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Hindawi Publishing CorporationInternational Journal of Differential EquationsVolume 2013 Article ID 890281 13 pageshttpdxdoiorg1011552013890281
Research ArticleExistence of Positive Periodic Solutions for Periodic NeutralLotka-Volterra System with Distributed Delays and Impulses
Zhenguo Luo12 and Liping Luo2
1 Department of Mathematics National University of Defense Technology Changsha 410073 China2Department of Mathematics Hengyang Normal University Hengyang Hunan 421008 China
Correspondence should be addressed to Zhenguo Luo robert186163com
Received 20 April 2013 Accepted 13 May 2013
Academic Editor Norio Yoshida
Copyright copy 2013 Z Luo and L Luo This 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
By using a fixed-point theorem of strict-set-contraction we investigate the existence of positive periodic solutions for aclass of the following impulsive neutral Lotka-Volterra system with distributed delays 1199091015840
119894(119905) = 119909
119894(119905)[119903119894(119905) minus sum
119899
119895=1119886119894119895(119905)119909119895(119905) minus
It is well known that natural environments are physicallyhighly variable and in response birth rates death ratesand other vital rates of populations vary greatly in timeTheoretical evidence suggests that many population andcommunity patterns represent intricate interactions betweenbiology and variation in the physical environment (see [1ndash4]) Thus the focus in theoretical models of populationand community dynamics must be not only on how pop-ulations depend on their own population densities or thepopulation densities of other organisms but also on howpopulations change in response to the physical environmentIt is reasonable to study the models of population withperiodic coefficients In addition to the theoretical andpractical significance the Lotka-Volterra model is one of thefamous models for dynamics of population therefore it hasbeen studied extensively [5ndash9] In view of the above effectsby applying a fixed-point theoremof strict-set-contraction Li[10] established criteria to guarantee the existence of positiveperiodic solutions of the following neutral Lotka-Volterrasystem with distributed delays
119894119895(120579)119889120579 119894 119895 = 1 2 119899 On the other side birth
of many species is an annual birth pulse or harvesting Tohave a more accurate description of many mathematicalecology systems we need to consider the use of impulsivedifferential equations [11ndash13] Some qualitative propertiessuch as oscillation periodicity asymptotic behavior andstability properties have been investigated extensively bymany authors over the past few years [14ndash18] However toour knowledge there are few published papers discussingthe existence of periodic solutions for neutral Lotka-Volterrasystem with distributed delays and impulses In this paperwe are concerned with the following neutral Lotka-Volterrasystem with distributed delays and impulses
119896) represents the right limit of 119909(119905)
at the point 119905119896) 119868119894119896
isin 119862(119877+ 119877+) that is 119909 changes
decreasingly suddenly at times 119905119896 120596 gt 0 is a constant 119877 =
(minusinfin +infin) 119877+= [0 +infin) We assume that there exists an
integer 119902 gt 0 such that 119905119896+119902
= 119905119896+ 120596 119868
119894(119896+119902)= 119868119894119896 where
0 lt 1199051lt 1199052lt sdot sdot sdot lt 119905
119902lt 120596
The main purpose of this paper is by using a fixed-point theorem of strict-set-contraction [19 20] to establishnew criteria to guarantee the existence of positive periodicsolutions of the system (2)
For convenience we introduce the notation
119891119872= max119905isin[0120596]
1003816100381610038161003816119891 (119905)
1003816100381610038161003816 119891
119871= min119905isin[0120596]
1003816100381610038161003816119891 (119905)
1003816100381610038161003816
120574119894= lim119906rarr0
sup sum
119905le119905119896le119905+120596
119868119894119896(119906)
119906
119894 = 1 2 119899
120575119894= 119890minusint120596
0119903119894(119905)119889119905
119894 = 1 2 119899
120572119894119895= int
120596
0
[120575119895119886119894119895(119905) minus 119887
The paper is organized as follows In Section 2 we givesome definitions and lemmas to prove themain results of thispaper In Section 3 by using a fixed-point theorem of strict-set-contraction we established some criteria to guarantee theexistence of at least one positive periodic solution of system(2) Finally in Section 4 we give an example to show thevalidity of our result
2 Preliminaries
In order to obtain the existence of a periodic solution ofsystem (2) we first introduce some definitions and lemmas
Definition 1 (see [13]) A function 119909119894 119877 rarr (0 +infin) is said
to be a positive solution of (2) if the following conditions aresatisfied
(a) 119909119894(119905) is absolutely continuous on each (119905
119896 119905119896+1)
(b) for each 119896 isin 119885+ 119909119894(119905+
119896) and 119909
119894(119905minus
119896) exist and 119909
119894(119905minus
119896) =
119909119894(119905119896)
(c) 119909119894(119905) satisfies the first equation of (2) for almost every-
where in 119877 and 119909119894(119905119896) satisfies the second equation of
(2) at impulsive point 119905119896 119896 isin 119885
+
Definition 2 (see [21]) Let 119883 be a real Banach space and 119864 aclosed nonempty subset of119883 119864 is a cone provided
(i) 120572119909 + 120573119910 isin 119864 for all 119909 119910 isin 119864 and all 120572 120573 ge 0(ii) 119909 minus119909 isin 119864 imply 119909 = 0
Definition 3 (see [21]) Let 119860 be a bounded subset in 119883Define
120572119883(119860) = inf 120575 gt 0 there is a finite number
of subsets 119860119894sub 119860 such that
119860 = ⋃
119894
119860119894and diam (119860
119894) le 120575
(4)
where diam(119860119894) denotes the diameter of the set 119860
119894 obvi-
ously 0 le 120572119883(119860) lt infin So 120572
119883(119860) is called the Kuratowski
measure of noncompactness of 119883
Definition 4 (see [21]) Let 119883 119884 be two Banach spaces and119863 sub 119883 a continuous and boundedmap 119879 119863 rarr 119884 is called119896-set contractive if for any bounded set 119878 sub 119863 we have
120572119884(119879 (119878)) le 119896120572
119883(119878) (5)
120601 is called strict-set-contractive if it is 119896-set-contractive forsome 0 le 119896 lt 1
Definition 5 (see [22]) The set 119865 isin 119875119862120596is said to be quasi-
equicontinuous in [0 120596] if for any 120598 gt 0 there exists 120575 gt 0
such that if 119909 isin 119865 119896 isin 119873+ 1199051 1199052isin (119905119896minus1 119905119896)cap[0 120596] and |119905
1minus
1199052| lt 120575 then |119909(119905
1) minus 119909(119905
2)| lt 120598
International Journal of Differential Equations 3
Lemma 6 (see [22]) The set 119865 sub 119875119862120596is relatively compact if
and only if
(1) 119865 is bounded that is 119909 le 119872 for each 119909 isin 119865 andsome119872 gt 0
(2) 119865 is quasi-equicontinuous in [0 120596]
Lemma 7 119909119894(119905) is an 120596-periodic solution of (2) is equivalent
to 119909119894(119905) is an 120596-periodic solution of the following equation
10 International Journal of Differential Equations
From (36) and (37) we obtain
1003817100381710038171003817119879119909 minus 119879119910
10038171003817100381710038171le (
119899
sum
119894=1
1003816100381610038161003816119909119894
10038161003816100381610038160
119899
sum
119895=1
119888119872
119894119895)120578 + 120598
119899
sum
119894=1
119861119894
le 119877max1le119894le119899
119899
sum
119895=1
119888119872
119894119895
120578 + 120598
119899
sum
119894=1
119861119894 for any 119909 119910 isin 119878
119894119895119896
(39)
As 120598 is arbitrarily small it follows that
120572119884(119879 (119878)) le 119877max
1le119895le119899
119899
sum
119895=1
119888119872
119894119895
120572119884(119878) (40)
Therefore 119879 is strict-set-contractive The proof of Lemma 10is complete
3 Main Results
Our main result of this paper is as follows
Theorem 11 Assume that (1198601)ndash(1198603) (1198605) hold
(i) If max119903119872119894 119894 = 1 2 119899 le 1 then system (2) has at
least one positive 120596-periodic solution
(ii) If (1198604) holds and min119903119872
119894 119894 = 1 2 119899 gt 1 then
system (2) has at least one positive 120596-periodic solution
Proof We only need to prove (i) since the proof of (ii) issimilar Let
119877 = (min119894isin[1119899]
1205752
119894
1 minus 120575119894
min119895isin[1119899]
120572119894119895)
minus1
0 lt 119903 lt min119894isin[1119899]
120575119894(1 minus 120575
119894) minus 120574119894
max119895isin[1119899]
120573119894119895
(41)
Then it is easy to see that 0 lt 119903 lt 119877 From Lemmas 9 and10 we know that 119879 is strict-set-contractive on 119864
119903119877 In view
of (26) we see that if there exists 119909lowast isin 119864 such that 119879119909lowast =119909lowast then 119909lowast is one positive 120596-periodic solution of system (2)
Nowwewill prove that condition (ii) of Lemma 7holds Firstwe prove that 119879119909 ge 119909 for all 119909 isin 119864 119909
1lt 119903 Otherwise
there exists 119909 isin 119864 1199091lt 119903 such that 119879119909 ge 119909 So 119909
0gt 0
and 119879119909 minus 119909 ge 0 which implies that
119879119894119909 (119905) minus 119909
119894(119905) ge 120575
119894
1003816100381610038161003816119879119894119909 minus 119909119894
10038161003816100381610038161ge 0
for any 119905 isin [0 120596] 119894 = 1 2 119899(42)
Moreover for 119905 isin [0 120596] we have(119879119894119909) (119905)
which is a contradictionTherefore condition (ii) of Lemma 7holds By Lemma 7 we see that 119879 has at least one nonzerofixed point in 119864 Thus the system (6) has at least one positive120596-periodic solutionTherefore it follows from Lemma 6 thatsystem (2) has a positive 120596-periodic solution The proof ofTheorem 11 is complete
Remark 12 If 119868119894119896(119909(119905119896)) = 0 we can easily derive the
corresponding results in [10] Sowe extend the correspondingresults in [10]
4 Example
In this section we give an example to show the effectivenessof our result
Example 1 Consider the following nonimpulsive system
1199091015840
1(119905)
= 1199091(119905) [
1 minus sin 11990524
minus (10 + sin 119905) 1199091(119905)
minus (9 minus cos 119905) 1199092(119905) minus (3 minus 2 cos 119905)
times int
0
minus12059111
11989111(120585) 1199091(119905 + 120585) 119889120585 minus
5 + sin 11990515
times int
0
minus12059112
11989112(120585) 1199092(119905 + 120585) 119889120585 minus (1 + sin 119905)
times int
0
minus12059011
11989211(120585) 1199091015840
1(119905 + 120585) 119889120585 minus
4 minus 3 cos 1199056
timesint
0
minus12059012
11989212(120585) 1199091015840
2(119905 + 120585) 119889120585]
1199091015840
2(119905)
= 1199092(119905) [
1 + cos 1199058120587
minus (12 + 3 cos 119905) 1199091(119905)
minus (7 minus sin 119905) 1199092(119905) minus (2 + 3 sin 119905)
times int
0
minus12059121
11989121(120585) 1199092(119905 + 120585) 119889120585 minus (1 minus cos 119905)
times int
0
minus12059122
11989122(120585) 1199092(119905 + 120585) 119889120585 minus
12 International Journal of Differential Equations
min1le119895le2
1205721119895 = 12057211= 20120587120575
1minus 8120587 asymp 238007
min1le119895le2
1205722119895 = 12057222= 14120587120575
2minus 3120587 asymp 244284
120575111988611(119905) minus 119887
11(119905) minus 119888
11(119905) gt 0009 gt 0
120575211988612(119905) minus 119887
12(119905) minus 119888
12(119905) gt 1 gt 0
120575111988621(119905) minus 119887
21(119905) minus 119888
21(119905) gt 4 gt 0
120575211988622(119905) minus 119887
22(119905) minus 119888
22(119905) gt 2 gt 0
(1 + 119903119871
1)
1205752
112057211
1 minus 1205751
=
119890minus(1205876)
(201205871205751minus 8120587)
1 minus 119890minus(12058712)
asymp 652614 gt 18
ge max0le119905le2120587
11988611(119905) + 119887
11(119905) + 119888
11(119905)
(1 + 119903119871
1)
1205752
112057212
1 minus 1205751
=
119890minus(1205876)
(181205871205752minus 2120587)
1 minus 119890minus(12058712)
asymp 1021184 gt
347
30
ge max0le119905le2120587
11988612(119905) + 119887
12(119905) + 119888
12(119905)
(1 + 119903119871
2)
1205752
212057221
1 minus 1205752
=
119890minus(12)
(241205871205751minus (141205873))
1 minus 119890minus(14)
asymp 1133411 gt
245
12
gt max0le119905le2120587
11988621(119905) + 119887
21(119905) + 119888
21(119905)
(1 + 119903119871
2)
1205752
212057222
1 minus 1205752
=
119890minus(12)
(141205871205752minus 3120587)
1 minus 119890minus(14)
asymp 628411 gt
53
5
gt max0le119905le2120587
11988622(119905) + 119887
22(119905) + 119888
22(119905)
1205752
1min1le119895le2
1205721119895
1 minus 1205751
=
119890minus(1205876)
(201205871205751minus 6120587)
1 minus 119890minus(12058712)
asymp 652614
1205752
2min1le119895le2
1205722119895
1 minus 1205752
=
119890minus(12)
(141205871205752minus 3120587)
1 minus 119890minus(14)
asymp 628411
min1le119894le2
1205752
119894min1le119895le2
120572119894119895
1 minus 120575119894
=
1205752
2min1le119895le2
1205722119895
1 minus 1205751
asymp 628411
119888119872
11+ 119888119872
12=
19
6
119888119872
21+ 119888119872
22=
61
60
max1le119894le2
2
sum
119895=1
119889119872
119894119895
=
19
6
(51)
Therefore
19
6
= max1le119894le2
2
sum
119894=1
119889119872
119894119895 lt min1le119894le2
1205752
119894min1le119895le2
120572119894119895
1 minus 120575119894
asymp 628411
(52)
Hence (1198601)ndash(1198603) (1198605) hold and 119886
119872
119894le 1 119894 = 1 2
According toTheorem 11 system (49) has at least one positive2120587-periodic solution
Acknowledgments
The authors are thankful to the referees and editor for theimprovement of the paper This work was supported bythe Construct Program of the Key Discipline in HunanProvince Research was supported by the National NaturalScience Foundation of China (10971229 11161015) the ChinaPostdoctoral Science Foundation (2012M512162) and HunanProvincial Natural Science-Hengyang United Foundation ofChina (11JJ9002)
References
[1] P Chesson ldquoUnderstanding the role of environmental variationin population and community dynamicsrdquo Theoretical Popula-tion Biology vol 64 no 3 pp 253ndash254 2003
[2] J Zhen and Z E Ma ldquoPeriodic solutions for delay differentialequations model of plankton allelopathyrdquo Computers amp Mathe-matics with Applications vol 44 no 3-4 pp 491ndash500 2002
[3] X Liu and G Ballinger ldquoBoundedness for impulsive delaydifferential equations and applications to population growthmodelsrdquo Nonlinear Analysis Theory Methods amp Applicationsvol 53 no 7-8 pp 1041ndash1062 2003
[4] Y Kuang Delay Differential Equations with Applications inPopulation Dynamics Academic Press New York NY USA1993
[5] Y K Li and Y Kuang ldquoPeriodic solutions of periodic delayLotka-Volterra equations and systemsrdquo Journal of MathematicalAnalysis and Applications vol 255 no 1 pp 260ndash280 2001
[6] Y K Li and L F Zhu ldquoExistence of periodic solutions of discreteLotka-Volterra systems with delaysrdquo Bulletin of the Institute ofMathematics Academia Sinica vol 33 no 4 pp 369ndash380 2005
[7] H Zhao and N Ding ldquoExistence and global attractivity ofpositive periodic solution for competition-predator systemwithvariable delaysrdquoChaos SolitonsampFractals vol 29 no 1 pp 162ndash170 2006
[8] Y Song M Han and Y Peng ldquoStability and Hopf bifurcationsin a competitive Lotka-Volterra system with two delaysrdquo ChaosSolitons amp Fractals vol 22 no 5 pp 1139ndash1148 2004
[9] Z Yang and J Cao ldquoPositive periodic solutions of neutral Lotka-Volterra system with periodic delaysrdquo Applied Mathematics andComputation vol 149 no 3 pp 661ndash687 2004
[10] Y K Li ldquoPositive periodic solutions of periodic neutral Lotka-Volterra system with distributed delaysrdquo Chaos Solitons ampFractals vol 37 no 1 pp 288ndash298 2008
[11] A M Samoikleno and N A Perestyuk Impulsive DifferentialEquations World Scientific Singapore 1995
[12] S T Zavalishchin and A N SesekinDynamic Impulse SystemsTheory and Applications Kluwer Academic Publishers Dor-drecht The Netherlands 1997
International Journal of Differential Equations 13
[13] V Lakshmikantham D D Bainov and P S Simenov Theoryof Impulsive Differential Equations World Science Singapore1989
[14] J R Yan A P Zhao and J J Nieto ldquoExistence and globalattractivity of positive periodic solution of periodic single-species impulsive Lotka-Volterra systemsrdquo Mathematical andComputer Modelling vol 40 no 5-6 pp 509ndash518 2004
[15] X Liu and G Ballinger ldquoBoundedness for impulsive delaydifferential equations and applications to population growthmodelsrdquo Nonlinear Analysis Theory Methods amp Applicationsvol 53 no 7-8 pp 1041ndash1062 2003
[16] W Zhang and M Fan ldquoPeriodicity in a generalized ecologicalcompetition system governed by impulsive differential equa-tions with delaysrdquo Mathematical and Computer Modelling vol39 no 4-5 pp 479ndash493 2004
[17] S L Sun and L S Chen ldquoExistence of positive periodic solutionof an impulsive delay logistic modelrdquo Applied Mathematics andComputation vol 184 no 2 pp 617ndash623 2007
