Discrete Dynamics in Nature and Society
Volume 2010 (2010), Article ID 839474, 19 pages
doi:10.1155/2010/839474
Research Article

Existence of Solutions of a Discrete Fourth-Order Boundary Value Problem

Ruyun Ma, Chenghua Gao, and Yongkui Chang

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

Received 4 December 2009; Revised 30 January 2010; Accepted 25 March 2010

Academic Editor: Leonid Shaikhet

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 a,b be two integers with b-a≥5 and let 𝕋2={a+2,a+3,…,b-2}. We show the existence of solutions for nonlinear fourth-order discrete boundary value problem Δ4u(t-2)=f(t,u(t), Δ2u(t-1)), t∈𝕋2, u(a+1)=u(b-1)=Δ2u(a)=Δ2u(b-2)=0 under a nonresonance condition involving two-parameter linear eigenvalue problem. We also study the existence and multiplicity of solutions of nonlinear perturbation of a resonant linear problem.