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| Main Authors: | , , |
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| Format: | Preprint |
| Published: |
2025
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| Subjects: | |
| Online Access: | https://arxiv.org/abs/2507.00859 |
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| _version_ | 1866909670243827712 |
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| author | Barroso, Cleon S. Bedoya, Jeimer V. da Silva, Carlos S. R. |
| author_facet | Barroso, Cleon S. Bedoya, Jeimer V. da Silva, Carlos S. R. |
| contents | We consider $L$-$ω$-nonexpansive maps $T\colon K\to K$ on a convex subset $K$ of a Banach space $X$, i.e., maps in which $ω_T(δ)\leq Lδ+ω(δ)$ with $L\in [0,1]$, $ω$ being a modulus of continuity and $ω_T$ is the minimal modulus of continuity of $T$. Both AFPP and FPP are studied. For moduli $ω$ with $ω'(0)=\infty$, we show that if $X$ contains an isomorphic copy of $\co$ then it fails the FPP for $0$-$ω$-nonexpansive maps with minimal displacement zero. In the affirmative direction, we prove for certain class of moduli $ω$ that $0$-$ω$-nonexpansive maps are constant on certain domains. Also, when $ω'(0)\leq 1-L$ we show that AFPP works and FPP also works under a monotonicity condition on $ω$. Further related results and examples are given. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2507_00859 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | On $L$-$ω$-nonexpansive maps Barroso, Cleon S. Bedoya, Jeimer V. da Silva, Carlos S. R. Functional Analysis We consider $L$-$ω$-nonexpansive maps $T\colon K\to K$ on a convex subset $K$ of a Banach space $X$, i.e., maps in which $ω_T(δ)\leq Lδ+ω(δ)$ with $L\in [0,1]$, $ω$ being a modulus of continuity and $ω_T$ is the minimal modulus of continuity of $T$. Both AFPP and FPP are studied. For moduli $ω$ with $ω'(0)=\infty$, we show that if $X$ contains an isomorphic copy of $\co$ then it fails the FPP for $0$-$ω$-nonexpansive maps with minimal displacement zero. In the affirmative direction, we prove for certain class of moduli $ω$ that $0$-$ω$-nonexpansive maps are constant on certain domains. Also, when $ω'(0)\leq 1-L$ we show that AFPP works and FPP also works under a monotonicity condition on $ω$. Further related results and examples are given. |
| title | On $L$-$ω$-nonexpansive maps |
| topic | Functional Analysis |
| url | https://arxiv.org/abs/2507.00859 |