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.