Existence of multiple solutions for second-order discrete boundary value problems

We give conditions on ƒ involving pairs of discrete lower and discrete upper solutions which lead to the existence of at least three solutions of the discrete two-point boundary value problem yk+1 − 2yk + yk−1 + ƒ(k,yk,vk) = 0, for k = 1,…, n − 1, y0 = 0 = yn, where ƒ is continuous and vk = yk − yk−1, for k = 1,…,n. In the special case ƒ(k,t,p) = ƒ(t) ≥ 0, we give growth conditions on ƒ and apply our general result to show the existence of three positive solutions. We give an example showing this latter result is sharp. Our results extend those of Avery and Peterson and are in the spirit of our results for the continuous analogue.