Periodic orbits for multivalued maps with continuous margins of intervals

Abstract

Let $I$ be a bounded connected subset of $ \mathbb{R}$ containing more than one point, and $\mathcal L(I)$ be the family of all nonempty connected subsets of $I$. Each map from $I$ to $\mathcal L(I)$ is called a multivalued map. A multivalued map $F\colon I\rightarrow\mathcal L(I)$ is called a multivalued map with continuous margins if both the left endpoint and the right endpoint functions of $F$ are continuous. We show that the well-known Sharkovskiĭ theorem for interval maps also holds for every multivalued map with continuous margins $F\colon I\rightarrow \mathcal L(I)$, that is, if $F$ has an $n$-periodic orbit and $n\succ m$ (in the Sharkovskiĭ ordering), then $F$ also has an $m$-periodic orbit.