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Research Article Periodic Solutions for Gause-Type Ratio...
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Hindawi Publishing Corporation Discrete Dynamics in Nature and Society Volume 2013, Article ID 368176, 8 pages http://dx.doi.org/10.1155/2013/368176 Research Article Periodic Solutions for Gause-Type Ratio-Dependent Predator-Prey Systems with Delays on Time Scales Xiaoquan Ding, Hongyuan Liu, and Fengye Wang School of Mathematics and Statistics, Henan University of Science and Technology, Luoyang, Henan 471023, China Correspondence should be addressed to Xiaoquan Ding; [email protected] Received 28 February 2013; Accepted 27 May 2013 Academic Editor: Qiru Wang Copyright © 2013 Xiaoquan Ding et al. is is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. is paper is devoted to periodic Gause-type ratio-dependent predator-prey systems with monotonic or nonmonotonic numerical responses on time scales. By using a continuation theorem based on coincidence degree theory, we establish easily verifiable criteria for the existence of periodic solutions. In particular, our results improve and generalize some known ones. 1. Introduction In this paper, we consider the following periodic Gause-type ratio-dependent predator-prey system with time delays on a time scale T : Δ 1 () = (, e 1 (−()) ) − (, e 1 ()− 2 () ), Δ 2 () = − () + ℎ (, e 1 (−())− 2 (−()) ). (1) Here T is a periodic time scale which has the subspace topology inherited from the standard topology on R. e symbol Δ stands for the delta-derivative which gives the ordinary derivative if T = R, and the forward difference operator if T = Z. e theory of dynamic equations on time scales, which was introduced in order to unify differential and difference equations, has recently received a lot of attention, and many results on the issue have been documented in monographs [1, 2] and papers [3–12]. In system (1), set () = exp[ 1 ()], () = exp[ 2 ()]. If T = R, then system (1) reduces to the standard Gause- type ratio-dependent predator-prey system governed by the ordinary differential equations () = () (, ( − ())) − () (, () () ), () = () [− () + ℎ (, ( − ()) ( − ()) )] , (2) where () and () stand for the population of the prey and the predator, respectively. e function (, V) is the growth rate of the prey in the absence of the predator. e function () is the death rate of the predator. e function (, V) =: V(, V), called functional response of predator to prey, describes the change in the rate of exploitation of prey by a predator as a result of a change in the prey density. e function ℎ(, V), called numerical response of predator to prey, describes the change in reproduction rate with changing prey density. If T = Z, then system (1) is reformulated as ( + 1) = () exp[ (, ( − ())) − (, () () )] , ( + 1) = () exp[− () + ℎ (, ( − ()) ( − ()) )] , (3) which is the discrete time ratio-dependent predator-prey system and is a discrete analogue of (2). We note that Ding [13], Ding and Jiang [14], Fan and Li [15], Fan et al. [16], Fan and Wang [17], Fan et al. [18], Hsu et al. [19], Kuang [20], Wang and Li [21], and Xia and Han [22] considered some special cases of system (2). Fan and Wang [23], and Xia et al. [24] discussed some special cases of system (3). Bohner et al. [4], and Shao [10] investigated some special cases of system (1). So far as we know, there is no published paper concerned system (1). e main purpose of this paper is, by using the coinci- dence degree theory developed by Gaines and Mawhin [25],
Transcript of Research Article Periodic Solutions for Gause-Type Ratio...
Hindawi Publishing CorporationDiscrete Dynamics in Nature and SocietyVolume 2013 Article ID 368176 8 pageshttpdxdoiorg1011552013368176
Research ArticlePeriodic Solutions for Gause-Type Ratio-DependentPredator-Prey Systems with Delays on Time Scales
Xiaoquan Ding Hongyuan Liu and Fengye Wang
School of Mathematics and Statistics Henan University of Science and Technology Luoyang Henan 471023 China
Correspondence should be addressed to Xiaoquan Ding xqdinghausteducn
Received 28 February 2013 Accepted 27 May 2013
Academic Editor Qiru Wang
Copyright copy 2013 Xiaoquan Ding et alThis is an open access article distributed under the Creative Commons Attribution Licensewhich permits unrestricted use distribution and reproduction in any medium provided the original work is properly cited
This paper is devoted to periodic Gause-type ratio-dependent predator-prey systems with monotonic or nonmonotonic numericalresponses on time scales By using a continuation theorem based on coincidence degree theory we establish easily verifiable criteriafor the existence of periodic solutions In particular our results improve and generalize some known ones
1 Introduction
In this paper we consider the following periodic Gause-typeratio-dependent predator-prey system with time delays on atime scale T
119906Δ
1(119905) = 119891 (119905 e1199061(119905minus120590(119905))) minus 119901 (119905 e1199061(119905)minus1199062(119905))
119906Δ
2(119905) = minus 119889 (119905) + ℎ (119905 e1199061(119905minus120591(119905))minus1199062(119905minus120591(119905)))
(1)
Here T is a periodic time scale which has the subspacetopology inherited from the standard topology on R Thesymbol Δ stands for the delta-derivative which gives theordinary derivative if T = R and the forward differenceoperator if T = Z
The theory of dynamic equations on time scales whichwas introduced in order to unify differential and differenceequations has recently received a lot of attention and manyresults on the issue have been documented in monographs[1 2] and papers [3ndash12]
In system (1) set 119909(119905) = exp[1199061(119905)] 119910(119905) = exp[119906
2(119905)]
If T = R then system (1) reduces to the standard Gause-type ratio-dependent predator-prey system governed by theordinary differential equations
1199091015840(119905) = 119909 (119905) 119891 (119905 119909 (119905 minus 120590 (119905))) minus 119910 (119905) 119892 (119905
119909 (119905)
119910 (119905)
)
1199101015840(119905) = 119910 (119905) [minus119889 (119905) + ℎ(119905
119909 (119905 minus 120591 (119905))
119910 (119905 minus 120591 (119905))
)]
(2)
where 119909(119905) and 119910(119905) stand for the population of the preyand the predator respectively The function 119891(119905 V) is thegrowth rate of the prey in the absence of the predator Thefunction 119889(119905) is the death rate of the predator The function119892(119905 V) = V119901(119905 V) called functional response of predator toprey describes the change in the rate of exploitation of preyby a predator as a result of a change in the prey densityThe function ℎ(119905 V) called numerical response of predator toprey describes the change in reproduction rate with changingprey density If T = Z then system (1) is reformulated as
119909 (119896 + 1) = 119909 (119896) exp[119891 (119896 119909 (119896 minus 120590 (119896))) minus 119901(119896
119909 (119896)
119910 (119896)
)]
119910 (119896 + 1) = 119910 (119896) exp[minus119889 (119896) + ℎ(119896
119909 (119896 minus 120591 (119896))
119910 (119896 minus 120591 (119896))
)]
(3)which is the discrete time ratio-dependent predator-preysystem and is a discrete analogue of (2)
We note that Ding [13] Ding and Jiang [14] Fan and Li[15] Fan et al [16] Fan and Wang [17] Fan et al [18] Hsu etal [19] Kuang [20] Wang and Li [21] and Xia and Han [22]considered some special cases of system (2) Fan and Wang[23] andXia et al [24] discussed some special cases of system(3) Bohner et al [4] and Shao [10] investigated some specialcases of system (1) So far as we know there is no publishedpaper concerned system (1)
The main purpose of this paper is by using the coinci-dence degree theory developed by Gaines and Mawhin [25]
2 Discrete Dynamics in Nature and Society
to derive sufficient conditions for the existence of periodicsolutions of system (1) Furthermore we will see that ourresults for the above system can be easily extended to theone with distributed or state-dependent delays Our resultsimprove or generalize some theorems in [10 13ndash18 21 23]
2 Preliminaries
In this section we briefly give some elements of the time scalecalculus recall the continuation theorem from coincidencedegree theory and state an auxiliary result that will be usedin this paper
First let us present some foundational definitions andresults from the calculus on time scales so that the paper isself-contained For more details we refer the reader to [1 2]
A time scale T is an arbitrary nonempty closed subset T ofthe real numbers R which inherits the standard topology ofR Thus the real numbers R the integers Z and the naturalnumbers N are examples of time scales while the rationalnumbersQ and the open interval (1 2) are no time scales
Let 120596 gt 0 Throughout this paper the time scale T isassumed to be 120596-periodic that is 119905 isin T implies 119905 + 120596 isin T Inparticular the time scaleT under consideration is unboundedabove and below
For 119905 isin T the forward and backward jumpoperators120590 120588
T rarr R are defined by
120590 (119905) = inf119904 isin T 119904 gt 119905 120588 (119905) = sup119904 isin T 119904 lt 119905
(4)
respectivelyIf 120590(119905) = 119905 119905 is called right-dense (otherwise right-
scattered) and if 120588(119905) = 119905 then 119905 is called left-dense(otherwise left-scattered)
A function 119891 T rarr R is said to be rd-continuous if it iscontinuous at right-dense points in T and its left-sided limitsexist (finite) at left-dense points in T The set of rd-continuousfunctions is denoted by 119862rd(T)
For 119891 T rarr R and 119905 isin T we define 119891Δ(119905) the delta-
derivative of 119891 at 119905 to be the number (provided it exists) withthe property that given any 120576 gt 0 there is a neighborhood119880
of 119905 (ie 119880 = (119905 minus 120575 119905 + 120575) cap T for some 120575 gt 0) in T such that
10038161003816100381610038161003816[119891 (120590 (119905)) minus 119891 (119904)] minus 119891
Δ(119905) [120590 (119905) minus 119904]
10038161003816100381610038161003816
le 120576 |120590 (119905) minus 119904| forall119904 isin 119880
(5)
119891 is said to be delta-differentiable if its delta-derivative existsfor all 119905 isin T The set of functions 119891 T rarr R that are delta-differentiable and whose delta-derivative are rd-continuousfunctions is denoted by 119862
1
rd(T)A function 119865 T rarr R is called a delta-antiderivative of
119891 T rarr R provided 119865Δ(119905) = 119891(119905) for all 119905 isin T Then we
define the delta integral by
int
119887
119886
119891 (119905) Δ119905 = 119865 (119887) minus 119865 (119886) forall119886 119887 isin T (6)
If 119891 is continuous then 119891 is rd-continuous and if 119891 isdelta-differentiable at 119905 then 119891 is continuous at 119905 Moreoverevery rd-continuous function on T has a delta-antiderivative
Next let us recall the continuation theorem in coinci-dence degree theory To do so we need to introduce thefollowing notation
Let 119883 119884 be real Banach spaces 119871 Dom119871 sub 119883 rarr 119884 bea linear mapping and119873 119883 rarr 119884 be a continuous mapping
Themapping 119871 is said to be a Fredholmmapping of indexzero if dim Ker 119871 = codim Im 119871 lt +infin and Im 119871 is closed in119884
If 119871 is a Fredholmmapping of index zero then there existcontinuous projectors 119875 119883 rarr 119883 and 119876 119884 rarr 119884 suchthat Im119875 = Ker 119871 Ker119876 = Im 119871 = Im(119868 minus 119876) It follows thatthe restriction 119871
119875of 119871 to Dom119871 cap Ker119875 (119868 minus 119875)119883 rarr Im 119871
is invertible Denote the inverse of 119871119875by 119870119875
The mapping 119873 is said to be 119871-compact on Ω if Ω is anopen bounded subset of 119883 119876119873(Ω) is bounded and 119870
119875(119868 minus
119876)119873 Ω rarr 119883 is compactSince Im119876 is isomorphic to Ker 119871 there exists an isomor-
phism 119869 Im119876 rarr Ker 119871Here we state the Gaines-Mawhin theorem which is a
main tool in the proof of our main result
Lemma 1 (ContinuationTheorem [25 page 40]) Let Ω sub 119883
be an open bounded set 119871 a Fredholm mapping of index zeroand 119873 119871-compact on Ω Assume
(a) for each 120582 isin (0 1) 119909 isin 120597Ω cap Dom119871 119871119909 = 120582119873119909
(b) for each 119909 isin 120597Ω cap Ker 119871119876119873119909 = 0
(c) deg(119869119876119873Ω cap Ker 119871 0) = 0
Then 119871119909 = 119873119909 has at least one solution in Ω cap Dom119871
For convenience and simplicity in the following discus-sion we always use the following notation
120581 = min[0 +infin) cap T 119868120596
= [120581 120581 + 120596] cap T
119886 =
1
120596
int
120581+120596
120581
119886 (119905) Δ119905 119860 =
1
120596
int
120581+120596
120581
|119886 (119905)| Δ119905
119887 (119909) =
1
120596
int
120581+120596
120581
119887 (119905 119909) Δ119905
119861 (119909) =
1
120596
int
120581+120596
120581
|119887 (119905 119909)| Δ119905
(7)
where 119886 isin 119862rd(T) is an 120596-periodic function 119887 T times R rarr R
is rd-continuous and 120596-periodic in its first variableIn order to achieve the prior estimation in the case of
dynamic equations on a time scale T we also require thefollowing inequality which is proved in [11 Theorem 21]
Discrete Dynamics in Nature and Society 3
Lemma 2 Let 1199051 1199052
isin 119868120596 and 119905 isin T If 120593 isin 119862
1
rd(T) is an120596-periodic real function then
120593 (119905) le 120593 (1199051) +
1
2
int
120581+120596
120581
10038161003816100381610038161003816120593Δ(119905)
10038161003816100381610038161003816Δ119905
120593 (119905) ge 120593 (1199052) minus
1
2
int
120581+120596
120581
10038161003816100381610038161003816120593Δ(119905)
10038161003816100381610038161003816Δ119905
(8)
3 Existence of Periodic Solutions
In this section we study the existence of periodic solutions ofsystem (1) For the sake of generality we make the followingfundamental assumptions for system (1)
(H1) 120590 120591 T rarr R+ are rd-continuous and 120596-periodicsuch that 119905 minus 120590(119905) 119905 minus 120591(119905) isin T for all 119905 isin T
(H2) 119889 T rarr R is rd-continuous and 120596-periodic suchthat
119889 gt 0(H3) 119891 TtimesR rarr R is rd-continuous and120596-periodic in thefirst variable and is continuously differentiable in thesecond variable such that
119891(0) gt 0 lim119909rarr+infin
119891(119909) lt
0 and (120597119891120597119909)(119905 119909) lt 0 for all 119905 isin T 119909 gt 0(H4) 119901 T times R rarr R+ is rd-continuous and 120596-periodic inthe first variable and is continuously differentiable inthe second variable
(H5) ℎ T times R rarr R+ is rd-continuous and 120596-periodic inthe first variable and is continuously differentiable inthe second variable such that ℎ(119905 0) = 0 for all 119905 isin T
(H6) 119891(0) gt 119888 and sup
119909ge0ℎ(119909) gt
119889 where 119888(119905) =
sup119909gt0
119901(119905 119909) for all 119905 isin T
From (H3) we have
1198911015840(119909) =
1
120596
int
120581+120596
120581
120597119891
120597119909
(119905 119909) Δ119905 lt 0 (9)
Thus 119891(119909) is strictly decreasing on [0 +infin) From this (H
3)
and (H6) one can easily see that each of the equations
119891 (119909) =
1
120596
int
120581+120596
120581
sup119909gt0
119901 (119905 119909) Δ119905119891 (119909) = 0 (10)
has a unique positive solution Let 1198811 1198812be their solutions
respectively then 1198811lt 1198812
31 Monotone Case In this section we study the existenceof 120596-periodic solution of system (1) when the numericalresponse function ℎ(119905 119909) satisfies the following monotonecondition
(H7) (120597ℎ120597119909)(119905 119909) gt 0 for 119905 isin T 119909 gt 0
From (H5) and (H
7) we have
ℎ1015840(119909) =
1
120596
int
120581+120596
120581
120597ℎ
120597119909
(119905 119909) Δ119905 gt 0 (11)
Then ℎ(119909) is strictly increasing on [0 +infin) From this (H
5)
and (H6) one can easily see that the equation
ℎ(119909) =119889 has a
unique positive solution Let V0be its solution
Theorem 3 Suppose that (H1)ndash(H7) holdThen system (1) has
at least one 120596-periodic solution
Proof Take
119883 = 119884 = 119906 = (1199061 (
119905) 1199062 (119905))
T| 119906119894isin 119862 (T)
119906119894 (
119905 + 120596) = 119906119894 (
119905) 119894 = 1 2
119906 =
100381710038171003817100381710038171003817
(1199061 (
119905) 1199062 (119905))
T100381710038171003817100381710038171003817
= max119905isin119868120596
10038161003816100381610038161199061 (
119905)1003816100381610038161003816+ max119905isin119868120596
10038161003816100381610038161199062 (
119905)1003816100381610038161003816
(12)
then 119883 and 119884 are Banach spaces with the norm sdot Set
119871 Dom119871 sub 119883 997888rarr 119884 119871(
1199061 (
119905)
1199062 (
119905)) = (
119906Δ
1(119905)
119906Δ
2(119905)
) (13)
where Dom119871 = 119906 = (1199061(119905) 1199062(119905))
Tisin 119883 | 119906
119894isin 1198621
rd(T) 119894 =
1 2 and
119873 119883 rarr 119884
119873(
1199061 (
119905)
1199062 (
119905)) = (
119891 (119905 e1199061(119905minus120590(119905))) minus 119901 (119905 e1199061(119905)minus1199062(119905))
minus119889 (119905) + ℎ (119905 e1199061(119905minus120591(119905))minus1199062(119905minus120591(119905))) )
(14)
With these notations system (1) can be written in the form
119871119906 = 119873119906 119906 isin 119883 (15)
Obviously Ker 119871 = R2 Im 119871 = (1199061(119905) 1199062(119905))
Tisin 119884
119894=
0 119894 = 1 2 is closed in 119884 and dimKer119871 = codimIm119871 = 2Therefore 119871 is a Fredholmmapping of index zero Now definetwo projectors 119875 119883 rarr 119883 and 119876 119884 rarr 119884 as
119875(
1199061 (
119905)
1199062 (
119905)) = 119876(
1199061 (
119905)
1199062 (
119905)) = (
1
2
) (
1199061 (
119905)
1199062 (
119905)) isin 119883 = 119884
(16)
then 119875 and 119876 are continuous projectors such that
Im119875 = Ker 119871 Ker119876 = Im 119871 = Im(119868 minus 119876) (17)
Furthermore through an easy computation we find that thegeneralized inverse 119870
119875of 119871119875has the form
119870119875
Im 119871 997888rarr Dom119871 cap Ker119875
119870119875 (
119906) = int
119905
120581
119906 (119904) Δ119904 minus
1
120596
int
120581+120596
120581
int
119905
120581
119906 (119904) Δ119904Δ119905
(18)
4 Discrete Dynamics in Nature and Society
Then 119876119873 119883 rarr 119884 and 119870119875(119868 minus 119876)119873 119883 rarr 119883 read as
119876119873119906
=
1
120596
(
int
120581+120596
120581
[119891 (119905 e1199061(119905minus120590(119905))) minus 119901 (119905 e1199061(119905)minus1199062(119905))] Δ119905
int
120581+120596
120581
[minus119889 (119905) + ℎ (119905 e1199061(119905minus120591(119905))minus1199062(119905minus120591(119905)))] Δ119905
)
119870119875 (
119868 minus 119876)119873119906
= int
119905
120581
119873119906 (119904) Δ119904 minus
1
120596
int
120581+120596
120581
int
119905
120581
119873119906 (119904) Δ119904Δ119905
minus
1
120596
(119905 minus 120581 minus
1
120596
int
120581+120596
120581
(119904 minus 120581) Δ119904)int
120581+120596
120581
119873119906 (119904) Δ119904
(19)
Clearly 119876119873 and 119870119875(119868 minus 119876)119873 are continuous By using
the Arzela-Ascoli theorem it is not difficult to prove that119870119875(119868 minus 119876)119873(Ω) is compact for any open bounded setΩ sub 119883
Moreover 119876119873(Ω) is bounded Therefore 119873 is 119871-compact onΩ with any open bounded set Ω sub 119883
In order to apply Lemma 1 we need to find an appropriateopen bounded subset in 119883 Corresponding to the operatorequation 119871119906 = 120582119873119906 120582 isin (0 1) we have
119906Δ
1(119905) = 120582 [119891 (119905 e1199061(119905minus120590(119905))) minus 119901 (119905 e1199061(119905)minus1199062(119905))]
119906Δ
2(119905) = 120582 [minus119889 (119905) + ℎ (119905 e1199061(119905minus120591(119905))minus1199062(119905minus120591(119905)))]
(20)
Suppose that (1199061(119905) 1199062(119905))
Tisin 119883 is a solution of (20) for a
certain 120582 isin (0 1) Integrating (20) on both sides from 120581 to120581 + 120596 leads to
int
120581+120596
120581
120582 [119891 (119905 e1199061(119905minus120590(119905))) minus 119901 (119905 e1199061(119905)minus1199062(119905))] Δ119905 = 0
int
120581+120596
120581
120582 [minus119889 (119905) + ℎ (119905 e1199061(119905minus120591(119905))minus1199062(119905minus120591(119905)))] Δ119905 = 0
(21)
That is
int
120581+120596
120581
119891 (119905 e1199061(119905minus120590(119905))) Δ119905 = int
120581+120596
120581
119901 (119905 e1199061(119905)minus1199062(119905)) Δ119905 (22)
int
120581+120596
120581
ℎ (119905 e1199061(119905minus120591(119905))minus1199062(119905minus120591(119905))) Δ119905 = int
120581+120596
120581
119889 (119905) Δ119905 =119889120596
(23)
From (22) we have
119891 (0) 120596 = int
120581+120596
120581
