Perturbation Theory for Dual Semigroups 2. Time-Dependent Perturbations in the Sun-Reflexive Case

摘要

Time-dependent perturbations of generators of strongly continuous semigroups on a Banach space are considered. 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, the authors show that the solutions of a generalized variation-of-constants formula define an evolutionary system. The authors investigate continuity and differentiability properties of the 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.