Boundary Value Problems
Volume 2009 (2009), Article ID 572512, 18 pages
Research Article

The Existence of Countably Many Positive Solutions for Nonlinear nth-Order Three-Point Boundary Value Problems

College of Sciences, Hebei University of Science and Technology, Shijiazhuang, Hebei 050018, China

Received 5 July 2009; Revised 30 August 2009; Accepted 30 October 2009

Academic Editor: Kanishka Perera

Copyright © 2009 Yude Ji and Yanping 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.


We consider the existence of countably many positive solutions for nonlinear nth-order three-point boundary value problem u(n)(t)+a(t)f(u(t))=0, t(0,1), u(0)=αu(η), u(0)==u(n2)(0)=0, u(1)=βu(η), where n2,α0,β0,0<η<1,α+(βα)ηn1<1, a(t)Lp[0,1] for some p1 and has countably many singularities in [0,1/2). The associated Green's function for the nth-order three-point boundary value problem is first given, and growth conditions are imposed on nonlinearity f which yield the existence of countably many positive solutions by using the Krasnosel'skii fixed point theorem and Leggett-Williams fixed point theorem for operators on a cone.