Stability of asymptotic properties of Hille-yosida operators under perturbations and retarded differential equations

Abstract

We give conditions on the strongly continuous semigroup (T ₀(t)) _t≥0, generated by the part of a Hille-Yosida operator A on X ₀ := D(A), and a non-autonomous family of operators (B(t)) _t≥0 such that the evolution family (U(t,s)) _t≥s≥0, generated by the part of (A + B(t)) _t≥0 in X ₀, inherits some asymptotic properties of (T ₀(t)) _t≥0 as boundedness, asymptotic almost periodicity, uniform ergodicity and total uniform ergodicity. Our main result is obtained via an extended result of Batty-Chill [7], which we prove here. We illustrate our abstract hypotheses and results by an example from population dynamics. Also, an application to non-autonomous retarded differential equations is given.