On a generalized Poincaré-Hopf formula in infinite dimensions

Aleksander Ćwiszewski; Wojciech Kryszewski

doi:10.3934/dcds.2011.29.953

Article Contents

2011, Volume 29, Issue 3: 953-978. Doi: 10.3934/dcds.2011.29.953

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On a generalized Poincaré-Hopf formula in infinite dimensions

Aleksander Ćwiszewski^1, and
Wojciech Kryszewski^1,

1.
Faculty of Mathematics and Computer Science, Nicholas Copernicus University, ul. Chopina 12/18, 87-100 Toruń

Received: December 31, 2009

Revised: May 31, 2010

Published: August 2011

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Abstract

We prove a formula relating the fixed point index of rest points of a completely continuous semiflow defined on a (not necessarily locally compact) metric space in the interior of an isolating block $B$ to the Euler characteristic of the pair $(B,B^-)$, where $B^-$ is the exit set. The proof relies on a general concept of an approximate neighborhood extension space and a full fixed point index theory for self-maps of such spaces. As a consequence, a generalized Poincaré-Hopf type formula for the differential equation determined by a perturbation of the generator of a compact $C_0$ semigroup is obtained.

Keywords:

Semiflow,
Poincaré-Hopf formula,
fixed point index,
Conley index,
approximate neighborhood extensors,
Euler characteristic,
perturbations of $C_0$-semigroup generators.

Mathematics Subject Classification: Primary: 37C25, 37B30, 55M15, 55M20; Secondary: 47D03, 47H11.

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