Semigroup properties and the Crandall Liggett approximation for a class of differential equations with state-dependent delays

Abstract

We present an approach for the resolution of a class of differential equations with state-dependent delays by the theory of strongly continuous nonlinear semigroups. We show that this class determines a strongly continuous semigroup in a closed subset of C0,1. We characterize the infinitesimal generator of this semigroup through its domain. Finally, an approximation of the Crandall-Liggett type for the semigroup is obtained in a dense subset of (C, ∥ · ∥∞). As far as we know this approach is new in the context of state-dependent delay equations while it is classical in the case of constant delay differential equations. © 2002 Elsevier Science (USA).