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Zeitschrift für Analysis und ihre Anwendungen


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Volume 34, Issue 4, 2015, pp. 477–484
DOI: 10.4171/ZAA/1550

Published online: 2015-10-29

Uniform Exponential Stability of Discrete Semigroup and Space of Asymptotically Almost Periodic Sequences

Nisar Ahmad[1], Habiba Khalid[2] and Akbar Zada[3]

(1) University of Peshawar, Pakistan
(2) University of Peshawar, Pakistan
(3) University of Peshawar, Pakistan

We prove that the discrete semigroup $\mathbb{T}= \{\mathcal{T}(n): n \in \mathbb{Z_+}\}$ is uniformly exponentially stable if and only if for each $z(n)\in \mathbb{AAP}_0(\mathbb{Z}_+,\mathcal{X})$ the solution of the Cauchy problem \begin{equation*} \left\{ \begin{aligned} % \nonumber to remove numbering (before each equation) y_{n+1} &= \mathcal{T}(1)y_{n}+ z(n+1), \\ y(0)&= 0 \end{aligned} \right. \end{equation*} belongs to $\mathbb{AAP}_0(\mathbb{Z}_+,\mathcal{X})$. Where $\mathcal{T}(1)$ is the algebraic generator of $\mathbb{T}$, $\mathbb{Z}_+$ is the set of all non-negative integers and $\mathcal{X}$ is a complex Banach space. Our proof uses the approach of discrete evolution semigroups.

Keywords: Exponential stability, discrete semigroups, periodic sequences, almost periodic sequences

Ahmad Nisar, Khalid Habiba, Zada Akbar: Uniform Exponential Stability of Discrete Semigroup and Space of Asymptotically Almost Periodic Sequences. Z. Anal. Anwend. 34 (2015), 477-484. doi: 10.4171/ZAA/1550