Semigroup properties and the Crandall Liggett approximation for a class of differential equations with state-dependent delays (Q1604631)

The following differential equation with state-dependent delays is considered \[ x'(t)= f(x(t- r(x_t))),\quad t\geq 0,\tag{1} \] with the initial condition \[ x_0= \varphi.\tag{2} \] It is assumed that \(f: \mathbb{R}\to \mathbb{R}\) is Lipschitzian, \(r\) is a functional acting from the function space \(C= C([- M,0],\mathbb{R})\) into \([0,M]\) and by \(x_t\) we denote the function from the space \(C\) defined by the equatility \(x_t(\Theta)= x(t+\Theta)\) for \(\Theta\in [-M,0]\). Here, it is shown that problem (1)--(2) generates a strongly continuous semigroup in a closed subset of the space \(C_{0,1}\) consisting of functions \(x: [-M,0]\to \mathbb{R}\) being Lipschitz continuous. The infinitesimal generator of that semigroup is characterized in terms of its domain. Moreover, an approximation of the Crandall-Liggett type of that semigroup is obtained in a dense subset of \(C\).