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Research ArticlePeriodic Solutions for Nonlinear Integro-Differential Systemswith Piecewise Constant Argument
Kuo-Shou Chiu
Departamento de Matematica Facultad de Ciencias Basicas Universidad Metropolitana de Ciencias de la Educacion Santiago Chile
Correspondence should be addressed to Kuo-Shou Chiu kschiuumcecl
Received 31 August 2013 Accepted 10 October 2013 Published 12 January 2014
Academic Editors R Adams A Ibeas and M Inc
Copyright copy 2014 Kuo-Shou Chiu 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
We investigate the existence of the periodic solutions of a nonlinear integro-differential system with piecewise alternately advancedand retarded argument of generalized type in short DEPCAG that is the argument is a general step function We consider thecritical case when associated linear homogeneous system admits nontrivial periodic solutions Criteria of existence of periodicsolutions of such equations are obtained In the process we use Greenrsquos function for periodic solutions and convert the givenDEPCAG into an equivalent integral equation Then we construct appropriate mappings and employ Krasnoselskiirsquos fixed pointtheorem to show the existence of a periodic solution of this type of nonlinear differential equations We also use the contractionmapping principle to show the existence of a unique periodic solution Appropriate examples are given to show the feasibility ofour results
1 Introduction
Among the functional differential equationsMyshkis [1] pro-posed to study differential equations with piecewise constantarguments DEPCAThe theory of scalar DEPCA of the type
where [sdot] signifies the greatest integer function was initiatedin [2ndash4] inWiener [5] the first book inDEPCA andhas beendeveloped by many authors [6ndash21] Applications of DEPCAare discussed in [5 22ndash25] They are hybrid equations theycombine the properties of both continuous systems anddiscrete equations Over the years great attention has beenpaid to the study of the existence of periodic solutions ofseveral different types of differential equations For specificreferences see [5ndash7 13 17 18 23 24 26ndash34]
Let Z N R and C be the set of all integers natural realand complex numbers respectively Denote by | sdot | a norm inR119899 119899 isin N Fix two real sequences 119905
119894 120574119894 119894 isin Z such that
119905119894lt 119905119894+1
and 119905119894le 120574119894le 119905119894+1
for all 119894 isin Z and 119905119894rarr plusmninfin as
119894 rarr plusmninfinLet 120574 R rarr R be a step function given by 120574(119905) = 120574
119894for
119905 isin 119868119894= [119905119894 119905119894+1) and consider theDEPCA (1) with this general
120574 In this case we speak of DEPCA of general type in shortDEPCAG Indeed 120574(119905) = [119905] corresponds to 120574
119894= 119905119894= 119894 isin Z
and 120574(119905) = 2[(119905 + 1)2] corresponds to 119905119894= 2119894 minus 1 120574
119894= 2119894
119894 isin Z The particular case of DEPCAG when 120574119894= 119905119894 119894 isin
Z an only delayed situation is considered by first time inAkhmet [8] The other extreme case is the only advancedsituation 120574
119894= 119905119894+1
Any other situation means an alternatelyadvanced anddelayed situationwith 119868+
119894= [119905119894 120574119894] the advanced
intervals and 119868minus119894= [120574119894 119905119894+1) the delayed intervals In [15 16]
Pinto has cleared the importance of the advanced and delayedintervalsThis decompositionwill be present in all our resultsSee [12 13 15 16 23 24 35] The integration or solutionof a DEPCA as proposed by its founders [2ndash4 6] is basedon the reduction of DEPCA to discrete equations To studynonlinear DEPCAG we will use the approach proposed byAkhmet in [9] based on the construction of an equivalentintegral equation but we also remark the clear influence ofthe discrete part and the corresponding difference equationswill be fundamental
Hindawi Publishing Corporatione Scientific World JournalVolume 2014 Article ID 514854 14 pageshttpdxdoiorg1011552014514854
2 The Scientific World Journal
In 2008 Akhmet et al [10] obtained some sufficientconditions for the existence and uniqueness of periodicsolutions for the following system
a small parameter belonging to an interval 119868 sub R with 0 isin 119868Recently Chiu and Pinto [23] using Poincare operator a
new Gronwall type lemma and fixed point theory obtainedsome sufficient conditions for the existence and uniquenessof periodic (or harmonic) and subharmonic solutions ofquasilinear differential equation with a general piecewiseconstant argument of the form
where 119905 isin R 119910 isin C119901 119860(119905) is a 119901 times 119901 matrix for 119901 isin N119891(119905 119909 119910) is a 119901 dimensional vector and 119891 is continuous inthe first argument and 120574(119905) = 120574
119894 if 119905119894le 119905 lt 119905
119894+1 119894 isin Z
In this paper comparing the three DEPCAG inequalities ofGronwall type and remarked new Gronwall lemma not onlyrequests a weaker condition than the other Gronwall lemmasbut also has a better estimate
It is well-known that there are many subjects in physicsand technology using mathematical methods that depend onthe linear and nonlinear integro-differential equations andit became clear that the existence of the periodic solutionsand its algorithm structure from more important problemsin the present time Where many of studies and researches[36ndash40] dedicates for treatment the autonomous and non-autonomous periodic systems and specially with the integralequations and differential equations and the linear andnonlinear differential and which is dealing in general shapewith the problems about periodic solutions theory and themodern methods in its quality treatment for the periodicdifferential equations
Samoilenko and Ronto [41] assume the numerical-analytic method to study the periodic solutions for ordinarydifferential equations and its algorithm structure and thismethod includes uniform sequences of periodic functionsand the results of that study are using the periodic solutionsonwide range in the difference of new processes industry andtechnology For example Samoilenko and Ronto [41] inves-tigated the existence and approximation of periodic solutionfor nonlinear system of integro-differential equations whichhas the form
1199091015840
(119905) = 119891(119905 119909 (119905) int
119905+119879
119905
119892 (119904 119909 (119904)) 119889119904) (4)
where 119909 isin 119863 sub R119899 119863 is a closed and bounded domainThe vectors functions 119891(119905 119909 119910) and 119892(119905 119909) are continuousfunctions in 119905 119909 119910 and periodic in 119905 of period 119879
Butris [42] investigated the periodic solution of nonlinearsystem of integro-differential equations depending on the
gamma distribution by using the numerical analyticmethodwhich has the form
1199091015840
(119905) = 119891(119905 120573 (119905 120572) 119909 (119905) int
where 119909 isin 119863 sub R119899 119863 is a closed and bounded domain Thevector functions119891(119905 120573(119905 120572) 119909) and119892(119905 120573(119905 120572) 119909) are definedon the domain (119905 120573(119905 120572) 119909) isin R times [0 119879] times 119863 times 119863
1
In the current paper we study the existence of periodicsolutions of a nonlinear integro-differential system withpiecewise alternately advanced and retarded argument
where 119860 R rarr R119899times119899 119891 R times R119899 times R119899 rarr R119899 and119892 R times R119899 times R119899 rarr R119899 are continuous in their respectivearguments In the analysis we use the idea of Greenrsquos functionfor periodic solutions and convert the nonlinear integro-differential systems with DEPCAG (6) into an equivalentintegral equationThen we employ Krasnoselskiirsquos fixed pointtheorem and show the existence of a periodic solution of thenonlinear integro-differential systems with DEPCAG (6) inTheorem 12We also obtain the existence of a unique periodicsolution in Theorem 14 employing the contraction mappingprinciple as the basicmathematical tool Furthermore appro-priate examples are given to show the feasibility of our results
In our paperwe assume that the solutions of the nonlinearintegro-differential systems with DEPCAG (6) are continu-ous functions But the deviating argument 120574(119905) is discontin-uous Thus in general the right-hand side of the DEPCAGsystem (6) has discontinuities at moments 119905
119894isin R 119894 isin Z As a
result we consider the solutions of theDEPCAGas functionswhich are continuous and continuously differentiable withinintervals [119905
119894 119905119894+1) 119894 isin Z In other words by a solution 119911(119905)
of the DEPCAG system (6) we mean a continuous functionon R such that the derivative 1199111015840(119905) exists at each point 119905 isinR with the possible exception of the points 119905
119894isin R 119894 isin
Z where a one-sided derivative exists and the nonlinearintegro-differential systemswithDEPCAG (6) are satisfied by119911(119905) on each interval (119905
119894 119905119894+1) 119894 isin Z as well
The rest of the paper is organized as follows In Section 2some definitions and preliminary results are introduced Weshow double 120596-periodicity of Greenrsquos function Section 3 isdevoted to establishing some criteria for the existence anduniqueness of periodic solutions of the DEPCAG system (6)Greenrsquos function and Banach Schauder and Krasnoselskiirsquosfixed point theorems below are fundamental to obtain themain results Furthermore appropriate examples are pro-vided in Section 4 to show the feasibility of our results
The Scientific World Journal 3
2 Greenrsquos Function and Periodicity
In this section we state and define Greenrsquos function for peri-odic solutions of the nonlinear integro-differential systemwith piecewise alternately advanced and retarded argument(6)
Let 119868 be the 119899 times 119899 identity matrix Denote byΦ(119905 119904) Φ(119904 119904) = 119868 119905 119904 isin R the fundamental matrixof solutions of the homogeneous system (7)
For every 119905 isin R let 119894 = 119894(119905) isin Z be the unique integersuch that 119905 isin 119868
119894= [119905119894 119905119894+1)
From now on the following assumption will be needed
(119873120596) The homogenous equation
1199101015840
(119905) = 119860 (119905) 119910 (119905) (7)
does not admit any nontrivial 120596-periodic solution
Remark 1 For 120591 isin R the condition (119873120596) equivalent to the
matrix (119868 minus Φ(120591 + 120596 120591)) is nonsingular
Now we solve the DEPCAG system (6) on 119868119894(120591)
In such a case the DEPCAG system (6) has 120596-periodicsolution 119911(119905) given by the integral equation (17) Beforestudying the existence of solutions of integral equation (17)in the next section firstly we define Greenrsquos function forperiodic solutions of the DEPCAG system (6)
Definition 2 Suppose that the condition (119873120596) holds For each
119905 119904 isin [120591 120591 + 120596] Greenrsquos function for (6) is given by
(119904) 120591 le 119904 le 119905 le 120591 + 120596
Φ (119905)119863Φminus1
(119904) 120591 le 119905 lt 119904 le 120591 + 120596(18)
where Φ(119905) is a fundamental solution of (7) and
119863 = ((Φminus1
(120591)Φ (120591 + 120596))minus1
minus 119868)minus1
(19)
We note that the condition (119873120596) implies the existence of
the matrix119863To prove double 120596-periodicity of Greenrsquos function119866(119905 119904)
we first give the following lemma
Lemma 3 Suppose that the condition (119873120596) holds Let the
matrix 119863 be defined by (19) and then one has the followingidentities
(119868 + 119863) = (Φminus1
(120591)Φ (120591 + 120596))minus1
119863
= (119868 minus Φminus1
(120591)Φ (120591 + 120596))minus1
Φ (119905)119863Φminus1
(119905 + 120596) minus Φ (119905)119863Φminus1
(119905) = 119868
(119868 + 119863)Φminus1
(120591 minus 120596)Φ (120591) = 119863
(20)
The Scientific World Journal 5
For Lemma 3we can prove an important property double120596-periodicity of Greenrsquos function 119866(119905 119904) to study after theexistence of periodic solutions
Lemma 4 Suppose that the condition (119873120596) holds Then
Greenrsquos function119866(119905 119904) is double120596-periodic that is119866(119905+120596 119904+120596) = 119866(119905 119904)
3 Existence of Periodic Solutions
In this section we prove the main theorems of this paperso we recall the nonlinear integro-differential systems withDEPCAG (6)
Using Definition 2 Remark 5 and double 120596-periodicityof Greenrsquos function similar formula is given by (17) So wehave obtained the following result
Proposition 6 Suppose that the conditions (119873120596) and (P) hold
Let (120591 119911(120591)) isin R times R119899 Then 119911(119905) = 119911(119905 120591 119911(120591)) is 120596-periodicsolution on R of the DEPCAG system (6) if and only if 119911(119905) is120596-periodic solution of the integral equation
119911 (119905) = int
120591+120596
120591
119866 (119905 119904) [119891(119904 119911 (119904) int
It is easy to see that the DEPCAG system (6) has 120596-periodic solution if and only if the operator I has one fixedpoint in P
120596
To prove some existence criteria for 120596-periodic solutionsof the DEPCAG system (6) we use the Banach Schauder andKrasnoselskiirsquos fixed point theorems
Next we state first Krasnoselskiirsquos fixed point theoremwhich enables us to prove the existence of a periodic solutionFor the proof of Krasnoselskiirsquos fixed point theorem we referthe reader to [43]
Theorem A (Krasnoselskiirsquos fixed point theorem) Let 119878 bea closed convex nonempty subset of a Banach space (119864 sdot )Suppose thatA andBmap 119878 into 119864 such that
Then there exists 119911 isin 119878 with 119911 = A119911 +B119911
Remark 7 Krasnoselskiirsquos theorem may be combined withBanach and Schauderrsquos fixed point theorems In a certainsense we can interpret this as follows if a compact operatorhas the fixed point property under a small perturbationthen this property can be inherited The theorem is usefulin establishing the existence results for perturbed operatorequations It also has awide range of applications to nonlinearintegral equations of mixed type for proving the existence ofsolutions Thus the existence of fixed points for the sum oftwo operators has attracted tremendous interest and theirapplications are frequent in nonlinear analysis See [32 3343ndash46]
We note that to apply Krasnoselskiirsquos fixed point theoremwe need to construct two mappings one is contraction andthe other is compact Therefore we express (38) as
times1003817100381710038171003817120593 minus 120577
1003817100381710038171003817
le (119888119866int
120591+120596
120591
[1199011(119904) + 119901
2(119904)] 119889119904)
1003817100381710038171003817120593 minus 1205771003817100381710038171003817
le 1198881198661198711
1003817100381710038171003817120593 minus 1205771003817100381710038171003817
(44)
HenceA defines a contraction mapping
Similarly B is given by (41) which may be also acontraction operator
Lemma 9 If (119873120596) (119875) and (119871
119892) holdB is given by (41)with
1198881198661198712lt 1 and thenB is a contraction mapping
Lemma 10 If (119873120596) holdsB is defined by (41) and thenB is
completely continuous that isB is continuous and the imageofB is contained in a compact set
Proof Step 1 First we prove that B P120596
rarr P120596is
continuousAs the operator A a change of variable in (41) we have
(B120593)(119905 + 120596) = (B120593)(119905) Now we want to show B iscontinuous
The function 119892(119905 119909 119910) is uniformly continuous on [120591 120591 +120596] times CR times CR and by the periodicity in 119905 the function119892(119905 119909 119910) is uniformly continuous on R times CR times CR Thusfor any 1205981015840 = (120598119888
119866120596) gt 0 there exists 120575 = 120575(120598) gt 0 such
that 1199111 1199112isin S 119911
1minus 1199112 le 120575 implies |119892(119905 119911
1(119905) 1199111(120574(119905))) minus
119892(119905 1199112(119905) 1199112(120574(119905)))| le 120598
1015840 for 119905 isin [120591 120591 + 120596] Then B1199111minus
B1199112 le 120598 In fact by the continuity of 119892
1003816100381610038161003816119892 (119905 1199111 (119905) 1199111 (120574 (119905))) minus 119892 (119905 1199112 (119905) 1199112 (120574 (119905)))1003816100381610038161003816 le 1205981015840
for 119905 isin [120591 120591 + 120596](45)
8 The Scientific World Journal
and then1003817100381710038171003817B1199111minusB1199112
Step 2 We show that the image of B is contained in acompact set
Let 119909 119910 isin CR and 119904 isin [120591 120591 + 120596] for the continuity of thefunction 119892(119904 119909 119910) there exists119872 gt 0 such that |119892(119904 119909 119910)| le119872 Let 120593
119899isin S where 119899 is a positive integer then we have
We now see that all the conditions of Krasnoselskiirsquostheorem are satisfied Thus there exists a fixed point 119911 in S
such that 119911 = A119911 +B119911 By Proposition 6 this fixed point isa solution of the DEPCAG system (6) Hence the DEPCAGsystem (6) has 120596-periodic solution
By the symmetry of the conditions we will obtain asTheorem 12
Theorem 13 Suppose the hypothesis (119873120596) (119875) (119871
119892) (119872119891)
(119872119862) (119862) hold Let 119877 be a positive constant satisfying the
inequality
119888119866(1198621+ 1198712) 119877 + 119888
119866(120573 + 120578
2+ 1205881) 120596 le 119877 (51)
Then the DEPCAG system (6) has at least one 120596-periodicsolution in S
By Lemma 9 the mapping B is a contraction and it isclear that B P
120596rarr P
120596 Also from Lemma 11 A is
completely continuousNext we prove that if 120593 120577 isin S with 120593 le 119877 and 120577 le 119877
then A120601 +B120577 le 119877
Let 120593 120577 isin S with 120593 le 119877 and 120577 le 119877 Then
Krasnoselskii (see [47]) proved that if 119860 is a stable constantmatrix without piecewise alternately advanced and retardedargument and lim
|119909|+|119910|rarr+infin(|119892(119905 119909 119910)|(|119909|+|119910|)) = 0 then
the system (56) has at least one periodic solution In our caseapplyingTheorem 12 this result is also valid for the DEPCAGsystem (56) requiring the hypothesis (119873
120596) (1198753) (119872119892) 119860(119905)
and 119892(119905 119909 119910) are periodic functions in 119905 with a period 120596 forall 119905 ge 120591 and |119892(119905 119909 119910)| le 119888
1(|119909| + |119910|) + 120588
2 with 2119888
1120596 lt 119862
2
and 1205882constants for |119909| + |119910| le 119877 119877 gt 0
Remark 17 Considering the nonlinear system of differentialequations with a general piecewise alternately advanced andretarded argument
In this particular case applyingTheorem 13 this result is alsovalid for the DEPCAG system (57) requiring the hypothesis(119873120596) (1198751) (1198753) (119871119892) (119872119891) and 119888
119866(1198621+1198712)119877+119888119866(120573+1205881)120596 le 119877
with = 1
Remark 18 Suppose that (1198751) is satisfied by 120596 = 120596
1
(1198752) by 120596 = 120596
2 and (119875
3) by 120596 = 120596
3 if 120596
119894120596119895is a
rational number for all 119894 119895 = 1 2 3 then (1198751) (1198752) and
(1198753) are simultaneously satisfied by 120596 = lcm120596
1 1205962 1205963
where lcm1205961 1205962 1205963 denotes the least common multiple
between 1205961 1205962and 120596
3 In the general case it is possible
that there exist five possible periods 1205961for 119860 120596
2for 119891 120596
3
for 119892 1205964for 119862 and the sequences 119905
119894119894isinZ 120574119894119894isinZ satisfy the
(1205965 119901) condition If 120596
119894120596119895is a rational number for all 119894 119895 =
1 2 3 4 5 so in this situation our results insure the existenceof 120596-periodic solution with 120596 = lcm120596
1 1205962 1205963 1205964 1205965
Therefore the above results insure the existence of 120596-periodicsolutions of the DEPCAG system (6) These solutions arecalled subharmonic solutions See Corollaries 19ndash22
To determine criteria for the existence and uniqueness ofsubharmonic solutions of theDEPCAG system (6) fromnowon we make the following assumption
(119875120596) There exists 120596 = lcm120596
1 1205962 1205963 1205964 1205965 gt 0 120596
119894120596119895
which is a rational number for all 119894 119895 = 1 2 3 4 5
such that
(1) 119860(119905) 119891(119905 1199091 1199101) and 119892(119905 119909
2 1199102) are periodic
functions in 119905 with a period 1205961 1205962 and 120596
3
respectively for all 119905 ge 120591(2) 119862(119905 + 120596
4 119904 + 120596
4 1199093) = 119862(119905 119904 119909
3) for all 119905 ge 120591
(3) There exists 119901 isin Z+ for which the sequences119905119894119894isinZ 120574119894119894isinZ satisfy the (120596
5 119901) condition
As immediate corollaries of Theorems 12ndash15 and Remark 18the following results are true
The Scientific World Journal 11
Corollary 19 Suppose the hypotheses (119873120596) (119875120596) (119871119891) (119871119862)
(119872119892) and (49) hold Then the DEPCAG system (6) has at least
one subharmonic solution in S
Corollary 20 Suppose the hypotheses (119873120596) (119875120596) (119871119892) (119872119891)
(119872119862) (119862) and (51) hold Then the DEPCAG system (6) has at
least one subharmonic solution in S
Corollary 21 Suppose the hypotheses (119873120596) (119875120596) (119871119891) (119871119862)
(119871119892) and (53) holdThen theDEPCAG system (6) has a unique
subharmonic solution
Corollary 22 Suppose the hypotheses (119873120596) (119875120596) (119872
119891)
(119872119862) (119872119892) (119862) and (55) hold Then the DEPCAG system (6)
has at least one subharmonic solution in S
4 Applications and Illustrative Examples
Wewill introduce appropriate examples in this sectionTheseexamples will show the feasibility of our theory
Mathematical modelling of real-life problems usuallyresults in functional equations like ordinary or partial differ-ential equations integral and integro-differential equationsand stochastic equations Many mathematical formulationsof physical phenomena contain integro-differential equa-tions these equations arise inmany fields like fluid dynamicsbiologicalmodels and chemical kinetics So we first considernonlinear integro-differential equations with a general piece-wise constant argument mentioned in the introduction andobtain some new sufficient conditions for the existence of theperiodic solutions of these systems
Example 1 Let 120582 R2 rarr [0infin) and ℎ R2 rarr [0infin) betwo functions satisfying
sup119905isinR
int
119905+2120587
119905
120582 (119905 119904) 119889119904 le sup119905isinR
int
119905+2120587
119905
ℎ (119905 119904) 119889119904 le ℎ (58)
Consider the following nonlinear integro-differential equa-tions with piecewise alternately advanced and retarded argu-ment of generalized type
|119862(119905 119904 119910)| le 1198882(119877)120582(119905 119904)|119910| +ℎ(119905 119904) for |119910| le 119877 where
2120587 le 1198621
(iv) 119892 R times R119899 times R119899 rarr R119899 is continuous and 119862(119905 119904 119910)satisfies (119862)
Indeed for any 120576 gt 0 there exists 120575 gt 0 such that |1199101minus1199102| le 120575
implies
1003816100381610038161003816119862 (119905 119904 1199101) minus 119862 (119905 119904 1199102)1003816100381610038161003816 le 1205761198882 (119877) 120582 (119905 119904) for 119905 119904 isin R
(60)
Furthermore there exists 119877 such that
exp (2119886lowast120587)exp (2119886
lowast120587) minus 1
[(1198621+ 1198622) 119877 + 2120587 (120588
2+ ℎ)] le 119877 (61)
where 119886lowast = sup119905isinR |119886(119905)| and 119886
lowast= inf119905isinR|119886(119905)|
Then by Theorem 15 the DEPCAG system (59) has atleast one 2120587-periodic solution
Example 2 Thus many examples can be constructed whereour results can be applied
Let Λ R rarr R119899times119899 and 120583 R rarr R119899 be two functionssatisfying
sup119905isinR
int
119905+120596
119905
|Λ (119904)| 119889119904 = Λ lt infin
sup119905isinR
int
119905+120596
119905
1003816100381610038161003816120583 (119905 minus 119904)1003816100381610038161003816 119889119904 = 120583 lt infin
(62)
Now consider the integro-differential system with piecewisealternately advanced and retarded argument
Theorem 12 implies that there exists at least a 120596-periodicsolution of the DEPCAG system (63)
12 The Scientific World Journal
Note that similar results can be obtained under (119871119892)
and (119872119862) On the other hand the periodic situation of the
DEPCAGsystem (56) and (57) can be treated in the samewayLet us consider another example for second-order differ-
ential equations with a general piecewise constant argumentIn this case we can show the existence and uniquenessof periodic solutions of the following nonlinear DEPCAGsystem
Example 3 Consider the following nonlinear DEPCAG sys-tem
11991010158401015840
(119905) + (12058121199102
(119905) minus 2) 1199101015840
(119905) minus 8119910 (119905)
minus 1205811sin (120596119905) 1199102 (120574 (119905)) minus 120581
2cos (120596119905) = 0
(65)
where 1205811 1205812isin R 120574(119905) = 120574
Then for 120593 120595 isin S we have1003817100381710038171003817119892 (sdot 120593 (sdot) 120593 (120574 (sdot))) minus 119892 (sdot 120595 (sdot) 120595 (120574 (sdot)))
le 212058121198772 sup119905isin[120591120591+120596]
1003816100381610038161003816100381610038161003816(1205951 (119905) minus 1205931 (119905)
1205952 (119905) minus 1205932 (119905)
)
1003816100381610038161003816100381610038161003816= 212058121198772 1003817100381710038171003817120593 minus 120595
1003817100381710038171003817
(69)
In a similar way for 119891 we have1003817100381710038171003817119891 (sdot 120593 (sdot) 120593 (120574 (sdot))) minus 119891 (sdot 120595 (sdot) 120595 (120574 (sdot)))
1003817100381710038171003817
le 212058111198771003817100381710038171003817120593 minus 120595
1003817100381710038171003817
(70)
ByTheorem 14 the DEPCAG system (65) has a unique 2120587120596-periodic solution in S
Conflict of Interests
The author declares that there is no conflict of interests re-garding the publication of this paper
Acknowledgment
This research was in part supported by FIBE 01-12 DIUMCE
References
[1] ADMyshkis ldquoOn certain problems in the theory of differentialequations with deviating argumentsrdquo Uspekhi Matematich-eskikh Nauk vol 32 pp 173ndash202 1977
[2] K L Cooke and JWiener ldquoRetarded differential equations withpiecewise constant delaysrdquo Journal of Mathematical Analysisand Applications vol 99 no 1 pp 265ndash297 1984
[3] K L Cooke and J Wiener ldquoAn equation alternately of retardedand advanced typerdquo Proceedings of the American MathematicalSociety vol 99 pp 726ndash732 1987
[4] S M Shah and J Wiener ldquoAdvanced differential equations withpiecewise constant argument deviationsrdquo International JournalofMathematics andMathematical Sciences vol 6 no 4 pp 671ndash703 1983
[5] J WienerGeneralized Solutions of Functional Differential Equa-tions World Scientific Singapore 1993
[6] A R Aftabizadeh and J Wiener ldquoOscillatory and periodicsolutions of an equation alternately of retarded and advancedtypesrdquo Applicable Analysis vol 23 no 3 pp 219ndash231 1986
[7] A R Aftabizadeh J Wiener and J M Xu ldquoOscillatory andperiodic solutions of delay differential equations with piecewiseconstant argumentrdquo Proceedings of the American MathematicalSociety vol 99 pp 673ndash679 1987
[8] M U Akhmet ldquoIntegral manifolds of differential equationswith piecewise constant argument of generalized typerdquo Nonlin-ear Analysis Theory Methods and Applications vol 66 no 2pp 367ndash383 2007
[9] M U Akhmet Nonlinear Hybrid ContinuousDiscrete-TimeModels Atlantis Press Amsterdam The Netherlands 2011
[10] M U Akhmet C Buyukadali and T Ergenc ldquoPeriodic solu-tions of the hybrid system with small parameterrdquo NonlinearAnalysis Hybrid Systems vol 2 no 2 pp 532ndash543 2008
[11] A Cabada J B Ferreiro and J J Nieto ldquoGreenrsquos functionand comparison principles for first order periodic differentialequations with piecewise constant argumentsrdquo Journal of Math-ematical Analysis and Applications vol 291 no 2 pp 690ndash6972004
[12] K S Chiu and M Pinto ldquoVariation of parameters formula andGronwall inequality for differential equations with a generalpiecewise constant argumentrdquo Acta Mathematicae ApplicataeSinica vol 27 no 4 pp 561ndash568 2011
The Scientific World Journal 13
[13] K S Chiu and M Pinto ldquoOscillatory and periodic solutionsin alternately advanced and delayed differential equationsrdquoCarpathian Journal of Mathematics vol 29 no 2 pp 149ndash1582013
[14] J J Nieto and R Rodrıguez-Lopez ldquoGreenrsquos function forsecond-order periodic boundary value problemswith piecewiseconstant argumentsrdquo Journal of Mathematical Analysis andApplications vol 304 no 1 pp 33ndash57 2005
[15] M Pinto ldquoAsymptotic equivalence of nonlinear and quasi lin-ear differential equations with piecewise constant argumentsrdquoMathematical and Computer Modelling vol 49 no 9-10 pp1750ndash1758 2009
[16] M Pinto ldquoCauchy and Green matrices type and stability inalternately advanced and delayed differential systemsrdquo Journalof Difference Equations and Applications vol 17 no 2 pp 235ndash254 2011
[17] G Q Wang ldquoExistence theorem of periodic solutions for adelay nonlinear differential equation with piecewise constantargumentsrdquo Journal of Mathematical Analysis and Applicationsvol 298 no 1 pp 298ndash307 2004
[18] G Q Wang and S S Cheng ldquoExistence and uniqueness ofperiodic solutions for a second-order nonlinear differentialequationwith piecewise constant argumentrdquo International Jour-nal ofMathematics andMathematical Sciences vol 2009 ArticleID 950797 14 pages 2009
[19] Y H Xia Z Huang and M Han ldquoExistence of almost periodicsolutions for forced perturbed systems with piecewise constantargumentrdquo Journal of Mathematical Analysis and Applicationsvol 333 no 2 pp 798ndash816 2007
[20] P Yang Y Liu and W Ge ldquoGreenrsquos function for second orderdifferential equations with piecewise constant argumentsrdquoNon-linear Analysis Theory Methods and Applications vol 64 no 8pp 1812ndash1830 2006
[21] R Yuan ldquoThe existence of almost periodic solutions of retardeddifferential equations with piecewise constant argumentrdquo Non-linear Analysis Theory Methods and Applications vol 48 no 7pp 1013ndash1032 2002
[22] A I Alonso J Hong and R Obaya ldquoAlmost periodic typesolutions of differential equations with piecewise constant argu-ment via almost periodic type sequencesrdquo Applied MathematicsLetters vol 13 no 2 pp 131ndash137 2000
[23] K S Chiu and M Pinto ldquoPeriodic solutions of differentialequations with a general piecewise constant argument andapplicationsrdquo Electronic Journal of Qualitative Theory of Differ-ential Equations vol 46 pp 1ndash19 2010
[24] K S Chiu M Pinto and J C Jeng ldquoExistence and globalconvergence of periodic solutions in recurrent neural networkmodels with a general piecewise alternately advanced andretarded argumentrdquo Acta Applicandae Mathematicae 2013
[25] L Dai Nonlinear Dynamics of Piecewise of Constant Systemsand Implememtation of Piecewise Constants Arguments WorldScientific Singapore 2008
[26] T A Burton Stability and Periodic Solutions of Ordinary andFunctional Differential Equations Academic Press New YorkNY USA 1985
[27] T A Burton and T Furumochi ldquoPeriodic and asymptoticallyperiodic solutions of neutral integral equationsrdquo ElectronicJournal of Qualitative Theory of Differential Equations vol 10pp 1ndash19 2000
[28] T A Burton and T Furumochi ldquoExistence theorems andperiodic solutions of neutral integral equationsrdquo NonlinearAnalysis Theory Methods and Applications vol 43 no 4 pp527ndash546 2001
[29] F Karakoc H Bereketoglu and G Seyhan ldquoOscillatory andperiodic solutions of impulsive differential equations withpiecewise constant argumentrdquoActa ApplicandaeMathematicaevol 110 no 1 pp 499ndash510 2010
[30] Y Liu P Yang and W Ge ldquoPeriodic solutions of higher-order delay differential equationsrdquo Nonlinear Analysis TheoryMethods and Applications vol 63 no 1 pp 136ndash152 2005
[31] X L Liu and W T Li ldquoPeriodic solutions for dynamic equa-tions on time scalesrdquo Nonlinear Analysis Theory Methods andApplications vol 67 no 5 pp 1457ndash1463 2007
[32] M Pinto ldquoDichotomy and existence of periodic solutions ofquasilinear functional differential equationsrdquo Nonlinear Analy-sis Theory Methods and Applications vol 72 no 3-4 pp 1227ndash1234 2010
[33] Y N Raffoul ldquoPeriodic solutions for neutral nonlinear differ-ential equations with functional delayrdquo Electronic Journal ofDifferential Equations vol 2003 no 102 pp 1ndash7 2003
[34] G Q Wang and S S Cheng ldquoPeriodic solutions of discreteRayleigh equations with deviating argumentsrdquo Taiwanese Jour-nal of Mathematics vol 13 no 6 pp 2051ndash2067 2009
[35] K S Chiu ldquoStability of oscillatory solutions of differential equa-tions with a general piecewise constant argumentrdquo ElectronicJournal of Qualitative Theory of Differential Equations vol 88pp 1ndash15 2011
[36] R N Butris and M A Aziz ldquoSome theorems in the existenceand uniqueness for system of nonlinear integro-differentialequationsrdquo Journal Education and Science vol 18 pp 76ndash892006
[37] A YMitropolsky andD IMortynyukPeriodic Solutions for theOscillations Systems with Retarded Argument General SchoolKiev Ukraine 1979
