Journal of Inequalities and Applications
Volume 2006 (2006), Article ID 79653, 17 pages
doi:10.1155/JIA/2006/79653

Existence and multiplicity of solutions for some three-point nonlinear boundary value problems

Xu Xian1 and Donal O'Regan2

1Department of Mathematics, Xuzhou Normal University, Xuzhou, Jiangsu 221116, China
2Department of Mathematics, National University of Ireland, Galway, University Road, Galway, Ireland

Received 30 September 2004; Accepted 20 October 2004

Copyright © 2006 Xu Xian and Donal O'Regan. 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 study the existence and multiplicity of solutions for the three-point nonlinear boundary value problem u″(t)+λa(t)f(u)=0, 0<t<1; u(0)=0=u(1)−γu(η), where η∈(0,1), γ∈[0,1), a(t) and f(u) are assumed to be positive and have some singularities, and λ is a positive parameter. Under certain conditions, we prove that there exists λ∗>0 such that the three-point nonlinear boundary value problem has at least two positive solutions for 0<λ<λ∗, at least one solution for λ=λ∗, and no solution for λ>λ∗.