Topological Methods in Nonlinear Analysis

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

Hans-Otto Walther

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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.

Article information

Source
Topol. Methods Nonlinear Anal., Volume 48, Number 2 (2016), 507-537.

Dates
First available in Project Euclid: 21 December 2016

Permanent link to this document
https://projecteuclid.org/euclid.tmna/1482289227

Digital Object Identifier
doi:10.12775/TMNA.2016.056

Mathematical Reviews number (MathSciNet)
MR3642771

Zentralblatt MATH identifier
1371.34105

Citation

Walther, Hans-Otto. Semiflows for differential equations with locally bounded delay on solution manifolds in the space $C^1((-\infty,0],\mathbb R^n)$. Topol. Methods Nonlinear Anal. 48 (2016), no. 2, 507--537. doi:10.12775/TMNA.2016.056. https://projecteuclid.org/euclid.tmna/1482289227


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