A New Proof for a Rolewicz's Type Theorem: An Evolution Semigroup Approach

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.