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On Completeness and Fixed Point Theorems in Fuzzy Metric Spaces

Author

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  • Valentín Gregori

    (Instituto de Investigación para la Gestión Integrada de Zonas Costeras, Universitat Politècnica de València, C/Paranimf, 1, 46730 Grao de Gandia, Spain)

  • Juan-José Miñana

    (Departamento de Matemática Aplicada, Universitat Politècnica de València, C/Paranimf, 1, 46730 Grao de Gandia, Spain)

  • Bernardino Roig

    (Instituto de Investigación para la Gestión Integrada de Zonas Costeras, Universitat Politècnica de València, C/Paranimf, 1, 46730 Grao de Gandia, Spain)

  • Almanzor Sapena

    (Instituto de Investigación para la Gestión Integrada de Zonas Costeras, Universitat Politècnica de València, C/Paranimf, 1, 46730 Grao de Gandia, Spain)

Abstract

This paper is devoted to showing the relevance of the notion of completeness used to establish a fixed point theorem in fuzzy metric spaces introduced by Kramosil and Michalek. Specifically, we show that demanding a stronger notion of completeness, called p -completeness, it is possible to relax some extra conditions on the space to obtain a fixed point theorem in this framework. To this end, we focus on a fixed point result, proved by Mihet for complete non-Archimedean fuzzy metric spaces (Theorem 1). So, we define a weaker concept than the non-Archimedean fuzzy metric, called t -strong, and we establish an alternative version of Miheţ’s theorem for p -complete t -strong fuzzy metrics (Theorem 2). In addition, an example of t -strong fuzzy metric spaces that are not non-Archimedean is provided.

Suggested Citation

  • Valentín Gregori & Juan-José Miñana & Bernardino Roig & Almanzor Sapena, 2024. "On Completeness and Fixed Point Theorems in Fuzzy Metric Spaces," Mathematics, MDPI, vol. 12(2), pages 1-7, January.
    • Handle: RePEc:gam:jmathe:v:12:y:2024:i:2:p:287-:d:1320002
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    References listed on IDEAS

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    1. Olga Grigorenko & Alexander Šostak, 2023. "Fuzzy Metrics in Terms of Fuzzy Relations," Mathematics, MDPI, vol. 11(16), pages 1-13, August.
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