Advances in Difference Equations
Volume 2011 (2011), Article ID 894135, 20 pages
doi:10.1155/2011/894135
Research Article

Solutions to a Three-Point Boundary Value Problem

Jin Liang1 and Zhi-Wei Lv2,3

1Department of Mathematics, Shanghai Jiao Tong University, Shanghai 200240, China
2Department of Mathematics and Physics, Anyang Institute of Technology, Anyang, Henan 455000, China
3Department of Mathematics, University of Science and Technology of China, Hefei, Anhui 230026, China

Received 25 November 2010; Accepted 19 January 2011

Academic Editor: Toka Diagana

Copyright © 2011 Jin Liang and Zhi-Wei Lv. 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 the fixed-point index theory and Leggett-Williams fixed-point theorem, we study the existence of multiple solutions to the three-point boundary value problem 𝑢 ( 𝑡 ) + 𝑎 ( 𝑡 ) 𝑓 ( 𝑡 , 𝑢 ( 𝑡 ) , 𝑢 ( 𝑡 ) ) = 0 ,   0 < 𝑡 < 1 ; 𝑢 ( 0 ) = 𝑢 ( 0 ) = 0 ; 𝑢 ( 1 ) − 𝛼 𝑢 ( 𝜂 ) = 𝜆 , where 𝜂 ∈ ( 0 , 1 / 2 ] , 𝛼 ∈ [ 1 / 2 𝜂 , 1 / 𝜂 ) are constants, 𝜆 ∈ ( 0 , ∞ ) is a parameter, and 𝑎 , 𝑓 are given functions. New existence theorems are obtained, which extend and complement some existing results. Examples are also given to illustrate our results.