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

TL;DR

This paper constructs differentiable solution operators for delay differential equations on open domains in a function space, enabling a unified framework for autonomous and nonautonomous cases, with applications to Volterra integro-differential equations.

Contribution

It introduces a continuous semiflow and process of differentiable solution operators on open subsets of a Fréchet space for delay differential equations, extending the theory to nonautonomous cases and Volterra equations.

Findings

Constructed differentiable solution operators on open domains.

Established a continuous semiflow and process for autonomous and nonautonomous equations.

Applied the framework to 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 $x_0\mapsto x_t$, $t\ge0$, on open subsets of the Fr\'echet 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 $x'(t)=\int_0^tk(t,s)h(x(s))ds$