Uniform Exponential Stability of Discrete Evolution Families on Space of p-Periodic Sequences

Abstract

We prove that the discrete system ${\zeta }_{n+1}={\mathcal{A}}_n{\zeta }_n$ is uniformly exponentially stable if and only if the unique solution of the Cauchy problem ${\zeta }_{n+1}={\mathcal{A}}_n{\zeta }_n+e^{i\theta \left(n+1\right)}z\left(n+1\right)$, $\mathrm{ }n\in {\mathbb{Z}}_{+}$, ${\zeta }_0=0,$ is bounded for any real number $\theta$ and any $p$-periodic sequence $z(n)$ with $z(0)=0$. Here, ${\mathcal{A}}_n$ is a sequence of bounded linear operators on Banach space $X$.