Nonlinear Discrete Periodic Boundary Value Problems at Resonance

Ruyun Ma,Huili Ma

doi:10.1155/2009/360871

Abstract

Let be an integer with , and let . We study the existence of solutions of nonlinear discrete problems , where with is the th eigenvalue of the corresponding linear eigenvalue problem.

Highlights

Initialed by Lazer and Leach 1, much work has been devoted to the study of existence result for nonlinear periodic boundary value problem y x m2y x g x, y x e x, x ∈ 0, 2π, 1.1 y 0 y 2π, y 0 y 2π, where m ≥ 0 is an integer
If dim Mk 1, we assume that Mk : span{ψk} in which ψk is the eigenfunction of λk
If dim Mk 2, we assume that Mk : span{ψk, φk} in which ψk and φk are two linearly independent eigenfunctions of λk

Summary

Introduction

Initialed by Lazer and Leach 1 , much work has been devoted to the study of existence result for nonlinear periodic boundary value problem y x m2y x g x, y x e x , x ∈ 0, 2π , 1.1 y 0 y 2π , y 0 y 2π , where m ≥ 0 is an integer. If dim Mk 2, we assume that Mk : span{ψk, φk} in which ψk and φk are two linearly independent eigenfunctions of λk It is the purpose of this paper to prove the existence results for problem 1.2 when there occurs resonance at the eigenvalue λk and the nonlinear function g may “touching” the eigenvalue λk 1. In 12 , Iannacci and Nkashama proved the analogue of Theorem 1.1 for continuoustime nonlinear periodic boundary value problems 1.1. The existence of solutions of second-order discrete problem at resonance was studied by Rodriguez in , in which the nonlinearity is required to be bounded.

Then D is a Hilbert space under the inner product

Existence of Periodic Solutions