905) |119909 (120591 (119905))|1199011 + 120583119892 (119905) |119909 (120590 (119905))|
1199012 + 120588 |119890 (119905)|]
times |119909 (119905)|minus119901
= [1205781[120578minus1
1120582119891 (119905) |119909 (120591 (119905))|
1199011] + 120578
2[120578minus1
2120583119892 (119905) |119909 (120590 (119905))|
1199012]
+1205780[120578minus1
0120588 |119890 (119905)|]] |119909 (119905)|
minus119901
ge [120578minus1
1120582119891 (119905) |119909 (120591 (119905))|
1199011]1205781
[120578minus1
2120583119892 (119905) |119909 (120590 (119905))|
1199012]1205782
times [120578minus1
0120588 |119890 (119905)|]
1205780
|119909 (119905)|minus119901
= 120582120578112058312057821205881205780 |119890 (119905)|
1205780(119891 (119905))
1205781
(119892 (119905))1205782
times|119909 (120591 (119905))|
12057811199011
|119909 (119905)|12057811199011
|119909 (120590 (119905))|12057821199012
|119909 (119905)|12057821199012
2
prod
119894=0
120578minus120578119894
119894
= 119896 (120582 120583 120588) |119890 (119905)|1205780(119891 (119905))
1205781
(119892 (119905))1205782
times (119909 (120591 (119905))
119909 (119905))
12057811199011
(119909 (120590 (119905))
119909 (119905))
12057821199012 2
prod
119894=0
120578minus120578119894
119894
ge 119896 (120582 120583 120588) |119890 (119905)|1205780(119891 (119905))
1205781
(119892 (119905))1205782
(120591119899(119905))1205781(1199011119901)
times (120590119899(119905))1205782(1199012119901)
2
prod
119894=0
120578minus120578119894
119894
= 119896 (120582 120583 120588)119876119899(119905)
(114)
where 119896(120582 120583 120588) and 119876119899(119905) are given respectively in (24) and
(25) Thus it is shown that required condition (5) in thecases (i)ndash(iv) is fulfilled with respect to 119896(120582 120583 120588) and 119876
119899(119905)
determined by (18) (21) or (24) and (19) (22) or (25)In conclusion according to the previous observation we
see that all assumptions of Lemma 4 are fulfilled and henceLemma 4 proves Theorems 5 6 and 7
Conflict of Interests
The author declares that there is no conflict of interestsregarding the publication of this paper
References
[1] Y G Sun ldquoA note on Nasrrsquos and Wongrsquos papersrdquo Journal ofMathematical Analysis and Applications vol 286 no 1 pp 363ndash367 2003
[2] Y G Sun C H Ou and J S W Wong ldquoInterval oscillationtheorems for a second-order linear differential equationrdquo Com-puters amp Mathematics with Applications vol 48 no 10-11 pp1693ndash1699 2004
[3] S Murugadass E Thandapani and S Pinelas ldquoOscillationcriteria for forced second-order mixed type quasilinear delaydifferential equationsrdquo Electronic Journal of Differential Equa-tions vol 2010 article 73 9 pages 2010
[4] Y Bai and L Liu ldquoNew oscillation criteria for second-orderdelay differential equations with mixed nonlinearitiesrdquoDiscreteDynamics in Nature and Society vol 2010 Article ID 796256 9pages 2010
[5] A F Guvenilir andA Zafer ldquoSecond-order oscillation of forcedfunctional differential equations with oscillatory potentialsrdquoComputers amp Mathematics with Applications vol 51 no 9-10pp 1395ndash1404 2006
[6] A Zafer ldquoInterval oscillation criteria for second order super-half linear functional differential equations with delay andadvanced argumentsrdquoMathematische Nachrichten vol 282 no9 pp 1334ndash1341 2009
[7] A F Guvenilir ldquoInterval oscillation of second-order functionaldifferential equations with oscillatory potentialsrdquo NonlinearAnalysis Theory Methods amp Applications vol 71 no 12 ppe2849ndashe2854 2009
[8] T S Hassan L Erbe and A Peterson ldquoForced oscillation ofsecond order differential equations with mixed nonlinearitiesrdquoActa Mathematica Scientia B vol 31 no 2 pp 613ndash626 2011
[9] M Pasic ldquoNew oscillation criteria for second-order forcedquasilinear functional differential equationsrdquo Abstract andApplied Analysis vol 2013 Article ID 735360 12 pages 2013
[10] L H Erbe Q Kong and B G Zhang Oscillation Theory forFunctional-Differential Equations vol 190 Marcel Dekker NewYork NY USA 1995
[11] V Kolmanovskii and A Myshkis Introduction to the Theoryand Applications of Functional-Differential Equations vol 463Kluwer Academic Publishers Dordrecht The Netherlands1999
[12] R P Agarwal M Bohner and W-T Li Nonoscillation andOscillation Theory for Functional Differential Equations vol267 Marcel Dekker New York NY USA 2004
[13] L Erbe T Hassan and A Peterson ldquoOscillation of secondorder functional dynamic equationsrdquo International Journal ofDifference Equations vol 5 no 2 pp 175ndash193 2010
[14] B Baculıkova J Dzurina and Y V Rogovchenko ldquoOscillationof third order trinomial delay differential equationsrdquo AppliedMathematics and Computation vol 218 no 13 pp 7023ndash70332012
[15] R P Agarwal L Berezansky E Braverman and A Domoshnit-sky Nonoscillation Theory of Functional Differential Equationswith Applications Springer New York NY USA 2012
16 Discrete Dynamics in Nature and Society
[16] J Zhang ldquoVariational approach to solitary wave solution ofthe generalized Zakharov equationrdquo Computers ampMathematicswith Applications vol 54 no 7-8 pp 1043ndash1046 2007
[17] T Ozis and A Yıldırım ldquoApplication of Hersquos semi-inversemethod to the nonlinear Schrodinger equationrdquo Computers ampMathematics with Applications vol 54 no 7-8 pp 1039ndash10422007
[18] X-C Cai andM-S Li ldquoPeriodic solution of Jacobi elliptic equa-tions by Hersquos perturbation methodrdquo Computers amp Mathematicswith Applications vol 54 no 7-8 pp 1210ndash1212 2007
[19] S Lenci G Menditto and A M Tarantino ldquoHomoclinic andheteroclinic bifurcations in the non-linear dynamics of a beamresting on an elastic substraterdquo International Journal of Non-Linear Mechanics vol 34 no 4 pp 615ndash632 1999
[20] D-J Huang and H-Q Zhang ldquoLink between travelling wavesand first order nonlinear ordinary differential equation with asixth-degree nonlinear termrdquoChaos Solitons amp Fractals vol 29no 4 pp 928ndash941 2006
[21] A I Maimistov ldquoPropagation of an ultimately short electro-magnetic pulse in a nonlinear medium described by the fifth-order Duffing modelrdquo Optics and Spectroscopy vol 94 pp 251ndash257 2003
[22] M N Hamdan and N H Shabaneh ldquoOn the large amplitudefree vibrations of a restrained uniform beam carrying anintermediate lumpedmassrdquo Journal of Sound andVibration vol199 no 5 pp 711ndash736 1997
[23] J-H He ldquoSome asymptotic methods for strongly nonlinearequationsrdquo International Journal of Modern Physics B vol 20no 10 pp 1141ndash1199 2006
[24] J B Marion Classical Dynamics of Particles and Systems 1970[25] I Kovacic and M J Brennan The Duffing Equation Nonlinear
Oscillatos and their Behaviour John Wiley amp Sons 1st edition2011
[26] F C Moon Chaotic Vibrations An Introduction for AppliedScientists and Engineers John Wiley amp Sons New York NYUSA 2004
[27] J J Stoker Nonlinear Vibrations 1950[28] G Chen and Z Tao ldquoAmplitude-frequency relationship for the
relativistic oscillatorrdquoAASRI Procedia vol 1 pp 400ndash403 2012[29] R E Mickens Oscillations in Planar Dynamic Systems World
Scientific Publishing Singapore 1996[30] A Belendez T Belendez C Neipp A Hernandez and M
L Alvarez ldquoApproximate solutions of a nonlinear oscillatortypified as a mass attached to a stretched elastic wire by thehomotopy perturbation methodrdquo Chaos Solitions and Fractalsvol 39 pp 746ndash764 2009
[31] A Belendez E Fernandez R Fuentes J J Rodes and I PascualldquoHarmonic balancing approach to nonlinear oscillations of apunctual charge in the eletric field of charged ringrdquo PhysicsLetters A vol 373 pp 735ndash740 2009
[32] A Elıas-Zuniga ldquoExact solution of the cubic-quintic Duffingoscillatorrdquo Applied Mathematical Modelling vol 37 no 4 pp2574ndash2579 2013
[33] A Belendez M L Alvarez J Frances et al ldquoAnalytical approx-imate solutions for the cubic-quintic Duffing oscillator in termsof elementary functionsrdquo Journal of Applied Mathematics vol2012 Article ID 286290 16 pages 2012
[34] A Elıas-Zuniga OMartınez-Romero andR K Cordoba-DıazldquoApproximate solution for the Duffing-harmonic oscillator bythe enhanced cubication methodrdquo Mathematical Problems inEngineering vol 2012 Article ID 618750 12 pages 2012
[35] C W Lim B S Wu andW P Sun ldquoHigher accuracy analyticalapproximations to the Duffing-harmonic oscillatorrdquo Journal ofSound and Vibration vol 296 no 4-5 pp 1039ndash1045 2006
[36] J He ldquoSome new approaches to Duffing equation with stronglyand high order nonlinearity II parametrized perturbationtechniquerdquo Communications in Nonlinear Science amp NumericalSimulation vol 4 no 1 pp 81ndash83 1999
[37] V Marinca and N Herisanu ldquoPeriodic solutions for somestrongly nonlinear oscillations by Hersquos variational iterationmethodrdquo Computers amp Mathematics with Applications vol 54no 7-8 pp 1188ndash1196 2007
[38] W Lu and Y Liu ldquoVibration control for the primary resonanceof the Duffing oscillator by a time delay state feedbackrdquoInternational Journal of Nonlinear Science vol 8 no 3 pp 324ndash328 2009
[39] H Y Hu and Z H Wang Dynamics of Controlled MechanicalSystems with Delayed Feedback Springer 2002
[40] M Hamdi and M Belhaq ldquoControl of bistability in a delayedDuffing oscillatorrdquo Advances in Acoustics and Vibration vol2012 Article ID 872498 6 pages 2012
[41] V Ravichandran C Chinnathambi and S Rajasekar ldquoNonlin-ear resonance in Duffing oscillator with fixed and integrativetime-delayed feedbacksrdquoPramana Journal of Physics vol 78 pp347ndash360 2013
[42] X You and H Xu ldquoAnalytical approximations for the periodicmotion of theDuffing systemwith delayed feedbackrdquoNumericalAlgorithms vol 56 no 4 pp 561ndash576 2011
[43] Z Sun W Xu X Yang and T Fang ldquoInducing or suppressingchaos in a double-well Duffing oscillator by time delay feed-backrdquo Chaos Solitons and Fractals vol 27 pp 705ndash714 2006
[44] H Wang H Hu and Z Wang ldquoGlobal dynamics of a Duffingoscillator with delayed displacement feedbackrdquo InternationalJournal of Bifurcation and Chaos in Applied Sciences andEngineering vol 14 no 8 pp 2753ndash2775 2004
[45] J Chiasson and J J LoiseauApplications of Time Delay SystemsSpringer 2007
[46] M Lakshmanan andDV SenthilkumarDynamics of NonlinearTime-Delay Systems Springer 2010
[47] G Stepan T Insperger and R Szalai ldquoDelay parametricexcitation and the nonlinear dynamics of cutting processesrdquoInternational Journal of Bifurcation and Chaos in Applied Sci-ences and Engineering vol 15 no 9 pp 2783ndash2798 2005
[48] U van der Heiden and H-O Walther ldquoExistence of chaos incontrol systems with delayed feedbackrdquo Journal of DifferentialEquations vol 47 no 2 pp 273ndash295 1983
[49] Y G Sun and J S W Wong ldquoOscillation criteria for secondorder forced ordinary differential equations with mixed non-linearitiesrdquo Journal of Mathematical Analysis and Applicationsvol 334 no 1 pp 549ndash560 2007
[50] J Heagy and W L Ditto ldquoDynamics of a two-frequencyparametrically driven Duffing oscillatorrdquo Journal of NonlinearScience vol 1 no 4 pp 423ndash455 1991
[51] A B Belogortsev ldquoBifurcations of tori and chaos in thequasiperiodically forced Duffing oscillatorrdquoNonlinearity vol 5no 4 pp 889ndash897 1992
[52] M Belhaq and M Houssni ldquoQuasi-periodic oscillations chaosand suppression of chaos in a nonlinear oscillator driven byparametric and external excitationsrdquo Nonlinear Dynamics vol18 no 1 pp 1ndash24 1999
[53] S H Saker P Y H Pang and R P Agarwal ldquoOscillationtheorems for second order nonlinear functional differential
Discrete Dynamics in Nature and Society 17
equations with dampingrdquo Dynamic Systems and Applicationsvol 12 no 3-4 pp 307ndash321 2003
[54] I N Bronshtein K A Semendyayev G Musiol and HMuehligHandbook of Mathematics Springer 5th edition 2007
[55] M Pasic ldquoFite-Wintner-Leighton-type oscillation criteria forsecond-order differential equations with nonlinear dampingrdquoAbstract and Applied Analysis vol 2013 Article ID 852180 10pages 2013
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
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Differential EquationsInternational Journal of
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Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
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Decision SciencesAdvances in
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Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
2 Discrete Dynamics in Nature and Society
2 Main Assumptions and Results
Let R+= (0infin) For the functions 119903 = 119903(119905) and 119860 = 119860(V)
both appearing in the second-order differential operator of(1) we suppose the following
0 lt 119903 (119905) le 1199030on [1199050infin) 119903 isin 119862
1([1199050infin) R
+) (2)
and 119860(V) is odd increasing and
|119906|minus119901minus1
119860 (V) V ge 120572 (|119860 (V)| |119906|minus119901)
for some 119901 gt 0 and all 119906 = 0 V isin R
120572 = 120572 (119904) 120572 isin 1198621
([0infin) [0infin))
1
1 + 120572isin 1198711(R+R+) 120587
lowast= int
infin
0
2119889119904
1 + 120572 (119904)
120572 (119888119904) ge 119888120574120572 (119904) for some 120574 ge 1 and all 119888 gt 0 119904 ge 0
(3)
For instance it is simple to check that for 120572(119904) = 119904120574
hypothesis (3) is fulfilled in the next three most importantcases of 119860(V) the linear operator 119860(V) = V if 119901 = 1 and120574 = 2 the quasilinear 119901-Laplacian operator 119860(V) = |V|119901minus1V if119901 ge 1 and 120574 = (119901 + 1)119901 and the quasilinear mean curvatureoperator 119860(V) = V(1 + V2)minus12 if 119901 = 1 and 120574 = 2 Somedetails about the number 120587
lowastwhich depends on the function
120572(119904) given in (3) are presented in Section 6The damped term 119861 = 119861(119905 119906 V) satisfies the strong
condition
119861 (119905 119906 V) sgn (119906) ge 0 forall119905 ge 1199050 119906 = 0 V isin R (4)
We hope that (4) can be relaxed with some weaker conditionwhich is commented as an open problem in Section 5 below
In the following fundamental lemma which plays acrucial role in the proof of the main results we are workingwith such solutions 119909(119905) of (1) that satisfies the inequality
[120582119865 (119905 119909 (120591 (119905))) + 120583119866 (119905 119909 (120590 (119905))) minus 120588119890 (119905)]
times (|119909 (119905) |119901minus1
119909 (119905))minus1
ge 119896 (120582 120583 120588)119876 (119905)
(5)
for all 119905 isin 119869 and some interval 119869 where the functions 119896(120582 120583 120588)and 119876(119905) do not depend on 119909(119905) but only on 119865 119866 119890 and theyare determined in the process below The functions 119896(120582 120583 120588)and 119876(119905) present the key point in the parametrically excitedoscillations
Lemma 1 Let assumptions (2) (3) and (4) hold Let (119886 119887) and(119888 119889) be two disjoint open intervals such that 120590(119887) le 120591(119888) Letthe functions 119896 = 119896(120582 120583 120588) 119896 isin 119862([0infin)
3R+) and 119876 =
119876(119905) 119876 isin 119862( 119869 [0infin)) 119876(119905) equiv 0 on 119869 be such that
1199011120574
1199031minus1120574
0
[119896 (120582 120583 120588)]1minus1120574
120587lowast
int119869
119876 (119905) 119889119905 ge (max119905isin 119869
119876 (119905))
1120574
(6)
for both 119869 = (119886 119887) and 119869 = (119888 119889) for all 120582 ge 1205820 120583 ge 120583
0 and
120588 ge 1205880 and for some (120582
0 1205830 1205880) isin R3+ where 119903
0 119901 120574 and 120587
lowast
are constants defined respectively in (2) and (3) Let 119909(119905) be asolution of (1) satisfying the next two statements
119894119891 119909 (119905) gt 0 119909 (120591 (119905)) gt 0 119909 (120590 (119905)) gt 0
119900119899 (120591 (119886) 120590 (119887))
119905ℎ119890119899 119909 (119905) 119904119886119905119894119904119891119894119890119904 (5) 119900119899 (119886 119887)
(7)
119894119891 119909 (119905) lt 0 119909 (120591 (119905)) lt 0 119909 (120590 (119905)) lt 0
119900119899 (120591 (119888) 120590 (119889))
119905ℎ119890119899 (119905) 119904119886119905119894119904119891119894119890119904 (5) 119900119899 (119888 119889)
(8)
Then 119909(119905) has at least one zero point in (120591(120591(119886)) 120590(120590(119889)))
This lemma simultaneously holds for all three types offunctional arguments delay advanced and delay-advancedIt will be proved in Section 6 In Corollaries 9 10 and 11below we give some simple conditions on two functions119891(119905) 119892(119905) and two numbers 119901
1 1199012such that all solutions
of (1) with the functions 119865(119905 119906) = 119891(119905 119906)|119906|1199011 sgn(119906) and
119866(119905 119906) = 119892(119905 119906)|119906|1199012 sgn(119906) satisfy required statements
(7) and (8) with respect to some intervals (119886119899 119887119899) and the
explicitly given functions 119896(120582 120583 120588) and 119876119899(119905) satisfying (6)
where 119890(119905) satisfies a basic assumptionIn what follows (119886
119899 119887119899) 119899 isin N denotes a sequence of
disjoint open intervals such that 1199050
le 120591(119886119899) lt 120590(119887
119899) le
120591(119886119899+1
) lt 120590(119887119899+1
) le sdot sdot sdot and 119886119899rarr infin as 119899 rarr infin Now
we present several variations of Lemma 1 in which essentialinequality (6) is relaxed with some asymptotic assumptionsthat are simpler to be verified in several applications
Lemma 2 Let assumptions (2) (3) and (4) hold Let thecontinuous function 119896 = 119896(120582 120583 120588) gt 0 and the sequence offunctions 119876
119899= 119876119899(119905) 119876
119899isin 119862([119886
119899 119887119899] [0infin)) 119876
119899(119905) equiv 0
on (119886119899 119887119899) satisfy the next two inequalities there are constants
1198880gt 0 119899
0isin N and (120582
0 1205830 1205880) isin R3+such that
119896 (120582 120583 120588) ge 1199030119901minus1(120574minus1)
(2120587lowast
1198880
)
120574(120574minus1)
forall120582 ge 1205820 120583 ge 120583
0 120588 ge 120588
0
(9)
where the numbers 1199030 119901 120574 and 120587
lowastare respectively from (2)
and (3) and
lim119899rarrinfin
(1
(max119905isin[119886119899119887119899]119876119899(119905))1120574
int
119887119899
119886119899
119876119899(119905) 119889119905) ge 119888
0gt 0
(10)
If 119909(119905) is a solution of (1) that satisfies (7) and (8) with 119886 =
1198862119899minus1
119887 = 1198872119899minus1
119888 = 1198862119899 119889 = 119887
2119899 and 119876(119905) = 119876
119899(119905) then 119909(119905)
has at least one zero point in (120591(120591(1198862119899minus1
)) 120590(120590(1198872119899))) forall119899 ge 119899
0
In the following slightly simpler version of Lemma 2inequality (9) is replaced with an asymptotic condition andat the same time the limit in (10) is relaxed with the limitinferiorThus conditions (9) and (10) are replaced with morepractical ones
Discrete Dynamics in Nature and Society 3
Lemma 3 Let assumptions (2) (3) and (4) hold Let thecontinuous function 119896 = 119896(120582 120583 120588) gt 0 and the sequence offunctions 119876
119899= 119876119899(119905) 119876119899isin 119862([119886
119899 119887119899] [0infin)) 119876
119899(119905) equiv 0 on
(119886119899 119887119899) satisfy respectively
119896 (120582 120583 120588) 997888rarr infin 119886119904 120582 997888rarr infin
119900119903 120583 997888rarr infin 119900119903 120588 997888rarr infin
(11)
lim inf119899rarrinfin
(1
(max119905isin[119886119899119887119899]119876119899(119905))1120574
int
119887119899
119886119899
119876119899(119905) 119889119905) gt 0 (12)
If 119909(119905) is a solution of (1) that satisfies (7) and (8) with 119886 =
1198862119899minus1
119887 = 1198872119899minus1
119888 = 1198862119899 119889 = 119887
2119899 and 119876(119905) = 119876
119899(119905) then 119909(119905)
has at least one zero point in (120591(120591(1198862119899minus1
)) 120590(120590(1198872119899))) forall119899 ge 119899
0
and for some 1198990isin N
In some concrete cases we use the next version ofLemmas 2 and 3 where condition (10) or (12) is replaced withappropriate one that appears in (1) with periodic coefficients
Lemma 4 Let assumptions (2) (3) and (4) hold Let thecontinuous function 119896 = 119896(120582 120583 120588) gt 0 and the sequence offunctions 119876
119899= 119876119899(119905) 119876119899isin 119862([119886
119899 119887119899] [0infin)) 119876
119899(119905) equiv 0 on
(119886119899 119887119899) satisfy respectively (11) and for some 119862
0 1198881isin R
0 lt max119905isin[119886119899 119887119899]
119876119899(119905) le 119862
0 int
119887119899
119886119899
119876119899(119905) 119889119905 ge 119888
1gt 0
forall119899 ge 1198990
(13)
and some 1198990isin N If 119909(119905) is a solution of (1) that satisfies
(7) and (8) with 119886 = 1198862119899minus1
119887 = 1198872119899minus1
119888 = 1198862119899 119889 = 119887
2119899
and 119876(119905) = 119876119899(119905) then 119909(119905) has at least one zero point in
(120591(120591(1198862119899minus1
)) 120590(120590(1198872119899))) forall119899 ge 119899
0
Next we suppose that the coefficient 119903(119905) additionallysatisfies
119903 (119904) le 119903 (119905) in delay
119903 (119904) ge 119903 (119905) in advanced case forall119904 le 119905
119903 (119905) is a constant in delay-advanced case
(14)
and the forcing term 119890(119905) satisfies
119890 (119905) le 0 on [120591 (1198862119899minus1
) 120590 (1198872119899minus1
)]
119890 (119905) ge 0 on [120591 (1198862119899) 120590 (119887
2119899)] 119899 isin N
(15)
We remark that 119890(119905) remains arbitrary function outside theset⋃119899(120591(119886119899) 120590(119887119899))
The first result of the paper deals with delay equation (1)
Theorem 5 Let assumptions (2) (3) (4) (14) and (15) hold120591(119905) = 119905 minus 120591 120591 ge 0 120590(119905) equiv 119905 119866(119905 119906) equiv 0 and let 119865(119905 119906) satisfy
119865 (119905 119906) sgn (119906) ge 119891 (119905) |119906|119902
forall119906 = 0 119905 ge 1199050
119891 (119905) ge 0 119891 (119905) equiv 0 119900119899 [120591 (119886119899) 119887119899]
(16)
where 119902 ge 119901 number 119901 is from (3) sequence (119886119899 119887119899) is from
(15) and 119891 isin 119862([1199050infin)R) is a periodic function with period
119879lowastgt 0 such that
[1198862119899minus1
+ 119879lowast 1198872119899minus1
+ 119879lowast] sube [119886
2119899+1 1198872119899+1
]
[1198862119899+ 119879lowast 1198872119899+ 119879lowast] sube [119886
2119899+2 1198872119899+2
] 119899 isin N(17)
Then (1) is oscillatory in the next two cases 119902 = 119901 andparameter 120582 is large enough 119902 gt 119901 120582 gt 0 120588 gt 0 and at leastone of parameters 120582 and 120588 is large enough
The proof ofTheorem 5 is presented in Section 6 and it isbased on Lemma 4 where
119896 (120582 120583 120588) = 120582119901119902
1205881minus(119901119902)
119902 ge 119901 (18)
119876119899(119905)
=
119891 (119905) (119905 minus 119886119899
119905 minus 119886119899+ 120591
)
119901
if 119902 = 119901
119902
119901(
119901
119902 minus 119901)
(119902119901)minus1
|119890 (119905)|1minus(119901119902)
(119891 (119905))119901119902
times(119905 minus 119886119899
119905 minus 119886119899+ 120591
)
119901
if 119902 gt 119901
(19)
The second result deals with advanced equation (1)
Theorem 6 Let assumptions (2) (3) (4) (14) and (15) hold120591(119905) equiv 119905 120590(119905) = 119905 + 120590 120590 ge 0 119865(119905 119906) equiv 0 and let 119866(119905 119906) satisfy
119866 (119905 119906) sgn (119906) ge 119892 (119905) |119906|119902
forall119906 = 0 119905 ge 1199050
119892 (119905) ge 0 119892 (119905) equiv 0 119900119899 [119886119899 120590 (119887119899)]
(20)
where 119902 ge 119901 number 119901 is from (3) sequence (119886119899 119887119899) is from
(15) and 119892 isin 119862([1199050infin)R) is a periodic function with period
119879lowastgt 0 such that (17) is fulfilled Then (1) is oscillatory in the
next two cases 119902 = 119901 and parameter 120583 is large enough 119902 gt 119901120583 gt 0 120588 gt 0 and at least one of parameters 120583 and 120588 is largeenough
The proof of Theorem 6 is based on Lemma 4 (seeSection 6) where
119896 (120582 120583 120588) = 120583119901119902
1205881minus(119901119902)
119902 ge 119901 (21)
119876119899(119905)
=
119892 (119905) (119887119899minus 119905
119887119899minus 119905 + 120590
)
119901
if 119902 = 119901
119902
119901(
119901
119902 minus 119901)
(119902119901)minus1
|119890 (119905)|1minus(119901119902)
(119892 (119905))119901119902
times(119887119899minus 119905
119887119899minus 119905 + 120590
)
119901
if 119902 gt 119901
(22)
The third result deals with delay-advanced equation (1)
4 Discrete Dynamics in Nature and Society
Theorem 7 Let assumptions (2) (3) (4) (14) and (15) hold120591(119905) = 119905 minus 120591 120591 ge 0 120590(119905) = 119905 + 120590 120590 ge 0 and 119865(119905 119906) and 119866(119905 119906)satisfy
119865 (119905 119906) sgn (119906) ge 119891 (119905) |119906|1199011 119866 (119905 119906) sgn (119906) ge 119892 (119905) |119906|
1199012
forall119906 = 0 119905 ge 1199050
119891 (119905) ge 0 119891 (119905) equiv 0 119892 (119905) ge 0
119892 (119905) equiv 0 119900119899 [120591 (119886119899) 120590 (119887
119899)]
(23)
where additionally 119891 119892 119890 isin 119862([1199050infin)R) are three periodic
functions having a common period 119879lowast
gt 0 such that (17)is fulfilled where 119890(119905) is the forcing term in (1) Then (1) isoscillatory provided one of the next two cases is fulfilled wherethe number 119901 is from (3) (1) (in superlinear delay-advancedcase) 119901
1gt 119901 119901
2gt 119901 120588 gt 0 and either parameter 120588 is
large enough or at least one of 120582 and 120583 is large enough (2) (insupersublinear delay-advanced case) 119901
1gt 119901 gt 119901
2gt 0 120582 gt 0
120583 gt 0 120588 gt 0 and at least one of parameters 120582 120583 and 120588 is largeenough
The proof of Theorem 7 is based on Lemma 4 (seeSection 6) where
119896 (120582 120583 120588) =
min 12058211990111990111205881minus(1199011199011) 12058311990111990121205881minus(1199011199012) superlinear case
120582120578112058312057811205881205780
supersublinear case
(24)
119876119899(119905)
=
1199011
119901(
119901
2 (1199011minus 119901)
)
(1199011119901)minus1
times |119890 (119905)|1minus(119901119901
1)(119891 (119905))
1199011199011
120591119899(119905)
+1199012
119901(
119901
2 (1199012minus 119901)
)
(1199012119901)minus1
|119890 (119905)|1minus(119901119901
2)
times (119892 (119905))1199011199012
120590119899(119905)
superlinear case|119890 (119905)|1205780(119891 (119905))
1205781
(119892 (119905))1205782
(120591119899(119905))1205781(1199011119901)
times (120590119899(119905))1205782(1199012119901)
2
prod
119894=0
120578minus120578119894
119894
supersublinear case
(25)
where we denote
120591119899(119905) = (
119905 minus 119886119899
119905 minus 119886119899+ 120591
)
119901
120590119899(119905) = (
119887119899minus 119905
119887119899minus 119905 + 120590
)
119901
(26)
Here the numbers 1205780 1205781 1205782isin (0 1) are chosen such that 120578
0+
1205781+ 1205782= 1 and 119901
11205781+ 11990121205782= 119901 Let us mention that if
1199011= 52 119901 = 1 and 119901
2= 12 and 120578
0= 1205781= 1205782= 13 then
(1199011 1199012) and (120578
0 1205781 1205782) satisfy previous two equalities About
the existence of such (119873+1)-tuple (1205780 1205781 120578
119873) in a general
case we refer to [49]
Remark 8 A difference between assumptions ofTheorems 56 and 7 is that 119890(119905) in Theorems 5 and 6 is not necessarilyperiodic or bounded function as it is supposed inTheorem 7
Now we study an important class of second-order func-tional differential equations as a particular case of (1)
(119903 (119905) 119860 (1199091015840
(119905)))1015840
+ 119861 (119905 119909 (119905) 1199091015840
(119905))
+ 120582119891 (119905) |119909 (120591 (119905))|1199011 sgn (119909 (120591 (119905)))
+ 120583119892 (119905) |119909 (120590 (119905))|1199012 sgn (119909 (120590 (119905))) = 120588119890 (119905) 119905 ge 119905
0
(27)
where 120591(119905) = 119905 minus 120591 120590(119905) = 119905 + 120590 and 120582 120583 120588 120591 120590 ge 0 and1199011 1199012gt 0 Using previous theorems we are able to state
the main consequences showing the parametrically excitedoscillations in (27)
Corollary 9 (delay equation) Let assumptions (2) (3) (4)(14) and (15) hold Let 119892(119905) equiv 0 119891(119905) ge 0 and 119891(119905) equiv 0 on[119886119899minus120591 119887119899] where119891 isin 119862([119905
0infin)R) is a periodic function with
period 119879lowast satisfying (16) Equation (27) is oscillatory in the
following two cases 1199011= 119901 and parameter 120582 is large enough
1199011gt 119901 120582 gt 0 120588 gt 0 and at least one of parameters 120582 and 120588 is
large enough
Corollary 10 (advanced equation) Let assumptions (2) (3)(4) (14) and (15) hold Let 119891(119905) equiv 0 119892(119905) ge 0 and 119892(119905) equiv 0
on [119886119899 119887119899+ 120590] where 119892 isin 119862([119905
0infin)R) is a periodic function
with period 119879lowast satisfying (17) Then (27) is oscillatory in the
following two cases 1199012= 119901 and parameter 120583 is large enough
1199012gt 119901 120583 gt 0 120588 gt 0 and at least one of parameters 120583 and 120588 is
large enough
Corollary 11 (delay-advanced equation) Let assumptions (2)(3) (4) (14) and (15) hold and 119891(119905) and 119892(119905) satisfy
119891 (119905) ge 0 119891 (119905) equiv 0
119892 (119905) ge 0 119892 (119905) equiv 0
119900119899 [119886119899minus 120591 119887119899+ 120590]
(28)
where additionally 119891 119892 119890 isin 119862([1199050infin)R) are three periodic
functions having a common period 119879lowast
gt 0 such that (17)is fulfilled where 119890(119905) is the forcing term in (1) Then (27) isoscillatory in the next two cases where the number 119901 is from(3) (1) (superlinear case) 119901
1gt 119901 119901
2gt 119901 120588 gt 0 and either
parameter 120588 is large enough or at least one of 120582 and 120583 is largeenough (2) (supersublinear case) 119901
1gt 119901 gt 119901
2gt 0 120582 gt 0
120583 gt 0 120588 gt 0 and at least one of parameters 120582 120583 and 120588 is largeenough
According to previous corollaries we can derive thefollowing examples
Example 12 (delay case) Let 119901 ge 1 120576 ge 0 and 119898 isin N
be fixed and 120591119898
isin R 0 le 120591119898
lt 120587(4119898) With the help of
Discrete Dynamics in Nature and Society 5
Corollary 9 the following two different classes of quasilineardelay differential equations
(100381610038161003816100381610038161199091015840
(119905)10038161003816100381610038161003816
119901minus1
1199091015840
(119905))1015840
+ 120582 sin (119898119905) 1003816100381610038161003816119909 (119905 minus 120591119898)1003816100381610038161003816119901+120576
times sgn (119909 (119905 minus 120591119898)) = minus120588 cos (2119898119905)
(1199091015840(119905)
radic1 + 11990910158402
(119905)
)
1015840
+ 120582 cos (119898119905) 1003816100381610038161003816119909 (119905 minus 120591119898)10038161003816100381610038161+120576
times sgn (119909 (119905 minus 120591119898)) = minus120588 cos (2119898119905)
(29)
are oscillatory provided at least one of 120582 gt 0 and 120588 gt 0 is largeenough (the case 120588 = 0 is possible if 120576 = 0) It is because forall 119899 isin N we have
minus cos (2119898119905)
le 0 on [2119899120587
1198982119899120587
119898+
120587
4119898]
ge 0 on [2119899120587
119898+
120587
41198982119899120587
119898+
120587
2119898]
sin (119898119905) ge 0 cos (119898119905) ge 0 on [2119899120587
1198982119899120587
119898+
120587
2119898]
[2119899120587
119898+ 120591119898+ 119879lowast2119899120587
119898+
120587
4119898+ 119879lowast]
= [(2119899 + 2) 120587
119898+ 120591119898(2119899 + 2) 120587
119898+
120587
4119898]
[2119899120587
119898+
120587
4119898+ 120591119898+ 119879lowast2119899120587
119898+
120587
2119898+ 119879lowast]
= [(2119899 + 2) 120587
119898+
120587
4119898+ 120591119898(2119899 + 2) 120587
119898+
120587
2119898]
(30)
where 119879lowast= 2120587119898 is the common period of the functions
sin(119898119905) and cos(119898119905) Thus in order to apply Corollary 9 wecan choose 119886
2119899minus1= 2119899120587119898 + 120591
119898 1198872119899minus1
= 2119899120587119898 + 120587(4119898)1198862119899= 2119899120587119898 + 120587(4119898) + 120591
119898 and 119887
2119899= 2119899120587119898 + 120587(2119898)
Example 13 (advanced case) Let 119901 ge 1 120576 ge 0 and 119898 isin
N be fixed and 120590119898
isin R 0 le 120590119898
lt 120587(4119898) With thehelp of Corollary 10 the following two classes of quasilinearadvanced differential equations
(100381610038161003816100381610038161199091015840
(119905)10038161003816100381610038161003816
119901minus1
1199091015840
(119905))1015840
+ 120583 sin (119898119905) 1003816100381610038161003816119909 (119905 + 120590119898)1003816100381610038161003816119901+120576
times sgn (119909 (119905 + 120590119898)) = minus120588 cos (2119898119905)
(1199091015840(119905)
radic1 + 11990910158402
(119905)
)
1015840
+ 120583 cos (119898119905) 1003816100381610038161003816119909 (119905 + 120590119898)10038161003816100381610038161+120576
times sgn (119909 (119905 + 120590119898)) = minus120588 cos (2119898119905)
(31)
are oscillatory provided at least one of 120583 gt 0 and 120588 gt 0 islarge enough (the case 120588 = 0 is possible if 120576 = 0) In order toapply Corollary 10 we can choose 119886
2119899minus1= 2119899120587119898 119887
2119899minus1=
2119899120587119898 + 120587(4119898) minus 120590119898 1198862119899
= 2119899120587119898 + 120587(4119898) and 1198872119899
=
2119899120587119898 + 120587(2119898) minus 120590119898
Example 14 (delay-advanced case) Let 119901 ge 1 1205761gt 0 1205762gt 0
and 119898 isin N be fixed and 120591119898ge 0 and 120590
119898ge 0 0 le 120591
119898+ 120590119898lt
120587(4119898) With the help of Corollary 11 the following class ofquasilinear delay-advanced differential equations
(100381610038161003816100381610038161199091015840
(119905)10038161003816100381610038161003816
119901minus1
1199091015840
(119905))1015840
+ 120582 sin (119898119905)
times1003816100381610038161003816119909 (119905 minus 120591119898)
1003816100381610038161003816119901+1205761 sgn (119909 (119905 minus 120591
119898))
+ 120583 cos (119898119905) 1003816100381610038161003816119909 (119905 + 120590119898)1003816100381610038161003816119901+1205762
times sgn (119909 (119905 + 120590119898)) = minus120588 cos (2119898119905)
(32)
is oscillatory provided either 120588 gt 0 is large enough or at leastone of 120582 gt 0 and 120583 gt 0 is large enough In order to applyCorollary 11 we can choose 119886
2119899minus1= 2119899120587119898 + 120591
119898 1198872119899minus1
=
2119899120587119898 + 120587(4119898) minus 120590119898 1198862119899
= 2119899120587119898 + 120587(4119898) + 120591119898 and
1198872119899= 2119899120587119898 + 120587(2119898) minus 120590
119898
3 Application to Duffing Equations withTime Delay Feedback
Let 120582 ge 0 denote the control gain parameter (often calledldquodisplacement feedback coefficientrdquo) 120591 gt 0 the time delayand 120588 ge 0 and 120596 gt 0 the amplitude and frequency of theexternal force respectively Let the function Φ = Φ(119905 119906) thatwill appear in the delay feedback term Φ(119905 119909(119905 minus 120591)) satisfythe general condition
Φ (119905 119906) sgn (119906) ge 1206010|119906|119902
forall119905 ge 1199050
119906 = 0 and some 119902 ge 1 1206010gt 0
(33)
For instance Φ(119905 119906) = 1199062119898minus1 119898 isin N or more general
Φ(119905 119906) = sum119898
119896=11206011198961199062119896minus1 120601
119896gt 0119898 isin N
In this section we consider the following large class ofundamped possible nonautonomous and nonconservativeDuffing equations without or with the general time delayfeedback Φ(119905 119909(119905 minus 120591))
(10038161003816100381610038161003816119909101584010038161003816100381610038161003816
119901minus1
1199091015840)1015840
+ 1205962
0119909 +
1205831|119909|1199031 sgn (119909)
(1205832+ 12058331199092)1199032
+
119898
sum
119894=1
120573119894(119905) |119909|
120572119894minus1119909 + 120582Φ (119905 119909 (119905 minus 120591)) = 120588 cos (120596119905)
(34)
where 1205960is the natural frequency 120583
1ge 0 is the density of the
nonlinear potential (or rigidity coefficient) and 1205832 1205833 1199031 1199032
are nonnegative constants 120573119894(119905) ge 0 and 120572
119894ge 1
When 119901 = 1 120582 = 0 and 120573119894(119905) equiv 120573
119894= const
(34) contains many most important classes of undampedautonomous Duffing oscillators such as the following
(i) the strongly nonlinear Duffing oscillator with smoothodd nonlinearity is given in (34) provided 120583
1= 0 and
120572119894= 2119894 + 1 let us recall some of its known particular
cases
(a) the classic Duffing oscillator 11990910158401015840 +12059620119909+120573119909
3= 0
has been recently studied in the searching of
6 Discrete Dynamics in Nature and Society
solitarywave solutions of classic and generalizedZakharov equations of plasma physics (see [16])and of nonlinear Schrodinger equation (see[17]) also it is strongly connected with theJacobi elliptic equation (see [18])
(b) the cubic-quintic oscillator 11990910158401015840 + 1205962
0119909 + 120573
11199093+
12057321199095
= 0 is used as a model for the non-linear dynamics of a slender elastica (see [19])in nonlinear wave systems (see [20]) for thepropagation of a short electromagnetic pulsein a nonlinear medium (see [21]) and in theunimodal Duffing temporal problem (see [22])
(c) the cubic truly nonlinear oscillator 11990910158401015840 + 1205731199093=
0 models the motion of a ball bearing thatoscillates in a glass tube that is bent into acurve (see [23]) as well as the motion of a massattached to identical stretched elastic wires (see[24])
(d) the nonhomogeneous Duffing oscillator 11990910158401015840 +1205962
0119909 + 120573119909
3= 120588 cos(120596119905) describes various forced
vibrations of beams springs with nonlinearstiffness cables plates shells and optical fibresin electrical circuits in nonlinear isolators andso forth (see for instance [25 26])
(ii) the general Duffing-harmonic oscillator (with rationalor irrational nonlinear restoring-force) is given in(30) if 120583
1= 0 120573119894= 0 and 120588 = 0 the most known
subclasses of these oscillators are
(a) the classic Duffing-harmonic oscillator 11990910158401015840+
(12058311199093(1205832+ 12058331199092)) = 0 which models many
conservative nonlinear oscillatory systems see[27]
(b) the relativistic harmonic oscillator11990910158401015840+ (1205831119909radic1 + 1199092) = 0 see [28]
(c) the nonlinear oscillator11990910158401015840+119909minus(1205831119909radic1 + 1199092) =
0 1205831
isin [0 1] which is typified as a massattached to a stretched elastic wire see [29 30]
(d) the nonlinear oscillator 11990910158401015840
+
(1205831119909(radic(1 + 1199092 )
3
) = 0 which presentsnonlinear oscillations of a punctual charge inthe electric field of charged ring see [31]
Finding several explicit forms of periodic approximate solu-tions for these oscillators has been intensively studied lastyears by many authors see for instance [28 30 32ndash37] andalso the references therein
When 120582 = 0 and linear time delay feedbackΦ(119905 119909(119905minus120591)) =119909(119905 minus 120591) the following topics have been studied for varioustypes of Duffing oscillators with time delayed feedback in[38] authors constructed a low-order approximate solutionunder weak feedback gain parameter about the low- andhigh-order approximations see also [39] in [40] with 120588 = 0the Hopf bifurcation diagrams have been explored for theapproximate periodic solutions (amplitude versus time delay120591 and feedback gain 120582 versus time delay 120591) moreover in [41]
authors made an analysis on the effect of the control gainand time delay parameters on the amplitude of approximateperiod solution from the theoretical and numerical pointsof view see also [42] in [43] authors studied the chaoticbehaviour with respect to gains and time delay parameterssee also [44]
Equations under time delay control such as (34) (espe-cially with damped term) are used as a model for variouscontrolled physical mechanical and engineering systemswith time delays see for instance [39 45ndash48] and thereferences therein
Here (34) contains very general nonlinear time delayfeedback Φ(119905 119909(119905 minus 120591)) with Φ satisfying (33) and the lineartime delay feedback 119909(119905 minus 120591) is only a particular case ofit and to the best of our knowledge the previous topicsare not considered for (34) as yet Moreover with suchan Φ the oscillations of (34) can be taken under a doubteven with the linear time delay feedback (see the nature ofthe approximations given in [38 39]) Hence we can posethe following question under what conditions on equationrsquosparameters (34) is a nonlinear oscillator that is possessesonly oscillatory solutions An answer is given in the nextresult as an easy consequence of the parametrically excitedoscillations by Theorem 5
Theorem 15 Let 120591 isin (0 120587120596) and (33) hold Equation (34) isoscillatory in the next two cases
(i) 119902 = 119901 and 120582 is large enough(ii) 119902 gt 119901 120582 gt 0 120588 gt 0 and at least one of 120582 and 120588 is large
enough
Proof Let 119903(119905) equiv 1 119860(V) = |V|119901minus1 119865(119905 119906) = Φ(119905 119906) 119866(119905 119906) equiv0 119890(119905) = cos(120596119905) and
119861 (119905 119906 V) = 1205962
0119906 +
1205831|119906|1199031s119892119899 (119906)
(1205832+ 12058331199062)1199032
+
119898
sum
119894=1
120573119894(119905) |119906|
120572119894minus1119906 (35)
It is easy to check that all assumptions of Theorem 5 arefulfilled with respect to the sequence 119886
119899= minus1205872120596 + 119899120587120596 + 120591
and 119887119899= 1205872120596 + 119899120587120596 + 120591 where 119886
119899lt 119887119899since it is supposed
that 120591 lt 120587120596 Hence Theorem 5 proves this theorem
Remark 16 Even in the linear forced case (119890(119905) equiv 0) it isnot easy to establish the oscillations of all solutions since theoscillation and nonoscillation can occur simultaneously Themost simple and important example for the coincidence ofoscillation and nonoscillation is the following linear forceddifferential equation 11990910158401015840 + (2119905)119909
1015840+ 119909 = 2119905 119905 gt 0 that
allows an oscillatory solution 1199091(119905) = (3 sin 119905)119905 + 2119905 and a
nonoscillatory solution 1199092(119905) = 2119905 This is not possible in
the linear case with 119890(119905) equiv 0 because of Sturmrsquos separationtheorem
4 Parametrically Excited Oscillations andWell-Known Oscillation Criteria
In this section we would like to draw the readerrsquos attentionto the fact that the parametrically excited oscillations have
Discrete Dynamics in Nature and Society 7
been already appearing in some published papers on theoscillation of functional differential equations but only insome examples illustrating certain main oscillation criteriaHowever with the help of our main results in which theparametrically excited oscillations are studied in a generalsetting the equations from these examples are replaced withgeneral ones also having parameters 120582 and 120583
In [1] (see also [2 Example 31] with 120591 = 0 [3 Example31] and [4 Section 3]) the author considers the oscillationof the second-order delay differential equation
11990910158401015840
(119905) + 119891 (119905) |119909 (120591 (119905))|120574 sgn119909 (120591 (119905)) = 119890 (119905) (36)
in the linear case (120574 = 1) and the superlinear (120574 gt 1)In the linear case (analogously for the superlinear case see[1 Theorem 2]) the author proved the following oscillationcriterion In what follows we denote
119863 (119886 119887) = 119906 isin 1198621
([119886 119887] R) 119906 (119905) equiv 0 119906 (119886) = 119906 (119887) = 0
(37)
Theorem 17 ([1 Theorem 1]) Suppose that for any 119879 ge 0there exist constants 119886
1 1198871 1198862 1198872such that 119879 le 119886
1lt 1198871 119879 le
1198862lt 1198872 and 119891(119905) ge 0 on [120591(119886
1) 1198871] cup [120591(119886
2) 1198872] 119890(119905) le 0
on [120591(1198861) 1198871] and 119890(119905) ge 0 on [120591(119886
2) 1198872] If there exists 119906 isin
119863(119886119894 119887119894) 119894 = 1 2 such that
int
119887119894
119886119894
[1199062
(119905) 119891 (119905)120591 (119905) minus 120591 (119886
119894)
119905 minus 120591 (119886119894)
minus (1199061015840
(119905))2
]119889119905 ge 0 (38)
then (36) with 120574 = 1 is oscillatory
Previous criterion has been applied on the followingparticular equation
11990910158401015840
(119905) + 120582 sin (119905)1003816100381610038161003816100381610038161003816119909 (119905 minus
120587
4)
1003816100381610038161003816100381610038161003816
120574
times sgn 119909(119905 minus120587
4) = cos (119905) 119905 ge 0
(39)
where 120582 ge 0 and 120574 = 1 Applying Theorem 17 to (39) theauthor proved that (39) is oscillatory provided the followinginequality
120582int
119887119894
119886119894
sin2 (2119905) cos2 (2119905) sin (119905)119905 minus 119886119894
119905 minus 119886119894+ 1205874
119889119905 ge120587
2 (40)
holds for sufficiently large 120582 Thus the oscillation of (39) isexcited by the large enough parameter 120582 However accordingto Theorems 5 and 6 we are able to show that the nextparametric equation that corresponds to general equation(36)
11990910158401015840
(119905) + 120582119891 (119905) |119909 (120591 (119905))|120574 sgn119909 (120591 (119905)) = 119890 (119905) (41)
is oscillatory provided 120582 is large enough where 1199011= 1199012= 120574
120583 = 0 and 120588 = 1Next in [5] (see also [6ndash8]) the authors consider the
oscillation of the following class of second-order differentialequations with delay and advanced arguments
(119903 (119905) 1199091015840
(119905))1015840
+ 119891 (119905) |119909 (120591 (119905))|1199011 sgn119909 (120591 (119905))
+ 119892 (119905) |119909 (120590 (119905))|1199012 sgn119909 (120590 (119905)) = 119890 (119905) 119905 ge 0
(42)
where 1199011 1199012ge 1 When 119901
1= 1199012= 1 the authors prove the
following result (for other cases see [5Theorems 32 33 and34]
Theorem 18 ([5 Theorem 31]) Suppose that for any 119879 ge
0 there exist intervals [120591(1198861) 1198871] [120591(119886
2) 1198872] [1198881 120590(1198891)] and
[1198882 120590(1198892)] contained in [119879infin) such that 119886
1lt 1198871 1198862lt 1198872
1198881lt 1198891 1198882lt 1198892 and
119891 (119905) ge 0 119900119899 [120591 (1198861) 1198871] cup [120591 (119886
2) 1198872]
119892 (119905) ge 0 119900119899 [1198881 120590 (1198891)] cup [119888
2 120590 (1198892)]
119890 (119905) le 0 119900119899 [120591 (1198861) 1198871] cup [1198881 120590 (1198891)]
119890 (119905) ge 0 119900119899 [120591 (1198862) 1198872] cup [1198882 120590 (1198892)]
(43)
and 119888119894= 120591(119886
119894) 119889119894= 119886119894 and 119887
119894= 120590(119889
119894) 119894 = 1 2 If there exist
1199061isin 119863(119886
119894 119887119894) and 119906
2isin 119863(119888119894 119889119894) such that either
int
119887119894
119886119894
[1199062
1(119905) 119891 (119905)
120591 (119905) minus 120591 (119886119894)
119905 minus 120591 (119886119894)
minus (1199061015840
1(119905))2
119903 (119905)] 119889119905 ge 0 (44)
or
int
119889119894
119888119894
[1199062
2(119905) 119891 (119905)
120590 (119889119894) minus 120590 (119905)
120590 (119889119894) minus 119905
minus (1199061015840
2(119905))2
119903 (119905)] 119889119905 ge 0 (45)
for 119894 = 1 2 then (42) with 1199011= 1199012= 1 is oscillatory
As a consequence of this result it has been concluded thatthe particular equation
(119903 (119905) 1199091015840
(119905))1015840
+ 120582 sin (119905) 119909 (119905 minus 120587
12)
+ 120583 cos (119905) 119909 (119905 + 120587
6) = cos (2119905) 119905 ge 0
(46)
is oscillatory provided either 120582 or 120583 is large enough Howeverby following Theorems 5 and 6 one can obtain the sameconclusion for the following general equation associated with(42)
(119903 (119905) 1199091015840
(119905))1015840
+ 120582119891 (119905) |119909 (120591 (119905))|1199011 sgn119909 (120591 (119905))
+ 120583119892 (119905) |119909 (120590 (119905))|1199012 sgn119909 (120590 (119905)) = 119890 (119905)
(47)
Related observation can be done with [8 Example 33]and [9 Example 21] where the quasilinear second-orderfunctional differential equations have been considered It isleft to the reader
5 Some Open Questions and Comments
In this section we discuss some problems related to ourmainresults that are not studied here
(1) Quasiperiodic Case In the theory of nonlinear oscillatorsa particularly important case occurs when the periodiccoefficients in the oscillator do not have any common periodIt is called the quasiperiodic (or two-frequency) nonlinear
8 Discrete Dynamics in Nature and Society
oscillator and studied for instance in [50ndash52] Since inTheorems 5 6 and 7 we assume that the correspondingperiodic functions have a commonperiod it is natural to posethe next question
Open Question 1 Is it possible to derive sufficient conditionsfor the oscillation of (27) in the casewhen119891(119905) and119892(119905) (resp119891(119905) 119892(119905) and ℎ(119905)) are two (resp three) periodic functionsnot having a common period
(2) Equation with More Functional Arguments Next regard-ing some second-order functional differential equationsconsidered in the references of this paper more than twononlinear functional terms are appearing and thereforeinstead of main equation (1) and corresponding particularequation (27) considered inTheorems 5 6 and 7 we suggestthe following classes of equations
(119903 (119905) 119860 (1199091015840
(119905)))1015840
+ 119861 (119905 119909 (119905) 1199091015840
(119905))
+
1198981
sum
119896=1
120582119896119865119896(119905 119909 (120591
119896(119905)))
+
1198982
sum
119896=1
120583119896119866119896(119905 119909 (120590
119896(119905))) = 120588119890 (119905)
(48)
where 0 le 120591119896(119905) le 119905 lim
119905rarrinfin120591119896(119905) = infin 120590
119896(119905) ge 119905 119898
1 1198982isin
N and
(119903 (119905) 119860 (1199091015840
(119905)))1015840
+ 119861 (119905 119909 (119905) 1199091015840
(119905))
+
1198981
sum
119896=1
120582119896119891119896(119905)
1003816100381610038161003816119909 (119905 minus 120591119896)1003816100381610038161003816119901119896 sgn (119909 (119905 minus 120591
119896))
+
1198982
sum
119896=1
120583119896119892119896(119905)
1003816100381610038161003816119909 (119905 + 120590119896)1003816100381610038161003816119902119896 sgn (119909 (119905 + 120590
119896)) = 120588119890 (119905)
(49)
where 120582119896 120583119896 120588 120591119896 120590119896ge 0 and 119901
119896 119902119896gt 0
Comment We suggest the reader to enlarge the main resultsof this paper to (48) and (49)
(3) Damped Duffing Equation In the application the Duffingequation (34) is often appearing with the linear damped term1199091015840(119905) that is
11990910158401015840+ 11988901199091015840+ 1205962
0119909 + 120573119909
3+ 120582Φ (119909 (119905 minus 120591)) = 120588 cos (120596119905) (50)
where 1198890
is the damped coefficient which can in anactive way influence various behaviours of (50) Since119861(119905 119909(119905) 119909
1015840(119905)) = 119889
01199091015840(119905) does not satisfy the required
assumption (4) we are not able to apply our main results to(50) Hence we pose the following questionOpen Question 2 Is it possible to obtain the parametricallyexcited oscillation for (1) in the case when the damped term119861(119905 119906 V) satisfies a larger condition than (4) in which thelinear damped term 120573119909
1015840(119905) is especially included
(4) Functional Argument in Damped Term In a class of Duff-ing equations we have two time delayed feedback and hence
besides the control gain parameter 1205821another parameter 120582
2
appears the so-called velocity gain parameter Hence insteadof (34) one can consider
11990910158401015840+ 11988901199091015840+ 1205962
0119909 + 120573119909
3+ 1205821119909 (119905 minus 120591)
+ 12058221199091015840
(119905 minus 120591) = 120588 cos (120596119905) (51)
Therefore we suggest the following problem for further studyOpen Question 3 Is it possible to obtain the parametricallyexcited oscillation for the following more general functionaldifferential equation than (1) in which the functional argu-ment appears in the damped term too as follows
(119903 (119905) 119860 (1199091015840
(119905)))1015840
+ 119861 (119905 119909 (119905) 1199091015840
(119905)) + 1205821119865 (119905 119909 (120591 (119905)))
+ 1205822119867(119905 119909
1015840
(120591 (119905))) = 120588119890 (119905) 119905 ge 1199050
(52)or
(119903 (119905) 119860 (1199091015840
(119905)))1015840
+ 119861 (119905 119909 (119905) 1199091015840
(119905)) + 1205831119866 (119905 119909 (120590 (119905)))
+ 1205832119867(119905 119909
1015840
(120590 (119905))) = 120588119890 (119905) 119905 ge 1199050
(53)
About known oscillation criteria for the second-order func-tional differential equations having the functional argumentin the damped term we refer the reader to for instance [53]and the references therein
6 Proofs of Main Results
The proof of Lemma 1 is based on the following three stepstwo working forms of condition (6) (see Lemmas 19 and 20)the existence of an explosive solution of a suitable Riccatidifferential inequality (see Proposition 22) and a comparisonprinciple (see Proposition 24)
Lemma 19 (a necessary condition to (6)) Let 0 lt 119903(119905) le 1199030
on [1199050infin) If assumption (6) is fulfilled then there is a positive
real number 120576 such that1
120587lowast
int119869
min119901
(120576119903 (119905))120574minus1
120576119896 (120582 120583 120588)119876 (119905) 119889119905 ge 1 (54)
for all 120582 ge 1205820 120583 ge 120583
0 and 120588 ge 120588
0and some (120582
0 1205830 1205880) isin R3+
Proof Since 0 lt 119903(119905) le 1199030for 119905 ge 119905
0 we conclude that for
120576 = (119901
119903120574minus1
0119896 (120582 120583 120588)max
119905isin 119869119876 (119905)
)
1120574
(120582 120583 120588) isin R3
+
(55)
it holds that 119901(120576119903(119905))120574minus1
ge 119901(1205761199030)120574minus1
= 120576119896(120582 120583 120588)
max119905isin 119869119876(119905) ge 120576119896(120582 120583 120588)119876(119905) 119905 isin 119869 and hence
int119869
min119901
(120576119903 (119905))120574minus1
120576119896 (120582 120583 120588)119876 (119905) 119889119905
= 120576119896 (120582 120583 120588) int119869
119876 (119905) 119889119905
(56)
Discrete Dynamics in Nature and Society 9
On the other hand from (6) we observe
1
120587lowast
int119869
119876 (119905) 119889119905 ge1199031minus(1120574)
0
1199011120574[119896 (120582 120583 120588)]1minus(1120574)
(max119905isin 119869
119876 (119905))
1120574
(57)
which together with (55) and (56) gives
1
120587lowast
int119869
min119901
(120576119903 (119905))120574minus1
120576119896 (120582 120583 120588)119876 (119905) 119889119905
= 120576119896 (120582 120583 120588)1
120587lowast
int119869
119876 (119905) 119889119905
ge 1205761199031minus(1120574)
0
1199011120574[119896 (120582 120583 120588)]
1120574
(max119905isin 119869
119876 (119905))
1120574
= 1
(58)
for all 119899 ge 1198990 120582 ge 120582
0 120583 ge 120583
0 and 120588 ge 120588
0 It proves this
lemma
Lemma 20 (an equivalent condition to (54)) Assumption(54) is fulfilled if and only if there is a real number 120576 gt 0 and acontinuous function 119870(119905) ge 0 119905 isin 119869 such that
1198880= int119869
119870 (119905) 119889119905 gt 0119870 (119905)
1198880
le1
120587lowast
timesmin119901
(120576119903 (119905))120574minus1
120576119896 (120582 120583 120588)119876 (119905)
(59)
for all 119905 isin 119869 120582 ge 1205820 120583 ge 120583
0 and 120588 ge 120588
0and some (120582
0 1205830 1205880) isin
R3+
Proof This proof is very elementary Indeed if (54) holdsthen the function119870(119905) and number 119888
0 defined by
119870 (119905) =1
120587lowast
min119901
(120576119903 (119905))120574minus1
120576119896 (120582 120583 120588)119876 (119905)
1198880= int119869
119870 (119905) 119889119905
(60)
obviously satisfy 1198880
ge 1 and 119870(119905)1198880
le 119870(119905) = (1120587lowast)
min119901(120576119903(119905))120574minus1 120576119896(120582 120583 120588)119876(119905) which shows (59) Con-versely if (59) holds then integrating both sides of thesecond inequality in (59) we obtain
int119869
1
120587lowast
min119901
(120576119903 (119905))120574minus1
120576119896 (120582 120583 120588)119876 (119905) ge int119869
119870 (119905)
1198880
119889119905 = 1
(61)
which shows (54)
In conclusion according to previous two lemmas wesee that supposed condition (6) implies (59) which plays animportant role in the proof of the main results
The second step in the proof of Lemma 1 is to prove theexistence of a function 120596(119905) which blows up in the finitetime and satisfies a generalized Riccati differential lowerinequality we briefly present the existence and properties
of the so-called generalized tangent type function In whatfollows let 120587
lowastbe a positive real number defined in (3) Let us
remark that 120572(119904) = 119904120574 120574 gt 1 implies 120587
lowast= (2120587)(120574 sin(120587120574))
see for instance [54] and obviously for 120574 = 2wehave120587lowast= 120587
Lemma 21 Let 120572 [0infin) rarr [0infin) be a continuousfunction such that
int
infin
0
119889120591
1 + 120572 (120591)lt infin (62)
Then there is a real number 120587lowastgt 0 and a function 119911 = 119911(119904)
119911 isin 1198621((minus120587lowast2 120587lowast2)R) such that
119889119911
119889119904= 1 + 120572 (|119911 (119904)|) 119904 isin (minus
120587lowast
2120587lowast
2)
119911 (0) = 0
(63)
Moreover 119911(119904) is increasing and odd
lim119904rarr120587lowast2
119911 (119904) = infin 120587lowast=
2120587
120574 sin (120587120574)for 120572 (119904) = 119904
120574
120574 gt 1
(64)
In particular for 120572(119904) = 1199042 one can take 119911(119904) = tan(119904) and
120587lowast= 120587
Proof Let 119885 = 119885(119905) 119905 isin R be a function defined by
119885 (119905) = int
119905
0
1
1 + 120572 (|120591|)119889120591 119905 isin R (65)
The function 119885(119905) is well defined since 120572(119904) is positive andcontinuous on [0infin) 119885(119905) is increasing and odd functionand
119889119885
119889119905=
1
1 + 120572 (|119905|) 119905 isin R
119885 (0) = 0 119885 isin 1198621
(RR)
(66)
Moreover because of (62) there is a real number 120587lowastgt 0 such
that120587lowast
2= int
infin
0
119889120591
1 + 120572 (120591) (67)
Thus 119885 R rarr (minus120587lowast2 120587lowast2) and there exists an inverse
function 119885minus1 = 119885minus1(119904) of the original function 119885 = 119885(119905) and
119885minus1
(minus120587lowast2 120587lowast2) rarr R Also from 119885(119885
minus1(119904)) = 119904 and
119889119885119889119905 = 0 onR we also derive that119889119885minus1119889119904 = 0 on its domain(minus120587lowast2 120587lowast2) and
119889119885
119889119905(119885minus1
(119904)) =1
(119889119885minus1119889119904) 119904 isin (minus
120587lowast
2120587lowast
2) (68)
Putting 119905 = 119885minus1(119904) for 119904 isin (minus120587
lowast2 120587lowast2) into (66) and using
(68) we easily obtain
119889119885minus1
119889119904= 1 + 120572 (
10038161003816100381610038161003816119885minus1
(119904)10038161003816100381610038161003816) 119904 isin (minus
120587lowast
2120587lowast
2)
119885minus1
(0) = 0 119885minus1isin 1198621((minus
120587lowast
2120587lowast
2) R)
(69)
10 Discrete Dynamics in Nature and Society
Moreover from (67) we have lim119904rarr120587lowast2119885minus1(119904) = 119885
minus1
(lim119905rarrinfin
119885(119905)) = lim119905rarrinfin
119885minus1119885(119905) = lim
119905rarrinfin119905 = infin Thus
if we set 119911(119904) = 119885minus1(119904) then previous two statements and
(67) prove this lemma
Next we prove the main result of this section
Proposition 22 Let (2) and (6) hold where 119869 = (119886 119887) Let 120576 gt0 be a real number and let119870(119905) ge 0 119905 isin [119886 119887] be a continuousfunction both obtained in Lemma 20 Let 120587
lowastbe from (3) and
1198880from (59) and let 119877
119886isin R be an arbitrary real number If
119911 = 119911(119904) is the generalized tangens function defined in (63)and 119881(119905) is a function defined by
119881 (119905) =120587lowast
1198880
int
119905
119886
119870 (120591) 119889120591 + 119911minus1(119877119886) 119905 isin [119886 119887] (70)
then there is a 119879lowast119886isin [119886 119887) such that
119881 (119879lowast
119886) =
120587lowast
2 119881 ([119886 119879
lowast
119886)) sub (minus
120587lowast
2120587lowast
2) (71)
Moreover for a function 120596(119905) defined by120596 (119905) = 119911 (119881 (119905)) 119905 isin [119886 119879
lowast
119886) (72)
one has 120596(119886) = 119877119886 lim119905rarr119879
lowast
119886
120596(119905) = infin and
119889120596
119889119905le
119901
(120576119903 (119905))120574minus1
120572 (1003816100381610038161003816120596 (119905)
1003816100381610038161003816)
+ 120576119896 (120582 120583 120588)119876 (119905) 119905 isin (119886 119879lowast
119886)
(73)
where the numbers 119901 and 120574 are from (3) and the functions119896(120582 120583 120588) and 119876(119905) are from (6)
Proof Under assumptions (2) and (6) and because of Lem-mas 19 and 20 we obtain 120576 gt 0 and 119870(119905) gt 0 119905 isin [119886 119887]satisfying inequality (59)
Next since 119911minus1(119877119886) isin (minus120587
lowast2 120587lowast2) (see Lemma 21)
from (70) we directly obtain
119881 (119886) = 119911minus1(119877119886) lt
120587lowast
2 119881 (119887) = 120587
lowast+ 119911minus1(119877119886) gt
120587lowast
2
(74)Since 119870 isin 119862([119886 119887] [0infin)) we obtain 119881 isin 119862([119886 119887]R) cap
1198621((119886 119887)R) and from (74) we observe that there exist
numbers 119879lowast119886isin (119886 119887) such that119881(119879lowast
119886) = 120587lowast2 Also119870(119905)119888
0ge
0 gives 119881([119886 119879lowast119886)) sub (minus120587
lowast2 120587lowast2) which proves statement
(71) Moreover it together with Lemma 21 and (72) provesthat
lim119905rarr119879
lowast
119886
120596 (119905) = lim119905rarr119879
lowast
119886
119911 (119881 (119905)) = 119911 (120587lowast
2) = infin (75)
Next according to (59) (63) and (72) we make thefollowing calculation on the interval [119886 119879lowast
119886)
1205961015840
(119905) = 1199111015840
(119881 (119905)) 1198811015840
(119905) = [1 + 120572 (|119911 (119881 (119905))|)]120587lowast
1198880
119870 (119905)
= [1 + 120572 (1003816100381610038161003816120596 (119905)
1003816100381610038161003816)]120587lowast
1198880
119870 (119905)
le119901
(120576119903 (119905))120574minus1
120572 (1003816100381610038161003816120596 (119905)
1003816100381610038161003816) + 120576119896 (120582 120583 120588)119876 (119905)
(76)
Thus all assertions of this proposition are proved
Remark 23 In the proof of the main result the number 119877119886
is determined by 119877119886= 120596(119886) where 120596(119905) denotes a function
associated with a nonoscillatory solution and it is given by(84) below
The third step in the proof of Lemma 1 is to show thefollowing pointwise comparison principle for the functions120596and120596 satisfying respectively the lower and upper differentialinequalities (73) and
119889120596
119889119905ge
119901
(120576119903 (119905))120574minus1
120572 (|120596 (119905)|) + 120576119896 (120582 120583 120588)119876 (119905) 119905 isin (119886 119887)
(77)
Proposition 24 Let [119886 119887) sub [1199050infin) be an arbitrary inter-
val One supposes that all coefficients of Riccati differentialinequalities (73) and (77) are continuous and strictly positivefunctions Let 120596 120596 isin 119862
1([119886 119887)R) be two functions satisfying
respectively (73) and (77) on the interval [119886 119887) Then
120596 (119886) le 120596 (119886) 119894119898119901119897119894119890119904 120596 (119905) le 120596 (119905) forall119905 isin [119886 119887) (78)
Proof Let119867(119905 119906) be a function defined by
119867(119905 119906) =119901
(120576119903 (119905))120574minus1
120572 (|119906|) + 120576119896 (120582 120583 120588)119876 (119905)
119905 isin [119886 119887) 119906 isin R
(79)
Let 119868 sub [119886 119887) and 119872 gt 0 be arbitrary For any two 1199061
1199062 minus119872 le 119906
1lt 1199062le 119872 let 119868
12be an interval defined
by 11986812
= (min|1199061| |1199062|max|119906
1| |1199062|) Since 120572(119904) is a 1198621-
function on [0infin) we know by the Lagrange mean valuetheorem applied on 119868
12that there is a 120585 isin 119868
12such that
120572 (10038161003816100381610038161199062
1003816100381610038161003816) minus 120572 (10038161003816100381610038161199061
1003816100381610038161003816)
1199062minus 1199061
le
1003816100381610038161003816120572 (10038161003816100381610038161199062
1003816100381610038161003816) minus 120572 (10038161003816100381610038161199061
1003816100381610038161003816)1003816100381610038161003816
1199062minus 1199061
=100381610038161003816100381610038161205721015840
(120585)10038161003816100381610038161003816
100381610038161003816100381610038161003816100381610038161199062
1003816100381610038161003816 minus10038161003816100381610038161199061
10038161003816100381610038161003816100381610038161003816
1199062minus 1199061
le100381610038161003816100381610038161205721015840
(120585)10038161003816100381610038161003816
le max119904isin11986812
100381610038161003816100381610038161205721015840
(119904)10038161003816100381610038161003816
(80)
since ||1199062| minus |1199061|| le 119906
2minus 1199061 Hence for any 119905 isin 119868 and 119906
1 1199062
minus119872 le 1199061lt 1199062le 119872 we have
119867(119905 1199062) minus 119867 (119905 119906
1)
1199062minus 1199061
= 1205880(119905)
120572 (10038161003816100381610038161199062
1003816100381610038161003816) minus 120572 (10038161003816100381610038161199061
1003816100381610038161003816)
1199062minus 1199061
le 1205880(119905)max119904isin11986812
100381610038161003816100381610038161205721015840
(119904)10038161003816100381610038161003816= 1198710(119905)
(81)
Thus the function119867(119905 119906) from (79) satisfies required condi-tion of [55 Lemma 19] and applying it to (73) and (77) weprove this proposition
Proof of Lemma 1 On the contrary let 119909(119905) be a solution of(1) such that
119909 (119905) = 0 on (120591 (120591 (119886)) 120590 (120590 (119889))) (82)
Discrete Dynamics in Nature and Society 11
that is 119909(119905) gt 0 on (120591(120591(119886)) 120590(120590(119889))) or 119909(119905) lt 0 on(120591(120591(119886)) 120590(120590(119889))) since 119909(119905) is a continuous function on[1199050infin) Let for instance
119909 (119905) gt 0 on (120591 (120591 (119886)) 120590 (120590 (119889))) (83)
Another case can be analogously treated let us see thecomment at the end of this proof In particular from (83)we have 119909(119905) gt 0 on (120591(120591(119886)) 120590(120590(119887))) which implies (since120591(119905) and 120590(119905) are increasing functions) 119909(119904) gt 0 for all 119904 isin
(120591(119886) 120590(119887)) cup (120591(120591(119886)) 120591(120590(119887))) cup (120590(120591(119886)) 120590(120590(119887))) whichyields 119909(119905) gt 0 119909(120591(119905)) gt 0 and 119909(120590(119905)) gt 0 on (120591(119886) 120590(119887))Hence by assumption (7) we may use inequality (5) on theinterval (119886 119887)
Firstly we show that the following classic Riccati transfor-mation of 119909(119905)
120596 (119905) = minus120576119903 (119905) 119860 (119909
1015840(119905))
|119909 (119905)|119901minus1
119909 (119905) 119905 isin (119886 119887) 120576 gt 0 (84)
satisfies upper Riccati differential inequality (77) Let usremark that from (1) we have in particular
minus(119903 (119905) 119860 (1199091015840
(119905)))1015840
= 119861 (119905 119909 (119905) 1199091015840
(119905)) + 120582119865 (119905 119909 (120591 (119905)))
+ 120583119866 (119905 119909 (120590 (119905))) minus 120588119890 (119905) 119905 ge 1199050
(85)
Taking the first derivative on both sides of (84) and usingassumptions (3) (4) and (5) as well as equality (85) and(|119909(119905)|
119901minus1119909(119905))1015840
= 119901|119909(119905)|119901minus1
1199091015840(119905) we obtain
119889120596
119889119905= 120576119901 119903 (119905)
119860 (1199091015840(119905)) 1199091015840(119905)
|119909 (119905)|119901+1
minus 1205761
|119909 (119905)|119901minus1
119909 (119905)(119903 (119905) 119860 (119909
1015840
(119905)))1015840
= 120576119901119903 (119905)119860 (1199091015840(119905)) 1199091015840(119905)
|119909 (119905)|119901+1
+120576
|119909 (119905)|119901minus1
119909 (119905)
times [120582119861 (119905 119909 (119905) 1199091015840
(119905)) + 119865 (119905 119909 (120591 (119905)))
+120583119866 (119905 119909 (120590 (119905))) minus 120588119890 (119905) ]
ge 120576119901119903 (119905)119860 (1199091015840(119905)) 1199091015840(119905)
|119909 (119905)|119901+1
+120576
|119909 (119905)|119901minus1
119909 (119905)
times [120582119865 (119905 119909 (120591 (119905))) + 120583119866 (119905 119909 (120590 (119905))) minus 120588119890 (119905)]
ge 120576119901119903 (119905) 120572(
10038161003816100381610038161003816119860 (1199091015840(119905))
10038161003816100381610038161003816
|119909 (119905)|119901
) + 120576119896 (120582 120583 120588)119876 (119905)
= 120576119901119903 (119905) 120572 (|120596 (119905)|
120576119903 (119905)) + 120576119896 (120582 120583 120588)119876 (119905)
ge119901
(120576119903 (119905))120574minus1
120572 (|120596 (119905)|) + 120576119896 (120582 120583 120588)119876 (119905) 119905 isin (119886 119887)
(86)
Thus according to inequality (5) it is shown that if 119909(119905) isa solution of (1) which satisfies (83) then the function 120596(119905)
defined by (84) satisfies the Riccati differential inequality(77) and 120596 isin 119862((119886 119887)R) On the other hand let 119877
119886be a
real number defined by 119877119886= 120596(119886) According to (6) and
Lemma 19 we obtain (54) which together with Lemma 20ensures that we may use Proposition 22 for such chosen realnumber 119877
119886 Hence we obtain a function 120596(119905) defined by (72)
which satisfies the lower Riccati differential inequality (73) on[119886 119879lowast
119886) 119879lowast119886isin (119886 119887) such that 120596(a) = 119877
119886and lim
119905rarr119879lowast
119886
120596(119905) =
infin Therefore by 120596(119886) = 119877119886= 120596(119886) and Proposition 24 we
conclude that lim119905rarr119879
lowast
119886
120596(119905) = infin too which is a contradictionwith the above conclusion saying that 120596 isin 119862((119886 119887)R) Thushypothesis (82) is not true and consequently Lemma 1 isshown
For the analogous case 119909(119905) lt 0 on (120591(120591(119886)) 120590(120590(119889))) wealso have 119909(119905) lt 0 on (120591(120591(119888)) 120590(120590(119889))) which implies (since120591(119905) and 120590(119905) are increasing functions)
119909 (119904) lt 0 forall119904 isin (120591 (119888) 120590 (119889)) cup (120591 (120591 (119888)) 120591 (120590 (119889)))
cup (120590 (120591 (119888)) 120590 (120590 (119889)))
(87)
which yields 119909(119905) lt 0 119909(120591(119905)) lt 0 and 119909(120590(119905)) lt 0 on(120591(119888) 120590(119889)) Now we can repeat the preceding procedure buton interval (119888 119889) and using (8) instead of (119886 119887) and (7)
Proof of Lemma 2 From assumption (10) we obtain the exis-tence of an 119899
0isin N such that
int
119887119899
119886119899
119876119899(119905) 119889119905 ge
1198880
2( max119905isin[119886119899 119887119899]
119876119899(119905))
1120574
119899 ge 1198990 (88)
that is
2
1198880
int
119887119899
119886119899
119876119899(119905) 119889119905 ge ( max
119905isin[119886119899 119887119899]119876119899(119905))
1120574
119899 ge 1198990 (89)
Now from (9) and previous inequality we deduce that forlarge enough 120582 120583 120588 and 119899
1199011120574
1199031minus1120574
0
[119896 (120582 120583 120588)]1minus1120574
120587lowast
int
119887119899
119886119899
119876119899(119905) 119889119905
ge2
1198880
int
119887119899
119886119899
119876119899(119905) 119889119905 ge ( max
119905isin[119886119899 119887119899]119876119899(119905))
1120574
(90)
which shows (6) Thus all assumptions of Lemma 1 arefulfilled and hence Lemma 2 immediately follows fromLemma 1
Proof of Lemma 3 Obviously assumption (11) is a particularcase of assumption (9) Hence this proof is very similar tothe proof of Lemma 2 and so it is left to the reader
Proof of Lemma 4 It is clear that from assumption (13) weobtain
1
(max119905isin[119886119899119887119899]119876119899(119905))1120574
int
119887119899
119886119899
119876119899(119905) 119889119905 ge
1198881
1198621120574
0
gt 0 forall119899 ge 1198990
(91)
12 Discrete Dynamics in Nature and Society
Thus hypothesis (12) is fulfilled and therefore Lemma 3proves this lemma
Proof of Theorems 5 6 and 7 This proof is based onLemma 4 In order to simplify notation in many placesin this proof we set 120591(119905) = 119905 minus 120591 and 120590(119905) = 119905 + 120590 Sinceassumptions (2) (3) and (4) have been already supposed inTheorems 5 6 and 7 in order to prove these theorems byLemma 4 we are going to show that the functions 119896(120582 120583 120588)and 119876
119899(119905) explicitly given respectively in (18) (21) or (24)
and (19) (22) or (25) satisfy required conditions (11) and(13) respectively and that every solution 119909(119905) of (27) satisfiesconditions (7) and (8) with respect to functions 119896(120582 120583 120588)and 119876
119899(119905) where 119886 = 119886
2119899minus1 119887 = 119887
2119899minus1 119888 = 119886
2119899 and 119889 = 119887
2119899
The proof that the function 119896(120582 120583 120588) given in (18) (21) or(24) satisfies (11) Passing to the limit in (18) (21) or (24) it isvery simple to show (11)
The proof that the function 119876119899(119905) given in (19) (22) or
(25) satisfies the first claim in (13) From (25) we immediatelyobtain
1003816100381610038161003816120591119899 (119905)1003816100381610038161003816 =
100381610038161003816100381610038161003816100381610038161003816
(119905 minus 119886119899
119905 minus 119886119899+ 120591
)
119901100381610038161003816100381610038161003816100381610038161003816
le 1
1003816100381610038161003816120590119899 (119905)1003816100381610038161003816 =
100381610038161003816100381610038161003816100381610038161003816
(119887119899minus 119905
119887119899minus 119905 + 120590
)
119901100381610038161003816100381610038161003816100381610038161003816
le 1 forall119899 isin N
(92)
Next by assumptions of this corollary we can conclude thatthere are three positive constants 119891
0 1198920 1198900such that |119891(119905)| le
1198910and |119892(119905)| le 119892
0on [1199050infin) in cases (i) and (ii) and
|119890(119905)| le 1198900on [1199050infin) in cases (iii) and (iv) Putting previous
inequalities into (19) (22) or (25) for all 119899 isin N and 119905 isin
[1199050infin) it holds that
1003816100381610038161003816119876119899 (119905)1003816100381610038161003816 le
119902
119901(
119901
119902 minus 119901)
(119902119901)minus1
1198901minus(119901119902)
0119891119901119902
0
delay case with 119902 gt 119901
119902
119901(
119901
119902 minus 119901)
(119902119901)minus1
1198901minus(119901119902)
0119892119901119902
0
advanced case with 119902 gt 119901
1199011
119901(
119901
2 (1199011minus 119901)
)
(1199011119901)minus1
1198901minus(119901119901
1)
0119891119901119902
0+1199012
119901
times(119901
2 (1199012minus 119901)
)
(1199012119901)minus1
1198901minus(119901119901
2)
0119892119901119902
0
delay-advanced case (i)
1198901205780
01198911205781
01198921205782
0
2
prod
119894=0
120578minus120578119894
119894
delay-advanced case (ii) (93)
which shows the first claim in (13)
The proof that the function119876119899(119905) given in (19) (22) or (25)
satisfies the second claim in (13)Without loss of generality weprove this claim only in case (i) since for other cases the prooffollows analogously In this sense let119876
119899(119905) = 119891(119905)120591
119899(119905) Since
1198862119899+1
minus 1198862119899minus1
le 119879lowast 1198872119899+1
minus 1198872119899minus1
ge 119879lowast 1198862119899+2
minus 1198862119899le 119879lowast and
1198872119899+2
minus 1198872119899
ge 119879lowast where 119879
lowastgt 0 is the period of the function
119891(119905) we have 1198862119899minus1
le 1198861+(119899minus1)119879
lowastand 1198872119899minus1
ge 1198871+(119899minus1)119879
lowast
119899 isin N Hence
int
1198872119899minus1
1198862119899minus1
119876119899(119905) 119889119905
= int
1198872119899minus1
1198862119899minus1
119891 (119905) (119905 minus 1198862119899minus1
119905 minus 1198862119899minus1
+ 120591)
119901
119889119905
ge int
1198871+(119899minus1)119879
lowast
1198861+(119899minus1)119879lowast
119891 (119905) (119905 minus 1198861minus (119899 minus 1) 119879
lowast
119905 minus 1198861minus (119899 minus 1) 119879
lowast+ 120591
)
119901
119889119905
= int
1198871
1198861
119891 (119904 + (119899 minus 1) 119879lowast) (
119904 minus 1198861
119904 minus 1198861+ 120591
)
119901
119889119904
= int
1198871
1198861
119891 (119904) (119904 minus 1198861
119904 minus 1198861+ 120591
)
119901
119889119904
(94)
which proves that the integral on the left hand side does notdepend on 119899 isin N that is the second claim in (13) is shown on[1198862119899minus1
1198872119899minus1
] This claim follows in the same way on [1198862119899 1198872119899]
Thus the second claim in (13) is proved on [119886119899 119887119899]
Next to the end of this proof let 119909(119905) be a solu-tion of (1) In particular it implies that (119903(119905)119860(1199091015840(119905)))1015840 =
minus119861(119905 119909(119905) 1199091015840(119905)) minus 120582119865(119905 119909(120591(119905))) minus 120583119866(119905 119909(120590(119905))) + 120588119890(119905) It
together with assumptions (15) (16) (20) and (23) easilygives the next two statements
if 119909 (119905) gt 0 119909 (120591 (119905)) gt 0 119909 (120590 (119905)) gt 0
on (120591 (1198862119899minus1
) 120590 (1198872119899minus1
))
then 119909 (119905) satisfies 119903 (119905) 119860 (1199091015840
(119905)) le 0
on (120591 (1198862119899minus1
) 120590 (1198872119899minus1
)) 119899 ge 1198990
(95)
if 119909 (119905) lt 0 119909 (120591 (119905)) lt 0 119909 (120590 (119905)) lt 0
on (120591 (1198862119899) 120590 (119887
2119899))
then 119909 (119905) satisfies 119903 (119905) 119860 (1199091015840
(119905)) ge 0
on (120591 (1198862119899) 120590 (119887
2119899)) 119899 ge 119899
0
(96)
Now we need the following lemma
Discrete Dynamics in Nature and Society 13
Lemma 25 Let 120591119886119887(119905) and 120590
119886119887(119905) be defined by
120591119886119887(119905) = (
120591 (119905) minus 120591 (119886)
119905 minus 120591 (119886))
119901
120590119886119887(119905) = (
120590 (119887) minus 120590 (119905)
120590 (119887) minus 119905)
119901
119905 isin (119886 119887)
(97)
and let 119909 isin 1198622([1198790infin)R) be an arbitrary function If
(119903(119905)119860(1199091015840(119905)) le 0 for all 119905 isin (120591(119886) 120590(119887)) or (119903(119905)119860(1199091015840(119905)) ge 0
for all 119905 isin (120591(119886) 120590(119887)) then
119909 (120591 (119905))
119909 (119905)ge (120591119886119887(119905))1119901
119909 (120590 (119905))
119909 (119905)ge (120590119886119887(119905))1119901
119905 isin (119886 119887)
(98)
Since119860(V) is supposed to be odd and increasing functionjust before (3) and 119903(119905) satisfies (14) the proof of Lemma 25in the first case that is 119903(119905)119860(1199091015840(119905)) le 0 for all 119905 isin (120591(119886) 120590(119887))is the same as the proof of [9 Corollaries 17 and 18] But in thesecond case that is 119903(119905)119860(1199091015840(119905)) ge 0 for all 119905 isin (120591(119886) 120590(119887))the proof is as follows if previous inequality holds then119903(119905)119860(minus119909
1015840(119905)) le 0 for all 119905 isin (120591(119886) 120590(119887)) and therefore to
the function minus119909(119905) one can apply the first case of this lemmaand consequently one obtains
119909 (120591 (119905))
119909 (119905)=minus119909 (120591 (119905))
minus119909 (119905)ge (120591119886119887(119905))1119901
119909 (120590 (119905))
119909 (119905)=minus119909 (120590 (119905))
minus119909 (119905)ge (120590119886119887(119905))1119901
119905 isin (119886 119887)
(99)
which proves this lemma in the second caseNow combining statements (95) (96) and (98) one
easily obtains
if 119909 (119905) gt 0 119909 (120591 (119905)) gt 0 119909 (120590 (119905)) gt 0
on (120591 (1198862119899minus1
) 120590 (1198872119899minus1
))
then 119909 (119905) satisfies 119909 (120591 (119905))
119909 (119905)ge (120591119899(119905))1119901
on (1198862119899minus1
1198872119899minus1
) 119899 ge 1198990
(100)
if 119909 (119905) lt 0 119909 (120591 (119905)) lt 0 119909 (120590 (119905)) lt 0
on (120591 (1198862119899) 120590 (119887
2119899))
then 119909 (119905) satisfies 119909 (120590 (119905))
119909 (119905)ge (120590119899(119905))1119901
on (1198862119899 1198872119899) 119899 ge 119899
0
(101)
where 120591119899(119905) and 120590
119899(119905) are defined in (26)
The proof that 119909(119905) satisfies (7) and (8) In this proofwe frequently use assumptions (16) (20) and (23) andstatements (100) and (101) Also because of (15) and 119865(119905 119906) =
119891(119905)|119906|1199011 sgn(119906) 119866(119905 119906) = 119892(119905)|119906|
1199012 sgn(119906) in both cases
(100) and (101) we can simultaneously use
minus119890 (119905) (|119909 (119905)|119901minus1
119909 (119905))minus1
= |119890 (119905)| |119909 (119905)|minus119901
ge 0 on 119869119899
119865 (119905 119909 (120591 (119905))) (|119909 (119905)|119901minus1
119909 (119905))minus1
ge 119891 (119905) |119909 (120591 (119905))|1199011 |119909 (119905)|
minus119901ge 0 on 119869
119899
119866 (119905 119909 (120590 (119905))) (|119909 (119905)|119901minus1
119909 (119905))minus1
ge 119892 (119905) |119909 (120590 (119905))|1199012 |119909 (119905)|
minus119901ge 0 on 119869
119899
|119909 (120591 (119905))| |119909 (119905)|minus1=119909 (120591 (119905))
119909 (119905)
|119909 (120590 (119905))| |119909 (119905)|minus1=119909 (120590 (119905))
119909 (119905)on 119869119899
(102)
where 119869119899= (1198862119899minus1
1198872119899minus1
) in the case of (100) and 119869119899= (1198862119899 1198872119899)
in the case of (101)
(i) Delay or Advanced Case with 119902 = 119901 Since 119902 = 119901 we obtain
[120582119865 (119905 119909 (120591 (119905))) + 120583119866 (119905 119909 (120590 (119905))) minus 120588119890 (119905)]
times (|119909 (119905)|119901minus1
119909 (119905))minus1
ge [120582119891 (119905) |119909 (120591 (119905))|119901+ 120583119892 (119905) |119909 (120590 (119905))|
119901
+120588 |119890 (119905)| ] |119909 (119905)|minus119901
ge [120582119891 (119905) |119909 (120591 (119905))|119901+ 120583119892 (119905) |119909 (120590 (119905))|
119901] |119909 (119905)|
minus119901
= 120582119891 (119905) (119909 (120591 (119905))
119909 (119905))
119901
+ 120583119892 (119905) (119909 (120590 (119905))
119909 (119905))
119901
ge 120582119891 (119905) 120591119899(119905) + 120583119892 (119905) 120590
119899(119905) 119905 isin 119869
119899
(103)
where the functions 120591119899(119905) and 120590
119899(119905) are defined in (26)
(ii) Delay Case with 119902 gt 119901 In this part we use the nextelementary inequality
119883120574+ (120574 minus 1) 119884
120574ge 120574119883119884
120574minus1 120574 gt 1 119883 119884 ge 0 (104)
Since 119902 gt 119901 and using (104) especially for
120574 =119902
119901gt 1 119883 = (120582119891 (119905))
119901119902
|119909 (120591 (119905))|119901
119884 = (119901
119902 minus 119901120588 |119890 (119905)|)
119901119902
(105)
14 Discrete Dynamics in Nature and Society
for all 119905 isin 119869119899we obtain
[120582119865 (119905 119909 (120591 (119905))) + 120583119866 (119905 119909 (120590 (119905))) minus 120588119890 (119905)]
times (|119909 (119905)|119901minus1
119909 (119905))minus1
= [120582119865 (119905 119909 (120591 (119905))) minus 120588119890 (119905)] (|119909 (119905)|119901minus1
119909 (119905))minus1
ge [120582119891 (119905) |119909 (120591 (119905))|1199011 + 120588 |119890 (119905)|] |119909 (119905)|
minus119901
= [119883120574+ (120574 minus 1) 119884
120574] |119909 (119905)|
minus119901
ge119902
119901(120582119891 (119905))
119901119902
|119909 (120591 (119905))|119901
times (119901
119902 minus 119901120588 |119890 (119905)|)
(119901119902)((119902119901)minus1)
|119909 (119905)|minus119901
= 120582119901119902
1205881minus(119901119902)
119902
119901(
119901
119902 minus 119901)
1minus(119901119902)
|119890 (119905)|1minus(119901119902)
times (119891 (119905))119901119902
(119909 (120591 (119905))
119909 (119905))
119901
ge 119896 (120582 120583 120588)119902
119901(
119901
119902 minus 119901)
1minus(119901119902)
|119890 (119905)|1minus(119901119902)
times (119891 (119905))119901119902
120591119899(119905)
(106)
where the function 119896(120582 120583 120588) is from (18)
(iii) Advanced Case with 119902 gt 119901 Using the same line ofarguments as in the proof of the previous case for all 119905 isin 119869
119899
we obtain
[120582119865 (119905 119909 (120591 (119905))) + 120583119866 (119905 119909 (120590 (119905))) minus 120588119890 (119905)]
times (|119909 (119905)|119901minus1
119909 (119905))minus1
= [120583119866 (119905 119909 (120590 (119905))) minus 120588119890 (119905)] (|119909 (119905)|119901minus1
119909 (119905))minus1
ge 119896 (120582 120583 120588)119902
119901(
119901
119902 minus 119901)
1minus(119901119902)
|119890 (119905)|1minus(119901119902)
times (119892 (119905))119901119902
120590119899(119905)
(107)
where the function 119896(120582 120583 120588) is from (21)
(iv) Superlinear Delay-Advanced Case Since 1199011 1199012gt 119901 for
all 119905 isin 119869119899we obtain
[120582119865 (119905 119909 (120591 (119905))) + 120583119866 (119905 119909 (120590 (119905))) minus 120588119890 (119905)]
times (|119909 (119905)|119901minus1
119909 (119905))minus1
= [120582119865 (119905 119909 (120591 (119905))) minus120588
2119890 (119905)] (|119909 (119905)|
119901minus1119909 (119905))minus1
+ [120583119866 (119905 119909 (120590 (119905))) minus120588
2119890 (119905)] (|119909 (119905)|
119901minus1119909 (119905))minus1
ge [120582119891 (119905) |119909 (120591 (119905))|1199011 +
120588
2|119890 (119905)|] |119909 (119905)|
minus119901
+ [120583119892 (119905) |119909 (120590 (119905))|1199012 +
120588
2|119890 (119905)|] |119909 (119905)|
minus119901
(108)
Now just the same as in the proofs of previous delay andadvanced cases with 119902 gt 119901 and with the help of (104) inparticular for
120574 =1199011
119901gt 1 119883 = (120582119891 (119905))
1199011199011
|119909 (120591 (119905))|119901
119884 = (119901
1199011minus 119901
120588
2|119890 (119905)|)
1199011199011
(109)
we have
[120582119891 (119905) |119909 (120591 (119905))|1199011 +
120588
2 |119890 (119905)|] |119909 (119905)|
minus119901
= [119883120574+ (120574 minus 1) 119884
120574] |119909 (119905)|
minus119901
ge1199011
119901(120582119891 (119905))
1199011199011
|119909 (120591 (119905))|119901
times (119901
1199011minus 119901
120588
2|119890 (119905)|)
(1199011199011)((1199011119901)minus1)
|119909 (119905)|minus119901
= 12058211990111990111205881minus(119901119901
1)1199011
119901(
119901
2 (1199011minus 119901)
)
1minus(1199011199011)
times |119890 (119905)|1minus(119901119901
1)(119891 (119905))
1199011199011
(119909 (120591 (119905))
119909 (119905))
119901
ge 119896 (120582 120583 120588)1199011
119901(
119901
2 (1199011minus 119901)
)
1minus(1199011199011)
times |119890 (119905)|1minus(119901119901
1)(119891 (119905))
1199011199011
120591119899(119905)
(110)
where the function 119896(120582 120583 120588) is from (24) Analogously weshow that
[120583119892 (119905) |119909 (120590 (119905))|1199012 +
120588
2|119890 (119905)|] |119909 (119905)|
minus119901
ge 119896 (120582 120583 120588)1199012
119901(
119901
2 (1199012minus 119901)
)
1minus(1199011199012)
times |119890 (119905)|1minus(119901119901
2)(119891 (119905))
1199011199012
120590119899(119905)
(111)
Discrete Dynamics in Nature and Society 15
Summarizing previous calculation we obtain
[120582119865 (119905 119909 (120591 (119905))) + 120583119866 (119905 119909 (120590 (119905))) minus 120588119890 (119905)]
times (|119909 (119905)|119901minus1
119909 (119905))minus1
ge 119896 (120582 120583 120588)119876119899(119905) 119905 isin 119869
119899
(112)
where the function 119896(120582 120583 120588) is from (24)
(v) Supersublinear Delay-Advanced Case Since 1199011gt 119901 gt 119901
2
and the following well-known elementary inequality holds
12057801199060+ 12057811199061+ 12057821199062ge 1199061205780
01199061205781
11199061205782
2 120578119894ge 0 119906
119894ge 0 (113)
from 1205780 1205781 1205782isin (0 1) 120578
0+ 1205781+ 1205782= 1 and 119901
11205781+ 11990121205782= 119901
we obtain for all 119905 isin 119869119899 for all 119905 isin 119869
119899we obtain
[120582119865 (119905 119909 (120591 (119905))) + 120583119866 (119905 119909 (120590 (119905))) minus 120588119890 (119905)]
times (|119909 (119905)|119901minus1
119909 (119905))minus1
ge [120582119891 (119905) |119909 (120591 (119905))|1199011 + 120583119892 (119905) |119909 (120590 (119905))|
1199012 + 120588 |119890 (119905)|]
times |119909 (119905)|minus119901
= [1205781[120578minus1
1120582119891 (119905) |119909 (120591 (119905))|
1199011] + 120578
2[120578minus1
2120583119892 (119905) |119909 (120590 (119905))|
1199012]
+1205780[120578minus1
0120588 |119890 (119905)|]] |119909 (119905)|
minus119901
ge [120578minus1
1120582119891 (119905) |119909 (120591 (119905))|
1199011]1205781
[120578minus1
2120583119892 (119905) |119909 (120590 (119905))|
1199012]1205782
times [120578minus1
0120588 |119890 (119905)|]
1205780
|119909 (119905)|minus119901
= 120582120578112058312057821205881205780 |119890 (119905)|
1205780(119891 (119905))
1205781
(119892 (119905))1205782
times|119909 (120591 (119905))|
12057811199011
|119909 (119905)|12057811199011
|119909 (120590 (119905))|12057821199012
|119909 (119905)|12057821199012
2
prod
119894=0
120578minus120578119894
119894
= 119896 (120582 120583 120588) |119890 (119905)|1205780(119891 (119905))
1205781
(119892 (119905))1205782
times (119909 (120591 (119905))
119909 (119905))
12057811199011
(119909 (120590 (119905))
119909 (119905))
12057821199012 2
prod
119894=0
120578minus120578119894
119894
ge 119896 (120582 120583 120588) |119890 (119905)|1205780(119891 (119905))
1205781
(119892 (119905))1205782
(120591119899(119905))1205781(1199011119901)
times (120590119899(119905))1205782(1199012119901)
2
prod
119894=0
120578minus120578119894
119894
= 119896 (120582 120583 120588)119876119899(119905)
(114)
where 119896(120582 120583 120588) and 119876119899(119905) are given respectively in (24) and