[38] L L Tai ldquoNumerical-analytic method for the investigation ofautonomous systems of differential equationsrdquo Ukrainskii Ma-tematicheskii Zhurnal vol 30 no 3 pp 309ndash317 1978
[39] Yu D Shlapak ldquoPeriodic solutions of ordinary differentialequations of first order that are not solved with respect to thederivativerdquo Ukrainian Mathematical Journal vol 32 no 5 pp420ndash425 1980
[40] A M Samoilenko ldquoNumerical-analytic method for the investi-gation of periodic systems of ordinary differential equationsrdquoUkrainskii Matematicheskii Zhurnal vol 17 no 4 pp 16ndash231965
[41] AM Samoilenko andN I RontoNumerical-Analytic Methodsfor Investigations of Periodic Solutions Mir Publishers MoscowRussia 1979
[42] R N Butris ldquoPeriodic solution of nonlinear system of integro-differential equations depending on the gamma distributionrdquoGeneral Mathematics Notes vol 15 no 1 pp 56ndash71 2013
[43] D R Smart Fixed Points Theorems Cambridge UniversityPress Cambridge UK 1980
[44] T A Burton ldquoKrasnoselskiirsquos inversion principle and fixedpointsrdquo Nonlinear Analysis Theory Methods and Applicationsvol 30 no 7 pp 3975ndash3986 1997
[45] T A Burton ldquoA fixed-point theorem of Krasnoselskiirdquo AppliedMathematics Letters vol 11 no 1 pp 85ndash88 1998
14 The Scientific World Journal
[46] M Pinto and D Sepulveda ldquoh-asymptotic stability by fixedpoint in neutral nonlinear differential equations with delayrdquoNonlinear Analysis Theory Methods and Applications vol 74no 12 pp 3926ndash3933 2011
[47] M A Krasnoselskii Translation Along Trajectories of Differen-tial Equations Nauka Moscow Russia 1966 (Russian) Englishtranslation American Mathematical Society Providence RIUSA 1968
a small parameter belonging to an interval 119868 sub R with 0 isin 119868Recently Chiu and Pinto [23] using Poincare operator a
new Gronwall type lemma and fixed point theory obtainedsome sufficient conditions for the existence and uniquenessof periodic (or harmonic) and subharmonic solutions ofquasilinear differential equation with a general piecewiseconstant argument of the form
where 119905 isin R 119910 isin C119901 119860(119905) is a 119901 times 119901 matrix for 119901 isin N119891(119905 119909 119910) is a 119901 dimensional vector and 119891 is continuous inthe first argument and 120574(119905) = 120574
119894 if 119905119894le 119905 lt 119905
119894+1 119894 isin Z
In this paper comparing the three DEPCAG inequalities ofGronwall type and remarked new Gronwall lemma not onlyrequests a weaker condition than the other Gronwall lemmasbut also has a better estimate
It is well-known that there are many subjects in physicsand technology using mathematical methods that depend onthe linear and nonlinear integro-differential equations andit became clear that the existence of the periodic solutionsand its algorithm structure from more important problemsin the present time Where many of studies and researches[36ndash40] dedicates for treatment the autonomous and non-autonomous periodic systems and specially with the integralequations and differential equations and the linear andnonlinear differential and which is dealing in general shapewith the problems about periodic solutions theory and themodern methods in its quality treatment for the periodicdifferential equations
Samoilenko and Ronto [41] assume the numerical-analytic method to study the periodic solutions for ordinarydifferential equations and its algorithm structure and thismethod includes uniform sequences of periodic functionsand the results of that study are using the periodic solutionsonwide range in the difference of new processes industry andtechnology For example Samoilenko and Ronto [41] inves-tigated the existence and approximation of periodic solutionfor nonlinear system of integro-differential equations whichhas the form
1199091015840
(119905) = 119891(119905 119909 (119905) int
119905+119879
119905
119892 (119904 119909 (119904)) 119889119904) (4)
where 119909 isin 119863 sub R119899 119863 is a closed and bounded domainThe vectors functions 119891(119905 119909 119910) and 119892(119905 119909) are continuousfunctions in 119905 119909 119910 and periodic in 119905 of period 119879
Butris [42] investigated the periodic solution of nonlinearsystem of integro-differential equations depending on the
gamma distribution by using the numerical analyticmethodwhich has the form
1199091015840
(119905) = 119891(119905 120573 (119905 120572) 119909 (119905) int
where 119909 isin 119863 sub R119899 119863 is a closed and bounded domain Thevector functions119891(119905 120573(119905 120572) 119909) and119892(119905 120573(119905 120572) 119909) are definedon the domain (119905 120573(119905 120572) 119909) isin R times [0 119879] times 119863 times 119863
1
In the current paper we study the existence of periodicsolutions of a nonlinear integro-differential system withpiecewise alternately advanced and retarded argument
where 119860 R rarr R119899times119899 119891 R times R119899 times R119899 rarr R119899 and119892 R times R119899 times R119899 rarr R119899 are continuous in their respectivearguments In the analysis we use the idea of Greenrsquos functionfor periodic solutions and convert the nonlinear integro-differential systems with DEPCAG (6) into an equivalentintegral equationThen we employ Krasnoselskiirsquos fixed pointtheorem and show the existence of a periodic solution of thenonlinear integro-differential systems with DEPCAG (6) inTheorem 12We also obtain the existence of a unique periodicsolution in Theorem 14 employing the contraction mappingprinciple as the basicmathematical tool Furthermore appro-priate examples are given to show the feasibility of our results
In our paperwe assume that the solutions of the nonlinearintegro-differential systems with DEPCAG (6) are continu-ous functions But the deviating argument 120574(119905) is discontin-uous Thus in general the right-hand side of the DEPCAGsystem (6) has discontinuities at moments 119905
119894isin R 119894 isin Z As a
result we consider the solutions of theDEPCAGas functionswhich are continuous and continuously differentiable withinintervals [119905
119894 119905119894+1) 119894 isin Z In other words by a solution 119911(119905)
of the DEPCAG system (6) we mean a continuous functionon R such that the derivative 1199111015840(119905) exists at each point 119905 isinR with the possible exception of the points 119905
119894isin R 119894 isin
Z where a one-sided derivative exists and the nonlinearintegro-differential systemswithDEPCAG (6) are satisfied by119911(119905) on each interval (119905
119894 119905119894+1) 119894 isin Z as well
The rest of the paper is organized as follows In Section 2some definitions and preliminary results are introduced Weshow double 120596-periodicity of Greenrsquos function Section 3 isdevoted to establishing some criteria for the existence anduniqueness of periodic solutions of the DEPCAG system (6)Greenrsquos function and Banach Schauder and Krasnoselskiirsquosfixed point theorems below are fundamental to obtain themain results Furthermore appropriate examples are pro-vided in Section 4 to show the feasibility of our results
The Scientific World Journal 3
2 Greenrsquos Function and Periodicity
In this section we state and define Greenrsquos function for peri-odic solutions of the nonlinear integro-differential systemwith piecewise alternately advanced and retarded argument(6)
Let 119868 be the 119899 times 119899 identity matrix Denote byΦ(119905 119904) Φ(119904 119904) = 119868 119905 119904 isin R the fundamental matrixof solutions of the homogeneous system (7)
For every 119905 isin R let 119894 = 119894(119905) isin Z be the unique integersuch that 119905 isin 119868
119894= [119905119894 119905119894+1)
From now on the following assumption will be needed
(119873120596) The homogenous equation
1199101015840
(119905) = 119860 (119905) 119910 (119905) (7)
does not admit any nontrivial 120596-periodic solution
Remark 1 For 120591 isin R the condition (119873120596) equivalent to the
matrix (119868 minus Φ(120591 + 120596 120591)) is nonsingular
Now we solve the DEPCAG system (6) on 119868119894(120591)
In such a case the DEPCAG system (6) has 120596-periodicsolution 119911(119905) given by the integral equation (17) Beforestudying the existence of solutions of integral equation (17)in the next section firstly we define Greenrsquos function forperiodic solutions of the DEPCAG system (6)
Definition 2 Suppose that the condition (119873120596) holds For each
119905 119904 isin [120591 120591 + 120596] Greenrsquos function for (6) is given by
(119904) 120591 le 119904 le 119905 le 120591 + 120596
Φ (119905)119863Φminus1
(119904) 120591 le 119905 lt 119904 le 120591 + 120596(18)
where Φ(119905) is a fundamental solution of (7) and
119863 = ((Φminus1
(120591)Φ (120591 + 120596))minus1
minus 119868)minus1
(19)
We note that the condition (119873120596) implies the existence of
the matrix119863To prove double 120596-periodicity of Greenrsquos function119866(119905 119904)
we first give the following lemma
Lemma 3 Suppose that the condition (119873120596) holds Let the
matrix 119863 be defined by (19) and then one has the followingidentities
(119868 + 119863) = (Φminus1
(120591)Φ (120591 + 120596))minus1
119863
= (119868 minus Φminus1
(120591)Φ (120591 + 120596))minus1
Φ (119905)119863Φminus1
(119905 + 120596) minus Φ (119905)119863Φminus1
(119905) = 119868
(119868 + 119863)Φminus1
(120591 minus 120596)Φ (120591) = 119863
(20)
The Scientific World Journal 5
For Lemma 3we can prove an important property double120596-periodicity of Greenrsquos function 119866(119905 119904) to study after theexistence of periodic solutions
Lemma 4 Suppose that the condition (119873120596) holds Then
Greenrsquos function119866(119905 119904) is double120596-periodic that is119866(119905+120596 119904+120596) = 119866(119905 119904)
3 Existence of Periodic Solutions
In this section we prove the main theorems of this paperso we recall the nonlinear integro-differential systems withDEPCAG (6)
Using Definition 2 Remark 5 and double 120596-periodicityof Greenrsquos function similar formula is given by (17) So wehave obtained the following result
Proposition 6 Suppose that the conditions (119873120596) and (P) hold
Let (120591 119911(120591)) isin R times R119899 Then 119911(119905) = 119911(119905 120591 119911(120591)) is 120596-periodicsolution on R of the DEPCAG system (6) if and only if 119911(119905) is120596-periodic solution of the integral equation
119911 (119905) = int
120591+120596
120591
119866 (119905 119904) [119891(119904 119911 (119904) int
It is easy to see that the DEPCAG system (6) has 120596-periodic solution if and only if the operator I has one fixedpoint in P
120596
To prove some existence criteria for 120596-periodic solutionsof the DEPCAG system (6) we use the Banach Schauder andKrasnoselskiirsquos fixed point theorems
Next we state first Krasnoselskiirsquos fixed point theoremwhich enables us to prove the existence of a periodic solutionFor the proof of Krasnoselskiirsquos fixed point theorem we referthe reader to [43]
Theorem A (Krasnoselskiirsquos fixed point theorem) Let 119878 bea closed convex nonempty subset of a Banach space (119864 sdot )Suppose thatA andBmap 119878 into 119864 such that
Then there exists 119911 isin 119878 with 119911 = A119911 +B119911
Remark 7 Krasnoselskiirsquos theorem may be combined withBanach and Schauderrsquos fixed point theorems In a certainsense we can interpret this as follows if a compact operatorhas the fixed point property under a small perturbationthen this property can be inherited The theorem is usefulin establishing the existence results for perturbed operatorequations It also has awide range of applications to nonlinearintegral equations of mixed type for proving the existence ofsolutions Thus the existence of fixed points for the sum oftwo operators has attracted tremendous interest and theirapplications are frequent in nonlinear analysis See [32 3343ndash46]
We note that to apply Krasnoselskiirsquos fixed point theoremwe need to construct two mappings one is contraction andthe other is compact Therefore we express (38) as
times1003817100381710038171003817120593 minus 120577
1003817100381710038171003817
le (119888119866int
120591+120596
120591
[1199011(119904) + 119901
2(119904)] 119889119904)
1003817100381710038171003817120593 minus 1205771003817100381710038171003817
le 1198881198661198711
1003817100381710038171003817120593 minus 1205771003817100381710038171003817
(44)
HenceA defines a contraction mapping
Similarly B is given by (41) which may be also acontraction operator
Lemma 9 If (119873120596) (119875) and (119871
119892) holdB is given by (41)with
1198881198661198712lt 1 and thenB is a contraction mapping
Lemma 10 If (119873120596) holdsB is defined by (41) and thenB is
completely continuous that isB is continuous and the imageofB is contained in a compact set
Proof Step 1 First we prove that B P120596
rarr P120596is
continuousAs the operator A a change of variable in (41) we have
(B120593)(119905 + 120596) = (B120593)(119905) Now we want to show B iscontinuous
The function 119892(119905 119909 119910) is uniformly continuous on [120591 120591 +120596] times CR times CR and by the periodicity in 119905 the function119892(119905 119909 119910) is uniformly continuous on R times CR times CR Thusfor any 1205981015840 = (120598119888
119866120596) gt 0 there exists 120575 = 120575(120598) gt 0 such
that 1199111 1199112isin S 119911
1minus 1199112 le 120575 implies |119892(119905 119911
1(119905) 1199111(120574(119905))) minus
119892(119905 1199112(119905) 1199112(120574(119905)))| le 120598
1015840 for 119905 isin [120591 120591 + 120596] Then B1199111minus
B1199112 le 120598 In fact by the continuity of 119892
1003816100381610038161003816119892 (119905 1199111 (119905) 1199111 (120574 (119905))) minus 119892 (119905 1199112 (119905) 1199112 (120574 (119905)))1003816100381610038161003816 le 1205981015840
for 119905 isin [120591 120591 + 120596](45)
8 The Scientific World Journal
and then1003817100381710038171003817B1199111minusB1199112
Step 2 We show that the image of B is contained in acompact set
Let 119909 119910 isin CR and 119904 isin [120591 120591 + 120596] for the continuity of thefunction 119892(119904 119909 119910) there exists119872 gt 0 such that |119892(119904 119909 119910)| le119872 Let 120593
119899isin S where 119899 is a positive integer then we have
We now see that all the conditions of Krasnoselskiirsquostheorem are satisfied Thus there exists a fixed point 119911 in S
such that 119911 = A119911 +B119911 By Proposition 6 this fixed point isa solution of the DEPCAG system (6) Hence the DEPCAGsystem (6) has 120596-periodic solution
By the symmetry of the conditions we will obtain asTheorem 12
Theorem 13 Suppose the hypothesis (119873120596) (119875) (119871
119892) (119872119891)
(119872119862) (119862) hold Let 119877 be a positive constant satisfying the
inequality
119888119866(1198621+ 1198712) 119877 + 119888
119866(120573 + 120578
2+ 1205881) 120596 le 119877 (51)
Then the DEPCAG system (6) has at least one 120596-periodicsolution in S
By Lemma 9 the mapping B is a contraction and it isclear that B P
120596rarr P
120596 Also from Lemma 11 A is
completely continuousNext we prove that if 120593 120577 isin S with 120593 le 119877 and 120577 le 119877
then A120601 +B120577 le 119877
Let 120593 120577 isin S with 120593 le 119877 and 120577 le 119877 Then
Krasnoselskii (see [47]) proved that if 119860 is a stable constantmatrix without piecewise alternately advanced and retardedargument and lim
|119909|+|119910|rarr+infin(|119892(119905 119909 119910)|(|119909|+|119910|)) = 0 then
the system (56) has at least one periodic solution In our caseapplyingTheorem 12 this result is also valid for the DEPCAGsystem (56) requiring the hypothesis (119873
120596) (1198753) (119872119892) 119860(119905)
and 119892(119905 119909 119910) are periodic functions in 119905 with a period 120596 forall 119905 ge 120591 and |119892(119905 119909 119910)| le 119888
1(|119909| + |119910|) + 120588
2 with 2119888
1120596 lt 119862
2
and 1205882constants for |119909| + |119910| le 119877 119877 gt 0
Remark 17 Considering the nonlinear system of differentialequations with a general piecewise alternately advanced andretarded argument
In this particular case applyingTheorem 13 this result is alsovalid for the DEPCAG system (57) requiring the hypothesis(119873120596) (1198751) (1198753) (119871119892) (119872119891) and 119888
119866(1198621+1198712)119877+119888119866(120573+1205881)120596 le 119877
with = 1
Remark 18 Suppose that (1198751) is satisfied by 120596 = 120596
1
(1198752) by 120596 = 120596
2 and (119875
3) by 120596 = 120596
3 if 120596
119894120596119895is a
rational number for all 119894 119895 = 1 2 3 then (1198751) (1198752) and
(1198753) are simultaneously satisfied by 120596 = lcm120596
1 1205962 1205963
where lcm1205961 1205962 1205963 denotes the least common multiple
between 1205961 1205962and 120596
3 In the general case it is possible
that there exist five possible periods 1205961for 119860 120596
2for 119891 120596
3
for 119892 1205964for 119862 and the sequences 119905
119894119894isinZ 120574119894119894isinZ satisfy the
(1205965 119901) condition If 120596
119894120596119895is a rational number for all 119894 119895 =
1 2 3 4 5 so in this situation our results insure the existenceof 120596-periodic solution with 120596 = lcm120596
1 1205962 1205963 1205964 1205965
Therefore the above results insure the existence of 120596-periodicsolutions of the DEPCAG system (6) These solutions arecalled subharmonic solutions See Corollaries 19ndash22
To determine criteria for the existence and uniqueness ofsubharmonic solutions of theDEPCAG system (6) fromnowon we make the following assumption
(119875120596) There exists 120596 = lcm120596
1 1205962 1205963 1205964 1205965 gt 0 120596
119894120596119895
which is a rational number for all 119894 119895 = 1 2 3 4 5
such that
(1) 119860(119905) 119891(119905 1199091 1199101) and 119892(119905 119909
2 1199102) are periodic
functions in 119905 with a period 1205961 1205962 and 120596
3
respectively for all 119905 ge 120591(2) 119862(119905 + 120596
4 119904 + 120596
4 1199093) = 119862(119905 119904 119909
3) for all 119905 ge 120591
(3) There exists 119901 isin Z+ for which the sequences119905119894119894isinZ 120574119894119894isinZ satisfy the (120596
5 119901) condition
As immediate corollaries of Theorems 12ndash15 and Remark 18the following results are true
The Scientific World Journal 11
Corollary 19 Suppose the hypotheses (119873120596) (119875120596) (119871119891) (119871119862)
(119872119892) and (49) hold Then the DEPCAG system (6) has at least
one subharmonic solution in S
Corollary 20 Suppose the hypotheses (119873120596) (119875120596) (119871119892) (119872119891)
(119872119862) (119862) and (51) hold Then the DEPCAG system (6) has at
least one subharmonic solution in S
Corollary 21 Suppose the hypotheses (119873120596) (119875120596) (119871119891) (119871119862)
(119871119892) and (53) holdThen theDEPCAG system (6) has a unique
subharmonic solution
Corollary 22 Suppose the hypotheses (119873120596) (119875120596) (119872
119891)
(119872119862) (119872119892) (119862) and (55) hold Then the DEPCAG system (6)
has at least one subharmonic solution in S
4 Applications and Illustrative Examples
Wewill introduce appropriate examples in this sectionTheseexamples will show the feasibility of our theory
Mathematical modelling of real-life problems usuallyresults in functional equations like ordinary or partial differ-ential equations integral and integro-differential equationsand stochastic equations Many mathematical formulationsof physical phenomena contain integro-differential equa-tions these equations arise inmany fields like fluid dynamicsbiologicalmodels and chemical kinetics So we first considernonlinear integro-differential equations with a general piece-wise constant argument mentioned in the introduction andobtain some new sufficient conditions for the existence of theperiodic solutions of these systems
Example 1 Let 120582 R2 rarr [0infin) and ℎ R2 rarr [0infin) betwo functions satisfying
sup119905isinR
int
119905+2120587
119905
120582 (119905 119904) 119889119904 le sup119905isinR
int
119905+2120587
119905
ℎ (119905 119904) 119889119904 le ℎ (58)
Consider the following nonlinear integro-differential equa-tions with piecewise alternately advanced and retarded argu-ment of generalized type
|119862(119905 119904 119910)| le 1198882(119877)120582(119905 119904)|119910| +ℎ(119905 119904) for |119910| le 119877 where
2120587 le 1198621
(iv) 119892 R times R119899 times R119899 rarr R119899 is continuous and 119862(119905 119904 119910)satisfies (119862)
Indeed for any 120576 gt 0 there exists 120575 gt 0 such that |1199101minus1199102| le 120575
implies
1003816100381610038161003816119862 (119905 119904 1199101) minus 119862 (119905 119904 1199102)1003816100381610038161003816 le 1205761198882 (119877) 120582 (119905 119904) for 119905 119904 isin R
(60)
Furthermore there exists 119877 such that
exp (2119886lowast120587)exp (2119886
lowast120587) minus 1
[(1198621+ 1198622) 119877 + 2120587 (120588
2+ ℎ)] le 119877 (61)
where 119886lowast = sup119905isinR |119886(119905)| and 119886
lowast= inf119905isinR|119886(119905)|
Then by Theorem 15 the DEPCAG system (59) has atleast one 2120587-periodic solution
Example 2 Thus many examples can be constructed whereour results can be applied
Let Λ R rarr R119899times119899 and 120583 R rarr R119899 be two functionssatisfying
sup119905isinR
int
119905+120596
119905
|Λ (119904)| 119889119904 = Λ lt infin
sup119905isinR
int
119905+120596
119905
1003816100381610038161003816120583 (119905 minus 119904)1003816100381610038161003816 119889119904 = 120583 lt infin
(62)
Now consider the integro-differential system with piecewisealternately advanced and retarded argument
Theorem 12 implies that there exists at least a 120596-periodicsolution of the DEPCAG system (63)
12 The Scientific World Journal
Note that similar results can be obtained under (119871119892)
and (119872119862) On the other hand the periodic situation of the
DEPCAGsystem (56) and (57) can be treated in the samewayLet us consider another example for second-order differ-
ential equations with a general piecewise constant argumentIn this case we can show the existence and uniquenessof periodic solutions of the following nonlinear DEPCAGsystem
Example 3 Consider the following nonlinear DEPCAG sys-tem
11991010158401015840
(119905) + (12058121199102
(119905) minus 2) 1199101015840
(119905) minus 8119910 (119905)
minus 1205811sin (120596119905) 1199102 (120574 (119905)) minus 120581
2cos (120596119905) = 0
(65)
where 1205811 1205812isin R 120574(119905) = 120574
Then for 120593 120595 isin S we have1003817100381710038171003817119892 (sdot 120593 (sdot) 120593 (120574 (sdot))) minus 119892 (sdot 120595 (sdot) 120595 (120574 (sdot)))
le 212058121198772 sup119905isin[120591120591+120596]
1003816100381610038161003816100381610038161003816(1205951 (119905) minus 1205931 (119905)
1205952 (119905) minus 1205932 (119905)
)
1003816100381610038161003816100381610038161003816= 212058121198772 1003817100381710038171003817120593 minus 120595
1003817100381710038171003817
(69)
In a similar way for 119891 we have1003817100381710038171003817119891 (sdot 120593 (sdot) 120593 (120574 (sdot))) minus 119891 (sdot 120595 (sdot) 120595 (120574 (sdot)))
1003817100381710038171003817
le 212058111198771003817100381710038171003817120593 minus 120595
1003817100381710038171003817
(70)
ByTheorem 14 the DEPCAG system (65) has a unique 2120587120596-periodic solution in S
Conflict of Interests
The author declares that there is no conflict of interests re-garding the publication of this paper
Acknowledgment
This research was in part supported by FIBE 01-12 DIUMCE
References
[1] ADMyshkis ldquoOn certain problems in the theory of differentialequations with deviating argumentsrdquo Uspekhi Matematich-eskikh Nauk vol 32 pp 173ndash202 1977
[2] K L Cooke and JWiener ldquoRetarded differential equations withpiecewise constant delaysrdquo Journal of Mathematical Analysisand Applications vol 99 no 1 pp 265ndash297 1984
[3] K L Cooke and J Wiener ldquoAn equation alternately of retardedand advanced typerdquo Proceedings of the American MathematicalSociety vol 99 pp 726ndash732 1987
[4] S M Shah and J Wiener ldquoAdvanced differential equations withpiecewise constant argument deviationsrdquo International JournalofMathematics andMathematical Sciences vol 6 no 4 pp 671ndash703 1983
[5] J WienerGeneralized Solutions of Functional Differential Equa-tions World Scientific Singapore 1993
[6] A R Aftabizadeh and J Wiener ldquoOscillatory and periodicsolutions of an equation alternately of retarded and advancedtypesrdquo Applicable Analysis vol 23 no 3 pp 219ndash231 1986
[7] A R Aftabizadeh J Wiener and J M Xu ldquoOscillatory andperiodic solutions of delay differential equations with piecewiseconstant argumentrdquo Proceedings of the American MathematicalSociety vol 99 pp 673ndash679 1987
[8] M U Akhmet ldquoIntegral manifolds of differential equationswith piecewise constant argument of generalized typerdquo Nonlin-ear Analysis Theory Methods and Applications vol 66 no 2pp 367ndash383 2007
[9] M U Akhmet Nonlinear Hybrid ContinuousDiscrete-TimeModels Atlantis Press Amsterdam The Netherlands 2011
[10] M U Akhmet C Buyukadali and T Ergenc ldquoPeriodic solu-tions of the hybrid system with small parameterrdquo NonlinearAnalysis Hybrid Systems vol 2 no 2 pp 532ndash543 2008
[11] A Cabada J B Ferreiro and J J Nieto ldquoGreenrsquos functionand comparison principles for first order periodic differentialequations with piecewise constant argumentsrdquo Journal of Math-ematical Analysis and Applications vol 291 no 2 pp 690ndash6972004
[12] K S Chiu and M Pinto ldquoVariation of parameters formula andGronwall inequality for differential equations with a generalpiecewise constant argumentrdquo Acta Mathematicae ApplicataeSinica vol 27 no 4 pp 561ndash568 2011
The Scientific World Journal 13
[13] K S Chiu and M Pinto ldquoOscillatory and periodic solutionsin alternately advanced and delayed differential equationsrdquoCarpathian Journal of Mathematics vol 29 no 2 pp 149ndash1582013
[14] J J Nieto and R Rodrıguez-Lopez ldquoGreenrsquos function forsecond-order periodic boundary value problemswith piecewiseconstant argumentsrdquo Journal of Mathematical Analysis andApplications vol 304 no 1 pp 33ndash57 2005
[15] M Pinto ldquoAsymptotic equivalence of nonlinear and quasi lin-ear differential equations with piecewise constant argumentsrdquoMathematical and Computer Modelling vol 49 no 9-10 pp1750ndash1758 2009
[16] M Pinto ldquoCauchy and Green matrices type and stability inalternately advanced and delayed differential systemsrdquo Journalof Difference Equations and Applications vol 17 no 2 pp 235ndash254 2011
[17] G Q Wang ldquoExistence theorem of periodic solutions for adelay nonlinear differential equation with piecewise constantargumentsrdquo Journal of Mathematical Analysis and Applicationsvol 298 no 1 pp 298ndash307 2004
[18] G Q Wang and S S Cheng ldquoExistence and uniqueness ofperiodic solutions for a second-order nonlinear differentialequationwith piecewise constant argumentrdquo International Jour-nal ofMathematics andMathematical Sciences vol 2009 ArticleID 950797 14 pages 2009
[19] Y H Xia Z Huang and M Han ldquoExistence of almost periodicsolutions for forced perturbed systems with piecewise constantargumentrdquo Journal of Mathematical Analysis and Applicationsvol 333 no 2 pp 798ndash816 2007
[20] P Yang Y Liu and W Ge ldquoGreenrsquos function for second orderdifferential equations with piecewise constant argumentsrdquoNon-linear Analysis Theory Methods and Applications vol 64 no 8pp 1812ndash1830 2006
[21] R Yuan ldquoThe existence of almost periodic solutions of retardeddifferential equations with piecewise constant argumentrdquo Non-linear Analysis Theory Methods and Applications vol 48 no 7pp 1013ndash1032 2002
[22] A I Alonso J Hong and R Obaya ldquoAlmost periodic typesolutions of differential equations with piecewise constant argu-ment via almost periodic type sequencesrdquo Applied MathematicsLetters vol 13 no 2 pp 131ndash137 2000
[23] K S Chiu and M Pinto ldquoPeriodic solutions of differentialequations with a general piecewise constant argument andapplicationsrdquo Electronic Journal of Qualitative Theory of Differ-ential Equations vol 46 pp 1ndash19 2010
[24] K S Chiu M Pinto and J C Jeng ldquoExistence and globalconvergence of periodic solutions in recurrent neural networkmodels with a general piecewise alternately advanced andretarded argumentrdquo Acta Applicandae Mathematicae 2013
[25] L Dai Nonlinear Dynamics of Piecewise of Constant Systemsand Implememtation of Piecewise Constants Arguments WorldScientific Singapore 2008
[26] T A Burton Stability and Periodic Solutions of Ordinary andFunctional Differential Equations Academic Press New YorkNY USA 1985
[27] T A Burton and T Furumochi ldquoPeriodic and asymptoticallyperiodic solutions of neutral integral equationsrdquo ElectronicJournal of Qualitative Theory of Differential Equations vol 10pp 1ndash19 2000
[28] T A Burton and T Furumochi ldquoExistence theorems andperiodic solutions of neutral integral equationsrdquo NonlinearAnalysis Theory Methods and Applications vol 43 no 4 pp527ndash546 2001
[29] F Karakoc H Bereketoglu and G Seyhan ldquoOscillatory andperiodic solutions of impulsive differential equations withpiecewise constant argumentrdquoActa ApplicandaeMathematicaevol 110 no 1 pp 499ndash510 2010
[30] Y Liu P Yang and W Ge ldquoPeriodic solutions of higher-order delay differential equationsrdquo Nonlinear Analysis TheoryMethods and Applications vol 63 no 1 pp 136ndash152 2005
[31] X L Liu and W T Li ldquoPeriodic solutions for dynamic equa-tions on time scalesrdquo Nonlinear Analysis Theory Methods andApplications vol 67 no 5 pp 1457ndash1463 2007
[32] M Pinto ldquoDichotomy and existence of periodic solutions ofquasilinear functional differential equationsrdquo Nonlinear Analy-sis Theory Methods and Applications vol 72 no 3-4 pp 1227ndash1234 2010
[33] Y N Raffoul ldquoPeriodic solutions for neutral nonlinear differ-ential equations with functional delayrdquo Electronic Journal ofDifferential Equations vol 2003 no 102 pp 1ndash7 2003
[34] G Q Wang and S S Cheng ldquoPeriodic solutions of discreteRayleigh equations with deviating argumentsrdquo Taiwanese Jour-nal of Mathematics vol 13 no 6 pp 2051ndash2067 2009
[35] K S Chiu ldquoStability of oscillatory solutions of differential equa-tions with a general piecewise constant argumentrdquo ElectronicJournal of Qualitative Theory of Differential Equations vol 88pp 1ndash15 2011
[36] R N Butris and M A Aziz ldquoSome theorems in the existenceand uniqueness for system of nonlinear integro-differentialequationsrdquo Journal Education and Science vol 18 pp 76ndash892006
[37] A YMitropolsky andD IMortynyukPeriodic Solutions for theOscillations Systems with Retarded Argument General SchoolKiev Ukraine 1979
[38] L L Tai ldquoNumerical-analytic method for the investigation ofautonomous systems of differential equationsrdquo Ukrainskii Ma-tematicheskii Zhurnal vol 30 no 3 pp 309ndash317 1978
[39] Yu D Shlapak ldquoPeriodic solutions of ordinary differentialequations of first order that are not solved with respect to thederivativerdquo Ukrainian Mathematical Journal vol 32 no 5 pp420ndash425 1980
[40] A M Samoilenko ldquoNumerical-analytic method for the investi-gation of periodic systems of ordinary differential equationsrdquoUkrainskii Matematicheskii Zhurnal vol 17 no 4 pp 16ndash231965
[41] AM Samoilenko andN I RontoNumerical-Analytic Methodsfor Investigations of Periodic Solutions Mir Publishers MoscowRussia 1979
[42] R N Butris ldquoPeriodic solution of nonlinear system of integro-differential equations depending on the gamma distributionrdquoGeneral Mathematics Notes vol 15 no 1 pp 56ndash71 2013
[43] D R Smart Fixed Points Theorems Cambridge UniversityPress Cambridge UK 1980
[44] T A Burton ldquoKrasnoselskiirsquos inversion principle and fixedpointsrdquo Nonlinear Analysis Theory Methods and Applicationsvol 30 no 7 pp 3975ndash3986 1997
[45] T A Burton ldquoA fixed-point theorem of Krasnoselskiirdquo AppliedMathematics Letters vol 11 no 1 pp 85ndash88 1998
14 The Scientific World Journal
[46] M Pinto and D Sepulveda ldquoh-asymptotic stability by fixedpoint in neutral nonlinear differential equations with delayrdquoNonlinear Analysis Theory Methods and Applications vol 74no 12 pp 3926ndash3933 2011
[47] M A Krasnoselskii Translation Along Trajectories of Differen-tial Equations Nauka Moscow Russia 1966 (Russian) Englishtranslation American Mathematical Society Providence RIUSA 1968
In this section we state and define Greenrsquos function for peri-odic solutions of the nonlinear integro-differential systemwith piecewise alternately advanced and retarded argument(6)
Let 119868 be the 119899 times 119899 identity matrix Denote byΦ(119905 119904) Φ(119904 119904) = 119868 119905 119904 isin R the fundamental matrixof solutions of the homogeneous system (7)
For every 119905 isin R let 119894 = 119894(119905) isin Z be the unique integersuch that 119905 isin 119868
119894= [119905119894 119905119894+1)
From now on the following assumption will be needed
(119873120596) The homogenous equation
1199101015840
(119905) = 119860 (119905) 119910 (119905) (7)
does not admit any nontrivial 120596-periodic solution
Remark 1 For 120591 isin R the condition (119873120596) equivalent to the
matrix (119868 minus Φ(120591 + 120596 120591)) is nonsingular
Now we solve the DEPCAG system (6) on 119868119894(120591)
