Existence of positive solutions for nonlinear third-order three-point boundary value problems

This paper is concerned with the following nonlinear third-order three-point boundary value problem: u''' (t) + a(t) f (u(t)) = 0, t is an element of (0, 1), u(0) = u' (0) = 0, u'(1) = alpha u' (eta), where 0 < eta < 1 and 1 < alpha < 1/eta. First, the Green's function for the associated linear boundary Value problem is constructed , and then, some useful properties of the Green's function are obtained by a new method. Finally, existence results for at least one Positive Solution for the above problem are established when f is superlinear or sublinear. (C) 2008 Published by Elsevier Ltd.