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

Tom 171 / 2002

Marek Izydorek, Krzysztof P. Rybakowski Fundamenta Mathematicae 171 (2002), 31-52 MSC: 58E05, 34C35. DOI: 10.4064/fm171-1-2


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.


  • Marek IzydorekFaculty of Technical Physics
    and Applied Mathematics
    Technical University Gdańsk
    Narutowicza 11/12
    80-952 Gdańsk, Poland
  • Krzysztof P. RybakowskiFachbereich Mathematik
    Universität Rostock
    Universitätsplatz 1
    18055 Rostock, Germany

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