Existence Theorems of Periodic Solutions for Second-Order Nonlinear Difference Equations

Xiaochun Cai,Jianshe Yu

doi:10.1155/2008/247071

Abstract

The authors consider the second-order nonlinear difference equation of the type using critical point theory, and they obtain some new results on the existence of periodic solutions.

Highlights

We denote by N, Z, R the set of all natural numbers, integers, and real numbers, respectively
Δ pn Δxn−1 δ qnxnδ f n, xn, n ∈ Z, 1.1 where the forward difference operator Δ is defined by the equation Δxn xn 1 − xn and
In 1.1, the given real sequences {pn}, {qn} satisfy pn T pn > 0, qn T qn for any n ∈ Z, f : Z × R → R is continuous in the second variable, and f n T, z f n, z for a given positive integer T and for all n, z ∈ Z × R. −1 δ −1, δ > 0, and δ is the ratio of odd positive integers

Summary

Introduction

We denote by N, Z, R the set of all natural numbers, integers, and real numbers, respectively. In 1.1 , the given real sequences {pn}, {qn} satisfy pn T pn > 0, qn T qn for any n ∈ Z, f : Z × R → R is continuous in the second variable, and f n T, z f n, z for a given positive integer T and for all n, z ∈ Z × R. In 6 , by critical point method, the existence of periodic and subharmonic solutions of equation. It is interesting to study second-order nonlinear difference equations 1.1 because they are discrete analogues of differential equation ptφuft, u 0. They do have physical applications in the study of nuclear physics, gas aerodynamics, infiltrating medium theory, and plasma physics as evidenced in 12, 13. I possesses a critical value c ≥ ω and c inf max I h u , 1.8 h∈Γ u∈Bρ X1 where Γ {h ∈ C Bρ X1,X |h|∂Bρ X1 id}

Preliminaries

Main results