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

 

A version of Sharkovskii's theorem for differential equations


Authors: Jan Andres and Karel Pastor
Journal: Proc. Amer. Math. Soc. 133 (2005), 449-453
MSC (2000): Primary 34C25, 34A60, 37E05, 47H04
Published electronically: August 30, 2004
MathSciNet review: 2093067
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Abstract: We present a version of the Sharkovskii cycle coexistence theorem for differential equations. Our earlier applicable version is extended here to hold with the exception of at most two orbits. This result, which (because of counter-examples) cannot be improved, is then applied to ordinary differential equations and inclusions. In particular, if a time-periodic differential equation has $n$-periodic solutions with $n \not= 2^m$, for all $m \in {\mathbb N}$, then infinitely many subharmonics coexist.


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Additional Information

Jan Andres
Affiliation: Department of Mathematical Analysis, Faculty of Science, Palacký University, Tomkova 40, 779 00 Olomouc-Hejčín, Czech Republic
Email: andres@risc.upol.cz

Karel Pastor
Affiliation: Department of Mathematical Analysis, Faculty of Science, Palacký University, Tomkova 40, 779 00 Olomouc-Hejčín, Czech Republic
Email: pastor@inf.upol.cz

DOI: http://dx.doi.org/10.1090/S0002-9939-04-07627-0
PII: S 0002-9939(04)07627-0
Keywords: Sharkovskii's theorem, applicable (multivalued) version, $M$-maps, (primary) orbits, translation operators, subharmonics, multiplicity results
Received by editor(s): September 3, 2003
Published electronically: August 30, 2004
Additional Notes: Supported by the Council of Czech Government (J14/98:153100011)
Communicated by: Carmen C. Chicone
Article copyright: © Copyright 2004 American Mathematical Society