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Mathematics
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Topological Study on Fuzzy Metric Spaces and Their Generalizations
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Topological Study on Fuzzy Metric Spaces and Their Generalizations

A special issue of Mathematics (ISSN 2227-7390). This special issue belongs to the section "Fuzzy Sets, Systems and Decision Making".

Deadline for manuscript submissions: 31 December 2024 | Viewed by 7725

Special Issue Editor

Dr. Juan José Miñana

E-Mail Website
Guest Editor
Departamento de Matemática Aplicada, Universitat Politècnica de València, C/Paranimf, Grao de Gandia, 46730 Valencia, Spain
Interests: topology; aggregation operators; fixed point theory; multiagent systems; fuzzy metric space; fuzzy topology
Special Issues, Collections and Topics in MDPI journals

Special Issue Information

Dear Colleagues,

The concept of fuzzy metric space introduced by Kramosil and Michalek, and the slightly modified theory later given by George and Veeramani, constitute an intensive field of research in mathematics. The study of their topological properties has especially garnered interest. Although topological object fuzzy metrics and classical ones are similar, differences between them can be found from a purely metric point of view. Indeed, fixed-point theory in fuzzy metric spaces has demonstrated many differences when comparing these spaces with their classical counterpart. Additionally, both theories of fuzzy metrics include in their definition a parameter which has allowed for the introduction of novel concepts which would not make sense in the classical context.

This Special Issue is devoted to publishing high-quality papers delving into the study of the topological properties of fuzzy metric spaces, as well as their generalizations as fuzzy quasi-metrics, fuzzy partial metrics, modular indistinguishability operators, etc. Moreover, papers addressing the aggregation of the abovementioned fuzzy measurements or providing new methods of constructing such examples will be taken into consideration.

Prof. Dr. Juan José Miñana
Guest Editor

Manuscript Submission Information

Manuscripts should be submitted online at www.mdpi.com by registering and logging in to this website. Once you are registered, click here to go to the submission form. Manuscripts can be submitted until the deadline. All submissions that pass pre-check are peer-reviewed. Accepted papers will be published continuously in the journal (as soon as accepted) and will be listed together on the special issue website. Research articles, review articles as well as short communications are invited. For planned papers, a title and short abstract (about 100 words) can be sent to the Editorial Office for announcement on this website.

Submitted manuscripts should not have been published previously, nor be under consideration for publication elsewhere (except conference proceedings papers). All manuscripts are thoroughly refereed through a single-blind peer-review process. A guide for authors and other relevant information for submission of manuscripts is available on the Instructions for Authors page. Mathematics is an international peer-reviewed open access semimonthly journal published by MDPI.

Please visit the Instructions for Authors page before submitting a manuscript. The Article Processing Charge (APC) for publication in this open access journal is 2600 CHF (Swiss Francs). Submitted papers should be well formatted and use good English. Authors may use MDPI's English editing service prior to publication or during author revisions.

Keywords

  • fuzzy (quasi-)metric space
  • fuzzy partial metric space
  • modular indistinguishability operator
  • aggregation
  • generating fuzzy metrics
  • fixed-point theory

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Published Papers (6 papers)

