International Journal of Mathematics and Mathematical Sciences
Volume 2010 (2010), Article ID 907453, 21 pages
Research Article

Existence of Multiple Solutions of a Second-Order Difference Boundary Value Problem

College of Mathematics and Information Sciences, Guangzhou University, Guangzhou 510006, China

Received 3 July 2009; Revised 5 February 2010; Accepted 7 March 2010

Academic Editor: Martin Bohner

Copyright © 2010 Bo Zheng and Huafeng Xiao. 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.


This paper studies the existence of multiple solutions of the second-order difference boundary value problem Δ2u(n1)+V(u(n))=0, n(1,T), u(0)=0=u(T+1). By applying Morse theory, critical groups, and the mountain pass theorem, we prove that the previous equation has at least three nontrivial solutions when the problem is resonant at the eigenvalue λk  (k2) of linear difference problem Δ2u(n1)+λu(n)=0, n(1,T), u(0)=0=u(T+1) near infinity and the trivial solution of the first equation is a local minimizer under some assumptions on V.