Fr\'echet differentiability in Fr\'echet spaces, and differential equations with unbounded variable delay

TL;DR

This paper develops a framework for Fréchet differentiability in Fréchet spaces and applies it to delay differential equations with unbounded delays, constructing solution semiflows and invariant manifolds.

Contribution

It introduces Fréchet differentiability for maps between Fréchet spaces and constructs solution semiflows for delay differential equations with unbounded delays.

Findings

Constructed a continuous semiflow of solution operators for delay equations

Established local invariant manifolds at stationary points

Applied results to equations with unbounded, locally bounded delay

Abstract

We introduce and discuss Fr\'echet differentiability for maps between Fr\'echet spaces. For 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 submanifolds of the Fr\'echet space $C^1((-\infty,0],\mathbb{R}^n)$, and establish local invariant manifolds at stationary points by means of transversality and embedding properties. The results apply to examples with unbounded but locally bounded delay.