(25) Thus it is shown that required condition (5) in thecases (i)ndash(iv) is fulfilled with respect to 119896(120582 120583 120588) and 119876
119899(119905)
determined by (18) (21) or (24) and (19) (22) or (25)In conclusion according to the previous observation we
see that all assumptions of Lemma 4 are fulfilled and henceLemma 4 proves Theorems 5 6 and 7
Conflict of Interests
The author declares that there is no conflict of interestsregarding the publication of this paper
References
[1] Y G Sun ldquoA note on Nasrrsquos and Wongrsquos papersrdquo Journal ofMathematical Analysis and Applications vol 286 no 1 pp 363ndash367 2003
[2] Y G Sun C H Ou and J S W Wong ldquoInterval oscillationtheorems for a second-order linear differential equationrdquo Com-puters amp Mathematics with Applications vol 48 no 10-11 pp1693ndash1699 2004
[3] S Murugadass E Thandapani and S Pinelas ldquoOscillationcriteria for forced second-order mixed type quasilinear delaydifferential equationsrdquo Electronic Journal of Differential Equa-tions vol 2010 article 73 9 pages 2010
[4] Y Bai and L Liu ldquoNew oscillation criteria for second-orderdelay differential equations with mixed nonlinearitiesrdquoDiscreteDynamics in Nature and Society vol 2010 Article ID 796256 9pages 2010
[5] A F Guvenilir andA Zafer ldquoSecond-order oscillation of forcedfunctional differential equations with oscillatory potentialsrdquoComputers amp Mathematics with Applications vol 51 no 9-10pp 1395ndash1404 2006
[6] A Zafer ldquoInterval oscillation criteria for second order super-half linear functional differential equations with delay andadvanced argumentsrdquoMathematische Nachrichten vol 282 no9 pp 1334ndash1341 2009
[7] A F Guvenilir ldquoInterval oscillation of second-order functionaldifferential equations with oscillatory potentialsrdquo NonlinearAnalysis Theory Methods amp Applications vol 71 no 12 ppe2849ndashe2854 2009
[8] T S Hassan L Erbe and A Peterson ldquoForced oscillation ofsecond order differential equations with mixed nonlinearitiesrdquoActa Mathematica Scientia B vol 31 no 2 pp 613ndash626 2011
[9] M Pasic ldquoNew oscillation criteria for second-order forcedquasilinear functional differential equationsrdquo Abstract andApplied Analysis vol 2013 Article ID 735360 12 pages 2013
[10] L H Erbe Q Kong and B G Zhang Oscillation Theory forFunctional-Differential Equations vol 190 Marcel Dekker NewYork NY USA 1995
[11] V Kolmanovskii and A Myshkis Introduction to the Theoryand Applications of Functional-Differential Equations vol 463Kluwer Academic Publishers Dordrecht The Netherlands1999
[12] R P Agarwal M Bohner and W-T Li Nonoscillation andOscillation Theory for Functional Differential Equations vol267 Marcel Dekker New York NY USA 2004
[13] L Erbe T Hassan and A Peterson ldquoOscillation of secondorder functional dynamic equationsrdquo International Journal ofDifference Equations vol 5 no 2 pp 175ndash193 2010
[14] B Baculıkova J Dzurina and Y V Rogovchenko ldquoOscillationof third order trinomial delay differential equationsrdquo AppliedMathematics and Computation vol 218 no 13 pp 7023ndash70332012
[15] R P Agarwal L Berezansky E Braverman and A Domoshnit-sky Nonoscillation Theory of Functional Differential Equationswith Applications Springer New York NY USA 2012
16 Discrete Dynamics in Nature and Society
[16] J Zhang ldquoVariational approach to solitary wave solution ofthe generalized Zakharov equationrdquo Computers ampMathematicswith Applications vol 54 no 7-8 pp 1043ndash1046 2007
[17] T Ozis and A Yıldırım ldquoApplication of Hersquos semi-inversemethod to the nonlinear Schrodinger equationrdquo Computers ampMathematics with Applications vol 54 no 7-8 pp 1039ndash10422007
[18] X-C Cai andM-S Li ldquoPeriodic solution of Jacobi elliptic equa-tions by Hersquos perturbation methodrdquo Computers amp Mathematicswith Applications vol 54 no 7-8 pp 1210ndash1212 2007
[19] S Lenci G Menditto and A M Tarantino ldquoHomoclinic andheteroclinic bifurcations in the non-linear dynamics of a beamresting on an elastic substraterdquo International Journal of Non-Linear Mechanics vol 34 no 4 pp 615ndash632 1999
[20] D-J Huang and H-Q Zhang ldquoLink between travelling wavesand first order nonlinear ordinary differential equation with asixth-degree nonlinear termrdquoChaos Solitons amp Fractals vol 29no 4 pp 928ndash941 2006
[21] A I Maimistov ldquoPropagation of an ultimately short electro-magnetic pulse in a nonlinear medium described by the fifth-order Duffing modelrdquo Optics and Spectroscopy vol 94 pp 251ndash257 2003
[22] M N Hamdan and N H Shabaneh ldquoOn the large amplitudefree vibrations of a restrained uniform beam carrying anintermediate lumpedmassrdquo Journal of Sound andVibration vol199 no 5 pp 711ndash736 1997
[23] J-H He ldquoSome asymptotic methods for strongly nonlinearequationsrdquo International Journal of Modern Physics B vol 20no 10 pp 1141ndash1199 2006
[24] J B Marion Classical Dynamics of Particles and Systems 1970[25] I Kovacic and M J Brennan The Duffing Equation Nonlinear
Oscillatos and their Behaviour John Wiley amp Sons 1st edition2011
[26] F C Moon Chaotic Vibrations An Introduction for AppliedScientists and Engineers John Wiley amp Sons New York NYUSA 2004
[27] J J Stoker Nonlinear Vibrations 1950[28] G Chen and Z Tao ldquoAmplitude-frequency relationship for the
relativistic oscillatorrdquoAASRI Procedia vol 1 pp 400ndash403 2012[29] R E Mickens Oscillations in Planar Dynamic Systems World
Scientific Publishing Singapore 1996[30] A Belendez T Belendez C Neipp A Hernandez and M
L Alvarez ldquoApproximate solutions of a nonlinear oscillatortypified as a mass attached to a stretched elastic wire by thehomotopy perturbation methodrdquo Chaos Solitions and Fractalsvol 39 pp 746ndash764 2009
[31] A Belendez E Fernandez R Fuentes J J Rodes and I PascualldquoHarmonic balancing approach to nonlinear oscillations of apunctual charge in the eletric field of charged ringrdquo PhysicsLetters A vol 373 pp 735ndash740 2009
[32] A Elıas-Zuniga ldquoExact solution of the cubic-quintic Duffingoscillatorrdquo Applied Mathematical Modelling vol 37 no 4 pp2574ndash2579 2013
[33] A Belendez M L Alvarez J Frances et al ldquoAnalytical approx-imate solutions for the cubic-quintic Duffing oscillator in termsof elementary functionsrdquo Journal of Applied Mathematics vol2012 Article ID 286290 16 pages 2012
[34] A Elıas-Zuniga OMartınez-Romero andR K Cordoba-DıazldquoApproximate solution for the Duffing-harmonic oscillator bythe enhanced cubication methodrdquo Mathematical Problems inEngineering vol 2012 Article ID 618750 12 pages 2012
[35] C W Lim B S Wu andW P Sun ldquoHigher accuracy analyticalapproximations to the Duffing-harmonic oscillatorrdquo Journal ofSound and Vibration vol 296 no 4-5 pp 1039ndash1045 2006
[36] J He ldquoSome new approaches to Duffing equation with stronglyand high order nonlinearity II parametrized perturbationtechniquerdquo Communications in Nonlinear Science amp NumericalSimulation vol 4 no 1 pp 81ndash83 1999
[37] V Marinca and N Herisanu ldquoPeriodic solutions for somestrongly nonlinear oscillations by Hersquos variational iterationmethodrdquo Computers amp Mathematics with Applications vol 54no 7-8 pp 1188ndash1196 2007
[38] W Lu and Y Liu ldquoVibration control for the primary resonanceof the Duffing oscillator by a time delay state feedbackrdquoInternational Journal of Nonlinear Science vol 8 no 3 pp 324ndash328 2009
[39] H Y Hu and Z H Wang Dynamics of Controlled MechanicalSystems with Delayed Feedback Springer 2002
[40] M Hamdi and M Belhaq ldquoControl of bistability in a delayedDuffing oscillatorrdquo Advances in Acoustics and Vibration vol2012 Article ID 872498 6 pages 2012
[41] V Ravichandran C Chinnathambi and S Rajasekar ldquoNonlin-ear resonance in Duffing oscillator with fixed and integrativetime-delayed feedbacksrdquoPramana Journal of Physics vol 78 pp347ndash360 2013
[42] X You and H Xu ldquoAnalytical approximations for the periodicmotion of theDuffing systemwith delayed feedbackrdquoNumericalAlgorithms vol 56 no 4 pp 561ndash576 2011
[43] Z Sun W Xu X Yang and T Fang ldquoInducing or suppressingchaos in a double-well Duffing oscillator by time delay feed-backrdquo Chaos Solitons and Fractals vol 27 pp 705ndash714 2006
[44] H Wang H Hu and Z Wang ldquoGlobal dynamics of a Duffingoscillator with delayed displacement feedbackrdquo InternationalJournal of Bifurcation and Chaos in Applied Sciences andEngineering vol 14 no 8 pp 2753ndash2775 2004
[45] J Chiasson and J J LoiseauApplications of Time Delay SystemsSpringer 2007
[46] M Lakshmanan andDV SenthilkumarDynamics of NonlinearTime-Delay Systems Springer 2010
[47] G Stepan T Insperger and R Szalai ldquoDelay parametricexcitation and the nonlinear dynamics of cutting processesrdquoInternational Journal of Bifurcation and Chaos in Applied Sci-ences and Engineering vol 15 no 9 pp 2783ndash2798 2005
[48] U van der Heiden and H-O Walther ldquoExistence of chaos incontrol systems with delayed feedbackrdquo Journal of DifferentialEquations vol 47 no 2 pp 273ndash295 1983
[49] Y G Sun and J S W Wong ldquoOscillation criteria for secondorder forced ordinary differential equations with mixed non-linearitiesrdquo Journal of Mathematical Analysis and Applicationsvol 334 no 1 pp 549ndash560 2007
[50] J Heagy and W L Ditto ldquoDynamics of a two-frequencyparametrically driven Duffing oscillatorrdquo Journal of NonlinearScience vol 1 no 4 pp 423ndash455 1991
[51] A B Belogortsev ldquoBifurcations of tori and chaos in thequasiperiodically forced Duffing oscillatorrdquoNonlinearity vol 5no 4 pp 889ndash897 1992
[52] M Belhaq and M Houssni ldquoQuasi-periodic oscillations chaosand suppression of chaos in a nonlinear oscillator driven byparametric and external excitationsrdquo Nonlinear Dynamics vol18 no 1 pp 1ndash24 1999
[53] S H Saker P Y H Pang and R P Agarwal ldquoOscillationtheorems for second order nonlinear functional differential
Discrete Dynamics in Nature and Society 17
equations with dampingrdquo Dynamic Systems and Applicationsvol 12 no 3-4 pp 307ndash321 2003
[54] I N Bronshtein K A Semendyayev G Musiol and HMuehligHandbook of Mathematics Springer 5th edition 2007
[55] M Pasic ldquoFite-Wintner-Leighton-type oscillation criteria forsecond-order differential equations with nonlinear dampingrdquoAbstract and Applied Analysis vol 2013 Article ID 852180 10pages 2013
Submit your manuscripts athttpwwwhindawicom
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Differential EquationsInternational Journal of
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Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
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Decision SciencesAdvances in
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Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
Discrete Dynamics in Nature and Society 3
Lemma 3 Let assumptions (2) (3) and (4) hold Let thecontinuous function 119896 = 119896(120582 120583 120588) gt 0 and the sequence offunctions 119876
119899= 119876119899(119905) 119876119899isin 119862([119886
119899 119887119899] [0infin)) 119876
119899(119905) equiv 0 on
(119886119899 119887119899) satisfy respectively
119896 (120582 120583 120588) 997888rarr infin 119886119904 120582 997888rarr infin
119900119903 120583 997888rarr infin 119900119903 120588 997888rarr infin
(11)
lim inf119899rarrinfin
(1
(max119905isin[119886119899119887119899]119876119899(119905))1120574
int
119887119899
119886119899
119876119899(119905) 119889119905) gt 0 (12)
If 119909(119905) is a solution of (1) that satisfies (7) and (8) with 119886 =
1198862119899minus1
119887 = 1198872119899minus1
119888 = 1198862119899 119889 = 119887
2119899 and 119876(119905) = 119876
119899(119905) then 119909(119905)
has at least one zero point in (120591(120591(1198862119899minus1
)) 120590(120590(1198872119899))) forall119899 ge 119899
0
and for some 1198990isin N
In some concrete cases we use the next version ofLemmas 2 and 3 where condition (10) or (12) is replaced withappropriate one that appears in (1) with periodic coefficients
Lemma 4 Let assumptions (2) (3) and (4) hold Let thecontinuous function 119896 = 119896(120582 120583 120588) gt 0 and the sequence offunctions 119876
119899= 119876119899(119905) 119876119899isin 119862([119886
119899 119887119899] [0infin)) 119876
119899(119905) equiv 0 on
(119886119899 119887119899) satisfy respectively (11) and for some 119862
0 1198881isin R
0 lt max119905isin[119886119899 119887119899]
119876119899(119905) le 119862
0 int
119887119899
119886119899
119876119899(119905) 119889119905 ge 119888
1gt 0
forall119899 ge 1198990
(13)
and some 1198990isin N If 119909(119905) is a solution of (1) that satisfies
(7) and (8) with 119886 = 1198862119899minus1
119887 = 1198872119899minus1
119888 = 1198862119899 119889 = 119887
2119899
and 119876(119905) = 119876119899(119905) then 119909(119905) has at least one zero point in
(120591(120591(1198862119899minus1
)) 120590(120590(1198872119899))) forall119899 ge 119899
0
Next we suppose that the coefficient 119903(119905) additionallysatisfies
119903 (119904) le 119903 (119905) in delay
119903 (119904) ge 119903 (119905) in advanced case forall119904 le 119905
119903 (119905) is a constant in delay-advanced case
(14)
and the forcing term 119890(119905) satisfies
119890 (119905) le 0 on [120591 (1198862119899minus1
) 120590 (1198872119899minus1
)]
119890 (119905) ge 0 on [120591 (1198862119899) 120590 (119887
2119899)] 119899 isin N
(15)
We remark that 119890(119905) remains arbitrary function outside theset⋃119899(120591(119886119899) 120590(119887119899))
The first result of the paper deals with delay equation (1)
Theorem 5 Let assumptions (2) (3) (4) (14) and (15) hold120591(119905) = 119905 minus 120591 120591 ge 0 120590(119905) equiv 119905 119866(119905 119906) equiv 0 and let 119865(119905 119906) satisfy
119865 (119905 119906) sgn (119906) ge 119891 (119905) |119906|119902
forall119906 = 0 119905 ge 1199050
119891 (119905) ge 0 119891 (119905) equiv 0 119900119899 [120591 (119886119899) 119887119899]
(16)
where 119902 ge 119901 number 119901 is from (3) sequence (119886119899 119887119899) is from
(15) and 119891 isin 119862([1199050infin)R) is a periodic function with period
119879lowastgt 0 such that
[1198862119899minus1
+ 119879lowast 1198872119899minus1
+ 119879lowast] sube [119886
2119899+1 1198872119899+1
]
[1198862119899+ 119879lowast 1198872119899+ 119879lowast] sube [119886
2119899+2 1198872119899+2
] 119899 isin N(17)
Then (1) is oscillatory in the next two cases 119902 = 119901 andparameter 120582 is large enough 119902 gt 119901 120582 gt 0 120588 gt 0 and at leastone of parameters 120582 and 120588 is large enough
The proof ofTheorem 5 is presented in Section 6 and it isbased on Lemma 4 where
119896 (120582 120583 120588) = 120582119901119902
1205881minus(119901119902)
119902 ge 119901 (18)
119876119899(119905)
=
119891 (119905) (119905 minus 119886119899
119905 minus 119886119899+ 120591
)
119901
if 119902 = 119901
119902
119901(
119901
119902 minus 119901)
(119902119901)minus1
|119890 (119905)|1minus(119901119902)
(119891 (119905))119901119902
times(119905 minus 119886119899
119905 minus 119886119899+ 120591
)
119901
if 119902 gt 119901
(19)
The second result deals with advanced equation (1)
Theorem 6 Let assumptions (2) (3) (4) (14) and (15) hold120591(119905) equiv 119905 120590(119905) = 119905 + 120590 120590 ge 0 119865(119905 119906) equiv 0 and let 119866(119905 119906) satisfy
119866 (119905 119906) sgn (119906) ge 119892 (119905) |119906|119902
forall119906 = 0 119905 ge 1199050
119892 (119905) ge 0 119892 (119905) equiv 0 119900119899 [119886119899 120590 (119887119899)]
(20)
where 119902 ge 119901 number 119901 is from (3) sequence (119886119899 119887119899) is from
(15) and 119892 isin 119862([1199050infin)R) is a periodic function with period
119879lowastgt 0 such that (17) is fulfilled Then (1) is oscillatory in the
next two cases 119902 = 119901 and parameter 120583 is large enough 119902 gt 119901120583 gt 0 120588 gt 0 and at least one of parameters 120583 and 120588 is largeenough
The proof of Theorem 6 is based on Lemma 4 (seeSection 6) where
119896 (120582 120583 120588) = 120583119901119902
1205881minus(119901119902)
119902 ge 119901 (21)
119876119899(119905)
=
119892 (119905) (119887119899minus 119905
119887119899minus 119905 + 120590
)
119901
if 119902 = 119901
119902
119901(
119901
119902 minus 119901)
(119902119901)minus1
|119890 (119905)|1minus(119901119902)
(119892 (119905))119901119902
times(119887119899minus 119905
119887119899minus 119905 + 120590
)
119901
if 119902 gt 119901
(22)
The third result deals with delay-advanced equation (1)
4 Discrete Dynamics in Nature and Society
Theorem 7 Let assumptions (2) (3) (4) (14) and (15) hold120591(119905) = 119905 minus 120591 120591 ge 0 120590(119905) = 119905 + 120590 120590 ge 0 and 119865(119905 119906) and 119866(119905 119906)satisfy
119865 (119905 119906) sgn (119906) ge 119891 (119905) |119906|1199011 119866 (119905 119906) sgn (119906) ge 119892 (119905) |119906|
1199012
forall119906 = 0 119905 ge 1199050
119891 (119905) ge 0 119891 (119905) equiv 0 119892 (119905) ge 0
119892 (119905) equiv 0 119900119899 [120591 (119886119899) 120590 (119887
119899)]
(23)
where additionally 119891 119892 119890 isin 119862([1199050infin)R) are three periodic
functions having a common period 119879lowast
gt 0 such that (17)is fulfilled where 119890(119905) is the forcing term in (1) Then (1) isoscillatory provided one of the next two cases is fulfilled wherethe number 119901 is from (3) (1) (in superlinear delay-advancedcase) 119901
1gt 119901 119901
2gt 119901 120588 gt 0 and either parameter 120588 is
large enough or at least one of 120582 and 120583 is large enough (2) (insupersublinear delay-advanced case) 119901
1gt 119901 gt 119901
2gt 0 120582 gt 0
120583 gt 0 120588 gt 0 and at least one of parameters 120582 120583 and 120588 is largeenough
The proof of Theorem 7 is based on Lemma 4 (seeSection 6) where
119896 (120582 120583 120588) =
min 12058211990111990111205881minus(1199011199011) 12058311990111990121205881minus(1199011199012) superlinear case
120582120578112058312057811205881205780
supersublinear case
(24)
119876119899(119905)
=
1199011
119901(
119901
2 (1199011minus 119901)
)
(1199011119901)minus1
times |119890 (119905)|1minus(119901119901
1)(119891 (119905))
1199011199011
120591119899(119905)
+1199012
119901(
119901
2 (1199012minus 119901)
)
(1199012119901)minus1
|119890 (119905)|1minus(119901119901
2)
times (119892 (119905))1199011199012
120590119899(119905)
superlinear case|119890 (119905)|1205780(119891 (119905))
1205781
(119892 (119905))1205782
(120591119899(119905))1205781(1199011119901)
times (120590119899(119905))1205782(1199012119901)
2
prod
119894=0
120578minus120578119894
119894
supersublinear case
(25)
where we denote
120591119899(119905) = (
119905 minus 119886119899
119905 minus 119886119899+ 120591
)
119901
120590119899(119905) = (
119887119899minus 119905
119887119899minus 119905 + 120590
)
119901
(26)
Here the numbers 1205780 1205781 1205782isin (0 1) are chosen such that 120578
0+
1205781+ 1205782= 1 and 119901
11205781+ 11990121205782= 119901 Let us mention that if
1199011= 52 119901 = 1 and 119901
2= 12 and 120578
0= 1205781= 1205782= 13 then
(1199011 1199012) and (120578
0 1205781 1205782) satisfy previous two equalities About
the existence of such (119873+1)-tuple (1205780 1205781 120578
119873) in a general
case we refer to [49]
Remark 8 A difference between assumptions ofTheorems 56 and 7 is that 119890(119905) in Theorems 5 and 6 is not necessarilyperiodic or bounded function as it is supposed inTheorem 7
Now we study an important class of second-order func-tional differential equations as a particular case of (1)
(119903 (119905) 119860 (1199091015840
(119905)))1015840
+ 119861 (119905 119909 (119905) 1199091015840
(119905))
+ 120582119891 (119905) |119909 (120591 (119905))|1199011 sgn (119909 (120591 (119905)))
+ 120583119892 (119905) |119909 (120590 (119905))|1199012 sgn (119909 (120590 (119905))) = 120588119890 (119905) 119905 ge 119905
0
(27)
where 120591(119905) = 119905 minus 120591 120590(119905) = 119905 + 120590 and 120582 120583 120588 120591 120590 ge 0 and1199011 1199012gt 0 Using previous theorems we are able to state
the main consequences showing the parametrically excitedoscillations in (27)
Corollary 9 (delay equation) Let assumptions (2) (3) (4)(14) and (15) hold Let 119892(119905) equiv 0 119891(119905) ge 0 and 119891(119905) equiv 0 on[119886119899minus120591 119887119899] where119891 isin 119862([119905
0infin)R) is a periodic function with
period 119879lowast satisfying (16) Equation (27) is oscillatory in the
following two cases 1199011= 119901 and parameter 120582 is large enough
1199011gt 119901 120582 gt 0 120588 gt 0 and at least one of parameters 120582 and 120588 is
large enough
Corollary 10 (advanced equation) Let assumptions (2) (3)(4) (14) and (15) hold Let 119891(119905) equiv 0 119892(119905) ge 0 and 119892(119905) equiv 0
on [119886119899 119887119899+ 120590] where 119892 isin 119862([119905
0infin)R) is a periodic function
with period 119879lowast satisfying (17) Then (27) is oscillatory in the
following two cases 1199012= 119901 and parameter 120583 is large enough
1199012gt 119901 120583 gt 0 120588 gt 0 and at least one of parameters 120583 and 120588 is
large enough
Corollary 11 (delay-advanced equation) Let assumptions (2)(3) (4) (14) and (15) hold and 119891(119905) and 119892(119905) satisfy
119891 (119905) ge 0 119891 (119905) equiv 0
119892 (119905) ge 0 119892 (119905) equiv 0
119900119899 [119886119899minus 120591 119887119899+ 120590]
(28)
where additionally 119891 119892 119890 isin 119862([1199050infin)R) are three periodic
functions having a common period 119879lowast
gt 0 such that (17)is fulfilled where 119890(119905) is the forcing term in (1) Then (27) isoscillatory in the next two cases where the number 119901 is from(3) (1) (superlinear case) 119901
1gt 119901 119901
2gt 119901 120588 gt 0 and either
parameter 120588 is large enough or at least one of 120582 and 120583 is largeenough (2) (supersublinear case) 119901
1gt 119901 gt 119901
2gt 0 120582 gt 0
120583 gt 0 120588 gt 0 and at least one of parameters 120582 120583 and 120588 is largeenough
According to previous corollaries we can derive thefollowing examples
Example 12 (delay case) Let 119901 ge 1 120576 ge 0 and 119898 isin N
be fixed and 120591119898
isin R 0 le 120591119898
lt 120587(4119898) With the help of
Discrete Dynamics in Nature and Society 5
Corollary 9 the following two different classes of quasilineardelay differential equations
(100381610038161003816100381610038161199091015840
(119905)10038161003816100381610038161003816
119901minus1
1199091015840
(119905))1015840
+ 120582 sin (119898119905) 1003816100381610038161003816119909 (119905 minus 120591119898)1003816100381610038161003816119901+120576
times sgn (119909 (119905 minus 120591119898)) = minus120588 cos (2119898119905)
(1199091015840(119905)
radic1 + 11990910158402
(119905)
)
1015840
+ 120582 cos (119898119905) 1003816100381610038161003816119909 (119905 minus 120591119898)10038161003816100381610038161+120576
times sgn (119909 (119905 minus 120591119898)) = minus120588 cos (2119898119905)
(29)
are oscillatory provided at least one of 120582 gt 0 and 120588 gt 0 is largeenough (the case 120588 = 0 is possible if 120576 = 0) It is because forall 119899 isin N we have
minus cos (2119898119905)
le 0 on [2119899120587
1198982119899120587
119898+
120587
4119898]
ge 0 on [2119899120587
119898+
120587
41198982119899120587
119898+
120587
2119898]
sin (119898119905) ge 0 cos (119898119905) ge 0 on [2119899120587
1198982119899120587
119898+
120587
2119898]
[2119899120587
119898+ 120591119898+ 119879lowast2119899120587
119898+
120587
4119898+ 119879lowast]
= [(2119899 + 2) 120587
119898+ 120591119898(2119899 + 2) 120587
119898+
120587
4119898]
[2119899120587
119898+
120587
4119898+ 120591119898+ 119879lowast2119899120587
119898+
120587
2119898+ 119879lowast]
= [(2119899 + 2) 120587
119898+
120587
4119898+ 120591119898(2119899 + 2) 120587
119898+
120587
2119898]
(30)
where 119879lowast= 2120587119898 is the common period of the functions
sin(119898119905) and cos(119898119905) Thus in order to apply Corollary 9 wecan choose 119886
2119899minus1= 2119899120587119898 + 120591
119898 1198872119899minus1
= 2119899120587119898 + 120587(4119898)1198862119899= 2119899120587119898 + 120587(4119898) + 120591
119898 and 119887
2119899= 2119899120587119898 + 120587(2119898)
Example 13 (advanced case) Let 119901 ge 1 120576 ge 0 and 119898 isin
N be fixed and 120590119898
isin R 0 le 120590119898
lt 120587(4119898) With thehelp of Corollary 10 the following two classes of quasilinearadvanced differential equations
(100381610038161003816100381610038161199091015840
(119905)10038161003816100381610038161003816
119901minus1
1199091015840
(119905))1015840
+ 120583 sin (119898119905) 1003816100381610038161003816119909 (119905 + 120590119898)1003816100381610038161003816119901+120576
times sgn (119909 (119905 + 120590119898)) = minus120588 cos (2119898119905)
(1199091015840(119905)
radic1 + 11990910158402
(119905)
)
1015840
+ 120583 cos (119898119905) 1003816100381610038161003816119909 (119905 + 120590119898)10038161003816100381610038161+120576
times sgn (119909 (119905 + 120590119898)) = minus120588 cos (2119898119905)
(31)
are oscillatory provided at least one of 120583 gt 0 and 120588 gt 0 islarge enough (the case 120588 = 0 is possible if 120576 = 0) In order toapply Corollary 10 we can choose 119886
2119899minus1= 2119899120587119898 119887
2119899minus1=
2119899120587119898 + 120587(4119898) minus 120590119898 1198862119899
= 2119899120587119898 + 120587(4119898) and 1198872119899
=
2119899120587119898 + 120587(2119898) minus 120590119898
Example 14 (delay-advanced case) Let 119901 ge 1 1205761gt 0 1205762gt 0
and 119898 isin N be fixed and 120591119898ge 0 and 120590
119898ge 0 0 le 120591
119898+ 120590119898lt
120587(4119898) With the help of Corollary 11 the following class ofquasilinear delay-advanced differential equations
(100381610038161003816100381610038161199091015840
(119905)10038161003816100381610038161003816
119901minus1
1199091015840
(119905))1015840
+ 120582 sin (119898119905)
times1003816100381610038161003816119909 (119905 minus 120591119898)
1003816100381610038161003816119901+1205761 sgn (119909 (119905 minus 120591
119898))
+ 120583 cos (119898119905) 1003816100381610038161003816119909 (119905 + 120590119898)1003816100381610038161003816119901+1205762
times sgn (119909 (119905 + 120590119898)) = minus120588 cos (2119898119905)
(32)
is oscillatory provided either 120588 gt 0 is large enough or at leastone of 120582 gt 0 and 120583 gt 0 is large enough In order to applyCorollary 11 we can choose 119886
2119899minus1= 2119899120587119898 + 120591
119898 1198872119899minus1
=
2119899120587119898 + 120587(4119898) minus 120590119898 1198862119899
= 2119899120587119898 + 120587(4119898) + 120591119898 and
1198872119899= 2119899120587119898 + 120587(2119898) minus 120590
119898
3 Application to Duffing Equations withTime Delay Feedback
Let 120582 ge 0 denote the control gain parameter (often calledldquodisplacement feedback coefficientrdquo) 120591 gt 0 the time delayand 120588 ge 0 and 120596 gt 0 the amplitude and frequency of theexternal force respectively Let the function Φ = Φ(119905 119906) thatwill appear in the delay feedback term Φ(119905 119909(119905 minus 120591)) satisfythe general condition
Φ (119905 119906) sgn (119906) ge 1206010|119906|119902
forall119905 ge 1199050
119906 = 0 and some 119902 ge 1 1206010gt 0
(33)
For instance Φ(119905 119906) = 1199062119898minus1 119898 isin N or more general
Φ(119905 119906) = sum119898
119896=11206011198961199062119896minus1 120601
119896gt 0119898 isin N
In this section we consider the following large class ofundamped possible nonautonomous and nonconservativeDuffing equations without or with the general time delayfeedback Φ(119905 119909(119905 minus 120591))
(10038161003816100381610038161003816119909101584010038161003816100381610038161003816
119901minus1
1199091015840)1015840
+ 1205962
0119909 +
1205831|119909|1199031 sgn (119909)
(1205832+ 12058331199092)1199032
+
119898
sum
119894=1
120573119894(119905) |119909|
120572119894minus1119909 + 120582Φ (119905 119909 (119905 minus 120591)) = 120588 cos (120596119905)
(34)
where 1205960is the natural frequency 120583
1ge 0 is the density of the
nonlinear potential (or rigidity coefficient) and 1205832 1205833 1199031 1199032
are nonnegative constants 120573119894(119905) ge 0 and 120572
119894ge 1
When 119901 = 1 120582 = 0 and 120573119894(119905) equiv 120573
119894= const
(34) contains many most important classes of undampedautonomous Duffing oscillators such as the following
(i) the strongly nonlinear Duffing oscillator with smoothodd nonlinearity is given in (34) provided 120583
1= 0 and
120572119894= 2119894 + 1 let us recall some of its known particular
cases
(a) the classic Duffing oscillator 11990910158401015840 +12059620119909+120573119909
3= 0
has been recently studied in the searching of
6 Discrete Dynamics in Nature and Society
solitarywave solutions of classic and generalizedZakharov equations of plasma physics (see [16])and of nonlinear Schrodinger equation (see[17]) also it is strongly connected with theJacobi elliptic equation (see [18])
(b) the cubic-quintic oscillator 11990910158401015840 + 1205962
0119909 + 120573
11199093+
12057321199095
= 0 is used as a model for the non-linear dynamics of a slender elastica (see [19])in nonlinear wave systems (see [20]) for thepropagation of a short electromagnetic pulsein a nonlinear medium (see [21]) and in theunimodal Duffing temporal problem (see [22])
(c) the cubic truly nonlinear oscillator 11990910158401015840 + 1205731199093=
0 models the motion of a ball bearing thatoscillates in a glass tube that is bent into acurve (see [23]) as well as the motion of a massattached to identical stretched elastic wires (see[24])
(d) the nonhomogeneous Duffing oscillator 11990910158401015840 +1205962
0119909 + 120573119909
3= 120588 cos(120596119905) describes various forced
vibrations of beams springs with nonlinearstiffness cables plates shells and optical fibresin electrical circuits in nonlinear isolators andso forth (see for instance [25 26])
(ii) the general Duffing-harmonic oscillator (with rationalor irrational nonlinear restoring-force) is given in(30) if 120583
1= 0 120573119894= 0 and 120588 = 0 the most known
subclasses of these oscillators are
(a) the classic Duffing-harmonic oscillator 11990910158401015840+
(12058311199093(1205832+ 12058331199092)) = 0 which models many
conservative nonlinear oscillatory systems see[27]
(b) the relativistic harmonic oscillator11990910158401015840+ (1205831119909radic1 + 1199092) = 0 see [28]
(c) the nonlinear oscillator11990910158401015840+119909minus(1205831119909radic1 + 1199092) =
0 1205831
isin [0 1] which is typified as a massattached to a stretched elastic wire see [29 30]
(d) the nonlinear oscillator 11990910158401015840
+
(1205831119909(radic(1 + 1199092 )
3
) = 0 which presentsnonlinear oscillations of a punctual charge inthe electric field of charged ring see [31]
Finding several explicit forms of periodic approximate solu-tions for these oscillators has been intensively studied lastyears by many authors see for instance [28 30 32ndash37] andalso the references therein
When 120582 = 0 and linear time delay feedbackΦ(119905 119909(119905minus120591)) =119909(119905 minus 120591) the following topics have been studied for varioustypes of Duffing oscillators with time delayed feedback in[38] authors constructed a low-order approximate solutionunder weak feedback gain parameter about the low- andhigh-order approximations see also [39] in [40] with 120588 = 0the Hopf bifurcation diagrams have been explored for theapproximate periodic solutions (amplitude versus time delay120591 and feedback gain 120582 versus time delay 120591) moreover in [41]
authors made an analysis on the effect of the control gainand time delay parameters on the amplitude of approximateperiod solution from the theoretical and numerical pointsof view see also [42] in [43] authors studied the chaoticbehaviour with respect to gains and time delay parameterssee also [44]
Equations under time delay control such as (34) (espe-cially with damped term) are used as a model for variouscontrolled physical mechanical and engineering systemswith time delays see for instance [39 45ndash48] and thereferences therein
Here (34) contains very general nonlinear time delayfeedback Φ(119905 119909(119905 minus 120591)) with Φ satisfying (33) and the lineartime delay feedback 119909(119905 minus 120591) is only a particular case ofit and to the best of our knowledge the previous topicsare not considered for (34) as yet Moreover with suchan Φ the oscillations of (34) can be taken under a doubteven with the linear time delay feedback (see the nature ofthe approximations given in [38 39]) Hence we can posethe following question under what conditions on equationrsquosparameters (34) is a nonlinear oscillator that is possessesonly oscillatory solutions An answer is given in the nextresult as an easy consequence of the parametrically excitedoscillations by Theorem 5
Theorem 15 Let 120591 isin (0 120587120596) and (33) hold Equation (34) isoscillatory in the next two cases
(i) 119902 = 119901 and 120582 is large enough(ii) 119902 gt 119901 120582 gt 0 120588 gt 0 and at least one of 120582 and 120588 is large
enough
Proof Let 119903(119905) equiv 1 119860(V) = |V|119901minus1 119865(119905 119906) = Φ(119905 119906) 119866(119905 119906) equiv0 119890(119905) = cos(120596119905) and
119861 (119905 119906 V) = 1205962
0119906 +
1205831|119906|1199031s119892119899 (119906)
(1205832+ 12058331199062)1199032
+
119898
sum
119894=1
120573119894(119905) |119906|
120572119894minus1119906 (35)
It is easy to check that all assumptions of Theorem 5 arefulfilled with respect to the sequence 119886
119899= minus1205872120596 + 119899120587120596 + 120591
and 119887119899= 1205872120596 + 119899120587120596 + 120591 where 119886
119899lt 119887119899since it is supposed
that 120591 lt 120587120596 Hence Theorem 5 proves this theorem
Remark 16 Even in the linear forced case (119890(119905) equiv 0) it isnot easy to establish the oscillations of all solutions since theoscillation and nonoscillation can occur simultaneously Themost simple and important example for the coincidence ofoscillation and nonoscillation is the following linear forceddifferential equation 11990910158401015840 + (2119905)119909
1015840+ 119909 = 2119905 119905 gt 0 that
allows an oscillatory solution 1199091(119905) = (3 sin 119905)119905 + 2119905 and a
nonoscillatory solution 1199092(119905) = 2119905 This is not possible in
the linear case with 119890(119905) equiv 0 because of Sturmrsquos separationtheorem
4 Parametrically Excited Oscillations andWell-Known Oscillation Criteria
In this section we would like to draw the readerrsquos attentionto the fact that the parametrically excited oscillations have
Discrete Dynamics in Nature and Society 7
been already appearing in some published papers on theoscillation of functional differential equations but only insome examples illustrating certain main oscillation criteriaHowever with the help of our main results in which theparametrically excited oscillations are studied in a generalsetting the equations from these examples are replaced withgeneral ones also having parameters 120582 and 120583
In [1] (see also [2 Example 31] with 120591 = 0 [3 Example31] and [4 Section 3]) the author considers the oscillationof the second-order delay differential equation
11990910158401015840
(119905) + 119891 (119905) |119909 (120591 (119905))|120574 sgn119909 (120591 (119905)) = 119890 (119905) (36)
in the linear case (120574 = 1) and the superlinear (120574 gt 1)In the linear case (analogously for the superlinear case see[1 Theorem 2]) the author proved the following oscillationcriterion In what follows we denote
119863 (119886 119887) = 119906 isin 1198621
([119886 119887] R) 119906 (119905) equiv 0 119906 (119886) = 119906 (119887) = 0
(37)
Theorem 17 ([1 Theorem 1]) Suppose that for any 119879 ge 0there exist constants 119886
1 1198871 1198862 1198872such that 119879 le 119886
1lt 1198871 119879 le
1198862lt 1198872 and 119891(119905) ge 0 on [120591(119886
1) 1198871] cup [120591(119886
2) 1198872] 119890(119905) le 0
on [120591(1198861) 1198871] and 119890(119905) ge 0 on [120591(119886
2) 1198872] If there exists 119906 isin
119863(119886119894 119887119894) 119894 = 1 2 such that
int
119887119894
119886119894
[1199062
(119905) 119891 (119905)120591 (119905) minus 120591 (119886
119894)
119905 minus 120591 (119886119894)
minus (1199061015840
(119905))2
]119889119905 ge 0 (38)
then (36) with 120574 = 1 is oscillatory
Previous criterion has been applied on the followingparticular equation
11990910158401015840
(119905) + 120582 sin (119905)1003816100381610038161003816100381610038161003816119909 (119905 minus
120587
4)
1003816100381610038161003816100381610038161003816
120574
times sgn 119909(119905 minus120587
4) = cos (119905) 119905 ge 0
(39)
where 120582 ge 0 and 120574 = 1 Applying Theorem 17 to (39) theauthor proved that (39) is oscillatory provided the followinginequality
120582int
119887119894
119886119894
sin2 (2119905) cos2 (2119905) sin (119905)119905 minus 119886119894
119905 minus 119886119894+ 1205874
119889119905 ge120587
2 (40)
holds for sufficiently large 120582 Thus the oscillation of (39) isexcited by the large enough parameter 120582 However accordingto Theorems 5 and 6 we are able to show that the nextparametric equation that corresponds to general equation(36)
11990910158401015840
(119905) + 120582119891 (119905) |119909 (120591 (119905))|120574 sgn119909 (120591 (119905)) = 119890 (119905) (41)
is oscillatory provided 120582 is large enough where 1199011= 1199012= 120574
120583 = 0 and 120588 = 1Next in [5] (see also [6ndash8]) the authors consider the
oscillation of the following class of second-order differentialequations with delay and advanced arguments
(119903 (119905) 1199091015840
(119905))1015840
+ 119891 (119905) |119909 (120591 (119905))|1199011 sgn119909 (120591 (119905))
+ 119892 (119905) |119909 (120590 (119905))|1199012 sgn119909 (120590 (119905)) = 119890 (119905) 119905 ge 0
(42)
where 1199011 1199012ge 1 When 119901
1= 1199012= 1 the authors prove the
following result (for other cases see [5Theorems 32 33 and34]
Theorem 18 ([5 Theorem 31]) Suppose that for any 119879 ge
0 there exist intervals [120591(1198861) 1198871] [120591(119886
2) 1198872] [1198881 120590(1198891)] and
[1198882 120590(1198892)] contained in [119879infin) such that 119886
1lt 1198871 1198862lt 1198872
1198881lt 1198891 1198882lt 1198892 and
119891 (119905) ge 0 119900119899 [120591 (1198861) 1198871] cup [120591 (119886
2) 1198872]
119892 (119905) ge 0 119900119899 [1198881 120590 (1198891)] cup [119888
2 120590 (1198892)]
119890 (119905) le 0 119900119899 [120591 (1198861) 1198871] cup [1198881 120590 (1198891)]
119890 (119905) ge 0 119900119899 [120591 (1198862) 1198872] cup [1198882 120590 (1198892)]
(43)
and 119888119894= 120591(119886
119894) 119889119894= 119886119894 and 119887
119894= 120590(119889
119894) 119894 = 1 2 If there exist
1199061isin 119863(119886
119894 119887119894) and 119906
2isin 119863(119888119894 119889119894) such that either
int
119887119894
119886119894
[1199062
1(119905) 119891 (119905)
120591 (119905) minus 120591 (119886119894)
119905 minus 120591 (119886119894)
minus (1199061015840
1(119905))2
119903 (119905)] 119889119905 ge 0 (44)
or
int
119889119894
119888119894
[1199062
2(119905) 119891 (119905)
120590 (119889119894) minus 120590 (119905)
120590 (119889119894) minus 119905
minus (1199061015840
2(119905))2
119903 (119905)] 119889119905 ge 0 (45)
for 119894 = 1 2 then (42) with 1199011= 1199012= 1 is oscillatory
As a consequence of this result it has been concluded thatthe particular equation
(119903 (119905) 1199091015840
(119905))1015840
+ 120582 sin (119905) 119909 (119905 minus 120587
12)
+ 120583 cos (119905) 119909 (119905 + 120587
6) = cos (2119905) 119905 ge 0
(46)
is oscillatory provided either 120582 or 120583 is large enough Howeverby following Theorems 5 and 6 one can obtain the sameconclusion for the following general equation associated with(42)
(119903 (119905) 1199091015840
(119905))1015840
+ 120582119891 (119905) |119909 (120591 (119905))|1199011 sgn119909 (120591 (119905))
+ 120583119892 (119905) |119909 (120590 (119905))|1199012 sgn119909 (120590 (119905)) = 119890 (119905)
(47)
Related observation can be done with [8 Example 33]and [9 Example 21] where the quasilinear second-orderfunctional differential equations have been considered It isleft to the reader
5 Some Open Questions and Comments
In this section we discuss some problems related to ourmainresults that are not studied here
(1) Quasiperiodic Case In the theory of nonlinear oscillatorsa particularly important case occurs when the periodiccoefficients in the oscillator do not have any common periodIt is called the quasiperiodic (or two-frequency) nonlinear
8 Discrete Dynamics in Nature and Society
oscillator and studied for instance in [50ndash52] Since inTheorems 5 6 and 7 we assume that the correspondingperiodic functions have a commonperiod it is natural to posethe next question
Open Question 1 Is it possible to derive sufficient conditionsfor the oscillation of (27) in the casewhen119891(119905) and119892(119905) (resp119891(119905) 119892(119905) and ℎ(119905)) are two (resp three) periodic functionsnot having a common period
(2) Equation with More Functional Arguments Next regard-ing some second-order functional differential equationsconsidered in the references of this paper more than twononlinear functional terms are appearing and thereforeinstead of main equation (1) and corresponding particularequation (27) considered inTheorems 5 6 and 7 we suggestthe following classes of equations
(119903 (119905) 119860 (1199091015840
(119905)))1015840
+ 119861 (119905 119909 (119905) 1199091015840
(119905))
+
1198981
sum
119896=1
120582119896119865119896(119905 119909 (120591
119896(119905)))
+
1198982
sum
119896=1
120583119896119866119896(119905 119909 (120590
119896(119905))) = 120588119890 (119905)
(48)
where 0 le 120591119896(119905) le 119905 lim
119905rarrinfin120591119896(119905) = infin 120590
119896(119905) ge 119905 119898
1 1198982isin
N and
(119903 (119905) 119860 (1199091015840
(119905)))1015840
+ 119861 (119905 119909 (119905) 1199091015840
(119905))
+
1198981
sum
119896=1
120582119896119891119896(119905)
1003816100381610038161003816119909 (119905 minus 120591119896)1003816100381610038161003816119901119896 sgn (119909 (119905 minus 120591
119896))
+
1198982
sum
119896=1
120583119896119892119896(119905)
1003816100381610038161003816119909 (119905 + 120590119896)1003816100381610038161003816119902119896 sgn (119909 (119905 + 120590
119896)) = 120588119890 (119905)
(49)
where 120582119896 120583119896 120588 120591119896 120590119896ge 0 and 119901
119896 119902119896gt 0
Comment We suggest the reader to enlarge the main resultsof this paper to (48) and (49)
(3) Damped Duffing Equation In the application the Duffingequation (34) is often appearing with the linear damped term1199091015840(119905) that is
11990910158401015840+ 11988901199091015840+ 1205962
0119909 + 120573119909
3+ 120582Φ (119909 (119905 minus 120591)) = 120588 cos (120596119905) (50)
where 1198890
is the damped coefficient which can in anactive way influence various behaviours of (50) Since119861(119905 119909(119905) 119909
1015840(119905)) = 119889
01199091015840(119905) does not satisfy the required
assumption (4) we are not able to apply our main results to(50) Hence we pose the following questionOpen Question 2 Is it possible to obtain the parametricallyexcited oscillation for (1) in the case when the damped term119861(119905 119906 V) satisfies a larger condition than (4) in which thelinear damped term 120573119909
1015840(119905) is especially included
(4) Functional Argument in Damped Term In a class of Duff-ing equations we have two time delayed feedback and hence
besides the control gain parameter 1205821another parameter 120582
2
appears the so-called velocity gain parameter Hence insteadof (34) one can consider
11990910158401015840+ 11988901199091015840+ 1205962
0119909 + 120573119909
3+ 1205821119909 (119905 minus 120591)
+ 12058221199091015840
(119905 minus 120591) = 120588 cos (120596119905) (51)
Therefore we suggest the following problem for further studyOpen Question 3 Is it possible to obtain the parametricallyexcited oscillation for the following more general functionaldifferential equation than (1) in which the functional argu-ment appears in the damped term too as follows
(119903 (119905) 119860 (1199091015840
(119905)))1015840
+ 119861 (119905 119909 (119905) 1199091015840
(119905)) + 1205821119865 (119905 119909 (120591 (119905)))
+ 1205822119867(119905 119909
1015840
(120591 (119905))) = 120588119890 (119905) 119905 ge 1199050
(52)or
(119903 (119905) 119860 (1199091015840
(119905)))1015840
+ 119861 (119905 119909 (119905) 1199091015840
(119905)) + 1205831119866 (119905 119909 (120590 (119905)))
+ 1205832119867(119905 119909
1015840
(120590 (119905))) = 120588119890 (119905) 119905 ge 1199050
(53)
About known oscillation criteria for the second-order func-tional differential equations having the functional argumentin the damped term we refer the reader to for instance [53]and the references therein
6 Proofs of Main Results
The proof of Lemma 1 is based on the following three stepstwo working forms of condition (6) (see Lemmas 19 and 20)the existence of an explosive solution of a suitable Riccatidifferential inequality (see Proposition 22) and a comparisonprinciple (see Proposition 24)
Lemma 19 (a necessary condition to (6)) Let 0 lt 119903(119905) le 1199030
on [1199050infin) If assumption (6) is fulfilled then there is a positive
real number 120576 such that1
120587lowast
int119869
min119901
(120576119903 (119905))120574minus1
120576119896 (120582 120583 120588)119876 (119905) 119889119905 ge 1 (54)
for all 120582 ge 1205820 120583 ge 120583
0 and 120588 ge 120588
0and some (120582
0 1205830 1205880) isin R3+
Proof Since 0 lt 119903(119905) le 1199030for 119905 ge 119905
0 we conclude that for
120576 = (119901
119903120574minus1
0119896 (120582 120583 120588)max
119905isin 119869119876 (119905)
)
1120574
(120582 120583 120588) isin R3
+
(55)
it holds that 119901(120576119903(119905))120574minus1
ge 119901(1205761199030)120574minus1
= 120576119896(120582 120583 120588)
max119905isin 119869119876(119905) ge 120576119896(120582 120583 120588)119876(119905) 119905 isin 119869 and hence
int119869
min119901
(120576119903 (119905))120574minus1
120576119896 (120582 120583 120588)119876 (119905) 119889119905
= 120576119896 (120582 120583 120588) int119869
119876 (119905) 119889119905
(56)
Discrete Dynamics in Nature and Society 9
On the other hand from (6) we observe
1
120587lowast
int119869
119876 (119905) 119889119905 ge1199031minus(1120574)
0
1199011120574[119896 (120582 120583 120588)]1minus(1120574)
(max119905isin 119869
119876 (119905))
1120574
(57)
which together with (55) and (56) gives
1
120587lowast
int119869
min119901
(120576119903 (119905))120574minus1
120576119896 (120582 120583 120588)119876 (119905) 119889119905
= 120576119896 (120582 120583 120588)1
120587lowast
int119869
119876 (119905) 119889119905
ge 1205761199031minus(1120574)
0
1199011120574[119896 (120582 120583 120588)]
1120574
(max119905isin 119869
119876 (119905))
1120574
= 1
(58)
for all 119899 ge 1198990 120582 ge 120582
0 120583 ge 120583
0 and 120588 ge 120588
0 It proves this
lemma
Lemma 20 (an equivalent condition to (54)) Assumption(54) is fulfilled if and only if there is a real number 120576 gt 0 and acontinuous function 119870(119905) ge 0 119905 isin 119869 such that
1198880= int119869
119870 (119905) 119889119905 gt 0119870 (119905)
1198880
le1
120587lowast
timesmin119901
(120576119903 (119905))120574minus1
120576119896 (120582 120583 120588)119876 (119905)
(59)
for all 119905 isin 119869 120582 ge 1205820 120583 ge 120583
0 and 120588 ge 120588
0and some (120582
0 1205830 1205880) isin
R3+
Proof This proof is very elementary Indeed if (54) holdsthen the function119870(119905) and number 119888
0 defined by
119870 (119905) =1
120587lowast
min119901
(120576119903 (119905))120574minus1
120576119896 (120582 120583 120588)119876 (119905)
1198880= int119869
119870 (119905) 119889119905
(60)
obviously satisfy 1198880
ge 1 and 119870(119905)1198880
le 119870(119905) = (1120587lowast)
min119901(120576119903(119905))120574minus1 120576119896(120582 120583 120588)119876(119905) which shows (59) Con-versely if (59) holds then integrating both sides of thesecond inequality in (59) we obtain
int119869
1
120587lowast
min119901
(120576119903 (119905))120574minus1
120576119896 (120582 120583 120588)119876 (119905) ge int119869
119870 (119905)
1198880
119889119905 = 1
(61)
which shows (54)
In conclusion according to previous two lemmas wesee that supposed condition (6) implies (59) which plays animportant role in the proof of the main results
The second step in the proof of Lemma 1 is to prove theexistence of a function 120596(119905) which blows up in the finitetime and satisfies a generalized Riccati differential lowerinequality we briefly present the existence and properties
of the so-called generalized tangent type function In whatfollows let 120587
lowastbe a positive real number defined in (3) Let us
remark that 120572(119904) = 119904120574 120574 gt 1 implies 120587
lowast= (2120587)(120574 sin(120587120574))
see for instance [54] and obviously for 120574 = 2wehave120587lowast= 120587
Lemma 21 Let 120572 [0infin) rarr [0infin) be a continuousfunction such that
int
infin
0
119889120591
1 + 120572 (120591)lt infin (62)
Then there is a real number 120587lowastgt 0 and a function 119911 = 119911(119904)
119911 isin 1198621((minus120587lowast2 120587lowast2)R) such that
119889119911
119889119904= 1 + 120572 (|119911 (119904)|) 119904 isin (minus
120587lowast
2120587lowast
2)
119911 (0) = 0
(63)
Moreover 119911(119904) is increasing and odd
lim119904rarr120587lowast2
119911 (119904) = infin 120587lowast=
2120587
120574 sin (120587120574)for 120572 (119904) = 119904
120574
120574 gt 1
(64)
In particular for 120572(119904) = 1199042 one can take 119911(119904) = tan(119904) and
120587lowast= 120587
Proof Let 119885 = 119885(119905) 119905 isin R be a function defined by
119885 (119905) = int
119905
0
1
1 + 120572 (|120591|)119889120591 119905 isin R (65)
The function 119885(119905) is well defined since 120572(119904) is positive andcontinuous on [0infin) 119885(119905) is increasing and odd functionand
119889119885
119889119905=
1
1 + 120572 (|119905|) 119905 isin R
119885 (0) = 0 119885 isin 1198621
(RR)
(66)
Moreover because of (62) there is a real number 120587lowastgt 0 such
that120587lowast
2= int
infin
0
119889120591
1 + 120572 (120591) (67)
Thus 119885 R rarr (minus120587lowast2 120587lowast2) and there exists an inverse
function 119885minus1 = 119885minus1(119904) of the original function 119885 = 119885(119905) and
119885minus1
(minus120587lowast2 120587lowast2) rarr R Also from 119885(119885
minus1(119904)) = 119904 and
119889119885119889119905 = 0 onR we also derive that119889119885minus1119889119904 = 0 on its domain(minus120587lowast2 120587lowast2) and
119889119885
119889119905(119885minus1
(119904)) =1
(119889119885minus1119889119904) 119904 isin (minus
120587lowast
2120587lowast
2) (68)
Putting 119905 = 119885minus1(119904) for 119904 isin (minus120587
lowast2 120587lowast2) into (66) and using
(68) we easily obtain
119889119885minus1
119889119904= 1 + 120572 (
10038161003816100381610038161003816119885minus1
(119904)10038161003816100381610038161003816) 119904 isin (minus
120587lowast
2120587lowast
2)
119885minus1
(0) = 0 119885minus1isin 1198621((minus
120587lowast
2120587lowast
2) R)
(69)
10 Discrete Dynamics in Nature and Society
Moreover from (67) we have lim119904rarr120587lowast2119885minus1(119904) = 119885
minus1
(lim119905rarrinfin
119885(119905)) = lim119905rarrinfin
119885minus1119885(119905) = lim
119905rarrinfin119905 = infin Thus
if we set 119911(119904) = 119885minus1(119904) then previous two statements and
(67) prove this lemma
Next we prove the main result of this section
Proposition 22 Let (2) and (6) hold where 119869 = (119886 119887) Let 120576 gt0 be a real number and let119870(119905) ge 0 119905 isin [119886 119887] be a continuousfunction both obtained in Lemma 20 Let 120587
lowastbe from (3) and
1198880from (59) and let 119877
119886isin R be an arbitrary real number If
119911 = 119911(119904) is the generalized tangens function defined in (63)and 119881(119905) is a function defined by
119881 (119905) =120587lowast
1198880
int
119905
119886
119870 (120591) 119889120591 + 119911minus1(119877119886) 119905 isin [119886 119887] (70)
then there is a 119879lowast119886isin [119886 119887) such that
119881 (119879lowast
119886) =
120587lowast
2 119881 ([119886 119879
lowast
119886)) sub (minus
120587lowast
2120587lowast
2) (71)
Moreover for a function 120596(119905) defined by120596 (119905) = 119911 (119881 (119905)) 119905 isin [119886 119879
lowast
119886) (72)
one has 120596(119886) = 119877119886 lim119905rarr119879
lowast
119886
120596(119905) = infin and
119889120596
119889119905le
119901
(120576119903 (119905))120574minus1
120572 (1003816100381610038161003816120596 (119905)
1003816100381610038161003816)
+ 120576119896 (120582 120583 120588)119876 (119905) 119905 isin (119886 119879lowast
119886)
(73)
where the numbers 119901 and 120574 are from (3) and the functions119896(120582 120583 120588) and 119876(119905) are from (6)
Proof Under assumptions (2) and (6) and because of Lem-mas 19 and 20 we obtain 120576 gt 0 and 119870(119905) gt 0 119905 isin [119886 119887]satisfying inequality (59)
Next since 119911minus1(119877119886) isin (minus120587
lowast2 120587lowast2) (see Lemma 21)
from (70) we directly obtain
119881 (119886) = 119911minus1(119877119886) lt
120587lowast
2 119881 (119887) = 120587
lowast+ 119911minus1(119877119886) gt
120587lowast
2
(74)Since 119870 isin 119862([119886 119887] [0infin)) we obtain 119881 isin 119862([119886 119887]R) cap
1198621((119886 119887)R) and from (74) we observe that there exist
numbers 119879lowast119886isin (119886 119887) such that119881(119879lowast
119886) = 120587lowast2 Also119870(119905)119888
0ge
0 gives 119881([119886 119879lowast119886)) sub (minus120587
lowast2 120587lowast2) which proves statement
(71) Moreover it together with Lemma 21 and (72) provesthat
lim119905rarr119879
lowast
119886
120596 (119905) = lim119905rarr119879
lowast
119886
119911 (119881 (119905)) = 119911 (120587lowast
2) = infin (75)
Next according to (59) (63) and (72) we make thefollowing calculation on the interval [119886 119879lowast
119886)
1205961015840
(119905) = 1199111015840
(119881 (119905)) 1198811015840
(119905) = [1 + 120572 (|119911 (119881 (119905))|)]120587lowast
1198880
119870 (119905)
= [1 + 120572 (1003816100381610038161003816120596 (119905)
1003816100381610038161003816)]120587lowast
1198880
119870 (119905)
le119901
(120576119903 (119905))120574minus1
120572 (1003816100381610038161003816120596 (119905)
1003816100381610038161003816) + 120576119896 (120582 120583 120588)119876 (119905)
(76)
Thus all assertions of this proposition are proved
Remark 23 In the proof of the main result the number 119877119886
is determined by 119877119886= 120596(119886) where 120596(119905) denotes a function
associated with a nonoscillatory solution and it is given by(84) below
The third step in the proof of Lemma 1 is to show thefollowing pointwise comparison principle for the functions120596and120596 satisfying respectively the lower and upper differentialinequalities (73) and
119889120596
119889119905ge
119901
(120576119903 (119905))120574minus1
120572 (|120596 (119905)|) + 120576119896 (120582 120583 120588)119876 (119905) 119905 isin (119886 119887)
(77)
Proposition 24 Let [119886 119887) sub [1199050infin) be an arbitrary inter-
val One supposes that all coefficients of Riccati differentialinequalities (73) and (77) are continuous and strictly positivefunctions Let 120596 120596 isin 119862
1([119886 119887)R) be two functions satisfying
respectively (73) and (77) on the interval [119886 119887) Then
120596 (119886) le 120596 (119886) 119894119898119901119897119894119890119904 120596 (119905) le 120596 (119905) forall119905 isin [119886 119887) (78)
Proof Let119867(119905 119906) be a function defined by
119867(119905 119906) =119901
(120576119903 (119905))120574minus1
120572 (|119906|) + 120576119896 (120582 120583 120588)119876 (119905)
119905 isin [119886 119887) 119906 isin R
(79)
Let 119868 sub [119886 119887) and 119872 gt 0 be arbitrary For any two 1199061
1199062 minus119872 le 119906
1lt 1199062le 119872 let 119868
12be an interval defined
by 11986812
= (min|1199061| |1199062|max|119906
1| |1199062|) Since 120572(119904) is a 1198621-
function on [0infin) we know by the Lagrange mean valuetheorem applied on 119868
12that there is a 120585 isin 119868
12such that
120572 (10038161003816100381610038161199062
1003816100381610038161003816) minus 120572 (10038161003816100381610038161199061
1003816100381610038161003816)
1199062minus 1199061
le
1003816100381610038161003816120572 (10038161003816100381610038161199062
1003816100381610038161003816) minus 120572 (10038161003816100381610038161199061
1003816100381610038161003816)1003816100381610038161003816
1199062minus 1199061
=100381610038161003816100381610038161205721015840
(120585)10038161003816100381610038161003816
100381610038161003816100381610038161003816100381610038161199062
1003816100381610038161003816 minus10038161003816100381610038161199061
10038161003816100381610038161003816100381610038161003816
1199062minus 1199061
le100381610038161003816100381610038161205721015840
(120585)10038161003816100381610038161003816
le max119904isin11986812
100381610038161003816100381610038161205721015840
(119904)10038161003816100381610038161003816
(80)
since ||1199062| minus |1199061|| le 119906
2minus 1199061 Hence for any 119905 isin 119868 and 119906
1 1199062
minus119872 le 1199061lt 1199062le 119872 we have
119867(119905 1199062) minus 119867 (119905 119906
1)
1199062minus 1199061
= 1205880(119905)
120572 (10038161003816100381610038161199062
1003816100381610038161003816) minus 120572 (10038161003816100381610038161199061
1003816100381610038161003816)
1199062minus 1199061
le 1205880(119905)max119904isin11986812
100381610038161003816100381610038161205721015840
(119904)10038161003816100381610038161003816= 1198710(119905)
(81)
Thus the function119867(119905 119906) from (79) satisfies required condi-tion of [55 Lemma 19] and applying it to (73) and (77) weprove this proposition
Proof of Lemma 1 On the contrary let 119909(119905) be a solution of(1) such that
119909 (119905) = 0 on (120591 (120591 (119886)) 120590 (120590 (119889))) (82)
Discrete Dynamics in Nature and Society 11
that is 119909(119905) gt 0 on (120591(120591(119886)) 120590(120590(119889))) or 119909(119905) lt 0 on(120591(120591(119886)) 120590(120590(119889))) since 119909(119905) is a continuous function on[1199050infin) Let for instance
119909 (119905) gt 0 on (120591 (120591 (119886)) 120590 (120590 (119889))) (83)
Another case can be analogously treated let us see thecomment at the end of this proof In particular from (83)we have 119909(119905) gt 0 on (120591(120591(119886)) 120590(120590(119887))) which implies (since120591(119905) and 120590(119905) are increasing functions) 119909(119904) gt 0 for all 119904 isin
(120591(119886) 120590(119887)) cup (120591(120591(119886)) 120591(120590(119887))) cup (120590(120591(119886)) 120590(120590(119887))) whichyields 119909(119905) gt 0 119909(120591(119905)) gt 0 and 119909(120590(119905)) gt 0 on (120591(119886) 120590(119887))Hence by assumption (7) we may use inequality (5) on theinterval (119886 119887)
Firstly we show that the following classic Riccati transfor-mation of 119909(119905)
120596 (119905) = minus120576119903 (119905) 119860 (119909
1015840(119905))
|119909 (119905)|119901minus1
119909 (119905) 119905 isin (119886 119887) 120576 gt 0 (84)
satisfies upper Riccati differential inequality (77) Let usremark that from (1) we have in particular
minus(119903 (119905) 119860 (1199091015840
(119905)))1015840
= 119861 (119905 119909 (119905) 1199091015840
(119905)) + 120582119865 (119905 119909 (120591 (119905)))
+ 120583119866 (119905 119909 (120590 (119905))) minus 120588119890 (119905) 119905 ge 1199050
(85)
Taking the first derivative on both sides of (84) and usingassumptions (3) (4) and (5) as well as equality (85) and(|119909(119905)|
119901minus1119909(119905))1015840
= 119901|119909(119905)|119901minus1
1199091015840(119905) we obtain
119889120596
119889119905= 120576119901 119903 (119905)
119860 (1199091015840(119905)) 1199091015840(119905)
|119909 (119905)|119901+1
minus 1205761
|119909 (119905)|119901minus1
119909 (119905)(119903 (119905) 119860 (119909
1015840
(119905)))1015840
= 120576119901119903 (119905)119860 (1199091015840(119905)) 1199091015840(119905)
|119909 (119905)|119901+1
+120576
|119909 (119905)|119901minus1
119909 (119905)
times [120582119861 (119905 119909 (119905) 1199091015840
(119905)) + 119865 (119905 119909 (120591 (119905)))
+120583119866 (119905 119909 (120590 (119905))) minus 120588119890 (119905) ]
ge 120576119901119903 (119905)119860 (1199091015840(119905)) 1199091015840(119905)
|119909 (119905)|119901+1
+120576
|119909 (119905)|119901minus1
119909 (119905)
times [120582119865 (119905 119909 (120591 (119905))) + 120583119866 (119905 119909 (120590 (119905))) minus 120588119890 (119905)]
ge 120576119901119903 (119905) 120572(
10038161003816100381610038161003816119860 (1199091015840(119905))
10038161003816100381610038161003816
|119909 (119905)|119901
) + 120576119896 (120582 120583 120588)119876 (119905)
= 120576119901119903 (119905) 120572 (|120596 (119905)|
120576119903 (119905)) + 120576119896 (120582 120583 120588)119876 (119905)
ge119901
(120576119903 (119905))120574minus1
120572 (|120596 (119905)|) + 120576119896 (120582 120583 120588)119876 (119905) 119905 isin (119886 119887)
(86)
Thus according to inequality (5) it is shown that if 119909(119905) isa solution of (1) which satisfies (83) then the function 120596(119905)
defined by (84) satisfies the Riccati differential inequality(77) and 120596 isin 119862((119886 119887)R) On the other hand let 119877
119886be a
real number defined by 119877119886= 120596(119886) According to (6) and
Lemma 19 we obtain (54) which together with Lemma 20ensures that we may use Proposition 22 for such chosen realnumber 119877
119886 Hence we obtain a function 120596(119905) defined by (72)
which satisfies the lower Riccati differential inequality (73) on[119886 119879lowast
119886) 119879lowast119886isin (119886 119887) such that 120596(a) = 119877
119886and lim
119905rarr119879lowast
119886
120596(119905) =
infin Therefore by 120596(119886) = 119877119886= 120596(119886) and Proposition 24 we
conclude that lim119905rarr119879
lowast
119886
120596(119905) = infin too which is a contradictionwith the above conclusion saying that 120596 isin 119862((119886 119887)R) Thushypothesis (82) is not true and consequently Lemma 1 isshown
For the analogous case 119909(119905) lt 0 on (120591(120591(119886)) 120590(120590(119889))) wealso have 119909(119905) lt 0 on (120591(120591(119888)) 120590(120590(119889))) which implies (since120591(119905) and 120590(119905) are increasing functions)
119909 (119904) lt 0 forall119904 isin (120591 (119888) 120590 (119889)) cup (120591 (120591 (119888)) 120591 (120590 (119889)))
cup (120590 (120591 (119888)) 120590 (120590 (119889)))
(87)
which yields 119909(119905) lt 0 119909(120591(119905)) lt 0 and 119909(120590(119905)) lt 0 on(120591(119888) 120590(119889)) Now we can repeat the preceding procedure buton interval (119888 119889) and using (8) instead of (119886 119887) and (7)
Proof of Lemma 2 From assumption (10) we obtain the exis-tence of an 119899
0isin N such that
int
119887119899
119886119899
119876119899(119905) 119889119905 ge
1198880
2( max119905isin[119886119899 119887119899]
119876119899(119905))
1120574
119899 ge 1198990 (88)
that is
2
1198880
int
119887119899
119886119899
119876119899(119905) 119889119905 ge ( max
119905isin[119886119899 119887119899]119876119899(119905))
1120574
119899 ge 1198990 (89)
Now from (9) and previous inequality we deduce that forlarge enough 120582 120583 120588 and 119899
1199011120574
1199031minus1120574
0
[119896 (120582 120583 120588)]1minus1120574
120587lowast
int
119887119899
119886119899
119876119899(119905) 119889119905
ge2
1198880
int
119887119899
119886119899
119876119899(119905) 119889119905 ge ( max
119905isin[119886119899 119887119899]119876119899(119905))
1120574
(90)
which shows (6) Thus all assumptions of Lemma 1 arefulfilled and hence Lemma 2 immediately follows fromLemma 1
Proof of Lemma 3 Obviously assumption (11) is a particularcase of assumption (9) Hence this proof is very similar tothe proof of Lemma 2 and so it is left to the reader
Proof of Lemma 4 It is clear that from assumption (13) weobtain
1
(max119905isin[119886119899119887119899]119876119899(119905))1120574
int
119887119899
119886119899
119876119899(119905) 119889119905 ge
1198881
1198621120574
0
gt 0 forall119899 ge 1198990
(91)
12 Discrete Dynamics in Nature and Society
Thus hypothesis (12) is fulfilled and therefore Lemma 3proves this lemma
Proof of Theorems 5 6 and 7 This proof is based onLemma 4 In order to simplify notation in many placesin this proof we set 120591(119905) = 119905 minus 120591 and 120590(119905) = 119905 + 120590 Sinceassumptions (2) (3) and (4) have been already supposed inTheorems 5 6 and 7 in order to prove these theorems byLemma 4 we are going to show that the functions 119896(120582 120583 120588)and 119876
119899(119905) explicitly given respectively in (18) (21) or (24)
and (19) (22) or (25) satisfy required conditions (11) and(13) respectively and that every solution 119909(119905) of (27) satisfiesconditions (7) and (8) with respect to functions 119896(120582 120583 120588)and 119876
119899(119905) where 119886 = 119886
2119899minus1 119887 = 119887
2119899minus1 119888 = 119886
2119899 and 119889 = 119887
2119899
The proof that the function 119896(120582 120583 120588) given in (18) (21) or(24) satisfies (11) Passing to the limit in (18) (21) or (24) it isvery simple to show (11)
The proof that the function 119876119899(119905) given in (19) (22) or
(25) satisfies the first claim in (13) From (25) we immediatelyobtain
1003816100381610038161003816120591119899 (119905)1003816100381610038161003816 =
100381610038161003816100381610038161003816100381610038161003816
(119905 minus 119886119899
119905 minus 119886119899+ 120591
)
119901100381610038161003816100381610038161003816100381610038161003816
le 1
1003816100381610038161003816120590119899 (119905)1003816100381610038161003816 =
100381610038161003816100381610038161003816100381610038161003816
(119887119899minus 119905
119887119899minus 119905 + 120590
)
119901100381610038161003816100381610038161003816100381610038161003816
le 1 forall119899 isin N
(92)
Next by assumptions of this corollary we can conclude thatthere are three positive constants 119891
0 1198920 1198900such that |119891(119905)| le
1198910and |119892(119905)| le 119892
0on [1199050infin) in cases (i) and (ii) and
|119890(119905)| le 1198900on [1199050infin) in cases (iii) and (iv) Putting previous
inequalities into (19) (22) or (25) for all 119899 isin N and 119905 isin
[1199050infin) it holds that
1003816100381610038161003816119876119899 (119905)1003816100381610038161003816 le
119902
119901(
119901
119902 minus 119901)
(119902119901)minus1
1198901minus(119901119902)
0119891119901119902
0
delay case with 119902 gt 119901
119902
119901(
119901
119902 minus 119901)
(119902119901)minus1
1198901minus(119901119902)
0119892119901119902
0
advanced case with 119902 gt 119901
1199011
119901(
119901
2 (1199011minus 119901)
)
(1199011119901)minus1
1198901minus(119901119901
1)
0119891119901119902
0+1199012
119901
times(119901
2 (1199012minus 119901)
)
(1199012119901)minus1
1198901minus(119901119901
2)
0119892119901119902
0
delay-advanced case (i)
1198901205780
01198911205781
01198921205782
0
2
prod
119894=0
120578minus120578119894
119894
delay-advanced case (ii) (93)
which shows the first claim in (13)
The proof that the function119876119899(119905) given in (19) (22) or (25)
satisfies the second claim in (13)Without loss of generality weprove this claim only in case (i) since for other cases the prooffollows analogously In this sense let119876
119899(119905) = 119891(119905)120591
119899(119905) Since
1198862119899+1
minus 1198862119899minus1
le 119879lowast 1198872119899+1
minus 1198872119899minus1
ge 119879lowast 1198862119899+2
minus 1198862119899le 119879lowast and
1198872119899+2
minus 1198872119899
ge 119879lowast where 119879
lowastgt 0 is the period of the function
119891(119905) we have 1198862119899minus1
le 1198861+(119899minus1)119879
lowastand 1198872119899minus1
ge 1198871+(119899minus1)119879
lowast
119899 isin N Hence
int
1198872119899minus1
1198862119899minus1
119876119899(119905) 119889119905
= int
1198872119899minus1
1198862119899minus1
119891 (119905) (119905 minus 1198862119899minus1
119905 minus 1198862119899minus1
+ 120591)
119901
119889119905
ge int
1198871+(119899minus1)119879
lowast
1198861+(119899minus1)119879lowast
119891 (119905) (119905 minus 1198861minus (119899 minus 1) 119879
lowast
119905 minus 1198861minus (119899 minus 1) 119879
lowast+ 120591
)
119901
119889119905
= int
1198871
1198861
119891 (119904 + (119899 minus 1) 119879lowast) (
119904 minus 1198861
119904 minus 1198861+ 120591
)
119901
119889119904
= int
1198871
1198861
119891 (119904) (119904 minus 1198861
119904 minus 1198861+ 120591
)
119901
119889119904
(94)
which proves that the integral on the left hand side does notdepend on 119899 isin N that is the second claim in (13) is shown on[1198862119899minus1
1198872119899minus1
] This claim follows in the same way on [1198862119899 1198872119899]
Thus the second claim in (13) is proved on [119886119899 119887119899]
Next to the end of this proof let 119909(119905) be a solu-tion of (1) In particular it implies that (119903(119905)119860(1199091015840(119905)))1015840 =
minus119861(119905 119909(119905) 1199091015840(119905)) minus 120582119865(119905 119909(120591(119905))) minus 120583119866(119905 119909(120590(119905))) + 120588119890(119905) It
together with assumptions (15) (16) (20) and (23) easilygives the next two statements
if 119909 (119905) gt 0 119909 (120591 (119905)) gt 0 119909 (120590 (119905)) gt 0
on (120591 (1198862119899minus1
) 120590 (1198872119899minus1
))
then 119909 (119905) satisfies 119903 (119905) 119860 (1199091015840
(119905)) le 0
on (120591 (1198862119899minus1
) 120590 (1198872119899minus1
)) 119899 ge 1198990
(95)
if 119909 (119905) lt 0 119909 (120591 (119905)) lt 0 119909 (120590 (119905)) lt 0
on (120591 (1198862119899) 120590 (119887
2119899))
then 119909 (119905) satisfies 119903 (119905) 119860 (1199091015840
(119905)) ge 0
on (120591 (1198862119899) 120590 (119887
2119899)) 119899 ge 119899
0
(96)
Now we need the following lemma
Discrete Dynamics in Nature and Society 13
Lemma 25 Let 120591119886119887(119905) and 120590
119886119887(119905) be defined by
120591119886119887(119905) = (
120591 (119905) minus 120591 (119886)
119905 minus 120591 (119886))
119901
120590119886119887(119905) = (
120590 (119887) minus 120590 (119905)
120590 (119887) minus 119905)
119901
119905 isin (119886 119887)
(97)
and let 119909 isin 1198622([1198790infin)R) be an arbitrary function If
(119903(119905)119860(1199091015840(119905)) le 0 for all 119905 isin (120591(119886) 120590(119887)) or (119903(119905)119860(1199091015840(119905)) ge 0
for all 119905 isin (120591(119886) 120590(119887)) then
119909 (120591 (119905))
119909 (119905)ge (120591119886119887(119905))1119901
119909 (120590 (119905))
119909 (119905)ge (120590119886119887(119905))1119901
119905 isin (119886 119887)
(98)
Since119860(V) is supposed to be odd and increasing functionjust before (3) and 119903(119905) satisfies (14) the proof of Lemma 25in the first case that is 119903(119905)119860(1199091015840(119905)) le 0 for all 119905 isin (120591(119886) 120590(119887))is the same as the proof of [9 Corollaries 17 and 18] But in thesecond case that is 119903(119905)119860(1199091015840(119905)) ge 0 for all 119905 isin (120591(119886) 120590(119887))the proof is as follows if previous inequality holds then119903(119905)119860(minus119909
1015840(119905)) le 0 for all 119905 isin (120591(119886) 120590(119887)) and therefore to
the function minus119909(119905) one can apply the first case of this lemmaand consequently one obtains
119909 (120591 (119905))
119909 (119905)=minus119909 (120591 (119905))
minus119909 (119905)ge (120591119886119887(119905))1119901
119909 (120590 (119905))
119909 (119905)=minus119909 (120590 (119905))
minus119909 (119905)ge (120590119886119887(119905))1119901
119905 isin (119886 119887)
(99)
which proves this lemma in the second caseNow combining statements (95) (96) and (98) one
easily obtains
if 119909 (119905) gt 0 119909 (120591 (119905)) gt 0 119909 (120590 (119905)) gt 0
on (120591 (1198862119899minus1
) 120590 (1198872119899minus1
))
then 119909 (119905) satisfies 119909 (120591 (119905))
119909 (119905)ge (120591119899(119905))1119901
on (1198862119899minus1
1198872119899minus1
) 119899 ge 1198990
(100)
if 119909 (119905) lt 0 119909 (120591 (119905)) lt 0 119909 (120590 (119905)) lt 0
on (120591 (1198862119899) 120590 (119887
2119899))
then 119909 (119905) satisfies 119909 (120590 (119905))
119909 (119905)ge (120590119899(119905))1119901
on (1198862119899 1198872119899) 119899 ge 119899
0
(101)
where 120591119899(119905) and 120590
119899(119905) are defined in (26)
The proof that 119909(119905) satisfies (7) and (8) In this proofwe frequently use assumptions (16) (20) and (23) andstatements (100) and (101) Also because of (15) and 119865(119905 119906) =
119891(119905)|119906|1199011 sgn(119906) 119866(119905 119906) = 119892(119905)|119906|
1199012 sgn(119906) in both cases
(100) and (101) we can simultaneously use
minus119890 (119905) (|119909 (119905)|119901minus1
119909 (119905))minus1
= |119890 (119905)| |119909 (119905)|minus119901
ge 0 on 119869119899
119865 (119905 119909 (120591 (119905))) (|119909 (119905)|119901minus1
119909 (119905))minus1
ge 119891 (119905) |119909 (120591 (119905))|1199011 |119909 (119905)|
minus119901ge 0 on 119869
119899
119866 (119905 119909 (120590 (119905))) (|119909 (119905)|119901minus1
119909 (119905))minus1
ge 119892 (119905) |119909 (120590 (119905))|1199012 |119909 (119905)|
minus119901ge 0 on 119869
119899
|119909 (120591 (119905))| |119909 (119905)|minus1=119909 (120591 (119905))
119909 (119905)
|119909 (120590 (119905))| |119909 (119905)|minus1=119909 (120590 (119905))
119909 (119905)on 119869119899
(102)
where 119869119899= (1198862119899minus1
1198872119899minus1
) in the case of (100) and 119869119899= (1198862119899 1198872119899)
in the case of (101)
(i) Delay or Advanced Case with 119902 = 119901 Since 119902 = 119901 we obtain
[120582119865 (119905 119909 (120591 (119905))) + 120583119866 (119905 119909 (120590 (119905))) minus 120588119890 (119905)]
times (|119909 (119905)|119901minus1
119909 (119905))minus1
ge [120582119891 (119905) |119909 (120591 (119905))|119901+ 120583119892 (119905) |119909 (120590 (119905))|
119901
+120588 |119890 (119905)| ] |119909 (119905)|minus119901
ge [120582119891 (119905) |119909 (120591 (119905))|119901+ 120583119892 (119905) |119909 (120590 (119905))|
119901] |119909 (119905)|
minus119901
= 120582119891 (119905) (119909 (120591 (119905))
119909 (119905))
119901
+ 120583119892 (119905) (119909 (120590 (119905))
119909 (119905))
119901
ge 120582119891 (119905) 120591119899(119905) + 120583119892 (119905) 120590
119899(119905) 119905 isin 119869
119899
(103)
where the functions 120591119899(119905) and 120590
119899(119905) are defined in (26)
(ii) Delay Case with 119902 gt 119901 In this part we use the nextelementary inequality
119883120574+ (120574 minus 1) 119884
120574ge 120574119883119884
120574minus1 120574 gt 1 119883 119884 ge 0 (104)
Since 119902 gt 119901 and using (104) especially for
120574 =119902
119901gt 1 119883 = (120582119891 (119905))
119901119902
|119909 (120591 (119905))|119901
119884 = (119901
119902 minus 119901120588 |119890 (119905)|)
119901119902
(105)
14 Discrete Dynamics in Nature and Society
for all 119905 isin 119869119899we obtain
[120582119865 (119905 119909 (120591 (119905))) + 120583119866 (119905 119909 (120590 (119905))) minus 120588119890 (119905)]
times (|119909 (119905)|119901minus1
119909 (119905))minus1
= [120582119865 (119905 119909 (120591 (119905))) minus 120588119890 (119905)] (|119909 (119905)|119901minus1
119909 (119905))minus1
ge [120582119891 (119905) |119909 (120591 (119905))|1199011 + 120588 |119890 (119905)|] |119909 (119905)|
minus119901
= [119883120574+ (120574 minus 1) 119884
120574] |119909 (119905)|
minus119901
ge119902
119901(120582119891 (119905))
119901119902
|119909 (120591 (119905))|119901
times (119901
119902 minus 119901120588 |119890 (119905)|)
(119901119902)((119902119901)minus1)
|119909 (119905)|minus119901
= 120582119901119902
1205881minus(119901119902)
119902
119901(
119901
119902 minus 119901)
1minus(119901119902)
|119890 (119905)|1minus(119901119902)
times (119891 (119905))119901119902
(119909 (120591 (119905))
119909 (119905))
119901
ge 119896 (120582 120583 120588)119902
119901(
119901
119902 minus 119901)
1minus(119901119902)
|119890 (119905)|1minus(119901119902)
times (119891 (119905))119901119902
120591119899(119905)
(106)
where the function 119896(120582 120583 120588) is from (18)
(iii) Advanced Case with 119902 gt 119901 Using the same line ofarguments as in the proof of the previous case for all 119905 isin 119869
119899
we obtain
[120582119865 (119905 119909 (120591 (119905))) + 120583119866 (119905 119909 (120590 (119905))) minus 120588119890 (119905)]
times (|119909 (119905)|119901minus1
119909 (119905))minus1
= [120583119866 (119905 119909 (120590 (119905))) minus 120588119890 (119905)] (|119909 (119905)|119901minus1
119909 (119905))minus1
ge 119896 (120582 120583 120588)119902
119901(
119901
119902 minus 119901)
1minus(119901119902)
|119890 (119905)|1minus(119901119902)
times (119892 (119905))119901119902
120590119899(119905)
(107)
where the function 119896(120582 120583 120588) is from (21)
(iv) Superlinear Delay-Advanced Case Since 1199011 1199012gt 119901 for
all 119905 isin 119869119899we obtain
[120582119865 (119905 119909 (120591 (119905))) + 120583119866 (119905 119909 (120590 (119905))) minus 120588119890 (119905)]
times (|119909 (119905)|119901minus1
119909 (119905))minus1
= [120582119865 (119905 119909 (120591 (119905))) minus120588
2119890 (119905)] (|119909 (119905)|
119901minus1119909 (119905))minus1
+ [120583119866 (119905 119909 (120590 (119905))) minus120588
2119890 (119905)] (|119909 (119905)|
119901minus1119909 (119905))minus1
ge [120582119891 (119905) |119909 (120591 (119905))|1199011 +
120588
2|119890 (119905)|] |119909 (119905)|
minus119901
+ [120583119892 (119905) |119909 (120590 (119905))|1199012 +
120588
2|119890 (119905)|] |119909 (119905)|
minus119901
(108)
Now just the same as in the proofs of previous delay andadvanced cases with 119902 gt 119901 and with the help of (104) inparticular for
120574 =1199011
119901gt 1 119883 = (120582119891 (119905))
1199011199011
|119909 (120591 (119905))|119901
119884 = (119901
1199011minus 119901
120588
2|119890 (119905)|)
1199011199011
(109)
we have
[120582119891 (119905) |119909 (120591 (119905))|1199011 +
120588
2 |119890 (119905)|] |119909 (119905)|
minus119901
= [119883120574+ (120574 minus 1) 119884
120574] |119909 (119905)|
minus119901
ge1199011
119901(120582119891 (119905))
1199011199011
|119909 (120591 (119905))|119901
times (119901
1199011minus 119901
120588
2|119890 (119905)|)
(1199011199011)((1199011119901)minus1)
|119909 (119905)|minus119901
= 12058211990111990111205881minus(119901119901
1)1199011
119901(
119901
2 (1199011minus 119901)
)
1minus(1199011199011)
times |119890 (119905)|1minus(119901119901
1)(119891 (119905))
1199011199011
(119909 (120591 (119905))
119909 (119905))
119901
ge 119896 (120582 120583 120588)1199011
119901(
119901
2 (1199011minus 119901)
)
1minus(1199011199011)
times |119890 (119905)|1minus(119901119901
1)(119891 (119905))
1199011199011
120591119899(119905)
(110)
where the function 119896(120582 120583 120588) is from (24) Analogously weshow that
[120583119892 (119905) |119909 (120590 (119905))|1199012 +
120588
2|119890 (119905)|] |119909 (119905)|
minus119901
ge 119896 (120582 120583 120588)1199012
119901(
119901
2 (1199012minus 119901)
)
1minus(1199011199012)
times |119890 (119905)|1minus(119901119901
2)(119891 (119905))
1199011199012
120590119899(119905)
(111)
Discrete Dynamics in Nature and Society 15
Summarizing previous calculation we obtain
[120582119865 (119905 119909 (120591 (119905))) + 120583119866 (119905 119909 (120590 (119905))) minus 120588119890 (119905)]
times (|119909 (119905)|119901minus1
119909 (119905))minus1
ge 119896 (120582 120583 120588)119876119899(119905) 119905 isin 119869
119899
(112)
where the function 119896(120582 120583 120588) is from (24)
(v) Supersublinear Delay-Advanced Case Since 1199011gt 119901 gt 119901
2
and the following well-known elementary inequality holds
12057801199060+ 12057811199061+ 12057821199062ge 1199061205780
01199061205781
11199061205782
2 120578119894ge 0 119906
119894ge 0 (113)
from 1205780 1205781 1205782isin (0 1) 120578
0+ 1205781+ 1205782= 1 and 119901
11205781+ 11990121205782= 119901
we obtain for all 119905 isin 119869119899 for all 119905 isin 119869
119899we obtain
[120582119865 (119905 119909 (120591 (119905))) + 120583119866 (119905 119909 (120590 (119905))) minus 120588119890 (119905)]
times (|119909 (119905)|119901minus1
119909 (119905))minus1
ge [120582119891 (119905) |119909 (120591 (119905))|1199011 + 120583119892 (119905) |119909 (120590 (119905))|
1199012 + 120588 |119890 (119905)|]
times |119909 (119905)|minus119901
= [1205781[120578minus1
1120582119891 (119905) |119909 (120591 (119905))|
1199011] + 120578
2[120578minus1
2120583119892 (119905) |119909 (120590 (119905))|
1199012]
+1205780[120578minus1
0120588 |119890 (119905)|]] |119909 (119905)|
minus119901
ge [120578minus1
1120582119891 (119905) |119909 (120591 (119905))|
1199011]1205781
[120578minus1
2120583119892 (119905) |119909 (120590 (119905))|
1199012]1205782
times [120578minus1
0120588 |119890 (119905)|]
1205780
|119909 (119905)|minus119901
= 120582120578112058312057821205881205780 |119890 (119905)|
1205780(119891 (119905))
1205781
(119892 (119905))1205782
times|119909 (120591 (119905))|
12057811199011
|119909 (119905)|12057811199011
|119909 (120590 (119905))|12057821199012
|119909 (119905)|12057821199012
2
prod
119894=0
120578minus120578119894
119894
= 119896 (120582 120583 120588) |119890 (119905)|1205780(119891 (119905))
1205781
(119892 (119905))1205782
times (119909 (120591 (119905))
119909 (119905))
12057811199011
(119909 (120590 (119905))
119909 (119905))
12057821199012 2
prod
119894=0
120578minus120578119894
119894
ge 119896 (120582 120583 120588) |119890 (119905)|1205780(119891 (119905))
1205781
(119892 (119905))1205782
(120591119899(119905))1205781(1199011119901)
times (120590119899(119905))1205782(1199012119901)
2
prod
119894=0
120578minus120578119894
119894
= 119896 (120582 120583 120588)119876119899(119905)
(114)
where 119896(120582 120583 120588) and 119876119899(119905) are given respectively in (24) and
(25) Thus it is shown that required condition (5) in thecases (i)ndash(iv) is fulfilled with respect to 119896(120582 120583 120588) and 119876
119899(119905)
determined by (18) (21) or (24) and (19) (22) or (25)In conclusion according to the previous observation we
see that all assumptions of Lemma 4 are fulfilled and henceLemma 4 proves Theorems 5 6 and 7
Conflict of Interests
The author declares that there is no conflict of interestsregarding the publication of this paper
References
[1] Y G Sun ldquoA note on Nasrrsquos and Wongrsquos papersrdquo Journal ofMathematical Analysis and Applications vol 286 no 1 pp 363ndash367 2003
[2] Y G Sun C H Ou and J S W Wong ldquoInterval oscillationtheorems for a second-order linear differential equationrdquo Com-puters amp Mathematics with Applications vol 48 no 10-11 pp1693ndash1699 2004
[3] S Murugadass E Thandapani and S Pinelas ldquoOscillationcriteria for forced second-order mixed type quasilinear delaydifferential equationsrdquo Electronic Journal of Differential Equa-tions vol 2010 article 73 9 pages 2010
[4] Y Bai and L Liu ldquoNew oscillation criteria for second-orderdelay differential equations with mixed nonlinearitiesrdquoDiscreteDynamics in Nature and Society vol 2010 Article ID 796256 9pages 2010
[5] A F Guvenilir andA Zafer ldquoSecond-order oscillation of forcedfunctional differential equations with oscillatory potentialsrdquoComputers amp Mathematics with Applications vol 51 no 9-10pp 1395ndash1404 2006
[6] A Zafer ldquoInterval oscillation criteria for second order super-half linear functional differential equations with delay andadvanced argumentsrdquoMathematische Nachrichten vol 282 no9 pp 1334ndash1341 2009
[7] A F Guvenilir ldquoInterval oscillation of second-order functionaldifferential equations with oscillatory potentialsrdquo NonlinearAnalysis Theory Methods amp Applications vol 71 no 12 ppe2849ndashe2854 2009
[8] T S Hassan L Erbe and A Peterson ldquoForced oscillation ofsecond order differential equations with mixed nonlinearitiesrdquoActa Mathematica Scientia B vol 31 no 2 pp 613ndash626 2011
[9] M Pasic ldquoNew oscillation criteria for second-order forcedquasilinear functional differential equationsrdquo Abstract andApplied Analysis vol 2013 Article ID 735360 12 pages 2013
[10] L H Erbe Q Kong and B G Zhang Oscillation Theory forFunctional-Differential Equations vol 190 Marcel Dekker NewYork NY USA 1995
[11] V Kolmanovskii and A Myshkis Introduction to the Theoryand Applications of Functional-Differential Equations vol 463Kluwer Academic Publishers Dordrecht The Netherlands1999
[12] R P Agarwal M Bohner and W-T Li Nonoscillation andOscillation Theory for Functional Differential Equations vol267 Marcel Dekker New York NY USA 2004
[13] L Erbe T Hassan and A Peterson ldquoOscillation of secondorder functional dynamic equationsrdquo International Journal ofDifference Equations vol 5 no 2 pp 175ndash193 2010
[14] B Baculıkova J Dzurina and Y V Rogovchenko ldquoOscillationof third order trinomial delay differential equationsrdquo AppliedMathematics and Computation vol 218 no 13 pp 7023ndash70332012
[15] R P Agarwal L Berezansky E Braverman and A Domoshnit-sky Nonoscillation Theory of Functional Differential Equationswith Applications Springer New York NY USA 2012
16 Discrete Dynamics in Nature and Society
[16] J Zhang ldquoVariational approach to solitary wave solution ofthe generalized Zakharov equationrdquo Computers ampMathematicswith Applications vol 54 no 7-8 pp 1043ndash1046 2007
[17] T Ozis and A Yıldırım ldquoApplication of Hersquos semi-inversemethod to the nonlinear Schrodinger equationrdquo Computers ampMathematics with Applications vol 54 no 7-8 pp 1039ndash10422007
[18] X-C Cai andM-S Li ldquoPeriodic solution of Jacobi elliptic equa-tions by Hersquos perturbation methodrdquo Computers amp Mathematicswith Applications vol 54 no 7-8 pp 1210ndash1212 2007
[19] S Lenci G Menditto and A M Tarantino ldquoHomoclinic andheteroclinic bifurcations in the non-linear dynamics of a beamresting on an elastic substraterdquo International Journal of Non-Linear Mechanics vol 34 no 4 pp 615ndash632 1999
[20] D-J Huang and H-Q Zhang ldquoLink between travelling wavesand first order nonlinear ordinary differential equation with asixth-degree nonlinear termrdquoChaos Solitons amp Fractals vol 29no 4 pp 928ndash941 2006
[21] A I Maimistov ldquoPropagation of an ultimately short electro-magnetic pulse in a nonlinear medium described by the fifth-order Duffing modelrdquo Optics and Spectroscopy vol 94 pp 251ndash257 2003
[22] M N Hamdan and N H Shabaneh ldquoOn the large amplitudefree vibrations of a restrained uniform beam carrying anintermediate lumpedmassrdquo Journal of Sound andVibration vol199 no 5 pp 711ndash736 1997
[23] J-H He ldquoSome asymptotic methods for strongly nonlinearequationsrdquo International Journal of Modern Physics B vol 20no 10 pp 1141ndash1199 2006
[24] J B Marion Classical Dynamics of Particles and Systems 1970[25] I Kovacic and M J Brennan The Duffing Equation Nonlinear
Oscillatos and their Behaviour John Wiley amp Sons 1st edition2011
[26] F C Moon Chaotic Vibrations An Introduction for AppliedScientists and Engineers John Wiley amp Sons New York NYUSA 2004
[27] J J Stoker Nonlinear Vibrations 1950[28] G Chen and Z Tao ldquoAmplitude-frequency relationship for the
relativistic oscillatorrdquoAASRI Procedia vol 1 pp 400ndash403 2012[29] R E Mickens Oscillations in Planar Dynamic Systems World
Scientific Publishing Singapore 1996[30] A Belendez T Belendez C Neipp A Hernandez and M
L Alvarez ldquoApproximate solutions of a nonlinear oscillatortypified as a mass attached to a stretched elastic wire by thehomotopy perturbation methodrdquo Chaos Solitions and Fractalsvol 39 pp 746ndash764 2009
[31] A Belendez E Fernandez R Fuentes J J Rodes and I PascualldquoHarmonic balancing approach to nonlinear oscillations of apunctual charge in the eletric field of charged ringrdquo PhysicsLetters A vol 373 pp 735ndash740 2009
[32] A Elıas-Zuniga ldquoExact solution of the cubic-quintic Duffingoscillatorrdquo Applied Mathematical Modelling vol 37 no 4 pp2574ndash2579 2013
[33] A Belendez M L Alvarez J Frances et al ldquoAnalytical approx-imate solutions for the cubic-quintic Duffing oscillator in termsof elementary functionsrdquo Journal of Applied Mathematics vol2012 Article ID 286290 16 pages 2012
[34] A Elıas-Zuniga OMartınez-Romero andR K Cordoba-DıazldquoApproximate solution for the Duffing-harmonic oscillator bythe enhanced cubication methodrdquo Mathematical Problems inEngineering vol 2012 Article ID 618750 12 pages 2012
[35] C W Lim B S Wu andW P Sun ldquoHigher accuracy analyticalapproximations to the Duffing-harmonic oscillatorrdquo Journal ofSound and Vibration vol 296 no 4-5 pp 1039ndash1045 2006
[36] J He ldquoSome new approaches to Duffing equation with stronglyand high order nonlinearity II parametrized perturbationtechniquerdquo Communications in Nonlinear Science amp NumericalSimulation vol 4 no 1 pp 81ndash83 1999
[37] V Marinca and N Herisanu ldquoPeriodic solutions for somestrongly nonlinear oscillations by Hersquos variational iterationmethodrdquo Computers amp Mathematics with Applications vol 54no 7-8 pp 1188ndash1196 2007
[38] W Lu and Y Liu ldquoVibration control for the primary resonanceof the Duffing oscillator by a time delay state feedbackrdquoInternational Journal of Nonlinear Science vol 8 no 3 pp 324ndash328 2009
[39] H Y Hu and Z H Wang Dynamics of Controlled MechanicalSystems with Delayed Feedback Springer 2002
[40] M Hamdi and M Belhaq ldquoControl of bistability in a delayedDuffing oscillatorrdquo Advances in Acoustics and Vibration vol2012 Article ID 872498 6 pages 2012
[41] V Ravichandran C Chinnathambi and S Rajasekar ldquoNonlin-ear resonance in Duffing oscillator with fixed and integrativetime-delayed feedbacksrdquoPramana Journal of Physics vol 78 pp347ndash360 2013
[42] X You and H Xu ldquoAnalytical approximations for the periodicmotion of theDuffing systemwith delayed feedbackrdquoNumericalAlgorithms vol 56 no 4 pp 561ndash576 2011
[43] Z Sun W Xu X Yang and T Fang ldquoInducing or suppressingchaos in a double-well Duffing oscillator by time delay feed-backrdquo Chaos Solitons and Fractals vol 27 pp 705ndash714 2006
