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.

Abstract

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(n−2)(0)=0, u(1)=βu(η), where n≥2,α≥0,β≥0,0<η<1,α+(β−α)ηn−1<1, a(t)∈Lp[0,1] for some p≥1 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.