On the Conley index in Hilbert spaces in the absence of uniqueness

Marek Izydorek¹ ; Krzysztof P. Rybakowski²

¹ Faculty of Technical Physics and Applied Mathematics Technical University Gdańsk Narutowicza 11/12 80-952 Gdańsk, Poland
² Fachbereich Mathematik Universität Rostock Universitätsplatz 1 18055 Rostock, Germany

Fundamenta Mathematicae, Tome 171 (2002) no. 1, pp. 31-52

Résumé

Consider the ordinary differential equation $$\dot x=Lx+K(x)\tag 1 $$ on an infinite-dimensional Hilbert space $E$, where $L$ is a bounded linear operator on $E$ which is assumed to be strongly indefinite and $K : E\to E$ is a completely continuous but not necessarily locally Lipschitzian map. Given any isolating neighborhood $N$ relative to equation (1) we define a Conley-type index of $N$. This index is based on Galerkin approximation of equation (1) by finite-dimensional ODEs and extends to the non-Lipschitzian case the ${\cal L}{\cal S}$-Conley index theory introduced in [9]. This extended ${\cal L}{\cal S}$-Conley index allows applications to strongly indefinite variational problems $\nabla {\mit \Phi }(x)=0$ where ${\mit \Phi } : E\to {\mathbb R}$ is merely a $C^1$-function.

Détail

DOI : 10.4064/fm171-1-2

Keywords: consider ordinary differential equation dot tag infinite dimensional hilbert space where bounded linear operator which assumed strongly indefinite completely continuous necessarily locally lipschitzian map given isolating neighborhood relative equation define conley type index index based galerkin approximation equation finite dimensional odes extends non lipschitzian cal cal conley index theory introduced extended cal cal conley index allows applications strongly indefinite variational problems nabla mit phi where mit phi mathbb merely function