Solvability for a nonlinear second-order three-point boundary value problem

This paper investigates the existence of nontrivial solution for the three-point boundary value problem u + f (t,u) = 0, 0 < t <1, u (0) = 0, u(1) = αu(η), where η ∈ (0, 1), α ∈ R, α = 1, f ∈ C([0, 1] × R,R). Under certain growth conditions on the nonlinearity f , several sufficient conditions for the existence of nontrivial solution are obtained by using Leray–Schauder nonlinear alternative. As an application, some examples to demonstrate our results are given.  2004 Elsevier Inc. All rights reserved