A Theorem of Rolewicz's Type for Measurable Evolution Families in Banach Spaces

Abstract

Let φ be a positive and non-decreasing function defined on the real half-line and U be a strongly measurable, exponentially bounded evolution family of bounded linear operators acting on a Banach space and satisfying a certain measurability condition as in Theorem 1 below. We prove that if φ and U satisfy a certain integral condition (see the relation 1 from Theorem 1 below) then U is uniformly exponentially stable. For φ continuous and U strongly continuous and exponentially bounded, this result is due to Rolewicz. The proofs uses the relatively recent techniques involving evolution semigroup theory.