Multiple positive solutions to a third-order discrete focal boundary value problem

We are concerned with the discrete focal boundary value problem Δ3x(t − k) = f(x(t)), x(a) = Δx(t2) = Δ2x(b + 1 = 0. Under various assumptions on f and the integers a, t2, and b we prove the existence of three positive solutions of this boundary value problem. To prove our results, we will apply a generalization of the Leggett-Williams fixed-point theorem.