Some extensions of the Brouwer fixed point theorem
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arXiv
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| Autori principali: | , , , |
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| Natura: | Preprint |
| Pubblicazione: |
2024
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| author | Mai, Jiehua Shi, Enhui Yan, Kesong Zeng, Fanping |
| author_facet | Mai, Jiehua Shi, Enhui Yan, Kesong Zeng, Fanping |
| contents | We study the existence of fixed points for continuous maps $f$ from an $n$-ball $X$ in $\mathbb R^n$ to $\mathbb R^n$ with $n\geq 1$. We show that $f$ has a fixed point if, for some absolute retract $Y\subset\partial X$, $f(Y)\subset X$ and $\partial X-Y$ is an $(f, X)$-blockading set. For $n\geq 2$, let $D$ be an $n$-ball in $X$ and $Y$ be an $(n-1)$-ball in $\partial X$. Relying on the result just mentioned, we show the existence of a fixed point of $f$, if $D$ and $Y$ are well placed and behave well under $f$, and ${\rm deg}(f_D)=-{\rm deg}(f_{\partial Y})$, where $f_D=f|D: D \rightarrow \mathbb{R}^n$ and $f_{\partial Y}=f|\partial Y: \partial Y \rightarrow \partial Y$. The degree ${\rm deg}(f_D)$ of $f_D$ is explicitly defined and some elementary properties of which are investigated. These results extend the Brouwer fixed point theorem. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2404_05248 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Some extensions of the Brouwer fixed point theorem Mai, Jiehua Shi, Enhui Yan, Kesong Zeng, Fanping Dynamical Systems Algebraic Topology Geometric Topology 55M20, 55M25, 54H20 We study the existence of fixed points for continuous maps $f$ from an $n$-ball $X$ in $\mathbb R^n$ to $\mathbb R^n$ with $n\geq 1$. We show that $f$ has a fixed point if, for some absolute retract $Y\subset\partial X$, $f(Y)\subset X$ and $\partial X-Y$ is an $(f, X)$-blockading set. For $n\geq 2$, let $D$ be an $n$-ball in $X$ and $Y$ be an $(n-1)$-ball in $\partial X$. Relying on the result just mentioned, we show the existence of a fixed point of $f$, if $D$ and $Y$ are well placed and behave well under $f$, and ${\rm deg}(f_D)=-{\rm deg}(f_{\partial Y})$, where $f_D=f|D: D \rightarrow \mathbb{R}^n$ and $f_{\partial Y}=f|\partial Y: \partial Y \rightarrow \partial Y$. The degree ${\rm deg}(f_D)$ of $f_D$ is explicitly defined and some elementary properties of which are investigated. These results extend the Brouwer fixed point theorem. |
| title | Some extensions of the Brouwer fixed point theorem |
| topic | Dynamical Systems Algebraic Topology Geometric Topology 55M20, 55M25, 54H20 |
| url | https://arxiv.org/abs/2404.05248 |