[119891 (119905 0) minus 119891 (119905 e1199061(119905minus120590(119905)))
+119901 (119905 e1199061(119905)minus1199062(119905))] Δ119905
(24)
It follows from (20) (22) (23) (24) and (H3)ndash(H5) that
int
120581+120596
120581
10038161003816100381610038161003816119906Δ
1(119905)
10038161003816100381610038161003816Δ119905
le 120582int
120581+120596
120581
10038161003816100381610038161003816119891 (119905 e1199061(119905minus120590(119905))) minus 119898 (119905 e1199061(119905)minus1199062(119905))1003816100381610038161003816
1003816Δ119905
lt int
120581+120596
120581
[119891 (119905 0) minus 119891 (119905 e1199061(119905minus120590(119905)))] + 119901 (119905 e1199061(119905)minus1199062(119905))Δ119905
+ int
120581+120596
120581
1003816100381610038161003816119891 (119905 0)
1003816100381610038161003816Δ119905 = [119865 (0) +
119891 (0)] 120596
(25)
int
120581+120596
120581
10038161003816100381610038161003816119906Δ
2(119905)
10038161003816100381610038161003816Δ119905
le 120582int
120581+120596
120581
10038161003816100381610038161003816minus119889 (119905) + ℎ (119905 e1199061(119905minus120591(119905))minus1199062(119905minus120591(119905)))1003816100381610038161003816
1003816Δ119905
lt int
120581+120596
120581
|119889 (119905)| Δ119905 + int
120581+120596
120581
ℎ (119905 e1199061(119905minus120591(119905))minus1199062(119905minus120591(119905))) Δ119905
= (119863 +119889)120596
(26)
Since (1199061(119905) 1199062(119905))
Tisin 119883 there exist 120585
119894 120578119894isin 119868120596such that
119906119894(120585119894) = max119905isin119868120596
119906119894 (
119905) 119906119894(120578119894) = min119905isin119868120596
119906i (119905) 119894 = 1 2
(27)
Then from (22) (27) (H3) (H4) and the monotonicity of
function 119891 we have
119891 (e1199061(1205851)) =
1
120596
int
120581+120596
120581
119891 (119905 e1199061(1205851)) Δ119905
le
1
120596
int
120581+120596
120581
119891 (119905 e1199061(119905minus120590(119905))) Δ119905
le
1
120596
int
120581+120596
120581
sup119909gt0
119901 (119905 119909) Δ119905
(28)
which together with the monotonicity of function 119891 leads to
1199061(1205851) ge ln119881
1 (29)
By Lemma 2 we obtain from (25) and (29) that for all 119905 isin 119868120596
1199061 (
119905) ge 1199061(1205851) minus
1
2
int
120581+120596
120581
10038161003816100381610038161003816119906Δ
1(119905)
10038161003816100381610038161003816Δ119905
gt ln1198811minus
119865 (0) +119891 (0)
2
120596 = 1198611
(30)
From (22) (27) (H3) and themonotonicity of function119891 we
have
119891 (e1199061(1205781)) =
1
120596
int
120581+120596
120581
119891 (119905 e1199061(1205781)) Δ119905
ge
1
120596
int
120581+120596
120581
119891 (119905 e1199061(119905minus120590(119905))) Δ119905 gt 0
(31)
Discrete Dynamics in Nature and Society 5
which together with the monotonicity of function 119891 leads to
1199061(1205781) le ln119881
2 (32)
By Lemma 2 we get from (25) and (32) that for all 119905 isin 119868120596
1199061 (
119905) le 1199061(1205781) +
1
2
int
120581+120596
120581
10038161003816100381610038161003816119906Δ
1(119905)
10038161003816100381610038161003816Δ119905
lt ln1198812+
119865 (0) +119891 (0)
2
120596 = 1198612
(33)
From (23) (27) (H5) and themonotonicity of function ℎ we
have
ℎ (e1199061(1205851)minus1199062(1205782)) ge
1
120596
int
120581+120596
120581
ℎ (119905 e1199061(119905minus120591(119905))minus1199062(119905minus120591(119905))) Δ119905 =119889
ℎ (e1199061(1205781)minus1199062(1205852)) le
1
120596
int
120581+120596
120581
ℎ (119905 e1199061(119905minus120591(119905))minus1199062(119905minus120591(119905))) Δ119905 =119889
(34)
which together with the monotonicity of function ℎ lead to
e1199061(1205851)minus1199062(1205782) ge V0 e1199061(1205781)minus1199062(1205852) le V
0 (35)
That is
1199062(1205782) le 1199061(1205851) minus ln V
0
1199062(1205852) ge 1199061(1205781) minus ln V
0
(36)
By Lemma 2 we get from (26) (30) (33) and (36) that for all119905 isin 119868120596
1199062 (
119905) le 1199062(1205782) +
1
2
int
120581+120596
120581
10038161003816100381610038161003816119906Δ
2(119905)
10038161003816100381610038161003816Δ119905
lt 1198611minus ln V0+
119863 +119889
2
120596 = 1198613
1199062 (
119905) ge 1199062(1205852) minus
1
2
int
120581+120596
120581
10038161003816100381610038161003816119906Δ
2(119905)
10038161003816100381610038161003816Δ119905
gt 1198612minus ln V0minus
119863 +119889
2
120596 = 1198614
(37)
In view of (30) (33) (37) we obtain
max119905isin119868120596
10038161003816100381610038161199061 (
119905)1003816100381610038161003816le max100381610038161003816
10038161198611
100381610038161003816100381610038161003816100381610038161198612
1003816100381610038161003816 = 119861
5 (38)
max119905isin119868120596
10038161003816100381610038161199062 (
119905)1003816100381610038161003816le max100381610038161003816
10038161198613
100381610038161003816100381610038161003816100381610038161198614
1003816100381610038161003816 = 119861
6 (39)
Clearly 1198615and 119861
6are independent of 120582
From (H3) (H6) and the monotonicity of functions
119891
and ℎ it is easy to show that the algebraic system
119891 (e1199061) minus 119901 (e1199061minus1199062) = 0
minus119889 +
ℎ (e1199061minus1199062) = 0
(40)
has a unique solution (119906lowast
1 119906lowast
2)T
isin R2
We now take Ω = (1199061(119905) 1199062(119905))
Tisin 119883 (119906
1
(119905) 1199062(119905))
T lt 119861
5+ 1198616+ 1198610 where 119861
0is taken sufficiently
large such that (119906lowast1 119906lowast
2)T = |119906
lowast
1| + |119906lowast
2| lt 1198610 With the help
of (38) and (39) it is easy to see that Ω satisfies condition (a)in Lemma 1 When (119906
1(119905) 1199062(119905))
Tisin 120597Ω cap Ker 119871 = 120597Ω cap R2
(1199061(119905) 1199062(119905))
T is a constant vector in R2 with |1199061| + |119906
2| =
1198615+ 1198616+ 1198610 Thus we have
119876119873(
1199061
1199062
) = (
119891 (e1199061) minus 119901 (e1199061minus1199062)
minus119889 +
ℎ (e1199061minus1199062)
) = (
0
0) (41)
This proves that condition (b) in Lemma 1 is satisfiedTaking 119869 = 119868 Im119876 rarr Ker 119871 (119906
1 1199062)T
rarr (1199061 1199062)T a
direct calculation shows that
deg (119869119876119873Ω cap Ker 119871 0)
= signminus 1198911015840(e119906lowast
1)ℎ1015840(e119906lowast
1minus119906lowast
2) e2119906
lowast
1minus119906lowast
2
= 1 = 0
(42)
By now we have proved that Ω satisfies all the requirementsin Lemma 1 Hence (1) has at least one 120596-periodic solutionThis completes the proof
Noticing that both systems (2) and (3) are special cases ofsystem (1) byTheorem 3 we can obtain the following results
Theorem4 Suppose that (H1)ndash(H7) holdThen system (2) has
at least one positive 120596-periodic solution
Theorem5 Suppose that (H1)ndash(H7) holdThen system (3) has
at least one positive 120596-periodic solution
32 Nonmonotone Case In this section we study the exis-tence of120596-periodic solution of system (1) when the numericalresponse function ℎ(119905 119909) satisfies the following nonmono-tone condition
(H8) limVrarr+infinℎ(119905 119909) = 0 for 119905 isin R and there exists apositive constant V such that
(119909 minus V)120597ℎ
120597119909
(119905 119909) lt 0 for 119905 isin T 119909 gt 0 119909 = V (43)
By (H5) and (H
8) we have
(119909 minus V) ℎ1015840 (119909) =
1
120596
int
120581+120596
120581
(119909 minus V)120597ℎ
120597119909
(119905 119909) Δ119905 lt 0
limVrarr+infin
ℎ (119909) =
1
120596
int
120581+120596
120581
lim119909rarr+infin
ℎ (119905 119909) Δ119905 = 0
(44)
then ℎ(119909) is strictly increasing on [0 V] and strictly decreasing
on [V +infin) By these (H5) and (H
6) one can easily see that
the equation
ℎ (119909) =
119889 (45)
has two distinct positive solutions namely V1 V2 Without
loss of generality we suppose that V1lt V2 then V
1lt V lt V
2
6 Discrete Dynamics in Nature and Society
Theorem 6 In addition to (H1)ndash(H6) and (H
8) suppose
further that the following condition holds
(H9) (11988121198811)119890[119865(0)+119891(0)++119889]120596
lt V2V1
Then system (1) has at least two 120596-periodic solutions
Proof The proof is similar to that of Theorem 3 To completethe proof we need to find two disjoint open bounded subsetsin 119883 Under condition (H
9) (34) is no longer valid So we
need to make a corresponding changeDesignating 119911(119905) = 119906
1(119905)minus1199062(119905) there also exist 120585
0 1205780isin 119868120596
such that
119911 (1205850) = max119905isin119868120596
119911 (119905) 119911 (1205780) = min119905isin119868120596
119911 (119905) (46)
From (23) (27) and the monotonicity of functions ℎ and ℎ
we will show that 119911(1205850) and 119911(120578
0) can not simultaneously lie
in (minusinfin ln V1) (ln V
1 ln V2) or (ln V
2 +infin) In fact if 119911(120578
0) le
119911(1205850) lt ln V
1 then
119889 =
1
120596
int
120581+120596
120581
ℎ (119905 e119911(119905minus120591(119905))) Δ119905
leℎ (e119911(1205850)) lt
ℎ (V1) =
119889
(47)
This is a contradiction If ln V2lt 119911(1205780) le 119911(120585
0) then
119889 =
1
120596
int
120581+120596
120581
ℎ (119905 e119911(119905minus120591(119905))) Δ119905
leℎ (e119911(1205780)) lt
ℎ (V2) =
119889
(48)
This is also a contradiction If ln V1
lt 119911(1205780) le 119911(120585
0) lt ln V
2
then
119889 =
1
120596
int
120581+120596
120581
ℎ (119905 e119911(119905minus120591(119905))) Δ119905
ge minℎ (e119911(1205850)) ℎ (e119911(1205780))
gtℎ (V1) =
ℎ (V2) =
119889
(49)
This is also a contradiction Consequently the distribution of119911(1205850) and 119911(120578
0) only have the following two cases
Case 1 119911(1205780) le ln V
2le 119911(1205850) Noticing that
119911 (1205780) ge 1199061(1205781) minus 1199062(1205852)
119911 (1205850) le 1199061(1205851) minus 1199062(1205782)
(50)
from (27) (30) and (33) we obtain
1199062(1205852) ge 1199061(1205781) minus 119911 (120578
0) ge 1199061(1205781) minus ln V
2gt 1198611minus ln V2
1199062(1205782) le 1199061(1205851) minus 119911 (120585
0) le 1199061(1205851) minus ln V
2lt 1198612minus ln V2
(51)
By Lemma 2 we find from (26) and (51) that for all 119905 isin 119868120596
1199062 (
119905) ge 1199062(1205852) minus
1
2
int
120581+120596
120581
10038161003816100381610038161003816119906Δ
2(119905)
10038161003816100381610038161003816Δ119905
gt 1198611minus ln V2minus
119863 +119889
2
120596 = 1198617
1199062 (
119905) le 1199062(1205782) +
1
2
int
120581+120596
120581
10038161003816100381610038161003816119906Δ
2(119905)
10038161003816100381610038161003816Δ119905
lt 1198612minus ln V2+
119863 +119889
2
120596 = 1198618
(52)
Case 2 119911(1205780) le ln V
1le 119911(120585
0) From (27) (30) and (33) we
also obtain
1199062(1205852) ge 1199061(1205781) minus 119911 (120578
0) ge 1199061(1205781) minus ln V
1gt 1198611minus ln V1
1199062(1205782) le 1199061(1205851) minus 119911 (120585
0) le 1199061(1205851) minus ln V
1lt 1198612minus ln V1
(53)
By Lemma 2 we find from (26) and (53) that for all 119905 isin 119868120596
1199062 (
119905) ge 1199062(1205852) minus
1
2
int
120581+120596
120581
10038161003816100381610038161003816119906Δ
2(119905)
10038161003816100381610038161003816Δ119905
gt 1198611minus ln V1minus
119863 +119889
2
120596 = 1198619
1199062 (
119905) le 1199062(1205782) +
1
2
int
120581+120596
120581
10038161003816100381610038161003816119906Δ
2(119905)
10038161003816100381610038161003816Δ119905
lt 1198612minus ln V1+
119863 +119889
2
120596 = 11986110
(54)
By (H9) we know that
1198617lt 1198618lt 1198619lt 11986110 (55)
Clearly 1198617 1198618 1198619 and 119861
10are independent of 120582
By themonotonicity of functions 119891 and
ℎ (H3) (H5) and
(H6) it is easy to show that algebraic system (40) has two
distinct solutions
119906(119894)
= (ln119881(119894)
0 ln119881(119894)
0minus ln V119894)
T 119894 = 1 2 (56)
where119881(119894)
0is the unique solution of equation
119891(V)minus119901(V119894) = 0
Obviously
1198811le 119881(119894)
0lt 1198812 (57)
We now take
Ω1= (119906
1 (119905) 1199062 (
119905))T
isin 119883 10038161003816100381610038161199061 (
119905)1003816100381610038161003816
lt 1198615+ 1205730 1199062 (
119905) isin (1198617 1198618)
Ω2= (119906
1 (119905) 1199062 (
119905))T
isin 119883 10038161003816100381610038161199061 (
119905)1003816100381610038161003816
lt 1198615+ 1205730 1199062 (
119905) isin (1198619 11986110)
(58)
Discrete Dynamics in Nature and Society 7
where 1205730is taken sufficiently large such that
1205730gt max
10038161003816100381610038161003816ln119881(1)
0
10038161003816100381610038161003816
10038161003816100381610038161003816ln119881(2)
0
10038161003816100381610038161003816 (59)
Then both Ω1and Ω
2are bounded open subsets of 119883 It
follows from (55) (57) and (59) that 119906(1) isin Ω1 119906(2) isin Ω
2 and
Ω1cap Ω2= With the help of (38) (52) (54) and (59) it is
easy to see that Ω1and Ω
2satisfy condition (a) in Lemma 1
When (1199061(119905) 1199062(119905))
Tisin 120597Ω119894cap Ker 119871 = 120597Ω
119894cap R2 (119894 = 1 2)
(1199061(119905) 1199062(119905))
T is a constant vector in R2 Thus we have
119876119873(
1199061
1199062
) = (
119891 (e1199061) minus 119901 (e1199061minus1199062)
minus119889 +
ℎ (e1199061minus1199062)
) = (
0
0) (60)
This proves that condition (b) in Lemma 1 is satisfied A directcalculation shows that
deg (119869119876119873Ω119894cap Ker 119871 0)
= signminusV119894119881(119894)
01198911015840(119881(119894)
0)ℎ1015840(V119894)
= (minus1)119894minus1
= 0 119894 = 1 2
(61)
By now we have proved that Ω1and Ω
2satisfy all the
requirements in Lemma 1 Hence system (1) has at least two120596-periodic solutions 119906
lowast(119905) and 119906
dagger(119905) in Dom119871 cap Ω
1and
Dom119871 cap Ω2 respectively This completes the proof
Noticing that both systems (2) and (3) are special cases ofsystem (1) byTheorem 6 we can obtain the following results
Theorem 7 Suppose that (H1)ndash(H6) and (H
8)-(H9) hold
Then system (2) has at least two positive 120596-periodic solutions
Theorem 8 Suppose that (H1)ndash(H6) and (H
8)-(H9) hold
Then system (3) has at least two positive 120596-periodic solutions
Remark 9 The proof of Theorems 3 and 6 shows that allthe above results also remain valid if some delay terms arereplaced by terms with distributed or state-dependent delays
Remark 10 In their Theorem 22 Ding and Jiang [14] provedthat (2) has at least two positive 120596-periodic solutions if (H
1)ndash
(H6) (H8) and
(H10158409) (11988121198811)e2[119865(0)+119891(0)++119889]120596 lt V
2V1
hold Obviously the condition (H10158409) implies (H
9) Hence our
Theorem 7 improves Theorem 22 of [14]
Remark 11 Shao [10] studied a special case of system (1) for119891(119905 119909) = 119886(119905) minus 119887(119905)119909 Therefore our Theorem 3 generalizesTheorem 1 of [10]
Remark 12 Fan and Wang [17] Wang and Li [21] and Fanet al [16 18] studied some special cases of system (2) for119891(119905 119909) = 119886(119905) minus 119887(119905)119909 Therefore our Theorem 4 generalizesTheorem 21 of [17] Theorem 31 of [16] Theorem 35 of [18]andTheorem 21 of [21]
Remark 13 Fan and Wang [23] and Xia et al [24] studiedsome special cases of system (3) for 119891(119905 119909) = 119886(119905) minus 119887(119905)119909Therefore our Theorem 5 generalizes Theorem 21 of [23]andTheorem 31 of [24]
Acknowledgments
This work is supported by the National Natural ScienceFoundation of China under Grant 11271110 the Key Programsfor Science and Technology of the Education Department ofHenan Province under Grant 12A110007 and the ScientificResearch Start-up Funds of Henan University of Science andTechnology
References
[1] M Bohner and A PetersonDynamic Equations on Time ScalesAn Introduction with Applications Birkhauser Boston MassUSA 2001
[2] M Bohner and A Peterson Advances in Dynamic Equations onTime Scales Birkhauser Boston Mass USA 2003
[3] M Bohner L Erbe and A Peterson ldquoOscillation for nonlinearsecond order dynamic equations on a time scalerdquo Journal ofMathematical Analysis and Applications vol 301 no 2 pp 491ndash507 2005
[4] M Bohner M Fan and J Zhang ldquoExistence of periodicsolutions in predator-prey and competition dynamic systemsrdquoNonlinear Analysis Real World Applications vol 7 no 5 pp1193ndash1204 2006
[5] X Ding and G Zhao ldquoPeriodic solutions for a semi-ratio-dependent predator-prey system with delays on time scalesrdquoDiscrete Dynamics in Nature and Society vol 2012 Article ID928704 15 pages 2012
[6] M Fazly and M Hesaaraki ldquoPeriodic solutions for a semi-ratio-dependent predator-prey dynamical systemwith a class offunctional responses on time scalesrdquo Discrete and ContinuousDynamical Systems B vol 9 no 2 pp 267ndash279 2008
[7] J Hoffacker and C C Tisdell ldquoStability and instability fordynamic equations on time scalesrdquo Computers amp Mathematicswith Applications vol 49 no 9-10 pp 1327ndash1334 2005
[8] Y Li and H Zhang ldquoExistence of periodic solutions for a peri-odic mutualism model on time scalesrdquo Journal of MathematicalAnalysis and Applications vol 343 no 2 pp 818ndash825 2008
[9] J Liu Y Li and L Zhao ldquoOn a periodic predator-prey systemwith time delays on time scalesrdquo Communications in NonlinearScience and Numerical Simulation vol 14 no 8 pp 3432ndash34382009
[10] Y Shao ldquoExistence of positive periodic solution of a ratio-dependent predator-prey model on time scale with monotonicfunctional responserdquo Advances in Mathematics vol 39 no 2pp 224ndash232 2010
[11] B B Zhang and M Fan ldquoA remark on the application ofcoincidence degree to periodicity of dynamic equtions on timescalesrdquo Journal of Northeast Normal University vol 39 no 4 pp1ndash3 2007
[12] L Zhang H-X Li and X-B Zhang ldquoPeriodic solutions ofcompetition Lotka-Volterra dynamic system on time scalesrdquoComputers amp Mathematics with Applications vol 57 no 7 pp1204ndash1211 2009
8 Discrete Dynamics in Nature and Society
[13] X Ding ldquoPositive periodic solutions in generalized ratio-dependent predator-prey systemsrdquo Nonlinear Analysis RealWorld Applications vol 9 no 2 pp 394ndash402 2008
[14] X Ding and J Jiang ldquoMultiple periodic solutions in delayedGause-type ratio-dependent predator-prey systems with non-monotonic numerical responsesrdquo Mathematical and ComputerModelling vol 47 no 11-12 pp 1323ndash1331 2008
[15] Y H Fan andW T Li ldquoPermanence in delayed ratio-dependentpredator-prey models with monotonic functional responsesrdquoNonlinear Analysis Real World Applications vol 8 no 2 pp424ndash434 2007
[16] Y H Fan W T Li and L L Wang ldquoPeriodic solutions ofdelayed ratio-dependent predator-preymodels withmonotonicor nonmonotonic functional responsesrdquo Nonlinear AnalysisReal World Applications vol 5 no 2 pp 247ndash263 2004
[17] M Fan and K Wang ldquoPeriodicity in a delayed ratio-dependentpredator-prey systemrdquo Journal of Mathematical Analysis andApplications vol 262 no 1 pp 179ndash190 2001
[18] M Fan QWang and X Zou ldquoDynamics of a non-autonomousratio-dependent predator-prey systemrdquo Proceedings of the RoyalSociety of Edinburgh A vol 133 no 1 pp 97ndash118 2003
[19] S-B Hsu T-W Hwang and Y Kuang ldquoGlobal analysis ofthe Michaelis-Menten-type ratio-dependent predator-prey sys-temrdquo Journal of Mathematical Biology vol 42 no 6 pp 489ndash506 2001
[20] Y Kuang ldquoRich dynamics of Gauss-type ratio-dependentpredator-prey systemsrdquo Fields Institute Communications vol 21pp 325ndash337 1999
[21] L-L Wang and W-T Li ldquoPeriodic solutions and permanencefor a delayed nonautonomous ratio-dependent predator-preymodel with Holling type functional responserdquo Journal of Com-putational and Applied Mathematics vol 162 no 2 pp 341ndash3572004
[22] Y Xia and M Han ldquoMultiple periodic solutions of a ratio-dependent predator-prey modelrdquo Chaos Solitons amp Fractalsvol 39 no 3 pp 1100ndash1108 2009
[23] M Fan and K Wang ldquoPeriodic solutions of a discrete timenonautonomous ratio-dependent predator-prey systemrdquoMath-ematical and Computer Modelling vol 35 no 9-10 pp 951ndash9612002
[24] Y Xia J Cao andM Lin ldquoDiscrete-time analogues of predator-prey models with monotonic or nonmonotonic functionalresponsesrdquo Nonlinear Analysis Real World Applications vol 8no 4 pp 1079ndash1095 2007
[25] R E Gaines and J L Mawhin Coincidence Degree andNonlinear Differential Equations vol 568 of Lecture Notes inMathematics Springer Berlin Germany 1977
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Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
2 Discrete Dynamics in Nature and Society
to derive sufficient conditions for the existence of periodicsolutions of system (1) Furthermore we will see that ourresults for the above system can be easily extended to theone with distributed or state-dependent delays Our resultsimprove or generalize some theorems in [10 13ndash18 21 23]
2 Preliminaries
In this section we briefly give some elements of the time scalecalculus recall the continuation theorem from coincidencedegree theory and state an auxiliary result that will be usedin this paper
