Oscillation Criteria of Third-Order Nonlinear Impulsive ... · HindawiPublishingCorporation...
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Hindawi Publishing CorporationAbstract and Applied AnalysisVolume 2013 Article ID 405397 8 pageshttpdxdoiorg1011552013405397
Research ArticleOscillation Criteria of Third-Order Nonlinear ImpulsiveDifferential Equations with Delay
Xiuxiang Liu
School of Mathematical Sciences South China Normal University Guangzhou Guangdong 510631 China
Correspondence should be addressed to Xiuxiang Liu liuxxscnueducn
Received 14 November 2012 Accepted 10 January 2013
Academic Editor Norio Yoshida
Copyright copy 2013 Xiuxiang LiuThis 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
This paper deals with the oscillation of third-order nonlinear impulsive equations with delayThe results in this paper improve andextend some results for the equations without impulses Some examples are given to illustrate the main results
1 Introduction
In this paper we are concerned with oscillation of the third-order nonlinear impulsive equations with delay
119909101584010158401015840
(119905) + 119891 (119905 119909 (119905) 119909 (119905 minus 120590)) = 0 119905 ge 1199050 119905 = 120591
119896
119909 (120591+
119896) = 119886119896119909 (120591119896) 119909
1015840(120591+
119896) = 1198871198961199091015840(120591119896)
11990910158401015840(120591+
119896) = 11988811989611990910158401015840(120591119896) 119896 = 1 2
(1)
where 120590 gt 0 is the delay 120591119896 is the sequence of impulsive
moments which satisfies 0 le 1199050
lt 1205911lt sdot sdot sdot lt 120591
119896lt sdot sdot sdot and
lim119896rarrinfin
120591119896= infin 120591
119896+1minus 120591119896ge 120590
Throughout this paper we will assume that the followingassumptions are satisfied
(H1) 119891(119905 119906 119907) is continuous in [1199050
minus 120590infin) times R times R119906119891(119905 119906 119907) gt 0 for 119906119907 gt 0
(H2) 119891(119905 119906 119907)120593(119907) ge 119901(119905) for 119907 = 0 where 119901(119905) is continu-ous in [119905
0minus120590infin) 119901(119905) ge 0( equiv 0) 120593(119909)119909 ge 120583 gt 0 for
all 119909 = 0(H3) 119886
119896 119887119896 and 119888
119896are positive constants
Our attention is restricted to those solutions of (1) whichexist on half line [119905
0infin) and satisfy sup|119909(119905) 119905 gt 119879 gt
0 for any 119879 ge 119905119909 For the general theory of impulsive
differential equations withwithout delay we refer the readersto monographs or papers [1ndash4] A solution of (1) is saidto be oscillatory if it has arbitrarily large zeros otherwise
it is nonoscillatory It is well known that there is a drasticdifference in the behavior of solutions between differen-tial equations with impulses and those without impulsesSome differential equations are nonoscillatory but they maybecome oscillatory if some proper impulse controls areadded to them see [5] and Example 13 in Section 4 In therecent years the oscillation theory and asymptotic behaviorof impulsive differential equations and their applicationshave been and still are receiving intensive attention Forcontribution we refer to the recent survey paper by Agarwalet al [6] and the references cited therein But to the best ofthe authorsrsquo knowledge it seems that little has been done foroscillation of third-order impulsive differential equations [7]
When 119886119896
= 119887119896
= 119888119896
= 1 (1) reduces third-order delayequation withwithout delay which oscillatory theory hasbeen studied by many researchers see [8ndash12]
Our aim in this paper is to establish some new sufficientconditions which ensure that the solutions of (1) oscillate orconverge to a finite limit as 119905 tends to infinity In particular weextend some results in [9 11] to impulsive delay differentialequations The results in this paper are more general com-pared by those obtained by Mao and Wan [7] and improvesome of the results in [7] (see Example 13 in Section 4) Thenew results will be proved by making use of the techniquesused in [9 11]
The paper is organized as follows In Section 2 we provesome lemmas which play important roles in the proof ofthe main results In Section 3 some new sufficient condi-tions which guarantee that the solution of (1) oscillates or
2 Abstract and Applied Analysis
converges to a finite limit are established In Section 4 twoexamples are given to illustrate the main results
2 Preliminary Results
In this section we state and prove some lemmas which wewill need in the proofs of the main results First of all weintroduce the following notations R+ and N are the sets ofreal numbers and positive integer numbers respectively PC1is defined by
PC1 (R+R)
= 119909 R+997888rarr R 119909 (119905) is differentiable for
119905 ge 0 and 119905 = 120591119896 119909 (120591+
119896) and 119909
1015840(120591+
119896) exist
and 119909 (120591minus
119896) = 119909 (120591
119896) 1199091015840(120591minus
119896) = 1199091015840(120591119896)
(2)
The following lemma is from Lakshmikantham et al [3 Page32 Theorem 141]
Lemma 1 Assume that
(i) 120591119896119896isinN is the impulse moments sequence with 0 le 119905
0lt
1205911lt sdot sdot sdot lt 120591
119896lt sdot sdot sdot lim
119896rarrinfin120591119896= infin
(ii) 119898 isin PC1(R+R) and for 119905 ge 1199050 119896 isin N it holds that
1198981015840
(119905) le 119906 (119905)119898 (119905) + 119907 (119905) 119905 = 120591119896
119898 (120591+
119896) le 119889119896119898(120591119896) + 119890119896
(3)
where 119906 119907 isin 119862(R+R) 119889119896ge 0 and 119890
119896are real constantsThen
119898(119905) le 119898 (1199050) prod
1199050lt120591119896lt119905
119889119896exp(int
119905
1199050
119906 (119904)d119904)
+ int
119905
1199050
prod
119904lt120591119896lt119905
119889119896exp(int
119905
119904
119906 (120590)d120590) 119907 (119904)d119904
+ sum
1199050lt120591119896lt119905
prod
120591119896lt120591119895lt119905
119889119895exp(int
119905
120591119896
119906 (119904)d119904) 119890119896
(4)
Motivated by the ideas of Chen and Feng [5] we presentthe following key lemma which determines the sign of 1199091015840(119905)and 119909
10158401015840(119905) of the nonoscillation solution 119909(119905) of (1)
Lemma 2 Suppose that 119909(119905) is an eventually positive solutionof (1) and
lim119905rarrinfin
int
119905
1199050
prod
1199050lt120591119896lt119904
119887119896
119886119896
d119904 = infin (5)
lim119905rarrinfin
int
119905
1199050
prod
1199050lt120591119896lt119904
119888119896
119887119896
d119904 = infin (6)
Then it holds that one of the following two cases for sufficientlylarge 119879
(i) 11990910158401015840(120591+
119896) gt 0 11990910158401015840(119905) gt 0 and 119909
1015840(120591+
119896) gt 0 119909
1015840(119905) gt 0
(ii) 11990910158401015840(120591+
119896) gt 0 11990910158401015840(119905) gt 0 and 119909
1015840(120591+
119896) lt 0 119909
1015840(119905) lt 0
with 119905 isin (120591119896 120591119896+1
] and 120591119896ge 119879
Proof Assume that 119909(119905) is an eventually positive solution of(1)Wemay assume that there exists 119905
1gt 1199050such that 119909(119905) gt 0
and 119909(119905 minus 120590) gt 0 for 119905 ge 1199051 First we assert that 11990910158401015840(120591
119896) gt 0 for
any 119896 isin N Suppose not there exists some 120591119895ge 1199051such that
11990910158401015840(120591119895) le 0 By
119909101584010158401015840
(119905) = minus119891 (119905 119909 (119905) 119909 (119905 minus 120590)) le 0 equiv 0 (7)
we have 11990910158401015840(119905)monotonically decreasing in (120591119894 120591119894+1
] 119894 = 119895 119895+
1 Thus 11990910158401015840(120591+119894) = 11988811989411990910158401015840(120591119894) lt 0 119894 = 119895+1 119895+2 Consider
the impulsive differential inequalities
119909101584010158401015840
(119905) lt 0 119905 ge 120591119895+1
119905 = 120591119896
11990910158401015840(120591+
119896) le 11988811989611990910158401015840(120591119896) 119896 = 119895 + 1 119895 + 2
(8)
By Lemma 1 we have
11990910158401015840
(119905) le 11990910158401015840(120591+
119895+1) prod
120591119895+1lt120591119896lt119905
119888119896= minus120572 prod
120591119895+1lt120591119896lt119905
119888119896lt 0 (9)
There are two cases of the sign of 1199091015840(120591119896)
Case 1 If there exists some 120591119899
ge 120591119895+1
such that 1199091015840(120591119899) le 0
since 11990910158401015840(119905) lt 0 then 119909
1015840(120591119899+1
) lt 1199091015840(120591+
119899) le 0 and 119909
1015840(120591+
119899+1) =
120573 = 119887119899+1
1199091015840(120591119899+1
) lt 0 By induction it easily show 1199091015840(120591119896) lt 0
and hence 1199091015840(120591+
119896) le 1198871198961199091015840(120591119896) lt 0 for 119896 = 119899 + 1 119899 + 2 So
we obtain the following impulsive differential inequalities
11990910158401015840
(119905) le 0 119905 ge 120591119899+1
119905 = 120591119896
1199091015840(120591+
119896) le 119887lowast
1198961199091015840(120591119896) 119896 = 119899 + 1 119899 + 2
(10)
which follows from Lemma 1 that
1199091015840
(119905) le 1199091015840(120591+
119899+1) prod
120591119899+1lt120591119896lt119905
119887119896= minus120573 prod
120591119899+1lt120591119896lt119905
119887119896 (11)
From (11) and applying Lemma 1 noting that 119909(119905) gt 0 for119905 gt 1199051and 119909(120591
+
119896) le 119886119896119909(120591119896) for any 119896 isin N we have
119909 (119905) le 119909 (120591+
119899+1) prod
120591119899+1lt120591119896lt119905
119886119896minus 120573int
119905
120591119899+1
prod
119904lt120591119896lt119905
119886119896
prod
120591119899+1lt120591119896lt119904
119887119896d119904
= prod
120591119899+1lt120591119896lt119905
119886119896(119909 (120591
+
119899+1) minus 120573int
119905
120591119899+1
prod
120591119899+1lt120591119896lt119904
119887119896
119886119896
d119904)
(12)
Thus by (5) we have 119909(119905) lt 0 for 119905 sufficiently large which isa contradiction
Case 2 If 1199091015840(120591119896) gt 0 for any 119896 ge 119895 noting that 11990910158401015840(119905) lt 0 for
119905 isin (120591119895+1
120591119895+2
] we have 1199091015840(119905) gt 119909
1015840(120591119895+1
) gt 0 By induction
Abstract and Applied Analysis 3
we get that 1199091015840(119905) gt 0 for any 119905 isin (120591119896 120591119896+1
] 119896 = 119895 + 1 119895 + 2 So the following impulsive differential inequalities hold
11990910158401015840
(119905) le minus120572 prod
120591119895+1lt120591119896lt119905
119888119896 119905 gt 120591
119895+1 119905 = 120591
119896
1199091015840(120591+
119896) le 119887119896119909 (120591119896) 119896 = 119895 + 1 119895 + 2
(13)
According to Lemma 1 we get
1199091015840
(119905) le 1199091015840(120591+
119895+1) prod
120591119895+1lt120591119896lt119905
119887119896minus 120572int
119905
120591119895+1
prod
119904lt120591119896lt119905
119887119896
prod
120591119895+1lt120591119896lt119904
119888119896d119904
= prod
120591119895+1lt120591119896lt119905
119887119896(1199091015840(120591+
119895+1) minus 120572int
119905
120591119895+1
prod
120591119895+1lt120591119896lt119904
119888119896
119887119896
d119904)
(14)
Hence the condition (6) implies that 1199091015840(119905) lt 0 when 119905 is
sufficiently large which contradicts to 1199091015840(119905) gt 0 for 119905 gt 120591
119895+1
again In terms of the above discussion we see that11990910158401015840(120591119896) gt 0
for any 120591119896gt 119879 with sufficiently large 119879 Consequently noting
that 119909101584010158401015840(119905) lt 0 for any 119905 isin (120591
119896 120591119896+1
] we have 11990910158401015840(119905) gt
11990910158401015840(120591119896+1
) gt 0Next if there exists a 120591
119895gt 119879 such that 1199091015840(120591
119895) ge 0 then
1199091015840(120591+
119895) = 119887119895(119909(120591+
119895) ge 0 1199091015840(120591
119895+1) gt 119909
1015840(120591+
119895) ge 0 Therefore by
induction we have 1199091015840(119905) gt 1199091015840(120591+
119894) gt 0 for 119905 isin (120591
119894 120591119894+1
] 119894 = 119895 +
1 119895 + 2 So case (i) is satisfied Otherwise if 1199091015840(120591119896) lt 0 for
all 120591119896ge 119879 then 119909
1015840(120591+
119896) = 119887119896119909(120591+
119896) lt 0 Thus for 119905 isin (120591
119896 120591119896+1
]using 119909
10158401015840(119905) gt 0 we have 1199091015840(119905) lt 119909
1015840(120591119896+1
) lt 0 hence case (ii)is satisfied This completes the proof
Remark 3 Suppose that 119909(119905) is an eventually negative solu-tion of (1) If (5) and (6) hold one can prove it holds that oneof the following two cases in a similar way as Lemma 2
(i) 11990910158401015840(120591+
119896) lt 0 11990910158401015840(119905) lt 0 and 119909
1015840(120591+
119896) lt 0 1199091015840(119905) lt 0
(ii) 11990910158401015840(120591+
119896) lt 0 11990910158401015840(119905) lt 0 and 119909
1015840(120591+
119896) gt 0 1199091015840(119905) gt 0 with
119905 isin (120591119896 120591119896+1
] and 120591119896ge 119879
Lemma 4 Let 119909(119905) be a piecewise continuous function oncup119896isinN(120591119896 120591119896+1] which is continuous at 119905 = 120591
119896and is left contin-
uous at 119905 = 120591119896 If
(1) 119909(119905) ge 0 (le 0) for 119905 ge 1199050
(2) 119909(119905) is monotone nonincreasing (monotone nonde-creasing) on (120591
119896 120591119896+1
] (120591119896ge 119879) for 119879 large enough
(3) suminfin
119896=1[119909(120591+
119896) minus 119909(120591
119896)] converges
then lim119905rarrinfin
119909(119905) = 119886 ge 0 (le 0)
The proof of Lemma 4 is similar to that of [13 Theorem5] and hence is omitted
Lemma 5 Assume that 119909(119905) is a solution of (1) which satisfiescase (ii) in Lemma 2 In addition if
infin
sum
119896=1
1003816100381610038161003816119886119896minus 1
1003816100381610038161003816lt infin
infin
prod
119896=1
119886119896119894119904 119887119900119906119899119889119890119889 (15)
then lim119905rarrinfin
119909(119905) exists (finite)
Proof First we claim thatsuminfin119896=1
[119909(120591+
119896)minus119909(120591
119896)] is convergence
In fact since 119909(119905) is decreasing on (120591119896 120591119896+1
] (120591119896ge 119879 and 119879 is
defined in Lemma 2) then
119909 (120591119896+1
) le 119909 (120591+
119896) 119909 (120591
+
119896+1) le 119886119896+1
119909 (120591119896+1
) le 119886119896+1
119909 (120591+
119896)
(16)
Obviously by induction we can get
119909 (120591119896+119899
) le 119886119896+119899minus1
sdot sdot sdot 119886119896+1
119909 (120591+
119896)
119909 (120591+
119896+119899) le 119886119896+119899
sdot sdot sdot 119886119896+1
119909 (120591+
119896)
(17)
Since prodinfin
119896=1119886119896
is bounded we conclude that 119909(120591119896) is
bounded which follows that there exists1198721gt 0 such that
1003816100381610038161003816119909 (120591+
119896) minus 119909 (120591
119896)1003816100381610038161003816=
1003816100381610038161003816119886119896minus 1
1003816100381610038161003816119909 (120591119896) le 119872
1
1003816100381610038161003816119886119896minus 1
1003816100381610038161003816 (18)
Hencesuminfin119896=1
[119909(120591+
119896)minus119909(120591
119896)] is convergence sincesuminfin
119896=1|119886119896minus1| is
convergence which follows from Lemma 4 that lim119905rarrinfin
119909(119905)
exists The proof is complete
3 Main Results
In this section we establish some sufficient conditions whichguarantee that every solution 119909(119905) of (1) either oscillates orhas a finite limit Occasionally we will make the additionalassumption
lim sup119905rarrinfin
int
119905
120578
int
infin
119906
int
infin
119904
119901 (120579) d120579 d119904 d119906 = infin (19)
where here it is understood that
int
infin
1199050
119901 (119905) d119905 lt infin int
infin
1199050
int
infin
119906
119901 (120579) d120579 d119906 lt infin (20)
Now we are ready to state and prove the main results inthis paper The results will be proved by making use of thetechnique in [11]
Theorem 6 Assume that (5) (6) and (19) hold and 119909(119905) is asolution of (1) Furthermore assume that 119886
119896le 1 119887
119896ge 1 and
119888119896le 1 for 119896 isin N If there exists a positive differentiable function
119903 such that
lim119905rarrinfin
int
119905
1199050
[119901 (119904) 119903 (119904) minus 119860 (119904 120591119896(119904)
)] prod
1199050lt120591119896lt119904
1
119888119896
prod
1199050lt120591119896+120590lt119904
119886119896d119904 = infin
(21)
where 119896(119904) = max119896 120591119896lt 119904 and
119860 (119904 120591119896(119904)
)
=
119888119896(119904)[1199031015840(119904)]2
4120583119903 (119904) (119904 minus 120591119896(119904))
119904 isin (120591119896 120591119896+ 120590]
[1199031015840(119904)]2
4120583119903 (119904) (119904 minus 120591119896(119904)minus ((119888119896(119904)minus 1) 119888
119896(119904)) 120590)
119904 isin (120591119896+ 120590 120591119896+1]
(22)
Then 119909(119905) is oscillatory
4 Abstract and Applied Analysis
Proof Let 119909(119905) be a nonoscillatory solution of (1) withoutloss of generality we may assume that 119909(119905) gt 0 eventually(119909(119905) lt 0 eventually can be achieved in the similar way)By Lemma 2 either case (i) or case (ii) in Lemma 2 holdsAssume that 119909(119905) satisfies case (i) then 119909
10158401015840(119905) gt 0 1199091015840(119905) gt 0
for 119905 isin (120591119895 120591119895+1
] 120591119895ge 119879 (119879 is defined in Lemma 2) Define the
Riccati transformation 119906 by
119906 (119905) =
119903 (119905) 11990910158401015840(119905)
120593 (119909 (119905 minus 120590))
119905 ge 120591119895 119905 = 120591
119896 (23)
Thus 119906(119905) gt 0 for 119905 isin (120591119896 120591119896+1
] 119896 = 119895 119895 + 1 and
1199061015840
(119905) =
119903 (119905)
120593 (119909 (119905 minus 120590))
119909101584010158401015840
(119905)
+
1199031015840(119905) 120593 (119909 (119905 minus 120590)) minus 119903 (119905) 120593
1015840(119909 (119905 minus 120590)) 119909
1015840(119905 minus 120590)
1205932(119909 (119905 minus 120590))
times 11990910158401015840
(119905)
le minus 119901 (119905) 119903 (119905) +
1199031015840(119905)
119903 (119905)
119906 (119905)
minus
119903 (119905) 1205831199091015840(119905 minus 120590)
1205932(119909 (119905 minus 120590))
11990910158401015840
(119905)
(24)
If 119905 isin (120591119896 120591119896+120590] sub (120591
119896 120591119896+1
] namely 119905 minus120590 le 120591119896lt 119905 11990910158401015840(119905)
is decreasing in (119905 minus 120590 120591119896] and (120591
119896 119905] respectively In view of
the following
11990910158401015840
(119905 minus 120590) gt 11990910158401015840(120591119896) ge
11990910158401015840(120591+
119896)
119888119896
gt
11990910158401015840(119905)
119888119896
(25)
we have
1199091015840
(119905 minus 120590) = 1199091015840(120591119896minus 120590) + int
119905minus120590
120591119896minus120590
11990910158401015840
(119904) d119904
gt 11990910158401015840
(119905 minus 120590) (119905 minus 120591119896) gt
11990910158401015840(119905)
119888119896
(119905 minus 120591119896)
(26)
Thus
1199061015840
(119905) le minus 119901 (119905) 119903 (119905) +
1199031015840(119905)
119903 (119905)
119906 (119905) minus
120583119903 (119905) (119905 minus 120591119896)
1198881198961205932(119909 (119905 minus 120590))
[11990910158401015840
(119905)]
2
= minus 119901 (119905) 119903 (119905) +
1199031015840(119905)
119903 (119905)
119906 (119905) minus
120583 (119905 minus 120591119896)
119888119896119903 (119905)
[119906 (119905)]2
= minus (radic
120583 (119905 minus 120591119896)
119888119896119903 (119905)
119906 (119905) minus radic
119888119896
4120583119903 (119905) (119905 minus 120591119896)
1199031015840
(119905))
2
minus[
[
119901 (119905) 119903 (119905) minus
119888119896[1199031015840(119905)]
2
4120583119903 (119905) (119905 minus 120591119896)
]
]
le minus[
[
119901 (119905) 119903 (119905) minus
119888119896[1199031015840(119905)]
2
4120583119903 (119905) (119905 minus 120591119896)
]
]
(27)
If 119905 isin (120591119896+120590 120591119896+1
] sub (120591119896 120591119896+1
] that is 120591119896lt 119905minus120590 lt 119905 le 120591
119896+1
then
1199091015840
(119905 minus 120590) = 1199091015840(120591119896minus 120590) + int
120591119896
120591119896minus120590
11990910158401015840
(119904) d119904 + int
119905minus120590
120591119896
11990910158401015840
(119904) d119904
ge 11990910158401015840(120591119896) 120590 + 119909
10158401015840
(119905 minus 120590) (119905 minus 120591119896minus 120590)
ge
11990910158401015840(119905)
119888119896
120590 + 11990910158401015840
(119905) (119905 minus 120591119896minus 120590)
ge 11990910158401015840
(119905) (119905 minus 120591119896minus
119888119896minus 1
119888119896
120590)
(28)
Similarly we have
1199061015840
(119905) le minus[
[
119901 (119905) 119903 (119905) minus
[1199031015840(119905)]
2
4120583119903 (119905) (119905 minus 120591119896minus ((119888119896minus 1) 119888
119896) 120590)
]
]
