Existence of solutions for a three-point boundary value problem at resonance

This paper discusses the existence of solutions for the second-order three-point boundary value problem u″(t)=f(t,u(t)),t∈(0,1),u(0)=ϵu′(0),u(1)=αu(η), where f:[0,1]×R→Rf:[0,1]×R→R is continuous and ϵ∈[0,∞)ϵ∈[0,∞), α∈(0,∞)α∈(0,∞) and η∈(0,1)η∈(0,1) are given constants such that α(η+ϵ)=1+ϵα(η+ϵ)=1+ϵ. The proof of our main result is based upon the methods of lower and upper solutions using the connectivity properties of the solution sets of parameterized families of compact vector fields.