First let us present some foundational definitions andresults from the calculus on time scales so that the paper isself-contained For more details we refer the reader to [1 2]
A time scale T is an arbitrary nonempty closed subset T ofthe real numbers R which inherits the standard topology ofR Thus the real numbers R the integers Z and the naturalnumbers N are examples of time scales while the rationalnumbersQ and the open interval (1 2) are no time scales
Let 120596 gt 0 Throughout this paper the time scale T isassumed to be 120596-periodic that is 119905 isin T implies 119905 + 120596 isin T Inparticular the time scaleT under consideration is unboundedabove and below
For 119905 isin T the forward and backward jumpoperators120590 120588
T rarr R are defined by
120590 (119905) = inf119904 isin T 119904 gt 119905 120588 (119905) = sup119904 isin T 119904 lt 119905
(4)
respectivelyIf 120590(119905) = 119905 119905 is called right-dense (otherwise right-
scattered) and if 120588(119905) = 119905 then 119905 is called left-dense(otherwise left-scattered)
A function 119891 T rarr R is said to be rd-continuous if it iscontinuous at right-dense points in T and its left-sided limitsexist (finite) at left-dense points in T The set of rd-continuousfunctions is denoted by 119862rd(T)
For 119891 T rarr R and 119905 isin T we define 119891Δ(119905) the delta-
derivative of 119891 at 119905 to be the number (provided it exists) withthe property that given any 120576 gt 0 there is a neighborhood119880
of 119905 (ie 119880 = (119905 minus 120575 119905 + 120575) cap T for some 120575 gt 0) in T such that
10038161003816100381610038161003816[119891 (120590 (119905)) minus 119891 (119904)] minus 119891
Δ(119905) [120590 (119905) minus 119904]
10038161003816100381610038161003816
le 120576 |120590 (119905) minus 119904| forall119904 isin 119880
(5)
119891 is said to be delta-differentiable if its delta-derivative existsfor all 119905 isin T The set of functions 119891 T rarr R that are delta-differentiable and whose delta-derivative are rd-continuousfunctions is denoted by 119862
1
rd(T)A function 119865 T rarr R is called a delta-antiderivative of
119891 T rarr R provided 119865Δ(119905) = 119891(119905) for all 119905 isin T Then we
define the delta integral by
int
119887
119886
119891 (119905) Δ119905 = 119865 (119887) minus 119865 (119886) forall119886 119887 isin T (6)
If 119891 is continuous then 119891 is rd-continuous and if 119891 isdelta-differentiable at 119905 then 119891 is continuous at 119905 Moreoverevery rd-continuous function on T has a delta-antiderivative
Next let us recall the continuation theorem in coinci-dence degree theory To do so we need to introduce thefollowing notation
Let 119883 119884 be real Banach spaces 119871 Dom119871 sub 119883 rarr 119884 bea linear mapping and119873 119883 rarr 119884 be a continuous mapping
Themapping 119871 is said to be a Fredholmmapping of indexzero if dim Ker 119871 = codim Im 119871 lt +infin and Im 119871 is closed in119884
If 119871 is a Fredholmmapping of index zero then there existcontinuous projectors 119875 119883 rarr 119883 and 119876 119884 rarr 119884 suchthat Im119875 = Ker 119871 Ker119876 = Im 119871 = Im(119868 minus 119876) It follows thatthe restriction 119871
119875of 119871 to Dom119871 cap Ker119875 (119868 minus 119875)119883 rarr Im 119871
is invertible Denote the inverse of 119871119875by 119870119875
The mapping 119873 is said to be 119871-compact on Ω if Ω is anopen bounded subset of 119883 119876119873(Ω) is bounded and 119870
119875(119868 minus
119876)119873 Ω rarr 119883 is compactSince Im119876 is isomorphic to Ker 119871 there exists an isomor-
phism 119869 Im119876 rarr Ker 119871Here we state the Gaines-Mawhin theorem which is a
main tool in the proof of our main result
Lemma 1 (ContinuationTheorem [25 page 40]) Let Ω sub 119883
be an open bounded set 119871 a Fredholm mapping of index zeroand 119873 119871-compact on Ω Assume
(a) for each 120582 isin (0 1) 119909 isin 120597Ω cap Dom119871 119871119909 = 120582119873119909
(b) for each 119909 isin 120597Ω cap Ker 119871119876119873119909 = 0
(c) deg(119869119876119873Ω cap Ker 119871 0) = 0
Then 119871119909 = 119873119909 has at least one solution in Ω cap Dom119871
For convenience and simplicity in the following discus-sion we always use the following notation
120581 = min[0 +infin) cap T 119868120596
= [120581 120581 + 120596] cap T
119886 =
1
120596
int
120581+120596
120581
119886 (119905) Δ119905 119860 =
1
120596
int
120581+120596
120581
|119886 (119905)| Δ119905
119887 (119909) =
1
120596
int
120581+120596
120581
119887 (119905 119909) Δ119905
119861 (119909) =
1
120596
int
120581+120596
120581
|119887 (119905 119909)| Δ119905
(7)
where 119886 isin 119862rd(T) is an 120596-periodic function 119887 T times R rarr R
is rd-continuous and 120596-periodic in its first variableIn order to achieve the prior estimation in the case of
dynamic equations on a time scale T we also require thefollowing inequality which is proved in [11 Theorem 21]
Discrete Dynamics in Nature and Society 3
Lemma 2 Let 1199051 1199052
isin 119868120596 and 119905 isin T If 120593 isin 119862
1
rd(T) is an120596-periodic real function then
120593 (119905) le 120593 (1199051) +
1
2
int
120581+120596
120581
10038161003816100381610038161003816120593Δ(119905)
10038161003816100381610038161003816Δ119905
120593 (119905) ge 120593 (1199052) minus
1
2
int
120581+120596
120581
10038161003816100381610038161003816120593Δ(119905)
10038161003816100381610038161003816Δ119905
(8)
3 Existence of Periodic Solutions
In this section we study the existence of periodic solutions ofsystem (1) For the sake of generality we make the followingfundamental assumptions for system (1)
(H1) 120590 120591 T rarr R+ are rd-continuous and 120596-periodicsuch that 119905 minus 120590(119905) 119905 minus 120591(119905) isin T for all 119905 isin T
(H2) 119889 T rarr R is rd-continuous and 120596-periodic suchthat
119889 gt 0(H3) 119891 TtimesR rarr R is rd-continuous and120596-periodic in thefirst variable and is continuously differentiable in thesecond variable such that
119891(0) gt 0 lim119909rarr+infin
119891(119909) lt
0 and (120597119891120597119909)(119905 119909) lt 0 for all 119905 isin T 119909 gt 0(H4) 119901 T times R rarr R+ is rd-continuous and 120596-periodic inthe first variable and is continuously differentiable inthe second variable
(H5) ℎ T times R rarr R+ is rd-continuous and 120596-periodic inthe first variable and is continuously differentiable inthe second variable such that ℎ(119905 0) = 0 for all 119905 isin T
(H6) 119891(0) gt 119888 and sup
119909ge0ℎ(119909) gt
119889 where 119888(119905) =
sup119909gt0
119901(119905 119909) for all 119905 isin T
From (H3) we have
1198911015840(119909) =
1
120596
int
120581+120596
120581
120597119891
120597119909
(119905 119909) Δ119905 lt 0 (9)
Thus 119891(119909) is strictly decreasing on [0 +infin) From this (H
3)
and (H6) one can easily see that each of the equations
119891 (119909) =
1
120596
int
120581+120596
120581
sup119909gt0
119901 (119905 119909) Δ119905119891 (119909) = 0 (10)
has a unique positive solution Let 1198811 1198812be their solutions
respectively then 1198811lt 1198812
31 Monotone Case In this section we study the existenceof 120596-periodic solution of system (1) when the numericalresponse function ℎ(119905 119909) satisfies the following monotonecondition
(H7) (120597ℎ120597119909)(119905 119909) gt 0 for 119905 isin T 119909 gt 0
From (H5) and (H
7) we have
ℎ1015840(119909) =
1
120596
int
120581+120596
120581
120597ℎ
120597119909
(119905 119909) Δ119905 gt 0 (11)
Then ℎ(119909) is strictly increasing on [0 +infin) From this (H
5)
and (H6) one can easily see that the equation
ℎ(119909) =119889 has a
unique positive solution Let V0be its solution
Theorem 3 Suppose that (H1)ndash(H7) holdThen system (1) has
at least one 120596-periodic solution
Proof Take
119883 = 119884 = 119906 = (1199061 (
119905) 1199062 (119905))
T| 119906119894isin 119862 (T)
119906119894 (
119905 + 120596) = 119906119894 (
119905) 119894 = 1 2
119906 =
100381710038171003817100381710038171003817
(1199061 (
119905) 1199062 (119905))
T100381710038171003817100381710038171003817
= max119905isin119868120596
10038161003816100381610038161199061 (
119905)1003816100381610038161003816+ max119905isin119868120596
10038161003816100381610038161199062 (
119905)1003816100381610038161003816
(12)
then 119883 and 119884 are Banach spaces with the norm sdot Set
119871 Dom119871 sub 119883 997888rarr 119884 119871(
1199061 (
119905)
1199062 (
119905)) = (
119906Δ
1(119905)
119906Δ
2(119905)
) (13)
where Dom119871 = 119906 = (1199061(119905) 1199062(119905))
Tisin 119883 | 119906
119894isin 1198621
rd(T) 119894 =
1 2 and
119873 119883 rarr 119884
119873(
1199061 (
119905)
1199062 (
119905)) = (
119891 (119905 e1199061(119905minus120590(119905))) minus 119901 (119905 e1199061(119905)minus1199062(119905))
minus119889 (119905) + ℎ (119905 e1199061(119905minus120591(119905))minus1199062(119905minus120591(119905))) )
(14)
With these notations system (1) can be written in the form
119871119906 = 119873119906 119906 isin 119883 (15)
Obviously Ker 119871 = R2 Im 119871 = (1199061(119905) 1199062(119905))
Tisin 119884
119894=
0 119894 = 1 2 is closed in 119884 and dimKer119871 = codimIm119871 = 2Therefore 119871 is a Fredholmmapping of index zero Now definetwo projectors 119875 119883 rarr 119883 and 119876 119884 rarr 119884 as
119875(
1199061 (
119905)
1199062 (
119905)) = 119876(
1199061 (
119905)
1199062 (
119905)) = (
1
2
) (
1199061 (
119905)
1199062 (
119905)) isin 119883 = 119884
(16)
then 119875 and 119876 are continuous projectors such that
Im119875 = Ker 119871 Ker119876 = Im 119871 = Im(119868 minus 119876) (17)
Furthermore through an easy computation we find that thegeneralized inverse 119870
119875of 119871119875has the form
119870119875
Im 119871 997888rarr Dom119871 cap Ker119875
119870119875 (
119906) = int
119905
120581
119906 (119904) Δ119904 minus
1
120596
int
120581+120596
120581
int
119905
120581
119906 (119904) Δ119904Δ119905
(18)
4 Discrete Dynamics in Nature and Society
Then 119876119873 119883 rarr 119884 and 119870119875(119868 minus 119876)119873 119883 rarr 119883 read as
119876119873119906
=
1
120596
(
int
120581+120596
120581
[119891 (119905 e1199061(119905minus120590(119905))) minus 119901 (119905 e1199061(119905)minus1199062(119905))] Δ119905
int
120581+120596
120581
[minus119889 (119905) + ℎ (119905 e1199061(119905minus120591(119905))minus1199062(119905minus120591(119905)))] Δ119905
)
119870119875 (
119868 minus 119876)119873119906
= int
119905
120581
119873119906 (119904) Δ119904 minus
1
120596
int
120581+120596
120581
int
119905
120581
119873119906 (119904) Δ119904Δ119905
minus
1
120596
(119905 minus 120581 minus
1
120596
int
120581+120596
120581
(119904 minus 120581) Δ119904)int
120581+120596
120581
119873119906 (119904) Δ119904
(19)
Clearly 119876119873 and 119870119875(119868 minus 119876)119873 are continuous By using
the Arzela-Ascoli theorem it is not difficult to prove that119870119875(119868 minus 119876)119873(Ω) is compact for any open bounded setΩ sub 119883
Moreover 119876119873(Ω) is bounded Therefore 119873 is 119871-compact onΩ with any open bounded set Ω sub 119883
In order to apply Lemma 1 we need to find an appropriateopen bounded subset in 119883 Corresponding to the operatorequation 119871119906 = 120582119873119906 120582 isin (0 1) we have
119906Δ
1(119905) = 120582 [119891 (119905 e1199061(119905minus120590(119905))) minus 119901 (119905 e1199061(119905)minus1199062(119905))]
119906Δ
2(119905) = 120582 [minus119889 (119905) + ℎ (119905 e1199061(119905minus120591(119905))minus1199062(119905minus120591(119905)))]
(20)
Suppose that (1199061(119905) 1199062(119905))
Tisin 119883 is a solution of (20) for a
certain 120582 isin (0 1) Integrating (20) on both sides from 120581 to120581 + 120596 leads to
int
120581+120596
120581
120582 [119891 (119905 e1199061(119905minus120590(119905))) minus 119901 (119905 e1199061(119905)minus1199062(119905))] Δ119905 = 0
int
120581+120596
120581
120582 [minus119889 (119905) + ℎ (119905 e1199061(119905minus120591(119905))minus1199062(119905minus120591(119905)))] Δ119905 = 0
(21)
That is
int
120581+120596
120581
119891 (119905 e1199061(119905minus120590(119905))) Δ119905 = int
120581+120596
120581
119901 (119905 e1199061(119905)minus1199062(119905)) Δ119905 (22)
int
120581+120596
120581
ℎ (119905 e1199061(119905minus120591(119905))minus1199062(119905minus120591(119905))) Δ119905 = int
120581+120596
120581
119889 (119905) Δ119905 =119889120596
(23)
From (22) we have
119891 (0) 120596 = int
120581+120596
120581
[119891 (119905 0) minus 119891 (119905 e1199061(119905minus120590(119905)))
+119901 (119905 e1199061(119905)minus1199062(119905))] Δ119905
(24)
It follows from (20) (22) (23) (24) and (H3)ndash(H5) that
int
120581+120596
120581
10038161003816100381610038161003816119906Δ
1(119905)
10038161003816100381610038161003816Δ119905
le 120582int
120581+120596
120581
10038161003816100381610038161003816119891 (119905 e1199061(119905minus120590(119905))) minus 119898 (119905 e1199061(119905)minus1199062(119905))1003816100381610038161003816
1003816Δ119905
lt int
120581+120596
120581
[119891 (119905 0) minus 119891 (119905 e1199061(119905minus120590(119905)))] + 119901 (119905 e1199061(119905)minus1199062(119905))Δ119905
+ int
120581+120596
120581
1003816100381610038161003816119891 (119905 0)
1003816100381610038161003816Δ119905 = [119865 (0) +
119891 (0)] 120596
(25)
int
120581+120596
120581
10038161003816100381610038161003816119906Δ
2(119905)
10038161003816100381610038161003816Δ119905
le 120582int
120581+120596
120581
10038161003816100381610038161003816minus119889 (119905) + ℎ (119905 e1199061(119905minus120591(119905))minus1199062(119905minus120591(119905)))1003816100381610038161003816
1003816Δ119905
lt int
120581+120596
120581
|119889 (119905)| Δ119905 + int
120581+120596
120581
ℎ (119905 e1199061(119905minus120591(119905))minus1199062(119905minus120591(119905))) Δ119905
= (119863 +119889)120596
(26)
Since (1199061(119905) 1199062(119905))
Tisin 119883 there exist 120585
119894 120578119894isin 119868120596such that
119906119894(120585119894) = max119905isin119868120596
119906119894 (
119905) 119906119894(120578119894) = min119905isin119868120596
119906i (119905) 119894 = 1 2
(27)
Then from (22) (27) (H3) (H4) and the monotonicity of
function 119891 we have
119891 (e1199061(1205851)) =
1
120596
int
120581+120596
120581
119891 (119905 e1199061(1205851)) Δ119905
le
1
120596
int
120581+120596
120581
119891 (119905 e1199061(119905minus120590(119905))) Δ119905
le
1
120596
int
120581+120596
120581
sup119909gt0
119901 (119905 119909) Δ119905
(28)
which together with the monotonicity of function 119891 leads to
1199061(1205851) ge ln119881
1 (29)
By Lemma 2 we obtain from (25) and (29) that for all 119905 isin 119868120596
1199061 (
119905) ge 1199061(1205851) minus
1
2
int
120581+120596
120581
10038161003816100381610038161003816119906Δ
1(119905)
10038161003816100381610038161003816Δ119905
gt ln1198811minus
119865 (0) +119891 (0)
2
120596 = 1198611
(30)
From (22) (27) (H3) and themonotonicity of function119891 we
have
119891 (e1199061(1205781)) =
1
120596
int
120581+120596
120581
119891 (119905 e1199061(1205781)) Δ119905
ge
1
120596
int
120581+120596
120581
119891 (119905 e1199061(119905minus120590(119905))) Δ119905 gt 0
(31)
Discrete Dynamics in Nature and Society 5
which together with the monotonicity of function 119891 leads to
1199061(1205781) le ln119881
2 (32)
By Lemma 2 we get from (25) and (32) that for all 119905 isin 119868120596
1199061 (
119905) le 1199061(1205781) +
1
2
int
120581+120596
120581
10038161003816100381610038161003816119906Δ
1(119905)
10038161003816100381610038161003816Δ119905
lt ln1198812+
119865 (0) +119891 (0)
2
120596 = 1198612
(33)
From (23) (27) (H5) and themonotonicity of function ℎ we
have
ℎ (e1199061(1205851)minus1199062(1205782)) ge
1
120596
int
120581+120596
120581
ℎ (119905 e1199061(119905minus120591(119905))minus1199062(119905minus120591(119905))) Δ119905 =119889
ℎ (e1199061(1205781)minus1199062(1205852)) le
1
120596
int
120581+120596
120581
ℎ (119905 e1199061(119905minus120591(119905))minus1199062(119905minus120591(119905))) Δ119905 =119889
(34)
which together with the monotonicity of function ℎ lead to
e1199061(1205851)minus1199062(1205782) ge V0 e1199061(1205781)minus1199062(1205852) le V
0 (35)
That is
1199062(1205782) le 1199061(1205851) minus ln V
0
1199062(1205852) ge 1199061(1205781) minus ln V
0
(36)
By Lemma 2 we get from (26) (30) (33) and (36) that for all119905 isin 119868120596
1199062 (
119905) le 1199062(1205782) +
1
2
int
120581+120596
120581
10038161003816100381610038161003816119906Δ
2(119905)
10038161003816100381610038161003816Δ119905
lt 1198611minus ln V0+
119863 +119889
2
120596 = 1198613
1199062 (
119905) ge 1199062(1205852) minus
1
2
int
120581+120596
120581
10038161003816100381610038161003816119906Δ
2(119905)
10038161003816100381610038161003816Δ119905
gt 1198612minus ln V0minus
119863 +119889
2
120596 = 1198614
(37)
In view of (30) (33) (37) we obtain
max119905isin119868120596
10038161003816100381610038161199061 (
119905)1003816100381610038161003816le max100381610038161003816
10038161198611
100381610038161003816100381610038161003816100381610038161198612
1003816100381610038161003816 = 119861
5 (38)
max119905isin119868120596
10038161003816100381610038161199062 (
119905)1003816100381610038161003816le max100381610038161003816
10038161198613
100381610038161003816100381610038161003816100381610038161198614
1003816100381610038161003816 = 119861
6 (39)
Clearly 1198615and 119861
6are independent of 120582
From (H3) (H6) and the monotonicity of functions
119891
and ℎ it is easy to show that the algebraic system
119891 (e1199061) minus 119901 (e1199061minus1199062) = 0
minus119889 +
ℎ (e1199061minus1199062) = 0
(40)
has a unique solution (119906lowast
1 119906lowast
2)T
isin R2
We now take Ω = (1199061(119905) 1199062(119905))
Tisin 119883 (119906
1
(119905) 1199062(119905))
T lt 119861
5+ 1198616+ 1198610 where 119861
0is taken sufficiently
large such that (119906lowast1 119906lowast
2)T = |119906
lowast
1| + |119906lowast
2| lt 1198610 With the help
of (38) and (39) it is easy to see that Ω satisfies condition (a)in Lemma 1 When (119906
1(119905) 1199062(119905))
Tisin 120597Ω cap Ker 119871 = 120597Ω cap R2
(1199061(119905) 1199062(119905))
T is a constant vector in R2 with |1199061| + |119906
2| =
1198615+ 1198616+ 1198610 Thus we have
119876119873(
1199061
1199062
) = (
119891 (e1199061) minus 119901 (e1199061minus1199062)
minus119889 +
ℎ (e1199061minus1199062)
) = (
0
0) (41)
This proves that condition (b) in Lemma 1 is satisfiedTaking 119869 = 119868 Im119876 rarr Ker 119871 (119906
1 1199062)T
rarr (1199061 1199062)T a
direct calculation shows that
deg (119869119876119873Ω cap Ker 119871 0)
= signminus 1198911015840(e119906lowast
1)ℎ1015840(e119906lowast
1minus119906lowast
2) e2119906
lowast
1minus119906lowast
2
= 1 = 0
(42)
By now we have proved that Ω satisfies all the requirementsin Lemma 1 Hence (1) has at least one 120596-periodic solutionThis completes the proof
Noticing that both systems (2) and (3) are special cases ofsystem (1) byTheorem 3 we can obtain the following results
Theorem4 Suppose that (H1)ndash(H7) holdThen system (2) has
at least one positive 120596-periodic solution
Theorem5 Suppose that (H1)ndash(H7) holdThen system (3) has
at least one positive 120596-periodic solution
32 Nonmonotone Case In this section we study the exis-tence of120596-periodic solution of system (1) when the numericalresponse function ℎ(119905 119909) satisfies the following nonmono-tone condition
(H8) limVrarr+infinℎ(119905 119909) = 0 for 119905 isin R and there exists apositive constant V such that
(119909 minus V)120597ℎ
120597119909
(119905 119909) lt 0 for 119905 isin T 119909 gt 0 119909 = V (43)
By (H5) and (H
8) we have
(119909 minus V) ℎ1015840 (119909) =
1
120596
int
120581+120596
120581
(119909 minus V)120597ℎ
120597119909
(119905 119909) Δ119905 lt 0
limVrarr+infin
ℎ (119909) =
1
120596
int
120581+120596
120581
lim119909rarr+infin
ℎ (119905 119909) Δ119905 = 0
(44)
then ℎ(119909) is strictly increasing on [0 V] and strictly decreasing
on [V +infin) By these (H5) and (H
6) one can easily see that
the equation
ℎ (119909) =
119889 (45)
has two distinct positive solutions namely V1 V2 Without
loss of generality we suppose that V1lt V2 then V
1lt V lt V
2
6 Discrete Dynamics in Nature and Society
Theorem 6 In addition to (H1)ndash(H6) and (H
8) suppose
further that the following condition holds
(H9) (11988121198811)119890[119865(0)+119891(0)++119889]120596
lt V2V1
Then system (1) has at least two 120596-periodic solutions
Proof The proof is similar to that of Theorem 3 To completethe proof we need to find two disjoint open bounded subsetsin 119883 Under condition (H
9) (34) is no longer valid So we
need to make a corresponding changeDesignating 119911(119905) = 119906
1(119905)minus1199062(119905) there also exist 120585
0 1205780isin 119868120596