[18] X Li X Lin D Jiang and X Zhang ldquoExistence andmultiplicityof positive periodic solutions to functional differential equa-tions with impulse effectsrdquoNonlinear Analysis Theory Methodsamp Applications vol 62 no 4 pp 683ndash701 2005
[19] R E Gaines and J L Mawhin Coincidence Degree Theory andNonlinear Differential Equations Springer Berlin Germany1977
[20] N P Cac and J A Gatica ldquoFixed point theorems for mappingsin orderedBanach spacesrdquo Journal ofMathematical Analysis andApplications vol 71 no 2 pp 547ndash557 1979
[21] D J GuoNonlinear Functional Analysis in Chinese ShanDongScience and Technology Press 2001
[22] D Baınov and P Simeonov Impulsive Differential EquationsPeriodic Solutions and Applications vol 66 of Pitman Mono-graphs and Surveys in Pure and Applied Mathematics 1993
[23] D J Guo ldquoPositive solutions to nonlinear operator equa-tions and their applications to nonlinear integral equationsrdquoAdvances in Mathematics vol 13 no 4 pp 294ndash310 1984(Chinese)
119896) represents the right limit of 119909(119905)
at the point 119905119896) 119868119894119896
isin 119862(119877+ 119877+) that is 119909 changes
decreasingly suddenly at times 119905119896 120596 gt 0 is a constant 119877 =
(minusinfin +infin) 119877+= [0 +infin) We assume that there exists an
integer 119902 gt 0 such that 119905119896+119902
= 119905119896+ 120596 119868
119894(119896+119902)= 119868119894119896 where
0 lt 1199051lt 1199052lt sdot sdot sdot lt 119905
119902lt 120596
The main purpose of this paper is by using a fixed-point theorem of strict-set-contraction [19 20] to establishnew criteria to guarantee the existence of positive periodicsolutions of the system (2)
For convenience we introduce the notation
119891119872= max119905isin[0120596]
1003816100381610038161003816119891 (119905)
1003816100381610038161003816 119891
119871= min119905isin[0120596]
1003816100381610038161003816119891 (119905)
1003816100381610038161003816
120574119894= lim119906rarr0
sup sum
119905le119905119896le119905+120596
119868119894119896(119906)
119906
119894 = 1 2 119899
120575119894= 119890minusint120596
0119903119894(119905)119889119905
119894 = 1 2 119899
120572119894119895= int
120596
0
[120575119895119886119894119895(119905) minus 119887
The paper is organized as follows In Section 2 we givesome definitions and lemmas to prove themain results of thispaper In Section 3 by using a fixed-point theorem of strict-set-contraction we established some criteria to guarantee theexistence of at least one positive periodic solution of system(2) Finally in Section 4 we give an example to show thevalidity of our result
2 Preliminaries
In order to obtain the existence of a periodic solution ofsystem (2) we first introduce some definitions and lemmas
Definition 1 (see [13]) A function 119909119894 119877 rarr (0 +infin) is said
to be a positive solution of (2) if the following conditions aresatisfied
(a) 119909119894(119905) is absolutely continuous on each (119905
119896 119905119896+1)
(b) for each 119896 isin 119885+ 119909119894(119905+
119896) and 119909
119894(119905minus
119896) exist and 119909
119894(119905minus
119896) =
119909119894(119905119896)
(c) 119909119894(119905) satisfies the first equation of (2) for almost every-
where in 119877 and 119909119894(119905119896) satisfies the second equation of
(2) at impulsive point 119905119896 119896 isin 119885
+
Definition 2 (see [21]) Let 119883 be a real Banach space and 119864 aclosed nonempty subset of119883 119864 is a cone provided
(i) 120572119909 + 120573119910 isin 119864 for all 119909 119910 isin 119864 and all 120572 120573 ge 0(ii) 119909 minus119909 isin 119864 imply 119909 = 0
Definition 3 (see [21]) Let 119860 be a bounded subset in 119883Define
120572119883(119860) = inf 120575 gt 0 there is a finite number
of subsets 119860119894sub 119860 such that
119860 = ⋃
119894
119860119894and diam (119860
119894) le 120575
(4)
where diam(119860119894) denotes the diameter of the set 119860
119894 obvi-
ously 0 le 120572119883(119860) lt infin So 120572
119883(119860) is called the Kuratowski
measure of noncompactness of 119883
Definition 4 (see [21]) Let 119883 119884 be two Banach spaces and119863 sub 119883 a continuous and boundedmap 119879 119863 rarr 119884 is called119896-set contractive if for any bounded set 119878 sub 119863 we have
120572119884(119879 (119878)) le 119896120572
119883(119878) (5)
120601 is called strict-set-contractive if it is 119896-set-contractive forsome 0 le 119896 lt 1
Definition 5 (see [22]) The set 119865 isin 119875119862120596is said to be quasi-
equicontinuous in [0 120596] if for any 120598 gt 0 there exists 120575 gt 0
such that if 119909 isin 119865 119896 isin 119873+ 1199051 1199052isin (119905119896minus1 119905119896)cap[0 120596] and |119905
1minus
1199052| lt 120575 then |119909(119905
1) minus 119909(119905
2)| lt 120598
International Journal of Differential Equations 3
Lemma 6 (see [22]) The set 119865 sub 119875119862120596is relatively compact if
and only if
(1) 119865 is bounded that is 119909 le 119872 for each 119909 isin 119865 andsome119872 gt 0
(2) 119865 is quasi-equicontinuous in [0 120596]
Lemma 7 119909119894(119905) is an 120596-periodic solution of (2) is equivalent
to 119909119894(119905) is an 120596-periodic solution of the following equation
10 International Journal of Differential Equations
From (36) and (37) we obtain
1003817100381710038171003817119879119909 minus 119879119910
10038171003817100381710038171le (
119899
sum
119894=1
1003816100381610038161003816119909119894
10038161003816100381610038160
119899
sum
119895=1
119888119872
119894119895)120578 + 120598
119899
sum
119894=1
119861119894
le 119877max1le119894le119899
119899
sum
119895=1
119888119872
119894119895
120578 + 120598
119899
sum
119894=1
119861119894 for any 119909 119910 isin 119878
119894119895119896
(39)
As 120598 is arbitrarily small it follows that
120572119884(119879 (119878)) le 119877max
1le119895le119899
119899
sum
119895=1
119888119872
119894119895
120572119884(119878) (40)
Therefore 119879 is strict-set-contractive The proof of Lemma 10is complete
3 Main Results
Our main result of this paper is as follows
Theorem 11 Assume that (1198601)ndash(1198603) (1198605) hold
(i) If max119903119872119894 119894 = 1 2 119899 le 1 then system (2) has at
least one positive 120596-periodic solution
(ii) If (1198604) holds and min119903119872
119894 119894 = 1 2 119899 gt 1 then
system (2) has at least one positive 120596-periodic solution
Proof We only need to prove (i) since the proof of (ii) issimilar Let
119877 = (min119894isin[1119899]
1205752
119894
1 minus 120575119894
min119895isin[1119899]
120572119894119895)
minus1
0 lt 119903 lt min119894isin[1119899]
120575119894(1 minus 120575
119894) minus 120574119894
max119895isin[1119899]
120573119894119895
(41)
Then it is easy to see that 0 lt 119903 lt 119877 From Lemmas 9 and10 we know that 119879 is strict-set-contractive on 119864
119903119877 In view
of (26) we see that if there exists 119909lowast isin 119864 such that 119879119909lowast =119909lowast then 119909lowast is one positive 120596-periodic solution of system (2)
Nowwewill prove that condition (ii) of Lemma 7holds Firstwe prove that 119879119909 ge 119909 for all 119909 isin 119864 119909
1lt 119903 Otherwise
there exists 119909 isin 119864 1199091lt 119903 such that 119879119909 ge 119909 So 119909
0gt 0
and 119879119909 minus 119909 ge 0 which implies that
119879119894119909 (119905) minus 119909
119894(119905) ge 120575
119894
1003816100381610038161003816119879119894119909 minus 119909119894
10038161003816100381610038161ge 0
for any 119905 isin [0 120596] 119894 = 1 2 119899(42)
Moreover for 119905 isin [0 120596] we have(119879119894119909) (119905)
which is a contradictionTherefore condition (ii) of Lemma 7holds By Lemma 7 we see that 119879 has at least one nonzerofixed point in 119864 Thus the system (6) has at least one positive120596-periodic solutionTherefore it follows from Lemma 6 thatsystem (2) has a positive 120596-periodic solution The proof ofTheorem 11 is complete
Remark 12 If 119868119894119896(119909(119905119896)) = 0 we can easily derive the
corresponding results in [10] Sowe extend the correspondingresults in [10]
4 Example
In this section we give an example to show the effectivenessof our result
Example 1 Consider the following nonimpulsive system
1199091015840
1(119905)
= 1199091(119905) [
1 minus sin 11990524
minus (10 + sin 119905) 1199091(119905)
minus (9 minus cos 119905) 1199092(119905) minus (3 minus 2 cos 119905)
times int
0
minus12059111
11989111(120585) 1199091(119905 + 120585) 119889120585 minus
5 + sin 11990515
times int
0
minus12059112
11989112(120585) 1199092(119905 + 120585) 119889120585 minus (1 + sin 119905)
times int
0
minus12059011
11989211(120585) 1199091015840
1(119905 + 120585) 119889120585 minus
4 minus 3 cos 1199056
timesint
0
minus12059012
11989212(120585) 1199091015840
2(119905 + 120585) 119889120585]
1199091015840
2(119905)
= 1199092(119905) [
1 + cos 1199058120587
minus (12 + 3 cos 119905) 1199091(119905)
minus (7 minus sin 119905) 1199092(119905) minus (2 + 3 sin 119905)
times int
0
minus12059121
11989121(120585) 1199092(119905 + 120585) 119889120585 minus (1 minus cos 119905)
times int
0
minus12059122
11989122(120585) 1199092(119905 + 120585) 119889120585 minus
12 International Journal of Differential Equations
min1le119895le2
1205721119895 = 12057211= 20120587120575
1minus 8120587 asymp 238007
min1le119895le2
1205722119895 = 12057222= 14120587120575
2minus 3120587 asymp 244284
120575111988611(119905) minus 119887
11(119905) minus 119888
11(119905) gt 0009 gt 0
120575211988612(119905) minus 119887
12(119905) minus 119888
12(119905) gt 1 gt 0
120575111988621(119905) minus 119887
21(119905) minus 119888
21(119905) gt 4 gt 0
120575211988622(119905) minus 119887
22(119905) minus 119888
22(119905) gt 2 gt 0
(1 + 119903119871
1)
1205752
112057211
1 minus 1205751
=
119890minus(1205876)
(201205871205751minus 8120587)
1 minus 119890minus(12058712)
asymp 652614 gt 18
ge max0le119905le2120587
11988611(119905) + 119887
11(119905) + 119888
11(119905)
(1 + 119903119871
1)
1205752
112057212
1 minus 1205751
=
119890minus(1205876)
(181205871205752minus 2120587)
1 minus 119890minus(12058712)
asymp 1021184 gt
347
30
ge max0le119905le2120587
11988612(119905) + 119887
12(119905) + 119888
12(119905)
(1 + 119903119871
2)
1205752
212057221
1 minus 1205752
=
119890minus(12)
(241205871205751minus (141205873))
1 minus 119890minus(14)
asymp 1133411 gt
245
12
gt max0le119905le2120587
11988621(119905) + 119887
21(119905) + 119888
21(119905)
(1 + 119903119871
2)
1205752
212057222
1 minus 1205752
=
119890minus(12)
(141205871205752minus 3120587)
1 minus 119890minus(14)
asymp 628411 gt
53
5
gt max0le119905le2120587
11988622(119905) + 119887
22(119905) + 119888
22(119905)
1205752
1min1le119895le2
1205721119895
1 minus 1205751
=
119890minus(1205876)
(201205871205751minus 6120587)
1 minus 119890minus(12058712)
asymp 652614
1205752
2min1le119895le2
1205722119895
1 minus 1205752
=
119890minus(12)
(141205871205752minus 3120587)
1 minus 119890minus(14)
asymp 628411
min1le119894le2
1205752
119894min1le119895le2
120572119894119895
1 minus 120575119894
=
1205752
2min1le119895le2
1205722119895
1 minus 1205751
asymp 628411
119888119872
11+ 119888119872
12=
19
6
119888119872
21+ 119888119872
22=
61
60
max1le119894le2
2
sum
119895=1
119889119872
119894119895
=
19
6
(51)
Therefore
19
6
= max1le119894le2
2
sum
119894=1
119889119872
119894119895 lt min1le119894le2
1205752
119894min1le119895le2
120572119894119895
1 minus 120575119894
asymp 628411
(52)
Hence (1198601)ndash(1198603) (1198605) hold and 119886
119872
119894le 1 119894 = 1 2
According toTheorem 11 system (49) has at least one positive2120587-periodic solution
Acknowledgments
The authors are thankful to the referees and editor for theimprovement of the paper This work was supported bythe Construct Program of the Key Discipline in HunanProvince Research was supported by the National NaturalScience Foundation of China (10971229 11161015) the ChinaPostdoctoral Science Foundation (2012M512162) and HunanProvincial Natural Science-Hengyang United Foundation ofChina (11JJ9002)
References
[1] P Chesson ldquoUnderstanding the role of environmental variationin population and community dynamicsrdquo Theoretical Popula-tion Biology vol 64 no 3 pp 253ndash254 2003
[2] J Zhen and Z E Ma ldquoPeriodic solutions for delay differentialequations model of plankton allelopathyrdquo Computers amp Mathe-matics with Applications vol 44 no 3-4 pp 491ndash500 2002
[3] X Liu and G Ballinger ldquoBoundedness for impulsive delaydifferential equations and applications to population growthmodelsrdquo Nonlinear Analysis Theory Methods amp Applicationsvol 53 no 7-8 pp 1041ndash1062 2003
[4] Y Kuang Delay Differential Equations with Applications inPopulation Dynamics Academic Press New York NY USA1993
[5] Y K Li and Y Kuang ldquoPeriodic solutions of periodic delayLotka-Volterra equations and systemsrdquo Journal of MathematicalAnalysis and Applications vol 255 no 1 pp 260ndash280 2001
[6] Y K Li and L F Zhu ldquoExistence of periodic solutions of discreteLotka-Volterra systems with delaysrdquo Bulletin of the Institute ofMathematics Academia Sinica vol 33 no 4 pp 369ndash380 2005
[7] H Zhao and N Ding ldquoExistence and global attractivity ofpositive periodic solution for competition-predator systemwithvariable delaysrdquoChaos SolitonsampFractals vol 29 no 1 pp 162ndash170 2006
[8] Y Song M Han and Y Peng ldquoStability and Hopf bifurcationsin a competitive Lotka-Volterra system with two delaysrdquo ChaosSolitons amp Fractals vol 22 no 5 pp 1139ndash1148 2004
[9] Z Yang and J Cao ldquoPositive periodic solutions of neutral Lotka-Volterra system with periodic delaysrdquo Applied Mathematics andComputation vol 149 no 3 pp 661ndash687 2004
[10] Y K Li ldquoPositive periodic solutions of periodic neutral Lotka-Volterra system with distributed delaysrdquo Chaos Solitons ampFractals vol 37 no 1 pp 288ndash298 2008
[11] A M Samoikleno and N A Perestyuk Impulsive DifferentialEquations World Scientific Singapore 1995
[12] S T Zavalishchin and A N SesekinDynamic Impulse SystemsTheory and Applications Kluwer Academic Publishers Dor-drecht The Netherlands 1997
International Journal of Differential Equations 13
[13] V Lakshmikantham D D Bainov and P S Simenov Theoryof Impulsive Differential Equations World Science Singapore1989
[14] J R Yan A P Zhao and J J Nieto ldquoExistence and globalattractivity of positive periodic solution of periodic single-species impulsive Lotka-Volterra systemsrdquo Mathematical andComputer Modelling vol 40 no 5-6 pp 509ndash518 2004
[15] X Liu and G Ballinger ldquoBoundedness for impulsive delaydifferential equations and applications to population growthmodelsrdquo Nonlinear Analysis Theory Methods amp Applicationsvol 53 no 7-8 pp 1041ndash1062 2003
[16] W Zhang and M Fan ldquoPeriodicity in a generalized ecologicalcompetition system governed by impulsive differential equa-tions with delaysrdquo Mathematical and Computer Modelling vol39 no 4-5 pp 479ndash493 2004
[17] S L Sun and L S Chen ldquoExistence of positive periodic solutionof an impulsive delay logistic modelrdquo Applied Mathematics andComputation vol 184 no 2 pp 617ndash623 2007
[18] X Li X Lin D Jiang and X Zhang ldquoExistence andmultiplicityof positive periodic solutions to functional differential equa-tions with impulse effectsrdquoNonlinear Analysis Theory Methodsamp Applications vol 62 no 4 pp 683ndash701 2005
[19] R E Gaines and J L Mawhin Coincidence Degree Theory andNonlinear Differential Equations Springer Berlin Germany1977
[20] N P Cac and J A Gatica ldquoFixed point theorems for mappingsin orderedBanach spacesrdquo Journal ofMathematical Analysis andApplications vol 71 no 2 pp 547ndash557 1979
[21] D J GuoNonlinear Functional Analysis in Chinese ShanDongScience and Technology Press 2001
[22] D Baınov and P Simeonov Impulsive Differential EquationsPeriodic Solutions and Applications vol 66 of Pitman Mono-graphs and Surveys in Pure and Applied Mathematics 1993
[23] D J Guo ldquoPositive solutions to nonlinear operator equa-tions and their applications to nonlinear integral equationsrdquoAdvances in Mathematics vol 13 no 4 pp 294ndash310 1984(Chinese)
10 International Journal of Differential Equations
From (36) and (37) we obtain
1003817100381710038171003817119879119909 minus 119879119910
10038171003817100381710038171le (
119899
sum
119894=1
1003816100381610038161003816119909119894
10038161003816100381610038160
119899
sum
119895=1
119888119872
119894119895)120578 + 120598
119899
sum
119894=1
119861119894
le 119877max1le119894le119899
119899
sum
119895=1
119888119872
119894119895
120578 + 120598
119899
sum
119894=1
119861119894 for any 119909 119910 isin 119878
119894119895119896
(39)
As 120598 is arbitrarily small it follows that
120572119884(119879 (119878)) le 119877max
1le119895le119899
119899
sum
119895=1
119888119872
119894119895
120572119884(119878) (40)
Therefore 119879 is strict-set-contractive The proof of Lemma 10is complete
3 Main Results
Our main result of this paper is as follows
Theorem 11 Assume that (1198601)ndash(1198603) (1198605) hold
(i) If max119903119872119894 119894 = 1 2 119899 le 1 then system (2) has at
least one positive 120596-periodic solution
(ii) If (1198604) holds and min119903119872
119894 119894 = 1 2 119899 gt 1 then
