Connections between Hyers-Ulam stability and uniform exponential stability of discrete evolution families of bounded linear operators over Banach spaces

Abstract

In this article, we prove that the ω-periodic discrete evolution family $\Gamma:= \{\rho(n,k): n, k \in\mathbb{Z}_{+}, n\geq k\}$ of bounded linear operators is Hyers-Ulam stable if and only if it is uniformly exponentially stable under certain conditions. More precisely, we prove that if for each real number γ and each sequence $(\xi(n))$ taken from some Banach space, the approximate solution of the nonautonomous ω-periodic discrete system $\theta _{n+1} = \Lambda_{n}\theta_{n}$ , $n\in\mathbb{Z}_{+}$ is represented by $\phi _{n+1}=\Lambda_{n}\phi_{n}+e^{i\gamma(n+1)}\xi(n+1)$ , $n\in\mathbb{Z}_{+}$ ; $\phi_{0}=\theta_{0}$ , then the Hyers-Ulam stability of the nonautonomous ω-periodic discrete system $\theta_{n+1} = \Lambda_{n}\theta_{n}$ , $n\in\mathbb{Z}_{+}$ is equivalent to its uniform exponential stability.