Fréchet differentiability in Fréchet spaces, and differential equations with unbounded variable delay

Abstract

We introduce and discuss Fréchet differentiability for maps between Fréchet spaces. For delay differential equations $x^{\prime }(t)=f(x_t)$ we construct a continuous semiflow of continuously differentiable solution operators $x_0\mapsto x_t$, $t\ge 0$, on submanifolds of the Fréchet 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.

45 pages