system (2) has at least one positive 120596-periodic solution
Proof We only need to prove (i) since the proof of (ii) issimilar Let
119877 = (min119894isin[1119899]
1205752
119894
1 minus 120575119894
min119895isin[1119899]
120572119894119895)
minus1
0 lt 119903 lt min119894isin[1119899]
120575119894(1 minus 120575
119894) minus 120574119894
max119895isin[1119899]
120573119894119895
(41)
Then it is easy to see that 0 lt 119903 lt 119877 From Lemmas 9 and10 we know that 119879 is strict-set-contractive on 119864
119903119877 In view
of (26) we see that if there exists 119909lowast isin 119864 such that 119879119909lowast =119909lowast then 119909lowast is one positive 120596-periodic solution of system (2)
Nowwewill prove that condition (ii) of Lemma 7holds Firstwe prove that 119879119909 ge 119909 for all 119909 isin 119864 119909
1lt 119903 Otherwise
there exists 119909 isin 119864 1199091lt 119903 such that 119879119909 ge 119909 So 119909
0gt 0
and 119879119909 minus 119909 ge 0 which implies that
119879119894119909 (119905) minus 119909
119894(119905) ge 120575
119894
1003816100381610038161003816119879119894119909 minus 119909119894
10038161003816100381610038161ge 0
for any 119905 isin [0 120596] 119894 = 1 2 119899(42)
Moreover for 119905 isin [0 120596] we have(119879119894119909) (119905)
which is a contradictionTherefore condition (ii) of Lemma 7holds By Lemma 7 we see that 119879 has at least one nonzerofixed point in 119864 Thus the system (6) has at least one positive120596-periodic solutionTherefore it follows from Lemma 6 thatsystem (2) has a positive 120596-periodic solution The proof ofTheorem 11 is complete
Remark 12 If 119868119894119896(119909(119905119896)) = 0 we can easily derive the
corresponding results in [10] Sowe extend the correspondingresults in [10]
4 Example
In this section we give an example to show the effectivenessof our result
Example 1 Consider the following nonimpulsive system
1199091015840
1(119905)
= 1199091(119905) [
1 minus sin 11990524
minus (10 + sin 119905) 1199091(119905)
minus (9 minus cos 119905) 1199092(119905) minus (3 minus 2 cos 119905)
times int
0
minus12059111
11989111(120585) 1199091(119905 + 120585) 119889120585 minus
5 + sin 11990515
times int
0
minus12059112
11989112(120585) 1199092(119905 + 120585) 119889120585 minus (1 + sin 119905)
times int
0
minus12059011
11989211(120585) 1199091015840
1(119905 + 120585) 119889120585 minus
4 minus 3 cos 1199056
timesint
0
minus12059012
11989212(120585) 1199091015840
2(119905 + 120585) 119889120585]
1199091015840
2(119905)
= 1199092(119905) [
1 + cos 1199058120587
minus (12 + 3 cos 119905) 1199091(119905)
minus (7 minus sin 119905) 1199092(119905) minus (2 + 3 sin 119905)
times int
0
minus12059121
11989121(120585) 1199092(119905 + 120585) 119889120585 minus (1 minus cos 119905)
times int
0
minus12059122
11989122(120585) 1199092(119905 + 120585) 119889120585 minus
12 International Journal of Differential Equations
min1le119895le2
1205721119895 = 12057211= 20120587120575
1minus 8120587 asymp 238007
min1le119895le2
1205722119895 = 12057222= 14120587120575
2minus 3120587 asymp 244284
120575111988611(119905) minus 119887
11(119905) minus 119888
11(119905) gt 0009 gt 0
120575211988612(119905) minus 119887
12(119905) minus 119888
12(119905) gt 1 gt 0
120575111988621(119905) minus 119887
21(119905) minus 119888
21(119905) gt 4 gt 0
120575211988622(119905) minus 119887
22(119905) minus 119888
22(119905) gt 2 gt 0
(1 + 119903119871
1)
1205752
112057211
1 minus 1205751
=
119890minus(1205876)
(201205871205751minus 8120587)
1 minus 119890minus(12058712)
asymp 652614 gt 18
ge max0le119905le2120587
11988611(119905) + 119887
11(119905) + 119888
11(119905)
(1 + 119903119871
1)
1205752
112057212
1 minus 1205751
=
119890minus(1205876)
(181205871205752minus 2120587)
1 minus 119890minus(12058712)
asymp 1021184 gt
347
30
ge max0le119905le2120587
11988612(119905) + 119887
12(119905) + 119888
12(119905)
(1 + 119903119871
2)
1205752
212057221
1 minus 1205752
=
119890minus(12)
(241205871205751minus (141205873))
1 minus 119890minus(14)
asymp 1133411 gt
245
12
gt max0le119905le2120587
11988621(119905) + 119887
21(119905) + 119888
21(119905)
(1 + 119903119871
2)
1205752
212057222
1 minus 1205752
=
119890minus(12)
(141205871205752minus 3120587)
1 minus 119890minus(14)
asymp 628411 gt
53
5
gt max0le119905le2120587
11988622(119905) + 119887
22(119905) + 119888
22(119905)
1205752
1min1le119895le2
1205721119895
1 minus 1205751
=
119890minus(1205876)
(201205871205751minus 6120587)
1 minus 119890minus(12058712)
asymp 652614
1205752
2min1le119895le2
1205722119895
1 minus 1205752
=
119890minus(12)
(141205871205752minus 3120587)
1 minus 119890minus(14)
asymp 628411
min1le119894le2
1205752
119894min1le119895le2
120572119894119895
1 minus 120575119894
=
1205752
2min1le119895le2
1205722119895
1 minus 1205751
asymp 628411
119888119872
11+ 119888119872
12=
19
6
119888119872
21+ 119888119872
22=
61
60
max1le119894le2
2
sum
119895=1
119889119872
119894119895
=
19
6
(51)
Therefore
19
6
= max1le119894le2
2
sum
119894=1
119889119872
119894119895 lt min1le119894le2
1205752
119894min1le119895le2
120572119894119895
1 minus 120575119894
asymp 628411
(52)
Hence (1198601)ndash(1198603) (1198605) hold and 119886
119872
119894le 1 119894 = 1 2
According toTheorem 11 system (49) has at least one positive2120587-periodic solution
Acknowledgments
The authors are thankful to the referees and editor for theimprovement of the paper This work was supported bythe Construct Program of the Key Discipline in HunanProvince Research was supported by the National NaturalScience Foundation of China (10971229 11161015) the ChinaPostdoctoral Science Foundation (2012M512162) and HunanProvincial Natural Science-Hengyang United Foundation ofChina (11JJ9002)
References
[1] P Chesson ldquoUnderstanding the role of environmental variationin population and community dynamicsrdquo Theoretical Popula-tion Biology vol 64 no 3 pp 253ndash254 2003
[2] J Zhen and Z E Ma ldquoPeriodic solutions for delay differentialequations model of plankton allelopathyrdquo Computers amp Mathe-matics with Applications vol 44 no 3-4 pp 491ndash500 2002
[3] X Liu and G Ballinger ldquoBoundedness for impulsive delaydifferential equations and applications to population growthmodelsrdquo Nonlinear Analysis Theory Methods amp Applicationsvol 53 no 7-8 pp 1041ndash1062 2003
[4] Y Kuang Delay Differential Equations with Applications inPopulation Dynamics Academic Press New York NY USA1993
[5] Y K Li and Y Kuang ldquoPeriodic solutions of periodic delayLotka-Volterra equations and systemsrdquo Journal of MathematicalAnalysis and Applications vol 255 no 1 pp 260ndash280 2001
[6] Y K Li and L F Zhu ldquoExistence of periodic solutions of discreteLotka-Volterra systems with delaysrdquo Bulletin of the Institute ofMathematics Academia Sinica vol 33 no 4 pp 369ndash380 2005
[7] H Zhao and N Ding ldquoExistence and global attractivity ofpositive periodic solution for competition-predator systemwithvariable delaysrdquoChaos SolitonsampFractals vol 29 no 1 pp 162ndash170 2006
[8] Y Song M Han and Y Peng ldquoStability and Hopf bifurcationsin a competitive Lotka-Volterra system with two delaysrdquo ChaosSolitons amp Fractals vol 22 no 5 pp 1139ndash1148 2004
[9] Z Yang and J Cao ldquoPositive periodic solutions of neutral Lotka-Volterra system with periodic delaysrdquo Applied Mathematics andComputation vol 149 no 3 pp 661ndash687 2004
[10] Y K Li ldquoPositive periodic solutions of periodic neutral Lotka-Volterra system with distributed delaysrdquo Chaos Solitons ampFractals vol 37 no 1 pp 288ndash298 2008
[11] A M Samoikleno and N A Perestyuk Impulsive DifferentialEquations World Scientific Singapore 1995
[12] S T Zavalishchin and A N SesekinDynamic Impulse SystemsTheory and Applications Kluwer Academic Publishers Dor-drecht The Netherlands 1997
International Journal of Differential Equations 13
[13] V Lakshmikantham D D Bainov and P S Simenov Theoryof Impulsive Differential Equations World Science Singapore1989
[14] J R Yan A P Zhao and J J Nieto ldquoExistence and globalattractivity of positive periodic solution of periodic single-species impulsive Lotka-Volterra systemsrdquo Mathematical andComputer Modelling vol 40 no 5-6 pp 509ndash518 2004
[15] X Liu and G Ballinger ldquoBoundedness for impulsive delaydifferential equations and applications to population growthmodelsrdquo Nonlinear Analysis Theory Methods amp Applicationsvol 53 no 7-8 pp 1041ndash1062 2003
[16] W Zhang and M Fan ldquoPeriodicity in a generalized ecologicalcompetition system governed by impulsive differential equa-tions with delaysrdquo Mathematical and Computer Modelling vol39 no 4-5 pp 479ndash493 2004
[17] S L Sun and L S Chen ldquoExistence of positive periodic solutionof an impulsive delay logistic modelrdquo Applied Mathematics andComputation vol 184 no 2 pp 617ndash623 2007
[18] X Li X Lin D Jiang and X Zhang ldquoExistence andmultiplicityof positive periodic solutions to functional differential equa-tions with impulse effectsrdquoNonlinear Analysis Theory Methodsamp Applications vol 62 no 4 pp 683ndash701 2005
[19] R E Gaines and J L Mawhin Coincidence Degree Theory andNonlinear Differential Equations Springer Berlin Germany1977
[20] N P Cac and J A Gatica ldquoFixed point theorems for mappingsin orderedBanach spacesrdquo Journal ofMathematical Analysis andApplications vol 71 no 2 pp 547ndash557 1979
[21] D J GuoNonlinear Functional Analysis in Chinese ShanDongScience and Technology Press 2001
[22] D Baınov and P Simeonov Impulsive Differential EquationsPeriodic Solutions and Applications vol 66 of Pitman Mono-graphs and Surveys in Pure and Applied Mathematics 1993
[23] D J Guo ldquoPositive solutions to nonlinear operator equa-tions and their applications to nonlinear integral equationsrdquoAdvances in Mathematics vol 13 no 4 pp 294ndash310 1984(Chinese)
10 International Journal of Differential Equations
From (36) and (37) we obtain
1003817100381710038171003817119879119909 minus 119879119910
10038171003817100381710038171le (
119899
sum
119894=1
1003816100381610038161003816119909119894
10038161003816100381610038160
119899
sum
119895=1
119888119872
119894119895)120578 + 120598
119899
sum
119894=1
119861119894
le 119877max1le119894le119899
119899
sum
119895=1
119888119872
119894119895
120578 + 120598
119899
sum
119894=1
119861119894 for any 119909 119910 isin 119878
119894119895119896
(39)
As 120598 is arbitrarily small it follows that
120572119884(119879 (119878)) le 119877max
1le119895le119899
119899
sum
119895=1
119888119872
119894119895
120572119884(119878) (40)
Therefore 119879 is strict-set-contractive The proof of Lemma 10is complete
3 Main Results
Our main result of this paper is as follows
Theorem 11 Assume that (1198601)ndash(1198603) (1198605) hold
(i) If max119903119872119894 119894 = 1 2 119899 le 1 then system (2) has at
least one positive 120596-periodic solution
(ii) If (1198604) holds and min119903119872
119894 119894 = 1 2 119899 gt 1 then
system (2) has at least one positive 120596-periodic solution
Proof We only need to prove (i) since the proof of (ii) issimilar Let
119877 = (min119894isin[1119899]
1205752
119894
1 minus 120575119894
min119895isin[1119899]
120572119894119895)
minus1
0 lt 119903 lt min119894isin[1119899]
120575119894(1 minus 120575
119894) minus 120574119894
max119895isin[1119899]
120573119894119895
(41)
Then it is easy to see that 0 lt 119903 lt 119877 From Lemmas 9 and10 we know that 119879 is strict-set-contractive on 119864
119903119877 In view
of (26) we see that if there exists 119909lowast isin 119864 such that 119879119909lowast =119909lowast then 119909lowast is one positive 120596-periodic solution of system (2)
Nowwewill prove that condition (ii) of Lemma 7holds Firstwe prove that 119879119909 ge 119909 for all 119909 isin 119864 119909
1lt 119903 Otherwise
there exists 119909 isin 119864 1199091lt 119903 such that 119879119909 ge 119909 So 119909
0gt 0
and 119879119909 minus 119909 ge 0 which implies that
119879119894119909 (119905) minus 119909
119894(119905) ge 120575
119894
1003816100381610038161003816119879119894119909 minus 119909119894
10038161003816100381610038161ge 0
for any 119905 isin [0 120596] 119894 = 1 2 119899(42)
Moreover for 119905 isin [0 120596] we have(119879119894119909) (119905)
which is a contradictionTherefore condition (ii) of Lemma 7holds By Lemma 7 we see that 119879 has at least one nonzerofixed point in 119864 Thus the system (6) has at least one positive120596-periodic solutionTherefore it follows from Lemma 6 thatsystem (2) has a positive 120596-periodic solution The proof ofTheorem 11 is complete
Remark 12 If 119868119894119896(119909(119905119896)) = 0 we can easily derive the
corresponding results in [10] Sowe extend the correspondingresults in [10]
4 Example
In this section we give an example to show the effectivenessof our result
Example 1 Consider the following nonimpulsive system
1199091015840
1(119905)
= 1199091(119905) [
1 minus sin 11990524
minus (10 + sin 119905) 1199091(119905)
minus (9 minus cos 119905) 1199092(119905) minus (3 minus 2 cos 119905)
times int
0
minus12059111
11989111(120585) 1199091(119905 + 120585) 119889120585 minus
5 + sin 11990515
times int
0
minus12059112
11989112(120585) 1199092(119905 + 120585) 119889120585 minus (1 + sin 119905)
times int
0
minus12059011
11989211(120585) 1199091015840
1(119905 + 120585) 119889120585 minus
4 minus 3 cos 1199056
timesint
0
minus12059012
11989212(120585) 1199091015840
2(119905 + 120585) 119889120585]
1199091015840
2(119905)
= 1199092(119905) [
1 + cos 1199058120587
minus (12 + 3 cos 119905) 1199091(119905)
minus (7 minus sin 119905) 1199092(119905) minus (2 + 3 sin 119905)
times int
0
minus12059121
11989121(120585) 1199092(119905 + 120585) 119889120585 minus (1 minus cos 119905)
times int
0
minus12059122
11989122(120585) 1199092(119905 + 120585) 119889120585 minus
12 International Journal of Differential Equations
min1le119895le2
1205721119895 = 12057211= 20120587120575
1minus 8120587 asymp 238007
min1le119895le2
1205722119895 = 12057222= 14120587120575
2minus 3120587 asymp 244284
120575111988611(119905) minus 119887
11(119905) minus 119888
11(119905) gt 0009 gt 0
120575211988612(119905) minus 119887
12(119905) minus 119888
12(119905) gt 1 gt 0
120575111988621(119905) minus 119887
21(119905) minus 119888
21(119905) gt 4 gt 0
120575211988622(119905) minus 119887
22(119905) minus 119888
22(119905) gt 2 gt 0
(1 + 119903119871
1)
1205752
112057211
1 minus 1205751
=
119890minus(1205876)
(201205871205751minus 8120587)
1 minus 119890minus(12058712)
asymp 652614 gt 18
ge max0le119905le2120587
11988611(119905) + 119887
11(119905) + 119888
11(119905)
(1 + 119903119871
1)
1205752
112057212
1 minus 1205751
=
119890minus(1205876)
(181205871205752minus 2120587)
1 minus 119890minus(12058712)
asymp 1021184 gt
347
30
ge max0le119905le2120587
11988612(119905) + 119887
12(119905) + 119888
12(119905)
(1 + 119903119871
2)
1205752
212057221
1 minus 1205752
=
119890minus(12)
(241205871205751minus (141205873))
1 minus 119890minus(14)
asymp 1133411 gt
245
12
gt max0le119905le2120587
11988621(119905) + 119887
21(119905) + 119888
21(119905)
(1 + 119903119871
2)
1205752
212057222
1 minus 1205752
=
119890minus(12)
(141205871205752minus 3120587)
1 minus 119890minus(14)
asymp 628411 gt
53
5
gt max0le119905le2120587
11988622(119905) + 119887
22(119905) + 119888
22(119905)
1205752
1min1le119895le2
1205721119895
1 minus 1205751
=
119890minus(1205876)
(201205871205751minus 6120587)
1 minus 119890minus(12058712)
asymp 652614
1205752
2min1le119895le2
1205722119895
1 minus 1205752
=
119890minus(12)
(141205871205752minus 3120587)
1 minus 119890minus(14)
asymp 628411
min1le119894le2
1205752
119894min1le119895le2
120572119894119895
1 minus 120575119894
=
1205752
2min1le119895le2
1205722119895
1 minus 1205751
asymp 628411
119888119872
11+ 119888119872
12=
19
6
119888119872
21+ 119888119872
22=
61
60
max1le119894le2
2
sum
119895=1
119889119872
119894119895
=
19
6
(51)
Therefore
19
6
= max1le119894le2
2
sum
119894=1
119889119872
119894119895 lt min1le119894le2
1205752
119894min1le119895le2
120572119894119895
1 minus 120575119894
asymp 628411
(52)
Hence (1198601)ndash(1198603) (1198605) hold and 119886
119872
119894le 1 119894 = 1 2
According toTheorem 11 system (49) has at least one positive2120587-periodic solution
Acknowledgments
The authors are thankful to the referees and editor for theimprovement of the paper This work was supported bythe Construct Program of the Key Discipline in HunanProvince Research was supported by the National NaturalScience Foundation of China (10971229 11161015) the ChinaPostdoctoral Science Foundation (2012M512162) and HunanProvincial Natural Science-Hengyang United Foundation ofChina (11JJ9002)
References
[1] P Chesson ldquoUnderstanding the role of environmental variationin population and community dynamicsrdquo Theoretical Popula-tion Biology vol 64 no 3 pp 253ndash254 2003
[2] J Zhen and Z E Ma ldquoPeriodic solutions for delay differentialequations model of plankton allelopathyrdquo Computers amp Mathe-matics with Applications vol 44 no 3-4 pp 491ndash500 2002
[3] X Liu and G Ballinger ldquoBoundedness for impulsive delaydifferential equations and applications to population growthmodelsrdquo Nonlinear Analysis Theory Methods amp Applicationsvol 53 no 7-8 pp 1041ndash1062 2003
[4] Y Kuang Delay Differential Equations with Applications inPopulation Dynamics Academic Press New York NY USA1993
