Consider the ordinary differential equation (1) ẋ = Lx + K(x) 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 → 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 ℒ𝓢-Conley index theory introduced in [9]. This extended ℒ𝓢-Conley index allows applications to strongly indefinite variational problems ∇Φ(x) = 0 where Φ: E → ℝ is merely a C¹-function.
@article{bwmeta1.element.bwnjournal-article-doi-10_4064-fm171-1-2,
author = {Marek Izydorek and Krzysztof P. Rybakowski},
title = {On the Conley index in Hilbert spaces in the absence of uniqueness},
journal = {Fundamenta Mathematicae},
volume = {173},
year = {2002},
pages = {31-52},
zbl = {0994.58006},
language = {en},
url = {http://dml.mathdoc.fr/item/bwmeta1.element.bwnjournal-article-doi-10_4064-fm171-1-2}
}
Marek Izydorek; Krzysztof P. Rybakowski. On the Conley index in Hilbert spaces in the absence of uniqueness. Fundamenta Mathematicae, Tome 173 (2002) pp. 31-52. http://gdmltest.u-ga.fr/item/bwmeta1.element.bwnjournal-article-doi-10_4064-fm171-1-2/