Conley index in Hilbert spaces and the Leray-Schauder degree
Keywords
Conley index in Hilbert spaces, Leray-Schauder degree, LS-vector field, strongly indefinite operatorAbstract
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 $\widehat f(x)=x+L^{-1}K(x)$.Downloads
Published
2009-03-01
How to Cite
1.
STYBORSKI, Marcin. Conley index in Hilbert spaces and the Leray-Schauder degree. Topological Methods in Nonlinear Analysis. Online. 1 March 2009. Vol. 33, no. 1, pp. 131 - 148. [Accessed 23 April 2024].
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