Existence of Positive Periodic Solutions for Higher Order Singular Functional Difference Equations

We study a higher order singular functional difference equation on Z. Sufficient conditions are obtained for the existence of at least one positive periodic solution of the equation. Our proof utilizes the nonlinear alternative of Leray–Schauder.


Introduction
Nonlinear difference equations have numerous applications in modeling processes in biology, physics, statistics, and many other areas.For this reason, the existence of positive solutions to these equations is of great interest to many researchers.We refer the reader to [6][7][8][9][10][11][12][15][16][17][18] for some recent work on this subject.In this paper, we are concerned with a higher order functional difference equation.To introduce our equation, we let a = 1, b = 1 be any fixed positive numbers, and m, k, ω be any fixed positive integers, and for any u : Z → R, define Here, we study the existence of positive periodic solutions of the higher order functional difference equation where f : Z × (0, ∞) → R, τ : Z → Z, and r : Z → R are ω-periodic on Z, and f (n, x) is continuous in x.
Equation (1.1) with r(n) ≡ 0, i.e., the equation Corresponding author.Email: Lingju-Kong@utc.edu has been recently studied by Wang and Chen in [18] using Krasnosel'skii's fixed point theorem.When f (n, x) is nonsingular at x = 0, sufficient conditions were found there for the existence of positive periodic solutions.In this paper, we will establish a new existence criterion for equation (1.1).The nonlinear term f (n, x) is allowed to be singular at x = 0 in (1.1).The proof will employ a nonlinear alternative of Leray-Schauder.Our approach involves examining a one-parameter family of nonsingular problems constructed from a sequence of nonsingular perturbations of f .For each of these nonsingular problems, we will apply the nonlinear alternative of Leray-Schauder to obtain the existence of at least one positive periodic solution.
From this sequence of solutions, we will extract a subsequence that converges to a positive periodic solution of (1.1).This type of technique has been successfully used in obtaining positive solutions for several classes of singular problems, see, for example, [1,2,6,12,13].Our proofs are partly motivated by these works.Other results on singular problems can be found in [3-5, 7, 14].
As a simple application of our general existence theorem, we also derive some sufficient conditions for the existence of at least one positive periodic solution of the functional difference equation where α ≥ 0 and β ≥ 0 are constants, c, d, and r are ω-periodic functions on Z with c(n) > 0 and d(n) ≥ 0 on Z, and µ > 0 is a parameter.The remainder of this paper is laid out as follows.In Section 2, we present our assumptions and main results.Some preliminary lemmas as well as the proofs are given in Section 3.

In
where and Then, (2.2) implies that We make the following assumptions.
As a consequence of Theorem 2.1, we have the following corollary.

Proofs of the main results
Throughout this section, let X be the set of all real ω-periodic functions on Z.Then, equipped with the maximum norm u = max n∈[1,ω] Z |u(n)|, X is a Banach space.Lemma 3.1 below can be proved using [18, Lemma 2.1].
Lemma 3.1.Assume (H1) holds.Then for any h ∈ X, u(n) is a periodic solution of the equation if and only if u(n) is a solution of the summation equation where G(i, j) and n ij are given by (2.2) and (2.3) respectively.
We refer the reader to [1, Theorem 1.2.3] for the following version of the well known nonlinear alternative of Leray-Schauder.Lemma 3.2.Let K be a convex subset of a normed linear space X, and let Ω be a bounded open subset with p ∈ Ω.Then every compact map N : Ω → K has at least one of the following properties: (i) N has at least one fixed point in Ω; (ii) there is u ∈ ∂Ω and λ ∈ (0, 1) such that u = (1 − λ) p + λNu.
For any h ∈ X, it is easy to see that We will use this identity in the proof of Theorem 2.1.Now, we are ready to prove our results.
Proof of Theorem 2.1.We show the case where (H2)(a) holds.The proof for (H2)(b) is similar.
this paper, for any c, d ∈ Z with c ≤ d, let [c, d] Z denote the discrete interval {c, . . ., d}.For the function r(n) given in equation (1.1), define a function γ : Z → R by