such that
119911 (1205850) = max119905isin119868120596
119911 (119905) 119911 (1205780) = min119905isin119868120596
119911 (119905) (46)
From (23) (27) and the monotonicity of functions ℎ and ℎ
we will show that 119911(1205850) and 119911(120578
0) can not simultaneously lie
in (minusinfin ln V1) (ln V
1 ln V2) or (ln V
2 +infin) In fact if 119911(120578
0) le
119911(1205850) lt ln V
1 then
119889 =
1
120596
int
120581+120596
120581
ℎ (119905 e119911(119905minus120591(119905))) Δ119905
leℎ (e119911(1205850)) lt
ℎ (V1) =
119889
(47)
This is a contradiction If ln V2lt 119911(1205780) le 119911(120585
0) then
119889 =
1
120596
int
120581+120596
120581
ℎ (119905 e119911(119905minus120591(119905))) Δ119905
leℎ (e119911(1205780)) lt
ℎ (V2) =
119889
(48)
This is also a contradiction If ln V1
lt 119911(1205780) le 119911(120585
0) lt ln V
2
then
119889 =
1
120596
int
120581+120596
120581
ℎ (119905 e119911(119905minus120591(119905))) Δ119905
ge minℎ (e119911(1205850)) ℎ (e119911(1205780))
gtℎ (V1) =
ℎ (V2) =
119889
(49)
This is also a contradiction Consequently the distribution of119911(1205850) and 119911(120578
0) only have the following two cases
Case 1 119911(1205780) le ln V
2le 119911(1205850) Noticing that
119911 (1205780) ge 1199061(1205781) minus 1199062(1205852)
119911 (1205850) le 1199061(1205851) minus 1199062(1205782)
(50)
from (27) (30) and (33) we obtain
1199062(1205852) ge 1199061(1205781) minus 119911 (120578
0) ge 1199061(1205781) minus ln V
2gt 1198611minus ln V2
1199062(1205782) le 1199061(1205851) minus 119911 (120585
0) le 1199061(1205851) minus ln V
2lt 1198612minus ln V2
(51)
By Lemma 2 we find from (26) and (51) that for all 119905 isin 119868120596
1199062 (
119905) ge 1199062(1205852) minus
1
2
int
120581+120596
120581
10038161003816100381610038161003816119906Δ
2(119905)
10038161003816100381610038161003816Δ119905
gt 1198611minus ln V2minus
119863 +119889
2
120596 = 1198617
1199062 (
119905) le 1199062(1205782) +
1
2
int
120581+120596
120581
10038161003816100381610038161003816119906Δ
2(119905)
10038161003816100381610038161003816Δ119905
lt 1198612minus ln V2+
119863 +119889
2
120596 = 1198618
(52)
Case 2 119911(1205780) le ln V
1le 119911(120585
0) From (27) (30) and (33) we
also obtain
1199062(1205852) ge 1199061(1205781) minus 119911 (120578
0) ge 1199061(1205781) minus ln V
1gt 1198611minus ln V1
1199062(1205782) le 1199061(1205851) minus 119911 (120585
0) le 1199061(1205851) minus ln V
1lt 1198612minus ln V1
(53)
By Lemma 2 we find from (26) and (53) that for all 119905 isin 119868120596
1199062 (
119905) ge 1199062(1205852) minus
1
2
int
120581+120596
120581
10038161003816100381610038161003816119906Δ
2(119905)
10038161003816100381610038161003816Δ119905
gt 1198611minus ln V1minus
119863 +119889
2
120596 = 1198619
1199062 (
119905) le 1199062(1205782) +
1
2
int
120581+120596
120581
10038161003816100381610038161003816119906Δ
2(119905)
10038161003816100381610038161003816Δ119905
lt 1198612minus ln V1+
119863 +119889
2
120596 = 11986110
(54)
By (H9) we know that
1198617lt 1198618lt 1198619lt 11986110 (55)
Clearly 1198617 1198618 1198619 and 119861
10are independent of 120582
By themonotonicity of functions 119891 and
ℎ (H3) (H5) and
(H6) it is easy to show that algebraic system (40) has two
distinct solutions
119906(119894)
= (ln119881(119894)
0 ln119881(119894)
0minus ln V119894)
T 119894 = 1 2 (56)
where119881(119894)
0is the unique solution of equation
119891(V)minus119901(V119894) = 0
Obviously
1198811le 119881(119894)
0lt 1198812 (57)
We now take
Ω1= (119906
1 (119905) 1199062 (
119905))T
isin 119883 10038161003816100381610038161199061 (
119905)1003816100381610038161003816
lt 1198615+ 1205730 1199062 (
119905) isin (1198617 1198618)
Ω2= (119906
1 (119905) 1199062 (
119905))T
isin 119883 10038161003816100381610038161199061 (
119905)1003816100381610038161003816
lt 1198615+ 1205730 1199062 (
119905) isin (1198619 11986110)
(58)
Discrete Dynamics in Nature and Society 7
where 1205730is taken sufficiently large such that
1205730gt max
10038161003816100381610038161003816ln119881(1)
0
10038161003816100381610038161003816
10038161003816100381610038161003816ln119881(2)
0
10038161003816100381610038161003816 (59)
Then both Ω1and Ω
2are bounded open subsets of 119883 It
follows from (55) (57) and (59) that 119906(1) isin Ω1 119906(2) isin Ω
2 and
Ω1cap Ω2= With the help of (38) (52) (54) and (59) it is
easy to see that Ω1and Ω
2satisfy condition (a) in Lemma 1
When (1199061(119905) 1199062(119905))
Tisin 120597Ω119894cap Ker 119871 = 120597Ω
119894cap R2 (119894 = 1 2)
(1199061(119905) 1199062(119905))
T is a constant vector in R2 Thus we have
119876119873(
1199061
1199062
) = (
119891 (e1199061) minus 119901 (e1199061minus1199062)
minus119889 +
ℎ (e1199061minus1199062)
) = (
0
0) (60)
This proves that condition (b) in Lemma 1 is satisfied A directcalculation shows that
deg (119869119876119873Ω119894cap Ker 119871 0)
= signminusV119894119881(119894)
01198911015840(119881(119894)
0)ℎ1015840(V119894)
= (minus1)119894minus1
= 0 119894 = 1 2
(61)
By now we have proved that Ω1and Ω
2satisfy all the
requirements in Lemma 1 Hence system (1) has at least two120596-periodic solutions 119906
lowast(119905) and 119906
dagger(119905) in Dom119871 cap Ω
1and
Dom119871 cap Ω2 respectively This completes the proof
Noticing that both systems (2) and (3) are special cases ofsystem (1) byTheorem 6 we can obtain the following results
Theorem 7 Suppose that (H1)ndash(H6) and (H
8)-(H9) hold
Then system (2) has at least two positive 120596-periodic solutions
Theorem 8 Suppose that (H1)ndash(H6) and (H
8)-(H9) hold
Then system (3) has at least two positive 120596-periodic solutions
Remark 9 The proof of Theorems 3 and 6 shows that allthe above results also remain valid if some delay terms arereplaced by terms with distributed or state-dependent delays
Remark 10 In their Theorem 22 Ding and Jiang [14] provedthat (2) has at least two positive 120596-periodic solutions if (H
1)ndash
(H6) (H8) and
(H10158409) (11988121198811)e2[119865(0)+119891(0)++119889]120596 lt V
2V1
hold Obviously the condition (H10158409) implies (H
9) Hence our
Theorem 7 improves Theorem 22 of [14]
Remark 11 Shao [10] studied a special case of system (1) for119891(119905 119909) = 119886(119905) minus 119887(119905)119909 Therefore our Theorem 3 generalizesTheorem 1 of [10]
Remark 12 Fan and Wang [17] Wang and Li [21] and Fanet al [16 18] studied some special cases of system (2) for119891(119905 119909) = 119886(119905) minus 119887(119905)119909 Therefore our Theorem 4 generalizesTheorem 21 of [17] Theorem 31 of [16] Theorem 35 of [18]andTheorem 21 of [21]
Remark 13 Fan and Wang [23] and Xia et al [24] studiedsome special cases of system (3) for 119891(119905 119909) = 119886(119905) minus 119887(119905)119909Therefore our Theorem 5 generalizes Theorem 21 of [23]andTheorem 31 of [24]
Acknowledgments
This work is supported by the National Natural ScienceFoundation of China under Grant 11271110 the Key Programsfor Science and Technology of the Education Department ofHenan Province under Grant 12A110007 and the ScientificResearch Start-up Funds of Henan University of Science andTechnology
References
[1] M Bohner and A PetersonDynamic Equations on Time ScalesAn Introduction with Applications Birkhauser Boston MassUSA 2001
[2] M Bohner and A Peterson Advances in Dynamic Equations onTime Scales Birkhauser Boston Mass USA 2003
[3] M Bohner L Erbe and A Peterson ldquoOscillation for nonlinearsecond order dynamic equations on a time scalerdquo Journal ofMathematical Analysis and Applications vol 301 no 2 pp 491ndash507 2005
[4] M Bohner M Fan and J Zhang ldquoExistence of periodicsolutions in predator-prey and competition dynamic systemsrdquoNonlinear Analysis Real World Applications vol 7 no 5 pp1193ndash1204 2006
[5] X Ding and G Zhao ldquoPeriodic solutions for a semi-ratio-dependent predator-prey system with delays on time scalesrdquoDiscrete Dynamics in Nature and Society vol 2012 Article ID928704 15 pages 2012
[6] M Fazly and M Hesaaraki ldquoPeriodic solutions for a semi-ratio-dependent predator-prey dynamical systemwith a class offunctional responses on time scalesrdquo Discrete and ContinuousDynamical Systems B vol 9 no 2 pp 267ndash279 2008
[7] J Hoffacker and C C Tisdell ldquoStability and instability fordynamic equations on time scalesrdquo Computers amp Mathematicswith Applications vol 49 no 9-10 pp 1327ndash1334 2005
[8] Y Li and H Zhang ldquoExistence of periodic solutions for a peri-odic mutualism model on time scalesrdquo Journal of MathematicalAnalysis and Applications vol 343 no 2 pp 818ndash825 2008
[9] J Liu Y Li and L Zhao ldquoOn a periodic predator-prey systemwith time delays on time scalesrdquo Communications in NonlinearScience and Numerical Simulation vol 14 no 8 pp 3432ndash34382009
[10] Y Shao ldquoExistence of positive periodic solution of a ratio-dependent predator-prey model on time scale with monotonicfunctional responserdquo Advances in Mathematics vol 39 no 2pp 224ndash232 2010
[11] B B Zhang and M Fan ldquoA remark on the application ofcoincidence degree to periodicity of dynamic equtions on timescalesrdquo Journal of Northeast Normal University vol 39 no 4 pp1ndash3 2007
[12] L Zhang H-X Li and X-B Zhang ldquoPeriodic solutions ofcompetition Lotka-Volterra dynamic system on time scalesrdquoComputers amp Mathematics with Applications vol 57 no 7 pp1204ndash1211 2009
8 Discrete Dynamics in Nature and Society
[13] X Ding ldquoPositive periodic solutions in generalized ratio-dependent predator-prey systemsrdquo Nonlinear Analysis RealWorld Applications vol 9 no 2 pp 394ndash402 2008
[14] X Ding and J Jiang ldquoMultiple periodic solutions in delayedGause-type ratio-dependent predator-prey systems with non-monotonic numerical responsesrdquo Mathematical and ComputerModelling vol 47 no 11-12 pp 1323ndash1331 2008
[15] Y H Fan andW T Li ldquoPermanence in delayed ratio-dependentpredator-prey models with monotonic functional responsesrdquoNonlinear Analysis Real World Applications vol 8 no 2 pp424ndash434 2007
[16] Y H Fan W T Li and L L Wang ldquoPeriodic solutions ofdelayed ratio-dependent predator-preymodels withmonotonicor nonmonotonic functional responsesrdquo Nonlinear AnalysisReal World Applications vol 5 no 2 pp 247ndash263 2004
[17] M Fan and K Wang ldquoPeriodicity in a delayed ratio-dependentpredator-prey systemrdquo Journal of Mathematical Analysis andApplications vol 262 no 1 pp 179ndash190 2001
[18] M Fan QWang and X Zou ldquoDynamics of a non-autonomousratio-dependent predator-prey systemrdquo Proceedings of the RoyalSociety of Edinburgh A vol 133 no 1 pp 97ndash118 2003
[19] S-B Hsu T-W Hwang and Y Kuang ldquoGlobal analysis ofthe Michaelis-Menten-type ratio-dependent predator-prey sys-temrdquo Journal of Mathematical Biology vol 42 no 6 pp 489ndash506 2001
[20] Y Kuang ldquoRich dynamics of Gauss-type ratio-dependentpredator-prey systemsrdquo Fields Institute Communications vol 21pp 325ndash337 1999
[21] L-L Wang and W-T Li ldquoPeriodic solutions and permanencefor a delayed nonautonomous ratio-dependent predator-preymodel with Holling type functional responserdquo Journal of Com-putational and Applied Mathematics vol 162 no 2 pp 341ndash3572004
[22] Y Xia and M Han ldquoMultiple periodic solutions of a ratio-dependent predator-prey modelrdquo Chaos Solitons amp Fractalsvol 39 no 3 pp 1100ndash1108 2009
[23] M Fan and K Wang ldquoPeriodic solutions of a discrete timenonautonomous ratio-dependent predator-prey systemrdquoMath-ematical and Computer Modelling vol 35 no 9-10 pp 951ndash9612002
[24] Y Xia J Cao andM Lin ldquoDiscrete-time analogues of predator-prey models with monotonic or nonmonotonic functionalresponsesrdquo Nonlinear Analysis Real World Applications vol 8no 4 pp 1079ndash1095 2007
[25] R E Gaines and J L Mawhin Coincidence Degree andNonlinear Differential Equations vol 568 of Lecture Notes inMathematics Springer Berlin Germany 1977
Submit your manuscripts athttpwwwhindawicom
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Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
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Discrete Dynamics in Nature and Society
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Decision SciencesAdvances in
Discrete MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
Discrete Dynamics in Nature and Society 3
Lemma 2 Let 1199051 1199052
isin 119868120596 and 119905 isin T If 120593 isin 119862
1
rd(T) is an120596-periodic real function then
120593 (119905) le 120593 (1199051) +
1
2
int
120581+120596
120581
10038161003816100381610038161003816120593Δ(119905)
10038161003816100381610038161003816Δ119905
120593 (119905) ge 120593 (1199052) minus
1
2
int
120581+120596
120581
10038161003816100381610038161003816120593Δ(119905)
10038161003816100381610038161003816Δ119905
(8)
3 Existence of Periodic Solutions
In this section we study the existence of periodic solutions ofsystem (1) For the sake of generality we make the followingfundamental assumptions for system (1)
(H1) 120590 120591 T rarr R+ are rd-continuous and 120596-periodicsuch that 119905 minus 120590(119905) 119905 minus 120591(119905) isin T for all 119905 isin T
(H2) 119889 T rarr R is rd-continuous and 120596-periodic suchthat
119889 gt 0(H3) 119891 TtimesR rarr R is rd-continuous and120596-periodic in thefirst variable and is continuously differentiable in thesecond variable such that
119891(0) gt 0 lim119909rarr+infin
119891(119909) lt
0 and (120597119891120597119909)(119905 119909) lt 0 for all 119905 isin T 119909 gt 0(H4) 119901 T times R rarr R+ is rd-continuous and 120596-periodic inthe first variable and is continuously differentiable inthe second variable
(H5) ℎ T times R rarr R+ is rd-continuous and 120596-periodic inthe first variable and is continuously differentiable inthe second variable such that ℎ(119905 0) = 0 for all 119905 isin T
(H6) 119891(0) gt 119888 and sup
119909ge0ℎ(119909) gt
119889 where 119888(119905) =
sup119909gt0
119901(119905 119909) for all 119905 isin T
From (H3) we have
1198911015840(119909) =
1
120596
int
120581+120596
120581
120597119891
120597119909
(119905 119909) Δ119905 lt 0 (9)
Thus 119891(119909) is strictly decreasing on [0 +infin) From this (H
3)
and (H6) one can easily see that each of the equations
119891 (119909) =
1
120596
int
120581+120596
120581
sup119909gt0
119901 (119905 119909) Δ119905119891 (119909) = 0 (10)
has a unique positive solution Let 1198811 1198812be their solutions
respectively then 1198811lt 1198812
31 Monotone Case In this section we study the existenceof 120596-periodic solution of system (1) when the numericalresponse function ℎ(119905 119909) satisfies the following monotonecondition
(H7) (120597ℎ120597119909)(119905 119909) gt 0 for 119905 isin T 119909 gt 0
From (H5) and (H
7) we have
ℎ1015840(119909) =
1
120596
int
120581+120596
120581
120597ℎ
120597119909
(119905 119909) Δ119905 gt 0 (11)
Then ℎ(119909) is strictly increasing on [0 +infin) From this (H
5)
and (H6) one can easily see that the equation
ℎ(119909) =119889 has a
unique positive solution Let V0be its solution
Theorem 3 Suppose that (H1)ndash(H7) holdThen system (1) has
at least one 120596-periodic solution
Proof Take
119883 = 119884 = 119906 = (1199061 (
119905) 1199062 (119905))
T| 119906119894isin 119862 (T)
119906119894 (
119905 + 120596) = 119906119894 (
119905) 119894 = 1 2
119906 =
100381710038171003817100381710038171003817
(1199061 (
119905) 1199062 (119905))
T100381710038171003817100381710038171003817
= max119905isin119868120596
10038161003816100381610038161199061 (
119905)1003816100381610038161003816+ max119905isin119868120596
10038161003816100381610038161199062 (
119905)1003816100381610038161003816
(12)
then 119883 and 119884 are Banach spaces with the norm sdot Set
119871 Dom119871 sub 119883 997888rarr 119884 119871(
1199061 (
119905)
1199062 (
119905)) = (
119906Δ
1(119905)
119906Δ
2(119905)
) (13)
where Dom119871 = 119906 = (1199061(119905) 1199062(119905))
Tisin 119883 | 119906
119894isin 1198621
rd(T) 119894 =
1 2 and
119873 119883 rarr 119884
119873(
1199061 (
119905)
1199062 (
119905)) = (
119891 (119905 e1199061(119905minus120590(119905))) minus 119901 (119905 e1199061(119905)minus1199062(119905))
minus119889 (119905) + ℎ (119905 e1199061(119905minus120591(119905))minus1199062(119905minus120591(119905))) )
(14)
With these notations system (1) can be written in the form
119871119906 = 119873119906 119906 isin 119883 (15)
Obviously Ker 119871 = R2 Im 119871 = (1199061(119905) 1199062(119905))
Tisin 119884
119894=
0 119894 = 1 2 is closed in 119884 and dimKer119871 = codimIm119871 = 2Therefore 119871 is a Fredholmmapping of index zero Now definetwo projectors 119875 119883 rarr 119883 and 119876 119884 rarr 119884 as
119875(
1199061 (
119905)
1199062 (
119905)) = 119876(
1199061 (
119905)
1199062 (
119905)) = (
1
2
) (
1199061 (
119905)
1199062 (
119905)) isin 119883 = 119884
(16)
then 119875 and 119876 are continuous projectors such that
Im119875 = Ker 119871 Ker119876 = Im 119871 = Im(119868 minus 119876) (17)
Furthermore through an easy computation we find that thegeneralized inverse 119870
119875of 119871119875has the form
119870119875
Im 119871 997888rarr Dom119871 cap Ker119875
119870119875 (
119906) = int
119905
120581
119906 (119904) Δ119904 minus
1
120596
int
120581+120596
120581
int
119905
120581
119906 (119904) Δ119904Δ119905
(18)
4 Discrete Dynamics in Nature and Society
Then 119876119873 119883 rarr 119884 and 119870119875(119868 minus 119876)119873 119883 rarr 119883 read as
119876119873119906
=
1
120596
(
int
120581+120596
120581
[119891 (119905 e1199061(119905minus120590(119905))) minus 119901 (119905 e1199061(119905)minus1199062(119905))] Δ119905
int
120581+120596
120581
[minus119889 (119905) + ℎ (119905 e1199061(119905minus120591(119905))minus1199062(119905minus120591(119905)))] Δ119905
)
119870119875 (
119868 minus 119876)119873119906
= int
119905
120581
119873119906 (119904) Δ119904 minus
1
120596
int
120581+120596
120581
int
119905
120581
119873119906 (119904) Δ119904Δ119905
minus
1
120596
(119905 minus 120581 minus
1
120596
int
120581+120596
120581
(119904 minus 120581) Δ119904)int
120581+120596
120581
119873119906 (119904) Δ119904
(19)
Clearly 119876119873 and 119870119875(119868 minus 119876)119873 are continuous By using
the Arzela-Ascoli theorem it is not difficult to prove that119870119875(119868 minus 119876)119873(Ω) is compact for any open bounded setΩ sub 119883
Moreover 119876119873(Ω) is bounded Therefore 119873 is 119871-compact onΩ with any open bounded set Ω sub 119883
In order to apply Lemma 1 we need to find an appropriateopen bounded subset in 119883 Corresponding to the operatorequation 119871119906 = 120582119873119906 120582 isin (0 1) we have
119906Δ
1(119905) = 120582 [119891 (119905 e1199061(119905minus120590(119905))) minus 119901 (119905 e1199061(119905)minus1199062(119905))]
119906Δ
2(119905) = 120582 [minus119889 (119905) + ℎ (119905 e1199061(119905minus120591(119905))minus1199062(119905minus120591(119905)))]
(20)
Suppose that (1199061(119905) 1199062(119905))
Tisin 119883 is a solution of (20) for a
certain 120582 isin (0 1) Integrating (20) on both sides from 120581 to120581 + 120596 leads to
int
120581+120596
120581
120582 [119891 (119905 e1199061(119905minus120590(119905))) minus 119901 (119905 e1199061(119905)minus1199062(119905))] Δ119905 = 0
int
120581+120596
120581
120582 [minus119889 (119905) + ℎ (119905 e1199061(119905minus120591(119905))minus1199062(119905minus120591(119905)))] Δ119905 = 0
(21)
That is
int
120581+120596
120581
119891 (119905 e1199061(119905minus120590(119905))) Δ119905 = int
120581+120596
120581
119901 (119905 e1199061(119905)minus1199062(119905)) Δ119905 (22)
int
120581+120596
120581
ℎ (119905 e1199061(119905minus120591(119905))minus1199062(119905minus120591(119905))) Δ119905 = int
120581+120596
120581
119889 (119905) Δ119905 =119889120596
(23)
From (22) we have
119891 (0) 120596 = int
120581+120596
120581
[119891 (119905 0) minus 119891 (119905 e1199061(119905minus120590(119905)))
+119901 (119905 e1199061(119905)minus1199062(119905))] Δ119905
(24)
It follows from (20) (22) (23) (24) and (H3)ndash(H5) that
int
120581+120596
120581
10038161003816100381610038161003816119906Δ
1(119905)
10038161003816100381610038161003816Δ119905
le 120582int
120581+120596
120581
10038161003816100381610038161003816119891 (119905 e1199061(119905minus120590(119905))) minus 119898 (119905 e1199061(119905)minus1199062(119905))1003816100381610038161003816
1003816Δ119905
lt int
120581+120596
120581
[119891 (119905 0) minus 119891 (119905 e1199061(119905minus120590(119905)))] + 119901 (119905 e1199061(119905)minus1199062(119905))Δ119905
+ int
120581+120596
120581
1003816100381610038161003816119891 (119905 0)
1003816100381610038161003816Δ119905 = [119865 (0) +
119891 (0)] 120596
(25)
int
120581+120596
120581
10038161003816100381610038161003816119906Δ
2(119905)
10038161003816100381610038161003816Δ119905
le 120582int
120581+120596
120581
