Periodic Solutions for Nonlinear Integro-Differential Systems with Piecewise Constant Argument

We investigate the existence of the periodic solutions of a nonlinear integro-differential system with piecewise alternately advanced and retarded argument of generalized type, in short DEPCAG; that is, the argument is a general step function. We consider the critical case, when associated linear homogeneous system admits nontrivial periodic solutions. Criteria of existence of periodic solutions of such equations are obtained. In the process we use Green's function for periodic solutions and convert the given DEPCAG into an equivalent integral equation. Then we construct appropriate mappings and employ Krasnoselskii's fixed point theorem to show the existence of a periodic solution of this type of nonlinear differential equations. We also use the contraction mapping principle to show the existence of a unique periodic solution. Appropriate examples are given to show the feasibility of our results.

Let : R → R be a step function given by ( ) = for ∈ = [ , +1 ) and consider the DEPCA (1) with this general . In this case we speak of DEPCA of general type, in short DEPCAG. Indeed, ( ) = [ ] corresponds to = = ∈ Z, and ( ) = 2[( + 1)/2] corresponds to = 2 − 1, = 2 , ∈ Z. The particular case of DEPCAG, when = , ∈ Z, an only delayed situation, is considered by first time in Akhmet [8]. The other extreme case is the only advanced situation = +1 . Any other situation means an alternately advanced and delayed situation with + = [ , ] the advanced intervals and − = [ , +1 ) the delayed intervals. In [15,16], Pinto has cleared the importance of the advanced and delayed intervals. This decomposition will be present in all our results. See [12,13,15,16,23,24,35]. The integration or solution of a DEPCA, as proposed by its founders [2][3][4]6], is based on the reduction of DEPCA to discrete equations. To study nonlinear DEPCAG, we will use the approach proposed by Akhmet in [9], based on the construction of an equivalent integral equation, but we also remark the clear influence of the discrete part and the corresponding difference equations will be fundamental. 2 The Scientific World Journal In 2008, Akhmet et al. [10] obtained some sufficient conditions for the existence and uniqueness of periodic solutions for the following system: ( ) = ( ) ( ) + ℎ ( ) + ( , ( ) , ( ( )) , ) , (2) where : R → R × , ℎ : R → R, and : R×R ×R × → R are continuous functions, ( ) = if ≤ < +1 , and is a small parameter belonging to an interval ⊂ R with 0 ∈ .
Recently, Chiu and Pinto [23], using Poincaré operator, a new Gronwall type lemma and fixed point theory, obtained some sufficient conditions for the existence and uniqueness of periodic (or harmonic) and subharmonic solutions of quasilinear differential equation with a general piecewise constant argument of the form where ∈ R, ∈ C , ( ) is a × matrix for ∈ N, ( , , ) is a dimensional vector and is continuous in the first argument, and ( ) = , if ≤ < +1 , ∈ Z. In this paper, comparing the three DEPCAG inequalities of Gronwall type and remarked new Gronwall lemma not only requests a weaker condition than the other Gronwall lemmas but also has a better estimate.
It is well-known that there are many subjects in physics and technology using mathematical methods that depend on the linear and nonlinear integro-differential equations, and it became clear that the existence of the periodic solutions and its algorithm structure from more important problems in the present time. Where many of studies and researches [36][37][38][39][40] dedicates for treatment the autonomous and nonautonomous periodic systems and specially with the integral equations and differential equations and the linear and nonlinear differential and which is dealing in general shape with the problems about periodic solutions theory and the modern methods in its quality treatment for the periodic differential equations.
Samoilenko and Ronto [41] assume the numericalanalytic method to study the periodic solutions for ordinary differential equations and its algorithm structure and this method includes uniform sequences of periodic functions and the results of that study are using the periodic solutions on wide range in the difference of new processes industry and technology. For example, Samoilenko and Ronto [41] investigated the existence and approximation of periodic solution for nonlinear system of integro-differential equations which has the form where ∈ ⊂ R ; is a closed and bounded domain. The vectors functions ( , , ) and ( , ) are continuous functions in , , and periodic in of period .
In the current paper, we study the existence of periodic solutions of a nonlinear integro-differential system with piecewise alternately advanced and retarded argument: where In the analysis we use the idea of Green's function for periodic solutions and convert the nonlinear integrodifferential systems with DEPCAG (6) into an equivalent integral equation. Then we employ Krasnoselskii's fixed point theorem and show the existence of a periodic solution of the nonlinear integro-differential systems with DEPCAG (6) in Theorem 12. We also obtain the existence of a unique periodic solution in Theorem 14 employing the contraction mapping principle as the basic mathematical tool. Furthermore, appropriate examples are given to show the feasibility of our results.
In our paper we assume that the solutions of the nonlinear integro-differential systems with DEPCAG (6) are continuous functions. But the deviating argument ( ) is discontinuous. Thus, in general, the right-hand side of the DEPCAG system (6) has discontinuities at moments ∈ R, ∈ Z. As a result, we consider the solutions of the DEPCAG as functions, which are continuous and continuously differentiable within intervals [ , +1 ), ∈ Z. In other words, by a solution ( ) of the DEPCAG system (6) we mean a continuous function on R such that the derivative ( ) exists at each point ∈ R, with the possible exception of the points ∈ R, ∈ Z, where a one-sided derivative exists, and the nonlinear integro-differential systems with DEPCAG (6) are satisfied by ( ) on each interval ( , +1 ), ∈ Z as well. The rest of the paper is organized as follows. In Section 2, some definitions and preliminary results are introduced. We show double -periodicity of Green's function. Section 3 is devoted to establishing some criteria for the existence and uniqueness of periodic solutions of the DEPCAG system (6

