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Mathematics 2019, 7(3), 303; https://doi.org/10.3390/math7030303

Extended Fuzzy Metrics and Fixed Point Theorems

1
Instituto de Investigación para la Gestión Integrada de Zonas Costeras, Universitat Politècnica de València, Campus de Gandia, Calle Paranimf 1, 46730 Gandia, Spain
2
Departament de Ciències Matemàtiques i Informàtica, Universitat de les Illes Balears, Carretera de Valldemossa km 7.5, 07122 Palma, Spain
*
Author to whom correspondence should be addressed.
Received: 31 December 2018 / Revised: 18 February 2019 / Accepted: 20 March 2019 / Published: 25 March 2019
(This article belongs to the Special Issue Fuzziness and Mathematical Logic)
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PDF [296 KB, uploaded 25 March 2019]

Abstract

In this paper, we study those fuzzy metrics M on X, in the George and Veeramani’s sense, such that t > 0 M ( x , y , t ) > 0 . The continuous extension M 0 of M to X 2 × 0 , + is called extended fuzzy metric. We prove that M 0 generates a metrizable topology on X, which can be described in a similar way to a classical metric. M 0 can be used for simplifying or improving questions concerning M; in particular, we expose the interest of this kind of fuzzy metrics to obtain generalizations of fixed point theorems given in fuzzy metric spaces. View Full-Text
Keywords: fuzzy metric space; fuzzy contractive mapping; fixed point fuzzy metric space; fuzzy contractive mapping; fixed point
This is an open access article distributed under the Creative Commons Attribution License which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited (CC BY 4.0).
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Gregori, V.; Miñana, J.-J.; Miravet, D. Extended Fuzzy Metrics and Fixed Point Theorems. Mathematics 2019, 7, 303.

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