Advances in Difference Equations
Volume 2009 (2009), Article ID 730484, 10 pages
Research Article

Solutions of 2nth-Order Boundary Value Problem for Difference Equation via Variational Method

School of Mathematics, South China Normal University, Guangzhou 510631, China

Received 7 July 2009; Accepted 15 October 2009

Academic Editor: Kanishka Perera

Copyright © 2009 Qingrong Zou and Peixuan Weng. 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.


The variational method and critical point theory are employed to investigate the existence of solutions for 2nth-order difference equation Δn(pknΔnykn)+(1)n+1f(k,yk)=0 for k[1,N] with boundary value condition y1n=y2n==y0=0,  yN+1==yN+n=0 by constructing a functional, which transforms the existence of solutions of the boundary value problem (BVP) to the existence of critical points for the functional. Some criteria for the existence of at least one solution and two solutions are established which is the generalization for BVP of the even-order difference equations.