Existence of homoclinic constant sign solutions for a difference equation on the integers

We consider a difference equation involving the discrete p-Laplacian operator, depending on a positive real parameter k. We prove, under convenient assumptions, that for k big enough the equations admit at least one homoclinic constant sign solution in Z. Our method consists in two parts: first, we prove the existence of two Dirichlet-type solutions for the equation in the discrete interval ½Àn; nŠ, for all n 2 N big enough; then, we show that such solutions converge to a homoclinic solution in Z, as n ! 1. In the present paper we will deal with the following difference equation on Z, with homoclinic asymptotic conditions, depending on a real parameter k > 0: