Semiflows for differential equations with locally bounded delay on solution manifolds in the space $C^1((-∞,0],\mathbb R^n)$ - Walther, Hans-Otto (Topological Methods in Nonlinear Analysis, 2016)
We construct a semiflow of continuously differentiable solutionoperators 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 advantagethat it contains all histories $x_t=x(t+\cdot)$, $t\in\mathbb R$, ofevery possible entire solution of the delay differential equation,in contrast to a Banach space of maps $(-\infty,0]\to\mathbb R^n$ whosenorm would impose growth conditions at $-\infty$. The semiflowlives on the set $X_f=\{\phi\in U:\phi'(0)=f(\phi)\}$ which is asubmanifold of finite codimension in $C^1$. The hypotheses arethat the functional $f$ is continuously differentiable (in theMichal-Bastiani sense) and that the derivatives have a mildextension property. The result applies to autonomous differentialequations with state-dependent delay which may be unboundedbut which is locally bounded. The case of constant bounded delay,distributed or not, is included.