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Research

14 pages, 342 KiB  
Article
Concerning Fuzzy b-Metric Spaces †
by Salvador Romaguera
Mathematics 2023, 11(22), 4625; https://doi.org/10.3390/math11224625 - 12 Nov 2023
Viewed by 1075
Abstract
In an article published in 2015, Hussain et al. introduced a notion of a fuzzy b-metric space and obtained some fixed point theorems for this kind of space. Shortly thereafter, Nădăban presented a notion of a fuzzy b-metric space that is [...] Read more.
In an article published in 2015, Hussain et al. introduced a notion of a fuzzy b-metric space and obtained some fixed point theorems for this kind of space. Shortly thereafter, Nădăban presented a notion of a fuzzy b-metric space that is slightly different from the one given by Hussain et al., and explored some of its topological properties. Related to Nădăban’s study, Sedghi and Shobe, Saadati, and Šostak independently conducted investigations in articles published in 2012, 2015, and 2018, respectively, about another class of spaces that Sedgi and Shobe called b-fuzzy metric spaces, Saadati, fuzzy metric type spaces, and Šostak, fuzzy k-metric spaces. The main contributions of our paper are the following: First, we propose a notion of fuzzy b-metric space that encompasses and unifies the aforementioned types of spaces. Our approach, which is based on Gabriec’s notion of a fuzzy metric space, allows us to simultaneously cover two interesting classes of spaces, namely, the 01-fuzzy b-metric spaces and the K-stationary fuzzy b-metric spaces. Second, we show that each fuzzy b-metric space, in our sense, admits uniformity with a countable base. From this fact, we derive, among other consequences, that the topology induced by means of its “open” balls is metrizable. Finally, we obtain a characterization of complete fuzzy b-metric spaces with the help of a fixed point result which is also proved here. In support of our approach, several examples, including an application to a type of difference equations, are discussed. Full article
(This article belongs to the Special Issue Topological Study on Fuzzy Metric Spaces and Their Generalizations)
12 pages, 302 KiB  
Article
On Topological and Metric Properties of ⊕-sb-Metric Spaces
by Alexander Šostak, Tarkan Öner and İlyas Can Duman
Mathematics 2023, 11(19), 4090; https://doi.org/10.3390/math11194090 - 27 Sep 2023
Cited by 1 | Viewed by 893
Abstract
In this paper, we study ⊕-sb-metric spaces, which were introduced to generalize the concept of strong b-metric spaces. In particular, we study the properties of the topology induced via an ⊕-sb metric (separation properties, countability axioms, etc.), prove the continuity of the ⊕-sb-metric, [...] Read more.
In this paper, we study ⊕-sb-metric spaces, which were introduced to generalize the concept of strong b-metric spaces. In particular, we study the properties of the topology induced via an ⊕-sb metric (separation properties, countability axioms, etc.), prove the continuity of the ⊕-sb-metric, establish the metrizability of the ⊕-sb-metric spaces of countable weight, discuss the convergence structure of an ⊕-sb-metric space and prove the Baire category type theorem for such spaces. Most of the results obtained here are new already for strong b-metric spaces, i.e., in the case where an arithmetic sum “+” is taken in the role of ⊕. Full article
(This article belongs to the Special Issue Topological Study on Fuzzy Metric Spaces and Their Generalizations)
35 pages, 451 KiB  
Article
On a New Approach for Stability and Controllability Analysis of Functional Equations
by Safoura Rezaei Aderyani, Reza Saadati, Donal O’Regan and Chenkuan Li
Mathematics 2023, 11(16), 3458; https://doi.org/10.3390/math11163458 - 9 Aug 2023
Viewed by 952
Abstract
We consider a new approach to approximate stability analysis for a tri-additive functional inequality and to obtain the optimal approximation for permuting tri-derivations and tri-homomorphisms in unital matrix algebras via the vector-valued alternative fixed-point theorem, which is a popular technique of proving the [...] Read more.
We consider a new approach to approximate stability analysis for a tri-additive functional inequality and to obtain the optimal approximation for permuting tri-derivations and tri-homomorphisms in unital matrix algebras via the vector-valued alternative fixed-point theorem, which is a popular technique of proving the stability of functional equations. We also present a small list of aggregation functions on the classical, well-known special functions to investigate the best approximation error estimates using a different concept of perturbation stability. Full article