(29)
Thus we obtain
1199061015840
(119905) le minus [119901 (119905) 119903 (119905) minus 119860 (119905 120591119896(119905)
)] for 119905 isin (120591119896 120591119896+1
]
(30)
On the other hand
119906 (120591+
119896) =
119903 (120591119896) 11990910158401015840(120591+
119896)
120593 (119909 (120591119896minus 120590))
le 119888119896119906 (120591119896) (31)
Observing that 120593(119906) ge 120583119906 we have
119906 (120591+
119896+ 120590) =
119903 (120591119896+ 120590) 119909
10158401015840(120591119896+ 120590)
120593 (119909 (120591+
119896))
le
119903 (120591119896+ 120590) 119909
10158401015840(120591119896+ 120590)
120593 (119886119896119909 (120591119896))
le
119906 (120591119896+ 120590)
120583119886119896
(32)
Applying Lemma 1 it follows from (30) (31) and (32) that
119906 (119905) le 119906 (120591+
119895) prod
120591119895lt120591119896lt119905
119888119896
prod
120591119895lt120591119896+120590lt119905
1
120583119886119896
minus int
119905
120591119895
[119901 (119904) 119903 (119904) minus 119860 (119904 120591119896(119904)
)]
sdot prod
119904lt120591119896lt119905
119888119896
prod
119904lt120591119896+120590lt119905
1
120583119886119896
d119904
le prod
120591119895lt120591119896lt119905
119888119896
prod
120591119895lt120591119896+120590lt119905
1
120583119886119896
times (119906 (120591+
119895) minus 120583int
119905
120591119895
[119901 (119904) 119903 (119904) minus 119860 (119904 120591119896(119904)
)]
sdot prod
120591119895lt120591119896lt119904
1
119888119896
prod
120591119895lt120591119896+120590lt119904
119886119896d119904)
(33)
Abstract and Applied Analysis 5
which yields 119906(119905) lt 0 for all large 119905 This is contrary to 119906(119905) gt
0 and so case (i) in Lemma 2 is not possibleIf 119909(119905) satisfies the case (ii) in Lemma 2 that is 11990910158401015840(120591+
119896) gt
0 11990910158401015840(119905) gt 0 and 1199091015840(120591+
119896) lt 0 1199091015840(119905) lt 0 which proves that the
solution 119909(119905) is positive and decreasing Integrating (1) from119904 to 119905 (119905 ge 119904 ge 119879) we obtain
11990910158401015840
(119905) minus sum
119904lt120591119896lt119905
(119888119896minus 1) 119909
10158401015840(120591119896) minus 11990910158401015840
(119904)
+ int
119905
119904
119901 (120579) 120593 (119909 (120579 minus 120590)) d120579 le 0
(34)
Noting 119888119896le 1 and 119909
10158401015840(119905) gt 0 then it holds that
11990910158401015840
(119905) minus 11990910158401015840
(119904) + int
119905
119904
119901 (120579) 120593 (119909 (120579 minus 120590)) d120579 le 0 (35)
which leads to
minus11990910158401015840
(119904) + int
119905
119904
119901 (120579) 120593 (119909 (120579 minus 120590)) d120579 le 0 (36)
and hence
minus11990910158401015840
(119904) + int
infin
119904
119901 (120579) 120593 (119909 (120579 minus 120590)) d120579 le 0 (37)
Integrating the above inequality again from 119906 to 119905 (119905 ge 119906 ge 119879)one has
minus 1199091015840
(119905) + sum
119904lt120591119896lt119905
(119887119896minus 1) 119909
1015840(120591119896) + 1199091015840
(119906)
+ int
119905
119906
int
infin
119904
119901 (120579) 120593 (119909 (120579 minus 120590)) d120579 d119904 le 0
(38)
Using 1199091015840(119905) lt 0 and 119887
119896ge 1 we have
1199091015840
(119906) + int
infin
119906
int
infin
119904
119901 (120579) 120593 (119909 (120579 minus 120590)) d120579 d119904 le 0 (39)
Now we integrate the last inequality from 120578 to 119905 (119905 ge 120578 ge 119879)to obtain
119909 (119905) minus sum
120578lt120591119896lt119905
(119886119896minus 1) 119909 (120591
119896) minus 119909 (120578)
+ int
119905
120578
int
infin
119906
int
infin
119904
119901 (120579) 120593 (119909 (120579 minus 120590)) d120579 d119904 d119906 le 0
(40)
Since 119886119896
le 1 (119896 isin N) and 119909(119905) is decreasing then for 119905 isin
(120591119896 120591119896+1
] 119909(120591119896+1
) le 119909(119905) le 119909(120591+
119896) = 119886119896119909(120591119896) le 119909(120591
119896) 119896 isin N
Thus we get
120583119909 (120578) int
119905
120578
int
infin
119906
int
infin
119904
119901 (120579) d120579 d119904 d119906 le minus119909 (119905) + 119909 (120578) le 119909 (120578)
(41)
and then
int
119905
120578
int
infin
119906
int
infin
119904
119901 (120579) d120579 d119904 d119906 le
1
120583
(42)
which contradicts the condition (19) The proof is complete
Replace the condition (19) with (15) we may obtain thefollowing asymptotic results
Theorem 7 Assume that (5) (6) and (15) hold and 119909(119905) is asolution of (1) If there exists a positive differentiable function 119903
such that (21) hold then 119909(119905) is either oscillatory or has a finitelimit
Proof By the proof of Theorem 6 we know the case (i) inLemma 2 is not possible too since the condition (19) is notrequired to prove it So it suffices to show if there is a solutionsatisfying case (ii) in Lemma 2 that is if
11990910158401015840(120591+
119896) gt 0 119909
10158401015840
(119905) gt 0 1199091015840(120591+
119896) lt 0 119909
1015840
(119905) lt 0 (43)
with 119905 isin (120591119896 120591119896+1
] and 120591119896ge 119879 then lim
119905rarrinfin119909(119905) exists This
is obtained by applying Lemma 5 which leads to lim119905rarrinfin
119909(119905)
exists The proof is complete
Corollary 8 In addition to the assumption of Theorem 7assume that
lim119905rarrinfin
int
119905
1199050
prod
1199050lt120591119896lt119904
1
119888119896
119901 (119904) d119904 = infin (44)
Then solution 119909(119905) of (1) either oscillates or satisfieslim119905rarrinfin
119909(119905) = 0
Proof By the proof ofTheorem 7 lim119905rarrinfin
119909(119905) exists and wedefine it by lim
119905rarrinfin119909(119905) = 120574 ge 0 We now show 120574 = 0 If not
then 120574 gt 0 So lim119905rarrinfin
120593(119909(119905 minus 120590)) = 120593(120574) = 120581 gt 0 Hencethere exists 120591
119895gt 119879 such that 120593(119909(119905minus120590)) gt 1205812 for 119905 gt 120591
119895Then
119909101584010158401015840
(119905) = minus119891 (119905 119909 (119905) 119909 (119905 minus 120590))
le minus119901 (119905) 120593 (119909 (119905 minus 120590)) le minus
120581
2
119901 (119905) 119905 ge 120591119895
(45)
and note that 11990910158401015840(120591+119896) le 11988811989611990910158401015840(120591119896) since 11990910158401015840(119905) gt 0 which imply
that
11990910158401015840
(119905) le 11990910158401015840(120591+
119895) prod
120591119895lt120591119896lt119905
119888119896minus
120581
2
int
119905
120591119895
prod
119904lt120591119896lt119905
119888119896119901 (119904) d119904
le prod
120591119895lt120591119896lt119905
119888119896[
[
11990910158401015840(120591+
119895) minus
120581
2
int
119905
120591119895
prod
120591119895lt120591119896lt119904
1
119888119896
119901 (119904) d119904]
]
(46)
Thus in virtue of (44) it holds that 11990910158401015840(119905) lt 0 and contradicts11990910158401015840(119905) gt 0 for 119905 large enough the proof is complete
Remark 9 Theorem 6 and Corollary 8 extend the results in[11 Theorem 31] and [9 Corollary 1] respectively In factwhen 119886
119896= 119887119896
= 119888119896
= 1 for 119896 isin N which implies that theimpulses in (1) disappear In such a case (5) and (6) holdnaturally and (21) and (44) are reduced to
lim119905rarrinfin
int
119905
1199050
[
[
119901 (119904) 119903 (119904) minus
[1199031015840(119904)]
2
4120583119903 (119904) (119904 minus 119879)
]
]
d119904 = infin
lim119905rarrinfin
int
119905
1199050
119901 (119904) d119904 = infin
(47)
6 Abstract and Applied Analysis
which are similar to those in [11 Theorem 31] and [9Corollary 1] respectively
Next we present some new oscillation results for (1) byusing an integral averaging condition of Kamenevrsquos type
Theorem 10 Assume (5) (6) and (19) hold Furthermore119886119896
le 1120583 le 1 119887119896
ge 1 and 119888119896
le 1 119896 isin N If there exists apositive differentiable function 119903 such that
lim sup119905rarrinfin
1
119905119898
int
119905
119879
(119905 minus 119904)119898[119901 (119904) 119903 (119904) minus 119860 (119904 120591
119896(119904))] d119904 = infin
(48)
where 119860(119904 120591119896(119904)
) is defined by (21) and 119898 ge 1 Then everysolution of (1) is oscillatory
Proof We choose 119879 large enough such that Lemma 2 holdsBy Lemma 2 there are two possible cases First if the case (i)holds proceeding as in the proof of Theorem 6 we will endup with (32) By (30) we have
119901 (119905) 119903 (119905) minus 119860 (119905 120591119896(119905)
) le minus1199061015840
(119905) 119905 ge 119879 119905 = 120591119896 (49)
If 119905 isin (120591119896+ 120590 120591119896+1
] sub (120591119896 120591119896+1
] for 120591119895ge 119879 we obtain
int
119905
120591119895
(119905 minus 119904)119898[119901 (119904) 119903 (119904) minus 119860 (119904 120591
119896(119904))] d119904
le minusint
119905
120591119895
(119905 minus 119904)1198981199061015840
(119904) d119904(50)
An integration by parts of the right-hand side leads to
int
119905
120591119895
(119905 minus 119904)1198981199061015840
(119904) d119904
= (int
120591119895+120590
120591119895
+int
120591119895+1
120591119895+120590
+ sdot sdot sdot + int
120591119896+120590
120591119896
+int
119905
120591119896+120590
) (119905 minus 119904)1198981199061015840
(119904) d119904
= int
119905
120591119895
119906 (119904)
119898
(119905 minus 119904)119898minus1d119904
+
119896
sum
119894=119895
[119905 minus (120591119894+ 120590)]119898
[119906 (120591119894+ 120590) minus 119906 (120591
+
119894+ 120590)]
+
119896
sum
119894=119895+1
(119905 minus 120591119894)119898
[119906 (120591119894) minus 119906 (120591
+
119894)] minus (119905 minus 120591
119895)
119898
119906 (120591+
119895)
(51)
Take into account (31) (32) 119886119896le 1120583 and 119888
119896le 1 we have
int
119905
120591119895
(119905 minus 119904)1198981199061015840
(119904) d119904
ge
119896
sum
119894=119895+1
(119905 minus 120591119894)119898
(1 minus 119888119894) 119906 (120591119894) minus (119905 minus 120591
119895)
119898
119906 (120591+
119895)
ge minus(119905 minus 120591119895)
119898
119906 (120591+
119895)
(52)
If 119905 isin (120591119896 120591119896+ 120590] similarly we also get
int
119905
120591119895
(119905 minus 119904)1198981199061015840
(119904) d119904 ge minus(119905 minus 120591119895)
119898
119906 (120591+
119895) (53)
So it yields
int
119905
120591119895
(119905 minus 119904)119898[119901 (119904) 119903 (119904) minus 119860 (119904 120591
119896(119904))] d119904 le (119905 minus 120591
119895)
119898
119906 (120591+
119895)
(54)
which follows that1
119905119898
int
119905
120591119895
(119905 minus 119904)119898[119901 (119904) 119903 (119904) minus 119860 (119904 120591
119896(119904))] d119904
le (
119905 minus 120591119895
119905
)
119898
119906 (120591+
119895)
(55)
Hence
lim sup119905rarrinfin
1
119905119898
int
119905
120591119895
(119905 minus 119904)119898[119901 (119904) 119903 (119904) minus 119860 (119904 120591
119896(119904))] d119904 le 119906 (120591
+
119895)
(56)
which is a contradiction of (48)If case (ii) holds then as a manner with case (ii) in
Theorem 6 it is not possible too The proof is complete
Corollary 11 Assume (19) holds and 119886119896
= 119887119896
= 119888119896
= 1 for119896 isin N If there exists a positive differential function 119903 such that
lim sup119905rarrinfin
1
119905119898
int
119905
119879
(119905 minus 119904)119898[
[
119901 (119904) 119903 (119904) minus
[1199031015840(119904)]
2
4120583119903 (119904) (119904 minus 119879)
]
]
d119904 = infin
(57)
where 119879 is large enough such that Lemma 2 holds Then everysolution of (1) is oscillatory
Remark 12 Corollary 11 is an extension of [11 Theorem 32]into impulsive case Especially let 119903(119905) equiv 1 in (48) it reducesto
lim sup119905rarrinfin
1
119905119898
int
119905
119879
(119905 minus 119904)119898119901 (119904) d119904 = infin (58)
naturally which can be considered as the extension ofKamenev-type oscillation criteria for third-order impulsivedifferential equations with delay (see [8 14 15])
4 Examples
Example 13 Consider the third-order impulsive differentialequation with delay
119909101584010158401015840
(119905) + (1 + 1205721199092
(119905)) 119909 (119905 minus 120590) = 0 119905 gt 1199050 119905 = 120591
119896
119909 (120591+
119896) = 119886119896119909 (120591119896) 119909
1015840(120591+
119896) = 1198871198961199091015840(120591119896)
11990910158401015840(120591+
119896) = 11988811989611990910158401015840(120591119896) 119896 isin N
(59)
where 120590 gt 0 120572 ge 0 are constants 120591119896minus 120591119896minus1
gt 120590 for any 119896 isin N
Abstract and Applied Analysis 7
When 119886119896
= 119887119896
= 119888119896
= 1 for any 119896 isin N the impulses in(59) disappear by [16 Theorem 4] (59) is nonoscillatory if120590 lt e3 and 120572 = 0 However we may change its oscillation byproper impulsive control In fact let 120590 lt e3 and 120572 = 0 and120591119896= 1199050+ 119896120590 (119896 isin N) in (59) choose 120593(119909) = 119909 119901(119905) equiv 1 and
119903(119905) = 1 a simple calculation leads to
int
120591119899
1199050
prod
1199050lt120591119896lt119904
1
119888119896
prod
1199050lt120591119896+120590lt119904
119886119896119901 (119904) d119904
= [int
1205911
1199050
+(int
1205911+120590
1205911
+int
1205912
1205911+120590
) + + (int
120591119899minus1+120590
120591119899minus1
+int
120591119899
120591119899minus1+120590
)]
sdot prod
1199050lt120591119896lt119904
1
119888119896
prod
1199050lt120591119896+120590lt119904
119886119896d119904
= (1205911minus 1199050) + [
120590
1198881
+
1198861
1198881
(1205912minus 1205911minus 120590)] + sdot sdot sdot +
11988611198862sdot sdot sdot 119886119899minus2
120590
11988811198882sdot sdot sdot 119888119899minus1
+
11988611198862sdot sdot sdot 119886119899minus1
11988811198882sdot sdot sdot 119888119899minus1
(120591119899minus 120591119899minus1
minus 120590)
= 120590(1 +
1
1198881
+
1
1198882
+ sdot sdot sdot +
1
119888119899minus1
)
(60)
Then let
119886119896= 119888119896=
119896
119896 + 1
119887119896= 1 119896 isin N (61)
Obviously (5) (6) and (19) hold and
int
120591119899
1199050
prod
1199050lt120591119896lt119904
1
119888119896
prod
1199050lt120591119896+120590lt119904
119886119896119901 (119904) d119904
= 120590 (1 +
2
1
+
3
2
+ sdot sdot sdot +
119899
119899 minus 1
) 997888rarr infin (119899 997888rarr infin)
(62)
Thus (21) is also satisfied By Theorem 6 every solution of(59) is oscillatory
If we let
119886119896= 1 +
1
1198962 119887119896= 119888119896=
119896 + 1
119896
119896 isin N (63)
In this case it is easily to verify (5) (6) (15) (44) and(21) hold By Corollary 8 every solution of (59) is eitheroscillatory or tends to zero
Remark 14 It is easy to verify that in [7 Theorems 1 2 and3] cannot be applied to (59) On the other hand Theorem 7is not applicable for the condition (61) sincesuminfin
119899=1|119886119896minus1| does
not convergence
Example 15 Consider the third-order impulsive differentialequation with delay
119909101584010158401015840
(119905) + e119905 cosh (|119909 (119905)|120572minus1
119909 (119905)) 119909 (119905 minus 1) = 0
119905 ge 0 119905 = 2119896
119909 (120591+
119896) = 119886119896119909 (120591119896) 119909
1015840(120591+
119896) = 1198871198961199091015840(120591119896)
11990910158401015840(120591+
119896) = 11988811989611990910158401015840(120591119896) 120591
119896= 2119896 119896 isin N
(64)
where 120572 gt 0 119886119896= 1 119887119896= 119888119896= 119896(119896 + 1) 119896 isin N
Let 120593(119909) = 119909 119901(119905) = e119905 it is easy to verify that (5) (6)and (19) hold Choose 119903(119905) equiv 1 and119898 = 1 we have
1
119905
int
119905
119879
(119905 minus 119904) e119904d119904 =
e119905
119905
+ (
119879 minus 1
119905
minus 1) e119879 rarr infin (119905 rarr infin)
(65)
Acknowledgments
Theauthor is very grateful to ProfessorH Saker who presentsthe references [8ndash12 17] and gives many helpful suggestionswhich leads to an improvement of this paper The workis supported in part by the NSF of Guangdong province(S2012010010034)
References
[1] D D Bainov and P S Simeonov Impulsive Differential Equa-tions Periodic Solutions and Applications Longman Scientificand Technical 1993
[2] G Ballinger and X Liu ldquoExistence uniqueness and bounded-ness results for impulsive delay differential equationsrdquo Applica-ble Analysis vol 74 no 1-2 pp 71ndash93 2000
[3] V Lakshmikantham D D Baınov and P S Simeonov Theoryof Impulsive Differential Equations vol 6 World ScientificPublishing New Jersey NJ USA 1989
[4] A M Samoılenko and N A Perestyuk Impulsive DifferentialEquations vol 14 World Scientific Publishing New Jersey NJUSA 1995
[5] Y S Chen andW Z Feng ldquoOscillations of second order nonlin-ear ODE with impulsesrdquo Journal of Mathematical Analysis andApplications vol 210 no 1 pp 150ndash169 1997
[6] R P Agarwal F Karakoc and A Zafer ldquoA survey on oscilla-tion of impulsive ordinary differential equationsrdquo Advances inDifferential Equations vol 2010 pp 1ndash52 2010
[7] W-H Mao and A-HWan ldquoOscillatory and asymptotic behav-ior of solutions for nonlinear impulsive delay differential equa-tionsrdquo Acta Mathematicae Applicatae Sinica vol 22 EnglishSeries no 3 pp 387ndash396 2006
[8] L Erbe A Peterson and S H Saker ldquoAsymptotic behavior ofsolutions of a third-order nonlinear dynamic equation on timescalesrdquo Journal of Computational and Applied Mathematics vol181 no 1 pp 92ndash102 2005
[9] L Erbe A Peterson and S H Saker ldquoOscillation and asymp-totic behavior of a third-order nonlinear dynamic equationrdquoThe Canadian Applied Mathematics Quarterly vol 14 no 2 pp124ndash147 2006
8 Abstract and Applied Analysis
[10] L Erbe A Peterson and S H Saker ldquoHille and Nehari typecriteria for third-order dynamic equationsrdquo Journal of Mathe-matical Analysis and Applications vol 329 no 1 pp 112ndash1312007
[11] S H Saker ldquoOscillation criteria of third-order nonlinear delaydifferential equationsrdquo Mathematica Slovaca vol 56 no 4 pp433ndash450 2006
[12] S H Saker and J Dzurina ldquoOn the oscillation of certain classof third-order nonlinear delay differential equationsrdquo Mathe-matica Bohemica vol 135 no 3 pp 225ndash237 2010
[13] J Yu and J Yan ldquoPositive solutions and asymptotic behavior ofdelay differential equations with nonlinear impulsesrdquo Journal ofMathematical Analysis and Applications vol 207 no 2 pp 388ndash396 1997
[14] T Candan and R S Dahiya ldquoOscillation of third order func-tional differential equations with delayrdquo in Proceedings of the5th Mississippi State Conference on Differential Equations andComputational Simulations vol 10 p 7988 2003
[15] I V Kamenev ldquoAn integral criterion for oscillation of linear dif-ferential equations of second orderrdquo Matematicheskie Zametkivol 23 pp 136ndash138 1978
[16] G Ladas Y G Sficas and I P Stavroulakis ldquoNecessary and suf-ficient conditions for oscillations of higher order delay differ-ential equationsrdquo Transactions of the American MathematicalSociety vol 285 no 1 pp 81ndash90 1984
[17] B Baculıkova E M Elabbasy S H Saker and J DzurinaldquoOscillation criteria for third-order nonlinear differential equa-tionsrdquoMathematica Slovaca vol 58 no 2 pp 201ndash220 2008
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Stochastic AnalysisInternational Journal of
2 Abstract and Applied Analysis
converges to a finite limit are established In Section 4 twoexamples are given to illustrate the main results
2 Preliminary Results
In this section we state and prove some lemmas which wewill need in the proofs of the main results First of all weintroduce the following notations R+ and N are the sets ofreal numbers and positive integer numbers respectively PC1is defined by
PC1 (R+R)
= 119909 R+997888rarr R 119909 (119905) is differentiable for
119905 ge 0 and 119905 = 120591119896 119909 (120591+
119896) and 119909
1015840(120591+
119896) exist
and 119909 (120591minus
119896) = 119909 (120591
119896) 1199091015840(120591minus
119896) = 1199091015840(120591119896)
(2)
The following lemma is from Lakshmikantham et al [3 Page32 Theorem 141]
Lemma 1 Assume that
(i) 120591119896119896isinN is the impulse moments sequence with 0 le 119905
0lt
1205911lt sdot sdot sdot lt 120591
119896lt sdot sdot sdot lim
119896rarrinfin120591119896= infin
(ii) 119898 isin PC1(R+R) and for 119905 ge 1199050 119896 isin N it holds that
1198981015840
(119905) le 119906 (119905)119898 (119905) + 119907 (119905) 119905 = 120591119896
119898 (120591+
119896) le 119889119896119898(120591119896) + 119890119896
(3)
where 119906 119907 isin 119862(R+R) 119889119896ge 0 and 119890
119896are real constantsThen
119898(119905) le 119898 (1199050) prod
1199050lt120591119896lt119905
119889119896exp(int
119905
1199050
119906 (119904)d119904)
+ int
119905
1199050
prod
119904lt120591119896lt119905
119889119896exp(int
119905
119904
119906 (120590)d120590) 119907 (119904)d119904
+ sum
1199050lt120591119896lt119905
prod
120591119896lt120591119895lt119905
119889119895exp(int
119905
120591119896
