The large contraction principle and existence of periodic solutions for infinite delay Volterra difference equations

: In this article, we establish sufficient conditions for the existence of periodic solutions of a nonlinear infinite delay Volterra difference equation: We employ a Krasnosel’skiĭ type fixed point theorem, originally proved by Burton. The primary sufficient condition is not verifiable in terms of the parameters of the difference equation, and so we provide three applications in which the primary sufficient condition is verified.


Introduction
Krasnosel'skiĭ [11,19] is credited with a fixed point theorem in Banach spaces in which the fixed point operator is expressed as the sum of a compact operator and a contraction. This theorem has been generalized in different ways [2,3,5,13,14,20]. In [5], Burton introduced the concept of a large contraction and proved an extension of the Krasnosel'skiĭ fixed point theorem to the case in which the fixed point operator is expressed as the sum of a compact operator and a large contraction. Burton's theorem has proved to be quite useful in the study of both delay differential equations and delay Volterra difference equations [1, 4-6, 10, 15-18], as well as other functional or fractional equations [7][8][9]12].
In this paper we shall apply Burton's extended Krasnosel'skiĭ fixed point theorem to ∆x(n) = p(n) + b(n)h(x(n)) + n ∑ k=−∞ B(n, k)g(x(k)), n ∈ Z, (1.1) In Section 2, we shall provide the definition of a large contraction and state without proof Burton's extended Krasnosel'skiĭ fixed point theorem. In Section 3, we shall employ Burton's theorem and obtain sufficient conditions for the existence of a nontrivial periodic solution of (1.1). We shall close in Section 4 with three specific applications in which the primary sufficient condition is realized. The three applications are motivated by two specific large contractions, one observed by Burton [5] and one observed by Raffoul [17,18].

Preliminaries
Denote the set of all integers and real numbers by Z and R, respectively. Define N a = {a, a + 1, a + 2, · · · } for any a ∈ R. Assume that empty sums and products are taken to be 0 and 1, respectively. First, we introduce the concept of large contraction.
Definition 2.1 [5] Let (M, d) be a metric space and B : M → M . B is said to be a large contraction mapping

Remark 2.2
It is clear from the definition that if δ serves as a large contraction coefficient for ε 1 > 0 and ε 2 > ε 1 , then δ serves as a large contraction coefficient for ε 2 > 0. This comment is made since throughout Section 4, it is assumed that 0 < ε 1 < 1.
The following theorem is Burton's extended Krasnosel'skiĭ fixed point theorem. Then there exists z ∈ M with z = Az + Bz .

Existence of periodic solutions
In this section, we establish sufficient conditions for the existence of periodic solutions of the Volterra difference equation (1.1). We assume that there exists a least positive real number T such that and B(n + T, k + T ) = B(n, k), for all (n, k) ∈ Z 2 .
We assume that h and g are real valued mappings.
Let P T be the set of all T -periodic sequences {x(n)}, periodic in n . Then (P T , ∥.∥) is a Banach space with respect to the maximum norm Define a(n) = 1 + b(n) and assume throughout that Define the mapping H by We begin with the following lemma.
Proof Let x ∈ P T . Rewrite (1.1) in the form Thus, x is a solution of (1.1) if and only if x is a solution of (3.5). To obtain the representation in (3.3), note that x ∈ P T , which implies x(n − T ) = x(n), ) , . For simplicity, set ) −1 and note that η is independent of n due to the periodicity of a(n). Set Then for n ∈ {0, 1, 2, · · · , T − 1} and l ∈ {n − T, n − T + 1, · · · , n − 1}, we have Let J be a positive constant. This constant J will be carefully chosen in the applications. Define the set and note that M J is a bounded and convex subset of the Banach space P T . Let the mapping A : for n ∈ Z. Similarly, we set the map B : Assume that g(x) satisfies a local Lipschitz condition in x ; in particular, assume there exists a positive constant Then, for φ ∈ M J , we obtain Lemma 3.2 Assume that g satisfies the Lipschitz condition given in (3.10). Suppose that there exists a positive constant Λ such that

12)
for all n . Then the mapping A : M J → P T is continuous.
for all n ∈ Z.
which implies thatĀ(M J ) is uniformly bounded. 2 The following lemma gives a relationship between the mappings H and B, where B is the mapping defined in (3.9), in the sense of a large contraction. We first note that under the assumption, b(n) ≥ 0 for n ∈ Z, it follows that Let c(l) = ∏ n−1 s=l+1 a(s) . Then, Consequently, ) .

Now, consider
(3.14) Thus, we have Let ε > 0 and assume δ < 1 is such that if ∥x − y∥ ≥ ε then ∥Hx − Hy∥ < δ∥x − y∥. Since  in particular, B is a large contraction. 2 We state the main theorem, which provides sufficient conditions for the existence of a T periodic solution of (1.1). In particular, Thus, Theorem 2.3 applies and the theorem is proved.

Applications
We have seen that (3.15) is the primary sufficient condition. In this section we present three specific applications of (1.1) and verify (3.15) in each application. Throughout this section it is assumed that 0 < ε 1 < 1.
Burton [5] defined the concept of large contraction and showed that is a large contraction in a neighborhood of x = 0. Later, Raffoul [18] extended Burton's arguments and showed that x − x 5 is a large contraction in a neighborhood of x = 0. In particular, Raffoul [18] established the following lemma.
and then he showed that H is a large contraction on M J .
We employ Lemma 4.1 and Theorem 3.5 and find conditions on c > 0 such that the following Volterra difference equation has a 4-periodic solution. Let c > 0 and consider the Volterra difference equation where h(x) = x 5 . We calculate the estimates required to apply Theorem 3.5 for (4.
is a large contraction on the set M J for and then showed that Then there exists c between x 5 and y 5 such that Thus, it follows that H(x) = x − f (x 5 ) is a large contraction on [−J, J]; the estimates in Example 1 remain valid and so the following result is obtained.
In [18], Raffoul derived the analogous (4.4). At this time, it is not clear to us how to generalize these derivations.