Positive solutions of singular third-order three-point boundary value problem

In this paper we investigate the problem of existence of positive solutions for the nonlinear singular third-order three-point boundary value problem u''' (t) - lambda a(t)F(t, u(t)) = 0, 0 < T < 1, u(0) = u' (eta) = u"(1) = 0, where lambda is a positive parameter and eta epsilon [1/2, 1) is a constant. By using a fixed point theorem of cone expansion-compression type due to Krasnosel'skii, we establish various results on the existence of single and multiple positive solutions to the boundary value problem. Under various assumptions on functions F and a, we give explicitly the intervals for parameter lambda in which the existence of positive solutions is guaranteed. Especially, we allow the function a(t) of nonlinear term to have suitable singularities. (c) 2004 Elsevier Inc. All rights reserved.