Fixed Point Theory and Applications 
Volume 2006 (2006), Article ID 65195, 12 pages
doi:10.1155/FPTA/2006/65195

Periodic solutions of dissipative systems revisited

Jan Andres1 and Lech Górniewicz2

 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

Abstract

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.