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Periodic segment implies infinitely many periodic solutions
Author(s):
Waclaw
Marzantowicz;
Klaudiusz
Wójcik
Journal:
Proc. Amer. Math. Soc.
135
(2007),
2637-2647.
MSC (2000):
Primary 54H25, 37B35, 37B55
Posted:
March 21, 2007
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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 Poznan, Umultowska 67, 61-614 Poznan, Poland
Klaudiusz
Wójcik
Affiliation:
PWSZ Nowy Sacz, Institute of Pedagogy, Ul. Chruslicka 6, 33-300 Nowy Sacz, Poland and Institute of Mathematics, Jagiellonian University, Reymonta 4, 30-059 Kraków, Poland
DOI:
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
Posted:
March 21, 2007
Additional Notes:
The first author's research was supported by KBN grant 2P03A 03929
Communicated by:
Carmen C. Chicone
Copyright of article:
Copyright
2007,
American Mathematical Society
The copyright for this article reverts to public domain after 28 years from publication.
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