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

Abstract

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