### 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.

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.

#### Keywords

Multivalued map; interval map; periodic orbit; period; Sharkovskiĭ's order

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