Semiflows for differential equations with locally bounded delay on solution manifolds in the space $C^1((-\infty,0],\mathbb R^n)$

Abstract

We construct a semiflow of continuously differentiable solution operators for delay differential equations $x'(t)=f(x_t)$ with $f$ defined on an open subset of the Fréchet space $C^1=C^1((-\infty,0],\mathbb{R}^n)$. This space has the advantage that it contains all histories $x_t=x(t+\cdot)$, $t\in\mathbb R$, of every possible entire solution of the delay differential equation, in contrast to a Banach space of maps $(-\infty,0]\to\mathbb R^n$ whose norm would impose growth conditions at $-\infty$. The semiflow lives on the set $X_f=\{\phi\in U:\phi'(0)=f(\phi)\}$ which is a submanifold of finite codimension in $C^1$. The hypotheses are that the functional $f$ is continuously differentiable (in the Michal-Bastiani sense) and that the derivatives have a mild extension property. The result applies to autonomous differential equations with state-dependent delay which may be unbounded but which is locally bounded. The case of constant bounded delay, distributed or not, is included.