Description

• Title:
  Conley index in Hilbert spaces and the Leray-Schauder degree
• Creator:
  Styborski, Marcin
• Description:
  Let $H$ be a real infinite dimensional and separable Hilbert space. With an isolated invariant set $\inv(N)$ of a flow $\phi^t$ generated by an $\mathcal L\mathcal S$-vector field $f\:H\supseteq \Omega\to H$, $f(x)=Lx+K(x)$, where $L\:H\to H$ is strongly indefinite linear operator and $K\:H\supseteq \Omega\to H$ is completely continuous, one can associate a homotopy invariant $h_{\mathcal L\mathcal S}(\inv(N),\phi^t)$ called the $\mathcal L\mathcal S$-Conley index. In fact, this is a homotopy type of a finite CW-complex. We define the Betti numbers and hence the Euler characteristic of such index and prove the formula relating these numbers to the Leray-Schauder degree $\deg_{\mathcal L\mathcal S}(\widehat{f},N,0)$, where $\widehat f:H\supseteq \Omega\to H$ is defined as $\widehat f(x)=x+L^{-1}K(x)$.
• Object availability:
  free access
• Rights:
  Prawa autorskie (c) 2016 Topological Methods in Nonlinear Analysis
• Date:
  2009.03.01
• Type:
  info:eu-repo/semantics/article;
  info:eu-repo/semantics/publishedVersion;
  Recenzowany artykuł
• Language:
  English
• Source:
  Topological Methods in Nonlinear Analysis; Vol 33, No 1 (March 2009); 131 - 148;
  1230-3429
• Publisher:
  Nicolaus Copernicus University in Toruń
• Relation: https://apcz.umk.pl/TMNA/article/view/TMNA.2009.010/8649
• Subject:
  Conley index in Hilbert spaces;
  Leray-Schauder degree;
  LS-vector field;
  strongly indefinite operator;
  37B30;
  47H11
• Identifier: https://apcz.umk.pl/TMNA/article/view/TMNA.2009.010
• Data provider:
  Akademicka Platforma Czasopism
• Can I use it?:
  seek permission
• Type:
  other