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

This is the authors' abstract. We don't add key points for this paper.

In this article, we prove that the ω-periodic discrete evolution family Γ:= \ρ(n,k): n, k ₊, n≥ 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 (ξ(n)) taken from some Banach space, the approximate solution of the nonautonomous ω-periodic discrete system θ ₙ₊₁ = Λₙθₙ , n₊ is represented by φ ₙ₊₁=Λₙφₙ+eiγ(n+1)ξ(n+1) , n₊ ; φ₀=θ₀ , then the Hyers-Ulam stability of the nonautonomous ω-periodic discrete system θₙ₊₁ = Λₙθₙ , n₊ is equivalent to its uniform exponential stability.