Electron. J. Diff. Eqns., Vol. 2001(2001), No. 45, pp. 1-5.

A new proof for a Rolewicz's type theorem: An evolution semigroup approach

C. Buse & S. S. Dragomir

Let $\varphi$ be a positive and non-decreasing function defined on the real half-line and $\cal U$ be a strongly continuous and exponentially bounded evolution family of bounded linear operators acting on a Banach space. We prove that if $\varphi$ and $\cal U$ satisfy a certain integral condition (see the relation (2) below) then $\cal U$ is uniformly exponentially stable. For $\varphi$ continuous, this result is due to S. Rolewicz.

Submitted May 14, 2001. Published June 20, 2001.
Math Subject Classifications: 47A30, 93D05, 35B35, 35B40, 46A30.
Key Words: Evolution family of bounded linear operators, evolution operator semigroup, Rolewicz's theorem.

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Constantin Buse
Department of Mathematics
West University of Timisoara
Bd. V. Parvan 4
1900 Timisoara, Romania
e-mail: buse@hilbert.math.uvt.ro
Sever S. Dragomir
School of Communications and Informatics
Victoria University of Technology
PO Box 14428
Melburne City MC 8001
Victoria, Australia
e-mail: sever@matilda.vu.edu.au

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