(This article belongs to the Special Issue Topological Study on Fuzzy Metric Spaces and Their Generalizations)
11 pages, 293 KiB  
Article
Fuzzy Approximating Metrics, Approximating Parametrized Metrics and Their Relations with Fuzzy Partial Metrics
by Raivis Bēts and Alexander Šostak
Mathematics 2023, 11(15), 3313; https://doi.org/10.3390/math11153313 - 27 Jul 2023
Viewed by 902
Abstract
We generalize the concept of a fuzzy metric by introducing its approximating counterpart in order to make it more appropriate for the study of some problems related to combinatorics on words. We establish close relations between fuzzy approximating metrics in the case of [...] Read more.
We generalize the concept of a fuzzy metric by introducing its approximating counterpart in order to make it more appropriate for the study of some problems related to combinatorics on words. We establish close relations between fuzzy approximating metrics in the case of special t-norms and approximating parametrized metrics, discuss some relations between fuzzy approximating metrics and fuzzy partial metrics, as well as showing some possible applications of approximating parametrized metrics in the problems of combinatorics on words. Full article
(This article belongs to the Special Issue Topological Study on Fuzzy Metric Spaces and Their Generalizations)
11 pages, 293 KiB  
Article
Fuzzy Mittag–Leffler–Hyers–Ulam–Rassias Stability of Stochastic Differential Equations
by Reza Chaharpashlou, Reza Saadati and António M. Lopes
Mathematics 2023, 11(9), 2154; https://doi.org/10.3390/math11092154 - 4 May 2023
Cited by 1 | Viewed by 942
Abstract
Stability is the most relevant property of dynamical systems. The stability of stochastic differential equations is a challenging and still open problem. In this article, using a fuzzy Mittag–Leffler function, we introduce a new fuzzy controller function to stabilize the stochastic differential equation [...] Read more.
Stability is the most relevant property of dynamical systems. The stability of stochastic differential equations is a challenging and still open problem. In this article, using a fuzzy Mittag–Leffler function, we introduce a new fuzzy controller function to stabilize the stochastic differential equation (SDE) ν′(γ,μ)=Fγ,μ,ν(γ,μ). By adopting the fixed point technique, we are able to prove the fuzzy Mittag–Leffler–Hyers–Ulam–Rassias stability of the SDE. Full article
(This article belongs to the Special Issue Topological Study on Fuzzy Metric Spaces and Their Generalizations)
15 pages, 349 KiB  
Article
Fuzzy Partial Metric Spaces and Fixed Point Theorems
by Halis Aygün, Elif Güner, Juan-José Miñana and Oscar Valero
Mathematics 2022, 10(17), 3092; https://doi.org/10.3390/math10173092 - 28 Aug 2022
Cited by 3 | Viewed by 2177
Abstract
Partial metrics constitute a generalization of classical metrics for which self-distance may not be zero. They were introduced by S.G. Matthews in 1994 in order to provide an adequate mathematical framework for the denotational semantics of programming languages. Since then, different works were [...] Read more.
Partial metrics constitute a generalization of classical metrics for which self-distance may not be zero. They were introduced by S.G. Matthews in 1994 in order to provide an adequate mathematical framework for the denotational semantics of programming languages. Since then, different works were devoted to obtaining counterparts of metric fixed-point results in the more general context of partial metrics. Nevertheless, in the literature was shown that many of these generalizations are actually obtained as a corollary of their aforementioned classical counterparts. Recently, two fuzzy versions of partial metrics have been introduced in the literature. Such notions may constitute a future framework to extend already established fuzzy metric fixed point results to the partial metric context. The goal of this paper is to retrieve the conclusion drawn in the aforementioned paper by Haghia et al. to the fuzzy partial metric context. To achieve this goal, we construct a fuzzy metric from a fuzzy partial metric. The topology, Cauchy sequences, and completeness associated with this fuzzy metric are studied, and their relationships with the same notions associated to the fuzzy partial metric are provided. Moreover, this fuzzy metric helps us to show that many fixed point results stated in fuzzy metric spaces can be extended directly to the fuzzy partial metric framework. An outstanding difference between our approach and the classical technique introduced by Haghia et al. is shown. Full article
(This article belongs to the Special Issue Topological Study on Fuzzy Metric Spaces and Their Generalizations)
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