Existence of solution for nonlinear fourth-order three-point boundary value problem

Abstract

In this paper, we study the existence of nontrivial solution for the fourth-order three- point boundary value problem having the following form

u(4) (t) + f (t, u(t)) = 0, 0 < t < 1,

u(0) = α(η), u'(0) = u''(0) = 0, u(1) = βu(η),

where η ∈ (0, 1), α, β ∈ R, f ∈ C ([0, 1] × R, R). We give sufficient conditions that allow us to obtain the existence of a nontrivial solution. And by using the Leray-Schauder nonlinear alternative we prove the existence of at least one solution of the posed problem. As an application, we also given some examples to illustrate the results obtained.