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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 | References | Similar Articles | Additional Information


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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  • [AGJ1] J. ANDRES, L. G´ORNIEWICZ AND J. JEZIERSKI: A generalized Nielsen number and multiplicity results for differential inclusions. To appear in Topol. Appl. 12 (1999).
  • [AGJ2] J. ANDRES, L. G´ORNIEWICZ AND J. JEZIERSKI: Noncompact version of the multivalued Nielsen theory and its application to differential inclusions. Lecture Notes of the Schauder Center 2: ``Differential Inclusions and Optimal Control'' (Proceedings of the Banach Center Workshop held in Warsaw, September 27-October 3, 1997), (J. Andres, L. Górniewicz and P. Nistri, eds.), 33-50.
  • [BKM] A. YU. BORISOVICH, Z. KUCHARSKI AND W. MARZANTOWICZ: A multiplicity result for a system of real integral equations by use of the Nielsen number. Preprint (1997).
  • [Bo1] Bo Ju Jiang, Lectures on Nielsen fixed point theory, Contemporary Mathematics, vol. 14, American Mathematical Society, Providence, R.I., 1983. MR 685755
  • [Bo2] Bo Ju Jiang, Nielsen theory for periodic orbits and applications to dynamical systems, Nielsen theory and dynamical systems (South Hadley, MA, 1992) Contemp. Math., vol. 152, Amer. Math. Soc., Providence, RI, 1993, pp. 183–202. MR 1243475,
  • [Br1] Robert F. Brown, The Lefschetz fixed point theorem, Scott, Foresman and Co., Glenview, Ill.-London, 1971. MR 0283793
  • [Br2] Robert F. Brown, Multiple solutions to parametrized nonlinear differential systems from Nielsen fixed point theory, Nonlinear analysis, World Sci. Publishing, Singapore, 1987, pp. 89–98. MR 934101
  • [Br3] Robert F. Brown, Topological identification of multiple solutions to parametrized nonlinear equations, Pacific J. Math. 131 (1988), no. 1, 51–69. MR 917865
  • [Br4] R. F. Brown (ed.), Fixed point theory and its applications, Contemporary Mathematics, vol. 72, American Mathematical Society, Providence, RI, 1988. MR 956473
  • [CFM] M. Cecchi, M. Furi, and M. Marini, About the solvability of ordinary differential equations with asymptotic boundary conditions, Boll. Un. Mat. Ital. C (6) 4 (1985), no. 1, 329–345 (English, with Italian summary). MR 805224
  • [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
  • [F1] Michal Fečkan, Nielsen fixed point theory and nonlinear equations, J. Differential Equations 106 (1993), no. 2, 312–331. MR 1251856,
  • [F2] Michal Fečkan, Multiple periodic solutions of small vector fields on differentiable manifolds, J. Differential Equations 113 (1994), no. 1, 189–200. MR 1296167,
  • [G] Andrzej Granas, The Leray-Schauder index and the fixed point theory for arbitrary ANRs, Bull. Soc. Math. France 100 (1972), 209–228. MR 0309102
  • [HJ] Hai Hua Huang and Bo Ju Jiang, Braids and periodic solutions, Topological fixed point theory and applications (Tianjin, 1988) Lecture Notes in Math., vol. 1411, Springer, Berlin, 1989, pp. 107–123. MR 1031787,
  • [J] J. JEZIERSKI: Private communication.
  • [K] M. A. Krasnosel′skiĭ, The operator of translation along the trajectories of differential equations, Translations of Mathematical Monographs, Vol. 19. Translated from the Russian by Scripta Technica, American Mathematical Society, Providence, R.I., 1968. MR 0223640
  • [L] J. LERAY: La theorie des points fixes et ses applications en analyse. In: Proc. International Congres of Math., 1950, Vol. 2, Amer. Math. Soc., 1952. MR 13:859a
  • [M1] Takashi Matsuoka, The number and linking of periodic solutions of periodic systems, Invent. Math. 70 (1982/83), no. 3, 319–340. MR 683687,
  • [M2] Takashi Matsuoka, Waveform in the dynamical study of ordinary differential equations, Japan J. Appl. Math. 1 (1984), no. 2, 417–434. MR 840805,
  • [N] J. NIELSEN: Untersuchungen zur Topologie der geschlossenen zweiseitigen Flächen. Acta Math. 50 (1927), 189-358.
  • [R] L. Rybiński, On Carathéodory type selections, Fund. Math. 125 (1985), no. 3, 187–193. MR 813756
  • [S] K. SCHOLZ: The Nielsen fixed point theory for non-compact spaces. Rocky Mount. J. Math. 4 (1974), 81-87.
  • [W] F. WECKEN: Fixpunktklassen. I, Math. Ann. 117 (1941), 659-671; II, 118 (1942), 216-234; III, 118 (1942), 544-577. MR 3:140b; MR 5:275a; MR 5:275b

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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

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