[5] Y K Li and Y Kuang ldquoPeriodic solutions of periodic delayLotka-Volterra equations and systemsrdquo Journal of MathematicalAnalysis and Applications vol 255 no 1 pp 260ndash280 2001
[6] Y K Li and L F Zhu ldquoExistence of periodic solutions of discreteLotka-Volterra systems with delaysrdquo Bulletin of the Institute ofMathematics Academia Sinica vol 33 no 4 pp 369ndash380 2005
[7] H Zhao and N Ding ldquoExistence and global attractivity ofpositive periodic solution for competition-predator systemwithvariable delaysrdquoChaos SolitonsampFractals vol 29 no 1 pp 162ndash170 2006
[8] Y Song M Han and Y Peng ldquoStability and Hopf bifurcationsin a competitive Lotka-Volterra system with two delaysrdquo ChaosSolitons amp Fractals vol 22 no 5 pp 1139ndash1148 2004
[9] Z Yang and J Cao ldquoPositive periodic solutions of neutral Lotka-Volterra system with periodic delaysrdquo Applied Mathematics andComputation vol 149 no 3 pp 661ndash687 2004
[10] Y K Li ldquoPositive periodic solutions of periodic neutral Lotka-Volterra system with distributed delaysrdquo Chaos Solitons ampFractals vol 37 no 1 pp 288ndash298 2008
[11] A M Samoikleno and N A Perestyuk Impulsive DifferentialEquations World Scientific Singapore 1995
[12] S T Zavalishchin and A N SesekinDynamic Impulse SystemsTheory and Applications Kluwer Academic Publishers Dor-drecht The Netherlands 1997
International Journal of Differential Equations 13
[13] V Lakshmikantham D D Bainov and P S Simenov Theoryof Impulsive Differential Equations World Science Singapore1989
[14] J R Yan A P Zhao and J J Nieto ldquoExistence and globalattractivity of positive periodic solution of periodic single-species impulsive Lotka-Volterra systemsrdquo Mathematical andComputer Modelling vol 40 no 5-6 pp 509ndash518 2004
[15] X Liu and G Ballinger ldquoBoundedness for impulsive delaydifferential equations and applications to population growthmodelsrdquo Nonlinear Analysis Theory Methods amp Applicationsvol 53 no 7-8 pp 1041ndash1062 2003
[16] W Zhang and M Fan ldquoPeriodicity in a generalized ecologicalcompetition system governed by impulsive differential equa-tions with delaysrdquo Mathematical and Computer Modelling vol39 no 4-5 pp 479ndash493 2004
[17] S L Sun and L S Chen ldquoExistence of positive periodic solutionof an impulsive delay logistic modelrdquo Applied Mathematics andComputation vol 184 no 2 pp 617ndash623 2007
[18] X Li X Lin D Jiang and X Zhang ldquoExistence andmultiplicityof positive periodic solutions to functional differential equa-tions with impulse effectsrdquoNonlinear Analysis Theory Methodsamp Applications vol 62 no 4 pp 683ndash701 2005
[19] R E Gaines and J L Mawhin Coincidence Degree Theory andNonlinear Differential Equations Springer Berlin Germany1977
[20] N P Cac and J A Gatica ldquoFixed point theorems for mappingsin orderedBanach spacesrdquo Journal ofMathematical Analysis andApplications vol 71 no 2 pp 547ndash557 1979
[21] D J GuoNonlinear Functional Analysis in Chinese ShanDongScience and Technology Press 2001
[22] D Baınov and P Simeonov Impulsive Differential EquationsPeriodic Solutions and Applications vol 66 of Pitman Mono-graphs and Surveys in Pure and Applied Mathematics 1993
[23] D J Guo ldquoPositive solutions to nonlinear operator equa-tions and their applications to nonlinear integral equationsrdquoAdvances in Mathematics vol 13 no 4 pp 294ndash310 1984(Chinese)
10 International Journal of Differential Equations
From (36) and (37) we obtain
1003817100381710038171003817119879119909 minus 119879119910
10038171003817100381710038171le (
119899
sum
119894=1
1003816100381610038161003816119909119894
10038161003816100381610038160
119899
sum
119895=1
119888119872
119894119895)120578 + 120598
119899
sum
119894=1
119861119894
le 119877max1le119894le119899
119899
sum
119895=1
119888119872
119894119895
120578 + 120598
119899
sum
119894=1
119861119894 for any 119909 119910 isin 119878
119894119895119896
(39)
As 120598 is arbitrarily small it follows that
120572119884(119879 (119878)) le 119877max
1le119895le119899
119899
sum
119895=1
119888119872
119894119895
120572119884(119878) (40)
Therefore 119879 is strict-set-contractive The proof of Lemma 10is complete
3 Main Results
Our main result of this paper is as follows
Theorem 11 Assume that (1198601)ndash(1198603) (1198605) hold
(i) If max119903119872119894 119894 = 1 2 119899 le 1 then system (2) has at
least one positive 120596-periodic solution
(ii) If (1198604) holds and min119903119872
119894 119894 = 1 2 119899 gt 1 then
system (2) has at least one positive 120596-periodic solution
Proof We only need to prove (i) since the proof of (ii) issimilar Let
119877 = (min119894isin[1119899]
1205752
119894
1 minus 120575119894
min119895isin[1119899]
120572119894119895)
minus1
0 lt 119903 lt min119894isin[1119899]
120575119894(1 minus 120575
119894) minus 120574119894
max119895isin[1119899]
120573119894119895
(41)
Then it is easy to see that 0 lt 119903 lt 119877 From Lemmas 9 and10 we know that 119879 is strict-set-contractive on 119864
119903119877 In view
of (26) we see that if there exists 119909lowast isin 119864 such that 119879119909lowast =119909lowast then 119909lowast is one positive 120596-periodic solution of system (2)
Nowwewill prove that condition (ii) of Lemma 7holds Firstwe prove that 119879119909 ge 119909 for all 119909 isin 119864 119909
1lt 119903 Otherwise
there exists 119909 isin 119864 1199091lt 119903 such that 119879119909 ge 119909 So 119909
0gt 0
and 119879119909 minus 119909 ge 0 which implies that
119879119894119909 (119905) minus 119909
119894(119905) ge 120575
119894
1003816100381610038161003816119879119894119909 minus 119909119894
10038161003816100381610038161ge 0
for any 119905 isin [0 120596] 119894 = 1 2 119899(42)
Moreover for 119905 isin [0 120596] we have(119879119894119909) (119905)
which is a contradictionTherefore condition (ii) of Lemma 7holds By Lemma 7 we see that 119879 has at least one nonzerofixed point in 119864 Thus the system (6) has at least one positive120596-periodic solutionTherefore it follows from Lemma 6 thatsystem (2) has a positive 120596-periodic solution The proof ofTheorem 11 is complete
Remark 12 If 119868119894119896(119909(119905119896)) = 0 we can easily derive the
corresponding results in [10] Sowe extend the correspondingresults in [10]
4 Example
In this section we give an example to show the effectivenessof our result
Example 1 Consider the following nonimpulsive system
1199091015840
1(119905)
= 1199091(119905) [
1 minus sin 11990524
minus (10 + sin 119905) 1199091(119905)
minus (9 minus cos 119905) 1199092(119905) minus (3 minus 2 cos 119905)
times int
0
minus12059111
11989111(120585) 1199091(119905 + 120585) 119889120585 minus
5 + sin 11990515
times int
0
minus12059112
11989112(120585) 1199092(119905 + 120585) 119889120585 minus (1 + sin 119905)
times int
0
minus12059011
11989211(120585) 1199091015840
1(119905 + 120585) 119889120585 minus
4 minus 3 cos 1199056
timesint
0
minus12059012
11989212(120585) 1199091015840
2(119905 + 120585) 119889120585]
1199091015840
2(119905)
= 1199092(119905) [
1 + cos 1199058120587
minus (12 + 3 cos 119905) 1199091(119905)
minus (7 minus sin 119905) 1199092(119905) minus (2 + 3 sin 119905)
times int
0
minus12059121
11989121(120585) 1199092(119905 + 120585) 119889120585 minus (1 minus cos 119905)
times int
0
minus12059122
11989122(120585) 1199092(119905 + 120585) 119889120585 minus
12 International Journal of Differential Equations
min1le119895le2
1205721119895 = 12057211= 20120587120575
1minus 8120587 asymp 238007
min1le119895le2
1205722119895 = 12057222= 14120587120575
2minus 3120587 asymp 244284
120575111988611(119905) minus 119887
11(119905) minus 119888
11(119905) gt 0009 gt 0
120575211988612(119905) minus 119887
12(119905) minus 119888
12(119905) gt 1 gt 0
120575111988621(119905) minus 119887
21(119905) minus 119888
21(119905) gt 4 gt 0
120575211988622(119905) minus 119887
22(119905) minus 119888
22(119905) gt 2 gt 0
(1 + 119903119871
1)
1205752
112057211
1 minus 1205751
=
119890minus(1205876)
(201205871205751minus 8120587)
1 minus 119890minus(12058712)
asymp 652614 gt 18
ge max0le119905le2120587
11988611(119905) + 119887
11(119905) + 119888
11(119905)
(1 + 119903119871
1)
1205752
112057212
1 minus 1205751
=
119890minus(1205876)
(181205871205752minus 2120587)
1 minus 119890minus(12058712)
asymp 1021184 gt
347
30
ge max0le119905le2120587
11988612(119905) + 119887
12(119905) + 119888
12(119905)
(1 + 119903119871
2)
1205752
212057221
1 minus 1205752
=
119890minus(12)
(241205871205751minus (141205873))
1 minus 119890minus(14)
asymp 1133411 gt
245
12
gt max0le119905le2120587
11988621(119905) + 119887
21(119905) + 119888
21(119905)
(1 + 119903119871
2)
1205752
212057222
1 minus 1205752
=
119890minus(12)
(141205871205752minus 3120587)
1 minus 119890minus(14)
asymp 628411 gt
53
5
gt max0le119905le2120587
11988622(119905) + 119887
22(119905) + 119888
22(119905)
1205752
1min1le119895le2
1205721119895
1 minus 1205751
=
119890minus(1205876)
(201205871205751minus 6120587)
1 minus 119890minus(12058712)
asymp 652614
1205752
2min1le119895le2
1205722119895
1 minus 1205752
=
119890minus(12)
(141205871205752minus 3120587)
1 minus 119890minus(14)
asymp 628411
min1le119894le2
1205752
119894min1le119895le2
120572119894119895
1 minus 120575119894
=
1205752
2min1le119895le2
1205722119895
1 minus 1205751
asymp 628411
119888119872
11+ 119888119872
12=
19
6
119888119872
21+ 119888119872
22=
61
60
max1le119894le2
2
sum
119895=1
119889119872
119894119895
=
19
6
(51)
Therefore
19
6
= max1le119894le2
2
sum
119894=1
119889119872
119894119895 lt min1le119894le2
1205752
119894min1le119895le2
120572119894119895
1 minus 120575119894
asymp 628411
(52)
Hence (1198601)ndash(1198603) (1198605) hold and 119886
119872
119894le 1 119894 = 1 2
According toTheorem 11 system (49) has at least one positive2120587-periodic solution
Acknowledgments
The authors are thankful to the referees and editor for theimprovement of the paper This work was supported bythe Construct Program of the Key Discipline in HunanProvince Research was supported by the National NaturalScience Foundation of China (10971229 11161015) the ChinaPostdoctoral Science Foundation (2012M512162) and HunanProvincial Natural Science-Hengyang United Foundation ofChina (11JJ9002)
References
[1] P Chesson ldquoUnderstanding the role of environmental variationin population and community dynamicsrdquo Theoretical Popula-tion Biology vol 64 no 3 pp 253ndash254 2003
[2] J Zhen and Z E Ma ldquoPeriodic solutions for delay differentialequations model of plankton allelopathyrdquo Computers amp Mathe-matics with Applications vol 44 no 3-4 pp 491ndash500 2002
[3] X Liu and G Ballinger ldquoBoundedness for impulsive delaydifferential equations and applications to population growthmodelsrdquo Nonlinear Analysis Theory Methods amp Applicationsvol 53 no 7-8 pp 1041ndash1062 2003
[4] Y Kuang Delay Differential Equations with Applications inPopulation Dynamics Academic Press New York NY USA1993
[5] Y K Li and Y Kuang ldquoPeriodic solutions of periodic delayLotka-Volterra equations and systemsrdquo Journal of MathematicalAnalysis and Applications vol 255 no 1 pp 260ndash280 2001
[6] Y K Li and L F Zhu ldquoExistence of periodic solutions of discreteLotka-Volterra systems with delaysrdquo Bulletin of the Institute ofMathematics Academia Sinica vol 33 no 4 pp 369ndash380 2005
[7] H Zhao and N Ding ldquoExistence and global attractivity ofpositive periodic solution for competition-predator systemwithvariable delaysrdquoChaos SolitonsampFractals vol 29 no 1 pp 162ndash170 2006
[8] Y Song M Han and Y Peng ldquoStability and Hopf bifurcationsin a competitive Lotka-Volterra system with two delaysrdquo ChaosSolitons amp Fractals vol 22 no 5 pp 1139ndash1148 2004
[9] Z Yang and J Cao ldquoPositive periodic solutions of neutral Lotka-Volterra system with periodic delaysrdquo Applied Mathematics andComputation vol 149 no 3 pp 661ndash687 2004
[10] Y K Li ldquoPositive periodic solutions of periodic neutral Lotka-Volterra system with distributed delaysrdquo Chaos Solitons ampFractals vol 37 no 1 pp 288ndash298 2008
[11] A M Samoikleno and N A Perestyuk Impulsive DifferentialEquations World Scientific Singapore 1995
[12] S T Zavalishchin and A N SesekinDynamic Impulse SystemsTheory and Applications Kluwer Academic Publishers Dor-drecht The Netherlands 1997
International Journal of Differential Equations 13
[13] V Lakshmikantham D D Bainov and P S Simenov Theoryof Impulsive Differential Equations World Science Singapore1989
[14] J R Yan A P Zhao and J J Nieto ldquoExistence and globalattractivity of positive periodic solution of periodic single-species impulsive Lotka-Volterra systemsrdquo Mathematical andComputer Modelling vol 40 no 5-6 pp 509ndash518 2004
[15] X Liu and G Ballinger ldquoBoundedness for impulsive delaydifferential equations and applications to population growthmodelsrdquo Nonlinear Analysis Theory Methods amp Applicationsvol 53 no 7-8 pp 1041ndash1062 2003
[16] W Zhang and M Fan ldquoPeriodicity in a generalized ecologicalcompetition system governed by impulsive differential equa-tions with delaysrdquo Mathematical and Computer Modelling vol39 no 4-5 pp 479ndash493 2004
[17] S L Sun and L S Chen ldquoExistence of positive periodic solutionof an impulsive delay logistic modelrdquo Applied Mathematics andComputation vol 184 no 2 pp 617ndash623 2007
[18] X Li X Lin D Jiang and X Zhang ldquoExistence andmultiplicityof positive periodic solutions to functional differential equa-tions with impulse effectsrdquoNonlinear Analysis Theory Methodsamp Applications vol 62 no 4 pp 683ndash701 2005
[19] R E Gaines and J L Mawhin Coincidence Degree Theory andNonlinear Differential Equations Springer Berlin Germany1977
[20] N P Cac and J A Gatica ldquoFixed point theorems for mappingsin orderedBanach spacesrdquo Journal ofMathematical Analysis andApplications vol 71 no 2 pp 547ndash557 1979
[21] D J GuoNonlinear Functional Analysis in Chinese ShanDongScience and Technology Press 2001
[22] D Baınov and P Simeonov Impulsive Differential EquationsPeriodic Solutions and Applications vol 66 of Pitman Mono-graphs and Surveys in Pure and Applied Mathematics 1993
[23] D J Guo ldquoPositive solutions to nonlinear operator equa-tions and their applications to nonlinear integral equationsrdquoAdvances in Mathematics vol 13 no 4 pp 294ndash310 1984(Chinese)
10 International Journal of Differential Equations
From (36) and (37) we obtain
1003817100381710038171003817119879119909 minus 119879119910
10038171003817100381710038171le (
119899
sum
119894=1
1003816100381610038161003816119909119894
10038161003816100381610038160
119899
sum
119895=1
119888119872
119894119895)120578 + 120598
119899
sum
119894=1
119861119894
le 119877max1le119894le119899
119899
sum
119895=1
119888119872
119894119895
120578 + 120598
119899
sum
119894=1
119861119894 for any 119909 119910 isin 119878
119894119895119896
(39)
As 120598 is arbitrarily small it follows that
120572119884(119879 (119878)) le 119877max
1le119895le119899
119899
sum
119895=1
119888119872
119894119895
120572119884(119878) (40)
Therefore 119879 is strict-set-contractive The proof of Lemma 10is complete
3 Main Results
Our main result of this paper is as follows
Theorem 11 Assume that (1198601)ndash(1198603) (1198605) hold
(i) If max119903119872119894 119894 = 1 2 119899 le 1 then system (2) has at
least one positive 120596-periodic solution
(ii) If (1198604) holds and min119903119872
119894 119894 = 1 2 119899 gt 1 then
system (2) has at least one positive 120596-periodic solution
Proof We only need to prove (i) since the proof of (ii) issimilar Let
119877 = (min119894isin[1119899]
1205752
119894
1 minus 120575119894
min119895isin[1119899]
120572119894119895)
minus1
0 lt 119903 lt min119894isin[1119899]
120575119894(1 minus 120575
119894) minus 120574119894
max119895isin[1119899]
120573119894119895
(41)
Then it is easy to see that 0 lt 119903 lt 119877 From Lemmas 9 and10 we know that 119879 is strict-set-contractive on 119864
119903119877 In view
of (26) we see that if there exists 119909lowast isin 119864 such that 119879119909lowast =119909lowast then 119909lowast is one positive 120596-periodic solution of system (2)
Nowwewill prove that condition (ii) of Lemma 7holds Firstwe prove that 119879119909 ge 119909 for all 119909 isin 119864 119909
1lt 119903 Otherwise
there exists 119909 isin 119864 1199091lt 119903 such that 119879119909 ge 119909 So 119909
0gt 0
and 119879119909 minus 119909 ge 0 which implies that
119879119894119909 (119905) minus 119909
119894(119905) ge 120575
119894
1003816100381610038161003816119879119894119909 minus 119909119894
10038161003816100381610038161ge 0
for any 119905 isin [0 120596] 119894 = 1 2 119899(42)
Moreover for 119905 isin [0 120596] we have(119879119894119909) (119905)
which is a contradictionTherefore condition (ii) of Lemma 7holds By Lemma 7 we see that 119879 has at least one nonzerofixed point in 119864 Thus the system (6) has at least one positive120596-periodic solutionTherefore it follows from Lemma 6 thatsystem (2) has a positive 120596-periodic solution The proof ofTheorem 11 is complete
Remark 12 If 119868119894119896(119909(119905119896)) = 0 we can easily derive the
corresponding results in [10] Sowe extend the correspondingresults in [10]
4 Example
