Abstract
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.
Citation
Jan Andres. Tomáš Fürst. Karel Pastor. "Sharkovskii's theorem, differential inclusions, and beyond." Topol. Methods Nonlinear Anal. 33 (1) 149 - 168, 2009.
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