[44] H Wang H Hu and Z Wang ldquoGlobal dynamics of a Duffingoscillator with delayed displacement feedbackrdquo InternationalJournal of Bifurcation and Chaos in Applied Sciences andEngineering vol 14 no 8 pp 2753ndash2775 2004
[45] J Chiasson and J J LoiseauApplications of Time Delay SystemsSpringer 2007
[46] M Lakshmanan andDV SenthilkumarDynamics of NonlinearTime-Delay Systems Springer 2010
[47] G Stepan T Insperger and R Szalai ldquoDelay parametricexcitation and the nonlinear dynamics of cutting processesrdquoInternational Journal of Bifurcation and Chaos in Applied Sci-ences and Engineering vol 15 no 9 pp 2783ndash2798 2005
[48] U van der Heiden and H-O Walther ldquoExistence of chaos incontrol systems with delayed feedbackrdquo Journal of DifferentialEquations vol 47 no 2 pp 273ndash295 1983
[49] Y G Sun and J S W Wong ldquoOscillation criteria for secondorder forced ordinary differential equations with mixed non-linearitiesrdquo Journal of Mathematical Analysis and Applicationsvol 334 no 1 pp 549ndash560 2007
[50] J Heagy and W L Ditto ldquoDynamics of a two-frequencyparametrically driven Duffing oscillatorrdquo Journal of NonlinearScience vol 1 no 4 pp 423ndash455 1991
[51] A B Belogortsev ldquoBifurcations of tori and chaos in thequasiperiodically forced Duffing oscillatorrdquoNonlinearity vol 5no 4 pp 889ndash897 1992
[52] M Belhaq and M Houssni ldquoQuasi-periodic oscillations chaosand suppression of chaos in a nonlinear oscillator driven byparametric and external excitationsrdquo Nonlinear Dynamics vol18 no 1 pp 1ndash24 1999
[53] S H Saker P Y H Pang and R P Agarwal ldquoOscillationtheorems for second order nonlinear functional differential
Discrete Dynamics in Nature and Society 17
equations with dampingrdquo Dynamic Systems and Applicationsvol 12 no 3-4 pp 307ndash321 2003
[54] I N Bronshtein K A Semendyayev G Musiol and HMuehligHandbook of Mathematics Springer 5th edition 2007
[55] M Pasic ldquoFite-Wintner-Leighton-type oscillation criteria forsecond-order differential equations with nonlinear dampingrdquoAbstract and Applied Analysis vol 2013 Article ID 852180 10pages 2013
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
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Mathematical Problems in Engineering
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Differential EquationsInternational Journal of
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Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
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Operations ResearchAdvances in
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Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
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Algebra
Discrete Dynamics in Nature and Society
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
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Decision SciencesAdvances in
Discrete MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
4 Discrete Dynamics in Nature and Society
Theorem 7 Let assumptions (2) (3) (4) (14) and (15) hold120591(119905) = 119905 minus 120591 120591 ge 0 120590(119905) = 119905 + 120590 120590 ge 0 and 119865(119905 119906) and 119866(119905 119906)satisfy
119865 (119905 119906) sgn (119906) ge 119891 (119905) |119906|1199011 119866 (119905 119906) sgn (119906) ge 119892 (119905) |119906|
1199012
forall119906 = 0 119905 ge 1199050
119891 (119905) ge 0 119891 (119905) equiv 0 119892 (119905) ge 0
119892 (119905) equiv 0 119900119899 [120591 (119886119899) 120590 (119887
119899)]
(23)
where additionally 119891 119892 119890 isin 119862([1199050infin)R) are three periodic
functions having a common period 119879lowast
gt 0 such that (17)is fulfilled where 119890(119905) is the forcing term in (1) Then (1) isoscillatory provided one of the next two cases is fulfilled wherethe number 119901 is from (3) (1) (in superlinear delay-advancedcase) 119901
1gt 119901 119901
2gt 119901 120588 gt 0 and either parameter 120588 is
large enough or at least one of 120582 and 120583 is large enough (2) (insupersublinear delay-advanced case) 119901
1gt 119901 gt 119901
2gt 0 120582 gt 0
120583 gt 0 120588 gt 0 and at least one of parameters 120582 120583 and 120588 is largeenough
The proof of Theorem 7 is based on Lemma 4 (seeSection 6) where
119896 (120582 120583 120588) =
min 12058211990111990111205881minus(1199011199011) 12058311990111990121205881minus(1199011199012) superlinear case
120582120578112058312057811205881205780
supersublinear case
(24)
119876119899(119905)
=
1199011
119901(
119901
2 (1199011minus 119901)
)
(1199011119901)minus1
times |119890 (119905)|1minus(119901119901
1)(119891 (119905))
1199011199011
120591119899(119905)
+1199012
119901(
119901
2 (1199012minus 119901)
)
(1199012119901)minus1
|119890 (119905)|1minus(119901119901
2)
times (119892 (119905))1199011199012
120590119899(119905)
superlinear case|119890 (119905)|1205780(119891 (119905))
1205781
(119892 (119905))1205782
(120591119899(119905))1205781(1199011119901)
times (120590119899(119905))1205782(1199012119901)
2
prod
119894=0
120578minus120578119894
119894
supersublinear case
(25)
where we denote
120591119899(119905) = (
119905 minus 119886119899
119905 minus 119886119899+ 120591
)
119901
120590119899(119905) = (
119887119899minus 119905
119887119899minus 119905 + 120590
)
119901
(26)
Here the numbers 1205780 1205781 1205782isin (0 1) are chosen such that 120578
0+
1205781+ 1205782= 1 and 119901
11205781+ 11990121205782= 119901 Let us mention that if
1199011= 52 119901 = 1 and 119901
2= 12 and 120578
0= 1205781= 1205782= 13 then
(1199011 1199012) and (120578
0 1205781 1205782) satisfy previous two equalities About
the existence of such (119873+1)-tuple (1205780 1205781 120578
119873) in a general
case we refer to [49]
Remark 8 A difference between assumptions ofTheorems 56 and 7 is that 119890(119905) in Theorems 5 and 6 is not necessarilyperiodic or bounded function as it is supposed inTheorem 7
Now we study an important class of second-order func-tional differential equations as a particular case of (1)
(119903 (119905) 119860 (1199091015840
(119905)))1015840
+ 119861 (119905 119909 (119905) 1199091015840
(119905))
+ 120582119891 (119905) |119909 (120591 (119905))|1199011 sgn (119909 (120591 (119905)))
+ 120583119892 (119905) |119909 (120590 (119905))|1199012 sgn (119909 (120590 (119905))) = 120588119890 (119905) 119905 ge 119905
0
(27)
where 120591(119905) = 119905 minus 120591 120590(119905) = 119905 + 120590 and 120582 120583 120588 120591 120590 ge 0 and1199011 1199012gt 0 Using previous theorems we are able to state
the main consequences showing the parametrically excitedoscillations in (27)
Corollary 9 (delay equation) Let assumptions (2) (3) (4)(14) and (15) hold Let 119892(119905) equiv 0 119891(119905) ge 0 and 119891(119905) equiv 0 on[119886119899minus120591 119887119899] where119891 isin 119862([119905
0infin)R) is a periodic function with
period 119879lowast satisfying (16) Equation (27) is oscillatory in the
following two cases 1199011= 119901 and parameter 120582 is large enough
1199011gt 119901 120582 gt 0 120588 gt 0 and at least one of parameters 120582 and 120588 is
large enough
Corollary 10 (advanced equation) Let assumptions (2) (3)(4) (14) and (15) hold Let 119891(119905) equiv 0 119892(119905) ge 0 and 119892(119905) equiv 0
on [119886119899 119887119899+ 120590] where 119892 isin 119862([119905
0infin)R) is a periodic function
with period 119879lowast satisfying (17) Then (27) is oscillatory in the
following two cases 1199012= 119901 and parameter 120583 is large enough
1199012gt 119901 120583 gt 0 120588 gt 0 and at least one of parameters 120583 and 120588 is
large enough
Corollary 11 (delay-advanced equation) Let assumptions (2)(3) (4) (14) and (15) hold and 119891(119905) and 119892(119905) satisfy
119891 (119905) ge 0 119891 (119905) equiv 0
119892 (119905) ge 0 119892 (119905) equiv 0
119900119899 [119886119899minus 120591 119887119899+ 120590]
(28)
where additionally 119891 119892 119890 isin 119862([1199050infin)R) are three periodic
functions having a common period 119879lowast
gt 0 such that (17)is fulfilled where 119890(119905) is the forcing term in (1) Then (27) isoscillatory in the next two cases where the number 119901 is from(3) (1) (superlinear case) 119901
1gt 119901 119901
2gt 119901 120588 gt 0 and either
parameter 120588 is large enough or at least one of 120582 and 120583 is largeenough (2) (supersublinear case) 119901
1gt 119901 gt 119901
2gt 0 120582 gt 0
120583 gt 0 120588 gt 0 and at least one of parameters 120582 120583 and 120588 is largeenough
According to previous corollaries we can derive thefollowing examples
Example 12 (delay case) Let 119901 ge 1 120576 ge 0 and 119898 isin N
be fixed and 120591119898
isin R 0 le 120591119898
lt 120587(4119898) With the help of
Discrete Dynamics in Nature and Society 5
Corollary 9 the following two different classes of quasilineardelay differential equations
(100381610038161003816100381610038161199091015840
(119905)10038161003816100381610038161003816
119901minus1
1199091015840
(119905))1015840
+ 120582 sin (119898119905) 1003816100381610038161003816119909 (119905 minus 120591119898)1003816100381610038161003816119901+120576
times sgn (119909 (119905 minus 120591119898)) = minus120588 cos (2119898119905)
(1199091015840(119905)
radic1 + 11990910158402
(119905)
)
1015840
+ 120582 cos (119898119905) 1003816100381610038161003816119909 (119905 minus 120591119898)10038161003816100381610038161+120576
times sgn (119909 (119905 minus 120591119898)) = minus120588 cos (2119898119905)
(29)
are oscillatory provided at least one of 120582 gt 0 and 120588 gt 0 is largeenough (the case 120588 = 0 is possible if 120576 = 0) It is because forall 119899 isin N we have
minus cos (2119898119905)
le 0 on [2119899120587
1198982119899120587
119898+
120587
4119898]
ge 0 on [2119899120587
119898+
120587
41198982119899120587
119898+
120587
2119898]
sin (119898119905) ge 0 cos (119898119905) ge 0 on [2119899120587
1198982119899120587
119898+
120587
2119898]
[2119899120587
119898+ 120591119898+ 119879lowast2119899120587
119898+
120587
4119898+ 119879lowast]
= [(2119899 + 2) 120587
119898+ 120591119898(2119899 + 2) 120587
119898+
120587
4119898]
[2119899120587
119898+
120587
4119898+ 120591119898+ 119879lowast2119899120587
119898+
120587
2119898+ 119879lowast]
= [(2119899 + 2) 120587
119898+
120587
4119898+ 120591119898(2119899 + 2) 120587
119898+
120587
2119898]
(30)
where 119879lowast= 2120587119898 is the common period of the functions
sin(119898119905) and cos(119898119905) Thus in order to apply Corollary 9 wecan choose 119886
2119899minus1= 2119899120587119898 + 120591
119898 1198872119899minus1
= 2119899120587119898 + 120587(4119898)1198862119899= 2119899120587119898 + 120587(4119898) + 120591
119898 and 119887
2119899= 2119899120587119898 + 120587(2119898)
Example 13 (advanced case) Let 119901 ge 1 120576 ge 0 and 119898 isin
N be fixed and 120590119898
isin R 0 le 120590119898
lt 120587(4119898) With thehelp of Corollary 10 the following two classes of quasilinearadvanced differential equations
(100381610038161003816100381610038161199091015840
(119905)10038161003816100381610038161003816
119901minus1
1199091015840
(119905))1015840
+ 120583 sin (119898119905) 1003816100381610038161003816119909 (119905 + 120590119898)1003816100381610038161003816119901+120576
times sgn (119909 (119905 + 120590119898)) = minus120588 cos (2119898119905)
(1199091015840(119905)
radic1 + 11990910158402
(119905)
)
1015840
+ 120583 cos (119898119905) 1003816100381610038161003816119909 (119905 + 120590119898)10038161003816100381610038161+120576
times sgn (119909 (119905 + 120590119898)) = minus120588 cos (2119898119905)
(31)
are oscillatory provided at least one of 120583 gt 0 and 120588 gt 0 islarge enough (the case 120588 = 0 is possible if 120576 = 0) In order toapply Corollary 10 we can choose 119886
2119899minus1= 2119899120587119898 119887
2119899minus1=
2119899120587119898 + 120587(4119898) minus 120590119898 1198862119899
= 2119899120587119898 + 120587(4119898) and 1198872119899
=
2119899120587119898 + 120587(2119898) minus 120590119898
Example 14 (delay-advanced case) Let 119901 ge 1 1205761gt 0 1205762gt 0
and 119898 isin N be fixed and 120591119898ge 0 and 120590
119898ge 0 0 le 120591
119898+ 120590119898lt
120587(4119898) With the help of Corollary 11 the following class ofquasilinear delay-advanced differential equations
(100381610038161003816100381610038161199091015840
(119905)10038161003816100381610038161003816
119901minus1
1199091015840
(119905))1015840
+ 120582 sin (119898119905)
times1003816100381610038161003816119909 (119905 minus 120591119898)
1003816100381610038161003816119901+1205761 sgn (119909 (119905 minus 120591
119898))
+ 120583 cos (119898119905) 1003816100381610038161003816119909 (119905 + 120590119898)1003816100381610038161003816119901+1205762
times sgn (119909 (119905 + 120590119898)) = minus120588 cos (2119898119905)
(32)
is oscillatory provided either 120588 gt 0 is large enough or at leastone of 120582 gt 0 and 120583 gt 0 is large enough In order to applyCorollary 11 we can choose 119886
2119899minus1= 2119899120587119898 + 120591
119898 1198872119899minus1
=
2119899120587119898 + 120587(4119898) minus 120590119898 1198862119899
= 2119899120587119898 + 120587(4119898) + 120591119898 and
1198872119899= 2119899120587119898 + 120587(2119898) minus 120590
119898
3 Application to Duffing Equations withTime Delay Feedback
Let 120582 ge 0 denote the control gain parameter (often calledldquodisplacement feedback coefficientrdquo) 120591 gt 0 the time delayand 120588 ge 0 and 120596 gt 0 the amplitude and frequency of theexternal force respectively Let the function Φ = Φ(119905 119906) thatwill appear in the delay feedback term Φ(119905 119909(119905 minus 120591)) satisfythe general condition
Φ (119905 119906) sgn (119906) ge 1206010|119906|119902
forall119905 ge 1199050
119906 = 0 and some 119902 ge 1 1206010gt 0
(33)
For instance Φ(119905 119906) = 1199062119898minus1 119898 isin N or more general
Φ(119905 119906) = sum119898
119896=11206011198961199062119896minus1 120601
119896gt 0119898 isin N
In this section we consider the following large class ofundamped possible nonautonomous and nonconservativeDuffing equations without or with the general time delayfeedback Φ(119905 119909(119905 minus 120591))
(10038161003816100381610038161003816119909101584010038161003816100381610038161003816
119901minus1
1199091015840)1015840
+ 1205962
0119909 +
1205831|119909|1199031 sgn (119909)
(1205832+ 12058331199092)1199032
+
119898
sum
119894=1
120573119894(119905) |119909|
120572119894minus1119909 + 120582Φ (119905 119909 (119905 minus 120591)) = 120588 cos (120596119905)
(34)
where 1205960is the natural frequency 120583
1ge 0 is the density of the
nonlinear potential (or rigidity coefficient) and 1205832 1205833 1199031 1199032
are nonnegative constants 120573119894(119905) ge 0 and 120572
119894ge 1
When 119901 = 1 120582 = 0 and 120573119894(119905) equiv 120573
119894= const
(34) contains many most important classes of undampedautonomous Duffing oscillators such as the following
(i) the strongly nonlinear Duffing oscillator with smoothodd nonlinearity is given in (34) provided 120583
1= 0 and
120572119894= 2119894 + 1 let us recall some of its known particular
cases
(a) the classic Duffing oscillator 11990910158401015840 +12059620119909+120573119909
3= 0
has been recently studied in the searching of
6 Discrete Dynamics in Nature and Society
solitarywave solutions of classic and generalizedZakharov equations of plasma physics (see [16])and of nonlinear Schrodinger equation (see[17]) also it is strongly connected with theJacobi elliptic equation (see [18])
(b) the cubic-quintic oscillator 11990910158401015840 + 1205962
0119909 + 120573
11199093+
12057321199095
= 0 is used as a model for the non-linear dynamics of a slender elastica (see [19])in nonlinear wave systems (see [20]) for thepropagation of a short electromagnetic pulsein a nonlinear medium (see [21]) and in theunimodal Duffing temporal problem (see [22])
(c) the cubic truly nonlinear oscillator 11990910158401015840 + 1205731199093=
0 models the motion of a ball bearing thatoscillates in a glass tube that is bent into acurve (see [23]) as well as the motion of a massattached to identical stretched elastic wires (see[24])
(d) the nonhomogeneous Duffing oscillator 11990910158401015840 +1205962
0119909 + 120573119909
3= 120588 cos(120596119905) describes various forced
vibrations of beams springs with nonlinearstiffness cables plates shells and optical fibresin electrical circuits in nonlinear isolators andso forth (see for instance [25 26])
(ii) the general Duffing-harmonic oscillator (with rationalor irrational nonlinear restoring-force) is given in(30) if 120583
1= 0 120573119894= 0 and 120588 = 0 the most known
subclasses of these oscillators are
(a) the classic Duffing-harmonic oscillator 11990910158401015840+
(12058311199093(1205832+ 12058331199092)) = 0 which models many
conservative nonlinear oscillatory systems see[27]
(b) the relativistic harmonic oscillator11990910158401015840+ (1205831119909radic1 + 1199092) = 0 see [28]
(c) the nonlinear oscillator11990910158401015840+119909minus(1205831119909radic1 + 1199092) =
0 1205831
isin [0 1] which is typified as a massattached to a stretched elastic wire see [29 30]
(d) the nonlinear oscillator 11990910158401015840
+
(1205831119909(radic(1 + 1199092 )
3
) = 0 which presentsnonlinear oscillations of a punctual charge inthe electric field of charged ring see [31]
Finding several explicit forms of periodic approximate solu-tions for these oscillators has been intensively studied lastyears by many authors see for instance [28 30 32ndash37] andalso the references therein
When 120582 = 0 and linear time delay feedbackΦ(119905 119909(119905minus120591)) =119909(119905 minus 120591) the following topics have been studied for varioustypes of Duffing oscillators with time delayed feedback in[38] authors constructed a low-order approximate solutionunder weak feedback gain parameter about the low- andhigh-order approximations see also [39] in [40] with 120588 = 0the Hopf bifurcation diagrams have been explored for theapproximate periodic solutions (amplitude versus time delay120591 and feedback gain 120582 versus time delay 120591) moreover in [41]
authors made an analysis on the effect of the control gainand time delay parameters on the amplitude of approximateperiod solution from the theoretical and numerical pointsof view see also [42] in [43] authors studied the chaoticbehaviour with respect to gains and time delay parameterssee also [44]
Equations under time delay control such as (34) (espe-cially with damped term) are used as a model for variouscontrolled physical mechanical and engineering systemswith time delays see for instance [39 45ndash48] and thereferences therein
Here (34) contains very general nonlinear time delayfeedback Φ(119905 119909(119905 minus 120591)) with Φ satisfying (33) and the lineartime delay feedback 119909(119905 minus 120591) is only a particular case ofit and to the best of our knowledge the previous topicsare not considered for (34) as yet Moreover with suchan Φ the oscillations of (34) can be taken under a doubteven with the linear time delay feedback (see the nature ofthe approximations given in [38 39]) Hence we can posethe following question under what conditions on equationrsquosparameters (34) is a nonlinear oscillator that is possessesonly oscillatory solutions An answer is given in the nextresult as an easy consequence of the parametrically excitedoscillations by Theorem 5
Theorem 15 Let 120591 isin (0 120587120596) and (33) hold Equation (34) isoscillatory in the next two cases
(i) 119902 = 119901 and 120582 is large enough(ii) 119902 gt 119901 120582 gt 0 120588 gt 0 and at least one of 120582 and 120588 is large
enough
Proof Let 119903(119905) equiv 1 119860(V) = |V|119901minus1 119865(119905 119906) = Φ(119905 119906) 119866(119905 119906) equiv0 119890(119905) = cos(120596119905) and
119861 (119905 119906 V) = 1205962
0119906 +
1205831|119906|1199031s119892119899 (119906)
(1205832+ 12058331199062)1199032
+
119898
sum
119894=1
120573119894(119905) |119906|
120572119894minus1119906 (35)
It is easy to check that all assumptions of Theorem 5 arefulfilled with respect to the sequence 119886
119899= minus1205872120596 + 119899120587120596 + 120591
and 119887119899= 1205872120596 + 119899120587120596 + 120591 where 119886
119899lt 119887119899since it is supposed
that 120591 lt 120587120596 Hence Theorem 5 proves this theorem
Remark 16 Even in the linear forced case (119890(119905) equiv 0) it isnot easy to establish the oscillations of all solutions since theoscillation and nonoscillation can occur simultaneously Themost simple and important example for the coincidence ofoscillation and nonoscillation is the following linear forceddifferential equation 11990910158401015840 + (2119905)119909
1015840+ 119909 = 2119905 119905 gt 0 that
allows an oscillatory solution 1199091(119905) = (3 sin 119905)119905 + 2119905 and a
nonoscillatory solution 1199092(119905) = 2119905 This is not possible in
the linear case with 119890(119905) equiv 0 because of Sturmrsquos separationtheorem
4 Parametrically Excited Oscillations andWell-Known Oscillation Criteria
In this section we would like to draw the readerrsquos attentionto the fact that the parametrically excited oscillations have
Discrete Dynamics in Nature and Society 7
been already appearing in some published papers on theoscillation of functional differential equations but only insome examples illustrating certain main oscillation criteriaHowever with the help of our main results in which theparametrically excited oscillations are studied in a generalsetting the equations from these examples are replaced withgeneral ones also having parameters 120582 and 120583
In [1] (see also [2 Example 31] with 120591 = 0 [3 Example31] and [4 Section 3]) the author considers the oscillationof the second-order delay differential equation
11990910158401015840
(119905) + 119891 (119905) |119909 (120591 (119905))|120574 sgn119909 (120591 (119905)) = 119890 (119905) (36)
in the linear case (120574 = 1) and the superlinear (120574 gt 1)In the linear case (analogously for the superlinear case see[1 Theorem 2]) the author proved the following oscillationcriterion In what follows we denote
119863 (119886 119887) = 119906 isin 1198621
([119886 119887] R) 119906 (119905) equiv 0 119906 (119886) = 119906 (119887) = 0
(37)
Theorem 17 ([1 Theorem 1]) Suppose that for any 119879 ge 0there exist constants 119886
1 1198871 1198862 1198872such that 119879 le 119886
1lt 1198871 119879 le
1198862lt 1198872 and 119891(119905) ge 0 on [120591(119886
1) 1198871] cup [120591(119886
2) 1198872] 119890(119905) le 0
on [120591(1198861) 1198871] and 119890(119905) ge 0 on [120591(119886
2) 1198872] If there exists 119906 isin
119863(119886119894 119887119894) 119894 = 1 2 such that
int
119887119894
119886119894
[1199062
(119905) 119891 (119905)120591 (119905) minus 120591 (119886
119894)
119905 minus 120591 (119886119894)
minus (1199061015840
(119905))2
]119889119905 ge 0 (38)
then (36) with 120574 = 1 is oscillatory
Previous criterion has been applied on the followingparticular equation
11990910158401015840
(119905) + 120582 sin (119905)1003816100381610038161003816100381610038161003816119909 (119905 minus
120587
4)
1003816100381610038161003816100381610038161003816
120574
times sgn 119909(119905 minus120587
4) = cos (119905) 119905 ge 0
(39)
where 120582 ge 0 and 120574 = 1 Applying Theorem 17 to (39) theauthor proved that (39) is oscillatory provided the followinginequality
120582int
119887119894
119886119894
sin2 (2119905) cos2 (2119905) sin (119905)119905 minus 119886119894
119905 minus 119886119894+ 1205874
119889119905 ge120587
2 (40)
holds for sufficiently large 120582 Thus the oscillation of (39) isexcited by the large enough parameter 120582 However accordingto Theorems 5 and 6 we are able to show that the nextparametric equation that corresponds to general equation(36)
11990910158401015840
(119905) + 120582119891 (119905) |119909 (120591 (119905))|120574 sgn119909 (120591 (119905)) = 119890 (119905) (41)
is oscillatory provided 120582 is large enough where 1199011= 1199012= 120574
120583 = 0 and 120588 = 1Next in [5] (see also [6ndash8]) the authors consider the
oscillation of the following class of second-order differentialequations with delay and advanced arguments
(119903 (119905) 1199091015840
(119905))1015840
+ 119891 (119905) |119909 (120591 (119905))|1199011 sgn119909 (120591 (119905))
+ 119892 (119905) |119909 (120590 (119905))|1199012 sgn119909 (120590 (119905)) = 119890 (119905) 119905 ge 0
(42)
where 1199011 1199012ge 1 When 119901
1= 1199012= 1 the authors prove the
following result (for other cases see [5Theorems 32 33 and34]
Theorem 18 ([5 Theorem 31]) Suppose that for any 119879 ge
0 there exist intervals [120591(1198861) 1198871] [120591(119886
2) 1198872] [1198881 120590(1198891)] and
[1198882 120590(1198892)] contained in [119879infin) such that 119886
1lt 1198871 1198862lt 1198872
1198881lt 1198891 1198882lt 1198892 and
119891 (119905) ge 0 119900119899 [120591 (1198861) 1198871] cup [120591 (119886
2) 1198872]
119892 (119905) ge 0 119900119899 [1198881 120590 (1198891)] cup [119888
2 120590 (1198892)]
119890 (119905) le 0 119900119899 [120591 (1198861) 1198871] cup [1198881 120590 (1198891)]
119890 (119905) ge 0 119900119899 [120591 (1198862) 1198872] cup [1198882 120590 (1198892)]
(43)
and 119888119894= 120591(119886
119894) 119889119894= 119886119894 and 119887
119894= 120590(119889
119894) 119894 = 1 2 If there exist
1199061isin 119863(119886
119894 119887119894) and 119906
2isin 119863(119888119894 119889119894) such that either
int
119887119894
119886119894
[1199062
1(119905) 119891 (119905)
120591 (119905) minus 120591 (119886119894)
119905 minus 120591 (119886119894)
minus (1199061015840
1(119905))2
119903 (119905)] 119889119905 ge 0 (44)
or
int
119889119894
119888119894
[1199062
2(119905) 119891 (119905)
120590 (119889119894) minus 120590 (119905)
120590 (119889119894) minus 119905
minus (1199061015840
2(119905))2
119903 (119905)] 119889119905 ge 0 (45)
for 119894 = 1 2 then (42) with 1199011= 1199012= 1 is oscillatory
As a consequence of this result it has been concluded thatthe particular equation
(119903 (119905) 1199091015840
(119905))1015840
+ 120582 sin (119905) 119909 (119905 minus 120587
12)
+ 120583 cos (119905) 119909 (119905 + 120587
6) = cos (2119905) 119905 ge 0
(46)
is oscillatory provided either 120582 or 120583 is large enough Howeverby following Theorems 5 and 6 one can obtain the sameconclusion for the following general equation associated with(42)
(119903 (119905) 1199091015840
(119905))1015840
+ 120582119891 (119905) |119909 (120591 (119905))|1199011 sgn119909 (120591 (119905))
+ 120583119892 (119905) |119909 (120590 (119905))|1199012 sgn119909 (120590 (119905)) = 119890 (119905)
(47)
Related observation can be done with [8 Example 33]and [9 Example 21] where the quasilinear second-orderfunctional differential equations have been considered It isleft to the reader
5 Some Open Questions and Comments
In this section we discuss some problems related to ourmainresults that are not studied here
(1) Quasiperiodic Case In the theory of nonlinear oscillatorsa particularly important case occurs when the periodiccoefficients in the oscillator do not have any common periodIt is called the quasiperiodic (or two-frequency) nonlinear
8 Discrete Dynamics in Nature and Society
oscillator and studied for instance in [50ndash52] Since inTheorems 5 6 and 7 we assume that the correspondingperiodic functions have a commonperiod it is natural to posethe next question
Open Question 1 Is it possible to derive sufficient conditionsfor the oscillation of (27) in the casewhen119891(119905) and119892(119905) (resp119891(119905) 119892(119905) and ℎ(119905)) are two (resp three) periodic functionsnot having a common period
(2) Equation with More Functional Arguments Next regard-ing some second-order functional differential equationsconsidered in the references of this paper more than twononlinear functional terms are appearing and thereforeinstead of main equation (1) and corresponding particularequation (27) considered inTheorems 5 6 and 7 we suggestthe following classes of equations
(119903 (119905) 119860 (1199091015840
(119905)))1015840
+ 119861 (119905 119909 (119905) 1199091015840
(119905))
+
1198981
sum
119896=1
120582119896119865119896(119905 119909 (120591
119896(119905)))
+
1198982
sum
119896=1
120583119896119866119896(119905 119909 (120590
119896(119905))) = 120588119890 (119905)
(48)
where 0 le 120591119896(119905) le 119905 lim
119905rarrinfin120591119896(119905) = infin 120590
119896(119905) ge 119905 119898
1 1198982isin
N and
(119903 (119905) 119860 (1199091015840
(119905)))1015840
+ 119861 (119905 119909 (119905) 1199091015840
(119905))
+
1198981
sum
119896=1
120582119896119891119896(119905)
1003816100381610038161003816119909 (119905 minus 120591119896)1003816100381610038161003816119901119896 sgn (119909 (119905 minus 120591
119896))
+
1198982
sum
119896=1
120583119896119892119896(119905)
1003816100381610038161003816119909 (119905 + 120590119896)1003816100381610038161003816119902119896 sgn (119909 (119905 + 120590
119896)) = 120588119890 (119905)
(49)
where 120582119896 120583119896 120588 120591119896 120590119896ge 0 and 119901
119896 119902119896gt 0
Comment We suggest the reader to enlarge the main resultsof this paper to (48) and (49)
(3) Damped Duffing Equation In the application the Duffingequation (34) is often appearing with the linear damped term1199091015840(119905) that is
11990910158401015840+ 11988901199091015840+ 1205962
0119909 + 120573119909
3+ 120582Φ (119909 (119905 minus 120591)) = 120588 cos (120596119905) (50)
where 1198890
is the damped coefficient which can in anactive way influence various behaviours of (50) Since119861(119905 119909(119905) 119909
1015840(119905)) = 119889
01199091015840(119905) does not satisfy the required
assumption (4) we are not able to apply our main results to(50) Hence we pose the following questionOpen Question 2 Is it possible to obtain the parametricallyexcited oscillation for (1) in the case when the damped term119861(119905 119906 V) satisfies a larger condition than (4) in which thelinear damped term 120573119909
1015840(119905) is especially included
(4) Functional Argument in Damped Term In a class of Duff-ing equations we have two time delayed feedback and hence
besides the control gain parameter 1205821another parameter 120582
2
appears the so-called velocity gain parameter Hence insteadof (34) one can consider
11990910158401015840+ 11988901199091015840+ 1205962
0119909 + 120573119909
3+ 1205821119909 (119905 minus 120591)
+ 12058221199091015840
(119905 minus 120591) = 120588 cos (120596119905) (51)
Therefore we suggest the following problem for further studyOpen Question 3 Is it possible to obtain the parametricallyexcited oscillation for the following more general functionaldifferential equation than (1) in which the functional argu-ment appears in the damped term too as follows
(119903 (119905) 119860 (1199091015840
(119905)))1015840
+ 119861 (119905 119909 (119905) 1199091015840
(119905)) + 1205821119865 (119905 119909 (120591 (119905)))
+ 1205822119867(119905 119909
1015840
(120591 (119905))) = 120588119890 (119905) 119905 ge 1199050
(52)or
(119903 (119905) 119860 (1199091015840
(119905)))1015840
+ 119861 (119905 119909 (119905) 1199091015840
(119905)) + 1205831119866 (119905 119909 (120590 (119905)))
+ 1205832119867(119905 119909
1015840
(120590 (119905))) = 120588119890 (119905) 119905 ge 1199050
(53)
About known oscillation criteria for the second-order func-tional differential equations having the functional argumentin the damped term we refer the reader to for instance [53]and the references therein
6 Proofs of Main Results
The proof of Lemma 1 is based on the following three stepstwo working forms of condition (6) (see Lemmas 19 and 20)the existence of an explosive solution of a suitable Riccatidifferential inequality (see Proposition 22) and a comparisonprinciple (see Proposition 24)
Lemma 19 (a necessary condition to (6)) Let 0 lt 119903(119905) le 1199030
on [1199050infin) If assumption (6) is fulfilled then there is a positive
real number 120576 such that1
120587lowast
int119869
min119901
(120576119903 (119905))120574minus1
120576119896 (120582 120583 120588)119876 (119905) 119889119905 ge 1 (54)
for all 120582 ge 1205820 120583 ge 120583
0 and 120588 ge 120588
0and some (120582
0 1205830 1205880) isin R3+
Proof Since 0 lt 119903(119905) le 1199030for 119905 ge 119905
0 we conclude that for
120576 = (119901
119903120574minus1
0119896 (120582 120583 120588)max
119905isin 119869119876 (119905)
)
1120574
(120582 120583 120588) isin R3
+
(55)
it holds that 119901(120576119903(119905))120574minus1
ge 119901(1205761199030)120574minus1
= 120576119896(120582 120583 120588)
max119905isin 119869119876(119905) ge 120576119896(120582 120583 120588)119876(119905) 119905 isin 119869 and hence
int119869
min119901
(120576119903 (119905))120574minus1
120576119896 (120582 120583 120588)119876 (119905) 119889119905
= 120576119896 (120582 120583 120588) int119869
119876 (119905) 119889119905
(56)
Discrete Dynamics in Nature and Society 9
On the other hand from (6) we observe
1
120587lowast
int119869
119876 (119905) 119889119905 ge1199031minus(1120574)
0
1199011120574[119896 (120582 120583 120588)]1minus(1120574)
(max119905isin 119869
119876 (119905))
1120574
(57)
which together with (55) and (56) gives
1
120587lowast
int119869
min119901
(120576119903 (119905))120574minus1
120576119896 (120582 120583 120588)119876 (119905) 119889119905
= 120576119896 (120582 120583 120588)1
120587lowast
int119869
119876 (119905) 119889119905
ge 1205761199031minus(1120574)
0
1199011120574[119896 (120582 120583 120588)]
1120574
(max119905isin 119869
119876 (119905))
1120574
= 1
(58)
for all 119899 ge 1198990 120582 ge 120582
0 120583 ge 120583
0 and 120588 ge 120588
0 It proves this
lemma
Lemma 20 (an equivalent condition to (54)) Assumption(54) is fulfilled if and only if there is a real number 120576 gt 0 and acontinuous function 119870(119905) ge 0 119905 isin 119869 such that
1198880= int119869
119870 (119905) 119889119905 gt 0119870 (119905)
1198880
le1
120587lowast
timesmin119901
(120576119903 (119905))120574minus1
120576119896 (120582 120583 120588)119876 (119905)
(59)
for all 119905 isin 119869 120582 ge 1205820 120583 ge 120583
0 and 120588 ge 120588
0and some (120582
0 1205830 1205880) isin
R3+
Proof This proof is very elementary Indeed if (54) holdsthen the function119870(119905) and number 119888
0 defined by
119870 (119905) =1
120587lowast
min119901
(120576119903 (119905))120574minus1
120576119896 (120582 120583 120588)119876 (119905)
1198880= int119869
119870 (119905) 119889119905
(60)
obviously satisfy 1198880
ge 1 and 119870(119905)1198880
le 119870(119905) = (1120587lowast)
min119901(120576119903(119905))120574minus1 120576119896(120582 120583 120588)119876(119905) which shows (59) Con-versely if (59) holds then integrating both sides of thesecond inequality in (59) we obtain
int119869
1
120587lowast
min119901
(120576119903 (119905))120574minus1
120576119896 (120582 120583 120588)119876 (119905) ge int119869
119870 (119905)
1198880
119889119905 = 1
(61)
which shows (54)
In conclusion according to previous two lemmas wesee that supposed condition (6) implies (59) which plays animportant role in the proof of the main results
The second step in the proof of Lemma 1 is to prove theexistence of a function 120596(119905) which blows up in the finitetime and satisfies a generalized Riccati differential lowerinequality we briefly present the existence and properties
of the so-called generalized tangent type function In whatfollows let 120587
lowastbe a positive real number defined in (3) Let us
remark that 120572(119904) = 119904120574 120574 gt 1 implies 120587
lowast= (2120587)(120574 sin(120587120574))
see for instance [54] and obviously for 120574 = 2wehave120587lowast= 120587
Lemma 21 Let 120572 [0infin) rarr [0infin) be a continuousfunction such that
int
infin
0
119889120591
1 + 120572 (120591)lt infin (62)
Then there is a real number 120587lowastgt 0 and a function 119911 = 119911(119904)
119911 isin 1198621((minus120587lowast2 120587lowast2)R) such that
119889119911
119889119904= 1 + 120572 (|119911 (119904)|) 119904 isin (minus
120587lowast
2120587lowast
2)
119911 (0) = 0
(63)
Moreover 119911(119904) is increasing and odd
lim119904rarr120587lowast2
119911 (119904) = infin 120587lowast=
2120587
120574 sin (120587120574)for 120572 (119904) = 119904
120574
120574 gt 1
(64)
In particular for 120572(119904) = 1199042 one can take 119911(119904) = tan(119904) and
120587lowast= 120587
Proof Let 119885 = 119885(119905) 119905 isin R be a function defined by
119885 (119905) = int
119905
0
1
1 + 120572 (|120591|)119889120591 119905 isin R (65)
The function 119885(119905) is well defined since 120572(119904) is positive andcontinuous on [0infin) 119885(119905) is increasing and odd functionand
119889119885
119889119905=
1
1 + 120572 (|119905|) 119905 isin R
119885 (0) = 0 119885 isin 1198621
(RR)
(66)
Moreover because of (62) there is a real number 120587lowastgt 0 such
that120587lowast
2= int
infin
0
119889120591
1 + 120572 (120591) (67)
Thus 119885 R rarr (minus120587lowast2 120587lowast2) and there exists an inverse
function 119885minus1 = 119885minus1(119904) of the original function 119885 = 119885(119905) and
119885minus1
(minus120587lowast2 120587lowast2) rarr R Also from 119885(119885
minus1(119904)) = 119904 and
119889119885119889119905 = 0 onR we also derive that119889119885minus1119889119904 = 0 on its domain(minus120587lowast2 120587lowast2) and
119889119885
119889119905(119885minus1
(119904)) =1
(119889119885minus1119889119904) 119904 isin (minus
120587lowast
2120587lowast
2) (68)
Putting 119905 = 119885minus1(119904) for 119904 isin (minus120587
lowast2 120587lowast2) into (66) and using
(68) we easily obtain
119889119885minus1
119889119904= 1 + 120572 (
10038161003816100381610038161003816119885minus1
(119904)10038161003816100381610038161003816) 119904 isin (minus
120587lowast
2120587lowast
2)
119885minus1
(0) = 0 119885minus1isin 1198621((minus
120587lowast
2120587lowast
2) R)
(69)
10 Discrete Dynamics in Nature and Society
Moreover from (67) we have lim119904rarr120587lowast2119885minus1(119904) = 119885
minus1
(lim119905rarrinfin
119885(119905)) = lim119905rarrinfin
119885minus1119885(119905) = lim
119905rarrinfin119905 = infin Thus
if we set 119911(119904) = 119885minus1(119904) then previous two statements and
(67) prove this lemma
Next we prove the main result of this section
Proposition 22 Let (2) and (6) hold where 119869 = (119886 119887) Let 120576 gt0 be a real number and let119870(119905) ge 0 119905 isin [119886 119887] be a continuousfunction both obtained in Lemma 20 Let 120587
lowastbe from (3) and
1198880from (59) and let 119877
119886isin R be an arbitrary real number If
119911 = 119911(119904) is the generalized tangens function defined in (63)and 119881(119905) is a function defined by
119881 (119905) =120587lowast
1198880
int
119905
119886
119870 (120591) 119889120591 + 119911minus1(119877119886) 119905 isin [119886 119887] (70)
then there is a 119879lowast119886isin [119886 119887) such that
119881 (119879lowast
119886) =
120587lowast
2 119881 ([119886 119879
lowast
119886)) sub (minus
120587lowast
2120587lowast
2) (71)
Moreover for a function 120596(119905) defined by120596 (119905) = 119911 (119881 (119905)) 119905 isin [119886 119879
lowast
119886) (72)
one has 120596(119886) = 119877119886 lim119905rarr119879
lowast
119886
120596(119905) = infin and
119889120596
119889119905le
119901
(120576119903 (119905))120574minus1
120572 (1003816100381610038161003816120596 (119905)
1003816100381610038161003816)
+ 120576119896 (120582 120583 120588)119876 (119905) 119905 isin (119886 119879lowast
119886)
(73)
where the numbers 119901 and 120574 are from (3) and the functions119896(120582 120583 120588) and 119876(119905) are from (6)
Proof Under assumptions (2) and (6) and because of Lem-mas 19 and 20 we obtain 120576 gt 0 and 119870(119905) gt 0 119905 isin [119886 119887]satisfying inequality (59)
Next since 119911minus1(119877119886) isin (minus120587
lowast2 120587lowast2) (see Lemma 21)
from (70) we directly obtain
119881 (119886) = 119911minus1(119877119886) lt
120587lowast
2 119881 (119887) = 120587
lowast+ 119911minus1(119877119886) gt
120587lowast
2
(74)Since 119870 isin 119862([119886 119887] [0infin)) we obtain 119881 isin 119862([119886 119887]R) cap
1198621((119886 119887)R) and from (74) we observe that there exist
numbers 119879lowast119886isin (119886 119887) such that119881(119879lowast
119886) = 120587lowast2 Also119870(119905)119888
0ge
0 gives 119881([119886 119879lowast119886)) sub (minus120587
lowast2 120587lowast2) which proves statement
(71) Moreover it together with Lemma 21 and (72) provesthat
lim119905rarr119879
lowast
119886
120596 (119905) = lim119905rarr119879
lowast
119886
119911 (119881 (119905)) = 119911 (120587lowast
2) = infin (75)
Next according to (59) (63) and (72) we make thefollowing calculation on the interval [119886 119879lowast
119886)
1205961015840
(119905) = 1199111015840
(119881 (119905)) 1198811015840
(119905) = [1 + 120572 (|119911 (119881 (119905))|)]120587lowast
1198880
119870 (119905)
= [1 + 120572 (1003816100381610038161003816120596 (119905)
1003816100381610038161003816)]120587lowast
1198880
119870 (119905)
le119901
(120576119903 (119905))120574minus1
120572 (1003816100381610038161003816120596 (119905)
1003816100381610038161003816) + 120576119896 (120582 120583 120588)119876 (119905)
(76)
Thus all assertions of this proposition are proved
Remark 23 In the proof of the main result the number 119877119886
is determined by 119877119886= 120596(119886) where 120596(119905) denotes a function
associated with a nonoscillatory solution and it is given by(84) below
The third step in the proof of Lemma 1 is to show thefollowing pointwise comparison principle for the functions120596and120596 satisfying respectively the lower and upper differentialinequalities (73) and
119889120596
119889119905ge
119901
(120576119903 (119905))120574minus1
120572 (|120596 (119905)|) + 120576119896 (120582 120583 120588)119876 (119905) 119905 isin (119886 119887)
(77)
Proposition 24 Let [119886 119887) sub [1199050infin) be an arbitrary inter-
val One supposes that all coefficients of Riccati differentialinequalities (73) and (77) are continuous and strictly positivefunctions Let 120596 120596 isin 119862
1([119886 119887)R) be two functions satisfying
respectively (73) and (77) on the interval [119886 119887) Then
120596 (119886) le 120596 (119886) 119894119898119901119897119894119890119904 120596 (119905) le 120596 (119905) forall119905 isin [119886 119887) (78)
Proof Let119867(119905 119906) be a function defined by
119867(119905 119906) =119901
(120576119903 (119905))120574minus1
120572 (|119906|) + 120576119896 (120582 120583 120588)119876 (119905)
119905 isin [119886 119887) 119906 isin R
(79)
Let 119868 sub [119886 119887) and 119872 gt 0 be arbitrary For any two 1199061
1199062 minus119872 le 119906
1lt 1199062le 119872 let 119868
12be an interval defined
by 11986812
= (min|1199061| |1199062|max|119906
1| |1199062|) Since 120572(119904) is a 1198621-
function on [0infin) we know by the Lagrange mean valuetheorem applied on 119868
12that there is a 120585 isin 119868
12such that
120572 (10038161003816100381610038161199062
1003816100381610038161003816) minus 120572 (10038161003816100381610038161199061
1003816100381610038161003816)
1199062minus 1199061
le
1003816100381610038161003816120572 (10038161003816100381610038161199062
1003816100381610038161003816) minus 120572 (10038161003816100381610038161199061
1003816100381610038161003816)1003816100381610038161003816
1199062minus 1199061
=100381610038161003816100381610038161205721015840
(120585)10038161003816100381610038161003816
100381610038161003816100381610038161003816100381610038161199062
1003816100381610038161003816 minus10038161003816100381610038161199061
10038161003816100381610038161003816100381610038161003816
1199062minus 1199061
le100381610038161003816100381610038161205721015840
(120585)10038161003816100381610038161003816
le max119904isin11986812
100381610038161003816100381610038161205721015840
(119904)10038161003816100381610038161003816
(80)
since ||1199062| minus |1199061|| le 119906
2minus 1199061 Hence for any 119905 isin 119868 and 119906
1 1199062
minus119872 le 1199061lt 1199062le 119872 we have
119867(119905 1199062) minus 119867 (119905 119906
1)
1199062minus 1199061
= 1205880(119905)
120572 (10038161003816100381610038161199062
1003816100381610038161003816) minus 120572 (10038161003816100381610038161199061
1003816100381610038161003816)
1199062minus 1199061
le 1205880(119905)max119904isin11986812
100381610038161003816100381610038161205721015840
(119904)10038161003816100381610038161003816= 1198710(119905)
(81)
Thus the function119867(119905 119906) from (79) satisfies required condi-tion of [55 Lemma 19] and applying it to (73) and (77) weprove this proposition
Proof of Lemma 1 On the contrary let 119909(119905) be a solution of(1) such that
119909 (119905) = 0 on (120591 (120591 (119886)) 120590 (120590 (119889))) (82)
Discrete Dynamics in Nature and Society 11
that is 119909(119905) gt 0 on (120591(120591(119886)) 120590(120590(119889))) or 119909(119905) lt 0 on(120591(120591(119886)) 120590(120590(119889))) since 119909(119905) is a continuous function on[1199050infin) Let for instance
119909 (119905) gt 0 on (120591 (120591 (119886)) 120590 (120590 (119889))) (83)
Another case can be analogously treated let us see thecomment at the end of this proof In particular from (83)we have 119909(119905) gt 0 on (120591(120591(119886)) 120590(120590(119887))) which implies (since120591(119905) and 120590(119905) are increasing functions) 119909(119904) gt 0 for all 119904 isin
(120591(119886) 120590(119887)) cup (120591(120591(119886)) 120591(120590(119887))) cup (120590(120591(119886)) 120590(120590(119887))) whichyields 119909(119905) gt 0 119909(120591(119905)) gt 0 and 119909(120590(119905)) gt 0 on (120591(119886) 120590(119887))Hence by assumption (7) we may use inequality (5) on theinterval (119886 119887)
Firstly we show that the following classic Riccati transfor-mation of 119909(119905)
120596 (119905) = minus120576119903 (119905) 119860 (119909
1015840(119905))
|119909 (119905)|119901minus1
119909 (119905) 119905 isin (119886 119887) 120576 gt 0 (84)
satisfies upper Riccati differential inequality (77) Let usremark that from (1) we have in particular
minus(119903 (119905) 119860 (1199091015840
(119905)))1015840
= 119861 (119905 119909 (119905) 1199091015840
(119905)) + 120582119865 (119905 119909 (120591 (119905)))
+ 120583119866 (119905 119909 (120590 (119905))) minus 120588119890 (119905) 119905 ge 1199050
(85)
Taking the first derivative on both sides of (84) and usingassumptions (3) (4) and (5) as well as equality (85) and(|119909(119905)|
119901minus1119909(119905))1015840
= 119901|119909(119905)|119901minus1
1199091015840(119905) we obtain
119889120596
119889119905= 120576119901 119903 (119905)
119860 (1199091015840(119905)) 1199091015840(119905)
|119909 (119905)|119901+1
minus 1205761
|119909 (119905)|119901minus1
119909 (119905)(119903 (119905) 119860 (119909
1015840
(119905)))1015840
= 120576119901119903 (119905)119860 (1199091015840(119905)) 1199091015840(119905)
|119909 (119905)|119901+1
+120576
|119909 (119905)|119901minus1
119909 (119905)
times [120582119861 (119905 119909 (119905) 1199091015840
(119905)) + 119865 (119905 119909 (120591 (119905)))
+120583119866 (119905 119909 (120590 (119905))) minus 120588119890 (119905) ]
ge 120576119901119903 (119905)119860 (1199091015840(119905)) 1199091015840(119905)
|119909 (119905)|119901+1
+120576
|119909 (119905)|119901minus1
119909 (119905)
times [120582119865 (119905 119909 (120591 (119905))) + 120583119866 (119905 119909 (120590 (119905))) minus 120588119890 (119905)]
ge 120576119901119903 (119905) 120572(
10038161003816100381610038161003816119860 (1199091015840(119905))
10038161003816100381610038161003816
|119909 (119905)|119901
) + 120576119896 (120582 120583 120588)119876 (119905)
= 120576119901119903 (119905) 120572 (|120596 (119905)|
120576119903 (119905)) + 120576119896 (120582 120583 120588)119876 (119905)
ge119901
(120576119903 (119905))120574minus1
120572 (|120596 (119905)|) + 120576119896 (120582 120583 120588)119876 (119905) 119905 isin (119886 119887)
(86)
Thus according to inequality (5) it is shown that if 119909(119905) isa solution of (1) which satisfies (83) then the function 120596(119905)
defined by (84) satisfies the Riccati differential inequality(77) and 120596 isin 119862((119886 119887)R) On the other hand let 119877
119886be a
real number defined by 119877119886= 120596(119886) According to (6) and
Lemma 19 we obtain (54) which together with Lemma 20ensures that we may use Proposition 22 for such chosen realnumber 119877
119886 Hence we obtain a function 120596(119905) defined by (72)
which satisfies the lower Riccati differential inequality (73) on[119886 119879lowast
119886) 119879lowast119886isin (119886 119887) such that 120596(a) = 119877
119886and lim
119905rarr119879lowast
119886
120596(119905) =
infin Therefore by 120596(119886) = 119877119886= 120596(119886) and Proposition 24 we
conclude that lim119905rarr119879
lowast
119886
120596(119905) = infin too which is a contradictionwith the above conclusion saying that 120596 isin 119862((119886 119887)R) Thushypothesis (82) is not true and consequently Lemma 1 isshown
For the analogous case 119909(119905) lt 0 on (120591(120591(119886)) 120590(120590(119889))) wealso have 119909(119905) lt 0 on (120591(120591(119888)) 120590(120590(119889))) which implies (since120591(119905) and 120590(119905) are increasing functions)
119909 (119904) lt 0 forall119904 isin (120591 (119888) 120590 (119889)) cup (120591 (120591 (119888)) 120591 (120590 (119889)))
cup (120590 (120591 (119888)) 120590 (120590 (119889)))
(87)
which yields 119909(119905) lt 0 119909(120591(119905)) lt 0 and 119909(120590(119905)) lt 0 on(120591(119888) 120590(119889)) Now we can repeat the preceding procedure buton interval (119888 119889) and using (8) instead of (119886 119887) and (7)
Proof of Lemma 2 From assumption (10) we obtain the exis-tence of an 119899
0isin N such that
int
119887119899
119886119899
119876119899(119905) 119889119905 ge
1198880
2( max119905isin[119886119899 119887119899]
119876119899(119905))
1120574
119899 ge 1198990 (88)
that is
2
1198880
int
119887119899
119886119899
119876119899(119905) 119889119905 ge ( max
119905isin[119886119899 119887119899]119876119899(119905))
1120574
119899 ge 1198990 (89)
Now from (9) and previous inequality we deduce that forlarge enough 120582 120583 120588 and 119899
1199011120574
1199031minus1120574
0
[119896 (120582 120583 120588)]1minus1120574
120587lowast
int
119887119899
119886119899
119876119899(119905) 119889119905
ge2
1198880
int
119887119899
119886119899
119876119899(119905) 119889119905 ge ( max
119905isin[119886119899 119887119899]119876119899(119905))
1120574
(90)
which shows (6) Thus all assumptions of Lemma 1 arefulfilled and hence Lemma 2 immediately follows fromLemma 1
Proof of Lemma 3 Obviously assumption (11) is a particularcase of assumption (9) Hence this proof is very similar tothe proof of Lemma 2 and so it is left to the reader
Proof of Lemma 4 It is clear that from assumption (13) weobtain
1
(max119905isin[119886119899119887119899]119876119899(119905))1120574
int
119887119899
119886119899
119876119899(119905) 119889119905 ge
1198881
1198621120574
0
gt 0 forall119899 ge 1198990
(91)
12 Discrete Dynamics in Nature and Society
Thus hypothesis (12) is fulfilled and therefore Lemma 3proves this lemma
Proof of Theorems 5 6 and 7 This proof is based onLemma 4 In order to simplify notation in many placesin this proof we set 120591(119905) = 119905 minus 120591 and 120590(119905) = 119905 + 120590 Sinceassumptions (2) (3) and (4) have been already supposed inTheorems 5 6 and 7 in order to prove these theorems byLemma 4 we are going to show that the functions 119896(120582 120583 120588)and 119876
119899(119905) explicitly given respectively in (18) (21) or (24)
and (19) (22) or (25) satisfy required conditions (11) and(13) respectively and that every solution 119909(119905) of (27) satisfiesconditions (7) and (8) with respect to functions 119896(120582 120583 120588)and 119876
119899(119905) where 119886 = 119886
2119899minus1 119887 = 119887
2119899minus1 119888 = 119886
2119899 and 119889 = 119887
2119899
The proof that the function 119896(120582 120583 120588) given in (18) (21) or(24) satisfies (11) Passing to the limit in (18) (21) or (24) it isvery simple to show (11)
The proof that the function 119876119899(119905) given in (19) (22) or
(25) satisfies the first claim in (13) From (25) we immediatelyobtain
1003816100381610038161003816120591119899 (119905)1003816100381610038161003816 =
100381610038161003816100381610038161003816100381610038161003816
(119905 minus 119886119899
119905 minus 119886119899+ 120591
)
119901100381610038161003816100381610038161003816100381610038161003816
le 1
1003816100381610038161003816120590119899 (119905)1003816100381610038161003816 =
100381610038161003816100381610038161003816100381610038161003816
(119887119899minus 119905
119887119899minus 119905 + 120590
)
119901100381610038161003816100381610038161003816100381610038161003816
le 1 forall119899 isin N
(92)
Next by assumptions of this corollary we can conclude thatthere are three positive constants 119891
0 1198920 1198900such that |119891(119905)| le
1198910and |119892(119905)| le 119892
0on [1199050infin) in cases (i) and (ii) and
|119890(119905)| le 1198900on [1199050infin) in cases (iii) and (iv) Putting previous
inequalities into (19) (22) or (25) for all 119899 isin N and 119905 isin
[1199050infin) it holds that
1003816100381610038161003816119876119899 (119905)1003816100381610038161003816 le
119902
119901(
119901
119902 minus 119901)
(119902119901)minus1
1198901minus(119901119902)
0119891119901119902
0
delay case with 119902 gt 119901
119902
119901(
119901
119902 minus 119901)
(119902119901)minus1
1198901minus(119901119902)
0119892119901119902
0
advanced case with 119902 gt 119901
1199011
119901(
119901
2 (1199011minus 119901)
)
(1199011119901)minus1
1198901minus(119901119901
1)
0119891119901119902
0+1199012
119901
times(119901
2 (1199012minus 119901)
)
(1199012119901)minus1
1198901minus(119901119901
2)
0119892119901119902
0
delay-advanced case (i)
1198901205780
01198911205781
01198921205782
0
2
prod
119894=0
120578minus120578119894
119894
delay-advanced case (ii) (93)
which shows the first claim in (13)
The proof that the function119876119899(119905) given in (19) (22) or (25)
satisfies the second claim in (13)Without loss of generality weprove this claim only in case (i) since for other cases the prooffollows analogously In this sense let119876
119899(119905) = 119891(119905)120591
119899(119905) Since
1198862119899+1
minus 1198862119899minus1
le 119879lowast 1198872119899+1
minus 1198872119899minus1
ge 119879lowast 1198862119899+2
minus 1198862119899le 119879lowast and
1198872119899+2
minus 1198872119899
ge 119879lowast where 119879
lowastgt 0 is the period of the function
119891(119905) we have 1198862119899minus1
le 1198861+(119899minus1)119879
lowastand 1198872119899minus1
ge 1198871+(119899minus1)119879
lowast
119899 isin N Hence
int
1198872119899minus1
1198862119899minus1
119876119899(119905) 119889119905
= int
1198872119899minus1
1198862119899minus1
119891 (119905) (119905 minus 1198862119899minus1
119905 minus 1198862119899minus1
+ 120591)
119901
119889119905
ge int
1198871+(119899minus1)119879
lowast
1198861+(119899minus1)119879lowast
119891 (119905) (119905 minus 1198861minus (119899 minus 1) 119879
lowast
119905 minus 1198861minus (119899 minus 1) 119879
lowast+ 120591
)
119901
119889119905
= int
1198871
1198861
119891 (119904 + (119899 minus 1) 119879lowast) (
119904 minus 1198861
119904 minus 1198861+ 120591
)
119901
119889119904
= int
1198871
1198861
119891 (119904) (119904 minus 1198861
119904 minus 1198861+ 120591
)
119901
119889119904
(94)
which proves that the integral on the left hand side does notdepend on 119899 isin N that is the second claim in (13) is shown on[1198862119899minus1
1198872119899minus1
] This claim follows in the same way on [1198862119899 1198872119899]
Thus the second claim in (13) is proved on [119886119899 119887119899]
Next to the end of this proof let 119909(119905) be a solu-tion of (1) In particular it implies that (119903(119905)119860(1199091015840(119905)))1015840 =
minus119861(119905 119909(119905) 1199091015840(119905)) minus 120582119865(119905 119909(120591(119905))) minus 120583119866(119905 119909(120590(119905))) + 120588119890(119905) It
together with assumptions (15) (16) (20) and (23) easilygives the next two statements
if 119909 (119905) gt 0 119909 (120591 (119905)) gt 0 119909 (120590 (119905)) gt 0
on (120591 (1198862119899minus1
) 120590 (1198872119899minus1
))
then 119909 (119905) satisfies 119903 (119905) 119860 (1199091015840
(119905)) le 0
on (120591 (1198862119899minus1
) 120590 (1198872119899minus1
)) 119899 ge 1198990
(95)
if 119909 (119905) lt 0 119909 (120591 (119905)) lt 0 119909 (120590 (119905)) lt 0
on (120591 (1198862119899) 120590 (119887
2119899))
then 119909 (119905) satisfies 119903 (119905) 119860 (1199091015840
(119905)) ge 0
on (120591 (1198862119899) 120590 (119887
2119899)) 119899 ge 119899
0
(96)
Now we need the following lemma
Discrete Dynamics in Nature and Society 13
Lemma 25 Let 120591119886119887(119905) and 120590
119886119887(119905) be defined by
120591119886119887(119905) = (
120591 (119905) minus 120591 (119886)
119905 minus 120591 (119886))
119901
120590119886119887(119905) = (
120590 (119887) minus 120590 (119905)
120590 (119887) minus 119905)
119901
119905 isin (119886 119887)
(97)
and let 119909 isin 1198622([1198790infin)R) be an arbitrary function If
(119903(119905)119860(1199091015840(119905)) le 0 for all 119905 isin (120591(119886) 120590(119887)) or (119903(119905)119860(1199091015840(119905)) ge 0
for all 119905 isin (120591(119886) 120590(119887)) then
119909 (120591 (119905))
119909 (119905)ge (120591119886119887(119905))1119901
119909 (120590 (119905))
119909 (119905)ge (120590119886119887(119905))1119901
119905 isin (119886 119887)
(98)
Since119860(V) is supposed to be odd and increasing functionjust before (3) and 119903(119905) satisfies (14) the proof of Lemma 25in the first case that is 119903(119905)119860(1199091015840(119905)) le 0 for all 119905 isin (120591(119886) 120590(119887))is the same as the proof of [9 Corollaries 17 and 18] But in thesecond case that is 119903(119905)119860(1199091015840(119905)) ge 0 for all 119905 isin (120591(119886) 120590(119887))the proof is as follows if previous inequality holds then119903(119905)119860(minus119909
1015840(119905)) le 0 for all 119905 isin (120591(119886) 120590(119887)) and therefore to
the function minus119909(119905) one can apply the first case of this lemmaand consequently one obtains
119909 (120591 (119905))
119909 (119905)=minus119909 (120591 (119905))
minus119909 (119905)ge (120591119886119887(119905))1119901
119909 (120590 (119905))
119909 (119905)=minus119909 (120590 (119905))
minus119909 (119905)ge (120590119886119887(119905))1119901
119905 isin (119886 119887)
(99)
which proves this lemma in the second caseNow combining statements (95) (96) and (98) one
easily obtains
if 119909 (119905) gt 0 119909 (120591 (119905)) gt 0 119909 (120590 (119905)) gt 0
on (120591 (1198862119899minus1
) 120590 (1198872119899minus1
))
then 119909 (119905) satisfies 119909 (120591 (119905))
119909 (119905)ge (120591119899(119905))1119901
on (1198862119899minus1
1198872119899minus1
) 119899 ge 1198990
(100)
if 119909 (119905) lt 0 119909 (120591 (119905)) lt 0 119909 (120590 (119905)) lt 0
on (120591 (1198862119899) 120590 (119887
2119899))
then 119909 (119905) satisfies 119909 (120590 (119905))
119909 (119905)ge (120590119899(119905))1119901
on (1198862119899 1198872119899) 119899 ge 119899
0
(101)
where 120591119899(119905) and 120590
119899(119905) are defined in (26)
The proof that 119909(119905) satisfies (7) and (8) In this proofwe frequently use assumptions (16) (20) and (23) andstatements (100) and (101) Also because of (15) and 119865(119905 119906) =
119891(119905)|119906|1199011 sgn(119906) 119866(119905 119906) = 119892(119905)|119906|
1199012 sgn(119906) in both cases
(100) and (101) we can simultaneously use
minus119890 (119905) (|119909 (119905)|119901minus1
119909 (119905))minus1
= |119890 (119905)| |119909 (119905)|minus119901
ge 0 on 119869119899
119865 (119905 119909 (120591 (119905))) (|119909 (119905)|119901minus1
119909 (119905))minus1
ge 119891 (119905) |119909 (120591 (119905))|1199011 |119909 (119905)|
minus119901ge 0 on 119869
119899
119866 (119905 119909 (120590 (119905))) (|119909 (119905)|119901minus1
119909 (119905))minus1
ge 119892 (119905) |119909 (120590 (119905))|1199012 |119909 (119905)|
minus119901ge 0 on 119869
119899
|119909 (120591 (119905))| |119909 (119905)|minus1=119909 (120591 (119905))
119909 (119905)
|119909 (120590 (119905))| |119909 (119905)|minus1=119909 (120590 (119905))
119909 (119905)on 119869119899
(102)
where 119869119899= (1198862119899minus1
1198872119899minus1
) in the case of (100) and 119869119899= (1198862119899 1198872119899)
in the case of (101)
(i) Delay or Advanced Case with 119902 = 119901 Since 119902 = 119901 we obtain
[120582119865 (119905 119909 (120591 (119905))) + 120583119866 (119905 119909 (120590 (119905))) minus 120588119890 (119905)]
times (|119909 (119905)|119901minus1
119909 (119905))minus1
ge [120582119891 (119905) |119909 (120591 (119905))|119901+ 120583119892 (119905) |119909 (120590 (119905))|
119901
+120588 |119890 (119905)| ] |119909 (119905)|minus119901
ge [120582119891 (119905) |119909 (120591 (119905))|119901+ 120583119892 (119905) |119909 (120590 (119905))|
119901] |119909 (119905)|
minus119901
= 120582119891 (119905) (119909 (120591 (119905))
119909 (119905))
119901
+ 120583119892 (119905) (119909 (120590 (119905))
119909 (119905))
119901
ge 120582119891 (119905) 120591119899(119905) + 120583119892 (119905) 120590
119899(119905) 119905 isin 119869
119899
(103)
where the functions 120591119899(119905) and 120590
119899(119905) are defined in (26)
(ii) Delay Case with 119902 gt 119901 In this part we use the nextelementary inequality
119883120574+ (120574 minus 1) 119884
120574ge 120574119883119884
120574minus1 120574 gt 1 119883 119884 ge 0 (104)
Since 119902 gt 119901 and using (104) especially for
120574 =119902
119901gt 1 119883 = (120582119891 (119905))
119901119902
|119909 (120591 (119905))|119901
119884 = (119901
119902 minus 119901120588 |119890 (119905)|)
119901119902
(105)
14 Discrete Dynamics in Nature and Society
for all 119905 isin 119869119899we obtain
[120582119865 (119905 119909 (120591 (119905))) + 120583119866 (119905 119909 (120590 (119905))) minus 120588119890 (119905)]
times (|119909 (119905)|119901minus1
119909 (119905))minus1
= [120582119865 (119905 119909 (120591 (119905))) minus 120588119890 (119905)] (|119909 (119905)|119901minus1
119909 (119905))minus1
ge [120582119891 (119905) |119909 (120591 (119905))|1199011 + 120588 |119890 (119905)|] |119909 (119905)|
minus119901
= [119883120574+ (120574 minus 1) 119884
120574] |119909 (119905)|
minus119901
ge119902
119901(120582119891 (119905))
119901119902
|119909 (120591 (119905))|119901
times (119901
119902 minus 119901120588 |119890 (119905)|)
(119901119902)((119902119901)minus1)
|119909 (119905)|minus119901
= 120582119901119902
1205881minus(119901119902)
119902
119901(
119901
119902 minus 119901)
1minus(119901119902)
|119890 (119905)|1minus(119901119902)
times (119891 (119905))119901119902
(119909 (120591 (119905))
119909 (119905))
119901
ge 119896 (120582 120583 120588)119902
119901(
119901
119902 minus 119901)
1minus(119901119902)
|119890 (119905)|1minus(119901119902)
times (119891 (119905))119901119902
120591119899(119905)
(106)
where the function 119896(120582 120583 120588) is from (18)
(iii) Advanced Case with 119902 gt 119901 Using the same line ofarguments as in the proof of the previous case for all 119905 isin 119869
119899
we obtain
[120582119865 (119905 119909 (120591 (119905))) + 120583119866 (119905 119909 (120590 (119905))) minus 120588119890 (119905)]
times (|119909 (119905)|119901minus1
119909 (119905))minus1
= [120583119866 (119905 119909 (120590 (119905))) minus 120588119890 (119905)] (|119909 (119905)|119901minus1
119909 (119905))minus1
ge 119896 (120582 120583 120588)119902
119901(
119901
119902 minus 119901)
1minus(119901119902)
|119890 (119905)|1minus(119901119902)
times (119892 (119905))119901119902
120590119899(119905)
(107)
where the function 119896(120582 120583 120588) is from (21)
(iv) Superlinear Delay-Advanced Case Since 1199011 1199012gt 119901 for
all 119905 isin 119869119899we obtain
[120582119865 (119905 119909 (120591 (119905))) + 120583119866 (119905 119909 (120590 (119905))) minus 120588119890 (119905)]
times (|119909 (119905)|119901minus1
119909 (119905))minus1
= [120582119865 (119905 119909 (120591 (119905))) minus120588
2119890 (119905)] (|119909 (119905)|
119901minus1119909 (119905))minus1
+ [120583119866 (119905 119909 (120590 (119905))) minus120588
2119890 (119905)] (|119909 (119905)|
119901minus1119909 (119905))minus1
ge [120582119891 (119905) |119909 (120591 (119905))|1199011 +
120588
2|119890 (119905)|] |119909 (119905)|
minus119901
+ [120583119892 (119905) |119909 (120590 (119905))|1199012 +
120588
2|119890 (119905)|] |119909 (119905)|
minus119901
(108)
Now just the same as in the proofs of previous delay andadvanced cases with 119902 gt 119901 and with the help of (104) inparticular for
120574 =1199011
119901gt 1 119883 = (120582119891 (119905))
1199011199011
|119909 (120591 (119905))|119901
119884 = (119901
1199011minus 119901
120588
2|119890 (119905)|)
1199011199011
(109)
we have
[120582119891 (119905) |119909 (120591 (119905))|1199011 +
120588
2 |119890 (119905)|] |119909 (119905)|
minus119901
= [119883120574+ (120574 minus 1) 119884
120574] |119909 (119905)|
minus119901
ge1199011
119901(120582119891 (119905))
1199011199011
|119909 (120591 (119905))|119901
times (119901
1199011minus 119901
120588
2|119890 (119905)|)
(1199011199011)((1199011119901)minus1)
|119909 (119905)|minus119901
= 12058211990111990111205881minus(119901119901
1)1199011
119901(
119901
2 (1199011minus 119901)
)
1minus(1199011199011)
times |119890 (119905)|1minus(119901119901
1)(119891 (119905))
1199011199011
(119909 (120591 (119905))
119909 (119905))
119901
ge 119896 (120582 120583 120588)1199011
119901(
119901
2 (1199011minus 119901)
)
1minus(1199011199011)
times |119890 (119905)|1minus(119901119901
1)(119891 (119905))
1199011199011
120591119899(119905)
(110)
where the function 119896(120582 120583 120588) is from (24) Analogously weshow that
[120583119892 (119905) |119909 (120590 (119905))|1199012 +
120588
2|119890 (119905)|] |119909 (119905)|
minus119901
ge 119896 (120582 120583 120588)1199012
119901(
119901
2 (1199012minus 119901)
)
1minus(1199011199012)
times |119890 (119905)|1minus(119901119901
2)(119891 (119905))
1199011199012
120590119899(119905)
(111)
Discrete Dynamics in Nature and Society 15
Summarizing previous calculation we obtain
[120582119865 (119905 119909 (120591 (119905))) + 120583119866 (119905 119909 (120590 (119905))) minus 120588119890 (119905)]
times (|119909 (119905)|119901minus1
119909 (119905))minus1
ge 119896 (120582 120583 120588)119876119899(119905) 119905 isin 119869
119899
(112)
where the function 119896(120582 120583 120588) is from (24)
(v) Supersublinear Delay-Advanced Case Since 1199011gt 119901 gt 119901
2
and the following well-known elementary inequality holds
12057801199060+ 12057811199061+ 12057821199062ge 1199061205780
01199061205781
11199061205782
2 120578119894ge 0 119906
119894ge 0 (113)
from 1205780 1205781 1205782isin (0 1) 120578
0+ 1205781+ 1205782= 1 and 119901
11205781+ 11990121205782= 119901
we obtain for all 119905 isin 119869119899 for all 119905 isin 119869
119899we obtain
[120582119865 (119905 119909 (120591 (119905))) + 120583119866 (119905 119909 (120590 (119905))) minus 120588119890 (119905)]
times (|119909 (119905)|119901minus1
119909 (119905))minus1
ge [120582119891 (119905) |119909 (120591 (119905))|1199011 + 120583119892 (119905) |119909 (120590 (119905))|
1199012 + 120588 |119890 (119905)|]
times |119909 (119905)|minus119901
= [1205781[120578minus1
1120582119891 (119905) |119909 (120591 (119905))|
1199011] + 120578
2[120578minus1
2120583119892 (119905) |119909 (120590 (119905))|
1199012]
+1205780[120578minus1
0120588 |119890 (119905)|]] |119909 (119905)|
minus119901
ge [120578minus1
1120582119891 (119905) |119909 (120591 (119905))|
1199011]1205781
[120578minus1
2120583119892 (119905) |119909 (120590 (119905))|
1199012]1205782
times [120578minus1
0120588 |119890 (119905)|]
1205780
|119909 (119905)|minus119901
= 120582120578112058312057821205881205780 |119890 (119905)|
1205780(119891 (119905))
1205781
(119892 (119905))1205782
times|119909 (120591 (119905))|
12057811199011
|119909 (119905)|12057811199011
|119909 (120590 (119905))|12057821199012
|119909 (119905)|12057821199012
2
prod
119894=0
120578minus120578119894
119894
= 119896 (120582 120583 120588) |119890 (119905)|1205780(119891 (119905))
1205781
(119892 (119905))1205782
times (119909 (120591 (119905))
119909 (119905))
12057811199011
(119909 (120590 (119905))
119909 (119905))
12057821199012 2
prod
119894=0
120578minus120578119894
119894
ge 119896 (120582 120583 120588) |119890 (119905)|1205780(119891 (119905))
1205781
(119892 (119905))1205782
(120591119899(119905))1205781(1199011119901)
times (120590119899(119905))1205782(1199012119901)
2
prod
119894=0
120578minus120578119894
119894
= 119896 (120582 120583 120588)119876119899(119905)
(114)
where 119896(120582 120583 120588) and 119876119899(119905) are given respectively in (24) and
(25) Thus it is shown that required condition (5) in thecases (i)ndash(iv) is fulfilled with respect to 119896(120582 120583 120588) and 119876
119899(119905)
determined by (18) (21) or (24) and (19) (22) or (25)In conclusion according to the previous observation we
see that all assumptions of Lemma 4 are fulfilled and henceLemma 4 proves Theorems 5 6 and 7
Conflict of Interests
The author declares that there is no conflict of interestsregarding the publication of this paper
References
[1] Y G Sun ldquoA note on Nasrrsquos and Wongrsquos papersrdquo Journal ofMathematical Analysis and Applications vol 286 no 1 pp 363ndash367 2003
[2] Y G Sun C H Ou and J S W Wong ldquoInterval oscillationtheorems for a second-order linear differential equationrdquo Com-puters amp Mathematics with Applications vol 48 no 10-11 pp1693ndash1699 2004
[3] S Murugadass E Thandapani and S Pinelas ldquoOscillationcriteria for forced second-order mixed type quasilinear delaydifferential equationsrdquo Electronic Journal of Differential Equa-tions vol 2010 article 73 9 pages 2010
[4] Y Bai and L Liu ldquoNew oscillation criteria for second-orderdelay differential equations with mixed nonlinearitiesrdquoDiscreteDynamics in Nature and Society vol 2010 Article ID 796256 9pages 2010
[5] A F Guvenilir andA Zafer ldquoSecond-order oscillation of forcedfunctional differential equations with oscillatory potentialsrdquoComputers amp Mathematics with Applications vol 51 no 9-10pp 1395ndash1404 2006
[6] A Zafer ldquoInterval oscillation criteria for second order super-half linear functional differential equations with delay andadvanced argumentsrdquoMathematische Nachrichten vol 282 no9 pp 1334ndash1341 2009
[7] A F Guvenilir ldquoInterval oscillation of second-order functionaldifferential equations with oscillatory potentialsrdquo NonlinearAnalysis Theory Methods amp Applications vol 71 no 12 ppe2849ndashe2854 2009
[8] T S Hassan L Erbe and A Peterson ldquoForced oscillation ofsecond order differential equations with mixed nonlinearitiesrdquoActa Mathematica Scientia B vol 31 no 2 pp 613ndash626 2011
[9] M Pasic ldquoNew oscillation criteria for second-order forcedquasilinear functional differential equationsrdquo Abstract andApplied Analysis vol 2013 Article ID 735360 12 pages 2013
[10] L H Erbe Q Kong and B G Zhang Oscillation Theory forFunctional-Differential Equations vol 190 Marcel Dekker NewYork NY USA 1995
[11] V Kolmanovskii and A Myshkis Introduction to the Theoryand Applications of Functional-Differential Equations vol 463Kluwer Academic Publishers Dordrecht The Netherlands1999
[12] R P Agarwal M Bohner and W-T Li Nonoscillation andOscillation Theory for Functional Differential Equations vol267 Marcel Dekker New York NY USA 2004
[13] L Erbe T Hassan and A Peterson ldquoOscillation of secondorder functional dynamic equationsrdquo International Journal ofDifference Equations vol 5 no 2 pp 175ndash193 2010
[14] B Baculıkova J Dzurina and Y V Rogovchenko ldquoOscillationof third order trinomial delay differential equationsrdquo AppliedMathematics and Computation vol 218 no 13 pp 7023ndash70332012
[15] R P Agarwal L Berezansky E Braverman and A Domoshnit-sky Nonoscillation Theory of Functional Differential Equationswith Applications Springer New York NY USA 2012
16 Discrete Dynamics in Nature and Society
[16] J Zhang ldquoVariational approach to solitary wave solution ofthe generalized Zakharov equationrdquo Computers ampMathematicswith Applications vol 54 no 7-8 pp 1043ndash1046 2007
[17] T Ozis and A Yıldırım ldquoApplication of Hersquos semi-inversemethod to the nonlinear Schrodinger equationrdquo Computers ampMathematics with Applications vol 54 no 7-8 pp 1039ndash10422007
[18] X-C Cai andM-S Li ldquoPeriodic solution of Jacobi elliptic equa-tions by Hersquos perturbation methodrdquo Computers amp Mathematicswith Applications vol 54 no 7-8 pp 1210ndash1212 2007
[19] S Lenci G Menditto and A M Tarantino ldquoHomoclinic andheteroclinic bifurcations in the non-linear dynamics of a beamresting on an elastic substraterdquo International Journal of Non-Linear Mechanics vol 34 no 4 pp 615ndash632 1999
[20] D-J Huang and H-Q Zhang ldquoLink between travelling wavesand first order nonlinear ordinary differential equation with asixth-degree nonlinear termrdquoChaos Solitons amp Fractals vol 29no 4 pp 928ndash941 2006
[21] A I Maimistov ldquoPropagation of an ultimately short electro-magnetic pulse in a nonlinear medium described by the fifth-order Duffing modelrdquo Optics and Spectroscopy vol 94 pp 251ndash257 2003
[22] M N Hamdan and N H Shabaneh ldquoOn the large amplitudefree vibrations of a restrained uniform beam carrying anintermediate lumpedmassrdquo Journal of Sound andVibration vol199 no 5 pp 711ndash736 1997
[23] J-H He ldquoSome asymptotic methods for strongly nonlinearequationsrdquo International Journal of Modern Physics B vol 20no 10 pp 1141ndash1199 2006
[24] J B Marion Classical Dynamics of Particles and Systems 1970[25] I Kovacic and M J Brennan The Duffing Equation Nonlinear
Oscillatos and their Behaviour John Wiley amp Sons 1st edition2011
[26] F C Moon Chaotic Vibrations An Introduction for AppliedScientists and Engineers John Wiley amp Sons New York NYUSA 2004
[27] J J Stoker Nonlinear Vibrations 1950[28] G Chen and Z Tao ldquoAmplitude-frequency relationship for the
relativistic oscillatorrdquoAASRI Procedia vol 1 pp 400ndash403 2012[29] R E Mickens Oscillations in Planar Dynamic Systems World
Scientific Publishing Singapore 1996[30] A Belendez T Belendez C Neipp A Hernandez and M
L Alvarez ldquoApproximate solutions of a nonlinear oscillatortypified as a mass attached to a stretched elastic wire by thehomotopy perturbation methodrdquo Chaos Solitions and Fractalsvol 39 pp 746ndash764 2009
[31] A Belendez E Fernandez R Fuentes J J Rodes and I PascualldquoHarmonic balancing approach to nonlinear oscillations of apunctual charge in the eletric field of charged ringrdquo PhysicsLetters A vol 373 pp 735ndash740 2009
[32] A Elıas-Zuniga ldquoExact solution of the cubic-quintic Duffingoscillatorrdquo Applied Mathematical Modelling vol 37 no 4 pp2574ndash2579 2013
[33] A Belendez M L Alvarez J Frances et al ldquoAnalytical approx-imate solutions for the cubic-quintic Duffing oscillator in termsof elementary functionsrdquo Journal of Applied Mathematics vol2012 Article ID 286290 16 pages 2012
[34] A Elıas-Zuniga OMartınez-Romero andR K Cordoba-DıazldquoApproximate solution for the Duffing-harmonic oscillator bythe enhanced cubication methodrdquo Mathematical Problems inEngineering vol 2012 Article ID 618750 12 pages 2012
[35] C W Lim B S Wu andW P Sun ldquoHigher accuracy analyticalapproximations to the Duffing-harmonic oscillatorrdquo Journal ofSound and Vibration vol 296 no 4-5 pp 1039ndash1045 2006
[36] J He ldquoSome new approaches to Duffing equation with stronglyand high order nonlinearity II parametrized perturbationtechniquerdquo Communications in Nonlinear Science amp NumericalSimulation vol 4 no 1 pp 81ndash83 1999
[37] V Marinca and N Herisanu ldquoPeriodic solutions for somestrongly nonlinear oscillations by Hersquos variational iterationmethodrdquo Computers amp Mathematics with Applications vol 54no 7-8 pp 1188ndash1196 2007
[38] W Lu and Y Liu ldquoVibration control for the primary resonanceof the Duffing oscillator by a time delay state feedbackrdquoInternational Journal of Nonlinear Science vol 8 no 3 pp 324ndash328 2009
[39] H Y Hu and Z H Wang Dynamics of Controlled MechanicalSystems with Delayed Feedback Springer 2002
[40] M Hamdi and M Belhaq ldquoControl of bistability in a delayedDuffing oscillatorrdquo Advances in Acoustics and Vibration vol2012 Article ID 872498 6 pages 2012
[41] V Ravichandran C Chinnathambi and S Rajasekar ldquoNonlin-ear resonance in Duffing oscillator with fixed and integrativetime-delayed feedbacksrdquoPramana Journal of Physics vol 78 pp347ndash360 2013
[42] X You and H Xu ldquoAnalytical approximations for the periodicmotion of theDuffing systemwith delayed feedbackrdquoNumericalAlgorithms vol 56 no 4 pp 561ndash576 2011
[43] Z Sun W Xu X Yang and T Fang ldquoInducing or suppressingchaos in a double-well Duffing oscillator by time delay feed-backrdquo Chaos Solitons and Fractals vol 27 pp 705ndash714 2006
[44] H Wang H Hu and Z Wang ldquoGlobal dynamics of a Duffingoscillator with delayed displacement feedbackrdquo InternationalJournal of Bifurcation and Chaos in Applied Sciences andEngineering vol 14 no 8 pp 2753ndash2775 2004
[45] J Chiasson and J J LoiseauApplications of Time Delay SystemsSpringer 2007
[46] M Lakshmanan andDV SenthilkumarDynamics of NonlinearTime-Delay Systems Springer 2010
[47] G Stepan T Insperger and R Szalai ldquoDelay parametricexcitation and the nonlinear dynamics of cutting processesrdquoInternational Journal of Bifurcation and Chaos in Applied Sci-ences and Engineering vol 15 no 9 pp 2783ndash2798 2005
[48] U van der Heiden and H-O Walther ldquoExistence of chaos incontrol systems with delayed feedbackrdquo Journal of DifferentialEquations vol 47 no 2 pp 273ndash295 1983
[49] Y G Sun and J S W Wong ldquoOscillation criteria for secondorder forced ordinary differential equations with mixed non-linearitiesrdquo Journal of Mathematical Analysis and Applicationsvol 334 no 1 pp 549ndash560 2007
[50] J Heagy and W L Ditto ldquoDynamics of a two-frequencyparametrically driven Duffing oscillatorrdquo Journal of NonlinearScience vol 1 no 4 pp 423ndash455 1991
[51] A B Belogortsev ldquoBifurcations of tori and chaos in thequasiperiodically forced Duffing oscillatorrdquoNonlinearity vol 5no 4 pp 889ndash897 1992
[52] M Belhaq and M Houssni ldquoQuasi-periodic oscillations chaosand suppression of chaos in a nonlinear oscillator driven byparametric and external excitationsrdquo Nonlinear Dynamics vol18 no 1 pp 1ndash24 1999
[53] S H Saker P Y H Pang and R P Agarwal ldquoOscillationtheorems for second order nonlinear functional differential
Discrete Dynamics in Nature and Society 17
equations with dampingrdquo Dynamic Systems and Applicationsvol 12 no 3-4 pp 307ndash321 2003
[54] I N Bronshtein K A Semendyayev G Musiol and HMuehligHandbook of Mathematics Springer 5th edition 2007
[55] M Pasic ldquoFite-Wintner-Leighton-type oscillation criteria forsecond-order differential equations with nonlinear dampingrdquoAbstract and Applied Analysis vol 2013 Article ID 852180 10pages 2013
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical Problems in Engineering
Hindawi Publishing Corporationhttpwwwhindawicom
Differential EquationsInternational Journal of
Volume 2014
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Mathematical PhysicsAdvances in
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Operations ResearchAdvances in
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Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
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Algebra
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Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
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Decision SciencesAdvances in
Discrete MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
Discrete Dynamics in Nature and Society 5
Corollary 9 the following two different classes of quasilineardelay differential equations
(100381610038161003816100381610038161199091015840
(119905)10038161003816100381610038161003816
119901minus1
1199091015840
(119905))1015840
+ 120582 sin (119898119905) 1003816100381610038161003816119909 (119905 minus 120591119898)1003816100381610038161003816119901+120576
times sgn (119909 (119905 minus 120591119898)) = minus120588 cos (2119898119905)
(1199091015840(119905)
radic1 + 11990910158402
(119905)
)
1015840
+ 120582 cos (119898119905) 1003816100381610038161003816119909 (119905 minus 120591119898)10038161003816100381610038161+120576
times sgn (119909 (119905 minus 120591119898)) = minus120588 cos (2119898119905)
(29)
are oscillatory provided at least one of 120582 gt 0 and 120588 gt 0 is largeenough (the case 120588 = 0 is possible if 120576 = 0) It is because forall 119899 isin N we have
minus cos (2119898119905)
le 0 on [2119899120587
1198982119899120587
119898+
120587
4119898]
ge 0 on [2119899120587
119898+
120587
41198982119899120587
119898+
120587
2119898]
sin (119898119905) ge 0 cos (119898119905) ge 0 on [2119899120587
1198982119899120587
119898+
120587
2119898]
[2119899120587
119898+ 120591119898+ 119879lowast2119899120587
119898+
120587
4119898+ 119879lowast]
= [(2119899 + 2) 120587
119898+ 120591119898(2119899 + 2) 120587
119898+
120587
4119898]
[2119899120587
119898+
120587
4119898+ 120591119898+ 119879lowast2119899120587
119898+
120587
2119898+ 119879lowast]
= [(2119899 + 2) 120587
119898+
120587
4119898+ 120591119898(2119899 + 2) 120587
119898+
120587
2119898]
(30)
where 119879lowast= 2120587119898 is the common period of the functions
sin(119898119905) and cos(119898119905) Thus in order to apply Corollary 9 wecan choose 119886
2119899minus1= 2119899120587119898 + 120591
119898 1198872119899minus1
= 2119899120587119898 + 120587(4119898)1198862119899= 2119899120587119898 + 120587(4119898) + 120591
119898 and 119887
2119899= 2119899120587119898 + 120587(2119898)
Example 13 (advanced case) Let 119901 ge 1 120576 ge 0 and 119898 isin
N be fixed and 120590119898
isin R 0 le 120590119898
lt 120587(4119898) With thehelp of Corollary 10 the following two classes of quasilinearadvanced differential equations
(100381610038161003816100381610038161199091015840
(119905)10038161003816100381610038161003816
119901minus1
1199091015840
(119905))1015840
+ 120583 sin (119898119905) 1003816100381610038161003816119909 (119905 + 120590119898)1003816100381610038161003816119901+120576
times sgn (119909 (119905 + 120590119898)) = minus120588 cos (2119898119905)
(1199091015840(119905)
radic1 + 11990910158402
(119905)
)
1015840
+ 120583 cos (119898119905) 1003816100381610038161003816119909 (119905 + 120590119898)10038161003816100381610038161+120576
times sgn (119909 (119905 + 120590119898)) = minus120588 cos (2119898119905)
(31)
are oscillatory provided at least one of 120583 gt 0 and 120588 gt 0 islarge enough (the case 120588 = 0 is possible if 120576 = 0) In order toapply Corollary 10 we can choose 119886
2119899minus1= 2119899120587119898 119887
2119899minus1=
2119899120587119898 + 120587(4119898) minus 120590119898 1198862119899
= 2119899120587119898 + 120587(4119898) and 1198872119899
=
2119899120587119898 + 120587(2119898) minus 120590119898
Example 14 (delay-advanced case) Let 119901 ge 1 1205761gt 0 1205762gt 0
and 119898 isin N be fixed and 120591119898ge 0 and 120590
119898ge 0 0 le 120591
119898+ 120590119898lt
120587(4119898) With the help of Corollary 11 the following class ofquasilinear delay-advanced differential equations
(100381610038161003816100381610038161199091015840
(119905)10038161003816100381610038161003816
119901minus1
1199091015840
(119905))1015840
+ 120582 sin (119898119905)
times1003816100381610038161003816119909 (119905 minus 120591119898)
1003816100381610038161003816119901+1205761 sgn (119909 (119905 minus 120591
119898))
+ 120583 cos (119898119905) 1003816100381610038161003816119909 (119905 + 120590119898)1003816100381610038161003816119901+1205762
times sgn (119909 (119905 + 120590119898)) = minus120588 cos (2119898119905)
(32)
is oscillatory provided either 120588 gt 0 is large enough or at leastone of 120582 gt 0 and 120583 gt 0 is large enough In order to applyCorollary 11 we can choose 119886
2119899minus1= 2119899120587119898 + 120591
119898 1198872119899minus1
=
2119899120587119898 + 120587(4119898) minus 120590119898 1198862119899
= 2119899120587119898 + 120587(4119898) + 120591119898 and
1198872119899= 2119899120587119898 + 120587(2119898) minus 120590
119898
3 Application to Duffing Equations withTime Delay Feedback
Let 120582 ge 0 denote the control gain parameter (often calledldquodisplacement feedback coefficientrdquo) 120591 gt 0 the time delayand 120588 ge 0 and 120596 gt 0 the amplitude and frequency of theexternal force respectively Let the function Φ = Φ(119905 119906) thatwill appear in the delay feedback term Φ(119905 119909(119905 minus 120591)) satisfythe general condition
Φ (119905 119906) sgn (119906) ge 1206010|119906|119902
forall119905 ge 1199050
119906 = 0 and some 119902 ge 1 1206010gt 0
(33)
For instance Φ(119905 119906) = 1199062119898minus1 119898 isin N or more general
Φ(119905 119906) = sum119898
119896=11206011198961199062119896minus1 120601
119896gt 0119898 isin N
In this section we consider the following large class ofundamped possible nonautonomous and nonconservativeDuffing equations without or with the general time delayfeedback Φ(119905 119909(119905 minus 120591))
(10038161003816100381610038161003816119909101584010038161003816100381610038161003816
119901minus1
1199091015840)1015840
+ 1205962
0119909 +
1205831|119909|1199031 sgn (119909)
(1205832+ 12058331199092)1199032
+
119898
sum
119894=1
120573119894(119905) |119909|
120572119894minus1119909 + 120582Φ (119905 119909 (119905 minus 120591)) = 120588 cos (120596119905)
(34)
where 1205960is the natural frequency 120583
1ge 0 is the density of the
nonlinear potential (or rigidity coefficient) and 1205832 1205833 1199031 1199032
are nonnegative constants 120573119894(119905) ge 0 and 120572
119894ge 1
When 119901 = 1 120582 = 0 and 120573119894(119905) equiv 120573
119894= const
(34) contains many most important classes of undampedautonomous Duffing oscillators such as the following
(i) the strongly nonlinear Duffing oscillator with smoothodd nonlinearity is given in (34) provided 120583
1= 0 and
120572119894= 2119894 + 1 let us recall some of its known particular
cases
(a) the classic Duffing oscillator 11990910158401015840 +12059620119909+120573119909
3= 0
has been recently studied in the searching of
6 Discrete Dynamics in Nature and Society
solitarywave solutions of classic and generalizedZakharov equations of plasma physics (see [16])and of nonlinear Schrodinger equation (see[17]) also it is strongly connected with theJacobi elliptic equation (see [18])
(b) the cubic-quintic oscillator 11990910158401015840 + 1205962
0119909 + 120573
11199093+
12057321199095
= 0 is used as a model for the non-linear dynamics of a slender elastica (see [19])in nonlinear wave systems (see [20]) for thepropagation of a short electromagnetic pulsein a nonlinear medium (see [21]) and in theunimodal Duffing temporal problem (see [22])
(c) the cubic truly nonlinear oscillator 11990910158401015840 + 1205731199093=
0 models the motion of a ball bearing thatoscillates in a glass tube that is bent into acurve (see [23]) as well as the motion of a massattached to identical stretched elastic wires (see[24])
(d) the nonhomogeneous Duffing oscillator 11990910158401015840 +1205962
0119909 + 120573119909
3= 120588 cos(120596119905) describes various forced
vibrations of beams springs with nonlinearstiffness cables plates shells and optical fibresin electrical circuits in nonlinear isolators andso forth (see for instance [25 26])
(ii) the general Duffing-harmonic oscillator (with rationalor irrational nonlinear restoring-force) is given in(30) if 120583
1= 0 120573119894= 0 and 120588 = 0 the most known
subclasses of these oscillators are
(a) the classic Duffing-harmonic oscillator 11990910158401015840+
(12058311199093(1205832+ 12058331199092)) = 0 which models many
conservative nonlinear oscillatory systems see[27]
(b) the relativistic harmonic oscillator11990910158401015840+ (1205831119909radic1 + 1199092) = 0 see [28]
(c) the nonlinear oscillator11990910158401015840+119909minus(1205831119909radic1 + 1199092) =
0 1205831
isin [0 1] which is typified as a massattached to a stretched elastic wire see [29 30]
(d) the nonlinear oscillator 11990910158401015840
+
(1205831119909(radic(1 + 1199092 )
3
) = 0 which presentsnonlinear oscillations of a punctual charge inthe electric field of charged ring see [31]
Finding several explicit forms of periodic approximate solu-tions for these oscillators has been intensively studied lastyears by many authors see for instance [28 30 32ndash37] andalso the references therein
When 120582 = 0 and linear time delay feedbackΦ(119905 119909(119905minus120591)) =119909(119905 minus 120591) the following topics have been studied for varioustypes of Duffing oscillators with time delayed feedback in[38] authors constructed a low-order approximate solutionunder weak feedback gain parameter about the low- andhigh-order approximations see also [39] in [40] with 120588 = 0the Hopf bifurcation diagrams have been explored for theapproximate periodic solutions (amplitude versus time delay120591 and feedback gain 120582 versus time delay 120591) moreover in [41]
authors made an analysis on the effect of the control gainand time delay parameters on the amplitude of approximateperiod solution from the theoretical and numerical pointsof view see also [42] in [43] authors studied the chaoticbehaviour with respect to gains and time delay parameterssee also [44]
Equations under time delay control such as (34) (espe-cially with damped term) are used as a model for variouscontrolled physical mechanical and engineering systemswith time delays see for instance [39 45ndash48] and thereferences therein
Here (34) contains very general nonlinear time delayfeedback Φ(119905 119909(119905 minus 120591)) with Φ satisfying (33) and the lineartime delay feedback 119909(119905 minus 120591) is only a particular case ofit and to the best of our knowledge the previous topicsare not considered for (34) as yet Moreover with suchan Φ the oscillations of (34) can be taken under a doubteven with the linear time delay feedback (see the nature ofthe approximations given in [38 39]) Hence we can posethe following question under what conditions on equationrsquosparameters (34) is a nonlinear oscillator that is possessesonly oscillatory solutions An answer is given in the nextresult as an easy consequence of the parametrically excitedoscillations by Theorem 5
Theorem 15 Let 120591 isin (0 120587120596) and (33) hold Equation (34) isoscillatory in the next two cases
