Sharkovskii's theorem, differential inclusions, and beyond
Keywords
Sharkovskiĭ-type theorems, multivalued maps with monotone margins, Poincarĭé translation operators, coexistence of infinitely many periodic solutions, no exceptions, deterministic and random differential inclusions, random periodic solutionsAbstract
We explain why the Poincaré translation operators along the trajectories of upper-Carathéodory differential inclusions do not satisfy the exceptional cases, described in our earlier counter-examples, for upper semicontinuous maps. Such a discussion was stimulated by a recent paper of F. Obersnel and P. Omari, where they show that, for Carathéodory scalar differential equations, the existence of just one subharmonic solution (e.g of order $2$) implies the existence of subharmonics of all orders. We reprove this result alternatively just via a multivalued Poincaré translation operator approach. We also establish its randomized version on the basis of a universal randomization scheme developed recently by the first author.Downloads
Published
2009-03-01
How to Cite
1.
ANDRES, Jan, FÜRST, Tomáš and PASTOR, Karel. Sharkovskii’s theorem, differential inclusions, and beyond. Topological Methods in Nonlinear Analysis. Online. 1 March 2009. Vol. 33, no. 1, pp. 149 - 168. [Accessed 26 April 2024].
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