119906 (119904)d119904) 119890119896
(4)
Motivated by the ideas of Chen and Feng [5] we presentthe following key lemma which determines the sign of 1199091015840(119905)and 119909
10158401015840(119905) of the nonoscillation solution 119909(119905) of (1)
Lemma 2 Suppose that 119909(119905) is an eventually positive solutionof (1) and
lim119905rarrinfin
int
119905
1199050
prod
1199050lt120591119896lt119904
119887119896
119886119896
d119904 = infin (5)
lim119905rarrinfin
int
119905
1199050
prod
1199050lt120591119896lt119904
119888119896
119887119896
d119904 = infin (6)
Then it holds that one of the following two cases for sufficientlylarge 119879
(i) 11990910158401015840(120591+
119896) gt 0 11990910158401015840(119905) gt 0 and 119909
1015840(120591+
119896) gt 0 119909
1015840(119905) gt 0
(ii) 11990910158401015840(120591+
119896) gt 0 11990910158401015840(119905) gt 0 and 119909
1015840(120591+
119896) lt 0 119909
1015840(119905) lt 0
with 119905 isin (120591119896 120591119896+1
] and 120591119896ge 119879
Proof Assume that 119909(119905) is an eventually positive solution of(1)Wemay assume that there exists 119905
1gt 1199050such that 119909(119905) gt 0
and 119909(119905 minus 120590) gt 0 for 119905 ge 1199051 First we assert that 11990910158401015840(120591
119896) gt 0 for
any 119896 isin N Suppose not there exists some 120591119895ge 1199051such that
11990910158401015840(120591119895) le 0 By
119909101584010158401015840
(119905) = minus119891 (119905 119909 (119905) 119909 (119905 minus 120590)) le 0 equiv 0 (7)
we have 11990910158401015840(119905)monotonically decreasing in (120591119894 120591119894+1
] 119894 = 119895 119895+
1 Thus 11990910158401015840(120591+119894) = 11988811989411990910158401015840(120591119894) lt 0 119894 = 119895+1 119895+2 Consider
the impulsive differential inequalities
119909101584010158401015840
(119905) lt 0 119905 ge 120591119895+1
119905 = 120591119896
11990910158401015840(120591+
119896) le 11988811989611990910158401015840(120591119896) 119896 = 119895 + 1 119895 + 2
(8)
By Lemma 1 we have
11990910158401015840
(119905) le 11990910158401015840(120591+
119895+1) prod
120591119895+1lt120591119896lt119905
119888119896= minus120572 prod
120591119895+1lt120591119896lt119905
119888119896lt 0 (9)
There are two cases of the sign of 1199091015840(120591119896)
Case 1 If there exists some 120591119899
ge 120591119895+1
such that 1199091015840(120591119899) le 0
since 11990910158401015840(119905) lt 0 then 119909
1015840(120591119899+1
) lt 1199091015840(120591+
119899) le 0 and 119909
1015840(120591+
119899+1) =
120573 = 119887119899+1
1199091015840(120591119899+1
) lt 0 By induction it easily show 1199091015840(120591119896) lt 0
and hence 1199091015840(120591+
119896) le 1198871198961199091015840(120591119896) lt 0 for 119896 = 119899 + 1 119899 + 2 So
we obtain the following impulsive differential inequalities
11990910158401015840
(119905) le 0 119905 ge 120591119899+1
119905 = 120591119896
1199091015840(120591+
119896) le 119887lowast
1198961199091015840(120591119896) 119896 = 119899 + 1 119899 + 2
(10)
which follows from Lemma 1 that
1199091015840
(119905) le 1199091015840(120591+
119899+1) prod
120591119899+1lt120591119896lt119905
119887119896= minus120573 prod
120591119899+1lt120591119896lt119905
119887119896 (11)
From (11) and applying Lemma 1 noting that 119909(119905) gt 0 for119905 gt 1199051and 119909(120591
+
119896) le 119886119896119909(120591119896) for any 119896 isin N we have
119909 (119905) le 119909 (120591+
119899+1) prod
120591119899+1lt120591119896lt119905
119886119896minus 120573int
119905
120591119899+1
prod
119904lt120591119896lt119905
119886119896
prod
120591119899+1lt120591119896lt119904
119887119896d119904
= prod
120591119899+1lt120591119896lt119905
119886119896(119909 (120591
+
119899+1) minus 120573int
119905
120591119899+1
prod
120591119899+1lt120591119896lt119904
119887119896
119886119896
d119904)
(12)
Thus by (5) we have 119909(119905) lt 0 for 119905 sufficiently large which isa contradiction
Case 2 If 1199091015840(120591119896) gt 0 for any 119896 ge 119895 noting that 11990910158401015840(119905) lt 0 for
119905 isin (120591119895+1
120591119895+2
] we have 1199091015840(119905) gt 119909
1015840(120591119895+1
) gt 0 By induction
Abstract and Applied Analysis 3
we get that 1199091015840(119905) gt 0 for any 119905 isin (120591119896 120591119896+1
] 119896 = 119895 + 1 119895 + 2 So the following impulsive differential inequalities hold
11990910158401015840
(119905) le minus120572 prod
120591119895+1lt120591119896lt119905
119888119896 119905 gt 120591
119895+1 119905 = 120591
119896
1199091015840(120591+
119896) le 119887119896119909 (120591119896) 119896 = 119895 + 1 119895 + 2
(13)
According to Lemma 1 we get
1199091015840
(119905) le 1199091015840(120591+
119895+1) prod
120591119895+1lt120591119896lt119905
119887119896minus 120572int
119905
120591119895+1
prod
119904lt120591119896lt119905
119887119896
prod
120591119895+1lt120591119896lt119904
119888119896d119904
= prod
120591119895+1lt120591119896lt119905
119887119896(1199091015840(120591+
119895+1) minus 120572int
119905
120591119895+1
prod
120591119895+1lt120591119896lt119904
119888119896
119887119896
d119904)
(14)
Hence the condition (6) implies that 1199091015840(119905) lt 0 when 119905 is
sufficiently large which contradicts to 1199091015840(119905) gt 0 for 119905 gt 120591
119895+1
again In terms of the above discussion we see that11990910158401015840(120591119896) gt 0
for any 120591119896gt 119879 with sufficiently large 119879 Consequently noting
that 119909101584010158401015840(119905) lt 0 for any 119905 isin (120591
119896 120591119896+1
] we have 11990910158401015840(119905) gt
11990910158401015840(120591119896+1
) gt 0Next if there exists a 120591
119895gt 119879 such that 1199091015840(120591
119895) ge 0 then
1199091015840(120591+
119895) = 119887119895(119909(120591+
119895) ge 0 1199091015840(120591
119895+1) gt 119909
1015840(120591+
119895) ge 0 Therefore by
induction we have 1199091015840(119905) gt 1199091015840(120591+
119894) gt 0 for 119905 isin (120591
119894 120591119894+1
] 119894 = 119895 +
1 119895 + 2 So case (i) is satisfied Otherwise if 1199091015840(120591119896) lt 0 for
all 120591119896ge 119879 then 119909
1015840(120591+
119896) = 119887119896119909(120591+
119896) lt 0 Thus for 119905 isin (120591
119896 120591119896+1
]using 119909
10158401015840(119905) gt 0 we have 1199091015840(119905) lt 119909
1015840(120591119896+1
) lt 0 hence case (ii)is satisfied This completes the proof
Remark 3 Suppose that 119909(119905) is an eventually negative solu-tion of (1) If (5) and (6) hold one can prove it holds that oneof the following two cases in a similar way as Lemma 2
(i) 11990910158401015840(120591+
119896) lt 0 11990910158401015840(119905) lt 0 and 119909
1015840(120591+
119896) lt 0 1199091015840(119905) lt 0
(ii) 11990910158401015840(120591+
119896) lt 0 11990910158401015840(119905) lt 0 and 119909
1015840(120591+
119896) gt 0 1199091015840(119905) gt 0 with
119905 isin (120591119896 120591119896+1
] and 120591119896ge 119879
Lemma 4 Let 119909(119905) be a piecewise continuous function oncup119896isinN(120591119896 120591119896+1] which is continuous at 119905 = 120591
119896and is left contin-
uous at 119905 = 120591119896 If
(1) 119909(119905) ge 0 (le 0) for 119905 ge 1199050
(2) 119909(119905) is monotone nonincreasing (monotone nonde-creasing) on (120591
119896 120591119896+1
] (120591119896ge 119879) for 119879 large enough
(3) suminfin
119896=1[119909(120591+
119896) minus 119909(120591
119896)] converges
then lim119905rarrinfin
119909(119905) = 119886 ge 0 (le 0)
The proof of Lemma 4 is similar to that of [13 Theorem5] and hence is omitted
Lemma 5 Assume that 119909(119905) is a solution of (1) which satisfiescase (ii) in Lemma 2 In addition if
infin
sum
119896=1
1003816100381610038161003816119886119896minus 1
1003816100381610038161003816lt infin
infin
prod
119896=1
119886119896119894119904 119887119900119906119899119889119890119889 (15)
then lim119905rarrinfin
119909(119905) exists (finite)
Proof First we claim thatsuminfin119896=1
[119909(120591+
119896)minus119909(120591
119896)] is convergence
In fact since 119909(119905) is decreasing on (120591119896 120591119896+1
] (120591119896ge 119879 and 119879 is
defined in Lemma 2) then
119909 (120591119896+1
) le 119909 (120591+
119896) 119909 (120591
+
119896+1) le 119886119896+1
119909 (120591119896+1
) le 119886119896+1
119909 (120591+
119896)
(16)
Obviously by induction we can get
119909 (120591119896+119899
) le 119886119896+119899minus1
sdot sdot sdot 119886119896+1
119909 (120591+
119896)
119909 (120591+
119896+119899) le 119886119896+119899
sdot sdot sdot 119886119896+1
119909 (120591+
119896)
(17)
Since prodinfin
119896=1119886119896
is bounded we conclude that 119909(120591119896) is
bounded which follows that there exists1198721gt 0 such that
1003816100381610038161003816119909 (120591+
119896) minus 119909 (120591
119896)1003816100381610038161003816=
1003816100381610038161003816119886119896minus 1
1003816100381610038161003816119909 (120591119896) le 119872
1
1003816100381610038161003816119886119896minus 1
1003816100381610038161003816 (18)
Hencesuminfin119896=1
[119909(120591+
119896)minus119909(120591
119896)] is convergence sincesuminfin
119896=1|119886119896minus1| is
convergence which follows from Lemma 4 that lim119905rarrinfin
119909(119905)
exists The proof is complete
3 Main Results
In this section we establish some sufficient conditions whichguarantee that every solution 119909(119905) of (1) either oscillates orhas a finite limit Occasionally we will make the additionalassumption
lim sup119905rarrinfin
int
119905
120578
int
infin
119906
int
infin
119904
119901 (120579) d120579 d119904 d119906 = infin (19)
where here it is understood that
int
infin
1199050
119901 (119905) d119905 lt infin int
infin
1199050
int
infin
119906
119901 (120579) d120579 d119906 lt infin (20)
Now we are ready to state and prove the main results inthis paper The results will be proved by making use of thetechnique in [11]
Theorem 6 Assume that (5) (6) and (19) hold and 119909(119905) is asolution of (1) Furthermore assume that 119886
119896le 1 119887
119896ge 1 and
119888119896le 1 for 119896 isin N If there exists a positive differentiable function
119903 such that
lim119905rarrinfin
int
119905
1199050
[119901 (119904) 119903 (119904) minus 119860 (119904 120591119896(119904)
)] prod
1199050lt120591119896lt119904
1
119888119896
prod
1199050lt120591119896+120590lt119904
119886119896d119904 = infin
(21)
where 119896(119904) = max119896 120591119896lt 119904 and
119860 (119904 120591119896(119904)
)
=
119888119896(119904)[1199031015840(119904)]2
4120583119903 (119904) (119904 minus 120591119896(119904))
119904 isin (120591119896 120591119896+ 120590]
[1199031015840(119904)]2
4120583119903 (119904) (119904 minus 120591119896(119904)minus ((119888119896(119904)minus 1) 119888
119896(119904)) 120590)
119904 isin (120591119896+ 120590 120591119896+1]
(22)
Then 119909(119905) is oscillatory
4 Abstract and Applied Analysis
Proof Let 119909(119905) be a nonoscillatory solution of (1) withoutloss of generality we may assume that 119909(119905) gt 0 eventually(119909(119905) lt 0 eventually can be achieved in the similar way)By Lemma 2 either case (i) or case (ii) in Lemma 2 holdsAssume that 119909(119905) satisfies case (i) then 119909
10158401015840(119905) gt 0 1199091015840(119905) gt 0
for 119905 isin (120591119895 120591119895+1
] 120591119895ge 119879 (119879 is defined in Lemma 2) Define the
Riccati transformation 119906 by
119906 (119905) =
119903 (119905) 11990910158401015840(119905)
120593 (119909 (119905 minus 120590))
119905 ge 120591119895 119905 = 120591
119896 (23)
Thus 119906(119905) gt 0 for 119905 isin (120591119896 120591119896+1
] 119896 = 119895 119895 + 1 and
1199061015840
(119905) =
119903 (119905)
120593 (119909 (119905 minus 120590))
119909101584010158401015840
(119905)
+
1199031015840(119905) 120593 (119909 (119905 minus 120590)) minus 119903 (119905) 120593
1015840(119909 (119905 minus 120590)) 119909
1015840(119905 minus 120590)
1205932(119909 (119905 minus 120590))
times 11990910158401015840
(119905)
le minus 119901 (119905) 119903 (119905) +
1199031015840(119905)
119903 (119905)
119906 (119905)
minus
119903 (119905) 1205831199091015840(119905 minus 120590)
1205932(119909 (119905 minus 120590))
11990910158401015840
(119905)
(24)
If 119905 isin (120591119896 120591119896+120590] sub (120591
119896 120591119896+1
] namely 119905 minus120590 le 120591119896lt 119905 11990910158401015840(119905)
is decreasing in (119905 minus 120590 120591119896] and (120591
119896 119905] respectively In view of
the following
11990910158401015840
(119905 minus 120590) gt 11990910158401015840(120591119896) ge
11990910158401015840(120591+
119896)
119888119896
gt
11990910158401015840(119905)
119888119896
(25)
we have
1199091015840
(119905 minus 120590) = 1199091015840(120591119896minus 120590) + int
119905minus120590
120591119896minus120590
11990910158401015840
(119904) d119904
gt 11990910158401015840
(119905 minus 120590) (119905 minus 120591119896) gt
11990910158401015840(119905)
119888119896
(119905 minus 120591119896)
(26)
Thus
1199061015840
(119905) le minus 119901 (119905) 119903 (119905) +
1199031015840(119905)
119903 (119905)
119906 (119905) minus
120583119903 (119905) (119905 minus 120591119896)
1198881198961205932(119909 (119905 minus 120590))
[11990910158401015840
(119905)]
2
= minus 119901 (119905) 119903 (119905) +
1199031015840(119905)
119903 (119905)
119906 (119905) minus
120583 (119905 minus 120591119896)
119888119896119903 (119905)
[119906 (119905)]2
= minus (radic
120583 (119905 minus 120591119896)
119888119896119903 (119905)
119906 (119905) minus radic
119888119896
4120583119903 (119905) (119905 minus 120591119896)
1199031015840
(119905))
2
minus[
[
119901 (119905) 119903 (119905) minus
119888119896[1199031015840(119905)]
2
4120583119903 (119905) (119905 minus 120591119896)
]
]
le minus[
[
119901 (119905) 119903 (119905) minus
119888119896[1199031015840(119905)]
2
4120583119903 (119905) (119905 minus 120591119896)
]
]
(27)
If 119905 isin (120591119896+120590 120591119896+1
] sub (120591119896 120591119896+1
] that is 120591119896lt 119905minus120590 lt 119905 le 120591
119896+1
then
1199091015840
(119905 minus 120590) = 1199091015840(120591119896minus 120590) + int
120591119896
120591119896minus120590
11990910158401015840
(119904) d119904 + int
119905minus120590
120591119896
11990910158401015840
(119904) d119904
ge 11990910158401015840(120591119896) 120590 + 119909
10158401015840
(119905 minus 120590) (119905 minus 120591119896minus 120590)
ge
11990910158401015840(119905)
119888119896
120590 + 11990910158401015840
(119905) (119905 minus 120591119896minus 120590)
ge 11990910158401015840
(119905) (119905 minus 120591119896minus
119888119896minus 1
119888119896
120590)
(28)
Similarly we have
1199061015840
(119905) le minus[
[
119901 (119905) 119903 (119905) minus
[1199031015840(119905)]
2
4120583119903 (119905) (119905 minus 120591119896minus ((119888119896minus 1) 119888
119896) 120590)
]
]
(29)
Thus we obtain
1199061015840
(119905) le minus [119901 (119905) 119903 (119905) minus 119860 (119905 120591119896(119905)
)] for 119905 isin (120591119896 120591119896+1
]
(30)
On the other hand
119906 (120591+
119896) =
119903 (120591119896) 11990910158401015840(120591+
119896)
120593 (119909 (120591119896minus 120590))
le 119888119896119906 (120591119896) (31)
Observing that 120593(119906) ge 120583119906 we have
119906 (120591+
119896+ 120590) =
119903 (120591119896+ 120590) 119909
10158401015840(120591119896+ 120590)
120593 (119909 (120591+
119896))
le
119903 (120591119896+ 120590) 119909
10158401015840(120591119896+ 120590)
120593 (119886119896119909 (120591119896))
le
119906 (120591119896+ 120590)
120583119886119896
(32)
Applying Lemma 1 it follows from (30) (31) and (32) that
119906 (119905) le 119906 (120591+
119895) prod
120591119895lt120591119896lt119905
119888119896
prod
120591119895lt120591119896+120590lt119905
1
120583119886119896
minus int
119905
120591119895
[119901 (119904) 119903 (119904) minus 119860 (119904 120591119896(119904)
)]
sdot prod
119904lt120591119896lt119905
119888119896
prod
119904lt120591119896+120590lt119905
1
120583119886119896
d119904
le prod
120591119895lt120591119896lt119905
119888119896
prod
120591119895lt120591119896+120590lt119905
1
120583119886119896
times (119906 (120591+
119895) minus 120583int
119905
120591119895
[119901 (119904) 119903 (119904) minus 119860 (119904 120591119896(119904)
)]
sdot prod
120591119895lt120591119896lt119904
1
119888119896
prod
120591119895lt120591119896+120590lt119904
119886119896d119904)
(33)
Abstract and Applied Analysis 5
which yields 119906(119905) lt 0 for all large 119905 This is contrary to 119906(119905) gt
0 and so case (i) in Lemma 2 is not possibleIf 119909(119905) satisfies the case (ii) in Lemma 2 that is 11990910158401015840(120591+
119896) gt
0 11990910158401015840(119905) gt 0 and 1199091015840(120591+
119896) lt 0 1199091015840(119905) lt 0 which proves that the
solution 119909(119905) is positive and decreasing Integrating (1) from119904 to 119905 (119905 ge 119904 ge 119879) we obtain
11990910158401015840
(119905) minus sum
119904lt120591119896lt119905
(119888119896minus 1) 119909
10158401015840(120591119896) minus 11990910158401015840
(119904)
+ int
119905
119904
119901 (120579) 120593 (119909 (120579 minus 120590)) d120579 le 0
(34)
Noting 119888119896le 1 and 119909
10158401015840(119905) gt 0 then it holds that
11990910158401015840
(119905) minus 11990910158401015840
(119904) + int
119905
119904
119901 (120579) 120593 (119909 (120579 minus 120590)) d120579 le 0 (35)
which leads to
minus11990910158401015840
(119904) + int
119905
119904
119901 (120579) 120593 (119909 (120579 minus 120590)) d120579 le 0 (36)
and hence
minus11990910158401015840
(119904) + int
infin
119904
119901 (120579) 120593 (119909 (120579 minus 120590)) d120579 le 0 (37)
Integrating the above inequality again from 119906 to 119905 (119905 ge 119906 ge 119879)one has
minus 1199091015840
(119905) + sum
119904lt120591119896lt119905
(119887119896minus 1) 119909
1015840(120591119896) + 1199091015840
(119906)
+ int
119905
119906
int
infin
119904
119901 (120579) 120593 (119909 (120579 minus 120590)) d120579 d119904 le 0
(38)
Using 1199091015840(119905) lt 0 and 119887
119896ge 1 we have
1199091015840
(119906) + int
infin
119906
int
infin
119904
119901 (120579) 120593 (119909 (120579 minus 120590)) d120579 d119904 le 0 (39)
Now we integrate the last inequality from 120578 to 119905 (119905 ge 120578 ge 119879)to obtain
119909 (119905) minus sum
120578lt120591119896lt119905
(119886119896minus 1) 119909 (120591
119896) minus 119909 (120578)
+ int
119905
120578
int
infin
119906
int
infin
119904
119901 (120579) 120593 (119909 (120579 minus 120590)) d120579 d119904 d119906 le 0
(40)
Since 119886119896
le 1 (119896 isin N) and 119909(119905) is decreasing then for 119905 isin
(120591119896 120591119896+1
] 119909(120591119896+1
) le 119909(119905) le 119909(120591+
119896) = 119886119896119909(120591119896) le 119909(120591
119896) 119896 isin N
Thus we get
120583119909 (120578) int
119905
120578
int
infin
119906
int
infin
119904
119901 (120579) d120579 d119904 d119906 le minus119909 (119905) + 119909 (120578) le 119909 (120578)
(41)
and then
int
119905
120578
int
infin
119906
int
infin
119904
119901 (120579) d120579 d119904 d119906 le
1
120583
(42)
which contradicts the condition (19) The proof is complete
Replace the condition (19) with (15) we may obtain thefollowing asymptotic results
Theorem 7 Assume that (5) (6) and (15) hold and 119909(119905) is asolution of (1) If there exists a positive differentiable function 119903
such that (21) hold then 119909(119905) is either oscillatory or has a finitelimit
Proof By the proof of Theorem 6 we know the case (i) inLemma 2 is not possible too since the condition (19) is notrequired to prove it So it suffices to show if there is a solutionsatisfying case (ii) in Lemma 2 that is if
11990910158401015840(120591+
119896) gt 0 119909
10158401015840
(119905) gt 0 1199091015840(120591+
119896) lt 0 119909
1015840
(119905) lt 0 (43)
with 119905 isin (120591119896 120591119896+1
] and 120591119896ge 119879 then lim
119905rarrinfin119909(119905) exists This
is obtained by applying Lemma 5 which leads to lim119905rarrinfin
119909(119905)
exists The proof is complete
Corollary 8 In addition to the assumption of Theorem 7assume that
lim119905rarrinfin
int
119905
1199050
prod
1199050lt120591119896lt119904
1
119888119896
119901 (119904) d119904 = infin (44)
Then solution 119909(119905) of (1) either oscillates or satisfieslim119905rarrinfin
119909(119905) = 0
Proof By the proof ofTheorem 7 lim119905rarrinfin
119909(119905) exists and wedefine it by lim
119905rarrinfin119909(119905) = 120574 ge 0 We now show 120574 = 0 If not
then 120574 gt 0 So lim119905rarrinfin
120593(119909(119905 minus 120590)) = 120593(120574) = 120581 gt 0 Hencethere exists 120591
119895gt 119879 such that 120593(119909(119905minus120590)) gt 1205812 for 119905 gt 120591
119895Then
