Period two implies all periods for a class of ODEs: A multivalued map approach

Authors:
Jan Andres, Tomás Fürst and Karel Pastor

Journal:
Proc. Amer. Math. Soc. **135** (2007), 3187-3191

MSC (2000):
Primary 34A60, 34C25, 37E05, 47H04

Published electronically:
February 28, 2007

MathSciNet review:
2322749

Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: We present an elementary proof that, for a multivalued map with nonempty connected values and monotone margins, the existence of a periodic orbit of any order implies the existence of periodic orbits of all orders. This generalizes a very recent result of this type in terms of scalar ordinary differential equations without uniqueness, due to F. Obersnel and P. Omari, obtained by means of lower and upper solutions techniques.

**[AFJ]**J. Andres, J. Fišer, and L. Jüttner,*On a multivalued version of the Sharkovskii theorem and its application to differential inclusions*, Set-Valued Anal.**10**(2002), no. 1, 1–14. MR**1888453**, 10.1023/A:1014488216807**[AG]**Jan Andres and Lech Górniewicz,*Topological fixed point principles for boundary value problems*, Topological Fixed Point Theory and Its Applications, vol. 1, Kluwer Academic Publishers, Dordrecht, 2003. MR**1998968****[AJP]**Jan Andres, Libor Jüttner, and Karel Pastor,*On a multivalued version of the Sharkovskii theorem and its application to differential inclusions. II*, Set-Valued Anal.**13**(2005), no. 1, 47–68. MR**2128697**, 10.1007/s11228-004-8200-z**[AP]**Jan Andres and Karel Pastor,*A version of Sharkovskii’s theorem for differential equations*, Proc. Amer. Math. Soc.**133**(2005), no. 2, 449–453. MR**2093067**, 10.1090/S0002-9939-04-07627-0**[AS]**J. Andres and P. Šnyrychová:*Hyperchaos induced by multivalued maps*, in preparation.**[F]**A. F. Filippov,*Differential equations with discontinuous righthand sides*, Mathematics and its Applications (Soviet Series), vol. 18, Kluwer Academic Publishers Group, Dordrecht, 1988. Translated from the Russian. MR**1028776****[LY]**Tien Yien Li and James A. Yorke,*Period three implies chaos*, Amer. Math. Monthly**82**(1975), no. 10, 985–992. MR**0385028****[OO1]**Franco Obersnel and Pierpaolo Omari,*Old and new results for first order periodic ODEs without uniqueness: a comprehensive study by lower and upper solutions*, Adv. Nonlinear Stud.**4**(2004), no. 3, 323–376. MR**2079818****[OO2]**F. Obersnel and P. Omari:*Period two implies chaos for a class of ODEs*, Proc. Amer. Math. Soc., posted on January 9, 2007, PII: 5002-9937(07)08700-X (to appear in print).**[S]**A. N. Sharkovskiĭ,*Coexistence of cycles of a continuous map of the line into itself*, Proceedings of the Conference “Thirty Years after Sharkovskiĭ’s Theorem: New Perspectives” (Murcia, 1994), 1995, pp. 1263–1273. Translated from the Russian [Ukrain. Mat. Zh. 16 (1964), no. 1, 61–71; MR0159905 (28 #3121)] by J. Tolosa. MR**1361914**, 10.1142/S0218127495000934

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Additional Information

**Jan Andres**

Affiliation:
Department of Mathematical Analysis and Applications of Mathematics, Faculty of Science, Palacký University, Tomkova 40, 779 00 Olomouc-Hejčín, Czech Republic

Email:
andres@inf.upol.cz

**Tomás Fürst**

Affiliation:
Department of Mathematical Analysis and Applications of Mathematics, Faculty of Science, Palacký University, Tomkova 40, 779 00 Olomouc-Hejčín, Czech Republic

Email:
tomas.furst@seznam.cz

**Karel Pastor**

Affiliation:
Department of Mathematical Analysis and Applications of Mathematics, Faculty of Science, Palacký University, Tomkova 40, 779 00 Olomouc-Hejčín, Czech Republic

Email:
pastor@inf.upol.cz

DOI:
https://doi.org/10.1090/S0002-9939-07-08885-5

Keywords:
Periodic orbits,
multivalued maps,
monotone margins,
Sharkovskii's theorem,
ordinary differential equations without uniqueness,
subharmonic solutions.

Received by editor(s):
June 14, 2006

Published electronically:
February 28, 2007

Additional Notes:
This work was supported by the Council of Czech Government (MSM 6198959214).

Communicated by:
Carmen C. Chicone

Article copyright:
© Copyright 2007
American Mathematical Society