Differentiability in Fréchet spaces and delay differential equations

Abstract

In infinite-dimensional spaces there are non-equivalent notions of continuous differentiability which can be used to derive the familiar results of calculus up to the Implicit Function Theorem and beyond. For autonomous differential equations with variable delay, not necessarily bounded, the search for a state space in which solutions are unique and differentiable with respect to initial data leads to smoothness hypotheses on the vector functional $f$ in an equation of the general form \begin{equation*} x'(t)=f(x_t)\in\mathbb{R}^n,\quad\text{with}\ x_t(s)=x(t+s)\quad\text{for}\ s\leq 0, \end{equation*} which have implications (a) on the nature of the delay (which is hidden in $f$) and (b) on the type of continuous differentiability which is present. We find the appropriate {\it strong} kind of continuous differentiability and show that there is a continuous semiflow of continuously differentiable solution operators on a Fréchet manifold, with local invariant manifolds at equilibria.

Keywords

• fréchet space
• frechet differentiability
• delay differential equation
• unbounded delay
• semiflow
• invariant manifolds