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