Discrete Dynamics in Nature and Society
Volume 2010 (2010), Article ID 757416, 27 pages
doi:10.1155/2010/757416
Research Article

Spectrum of Linear Difference Operators and the Solvability of Nonlinear Discrete Problems

Ruyun Ma, Youji Xu, and Chenghua Gao

Department of Mathematics, Northwest Normal University, Lanzhou 730070, China

Received 28 June 2010; Revised 15 October 2010; Accepted 16 October 2010

Academic Editor: Binggen Zhang

Copyright © 2010 Ruyun Ma et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

Let T∈ℕ with T>5. Let 𝕋:={1,…,T}. We study the Fučik spectrum Σ of the discrete problem Δ2u(t-1)+λu+(t)-μu-(t)=0, t∈𝕋, u(0)=u(T+1)=0, where u+(t)=max⁡{u(t),0}, u-(t)=max⁡{-u(t),0}. We give an expression of Σ via the matching-extension method. We also use such discrete spectrum theory to study nonlinear boundary value problems of difference equations at resonance.