In such a case the DEPCAG system (6) has 120596-periodicsolution 119911(119905) given by the integral equation (17) Beforestudying the existence of solutions of integral equation (17)in the next section firstly we define Greenrsquos function forperiodic solutions of the DEPCAG system (6)
Definition 2 Suppose that the condition (119873120596) holds For each
119905 119904 isin [120591 120591 + 120596] Greenrsquos function for (6) is given by
(119904) 120591 le 119904 le 119905 le 120591 + 120596
Φ (119905)119863Φminus1
(119904) 120591 le 119905 lt 119904 le 120591 + 120596(18)
where Φ(119905) is a fundamental solution of (7) and
119863 = ((Φminus1
(120591)Φ (120591 + 120596))minus1
minus 119868)minus1
(19)
We note that the condition (119873120596) implies the existence of
the matrix119863To prove double 120596-periodicity of Greenrsquos function119866(119905 119904)
we first give the following lemma
Lemma 3 Suppose that the condition (119873120596) holds Let the
matrix 119863 be defined by (19) and then one has the followingidentities
(119868 + 119863) = (Φminus1
(120591)Φ (120591 + 120596))minus1
119863
= (119868 minus Φminus1
(120591)Φ (120591 + 120596))minus1
Φ (119905)119863Φminus1
(119905 + 120596) minus Φ (119905)119863Φminus1
(119905) = 119868
(119868 + 119863)Φminus1
(120591 minus 120596)Φ (120591) = 119863
(20)
The Scientific World Journal 5
For Lemma 3we can prove an important property double120596-periodicity of Greenrsquos function 119866(119905 119904) to study after theexistence of periodic solutions
Lemma 4 Suppose that the condition (119873120596) holds Then
Greenrsquos function119866(119905 119904) is double120596-periodic that is119866(119905+120596 119904+120596) = 119866(119905 119904)
3 Existence of Periodic Solutions
In this section we prove the main theorems of this paperso we recall the nonlinear integro-differential systems withDEPCAG (6)
Using Definition 2 Remark 5 and double 120596-periodicityof Greenrsquos function similar formula is given by (17) So wehave obtained the following result
Proposition 6 Suppose that the conditions (119873120596) and (P) hold
Let (120591 119911(120591)) isin R times R119899 Then 119911(119905) = 119911(119905 120591 119911(120591)) is 120596-periodicsolution on R of the DEPCAG system (6) if and only if 119911(119905) is120596-periodic solution of the integral equation
119911 (119905) = int
120591+120596
120591
119866 (119905 119904) [119891(119904 119911 (119904) int
It is easy to see that the DEPCAG system (6) has 120596-periodic solution if and only if the operator I has one fixedpoint in P
120596
To prove some existence criteria for 120596-periodic solutionsof the DEPCAG system (6) we use the Banach Schauder andKrasnoselskiirsquos fixed point theorems
Next we state first Krasnoselskiirsquos fixed point theoremwhich enables us to prove the existence of a periodic solutionFor the proof of Krasnoselskiirsquos fixed point theorem we referthe reader to [43]
Theorem A (Krasnoselskiirsquos fixed point theorem) Let 119878 bea closed convex nonempty subset of a Banach space (119864 sdot )Suppose thatA andBmap 119878 into 119864 such that
Then there exists 119911 isin 119878 with 119911 = A119911 +B119911
Remark 7 Krasnoselskiirsquos theorem may be combined withBanach and Schauderrsquos fixed point theorems In a certainsense we can interpret this as follows if a compact operatorhas the fixed point property under a small perturbationthen this property can be inherited The theorem is usefulin establishing the existence results for perturbed operatorequations It also has awide range of applications to nonlinearintegral equations of mixed type for proving the existence ofsolutions Thus the existence of fixed points for the sum oftwo operators has attracted tremendous interest and theirapplications are frequent in nonlinear analysis See [32 3343ndash46]
We note that to apply Krasnoselskiirsquos fixed point theoremwe need to construct two mappings one is contraction andthe other is compact Therefore we express (38) as
times1003817100381710038171003817120593 minus 120577
1003817100381710038171003817
le (119888119866int
120591+120596
120591
[1199011(119904) + 119901
2(119904)] 119889119904)
1003817100381710038171003817120593 minus 1205771003817100381710038171003817
le 1198881198661198711
1003817100381710038171003817120593 minus 1205771003817100381710038171003817
(44)
HenceA defines a contraction mapping
Similarly B is given by (41) which may be also acontraction operator
Lemma 9 If (119873120596) (119875) and (119871
119892) holdB is given by (41)with
1198881198661198712lt 1 and thenB is a contraction mapping
Lemma 10 If (119873120596) holdsB is defined by (41) and thenB is
completely continuous that isB is continuous and the imageofB is contained in a compact set
Proof Step 1 First we prove that B P120596
rarr P120596is
continuousAs the operator A a change of variable in (41) we have
(B120593)(119905 + 120596) = (B120593)(119905) Now we want to show B iscontinuous
The function 119892(119905 119909 119910) is uniformly continuous on [120591 120591 +120596] times CR times CR and by the periodicity in 119905 the function119892(119905 119909 119910) is uniformly continuous on R times CR times CR Thusfor any 1205981015840 = (120598119888
119866120596) gt 0 there exists 120575 = 120575(120598) gt 0 such
that 1199111 1199112isin S 119911
1minus 1199112 le 120575 implies |119892(119905 119911
1(119905) 1199111(120574(119905))) minus
119892(119905 1199112(119905) 1199112(120574(119905)))| le 120598
1015840 for 119905 isin [120591 120591 + 120596] Then B1199111minus
B1199112 le 120598 In fact by the continuity of 119892
1003816100381610038161003816119892 (119905 1199111 (119905) 1199111 (120574 (119905))) minus 119892 (119905 1199112 (119905) 1199112 (120574 (119905)))1003816100381610038161003816 le 1205981015840
for 119905 isin [120591 120591 + 120596](45)
8 The Scientific World Journal
and then1003817100381710038171003817B1199111minusB1199112
Step 2 We show that the image of B is contained in acompact set
Let 119909 119910 isin CR and 119904 isin [120591 120591 + 120596] for the continuity of thefunction 119892(119904 119909 119910) there exists119872 gt 0 such that |119892(119904 119909 119910)| le119872 Let 120593
119899isin S where 119899 is a positive integer then we have
We now see that all the conditions of Krasnoselskiirsquostheorem are satisfied Thus there exists a fixed point 119911 in S
such that 119911 = A119911 +B119911 By Proposition 6 this fixed point isa solution of the DEPCAG system (6) Hence the DEPCAGsystem (6) has 120596-periodic solution
By the symmetry of the conditions we will obtain asTheorem 12
Theorem 13 Suppose the hypothesis (119873120596) (119875) (119871
119892) (119872119891)
(119872119862) (119862) hold Let 119877 be a positive constant satisfying the
inequality
119888119866(1198621+ 1198712) 119877 + 119888
119866(120573 + 120578
2+ 1205881) 120596 le 119877 (51)
Then the DEPCAG system (6) has at least one 120596-periodicsolution in S
By Lemma 9 the mapping B is a contraction and it isclear that B P
120596rarr P
120596 Also from Lemma 11 A is
completely continuousNext we prove that if 120593 120577 isin S with 120593 le 119877 and 120577 le 119877
then A120601 +B120577 le 119877
Let 120593 120577 isin S with 120593 le 119877 and 120577 le 119877 Then
Krasnoselskii (see [47]) proved that if 119860 is a stable constantmatrix without piecewise alternately advanced and retardedargument and lim
|119909|+|119910|rarr+infin(|119892(119905 119909 119910)|(|119909|+|119910|)) = 0 then
the system (56) has at least one periodic solution In our caseapplyingTheorem 12 this result is also valid for the DEPCAGsystem (56) requiring the hypothesis (119873
120596) (1198753) (119872119892) 119860(119905)
and 119892(119905 119909 119910) are periodic functions in 119905 with a period 120596 forall 119905 ge 120591 and |119892(119905 119909 119910)| le 119888
1(|119909| + |119910|) + 120588
2 with 2119888
1120596 lt 119862
2
and 1205882constants for |119909| + |119910| le 119877 119877 gt 0
Remark 17 Considering the nonlinear system of differentialequations with a general piecewise alternately advanced andretarded argument
In this particular case applyingTheorem 13 this result is alsovalid for the DEPCAG system (57) requiring the hypothesis(119873120596) (1198751) (1198753) (119871119892) (119872119891) and 119888
119866(1198621+1198712)119877+119888119866(120573+1205881)120596 le 119877
with = 1
Remark 18 Suppose that (1198751) is satisfied by 120596 = 120596
1
(1198752) by 120596 = 120596
2 and (119875
3) by 120596 = 120596
3 if 120596
119894120596119895is a
rational number for all 119894 119895 = 1 2 3 then (1198751) (1198752) and
(1198753) are simultaneously satisfied by 120596 = lcm120596
1 1205962 1205963
where lcm1205961 1205962 1205963 denotes the least common multiple
between 1205961 1205962and 120596
3 In the general case it is possible
that there exist five possible periods 1205961for 119860 120596
2for 119891 120596
3
for 119892 1205964for 119862 and the sequences 119905
119894119894isinZ 120574119894119894isinZ satisfy the
(1205965 119901) condition If 120596
119894120596119895is a rational number for all 119894 119895 =
1 2 3 4 5 so in this situation our results insure the existenceof 120596-periodic solution with 120596 = lcm120596
1 1205962 1205963 1205964 1205965
Therefore the above results insure the existence of 120596-periodicsolutions of the DEPCAG system (6) These solutions arecalled subharmonic solutions See Corollaries 19ndash22
To determine criteria for the existence and uniqueness ofsubharmonic solutions of theDEPCAG system (6) fromnowon we make the following assumption
(119875120596) There exists 120596 = lcm120596
1 1205962 1205963 1205964 1205965 gt 0 120596
119894120596119895
which is a rational number for all 119894 119895 = 1 2 3 4 5
such that
(1) 119860(119905) 119891(119905 1199091 1199101) and 119892(119905 119909
2 1199102) are periodic
functions in 119905 with a period 1205961 1205962 and 120596
3
respectively for all 119905 ge 120591(2) 119862(119905 + 120596
4 119904 + 120596
4 1199093) = 119862(119905 119904 119909
3) for all 119905 ge 120591
(3) There exists 119901 isin Z+ for which the sequences119905119894119894isinZ 120574119894119894isinZ satisfy the (120596
5 119901) condition
As immediate corollaries of Theorems 12ndash15 and Remark 18the following results are true
The Scientific World Journal 11
Corollary 19 Suppose the hypotheses (119873120596) (119875120596) (119871119891) (119871119862)
(119872119892) and (49) hold Then the DEPCAG system (6) has at least
one subharmonic solution in S
Corollary 20 Suppose the hypotheses (119873120596) (119875120596) (119871119892) (119872119891)
(119872119862) (119862) and (51) hold Then the DEPCAG system (6) has at
least one subharmonic solution in S
Corollary 21 Suppose the hypotheses (119873120596) (119875120596) (119871119891) (119871119862)
(119871119892) and (53) holdThen theDEPCAG system (6) has a unique
subharmonic solution
Corollary 22 Suppose the hypotheses (119873120596) (119875120596) (119872
119891)
(119872119862) (119872119892) (119862) and (55) hold Then the DEPCAG system (6)
has at least one subharmonic solution in S
4 Applications and Illustrative Examples
Wewill introduce appropriate examples in this sectionTheseexamples will show the feasibility of our theory
Mathematical modelling of real-life problems usuallyresults in functional equations like ordinary or partial differ-ential equations integral and integro-differential equationsand stochastic equations Many mathematical formulationsof physical phenomena contain integro-differential equa-tions these equations arise inmany fields like fluid dynamicsbiologicalmodels and chemical kinetics So we first considernonlinear integro-differential equations with a general piece-wise constant argument mentioned in the introduction andobtain some new sufficient conditions for the existence of theperiodic solutions of these systems
Example 1 Let 120582 R2 rarr [0infin) and ℎ R2 rarr [0infin) betwo functions satisfying
sup119905isinR
int
119905+2120587
119905
120582 (119905 119904) 119889119904 le sup119905isinR
int
119905+2120587
119905
ℎ (119905 119904) 119889119904 le ℎ (58)
Consider the following nonlinear integro-differential equa-tions with piecewise alternately advanced and retarded argu-ment of generalized type
|119862(119905 119904 119910)| le 1198882(119877)120582(119905 119904)|119910| +ℎ(119905 119904) for |119910| le 119877 where
2120587 le 1198621
(iv) 119892 R times R119899 times R119899 rarr R119899 is continuous and 119862(119905 119904 119910)satisfies (119862)
Indeed for any 120576 gt 0 there exists 120575 gt 0 such that |1199101minus1199102| le 120575
implies
1003816100381610038161003816119862 (119905 119904 1199101) minus 119862 (119905 119904 1199102)1003816100381610038161003816 le 1205761198882 (119877) 120582 (119905 119904) for 119905 119904 isin R
(60)
Furthermore there exists 119877 such that
exp (2119886lowast120587)exp (2119886
lowast120587) minus 1
[(1198621+ 1198622) 119877 + 2120587 (120588
2+ ℎ)] le 119877 (61)
where 119886lowast = sup119905isinR |119886(119905)| and 119886
lowast= inf119905isinR|119886(119905)|
Then by Theorem 15 the DEPCAG system (59) has atleast one 2120587-periodic solution
Example 2 Thus many examples can be constructed whereour results can be applied
Let Λ R rarr R119899times119899 and 120583 R rarr R119899 be two functionssatisfying
sup119905isinR
int
119905+120596
119905
|Λ (119904)| 119889119904 = Λ lt infin
sup119905isinR
int
119905+120596
119905
1003816100381610038161003816120583 (119905 minus 119904)1003816100381610038161003816 119889119904 = 120583 lt infin
(62)
Now consider the integro-differential system with piecewisealternately advanced and retarded argument
Theorem 12 implies that there exists at least a 120596-periodicsolution of the DEPCAG system (63)
12 The Scientific World Journal
Note that similar results can be obtained under (119871119892)
and (119872119862) On the other hand the periodic situation of the
DEPCAGsystem (56) and (57) can be treated in the samewayLet us consider another example for second-order differ-
ential equations with a general piecewise constant argumentIn this case we can show the existence and uniquenessof periodic solutions of the following nonlinear DEPCAGsystem
Example 3 Consider the following nonlinear DEPCAG sys-tem
11991010158401015840
(119905) + (12058121199102
(119905) minus 2) 1199101015840
(119905) minus 8119910 (119905)
minus 1205811sin (120596119905) 1199102 (120574 (119905)) minus 120581
2cos (120596119905) = 0
(65)
where 1205811 1205812isin R 120574(119905) = 120574
Then for 120593 120595 isin S we have1003817100381710038171003817119892 (sdot 120593 (sdot) 120593 (120574 (sdot))) minus 119892 (sdot 120595 (sdot) 120595 (120574 (sdot)))
le 212058121198772 sup119905isin[120591120591+120596]
1003816100381610038161003816100381610038161003816(1205951 (119905) minus 1205931 (119905)
1205952 (119905) minus 1205932 (119905)
)
1003816100381610038161003816100381610038161003816= 212058121198772 1003817100381710038171003817120593 minus 120595
1003817100381710038171003817
(69)
In a similar way for 119891 we have1003817100381710038171003817119891 (sdot 120593 (sdot) 120593 (120574 (sdot))) minus 119891 (sdot 120595 (sdot) 120595 (120574 (sdot)))
1003817100381710038171003817
le 212058111198771003817100381710038171003817120593 minus 120595
1003817100381710038171003817
(70)
ByTheorem 14 the DEPCAG system (65) has a unique 2120587120596-periodic solution in S
Conflict of Interests
The author declares that there is no conflict of interests re-garding the publication of this paper
Acknowledgment
This research was in part supported by FIBE 01-12 DIUMCE
References
[1] ADMyshkis ldquoOn certain problems in the theory of differentialequations with deviating argumentsrdquo Uspekhi Matematich-eskikh Nauk vol 32 pp 173ndash202 1977
[2] K L Cooke and JWiener ldquoRetarded differential equations withpiecewise constant delaysrdquo Journal of Mathematical Analysisand Applications vol 99 no 1 pp 265ndash297 1984
[3] K L Cooke and J Wiener ldquoAn equation alternately of retardedand advanced typerdquo Proceedings of the American MathematicalSociety vol 99 pp 726ndash732 1987
[4] S M Shah and J Wiener ldquoAdvanced differential equations withpiecewise constant argument deviationsrdquo International JournalofMathematics andMathematical Sciences vol 6 no 4 pp 671ndash703 1983
[5] J WienerGeneralized Solutions of Functional Differential Equa-tions World Scientific Singapore 1993
[6] A R Aftabizadeh and J Wiener ldquoOscillatory and periodicsolutions of an equation alternately of retarded and advancedtypesrdquo Applicable Analysis vol 23 no 3 pp 219ndash231 1986
[7] A R Aftabizadeh J Wiener and J M Xu ldquoOscillatory andperiodic solutions of delay differential equations with piecewiseconstant argumentrdquo Proceedings of the American MathematicalSociety vol 99 pp 673ndash679 1987
[8] M U Akhmet ldquoIntegral manifolds of differential equationswith piecewise constant argument of generalized typerdquo Nonlin-ear Analysis Theory Methods and Applications vol 66 no 2pp 367ndash383 2007
[9] M U Akhmet Nonlinear Hybrid ContinuousDiscrete-TimeModels Atlantis Press Amsterdam The Netherlands 2011
[10] M U Akhmet C Buyukadali and T Ergenc ldquoPeriodic solu-tions of the hybrid system with small parameterrdquo NonlinearAnalysis Hybrid Systems vol 2 no 2 pp 532ndash543 2008
[11] A Cabada J B Ferreiro and J J Nieto ldquoGreenrsquos functionand comparison principles for first order periodic differentialequations with piecewise constant argumentsrdquo Journal of Math-ematical Analysis and Applications vol 291 no 2 pp 690ndash6972004
[12] K S Chiu and M Pinto ldquoVariation of parameters formula andGronwall inequality for differential equations with a generalpiecewise constant argumentrdquo Acta Mathematicae ApplicataeSinica vol 27 no 4 pp 561ndash568 2011
The Scientific World Journal 13
[13] K S Chiu and M Pinto ldquoOscillatory and periodic solutionsin alternately advanced and delayed differential equationsrdquoCarpathian Journal of Mathematics vol 29 no 2 pp 149ndash1582013
[14] J J Nieto and R Rodrıguez-Lopez ldquoGreenrsquos function forsecond-order periodic boundary value problemswith piecewiseconstant argumentsrdquo Journal of Mathematical Analysis andApplications vol 304 no 1 pp 33ndash57 2005
[15] M Pinto ldquoAsymptotic equivalence of nonlinear and quasi lin-ear differential equations with piecewise constant argumentsrdquoMathematical and Computer Modelling vol 49 no 9-10 pp1750ndash1758 2009
[16] M Pinto ldquoCauchy and Green matrices type and stability inalternately advanced and delayed differential systemsrdquo Journalof Difference Equations and Applications vol 17 no 2 pp 235ndash254 2011
[17] G Q Wang ldquoExistence theorem of periodic solutions for adelay nonlinear differential equation with piecewise constantargumentsrdquo Journal of Mathematical Analysis and Applicationsvol 298 no 1 pp 298ndash307 2004
[18] G Q Wang and S S Cheng ldquoExistence and uniqueness ofperiodic solutions for a second-order nonlinear differentialequationwith piecewise constant argumentrdquo International Jour-nal ofMathematics andMathematical Sciences vol 2009 ArticleID 950797 14 pages 2009
[19] Y H Xia Z Huang and M Han ldquoExistence of almost periodicsolutions for forced perturbed systems with piecewise constantargumentrdquo Journal of Mathematical Analysis and Applicationsvol 333 no 2 pp 798ndash816 2007
[20] P Yang Y Liu and W Ge ldquoGreenrsquos function for second orderdifferential equations with piecewise constant argumentsrdquoNon-linear Analysis Theory Methods and Applications vol 64 no 8pp 1812ndash1830 2006
[21] R Yuan ldquoThe existence of almost periodic solutions of retardeddifferential equations with piecewise constant argumentrdquo Non-linear Analysis Theory Methods and Applications vol 48 no 7pp 1013ndash1032 2002
[22] A I Alonso J Hong and R Obaya ldquoAlmost periodic typesolutions of differential equations with piecewise constant argu-ment via almost periodic type sequencesrdquo Applied MathematicsLetters vol 13 no 2 pp 131ndash137 2000
[23] K S Chiu and M Pinto ldquoPeriodic solutions of differentialequations with a general piecewise constant argument andapplicationsrdquo Electronic Journal of Qualitative Theory of Differ-ential Equations vol 46 pp 1ndash19 2010
[24] K S Chiu M Pinto and J C Jeng ldquoExistence and globalconvergence of periodic solutions in recurrent neural networkmodels with a general piecewise alternately advanced andretarded argumentrdquo Acta Applicandae Mathematicae 2013
[25] L Dai Nonlinear Dynamics of Piecewise of Constant Systemsand Implememtation of Piecewise Constants Arguments WorldScientific Singapore 2008
[26] T A Burton Stability and Periodic Solutions of Ordinary andFunctional Differential Equations Academic Press New YorkNY USA 1985
[27] T A Burton and T Furumochi ldquoPeriodic and asymptoticallyperiodic solutions of neutral integral equationsrdquo ElectronicJournal of Qualitative Theory of Differential Equations vol 10pp 1ndash19 2000
[28] T A Burton and T Furumochi ldquoExistence theorems andperiodic solutions of neutral integral equationsrdquo NonlinearAnalysis Theory Methods and Applications vol 43 no 4 pp527ndash546 2001
[29] F Karakoc H Bereketoglu and G Seyhan ldquoOscillatory andperiodic solutions of impulsive differential equations withpiecewise constant argumentrdquoActa ApplicandaeMathematicaevol 110 no 1 pp 499ndash510 2010
[30] Y Liu P Yang and W Ge ldquoPeriodic solutions of higher-order delay differential equationsrdquo Nonlinear Analysis TheoryMethods and Applications vol 63 no 1 pp 136ndash152 2005
[31] X L Liu and W T Li ldquoPeriodic solutions for dynamic equa-tions on time scalesrdquo Nonlinear Analysis Theory Methods andApplications vol 67 no 5 pp 1457ndash1463 2007
[32] M Pinto ldquoDichotomy and existence of periodic solutions ofquasilinear functional differential equationsrdquo Nonlinear Analy-sis Theory Methods and Applications vol 72 no 3-4 pp 1227ndash1234 2010
[33] Y N Raffoul ldquoPeriodic solutions for neutral nonlinear differ-ential equations with functional delayrdquo Electronic Journal ofDifferential Equations vol 2003 no 102 pp 1ndash7 2003
[34] G Q Wang and S S Cheng ldquoPeriodic solutions of discreteRayleigh equations with deviating argumentsrdquo Taiwanese Jour-nal of Mathematics vol 13 no 6 pp 2051ndash2067 2009
[35] K S Chiu ldquoStability of oscillatory solutions of differential equa-tions with a general piecewise constant argumentrdquo ElectronicJournal of Qualitative Theory of Differential Equations vol 88pp 1ndash15 2011
[36] R N Butris and M A Aziz ldquoSome theorems in the existenceand uniqueness for system of nonlinear integro-differentialequationsrdquo Journal Education and Science vol 18 pp 76ndash892006
[37] A YMitropolsky andD IMortynyukPeriodic Solutions for theOscillations Systems with Retarded Argument General SchoolKiev Ukraine 1979
[38] L L Tai ldquoNumerical-analytic method for the investigation ofautonomous systems of differential equationsrdquo Ukrainskii Ma-tematicheskii Zhurnal vol 30 no 3 pp 309ndash317 1978
[39] Yu D Shlapak ldquoPeriodic solutions of ordinary differentialequations of first order that are not solved with respect to thederivativerdquo Ukrainian Mathematical Journal vol 32 no 5 pp420ndash425 1980
[40] A M Samoilenko ldquoNumerical-analytic method for the investi-gation of periodic systems of ordinary differential equationsrdquoUkrainskii Matematicheskii Zhurnal vol 17 no 4 pp 16ndash231965
[41] AM Samoilenko andN I RontoNumerical-Analytic Methodsfor Investigations of Periodic Solutions Mir Publishers MoscowRussia 1979
[42] R N Butris ldquoPeriodic solution of nonlinear system of integro-differential equations depending on the gamma distributionrdquoGeneral Mathematics Notes vol 15 no 1 pp 56ndash71 2013
[43] D R Smart Fixed Points Theorems Cambridge UniversityPress Cambridge UK 1980
[44] T A Burton ldquoKrasnoselskiirsquos inversion principle and fixedpointsrdquo Nonlinear Analysis Theory Methods and Applicationsvol 30 no 7 pp 3975ndash3986 1997
[45] T A Burton ldquoA fixed-point theorem of Krasnoselskiirdquo AppliedMathematics Letters vol 11 no 1 pp 85ndash88 1998
14 The Scientific World Journal
[46] M Pinto and D Sepulveda ldquoh-asymptotic stability by fixedpoint in neutral nonlinear differential equations with delayrdquoNonlinear Analysis Theory Methods and Applications vol 74no 12 pp 3926ndash3933 2011
[47] M A Krasnoselskii Translation Along Trajectories of Differen-tial Equations Nauka Moscow Russia 1966 (Russian) Englishtranslation American Mathematical Society Providence RIUSA 1968
In such a case the DEPCAG system (6) has 120596-periodicsolution 119911(119905) given by the integral equation (17) Beforestudying the existence of solutions of integral equation (17)in the next section firstly we define Greenrsquos function forperiodic solutions of the DEPCAG system (6)
Definition 2 Suppose that the condition (119873120596) holds For each
119905 119904 isin [120591 120591 + 120596] Greenrsquos function for (6) is given by
(119904) 120591 le 119904 le 119905 le 120591 + 120596
Φ (119905)119863Φminus1
(119904) 120591 le 119905 lt 119904 le 120591 + 120596(18)
where Φ(119905) is a fundamental solution of (7) and
119863 = ((Φminus1
(120591)Φ (120591 + 120596))minus1
minus 119868)minus1
(19)
We note that the condition (119873120596) implies the existence of
the matrix119863To prove double 120596-periodicity of Greenrsquos function119866(119905 119904)
we first give the following lemma
Lemma 3 Suppose that the condition (119873120596) holds Let the
matrix 119863 be defined by (19) and then one has the followingidentities
(119868 + 119863) = (Φminus1
(120591)Φ (120591 + 120596))minus1
119863
= (119868 minus Φminus1
(120591)Φ (120591 + 120596))minus1
Φ (119905)119863Φminus1
(119905 + 120596) minus Φ (119905)119863Φminus1
(119905) = 119868
(119868 + 119863)Φminus1
(120591 minus 120596)Φ (120591) = 119863
(20)
The Scientific World Journal 5
For Lemma 3we can prove an important property double120596-periodicity of Greenrsquos function 119866(119905 119904) to study after theexistence of periodic solutions
Lemma 4 Suppose that the condition (119873120596) holds Then
Greenrsquos function119866(119905 119904) is double120596-periodic that is119866(119905+120596 119904+120596) = 119866(119905 119904)
3 Existence of Periodic Solutions
In this section we prove the main theorems of this paperso we recall the nonlinear integro-differential systems withDEPCAG (6)
Using Definition 2 Remark 5 and double 120596-periodicityof Greenrsquos function similar formula is given by (17) So wehave obtained the following result
Proposition 6 Suppose that the conditions (119873120596) and (P) hold
Let (120591 119911(120591)) isin R times R119899 Then 119911(119905) = 119911(119905 120591 119911(120591)) is 120596-periodicsolution on R of the DEPCAG system (6) if and only if 119911(119905) is120596-periodic solution of the integral equation
119911 (119905) = int
120591+120596
120591
119866 (119905 119904) [119891(119904 119911 (119904) int
It is easy to see that the DEPCAG system (6) has 120596-periodic solution if and only if the operator I has one fixedpoint in P
120596
To prove some existence criteria for 120596-periodic solutionsof the DEPCAG system (6) we use the Banach Schauder andKrasnoselskiirsquos fixed point theorems
Next we state first Krasnoselskiirsquos fixed point theoremwhich enables us to prove the existence of a periodic solutionFor the proof of Krasnoselskiirsquos fixed point theorem we referthe reader to [43]
Theorem A (Krasnoselskiirsquos fixed point theorem) Let 119878 bea closed convex nonempty subset of a Banach space (119864 sdot )Suppose thatA andBmap 119878 into 119864 such that
Then there exists 119911 isin 119878 with 119911 = A119911 +B119911
Remark 7 Krasnoselskiirsquos theorem may be combined withBanach and Schauderrsquos fixed point theorems In a certainsense we can interpret this as follows if a compact operatorhas the fixed point property under a small perturbationthen this property can be inherited The theorem is usefulin establishing the existence results for perturbed operatorequations It also has awide range of applications to nonlinearintegral equations of mixed type for proving the existence ofsolutions Thus the existence of fixed points for the sum oftwo operators has attracted tremendous interest and theirapplications are frequent in nonlinear analysis See [32 3343ndash46]
We note that to apply Krasnoselskiirsquos fixed point theoremwe need to construct two mappings one is contraction andthe other is compact Therefore we express (38) as
times1003817100381710038171003817120593 minus 120577
1003817100381710038171003817
le (119888119866int
120591+120596
120591
[1199011(119904) + 119901
2(119904)] 119889119904)
1003817100381710038171003817120593 minus 1205771003817100381710038171003817
le 1198881198661198711
1003817100381710038171003817120593 minus 1205771003817100381710038171003817
(44)
HenceA defines a contraction mapping
Similarly B is given by (41) which may be also acontraction operator
Lemma 9 If (119873120596) (119875) and (119871
119892) holdB is given by (41)with
1198881198661198712lt 1 and thenB is a contraction mapping
Lemma 10 If (119873120596) holdsB is defined by (41) and thenB is
completely continuous that isB is continuous and the imageofB is contained in a compact set
Proof Step 1 First we prove that B P120596
rarr P120596is
continuousAs the operator A a change of variable in (41) we have
(B120593)(119905 + 120596) = (B120593)(119905) Now we want to show B iscontinuous
The function 119892(119905 119909 119910) is uniformly continuous on [120591 120591 +120596] times CR times CR and by the periodicity in 119905 the function119892(119905 119909 119910) is uniformly continuous on R times CR times CR Thusfor any 1205981015840 = (120598119888
119866120596) gt 0 there exists 120575 = 120575(120598) gt 0 such
that 1199111 1199112isin S 119911
1minus 1199112 le 120575 implies |119892(119905 119911
1(119905) 1199111(120574(119905))) minus
119892(119905 1199112(119905) 1199112(120574(119905)))| le 120598
1015840 for 119905 isin [120591 120591 + 120596] Then B1199111minus
B1199112 le 120598 In fact by the continuity of 119892
1003816100381610038161003816119892 (119905 1199111 (119905) 1199111 (120574 (119905))) minus 119892 (119905 1199112 (119905) 1199112 (120574 (119905)))1003816100381610038161003816 le 1205981015840
for 119905 isin [120591 120591 + 120596](45)
8 The Scientific World Journal
and then1003817100381710038171003817B1199111minusB1199112
Step 2 We show that the image of B is contained in acompact set
Let 119909 119910 isin CR and 119904 isin [120591 120591 + 120596] for the continuity of thefunction 119892(119904 119909 119910) there exists119872 gt 0 such that |119892(119904 119909 119910)| le119872 Let 120593
119899isin S where 119899 is a positive integer then we have
We now see that all the conditions of Krasnoselskiirsquostheorem are satisfied Thus there exists a fixed point 119911 in S
such that 119911 = A119911 +B119911 By Proposition 6 this fixed point isa solution of the DEPCAG system (6) Hence the DEPCAGsystem (6) has 120596-periodic solution
By the symmetry of the conditions we will obtain asTheorem 12
Theorem 13 Suppose the hypothesis (119873120596) (119875) (119871
119892) (119872119891)
(119872119862) (119862) hold Let 119877 be a positive constant satisfying the
inequality
119888119866(1198621+ 1198712) 119877 + 119888
119866(120573 + 120578
2+ 1205881) 120596 le 119877 (51)
Then the DEPCAG system (6) has at least one 120596-periodicsolution in S
By Lemma 9 the mapping B is a contraction and it isclear that B P
120596rarr P
120596 Also from Lemma 11 A is
completely continuousNext we prove that if 120593 120577 isin S with 120593 le 119877 and 120577 le 119877
then A120601 +B120577 le 119877
Let 120593 120577 isin S with 120593 le 119877 and 120577 le 119877 Then
Krasnoselskii (see [47]) proved that if 119860 is a stable constantmatrix without piecewise alternately advanced and retardedargument and lim
|119909|+|119910|rarr+infin(|119892(119905 119909 119910)|(|119909|+|119910|)) = 0 then
the system (56) has at least one periodic solution In our caseapplyingTheorem 12 this result is also valid for the DEPCAGsystem (56) requiring the hypothesis (119873
120596) (1198753) (119872119892) 119860(119905)
and 119892(119905 119909 119910) are periodic functions in 119905 with a period 120596 forall 119905 ge 120591 and |119892(119905 119909 119910)| le 119888
1(|119909| + |119910|) + 120588
2 with 2119888
1120596 lt 119862
2
and 1205882constants for |119909| + |119910| le 119877 119877 gt 0
Remark 17 Considering the nonlinear system of differentialequations with a general piecewise alternately advanced andretarded argument
In this particular case applyingTheorem 13 this result is alsovalid for the DEPCAG system (57) requiring the hypothesis(119873120596) (1198751) (1198753) (119871119892) (119872119891) and 119888
119866(1198621+1198712)119877+119888119866(120573+1205881)120596 le 119877
with = 1
Remark 18 Suppose that (1198751) is satisfied by 120596 = 120596
1
(1198752) by 120596 = 120596
2 and (119875
3) by 120596 = 120596
3 if 120596
119894120596119895is a
rational number for all 119894 119895 = 1 2 3 then (1198751) (1198752) and
(1198753) are simultaneously satisfied by 120596 = lcm120596
1 1205962 1205963
where lcm1205961 1205962 1205963 denotes the least common multiple
between 1205961 1205962and 120596
3 In the general case it is possible
that there exist five possible periods 1205961for 119860 120596
2for 119891 120596
3
for 119892 1205964for 119862 and the sequences 119905
119894119894isinZ 120574119894119894isinZ satisfy the
(1205965 119901) condition If 120596
119894120596119895is a rational number for all 119894 119895 =
