Generic nonexpansive Hilbert space mappings
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arXiv
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| Format: | Preprint |
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2024
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| _version_ | 1866908490266574848 |
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| author | Ravasini, Davide Thimm, Daylen K. |
| author_facet | Ravasini, Davide Thimm, Daylen K. |
| contents | We consider a closed convex set $C$ in a separable, infinite-dimensional Hilbert space and endow the set $\mathcal{N}(C)$ of nonexpansive self-mappings on $C$ with the topology of pointwise convergence. We introduce the notion of a somewhat bounded set and establish a strong connection between this property and the existence of fixed points for the generic $f\in\mathcal{N}(C)$, in the sense of Baire categories. Namely, if $C$ is somewhat bounded, the generic nonexpansive mapping on $C$ admits a fixed point, whereas if $C$ is not somewhat bounded, the generic nonexpansive mapping on $C$ does not have any fixed points. This results in a topological 0-1 law: the set of all $f\in\mathcal{N}(C)$ with a fixed point is either meager or residual. We further prove that, generically, there are no fixed points in the interior of $C$ and, under additional geometric assumptions, we show the uniqueness of such fixed points for the generic $f\in\mathcal{N}(C)$ and the convergence of the iterates of $f$ to its fixed point. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2407_03881 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Generic nonexpansive Hilbert space mappings Ravasini, Davide Thimm, Daylen K. Functional Analysis 46C05, 54E52 We consider a closed convex set $C$ in a separable, infinite-dimensional Hilbert space and endow the set $\mathcal{N}(C)$ of nonexpansive self-mappings on $C$ with the topology of pointwise convergence. We introduce the notion of a somewhat bounded set and establish a strong connection between this property and the existence of fixed points for the generic $f\in\mathcal{N}(C)$, in the sense of Baire categories. Namely, if $C$ is somewhat bounded, the generic nonexpansive mapping on $C$ admits a fixed point, whereas if $C$ is not somewhat bounded, the generic nonexpansive mapping on $C$ does not have any fixed points. This results in a topological 0-1 law: the set of all $f\in\mathcal{N}(C)$ with a fixed point is either meager or residual. We further prove that, generically, there are no fixed points in the interior of $C$ and, under additional geometric assumptions, we show the uniqueness of such fixed points for the generic $f\in\mathcal{N}(C)$ and the convergence of the iterates of $f$ to its fixed point. |
| title | Generic nonexpansive Hilbert space mappings |
| topic | Functional Analysis 46C05, 54E52 |
| url | https://arxiv.org/abs/2407.03881 |