Randomized Sharkovsky-type results and random subharmonic solutions of differential inclusions

Andres, Jan; Barbarski, Paweł

doi:10.1090/proc/13014

Randomized Sharkovsky-type results and random subharmonic solutions of differential inclusions
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by Jan Andres and Paweł Barbarski PDF

Proc. Amer. Math. Soc. 144 (2016), 1971-1983 Request permission

Abstract:

Two multivalued deterministic versions of the celebrated Sharkovsky cycle coexistence theorem are randomized in terms of very general random periodic orbits. It is also shown that nontrivial subharmonics of scalar random upper-Carathéodory differential inclusions imply the coexistence of random subharmonics of all orders.

References

J. Andres, Randomization of Sharkovskii-type theorems, Proc. Amer. Math. Soc. 136 (2008), no. 4, 1385–1395, Erratum, Proc. Amer. Math. Soc. 136 (2008), no. 10, 3733–3734.
Jan Andres, Tomáš Fürst, and Karel Pastor, Period two implies all periods for a class of ODEs: a multivalued map approach, Proc. Amer. Math. Soc. 135 (2007), no. 10, 3187–3191. MR 2322749, DOI 10.1090/S0002-9939-07-08885-5
Jan Andres, Tomáš Fürst, and Karel Pastor, Full analogy of Sharkovsky’s theorem for lower semicontinuous maps, J. Math. Anal. Appl. 340 (2008), no. 2, 1132–1144. MR 2390917, DOI 10.1016/j.jmaa.2007.09.046
Jan Andres, Tomáš Fürst, and Karel Pastor, Sharkovskii’s theorem, differential inclusions, and beyond, Topol. Methods Nonlinear Anal. 33 (2009), no. 1, 149–168. MR 2512960, DOI 10.12775/TMNA.2009.011
Jan Andres and Lech Górniewicz, Topological fixed point principles for boundary value problems, Topological Fixed Point Theory and Its Applications, vol. 1, Kluwer Academic Publishers, Dordrecht, 2003. MR 1998968, DOI 10.1007/978-94-017-0407-6
Jan Andres and Lech Górniewicz, Random topological degree and random differential inclusions, Topol. Methods Nonlinear Anal. 40 (2012), no. 2, 337–358. MR 3074469, DOI 10.12775/TMNA.2001.038
Jan Andres, Karel Pastor, and Pavla Šnyrychová, A multivalued version of Sharkovskiĭ’s theorem holds with at most two exceptions, J. Fixed Point Theory Appl. 2 (2007), no. 1, 153–170. MR 2336505, DOI 10.1007/s11784-007-0029-2
PawełBarbarski, The Sharkovskiĭ theorem for spaces of measurable functions, J. Math. Anal. Appl. 373 (2011), no. 2, 414–421. MR 2720692, DOI 10.1016/j.jmaa.2010.07.033
Ismat Beg and Mujahid Abbas, Periodic points of random multivalued operators, Nonlinear Stud. 19 (2012), no. 1, 87–92. MR 2932799
Ismat Beg, Mujahid Abbas, and Akbar Azam, Periodic fixed points of random operators, Ann. Math. Inform. 37 (2010), 39–49. MR 2753025
Chunrong Feng and Huaizhong Zhao, Random periodic solutions of SPDEs via integral equations and Wiener-Sobolev compact embedding, J. Funct. Anal. 262 (2012), no. 10, 4377–4422. MR 2900470, DOI 10.1016/j.jfa.2012.02.024
Chunrong Feng, Huaizhong Zhao, and Bo Zhou, Pathwise random periodic solutions of stochastic differential equations, J. Differential Equations 251 (2011), no. 1, 119–149. MR 2793266, DOI 10.1016/j.jde.2011.03.019
Lech Górniewicz, Topological fixed point theory of multivalued mappings, 2nd ed., Topological Fixed Point Theory and Its Applications, vol. 4, Springer, Dordrecht, 2006. MR 2238622
C. J. Himmelberg, Measurable relations, Fund. Math. 87 (1975), 53–72. MR 367142, DOI 10.4064/fm-87-1-53-72
Shouchuan Hu and Nikolas S. Papageorgiou, Handbook of multivalued analysis. Vol. I, Mathematics and its Applications, vol. 419, Kluwer Academic Publishers, Dordrecht, 1997. Theory. MR 1485775, DOI 10.1007/978-1-4615-6359-4
Marc Klünger, Periodicity and Sharkovsky’s theorem for random dynamical systems, Stoch. Dyn. 1 (2001), no. 3, 299–338. MR 1859009, DOI 10.1142/S0219493701000199
S. J. Leese, Multifunctions of Souslin type, Bull. Austral. Math. Soc. 11 (1974), 395–411. MR 417365, DOI 10.1017/S0004972700044038
Franco Obersnel and Pierpaolo Omari, Period two implies chaos for a class of ODEs, Proc. Amer. Math. Soc. 135 (2007), no. 7, 2055–2058. MR 2299480, DOI 10.1090/S0002-9939-07-08700-X
Marina Pireddu, Period two implies chaos for a class of ODEs: a dynamical system approach, Rend. Istit. Mat. Univ. Trieste 41 (2009), 43–54 (2010). MR 2676964
A. N. Sharkovskiĭ, Coexistence of cycles of a continuous map of the line into itself, Proceedings of the Conference “Thirty Years after Sharkovskiĭ’s Theorem: New Perspectives” (Murcia, 1994), 1995, pp. 1263–1273. Translated from the Russian [Ukrain. Mat. Zh. 16 (1964), no. 1, 61–71; MR0159905 (28 #3121)] by J. Tolosa. MR 1361914, DOI 10.1142/S0218127495000934
Stanisław Sȩdziwy, Periodic solutions of scalar differential equations without uniqueness, Boll. Unione Mat. Ital. (9) 2 (2009), no. 2, 445–448. MR 2537280
Huaizhong Zhao and Zuo-Huan Zheng, Random periodic solutions of random dynamical systems, J. Differential Equations 246 (2009), no. 5, 2020–2038. MR 2494697, DOI 10.1016/j.jde.2008.10.011
Hicham Zmarrou and Ale Jan Homburg, Dynamics and bifurcations of random circle diffeomorphisms, Discrete Contin. Dyn. Syst. Ser. B 10 (2008), no. 2-3, 719–731. MR 2425065, DOI 10.3934/dcdsb.2008.10.719