Quasi-intermediate value theorem and outflanking arc theorem for plane maps
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arXiv
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| Auteurs principaux: | , , , |
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| Format: | Preprint |
| Publié: |
2024
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| _version_ | 1866929256547745792 |
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| author | Mai, Jiehua Shi, Enhui Yan, Kesong Zeng, Fanping |
| author_facet | Mai, Jiehua Shi, Enhui Yan, Kesong Zeng, Fanping |
| contents | For a disk $D$ in the plane $\mathbb R^2$ and a plane map $f$, we give several conditions on the restriction of $f$ to the boundary $\partial D$ of $D$ which imply the existence of a fixed point of $f$ in some specified domain in $D$. These conditions are similar to those appeared in the intermediate value theorem for maps on the real line. As an application of the main results, we establish a fixed point theorem for plane maps having an outflanking arc, which extends the famous theorem due to Brouwer: if $f$ is an orientation-preserving homeomorphism on the plane and has a periodic point, then it has a fixed point. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2402_16076 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Quasi-intermediate value theorem and outflanking arc theorem for plane maps Mai, Jiehua Shi, Enhui Yan, Kesong Zeng, Fanping Dynamical Systems For a disk $D$ in the plane $\mathbb R^2$ and a plane map $f$, we give several conditions on the restriction of $f$ to the boundary $\partial D$ of $D$ which imply the existence of a fixed point of $f$ in some specified domain in $D$. These conditions are similar to those appeared in the intermediate value theorem for maps on the real line. As an application of the main results, we establish a fixed point theorem for plane maps having an outflanking arc, which extends the famous theorem due to Brouwer: if $f$ is an orientation-preserving homeomorphism on the plane and has a periodic point, then it has a fixed point. |
| title | Quasi-intermediate value theorem and outflanking arc theorem for plane maps |
| topic | Dynamical Systems |
| url | https://arxiv.org/abs/2402.16076 |