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

摘要

In this article, we prove that the ω-periodic discrete evolution family Γ : = { ρ ( n , k ) : n , k ∈ Z + , n ≥ k } $Gamma:= {ho(n,k): n, k inmathbb{Z}_{+}, ngeq k}$ 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 ) ) $(xi(n))$ taken from some Banach space, the approximate solution of the nonautonomous ω-periodic discrete system θ n + 1 = Λ n θ n $heta _{n+1} = Lambda_{n}heta_{n}$ , n ∈ Z + $ninmathbb{Z}_{+}$ is represented by ϕ n + 1 = Λ n ϕ n + e i γ ( n + 1 ) ξ ( n + 1 ) $phi _{n+1}=Lambda_{n}phi_{n}+e^{igamma(n+1)}xi(n+1)$ , n ∈ Z +