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) −λa(t)F t,u(t) = 0, 0 < t <1, u(0) = u (η) = u (1) = 0, where λ is a positive parameter and η ∈ [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 λ in which the existence of positive solutions is guaranteed. Especially, we allow the function a(t) of nonlinear term to have suitable singularities.  2004 Elsevier Inc. All rights reserved