文章基本信息

标题：Connections between Hyers-Ulam stability and uniform exponential stability of discrete evolution families of bounded linear operators over Banach spaces
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作者：Tongxing Li ; Akbar Zada
期刊名称：Advances in Difference Equations
印刷版ISSN：1687-1839
电子版ISSN：1687-1847
出版年度：2016
卷号：2016
期号：1
页码：153
DOI：10.1186/s13662-016-0881-8
语种：English
出版社：Hindawi Publishing Corporation
摘要：In this article, we prove that the ω-periodic discrete evolution family Γ : = { ρ ( n , k ) : n , k ∈ Z + , n ≥ k } $\Gamma:= \{\rho(n,k): n, k \in\mathbb{Z}_{+}, n\geq k\}$ 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 ( 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 $\theta _{n+1} = \Lambda_{n}\theta_{n}$ , n ∈ Z + $n\in\mathbb{Z}_{+}$ is represented by ϕ n + 1 = Λ n ϕ n + e i γ ( n + 1 ) ξ ( n + 1 ) $\phi _{n+1}=\Lambda_{n}\phi_{n}+e^{i\gamma(n+1)}\xi(n+1)$ , n ∈ Z + $n\in\mathbb{Z}_{+}$ ; ϕ 0 = θ 0 $\phi_(=\theta_($ , then the Hyers-Ulam stability of the nonautonomous ω-periodic discrete system θ n + 1 = Λ n θ n $\theta_{n+1} = \Lambda_{n}\theta_{n}$ , n ∈ Z + $n\in\mathbb{Z}_{+}$ is equivalent to its uniform exponential stability.
关键词：Hyers-Ulam stability ; uniform exponential stability ; discrete evolution family of bounded linear operators ; periodic sequence