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Relative asymptotic equivalence between difference equations

Abstract:

In this paper we study the relative asymptotic equivalence between the solutions of the following two difference equations in a Banach space Z (Formula presented.) where (Formula presented.) , (Formula presented.) and the function (Formula presented.) is small enough in some sense. The discrete dichotomy definition and a discrete version of Rodrigues Inequality are the main tools in obtaining our results: Given a solution (Formula presented.) of the unperturbed system, we provide sufficient conditions to prove that there exists a family of solutions (Formula presented.) for the perturbed system such that (Formula presented.) Conversely, given a solution (Formula presented.) of the perturbed system having Lyapunov number (Formula presented.) , we prove that under certain conditions, there exists a family of solutions (Formula presented.) for the unperturbed system, such that (Formula presented.)