A Characterization of Lipschitz Continuous Evolution Families on Banach Spaces

Abstract

We consider evolution families (U(t, s)) of bounded, linear operators on a Banach space X and associate to it an evolution semigroup (T(t)) t≥0 defined by $$ T(t)f(s) = U(s,s - t)f(s - t) $$on the weighted function space C υ0(ℝ, X). In the case of Lipschitz continuity of (U(t, s)) we characterize the generator of this semigroup (T(t)) t≥0 and thus obtain a family of operators A(t) ∈ ℒ(X) such that (U(t,s)) solves the non-autonomous Cauchy problem $$ \dot{u} = A(t)u(t),u(s) = {{u}_{0}}. $$