Nontrivial solution for a three-point boundary-value problem

Abstract

In this paper, we study the existence of nontrivial solutions for the second-order three-point boundary-value problem u'' + ƒ(t, u) = 0, 0 < t < 1, u'(0) = 0, u(1) = ɑu'(η). where η ∈ (0, 1), ɑ ∈ ℝ, ƒ ∈ C([0, 1] x ℝ, ℝ). Under certain growth conditions on the nonlinearity ƒ and by using Leray-Schauder nonlinear alternative, sufficient conditions for the existence of nontrivial solution are obtained. We illustrate the results obtained with some examples.