### Poincaré-Hopf type formulas on convex sets of Banach spaces

DOI: http://dx.doi.org/10.12775/TMNA.2009.039

#### Abstract

We consider locally Lipschitz and completely continuous maps

$A\colon C\to C$ defined on a closed convex subset $C\subset X$ of

a Banach space $X$. The main interest lies in the case when $C$

has empty interior. We establish Poincaré-Hopf type formulas

relating fixed point index information about $A$ with homology

Conley index information about the semiflow on $C$ induced by

$-{\rm id}+A$. If $A$ is a gradient we also obtain results on the

critical groups of isolated fixed points of $A$ in $C$.

$A\colon C\to C$ defined on a closed convex subset $C\subset X$ of

a Banach space $X$. The main interest lies in the case when $C$

has empty interior. We establish Poincaré-Hopf type formulas

relating fixed point index information about $A$ with homology

Conley index information about the semiflow on $C$ induced by

$-{\rm id}+A$. If $A$ is a gradient we also obtain results on the

critical groups of isolated fixed points of $A$ in $C$.

#### Keywords

Fixed point index on convex sets; Conley index on convex sets; Poincaré-Hopf formula; critical groups

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