Properties of the semigroup and the approximation of Crandall–Liggett associated with a class of differential equations with delays depending implicitly on the state

In this paper we are concerned with the semigroup properties and the Crandall–Liggett approximation associated with a class of differential equations with delays depending implicitly on the state. The existence of a strongly continuous nonlinear semigroup associated with the solutions of the equations is proved. Various properties of this semigroup have been established, such as the continuous dependence, the compactness and the infinitesimal generators. An approximation of Crandall–Liggett type has also been established.