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

Wang, Yongfang; Zada, Akbar; Ahmad, Nisar; Lassoued, Dhaou; Li, Tongxing

Abstr. Appl. Anal. 2014, Article ID 784289, 4 p. (2014).

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 \(X\).

MSC:

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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