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On Uniform Exponential Stability of Exponentially Bounded Evolution Families

Buse, C and Choudary, A. D. R and Dragomir, Sever S and Prajea, M. S (2008) On Uniform Exponential Stability of Exponentially Bounded Evolution Families. Integral Equations and Operator Theory, 61 (3). pp. 325-340. ISSN 0378-620X

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Abstract

A result of Barbashin ([1], [15]) states that an exponentially bounded evolution family {U(t, s)}t≥s≥0 defined on a Banach space and satisfying some measurability conditions is uniformly exponentially stable if and only if for some 1 ≤ p < ∞, we have that: sup t≥0  t 0 ||U(t, s)||pds < ∞. Actually the Barbashin result was formulated for non-autonomous differential equations in the framework of finite dimensional spaces. Here we replace the above ”uniform” condition be a ”strong” one. Among others we shall prove that the evolution family {U(t, s)}t≥s≥0 is uniformly exponentially stable if there exists a non-decreasing function φ : R+ → R+ with φ(r) > 0 for all r > 0 such that for each x∗ ∈ X∗, one has: sup t≥0 t 0 φ(||U(t, s) ∗ x ∗||)ds < ∞. In particular, the family U is uniformly exponentially stable if and only if for some 0 < p < ∞ and each x ∗ ∈ X ∗ , the inequality sup t≥0  t 0 ||U(t, s) ∗ x ∗||pds < ∞ is fulfilled. The latter result extends a similar one from the recent paper [4]. Related results for periodic evolution families are also obtained.

Item Type: Article
Uncontrolled Keywords: ResPubID15209. operator semigroups, rearrangement function space, evolution families of bounded linear operators, uniform exponential stability
Subjects: Faculty/School/Research Centre/Department > School of Engineering and Science
FOR Classification > 0101 Pure Mathematics
SEO Classification > 970101 Expanding Knowledge in the Mathematical Sciences
Depositing User: VUIR
Date Deposited: 30 Aug 2011 06:36
Last Modified: 28 Jan 2015 22:43
URI: http://vuir.vu.edu.au/id/eprint/3546
DOI: 10.1007/s00020-008-1592-7
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Citations in Scopus: 6 - View on Scopus

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