### Sharkovskii's theorem, differential inclusions, and beyond

DOI: http://dx.doi.org/10.12775/TMNA.2009.011

#### 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.

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.

#### 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 solutions

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