Green's Function and Periodicity
In this section we state and define Green's function for periodic solutions of the nonlinear integro-differential system with piecewise alternately advanced and retarded argument (6).
From now on the following assumption will be needed.
( ) The homogenous equation does not admit any nontrivial -periodic solution.

Definition 2.
Suppose that the condition ( ) holds. For each , ∈ [ , + ], Green's function for (6) is given by where Φ( ) is a fundamental solution of (7) and We note that the condition ( ) implies the existence of the matrix .
To prove double -periodicity of Green's function ( , ), we first give the following lemma.
Lemma 3. Suppose that the condition ( ) holds. Let the matrix be defined by (19), and then one has the following identities: The Scientific World Journal 5 For Lemma 3 we can prove an important property, double -periodicity, of Green's function ( , ) to study after the existence of periodic solutions.

Existence of Periodic Solutions
In this section, we prove the main theorems of this paper, so we recall the nonlinear integro-differential systems with DEPCAG (6): For this, a natural Banach space is with the supremum norm Consider : R × R × R → R and : R × R × R → R are continuous functions. Moreover, we will refer to the following specific conditions.

Periodic Conditions
( ) There exists > 0 such that Remark 5. Note that ( , ) condition is a discrete relation, which moves the interval into + . Then we have the following consequences.
(i) For any ∈ R, the interval [ , + ] can be decomposed as follows: (ii) For ∈ [ , +1 ), we have Then, Using Definition 2, Remark 5, and double -periodicity of Green's function, similar formula is given by (17). So, we have obtained the following result.
Consider the operator I : P → P by It is easy to see that the DEPCAG system (6) hasperiodic solution if and only if the operator I has one fixed point in P .
To prove some existence criteria for -periodic solutions of the DEPCAG system (6) we use the Banach, Schauder, and Krasnoselskii's fixed point theorems.
Next we state first Krasnoselskii's fixed point theorem which enables us to prove the existence of a periodic solution. For the proof of Krasnoselskii's fixed point theorem we refer the reader to [43].
Theorem A (Krasnoselskii's fixed point theorem). Let be a closed convex nonempty subset of a Banach space ( , ‖ ⋅ ‖). Suppose that A and B map into such that Then there exists ∈ with = A + B .
Remark 7. Krasnoselskii's theorem may be combined with Banach and Schauder's fixed point theorems. In a certain sense, we can interpret this as follows: if a compact operator has the fixed point property, under a small perturbation, then this property can be inherited. The theorem is useful in establishing the existence results for perturbed operator equations. It also has a wide range of applications to nonlinear integral equations of mixed type for proving the existence of solutions. Thus the existence of fixed points for the sum of two operators has attracted tremendous interest, and their applications are frequent in nonlinear analysis. See [32,33,[43][44][45][46].
We note that to apply Krasnoselskii's fixed point theorem we need to construct two mappings; one is contraction and the other is compact. Therefore, we express (38) as where A, B : P → P are given by To simplify notations, we introduce the following constant and sets: Secondly, we show that A is a contraction mapping. Let , ∈ P ; then we have Hence A defines a contraction mapping.
Similarly, B is given by (41), which may be also a contraction operator.

