On a Multivalued Version of the Sharkovskii Theorem and its Application to Differential Inclusions
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Motivated by the applications to differential equations without uniqueness conditions, we separately prove multivalued versions of the celebrated Sharkovskii and Li–Yorke theorems. These are then applied, via multivalued Poincaré operators, to Carathéodory differential inclusions. Thus, besides another, infinitely many subharmonics of all integer orders can be obtained. Unlike in the single-valued case, for example, period three brings serious obstructions. Three counter-examples, related to these complications, are therefore presented as well. In a multivalued setting, new phenomena are so exhibited.
- Andres, J.: On the multivalued Poincaré operators. Topol. Methods Nonlinear Anal. 10(1) (1997), 171-182.
- Alseda, L., Balibrea, F., Llibre, J. and Misiurewicz, M. (eds): Thirty Years after Sharkovskii's Theorem: New Perspectives, Proc. Conf. Murcia, Spain, 13–18 June, 1994. Reprint of the Internat. J. Bifur. Chaos Appl. Sci. Engrg. 5(5) (1995).
- Bader, R., Gabor, G. and Kryszewski, W.: On the extension of approximations for set-valued maps and the repulsive fixed points, Boll. Un. Mat. Ital. B (7) 10 (1996), 399-416.
- Bielawski, R., Gorniewicz, L. and Plaskacz, S.: Topological approach to differential inclusions on closed subsets of ℝn, Dynam. Report. Expositions Dynam. Systems (N.S.) 1 (1992), 225-250.
- Gó rniewicz, L.: Homological methods in fixed-point theory of multi-valued maps. Diss. Math. 129 (1976), 1-71.
- Hartman, P.: Ordinary Differential Equations, Wiley, New York, 1964.
- Jarník, J. and Kurzweil, J.: Integral of multivalued mappings and its connection with differential relations, Časopis Pěst. Mat. 108 (1983), 8-28.
- Krasnosel'skii, M. A.: The Operator of Translation Along Trajectories of Ordinary Differential Equations, Amer. Math. Soc., Providence, 1968.
- Li, T. and Yorke, J.: Period three implies chaos, Amer. Math. Monthly 82 (1975), 985-992.
- Pliss, V. A.: Nonlocal Problems of the Oscillation Theory, Nauka, Moscow, 1964 (Russian).
- Robinson, C.: Dynamical Systems (Stability, Symbolic Dynamics, and Chaos), CRC Press, Boca Raton, 1995.
- Sharkovskii, A. N.: Coexistence of cycles of a continuous map of a line into itself, Ukrainian Math. J. 16 (1964), 61-71 (Russian).
- Yoshizawa, T.: Stability Theory and the Existence of Periodic and Almost Periodic Solutions, Springer, Berlin, 1975.
- On a Multivalued Version of the Sharkovskii Theorem and its Application to Differential Inclusions
Volume 10, Issue 1 , pp 1-14
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- Sharkovskii theorem
- Li–Yorke theorem
- periodic orbits
- multivalued version
- differential inclusions
- multiplicity results