Electron. J. Diff. Eqns., Vol. 2001(2001), No. 70, pp. 1-5.

A theorem of Rolewicz's type for measurable evolution families in Banach spaces

Constantin Buse & Sever S. Dragomir

Let $\varphi$ be a positive and non-decreasing function defined on the real half-line and ${\cal U}$ be a strongly measurable, exponentially bounded evolution family of bounded linear operators acting on a Banach space and satisfing a certain measurability condition as in Theorem 1 below. We prove that if $\varphi$ and ${\cal U}$ satisfy a certain integral condition (see the relation \ref{0.1} from Theorem 1 below) then ${\cal U}$ is uniformly exponentially stable. For $\varphi$ continuous and \mathcal U strongly continuous and exponentially bounded, this result is due to Rolewicz. The proofs uses the relatively recent techniques involving evolution semigroup theory.

Submitted September 2, 2001. Published November 23, 2001.
Math Subject Classifications: 47A30, 93D05, 35B35, 35B40, 46A30.
Key Words: Evolution family of bounded linear operators, evolution operator semigroup, Rolewicz's theorem, exponential stability.

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Constantin Buse
Department of Mathematics
West University of Timisoara
Bd. V. Parvan 4
1900 Timisoara, Romania
e-mail: buse@tim1.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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