Constantin Buse, Aftab Khan, Gul Rahmat and Afshan Tabassum: Uniform exponential stability for discrete non-autonomous systems via discrete evolution semigroups, p.193-205

Abstract:

We prove that a discrete evolution family

$$\mathcal{U}=\{U(m, n)\}_{m\ge n\in\mathbb{Z}_+}$$

of bounded linear operators acting on a complex Banach space $X$ is uniformly exponentially stable if and only if it is admissible in respect to the pair $(c_{00}(\mathbb{Z}_+, X), c_{00}(\mathbb{Z}_+, X)),$ (i. e. the sequence $n\mapsto\sum\limits_{k=0}^nU(n, k)f_k:\mathbb{Z}_+\to X$ belongs to $c_{00}(\mathbb{Z}_+, X)$ for each $(f_k)\in c_{00}(\mathbb{Z}_+, X)$). The approach is based on the theory of discrete evolution semigroups associated to such families.

Key Words: Non-autonomous discrete problems; discrete evolution families of bounded linear operators, discrete evolution semigroups, spectrum of bounded linear operators.

2000 Mathematics Subject Classification: Primary: 35B35;
Secondary: 39B42.

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