Copyright © 2012 Xiaojie Lin and Zhengmin Fu. 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 investigate the problem of existence of positive solutions for the nonlinear third-order three-point boundary value problem u‴(t)+λa(t)f(u(t))=0, 0<t<1, u(0)=u′(0)=0, u″(1)=∝u″(η), where λ is a positive parameter, ∝∈(0,1), η∈(0,1), f:(0,∞)→(0,∞), a:(0,1)→(0,∞) are continuous. Using a specially constructed cone, the fixed point index theorems and Leray-Schauder degree, this work shows the existence and multiplicities of positive solutions for the nonlinear third-order boundary value problem. Some examples are given to demonstrate the main results.