Period three plays a negative role in a multivalued version of Sharkovskii's theorem
This paper deals with periodic orbits of iterated multivalued maps \({\mathbb{R}}\multimap {\mathbb{R}}\) with nonempty values. For such maps, opposite to the case of the classical Sharkovskii's theorem, a \(3\)-orbit does not necessarily imply the existence of a \(2\)-orbit. However, the authors prove the following proposition: Assume that a multivalued upper semicontinuous mapping \(\varphi:{\mathbb{R}}\to 2^{\mathbb{R}}\setminus \{\emptyset\}\), whose values are either single points or closed intervals, has no primary \(3\)-orbit and admits an \(n\)-orbit for \(n\neq 1\). Then \(\varphi\) admits a \(2\)-orbit.
Sharkowskii's theorem: multivalued version
scientific article; zbMATH DE number 1836633