| Contents |
| In this paper we study the relative asymptotic equivalence between the solutions of the following two difference equations in a Banach space $Z$
\[
y(n+1)=A(n)y(n),\quad x(n+1)=A(n)x(n)+f(n,x(n)),
\]
where $y(n),x(n) \in Z$, $A\in l^{\infty}(\N,L(Z))$ and the function $f:\N\times Z \rightarrow Z$ is small enough in some sense. The generalized discrete dichotomy definition and a discrete version of Rodrigues Inequality play the main toll reaching our results, which is the following: Given a solution $y(n)$ of the unperturbed system, we provide sufficient conditions to prove that there exist a family of solutions $x(n)$ for the perturbed system such that
$$
\|y(n)-x(n)\|=\circ(\|y(n)\|), \quad \mbox{as} \quad n\rightarrow \infty.
$$
Conversely, given a solution $x(n)$ of the perturbed system having Lyapunov number $\alpha\in \R$, we prove that, under certain conditions, there exist a family of solutions $y(n)$ for the unperturbed system, such that
\[
\|y(n)-x(n)\|=\circ(\|x(n)\|),\quad \mbox{as} \quad n\rightarrow \infty.
\] |
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