Perturbation theory for dual semigroups II. Time-dependent perturbations in the sun-reflexive case

Abstract

SynopsisWe consider time-dependent perturbations of generators of strongly continuous semigroups on a Banach space. The perturbations map the Banach space into a bigger space, which is the second dual of the original space in a specific semigroup sense. Using the theory of dual semigroups we show that the solutions of a generalised variation-of-constants formuladefine an evolutionary system. We investigate continuity and differentiability propertiesof this evolutionary system and its dual system and examine in what sense the perturbed generator and its adjoint generate these evolutionary systems. It is shown that the results apply naturally to retarded functional differential equations and age structured population dynamics.