Uniform Exponential Stability of Discrete Semigroup and Space of Asymptotically Almost Periodic Sequences

Abstract

We prove that the discrete semigroup T = {T (n) : n ∈ Z+} is uniformly exponentially stable if and only if for each z(n) ∈ AAP0(Z+,X ) the solution of the Cauchy problem { yn+1 = T (1)yn + z(n + 1), y(0) = 0 belongs to AAP0(Z+,X ). Where T (1) is the algebraic generator of T, Z+ is the set of all non-negative integers and X is a complex Banach space. Our proof uses the approach of discrete evolution semigroups.