Fixed Point Theory and Applications
Volume 2005 (2005), Issue 2, Pages 137-167

Nielsen number and differential equations

Jan Andres

Department of Mathematical Analysis and Mathematical Applications, Faculty of Science, Palacký University, Tomkova 40, Olomouc-Hejčín 779 00, Czech Republic

Received 19 July 2004; Revised 7 December 2004

Copyright © 2005 Jan Andres. 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.


In reply to a problem of Jean Leray (application of the Nielsen theory to differential equations), two main approaches are presented. The first is via Poincaré's translation operator, while the second one is based on the Hammerstein-type solution operator. The applicability of various Nielsen theories is discussed with respect to several sorts of differential equations and inclusions. Links with the Sharkovskii-like theorems (a finite number of periodic solutions imply infinitely many subharmonics) are indicated, jointly with some further consequences like the nontrivial Rδ-structure of solutions of initial value problems. Some illustrating examples are supplied and open problems are formulated.