Description

• Title:
  Semiflows for differential equations with locally bounded delay on solution manifolds in the space $C^1((-\infty,0],\mathbb R^n)$
• Creator:
  Walther, Hans-Otto
• Description:
  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.
• Object availability:
  free access
• Rights:
  Prawa autorskie (c) 2016 Topological Methods in Nonlinear Analysis
• Date:
  2016.08.20
• Type:
  info:eu-repo/semantics/article;
  info:eu-repo/semantics/publishedVersion;
  Recenzowany artykuł
• Language:
  English
• Source:
  Topological Methods in Nonlinear Analysis; Vol 48, No 2 (December 2016); 507 - 537;
  1230-3429
• Publisher:
  Nicolaus Copernicus University in Toruń
• Relation: https://apcz.umk.pl/TMNA/article/view/TMNA.2016.056/9573 ;https://apcz.umk.pl/TMNA/article/view/TMNA.2016.056/9574
• Subject:
  Delay differential equation;
  state-dependent delay;
  unbounded delay;
  Fréchet space;
  34K05;
  37L05
• Identifier: https://apcz.umk.pl/TMNA/article/view/TMNA.2016.056
• Data provider:
  Akademicka Platforma Czasopism
• Can I use it?:
  seek permission
• Type:
  other