Periodic solutions for nonlinear second-order difference equations

Abstract

This paper is devoted to introduce a new approach to investigate the existence of periodic solutions for nonlinear difference equations. Transforming a second-order nonlinear difference equation on a finite discrete segment with periodic boundary conditions into a continuous system, we obtain the existence conditions of periodic solutions of the equation. The proof relies on coincidence degree theory and matrix theory without using Green’s function.

Introduction

The aim of this paper is to establish the existence conditions for periodic solutions of the below equation:

$$\mathrm{\Delta }(r_n\mathrm{\Delta }{x}_{n-1})+f(n\text{,}{x}_n)=0\text{,}n\in Z\text{,}$$

where $r_n:Z\to R$, for each n ∈ Z, and r_n+T = r_n; $f(n\text{,}z):Z\times R\to R$ is continuous in the second variable, and for any (n, z) ∈ Z × R, f(n + T, z) = f(n, z); T is a positive integer. As usual, R and Z denote the sets of all real numbers and integers, respectively. Given a < b in Z, let N[a, b] = {a, a + 1, … , b}. The forward difference operator Δ is defined by Δx_n = x_n+1 − x_n.

The research into the periodic solutions for differential equations has been always a very active subject. Rich results have been obtained due to the various powerful devices such as coincidence degree theory, cone theory and critical point theory, etc., see for example [1], [2], [3], [4], [5], [6], [7], [8], [9], [10], [11], [12], [13]. However, for the periodic problem of nonlinear difference equations, only in recent years there have been some research activities due to the realization that nonlinear difference equations are important in applications. Its new applications continue to arise with increasing frequency in the modelling of diverse phenomena in physics, biology, ecology and physiology. On the other hand, scarce techniques for studying the existence of periodic solutions of difference equations lead to the fact that progress in this direction has been rather slow, and only very recently papers [14], [15], [16], [17], [18], [19], [20] are known to us. The proof in [14], [17], [18], [19] is based on an application of a fixed-point theorem for the completely continuous operators in cones and uses properties of the Green’s function. It is necessary to perform a study of the sign of Green’s function for the corresponding linear equations. For practical purposes, serious difficulties arise in the search for a Green’s function and the judgement of its sign. In papers [15], [16], [20], it is the first time that the critical point theory is introduced to investigate the periodic problem for difference equations by Yu and Guo. But this technique is very fruitful only to the models which are of variational structure.

In this paper, we develop a new technique to study (1.1) by converting (1.1) into an continuous operator equation, and then apply matrix theory and coincidence degree theory to establish the existence conditions for periodic solutions of (1.1). In this way, we succeed in overcoming all difficulties mentioned above.

It is interesting to study Eq. (1.1) because it can be considered as a discrete analogue of the following second-order differential equation:

$$(p(t)y^{\prime }{)}^{\prime }+f(t\text{,}y)=0\text{,}$$

which is the general form of the Emden–Fowler equation arising in the study of astrophysics, gas dynamics, fluid mechanics, relativistic mechanics, nuclear physics and chemically reacting systems in terms of various explicit forms of f(t, y), see the survey paper of Wong [21]. So Eq. (1.2) has been extensively investigated, for instance, see [22], [23], [24], [25], [26], [27] and the reference therein. In addition, Eq. (1.1) does have its applicable setting as evidenced by the monograph [28], [29].

Most reference has paid attention to oscillation, nonoscillation, growth and asymptotic behavior of Eq. (1.1), for example, see [30], [31], [32], [33], [34]. However, there has never been discussion about the existence of periodic solutions.

Throughout this paper, R^T stands for the T-dimensional real vector space. Let A⁻¹ denote the inverse matrix of the matrix A and R(A) denote the rank of A. ∗ denotes transpose of a matrix or a vector. (·, ·) denotes the inner product of two elements.

This paper is organized as follows: Section 2 is devoted to show some preliminaries which play a fundamental role in the proof of the main results of this paper given in Section 3. In Section 4, we give the proof of the main results for (1.1) to possess at least one periodic solution.