Poincaré-Hopf type formulas on convex sets of Banach spaces
type: Item
instance of: scholarly article
title: Poincaré-Hopf type formulas on convex sets of Banach spaces
zbMATH Open document ID: 1196.37034
DOI: 10.12775/TMNA.2009.039
published in: Topological Methods in Nonlinear Analysis
publication date: 2010-07-09T00:00:00Z
review text: Let \(X\) be a Banach space and \(C \subset X\) a closed convex subset. It is supposed that \(C\) has empty interior as, for instance, is the case for order intervals in Sobolev spaces. Locally Lipschitz and completely continuous maps \(A: C\to C\) are considered for which the vector field \(-\text{id}+A\) induces a semiflow \(\varphi: D(\varphi)\subset [0,\infty)\times C\to C\) on \(C\). The authors establish Poincaré-Hopf type formulae relating a fixed point index information about \(A\) with homology Conley index information about the semiflow \(\varphi\) on \(C\). When \(A\) is a gradient some results on the critical groups of isolated fixed points of \(A\) in \(C\) are obtained.
Mathematics Subject Classification ID: 37B30
Mathematics Subject Classification ID: 47H10
Mathematics Subject Classification ID: 58E05
zbMATH DE Number: 5734952
zbMATH Keywords: fixed point index on convex sets
zbMATH Keywords: Conley index on convex sets
zbMATH Keywords: Poincaré-Hopf type formula
zbMATH Keywords: critical groups
author: Thomas J. Bartsch
author: E. N. Dancer
reviewed by: Boris V. Loginov
MaRDI profile type: Publication
full work available at URL: tmna.2009.039
OpenAlex ID: W2495768120
description: scientific article; zbMATH DE number 5734952