(i) 119902 = 119901 and 120582 is large enough(ii) 119902 gt 119901 120582 gt 0 120588 gt 0 and at least one of 120582 and 120588 is large
enough
Proof Let 119903(119905) equiv 1 119860(V) = |V|119901minus1 119865(119905 119906) = Φ(119905 119906) 119866(119905 119906) equiv0 119890(119905) = cos(120596119905) and
119861 (119905 119906 V) = 1205962
0119906 +
1205831|119906|1199031s119892119899 (119906)
(1205832+ 12058331199062)1199032
+
119898
sum
119894=1
120573119894(119905) |119906|
120572119894minus1119906 (35)
It is easy to check that all assumptions of Theorem 5 arefulfilled with respect to the sequence 119886
119899= minus1205872120596 + 119899120587120596 + 120591
and 119887119899= 1205872120596 + 119899120587120596 + 120591 where 119886
119899lt 119887119899since it is supposed
that 120591 lt 120587120596 Hence Theorem 5 proves this theorem
Remark 16 Even in the linear forced case (119890(119905) equiv 0) it isnot easy to establish the oscillations of all solutions since theoscillation and nonoscillation can occur simultaneously Themost simple and important example for the coincidence ofoscillation and nonoscillation is the following linear forceddifferential equation 11990910158401015840 + (2119905)119909
1015840+ 119909 = 2119905 119905 gt 0 that
allows an oscillatory solution 1199091(119905) = (3 sin 119905)119905 + 2119905 and a
nonoscillatory solution 1199092(119905) = 2119905 This is not possible in
the linear case with 119890(119905) equiv 0 because of Sturmrsquos separationtheorem
4 Parametrically Excited Oscillations andWell-Known Oscillation Criteria
In this section we would like to draw the readerrsquos attentionto the fact that the parametrically excited oscillations have
Discrete Dynamics in Nature and Society 7
been already appearing in some published papers on theoscillation of functional differential equations but only insome examples illustrating certain main oscillation criteriaHowever with the help of our main results in which theparametrically excited oscillations are studied in a generalsetting the equations from these examples are replaced withgeneral ones also having parameters 120582 and 120583
In [1] (see also [2 Example 31] with 120591 = 0 [3 Example31] and [4 Section 3]) the author considers the oscillationof the second-order delay differential equation
11990910158401015840
(119905) + 119891 (119905) |119909 (120591 (119905))|120574 sgn119909 (120591 (119905)) = 119890 (119905) (36)
in the linear case (120574 = 1) and the superlinear (120574 gt 1)In the linear case (analogously for the superlinear case see[1 Theorem 2]) the author proved the following oscillationcriterion In what follows we denote
119863 (119886 119887) = 119906 isin 1198621
([119886 119887] R) 119906 (119905) equiv 0 119906 (119886) = 119906 (119887) = 0
(37)
Theorem 17 ([1 Theorem 1]) Suppose that for any 119879 ge 0there exist constants 119886
1 1198871 1198862 1198872such that 119879 le 119886
1lt 1198871 119879 le
1198862lt 1198872 and 119891(119905) ge 0 on [120591(119886
1) 1198871] cup [120591(119886
2) 1198872] 119890(119905) le 0
on [120591(1198861) 1198871] and 119890(119905) ge 0 on [120591(119886
2) 1198872] If there exists 119906 isin
119863(119886119894 119887119894) 119894 = 1 2 such that
int
119887119894
119886119894
[1199062
(119905) 119891 (119905)120591 (119905) minus 120591 (119886
119894)
119905 minus 120591 (119886119894)
minus (1199061015840
(119905))2
]119889119905 ge 0 (38)
then (36) with 120574 = 1 is oscillatory
Previous criterion has been applied on the followingparticular equation
11990910158401015840
(119905) + 120582 sin (119905)1003816100381610038161003816100381610038161003816119909 (119905 minus
120587
4)
1003816100381610038161003816100381610038161003816
120574
times sgn 119909(119905 minus120587
4) = cos (119905) 119905 ge 0
(39)
where 120582 ge 0 and 120574 = 1 Applying Theorem 17 to (39) theauthor proved that (39) is oscillatory provided the followinginequality
120582int
119887119894
119886119894
sin2 (2119905) cos2 (2119905) sin (119905)119905 minus 119886119894
119905 minus 119886119894+ 1205874
119889119905 ge120587
2 (40)
holds for sufficiently large 120582 Thus the oscillation of (39) isexcited by the large enough parameter 120582 However accordingto Theorems 5 and 6 we are able to show that the nextparametric equation that corresponds to general equation(36)
11990910158401015840
(119905) + 120582119891 (119905) |119909 (120591 (119905))|120574 sgn119909 (120591 (119905)) = 119890 (119905) (41)
is oscillatory provided 120582 is large enough where 1199011= 1199012= 120574
120583 = 0 and 120588 = 1Next in [5] (see also [6ndash8]) the authors consider the
oscillation of the following class of second-order differentialequations with delay and advanced arguments
(119903 (119905) 1199091015840
(119905))1015840
+ 119891 (119905) |119909 (120591 (119905))|1199011 sgn119909 (120591 (119905))
+ 119892 (119905) |119909 (120590 (119905))|1199012 sgn119909 (120590 (119905)) = 119890 (119905) 119905 ge 0
(42)
where 1199011 1199012ge 1 When 119901
1= 1199012= 1 the authors prove the
following result (for other cases see [5Theorems 32 33 and34]
Theorem 18 ([5 Theorem 31]) Suppose that for any 119879 ge
0 there exist intervals [120591(1198861) 1198871] [120591(119886
2) 1198872] [1198881 120590(1198891)] and
[1198882 120590(1198892)] contained in [119879infin) such that 119886
1lt 1198871 1198862lt 1198872
1198881lt 1198891 1198882lt 1198892 and
119891 (119905) ge 0 119900119899 [120591 (1198861) 1198871] cup [120591 (119886
2) 1198872]
119892 (119905) ge 0 119900119899 [1198881 120590 (1198891)] cup [119888
2 120590 (1198892)]
119890 (119905) le 0 119900119899 [120591 (1198861) 1198871] cup [1198881 120590 (1198891)]
119890 (119905) ge 0 119900119899 [120591 (1198862) 1198872] cup [1198882 120590 (1198892)]
(43)
and 119888119894= 120591(119886
119894) 119889119894= 119886119894 and 119887
119894= 120590(119889
119894) 119894 = 1 2 If there exist
1199061isin 119863(119886
119894 119887119894) and 119906
2isin 119863(119888119894 119889119894) such that either
int
119887119894
119886119894
[1199062
1(119905) 119891 (119905)
120591 (119905) minus 120591 (119886119894)
119905 minus 120591 (119886119894)
minus (1199061015840
1(119905))2
119903 (119905)] 119889119905 ge 0 (44)
or
int
119889119894
119888119894
[1199062
2(119905) 119891 (119905)
120590 (119889119894) minus 120590 (119905)
120590 (119889119894) minus 119905
minus (1199061015840
2(119905))2
119903 (119905)] 119889119905 ge 0 (45)
for 119894 = 1 2 then (42) with 1199011= 1199012= 1 is oscillatory
As a consequence of this result it has been concluded thatthe particular equation
(119903 (119905) 1199091015840
(119905))1015840
+ 120582 sin (119905) 119909 (119905 minus 120587
12)
+ 120583 cos (119905) 119909 (119905 + 120587
6) = cos (2119905) 119905 ge 0
(46)
is oscillatory provided either 120582 or 120583 is large enough Howeverby following Theorems 5 and 6 one can obtain the sameconclusion for the following general equation associated with(42)
(119903 (119905) 1199091015840
(119905))1015840
+ 120582119891 (119905) |119909 (120591 (119905))|1199011 sgn119909 (120591 (119905))
+ 120583119892 (119905) |119909 (120590 (119905))|1199012 sgn119909 (120590 (119905)) = 119890 (119905)
(47)
Related observation can be done with [8 Example 33]and [9 Example 21] where the quasilinear second-orderfunctional differential equations have been considered It isleft to the reader
5 Some Open Questions and Comments
In this section we discuss some problems related to ourmainresults that are not studied here
(1) Quasiperiodic Case In the theory of nonlinear oscillatorsa particularly important case occurs when the periodiccoefficients in the oscillator do not have any common periodIt is called the quasiperiodic (or two-frequency) nonlinear
8 Discrete Dynamics in Nature and Society
oscillator and studied for instance in [50ndash52] Since inTheorems 5 6 and 7 we assume that the correspondingperiodic functions have a commonperiod it is natural to posethe next question
Open Question 1 Is it possible to derive sufficient conditionsfor the oscillation of (27) in the casewhen119891(119905) and119892(119905) (resp119891(119905) 119892(119905) and ℎ(119905)) are two (resp three) periodic functionsnot having a common period
(2) Equation with More Functional Arguments Next regard-ing some second-order functional differential equationsconsidered in the references of this paper more than twononlinear functional terms are appearing and thereforeinstead of main equation (1) and corresponding particularequation (27) considered inTheorems 5 6 and 7 we suggestthe following classes of equations
(119903 (119905) 119860 (1199091015840
(119905)))1015840
+ 119861 (119905 119909 (119905) 1199091015840
(119905))
+
1198981
sum
119896=1
120582119896119865119896(119905 119909 (120591
119896(119905)))
+
1198982
sum
119896=1
120583119896119866119896(119905 119909 (120590
119896(119905))) = 120588119890 (119905)
(48)
where 0 le 120591119896(119905) le 119905 lim
119905rarrinfin120591119896(119905) = infin 120590
119896(119905) ge 119905 119898
1 1198982isin
N and
(119903 (119905) 119860 (1199091015840
(119905)))1015840
+ 119861 (119905 119909 (119905) 1199091015840
(119905))
+
1198981
sum
119896=1
120582119896119891119896(119905)
1003816100381610038161003816119909 (119905 minus 120591119896)1003816100381610038161003816119901119896 sgn (119909 (119905 minus 120591
119896))
+
1198982
sum
119896=1
120583119896119892119896(119905)
1003816100381610038161003816119909 (119905 + 120590119896)1003816100381610038161003816119902119896 sgn (119909 (119905 + 120590
119896)) = 120588119890 (119905)
(49)
where 120582119896 120583119896 120588 120591119896 120590119896ge 0 and 119901
119896 119902119896gt 0
Comment We suggest the reader to enlarge the main resultsof this paper to (48) and (49)
(3) Damped Duffing Equation In the application the Duffingequation (34) is often appearing with the linear damped term1199091015840(119905) that is
11990910158401015840+ 11988901199091015840+ 1205962
0119909 + 120573119909
3+ 120582Φ (119909 (119905 minus 120591)) = 120588 cos (120596119905) (50)
where 1198890
is the damped coefficient which can in anactive way influence various behaviours of (50) Since119861(119905 119909(119905) 119909
1015840(119905)) = 119889
01199091015840(119905) does not satisfy the required
assumption (4) we are not able to apply our main results to(50) Hence we pose the following questionOpen Question 2 Is it possible to obtain the parametricallyexcited oscillation for (1) in the case when the damped term119861(119905 119906 V) satisfies a larger condition than (4) in which thelinear damped term 120573119909
1015840(119905) is especially included
(4) Functional Argument in Damped Term In a class of Duff-ing equations we have two time delayed feedback and hence
besides the control gain parameter 1205821another parameter 120582
2
appears the so-called velocity gain parameter Hence insteadof (34) one can consider
11990910158401015840+ 11988901199091015840+ 1205962
0119909 + 120573119909
3+ 1205821119909 (119905 minus 120591)
+ 12058221199091015840
(119905 minus 120591) = 120588 cos (120596119905) (51)
Therefore we suggest the following problem for further studyOpen Question 3 Is it possible to obtain the parametricallyexcited oscillation for the following more general functionaldifferential equation than (1) in which the functional argu-ment appears in the damped term too as follows
(119903 (119905) 119860 (1199091015840
(119905)))1015840
+ 119861 (119905 119909 (119905) 1199091015840
(119905)) + 1205821119865 (119905 119909 (120591 (119905)))
+ 1205822119867(119905 119909
1015840
(120591 (119905))) = 120588119890 (119905) 119905 ge 1199050
(52)or
(119903 (119905) 119860 (1199091015840
(119905)))1015840
+ 119861 (119905 119909 (119905) 1199091015840
(119905)) + 1205831119866 (119905 119909 (120590 (119905)))
+ 1205832119867(119905 119909
1015840
(120590 (119905))) = 120588119890 (119905) 119905 ge 1199050
(53)
About known oscillation criteria for the second-order func-tional differential equations having the functional argumentin the damped term we refer the reader to for instance [53]and the references therein
6 Proofs of Main Results
The proof of Lemma 1 is based on the following three stepstwo working forms of condition (6) (see Lemmas 19 and 20)the existence of an explosive solution of a suitable Riccatidifferential inequality (see Proposition 22) and a comparisonprinciple (see Proposition 24)
Lemma 19 (a necessary condition to (6)) Let 0 lt 119903(119905) le 1199030
on [1199050infin) If assumption (6) is fulfilled then there is a positive
real number 120576 such that1
120587lowast
int119869
min119901
(120576119903 (119905))120574minus1
120576119896 (120582 120583 120588)119876 (119905) 119889119905 ge 1 (54)
for all 120582 ge 1205820 120583 ge 120583
0 and 120588 ge 120588
0and some (120582
0 1205830 1205880) isin R3+
Proof Since 0 lt 119903(119905) le 1199030for 119905 ge 119905
0 we conclude that for
120576 = (119901
119903120574minus1
0119896 (120582 120583 120588)max
119905isin 119869119876 (119905)
)
1120574
(120582 120583 120588) isin R3
+
(55)
it holds that 119901(120576119903(119905))120574minus1
ge 119901(1205761199030)120574minus1
= 120576119896(120582 120583 120588)
max119905isin 119869119876(119905) ge 120576119896(120582 120583 120588)119876(119905) 119905 isin 119869 and hence
int119869
min119901
(120576119903 (119905))120574minus1
120576119896 (120582 120583 120588)119876 (119905) 119889119905
= 120576119896 (120582 120583 120588) int119869
119876 (119905) 119889119905
(56)
Discrete Dynamics in Nature and Society 9
On the other hand from (6) we observe
1
120587lowast
int119869
119876 (119905) 119889119905 ge1199031minus(1120574)
0
1199011120574[119896 (120582 120583 120588)]1minus(1120574)
(max119905isin 119869
119876 (119905))
1120574
(57)
which together with (55) and (56) gives
1
120587lowast
int119869
min119901
(120576119903 (119905))120574minus1
120576119896 (120582 120583 120588)119876 (119905) 119889119905
= 120576119896 (120582 120583 120588)1
120587lowast
int119869
119876 (119905) 119889119905
ge 1205761199031minus(1120574)
0
1199011120574[119896 (120582 120583 120588)]
1120574
(max119905isin 119869
119876 (119905))
1120574
= 1
(58)
for all 119899 ge 1198990 120582 ge 120582
0 120583 ge 120583
0 and 120588 ge 120588
0 It proves this
lemma
Lemma 20 (an equivalent condition to (54)) Assumption(54) is fulfilled if and only if there is a real number 120576 gt 0 and acontinuous function 119870(119905) ge 0 119905 isin 119869 such that
1198880= int119869
119870 (119905) 119889119905 gt 0119870 (119905)
1198880
le1
120587lowast
timesmin119901
(120576119903 (119905))120574minus1
120576119896 (120582 120583 120588)119876 (119905)
(59)
for all 119905 isin 119869 120582 ge 1205820 120583 ge 120583
0 and 120588 ge 120588
0and some (120582
0 1205830 1205880) isin
R3+
Proof This proof is very elementary Indeed if (54) holdsthen the function119870(119905) and number 119888
0 defined by
119870 (119905) =1
120587lowast
min119901
(120576119903 (119905))120574minus1
120576119896 (120582 120583 120588)119876 (119905)
1198880= int119869
119870 (119905) 119889119905
(60)
obviously satisfy 1198880
ge 1 and 119870(119905)1198880
le 119870(119905) = (1120587lowast)
min119901(120576119903(119905))120574minus1 120576119896(120582 120583 120588)119876(119905) which shows (59) Con-versely if (59) holds then integrating both sides of thesecond inequality in (59) we obtain
int119869
1
120587lowast
min119901
(120576119903 (119905))120574minus1
120576119896 (120582 120583 120588)119876 (119905) ge int119869
119870 (119905)
1198880
119889119905 = 1
(61)
which shows (54)
In conclusion according to previous two lemmas wesee that supposed condition (6) implies (59) which plays animportant role in the proof of the main results
The second step in the proof of Lemma 1 is to prove theexistence of a function 120596(119905) which blows up in the finitetime and satisfies a generalized Riccati differential lowerinequality we briefly present the existence and properties
of the so-called generalized tangent type function In whatfollows let 120587
lowastbe a positive real number defined in (3) Let us
remark that 120572(119904) = 119904120574 120574 gt 1 implies 120587
lowast= (2120587)(120574 sin(120587120574))
see for instance [54] and obviously for 120574 = 2wehave120587lowast= 120587
Lemma 21 Let 120572 [0infin) rarr [0infin) be a continuousfunction such that
int
infin
0
119889120591
1 + 120572 (120591)lt infin (62)
Then there is a real number 120587lowastgt 0 and a function 119911 = 119911(119904)
119911 isin 1198621((minus120587lowast2 120587lowast2)R) such that
119889119911
119889119904= 1 + 120572 (|119911 (119904)|) 119904 isin (minus
120587lowast
2120587lowast
2)
119911 (0) = 0
(63)
Moreover 119911(119904) is increasing and odd
lim119904rarr120587lowast2
119911 (119904) = infin 120587lowast=
2120587
120574 sin (120587120574)for 120572 (119904) = 119904
120574
120574 gt 1
(64)
In particular for 120572(119904) = 1199042 one can take 119911(119904) = tan(119904) and
120587lowast= 120587
Proof Let 119885 = 119885(119905) 119905 isin R be a function defined by
119885 (119905) = int
119905
0
1
1 + 120572 (|120591|)119889120591 119905 isin R (65)
The function 119885(119905) is well defined since 120572(119904) is positive andcontinuous on [0infin) 119885(119905) is increasing and odd functionand
119889119885
119889119905=
1
1 + 120572 (|119905|) 119905 isin R
119885 (0) = 0 119885 isin 1198621
(RR)
(66)
Moreover because of (62) there is a real number 120587lowastgt 0 such
that120587lowast
2= int
infin
0
119889120591
1 + 120572 (120591) (67)
Thus 119885 R rarr (minus120587lowast2 120587lowast2) and there exists an inverse
function 119885minus1 = 119885minus1(119904) of the original function 119885 = 119885(119905) and
119885minus1
(minus120587lowast2 120587lowast2) rarr R Also from 119885(119885
minus1(119904)) = 119904 and
119889119885119889119905 = 0 onR we also derive that119889119885minus1119889119904 = 0 on its domain(minus120587lowast2 120587lowast2) and
119889119885
119889119905(119885minus1
(119904)) =1
(119889119885minus1119889119904) 119904 isin (minus
120587lowast
2120587lowast
2) (68)
Putting 119905 = 119885minus1(119904) for 119904 isin (minus120587
lowast2 120587lowast2) into (66) and using
(68) we easily obtain
119889119885minus1
119889119904= 1 + 120572 (
10038161003816100381610038161003816119885minus1
(119904)10038161003816100381610038161003816) 119904 isin (minus
120587lowast
2120587lowast
2)
119885minus1
(0) = 0 119885minus1isin 1198621((minus
120587lowast
2120587lowast
2) R)
(69)
10 Discrete Dynamics in Nature and Society
Moreover from (67) we have lim119904rarr120587lowast2119885minus1(119904) = 119885
minus1
(lim119905rarrinfin
119885(119905)) = lim119905rarrinfin
119885minus1119885(119905) = lim
119905rarrinfin119905 = infin Thus
if we set 119911(119904) = 119885minus1(119904) then previous two statements and
(67) prove this lemma
Next we prove the main result of this section
Proposition 22 Let (2) and (6) hold where 119869 = (119886 119887) Let 120576 gt0 be a real number and let119870(119905) ge 0 119905 isin [119886 119887] be a continuousfunction both obtained in Lemma 20 Let 120587
lowastbe from (3) and
1198880from (59) and let 119877
119886isin R be an arbitrary real number If
119911 = 119911(119904) is the generalized tangens function defined in (63)and 119881(119905) is a function defined by
119881 (119905) =120587lowast
1198880
int
119905
119886
119870 (120591) 119889120591 + 119911minus1(119877119886) 119905 isin [119886 119887] (70)
then there is a 119879lowast119886isin [119886 119887) such that
119881 (119879lowast
119886) =
120587lowast
2 119881 ([119886 119879
lowast
119886)) sub (minus
120587lowast
2120587lowast
2) (71)
Moreover for a function 120596(119905) defined by120596 (119905) = 119911 (119881 (119905)) 119905 isin [119886 119879
lowast
119886) (72)
one has 120596(119886) = 119877119886 lim119905rarr119879
lowast
119886
120596(119905) = infin and
119889120596
119889119905le
119901
(120576119903 (119905))120574minus1
120572 (1003816100381610038161003816120596 (119905)
1003816100381610038161003816)
+ 120576119896 (120582 120583 120588)119876 (119905) 119905 isin (119886 119879lowast
119886)
(73)
where the numbers 119901 and 120574 are from (3) and the functions119896(120582 120583 120588) and 119876(119905) are from (6)
Proof Under assumptions (2) and (6) and because of Lem-mas 19 and 20 we obtain 120576 gt 0 and 119870(119905) gt 0 119905 isin [119886 119887]satisfying inequality (59)
Next since 119911minus1(119877119886) isin (minus120587
lowast2 120587lowast2) (see Lemma 21)
from (70) we directly obtain
119881 (119886) = 119911minus1(119877119886) lt
120587lowast
2 119881 (119887) = 120587
lowast+ 119911minus1(119877119886) gt
120587lowast
2
(74)Since 119870 isin 119862([119886 119887] [0infin)) we obtain 119881 isin 119862([119886 119887]R) cap
1198621((119886 119887)R) and from (74) we observe that there exist
numbers 119879lowast119886isin (119886 119887) such that119881(119879lowast
119886) = 120587lowast2 Also119870(119905)119888
0ge
0 gives 119881([119886 119879lowast119886)) sub (minus120587
lowast2 120587lowast2) which proves statement
(71) Moreover it together with Lemma 21 and (72) provesthat
lim119905rarr119879
lowast
119886
120596 (119905) = lim119905rarr119879
lowast
119886
119911 (119881 (119905)) = 119911 (120587lowast
2) = infin (75)
Next according to (59) (63) and (72) we make thefollowing calculation on the interval [119886 119879lowast
119886)
1205961015840
(119905) = 1199111015840
(119881 (119905)) 1198811015840
(119905) = [1 + 120572 (|119911 (119881 (119905))|)]120587lowast
1198880
119870 (119905)
= [1 + 120572 (1003816100381610038161003816120596 (119905)
1003816100381610038161003816)]120587lowast
1198880
119870 (119905)
le119901
(120576119903 (119905))120574minus1
120572 (1003816100381610038161003816120596 (119905)
1003816100381610038161003816) + 120576119896 (120582 120583 120588)119876 (119905)
(76)
Thus all assertions of this proposition are proved
Remark 23 In the proof of the main result the number 119877119886
is determined by 119877119886= 120596(119886) where 120596(119905) denotes a function
associated with a nonoscillatory solution and it is given by(84) below
The third step in the proof of Lemma 1 is to show thefollowing pointwise comparison principle for the functions120596and120596 satisfying respectively the lower and upper differentialinequalities (73) and
119889120596
119889119905ge
119901
(120576119903 (119905))120574minus1
120572 (|120596 (119905)|) + 120576119896 (120582 120583 120588)119876 (119905) 119905 isin (119886 119887)
(77)
Proposition 24 Let [119886 119887) sub [1199050infin) be an arbitrary inter-
val One supposes that all coefficients of Riccati differentialinequalities (73) and (77) are continuous and strictly positivefunctions Let 120596 120596 isin 119862
1([119886 119887)R) be two functions satisfying
respectively (73) and (77) on the interval [119886 119887) Then
120596 (119886) le 120596 (119886) 119894119898119901119897119894119890119904 120596 (119905) le 120596 (119905) forall119905 isin [119886 119887) (78)
Proof Let119867(119905 119906) be a function defined by
119867(119905 119906) =119901
(120576119903 (119905))120574minus1
120572 (|119906|) + 120576119896 (120582 120583 120588)119876 (119905)
119905 isin [119886 119887) 119906 isin R
(79)
Let 119868 sub [119886 119887) and 119872 gt 0 be arbitrary For any two 1199061
1199062 minus119872 le 119906
1lt 1199062le 119872 let 119868
12be an interval defined
by 11986812
= (min|1199061| |1199062|max|119906
1| |1199062|) Since 120572(119904) is a 1198621-
function on [0infin) we know by the Lagrange mean valuetheorem applied on 119868
12that there is a 120585 isin 119868
12such that
120572 (10038161003816100381610038161199062
1003816100381610038161003816) minus 120572 (10038161003816100381610038161199061
1003816100381610038161003816)
1199062minus 1199061
le
1003816100381610038161003816120572 (10038161003816100381610038161199062
1003816100381610038161003816) minus 120572 (10038161003816100381610038161199061
1003816100381610038161003816)1003816100381610038161003816
1199062minus 1199061
=100381610038161003816100381610038161205721015840
(120585)10038161003816100381610038161003816
100381610038161003816100381610038161003816100381610038161199062
1003816100381610038161003816 minus10038161003816100381610038161199061
10038161003816100381610038161003816100381610038161003816
1199062minus 1199061
le100381610038161003816100381610038161205721015840
(120585)10038161003816100381610038161003816
le max119904isin11986812
100381610038161003816100381610038161205721015840
(119904)10038161003816100381610038161003816
(80)
since ||1199062| minus |1199061|| le 119906
2minus 1199061 Hence for any 119905 isin 119868 and 119906
1 1199062
minus119872 le 1199061lt 1199062le 119872 we have
119867(119905 1199062) minus 119867 (119905 119906
1)
1199062minus 1199061
= 1205880(119905)
120572 (10038161003816100381610038161199062
1003816100381610038161003816) minus 120572 (10038161003816100381610038161199061
1003816100381610038161003816)
1199062minus 1199061
le 1205880(119905)max119904isin11986812
100381610038161003816100381610038161205721015840
(119904)10038161003816100381610038161003816= 1198710(119905)
(81)
Thus the function119867(119905 119906) from (79) satisfies required condi-tion of [55 Lemma 19] and applying it to (73) and (77) weprove this proposition
Proof of Lemma 1 On the contrary let 119909(119905) be a solution of(1) such that
119909 (119905) = 0 on (120591 (120591 (119886)) 120590 (120590 (119889))) (82)
Discrete Dynamics in Nature and Society 11
that is 119909(119905) gt 0 on (120591(120591(119886)) 120590(120590(119889))) or 119909(119905) lt 0 on(120591(120591(119886)) 120590(120590(119889))) since 119909(119905) is a continuous function on[1199050infin) Let for instance
119909 (119905) gt 0 on (120591 (120591 (119886)) 120590 (120590 (119889))) (83)
Another case can be analogously treated let us see thecomment at the end of this proof In particular from (83)we have 119909(119905) gt 0 on (120591(120591(119886)) 120590(120590(119887))) which implies (since120591(119905) and 120590(119905) are increasing functions) 119909(119904) gt 0 for all 119904 isin
(120591(119886) 120590(119887)) cup (120591(120591(119886)) 120591(120590(119887))) cup (120590(120591(119886)) 120590(120590(119887))) whichyields 119909(119905) gt 0 119909(120591(119905)) gt 0 and 119909(120590(119905)) gt 0 on (120591(119886) 120590(119887))Hence by assumption (7) we may use inequality (5) on theinterval (119886 119887)
Firstly we show that the following classic Riccati transfor-mation of 119909(119905)
120596 (119905) = minus120576119903 (119905) 119860 (119909
1015840(119905))
|119909 (119905)|119901minus1
119909 (119905) 119905 isin (119886 119887) 120576 gt 0 (84)
satisfies upper Riccati differential inequality (77) Let usremark that from (1) we have in particular
minus(119903 (119905) 119860 (1199091015840
(119905)))1015840
= 119861 (119905 119909 (119905) 1199091015840
(119905)) + 120582119865 (119905 119909 (120591 (119905)))
+ 120583119866 (119905 119909 (120590 (119905))) minus 120588119890 (119905) 119905 ge 1199050
(85)
Taking the first derivative on both sides of (84) and usingassumptions (3) (4) and (5) as well as equality (85) and(|119909(119905)|
119901minus1119909(119905))1015840
= 119901|119909(119905)|119901minus1
1199091015840(119905) we obtain
119889120596
119889119905= 120576119901 119903 (119905)
119860 (1199091015840(119905)) 1199091015840(119905)
|119909 (119905)|119901+1
minus 1205761
|119909 (119905)|119901minus1
119909 (119905)(119903 (119905) 119860 (119909
1015840
(119905)))1015840
= 120576119901119903 (119905)119860 (1199091015840(119905)) 1199091015840(119905)
|119909 (119905)|119901+1
+120576
|119909 (119905)|119901minus1
119909 (119905)
times [120582119861 (119905 119909 (119905) 1199091015840
(119905)) + 119865 (119905 119909 (120591 (119905)))
+120583119866 (119905 119909 (120590 (119905))) minus 120588119890 (119905) ]
ge 120576119901119903 (119905)119860 (1199091015840(119905)) 1199091015840(119905)
|119909 (119905)|119901+1
+120576
|119909 (119905)|119901minus1
119909 (119905)
times [120582119865 (119905 119909 (120591 (119905))) + 120583119866 (119905 119909 (120590 (119905))) minus 120588119890 (119905)]
ge 120576119901119903 (119905) 120572(
10038161003816100381610038161003816119860 (1199091015840(119905))
10038161003816100381610038161003816
|119909 (119905)|119901
) + 120576119896 (120582 120583 120588)119876 (119905)
= 120576119901119903 (119905) 120572 (|120596 (119905)|
120576119903 (119905)) + 120576119896 (120582 120583 120588)119876 (119905)
ge119901
(120576119903 (119905))120574minus1
120572 (|120596 (119905)|) + 120576119896 (120582 120583 120588)119876 (119905) 119905 isin (119886 119887)
(86)
Thus according to inequality (5) it is shown that if 119909(119905) isa solution of (1) which satisfies (83) then the function 120596(119905)
defined by (84) satisfies the Riccati differential inequality(77) and 120596 isin 119862((119886 119887)R) On the other hand let 119877
119886be a
real number defined by 119877119886= 120596(119886) According to (6) and
Lemma 19 we obtain (54) which together with Lemma 20ensures that we may use Proposition 22 for such chosen realnumber 119877
119886 Hence we obtain a function 120596(119905) defined by (72)
which satisfies the lower Riccati differential inequality (73) on[119886 119879lowast
119886) 119879lowast119886isin (119886 119887) such that 120596(a) = 119877
119886and lim
119905rarr119879lowast
119886
120596(119905) =
infin Therefore by 120596(119886) = 119877119886= 120596(119886) and Proposition 24 we
conclude that lim119905rarr119879
lowast
119886
120596(119905) = infin too which is a contradictionwith the above conclusion saying that 120596 isin 119862((119886 119887)R) Thushypothesis (82) is not true and consequently Lemma 1 isshown
For the analogous case 119909(119905) lt 0 on (120591(120591(119886)) 120590(120590(119889))) wealso have 119909(119905) lt 0 on (120591(120591(119888)) 120590(120590(119889))) which implies (since120591(119905) and 120590(119905) are increasing functions)
119909 (119904) lt 0 forall119904 isin (120591 (119888) 120590 (119889)) cup (120591 (120591 (119888)) 120591 (120590 (119889)))
cup (120590 (120591 (119888)) 120590 (120590 (119889)))
(87)
which yields 119909(119905) lt 0 119909(120591(119905)) lt 0 and 119909(120590(119905)) lt 0 on(120591(119888) 120590(119889)) Now we can repeat the preceding procedure buton interval (119888 119889) and using (8) instead of (119886 119887) and (7)
Proof of Lemma 2 From assumption (10) we obtain the exis-tence of an 119899
0isin N such that
int
119887119899
119886119899
119876119899(119905) 119889119905 ge
1198880
2( max119905isin[119886119899 119887119899]
119876119899(119905))
1120574
119899 ge 1198990 (88)
that is
2
1198880
int
119887119899
119886119899
119876119899(119905) 119889119905 ge ( max
119905isin[119886119899 119887119899]119876119899(119905))
1120574
119899 ge 1198990 (89)
Now from (9) and previous inequality we deduce that forlarge enough 120582 120583 120588 and 119899
1199011120574
1199031minus1120574
0
[119896 (120582 120583 120588)]1minus1120574
120587lowast
int
119887119899
119886119899
119876119899(119905) 119889119905
ge2
1198880
int
119887119899
119886119899
119876119899(119905) 119889119905 ge ( max
119905isin[119886119899 119887119899]119876119899(119905))
1120574
(90)
which shows (6) Thus all assumptions of Lemma 1 arefulfilled and hence Lemma 2 immediately follows fromLemma 1
Proof of Lemma 3 Obviously assumption (11) is a particularcase of assumption (9) Hence this proof is very similar tothe proof of Lemma 2 and so it is left to the reader
Proof of Lemma 4 It is clear that from assumption (13) weobtain
1
(max119905isin[119886119899119887119899]119876119899(119905))1120574
int
119887119899
119886119899
119876119899(119905) 119889119905 ge
1198881
1198621120574
0
gt 0 forall119899 ge 1198990
(91)
12 Discrete Dynamics in Nature and Society
Thus hypothesis (12) is fulfilled and therefore Lemma 3proves this lemma
Proof of Theorems 5 6 and 7 This proof is based onLemma 4 In order to simplify notation in many placesin this proof we set 120591(119905) = 119905 minus 120591 and 120590(119905) = 119905 + 120590 Sinceassumptions (2) (3) and (4) have been already supposed inTheorems 5 6 and 7 in order to prove these theorems byLemma 4 we are going to show that the functions 119896(120582 120583 120588)and 119876
119899(119905) explicitly given respectively in (18) (21) or (24)
and (19) (22) or (25) satisfy required conditions (11) and(13) respectively and that every solution 119909(119905) of (27) satisfiesconditions (7) and (8) with respect to functions 119896(120582 120583 120588)and 119876
119899(119905) where 119886 = 119886
2119899minus1 119887 = 119887
2119899minus1 119888 = 119886
2119899 and 119889 = 119887
2119899
The proof that the function 119896(120582 120583 120588) given in (18) (21) or(24) satisfies (11) Passing to the limit in (18) (21) or (24) it isvery simple to show (11)
The proof that the function 119876119899(119905) given in (19) (22) or
(25) satisfies the first claim in (13) From (25) we immediatelyobtain
1003816100381610038161003816120591119899 (119905)1003816100381610038161003816 =
100381610038161003816100381610038161003816100381610038161003816
(119905 minus 119886119899
119905 minus 119886119899+ 120591
)
119901100381610038161003816100381610038161003816100381610038161003816
le 1
1003816100381610038161003816120590119899 (119905)1003816100381610038161003816 =
100381610038161003816100381610038161003816100381610038161003816
(119887119899minus 119905
119887119899minus 119905 + 120590
)
119901100381610038161003816100381610038161003816100381610038161003816
le 1 forall119899 isin N
(92)
Next by assumptions of this corollary we can conclude thatthere are three positive constants 119891
0 1198920 1198900such that |119891(119905)| le
1198910and |119892(119905)| le 119892
0on [1199050infin) in cases (i) and (ii) and
|119890(119905)| le 1198900on [1199050infin) in cases (iii) and (iv) Putting previous
inequalities into (19) (22) or (25) for all 119899 isin N and 119905 isin
[1199050infin) it holds that
1003816100381610038161003816119876119899 (119905)1003816100381610038161003816 le
119902
119901(
119901
119902 minus 119901)
(119902119901)minus1
1198901minus(119901119902)
0119891119901119902
0
delay case with 119902 gt 119901
119902
119901(
119901
119902 minus 119901)
(119902119901)minus1
1198901minus(119901119902)
0119892119901119902
0
advanced case with 119902 gt 119901
1199011
119901(
119901
2 (1199011minus 119901)
)
(1199011119901)minus1
1198901minus(119901119901
1)
0119891119901119902
0+1199012
119901
times(119901
2 (1199012minus 119901)
)
(1199012119901)minus1
1198901minus(119901119901
2)
0119892119901119902
0
delay-advanced case (i)
1198901205780
01198911205781
01198921205782
0
2
prod
119894=0
120578minus120578119894
119894
delay-advanced case (ii) (93)
which shows the first claim in (13)
The proof that the function119876119899(119905) given in (19) (22) or (25)
satisfies the second claim in (13)Without loss of generality weprove this claim only in case (i) since for other cases the prooffollows analogously In this sense let119876
119899(119905) = 119891(119905)120591
119899(119905) Since
1198862119899+1
minus 1198862119899minus1
le 119879lowast 1198872119899+1
minus 1198872119899minus1
ge 119879lowast 1198862119899+2
minus 1198862119899le 119879lowast and
1198872119899+2
minus 1198872119899
ge 119879lowast where 119879
lowastgt 0 is the period of the function
119891(119905) we have 1198862119899minus1
le 1198861+(119899minus1)119879
lowastand 1198872119899minus1
ge 1198871+(119899minus1)119879
lowast
119899 isin N Hence
int
1198872119899minus1
1198862119899minus1
119876119899(119905) 119889119905
= int
1198872119899minus1
1198862119899minus1
119891 (119905) (119905 minus 1198862119899minus1
119905 minus 1198862119899minus1
+ 120591)
119901
119889119905
ge int
1198871+(119899minus1)119879
lowast
1198861+(119899minus1)119879lowast
119891 (119905) (119905 minus 1198861minus (119899 minus 1) 119879
lowast
119905 minus 1198861minus (119899 minus 1) 119879
lowast+ 120591
)
119901
119889119905
= int
1198871
1198861
119891 (119904 + (119899 minus 1) 119879lowast) (
119904 minus 1198861
119904 minus 1198861+ 120591
)
119901
119889119904
= int
1198871
1198861
119891 (119904) (119904 minus 1198861
119904 minus 1198861+ 120591
)
119901
119889119904
(94)
which proves that the integral on the left hand side does notdepend on 119899 isin N that is the second claim in (13) is shown on[1198862119899minus1
1198872119899minus1
] This claim follows in the same way on [1198862119899 1198872119899]
Thus the second claim in (13) is proved on [119886119899 119887119899]
Next to the end of this proof let 119909(119905) be a solu-tion of (1) In particular it implies that (119903(119905)119860(1199091015840(119905)))1015840 =
minus119861(119905 119909(119905) 1199091015840(119905)) minus 120582119865(119905 119909(120591(119905))) minus 120583119866(119905 119909(120590(119905))) + 120588119890(119905) It
together with assumptions (15) (16) (20) and (23) easilygives the next two statements
if 119909 (119905) gt 0 119909 (120591 (119905)) gt 0 119909 (120590 (119905)) gt 0
on (120591 (1198862119899minus1
) 120590 (1198872119899minus1
))
then 119909 (119905) satisfies 119903 (119905) 119860 (1199091015840
(119905)) le 0
on (120591 (1198862119899minus1
) 120590 (1198872119899minus1
)) 119899 ge 1198990
(95)
if 119909 (119905) lt 0 119909 (120591 (119905)) lt 0 119909 (120590 (119905)) lt 0
on (120591 (1198862119899) 120590 (119887
2119899))
then 119909 (119905) satisfies 119903 (119905) 119860 (1199091015840
(119905)) ge 0
on (120591 (1198862119899) 120590 (119887
2119899)) 119899 ge 119899
0
(96)
Now we need the following lemma
Discrete Dynamics in Nature and Society 13
Lemma 25 Let 120591119886119887(119905) and 120590
119886119887(119905) be defined by
120591119886119887(119905) = (
120591 (119905) minus 120591 (119886)
119905 minus 120591 (119886))
119901
120590119886119887(119905) = (
120590 (119887) minus 120590 (119905)
120590 (119887) minus 119905)
119901
119905 isin (119886 119887)
(97)
and let 119909 isin 1198622([1198790infin)R) be an arbitrary function If
(119903(119905)119860(1199091015840(119905)) le 0 for all 119905 isin (120591(119886) 120590(119887)) or (119903(119905)119860(1199091015840(119905)) ge 0
for all 119905 isin (120591(119886) 120590(119887)) then
119909 (120591 (119905))
119909 (119905)ge (120591119886119887(119905))1119901
119909 (120590 (119905))
119909 (119905)ge (120590119886119887(119905))1119901
119905 isin (119886 119887)
(98)
Since119860(V) is supposed to be odd and increasing functionjust before (3) and 119903(119905) satisfies (14) the proof of Lemma 25in the first case that is 119903(119905)119860(1199091015840(119905)) le 0 for all 119905 isin (120591(119886) 120590(119887))is the same as the proof of [9 Corollaries 17 and 18] But in thesecond case that is 119903(119905)119860(1199091015840(119905)) ge 0 for all 119905 isin (120591(119886) 120590(119887))the proof is as follows if previous inequality holds then119903(119905)119860(minus119909
1015840(119905)) le 0 for all 119905 isin (120591(119886) 120590(119887)) and therefore to
the function minus119909(119905) one can apply the first case of this lemmaand consequently one obtains
119909 (120591 (119905))
119909 (119905)=minus119909 (120591 (119905))
minus119909 (119905)ge (120591119886119887(119905))1119901
119909 (120590 (119905))
119909 (119905)=minus119909 (120590 (119905))
minus119909 (119905)ge (120590119886119887(119905))1119901
119905 isin (119886 119887)
(99)
which proves this lemma in the second caseNow combining statements (95) (96) and (98) one
easily obtains
if 119909 (119905) gt 0 119909 (120591 (119905)) gt 0 119909 (120590 (119905)) gt 0
on (120591 (1198862119899minus1
) 120590 (1198872119899minus1
))
then 119909 (119905) satisfies 119909 (120591 (119905))
119909 (119905)ge (120591119899(119905))1119901
on (1198862119899minus1
1198872119899minus1
) 119899 ge 1198990
(100)
if 119909 (119905) lt 0 119909 (120591 (119905)) lt 0 119909 (120590 (119905)) lt 0
on (120591 (1198862119899) 120590 (119887
2119899))
then 119909 (119905) satisfies 119909 (120590 (119905))
119909 (119905)ge (120590119899(119905))1119901
on (1198862119899 1198872119899) 119899 ge 119899
0
(101)
where 120591119899(119905) and 120590
119899(119905) are defined in (26)
The proof that 119909(119905) satisfies (7) and (8) In this proofwe frequently use assumptions (16) (20) and (23) andstatements (100) and (101) Also because of (15) and 119865(119905 119906) =
119891(119905)|119906|1199011 sgn(119906) 119866(119905 119906) = 119892(119905)|119906|
1199012 sgn(119906) in both cases
(100) and (101) we can simultaneously use
minus119890 (119905) (|119909 (119905)|119901minus1
119909 (119905))minus1
= |119890 (119905)| |119909 (119905)|minus119901
ge 0 on 119869119899
119865 (119905 119909 (120591 (119905))) (|119909 (119905)|119901minus1
119909 (119905))minus1
ge 119891 (119905) |119909 (120591 (119905))|1199011 |119909 (119905)|
minus119901ge 0 on 119869
119899
119866 (119905 119909 (120590 (119905))) (|119909 (119905)|119901minus1
119909 (119905))minus1
ge 119892 (119905) |119909 (120590 (119905))|1199012 |119909 (119905)|
minus119901ge 0 on 119869
119899
|119909 (120591 (119905))| |119909 (119905)|minus1=119909 (120591 (119905))
119909 (119905)
|119909 (120590 (119905))| |119909 (119905)|minus1=119909 (120590 (119905))
119909 (119905)on 119869119899
(102)
where 119869119899= (1198862119899minus1
1198872119899minus1
) in the case of (100) and 119869119899= (1198862119899 1198872119899)
in the case of (101)
(i) Delay or Advanced Case with 119902 = 119901 Since 119902 = 119901 we obtain
[120582119865 (119905 119909 (120591 (119905))) + 120583119866 (119905 119909 (120590 (119905))) minus 120588119890 (119905)]
times (|119909 (119905)|119901minus1
119909 (119905))minus1
ge [120582119891 (119905) |119909 (120591 (119905))|119901+ 120583119892 (119905) |119909 (120590 (119905))|
119901
+120588 |119890 (119905)| ] |119909 (119905)|minus119901
ge [120582119891 (119905) |119909 (120591 (119905))|119901+ 120583119892 (119905) |119909 (120590 (119905))|
119901] |119909 (119905)|
minus119901
= 120582119891 (119905) (119909 (120591 (119905))
119909 (119905))
119901
+ 120583119892 (119905) (119909 (120590 (119905))
119909 (119905))
119901
ge 120582119891 (119905) 120591119899(119905) + 120583119892 (119905) 120590
119899(119905) 119905 isin 119869
119899
(103)
where the functions 120591119899(119905) and 120590
119899(119905) are defined in (26)
(ii) Delay Case with 119902 gt 119901 In this part we use the nextelementary inequality
119883120574+ (120574 minus 1) 119884
120574ge 120574119883119884
120574minus1 120574 gt 1 119883 119884 ge 0 (104)
Since 119902 gt 119901 and using (104) especially for
120574 =119902
119901gt 1 119883 = (120582119891 (119905))
119901119902
|119909 (120591 (119905))|119901
119884 = (119901
119902 minus 119901120588 |119890 (119905)|)
119901119902
(105)
14 Discrete Dynamics in Nature and Society
for all 119905 isin 119869119899we obtain
[120582119865 (119905 119909 (120591 (119905))) + 120583119866 (119905 119909 (120590 (119905))) minus 120588119890 (119905)]
times (|119909 (119905)|119901minus1
119909 (119905))minus1
= [120582119865 (119905 119909 (120591 (119905))) minus 120588119890 (119905)] (|119909 (119905)|119901minus1
119909 (119905))minus1
ge [120582119891 (119905) |119909 (120591 (119905))|1199011 + 120588 |119890 (119905)|] |119909 (119905)|
minus119901
= [119883120574+ (120574 minus 1) 119884
120574] |119909 (119905)|
minus119901
ge119902
119901(120582119891 (119905))
119901119902
|119909 (120591 (119905))|119901
times (119901
119902 minus 119901120588 |119890 (119905)|)
(119901119902)((119902119901)minus1)
|119909 (119905)|minus119901
= 120582119901119902
1205881minus(119901119902)
119902
119901(
119901
119902 minus 119901)
1minus(119901119902)
|119890 (119905)|1minus(119901119902)
times (119891 (119905))119901119902
(119909 (120591 (119905))
119909 (119905))
119901
ge 119896 (120582 120583 120588)119902
119901(
119901
119902 minus 119901)
1minus(119901119902)
|119890 (119905)|1minus(119901119902)
times (119891 (119905))119901119902
120591119899(119905)
(106)
where the function 119896(120582 120583 120588) is from (18)
(iii) Advanced Case with 119902 gt 119901 Using the same line ofarguments as in the proof of the previous case for all 119905 isin 119869
119899
we obtain
[120582119865 (119905 119909 (120591 (119905))) + 120583119866 (119905 119909 (120590 (119905))) minus 120588119890 (119905)]
times (|119909 (119905)|119901minus1
119909 (119905))minus1
= [120583119866 (119905 119909 (120590 (119905))) minus 120588119890 (119905)] (|119909 (119905)|119901minus1
119909 (119905))minus1
ge 119896 (120582 120583 120588)119902
119901(
119901
119902 minus 119901)
1minus(119901119902)
|119890 (119905)|1minus(119901119902)
times (119892 (119905))119901119902
120590119899(119905)
(107)
where the function 119896(120582 120583 120588) is from (21)
(iv) Superlinear Delay-Advanced Case Since 1199011 1199012gt 119901 for
all 119905 isin 119869119899we obtain
[120582119865 (119905 119909 (120591 (119905))) + 120583119866 (119905 119909 (120590 (119905))) minus 120588119890 (119905)]
times (|119909 (119905)|119901minus1
119909 (119905))minus1
= [120582119865 (119905 119909 (120591 (119905))) minus120588
2119890 (119905)] (|119909 (119905)|
119901minus1119909 (119905))minus1
+ [120583119866 (119905 119909 (120590 (119905))) minus120588
2119890 (119905)] (|119909 (119905)|
119901minus1119909 (119905))minus1
ge [120582119891 (119905) |119909 (120591 (119905))|1199011 +
120588
2|119890 (119905)|] |119909 (119905)|
minus119901
+ [120583119892 (119905) |119909 (120590 (119905))|1199012 +
120588
2|119890 (119905)|] |119909 (119905)|
minus119901
(108)
Now just the same as in the proofs of previous delay andadvanced cases with 119902 gt 119901 and with the help of (104) inparticular for
120574 =1199011
119901gt 1 119883 = (120582119891 (119905))
1199011199011
|119909 (120591 (119905))|119901
119884 = (119901
1199011minus 119901
120588
2|119890 (119905)|)
1199011199011
(109)
we have
[120582119891 (119905) |119909 (120591 (119905))|1199011 +
120588
2 |119890 (119905)|] |119909 (119905)|
minus119901
= [119883120574+ (120574 minus 1) 119884
120574] |119909 (119905)|
minus119901
ge1199011
119901(120582119891 (119905))
1199011199011
|119909 (120591 (119905))|119901
times (119901
1199011minus 119901
120588
2|119890 (119905)|)
(1199011199011)((1199011119901)minus1)
|119909 (119905)|minus119901
= 12058211990111990111205881minus(119901119901
1)1199011
119901(
119901
2 (1199011minus 119901)
)
1minus(1199011199011)
times |119890 (119905)|1minus(119901119901
1)(119891 (119905))
1199011199011
(119909 (120591 (119905))
119909 (119905))
119901
ge 119896 (120582 120583 120588)1199011
119901(
119901
2 (1199011minus 119901)
)
1minus(1199011199011)
times |119890 (119905)|1minus(119901119901
1)(119891 (119905))
1199011199011
120591119899(119905)
(110)
where the function 119896(120582 120583 120588) is from (24) Analogously weshow that
[120583119892 (119905) |119909 (120590 (119905))|1199012 +
120588
2|119890 (119905)|] |119909 (119905)|
minus119901
ge 119896 (120582 120583 120588)1199012
119901(
119901
2 (1199012minus 119901)
)
1minus(1199011199012)
times |119890 (119905)|1minus(119901119901
2)(119891 (119905))
1199011199012
120590119899(119905)
(111)
Discrete Dynamics in Nature and Society 15
Summarizing previous calculation we obtain
[120582119865 (119905 119909 (120591 (119905))) + 120583119866 (119905 119909 (120590 (119905))) minus 120588119890 (119905)]
times (|119909 (119905)|119901minus1
119909 (119905))minus1
ge 119896 (120582 120583 120588)119876119899(119905) 119905 isin 119869
119899
(112)
where the function 119896(120582 120583 120588) is from (24)
(v) Supersublinear Delay-Advanced Case Since 1199011gt 119901 gt 119901
2
and the following well-known elementary inequality holds
12057801199060+ 12057811199061+ 12057821199062ge 1199061205780
01199061205781
11199061205782
2 120578119894ge 0 119906
119894ge 0 (113)
from 1205780 1205781 1205782isin (0 1) 120578
0+ 1205781+ 1205782= 1 and 119901
11205781+ 11990121205782= 119901
we obtain for all 119905 isin 119869119899 for all 119905 isin 119869
119899we obtain
[120582119865 (119905 119909 (120591 (119905))) + 120583119866 (119905 119909 (120590 (119905))) minus 120588119890 (119905)]
times (|119909 (119905)|119901minus1
119909 (119905))minus1
ge [120582119891 (119905) |119909 (120591 (119905))|1199011 + 120583119892 (119905) |119909 (120590 (119905))|
1199012 + 120588 |119890 (119905)|]
times |119909 (119905)|minus119901
= [1205781[120578minus1
1120582119891 (119905) |119909 (120591 (119905))|
1199011] + 120578
2[120578minus1
2120583119892 (119905) |119909 (120590 (119905))|
1199012]
+1205780[120578minus1
0120588 |119890 (119905)|]] |119909 (119905)|
minus119901
ge [120578minus1
1120582119891 (119905) |119909 (120591 (119905))|
1199011]1205781
[120578minus1
2120583119892 (119905) |119909 (120590 (119905))|
1199012]1205782
times [120578minus1
0120588 |119890 (119905)|]
1205780
|119909 (119905)|minus119901
= 120582120578112058312057821205881205780 |119890 (119905)|
1205780(119891 (119905))
1205781
(119892 (119905))1205782
times|119909 (120591 (119905))|
12057811199011
|119909 (119905)|12057811199011
|119909 (120590 (119905))|12057821199012
|119909 (119905)|12057821199012
2
prod
119894=0
120578minus120578119894
119894
= 119896 (120582 120583 120588) |119890 (119905)|1205780(119891 (119905))
1205781
(119892 (119905))1205782
times (119909 (120591 (119905))
119909 (119905))
12057811199011
(119909 (120590 (119905))
119909 (119905))
12057821199012 2
prod
119894=0
120578minus120578119894
119894
ge 119896 (120582 120583 120588) |119890 (119905)|1205780(119891 (119905))
1205781
(119892 (119905))1205782
(120591119899(119905))1205781(1199011119901)
times (120590119899(119905))1205782(1199012119901)
2
prod
119894=0
120578minus120578119894
119894
= 119896 (120582 120583 120588)119876119899(119905)
(114)
where 119896(120582 120583 120588) and 119876119899(119905) are given respectively in (24) and
(25) Thus it is shown that required condition (5) in thecases (i)ndash(iv) is fulfilled with respect to 119896(120582 120583 120588) and 119876
119899(119905)
determined by (18) (21) or (24) and (19) (22) or (25)In conclusion according to the previous observation we
see that all assumptions of Lemma 4 are fulfilled and henceLemma 4 proves Theorems 5 6 and 7
Conflict of Interests
The author declares that there is no conflict of interestsregarding the publication of this paper
References
[1] Y G Sun ldquoA note on Nasrrsquos and Wongrsquos papersrdquo Journal ofMathematical Analysis and Applications vol 286 no 1 pp 363ndash367 2003
[2] Y G Sun C H Ou and J S W Wong ldquoInterval oscillationtheorems for a second-order linear differential equationrdquo Com-puters amp Mathematics with Applications vol 48 no 10-11 pp1693ndash1699 2004
[3] S Murugadass E Thandapani and S Pinelas ldquoOscillationcriteria for forced second-order mixed type quasilinear delaydifferential equationsrdquo Electronic Journal of Differential Equa-tions vol 2010 article 73 9 pages 2010
[4] Y Bai and L Liu ldquoNew oscillation criteria for second-orderdelay differential equations with mixed nonlinearitiesrdquoDiscreteDynamics in Nature and Society vol 2010 Article ID 796256 9pages 2010
[5] A F Guvenilir andA Zafer ldquoSecond-order oscillation of forcedfunctional differential equations with oscillatory potentialsrdquoComputers amp Mathematics with Applications vol 51 no 9-10pp 1395ndash1404 2006
[6] A Zafer ldquoInterval oscillation criteria for second order super-half linear functional differential equations with delay andadvanced argumentsrdquoMathematische Nachrichten vol 282 no9 pp 1334ndash1341 2009
[7] A F Guvenilir ldquoInterval oscillation of second-order functionaldifferential equations with oscillatory potentialsrdquo NonlinearAnalysis Theory Methods amp Applications vol 71 no 12 ppe2849ndashe2854 2009
[8] T S Hassan L Erbe and A Peterson ldquoForced oscillation ofsecond order differential equations with mixed nonlinearitiesrdquoActa Mathematica Scientia B vol 31 no 2 pp 613ndash626 2011
[9] M Pasic ldquoNew oscillation criteria for second-order forcedquasilinear functional differential equationsrdquo Abstract andApplied Analysis vol 2013 Article ID 735360 12 pages 2013
[10] L H Erbe Q Kong and B G Zhang Oscillation Theory forFunctional-Differential Equations vol 190 Marcel Dekker NewYork NY USA 1995
[11] V Kolmanovskii and A Myshkis Introduction to the Theoryand Applications of Functional-Differential Equations vol 463Kluwer Academic Publishers Dordrecht The Netherlands1999
[12] R P Agarwal M Bohner and W-T Li Nonoscillation andOscillation Theory for Functional Differential Equations vol267 Marcel Dekker New York NY USA 2004
[13] L Erbe T Hassan and A Peterson ldquoOscillation of secondorder functional dynamic equationsrdquo International Journal ofDifference Equations vol 5 no 2 pp 175ndash193 2010
[14] B Baculıkova J Dzurina and Y V Rogovchenko ldquoOscillationof third order trinomial delay differential equationsrdquo AppliedMathematics and Computation vol 218 no 13 pp 7023ndash70332012
[15] R P Agarwal L Berezansky E Braverman and A Domoshnit-sky Nonoscillation Theory of Functional Differential Equationswith Applications Springer New York NY USA 2012
16 Discrete Dynamics in Nature and Society
[16] J Zhang ldquoVariational approach to solitary wave solution ofthe generalized Zakharov equationrdquo Computers ampMathematicswith Applications vol 54 no 7-8 pp 1043ndash1046 2007
[17] T Ozis and A Yıldırım ldquoApplication of Hersquos semi-inversemethod to the nonlinear Schrodinger equationrdquo Computers ampMathematics with Applications vol 54 no 7-8 pp 1039ndash10422007
[18] X-C Cai andM-S Li ldquoPeriodic solution of Jacobi elliptic equa-tions by Hersquos perturbation methodrdquo Computers amp Mathematicswith Applications vol 54 no 7-8 pp 1210ndash1212 2007
[19] S Lenci G Menditto and A M Tarantino ldquoHomoclinic andheteroclinic bifurcations in the non-linear dynamics of a beamresting on an elastic substraterdquo International Journal of Non-Linear Mechanics vol 34 no 4 pp 615ndash632 1999
[20] D-J Huang and H-Q Zhang ldquoLink between travelling wavesand first order nonlinear ordinary differential equation with asixth-degree nonlinear termrdquoChaos Solitons amp Fractals vol 29no 4 pp 928ndash941 2006
[21] A I Maimistov ldquoPropagation of an ultimately short electro-magnetic pulse in a nonlinear medium described by the fifth-order Duffing modelrdquo Optics and Spectroscopy vol 94 pp 251ndash257 2003
[22] M N Hamdan and N H Shabaneh ldquoOn the large amplitudefree vibrations of a restrained uniform beam carrying anintermediate lumpedmassrdquo Journal of Sound andVibration vol199 no 5 pp 711ndash736 1997
[23] J-H He ldquoSome asymptotic methods for strongly nonlinearequationsrdquo International Journal of Modern Physics B vol 20no 10 pp 1141ndash1199 2006
[24] J B Marion Classical Dynamics of Particles and Systems 1970[25] I Kovacic and M J Brennan The Duffing Equation Nonlinear
Oscillatos and their Behaviour John Wiley amp Sons 1st edition2011
[26] F C Moon Chaotic Vibrations An Introduction for AppliedScientists and Engineers John Wiley amp Sons New York NYUSA 2004
[27] J J Stoker Nonlinear Vibrations 1950[28] G Chen and Z Tao ldquoAmplitude-frequency relationship for the
relativistic oscillatorrdquoAASRI Procedia vol 1 pp 400ndash403 2012[29] R E Mickens Oscillations in Planar Dynamic Systems World
Scientific Publishing Singapore 1996[30] A Belendez T Belendez C Neipp A Hernandez and M
L Alvarez ldquoApproximate solutions of a nonlinear oscillatortypified as a mass attached to a stretched elastic wire by thehomotopy perturbation methodrdquo Chaos Solitions and Fractalsvol 39 pp 746ndash764 2009
[31] A Belendez E Fernandez R Fuentes J J Rodes and I PascualldquoHarmonic balancing approach to nonlinear oscillations of apunctual charge in the eletric field of charged ringrdquo PhysicsLetters A vol 373 pp 735ndash740 2009
[32] A Elıas-Zuniga ldquoExact solution of the cubic-quintic Duffingoscillatorrdquo Applied Mathematical Modelling vol 37 no 4 pp2574ndash2579 2013
[33] A Belendez M L Alvarez J Frances et al ldquoAnalytical approx-imate solutions for the cubic-quintic Duffing oscillator in termsof elementary functionsrdquo Journal of Applied Mathematics vol2012 Article ID 286290 16 pages 2012
[34] A Elıas-Zuniga OMartınez-Romero andR K Cordoba-DıazldquoApproximate solution for the Duffing-harmonic oscillator bythe enhanced cubication methodrdquo Mathematical Problems inEngineering vol 2012 Article ID 618750 12 pages 2012
[35] C W Lim B S Wu andW P Sun ldquoHigher accuracy analyticalapproximations to the Duffing-harmonic oscillatorrdquo Journal ofSound and Vibration vol 296 no 4-5 pp 1039ndash1045 2006
[36] J He ldquoSome new approaches to Duffing equation with stronglyand high order nonlinearity II parametrized perturbationtechniquerdquo Communications in Nonlinear Science amp NumericalSimulation vol 4 no 1 pp 81ndash83 1999
[37] V Marinca and N Herisanu ldquoPeriodic solutions for somestrongly nonlinear oscillations by Hersquos variational iterationmethodrdquo Computers amp Mathematics with Applications vol 54no 7-8 pp 1188ndash1196 2007
[38] W Lu and Y Liu ldquoVibration control for the primary resonanceof the Duffing oscillator by a time delay state feedbackrdquoInternational Journal of Nonlinear Science vol 8 no 3 pp 324ndash328 2009
[39] H Y Hu and Z H Wang Dynamics of Controlled MechanicalSystems with Delayed Feedback Springer 2002
[40] M Hamdi and M Belhaq ldquoControl of bistability in a delayedDuffing oscillatorrdquo Advances in Acoustics and Vibration vol2012 Article ID 872498 6 pages 2012
[41] V Ravichandran C Chinnathambi and S Rajasekar ldquoNonlin-ear resonance in Duffing oscillator with fixed and integrativetime-delayed feedbacksrdquoPramana Journal of Physics vol 78 pp347ndash360 2013
[42] X You and H Xu ldquoAnalytical approximations for the periodicmotion of theDuffing systemwith delayed feedbackrdquoNumericalAlgorithms vol 56 no 4 pp 561ndash576 2011
[43] Z Sun W Xu X Yang and T Fang ldquoInducing or suppressingchaos in a double-well Duffing oscillator by time delay feed-backrdquo Chaos Solitons and Fractals vol 27 pp 705ndash714 2006
[44] H Wang H Hu and Z Wang ldquoGlobal dynamics of a Duffingoscillator with delayed displacement feedbackrdquo InternationalJournal of Bifurcation and Chaos in Applied Sciences andEngineering vol 14 no 8 pp 2753ndash2775 2004
[45] J Chiasson and J J LoiseauApplications of Time Delay SystemsSpringer 2007
[46] M Lakshmanan andDV SenthilkumarDynamics of NonlinearTime-Delay Systems Springer 2010
[47] G Stepan T Insperger and R Szalai ldquoDelay parametricexcitation and the nonlinear dynamics of cutting processesrdquoInternational Journal of Bifurcation and Chaos in Applied Sci-ences and Engineering vol 15 no 9 pp 2783ndash2798 2005
[48] U van der Heiden and H-O Walther ldquoExistence of chaos incontrol systems with delayed feedbackrdquo Journal of DifferentialEquations vol 47 no 2 pp 273ndash295 1983
[49] Y G Sun and J S W Wong ldquoOscillation criteria for secondorder forced ordinary differential equations with mixed non-linearitiesrdquo Journal of Mathematical Analysis and Applicationsvol 334 no 1 pp 549ndash560 2007
[50] J Heagy and W L Ditto ldquoDynamics of a two-frequencyparametrically driven Duffing oscillatorrdquo Journal of NonlinearScience vol 1 no 4 pp 423ndash455 1991
[51] A B Belogortsev ldquoBifurcations of tori and chaos in thequasiperiodically forced Duffing oscillatorrdquoNonlinearity vol 5no 4 pp 889ndash897 1992
[52] M Belhaq and M Houssni ldquoQuasi-periodic oscillations chaosand suppression of chaos in a nonlinear oscillator driven byparametric and external excitationsrdquo Nonlinear Dynamics vol18 no 1 pp 1ndash24 1999
[53] S H Saker P Y H Pang and R P Agarwal ldquoOscillationtheorems for second order nonlinear functional differential
Discrete Dynamics in Nature and Society 17
equations with dampingrdquo Dynamic Systems and Applicationsvol 12 no 3-4 pp 307ndash321 2003
[54] I N Bronshtein K A Semendyayev G Musiol and HMuehligHandbook of Mathematics Springer 5th edition 2007
[55] M Pasic ldquoFite-Wintner-Leighton-type oscillation criteria forsecond-order differential equations with nonlinear dampingrdquoAbstract and Applied Analysis vol 2013 Article ID 852180 10pages 2013
Submit your manuscripts athttpwwwhindawicom
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Differential EquationsInternational Journal of
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Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
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Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
6 Discrete Dynamics in Nature and Society
solitarywave solutions of classic and generalizedZakharov equations of plasma physics (see [16])and of nonlinear Schrodinger equation (see[17]) also it is strongly connected with theJacobi elliptic equation (see [18])
(b) the cubic-quintic oscillator 11990910158401015840 + 1205962
0119909 + 120573
11199093+
12057321199095
= 0 is used as a model for the non-linear dynamics of a slender elastica (see [19])in nonlinear wave systems (see [20]) for thepropagation of a short electromagnetic pulsein a nonlinear medium (see [21]) and in theunimodal Duffing temporal problem (see [22])
(c) the cubic truly nonlinear oscillator 11990910158401015840 + 1205731199093=
0 models the motion of a ball bearing thatoscillates in a glass tube that is bent into acurve (see [23]) as well as the motion of a massattached to identical stretched elastic wires (see[24])
(d) the nonhomogeneous Duffing oscillator 11990910158401015840 +1205962
0119909 + 120573119909
3= 120588 cos(120596119905) describes various forced
vibrations of beams springs with nonlinearstiffness cables plates shells and optical fibresin electrical circuits in nonlinear isolators andso forth (see for instance [25 26])
(ii) the general Duffing-harmonic oscillator (with rationalor irrational nonlinear restoring-force) is given in(30) if 120583
1= 0 120573119894= 0 and 120588 = 0 the most known
subclasses of these oscillators are
(a) the classic Duffing-harmonic oscillator 11990910158401015840+
(12058311199093(1205832+ 12058331199092)) = 0 which models many
conservative nonlinear oscillatory systems see[27]
(b) the relativistic harmonic oscillator11990910158401015840+ (1205831119909radic1 + 1199092) = 0 see [28]
(c) the nonlinear oscillator11990910158401015840+119909minus(1205831119909radic1 + 1199092) =
0 1205831
isin [0 1] which is typified as a massattached to a stretched elastic wire see [29 30]
(d) the nonlinear oscillator 11990910158401015840
+
(1205831119909(radic(1 + 1199092 )
3
) = 0 which presentsnonlinear oscillations of a punctual charge inthe electric field of charged ring see [31]
Finding several explicit forms of periodic approximate solu-tions for these oscillators has been intensively studied lastyears by many authors see for instance [28 30 32ndash37] andalso the references therein
When 120582 = 0 and linear time delay feedbackΦ(119905 119909(119905minus120591)) =119909(119905 minus 120591) the following topics have been studied for varioustypes of Duffing oscillators with time delayed feedback in[38] authors constructed a low-order approximate solutionunder weak feedback gain parameter about the low- andhigh-order approximations see also [39] in [40] with 120588 = 0the Hopf bifurcation diagrams have been explored for theapproximate periodic solutions (amplitude versus time delay120591 and feedback gain 120582 versus time delay 120591) moreover in [41]
authors made an analysis on the effect of the control gainand time delay parameters on the amplitude of approximateperiod solution from the theoretical and numerical pointsof view see also [42] in [43] authors studied the chaoticbehaviour with respect to gains and time delay parameterssee also [44]
Equations under time delay control such as (34) (espe-cially with damped term) are used as a model for variouscontrolled physical mechanical and engineering systemswith time delays see for instance [39 45ndash48] and thereferences therein
Here (34) contains very general nonlinear time delayfeedback Φ(119905 119909(119905 minus 120591)) with Φ satisfying (33) and the lineartime delay feedback 119909(119905 minus 120591) is only a particular case ofit and to the best of our knowledge the previous topicsare not considered for (34) as yet Moreover with suchan Φ the oscillations of (34) can be taken under a doubteven with the linear time delay feedback (see the nature ofthe approximations given in [38 39]) Hence we can posethe following question under what conditions on equationrsquosparameters (34) is a nonlinear oscillator that is possessesonly oscillatory solutions An answer is given in the nextresult as an easy consequence of the parametrically excitedoscillations by Theorem 5
Theorem 15 Let 120591 isin (0 120587120596) and (33) hold Equation (34) isoscillatory in the next two cases
(i) 119902 = 119901 and 120582 is large enough(ii) 119902 gt 119901 120582 gt 0 120588 gt 0 and at least one of 120582 and 120588 is large
enough
Proof Let 119903(119905) equiv 1 119860(V) = |V|119901minus1 119865(119905 119906) = Φ(119905 119906) 119866(119905 119906) equiv0 119890(119905) = cos(120596119905) and
119861 (119905 119906 V) = 1205962
0119906 +
1205831|119906|1199031s119892119899 (119906)
(1205832+ 12058331199062)1199032
+
119898
sum
119894=1
120573119894(119905) |119906|
120572119894minus1119906 (35)
It is easy to check that all assumptions of Theorem 5 arefulfilled with respect to the sequence 119886
119899= minus1205872120596 + 119899120587120596 + 120591
and 119887119899= 1205872120596 + 119899120587120596 + 120591 where 119886
119899lt 119887119899since it is supposed
that 120591 lt 120587120596 Hence Theorem 5 proves this theorem
Remark 16 Even in the linear forced case (119890(119905) equiv 0) it isnot easy to establish the oscillations of all solutions since theoscillation and nonoscillation can occur simultaneously Themost simple and important example for the coincidence ofoscillation and nonoscillation is the following linear forceddifferential equation 11990910158401015840 + (2119905)119909
1015840+ 119909 = 2119905 119905 gt 0 that
allows an oscillatory solution 1199091(119905) = (3 sin 119905)119905 + 2119905 and a
nonoscillatory solution 1199092(119905) = 2119905 This is not possible in
the linear case with 119890(119905) equiv 0 because of Sturmrsquos separationtheorem
4 Parametrically Excited Oscillations andWell-Known Oscillation Criteria
In this section we would like to draw the readerrsquos attentionto the fact that the parametrically excited oscillations have
Discrete Dynamics in Nature and Society 7
been already appearing in some published papers on theoscillation of functional differential equations but only insome examples illustrating certain main oscillation criteriaHowever with the help of our main results in which theparametrically excited oscillations are studied in a generalsetting the equations from these examples are replaced withgeneral ones also having parameters 120582 and 120583
In [1] (see also [2 Example 31] with 120591 = 0 [3 Example31] and [4 Section 3]) the author considers the oscillationof the second-order delay differential equation
11990910158401015840
(119905) + 119891 (119905) |119909 (120591 (119905))|120574 sgn119909 (120591 (119905)) = 119890 (119905) (36)
in the linear case (120574 = 1) and the superlinear (120574 gt 1)In the linear case (analogously for the superlinear case see[1 Theorem 2]) the author proved the following oscillationcriterion In what follows we denote
119863 (119886 119887) = 119906 isin 1198621
([119886 119887] R) 119906 (119905) equiv 0 119906 (119886) = 119906 (119887) = 0
(37)
Theorem 17 ([1 Theorem 1]) Suppose that for any 119879 ge 0there exist constants 119886
1 1198871 1198862 1198872such that 119879 le 119886
1lt 1198871 119879 le
1198862lt 1198872 and 119891(119905) ge 0 on [120591(119886
1) 1198871] cup [120591(119886
2) 1198872] 119890(119905) le 0
on [120591(1198861) 1198871] and 119890(119905) ge 0 on [120591(119886
2) 1198872] If there exists 119906 isin
119863(119886119894 119887119894) 119894 = 1 2 such that
int
119887119894
119886119894
[1199062
(119905) 119891 (119905)120591 (119905) minus 120591 (119886
119894)
119905 minus 120591 (119886119894)
minus (1199061015840
(119905))2
]119889119905 ge 0 (38)
then (36) with 120574 = 1 is oscillatory
Previous criterion has been applied on the followingparticular equation
11990910158401015840
(119905) + 120582 sin (119905)1003816100381610038161003816100381610038161003816119909 (119905 minus
120587
4)
1003816100381610038161003816100381610038161003816
120574
times sgn 119909(119905 minus120587
4) = cos (119905) 119905 ge 0
(39)
where 120582 ge 0 and 120574 = 1 Applying Theorem 17 to (39) theauthor proved that (39) is oscillatory provided the followinginequality
120582int
119887119894
119886119894
sin2 (2119905) cos2 (2119905) sin (119905)119905 minus 119886119894
119905 minus 119886119894+ 1205874
119889119905 ge120587
2 (40)
holds for sufficiently large 120582 Thus the oscillation of (39) isexcited by the large enough parameter 120582 However accordingto Theorems 5 and 6 we are able to show that the nextparametric equation that corresponds to general equation(36)
11990910158401015840
(119905) + 120582119891 (119905) |119909 (120591 (119905))|120574 sgn119909 (120591 (119905)) = 119890 (119905) (41)
is oscillatory provided 120582 is large enough where 1199011= 1199012= 120574
120583 = 0 and 120588 = 1Next in [5] (see also [6ndash8]) the authors consider the
oscillation of the following class of second-order differentialequations with delay and advanced arguments
(119903 (119905) 1199091015840
(119905))1015840
+ 119891 (119905) |119909 (120591 (119905))|1199011 sgn119909 (120591 (119905))
+ 119892 (119905) |119909 (120590 (119905))|1199012 sgn119909 (120590 (119905)) = 119890 (119905) 119905 ge 0
(42)
where 1199011 1199012ge 1 When 119901
1= 1199012= 1 the authors prove the
following result (for other cases see [5Theorems 32 33 and34]
Theorem 18 ([5 Theorem 31]) Suppose that for any 119879 ge
0 there exist intervals [120591(1198861) 1198871] [120591(119886
2) 1198872] [1198881 120590(1198891)] and
[1198882 120590(1198892)] contained in [119879infin) such that 119886
1lt 1198871 1198862lt 1198872
1198881lt 1198891 1198882lt 1198892 and
119891 (119905) ge 0 119900119899 [120591 (1198861) 1198871] cup [120591 (119886
2) 1198872]
119892 (119905) ge 0 119900119899 [1198881 120590 (1198891)] cup [119888
2 120590 (1198892)]
119890 (119905) le 0 119900119899 [120591 (1198861) 1198871] cup [1198881 120590 (1198891)]
119890 (119905) ge 0 119900119899 [120591 (1198862) 1198872] cup [1198882 120590 (1198892)]
(43)
and 119888119894= 120591(119886
119894) 119889119894= 119886119894 and 119887
119894= 120590(119889
119894) 119894 = 1 2 If there exist
1199061isin 119863(119886
119894 119887119894) and 119906
2isin 119863(119888119894 119889119894) such that either
int
119887119894
119886119894
[1199062
1(119905) 119891 (119905)
120591 (119905) minus 120591 (119886119894)
119905 minus 120591 (119886119894)
minus (1199061015840
1(119905))2
119903 (119905)] 119889119905 ge 0 (44)
or
int
119889119894
119888119894
[1199062
2(119905) 119891 (119905)
120590 (119889119894) minus 120590 (119905)
120590 (119889119894) minus 119905
minus (1199061015840
2(119905))2
119903 (119905)] 119889119905 ge 0 (45)
for 119894 = 1 2 then (42) with 1199011= 1199012= 1 is oscillatory
As a consequence of this result it has been concluded thatthe particular equation
(119903 (119905) 1199091015840
(119905))1015840
+ 120582 sin (119905) 119909 (119905 minus 120587
12)
+ 120583 cos (119905) 119909 (119905 + 120587
6) = cos (2119905) 119905 ge 0
(46)
is oscillatory provided either 120582 or 120583 is large enough Howeverby following Theorems 5 and 6 one can obtain the sameconclusion for the following general equation associated with(42)
(119903 (119905) 1199091015840
(119905))1015840
+ 120582119891 (119905) |119909 (120591 (119905))|1199011 sgn119909 (120591 (119905))
+ 120583119892 (119905) |119909 (120590 (119905))|1199012 sgn119909 (120590 (119905)) = 119890 (119905)
(47)
Related observation can be done with [8 Example 33]and [9 Example 21] where the quasilinear second-orderfunctional differential equations have been considered It isleft to the reader
5 Some Open Questions and Comments
In this section we discuss some problems related to ourmainresults that are not studied here
(1) Quasiperiodic Case In the theory of nonlinear oscillatorsa particularly important case occurs when the periodiccoefficients in the oscillator do not have any common periodIt is called the quasiperiodic (or two-frequency) nonlinear
8 Discrete Dynamics in Nature and Society
oscillator and studied for instance in [50ndash52] Since inTheorems 5 6 and 7 we assume that the correspondingperiodic functions have a commonperiod it is natural to posethe next question
Open Question 1 Is it possible to derive sufficient conditionsfor the oscillation of (27) in the casewhen119891(119905) and119892(119905) (resp119891(119905) 119892(119905) and ℎ(119905)) are two (resp three) periodic functionsnot having a common period
(2) Equation with More Functional Arguments Next regard-ing some second-order functional differential equationsconsidered in the references of this paper more than twononlinear functional terms are appearing and thereforeinstead of main equation (1) and corresponding particularequation (27) considered inTheorems 5 6 and 7 we suggestthe following classes of equations
(119903 (119905) 119860 (1199091015840
(119905)))1015840
+ 119861 (119905 119909 (119905) 1199091015840
(119905))
+
1198981
sum
119896=1
120582119896119865119896(119905 119909 (120591
119896(119905)))
+
1198982
sum
119896=1
120583119896119866119896(119905 119909 (120590
119896(119905))) = 120588119890 (119905)
(48)
where 0 le 120591119896(119905) le 119905 lim
119905rarrinfin120591119896(119905) = infin 120590
119896(119905) ge 119905 119898
1 1198982isin
N and
(119903 (119905) 119860 (1199091015840
(119905)))1015840
+ 119861 (119905 119909 (119905) 1199091015840
(119905))
+
1198981
sum
119896=1
120582119896119891119896(119905)
1003816100381610038161003816119909 (119905 minus 120591119896)1003816100381610038161003816119901119896 sgn (119909 (119905 minus 120591
119896))
+
1198982
sum
119896=1
120583119896119892119896(119905)
1003816100381610038161003816119909 (119905 + 120590119896)1003816100381610038161003816119902119896 sgn (119909 (119905 + 120590
119896)) = 120588119890 (119905)
(49)
where 120582119896 120583119896 120588 120591119896 120590119896ge 0 and 119901
119896 119902119896gt 0
Comment We suggest the reader to enlarge the main resultsof this paper to (48) and (49)
(3) Damped Duffing Equation In the application the Duffingequation (34) is often appearing with the linear damped term1199091015840(119905) that is
11990910158401015840+ 11988901199091015840+ 1205962
0119909 + 120573119909
3+ 120582Φ (119909 (119905 minus 120591)) = 120588 cos (120596119905) (50)
where 1198890
is the damped coefficient which can in anactive way influence various behaviours of (50) Since119861(119905 119909(119905) 119909
1015840(119905)) = 119889
01199091015840(119905) does not satisfy the required
assumption (4) we are not able to apply our main results to(50) Hence we pose the following questionOpen Question 2 Is it possible to obtain the parametricallyexcited oscillation for (1) in the case when the damped term119861(119905 119906 V) satisfies a larger condition than (4) in which thelinear damped term 120573119909
1015840(119905) is especially included
(4) Functional Argument in Damped Term In a class of Duff-ing equations we have two time delayed feedback and hence
besides the control gain parameter 1205821another parameter 120582
2
appears the so-called velocity gain parameter Hence insteadof (34) one can consider
11990910158401015840+ 11988901199091015840+ 1205962
0119909 + 120573119909
3+ 1205821119909 (119905 minus 120591)
+ 12058221199091015840
(119905 minus 120591) = 120588 cos (120596119905) (51)
Therefore we suggest the following problem for further studyOpen Question 3 Is it possible to obtain the parametricallyexcited oscillation for the following more general functionaldifferential equation than (1) in which the functional argu-ment appears in the damped term too as follows
(119903 (119905) 119860 (1199091015840
(119905)))1015840
+ 119861 (119905 119909 (119905) 1199091015840
(119905)) + 1205821119865 (119905 119909 (120591 (119905)))
+ 1205822119867(119905 119909
1015840
(120591 (119905))) = 120588119890 (119905) 119905 ge 1199050
(52)or
(119903 (119905) 119860 (1199091015840
(119905)))1015840
+ 119861 (119905 119909 (119905) 1199091015840
(119905)) + 1205831119866 (119905 119909 (120590 (119905)))
+ 1205832119867(119905 119909
1015840
(120590 (119905))) = 120588119890 (119905) 119905 ge 1199050
(53)
About known oscillation criteria for the second-order func-tional differential equations having the functional argumentin the damped term we refer the reader to for instance [53]and the references therein
6 Proofs of Main Results
The proof of Lemma 1 is based on the following three stepstwo working forms of condition (6) (see Lemmas 19 and 20)the existence of an explosive solution of a suitable Riccatidifferential inequality (see Proposition 22) and a comparisonprinciple (see Proposition 24)
Lemma 19 (a necessary condition to (6)) Let 0 lt 119903(119905) le 1199030
on [1199050infin) If assumption (6) is fulfilled then there is a positive
real number 120576 such that1
120587lowast
int119869
min119901
(120576119903 (119905))120574minus1
120576119896 (120582 120583 120588)119876 (119905) 119889119905 ge 1 (54)
for all 120582 ge 1205820 120583 ge 120583
0 and 120588 ge 120588
0and some (120582
0 1205830 1205880) isin R3+
Proof Since 0 lt 119903(119905) le 1199030for 119905 ge 119905
0 we conclude that for
120576 = (119901
119903120574minus1
0119896 (120582 120583 120588)max
119905isin 119869119876 (119905)
)
1120574
(120582 120583 120588) isin R3
+
(55)
it holds that 119901(120576119903(119905))120574minus1
ge 119901(1205761199030)120574minus1
= 120576119896(120582 120583 120588)
max119905isin 119869119876(119905) ge 120576119896(120582 120583 120588)119876(119905) 119905 isin 119869 and hence
int119869
min119901
(120576119903 (119905))120574minus1
120576119896 (120582 120583 120588)119876 (119905) 119889119905
= 120576119896 (120582 120583 120588) int119869
119876 (119905) 119889119905
(56)
Discrete Dynamics in Nature and Society 9
On the other hand from (6) we observe
1
120587lowast
int119869
119876 (119905) 119889119905 ge1199031minus(1120574)
0
1199011120574[119896 (120582 120583 120588)]1minus(1120574)
(max119905isin 119869
119876 (119905))
1120574
(57)
which together with (55) and (56) gives
1
120587lowast
int119869
min119901
(120576119903 (119905))120574minus1
120576119896 (120582 120583 120588)119876 (119905) 119889119905
= 120576119896 (120582 120583 120588)1
120587lowast
int119869
119876 (119905) 119889119905
ge 1205761199031minus(1120574)
0
1199011120574[119896 (120582 120583 120588)]
1120574
(max119905isin 119869
119876 (119905))
1120574
= 1
(58)
for all 119899 ge 1198990 120582 ge 120582
0 120583 ge 120583
0 and 120588 ge 120588
0 It proves this
lemma
Lemma 20 (an equivalent condition to (54)) Assumption(54) is fulfilled if and only if there is a real number 120576 gt 0 and acontinuous function 119870(119905) ge 0 119905 isin 119869 such that
1198880= int119869
119870 (119905) 119889119905 gt 0119870 (119905)
1198880
le1
120587lowast
timesmin119901
(120576119903 (119905))120574minus1
120576119896 (120582 120583 120588)119876 (119905)
(59)
for all 119905 isin 119869 120582 ge 1205820 120583 ge 120583
0 and 120588 ge 120588
0and some (120582
0 1205830 1205880) isin
R3+
Proof This proof is very elementary Indeed if (54) holdsthen the function119870(119905) and number 119888
0 defined by
119870 (119905) =1
120587lowast
min119901
(120576119903 (119905))120574minus1
120576119896 (120582 120583 120588)119876 (119905)
1198880= int119869
119870 (119905) 119889119905
(60)
obviously satisfy 1198880
ge 1 and 119870(119905)1198880
le 119870(119905) = (1120587lowast)
min119901(120576119903(119905))120574minus1 120576119896(120582 120583 120588)119876(119905) which shows (59) Con-versely if (59) holds then integrating both sides of thesecond inequality in (59) we obtain
int119869
1
120587lowast
min119901
(120576119903 (119905))120574minus1
120576119896 (120582 120583 120588)119876 (119905) ge int119869
119870 (119905)
1198880
119889119905 = 1
(61)
which shows (54)
In conclusion according to previous two lemmas wesee that supposed condition (6) implies (59) which plays animportant role in the proof of the main results
The second step in the proof of Lemma 1 is to prove theexistence of a function 120596(119905) which blows up in the finitetime and satisfies a generalized Riccati differential lowerinequality we briefly present the existence and properties
of the so-called generalized tangent type function In whatfollows let 120587
lowastbe a positive real number defined in (3) Let us
remark that 120572(119904) = 119904120574 120574 gt 1 implies 120587
lowast= (2120587)(120574 sin(120587120574))
see for instance [54] and obviously for 120574 = 2wehave120587lowast= 120587
Lemma 21 Let 120572 [0infin) rarr [0infin) be a continuousfunction such that
int
infin
0
119889120591
1 + 120572 (120591)lt infin (62)
Then there is a real number 120587lowastgt 0 and a function 119911 = 119911(119904)
119911 isin 1198621((minus120587lowast2 120587lowast2)R) such that
119889119911
119889119904= 1 + 120572 (|119911 (119904)|) 119904 isin (minus
120587lowast
2120587lowast
2)
119911 (0) = 0
(63)
Moreover 119911(119904) is increasing and odd
lim119904rarr120587lowast2
119911 (119904) = infin 120587lowast=
2120587
120574 sin (120587120574)for 120572 (119904) = 119904
120574
120574 gt 1
(64)
In particular for 120572(119904) = 1199042 one can take 119911(119904) = tan(119904) and
120587lowast= 120587
Proof Let 119885 = 119885(119905) 119905 isin R be a function defined by
119885 (119905) = int
119905
0
1
1 + 120572 (|120591|)119889120591 119905 isin R (65)
The function 119885(119905) is well defined since 120572(119904) is positive andcontinuous on [0infin) 119885(119905) is increasing and odd functionand
119889119885
119889119905=
1
1 + 120572 (|119905|) 119905 isin R
119885 (0) = 0 119885 isin 1198621
(RR)
(66)
Moreover because of (62) there is a real number 120587lowastgt 0 such
that120587lowast
2= int
infin
0
119889120591
1 + 120572 (120591) (67)
Thus 119885 R rarr (minus120587lowast2 120587lowast2) and there exists an inverse
function 119885minus1 = 119885minus1(119904) of the original function 119885 = 119885(119905) and
119885minus1
(minus120587lowast2 120587lowast2) rarr R Also from 119885(119885
minus1(119904)) = 119904 and
119889119885119889119905 = 0 onR we also derive that119889119885minus1119889119904 = 0 on its domain(minus120587lowast2 120587lowast2) and
119889119885
119889119905(119885minus1
(119904)) =1
(119889119885minus1119889119904) 119904 isin (minus
120587lowast
2120587lowast
2) (68)
Putting 119905 = 119885minus1(119904) for 119904 isin (minus120587
lowast2 120587lowast2) into (66) and using
(68) we easily obtain
119889119885minus1
119889119904= 1 + 120572 (
10038161003816100381610038161003816119885minus1
(119904)10038161003816100381610038161003816) 119904 isin (minus
120587lowast
2120587lowast
2)
119885minus1
(0) = 0 119885minus1isin 1198621((minus
120587lowast
2120587lowast
2) R)
(69)
10 Discrete Dynamics in Nature and Society
Moreover from (67) we have lim119904rarr120587lowast2119885minus1(119904) = 119885
minus1
(lim119905rarrinfin
119885(119905)) = lim119905rarrinfin
119885minus1119885(119905) = lim
119905rarrinfin119905 = infin Thus
if we set 119911(119904) = 119885minus1(119904) then previous two statements and
(67) prove this lemma
Next we prove the main result of this section
Proposition 22 Let (2) and (6) hold where 119869 = (119886 119887) Let 120576 gt0 be a real number and let119870(119905) ge 0 119905 isin [119886 119887] be a continuousfunction both obtained in Lemma 20 Let 120587
lowastbe from (3) and
1198880from (59) and let 119877
119886isin R be an arbitrary real number If
119911 = 119911(119904) is the generalized tangens function defined in (63)and 119881(119905) is a function defined by
119881 (119905) =120587lowast
1198880
int
119905
119886
119870 (120591) 119889120591 + 119911minus1(119877119886) 119905 isin [119886 119887] (70)
then there is a 119879lowast119886isin [119886 119887) such that
119881 (119879lowast
119886) =
120587lowast
2 119881 ([119886 119879
lowast
119886)) sub (minus
120587lowast
2120587lowast
2) (71)
Moreover for a function 120596(119905) defined by120596 (119905) = 119911 (119881 (119905)) 119905 isin [119886 119879
lowast
119886) (72)
one has 120596(119886) = 119877119886 lim119905rarr119879
lowast
119886
120596(119905) = infin and
119889120596
119889119905le
119901
(120576119903 (119905))120574minus1
120572 (1003816100381610038161003816120596 (119905)
1003816100381610038161003816)
+ 120576119896 (120582 120583 120588)119876 (119905) 119905 isin (119886 119879lowast
119886)
(73)
where the numbers 119901 and 120574 are from (3) and the functions119896(120582 120583 120588) and 119876(119905) are from (6)
Proof Under assumptions (2) and (6) and because of Lem-mas 19 and 20 we obtain 120576 gt 0 and 119870(119905) gt 0 119905 isin [119886 119887]satisfying inequality (59)
Next since 119911minus1(119877119886) isin (minus120587
lowast2 120587lowast2) (see Lemma 21)
from (70) we directly obtain
119881 (119886) = 119911minus1(119877119886) lt
120587lowast
2 119881 (119887) = 120587
lowast+ 119911minus1(119877119886) gt
120587lowast
2
(74)Since 119870 isin 119862([119886 119887] [0infin)) we obtain 119881 isin 119862([119886 119887]R) cap
1198621((119886 119887)R) and from (74) we observe that there exist
numbers 119879lowast119886isin (119886 119887) such that119881(119879lowast
119886) = 120587lowast2 Also119870(119905)119888
0ge
0 gives 119881([119886 119879lowast119886)) sub (minus120587
lowast2 120587lowast2) which proves statement
(71) Moreover it together with Lemma 21 and (72) provesthat
lim119905rarr119879
lowast
119886
120596 (119905) = lim119905rarr119879
lowast
119886
119911 (119881 (119905)) = 119911 (120587lowast
2) = infin (75)
Next according to (59) (63) and (72) we make thefollowing calculation on the interval [119886 119879lowast
119886)
1205961015840
(119905) = 1199111015840
(119881 (119905)) 1198811015840
(119905) = [1 + 120572 (|119911 (119881 (119905))|)]120587lowast
1198880
119870 (119905)
= [1 + 120572 (1003816100381610038161003816120596 (119905)
1003816100381610038161003816)]120587lowast
1198880
119870 (119905)
le119901
(120576119903 (119905))120574minus1
120572 (1003816100381610038161003816120596 (119905)
1003816100381610038161003816) + 120576119896 (120582 120583 120588)119876 (119905)
(76)
Thus all assertions of this proposition are proved
Remark 23 In the proof of the main result the number 119877119886
is determined by 119877119886= 120596(119886) where 120596(119905) denotes a function
associated with a nonoscillatory solution and it is given by(84) below
The third step in the proof of Lemma 1 is to show thefollowing pointwise comparison principle for the functions120596and120596 satisfying respectively the lower and upper differentialinequalities (73) and
119889120596
119889119905ge
119901
(120576119903 (119905))120574minus1
120572 (|120596 (119905)|) + 120576119896 (120582 120583 120588)119876 (119905) 119905 isin (119886 119887)
(77)
Proposition 24 Let [119886 119887) sub [1199050infin) be an arbitrary inter-
val One supposes that all coefficients of Riccati differentialinequalities (73) and (77) are continuous and strictly positivefunctions Let 120596 120596 isin 119862
1([119886 119887)R) be two functions satisfying
respectively (73) and (77) on the interval [119886 119887) Then
120596 (119886) le 120596 (119886) 119894119898119901119897119894119890119904 120596 (119905) le 120596 (119905) forall119905 isin [119886 119887) (78)
Proof Let119867(119905 119906) be a function defined by
119867(119905 119906) =119901
(120576119903 (119905))120574minus1
120572 (|119906|) + 120576119896 (120582 120583 120588)119876 (119905)
119905 isin [119886 119887) 119906 isin R
(79)
Let 119868 sub [119886 119887) and 119872 gt 0 be arbitrary For any two 1199061
1199062 minus119872 le 119906
1lt 1199062le 119872 let 119868
12be an interval defined
by 11986812
= (min|1199061| |1199062|max|119906
1| |1199062|) Since 120572(119904) is a 1198621-
function on [0infin) we know by the Lagrange mean valuetheorem applied on 119868
12that there is a 120585 isin 119868
12such that
120572 (10038161003816100381610038161199062
1003816100381610038161003816) minus 120572 (10038161003816100381610038161199061
1003816100381610038161003816)
1199062minus 1199061
le
1003816100381610038161003816120572 (10038161003816100381610038161199062
1003816100381610038161003816) minus 120572 (10038161003816100381610038161199061
1003816100381610038161003816)1003816100381610038161003816
1199062minus 1199061
=100381610038161003816100381610038161205721015840
(120585)10038161003816100381610038161003816
100381610038161003816100381610038161003816100381610038161199062
1003816100381610038161003816 minus10038161003816100381610038161199061
10038161003816100381610038161003816100381610038161003816
1199062minus 1199061
le100381610038161003816100381610038161205721015840
(120585)10038161003816100381610038161003816
le max119904isin11986812
100381610038161003816100381610038161205721015840
(119904)10038161003816100381610038161003816
(80)
since ||1199062| minus |1199061|| le 119906
2minus 1199061 Hence for any 119905 isin 119868 and 119906
1 1199062
minus119872 le 1199061lt 1199062le 119872 we have
119867(119905 1199062) minus 119867 (119905 119906
1)
1199062minus 1199061
= 1205880(119905)
120572 (10038161003816100381610038161199062
1003816100381610038161003816) minus 120572 (10038161003816100381610038161199061
1003816100381610038161003816)
1199062minus 1199061
le 1205880(119905)max119904isin11986812
100381610038161003816100381610038161205721015840
(119904)10038161003816100381610038161003816= 1198710(119905)
(81)
Thus the function119867(119905 119906) from (79) satisfies required condi-tion of [55 Lemma 19] and applying it to (73) and (77) weprove this proposition
Proof of Lemma 1 On the contrary let 119909(119905) be a solution of(1) such that
119909 (119905) = 0 on (120591 (120591 (119886)) 120590 (120590 (119889))) (82)
Discrete Dynamics in Nature and Society 11
that is 119909(119905) gt 0 on (120591(120591(119886)) 120590(120590(119889))) or 119909(119905) lt 0 on(120591(120591(119886)) 120590(120590(119889))) since 119909(119905) is a continuous function on[1199050infin) Let for instance
119909 (119905) gt 0 on (120591 (120591 (119886)) 120590 (120590 (119889))) (83)
Another case can be analogously treated let us see thecomment at the end of this proof In particular from (83)we have 119909(119905) gt 0 on (120591(120591(119886)) 120590(120590(119887))) which implies (since120591(119905) and 120590(119905) are increasing functions) 119909(119904) gt 0 for all 119904 isin
(120591(119886) 120590(119887)) cup (120591(120591(119886)) 120591(120590(119887))) cup (120590(120591(119886)) 120590(120590(119887))) whichyields 119909(119905) gt 0 119909(120591(119905)) gt 0 and 119909(120590(119905)) gt 0 on (120591(119886) 120590(119887))Hence by assumption (7) we may use inequality (5) on theinterval (119886 119887)
