On Uniform Exponential Stability and Exact Admissibility of Discrete Semigroups

On Uniform Exponential Stability and Exact Admissibility of Discrete Semigroups

We prove that a discrete semigroup 𝕋={T(n):n∈ℤ+} of bounded linear operators acting on a complex Banach space X is uniformly exponentially stable if and only if, for each x∈AP0(ℤ+,X), the sequence n↦∑k=0n‍T(n-k)x(k):ℤ+→X belongs to AP0(ℤ+,X). Similar results for periodic discrete evolution families...

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Bibliographic Details
Main Authors: Aftab Khan, Gul Rahmat, Akbar Zada
Format: Article
Language:English
Published: Hindawi Limited 2013-01-01
Series:International Journal of Differential Equations
Online Access:http://dx.doi.org/10.1155/2013/268309
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spelling doaj-5ad0a0d2eaea41ab9cb603b692f789ef2020-11-24T22:40:01ZengHindawi LimitedInternational Journal of Differential Equations1687-96431687-96512013-01-01201310.1155/2013/268309268309On Uniform Exponential Stability and Exact Admissibility of Discrete SemigroupsAftab Khan0Gul Rahmat1Akbar Zada2Shaheed Benazir Bhutto University Sheringal, Dir Upper 18000, PakistanGovernment College University, Abdus Salam School of Mathematical Sciences (ASSMS), Lahore 54600, PakistanDepartment of Mathematics, University of Peshawar, Peshawar 25000, PakistanWe prove that a discrete semigroup 𝕋={T(n):n∈ℤ+} of bounded linear operators acting on a complex Banach space X is uniformly exponentially stable if and only if, for each x∈AP0(ℤ+,X), the sequence n↦∑k=0n‍T(n-k)x(k):ℤ+→X belongs to AP0(ℤ+,X). Similar results for periodic discrete evolution families are also stated.http://dx.doi.org/10.1155/2013/268309
collection DOAJ
language English
format Article
sources DOAJ
author Aftab Khan
Gul Rahmat
Akbar Zada
spellingShingle Aftab Khan
Gul Rahmat
Akbar Zada
On Uniform Exponential Stability and Exact Admissibility of Discrete Semigroups
International Journal of Differential Equations
author_facet Aftab Khan
Gul Rahmat
Akbar Zada
author_sort Aftab Khan
title On Uniform Exponential Stability and Exact Admissibility of Discrete Semigroups
title_short On Uniform Exponential Stability and Exact Admissibility of Discrete Semigroups
title_full On Uniform Exponential Stability and Exact Admissibility of Discrete Semigroups
title_fullStr On Uniform Exponential Stability and Exact Admissibility of Discrete Semigroups
title_full_unstemmed On Uniform Exponential Stability and Exact Admissibility of Discrete Semigroups
title_sort on uniform exponential stability and exact admissibility of discrete semigroups
publisher Hindawi Limited
series International Journal of Differential Equations
issn 1687-9643
1687-9651
publishDate 2013-01-01
description We prove that a discrete semigroup 𝕋={T(n):n∈ℤ+} of bounded linear operators acting on a complex Banach space X is uniformly exponentially stable if and only if, for each x∈AP0(ℤ+,X), the sequence n↦∑k=0n‍T(n-k)x(k):ℤ+→X belongs to AP0(ℤ+,X). Similar results for periodic discrete evolution families are also stated.
url http://dx.doi.org/10.1155/2013/268309
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AT gulrahmat onuniformexponentialstabilityandexactadmissibilityofdiscretesemigroups
AT akbarzada onuniformexponentialstabilityandexactadmissibilityofdiscretesemigroups
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