119909101584010158401015840
(119905) = minus119891 (119905 119909 (119905) 119909 (119905 minus 120590))
le minus119901 (119905) 120593 (119909 (119905 minus 120590)) le minus
120581
2
119901 (119905) 119905 ge 120591119895
(45)
and note that 11990910158401015840(120591+119896) le 11988811989611990910158401015840(120591119896) since 11990910158401015840(119905) gt 0 which imply
that
11990910158401015840
(119905) le 11990910158401015840(120591+
119895) prod
120591119895lt120591119896lt119905
119888119896minus
120581
2
int
119905
120591119895
prod
119904lt120591119896lt119905
119888119896119901 (119904) d119904
le prod
120591119895lt120591119896lt119905
119888119896[
[
11990910158401015840(120591+
119895) minus
120581
2
int
119905
120591119895
prod
120591119895lt120591119896lt119904
1
119888119896
119901 (119904) d119904]
]
(46)
Thus in virtue of (44) it holds that 11990910158401015840(119905) lt 0 and contradicts11990910158401015840(119905) gt 0 for 119905 large enough the proof is complete
Remark 9 Theorem 6 and Corollary 8 extend the results in[11 Theorem 31] and [9 Corollary 1] respectively In factwhen 119886
119896= 119887119896
= 119888119896
= 1 for 119896 isin N which implies that theimpulses in (1) disappear In such a case (5) and (6) holdnaturally and (21) and (44) are reduced to
lim119905rarrinfin
int
119905
1199050
[
[
119901 (119904) 119903 (119904) minus
[1199031015840(119904)]
2
4120583119903 (119904) (119904 minus 119879)
]
]
d119904 = infin
lim119905rarrinfin
int
119905
1199050
119901 (119904) d119904 = infin
(47)
6 Abstract and Applied Analysis
which are similar to those in [11 Theorem 31] and [9Corollary 1] respectively
Next we present some new oscillation results for (1) byusing an integral averaging condition of Kamenevrsquos type
Theorem 10 Assume (5) (6) and (19) hold Furthermore119886119896
le 1120583 le 1 119887119896
ge 1 and 119888119896
le 1 119896 isin N If there exists apositive differentiable function 119903 such that
lim sup119905rarrinfin
1
119905119898
int
119905
119879
(119905 minus 119904)119898[119901 (119904) 119903 (119904) minus 119860 (119904 120591
119896(119904))] d119904 = infin
(48)
where 119860(119904 120591119896(119904)
) is defined by (21) and 119898 ge 1 Then everysolution of (1) is oscillatory
Proof We choose 119879 large enough such that Lemma 2 holdsBy Lemma 2 there are two possible cases First if the case (i)holds proceeding as in the proof of Theorem 6 we will endup with (32) By (30) we have
119901 (119905) 119903 (119905) minus 119860 (119905 120591119896(119905)
) le minus1199061015840
(119905) 119905 ge 119879 119905 = 120591119896 (49)
If 119905 isin (120591119896+ 120590 120591119896+1
] sub (120591119896 120591119896+1
] for 120591119895ge 119879 we obtain
int
119905
120591119895
(119905 minus 119904)119898[119901 (119904) 119903 (119904) minus 119860 (119904 120591
119896(119904))] d119904
le minusint
119905
120591119895
(119905 minus 119904)1198981199061015840
(119904) d119904(50)
An integration by parts of the right-hand side leads to
int
119905
120591119895
(119905 minus 119904)1198981199061015840
(119904) d119904
= (int
120591119895+120590
120591119895
+int
120591119895+1
120591119895+120590
+ sdot sdot sdot + int
120591119896+120590
120591119896
+int
119905
120591119896+120590
) (119905 minus 119904)1198981199061015840
(119904) d119904
= int
119905
120591119895
119906 (119904)
119898
(119905 minus 119904)119898minus1d119904
+
119896
sum
119894=119895
[119905 minus (120591119894+ 120590)]119898
[119906 (120591119894+ 120590) minus 119906 (120591
+
119894+ 120590)]
+
119896
sum
119894=119895+1
(119905 minus 120591119894)119898
[119906 (120591119894) minus 119906 (120591
+
119894)] minus (119905 minus 120591
119895)
119898
119906 (120591+
119895)
(51)
Take into account (31) (32) 119886119896le 1120583 and 119888
119896le 1 we have
int
119905
120591119895
(119905 minus 119904)1198981199061015840
(119904) d119904
ge
119896
sum
119894=119895+1
(119905 minus 120591119894)119898
(1 minus 119888119894) 119906 (120591119894) minus (119905 minus 120591
119895)
119898
119906 (120591+
119895)
ge minus(119905 minus 120591119895)
119898
119906 (120591+
119895)
(52)
If 119905 isin (120591119896 120591119896+ 120590] similarly we also get
int
119905
120591119895
(119905 minus 119904)1198981199061015840
(119904) d119904 ge minus(119905 minus 120591119895)
119898
119906 (120591+
119895) (53)
So it yields
int
119905
120591119895
(119905 minus 119904)119898[119901 (119904) 119903 (119904) minus 119860 (119904 120591
119896(119904))] d119904 le (119905 minus 120591
119895)
119898
119906 (120591+
119895)
(54)
which follows that1
119905119898
int
119905
120591119895
(119905 minus 119904)119898[119901 (119904) 119903 (119904) minus 119860 (119904 120591
119896(119904))] d119904
le (
119905 minus 120591119895
119905
)
119898
119906 (120591+
119895)
(55)
Hence
lim sup119905rarrinfin
1
119905119898
int
119905
120591119895
(119905 minus 119904)119898[119901 (119904) 119903 (119904) minus 119860 (119904 120591
119896(119904))] d119904 le 119906 (120591
+
119895)
(56)
which is a contradiction of (48)If case (ii) holds then as a manner with case (ii) in
Theorem 6 it is not possible too The proof is complete
Corollary 11 Assume (19) holds and 119886119896
= 119887119896
= 119888119896
= 1 for119896 isin N If there exists a positive differential function 119903 such that
lim sup119905rarrinfin
1
119905119898
int
119905
119879
(119905 minus 119904)119898[
[
119901 (119904) 119903 (119904) minus
[1199031015840(119904)]
2
4120583119903 (119904) (119904 minus 119879)
]
]
d119904 = infin
(57)
where 119879 is large enough such that Lemma 2 holds Then everysolution of (1) is oscillatory
Remark 12 Corollary 11 is an extension of [11 Theorem 32]into impulsive case Especially let 119903(119905) equiv 1 in (48) it reducesto
lim sup119905rarrinfin
1
119905119898
int
119905
119879
(119905 minus 119904)119898119901 (119904) d119904 = infin (58)
naturally which can be considered as the extension ofKamenev-type oscillation criteria for third-order impulsivedifferential equations with delay (see [8 14 15])
4 Examples
Example 13 Consider the third-order impulsive differentialequation with delay
119909101584010158401015840
(119905) + (1 + 1205721199092
(119905)) 119909 (119905 minus 120590) = 0 119905 gt 1199050 119905 = 120591
119896
119909 (120591+
119896) = 119886119896119909 (120591119896) 119909
1015840(120591+
119896) = 1198871198961199091015840(120591119896)
11990910158401015840(120591+
119896) = 11988811989611990910158401015840(120591119896) 119896 isin N
(59)
where 120590 gt 0 120572 ge 0 are constants 120591119896minus 120591119896minus1
gt 120590 for any 119896 isin N
Abstract and Applied Analysis 7
When 119886119896
= 119887119896
= 119888119896
= 1 for any 119896 isin N the impulses in(59) disappear by [16 Theorem 4] (59) is nonoscillatory if120590 lt e3 and 120572 = 0 However we may change its oscillation byproper impulsive control In fact let 120590 lt e3 and 120572 = 0 and120591119896= 1199050+ 119896120590 (119896 isin N) in (59) choose 120593(119909) = 119909 119901(119905) equiv 1 and
119903(119905) = 1 a simple calculation leads to
int
120591119899
1199050
prod
1199050lt120591119896lt119904
1
119888119896
prod
1199050lt120591119896+120590lt119904
119886119896119901 (119904) d119904
= [int
1205911
1199050
+(int
1205911+120590
1205911
+int
1205912
1205911+120590
) + + (int
120591119899minus1+120590
120591119899minus1
+int
120591119899
120591119899minus1+120590
)]
sdot prod
1199050lt120591119896lt119904
1
119888119896
prod
1199050lt120591119896+120590lt119904
119886119896d119904
= (1205911minus 1199050) + [
120590
1198881
+
1198861
1198881
(1205912minus 1205911minus 120590)] + sdot sdot sdot +
11988611198862sdot sdot sdot 119886119899minus2
120590
11988811198882sdot sdot sdot 119888119899minus1
+
11988611198862sdot sdot sdot 119886119899minus1
11988811198882sdot sdot sdot 119888119899minus1
(120591119899minus 120591119899minus1
minus 120590)
= 120590(1 +
1
1198881
+
1
1198882
+ sdot sdot sdot +
1
119888119899minus1
)
(60)
Then let
119886119896= 119888119896=
119896
119896 + 1
119887119896= 1 119896 isin N (61)
Obviously (5) (6) and (19) hold and
int
120591119899
1199050
prod
1199050lt120591119896lt119904
1
119888119896
prod
1199050lt120591119896+120590lt119904
119886119896119901 (119904) d119904
= 120590 (1 +
2
1
+
3
2
+ sdot sdot sdot +
119899
119899 minus 1
) 997888rarr infin (119899 997888rarr infin)
(62)
Thus (21) is also satisfied By Theorem 6 every solution of(59) is oscillatory
If we let
119886119896= 1 +
1
1198962 119887119896= 119888119896=
119896 + 1
119896
119896 isin N (63)
In this case it is easily to verify (5) (6) (15) (44) and(21) hold By Corollary 8 every solution of (59) is eitheroscillatory or tends to zero
Remark 14 It is easy to verify that in [7 Theorems 1 2 and3] cannot be applied to (59) On the other hand Theorem 7is not applicable for the condition (61) sincesuminfin
119899=1|119886119896minus1| does
not convergence
Example 15 Consider the third-order impulsive differentialequation with delay
119909101584010158401015840
(119905) + e119905 cosh (|119909 (119905)|120572minus1
119909 (119905)) 119909 (119905 minus 1) = 0
119905 ge 0 119905 = 2119896
119909 (120591+
119896) = 119886119896119909 (120591119896) 119909
1015840(120591+
119896) = 1198871198961199091015840(120591119896)
11990910158401015840(120591+
119896) = 11988811989611990910158401015840(120591119896) 120591
119896= 2119896 119896 isin N
(64)
where 120572 gt 0 119886119896= 1 119887119896= 119888119896= 119896(119896 + 1) 119896 isin N
Let 120593(119909) = 119909 119901(119905) = e119905 it is easy to verify that (5) (6)and (19) hold Choose 119903(119905) equiv 1 and119898 = 1 we have
1
119905
int
119905
119879
(119905 minus 119904) e119904d119904 =
e119905
119905
+ (
119879 minus 1
119905
minus 1) e119879 rarr infin (119905 rarr infin)
(65)
Acknowledgments
Theauthor is very grateful to ProfessorH Saker who presentsthe references [8ndash12 17] and gives many helpful suggestionswhich leads to an improvement of this paper The workis supported in part by the NSF of Guangdong province(S2012010010034)
References
[1] D D Bainov and P S Simeonov Impulsive Differential Equa-tions Periodic Solutions and Applications Longman Scientificand Technical 1993
[2] G Ballinger and X Liu ldquoExistence uniqueness and bounded-ness results for impulsive delay differential equationsrdquo Applica-ble Analysis vol 74 no 1-2 pp 71ndash93 2000
[3] V Lakshmikantham D D Baınov and P S Simeonov Theoryof Impulsive Differential Equations vol 6 World ScientificPublishing New Jersey NJ USA 1989
[4] A M Samoılenko and N A Perestyuk Impulsive DifferentialEquations vol 14 World Scientific Publishing New Jersey NJUSA 1995
[5] Y S Chen andW Z Feng ldquoOscillations of second order nonlin-ear ODE with impulsesrdquo Journal of Mathematical Analysis andApplications vol 210 no 1 pp 150ndash169 1997
[6] R P Agarwal F Karakoc and A Zafer ldquoA survey on oscilla-tion of impulsive ordinary differential equationsrdquo Advances inDifferential Equations vol 2010 pp 1ndash52 2010
[7] W-H Mao and A-HWan ldquoOscillatory and asymptotic behav-ior of solutions for nonlinear impulsive delay differential equa-tionsrdquo Acta Mathematicae Applicatae Sinica vol 22 EnglishSeries no 3 pp 387ndash396 2006
[8] L Erbe A Peterson and S H Saker ldquoAsymptotic behavior ofsolutions of a third-order nonlinear dynamic equation on timescalesrdquo Journal of Computational and Applied Mathematics vol181 no 1 pp 92ndash102 2005
[9] L Erbe A Peterson and S H Saker ldquoOscillation and asymp-totic behavior of a third-order nonlinear dynamic equationrdquoThe Canadian Applied Mathematics Quarterly vol 14 no 2 pp124ndash147 2006
8 Abstract and Applied Analysis
[10] L Erbe A Peterson and S H Saker ldquoHille and Nehari typecriteria for third-order dynamic equationsrdquo Journal of Mathe-matical Analysis and Applications vol 329 no 1 pp 112ndash1312007
[11] S H Saker ldquoOscillation criteria of third-order nonlinear delaydifferential equationsrdquo Mathematica Slovaca vol 56 no 4 pp433ndash450 2006
[12] S H Saker and J Dzurina ldquoOn the oscillation of certain classof third-order nonlinear delay differential equationsrdquo Mathe-matica Bohemica vol 135 no 3 pp 225ndash237 2010
[13] J Yu and J Yan ldquoPositive solutions and asymptotic behavior ofdelay differential equations with nonlinear impulsesrdquo Journal ofMathematical Analysis and Applications vol 207 no 2 pp 388ndash396 1997
[14] T Candan and R S Dahiya ldquoOscillation of third order func-tional differential equations with delayrdquo in Proceedings of the5th Mississippi State Conference on Differential Equations andComputational Simulations vol 10 p 7988 2003
[15] I V Kamenev ldquoAn integral criterion for oscillation of linear dif-ferential equations of second orderrdquo Matematicheskie Zametkivol 23 pp 136ndash138 1978
[16] G Ladas Y G Sficas and I P Stavroulakis ldquoNecessary and suf-ficient conditions for oscillations of higher order delay differ-ential equationsrdquo Transactions of the American MathematicalSociety vol 285 no 1 pp 81ndash90 1984
[17] B Baculıkova E M Elabbasy S H Saker and J DzurinaldquoOscillation criteria for third-order nonlinear differential equa-tionsrdquoMathematica Slovaca vol 58 no 2 pp 201ndash220 2008
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Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical Problems in Engineering
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Differential EquationsInternational Journal of
Volume 2014
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Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
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Abstract and Applied Analysis 3
we get that 1199091015840(119905) gt 0 for any 119905 isin (120591119896 120591119896+1
] 119896 = 119895 + 1 119895 + 2 So the following impulsive differential inequalities hold
11990910158401015840
(119905) le minus120572 prod
120591119895+1lt120591119896lt119905
119888119896 119905 gt 120591
119895+1 119905 = 120591
119896
1199091015840(120591+
119896) le 119887119896119909 (120591119896) 119896 = 119895 + 1 119895 + 2
(13)
According to Lemma 1 we get
1199091015840
(119905) le 1199091015840(120591+
119895+1) prod
120591119895+1lt120591119896lt119905
119887119896minus 120572int
119905
120591119895+1
prod
119904lt120591119896lt119905
119887119896
prod
120591119895+1lt120591119896lt119904
119888119896d119904
= prod
120591119895+1lt120591119896lt119905
119887119896(1199091015840(120591+
119895+1) minus 120572int
119905
120591119895+1
prod
120591119895+1lt120591119896lt119904
119888119896
119887119896
d119904)
(14)
Hence the condition (6) implies that 1199091015840(119905) lt 0 when 119905 is
sufficiently large which contradicts to 1199091015840(119905) gt 0 for 119905 gt 120591
119895+1
again In terms of the above discussion we see that11990910158401015840(120591119896) gt 0
for any 120591119896gt 119879 with sufficiently large 119879 Consequently noting
that 119909101584010158401015840(119905) lt 0 for any 119905 isin (120591
119896 120591119896+1
] we have 11990910158401015840(119905) gt
11990910158401015840(120591119896+1
) gt 0Next if there exists a 120591
119895gt 119879 such that 1199091015840(120591
119895) ge 0 then
1199091015840(120591+
119895) = 119887119895(119909(120591+
119895) ge 0 1199091015840(120591
119895+1) gt 119909
1015840(120591+
119895) ge 0 Therefore by
induction we have 1199091015840(119905) gt 1199091015840(120591+
119894) gt 0 for 119905 isin (120591
119894 120591119894+1
] 119894 = 119895 +
1 119895 + 2 So case (i) is satisfied Otherwise if 1199091015840(120591119896) lt 0 for
all 120591119896ge 119879 then 119909
1015840(120591+
119896) = 119887119896119909(120591+
119896) lt 0 Thus for 119905 isin (120591
119896 120591119896+1
]using 119909
10158401015840(119905) gt 0 we have 1199091015840(119905) lt 119909
1015840(120591119896+1
) lt 0 hence case (ii)is satisfied This completes the proof
Remark 3 Suppose that 119909(119905) is an eventually negative solu-tion of (1) If (5) and (6) hold one can prove it holds that oneof the following two cases in a similar way as Lemma 2
(i) 11990910158401015840(120591+
119896) lt 0 11990910158401015840(119905) lt 0 and 119909
1015840(120591+
119896) lt 0 1199091015840(119905) lt 0
(ii) 11990910158401015840(120591+
119896) lt 0 11990910158401015840(119905) lt 0 and 119909
1015840(120591+
119896) gt 0 1199091015840(119905) gt 0 with
119905 isin (120591119896 120591119896+1
] and 120591119896ge 119879
Lemma 4 Let 119909(119905) be a piecewise continuous function oncup119896isinN(120591119896 120591119896+1] which is continuous at 119905 = 120591
119896and is left contin-
uous at 119905 = 120591119896 If
(1) 119909(119905) ge 0 (le 0) for 119905 ge 1199050
(2) 119909(119905) is monotone nonincreasing (monotone nonde-creasing) on (120591
119896 120591119896+1
] (120591119896ge 119879) for 119879 large enough
(3) suminfin
119896=1[119909(120591+
119896) minus 119909(120591
119896)] converges
then lim119905rarrinfin
119909(119905) = 119886 ge 0 (le 0)
The proof of Lemma 4 is similar to that of [13 Theorem5] and hence is omitted
Lemma 5 Assume that 119909(119905) is a solution of (1) which satisfiescase (ii) in Lemma 2 In addition if
infin
sum
119896=1
1003816100381610038161003816119886119896minus 1
1003816100381610038161003816lt infin
infin
prod
119896=1
119886119896119894119904 119887119900119906119899119889119890119889 (15)
then lim119905rarrinfin
119909(119905) exists (finite)
Proof First we claim thatsuminfin119896=1
[119909(120591+
119896)minus119909(120591
119896)] is convergence
In fact since 119909(119905) is decreasing on (120591119896 120591119896+1
] (120591119896ge 119879 and 119879 is
defined in Lemma 2) then
119909 (120591119896+1
) le 119909 (120591+
119896) 119909 (120591
+
119896+1) le 119886119896+1
119909 (120591119896+1
) le 119886119896+1
119909 (120591+
119896)
(16)
Obviously by induction we can get
119909 (120591119896+119899
) le 119886119896+119899minus1
sdot sdot sdot 119886119896+1
119909 (120591+
119896)
119909 (120591+
119896+119899) le 119886119896+119899
sdot sdot sdot 119886119896+1
119909 (120591+
119896)
(17)
Since prodinfin
119896=1119886119896
is bounded we conclude that 119909(120591119896) is
bounded which follows that there exists1198721gt 0 such that
1003816100381610038161003816119909 (120591+
119896) minus 119909 (120591
119896)1003816100381610038161003816=
1003816100381610038161003816119886119896minus 1
1003816100381610038161003816119909 (120591119896) le 119872
1
1003816100381610038161003816119886119896minus 1
1003816100381610038161003816 (18)
Hencesuminfin119896=1
[119909(120591+
119896)minus119909(120591
119896)] is convergence sincesuminfin
119896=1|119886119896minus1| is
convergence which follows from Lemma 4 that lim119905rarrinfin
119909(119905)
exists The proof is complete
3 Main Results
In this section we establish some sufficient conditions whichguarantee that every solution 119909(119905) of (1) either oscillates orhas a finite limit Occasionally we will make the additionalassumption
lim sup119905rarrinfin
int
119905
120578
int
infin
119906
int
infin
119904
119901 (120579) d120579 d119904 d119906 = infin (19)
where here it is understood that
int
infin
1199050
119901 (119905) d119905 lt infin int
infin
1199050
int
infin
119906
119901 (120579) d120579 d119906 lt infin (20)
Now we are ready to state and prove the main results inthis paper The results will be proved by making use of thetechnique in [11]
Theorem 6 Assume that (5) (6) and (19) hold and 119909(119905) is asolution of (1) Furthermore assume that 119886
119896le 1 119887
119896ge 1 and
119888119896le 1 for 119896 isin N If there exists a positive differentiable function
119903 such that
lim119905rarrinfin
int
119905
1199050
[119901 (119904) 119903 (119904) minus 119860 (119904 120591119896(119904)
)] prod
1199050lt120591119896lt119904
1
119888119896
prod
1199050lt120591119896+120590lt119904
119886119896d119904 = infin
(21)
where 119896(119904) = max119896 120591119896lt 119904 and
119860 (119904 120591119896(119904)
)
=
119888119896(119904)[1199031015840(119904)]2
4120583119903 (119904) (119904 minus 120591119896(119904))
119904 isin (120591119896 120591119896+ 120590]
[1199031015840(119904)]2
4120583119903 (119904) (119904 minus 120591119896(119904)minus ((119888119896(119904)minus 1) 119888
119896(119904)) 120590)
119904 isin (120591119896+ 120590 120591119896+1]
(22)
Then 119909(119905) is oscillatory
4 Abstract and Applied Analysis
Proof Let 119909(119905) be a nonoscillatory solution of (1) withoutloss of generality we may assume that 119909(119905) gt 0 eventually(119909(119905) lt 0 eventually can be achieved in the similar way)By Lemma 2 either case (i) or case (ii) in Lemma 2 holdsAssume that 119909(119905) satisfies case (i) then 119909
10158401015840(119905) gt 0 1199091015840(119905) gt 0
for 119905 isin (120591119895 120591119895+1
] 120591119895ge 119879 (119879 is defined in Lemma 2) Define the
Riccati transformation 119906 by
119906 (119905) =
119903 (119905) 11990910158401015840(119905)
120593 (119909 (119905 minus 120590))
