Fixed Point Theory and Applications
Volume 2011 (2011), Article ID 604046, 11 pages
doi:10.1155/2011/604046
Research Article

Existence of Positive Solutions for Nonlocal Fourth-Order Boundary Value Problem with Variable Parameter

Xiaoling Han, Hongliang Gao, and Jia Xu

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

Received 26 November 2010; Accepted 14 January 2011

Academic Editor: M. Furi

Copyright © 2011 Xiaoling Han et al. 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 Krasnoselskii's fixed point theorem and operator spectral theorem, the existence of positive solutions for the nonlocal fourth-order boundary value problem with variable parameter 𝑢 ( 4 ) ( 𝑡 ) + 𝐵 ( 𝑡 ) 𝑢 ″ ( 𝑡 ) = 𝜆 𝑓 ( 𝑡 , 𝑢 ( 𝑡 ) , 𝑢 ″ ( 𝑡 ) ) , 0 < 𝑡 < 1 , ∫ 𝑢 ( 0 ) = 𝑢 ( 1 ) = 1 0 𝑝 ( 𝑠 ) 𝑢 ( 𝑠 ) 𝑑 𝑠 , ∫ 𝑢 ″ ( 0 ) = 𝑢 ″ ( 1 ) = 1 0 𝑞 ( 𝑠 ) 𝑢 ″ ( 𝑠 ) 𝑑 𝑠 is considered, where 𝑝 , 𝑞 ∈ 𝐿 1 [ 0 , 1 ] , 𝜆 > 0 is a parameter, and 𝐵 ∈ 𝐶 [ 0 , 1 ] , 𝑓 ∈ 𝐶 ( [ 0 , 1 ] × [ 0 , ∞ ) × ( − ∞ , 0 ] , [ 0 , ∞ ) ) .