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

For autonomous delay differential equations $x'(t)=f(x_t)$ we construct a continuous semiflow of continuously differentiable solution operators $x_0\mapsto x_t$, $t\ge0$, 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 $x'(t)=\int_0^tk(t,s)h(x(s))ds$.