In this paper we study the relative asymptotic equivalence between the solutions of the following two difference equations in a Banach space Zy(n+1)=A(n)y(n),x(n+1)=A(n)x(n)+f(n,x(n)),where y(n),x(n)∈Z, A∈l∞(N,L(Z)) and the function f:N×Z→Z 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 y(n) of the unperturbed system, we provide sufficient conditions to prove that there exists a family of solutions x(n) for the perturbed system such that|y(n)−x(n)|=o(|y(n)|),asn→∞.Conversely, given a solution x(n) of the perturbed system having Lyapunov number α∈R, we prove that under certain conditions, there exists a family of solutions y(n) for the unperturbed system, such that|y(n)−x(n)|=o(|x(n)|),asn→∞.