Applied Mathematics & Information Sciences

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We prove that a discrete evolution family U := {U(n,m) : n ≥ m ∈ 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∈Z+ belonging to AP0(Z+,X), the solution of the discrete Cauchy Problem x(n+1) = A(n)x(n)+ f (n), n ∈ Z+ x(0) = 0 belongs also to AP0(Z+,X), where the operators-valued sequence (A(n))n∈Z+ generates the evolution family U. The approach we use is based on the theory of discrete evolution semigroups associated to this family.