Existence of Local Analytic Solutions for Systems of Difference Equations with Small Step Size

Abstract

We study the existence of analytic solutions of systems of difference equations where we have vector valued functions y of ε and (ε + x) equals to vector valued analytic functions F of ε, x and y in a neighborhood of (0, x^*, y^*) with y^* = F(0, x^*, y^*). Under the assumption that the Jacobian of F with respect to y at (0, x^*, y^*) minus the idendity is invertible, we first show the existence of a unique formal solution that is Gevrey-1. We also show, by applying a fixed point theorem, the existence of analytic solutions having a Gevrey-1 asymptotic expansion in small ε-sectors. This requires the construction of some bounded right inverse operators on a certain Banach space.