Firstly we show that the following classic Riccati transfor-mation of 119909(119905)
120596 (119905) = minus120576119903 (119905) 119860 (119909
1015840(119905))
|119909 (119905)|119901minus1
119909 (119905) 119905 isin (119886 119887) 120576 gt 0 (84)
satisfies upper Riccati differential inequality (77) Let usremark that from (1) we have in particular
minus(119903 (119905) 119860 (1199091015840
(119905)))1015840
= 119861 (119905 119909 (119905) 1199091015840
(119905)) + 120582119865 (119905 119909 (120591 (119905)))
+ 120583119866 (119905 119909 (120590 (119905))) minus 120588119890 (119905) 119905 ge 1199050
(85)
Taking the first derivative on both sides of (84) and usingassumptions (3) (4) and (5) as well as equality (85) and(|119909(119905)|
119901minus1119909(119905))1015840
= 119901|119909(119905)|119901minus1
1199091015840(119905) we obtain
119889120596
119889119905= 120576119901 119903 (119905)
119860 (1199091015840(119905)) 1199091015840(119905)
|119909 (119905)|119901+1
minus 1205761
|119909 (119905)|119901minus1
119909 (119905)(119903 (119905) 119860 (119909
1015840
(119905)))1015840
= 120576119901119903 (119905)119860 (1199091015840(119905)) 1199091015840(119905)
|119909 (119905)|119901+1
+120576
|119909 (119905)|119901minus1
119909 (119905)
times [120582119861 (119905 119909 (119905) 1199091015840
(119905)) + 119865 (119905 119909 (120591 (119905)))
+120583119866 (119905 119909 (120590 (119905))) minus 120588119890 (119905) ]
ge 120576119901119903 (119905)119860 (1199091015840(119905)) 1199091015840(119905)
|119909 (119905)|119901+1
+120576
|119909 (119905)|119901minus1
119909 (119905)
times [120582119865 (119905 119909 (120591 (119905))) + 120583119866 (119905 119909 (120590 (119905))) minus 120588119890 (119905)]
ge 120576119901119903 (119905) 120572(
10038161003816100381610038161003816119860 (1199091015840(119905))
10038161003816100381610038161003816
|119909 (119905)|119901
) + 120576119896 (120582 120583 120588)119876 (119905)
= 120576119901119903 (119905) 120572 (|120596 (119905)|
120576119903 (119905)) + 120576119896 (120582 120583 120588)119876 (119905)
ge119901
(120576119903 (119905))120574minus1
120572 (|120596 (119905)|) + 120576119896 (120582 120583 120588)119876 (119905) 119905 isin (119886 119887)
(86)
Thus according to inequality (5) it is shown that if 119909(119905) isa solution of (1) which satisfies (83) then the function 120596(119905)
defined by (84) satisfies the Riccati differential inequality(77) and 120596 isin 119862((119886 119887)R) On the other hand let 119877
119886be a
real number defined by 119877119886= 120596(119886) According to (6) and
Lemma 19 we obtain (54) which together with Lemma 20ensures that we may use Proposition 22 for such chosen realnumber 119877
119886 Hence we obtain a function 120596(119905) defined by (72)
which satisfies the lower Riccati differential inequality (73) on[119886 119879lowast
119886) 119879lowast119886isin (119886 119887) such that 120596(a) = 119877
119886and lim
119905rarr119879lowast
119886
120596(119905) =
infin Therefore by 120596(119886) = 119877119886= 120596(119886) and Proposition 24 we
conclude that lim119905rarr119879
lowast
119886
120596(119905) = infin too which is a contradictionwith the above conclusion saying that 120596 isin 119862((119886 119887)R) Thushypothesis (82) is not true and consequently Lemma 1 isshown
For the analogous case 119909(119905) lt 0 on (120591(120591(119886)) 120590(120590(119889))) wealso have 119909(119905) lt 0 on (120591(120591(119888)) 120590(120590(119889))) which implies (since120591(119905) and 120590(119905) are increasing functions)
119909 (119904) lt 0 forall119904 isin (120591 (119888) 120590 (119889)) cup (120591 (120591 (119888)) 120591 (120590 (119889)))
cup (120590 (120591 (119888)) 120590 (120590 (119889)))
(87)
which yields 119909(119905) lt 0 119909(120591(119905)) lt 0 and 119909(120590(119905)) lt 0 on(120591(119888) 120590(119889)) Now we can repeat the preceding procedure buton interval (119888 119889) and using (8) instead of (119886 119887) and (7)
Proof of Lemma 2 From assumption (10) we obtain the exis-tence of an 119899
0isin N such that
int
119887119899
119886119899
119876119899(119905) 119889119905 ge
1198880
2( max119905isin[119886119899 119887119899]
119876119899(119905))
1120574
119899 ge 1198990 (88)
that is
2
1198880
int
119887119899
119886119899
119876119899(119905) 119889119905 ge ( max
119905isin[119886119899 119887119899]119876119899(119905))
1120574
119899 ge 1198990 (89)
Now from (9) and previous inequality we deduce that forlarge enough 120582 120583 120588 and 119899
1199011120574
1199031minus1120574
0
[119896 (120582 120583 120588)]1minus1120574
120587lowast
int
119887119899
119886119899
119876119899(119905) 119889119905
ge2
1198880
int
119887119899
119886119899
119876119899(119905) 119889119905 ge ( max
119905isin[119886119899 119887119899]119876119899(119905))
1120574
(90)
which shows (6) Thus all assumptions of Lemma 1 arefulfilled and hence Lemma 2 immediately follows fromLemma 1
Proof of Lemma 3 Obviously assumption (11) is a particularcase of assumption (9) Hence this proof is very similar tothe proof of Lemma 2 and so it is left to the reader
Proof of Lemma 4 It is clear that from assumption (13) weobtain
1
(max119905isin[119886119899119887119899]119876119899(119905))1120574
int
119887119899
119886119899
119876119899(119905) 119889119905 ge
1198881
1198621120574
0
gt 0 forall119899 ge 1198990
(91)
12 Discrete Dynamics in Nature and Society
Thus hypothesis (12) is fulfilled and therefore Lemma 3proves this lemma
Proof of Theorems 5 6 and 7 This proof is based onLemma 4 In order to simplify notation in many placesin this proof we set 120591(119905) = 119905 minus 120591 and 120590(119905) = 119905 + 120590 Sinceassumptions (2) (3) and (4) have been already supposed inTheorems 5 6 and 7 in order to prove these theorems byLemma 4 we are going to show that the functions 119896(120582 120583 120588)and 119876
119899(119905) explicitly given respectively in (18) (21) or (24)
and (19) (22) or (25) satisfy required conditions (11) and(13) respectively and that every solution 119909(119905) of (27) satisfiesconditions (7) and (8) with respect to functions 119896(120582 120583 120588)and 119876
119899(119905) where 119886 = 119886
2119899minus1 119887 = 119887
2119899minus1 119888 = 119886
2119899 and 119889 = 119887
2119899
The proof that the function 119896(120582 120583 120588) given in (18) (21) or(24) satisfies (11) Passing to the limit in (18) (21) or (24) it isvery simple to show (11)
The proof that the function 119876119899(119905) given in (19) (22) or
(25) satisfies the first claim in (13) From (25) we immediatelyobtain
1003816100381610038161003816120591119899 (119905)1003816100381610038161003816 =
100381610038161003816100381610038161003816100381610038161003816
(119905 minus 119886119899
119905 minus 119886119899+ 120591
)
119901100381610038161003816100381610038161003816100381610038161003816
le 1
1003816100381610038161003816120590119899 (119905)1003816100381610038161003816 =
100381610038161003816100381610038161003816100381610038161003816
(119887119899minus 119905
119887119899minus 119905 + 120590
)
119901100381610038161003816100381610038161003816100381610038161003816
le 1 forall119899 isin N
(92)
Next by assumptions of this corollary we can conclude thatthere are three positive constants 119891
0 1198920 1198900such that |119891(119905)| le
1198910and |119892(119905)| le 119892
0on [1199050infin) in cases (i) and (ii) and
|119890(119905)| le 1198900on [1199050infin) in cases (iii) and (iv) Putting previous
inequalities into (19) (22) or (25) for all 119899 isin N and 119905 isin
[1199050infin) it holds that
1003816100381610038161003816119876119899 (119905)1003816100381610038161003816 le
119902
119901(
119901
119902 minus 119901)
(119902119901)minus1
1198901minus(119901119902)
0119891119901119902
0
delay case with 119902 gt 119901
119902
119901(
119901
119902 minus 119901)
(119902119901)minus1
1198901minus(119901119902)
0119892119901119902
0
advanced case with 119902 gt 119901
1199011
119901(
119901
2 (1199011minus 119901)
)
(1199011119901)minus1
1198901minus(119901119901
1)
0119891119901119902
0+1199012
119901
times(119901
2 (1199012minus 119901)
)
(1199012119901)minus1
1198901minus(119901119901
2)
0119892119901119902
0
delay-advanced case (i)
1198901205780
01198911205781
01198921205782
0
2
prod
119894=0
120578minus120578119894
119894
delay-advanced case (ii) (93)
which shows the first claim in (13)
The proof that the function119876119899(119905) given in (19) (22) or (25)
satisfies the second claim in (13)Without loss of generality weprove this claim only in case (i) since for other cases the prooffollows analogously In this sense let119876
119899(119905) = 119891(119905)120591
119899(119905) Since
1198862119899+1
minus 1198862119899minus1
le 119879lowast 1198872119899+1
minus 1198872119899minus1
ge 119879lowast 1198862119899+2
minus 1198862119899le 119879lowast and
1198872119899+2
minus 1198872119899
ge 119879lowast where 119879
lowastgt 0 is the period of the function
119891(119905) we have 1198862119899minus1
le 1198861+(119899minus1)119879
lowastand 1198872119899minus1
ge 1198871+(119899minus1)119879
lowast
119899 isin N Hence
int
1198872119899minus1
1198862119899minus1
119876119899(119905) 119889119905
= int
1198872119899minus1
1198862119899minus1
119891 (119905) (119905 minus 1198862119899minus1
119905 minus 1198862119899minus1
+ 120591)
119901
119889119905
ge int
1198871+(119899minus1)119879
lowast
1198861+(119899minus1)119879lowast
119891 (119905) (119905 minus 1198861minus (119899 minus 1) 119879
lowast
119905 minus 1198861minus (119899 minus 1) 119879
lowast+ 120591
)
119901
119889119905
= int
1198871
1198861
119891 (119904 + (119899 minus 1) 119879lowast) (
119904 minus 1198861
119904 minus 1198861+ 120591
)
119901
119889119904
= int
1198871
1198861
119891 (119904) (119904 minus 1198861
119904 minus 1198861+ 120591
)
119901
119889119904
(94)
which proves that the integral on the left hand side does notdepend on 119899 isin N that is the second claim in (13) is shown on[1198862119899minus1
1198872119899minus1
] This claim follows in the same way on [1198862119899 1198872119899]
Thus the second claim in (13) is proved on [119886119899 119887119899]
Next to the end of this proof let 119909(119905) be a solu-tion of (1) In particular it implies that (119903(119905)119860(1199091015840(119905)))1015840 =
minus119861(119905 119909(119905) 1199091015840(119905)) minus 120582119865(119905 119909(120591(119905))) minus 120583119866(119905 119909(120590(119905))) + 120588119890(119905) It
together with assumptions (15) (16) (20) and (23) easilygives the next two statements
if 119909 (119905) gt 0 119909 (120591 (119905)) gt 0 119909 (120590 (119905)) gt 0
on (120591 (1198862119899minus1
) 120590 (1198872119899minus1
))
then 119909 (119905) satisfies 119903 (119905) 119860 (1199091015840
(119905)) le 0
on (120591 (1198862119899minus1
) 120590 (1198872119899minus1
)) 119899 ge 1198990
(95)
if 119909 (119905) lt 0 119909 (120591 (119905)) lt 0 119909 (120590 (119905)) lt 0
on (120591 (1198862119899) 120590 (119887
2119899))
then 119909 (119905) satisfies 119903 (119905) 119860 (1199091015840
(119905)) ge 0
on (120591 (1198862119899) 120590 (119887
2119899)) 119899 ge 119899
0
(96)
Now we need the following lemma
Discrete Dynamics in Nature and Society 13
Lemma 25 Let 120591119886119887(119905) and 120590
119886119887(119905) be defined by
120591119886119887(119905) = (
120591 (119905) minus 120591 (119886)
119905 minus 120591 (119886))
119901
120590119886119887(119905) = (
120590 (119887) minus 120590 (119905)
120590 (119887) minus 119905)
119901
119905 isin (119886 119887)
(97)
and let 119909 isin 1198622([1198790infin)R) be an arbitrary function If
(119903(119905)119860(1199091015840(119905)) le 0 for all 119905 isin (120591(119886) 120590(119887)) or (119903(119905)119860(1199091015840(119905)) ge 0
for all 119905 isin (120591(119886) 120590(119887)) then
119909 (120591 (119905))
119909 (119905)ge (120591119886119887(119905))1119901
119909 (120590 (119905))
119909 (119905)ge (120590119886119887(119905))1119901
119905 isin (119886 119887)
(98)
Since119860(V) is supposed to be odd and increasing functionjust before (3) and 119903(119905) satisfies (14) the proof of Lemma 25in the first case that is 119903(119905)119860(1199091015840(119905)) le 0 for all 119905 isin (120591(119886) 120590(119887))is the same as the proof of [9 Corollaries 17 and 18] But in thesecond case that is 119903(119905)119860(1199091015840(119905)) ge 0 for all 119905 isin (120591(119886) 120590(119887))the proof is as follows if previous inequality holds then119903(119905)119860(minus119909
1015840(119905)) le 0 for all 119905 isin (120591(119886) 120590(119887)) and therefore to
the function minus119909(119905) one can apply the first case of this lemmaand consequently one obtains
119909 (120591 (119905))
119909 (119905)=minus119909 (120591 (119905))
minus119909 (119905)ge (120591119886119887(119905))1119901
119909 (120590 (119905))
119909 (119905)=minus119909 (120590 (119905))
minus119909 (119905)ge (120590119886119887(119905))1119901
119905 isin (119886 119887)
(99)
which proves this lemma in the second caseNow combining statements (95) (96) and (98) one
easily obtains
if 119909 (119905) gt 0 119909 (120591 (119905)) gt 0 119909 (120590 (119905)) gt 0
on (120591 (1198862119899minus1
) 120590 (1198872119899minus1
))
then 119909 (119905) satisfies 119909 (120591 (119905))
119909 (119905)ge (120591119899(119905))1119901
on (1198862119899minus1
1198872119899minus1
) 119899 ge 1198990
(100)
if 119909 (119905) lt 0 119909 (120591 (119905)) lt 0 119909 (120590 (119905)) lt 0
on (120591 (1198862119899) 120590 (119887
2119899))
then 119909 (119905) satisfies 119909 (120590 (119905))
119909 (119905)ge (120590119899(119905))1119901
on (1198862119899 1198872119899) 119899 ge 119899
0
(101)
where 120591119899(119905) and 120590
119899(119905) are defined in (26)
The proof that 119909(119905) satisfies (7) and (8) In this proofwe frequently use assumptions (16) (20) and (23) andstatements (100) and (101) Also because of (15) and 119865(119905 119906) =
119891(119905)|119906|1199011 sgn(119906) 119866(119905 119906) = 119892(119905)|119906|
1199012 sgn(119906) in both cases
(100) and (101) we can simultaneously use
minus119890 (119905) (|119909 (119905)|119901minus1
119909 (119905))minus1
= |119890 (119905)| |119909 (119905)|minus119901
ge 0 on 119869119899
119865 (119905 119909 (120591 (119905))) (|119909 (119905)|119901minus1
119909 (119905))minus1
ge 119891 (119905) |119909 (120591 (119905))|1199011 |119909 (119905)|
minus119901ge 0 on 119869
119899
119866 (119905 119909 (120590 (119905))) (|119909 (119905)|119901minus1
119909 (119905))minus1
ge 119892 (119905) |119909 (120590 (119905))|1199012 |119909 (119905)|
minus119901ge 0 on 119869
119899
|119909 (120591 (119905))| |119909 (119905)|minus1=119909 (120591 (119905))
119909 (119905)
|119909 (120590 (119905))| |119909 (119905)|minus1=119909 (120590 (119905))
119909 (119905)on 119869119899
(102)
where 119869119899= (1198862119899minus1
1198872119899minus1
) in the case of (100) and 119869119899= (1198862119899 1198872119899)
in the case of (101)
(i) Delay or Advanced Case with 119902 = 119901 Since 119902 = 119901 we obtain
[120582119865 (119905 119909 (120591 (119905))) + 120583119866 (119905 119909 (120590 (119905))) minus 120588119890 (119905)]
times (|119909 (119905)|119901minus1
119909 (119905))minus1
ge [120582119891 (119905) |119909 (120591 (119905))|119901+ 120583119892 (119905) |119909 (120590 (119905))|
119901
+120588 |119890 (119905)| ] |119909 (119905)|minus119901
ge [120582119891 (119905) |119909 (120591 (119905))|119901+ 120583119892 (119905) |119909 (120590 (119905))|
119901] |119909 (119905)|
minus119901
= 120582119891 (119905) (119909 (120591 (119905))
119909 (119905))
119901
+ 120583119892 (119905) (119909 (120590 (119905))
119909 (119905))
119901
ge 120582119891 (119905) 120591119899(119905) + 120583119892 (119905) 120590
119899(119905) 119905 isin 119869
119899
(103)
where the functions 120591119899(119905) and 120590
119899(119905) are defined in (26)
(ii) Delay Case with 119902 gt 119901 In this part we use the nextelementary inequality
119883120574+ (120574 minus 1) 119884
120574ge 120574119883119884
120574minus1 120574 gt 1 119883 119884 ge 0 (104)
Since 119902 gt 119901 and using (104) especially for
120574 =119902
119901gt 1 119883 = (120582119891 (119905))
119901119902
|119909 (120591 (119905))|119901
119884 = (119901
119902 minus 119901120588 |119890 (119905)|)
119901119902
(105)
14 Discrete Dynamics in Nature and Society
for all 119905 isin 119869119899we obtain
[120582119865 (119905 119909 (120591 (119905))) + 120583119866 (119905 119909 (120590 (119905))) minus 120588119890 (119905)]
times (|119909 (119905)|119901minus1
119909 (119905))minus1
= [120582119865 (119905 119909 (120591 (119905))) minus 120588119890 (119905)] (|119909 (119905)|119901minus1
119909 (119905))minus1
ge [120582119891 (119905) |119909 (120591 (119905))|1199011 + 120588 |119890 (119905)|] |119909 (119905)|
minus119901
= [119883120574+ (120574 minus 1) 119884
120574] |119909 (119905)|
minus119901
ge119902
119901(120582119891 (119905))
119901119902
|119909 (120591 (119905))|119901
times (119901
119902 minus 119901120588 |119890 (119905)|)
(119901119902)((119902119901)minus1)
|119909 (119905)|minus119901
= 120582119901119902
1205881minus(119901119902)
119902
119901(
119901
119902 minus 119901)
1minus(119901119902)
|119890 (119905)|1minus(119901119902)
times (119891 (119905))119901119902
(119909 (120591 (119905))
119909 (119905))
119901
ge 119896 (120582 120583 120588)119902
119901(
119901
119902 minus 119901)
1minus(119901119902)
|119890 (119905)|1minus(119901119902)
times (119891 (119905))119901119902
120591119899(119905)
(106)
where the function 119896(120582 120583 120588) is from (18)
(iii) Advanced Case with 119902 gt 119901 Using the same line ofarguments as in the proof of the previous case for all 119905 isin 119869
119899
we obtain
[120582119865 (119905 119909 (120591 (119905))) + 120583119866 (119905 119909 (120590 (119905))) minus 120588119890 (119905)]
times (|119909 (119905)|119901minus1
119909 (119905))minus1
= [120583119866 (119905 119909 (120590 (119905))) minus 120588119890 (119905)] (|119909 (119905)|119901minus1
119909 (119905))minus1
ge 119896 (120582 120583 120588)119902
119901(
119901
119902 minus 119901)
1minus(119901119902)
|119890 (119905)|1minus(119901119902)
times (119892 (119905))119901119902
120590119899(119905)
(107)
where the function 119896(120582 120583 120588) is from (21)
(iv) Superlinear Delay-Advanced Case Since 1199011 1199012gt 119901 for
all 119905 isin 119869119899we obtain
[120582119865 (119905 119909 (120591 (119905))) + 120583119866 (119905 119909 (120590 (119905))) minus 120588119890 (119905)]
times (|119909 (119905)|119901minus1
119909 (119905))minus1
= [120582119865 (119905 119909 (120591 (119905))) minus120588
2119890 (119905)] (|119909 (119905)|
119901minus1119909 (119905))minus1
+ [120583119866 (119905 119909 (120590 (119905))) minus120588
2119890 (119905)] (|119909 (119905)|
119901minus1119909 (119905))minus1
ge [120582119891 (119905) |119909 (120591 (119905))|1199011 +
120588
2|119890 (119905)|] |119909 (119905)|
minus119901
+ [120583119892 (119905) |119909 (120590 (119905))|1199012 +
120588
2|119890 (119905)|] |119909 (119905)|
minus119901
(108)
Now just the same as in the proofs of previous delay andadvanced cases with 119902 gt 119901 and with the help of (104) inparticular for
120574 =1199011
119901gt 1 119883 = (120582119891 (119905))
1199011199011
|119909 (120591 (119905))|119901
119884 = (119901
1199011minus 119901
120588
2|119890 (119905)|)
1199011199011
(109)
we have
[120582119891 (119905) |119909 (120591 (119905))|1199011 +
120588
2 |119890 (119905)|] |119909 (119905)|
minus119901
= [119883120574+ (120574 minus 1) 119884
120574] |119909 (119905)|
minus119901
ge1199011
119901(120582119891 (119905))
1199011199011
|119909 (120591 (119905))|119901
times (119901
1199011minus 119901
120588
2|119890 (119905)|)
(1199011199011)((1199011119901)minus1)
|119909 (119905)|minus119901
= 12058211990111990111205881minus(119901119901
1)1199011
119901(
119901
2 (1199011minus 119901)
)
1minus(1199011199011)
times |119890 (119905)|1minus(119901119901
1)(119891 (119905))
1199011199011
(119909 (120591 (119905))
119909 (119905))
119901
ge 119896 (120582 120583 120588)1199011
119901(
119901
2 (1199011minus 119901)
)
1minus(1199011199011)
times |119890 (119905)|1minus(119901119901
1)(119891 (119905))
1199011199011
120591119899(119905)
(110)
where the function 119896(120582 120583 120588) is from (24) Analogously weshow that
[120583119892 (119905) |119909 (120590 (119905))|1199012 +
120588
2|119890 (119905)|] |119909 (119905)|
minus119901
ge 119896 (120582 120583 120588)1199012
119901(
119901
2 (1199012minus 119901)
)
1minus(1199011199012)
times |119890 (119905)|1minus(119901119901
2)(119891 (119905))
1199011199012
120590119899(119905)
(111)
Discrete Dynamics in Nature and Society 15
Summarizing previous calculation we obtain
[120582119865 (119905 119909 (120591 (119905))) + 120583119866 (119905 119909 (120590 (119905))) minus 120588119890 (119905)]
times (|119909 (119905)|119901minus1
119909 (119905))minus1
ge 119896 (120582 120583 120588)119876119899(119905) 119905 isin 119869
119899
(112)
where the function 119896(120582 120583 120588) is from (24)
(v) Supersublinear Delay-Advanced Case Since 1199011gt 119901 gt 119901
2
and the following well-known elementary inequality holds
12057801199060+ 12057811199061+ 12057821199062ge 1199061205780
01199061205781
11199061205782
2 120578119894ge 0 119906
119894ge 0 (113)
from 1205780 1205781 1205782isin (0 1) 120578
0+ 1205781+ 1205782= 1 and 119901
11205781+ 11990121205782= 119901
we obtain for all 119905 isin 119869119899 for all 119905 isin 119869
119899we obtain
[120582119865 (119905 119909 (120591 (119905))) + 120583119866 (119905 119909 (120590 (119905))) minus 120588119890 (119905)]
times (|119909 (119905)|119901minus1
119909 (119905))minus1
ge [120582119891 (119905) |119909 (120591 (119905))|1199011 + 120583119892 (119905) |119909 (120590 (119905))|
1199012 + 120588 |119890 (119905)|]
times |119909 (119905)|minus119901
= [1205781[120578minus1
1120582119891 (119905) |119909 (120591 (119905))|
1199011] + 120578
2[120578minus1
2120583119892 (119905) |119909 (120590 (119905))|
1199012]
+1205780[120578minus1
0120588 |119890 (119905)|]] |119909 (119905)|
minus119901
ge [120578minus1
1120582119891 (119905) |119909 (120591 (119905))|
1199011]1205781
[120578minus1
2120583119892 (119905) |119909 (120590 (119905))|
1199012]1205782
times [120578minus1
0120588 |119890 (119905)|]
1205780
|119909 (119905)|minus119901
= 120582120578112058312057821205881205780 |119890 (119905)|
1205780(119891 (119905))
1205781
(119892 (119905))1205782
times|119909 (120591 (119905))|
12057811199011
|119909 (119905)|12057811199011
|119909 (120590 (119905))|12057821199012
|119909 (119905)|12057821199012
2
prod
119894=0
120578minus120578119894
119894
= 119896 (120582 120583 120588) |119890 (119905)|1205780(119891 (119905))
1205781
(119892 (119905))1205782
times (119909 (120591 (119905))
119909 (119905))
12057811199011
(119909 (120590 (119905))
119909 (119905))
12057821199012 2
prod
119894=0
120578minus120578119894
119894
ge 119896 (120582 120583 120588) |119890 (119905)|1205780(119891 (119905))
1205781
(119892 (119905))1205782
(120591119899(119905))1205781(1199011119901)
times (120590119899(119905))1205782(1199012119901)
2
prod
119894=0
120578minus120578119894
119894
= 119896 (120582 120583 120588)119876119899(119905)
(114)
where 119896(120582 120583 120588) and 119876119899(119905) are given respectively in (24) and
(25) Thus it is shown that required condition (5) in thecases (i)ndash(iv) is fulfilled with respect to 119896(120582 120583 120588) and 119876
119899(119905)
determined by (18) (21) or (24) and (19) (22) or (25)In conclusion according to the previous observation we
see that all assumptions of Lemma 4 are fulfilled and henceLemma 4 proves Theorems 5 6 and 7
Conflict of Interests
The author declares that there is no conflict of interestsregarding the publication of this paper
References
[1] Y G Sun ldquoA note on Nasrrsquos and Wongrsquos papersrdquo Journal ofMathematical Analysis and Applications vol 286 no 1 pp 363ndash367 2003
[2] Y G Sun C H Ou and J S W Wong ldquoInterval oscillationtheorems for a second-order linear differential equationrdquo Com-puters amp Mathematics with Applications vol 48 no 10-11 pp1693ndash1699 2004
[3] S Murugadass E Thandapani and S Pinelas ldquoOscillationcriteria for forced second-order mixed type quasilinear delaydifferential equationsrdquo Electronic Journal of Differential Equa-tions vol 2010 article 73 9 pages 2010
[4] Y Bai and L Liu ldquoNew oscillation criteria for second-orderdelay differential equations with mixed nonlinearitiesrdquoDiscreteDynamics in Nature and Society vol 2010 Article ID 796256 9pages 2010
[5] A F Guvenilir andA Zafer ldquoSecond-order oscillation of forcedfunctional differential equations with oscillatory potentialsrdquoComputers amp Mathematics with Applications vol 51 no 9-10pp 1395ndash1404 2006
[6] A Zafer ldquoInterval oscillation criteria for second order super-half linear functional differential equations with delay andadvanced argumentsrdquoMathematische Nachrichten vol 282 no9 pp 1334ndash1341 2009
[7] A F Guvenilir ldquoInterval oscillation of second-order functionaldifferential equations with oscillatory potentialsrdquo NonlinearAnalysis Theory Methods amp Applications vol 71 no 12 ppe2849ndashe2854 2009
[8] T S Hassan L Erbe and A Peterson ldquoForced oscillation ofsecond order differential equations with mixed nonlinearitiesrdquoActa Mathematica Scientia B vol 31 no 2 pp 613ndash626 2011
[9] M Pasic ldquoNew oscillation criteria for second-order forcedquasilinear functional differential equationsrdquo Abstract andApplied Analysis vol 2013 Article ID 735360 12 pages 2013
[10] L H Erbe Q Kong and B G Zhang Oscillation Theory forFunctional-Differential Equations vol 190 Marcel Dekker NewYork NY USA 1995
[11] V Kolmanovskii and A Myshkis Introduction to the Theoryand Applications of Functional-Differential Equations vol 463Kluwer Academic Publishers Dordrecht The Netherlands1999
[12] R P Agarwal M Bohner and W-T Li Nonoscillation andOscillation Theory for Functional Differential Equations vol267 Marcel Dekker New York NY USA 2004
[13] L Erbe T Hassan and A Peterson ldquoOscillation of secondorder functional dynamic equationsrdquo International Journal ofDifference Equations vol 5 no 2 pp 175ndash193 2010
[14] B Baculıkova J Dzurina and Y V Rogovchenko ldquoOscillationof third order trinomial delay differential equationsrdquo AppliedMathematics and Computation vol 218 no 13 pp 7023ndash70332012
[15] R P Agarwal L Berezansky E Braverman and A Domoshnit-sky Nonoscillation Theory of Functional Differential Equationswith Applications Springer New York NY USA 2012
16 Discrete Dynamics in Nature and Society
[16] J Zhang ldquoVariational approach to solitary wave solution ofthe generalized Zakharov equationrdquo Computers ampMathematicswith Applications vol 54 no 7-8 pp 1043ndash1046 2007
[17] T Ozis and A Yıldırım ldquoApplication of Hersquos semi-inversemethod to the nonlinear Schrodinger equationrdquo Computers ampMathematics with Applications vol 54 no 7-8 pp 1039ndash10422007
[18] X-C Cai andM-S Li ldquoPeriodic solution of Jacobi elliptic equa-tions by Hersquos perturbation methodrdquo Computers amp Mathematicswith Applications vol 54 no 7-8 pp 1210ndash1212 2007
[19] S Lenci G Menditto and A M Tarantino ldquoHomoclinic andheteroclinic bifurcations in the non-linear dynamics of a beamresting on an elastic substraterdquo International Journal of Non-Linear Mechanics vol 34 no 4 pp 615ndash632 1999
[20] D-J Huang and H-Q Zhang ldquoLink between travelling wavesand first order nonlinear ordinary differential equation with asixth-degree nonlinear termrdquoChaos Solitons amp Fractals vol 29no 4 pp 928ndash941 2006
[21] A I Maimistov ldquoPropagation of an ultimately short electro-magnetic pulse in a nonlinear medium described by the fifth-order Duffing modelrdquo Optics and Spectroscopy vol 94 pp 251ndash257 2003
[22] M N Hamdan and N H Shabaneh ldquoOn the large amplitudefree vibrations of a restrained uniform beam carrying anintermediate lumpedmassrdquo Journal of Sound andVibration vol199 no 5 pp 711ndash736 1997
[23] J-H He ldquoSome asymptotic methods for strongly nonlinearequationsrdquo International Journal of Modern Physics B vol 20no 10 pp 1141ndash1199 2006
[24] J B Marion Classical Dynamics of Particles and Systems 1970[25] I Kovacic and M J Brennan The Duffing Equation Nonlinear
Oscillatos and their Behaviour John Wiley amp Sons 1st edition2011
[26] F C Moon Chaotic Vibrations An Introduction for AppliedScientists and Engineers John Wiley amp Sons New York NYUSA 2004
[27] J J Stoker Nonlinear Vibrations 1950[28] G Chen and Z Tao ldquoAmplitude-frequency relationship for the
relativistic oscillatorrdquoAASRI Procedia vol 1 pp 400ndash403 2012[29] R E Mickens Oscillations in Planar Dynamic Systems World
Scientific Publishing Singapore 1996[30] A Belendez T Belendez C Neipp A Hernandez and M
L Alvarez ldquoApproximate solutions of a nonlinear oscillatortypified as a mass attached to a stretched elastic wire by thehomotopy perturbation methodrdquo Chaos Solitions and Fractalsvol 39 pp 746ndash764 2009
[31] A Belendez E Fernandez R Fuentes J J Rodes and I PascualldquoHarmonic balancing approach to nonlinear oscillations of apunctual charge in the eletric field of charged ringrdquo PhysicsLetters A vol 373 pp 735ndash740 2009
[32] A Elıas-Zuniga ldquoExact solution of the cubic-quintic Duffingoscillatorrdquo Applied Mathematical Modelling vol 37 no 4 pp2574ndash2579 2013
[33] A Belendez M L Alvarez J Frances et al ldquoAnalytical approx-imate solutions for the cubic-quintic Duffing oscillator in termsof elementary functionsrdquo Journal of Applied Mathematics vol2012 Article ID 286290 16 pages 2012
[34] A Elıas-Zuniga OMartınez-Romero andR K Cordoba-DıazldquoApproximate solution for the Duffing-harmonic oscillator bythe enhanced cubication methodrdquo Mathematical Problems inEngineering vol 2012 Article ID 618750 12 pages 2012
[35] C W Lim B S Wu andW P Sun ldquoHigher accuracy analyticalapproximations to the Duffing-harmonic oscillatorrdquo Journal ofSound and Vibration vol 296 no 4-5 pp 1039ndash1045 2006
[36] J He ldquoSome new approaches to Duffing equation with stronglyand high order nonlinearity II parametrized perturbationtechniquerdquo Communications in Nonlinear Science amp NumericalSimulation vol 4 no 1 pp 81ndash83 1999
[37] V Marinca and N Herisanu ldquoPeriodic solutions for somestrongly nonlinear oscillations by Hersquos variational iterationmethodrdquo Computers amp Mathematics with Applications vol 54no 7-8 pp 1188ndash1196 2007
[38] W Lu and Y Liu ldquoVibration control for the primary resonanceof the Duffing oscillator by a time delay state feedbackrdquoInternational Journal of Nonlinear Science vol 8 no 3 pp 324ndash328 2009
[39] H Y Hu and Z H Wang Dynamics of Controlled MechanicalSystems with Delayed Feedback Springer 2002
[40] M Hamdi and M Belhaq ldquoControl of bistability in a delayedDuffing oscillatorrdquo Advances in Acoustics and Vibration vol2012 Article ID 872498 6 pages 2012
[41] V Ravichandran C Chinnathambi and S Rajasekar ldquoNonlin-ear resonance in Duffing oscillator with fixed and integrativetime-delayed feedbacksrdquoPramana Journal of Physics vol 78 pp347ndash360 2013
[42] X You and H Xu ldquoAnalytical approximations for the periodicmotion of theDuffing systemwith delayed feedbackrdquoNumericalAlgorithms vol 56 no 4 pp 561ndash576 2011
[43] Z Sun W Xu X Yang and T Fang ldquoInducing or suppressingchaos in a double-well Duffing oscillator by time delay feed-backrdquo Chaos Solitons and Fractals vol 27 pp 705ndash714 2006
[44] H Wang H Hu and Z Wang ldquoGlobal dynamics of a Duffingoscillator with delayed displacement feedbackrdquo InternationalJournal of Bifurcation and Chaos in Applied Sciences andEngineering vol 14 no 8 pp 2753ndash2775 2004
[45] J Chiasson and J J LoiseauApplications of Time Delay SystemsSpringer 2007
[46] M Lakshmanan andDV SenthilkumarDynamics of NonlinearTime-Delay Systems Springer 2010
[47] G Stepan T Insperger and R Szalai ldquoDelay parametricexcitation and the nonlinear dynamics of cutting processesrdquoInternational Journal of Bifurcation and Chaos in Applied Sci-ences and Engineering vol 15 no 9 pp 2783ndash2798 2005
[48] U van der Heiden and H-O Walther ldquoExistence of chaos incontrol systems with delayed feedbackrdquo Journal of DifferentialEquations vol 47 no 2 pp 273ndash295 1983
[49] Y G Sun and J S W Wong ldquoOscillation criteria for secondorder forced ordinary differential equations with mixed non-linearitiesrdquo Journal of Mathematical Analysis and Applicationsvol 334 no 1 pp 549ndash560 2007
[50] J Heagy and W L Ditto ldquoDynamics of a two-frequencyparametrically driven Duffing oscillatorrdquo Journal of NonlinearScience vol 1 no 4 pp 423ndash455 1991
[51] A B Belogortsev ldquoBifurcations of tori and chaos in thequasiperiodically forced Duffing oscillatorrdquoNonlinearity vol 5no 4 pp 889ndash897 1992
[52] M Belhaq and M Houssni ldquoQuasi-periodic oscillations chaosand suppression of chaos in a nonlinear oscillator driven byparametric and external excitationsrdquo Nonlinear Dynamics vol18 no 1 pp 1ndash24 1999
[53] S H Saker P Y H Pang and R P Agarwal ldquoOscillationtheorems for second order nonlinear functional differential
Discrete Dynamics in Nature and Society 17
equations with dampingrdquo Dynamic Systems and Applicationsvol 12 no 3-4 pp 307ndash321 2003
[54] I N Bronshtein K A Semendyayev G Musiol and HMuehligHandbook of Mathematics Springer 5th edition 2007
[55] M Pasic ldquoFite-Wintner-Leighton-type oscillation criteria forsecond-order differential equations with nonlinear dampingrdquoAbstract and Applied Analysis vol 2013 Article ID 852180 10pages 2013
Submit your manuscripts athttpwwwhindawicom
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Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
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Decision SciencesAdvances in
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Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
Discrete Dynamics in Nature and Society 7
been already appearing in some published papers on theoscillation of functional differential equations but only insome examples illustrating certain main oscillation criteriaHowever with the help of our main results in which theparametrically excited oscillations are studied in a generalsetting the equations from these examples are replaced withgeneral ones also having parameters 120582 and 120583
In [1] (see also [2 Example 31] with 120591 = 0 [3 Example31] and [4 Section 3]) the author considers the oscillationof the second-order delay differential equation
11990910158401015840
(119905) + 119891 (119905) |119909 (120591 (119905))|120574 sgn119909 (120591 (119905)) = 119890 (119905) (36)
in the linear case (120574 = 1) and the superlinear (120574 gt 1)In the linear case (analogously for the superlinear case see[1 Theorem 2]) the author proved the following oscillationcriterion In what follows we denote
119863 (119886 119887) = 119906 isin 1198621
([119886 119887] R) 119906 (119905) equiv 0 119906 (119886) = 119906 (119887) = 0
(37)
Theorem 17 ([1 Theorem 1]) Suppose that for any 119879 ge 0there exist constants 119886
1 1198871 1198862 1198872such that 119879 le 119886
1lt 1198871 119879 le
1198862lt 1198872 and 119891(119905) ge 0 on [120591(119886
1) 1198871] cup [120591(119886
2) 1198872] 119890(119905) le 0
on [120591(1198861) 1198871] and 119890(119905) ge 0 on [120591(119886
2) 1198872] If there exists 119906 isin
119863(119886119894 119887119894) 119894 = 1 2 such that
int
119887119894
119886119894
[1199062
(119905) 119891 (119905)120591 (119905) minus 120591 (119886
119894)
119905 minus 120591 (119886119894)
minus (1199061015840
(119905))2
]119889119905 ge 0 (38)
then (36) with 120574 = 1 is oscillatory
Previous criterion has been applied on the followingparticular equation
11990910158401015840
(119905) + 120582 sin (119905)1003816100381610038161003816100381610038161003816119909 (119905 minus
120587
4)
1003816100381610038161003816100381610038161003816
120574
times sgn 119909(119905 minus120587
4) = cos (119905) 119905 ge 0
(39)
where 120582 ge 0 and 120574 = 1 Applying Theorem 17 to (39) theauthor proved that (39) is oscillatory provided the followinginequality
120582int
119887119894
119886119894
sin2 (2119905) cos2 (2119905) sin (119905)119905 minus 119886119894
119905 minus 119886119894+ 1205874
119889119905 ge120587
2 (40)
holds for sufficiently large 120582 Thus the oscillation of (39) isexcited by the large enough parameter 120582 However accordingto Theorems 5 and 6 we are able to show that the nextparametric equation that corresponds to general equation(36)
11990910158401015840
(119905) + 120582119891 (119905) |119909 (120591 (119905))|120574 sgn119909 (120591 (119905)) = 119890 (119905) (41)
is oscillatory provided 120582 is large enough where 1199011= 1199012= 120574
120583 = 0 and 120588 = 1Next in [5] (see also [6ndash8]) the authors consider the
oscillation of the following class of second-order differentialequations with delay and advanced arguments
(119903 (119905) 1199091015840
(119905))1015840
+ 119891 (119905) |119909 (120591 (119905))|1199011 sgn119909 (120591 (119905))
+ 119892 (119905) |119909 (120590 (119905))|1199012 sgn119909 (120590 (119905)) = 119890 (119905) 119905 ge 0
(42)
where 1199011 1199012ge 1 When 119901
1= 1199012= 1 the authors prove the
following result (for other cases see [5Theorems 32 33 and34]
Theorem 18 ([5 Theorem 31]) Suppose that for any 119879 ge
0 there exist intervals [120591(1198861) 1198871] [120591(119886
2) 1198872] [1198881 120590(1198891)] and
[1198882 120590(1198892)] contained in [119879infin) such that 119886
1lt 1198871 1198862lt 1198872
1198881lt 1198891 1198882lt 1198892 and
119891 (119905) ge 0 119900119899 [120591 (1198861) 1198871] cup [120591 (119886
2) 1198872]
119892 (119905) ge 0 119900119899 [1198881 120590 (1198891)] cup [119888
2 120590 (1198892)]
119890 (119905) le 0 119900119899 [120591 (1198861) 1198871] cup [1198881 120590 (1198891)]
119890 (119905) ge 0 119900119899 [120591 (1198862) 1198872] cup [1198882 120590 (1198892)]
(43)
and 119888119894= 120591(119886
119894) 119889119894= 119886119894 and 119887
119894= 120590(119889
119894) 119894 = 1 2 If there exist
1199061isin 119863(119886
119894 119887119894) and 119906
2isin 119863(119888119894 119889119894) such that either
int
119887119894
119886119894
[1199062
1(119905) 119891 (119905)
120591 (119905) minus 120591 (119886119894)
119905 minus 120591 (119886119894)
minus (1199061015840
1(119905))2
119903 (119905)] 119889119905 ge 0 (44)
or
int
119889119894
119888119894
[1199062
2(119905) 119891 (119905)
120590 (119889119894) minus 120590 (119905)
120590 (119889119894) minus 119905
minus (1199061015840
2(119905))2
119903 (119905)] 119889119905 ge 0 (45)
for 119894 = 1 2 then (42) with 1199011= 1199012= 1 is oscillatory
As a consequence of this result it has been concluded thatthe particular equation
(119903 (119905) 1199091015840
(119905))1015840
+ 120582 sin (119905) 119909 (119905 minus 120587
12)
+ 120583 cos (119905) 119909 (119905 + 120587
6) = cos (2119905) 119905 ge 0
(46)
is oscillatory provided either 120582 or 120583 is large enough Howeverby following Theorems 5 and 6 one can obtain the sameconclusion for the following general equation associated with(42)
(119903 (119905) 1199091015840
(119905))1015840
+ 120582119891 (119905) |119909 (120591 (119905))|1199011 sgn119909 (120591 (119905))
+ 120583119892 (119905) |119909 (120590 (119905))|1199012 sgn119909 (120590 (119905)) = 119890 (119905)
(47)
Related observation can be done with [8 Example 33]and [9 Example 21] where the quasilinear second-orderfunctional differential equations have been considered It isleft to the reader
5 Some Open Questions and Comments
In this section we discuss some problems related to ourmainresults that are not studied here
(1) Quasiperiodic Case In the theory of nonlinear oscillatorsa particularly important case occurs when the periodiccoefficients in the oscillator do not have any common periodIt is called the quasiperiodic (or two-frequency) nonlinear
8 Discrete Dynamics in Nature and Society
oscillator and studied for instance in [50ndash52] Since inTheorems 5 6 and 7 we assume that the correspondingperiodic functions have a commonperiod it is natural to posethe next question
Open Question 1 Is it possible to derive sufficient conditionsfor the oscillation of (27) in the casewhen119891(119905) and119892(119905) (resp119891(119905) 119892(119905) and ℎ(119905)) are two (resp three) periodic functionsnot having a common period
(2) Equation with More Functional Arguments Next regard-ing some second-order functional differential equationsconsidered in the references of this paper more than twononlinear functional terms are appearing and thereforeinstead of main equation (1) and corresponding particularequation (27) considered inTheorems 5 6 and 7 we suggestthe following classes of equations
(119903 (119905) 119860 (1199091015840
(119905)))1015840
+ 119861 (119905 119909 (119905) 1199091015840
(119905))
+
1198981
sum
119896=1
120582119896119865119896(119905 119909 (120591
119896(119905)))
+
1198982
sum
119896=1
120583119896119866119896(119905 119909 (120590
119896(119905))) = 120588119890 (119905)
(48)
where 0 le 120591119896(119905) le 119905 lim
119905rarrinfin120591119896(119905) = infin 120590
119896(119905) ge 119905 119898
1 1198982isin
N and
(119903 (119905) 119860 (1199091015840
(119905)))1015840
+ 119861 (119905 119909 (119905) 1199091015840
(119905))
+
1198981
sum
119896=1
120582119896119891119896(119905)
1003816100381610038161003816119909 (119905 minus 120591119896)1003816100381610038161003816119901119896 sgn (119909 (119905 minus 120591
119896))
+
1198982
sum
119896=1
120583119896119892119896(119905)
1003816100381610038161003816119909 (119905 + 120590119896)1003816100381610038161003816119902119896 sgn (119909 (119905 + 120590
119896)) = 120588119890 (119905)
(49)
where 120582119896 120583119896 120588 120591119896 120590119896ge 0 and 119901
119896 119902119896gt 0
Comment We suggest the reader to enlarge the main resultsof this paper to (48) and (49)
(3) Damped Duffing Equation In the application the Duffingequation (34) is often appearing with the linear damped term1199091015840(119905) that is
11990910158401015840+ 11988901199091015840+ 1205962
0119909 + 120573119909
3+ 120582Φ (119909 (119905 minus 120591)) = 120588 cos (120596119905) (50)
where 1198890
is the damped coefficient which can in anactive way influence various behaviours of (50) Since119861(119905 119909(119905) 119909
1015840(119905)) = 119889
01199091015840(119905) does not satisfy the required
assumption (4) we are not able to apply our main results to(50) Hence we pose the following questionOpen Question 2 Is it possible to obtain the parametricallyexcited oscillation for (1) in the case when the damped term119861(119905 119906 V) satisfies a larger condition than (4) in which thelinear damped term 120573119909
1015840(119905) is especially included
(4) Functional Argument in Damped Term In a class of Duff-ing equations we have two time delayed feedback and hence
besides the control gain parameter 1205821another parameter 120582
2
appears the so-called velocity gain parameter Hence insteadof (34) one can consider
11990910158401015840+ 11988901199091015840+ 1205962
0119909 + 120573119909
3+ 1205821119909 (119905 minus 120591)
+ 12058221199091015840
(119905 minus 120591) = 120588 cos (120596119905) (51)
Therefore we suggest the following problem for further studyOpen Question 3 Is it possible to obtain the parametricallyexcited oscillation for the following more general functionaldifferential equation than (1) in which the functional argu-ment appears in the damped term too as follows
(119903 (119905) 119860 (1199091015840
(119905)))1015840
+ 119861 (119905 119909 (119905) 1199091015840
(119905)) + 1205821119865 (119905 119909 (120591 (119905)))
+ 1205822119867(119905 119909
1015840
(120591 (119905))) = 120588119890 (119905) 119905 ge 1199050
(52)or
(119903 (119905) 119860 (1199091015840
(119905)))1015840
+ 119861 (119905 119909 (119905) 1199091015840
(119905)) + 1205831119866 (119905 119909 (120590 (119905)))
+ 1205832119867(119905 119909
1015840
(120590 (119905))) = 120588119890 (119905) 119905 ge 1199050
(53)
About known oscillation criteria for the second-order func-tional differential equations having the functional argumentin the damped term we refer the reader to for instance [53]and the references therein
6 Proofs of Main Results
The proof of Lemma 1 is based on the following three stepstwo working forms of condition (6) (see Lemmas 19 and 20)the existence of an explosive solution of a suitable Riccatidifferential inequality (see Proposition 22) and a comparisonprinciple (see Proposition 24)
Lemma 19 (a necessary condition to (6)) Let 0 lt 119903(119905) le 1199030
on [1199050infin) If assumption (6) is fulfilled then there is a positive
real number 120576 such that1
120587lowast
int119869
min119901
(120576119903 (119905))120574minus1
120576119896 (120582 120583 120588)119876 (119905) 119889119905 ge 1 (54)
for all 120582 ge 1205820 120583 ge 120583
0 and 120588 ge 120588
0and some (120582
0 1205830 1205880) isin R3+
Proof Since 0 lt 119903(119905) le 1199030for 119905 ge 119905
0 we conclude that for
120576 = (119901
119903120574minus1
0119896 (120582 120583 120588)max
119905isin 119869119876 (119905)
)
1120574
(120582 120583 120588) isin R3
+
(55)
it holds that 119901(120576119903(119905))120574minus1
ge 119901(1205761199030)120574minus1
= 120576119896(120582 120583 120588)
max119905isin 119869119876(119905) ge 120576119896(120582 120583 120588)119876(119905) 119905 isin 119869 and hence
int119869
min119901
(120576119903 (119905))120574minus1
120576119896 (120582 120583 120588)119876 (119905) 119889119905
= 120576119896 (120582 120583 120588) int119869
119876 (119905) 119889119905
(56)
Discrete Dynamics in Nature and Society 9
On the other hand from (6) we observe
1
120587lowast
int119869
119876 (119905) 119889119905 ge1199031minus(1120574)
0
1199011120574[119896 (120582 120583 120588)]1minus(1120574)
(max119905isin 119869
119876 (119905))
1120574
(57)
which together with (55) and (56) gives
1
120587lowast
int119869
min119901
(120576119903 (119905))120574minus1
120576119896 (120582 120583 120588)119876 (119905) 119889119905
= 120576119896 (120582 120583 120588)1
120587lowast
int119869
119876 (119905) 119889119905
ge 1205761199031minus(1120574)
0
1199011120574[119896 (120582 120583 120588)]
1120574
(max119905isin 119869
119876 (119905))
1120574
= 1
(58)
for all 119899 ge 1198990 120582 ge 120582
0 120583 ge 120583
0 and 120588 ge 120588
0 It proves this
lemma
Lemma 20 (an equivalent condition to (54)) Assumption(54) is fulfilled if and only if there is a real number 120576 gt 0 and acontinuous function 119870(119905) ge 0 119905 isin 119869 such that
1198880= int119869
119870 (119905) 119889119905 gt 0119870 (119905)
1198880
le1
120587lowast
timesmin119901
(120576119903 (119905))120574minus1
120576119896 (120582 120583 120588)119876 (119905)
(59)
for all 119905 isin 119869 120582 ge 1205820 120583 ge 120583
0 and 120588 ge 120588
0and some (120582
0 1205830 1205880) isin
R3+
Proof This proof is very elementary Indeed if (54) holdsthen the function119870(119905) and number 119888
0 defined by
119870 (119905) =1
120587lowast
min119901
(120576119903 (119905))120574minus1
120576119896 (120582 120583 120588)119876 (119905)
1198880= int119869
119870 (119905) 119889119905
(60)
obviously satisfy 1198880
ge 1 and 119870(119905)1198880
le 119870(119905) = (1120587lowast)
min119901(120576119903(119905))120574minus1 120576119896(120582 120583 120588)119876(119905) which shows (59) Con-versely if (59) holds then integrating both sides of thesecond inequality in (59) we obtain
int119869
1
120587lowast
min119901
(120576119903 (119905))120574minus1
120576119896 (120582 120583 120588)119876 (119905) ge int119869
119870 (119905)
1198880
119889119905 = 1
(61)
which shows (54)
In conclusion according to previous two lemmas wesee that supposed condition (6) implies (59) which plays animportant role in the proof of the main results
The second step in the proof of Lemma 1 is to prove theexistence of a function 120596(119905) which blows up in the finitetime and satisfies a generalized Riccati differential lowerinequality we briefly present the existence and properties
of the so-called generalized tangent type function In whatfollows let 120587
lowastbe a positive real number defined in (3) Let us
remark that 120572(119904) = 119904120574 120574 gt 1 implies 120587
lowast= (2120587)(120574 sin(120587120574))
see for instance [54] and obviously for 120574 = 2wehave120587lowast= 120587
Lemma 21 Let 120572 [0infin) rarr [0infin) be a continuousfunction such that
int
infin
0
119889120591
1 + 120572 (120591)lt infin (62)
Then there is a real number 120587lowastgt 0 and a function 119911 = 119911(119904)
119911 isin 1198621((minus120587lowast2 120587lowast2)R) such that
119889119911
119889119904= 1 + 120572 (|119911 (119904)|) 119904 isin (minus
120587lowast
2120587lowast
2)
119911 (0) = 0
(63)
Moreover 119911(119904) is increasing and odd
lim119904rarr120587lowast2
119911 (119904) = infin 120587lowast=
2120587
120574 sin (120587120574)for 120572 (119904) = 119904
120574
120574 gt 1
(64)
In particular for 120572(119904) = 1199042 one can take 119911(119904) = tan(119904) and
120587lowast= 120587
Proof Let 119885 = 119885(119905) 119905 isin R be a function defined by
119885 (119905) = int
119905
0
1
1 + 120572 (|120591|)119889120591 119905 isin R (65)
The function 119885(119905) is well defined since 120572(119904) is positive andcontinuous on [0infin) 119885(119905) is increasing and odd functionand
119889119885
119889119905=
1
1 + 120572 (|119905|) 119905 isin R
119885 (0) = 0 119885 isin 1198621
(RR)
(66)
Moreover because of (62) there is a real number 120587lowastgt 0 such
that120587lowast
2= int
infin
0
119889120591
1 + 120572 (120591) (67)
Thus 119885 R rarr (minus120587lowast2 120587lowast2) and there exists an inverse
function 119885minus1 = 119885minus1(119904) of the original function 119885 = 119885(119905) and
119885minus1
(minus120587lowast2 120587lowast2) rarr R Also from 119885(119885
minus1(119904)) = 119904 and
119889119885119889119905 = 0 onR we also derive that119889119885minus1119889119904 = 0 on its domain(minus120587lowast2 120587lowast2) and
119889119885
119889119905(119885minus1
(119904)) =1
(119889119885minus1119889119904) 119904 isin (minus
120587lowast
2120587lowast
2) (68)
Putting 119905 = 119885minus1(119904) for 119904 isin (minus120587
lowast2 120587lowast2) into (66) and using
(68) we easily obtain
119889119885minus1
119889119904= 1 + 120572 (
10038161003816100381610038161003816119885minus1
(119904)10038161003816100381610038161003816) 119904 isin (minus
120587lowast
2120587lowast
2)
119885minus1
(0) = 0 119885minus1isin 1198621((minus
120587lowast
2120587lowast
2) R)
(69)
10 Discrete Dynamics in Nature and Society
Moreover from (67) we have lim119904rarr120587lowast2119885minus1(119904) = 119885
minus1
(lim119905rarrinfin
119885(119905)) = lim119905rarrinfin
119885minus1119885(119905) = lim
119905rarrinfin119905 = infin Thus
if we set 119911(119904) = 119885minus1(119904) then previous two statements and
(67) prove this lemma
Next we prove the main result of this section
Proposition 22 Let (2) and (6) hold where 119869 = (119886 119887) Let 120576 gt0 be a real number and let119870(119905) ge 0 119905 isin [119886 119887] be a continuousfunction both obtained in Lemma 20 Let 120587
lowastbe from (3) and
1198880from (59) and let 119877
119886isin R be an arbitrary real number If
119911 = 119911(119904) is the generalized tangens function defined in (63)and 119881(119905) is a function defined by
119881 (119905) =120587lowast
1198880
int
119905
119886
119870 (120591) 119889120591 + 119911minus1(119877119886) 119905 isin [119886 119887] (70)
then there is a 119879lowast119886isin [119886 119887) such that
119881 (119879lowast
119886) =
120587lowast
2 119881 ([119886 119879
lowast
119886)) sub (minus
120587lowast
2120587lowast
2) (71)
Moreover for a function 120596(119905) defined by120596 (119905) = 119911 (119881 (119905)) 119905 isin [119886 119879
lowast
119886) (72)
one has 120596(119886) = 119877119886 lim119905rarr119879
lowast
119886
120596(119905) = infin and
119889120596
119889119905le
119901
(120576119903 (119905))120574minus1
120572 (1003816100381610038161003816120596 (119905)
1003816100381610038161003816)
+ 120576119896 (120582 120583 120588)119876 (119905) 119905 isin (119886 119879lowast
119886)
(73)
where the numbers 119901 and 120574 are from (3) and the functions119896(120582 120583 120588) and 119876(119905) are from (6)
Proof Under assumptions (2) and (6) and because of Lem-mas 19 and 20 we obtain 120576 gt 0 and 119870(119905) gt 0 119905 isin [119886 119887]satisfying inequality (59)
Next since 119911minus1(119877119886) isin (minus120587
lowast2 120587lowast2) (see Lemma 21)
from (70) we directly obtain
119881 (119886) = 119911minus1(119877119886) lt
120587lowast
2 119881 (119887) = 120587
lowast+ 119911minus1(119877119886) gt
120587lowast
2
(74)Since 119870 isin 119862([119886 119887] [0infin)) we obtain 119881 isin 119862([119886 119887]R) cap
1198621((119886 119887)R) and from (74) we observe that there exist
numbers 119879lowast119886isin (119886 119887) such that119881(119879lowast
119886) = 120587lowast2 Also119870(119905)119888
0ge
0 gives 119881([119886 119879lowast119886)) sub (minus120587
lowast2 120587lowast2) which proves statement
(71) Moreover it together with Lemma 21 and (72) provesthat
lim119905rarr119879
lowast
119886
120596 (119905) = lim119905rarr119879
lowast
119886
119911 (119881 (119905)) = 119911 (120587lowast
2) = infin (75)
Next according to (59) (63) and (72) we make thefollowing calculation on the interval [119886 119879lowast
119886)
1205961015840
(119905) = 1199111015840
(119881 (119905)) 1198811015840
(119905) = [1 + 120572 (|119911 (119881 (119905))|)]120587lowast
1198880
119870 (119905)
= [1 + 120572 (1003816100381610038161003816120596 (119905)
1003816100381610038161003816)]120587lowast
1198880
119870 (119905)
le119901
(120576119903 (119905))120574minus1
120572 (1003816100381610038161003816120596 (119905)
1003816100381610038161003816) + 120576119896 (120582 120583 120588)119876 (119905)
(76)
Thus all assertions of this proposition are proved
Remark 23 In the proof of the main result the number 119877119886
is determined by 119877119886= 120596(119886) where 120596(119905) denotes a function
associated with a nonoscillatory solution and it is given by(84) below
The third step in the proof of Lemma 1 is to show thefollowing pointwise comparison principle for the functions120596and120596 satisfying respectively the lower and upper differentialinequalities (73) and
119889120596
119889119905ge
119901
(120576119903 (119905))120574minus1
120572 (|120596 (119905)|) + 120576119896 (120582 120583 120588)119876 (119905) 119905 isin (119886 119887)
(77)
Proposition 24 Let [119886 119887) sub [1199050infin) be an arbitrary inter-
val One supposes that all coefficients of Riccati differentialinequalities (73) and (77) are continuous and strictly positivefunctions Let 120596 120596 isin 119862
1([119886 119887)R) be two functions satisfying
respectively (73) and (77) on the interval [119886 119887) Then
120596 (119886) le 120596 (119886) 119894119898119901119897119894119890119904 120596 (119905) le 120596 (119905) forall119905 isin [119886 119887) (78)
Proof Let119867(119905 119906) be a function defined by
119867(119905 119906) =119901
(120576119903 (119905))120574minus1
120572 (|119906|) + 120576119896 (120582 120583 120588)119876 (119905)
119905 isin [119886 119887) 119906 isin R
(79)
Let 119868 sub [119886 119887) and 119872 gt 0 be arbitrary For any two 1199061
1199062 minus119872 le 119906
1lt 1199062le 119872 let 119868
12be an interval defined
by 11986812
= (min|1199061| |1199062|max|119906
1| |1199062|) Since 120572(119904) is a 1198621-
function on [0infin) we know by the Lagrange mean valuetheorem applied on 119868
12that there is a 120585 isin 119868
12such that
120572 (10038161003816100381610038161199062
1003816100381610038161003816) minus 120572 (10038161003816100381610038161199061
1003816100381610038161003816)
1199062minus 1199061
le
1003816100381610038161003816120572 (10038161003816100381610038161199062
1003816100381610038161003816) minus 120572 (10038161003816100381610038161199061
1003816100381610038161003816)1003816100381610038161003816
1199062minus 1199061
=100381610038161003816100381610038161205721015840
(120585)10038161003816100381610038161003816
100381610038161003816100381610038161003816100381610038161199062
1003816100381610038161003816 minus10038161003816100381610038161199061
10038161003816100381610038161003816100381610038161003816
1199062minus 1199061
le100381610038161003816100381610038161205721015840
(120585)10038161003816100381610038161003816
le max119904isin11986812
100381610038161003816100381610038161205721015840
(119904)10038161003816100381610038161003816
(80)
since ||1199062| minus |1199061|| le 119906
2minus 1199061 Hence for any 119905 isin 119868 and 119906
1 1199062
minus119872 le 1199061lt 1199062le 119872 we have
119867(119905 1199062) minus 119867 (119905 119906
1)
1199062minus 1199061
= 1205880(119905)
120572 (10038161003816100381610038161199062
1003816100381610038161003816) minus 120572 (10038161003816100381610038161199061
1003816100381610038161003816)
1199062minus 1199061
le 1205880(119905)max119904isin11986812
100381610038161003816100381610038161205721015840
(119904)10038161003816100381610038161003816= 1198710(119905)
(81)
Thus the function119867(119905 119906) from (79) satisfies required condi-tion of [55 Lemma 19] and applying it to (73) and (77) weprove this proposition
Proof of Lemma 1 On the contrary let 119909(119905) be a solution of(1) such that
119909 (119905) = 0 on (120591 (120591 (119886)) 120590 (120590 (119889))) (82)
Discrete Dynamics in Nature and Society 11
that is 119909(119905) gt 0 on (120591(120591(119886)) 120590(120590(119889))) or 119909(119905) lt 0 on(120591(120591(119886)) 120590(120590(119889))) since 119909(119905) is a continuous function on[1199050infin) Let for instance
119909 (119905) gt 0 on (120591 (120591 (119886)) 120590 (120590 (119889))) (83)
Another case can be analogously treated let us see thecomment at the end of this proof In particular from (83)we have 119909(119905) gt 0 on (120591(120591(119886)) 120590(120590(119887))) which implies (since120591(119905) and 120590(119905) are increasing functions) 119909(119904) gt 0 for all 119904 isin
(120591(119886) 120590(119887)) cup (120591(120591(119886)) 120591(120590(119887))) cup (120590(120591(119886)) 120590(120590(119887))) whichyields 119909(119905) gt 0 119909(120591(119905)) gt 0 and 119909(120590(119905)) gt 0 on (120591(119886) 120590(119887))Hence by assumption (7) we may use inequality (5) on theinterval (119886 119887)
Firstly we show that the following classic Riccati transfor-mation of 119909(119905)
120596 (119905) = minus120576119903 (119905) 119860 (119909
1015840(119905))
|119909 (119905)|119901minus1
119909 (119905) 119905 isin (119886 119887) 120576 gt 0 (84)
satisfies upper Riccati differential inequality (77) Let usremark that from (1) we have in particular
minus(119903 (119905) 119860 (1199091015840
(119905)))1015840
= 119861 (119905 119909 (119905) 1199091015840
(119905)) + 120582119865 (119905 119909 (120591 (119905)))
+ 120583119866 (119905 119909 (120590 (119905))) minus 120588119890 (119905) 119905 ge 1199050
(85)
Taking the first derivative on both sides of (84) and usingassumptions (3) (4) and (5) as well as equality (85) and(|119909(119905)|
119901minus1119909(119905))1015840
= 119901|119909(119905)|119901minus1
1199091015840(119905) we obtain
119889120596
119889119905= 120576119901 119903 (119905)
119860 (1199091015840(119905)) 1199091015840(119905)
|119909 (119905)|119901+1
minus 1205761
|119909 (119905)|119901minus1
119909 (119905)(119903 (119905) 119860 (119909
1015840
(119905)))1015840
= 120576119901119903 (119905)119860 (1199091015840(119905)) 1199091015840(119905)
|119909 (119905)|119901+1
+120576
|119909 (119905)|119901minus1
119909 (119905)
times [120582119861 (119905 119909 (119905) 1199091015840
(119905)) + 119865 (119905 119909 (120591 (119905)))
+120583119866 (119905 119909 (120590 (119905))) minus 120588119890 (119905) ]
ge 120576119901119903 (119905)119860 (1199091015840(119905)) 1199091015840(119905)
|119909 (119905)|119901+1
+120576
|119909 (119905)|119901minus1
119909 (119905)
times [120582119865 (119905 119909 (120591 (119905))) + 120583119866 (119905 119909 (120590 (119905))) minus 120588119890 (119905)]
ge 120576119901119903 (119905) 120572(
10038161003816100381610038161003816119860 (1199091015840(119905))
10038161003816100381610038161003816
|119909 (119905)|119901
) + 120576119896 (120582 120583 120588)119876 (119905)
= 120576119901119903 (119905) 120572 (|120596 (119905)|
120576119903 (119905)) + 120576119896 (120582 120583 120588)119876 (119905)
ge119901
(120576119903 (119905))120574minus1
120572 (|120596 (119905)|) + 120576119896 (120582 120583 120588)119876 (119905) 119905 isin (119886 119887)
(86)
Thus according to inequality (5) it is shown that if 119909(119905) isa solution of (1) which satisfies (83) then the function 120596(119905)
defined by (84) satisfies the Riccati differential inequality(77) and 120596 isin 119862((119886 119887)R) On the other hand let 119877
119886be a
real number defined by 119877119886= 120596(119886) According to (6) and
Lemma 19 we obtain (54) which together with Lemma 20ensures that we may use Proposition 22 for such chosen realnumber 119877
119886 Hence we obtain a function 120596(119905) defined by (72)
which satisfies the lower Riccati differential inequality (73) on[119886 119879lowast
119886) 119879lowast119886isin (119886 119887) such that 120596(a) = 119877
119886and lim
119905rarr119879lowast
119886
120596(119905) =
infin Therefore by 120596(119886) = 119877119886= 120596(119886) and Proposition 24 we
conclude that lim119905rarr119879
lowast
119886
120596(119905) = infin too which is a contradictionwith the above conclusion saying that 120596 isin 119862((119886 119887)R) Thushypothesis (82) is not true and consequently Lemma 1 isshown
For the analogous case 119909(119905) lt 0 on (120591(120591(119886)) 120590(120590(119889))) wealso have 119909(119905) lt 0 on (120591(120591(119888)) 120590(120590(119889))) which implies (since120591(119905) and 120590(119905) are increasing functions)
119909 (119904) lt 0 forall119904 isin (120591 (119888) 120590 (119889)) cup (120591 (120591 (119888)) 120591 (120590 (119889)))
cup (120590 (120591 (119888)) 120590 (120590 (119889)))
(87)
which yields 119909(119905) lt 0 119909(120591(119905)) lt 0 and 119909(120590(119905)) lt 0 on(120591(119888) 120590(119889)) Now we can repeat the preceding procedure buton interval (119888 119889) and using (8) instead of (119886 119887) and (7)
Proof of Lemma 2 From assumption (10) we obtain the exis-tence of an 119899
0isin N such that
int
119887119899
119886119899
119876119899(119905) 119889119905 ge
1198880
2( max119905isin[119886119899 119887119899]
119876119899(119905))
1120574
119899 ge 1198990 (88)
that is
2
1198880
int
119887119899
119886119899
119876119899(119905) 119889119905 ge ( max
119905isin[119886119899 119887119899]119876119899(119905))
1120574
119899 ge 1198990 (89)
Now from (9) and previous inequality we deduce that forlarge enough 120582 120583 120588 and 119899
1199011120574
1199031minus1120574
0
[119896 (120582 120583 120588)]1minus1120574
120587lowast
int
119887119899
119886119899
119876119899(119905) 119889119905
ge2
1198880
int
119887119899
119886119899
119876119899(119905) 119889119905 ge ( max
119905isin[119886119899 119887119899]119876119899(119905))
1120574
(90)
which shows (6) Thus all assumptions of Lemma 1 arefulfilled and hence Lemma 2 immediately follows fromLemma 1
Proof of Lemma 3 Obviously assumption (11) is a particularcase of assumption (9) Hence this proof is very similar tothe proof of Lemma 2 and so it is left to the reader
Proof of Lemma 4 It is clear that from assumption (13) weobtain
1
(max119905isin[119886119899119887119899]119876119899(119905))1120574
int
119887119899
119886119899
119876119899(119905) 119889119905 ge
1198881
1198621120574
0
gt 0 forall119899 ge 1198990
(91)
12 Discrete Dynamics in Nature and Society
Thus hypothesis (12) is fulfilled and therefore Lemma 3proves this lemma
Proof of Theorems 5 6 and 7 This proof is based onLemma 4 In order to simplify notation in many placesin this proof we set 120591(119905) = 119905 minus 120591 and 120590(119905) = 119905 + 120590 Sinceassumptions (2) (3) and (4) have been already supposed inTheorems 5 6 and 7 in order to prove these theorems byLemma 4 we are going to show that the functions 119896(120582 120583 120588)and 119876
119899(119905) explicitly given respectively in (18) (21) or (24)
and (19) (22) or (25) satisfy required conditions (11) and(13) respectively and that every solution 119909(119905) of (27) satisfiesconditions (7) and (8) with respect to functions 119896(120582 120583 120588)and 119876
119899(119905) where 119886 = 119886
2119899minus1 119887 = 119887
2119899minus1 119888 = 119886
2119899 and 119889 = 119887
2119899
The proof that the function 119896(120582 120583 120588) given in (18) (21) or(24) satisfies (11) Passing to the limit in (18) (21) or (24) it isvery simple to show (11)
The proof that the function 119876119899(119905) given in (19) (22) or
(25) satisfies the first claim in (13) From (25) we immediatelyobtain
1003816100381610038161003816120591119899 (119905)1003816100381610038161003816 =
100381610038161003816100381610038161003816100381610038161003816
(119905 minus 119886119899
119905 minus 119886119899+ 120591
)
119901100381610038161003816100381610038161003816100381610038161003816
le 1
1003816100381610038161003816120590119899 (119905)1003816100381610038161003816 =
100381610038161003816100381610038161003816100381610038161003816
(119887119899minus 119905
119887119899minus 119905 + 120590
)
119901100381610038161003816100381610038161003816100381610038161003816
le 1 forall119899 isin N
(92)
Next by assumptions of this corollary we can conclude thatthere are three positive constants 119891
0 1198920 1198900such that |119891(119905)| le
1198910and |119892(119905)| le 119892
0on [1199050infin) in cases (i) and (ii) and
|119890(119905)| le 1198900on [1199050infin) in cases (iii) and (iv) Putting previous
inequalities into (19) (22) or (25) for all 119899 isin N and 119905 isin
[1199050infin) it holds that
1003816100381610038161003816119876119899 (119905)1003816100381610038161003816 le
119902
119901(
119901
119902 minus 119901)
(119902119901)minus1
1198901minus(119901119902)
0119891119901119902
0
delay case with 119902 gt 119901
119902
119901(
119901
119902 minus 119901)
(119902119901)minus1
1198901minus(119901119902)
0119892119901119902
0
advanced case with 119902 gt 119901
1199011
119901(
119901
2 (1199011minus 119901)
)
(1199011119901)minus1
1198901minus(119901119901
1)
0119891119901119902
0+1199012
119901
times(119901
2 (1199012minus 119901)
)
(1199012119901)minus1
1198901minus(119901119901
2)
0119892119901119902
0
delay-advanced case (i)
1198901205780
01198911205781
01198921205782
0
2
prod
119894=0
120578minus120578119894
119894
delay-advanced case (ii) (93)
which shows the first claim in (13)
The proof that the function119876119899(119905) given in (19) (22) or (25)
satisfies the second claim in (13)Without loss of generality weprove this claim only in case (i) since for other cases the prooffollows analogously In this sense let119876
119899(119905) = 119891(119905)120591
119899(119905) Since
1198862119899+1
minus 1198862119899minus1
le 119879lowast 1198872119899+1
minus 1198872119899minus1
ge 119879lowast 1198862119899+2
minus 1198862119899le 119879lowast and
1198872119899+2
minus 1198872119899
ge 119879lowast where 119879
lowastgt 0 is the period of the function
119891(119905) we have 1198862119899minus1
le 1198861+(119899minus1)119879
lowastand 1198872119899minus1
ge 1198871+(119899minus1)119879
lowast
119899 isin N Hence
int
1198872119899minus1
1198862119899minus1
119876119899(119905) 119889119905
= int
1198872119899minus1
1198862119899minus1
119891 (119905) (119905 minus 1198862119899minus1
119905 minus 1198862119899minus1
+ 120591)
119901
119889119905
ge int
1198871+(119899minus1)119879
lowast
1198861+(119899minus1)119879lowast
119891 (119905) (119905 minus 1198861minus (119899 minus 1) 119879
lowast
119905 minus 1198861minus (119899 minus 1) 119879
lowast+ 120591
)
119901
119889119905
= int
1198871
1198861
119891 (119904 + (119899 minus 1) 119879lowast) (
119904 minus 1198861
119904 minus 1198861+ 120591
)
119901
119889119904
= int
1198871
1198861
119891 (119904) (119904 minus 1198861
119904 minus 1198861+ 120591
)
119901
119889119904
(94)
which proves that the integral on the left hand side does notdepend on 119899 isin N that is the second claim in (13) is shown on[1198862119899minus1
1198872119899minus1
] This claim follows in the same way on [1198862119899 1198872119899]
Thus the second claim in (13) is proved on [119886119899 119887119899]
Next to the end of this proof let 119909(119905) be a solu-tion of (1) In particular it implies that (119903(119905)119860(1199091015840(119905)))1015840 =
minus119861(119905 119909(119905) 1199091015840(119905)) minus 120582119865(119905 119909(120591(119905))) minus 120583119866(119905 119909(120590(119905))) + 120588119890(119905) It
together with assumptions (15) (16) (20) and (23) easilygives the next two statements
if 119909 (119905) gt 0 119909 (120591 (119905)) gt 0 119909 (120590 (119905)) gt 0
on (120591 (1198862119899minus1
) 120590 (1198872119899minus1
))
then 119909 (119905) satisfies 119903 (119905) 119860 (1199091015840
(119905)) le 0
on (120591 (1198862119899minus1
) 120590 (1198872119899minus1
)) 119899 ge 1198990
(95)
if 119909 (119905) lt 0 119909 (120591 (119905)) lt 0 119909 (120590 (119905)) lt 0
on (120591 (1198862119899) 120590 (119887
2119899))
then 119909 (119905) satisfies 119903 (119905) 119860 (1199091015840
(119905)) ge 0
on (120591 (1198862119899) 120590 (119887
2119899)) 119899 ge 119899
0
(96)
Now we need the following lemma
Discrete Dynamics in Nature and Society 13
Lemma 25 Let 120591119886119887(119905) and 120590
119886119887(119905) be defined by
120591119886119887(119905) = (
120591 (119905) minus 120591 (119886)
119905 minus 120591 (119886))
119901
120590119886119887(119905) = (
120590 (119887) minus 120590 (119905)
120590 (119887) minus 119905)
119901
119905 isin (119886 119887)
(97)
and let 119909 isin 1198622([1198790infin)R) be an arbitrary function If
(119903(119905)119860(1199091015840(119905)) le 0 for all 119905 isin (120591(119886) 120590(119887)) or (119903(119905)119860(1199091015840(119905)) ge 0
for all 119905 isin (120591(119886) 120590(119887)) then
119909 (120591 (119905))
119909 (119905)ge (120591119886119887(119905))1119901
119909 (120590 (119905))
119909 (119905)ge (120590119886119887(119905))1119901
119905 isin (119886 119887)
(98)
Since119860(V) is supposed to be odd and increasing functionjust before (3) and 119903(119905) satisfies (14) the proof of Lemma 25in the first case that is 119903(119905)119860(1199091015840(119905)) le 0 for all 119905 isin (120591(119886) 120590(119887))is the same as the proof of [9 Corollaries 17 and 18] But in thesecond case that is 119903(119905)119860(1199091015840(119905)) ge 0 for all 119905 isin (120591(119886) 120590(119887))the proof is as follows if previous inequality holds then119903(119905)119860(minus119909
1015840(119905)) le 0 for all 119905 isin (120591(119886) 120590(119887)) and therefore to
the function minus119909(119905) one can apply the first case of this lemmaand consequently one obtains
119909 (120591 (119905))
119909 (119905)=minus119909 (120591 (119905))
minus119909 (119905)ge (120591119886119887(119905))1119901
119909 (120590 (119905))
119909 (119905)=minus119909 (120590 (119905))
minus119909 (119905)ge (120590119886119887(119905))1119901
119905 isin (119886 119887)
(99)
which proves this lemma in the second caseNow combining statements (95) (96) and (98) one
easily obtains
if 119909 (119905) gt 0 119909 (120591 (119905)) gt 0 119909 (120590 (119905)) gt 0
on (120591 (1198862119899minus1
) 120590 (1198872119899minus1
))
then 119909 (119905) satisfies 119909 (120591 (119905))
119909 (119905)ge (120591119899(119905))1119901
on (1198862119899minus1
1198872119899minus1
) 119899 ge 1198990
(100)
if 119909 (119905) lt 0 119909 (120591 (119905)) lt 0 119909 (120590 (119905)) lt 0
on (120591 (1198862119899) 120590 (119887
2119899))
then 119909 (119905) satisfies 119909 (120590 (119905))
119909 (119905)ge (120590119899(119905))1119901
on (1198862119899 1198872119899) 119899 ge 119899
0
(101)
where 120591119899(119905) and 120590
119899(119905) are defined in (26)
The proof that 119909(119905) satisfies (7) and (8) In this proofwe frequently use assumptions (16) (20) and (23) andstatements (100) and (101) Also because of (15) and 119865(119905 119906) =
119891(119905)|119906|1199011 sgn(119906) 119866(119905 119906) = 119892(119905)|119906|
1199012 sgn(119906) in both cases
(100) and (101) we can simultaneously use
minus119890 (119905) (|119909 (119905)|119901minus1
119909 (119905))minus1
= |119890 (119905)| |119909 (119905)|minus119901
ge 0 on 119869119899
119865 (119905 119909 (120591 (119905))) (|119909 (119905)|119901minus1
119909 (119905))minus1
ge 119891 (119905) |119909 (120591 (119905))|1199011 |119909 (119905)|
minus119901ge 0 on 119869
119899
119866 (119905 119909 (120590 (119905))) (|119909 (119905)|119901minus1
119909 (119905))minus1
ge 119892 (119905) |119909 (120590 (119905))|1199012 |119909 (119905)|
minus119901ge 0 on 119869
119899
|119909 (120591 (119905))| |119909 (119905)|minus1=119909 (120591 (119905))
119909 (119905)
|119909 (120590 (119905))| |119909 (119905)|minus1=119909 (120590 (119905))
119909 (119905)on 119869119899
(102)
where 119869119899= (1198862119899minus1
1198872119899minus1
) in the case of (100) and 119869119899= (1198862119899 1198872119899)
in the case of (101)
(i) Delay or Advanced Case with 119902 = 119901 Since 119902 = 119901 we obtain
[120582119865 (119905 119909 (120591 (119905))) + 120583119866 (119905 119909 (120590 (119905))) minus 120588119890 (119905)]
times (|119909 (119905)|119901minus1
119909 (119905))minus1
ge [120582119891 (119905) |119909 (120591 (119905))|119901+ 120583119892 (119905) |119909 (120590 (119905))|
119901
+120588 |119890 (119905)| ] |119909 (119905)|minus119901
ge [120582119891 (119905) |119909 (120591 (119905))|119901+ 120583119892 (119905) |119909 (120590 (119905))|
119901] |119909 (119905)|
minus119901
= 120582119891 (119905) (119909 (120591 (119905))
119909 (119905))
119901
+ 120583119892 (119905) (119909 (120590 (119905))
119909 (119905))
119901
ge 120582119891 (119905) 120591119899(119905) + 120583119892 (119905) 120590
119899(119905) 119905 isin 119869
119899
(103)
where the functions 120591119899(119905) and 120590
119899(119905) are defined in (26)
(ii) Delay Case with 119902 gt 119901 In this part we use the nextelementary inequality
119883120574+ (120574 minus 1) 119884
120574ge 120574119883119884
120574minus1 120574 gt 1 119883 119884 ge 0 (104)
Since 119902 gt 119901 and using (104) especially for
120574 =119902
119901gt 1 119883 = (120582119891 (119905))
119901119902
|119909 (120591 (119905))|119901
119884 = (119901
119902 minus 119901120588 |119890 (119905)|)
119901119902
(105)
14 Discrete Dynamics in Nature and Society
for all 119905 isin 119869119899we obtain
[120582119865 (119905 119909 (120591 (119905))) + 120583119866 (119905 119909 (120590 (119905))) minus 120588119890 (119905)]
times (|119909 (119905)|119901minus1
119909 (119905))minus1
= [120582119865 (119905 119909 (120591 (119905))) minus 120588119890 (119905)] (|119909 (119905)|119901minus1
119909 (119905))minus1
ge [120582119891 (119905) |119909 (120591 (119905))|1199011 + 120588 |119890 (119905)|] |119909 (119905)|
minus119901
= [119883120574+ (120574 minus 1) 119884
120574] |119909 (119905)|
minus119901
ge119902
119901(120582119891 (119905))
119901119902
|119909 (120591 (119905))|119901
times (119901
119902 minus 119901120588 |119890 (119905)|)
(119901119902)((119902119901)minus1)
|119909 (119905)|minus119901
= 120582119901119902
1205881minus(119901119902)
119902
119901(
119901
119902 minus 119901)
1minus(119901119902)
|119890 (119905)|1minus(119901119902)
times (119891 (119905))119901119902
(119909 (120591 (119905))
119909 (119905))
119901
ge 119896 (120582 120583 120588)119902
119901(
119901
119902 minus 119901)
1minus(119901119902)
|119890 (119905)|1minus(119901119902)
times (119891 (119905))119901119902
120591119899(119905)
(106)
where the function 119896(120582 120583 120588) is from (18)
(iii) Advanced Case with 119902 gt 119901 Using the same line ofarguments as in the proof of the previous case for all 119905 isin 119869
119899
we obtain
[120582119865 (119905 119909 (120591 (119905))) + 120583119866 (119905 119909 (120590 (119905))) minus 120588119890 (119905)]
times (|119909 (119905)|119901minus1
119909 (119905))minus1
= [120583119866 (119905 119909 (120590 (119905))) minus 120588119890 (119905)] (|119909 (119905)|119901minus1
119909 (119905))minus1
ge 119896 (120582 120583 120588)119902
119901(
119901
119902 minus 119901)
1minus(119901119902)
|119890 (119905)|1minus(119901119902)
times (119892 (119905))119901119902
120590119899(119905)
(107)
where the function 119896(120582 120583 120588) is from (21)
(iv) Superlinear Delay-Advanced Case Since 1199011 1199012gt 119901 for
all 119905 isin 119869119899we obtain
[120582119865 (119905 119909 (120591 (119905))) + 120583119866 (119905 119909 (120590 (119905))) minus 120588119890 (119905)]
times (|119909 (119905)|119901minus1
119909 (119905))minus1
= [120582119865 (119905 119909 (120591 (119905))) minus120588
2119890 (119905)] (|119909 (119905)|
119901minus1119909 (119905))minus1
+ [120583119866 (119905 119909 (120590 (119905))) minus120588
2119890 (119905)] (|119909 (119905)|
119901minus1119909 (119905))minus1
ge [120582119891 (119905) |119909 (120591 (119905))|1199011 +
120588
2|119890 (119905)|] |119909 (119905)|
minus119901
+ [120583119892 (119905) |119909 (120590 (119905))|1199012 +
120588
2|119890 (119905)|] |119909 (119905)|
minus119901
(108)
Now just the same as in the proofs of previous delay andadvanced cases with 119902 gt 119901 and with the help of (104) inparticular for
120574 =1199011
119901gt 1 119883 = (120582119891 (119905))
1199011199011
|119909 (120591 (119905))|119901
119884 = (119901
1199011minus 119901
120588
2|119890 (119905)|)
1199011199011
(109)
we have
[120582119891 (119905) |119909 (120591 (119905))|1199011 +
120588
2 |119890 (119905)|] |119909 (119905)|
minus119901
= [119883120574+ (120574 minus 1) 119884
120574] |119909 (119905)|
minus119901
ge1199011
119901(120582119891 (119905))
1199011199011
|119909 (120591 (119905))|119901
times (119901
1199011minus 119901
120588
2|119890 (119905)|)
(1199011199011)((1199011119901)minus1)
|119909 (119905)|minus119901
= 12058211990111990111205881minus(119901119901
1)1199011
119901(
119901
2 (1199011minus 119901)
)
1minus(1199011199011)
times |119890 (119905)|1minus(119901119901
1)(119891 (119905))
1199011199011
(119909 (120591 (119905))
119909 (119905))
119901
ge 119896 (120582 120583 120588)1199011
119901(
119901
2 (1199011minus 119901)
)
1minus(1199011199011)
times |119890 (119905)|1minus(119901119901
1)(119891 (119905))
1199011199011
120591119899(119905)
(110)
where the function 119896(120582 120583 120588) is from (24) Analogously weshow that
[120583119892 (119905) |119909 (120590 (119905))|1199012 +
120588
2|119890 (119905)|] |119909 (119905)|
minus119901
ge 119896 (120582 120583 120588)1199012
119901(
119901
2 (1199012minus 119901)
)
1minus(1199011199012)
times |119890 (119905)|1minus(119901119901
2)(119891 (119905))
1199011199012
120590119899(119905)
(111)
Discrete Dynamics in Nature and Society 15
Summarizing previous calculation we obtain
[120582119865 (119905 119909 (120591 (119905))) + 120583119866 (119905 119909 (120590 (119905))) minus 120588119890 (119905)]
times (|119909 (119905)|119901minus1
119909 (119905))minus1
ge 119896 (120582 120583 120588)119876119899(119905) 119905 isin 119869
119899
(112)
where the function 119896(120582 120583 120588) is from (24)
(v) Supersublinear Delay-Advanced Case Since 1199011gt 119901 gt 119901
2
and the following well-known elementary inequality holds
12057801199060+ 12057811199061+ 12057821199062ge 1199061205780
01199061205781
11199061205782
2 120578119894ge 0 119906
119894ge 0 (113)
from 1205780 1205781 1205782isin (0 1) 120578
0+ 1205781+ 1205782= 1 and 119901
11205781+ 11990121205782= 119901
we obtain for all 119905 isin 119869119899 for all 119905 isin 119869
119899we obtain
[120582119865 (119905 119909 (120591 (119905))) + 120583119866 (119905 119909 (120590 (119905))) minus 120588119890 (119905)]
times (|119909 (119905)|119901minus1
119909 (119905))minus1
ge [120582119891 (119905) |119909 (120591 (119905))|1199011 + 120583119892 (119905) |119909 (120590 (119905))|
1199012 + 120588 |119890 (119905)|]
times |119909 (119905)|minus119901
= [1205781[120578minus1
1120582119891 (119905) |119909 (120591 (119905))|
1199011] + 120578
2[120578minus1
2120583119892 (119905) |119909 (120590 (119905))|
1199012]
+1205780[120578minus1
0120588 |119890 (119905)|]] |119909 (119905)|
minus119901
ge [120578minus1
1120582119891 (119905) |119909 (120591 (119905))|
1199011]1205781
[120578minus1
2120583119892 (119905) |119909 (120590 (119905))|
1199012]1205782
times [120578minus1
0120588 |119890 (119905)|]
1205780
|119909 (119905)|minus119901
= 120582120578112058312057821205881205780 |119890 (119905)|
1205780(119891 (119905))
1205781
(119892 (119905))1205782
times|119909 (120591 (119905))|
12057811199011
|119909 (119905)|12057811199011
|119909 (120590 (119905))|12057821199012
|119909 (119905)|12057821199012
2
prod
119894=0
120578minus120578119894
119894
= 119896 (120582 120583 120588) |119890 (119905)|1205780(119891 (119905))
1205781
(119892 (119905))1205782
times (119909 (120591 (119905))
119909 (119905))
12057811199011
(119909 (120590 (119905))
119909 (119905))
12057821199012 2
prod
119894=0
120578minus120578119894
119894
ge 119896 (120582 120583 120588) |119890 (119905)|1205780(119891 (119905))
1205781
(119892 (119905))1205782
(120591119899(119905))1205781(1199011119901)
times (120590119899(119905))1205782(1199012119901)
2
prod
119894=0
120578minus120578119894
119894
= 119896 (120582 120583 120588)119876119899(119905)
(114)
where 119896(120582 120583 120588) and 119876119899(119905) are given respectively in (24) and
(25) Thus it is shown that required condition (5) in thecases (i)ndash(iv) is fulfilled with respect to 119896(120582 120583 120588) and 119876
119899(119905)
determined by (18) (21) or (24) and (19) (22) or (25)In conclusion according to the previous observation we
see that all assumptions of Lemma 4 are fulfilled and henceLemma 4 proves Theorems 5 6 and 7
Conflict of Interests
The author declares that there is no conflict of interestsregarding the publication of this paper
References
[1] Y G Sun ldquoA note on Nasrrsquos and Wongrsquos papersrdquo Journal ofMathematical Analysis and Applications vol 286 no 1 pp 363ndash367 2003
[2] Y G Sun C H Ou and J S W Wong ldquoInterval oscillationtheorems for a second-order linear differential equationrdquo Com-puters amp Mathematics with Applications vol 48 no 10-11 pp1693ndash1699 2004
[3] S Murugadass E Thandapani and S Pinelas ldquoOscillationcriteria for forced second-order mixed type quasilinear delaydifferential equationsrdquo Electronic Journal of Differential Equa-tions vol 2010 article 73 9 pages 2010
[4] Y Bai and L Liu ldquoNew oscillation criteria for second-orderdelay differential equations with mixed nonlinearitiesrdquoDiscreteDynamics in Nature and Society vol 2010 Article ID 796256 9pages 2010
[5] A F Guvenilir andA Zafer ldquoSecond-order oscillation of forcedfunctional differential equations with oscillatory potentialsrdquoComputers amp Mathematics with Applications vol 51 no 9-10pp 1395ndash1404 2006
[6] A Zafer ldquoInterval oscillation criteria for second order super-half linear functional differential equations with delay andadvanced argumentsrdquoMathematische Nachrichten vol 282 no9 pp 1334ndash1341 2009
[7] A F Guvenilir ldquoInterval oscillation of second-order functionaldifferential equations with oscillatory potentialsrdquo NonlinearAnalysis Theory Methods amp Applications vol 71 no 12 ppe2849ndashe2854 2009
[8] T S Hassan L Erbe and A Peterson ldquoForced oscillation ofsecond order differential equations with mixed nonlinearitiesrdquoActa Mathematica Scientia B vol 31 no 2 pp 613ndash626 2011
[9] M Pasic ldquoNew oscillation criteria for second-order forcedquasilinear functional differential equationsrdquo Abstract andApplied Analysis vol 2013 Article ID 735360 12 pages 2013
[10] L H Erbe Q Kong and B G Zhang Oscillation Theory forFunctional-Differential Equations vol 190 Marcel Dekker NewYork NY USA 1995
[11] V Kolmanovskii and A Myshkis Introduction to the Theoryand Applications of Functional-Differential Equations vol 463Kluwer Academic Publishers Dordrecht The Netherlands1999
[12] R P Agarwal M Bohner and W-T Li Nonoscillation andOscillation Theory for Functional Differential Equations vol267 Marcel Dekker New York NY USA 2004
[13] L Erbe T Hassan and A Peterson ldquoOscillation of secondorder functional dynamic equationsrdquo International Journal ofDifference Equations vol 5 no 2 pp 175ndash193 2010
[14] B Baculıkova J Dzurina and Y V Rogovchenko ldquoOscillationof third order trinomial delay differential equationsrdquo AppliedMathematics and Computation vol 218 no 13 pp 7023ndash70332012
[15] R P Agarwal L Berezansky E Braverman and A Domoshnit-sky Nonoscillation Theory of Functional Differential Equationswith Applications Springer New York NY USA 2012
16 Discrete Dynamics in Nature and Society
[16] J Zhang ldquoVariational approach to solitary wave solution ofthe generalized Zakharov equationrdquo Computers ampMathematicswith Applications vol 54 no 7-8 pp 1043ndash1046 2007
[17] T Ozis and A Yıldırım ldquoApplication of Hersquos semi-inversemethod to the nonlinear Schrodinger equationrdquo Computers ampMathematics with Applications vol 54 no 7-8 pp 1039ndash10422007
[18] X-C Cai andM-S Li ldquoPeriodic solution of Jacobi elliptic equa-tions by Hersquos perturbation methodrdquo Computers amp Mathematicswith Applications vol 54 no 7-8 pp 1210ndash1212 2007
[19] S Lenci G Menditto and A M Tarantino ldquoHomoclinic andheteroclinic bifurcations in the non-linear dynamics of a beamresting on an elastic substraterdquo International Journal of Non-Linear Mechanics vol 34 no 4 pp 615ndash632 1999
[20] D-J Huang and H-Q Zhang ldquoLink between travelling wavesand first order nonlinear ordinary differential equation with asixth-degree nonlinear termrdquoChaos Solitons amp Fractals vol 29no 4 pp 928ndash941 2006
[21] A I Maimistov ldquoPropagation of an ultimately short electro-magnetic pulse in a nonlinear medium described by the fifth-order Duffing modelrdquo Optics and Spectroscopy vol 94 pp 251ndash257 2003
[22] M N Hamdan and N H Shabaneh ldquoOn the large amplitudefree vibrations of a restrained uniform beam carrying anintermediate lumpedmassrdquo Journal of Sound andVibration vol199 no 5 pp 711ndash736 1997
[23] J-H He ldquoSome asymptotic methods for strongly nonlinearequationsrdquo International Journal of Modern Physics B vol 20no 10 pp 1141ndash1199 2006
[24] J B Marion Classical Dynamics of Particles and Systems 1970[25] I Kovacic and M J Brennan The Duffing Equation Nonlinear
Oscillatos and their Behaviour John Wiley amp Sons 1st edition2011
[26] F C Moon Chaotic Vibrations An Introduction for AppliedScientists and Engineers John Wiley amp Sons New York NYUSA 2004
[27] J J Stoker Nonlinear Vibrations 1950[28] G Chen and Z Tao ldquoAmplitude-frequency relationship for the
relativistic oscillatorrdquoAASRI Procedia vol 1 pp 400ndash403 2012[29] R E Mickens Oscillations in Planar Dynamic Systems World
Scientific Publishing Singapore 1996[30] A Belendez T Belendez C Neipp A Hernandez and M
L Alvarez ldquoApproximate solutions of a nonlinear oscillatortypified as a mass attached to a stretched elastic wire by thehomotopy perturbation methodrdquo Chaos Solitions and Fractalsvol 39 pp 746ndash764 2009
[31] A Belendez E Fernandez R Fuentes J J Rodes and I PascualldquoHarmonic balancing approach to nonlinear oscillations of apunctual charge in the eletric field of charged ringrdquo PhysicsLetters A vol 373 pp 735ndash740 2009
[32] A Elıas-Zuniga ldquoExact solution of the cubic-quintic Duffingoscillatorrdquo Applied Mathematical Modelling vol 37 no 4 pp2574ndash2579 2013
[33] A Belendez M L Alvarez J Frances et al ldquoAnalytical approx-imate solutions for the cubic-quintic Duffing oscillator in termsof elementary functionsrdquo Journal of Applied Mathematics vol2012 Article ID 286290 16 pages 2012
[34] A Elıas-Zuniga OMartınez-Romero andR K Cordoba-DıazldquoApproximate solution for the Duffing-harmonic oscillator bythe enhanced cubication methodrdquo Mathematical Problems inEngineering vol 2012 Article ID 618750 12 pages 2012
[35] C W Lim B S Wu andW P Sun ldquoHigher accuracy analyticalapproximations to the Duffing-harmonic oscillatorrdquo Journal ofSound and Vibration vol 296 no 4-5 pp 1039ndash1045 2006
[36] J He ldquoSome new approaches to Duffing equation with stronglyand high order nonlinearity II parametrized perturbationtechniquerdquo Communications in Nonlinear Science amp NumericalSimulation vol 4 no 1 pp 81ndash83 1999
[37] V Marinca and N Herisanu ldquoPeriodic solutions for somestrongly nonlinear oscillations by Hersquos variational iterationmethodrdquo Computers amp Mathematics with Applications vol 54no 7-8 pp 1188ndash1196 2007
[38] W Lu and Y Liu ldquoVibration control for the primary resonanceof the Duffing oscillator by a time delay state feedbackrdquoInternational Journal of Nonlinear Science vol 8 no 3 pp 324ndash328 2009
[39] H Y Hu and Z H Wang Dynamics of Controlled MechanicalSystems with Delayed Feedback Springer 2002
[40] M Hamdi and M Belhaq ldquoControl of bistability in a delayedDuffing oscillatorrdquo Advances in Acoustics and Vibration vol2012 Article ID 872498 6 pages 2012
[41] V Ravichandran C Chinnathambi and S Rajasekar ldquoNonlin-ear resonance in Duffing oscillator with fixed and integrativetime-delayed feedbacksrdquoPramana Journal of Physics vol 78 pp347ndash360 2013
[42] X You and H Xu ldquoAnalytical approximations for the periodicmotion of theDuffing systemwith delayed feedbackrdquoNumericalAlgorithms vol 56 no 4 pp 561ndash576 2011
[43] Z Sun W Xu X Yang and T Fang ldquoInducing or suppressingchaos in a double-well Duffing oscillator by time delay feed-backrdquo Chaos Solitons and Fractals vol 27 pp 705ndash714 2006
[44] H Wang H Hu and Z Wang ldquoGlobal dynamics of a Duffingoscillator with delayed displacement feedbackrdquo InternationalJournal of Bifurcation and Chaos in Applied Sciences andEngineering vol 14 no 8 pp 2753ndash2775 2004
[45] J Chiasson and J J LoiseauApplications of Time Delay SystemsSpringer 2007
[46] M Lakshmanan andDV SenthilkumarDynamics of NonlinearTime-Delay Systems Springer 2010
[47] G Stepan T Insperger and R Szalai ldquoDelay parametricexcitation and the nonlinear dynamics of cutting processesrdquoInternational Journal of Bifurcation and Chaos in Applied Sci-ences and Engineering vol 15 no 9 pp 2783ndash2798 2005
