On Completeness and Fixed Point Theorems in Fuzzy Metric Spaces

Valentín Gregori (), Juan-José Miñana, Bernardino Roig and Almanzor Sapena
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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

Mathematics, 2024, vol. 12, issue 2, 1-7

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.

Keywords: fuzzy metric; cauchy sequence; p -Cauchy sequence; completeness; fixed point (search for similar items in EconPapers)
JEL-codes: C (search for similar items in EconPapers)
Date: 2024
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