## Topological Methods in Nonlinear Analysis

### 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.

#### Article information

Source
Topol. Methods Nonlinear Anal., Volume 48, Number 2 (2016), 453-464.

Dates
First available in Project Euclid: 21 December 2016

https://projecteuclid.org/euclid.tmna/1482289223

Digital Object Identifier
doi:10.12775/TMNA.2016.052

Mathematical Reviews number (MathSciNet)
MR3642767

Zentralblatt MATH identifier
1373.54027

#### Citation

Mai, Jiehua; Sun, Taixiang. Periodic orbits for multivalued maps with continuous margins of intervals. Topol. Methods Nonlinear Anal. 48 (2016), no. 2, 453--464. doi:10.12775/TMNA.2016.052. https://projecteuclid.org/euclid.tmna/1482289223

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