Periodic Solutions for a System of Nonlinear Neutral Functional Difference Equations with Two Functional Delays

In this paper, we study the existence and uniqueness of periodic solutions of the system of nonlinear neutral difference equations ∆x (n) = A (n)x (n− τ (n)) + ∆Q (n, x (n− g (n))) +G (n, x (n) , x (n− g (n))) . By using Krasnoselski’s fixed point theorem we obtain the existence of periodic solution and by contraction mapping principle we obtain the uniqueness. An example is given to illustrate our result. Our results extend and generalize the work [13].


Introduction
A qualitative analysis such as periodicity and stability of solutions of neutral difference equations which the delay has been studied extensively by many authors, we refer the readers to [1]- [5], [7]- [9], [10,12,13] and references therein for a wealth of reference materials on the subject.
By employing the Krasnoselskii's fixed point theorem, the author obtained existence results for periodic solutions. Also, the author used the contraction mapping principle to show the uniqueness of periodic solutions of (1).
In the current paper, we study the existence and uniqueness of periodic solutions of the system of nonlinear neutral difference equations (2) ∆x (n) = A (n) x (n − τ (n)) + ∆Q (n, x (n − g (n))) + G (n, x (n) , x (n − g (n))) , where A(·) is N × N matrix with sequences as its elements, τ , g : Z → Z + are scalar and the functions Q : Z × R N → R N and G : Z × R N × R N → R N are continuous in x. The sets Z and Z + denote the integers and the nonnegative integers, respectively. In the analysis we use the fundamental matrix solution of ∆x (n) = A (n) x (n) to invert the system (2). Then we employ the Krasnoselskii's fixed point theorem to show the existence of periodic solutions of system (2). The obtained mapping is the sum of two mappings, one is a compact operator and the other is a contraction. Also, transforming system (2) to a fixed point problem enables us to show the uniqueness of the periodic solution by appealing to the contraction mapping principle. The organization of this paper is as follows. In Section 2, we present the inversion of (2) and the fixed point theorems that we employ to help us show the existence and uniqueness of periodic solutions to system (2). In Section 3, we present our main results with an example.

Preliminaries
For the definitions of the different notions used throughout this paper we refer, for example [6,7,10,11,14].
For T > 1 define where C Z, R N is the space of all N -vector continuous functions. Then C T is a Banach space when it is endowed with the supremum norm Note that C T is equivalent to the Euclidean space R N T , where |·| denotes the infinity norm for x ∈ R N . Also, if A is an N × N real matrix, then we define the norm of A by Definition 2.1. If the matrix B (·) is periodic of period T , then the linear system is said to be noncritical with respect to T , if it has no periodic solution of period T except the trivial solution y = 0.
In this paper we assume that (4) where τ * , g * are constant. For n ∈ Z, x, y, z, w ∈ R N , the functions Q (n, x) and G (n, x, y) are periodic in n of period T , they are also globally Lipschitz continuous in x and in x and y, respectively. That is and there are positive constants k 1 , k 2 , k 3 such that Throughout this paper it is assumed that the matrix B (n) = I + A (n) is nonsingular and the system (3) is noncritical, where I is the N × N identity matrix. Also, if x (·) is a sequence, then the forward operator E is defined as Ex (n) = x (n + 1). Now, we state some known results about system (3). Let K (n) represent the fundamental matrix of (3) with K (0) = I, then: a. det K (n) = 0. b. K (n + T ) = B (n) K (n) and K −1 (n + T ) = K −1 (n) B −1 (n). c. System (3) is noncritical if and only if det (I − K (T )) = 0. d. There exists a nonsingular matrix L such that K (n + T ) = B (n) K (n) L T and K −1 (n + T ) = L −T K −1 (n). The following lemma is fundamental to our results.
Proof. Let x ∈ C T be a solution of (2) and K (·) is a fundamental matrix of solutions for (3). Rewrite the equation (2) as

This implies
Mouataz Billah Mesmouli, Abdelouaheb Ardjouni, and Ahcene Djoudi 61 Summing of the above equation from 0 to n − 1 yields A substitution of (12) into (11) yields Now, we will show that (13) is equivalent to (8). Since Then the equations (13) becomes For the sake of simplicity, we let , the above expression yields (14) x (n) = Q (n, x (n − g (n))) − n−1 By (d) we have K (n − T ) = K (n) L −T and K (T ) = L T . Hence,
We end this section by stating the fixed point theorems that we employ to help us show the existence and uniqueness of periodic solutions to equation (2); see [6,14].
Theorem 2.1 (Contraction Mapping Principle). Let (X , ρ) a complete metric space and let P : X → X . If there is a constant α < 1 such that for x, y ∈ X we have ρ (P x, P y) ≤ αρ (x, y) , then there is one and only one point z ∈ X with P z = z.
Krasnoselskii (see [14]) combined the contraction mapping theorem and Shauder's theorem and formulated the following hybrid result. Theorem 2.2 (Krasnoselskii). Let M be a closed bounded convex nonempty subset of a Banach space (X , · ). Suppose that R and S map M into X such that (i) R is compact and continuous, (ii) S is a contraction mapping, (iii) x, y ∈ M, implies Rx + Sy ∈ M, then there exists z ∈ M with z = Rz + Sz.

Existence and Uniqueness of Periodic Solution
By applying Theorems 2.1 and 2.2, we obtain in this Section the existence and the uniqueness of the periodic solution of (2). So, let a Banach space (C T , · ), a closed bounded convex subset of C T , with L > 0, and by the Lemma 2.1, let a mapping H given by Therefore, we express equation (17) as where R and S are given by and By a series of steps we will prove the fulfillment of (i), (ii) and (iii) in Theorem 2.2. So that, since ϕ ∈ C T , (4) and (5) (s))) − G (s, ϕ (s) , ϕ (s − g (s)))| .
Let ϕ N ∈ M where N is a positive integer, then by (18) we obtain Second, we show that R maps bounded subsets into compact sets. As M is bounded and R is continuous, then RM is a subset of R N T which is bounded. Thus RM is contained in a compact subset of M. Therefore R is continuous in M and RM is contained in a compact subset of M.
Then (2) has a T -periodic solution.
Proof. By Lemma 3.1, R : M → C T is continuous and R(M) is contained in a compact set. Also, from Lemma 3.2, the mapping S : M → C T is a contraction. Next, we show that if ϕ, φ ∈ M, we have Rϕ + Sφ ≤ L. Let ϕ, φ ∈ M with ϕ , φ ≤ L. Then Clearly, all the hypotheses of the Krasnoselskii's theorem are satisfied. Thus there exists a fixed point z ∈ M such that z = Rz + Sz. By Lemma 2.1 this fixed point is a solution of (2). Hence (2) has a T -periodic solution.
then equation (2) has a unique T -periodic solution.

then (2) has a unique solution in M.
Proof. Let the mapping H defined by (17). Then the proof follow immediately from Theorem 3.1 and Theorem 3.2.