Delay differential equations with differentiable solution operators on open domains in $C((-\infty ,0],{\mathbb{R}}^n)$, and processes for Volterra integro-differential equations

Abstract

For autonomous delay differential equations $x^{\prime }(t)=f(x_t)$ we construct a continuous semiflow of continuously differentiable solution operators $x_0\mapsto x_t$, $t\ge 0$, on open subsets of the Fréchet space $C((-\infty ,0],{\mathbb{R}}^n)$. For nonautonomous equations this yields a continuous process of differentiable solution operators. As an application we obtain processes which incorporate all solutions of Volterra integro-differential equations $x^{\prime }(t)={\int }_0^{t}k(t,s)h(x(s))ds$

26 pages