Topological Equivalence of Nonautonomous Difference Equations with a Family of Dichotomies on the Half Line

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 Lipschitzness conditions. As a consequence, we study the asymptotical stability and its preservation by topological equivalence.