Metrization of probabilistic metric spaces. Applications to fixed point theory and Arzela-Ascoli type theorem

1 Dynamique et contrôl Optimal
SAMM - Statistique, Analyse et Modélisation Multidisciplinaire (SAmos-Marin Mersenne)
Abstract : Schweizer, Sklar and Thorp proved in 1960 that a Menger space $(G,D,T)$ under a continuous $t$-norm $T$, induce a natural topology $\tau$ wich is metrizable. We extend this result to any probabilistic metric space $(G,D,\star)$ provided that the triangle function $\star$ is continuous. We prove in this case, that the topological space $(G,\tau)$ is uniformly homeomorphic to a (deterministic) metric space $(G,\sigma_D)$ for some canonical metric $\sigma_D$ on $G$. As applications, we extend the fixed point theorem of Hicks to probabilistic metric spaces which are not necessarily Menger spaces and we prove a probabilistic Arzela-Ascoli type theorem.
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Submitted on : Sunday, July 7, 2019 - 10:57:58 PM
Last modification on : Wednesday, September 23, 2020 - 4:31:29 AM

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• HAL Id : hal-02112322, version 2
• ARXIV : 1904.12514

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Mohammed Bachir, Bruno Nazaret. Metrization of probabilistic metric spaces. Applications to fixed point theory and Arzela-Ascoli type theorem. 2019. ⟨hal-02112322v2⟩

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