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The existence of periodic solutions to nonautonomous differential inclusions

Authors: Jack W. Macki, Paolo Nistri and Pietro Zecca
Journal: Proc. Amer. Math. Soc. 104 (1988), 840-844
MSC: Primary 34A60; Secondary 34A10, 34A25
MathSciNet review: 931741
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Abstract: For an $ m$-dimensional differential inclusion of the form

$\displaystyle \dot x \in A(t)x(t) + F[t,x(t)],$

with $ A$ and $ F$ $ T$-periodic in $ t$, we prove the existence of a nonconstant periodic solution. Our hypotheses require $ m$ to be odd, and require $ F$ to have different growth behavior for $ \left\vert x \right\vert$ small and $ \left\vert x \right\vert$ large (often the case in applications). The idea is to guarantee that the topological degree associated with the system has different values on two distinct neighborhoods of the origin.

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  • [1] Arrigo Cellina and Andrzej Lasota, A new approach to the definition of topological degree for multi-valued mappings, Atti Accad. Naz. Lincei Rend. Cl. Sci. Fis. Mat. Natur. (8) 47 (1969), 434–440 (1970) (English, with Italian summary). MR 0276937
  • [2] M. Furi, P. Nistri, M. P. Pera, and P. L. Zezza, Topological methods for the global controllability of nonlinear systems, J. Optim. Theory Appl. 45 (1985), no. 2, 231–256. MR 778146, 10.1007/BF00939979
  • [3] M. A. Krasnosel′skiĭ and P. P. Zabreĭko, Geometrical methods of nonlinear analysis, Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 263, Springer-Verlag, Berlin, 1984. Translated from the Russian by Christian C. Fenske. MR 736839
  • [4] J. M. Lasry and R. Robert, Analyse nonlineaire multivoque, Cahiers de Math. de la Decision #7611, Equipe de Recherche de Math. de la Decision, Equipe de Recherche Associe de CNRS #249, Ceremade.
  • [5] Noel G. Lloyd, A survey of degree theory: basis and development, IEEE Trans. Circuits and Systems 30 (1983), no. 8, 607–616. MR 715516, 10.1109/TCS.1983.1085396
  • [6] Paolo Nistri, Periodic control problems for a class of nonlinear periodic differential systems, Nonlinear Anal. 7 (1983), no. 1, 79–90. MR 687032, 10.1016/0362-546X(83)90105-0
  • [7] -, Nonlinear boundary value control problems, Proc. 25th IEEE Conf. on Decision and Control, Athens, Dec. 1986, pp. 600-601.

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Keywords: Periodic solution, differential inclusions
Article copyright: © Copyright 1988 American Mathematical Society