Existence of Homoclinic Solutions for a Class of Nonlinear Difference Equations

Peng Chen,X H Tang

doi:10.1155/2010/470375

Abstract

By using the critical point theory, we establish some existence criteria to guarantee that the nonlinear difference equation has at least one homoclinic solution, where , and is non periodic in . Our conditions on the nonlinear term are rather relaxed, and we generalize some existing results in the literature.

Highlights

Consider the nonlinear difference equation of the formΔ p n Δu n − 1 δ − qnxnδfn, u n, n ∈ Z, 1.1 where Δ is the forward difference operator defined by Δu n u n 1 − u n, Δ2u n Δ Δu n, δ > 0 is the ratio of odd positive integers, {p n } and {q n } are real sequences, {p n } / 0. f : Z × R → R
Difference equations have attracted the interest of many researchers in the past twenty years since they provided a natural description of several discrete models
There are some new results on periodic solutions of nonlinear difference equations by using the critical point theory in the literature; see 1–3

Summary

Introduction

Δ p n Δu n − 1 δ − qnxnδfn, u n , n ∈ Z, 1.1 where Δ is the forward difference operator defined by Δu n u n 1 − u n , Δ2u n Δ Δu n , δ > 0 is the ratio of odd positive integers, {p n } and {q n } are real sequences, {p n } / 0. f : Z × R → R. AR For every n ∈ Z, W is continuously differentiable in x, and there is a constant μ > 2 such that 0 < μW n, x ≤ ∇W n, x , x , ∀ n, x ∈ Z × RN \ {0} It seems that results on the existence of homoclinic solutions of 1.1 by critical point method have not been considered in the literature. Motivated by the paper 16 , the intention of this paper is that, under the assumption that F n, x is indefinite sign and subquadratic as |n| → ∞, we will establish some existence criteria to guarantee that 1.1 has at least one homoclinic solution by using minimization theorem in critical point theory.

Preliminaries

Proofs of Theorems

Examples