Electronic Journal of Differential Equations,
Vol. 2001(2001), No. 70, pp. 1-5.
Title: A theorem of Rolewicz's type for measurable
evolution families in Banach spaces
Authors: Constantin Buse (West Univ. of Timisoara, Romania)
Sever 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 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 ${\mathcal U}$ satisfy a certain
integral condition (see the relation \ref{0.1} from Theorem 1 below)
then ${\mathcal U}$ 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.