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

Abstract

We prove that the discrete semigroup $\mathbb{T}=\{\mathcal{T}(n):n\in {\mathbb{Z}}_{+}\}$ is uniformly exponentially stable if and only if for each $z(n)\in {\mathbb{AAP}}_0({\mathbb{Z}}_{+},\mathcal{X})$ the solution of the Cauchy problem

belongs to ${\mathbb{AAP}}_0({\mathbb{Z}}_{+},\mathcal{X})$. Where $\mathcal{T}(1)$ is the algebraic generator of $\mathbb{T}$, ${\mathbb{Z}}_{+}$ is the set of all non-negative integers and $\mathcal{X}$ is a complex Banach space. Our proof uses the approach of discrete evolution semigroups.