10038161003816100381610038161003816minus119889 (119905) + ℎ (119905 e1199061(119905minus120591(119905))minus1199062(119905minus120591(119905)))1003816100381610038161003816
1003816Δ119905
lt int
120581+120596
120581
|119889 (119905)| Δ119905 + int
120581+120596
120581
ℎ (119905 e1199061(119905minus120591(119905))minus1199062(119905minus120591(119905))) Δ119905
= (119863 +119889)120596
(26)
Since (1199061(119905) 1199062(119905))
Tisin 119883 there exist 120585
119894 120578119894isin 119868120596such that
119906119894(120585119894) = max119905isin119868120596
119906119894 (
119905) 119906119894(120578119894) = min119905isin119868120596
119906i (119905) 119894 = 1 2
(27)
Then from (22) (27) (H3) (H4) and the monotonicity of
function 119891 we have
119891 (e1199061(1205851)) =
1
120596
int
120581+120596
120581
119891 (119905 e1199061(1205851)) Δ119905
le
1
120596
int
120581+120596
120581
119891 (119905 e1199061(119905minus120590(119905))) Δ119905
le
1
120596
int
120581+120596
120581
sup119909gt0
119901 (119905 119909) Δ119905
(28)
which together with the monotonicity of function 119891 leads to
1199061(1205851) ge ln119881
1 (29)
By Lemma 2 we obtain from (25) and (29) that for all 119905 isin 119868120596
1199061 (
119905) ge 1199061(1205851) minus
1
2
int
120581+120596
120581
10038161003816100381610038161003816119906Δ
1(119905)
10038161003816100381610038161003816Δ119905
gt ln1198811minus
119865 (0) +119891 (0)
2
120596 = 1198611
(30)
From (22) (27) (H3) and themonotonicity of function119891 we
have
119891 (e1199061(1205781)) =
1
120596
int
120581+120596
120581
119891 (119905 e1199061(1205781)) Δ119905
ge
1
120596
int
120581+120596
120581
119891 (119905 e1199061(119905minus120590(119905))) Δ119905 gt 0
(31)
Discrete Dynamics in Nature and Society 5
which together with the monotonicity of function 119891 leads to
1199061(1205781) le ln119881
2 (32)
By Lemma 2 we get from (25) and (32) that for all 119905 isin 119868120596
1199061 (
119905) le 1199061(1205781) +
1
2
int
120581+120596
120581
10038161003816100381610038161003816119906Δ
1(119905)
10038161003816100381610038161003816Δ119905
lt ln1198812+
119865 (0) +119891 (0)
2
120596 = 1198612
(33)
From (23) (27) (H5) and themonotonicity of function ℎ we
have
ℎ (e1199061(1205851)minus1199062(1205782)) ge
1
120596
int
120581+120596
120581
ℎ (119905 e1199061(119905minus120591(119905))minus1199062(119905minus120591(119905))) Δ119905 =119889
ℎ (e1199061(1205781)minus1199062(1205852)) le
1
120596
int
120581+120596
120581
ℎ (119905 e1199061(119905minus120591(119905))minus1199062(119905minus120591(119905))) Δ119905 =119889
(34)
which together with the monotonicity of function ℎ lead to
e1199061(1205851)minus1199062(1205782) ge V0 e1199061(1205781)minus1199062(1205852) le V
0 (35)
That is
1199062(1205782) le 1199061(1205851) minus ln V
0
1199062(1205852) ge 1199061(1205781) minus ln V
0
(36)
By Lemma 2 we get from (26) (30) (33) and (36) that for all119905 isin 119868120596
1199062 (
119905) le 1199062(1205782) +
1
2
int
120581+120596
120581
10038161003816100381610038161003816119906Δ
2(119905)
10038161003816100381610038161003816Δ119905
lt 1198611minus ln V0+
119863 +119889
2
120596 = 1198613
1199062 (
119905) ge 1199062(1205852) minus
1
2
int
120581+120596
120581
10038161003816100381610038161003816119906Δ
2(119905)
10038161003816100381610038161003816Δ119905
gt 1198612minus ln V0minus
119863 +119889
2
120596 = 1198614
(37)
In view of (30) (33) (37) we obtain
max119905isin119868120596
10038161003816100381610038161199061 (
119905)1003816100381610038161003816le max100381610038161003816
10038161198611
100381610038161003816100381610038161003816100381610038161198612
1003816100381610038161003816 = 119861
5 (38)
max119905isin119868120596
10038161003816100381610038161199062 (
119905)1003816100381610038161003816le max100381610038161003816
10038161198613
100381610038161003816100381610038161003816100381610038161198614
1003816100381610038161003816 = 119861
6 (39)
Clearly 1198615and 119861
6are independent of 120582
From (H3) (H6) and the monotonicity of functions
119891
and ℎ it is easy to show that the algebraic system
119891 (e1199061) minus 119901 (e1199061minus1199062) = 0
minus119889 +
ℎ (e1199061minus1199062) = 0
(40)
has a unique solution (119906lowast
1 119906lowast
2)T
isin R2
We now take Ω = (1199061(119905) 1199062(119905))
Tisin 119883 (119906
1
(119905) 1199062(119905))
T lt 119861
5+ 1198616+ 1198610 where 119861
0is taken sufficiently
large such that (119906lowast1 119906lowast
2)T = |119906
lowast
1| + |119906lowast
2| lt 1198610 With the help
of (38) and (39) it is easy to see that Ω satisfies condition (a)in Lemma 1 When (119906
1(119905) 1199062(119905))
Tisin 120597Ω cap Ker 119871 = 120597Ω cap R2
(1199061(119905) 1199062(119905))
T is a constant vector in R2 with |1199061| + |119906
2| =
1198615+ 1198616+ 1198610 Thus we have
119876119873(
1199061
1199062
) = (
119891 (e1199061) minus 119901 (e1199061minus1199062)
minus119889 +
ℎ (e1199061minus1199062)
) = (
0
0) (41)
This proves that condition (b) in Lemma 1 is satisfiedTaking 119869 = 119868 Im119876 rarr Ker 119871 (119906
1 1199062)T
rarr (1199061 1199062)T a
direct calculation shows that
deg (119869119876119873Ω cap Ker 119871 0)
= signminus 1198911015840(e119906lowast
1)ℎ1015840(e119906lowast
1minus119906lowast
2) e2119906
lowast
1minus119906lowast
2
= 1 = 0
(42)
By now we have proved that Ω satisfies all the requirementsin Lemma 1 Hence (1) has at least one 120596-periodic solutionThis completes the proof
Noticing that both systems (2) and (3) are special cases ofsystem (1) byTheorem 3 we can obtain the following results
Theorem4 Suppose that (H1)ndash(H7) holdThen system (2) has
at least one positive 120596-periodic solution
Theorem5 Suppose that (H1)ndash(H7) holdThen system (3) has
at least one positive 120596-periodic solution
32 Nonmonotone Case In this section we study the exis-tence of120596-periodic solution of system (1) when the numericalresponse function ℎ(119905 119909) satisfies the following nonmono-tone condition
(H8) limVrarr+infinℎ(119905 119909) = 0 for 119905 isin R and there exists apositive constant V such that
(119909 minus V)120597ℎ
120597119909
(119905 119909) lt 0 for 119905 isin T 119909 gt 0 119909 = V (43)
By (H5) and (H
8) we have
(119909 minus V) ℎ1015840 (119909) =
1
120596
int
120581+120596
120581
(119909 minus V)120597ℎ
120597119909
(119905 119909) Δ119905 lt 0
limVrarr+infin
ℎ (119909) =
1
120596
int
120581+120596
120581
lim119909rarr+infin
ℎ (119905 119909) Δ119905 = 0
(44)
then ℎ(119909) is strictly increasing on [0 V] and strictly decreasing
on [V +infin) By these (H5) and (H
6) one can easily see that
the equation
ℎ (119909) =
119889 (45)
has two distinct positive solutions namely V1 V2 Without
loss of generality we suppose that V1lt V2 then V
1lt V lt V
2
6 Discrete Dynamics in Nature and Society
Theorem 6 In addition to (H1)ndash(H6) and (H
8) suppose
further that the following condition holds
(H9) (11988121198811)119890[119865(0)+119891(0)++119889]120596
lt V2V1
Then system (1) has at least two 120596-periodic solutions
Proof The proof is similar to that of Theorem 3 To completethe proof we need to find two disjoint open bounded subsetsin 119883 Under condition (H
9) (34) is no longer valid So we
need to make a corresponding changeDesignating 119911(119905) = 119906
1(119905)minus1199062(119905) there also exist 120585
0 1205780isin 119868120596
such that
119911 (1205850) = max119905isin119868120596
119911 (119905) 119911 (1205780) = min119905isin119868120596
119911 (119905) (46)
From (23) (27) and the monotonicity of functions ℎ and ℎ
we will show that 119911(1205850) and 119911(120578
0) can not simultaneously lie
in (minusinfin ln V1) (ln V
1 ln V2) or (ln V
2 +infin) In fact if 119911(120578
0) le
119911(1205850) lt ln V
1 then
119889 =
1
120596
int
120581+120596
120581
ℎ (119905 e119911(119905minus120591(119905))) Δ119905
leℎ (e119911(1205850)) lt
ℎ (V1) =
119889
(47)
This is a contradiction If ln V2lt 119911(1205780) le 119911(120585
0) then
119889 =
1
120596
int
120581+120596
120581
ℎ (119905 e119911(119905minus120591(119905))) Δ119905
leℎ (e119911(1205780)) lt
ℎ (V2) =
119889
(48)
This is also a contradiction If ln V1
lt 119911(1205780) le 119911(120585
0) lt ln V
2
then
119889 =
1
120596
int
120581+120596
120581
ℎ (119905 e119911(119905minus120591(119905))) Δ119905
ge minℎ (e119911(1205850)) ℎ (e119911(1205780))
gtℎ (V1) =
ℎ (V2) =
119889
(49)
This is also a contradiction Consequently the distribution of119911(1205850) and 119911(120578
0) only have the following two cases
Case 1 119911(1205780) le ln V
2le 119911(1205850) Noticing that
119911 (1205780) ge 1199061(1205781) minus 1199062(1205852)
119911 (1205850) le 1199061(1205851) minus 1199062(1205782)
(50)
from (27) (30) and (33) we obtain
1199062(1205852) ge 1199061(1205781) minus 119911 (120578
0) ge 1199061(1205781) minus ln V
2gt 1198611minus ln V2
1199062(1205782) le 1199061(1205851) minus 119911 (120585
0) le 1199061(1205851) minus ln V
2lt 1198612minus ln V2
(51)
By Lemma 2 we find from (26) and (51) that for all 119905 isin 119868120596
1199062 (
119905) ge 1199062(1205852) minus
1
2
int
120581+120596
120581
10038161003816100381610038161003816119906Δ
2(119905)
10038161003816100381610038161003816Δ119905
gt 1198611minus ln V2minus
119863 +119889
2
120596 = 1198617
1199062 (
119905) le 1199062(1205782) +
1
2
int
120581+120596
120581
10038161003816100381610038161003816119906Δ
2(119905)
10038161003816100381610038161003816Δ119905
lt 1198612minus ln V2+
119863 +119889
2
120596 = 1198618
(52)
Case 2 119911(1205780) le ln V
1le 119911(120585
0) From (27) (30) and (33) we
also obtain
1199062(1205852) ge 1199061(1205781) minus 119911 (120578
0) ge 1199061(1205781) minus ln V
1gt 1198611minus ln V1
1199062(1205782) le 1199061(1205851) minus 119911 (120585
0) le 1199061(1205851) minus ln V
1lt 1198612minus ln V1
(53)
By Lemma 2 we find from (26) and (53) that for all 119905 isin 119868120596
1199062 (
119905) ge 1199062(1205852) minus
1
2
int
120581+120596
120581
10038161003816100381610038161003816119906Δ
2(119905)
10038161003816100381610038161003816Δ119905
gt 1198611minus ln V1minus
119863 +119889
2
120596 = 1198619
1199062 (
119905) le 1199062(1205782) +
1
2
int
120581+120596
120581
10038161003816100381610038161003816119906Δ
2(119905)
10038161003816100381610038161003816Δ119905
lt 1198612minus ln V1+
119863 +119889
2
120596 = 11986110
(54)
By (H9) we know that
1198617lt 1198618lt 1198619lt 11986110 (55)
Clearly 1198617 1198618 1198619 and 119861
10are independent of 120582
By themonotonicity of functions 119891 and
ℎ (H3) (H5) and
(H6) it is easy to show that algebraic system (40) has two
distinct solutions
119906(119894)
= (ln119881(119894)
0 ln119881(119894)
0minus ln V119894)
T 119894 = 1 2 (56)
where119881(119894)
0is the unique solution of equation
119891(V)minus119901(V119894) = 0
Obviously
1198811le 119881(119894)
0lt 1198812 (57)
We now take
Ω1= (119906
1 (119905) 1199062 (
119905))T
isin 119883 10038161003816100381610038161199061 (
119905)1003816100381610038161003816
lt 1198615+ 1205730 1199062 (
119905) isin (1198617 1198618)
Ω2= (119906
1 (119905) 1199062 (
119905))T
isin 119883 10038161003816100381610038161199061 (
119905)1003816100381610038161003816
lt 1198615+ 1205730 1199062 (
119905) isin (1198619 11986110)
(58)
Discrete Dynamics in Nature and Society 7
where 1205730is taken sufficiently large such that
1205730gt max
10038161003816100381610038161003816ln119881(1)
0
10038161003816100381610038161003816
10038161003816100381610038161003816ln119881(2)
0
10038161003816100381610038161003816 (59)
Then both Ω1and Ω
2are bounded open subsets of 119883 It
follows from (55) (57) and (59) that 119906(1) isin Ω1 119906(2) isin Ω
2 and
Ω1cap Ω2= With the help of (38) (52) (54) and (59) it is
easy to see that Ω1and Ω
2satisfy condition (a) in Lemma 1
When (1199061(119905) 1199062(119905))
Tisin 120597Ω119894cap Ker 119871 = 120597Ω
119894cap R2 (119894 = 1 2)
(1199061(119905) 1199062(119905))
T is a constant vector in R2 Thus we have
119876119873(
1199061
1199062
) = (
119891 (e1199061) minus 119901 (e1199061minus1199062)
minus119889 +
ℎ (e1199061minus1199062)
) = (
0
0) (60)
This proves that condition (b) in Lemma 1 is satisfied A directcalculation shows that
deg (119869119876119873Ω119894cap Ker 119871 0)
= signminusV119894119881(119894)
01198911015840(119881(119894)
0)ℎ1015840(V119894)
= (minus1)119894minus1
= 0 119894 = 1 2
(61)
By now we have proved that Ω1and Ω
2satisfy all the
requirements in Lemma 1 Hence system (1) has at least two120596-periodic solutions 119906
lowast(119905) and 119906
dagger(119905) in Dom119871 cap Ω
1and
Dom119871 cap Ω2 respectively This completes the proof
Noticing that both systems (2) and (3) are special cases ofsystem (1) byTheorem 6 we can obtain the following results
Theorem 7 Suppose that (H1)ndash(H6) and (H
8)-(H9) hold
Then system (2) has at least two positive 120596-periodic solutions
Theorem 8 Suppose that (H1)ndash(H6) and (H
8)-(H9) hold
Then system (3) has at least two positive 120596-periodic solutions
Remark 9 The proof of Theorems 3 and 6 shows that allthe above results also remain valid if some delay terms arereplaced by terms with distributed or state-dependent delays
Remark 10 In their Theorem 22 Ding and Jiang [14] provedthat (2) has at least two positive 120596-periodic solutions if (H
1)ndash
(H6) (H8) and
(H10158409) (11988121198811)e2[119865(0)+119891(0)++119889]120596 lt V
2V1
hold Obviously the condition (H10158409) implies (H
9) Hence our
Theorem 7 improves Theorem 22 of [14]
Remark 11 Shao [10] studied a special case of system (1) for119891(119905 119909) = 119886(119905) minus 119887(119905)119909 Therefore our Theorem 3 generalizesTheorem 1 of [10]
Remark 12 Fan and Wang [17] Wang and Li [21] and Fanet al [16 18] studied some special cases of system (2) for119891(119905 119909) = 119886(119905) minus 119887(119905)119909 Therefore our Theorem 4 generalizesTheorem 21 of [17] Theorem 31 of [16] Theorem 35 of [18]andTheorem 21 of [21]
Remark 13 Fan and Wang [23] and Xia et al [24] studiedsome special cases of system (3) for 119891(119905 119909) = 119886(119905) minus 119887(119905)119909Therefore our Theorem 5 generalizes Theorem 21 of [23]andTheorem 31 of [24]
Acknowledgments
This work is supported by the National Natural ScienceFoundation of China under Grant 11271110 the Key Programsfor Science and Technology of the Education Department ofHenan Province under Grant 12A110007 and the ScientificResearch Start-up Funds of Henan University of Science andTechnology
References
[1] M Bohner and A PetersonDynamic Equations on Time ScalesAn Introduction with Applications Birkhauser Boston MassUSA 2001
[2] M Bohner and A Peterson Advances in Dynamic Equations onTime Scales Birkhauser Boston Mass USA 2003
[3] M Bohner L Erbe and A Peterson ldquoOscillation for nonlinearsecond order dynamic equations on a time scalerdquo Journal ofMathematical Analysis and Applications vol 301 no 2 pp 491ndash507 2005
[4] M Bohner M Fan and J Zhang ldquoExistence of periodicsolutions in predator-prey and competition dynamic systemsrdquoNonlinear Analysis Real World Applications vol 7 no 5 pp1193ndash1204 2006
[5] X Ding and G Zhao ldquoPeriodic solutions for a semi-ratio-dependent predator-prey system with delays on time scalesrdquoDiscrete Dynamics in Nature and Society vol 2012 Article ID928704 15 pages 2012
[6] M Fazly and M Hesaaraki ldquoPeriodic solutions for a semi-ratio-dependent predator-prey dynamical systemwith a class offunctional responses on time scalesrdquo Discrete and ContinuousDynamical Systems B vol 9 no 2 pp 267ndash279 2008
[7] J Hoffacker and C C Tisdell ldquoStability and instability fordynamic equations on time scalesrdquo Computers amp Mathematicswith Applications vol 49 no 9-10 pp 1327ndash1334 2005
[8] Y Li and H Zhang ldquoExistence of periodic solutions for a peri-odic mutualism model on time scalesrdquo Journal of MathematicalAnalysis and Applications vol 343 no 2 pp 818ndash825 2008
[9] J Liu Y Li and L Zhao ldquoOn a periodic predator-prey systemwith time delays on time scalesrdquo Communications in NonlinearScience and Numerical Simulation vol 14 no 8 pp 3432ndash34382009
[10] Y Shao ldquoExistence of positive periodic solution of a ratio-dependent predator-prey model on time scale with monotonicfunctional responserdquo Advances in Mathematics vol 39 no 2pp 224ndash232 2010
[11] B B Zhang and M Fan ldquoA remark on the application ofcoincidence degree to periodicity of dynamic equtions on timescalesrdquo Journal of Northeast Normal University vol 39 no 4 pp1ndash3 2007
[12] L Zhang H-X Li and X-B Zhang ldquoPeriodic solutions ofcompetition Lotka-Volterra dynamic system on time scalesrdquoComputers amp Mathematics with Applications vol 57 no 7 pp1204ndash1211 2009
8 Discrete Dynamics in Nature and Society
[13] X Ding ldquoPositive periodic solutions in generalized ratio-dependent predator-prey systemsrdquo Nonlinear Analysis RealWorld Applications vol 9 no 2 pp 394ndash402 2008
[14] X Ding and J Jiang ldquoMultiple periodic solutions in delayedGause-type ratio-dependent predator-prey systems with non-monotonic numerical responsesrdquo Mathematical and ComputerModelling vol 47 no 11-12 pp 1323ndash1331 2008
[15] Y H Fan andW T Li ldquoPermanence in delayed ratio-dependentpredator-prey models with monotonic functional responsesrdquoNonlinear Analysis Real World Applications vol 8 no 2 pp424ndash434 2007
[16] Y H Fan W T Li and L L Wang ldquoPeriodic solutions ofdelayed ratio-dependent predator-preymodels withmonotonicor nonmonotonic functional responsesrdquo Nonlinear AnalysisReal World Applications vol 5 no 2 pp 247ndash263 2004
[17] M Fan and K Wang ldquoPeriodicity in a delayed ratio-dependentpredator-prey systemrdquo Journal of Mathematical Analysis andApplications vol 262 no 1 pp 179ndash190 2001
[18] M Fan QWang and X Zou ldquoDynamics of a non-autonomousratio-dependent predator-prey systemrdquo Proceedings of the RoyalSociety of Edinburgh A vol 133 no 1 pp 97ndash118 2003
[19] S-B Hsu T-W Hwang and Y Kuang ldquoGlobal analysis ofthe Michaelis-Menten-type ratio-dependent predator-prey sys-temrdquo Journal of Mathematical Biology vol 42 no 6 pp 489ndash506 2001
[20] Y Kuang ldquoRich dynamics of Gauss-type ratio-dependentpredator-prey systemsrdquo Fields Institute Communications vol 21pp 325ndash337 1999
[21] L-L Wang and W-T Li ldquoPeriodic solutions and permanencefor a delayed nonautonomous ratio-dependent predator-preymodel with Holling type functional responserdquo Journal of Com-putational and Applied Mathematics vol 162 no 2 pp 341ndash3572004
[22] Y Xia and M Han ldquoMultiple periodic solutions of a ratio-dependent predator-prey modelrdquo Chaos Solitons amp Fractalsvol 39 no 3 pp 1100ndash1108 2009
[23] M Fan and K Wang ldquoPeriodic solutions of a discrete timenonautonomous ratio-dependent predator-prey systemrdquoMath-ematical and Computer Modelling vol 35 no 9-10 pp 951ndash9612002
[24] Y Xia J Cao andM Lin ldquoDiscrete-time analogues of predator-prey models with monotonic or nonmonotonic functionalresponsesrdquo Nonlinear Analysis Real World Applications vol 8no 4 pp 1079ndash1095 2007
[25] R E Gaines and J L Mawhin Coincidence Degree andNonlinear Differential Equations vol 568 of Lecture Notes inMathematics Springer Berlin Germany 1977
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Stochastic AnalysisInternational Journal of
4 Discrete Dynamics in Nature and Society
Then 119876119873 119883 rarr 119884 and 119870119875(119868 minus 119876)119873 119883 rarr 119883 read as
119876119873119906
=
1
120596
(
int
120581+120596
120581
[119891 (119905 e1199061(119905minus120590(119905))) minus 119901 (119905 e1199061(119905)minus1199062(119905))] Δ119905
int
120581+120596
120581
[minus119889 (119905) + ℎ (119905 e1199061(119905minus120591(119905))minus1199062(119905minus120591(119905)))] Δ119905
)
119870119875 (
119868 minus 119876)119873119906
= int
119905
120581
119873119906 (119904) Δ119904 minus
1
120596
int
120581+120596
120581
int
119905
120581
119873119906 (119904) Δ119904Δ119905
minus
1
120596
(119905 minus 120581 minus
1
120596
int
120581+120596
120581
(119904 minus 120581) Δ119904)int
120581+120596
120581
119873119906 (119904) Δ119904
(19)
Clearly 119876119873 and 119870119875(119868 minus 119876)119873 are continuous By using
the Arzela-Ascoli theorem it is not difficult to prove that119870119875(119868 minus 119876)119873(Ω) is compact for any open bounded setΩ sub 119883
Moreover 119876119873(Ω) is bounded Therefore 119873 is 119871-compact onΩ with any open bounded set Ω sub 119883
In order to apply Lemma 1 we need to find an appropriateopen bounded subset in 119883 Corresponding to the operatorequation 119871119906 = 120582119873119906 120582 isin (0 1) we have
119906Δ
1(119905) = 120582 [119891 (119905 e1199061(119905minus120590(119905))) minus 119901 (119905 e1199061(119905)minus1199062(119905))]
119906Δ
2(119905) = 120582 [minus119889 (119905) + ℎ (119905 e1199061(119905minus120591(119905))minus1199062(119905minus120591(119905)))]
(20)
Suppose that (1199061(119905) 1199062(119905))
