The existence of solutions are discussed for the secondorder difference equation threepoint boundary value problems
Δ^{2}u(t-1)=f(t, u(t)),t∈T,
u(0)=εΔu(0), u(T+1)=αu(η),
where f: T×R→R is continuous, T is a fixed positive integer, T:=｛1, 2,…,T｝. And ε∈［0,∞), α∈(0,∞), η∈T are given constants such that α(η+ε)=T+1+ε. The proof of our main results is based on the method of lower and upper solutions using the connectivity properties of the solution sets of parameterized families of compact vector fields.