Sharkovskii's theorem, differential inclusions, and beyond

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.