Discrete third-order three-point boundary value problem

We are concerned with the discrete boundary value problem Lx(t) := Delta(3)x(t) = f (t, x(t + 1)) t(1) <= t <= t(3) - 1, x(t(1)) =0, alpha x(t(2))-beta Delta x(t(2))=0, gamma x(t(3))+delta Delta(2)x(t(3))=0, and the eigenvalue problem Delta(3)x(t) =)lambda f(t, x(t + 1)) with the same boundary conditions where t(1) < t(2) < t(3) are distinct integers. Under various assumptions on f and),, we prove the existence of positive solutions of both problems by applying a fixed point theorem. (C) 2006 Elsevier B.V. All rights reserved.