Discrete third-order three-point right-focal boundary value problems

We are concerned with the discrete right-focal boundary value problem Δ3x(t) = f(t, x(t+1)), x(t1) = Δx(t2) = Δ2x(t3) = 0, and the eigenvalue problem Δ3x(t) = λa(t)f(x(t+1)) with the same boundary conditions, where t1 < t2 < t3. Under various assumptions on f, a, and λ, we prove the existence of positive solutions of both problems by applying a fixed-point theorem.