Tisin 119883 is a solution of (20) for a
certain 120582 isin (0 1) Integrating (20) on both sides from 120581 to120581 + 120596 leads to
int
120581+120596
120581
120582 [119891 (119905 e1199061(119905minus120590(119905))) minus 119901 (119905 e1199061(119905)minus1199062(119905))] Δ119905 = 0
int
120581+120596
120581
120582 [minus119889 (119905) + ℎ (119905 e1199061(119905minus120591(119905))minus1199062(119905minus120591(119905)))] Δ119905 = 0
(21)
That is
int
120581+120596
120581
119891 (119905 e1199061(119905minus120590(119905))) Δ119905 = int
120581+120596
120581
119901 (119905 e1199061(119905)minus1199062(119905)) Δ119905 (22)
int
120581+120596
120581
ℎ (119905 e1199061(119905minus120591(119905))minus1199062(119905minus120591(119905))) Δ119905 = int
120581+120596
120581
119889 (119905) Δ119905 =119889120596
(23)
From (22) we have
119891 (0) 120596 = int
120581+120596
120581
[119891 (119905 0) minus 119891 (119905 e1199061(119905minus120590(119905)))
+119901 (119905 e1199061(119905)minus1199062(119905))] Δ119905
(24)
It follows from (20) (22) (23) (24) and (H3)ndash(H5) that
int
120581+120596
120581
10038161003816100381610038161003816119906Δ
1(119905)
10038161003816100381610038161003816Δ119905
le 120582int
120581+120596
120581
10038161003816100381610038161003816119891 (119905 e1199061(119905minus120590(119905))) minus 119898 (119905 e1199061(119905)minus1199062(119905))1003816100381610038161003816
1003816Δ119905
lt int
120581+120596
120581
[119891 (119905 0) minus 119891 (119905 e1199061(119905minus120590(119905)))] + 119901 (119905 e1199061(119905)minus1199062(119905))Δ119905
+ int
120581+120596
120581
1003816100381610038161003816119891 (119905 0)
1003816100381610038161003816Δ119905 = [119865 (0) +
119891 (0)] 120596
(25)
int
120581+120596
120581
10038161003816100381610038161003816119906Δ
2(119905)
10038161003816100381610038161003816Δ119905
le 120582int
120581+120596
120581
10038161003816100381610038161003816minus119889 (119905) + ℎ (119905 e1199061(119905minus120591(119905))minus1199062(119905minus120591(119905)))1003816100381610038161003816
1003816Δ119905
lt int
120581+120596
120581
|119889 (119905)| Δ119905 + int
120581+120596
120581
ℎ (119905 e1199061(119905minus120591(119905))minus1199062(119905minus120591(119905))) Δ119905
= (119863 +119889)120596
(26)
Since (1199061(119905) 1199062(119905))
Tisin 119883 there exist 120585
119894 120578119894isin 119868120596such that
119906119894(120585119894) = max119905isin119868120596
119906119894 (
119905) 119906119894(120578119894) = min119905isin119868120596
119906i (119905) 119894 = 1 2
(27)
Then from (22) (27) (H3) (H4) and the monotonicity of
function 119891 we have
119891 (e1199061(1205851)) =
1
120596
int
120581+120596
120581
119891 (119905 e1199061(1205851)) Δ119905
le
1
120596
int
120581+120596
120581
119891 (119905 e1199061(119905minus120590(119905))) Δ119905
le
1
120596
int
120581+120596
120581
sup119909gt0
119901 (119905 119909) Δ119905
(28)
which together with the monotonicity of function 119891 leads to
1199061(1205851) ge ln119881
1 (29)
By Lemma 2 we obtain from (25) and (29) that for all 119905 isin 119868120596
1199061 (
119905) ge 1199061(1205851) minus
1
2
int
120581+120596
120581
10038161003816100381610038161003816119906Δ
1(119905)
10038161003816100381610038161003816Δ119905
gt ln1198811minus
119865 (0) +119891 (0)
2
120596 = 1198611
(30)
From (22) (27) (H3) and themonotonicity of function119891 we
have
119891 (e1199061(1205781)) =
1
120596
int
120581+120596
120581
119891 (119905 e1199061(1205781)) Δ119905
ge
1
120596
int
120581+120596
120581
119891 (119905 e1199061(119905minus120590(119905))) Δ119905 gt 0
(31)
Discrete Dynamics in Nature and Society 5
which together with the monotonicity of function 119891 leads to
1199061(1205781) le ln119881
2 (32)
By Lemma 2 we get from (25) and (32) that for all 119905 isin 119868120596
1199061 (
119905) le 1199061(1205781) +
1
2
int
120581+120596
120581
10038161003816100381610038161003816119906Δ
1(119905)
10038161003816100381610038161003816Δ119905
lt ln1198812+
119865 (0) +119891 (0)
2
120596 = 1198612
(33)
From (23) (27) (H5) and themonotonicity of function ℎ we
have
ℎ (e1199061(1205851)minus1199062(1205782)) ge
1
120596
int
120581+120596
120581
ℎ (119905 e1199061(119905minus120591(119905))minus1199062(119905minus120591(119905))) Δ119905 =119889
ℎ (e1199061(1205781)minus1199062(1205852)) le
1
120596
int
120581+120596
120581
ℎ (119905 e1199061(119905minus120591(119905))minus1199062(119905minus120591(119905))) Δ119905 =119889
(34)
which together with the monotonicity of function ℎ lead to
e1199061(1205851)minus1199062(1205782) ge V0 e1199061(1205781)minus1199062(1205852) le V
0 (35)
That is
1199062(1205782) le 1199061(1205851) minus ln V
0
1199062(1205852) ge 1199061(1205781) minus ln V
0
(36)
By Lemma 2 we get from (26) (30) (33) and (36) that for all119905 isin 119868120596
1199062 (
119905) le 1199062(1205782) +
1
2
int
120581+120596
120581
10038161003816100381610038161003816119906Δ
2(119905)
10038161003816100381610038161003816Δ119905
lt 1198611minus ln V0+
119863 +119889
2
120596 = 1198613
1199062 (
119905) ge 1199062(1205852) minus
1
2
int
120581+120596
120581
10038161003816100381610038161003816119906Δ
2(119905)
10038161003816100381610038161003816Δ119905
gt 1198612minus ln V0minus
119863 +119889
2
120596 = 1198614
(37)
In view of (30) (33) (37) we obtain
max119905isin119868120596
10038161003816100381610038161199061 (
119905)1003816100381610038161003816le max100381610038161003816
10038161198611
100381610038161003816100381610038161003816100381610038161198612
1003816100381610038161003816 = 119861
5 (38)
max119905isin119868120596
10038161003816100381610038161199062 (
119905)1003816100381610038161003816le max100381610038161003816
10038161198613
100381610038161003816100381610038161003816100381610038161198614
1003816100381610038161003816 = 119861
6 (39)
Clearly 1198615and 119861
6are independent of 120582
From (H3) (H6) and the monotonicity of functions
119891
and ℎ it is easy to show that the algebraic system
119891 (e1199061) minus 119901 (e1199061minus1199062) = 0
minus119889 +
ℎ (e1199061minus1199062) = 0
(40)
has a unique solution (119906lowast
1 119906lowast
2)T
isin R2
We now take Ω = (1199061(119905) 1199062(119905))
Tisin 119883 (119906
1
(119905) 1199062(119905))
T lt 119861
5+ 1198616+ 1198610 where 119861
0is taken sufficiently
large such that (119906lowast1 119906lowast
2)T = |119906
lowast
1| + |119906lowast
2| lt 1198610 With the help
of (38) and (39) it is easy to see that Ω satisfies condition (a)in Lemma 1 When (119906
1(119905) 1199062(119905))
Tisin 120597Ω cap Ker 119871 = 120597Ω cap R2
(1199061(119905) 1199062(119905))
T is a constant vector in R2 with |1199061| + |119906
2| =
1198615+ 1198616+ 1198610 Thus we have
119876119873(
1199061
1199062
) = (
119891 (e1199061) minus 119901 (e1199061minus1199062)
minus119889 +
ℎ (e1199061minus1199062)
) = (
0
0) (41)
This proves that condition (b) in Lemma 1 is satisfiedTaking 119869 = 119868 Im119876 rarr Ker 119871 (119906
1 1199062)T
rarr (1199061 1199062)T a
direct calculation shows that
deg (119869119876119873Ω cap Ker 119871 0)
= signminus 1198911015840(e119906lowast
1)ℎ1015840(e119906lowast
1minus119906lowast
2) e2119906
lowast
1minus119906lowast
2
= 1 = 0
(42)
By now we have proved that Ω satisfies all the requirementsin Lemma 1 Hence (1) has at least one 120596-periodic solutionThis completes the proof
Noticing that both systems (2) and (3) are special cases ofsystem (1) byTheorem 3 we can obtain the following results
Theorem4 Suppose that (H1)ndash(H7) holdThen system (2) has
at least one positive 120596-periodic solution
Theorem5 Suppose that (H1)ndash(H7) holdThen system (3) has
at least one positive 120596-periodic solution
32 Nonmonotone Case In this section we study the exis-tence of120596-periodic solution of system (1) when the numericalresponse function ℎ(119905 119909) satisfies the following nonmono-tone condition
(H8) limVrarr+infinℎ(119905 119909) = 0 for 119905 isin R and there exists apositive constant V such that
(119909 minus V)120597ℎ
120597119909
(119905 119909) lt 0 for 119905 isin T 119909 gt 0 119909 = V (43)
By (H5) and (H
8) we have
(119909 minus V) ℎ1015840 (119909) =
1
120596
int
120581+120596
120581
(119909 minus V)120597ℎ
120597119909
(119905 119909) Δ119905 lt 0
limVrarr+infin
ℎ (119909) =
1
120596
int
120581+120596
120581
lim119909rarr+infin
ℎ (119905 119909) Δ119905 = 0
(44)
then ℎ(119909) is strictly increasing on [0 V] and strictly decreasing
on [V +infin) By these (H5) and (H
6) one can easily see that
the equation
ℎ (119909) =
119889 (45)
has two distinct positive solutions namely V1 V2 Without
loss of generality we suppose that V1lt V2 then V
1lt V lt V
2
6 Discrete Dynamics in Nature and Society
Theorem 6 In addition to (H1)ndash(H6) and (H
8) suppose
further that the following condition holds
(H9) (11988121198811)119890[119865(0)+119891(0)++119889]120596
lt V2V1
Then system (1) has at least two 120596-periodic solutions
Proof The proof is similar to that of Theorem 3 To completethe proof we need to find two disjoint open bounded subsetsin 119883 Under condition (H
9) (34) is no longer valid So we
need to make a corresponding changeDesignating 119911(119905) = 119906
1(119905)minus1199062(119905) there also exist 120585
0 1205780isin 119868120596
such that
119911 (1205850) = max119905isin119868120596
119911 (119905) 119911 (1205780) = min119905isin119868120596
119911 (119905) (46)
From (23) (27) and the monotonicity of functions ℎ and ℎ
we will show that 119911(1205850) and 119911(120578
0) can not simultaneously lie
in (minusinfin ln V1) (ln V
1 ln V2) or (ln V
2 +infin) In fact if 119911(120578
0) le
119911(1205850) lt ln V
1 then
119889 =
1
120596
int
120581+120596
120581
ℎ (119905 e119911(119905minus120591(119905))) Δ119905
leℎ (e119911(1205850)) lt
ℎ (V1) =
119889
(47)
This is a contradiction If ln V2lt 119911(1205780) le 119911(120585
0) then
119889 =
1
120596
int
120581+120596
120581
ℎ (119905 e119911(119905minus120591(119905))) Δ119905
leℎ (e119911(1205780)) lt
ℎ (V2) =
119889
(48)
This is also a contradiction If ln V1
lt 119911(1205780) le 119911(120585
0) lt ln V
2
then
119889 =
1
120596
int
120581+120596
120581
ℎ (119905 e119911(119905minus120591(119905))) Δ119905
ge minℎ (e119911(1205850)) ℎ (e119911(1205780))
gtℎ (V1) =
ℎ (V2) =
119889
(49)
This is also a contradiction Consequently the distribution of119911(1205850) and 119911(120578
0) only have the following two cases
Case 1 119911(1205780) le ln V
2le 119911(1205850) Noticing that
119911 (1205780) ge 1199061(1205781) minus 1199062(1205852)
119911 (1205850) le 1199061(1205851) minus 1199062(1205782)
(50)
from (27) (30) and (33) we obtain
1199062(1205852) ge 1199061(1205781) minus 119911 (120578
0) ge 1199061(1205781) minus ln V
2gt 1198611minus ln V2
1199062(1205782) le 1199061(1205851) minus 119911 (120585
0) le 1199061(1205851) minus ln V
2lt 1198612minus ln V2
(51)
By Lemma 2 we find from (26) and (51) that for all 119905 isin 119868120596
1199062 (
119905) ge 1199062(1205852) minus
1
2
int
120581+120596
120581
10038161003816100381610038161003816119906Δ
2(119905)
10038161003816100381610038161003816Δ119905
gt 1198611minus ln V2minus
119863 +119889
2
120596 = 1198617
1199062 (
119905) le 1199062(1205782) +
1
2
int
120581+120596
120581
10038161003816100381610038161003816119906Δ
2(119905)
10038161003816100381610038161003816Δ119905
lt 1198612minus ln V2+
119863 +119889
2
120596 = 1198618
(52)
Case 2 119911(1205780) le ln V
1le 119911(120585
0) From (27) (30) and (33) we
also obtain
1199062(1205852) ge 1199061(1205781) minus 119911 (120578
0) ge 1199061(1205781) minus ln V
1gt 1198611minus ln V1
1199062(1205782) le 1199061(1205851) minus 119911 (120585
0) le 1199061(1205851) minus ln V
1lt 1198612minus ln V1
(53)
By Lemma 2 we find from (26) and (53) that for all 119905 isin 119868120596
1199062 (
119905) ge 1199062(1205852) minus
1
2
int
120581+120596
120581
10038161003816100381610038161003816119906Δ
2(119905)
10038161003816100381610038161003816Δ119905
gt 1198611minus ln V1minus
119863 +119889
2
120596 = 1198619
1199062 (
119905) le 1199062(1205782) +
1
2
int
120581+120596
120581
10038161003816100381610038161003816119906Δ
2(119905)
10038161003816100381610038161003816Δ119905
lt 1198612minus ln V1+
119863 +119889
2
120596 = 11986110
(54)
By (H9) we know that
1198617lt 1198618lt 1198619lt 11986110 (55)
Clearly 1198617 1198618 1198619 and 119861
10are independent of 120582
By themonotonicity of functions 119891 and
ℎ (H3) (H5) and
(H6) it is easy to show that algebraic system (40) has two
distinct solutions
119906(119894)
= (ln119881(119894)
0 ln119881(119894)
0minus ln V119894)
T 119894 = 1 2 (56)
where119881(119894)
0is the unique solution of equation
119891(V)minus119901(V119894) = 0
Obviously
1198811le 119881(119894)
0lt 1198812 (57)
We now take
Ω1= (119906
1 (119905) 1199062 (
119905))T
isin 119883 10038161003816100381610038161199061 (
119905)1003816100381610038161003816
lt 1198615+ 1205730 1199062 (
119905) isin (1198617 1198618)
Ω2= (119906
1 (119905) 1199062 (
119905))T
isin 119883 10038161003816100381610038161199061 (
119905)1003816100381610038161003816
lt 1198615+ 1205730 1199062 (
119905) isin (1198619 11986110)
(58)
Discrete Dynamics in Nature and Society 7
where 1205730is taken sufficiently large such that
1205730gt max
10038161003816100381610038161003816ln119881(1)
0
10038161003816100381610038161003816
10038161003816100381610038161003816ln119881(2)
0
10038161003816100381610038161003816 (59)
Then both Ω1and Ω
2are bounded open subsets of 119883 It
follows from (55) (57) and (59) that 119906(1) isin Ω1 119906(2) isin Ω
2 and
Ω1cap Ω2= With the help of (38) (52) (54) and (59) it is
easy to see that Ω1and Ω
2satisfy condition (a) in Lemma 1
When (1199061(119905) 1199062(119905))
Tisin 120597Ω119894cap Ker 119871 = 120597Ω
119894cap R2 (119894 = 1 2)
(1199061(119905) 1199062(119905))
T is a constant vector in R2 Thus we have
119876119873(
1199061
1199062
) = (
119891 (e1199061) minus 119901 (e1199061minus1199062)
minus119889 +
ℎ (e1199061minus1199062)
) = (
0
0) (60)
This proves that condition (b) in Lemma 1 is satisfied A directcalculation shows that
deg (119869119876119873Ω119894cap Ker 119871 0)
= signminusV119894119881(119894)
01198911015840(119881(119894)
0)ℎ1015840(V119894)
= (minus1)119894minus1
= 0 119894 = 1 2
(61)
By now we have proved that Ω1and Ω
2satisfy all the
requirements in Lemma 1 Hence system (1) has at least two120596-periodic solutions 119906
lowast(119905) and 119906
dagger(119905) in Dom119871 cap Ω
1and
Dom119871 cap Ω2 respectively This completes the proof
Noticing that both systems (2) and (3) are special cases ofsystem (1) byTheorem 6 we can obtain the following results
Theorem 7 Suppose that (H1)ndash(H6) and (H
8)-(H9) hold
Then system (2) has at least two positive 120596-periodic solutions
Theorem 8 Suppose that (H1)ndash(H6) and (H
8)-(H9) hold
Then system (3) has at least two positive 120596-periodic solutions
Remark 9 The proof of Theorems 3 and 6 shows that allthe above results also remain valid if some delay terms arereplaced by terms with distributed or state-dependent delays
Remark 10 In their Theorem 22 Ding and Jiang [14] provedthat (2) has at least two positive 120596-periodic solutions if (H
1)ndash
(H6) (H8) and
(H10158409) (11988121198811)e2[119865(0)+119891(0)++119889]120596 lt V
2V1
hold Obviously the condition (H10158409) implies (H
9) Hence our
Theorem 7 improves Theorem 22 of [14]
Remark 11 Shao [10] studied a special case of system (1) for119891(119905 119909) = 119886(119905) minus 119887(119905)119909 Therefore our Theorem 3 generalizesTheorem 1 of [10]
Remark 12 Fan and Wang [17] Wang and Li [21] and Fanet al [16 18] studied some special cases of system (2) for119891(119905 119909) = 119886(119905) minus 119887(119905)119909 Therefore our Theorem 4 generalizesTheorem 21 of [17] Theorem 31 of [16] Theorem 35 of [18]andTheorem 21 of [21]
Remark 13 Fan and Wang [23] and Xia et al [24] studiedsome special cases of system (3) for 119891(119905 119909) = 119886(119905) minus 119887(119905)119909Therefore our Theorem 5 generalizes Theorem 21 of [23]andTheorem 31 of [24]
Acknowledgments
This work is supported by the National Natural ScienceFoundation of China under Grant 11271110 the Key Programsfor Science and Technology of the Education Department ofHenan Province under Grant 12A110007 and the ScientificResearch Start-up Funds of Henan University of Science andTechnology
References
[1] M Bohner and A PetersonDynamic Equations on Time ScalesAn Introduction with Applications Birkhauser Boston MassUSA 2001
[2] M Bohner and A Peterson Advances in Dynamic Equations onTime Scales Birkhauser Boston Mass USA 2003
[3] M Bohner L Erbe and A Peterson ldquoOscillation for nonlinearsecond order dynamic equations on a time scalerdquo Journal ofMathematical Analysis and Applications vol 301 no 2 pp 491ndash507 2005
[4] M Bohner M Fan and J Zhang ldquoExistence of periodicsolutions in predator-prey and competition dynamic systemsrdquoNonlinear Analysis Real World Applications vol 7 no 5 pp1193ndash1204 2006
[5] X Ding and G Zhao ldquoPeriodic solutions for a semi-ratio-dependent predator-prey system with delays on time scalesrdquoDiscrete Dynamics in Nature and Society vol 2012 Article ID928704 15 pages 2012
[6] M Fazly and M Hesaaraki ldquoPeriodic solutions for a semi-ratio-dependent predator-prey dynamical systemwith a class offunctional responses on time scalesrdquo Discrete and ContinuousDynamical Systems B vol 9 no 2 pp 267ndash279 2008
[7] J Hoffacker and C C Tisdell ldquoStability and instability fordynamic equations on time scalesrdquo Computers amp Mathematicswith Applications vol 49 no 9-10 pp 1327ndash1334 2005
[8] Y Li and H Zhang ldquoExistence of periodic solutions for a peri-odic mutualism model on time scalesrdquo Journal of MathematicalAnalysis and Applications vol 343 no 2 pp 818ndash825 2008
[9] J Liu Y Li and L Zhao ldquoOn a periodic predator-prey systemwith time delays on time scalesrdquo Communications in NonlinearScience and Numerical Simulation vol 14 no 8 pp 3432ndash34382009
[10] Y Shao ldquoExistence of positive periodic solution of a ratio-dependent predator-prey model on time scale with monotonicfunctional responserdquo Advances in Mathematics vol 39 no 2pp 224ndash232 2010
[11] B B Zhang and M Fan ldquoA remark on the application ofcoincidence degree to periodicity of dynamic equtions on timescalesrdquo Journal of Northeast Normal University vol 39 no 4 pp1ndash3 2007
[12] L Zhang H-X Li and X-B Zhang ldquoPeriodic solutions ofcompetition Lotka-Volterra dynamic system on time scalesrdquoComputers amp Mathematics with Applications vol 57 no 7 pp1204ndash1211 2009
8 Discrete Dynamics in Nature and Society
[13] X Ding ldquoPositive periodic solutions in generalized ratio-dependent predator-prey systemsrdquo Nonlinear Analysis RealWorld Applications vol 9 no 2 pp 394ndash402 2008
[14] X Ding and J Jiang ldquoMultiple periodic solutions in delayedGause-type ratio-dependent predator-prey systems with non-monotonic numerical responsesrdquo Mathematical and ComputerModelling vol 47 no 11-12 pp 1323ndash1331 2008
[15] Y H Fan andW T Li ldquoPermanence in delayed ratio-dependentpredator-prey models with monotonic functional responsesrdquoNonlinear Analysis Real World Applications vol 8 no 2 pp424ndash434 2007
[16] Y H Fan W T Li and L L Wang ldquoPeriodic solutions ofdelayed ratio-dependent predator-preymodels withmonotonicor nonmonotonic functional responsesrdquo Nonlinear AnalysisReal World Applications vol 5 no 2 pp 247ndash263 2004
[17] M Fan and K Wang ldquoPeriodicity in a delayed ratio-dependentpredator-prey systemrdquo Journal of Mathematical Analysis andApplications vol 262 no 1 pp 179ndash190 2001
[18] M Fan QWang and X Zou ldquoDynamics of a non-autonomousratio-dependent predator-prey systemrdquo Proceedings of the RoyalSociety of Edinburgh A vol 133 no 1 pp 97ndash118 2003
[19] S-B Hsu T-W Hwang and Y Kuang ldquoGlobal analysis ofthe Michaelis-Menten-type ratio-dependent predator-prey sys-temrdquo Journal of Mathematical Biology vol 42 no 6 pp 489ndash506 2001
[20] Y Kuang ldquoRich dynamics of Gauss-type ratio-dependentpredator-prey systemsrdquo Fields Institute Communications vol 21pp 325ndash337 1999
[21] L-L Wang and W-T Li ldquoPeriodic solutions and permanencefor a delayed nonautonomous ratio-dependent predator-preymodel with Holling type functional responserdquo Journal of Com-putational and Applied Mathematics vol 162 no 2 pp 341ndash3572004
[22] Y Xia and M Han ldquoMultiple periodic solutions of a ratio-dependent predator-prey modelrdquo Chaos Solitons amp Fractalsvol 39 no 3 pp 1100ndash1108 2009
[23] M Fan and K Wang ldquoPeriodic solutions of a discrete timenonautonomous ratio-dependent predator-prey systemrdquoMath-ematical and Computer Modelling vol 35 no 9-10 pp 951ndash9612002
[24] Y Xia J Cao andM Lin ldquoDiscrete-time analogues of predator-prey models with monotonic or nonmonotonic functionalresponsesrdquo Nonlinear Analysis Real World Applications vol 8no 4 pp 1079ndash1095 2007
