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Proceedings of the American Mathematical Society
Proceedings of the American Mathematical Society
ISSN 1088-6826(online) ISSN 0002-9939(print)

 

Periodic segment implies infinitely many periodic solutions


Authors: Waclaw Marzantowicz and Klaudiusz Wójcik
Journal: Proc. Amer. Math. Soc. 135 (2007), 2637-2647
MSC (2000): Primary 54H25, 37B35, 37B55
Published electronically: March 21, 2007
MathSciNet review: 2302587
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Abstract: In this note we show that the existence of a periodic segment for a non-autonomous ODE with periodic coefficients implies the existence of infinitely many periodic solutions inside this segment provided that a sequence of Lefschetz numbers of iterations of an associated map is not constant. In the case when this sequence is bounded we have to impose a geometric condition on the segment to get solutions by use of symbolic dynamics.


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Additional Information

Waclaw Marzantowicz
Affiliation: Faculty of Mathematics and Computer Science, Adam Mickiewicz Universiy of Poznań, Umultowska 67, 61-614 Poznań, Poland

Klaudiusz Wójcik
Affiliation: PWSZ Nowy Sacz, Institute of Pedagogy, Ul. Chruślicka 6, 33-300 Nowy Sacz, Poland and Institute of Mathematics, Jagiellonian University, Reymonta 4, 30-059 Kraków, Poland

DOI: http://dx.doi.org/10.1090/S0002-9939-07-08750-3
PII: S 0002-9939(07)08750-3
Keywords: Periodic solution, periodic segment, Lefschetz number, fixed point index.
Received by editor(s): January 2, 2006
Received by editor(s) in revised form: April 7, 2006
Published electronically: March 21, 2007
Additional Notes: The first author’s research was supported by KBN grant 2P03A 03929
Communicated by: Carmen C. Chicone
Article copyright: © Copyright 2007 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.