[48] U van der Heiden and H-O Walther ldquoExistence of chaos incontrol systems with delayed feedbackrdquo Journal of DifferentialEquations vol 47 no 2 pp 273ndash295 1983
[49] Y G Sun and J S W Wong ldquoOscillation criteria for secondorder forced ordinary differential equations with mixed non-linearitiesrdquo Journal of Mathematical Analysis and Applicationsvol 334 no 1 pp 549ndash560 2007
[50] J Heagy and W L Ditto ldquoDynamics of a two-frequencyparametrically driven Duffing oscillatorrdquo Journal of NonlinearScience vol 1 no 4 pp 423ndash455 1991
[51] A B Belogortsev ldquoBifurcations of tori and chaos in thequasiperiodically forced Duffing oscillatorrdquoNonlinearity vol 5no 4 pp 889ndash897 1992
[52] M Belhaq and M Houssni ldquoQuasi-periodic oscillations chaosand suppression of chaos in a nonlinear oscillator driven byparametric and external excitationsrdquo Nonlinear Dynamics vol18 no 1 pp 1ndash24 1999
[53] S H Saker P Y H Pang and R P Agarwal ldquoOscillationtheorems for second order nonlinear functional differential
Discrete Dynamics in Nature and Society 17
equations with dampingrdquo Dynamic Systems and Applicationsvol 12 no 3-4 pp 307ndash321 2003
[54] I N Bronshtein K A Semendyayev G Musiol and HMuehligHandbook of Mathematics Springer 5th edition 2007
[55] M Pasic ldquoFite-Wintner-Leighton-type oscillation criteria forsecond-order differential equations with nonlinear dampingrdquoAbstract and Applied Analysis vol 2013 Article ID 852180 10pages 2013
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
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Mathematical Problems in Engineering
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Differential EquationsInternational Journal of
Volume 2014
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International Journal of
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Operations ResearchAdvances in
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Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
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Discrete Dynamics in Nature and Society
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Decision SciencesAdvances in
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Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
8 Discrete Dynamics in Nature and Society
oscillator and studied for instance in [50ndash52] Since inTheorems 5 6 and 7 we assume that the correspondingperiodic functions have a commonperiod it is natural to posethe next question
Open Question 1 Is it possible to derive sufficient conditionsfor the oscillation of (27) in the casewhen119891(119905) and119892(119905) (resp119891(119905) 119892(119905) and ℎ(119905)) are two (resp three) periodic functionsnot having a common period
(2) Equation with More Functional Arguments Next regard-ing some second-order functional differential equationsconsidered in the references of this paper more than twononlinear functional terms are appearing and thereforeinstead of main equation (1) and corresponding particularequation (27) considered inTheorems 5 6 and 7 we suggestthe following classes of equations
(119903 (119905) 119860 (1199091015840
(119905)))1015840
+ 119861 (119905 119909 (119905) 1199091015840
(119905))
+
1198981
sum
119896=1
120582119896119865119896(119905 119909 (120591
119896(119905)))
+
1198982
sum
119896=1
120583119896119866119896(119905 119909 (120590
119896(119905))) = 120588119890 (119905)
(48)
where 0 le 120591119896(119905) le 119905 lim
119905rarrinfin120591119896(119905) = infin 120590
119896(119905) ge 119905 119898
1 1198982isin
N and
(119903 (119905) 119860 (1199091015840
(119905)))1015840
+ 119861 (119905 119909 (119905) 1199091015840
(119905))
+
1198981
sum
119896=1
120582119896119891119896(119905)
1003816100381610038161003816119909 (119905 minus 120591119896)1003816100381610038161003816119901119896 sgn (119909 (119905 minus 120591
119896))
+
1198982
sum
119896=1
120583119896119892119896(119905)
1003816100381610038161003816119909 (119905 + 120590119896)1003816100381610038161003816119902119896 sgn (119909 (119905 + 120590
119896)) = 120588119890 (119905)
(49)
where 120582119896 120583119896 120588 120591119896 120590119896ge 0 and 119901
119896 119902119896gt 0
Comment We suggest the reader to enlarge the main resultsof this paper to (48) and (49)
(3) Damped Duffing Equation In the application the Duffingequation (34) is often appearing with the linear damped term1199091015840(119905) that is
11990910158401015840+ 11988901199091015840+ 1205962
0119909 + 120573119909
3+ 120582Φ (119909 (119905 minus 120591)) = 120588 cos (120596119905) (50)
where 1198890
is the damped coefficient which can in anactive way influence various behaviours of (50) Since119861(119905 119909(119905) 119909
1015840(119905)) = 119889
01199091015840(119905) does not satisfy the required
assumption (4) we are not able to apply our main results to(50) Hence we pose the following questionOpen Question 2 Is it possible to obtain the parametricallyexcited oscillation for (1) in the case when the damped term119861(119905 119906 V) satisfies a larger condition than (4) in which thelinear damped term 120573119909
1015840(119905) is especially included
(4) Functional Argument in Damped Term In a class of Duff-ing equations we have two time delayed feedback and hence
besides the control gain parameter 1205821another parameter 120582
2
appears the so-called velocity gain parameter Hence insteadof (34) one can consider
11990910158401015840+ 11988901199091015840+ 1205962
0119909 + 120573119909
3+ 1205821119909 (119905 minus 120591)
+ 12058221199091015840
(119905 minus 120591) = 120588 cos (120596119905) (51)
Therefore we suggest the following problem for further studyOpen Question 3 Is it possible to obtain the parametricallyexcited oscillation for the following more general functionaldifferential equation than (1) in which the functional argu-ment appears in the damped term too as follows
(119903 (119905) 119860 (1199091015840
(119905)))1015840
+ 119861 (119905 119909 (119905) 1199091015840
(119905)) + 1205821119865 (119905 119909 (120591 (119905)))
+ 1205822119867(119905 119909
1015840
(120591 (119905))) = 120588119890 (119905) 119905 ge 1199050
(52)or
(119903 (119905) 119860 (1199091015840
(119905)))1015840
+ 119861 (119905 119909 (119905) 1199091015840
(119905)) + 1205831119866 (119905 119909 (120590 (119905)))
+ 1205832119867(119905 119909
1015840
(120590 (119905))) = 120588119890 (119905) 119905 ge 1199050
(53)
About known oscillation criteria for the second-order func-tional differential equations having the functional argumentin the damped term we refer the reader to for instance [53]and the references therein
6 Proofs of Main Results
The proof of Lemma 1 is based on the following three stepstwo working forms of condition (6) (see Lemmas 19 and 20)the existence of an explosive solution of a suitable Riccatidifferential inequality (see Proposition 22) and a comparisonprinciple (see Proposition 24)
Lemma 19 (a necessary condition to (6)) Let 0 lt 119903(119905) le 1199030
on [1199050infin) If assumption (6) is fulfilled then there is a positive
real number 120576 such that1
120587lowast
int119869
min119901
(120576119903 (119905))120574minus1
120576119896 (120582 120583 120588)119876 (119905) 119889119905 ge 1 (54)
for all 120582 ge 1205820 120583 ge 120583
0 and 120588 ge 120588
0and some (120582
0 1205830 1205880) isin R3+
Proof Since 0 lt 119903(119905) le 1199030for 119905 ge 119905
0 we conclude that for
120576 = (119901
119903120574minus1
0119896 (120582 120583 120588)max
119905isin 119869119876 (119905)
)
1120574
(120582 120583 120588) isin R3
+
(55)
it holds that 119901(120576119903(119905))120574minus1
ge 119901(1205761199030)120574minus1
= 120576119896(120582 120583 120588)
max119905isin 119869119876(119905) ge 120576119896(120582 120583 120588)119876(119905) 119905 isin 119869 and hence
int119869
min119901
(120576119903 (119905))120574minus1
120576119896 (120582 120583 120588)119876 (119905) 119889119905
= 120576119896 (120582 120583 120588) int119869
119876 (119905) 119889119905
(56)
Discrete Dynamics in Nature and Society 9
On the other hand from (6) we observe
1
120587lowast
int119869
119876 (119905) 119889119905 ge1199031minus(1120574)
0
1199011120574[119896 (120582 120583 120588)]1minus(1120574)
(max119905isin 119869
119876 (119905))
1120574
(57)
which together with (55) and (56) gives
1
120587lowast
int119869
min119901
(120576119903 (119905))120574minus1
120576119896 (120582 120583 120588)119876 (119905) 119889119905
= 120576119896 (120582 120583 120588)1
120587lowast
int119869
119876 (119905) 119889119905
ge 1205761199031minus(1120574)
0
1199011120574[119896 (120582 120583 120588)]
1120574
(max119905isin 119869
119876 (119905))
1120574
= 1
(58)
for all 119899 ge 1198990 120582 ge 120582
0 120583 ge 120583
0 and 120588 ge 120588
0 It proves this
lemma
Lemma 20 (an equivalent condition to (54)) Assumption(54) is fulfilled if and only if there is a real number 120576 gt 0 and acontinuous function 119870(119905) ge 0 119905 isin 119869 such that
1198880= int119869
119870 (119905) 119889119905 gt 0119870 (119905)
1198880
le1
120587lowast
timesmin119901
(120576119903 (119905))120574minus1
120576119896 (120582 120583 120588)119876 (119905)
(59)
for all 119905 isin 119869 120582 ge 1205820 120583 ge 120583
0 and 120588 ge 120588
0and some (120582
0 1205830 1205880) isin
R3+
Proof This proof is very elementary Indeed if (54) holdsthen the function119870(119905) and number 119888
0 defined by
119870 (119905) =1
120587lowast
min119901
(120576119903 (119905))120574minus1
120576119896 (120582 120583 120588)119876 (119905)
1198880= int119869
119870 (119905) 119889119905
(60)
obviously satisfy 1198880
ge 1 and 119870(119905)1198880
le 119870(119905) = (1120587lowast)
min119901(120576119903(119905))120574minus1 120576119896(120582 120583 120588)119876(119905) which shows (59) Con-versely if (59) holds then integrating both sides of thesecond inequality in (59) we obtain
int119869
1
120587lowast
min119901
(120576119903 (119905))120574minus1
120576119896 (120582 120583 120588)119876 (119905) ge int119869
119870 (119905)
1198880
119889119905 = 1
(61)
which shows (54)
In conclusion according to previous two lemmas wesee that supposed condition (6) implies (59) which plays animportant role in the proof of the main results
The second step in the proof of Lemma 1 is to prove theexistence of a function 120596(119905) which blows up in the finitetime and satisfies a generalized Riccati differential lowerinequality we briefly present the existence and properties
of the so-called generalized tangent type function In whatfollows let 120587
lowastbe a positive real number defined in (3) Let us
remark that 120572(119904) = 119904120574 120574 gt 1 implies 120587
lowast= (2120587)(120574 sin(120587120574))
see for instance [54] and obviously for 120574 = 2wehave120587lowast= 120587
Lemma 21 Let 120572 [0infin) rarr [0infin) be a continuousfunction such that
int
infin
0
119889120591
1 + 120572 (120591)lt infin (62)
Then there is a real number 120587lowastgt 0 and a function 119911 = 119911(119904)
119911 isin 1198621((minus120587lowast2 120587lowast2)R) such that
119889119911
119889119904= 1 + 120572 (|119911 (119904)|) 119904 isin (minus
120587lowast
2120587lowast
2)
119911 (0) = 0
(63)
Moreover 119911(119904) is increasing and odd
lim119904rarr120587lowast2
119911 (119904) = infin 120587lowast=
2120587
120574 sin (120587120574)for 120572 (119904) = 119904
120574
120574 gt 1
(64)
In particular for 120572(119904) = 1199042 one can take 119911(119904) = tan(119904) and
120587lowast= 120587
Proof Let 119885 = 119885(119905) 119905 isin R be a function defined by
119885 (119905) = int
119905
0
1
1 + 120572 (|120591|)119889120591 119905 isin R (65)
The function 119885(119905) is well defined since 120572(119904) is positive andcontinuous on [0infin) 119885(119905) is increasing and odd functionand
119889119885
119889119905=
1
1 + 120572 (|119905|) 119905 isin R
119885 (0) = 0 119885 isin 1198621
(RR)
(66)
Moreover because of (62) there is a real number 120587lowastgt 0 such
that120587lowast
2= int
infin
0
119889120591
1 + 120572 (120591) (67)
Thus 119885 R rarr (minus120587lowast2 120587lowast2) and there exists an inverse
function 119885minus1 = 119885minus1(119904) of the original function 119885 = 119885(119905) and
119885minus1
(minus120587lowast2 120587lowast2) rarr R Also from 119885(119885
minus1(119904)) = 119904 and
119889119885119889119905 = 0 onR we also derive that119889119885minus1119889119904 = 0 on its domain(minus120587lowast2 120587lowast2) and
119889119885
119889119905(119885minus1
(119904)) =1
(119889119885minus1119889119904) 119904 isin (minus
120587lowast
2120587lowast
2) (68)
Putting 119905 = 119885minus1(119904) for 119904 isin (minus120587
lowast2 120587lowast2) into (66) and using
(68) we easily obtain
119889119885minus1
119889119904= 1 + 120572 (
10038161003816100381610038161003816119885minus1
(119904)10038161003816100381610038161003816) 119904 isin (minus
120587lowast
2120587lowast
2)
119885minus1
(0) = 0 119885minus1isin 1198621((minus
120587lowast
2120587lowast
2) R)
(69)
10 Discrete Dynamics in Nature and Society
Moreover from (67) we have lim119904rarr120587lowast2119885minus1(119904) = 119885
minus1
(lim119905rarrinfin
119885(119905)) = lim119905rarrinfin
119885minus1119885(119905) = lim
119905rarrinfin119905 = infin Thus
if we set 119911(119904) = 119885minus1(119904) then previous two statements and
(67) prove this lemma
Next we prove the main result of this section
Proposition 22 Let (2) and (6) hold where 119869 = (119886 119887) Let 120576 gt0 be a real number and let119870(119905) ge 0 119905 isin [119886 119887] be a continuousfunction both obtained in Lemma 20 Let 120587
lowastbe from (3) and
1198880from (59) and let 119877
119886isin R be an arbitrary real number If
119911 = 119911(119904) is the generalized tangens function defined in (63)and 119881(119905) is a function defined by
119881 (119905) =120587lowast
1198880
int
119905
119886
119870 (120591) 119889120591 + 119911minus1(119877119886) 119905 isin [119886 119887] (70)
then there is a 119879lowast119886isin [119886 119887) such that
119881 (119879lowast
119886) =
120587lowast
2 119881 ([119886 119879
lowast
119886)) sub (minus
120587lowast
2120587lowast
2) (71)
Moreover for a function 120596(119905) defined by120596 (119905) = 119911 (119881 (119905)) 119905 isin [119886 119879
lowast
119886) (72)
one has 120596(119886) = 119877119886 lim119905rarr119879
lowast
119886
120596(119905) = infin and
119889120596
119889119905le
119901
(120576119903 (119905))120574minus1
120572 (1003816100381610038161003816120596 (119905)
1003816100381610038161003816)
+ 120576119896 (120582 120583 120588)119876 (119905) 119905 isin (119886 119879lowast
119886)
(73)
where the numbers 119901 and 120574 are from (3) and the functions119896(120582 120583 120588) and 119876(119905) are from (6)
Proof Under assumptions (2) and (6) and because of Lem-mas 19 and 20 we obtain 120576 gt 0 and 119870(119905) gt 0 119905 isin [119886 119887]satisfying inequality (59)
Next since 119911minus1(119877119886) isin (minus120587
lowast2 120587lowast2) (see Lemma 21)
from (70) we directly obtain
119881 (119886) = 119911minus1(119877119886) lt
120587lowast
2 119881 (119887) = 120587
lowast+ 119911minus1(119877119886) gt
120587lowast
2
(74)Since 119870 isin 119862([119886 119887] [0infin)) we obtain 119881 isin 119862([119886 119887]R) cap
1198621((119886 119887)R) and from (74) we observe that there exist
numbers 119879lowast119886isin (119886 119887) such that119881(119879lowast
119886) = 120587lowast2 Also119870(119905)119888
0ge
0 gives 119881([119886 119879lowast119886)) sub (minus120587
lowast2 120587lowast2) which proves statement
(71) Moreover it together with Lemma 21 and (72) provesthat
lim119905rarr119879
lowast
119886
120596 (119905) = lim119905rarr119879
lowast
119886
119911 (119881 (119905)) = 119911 (120587lowast
2) = infin (75)
Next according to (59) (63) and (72) we make thefollowing calculation on the interval [119886 119879lowast
119886)
1205961015840
(119905) = 1199111015840
(119881 (119905)) 1198811015840
(119905) = [1 + 120572 (|119911 (119881 (119905))|)]120587lowast
1198880
119870 (119905)
= [1 + 120572 (1003816100381610038161003816120596 (119905)
1003816100381610038161003816)]120587lowast
1198880
119870 (119905)
le119901
(120576119903 (119905))120574minus1
120572 (1003816100381610038161003816120596 (119905)
1003816100381610038161003816) + 120576119896 (120582 120583 120588)119876 (119905)
(76)
Thus all assertions of this proposition are proved
Remark 23 In the proof of the main result the number 119877119886
is determined by 119877119886= 120596(119886) where 120596(119905) denotes a function
associated with a nonoscillatory solution and it is given by(84) below
The third step in the proof of Lemma 1 is to show thefollowing pointwise comparison principle for the functions120596and120596 satisfying respectively the lower and upper differentialinequalities (73) and
119889120596
119889119905ge
119901
(120576119903 (119905))120574minus1
120572 (|120596 (119905)|) + 120576119896 (120582 120583 120588)119876 (119905) 119905 isin (119886 119887)
(77)
Proposition 24 Let [119886 119887) sub [1199050infin) be an arbitrary inter-
val One supposes that all coefficients of Riccati differentialinequalities (73) and (77) are continuous and strictly positivefunctions Let 120596 120596 isin 119862
1([119886 119887)R) be two functions satisfying
respectively (73) and (77) on the interval [119886 119887) Then
120596 (119886) le 120596 (119886) 119894119898119901119897119894119890119904 120596 (119905) le 120596 (119905) forall119905 isin [119886 119887) (78)
Proof Let119867(119905 119906) be a function defined by
119867(119905 119906) =119901
(120576119903 (119905))120574minus1
120572 (|119906|) + 120576119896 (120582 120583 120588)119876 (119905)
119905 isin [119886 119887) 119906 isin R
(79)
Let 119868 sub [119886 119887) and 119872 gt 0 be arbitrary For any two 1199061
1199062 minus119872 le 119906
1lt 1199062le 119872 let 119868
12be an interval defined
by 11986812
= (min|1199061| |1199062|max|119906
1| |1199062|) Since 120572(119904) is a 1198621-
function on [0infin) we know by the Lagrange mean valuetheorem applied on 119868
12that there is a 120585 isin 119868
12such that
120572 (10038161003816100381610038161199062
1003816100381610038161003816) minus 120572 (10038161003816100381610038161199061
1003816100381610038161003816)
1199062minus 1199061
le
1003816100381610038161003816120572 (10038161003816100381610038161199062
1003816100381610038161003816) minus 120572 (10038161003816100381610038161199061
1003816100381610038161003816)1003816100381610038161003816
1199062minus 1199061
=100381610038161003816100381610038161205721015840
(120585)10038161003816100381610038161003816
100381610038161003816100381610038161003816100381610038161199062
1003816100381610038161003816 minus10038161003816100381610038161199061
10038161003816100381610038161003816100381610038161003816
1199062minus 1199061
le100381610038161003816100381610038161205721015840
(120585)10038161003816100381610038161003816
le max119904isin11986812
100381610038161003816100381610038161205721015840
(119904)10038161003816100381610038161003816
(80)
since ||1199062| minus |1199061|| le 119906
2minus 1199061 Hence for any 119905 isin 119868 and 119906
1 1199062
minus119872 le 1199061lt 1199062le 119872 we have
119867(119905 1199062) minus 119867 (119905 119906
1)
1199062minus 1199061
= 1205880(119905)
120572 (10038161003816100381610038161199062
1003816100381610038161003816) minus 120572 (10038161003816100381610038161199061
1003816100381610038161003816)
1199062minus 1199061
le 1205880(119905)max119904isin11986812
100381610038161003816100381610038161205721015840
(119904)10038161003816100381610038161003816= 1198710(119905)
(81)
Thus the function119867(119905 119906) from (79) satisfies required condi-tion of [55 Lemma 19] and applying it to (73) and (77) weprove this proposition
Proof of Lemma 1 On the contrary let 119909(119905) be a solution of(1) such that
119909 (119905) = 0 on (120591 (120591 (119886)) 120590 (120590 (119889))) (82)
Discrete Dynamics in Nature and Society 11
that is 119909(119905) gt 0 on (120591(120591(119886)) 120590(120590(119889))) or 119909(119905) lt 0 on(120591(120591(119886)) 120590(120590(119889))) since 119909(119905) is a continuous function on[1199050infin) Let for instance
119909 (119905) gt 0 on (120591 (120591 (119886)) 120590 (120590 (119889))) (83)
Another case can be analogously treated let us see thecomment at the end of this proof In particular from (83)we have 119909(119905) gt 0 on (120591(120591(119886)) 120590(120590(119887))) which implies (since120591(119905) and 120590(119905) are increasing functions) 119909(119904) gt 0 for all 119904 isin
(120591(119886) 120590(119887)) cup (120591(120591(119886)) 120591(120590(119887))) cup (120590(120591(119886)) 120590(120590(119887))) whichyields 119909(119905) gt 0 119909(120591(119905)) gt 0 and 119909(120590(119905)) gt 0 on (120591(119886) 120590(119887))Hence by assumption (7) we may use inequality (5) on theinterval (119886 119887)
Firstly we show that the following classic Riccati transfor-mation of 119909(119905)
120596 (119905) = minus120576119903 (119905) 119860 (119909
1015840(119905))
|119909 (119905)|119901minus1
119909 (119905) 119905 isin (119886 119887) 120576 gt 0 (84)
satisfies upper Riccati differential inequality (77) Let usremark that from (1) we have in particular
minus(119903 (119905) 119860 (1199091015840
(119905)))1015840
= 119861 (119905 119909 (119905) 1199091015840
(119905)) + 120582119865 (119905 119909 (120591 (119905)))
+ 120583119866 (119905 119909 (120590 (119905))) minus 120588119890 (119905) 119905 ge 1199050
(85)
Taking the first derivative on both sides of (84) and usingassumptions (3) (4) and (5) as well as equality (85) and(|119909(119905)|
119901minus1119909(119905))1015840
= 119901|119909(119905)|119901minus1
1199091015840(119905) we obtain
119889120596
119889119905= 120576119901 119903 (119905)
119860 (1199091015840(119905)) 1199091015840(119905)
|119909 (119905)|119901+1
minus 1205761
|119909 (119905)|119901minus1
119909 (119905)(119903 (119905) 119860 (119909
1015840
(119905)))1015840
= 120576119901119903 (119905)119860 (1199091015840(119905)) 1199091015840(119905)
|119909 (119905)|119901+1
+120576
|119909 (119905)|119901minus1
119909 (119905)
times [120582119861 (119905 119909 (119905) 1199091015840
(119905)) + 119865 (119905 119909 (120591 (119905)))
+120583119866 (119905 119909 (120590 (119905))) minus 120588119890 (119905) ]
ge 120576119901119903 (119905)119860 (1199091015840(119905)) 1199091015840(119905)
|119909 (119905)|119901+1
+120576
|119909 (119905)|119901minus1
119909 (119905)
times [120582119865 (119905 119909 (120591 (119905))) + 120583119866 (119905 119909 (120590 (119905))) minus 120588119890 (119905)]
ge 120576119901119903 (119905) 120572(
10038161003816100381610038161003816119860 (1199091015840(119905))
10038161003816100381610038161003816
|119909 (119905)|119901
) + 120576119896 (120582 120583 120588)119876 (119905)
= 120576119901119903 (119905) 120572 (|120596 (119905)|
120576119903 (119905)) + 120576119896 (120582 120583 120588)119876 (119905)
ge119901
(120576119903 (119905))120574minus1
120572 (|120596 (119905)|) + 120576119896 (120582 120583 120588)119876 (119905) 119905 isin (119886 119887)
(86)
Thus according to inequality (5) it is shown that if 119909(119905) isa solution of (1) which satisfies (83) then the function 120596(119905)
defined by (84) satisfies the Riccati differential inequality(77) and 120596 isin 119862((119886 119887)R) On the other hand let 119877
119886be a
real number defined by 119877119886= 120596(119886) According to (6) and
Lemma 19 we obtain (54) which together with Lemma 20ensures that we may use Proposition 22 for such chosen realnumber 119877
119886 Hence we obtain a function 120596(119905) defined by (72)
which satisfies the lower Riccati differential inequality (73) on[119886 119879lowast
119886) 119879lowast119886isin (119886 119887) such that 120596(a) = 119877
119886and lim
119905rarr119879lowast
119886
120596(119905) =
infin Therefore by 120596(119886) = 119877119886= 120596(119886) and Proposition 24 we
conclude that lim119905rarr119879
lowast
119886
120596(119905) = infin too which is a contradictionwith the above conclusion saying that 120596 isin 119862((119886 119887)R) Thushypothesis (82) is not true and consequently Lemma 1 isshown
For the analogous case 119909(119905) lt 0 on (120591(120591(119886)) 120590(120590(119889))) wealso have 119909(119905) lt 0 on (120591(120591(119888)) 120590(120590(119889))) which implies (since120591(119905) and 120590(119905) are increasing functions)
119909 (119904) lt 0 forall119904 isin (120591 (119888) 120590 (119889)) cup (120591 (120591 (119888)) 120591 (120590 (119889)))
cup (120590 (120591 (119888)) 120590 (120590 (119889)))
(87)
which yields 119909(119905) lt 0 119909(120591(119905)) lt 0 and 119909(120590(119905)) lt 0 on(120591(119888) 120590(119889)) Now we can repeat the preceding procedure buton interval (119888 119889) and using (8) instead of (119886 119887) and (7)
Proof of Lemma 2 From assumption (10) we obtain the exis-tence of an 119899
0isin N such that
int
119887119899
119886119899
119876119899(119905) 119889119905 ge
1198880
2( max119905isin[119886119899 119887119899]
119876119899(119905))
1120574
119899 ge 1198990 (88)
that is
2
1198880
int
119887119899
119886119899
119876119899(119905) 119889119905 ge ( max
119905isin[119886119899 119887119899]119876119899(119905))
1120574
119899 ge 1198990 (89)
Now from (9) and previous inequality we deduce that forlarge enough 120582 120583 120588 and 119899
1199011120574
1199031minus1120574
0
[119896 (120582 120583 120588)]1minus1120574
120587lowast
int
119887119899
119886119899
119876119899(119905) 119889119905
ge2
1198880
int
119887119899
119886119899
119876119899(119905) 119889119905 ge ( max
119905isin[119886119899 119887119899]119876119899(119905))
1120574
(90)
which shows (6) Thus all assumptions of Lemma 1 arefulfilled and hence Lemma 2 immediately follows fromLemma 1
Proof of Lemma 3 Obviously assumption (11) is a particularcase of assumption (9) Hence this proof is very similar tothe proof of Lemma 2 and so it is left to the reader
Proof of Lemma 4 It is clear that from assumption (13) weobtain
1
(max119905isin[119886119899119887119899]119876119899(119905))1120574
int
119887119899
119886119899
119876119899(119905) 119889119905 ge
1198881
1198621120574
0
gt 0 forall119899 ge 1198990
(91)
12 Discrete Dynamics in Nature and Society
Thus hypothesis (12) is fulfilled and therefore Lemma 3proves this lemma
Proof of Theorems 5 6 and 7 This proof is based onLemma 4 In order to simplify notation in many placesin this proof we set 120591(119905) = 119905 minus 120591 and 120590(119905) = 119905 + 120590 Sinceassumptions (2) (3) and (4) have been already supposed inTheorems 5 6 and 7 in order to prove these theorems byLemma 4 we are going to show that the functions 119896(120582 120583 120588)and 119876
119899(119905) explicitly given respectively in (18) (21) or (24)
and (19) (22) or (25) satisfy required conditions (11) and(13) respectively and that every solution 119909(119905) of (27) satisfiesconditions (7) and (8) with respect to functions 119896(120582 120583 120588)and 119876
119899(119905) where 119886 = 119886
2119899minus1 119887 = 119887
2119899minus1 119888 = 119886
2119899 and 119889 = 119887
2119899
The proof that the function 119896(120582 120583 120588) given in (18) (21) or(24) satisfies (11) Passing to the limit in (18) (21) or (24) it isvery simple to show (11)
The proof that the function 119876119899(119905) given in (19) (22) or
(25) satisfies the first claim in (13) From (25) we immediatelyobtain
1003816100381610038161003816120591119899 (119905)1003816100381610038161003816 =
100381610038161003816100381610038161003816100381610038161003816
(119905 minus 119886119899
119905 minus 119886119899+ 120591
)
119901100381610038161003816100381610038161003816100381610038161003816
le 1
1003816100381610038161003816120590119899 (119905)1003816100381610038161003816 =
100381610038161003816100381610038161003816100381610038161003816
(119887119899minus 119905
119887119899minus 119905 + 120590
)
119901100381610038161003816100381610038161003816100381610038161003816
le 1 forall119899 isin N
(92)
Next by assumptions of this corollary we can conclude thatthere are three positive constants 119891
0 1198920 1198900such that |119891(119905)| le
1198910and |119892(119905)| le 119892
0on [1199050infin) in cases (i) and (ii) and
|119890(119905)| le 1198900on [1199050infin) in cases (iii) and (iv) Putting previous
inequalities into (19) (22) or (25) for all 119899 isin N and 119905 isin
[1199050infin) it holds that
1003816100381610038161003816119876119899 (119905)1003816100381610038161003816 le
119902
119901(
119901
119902 minus 119901)
(119902119901)minus1
1198901minus(119901119902)
0119891119901119902
0
delay case with 119902 gt 119901
119902
119901(
119901
119902 minus 119901)
(119902119901)minus1
1198901minus(119901119902)
0119892119901119902
0
advanced case with 119902 gt 119901
1199011
119901(
119901
2 (1199011minus 119901)
)
(1199011119901)minus1
1198901minus(119901119901
1)
0119891119901119902
0+1199012
119901
times(119901
2 (1199012minus 119901)
)
(1199012119901)minus1
1198901minus(119901119901
2)
0119892119901119902
0
delay-advanced case (i)
1198901205780
01198911205781
01198921205782
0
2
prod
119894=0
120578minus120578119894
119894
delay-advanced case (ii) (93)
which shows the first claim in (13)
The proof that the function119876119899(119905) given in (19) (22) or (25)
satisfies the second claim in (13)Without loss of generality weprove this claim only in case (i) since for other cases the prooffollows analogously In this sense let119876
119899(119905) = 119891(119905)120591
119899(119905) Since
1198862119899+1
minus 1198862119899minus1
le 119879lowast 1198872119899+1
minus 1198872119899minus1
ge 119879lowast 1198862119899+2
minus 1198862119899le 119879lowast and
1198872119899+2
minus 1198872119899
ge 119879lowast where 119879
lowastgt 0 is the period of the function
119891(119905) we have 1198862119899minus1
le 1198861+(119899minus1)119879
lowastand 1198872119899minus1
ge 1198871+(119899minus1)119879
lowast
119899 isin N Hence
int
1198872119899minus1
1198862119899minus1
119876119899(119905) 119889119905
= int
1198872119899minus1
1198862119899minus1
119891 (119905) (119905 minus 1198862119899minus1
119905 minus 1198862119899minus1
+ 120591)
119901
119889119905
ge int
1198871+(119899minus1)119879
lowast
1198861+(119899minus1)119879lowast
119891 (119905) (119905 minus 1198861minus (119899 minus 1) 119879
lowast
119905 minus 1198861minus (119899 minus 1) 119879
lowast+ 120591
)
119901
119889119905
= int
1198871
1198861
119891 (119904 + (119899 minus 1) 119879lowast) (
119904 minus 1198861
119904 minus 1198861+ 120591
)
119901
119889119904
= int
1198871
1198861
119891 (119904) (119904 minus 1198861
119904 minus 1198861+ 120591
)
119901
119889119904
(94)
which proves that the integral on the left hand side does notdepend on 119899 isin N that is the second claim in (13) is shown on[1198862119899minus1
1198872119899minus1
] This claim follows in the same way on [1198862119899 1198872119899]
Thus the second claim in (13) is proved on [119886119899 119887119899]
Next to the end of this proof let 119909(119905) be a solu-tion of (1) In particular it implies that (119903(119905)119860(1199091015840(119905)))1015840 =
minus119861(119905 119909(119905) 1199091015840(119905)) minus 120582119865(119905 119909(120591(119905))) minus 120583119866(119905 119909(120590(119905))) + 120588119890(119905) It
together with assumptions (15) (16) (20) and (23) easilygives the next two statements
if 119909 (119905) gt 0 119909 (120591 (119905)) gt 0 119909 (120590 (119905)) gt 0
on (120591 (1198862119899minus1
) 120590 (1198872119899minus1
))
then 119909 (119905) satisfies 119903 (119905) 119860 (1199091015840
(119905)) le 0
on (120591 (1198862119899minus1
) 120590 (1198872119899minus1
)) 119899 ge 1198990
(95)
if 119909 (119905) lt 0 119909 (120591 (119905)) lt 0 119909 (120590 (119905)) lt 0
on (120591 (1198862119899) 120590 (119887
2119899))
then 119909 (119905) satisfies 119903 (119905) 119860 (1199091015840
(119905)) ge 0
on (120591 (1198862119899) 120590 (119887
2119899)) 119899 ge 119899
0
(96)
Now we need the following lemma
Discrete Dynamics in Nature and Society 13
Lemma 25 Let 120591119886119887(119905) and 120590
119886119887(119905) be defined by
120591119886119887(119905) = (
120591 (119905) minus 120591 (119886)
119905 minus 120591 (119886))
119901
120590119886119887(119905) = (
120590 (119887) minus 120590 (119905)
120590 (119887) minus 119905)
119901
119905 isin (119886 119887)
(97)
and let 119909 isin 1198622([1198790infin)R) be an arbitrary function If
(119903(119905)119860(1199091015840(119905)) le 0 for all 119905 isin (120591(119886) 120590(119887)) or (119903(119905)119860(1199091015840(119905)) ge 0
for all 119905 isin (120591(119886) 120590(119887)) then
119909 (120591 (119905))
119909 (119905)ge (120591119886119887(119905))1119901
119909 (120590 (119905))
119909 (119905)ge (120590119886119887(119905))1119901
119905 isin (119886 119887)
(98)
Since119860(V) is supposed to be odd and increasing functionjust before (3) and 119903(119905) satisfies (14) the proof of Lemma 25in the first case that is 119903(119905)119860(1199091015840(119905)) le 0 for all 119905 isin (120591(119886) 120590(119887))is the same as the proof of [9 Corollaries 17 and 18] But in thesecond case that is 119903(119905)119860(1199091015840(119905)) ge 0 for all 119905 isin (120591(119886) 120590(119887))the proof is as follows if previous inequality holds then119903(119905)119860(minus119909
1015840(119905)) le 0 for all 119905 isin (120591(119886) 120590(119887)) and therefore to
the function minus119909(119905) one can apply the first case of this lemmaand consequently one obtains
119909 (120591 (119905))
119909 (119905)=minus119909 (120591 (119905))
minus119909 (119905)ge (120591119886119887(119905))1119901
119909 (120590 (119905))
119909 (119905)=minus119909 (120590 (119905))
minus119909 (119905)ge (120590119886119887(119905))1119901
119905 isin (119886 119887)
(99)
which proves this lemma in the second caseNow combining statements (95) (96) and (98) one
easily obtains
if 119909 (119905) gt 0 119909 (120591 (119905)) gt 0 119909 (120590 (119905)) gt 0
on (120591 (1198862119899minus1
) 120590 (1198872119899minus1
))
then 119909 (119905) satisfies 119909 (120591 (119905))
119909 (119905)ge (120591119899(119905))1119901
on (1198862119899minus1
1198872119899minus1
) 119899 ge 1198990
(100)
if 119909 (119905) lt 0 119909 (120591 (119905)) lt 0 119909 (120590 (119905)) lt 0
on (120591 (1198862119899) 120590 (119887
2119899))
then 119909 (119905) satisfies 119909 (120590 (119905))
119909 (119905)ge (120590119899(119905))1119901
on (1198862119899 1198872119899) 119899 ge 119899
0
(101)
where 120591119899(119905) and 120590
119899(119905) are defined in (26)
The proof that 119909(119905) satisfies (7) and (8) In this proofwe frequently use assumptions (16) (20) and (23) andstatements (100) and (101) Also because of (15) and 119865(119905 119906) =
119891(119905)|119906|1199011 sgn(119906) 119866(119905 119906) = 119892(119905)|119906|
1199012 sgn(119906) in both cases
(100) and (101) we can simultaneously use
minus119890 (119905) (|119909 (119905)|119901minus1
119909 (119905))minus1
= |119890 (119905)| |119909 (119905)|minus119901
ge 0 on 119869119899
119865 (119905 119909 (120591 (119905))) (|119909 (119905)|119901minus1
119909 (119905))minus1
ge 119891 (119905) |119909 (120591 (119905))|1199011 |119909 (119905)|
minus119901ge 0 on 119869
119899
119866 (119905 119909 (120590 (119905))) (|119909 (119905)|119901minus1
119909 (119905))minus1
ge 119892 (119905) |119909 (120590 (119905))|1199012 |119909 (119905)|
minus119901ge 0 on 119869
119899
|119909 (120591 (119905))| |119909 (119905)|minus1=119909 (120591 (119905))
119909 (119905)
|119909 (120590 (119905))| |119909 (119905)|minus1=119909 (120590 (119905))
119909 (119905)on 119869119899
(102)
where 119869119899= (1198862119899minus1
1198872119899minus1
) in the case of (100) and 119869119899= (1198862119899 1198872119899)
in the case of (101)
(i) Delay or Advanced Case with 119902 = 119901 Since 119902 = 119901 we obtain
[120582119865 (119905 119909 (120591 (119905))) + 120583119866 (119905 119909 (120590 (119905))) minus 120588119890 (119905)]
times (|119909 (119905)|119901minus1
119909 (119905))minus1
ge [120582119891 (119905) |119909 (120591 (119905))|119901+ 120583119892 (119905) |119909 (120590 (119905))|
119901
+120588 |119890 (119905)| ] |119909 (119905)|minus119901
ge [120582119891 (119905) |119909 (120591 (119905))|119901+ 120583119892 (119905) |119909 (120590 (119905))|
119901] |119909 (119905)|
minus119901
= 120582119891 (119905) (119909 (120591 (119905))
119909 (119905))
119901
+ 120583119892 (119905) (119909 (120590 (119905))
119909 (119905))
119901
ge 120582119891 (119905) 120591119899(119905) + 120583119892 (119905) 120590
119899(119905) 119905 isin 119869
119899
(103)
where the functions 120591119899(119905) and 120590
119899(119905) are defined in (26)
(ii) Delay Case with 119902 gt 119901 In this part we use the nextelementary inequality
119883120574+ (120574 minus 1) 119884
120574ge 120574119883119884
120574minus1 120574 gt 1 119883 119884 ge 0 (104)
Since 119902 gt 119901 and using (104) especially for
120574 =119902
119901gt 1 119883 = (120582119891 (119905))
119901119902
|119909 (120591 (119905))|119901
119884 = (119901
119902 minus 119901120588 |119890 (119905)|)
119901119902
(105)
14 Discrete Dynamics in Nature and Society
for all 119905 isin 119869119899we obtain
[120582119865 (119905 119909 (120591 (119905))) + 120583119866 (119905 119909 (120590 (119905))) minus 120588119890 (119905)]
times (|119909 (119905)|119901minus1
119909 (119905))minus1
= [120582119865 (119905 119909 (120591 (119905))) minus 120588119890 (119905)] (|119909 (119905)|119901minus1
119909 (119905))minus1
ge [120582119891 (119905) |119909 (120591 (119905))|1199011 + 120588 |119890 (119905)|] |119909 (119905)|
minus119901
= [119883120574+ (120574 minus 1) 119884
120574] |119909 (119905)|
minus119901
ge119902
119901(120582119891 (119905))
119901119902
|119909 (120591 (119905))|119901
times (119901
119902 minus 119901120588 |119890 (119905)|)
(119901119902)((119902119901)minus1)
|119909 (119905)|minus119901
= 120582119901119902
1205881minus(119901119902)
119902
119901(
119901
119902 minus 119901)
1minus(119901119902)
|119890 (119905)|1minus(119901119902)
times (119891 (119905))119901119902
(119909 (120591 (119905))
119909 (119905))
119901
ge 119896 (120582 120583 120588)119902
119901(
119901
119902 minus 119901)
1minus(119901119902)
|119890 (119905)|1minus(119901119902)
times (119891 (119905))119901119902
120591119899(119905)
(106)
where the function 119896(120582 120583 120588) is from (18)
(iii) Advanced Case with 119902 gt 119901 Using the same line ofarguments as in the proof of the previous case for all 119905 isin 119869
119899
we obtain
[120582119865 (119905 119909 (120591 (119905))) + 120583119866 (119905 119909 (120590 (119905))) minus 120588119890 (119905)]
times (|119909 (119905)|119901minus1
119909 (119905))minus1
= [120583119866 (119905 119909 (120590 (119905))) minus 120588119890 (119905)] (|119909 (119905)|119901minus1
119909 (119905))minus1
ge 119896 (120582 120583 120588)119902
119901(
119901
119902 minus 119901)
1minus(119901119902)
|119890 (119905)|1minus(119901119902)
times (119892 (119905))119901119902
120590119899(119905)
(107)
where the function 119896(120582 120583 120588) is from (21)
(iv) Superlinear Delay-Advanced Case Since 1199011 1199012gt 119901 for
all 119905 isin 119869119899we obtain
[120582119865 (119905 119909 (120591 (119905))) + 120583119866 (119905 119909 (120590 (119905))) minus 120588119890 (119905)]
times (|119909 (119905)|119901minus1
119909 (119905))minus1
= [120582119865 (119905 119909 (120591 (119905))) minus120588
2119890 (119905)] (|119909 (119905)|
119901minus1119909 (119905))minus1
+ [120583119866 (119905 119909 (120590 (119905))) minus120588
2119890 (119905)] (|119909 (119905)|
119901minus1119909 (119905))minus1
ge [120582119891 (119905) |119909 (120591 (119905))|1199011 +
120588
2|119890 (119905)|] |119909 (119905)|
minus119901
+ [120583119892 (119905) |119909 (120590 (119905))|1199012 +
120588
2|119890 (119905)|] |119909 (119905)|
minus119901
(108)
Now just the same as in the proofs of previous delay andadvanced cases with 119902 gt 119901 and with the help of (104) inparticular for
120574 =1199011
119901gt 1 119883 = (120582119891 (119905))
1199011199011
|119909 (120591 (119905))|119901
119884 = (119901
1199011minus 119901
120588
2|119890 (119905)|)
1199011199011
(109)
we have
[120582119891 (119905) |119909 (120591 (119905))|1199011 +
120588
2 |119890 (119905)|] |119909 (119905)|
minus119901
= [119883120574+ (120574 minus 1) 119884
120574] |119909 (119905)|
minus119901
ge1199011
119901(120582119891 (119905))
1199011199011
|119909 (120591 (119905))|119901
times (119901
1199011minus 119901
120588
2|119890 (119905)|)
(1199011199011)((1199011119901)minus1)
|119909 (119905)|minus119901
= 12058211990111990111205881minus(119901119901
1)1199011
119901(
119901
2 (1199011minus 119901)
)
1minus(1199011199011)
times |119890 (119905)|1minus(119901119901
1)(119891 (119905))
1199011199011
(119909 (120591 (119905))
119909 (119905))
119901
ge 119896 (120582 120583 120588)1199011
119901(
119901
2 (1199011minus 119901)
)
1minus(1199011199011)
times |119890 (119905)|1minus(119901119901
1)(119891 (119905))
1199011199011
120591119899(119905)
(110)
where the function 119896(120582 120583 120588) is from (24) Analogously weshow that
[120583119892 (119905) |119909 (120590 (119905))|1199012 +
120588
2|119890 (119905)|] |119909 (119905)|
minus119901
ge 119896 (120582 120583 120588)1199012
119901(
119901
2 (1199012minus 119901)
)
1minus(1199011199012)
times |119890 (119905)|1minus(119901119901
2)(119891 (119905))
1199011199012
120590119899(119905)
(111)
Discrete Dynamics in Nature and Society 15
Summarizing previous calculation we obtain
[120582119865 (119905 119909 (120591 (119905))) + 120583119866 (119905 119909 (120590 (119905))) minus 120588119890 (119905)]
times (|119909 (119905)|119901minus1
119909 (119905))minus1
ge 119896 (120582 120583 120588)119876119899(119905) 119905 isin 119869
119899
(112)
where the function 119896(120582 120583 120588) is from (24)
(v) Supersublinear Delay-Advanced Case Since 1199011gt 119901 gt 119901
2
and the following well-known elementary inequality holds
12057801199060+ 12057811199061+ 12057821199062ge 1199061205780
01199061205781
11199061205782
2 120578119894ge 0 119906
119894ge 0 (113)
from 1205780 1205781 1205782isin (0 1) 120578
0+ 1205781+ 1205782= 1 and 119901
11205781+ 11990121205782= 119901
we obtain for all 119905 isin 119869119899 for all 119905 isin 119869
119899we obtain
[120582119865 (119905 119909 (120591 (119905))) + 120583119866 (119905 119909 (120590 (119905))) minus 120588119890 (119905)]
times (|119909 (119905)|119901minus1
119909 (119905))minus1
ge [120582119891 (119905) |119909 (120591 (119905))|1199011 + 120583119892 (119905) |119909 (120590 (119905))|
1199012 + 120588 |119890 (119905)|]
times |119909 (119905)|minus119901
= [1205781[120578minus1
1120582119891 (119905) |119909 (120591 (119905))|
1199011] + 120578
2[120578minus1
2120583119892 (119905) |119909 (120590 (119905))|
1199012]
+1205780[120578minus1
0120588 |119890 (119905)|]] |119909 (119905)|
minus119901
ge [120578minus1
1120582119891 (119905) |119909 (120591 (119905))|
1199011]1205781
[120578minus1
2120583119892 (119905) |119909 (120590 (119905))|
1199012]1205782
times [120578minus1
0120588 |119890 (119905)|]
1205780
|119909 (119905)|minus119901
= 120582120578112058312057821205881205780 |119890 (119905)|
1205780(119891 (119905))
1205781
(119892 (119905))1205782
times|119909 (120591 (119905))|
12057811199011
|119909 (119905)|12057811199011
|119909 (120590 (119905))|12057821199012
|119909 (119905)|12057821199012
2
prod
119894=0
120578minus120578119894
119894
= 119896 (120582 120583 120588) |119890 (119905)|1205780(119891 (119905))
1205781
(119892 (119905))1205782
times (119909 (120591 (119905))
119909 (119905))
12057811199011
(119909 (120590 (119905))
119909 (119905))
12057821199012 2
prod
119894=0
120578minus120578119894
119894
ge 119896 (120582 120583 120588) |119890 (119905)|1205780(119891 (119905))
1205781
(119892 (119905))1205782
(120591119899(119905))1205781(1199011119901)
times (120590119899(119905))1205782(1199012119901)
2
prod
119894=0
120578minus120578119894
119894
= 119896 (120582 120583 120588)119876119899(119905)
(114)
where 119896(120582 120583 120588) and 119876119899(119905) are given respectively in (24) and
(25) Thus it is shown that required condition (5) in thecases (i)ndash(iv) is fulfilled with respect to 119896(120582 120583 120588) and 119876
119899(119905)
determined by (18) (21) or (24) and (19) (22) or (25)In conclusion according to the previous observation we
see that all assumptions of Lemma 4 are fulfilled and henceLemma 4 proves Theorems 5 6 and 7
Conflict of Interests
The author declares that there is no conflict of interestsregarding the publication of this paper
References
[1] Y G Sun ldquoA note on Nasrrsquos and Wongrsquos papersrdquo Journal ofMathematical Analysis and Applications vol 286 no 1 pp 363ndash367 2003
[2] Y G Sun C H Ou and J S W Wong ldquoInterval oscillationtheorems for a second-order linear differential equationrdquo Com-puters amp Mathematics with Applications vol 48 no 10-11 pp1693ndash1699 2004
[3] S Murugadass E Thandapani and S Pinelas ldquoOscillationcriteria for forced second-order mixed type quasilinear delaydifferential equationsrdquo Electronic Journal of Differential Equa-tions vol 2010 article 73 9 pages 2010
[4] Y Bai and L Liu ldquoNew oscillation criteria for second-orderdelay differential equations with mixed nonlinearitiesrdquoDiscreteDynamics in Nature and Society vol 2010 Article ID 796256 9pages 2010
[5] A F Guvenilir andA Zafer ldquoSecond-order oscillation of forcedfunctional differential equations with oscillatory potentialsrdquoComputers amp Mathematics with Applications vol 51 no 9-10pp 1395ndash1404 2006
[6] A Zafer ldquoInterval oscillation criteria for second order super-half linear functional differential equations with delay andadvanced argumentsrdquoMathematische Nachrichten vol 282 no9 pp 1334ndash1341 2009
[7] A F Guvenilir ldquoInterval oscillation of second-order functionaldifferential equations with oscillatory potentialsrdquo NonlinearAnalysis Theory Methods amp Applications vol 71 no 12 ppe2849ndashe2854 2009
[8] T S Hassan L Erbe and A Peterson ldquoForced oscillation ofsecond order differential equations with mixed nonlinearitiesrdquoActa Mathematica Scientia B vol 31 no 2 pp 613ndash626 2011
[9] M Pasic ldquoNew oscillation criteria for second-order forcedquasilinear functional differential equationsrdquo Abstract andApplied Analysis vol 2013 Article ID 735360 12 pages 2013
[10] L H Erbe Q Kong and B G Zhang Oscillation Theory forFunctional-Differential Equations vol 190 Marcel Dekker NewYork NY USA 1995
[11] V Kolmanovskii and A Myshkis Introduction to the Theoryand Applications of Functional-Differential Equations vol 463Kluwer Academic Publishers Dordrecht The Netherlands1999
[12] R P Agarwal M Bohner and W-T Li Nonoscillation andOscillation Theory for Functional Differential Equations vol267 Marcel Dekker New York NY USA 2004
[13] L Erbe T Hassan and A Peterson ldquoOscillation of secondorder functional dynamic equationsrdquo International Journal ofDifference Equations vol 5 no 2 pp 175ndash193 2010
[14] B Baculıkova J Dzurina and Y V Rogovchenko ldquoOscillationof third order trinomial delay differential equationsrdquo AppliedMathematics and Computation vol 218 no 13 pp 7023ndash70332012
[15] R P Agarwal L Berezansky E Braverman and A Domoshnit-sky Nonoscillation Theory of Functional Differential Equationswith Applications Springer New York NY USA 2012
16 Discrete Dynamics in Nature and Society
[16] J Zhang ldquoVariational approach to solitary wave solution ofthe generalized Zakharov equationrdquo Computers ampMathematicswith Applications vol 54 no 7-8 pp 1043ndash1046 2007
[17] T Ozis and A Yıldırım ldquoApplication of Hersquos semi-inversemethod to the nonlinear Schrodinger equationrdquo Computers ampMathematics with Applications vol 54 no 7-8 pp 1039ndash10422007
[18] X-C Cai andM-S Li ldquoPeriodic solution of Jacobi elliptic equa-tions by Hersquos perturbation methodrdquo Computers amp Mathematicswith Applications vol 54 no 7-8 pp 1210ndash1212 2007
[19] S Lenci G Menditto and A M Tarantino ldquoHomoclinic andheteroclinic bifurcations in the non-linear dynamics of a beamresting on an elastic substraterdquo International Journal of Non-Linear Mechanics vol 34 no 4 pp 615ndash632 1999
[20] D-J Huang and H-Q Zhang ldquoLink between travelling wavesand first order nonlinear ordinary differential equation with asixth-degree nonlinear termrdquoChaos Solitons amp Fractals vol 29no 4 pp 928ndash941 2006
[21] A I Maimistov ldquoPropagation of an ultimately short electro-magnetic pulse in a nonlinear medium described by the fifth-order Duffing modelrdquo Optics and Spectroscopy vol 94 pp 251ndash257 2003
[22] M N Hamdan and N H Shabaneh ldquoOn the large amplitudefree vibrations of a restrained uniform beam carrying anintermediate lumpedmassrdquo Journal of Sound andVibration vol199 no 5 pp 711ndash736 1997
[23] J-H He ldquoSome asymptotic methods for strongly nonlinearequationsrdquo International Journal of Modern Physics B vol 20no 10 pp 1141ndash1199 2006
[24] J B Marion Classical Dynamics of Particles and Systems 1970[25] I Kovacic and M J Brennan The Duffing Equation Nonlinear
Oscillatos and their Behaviour John Wiley amp Sons 1st edition2011
[26] F C Moon Chaotic Vibrations An Introduction for AppliedScientists and Engineers John Wiley amp Sons New York NYUSA 2004
[27] J J Stoker Nonlinear Vibrations 1950[28] G Chen and Z Tao ldquoAmplitude-frequency relationship for the
relativistic oscillatorrdquoAASRI Procedia vol 1 pp 400ndash403 2012[29] R E Mickens Oscillations in Planar Dynamic Systems World
Scientific Publishing Singapore 1996[30] A Belendez T Belendez C Neipp A Hernandez and M
L Alvarez ldquoApproximate solutions of a nonlinear oscillatortypified as a mass attached to a stretched elastic wire by thehomotopy perturbation methodrdquo Chaos Solitions and Fractalsvol 39 pp 746ndash764 2009
[31] A Belendez E Fernandez R Fuentes J J Rodes and I PascualldquoHarmonic balancing approach to nonlinear oscillations of apunctual charge in the eletric field of charged ringrdquo PhysicsLetters A vol 373 pp 735ndash740 2009
[32] A Elıas-Zuniga ldquoExact solution of the cubic-quintic Duffingoscillatorrdquo Applied Mathematical Modelling vol 37 no 4 pp2574ndash2579 2013
[33] A Belendez M L Alvarez J Frances et al ldquoAnalytical approx-imate solutions for the cubic-quintic Duffing oscillator in termsof elementary functionsrdquo Journal of Applied Mathematics vol2012 Article ID 286290 16 pages 2012
[34] A Elıas-Zuniga OMartınez-Romero andR K Cordoba-DıazldquoApproximate solution for the Duffing-harmonic oscillator bythe enhanced cubication methodrdquo Mathematical Problems inEngineering vol 2012 Article ID 618750 12 pages 2012
[35] C W Lim B S Wu andW P Sun ldquoHigher accuracy analyticalapproximations to the Duffing-harmonic oscillatorrdquo Journal ofSound and Vibration vol 296 no 4-5 pp 1039ndash1045 2006
[36] J He ldquoSome new approaches to Duffing equation with stronglyand high order nonlinearity II parametrized perturbationtechniquerdquo Communications in Nonlinear Science amp NumericalSimulation vol 4 no 1 pp 81ndash83 1999
[37] V Marinca and N Herisanu ldquoPeriodic solutions for somestrongly nonlinear oscillations by Hersquos variational iterationmethodrdquo Computers amp Mathematics with Applications vol 54no 7-8 pp 1188ndash1196 2007
[38] W Lu and Y Liu ldquoVibration control for the primary resonanceof the Duffing oscillator by a time delay state feedbackrdquoInternational Journal of Nonlinear Science vol 8 no 3 pp 324ndash328 2009
[39] H Y Hu and Z H Wang Dynamics of Controlled MechanicalSystems with Delayed Feedback Springer 2002
[40] M Hamdi and M Belhaq ldquoControl of bistability in a delayedDuffing oscillatorrdquo Advances in Acoustics and Vibration vol2012 Article ID 872498 6 pages 2012
[41] V Ravichandran C Chinnathambi and S Rajasekar ldquoNonlin-ear resonance in Duffing oscillator with fixed and integrativetime-delayed feedbacksrdquoPramana Journal of Physics vol 78 pp347ndash360 2013
[42] X You and H Xu ldquoAnalytical approximations for the periodicmotion of theDuffing systemwith delayed feedbackrdquoNumericalAlgorithms vol 56 no 4 pp 561ndash576 2011
[43] Z Sun W Xu X Yang and T Fang ldquoInducing or suppressingchaos in a double-well Duffing oscillator by time delay feed-backrdquo Chaos Solitons and Fractals vol 27 pp 705ndash714 2006
[44] H Wang H Hu and Z Wang ldquoGlobal dynamics of a Duffingoscillator with delayed displacement feedbackrdquo InternationalJournal of Bifurcation and Chaos in Applied Sciences andEngineering vol 14 no 8 pp 2753ndash2775 2004
[45] J Chiasson and J J LoiseauApplications of Time Delay SystemsSpringer 2007
[46] M Lakshmanan andDV SenthilkumarDynamics of NonlinearTime-Delay Systems Springer 2010
[47] G Stepan T Insperger and R Szalai ldquoDelay parametricexcitation and the nonlinear dynamics of cutting processesrdquoInternational Journal of Bifurcation and Chaos in Applied Sci-ences and Engineering vol 15 no 9 pp 2783ndash2798 2005
[48] U van der Heiden and H-O Walther ldquoExistence of chaos incontrol systems with delayed feedbackrdquo Journal of DifferentialEquations vol 47 no 2 pp 273ndash295 1983
[49] Y G Sun and J S W Wong ldquoOscillation criteria for secondorder forced ordinary differential equations with mixed non-linearitiesrdquo Journal of Mathematical Analysis and Applicationsvol 334 no 1 pp 549ndash560 2007
[50] J Heagy and W L Ditto ldquoDynamics of a two-frequencyparametrically driven Duffing oscillatorrdquo Journal of NonlinearScience vol 1 no 4 pp 423ndash455 1991
[51] A B Belogortsev ldquoBifurcations of tori and chaos in thequasiperiodically forced Duffing oscillatorrdquoNonlinearity vol 5no 4 pp 889ndash897 1992
[52] M Belhaq and M Houssni ldquoQuasi-periodic oscillations chaosand suppression of chaos in a nonlinear oscillator driven byparametric and external excitationsrdquo Nonlinear Dynamics vol18 no 1 pp 1ndash24 1999
[53] S H Saker P Y H Pang and R P Agarwal ldquoOscillationtheorems for second order nonlinear functional differential
Discrete Dynamics in Nature and Society 17
equations with dampingrdquo Dynamic Systems and Applicationsvol 12 no 3-4 pp 307ndash321 2003
[54] I N Bronshtein K A Semendyayev G Musiol and HMuehligHandbook of Mathematics Springer 5th edition 2007
[55] M Pasic ldquoFite-Wintner-Leighton-type oscillation criteria forsecond-order differential equations with nonlinear dampingrdquoAbstract and Applied Analysis vol 2013 Article ID 852180 10pages 2013
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
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Mathematical Problems in Engineering
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Differential EquationsInternational Journal of
Volume 2014
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Mathematical PhysicsAdvances in
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Operations ResearchAdvances in
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Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
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Algebra
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Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
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Decision SciencesAdvances in
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Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
Discrete Dynamics in Nature and Society 9
On the other hand from (6) we observe
1
120587lowast
int119869
119876 (119905) 119889119905 ge1199031minus(1120574)
0
1199011120574[119896 (120582 120583 120588)]1minus(1120574)
(max119905isin 119869
119876 (119905))
1120574
(57)
which together with (55) and (56) gives
1
120587lowast
int119869
min119901
(120576119903 (119905))120574minus1
120576119896 (120582 120583 120588)119876 (119905) 119889119905
= 120576119896 (120582 120583 120588)1
120587lowast
int119869
119876 (119905) 119889119905
ge 1205761199031minus(1120574)
0
1199011120574[119896 (120582 120583 120588)]
1120574
(max119905isin 119869
119876 (119905))
1120574
= 1
(58)
for all 119899 ge 1198990 120582 ge 120582
0 120583 ge 120583
0 and 120588 ge 120588
0 It proves this
lemma
Lemma 20 (an equivalent condition to (54)) Assumption(54) is fulfilled if and only if there is a real number 120576 gt 0 and acontinuous function 119870(119905) ge 0 119905 isin 119869 such that
1198880= int119869
119870 (119905) 119889119905 gt 0119870 (119905)
1198880
le1
120587lowast
timesmin119901
(120576119903 (119905))120574minus1
120576119896 (120582 120583 120588)119876 (119905)
(59)
for all 119905 isin 119869 120582 ge 1205820 120583 ge 120583
0 and 120588 ge 120588
0and some (120582
0 1205830 1205880) isin
R3+
Proof This proof is very elementary Indeed if (54) holdsthen the function119870(119905) and number 119888
0 defined by
119870 (119905) =1
120587lowast
min119901
(120576119903 (119905))120574minus1
120576119896 (120582 120583 120588)119876 (119905)
1198880= int119869
119870 (119905) 119889119905
(60)
obviously satisfy 1198880
ge 1 and 119870(119905)1198880
le 119870(119905) = (1120587lowast)
min119901(120576119903(119905))120574minus1 120576119896(120582 120583 120588)119876(119905) which shows (59) Con-versely if (59) holds then integrating both sides of thesecond inequality in (59) we obtain
int119869
1
120587lowast
min119901
(120576119903 (119905))120574minus1
120576119896 (120582 120583 120588)119876 (119905) ge int119869
119870 (119905)
1198880
119889119905 = 1
(61)
which shows (54)
In conclusion according to previous two lemmas wesee that supposed condition (6) implies (59) which plays animportant role in the proof of the main results
The second step in the proof of Lemma 1 is to prove theexistence of a function 120596(119905) which blows up in the finitetime and satisfies a generalized Riccati differential lowerinequality we briefly present the existence and properties
of the so-called generalized tangent type function In whatfollows let 120587
lowastbe a positive real number defined in (3) Let us
remark that 120572(119904) = 119904120574 120574 gt 1 implies 120587
lowast= (2120587)(120574 sin(120587120574))
see for instance [54] and obviously for 120574 = 2wehave120587lowast= 120587
Lemma 21 Let 120572 [0infin) rarr [0infin) be a continuousfunction such that
int
infin
0
119889120591
1 + 120572 (120591)lt infin (62)
Then there is a real number 120587lowastgt 0 and a function 119911 = 119911(119904)
119911 isin 1198621((minus120587lowast2 120587lowast2)R) such that
119889119911
119889119904= 1 + 120572 (|119911 (119904)|) 119904 isin (minus
120587lowast
2120587lowast
2)
119911 (0) = 0
(63)
Moreover 119911(119904) is increasing and odd
lim119904rarr120587lowast2
119911 (119904) = infin 120587lowast=
2120587
120574 sin (120587120574)for 120572 (119904) = 119904
120574
120574 gt 1
(64)
In particular for 120572(119904) = 1199042 one can take 119911(119904) = tan(119904) and
120587lowast= 120587
Proof Let 119885 = 119885(119905) 119905 isin R be a function defined by
119885 (119905) = int
119905
0
1
1 + 120572 (|120591|)119889120591 119905 isin R (65)
The function 119885(119905) is well defined since 120572(119904) is positive andcontinuous on [0infin) 119885(119905) is increasing and odd functionand
119889119885
119889119905=
1
1 + 120572 (|119905|) 119905 isin R
119885 (0) = 0 119885 isin 1198621
(RR)
(66)
Moreover because of (62) there is a real number 120587lowastgt 0 such
that120587lowast
2= int
infin
0
119889120591
1 + 120572 (120591) (67)
Thus 119885 R rarr (minus120587lowast2 120587lowast2) and there exists an inverse
function 119885minus1 = 119885minus1(119904) of the original function 119885 = 119885(119905) and
119885minus1
(minus120587lowast2 120587lowast2) rarr R Also from 119885(119885
minus1(119904)) = 119904 and
119889119885119889119905 = 0 onR we also derive that119889119885minus1119889119904 = 0 on its domain(minus120587lowast2 120587lowast2) and
119889119885
119889119905(119885minus1
(119904)) =1
(119889119885minus1119889119904) 119904 isin (minus
120587lowast
2120587lowast
2) (68)
Putting 119905 = 119885minus1(119904) for 119904 isin (minus120587
lowast2 120587lowast2) into (66) and using
(68) we easily obtain
119889119885minus1
119889119904= 1 + 120572 (
10038161003816100381610038161003816119885minus1
(119904)10038161003816100381610038161003816) 119904 isin (minus
120587lowast
2120587lowast
2)
119885minus1
(0) = 0 119885minus1isin 1198621((minus
120587lowast
2120587lowast
2) R)
(69)
10 Discrete Dynamics in Nature and Society
Moreover from (67) we have lim119904rarr120587lowast2119885minus1(119904) = 119885
minus1
(lim119905rarrinfin
119885(119905)) = lim119905rarrinfin
119885minus1119885(119905) = lim
119905rarrinfin119905 = infin Thus
if we set 119911(119904) = 119885minus1(119904) then previous two statements and
(67) prove this lemma
Next we prove the main result of this section
Proposition 22 Let (2) and (6) hold where 119869 = (119886 119887) Let 120576 gt0 be a real number and let119870(119905) ge 0 119905 isin [119886 119887] be a continuousfunction both obtained in Lemma 20 Let 120587
lowastbe from (3) and
1198880from (59) and let 119877
119886isin R be an arbitrary real number If
119911 = 119911(119904) is the generalized tangens function defined in (63)and 119881(119905) is a function defined by
119881 (119905) =120587lowast
1198880
int
119905
119886
119870 (120591) 119889120591 + 119911minus1(119877119886) 119905 isin [119886 119887] (70)
then there is a 119879lowast119886isin [119886 119887) such that
119881 (119879lowast
119886) =
120587lowast
2 119881 ([119886 119879
lowast
119886)) sub (minus
120587lowast
2120587lowast
2) (71)
Moreover for a function 120596(119905) defined by120596 (119905) = 119911 (119881 (119905)) 119905 isin [119886 119879
lowast
119886) (72)
one has 120596(119886) = 119877119886 lim119905rarr119879
lowast
119886
120596(119905) = infin and
119889120596
119889119905le
119901
(120576119903 (119905))120574minus1
120572 (1003816100381610038161003816120596 (119905)
1003816100381610038161003816)
+ 120576119896 (120582 120583 120588)119876 (119905) 119905 isin (119886 119879lowast
119886)
(73)
where the numbers 119901 and 120574 are from (3) and the functions119896(120582 120583 120588) and 119876(119905) are from (6)
Proof Under assumptions (2) and (6) and because of Lem-mas 19 and 20 we obtain 120576 gt 0 and 119870(119905) gt 0 119905 isin [119886 119887]satisfying inequality (59)
Next since 119911minus1(119877119886) isin (minus120587
lowast2 120587lowast2) (see Lemma 21)
from (70) we directly obtain
119881 (119886) = 119911minus1(119877119886) lt
120587lowast
2 119881 (119887) = 120587
lowast+ 119911minus1(119877119886) gt
120587lowast
2
(74)Since 119870 isin 119862([119886 119887] [0infin)) we obtain 119881 isin 119862([119886 119887]R) cap
1198621((119886 119887)R) and from (74) we observe that there exist
numbers 119879lowast119886isin (119886 119887) such that119881(119879lowast
119886) = 120587lowast2 Also119870(119905)119888
0ge
0 gives 119881([119886 119879lowast119886)) sub (minus120587
lowast2 120587lowast2) which proves statement
(71) Moreover it together with Lemma 21 and (72) provesthat
lim119905rarr119879
lowast
119886
120596 (119905) = lim119905rarr119879
lowast
119886
119911 (119881 (119905)) = 119911 (120587lowast
2) = infin (75)
Next according to (59) (63) and (72) we make thefollowing calculation on the interval [119886 119879lowast
119886)
1205961015840
(119905) = 1199111015840
(119881 (119905)) 1198811015840
(119905) = [1 + 120572 (|119911 (119881 (119905))|)]120587lowast
1198880
119870 (119905)
= [1 + 120572 (1003816100381610038161003816120596 (119905)
1003816100381610038161003816)]120587lowast
1198880
119870 (119905)
le119901
(120576119903 (119905))120574minus1
120572 (1003816100381610038161003816120596 (119905)
1003816100381610038161003816) + 120576119896 (120582 120583 120588)119876 (119905)
(76)
Thus all assertions of this proposition are proved
Remark 23 In the proof of the main result the number 119877119886
is determined by 119877119886= 120596(119886) where 120596(119905) denotes a function
associated with a nonoscillatory solution and it is given by(84) below
The third step in the proof of Lemma 1 is to show thefollowing pointwise comparison principle for the functions120596and120596 satisfying respectively the lower and upper differentialinequalities (73) and
119889120596
119889119905ge
119901
(120576119903 (119905))120574minus1
120572 (|120596 (119905)|) + 120576119896 (120582 120583 120588)119876 (119905) 119905 isin (119886 119887)
(77)
Proposition 24 Let [119886 119887) sub [1199050infin) be an arbitrary inter-
val One supposes that all coefficients of Riccati differentialinequalities (73) and (77) are continuous and strictly positivefunctions Let 120596 120596 isin 119862
1([119886 119887)R) be two functions satisfying
respectively (73) and (77) on the interval [119886 119887) Then
120596 (119886) le 120596 (119886) 119894119898119901119897119894119890119904 120596 (119905) le 120596 (119905) forall119905 isin [119886 119887) (78)
Proof Let119867(119905 119906) be a function defined by
119867(119905 119906) =119901
(120576119903 (119905))120574minus1
120572 (|119906|) + 120576119896 (120582 120583 120588)119876 (119905)
119905 isin [119886 119887) 119906 isin R
(79)
Let 119868 sub [119886 119887) and 119872 gt 0 be arbitrary For any two 1199061
1199062 minus119872 le 119906
1lt 1199062le 119872 let 119868
12be an interval defined
by 11986812
= (min|1199061| |1199062|max|119906
1| |1199062|) Since 120572(119904) is a 1198621-
function on [0infin) we know by the Lagrange mean valuetheorem applied on 119868
12that there is a 120585 isin 119868
12such that
120572 (10038161003816100381610038161199062
1003816100381610038161003816) minus 120572 (10038161003816100381610038161199061
1003816100381610038161003816)
1199062minus 1199061
le
1003816100381610038161003816120572 (10038161003816100381610038161199062
1003816100381610038161003816) minus 120572 (10038161003816100381610038161199061
1003816100381610038161003816)1003816100381610038161003816
1199062minus 1199061
=100381610038161003816100381610038161205721015840
(120585)10038161003816100381610038161003816
100381610038161003816100381610038161003816100381610038161199062
1003816100381610038161003816 minus10038161003816100381610038161199061
10038161003816100381610038161003816100381610038161003816
1199062minus 1199061
le100381610038161003816100381610038161205721015840
(120585)10038161003816100381610038161003816
le max119904isin11986812
100381610038161003816100381610038161205721015840
(119904)10038161003816100381610038161003816
(80)
since ||1199062| minus |1199061|| le 119906
2minus 1199061 Hence for any 119905 isin 119868 and 119906
1 1199062
minus119872 le 1199061lt 1199062le 119872 we have
119867(119905 1199062) minus 119867 (119905 119906
1)
1199062minus 1199061
= 1205880(119905)
120572 (10038161003816100381610038161199062
1003816100381610038161003816) minus 120572 (10038161003816100381610038161199061
1003816100381610038161003816)
1199062minus 1199061
le 1205880(119905)max119904isin11986812
100381610038161003816100381610038161205721015840
(119904)10038161003816100381610038161003816= 1198710(119905)
(81)
Thus the function119867(119905 119906) from (79) satisfies required condi-tion of [55 Lemma 19] and applying it to (73) and (77) weprove this proposition
Proof of Lemma 1 On the contrary let 119909(119905) be a solution of(1) such that
119909 (119905) = 0 on (120591 (120591 (119886)) 120590 (120590 (119889))) (82)
Discrete Dynamics in Nature and Society 11
that is 119909(119905) gt 0 on (120591(120591(119886)) 120590(120590(119889))) or 119909(119905) lt 0 on(120591(120591(119886)) 120590(120590(119889))) since 119909(119905) is a continuous function on[1199050infin) Let for instance
119909 (119905) gt 0 on (120591 (120591 (119886)) 120590 (120590 (119889))) (83)
Another case can be analogously treated let us see thecomment at the end of this proof In particular from (83)we have 119909(119905) gt 0 on (120591(120591(119886)) 120590(120590(119887))) which implies (since120591(119905) and 120590(119905) are increasing functions) 119909(119904) gt 0 for all 119904 isin
(120591(119886) 120590(119887)) cup (120591(120591(119886)) 120591(120590(119887))) cup (120590(120591(119886)) 120590(120590(119887))) whichyields 119909(119905) gt 0 119909(120591(119905)) gt 0 and 119909(120590(119905)) gt 0 on (120591(119886) 120590(119887))Hence by assumption (7) we may use inequality (5) on theinterval (119886 119887)
Firstly we show that the following classic Riccati transfor-mation of 119909(119905)
120596 (119905) = minus120576119903 (119905) 119860 (119909
1015840(119905))
|119909 (119905)|119901minus1
119909 (119905) 119905 isin (119886 119887) 120576 gt 0 (84)
satisfies upper Riccati differential inequality (77) Let usremark that from (1) we have in particular
minus(119903 (119905) 119860 (1199091015840
(119905)))1015840
= 119861 (119905 119909 (119905) 1199091015840
(119905)) + 120582119865 (119905 119909 (120591 (119905)))
+ 120583119866 (119905 119909 (120590 (119905))) minus 120588119890 (119905) 119905 ge 1199050
(85)
Taking the first derivative on both sides of (84) and usingassumptions (3) (4) and (5) as well as equality (85) and(|119909(119905)|
119901minus1119909(119905))1015840
= 119901|119909(119905)|119901minus1
1199091015840(119905) we obtain
119889120596
119889119905= 120576119901 119903 (119905)
119860 (1199091015840(119905)) 1199091015840(119905)
|119909 (119905)|119901+1
minus 1205761
|119909 (119905)|119901minus1
119909 (119905)(119903 (119905) 119860 (119909
1015840
(119905)))1015840
= 120576119901119903 (119905)119860 (1199091015840(119905)) 1199091015840(119905)
|119909 (119905)|119901+1
+120576
|119909 (119905)|119901minus1
119909 (119905)
times [120582119861 (119905 119909 (119905) 1199091015840
(119905)) + 119865 (119905 119909 (120591 (119905)))
+120583119866 (119905 119909 (120590 (119905))) minus 120588119890 (119905) ]
ge 120576119901119903 (119905)119860 (1199091015840(119905)) 1199091015840(119905)
|119909 (119905)|119901+1
+120576
|119909 (119905)|119901minus1
119909 (119905)
times [120582119865 (119905 119909 (120591 (119905))) + 120583119866 (119905 119909 (120590 (119905))) minus 120588119890 (119905)]
ge 120576119901119903 (119905) 120572(
10038161003816100381610038161003816119860 (1199091015840(119905))
10038161003816100381610038161003816
|119909 (119905)|119901
) + 120576119896 (120582 120583 120588)119876 (119905)
= 120576119901119903 (119905) 120572 (|120596 (119905)|
120576119903 (119905)) + 120576119896 (120582 120583 120588)119876 (119905)
ge119901
(120576119903 (119905))120574minus1
120572 (|120596 (119905)|) + 120576119896 (120582 120583 120588)119876 (119905) 119905 isin (119886 119887)
(86)
Thus according to inequality (5) it is shown that if 119909(119905) isa solution of (1) which satisfies (83) then the function 120596(119905)
defined by (84) satisfies the Riccati differential inequality(77) and 120596 isin 119862((119886 119887)R) On the other hand let 119877
119886be a
real number defined by 119877119886= 120596(119886) According to (6) and
Lemma 19 we obtain (54) which together with Lemma 20ensures that we may use Proposition 22 for such chosen realnumber 119877
119886 Hence we obtain a function 120596(119905) defined by (72)
which satisfies the lower Riccati differential inequality (73) on[119886 119879lowast
119886) 119879lowast119886isin (119886 119887) such that 120596(a) = 119877
119886and lim
119905rarr119879lowast
119886
120596(119905) =
infin Therefore by 120596(119886) = 119877119886= 120596(119886) and Proposition 24 we
conclude that lim119905rarr119879
lowast
119886
120596(119905) = infin too which is a contradictionwith the above conclusion saying that 120596 isin 119862((119886 119887)R) Thushypothesis (82) is not true and consequently Lemma 1 isshown
For the analogous case 119909(119905) lt 0 on (120591(120591(119886)) 120590(120590(119889))) wealso have 119909(119905) lt 0 on (120591(120591(119888)) 120590(120590(119889))) which implies (since120591(119905) and 120590(119905) are increasing functions)
119909 (119904) lt 0 forall119904 isin (120591 (119888) 120590 (119889)) cup (120591 (120591 (119888)) 120591 (120590 (119889)))
cup (120590 (120591 (119888)) 120590 (120590 (119889)))
(87)
which yields 119909(119905) lt 0 119909(120591(119905)) lt 0 and 119909(120590(119905)) lt 0 on(120591(119888) 120590(119889)) Now we can repeat the preceding procedure buton interval (119888 119889) and using (8) instead of (119886 119887) and (7)
Proof of Lemma 2 From assumption (10) we obtain the exis-tence of an 119899
0isin N such that
int
119887119899
119886119899
119876119899(119905) 119889119905 ge
1198880
2( max119905isin[119886119899 119887119899]
119876119899(119905))
1120574
119899 ge 1198990 (88)
that is
2
1198880
int
119887119899
119886119899
119876119899(119905) 119889119905 ge ( max
119905isin[119886119899 119887119899]119876119899(119905))
1120574
119899 ge 1198990 (89)
Now from (9) and previous inequality we deduce that forlarge enough 120582 120583 120588 and 119899
1199011120574
1199031minus1120574
0
[119896 (120582 120583 120588)]1minus1120574
120587lowast
int
119887119899
119886119899
119876119899(119905) 119889119905
ge2
1198880
int
119887119899
119886119899
119876119899(119905) 119889119905 ge ( max
119905isin[119886119899 119887119899]119876119899(119905))
1120574
(90)
which shows (6) Thus all assumptions of Lemma 1 arefulfilled and hence Lemma 2 immediately follows fromLemma 1
Proof of Lemma 3 Obviously assumption (11) is a particularcase of assumption (9) Hence this proof is very similar tothe proof of Lemma 2 and so it is left to the reader
Proof of Lemma 4 It is clear that from assumption (13) weobtain
1
(max119905isin[119886119899119887119899]119876119899(119905))1120574
int
119887119899
119886119899
119876119899(119905) 119889119905 ge
1198881
1198621120574
0
gt 0 forall119899 ge 1198990
(91)
12 Discrete Dynamics in Nature and Society
Thus hypothesis (12) is fulfilled and therefore Lemma 3proves this lemma
Proof of Theorems 5 6 and 7 This proof is based onLemma 4 In order to simplify notation in many placesin this proof we set 120591(119905) = 119905 minus 120591 and 120590(119905) = 119905 + 120590 Sinceassumptions (2) (3) and (4) have been already supposed inTheorems 5 6 and 7 in order to prove these theorems byLemma 4 we are going to show that the functions 119896(120582 120583 120588)and 119876
119899(119905) explicitly given respectively in (18) (21) or (24)
and (19) (22) or (25) satisfy required conditions (11) and(13) respectively and that every solution 119909(119905) of (27) satisfiesconditions (7) and (8) with respect to functions 119896(120582 120583 120588)and 119876
119899(119905) where 119886 = 119886
2119899minus1 119887 = 119887
2119899minus1 119888 = 119886
2119899 and 119889 = 119887
2119899
The proof that the function 119896(120582 120583 120588) given in (18) (21) or(24) satisfies (11) Passing to the limit in (18) (21) or (24) it isvery simple to show (11)
The proof that the function 119876119899(119905) given in (19) (22) or
(25) satisfies the first claim in (13) From (25) we immediatelyobtain
1003816100381610038161003816120591119899 (119905)1003816100381610038161003816 =
100381610038161003816100381610038161003816100381610038161003816
(119905 minus 119886119899
119905 minus 119886119899+ 120591
)
119901100381610038161003816100381610038161003816100381610038161003816
le 1
1003816100381610038161003816120590119899 (119905)1003816100381610038161003816 =
100381610038161003816100381610038161003816100381610038161003816
(119887119899minus 119905
119887119899minus 119905 + 120590
)
119901100381610038161003816100381610038161003816100381610038161003816
le 1 forall119899 isin N
(92)
Next by assumptions of this corollary we can conclude thatthere are three positive constants 119891
0 1198920 1198900such that |119891(119905)| le
1198910and |119892(119905)| le 119892
0on [1199050infin) in cases (i) and (ii) and
|119890(119905)| le 1198900on [1199050infin) in cases (iii) and (iv) Putting previous
inequalities into (19) (22) or (25) for all 119899 isin N and 119905 isin
[1199050infin) it holds that
1003816100381610038161003816119876119899 (119905)1003816100381610038161003816 le
119902
119901(
119901
119902 minus 119901)
(119902119901)minus1
1198901minus(119901119902)
0119891119901119902
0
delay case with 119902 gt 119901
119902
119901(
119901
119902 minus 119901)
(119902119901)minus1
1198901minus(119901119902)
0119892119901119902
0
advanced case with 119902 gt 119901
1199011
119901(
119901
2 (1199011minus 119901)
)
(1199011119901)minus1
1198901minus(119901119901
1)
0119891119901119902
0+1199012
119901
times(119901
2 (1199012minus 119901)
)
(1199012119901)minus1
1198901minus(119901119901
2)
0119892119901119902
0
delay-advanced case (i)
1198901205780
01198911205781
01198921205782
0
2
prod
119894=0
120578minus120578119894
119894
delay-advanced case (ii) (93)
which shows the first claim in (13)
The proof that the function119876119899(119905) given in (19) (22) or (25)
satisfies the second claim in (13)Without loss of generality weprove this claim only in case (i) since for other cases the prooffollows analogously In this sense let119876
119899(119905) = 119891(119905)120591
119899(119905) Since
1198862119899+1
minus 1198862119899minus1
le 119879lowast 1198872119899+1
minus 1198872119899minus1
ge 119879lowast 1198862119899+2
minus 1198862119899le 119879lowast and
1198872119899+2
minus 1198872119899
ge 119879lowast where 119879
lowastgt 0 is the period of the function
119891(119905) we have 1198862119899minus1
le 1198861+(119899minus1)119879
lowastand 1198872119899minus1
ge 1198871+(119899minus1)119879
lowast
119899 isin N Hence
int
1198872119899minus1
1198862119899minus1
119876119899(119905) 119889119905
= int
1198872119899minus1
1198862119899minus1
119891 (119905) (119905 minus 1198862119899minus1
119905 minus 1198862119899minus1
+ 120591)
119901
119889119905
ge int
1198871+(119899minus1)119879
lowast
1198861+(119899minus1)119879lowast
119891 (119905) (119905 minus 1198861minus (119899 minus 1) 119879
lowast
119905 minus 1198861minus (119899 minus 1) 119879
lowast+ 120591
)
119901
119889119905
= int
1198871
1198861
119891 (119904 + (119899 minus 1) 119879lowast) (
119904 minus 1198861
119904 minus 1198861+ 120591
)
119901
119889119904
= int
1198871
1198861
119891 (119904) (119904 minus 1198861
119904 minus 1198861+ 120591
)
119901
119889119904
(94)
which proves that the integral on the left hand side does notdepend on 119899 isin N that is the second claim in (13) is shown on[1198862119899minus1
1198872119899minus1
] This claim follows in the same way on [1198862119899 1198872119899]
Thus the second claim in (13) is proved on [119886119899 119887119899]
Next to the end of this proof let 119909(119905) be a solu-tion of (1) In particular it implies that (119903(119905)119860(1199091015840(119905)))1015840 =
minus119861(119905 119909(119905) 1199091015840(119905)) minus 120582119865(119905 119909(120591(119905))) minus 120583119866(119905 119909(120590(119905))) + 120588119890(119905) It
together with assumptions (15) (16) (20) and (23) easilygives the next two statements
if 119909 (119905) gt 0 119909 (120591 (119905)) gt 0 119909 (120590 (119905)) gt 0
on (120591 (1198862119899minus1
) 120590 (1198872119899minus1
))
then 119909 (119905) satisfies 119903 (119905) 119860 (1199091015840
(119905)) le 0
on (120591 (1198862119899minus1
) 120590 (1198872119899minus1
)) 119899 ge 1198990
(95)
if 119909 (119905) lt 0 119909 (120591 (119905)) lt 0 119909 (120590 (119905)) lt 0
on (120591 (1198862119899) 120590 (119887
2119899))
then 119909 (119905) satisfies 119903 (119905) 119860 (1199091015840
(119905)) ge 0
on (120591 (1198862119899) 120590 (119887
2119899)) 119899 ge 119899
0
(96)
Now we need the following lemma
Discrete Dynamics in Nature and Society 13
Lemma 25 Let 120591119886119887(119905) and 120590
119886119887(119905) be defined by
120591119886119887(119905) = (
120591 (119905) minus 120591 (119886)
119905 minus 120591 (119886))
119901
120590119886119887(119905) = (
120590 (119887) minus 120590 (119905)
120590 (119887) minus 119905)
119901
119905 isin (119886 119887)
(97)
and let 119909 isin 1198622([1198790infin)R) be an arbitrary function If
(119903(119905)119860(1199091015840(119905)) le 0 for all 119905 isin (120591(119886) 120590(119887)) or (119903(119905)119860(1199091015840(119905)) ge 0
for all 119905 isin (120591(119886) 120590(119887)) then
119909 (120591 (119905))
119909 (119905)ge (120591119886119887(119905))1119901
119909 (120590 (119905))
119909 (119905)ge (120590119886119887(119905))1119901
119905 isin (119886 119887)
(98)
Since119860(V) is supposed to be odd and increasing functionjust before (3) and 119903(119905) satisfies (14) the proof of Lemma 25in the first case that is 119903(119905)119860(1199091015840(119905)) le 0 for all 119905 isin (120591(119886) 120590(119887))is the same as the proof of [9 Corollaries 17 and 18] But in thesecond case that is 119903(119905)119860(1199091015840(119905)) ge 0 for all 119905 isin (120591(119886) 120590(119887))the proof is as follows if previous inequality holds then119903(119905)119860(minus119909
1015840(119905)) le 0 for all 119905 isin (120591(119886) 120590(119887)) and therefore to
the function minus119909(119905) one can apply the first case of this lemmaand consequently one obtains
119909 (120591 (119905))
119909 (119905)=minus119909 (120591 (119905))
minus119909 (119905)ge (120591119886119887(119905))1119901
119909 (120590 (119905))
119909 (119905)=minus119909 (120590 (119905))
minus119909 (119905)ge (120590119886119887(119905))1119901
119905 isin (119886 119887)
(99)
which proves this lemma in the second caseNow combining statements (95) (96) and (98) one
easily obtains
if 119909 (119905) gt 0 119909 (120591 (119905)) gt 0 119909 (120590 (119905)) gt 0
on (120591 (1198862119899minus1
) 120590 (1198872119899minus1
))
then 119909 (119905) satisfies 119909 (120591 (119905))
119909 (119905)ge (120591119899(119905))1119901
on (1198862119899minus1
1198872119899minus1
) 119899 ge 1198990
(100)
if 119909 (119905) lt 0 119909 (120591 (119905)) lt 0 119909 (120590 (119905)) lt 0
on (120591 (1198862119899) 120590 (119887
2119899))
then 119909 (119905) satisfies 119909 (120590 (119905))
119909 (119905)ge (120590119899(119905))1119901
on (1198862119899 1198872119899) 119899 ge 119899
0
(101)
where 120591119899(119905) and 120590
119899(119905) are defined in (26)
The proof that 119909(119905) satisfies (7) and (8) In this proofwe frequently use assumptions (16) (20) and (23) andstatements (100) and (101) Also because of (15) and 119865(119905 119906) =
119891(119905)|119906|1199011 sgn(119906) 119866(119905 119906) = 119892(119905)|119906|
1199012 sgn(119906) in both cases
(100) and (101) we can simultaneously use
minus119890 (119905) (|119909 (119905)|119901minus1
119909 (119905))minus1
= |119890 (119905)| |119909 (119905)|minus119901
ge 0 on 119869119899
119865 (119905 119909 (120591 (119905))) (|119909 (119905)|119901minus1
119909 (119905))minus1
ge 119891 (119905) |119909 (120591 (119905))|1199011 |119909 (119905)|
minus119901ge 0 on 119869
119899
119866 (119905 119909 (120590 (119905))) (|119909 (119905)|119901minus1
119909 (119905))minus1
ge 119892 (119905) |119909 (120590 (119905))|1199012 |119909 (119905)|
minus119901ge 0 on 119869
119899
|119909 (120591 (119905))| |119909 (119905)|minus1=119909 (120591 (119905))
119909 (119905)
|119909 (120590 (119905))| |119909 (119905)|minus1=119909 (120590 (119905))
119909 (119905)on 119869119899
(102)
where 119869119899= (1198862119899minus1
1198872119899minus1
) in the case of (100) and 119869119899= (1198862119899 1198872119899)
in the case of (101)
(i) Delay or Advanced Case with 119902 = 119901 Since 119902 = 119901 we obtain
[120582119865 (119905 119909 (120591 (119905))) + 120583119866 (119905 119909 (120590 (119905))) minus 120588119890 (119905)]
times (|119909 (119905)|119901minus1
119909 (119905))minus1
ge [120582119891 (119905) |119909 (120591 (119905))|119901+ 120583119892 (119905) |119909 (120590 (119905))|
119901
+120588 |119890 (119905)| ] |119909 (119905)|minus119901
ge [120582119891 (119905) |119909 (120591 (119905))|119901+ 120583119892 (119905) |119909 (120590 (119905))|
119901] |119909 (119905)|
minus119901
= 120582119891 (119905) (119909 (120591 (119905))
119909 (119905))
119901
+ 120583119892 (119905) (119909 (120590 (119905))
119909 (119905))
119901
ge 120582119891 (119905) 120591119899(119905) + 120583119892 (119905) 120590
119899(119905) 119905 isin 119869
119899
(103)
where the functions 120591119899(119905) and 120590
119899(119905) are defined in (26)
(ii) Delay Case with 119902 gt 119901 In this part we use the nextelementary inequality
119883120574+ (120574 minus 1) 119884
120574ge 120574119883119884
120574minus1 120574 gt 1 119883 119884 ge 0 (104)
Since 119902 gt 119901 and using (104) especially for
120574 =119902
119901gt 1 119883 = (120582119891 (119905))
119901119902
|119909 (120591 (119905))|119901
119884 = (119901
119902 minus 119901120588 |119890 (119905)|)
119901119902
(105)
14 Discrete Dynamics in Nature and Society
for all 119905 isin 119869119899we obtain
[120582119865 (119905 119909 (120591 (119905))) + 120583119866 (119905 119909 (120590 (119905))) minus 120588119890 (119905)]
times (|119909 (119905)|119901minus1
119909 (119905))minus1
= [120582119865 (119905 119909 (120591 (119905))) minus 120588119890 (119905)] (|119909 (119905)|119901minus1
119909 (119905))minus1
ge [120582119891 (119905) |119909 (120591 (119905))|1199011 + 120588 |119890 (119905)|] |119909 (119905)|
minus119901
= [119883120574+ (120574 minus 1) 119884
120574] |119909 (119905)|
minus119901
ge119902
119901(120582119891 (119905))
119901119902
|119909 (120591 (119905))|119901
times (119901
119902 minus 119901120588 |119890 (119905)|)
(119901119902)((119902119901)minus1)
|119909 (119905)|minus119901
= 120582119901119902
1205881minus(119901119902)
119902
119901(
119901
119902 minus 119901)
1minus(119901119902)
|119890 (119905)|1minus(119901119902)
times (119891 (119905))119901119902
(119909 (120591 (119905))
119909 (119905))
119901
ge 119896 (120582 120583 120588)119902
119901(
119901
119902 minus 119901)
1minus(119901119902)
|119890 (119905)|1minus(119901119902)
times (119891 (119905))119901119902
120591119899(119905)
(106)
where the function 119896(120582 120583 120588) is from (18)
(iii) Advanced Case with 119902 gt 119901 Using the same line ofarguments as in the proof of the previous case for all 119905 isin 119869
119899
we obtain
[120582119865 (119905 119909 (120591 (119905))) + 120583119866 (119905 119909 (120590 (119905))) minus 120588119890 (119905)]
times (|119909 (119905)|119901minus1
119909 (119905))minus1
= [120583119866 (119905 119909 (120590 (119905))) minus 120588119890 (119905)] (|119909 (119905)|119901minus1
119909 (119905))minus1
ge 119896 (120582 120583 120588)119902
119901(
119901
119902 minus 119901)
1minus(119901119902)
|119890 (119905)|1minus(119901119902)
times (119892 (119905))119901119902
120590119899(119905)
(107)
where the function 119896(120582 120583 120588) is from (21)
(iv) Superlinear Delay-Advanced Case Since 1199011 1199012gt 119901 for
all 119905 isin 119869119899we obtain
[120582119865 (119905 119909 (120591 (119905))) + 120583119866 (119905 119909 (120590 (119905))) minus 120588119890 (119905)]
times (|119909 (119905)|119901minus1
119909 (119905))minus1
= [120582119865 (119905 119909 (120591 (119905))) minus120588
2119890 (119905)] (|119909 (119905)|
119901minus1119909 (119905))minus1
+ [120583119866 (119905 119909 (120590 (119905))) minus120588
2119890 (119905)] (|119909 (119905)|
119901minus1119909 (119905))minus1
ge [120582119891 (119905) |119909 (120591 (119905))|1199011 +
120588
2|119890 (119905)|] |119909 (119905)|
minus119901
+ [120583119892 (119905) |119909 (120590 (119905))|1199012 +
120588
2|119890 (119905)|] |119909 (119905)|
minus119901
(108)
Now just the same as in the proofs of previous delay andadvanced cases with 119902 gt 119901 and with the help of (104) inparticular for
120574 =1199011
119901gt 1 119883 = (120582119891 (119905))
1199011199011
|119909 (120591 (119905))|119901
119884 = (119901
1199011minus 119901
120588
2|119890 (119905)|)
1199011199011
(109)
we have
[120582119891 (119905) |119909 (120591 (119905))|1199011 +
120588
2 |119890 (119905)|] |119909 (119905)|
minus119901
= [119883120574+ (120574 minus 1) 119884
120574] |119909 (119905)|
minus119901
ge1199011
119901(120582119891 (119905))
1199011199011
|119909 (120591 (119905))|119901
times (119901
1199011minus 119901
120588
2|119890 (119905)|)
(1199011199011)((1199011119901)minus1)
|119909 (119905)|minus119901
= 12058211990111990111205881minus(119901119901
1)1199011
119901(
119901
2 (1199011minus 119901)
)
1minus(1199011199011)
times |119890 (119905)|1minus(119901119901
1)(119891 (119905))
1199011199011
(119909 (120591 (119905))
119909 (119905))
119901
ge 119896 (120582 120583 120588)1199011
119901(
119901
2 (1199011minus 119901)
)
1minus(1199011199011)
times |119890 (119905)|1minus(119901119901
1)(119891 (119905))
1199011199011
120591119899(119905)
(110)
where the function 119896(120582 120583 120588) is from (24) Analogously weshow that
[120583119892 (119905) |119909 (120590 (119905))|1199012 +
120588
2|119890 (119905)|] |119909 (119905)|
minus119901
ge 119896 (120582 120583 120588)1199012
119901(
119901
2 (1199012minus 119901)
)
1minus(1199011199012)
times |119890 (119905)|1minus(119901119901
2)(119891 (119905))
1199011199012
120590119899(119905)
(111)
Discrete Dynamics in Nature and Society 15
Summarizing previous calculation we obtain
[120582119865 (119905 119909 (120591 (119905))) + 120583119866 (119905 119909 (120590 (119905))) minus 120588119890 (119905)]
times (|119909 (119905)|119901minus1
119909 (119905))minus1
ge 119896 (120582 120583 120588)119876119899(119905) 119905 isin 119869
119899
(112)
where the function 119896(120582 120583 120588) is from (24)
(v) Supersublinear Delay-Advanced Case Since 1199011gt 119901 gt 119901
2
and the following well-known elementary inequality holds
12057801199060+ 12057811199061+ 12057821199062ge 1199061205780
01199061205781
11199061205782
2 120578119894ge 0 119906
119894ge 0 (113)
from 1205780 1205781 1205782isin (0 1) 120578
0+ 1205781+ 1205782= 1 and 119901
11205781+ 11990121205782= 119901
we obtain for all 119905 isin 119869119899 for all 119905 isin 119869
119899we obtain
[120582119865 (119905 119909 (120591 (119905))) + 120583119866 (119905 119909 (120590 (119905))) minus 120588119890 (119905)]
times (|119909 (119905)|119901minus1
119909 (119905))minus1
ge [120582119891 (119905) |119909 (120591 (119905))|1199011 + 120583119892 (119905) |119909 (120590 (119905))|
1199012 + 120588 |119890 (119905)|]
times |119909 (119905)|minus119901
= [1205781[120578minus1
1120582119891 (119905) |119909 (120591 (119905))|
1199011] + 120578
2[120578minus1
2120583119892 (119905) |119909 (120590 (119905))|
1199012]
+1205780[120578minus1
0120588 |119890 (119905)|]] |119909 (119905)|
minus119901
ge [120578minus1
1120582119891 (119905) |119909 (120591 (119905))|
1199011]1205781
[120578minus1
2120583119892 (119905) |119909 (120590 (119905))|
1199012]1205782
times [120578minus1
0120588 |119890 (119905)|]
1205780
|119909 (119905)|minus119901
= 120582120578112058312057821205881205780 |119890 (119905)|
1205780(119891 (119905))
1205781
(119892 (119905))1205782
times|119909 (120591 (119905))|