119905 ge 120591119895 119905 = 120591
119896 (23)
Thus 119906(119905) gt 0 for 119905 isin (120591119896 120591119896+1
] 119896 = 119895 119895 + 1 and
1199061015840
(119905) =
119903 (119905)
120593 (119909 (119905 minus 120590))
119909101584010158401015840
(119905)
+
1199031015840(119905) 120593 (119909 (119905 minus 120590)) minus 119903 (119905) 120593
1015840(119909 (119905 minus 120590)) 119909
1015840(119905 minus 120590)
1205932(119909 (119905 minus 120590))
times 11990910158401015840
(119905)
le minus 119901 (119905) 119903 (119905) +
1199031015840(119905)
119903 (119905)
119906 (119905)
minus
119903 (119905) 1205831199091015840(119905 minus 120590)
1205932(119909 (119905 minus 120590))
11990910158401015840
(119905)
(24)
If 119905 isin (120591119896 120591119896+120590] sub (120591
119896 120591119896+1
] namely 119905 minus120590 le 120591119896lt 119905 11990910158401015840(119905)
is decreasing in (119905 minus 120590 120591119896] and (120591
119896 119905] respectively In view of
the following
11990910158401015840
(119905 minus 120590) gt 11990910158401015840(120591119896) ge
11990910158401015840(120591+
119896)
119888119896
gt
11990910158401015840(119905)
119888119896
(25)
we have
1199091015840
(119905 minus 120590) = 1199091015840(120591119896minus 120590) + int
119905minus120590
120591119896minus120590
11990910158401015840
(119904) d119904
gt 11990910158401015840
(119905 minus 120590) (119905 minus 120591119896) gt
11990910158401015840(119905)
119888119896
(119905 minus 120591119896)
(26)
Thus
1199061015840
(119905) le minus 119901 (119905) 119903 (119905) +
1199031015840(119905)
119903 (119905)
119906 (119905) minus
120583119903 (119905) (119905 minus 120591119896)
1198881198961205932(119909 (119905 minus 120590))
[11990910158401015840
(119905)]
2
= minus 119901 (119905) 119903 (119905) +
1199031015840(119905)
119903 (119905)
119906 (119905) minus
120583 (119905 minus 120591119896)
119888119896119903 (119905)
[119906 (119905)]2
= minus (radic
120583 (119905 minus 120591119896)
119888119896119903 (119905)
119906 (119905) minus radic
119888119896
4120583119903 (119905) (119905 minus 120591119896)
1199031015840
(119905))
2
minus[
[
119901 (119905) 119903 (119905) minus
119888119896[1199031015840(119905)]
2
4120583119903 (119905) (119905 minus 120591119896)
]
]
le minus[
[
119901 (119905) 119903 (119905) minus
119888119896[1199031015840(119905)]
2
4120583119903 (119905) (119905 minus 120591119896)
]
]
(27)
If 119905 isin (120591119896+120590 120591119896+1
] sub (120591119896 120591119896+1
] that is 120591119896lt 119905minus120590 lt 119905 le 120591
119896+1
then
1199091015840
(119905 minus 120590) = 1199091015840(120591119896minus 120590) + int
120591119896
120591119896minus120590
11990910158401015840
(119904) d119904 + int
119905minus120590
120591119896
11990910158401015840
(119904) d119904
ge 11990910158401015840(120591119896) 120590 + 119909
10158401015840
(119905 minus 120590) (119905 minus 120591119896minus 120590)
ge
11990910158401015840(119905)
119888119896
120590 + 11990910158401015840
(119905) (119905 minus 120591119896minus 120590)
ge 11990910158401015840
(119905) (119905 minus 120591119896minus
119888119896minus 1
119888119896
120590)
(28)
Similarly we have
1199061015840
(119905) le minus[
[
119901 (119905) 119903 (119905) minus
[1199031015840(119905)]
2
4120583119903 (119905) (119905 minus 120591119896minus ((119888119896minus 1) 119888
119896) 120590)
]
]
(29)
Thus we obtain
1199061015840
(119905) le minus [119901 (119905) 119903 (119905) minus 119860 (119905 120591119896(119905)
)] for 119905 isin (120591119896 120591119896+1
]
(30)
On the other hand
119906 (120591+
119896) =
119903 (120591119896) 11990910158401015840(120591+
119896)
120593 (119909 (120591119896minus 120590))
le 119888119896119906 (120591119896) (31)
Observing that 120593(119906) ge 120583119906 we have
119906 (120591+
119896+ 120590) =
119903 (120591119896+ 120590) 119909
10158401015840(120591119896+ 120590)
120593 (119909 (120591+
119896))
le
119903 (120591119896+ 120590) 119909
10158401015840(120591119896+ 120590)
120593 (119886119896119909 (120591119896))
le
119906 (120591119896+ 120590)
120583119886119896
(32)
Applying Lemma 1 it follows from (30) (31) and (32) that
119906 (119905) le 119906 (120591+
119895) prod
120591119895lt120591119896lt119905
119888119896
prod
120591119895lt120591119896+120590lt119905
1
120583119886119896
minus int
119905
120591119895
[119901 (119904) 119903 (119904) minus 119860 (119904 120591119896(119904)
)]
sdot prod
119904lt120591119896lt119905
119888119896
prod
119904lt120591119896+120590lt119905
1
120583119886119896
d119904
le prod
120591119895lt120591119896lt119905
119888119896
prod
120591119895lt120591119896+120590lt119905
1
120583119886119896
times (119906 (120591+
119895) minus 120583int
119905
120591119895
[119901 (119904) 119903 (119904) minus 119860 (119904 120591119896(119904)
)]
sdot prod
120591119895lt120591119896lt119904
1
119888119896
prod
120591119895lt120591119896+120590lt119904
119886119896d119904)
(33)
Abstract and Applied Analysis 5
which yields 119906(119905) lt 0 for all large 119905 This is contrary to 119906(119905) gt
0 and so case (i) in Lemma 2 is not possibleIf 119909(119905) satisfies the case (ii) in Lemma 2 that is 11990910158401015840(120591+
119896) gt
0 11990910158401015840(119905) gt 0 and 1199091015840(120591+
119896) lt 0 1199091015840(119905) lt 0 which proves that the
solution 119909(119905) is positive and decreasing Integrating (1) from119904 to 119905 (119905 ge 119904 ge 119879) we obtain
11990910158401015840
(119905) minus sum
119904lt120591119896lt119905
(119888119896minus 1) 119909
10158401015840(120591119896) minus 11990910158401015840
(119904)
+ int
119905
119904
119901 (120579) 120593 (119909 (120579 minus 120590)) d120579 le 0
(34)
Noting 119888119896le 1 and 119909
10158401015840(119905) gt 0 then it holds that
11990910158401015840
(119905) minus 11990910158401015840
(119904) + int
119905
119904
119901 (120579) 120593 (119909 (120579 minus 120590)) d120579 le 0 (35)
which leads to
minus11990910158401015840
(119904) + int
119905
119904
119901 (120579) 120593 (119909 (120579 minus 120590)) d120579 le 0 (36)
and hence
minus11990910158401015840
(119904) + int
infin
119904
119901 (120579) 120593 (119909 (120579 minus 120590)) d120579 le 0 (37)
Integrating the above inequality again from 119906 to 119905 (119905 ge 119906 ge 119879)one has
minus 1199091015840
(119905) + sum
119904lt120591119896lt119905
(119887119896minus 1) 119909
1015840(120591119896) + 1199091015840
(119906)
+ int
119905
119906
int
infin
119904
119901 (120579) 120593 (119909 (120579 minus 120590)) d120579 d119904 le 0
(38)
Using 1199091015840(119905) lt 0 and 119887
119896ge 1 we have
1199091015840
(119906) + int
infin
119906
int
infin
119904
119901 (120579) 120593 (119909 (120579 minus 120590)) d120579 d119904 le 0 (39)
Now we integrate the last inequality from 120578 to 119905 (119905 ge 120578 ge 119879)to obtain
119909 (119905) minus sum
120578lt120591119896lt119905
(119886119896minus 1) 119909 (120591
119896) minus 119909 (120578)
+ int
119905
120578
int
infin
119906
int
infin
119904
119901 (120579) 120593 (119909 (120579 minus 120590)) d120579 d119904 d119906 le 0
(40)
Since 119886119896
le 1 (119896 isin N) and 119909(119905) is decreasing then for 119905 isin
(120591119896 120591119896+1
] 119909(120591119896+1
) le 119909(119905) le 119909(120591+
119896) = 119886119896119909(120591119896) le 119909(120591
119896) 119896 isin N
Thus we get
120583119909 (120578) int
119905
120578
int
infin
119906
int
infin
119904
119901 (120579) d120579 d119904 d119906 le minus119909 (119905) + 119909 (120578) le 119909 (120578)
(41)
and then
int
119905
120578
int
infin
119906
int
infin
119904
119901 (120579) d120579 d119904 d119906 le
1
120583
(42)
which contradicts the condition (19) The proof is complete
Replace the condition (19) with (15) we may obtain thefollowing asymptotic results
Theorem 7 Assume that (5) (6) and (15) hold and 119909(119905) is asolution of (1) If there exists a positive differentiable function 119903
such that (21) hold then 119909(119905) is either oscillatory or has a finitelimit
Proof By the proof of Theorem 6 we know the case (i) inLemma 2 is not possible too since the condition (19) is notrequired to prove it So it suffices to show if there is a solutionsatisfying case (ii) in Lemma 2 that is if
11990910158401015840(120591+
119896) gt 0 119909
10158401015840
(119905) gt 0 1199091015840(120591+
119896) lt 0 119909
1015840
(119905) lt 0 (43)
with 119905 isin (120591119896 120591119896+1
] and 120591119896ge 119879 then lim
119905rarrinfin119909(119905) exists This
is obtained by applying Lemma 5 which leads to lim119905rarrinfin
119909(119905)
exists The proof is complete
Corollary 8 In addition to the assumption of Theorem 7assume that
lim119905rarrinfin
int
119905
1199050
prod
1199050lt120591119896lt119904
1
119888119896
119901 (119904) d119904 = infin (44)
Then solution 119909(119905) of (1) either oscillates or satisfieslim119905rarrinfin
119909(119905) = 0
Proof By the proof ofTheorem 7 lim119905rarrinfin
119909(119905) exists and wedefine it by lim
119905rarrinfin119909(119905) = 120574 ge 0 We now show 120574 = 0 If not
then 120574 gt 0 So lim119905rarrinfin
120593(119909(119905 minus 120590)) = 120593(120574) = 120581 gt 0 Hencethere exists 120591
119895gt 119879 such that 120593(119909(119905minus120590)) gt 1205812 for 119905 gt 120591
119895Then
119909101584010158401015840
(119905) = minus119891 (119905 119909 (119905) 119909 (119905 minus 120590))
le minus119901 (119905) 120593 (119909 (119905 minus 120590)) le minus
120581
2
119901 (119905) 119905 ge 120591119895
(45)
and note that 11990910158401015840(120591+119896) le 11988811989611990910158401015840(120591119896) since 11990910158401015840(119905) gt 0 which imply
that
11990910158401015840
(119905) le 11990910158401015840(120591+
119895) prod
120591119895lt120591119896lt119905
119888119896minus
120581
2
int
119905
120591119895
prod
119904lt120591119896lt119905
119888119896119901 (119904) d119904
le prod
120591119895lt120591119896lt119905
119888119896[
[
11990910158401015840(120591+
119895) minus
120581
2
int
119905
120591119895
prod
120591119895lt120591119896lt119904
1
119888119896
119901 (119904) d119904]
]
(46)
Thus in virtue of (44) it holds that 11990910158401015840(119905) lt 0 and contradicts11990910158401015840(119905) gt 0 for 119905 large enough the proof is complete
Remark 9 Theorem 6 and Corollary 8 extend the results in[11 Theorem 31] and [9 Corollary 1] respectively In factwhen 119886
119896= 119887119896
= 119888119896
= 1 for 119896 isin N which implies that theimpulses in (1) disappear In such a case (5) and (6) holdnaturally and (21) and (44) are reduced to
lim119905rarrinfin
int
119905
1199050
[
[
119901 (119904) 119903 (119904) minus
[1199031015840(119904)]
2
4120583119903 (119904) (119904 minus 119879)
]
]
d119904 = infin
lim119905rarrinfin
int
119905
1199050
119901 (119904) d119904 = infin
(47)
6 Abstract and Applied Analysis
which are similar to those in [11 Theorem 31] and [9Corollary 1] respectively
Next we present some new oscillation results for (1) byusing an integral averaging condition of Kamenevrsquos type
Theorem 10 Assume (5) (6) and (19) hold Furthermore119886119896
le 1120583 le 1 119887119896
ge 1 and 119888119896
le 1 119896 isin N If there exists apositive differentiable function 119903 such that
lim sup119905rarrinfin
1
119905119898
int
119905
119879
(119905 minus 119904)119898[119901 (119904) 119903 (119904) minus 119860 (119904 120591
119896(119904))] d119904 = infin
(48)
where 119860(119904 120591119896(119904)
) is defined by (21) and 119898 ge 1 Then everysolution of (1) is oscillatory
Proof We choose 119879 large enough such that Lemma 2 holdsBy Lemma 2 there are two possible cases First if the case (i)holds proceeding as in the proof of Theorem 6 we will endup with (32) By (30) we have
119901 (119905) 119903 (119905) minus 119860 (119905 120591119896(119905)
) le minus1199061015840
(119905) 119905 ge 119879 119905 = 120591119896 (49)
If 119905 isin (120591119896+ 120590 120591119896+1
] sub (120591119896 120591119896+1
] for 120591119895ge 119879 we obtain
int
119905
120591119895
(119905 minus 119904)119898[119901 (119904) 119903 (119904) minus 119860 (119904 120591
119896(119904))] d119904
le minusint
119905
120591119895
(119905 minus 119904)1198981199061015840
(119904) d119904(50)
An integration by parts of the right-hand side leads to
int
119905
120591119895
(119905 minus 119904)1198981199061015840
(119904) d119904
= (int
120591119895+120590
120591119895
+int
120591119895+1
120591119895+120590
+ sdot sdot sdot + int
120591119896+120590
120591119896
+int
119905
120591119896+120590
) (119905 minus 119904)1198981199061015840
(119904) d119904
= int
119905
120591119895
119906 (119904)
119898
(119905 minus 119904)119898minus1d119904
+
119896
sum
119894=119895
[119905 minus (120591119894+ 120590)]119898
[119906 (120591119894+ 120590) minus 119906 (120591
+
119894+ 120590)]
+
119896
sum
119894=119895+1
(119905 minus 120591119894)119898
[119906 (120591119894) minus 119906 (120591
+
119894)] minus (119905 minus 120591
119895)
119898
119906 (120591+
119895)
(51)
Take into account (31) (32) 119886119896le 1120583 and 119888
119896le 1 we have
int
119905
120591119895
(119905 minus 119904)1198981199061015840
(119904) d119904
ge
119896
sum
119894=119895+1
(119905 minus 120591119894)119898
(1 minus 119888119894) 119906 (120591119894) minus (119905 minus 120591
119895)
119898
119906 (120591+
119895)
ge minus(119905 minus 120591119895)
119898
119906 (120591+
119895)
(52)
If 119905 isin (120591119896 120591119896+ 120590] similarly we also get
int
119905
120591119895
(119905 minus 119904)1198981199061015840
(119904) d119904 ge minus(119905 minus 120591119895)
119898
119906 (120591+
119895) (53)
So it yields
int
119905
120591119895
(119905 minus 119904)119898[119901 (119904) 119903 (119904) minus 119860 (119904 120591
119896(119904))] d119904 le (119905 minus 120591
119895)
119898
119906 (120591+
119895)
(54)
which follows that1
119905119898
int
119905
120591119895
(119905 minus 119904)119898[119901 (119904) 119903 (119904) minus 119860 (119904 120591
119896(119904))] d119904
le (
119905 minus 120591119895
119905
)
119898
119906 (120591+
119895)
(55)
Hence
lim sup119905rarrinfin
1
119905119898
int
119905
120591119895
(119905 minus 119904)119898[119901 (119904) 119903 (119904) minus 119860 (119904 120591
119896(119904))] d119904 le 119906 (120591
+
119895)
(56)
which is a contradiction of (48)If case (ii) holds then as a manner with case (ii) in
Theorem 6 it is not possible too The proof is complete
Corollary 11 Assume (19) holds and 119886119896
= 119887119896
= 119888119896
= 1 for119896 isin N If there exists a positive differential function 119903 such that
lim sup119905rarrinfin
1
119905119898
int
119905
119879
(119905 minus 119904)119898[
[
119901 (119904) 119903 (119904) minus
[1199031015840(119904)]
2
4120583119903 (119904) (119904 minus 119879)
]
]
d119904 = infin
(57)
where 119879 is large enough such that Lemma 2 holds Then everysolution of (1) is oscillatory
Remark 12 Corollary 11 is an extension of [11 Theorem 32]into impulsive case Especially let 119903(119905) equiv 1 in (48) it reducesto
lim sup119905rarrinfin
1
119905119898
int
119905
119879
(119905 minus 119904)119898119901 (119904) d119904 = infin (58)
naturally which can be considered as the extension ofKamenev-type oscillation criteria for third-order impulsivedifferential equations with delay (see [8 14 15])
4 Examples
Example 13 Consider the third-order impulsive differentialequation with delay
119909101584010158401015840
(119905) + (1 + 1205721199092
(119905)) 119909 (119905 minus 120590) = 0 119905 gt 1199050 119905 = 120591
119896
119909 (120591+
119896) = 119886119896119909 (120591119896) 119909
1015840(120591+
119896) = 1198871198961199091015840(120591119896)
11990910158401015840(120591+
119896) = 11988811989611990910158401015840(120591119896) 119896 isin N
(59)
where 120590 gt 0 120572 ge 0 are constants 120591119896minus 120591119896minus1
gt 120590 for any 119896 isin N
Abstract and Applied Analysis 7
When 119886119896
= 119887119896
= 119888119896
= 1 for any 119896 isin N the impulses in(59) disappear by [16 Theorem 4] (59) is nonoscillatory if120590 lt e3 and 120572 = 0 However we may change its oscillation byproper impulsive control In fact let 120590 lt e3 and 120572 = 0 and120591119896= 1199050+ 119896120590 (119896 isin N) in (59) choose 120593(119909) = 119909 119901(119905) equiv 1 and
119903(119905) = 1 a simple calculation leads to
int
120591119899
1199050
prod
1199050lt120591119896lt119904
1
119888119896
prod
1199050lt120591119896+120590lt119904
119886119896119901 (119904) d119904
= [int
1205911
1199050
+(int
1205911+120590
1205911
+int
1205912
1205911+120590
) + + (int
120591119899minus1+120590
120591119899minus1
+int
120591119899
120591119899minus1+120590
)]
sdot prod
1199050lt120591119896lt119904
1
119888119896
prod
1199050lt120591119896+120590lt119904
119886119896d119904
= (1205911minus 1199050) + [
120590
1198881
+
1198861
1198881
(1205912minus 1205911minus 120590)] + sdot sdot sdot +
11988611198862sdot sdot sdot 119886119899minus2
120590
11988811198882sdot sdot sdot 119888119899minus1
+
11988611198862sdot sdot sdot 119886119899minus1
11988811198882sdot sdot sdot 119888119899minus1
(120591119899minus 120591119899minus1
minus 120590)
= 120590(1 +
1
1198881
+
1
1198882
+ sdot sdot sdot +
1
119888119899minus1
)
(60)
Then let
119886119896= 119888119896=
119896
119896 + 1
119887119896= 1 119896 isin N (61)
Obviously (5) (6) and (19) hold and
int
120591119899
1199050
prod
1199050lt120591119896lt119904
1
119888119896
prod
1199050lt120591119896+120590lt119904
119886119896119901 (119904) d119904
= 120590 (1 +
2
1
+
3
2
+ sdot sdot sdot +
119899
119899 minus 1
) 997888rarr infin (119899 997888rarr infin)
(62)
Thus (21) is also satisfied By Theorem 6 every solution of(59) is oscillatory
If we let
119886119896= 1 +
1
1198962 119887119896= 119888119896=
119896 + 1
119896
119896 isin N (63)
In this case it is easily to verify (5) (6) (15) (44) and(21) hold By Corollary 8 every solution of (59) is eitheroscillatory or tends to zero
Remark 14 It is easy to verify that in [7 Theorems 1 2 and3] cannot be applied to (59) On the other hand Theorem 7is not applicable for the condition (61) sincesuminfin
119899=1|119886119896minus1| does
not convergence
Example 15 Consider the third-order impulsive differentialequation with delay
119909101584010158401015840
(119905) + e119905 cosh (|119909 (119905)|120572minus1
119909 (119905)) 119909 (119905 minus 1) = 0
119905 ge 0 119905 = 2119896
119909 (120591+
119896) = 119886119896119909 (120591119896) 119909
1015840(120591+
119896) = 1198871198961199091015840(120591119896)
11990910158401015840(120591+
119896) = 11988811989611990910158401015840(120591119896) 120591
119896= 2119896 119896 isin N
(64)
where 120572 gt 0 119886119896= 1 119887119896= 119888119896= 119896(119896 + 1) 119896 isin N
Let 120593(119909) = 119909 119901(119905) = e119905 it is easy to verify that (5) (6)and (19) hold Choose 119903(119905) equiv 1 and119898 = 1 we have
1
119905
int
119905
119879
(119905 minus 119904) e119904d119904 =
e119905
119905
+ (
119879 minus 1
119905
minus 1) e119879 rarr infin (119905 rarr infin)
(65)
Acknowledgments
Theauthor is very grateful to ProfessorH Saker who presentsthe references [8ndash12 17] and gives many helpful suggestionswhich leads to an improvement of this paper The workis supported in part by the NSF of Guangdong province(S2012010010034)
References
[1] D D Bainov and P S Simeonov Impulsive Differential Equa-tions Periodic Solutions and Applications Longman Scientificand Technical 1993
[2] G Ballinger and X Liu ldquoExistence uniqueness and bounded-ness results for impulsive delay differential equationsrdquo Applica-ble Analysis vol 74 no 1-2 pp 71ndash93 2000
[3] V Lakshmikantham D D Baınov and P S Simeonov Theoryof Impulsive Differential Equations vol 6 World ScientificPublishing New Jersey NJ USA 1989
[4] A M Samoılenko and N A Perestyuk Impulsive DifferentialEquations vol 14 World Scientific Publishing New Jersey NJUSA 1995
[5] Y S Chen andW Z Feng ldquoOscillations of second order nonlin-ear ODE with impulsesrdquo Journal of Mathematical Analysis andApplications vol 210 no 1 pp 150ndash169 1997
[6] R P Agarwal F Karakoc and A Zafer ldquoA survey on oscilla-tion of impulsive ordinary differential equationsrdquo Advances inDifferential Equations vol 2010 pp 1ndash52 2010
[7] W-H Mao and A-HWan ldquoOscillatory and asymptotic behav-ior of solutions for nonlinear impulsive delay differential equa-tionsrdquo Acta Mathematicae Applicatae Sinica vol 22 EnglishSeries no 3 pp 387ndash396 2006
[8] L Erbe A Peterson and S H Saker ldquoAsymptotic behavior ofsolutions of a third-order nonlinear dynamic equation on timescalesrdquo Journal of Computational and Applied Mathematics vol181 no 1 pp 92ndash102 2005
[9] L Erbe A Peterson and S H Saker ldquoOscillation and asymp-totic behavior of a third-order nonlinear dynamic equationrdquoThe Canadian Applied Mathematics Quarterly vol 14 no 2 pp124ndash147 2006
