Abstract and Applied Analysis
Volume 2012 (2012), Article ID 918082, 17 pages
http://dx.doi.org/10.1155/2012/918082
Research Article

Existence and Multiplicity of Positive Solutions of a Nonlinear Discrete Fourth-Order Boundary Value Problem

Ruyun Ma and Yanqiong Lu

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

Received 16 September 2011; Revised 14 December 2011; Accepted 25 December 2011

Academic Editor: Yuriy Rogovchenko

Copyright © 2012 Ruyun Ma and Yanqiong Lu. 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

we show the existence and multiplicity of positive solutions of the nonlinear discrete fourth-order boundary value problem Δ 4 𝑢 ( 𝑡 − 2 ) = 𝜆 ℎ ( 𝑡 ) 𝑓 ( 𝑢 ( 𝑡 ) ) , 𝑡 ∈ 𝕋 2 , 𝑢 ( 1 ) = 𝑢 ( 𝑇 + 1 ) = Δ 2 𝑢 ( 0 ) = Δ 2 𝑢 ( 𝑇 ) = 0 , where 𝜆 > 0 , ℎ ∶ 𝕋 2 → ( 0 , ∞ ) is continuous, and 𝑓 ∶ ℝ → [ 0 , ∞ ) is continuous, 𝑇 > 4 , 𝕋 2 = { 2 , 3 , … , 𝑇 } . The main tool is the Dancer's global bifurcation theorem.