In this section we give an example to show the effectivenessof our result
Example 1 Consider the following nonimpulsive system
1199091015840
1(119905)
= 1199091(119905) [
1 minus sin 11990524
minus (10 + sin 119905) 1199091(119905)
minus (9 minus cos 119905) 1199092(119905) minus (3 minus 2 cos 119905)
times int
0
minus12059111
11989111(120585) 1199091(119905 + 120585) 119889120585 minus
5 + sin 11990515
times int
0
minus12059112
11989112(120585) 1199092(119905 + 120585) 119889120585 minus (1 + sin 119905)
times int
0
minus12059011
11989211(120585) 1199091015840
1(119905 + 120585) 119889120585 minus
4 minus 3 cos 1199056
timesint
0
minus12059012
11989212(120585) 1199091015840
2(119905 + 120585) 119889120585]
1199091015840
2(119905)
= 1199092(119905) [
1 + cos 1199058120587
minus (12 + 3 cos 119905) 1199091(119905)
minus (7 minus sin 119905) 1199092(119905) minus (2 + 3 sin 119905)
times int
0
minus12059121
11989121(120585) 1199092(119905 + 120585) 119889120585 minus (1 minus cos 119905)
times int
0
minus12059122
11989122(120585) 1199092(119905 + 120585) 119889120585 minus
12 International Journal of Differential Equations
min1le119895le2
1205721119895 = 12057211= 20120587120575
1minus 8120587 asymp 238007
min1le119895le2
1205722119895 = 12057222= 14120587120575
2minus 3120587 asymp 244284
120575111988611(119905) minus 119887
11(119905) minus 119888
11(119905) gt 0009 gt 0
120575211988612(119905) minus 119887
12(119905) minus 119888
12(119905) gt 1 gt 0
120575111988621(119905) minus 119887
21(119905) minus 119888
21(119905) gt 4 gt 0
120575211988622(119905) minus 119887
22(119905) minus 119888
22(119905) gt 2 gt 0
(1 + 119903119871
1)
1205752
112057211
1 minus 1205751
=
119890minus(1205876)
(201205871205751minus 8120587)
1 minus 119890minus(12058712)
asymp 652614 gt 18
ge max0le119905le2120587
11988611(119905) + 119887
11(119905) + 119888
11(119905)
(1 + 119903119871
1)
1205752
112057212
1 minus 1205751
=
119890minus(1205876)
(181205871205752minus 2120587)
1 minus 119890minus(12058712)
asymp 1021184 gt
347
30
ge max0le119905le2120587
11988612(119905) + 119887
12(119905) + 119888
12(119905)
(1 + 119903119871
2)
1205752
212057221
1 minus 1205752
=
119890minus(12)
(241205871205751minus (141205873))
1 minus 119890minus(14)
asymp 1133411 gt
245
12
gt max0le119905le2120587
11988621(119905) + 119887
21(119905) + 119888
21(119905)
(1 + 119903119871
2)
1205752
212057222
1 minus 1205752
=
119890minus(12)
(141205871205752minus 3120587)
1 minus 119890minus(14)
asymp 628411 gt
53
5
gt max0le119905le2120587
11988622(119905) + 119887
22(119905) + 119888
22(119905)
1205752
1min1le119895le2
1205721119895
1 minus 1205751
=
119890minus(1205876)
(201205871205751minus 6120587)
1 minus 119890minus(12058712)
asymp 652614
1205752
2min1le119895le2
1205722119895
1 minus 1205752
=
119890minus(12)
(141205871205752minus 3120587)
1 minus 119890minus(14)
asymp 628411
min1le119894le2
1205752
119894min1le119895le2
120572119894119895
1 minus 120575119894
=
1205752
2min1le119895le2
1205722119895
1 minus 1205751
asymp 628411
119888119872
11+ 119888119872
12=
19
6
119888119872
21+ 119888119872
22=
61
60
max1le119894le2
2
sum
119895=1
119889119872
119894119895
=
19
6
(51)
Therefore
19
6
= max1le119894le2
2
sum
119894=1
119889119872
119894119895 lt min1le119894le2
1205752
119894min1le119895le2
120572119894119895
1 minus 120575119894
asymp 628411
(52)
Hence (1198601)ndash(1198603) (1198605) hold and 119886
119872
119894le 1 119894 = 1 2
According toTheorem 11 system (49) has at least one positive2120587-periodic solution
Acknowledgments
The authors are thankful to the referees and editor for theimprovement of the paper This work was supported bythe Construct Program of the Key Discipline in HunanProvince Research was supported by the National NaturalScience Foundation of China (10971229 11161015) the ChinaPostdoctoral Science Foundation (2012M512162) and HunanProvincial Natural Science-Hengyang United Foundation ofChina (11JJ9002)
References
[1] P Chesson ldquoUnderstanding the role of environmental variationin population and community dynamicsrdquo Theoretical Popula-tion Biology vol 64 no 3 pp 253ndash254 2003
[2] J Zhen and Z E Ma ldquoPeriodic solutions for delay differentialequations model of plankton allelopathyrdquo Computers amp Mathe-matics with Applications vol 44 no 3-4 pp 491ndash500 2002
[3] X Liu and G Ballinger ldquoBoundedness for impulsive delaydifferential equations and applications to population growthmodelsrdquo Nonlinear Analysis Theory Methods amp Applicationsvol 53 no 7-8 pp 1041ndash1062 2003
[4] Y Kuang Delay Differential Equations with Applications inPopulation Dynamics Academic Press New York NY USA1993
[5] Y K Li and Y Kuang ldquoPeriodic solutions of periodic delayLotka-Volterra equations and systemsrdquo Journal of MathematicalAnalysis and Applications vol 255 no 1 pp 260ndash280 2001
[6] Y K Li and L F Zhu ldquoExistence of periodic solutions of discreteLotka-Volterra systems with delaysrdquo Bulletin of the Institute ofMathematics Academia Sinica vol 33 no 4 pp 369ndash380 2005
[7] H Zhao and N Ding ldquoExistence and global attractivity ofpositive periodic solution for competition-predator systemwithvariable delaysrdquoChaos SolitonsampFractals vol 29 no 1 pp 162ndash170 2006
[8] Y Song M Han and Y Peng ldquoStability and Hopf bifurcationsin a competitive Lotka-Volterra system with two delaysrdquo ChaosSolitons amp Fractals vol 22 no 5 pp 1139ndash1148 2004
[9] Z Yang and J Cao ldquoPositive periodic solutions of neutral Lotka-Volterra system with periodic delaysrdquo Applied Mathematics andComputation vol 149 no 3 pp 661ndash687 2004
[10] Y K Li ldquoPositive periodic solutions of periodic neutral Lotka-Volterra system with distributed delaysrdquo Chaos Solitons ampFractals vol 37 no 1 pp 288ndash298 2008
[11] A M Samoikleno and N A Perestyuk Impulsive DifferentialEquations World Scientific Singapore 1995
[12] S T Zavalishchin and A N SesekinDynamic Impulse SystemsTheory and Applications Kluwer Academic Publishers Dor-drecht The Netherlands 1997
International Journal of Differential Equations 13
[13] V Lakshmikantham D D Bainov and P S Simenov Theoryof Impulsive Differential Equations World Science Singapore1989
[14] J R Yan A P Zhao and J J Nieto ldquoExistence and globalattractivity of positive periodic solution of periodic single-species impulsive Lotka-Volterra systemsrdquo Mathematical andComputer Modelling vol 40 no 5-6 pp 509ndash518 2004
[15] X Liu and G Ballinger ldquoBoundedness for impulsive delaydifferential equations and applications to population growthmodelsrdquo Nonlinear Analysis Theory Methods amp Applicationsvol 53 no 7-8 pp 1041ndash1062 2003
[16] W Zhang and M Fan ldquoPeriodicity in a generalized ecologicalcompetition system governed by impulsive differential equa-tions with delaysrdquo Mathematical and Computer Modelling vol39 no 4-5 pp 479ndash493 2004
[17] S L Sun and L S Chen ldquoExistence of positive periodic solutionof an impulsive delay logistic modelrdquo Applied Mathematics andComputation vol 184 no 2 pp 617ndash623 2007
[18] X Li X Lin D Jiang and X Zhang ldquoExistence andmultiplicityof positive periodic solutions to functional differential equa-tions with impulse effectsrdquoNonlinear Analysis Theory Methodsamp Applications vol 62 no 4 pp 683ndash701 2005
[19] R E Gaines and J L Mawhin Coincidence Degree Theory andNonlinear Differential Equations Springer Berlin Germany1977
[20] N P Cac and J A Gatica ldquoFixed point theorems for mappingsin orderedBanach spacesrdquo Journal ofMathematical Analysis andApplications vol 71 no 2 pp 547ndash557 1979
[21] D J GuoNonlinear Functional Analysis in Chinese ShanDongScience and Technology Press 2001
[22] D Baınov and P Simeonov Impulsive Differential EquationsPeriodic Solutions and Applications vol 66 of Pitman Mono-graphs and Surveys in Pure and Applied Mathematics 1993
[23] D J Guo ldquoPositive solutions to nonlinear operator equa-tions and their applications to nonlinear integral equationsrdquoAdvances in Mathematics vol 13 no 4 pp 294ndash310 1984(Chinese)
10 International Journal of Differential Equations
From (36) and (37) we obtain
1003817100381710038171003817119879119909 minus 119879119910
10038171003817100381710038171le (
119899
sum
119894=1
1003816100381610038161003816119909119894
10038161003816100381610038160
119899
sum
119895=1
119888119872
119894119895)120578 + 120598
119899
sum
119894=1
119861119894
le 119877max1le119894le119899
119899
sum
119895=1
119888119872
119894119895
120578 + 120598
119899
sum
119894=1
119861119894 for any 119909 119910 isin 119878
119894119895119896
(39)
As 120598 is arbitrarily small it follows that
120572119884(119879 (119878)) le 119877max
1le119895le119899
119899
sum
119895=1
119888119872
119894119895
120572119884(119878) (40)
Therefore 119879 is strict-set-contractive The proof of Lemma 10is complete
3 Main Results
Our main result of this paper is as follows
Theorem 11 Assume that (1198601)ndash(1198603) (1198605) hold
(i) If max119903119872119894 119894 = 1 2 119899 le 1 then system (2) has at
least one positive 120596-periodic solution
(ii) If (1198604) holds and min119903119872
119894 119894 = 1 2 119899 gt 1 then
system (2) has at least one positive 120596-periodic solution
Proof We only need to prove (i) since the proof of (ii) issimilar Let
119877 = (min119894isin[1119899]
1205752
119894
1 minus 120575119894
min119895isin[1119899]
120572119894119895)
minus1
0 lt 119903 lt min119894isin[1119899]
120575119894(1 minus 120575
119894) minus 120574119894
max119895isin[1119899]
120573119894119895
(41)
Then it is easy to see that 0 lt 119903 lt 119877 From Lemmas 9 and10 we know that 119879 is strict-set-contractive on 119864
119903119877 In view
of (26) we see that if there exists 119909lowast isin 119864 such that 119879119909lowast =119909lowast then 119909lowast is one positive 120596-periodic solution of system (2)
Nowwewill prove that condition (ii) of Lemma 7holds Firstwe prove that 119879119909 ge 119909 for all 119909 isin 119864 119909
1lt 119903 Otherwise
there exists 119909 isin 119864 1199091lt 119903 such that 119879119909 ge 119909 So 119909
0gt 0
and 119879119909 minus 119909 ge 0 which implies that
119879119894119909 (119905) minus 119909
119894(119905) ge 120575
119894
1003816100381610038161003816119879119894119909 minus 119909119894
10038161003816100381610038161ge 0
for any 119905 isin [0 120596] 119894 = 1 2 119899(42)
Moreover for 119905 isin [0 120596] we have(119879119894119909) (119905)
which is a contradictionTherefore condition (ii) of Lemma 7holds By Lemma 7 we see that 119879 has at least one nonzerofixed point in 119864 Thus the system (6) has at least one positive120596-periodic solutionTherefore it follows from Lemma 6 thatsystem (2) has a positive 120596-periodic solution The proof ofTheorem 11 is complete
Remark 12 If 119868119894119896(119909(119905119896)) = 0 we can easily derive the
corresponding results in [10] Sowe extend the correspondingresults in [10]
4 Example
In this section we give an example to show the effectivenessof our result
Example 1 Consider the following nonimpulsive system
1199091015840
1(119905)
= 1199091(119905) [
1 minus sin 11990524
minus (10 + sin 119905) 1199091(119905)
minus (9 minus cos 119905) 1199092(119905) minus (3 minus 2 cos 119905)
times int
0
minus12059111
11989111(120585) 1199091(119905 + 120585) 119889120585 minus
5 + sin 11990515
times int
0
minus12059112
11989112(120585) 1199092(119905 + 120585) 119889120585 minus (1 + sin 119905)
times int
0
minus12059011
11989211(120585) 1199091015840
1(119905 + 120585) 119889120585 minus
4 minus 3 cos 1199056
timesint
0
minus12059012
11989212(120585) 1199091015840
2(119905 + 120585) 119889120585]
1199091015840
2(119905)
= 1199092(119905) [
1 + cos 1199058120587
minus (12 + 3 cos 119905) 1199091(119905)
minus (7 minus sin 119905) 1199092(119905) minus (2 + 3 sin 119905)
times int
0
minus12059121
11989121(120585) 1199092(119905 + 120585) 119889120585 minus (1 minus cos 119905)
times int
0
minus12059122
11989122(120585) 1199092(119905 + 120585) 119889120585 minus
12 International Journal of Differential Equations
min1le119895le2
1205721119895 = 12057211= 20120587120575
1minus 8120587 asymp 238007
min1le119895le2
1205722119895 = 12057222= 14120587120575
2minus 3120587 asymp 244284
120575111988611(119905) minus 119887
11(119905) minus 119888
11(119905) gt 0009 gt 0
120575211988612(119905) minus 119887
12(119905) minus 119888
12(119905) gt 1 gt 0
120575111988621(119905) minus 119887
21(119905) minus 119888
21(119905) gt 4 gt 0
120575211988622(119905) minus 119887
22(119905) minus 119888
22(119905) gt 2 gt 0
(1 + 119903119871
1)
1205752
112057211
1 minus 1205751
=
119890minus(1205876)
(201205871205751minus 8120587)
1 minus 119890minus(12058712)
asymp 652614 gt 18
ge max0le119905le2120587
11988611(119905) + 119887
11(119905) + 119888
11(119905)
(1 + 119903119871
1)
1205752
112057212
1 minus 1205751
=
119890minus(1205876)
(181205871205752minus 2120587)
1 minus 119890minus(12058712)
asymp 1021184 gt
347
30
ge max0le119905le2120587
11988612(119905) + 119887
12(119905) + 119888
12(119905)
(1 + 119903119871
2)
1205752
212057221
1 minus 1205752
=
119890minus(12)
(241205871205751minus (141205873))
1 minus 119890minus(14)
asymp 1133411 gt
245
12
gt max0le119905le2120587
11988621(119905) + 119887
21(119905) + 119888
21(119905)
(1 + 119903119871
2)
1205752
212057222
1 minus 1205752
=
119890minus(12)
(141205871205752minus 3120587)
1 minus 119890minus(14)
asymp 628411 gt
53
5
gt max0le119905le2120587
11988622(119905) + 119887
22(119905) + 119888
22(119905)
1205752
1min1le119895le2
1205721119895
1 minus 1205751
=
119890minus(1205876)
(201205871205751minus 6120587)
1 minus 119890minus(12058712)
asymp 652614
1205752
2min1le119895le2
1205722119895
1 minus 1205752
=
119890minus(12)
(141205871205752minus 3120587)
1 minus 119890minus(14)
asymp 628411
min1le119894le2
1205752
119894min1le119895le2
120572119894119895
1 minus 120575119894
=
1205752
2min1le119895le2
1205722119895
1 minus 1205751
asymp 628411
119888119872
11+ 119888119872
12=
19
6
119888119872
21+ 119888119872
22=
61
60
max1le119894le2
2
sum
119895=1
119889119872
119894119895
=
19
6
(51)
Therefore
19
6
= max1le119894le2
2
sum
119894=1
119889119872
119894119895 lt min1le119894le2
1205752
119894min1le119895le2
120572119894119895
1 minus 120575119894
asymp 628411
(52)
Hence (1198601)ndash(1198603) (1198605) hold and 119886
119872
119894le 1 119894 = 1 2
According toTheorem 11 system (49) has at least one positive2120587-periodic solution
Acknowledgments
The authors are thankful to the referees and editor for theimprovement of the paper This work was supported bythe Construct Program of the Key Discipline in HunanProvince Research was supported by the National NaturalScience Foundation of China (10971229 11161015) the ChinaPostdoctoral Science Foundation (2012M512162) and HunanProvincial Natural Science-Hengyang United Foundation ofChina (11JJ9002)
References
[1] P Chesson ldquoUnderstanding the role of environmental variationin population and community dynamicsrdquo Theoretical Popula-tion Biology vol 64 no 3 pp 253ndash254 2003
[2] J Zhen and Z E Ma ldquoPeriodic solutions for delay differentialequations model of plankton allelopathyrdquo Computers amp Mathe-matics with Applications vol 44 no 3-4 pp 491ndash500 2002
[3] X Liu and G Ballinger ldquoBoundedness for impulsive delaydifferential equations and applications to population growthmodelsrdquo Nonlinear Analysis Theory Methods amp Applicationsvol 53 no 7-8 pp 1041ndash1062 2003
[4] Y Kuang Delay Differential Equations with Applications inPopulation Dynamics Academic Press New York NY USA1993
[5] Y K Li and Y Kuang ldquoPeriodic solutions of periodic delayLotka-Volterra equations and systemsrdquo Journal of MathematicalAnalysis and Applications vol 255 no 1 pp 260ndash280 2001
[6] Y K Li and L F Zhu ldquoExistence of periodic solutions of discreteLotka-Volterra systems with delaysrdquo Bulletin of the Institute ofMathematics Academia Sinica vol 33 no 4 pp 369ndash380 2005
[7] H Zhao and N Ding ldquoExistence and global attractivity ofpositive periodic solution for competition-predator systemwithvariable delaysrdquoChaos SolitonsampFractals vol 29 no 1 pp 162ndash170 2006
[8] Y Song M Han and Y Peng ldquoStability and Hopf bifurcationsin a competitive Lotka-Volterra system with two delaysrdquo ChaosSolitons amp Fractals vol 22 no 5 pp 1139ndash1148 2004
[9] Z Yang and J Cao ldquoPositive periodic solutions of neutral Lotka-Volterra system with periodic delaysrdquo Applied Mathematics andComputation vol 149 no 3 pp 661ndash687 2004
[10] Y K Li ldquoPositive periodic solutions of periodic neutral Lotka-Volterra system with distributed delaysrdquo Chaos Solitons ampFractals vol 37 no 1 pp 288ndash298 2008
[11] A M Samoikleno and N A Perestyuk Impulsive DifferentialEquations World Scientific Singapore 1995
[12] S T Zavalishchin and A N SesekinDynamic Impulse SystemsTheory and Applications Kluwer Academic Publishers Dor-drecht The Netherlands 1997
International Journal of Differential Equations 13
[13] V Lakshmikantham D D Bainov and P S Simenov Theoryof Impulsive Differential Equations World Science Singapore1989
[14] J R Yan A P Zhao and J J Nieto ldquoExistence and globalattractivity of positive periodic solution of periodic single-species impulsive Lotka-Volterra systemsrdquo Mathematical andComputer Modelling vol 40 no 5-6 pp 509ndash518 2004
