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

Andres, Jan

doi:10.1090/S0002-9939-00-05324-7

A nontrivial example of application of the Nielsen fixed-point theory to differential systems: Problem of Jean Leray
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Proc. Amer. Math. Soc. 128 (2000), 2921-2931 Request permission

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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