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

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 gi ven. 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 Ni elsen number. Subsequently, in view of T. Matsuoka's results in Invent. Mat h. (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 syst em. Unlike in other papers on this topic, no parameters are involved and no simple alternative approach can be used for the same goal.