Linearization for difference equations with infinite delay

TL;DR

This paper develops a conjugacy map for linear difference equations with infinite delay and nonlinear perturbations, establishing conditions for its invertibility and applying it to exponential dichotomy cases.

Contribution

It introduces a conjugacy map for infinite delay difference equations and proves its invertibility under certain conditions, extending the understanding of their dynamics.

Findings

Conjugacy map constructed for linear difference equations with infinite delay.

Proved the conjugacy map is one-to-one under additional conditions.

Applied results to cases of exponential dichotomy.

Abstract

In this article, we construct a conjugacy map for a linear difference equation with infinite delay and corresponding nonlinear perturbation. We also prove that the conjugacy map is one-one with some additional conditions. As an application of our result, we show that the cases of (uniform) exponential dichotomy follow from our result.