Lemma 10. If ( ) holds, B is defined by (41), and then B is completely continuous; that is, B is continuous and the image of B is contained in a compact set.
Proof. Step 1. First we prove that B : P → P is continuous.
As the operator A, a change of variable in (41) and the continuity of B is proved.
Step 2. We show that the image of B is contained in a compact set.
Let In a similar way, for A we obtain the following. (40), and then A is completely continuous.

Lemma 11. If ( ) and ( ) hold, A is defined by
(49) Then the DEPCAG system (6) has at least one -periodic solution in S.
Proof. By Lemma 8, the mapping A is a contraction and it is clear that A : P → P . Also, from Lemma 10, B is completely continuous.
Let , ∈ S with ‖ ‖ ≤ and ‖ ‖ ≤ . Then We now see that all the conditions of Krasnoselskii's theorem are satisfied. Thus there exists a fixed point in S such that = A + B . By Proposition 6, this fixed point is a solution of the DEPCAG system (6). Hence the DEPCAG system (6) has -periodic solution.
By the symmetry of the conditions, we will obtain as Theorem 12.
By Lemma 9, the mapping B is a contraction and it is clear that B : P → P . Also, from Lemma 11, A is completely continuous.
Let , ∈ S with ‖ ‖ ≤ and ‖ ‖ ≤ . Then Applying Banach's fixed point theorem we have the following.
then the DEPCAG system (6) has a unique -periodic solution.
Proof. Let the mapping I be given by (38). For , ∈ P , in view of (38), we have This completes the proof by invoking the contraction mapping principle.
As a direct consequence of the method, Schauder's theorem implies the following.
Then the DEPCAG system (6) has at least one -periodic solution in S.
if / is a rational number for all , = 1, 2, 3; then ( 1 ), ( 2 ), and ( 3 ) are simultaneously satisfied by = l.c.m.{ 1 , 2 , 3 }, where l.c.m.{ 1 , 2 , 3 } denotes the least common multiple between 1 , 2 and 3 . In the general case it is possible that there exist five possible periods: 1 for , 2 for , 3 for , 4 for , and the sequences { } ∈Z , { } ∈Z satisfy the ( 5 , ) condition. If / is a rational number for all , = 1, 2, 3, 4, 5, so, in this situation our results insure the existence of -periodic solution with = l.c.m.{ 1 , 2 , 3 , 4 , 5 }. Therefore the above results insure the existence of -periodic solutions of the DEPCAG system (6). These solutions are called subharmonic solutions. See  To determine criteria for the existence and uniqueness of subharmonic solutions of the DEPCAG system (6), from now on we make the following assumption.

Applications and Illustrative Examples
We will introduce appropriate examples in this section. These examples will show the feasibility of our theory. Mathematical modelling of real-life problems usually results in functional equations, like ordinary or partial differential equations, integral and integro-differential equations, and stochastic equations. Many mathematical formulations of physical phenomena contain integro-differential equations; these equations arise in many fields like fluid dynamics, biological models, and chemical kinetics. So, we first consider nonlinear integro-differential equations with a general piecewise constant argument mentioned in the introduction and obtain some new sufficient conditions for the existence of the periodic solutions of these systems. (iv) : R × R × R → R is continuous and ( , , ) satisfies ( ).

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The Scientific World Journal Note that similar results can be obtained under ( ) and ( ). On the other hand, the periodic situation of the DEPCAG system (56) and (57) can be treated in the same way.
Let us consider another example for second-order differential equations with a general piecewise constant argument. In this case, we can show the existence and uniqueness of periodic solutions of the following nonlinear DEPCAG system.
By Theorem 14, the DEPCAG system (65) has a unique 2 /periodic solution in S.

Conflict of Interests
The author declares that there is no conflict of interests regarding the publication of this paper.