Delay Differential Equations with Differentiable Solution Operators on Open Domains in C((−∞, 0], ℝⁿ) 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 x₀ ↦ x_t, t ≥ 0, on open subsets of the Fréchet space C((−∞, 0], ℝⁿ). 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 \({x}{\prime}\left(t\right)=\underset{0}{\overset{t}{\int }}k\left(t,s\right)h\left(x\left(s\right)\right)ds\) $x\prime \left(t\right)=\underset{0}{\stackrel{t}{\int }}k\left(t,,,s\right)h\left(x,\left(s\right)\right)ds$ .