On the Conley index in Hilbert spaces in the absence of uniqueness
Tom 171 / 2002
Fundamenta Mathematicae 171 (2002), 31-52
MSC: 58E05, 34C35.
DOI: 10.4064/fm171-1-2
Streszczenie
Consider the ordinary differential equation 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.
