New aspects of nonautonomous discrete systems stability
Résumé
We prove that a discrete evolution family ${\bf U}:=\{U(n,m):\; n\geq m\in \mathbb{Z}_+\}$ of bounded linear operators acting on a complex Banach space $X$ is uniformly esponentially stable if and only if for each forcing term $(f(n))_{n\in \mathbb{Z}_+}$ belonging to $AP_0(\mathbb{Z}_+, X)$, the solution of the discrete Cauchy Problem
$$
\left\{
\begin{array}{lc}
x(n+1)=A(n)x(n)+f(n), n\in \mathbb{Z}_+ \\
x(0)=0
\end{array}
\right.
$$
belongs to $AP_0(\mathbb{Z}_+, X)$, where the operators-valued sequence $(A(n))_{n\in \mathbb{Z}_+}$ generates the evolution family ${\bf U}$. The approach we use is based on the theory of
discrete evolution semigroups associated to this family.