Existence of Periodic Solutions to Nonlinear Difference Equations at Full Resonance

Abstract

The purpose of this paper is to search for periodic solutions to a system of nonlinear difference equations of the form \[\Delta x(t) = f(\epsilon,t,x(t)).\] The corresponding linear homogeneous system has an $n$-dimensional kernel, i.e. the system is at full resonance. We provide sufficient conditions for the existence of periodic solutions based on asymptotic properties of the nonlinearity $f$ when $\epsilon=0$. To this end, we employ a projection method using the Lyapunov-Schmidt procedure together with Brouwer's fixed point theorem.