Fixed Point Theory and Applications
Volume 2006 (2006), Article ID 65195, 12 pages
Periodic solutions of dissipative systems revisited
1Department of Mathematics Analysis, Faculty of Science, Palacký University, Tomkova 40, Olomouc-Hejčín 779 00, Czech Republic
2Faculty of Mathematics and Informatics, Nicolaus Copernicus University, Chopina 12/18, Torun 87-100, Poland
Received 23 June 2005; Revised 4 October 2005; Accepted 17 October 2005
Copyright © 2006 Jan Andres and Lech Górniewicz. 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.
We reprove in an extremely simple way the classical theorem that
time periodic dissipative systems imply the existence of harmonic
periodic solutions, in the case of uniqueness. We will also show
that, in the lack of uniqueness, the existence of harmonics is
implied by uniform dissipativity. The localization of starting
points and multiplicity of periodic solutions will be established,
under suitable additional assumptions, as well. The arguments are
based on the application of various asymptotic fixed point
theorems of the Lefschetz and Nielsen type.