A nonlinear semigroup for a functional differential equation

Abstract

A representation theorem is obtained for solutions of the nonlinear functional differential equation ( 1 ) u ′ ( t ) = F ( u t ) , t ⩾ 0 , u ( t ) = ϕ ( t ) , t ⩽ 0 , \begin{equation}\tag {$1$} u’(t) = F({u_t}), t \geqslant 0,\quad u(t) = \phi (t), t \leqslant 0,\end{equation} as a semigroup of nonlinear operators on a space of initial data X of “fading memory type.” Equation (1) is studied in the abstract setting of a Banach space E. The nonlinear functional F is a uniformly Lipschitz continuous mapping from X to E. The semigroup is constructed by transforming (1) to an abstract Cauchy problem ( C P ) w ′ ( t ) + A w ( t ) = 0 , w ( 0 ) = ϕ , \begin{equation}\tag {$CP$} w’(t) + Aw(t) = 0,\quad w(0) = \phi ,\end{equation} in the space X and applying a generation theorem of M. Crandall and T. Liggett to the operator A in X. The case when (1) is a nonlinear Volterra integrodifferential equation of infinite delay is given special consideration. The semigroup representation is used to obtain finite difference approximations for solutions of (CP) and to study the continuity of solutions of (1) with respect to perturbations of F and ϕ \phi .