[15] X Liu and G Ballinger ldquoBoundedness for impulsive delaydifferential equations and applications to population growthmodelsrdquo Nonlinear Analysis Theory Methods amp Applicationsvol 53 no 7-8 pp 1041ndash1062 2003
[16] W Zhang and M Fan ldquoPeriodicity in a generalized ecologicalcompetition system governed by impulsive differential equa-tions with delaysrdquo Mathematical and Computer Modelling vol39 no 4-5 pp 479ndash493 2004
[17] S L Sun and L S Chen ldquoExistence of positive periodic solutionof an impulsive delay logistic modelrdquo Applied Mathematics andComputation vol 184 no 2 pp 617ndash623 2007
[18] X Li X Lin D Jiang and X Zhang ldquoExistence andmultiplicityof positive periodic solutions to functional differential equa-tions with impulse effectsrdquoNonlinear Analysis Theory Methodsamp Applications vol 62 no 4 pp 683ndash701 2005
[19] R E Gaines and J L Mawhin Coincidence Degree Theory andNonlinear Differential Equations Springer Berlin Germany1977
[20] N P Cac and J A Gatica ldquoFixed point theorems for mappingsin orderedBanach spacesrdquo Journal ofMathematical Analysis andApplications vol 71 no 2 pp 547ndash557 1979
[21] D J GuoNonlinear Functional Analysis in Chinese ShanDongScience and Technology Press 2001
[22] D Baınov and P Simeonov Impulsive Differential EquationsPeriodic Solutions and Applications vol 66 of Pitman Mono-graphs and Surveys in Pure and Applied Mathematics 1993
[23] D J Guo ldquoPositive solutions to nonlinear operator equa-tions and their applications to nonlinear integral equationsrdquoAdvances in Mathematics vol 13 no 4 pp 294ndash310 1984(Chinese)
10 International Journal of Differential Equations
From (36) and (37) we obtain
1003817100381710038171003817119879119909 minus 119879119910
10038171003817100381710038171le (
119899
sum
119894=1
1003816100381610038161003816119909119894
10038161003816100381610038160
119899
sum
119895=1
119888119872
119894119895)120578 + 120598
119899
sum
119894=1
119861119894
le 119877max1le119894le119899
119899
sum
119895=1
119888119872
119894119895
120578 + 120598
119899
sum
119894=1
119861119894 for any 119909 119910 isin 119878
119894119895119896
(39)
As 120598 is arbitrarily small it follows that
120572119884(119879 (119878)) le 119877max
1le119895le119899
119899
sum
119895=1
119888119872
119894119895
120572119884(119878) (40)
Therefore 119879 is strict-set-contractive The proof of Lemma 10is complete
3 Main Results
Our main result of this paper is as follows
Theorem 11 Assume that (1198601)ndash(1198603) (1198605) hold
(i) If max119903119872119894 119894 = 1 2 119899 le 1 then system (2) has at
least one positive 120596-periodic solution
(ii) If (1198604) holds and min119903119872
119894 119894 = 1 2 119899 gt 1 then
system (2) has at least one positive 120596-periodic solution
Proof We only need to prove (i) since the proof of (ii) issimilar Let
119877 = (min119894isin[1119899]
1205752
119894
1 minus 120575119894
min119895isin[1119899]
120572119894119895)
minus1
0 lt 119903 lt min119894isin[1119899]
120575119894(1 minus 120575
119894) minus 120574119894
max119895isin[1119899]
120573119894119895
(41)
Then it is easy to see that 0 lt 119903 lt 119877 From Lemmas 9 and10 we know that 119879 is strict-set-contractive on 119864
119903119877 In view
of (26) we see that if there exists 119909lowast isin 119864 such that 119879119909lowast =119909lowast then 119909lowast is one positive 120596-periodic solution of system (2)
Nowwewill prove that condition (ii) of Lemma 7holds Firstwe prove that 119879119909 ge 119909 for all 119909 isin 119864 119909
1lt 119903 Otherwise
there exists 119909 isin 119864 1199091lt 119903 such that 119879119909 ge 119909 So 119909
0gt 0
and 119879119909 minus 119909 ge 0 which implies that
119879119894119909 (119905) minus 119909
119894(119905) ge 120575
119894
1003816100381610038161003816119879119894119909 minus 119909119894
10038161003816100381610038161ge 0
for any 119905 isin [0 120596] 119894 = 1 2 119899(42)
Moreover for 119905 isin [0 120596] we have(119879119894119909) (119905)
which is a contradictionTherefore condition (ii) of Lemma 7holds By Lemma 7 we see that 119879 has at least one nonzerofixed point in 119864 Thus the system (6) has at least one positive120596-periodic solutionTherefore it follows from Lemma 6 thatsystem (2) has a positive 120596-periodic solution The proof ofTheorem 11 is complete
Remark 12 If 119868119894119896(119909(119905119896)) = 0 we can easily derive the
corresponding results in [10] Sowe extend the correspondingresults in [10]
4 Example
In this section we give an example to show the effectivenessof our result
Example 1 Consider the following nonimpulsive system
1199091015840
1(119905)
= 1199091(119905) [
1 minus sin 11990524
minus (10 + sin 119905) 1199091(119905)
minus (9 minus cos 119905) 1199092(119905) minus (3 minus 2 cos 119905)
times int
0
minus12059111
11989111(120585) 1199091(119905 + 120585) 119889120585 minus
5 + sin 11990515
times int
0
minus12059112
11989112(120585) 1199092(119905 + 120585) 119889120585 minus (1 + sin 119905)
times int
0
minus12059011
11989211(120585) 1199091015840
1(119905 + 120585) 119889120585 minus
4 minus 3 cos 1199056
timesint
0
minus12059012
11989212(120585) 1199091015840
2(119905 + 120585) 119889120585]
1199091015840
2(119905)
= 1199092(119905) [
1 + cos 1199058120587
minus (12 + 3 cos 119905) 1199091(119905)
minus (7 minus sin 119905) 1199092(119905) minus (2 + 3 sin 119905)
times int
0
minus12059121
11989121(120585) 1199092(119905 + 120585) 119889120585 minus (1 minus cos 119905)
times int
0
minus12059122
11989122(120585) 1199092(119905 + 120585) 119889120585 minus
12 International Journal of Differential Equations
min1le119895le2
1205721119895 = 12057211= 20120587120575
1minus 8120587 asymp 238007
min1le119895le2
1205722119895 = 12057222= 14120587120575
2minus 3120587 asymp 244284
120575111988611(119905) minus 119887
11(119905) minus 119888
11(119905) gt 0009 gt 0
120575211988612(119905) minus 119887
12(119905) minus 119888
12(119905) gt 1 gt 0
120575111988621(119905) minus 119887
21(119905) minus 119888
21(119905) gt 4 gt 0
120575211988622(119905) minus 119887
22(119905) minus 119888
22(119905) gt 2 gt 0
(1 + 119903119871
1)
1205752
112057211
1 minus 1205751
=
119890minus(1205876)
(201205871205751minus 8120587)
1 minus 119890minus(12058712)
asymp 652614 gt 18
ge max0le119905le2120587
11988611(119905) + 119887
11(119905) + 119888
11(119905)
(1 + 119903119871
1)
1205752
112057212
1 minus 1205751
=
119890minus(1205876)
(181205871205752minus 2120587)
1 minus 119890minus(12058712)
asymp 1021184 gt
347
30
ge max0le119905le2120587
11988612(119905) + 119887
12(119905) + 119888
12(119905)
(1 + 119903119871
2)
1205752
212057221
1 minus 1205752
=
119890minus(12)
(241205871205751minus (141205873))
1 minus 119890minus(14)
asymp 1133411 gt
245
12
gt max0le119905le2120587
11988621(119905) + 119887
21(119905) + 119888
21(119905)
(1 + 119903119871
2)
1205752
212057222
1 minus 1205752
=
119890minus(12)
(141205871205752minus 3120587)
1 minus 119890minus(14)
asymp 628411 gt
53
5
gt max0le119905le2120587
11988622(119905) + 119887
22(119905) + 119888
22(119905)
1205752
1min1le119895le2
1205721119895
1 minus 1205751
=
119890minus(1205876)
(201205871205751minus 6120587)
1 minus 119890minus(12058712)
asymp 652614
1205752
2min1le119895le2
1205722119895
1 minus 1205752
=
119890minus(12)
(141205871205752minus 3120587)
1 minus 119890minus(14)
asymp 628411
min1le119894le2
1205752
119894min1le119895le2
120572119894119895
1 minus 120575119894
=
1205752
2min1le119895le2
1205722119895
1 minus 1205751
asymp 628411
119888119872
11+ 119888119872
12=
19
6
119888119872
21+ 119888119872
22=
61
60
max1le119894le2
2
sum
119895=1
119889119872
119894119895
=
19
6
(51)
Therefore
19
6
= max1le119894le2
2
sum
119894=1
119889119872
119894119895 lt min1le119894le2
1205752
119894min1le119895le2
120572119894119895
1 minus 120575119894
asymp 628411
(52)
Hence (1198601)ndash(1198603) (1198605) hold and 119886
119872
119894le 1 119894 = 1 2
According toTheorem 11 system (49) has at least one positive2120587-periodic solution
Acknowledgments
The authors are thankful to the referees and editor for theimprovement of the paper This work was supported bythe Construct Program of the Key Discipline in HunanProvince Research was supported by the National NaturalScience Foundation of China (10971229 11161015) the ChinaPostdoctoral Science Foundation (2012M512162) and HunanProvincial Natural Science-Hengyang United Foundation ofChina (11JJ9002)
References
[1] P Chesson ldquoUnderstanding the role of environmental variationin population and community dynamicsrdquo Theoretical Popula-tion Biology vol 64 no 3 pp 253ndash254 2003
[2] J Zhen and Z E Ma ldquoPeriodic solutions for delay differentialequations model of plankton allelopathyrdquo Computers amp Mathe-matics with Applications vol 44 no 3-4 pp 491ndash500 2002
[3] X Liu and G Ballinger ldquoBoundedness for impulsive delaydifferential equations and applications to population growthmodelsrdquo Nonlinear Analysis Theory Methods amp Applicationsvol 53 no 7-8 pp 1041ndash1062 2003
[4] Y Kuang Delay Differential Equations with Applications inPopulation Dynamics Academic Press New York NY USA1993
[5] Y K Li and Y Kuang ldquoPeriodic solutions of periodic delayLotka-Volterra equations and systemsrdquo Journal of MathematicalAnalysis and Applications vol 255 no 1 pp 260ndash280 2001
[6] Y K Li and L F Zhu ldquoExistence of periodic solutions of discreteLotka-Volterra systems with delaysrdquo Bulletin of the Institute ofMathematics Academia Sinica vol 33 no 4 pp 369ndash380 2005
[7] H Zhao and N Ding ldquoExistence and global attractivity ofpositive periodic solution for competition-predator systemwithvariable delaysrdquoChaos SolitonsampFractals vol 29 no 1 pp 162ndash170 2006
[8] Y Song M Han and Y Peng ldquoStability and Hopf bifurcationsin a competitive Lotka-Volterra system with two delaysrdquo ChaosSolitons amp Fractals vol 22 no 5 pp 1139ndash1148 2004
[9] Z Yang and J Cao ldquoPositive periodic solutions of neutral Lotka-Volterra system with periodic delaysrdquo Applied Mathematics andComputation vol 149 no 3 pp 661ndash687 2004
[10] Y K Li ldquoPositive periodic solutions of periodic neutral Lotka-Volterra system with distributed delaysrdquo Chaos Solitons ampFractals vol 37 no 1 pp 288ndash298 2008
[11] A M Samoikleno and N A Perestyuk Impulsive DifferentialEquations World Scientific Singapore 1995
[12] S T Zavalishchin and A N SesekinDynamic Impulse SystemsTheory and Applications Kluwer Academic Publishers Dor-drecht The Netherlands 1997
International Journal of Differential Equations 13
[13] V Lakshmikantham D D Bainov and P S Simenov Theoryof Impulsive Differential Equations World Science Singapore1989
[14] J R Yan A P Zhao and J J Nieto ldquoExistence and globalattractivity of positive periodic solution of periodic single-species impulsive Lotka-Volterra systemsrdquo Mathematical andComputer Modelling vol 40 no 5-6 pp 509ndash518 2004
[15] X Liu and G Ballinger ldquoBoundedness for impulsive delaydifferential equations and applications to population growthmodelsrdquo Nonlinear Analysis Theory Methods amp Applicationsvol 53 no 7-8 pp 1041ndash1062 2003
[16] W Zhang and M Fan ldquoPeriodicity in a generalized ecologicalcompetition system governed by impulsive differential equa-tions with delaysrdquo Mathematical and Computer Modelling vol39 no 4-5 pp 479ndash493 2004
[17] S L Sun and L S Chen ldquoExistence of positive periodic solutionof an impulsive delay logistic modelrdquo Applied Mathematics andComputation vol 184 no 2 pp 617ndash623 2007
[18] X Li X Lin D Jiang and X Zhang ldquoExistence andmultiplicityof positive periodic solutions to functional differential equa-tions with impulse effectsrdquoNonlinear Analysis Theory Methodsamp Applications vol 62 no 4 pp 683ndash701 2005
[19] R E Gaines and J L Mawhin Coincidence Degree Theory andNonlinear Differential Equations Springer Berlin Germany1977
[20] N P Cac and J A Gatica ldquoFixed point theorems for mappingsin orderedBanach spacesrdquo Journal ofMathematical Analysis andApplications vol 71 no 2 pp 547ndash557 1979
[21] D J GuoNonlinear Functional Analysis in Chinese ShanDongScience and Technology Press 2001
[22] D Baınov and P Simeonov Impulsive Differential EquationsPeriodic Solutions and Applications vol 66 of Pitman Mono-graphs and Surveys in Pure and Applied Mathematics 1993
[23] D J Guo ldquoPositive solutions to nonlinear operator equa-tions and their applications to nonlinear integral equationsrdquoAdvances in Mathematics vol 13 no 4 pp 294ndash310 1984(Chinese)
10 International Journal of Differential Equations
From (36) and (37) we obtain
1003817100381710038171003817119879119909 minus 119879119910
10038171003817100381710038171le (
119899
sum
119894=1
1003816100381610038161003816119909119894
10038161003816100381610038160
119899
sum
119895=1
119888119872
119894119895)120578 + 120598
119899
sum
119894=1
119861119894
le 119877max1le119894le119899
119899
sum
119895=1
119888119872
119894119895
120578 + 120598
119899
sum
119894=1
119861119894 for any 119909 119910 isin 119878
119894119895119896
(39)
As 120598 is arbitrarily small it follows that
120572119884(119879 (119878)) le 119877max
1le119895le119899
119899
sum
119895=1
119888119872
119894119895
120572119884(119878) (40)
Therefore 119879 is strict-set-contractive The proof of Lemma 10is complete
3 Main Results
Our main result of this paper is as follows
Theorem 11 Assume that (1198601)ndash(1198603) (1198605) hold
(i) If max119903119872119894 119894 = 1 2 119899 le 1 then system (2) has at
least one positive 120596-periodic solution
(ii) If (1198604) holds and min119903119872
119894 119894 = 1 2 119899 gt 1 then
system (2) has at least one positive 120596-periodic solution
Proof We only need to prove (i) since the proof of (ii) issimilar Let
119877 = (min119894isin[1119899]
1205752
119894
1 minus 120575119894
min119895isin[1119899]
120572119894119895)
minus1
0 lt 119903 lt min119894isin[1119899]
120575119894(1 minus 120575
119894) minus 120574119894
max119895isin[1119899]
120573119894119895
(41)
Then it is easy to see that 0 lt 119903 lt 119877 From Lemmas 9 and10 we know that 119879 is strict-set-contractive on 119864
119903119877 In view
of (26) we see that if there exists 119909lowast isin 119864 such that 119879119909lowast =119909lowast then 119909lowast is one positive 120596-periodic solution of system (2)
Nowwewill prove that condition (ii) of Lemma 7holds Firstwe prove that 119879119909 ge 119909 for all 119909 isin 119864 119909
1lt 119903 Otherwise
there exists 119909 isin 119864 1199091lt 119903 such that 119879119909 ge 119909 So 119909
0gt 0
and 119879119909 minus 119909 ge 0 which implies that
119879119894119909 (119905) minus 119909
119894(119905) ge 120575
119894
1003816100381610038161003816119879119894119909 minus 119909119894
10038161003816100381610038161ge 0
for any 119905 isin [0 120596] 119894 = 1 2 119899(42)
Moreover for 119905 isin [0 120596] we have(119879119894119909) (119905)
which is a contradictionTherefore condition (ii) of Lemma 7holds By Lemma 7 we see that 119879 has at least one nonzerofixed point in 119864 Thus the system (6) has at least one positive120596-periodic solutionTherefore it follows from Lemma 6 thatsystem (2) has a positive 120596-periodic solution The proof ofTheorem 11 is complete
Remark 12 If 119868119894119896(119909(119905119896)) = 0 we can easily derive the
corresponding results in [10] Sowe extend the correspondingresults in [10]
4 Example
In this section we give an example to show the effectivenessof our result
Example 1 Consider the following nonimpulsive system
1199091015840
1(119905)
= 1199091(119905) [
1 minus sin 11990524
minus (10 + sin 119905) 1199091(119905)
minus (9 minus cos 119905) 1199092(119905) minus (3 minus 2 cos 119905)
times int
0
minus12059111
11989111(120585) 1199091(119905 + 120585) 119889120585 minus
5 + sin 11990515
times int
0
minus12059112
11989112(120585) 1199092(119905 + 120585) 119889120585 minus (1 + sin 119905)
times int
0
minus12059011
11989211(120585) 1199091015840
1(119905 + 120585) 119889120585 minus
4 minus 3 cos 1199056
timesint
0
minus12059012
11989212(120585) 1199091015840
2(119905 + 120585) 119889120585]
1199091015840
2(119905)
= 1199092(119905) [
1 + cos 1199058120587
minus (12 + 3 cos 119905) 1199091(119905)
minus (7 minus sin 119905) 1199092(119905) minus (2 + 3 sin 119905)
times int
0
minus12059121
11989121(120585) 1199092(119905 + 120585) 119889120585 minus (1 minus cos 119905)
times int
0
minus12059122
11989122(120585) 1199092(119905 + 120585) 119889120585 minus
12 International Journal of Differential Equations
min1le119895le2
1205721119895 = 12057211= 20120587120575
1minus 8120587 asymp 238007
min1le119895le2
1205722119895 = 12057222= 14120587120575
2minus 3120587 asymp 244284
120575111988611(119905) minus 119887
11(119905) minus 119888
11(119905) gt 0009 gt 0
120575211988612(119905) minus 119887
12(119905) minus 119888
12(119905) gt 1 gt 0
120575111988621(119905) minus 119887
21(119905) minus 119888
21(119905) gt 4 gt 0
120575211988622(119905) minus 119887
22(119905) minus 119888
22(119905) gt 2 gt 0
(1 + 119903119871
1)
1205752
112057211
1 minus 1205751
=
119890minus(1205876)
(201205871205751minus 8120587)
1 minus 119890minus(12058712)
asymp 652614 gt 18
ge max0le119905le2120587