1 2 3 4 5 so in this situation our results insure the existenceof 120596-periodic solution with 120596 = lcm120596
1 1205962 1205963 1205964 1205965
Therefore the above results insure the existence of 120596-periodicsolutions of the DEPCAG system (6) These solutions arecalled subharmonic solutions See Corollaries 19ndash22
To determine criteria for the existence and uniqueness ofsubharmonic solutions of theDEPCAG system (6) fromnowon we make the following assumption
(119875120596) There exists 120596 = lcm120596
1 1205962 1205963 1205964 1205965 gt 0 120596
119894120596119895
which is a rational number for all 119894 119895 = 1 2 3 4 5
such that
(1) 119860(119905) 119891(119905 1199091 1199101) and 119892(119905 119909
2 1199102) are periodic
functions in 119905 with a period 1205961 1205962 and 120596
3
respectively for all 119905 ge 120591(2) 119862(119905 + 120596
4 119904 + 120596
4 1199093) = 119862(119905 119904 119909
3) for all 119905 ge 120591
(3) There exists 119901 isin Z+ for which the sequences119905119894119894isinZ 120574119894119894isinZ satisfy the (120596
5 119901) condition
As immediate corollaries of Theorems 12ndash15 and Remark 18the following results are true
The Scientific World Journal 11
Corollary 19 Suppose the hypotheses (119873120596) (119875120596) (119871119891) (119871119862)
(119872119892) and (49) hold Then the DEPCAG system (6) has at least
one subharmonic solution in S
Corollary 20 Suppose the hypotheses (119873120596) (119875120596) (119871119892) (119872119891)
(119872119862) (119862) and (51) hold Then the DEPCAG system (6) has at
least one subharmonic solution in S
Corollary 21 Suppose the hypotheses (119873120596) (119875120596) (119871119891) (119871119862)
(119871119892) and (53) holdThen theDEPCAG system (6) has a unique
subharmonic solution
Corollary 22 Suppose the hypotheses (119873120596) (119875120596) (119872
119891)
(119872119862) (119872119892) (119862) and (55) hold Then the DEPCAG system (6)
has at least one subharmonic solution in S
4 Applications and Illustrative Examples
Wewill introduce appropriate examples in this sectionTheseexamples will show the feasibility of our theory
Mathematical modelling of real-life problems usuallyresults in functional equations like ordinary or partial differ-ential equations integral and integro-differential equationsand stochastic equations Many mathematical formulationsof physical phenomena contain integro-differential equa-tions these equations arise inmany fields like fluid dynamicsbiologicalmodels and chemical kinetics So we first considernonlinear integro-differential equations with a general piece-wise constant argument mentioned in the introduction andobtain some new sufficient conditions for the existence of theperiodic solutions of these systems
Example 1 Let 120582 R2 rarr [0infin) and ℎ R2 rarr [0infin) betwo functions satisfying
sup119905isinR
int
119905+2120587
119905
120582 (119905 119904) 119889119904 le sup119905isinR
int
119905+2120587
119905
ℎ (119905 119904) 119889119904 le ℎ (58)
Consider the following nonlinear integro-differential equa-tions with piecewise alternately advanced and retarded argu-ment of generalized type
|119862(119905 119904 119910)| le 1198882(119877)120582(119905 119904)|119910| +ℎ(119905 119904) for |119910| le 119877 where
2120587 le 1198621
(iv) 119892 R times R119899 times R119899 rarr R119899 is continuous and 119862(119905 119904 119910)satisfies (119862)
Indeed for any 120576 gt 0 there exists 120575 gt 0 such that |1199101minus1199102| le 120575
implies
1003816100381610038161003816119862 (119905 119904 1199101) minus 119862 (119905 119904 1199102)1003816100381610038161003816 le 1205761198882 (119877) 120582 (119905 119904) for 119905 119904 isin R
(60)
Furthermore there exists 119877 such that
exp (2119886lowast120587)exp (2119886
lowast120587) minus 1
[(1198621+ 1198622) 119877 + 2120587 (120588
2+ ℎ)] le 119877 (61)
where 119886lowast = sup119905isinR |119886(119905)| and 119886
lowast= inf119905isinR|119886(119905)|
Then by Theorem 15 the DEPCAG system (59) has atleast one 2120587-periodic solution
Example 2 Thus many examples can be constructed whereour results can be applied
Let Λ R rarr R119899times119899 and 120583 R rarr R119899 be two functionssatisfying
sup119905isinR
int
119905+120596
119905
|Λ (119904)| 119889119904 = Λ lt infin
sup119905isinR
int
119905+120596
119905
1003816100381610038161003816120583 (119905 minus 119904)1003816100381610038161003816 119889119904 = 120583 lt infin
(62)
Now consider the integro-differential system with piecewisealternately advanced and retarded argument
Theorem 12 implies that there exists at least a 120596-periodicsolution of the DEPCAG system (63)
12 The Scientific World Journal
Note that similar results can be obtained under (119871119892)
and (119872119862) On the other hand the periodic situation of the
DEPCAGsystem (56) and (57) can be treated in the samewayLet us consider another example for second-order differ-
ential equations with a general piecewise constant argumentIn this case we can show the existence and uniquenessof periodic solutions of the following nonlinear DEPCAGsystem
Example 3 Consider the following nonlinear DEPCAG sys-tem
11991010158401015840
(119905) + (12058121199102
(119905) minus 2) 1199101015840
(119905) minus 8119910 (119905)
minus 1205811sin (120596119905) 1199102 (120574 (119905)) minus 120581
2cos (120596119905) = 0
(65)
where 1205811 1205812isin R 120574(119905) = 120574
Then for 120593 120595 isin S we have1003817100381710038171003817119892 (sdot 120593 (sdot) 120593 (120574 (sdot))) minus 119892 (sdot 120595 (sdot) 120595 (120574 (sdot)))
le 212058121198772 sup119905isin[120591120591+120596]
1003816100381610038161003816100381610038161003816(1205951 (119905) minus 1205931 (119905)
1205952 (119905) minus 1205932 (119905)
)
1003816100381610038161003816100381610038161003816= 212058121198772 1003817100381710038171003817120593 minus 120595
1003817100381710038171003817
(69)
In a similar way for 119891 we have1003817100381710038171003817119891 (sdot 120593 (sdot) 120593 (120574 (sdot))) minus 119891 (sdot 120595 (sdot) 120595 (120574 (sdot)))
1003817100381710038171003817
le 212058111198771003817100381710038171003817120593 minus 120595
1003817100381710038171003817
(70)
ByTheorem 14 the DEPCAG system (65) has a unique 2120587120596-periodic solution in S
Conflict of Interests
The author declares that there is no conflict of interests re-garding the publication of this paper
Acknowledgment
This research was in part supported by FIBE 01-12 DIUMCE
References
[1] ADMyshkis ldquoOn certain problems in the theory of differentialequations with deviating argumentsrdquo Uspekhi Matematich-eskikh Nauk vol 32 pp 173ndash202 1977
[2] K L Cooke and JWiener ldquoRetarded differential equations withpiecewise constant delaysrdquo Journal of Mathematical Analysisand Applications vol 99 no 1 pp 265ndash297 1984
[3] K L Cooke and J Wiener ldquoAn equation alternately of retardedand advanced typerdquo Proceedings of the American MathematicalSociety vol 99 pp 726ndash732 1987
[4] S M Shah and J Wiener ldquoAdvanced differential equations withpiecewise constant argument deviationsrdquo International JournalofMathematics andMathematical Sciences vol 6 no 4 pp 671ndash703 1983
[5] J WienerGeneralized Solutions of Functional Differential Equa-tions World Scientific Singapore 1993
[6] A R Aftabizadeh and J Wiener ldquoOscillatory and periodicsolutions of an equation alternately of retarded and advancedtypesrdquo Applicable Analysis vol 23 no 3 pp 219ndash231 1986
[7] A R Aftabizadeh J Wiener and J M Xu ldquoOscillatory andperiodic solutions of delay differential equations with piecewiseconstant argumentrdquo Proceedings of the American MathematicalSociety vol 99 pp 673ndash679 1987
[8] M U Akhmet ldquoIntegral manifolds of differential equationswith piecewise constant argument of generalized typerdquo Nonlin-ear Analysis Theory Methods and Applications vol 66 no 2pp 367ndash383 2007
[9] M U Akhmet Nonlinear Hybrid ContinuousDiscrete-TimeModels Atlantis Press Amsterdam The Netherlands 2011
[10] M U Akhmet C Buyukadali and T Ergenc ldquoPeriodic solu-tions of the hybrid system with small parameterrdquo NonlinearAnalysis Hybrid Systems vol 2 no 2 pp 532ndash543 2008
[11] A Cabada J B Ferreiro and J J Nieto ldquoGreenrsquos functionand comparison principles for first order periodic differentialequations with piecewise constant argumentsrdquo Journal of Math-ematical Analysis and Applications vol 291 no 2 pp 690ndash6972004
[12] K S Chiu and M Pinto ldquoVariation of parameters formula andGronwall inequality for differential equations with a generalpiecewise constant argumentrdquo Acta Mathematicae ApplicataeSinica vol 27 no 4 pp 561ndash568 2011
The Scientific World Journal 13
[13] K S Chiu and M Pinto ldquoOscillatory and periodic solutionsin alternately advanced and delayed differential equationsrdquoCarpathian Journal of Mathematics vol 29 no 2 pp 149ndash1582013
[14] J J Nieto and R Rodrıguez-Lopez ldquoGreenrsquos function forsecond-order periodic boundary value problemswith piecewiseconstant argumentsrdquo Journal of Mathematical Analysis andApplications vol 304 no 1 pp 33ndash57 2005
[15] M Pinto ldquoAsymptotic equivalence of nonlinear and quasi lin-ear differential equations with piecewise constant argumentsrdquoMathematical and Computer Modelling vol 49 no 9-10 pp1750ndash1758 2009
[16] M Pinto ldquoCauchy and Green matrices type and stability inalternately advanced and delayed differential systemsrdquo Journalof Difference Equations and Applications vol 17 no 2 pp 235ndash254 2011
[17] G Q Wang ldquoExistence theorem of periodic solutions for adelay nonlinear differential equation with piecewise constantargumentsrdquo Journal of Mathematical Analysis and Applicationsvol 298 no 1 pp 298ndash307 2004
[18] G Q Wang and S S Cheng ldquoExistence and uniqueness ofperiodic solutions for a second-order nonlinear differentialequationwith piecewise constant argumentrdquo International Jour-nal ofMathematics andMathematical Sciences vol 2009 ArticleID 950797 14 pages 2009
[19] Y H Xia Z Huang and M Han ldquoExistence of almost periodicsolutions for forced perturbed systems with piecewise constantargumentrdquo Journal of Mathematical Analysis and Applicationsvol 333 no 2 pp 798ndash816 2007
[20] P Yang Y Liu and W Ge ldquoGreenrsquos function for second orderdifferential equations with piecewise constant argumentsrdquoNon-linear Analysis Theory Methods and Applications vol 64 no 8pp 1812ndash1830 2006
[21] R Yuan ldquoThe existence of almost periodic solutions of retardeddifferential equations with piecewise constant argumentrdquo Non-linear Analysis Theory Methods and Applications vol 48 no 7pp 1013ndash1032 2002
[22] A I Alonso J Hong and R Obaya ldquoAlmost periodic typesolutions of differential equations with piecewise constant argu-ment via almost periodic type sequencesrdquo Applied MathematicsLetters vol 13 no 2 pp 131ndash137 2000
[23] K S Chiu and M Pinto ldquoPeriodic solutions of differentialequations with a general piecewise constant argument andapplicationsrdquo Electronic Journal of Qualitative Theory of Differ-ential Equations vol 46 pp 1ndash19 2010
[24] K S Chiu M Pinto and J C Jeng ldquoExistence and globalconvergence of periodic solutions in recurrent neural networkmodels with a general piecewise alternately advanced andretarded argumentrdquo Acta Applicandae Mathematicae 2013
[25] L Dai Nonlinear Dynamics of Piecewise of Constant Systemsand Implememtation of Piecewise Constants Arguments WorldScientific Singapore 2008
[26] T A Burton Stability and Periodic Solutions of Ordinary andFunctional Differential Equations Academic Press New YorkNY USA 1985
[27] T A Burton and T Furumochi ldquoPeriodic and asymptoticallyperiodic solutions of neutral integral equationsrdquo ElectronicJournal of Qualitative Theory of Differential Equations vol 10pp 1ndash19 2000
[28] T A Burton and T Furumochi ldquoExistence theorems andperiodic solutions of neutral integral equationsrdquo NonlinearAnalysis Theory Methods and Applications vol 43 no 4 pp527ndash546 2001
[29] F Karakoc H Bereketoglu and G Seyhan ldquoOscillatory andperiodic solutions of impulsive differential equations withpiecewise constant argumentrdquoActa ApplicandaeMathematicaevol 110 no 1 pp 499ndash510 2010
[30] Y Liu P Yang and W Ge ldquoPeriodic solutions of higher-order delay differential equationsrdquo Nonlinear Analysis TheoryMethods and Applications vol 63 no 1 pp 136ndash152 2005
[31] X L Liu and W T Li ldquoPeriodic solutions for dynamic equa-tions on time scalesrdquo Nonlinear Analysis Theory Methods andApplications vol 67 no 5 pp 1457ndash1463 2007
[32] M Pinto ldquoDichotomy and existence of periodic solutions ofquasilinear functional differential equationsrdquo Nonlinear Analy-sis Theory Methods and Applications vol 72 no 3-4 pp 1227ndash1234 2010
[33] Y N Raffoul ldquoPeriodic solutions for neutral nonlinear differ-ential equations with functional delayrdquo Electronic Journal ofDifferential Equations vol 2003 no 102 pp 1ndash7 2003
[34] G Q Wang and S S Cheng ldquoPeriodic solutions of discreteRayleigh equations with deviating argumentsrdquo Taiwanese Jour-nal of Mathematics vol 13 no 6 pp 2051ndash2067 2009
[35] K S Chiu ldquoStability of oscillatory solutions of differential equa-tions with a general piecewise constant argumentrdquo ElectronicJournal of Qualitative Theory of Differential Equations vol 88pp 1ndash15 2011
[36] R N Butris and M A Aziz ldquoSome theorems in the existenceand uniqueness for system of nonlinear integro-differentialequationsrdquo Journal Education and Science vol 18 pp 76ndash892006
[37] A YMitropolsky andD IMortynyukPeriodic Solutions for theOscillations Systems with Retarded Argument General SchoolKiev Ukraine 1979
[38] L L Tai ldquoNumerical-analytic method for the investigation ofautonomous systems of differential equationsrdquo Ukrainskii Ma-tematicheskii Zhurnal vol 30 no 3 pp 309ndash317 1978
[39] Yu D Shlapak ldquoPeriodic solutions of ordinary differentialequations of first order that are not solved with respect to thederivativerdquo Ukrainian Mathematical Journal vol 32 no 5 pp420ndash425 1980
[40] A M Samoilenko ldquoNumerical-analytic method for the investi-gation of periodic systems of ordinary differential equationsrdquoUkrainskii Matematicheskii Zhurnal vol 17 no 4 pp 16ndash231965
[41] AM Samoilenko andN I RontoNumerical-Analytic Methodsfor Investigations of Periodic Solutions Mir Publishers MoscowRussia 1979
[42] R N Butris ldquoPeriodic solution of nonlinear system of integro-differential equations depending on the gamma distributionrdquoGeneral Mathematics Notes vol 15 no 1 pp 56ndash71 2013
[43] D R Smart Fixed Points Theorems Cambridge UniversityPress Cambridge UK 1980
[44] T A Burton ldquoKrasnoselskiirsquos inversion principle and fixedpointsrdquo Nonlinear Analysis Theory Methods and Applicationsvol 30 no 7 pp 3975ndash3986 1997
[45] T A Burton ldquoA fixed-point theorem of Krasnoselskiirdquo AppliedMathematics Letters vol 11 no 1 pp 85ndash88 1998
14 The Scientific World Journal
[46] M Pinto and D Sepulveda ldquoh-asymptotic stability by fixedpoint in neutral nonlinear differential equations with delayrdquoNonlinear Analysis Theory Methods and Applications vol 74no 12 pp 3926ndash3933 2011
[47] M A Krasnoselskii Translation Along Trajectories of Differen-tial Equations Nauka Moscow Russia 1966 (Russian) Englishtranslation American Mathematical Society Providence RIUSA 1968
For Lemma 3we can prove an important property double120596-periodicity of Greenrsquos function 119866(119905 119904) to study after theexistence of periodic solutions
Lemma 4 Suppose that the condition (119873120596) holds Then
Greenrsquos function119866(119905 119904) is double120596-periodic that is119866(119905+120596 119904+120596) = 119866(119905 119904)
3 Existence of Periodic Solutions
In this section we prove the main theorems of this paperso we recall the nonlinear integro-differential systems withDEPCAG (6)
Using Definition 2 Remark 5 and double 120596-periodicityof Greenrsquos function similar formula is given by (17) So wehave obtained the following result
Proposition 6 Suppose that the conditions (119873120596) and (P) hold
Let (120591 119911(120591)) isin R times R119899 Then 119911(119905) = 119911(119905 120591 119911(120591)) is 120596-periodicsolution on R of the DEPCAG system (6) if and only if 119911(119905) is120596-periodic solution of the integral equation
119911 (119905) = int
120591+120596
120591
119866 (119905 119904) [119891(119904 119911 (119904) int
It is easy to see that the DEPCAG system (6) has 120596-periodic solution if and only if the operator I has one fixedpoint in P
120596
To prove some existence criteria for 120596-periodic solutionsof the DEPCAG system (6) we use the Banach Schauder andKrasnoselskiirsquos fixed point theorems
Next we state first Krasnoselskiirsquos fixed point theoremwhich enables us to prove the existence of a periodic solutionFor the proof of Krasnoselskiirsquos fixed point theorem we referthe reader to [43]
Theorem A (Krasnoselskiirsquos fixed point theorem) Let 119878 bea closed convex nonempty subset of a Banach space (119864 sdot )Suppose thatA andBmap 119878 into 119864 such that
Then there exists 119911 isin 119878 with 119911 = A119911 +B119911
Remark 7 Krasnoselskiirsquos theorem may be combined withBanach and Schauderrsquos fixed point theorems In a certainsense we can interpret this as follows if a compact operatorhas the fixed point property under a small perturbationthen this property can be inherited The theorem is usefulin establishing the existence results for perturbed operatorequations It also has awide range of applications to nonlinearintegral equations of mixed type for proving the existence ofsolutions Thus the existence of fixed points for the sum oftwo operators has attracted tremendous interest and theirapplications are frequent in nonlinear analysis See [32 3343ndash46]
We note that to apply Krasnoselskiirsquos fixed point theoremwe need to construct two mappings one is contraction andthe other is compact Therefore we express (38) as
times1003817100381710038171003817120593 minus 120577
1003817100381710038171003817
le (119888119866int
120591+120596
120591
[1199011(119904) + 119901
2(119904)] 119889119904)
1003817100381710038171003817120593 minus 1205771003817100381710038171003817
le 1198881198661198711
1003817100381710038171003817120593 minus 1205771003817100381710038171003817
(44)
HenceA defines a contraction mapping
Similarly B is given by (41) which may be also acontraction operator
Lemma 9 If (119873120596) (119875) and (119871
119892) holdB is given by (41)with
1198881198661198712lt 1 and thenB is a contraction mapping
Lemma 10 If (119873120596) holdsB is defined by (41) and thenB is
completely continuous that isB is continuous and the imageofB is contained in a compact set
Proof Step 1 First we prove that B P120596
rarr P120596is
continuousAs the operator A a change of variable in (41) we have
(B120593)(119905 + 120596) = (B120593)(119905) Now we want to show B iscontinuous
The function 119892(119905 119909 119910) is uniformly continuous on [120591 120591 +120596] times CR times CR and by the periodicity in 119905 the function119892(119905 119909 119910) is uniformly continuous on R times CR times CR Thusfor any 1205981015840 = (120598119888
119866120596) gt 0 there exists 120575 = 120575(120598) gt 0 such
that 1199111 1199112isin S 119911
1minus 1199112 le 120575 implies |119892(119905 119911
1(119905) 1199111(120574(119905))) minus
119892(119905 1199112(119905) 1199112(120574(119905)))| le 120598
1015840 for 119905 isin [120591 120591 + 120596] Then B1199111minus
B1199112 le 120598 In fact by the continuity of 119892
1003816100381610038161003816119892 (119905 1199111 (119905) 1199111 (120574 (119905))) minus 119892 (119905 1199112 (119905) 1199112 (120574 (119905)))1003816100381610038161003816 le 1205981015840
for 119905 isin [120591 120591 + 120596](45)
8 The Scientific World Journal
and then1003817100381710038171003817B1199111minusB1199112
Step 2 We show that the image of B is contained in acompact set
Let 119909 119910 isin CR and 119904 isin [120591 120591 + 120596] for the continuity of thefunction 119892(119904 119909 119910) there exists119872 gt 0 such that |119892(119904 119909 119910)| le119872 Let 120593
119899isin S where 119899 is a positive integer then we have
We now see that all the conditions of Krasnoselskiirsquostheorem are satisfied Thus there exists a fixed point 119911 in S
such that 119911 = A119911 +B119911 By Proposition 6 this fixed point isa solution of the DEPCAG system (6) Hence the DEPCAGsystem (6) has 120596-periodic solution
By the symmetry of the conditions we will obtain asTheorem 12
Theorem 13 Suppose the hypothesis (119873120596) (119875) (119871
119892) (119872119891)
(119872119862) (119862) hold Let 119877 be a positive constant satisfying the
inequality
119888119866(1198621+ 1198712) 119877 + 119888
119866(120573 + 120578
2+ 1205881) 120596 le 119877 (51)
Then the DEPCAG system (6) has at least one 120596-periodicsolution in S
By Lemma 9 the mapping B is a contraction and it isclear that B P
120596rarr P
120596 Also from Lemma 11 A is
completely continuousNext we prove that if 120593 120577 isin S with 120593 le 119877 and 120577 le 119877
then A120601 +B120577 le 119877
Let 120593 120577 isin S with 120593 le 119877 and 120577 le 119877 Then
Krasnoselskii (see [47]) proved that if 119860 is a stable constantmatrix without piecewise alternately advanced and retardedargument and lim
|119909|+|119910|rarr+infin(|119892(119905 119909 119910)|(|119909|+|119910|)) = 0 then
the system (56) has at least one periodic solution In our caseapplyingTheorem 12 this result is also valid for the DEPCAGsystem (56) requiring the hypothesis (119873
120596) (1198753) (119872119892) 119860(119905)
and 119892(119905 119909 119910) are periodic functions in 119905 with a period 120596 forall 119905 ge 120591 and |119892(119905 119909 119910)| le 119888
1(|119909| + |119910|) + 120588
2 with 2119888
1120596 lt 119862
2
and 1205882constants for |119909| + |119910| le 119877 119877 gt 0
Remark 17 Considering the nonlinear system of differentialequations with a general piecewise alternately advanced andretarded argument
In this particular case applyingTheorem 13 this result is alsovalid for the DEPCAG system (57) requiring the hypothesis(119873120596) (1198751) (1198753) (119871119892) (119872119891) and 119888
119866(1198621+1198712)119877+119888119866(120573+1205881)120596 le 119877
with = 1
Remark 18 Suppose that (1198751) is satisfied by 120596 = 120596
1
(1198752) by 120596 = 120596
2 and (119875
3) by 120596 = 120596
3 if 120596
119894120596119895is a
rational number for all 119894 119895 = 1 2 3 then (1198751) (1198752) and
(1198753) are simultaneously satisfied by 120596 = lcm120596
1 1205962 1205963
where lcm1205961 1205962 1205963 denotes the least common multiple
between 1205961 1205962and 120596
3 In the general case it is possible
that there exist five possible periods 1205961for 119860 120596
2for 119891 120596
3
for 119892 1205964for 119862 and the sequences 119905
119894119894isinZ 120574119894119894isinZ satisfy the
(1205965 119901) condition If 120596
119894120596119895is a rational number for all 119894 119895 =
1 2 3 4 5 so in this situation our results insure the existenceof 120596-periodic solution with 120596 = lcm120596
1 1205962 1205963 1205964 1205965
Therefore the above results insure the existence of 120596-periodicsolutions of the DEPCAG system (6) These solutions arecalled subharmonic solutions See Corollaries 19ndash22
To determine criteria for the existence and uniqueness ofsubharmonic solutions of theDEPCAG system (6) fromnowon we make the following assumption
(119875120596) There exists 120596 = lcm120596
1 1205962 1205963 1205964 1205965 gt 0 120596
119894120596119895
which is a rational number for all 119894 119895 = 1 2 3 4 5
such that
(1) 119860(119905) 119891(119905 1199091 1199101) and 119892(119905 119909
2 1199102) are periodic
functions in 119905 with a period 1205961 1205962 and 120596
3
respectively for all 119905 ge 120591(2) 119862(119905 + 120596
4 119904 + 120596
4 1199093) = 119862(119905 119904 119909
3) for all 119905 ge 120591
(3) There exists 119901 isin Z+ for which the sequences119905119894119894isinZ 120574119894119894isinZ satisfy the (120596
5 119901) condition
As immediate corollaries of Theorems 12ndash15 and Remark 18the following results are true
The Scientific World Journal 11
Corollary 19 Suppose the hypotheses (119873120596) (119875120596) (119871119891) (119871119862)
(119872119892) and (49) hold Then the DEPCAG system (6) has at least
one subharmonic solution in S
Corollary 20 Suppose the hypotheses (119873120596) (119875120596) (119871119892) (119872119891)
(119872119862) (119862) and (51) hold Then the DEPCAG system (6) has at
least one subharmonic solution in S
Corollary 21 Suppose the hypotheses (119873120596) (119875120596) (119871119891) (119871119862)
(119871119892) and (53) holdThen theDEPCAG system (6) has a unique
subharmonic solution
Corollary 22 Suppose the hypotheses (119873120596) (119875120596) (119872
119891)
(119872119862) (119872119892) (119862) and (55) hold Then the DEPCAG system (6)
has at least one subharmonic solution in S
4 Applications and Illustrative Examples
Wewill introduce appropriate examples in this sectionTheseexamples will show the feasibility of our theory
Mathematical modelling of real-life problems usuallyresults in functional equations like ordinary or partial differ-ential equations integral and integro-differential equationsand stochastic equations Many mathematical formulationsof physical phenomena contain integro-differential equa-tions these equations arise inmany fields like fluid dynamicsbiologicalmodels and chemical kinetics So we first considernonlinear integro-differential equations with a general piece-wise constant argument mentioned in the introduction andobtain some new sufficient conditions for the existence of theperiodic solutions of these systems
Example 1 Let 120582 R2 rarr [0infin) and ℎ R2 rarr [0infin) betwo functions satisfying
sup119905isinR
int
119905+2120587
119905
120582 (119905 119904) 119889119904 le sup119905isinR
int
119905+2120587
119905
ℎ (119905 119904) 119889119904 le ℎ (58)
Consider the following nonlinear integro-differential equa-tions with piecewise alternately advanced and retarded argu-ment of generalized type
|119862(119905 119904 119910)| le 1198882(119877)120582(119905 119904)|119910| +ℎ(119905 119904) for |119910| le 119877 where
2120587 le 1198621
(iv) 119892 R times R119899 times R119899 rarr R119899 is continuous and 119862(119905 119904 119910)satisfies (119862)
Indeed for any 120576 gt 0 there exists 120575 gt 0 such that |1199101minus1199102| le 120575
implies
1003816100381610038161003816119862 (119905 119904 1199101) minus 119862 (119905 119904 1199102)1003816100381610038161003816 le 1205761198882 (119877) 120582 (119905 119904) for 119905 119904 isin R
(60)
Furthermore there exists 119877 such that
exp (2119886lowast120587)exp (2119886
lowast120587) minus 1
[(1198621+ 1198622) 119877 + 2120587 (120588
2+ ℎ)] le 119877 (61)
where 119886lowast = sup119905isinR |119886(119905)| and 119886
lowast= inf119905isinR|119886(119905)|
Then by Theorem 15 the DEPCAG system (59) has atleast one 2120587-periodic solution
Example 2 Thus many examples can be constructed whereour results can be applied
Let Λ R rarr R119899times119899 and 120583 R rarr R119899 be two functionssatisfying
sup119905isinR
int
119905+120596
119905
|Λ (119904)| 119889119904 = Λ lt infin
sup119905isinR
int
119905+120596
119905
1003816100381610038161003816120583 (119905 minus 119904)1003816100381610038161003816 119889119904 = 120583 lt infin
(62)
Now consider the integro-differential system with piecewisealternately advanced and retarded argument
Theorem 12 implies that there exists at least a 120596-periodicsolution of the DEPCAG system (63)
12 The Scientific World Journal
Note that similar results can be obtained under (119871119892)
and (119872119862) On the other hand the periodic situation of the
DEPCAGsystem (56) and (57) can be treated in the samewayLet us consider another example for second-order differ-
ential equations with a general piecewise constant argumentIn this case we can show the existence and uniquenessof periodic solutions of the following nonlinear DEPCAGsystem
Example 3 Consider the following nonlinear DEPCAG sys-tem
11991010158401015840
(119905) + (12058121199102
(119905) minus 2) 1199101015840
(119905) minus 8119910 (119905)
minus 1205811sin (120596119905) 1199102 (120574 (119905)) minus 120581
2cos (120596119905) = 0
(65)
where 1205811 1205812isin R 120574(119905) = 120574
Then for 120593 120595 isin S we have1003817100381710038171003817119892 (sdot 120593 (sdot) 120593 (120574 (sdot))) minus 119892 (sdot 120595 (sdot) 120595 (120574 (sdot)))
le 212058121198772 sup119905isin[120591120591+120596]
1003816100381610038161003816100381610038161003816(1205951 (119905) minus 1205931 (119905)
1205952 (119905) minus 1205932 (119905)
)
1003816100381610038161003816100381610038161003816= 212058121198772 1003817100381710038171003817120593 minus 120595
1003817100381710038171003817
(69)
In a similar way for 119891 we have1003817100381710038171003817119891 (sdot 120593 (sdot) 120593 (120574 (sdot))) minus 119891 (sdot 120595 (sdot) 120595 (120574 (sdot)))
1003817100381710038171003817
le 212058111198771003817100381710038171003817120593 minus 120595
1003817100381710038171003817
(70)
ByTheorem 14 the DEPCAG system (65) has a unique 2120587120596-periodic solution in S
Conflict of Interests
The author declares that there is no conflict of interests re-garding the publication of this paper
Acknowledgment
This research was in part supported by FIBE 01-12 DIUMCE
References
[1] ADMyshkis ldquoOn certain problems in the theory of differentialequations with deviating argumentsrdquo Uspekhi Matematich-eskikh Nauk vol 32 pp 173ndash202 1977
[2] K L Cooke and JWiener ldquoRetarded differential equations withpiecewise constant delaysrdquo Journal of Mathematical Analysisand Applications vol 99 no 1 pp 265ndash297 1984
[3] K L Cooke and J Wiener ldquoAn equation alternately of retardedand advanced typerdquo Proceedings of the American MathematicalSociety vol 99 pp 726ndash732 1987
[4] S M Shah and J Wiener ldquoAdvanced differential equations withpiecewise constant argument deviationsrdquo International JournalofMathematics andMathematical Sciences vol 6 no 4 pp 671ndash703 1983
[5] J WienerGeneralized Solutions of Functional Differential Equa-tions World Scientific Singapore 1993
[6] A R Aftabizadeh and J Wiener ldquoOscillatory and periodicsolutions of an equation alternately of retarded and advancedtypesrdquo Applicable Analysis vol 23 no 3 pp 219ndash231 1986
[7] A R Aftabizadeh J Wiener and J M Xu ldquoOscillatory andperiodic solutions of delay differential equations with piecewiseconstant argumentrdquo Proceedings of the American MathematicalSociety vol 99 pp 673ndash679 1987
[8] M U Akhmet ldquoIntegral manifolds of differential equationswith piecewise constant argument of generalized typerdquo Nonlin-ear Analysis Theory Methods and Applications vol 66 no 2pp 367ndash383 2007
[9] M U Akhmet Nonlinear Hybrid ContinuousDiscrete-TimeModels Atlantis Press Amsterdam The Netherlands 2011
[10] M U Akhmet C Buyukadali and T Ergenc ldquoPeriodic solu-tions of the hybrid system with small parameterrdquo NonlinearAnalysis Hybrid Systems vol 2 no 2 pp 532ndash543 2008
[11] A Cabada J B Ferreiro and J J Nieto ldquoGreenrsquos functionand comparison principles for first order periodic differentialequations with piecewise constant argumentsrdquo Journal of Math-ematical Analysis and Applications vol 291 no 2 pp 690ndash6972004
[12] K S Chiu and M Pinto ldquoVariation of parameters formula andGronwall inequality for differential equations with a generalpiecewise constant argumentrdquo Acta Mathematicae ApplicataeSinica vol 27 no 4 pp 561ndash568 2011
The Scientific World Journal 13
[13] K S Chiu and M Pinto ldquoOscillatory and periodic solutionsin alternately advanced and delayed differential equationsrdquoCarpathian Journal of Mathematics vol 29 no 2 pp 149ndash1582013
[14] J J Nieto and R Rodrıguez-Lopez ldquoGreenrsquos function forsecond-order periodic boundary value problemswith piecewiseconstant argumentsrdquo Journal of Mathematical Analysis andApplications vol 304 no 1 pp 33ndash57 2005
[15] M Pinto ldquoAsymptotic equivalence of nonlinear and quasi lin-ear differential equations with piecewise constant argumentsrdquoMathematical and Computer Modelling vol 49 no 9-10 pp1750ndash1758 2009
[16] M Pinto ldquoCauchy and Green matrices type and stability inalternately advanced and delayed differential systemsrdquo Journalof Difference Equations and Applications vol 17 no 2 pp 235ndash254 2011
[17] G Q Wang ldquoExistence theorem of periodic solutions for adelay nonlinear differential equation with piecewise constantargumentsrdquo Journal of Mathematical Analysis and Applicationsvol 298 no 1 pp 298ndash307 2004
[18] G Q Wang and S S Cheng ldquoExistence and uniqueness ofperiodic solutions for a second-order nonlinear differentialequationwith piecewise constant argumentrdquo International Jour-nal ofMathematics andMathematical Sciences vol 2009 ArticleID 950797 14 pages 2009
[19] Y H Xia Z Huang and M Han ldquoExistence of almost periodicsolutions for forced perturbed systems with piecewise constantargumentrdquo Journal of Mathematical Analysis and Applicationsvol 333 no 2 pp 798ndash816 2007
[20] P Yang Y Liu and W Ge ldquoGreenrsquos function for second orderdifferential equations with piecewise constant argumentsrdquoNon-linear Analysis Theory Methods and Applications vol 64 no 8pp 1812ndash1830 2006
