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


Author: Jan Andres
Journal: Proc. Amer. Math. Soc. 128 (2000), 2921-2931
MSC (1991): Primary 34B15, 47H10
Published electronically: March 2, 2000
MathSciNet review: 1664285
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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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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

DOI: https://doi.org/10.1090/S0002-9939-00-05324-7
Keywords: Nielsen number, lower estimate of fixed points, multiplicity results, Carath\'eodory systems, nontrivial application
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
Article copyright: © Copyright 2000 American Mathematical Society