11988611(119905) + 119887
11(119905) + 119888
11(119905)
(1 + 119903119871
1)
1205752
112057212
1 minus 1205751
=
119890minus(1205876)
(181205871205752minus 2120587)
1 minus 119890minus(12058712)
asymp 1021184 gt
347
30
ge max0le119905le2120587
11988612(119905) + 119887
12(119905) + 119888
12(119905)
(1 + 119903119871
2)
1205752
212057221
1 minus 1205752
=
119890minus(12)
(241205871205751minus (141205873))
1 minus 119890minus(14)
asymp 1133411 gt
245
12
gt max0le119905le2120587
11988621(119905) + 119887
21(119905) + 119888
21(119905)
(1 + 119903119871
2)
1205752
212057222
1 minus 1205752
=
119890minus(12)
(141205871205752minus 3120587)
1 minus 119890minus(14)
asymp 628411 gt
53
5
gt max0le119905le2120587
11988622(119905) + 119887
22(119905) + 119888
22(119905)
1205752
1min1le119895le2
1205721119895
1 minus 1205751
=
119890minus(1205876)
(201205871205751minus 6120587)
1 minus 119890minus(12058712)
asymp 652614
1205752
2min1le119895le2
1205722119895
1 minus 1205752
=
119890minus(12)
(141205871205752minus 3120587)
1 minus 119890minus(14)
asymp 628411
min1le119894le2
1205752
119894min1le119895le2
120572119894119895
1 minus 120575119894
=
1205752
2min1le119895le2
1205722119895
1 minus 1205751
asymp 628411
119888119872
11+ 119888119872
12=
19
6
119888119872
21+ 119888119872
22=
61
60
max1le119894le2
2
sum
119895=1
119889119872
119894119895
=
19
6
(51)
Therefore
19
6
= max1le119894le2
2
sum
119894=1
119889119872
119894119895 lt min1le119894le2
1205752
119894min1le119895le2
120572119894119895
1 minus 120575119894
asymp 628411
(52)
Hence (1198601)ndash(1198603) (1198605) hold and 119886
119872
119894le 1 119894 = 1 2
According toTheorem 11 system (49) has at least one positive2120587-periodic solution
Acknowledgments
The authors are thankful to the referees and editor for theimprovement of the paper This work was supported bythe Construct Program of the Key Discipline in HunanProvince Research was supported by the National NaturalScience Foundation of China (10971229 11161015) the ChinaPostdoctoral Science Foundation (2012M512162) and HunanProvincial Natural Science-Hengyang United Foundation ofChina (11JJ9002)
References
[1] P Chesson ldquoUnderstanding the role of environmental variationin population and community dynamicsrdquo Theoretical Popula-tion Biology vol 64 no 3 pp 253ndash254 2003
[2] J Zhen and Z E Ma ldquoPeriodic solutions for delay differentialequations model of plankton allelopathyrdquo Computers amp Mathe-matics with Applications vol 44 no 3-4 pp 491ndash500 2002
[3] X Liu and G Ballinger ldquoBoundedness for impulsive delaydifferential equations and applications to population growthmodelsrdquo Nonlinear Analysis Theory Methods amp Applicationsvol 53 no 7-8 pp 1041ndash1062 2003
[4] Y Kuang Delay Differential Equations with Applications inPopulation Dynamics Academic Press New York NY USA1993
[5] Y K Li and Y Kuang ldquoPeriodic solutions of periodic delayLotka-Volterra equations and systemsrdquo Journal of MathematicalAnalysis and Applications vol 255 no 1 pp 260ndash280 2001
[6] Y K Li and L F Zhu ldquoExistence of periodic solutions of discreteLotka-Volterra systems with delaysrdquo Bulletin of the Institute ofMathematics Academia Sinica vol 33 no 4 pp 369ndash380 2005
[7] H Zhao and N Ding ldquoExistence and global attractivity ofpositive periodic solution for competition-predator systemwithvariable delaysrdquoChaos SolitonsampFractals vol 29 no 1 pp 162ndash170 2006
[8] Y Song M Han and Y Peng ldquoStability and Hopf bifurcationsin a competitive Lotka-Volterra system with two delaysrdquo ChaosSolitons amp Fractals vol 22 no 5 pp 1139ndash1148 2004
[9] Z Yang and J Cao ldquoPositive periodic solutions of neutral Lotka-Volterra system with periodic delaysrdquo Applied Mathematics andComputation vol 149 no 3 pp 661ndash687 2004
[10] Y K Li ldquoPositive periodic solutions of periodic neutral Lotka-Volterra system with distributed delaysrdquo Chaos Solitons ampFractals vol 37 no 1 pp 288ndash298 2008
[11] A M Samoikleno and N A Perestyuk Impulsive DifferentialEquations World Scientific Singapore 1995
[12] S T Zavalishchin and A N SesekinDynamic Impulse SystemsTheory and Applications Kluwer Academic Publishers Dor-drecht The Netherlands 1997
International Journal of Differential Equations 13
[13] V Lakshmikantham D D Bainov and P S Simenov Theoryof Impulsive Differential Equations World Science Singapore1989
[14] J R Yan A P Zhao and J J Nieto ldquoExistence and globalattractivity of positive periodic solution of periodic single-species impulsive Lotka-Volterra systemsrdquo Mathematical andComputer Modelling vol 40 no 5-6 pp 509ndash518 2004
[15] X Liu and G Ballinger ldquoBoundedness for impulsive delaydifferential equations and applications to population growthmodelsrdquo Nonlinear Analysis Theory Methods amp Applicationsvol 53 no 7-8 pp 1041ndash1062 2003
[16] W Zhang and M Fan ldquoPeriodicity in a generalized ecologicalcompetition system governed by impulsive differential equa-tions with delaysrdquo Mathematical and Computer Modelling vol39 no 4-5 pp 479ndash493 2004
[17] S L Sun and L S Chen ldquoExistence of positive periodic solutionof an impulsive delay logistic modelrdquo Applied Mathematics andComputation vol 184 no 2 pp 617ndash623 2007
[18] X Li X Lin D Jiang and X Zhang ldquoExistence andmultiplicityof positive periodic solutions to functional differential equa-tions with impulse effectsrdquoNonlinear Analysis Theory Methodsamp Applications vol 62 no 4 pp 683ndash701 2005
[19] R E Gaines and J L Mawhin Coincidence Degree Theory andNonlinear Differential Equations Springer Berlin Germany1977
[20] N P Cac and J A Gatica ldquoFixed point theorems for mappingsin orderedBanach spacesrdquo Journal ofMathematical Analysis andApplications vol 71 no 2 pp 547ndash557 1979
[21] D J GuoNonlinear Functional Analysis in Chinese ShanDongScience and Technology Press 2001
[22] D Baınov and P Simeonov Impulsive Differential EquationsPeriodic Solutions and Applications vol 66 of Pitman Mono-graphs and Surveys in Pure and Applied Mathematics 1993
[23] D J Guo ldquoPositive solutions to nonlinear operator equa-tions and their applications to nonlinear integral equationsrdquoAdvances in Mathematics vol 13 no 4 pp 294ndash310 1984(Chinese)
10 International Journal of Differential Equations
From (36) and (37) we obtain
1003817100381710038171003817119879119909 minus 119879119910
10038171003817100381710038171le (
119899
sum
119894=1
1003816100381610038161003816119909119894
10038161003816100381610038160
119899
sum
119895=1
119888119872
119894119895)120578 + 120598
119899
sum
119894=1
119861119894
le 119877max1le119894le119899
119899
sum
119895=1
119888119872
119894119895
120578 + 120598
119899
sum
119894=1
119861119894 for any 119909 119910 isin 119878
119894119895119896
(39)
As 120598 is arbitrarily small it follows that
120572119884(119879 (119878)) le 119877max
1le119895le119899
119899
sum
119895=1
119888119872
119894119895
120572119884(119878) (40)
Therefore 119879 is strict-set-contractive The proof of Lemma 10is complete
3 Main Results
Our main result of this paper is as follows
Theorem 11 Assume that (1198601)ndash(1198603) (1198605) hold
(i) If max119903119872119894 119894 = 1 2 119899 le 1 then system (2) has at
least one positive 120596-periodic solution
(ii) If (1198604) holds and min119903119872
119894 119894 = 1 2 119899 gt 1 then
system (2) has at least one positive 120596-periodic solution
Proof We only need to prove (i) since the proof of (ii) issimilar Let
119877 = (min119894isin[1119899]
1205752
119894
1 minus 120575119894
min119895isin[1119899]
120572119894119895)
minus1
0 lt 119903 lt min119894isin[1119899]
120575119894(1 minus 120575
119894) minus 120574119894
max119895isin[1119899]
120573119894119895
(41)
Then it is easy to see that 0 lt 119903 lt 119877 From Lemmas 9 and10 we know that 119879 is strict-set-contractive on 119864
119903119877 In view
of (26) we see that if there exists 119909lowast isin 119864 such that 119879119909lowast =119909lowast then 119909lowast is one positive 120596-periodic solution of system (2)
Nowwewill prove that condition (ii) of Lemma 7holds Firstwe prove that 119879119909 ge 119909 for all 119909 isin 119864 119909
1lt 119903 Otherwise
there exists 119909 isin 119864 1199091lt 119903 such that 119879119909 ge 119909 So 119909
0gt 0
and 119879119909 minus 119909 ge 0 which implies that
119879119894119909 (119905) minus 119909
119894(119905) ge 120575
119894
1003816100381610038161003816119879119894119909 minus 119909119894
10038161003816100381610038161ge 0
for any 119905 isin [0 120596] 119894 = 1 2 119899(42)
Moreover for 119905 isin [0 120596] we have(119879119894119909) (119905)
which is a contradictionTherefore condition (ii) of Lemma 7holds By Lemma 7 we see that 119879 has at least one nonzerofixed point in 119864 Thus the system (6) has at least one positive120596-periodic solutionTherefore it follows from Lemma 6 thatsystem (2) has a positive 120596-periodic solution The proof ofTheorem 11 is complete
Remark 12 If 119868119894119896(119909(119905119896)) = 0 we can easily derive the
corresponding results in [10] Sowe extend the correspondingresults in [10]
4 Example
In this section we give an example to show the effectivenessof our result
Example 1 Consider the following nonimpulsive system
1199091015840
1(119905)
= 1199091(119905) [
1 minus sin 11990524
minus (10 + sin 119905) 1199091(119905)
minus (9 minus cos 119905) 1199092(119905) minus (3 minus 2 cos 119905)
times int
0
minus12059111
11989111(120585) 1199091(119905 + 120585) 119889120585 minus
5 + sin 11990515
times int
0
minus12059112
11989112(120585) 1199092(119905 + 120585) 119889120585 minus (1 + sin 119905)
times int
0
minus12059011
11989211(120585) 1199091015840
1(119905 + 120585) 119889120585 minus
4 minus 3 cos 1199056
timesint
0
minus12059012
11989212(120585) 1199091015840
2(119905 + 120585) 119889120585]
1199091015840
2(119905)
= 1199092(119905) [
1 + cos 1199058120587
minus (12 + 3 cos 119905) 1199091(119905)
minus (7 minus sin 119905) 1199092(119905) minus (2 + 3 sin 119905)
times int
0
minus12059121
11989121(120585) 1199092(119905 + 120585) 119889120585 minus (1 minus cos 119905)
times int
0
minus12059122
11989122(120585) 1199092(119905 + 120585) 119889120585 minus
12 International Journal of Differential Equations
min1le119895le2
1205721119895 = 12057211= 20120587120575
1minus 8120587 asymp 238007
min1le119895le2
1205722119895 = 12057222= 14120587120575
2minus 3120587 asymp 244284
120575111988611(119905) minus 119887
11(119905) minus 119888
11(119905) gt 0009 gt 0
120575211988612(119905) minus 119887
12(119905) minus 119888
12(119905) gt 1 gt 0
120575111988621(119905) minus 119887
21(119905) minus 119888
21(119905) gt 4 gt 0
120575211988622(119905) minus 119887
22(119905) minus 119888
22(119905) gt 2 gt 0
(1 + 119903119871
1)
1205752
112057211
1 minus 1205751
=
119890minus(1205876)
(201205871205751minus 8120587)
1 minus 119890minus(12058712)
asymp 652614 gt 18
ge max0le119905le2120587
11988611(119905) + 119887
11(119905) + 119888
11(119905)
(1 + 119903119871
1)
1205752
112057212
1 minus 1205751
=
119890minus(1205876)
(181205871205752minus 2120587)
1 minus 119890minus(12058712)
asymp 1021184 gt
347
30
ge max0le119905le2120587
11988612(119905) + 119887
12(119905) + 119888
12(119905)
(1 + 119903119871
2)
1205752
212057221
1 minus 1205752
=
119890minus(12)
(241205871205751minus (141205873))
1 minus 119890minus(14)
asymp 1133411 gt
245
12
gt max0le119905le2120587
11988621(119905) + 119887
21(119905) + 119888
21(119905)
(1 + 119903119871
2)
1205752
212057222
1 minus 1205752
=
119890minus(12)
(141205871205752minus 3120587)
1 minus 119890minus(14)
asymp 628411 gt
53
5
gt max0le119905le2120587
11988622(119905) + 119887
22(119905) + 119888
22(119905)
1205752
1min1le119895le2
1205721119895
1 minus 1205751
=
119890minus(1205876)
(201205871205751minus 6120587)
1 minus 119890minus(12058712)
asymp 652614
1205752
2min1le119895le2
1205722119895
1 minus 1205752
=
119890minus(12)
(141205871205752minus 3120587)
1 minus 119890minus(14)
asymp 628411
min1le119894le2
1205752
119894min1le119895le2
120572119894119895
1 minus 120575119894
=
1205752
2min1le119895le2
1205722119895
1 minus 1205751
asymp 628411
119888119872
11+ 119888119872
12=
19
6
119888119872
21+ 119888119872
22=
61
60
max1le119894le2
2
sum
119895=1
119889119872
119894119895
=
19
6
(51)
Therefore
19
6
= max1le119894le2
2
sum
119894=1
119889119872
119894119895 lt min1le119894le2
1205752
119894min1le119895le2
120572119894119895
1 minus 120575119894
asymp 628411
(52)
Hence (1198601)ndash(1198603) (1198605) hold and 119886
119872
119894le 1 119894 = 1 2
According toTheorem 11 system (49) has at least one positive2120587-periodic solution
Acknowledgments
The authors are thankful to the referees and editor for theimprovement of the paper This work was supported bythe Construct Program of the Key Discipline in HunanProvince Research was supported by the National NaturalScience Foundation of China (10971229 11161015) the ChinaPostdoctoral Science Foundation (2012M512162) and HunanProvincial Natural Science-Hengyang United Foundation ofChina (11JJ9002)
References
[1] P Chesson ldquoUnderstanding the role of environmental variationin population and community dynamicsrdquo Theoretical Popula-tion Biology vol 64 no 3 pp 253ndash254 2003
[2] J Zhen and Z E Ma ldquoPeriodic solutions for delay differentialequations model of plankton allelopathyrdquo Computers amp Mathe-matics with Applications vol 44 no 3-4 pp 491ndash500 2002
[3] X Liu and G Ballinger ldquoBoundedness for impulsive delaydifferential equations and applications to population growthmodelsrdquo Nonlinear Analysis Theory Methods amp Applicationsvol 53 no 7-8 pp 1041ndash1062 2003
[4] Y Kuang Delay Differential Equations with Applications inPopulation Dynamics Academic Press New York NY USA1993
[5] Y K Li and Y Kuang ldquoPeriodic solutions of periodic delayLotka-Volterra equations and systemsrdquo Journal of MathematicalAnalysis and Applications vol 255 no 1 pp 260ndash280 2001
[6] Y K Li and L F Zhu ldquoExistence of periodic solutions of discreteLotka-Volterra systems with delaysrdquo Bulletin of the Institute ofMathematics Academia Sinica vol 33 no 4 pp 369ndash380 2005
[7] H Zhao and N Ding ldquoExistence and global attractivity ofpositive periodic solution for competition-predator systemwithvariable delaysrdquoChaos SolitonsampFractals vol 29 no 1 pp 162ndash170 2006
[8] Y Song M Han and Y Peng ldquoStability and Hopf bifurcationsin a competitive Lotka-Volterra system with two delaysrdquo ChaosSolitons amp Fractals vol 22 no 5 pp 1139ndash1148 2004
[9] Z Yang and J Cao ldquoPositive periodic solutions of neutral Lotka-Volterra system with periodic delaysrdquo Applied Mathematics andComputation vol 149 no 3 pp 661ndash687 2004
[10] Y K Li ldquoPositive periodic solutions of periodic neutral Lotka-Volterra system with distributed delaysrdquo Chaos Solitons ampFractals vol 37 no 1 pp 288ndash298 2008
[11] A M Samoikleno and N A Perestyuk Impulsive DifferentialEquations World Scientific Singapore 1995
[12] S T Zavalishchin and A N SesekinDynamic Impulse SystemsTheory and Applications Kluwer Academic Publishers Dor-drecht The Netherlands 1997
International Journal of Differential Equations 13
[13] V Lakshmikantham D D Bainov and P S Simenov Theoryof Impulsive Differential Equations World Science Singapore1989
[14] J R Yan A P Zhao and J J Nieto ldquoExistence and globalattractivity of positive periodic solution of periodic single-species impulsive Lotka-Volterra systemsrdquo Mathematical andComputer Modelling vol 40 no 5-6 pp 509ndash518 2004
[15] X Liu and G Ballinger ldquoBoundedness for impulsive delaydifferential equations and applications to population growthmodelsrdquo Nonlinear Analysis Theory Methods amp Applicationsvol 53 no 7-8 pp 1041ndash1062 2003
[16] W Zhang and M Fan ldquoPeriodicity in a generalized ecologicalcompetition system governed by impulsive differential equa-tions with delaysrdquo Mathematical and Computer Modelling vol39 no 4-5 pp 479ndash493 2004
[17] S L Sun and L S Chen ldquoExistence of positive periodic solutionof an impulsive delay logistic modelrdquo Applied Mathematics andComputation vol 184 no 2 pp 617ndash623 2007
[18] X Li X Lin D Jiang and X Zhang ldquoExistence andmultiplicityof positive periodic solutions to functional differential equa-tions with impulse effectsrdquoNonlinear Analysis Theory Methodsamp Applications vol 62 no 4 pp 683ndash701 2005
[19] R E Gaines and J L Mawhin Coincidence Degree Theory andNonlinear Differential Equations Springer Berlin Germany1977
[20] N P Cac and J A Gatica ldquoFixed point theorems for mappingsin orderedBanach spacesrdquo Journal ofMathematical Analysis andApplications vol 71 no 2 pp 547ndash557 1979
[21] D J GuoNonlinear Functional Analysis in Chinese ShanDongScience and Technology Press 2001
[22] D Baınov and P Simeonov Impulsive Differential EquationsPeriodic Solutions and Applications vol 66 of Pitman Mono-graphs and Surveys in Pure and Applied Mathematics 1993
[23] D J Guo ldquoPositive solutions to nonlinear operator equa-tions and their applications to nonlinear integral equationsrdquoAdvances in Mathematics vol 13 no 4 pp 294ndash310 1984(Chinese)