[21] R Yuan ldquoThe existence of almost periodic solutions of retardeddifferential equations with piecewise constant argumentrdquo Non-linear Analysis Theory Methods and Applications vol 48 no 7pp 1013ndash1032 2002
[22] A I Alonso J Hong and R Obaya ldquoAlmost periodic typesolutions of differential equations with piecewise constant argu-ment via almost periodic type sequencesrdquo Applied MathematicsLetters vol 13 no 2 pp 131ndash137 2000
[23] K S Chiu and M Pinto ldquoPeriodic solutions of differentialequations with a general piecewise constant argument andapplicationsrdquo Electronic Journal of Qualitative Theory of Differ-ential Equations vol 46 pp 1ndash19 2010
[24] K S Chiu M Pinto and J C Jeng ldquoExistence and globalconvergence of periodic solutions in recurrent neural networkmodels with a general piecewise alternately advanced andretarded argumentrdquo Acta Applicandae Mathematicae 2013
[25] L Dai Nonlinear Dynamics of Piecewise of Constant Systemsand Implememtation of Piecewise Constants Arguments WorldScientific Singapore 2008
[26] T A Burton Stability and Periodic Solutions of Ordinary andFunctional Differential Equations Academic Press New YorkNY USA 1985
[27] T A Burton and T Furumochi ldquoPeriodic and asymptoticallyperiodic solutions of neutral integral equationsrdquo ElectronicJournal of Qualitative Theory of Differential Equations vol 10pp 1ndash19 2000
[28] T A Burton and T Furumochi ldquoExistence theorems andperiodic solutions of neutral integral equationsrdquo NonlinearAnalysis Theory Methods and Applications vol 43 no 4 pp527ndash546 2001
[29] F Karakoc H Bereketoglu and G Seyhan ldquoOscillatory andperiodic solutions of impulsive differential equations withpiecewise constant argumentrdquoActa ApplicandaeMathematicaevol 110 no 1 pp 499ndash510 2010
[30] Y Liu P Yang and W Ge ldquoPeriodic solutions of higher-order delay differential equationsrdquo Nonlinear Analysis TheoryMethods and Applications vol 63 no 1 pp 136ndash152 2005
[31] X L Liu and W T Li ldquoPeriodic solutions for dynamic equa-tions on time scalesrdquo Nonlinear Analysis Theory Methods andApplications vol 67 no 5 pp 1457ndash1463 2007
[32] M Pinto ldquoDichotomy and existence of periodic solutions ofquasilinear functional differential equationsrdquo Nonlinear Analy-sis Theory Methods and Applications vol 72 no 3-4 pp 1227ndash1234 2010
[33] Y N Raffoul ldquoPeriodic solutions for neutral nonlinear differ-ential equations with functional delayrdquo Electronic Journal ofDifferential Equations vol 2003 no 102 pp 1ndash7 2003
[34] G Q Wang and S S Cheng ldquoPeriodic solutions of discreteRayleigh equations with deviating argumentsrdquo Taiwanese Jour-nal of Mathematics vol 13 no 6 pp 2051ndash2067 2009
[35] K S Chiu ldquoStability of oscillatory solutions of differential equa-tions with a general piecewise constant argumentrdquo ElectronicJournal of Qualitative Theory of Differential Equations vol 88pp 1ndash15 2011
[36] R N Butris and M A Aziz ldquoSome theorems in the existenceand uniqueness for system of nonlinear integro-differentialequationsrdquo Journal Education and Science vol 18 pp 76ndash892006
[37] A YMitropolsky andD IMortynyukPeriodic Solutions for theOscillations Systems with Retarded Argument General SchoolKiev Ukraine 1979
[38] L L Tai ldquoNumerical-analytic method for the investigation ofautonomous systems of differential equationsrdquo Ukrainskii Ma-tematicheskii Zhurnal vol 30 no 3 pp 309ndash317 1978
[39] Yu D Shlapak ldquoPeriodic solutions of ordinary differentialequations of first order that are not solved with respect to thederivativerdquo Ukrainian Mathematical Journal vol 32 no 5 pp420ndash425 1980
[40] A M Samoilenko ldquoNumerical-analytic method for the investi-gation of periodic systems of ordinary differential equationsrdquoUkrainskii Matematicheskii Zhurnal vol 17 no 4 pp 16ndash231965
[41] AM Samoilenko andN I RontoNumerical-Analytic Methodsfor Investigations of Periodic Solutions Mir Publishers MoscowRussia 1979
[42] R N Butris ldquoPeriodic solution of nonlinear system of integro-differential equations depending on the gamma distributionrdquoGeneral Mathematics Notes vol 15 no 1 pp 56ndash71 2013
[43] D R Smart Fixed Points Theorems Cambridge UniversityPress Cambridge UK 1980
[44] T A Burton ldquoKrasnoselskiirsquos inversion principle and fixedpointsrdquo Nonlinear Analysis Theory Methods and Applicationsvol 30 no 7 pp 3975ndash3986 1997
[45] T A Burton ldquoA fixed-point theorem of Krasnoselskiirdquo AppliedMathematics Letters vol 11 no 1 pp 85ndash88 1998
14 The Scientific World Journal
[46] M Pinto and D Sepulveda ldquoh-asymptotic stability by fixedpoint in neutral nonlinear differential equations with delayrdquoNonlinear Analysis Theory Methods and Applications vol 74no 12 pp 3926ndash3933 2011
[47] M A Krasnoselskii Translation Along Trajectories of Differen-tial Equations Nauka Moscow Russia 1966 (Russian) Englishtranslation American Mathematical Society Providence RIUSA 1968
Using Definition 2 Remark 5 and double 120596-periodicityof Greenrsquos function similar formula is given by (17) So wehave obtained the following result
Proposition 6 Suppose that the conditions (119873120596) and (P) hold
Let (120591 119911(120591)) isin R times R119899 Then 119911(119905) = 119911(119905 120591 119911(120591)) is 120596-periodicsolution on R of the DEPCAG system (6) if and only if 119911(119905) is120596-periodic solution of the integral equation
119911 (119905) = int
120591+120596
120591
119866 (119905 119904) [119891(119904 119911 (119904) int
It is easy to see that the DEPCAG system (6) has 120596-periodic solution if and only if the operator I has one fixedpoint in P
120596
To prove some existence criteria for 120596-periodic solutionsof the DEPCAG system (6) we use the Banach Schauder andKrasnoselskiirsquos fixed point theorems
Next we state first Krasnoselskiirsquos fixed point theoremwhich enables us to prove the existence of a periodic solutionFor the proof of Krasnoselskiirsquos fixed point theorem we referthe reader to [43]
Theorem A (Krasnoselskiirsquos fixed point theorem) Let 119878 bea closed convex nonempty subset of a Banach space (119864 sdot )Suppose thatA andBmap 119878 into 119864 such that
Then there exists 119911 isin 119878 with 119911 = A119911 +B119911
Remark 7 Krasnoselskiirsquos theorem may be combined withBanach and Schauderrsquos fixed point theorems In a certainsense we can interpret this as follows if a compact operatorhas the fixed point property under a small perturbationthen this property can be inherited The theorem is usefulin establishing the existence results for perturbed operatorequations It also has awide range of applications to nonlinearintegral equations of mixed type for proving the existence ofsolutions Thus the existence of fixed points for the sum oftwo operators has attracted tremendous interest and theirapplications are frequent in nonlinear analysis See [32 3343ndash46]
We note that to apply Krasnoselskiirsquos fixed point theoremwe need to construct two mappings one is contraction andthe other is compact Therefore we express (38) as
times1003817100381710038171003817120593 minus 120577
1003817100381710038171003817
le (119888119866int
120591+120596
120591
[1199011(119904) + 119901
2(119904)] 119889119904)
1003817100381710038171003817120593 minus 1205771003817100381710038171003817
le 1198881198661198711
1003817100381710038171003817120593 minus 1205771003817100381710038171003817
(44)
HenceA defines a contraction mapping
Similarly B is given by (41) which may be also acontraction operator
Lemma 9 If (119873120596) (119875) and (119871
119892) holdB is given by (41)with
1198881198661198712lt 1 and thenB is a contraction mapping
Lemma 10 If (119873120596) holdsB is defined by (41) and thenB is
completely continuous that isB is continuous and the imageofB is contained in a compact set
Proof Step 1 First we prove that B P120596
rarr P120596is
continuousAs the operator A a change of variable in (41) we have
(B120593)(119905 + 120596) = (B120593)(119905) Now we want to show B iscontinuous
The function 119892(119905 119909 119910) is uniformly continuous on [120591 120591 +120596] times CR times CR and by the periodicity in 119905 the function119892(119905 119909 119910) is uniformly continuous on R times CR times CR Thusfor any 1205981015840 = (120598119888
119866120596) gt 0 there exists 120575 = 120575(120598) gt 0 such
that 1199111 1199112isin S 119911
1minus 1199112 le 120575 implies |119892(119905 119911
1(119905) 1199111(120574(119905))) minus
119892(119905 1199112(119905) 1199112(120574(119905)))| le 120598
1015840 for 119905 isin [120591 120591 + 120596] Then B1199111minus
B1199112 le 120598 In fact by the continuity of 119892
1003816100381610038161003816119892 (119905 1199111 (119905) 1199111 (120574 (119905))) minus 119892 (119905 1199112 (119905) 1199112 (120574 (119905)))1003816100381610038161003816 le 1205981015840
for 119905 isin [120591 120591 + 120596](45)
8 The Scientific World Journal
and then1003817100381710038171003817B1199111minusB1199112
Step 2 We show that the image of B is contained in acompact set
Let 119909 119910 isin CR and 119904 isin [120591 120591 + 120596] for the continuity of thefunction 119892(119904 119909 119910) there exists119872 gt 0 such that |119892(119904 119909 119910)| le119872 Let 120593
119899isin S where 119899 is a positive integer then we have
We now see that all the conditions of Krasnoselskiirsquostheorem are satisfied Thus there exists a fixed point 119911 in S
such that 119911 = A119911 +B119911 By Proposition 6 this fixed point isa solution of the DEPCAG system (6) Hence the DEPCAGsystem (6) has 120596-periodic solution
By the symmetry of the conditions we will obtain asTheorem 12
Theorem 13 Suppose the hypothesis (119873120596) (119875) (119871
119892) (119872119891)
(119872119862) (119862) hold Let 119877 be a positive constant satisfying the
inequality
119888119866(1198621+ 1198712) 119877 + 119888
119866(120573 + 120578
2+ 1205881) 120596 le 119877 (51)
Then the DEPCAG system (6) has at least one 120596-periodicsolution in S
By Lemma 9 the mapping B is a contraction and it isclear that B P
120596rarr P
120596 Also from Lemma 11 A is
completely continuousNext we prove that if 120593 120577 isin S with 120593 le 119877 and 120577 le 119877
then A120601 +B120577 le 119877
Let 120593 120577 isin S with 120593 le 119877 and 120577 le 119877 Then
Krasnoselskii (see [47]) proved that if 119860 is a stable constantmatrix without piecewise alternately advanced and retardedargument and lim
|119909|+|119910|rarr+infin(|119892(119905 119909 119910)|(|119909|+|119910|)) = 0 then
the system (56) has at least one periodic solution In our caseapplyingTheorem 12 this result is also valid for the DEPCAGsystem (56) requiring the hypothesis (119873
120596) (1198753) (119872119892) 119860(119905)
and 119892(119905 119909 119910) are periodic functions in 119905 with a period 120596 forall 119905 ge 120591 and |119892(119905 119909 119910)| le 119888
1(|119909| + |119910|) + 120588
2 with 2119888
1120596 lt 119862
2
and 1205882constants for |119909| + |119910| le 119877 119877 gt 0
Remark 17 Considering the nonlinear system of differentialequations with a general piecewise alternately advanced andretarded argument
In this particular case applyingTheorem 13 this result is alsovalid for the DEPCAG system (57) requiring the hypothesis(119873120596) (1198751) (1198753) (119871119892) (119872119891) and 119888
119866(1198621+1198712)119877+119888119866(120573+1205881)120596 le 119877
with = 1
Remark 18 Suppose that (1198751) is satisfied by 120596 = 120596
1
(1198752) by 120596 = 120596
2 and (119875
3) by 120596 = 120596
3 if 120596
119894120596119895is a
rational number for all 119894 119895 = 1 2 3 then (1198751) (1198752) and
(1198753) are simultaneously satisfied by 120596 = lcm120596
1 1205962 1205963
where lcm1205961 1205962 1205963 denotes the least common multiple
between 1205961 1205962and 120596
3 In the general case it is possible
that there exist five possible periods 1205961for 119860 120596
2for 119891 120596
3
for 119892 1205964for 119862 and the sequences 119905
119894119894isinZ 120574119894119894isinZ satisfy the
(1205965 119901) condition If 120596
119894120596119895is a rational number for all 119894 119895 =
1 2 3 4 5 so in this situation our results insure the existenceof 120596-periodic solution with 120596 = lcm120596
1 1205962 1205963 1205964 1205965
Therefore the above results insure the existence of 120596-periodicsolutions of the DEPCAG system (6) These solutions arecalled subharmonic solutions See Corollaries 19ndash22
To determine criteria for the existence and uniqueness ofsubharmonic solutions of theDEPCAG system (6) fromnowon we make the following assumption
(119875120596) There exists 120596 = lcm120596
1 1205962 1205963 1205964 1205965 gt 0 120596
119894120596119895
which is a rational number for all 119894 119895 = 1 2 3 4 5
such that
(1) 119860(119905) 119891(119905 1199091 1199101) and 119892(119905 119909
2 1199102) are periodic
functions in 119905 with a period 1205961 1205962 and 120596
3
respectively for all 119905 ge 120591(2) 119862(119905 + 120596
4 119904 + 120596
4 1199093) = 119862(119905 119904 119909
3) for all 119905 ge 120591
(3) There exists 119901 isin Z+ for which the sequences119905119894119894isinZ 120574119894119894isinZ satisfy the (120596
5 119901) condition
As immediate corollaries of Theorems 12ndash15 and Remark 18the following results are true
The Scientific World Journal 11
Corollary 19 Suppose the hypotheses (119873120596) (119875120596) (119871119891) (119871119862)
(119872119892) and (49) hold Then the DEPCAG system (6) has at least
one subharmonic solution in S
Corollary 20 Suppose the hypotheses (119873120596) (119875120596) (119871119892) (119872119891)
(119872119862) (119862) and (51) hold Then the DEPCAG system (6) has at
least one subharmonic solution in S
Corollary 21 Suppose the hypotheses (119873120596) (119875120596) (119871119891) (119871119862)
(119871119892) and (53) holdThen theDEPCAG system (6) has a unique
subharmonic solution
Corollary 22 Suppose the hypotheses (119873120596) (119875120596) (119872
119891)
(119872119862) (119872119892) (119862) and (55) hold Then the DEPCAG system (6)
has at least one subharmonic solution in S
4 Applications and Illustrative Examples
Wewill introduce appropriate examples in this sectionTheseexamples will show the feasibility of our theory
Mathematical modelling of real-life problems usuallyresults in functional equations like ordinary or partial differ-ential equations integral and integro-differential equationsand stochastic equations Many mathematical formulationsof physical phenomena contain integro-differential equa-tions these equations arise inmany fields like fluid dynamicsbiologicalmodels and chemical kinetics So we first considernonlinear integro-differential equations with a general piece-wise constant argument mentioned in the introduction andobtain some new sufficient conditions for the existence of theperiodic solutions of these systems
Example 1 Let 120582 R2 rarr [0infin) and ℎ R2 rarr [0infin) betwo functions satisfying
sup119905isinR
int
119905+2120587
119905
120582 (119905 119904) 119889119904 le sup119905isinR
int
119905+2120587
119905
ℎ (119905 119904) 119889119904 le ℎ (58)
Consider the following nonlinear integro-differential equa-tions with piecewise alternately advanced and retarded argu-ment of generalized type
|119862(119905 119904 119910)| le 1198882(119877)120582(119905 119904)|119910| +ℎ(119905 119904) for |119910| le 119877 where
2120587 le 1198621
(iv) 119892 R times R119899 times R119899 rarr R119899 is continuous and 119862(119905 119904 119910)satisfies (119862)
Indeed for any 120576 gt 0 there exists 120575 gt 0 such that |1199101minus1199102| le 120575
implies
1003816100381610038161003816119862 (119905 119904 1199101) minus 119862 (119905 119904 1199102)1003816100381610038161003816 le 1205761198882 (119877) 120582 (119905 119904) for 119905 119904 isin R
(60)
Furthermore there exists 119877 such that
exp (2119886lowast120587)exp (2119886
lowast120587) minus 1
[(1198621+ 1198622) 119877 + 2120587 (120588
2+ ℎ)] le 119877 (61)
where 119886lowast = sup119905isinR |119886(119905)| and 119886
lowast= inf119905isinR|119886(119905)|
Then by Theorem 15 the DEPCAG system (59) has atleast one 2120587-periodic solution
Example 2 Thus many examples can be constructed whereour results can be applied
Let Λ R rarr R119899times119899 and 120583 R rarr R119899 be two functionssatisfying
sup119905isinR
int
119905+120596
119905
|Λ (119904)| 119889119904 = Λ lt infin
sup119905isinR
int
119905+120596
119905
1003816100381610038161003816120583 (119905 minus 119904)1003816100381610038161003816 119889119904 = 120583 lt infin
(62)
Now consider the integro-differential system with piecewisealternately advanced and retarded argument
Theorem 12 implies that there exists at least a 120596-periodicsolution of the DEPCAG system (63)
12 The Scientific World Journal
Note that similar results can be obtained under (119871119892)
and (119872119862) On the other hand the periodic situation of the
DEPCAGsystem (56) and (57) can be treated in the samewayLet us consider another example for second-order differ-
ential equations with a general piecewise constant argumentIn this case we can show the existence and uniquenessof periodic solutions of the following nonlinear DEPCAGsystem
Example 3 Consider the following nonlinear DEPCAG sys-tem
11991010158401015840
(119905) + (12058121199102
(119905) minus 2) 1199101015840
(119905) minus 8119910 (119905)
minus 1205811sin (120596119905) 1199102 (120574 (119905)) minus 120581
2cos (120596119905) = 0
(65)
where 1205811 1205812isin R 120574(119905) = 120574
Then for 120593 120595 isin S we have1003817100381710038171003817119892 (sdot 120593 (sdot) 120593 (120574 (sdot))) minus 119892 (sdot 120595 (sdot) 120595 (120574 (sdot)))
le 212058121198772 sup119905isin[120591120591+120596]
1003816100381610038161003816100381610038161003816(1205951 (119905) minus 1205931 (119905)
1205952 (119905) minus 1205932 (119905)
)
1003816100381610038161003816100381610038161003816= 212058121198772 1003817100381710038171003817120593 minus 120595
1003817100381710038171003817
(69)
In a similar way for 119891 we have1003817100381710038171003817119891 (sdot 120593 (sdot) 120593 (120574 (sdot))) minus 119891 (sdot 120595 (sdot) 120595 (120574 (sdot)))
1003817100381710038171003817
le 212058111198771003817100381710038171003817120593 minus 120595
1003817100381710038171003817
(70)
ByTheorem 14 the DEPCAG system (65) has a unique 2120587120596-periodic solution in S
Conflict of Interests
The author declares that there is no conflict of interests re-garding the publication of this paper
Acknowledgment
This research was in part supported by FIBE 01-12 DIUMCE
References
[1] ADMyshkis ldquoOn certain problems in the theory of differentialequations with deviating argumentsrdquo Uspekhi Matematich-eskikh Nauk vol 32 pp 173ndash202 1977
[2] K L Cooke and JWiener ldquoRetarded differential equations withpiecewise constant delaysrdquo Journal of Mathematical Analysisand Applications vol 99 no 1 pp 265ndash297 1984
[3] K L Cooke and J Wiener ldquoAn equation alternately of retardedand advanced typerdquo Proceedings of the American MathematicalSociety vol 99 pp 726ndash732 1987
[4] S M Shah and J Wiener ldquoAdvanced differential equations withpiecewise constant argument deviationsrdquo International JournalofMathematics andMathematical Sciences vol 6 no 4 pp 671ndash703 1983
[5] J WienerGeneralized Solutions of Functional Differential Equa-tions World Scientific Singapore 1993
[6] A R Aftabizadeh and J Wiener ldquoOscillatory and periodicsolutions of an equation alternately of retarded and advancedtypesrdquo Applicable Analysis vol 23 no 3 pp 219ndash231 1986
[7] A R Aftabizadeh J Wiener and J M Xu ldquoOscillatory andperiodic solutions of delay differential equations with piecewiseconstant argumentrdquo Proceedings of the American MathematicalSociety vol 99 pp 673ndash679 1987
[8] M U Akhmet ldquoIntegral manifolds of differential equationswith piecewise constant argument of generalized typerdquo Nonlin-ear Analysis Theory Methods and Applications vol 66 no 2pp 367ndash383 2007
[9] M U Akhmet Nonlinear Hybrid ContinuousDiscrete-TimeModels Atlantis Press Amsterdam The Netherlands 2011
[10] M U Akhmet C Buyukadali and T Ergenc ldquoPeriodic solu-tions of the hybrid system with small parameterrdquo NonlinearAnalysis Hybrid Systems vol 2 no 2 pp 532ndash543 2008
[11] A Cabada J B Ferreiro and J J Nieto ldquoGreenrsquos functionand comparison principles for first order periodic differentialequations with piecewise constant argumentsrdquo Journal of Math-ematical Analysis and Applications vol 291 no 2 pp 690ndash6972004
[12] K S Chiu and M Pinto ldquoVariation of parameters formula andGronwall inequality for differential equations with a generalpiecewise constant argumentrdquo Acta Mathematicae ApplicataeSinica vol 27 no 4 pp 561ndash568 2011
The Scientific World Journal 13
[13] K S Chiu and M Pinto ldquoOscillatory and periodic solutionsin alternately advanced and delayed differential equationsrdquoCarpathian Journal of Mathematics vol 29 no 2 pp 149ndash1582013
[14] J J Nieto and R Rodrıguez-Lopez ldquoGreenrsquos function forsecond-order periodic boundary value problemswith piecewiseconstant argumentsrdquo Journal of Mathematical Analysis andApplications vol 304 no 1 pp 33ndash57 2005
[15] M Pinto ldquoAsymptotic equivalence of nonlinear and quasi lin-ear differential equations with piecewise constant argumentsrdquoMathematical and Computer Modelling vol 49 no 9-10 pp1750ndash1758 2009
[16] M Pinto ldquoCauchy and Green matrices type and stability inalternately advanced and delayed differential systemsrdquo Journalof Difference Equations and Applications vol 17 no 2 pp 235ndash254 2011
[17] G Q Wang ldquoExistence theorem of periodic solutions for adelay nonlinear differential equation with piecewise constantargumentsrdquo Journal of Mathematical Analysis and Applicationsvol 298 no 1 pp 298ndash307 2004
[18] G Q Wang and S S Cheng ldquoExistence and uniqueness ofperiodic solutions for a second-order nonlinear differentialequationwith piecewise constant argumentrdquo International Jour-nal ofMathematics andMathematical Sciences vol 2009 ArticleID 950797 14 pages 2009
[19] Y H Xia Z Huang and M Han ldquoExistence of almost periodicsolutions for forced perturbed systems with piecewise constantargumentrdquo Journal of Mathematical Analysis and Applicationsvol 333 no 2 pp 798ndash816 2007
[20] P Yang Y Liu and W Ge ldquoGreenrsquos function for second orderdifferential equations with piecewise constant argumentsrdquoNon-linear Analysis Theory Methods and Applications vol 64 no 8pp 1812ndash1830 2006
[21] R Yuan ldquoThe existence of almost periodic solutions of retardeddifferential equations with piecewise constant argumentrdquo Non-linear Analysis Theory Methods and Applications vol 48 no 7pp 1013ndash1032 2002
[22] A I Alonso J Hong and R Obaya ldquoAlmost periodic typesolutions of differential equations with piecewise constant argu-ment via almost periodic type sequencesrdquo Applied MathematicsLetters vol 13 no 2 pp 131ndash137 2000
[23] K S Chiu and M Pinto ldquoPeriodic solutions of differentialequations with a general piecewise constant argument andapplicationsrdquo Electronic Journal of Qualitative Theory of Differ-ential Equations vol 46 pp 1ndash19 2010
[24] K S Chiu M Pinto and J C Jeng ldquoExistence and globalconvergence of periodic solutions in recurrent neural networkmodels with a general piecewise alternately advanced andretarded argumentrdquo Acta Applicandae Mathematicae 2013
[25] L Dai Nonlinear Dynamics of Piecewise of Constant Systemsand Implememtation of Piecewise Constants Arguments WorldScientific Singapore 2008
[26] T A Burton Stability and Periodic Solutions of Ordinary andFunctional Differential Equations Academic Press New YorkNY USA 1985
[27] T A Burton and T Furumochi ldquoPeriodic and asymptoticallyperiodic solutions of neutral integral equationsrdquo ElectronicJournal of Qualitative Theory of Differential Equations vol 10pp 1ndash19 2000
[28] T A Burton and T Furumochi ldquoExistence theorems andperiodic solutions of neutral integral equationsrdquo NonlinearAnalysis Theory Methods and Applications vol 43 no 4 pp527ndash546 2001
[29] F Karakoc H Bereketoglu and G Seyhan ldquoOscillatory andperiodic solutions of impulsive differential equations withpiecewise constant argumentrdquoActa ApplicandaeMathematicaevol 110 no 1 pp 499ndash510 2010
[30] Y Liu P Yang and W Ge ldquoPeriodic solutions of higher-order delay differential equationsrdquo Nonlinear Analysis TheoryMethods and Applications vol 63 no 1 pp 136ndash152 2005
[31] X L Liu and W T Li ldquoPeriodic solutions for dynamic equa-tions on time scalesrdquo Nonlinear Analysis Theory Methods andApplications vol 67 no 5 pp 1457ndash1463 2007
[32] M Pinto ldquoDichotomy and existence of periodic solutions ofquasilinear functional differential equationsrdquo Nonlinear Analy-sis Theory Methods and Applications vol 72 no 3-4 pp 1227ndash1234 2010
[33] Y N Raffoul ldquoPeriodic solutions for neutral nonlinear differ-ential equations with functional delayrdquo Electronic Journal ofDifferential Equations vol 2003 no 102 pp 1ndash7 2003
[34] G Q Wang and S S Cheng ldquoPeriodic solutions of discreteRayleigh equations with deviating argumentsrdquo Taiwanese Jour-nal of Mathematics vol 13 no 6 pp 2051ndash2067 2009
[35] K S Chiu ldquoStability of oscillatory solutions of differential equa-tions with a general piecewise constant argumentrdquo ElectronicJournal of Qualitative Theory of Differential Equations vol 88pp 1ndash15 2011
[36] R N Butris and M A Aziz ldquoSome theorems in the existenceand uniqueness for system of nonlinear integro-differentialequationsrdquo Journal Education and Science vol 18 pp 76ndash892006
[37] A YMitropolsky andD IMortynyukPeriodic Solutions for theOscillations Systems with Retarded Argument General SchoolKiev Ukraine 1979
[38] L L Tai ldquoNumerical-analytic method for the investigation ofautonomous systems of differential equationsrdquo Ukrainskii Ma-tematicheskii Zhurnal vol 30 no 3 pp 309ndash317 1978
[39] Yu D Shlapak ldquoPeriodic solutions of ordinary differentialequations of first order that are not solved with respect to thederivativerdquo Ukrainian Mathematical Journal vol 32 no 5 pp420ndash425 1980
[40] A M Samoilenko ldquoNumerical-analytic method for the investi-gation of periodic systems of ordinary differential equationsrdquoUkrainskii Matematicheskii Zhurnal vol 17 no 4 pp 16ndash231965
[41] AM Samoilenko andN I RontoNumerical-Analytic Methodsfor Investigations of Periodic Solutions Mir Publishers MoscowRussia 1979
[42] R N Butris ldquoPeriodic solution of nonlinear system of integro-differential equations depending on the gamma distributionrdquoGeneral Mathematics Notes vol 15 no 1 pp 56ndash71 2013
[43] D R Smart Fixed Points Theorems Cambridge UniversityPress Cambridge UK 1980
[44] T A Burton ldquoKrasnoselskiirsquos inversion principle and fixedpointsrdquo Nonlinear Analysis Theory Methods and Applicationsvol 30 no 7 pp 3975ndash3986 1997
[45] T A Burton ldquoA fixed-point theorem of Krasnoselskiirdquo AppliedMathematics Letters vol 11 no 1 pp 85ndash88 1998
14 The Scientific World Journal
[46] M Pinto and D Sepulveda ldquoh-asymptotic stability by fixedpoint in neutral nonlinear differential equations with delayrdquoNonlinear Analysis Theory Methods and Applications vol 74no 12 pp 3926ndash3933 2011
[47] M A Krasnoselskii Translation Along Trajectories of Differen-tial Equations Nauka Moscow Russia 1966 (Russian) Englishtranslation American Mathematical Society Providence RIUSA 1968
times1003817100381710038171003817120593 minus 120577
1003817100381710038171003817
le (119888119866int
120591+120596
120591
[1199011(119904) + 119901
2(119904)] 119889119904)
1003817100381710038171003817120593 minus 1205771003817100381710038171003817
le 1198881198661198711
1003817100381710038171003817120593 minus 1205771003817100381710038171003817
(44)
HenceA defines a contraction mapping
Similarly B is given by (41) which may be also acontraction operator
Lemma 9 If (119873120596) (119875) and (119871
119892) holdB is given by (41)with
1198881198661198712lt 1 and thenB is a contraction mapping
Lemma 10 If (119873120596) holdsB is defined by (41) and thenB is
completely continuous that isB is continuous and the imageofB is contained in a compact set
Proof Step 1 First we prove that B P120596
rarr P120596is
continuousAs the operator A a change of variable in (41) we have
(B120593)(119905 + 120596) = (B120593)(119905) Now we want to show B iscontinuous
The function 119892(119905 119909 119910) is uniformly continuous on [120591 120591 +120596] times CR times CR and by the periodicity in 119905 the function119892(119905 119909 119910) is uniformly continuous on R times CR times CR Thusfor any 1205981015840 = (120598119888
119866120596) gt 0 there exists 120575 = 120575(120598) gt 0 such
that 1199111 1199112isin S 119911
1minus 1199112 le 120575 implies |119892(119905 119911
1(119905) 1199111(120574(119905))) minus
119892(119905 1199112(119905) 1199112(120574(119905)))| le 120598
1015840 for 119905 isin [120591 120591 + 120596] Then B1199111minus
B1199112 le 120598 In fact by the continuity of 119892
1003816100381610038161003816119892 (119905 1199111 (119905) 1199111 (120574 (119905))) minus 119892 (119905 1199112 (119905) 1199112 (120574 (119905)))1003816100381610038161003816 le 1205981015840
for 119905 isin [120591 120591 + 120596](45)
8 The Scientific World Journal
and then1003817100381710038171003817B1199111minusB1199112
Step 2 We show that the image of B is contained in acompact set
Let 119909 119910 isin CR and 119904 isin [120591 120591 + 120596] for the continuity of thefunction 119892(119904 119909 119910) there exists119872 gt 0 such that |119892(119904 119909 119910)| le119872 Let 120593
119899isin S where 119899 is a positive integer then we have
We now see that all the conditions of Krasnoselskiirsquostheorem are satisfied Thus there exists a fixed point 119911 in S
such that 119911 = A119911 +B119911 By Proposition 6 this fixed point isa solution of the DEPCAG system (6) Hence the DEPCAGsystem (6) has 120596-periodic solution
By the symmetry of the conditions we will obtain asTheorem 12
Theorem 13 Suppose the hypothesis (119873120596) (119875) (119871
119892) (119872119891)
(119872119862) (119862) hold Let 119877 be a positive constant satisfying the
inequality
119888119866(1198621+ 1198712) 119877 + 119888
119866(120573 + 120578
2+ 1205881) 120596 le 119877 (51)
Then the DEPCAG system (6) has at least one 120596-periodicsolution in S
By Lemma 9 the mapping B is a contraction and it isclear that B P
120596rarr P
120596 Also from Lemma 11 A is
completely continuousNext we prove that if 120593 120577 isin S with 120593 le 119877 and 120577 le 119877
then A120601 +B120577 le 119877
Let 120593 120577 isin S with 120593 le 119877 and 120577 le 119877 Then
Krasnoselskii (see [47]) proved that if 119860 is a stable constantmatrix without piecewise alternately advanced and retardedargument and lim
|119909|+|119910|rarr+infin(|119892(119905 119909 119910)|(|119909|+|119910|)) = 0 then
the system (56) has at least one periodic solution In our caseapplyingTheorem 12 this result is also valid for the DEPCAGsystem (56) requiring the hypothesis (119873
120596) (1198753) (119872119892) 119860(119905)
and 119892(119905 119909 119910) are periodic functions in 119905 with a period 120596 forall 119905 ge 120591 and |119892(119905 119909 119910)| le 119888
1(|119909| + |119910|) + 120588
2 with 2119888
1120596 lt 119862
2
and 1205882constants for |119909| + |119910| le 119877 119877 gt 0
Remark 17 Considering the nonlinear system of differentialequations with a general piecewise alternately advanced andretarded argument
In this particular case applyingTheorem 13 this result is alsovalid for the DEPCAG system (57) requiring the hypothesis(119873120596) (1198751) (1198753) (119871119892) (119872119891) and 119888
119866(1198621+1198712)119877+119888119866(120573+1205881)120596 le 119877
with = 1
Remark 18 Suppose that (1198751) is satisfied by 120596 = 120596
1
(1198752) by 120596 = 120596
2 and (119875
3) by 120596 = 120596
3 if 120596
119894120596119895is a
rational number for all 119894 119895 = 1 2 3 then (1198751) (1198752) and
(1198753) are simultaneously satisfied by 120596 = lcm120596
1 1205962 1205963
where lcm1205961 1205962 1205963 denotes the least common multiple
between 1205961 1205962and 120596
3 In the general case it is possible
that there exist five possible periods 1205961for 119860 120596
2for 119891 120596
3
for 119892 1205964for 119862 and the sequences 119905
119894119894isinZ 120574119894119894isinZ satisfy the
(1205965 119901) condition If 120596
119894120596119895is a rational number for all 119894 119895 =
1 2 3 4 5 so in this situation our results insure the existenceof 120596-periodic solution with 120596 = lcm120596
1 1205962 1205963 1205964 1205965
Therefore the above results insure the existence of 120596-periodicsolutions of the DEPCAG system (6) These solutions arecalled subharmonic solutions See Corollaries 19ndash22
To determine criteria for the existence and uniqueness ofsubharmonic solutions of theDEPCAG system (6) fromnowon we make the following assumption
(119875120596) There exists 120596 = lcm120596
1 1205962 1205963 1205964 1205965 gt 0 120596
119894120596119895
which is a rational number for all 119894 119895 = 1 2 3 4 5
such that
(1) 119860(119905) 119891(119905 1199091 1199101) and 119892(119905 119909
2 1199102) are periodic
