Electronic Journal of Differential Equations,
Vol. 2001(2001), No. 45, pp. 1-5.
Title: A new proof for a Rolewicz's type theorem:
An evolution semigroup approach
Authors: C. Buse (West Univ. of Timisoara, Romania)
S. S. Dragomir (Victoria Univ. of Technology, Australia)
Abstract:
Let $\varphi$ be a positive and non-decreasing function defined on the
real half-line and $\mathcal{U}$ be a strongly continuous and
exponentially bounded evolution family of bounded linear operators
acting on a Banach space. We prove that if $\varphi$ and $\mathcal{U}$
satisfy a certain integral condition (see the relation (2)
below) then $\mathcal{U}$ 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.