8 Abstract and Applied Analysis
[10] L Erbe A Peterson and S H Saker ldquoHille and Nehari typecriteria for third-order dynamic equationsrdquo Journal of Mathe-matical Analysis and Applications vol 329 no 1 pp 112ndash1312007
[11] S H Saker ldquoOscillation criteria of third-order nonlinear delaydifferential equationsrdquo Mathematica Slovaca vol 56 no 4 pp433ndash450 2006
[12] S H Saker and J Dzurina ldquoOn the oscillation of certain classof third-order nonlinear delay differential equationsrdquo Mathe-matica Bohemica vol 135 no 3 pp 225ndash237 2010
[13] J Yu and J Yan ldquoPositive solutions and asymptotic behavior ofdelay differential equations with nonlinear impulsesrdquo Journal ofMathematical Analysis and Applications vol 207 no 2 pp 388ndash396 1997
[14] T Candan and R S Dahiya ldquoOscillation of third order func-tional differential equations with delayrdquo in Proceedings of the5th Mississippi State Conference on Differential Equations andComputational Simulations vol 10 p 7988 2003
[15] I V Kamenev ldquoAn integral criterion for oscillation of linear dif-ferential equations of second orderrdquo Matematicheskie Zametkivol 23 pp 136ndash138 1978
[16] G Ladas Y G Sficas and I P Stavroulakis ldquoNecessary and suf-ficient conditions for oscillations of higher order delay differ-ential equationsrdquo Transactions of the American MathematicalSociety vol 285 no 1 pp 81ndash90 1984
[17] B Baculıkova E M Elabbasy S H Saker and J DzurinaldquoOscillation criteria for third-order nonlinear differential equa-tionsrdquoMathematica Slovaca vol 58 no 2 pp 201ndash220 2008
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Volume 2014
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Stochastic AnalysisInternational Journal of
4 Abstract and Applied Analysis
Proof Let 119909(119905) be a nonoscillatory solution of (1) withoutloss of generality we may assume that 119909(119905) gt 0 eventually(119909(119905) lt 0 eventually can be achieved in the similar way)By Lemma 2 either case (i) or case (ii) in Lemma 2 holdsAssume that 119909(119905) satisfies case (i) then 119909
10158401015840(119905) gt 0 1199091015840(119905) gt 0
for 119905 isin (120591119895 120591119895+1
] 120591119895ge 119879 (119879 is defined in Lemma 2) Define the
Riccati transformation 119906 by
119906 (119905) =
119903 (119905) 11990910158401015840(119905)
120593 (119909 (119905 minus 120590))
119905 ge 120591119895 119905 = 120591
119896 (23)
Thus 119906(119905) gt 0 for 119905 isin (120591119896 120591119896+1
] 119896 = 119895 119895 + 1 and
1199061015840
(119905) =
119903 (119905)
120593 (119909 (119905 minus 120590))
119909101584010158401015840
(119905)
+
1199031015840(119905) 120593 (119909 (119905 minus 120590)) minus 119903 (119905) 120593
1015840(119909 (119905 minus 120590)) 119909
1015840(119905 minus 120590)
1205932(119909 (119905 minus 120590))
times 11990910158401015840
(119905)
le minus 119901 (119905) 119903 (119905) +
1199031015840(119905)
119903 (119905)
119906 (119905)
minus
119903 (119905) 1205831199091015840(119905 minus 120590)
1205932(119909 (119905 minus 120590))
11990910158401015840
(119905)
(24)
If 119905 isin (120591119896 120591119896+120590] sub (120591
119896 120591119896+1
] namely 119905 minus120590 le 120591119896lt 119905 11990910158401015840(119905)
is decreasing in (119905 minus 120590 120591119896] and (120591
119896 119905] respectively In view of
the following
11990910158401015840
(119905 minus 120590) gt 11990910158401015840(120591119896) ge
11990910158401015840(120591+
119896)
119888119896
gt
11990910158401015840(119905)
119888119896
(25)
we have
1199091015840
(119905 minus 120590) = 1199091015840(120591119896minus 120590) + int
119905minus120590
120591119896minus120590
11990910158401015840
(119904) d119904
gt 11990910158401015840
(119905 minus 120590) (119905 minus 120591119896) gt
11990910158401015840(119905)
119888119896
(119905 minus 120591119896)
(26)
Thus
1199061015840
(119905) le minus 119901 (119905) 119903 (119905) +
1199031015840(119905)
119903 (119905)
119906 (119905) minus
120583119903 (119905) (119905 minus 120591119896)
1198881198961205932(119909 (119905 minus 120590))
[11990910158401015840
(119905)]
2
= minus 119901 (119905) 119903 (119905) +
1199031015840(119905)
119903 (119905)
119906 (119905) minus
120583 (119905 minus 120591119896)
119888119896119903 (119905)
[119906 (119905)]2
= minus (radic
120583 (119905 minus 120591119896)
119888119896119903 (119905)
119906 (119905) minus radic
119888119896
4120583119903 (119905) (119905 minus 120591119896)
1199031015840
(119905))
2
minus[
[
119901 (119905) 119903 (119905) minus
119888119896[1199031015840(119905)]
2
4120583119903 (119905) (119905 minus 120591119896)
]
]
le minus[
[
119901 (119905) 119903 (119905) minus
119888119896[1199031015840(119905)]
2
4120583119903 (119905) (119905 minus 120591119896)
]
]
(27)
If 119905 isin (120591119896+120590 120591119896+1
] sub (120591119896 120591119896+1
] that is 120591119896lt 119905minus120590 lt 119905 le 120591
119896+1
then
1199091015840
(119905 minus 120590) = 1199091015840(120591119896minus 120590) + int
120591119896
120591119896minus120590
11990910158401015840
(119904) d119904 + int
119905minus120590
120591119896
11990910158401015840
(119904) d119904
ge 11990910158401015840(120591119896) 120590 + 119909
10158401015840
(119905 minus 120590) (119905 minus 120591119896minus 120590)
ge
11990910158401015840(119905)
119888119896
120590 + 11990910158401015840
(119905) (119905 minus 120591119896minus 120590)
ge 11990910158401015840
(119905) (119905 minus 120591119896minus
119888119896minus 1
119888119896
120590)
(28)
Similarly we have
1199061015840
(119905) le minus[
[
119901 (119905) 119903 (119905) minus
[1199031015840(119905)]
2
4120583119903 (119905) (119905 minus 120591119896minus ((119888119896minus 1) 119888
119896) 120590)
]
]
(29)
Thus we obtain
1199061015840
(119905) le minus [119901 (119905) 119903 (119905) minus 119860 (119905 120591119896(119905)
)] for 119905 isin (120591119896 120591119896+1
]
(30)
On the other hand
119906 (120591+
119896) =
119903 (120591119896) 11990910158401015840(120591+
119896)
120593 (119909 (120591119896minus 120590))
le 119888119896119906 (120591119896) (31)
Observing that 120593(119906) ge 120583119906 we have
119906 (120591+
119896+ 120590) =
119903 (120591119896+ 120590) 119909
10158401015840(120591119896+ 120590)
120593 (119909 (120591+
119896))
le
119903 (120591119896+ 120590) 119909
10158401015840(120591119896+ 120590)
120593 (119886119896119909 (120591119896))
le
119906 (120591119896+ 120590)
120583119886119896
(32)
Applying Lemma 1 it follows from (30) (31) and (32) that
119906 (119905) le 119906 (120591+
119895) prod
120591119895lt120591119896lt119905
119888119896
prod
120591119895lt120591119896+120590lt119905
1
120583119886119896
minus int
119905
120591119895
[119901 (119904) 119903 (119904) minus 119860 (119904 120591119896(119904)
)]
sdot prod
119904lt120591119896lt119905
119888119896
prod
119904lt120591119896+120590lt119905
1
120583119886119896
d119904
le prod
120591119895lt120591119896lt119905
119888119896
prod
120591119895lt120591119896+120590lt119905
1
120583119886119896
times (119906 (120591+
119895) minus 120583int
119905
120591119895
[119901 (119904) 119903 (119904) minus 119860 (119904 120591119896(119904)
)]
sdot prod
120591119895lt120591119896lt119904
1
119888119896
prod
120591119895lt120591119896+120590lt119904
119886119896d119904)
(33)
Abstract and Applied Analysis 5
which yields 119906(119905) lt 0 for all large 119905 This is contrary to 119906(119905) gt
0 and so case (i) in Lemma 2 is not possibleIf 119909(119905) satisfies the case (ii) in Lemma 2 that is 11990910158401015840(120591+
119896) gt
0 11990910158401015840(119905) gt 0 and 1199091015840(120591+
119896) lt 0 1199091015840(119905) lt 0 which proves that the
solution 119909(119905) is positive and decreasing Integrating (1) from119904 to 119905 (119905 ge 119904 ge 119879) we obtain
11990910158401015840
(119905) minus sum
119904lt120591119896lt119905
(119888119896minus 1) 119909
10158401015840(120591119896) minus 11990910158401015840
(119904)
+ int
119905
119904
119901 (120579) 120593 (119909 (120579 minus 120590)) d120579 le 0
(34)
Noting 119888119896le 1 and 119909
10158401015840(119905) gt 0 then it holds that
11990910158401015840
(119905) minus 11990910158401015840
(119904) + int
119905
119904
119901 (120579) 120593 (119909 (120579 minus 120590)) d120579 le 0 (35)
which leads to
minus11990910158401015840
(119904) + int
119905
119904
119901 (120579) 120593 (119909 (120579 minus 120590)) d120579 le 0 (36)
and hence
minus11990910158401015840
(119904) + int
infin
119904
119901 (120579) 120593 (119909 (120579 minus 120590)) d120579 le 0 (37)
Integrating the above inequality again from 119906 to 119905 (119905 ge 119906 ge 119879)one has
minus 1199091015840
(119905) + sum
119904lt120591119896lt119905
(119887119896minus 1) 119909
1015840(120591119896) + 1199091015840
(119906)
+ int
119905
119906
int
infin
119904
119901 (120579) 120593 (119909 (120579 minus 120590)) d120579 d119904 le 0
(38)
Using 1199091015840(119905) lt 0 and 119887
119896ge 1 we have
1199091015840
(119906) + int
infin
119906
int
infin
119904
119901 (120579) 120593 (119909 (120579 minus 120590)) d120579 d119904 le 0 (39)
Now we integrate the last inequality from 120578 to 119905 (119905 ge 120578 ge 119879)to obtain
119909 (119905) minus sum
120578lt120591119896lt119905
(119886119896minus 1) 119909 (120591
119896) minus 119909 (120578)
+ int
119905
120578
int
infin
119906
int
infin
119904
119901 (120579) 120593 (119909 (120579 minus 120590)) d120579 d119904 d119906 le 0
(40)
Since 119886119896
le 1 (119896 isin N) and 119909(119905) is decreasing then for 119905 isin
(120591119896 120591119896+1
] 119909(120591119896+1
) le 119909(119905) le 119909(120591+
119896) = 119886119896119909(120591119896) le 119909(120591
119896) 119896 isin N
Thus we get
120583119909 (120578) int
119905
120578
int
infin
119906
int
infin
119904
119901 (120579) d120579 d119904 d119906 le minus119909 (119905) + 119909 (120578) le 119909 (120578)
(41)
and then
int
119905
120578
int
infin
119906
int
infin
119904
119901 (120579) d120579 d119904 d119906 le
1
120583
(42)
which contradicts the condition (19) The proof is complete
Replace the condition (19) with (15) we may obtain thefollowing asymptotic results
Theorem 7 Assume that (5) (6) and (15) hold and 119909(119905) is asolution of (1) If there exists a positive differentiable function 119903
such that (21) hold then 119909(119905) is either oscillatory or has a finitelimit
Proof By the proof of Theorem 6 we know the case (i) inLemma 2 is not possible too since the condition (19) is notrequired to prove it So it suffices to show if there is a solutionsatisfying case (ii) in Lemma 2 that is if
11990910158401015840(120591+
119896) gt 0 119909
10158401015840
(119905) gt 0 1199091015840(120591+
119896) lt 0 119909
1015840
(119905) lt 0 (43)
with 119905 isin (120591119896 120591119896+1
] and 120591119896ge 119879 then lim
119905rarrinfin119909(119905) exists This
is obtained by applying Lemma 5 which leads to lim119905rarrinfin
119909(119905)
exists The proof is complete
Corollary 8 In addition to the assumption of Theorem 7assume that
lim119905rarrinfin
int
119905
1199050
prod
1199050lt120591119896lt119904
1
119888119896
119901 (119904) d119904 = infin (44)
Then solution 119909(119905) of (1) either oscillates or satisfieslim119905rarrinfin
119909(119905) = 0
Proof By the proof ofTheorem 7 lim119905rarrinfin
119909(119905) exists and wedefine it by lim
119905rarrinfin119909(119905) = 120574 ge 0 We now show 120574 = 0 If not
then 120574 gt 0 So lim119905rarrinfin
120593(119909(119905 minus 120590)) = 120593(120574) = 120581 gt 0 Hencethere exists 120591
119895gt 119879 such that 120593(119909(119905minus120590)) gt 1205812 for 119905 gt 120591
119895Then
119909101584010158401015840
(119905) = minus119891 (119905 119909 (119905) 119909 (119905 minus 120590))
le minus119901 (119905) 120593 (119909 (119905 minus 120590)) le minus
120581
2
119901 (119905) 119905 ge 120591119895
(45)
and note that 11990910158401015840(120591+119896) le 11988811989611990910158401015840(120591119896) since 11990910158401015840(119905) gt 0 which imply
that
11990910158401015840
(119905) le 11990910158401015840(120591+
119895) prod
120591119895lt120591119896lt119905
119888119896minus
120581
2
int
119905
120591119895
prod
119904lt120591119896lt119905
119888119896119901 (119904) d119904
le prod
120591119895lt120591119896lt119905
119888119896[
[
11990910158401015840(120591+
119895) minus
120581
2
int
119905
120591119895
prod
120591119895lt120591119896lt119904
1
119888119896
119901 (119904) d119904]
]
(46)
Thus in virtue of (44) it holds that 11990910158401015840(119905) lt 0 and contradicts11990910158401015840(119905) gt 0 for 119905 large enough the proof is complete
Remark 9 Theorem 6 and Corollary 8 extend the results in[11 Theorem 31] and [9 Corollary 1] respectively In factwhen 119886
119896= 119887119896
= 119888119896
= 1 for 119896 isin N which implies that theimpulses in (1) disappear In such a case (5) and (6) holdnaturally and (21) and (44) are reduced to
lim119905rarrinfin
int
119905
1199050
[
[
119901 (119904) 119903 (119904) minus
[1199031015840(119904)]
2
4120583119903 (119904) (119904 minus 119879)
]
]
d119904 = infin
lim119905rarrinfin
int
119905
1199050
119901 (119904) d119904 = infin
(47)
6 Abstract and Applied Analysis
which are similar to those in [11 Theorem 31] and [9Corollary 1] respectively
Next we present some new oscillation results for (1) byusing an integral averaging condition of Kamenevrsquos type
Theorem 10 Assume (5) (6) and (19) hold Furthermore119886119896
le 1120583 le 1 119887119896
ge 1 and 119888119896
le 1 119896 isin N If there exists apositive differentiable function 119903 such that
lim sup119905rarrinfin
1
119905119898
int
119905
119879
(119905 minus 119904)119898[119901 (119904) 119903 (119904) minus 119860 (119904 120591
119896(119904))] d119904 = infin
(48)
where 119860(119904 120591119896(119904)
) is defined by (21) and 119898 ge 1 Then everysolution of (1) is oscillatory
Proof We choose 119879 large enough such that Lemma 2 holdsBy Lemma 2 there are two possible cases First if the case (i)holds proceeding as in the proof of Theorem 6 we will endup with (32) By (30) we have
119901 (119905) 119903 (119905) minus 119860 (119905 120591119896(119905)
) le minus1199061015840
(119905) 119905 ge 119879 119905 = 120591119896 (49)
If 119905 isin (120591119896+ 120590 120591119896+1
] sub (120591119896 120591119896+1
] for 120591119895ge 119879 we obtain
int
119905
120591119895
(119905 minus 119904)119898[119901 (119904) 119903 (119904) minus 119860 (119904 120591
119896(119904))] d119904
le minusint
119905
120591119895
(119905 minus 119904)1198981199061015840
(119904) d119904(50)
An integration by parts of the right-hand side leads to
int
119905
120591119895
(119905 minus 119904)1198981199061015840
(119904) d119904
= (int
120591119895+120590
120591119895
+int
120591119895+1
120591119895+120590
+ sdot sdot sdot + int
120591119896+120590
120591119896
+int
119905
120591119896+120590
) (119905 minus 119904)1198981199061015840
(119904) d119904
= int
119905
120591119895
119906 (119904)
119898
(119905 minus 119904)119898minus1d119904
+
119896
sum
119894=119895
[119905 minus (120591119894+ 120590)]119898
[119906 (120591119894+ 120590) minus 119906 (120591
+
119894+ 120590)]
+
119896
sum
119894=119895+1
(119905 minus 120591119894)119898
[119906 (120591119894) minus 119906 (120591
+
119894)] minus (119905 minus 120591
119895)
119898
119906 (120591+
119895)
(51)
Take into account (31) (32) 119886119896le 1120583 and 119888
119896le 1 we have
int
119905
120591119895
(119905 minus 119904)1198981199061015840
(119904) d119904
ge
119896
sum
119894=119895+1
(119905 minus 120591119894)119898
(1 minus 119888119894) 119906 (120591119894) minus (119905 minus 120591
119895)
119898
119906 (120591+
119895)
ge minus(119905 minus 120591119895)
119898
119906 (120591+
119895)
(52)
If 119905 isin (120591119896 120591119896+ 120590] similarly we also get
int
119905
120591119895
(119905 minus 119904)1198981199061015840
(119904) d119904 ge minus(119905 minus 120591119895)
119898
119906 (120591+
119895) (53)
So it yields
int
119905
120591119895
(119905 minus 119904)119898[119901 (119904) 119903 (119904) minus 119860 (119904 120591
119896(119904))] d119904 le (119905 minus 120591
119895)
119898
119906 (120591+
119895)
(54)
which follows that1
119905119898
int
119905
120591119895
(119905 minus 119904)119898[119901 (119904) 119903 (119904) minus 119860 (119904 120591
119896(119904))] d119904
le (
119905 minus 120591119895
119905
)
119898
119906 (120591+
119895)
(55)
Hence
lim sup119905rarrinfin
1
119905119898
int
119905
120591119895
(119905 minus 119904)119898[119901 (119904) 119903 (119904) minus 119860 (119904 120591
119896(119904))] d119904 le 119906 (120591
+
119895)
(56)
which is a contradiction of (48)If case (ii) holds then as a manner with case (ii) in
Theorem 6 it is not possible too The proof is complete
Corollary 11 Assume (19) holds and 119886119896
= 119887119896
= 119888119896
= 1 for119896 isin N If there exists a positive differential function 119903 such that
lim sup119905rarrinfin
1
119905119898
int
119905
119879
(119905 minus 119904)119898[
[
119901 (119904) 119903 (119904) minus
[1199031015840(119904)]
2
4120583119903 (119904) (119904 minus 119879)
]
]
d119904 = infin
(57)
where 119879 is large enough such that Lemma 2 holds Then everysolution of (1) is oscillatory
Remark 12 Corollary 11 is an extension of [11 Theorem 32]into impulsive case Especially let 119903(119905) equiv 1 in (48) it reducesto
lim sup119905rarrinfin
1
119905119898
int
119905
119879
(119905 minus 119904)119898119901 (119904) d119904 = infin (58)
naturally which can be considered as the extension ofKamenev-type oscillation criteria for third-order impulsivedifferential equations with delay (see [8 14 15])
4 Examples
Example 13 Consider the third-order impulsive differentialequation with delay
119909101584010158401015840
(119905) + (1 + 1205721199092
(119905)) 119909 (119905 minus 120590) = 0 119905 gt 1199050 119905 = 120591
119896
119909 (120591+
119896) = 119886119896119909 (120591119896) 119909
1015840(120591+
119896) = 1198871198961199091015840(120591119896)
11990910158401015840(120591+
119896) = 11988811989611990910158401015840(120591119896) 119896 isin N
(59)
where 120590 gt 0 120572 ge 0 are constants 120591119896minus 120591119896minus1
gt 120590 for any 119896 isin N
Abstract and Applied Analysis 7
When 119886119896
= 119887119896
= 119888119896
= 1 for any 119896 isin N the impulses in(59) disappear by [16 Theorem 4] (59) is nonoscillatory if120590 lt e3 and 120572 = 0 However we may change its oscillation byproper impulsive control In fact let 120590 lt e3 and 120572 = 0 and120591119896= 1199050+ 119896120590 (119896 isin N) in (59) choose 120593(119909) = 119909 119901(119905) equiv 1 and
119903(119905) = 1 a simple calculation leads to
int
120591119899
1199050
prod
1199050lt120591119896lt119904
1
119888119896
prod
1199050lt120591119896+120590lt119904
119886119896119901 (119904) d119904
= [int
1205911
1199050
+(int
1205911+120590
1205911
+int
1205912
1205911+120590
) + + (int
120591119899minus1+120590
120591119899minus1
+int
120591119899
120591119899minus1+120590
)]
sdot prod
1199050lt120591119896lt119904
1
119888119896
prod
1199050lt120591119896+120590lt119904
119886119896d119904
= (1205911minus 1199050) + [
120590
1198881
+
1198861
1198881
(1205912minus 1205911minus 120590)] + sdot sdot sdot +
11988611198862sdot sdot sdot 119886119899minus2
120590
11988811198882sdot sdot sdot 119888119899minus1
+
11988611198862sdot sdot sdot 119886119899minus1
11988811198882sdot sdot sdot 119888119899minus1
(120591119899minus 120591119899minus1
minus 120590)
= 120590(1 +
1
1198881
+
1
1198882
+ sdot sdot sdot +
1
119888119899minus1
)
(60)
Then let
119886119896= 119888119896=
119896
119896 + 1
119887119896= 1 119896 isin N (61)
Obviously (5) (6) and (19) hold and
int
120591119899
1199050
prod
1199050lt120591119896lt119904
1
119888119896
prod
1199050lt120591119896+120590lt119904
119886119896119901 (119904) d119904
= 120590 (1 +
2
1
+
3
2
+ sdot sdot sdot +
119899
119899 minus 1
) 997888rarr infin (119899 997888rarr infin)
(62)
Thus (21) is also satisfied By Theorem 6 every solution of(59) is oscillatory
If we let
119886119896= 1 +
1
1198962 119887119896= 119888119896=
119896 + 1
119896
119896 isin N (63)
In this case it is easily to verify (5) (6) (15) (44) and(21) hold By Corollary 8 every solution of (59) is eitheroscillatory or tends to zero
Remark 14 It is easy to verify that in [7 Theorems 1 2 and3] cannot be applied to (59) On the other hand Theorem 7is not applicable for the condition (61) sincesuminfin
119899=1|119886119896minus1| does
not convergence
Example 15 Consider the third-order impulsive differentialequation with delay
119909101584010158401015840
(119905) + e119905 cosh (|119909 (119905)|120572minus1
119909 (119905)) 119909 (119905 minus 1) = 0
119905 ge 0 119905 = 2119896
119909 (120591+
