Fixed Point Theory and Applications
Volume 2006 (2006), Article ID 36361, 30 pages
doi:10.1155/FPTA/2006/36361

Merging of degree and index theory

Martin Väth1,2

1Institute of Mathematics, University of W\"urzburg, Am Hubland, Würzburg 97074, Germany
2Department of Mathematics and Computer Science (WE1), Free University of Berlin, Arnimallee 2-6, Berlin 14195, Germany

Received 14 January 2006; Revised 19 April 2006; Accepted 24 April 2006

Copyright © 2006 Martin Väth. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

The topological approaches to find solutions of a coincidence equation f1(x)=f2(x) can roughly be divided into degree and index theories. We describe how these methods can be combined. We are led to a concept of an extended degree theory for function triples which turns out to be natural in many respects. In particular, this approach is useful to find solutions of inclusion problems F(x)∈Φ(x). As a side result, we obtain a necessary condition for a compact AR to be a topological group.