## A nontrivial example of application of the Nielsen fixed-point theory to differential systems: Problem of Jean Leray

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- by Jan Andres
- Proc. Amer. Math. Soc.
**128**(2000), 2921-2931 - DOI: https://doi.org/10.1090/S0002-9939-00-05324-7
- Published electronically: March 2, 2000
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## Abstract:

In reply to a problem posed by Jean Leray in 1950, a nontrivial example of application of the Nielsen fixed-point theory to differential systems is given. So the existence of two entirely bounded solutions or three periodic (harmonic) solutions of a planar system of ODEs is proved by means of the Nielsen number. Subsequently, in view of T. Matsuoka’s results in Invent. Math. (**70**(1983), 319-340) and Japan J. Appl. Math. (1 (1984), no. 2, 417–434), infinitely many subharmonics can be generically deduced for a smooth system. Unlike in other papers on this topic, no parameters are involved and no simple alternative approach can be used for the same goal.

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## Bibliographic Information

**Jan Andres**- Affiliation: Department of Mathematical Analysis, Faculty of Science, Palacký University, Tomkova 40, 779 00 Olomouc-Hejčín, Czech Republic
- MR Author ID: 222871
- Email: andres@risc.upol.cz
- Received by editor(s): May 4, 1998
- Received by editor(s) in revised form: November 6, 1998
- Published electronically: March 2, 2000
- Communicated by: Dale Alspach
- © Copyright 2000 American Mathematical Society
- Journal: Proc. Amer. Math. Soc.
**128**(2000), 2921-2931 - MSC (1991): Primary 34B15, 47H10
- DOI: https://doi.org/10.1090/S0002-9939-00-05324-7
- MathSciNet review: 1664285