Retraction methods and fixed point free maps with null minimal displacements on unit balls
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| Formato: | Preprint |
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2023
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| _version_ | 1866916378131300352 |
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| author | Barroso, C. S. Ferreira, V. |
| author_facet | Barroso, C. S. Ferreira, V. |
| contents | In this paper we consider the class of Lipschitz maps on the unit ball $B_X$ of a Banach space $X$, and the question we deal with is whether for any $λ>1$ there exists a $λ$-Lipschitz fixed-point free mapping $T\colon B_X\to B_X$ with $\mathrm{d}(T,B_X)=0$. We also consider its Hölder version. New related results are obtained. We show that if $X$ has a spreading Schauder basis then such mappings can always be built, answering a question posed by the first author in \cite{Bar}. In the general case, using a recent approach of R. Medina \cite{M} concerning Hölder retractions of $(r_n)$-flat closed convex sets, we show that for any decreasing null sequence $(r_n)\subset \mathbb{R}$ and $α\in (0,1)$, there exists a fixed-point free mapping $T$ on $B_X$ so that $\|T^nx - T^n y\|\leq r_n(\| x - y\|^α+1)$ for all $x, y\in B_X$ and $n\in\mathbb{N}$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2307_12958 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Retraction methods and fixed point free maps with null minimal displacements on unit balls Barroso, C. S. Ferreira, V. Functional Analysis In this paper we consider the class of Lipschitz maps on the unit ball $B_X$ of a Banach space $X$, and the question we deal with is whether for any $λ>1$ there exists a $λ$-Lipschitz fixed-point free mapping $T\colon B_X\to B_X$ with $\mathrm{d}(T,B_X)=0$. We also consider its Hölder version. New related results are obtained. We show that if $X$ has a spreading Schauder basis then such mappings can always be built, answering a question posed by the first author in \cite{Bar}. In the general case, using a recent approach of R. Medina \cite{M} concerning Hölder retractions of $(r_n)$-flat closed convex sets, we show that for any decreasing null sequence $(r_n)\subset \mathbb{R}$ and $α\in (0,1)$, there exists a fixed-point free mapping $T$ on $B_X$ so that $\|T^nx - T^n y\|\leq r_n(\| x - y\|^α+1)$ for all $x, y\in B_X$ and $n\in\mathbb{N}$. |
| title | Retraction methods and fixed point free maps with null minimal displacements on unit balls |
| topic | Functional Analysis |
| url | https://arxiv.org/abs/2307.12958 |