Semigroups of translations and functional differential equations
A retarded functional differential equation of the form \(\dot x(\)t)\(=f(t,x_ t)\), \(t\geq t_ 0\), \(x_{t_ 0}=\phi\) is studied and an appropriate Banach space of measurable functions \(\phi\) : (-\(\infty,0]\to {\mathbb{R}}^ n\) is introduced on which it is assumed that the equation has unique solutions which depend continuously on the initial data. Using the theory of nonlinear evolution operators, it is shown that the solution of the equation U(t,s)x for \(x\in X\) is an evolution operator. Finally it is shown that, under the conditions of a theorem of \textit{M. G. Crandall} and \textit{A. Pazy} [Isr. J. Math. 11, 57-94 (1972; Zbl 0249.34049)] this evolution operator is a translation.
retarded functional differential equation
scientific article; zbMATH DE number 3951046