functions in 119905 with a period 1205961 1205962 and 120596
3
respectively for all 119905 ge 120591(2) 119862(119905 + 120596
4 119904 + 120596
4 1199093) = 119862(119905 119904 119909
3) for all 119905 ge 120591
(3) There exists 119901 isin Z+ for which the sequences119905119894119894isinZ 120574119894119894isinZ satisfy the (120596
5 119901) condition
As immediate corollaries of Theorems 12ndash15 and Remark 18the following results are true
The Scientific World Journal 11
Corollary 19 Suppose the hypotheses (119873120596) (119875120596) (119871119891) (119871119862)
(119872119892) and (49) hold Then the DEPCAG system (6) has at least
one subharmonic solution in S
Corollary 20 Suppose the hypotheses (119873120596) (119875120596) (119871119892) (119872119891)
(119872119862) (119862) and (51) hold Then the DEPCAG system (6) has at
least one subharmonic solution in S
Corollary 21 Suppose the hypotheses (119873120596) (119875120596) (119871119891) (119871119862)
(119871119892) and (53) holdThen theDEPCAG system (6) has a unique
subharmonic solution
Corollary 22 Suppose the hypotheses (119873120596) (119875120596) (119872
119891)
(119872119862) (119872119892) (119862) and (55) hold Then the DEPCAG system (6)
has at least one subharmonic solution in S
4 Applications and Illustrative Examples
Wewill introduce appropriate examples in this sectionTheseexamples will show the feasibility of our theory
Mathematical modelling of real-life problems usuallyresults in functional equations like ordinary or partial differ-ential equations integral and integro-differential equationsand stochastic equations Many mathematical formulationsof physical phenomena contain integro-differential equa-tions these equations arise inmany fields like fluid dynamicsbiologicalmodels and chemical kinetics So we first considernonlinear integro-differential equations with a general piece-wise constant argument mentioned in the introduction andobtain some new sufficient conditions for the existence of theperiodic solutions of these systems
Example 1 Let 120582 R2 rarr [0infin) and ℎ R2 rarr [0infin) betwo functions satisfying
sup119905isinR
int
119905+2120587
119905
120582 (119905 119904) 119889119904 le sup119905isinR
int
119905+2120587
119905
ℎ (119905 119904) 119889119904 le ℎ (58)
Consider the following nonlinear integro-differential equa-tions with piecewise alternately advanced and retarded argu-ment of generalized type
|119862(119905 119904 119910)| le 1198882(119877)120582(119905 119904)|119910| +ℎ(119905 119904) for |119910| le 119877 where
2120587 le 1198621
(iv) 119892 R times R119899 times R119899 rarr R119899 is continuous and 119862(119905 119904 119910)satisfies (119862)
Indeed for any 120576 gt 0 there exists 120575 gt 0 such that |1199101minus1199102| le 120575
implies
1003816100381610038161003816119862 (119905 119904 1199101) minus 119862 (119905 119904 1199102)1003816100381610038161003816 le 1205761198882 (119877) 120582 (119905 119904) for 119905 119904 isin R
(60)
Furthermore there exists 119877 such that
exp (2119886lowast120587)exp (2119886
lowast120587) minus 1
[(1198621+ 1198622) 119877 + 2120587 (120588
2+ ℎ)] le 119877 (61)
where 119886lowast = sup119905isinR |119886(119905)| and 119886
lowast= inf119905isinR|119886(119905)|
Then by Theorem 15 the DEPCAG system (59) has atleast one 2120587-periodic solution
Example 2 Thus many examples can be constructed whereour results can be applied
Let Λ R rarr R119899times119899 and 120583 R rarr R119899 be two functionssatisfying
sup119905isinR
int
119905+120596
119905
|Λ (119904)| 119889119904 = Λ lt infin
sup119905isinR
int
119905+120596
119905
1003816100381610038161003816120583 (119905 minus 119904)1003816100381610038161003816 119889119904 = 120583 lt infin
(62)
Now consider the integro-differential system with piecewisealternately advanced and retarded argument
Theorem 12 implies that there exists at least a 120596-periodicsolution of the DEPCAG system (63)
12 The Scientific World Journal
Note that similar results can be obtained under (119871119892)
and (119872119862) On the other hand the periodic situation of the
DEPCAGsystem (56) and (57) can be treated in the samewayLet us consider another example for second-order differ-
ential equations with a general piecewise constant argumentIn this case we can show the existence and uniquenessof periodic solutions of the following nonlinear DEPCAGsystem
Example 3 Consider the following nonlinear DEPCAG sys-tem
11991010158401015840
(119905) + (12058121199102
(119905) minus 2) 1199101015840
(119905) minus 8119910 (119905)
minus 1205811sin (120596119905) 1199102 (120574 (119905)) minus 120581
2cos (120596119905) = 0
(65)
where 1205811 1205812isin R 120574(119905) = 120574
Then for 120593 120595 isin S we have1003817100381710038171003817119892 (sdot 120593 (sdot) 120593 (120574 (sdot))) minus 119892 (sdot 120595 (sdot) 120595 (120574 (sdot)))
le 212058121198772 sup119905isin[120591120591+120596]
1003816100381610038161003816100381610038161003816(1205951 (119905) minus 1205931 (119905)
1205952 (119905) minus 1205932 (119905)
)
1003816100381610038161003816100381610038161003816= 212058121198772 1003817100381710038171003817120593 minus 120595
1003817100381710038171003817
(69)
In a similar way for 119891 we have1003817100381710038171003817119891 (sdot 120593 (sdot) 120593 (120574 (sdot))) minus 119891 (sdot 120595 (sdot) 120595 (120574 (sdot)))
1003817100381710038171003817
le 212058111198771003817100381710038171003817120593 minus 120595
1003817100381710038171003817
(70)
ByTheorem 14 the DEPCAG system (65) has a unique 2120587120596-periodic solution in S
Conflict of Interests
The author declares that there is no conflict of interests re-garding the publication of this paper
Acknowledgment
This research was in part supported by FIBE 01-12 DIUMCE
References
[1] ADMyshkis ldquoOn certain problems in the theory of differentialequations with deviating argumentsrdquo Uspekhi Matematich-eskikh Nauk vol 32 pp 173ndash202 1977
[2] K L Cooke and JWiener ldquoRetarded differential equations withpiecewise constant delaysrdquo Journal of Mathematical Analysisand Applications vol 99 no 1 pp 265ndash297 1984
[3] K L Cooke and J Wiener ldquoAn equation alternately of retardedand advanced typerdquo Proceedings of the American MathematicalSociety vol 99 pp 726ndash732 1987
[4] S M Shah and J Wiener ldquoAdvanced differential equations withpiecewise constant argument deviationsrdquo International JournalofMathematics andMathematical Sciences vol 6 no 4 pp 671ndash703 1983
[5] J WienerGeneralized Solutions of Functional Differential Equa-tions World Scientific Singapore 1993
[6] A R Aftabizadeh and J Wiener ldquoOscillatory and periodicsolutions of an equation alternately of retarded and advancedtypesrdquo Applicable Analysis vol 23 no 3 pp 219ndash231 1986
[7] A R Aftabizadeh J Wiener and J M Xu ldquoOscillatory andperiodic solutions of delay differential equations with piecewiseconstant argumentrdquo Proceedings of the American MathematicalSociety vol 99 pp 673ndash679 1987
[8] M U Akhmet ldquoIntegral manifolds of differential equationswith piecewise constant argument of generalized typerdquo Nonlin-ear Analysis Theory Methods and Applications vol 66 no 2pp 367ndash383 2007
[9] M U Akhmet Nonlinear Hybrid ContinuousDiscrete-TimeModels Atlantis Press Amsterdam The Netherlands 2011
[10] M U Akhmet C Buyukadali and T Ergenc ldquoPeriodic solu-tions of the hybrid system with small parameterrdquo NonlinearAnalysis Hybrid Systems vol 2 no 2 pp 532ndash543 2008
[11] A Cabada J B Ferreiro and J J Nieto ldquoGreenrsquos functionand comparison principles for first order periodic differentialequations with piecewise constant argumentsrdquo Journal of Math-ematical Analysis and Applications vol 291 no 2 pp 690ndash6972004
[12] K S Chiu and M Pinto ldquoVariation of parameters formula andGronwall inequality for differential equations with a generalpiecewise constant argumentrdquo Acta Mathematicae ApplicataeSinica vol 27 no 4 pp 561ndash568 2011
The Scientific World Journal 13
[13] K S Chiu and M Pinto ldquoOscillatory and periodic solutionsin alternately advanced and delayed differential equationsrdquoCarpathian Journal of Mathematics vol 29 no 2 pp 149ndash1582013
[14] J J Nieto and R Rodrıguez-Lopez ldquoGreenrsquos function forsecond-order periodic boundary value problemswith piecewiseconstant argumentsrdquo Journal of Mathematical Analysis andApplications vol 304 no 1 pp 33ndash57 2005
[15] M Pinto ldquoAsymptotic equivalence of nonlinear and quasi lin-ear differential equations with piecewise constant argumentsrdquoMathematical and Computer Modelling vol 49 no 9-10 pp1750ndash1758 2009
[16] M Pinto ldquoCauchy and Green matrices type and stability inalternately advanced and delayed differential systemsrdquo Journalof Difference Equations and Applications vol 17 no 2 pp 235ndash254 2011
[17] G Q Wang ldquoExistence theorem of periodic solutions for adelay nonlinear differential equation with piecewise constantargumentsrdquo Journal of Mathematical Analysis and Applicationsvol 298 no 1 pp 298ndash307 2004
[18] G Q Wang and S S Cheng ldquoExistence and uniqueness ofperiodic solutions for a second-order nonlinear differentialequationwith piecewise constant argumentrdquo International Jour-nal ofMathematics andMathematical Sciences vol 2009 ArticleID 950797 14 pages 2009
[19] Y H Xia Z Huang and M Han ldquoExistence of almost periodicsolutions for forced perturbed systems with piecewise constantargumentrdquo Journal of Mathematical Analysis and Applicationsvol 333 no 2 pp 798ndash816 2007
[20] P Yang Y Liu and W Ge ldquoGreenrsquos function for second orderdifferential equations with piecewise constant argumentsrdquoNon-linear Analysis Theory Methods and Applications vol 64 no 8pp 1812ndash1830 2006
[21] R Yuan ldquoThe existence of almost periodic solutions of retardeddifferential equations with piecewise constant argumentrdquo Non-linear Analysis Theory Methods and Applications vol 48 no 7pp 1013ndash1032 2002
[22] A I Alonso J Hong and R Obaya ldquoAlmost periodic typesolutions of differential equations with piecewise constant argu-ment via almost periodic type sequencesrdquo Applied MathematicsLetters vol 13 no 2 pp 131ndash137 2000
[23] K S Chiu and M Pinto ldquoPeriodic solutions of differentialequations with a general piecewise constant argument andapplicationsrdquo Electronic Journal of Qualitative Theory of Differ-ential Equations vol 46 pp 1ndash19 2010
[24] K S Chiu M Pinto and J C Jeng ldquoExistence and globalconvergence of periodic solutions in recurrent neural networkmodels with a general piecewise alternately advanced andretarded argumentrdquo Acta Applicandae Mathematicae 2013
[25] L Dai Nonlinear Dynamics of Piecewise of Constant Systemsand Implememtation of Piecewise Constants Arguments WorldScientific Singapore 2008
[26] T A Burton Stability and Periodic Solutions of Ordinary andFunctional Differential Equations Academic Press New YorkNY USA 1985
[27] T A Burton and T Furumochi ldquoPeriodic and asymptoticallyperiodic solutions of neutral integral equationsrdquo ElectronicJournal of Qualitative Theory of Differential Equations vol 10pp 1ndash19 2000
[28] T A Burton and T Furumochi ldquoExistence theorems andperiodic solutions of neutral integral equationsrdquo NonlinearAnalysis Theory Methods and Applications vol 43 no 4 pp527ndash546 2001
[29] F Karakoc H Bereketoglu and G Seyhan ldquoOscillatory andperiodic solutions of impulsive differential equations withpiecewise constant argumentrdquoActa ApplicandaeMathematicaevol 110 no 1 pp 499ndash510 2010
[30] Y Liu P Yang and W Ge ldquoPeriodic solutions of higher-order delay differential equationsrdquo Nonlinear Analysis TheoryMethods and Applications vol 63 no 1 pp 136ndash152 2005
[31] X L Liu and W T Li ldquoPeriodic solutions for dynamic equa-tions on time scalesrdquo Nonlinear Analysis Theory Methods andApplications vol 67 no 5 pp 1457ndash1463 2007
[32] M Pinto ldquoDichotomy and existence of periodic solutions ofquasilinear functional differential equationsrdquo Nonlinear Analy-sis Theory Methods and Applications vol 72 no 3-4 pp 1227ndash1234 2010
[33] Y N Raffoul ldquoPeriodic solutions for neutral nonlinear differ-ential equations with functional delayrdquo Electronic Journal ofDifferential Equations vol 2003 no 102 pp 1ndash7 2003
[34] G Q Wang and S S Cheng ldquoPeriodic solutions of discreteRayleigh equations with deviating argumentsrdquo Taiwanese Jour-nal of Mathematics vol 13 no 6 pp 2051ndash2067 2009
[35] K S Chiu ldquoStability of oscillatory solutions of differential equa-tions with a general piecewise constant argumentrdquo ElectronicJournal of Qualitative Theory of Differential Equations vol 88pp 1ndash15 2011
[36] R N Butris and M A Aziz ldquoSome theorems in the existenceand uniqueness for system of nonlinear integro-differentialequationsrdquo Journal Education and Science vol 18 pp 76ndash892006
[37] A YMitropolsky andD IMortynyukPeriodic Solutions for theOscillations Systems with Retarded Argument General SchoolKiev Ukraine 1979
[38] L L Tai ldquoNumerical-analytic method for the investigation ofautonomous systems of differential equationsrdquo Ukrainskii Ma-tematicheskii Zhurnal vol 30 no 3 pp 309ndash317 1978
[39] Yu D Shlapak ldquoPeriodic solutions of ordinary differentialequations of first order that are not solved with respect to thederivativerdquo Ukrainian Mathematical Journal vol 32 no 5 pp420ndash425 1980
[40] A M Samoilenko ldquoNumerical-analytic method for the investi-gation of periodic systems of ordinary differential equationsrdquoUkrainskii Matematicheskii Zhurnal vol 17 no 4 pp 16ndash231965
[41] AM Samoilenko andN I RontoNumerical-Analytic Methodsfor Investigations of Periodic Solutions Mir Publishers MoscowRussia 1979
[42] R N Butris ldquoPeriodic solution of nonlinear system of integro-differential equations depending on the gamma distributionrdquoGeneral Mathematics Notes vol 15 no 1 pp 56ndash71 2013
[43] D R Smart Fixed Points Theorems Cambridge UniversityPress Cambridge UK 1980
[44] T A Burton ldquoKrasnoselskiirsquos inversion principle and fixedpointsrdquo Nonlinear Analysis Theory Methods and Applicationsvol 30 no 7 pp 3975ndash3986 1997
[45] T A Burton ldquoA fixed-point theorem of Krasnoselskiirdquo AppliedMathematics Letters vol 11 no 1 pp 85ndash88 1998
14 The Scientific World Journal
[46] M Pinto and D Sepulveda ldquoh-asymptotic stability by fixedpoint in neutral nonlinear differential equations with delayrdquoNonlinear Analysis Theory Methods and Applications vol 74no 12 pp 3926ndash3933 2011
[47] M A Krasnoselskii Translation Along Trajectories of Differen-tial Equations Nauka Moscow Russia 1966 (Russian) Englishtranslation American Mathematical Society Providence RIUSA 1968
Step 2 We show that the image of B is contained in acompact set
Let 119909 119910 isin CR and 119904 isin [120591 120591 + 120596] for the continuity of thefunction 119892(119904 119909 119910) there exists119872 gt 0 such that |119892(119904 119909 119910)| le119872 Let 120593
119899isin S where 119899 is a positive integer then we have
We now see that all the conditions of Krasnoselskiirsquostheorem are satisfied Thus there exists a fixed point 119911 in S
such that 119911 = A119911 +B119911 By Proposition 6 this fixed point isa solution of the DEPCAG system (6) Hence the DEPCAGsystem (6) has 120596-periodic solution
By the symmetry of the conditions we will obtain asTheorem 12
Theorem 13 Suppose the hypothesis (119873120596) (119875) (119871
119892) (119872119891)
(119872119862) (119862) hold Let 119877 be a positive constant satisfying the
inequality
119888119866(1198621+ 1198712) 119877 + 119888
119866(120573 + 120578
2+ 1205881) 120596 le 119877 (51)
Then the DEPCAG system (6) has at least one 120596-periodicsolution in S
By Lemma 9 the mapping B is a contraction and it isclear that B P
120596rarr P
120596 Also from Lemma 11 A is
completely continuousNext we prove that if 120593 120577 isin S with 120593 le 119877 and 120577 le 119877
then A120601 +B120577 le 119877
Let 120593 120577 isin S with 120593 le 119877 and 120577 le 119877 Then
Krasnoselskii (see [47]) proved that if 119860 is a stable constantmatrix without piecewise alternately advanced and retardedargument and lim
|119909|+|119910|rarr+infin(|119892(119905 119909 119910)|(|119909|+|119910|)) = 0 then
the system (56) has at least one periodic solution In our caseapplyingTheorem 12 this result is also valid for the DEPCAGsystem (56) requiring the hypothesis (119873
120596) (1198753) (119872119892) 119860(119905)
and 119892(119905 119909 119910) are periodic functions in 119905 with a period 120596 forall 119905 ge 120591 and |119892(119905 119909 119910)| le 119888
1(|119909| + |119910|) + 120588
2 with 2119888
1120596 lt 119862
2
and 1205882constants for |119909| + |119910| le 119877 119877 gt 0
Remark 17 Considering the nonlinear system of differentialequations with a general piecewise alternately advanced andretarded argument
In this particular case applyingTheorem 13 this result is alsovalid for the DEPCAG system (57) requiring the hypothesis(119873120596) (1198751) (1198753) (119871119892) (119872119891) and 119888
119866(1198621+1198712)119877+119888119866(120573+1205881)120596 le 119877
with = 1
Remark 18 Suppose that (1198751) is satisfied by 120596 = 120596
1
(1198752) by 120596 = 120596
2 and (119875
3) by 120596 = 120596
3 if 120596
119894120596119895is a
rational number for all 119894 119895 = 1 2 3 then (1198751) (1198752) and
(1198753) are simultaneously satisfied by 120596 = lcm120596
1 1205962 1205963
where lcm1205961 1205962 1205963 denotes the least common multiple
between 1205961 1205962and 120596
3 In the general case it is possible
that there exist five possible periods 1205961for 119860 120596
2for 119891 120596
3
for 119892 1205964for 119862 and the sequences 119905
119894119894isinZ 120574119894119894isinZ satisfy the
(1205965 119901) condition If 120596
119894120596119895is a rational number for all 119894 119895 =
1 2 3 4 5 so in this situation our results insure the existenceof 120596-periodic solution with 120596 = lcm120596
1 1205962 1205963 1205964 1205965
Therefore the above results insure the existence of 120596-periodicsolutions of the DEPCAG system (6) These solutions arecalled subharmonic solutions See Corollaries 19ndash22
To determine criteria for the existence and uniqueness ofsubharmonic solutions of theDEPCAG system (6) fromnowon we make the following assumption
(119875120596) There exists 120596 = lcm120596
1 1205962 1205963 1205964 1205965 gt 0 120596
119894120596119895
which is a rational number for all 119894 119895 = 1 2 3 4 5
such that
(1) 119860(119905) 119891(119905 1199091 1199101) and 119892(119905 119909
2 1199102) are periodic
functions in 119905 with a period 1205961 1205962 and 120596
3
respectively for all 119905 ge 120591(2) 119862(119905 + 120596
4 119904 + 120596
4 1199093) = 119862(119905 119904 119909
3) for all 119905 ge 120591
(3) There exists 119901 isin Z+ for which the sequences119905119894119894isinZ 120574119894119894isinZ satisfy the (120596
5 119901) condition
As immediate corollaries of Theorems 12ndash15 and Remark 18the following results are true
The Scientific World Journal 11
Corollary 19 Suppose the hypotheses (119873120596) (119875120596) (119871119891) (119871119862)
(119872119892) and (49) hold Then the DEPCAG system (6) has at least
one subharmonic solution in S
Corollary 20 Suppose the hypotheses (119873120596) (119875120596) (119871119892) (119872119891)
(119872119862) (119862) and (51) hold Then the DEPCAG system (6) has at
least one subharmonic solution in S
Corollary 21 Suppose the hypotheses (119873120596) (119875120596) (119871119891) (119871119862)
(119871119892) and (53) holdThen theDEPCAG system (6) has a unique
subharmonic solution
Corollary 22 Suppose the hypotheses (119873120596) (119875120596) (119872
119891)
(119872119862) (119872119892) (119862) and (55) hold Then the DEPCAG system (6)
has at least one subharmonic solution in S
4 Applications and Illustrative Examples
Wewill introduce appropriate examples in this sectionTheseexamples will show the feasibility of our theory
Mathematical modelling of real-life problems usuallyresults in functional equations like ordinary or partial differ-ential equations integral and integro-differential equationsand stochastic equations Many mathematical formulationsof physical phenomena contain integro-differential equa-tions these equations arise inmany fields like fluid dynamicsbiologicalmodels and chemical kinetics So we first considernonlinear integro-differential equations with a general piece-wise constant argument mentioned in the introduction andobtain some new sufficient conditions for the existence of theperiodic solutions of these systems
Example 1 Let 120582 R2 rarr [0infin) and ℎ R2 rarr [0infin) betwo functions satisfying
sup119905isinR
int
119905+2120587
119905
120582 (119905 119904) 119889119904 le sup119905isinR
int
119905+2120587
119905
ℎ (119905 119904) 119889119904 le ℎ (58)
Consider the following nonlinear integro-differential equa-tions with piecewise alternately advanced and retarded argu-ment of generalized type
|119862(119905 119904 119910)| le 1198882(119877)120582(119905 119904)|119910| +ℎ(119905 119904) for |119910| le 119877 where
2120587 le 1198621
(iv) 119892 R times R119899 times R119899 rarr R119899 is continuous and 119862(119905 119904 119910)satisfies (119862)
Indeed for any 120576 gt 0 there exists 120575 gt 0 such that |1199101minus1199102| le 120575
implies
1003816100381610038161003816119862 (119905 119904 1199101) minus 119862 (119905 119904 1199102)1003816100381610038161003816 le 1205761198882 (119877) 120582 (119905 119904) for 119905 119904 isin R
(60)
Furthermore there exists 119877 such that
exp (2119886lowast120587)exp (2119886
lowast120587) minus 1
[(1198621+ 1198622) 119877 + 2120587 (120588
2+ ℎ)] le 119877 (61)
where 119886lowast = sup119905isinR |119886(119905)| and 119886
lowast= inf119905isinR|119886(119905)|
Then by Theorem 15 the DEPCAG system (59) has atleast one 2120587-periodic solution
Example 2 Thus many examples can be constructed whereour results can be applied
Let Λ R rarr R119899times119899 and 120583 R rarr R119899 be two functionssatisfying
sup119905isinR
int
119905+120596
119905
|Λ (119904)| 119889119904 = Λ lt infin
sup119905isinR
int
119905+120596
119905
1003816100381610038161003816120583 (119905 minus 119904)1003816100381610038161003816 119889119904 = 120583 lt infin
(62)
Now consider the integro-differential system with piecewisealternately advanced and retarded argument
Theorem 12 implies that there exists at least a 120596-periodicsolution of the DEPCAG system (63)
12 The Scientific World Journal
Note that similar results can be obtained under (119871119892)
and (119872119862) On the other hand the periodic situation of the
DEPCAGsystem (56) and (57) can be treated in the samewayLet us consider another example for second-order differ-
ential equations with a general piecewise constant argumentIn this case we can show the existence and uniquenessof periodic solutions of the following nonlinear DEPCAGsystem
Example 3 Consider the following nonlinear DEPCAG sys-tem
11991010158401015840
(119905) + (12058121199102
(119905) minus 2) 1199101015840
(119905) minus 8119910 (119905)
minus 1205811sin (120596119905) 1199102 (120574 (119905)) minus 120581
2cos (120596119905) = 0
(65)
where 1205811 1205812isin R 120574(119905) = 120574
Then for 120593 120595 isin S we have1003817100381710038171003817119892 (sdot 120593 (sdot) 120593 (120574 (sdot))) minus 119892 (sdot 120595 (sdot) 120595 (120574 (sdot)))
le 212058121198772 sup119905isin[120591120591+120596]
1003816100381610038161003816100381610038161003816(1205951 (119905) minus 1205931 (119905)
1205952 (119905) minus 1205932 (119905)
)
1003816100381610038161003816100381610038161003816= 212058121198772 1003817100381710038171003817120593 minus 120595
1003817100381710038171003817
(69)
In a similar way for 119891 we have1003817100381710038171003817119891 (sdot 120593 (sdot) 120593 (120574 (sdot))) minus 119891 (sdot 120595 (sdot) 120595 (120574 (sdot)))
1003817100381710038171003817
le 212058111198771003817100381710038171003817120593 minus 120595
1003817100381710038171003817
(70)
ByTheorem 14 the DEPCAG system (65) has a unique 2120587120596-periodic solution in S
Conflict of Interests
The author declares that there is no conflict of interests re-garding the publication of this paper
Acknowledgment
This research was in part supported by FIBE 01-12 DIUMCE
References
[1] ADMyshkis ldquoOn certain problems in the theory of differentialequations with deviating argumentsrdquo Uspekhi Matematich-eskikh Nauk vol 32 pp 173ndash202 1977
[2] K L Cooke and JWiener ldquoRetarded differential equations withpiecewise constant delaysrdquo Journal of Mathematical Analysisand Applications vol 99 no 1 pp 265ndash297 1984
[3] K L Cooke and J Wiener ldquoAn equation alternately of retardedand advanced typerdquo Proceedings of the American MathematicalSociety vol 99 pp 726ndash732 1987
[4] S M Shah and J Wiener ldquoAdvanced differential equations withpiecewise constant argument deviationsrdquo International JournalofMathematics andMathematical Sciences vol 6 no 4 pp 671ndash703 1983
[5] J WienerGeneralized Solutions of Functional Differential Equa-tions World Scientific Singapore 1993
[6] A R Aftabizadeh and J Wiener ldquoOscillatory and periodicsolutions of an equation alternately of retarded and advancedtypesrdquo Applicable Analysis vol 23 no 3 pp 219ndash231 1986
[7] A R Aftabizadeh J Wiener and J M Xu ldquoOscillatory andperiodic solutions of delay differential equations with piecewiseconstant argumentrdquo Proceedings of the American MathematicalSociety vol 99 pp 673ndash679 1987
[8] M U Akhmet ldquoIntegral manifolds of differential equationswith piecewise constant argument of generalized typerdquo Nonlin-ear Analysis Theory Methods and Applications vol 66 no 2pp 367ndash383 2007
[9] M U Akhmet Nonlinear Hybrid ContinuousDiscrete-TimeModels Atlantis Press Amsterdam The Netherlands 2011
[10] M U Akhmet C Buyukadali and T Ergenc ldquoPeriodic solu-tions of the hybrid system with small parameterrdquo NonlinearAnalysis Hybrid Systems vol 2 no 2 pp 532ndash543 2008
[11] A Cabada J B Ferreiro and J J Nieto ldquoGreenrsquos functionand comparison principles for first order periodic differentialequations with piecewise constant argumentsrdquo Journal of Math-ematical Analysis and Applications vol 291 no 2 pp 690ndash6972004
[12] K S Chiu and M Pinto ldquoVariation of parameters formula andGronwall inequality for differential equations with a generalpiecewise constant argumentrdquo Acta Mathematicae ApplicataeSinica vol 27 no 4 pp 561ndash568 2011
The Scientific World Journal 13
[13] K S Chiu and M Pinto ldquoOscillatory and periodic solutionsin alternately advanced and delayed differential equationsrdquoCarpathian Journal of Mathematics vol 29 no 2 pp 149ndash1582013
[14] J J Nieto and R Rodrıguez-Lopez ldquoGreenrsquos function forsecond-order periodic boundary value problemswith piecewiseconstant argumentsrdquo Journal of Mathematical Analysis andApplications vol 304 no 1 pp 33ndash57 2005
[15] M Pinto ldquoAsymptotic equivalence of nonlinear and quasi lin-ear differential equations with piecewise constant argumentsrdquoMathematical and Computer Modelling vol 49 no 9-10 pp1750ndash1758 2009
[16] M Pinto ldquoCauchy and Green matrices type and stability inalternately advanced and delayed differential systemsrdquo Journalof Difference Equations and Applications vol 17 no 2 pp 235ndash254 2011
[17] G Q Wang ldquoExistence theorem of periodic solutions for adelay nonlinear differential equation with piecewise constantargumentsrdquo Journal of Mathematical Analysis and Applicationsvol 298 no 1 pp 298ndash307 2004
[18] G Q Wang and S S Cheng ldquoExistence and uniqueness ofperiodic solutions for a second-order nonlinear differentialequationwith piecewise constant argumentrdquo International Jour-nal ofMathematics andMathematical Sciences vol 2009 ArticleID 950797 14 pages 2009
[19] Y H Xia Z Huang and M Han ldquoExistence of almost periodicsolutions for forced perturbed systems with piecewise constantargumentrdquo Journal of Mathematical Analysis and Applicationsvol 333 no 2 pp 798ndash816 2007
[20] P Yang Y Liu and W Ge ldquoGreenrsquos function for second orderdifferential equations with piecewise constant argumentsrdquoNon-linear Analysis Theory Methods and Applications vol 64 no 8pp 1812ndash1830 2006
[21] R Yuan ldquoThe existence of almost periodic solutions of retardeddifferential equations with piecewise constant argumentrdquo Non-linear Analysis Theory Methods and Applications vol 48 no 7pp 1013ndash1032 2002
[22] A I Alonso J Hong and R Obaya ldquoAlmost periodic typesolutions of differential equations with piecewise constant argu-ment via almost periodic type sequencesrdquo Applied MathematicsLetters vol 13 no 2 pp 131ndash137 2000
[23] K S Chiu and M Pinto ldquoPeriodic solutions of differentialequations with a general piecewise constant argument andapplicationsrdquo Electronic Journal of Qualitative Theory of Differ-ential Equations vol 46 pp 1ndash19 2010
[24] K S Chiu M Pinto and J C Jeng ldquoExistence and globalconvergence of periodic solutions in recurrent neural networkmodels with a general piecewise alternately advanced andretarded argumentrdquo Acta Applicandae Mathematicae 2013
[25] L Dai Nonlinear Dynamics of Piecewise of Constant Systemsand Implememtation of Piecewise Constants Arguments WorldScientific Singapore 2008
[26] T A Burton Stability and Periodic Solutions of Ordinary andFunctional Differential Equations Academic Press New YorkNY USA 1985
[27] T A Burton and T Furumochi ldquoPeriodic and asymptoticallyperiodic solutions of neutral integral equationsrdquo ElectronicJournal of Qualitative Theory of Differential Equations vol 10pp 1ndash19 2000
[28] T A Burton and T Furumochi ldquoExistence theorems andperiodic solutions of neutral integral equationsrdquo NonlinearAnalysis Theory Methods and Applications vol 43 no 4 pp527ndash546 2001
[29] F Karakoc H Bereketoglu and G Seyhan ldquoOscillatory andperiodic solutions of impulsive differential equations withpiecewise constant argumentrdquoActa ApplicandaeMathematicaevol 110 no 1 pp 499ndash510 2010
[30] Y Liu P Yang and W Ge ldquoPeriodic solutions of higher-order delay differential equationsrdquo Nonlinear Analysis TheoryMethods and Applications vol 63 no 1 pp 136ndash152 2005
[31] X L Liu and W T Li ldquoPeriodic solutions for dynamic equa-tions on time scalesrdquo Nonlinear Analysis Theory Methods andApplications vol 67 no 5 pp 1457ndash1463 2007
[32] M Pinto ldquoDichotomy and existence of periodic solutions ofquasilinear functional differential equationsrdquo Nonlinear Analy-sis Theory Methods and Applications vol 72 no 3-4 pp 1227ndash1234 2010
[33] Y N Raffoul ldquoPeriodic solutions for neutral nonlinear differ-ential equations with functional delayrdquo Electronic Journal ofDifferential Equations vol 2003 no 102 pp 1ndash7 2003
[34] G Q Wang and S S Cheng ldquoPeriodic solutions of discreteRayleigh equations with deviating argumentsrdquo Taiwanese Jour-nal of Mathematics vol 13 no 6 pp 2051ndash2067 2009
[35] K S Chiu ldquoStability of oscillatory solutions of differential equa-tions with a general piecewise constant argumentrdquo ElectronicJournal of Qualitative Theory of Differential Equations vol 88pp 1ndash15 2011
[36] R N Butris and M A Aziz ldquoSome theorems in the existenceand uniqueness for system of nonlinear integro-differentialequationsrdquo Journal Education and Science vol 18 pp 76ndash892006
[37] A YMitropolsky andD IMortynyukPeriodic Solutions for theOscillations Systems with Retarded Argument General SchoolKiev Ukraine 1979
[38] L L Tai ldquoNumerical-analytic method for the investigation ofautonomous systems of differential equationsrdquo Ukrainskii Ma-tematicheskii Zhurnal vol 30 no 3 pp 309ndash317 1978
[39] Yu D Shlapak ldquoPeriodic solutions of ordinary differentialequations of first order that are not solved with respect to thederivativerdquo Ukrainian Mathematical Journal vol 32 no 5 pp420ndash425 1980
[40] A M Samoilenko ldquoNumerical-analytic method for the investi-gation of periodic systems of ordinary differential equationsrdquoUkrainskii Matematicheskii Zhurnal vol 17 no 4 pp 16ndash231965
[41] AM Samoilenko andN I RontoNumerical-Analytic Methodsfor Investigations of Periodic Solutions Mir Publishers MoscowRussia 1979
[42] R N Butris ldquoPeriodic solution of nonlinear system of integro-differential equations depending on the gamma distributionrdquoGeneral Mathematics Notes vol 15 no 1 pp 56ndash71 2013
[43] D R Smart Fixed Points Theorems Cambridge UniversityPress Cambridge UK 1980
[44] T A Burton ldquoKrasnoselskiirsquos inversion principle and fixedpointsrdquo Nonlinear Analysis Theory Methods and Applicationsvol 30 no 7 pp 3975ndash3986 1997
[45] T A Burton ldquoA fixed-point theorem of Krasnoselskiirdquo AppliedMathematics Letters vol 11 no 1 pp 85ndash88 1998
14 The Scientific World Journal
[46] M Pinto and D Sepulveda ldquoh-asymptotic stability by fixedpoint in neutral nonlinear differential equations with delayrdquoNonlinear Analysis Theory Methods and Applications vol 74no 12 pp 3926ndash3933 2011
[47] M A Krasnoselskii Translation Along Trajectories of Differen-tial Equations Nauka Moscow Russia 1966 (Russian) Englishtranslation American Mathematical Society Providence RIUSA 1968
We now see that all the conditions of Krasnoselskiirsquostheorem are satisfied Thus there exists a fixed point 119911 in S
such that 119911 = A119911 +B119911 By Proposition 6 this fixed point isa solution of the DEPCAG system (6) Hence the DEPCAGsystem (6) has 120596-periodic solution
By the symmetry of the conditions we will obtain asTheorem 12