12057811199011
|119909 (119905)|12057811199011
|119909 (120590 (119905))|12057821199012
|119909 (119905)|12057821199012
2
prod
119894=0
120578minus120578119894
119894
= 119896 (120582 120583 120588) |119890 (119905)|1205780(119891 (119905))
1205781
(119892 (119905))1205782
times (119909 (120591 (119905))
119909 (119905))
12057811199011
(119909 (120590 (119905))
119909 (119905))
12057821199012 2
prod
119894=0
120578minus120578119894
119894
ge 119896 (120582 120583 120588) |119890 (119905)|1205780(119891 (119905))
1205781
(119892 (119905))1205782
(120591119899(119905))1205781(1199011119901)
times (120590119899(119905))1205782(1199012119901)
2
prod
119894=0
120578minus120578119894
119894
= 119896 (120582 120583 120588)119876119899(119905)
(114)
where 119896(120582 120583 120588) and 119876119899(119905) are given respectively in (24) and
(25) Thus it is shown that required condition (5) in thecases (i)ndash(iv) is fulfilled with respect to 119896(120582 120583 120588) and 119876
119899(119905)
determined by (18) (21) or (24) and (19) (22) or (25)In conclusion according to the previous observation we
see that all assumptions of Lemma 4 are fulfilled and henceLemma 4 proves Theorems 5 6 and 7
Conflict of Interests
The author declares that there is no conflict of interestsregarding the publication of this paper
References
[1] Y G Sun ldquoA note on Nasrrsquos and Wongrsquos papersrdquo Journal ofMathematical Analysis and Applications vol 286 no 1 pp 363ndash367 2003
[2] Y G Sun C H Ou and J S W Wong ldquoInterval oscillationtheorems for a second-order linear differential equationrdquo Com-puters amp Mathematics with Applications vol 48 no 10-11 pp1693ndash1699 2004
[3] S Murugadass E Thandapani and S Pinelas ldquoOscillationcriteria for forced second-order mixed type quasilinear delaydifferential equationsrdquo Electronic Journal of Differential Equa-tions vol 2010 article 73 9 pages 2010
[4] Y Bai and L Liu ldquoNew oscillation criteria for second-orderdelay differential equations with mixed nonlinearitiesrdquoDiscreteDynamics in Nature and Society vol 2010 Article ID 796256 9pages 2010
[5] A F Guvenilir andA Zafer ldquoSecond-order oscillation of forcedfunctional differential equations with oscillatory potentialsrdquoComputers amp Mathematics with Applications vol 51 no 9-10pp 1395ndash1404 2006
[6] A Zafer ldquoInterval oscillation criteria for second order super-half linear functional differential equations with delay andadvanced argumentsrdquoMathematische Nachrichten vol 282 no9 pp 1334ndash1341 2009
[7] A F Guvenilir ldquoInterval oscillation of second-order functionaldifferential equations with oscillatory potentialsrdquo NonlinearAnalysis Theory Methods amp Applications vol 71 no 12 ppe2849ndashe2854 2009
[8] T S Hassan L Erbe and A Peterson ldquoForced oscillation ofsecond order differential equations with mixed nonlinearitiesrdquoActa Mathematica Scientia B vol 31 no 2 pp 613ndash626 2011
[9] M Pasic ldquoNew oscillation criteria for second-order forcedquasilinear functional differential equationsrdquo Abstract andApplied Analysis vol 2013 Article ID 735360 12 pages 2013
[10] L H Erbe Q Kong and B G Zhang Oscillation Theory forFunctional-Differential Equations vol 190 Marcel Dekker NewYork NY USA 1995
[11] V Kolmanovskii and A Myshkis Introduction to the Theoryand Applications of Functional-Differential Equations vol 463Kluwer Academic Publishers Dordrecht The Netherlands1999
[12] R P Agarwal M Bohner and W-T Li Nonoscillation andOscillation Theory for Functional Differential Equations vol267 Marcel Dekker New York NY USA 2004
[13] L Erbe T Hassan and A Peterson ldquoOscillation of secondorder functional dynamic equationsrdquo International Journal ofDifference Equations vol 5 no 2 pp 175ndash193 2010
[14] B Baculıkova J Dzurina and Y V Rogovchenko ldquoOscillationof third order trinomial delay differential equationsrdquo AppliedMathematics and Computation vol 218 no 13 pp 7023ndash70332012
[15] R P Agarwal L Berezansky E Braverman and A Domoshnit-sky Nonoscillation Theory of Functional Differential Equationswith Applications Springer New York NY USA 2012
16 Discrete Dynamics in Nature and Society
[16] J Zhang ldquoVariational approach to solitary wave solution ofthe generalized Zakharov equationrdquo Computers ampMathematicswith Applications vol 54 no 7-8 pp 1043ndash1046 2007
[17] T Ozis and A Yıldırım ldquoApplication of Hersquos semi-inversemethod to the nonlinear Schrodinger equationrdquo Computers ampMathematics with Applications vol 54 no 7-8 pp 1039ndash10422007
[18] X-C Cai andM-S Li ldquoPeriodic solution of Jacobi elliptic equa-tions by Hersquos perturbation methodrdquo Computers amp Mathematicswith Applications vol 54 no 7-8 pp 1210ndash1212 2007
[19] S Lenci G Menditto and A M Tarantino ldquoHomoclinic andheteroclinic bifurcations in the non-linear dynamics of a beamresting on an elastic substraterdquo International Journal of Non-Linear Mechanics vol 34 no 4 pp 615ndash632 1999
[20] D-J Huang and H-Q Zhang ldquoLink between travelling wavesand first order nonlinear ordinary differential equation with asixth-degree nonlinear termrdquoChaos Solitons amp Fractals vol 29no 4 pp 928ndash941 2006
[21] A I Maimistov ldquoPropagation of an ultimately short electro-magnetic pulse in a nonlinear medium described by the fifth-order Duffing modelrdquo Optics and Spectroscopy vol 94 pp 251ndash257 2003
[22] M N Hamdan and N H Shabaneh ldquoOn the large amplitudefree vibrations of a restrained uniform beam carrying anintermediate lumpedmassrdquo Journal of Sound andVibration vol199 no 5 pp 711ndash736 1997
[23] J-H He ldquoSome asymptotic methods for strongly nonlinearequationsrdquo International Journal of Modern Physics B vol 20no 10 pp 1141ndash1199 2006
[24] J B Marion Classical Dynamics of Particles and Systems 1970[25] I Kovacic and M J Brennan The Duffing Equation Nonlinear
Oscillatos and their Behaviour John Wiley amp Sons 1st edition2011
[26] F C Moon Chaotic Vibrations An Introduction for AppliedScientists and Engineers John Wiley amp Sons New York NYUSA 2004
[27] J J Stoker Nonlinear Vibrations 1950[28] G Chen and Z Tao ldquoAmplitude-frequency relationship for the
relativistic oscillatorrdquoAASRI Procedia vol 1 pp 400ndash403 2012[29] R E Mickens Oscillations in Planar Dynamic Systems World
Scientific Publishing Singapore 1996[30] A Belendez T Belendez C Neipp A Hernandez and M
L Alvarez ldquoApproximate solutions of a nonlinear oscillatortypified as a mass attached to a stretched elastic wire by thehomotopy perturbation methodrdquo Chaos Solitions and Fractalsvol 39 pp 746ndash764 2009
[31] A Belendez E Fernandez R Fuentes J J Rodes and I PascualldquoHarmonic balancing approach to nonlinear oscillations of apunctual charge in the eletric field of charged ringrdquo PhysicsLetters A vol 373 pp 735ndash740 2009
[32] A Elıas-Zuniga ldquoExact solution of the cubic-quintic Duffingoscillatorrdquo Applied Mathematical Modelling vol 37 no 4 pp2574ndash2579 2013
[33] A Belendez M L Alvarez J Frances et al ldquoAnalytical approx-imate solutions for the cubic-quintic Duffing oscillator in termsof elementary functionsrdquo Journal of Applied Mathematics vol2012 Article ID 286290 16 pages 2012
[34] A Elıas-Zuniga OMartınez-Romero andR K Cordoba-DıazldquoApproximate solution for the Duffing-harmonic oscillator bythe enhanced cubication methodrdquo Mathematical Problems inEngineering vol 2012 Article ID 618750 12 pages 2012
[35] C W Lim B S Wu andW P Sun ldquoHigher accuracy analyticalapproximations to the Duffing-harmonic oscillatorrdquo Journal ofSound and Vibration vol 296 no 4-5 pp 1039ndash1045 2006
[36] J He ldquoSome new approaches to Duffing equation with stronglyand high order nonlinearity II parametrized perturbationtechniquerdquo Communications in Nonlinear Science amp NumericalSimulation vol 4 no 1 pp 81ndash83 1999
[37] V Marinca and N Herisanu ldquoPeriodic solutions for somestrongly nonlinear oscillations by Hersquos variational iterationmethodrdquo Computers amp Mathematics with Applications vol 54no 7-8 pp 1188ndash1196 2007
[38] W Lu and Y Liu ldquoVibration control for the primary resonanceof the Duffing oscillator by a time delay state feedbackrdquoInternational Journal of Nonlinear Science vol 8 no 3 pp 324ndash328 2009
[39] H Y Hu and Z H Wang Dynamics of Controlled MechanicalSystems with Delayed Feedback Springer 2002
[40] M Hamdi and M Belhaq ldquoControl of bistability in a delayedDuffing oscillatorrdquo Advances in Acoustics and Vibration vol2012 Article ID 872498 6 pages 2012
[41] V Ravichandran C Chinnathambi and S Rajasekar ldquoNonlin-ear resonance in Duffing oscillator with fixed and integrativetime-delayed feedbacksrdquoPramana Journal of Physics vol 78 pp347ndash360 2013
[42] X You and H Xu ldquoAnalytical approximations for the periodicmotion of theDuffing systemwith delayed feedbackrdquoNumericalAlgorithms vol 56 no 4 pp 561ndash576 2011
[43] Z Sun W Xu X Yang and T Fang ldquoInducing or suppressingchaos in a double-well Duffing oscillator by time delay feed-backrdquo Chaos Solitons and Fractals vol 27 pp 705ndash714 2006
[44] H Wang H Hu and Z Wang ldquoGlobal dynamics of a Duffingoscillator with delayed displacement feedbackrdquo InternationalJournal of Bifurcation and Chaos in Applied Sciences andEngineering vol 14 no 8 pp 2753ndash2775 2004
[45] J Chiasson and J J LoiseauApplications of Time Delay SystemsSpringer 2007
[46] M Lakshmanan andDV SenthilkumarDynamics of NonlinearTime-Delay Systems Springer 2010
[47] G Stepan T Insperger and R Szalai ldquoDelay parametricexcitation and the nonlinear dynamics of cutting processesrdquoInternational Journal of Bifurcation and Chaos in Applied Sci-ences and Engineering vol 15 no 9 pp 2783ndash2798 2005
[48] U van der Heiden and H-O Walther ldquoExistence of chaos incontrol systems with delayed feedbackrdquo Journal of DifferentialEquations vol 47 no 2 pp 273ndash295 1983
[49] Y G Sun and J S W Wong ldquoOscillation criteria for secondorder forced ordinary differential equations with mixed non-linearitiesrdquo Journal of Mathematical Analysis and Applicationsvol 334 no 1 pp 549ndash560 2007
[50] J Heagy and W L Ditto ldquoDynamics of a two-frequencyparametrically driven Duffing oscillatorrdquo Journal of NonlinearScience vol 1 no 4 pp 423ndash455 1991
[51] A B Belogortsev ldquoBifurcations of tori and chaos in thequasiperiodically forced Duffing oscillatorrdquoNonlinearity vol 5no 4 pp 889ndash897 1992
[52] M Belhaq and M Houssni ldquoQuasi-periodic oscillations chaosand suppression of chaos in a nonlinear oscillator driven byparametric and external excitationsrdquo Nonlinear Dynamics vol18 no 1 pp 1ndash24 1999
[53] S H Saker P Y H Pang and R P Agarwal ldquoOscillationtheorems for second order nonlinear functional differential
Discrete Dynamics in Nature and Society 17
equations with dampingrdquo Dynamic Systems and Applicationsvol 12 no 3-4 pp 307ndash321 2003
[54] I N Bronshtein K A Semendyayev G Musiol and HMuehligHandbook of Mathematics Springer 5th edition 2007
[55] M Pasic ldquoFite-Wintner-Leighton-type oscillation criteria forsecond-order differential equations with nonlinear dampingrdquoAbstract and Applied Analysis vol 2013 Article ID 852180 10pages 2013
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical Problems in Engineering
Hindawi Publishing Corporationhttpwwwhindawicom
Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
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Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
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Mathematical PhysicsAdvances in
Complex AnalysisJournal of
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OptimizationJournal of
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CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
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Operations ResearchAdvances in
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Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
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Algebra
Discrete Dynamics in Nature and Society
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
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Decision SciencesAdvances in
Discrete MathematicsJournal of
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Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
10 Discrete Dynamics in Nature and Society
Moreover from (67) we have lim119904rarr120587lowast2119885minus1(119904) = 119885
minus1
(lim119905rarrinfin
119885(119905)) = lim119905rarrinfin
119885minus1119885(119905) = lim
119905rarrinfin119905 = infin Thus
if we set 119911(119904) = 119885minus1(119904) then previous two statements and
(67) prove this lemma
Next we prove the main result of this section
Proposition 22 Let (2) and (6) hold where 119869 = (119886 119887) Let 120576 gt0 be a real number and let119870(119905) ge 0 119905 isin [119886 119887] be a continuousfunction both obtained in Lemma 20 Let 120587
lowastbe from (3) and
1198880from (59) and let 119877
119886isin R be an arbitrary real number If
119911 = 119911(119904) is the generalized tangens function defined in (63)and 119881(119905) is a function defined by
119881 (119905) =120587lowast
1198880
int
119905
119886
119870 (120591) 119889120591 + 119911minus1(119877119886) 119905 isin [119886 119887] (70)
then there is a 119879lowast119886isin [119886 119887) such that
119881 (119879lowast
119886) =
120587lowast
2 119881 ([119886 119879
lowast
119886)) sub (minus
120587lowast
2120587lowast
2) (71)
Moreover for a function 120596(119905) defined by120596 (119905) = 119911 (119881 (119905)) 119905 isin [119886 119879
lowast
119886) (72)
one has 120596(119886) = 119877119886 lim119905rarr119879
lowast
119886
120596(119905) = infin and
119889120596
119889119905le
119901
(120576119903 (119905))120574minus1
120572 (1003816100381610038161003816120596 (119905)
1003816100381610038161003816)
+ 120576119896 (120582 120583 120588)119876 (119905) 119905 isin (119886 119879lowast
119886)
(73)
where the numbers 119901 and 120574 are from (3) and the functions119896(120582 120583 120588) and 119876(119905) are from (6)
Proof Under assumptions (2) and (6) and because of Lem-mas 19 and 20 we obtain 120576 gt 0 and 119870(119905) gt 0 119905 isin [119886 119887]satisfying inequality (59)
Next since 119911minus1(119877119886) isin (minus120587
lowast2 120587lowast2) (see Lemma 21)
from (70) we directly obtain
119881 (119886) = 119911minus1(119877119886) lt
120587lowast
2 119881 (119887) = 120587
lowast+ 119911minus1(119877119886) gt
120587lowast
2
(74)Since 119870 isin 119862([119886 119887] [0infin)) we obtain 119881 isin 119862([119886 119887]R) cap
1198621((119886 119887)R) and from (74) we observe that there exist
numbers 119879lowast119886isin (119886 119887) such that119881(119879lowast
119886) = 120587lowast2 Also119870(119905)119888
0ge
0 gives 119881([119886 119879lowast119886)) sub (minus120587
lowast2 120587lowast2) which proves statement
(71) Moreover it together with Lemma 21 and (72) provesthat
lim119905rarr119879
lowast
119886
120596 (119905) = lim119905rarr119879
lowast
119886
119911 (119881 (119905)) = 119911 (120587lowast
2) = infin (75)
Next according to (59) (63) and (72) we make thefollowing calculation on the interval [119886 119879lowast
119886)
1205961015840
(119905) = 1199111015840
(119881 (119905)) 1198811015840
(119905) = [1 + 120572 (|119911 (119881 (119905))|)]120587lowast
1198880
119870 (119905)
= [1 + 120572 (1003816100381610038161003816120596 (119905)
1003816100381610038161003816)]120587lowast
1198880
119870 (119905)
le119901
(120576119903 (119905))120574minus1
120572 (1003816100381610038161003816120596 (119905)
1003816100381610038161003816) + 120576119896 (120582 120583 120588)119876 (119905)
(76)
Thus all assertions of this proposition are proved
Remark 23 In the proof of the main result the number 119877119886
is determined by 119877119886= 120596(119886) where 120596(119905) denotes a function
associated with a nonoscillatory solution and it is given by(84) below
The third step in the proof of Lemma 1 is to show thefollowing pointwise comparison principle for the functions120596and120596 satisfying respectively the lower and upper differentialinequalities (73) and
119889120596
119889119905ge
119901
(120576119903 (119905))120574minus1
120572 (|120596 (119905)|) + 120576119896 (120582 120583 120588)119876 (119905) 119905 isin (119886 119887)
(77)
Proposition 24 Let [119886 119887) sub [1199050infin) be an arbitrary inter-
val One supposes that all coefficients of Riccati differentialinequalities (73) and (77) are continuous and strictly positivefunctions Let 120596 120596 isin 119862
1([119886 119887)R) be two functions satisfying
respectively (73) and (77) on the interval [119886 119887) Then
120596 (119886) le 120596 (119886) 119894119898119901119897119894119890119904 120596 (119905) le 120596 (119905) forall119905 isin [119886 119887) (78)
Proof Let119867(119905 119906) be a function defined by
119867(119905 119906) =119901
(120576119903 (119905))120574minus1
120572 (|119906|) + 120576119896 (120582 120583 120588)119876 (119905)
119905 isin [119886 119887) 119906 isin R
(79)
Let 119868 sub [119886 119887) and 119872 gt 0 be arbitrary For any two 1199061
1199062 minus119872 le 119906
1lt 1199062le 119872 let 119868
12be an interval defined
by 11986812
= (min|1199061| |1199062|max|119906
1| |1199062|) Since 120572(119904) is a 1198621-
function on [0infin) we know by the Lagrange mean valuetheorem applied on 119868
12that there is a 120585 isin 119868
12such that
120572 (10038161003816100381610038161199062
1003816100381610038161003816) minus 120572 (10038161003816100381610038161199061
1003816100381610038161003816)
1199062minus 1199061
le
1003816100381610038161003816120572 (10038161003816100381610038161199062
1003816100381610038161003816) minus 120572 (10038161003816100381610038161199061
1003816100381610038161003816)1003816100381610038161003816
1199062minus 1199061
=100381610038161003816100381610038161205721015840
(120585)10038161003816100381610038161003816
100381610038161003816100381610038161003816100381610038161199062
1003816100381610038161003816 minus10038161003816100381610038161199061
10038161003816100381610038161003816100381610038161003816
1199062minus 1199061
le100381610038161003816100381610038161205721015840
(120585)10038161003816100381610038161003816
le max119904isin11986812
100381610038161003816100381610038161205721015840
(119904)10038161003816100381610038161003816
(80)
since ||1199062| minus |1199061|| le 119906
2minus 1199061 Hence for any 119905 isin 119868 and 119906
1 1199062
minus119872 le 1199061lt 1199062le 119872 we have
119867(119905 1199062) minus 119867 (119905 119906
1)
1199062minus 1199061
= 1205880(119905)
120572 (10038161003816100381610038161199062
1003816100381610038161003816) minus 120572 (10038161003816100381610038161199061
1003816100381610038161003816)
1199062minus 1199061
le 1205880(119905)max119904isin11986812
100381610038161003816100381610038161205721015840
(119904)10038161003816100381610038161003816= 1198710(119905)
(81)
Thus the function119867(119905 119906) from (79) satisfies required condi-tion of [55 Lemma 19] and applying it to (73) and (77) weprove this proposition
Proof of Lemma 1 On the contrary let 119909(119905) be a solution of(1) such that
119909 (119905) = 0 on (120591 (120591 (119886)) 120590 (120590 (119889))) (82)
Discrete Dynamics in Nature and Society 11
that is 119909(119905) gt 0 on (120591(120591(119886)) 120590(120590(119889))) or 119909(119905) lt 0 on(120591(120591(119886)) 120590(120590(119889))) since 119909(119905) is a continuous function on[1199050infin) Let for instance
119909 (119905) gt 0 on (120591 (120591 (119886)) 120590 (120590 (119889))) (83)
Another case can be analogously treated let us see thecomment at the end of this proof In particular from (83)we have 119909(119905) gt 0 on (120591(120591(119886)) 120590(120590(119887))) which implies (since120591(119905) and 120590(119905) are increasing functions) 119909(119904) gt 0 for all 119904 isin
(120591(119886) 120590(119887)) cup (120591(120591(119886)) 120591(120590(119887))) cup (120590(120591(119886)) 120590(120590(119887))) whichyields 119909(119905) gt 0 119909(120591(119905)) gt 0 and 119909(120590(119905)) gt 0 on (120591(119886) 120590(119887))Hence by assumption (7) we may use inequality (5) on theinterval (119886 119887)
Firstly we show that the following classic Riccati transfor-mation of 119909(119905)
120596 (119905) = minus120576119903 (119905) 119860 (119909
1015840(119905))
|119909 (119905)|119901minus1
119909 (119905) 119905 isin (119886 119887) 120576 gt 0 (84)
satisfies upper Riccati differential inequality (77) Let usremark that from (1) we have in particular
minus(119903 (119905) 119860 (1199091015840
(119905)))1015840
= 119861 (119905 119909 (119905) 1199091015840
(119905)) + 120582119865 (119905 119909 (120591 (119905)))
+ 120583119866 (119905 119909 (120590 (119905))) minus 120588119890 (119905) 119905 ge 1199050
(85)
Taking the first derivative on both sides of (84) and usingassumptions (3) (4) and (5) as well as equality (85) and(|119909(119905)|
119901minus1119909(119905))1015840
= 119901|119909(119905)|119901minus1
1199091015840(119905) we obtain
119889120596
119889119905= 120576119901 119903 (119905)
119860 (1199091015840(119905)) 1199091015840(119905)
|119909 (119905)|119901+1
minus 1205761
|119909 (119905)|119901minus1
119909 (119905)(119903 (119905) 119860 (119909
1015840
(119905)))1015840
= 120576119901119903 (119905)119860 (1199091015840(119905)) 1199091015840(119905)
|119909 (119905)|119901+1
+120576
|119909 (119905)|119901minus1
119909 (119905)
times [120582119861 (119905 119909 (119905) 1199091015840
(119905)) + 119865 (119905 119909 (120591 (119905)))
+120583119866 (119905 119909 (120590 (119905))) minus 120588119890 (119905) ]
ge 120576119901119903 (119905)119860 (1199091015840(119905)) 1199091015840(119905)
|119909 (119905)|119901+1
+120576
|119909 (119905)|119901minus1
119909 (119905)
times [120582119865 (119905 119909 (120591 (119905))) + 120583119866 (119905 119909 (120590 (119905))) minus 120588119890 (119905)]
ge 120576119901119903 (119905) 120572(
10038161003816100381610038161003816119860 (1199091015840(119905))
10038161003816100381610038161003816
|119909 (119905)|119901
) + 120576119896 (120582 120583 120588)119876 (119905)
= 120576119901119903 (119905) 120572 (|120596 (119905)|
120576119903 (119905)) + 120576119896 (120582 120583 120588)119876 (119905)
ge119901
(120576119903 (119905))120574minus1
120572 (|120596 (119905)|) + 120576119896 (120582 120583 120588)119876 (119905) 119905 isin (119886 119887)
(86)
Thus according to inequality (5) it is shown that if 119909(119905) isa solution of (1) which satisfies (83) then the function 120596(119905)
defined by (84) satisfies the Riccati differential inequality(77) and 120596 isin 119862((119886 119887)R) On the other hand let 119877
119886be a
real number defined by 119877119886= 120596(119886) According to (6) and
Lemma 19 we obtain (54) which together with Lemma 20ensures that we may use Proposition 22 for such chosen realnumber 119877
119886 Hence we obtain a function 120596(119905) defined by (72)
which satisfies the lower Riccati differential inequality (73) on[119886 119879lowast
119886) 119879lowast119886isin (119886 119887) such that 120596(a) = 119877
119886and lim
119905rarr119879lowast
119886
120596(119905) =
infin Therefore by 120596(119886) = 119877119886= 120596(119886) and Proposition 24 we
conclude that lim119905rarr119879
lowast
119886
120596(119905) = infin too which is a contradictionwith the above conclusion saying that 120596 isin 119862((119886 119887)R) Thushypothesis (82) is not true and consequently Lemma 1 isshown
For the analogous case 119909(119905) lt 0 on (120591(120591(119886)) 120590(120590(119889))) wealso have 119909(119905) lt 0 on (120591(120591(119888)) 120590(120590(119889))) which implies (since120591(119905) and 120590(119905) are increasing functions)
119909 (119904) lt 0 forall119904 isin (120591 (119888) 120590 (119889)) cup (120591 (120591 (119888)) 120591 (120590 (119889)))
cup (120590 (120591 (119888)) 120590 (120590 (119889)))
(87)
which yields 119909(119905) lt 0 119909(120591(119905)) lt 0 and 119909(120590(119905)) lt 0 on(120591(119888) 120590(119889)) Now we can repeat the preceding procedure buton interval (119888 119889) and using (8) instead of (119886 119887) and (7)
Proof of Lemma 2 From assumption (10) we obtain the exis-tence of an 119899
0isin N such that
int
119887119899
119886119899
119876119899(119905) 119889119905 ge
1198880
2( max119905isin[119886119899 119887119899]
119876119899(119905))
1120574
119899 ge 1198990 (88)
that is
2
1198880
int
119887119899
119886119899
119876119899(119905) 119889119905 ge ( max
119905isin[119886119899 119887119899]119876119899(119905))
1120574
119899 ge 1198990 (89)
Now from (9) and previous inequality we deduce that forlarge enough 120582 120583 120588 and 119899
1199011120574
1199031minus1120574
0
[119896 (120582 120583 120588)]1minus1120574
120587lowast
int
119887119899
119886119899
119876119899(119905) 119889119905
ge2
1198880
int
119887119899
119886119899
119876119899(119905) 119889119905 ge ( max
119905isin[119886119899 119887119899]119876119899(119905))
1120574
(90)
which shows (6) Thus all assumptions of Lemma 1 arefulfilled and hence Lemma 2 immediately follows fromLemma 1
Proof of Lemma 3 Obviously assumption (11) is a particularcase of assumption (9) Hence this proof is very similar tothe proof of Lemma 2 and so it is left to the reader
Proof of Lemma 4 It is clear that from assumption (13) weobtain
1
(max119905isin[119886119899119887119899]119876119899(119905))1120574
int
119887119899
119886119899
119876119899(119905) 119889119905 ge
1198881
1198621120574
0
gt 0 forall119899 ge 1198990
(91)
12 Discrete Dynamics in Nature and Society
Thus hypothesis (12) is fulfilled and therefore Lemma 3proves this lemma
Proof of Theorems 5 6 and 7 This proof is based onLemma 4 In order to simplify notation in many placesin this proof we set 120591(119905) = 119905 minus 120591 and 120590(119905) = 119905 + 120590 Sinceassumptions (2) (3) and (4) have been already supposed inTheorems 5 6 and 7 in order to prove these theorems byLemma 4 we are going to show that the functions 119896(120582 120583 120588)and 119876
119899(119905) explicitly given respectively in (18) (21) or (24)
and (19) (22) or (25) satisfy required conditions (11) and(13) respectively and that every solution 119909(119905) of (27) satisfiesconditions (7) and (8) with respect to functions 119896(120582 120583 120588)and 119876
119899(119905) where 119886 = 119886
2119899minus1 119887 = 119887
2119899minus1 119888 = 119886
2119899 and 119889 = 119887
2119899
The proof that the function 119896(120582 120583 120588) given in (18) (21) or(24) satisfies (11) Passing to the limit in (18) (21) or (24) it isvery simple to show (11)
The proof that the function 119876119899(119905) given in (19) (22) or
(25) satisfies the first claim in (13) From (25) we immediatelyobtain
1003816100381610038161003816120591119899 (119905)1003816100381610038161003816 =
100381610038161003816100381610038161003816100381610038161003816
(119905 minus 119886119899
119905 minus 119886119899+ 120591
)
119901100381610038161003816100381610038161003816100381610038161003816
le 1
1003816100381610038161003816120590119899 (119905)1003816100381610038161003816 =
100381610038161003816100381610038161003816100381610038161003816
(119887119899minus 119905
119887119899minus 119905 + 120590
)
119901100381610038161003816100381610038161003816100381610038161003816
le 1 forall119899 isin N
(92)
Next by assumptions of this corollary we can conclude thatthere are three positive constants 119891
0 1198920 1198900such that |119891(119905)| le
1198910and |119892(119905)| le 119892
0on [1199050infin) in cases (i) and (ii) and
|119890(119905)| le 1198900on [1199050infin) in cases (iii) and (iv) Putting previous
inequalities into (19) (22) or (25) for all 119899 isin N and 119905 isin
[1199050infin) it holds that
1003816100381610038161003816119876119899 (119905)1003816100381610038161003816 le
119902
119901(
119901
119902 minus 119901)
(119902119901)minus1
1198901minus(119901119902)
0119891119901119902
0
delay case with 119902 gt 119901
119902
119901(
119901
119902 minus 119901)
(119902119901)minus1
1198901minus(119901119902)
0119892119901119902
0
advanced case with 119902 gt 119901
1199011
119901(
119901
2 (1199011minus 119901)
)
(1199011119901)minus1
1198901minus(119901119901
1)
0119891119901119902
0+1199012
119901
times(119901
2 (1199012minus 119901)
)
(1199012119901)minus1
1198901minus(119901119901
2)
0119892119901119902
0
delay-advanced case (i)
1198901205780
01198911205781
01198921205782
0
2
prod
119894=0
120578minus120578119894
119894
delay-advanced case (ii) (93)
which shows the first claim in (13)
The proof that the function119876119899(119905) given in (19) (22) or (25)
satisfies the second claim in (13)Without loss of generality weprove this claim only in case (i) since for other cases the prooffollows analogously In this sense let119876
119899(119905) = 119891(119905)120591
119899(119905) Since
1198862119899+1
minus 1198862119899minus1
le 119879lowast 1198872119899+1
minus 1198872119899minus1
ge 119879lowast 1198862119899+2
minus 1198862119899le 119879lowast and
1198872119899+2
minus 1198872119899
ge 119879lowast where 119879
lowastgt 0 is the period of the function
119891(119905) we have 1198862119899minus1
le 1198861+(119899minus1)119879
lowastand 1198872119899minus1
ge 1198871+(119899minus1)119879
lowast
119899 isin N Hence
int
1198872119899minus1
1198862119899minus1
119876119899(119905) 119889119905
= int
1198872119899minus1
1198862119899minus1
119891 (119905) (119905 minus 1198862119899minus1
119905 minus 1198862119899minus1
+ 120591)
119901
119889119905
ge int
1198871+(119899minus1)119879
lowast
1198861+(119899minus1)119879lowast
119891 (119905) (119905 minus 1198861minus (119899 minus 1) 119879
lowast
119905 minus 1198861minus (119899 minus 1) 119879
lowast+ 120591
)
119901
119889119905
= int
1198871
1198861
119891 (119904 + (119899 minus 1) 119879lowast) (
119904 minus 1198861
119904 minus 1198861+ 120591
)
119901
119889119904
= int
1198871
1198861
119891 (119904) (119904 minus 1198861
119904 minus 1198861+ 120591
)
119901
119889119904
(94)
which proves that the integral on the left hand side does notdepend on 119899 isin N that is the second claim in (13) is shown on[1198862119899minus1
1198872119899minus1
] This claim follows in the same way on [1198862119899 1198872119899]
Thus the second claim in (13) is proved on [119886119899 119887119899]
Next to the end of this proof let 119909(119905) be a solu-tion of (1) In particular it implies that (119903(119905)119860(1199091015840(119905)))1015840 =
minus119861(119905 119909(119905) 1199091015840(119905)) minus 120582119865(119905 119909(120591(119905))) minus 120583119866(119905 119909(120590(119905))) + 120588119890(119905) It
together with assumptions (15) (16) (20) and (23) easilygives the next two statements
if 119909 (119905) gt 0 119909 (120591 (119905)) gt 0 119909 (120590 (119905)) gt 0
on (120591 (1198862119899minus1
) 120590 (1198872119899minus1
))
then 119909 (119905) satisfies 119903 (119905) 119860 (1199091015840
(119905)) le 0
on (120591 (1198862119899minus1
) 120590 (1198872119899minus1
)) 119899 ge 1198990
(95)
if 119909 (119905) lt 0 119909 (120591 (119905)) lt 0 119909 (120590 (119905)) lt 0
on (120591 (1198862119899) 120590 (119887
2119899))
then 119909 (119905) satisfies 119903 (119905) 119860 (1199091015840
(119905)) ge 0
on (120591 (1198862119899) 120590 (119887
2119899)) 119899 ge 119899
0
(96)
Now we need the following lemma
Discrete Dynamics in Nature and Society 13
Lemma 25 Let 120591119886119887(119905) and 120590
119886119887(119905) be defined by
120591119886119887(119905) = (
120591 (119905) minus 120591 (119886)
119905 minus 120591 (119886))
119901
120590119886119887(119905) = (
120590 (119887) minus 120590 (119905)
120590 (119887) minus 119905)
119901
119905 isin (119886 119887)
(97)
and let 119909 isin 1198622([1198790infin)R) be an arbitrary function If
(119903(119905)119860(1199091015840(119905)) le 0 for all 119905 isin (120591(119886) 120590(119887)) or (119903(119905)119860(1199091015840(119905)) ge 0
for all 119905 isin (120591(119886) 120590(119887)) then
119909 (120591 (119905))
119909 (119905)ge (120591119886119887(119905))1119901
119909 (120590 (119905))
119909 (119905)ge (120590119886119887(119905))1119901
119905 isin (119886 119887)
(98)
Since119860(V) is supposed to be odd and increasing functionjust before (3) and 119903(119905) satisfies (14) the proof of Lemma 25in the first case that is 119903(119905)119860(1199091015840(119905)) le 0 for all 119905 isin (120591(119886) 120590(119887))is the same as the proof of [9 Corollaries 17 and 18] But in thesecond case that is 119903(119905)119860(1199091015840(119905)) ge 0 for all 119905 isin (120591(119886) 120590(119887))the proof is as follows if previous inequality holds then119903(119905)119860(minus119909
1015840(119905)) le 0 for all 119905 isin (120591(119886) 120590(119887)) and therefore to
the function minus119909(119905) one can apply the first case of this lemmaand consequently one obtains
119909 (120591 (119905))
119909 (119905)=minus119909 (120591 (119905))
minus119909 (119905)ge (120591119886119887(119905))1119901
119909 (120590 (119905))
119909 (119905)=minus119909 (120590 (119905))
minus119909 (119905)ge (120590119886119887(119905))1119901
119905 isin (119886 119887)
(99)
which proves this lemma in the second caseNow combining statements (95) (96) and (98) one
easily obtains
if 119909 (119905) gt 0 119909 (120591 (119905)) gt 0 119909 (120590 (119905)) gt 0
on (120591 (1198862119899minus1
) 120590 (1198872119899minus1
))
then 119909 (119905) satisfies 119909 (120591 (119905))
119909 (119905)ge (120591119899(119905))1119901
on (1198862119899minus1
1198872119899minus1
) 119899 ge 1198990
(100)
if 119909 (119905) lt 0 119909 (120591 (119905)) lt 0 119909 (120590 (119905)) lt 0
on (120591 (1198862119899) 120590 (119887
2119899))
then 119909 (119905) satisfies 119909 (120590 (119905))
119909 (119905)ge (120590119899(119905))1119901
on (1198862119899 1198872119899) 119899 ge 119899
0
(101)
where 120591119899(119905) and 120590
119899(119905) are defined in (26)
The proof that 119909(119905) satisfies (7) and (8) In this proofwe frequently use assumptions (16) (20) and (23) andstatements (100) and (101) Also because of (15) and 119865(119905 119906) =
119891(119905)|119906|1199011 sgn(119906) 119866(119905 119906) = 119892(119905)|119906|
1199012 sgn(119906) in both cases
(100) and (101) we can simultaneously use
minus119890 (119905) (|119909 (119905)|119901minus1
119909 (119905))minus1
= |119890 (119905)| |119909 (119905)|minus119901
ge 0 on 119869119899
119865 (119905 119909 (120591 (119905))) (|119909 (119905)|119901minus1
119909 (119905))minus1
ge 119891 (119905) |119909 (120591 (119905))|1199011 |119909 (119905)|
minus119901ge 0 on 119869
119899
119866 (119905 119909 (120590 (119905))) (|119909 (119905)|119901minus1
119909 (119905))minus1
ge 119892 (119905) |119909 (120590 (119905))|1199012 |119909 (119905)|
minus119901ge 0 on 119869
119899
|119909 (120591 (119905))| |119909 (119905)|minus1=119909 (120591 (119905))
119909 (119905)
|119909 (120590 (119905))| |119909 (119905)|minus1=119909 (120590 (119905))
119909 (119905)on 119869119899
(102)
where 119869119899= (1198862119899minus1
1198872119899minus1
) in the case of (100) and 119869119899= (1198862119899 1198872119899)
in the case of (101)
(i) Delay or Advanced Case with 119902 = 119901 Since 119902 = 119901 we obtain
[120582119865 (119905 119909 (120591 (119905))) + 120583119866 (119905 119909 (120590 (119905))) minus 120588119890 (119905)]
times (|119909 (119905)|119901minus1
119909 (119905))minus1
ge [120582119891 (119905) |119909 (120591 (119905))|119901+ 120583119892 (119905) |119909 (120590 (119905))|
119901
+120588 |119890 (119905)| ] |119909 (119905)|minus119901
ge [120582119891 (119905) |119909 (120591 (119905))|119901+ 120583119892 (119905) |119909 (120590 (119905))|
119901] |119909 (119905)|
minus119901
= 120582119891 (119905) (119909 (120591 (119905))
119909 (119905))
119901
+ 120583119892 (119905) (119909 (120590 (119905))
119909 (119905))
119901
ge 120582119891 (119905) 120591119899(119905) + 120583119892 (119905) 120590
119899(119905) 119905 isin 119869
119899
(103)
where the functions 120591119899(119905) and 120590
119899(119905) are defined in (26)
(ii) Delay Case with 119902 gt 119901 In this part we use the nextelementary inequality
119883120574+ (120574 minus 1) 119884
120574ge 120574119883119884
120574minus1 120574 gt 1 119883 119884 ge 0 (104)
Since 119902 gt 119901 and using (104) especially for
120574 =119902
119901gt 1 119883 = (120582119891 (119905))
119901119902
|119909 (120591 (119905))|119901
119884 = (119901
119902 minus 119901120588 |119890 (119905)|)
119901119902
(105)
14 Discrete Dynamics in Nature and Society
for all 119905 isin 119869119899we obtain
[120582119865 (119905 119909 (120591 (119905))) + 120583119866 (119905 119909 (120590 (119905))) minus 120588119890 (119905)]
times (|119909 (119905)|119901minus1
119909 (119905))minus1
= [120582119865 (119905 119909 (120591 (119905))) minus 120588119890 (119905)] (|119909 (119905)|119901minus1
119909 (119905))minus1
ge [120582119891 (119905) |119909 (120591 (119905))|1199011 + 120588 |119890 (119905)|] |119909 (119905)|
minus119901
= [119883120574+ (120574 minus 1) 119884
120574] |119909 (119905)|
minus119901
ge119902
119901(120582119891 (119905))
119901119902
|119909 (120591 (119905))|119901
times (119901
119902 minus 119901120588 |119890 (119905)|)
(119901119902)((119902119901)minus1)
|119909 (119905)|minus119901
= 120582119901119902
1205881minus(119901119902)
119902
119901(
119901
119902 minus 119901)
1minus(119901119902)
|119890 (119905)|1minus(119901119902)
times (119891 (119905))119901119902
(119909 (120591 (119905))
119909 (119905))
119901
ge 119896 (120582 120583 120588)119902
119901(
119901
119902 minus 119901)
1minus(119901119902)
|119890 (119905)|1minus(119901119902)
times (119891 (119905))119901119902
120591119899(119905)
(106)
where the function 119896(120582 120583 120588) is from (18)
(iii) Advanced Case with 119902 gt 119901 Using the same line ofarguments as in the proof of the previous case for all 119905 isin 119869
119899
we obtain
[120582119865 (119905 119909 (120591 (119905))) + 120583119866 (119905 119909 (120590 (119905))) minus 120588119890 (119905)]
times (|119909 (119905)|119901minus1
119909 (119905))minus1
= [120583119866 (119905 119909 (120590 (119905))) minus 120588119890 (119905)] (|119909 (119905)|119901minus1
119909 (119905))minus1
ge 119896 (120582 120583 120588)119902
119901(
119901
119902 minus 119901)
1minus(119901119902)
|119890 (119905)|1minus(119901119902)
times (119892 (119905))119901119902
120590119899(119905)
(107)
where the function 119896(120582 120583 120588) is from (21)
(iv) Superlinear Delay-Advanced Case Since 1199011 1199012gt 119901 for
all 119905 isin 119869119899we obtain
[120582119865 (119905 119909 (120591 (119905))) + 120583119866 (119905 119909 (120590 (119905))) minus 120588119890 (119905)]
times (|119909 (119905)|119901minus1
119909 (119905))minus1
= [120582119865 (119905 119909 (120591 (119905))) minus120588
2119890 (119905)] (|119909 (119905)|
119901minus1119909 (119905))minus1
+ [120583119866 (119905 119909 (120590 (119905))) minus120588
2119890 (119905)] (|119909 (119905)|
119901minus1119909 (119905))minus1
ge [120582119891 (119905) |119909 (120591 (119905))|1199011 +
120588
2|119890 (119905)|] |119909 (119905)|
minus119901
+ [120583119892 (119905) |119909 (120590 (119905))|1199012 +
120588
2|119890 (119905)|] |119909 (119905)|
minus119901
(108)
Now just the same as in the proofs of previous delay andadvanced cases with 119902 gt 119901 and with the help of (104) inparticular for
120574 =1199011
119901gt 1 119883 = (120582119891 (119905))
1199011199011
|119909 (120591 (119905))|119901
119884 = (119901
1199011minus 119901
120588
2|119890 (119905)|)
1199011199011
(109)
we have
[120582119891 (119905) |119909 (120591 (119905))|1199011 +
120588
2 |119890 (119905)|] |119909 (119905)|
minus119901
= [119883120574+ (120574 minus 1) 119884
120574] |119909 (119905)|
minus119901
ge1199011
119901(120582119891 (119905))
1199011199011
|119909 (120591 (119905))|119901
times (119901
1199011minus 119901
120588
2|119890 (119905)|)
(1199011199011)((1199011119901)minus1)
|119909 (119905)|minus119901
= 12058211990111990111205881minus(119901119901
1)1199011
119901(
119901
2 (1199011minus 119901)
)
1minus(1199011199011)
times |119890 (119905)|1minus(119901119901
1)(119891 (119905))
1199011199011
(119909 (120591 (119905))
119909 (119905))
119901
ge 119896 (120582 120583 120588)1199011
119901(
119901
2 (1199011minus 119901)
)
1minus(1199011199011)
times |119890 (119905)|1minus(119901119901
1)(119891 (119905))
1199011199011
120591119899(119905)
(110)
where the function 119896(120582 120583 120588) is from (24) Analogously weshow that
[120583119892 (119905) |119909 (120590 (119905))|1199012 +
120588
2|119890 (119905)|] |119909 (119905)|
minus119901
ge 119896 (120582 120583 120588)1199012
119901(
119901
2 (1199012minus 119901)
)
1minus(1199011199012)
times |119890 (119905)|1minus(119901119901
2)(119891 (119905))
1199011199012
120590119899(119905)
(111)
Discrete Dynamics in Nature and Society 15
Summarizing previous calculation we obtain
[120582119865 (119905 119909 (120591 (119905))) + 120583119866 (119905 119909 (120590 (119905))) minus 120588119890 (119905)]
times (|119909 (119905)|119901minus1
119909 (119905))minus1
ge 119896 (120582 120583 120588)119876119899(119905) 119905 isin 119869
119899
(112)
where the function 119896(120582 120583 120588) is from (24)
(v) Supersublinear Delay-Advanced Case Since 1199011gt 119901 gt 119901
2
and the following well-known elementary inequality holds
12057801199060+ 12057811199061+ 12057821199062ge 1199061205780
01199061205781
11199061205782
2 120578119894ge 0 119906
119894ge 0 (113)
from 1205780 1205781 1205782isin (0 1) 120578
0+ 1205781+ 1205782= 1 and 119901
11205781+ 11990121205782= 119901
we obtain for all 119905 isin 119869119899 for all 119905 isin 119869
119899we obtain
[120582119865 (119905 119909 (120591 (119905))) + 120583119866 (119905 119909 (120590 (119905))) minus 120588119890 (119905)]
times (|119909 (119905)|119901minus1
119909 (119905))minus1
ge [120582119891 (119905) |119909 (120591 (119905))|1199011 + 120583119892 (119905) |119909 (120590 (119905))|
1199012 + 120588 |119890 (119905)|]
times |119909 (119905)|minus119901
= [1205781[120578minus1
1120582119891 (119905) |119909 (120591 (119905))|
1199011] + 120578
2[120578minus1
2120583119892 (119905) |119909 (120590 (119905))|
1199012]
+1205780[120578minus1
0120588 |119890 (119905)|]] |119909 (119905)|
minus119901
ge [120578minus1
1120582119891 (119905) |119909 (120591 (119905))|
1199011]1205781
[120578minus1
2120583119892 (119905) |119909 (120590 (119905))|
1199012]1205782
times [120578minus1
0120588 |119890 (119905)|]
1205780
|119909 (119905)|minus119901
= 120582120578112058312057821205881205780 |119890 (119905)|
1205780(119891 (119905))
1205781
(119892 (119905))1205782
times|119909 (120591 (119905))|
12057811199011
|119909 (119905)|12057811199011
|119909 (120590 (119905))|12057821199012
|119909 (119905)|12057821199012
2
prod
119894=0
120578minus120578119894
119894
= 119896 (120582 120583 120588) |119890 (119905)|1205780(119891 (119905))
1205781
(119892 (119905))1205782
times (119909 (120591 (119905))
119909 (119905))
12057811199011
(119909 (120590 (119905))
119909 (119905))
12057821199012 2
prod
119894=0
120578minus120578119894
119894
ge 119896 (120582 120583 120588) |119890 (119905)|1205780(119891 (119905))
1205781
(119892 (119905))1205782
(120591119899(119905))1205781(1199011119901)
times (120590119899(119905))1205782(1199012119901)
2
prod
119894=0
120578minus120578119894
119894
= 119896 (120582 120583 120588)119876119899(119905)
(114)
where 119896(120582 120583 120588) and 119876119899(119905) are given respectively in (24) and
(25) Thus it is shown that required condition (5) in thecases (i)ndash(iv) is fulfilled with respect to 119896(120582 120583 120588) and 119876
119899(119905)
determined by (18) (21) or (24) and (19) (22) or (25)In conclusion according to the previous observation we
see that all assumptions of Lemma 4 are fulfilled and henceLemma 4 proves Theorems 5 6 and 7
Conflict of Interests
The author declares that there is no conflict of interestsregarding the publication of this paper
References
[1] Y G Sun ldquoA note on Nasrrsquos and Wongrsquos papersrdquo Journal ofMathematical Analysis and Applications vol 286 no 1 pp 363ndash367 2003
[2] Y G Sun C H Ou and J S W Wong ldquoInterval oscillationtheorems for a second-order linear differential equationrdquo Com-puters amp Mathematics with Applications vol 48 no 10-11 pp1693ndash1699 2004
[3] S Murugadass E Thandapani and S Pinelas ldquoOscillationcriteria for forced second-order mixed type quasilinear delaydifferential equationsrdquo Electronic Journal of Differential Equa-tions vol 2010 article 73 9 pages 2010
[4] Y Bai and L Liu ldquoNew oscillation criteria for second-orderdelay differential equations with mixed nonlinearitiesrdquoDiscreteDynamics in Nature and Society vol 2010 Article ID 796256 9pages 2010
[5] A F Guvenilir andA Zafer ldquoSecond-order oscillation of forcedfunctional differential equations with oscillatory potentialsrdquoComputers amp Mathematics with Applications vol 51 no 9-10pp 1395ndash1404 2006
[6] A Zafer ldquoInterval oscillation criteria for second order super-half linear functional differential equations with delay andadvanced argumentsrdquoMathematische Nachrichten vol 282 no9 pp 1334ndash1341 2009
[7] A F Guvenilir ldquoInterval oscillation of second-order functionaldifferential equations with oscillatory potentialsrdquo NonlinearAnalysis Theory Methods amp Applications vol 71 no 12 ppe2849ndashe2854 2009
[8] T S Hassan L Erbe and A Peterson ldquoForced oscillation ofsecond order differential equations with mixed nonlinearitiesrdquoActa Mathematica Scientia B vol 31 no 2 pp 613ndash626 2011
[9] M Pasic ldquoNew oscillation criteria for second-order forcedquasilinear functional differential equationsrdquo Abstract andApplied Analysis vol 2013 Article ID 735360 12 pages 2013
[10] L H Erbe Q Kong and B G Zhang Oscillation Theory forFunctional-Differential Equations vol 190 Marcel Dekker NewYork NY USA 1995
[11] V Kolmanovskii and A Myshkis Introduction to the Theoryand Applications of Functional-Differential Equations vol 463Kluwer Academic Publishers Dordrecht The Netherlands1999
[12] R P Agarwal M Bohner and W-T Li Nonoscillation andOscillation Theory for Functional Differential Equations vol267 Marcel Dekker New York NY USA 2004
[13] L Erbe T Hassan and A Peterson ldquoOscillation of secondorder functional dynamic equationsrdquo International Journal ofDifference Equations vol 5 no 2 pp 175ndash193 2010
[14] B Baculıkova J Dzurina and Y V Rogovchenko ldquoOscillationof third order trinomial delay differential equationsrdquo AppliedMathematics and Computation vol 218 no 13 pp 7023ndash70332012
[15] R P Agarwal L Berezansky E Braverman and A Domoshnit-sky Nonoscillation Theory of Functional Differential Equationswith Applications Springer New York NY USA 2012
16 Discrete Dynamics in Nature and Society
[16] J Zhang ldquoVariational approach to solitary wave solution ofthe generalized Zakharov equationrdquo Computers ampMathematicswith Applications vol 54 no 7-8 pp 1043ndash1046 2007
[17] T Ozis and A Yıldırım ldquoApplication of Hersquos semi-inversemethod to the nonlinear Schrodinger equationrdquo Computers ampMathematics with Applications vol 54 no 7-8 pp 1039ndash10422007
[18] X-C Cai andM-S Li ldquoPeriodic solution of Jacobi elliptic equa-tions by Hersquos perturbation methodrdquo Computers amp Mathematicswith Applications vol 54 no 7-8 pp 1210ndash1212 2007
[19] S Lenci G Menditto and A M Tarantino ldquoHomoclinic andheteroclinic bifurcations in the non-linear dynamics of a beamresting on an elastic substraterdquo International Journal of Non-Linear Mechanics vol 34 no 4 pp 615ndash632 1999
[20] D-J Huang and H-Q Zhang ldquoLink between travelling wavesand first order nonlinear ordinary differential equation with asixth-degree nonlinear termrdquoChaos Solitons amp Fractals vol 29no 4 pp 928ndash941 2006
[21] A I Maimistov ldquoPropagation of an ultimately short electro-magnetic pulse in a nonlinear medium described by the fifth-order Duffing modelrdquo Optics and Spectroscopy vol 94 pp 251ndash257 2003
[22] M N Hamdan and N H Shabaneh ldquoOn the large amplitudefree vibrations of a restrained uniform beam carrying anintermediate lumpedmassrdquo Journal of Sound andVibration vol199 no 5 pp 711ndash736 1997
[23] J-H He ldquoSome asymptotic methods for strongly nonlinearequationsrdquo International Journal of Modern Physics B vol 20no 10 pp 1141ndash1199 2006
[24] J B Marion Classical Dynamics of Particles and Systems 1970[25] I Kovacic and M J Brennan The Duffing Equation Nonlinear
Oscillatos and their Behaviour John Wiley amp Sons 1st edition2011
[26] F C Moon Chaotic Vibrations An Introduction for AppliedScientists and Engineers John Wiley amp Sons New York NYUSA 2004
[27] J J Stoker Nonlinear Vibrations 1950[28] G Chen and Z Tao ldquoAmplitude-frequency relationship for the
relativistic oscillatorrdquoAASRI Procedia vol 1 pp 400ndash403 2012[29] R E Mickens Oscillations in Planar Dynamic Systems World
Scientific Publishing Singapore 1996[30] A Belendez T Belendez C Neipp A Hernandez and M
L Alvarez ldquoApproximate solutions of a nonlinear oscillatortypified as a mass attached to a stretched elastic wire by thehomotopy perturbation methodrdquo Chaos Solitions and Fractalsvol 39 pp 746ndash764 2009
[31] A Belendez E Fernandez R Fuentes J J Rodes and I PascualldquoHarmonic balancing approach to nonlinear oscillations of apunctual charge in the eletric field of charged ringrdquo PhysicsLetters A vol 373 pp 735ndash740 2009
[32] A Elıas-Zuniga ldquoExact solution of the cubic-quintic Duffingoscillatorrdquo Applied Mathematical Modelling vol 37 no 4 pp2574ndash2579 2013
[33] A Belendez M L Alvarez J Frances et al ldquoAnalytical approx-imate solutions for the cubic-quintic Duffing oscillator in termsof elementary functionsrdquo Journal of Applied Mathematics vol2012 Article ID 286290 16 pages 2012
[34] A Elıas-Zuniga OMartınez-Romero andR K Cordoba-DıazldquoApproximate solution for the Duffing-harmonic oscillator bythe enhanced cubication methodrdquo Mathematical Problems inEngineering vol 2012 Article ID 618750 12 pages 2012
[35] C W Lim B S Wu andW P Sun ldquoHigher accuracy analyticalapproximations to the Duffing-harmonic oscillatorrdquo Journal ofSound and Vibration vol 296 no 4-5 pp 1039ndash1045 2006
[36] J He ldquoSome new approaches to Duffing equation with stronglyand high order nonlinearity II parametrized perturbationtechniquerdquo Communications in Nonlinear Science amp NumericalSimulation vol 4 no 1 pp 81ndash83 1999
[37] V Marinca and N Herisanu ldquoPeriodic solutions for somestrongly nonlinear oscillations by Hersquos variational iterationmethodrdquo Computers amp Mathematics with Applications vol 54no 7-8 pp 1188ndash1196 2007
[38] W Lu and Y Liu ldquoVibration control for the primary resonanceof the Duffing oscillator by a time delay state feedbackrdquoInternational Journal of Nonlinear Science vol 8 no 3 pp 324ndash328 2009
[39] H Y Hu and Z H Wang Dynamics of Controlled MechanicalSystems with Delayed Feedback Springer 2002
[40] M Hamdi and M Belhaq ldquoControl of bistability in a delayedDuffing oscillatorrdquo Advances in Acoustics and Vibration vol2012 Article ID 872498 6 pages 2012
[41] V Ravichandran C Chinnathambi and S Rajasekar ldquoNonlin-ear resonance in Duffing oscillator with fixed and integrativetime-delayed feedbacksrdquoPramana Journal of Physics vol 78 pp347ndash360 2013
[42] X You and H Xu ldquoAnalytical approximations for the periodicmotion of theDuffing systemwith delayed feedbackrdquoNumericalAlgorithms vol 56 no 4 pp 561ndash576 2011
[43] Z Sun W Xu X Yang and T Fang ldquoInducing or suppressingchaos in a double-well Duffing oscillator by time delay feed-backrdquo Chaos Solitons and Fractals vol 27 pp 705ndash714 2006
[44] H Wang H Hu and Z Wang ldquoGlobal dynamics of a Duffingoscillator with delayed displacement feedbackrdquo InternationalJournal of Bifurcation and Chaos in Applied Sciences andEngineering vol 14 no 8 pp 2753ndash2775 2004
[45] J Chiasson and J J LoiseauApplications of Time Delay SystemsSpringer 2007
[46] M Lakshmanan andDV SenthilkumarDynamics of NonlinearTime-Delay Systems Springer 2010
[47] G Stepan T Insperger and R Szalai ldquoDelay parametricexcitation and the nonlinear dynamics of cutting processesrdquoInternational Journal of Bifurcation and Chaos in Applied Sci-ences and Engineering vol 15 no 9 pp 2783ndash2798 2005
[48] U van der Heiden and H-O Walther ldquoExistence of chaos incontrol systems with delayed feedbackrdquo Journal of DifferentialEquations vol 47 no 2 pp 273ndash295 1983
[49] Y G Sun and J S W Wong ldquoOscillation criteria for secondorder forced ordinary differential equations with mixed non-linearitiesrdquo Journal of Mathematical Analysis and Applicationsvol 334 no 1 pp 549ndash560 2007
[50] J Heagy and W L Ditto ldquoDynamics of a two-frequencyparametrically driven Duffing oscillatorrdquo Journal of NonlinearScience vol 1 no 4 pp 423ndash455 1991
[51] A B Belogortsev ldquoBifurcations of tori and chaos in thequasiperiodically forced Duffing oscillatorrdquoNonlinearity vol 5no 4 pp 889ndash897 1992
[52] M Belhaq and M Houssni ldquoQuasi-periodic oscillations chaosand suppression of chaos in a nonlinear oscillator driven byparametric and external excitationsrdquo Nonlinear Dynamics vol18 no 1 pp 1ndash24 1999
[53] S H Saker P Y H Pang and R P Agarwal ldquoOscillationtheorems for second order nonlinear functional differential
Discrete Dynamics in Nature and Society 17
equations with dampingrdquo Dynamic Systems and Applicationsvol 12 no 3-4 pp 307ndash321 2003
[54] I N Bronshtein K A Semendyayev G Musiol and HMuehligHandbook of Mathematics Springer 5th edition 2007
[55] M Pasic ldquoFite-Wintner-Leighton-type oscillation criteria forsecond-order differential equations with nonlinear dampingrdquoAbstract and Applied Analysis vol 2013 Article ID 852180 10pages 2013
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
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Mathematical Problems in Engineering
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Differential EquationsInternational Journal of
Volume 2014
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Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
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Decision SciencesAdvances in
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Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
Discrete Dynamics in Nature and Society 11
that is 119909(119905) gt 0 on (120591(120591(119886)) 120590(120590(119889))) or 119909(119905) lt 0 on(120591(120591(119886)) 120590(120590(119889))) since 119909(119905) is a continuous function on[1199050infin) Let for instance
119909 (119905) gt 0 on (120591 (120591 (119886)) 120590 (120590 (119889))) (83)
Another case can be analogously treated let us see thecomment at the end of this proof In particular from (83)we have 119909(119905) gt 0 on (120591(120591(119886)) 120590(120590(119887))) which implies (since120591(119905) and 120590(119905) are increasing functions) 119909(119904) gt 0 for all 119904 isin
(120591(119886) 120590(119887)) cup (120591(120591(119886)) 120591(120590(119887))) cup (120590(120591(119886)) 120590(120590(119887))) whichyields 119909(119905) gt 0 119909(120591(119905)) gt 0 and 119909(120590(119905)) gt 0 on (120591(119886) 120590(119887))Hence by assumption (7) we may use inequality (5) on theinterval (119886 119887)
Firstly we show that the following classic Riccati transfor-mation of 119909(119905)
120596 (119905) = minus120576119903 (119905) 119860 (119909
1015840(119905))
|119909 (119905)|119901minus1
119909 (119905) 119905 isin (119886 119887) 120576 gt 0 (84)
satisfies upper Riccati differential inequality (77) Let usremark that from (1) we have in particular
minus(119903 (119905) 119860 (1199091015840
(119905)))1015840
= 119861 (119905 119909 (119905) 1199091015840
(119905)) + 120582119865 (119905 119909 (120591 (119905)))
+ 120583119866 (119905 119909 (120590 (119905))) minus 120588119890 (119905) 119905 ge 1199050
(85)
Taking the first derivative on both sides of (84) and usingassumptions (3) (4) and (5) as well as equality (85) and(|119909(119905)|
119901minus1119909(119905))1015840
= 119901|119909(119905)|119901minus1
1199091015840(119905) we obtain
119889120596
119889119905= 120576119901 119903 (119905)
119860 (1199091015840(119905)) 1199091015840(119905)
|119909 (119905)|119901+1
minus 1205761
|119909 (119905)|119901minus1
119909 (119905)(119903 (119905) 119860 (119909
1015840
(119905)))1015840
= 120576119901119903 (119905)119860 (1199091015840(119905)) 1199091015840(119905)
|119909 (119905)|119901+1
+120576
|119909 (119905)|119901minus1
119909 (119905)
times [120582119861 (119905 119909 (119905) 1199091015840
(119905)) + 119865 (119905 119909 (120591 (119905)))
+120583119866 (119905 119909 (120590 (119905))) minus 120588119890 (119905) ]
ge 120576119901119903 (119905)119860 (1199091015840(119905)) 1199091015840(119905)
|119909 (119905)|119901+1
+120576
|119909 (119905)|119901minus1
119909 (119905)
times [120582119865 (119905 119909 (120591 (119905))) + 120583119866 (119905 119909 (120590 (119905))) minus 120588119890 (119905)]
ge 120576119901119903 (119905) 120572(
10038161003816100381610038161003816119860 (1199091015840(119905))
10038161003816100381610038161003816
|119909 (119905)|119901
) + 120576119896 (120582 120583 120588)119876 (119905)
= 120576119901119903 (119905) 120572 (|120596 (119905)|
120576119903 (119905)) + 120576119896 (120582 120583 120588)119876 (119905)
ge119901
(120576119903 (119905))120574minus1
120572 (|120596 (119905)|) + 120576119896 (120582 120583 120588)119876 (119905) 119905 isin (119886 119887)
(86)
Thus according to inequality (5) it is shown that if 119909(119905) isa solution of (1) which satisfies (83) then the function 120596(119905)
defined by (84) satisfies the Riccati differential inequality(77) and 120596 isin 119862((119886 119887)R) On the other hand let 119877
119886be a
real number defined by 119877119886= 120596(119886) According to (6) and
Lemma 19 we obtain (54) which together with Lemma 20ensures that we may use Proposition 22 for such chosen realnumber 119877
119886 Hence we obtain a function 120596(119905) defined by (72)
which satisfies the lower Riccati differential inequality (73) on[119886 119879lowast
119886) 119879lowast119886isin (119886 119887) such that 120596(a) = 119877
119886and lim
119905rarr119879lowast
119886
120596(119905) =
infin Therefore by 120596(119886) = 119877119886= 120596(119886) and Proposition 24 we
conclude that lim119905rarr119879
lowast
119886
120596(119905) = infin too which is a contradictionwith the above conclusion saying that 120596 isin 119862((119886 119887)R) Thushypothesis (82) is not true and consequently Lemma 1 isshown
For the analogous case 119909(119905) lt 0 on (120591(120591(119886)) 120590(120590(119889))) wealso have 119909(119905) lt 0 on (120591(120591(119888)) 120590(120590(119889))) which implies (since120591(119905) and 120590(119905) are increasing functions)
119909 (119904) lt 0 forall119904 isin (120591 (119888) 120590 (119889)) cup (120591 (120591 (119888)) 120591 (120590 (119889)))
cup (120590 (120591 (119888)) 120590 (120590 (119889)))
(87)
which yields 119909(119905) lt 0 119909(120591(119905)) lt 0 and 119909(120590(119905)) lt 0 on(120591(119888) 120590(119889)) Now we can repeat the preceding procedure buton interval (119888 119889) and using (8) instead of (119886 119887) and (7)
Proof of Lemma 2 From assumption (10) we obtain the exis-tence of an 119899
0isin N such that
int
119887119899
119886119899
119876119899(119905) 119889119905 ge
1198880
2( max119905isin[119886119899 119887119899]
119876119899(119905))
1120574
119899 ge 1198990 (88)
that is
2
1198880
int
119887119899
119886119899
119876119899(119905) 119889119905 ge ( max
119905isin[119886119899 119887119899]119876119899(119905))
1120574
119899 ge 1198990 (89)
Now from (9) and previous inequality we deduce that forlarge enough 120582 120583 120588 and 119899
1199011120574
1199031minus1120574
0
[119896 (120582 120583 120588)]1minus1120574
120587lowast
int
119887119899
119886119899
119876119899(119905) 119889119905
ge2
1198880
int
119887119899
119886119899
119876119899(119905) 119889119905 ge ( max
119905isin[119886119899 119887119899]119876119899(119905))
1120574
(90)
which shows (6) Thus all assumptions of Lemma 1 arefulfilled and hence Lemma 2 immediately follows fromLemma 1
Proof of Lemma 3 Obviously assumption (11) is a particularcase of assumption (9) Hence this proof is very similar tothe proof of Lemma 2 and so it is left to the reader
Proof of Lemma 4 It is clear that from assumption (13) weobtain
1
(max119905isin[119886119899119887119899]119876119899(119905))1120574
int
119887119899
119886119899
119876119899(119905) 119889119905 ge
1198881
1198621120574
0
gt 0 forall119899 ge 1198990
(91)
12 Discrete Dynamics in Nature and Society
Thus hypothesis (12) is fulfilled and therefore Lemma 3proves this lemma
Proof of Theorems 5 6 and 7 This proof is based onLemma 4 In order to simplify notation in many placesin this proof we set 120591(119905) = 119905 minus 120591 and 120590(119905) = 119905 + 120590 Sinceassumptions (2) (3) and (4) have been already supposed inTheorems 5 6 and 7 in order to prove these theorems byLemma 4 we are going to show that the functions 119896(120582 120583 120588)and 119876
119899(119905) explicitly given respectively in (18) (21) or (24)
and (19) (22) or (25) satisfy required conditions (11) and(13) respectively and that every solution 119909(119905) of (27) satisfiesconditions (7) and (8) with respect to functions 119896(120582 120583 120588)and 119876
119899(119905) where 119886 = 119886
2119899minus1 119887 = 119887
2119899minus1 119888 = 119886
2119899 and 119889 = 119887
2119899
The proof that the function 119896(120582 120583 120588) given in (18) (21) or(24) satisfies (11) Passing to the limit in (18) (21) or (24) it isvery simple to show (11)
The proof that the function 119876119899(119905) given in (19) (22) or
(25) satisfies the first claim in (13) From (25) we immediatelyobtain
1003816100381610038161003816120591119899 (119905)1003816100381610038161003816 =
100381610038161003816100381610038161003816100381610038161003816
(119905 minus 119886119899
119905 minus 119886119899+ 120591
)
119901100381610038161003816100381610038161003816100381610038161003816
le 1
1003816100381610038161003816120590119899 (119905)1003816100381610038161003816 =
100381610038161003816100381610038161003816100381610038161003816
(119887119899minus 119905
119887119899minus 119905 + 120590
)
119901100381610038161003816100381610038161003816100381610038161003816
le 1 forall119899 isin N
(92)
Next by assumptions of this corollary we can conclude thatthere are three positive constants 119891
0 1198920 1198900such that |119891(119905)| le
1198910and |119892(119905)| le 119892
0on [1199050infin) in cases (i) and (ii) and
|119890(119905)| le 1198900on [1199050infin) in cases (iii) and (iv) Putting previous
inequalities into (19) (22) or (25) for all 119899 isin N and 119905 isin
[1199050infin) it holds that
1003816100381610038161003816119876119899 (119905)1003816100381610038161003816 le
119902
119901(
119901
119902 minus 119901)
(119902119901)minus1
1198901minus(119901119902)
0119891119901119902
0
delay case with 119902 gt 119901
119902
119901(
119901
119902 minus 119901)
(119902119901)minus1
1198901minus(119901119902)
0119892119901119902
0
advanced case with 119902 gt 119901
1199011
119901(
119901
2 (1199011minus 119901)
)
(1199011119901)minus1
1198901minus(119901119901
1)
0119891119901119902
0+1199012
119901
times(119901
2 (1199012minus 119901)
)
(1199012119901)minus1
1198901minus(119901119901
2)
0119892119901119902
0
delay-advanced case (i)
1198901205780
01198911205781
01198921205782
0
2
prod
119894=0
120578minus120578119894
119894
delay-advanced case (ii) (93)
which shows the first claim in (13)
The proof that the function119876119899(119905) given in (19) (22) or (25)
satisfies the second claim in (13)Without loss of generality weprove this claim only in case (i) since for other cases the prooffollows analogously In this sense let119876
119899(119905) = 119891(119905)120591
119899(119905) Since
1198862119899+1
minus 1198862119899minus1
le 119879lowast 1198872119899+1
minus 1198872119899minus1
ge 119879lowast 1198862119899+2
minus 1198862119899le 119879lowast and
1198872119899+2
minus 1198872119899
ge 119879lowast where 119879
lowastgt 0 is the period of the function
119891(119905) we have 1198862119899minus1
le 1198861+(119899minus1)119879
lowastand 1198872119899minus1
ge 1198871+(119899minus1)119879
lowast
119899 isin N Hence
int
1198872119899minus1
1198862119899minus1
119876119899(119905) 119889119905
= int
1198872119899minus1
1198862119899minus1
119891 (119905) (119905 minus 1198862119899minus1
119905 minus 1198862119899minus1
+ 120591)
119901
119889119905
ge int
1198871+(119899minus1)119879
lowast
1198861+(119899minus1)119879lowast
119891 (119905) (119905 minus 1198861minus (119899 minus 1) 119879
lowast
119905 minus 1198861minus (119899 minus 1) 119879
lowast+ 120591
)
119901
119889119905
= int
1198871
1198861
119891 (119904 + (119899 minus 1) 119879lowast) (
119904 minus 1198861
119904 minus 1198861+ 120591
)
119901
119889119904
= int
1198871
1198861
119891 (119904) (119904 minus 1198861
119904 minus 1198861+ 120591
)
119901
119889119904
(94)
which proves that the integral on the left hand side does notdepend on 119899 isin N that is the second claim in (13) is shown on[1198862119899minus1
1198872119899minus1
] This claim follows in the same way on [1198862119899 1198872119899]
Thus the second claim in (13) is proved on [119886119899 119887119899]
Next to the end of this proof let 119909(119905) be a solu-tion of (1) In particular it implies that (119903(119905)119860(1199091015840(119905)))1015840 =
minus119861(119905 119909(119905) 1199091015840(119905)) minus 120582119865(119905 119909(120591(119905))) minus 120583119866(119905 119909(120590(119905))) + 120588119890(119905) It
together with assumptions (15) (16) (20) and (23) easilygives the next two statements
if 119909 (119905) gt 0 119909 (120591 (119905)) gt 0 119909 (120590 (119905)) gt 0
on (120591 (1198862119899minus1
) 120590 (1198872119899minus1
))
then 119909 (119905) satisfies 119903 (119905) 119860 (1199091015840
(119905)) le 0
on (120591 (1198862119899minus1
) 120590 (1198872119899minus1
)) 119899 ge 1198990
(95)
if 119909 (119905) lt 0 119909 (120591 (119905)) lt 0 119909 (120590 (119905)) lt 0
on (120591 (1198862119899) 120590 (119887
2119899))
then 119909 (119905) satisfies 119903 (119905) 119860 (1199091015840
(119905)) ge 0
on (120591 (1198862119899) 120590 (119887
2119899)) 119899 ge 119899
0
(96)
Now we need the following lemma
Discrete Dynamics in Nature and Society 13
Lemma 25 Let 120591119886119887(119905) and 120590
119886119887(119905) be defined by
120591119886119887(119905) = (
120591 (119905) minus 120591 (119886)
119905 minus 120591 (119886))
119901
120590119886119887(119905) = (
120590 (119887) minus 120590 (119905)
120590 (119887) minus 119905)
119901
119905 isin (119886 119887)
(97)
and let 119909 isin 1198622([1198790infin)R) be an arbitrary function If
(119903(119905)119860(1199091015840(119905)) le 0 for all 119905 isin (120591(119886) 120590(119887)) or (119903(119905)119860(1199091015840(119905)) ge 0
for all 119905 isin (120591(119886) 120590(119887)) then
119909 (120591 (119905))
119909 (119905)ge (120591119886119887(119905))1119901
119909 (120590 (119905))
119909 (119905)ge (120590119886119887(119905))1119901
119905 isin (119886 119887)
(98)
Since119860(V) is supposed to be odd and increasing functionjust before (3) and 119903(119905) satisfies (14) the proof of Lemma 25in the first case that is 119903(119905)119860(1199091015840(119905)) le 0 for all 119905 isin (120591(119886) 120590(119887))is the same as the proof of [9 Corollaries 17 and 18] But in thesecond case that is 119903(119905)119860(1199091015840(119905)) ge 0 for all 119905 isin (120591(119886) 120590(119887))the proof is as follows if previous inequality holds then119903(119905)119860(minus119909
1015840(119905)) le 0 for all 119905 isin (120591(119886) 120590(119887)) and therefore to
the function minus119909(119905) one can apply the first case of this lemmaand consequently one obtains
119909 (120591 (119905))
119909 (119905)=minus119909 (120591 (119905))
minus119909 (119905)ge (120591119886119887(119905))1119901
119909 (120590 (119905))
119909 (119905)=minus119909 (120590 (119905))
minus119909 (119905)ge (120590119886119887(119905))1119901
119905 isin (119886 119887)
(99)
which proves this lemma in the second caseNow combining statements (95) (96) and (98) one
easily obtains
if 119909 (119905) gt 0 119909 (120591 (119905)) gt 0 119909 (120590 (119905)) gt 0
on (120591 (1198862119899minus1
) 120590 (1198872119899minus1
))
then 119909 (119905) satisfies 119909 (120591 (119905))
119909 (119905)ge (120591119899(119905))1119901
on (1198862119899minus1
1198872119899minus1
) 119899 ge 1198990
(100)
if 119909 (119905) lt 0 119909 (120591 (119905)) lt 0 119909 (120590 (119905)) lt 0
on (120591 (1198862119899) 120590 (119887
2119899))
then 119909 (119905) satisfies 119909 (120590 (119905))
119909 (119905)ge (120590119899(119905))1119901
on (1198862119899 1198872119899) 119899 ge 119899
0
(101)
where 120591119899(119905) and 120590
119899(119905) are defined in (26)
The proof that 119909(119905) satisfies (7) and (8) In this proofwe frequently use assumptions (16) (20) and (23) andstatements (100) and (101) Also because of (15) and 119865(119905 119906) =
119891(119905)|119906|1199011 sgn(119906) 119866(119905 119906) = 119892(119905)|119906|
1199012 sgn(119906) in both cases
(100) and (101) we can simultaneously use
minus119890 (119905) (|119909 (119905)|119901minus1
119909 (119905))minus1
= |119890 (119905)| |119909 (119905)|minus119901
ge 0 on 119869119899
119865 (119905 119909 (120591 (119905))) (|119909 (119905)|119901minus1
119909 (119905))minus1
ge 119891 (119905) |119909 (120591 (119905))|1199011 |119909 (119905)|
minus119901ge 0 on 119869
119899
119866 (119905 119909 (120590 (119905))) (|119909 (119905)|119901minus1
119909 (119905))minus1
ge 119892 (119905) |119909 (120590 (119905))|1199012 |119909 (119905)|
minus119901ge 0 on 119869
119899
|119909 (120591 (119905))| |119909 (119905)|minus1=119909 (120591 (119905))
119909 (119905)
|119909 (120590 (119905))| |119909 (119905)|minus1=119909 (120590 (119905))
119909 (119905)on 119869119899
(102)
where 119869119899= (1198862119899minus1
1198872119899minus1
) in the case of (100) and 119869119899= (1198862119899 1198872119899)
in the case of (101)
(i) Delay or Advanced Case with 119902 = 119901 Since 119902 = 119901 we obtain
[120582119865 (119905 119909 (120591 (119905))) + 120583119866 (119905 119909 (120590 (119905))) minus 120588119890 (119905)]
times (|119909 (119905)|119901minus1
119909 (119905))minus1
ge [120582119891 (119905) |119909 (120591 (119905))|119901+ 120583119892 (119905) |119909 (120590 (119905))|
119901
+120588 |119890 (119905)| ] |119909 (119905)|minus119901
ge [120582119891 (119905) |119909 (120591 (119905))|119901+ 120583119892 (119905) |119909 (120590 (119905))|
119901] |119909 (119905)|
minus119901
= 120582119891 (119905) (119909 (120591 (119905))
119909 (119905))
119901
+ 120583119892 (119905) (119909 (120590 (119905))
119909 (119905))
119901
ge 120582119891 (119905) 120591119899(119905) + 120583119892 (119905) 120590
119899(119905) 119905 isin 119869
119899
(103)
where the functions 120591119899(119905) and 120590
119899(119905) are defined in (26)
(ii) Delay Case with 119902 gt 119901 In this part we use the nextelementary inequality
119883120574+ (120574 minus 1) 119884
120574ge 120574119883119884
120574minus1 120574 gt 1 119883 119884 ge 0 (104)
Since 119902 gt 119901 and using (104) especially for
120574 =119902
119901gt 1 119883 = (120582119891 (119905))
119901119902
|119909 (120591 (119905))|119901
119884 = (119901
119902 minus 119901120588 |119890 (119905)|)
119901119902
(105)
14 Discrete Dynamics in Nature and Society
for all 119905 isin 119869119899we obtain
[120582119865 (119905 119909 (120591 (119905))) + 120583119866 (119905 119909 (120590 (119905))) minus 120588119890 (119905)]
times (|119909 (119905)|119901minus1
119909 (119905))minus1
= [120582119865 (119905 119909 (120591 (119905))) minus 120588119890 (119905)] (|119909 (119905)|119901minus1
119909 (119905))minus1
ge [120582119891 (119905) |119909 (120591 (119905))|1199011 + 120588 |119890 (119905)|] |119909 (119905)|
minus119901
= [119883120574+ (120574 minus 1) 119884
120574] |119909 (119905)|
minus119901
ge119902
119901(120582119891 (119905))
119901119902
|119909 (120591 (119905))|119901
times (119901
119902 minus 119901120588 |119890 (119905)|)
(119901119902)((119902119901)minus1)
|119909 (119905)|minus119901
= 120582119901119902
1205881minus(119901119902)
119902
119901(
119901
119902 minus 119901)
1minus(119901119902)
|119890 (119905)|1minus(119901119902)
times (119891 (119905))119901119902
(119909 (120591 (119905))
119909 (119905))
119901
ge 119896 (120582 120583 120588)119902
119901(
119901
119902 minus 119901)
1minus(119901119902)
|119890 (119905)|1minus(119901119902)
times (119891 (119905))119901119902
120591119899(119905)
(106)
where the function 119896(120582 120583 120588) is from (18)
(iii) Advanced Case with 119902 gt 119901 Using the same line ofarguments as in the proof of the previous case for all 119905 isin 119869
119899
we obtain
[120582119865 (119905 119909 (120591 (119905))) + 120583119866 (119905 119909 (120590 (119905))) minus 120588119890 (119905)]
times (|119909 (119905)|119901minus1
119909 (119905))minus1
= [120583119866 (119905 119909 (120590 (119905))) minus 120588119890 (119905)] (|119909 (119905)|119901minus1
119909 (119905))minus1
ge 119896 (120582 120583 120588)119902
119901(
119901
119902 minus 119901)
1minus(119901119902)
|119890 (119905)|1minus(119901119902)
times (119892 (119905))119901119902
120590119899(119905)
(107)
where the function 119896(120582 120583 120588) is from (21)
(iv) Superlinear Delay-Advanced Case Since 1199011 1199012gt 119901 for
all 119905 isin 119869119899we obtain
[120582119865 (119905 119909 (120591 (119905))) + 120583119866 (119905 119909 (120590 (119905))) minus 120588119890 (119905)]
times (|119909 (119905)|119901minus1
119909 (119905))minus1
= [120582119865 (119905 119909 (120591 (119905))) minus120588
2119890 (119905)] (|119909 (119905)|
119901minus1119909 (119905))minus1
+ [120583119866 (119905 119909 (120590 (119905))) minus120588
2119890 (119905)] (|119909 (119905)|
119901minus1119909 (119905))minus1
ge [120582119891 (119905) |119909 (120591 (119905))|1199011 +
120588
2|119890 (119905)|] |119909 (119905)|
minus119901
+ [120583119892 (119905) |119909 (120590 (119905))|1199012 +
120588
2|119890 (119905)|] |119909 (119905)|
minus119901
(108)
Now just the same as in the proofs of previous delay andadvanced cases with 119902 gt 119901 and with the help of (104) inparticular for
120574 =1199011
119901gt 1 119883 = (120582119891 (119905))
1199011199011
|119909 (120591 (119905))|119901
119884 = (119901
1199011minus 119901
120588
2|119890 (119905)|)
1199011199011
(109)
we have
[120582119891 (119905) |119909 (120591 (119905))|1199011 +
120588
2 |119890 (119905)|] |119909 (119905)|
minus119901
= [119883120574+ (120574 minus 1) 119884
120574] |119909 (119905)|
minus119901
ge1199011
119901(120582119891 (119905))
1199011199011
|119909 (120591 (119905))|119901
times (119901
1199011minus 119901
120588
2|119890 (119905)|)
(1199011199011)((1199011119901)minus1)
|119909 (119905)|minus119901
= 12058211990111990111205881minus(119901119901
1)1199011
119901(
119901
2 (1199011minus 119901)
)
1minus(1199011199011)
times |119890 (119905)|1minus(119901119901
1)(119891 (119905))
1199011199011
(119909 (120591 (119905))
119909 (119905))
119901
ge 119896 (120582 120583 120588)1199011
119901(
119901
2 (1199011minus 119901)
)
1minus(1199011199011)
times |119890 (119905)|1minus(119901119901
1)(119891 (119905))
1199011199011
120591119899(119905)
(110)
where the function 119896(120582 120583 120588) is from (24) Analogously weshow that
[120583119892 (119905) |119909 (120590 (119905))|1199012 +
120588
2|119890 (119905)|] |119909 (119905)|
minus119901
ge 119896 (120582 120583 120588)1199012
119901(
119901
2 (1199012minus 119901)
)
1minus(1199011199012)
times |119890 (119905)|1minus(119901119901
2)(119891 (119905))
1199011199012
120590119899(119905)
(111)
Discrete Dynamics in Nature and Society 15
Summarizing previous calculation we obtain
[120582119865 (119905 119909 (120591 (119905))) + 120583119866 (119905 119909 (120590 (119905))) minus 120588119890 (119905)]
times (|119909 (119905)|119901minus1
119909 (119905))minus1
ge 119896 (120582 120583 120588)119876119899(119905) 119905 isin 119869
119899
(112)
where the function 119896(120582 120583 120588) is from (24)
(v) Supersublinear Delay-Advanced Case Since 1199011gt 119901 gt 119901
2
and the following well-known elementary inequality holds
12057801199060+ 12057811199061+ 12057821199062ge 1199061205780
01199061205781
11199061205782
2 120578119894ge 0 119906
119894ge 0 (113)
from 1205780 1205781 1205782isin (0 1) 120578
0+ 1205781+ 1205782= 1 and 119901
11205781+ 11990121205782= 119901
we obtain for all 119905 isin 119869119899 for all 119905 isin 119869
119899we obtain
[120582119865 (119905 119909 (120591 (119905))) + 120583119866 (119905 119909 (120590 (119905))) minus 120588119890 (119905)]
times (|119909 (119905)|119901minus1
119909 (119905))minus1
ge [120582119891 (119905) |119909 (120591 (119905))|1199011 + 120583119892 (119905) |119909 (120590 (119905))|
1199012 + 120588 |119890 (119905)|]
times |119909 (119905)|minus119901
= [1205781[120578minus1
1120582119891 (119905) |119909 (120591 (119905))|
1199011] + 120578
2[120578minus1
2120583119892 (119905) |119909 (120590 (119905))|
1199012]
+1205780[120578minus1
0120588 |119890 (119905)|]] |119909 (119905)|
minus119901
ge [120578minus1
1120582119891 (119905) |119909 (120591 (119905))|
1199011]1205781
[120578minus1
2120583119892 (119905) |119909 (120590 (119905))|
1199012]1205782
times [120578minus1
0120588 |119890 (119905)|]
1205780
|119909 (119905)|minus119901
= 120582120578112058312057821205881205780 |119890 (119905)|
1205780(119891 (119905))
1205781
(119892 (119905))1205782
times|119909 (120591 (119905))|
12057811199011
|119909 (119905)|12057811199011
|119909 (120590 (119905))|12057821199012
|119909 (119905)|12057821199012
2
prod
119894=0
120578minus120578119894
119894
= 119896 (120582 120583 120588) |119890 (119905)|1205780(119891 (119905))
1205781
(119892 (119905))1205782
times (119909 (120591 (119905))
119909 (119905))
12057811199011
(119909 (120590 (119905))
119909 (119905))
12057821199012 2
prod
119894=0
120578minus120578119894
119894
ge 119896 (120582 120583 120588) |119890 (119905)|1205780(119891 (119905))
1205781
(119892 (119905))1205782
(120591119899(119905))1205781(1199011119901)
times (120590119899(119905))1205782(1199012119901)
2
prod
119894=0
120578minus120578119894
119894
= 119896 (120582 120583 120588)119876119899(119905)
(114)
where 119896(120582 120583 120588) and 119876119899(119905) are given respectively in (24) and
(25) Thus it is shown that required condition (5) in thecases (i)ndash(iv) is fulfilled with respect to 119896(120582 120583 120588) and 119876
119899(119905)
determined by (18) (21) or (24) and (19) (22) or (25)In conclusion according to the previous observation we
see that all assumptions of Lemma 4 are fulfilled and henceLemma 4 proves Theorems 5 6 and 7
Conflict of Interests
The author declares that there is no conflict of interestsregarding the publication of this paper
References
[1] Y G Sun ldquoA note on Nasrrsquos and Wongrsquos papersrdquo Journal ofMathematical Analysis and Applications vol 286 no 1 pp 363ndash367 2003
[2] Y G Sun C H Ou and J S W Wong ldquoInterval oscillationtheorems for a second-order linear differential equationrdquo Com-puters amp Mathematics with Applications vol 48 no 10-11 pp1693ndash1699 2004
[3] S Murugadass E Thandapani and S Pinelas ldquoOscillationcriteria for forced second-order mixed type quasilinear delaydifferential equationsrdquo Electronic Journal of Differential Equa-tions vol 2010 article 73 9 pages 2010
[4] Y Bai and L Liu ldquoNew oscillation criteria for second-orderdelay differential equations with mixed nonlinearitiesrdquoDiscreteDynamics in Nature and Society vol 2010 Article ID 796256 9pages 2010
[5] A F Guvenilir andA Zafer ldquoSecond-order oscillation of forcedfunctional differential equations with oscillatory potentialsrdquoComputers amp Mathematics with Applications vol 51 no 9-10pp 1395ndash1404 2006
[6] A Zafer ldquoInterval oscillation criteria for second order super-half linear functional differential equations with delay andadvanced argumentsrdquoMathematische Nachrichten vol 282 no9 pp 1334ndash1341 2009
[7] A F Guvenilir ldquoInterval oscillation of second-order functionaldifferential equations with oscillatory potentialsrdquo NonlinearAnalysis Theory Methods amp Applications vol 71 no 12 ppe2849ndashe2854 2009
[8] T S Hassan L Erbe and A Peterson ldquoForced oscillation ofsecond order differential equations with mixed nonlinearitiesrdquoActa Mathematica Scientia B vol 31 no 2 pp 613ndash626 2011
[9] M Pasic ldquoNew oscillation criteria for second-order forcedquasilinear functional differential equationsrdquo Abstract andApplied Analysis vol 2013 Article ID 735360 12 pages 2013
[10] L H Erbe Q Kong and B G Zhang Oscillation Theory forFunctional-Differential Equations vol 190 Marcel Dekker NewYork NY USA 1995
[11] V Kolmanovskii and A Myshkis Introduction to the Theoryand Applications of Functional-Differential Equations vol 463Kluwer Academic Publishers Dordrecht The Netherlands1999
[12] R P Agarwal M Bohner and W-T Li Nonoscillation andOscillation Theory for Functional Differential Equations vol267 Marcel Dekker New York NY USA 2004
[13] L Erbe T Hassan and A Peterson ldquoOscillation of secondorder functional dynamic equationsrdquo International Journal ofDifference Equations vol 5 no 2 pp 175ndash193 2010
[14] B Baculıkova J Dzurina and Y V Rogovchenko ldquoOscillationof third order trinomial delay differential equationsrdquo AppliedMathematics and Computation vol 218 no 13 pp 7023ndash70332012
[15] R P Agarwal L Berezansky E Braverman and A Domoshnit-sky Nonoscillation Theory of Functional Differential Equationswith Applications Springer New York NY USA 2012
16 Discrete Dynamics in Nature and Society
[16] J Zhang ldquoVariational approach to solitary wave solution ofthe generalized Zakharov equationrdquo Computers ampMathematicswith Applications vol 54 no 7-8 pp 1043ndash1046 2007
[17] T Ozis and A Yıldırım ldquoApplication of Hersquos semi-inversemethod to the nonlinear Schrodinger equationrdquo Computers ampMathematics with Applications vol 54 no 7-8 pp 1039ndash10422007
[18] X-C Cai andM-S Li ldquoPeriodic solution of Jacobi elliptic equa-tions by Hersquos perturbation methodrdquo Computers amp Mathematicswith Applications vol 54 no 7-8 pp 1210ndash1212 2007
[19] S Lenci G Menditto and A M Tarantino ldquoHomoclinic andheteroclinic bifurcations in the non-linear dynamics of a beamresting on an elastic substraterdquo International Journal of Non-Linear Mechanics vol 34 no 4 pp 615ndash632 1999
[20] D-J Huang and H-Q Zhang ldquoLink between travelling wavesand first order nonlinear ordinary differential equation with asixth-degree nonlinear termrdquoChaos Solitons amp Fractals vol 29no 4 pp 928ndash941 2006
[21] A I Maimistov ldquoPropagation of an ultimately short electro-magnetic pulse in a nonlinear medium described by the fifth-order Duffing modelrdquo Optics and Spectroscopy vol 94 pp 251ndash257 2003
[22] M N Hamdan and N H Shabaneh ldquoOn the large amplitudefree vibrations of a restrained uniform beam carrying anintermediate lumpedmassrdquo Journal of Sound andVibration vol199 no 5 pp 711ndash736 1997
[23] J-H He ldquoSome asymptotic methods for strongly nonlinearequationsrdquo International Journal of Modern Physics B vol 20no 10 pp 1141ndash1199 2006
[24] J B Marion Classical Dynamics of Particles and Systems 1970[25] I Kovacic and M J Brennan The Duffing Equation Nonlinear
Oscillatos and their Behaviour John Wiley amp Sons 1st edition2011
[26] F C Moon Chaotic Vibrations An Introduction for AppliedScientists and Engineers John Wiley amp Sons New York NYUSA 2004
[27] J J Stoker Nonlinear Vibrations 1950[28] G Chen and Z Tao ldquoAmplitude-frequency relationship for the
relativistic oscillatorrdquoAASRI Procedia vol 1 pp 400ndash403 2012[29] R E Mickens Oscillations in Planar Dynamic Systems World
Scientific Publishing Singapore 1996[30] A Belendez T Belendez C Neipp A Hernandez and M
L Alvarez ldquoApproximate solutions of a nonlinear oscillatortypified as a mass attached to a stretched elastic wire by thehomotopy perturbation methodrdquo Chaos Solitions and Fractalsvol 39 pp 746ndash764 2009
[31] A Belendez E Fernandez R Fuentes J J Rodes and I PascualldquoHarmonic balancing approach to nonlinear oscillations of apunctual charge in the eletric field of charged ringrdquo PhysicsLetters A vol 373 pp 735ndash740 2009
[32] A Elıas-Zuniga ldquoExact solution of the cubic-quintic Duffingoscillatorrdquo Applied Mathematical Modelling vol 37 no 4 pp2574ndash2579 2013
[33] A Belendez M L Alvarez J Frances et al ldquoAnalytical approx-imate solutions for the cubic-quintic Duffing oscillator in termsof elementary functionsrdquo Journal of Applied Mathematics vol2012 Article ID 286290 16 pages 2012
[34] A Elıas-Zuniga OMartınez-Romero andR K Cordoba-DıazldquoApproximate solution for the Duffing-harmonic oscillator bythe enhanced cubication methodrdquo Mathematical Problems inEngineering vol 2012 Article ID 618750 12 pages 2012
[35] C W Lim B S Wu andW P Sun ldquoHigher accuracy analyticalapproximations to the Duffing-harmonic oscillatorrdquo Journal ofSound and Vibration vol 296 no 4-5 pp 1039ndash1045 2006
[36] J He ldquoSome new approaches to Duffing equation with stronglyand high order nonlinearity II parametrized perturbationtechniquerdquo Communications in Nonlinear Science amp NumericalSimulation vol 4 no 1 pp 81ndash83 1999
[37] V Marinca and N Herisanu ldquoPeriodic solutions for somestrongly nonlinear oscillations by Hersquos variational iterationmethodrdquo Computers amp Mathematics with Applications vol 54no 7-8 pp 1188ndash1196 2007
[38] W Lu and Y Liu ldquoVibration control for the primary resonanceof the Duffing oscillator by a time delay state feedbackrdquoInternational Journal of Nonlinear Science vol 8 no 3 pp 324ndash328 2009
[39] H Y Hu and Z H Wang Dynamics of Controlled MechanicalSystems with Delayed Feedback Springer 2002
[40] M Hamdi and M Belhaq ldquoControl of bistability in a delayedDuffing oscillatorrdquo Advances in Acoustics and Vibration vol2012 Article ID 872498 6 pages 2012
[41] V Ravichandran C Chinnathambi and S Rajasekar ldquoNonlin-ear resonance in Duffing oscillator with fixed and integrativetime-delayed feedbacksrdquoPramana Journal of Physics vol 78 pp347ndash360 2013
[42] X You and H Xu ldquoAnalytical approximations for the periodicmotion of theDuffing systemwith delayed feedbackrdquoNumericalAlgorithms vol 56 no 4 pp 561ndash576 2011
[43] Z Sun W Xu X Yang and T Fang ldquoInducing or suppressingchaos in a double-well Duffing oscillator by time delay feed-backrdquo Chaos Solitons and Fractals vol 27 pp 705ndash714 2006
[44] H Wang H Hu and Z Wang ldquoGlobal dynamics of a Duffingoscillator with delayed displacement feedbackrdquo InternationalJournal of Bifurcation and Chaos in Applied Sciences andEngineering vol 14 no 8 pp 2753ndash2775 2004
[45] J Chiasson and J J LoiseauApplications of Time Delay SystemsSpringer 2007
[46] M Lakshmanan andDV SenthilkumarDynamics of NonlinearTime-Delay Systems Springer 2010
[47] G Stepan T Insperger and R Szalai ldquoDelay parametricexcitation and the nonlinear dynamics of cutting processesrdquoInternational Journal of Bifurcation and Chaos in Applied Sci-ences and Engineering vol 15 no 9 pp 2783ndash2798 2005
[48] U van der Heiden and H-O Walther ldquoExistence of chaos incontrol systems with delayed feedbackrdquo Journal of DifferentialEquations vol 47 no 2 pp 273ndash295 1983
[49] Y G Sun and J S W Wong ldquoOscillation criteria for secondorder forced ordinary differential equations with mixed non-linearitiesrdquo Journal of Mathematical Analysis and Applicationsvol 334 no 1 pp 549ndash560 2007
[50] J Heagy and W L Ditto ldquoDynamics of a two-frequencyparametrically driven Duffing oscillatorrdquo Journal of NonlinearScience vol 1 no 4 pp 423ndash455 1991
[51] A B Belogortsev ldquoBifurcations of tori and chaos in thequasiperiodically forced Duffing oscillatorrdquoNonlinearity vol 5no 4 pp 889ndash897 1992
[52] M Belhaq and M Houssni ldquoQuasi-periodic oscillations chaosand suppression of chaos in a nonlinear oscillator driven byparametric and external excitationsrdquo Nonlinear Dynamics vol18 no 1 pp 1ndash24 1999
[53] S H Saker P Y H Pang and R P Agarwal ldquoOscillationtheorems for second order nonlinear functional differential
Discrete Dynamics in Nature and Society 17
equations with dampingrdquo Dynamic Systems and Applicationsvol 12 no 3-4 pp 307ndash321 2003
[54] I N Bronshtein K A Semendyayev G Musiol and HMuehligHandbook of Mathematics Springer 5th edition 2007
[55] M Pasic ldquoFite-Wintner-Leighton-type oscillation criteria forsecond-order differential equations with nonlinear dampingrdquoAbstract and Applied Analysis vol 2013 Article ID 852180 10pages 2013
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
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Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical Problems in Engineering
Hindawi Publishing Corporationhttpwwwhindawicom
Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
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Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
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OptimizationJournal of
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CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
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Operations ResearchAdvances in
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Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Algebra
Discrete Dynamics in Nature and Society
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
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Decision SciencesAdvances in
Discrete MathematicsJournal of
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Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
12 Discrete Dynamics in Nature and Society
Thus hypothesis (12) is fulfilled and therefore Lemma 3proves this lemma
Proof of Theorems 5 6 and 7 This proof is based onLemma 4 In order to simplify notation in many placesin this proof we set 120591(119905) = 119905 minus 120591 and 120590(119905) = 119905 + 120590 Sinceassumptions (2) (3) and (4) have been already supposed inTheorems 5 6 and 7 in order to prove these theorems byLemma 4 we are going to show that the functions 119896(120582 120583 120588)and 119876
119899(119905) explicitly given respectively in (18) (21) or (24)
and (19) (22) or (25) satisfy required conditions (11) and(13) respectively and that every solution 119909(119905) of (27) satisfiesconditions (7) and (8) with respect to functions 119896(120582 120583 120588)and 119876
119899(119905) where 119886 = 119886
2119899minus1 119887 = 119887
