Topological Equivalence of nonautonomous difference equations with a family of dichotomies on the half line

Abstract

A linear system of difference equations and a nonlinear perturbation are considered, we obtain sufficient conditions to ensure the topological equivalence between them, namely, the linear part satisfies a property of dichotomy on the positive half–line while the nonlinearity has some boundedness and Lipschitz conditions. In addition, we provide new characterizations for the resulting homeomorphisms. When the linear system is asymptotically stable and the nonlinear system has a unique equilibrium, we deduce sharper results for the smoothness of the topological equivalence. Finally, we study the asymptotic stability and its preservation by topological equivalence.