119896) = 119886119896119909 (120591119896) 119909
1015840(120591+
119896) = 1198871198961199091015840(120591119896)
11990910158401015840(120591+
119896) = 11988811989611990910158401015840(120591119896) 120591
119896= 2119896 119896 isin N
(64)
where 120572 gt 0 119886119896= 1 119887119896= 119888119896= 119896(119896 + 1) 119896 isin N
Let 120593(119909) = 119909 119901(119905) = e119905 it is easy to verify that (5) (6)and (19) hold Choose 119903(119905) equiv 1 and119898 = 1 we have
1
119905
int
119905
119879
(119905 minus 119904) e119904d119904 =
e119905
119905
+ (
119879 minus 1
119905
minus 1) e119879 rarr infin (119905 rarr infin)
(65)
Acknowledgments
Theauthor is very grateful to ProfessorH Saker who presentsthe references [8ndash12 17] and gives many helpful suggestionswhich leads to an improvement of this paper The workis supported in part by the NSF of Guangdong province(S2012010010034)
References
[1] D D Bainov and P S Simeonov Impulsive Differential Equa-tions Periodic Solutions and Applications Longman Scientificand Technical 1993
[2] G Ballinger and X Liu ldquoExistence uniqueness and bounded-ness results for impulsive delay differential equationsrdquo Applica-ble Analysis vol 74 no 1-2 pp 71ndash93 2000
[3] V Lakshmikantham D D Baınov and P S Simeonov Theoryof Impulsive Differential Equations vol 6 World ScientificPublishing New Jersey NJ USA 1989
[4] A M Samoılenko and N A Perestyuk Impulsive DifferentialEquations vol 14 World Scientific Publishing New Jersey NJUSA 1995
[5] Y S Chen andW Z Feng ldquoOscillations of second order nonlin-ear ODE with impulsesrdquo Journal of Mathematical Analysis andApplications vol 210 no 1 pp 150ndash169 1997
[6] R P Agarwal F Karakoc and A Zafer ldquoA survey on oscilla-tion of impulsive ordinary differential equationsrdquo Advances inDifferential Equations vol 2010 pp 1ndash52 2010
[7] W-H Mao and A-HWan ldquoOscillatory and asymptotic behav-ior of solutions for nonlinear impulsive delay differential equa-tionsrdquo Acta Mathematicae Applicatae Sinica vol 22 EnglishSeries no 3 pp 387ndash396 2006
[8] L Erbe A Peterson and S H Saker ldquoAsymptotic behavior ofsolutions of a third-order nonlinear dynamic equation on timescalesrdquo Journal of Computational and Applied Mathematics vol181 no 1 pp 92ndash102 2005
[9] L Erbe A Peterson and S H Saker ldquoOscillation and asymp-totic behavior of a third-order nonlinear dynamic equationrdquoThe Canadian Applied Mathematics Quarterly vol 14 no 2 pp124ndash147 2006
8 Abstract and Applied Analysis
[10] L Erbe A Peterson and S H Saker ldquoHille and Nehari typecriteria for third-order dynamic equationsrdquo Journal of Mathe-matical Analysis and Applications vol 329 no 1 pp 112ndash1312007
[11] S H Saker ldquoOscillation criteria of third-order nonlinear delaydifferential equationsrdquo Mathematica Slovaca vol 56 no 4 pp433ndash450 2006
[12] S H Saker and J Dzurina ldquoOn the oscillation of certain classof third-order nonlinear delay differential equationsrdquo Mathe-matica Bohemica vol 135 no 3 pp 225ndash237 2010
[13] J Yu and J Yan ldquoPositive solutions and asymptotic behavior ofdelay differential equations with nonlinear impulsesrdquo Journal ofMathematical Analysis and Applications vol 207 no 2 pp 388ndash396 1997
[14] T Candan and R S Dahiya ldquoOscillation of third order func-tional differential equations with delayrdquo in Proceedings of the5th Mississippi State Conference on Differential Equations andComputational Simulations vol 10 p 7988 2003
[15] I V Kamenev ldquoAn integral criterion for oscillation of linear dif-ferential equations of second orderrdquo Matematicheskie Zametkivol 23 pp 136ndash138 1978
[16] G Ladas Y G Sficas and I P Stavroulakis ldquoNecessary and suf-ficient conditions for oscillations of higher order delay differ-ential equationsrdquo Transactions of the American MathematicalSociety vol 285 no 1 pp 81ndash90 1984
[17] B Baculıkova E M Elabbasy S H Saker and J DzurinaldquoOscillation criteria for third-order nonlinear differential equa-tionsrdquoMathematica Slovaca vol 58 no 2 pp 201ndash220 2008
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Stochastic AnalysisInternational Journal of
Abstract and Applied Analysis 5
which yields 119906(119905) lt 0 for all large 119905 This is contrary to 119906(119905) gt
0 and so case (i) in Lemma 2 is not possibleIf 119909(119905) satisfies the case (ii) in Lemma 2 that is 11990910158401015840(120591+
119896) gt
0 11990910158401015840(119905) gt 0 and 1199091015840(120591+
119896) lt 0 1199091015840(119905) lt 0 which proves that the
solution 119909(119905) is positive and decreasing Integrating (1) from119904 to 119905 (119905 ge 119904 ge 119879) we obtain
11990910158401015840
(119905) minus sum
119904lt120591119896lt119905
(119888119896minus 1) 119909
10158401015840(120591119896) minus 11990910158401015840
(119904)
+ int
119905
119904
119901 (120579) 120593 (119909 (120579 minus 120590)) d120579 le 0
(34)
Noting 119888119896le 1 and 119909
10158401015840(119905) gt 0 then it holds that
11990910158401015840
(119905) minus 11990910158401015840
(119904) + int
119905
119904
119901 (120579) 120593 (119909 (120579 minus 120590)) d120579 le 0 (35)
which leads to
minus11990910158401015840
(119904) + int
119905
119904
119901 (120579) 120593 (119909 (120579 minus 120590)) d120579 le 0 (36)
and hence
minus11990910158401015840
(119904) + int
infin
119904
119901 (120579) 120593 (119909 (120579 minus 120590)) d120579 le 0 (37)
Integrating the above inequality again from 119906 to 119905 (119905 ge 119906 ge 119879)one has
minus 1199091015840
(119905) + sum
119904lt120591119896lt119905
(119887119896minus 1) 119909
1015840(120591119896) + 1199091015840
(119906)
+ int
119905
119906
int
infin
119904
119901 (120579) 120593 (119909 (120579 minus 120590)) d120579 d119904 le 0
(38)
Using 1199091015840(119905) lt 0 and 119887
119896ge 1 we have
1199091015840
(119906) + int
infin
119906
int
infin
119904
119901 (120579) 120593 (119909 (120579 minus 120590)) d120579 d119904 le 0 (39)
Now we integrate the last inequality from 120578 to 119905 (119905 ge 120578 ge 119879)to obtain
119909 (119905) minus sum
120578lt120591119896lt119905
(119886119896minus 1) 119909 (120591
119896) minus 119909 (120578)
+ int
119905
120578
int
infin
119906
int
infin
119904
119901 (120579) 120593 (119909 (120579 minus 120590)) d120579 d119904 d119906 le 0
(40)
Since 119886119896
le 1 (119896 isin N) and 119909(119905) is decreasing then for 119905 isin
(120591119896 120591119896+1
] 119909(120591119896+1
) le 119909(119905) le 119909(120591+
119896) = 119886119896119909(120591119896) le 119909(120591
119896) 119896 isin N
Thus we get
120583119909 (120578) int
119905
120578
int
infin
119906
int
infin
119904
119901 (120579) d120579 d119904 d119906 le minus119909 (119905) + 119909 (120578) le 119909 (120578)
(41)
and then
int
119905
120578
int
infin
119906
int
infin
119904
119901 (120579) d120579 d119904 d119906 le
1
120583
(42)
which contradicts the condition (19) The proof is complete
Replace the condition (19) with (15) we may obtain thefollowing asymptotic results
Theorem 7 Assume that (5) (6) and (15) hold and 119909(119905) is asolution of (1) If there exists a positive differentiable function 119903
such that (21) hold then 119909(119905) is either oscillatory or has a finitelimit
Proof By the proof of Theorem 6 we know the case (i) inLemma 2 is not possible too since the condition (19) is notrequired to prove it So it suffices to show if there is a solutionsatisfying case (ii) in Lemma 2 that is if
11990910158401015840(120591+
119896) gt 0 119909
10158401015840
(119905) gt 0 1199091015840(120591+
119896) lt 0 119909
1015840
(119905) lt 0 (43)
with 119905 isin (120591119896 120591119896+1
] and 120591119896ge 119879 then lim
119905rarrinfin119909(119905) exists This
is obtained by applying Lemma 5 which leads to lim119905rarrinfin
119909(119905)
exists The proof is complete
Corollary 8 In addition to the assumption of Theorem 7assume that
lim119905rarrinfin
int
119905
1199050
prod
1199050lt120591119896lt119904
1
119888119896
119901 (119904) d119904 = infin (44)
Then solution 119909(119905) of (1) either oscillates or satisfieslim119905rarrinfin
119909(119905) = 0
Proof By the proof ofTheorem 7 lim119905rarrinfin
119909(119905) exists and wedefine it by lim
119905rarrinfin119909(119905) = 120574 ge 0 We now show 120574 = 0 If not
then 120574 gt 0 So lim119905rarrinfin
120593(119909(119905 minus 120590)) = 120593(120574) = 120581 gt 0 Hencethere exists 120591
119895gt 119879 such that 120593(119909(119905minus120590)) gt 1205812 for 119905 gt 120591
119895Then
119909101584010158401015840
(119905) = minus119891 (119905 119909 (119905) 119909 (119905 minus 120590))
le minus119901 (119905) 120593 (119909 (119905 minus 120590)) le minus
120581
2
119901 (119905) 119905 ge 120591119895
(45)
and note that 11990910158401015840(120591+119896) le 11988811989611990910158401015840(120591119896) since 11990910158401015840(119905) gt 0 which imply
that
11990910158401015840
(119905) le 11990910158401015840(120591+
119895) prod
120591119895lt120591119896lt119905
119888119896minus
120581
2
int
119905
120591119895
prod
119904lt120591119896lt119905
119888119896119901 (119904) d119904
le prod
120591119895lt120591119896lt119905
119888119896[
[
11990910158401015840(120591+
119895) minus
120581
2
int
119905
120591119895
prod
120591119895lt120591119896lt119904
1
119888119896
119901 (119904) d119904]
]
(46)
Thus in virtue of (44) it holds that 11990910158401015840(119905) lt 0 and contradicts11990910158401015840(119905) gt 0 for 119905 large enough the proof is complete
Remark 9 Theorem 6 and Corollary 8 extend the results in[11 Theorem 31] and [9 Corollary 1] respectively In factwhen 119886
119896= 119887119896
= 119888119896
= 1 for 119896 isin N which implies that theimpulses in (1) disappear In such a case (5) and (6) holdnaturally and (21) and (44) are reduced to
lim119905rarrinfin
int
119905
1199050
[
[
119901 (119904) 119903 (119904) minus
[1199031015840(119904)]
2
4120583119903 (119904) (119904 minus 119879)
]
]
d119904 = infin
lim119905rarrinfin
int
119905
1199050
119901 (119904) d119904 = infin
(47)
6 Abstract and Applied Analysis
which are similar to those in [11 Theorem 31] and [9Corollary 1] respectively
Next we present some new oscillation results for (1) byusing an integral averaging condition of Kamenevrsquos type
Theorem 10 Assume (5) (6) and (19) hold Furthermore119886119896
le 1120583 le 1 119887119896
ge 1 and 119888119896
le 1 119896 isin N If there exists apositive differentiable function 119903 such that
lim sup119905rarrinfin
1
119905119898
int
119905
119879
(119905 minus 119904)119898[119901 (119904) 119903 (119904) minus 119860 (119904 120591
119896(119904))] d119904 = infin
(48)
where 119860(119904 120591119896(119904)
) is defined by (21) and 119898 ge 1 Then everysolution of (1) is oscillatory
Proof We choose 119879 large enough such that Lemma 2 holdsBy Lemma 2 there are two possible cases First if the case (i)holds proceeding as in the proof of Theorem 6 we will endup with (32) By (30) we have
119901 (119905) 119903 (119905) minus 119860 (119905 120591119896(119905)
) le minus1199061015840
(119905) 119905 ge 119879 119905 = 120591119896 (49)
If 119905 isin (120591119896+ 120590 120591119896+1
] sub (120591119896 120591119896+1
] for 120591119895ge 119879 we obtain
int
119905
120591119895
(119905 minus 119904)119898[119901 (119904) 119903 (119904) minus 119860 (119904 120591
119896(119904))] d119904
le minusint
119905
120591119895
(119905 minus 119904)1198981199061015840
(119904) d119904(50)
An integration by parts of the right-hand side leads to
int
119905
120591119895
(119905 minus 119904)1198981199061015840
(119904) d119904
= (int
120591119895+120590
120591119895
+int
120591119895+1
120591119895+120590
+ sdot sdot sdot + int
120591119896+120590
120591119896
+int
119905
120591119896+120590
) (119905 minus 119904)1198981199061015840
(119904) d119904
= int
119905
120591119895
119906 (119904)
119898
(119905 minus 119904)119898minus1d119904
+
119896
sum
119894=119895
[119905 minus (120591119894+ 120590)]119898
[119906 (120591119894+ 120590) minus 119906 (120591
+
119894+ 120590)]
+
119896
sum
119894=119895+1
(119905 minus 120591119894)119898
[119906 (120591119894) minus 119906 (120591
+
119894)] minus (119905 minus 120591
119895)
119898
119906 (120591+
119895)
(51)
Take into account (31) (32) 119886119896le 1120583 and 119888
119896le 1 we have
int
119905
120591119895
(119905 minus 119904)1198981199061015840
(119904) d119904
ge
119896
sum
119894=119895+1
(119905 minus 120591119894)119898
(1 minus 119888119894) 119906 (120591119894) minus (119905 minus 120591
119895)
119898
119906 (120591+
119895)
ge minus(119905 minus 120591119895)
119898
119906 (120591+
119895)
(52)
If 119905 isin (120591119896 120591119896+ 120590] similarly we also get
int
119905
120591119895
(119905 minus 119904)1198981199061015840
(119904) d119904 ge minus(119905 minus 120591119895)
119898
119906 (120591+
119895) (53)
So it yields
int
119905
120591119895
(119905 minus 119904)119898[119901 (119904) 119903 (119904) minus 119860 (119904 120591
119896(119904))] d119904 le (119905 minus 120591
119895)
119898
119906 (120591+
119895)
(54)
which follows that1
119905119898
int
119905
120591119895
(119905 minus 119904)119898[119901 (119904) 119903 (119904) minus 119860 (119904 120591
119896(119904))] d119904
le (
119905 minus 120591119895
119905
)
119898
119906 (120591+
119895)
(55)
Hence
lim sup119905rarrinfin
1
119905119898
int
119905
120591119895
(119905 minus 119904)119898[119901 (119904) 119903 (119904) minus 119860 (119904 120591
119896(119904))] d119904 le 119906 (120591
+
119895)
(56)
which is a contradiction of (48)If case (ii) holds then as a manner with case (ii) in
Theorem 6 it is not possible too The proof is complete
Corollary 11 Assume (19) holds and 119886119896
= 119887119896
= 119888119896
= 1 for119896 isin N If there exists a positive differential function 119903 such that
lim sup119905rarrinfin
1
119905119898
int
119905
119879
(119905 minus 119904)119898[
[
119901 (119904) 119903 (119904) minus
[1199031015840(119904)]
2
4120583119903 (119904) (119904 minus 119879)
]
]
d119904 = infin
(57)
where 119879 is large enough such that Lemma 2 holds Then everysolution of (1) is oscillatory
Remark 12 Corollary 11 is an extension of [11 Theorem 32]into impulsive case Especially let 119903(119905) equiv 1 in (48) it reducesto
lim sup119905rarrinfin
1
119905119898
int
119905
119879
(119905 minus 119904)119898119901 (119904) d119904 = infin (58)
naturally which can be considered as the extension ofKamenev-type oscillation criteria for third-order impulsivedifferential equations with delay (see [8 14 15])
4 Examples
Example 13 Consider the third-order impulsive differentialequation with delay
119909101584010158401015840
(119905) + (1 + 1205721199092
(119905)) 119909 (119905 minus 120590) = 0 119905 gt 1199050 119905 = 120591
119896
119909 (120591+
119896) = 119886119896119909 (120591119896) 119909
1015840(120591+
119896) = 1198871198961199091015840(120591119896)
11990910158401015840(120591+
119896) = 11988811989611990910158401015840(120591119896) 119896 isin N
(59)
where 120590 gt 0 120572 ge 0 are constants 120591119896minus 120591119896minus1
gt 120590 for any 119896 isin N
Abstract and Applied Analysis 7
When 119886119896
= 119887119896
= 119888119896
= 1 for any 119896 isin N the impulses in(59) disappear by [16 Theorem 4] (59) is nonoscillatory if120590 lt e3 and 120572 = 0 However we may change its oscillation byproper impulsive control In fact let 120590 lt e3 and 120572 = 0 and120591119896= 1199050+ 119896120590 (119896 isin N) in (59) choose 120593(119909) = 119909 119901(119905) equiv 1 and
119903(119905) = 1 a simple calculation leads to
int
120591119899
1199050
prod
1199050lt120591119896lt119904
1
119888119896
prod
1199050lt120591119896+120590lt119904
119886119896119901 (119904) d119904
= [int
1205911
1199050
+(int
1205911+120590
1205911
+int
1205912
1205911+120590
) + + (int
120591119899minus1+120590
120591119899minus1
+int
120591119899
120591119899minus1+120590
)]
sdot prod
1199050lt120591119896lt119904
1
119888119896
prod
1199050lt120591119896+120590lt119904
119886119896d119904
= (1205911minus 1199050) + [
120590
1198881
+
1198861
1198881
(1205912minus 1205911minus 120590)] + sdot sdot sdot +
11988611198862sdot sdot sdot 119886119899minus2
120590
11988811198882sdot sdot sdot 119888119899minus1
+
11988611198862sdot sdot sdot 119886119899minus1
11988811198882sdot sdot sdot 119888119899minus1
(120591119899minus 120591119899minus1
minus 120590)
= 120590(1 +
1
1198881
+
1
1198882
+ sdot sdot sdot +
1
119888119899minus1
)
(60)
Then let
119886119896= 119888119896=
119896
119896 + 1
119887119896= 1 119896 isin N (61)
Obviously (5) (6) and (19) hold and
int
120591119899
1199050
prod
1199050lt120591119896lt119904
1
119888119896
prod
1199050lt120591119896+120590lt119904
119886119896119901 (119904) d119904
= 120590 (1 +
2
1
+
3
2
+ sdot sdot sdot +
119899
119899 minus 1
) 997888rarr infin (119899 997888rarr infin)
(62)
Thus (21) is also satisfied By Theorem 6 every solution of(59) is oscillatory
If we let
119886119896= 1 +
1
1198962 119887119896= 119888119896=
119896 + 1
119896
119896 isin N (63)
In this case it is easily to verify (5) (6) (15) (44) and(21) hold By Corollary 8 every solution of (59) is eitheroscillatory or tends to zero
Remark 14 It is easy to verify that in [7 Theorems 1 2 and3] cannot be applied to (59) On the other hand Theorem 7is not applicable for the condition (61) sincesuminfin
119899=1|119886119896minus1| does
not convergence
Example 15 Consider the third-order impulsive differentialequation with delay
119909101584010158401015840
(119905) + e119905 cosh (|119909 (119905)|120572minus1
119909 (119905)) 119909 (119905 minus 1) = 0
119905 ge 0 119905 = 2119896
119909 (120591+
119896) = 119886119896119909 (120591119896) 119909
1015840(120591+
119896) = 1198871198961199091015840(120591119896)
11990910158401015840(120591+
119896) = 11988811989611990910158401015840(120591119896) 120591
119896= 2119896 119896 isin N
(64)
where 120572 gt 0 119886119896= 1 119887119896= 119888119896= 119896(119896 + 1) 119896 isin N
Let 120593(119909) = 119909 119901(119905) = e119905 it is easy to verify that (5) (6)and (19) hold Choose 119903(119905) equiv 1 and119898 = 1 we have
1
119905
int
119905
119879
(119905 minus 119904) e119904d119904 =
e119905
119905
+ (
119879 minus 1
119905
minus 1) e119879 rarr infin (119905 rarr infin)
(65)
Acknowledgments
Theauthor is very grateful to ProfessorH Saker who presentsthe references [8ndash12 17] and gives many helpful suggestionswhich leads to an improvement of this paper The workis supported in part by the NSF of Guangdong province(S2012010010034)
References
[1] D D Bainov and P S Simeonov Impulsive Differential Equa-tions Periodic Solutions and Applications Longman Scientificand Technical 1993
[2] G Ballinger and X Liu ldquoExistence uniqueness and bounded-ness results for impulsive delay differential equationsrdquo Applica-ble Analysis vol 74 no 1-2 pp 71ndash93 2000
[3] V Lakshmikantham D D Baınov and P S Simeonov Theoryof Impulsive Differential Equations vol 6 World ScientificPublishing New Jersey NJ USA 1989
[4] A M Samoılenko and N A Perestyuk Impulsive DifferentialEquations vol 14 World Scientific Publishing New Jersey NJUSA 1995
[5] Y S Chen andW Z Feng ldquoOscillations of second order nonlin-ear ODE with impulsesrdquo Journal of Mathematical Analysis andApplications vol 210 no 1 pp 150ndash169 1997
[6] R P Agarwal F Karakoc and A Zafer ldquoA survey on oscilla-tion of impulsive ordinary differential equationsrdquo Advances inDifferential Equations vol 2010 pp 1ndash52 2010
[7] W-H Mao and A-HWan ldquoOscillatory and asymptotic behav-ior of solutions for nonlinear impulsive delay differential equa-tionsrdquo Acta Mathematicae Applicatae Sinica vol 22 EnglishSeries no 3 pp 387ndash396 2006
[8] L Erbe A Peterson and S H Saker ldquoAsymptotic behavior ofsolutions of a third-order nonlinear dynamic equation on timescalesrdquo Journal of Computational and Applied Mathematics vol181 no 1 pp 92ndash102 2005
[9] L Erbe A Peterson and S H Saker ldquoOscillation and asymp-totic behavior of a third-order nonlinear dynamic equationrdquoThe Canadian Applied Mathematics Quarterly vol 14 no 2 pp124ndash147 2006
8 Abstract and Applied Analysis
[10] L Erbe A Peterson and S H Saker ldquoHille and Nehari typecriteria for third-order dynamic equationsrdquo Journal of Mathe-matical Analysis and Applications vol 329 no 1 pp 112ndash1312007
[11] S H Saker ldquoOscillation criteria of third-order nonlinear delaydifferential equationsrdquo Mathematica Slovaca vol 56 no 4 pp433ndash450 2006
[12] S H Saker and J Dzurina ldquoOn the oscillation of certain classof third-order nonlinear delay differential equationsrdquo Mathe-matica Bohemica vol 135 no 3 pp 225ndash237 2010
[13] J Yu and J Yan ldquoPositive solutions and asymptotic behavior ofdelay differential equations with nonlinear impulsesrdquo Journal ofMathematical Analysis and Applications vol 207 no 2 pp 388ndash396 1997