Theorem 13 Suppose the hypothesis (119873120596) (119875) (119871
119892) (119872119891)
(119872119862) (119862) hold Let 119877 be a positive constant satisfying the
inequality
119888119866(1198621+ 1198712) 119877 + 119888
119866(120573 + 120578
2+ 1205881) 120596 le 119877 (51)
Then the DEPCAG system (6) has at least one 120596-periodicsolution in S
By Lemma 9 the mapping B is a contraction and it isclear that B P
120596rarr P
120596 Also from Lemma 11 A is
completely continuousNext we prove that if 120593 120577 isin S with 120593 le 119877 and 120577 le 119877
then A120601 +B120577 le 119877
Let 120593 120577 isin S with 120593 le 119877 and 120577 le 119877 Then
Krasnoselskii (see [47]) proved that if 119860 is a stable constantmatrix without piecewise alternately advanced and retardedargument and lim
|119909|+|119910|rarr+infin(|119892(119905 119909 119910)|(|119909|+|119910|)) = 0 then
the system (56) has at least one periodic solution In our caseapplyingTheorem 12 this result is also valid for the DEPCAGsystem (56) requiring the hypothesis (119873
120596) (1198753) (119872119892) 119860(119905)
and 119892(119905 119909 119910) are periodic functions in 119905 with a period 120596 forall 119905 ge 120591 and |119892(119905 119909 119910)| le 119888
1(|119909| + |119910|) + 120588
2 with 2119888
1120596 lt 119862
2
and 1205882constants for |119909| + |119910| le 119877 119877 gt 0
Remark 17 Considering the nonlinear system of differentialequations with a general piecewise alternately advanced andretarded argument
In this particular case applyingTheorem 13 this result is alsovalid for the DEPCAG system (57) requiring the hypothesis(119873120596) (1198751) (1198753) (119871119892) (119872119891) and 119888
119866(1198621+1198712)119877+119888119866(120573+1205881)120596 le 119877
with = 1
Remark 18 Suppose that (1198751) is satisfied by 120596 = 120596
1
(1198752) by 120596 = 120596
2 and (119875
3) by 120596 = 120596
3 if 120596
119894120596119895is a
rational number for all 119894 119895 = 1 2 3 then (1198751) (1198752) and
(1198753) are simultaneously satisfied by 120596 = lcm120596
1 1205962 1205963
where lcm1205961 1205962 1205963 denotes the least common multiple
between 1205961 1205962and 120596
3 In the general case it is possible
that there exist five possible periods 1205961for 119860 120596
2for 119891 120596
3
for 119892 1205964for 119862 and the sequences 119905
119894119894isinZ 120574119894119894isinZ satisfy the
(1205965 119901) condition If 120596
119894120596119895is a rational number for all 119894 119895 =
1 2 3 4 5 so in this situation our results insure the existenceof 120596-periodic solution with 120596 = lcm120596
1 1205962 1205963 1205964 1205965
Therefore the above results insure the existence of 120596-periodicsolutions of the DEPCAG system (6) These solutions arecalled subharmonic solutions See Corollaries 19ndash22
To determine criteria for the existence and uniqueness ofsubharmonic solutions of theDEPCAG system (6) fromnowon we make the following assumption
(119875120596) There exists 120596 = lcm120596
1 1205962 1205963 1205964 1205965 gt 0 120596
119894120596119895
which is a rational number for all 119894 119895 = 1 2 3 4 5
such that
(1) 119860(119905) 119891(119905 1199091 1199101) and 119892(119905 119909
2 1199102) are periodic
functions in 119905 with a period 1205961 1205962 and 120596
3
respectively for all 119905 ge 120591(2) 119862(119905 + 120596
4 119904 + 120596
4 1199093) = 119862(119905 119904 119909
3) for all 119905 ge 120591
(3) There exists 119901 isin Z+ for which the sequences119905119894119894isinZ 120574119894119894isinZ satisfy the (120596
5 119901) condition
As immediate corollaries of Theorems 12ndash15 and Remark 18the following results are true
The Scientific World Journal 11
Corollary 19 Suppose the hypotheses (119873120596) (119875120596) (119871119891) (119871119862)
(119872119892) and (49) hold Then the DEPCAG system (6) has at least
one subharmonic solution in S
Corollary 20 Suppose the hypotheses (119873120596) (119875120596) (119871119892) (119872119891)
(119872119862) (119862) and (51) hold Then the DEPCAG system (6) has at
least one subharmonic solution in S
Corollary 21 Suppose the hypotheses (119873120596) (119875120596) (119871119891) (119871119862)
(119871119892) and (53) holdThen theDEPCAG system (6) has a unique
subharmonic solution
Corollary 22 Suppose the hypotheses (119873120596) (119875120596) (119872
119891)
(119872119862) (119872119892) (119862) and (55) hold Then the DEPCAG system (6)
has at least one subharmonic solution in S
4 Applications and Illustrative Examples
Wewill introduce appropriate examples in this sectionTheseexamples will show the feasibility of our theory
Mathematical modelling of real-life problems usuallyresults in functional equations like ordinary or partial differ-ential equations integral and integro-differential equationsand stochastic equations Many mathematical formulationsof physical phenomena contain integro-differential equa-tions these equations arise inmany fields like fluid dynamicsbiologicalmodels and chemical kinetics So we first considernonlinear integro-differential equations with a general piece-wise constant argument mentioned in the introduction andobtain some new sufficient conditions for the existence of theperiodic solutions of these systems
Example 1 Let 120582 R2 rarr [0infin) and ℎ R2 rarr [0infin) betwo functions satisfying
sup119905isinR
int
119905+2120587
119905
120582 (119905 119904) 119889119904 le sup119905isinR
int
119905+2120587
119905
ℎ (119905 119904) 119889119904 le ℎ (58)
Consider the following nonlinear integro-differential equa-tions with piecewise alternately advanced and retarded argu-ment of generalized type
|119862(119905 119904 119910)| le 1198882(119877)120582(119905 119904)|119910| +ℎ(119905 119904) for |119910| le 119877 where
2120587 le 1198621
(iv) 119892 R times R119899 times R119899 rarr R119899 is continuous and 119862(119905 119904 119910)satisfies (119862)
Indeed for any 120576 gt 0 there exists 120575 gt 0 such that |1199101minus1199102| le 120575
implies
1003816100381610038161003816119862 (119905 119904 1199101) minus 119862 (119905 119904 1199102)1003816100381610038161003816 le 1205761198882 (119877) 120582 (119905 119904) for 119905 119904 isin R
(60)
Furthermore there exists 119877 such that
exp (2119886lowast120587)exp (2119886
lowast120587) minus 1
[(1198621+ 1198622) 119877 + 2120587 (120588
2+ ℎ)] le 119877 (61)
where 119886lowast = sup119905isinR |119886(119905)| and 119886
lowast= inf119905isinR|119886(119905)|
Then by Theorem 15 the DEPCAG system (59) has atleast one 2120587-periodic solution
Example 2 Thus many examples can be constructed whereour results can be applied
Let Λ R rarr R119899times119899 and 120583 R rarr R119899 be two functionssatisfying
sup119905isinR
int
119905+120596
119905
|Λ (119904)| 119889119904 = Λ lt infin
sup119905isinR
int
119905+120596
119905
1003816100381610038161003816120583 (119905 minus 119904)1003816100381610038161003816 119889119904 = 120583 lt infin
(62)
Now consider the integro-differential system with piecewisealternately advanced and retarded argument
Theorem 12 implies that there exists at least a 120596-periodicsolution of the DEPCAG system (63)
12 The Scientific World Journal
Note that similar results can be obtained under (119871119892)
and (119872119862) On the other hand the periodic situation of the
DEPCAGsystem (56) and (57) can be treated in the samewayLet us consider another example for second-order differ-
ential equations with a general piecewise constant argumentIn this case we can show the existence and uniquenessof periodic solutions of the following nonlinear DEPCAGsystem
Example 3 Consider the following nonlinear DEPCAG sys-tem
11991010158401015840
(119905) + (12058121199102
(119905) minus 2) 1199101015840
(119905) minus 8119910 (119905)
minus 1205811sin (120596119905) 1199102 (120574 (119905)) minus 120581
2cos (120596119905) = 0
(65)
where 1205811 1205812isin R 120574(119905) = 120574
Then for 120593 120595 isin S we have1003817100381710038171003817119892 (sdot 120593 (sdot) 120593 (120574 (sdot))) minus 119892 (sdot 120595 (sdot) 120595 (120574 (sdot)))
le 212058121198772 sup119905isin[120591120591+120596]
1003816100381610038161003816100381610038161003816(1205951 (119905) minus 1205931 (119905)
1205952 (119905) minus 1205932 (119905)
)
1003816100381610038161003816100381610038161003816= 212058121198772 1003817100381710038171003817120593 minus 120595
1003817100381710038171003817
(69)
In a similar way for 119891 we have1003817100381710038171003817119891 (sdot 120593 (sdot) 120593 (120574 (sdot))) minus 119891 (sdot 120595 (sdot) 120595 (120574 (sdot)))
1003817100381710038171003817
le 212058111198771003817100381710038171003817120593 minus 120595
1003817100381710038171003817
(70)
ByTheorem 14 the DEPCAG system (65) has a unique 2120587120596-periodic solution in S
Conflict of Interests
The author declares that there is no conflict of interests re-garding the publication of this paper
Acknowledgment
This research was in part supported by FIBE 01-12 DIUMCE
References
[1] ADMyshkis ldquoOn certain problems in the theory of differentialequations with deviating argumentsrdquo Uspekhi Matematich-eskikh Nauk vol 32 pp 173ndash202 1977
[2] K L Cooke and JWiener ldquoRetarded differential equations withpiecewise constant delaysrdquo Journal of Mathematical Analysisand Applications vol 99 no 1 pp 265ndash297 1984
[3] K L Cooke and J Wiener ldquoAn equation alternately of retardedand advanced typerdquo Proceedings of the American MathematicalSociety vol 99 pp 726ndash732 1987
[4] S M Shah and J Wiener ldquoAdvanced differential equations withpiecewise constant argument deviationsrdquo International JournalofMathematics andMathematical Sciences vol 6 no 4 pp 671ndash703 1983
[5] J WienerGeneralized Solutions of Functional Differential Equa-tions World Scientific Singapore 1993
[6] A R Aftabizadeh and J Wiener ldquoOscillatory and periodicsolutions of an equation alternately of retarded and advancedtypesrdquo Applicable Analysis vol 23 no 3 pp 219ndash231 1986
[7] A R Aftabizadeh J Wiener and J M Xu ldquoOscillatory andperiodic solutions of delay differential equations with piecewiseconstant argumentrdquo Proceedings of the American MathematicalSociety vol 99 pp 673ndash679 1987
[8] M U Akhmet ldquoIntegral manifolds of differential equationswith piecewise constant argument of generalized typerdquo Nonlin-ear Analysis Theory Methods and Applications vol 66 no 2pp 367ndash383 2007
[9] M U Akhmet Nonlinear Hybrid ContinuousDiscrete-TimeModels Atlantis Press Amsterdam The Netherlands 2011
[10] M U Akhmet C Buyukadali and T Ergenc ldquoPeriodic solu-tions of the hybrid system with small parameterrdquo NonlinearAnalysis Hybrid Systems vol 2 no 2 pp 532ndash543 2008
[11] A Cabada J B Ferreiro and J J Nieto ldquoGreenrsquos functionand comparison principles for first order periodic differentialequations with piecewise constant argumentsrdquo Journal of Math-ematical Analysis and Applications vol 291 no 2 pp 690ndash6972004
[12] K S Chiu and M Pinto ldquoVariation of parameters formula andGronwall inequality for differential equations with a generalpiecewise constant argumentrdquo Acta Mathematicae ApplicataeSinica vol 27 no 4 pp 561ndash568 2011
The Scientific World Journal 13
[13] K S Chiu and M Pinto ldquoOscillatory and periodic solutionsin alternately advanced and delayed differential equationsrdquoCarpathian Journal of Mathematics vol 29 no 2 pp 149ndash1582013
[14] J J Nieto and R Rodrıguez-Lopez ldquoGreenrsquos function forsecond-order periodic boundary value problemswith piecewiseconstant argumentsrdquo Journal of Mathematical Analysis andApplications vol 304 no 1 pp 33ndash57 2005
[15] M Pinto ldquoAsymptotic equivalence of nonlinear and quasi lin-ear differential equations with piecewise constant argumentsrdquoMathematical and Computer Modelling vol 49 no 9-10 pp1750ndash1758 2009
[16] M Pinto ldquoCauchy and Green matrices type and stability inalternately advanced and delayed differential systemsrdquo Journalof Difference Equations and Applications vol 17 no 2 pp 235ndash254 2011
[17] G Q Wang ldquoExistence theorem of periodic solutions for adelay nonlinear differential equation with piecewise constantargumentsrdquo Journal of Mathematical Analysis and Applicationsvol 298 no 1 pp 298ndash307 2004
[18] G Q Wang and S S Cheng ldquoExistence and uniqueness ofperiodic solutions for a second-order nonlinear differentialequationwith piecewise constant argumentrdquo International Jour-nal ofMathematics andMathematical Sciences vol 2009 ArticleID 950797 14 pages 2009
[19] Y H Xia Z Huang and M Han ldquoExistence of almost periodicsolutions for forced perturbed systems with piecewise constantargumentrdquo Journal of Mathematical Analysis and Applicationsvol 333 no 2 pp 798ndash816 2007
[20] P Yang Y Liu and W Ge ldquoGreenrsquos function for second orderdifferential equations with piecewise constant argumentsrdquoNon-linear Analysis Theory Methods and Applications vol 64 no 8pp 1812ndash1830 2006
[21] R Yuan ldquoThe existence of almost periodic solutions of retardeddifferential equations with piecewise constant argumentrdquo Non-linear Analysis Theory Methods and Applications vol 48 no 7pp 1013ndash1032 2002
[22] A I Alonso J Hong and R Obaya ldquoAlmost periodic typesolutions of differential equations with piecewise constant argu-ment via almost periodic type sequencesrdquo Applied MathematicsLetters vol 13 no 2 pp 131ndash137 2000
[23] K S Chiu and M Pinto ldquoPeriodic solutions of differentialequations with a general piecewise constant argument andapplicationsrdquo Electronic Journal of Qualitative Theory of Differ-ential Equations vol 46 pp 1ndash19 2010
[24] K S Chiu M Pinto and J C Jeng ldquoExistence and globalconvergence of periodic solutions in recurrent neural networkmodels with a general piecewise alternately advanced andretarded argumentrdquo Acta Applicandae Mathematicae 2013
[25] L Dai Nonlinear Dynamics of Piecewise of Constant Systemsand Implememtation of Piecewise Constants Arguments WorldScientific Singapore 2008
[26] T A Burton Stability and Periodic Solutions of Ordinary andFunctional Differential Equations Academic Press New YorkNY USA 1985
[27] T A Burton and T Furumochi ldquoPeriodic and asymptoticallyperiodic solutions of neutral integral equationsrdquo ElectronicJournal of Qualitative Theory of Differential Equations vol 10pp 1ndash19 2000
[28] T A Burton and T Furumochi ldquoExistence theorems andperiodic solutions of neutral integral equationsrdquo NonlinearAnalysis Theory Methods and Applications vol 43 no 4 pp527ndash546 2001
[29] F Karakoc H Bereketoglu and G Seyhan ldquoOscillatory andperiodic solutions of impulsive differential equations withpiecewise constant argumentrdquoActa ApplicandaeMathematicaevol 110 no 1 pp 499ndash510 2010
[30] Y Liu P Yang and W Ge ldquoPeriodic solutions of higher-order delay differential equationsrdquo Nonlinear Analysis TheoryMethods and Applications vol 63 no 1 pp 136ndash152 2005
[31] X L Liu and W T Li ldquoPeriodic solutions for dynamic equa-tions on time scalesrdquo Nonlinear Analysis Theory Methods andApplications vol 67 no 5 pp 1457ndash1463 2007
[32] M Pinto ldquoDichotomy and existence of periodic solutions ofquasilinear functional differential equationsrdquo Nonlinear Analy-sis Theory Methods and Applications vol 72 no 3-4 pp 1227ndash1234 2010
[33] Y N Raffoul ldquoPeriodic solutions for neutral nonlinear differ-ential equations with functional delayrdquo Electronic Journal ofDifferential Equations vol 2003 no 102 pp 1ndash7 2003
[34] G Q Wang and S S Cheng ldquoPeriodic solutions of discreteRayleigh equations with deviating argumentsrdquo Taiwanese Jour-nal of Mathematics vol 13 no 6 pp 2051ndash2067 2009
[35] K S Chiu ldquoStability of oscillatory solutions of differential equa-tions with a general piecewise constant argumentrdquo ElectronicJournal of Qualitative Theory of Differential Equations vol 88pp 1ndash15 2011
[36] R N Butris and M A Aziz ldquoSome theorems in the existenceand uniqueness for system of nonlinear integro-differentialequationsrdquo Journal Education and Science vol 18 pp 76ndash892006
[37] A YMitropolsky andD IMortynyukPeriodic Solutions for theOscillations Systems with Retarded Argument General SchoolKiev Ukraine 1979
[38] L L Tai ldquoNumerical-analytic method for the investigation ofautonomous systems of differential equationsrdquo Ukrainskii Ma-tematicheskii Zhurnal vol 30 no 3 pp 309ndash317 1978
[39] Yu D Shlapak ldquoPeriodic solutions of ordinary differentialequations of first order that are not solved with respect to thederivativerdquo Ukrainian Mathematical Journal vol 32 no 5 pp420ndash425 1980
[40] A M Samoilenko ldquoNumerical-analytic method for the investi-gation of periodic systems of ordinary differential equationsrdquoUkrainskii Matematicheskii Zhurnal vol 17 no 4 pp 16ndash231965
[41] AM Samoilenko andN I RontoNumerical-Analytic Methodsfor Investigations of Periodic Solutions Mir Publishers MoscowRussia 1979
[42] R N Butris ldquoPeriodic solution of nonlinear system of integro-differential equations depending on the gamma distributionrdquoGeneral Mathematics Notes vol 15 no 1 pp 56ndash71 2013
[43] D R Smart Fixed Points Theorems Cambridge UniversityPress Cambridge UK 1980
[44] T A Burton ldquoKrasnoselskiirsquos inversion principle and fixedpointsrdquo Nonlinear Analysis Theory Methods and Applicationsvol 30 no 7 pp 3975ndash3986 1997
[45] T A Burton ldquoA fixed-point theorem of Krasnoselskiirdquo AppliedMathematics Letters vol 11 no 1 pp 85ndash88 1998
14 The Scientific World Journal
[46] M Pinto and D Sepulveda ldquoh-asymptotic stability by fixedpoint in neutral nonlinear differential equations with delayrdquoNonlinear Analysis Theory Methods and Applications vol 74no 12 pp 3926ndash3933 2011
[47] M A Krasnoselskii Translation Along Trajectories of Differen-tial Equations Nauka Moscow Russia 1966 (Russian) Englishtranslation American Mathematical Society Providence RIUSA 1968
Krasnoselskii (see [47]) proved that if 119860 is a stable constantmatrix without piecewise alternately advanced and retardedargument and lim
|119909|+|119910|rarr+infin(|119892(119905 119909 119910)|(|119909|+|119910|)) = 0 then
the system (56) has at least one periodic solution In our caseapplyingTheorem 12 this result is also valid for the DEPCAGsystem (56) requiring the hypothesis (119873
120596) (1198753) (119872119892) 119860(119905)
and 119892(119905 119909 119910) are periodic functions in 119905 with a period 120596 forall 119905 ge 120591 and |119892(119905 119909 119910)| le 119888
1(|119909| + |119910|) + 120588
2 with 2119888
1120596 lt 119862
2
and 1205882constants for |119909| + |119910| le 119877 119877 gt 0
Remark 17 Considering the nonlinear system of differentialequations with a general piecewise alternately advanced andretarded argument
In this particular case applyingTheorem 13 this result is alsovalid for the DEPCAG system (57) requiring the hypothesis(119873120596) (1198751) (1198753) (119871119892) (119872119891) and 119888
119866(1198621+1198712)119877+119888119866(120573+1205881)120596 le 119877
with = 1
Remark 18 Suppose that (1198751) is satisfied by 120596 = 120596
1
(1198752) by 120596 = 120596
2 and (119875
3) by 120596 = 120596
3 if 120596
119894120596119895is a
rational number for all 119894 119895 = 1 2 3 then (1198751) (1198752) and
(1198753) are simultaneously satisfied by 120596 = lcm120596
1 1205962 1205963
where lcm1205961 1205962 1205963 denotes the least common multiple
between 1205961 1205962and 120596
3 In the general case it is possible
that there exist five possible periods 1205961for 119860 120596
2for 119891 120596
3
for 119892 1205964for 119862 and the sequences 119905
119894119894isinZ 120574119894119894isinZ satisfy the
(1205965 119901) condition If 120596
119894120596119895is a rational number for all 119894 119895 =
1 2 3 4 5 so in this situation our results insure the existenceof 120596-periodic solution with 120596 = lcm120596
1 1205962 1205963 1205964 1205965
Therefore the above results insure the existence of 120596-periodicsolutions of the DEPCAG system (6) These solutions arecalled subharmonic solutions See Corollaries 19ndash22
To determine criteria for the existence and uniqueness ofsubharmonic solutions of theDEPCAG system (6) fromnowon we make the following assumption
(119875120596) There exists 120596 = lcm120596
1 1205962 1205963 1205964 1205965 gt 0 120596
119894120596119895
which is a rational number for all 119894 119895 = 1 2 3 4 5
such that
(1) 119860(119905) 119891(119905 1199091 1199101) and 119892(119905 119909
2 1199102) are periodic
functions in 119905 with a period 1205961 1205962 and 120596
3
respectively for all 119905 ge 120591(2) 119862(119905 + 120596
4 119904 + 120596
4 1199093) = 119862(119905 119904 119909
3) for all 119905 ge 120591
(3) There exists 119901 isin Z+ for which the sequences119905119894119894isinZ 120574119894119894isinZ satisfy the (120596
5 119901) condition
As immediate corollaries of Theorems 12ndash15 and Remark 18the following results are true
The Scientific World Journal 11
Corollary 19 Suppose the hypotheses (119873120596) (119875120596) (119871119891) (119871119862)
(119872119892) and (49) hold Then the DEPCAG system (6) has at least
one subharmonic solution in S
Corollary 20 Suppose the hypotheses (119873120596) (119875120596) (119871119892) (119872119891)
(119872119862) (119862) and (51) hold Then the DEPCAG system (6) has at
least one subharmonic solution in S
Corollary 21 Suppose the hypotheses (119873120596) (119875120596) (119871119891) (119871119862)
(119871119892) and (53) holdThen theDEPCAG system (6) has a unique
subharmonic solution
Corollary 22 Suppose the hypotheses (119873120596) (119875120596) (119872
119891)
(119872119862) (119872119892) (119862) and (55) hold Then the DEPCAG system (6)
has at least one subharmonic solution in S
4 Applications and Illustrative Examples
Wewill introduce appropriate examples in this sectionTheseexamples will show the feasibility of our theory
Mathematical modelling of real-life problems usuallyresults in functional equations like ordinary or partial differ-ential equations integral and integro-differential equationsand stochastic equations Many mathematical formulationsof physical phenomena contain integro-differential equa-tions these equations arise inmany fields like fluid dynamicsbiologicalmodels and chemical kinetics So we first considernonlinear integro-differential equations with a general piece-wise constant argument mentioned in the introduction andobtain some new sufficient conditions for the existence of theperiodic solutions of these systems
Example 1 Let 120582 R2 rarr [0infin) and ℎ R2 rarr [0infin) betwo functions satisfying
sup119905isinR
int
119905+2120587
119905
120582 (119905 119904) 119889119904 le sup119905isinR
int
119905+2120587
119905
ℎ (119905 119904) 119889119904 le ℎ (58)
Consider the following nonlinear integro-differential equa-tions with piecewise alternately advanced and retarded argu-ment of generalized type
|119862(119905 119904 119910)| le 1198882(119877)120582(119905 119904)|119910| +ℎ(119905 119904) for |119910| le 119877 where
2120587 le 1198621
(iv) 119892 R times R119899 times R119899 rarr R119899 is continuous and 119862(119905 119904 119910)satisfies (119862)
Indeed for any 120576 gt 0 there exists 120575 gt 0 such that |1199101minus1199102| le 120575
implies
1003816100381610038161003816119862 (119905 119904 1199101) minus 119862 (119905 119904 1199102)1003816100381610038161003816 le 1205761198882 (119877) 120582 (119905 119904) for 119905 119904 isin R
(60)
Furthermore there exists 119877 such that
exp (2119886lowast120587)exp (2119886
lowast120587) minus 1
[(1198621+ 1198622) 119877 + 2120587 (120588
2+ ℎ)] le 119877 (61)
where 119886lowast = sup119905isinR |119886(119905)| and 119886
lowast= inf119905isinR|119886(119905)|
Then by Theorem 15 the DEPCAG system (59) has atleast one 2120587-periodic solution
Example 2 Thus many examples can be constructed whereour results can be applied
Let Λ R rarr R119899times119899 and 120583 R rarr R119899 be two functionssatisfying
sup119905isinR
int
119905+120596
119905
|Λ (119904)| 119889119904 = Λ lt infin
sup119905isinR
int
119905+120596
119905
1003816100381610038161003816120583 (119905 minus 119904)1003816100381610038161003816 119889119904 = 120583 lt infin
(62)
Now consider the integro-differential system with piecewisealternately advanced and retarded argument
Theorem 12 implies that there exists at least a 120596-periodicsolution of the DEPCAG system (63)
12 The Scientific World Journal
Note that similar results can be obtained under (119871119892)
and (119872119862) On the other hand the periodic situation of the
DEPCAGsystem (56) and (57) can be treated in the samewayLet us consider another example for second-order differ-
ential equations with a general piecewise constant argumentIn this case we can show the existence and uniquenessof periodic solutions of the following nonlinear DEPCAGsystem
Example 3 Consider the following nonlinear DEPCAG sys-tem
11991010158401015840
(119905) + (12058121199102
(119905) minus 2) 1199101015840
(119905) minus 8119910 (119905)
minus 1205811sin (120596119905) 1199102 (120574 (119905)) minus 120581
2cos (120596119905) = 0
(65)
where 1205811 1205812isin R 120574(119905) = 120574
Then for 120593 120595 isin S we have1003817100381710038171003817119892 (sdot 120593 (sdot) 120593 (120574 (sdot))) minus 119892 (sdot 120595 (sdot) 120595 (120574 (sdot)))
le 212058121198772 sup119905isin[120591120591+120596]
1003816100381610038161003816100381610038161003816(1205951 (119905) minus 1205931 (119905)
1205952 (119905) minus 1205932 (119905)
)
1003816100381610038161003816100381610038161003816= 212058121198772 1003817100381710038171003817120593 minus 120595
1003817100381710038171003817
(69)
In a similar way for 119891 we have1003817100381710038171003817119891 (sdot 120593 (sdot) 120593 (120574 (sdot))) minus 119891 (sdot 120595 (sdot) 120595 (120574 (sdot)))
1003817100381710038171003817
le 212058111198771003817100381710038171003817120593 minus 120595
1003817100381710038171003817
(70)
ByTheorem 14 the DEPCAG system (65) has a unique 2120587120596-periodic solution in S
Conflict of Interests
The author declares that there is no conflict of interests re-garding the publication of this paper
Acknowledgment
This research was in part supported by FIBE 01-12 DIUMCE
References
[1] ADMyshkis ldquoOn certain problems in the theory of differentialequations with deviating argumentsrdquo Uspekhi Matematich-eskikh Nauk vol 32 pp 173ndash202 1977
[2] K L Cooke and JWiener ldquoRetarded differential equations withpiecewise constant delaysrdquo Journal of Mathematical Analysisand Applications vol 99 no 1 pp 265ndash297 1984
[3] K L Cooke and J Wiener ldquoAn equation alternately of retardedand advanced typerdquo Proceedings of the American MathematicalSociety vol 99 pp 726ndash732 1987
[4] S M Shah and J Wiener ldquoAdvanced differential equations withpiecewise constant argument deviationsrdquo International JournalofMathematics andMathematical Sciences vol 6 no 4 pp 671ndash703 1983
[5] J WienerGeneralized Solutions of Functional Differential Equa-tions World Scientific Singapore 1993
[6] A R Aftabizadeh and J Wiener ldquoOscillatory and periodicsolutions of an equation alternately of retarded and advancedtypesrdquo Applicable Analysis vol 23 no 3 pp 219ndash231 1986
[7] A R Aftabizadeh J Wiener and J M Xu ldquoOscillatory andperiodic solutions of delay differential equations with piecewiseconstant argumentrdquo Proceedings of the American MathematicalSociety vol 99 pp 673ndash679 1987
[8] M U Akhmet ldquoIntegral manifolds of differential equationswith piecewise constant argument of generalized typerdquo Nonlin-ear Analysis Theory Methods and Applications vol 66 no 2pp 367ndash383 2007
[9] M U Akhmet Nonlinear Hybrid ContinuousDiscrete-TimeModels Atlantis Press Amsterdam The Netherlands 2011
[10] M U Akhmet C Buyukadali and T Ergenc ldquoPeriodic solu-tions of the hybrid system with small parameterrdquo NonlinearAnalysis Hybrid Systems vol 2 no 2 pp 532ndash543 2008
[11] A Cabada J B Ferreiro and J J Nieto ldquoGreenrsquos functionand comparison principles for first order periodic differentialequations with piecewise constant argumentsrdquo Journal of Math-ematical Analysis and Applications vol 291 no 2 pp 690ndash6972004
[12] K S Chiu and M Pinto ldquoVariation of parameters formula andGronwall inequality for differential equations with a generalpiecewise constant argumentrdquo Acta Mathematicae ApplicataeSinica vol 27 no 4 pp 561ndash568 2011
The Scientific World Journal 13
[13] K S Chiu and M Pinto ldquoOscillatory and periodic solutionsin alternately advanced and delayed differential equationsrdquoCarpathian Journal of Mathematics vol 29 no 2 pp 149ndash1582013
[14] J J Nieto and R Rodrıguez-Lopez ldquoGreenrsquos function forsecond-order periodic boundary value problemswith piecewiseconstant argumentsrdquo Journal of Mathematical Analysis andApplications vol 304 no 1 pp 33ndash57 2005
[15] M Pinto ldquoAsymptotic equivalence of nonlinear and quasi lin-ear differential equations with piecewise constant argumentsrdquoMathematical and Computer Modelling vol 49 no 9-10 pp1750ndash1758 2009
[16] M Pinto ldquoCauchy and Green matrices type and stability inalternately advanced and delayed differential systemsrdquo Journalof Difference Equations and Applications vol 17 no 2 pp 235ndash254 2011
[17] G Q Wang ldquoExistence theorem of periodic solutions for adelay nonlinear differential equation with piecewise constantargumentsrdquo Journal of Mathematical Analysis and Applicationsvol 298 no 1 pp 298ndash307 2004
[18] G Q Wang and S S Cheng ldquoExistence and uniqueness ofperiodic solutions for a second-order nonlinear differentialequationwith piecewise constant argumentrdquo International Jour-nal ofMathematics andMathematical Sciences vol 2009 ArticleID 950797 14 pages 2009
[19] Y H Xia Z Huang and M Han ldquoExistence of almost periodicsolutions for forced perturbed systems with piecewise constantargumentrdquo Journal of Mathematical Analysis and Applicationsvol 333 no 2 pp 798ndash816 2007
[20] P Yang Y Liu and W Ge ldquoGreenrsquos function for second orderdifferential equations with piecewise constant argumentsrdquoNon-linear Analysis Theory Methods and Applications vol 64 no 8pp 1812ndash1830 2006
[21] R Yuan ldquoThe existence of almost periodic solutions of retardeddifferential equations with piecewise constant argumentrdquo Non-linear Analysis Theory Methods and Applications vol 48 no 7pp 1013ndash1032 2002
[22] A I Alonso J Hong and R Obaya ldquoAlmost periodic typesolutions of differential equations with piecewise constant argu-ment via almost periodic type sequencesrdquo Applied MathematicsLetters vol 13 no 2 pp 131ndash137 2000
[23] K S Chiu and M Pinto ldquoPeriodic solutions of differentialequations with a general piecewise constant argument andapplicationsrdquo Electronic Journal of Qualitative Theory of Differ-ential Equations vol 46 pp 1ndash19 2010
[24] K S Chiu M Pinto and J C Jeng ldquoExistence and globalconvergence of periodic solutions in recurrent neural networkmodels with a general piecewise alternately advanced andretarded argumentrdquo Acta Applicandae Mathematicae 2013
[25] L Dai Nonlinear Dynamics of Piecewise of Constant Systemsand Implememtation of Piecewise Constants Arguments WorldScientific Singapore 2008
[26] T A Burton Stability and Periodic Solutions of Ordinary andFunctional Differential Equations Academic Press New YorkNY USA 1985
[27] T A Burton and T Furumochi ldquoPeriodic and asymptoticallyperiodic solutions of neutral integral equationsrdquo ElectronicJournal of Qualitative Theory of Differential Equations vol 10pp 1ndash19 2000
[28] T A Burton and T Furumochi ldquoExistence theorems andperiodic solutions of neutral integral equationsrdquo NonlinearAnalysis Theory Methods and Applications vol 43 no 4 pp527ndash546 2001
[29] F Karakoc H Bereketoglu and G Seyhan ldquoOscillatory andperiodic solutions of impulsive differential equations withpiecewise constant argumentrdquoActa ApplicandaeMathematicaevol 110 no 1 pp 499ndash510 2010
[30] Y Liu P Yang and W Ge ldquoPeriodic solutions of higher-order delay differential equationsrdquo Nonlinear Analysis TheoryMethods and Applications vol 63 no 1 pp 136ndash152 2005
[31] X L Liu and W T Li ldquoPeriodic solutions for dynamic equa-tions on time scalesrdquo Nonlinear Analysis Theory Methods andApplications vol 67 no 5 pp 1457ndash1463 2007
[32] M Pinto ldquoDichotomy and existence of periodic solutions ofquasilinear functional differential equationsrdquo Nonlinear Analy-sis Theory Methods and Applications vol 72 no 3-4 pp 1227ndash1234 2010
[33] Y N Raffoul ldquoPeriodic solutions for neutral nonlinear differ-ential equations with functional delayrdquo Electronic Journal ofDifferential Equations vol 2003 no 102 pp 1ndash7 2003
[34] G Q Wang and S S Cheng ldquoPeriodic solutions of discreteRayleigh equations with deviating argumentsrdquo Taiwanese Jour-nal of Mathematics vol 13 no 6 pp 2051ndash2067 2009
[35] K S Chiu ldquoStability of oscillatory solutions of differential equa-tions with a general piecewise constant argumentrdquo ElectronicJournal of Qualitative Theory of Differential Equations vol 88pp 1ndash15 2011
[36] R N Butris and M A Aziz ldquoSome theorems in the existenceand uniqueness for system of nonlinear integro-differentialequationsrdquo Journal Education and Science vol 18 pp 76ndash892006
[37] A YMitropolsky andD IMortynyukPeriodic Solutions for theOscillations Systems with Retarded Argument General SchoolKiev Ukraine 1979
