Delay Differential Equations with Differentiable Solution Operators on Open Domains in C((-∞, 0], Rn) and Processes for Volterra Integro-Differential Equations

Abstract

For autonomous delay differential equations ${x'(t)=f(x_t)}$ we construct a continuous semiflow of continuously differentiable solution operators ${x_0 \to x_t}$, ${t \le 0}$, on open subsets of the Fre´chet space ${C((-\infty, 0], R^n)}$. 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 ${x'(t)={\int_0}^t k(t,s) h(x(s)) ds}$.