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

DOI: http://dx.doi.org/10.12775/TMNA.2016.056

#### 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.

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.

#### Keywords

Delay differential equation; state-dependent delay; unbounded delay; Fréchet space

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