Periodic orbits for multivalued maps with continuous margins of intervals

Jiehua Mai, Taixiang Sun

DOI: http://dx.doi.org/10.12775/TMNA.2016.052

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.

Keywords


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

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