2119899minus1 119888 = 119886
2119899 and 119889 = 119887
2119899
The proof that the function 119896(120582 120583 120588) given in (18) (21) or(24) satisfies (11) Passing to the limit in (18) (21) or (24) it isvery simple to show (11)
The proof that the function 119876119899(119905) given in (19) (22) or
(25) satisfies the first claim in (13) From (25) we immediatelyobtain
1003816100381610038161003816120591119899 (119905)1003816100381610038161003816 =
100381610038161003816100381610038161003816100381610038161003816
(119905 minus 119886119899
119905 minus 119886119899+ 120591
)
119901100381610038161003816100381610038161003816100381610038161003816
le 1
1003816100381610038161003816120590119899 (119905)1003816100381610038161003816 =
100381610038161003816100381610038161003816100381610038161003816
(119887119899minus 119905
119887119899minus 119905 + 120590
)
119901100381610038161003816100381610038161003816100381610038161003816
le 1 forall119899 isin N
(92)
Next by assumptions of this corollary we can conclude thatthere are three positive constants 119891
0 1198920 1198900such that |119891(119905)| le
1198910and |119892(119905)| le 119892
0on [1199050infin) in cases (i) and (ii) and
|119890(119905)| le 1198900on [1199050infin) in cases (iii) and (iv) Putting previous
inequalities into (19) (22) or (25) for all 119899 isin N and 119905 isin
[1199050infin) it holds that
1003816100381610038161003816119876119899 (119905)1003816100381610038161003816 le
119902
119901(
119901
119902 minus 119901)
(119902119901)minus1
1198901minus(119901119902)
0119891119901119902
0
delay case with 119902 gt 119901
119902
119901(
119901
119902 minus 119901)
(119902119901)minus1
1198901minus(119901119902)
0119892119901119902
0
advanced case with 119902 gt 119901
1199011
119901(
119901
2 (1199011minus 119901)
)
(1199011119901)minus1
1198901minus(119901119901
1)
0119891119901119902
0+1199012
119901
times(119901
2 (1199012minus 119901)
)
(1199012119901)minus1
1198901minus(119901119901
2)
0119892119901119902
0
delay-advanced case (i)
1198901205780
01198911205781
01198921205782
0
2
prod
119894=0
120578minus120578119894
119894
delay-advanced case (ii) (93)
which shows the first claim in (13)
The proof that the function119876119899(119905) given in (19) (22) or (25)
satisfies the second claim in (13)Without loss of generality weprove this claim only in case (i) since for other cases the prooffollows analogously In this sense let119876
119899(119905) = 119891(119905)120591
119899(119905) Since
1198862119899+1
minus 1198862119899minus1
le 119879lowast 1198872119899+1
minus 1198872119899minus1
ge 119879lowast 1198862119899+2
minus 1198862119899le 119879lowast and
1198872119899+2
minus 1198872119899
ge 119879lowast where 119879
lowastgt 0 is the period of the function
119891(119905) we have 1198862119899minus1
le 1198861+(119899minus1)119879
lowastand 1198872119899minus1
ge 1198871+(119899minus1)119879
lowast
119899 isin N Hence
int
1198872119899minus1
1198862119899minus1
119876119899(119905) 119889119905
= int
1198872119899minus1
1198862119899minus1
119891 (119905) (119905 minus 1198862119899minus1
119905 minus 1198862119899minus1
+ 120591)
119901
119889119905
ge int
1198871+(119899minus1)119879
lowast
1198861+(119899minus1)119879lowast
119891 (119905) (119905 minus 1198861minus (119899 minus 1) 119879
lowast
119905 minus 1198861minus (119899 minus 1) 119879
lowast+ 120591
)
119901
119889119905
= int
1198871
1198861
119891 (119904 + (119899 minus 1) 119879lowast) (
119904 minus 1198861
119904 minus 1198861+ 120591
)
119901
119889119904
= int
1198871
1198861
119891 (119904) (119904 minus 1198861
119904 minus 1198861+ 120591
)
119901
119889119904
(94)
which proves that the integral on the left hand side does notdepend on 119899 isin N that is the second claim in (13) is shown on[1198862119899minus1
1198872119899minus1
] This claim follows in the same way on [1198862119899 1198872119899]
Thus the second claim in (13) is proved on [119886119899 119887119899]
Next to the end of this proof let 119909(119905) be a solu-tion of (1) In particular it implies that (119903(119905)119860(1199091015840(119905)))1015840 =
minus119861(119905 119909(119905) 1199091015840(119905)) minus 120582119865(119905 119909(120591(119905))) minus 120583119866(119905 119909(120590(119905))) + 120588119890(119905) It
together with assumptions (15) (16) (20) and (23) easilygives the next two statements
if 119909 (119905) gt 0 119909 (120591 (119905)) gt 0 119909 (120590 (119905)) gt 0
on (120591 (1198862119899minus1
) 120590 (1198872119899minus1
))
then 119909 (119905) satisfies 119903 (119905) 119860 (1199091015840
(119905)) le 0
on (120591 (1198862119899minus1
) 120590 (1198872119899minus1
)) 119899 ge 1198990
(95)
if 119909 (119905) lt 0 119909 (120591 (119905)) lt 0 119909 (120590 (119905)) lt 0
on (120591 (1198862119899) 120590 (119887
2119899))
then 119909 (119905) satisfies 119903 (119905) 119860 (1199091015840
(119905)) ge 0
on (120591 (1198862119899) 120590 (119887
2119899)) 119899 ge 119899
0
(96)
Now we need the following lemma
Discrete Dynamics in Nature and Society 13
Lemma 25 Let 120591119886119887(119905) and 120590
119886119887(119905) be defined by
120591119886119887(119905) = (
120591 (119905) minus 120591 (119886)
119905 minus 120591 (119886))
119901
120590119886119887(119905) = (
120590 (119887) minus 120590 (119905)
120590 (119887) minus 119905)
119901
119905 isin (119886 119887)
(97)
and let 119909 isin 1198622([1198790infin)R) be an arbitrary function If
(119903(119905)119860(1199091015840(119905)) le 0 for all 119905 isin (120591(119886) 120590(119887)) or (119903(119905)119860(1199091015840(119905)) ge 0
for all 119905 isin (120591(119886) 120590(119887)) then
119909 (120591 (119905))
119909 (119905)ge (120591119886119887(119905))1119901
119909 (120590 (119905))
119909 (119905)ge (120590119886119887(119905))1119901
119905 isin (119886 119887)
(98)
Since119860(V) is supposed to be odd and increasing functionjust before (3) and 119903(119905) satisfies (14) the proof of Lemma 25in the first case that is 119903(119905)119860(1199091015840(119905)) le 0 for all 119905 isin (120591(119886) 120590(119887))is the same as the proof of [9 Corollaries 17 and 18] But in thesecond case that is 119903(119905)119860(1199091015840(119905)) ge 0 for all 119905 isin (120591(119886) 120590(119887))the proof is as follows if previous inequality holds then119903(119905)119860(minus119909
1015840(119905)) le 0 for all 119905 isin (120591(119886) 120590(119887)) and therefore to
the function minus119909(119905) one can apply the first case of this lemmaand consequently one obtains
119909 (120591 (119905))
119909 (119905)=minus119909 (120591 (119905))
minus119909 (119905)ge (120591119886119887(119905))1119901
119909 (120590 (119905))
119909 (119905)=minus119909 (120590 (119905))
minus119909 (119905)ge (120590119886119887(119905))1119901
119905 isin (119886 119887)
(99)
which proves this lemma in the second caseNow combining statements (95) (96) and (98) one
easily obtains
if 119909 (119905) gt 0 119909 (120591 (119905)) gt 0 119909 (120590 (119905)) gt 0
on (120591 (1198862119899minus1
) 120590 (1198872119899minus1
))
then 119909 (119905) satisfies 119909 (120591 (119905))
119909 (119905)ge (120591119899(119905))1119901
on (1198862119899minus1
1198872119899minus1
) 119899 ge 1198990
(100)
if 119909 (119905) lt 0 119909 (120591 (119905)) lt 0 119909 (120590 (119905)) lt 0
on (120591 (1198862119899) 120590 (119887
2119899))
then 119909 (119905) satisfies 119909 (120590 (119905))
119909 (119905)ge (120590119899(119905))1119901
on (1198862119899 1198872119899) 119899 ge 119899
0
(101)
where 120591119899(119905) and 120590
119899(119905) are defined in (26)
The proof that 119909(119905) satisfies (7) and (8) In this proofwe frequently use assumptions (16) (20) and (23) andstatements (100) and (101) Also because of (15) and 119865(119905 119906) =
119891(119905)|119906|1199011 sgn(119906) 119866(119905 119906) = 119892(119905)|119906|
1199012 sgn(119906) in both cases
(100) and (101) we can simultaneously use
minus119890 (119905) (|119909 (119905)|119901minus1
119909 (119905))minus1
= |119890 (119905)| |119909 (119905)|minus119901
ge 0 on 119869119899
119865 (119905 119909 (120591 (119905))) (|119909 (119905)|119901minus1
119909 (119905))minus1
ge 119891 (119905) |119909 (120591 (119905))|1199011 |119909 (119905)|
minus119901ge 0 on 119869
119899
119866 (119905 119909 (120590 (119905))) (|119909 (119905)|119901minus1
119909 (119905))minus1
ge 119892 (119905) |119909 (120590 (119905))|1199012 |119909 (119905)|
minus119901ge 0 on 119869
119899
|119909 (120591 (119905))| |119909 (119905)|minus1=119909 (120591 (119905))
119909 (119905)
|119909 (120590 (119905))| |119909 (119905)|minus1=119909 (120590 (119905))
119909 (119905)on 119869119899
(102)
where 119869119899= (1198862119899minus1
1198872119899minus1
) in the case of (100) and 119869119899= (1198862119899 1198872119899)
in the case of (101)
(i) Delay or Advanced Case with 119902 = 119901 Since 119902 = 119901 we obtain
[120582119865 (119905 119909 (120591 (119905))) + 120583119866 (119905 119909 (120590 (119905))) minus 120588119890 (119905)]
times (|119909 (119905)|119901minus1
119909 (119905))minus1
ge [120582119891 (119905) |119909 (120591 (119905))|119901+ 120583119892 (119905) |119909 (120590 (119905))|
119901
+120588 |119890 (119905)| ] |119909 (119905)|minus119901
ge [120582119891 (119905) |119909 (120591 (119905))|119901+ 120583119892 (119905) |119909 (120590 (119905))|
119901] |119909 (119905)|
minus119901
= 120582119891 (119905) (119909 (120591 (119905))
119909 (119905))
119901
+ 120583119892 (119905) (119909 (120590 (119905))
119909 (119905))
119901
ge 120582119891 (119905) 120591119899(119905) + 120583119892 (119905) 120590
119899(119905) 119905 isin 119869
119899
(103)
where the functions 120591119899(119905) and 120590
119899(119905) are defined in (26)
(ii) Delay Case with 119902 gt 119901 In this part we use the nextelementary inequality
119883120574+ (120574 minus 1) 119884
120574ge 120574119883119884
120574minus1 120574 gt 1 119883 119884 ge 0 (104)
Since 119902 gt 119901 and using (104) especially for
120574 =119902
119901gt 1 119883 = (120582119891 (119905))
119901119902
|119909 (120591 (119905))|119901
119884 = (119901
119902 minus 119901120588 |119890 (119905)|)
119901119902
(105)
14 Discrete Dynamics in Nature and Society
for all 119905 isin 119869119899we obtain
[120582119865 (119905 119909 (120591 (119905))) + 120583119866 (119905 119909 (120590 (119905))) minus 120588119890 (119905)]
times (|119909 (119905)|119901minus1
119909 (119905))minus1
= [120582119865 (119905 119909 (120591 (119905))) minus 120588119890 (119905)] (|119909 (119905)|119901minus1
119909 (119905))minus1
ge [120582119891 (119905) |119909 (120591 (119905))|1199011 + 120588 |119890 (119905)|] |119909 (119905)|
minus119901
= [119883120574+ (120574 minus 1) 119884
120574] |119909 (119905)|
minus119901
ge119902
119901(120582119891 (119905))
119901119902
|119909 (120591 (119905))|119901
times (119901
119902 minus 119901120588 |119890 (119905)|)
(119901119902)((119902119901)minus1)
|119909 (119905)|minus119901
= 120582119901119902
1205881minus(119901119902)
119902
119901(
119901
119902 minus 119901)
1minus(119901119902)
|119890 (119905)|1minus(119901119902)
times (119891 (119905))119901119902
(119909 (120591 (119905))
119909 (119905))
119901
ge 119896 (120582 120583 120588)119902
119901(
119901
119902 minus 119901)
1minus(119901119902)
|119890 (119905)|1minus(119901119902)
times (119891 (119905))119901119902
120591119899(119905)
(106)
where the function 119896(120582 120583 120588) is from (18)
(iii) Advanced Case with 119902 gt 119901 Using the same line ofarguments as in the proof of the previous case for all 119905 isin 119869
119899
we obtain
[120582119865 (119905 119909 (120591 (119905))) + 120583119866 (119905 119909 (120590 (119905))) minus 120588119890 (119905)]
times (|119909 (119905)|119901minus1
119909 (119905))minus1
= [120583119866 (119905 119909 (120590 (119905))) minus 120588119890 (119905)] (|119909 (119905)|119901minus1
119909 (119905))minus1
ge 119896 (120582 120583 120588)119902
119901(
119901
119902 minus 119901)
1minus(119901119902)
|119890 (119905)|1minus(119901119902)
times (119892 (119905))119901119902
120590119899(119905)
(107)
where the function 119896(120582 120583 120588) is from (21)
(iv) Superlinear Delay-Advanced Case Since 1199011 1199012gt 119901 for
all 119905 isin 119869119899we obtain
[120582119865 (119905 119909 (120591 (119905))) + 120583119866 (119905 119909 (120590 (119905))) minus 120588119890 (119905)]
times (|119909 (119905)|119901minus1
119909 (119905))minus1
= [120582119865 (119905 119909 (120591 (119905))) minus120588
2119890 (119905)] (|119909 (119905)|
119901minus1119909 (119905))minus1
+ [120583119866 (119905 119909 (120590 (119905))) minus120588
2119890 (119905)] (|119909 (119905)|
119901minus1119909 (119905))minus1
ge [120582119891 (119905) |119909 (120591 (119905))|1199011 +
120588
2|119890 (119905)|] |119909 (119905)|
minus119901
+ [120583119892 (119905) |119909 (120590 (119905))|1199012 +
120588
2|119890 (119905)|] |119909 (119905)|
minus119901
(108)
Now just the same as in the proofs of previous delay andadvanced cases with 119902 gt 119901 and with the help of (104) inparticular for
120574 =1199011
119901gt 1 119883 = (120582119891 (119905))
1199011199011
|119909 (120591 (119905))|119901
119884 = (119901
1199011minus 119901
120588
2|119890 (119905)|)
1199011199011
(109)
we have
[120582119891 (119905) |119909 (120591 (119905))|1199011 +
120588
2 |119890 (119905)|] |119909 (119905)|
minus119901
= [119883120574+ (120574 minus 1) 119884
120574] |119909 (119905)|
minus119901
ge1199011
119901(120582119891 (119905))
1199011199011
|119909 (120591 (119905))|119901
times (119901
1199011minus 119901
120588
2|119890 (119905)|)
(1199011199011)((1199011119901)minus1)
|119909 (119905)|minus119901
= 12058211990111990111205881minus(119901119901
1)1199011
119901(
119901
2 (1199011minus 119901)
)
1minus(1199011199011)
times |119890 (119905)|1minus(119901119901
1)(119891 (119905))
1199011199011
(119909 (120591 (119905))
119909 (119905))
119901
ge 119896 (120582 120583 120588)1199011
119901(
119901
2 (1199011minus 119901)
)
1minus(1199011199011)
times |119890 (119905)|1minus(119901119901
1)(119891 (119905))
1199011199011
120591119899(119905)
(110)
where the function 119896(120582 120583 120588) is from (24) Analogously weshow that
[120583119892 (119905) |119909 (120590 (119905))|1199012 +
120588
2|119890 (119905)|] |119909 (119905)|
minus119901
ge 119896 (120582 120583 120588)1199012
119901(
119901
2 (1199012minus 119901)
)
1minus(1199011199012)
times |119890 (119905)|1minus(119901119901
2)(119891 (119905))
1199011199012
120590119899(119905)
(111)
Discrete Dynamics in Nature and Society 15
Summarizing previous calculation we obtain
[120582119865 (119905 119909 (120591 (119905))) + 120583119866 (119905 119909 (120590 (119905))) minus 120588119890 (119905)]
times (|119909 (119905)|119901minus1
119909 (119905))minus1
ge 119896 (120582 120583 120588)119876119899(119905) 119905 isin 119869
119899
(112)
where the function 119896(120582 120583 120588) is from (24)
(v) Supersublinear Delay-Advanced Case Since 1199011gt 119901 gt 119901
2
and the following well-known elementary inequality holds
12057801199060+ 12057811199061+ 12057821199062ge 1199061205780
01199061205781
11199061205782
2 120578119894ge 0 119906
119894ge 0 (113)
from 1205780 1205781 1205782isin (0 1) 120578
0+ 1205781+ 1205782= 1 and 119901
11205781+ 11990121205782= 119901
we obtain for all 119905 isin 119869119899 for all 119905 isin 119869
119899we obtain
[120582119865 (119905 119909 (120591 (119905))) + 120583119866 (119905 119909 (120590 (119905))) minus 120588119890 (119905)]
times (|119909 (119905)|119901minus1
119909 (119905))minus1
ge [120582119891 (119905) |119909 (120591 (119905))|1199011 + 120583119892 (119905) |119909 (120590 (119905))|
1199012 + 120588 |119890 (119905)|]
times |119909 (119905)|minus119901
= [1205781[120578minus1
1120582119891 (119905) |119909 (120591 (119905))|
1199011] + 120578
2[120578minus1
2120583119892 (119905) |119909 (120590 (119905))|
1199012]
+1205780[120578minus1
0120588 |119890 (119905)|]] |119909 (119905)|
minus119901
ge [120578minus1
1120582119891 (119905) |119909 (120591 (119905))|
1199011]1205781
[120578minus1
2120583119892 (119905) |119909 (120590 (119905))|
1199012]1205782
times [120578minus1
0120588 |119890 (119905)|]
1205780
|119909 (119905)|minus119901
= 120582120578112058312057821205881205780 |119890 (119905)|
1205780(119891 (119905))
1205781
(119892 (119905))1205782
times|119909 (120591 (119905))|
12057811199011
|119909 (119905)|12057811199011
|119909 (120590 (119905))|12057821199012
|119909 (119905)|12057821199012
2
prod
119894=0
120578minus120578119894
119894
= 119896 (120582 120583 120588) |119890 (119905)|1205780(119891 (119905))
1205781
(119892 (119905))1205782
times (119909 (120591 (119905))
119909 (119905))
12057811199011
(119909 (120590 (119905))
119909 (119905))
12057821199012 2
prod
119894=0
120578minus120578119894
119894
ge 119896 (120582 120583 120588) |119890 (119905)|1205780(119891 (119905))
1205781
(119892 (119905))1205782
(120591119899(119905))1205781(1199011119901)
times (120590119899(119905))1205782(1199012119901)
2
prod
119894=0
120578minus120578119894
119894
= 119896 (120582 120583 120588)119876119899(119905)
(114)
where 119896(120582 120583 120588) and 119876119899(119905) are given respectively in (24) and
(25) Thus it is shown that required condition (5) in thecases (i)ndash(iv) is fulfilled with respect to 119896(120582 120583 120588) and 119876
119899(119905)
determined by (18) (21) or (24) and (19) (22) or (25)In conclusion according to the previous observation we
see that all assumptions of Lemma 4 are fulfilled and henceLemma 4 proves Theorems 5 6 and 7
Conflict of Interests
The author declares that there is no conflict of interestsregarding the publication of this paper
References
[1] Y G Sun ldquoA note on Nasrrsquos and Wongrsquos papersrdquo Journal ofMathematical Analysis and Applications vol 286 no 1 pp 363ndash367 2003
[2] Y G Sun C H Ou and J S W Wong ldquoInterval oscillationtheorems for a second-order linear differential equationrdquo Com-puters amp Mathematics with Applications vol 48 no 10-11 pp1693ndash1699 2004
[3] S Murugadass E Thandapani and S Pinelas ldquoOscillationcriteria for forced second-order mixed type quasilinear delaydifferential equationsrdquo Electronic Journal of Differential Equa-tions vol 2010 article 73 9 pages 2010
[4] Y Bai and L Liu ldquoNew oscillation criteria for second-orderdelay differential equations with mixed nonlinearitiesrdquoDiscreteDynamics in Nature and Society vol 2010 Article ID 796256 9pages 2010
[5] A F Guvenilir andA Zafer ldquoSecond-order oscillation of forcedfunctional differential equations with oscillatory potentialsrdquoComputers amp Mathematics with Applications vol 51 no 9-10pp 1395ndash1404 2006
[6] A Zafer ldquoInterval oscillation criteria for second order super-half linear functional differential equations with delay andadvanced argumentsrdquoMathematische Nachrichten vol 282 no9 pp 1334ndash1341 2009
[7] A F Guvenilir ldquoInterval oscillation of second-order functionaldifferential equations with oscillatory potentialsrdquo NonlinearAnalysis Theory Methods amp Applications vol 71 no 12 ppe2849ndashe2854 2009
[8] T S Hassan L Erbe and A Peterson ldquoForced oscillation ofsecond order differential equations with mixed nonlinearitiesrdquoActa Mathematica Scientia B vol 31 no 2 pp 613ndash626 2011
[9] M Pasic ldquoNew oscillation criteria for second-order forcedquasilinear functional differential equationsrdquo Abstract andApplied Analysis vol 2013 Article ID 735360 12 pages 2013
[10] L H Erbe Q Kong and B G Zhang Oscillation Theory forFunctional-Differential Equations vol 190 Marcel Dekker NewYork NY USA 1995
[11] V Kolmanovskii and A Myshkis Introduction to the Theoryand Applications of Functional-Differential Equations vol 463Kluwer Academic Publishers Dordrecht The Netherlands1999
[12] R P Agarwal M Bohner and W-T Li Nonoscillation andOscillation Theory for Functional Differential Equations vol267 Marcel Dekker New York NY USA 2004
[13] L Erbe T Hassan and A Peterson ldquoOscillation of secondorder functional dynamic equationsrdquo International Journal ofDifference Equations vol 5 no 2 pp 175ndash193 2010
[14] B Baculıkova J Dzurina and Y V Rogovchenko ldquoOscillationof third order trinomial delay differential equationsrdquo AppliedMathematics and Computation vol 218 no 13 pp 7023ndash70332012
[15] R P Agarwal L Berezansky E Braverman and A Domoshnit-sky Nonoscillation Theory of Functional Differential Equationswith Applications Springer New York NY USA 2012
16 Discrete Dynamics in Nature and Society
[16] J Zhang ldquoVariational approach to solitary wave solution ofthe generalized Zakharov equationrdquo Computers ampMathematicswith Applications vol 54 no 7-8 pp 1043ndash1046 2007
[17] T Ozis and A Yıldırım ldquoApplication of Hersquos semi-inversemethod to the nonlinear Schrodinger equationrdquo Computers ampMathematics with Applications vol 54 no 7-8 pp 1039ndash10422007
[18] X-C Cai andM-S Li ldquoPeriodic solution of Jacobi elliptic equa-tions by Hersquos perturbation methodrdquo Computers amp Mathematicswith Applications vol 54 no 7-8 pp 1210ndash1212 2007
[19] S Lenci G Menditto and A M Tarantino ldquoHomoclinic andheteroclinic bifurcations in the non-linear dynamics of a beamresting on an elastic substraterdquo International Journal of Non-Linear Mechanics vol 34 no 4 pp 615ndash632 1999
[20] D-J Huang and H-Q Zhang ldquoLink between travelling wavesand first order nonlinear ordinary differential equation with asixth-degree nonlinear termrdquoChaos Solitons amp Fractals vol 29no 4 pp 928ndash941 2006
[21] A I Maimistov ldquoPropagation of an ultimately short electro-magnetic pulse in a nonlinear medium described by the fifth-order Duffing modelrdquo Optics and Spectroscopy vol 94 pp 251ndash257 2003
[22] M N Hamdan and N H Shabaneh ldquoOn the large amplitudefree vibrations of a restrained uniform beam carrying anintermediate lumpedmassrdquo Journal of Sound andVibration vol199 no 5 pp 711ndash736 1997
[23] J-H He ldquoSome asymptotic methods for strongly nonlinearequationsrdquo International Journal of Modern Physics B vol 20no 10 pp 1141ndash1199 2006
[24] J B Marion Classical Dynamics of Particles and Systems 1970[25] I Kovacic and M J Brennan The Duffing Equation Nonlinear
Oscillatos and their Behaviour John Wiley amp Sons 1st edition2011
[26] F C Moon Chaotic Vibrations An Introduction for AppliedScientists and Engineers John Wiley amp Sons New York NYUSA 2004
[27] J J Stoker Nonlinear Vibrations 1950[28] G Chen and Z Tao ldquoAmplitude-frequency relationship for the
relativistic oscillatorrdquoAASRI Procedia vol 1 pp 400ndash403 2012[29] R E Mickens Oscillations in Planar Dynamic Systems World
Scientific Publishing Singapore 1996[30] A Belendez T Belendez C Neipp A Hernandez and M
L Alvarez ldquoApproximate solutions of a nonlinear oscillatortypified as a mass attached to a stretched elastic wire by thehomotopy perturbation methodrdquo Chaos Solitions and Fractalsvol 39 pp 746ndash764 2009
[31] A Belendez E Fernandez R Fuentes J J Rodes and I PascualldquoHarmonic balancing approach to nonlinear oscillations of apunctual charge in the eletric field of charged ringrdquo PhysicsLetters A vol 373 pp 735ndash740 2009
[32] A Elıas-Zuniga ldquoExact solution of the cubic-quintic Duffingoscillatorrdquo Applied Mathematical Modelling vol 37 no 4 pp2574ndash2579 2013
[33] A Belendez M L Alvarez J Frances et al ldquoAnalytical approx-imate solutions for the cubic-quintic Duffing oscillator in termsof elementary functionsrdquo Journal of Applied Mathematics vol2012 Article ID 286290 16 pages 2012
[34] A Elıas-Zuniga OMartınez-Romero andR K Cordoba-DıazldquoApproximate solution for the Duffing-harmonic oscillator bythe enhanced cubication methodrdquo Mathematical Problems inEngineering vol 2012 Article ID 618750 12 pages 2012
[35] C W Lim B S Wu andW P Sun ldquoHigher accuracy analyticalapproximations to the Duffing-harmonic oscillatorrdquo Journal ofSound and Vibration vol 296 no 4-5 pp 1039ndash1045 2006
[36] J He ldquoSome new approaches to Duffing equation with stronglyand high order nonlinearity II parametrized perturbationtechniquerdquo Communications in Nonlinear Science amp NumericalSimulation vol 4 no 1 pp 81ndash83 1999
[37] V Marinca and N Herisanu ldquoPeriodic solutions for somestrongly nonlinear oscillations by Hersquos variational iterationmethodrdquo Computers amp Mathematics with Applications vol 54no 7-8 pp 1188ndash1196 2007
[38] W Lu and Y Liu ldquoVibration control for the primary resonanceof the Duffing oscillator by a time delay state feedbackrdquoInternational Journal of Nonlinear Science vol 8 no 3 pp 324ndash328 2009
[39] H Y Hu and Z H Wang Dynamics of Controlled MechanicalSystems with Delayed Feedback Springer 2002
[40] M Hamdi and M Belhaq ldquoControl of bistability in a delayedDuffing oscillatorrdquo Advances in Acoustics and Vibration vol2012 Article ID 872498 6 pages 2012
[41] V Ravichandran C Chinnathambi and S Rajasekar ldquoNonlin-ear resonance in Duffing oscillator with fixed and integrativetime-delayed feedbacksrdquoPramana Journal of Physics vol 78 pp347ndash360 2013
[42] X You and H Xu ldquoAnalytical approximations for the periodicmotion of theDuffing systemwith delayed feedbackrdquoNumericalAlgorithms vol 56 no 4 pp 561ndash576 2011
[43] Z Sun W Xu X Yang and T Fang ldquoInducing or suppressingchaos in a double-well Duffing oscillator by time delay feed-backrdquo Chaos Solitons and Fractals vol 27 pp 705ndash714 2006
[44] H Wang H Hu and Z Wang ldquoGlobal dynamics of a Duffingoscillator with delayed displacement feedbackrdquo InternationalJournal of Bifurcation and Chaos in Applied Sciences andEngineering vol 14 no 8 pp 2753ndash2775 2004
[45] J Chiasson and J J LoiseauApplications of Time Delay SystemsSpringer 2007
[46] M Lakshmanan andDV SenthilkumarDynamics of NonlinearTime-Delay Systems Springer 2010
[47] G Stepan T Insperger and R Szalai ldquoDelay parametricexcitation and the nonlinear dynamics of cutting processesrdquoInternational Journal of Bifurcation and Chaos in Applied Sci-ences and Engineering vol 15 no 9 pp 2783ndash2798 2005
[48] U van der Heiden and H-O Walther ldquoExistence of chaos incontrol systems with delayed feedbackrdquo Journal of DifferentialEquations vol 47 no 2 pp 273ndash295 1983
[49] Y G Sun and J S W Wong ldquoOscillation criteria for secondorder forced ordinary differential equations with mixed non-linearitiesrdquo Journal of Mathematical Analysis and Applicationsvol 334 no 1 pp 549ndash560 2007
[50] J Heagy and W L Ditto ldquoDynamics of a two-frequencyparametrically driven Duffing oscillatorrdquo Journal of NonlinearScience vol 1 no 4 pp 423ndash455 1991
[51] A B Belogortsev ldquoBifurcations of tori and chaos in thequasiperiodically forced Duffing oscillatorrdquoNonlinearity vol 5no 4 pp 889ndash897 1992
[52] M Belhaq and M Houssni ldquoQuasi-periodic oscillations chaosand suppression of chaos in a nonlinear oscillator driven byparametric and external excitationsrdquo Nonlinear Dynamics vol18 no 1 pp 1ndash24 1999
[53] S H Saker P Y H Pang and R P Agarwal ldquoOscillationtheorems for second order nonlinear functional differential
Discrete Dynamics in Nature and Society 17
equations with dampingrdquo Dynamic Systems and Applicationsvol 12 no 3-4 pp 307ndash321 2003
[54] I N Bronshtein K A Semendyayev G Musiol and HMuehligHandbook of Mathematics Springer 5th edition 2007
[55] M Pasic ldquoFite-Wintner-Leighton-type oscillation criteria forsecond-order differential equations with nonlinear dampingrdquoAbstract and Applied Analysis vol 2013 Article ID 852180 10pages 2013
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
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Mathematical Problems in Engineering
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Differential EquationsInternational Journal of
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Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
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Decision SciencesAdvances in
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Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
Discrete Dynamics in Nature and Society 13
Lemma 25 Let 120591119886119887(119905) and 120590
119886119887(119905) be defined by
120591119886119887(119905) = (
120591 (119905) minus 120591 (119886)
119905 minus 120591 (119886))
119901
120590119886119887(119905) = (
120590 (119887) minus 120590 (119905)
120590 (119887) minus 119905)
119901
119905 isin (119886 119887)
(97)
and let 119909 isin 1198622([1198790infin)R) be an arbitrary function If
(119903(119905)119860(1199091015840(119905)) le 0 for all 119905 isin (120591(119886) 120590(119887)) or (119903(119905)119860(1199091015840(119905)) ge 0
for all 119905 isin (120591(119886) 120590(119887)) then
119909 (120591 (119905))
119909 (119905)ge (120591119886119887(119905))1119901
119909 (120590 (119905))
119909 (119905)ge (120590119886119887(119905))1119901
119905 isin (119886 119887)
(98)
Since119860(V) is supposed to be odd and increasing functionjust before (3) and 119903(119905) satisfies (14) the proof of Lemma 25in the first case that is 119903(119905)119860(1199091015840(119905)) le 0 for all 119905 isin (120591(119886) 120590(119887))is the same as the proof of [9 Corollaries 17 and 18] But in thesecond case that is 119903(119905)119860(1199091015840(119905)) ge 0 for all 119905 isin (120591(119886) 120590(119887))the proof is as follows if previous inequality holds then119903(119905)119860(minus119909
1015840(119905)) le 0 for all 119905 isin (120591(119886) 120590(119887)) and therefore to
the function minus119909(119905) one can apply the first case of this lemmaand consequently one obtains
119909 (120591 (119905))
119909 (119905)=minus119909 (120591 (119905))
minus119909 (119905)ge (120591119886119887(119905))1119901
119909 (120590 (119905))
119909 (119905)=minus119909 (120590 (119905))
minus119909 (119905)ge (120590119886119887(119905))1119901
119905 isin (119886 119887)
(99)
which proves this lemma in the second caseNow combining statements (95) (96) and (98) one
easily obtains
if 119909 (119905) gt 0 119909 (120591 (119905)) gt 0 119909 (120590 (119905)) gt 0
on (120591 (1198862119899minus1
) 120590 (1198872119899minus1
))
then 119909 (119905) satisfies 119909 (120591 (119905))
119909 (119905)ge (120591119899(119905))1119901
on (1198862119899minus1
1198872119899minus1
) 119899 ge 1198990
(100)
if 119909 (119905) lt 0 119909 (120591 (119905)) lt 0 119909 (120590 (119905)) lt 0
on (120591 (1198862119899) 120590 (119887
2119899))
then 119909 (119905) satisfies 119909 (120590 (119905))
119909 (119905)ge (120590119899(119905))1119901
on (1198862119899 1198872119899) 119899 ge 119899
0
(101)
where 120591119899(119905) and 120590
119899(119905) are defined in (26)
The proof that 119909(119905) satisfies (7) and (8) In this proofwe frequently use assumptions (16) (20) and (23) andstatements (100) and (101) Also because of (15) and 119865(119905 119906) =
119891(119905)|119906|1199011 sgn(119906) 119866(119905 119906) = 119892(119905)|119906|
1199012 sgn(119906) in both cases
(100) and (101) we can simultaneously use
minus119890 (119905) (|119909 (119905)|119901minus1
119909 (119905))minus1
= |119890 (119905)| |119909 (119905)|minus119901
ge 0 on 119869119899
119865 (119905 119909 (120591 (119905))) (|119909 (119905)|119901minus1
119909 (119905))minus1
ge 119891 (119905) |119909 (120591 (119905))|1199011 |119909 (119905)|
minus119901ge 0 on 119869
119899
119866 (119905 119909 (120590 (119905))) (|119909 (119905)|119901minus1
119909 (119905))minus1
ge 119892 (119905) |119909 (120590 (119905))|1199012 |119909 (119905)|
minus119901ge 0 on 119869
119899
|119909 (120591 (119905))| |119909 (119905)|minus1=119909 (120591 (119905))
119909 (119905)
|119909 (120590 (119905))| |119909 (119905)|minus1=119909 (120590 (119905))
119909 (119905)on 119869119899
(102)
where 119869119899= (1198862119899minus1
1198872119899minus1
) in the case of (100) and 119869119899= (1198862119899 1198872119899)
in the case of (101)
(i) Delay or Advanced Case with 119902 = 119901 Since 119902 = 119901 we obtain
[120582119865 (119905 119909 (120591 (119905))) + 120583119866 (119905 119909 (120590 (119905))) minus 120588119890 (119905)]
times (|119909 (119905)|119901minus1
119909 (119905))minus1
ge [120582119891 (119905) |119909 (120591 (119905))|119901+ 120583119892 (119905) |119909 (120590 (119905))|
119901
+120588 |119890 (119905)| ] |119909 (119905)|minus119901
ge [120582119891 (119905) |119909 (120591 (119905))|119901+ 120583119892 (119905) |119909 (120590 (119905))|
119901] |119909 (119905)|
minus119901
= 120582119891 (119905) (119909 (120591 (119905))
119909 (119905))
119901
+ 120583119892 (119905) (119909 (120590 (119905))
119909 (119905))
119901
ge 120582119891 (119905) 120591119899(119905) + 120583119892 (119905) 120590
119899(119905) 119905 isin 119869
119899
(103)
where the functions 120591119899(119905) and 120590
119899(119905) are defined in (26)
(ii) Delay Case with 119902 gt 119901 In this part we use the nextelementary inequality
119883120574+ (120574 minus 1) 119884
120574ge 120574119883119884
120574minus1 120574 gt 1 119883 119884 ge 0 (104)
Since 119902 gt 119901 and using (104) especially for
120574 =119902
119901gt 1 119883 = (120582119891 (119905))
119901119902
|119909 (120591 (119905))|119901
119884 = (119901
119902 minus 119901120588 |119890 (119905)|)
119901119902
(105)
14 Discrete Dynamics in Nature and Society
for all 119905 isin 119869119899we obtain
[120582119865 (119905 119909 (120591 (119905))) + 120583119866 (119905 119909 (120590 (119905))) minus 120588119890 (119905)]
times (|119909 (119905)|119901minus1
119909 (119905))minus1
= [120582119865 (119905 119909 (120591 (119905))) minus 120588119890 (119905)] (|119909 (119905)|119901minus1
119909 (119905))minus1
ge [120582119891 (119905) |119909 (120591 (119905))|1199011 + 120588 |119890 (119905)|] |119909 (119905)|
minus119901
= [119883120574+ (120574 minus 1) 119884
120574] |119909 (119905)|
minus119901
ge119902
119901(120582119891 (119905))
119901119902
|119909 (120591 (119905))|119901
times (119901
119902 minus 119901120588 |119890 (119905)|)
(119901119902)((119902119901)minus1)
|119909 (119905)|minus119901
= 120582119901119902
1205881minus(119901119902)
119902
119901(
119901
119902 minus 119901)
1minus(119901119902)
|119890 (119905)|1minus(119901119902)
times (119891 (119905))119901119902
(119909 (120591 (119905))
119909 (119905))
119901
ge 119896 (120582 120583 120588)119902
119901(
119901
119902 minus 119901)
1minus(119901119902)
|119890 (119905)|1minus(119901119902)
times (119891 (119905))119901119902
120591119899(119905)
(106)
where the function 119896(120582 120583 120588) is from (18)
(iii) Advanced Case with 119902 gt 119901 Using the same line ofarguments as in the proof of the previous case for all 119905 isin 119869
119899
we obtain
[120582119865 (119905 119909 (120591 (119905))) + 120583119866 (119905 119909 (120590 (119905))) minus 120588119890 (119905)]
times (|119909 (119905)|119901minus1
119909 (119905))minus1
= [120583119866 (119905 119909 (120590 (119905))) minus 120588119890 (119905)] (|119909 (119905)|119901minus1
119909 (119905))minus1
ge 119896 (120582 120583 120588)119902
119901(
119901
119902 minus 119901)
1minus(119901119902)
|119890 (119905)|1minus(119901119902)
times (119892 (119905))119901119902
120590119899(119905)
(107)
where the function 119896(120582 120583 120588) is from (21)
(iv) Superlinear Delay-Advanced Case Since 1199011 1199012gt 119901 for
all 119905 isin 119869119899we obtain
[120582119865 (119905 119909 (120591 (119905))) + 120583119866 (119905 119909 (120590 (119905))) minus 120588119890 (119905)]
times (|119909 (119905)|119901minus1
119909 (119905))minus1
= [120582119865 (119905 119909 (120591 (119905))) minus120588
2119890 (119905)] (|119909 (119905)|
119901minus1119909 (119905))minus1
+ [120583119866 (119905 119909 (120590 (119905))) minus120588
2119890 (119905)] (|119909 (119905)|
119901minus1119909 (119905))minus1
ge [120582119891 (119905) |119909 (120591 (119905))|1199011 +
120588
2|119890 (119905)|] |119909 (119905)|
minus119901
+ [120583119892 (119905) |119909 (120590 (119905))|1199012 +
120588
2|119890 (119905)|] |119909 (119905)|
minus119901
(108)
Now just the same as in the proofs of previous delay andadvanced cases with 119902 gt 119901 and with the help of (104) inparticular for
120574 =1199011
119901gt 1 119883 = (120582119891 (119905))
1199011199011
|119909 (120591 (119905))|119901
119884 = (119901
1199011minus 119901
120588
2|119890 (119905)|)
1199011199011
(109)
we have
[120582119891 (119905) |119909 (120591 (119905))|1199011 +
120588
2 |119890 (119905)|] |119909 (119905)|
minus119901
= [119883120574+ (120574 minus 1) 119884
120574] |119909 (119905)|
minus119901
ge1199011
119901(120582119891 (119905))
1199011199011
|119909 (120591 (119905))|119901
times (119901
1199011minus 119901
120588
2|119890 (119905)|)
(1199011199011)((1199011119901)minus1)
|119909 (119905)|minus119901
= 12058211990111990111205881minus(119901119901
1)1199011
119901(
119901
2 (1199011minus 119901)
)
1minus(1199011199011)
times |119890 (119905)|1minus(119901119901
1)(119891 (119905))
1199011199011
(119909 (120591 (119905))
119909 (119905))
119901
ge 119896 (120582 120583 120588)1199011
119901(
119901
2 (1199011minus 119901)
)
1minus(1199011199011)
times |119890 (119905)|1minus(119901119901
1)(119891 (119905))
1199011199011
120591119899(119905)
(110)
where the function 119896(120582 120583 120588) is from (24) Analogously weshow that
[120583119892 (119905) |119909 (120590 (119905))|1199012 +
120588
2|119890 (119905)|] |119909 (119905)|
minus119901
ge 119896 (120582 120583 120588)1199012
119901(
119901
2 (1199012minus 119901)
)
1minus(1199011199012)
times |119890 (119905)|1minus(119901119901
2)(119891 (119905))
1199011199012
120590119899(119905)
(111)
Discrete Dynamics in Nature and Society 15
Summarizing previous calculation we obtain
[120582119865 (119905 119909 (120591 (119905))) + 120583119866 (119905 119909 (120590 (119905))) minus 120588119890 (119905)]
times (|119909 (119905)|119901minus1
119909 (119905))minus1
ge 119896 (120582 120583 120588)119876119899(119905) 119905 isin 119869
119899
(112)
where the function 119896(120582 120583 120588) is from (24)
(v) Supersublinear Delay-Advanced Case Since 1199011gt 119901 gt 119901
2
and the following well-known elementary inequality holds
12057801199060+ 12057811199061+ 12057821199062ge 1199061205780
01199061205781
11199061205782
2 120578119894ge 0 119906
119894ge 0 (113)
from 1205780 1205781 1205782isin (0 1) 120578
0+ 1205781+ 1205782= 1 and 119901
11205781+ 11990121205782= 119901
we obtain for all 119905 isin 119869119899 for all 119905 isin 119869
119899we obtain
[120582119865 (119905 119909 (120591 (119905))) + 120583119866 (119905 119909 (120590 (119905))) minus 120588119890 (119905)]
times (|119909 (119905)|119901minus1
119909 (119905))minus1
ge [120582119891 (119905) |119909 (120591 (119905))|1199011 + 120583119892 (119905) |119909 (120590 (119905))|
1199012 + 120588 |119890 (119905)|]
times |119909 (119905)|minus119901
= [1205781[120578minus1
1120582119891 (119905) |119909 (120591 (119905))|
1199011] + 120578
2[120578minus1
2120583119892 (119905) |119909 (120590 (119905))|
1199012]
+1205780[120578minus1
0120588 |119890 (119905)|]] |119909 (119905)|
minus119901
ge [120578minus1
1120582119891 (119905) |119909 (120591 (119905))|
1199011]1205781
[120578minus1
2120583119892 (119905) |119909 (120590 (119905))|
1199012]1205782
times [120578minus1
0120588 |119890 (119905)|]
1205780
|119909 (119905)|minus119901
= 120582120578112058312057821205881205780 |119890 (119905)|
1205780(119891 (119905))
1205781
(119892 (119905))1205782
times|119909 (120591 (119905))|
12057811199011
|119909 (119905)|12057811199011
|119909 (120590 (119905))|12057821199012
|119909 (119905)|12057821199012
2
prod
119894=0
120578minus120578119894
119894
= 119896 (120582 120583 120588) |119890 (119905)|1205780(119891 (119905))
1205781
(119892 (119905))1205782
times (119909 (120591 (119905))
119909 (119905))
12057811199011
(119909 (120590 (119905))
119909 (119905))
12057821199012 2
prod
119894=0
120578minus120578119894
119894
ge 119896 (120582 120583 120588) |119890 (119905)|1205780(119891 (119905))
1205781
(119892 (119905))1205782
(120591119899(119905))1205781(1199011119901)
times (120590119899(119905))1205782(1199012119901)
2
prod
119894=0
120578minus120578119894
119894
= 119896 (120582 120583 120588)119876119899(119905)
(114)
where 119896(120582 120583 120588) and 119876119899(119905) are given respectively in (24) and
(25) Thus it is shown that required condition (5) in thecases (i)ndash(iv) is fulfilled with respect to 119896(120582 120583 120588) and 119876
119899(119905)
determined by (18) (21) or (24) and (19) (22) or (25)In conclusion according to the previous observation we
see that all assumptions of Lemma 4 are fulfilled and henceLemma 4 proves Theorems 5 6 and 7
Conflict of Interests
The author declares that there is no conflict of interestsregarding the publication of this paper
References
[1] Y G Sun ldquoA note on Nasrrsquos and Wongrsquos papersrdquo Journal ofMathematical Analysis and Applications vol 286 no 1 pp 363ndash367 2003
[2] Y G Sun C H Ou and J S W Wong ldquoInterval oscillationtheorems for a second-order linear differential equationrdquo Com-puters amp Mathematics with Applications vol 48 no 10-11 pp1693ndash1699 2004
[3] S Murugadass E Thandapani and S Pinelas ldquoOscillationcriteria for forced second-order mixed type quasilinear delaydifferential equationsrdquo Electronic Journal of Differential Equa-tions vol 2010 article 73 9 pages 2010
[4] Y Bai and L Liu ldquoNew oscillation criteria for second-orderdelay differential equations with mixed nonlinearitiesrdquoDiscreteDynamics in Nature and Society vol 2010 Article ID 796256 9pages 2010
[5] A F Guvenilir andA Zafer ldquoSecond-order oscillation of forcedfunctional differential equations with oscillatory potentialsrdquoComputers amp Mathematics with Applications vol 51 no 9-10pp 1395ndash1404 2006
[6] A Zafer ldquoInterval oscillation criteria for second order super-half linear functional differential equations with delay andadvanced argumentsrdquoMathematische Nachrichten vol 282 no9 pp 1334ndash1341 2009
[7] A F Guvenilir ldquoInterval oscillation of second-order functionaldifferential equations with oscillatory potentialsrdquo NonlinearAnalysis Theory Methods amp Applications vol 71 no 12 ppe2849ndashe2854 2009
[8] T S Hassan L Erbe and A Peterson ldquoForced oscillation ofsecond order differential equations with mixed nonlinearitiesrdquoActa Mathematica Scientia B vol 31 no 2 pp 613ndash626 2011
[9] M Pasic ldquoNew oscillation criteria for second-order forcedquasilinear functional differential equationsrdquo Abstract andApplied Analysis vol 2013 Article ID 735360 12 pages 2013
[10] L H Erbe Q Kong and B G Zhang Oscillation Theory forFunctional-Differential Equations vol 190 Marcel Dekker NewYork NY USA 1995
[11] V Kolmanovskii and A Myshkis Introduction to the Theoryand Applications of Functional-Differential Equations vol 463Kluwer Academic Publishers Dordrecht The Netherlands1999
[12] R P Agarwal M Bohner and W-T Li Nonoscillation andOscillation Theory for Functional Differential Equations vol267 Marcel Dekker New York NY USA 2004
[13] L Erbe T Hassan and A Peterson ldquoOscillation of secondorder functional dynamic equationsrdquo International Journal ofDifference Equations vol 5 no 2 pp 175ndash193 2010
[14] B Baculıkova J Dzurina and Y V Rogovchenko ldquoOscillationof third order trinomial delay differential equationsrdquo AppliedMathematics and Computation vol 218 no 13 pp 7023ndash70332012
[15] R P Agarwal L Berezansky E Braverman and A Domoshnit-sky Nonoscillation Theory of Functional Differential Equationswith Applications Springer New York NY USA 2012
16 Discrete Dynamics in Nature and Society
[16] J Zhang ldquoVariational approach to solitary wave solution ofthe generalized Zakharov equationrdquo Computers ampMathematicswith Applications vol 54 no 7-8 pp 1043ndash1046 2007
[17] T Ozis and A Yıldırım ldquoApplication of Hersquos semi-inversemethod to the nonlinear Schrodinger equationrdquo Computers ampMathematics with Applications vol 54 no 7-8 pp 1039ndash10422007
[18] X-C Cai andM-S Li ldquoPeriodic solution of Jacobi elliptic equa-tions by Hersquos perturbation methodrdquo Computers amp Mathematicswith Applications vol 54 no 7-8 pp 1210ndash1212 2007
[19] S Lenci G Menditto and A M Tarantino ldquoHomoclinic andheteroclinic bifurcations in the non-linear dynamics of a beamresting on an elastic substraterdquo International Journal of Non-Linear Mechanics vol 34 no 4 pp 615ndash632 1999
[20] D-J Huang and H-Q Zhang ldquoLink between travelling wavesand first order nonlinear ordinary differential equation with asixth-degree nonlinear termrdquoChaos Solitons amp Fractals vol 29no 4 pp 928ndash941 2006
[21] A I Maimistov ldquoPropagation of an ultimately short electro-magnetic pulse in a nonlinear medium described by the fifth-order Duffing modelrdquo Optics and Spectroscopy vol 94 pp 251ndash257 2003
[22] M N Hamdan and N H Shabaneh ldquoOn the large amplitudefree vibrations of a restrained uniform beam carrying anintermediate lumpedmassrdquo Journal of Sound andVibration vol199 no 5 pp 711ndash736 1997
[23] J-H He ldquoSome asymptotic methods for strongly nonlinearequationsrdquo International Journal of Modern Physics B vol 20no 10 pp 1141ndash1199 2006
[24] J B Marion Classical Dynamics of Particles and Systems 1970[25] I Kovacic and M J Brennan The Duffing Equation Nonlinear
Oscillatos and their Behaviour John Wiley amp Sons 1st edition2011
[26] F C Moon Chaotic Vibrations An Introduction for AppliedScientists and Engineers John Wiley amp Sons New York NYUSA 2004
[27] J J Stoker Nonlinear Vibrations 1950[28] G Chen and Z Tao ldquoAmplitude-frequency relationship for the
relativistic oscillatorrdquoAASRI Procedia vol 1 pp 400ndash403 2012[29] R E Mickens Oscillations in Planar Dynamic Systems World
Scientific Publishing Singapore 1996[30] A Belendez T Belendez C Neipp A Hernandez and M
L Alvarez ldquoApproximate solutions of a nonlinear oscillatortypified as a mass attached to a stretched elastic wire by thehomotopy perturbation methodrdquo Chaos Solitions and Fractalsvol 39 pp 746ndash764 2009
[31] A Belendez E Fernandez R Fuentes J J Rodes and I PascualldquoHarmonic balancing approach to nonlinear oscillations of apunctual charge in the eletric field of charged ringrdquo PhysicsLetters A vol 373 pp 735ndash740 2009
[32] A Elıas-Zuniga ldquoExact solution of the cubic-quintic Duffingoscillatorrdquo Applied Mathematical Modelling vol 37 no 4 pp2574ndash2579 2013
[33] A Belendez M L Alvarez J Frances et al ldquoAnalytical approx-imate solutions for the cubic-quintic Duffing oscillator in termsof elementary functionsrdquo Journal of Applied Mathematics vol2012 Article ID 286290 16 pages 2012
[34] A Elıas-Zuniga OMartınez-Romero andR K Cordoba-DıazldquoApproximate solution for the Duffing-harmonic oscillator bythe enhanced cubication methodrdquo Mathematical Problems inEngineering vol 2012 Article ID 618750 12 pages 2012
[35] C W Lim B S Wu andW P Sun ldquoHigher accuracy analyticalapproximations to the Duffing-harmonic oscillatorrdquo Journal ofSound and Vibration vol 296 no 4-5 pp 1039ndash1045 2006
[36] J He ldquoSome new approaches to Duffing equation with stronglyand high order nonlinearity II parametrized perturbationtechniquerdquo Communications in Nonlinear Science amp NumericalSimulation vol 4 no 1 pp 81ndash83 1999
[37] V Marinca and N Herisanu ldquoPeriodic solutions for somestrongly nonlinear oscillations by Hersquos variational iterationmethodrdquo Computers amp Mathematics with Applications vol 54no 7-8 pp 1188ndash1196 2007
[38] W Lu and Y Liu ldquoVibration control for the primary resonanceof the Duffing oscillator by a time delay state feedbackrdquoInternational Journal of Nonlinear Science vol 8 no 3 pp 324ndash328 2009
[39] H Y Hu and Z H Wang Dynamics of Controlled MechanicalSystems with Delayed Feedback Springer 2002
[40] M Hamdi and M Belhaq ldquoControl of bistability in a delayedDuffing oscillatorrdquo Advances in Acoustics and Vibration vol2012 Article ID 872498 6 pages 2012
[41] V Ravichandran C Chinnathambi and S Rajasekar ldquoNonlin-ear resonance in Duffing oscillator with fixed and integrativetime-delayed feedbacksrdquoPramana Journal of Physics vol 78 pp347ndash360 2013
[42] X You and H Xu ldquoAnalytical approximations for the periodicmotion of theDuffing systemwith delayed feedbackrdquoNumericalAlgorithms vol 56 no 4 pp 561ndash576 2011
[43] Z Sun W Xu X Yang and T Fang ldquoInducing or suppressingchaos in a double-well Duffing oscillator by time delay feed-backrdquo Chaos Solitons and Fractals vol 27 pp 705ndash714 2006
[44] H Wang H Hu and Z Wang ldquoGlobal dynamics of a Duffingoscillator with delayed displacement feedbackrdquo InternationalJournal of Bifurcation and Chaos in Applied Sciences andEngineering vol 14 no 8 pp 2753ndash2775 2004
[45] J Chiasson and J J LoiseauApplications of Time Delay SystemsSpringer 2007
[46] M Lakshmanan andDV SenthilkumarDynamics of NonlinearTime-Delay Systems Springer 2010
[47] G Stepan T Insperger and R Szalai ldquoDelay parametricexcitation and the nonlinear dynamics of cutting processesrdquoInternational Journal of Bifurcation and Chaos in Applied Sci-ences and Engineering vol 15 no 9 pp 2783ndash2798 2005
[48] U van der Heiden and H-O Walther ldquoExistence of chaos incontrol systems with delayed feedbackrdquo Journal of DifferentialEquations vol 47 no 2 pp 273ndash295 1983
[49] Y G Sun and J S W Wong ldquoOscillation criteria for secondorder forced ordinary differential equations with mixed non-linearitiesrdquo Journal of Mathematical Analysis and Applicationsvol 334 no 1 pp 549ndash560 2007
[50] J Heagy and W L Ditto ldquoDynamics of a two-frequencyparametrically driven Duffing oscillatorrdquo Journal of NonlinearScience vol 1 no 4 pp 423ndash455 1991
[51] A B Belogortsev ldquoBifurcations of tori and chaos in thequasiperiodically forced Duffing oscillatorrdquoNonlinearity vol 5no 4 pp 889ndash897 1992
[52] M Belhaq and M Houssni ldquoQuasi-periodic oscillations chaosand suppression of chaos in a nonlinear oscillator driven byparametric and external excitationsrdquo Nonlinear Dynamics vol18 no 1 pp 1ndash24 1999
[53] S H Saker P Y H Pang and R P Agarwal ldquoOscillationtheorems for second order nonlinear functional differential
Discrete Dynamics in Nature and Society 17
equations with dampingrdquo Dynamic Systems and Applicationsvol 12 no 3-4 pp 307ndash321 2003
[54] I N Bronshtein K A Semendyayev G Musiol and HMuehligHandbook of Mathematics Springer 5th edition 2007
[55] M Pasic ldquoFite-Wintner-Leighton-type oscillation criteria forsecond-order differential equations with nonlinear dampingrdquoAbstract and Applied Analysis vol 2013 Article ID 852180 10pages 2013
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Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Algebra
Discrete Dynamics in Nature and Society
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Decision SciencesAdvances in
Discrete MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
14 Discrete Dynamics in Nature and Society
for all 119905 isin 119869119899we obtain
[120582119865 (119905 119909 (120591 (119905))) + 120583119866 (119905 119909 (120590 (119905))) minus 120588119890 (119905)]
times (|119909 (119905)|119901minus1
119909 (119905))minus1
= [120582119865 (119905 119909 (120591 (119905))) minus 120588119890 (119905)] (|119909 (119905)|119901minus1
119909 (119905))minus1
ge [120582119891 (119905) |119909 (120591 (119905))|1199011 + 120588 |119890 (119905)|] |119909 (119905)|
minus119901
= [119883120574+ (120574 minus 1) 119884
120574] |119909 (119905)|
minus119901
ge119902
119901(120582119891 (119905))
119901119902
|119909 (120591 (119905))|119901
times (119901
119902 minus 119901120588 |119890 (119905)|)
(119901119902)((119902119901)minus1)
|119909 (119905)|minus119901
= 120582119901119902
1205881minus(119901119902)
119902
119901(
119901
119902 minus 119901)
1minus(119901119902)
|119890 (119905)|1minus(119901119902)
times (119891 (119905))119901119902
(119909 (120591 (119905))
119909 (119905))
119901
ge 119896 (120582 120583 120588)119902
119901(
119901
119902 minus 119901)
1minus(119901119902)
|119890 (119905)|1minus(119901119902)
times (119891 (119905))119901119902
120591119899(119905)
(106)
where the function 119896(120582 120583 120588) is from (18)
(iii) Advanced Case with 119902 gt 119901 Using the same line ofarguments as in the proof of the previous case for all 119905 isin 119869
119899
we obtain
[120582119865 (119905 119909 (120591 (119905))) + 120583119866 (119905 119909 (120590 (119905))) minus 120588119890 (119905)]
times (|119909 (119905)|119901minus1
119909 (119905))minus1
= [120583119866 (119905 119909 (120590 (119905))) minus 120588119890 (119905)] (|119909 (119905)|119901minus1
119909 (119905))minus1
ge 119896 (120582 120583 120588)119902
119901(
119901
119902 minus 119901)
1minus(119901119902)
|119890 (119905)|1minus(119901119902)
times (119892 (119905))119901119902
120590119899(119905)
(107)
where the function 119896(120582 120583 120588) is from (21)
(iv) Superlinear Delay-Advanced Case Since 1199011 1199012gt 119901 for
all 119905 isin 119869119899we obtain
[120582119865 (119905 119909 (120591 (119905))) + 120583119866 (119905 119909 (120590 (119905))) minus 120588119890 (119905)]
times (|119909 (119905)|119901minus1
119909 (119905))minus1
= [120582119865 (119905 119909 (120591 (119905))) minus120588
2119890 (119905)] (|119909 (119905)|
119901minus1119909 (119905))minus1
+ [120583119866 (119905 119909 (120590 (119905))) minus120588
2119890 (119905)] (|119909 (119905)|
119901minus1119909 (119905))minus1
ge [120582119891 (119905) |119909 (120591 (119905))|1199011 +
120588
2|119890 (119905)|] |119909 (119905)|
minus119901
+ [120583119892 (119905) |119909 (120590 (119905))|1199012 +
120588
2|119890 (119905)|] |119909 (119905)|
minus119901
(108)
Now just the same as in the proofs of previous delay andadvanced cases with 119902 gt 119901 and with the help of (104) inparticular for
120574 =1199011
119901gt 1 119883 = (120582119891 (119905))
1199011199011
|119909 (120591 (119905))|119901
119884 = (119901
1199011minus 119901
120588
2|119890 (119905)|)
1199011199011
(109)
we have
[120582119891 (119905) |119909 (120591 (119905))|1199011 +
120588
2 |119890 (119905)|] |119909 (119905)|
minus119901
= [119883120574+ (120574 minus 1) 119884
120574] |119909 (119905)|
minus119901
ge1199011
119901(120582119891 (119905))
1199011199011
|119909 (120591 (119905))|119901
times (119901
1199011minus 119901
120588
2|119890 (119905)|)
(1199011199011)((1199011119901)minus1)
|119909 (119905)|minus119901
= 12058211990111990111205881minus(119901119901
1)1199011
119901(
119901
2 (1199011minus 119901)
)
1minus(1199011199011)
times |119890 (119905)|1minus(119901119901
1)(119891 (119905))
1199011199011
(119909 (120591 (119905))
119909 (119905))
119901
ge 119896 (120582 120583 120588)1199011
119901(
119901
2 (1199011minus 119901)
)
1minus(1199011199011)
times |119890 (119905)|1minus(119901119901
1)(119891 (119905))
1199011199011
120591119899(119905)
(110)
where the function 119896(120582 120583 120588) is from (24) Analogously weshow that
[120583119892 (119905) |119909 (120590 (119905))|1199012 +
120588
2|119890 (119905)|] |119909 (119905)|
minus119901
ge 119896 (120582 120583 120588)1199012
119901(
119901
2 (1199012minus 119901)
)
1minus(1199011199012)
times |119890 (119905)|1minus(119901119901
2)(119891 (119905))
1199011199012
120590119899(119905)
(111)
Discrete Dynamics in Nature and Society 15
Summarizing previous calculation we obtain
[120582119865 (119905 119909 (120591 (119905))) + 120583119866 (119905 119909 (120590 (119905))) minus 120588119890 (119905)]
times (|119909 (119905)|119901minus1
119909 (119905))minus1
ge 119896 (120582 120583 120588)119876119899(119905) 119905 isin 119869
119899
(112)
where the function 119896(120582 120583 120588) is from (24)
(v) Supersublinear Delay-Advanced Case Since 1199011gt 119901 gt 119901
2
and the following well-known elementary inequality holds
12057801199060+ 12057811199061+ 12057821199062ge 1199061205780
01199061205781
11199061205782
2 120578119894ge 0 119906
119894ge 0 (113)
from 1205780 1205781 1205782isin (0 1) 120578
0+ 1205781+ 1205782= 1 and 119901
11205781+ 11990121205782= 119901
we obtain for all 119905 isin 119869119899 for all 119905 isin 119869
119899we obtain
[120582119865 (119905 119909 (120591 (119905))) + 120583119866 (119905 119909 (120590 (119905))) minus 120588119890 (119905)]
times (|119909 (119905)|119901minus1
119909 (119905))minus1
ge [120582119891 (119905) |119909 (120591 (119905))|1199011 + 120583119892 (119905) |119909 (120590 (119905))|
1199012 + 120588 |119890 (119905)|]
times |119909 (119905)|minus119901
= [1205781[120578minus1
1120582119891 (119905) |119909 (120591 (119905))|
1199011] + 120578
2[120578minus1
2120583119892 (119905) |119909 (120590 (119905))|
1199012]
+1205780[120578minus1
0120588 |119890 (119905)|]] |119909 (119905)|
minus119901
ge [120578minus1
1120582119891 (119905) |119909 (120591 (119905))|
1199011]1205781
[120578minus1
2120583119892 (119905) |119909 (120590 (119905))|
1199012]1205782
times [120578minus1
0120588 |119890 (119905)|]
1205780
|119909 (119905)|minus119901
= 120582120578112058312057821205881205780 |119890 (119905)|
1205780(119891 (119905))
1205781
(119892 (119905))1205782
times|119909 (120591 (119905))|
12057811199011
|119909 (119905)|12057811199011
|119909 (120590 (119905))|12057821199012
|119909 (119905)|12057821199012
2
prod
119894=0
120578minus120578119894
119894
= 119896 (120582 120583 120588) |119890 (119905)|1205780(119891 (119905))
1205781
(119892 (119905))1205782
times (119909 (120591 (119905))
119909 (119905))
12057811199011
(119909 (120590 (119905))
119909 (119905))
12057821199012 2
prod
119894=0
120578minus120578119894
119894
ge 119896 (120582 120583 120588) |119890 (119905)|1205780(119891 (119905))
1205781
(119892 (119905))1205782
(120591119899(119905))1205781(1199011119901)
times (120590119899(119905))1205782(1199012119901)
2
prod
119894=0
120578minus120578119894
119894
= 119896 (120582 120583 120588)119876119899(119905)
(114)
where 119896(120582 120583 120588) and 119876119899(119905) are given respectively in (24) and
(25) Thus it is shown that required condition (5) in thecases (i)ndash(iv) is fulfilled with respect to 119896(120582 120583 120588) and 119876
119899(119905)
determined by (18) (21) or (24) and (19) (22) or (25)In conclusion according to the previous observation we
see that all assumptions of Lemma 4 are fulfilled and henceLemma 4 proves Theorems 5 6 and 7
Conflict of Interests
The author declares that there is no conflict of interestsregarding the publication of this paper
References
[1] Y G Sun ldquoA note on Nasrrsquos and Wongrsquos papersrdquo Journal ofMathematical Analysis and Applications vol 286 no 1 pp 363ndash367 2003
[2] Y G Sun C H Ou and J S W Wong ldquoInterval oscillationtheorems for a second-order linear differential equationrdquo Com-puters amp Mathematics with Applications vol 48 no 10-11 pp1693ndash1699 2004
[3] S Murugadass E Thandapani and S Pinelas ldquoOscillationcriteria for forced second-order mixed type quasilinear delaydifferential equationsrdquo Electronic Journal of Differential Equa-tions vol 2010 article 73 9 pages 2010
[4] Y Bai and L Liu ldquoNew oscillation criteria for second-orderdelay differential equations with mixed nonlinearitiesrdquoDiscreteDynamics in Nature and Society vol 2010 Article ID 796256 9pages 2010
[5] A F Guvenilir andA Zafer ldquoSecond-order oscillation of forcedfunctional differential equations with oscillatory potentialsrdquoComputers amp Mathematics with Applications vol 51 no 9-10pp 1395ndash1404 2006
[6] A Zafer ldquoInterval oscillation criteria for second order super-half linear functional differential equations with delay andadvanced argumentsrdquoMathematische Nachrichten vol 282 no9 pp 1334ndash1341 2009
[7] A F Guvenilir ldquoInterval oscillation of second-order functionaldifferential equations with oscillatory potentialsrdquo NonlinearAnalysis Theory Methods amp Applications vol 71 no 12 ppe2849ndashe2854 2009
[8] T S Hassan L Erbe and A Peterson ldquoForced oscillation ofsecond order differential equations with mixed nonlinearitiesrdquoActa Mathematica Scientia B vol 31 no 2 pp 613ndash626 2011
[9] M Pasic ldquoNew oscillation criteria for second-order forcedquasilinear functional differential equationsrdquo Abstract andApplied Analysis vol 2013 Article ID 735360 12 pages 2013
[10] L H Erbe Q Kong and B G Zhang Oscillation Theory forFunctional-Differential Equations vol 190 Marcel Dekker NewYork NY USA 1995
[11] V Kolmanovskii and A Myshkis Introduction to the Theoryand Applications of Functional-Differential Equations vol 463Kluwer Academic Publishers Dordrecht The Netherlands1999
[12] R P Agarwal M Bohner and W-T Li Nonoscillation andOscillation Theory for Functional Differential Equations vol267 Marcel Dekker New York NY USA 2004
[13] L Erbe T Hassan and A Peterson ldquoOscillation of secondorder functional dynamic equationsrdquo International Journal ofDifference Equations vol 5 no 2 pp 175ndash193 2010
[14] B Baculıkova J Dzurina and Y V Rogovchenko ldquoOscillationof third order trinomial delay differential equationsrdquo AppliedMathematics and Computation vol 218 no 13 pp 7023ndash70332012
[15] R P Agarwal L Berezansky E Braverman and A Domoshnit-sky Nonoscillation Theory of Functional Differential Equationswith Applications Springer New York NY USA 2012
16 Discrete Dynamics in Nature and Society
[16] J Zhang ldquoVariational approach to solitary wave solution ofthe generalized Zakharov equationrdquo Computers ampMathematicswith Applications vol 54 no 7-8 pp 1043ndash1046 2007
[17] T Ozis and A Yıldırım ldquoApplication of Hersquos semi-inversemethod to the nonlinear Schrodinger equationrdquo Computers ampMathematics with Applications vol 54 no 7-8 pp 1039ndash10422007
[18] X-C Cai andM-S Li ldquoPeriodic solution of Jacobi elliptic equa-tions by Hersquos perturbation methodrdquo Computers amp Mathematicswith Applications vol 54 no 7-8 pp 1210ndash1212 2007
[19] S Lenci G Menditto and A M Tarantino ldquoHomoclinic andheteroclinic bifurcations in the non-linear dynamics of a beamresting on an elastic substraterdquo International Journal of Non-Linear Mechanics vol 34 no 4 pp 615ndash632 1999
[20] D-J Huang and H-Q Zhang ldquoLink between travelling wavesand first order nonlinear ordinary differential equation with asixth-degree nonlinear termrdquoChaos Solitons amp Fractals vol 29no 4 pp 928ndash941 2006
[21] A I Maimistov ldquoPropagation of an ultimately short electro-magnetic pulse in a nonlinear medium described by the fifth-order Duffing modelrdquo Optics and Spectroscopy vol 94 pp 251ndash257 2003
[22] M N Hamdan and N H Shabaneh ldquoOn the large amplitudefree vibrations of a restrained uniform beam carrying anintermediate lumpedmassrdquo Journal of Sound andVibration vol199 no 5 pp 711ndash736 1997
[23] J-H He ldquoSome asymptotic methods for strongly nonlinearequationsrdquo International Journal of Modern Physics B vol 20no 10 pp 1141ndash1199 2006
[24] J B Marion Classical Dynamics of Particles and Systems 1970[25] I Kovacic and M J Brennan The Duffing Equation Nonlinear
Oscillatos and their Behaviour John Wiley amp Sons 1st edition2011
[26] F C Moon Chaotic Vibrations An Introduction for AppliedScientists and Engineers John Wiley amp Sons New York NYUSA 2004
[27] J J Stoker Nonlinear Vibrations 1950[28] G Chen and Z Tao ldquoAmplitude-frequency relationship for the
relativistic oscillatorrdquoAASRI Procedia vol 1 pp 400ndash403 2012[29] R E Mickens Oscillations in Planar Dynamic Systems World
Scientific Publishing Singapore 1996[30] A Belendez T Belendez C Neipp A Hernandez and M
L Alvarez ldquoApproximate solutions of a nonlinear oscillatortypified as a mass attached to a stretched elastic wire by thehomotopy perturbation methodrdquo Chaos Solitions and Fractalsvol 39 pp 746ndash764 2009
[31] A Belendez E Fernandez R Fuentes J J Rodes and I PascualldquoHarmonic balancing approach to nonlinear oscillations of apunctual charge in the eletric field of charged ringrdquo PhysicsLetters A vol 373 pp 735ndash740 2009
[32] A Elıas-Zuniga ldquoExact solution of the cubic-quintic Duffingoscillatorrdquo Applied Mathematical Modelling vol 37 no 4 pp2574ndash2579 2013
[33] A Belendez M L Alvarez J Frances et al ldquoAnalytical approx-imate solutions for the cubic-quintic Duffing oscillator in termsof elementary functionsrdquo Journal of Applied Mathematics vol2012 Article ID 286290 16 pages 2012
[34] A Elıas-Zuniga OMartınez-Romero andR K Cordoba-DıazldquoApproximate solution for the Duffing-harmonic oscillator bythe enhanced cubication methodrdquo Mathematical Problems inEngineering vol 2012 Article ID 618750 12 pages 2012
[35] C W Lim B S Wu andW P Sun ldquoHigher accuracy analyticalapproximations to the Duffing-harmonic oscillatorrdquo Journal ofSound and Vibration vol 296 no 4-5 pp 1039ndash1045 2006
[36] J He ldquoSome new approaches to Duffing equation with stronglyand high order nonlinearity II parametrized perturbationtechniquerdquo Communications in Nonlinear Science amp NumericalSimulation vol 4 no 1 pp 81ndash83 1999
[37] V Marinca and N Herisanu ldquoPeriodic solutions for somestrongly nonlinear oscillations by Hersquos variational iterationmethodrdquo Computers amp Mathematics with Applications vol 54no 7-8 pp 1188ndash1196 2007
[38] W Lu and Y Liu ldquoVibration control for the primary resonanceof the Duffing oscillator by a time delay state feedbackrdquoInternational Journal of Nonlinear Science vol 8 no 3 pp 324ndash328 2009
[39] H Y Hu and Z H Wang Dynamics of Controlled MechanicalSystems with Delayed Feedback Springer 2002
[40] M Hamdi and M Belhaq ldquoControl of bistability in a delayedDuffing oscillatorrdquo Advances in Acoustics and Vibration vol2012 Article ID 872498 6 pages 2012
[41] V Ravichandran C Chinnathambi and S Rajasekar ldquoNonlin-ear resonance in Duffing oscillator with fixed and integrativetime-delayed feedbacksrdquoPramana Journal of Physics vol 78 pp347ndash360 2013
[42] X You and H Xu ldquoAnalytical approximations for the periodicmotion of theDuffing systemwith delayed feedbackrdquoNumericalAlgorithms vol 56 no 4 pp 561ndash576 2011
[43] Z Sun W Xu X Yang and T Fang ldquoInducing or suppressingchaos in a double-well Duffing oscillator by time delay feed-backrdquo Chaos Solitons and Fractals vol 27 pp 705ndash714 2006
[44] H Wang H Hu and Z Wang ldquoGlobal dynamics of a Duffingoscillator with delayed displacement feedbackrdquo InternationalJournal of Bifurcation and Chaos in Applied Sciences andEngineering vol 14 no 8 pp 2753ndash2775 2004
[45] J Chiasson and J J LoiseauApplications of Time Delay SystemsSpringer 2007
[46] M Lakshmanan andDV SenthilkumarDynamics of NonlinearTime-Delay Systems Springer 2010
[47] G Stepan T Insperger and R Szalai ldquoDelay parametricexcitation and the nonlinear dynamics of cutting processesrdquoInternational Journal of Bifurcation and Chaos in Applied Sci-ences and Engineering vol 15 no 9 pp 2783ndash2798 2005
[48] U van der Heiden and H-O Walther ldquoExistence of chaos incontrol systems with delayed feedbackrdquo Journal of DifferentialEquations vol 47 no 2 pp 273ndash295 1983
[49] Y G Sun and J S W Wong ldquoOscillation criteria for secondorder forced ordinary differential equations with mixed non-linearitiesrdquo Journal of Mathematical Analysis and Applicationsvol 334 no 1 pp 549ndash560 2007
[50] J Heagy and W L Ditto ldquoDynamics of a two-frequencyparametrically driven Duffing oscillatorrdquo Journal of NonlinearScience vol 1 no 4 pp 423ndash455 1991
[51] A B Belogortsev ldquoBifurcations of tori and chaos in thequasiperiodically forced Duffing oscillatorrdquoNonlinearity vol 5no 4 pp 889ndash897 1992
[52] M Belhaq and M Houssni ldquoQuasi-periodic oscillations chaosand suppression of chaos in a nonlinear oscillator driven byparametric and external excitationsrdquo Nonlinear Dynamics vol18 no 1 pp 1ndash24 1999
[53] S H Saker P Y H Pang and R P Agarwal ldquoOscillationtheorems for second order nonlinear functional differential
Discrete Dynamics in Nature and Society 17
equations with dampingrdquo Dynamic Systems and Applicationsvol 12 no 3-4 pp 307ndash321 2003
[54] I N Bronshtein K A Semendyayev G Musiol and HMuehligHandbook of Mathematics Springer 5th edition 2007
[55] M Pasic ldquoFite-Wintner-Leighton-type oscillation criteria forsecond-order differential equations with nonlinear dampingrdquoAbstract and Applied Analysis vol 2013 Article ID 852180 10pages 2013
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical Problems in Engineering
Hindawi Publishing Corporationhttpwwwhindawicom
Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
OptimizationJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Operations ResearchAdvances in
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Algebra
Discrete Dynamics in Nature and Society
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Decision SciencesAdvances in
Discrete MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
Discrete Dynamics in Nature and Society 15
Summarizing previous calculation we obtain
[120582119865 (119905 119909 (120591 (119905))) + 120583119866 (119905 119909 (120590 (119905))) minus 120588119890 (119905)]
times (|119909 (119905)|119901minus1
119909 (119905))minus1
ge 119896 (120582 120583 120588)119876119899(119905) 119905 isin 119869
119899
(112)
where the function 119896(120582 120583 120588) is from (24)
(v) Supersublinear Delay-Advanced Case Since 1199011gt 119901 gt 119901
2
and the following well-known elementary inequality holds
12057801199060+ 12057811199061+ 12057821199062ge 1199061205780
01199061205781
11199061205782
2 120578119894ge 0 119906
119894ge 0 (113)
from 1205780 1205781 1205782isin (0 1) 120578
0+ 1205781+ 1205782= 1 and 119901
11205781+ 11990121205782= 119901
we obtain for all 119905 isin 119869119899 for all 119905 isin 119869
119899we obtain
[120582119865 (119905 119909 (120591 (119905))) + 120583119866 (119905 119909 (120590 (119905))) minus 120588119890 (119905)]
times (|119909 (119905)|119901minus1
119909 (119905))minus1
ge [120582119891 (119905) |119909 (120591 (119905))|1199011 + 120583119892 (119905) |119909 (120590 (119905))|
1199012 + 120588 |119890 (119905)|]
times |119909 (119905)|minus119901
= [1205781[120578minus1
1120582119891 (119905) |119909 (120591 (119905))|
1199011] + 120578
2[120578minus1
2120583119892 (119905) |119909 (120590 (119905))|
1199012]
+1205780[120578minus1
0120588 |119890 (119905)|]] |119909 (119905)|
minus119901
ge [120578minus1
1120582119891 (119905) |119909 (120591 (119905))|
1199011]1205781
[120578minus1
2120583119892 (119905) |119909 (120590 (119905))|
1199012]1205782
times [120578minus1
0120588 |119890 (119905)|]
1205780
|119909 (119905)|minus119901
= 120582120578112058312057821205881205780 |119890 (119905)|
1205780(119891 (119905))
1205781
(119892 (119905))1205782
times|119909 (120591 (119905))|
12057811199011
|119909 (119905)|12057811199011
|119909 (120590 (119905))|12057821199012
|119909 (119905)|12057821199012
2
prod
119894=0
120578minus120578119894
119894
= 119896 (120582 120583 120588) |119890 (119905)|1205780(119891 (119905))
1205781
(119892 (119905))1205782
times (119909 (120591 (119905))
119909 (119905))
12057811199011
(119909 (120590 (119905))
119909 (119905))
12057821199012 2
prod
119894=0
120578minus120578119894
119894
ge 119896 (120582 120583 120588) |119890 (119905)|1205780(119891 (119905))
1205781
(119892 (119905))1205782
(120591119899(119905))1205781(1199011119901)
times (120590119899(119905))1205782(1199012119901)
2
prod
119894=0
120578minus120578119894
119894
= 119896 (120582 120583 120588)119876119899(119905)
(114)
where 119896(120582 120583 120588) and 119876119899(119905) are given respectively in (24) and
(25) Thus it is shown that required condition (5) in thecases (i)ndash(iv) is fulfilled with respect to 119896(120582 120583 120588) and 119876
119899(119905)
determined by (18) (21) or (24) and (19) (22) or (25)In conclusion according to the previous observation we
see that all assumptions of Lemma 4 are fulfilled and henceLemma 4 proves Theorems 5 6 and 7
Conflict of Interests
The author declares that there is no conflict of interestsregarding the publication of this paper
References
[1] Y G Sun ldquoA note on Nasrrsquos and Wongrsquos papersrdquo Journal ofMathematical Analysis and Applications vol 286 no 1 pp 363ndash367 2003
[2] Y G Sun C H Ou and J S W Wong ldquoInterval oscillationtheorems for a second-order linear differential equationrdquo Com-puters amp Mathematics with Applications vol 48 no 10-11 pp1693ndash1699 2004
[3] S Murugadass E Thandapani and S Pinelas ldquoOscillationcriteria for forced second-order mixed type quasilinear delaydifferential equationsrdquo Electronic Journal of Differential Equa-tions vol 2010 article 73 9 pages 2010
[4] Y Bai and L Liu ldquoNew oscillation criteria for second-orderdelay differential equations with mixed nonlinearitiesrdquoDiscreteDynamics in Nature and Society vol 2010 Article ID 796256 9pages 2010
[5] A F Guvenilir andA Zafer ldquoSecond-order oscillation of forcedfunctional differential equations with oscillatory potentialsrdquoComputers amp Mathematics with Applications vol 51 no 9-10pp 1395ndash1404 2006
[6] A Zafer ldquoInterval oscillation criteria for second order super-half linear functional differential equations with delay andadvanced argumentsrdquoMathematische Nachrichten vol 282 no9 pp 1334ndash1341 2009
[7] A F Guvenilir ldquoInterval oscillation of second-order functionaldifferential equations with oscillatory potentialsrdquo NonlinearAnalysis Theory Methods amp Applications vol 71 no 12 ppe2849ndashe2854 2009
[8] T S Hassan L Erbe and A Peterson ldquoForced oscillation ofsecond order differential equations with mixed nonlinearitiesrdquoActa Mathematica Scientia B vol 31 no 2 pp 613ndash626 2011
[9] M Pasic ldquoNew oscillation criteria for second-order forcedquasilinear functional differential equationsrdquo Abstract andApplied Analysis vol 2013 Article ID 735360 12 pages 2013
[10] L H Erbe Q Kong and B G Zhang Oscillation Theory forFunctional-Differential Equations vol 190 Marcel Dekker NewYork NY USA 1995
[11] V Kolmanovskii and A Myshkis Introduction to the Theoryand Applications of Functional-Differential Equations vol 463Kluwer Academic Publishers Dordrecht The Netherlands1999
[12] R P Agarwal M Bohner and W-T Li Nonoscillation andOscillation Theory for Functional Differential Equations vol267 Marcel Dekker New York NY USA 2004
[13] L Erbe T Hassan and A Peterson ldquoOscillation of secondorder functional dynamic equationsrdquo International Journal ofDifference Equations vol 5 no 2 pp 175ndash193 2010
[14] B Baculıkova J Dzurina and Y V Rogovchenko ldquoOscillationof third order trinomial delay differential equationsrdquo AppliedMathematics and Computation vol 218 no 13 pp 7023ndash70332012
[15] R P Agarwal L Berezansky E Braverman and A Domoshnit-sky Nonoscillation Theory of Functional Differential Equationswith Applications Springer New York NY USA 2012
16 Discrete Dynamics in Nature and Society
[16] J Zhang ldquoVariational approach to solitary wave solution ofthe generalized Zakharov equationrdquo Computers ampMathematicswith Applications vol 54 no 7-8 pp 1043ndash1046 2007
[17] T Ozis and A Yıldırım ldquoApplication of Hersquos semi-inversemethod to the nonlinear Schrodinger equationrdquo Computers ampMathematics with Applications vol 54 no 7-8 pp 1039ndash10422007
[18] X-C Cai andM-S Li ldquoPeriodic solution of Jacobi elliptic equa-tions by Hersquos perturbation methodrdquo Computers amp Mathematicswith Applications vol 54 no 7-8 pp 1210ndash1212 2007
[19] S Lenci G Menditto and A M Tarantino ldquoHomoclinic andheteroclinic bifurcations in the non-linear dynamics of a beamresting on an elastic substraterdquo International Journal of Non-Linear Mechanics vol 34 no 4 pp 615ndash632 1999
[20] D-J Huang and H-Q Zhang ldquoLink between travelling wavesand first order nonlinear ordinary differential equation with asixth-degree nonlinear termrdquoChaos Solitons amp Fractals vol 29no 4 pp 928ndash941 2006
[21] A I Maimistov ldquoPropagation of an ultimately short electro-magnetic pulse in a nonlinear medium described by the fifth-order Duffing modelrdquo Optics and Spectroscopy vol 94 pp 251ndash257 2003
[22] M N Hamdan and N H Shabaneh ldquoOn the large amplitudefree vibrations of a restrained uniform beam carrying anintermediate lumpedmassrdquo Journal of Sound andVibration vol199 no 5 pp 711ndash736 1997
[23] J-H He ldquoSome asymptotic methods for strongly nonlinearequationsrdquo International Journal of Modern Physics B vol 20no 10 pp 1141ndash1199 2006
[24] J B Marion Classical Dynamics of Particles and Systems 1970[25] I Kovacic and M J Brennan The Duffing Equation Nonlinear
Oscillatos and their Behaviour John Wiley amp Sons 1st edition2011
[26] F C Moon Chaotic Vibrations An Introduction for AppliedScientists and Engineers John Wiley amp Sons New York NYUSA 2004
[27] J J Stoker Nonlinear Vibrations 1950[28] G Chen and Z Tao ldquoAmplitude-frequency relationship for the
relativistic oscillatorrdquoAASRI Procedia vol 1 pp 400ndash403 2012[29] R E Mickens Oscillations in Planar Dynamic Systems World
Scientific Publishing Singapore 1996[30] A Belendez T Belendez C Neipp A Hernandez and M
L Alvarez ldquoApproximate solutions of a nonlinear oscillatortypified as a mass attached to a stretched elastic wire by thehomotopy perturbation methodrdquo Chaos Solitions and Fractalsvol 39 pp 746ndash764 2009
[31] A Belendez E Fernandez R Fuentes J J Rodes and I PascualldquoHarmonic balancing approach to nonlinear oscillations of apunctual charge in the eletric field of charged ringrdquo PhysicsLetters A vol 373 pp 735ndash740 2009
[32] A Elıas-Zuniga ldquoExact solution of the cubic-quintic Duffingoscillatorrdquo Applied Mathematical Modelling vol 37 no 4 pp2574ndash2579 2013
[33] A Belendez M L Alvarez J Frances et al ldquoAnalytical approx-imate solutions for the cubic-quintic Duffing oscillator in termsof elementary functionsrdquo Journal of Applied Mathematics vol2012 Article ID 286290 16 pages 2012
[34] A Elıas-Zuniga OMartınez-Romero andR K Cordoba-DıazldquoApproximate solution for the Duffing-harmonic oscillator bythe enhanced cubication methodrdquo Mathematical Problems inEngineering vol 2012 Article ID 618750 12 pages 2012
[35] C W Lim B S Wu andW P Sun ldquoHigher accuracy analyticalapproximations to the Duffing-harmonic oscillatorrdquo Journal ofSound and Vibration vol 296 no 4-5 pp 1039ndash1045 2006
[36] J He ldquoSome new approaches to Duffing equation with stronglyand high order nonlinearity II parametrized perturbationtechniquerdquo Communications in Nonlinear Science amp NumericalSimulation vol 4 no 1 pp 81ndash83 1999
[37] V Marinca and N Herisanu ldquoPeriodic solutions for somestrongly nonlinear oscillations by Hersquos variational iterationmethodrdquo Computers amp Mathematics with Applications vol 54no 7-8 pp 1188ndash1196 2007
[38] W Lu and Y Liu ldquoVibration control for the primary resonanceof the Duffing oscillator by a time delay state feedbackrdquoInternational Journal of Nonlinear Science vol 8 no 3 pp 324ndash328 2009
[39] H Y Hu and Z H Wang Dynamics of Controlled MechanicalSystems with Delayed Feedback Springer 2002
[40] M Hamdi and M Belhaq ldquoControl of bistability in a delayedDuffing oscillatorrdquo Advances in Acoustics and Vibration vol2012 Article ID 872498 6 pages 2012
[41] V Ravichandran C Chinnathambi and S Rajasekar ldquoNonlin-ear resonance in Duffing oscillator with fixed and integrativetime-delayed feedbacksrdquoPramana Journal of Physics vol 78 pp347ndash360 2013
[42] X You and H Xu ldquoAnalytical approximations for the periodicmotion of theDuffing systemwith delayed feedbackrdquoNumericalAlgorithms vol 56 no 4 pp 561ndash576 2011
[43] Z Sun W Xu X Yang and T Fang ldquoInducing or suppressingchaos in a double-well Duffing oscillator by time delay feed-backrdquo Chaos Solitons and Fractals vol 27 pp 705ndash714 2006
[44] H Wang H Hu and Z Wang ldquoGlobal dynamics of a Duffingoscillator with delayed displacement feedbackrdquo InternationalJournal of Bifurcation and Chaos in Applied Sciences andEngineering vol 14 no 8 pp 2753ndash2775 2004
[45] J Chiasson and J J LoiseauApplications of Time Delay SystemsSpringer 2007
[46] M Lakshmanan andDV SenthilkumarDynamics of NonlinearTime-Delay Systems Springer 2010
[47] G Stepan T Insperger and R Szalai ldquoDelay parametricexcitation and the nonlinear dynamics of cutting processesrdquoInternational Journal of Bifurcation and Chaos in Applied Sci-ences and Engineering vol 15 no 9 pp 2783ndash2798 2005
[48] U van der Heiden and H-O Walther ldquoExistence of chaos incontrol systems with delayed feedbackrdquo Journal of DifferentialEquations vol 47 no 2 pp 273ndash295 1983
[49] Y G Sun and J S W Wong ldquoOscillation criteria for secondorder forced ordinary differential equations with mixed non-linearitiesrdquo Journal of Mathematical Analysis and Applicationsvol 334 no 1 pp 549ndash560 2007
[50] J Heagy and W L Ditto ldquoDynamics of a two-frequencyparametrically driven Duffing oscillatorrdquo Journal of NonlinearScience vol 1 no 4 pp 423ndash455 1991
[51] A B Belogortsev ldquoBifurcations of tori and chaos in thequasiperiodically forced Duffing oscillatorrdquoNonlinearity vol 5no 4 pp 889ndash897 1992
[52] M Belhaq and M Houssni ldquoQuasi-periodic oscillations chaosand suppression of chaos in a nonlinear oscillator driven byparametric and external excitationsrdquo Nonlinear Dynamics vol18 no 1 pp 1ndash24 1999
[53] S H Saker P Y H Pang and R P Agarwal ldquoOscillationtheorems for second order nonlinear functional differential
Discrete Dynamics in Nature and Society 17
equations with dampingrdquo Dynamic Systems and Applicationsvol 12 no 3-4 pp 307ndash321 2003
[54] I N Bronshtein K A Semendyayev G Musiol and HMuehligHandbook of Mathematics Springer 5th edition 2007
[55] M Pasic ldquoFite-Wintner-Leighton-type oscillation criteria forsecond-order differential equations with nonlinear dampingrdquoAbstract and Applied Analysis vol 2013 Article ID 852180 10pages 2013
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical Problems in Engineering
Hindawi Publishing Corporationhttpwwwhindawicom
Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
OptimizationJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Operations ResearchAdvances in
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Algebra
Discrete Dynamics in Nature and Society
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Decision SciencesAdvances in
Discrete MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
16 Discrete Dynamics in Nature and Society
[16] J Zhang ldquoVariational approach to solitary wave solution ofthe generalized Zakharov equationrdquo Computers ampMathematicswith Applications vol 54 no 7-8 pp 1043ndash1046 2007
[17] T Ozis and A Yıldırım ldquoApplication of Hersquos semi-inversemethod to the nonlinear Schrodinger equationrdquo Computers ampMathematics with Applications vol 54 no 7-8 pp 1039ndash10422007
[18] X-C Cai andM-S Li ldquoPeriodic solution of Jacobi elliptic equa-tions by Hersquos perturbation methodrdquo Computers amp Mathematicswith Applications vol 54 no 7-8 pp 1210ndash1212 2007
[19] S Lenci G Menditto and A M Tarantino ldquoHomoclinic andheteroclinic bifurcations in the non-linear dynamics of a beamresting on an elastic substraterdquo International Journal of Non-Linear Mechanics vol 34 no 4 pp 615ndash632 1999
[20] D-J Huang and H-Q Zhang ldquoLink between travelling wavesand first order nonlinear ordinary differential equation with asixth-degree nonlinear termrdquoChaos Solitons amp Fractals vol 29no 4 pp 928ndash941 2006
[21] A I Maimistov ldquoPropagation of an ultimately short electro-magnetic pulse in a nonlinear medium described by the fifth-order Duffing modelrdquo Optics and Spectroscopy vol 94 pp 251ndash257 2003
[22] M N Hamdan and N H Shabaneh ldquoOn the large amplitudefree vibrations of a restrained uniform beam carrying anintermediate lumpedmassrdquo Journal of Sound andVibration vol199 no 5 pp 711ndash736 1997
[23] J-H He ldquoSome asymptotic methods for strongly nonlinearequationsrdquo International Journal of Modern Physics B vol 20no 10 pp 1141ndash1199 2006
[24] J B Marion Classical Dynamics of Particles and Systems 1970[25] I Kovacic and M J Brennan The Duffing Equation Nonlinear
Oscillatos and their Behaviour John Wiley amp Sons 1st edition2011
[26] F C Moon Chaotic Vibrations An Introduction for AppliedScientists and Engineers John Wiley amp Sons New York NYUSA 2004
[27] J J Stoker Nonlinear Vibrations 1950[28] G Chen and Z Tao ldquoAmplitude-frequency relationship for the
relativistic oscillatorrdquoAASRI Procedia vol 1 pp 400ndash403 2012[29] R E Mickens Oscillations in Planar Dynamic Systems World
Scientific Publishing Singapore 1996[30] A Belendez T Belendez C Neipp A Hernandez and M
L Alvarez ldquoApproximate solutions of a nonlinear oscillatortypified as a mass attached to a stretched elastic wire by thehomotopy perturbation methodrdquo Chaos Solitions and Fractalsvol 39 pp 746ndash764 2009
[31] A Belendez E Fernandez R Fuentes J J Rodes and I PascualldquoHarmonic balancing approach to nonlinear oscillations of apunctual charge in the eletric field of charged ringrdquo PhysicsLetters A vol 373 pp 735ndash740 2009
[32] A Elıas-Zuniga ldquoExact solution of the cubic-quintic Duffingoscillatorrdquo Applied Mathematical Modelling vol 37 no 4 pp2574ndash2579 2013
[33] A Belendez M L Alvarez J Frances et al ldquoAnalytical approx-imate solutions for the cubic-quintic Duffing oscillator in termsof elementary functionsrdquo Journal of Applied Mathematics vol2012 Article ID 286290 16 pages 2012
[34] A Elıas-Zuniga OMartınez-Romero andR K Cordoba-DıazldquoApproximate solution for the Duffing-harmonic oscillator bythe enhanced cubication methodrdquo Mathematical Problems inEngineering vol 2012 Article ID 618750 12 pages 2012
[35] C W Lim B S Wu andW P Sun ldquoHigher accuracy analyticalapproximations to the Duffing-harmonic oscillatorrdquo Journal ofSound and Vibration vol 296 no 4-5 pp 1039ndash1045 2006
[36] J He ldquoSome new approaches to Duffing equation with stronglyand high order nonlinearity II parametrized perturbationtechniquerdquo Communications in Nonlinear Science amp NumericalSimulation vol 4 no 1 pp 81ndash83 1999
[37] V Marinca and N Herisanu ldquoPeriodic solutions for somestrongly nonlinear oscillations by Hersquos variational iterationmethodrdquo Computers amp Mathematics with Applications vol 54no 7-8 pp 1188ndash1196 2007
[38] W Lu and Y Liu ldquoVibration control for the primary resonanceof the Duffing oscillator by a time delay state feedbackrdquoInternational Journal of Nonlinear Science vol 8 no 3 pp 324ndash328 2009
[39] H Y Hu and Z H Wang Dynamics of Controlled MechanicalSystems with Delayed Feedback Springer 2002
[40] M Hamdi and M Belhaq ldquoControl of bistability in a delayedDuffing oscillatorrdquo Advances in Acoustics and Vibration vol2012 Article ID 872498 6 pages 2012
[41] V Ravichandran C Chinnathambi and S Rajasekar ldquoNonlin-ear resonance in Duffing oscillator with fixed and integrativetime-delayed feedbacksrdquoPramana Journal of Physics vol 78 pp347ndash360 2013
[42] X You and H Xu ldquoAnalytical approximations for the periodicmotion of theDuffing systemwith delayed feedbackrdquoNumericalAlgorithms vol 56 no 4 pp 561ndash576 2011
[43] Z Sun W Xu X Yang and T Fang ldquoInducing or suppressingchaos in a double-well Duffing oscillator by time delay feed-backrdquo Chaos Solitons and Fractals vol 27 pp 705ndash714 2006
[44] H Wang H Hu and Z Wang ldquoGlobal dynamics of a Duffingoscillator with delayed displacement feedbackrdquo InternationalJournal of Bifurcation and Chaos in Applied Sciences andEngineering vol 14 no 8 pp 2753ndash2775 2004
[45] J Chiasson and J J LoiseauApplications of Time Delay SystemsSpringer 2007
[46] M Lakshmanan andDV SenthilkumarDynamics of NonlinearTime-Delay Systems Springer 2010
[47] G Stepan T Insperger and R Szalai ldquoDelay parametricexcitation and the nonlinear dynamics of cutting processesrdquoInternational Journal of Bifurcation and Chaos in Applied Sci-ences and Engineering vol 15 no 9 pp 2783ndash2798 2005
[48] U van der Heiden and H-O Walther ldquoExistence of chaos incontrol systems with delayed feedbackrdquo Journal of DifferentialEquations vol 47 no 2 pp 273ndash295 1983
[49] Y G Sun and J S W Wong ldquoOscillation criteria for secondorder forced ordinary differential equations with mixed non-linearitiesrdquo Journal of Mathematical Analysis and Applicationsvol 334 no 1 pp 549ndash560 2007
[50] J Heagy and W L Ditto ldquoDynamics of a two-frequencyparametrically driven Duffing oscillatorrdquo Journal of NonlinearScience vol 1 no 4 pp 423ndash455 1991
[51] A B Belogortsev ldquoBifurcations of tori and chaos in thequasiperiodically forced Duffing oscillatorrdquoNonlinearity vol 5no 4 pp 889ndash897 1992
[52] M Belhaq and M Houssni ldquoQuasi-periodic oscillations chaosand suppression of chaos in a nonlinear oscillator driven byparametric and external excitationsrdquo Nonlinear Dynamics vol18 no 1 pp 1ndash24 1999
[53] S H Saker P Y H Pang and R P Agarwal ldquoOscillationtheorems for second order nonlinear functional differential
Discrete Dynamics in Nature and Society 17
equations with dampingrdquo Dynamic Systems and Applicationsvol 12 no 3-4 pp 307ndash321 2003
[54] I N Bronshtein K A Semendyayev G Musiol and HMuehligHandbook of Mathematics Springer 5th edition 2007
[55] M Pasic ldquoFite-Wintner-Leighton-type oscillation criteria forsecond-order differential equations with nonlinear dampingrdquoAbstract and Applied Analysis vol 2013 Article ID 852180 10pages 2013
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical Problems in Engineering
Hindawi Publishing Corporationhttpwwwhindawicom
Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
OptimizationJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Operations ResearchAdvances in
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Algebra
Discrete Dynamics in Nature and Society
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Decision SciencesAdvances in
Discrete MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
Discrete Dynamics in Nature and Society 17
equations with dampingrdquo Dynamic Systems and Applicationsvol 12 no 3-4 pp 307ndash321 2003
[54] I N Bronshtein K A Semendyayev G Musiol and HMuehligHandbook of Mathematics Springer 5th edition 2007
[55] M Pasic ldquoFite-Wintner-Leighton-type oscillation criteria forsecond-order differential equations with nonlinear dampingrdquoAbstract and Applied Analysis vol 2013 Article ID 852180 10pages 2013
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical Problems in Engineering
Hindawi Publishing Corporationhttpwwwhindawicom
Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
OptimizationJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Operations ResearchAdvances in
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Algebra
Discrete Dynamics in Nature and Society
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Decision SciencesAdvances in
Discrete MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical Problems in Engineering
Hindawi Publishing Corporationhttpwwwhindawicom
Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
OptimizationJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Operations ResearchAdvances in
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Algebra
Discrete Dynamics in Nature and Society
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Decision SciencesAdvances in
Discrete MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
