Conley index in Hilbert spaces and the Leray-Schauder degree

Marcin Styborski

DOI: http://dx.doi.org/10.12775/TMNA.2009.010

Abstract


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 $\widehat
f(x)=x+L^{-1}K(x)$.

Keywords


Conley index in Hilbert spaces; Leray-Schauder degree; LS-vector field; strongly indefinite operator

Full Text:

FULL TEXT

Refbacks

  • There are currently no refbacks.

Partnerzy platformy czasopism