Search

[EN] George and Veeramani characterized complete fuzzy metric spaces ¿ by means of nested sequences of closed sets of X which have fuzzy diameter zero. According to the concept of p-convergence due to D. Mihet, an appropriate concept of p-Cauchy sequence was given. In this paper we introduce for a concept of p-fuzzy diameter zero, which is according to the concept of p-convergence. Then, we characterize by means of certain nested sequences , which have p-fuzzy diameter zero, those fuzzy metric spaces in which p-Cauchy sequences are convergent (p-convergent), called p-complete spaces (w-p-complete spaces). As a consequence of our results we obtain the well-known characterization of a complete metric space by means of nested sequences of closed sets of (X,d).

[EN] In this paper we present an example of a fuzzy psi-contractive sequence in the sense of D. Mihet, which is not Cauchy in a fuzzy metric space in the sense of George and Veeramani. To overcome this drawback we introduce and study a concept of strictly fuzzy contractive sequence. Then, we also make an appropriate correction to Lemma 3.2 of Gregori and Minana (2016) [5]. (C) 2019 Elsevier B.V. All rights reserved., Valentín Gregori acknowledges the support of Ministry of Economy and Competitiveness of Spain under Grant MTM 2015-64373-P (MINECO/FEDER, UE). Juan José Miñana acknowledges the support of Programa Operatiu FEDER 2014 2020 de les Illes Balears (50%), by project ref. PROCOE/4/2017 (Direccio General d'Innovacio i Recerca, Govern de les Illes Balears), and of project ROBINS. The latter has received research funding from the EU H2020 framework under GA 779776. This publication reflects only the authors views and the European Union is not liable for any use that may be made of the information contained therein.

[EN] Different types of fuzzy uniformities have been introduced in the literature standing out the notions due to Hutton, Hohle and Lowen. The main purpose of this paper is to study several methods to endow a fuzzy metric space (X, M, *), in the sense of George and Veeramani, with a probabilistic uniformity and with a Hutton [0, 1](-quasi)-uniformity. We will show the functorial behavior of these constructions as well as its relation with respect to Lowen's functors and Katsaras's functors, which establish a relationship between the categories of probabilistic uniformities and Hutton [0, 1](-quasi)-uniformities with the category of classical uniformities respectively. Furthermore, we also study the fuzzy topologies associated with these fuzzy uniformities. (C) 2017 Elsevier B.V. All rights reserved., The first named author acknowledges the support of the grants MTM2015-63608-P (MINECO/FEDER, UE) and IT974-16 (Basque Government).The second named author is supported by the grant MTM2015-64373-P (MINECO/FEDER, UE).

In a recent paper Fang (2015) [1] , J.X. Fang generalized a crucial fixed point theorem for probabilistic φ -contractions on complete Menger spaces due to Jachymski (2010) [3] . In this note we show that actually Fang's theorem is an easy consequence of Jachymski's theorem. We also observe that the proof of a fixed point theorem for complete metric spaces deduced by Fang from his main result is not correct and present a new proof of it.

In this paper we give a characterization of the class of completable strong (non-Archimedean) fuzzy metric spaces, in the sense of George and Veeramani.

In this paper we give a fixed point theorem in the context of fuzzy metric spaces in the sense of George and Veeramani. As a consequence of our result we obtain a fixed point theorem due to D. Mihet and generalize a fixed point theorem due to D. Wardowski. Also, we answer in a positive way to a question posed by D. Wardowski, and solve partially an open question on Cauchyness and contractivity. (C) 2015 Elsevier B.V. All rights reserved., Juan Jose Minana acknowledges the support of Conselleria de Educacion, Formacion y Empleo of Generalitat Valenciana, Spain, by Programa Vali+d para investigadores en formacion under Grant ACIF/2012/040.

[EN] We introduce and study the notions of lower Wijsman topology, upper Wijsman topology and Wijsman topology of a fuzzy metric space in the sense of Kramosil and Michalek. In particular, quasi-uniformizability, uniformizability, quasi-metrizability and metrizability of these topologies are discussed. Their relations with other hypertopologies are also analyzed. Corresponding results to the Wijsman topology of a metric space are deduced from our approach with the help of the standard fuzzy metric., J. Gutierrez Garcia acknowledges the support of the Ministry of Economy and Competitiveness of Spain, Grant MTM2012-37894-C02-02. J. Rodriguez-Lopez, S. Romaguera and M. Sanchis also acknowledge the support of the Ministry of Economy and Competitiveness of Spain, Grant MTM2012-37894-C02-01.

In this paper we construct a non-completable fuzzy metric space in the sense of George and Veeramani which allows to answer an open question related to continuity on the real parameter t. In addition, the constructed space is not strong (non-Archimedean)., Juan Jose Minana acknowledges the support of Conselleria de Educacion, Formacion y Empleo (Programa Vali+d para investigadores en formacion) of Generalitat Valenciana, Spain and the support of Universitat Politecnica de Valencia under Grant PAID-06-12 SP20120471.

In this note we answer two recent questions posed by Morillas and Sapena [10] related to standard convergence in fuzzy metric spaces in the sense of George and Veeramani. The obtained results lead us to establish what conditions must satisfy a concept about sequential convergence to be considered compatible with a concept of Cauchyness., Juan Jose Minana acknowledges the support of Conselleria de Educacion, Formacion y Empleo (Programa Vali+d para investigadores en formacion) of Generalitat Valenciana, Spain, ACIF/2012/040, and the support of Universitat Politecnica de Valencia under Grant PAID-06-12 SP20120471.

We introduce a partial order (sic)(M) on the set BX of formal balls of a fuzzy metric space (X, M, Lambda) in the sense of Kramosil and Michalek, and discuss some of its properties. We also characterize when the poset (BX, (sic)(M)) is a continuous domain by means of a new notion of fuzzy metric completeness introduced here. The well-known theorem of Edalat and Heckmann that a metric space is complete if and only if its poset of formal balls is a continuous domain, is deduced from our characterization, Supported by the Ministry of Economy and Competitiveness of Spain, under grant MTM2012-37894-C02-01.

Answering a recent question posed by Gregori et al. [On a class of completable fuzzy metric spaces, Fuzzy Sets and Systems 161 (2010), 2193-2205] we present two examples of non-strong fuzzy metrics (in the sense of George and Veeramani). © 2010 Elsevier B.V. All rights reserved., This research was supported by the Ministry of Science and Innovation of Spain under Grants MTM2009-12872-C02-01 and MTM2009-12872-C02-02. J. Gutierrez Garcia also acknowledges financial support from the University of the Basque Country under Grant GIU07/27.