Discrete Dynamics in Nature and Society
Volume 2011 (2011), Article ID 319829, 15 pages
http://dx.doi.org/10.1155/2011/319829
Research Article

Existence and Multiplicity of Solutions for Discrete Nonlinear Two-Point Boundary Value Problems

Jianmin Guo and Caixia Guo

School of Mathematics and Computer Sciences, Shanxi Datong University, Datong, Shanxi 037008, China

Received 26 September 2010; Accepted 5 January 2011

Academic Editor: Chang-hong Wang

Copyright © 2011 Jianmin Guo and Caixia Guo. 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

By using Morse theory, the critical point theory, and the character of 𝐾 1 / 2 , we consider the existence and multiplicity results of solutions to the following discrete nonlinear two-point boundary value problem − Δ 2 𝑥 ( 𝑘 − 1 ) = 𝑓 ( 𝑘 , 𝑥 ( 𝑘 ) ) , 𝑘 ∈ ℤ ( 1 , 𝑇 ) subject to 𝑥 ( 0 ) = 0 = Δ 𝑥 ( 𝑇 ) , where 𝑇 is a positive integer, ℤ ( 1 , 𝑇 ) = { 1 , 2 , … , 𝑇 } , Δ is the forward difference operator defined by Δ 𝑥 ( 𝑘 ) = 𝑥 ( 𝑘 + 1 ) − 𝑥 ( 𝑘 ) , and 𝑓 ∶ ℤ ( 1 , 𝑇 ) × ℝ → ℝ is continuous. In argument, Morse inequalities play an important role.