Spectrum of Discrete Second-Order Difference Operator with Sign-Changing Weight and Its Applications

Ruyun Ma,Chenghua Gao

doi:10.1155/2014/590968

Abstract

LetT>1be an integer, and let𝕋=1,2,…,T. We discuss the spectrum of discrete linear second-order eigenvalue problemsΔ2ut-1+λmtut=0, t∈𝕋, u0=uT+1=0, whereλ≠0is a parameter,m:𝕋→ℝchanges sign andmt≠0on𝕋. At last, as an application of this spectrum result, we show the existence of sign-changing solutions of discrete nonlinear second-order problems by using bifurcate technique.

Highlights

Let T > 1 be an integer, T = {1, 2, . . . , T}
Let us consider the spectrum of the discrete second-order linear eigenvalue problem
Where λ ≠ 0 is a parameter, and m changes sign on T; that is, m satisfies the following

Summary

Introduction

Let us consider the spectrum of the discrete second-order linear eigenvalue problem. There are few results on the spectrum of discrete second-order linear eigenvalue problems when m(t) changes its sign on T. In 2008, Shi and Yan [13] discussed the spectral theory of left definite difference operators when m(t) may change its sign They provided no information about the sign of the eigenvalues and no information about the corresponding eigenfunctions. Ma et al [14] obtained that (1) and (2) have two principal eigenvalues λ1,− < 0 < λ1,+ and they studied some corresponding discrete nonlinear problems It is the purpose of this paper to establish the discrete analogue of Theorem A for the discrete problems (1) and (2). We apply our spectrum theory and the Rabinowitz’s bifurcation theorem to consider the existence of sign-changing solutions of discrete nonlinear problems. The weight m(t) > 0 in these papers

Preliminaries

Application