Fixed Point Theory and Applications
Volume 2004 (2004), Issue 1, Pages 49-69

Two topological definitions of a Nielsen number for coincidences of noncompact maps

Jan Andres1 and Martin Väth2,3

1Department of Mathematical Analysis, Faculty of Science, Palacký University, Tomkova 40, Olomouc-Hejčín 779 00, Czech Republic
2Department of Mathematics, University of Würzburg, Am Hubland, Würzburg D-97074, Germany
3Fachbereich Mathematik und Informatik (WE1), Freie Universität Berlin, Arnimallee 2-6, Berlin 14195, Germany

Received 25 August 2003

Copyright © 2004 Jan Andres and 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.


The Nielsen number is a homotopic invariant and a lower bound for the number of coincidences of a pair of continuous functions. We give two homotopic (topological) definitions of this number in general situations, based on the approaches of Wecken and Nielsen, respectively, and we discuss why these definitions do not coincide and correspond to two completely different approaches to coincidence theory.