Random Fixed Point Theorems for Relaxed Asymptotic Contractions in Random Normed Modules
Fuente:
arXiv
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| Natura: | Preprint |
| Pubblicazione: |
2026
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| _version_ | 1866918485873917952 |
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| author | Shi, Jie |
| author_facet | Shi, Jie |
| contents | We introduce the notion of a random relaxed asymptotic contraction in the setting of random normed modules. The contraction condition employs two quasi-metrics that are built directly from the random operator: a lower quasi-metric which adaptively switches between a four-point minimum and the ordinary one-step distance, and an upper quasi-metric which takes the maximum of four fundamental distances. The bounds are allowed to depend on the iteration index and are required to converge locally uniformly almost surely to a Boyd--Wong function. Using the fibre decomposition method based on \(σ\)-stability and the local property, we show that any such mapping defined on an essentially bounded, \(σ\)-stable and \(L^0\)-closed set admits a unique random fixed point, and all iterates converge in the \((ε,λ)\)-topology. Our result strictly generalizes the random analogue of Kirk's asymptotic contraction theorem and unifies several deterministic and random fixed point theorems under a single flexible framework. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2605_04432 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Random Fixed Point Theorems for Relaxed Asymptotic Contractions in Random Normed Modules Shi, Jie Functional Analysis We introduce the notion of a random relaxed asymptotic contraction in the setting of random normed modules. The contraction condition employs two quasi-metrics that are built directly from the random operator: a lower quasi-metric which adaptively switches between a four-point minimum and the ordinary one-step distance, and an upper quasi-metric which takes the maximum of four fundamental distances. The bounds are allowed to depend on the iteration index and are required to converge locally uniformly almost surely to a Boyd--Wong function. Using the fibre decomposition method based on \(σ\)-stability and the local property, we show that any such mapping defined on an essentially bounded, \(σ\)-stable and \(L^0\)-closed set admits a unique random fixed point, and all iterates converge in the \((ε,λ)\)-topology. Our result strictly generalizes the random analogue of Kirk's asymptotic contraction theorem and unifies several deterministic and random fixed point theorems under a single flexible framework. |
| title | Random Fixed Point Theorems for Relaxed Asymptotic Contractions in Random Normed Modules |
| topic | Functional Analysis |
| url | https://arxiv.org/abs/2605.04432 |