Uniform exponential stability of discrete evolution families on space of \(p\)-periodic sequences. (English) Zbl 1469.39005

Summary: 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 \(\zeta_{n + 1} = \mathcal{A}_n \zeta_n + e^{i \theta \left(n + 1\right)} z \left(n + 1\right)\), \( n \in \mathbb{Z}_+\), \(\zeta_0 = 0\), 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 \(X\).


39A12 Discrete version of topics in analysis
39B42 Matrix and operator functional equations
39B82 Stability, separation, extension, and related topics for functional equations
34G10 Linear differential equations in abstract spaces
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