which is a contradictionTherefore condition (ii) of Lemma 7holds By Lemma 7 we see that 119879 has at least one nonzerofixed point in 119864 Thus the system (6) has at least one positive120596-periodic solutionTherefore it follows from Lemma 6 thatsystem (2) has a positive 120596-periodic solution The proof ofTheorem 11 is complete
Remark 12 If 119868119894119896(119909(119905119896)) = 0 we can easily derive the
corresponding results in [10] Sowe extend the correspondingresults in [10]
4 Example
In this section we give an example to show the effectivenessof our result
Example 1 Consider the following nonimpulsive system
1199091015840
1(119905)
= 1199091(119905) [
1 minus sin 11990524
minus (10 + sin 119905) 1199091(119905)
minus (9 minus cos 119905) 1199092(119905) minus (3 minus 2 cos 119905)
times int
0
minus12059111
11989111(120585) 1199091(119905 + 120585) 119889120585 minus
5 + sin 11990515
times int
0
minus12059112
11989112(120585) 1199092(119905 + 120585) 119889120585 minus (1 + sin 119905)
times int
0
minus12059011
11989211(120585) 1199091015840
1(119905 + 120585) 119889120585 minus
4 minus 3 cos 1199056
timesint
0
minus12059012
11989212(120585) 1199091015840
2(119905 + 120585) 119889120585]
1199091015840
2(119905)
= 1199092(119905) [
1 + cos 1199058120587
minus (12 + 3 cos 119905) 1199091(119905)
minus (7 minus sin 119905) 1199092(119905) minus (2 + 3 sin 119905)
times int
0
minus12059121
11989121(120585) 1199092(119905 + 120585) 119889120585 minus (1 minus cos 119905)
times int
0
minus12059122
11989122(120585) 1199092(119905 + 120585) 119889120585 minus
12 International Journal of Differential Equations
min1le119895le2
1205721119895 = 12057211= 20120587120575
1minus 8120587 asymp 238007
min1le119895le2
1205722119895 = 12057222= 14120587120575
2minus 3120587 asymp 244284
120575111988611(119905) minus 119887
11(119905) minus 119888
11(119905) gt 0009 gt 0
120575211988612(119905) minus 119887
12(119905) minus 119888
12(119905) gt 1 gt 0
120575111988621(119905) minus 119887
21(119905) minus 119888
21(119905) gt 4 gt 0
120575211988622(119905) minus 119887
22(119905) minus 119888
22(119905) gt 2 gt 0
(1 + 119903119871
1)
1205752
112057211
1 minus 1205751
=
119890minus(1205876)
(201205871205751minus 8120587)
1 minus 119890minus(12058712)
asymp 652614 gt 18
ge max0le119905le2120587
11988611(119905) + 119887
11(119905) + 119888
11(119905)
(1 + 119903119871
1)
1205752
112057212
1 minus 1205751
=
119890minus(1205876)
(181205871205752minus 2120587)
1 minus 119890minus(12058712)
asymp 1021184 gt
347
30
ge max0le119905le2120587
11988612(119905) + 119887
12(119905) + 119888
12(119905)
(1 + 119903119871
2)
1205752
212057221
1 minus 1205752
=
119890minus(12)
(241205871205751minus (141205873))
1 minus 119890minus(14)
asymp 1133411 gt
245
12
gt max0le119905le2120587
11988621(119905) + 119887
21(119905) + 119888
21(119905)
(1 + 119903119871
2)
1205752
212057222
1 minus 1205752
=
119890minus(12)
(141205871205752minus 3120587)
1 minus 119890minus(14)
asymp 628411 gt
53
5
gt max0le119905le2120587
11988622(119905) + 119887
22(119905) + 119888
22(119905)
1205752
1min1le119895le2
1205721119895
1 minus 1205751
=
119890minus(1205876)
(201205871205751minus 6120587)
1 minus 119890minus(12058712)
asymp 652614
1205752
2min1le119895le2
1205722119895
1 minus 1205752
=
119890minus(12)
(141205871205752minus 3120587)
1 minus 119890minus(14)
asymp 628411
min1le119894le2
1205752
119894min1le119895le2
120572119894119895
1 minus 120575119894
=
1205752
2min1le119895le2
1205722119895
1 minus 1205751
asymp 628411
119888119872
11+ 119888119872
12=
19
6
119888119872
21+ 119888119872
22=
61
60
max1le119894le2
2
sum
119895=1
119889119872
119894119895
=
19
6
(51)
Therefore
19
6
= max1le119894le2
2
sum
119894=1
119889119872
119894119895 lt min1le119894le2
1205752
119894min1le119895le2
120572119894119895
1 minus 120575119894
asymp 628411
(52)
Hence (1198601)ndash(1198603) (1198605) hold and 119886
119872
119894le 1 119894 = 1 2
According toTheorem 11 system (49) has at least one positive2120587-periodic solution
Acknowledgments
The authors are thankful to the referees and editor for theimprovement of the paper This work was supported bythe Construct Program of the Key Discipline in HunanProvince Research was supported by the National NaturalScience Foundation of China (10971229 11161015) the ChinaPostdoctoral Science Foundation (2012M512162) and HunanProvincial Natural Science-Hengyang United Foundation ofChina (11JJ9002)
References
[1] P Chesson ldquoUnderstanding the role of environmental variationin population and community dynamicsrdquo Theoretical Popula-tion Biology vol 64 no 3 pp 253ndash254 2003
[2] J Zhen and Z E Ma ldquoPeriodic solutions for delay differentialequations model of plankton allelopathyrdquo Computers amp Mathe-matics with Applications vol 44 no 3-4 pp 491ndash500 2002
[3] X Liu and G Ballinger ldquoBoundedness for impulsive delaydifferential equations and applications to population growthmodelsrdquo Nonlinear Analysis Theory Methods amp Applicationsvol 53 no 7-8 pp 1041ndash1062 2003
[4] Y Kuang Delay Differential Equations with Applications inPopulation Dynamics Academic Press New York NY USA1993
[5] Y K Li and Y Kuang ldquoPeriodic solutions of periodic delayLotka-Volterra equations and systemsrdquo Journal of MathematicalAnalysis and Applications vol 255 no 1 pp 260ndash280 2001
[6] Y K Li and L F Zhu ldquoExistence of periodic solutions of discreteLotka-Volterra systems with delaysrdquo Bulletin of the Institute ofMathematics Academia Sinica vol 33 no 4 pp 369ndash380 2005
[7] H Zhao and N Ding ldquoExistence and global attractivity ofpositive periodic solution for competition-predator systemwithvariable delaysrdquoChaos SolitonsampFractals vol 29 no 1 pp 162ndash170 2006
[8] Y Song M Han and Y Peng ldquoStability and Hopf bifurcationsin a competitive Lotka-Volterra system with two delaysrdquo ChaosSolitons amp Fractals vol 22 no 5 pp 1139ndash1148 2004
[9] Z Yang and J Cao ldquoPositive periodic solutions of neutral Lotka-Volterra system with periodic delaysrdquo Applied Mathematics andComputation vol 149 no 3 pp 661ndash687 2004
[10] Y K Li ldquoPositive periodic solutions of periodic neutral Lotka-Volterra system with distributed delaysrdquo Chaos Solitons ampFractals vol 37 no 1 pp 288ndash298 2008
[11] A M Samoikleno and N A Perestyuk Impulsive DifferentialEquations World Scientific Singapore 1995
[12] S T Zavalishchin and A N SesekinDynamic Impulse SystemsTheory and Applications Kluwer Academic Publishers Dor-drecht The Netherlands 1997
International Journal of Differential Equations 13
[13] V Lakshmikantham D D Bainov and P S Simenov Theoryof Impulsive Differential Equations World Science Singapore1989
[14] J R Yan A P Zhao and J J Nieto ldquoExistence and globalattractivity of positive periodic solution of periodic single-species impulsive Lotka-Volterra systemsrdquo Mathematical andComputer Modelling vol 40 no 5-6 pp 509ndash518 2004
[15] X Liu and G Ballinger ldquoBoundedness for impulsive delaydifferential equations and applications to population growthmodelsrdquo Nonlinear Analysis Theory Methods amp Applicationsvol 53 no 7-8 pp 1041ndash1062 2003
[16] W Zhang and M Fan ldquoPeriodicity in a generalized ecologicalcompetition system governed by impulsive differential equa-tions with delaysrdquo Mathematical and Computer Modelling vol39 no 4-5 pp 479ndash493 2004
[17] S L Sun and L S Chen ldquoExistence of positive periodic solutionof an impulsive delay logistic modelrdquo Applied Mathematics andComputation vol 184 no 2 pp 617ndash623 2007
[18] X Li X Lin D Jiang and X Zhang ldquoExistence andmultiplicityof positive periodic solutions to functional differential equa-tions with impulse effectsrdquoNonlinear Analysis Theory Methodsamp Applications vol 62 no 4 pp 683ndash701 2005
[19] R E Gaines and J L Mawhin Coincidence Degree Theory andNonlinear Differential Equations Springer Berlin Germany1977
[20] N P Cac and J A Gatica ldquoFixed point theorems for mappingsin orderedBanach spacesrdquo Journal ofMathematical Analysis andApplications vol 71 no 2 pp 547ndash557 1979
[21] D J GuoNonlinear Functional Analysis in Chinese ShanDongScience and Technology Press 2001
[22] D Baınov and P Simeonov Impulsive Differential EquationsPeriodic Solutions and Applications vol 66 of Pitman Mono-graphs and Surveys in Pure and Applied Mathematics 1993
[23] D J Guo ldquoPositive solutions to nonlinear operator equa-tions and their applications to nonlinear integral equationsrdquoAdvances in Mathematics vol 13 no 4 pp 294ndash310 1984(Chinese)
12 International Journal of Differential Equations
min1le119895le2
1205721119895 = 12057211= 20120587120575
1minus 8120587 asymp 238007
min1le119895le2
1205722119895 = 12057222= 14120587120575
2minus 3120587 asymp 244284
120575111988611(119905) minus 119887
11(119905) minus 119888
11(119905) gt 0009 gt 0
120575211988612(119905) minus 119887
12(119905) minus 119888
12(119905) gt 1 gt 0
120575111988621(119905) minus 119887
21(119905) minus 119888
21(119905) gt 4 gt 0
120575211988622(119905) minus 119887
22(119905) minus 119888
22(119905) gt 2 gt 0
(1 + 119903119871
1)
1205752
112057211
1 minus 1205751
=
119890minus(1205876)
(201205871205751minus 8120587)
1 minus 119890minus(12058712)
asymp 652614 gt 18
ge max0le119905le2120587
11988611(119905) + 119887
11(119905) + 119888
11(119905)
(1 + 119903119871
1)
1205752
112057212
1 minus 1205751
=
119890minus(1205876)
(181205871205752minus 2120587)
1 minus 119890minus(12058712)
asymp 1021184 gt
347
30
ge max0le119905le2120587
11988612(119905) + 119887
12(119905) + 119888
12(119905)
(1 + 119903119871
2)
1205752
212057221
1 minus 1205752
=
119890minus(12)
(241205871205751minus (141205873))
1 minus 119890minus(14)
asymp 1133411 gt
245
12
gt max0le119905le2120587
11988621(119905) + 119887
21(119905) + 119888
21(119905)
(1 + 119903119871
2)
1205752
212057222
1 minus 1205752
=
119890minus(12)
(141205871205752minus 3120587)
1 minus 119890minus(14)
asymp 628411 gt
53
5
gt max0le119905le2120587
11988622(119905) + 119887
22(119905) + 119888
22(119905)
1205752
1min1le119895le2
1205721119895
1 minus 1205751
=
119890minus(1205876)
(201205871205751minus 6120587)
1 minus 119890minus(12058712)
asymp 652614
1205752
2min1le119895le2
1205722119895
1 minus 1205752
=
119890minus(12)
(141205871205752minus 3120587)
1 minus 119890minus(14)
asymp 628411
min1le119894le2
1205752
119894min1le119895le2
120572119894119895
1 minus 120575119894
=
1205752
2min1le119895le2
1205722119895
1 minus 1205751
asymp 628411
119888119872
11+ 119888119872
12=
19
6
119888119872
21+ 119888119872
22=
61
60
max1le119894le2
2
sum
119895=1
119889119872
119894119895
=
19
6
(51)
Therefore
19
6
= max1le119894le2
2
sum
119894=1
119889119872
119894119895 lt min1le119894le2
1205752
119894min1le119895le2
120572119894119895
1 minus 120575119894
asymp 628411
(52)
Hence (1198601)ndash(1198603) (1198605) hold and 119886
119872
119894le 1 119894 = 1 2
According toTheorem 11 system (49) has at least one positive2120587-periodic solution
Acknowledgments
The authors are thankful to the referees and editor for theimprovement of the paper This work was supported bythe Construct Program of the Key Discipline in HunanProvince Research was supported by the National NaturalScience Foundation of China (10971229 11161015) the ChinaPostdoctoral Science Foundation (2012M512162) and HunanProvincial Natural Science-Hengyang United Foundation ofChina (11JJ9002)
References
[1] P Chesson ldquoUnderstanding the role of environmental variationin population and community dynamicsrdquo Theoretical Popula-tion Biology vol 64 no 3 pp 253ndash254 2003
[2] J Zhen and Z E Ma ldquoPeriodic solutions for delay differentialequations model of plankton allelopathyrdquo Computers amp Mathe-matics with Applications vol 44 no 3-4 pp 491ndash500 2002
[3] X Liu and G Ballinger ldquoBoundedness for impulsive delaydifferential equations and applications to population growthmodelsrdquo Nonlinear Analysis Theory Methods amp Applicationsvol 53 no 7-8 pp 1041ndash1062 2003
[4] Y Kuang Delay Differential Equations with Applications inPopulation Dynamics Academic Press New York NY USA1993
[5] Y K Li and Y Kuang ldquoPeriodic solutions of periodic delayLotka-Volterra equations and systemsrdquo Journal of MathematicalAnalysis and Applications vol 255 no 1 pp 260ndash280 2001
[6] Y K Li and L F Zhu ldquoExistence of periodic solutions of discreteLotka-Volterra systems with delaysrdquo Bulletin of the Institute ofMathematics Academia Sinica vol 33 no 4 pp 369ndash380 2005
[7] H Zhao and N Ding ldquoExistence and global attractivity ofpositive periodic solution for competition-predator systemwithvariable delaysrdquoChaos SolitonsampFractals vol 29 no 1 pp 162ndash170 2006
[8] Y Song M Han and Y Peng ldquoStability and Hopf bifurcationsin a competitive Lotka-Volterra system with two delaysrdquo ChaosSolitons amp Fractals vol 22 no 5 pp 1139ndash1148 2004
[9] Z Yang and J Cao ldquoPositive periodic solutions of neutral Lotka-Volterra system with periodic delaysrdquo Applied Mathematics andComputation vol 149 no 3 pp 661ndash687 2004
[10] Y K Li ldquoPositive periodic solutions of periodic neutral Lotka-Volterra system with distributed delaysrdquo Chaos Solitons ampFractals vol 37 no 1 pp 288ndash298 2008
[11] A M Samoikleno and N A Perestyuk Impulsive DifferentialEquations World Scientific Singapore 1995
[12] S T Zavalishchin and A N SesekinDynamic Impulse SystemsTheory and Applications Kluwer Academic Publishers Dor-drecht The Netherlands 1997
International Journal of Differential Equations 13
[13] V Lakshmikantham D D Bainov and P S Simenov Theoryof Impulsive Differential Equations World Science Singapore1989
[14] J R Yan A P Zhao and J J Nieto ldquoExistence and globalattractivity of positive periodic solution of periodic single-species impulsive Lotka-Volterra systemsrdquo Mathematical andComputer Modelling vol 40 no 5-6 pp 509ndash518 2004
[15] X Liu and G Ballinger ldquoBoundedness for impulsive delaydifferential equations and applications to population growthmodelsrdquo Nonlinear Analysis Theory Methods amp Applicationsvol 53 no 7-8 pp 1041ndash1062 2003
[16] W Zhang and M Fan ldquoPeriodicity in a generalized ecologicalcompetition system governed by impulsive differential equa-tions with delaysrdquo Mathematical and Computer Modelling vol39 no 4-5 pp 479ndash493 2004
[17] S L Sun and L S Chen ldquoExistence of positive periodic solutionof an impulsive delay logistic modelrdquo Applied Mathematics andComputation vol 184 no 2 pp 617ndash623 2007
[18] X Li X Lin D Jiang and X Zhang ldquoExistence andmultiplicityof positive periodic solutions to functional differential equa-tions with impulse effectsrdquoNonlinear Analysis Theory Methodsamp Applications vol 62 no 4 pp 683ndash701 2005
[19] R E Gaines and J L Mawhin Coincidence Degree Theory andNonlinear Differential Equations Springer Berlin Germany1977
[20] N P Cac and J A Gatica ldquoFixed point theorems for mappingsin orderedBanach spacesrdquo Journal ofMathematical Analysis andApplications vol 71 no 2 pp 547ndash557 1979
[21] D J GuoNonlinear Functional Analysis in Chinese ShanDongScience and Technology Press 2001
[22] D Baınov and P Simeonov Impulsive Differential EquationsPeriodic Solutions and Applications vol 66 of Pitman Mono-graphs and Surveys in Pure and Applied Mathematics 1993
[23] D J Guo ldquoPositive solutions to nonlinear operator equa-tions and their applications to nonlinear integral equationsrdquoAdvances in Mathematics vol 13 no 4 pp 294ndash310 1984(Chinese)
International Journal of Differential Equations 13
[13] V Lakshmikantham D D Bainov and P S Simenov Theoryof Impulsive Differential Equations World Science Singapore1989
[14] J R Yan A P Zhao and J J Nieto ldquoExistence and globalattractivity of positive periodic solution of periodic single-species impulsive Lotka-Volterra systemsrdquo Mathematical andComputer Modelling vol 40 no 5-6 pp 509ndash518 2004
[15] X Liu and G Ballinger ldquoBoundedness for impulsive delaydifferential equations and applications to population growthmodelsrdquo Nonlinear Analysis Theory Methods amp Applicationsvol 53 no 7-8 pp 1041ndash1062 2003
[16] W Zhang and M Fan ldquoPeriodicity in a generalized ecologicalcompetition system governed by impulsive differential equa-tions with delaysrdquo Mathematical and Computer Modelling vol39 no 4-5 pp 479ndash493 2004
[17] S L Sun and L S Chen ldquoExistence of positive periodic solutionof an impulsive delay logistic modelrdquo Applied Mathematics andComputation vol 184 no 2 pp 617ndash623 2007
[18] X Li X Lin D Jiang and X Zhang ldquoExistence andmultiplicityof positive periodic solutions to functional differential equa-tions with impulse effectsrdquoNonlinear Analysis Theory Methodsamp Applications vol 62 no 4 pp 683ndash701 2005
[19] R E Gaines and J L Mawhin Coincidence Degree Theory andNonlinear Differential Equations Springer Berlin Germany1977
[20] N P Cac and J A Gatica ldquoFixed point theorems for mappingsin orderedBanach spacesrdquo Journal ofMathematical Analysis andApplications vol 71 no 2 pp 547ndash557 1979
[21] D J GuoNonlinear Functional Analysis in Chinese ShanDongScience and Technology Press 2001
[22] D Baınov and P Simeonov Impulsive Differential EquationsPeriodic Solutions and Applications vol 66 of Pitman Mono-graphs and Surveys in Pure and Applied Mathematics 1993
[23] D J Guo ldquoPositive solutions to nonlinear operator equa-tions and their applications to nonlinear integral equationsrdquoAdvances in Mathematics vol 13 no 4 pp 294ndash310 1984(Chinese)