A new proof for a Rolewicz's type theorem: An evolution semigroup approach

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