On variational methods for periodic discrete problems

Abstract

We prove existence results for a general 2n-th order discrete boundary value problem subjected to periodic conditions. In contrast to published work, we exploit the finite dimension of function spaces and some simple results from matrix theory to construct a nonlinear equation in which is eventually studied with the degree theory and simple variational methods. Existence and uniqueness theorems, which are obtained in this way, improve the existing results (even in the second-order case) by including non-continuous right-hand sides, higher order problems and non-invertible difference operators.