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

Marek Izydorek, and Krzysztof P. Rybakowski. "On the Conley index in Hilbert spaces in the absence of uniqueness." Fundamenta Mathematicae 171.1 (2002): 31-52. <http://eudml.org/doc/283016>.

@article{MarekIzydorek2002,
abstract = { 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. },
author = {Marek Izydorek, Krzysztof P. Rybakowski},
journal = {Fundamenta Mathematicae},
keywords = {Conley-type index; strongly indefinite variational problem},
language = {eng},
number = {1},
pages = {31-52},
title = {On the Conley index in Hilbert spaces in the absence of uniqueness},
url = {http://eudml.org/doc/283016},
volume = {171},
year = {2002},
}

TY - JOUR
AU - Marek Izydorek
AU - Krzysztof P. Rybakowski
TI - On the Conley index in Hilbert spaces in the absence of uniqueness
JO - Fundamenta Mathematicae
PY - 2002
VL - 171
IS - 1
SP - 31
EP - 52
AB - 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.
LA - eng
KW - Conley-type index; strongly indefinite variational problem
UR - http://eudml.org/doc/283016
ER -