[38] L L Tai ldquoNumerical-analytic method for the investigation ofautonomous systems of differential equationsrdquo Ukrainskii Ma-tematicheskii Zhurnal vol 30 no 3 pp 309ndash317 1978
[39] Yu D Shlapak ldquoPeriodic solutions of ordinary differentialequations of first order that are not solved with respect to thederivativerdquo Ukrainian Mathematical Journal vol 32 no 5 pp420ndash425 1980
[40] A M Samoilenko ldquoNumerical-analytic method for the investi-gation of periodic systems of ordinary differential equationsrdquoUkrainskii Matematicheskii Zhurnal vol 17 no 4 pp 16ndash231965
[41] AM Samoilenko andN I RontoNumerical-Analytic Methodsfor Investigations of Periodic Solutions Mir Publishers MoscowRussia 1979
[42] R N Butris ldquoPeriodic solution of nonlinear system of integro-differential equations depending on the gamma distributionrdquoGeneral Mathematics Notes vol 15 no 1 pp 56ndash71 2013
[43] D R Smart Fixed Points Theorems Cambridge UniversityPress Cambridge UK 1980
[44] T A Burton ldquoKrasnoselskiirsquos inversion principle and fixedpointsrdquo Nonlinear Analysis Theory Methods and Applicationsvol 30 no 7 pp 3975ndash3986 1997
[45] T A Burton ldquoA fixed-point theorem of Krasnoselskiirdquo AppliedMathematics Letters vol 11 no 1 pp 85ndash88 1998
14 The Scientific World Journal
[46] M Pinto and D Sepulveda ldquoh-asymptotic stability by fixedpoint in neutral nonlinear differential equations with delayrdquoNonlinear Analysis Theory Methods and Applications vol 74no 12 pp 3926ndash3933 2011
[47] M A Krasnoselskii Translation Along Trajectories of Differen-tial Equations Nauka Moscow Russia 1966 (Russian) Englishtranslation American Mathematical Society Providence RIUSA 1968
Corollary 19 Suppose the hypotheses (119873120596) (119875120596) (119871119891) (119871119862)
(119872119892) and (49) hold Then the DEPCAG system (6) has at least
one subharmonic solution in S
Corollary 20 Suppose the hypotheses (119873120596) (119875120596) (119871119892) (119872119891)
(119872119862) (119862) and (51) hold Then the DEPCAG system (6) has at
least one subharmonic solution in S
Corollary 21 Suppose the hypotheses (119873120596) (119875120596) (119871119891) (119871119862)
(119871119892) and (53) holdThen theDEPCAG system (6) has a unique
subharmonic solution
Corollary 22 Suppose the hypotheses (119873120596) (119875120596) (119872
119891)
(119872119862) (119872119892) (119862) and (55) hold Then the DEPCAG system (6)
has at least one subharmonic solution in S
4 Applications and Illustrative Examples
Wewill introduce appropriate examples in this sectionTheseexamples will show the feasibility of our theory
Mathematical modelling of real-life problems usuallyresults in functional equations like ordinary or partial differ-ential equations integral and integro-differential equationsand stochastic equations Many mathematical formulationsof physical phenomena contain integro-differential equa-tions these equations arise inmany fields like fluid dynamicsbiologicalmodels and chemical kinetics So we first considernonlinear integro-differential equations with a general piece-wise constant argument mentioned in the introduction andobtain some new sufficient conditions for the existence of theperiodic solutions of these systems
Example 1 Let 120582 R2 rarr [0infin) and ℎ R2 rarr [0infin) betwo functions satisfying
sup119905isinR
int
119905+2120587
119905
120582 (119905 119904) 119889119904 le sup119905isinR
int
119905+2120587
119905
ℎ (119905 119904) 119889119904 le ℎ (58)
Consider the following nonlinear integro-differential equa-tions with piecewise alternately advanced and retarded argu-ment of generalized type
|119862(119905 119904 119910)| le 1198882(119877)120582(119905 119904)|119910| +ℎ(119905 119904) for |119910| le 119877 where
2120587 le 1198621
(iv) 119892 R times R119899 times R119899 rarr R119899 is continuous and 119862(119905 119904 119910)satisfies (119862)
Indeed for any 120576 gt 0 there exists 120575 gt 0 such that |1199101minus1199102| le 120575
implies
1003816100381610038161003816119862 (119905 119904 1199101) minus 119862 (119905 119904 1199102)1003816100381610038161003816 le 1205761198882 (119877) 120582 (119905 119904) for 119905 119904 isin R
(60)
Furthermore there exists 119877 such that
exp (2119886lowast120587)exp (2119886
lowast120587) minus 1
[(1198621+ 1198622) 119877 + 2120587 (120588
2+ ℎ)] le 119877 (61)
where 119886lowast = sup119905isinR |119886(119905)| and 119886
lowast= inf119905isinR|119886(119905)|
Then by Theorem 15 the DEPCAG system (59) has atleast one 2120587-periodic solution
Example 2 Thus many examples can be constructed whereour results can be applied
Let Λ R rarr R119899times119899 and 120583 R rarr R119899 be two functionssatisfying
sup119905isinR
int
119905+120596
119905
|Λ (119904)| 119889119904 = Λ lt infin
sup119905isinR
int
119905+120596
119905
1003816100381610038161003816120583 (119905 minus 119904)1003816100381610038161003816 119889119904 = 120583 lt infin
(62)
Now consider the integro-differential system with piecewisealternately advanced and retarded argument
Theorem 12 implies that there exists at least a 120596-periodicsolution of the DEPCAG system (63)
12 The Scientific World Journal
Note that similar results can be obtained under (119871119892)
and (119872119862) On the other hand the periodic situation of the
DEPCAGsystem (56) and (57) can be treated in the samewayLet us consider another example for second-order differ-
ential equations with a general piecewise constant argumentIn this case we can show the existence and uniquenessof periodic solutions of the following nonlinear DEPCAGsystem
Example 3 Consider the following nonlinear DEPCAG sys-tem
11991010158401015840
(119905) + (12058121199102
(119905) minus 2) 1199101015840
(119905) minus 8119910 (119905)
minus 1205811sin (120596119905) 1199102 (120574 (119905)) minus 120581
2cos (120596119905) = 0
(65)
where 1205811 1205812isin R 120574(119905) = 120574
Then for 120593 120595 isin S we have1003817100381710038171003817119892 (sdot 120593 (sdot) 120593 (120574 (sdot))) minus 119892 (sdot 120595 (sdot) 120595 (120574 (sdot)))
le 212058121198772 sup119905isin[120591120591+120596]
1003816100381610038161003816100381610038161003816(1205951 (119905) minus 1205931 (119905)
1205952 (119905) minus 1205932 (119905)
)
1003816100381610038161003816100381610038161003816= 212058121198772 1003817100381710038171003817120593 minus 120595
1003817100381710038171003817
(69)
In a similar way for 119891 we have1003817100381710038171003817119891 (sdot 120593 (sdot) 120593 (120574 (sdot))) minus 119891 (sdot 120595 (sdot) 120595 (120574 (sdot)))
1003817100381710038171003817
le 212058111198771003817100381710038171003817120593 minus 120595
1003817100381710038171003817
(70)
ByTheorem 14 the DEPCAG system (65) has a unique 2120587120596-periodic solution in S
Conflict of Interests
The author declares that there is no conflict of interests re-garding the publication of this paper
Acknowledgment
This research was in part supported by FIBE 01-12 DIUMCE
References
[1] ADMyshkis ldquoOn certain problems in the theory of differentialequations with deviating argumentsrdquo Uspekhi Matematich-eskikh Nauk vol 32 pp 173ndash202 1977
[2] K L Cooke and JWiener ldquoRetarded differential equations withpiecewise constant delaysrdquo Journal of Mathematical Analysisand Applications vol 99 no 1 pp 265ndash297 1984
[3] K L Cooke and J Wiener ldquoAn equation alternately of retardedand advanced typerdquo Proceedings of the American MathematicalSociety vol 99 pp 726ndash732 1987
[4] S M Shah and J Wiener ldquoAdvanced differential equations withpiecewise constant argument deviationsrdquo International JournalofMathematics andMathematical Sciences vol 6 no 4 pp 671ndash703 1983
[5] J WienerGeneralized Solutions of Functional Differential Equa-tions World Scientific Singapore 1993
[6] A R Aftabizadeh and J Wiener ldquoOscillatory and periodicsolutions of an equation alternately of retarded and advancedtypesrdquo Applicable Analysis vol 23 no 3 pp 219ndash231 1986
[7] A R Aftabizadeh J Wiener and J M Xu ldquoOscillatory andperiodic solutions of delay differential equations with piecewiseconstant argumentrdquo Proceedings of the American MathematicalSociety vol 99 pp 673ndash679 1987
[8] M U Akhmet ldquoIntegral manifolds of differential equationswith piecewise constant argument of generalized typerdquo Nonlin-ear Analysis Theory Methods and Applications vol 66 no 2pp 367ndash383 2007
[9] M U Akhmet Nonlinear Hybrid ContinuousDiscrete-TimeModels Atlantis Press Amsterdam The Netherlands 2011
[10] M U Akhmet C Buyukadali and T Ergenc ldquoPeriodic solu-tions of the hybrid system with small parameterrdquo NonlinearAnalysis Hybrid Systems vol 2 no 2 pp 532ndash543 2008
[11] A Cabada J B Ferreiro and J J Nieto ldquoGreenrsquos functionand comparison principles for first order periodic differentialequations with piecewise constant argumentsrdquo Journal of Math-ematical Analysis and Applications vol 291 no 2 pp 690ndash6972004
[12] K S Chiu and M Pinto ldquoVariation of parameters formula andGronwall inequality for differential equations with a generalpiecewise constant argumentrdquo Acta Mathematicae ApplicataeSinica vol 27 no 4 pp 561ndash568 2011
The Scientific World Journal 13
[13] K S Chiu and M Pinto ldquoOscillatory and periodic solutionsin alternately advanced and delayed differential equationsrdquoCarpathian Journal of Mathematics vol 29 no 2 pp 149ndash1582013
[14] J J Nieto and R Rodrıguez-Lopez ldquoGreenrsquos function forsecond-order periodic boundary value problemswith piecewiseconstant argumentsrdquo Journal of Mathematical Analysis andApplications vol 304 no 1 pp 33ndash57 2005
[15] M Pinto ldquoAsymptotic equivalence of nonlinear and quasi lin-ear differential equations with piecewise constant argumentsrdquoMathematical and Computer Modelling vol 49 no 9-10 pp1750ndash1758 2009
[16] M Pinto ldquoCauchy and Green matrices type and stability inalternately advanced and delayed differential systemsrdquo Journalof Difference Equations and Applications vol 17 no 2 pp 235ndash254 2011
[17] G Q Wang ldquoExistence theorem of periodic solutions for adelay nonlinear differential equation with piecewise constantargumentsrdquo Journal of Mathematical Analysis and Applicationsvol 298 no 1 pp 298ndash307 2004
[18] G Q Wang and S S Cheng ldquoExistence and uniqueness ofperiodic solutions for a second-order nonlinear differentialequationwith piecewise constant argumentrdquo International Jour-nal ofMathematics andMathematical Sciences vol 2009 ArticleID 950797 14 pages 2009
[19] Y H Xia Z Huang and M Han ldquoExistence of almost periodicsolutions for forced perturbed systems with piecewise constantargumentrdquo Journal of Mathematical Analysis and Applicationsvol 333 no 2 pp 798ndash816 2007
[20] P Yang Y Liu and W Ge ldquoGreenrsquos function for second orderdifferential equations with piecewise constant argumentsrdquoNon-linear Analysis Theory Methods and Applications vol 64 no 8pp 1812ndash1830 2006
[21] R Yuan ldquoThe existence of almost periodic solutions of retardeddifferential equations with piecewise constant argumentrdquo Non-linear Analysis Theory Methods and Applications vol 48 no 7pp 1013ndash1032 2002
[22] A I Alonso J Hong and R Obaya ldquoAlmost periodic typesolutions of differential equations with piecewise constant argu-ment via almost periodic type sequencesrdquo Applied MathematicsLetters vol 13 no 2 pp 131ndash137 2000
[23] K S Chiu and M Pinto ldquoPeriodic solutions of differentialequations with a general piecewise constant argument andapplicationsrdquo Electronic Journal of Qualitative Theory of Differ-ential Equations vol 46 pp 1ndash19 2010
[24] K S Chiu M Pinto and J C Jeng ldquoExistence and globalconvergence of periodic solutions in recurrent neural networkmodels with a general piecewise alternately advanced andretarded argumentrdquo Acta Applicandae Mathematicae 2013
[25] L Dai Nonlinear Dynamics of Piecewise of Constant Systemsand Implememtation of Piecewise Constants Arguments WorldScientific Singapore 2008
[26] T A Burton Stability and Periodic Solutions of Ordinary andFunctional Differential Equations Academic Press New YorkNY USA 1985
[27] T A Burton and T Furumochi ldquoPeriodic and asymptoticallyperiodic solutions of neutral integral equationsrdquo ElectronicJournal of Qualitative Theory of Differential Equations vol 10pp 1ndash19 2000
[28] T A Burton and T Furumochi ldquoExistence theorems andperiodic solutions of neutral integral equationsrdquo NonlinearAnalysis Theory Methods and Applications vol 43 no 4 pp527ndash546 2001
[29] F Karakoc H Bereketoglu and G Seyhan ldquoOscillatory andperiodic solutions of impulsive differential equations withpiecewise constant argumentrdquoActa ApplicandaeMathematicaevol 110 no 1 pp 499ndash510 2010
[30] Y Liu P Yang and W Ge ldquoPeriodic solutions of higher-order delay differential equationsrdquo Nonlinear Analysis TheoryMethods and Applications vol 63 no 1 pp 136ndash152 2005
[31] X L Liu and W T Li ldquoPeriodic solutions for dynamic equa-tions on time scalesrdquo Nonlinear Analysis Theory Methods andApplications vol 67 no 5 pp 1457ndash1463 2007
[32] M Pinto ldquoDichotomy and existence of periodic solutions ofquasilinear functional differential equationsrdquo Nonlinear Analy-sis Theory Methods and Applications vol 72 no 3-4 pp 1227ndash1234 2010
[33] Y N Raffoul ldquoPeriodic solutions for neutral nonlinear differ-ential equations with functional delayrdquo Electronic Journal ofDifferential Equations vol 2003 no 102 pp 1ndash7 2003
[34] G Q Wang and S S Cheng ldquoPeriodic solutions of discreteRayleigh equations with deviating argumentsrdquo Taiwanese Jour-nal of Mathematics vol 13 no 6 pp 2051ndash2067 2009
[35] K S Chiu ldquoStability of oscillatory solutions of differential equa-tions with a general piecewise constant argumentrdquo ElectronicJournal of Qualitative Theory of Differential Equations vol 88pp 1ndash15 2011
[36] R N Butris and M A Aziz ldquoSome theorems in the existenceand uniqueness for system of nonlinear integro-differentialequationsrdquo Journal Education and Science vol 18 pp 76ndash892006
[37] A YMitropolsky andD IMortynyukPeriodic Solutions for theOscillations Systems with Retarded Argument General SchoolKiev Ukraine 1979
[38] L L Tai ldquoNumerical-analytic method for the investigation ofautonomous systems of differential equationsrdquo Ukrainskii Ma-tematicheskii Zhurnal vol 30 no 3 pp 309ndash317 1978
[39] Yu D Shlapak ldquoPeriodic solutions of ordinary differentialequations of first order that are not solved with respect to thederivativerdquo Ukrainian Mathematical Journal vol 32 no 5 pp420ndash425 1980
[40] A M Samoilenko ldquoNumerical-analytic method for the investi-gation of periodic systems of ordinary differential equationsrdquoUkrainskii Matematicheskii Zhurnal vol 17 no 4 pp 16ndash231965
[41] AM Samoilenko andN I RontoNumerical-Analytic Methodsfor Investigations of Periodic Solutions Mir Publishers MoscowRussia 1979
[42] R N Butris ldquoPeriodic solution of nonlinear system of integro-differential equations depending on the gamma distributionrdquoGeneral Mathematics Notes vol 15 no 1 pp 56ndash71 2013
[43] D R Smart Fixed Points Theorems Cambridge UniversityPress Cambridge UK 1980
[44] T A Burton ldquoKrasnoselskiirsquos inversion principle and fixedpointsrdquo Nonlinear Analysis Theory Methods and Applicationsvol 30 no 7 pp 3975ndash3986 1997
[45] T A Burton ldquoA fixed-point theorem of Krasnoselskiirdquo AppliedMathematics Letters vol 11 no 1 pp 85ndash88 1998
14 The Scientific World Journal
[46] M Pinto and D Sepulveda ldquoh-asymptotic stability by fixedpoint in neutral nonlinear differential equations with delayrdquoNonlinear Analysis Theory Methods and Applications vol 74no 12 pp 3926ndash3933 2011
[47] M A Krasnoselskii Translation Along Trajectories of Differen-tial Equations Nauka Moscow Russia 1966 (Russian) Englishtranslation American Mathematical Society Providence RIUSA 1968
Note that similar results can be obtained under (119871119892)
and (119872119862) On the other hand the periodic situation of the
DEPCAGsystem (56) and (57) can be treated in the samewayLet us consider another example for second-order differ-
ential equations with a general piecewise constant argumentIn this case we can show the existence and uniquenessof periodic solutions of the following nonlinear DEPCAGsystem
Example 3 Consider the following nonlinear DEPCAG sys-tem
11991010158401015840
(119905) + (12058121199102
(119905) minus 2) 1199101015840
(119905) minus 8119910 (119905)
minus 1205811sin (120596119905) 1199102 (120574 (119905)) minus 120581
2cos (120596119905) = 0
(65)
where 1205811 1205812isin R 120574(119905) = 120574
Then for 120593 120595 isin S we have1003817100381710038171003817119892 (sdot 120593 (sdot) 120593 (120574 (sdot))) minus 119892 (sdot 120595 (sdot) 120595 (120574 (sdot)))
le 212058121198772 sup119905isin[120591120591+120596]
1003816100381610038161003816100381610038161003816(1205951 (119905) minus 1205931 (119905)
1205952 (119905) minus 1205932 (119905)
)
1003816100381610038161003816100381610038161003816= 212058121198772 1003817100381710038171003817120593 minus 120595
1003817100381710038171003817
(69)
In a similar way for 119891 we have1003817100381710038171003817119891 (sdot 120593 (sdot) 120593 (120574 (sdot))) minus 119891 (sdot 120595 (sdot) 120595 (120574 (sdot)))
1003817100381710038171003817
le 212058111198771003817100381710038171003817120593 minus 120595
1003817100381710038171003817
(70)
ByTheorem 14 the DEPCAG system (65) has a unique 2120587120596-periodic solution in S
Conflict of Interests
The author declares that there is no conflict of interests re-garding the publication of this paper
Acknowledgment
This research was in part supported by FIBE 01-12 DIUMCE
References
[1] ADMyshkis ldquoOn certain problems in the theory of differentialequations with deviating argumentsrdquo Uspekhi Matematich-eskikh Nauk vol 32 pp 173ndash202 1977
[2] K L Cooke and JWiener ldquoRetarded differential equations withpiecewise constant delaysrdquo Journal of Mathematical Analysisand Applications vol 99 no 1 pp 265ndash297 1984
[3] K L Cooke and J Wiener ldquoAn equation alternately of retardedand advanced typerdquo Proceedings of the American MathematicalSociety vol 99 pp 726ndash732 1987
[4] S M Shah and J Wiener ldquoAdvanced differential equations withpiecewise constant argument deviationsrdquo International JournalofMathematics andMathematical Sciences vol 6 no 4 pp 671ndash703 1983
[5] J WienerGeneralized Solutions of Functional Differential Equa-tions World Scientific Singapore 1993
[6] A R Aftabizadeh and J Wiener ldquoOscillatory and periodicsolutions of an equation alternately of retarded and advancedtypesrdquo Applicable Analysis vol 23 no 3 pp 219ndash231 1986
[7] A R Aftabizadeh J Wiener and J M Xu ldquoOscillatory andperiodic solutions of delay differential equations with piecewiseconstant argumentrdquo Proceedings of the American MathematicalSociety vol 99 pp 673ndash679 1987
[8] M U Akhmet ldquoIntegral manifolds of differential equationswith piecewise constant argument of generalized typerdquo Nonlin-ear Analysis Theory Methods and Applications vol 66 no 2pp 367ndash383 2007
[9] M U Akhmet Nonlinear Hybrid ContinuousDiscrete-TimeModels Atlantis Press Amsterdam The Netherlands 2011
[10] M U Akhmet C Buyukadali and T Ergenc ldquoPeriodic solu-tions of the hybrid system with small parameterrdquo NonlinearAnalysis Hybrid Systems vol 2 no 2 pp 532ndash543 2008
[11] A Cabada J B Ferreiro and J J Nieto ldquoGreenrsquos functionand comparison principles for first order periodic differentialequations with piecewise constant argumentsrdquo Journal of Math-ematical Analysis and Applications vol 291 no 2 pp 690ndash6972004
[12] K S Chiu and M Pinto ldquoVariation of parameters formula andGronwall inequality for differential equations with a generalpiecewise constant argumentrdquo Acta Mathematicae ApplicataeSinica vol 27 no 4 pp 561ndash568 2011
The Scientific World Journal 13
[13] K S Chiu and M Pinto ldquoOscillatory and periodic solutionsin alternately advanced and delayed differential equationsrdquoCarpathian Journal of Mathematics vol 29 no 2 pp 149ndash1582013
[14] J J Nieto and R Rodrıguez-Lopez ldquoGreenrsquos function forsecond-order periodic boundary value problemswith piecewiseconstant argumentsrdquo Journal of Mathematical Analysis andApplications vol 304 no 1 pp 33ndash57 2005
[15] M Pinto ldquoAsymptotic equivalence of nonlinear and quasi lin-ear differential equations with piecewise constant argumentsrdquoMathematical and Computer Modelling vol 49 no 9-10 pp1750ndash1758 2009
[16] M Pinto ldquoCauchy and Green matrices type and stability inalternately advanced and delayed differential systemsrdquo Journalof Difference Equations and Applications vol 17 no 2 pp 235ndash254 2011
[17] G Q Wang ldquoExistence theorem of periodic solutions for adelay nonlinear differential equation with piecewise constantargumentsrdquo Journal of Mathematical Analysis and Applicationsvol 298 no 1 pp 298ndash307 2004
[18] G Q Wang and S S Cheng ldquoExistence and uniqueness ofperiodic solutions for a second-order nonlinear differentialequationwith piecewise constant argumentrdquo International Jour-nal ofMathematics andMathematical Sciences vol 2009 ArticleID 950797 14 pages 2009
[19] Y H Xia Z Huang and M Han ldquoExistence of almost periodicsolutions for forced perturbed systems with piecewise constantargumentrdquo Journal of Mathematical Analysis and Applicationsvol 333 no 2 pp 798ndash816 2007
[20] P Yang Y Liu and W Ge ldquoGreenrsquos function for second orderdifferential equations with piecewise constant argumentsrdquoNon-linear Analysis Theory Methods and Applications vol 64 no 8pp 1812ndash1830 2006
[21] R Yuan ldquoThe existence of almost periodic solutions of retardeddifferential equations with piecewise constant argumentrdquo Non-linear Analysis Theory Methods and Applications vol 48 no 7pp 1013ndash1032 2002
[22] A I Alonso J Hong and R Obaya ldquoAlmost periodic typesolutions of differential equations with piecewise constant argu-ment via almost periodic type sequencesrdquo Applied MathematicsLetters vol 13 no 2 pp 131ndash137 2000
[23] K S Chiu and M Pinto ldquoPeriodic solutions of differentialequations with a general piecewise constant argument andapplicationsrdquo Electronic Journal of Qualitative Theory of Differ-ential Equations vol 46 pp 1ndash19 2010
[24] K S Chiu M Pinto and J C Jeng ldquoExistence and globalconvergence of periodic solutions in recurrent neural networkmodels with a general piecewise alternately advanced andretarded argumentrdquo Acta Applicandae Mathematicae 2013
[25] L Dai Nonlinear Dynamics of Piecewise of Constant Systemsand Implememtation of Piecewise Constants Arguments WorldScientific Singapore 2008
[26] T A Burton Stability and Periodic Solutions of Ordinary andFunctional Differential Equations Academic Press New YorkNY USA 1985
[27] T A Burton and T Furumochi ldquoPeriodic and asymptoticallyperiodic solutions of neutral integral equationsrdquo ElectronicJournal of Qualitative Theory of Differential Equations vol 10pp 1ndash19 2000
[28] T A Burton and T Furumochi ldquoExistence theorems andperiodic solutions of neutral integral equationsrdquo NonlinearAnalysis Theory Methods and Applications vol 43 no 4 pp527ndash546 2001
[29] F Karakoc H Bereketoglu and G Seyhan ldquoOscillatory andperiodic solutions of impulsive differential equations withpiecewise constant argumentrdquoActa ApplicandaeMathematicaevol 110 no 1 pp 499ndash510 2010
[30] Y Liu P Yang and W Ge ldquoPeriodic solutions of higher-order delay differential equationsrdquo Nonlinear Analysis TheoryMethods and Applications vol 63 no 1 pp 136ndash152 2005
[31] X L Liu and W T Li ldquoPeriodic solutions for dynamic equa-tions on time scalesrdquo Nonlinear Analysis Theory Methods andApplications vol 67 no 5 pp 1457ndash1463 2007
[32] M Pinto ldquoDichotomy and existence of periodic solutions ofquasilinear functional differential equationsrdquo Nonlinear Analy-sis Theory Methods and Applications vol 72 no 3-4 pp 1227ndash1234 2010
[33] Y N Raffoul ldquoPeriodic solutions for neutral nonlinear differ-ential equations with functional delayrdquo Electronic Journal ofDifferential Equations vol 2003 no 102 pp 1ndash7 2003
[34] G Q Wang and S S Cheng ldquoPeriodic solutions of discreteRayleigh equations with deviating argumentsrdquo Taiwanese Jour-nal of Mathematics vol 13 no 6 pp 2051ndash2067 2009
[35] K S Chiu ldquoStability of oscillatory solutions of differential equa-tions with a general piecewise constant argumentrdquo ElectronicJournal of Qualitative Theory of Differential Equations vol 88pp 1ndash15 2011
[36] R N Butris and M A Aziz ldquoSome theorems in the existenceand uniqueness for system of nonlinear integro-differentialequationsrdquo Journal Education and Science vol 18 pp 76ndash892006
[37] A YMitropolsky andD IMortynyukPeriodic Solutions for theOscillations Systems with Retarded Argument General SchoolKiev Ukraine 1979
[38] L L Tai ldquoNumerical-analytic method for the investigation ofautonomous systems of differential equationsrdquo Ukrainskii Ma-tematicheskii Zhurnal vol 30 no 3 pp 309ndash317 1978
[39] Yu D Shlapak ldquoPeriodic solutions of ordinary differentialequations of first order that are not solved with respect to thederivativerdquo Ukrainian Mathematical Journal vol 32 no 5 pp420ndash425 1980
[40] A M Samoilenko ldquoNumerical-analytic method for the investi-gation of periodic systems of ordinary differential equationsrdquoUkrainskii Matematicheskii Zhurnal vol 17 no 4 pp 16ndash231965
[41] AM Samoilenko andN I RontoNumerical-Analytic Methodsfor Investigations of Periodic Solutions Mir Publishers MoscowRussia 1979
[42] R N Butris ldquoPeriodic solution of nonlinear system of integro-differential equations depending on the gamma distributionrdquoGeneral Mathematics Notes vol 15 no 1 pp 56ndash71 2013
[43] D R Smart Fixed Points Theorems Cambridge UniversityPress Cambridge UK 1980
[44] T A Burton ldquoKrasnoselskiirsquos inversion principle and fixedpointsrdquo Nonlinear Analysis Theory Methods and Applicationsvol 30 no 7 pp 3975ndash3986 1997
[45] T A Burton ldquoA fixed-point theorem of Krasnoselskiirdquo AppliedMathematics Letters vol 11 no 1 pp 85ndash88 1998
14 The Scientific World Journal
[46] M Pinto and D Sepulveda ldquoh-asymptotic stability by fixedpoint in neutral nonlinear differential equations with delayrdquoNonlinear Analysis Theory Methods and Applications vol 74no 12 pp 3926ndash3933 2011
[47] M A Krasnoselskii Translation Along Trajectories of Differen-tial Equations Nauka Moscow Russia 1966 (Russian) Englishtranslation American Mathematical Society Providence RIUSA 1968
[13] K S Chiu and M Pinto ldquoOscillatory and periodic solutionsin alternately advanced and delayed differential equationsrdquoCarpathian Journal of Mathematics vol 29 no 2 pp 149ndash1582013
[14] J J Nieto and R Rodrıguez-Lopez ldquoGreenrsquos function forsecond-order periodic boundary value problemswith piecewiseconstant argumentsrdquo Journal of Mathematical Analysis andApplications vol 304 no 1 pp 33ndash57 2005
[15] M Pinto ldquoAsymptotic equivalence of nonlinear and quasi lin-ear differential equations with piecewise constant argumentsrdquoMathematical and Computer Modelling vol 49 no 9-10 pp1750ndash1758 2009
[16] M Pinto ldquoCauchy and Green matrices type and stability inalternately advanced and delayed differential systemsrdquo Journalof Difference Equations and Applications vol 17 no 2 pp 235ndash254 2011
[17] G Q Wang ldquoExistence theorem of periodic solutions for adelay nonlinear differential equation with piecewise constantargumentsrdquo Journal of Mathematical Analysis and Applicationsvol 298 no 1 pp 298ndash307 2004
[18] G Q Wang and S S Cheng ldquoExistence and uniqueness ofperiodic solutions for a second-order nonlinear differentialequationwith piecewise constant argumentrdquo International Jour-nal ofMathematics andMathematical Sciences vol 2009 ArticleID 950797 14 pages 2009
[19] Y H Xia Z Huang and M Han ldquoExistence of almost periodicsolutions for forced perturbed systems with piecewise constantargumentrdquo Journal of Mathematical Analysis and Applicationsvol 333 no 2 pp 798ndash816 2007
[20] P Yang Y Liu and W Ge ldquoGreenrsquos function for second orderdifferential equations with piecewise constant argumentsrdquoNon-linear Analysis Theory Methods and Applications vol 64 no 8pp 1812ndash1830 2006
[21] R Yuan ldquoThe existence of almost periodic solutions of retardeddifferential equations with piecewise constant argumentrdquo Non-linear Analysis Theory Methods and Applications vol 48 no 7pp 1013ndash1032 2002
[22] A I Alonso J Hong and R Obaya ldquoAlmost periodic typesolutions of differential equations with piecewise constant argu-ment via almost periodic type sequencesrdquo Applied MathematicsLetters vol 13 no 2 pp 131ndash137 2000
[23] K S Chiu and M Pinto ldquoPeriodic solutions of differentialequations with a general piecewise constant argument andapplicationsrdquo Electronic Journal of Qualitative Theory of Differ-ential Equations vol 46 pp 1ndash19 2010
[24] K S Chiu M Pinto and J C Jeng ldquoExistence and globalconvergence of periodic solutions in recurrent neural networkmodels with a general piecewise alternately advanced andretarded argumentrdquo Acta Applicandae Mathematicae 2013
[25] L Dai Nonlinear Dynamics of Piecewise of Constant Systemsand Implememtation of Piecewise Constants Arguments WorldScientific Singapore 2008
[26] T A Burton Stability and Periodic Solutions of Ordinary andFunctional Differential Equations Academic Press New YorkNY USA 1985
[27] T A Burton and T Furumochi ldquoPeriodic and asymptoticallyperiodic solutions of neutral integral equationsrdquo ElectronicJournal of Qualitative Theory of Differential Equations vol 10pp 1ndash19 2000
[28] T A Burton and T Furumochi ldquoExistence theorems andperiodic solutions of neutral integral equationsrdquo NonlinearAnalysis Theory Methods and Applications vol 43 no 4 pp527ndash546 2001
[29] F Karakoc H Bereketoglu and G Seyhan ldquoOscillatory andperiodic solutions of impulsive differential equations withpiecewise constant argumentrdquoActa ApplicandaeMathematicaevol 110 no 1 pp 499ndash510 2010
[30] Y Liu P Yang and W Ge ldquoPeriodic solutions of higher-order delay differential equationsrdquo Nonlinear Analysis TheoryMethods and Applications vol 63 no 1 pp 136ndash152 2005
[31] X L Liu and W T Li ldquoPeriodic solutions for dynamic equa-tions on time scalesrdquo Nonlinear Analysis Theory Methods andApplications vol 67 no 5 pp 1457ndash1463 2007
[32] M Pinto ldquoDichotomy and existence of periodic solutions ofquasilinear functional differential equationsrdquo Nonlinear Analy-sis Theory Methods and Applications vol 72 no 3-4 pp 1227ndash1234 2010
[33] Y N Raffoul ldquoPeriodic solutions for neutral nonlinear differ-ential equations with functional delayrdquo Electronic Journal ofDifferential Equations vol 2003 no 102 pp 1ndash7 2003
[34] G Q Wang and S S Cheng ldquoPeriodic solutions of discreteRayleigh equations with deviating argumentsrdquo Taiwanese Jour-nal of Mathematics vol 13 no 6 pp 2051ndash2067 2009
[35] K S Chiu ldquoStability of oscillatory solutions of differential equa-tions with a general piecewise constant argumentrdquo ElectronicJournal of Qualitative Theory of Differential Equations vol 88pp 1ndash15 2011
[36] R N Butris and M A Aziz ldquoSome theorems in the existenceand uniqueness for system of nonlinear integro-differentialequationsrdquo Journal Education and Science vol 18 pp 76ndash892006
[37] A YMitropolsky andD IMortynyukPeriodic Solutions for theOscillations Systems with Retarded Argument General SchoolKiev Ukraine 1979
[38] L L Tai ldquoNumerical-analytic method for the investigation ofautonomous systems of differential equationsrdquo Ukrainskii Ma-tematicheskii Zhurnal vol 30 no 3 pp 309ndash317 1978
[39] Yu D Shlapak ldquoPeriodic solutions of ordinary differentialequations of first order that are not solved with respect to thederivativerdquo Ukrainian Mathematical Journal vol 32 no 5 pp420ndash425 1980
[40] A M Samoilenko ldquoNumerical-analytic method for the investi-gation of periodic systems of ordinary differential equationsrdquoUkrainskii Matematicheskii Zhurnal vol 17 no 4 pp 16ndash231965
[41] AM Samoilenko andN I RontoNumerical-Analytic Methodsfor Investigations of Periodic Solutions Mir Publishers MoscowRussia 1979
[42] R N Butris ldquoPeriodic solution of nonlinear system of integro-differential equations depending on the gamma distributionrdquoGeneral Mathematics Notes vol 15 no 1 pp 56ndash71 2013
[43] D R Smart Fixed Points Theorems Cambridge UniversityPress Cambridge UK 1980
[44] T A Burton ldquoKrasnoselskiirsquos inversion principle and fixedpointsrdquo Nonlinear Analysis Theory Methods and Applicationsvol 30 no 7 pp 3975ndash3986 1997
[45] T A Burton ldquoA fixed-point theorem of Krasnoselskiirdquo AppliedMathematics Letters vol 11 no 1 pp 85ndash88 1998
14 The Scientific World Journal
[46] M Pinto and D Sepulveda ldquoh-asymptotic stability by fixedpoint in neutral nonlinear differential equations with delayrdquoNonlinear Analysis Theory Methods and Applications vol 74no 12 pp 3926ndash3933 2011
[47] M A Krasnoselskii Translation Along Trajectories of Differen-tial Equations Nauka Moscow Russia 1966 (Russian) Englishtranslation American Mathematical Society Providence RIUSA 1968
[46] M Pinto and D Sepulveda ldquoh-asymptotic stability by fixedpoint in neutral nonlinear differential equations with delayrdquoNonlinear Analysis Theory Methods and Applications vol 74no 12 pp 3926ndash3933 2011
[47] M A Krasnoselskii Translation Along Trajectories of Differen-tial Equations Nauka Moscow Russia 1966 (Russian) Englishtranslation American Mathematical Society Providence RIUSA 1968