[14] T Candan and R S Dahiya ldquoOscillation of third order func-tional differential equations with delayrdquo in Proceedings of the5th Mississippi State Conference on Differential Equations andComputational Simulations vol 10 p 7988 2003
[15] I V Kamenev ldquoAn integral criterion for oscillation of linear dif-ferential equations of second orderrdquo Matematicheskie Zametkivol 23 pp 136ndash138 1978
[16] G Ladas Y G Sficas and I P Stavroulakis ldquoNecessary and suf-ficient conditions for oscillations of higher order delay differ-ential equationsrdquo Transactions of the American MathematicalSociety vol 285 no 1 pp 81ndash90 1984
[17] B Baculıkova E M Elabbasy S H Saker and J DzurinaldquoOscillation criteria for third-order nonlinear differential equa-tionsrdquoMathematica Slovaca vol 58 no 2 pp 201ndash220 2008
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Algebra
Discrete Dynamics in Nature and Society
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Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
6 Abstract and Applied Analysis
which are similar to those in [11 Theorem 31] and [9Corollary 1] respectively
Next we present some new oscillation results for (1) byusing an integral averaging condition of Kamenevrsquos type
Theorem 10 Assume (5) (6) and (19) hold Furthermore119886119896
le 1120583 le 1 119887119896
ge 1 and 119888119896
le 1 119896 isin N If there exists apositive differentiable function 119903 such that
lim sup119905rarrinfin
1
119905119898
int
119905
119879
(119905 minus 119904)119898[119901 (119904) 119903 (119904) minus 119860 (119904 120591
119896(119904))] d119904 = infin
(48)
where 119860(119904 120591119896(119904)
) is defined by (21) and 119898 ge 1 Then everysolution of (1) is oscillatory
Proof We choose 119879 large enough such that Lemma 2 holdsBy Lemma 2 there are two possible cases First if the case (i)holds proceeding as in the proof of Theorem 6 we will endup with (32) By (30) we have
119901 (119905) 119903 (119905) minus 119860 (119905 120591119896(119905)
) le minus1199061015840
(119905) 119905 ge 119879 119905 = 120591119896 (49)
If 119905 isin (120591119896+ 120590 120591119896+1
] sub (120591119896 120591119896+1
] for 120591119895ge 119879 we obtain
int
119905
120591119895
(119905 minus 119904)119898[119901 (119904) 119903 (119904) minus 119860 (119904 120591
119896(119904))] d119904
le minusint
119905
120591119895
(119905 minus 119904)1198981199061015840
(119904) d119904(50)
An integration by parts of the right-hand side leads to
int
119905
120591119895
(119905 minus 119904)1198981199061015840
(119904) d119904
= (int
120591119895+120590
120591119895
+int
120591119895+1
120591119895+120590
+ sdot sdot sdot + int
120591119896+120590
120591119896
+int
119905
120591119896+120590
) (119905 minus 119904)1198981199061015840
(119904) d119904
= int
119905
120591119895
119906 (119904)
119898
(119905 minus 119904)119898minus1d119904
+
119896
sum
119894=119895
[119905 minus (120591119894+ 120590)]119898
[119906 (120591119894+ 120590) minus 119906 (120591
+
119894+ 120590)]
+
119896
sum
119894=119895+1
(119905 minus 120591119894)119898
[119906 (120591119894) minus 119906 (120591
+
119894)] minus (119905 minus 120591
119895)
119898
119906 (120591+
119895)
(51)
Take into account (31) (32) 119886119896le 1120583 and 119888
119896le 1 we have
int
119905
120591119895
(119905 minus 119904)1198981199061015840
(119904) d119904
ge
119896
sum
119894=119895+1
(119905 minus 120591119894)119898
(1 minus 119888119894) 119906 (120591119894) minus (119905 minus 120591
119895)
119898
119906 (120591+
119895)
ge minus(119905 minus 120591119895)
119898
119906 (120591+
119895)
(52)
If 119905 isin (120591119896 120591119896+ 120590] similarly we also get
int
119905
120591119895
(119905 minus 119904)1198981199061015840
(119904) d119904 ge minus(119905 minus 120591119895)
119898
119906 (120591+
119895) (53)
So it yields
int
119905
120591119895
(119905 minus 119904)119898[119901 (119904) 119903 (119904) minus 119860 (119904 120591
119896(119904))] d119904 le (119905 minus 120591
119895)
119898
119906 (120591+
119895)
(54)
which follows that1
119905119898
int
119905
120591119895
(119905 minus 119904)119898[119901 (119904) 119903 (119904) minus 119860 (119904 120591
119896(119904))] d119904
le (
119905 minus 120591119895
119905
)
119898
119906 (120591+
119895)
(55)
Hence
lim sup119905rarrinfin
1
119905119898
int
119905
120591119895
(119905 minus 119904)119898[119901 (119904) 119903 (119904) minus 119860 (119904 120591
119896(119904))] d119904 le 119906 (120591
+
119895)
(56)
which is a contradiction of (48)If case (ii) holds then as a manner with case (ii) in
Theorem 6 it is not possible too The proof is complete
Corollary 11 Assume (19) holds and 119886119896
= 119887119896
= 119888119896
= 1 for119896 isin N If there exists a positive differential function 119903 such that
lim sup119905rarrinfin
1
119905119898
int
119905
119879
(119905 minus 119904)119898[
[
119901 (119904) 119903 (119904) minus
[1199031015840(119904)]
2
4120583119903 (119904) (119904 minus 119879)
]
]
d119904 = infin
(57)
where 119879 is large enough such that Lemma 2 holds Then everysolution of (1) is oscillatory
Remark 12 Corollary 11 is an extension of [11 Theorem 32]into impulsive case Especially let 119903(119905) equiv 1 in (48) it reducesto
lim sup119905rarrinfin
1
119905119898
int
119905
119879
(119905 minus 119904)119898119901 (119904) d119904 = infin (58)
naturally which can be considered as the extension ofKamenev-type oscillation criteria for third-order impulsivedifferential equations with delay (see [8 14 15])
4 Examples
Example 13 Consider the third-order impulsive differentialequation with delay
119909101584010158401015840
(119905) + (1 + 1205721199092
(119905)) 119909 (119905 minus 120590) = 0 119905 gt 1199050 119905 = 120591
119896
119909 (120591+
119896) = 119886119896119909 (120591119896) 119909
1015840(120591+
119896) = 1198871198961199091015840(120591119896)
11990910158401015840(120591+
119896) = 11988811989611990910158401015840(120591119896) 119896 isin N
(59)
where 120590 gt 0 120572 ge 0 are constants 120591119896minus 120591119896minus1
gt 120590 for any 119896 isin N
Abstract and Applied Analysis 7
When 119886119896
= 119887119896
= 119888119896
= 1 for any 119896 isin N the impulses in(59) disappear by [16 Theorem 4] (59) is nonoscillatory if120590 lt e3 and 120572 = 0 However we may change its oscillation byproper impulsive control In fact let 120590 lt e3 and 120572 = 0 and120591119896= 1199050+ 119896120590 (119896 isin N) in (59) choose 120593(119909) = 119909 119901(119905) equiv 1 and
119903(119905) = 1 a simple calculation leads to
int
120591119899
1199050
prod
1199050lt120591119896lt119904
1
119888119896
prod
1199050lt120591119896+120590lt119904
119886119896119901 (119904) d119904
= [int
1205911
1199050
+(int
1205911+120590
1205911
+int
1205912
1205911+120590
) + + (int
120591119899minus1+120590
120591119899minus1
+int
120591119899
120591119899minus1+120590
)]
sdot prod
1199050lt120591119896lt119904
1
119888119896
prod
1199050lt120591119896+120590lt119904
119886119896d119904
= (1205911minus 1199050) + [
120590
1198881
+
1198861
1198881
(1205912minus 1205911minus 120590)] + sdot sdot sdot +
11988611198862sdot sdot sdot 119886119899minus2
120590
11988811198882sdot sdot sdot 119888119899minus1
+
11988611198862sdot sdot sdot 119886119899minus1
11988811198882sdot sdot sdot 119888119899minus1
(120591119899minus 120591119899minus1
minus 120590)
= 120590(1 +
1
1198881
+
1
1198882
+ sdot sdot sdot +
1
119888119899minus1
)
(60)
Then let
119886119896= 119888119896=
119896
119896 + 1
119887119896= 1 119896 isin N (61)
Obviously (5) (6) and (19) hold and
int
120591119899
1199050
prod
1199050lt120591119896lt119904
1
119888119896
prod
1199050lt120591119896+120590lt119904
119886119896119901 (119904) d119904
= 120590 (1 +
2
1
+
3
2
+ sdot sdot sdot +
119899
119899 minus 1
) 997888rarr infin (119899 997888rarr infin)
(62)
Thus (21) is also satisfied By Theorem 6 every solution of(59) is oscillatory
If we let
119886119896= 1 +
1
1198962 119887119896= 119888119896=
119896 + 1
119896
119896 isin N (63)
In this case it is easily to verify (5) (6) (15) (44) and(21) hold By Corollary 8 every solution of (59) is eitheroscillatory or tends to zero
Remark 14 It is easy to verify that in [7 Theorems 1 2 and3] cannot be applied to (59) On the other hand Theorem 7is not applicable for the condition (61) sincesuminfin
119899=1|119886119896minus1| does
not convergence
Example 15 Consider the third-order impulsive differentialequation with delay
119909101584010158401015840
(119905) + e119905 cosh (|119909 (119905)|120572minus1
119909 (119905)) 119909 (119905 minus 1) = 0
119905 ge 0 119905 = 2119896
119909 (120591+
119896) = 119886119896119909 (120591119896) 119909
1015840(120591+
119896) = 1198871198961199091015840(120591119896)
11990910158401015840(120591+
119896) = 11988811989611990910158401015840(120591119896) 120591
119896= 2119896 119896 isin N
(64)
where 120572 gt 0 119886119896= 1 119887119896= 119888119896= 119896(119896 + 1) 119896 isin N
Let 120593(119909) = 119909 119901(119905) = e119905 it is easy to verify that (5) (6)and (19) hold Choose 119903(119905) equiv 1 and119898 = 1 we have
1
119905
int
119905
119879
(119905 minus 119904) e119904d119904 =
e119905
119905
+ (
119879 minus 1
119905
minus 1) e119879 rarr infin (119905 rarr infin)
(65)
Acknowledgments
Theauthor is very grateful to ProfessorH Saker who presentsthe references [8ndash12 17] and gives many helpful suggestionswhich leads to an improvement of this paper The workis supported in part by the NSF of Guangdong province(S2012010010034)
References
[1] D D Bainov and P S Simeonov Impulsive Differential Equa-tions Periodic Solutions and Applications Longman Scientificand Technical 1993
[2] G Ballinger and X Liu ldquoExistence uniqueness and bounded-ness results for impulsive delay differential equationsrdquo Applica-ble Analysis vol 74 no 1-2 pp 71ndash93 2000
[3] V Lakshmikantham D D Baınov and P S Simeonov Theoryof Impulsive Differential Equations vol 6 World ScientificPublishing New Jersey NJ USA 1989
[4] A M Samoılenko and N A Perestyuk Impulsive DifferentialEquations vol 14 World Scientific Publishing New Jersey NJUSA 1995
[5] Y S Chen andW Z Feng ldquoOscillations of second order nonlin-ear ODE with impulsesrdquo Journal of Mathematical Analysis andApplications vol 210 no 1 pp 150ndash169 1997
[6] R P Agarwal F Karakoc and A Zafer ldquoA survey on oscilla-tion of impulsive ordinary differential equationsrdquo Advances inDifferential Equations vol 2010 pp 1ndash52 2010
[7] W-H Mao and A-HWan ldquoOscillatory and asymptotic behav-ior of solutions for nonlinear impulsive delay differential equa-tionsrdquo Acta Mathematicae Applicatae Sinica vol 22 EnglishSeries no 3 pp 387ndash396 2006
[8] L Erbe A Peterson and S H Saker ldquoAsymptotic behavior ofsolutions of a third-order nonlinear dynamic equation on timescalesrdquo Journal of Computational and Applied Mathematics vol181 no 1 pp 92ndash102 2005
[9] L Erbe A Peterson and S H Saker ldquoOscillation and asymp-totic behavior of a third-order nonlinear dynamic equationrdquoThe Canadian Applied Mathematics Quarterly vol 14 no 2 pp124ndash147 2006
8 Abstract and Applied Analysis
[10] L Erbe A Peterson and S H Saker ldquoHille and Nehari typecriteria for third-order dynamic equationsrdquo Journal of Mathe-matical Analysis and Applications vol 329 no 1 pp 112ndash1312007
[11] S H Saker ldquoOscillation criteria of third-order nonlinear delaydifferential equationsrdquo Mathematica Slovaca vol 56 no 4 pp433ndash450 2006
[12] S H Saker and J Dzurina ldquoOn the oscillation of certain classof third-order nonlinear delay differential equationsrdquo Mathe-matica Bohemica vol 135 no 3 pp 225ndash237 2010
[13] J Yu and J Yan ldquoPositive solutions and asymptotic behavior ofdelay differential equations with nonlinear impulsesrdquo Journal ofMathematical Analysis and Applications vol 207 no 2 pp 388ndash396 1997
[14] T Candan and R S Dahiya ldquoOscillation of third order func-tional differential equations with delayrdquo in Proceedings of the5th Mississippi State Conference on Differential Equations andComputational Simulations vol 10 p 7988 2003
[15] I V Kamenev ldquoAn integral criterion for oscillation of linear dif-ferential equations of second orderrdquo Matematicheskie Zametkivol 23 pp 136ndash138 1978
[16] G Ladas Y G Sficas and I P Stavroulakis ldquoNecessary and suf-ficient conditions for oscillations of higher order delay differ-ential equationsrdquo Transactions of the American MathematicalSociety vol 285 no 1 pp 81ndash90 1984
[17] B Baculıkova E M Elabbasy S H Saker and J DzurinaldquoOscillation criteria for third-order nonlinear differential equa-tionsrdquoMathematica Slovaca vol 58 no 2 pp 201ndash220 2008
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
Abstract and Applied Analysis 7
When 119886119896
= 119887119896
= 119888119896
= 1 for any 119896 isin N the impulses in(59) disappear by [16 Theorem 4] (59) is nonoscillatory if120590 lt e3 and 120572 = 0 However we may change its oscillation byproper impulsive control In fact let 120590 lt e3 and 120572 = 0 and120591119896= 1199050+ 119896120590 (119896 isin N) in (59) choose 120593(119909) = 119909 119901(119905) equiv 1 and
119903(119905) = 1 a simple calculation leads to
int
120591119899
1199050
prod
1199050lt120591119896lt119904
1
119888119896
prod
1199050lt120591119896+120590lt119904
119886119896119901 (119904) d119904
= [int
1205911
1199050
+(int
1205911+120590
1205911
+int
1205912
1205911+120590
) + + (int
120591119899minus1+120590
120591119899minus1
+int
120591119899
120591119899minus1+120590
)]
sdot prod
1199050lt120591119896lt119904
1
119888119896
prod
1199050lt120591119896+120590lt119904
119886119896d119904
= (1205911minus 1199050) + [
120590
1198881
+
1198861
1198881
(1205912minus 1205911minus 120590)] + sdot sdot sdot +
11988611198862sdot sdot sdot 119886119899minus2
120590
11988811198882sdot sdot sdot 119888119899minus1
+
11988611198862sdot sdot sdot 119886119899minus1
11988811198882sdot sdot sdot 119888119899minus1
(120591119899minus 120591119899minus1
minus 120590)
= 120590(1 +
1
1198881
+
1
1198882
+ sdot sdot sdot +
1
119888119899minus1
)
(60)
Then let
119886119896= 119888119896=
119896
119896 + 1
119887119896= 1 119896 isin N (61)
Obviously (5) (6) and (19) hold and
int
120591119899
1199050
prod
1199050lt120591119896lt119904
1
119888119896
prod
1199050lt120591119896+120590lt119904
119886119896119901 (119904) d119904
= 120590 (1 +
2
1
+
3
2
+ sdot sdot sdot +
119899
119899 minus 1
) 997888rarr infin (119899 997888rarr infin)
(62)
Thus (21) is also satisfied By Theorem 6 every solution of(59) is oscillatory
If we let
119886119896= 1 +
1
1198962 119887119896= 119888119896=
119896 + 1
119896
119896 isin N (63)
In this case it is easily to verify (5) (6) (15) (44) and(21) hold By Corollary 8 every solution of (59) is eitheroscillatory or tends to zero
Remark 14 It is easy to verify that in [7 Theorems 1 2 and3] cannot be applied to (59) On the other hand Theorem 7is not applicable for the condition (61) sincesuminfin
119899=1|119886119896minus1| does
not convergence
Example 15 Consider the third-order impulsive differentialequation with delay
119909101584010158401015840
(119905) + e119905 cosh (|119909 (119905)|120572minus1
119909 (119905)) 119909 (119905 minus 1) = 0
119905 ge 0 119905 = 2119896
119909 (120591+
119896) = 119886119896119909 (120591119896) 119909
1015840(120591+
119896) = 1198871198961199091015840(120591119896)
11990910158401015840(120591+
119896) = 11988811989611990910158401015840(120591119896) 120591
119896= 2119896 119896 isin N
(64)
where 120572 gt 0 119886119896= 1 119887119896= 119888119896= 119896(119896 + 1) 119896 isin N
Let 120593(119909) = 119909 119901(119905) = e119905 it is easy to verify that (5) (6)and (19) hold Choose 119903(119905) equiv 1 and119898 = 1 we have
1
119905
int
119905
119879
(119905 minus 119904) e119904d119904 =
e119905
119905
+ (
119879 minus 1
119905
minus 1) e119879 rarr infin (119905 rarr infin)
(65)
Acknowledgments
Theauthor is very grateful to ProfessorH Saker who presentsthe references [8ndash12 17] and gives many helpful suggestionswhich leads to an improvement of this paper The workis supported in part by the NSF of Guangdong province(S2012010010034)
References
[1] D D Bainov and P S Simeonov Impulsive Differential Equa-tions Periodic Solutions and Applications Longman Scientificand Technical 1993
[2] G Ballinger and X Liu ldquoExistence uniqueness and bounded-ness results for impulsive delay differential equationsrdquo Applica-ble Analysis vol 74 no 1-2 pp 71ndash93 2000
[3] V Lakshmikantham D D Baınov and P S Simeonov Theoryof Impulsive Differential Equations vol 6 World ScientificPublishing New Jersey NJ USA 1989
[4] A M Samoılenko and N A Perestyuk Impulsive DifferentialEquations vol 14 World Scientific Publishing New Jersey NJUSA 1995
[5] Y S Chen andW Z Feng ldquoOscillations of second order nonlin-ear ODE with impulsesrdquo Journal of Mathematical Analysis andApplications vol 210 no 1 pp 150ndash169 1997
[6] R P Agarwal F Karakoc and A Zafer ldquoA survey on oscilla-tion of impulsive ordinary differential equationsrdquo Advances inDifferential Equations vol 2010 pp 1ndash52 2010
[7] W-H Mao and A-HWan ldquoOscillatory and asymptotic behav-ior of solutions for nonlinear impulsive delay differential equa-tionsrdquo Acta Mathematicae Applicatae Sinica vol 22 EnglishSeries no 3 pp 387ndash396 2006
[8] L Erbe A Peterson and S H Saker ldquoAsymptotic behavior ofsolutions of a third-order nonlinear dynamic equation on timescalesrdquo Journal of Computational and Applied Mathematics vol181 no 1 pp 92ndash102 2005
[9] L Erbe A Peterson and S H Saker ldquoOscillation and asymp-totic behavior of a third-order nonlinear dynamic equationrdquoThe Canadian Applied Mathematics Quarterly vol 14 no 2 pp124ndash147 2006
8 Abstract and Applied Analysis
[10] L Erbe A Peterson and S H Saker ldquoHille and Nehari typecriteria for third-order dynamic equationsrdquo Journal of Mathe-matical Analysis and Applications vol 329 no 1 pp 112ndash1312007
[11] S H Saker ldquoOscillation criteria of third-order nonlinear delaydifferential equationsrdquo Mathematica Slovaca vol 56 no 4 pp433ndash450 2006
[12] S H Saker and J Dzurina ldquoOn the oscillation of certain classof third-order nonlinear delay differential equationsrdquo Mathe-matica Bohemica vol 135 no 3 pp 225ndash237 2010
[13] J Yu and J Yan ldquoPositive solutions and asymptotic behavior ofdelay differential equations with nonlinear impulsesrdquo Journal ofMathematical Analysis and Applications vol 207 no 2 pp 388ndash396 1997
[14] T Candan and R S Dahiya ldquoOscillation of third order func-tional differential equations with delayrdquo in Proceedings of the5th Mississippi State Conference on Differential Equations andComputational Simulations vol 10 p 7988 2003
[15] I V Kamenev ldquoAn integral criterion for oscillation of linear dif-ferential equations of second orderrdquo Matematicheskie Zametkivol 23 pp 136ndash138 1978
[16] G Ladas Y G Sficas and I P Stavroulakis ldquoNecessary and suf-ficient conditions for oscillations of higher order delay differ-ential equationsrdquo Transactions of the American MathematicalSociety vol 285 no 1 pp 81ndash90 1984
[17] B Baculıkova E M Elabbasy S H Saker and J DzurinaldquoOscillation criteria for third-order nonlinear differential equa-tionsrdquoMathematica Slovaca vol 58 no 2 pp 201ndash220 2008
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
8 Abstract and Applied Analysis
[10] L Erbe A Peterson and S H Saker ldquoHille and Nehari typecriteria for third-order dynamic equationsrdquo Journal of Mathe-matical Analysis and Applications vol 329 no 1 pp 112ndash1312007
[11] S H Saker ldquoOscillation criteria of third-order nonlinear delaydifferential equationsrdquo Mathematica Slovaca vol 56 no 4 pp433ndash450 2006
[12] S H Saker and J Dzurina ldquoOn the oscillation of certain classof third-order nonlinear delay differential equationsrdquo Mathe-matica Bohemica vol 135 no 3 pp 225ndash237 2010
[13] J Yu and J Yan ldquoPositive solutions and asymptotic behavior ofdelay differential equations with nonlinear impulsesrdquo Journal ofMathematical Analysis and Applications vol 207 no 2 pp 388ndash396 1997
[14] T Candan and R S Dahiya ldquoOscillation of third order func-tional differential equations with delayrdquo in Proceedings of the5th Mississippi State Conference on Differential Equations andComputational Simulations vol 10 p 7988 2003
[15] I V Kamenev ldquoAn integral criterion for oscillation of linear dif-ferential equations of second orderrdquo Matematicheskie Zametkivol 23 pp 136ndash138 1978
[16] G Ladas Y G Sficas and I P Stavroulakis ldquoNecessary and suf-ficient conditions for oscillations of higher order delay differ-ential equationsrdquo Transactions of the American MathematicalSociety vol 285 no 1 pp 81ndash90 1984
[17] B Baculıkova E M Elabbasy S H Saker and J DzurinaldquoOscillation criteria for third-order nonlinear differential equa-tionsrdquoMathematica Slovaca vol 58 no 2 pp 201ndash220 2008
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