[25] R E Gaines and J L Mawhin Coincidence Degree andNonlinear Differential Equations vol 568 of Lecture Notes inMathematics Springer Berlin Germany 1977
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
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Differential EquationsInternational Journal of
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Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
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Algebra
Discrete Dynamics in Nature and Society
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
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Decision SciencesAdvances in
Discrete MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
Discrete Dynamics in Nature and Society 5
which together with the monotonicity of function 119891 leads to
1199061(1205781) le ln119881
2 (32)
By Lemma 2 we get from (25) and (32) that for all 119905 isin 119868120596
1199061 (
119905) le 1199061(1205781) +
1
2
int
120581+120596
120581
10038161003816100381610038161003816119906Δ
1(119905)
10038161003816100381610038161003816Δ119905
lt ln1198812+
119865 (0) +119891 (0)
2
120596 = 1198612
(33)
From (23) (27) (H5) and themonotonicity of function ℎ we
have
ℎ (e1199061(1205851)minus1199062(1205782)) ge
1
120596
int
120581+120596
120581
ℎ (119905 e1199061(119905minus120591(119905))minus1199062(119905minus120591(119905))) Δ119905 =119889
ℎ (e1199061(1205781)minus1199062(1205852)) le
1
120596
int
120581+120596
120581
ℎ (119905 e1199061(119905minus120591(119905))minus1199062(119905minus120591(119905))) Δ119905 =119889
(34)
which together with the monotonicity of function ℎ lead to
e1199061(1205851)minus1199062(1205782) ge V0 e1199061(1205781)minus1199062(1205852) le V
0 (35)
That is
1199062(1205782) le 1199061(1205851) minus ln V
0
1199062(1205852) ge 1199061(1205781) minus ln V
0
(36)
By Lemma 2 we get from (26) (30) (33) and (36) that for all119905 isin 119868120596
1199062 (
119905) le 1199062(1205782) +
1
2
int
120581+120596
120581
10038161003816100381610038161003816119906Δ
2(119905)
10038161003816100381610038161003816Δ119905
lt 1198611minus ln V0+
119863 +119889
2
120596 = 1198613
1199062 (
119905) ge 1199062(1205852) minus
1
2
int
120581+120596
120581
10038161003816100381610038161003816119906Δ
2(119905)
10038161003816100381610038161003816Δ119905
gt 1198612minus ln V0minus
119863 +119889
2
120596 = 1198614
(37)
In view of (30) (33) (37) we obtain
max119905isin119868120596
10038161003816100381610038161199061 (
119905)1003816100381610038161003816le max100381610038161003816
10038161198611
100381610038161003816100381610038161003816100381610038161198612
1003816100381610038161003816 = 119861
5 (38)
max119905isin119868120596
10038161003816100381610038161199062 (
119905)1003816100381610038161003816le max100381610038161003816
10038161198613
100381610038161003816100381610038161003816100381610038161198614
1003816100381610038161003816 = 119861
6 (39)
Clearly 1198615and 119861
6are independent of 120582
From (H3) (H6) and the monotonicity of functions
119891
and ℎ it is easy to show that the algebraic system
119891 (e1199061) minus 119901 (e1199061minus1199062) = 0
minus119889 +
ℎ (e1199061minus1199062) = 0
(40)
has a unique solution (119906lowast
1 119906lowast
2)T
isin R2
We now take Ω = (1199061(119905) 1199062(119905))
Tisin 119883 (119906
1
(119905) 1199062(119905))
T lt 119861
5+ 1198616+ 1198610 where 119861
0is taken sufficiently
large such that (119906lowast1 119906lowast
2)T = |119906
lowast
1| + |119906lowast
2| lt 1198610 With the help
of (38) and (39) it is easy to see that Ω satisfies condition (a)in Lemma 1 When (119906
1(119905) 1199062(119905))
Tisin 120597Ω cap Ker 119871 = 120597Ω cap R2
(1199061(119905) 1199062(119905))
T is a constant vector in R2 with |1199061| + |119906
2| =
1198615+ 1198616+ 1198610 Thus we have
119876119873(
1199061
1199062
) = (
119891 (e1199061) minus 119901 (e1199061minus1199062)
minus119889 +
ℎ (e1199061minus1199062)
) = (
0
0) (41)
This proves that condition (b) in Lemma 1 is satisfiedTaking 119869 = 119868 Im119876 rarr Ker 119871 (119906
1 1199062)T
rarr (1199061 1199062)T a
direct calculation shows that
deg (119869119876119873Ω cap Ker 119871 0)
= signminus 1198911015840(e119906lowast
1)ℎ1015840(e119906lowast
1minus119906lowast
2) e2119906
lowast
1minus119906lowast
2
= 1 = 0
(42)
By now we have proved that Ω satisfies all the requirementsin Lemma 1 Hence (1) has at least one 120596-periodic solutionThis completes the proof
Noticing that both systems (2) and (3) are special cases ofsystem (1) byTheorem 3 we can obtain the following results
Theorem4 Suppose that (H1)ndash(H7) holdThen system (2) has
at least one positive 120596-periodic solution
Theorem5 Suppose that (H1)ndash(H7) holdThen system (3) has
at least one positive 120596-periodic solution
32 Nonmonotone Case In this section we study the exis-tence of120596-periodic solution of system (1) when the numericalresponse function ℎ(119905 119909) satisfies the following nonmono-tone condition
(H8) limVrarr+infinℎ(119905 119909) = 0 for 119905 isin R and there exists apositive constant V such that
(119909 minus V)120597ℎ
120597119909
(119905 119909) lt 0 for 119905 isin T 119909 gt 0 119909 = V (43)
By (H5) and (H
8) we have
(119909 minus V) ℎ1015840 (119909) =
1
120596
int
120581+120596
120581
(119909 minus V)120597ℎ
120597119909
(119905 119909) Δ119905 lt 0
limVrarr+infin
ℎ (119909) =
1
120596
int
120581+120596
120581
lim119909rarr+infin
ℎ (119905 119909) Δ119905 = 0
(44)
then ℎ(119909) is strictly increasing on [0 V] and strictly decreasing
on [V +infin) By these (H5) and (H
6) one can easily see that
the equation
ℎ (119909) =
119889 (45)
has two distinct positive solutions namely V1 V2 Without
loss of generality we suppose that V1lt V2 then V
1lt V lt V
2
6 Discrete Dynamics in Nature and Society
Theorem 6 In addition to (H1)ndash(H6) and (H
8) suppose
further that the following condition holds
(H9) (11988121198811)119890[119865(0)+119891(0)++119889]120596
lt V2V1
Then system (1) has at least two 120596-periodic solutions
Proof The proof is similar to that of Theorem 3 To completethe proof we need to find two disjoint open bounded subsetsin 119883 Under condition (H
9) (34) is no longer valid So we
need to make a corresponding changeDesignating 119911(119905) = 119906
1(119905)minus1199062(119905) there also exist 120585
0 1205780isin 119868120596
such that
119911 (1205850) = max119905isin119868120596
119911 (119905) 119911 (1205780) = min119905isin119868120596
119911 (119905) (46)
From (23) (27) and the monotonicity of functions ℎ and ℎ
we will show that 119911(1205850) and 119911(120578
0) can not simultaneously lie
in (minusinfin ln V1) (ln V
1 ln V2) or (ln V
2 +infin) In fact if 119911(120578
0) le
119911(1205850) lt ln V
1 then
119889 =
1
120596
int
120581+120596
120581
ℎ (119905 e119911(119905minus120591(119905))) Δ119905
leℎ (e119911(1205850)) lt
ℎ (V1) =
119889
(47)
This is a contradiction If ln V2lt 119911(1205780) le 119911(120585
0) then
119889 =
1
120596
int
120581+120596
120581
ℎ (119905 e119911(119905minus120591(119905))) Δ119905
leℎ (e119911(1205780)) lt
ℎ (V2) =
119889
(48)
This is also a contradiction If ln V1
lt 119911(1205780) le 119911(120585
0) lt ln V
2
then
119889 =
1
120596
int
120581+120596
120581
ℎ (119905 e119911(119905minus120591(119905))) Δ119905
ge minℎ (e119911(1205850)) ℎ (e119911(1205780))
gtℎ (V1) =
ℎ (V2) =
119889
(49)
This is also a contradiction Consequently the distribution of119911(1205850) and 119911(120578
0) only have the following two cases
Case 1 119911(1205780) le ln V
2le 119911(1205850) Noticing that
119911 (1205780) ge 1199061(1205781) minus 1199062(1205852)
119911 (1205850) le 1199061(1205851) minus 1199062(1205782)
(50)
from (27) (30) and (33) we obtain
1199062(1205852) ge 1199061(1205781) minus 119911 (120578
0) ge 1199061(1205781) minus ln V
2gt 1198611minus ln V2
1199062(1205782) le 1199061(1205851) minus 119911 (120585
0) le 1199061(1205851) minus ln V
2lt 1198612minus ln V2
(51)
By Lemma 2 we find from (26) and (51) that for all 119905 isin 119868120596
1199062 (
119905) ge 1199062(1205852) minus
1
2
int
120581+120596
120581
10038161003816100381610038161003816119906Δ
2(119905)
10038161003816100381610038161003816Δ119905
gt 1198611minus ln V2minus
119863 +119889
2
120596 = 1198617
1199062 (
119905) le 1199062(1205782) +
1
2
int
120581+120596
120581
10038161003816100381610038161003816119906Δ
2(119905)
10038161003816100381610038161003816Δ119905
lt 1198612minus ln V2+
119863 +119889
2
120596 = 1198618
(52)
Case 2 119911(1205780) le ln V
1le 119911(120585
0) From (27) (30) and (33) we
also obtain
1199062(1205852) ge 1199061(1205781) minus 119911 (120578
0) ge 1199061(1205781) minus ln V
1gt 1198611minus ln V1
1199062(1205782) le 1199061(1205851) minus 119911 (120585
0) le 1199061(1205851) minus ln V
1lt 1198612minus ln V1
(53)
By Lemma 2 we find from (26) and (53) that for all 119905 isin 119868120596
1199062 (
119905) ge 1199062(1205852) minus
1
2
int
120581+120596
120581
10038161003816100381610038161003816119906Δ
2(119905)
10038161003816100381610038161003816Δ119905
gt 1198611minus ln V1minus
119863 +119889
2
120596 = 1198619
1199062 (
119905) le 1199062(1205782) +
1
2
int
120581+120596
120581
10038161003816100381610038161003816119906Δ
2(119905)
10038161003816100381610038161003816Δ119905
lt 1198612minus ln V1+
119863 +119889
2
120596 = 11986110
(54)
By (H9) we know that
1198617lt 1198618lt 1198619lt 11986110 (55)
Clearly 1198617 1198618 1198619 and 119861
10are independent of 120582
By themonotonicity of functions 119891 and
ℎ (H3) (H5) and
(H6) it is easy to show that algebraic system (40) has two
distinct solutions
119906(119894)
= (ln119881(119894)
0 ln119881(119894)
0minus ln V119894)
T 119894 = 1 2 (56)
where119881(119894)
0is the unique solution of equation
119891(V)minus119901(V119894) = 0
Obviously
1198811le 119881(119894)
0lt 1198812 (57)
We now take
Ω1= (119906
1 (119905) 1199062 (
119905))T
isin 119883 10038161003816100381610038161199061 (
119905)1003816100381610038161003816
lt 1198615+ 1205730 1199062 (
119905) isin (1198617 1198618)
Ω2= (119906
1 (119905) 1199062 (
119905))T
isin 119883 10038161003816100381610038161199061 (
119905)1003816100381610038161003816
lt 1198615+ 1205730 1199062 (
119905) isin (1198619 11986110)
(58)
Discrete Dynamics in Nature and Society 7
where 1205730is taken sufficiently large such that
1205730gt max
10038161003816100381610038161003816ln119881(1)
0
10038161003816100381610038161003816
10038161003816100381610038161003816ln119881(2)
0
10038161003816100381610038161003816 (59)
Then both Ω1and Ω
2are bounded open subsets of 119883 It
follows from (55) (57) and (59) that 119906(1) isin Ω1 119906(2) isin Ω
2 and
Ω1cap Ω2= With the help of (38) (52) (54) and (59) it is
easy to see that Ω1and Ω
2satisfy condition (a) in Lemma 1
When (1199061(119905) 1199062(119905))
Tisin 120597Ω119894cap Ker 119871 = 120597Ω
119894cap R2 (119894 = 1 2)
(1199061(119905) 1199062(119905))
T is a constant vector in R2 Thus we have
119876119873(
1199061
1199062
) = (
119891 (e1199061) minus 119901 (e1199061minus1199062)
minus119889 +
ℎ (e1199061minus1199062)
) = (
0
0) (60)
This proves that condition (b) in Lemma 1 is satisfied A directcalculation shows that
deg (119869119876119873Ω119894cap Ker 119871 0)
= signminusV119894119881(119894)
01198911015840(119881(119894)
0)ℎ1015840(V119894)
= (minus1)119894minus1
= 0 119894 = 1 2
(61)
By now we have proved that Ω1and Ω
2satisfy all the
requirements in Lemma 1 Hence system (1) has at least two120596-periodic solutions 119906
lowast(119905) and 119906
dagger(119905) in Dom119871 cap Ω
1and
Dom119871 cap Ω2 respectively This completes the proof
Noticing that both systems (2) and (3) are special cases ofsystem (1) byTheorem 6 we can obtain the following results
Theorem 7 Suppose that (H1)ndash(H6) and (H
8)-(H9) hold
Then system (2) has at least two positive 120596-periodic solutions
Theorem 8 Suppose that (H1)ndash(H6) and (H
8)-(H9) hold
Then system (3) has at least two positive 120596-periodic solutions
Remark 9 The proof of Theorems 3 and 6 shows that allthe above results also remain valid if some delay terms arereplaced by terms with distributed or state-dependent delays
Remark 10 In their Theorem 22 Ding and Jiang [14] provedthat (2) has at least two positive 120596-periodic solutions if (H
1)ndash
(H6) (H8) and
(H10158409) (11988121198811)e2[119865(0)+119891(0)++119889]120596 lt V
2V1
hold Obviously the condition (H10158409) implies (H
9) Hence our
Theorem 7 improves Theorem 22 of [14]
Remark 11 Shao [10] studied a special case of system (1) for119891(119905 119909) = 119886(119905) minus 119887(119905)119909 Therefore our Theorem 3 generalizesTheorem 1 of [10]
Remark 12 Fan and Wang [17] Wang and Li [21] and Fanet al [16 18] studied some special cases of system (2) for119891(119905 119909) = 119886(119905) minus 119887(119905)119909 Therefore our Theorem 4 generalizesTheorem 21 of [17] Theorem 31 of [16] Theorem 35 of [18]andTheorem 21 of [21]
Remark 13 Fan and Wang [23] and Xia et al [24] studiedsome special cases of system (3) for 119891(119905 119909) = 119886(119905) minus 119887(119905)119909Therefore our Theorem 5 generalizes Theorem 21 of [23]andTheorem 31 of [24]
Acknowledgments
This work is supported by the National Natural ScienceFoundation of China under Grant 11271110 the Key Programsfor Science and Technology of the Education Department ofHenan Province under Grant 12A110007 and the ScientificResearch Start-up Funds of Henan University of Science andTechnology
References
[1] M Bohner and A PetersonDynamic Equations on Time ScalesAn Introduction with Applications Birkhauser Boston MassUSA 2001
[2] M Bohner and A Peterson Advances in Dynamic Equations onTime Scales Birkhauser Boston Mass USA 2003
[3] M Bohner L Erbe and A Peterson ldquoOscillation for nonlinearsecond order dynamic equations on a time scalerdquo Journal ofMathematical Analysis and Applications vol 301 no 2 pp 491ndash507 2005
[4] M Bohner M Fan and J Zhang ldquoExistence of periodicsolutions in predator-prey and competition dynamic systemsrdquoNonlinear Analysis Real World Applications vol 7 no 5 pp1193ndash1204 2006
[5] X Ding and G Zhao ldquoPeriodic solutions for a semi-ratio-dependent predator-prey system with delays on time scalesrdquoDiscrete Dynamics in Nature and Society vol 2012 Article ID928704 15 pages 2012
[6] M Fazly and M Hesaaraki ldquoPeriodic solutions for a semi-ratio-dependent predator-prey dynamical systemwith a class offunctional responses on time scalesrdquo Discrete and ContinuousDynamical Systems B vol 9 no 2 pp 267ndash279 2008
[7] J Hoffacker and C C Tisdell ldquoStability and instability fordynamic equations on time scalesrdquo Computers amp Mathematicswith Applications vol 49 no 9-10 pp 1327ndash1334 2005
[8] Y Li and H Zhang ldquoExistence of periodic solutions for a peri-odic mutualism model on time scalesrdquo Journal of MathematicalAnalysis and Applications vol 343 no 2 pp 818ndash825 2008
[9] J Liu Y Li and L Zhao ldquoOn a periodic predator-prey systemwith time delays on time scalesrdquo Communications in NonlinearScience and Numerical Simulation vol 14 no 8 pp 3432ndash34382009
[10] Y Shao ldquoExistence of positive periodic solution of a ratio-dependent predator-prey model on time scale with monotonicfunctional responserdquo Advances in Mathematics vol 39 no 2pp 224ndash232 2010
[11] B B Zhang and M Fan ldquoA remark on the application ofcoincidence degree to periodicity of dynamic equtions on timescalesrdquo Journal of Northeast Normal University vol 39 no 4 pp1ndash3 2007
[12] L Zhang H-X Li and X-B Zhang ldquoPeriodic solutions ofcompetition Lotka-Volterra dynamic system on time scalesrdquoComputers amp Mathematics with Applications vol 57 no 7 pp1204ndash1211 2009
8 Discrete Dynamics in Nature and Society
[13] X Ding ldquoPositive periodic solutions in generalized ratio-dependent predator-prey systemsrdquo Nonlinear Analysis RealWorld Applications vol 9 no 2 pp 394ndash402 2008
[14] X Ding and J Jiang ldquoMultiple periodic solutions in delayedGause-type ratio-dependent predator-prey systems with non-monotonic numerical responsesrdquo Mathematical and ComputerModelling vol 47 no 11-12 pp 1323ndash1331 2008
[15] Y H Fan andW T Li ldquoPermanence in delayed ratio-dependentpredator-prey models with monotonic functional responsesrdquoNonlinear Analysis Real World Applications vol 8 no 2 pp424ndash434 2007
[16] Y H Fan W T Li and L L Wang ldquoPeriodic solutions ofdelayed ratio-dependent predator-preymodels withmonotonicor nonmonotonic functional responsesrdquo Nonlinear AnalysisReal World Applications vol 5 no 2 pp 247ndash263 2004
[17] M Fan and K Wang ldquoPeriodicity in a delayed ratio-dependentpredator-prey systemrdquo Journal of Mathematical Analysis andApplications vol 262 no 1 pp 179ndash190 2001
[18] M Fan QWang and X Zou ldquoDynamics of a non-autonomousratio-dependent predator-prey systemrdquo Proceedings of the RoyalSociety of Edinburgh A vol 133 no 1 pp 97ndash118 2003
[19] S-B Hsu T-W Hwang and Y Kuang ldquoGlobal analysis ofthe Michaelis-Menten-type ratio-dependent predator-prey sys-temrdquo Journal of Mathematical Biology vol 42 no 6 pp 489ndash506 2001
[20] Y Kuang ldquoRich dynamics of Gauss-type ratio-dependentpredator-prey systemsrdquo Fields Institute Communications vol 21pp 325ndash337 1999
[21] L-L Wang and W-T Li ldquoPeriodic solutions and permanencefor a delayed nonautonomous ratio-dependent predator-preymodel with Holling type functional responserdquo Journal of Com-putational and Applied Mathematics vol 162 no 2 pp 341ndash3572004
[22] Y Xia and M Han ldquoMultiple periodic solutions of a ratio-dependent predator-prey modelrdquo Chaos Solitons amp Fractalsvol 39 no 3 pp 1100ndash1108 2009
[23] M Fan and K Wang ldquoPeriodic solutions of a discrete timenonautonomous ratio-dependent predator-prey systemrdquoMath-ematical and Computer Modelling vol 35 no 9-10 pp 951ndash9612002
[24] Y Xia J Cao andM Lin ldquoDiscrete-time analogues of predator-prey models with monotonic or nonmonotonic functionalresponsesrdquo Nonlinear Analysis Real World Applications vol 8no 4 pp 1079ndash1095 2007
[25] R E Gaines and J L Mawhin Coincidence Degree andNonlinear Differential Equations vol 568 of Lecture Notes inMathematics Springer Berlin Germany 1977
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
6 Discrete Dynamics in Nature and Society
Theorem 6 In addition to (H1)ndash(H6) and (H
8) suppose
further that the following condition holds
(H9) (11988121198811)119890[119865(0)+119891(0)++119889]120596
lt V2V1
Then system (1) has at least two 120596-periodic solutions
Proof The proof is similar to that of Theorem 3 To completethe proof we need to find two disjoint open bounded subsetsin 119883 Under condition (H
9) (34) is no longer valid So we
need to make a corresponding changeDesignating 119911(119905) = 119906
1(119905)minus1199062(119905) there also exist 120585
0 1205780isin 119868120596
such that
119911 (1205850) = max119905isin119868120596
119911 (119905) 119911 (1205780) = min119905isin119868120596
119911 (119905) (46)
From (23) (27) and the monotonicity of functions ℎ and ℎ
we will show that 119911(1205850) and 119911(120578
0) can not simultaneously lie
in (minusinfin ln V1) (ln V
1 ln V2) or (ln V
2 +infin) In fact if 119911(120578
0) le
119911(1205850) lt ln V
1 then
119889 =
1
120596
int
120581+120596
120581
ℎ (119905 e119911(119905minus120591(119905))) Δ119905
leℎ (e119911(1205850)) lt
ℎ (V1) =
119889
(47)
This is a contradiction If ln V2lt 119911(1205780) le 119911(120585
0) then
119889 =
1
120596
int
120581+120596
120581
ℎ (119905 e119911(119905minus120591(119905))) Δ119905
leℎ (e119911(1205780)) lt
ℎ (V2) =
119889
(48)
This is also a contradiction If ln V1
lt 119911(1205780) le 119911(120585
0) lt ln V
2
then
119889 =
1
120596
int
120581+120596
120581
ℎ (119905 e119911(119905minus120591(119905))) Δ119905
ge minℎ (e119911(1205850)) ℎ (e119911(1205780))
gtℎ (V1) =
ℎ (V2) =
119889
(49)
This is also a contradiction Consequently the distribution of119911(1205850) and 119911(120578
0) only have the following two cases
Case 1 119911(1205780) le ln V
2le 119911(1205850) Noticing that
119911 (1205780) ge 1199061(1205781) minus 1199062(1205852)
119911 (1205850) le 1199061(1205851) minus 1199062(1205782)
(50)
from (27) (30) and (33) we obtain
1199062(1205852) ge 1199061(1205781) minus 119911 (120578
0) ge 1199061(1205781) minus ln V
2gt 1198611minus ln V2
1199062(1205782) le 1199061(1205851) minus 119911 (120585
0) le 1199061(1205851) minus ln V
2lt 1198612minus ln V2
(51)
By Lemma 2 we find from (26) and (51) that for all 119905 isin 119868120596
1199062 (
119905) ge 1199062(1205852) minus
1
2
int
120581+120596
120581
10038161003816100381610038161003816119906Δ
2(119905)
10038161003816100381610038161003816Δ119905
gt 1198611minus ln V2minus
119863 +119889
2
120596 = 1198617
1199062 (
119905) le 1199062(1205782) +
1
2
int
120581+120596
120581
10038161003816100381610038161003816119906Δ
2(119905)
10038161003816100381610038161003816Δ119905
