Three-point boundary value problems for second-order ordinary differential equations in Banach spaces

In this paper, by using the Sadovskii fixed point theorem, we study the existence of at least one solution for the second-order three-point boundary value problem u"(t) + f (t.u(t), u'(t)) = theta, 0 < t < 1, u(0) = theta, u(1) = alpha u(eta) in a Banach space E, where theta is the zero element of E, 0 < alpha < 1, 0 < eta < 1/alpha. We also obtain the existence of at least one positive solution. As an application, we give A, example to demonstrate our results. (C) 2008 Elsevier Ltd. All rights reserved.