lt 1198612minus ln V2+
119863 +119889
2
120596 = 1198618
(52)
Case 2 119911(1205780) le ln V
1le 119911(120585
0) From (27) (30) and (33) we
also obtain
1199062(1205852) ge 1199061(1205781) minus 119911 (120578
0) ge 1199061(1205781) minus ln V
1gt 1198611minus ln V1
1199062(1205782) le 1199061(1205851) minus 119911 (120585
0) le 1199061(1205851) minus ln V
1lt 1198612minus ln V1
(53)
By Lemma 2 we find from (26) and (53) that for all 119905 isin 119868120596
1199062 (
119905) ge 1199062(1205852) minus
1
2
int
120581+120596
120581
10038161003816100381610038161003816119906Δ
2(119905)
10038161003816100381610038161003816Δ119905
gt 1198611minus ln V1minus
119863 +119889
2
120596 = 1198619
1199062 (
119905) le 1199062(1205782) +
1
2
int
120581+120596
120581
10038161003816100381610038161003816119906Δ
2(119905)
10038161003816100381610038161003816Δ119905
lt 1198612minus ln V1+
119863 +119889
2
120596 = 11986110
(54)
By (H9) we know that
1198617lt 1198618lt 1198619lt 11986110 (55)
Clearly 1198617 1198618 1198619 and 119861
10are independent of 120582
By themonotonicity of functions 119891 and
ℎ (H3) (H5) and
(H6) it is easy to show that algebraic system (40) has two
distinct solutions
119906(119894)
= (ln119881(119894)
0 ln119881(119894)
0minus ln V119894)
T 119894 = 1 2 (56)
where119881(119894)
0is the unique solution of equation
119891(V)minus119901(V119894) = 0
Obviously
1198811le 119881(119894)
0lt 1198812 (57)
We now take
Ω1= (119906
1 (119905) 1199062 (
119905))T
isin 119883 10038161003816100381610038161199061 (
119905)1003816100381610038161003816
lt 1198615+ 1205730 1199062 (
119905) isin (1198617 1198618)
Ω2= (119906
1 (119905) 1199062 (
119905))T
isin 119883 10038161003816100381610038161199061 (
119905)1003816100381610038161003816
lt 1198615+ 1205730 1199062 (
119905) isin (1198619 11986110)
(58)
Discrete Dynamics in Nature and Society 7
where 1205730is taken sufficiently large such that
1205730gt max
10038161003816100381610038161003816ln119881(1)
0
10038161003816100381610038161003816
10038161003816100381610038161003816ln119881(2)
0
10038161003816100381610038161003816 (59)
Then both Ω1and Ω
2are bounded open subsets of 119883 It
follows from (55) (57) and (59) that 119906(1) isin Ω1 119906(2) isin Ω
2 and
Ω1cap Ω2= With the help of (38) (52) (54) and (59) it is
easy to see that Ω1and Ω
2satisfy condition (a) in Lemma 1
When (1199061(119905) 1199062(119905))
Tisin 120597Ω119894cap Ker 119871 = 120597Ω
119894cap R2 (119894 = 1 2)
(1199061(119905) 1199062(119905))
T is a constant vector in R2 Thus we have
119876119873(
1199061
1199062
) = (
119891 (e1199061) minus 119901 (e1199061minus1199062)
minus119889 +
ℎ (e1199061minus1199062)
) = (
0
0) (60)
This proves that condition (b) in Lemma 1 is satisfied A directcalculation shows that
deg (119869119876119873Ω119894cap Ker 119871 0)
= signminusV119894119881(119894)
01198911015840(119881(119894)
0)ℎ1015840(V119894)
= (minus1)119894minus1
= 0 119894 = 1 2
(61)
By now we have proved that Ω1and Ω
2satisfy all the
requirements in Lemma 1 Hence system (1) has at least two120596-periodic solutions 119906
lowast(119905) and 119906
dagger(119905) in Dom119871 cap Ω
1and
Dom119871 cap Ω2 respectively This completes the proof
Noticing that both systems (2) and (3) are special cases ofsystem (1) byTheorem 6 we can obtain the following results
Theorem 7 Suppose that (H1)ndash(H6) and (H
8)-(H9) hold
Then system (2) has at least two positive 120596-periodic solutions
Theorem 8 Suppose that (H1)ndash(H6) and (H
8)-(H9) hold
Then system (3) has at least two positive 120596-periodic solutions
Remark 9 The proof of Theorems 3 and 6 shows that allthe above results also remain valid if some delay terms arereplaced by terms with distributed or state-dependent delays
Remark 10 In their Theorem 22 Ding and Jiang [14] provedthat (2) has at least two positive 120596-periodic solutions if (H
1)ndash
(H6) (H8) and
(H10158409) (11988121198811)e2[119865(0)+119891(0)++119889]120596 lt V
2V1
hold Obviously the condition (H10158409) implies (H
9) Hence our
Theorem 7 improves Theorem 22 of [14]
Remark 11 Shao [10] studied a special case of system (1) for119891(119905 119909) = 119886(119905) minus 119887(119905)119909 Therefore our Theorem 3 generalizesTheorem 1 of [10]
Remark 12 Fan and Wang [17] Wang and Li [21] and Fanet al [16 18] studied some special cases of system (2) for119891(119905 119909) = 119886(119905) minus 119887(119905)119909 Therefore our Theorem 4 generalizesTheorem 21 of [17] Theorem 31 of [16] Theorem 35 of [18]andTheorem 21 of [21]
Remark 13 Fan and Wang [23] and Xia et al [24] studiedsome special cases of system (3) for 119891(119905 119909) = 119886(119905) minus 119887(119905)119909Therefore our Theorem 5 generalizes Theorem 21 of [23]andTheorem 31 of [24]
Acknowledgments
This work is supported by the National Natural ScienceFoundation of China under Grant 11271110 the Key Programsfor Science and Technology of the Education Department ofHenan Province under Grant 12A110007 and the ScientificResearch Start-up Funds of Henan University of Science andTechnology
References
[1] M Bohner and A PetersonDynamic Equations on Time ScalesAn Introduction with Applications Birkhauser Boston MassUSA 2001
[2] M Bohner and A Peterson Advances in Dynamic Equations onTime Scales Birkhauser Boston Mass USA 2003
[3] M Bohner L Erbe and A Peterson ldquoOscillation for nonlinearsecond order dynamic equations on a time scalerdquo Journal ofMathematical Analysis and Applications vol 301 no 2 pp 491ndash507 2005
[4] M Bohner M Fan and J Zhang ldquoExistence of periodicsolutions in predator-prey and competition dynamic systemsrdquoNonlinear Analysis Real World Applications vol 7 no 5 pp1193ndash1204 2006
[5] X Ding and G Zhao ldquoPeriodic solutions for a semi-ratio-dependent predator-prey system with delays on time scalesrdquoDiscrete Dynamics in Nature and Society vol 2012 Article ID928704 15 pages 2012
[6] M Fazly and M Hesaaraki ldquoPeriodic solutions for a semi-ratio-dependent predator-prey dynamical systemwith a class offunctional responses on time scalesrdquo Discrete and ContinuousDynamical Systems B vol 9 no 2 pp 267ndash279 2008
[7] J Hoffacker and C C Tisdell ldquoStability and instability fordynamic equations on time scalesrdquo Computers amp Mathematicswith Applications vol 49 no 9-10 pp 1327ndash1334 2005
[8] Y Li and H Zhang ldquoExistence of periodic solutions for a peri-odic mutualism model on time scalesrdquo Journal of MathematicalAnalysis and Applications vol 343 no 2 pp 818ndash825 2008
[9] J Liu Y Li and L Zhao ldquoOn a periodic predator-prey systemwith time delays on time scalesrdquo Communications in NonlinearScience and Numerical Simulation vol 14 no 8 pp 3432ndash34382009
[10] Y Shao ldquoExistence of positive periodic solution of a ratio-dependent predator-prey model on time scale with monotonicfunctional responserdquo Advances in Mathematics vol 39 no 2pp 224ndash232 2010
[11] B B Zhang and M Fan ldquoA remark on the application ofcoincidence degree to periodicity of dynamic equtions on timescalesrdquo Journal of Northeast Normal University vol 39 no 4 pp1ndash3 2007
[12] L Zhang H-X Li and X-B Zhang ldquoPeriodic solutions ofcompetition Lotka-Volterra dynamic system on time scalesrdquoComputers amp Mathematics with Applications vol 57 no 7 pp1204ndash1211 2009
8 Discrete Dynamics in Nature and Society
[13] X Ding ldquoPositive periodic solutions in generalized ratio-dependent predator-prey systemsrdquo Nonlinear Analysis RealWorld Applications vol 9 no 2 pp 394ndash402 2008
[14] X Ding and J Jiang ldquoMultiple periodic solutions in delayedGause-type ratio-dependent predator-prey systems with non-monotonic numerical responsesrdquo Mathematical and ComputerModelling vol 47 no 11-12 pp 1323ndash1331 2008
[15] Y H Fan andW T Li ldquoPermanence in delayed ratio-dependentpredator-prey models with monotonic functional responsesrdquoNonlinear Analysis Real World Applications vol 8 no 2 pp424ndash434 2007
[16] Y H Fan W T Li and L L Wang ldquoPeriodic solutions ofdelayed ratio-dependent predator-preymodels withmonotonicor nonmonotonic functional responsesrdquo Nonlinear AnalysisReal World Applications vol 5 no 2 pp 247ndash263 2004
[17] M Fan and K Wang ldquoPeriodicity in a delayed ratio-dependentpredator-prey systemrdquo Journal of Mathematical Analysis andApplications vol 262 no 1 pp 179ndash190 2001
[18] M Fan QWang and X Zou ldquoDynamics of a non-autonomousratio-dependent predator-prey systemrdquo Proceedings of the RoyalSociety of Edinburgh A vol 133 no 1 pp 97ndash118 2003
[19] S-B Hsu T-W Hwang and Y Kuang ldquoGlobal analysis ofthe Michaelis-Menten-type ratio-dependent predator-prey sys-temrdquo Journal of Mathematical Biology vol 42 no 6 pp 489ndash506 2001
[20] Y Kuang ldquoRich dynamics of Gauss-type ratio-dependentpredator-prey systemsrdquo Fields Institute Communications vol 21pp 325ndash337 1999
[21] L-L Wang and W-T Li ldquoPeriodic solutions and permanencefor a delayed nonautonomous ratio-dependent predator-preymodel with Holling type functional responserdquo Journal of Com-putational and Applied Mathematics vol 162 no 2 pp 341ndash3572004
[22] Y Xia and M Han ldquoMultiple periodic solutions of a ratio-dependent predator-prey modelrdquo Chaos Solitons amp Fractalsvol 39 no 3 pp 1100ndash1108 2009
[23] M Fan and K Wang ldquoPeriodic solutions of a discrete timenonautonomous ratio-dependent predator-prey systemrdquoMath-ematical and Computer Modelling vol 35 no 9-10 pp 951ndash9612002
[24] Y Xia J Cao andM Lin ldquoDiscrete-time analogues of predator-prey models with monotonic or nonmonotonic functionalresponsesrdquo Nonlinear Analysis Real World Applications vol 8no 4 pp 1079ndash1095 2007
[25] R E Gaines and J L Mawhin Coincidence Degree andNonlinear Differential Equations vol 568 of Lecture Notes inMathematics Springer Berlin Germany 1977
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical Problems in Engineering
Hindawi Publishing Corporationhttpwwwhindawicom
Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
OptimizationJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Operations ResearchAdvances in
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Algebra
Discrete Dynamics in Nature and Society
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Decision SciencesAdvances in
Discrete MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
Discrete Dynamics in Nature and Society 7
where 1205730is taken sufficiently large such that
1205730gt max
10038161003816100381610038161003816ln119881(1)
0
10038161003816100381610038161003816
10038161003816100381610038161003816ln119881(2)
0
10038161003816100381610038161003816 (59)
Then both Ω1and Ω
2are bounded open subsets of 119883 It
follows from (55) (57) and (59) that 119906(1) isin Ω1 119906(2) isin Ω
2 and
Ω1cap Ω2= With the help of (38) (52) (54) and (59) it is
easy to see that Ω1and Ω
2satisfy condition (a) in Lemma 1
When (1199061(119905) 1199062(119905))
Tisin 120597Ω119894cap Ker 119871 = 120597Ω
119894cap R2 (119894 = 1 2)
(1199061(119905) 1199062(119905))
T is a constant vector in R2 Thus we have
119876119873(
1199061
1199062
) = (
119891 (e1199061) minus 119901 (e1199061minus1199062)
minus119889 +
ℎ (e1199061minus1199062)
) = (
0
0) (60)
This proves that condition (b) in Lemma 1 is satisfied A directcalculation shows that
deg (119869119876119873Ω119894cap Ker 119871 0)
= signminusV119894119881(119894)
01198911015840(119881(119894)
0)ℎ1015840(V119894)
= (minus1)119894minus1
= 0 119894 = 1 2
(61)
By now we have proved that Ω1and Ω
2satisfy all the
requirements in Lemma 1 Hence system (1) has at least two120596-periodic solutions 119906
lowast(119905) and 119906
dagger(119905) in Dom119871 cap Ω
1and
Dom119871 cap Ω2 respectively This completes the proof
Noticing that both systems (2) and (3) are special cases ofsystem (1) byTheorem 6 we can obtain the following results
Theorem 7 Suppose that (H1)ndash(H6) and (H
8)-(H9) hold
Then system (2) has at least two positive 120596-periodic solutions
Theorem 8 Suppose that (H1)ndash(H6) and (H
8)-(H9) hold
Then system (3) has at least two positive 120596-periodic solutions
Remark 9 The proof of Theorems 3 and 6 shows that allthe above results also remain valid if some delay terms arereplaced by terms with distributed or state-dependent delays
Remark 10 In their Theorem 22 Ding and Jiang [14] provedthat (2) has at least two positive 120596-periodic solutions if (H
1)ndash
(H6) (H8) and
(H10158409) (11988121198811)e2[119865(0)+119891(0)++119889]120596 lt V
2V1
hold Obviously the condition (H10158409) implies (H
9) Hence our
Theorem 7 improves Theorem 22 of [14]
Remark 11 Shao [10] studied a special case of system (1) for119891(119905 119909) = 119886(119905) minus 119887(119905)119909 Therefore our Theorem 3 generalizesTheorem 1 of [10]
Remark 12 Fan and Wang [17] Wang and Li [21] and Fanet al [16 18] studied some special cases of system (2) for119891(119905 119909) = 119886(119905) minus 119887(119905)119909 Therefore our Theorem 4 generalizesTheorem 21 of [17] Theorem 31 of [16] Theorem 35 of [18]andTheorem 21 of [21]
Remark 13 Fan and Wang [23] and Xia et al [24] studiedsome special cases of system (3) for 119891(119905 119909) = 119886(119905) minus 119887(119905)119909Therefore our Theorem 5 generalizes Theorem 21 of [23]andTheorem 31 of [24]
Acknowledgments
This work is supported by the National Natural ScienceFoundation of China under Grant 11271110 the Key Programsfor Science and Technology of the Education Department ofHenan Province under Grant 12A110007 and the ScientificResearch Start-up Funds of Henan University of Science andTechnology
References
[1] M Bohner and A PetersonDynamic Equations on Time ScalesAn Introduction with Applications Birkhauser Boston MassUSA 2001
[2] M Bohner and A Peterson Advances in Dynamic Equations onTime Scales Birkhauser Boston Mass USA 2003
[3] M Bohner L Erbe and A Peterson ldquoOscillation for nonlinearsecond order dynamic equations on a time scalerdquo Journal ofMathematical Analysis and Applications vol 301 no 2 pp 491ndash507 2005
[4] M Bohner M Fan and J Zhang ldquoExistence of periodicsolutions in predator-prey and competition dynamic systemsrdquoNonlinear Analysis Real World Applications vol 7 no 5 pp1193ndash1204 2006
[5] X Ding and G Zhao ldquoPeriodic solutions for a semi-ratio-dependent predator-prey system with delays on time scalesrdquoDiscrete Dynamics in Nature and Society vol 2012 Article ID928704 15 pages 2012
[6] M Fazly and M Hesaaraki ldquoPeriodic solutions for a semi-ratio-dependent predator-prey dynamical systemwith a class offunctional responses on time scalesrdquo Discrete and ContinuousDynamical Systems B vol 9 no 2 pp 267ndash279 2008
[7] J Hoffacker and C C Tisdell ldquoStability and instability fordynamic equations on time scalesrdquo Computers amp Mathematicswith Applications vol 49 no 9-10 pp 1327ndash1334 2005
[8] Y Li and H Zhang ldquoExistence of periodic solutions for a peri-odic mutualism model on time scalesrdquo Journal of MathematicalAnalysis and Applications vol 343 no 2 pp 818ndash825 2008
[9] J Liu Y Li and L Zhao ldquoOn a periodic predator-prey systemwith time delays on time scalesrdquo Communications in NonlinearScience and Numerical Simulation vol 14 no 8 pp 3432ndash34382009
[10] Y Shao ldquoExistence of positive periodic solution of a ratio-dependent predator-prey model on time scale with monotonicfunctional responserdquo Advances in Mathematics vol 39 no 2pp 224ndash232 2010
[11] B B Zhang and M Fan ldquoA remark on the application ofcoincidence degree to periodicity of dynamic equtions on timescalesrdquo Journal of Northeast Normal University vol 39 no 4 pp1ndash3 2007
[12] L Zhang H-X Li and X-B Zhang ldquoPeriodic solutions ofcompetition Lotka-Volterra dynamic system on time scalesrdquoComputers amp Mathematics with Applications vol 57 no 7 pp1204ndash1211 2009
8 Discrete Dynamics in Nature and Society
[13] X Ding ldquoPositive periodic solutions in generalized ratio-dependent predator-prey systemsrdquo Nonlinear Analysis RealWorld Applications vol 9 no 2 pp 394ndash402 2008
[14] X Ding and J Jiang ldquoMultiple periodic solutions in delayedGause-type ratio-dependent predator-prey systems with non-monotonic numerical responsesrdquo Mathematical and ComputerModelling vol 47 no 11-12 pp 1323ndash1331 2008
[15] Y H Fan andW T Li ldquoPermanence in delayed ratio-dependentpredator-prey models with monotonic functional responsesrdquoNonlinear Analysis Real World Applications vol 8 no 2 pp424ndash434 2007
[16] Y H Fan W T Li and L L Wang ldquoPeriodic solutions ofdelayed ratio-dependent predator-preymodels withmonotonicor nonmonotonic functional responsesrdquo Nonlinear AnalysisReal World Applications vol 5 no 2 pp 247ndash263 2004
[17] M Fan and K Wang ldquoPeriodicity in a delayed ratio-dependentpredator-prey systemrdquo Journal of Mathematical Analysis andApplications vol 262 no 1 pp 179ndash190 2001
[18] M Fan QWang and X Zou ldquoDynamics of a non-autonomousratio-dependent predator-prey systemrdquo Proceedings of the RoyalSociety of Edinburgh A vol 133 no 1 pp 97ndash118 2003
[19] S-B Hsu T-W Hwang and Y Kuang ldquoGlobal analysis ofthe Michaelis-Menten-type ratio-dependent predator-prey sys-temrdquo Journal of Mathematical Biology vol 42 no 6 pp 489ndash506 2001
[20] Y Kuang ldquoRich dynamics of Gauss-type ratio-dependentpredator-prey systemsrdquo Fields Institute Communications vol 21pp 325ndash337 1999
[21] L-L Wang and W-T Li ldquoPeriodic solutions and permanencefor a delayed nonautonomous ratio-dependent predator-preymodel with Holling type functional responserdquo Journal of Com-putational and Applied Mathematics vol 162 no 2 pp 341ndash3572004
[22] Y Xia and M Han ldquoMultiple periodic solutions of a ratio-dependent predator-prey modelrdquo Chaos Solitons amp Fractalsvol 39 no 3 pp 1100ndash1108 2009
[23] M Fan and K Wang ldquoPeriodic solutions of a discrete timenonautonomous ratio-dependent predator-prey systemrdquoMath-ematical and Computer Modelling vol 35 no 9-10 pp 951ndash9612002
[24] Y Xia J Cao andM Lin ldquoDiscrete-time analogues of predator-prey models with monotonic or nonmonotonic functionalresponsesrdquo Nonlinear Analysis Real World Applications vol 8no 4 pp 1079ndash1095 2007
[25] R E Gaines and J L Mawhin Coincidence Degree andNonlinear Differential Equations vol 568 of Lecture Notes inMathematics Springer Berlin Germany 1977
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 Discrete Dynamics in Nature and Society
[13] X Ding ldquoPositive periodic solutions in generalized ratio-dependent predator-prey systemsrdquo Nonlinear Analysis RealWorld Applications vol 9 no 2 pp 394ndash402 2008
[14] X Ding and J Jiang ldquoMultiple periodic solutions in delayedGause-type ratio-dependent predator-prey systems with non-monotonic numerical responsesrdquo Mathematical and ComputerModelling vol 47 no 11-12 pp 1323ndash1331 2008
[15] Y H Fan andW T Li ldquoPermanence in delayed ratio-dependentpredator-prey models with monotonic functional responsesrdquoNonlinear Analysis Real World Applications vol 8 no 2 pp424ndash434 2007
[16] Y H Fan W T Li and L L Wang ldquoPeriodic solutions ofdelayed ratio-dependent predator-preymodels withmonotonicor nonmonotonic functional responsesrdquo Nonlinear AnalysisReal World Applications vol 5 no 2 pp 247ndash263 2004
[17] M Fan and K Wang ldquoPeriodicity in a delayed ratio-dependentpredator-prey systemrdquo Journal of Mathematical Analysis andApplications vol 262 no 1 pp 179ndash190 2001
[18] M Fan QWang and X Zou ldquoDynamics of a non-autonomousratio-dependent predator-prey systemrdquo Proceedings of the RoyalSociety of Edinburgh A vol 133 no 1 pp 97ndash118 2003
[19] S-B Hsu T-W Hwang and Y Kuang ldquoGlobal analysis ofthe Michaelis-Menten-type ratio-dependent predator-prey sys-temrdquo Journal of Mathematical Biology vol 42 no 6 pp 489ndash506 2001
[20] Y Kuang ldquoRich dynamics of Gauss-type ratio-dependentpredator-prey systemsrdquo Fields Institute Communications vol 21pp 325ndash337 1999
[21] L-L Wang and W-T Li ldquoPeriodic solutions and permanencefor a delayed nonautonomous ratio-dependent predator-preymodel with Holling type functional responserdquo Journal of Com-putational and Applied Mathematics vol 162 no 2 pp 341ndash3572004
[22] Y Xia and M Han ldquoMultiple periodic solutions of a ratio-dependent predator-prey modelrdquo Chaos Solitons amp Fractalsvol 39 no 3 pp 1100ndash1108 2009
[23] M Fan and K Wang ldquoPeriodic solutions of a discrete timenonautonomous ratio-dependent predator-prey systemrdquoMath-ematical and Computer Modelling vol 35 no 9-10 pp 951ndash9612002
[24] Y Xia J Cao andM Lin ldquoDiscrete-time analogues of predator-prey models with monotonic or nonmonotonic functionalresponsesrdquo Nonlinear Analysis Real World Applications vol 8no 4 pp 1079ndash1095 2007
[25] R E Gaines and J L Mawhin Coincidence Degree andNonlinear Differential Equations vol 568 of Lecture Notes inMathematics Springer Berlin Germany 1977
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
