New Advances in Fuzzy Metric Spaces, Soft Metric Spaces, and Other Related Structures, 2nd Edition

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: 14 June 2024 | Viewed by 4992

Special Issue Editors

Instituto Universitario de Matemática Pura y Aplicada-IUMPA, Universitat Politècnica de Valencia, 46022 Valencia, Spain
Interests: spaces from general topology with richer structures (metric spaces, quasi-metric spaces, and uniformities); fuzzy metric spaces; fixed point theory
Special Issues, Collections and Topics in MDPI journals
Institut de Matemàtiques i Aplicacions de Castelló (IMAC), Universitat Jaume I, Av. Vicent Sos Baynat s/n, C.P. 12071 Castelló de la Plana, Spain
Interests: mathematics; fuzzy set
Special Issues, Collections and Topics in MDPI journals

Special Issue Information

Dear Colleagues,

General topology constitutes a fundamental facet of the development of many branches of fuzzy mathematics, as well as other related disciplines that have received a strong research impulse in recent years, for instance soft set theory. In particular, fuzzy metric spaces, in their different versions and variants, as well as several related structures (Menger spaces, fuzzy normed spaces, Hutton quasi-uniformities, etc.) are a key tool in such developments.

The main purpose of this Special Issue is the publication of papers of high quality that gather new advances in the study of (non-elementary) topological properties, as well as completeness, completion, convergence, construction of hyperspaces, etc., in the realm of fuzzy metric spaces, soft metric spaces, and related structures. New and significant contributions to the theory of fixed point for these spaces and others related to them (quasi-metric, b-metric, partial metric spaces, etc.) are welcome. Furthermore, non-elementary applications to topological algebra, functional analysis, ordinary and partial differential equations, computer science, etc., will be also considered.

Prof. Dr. Salvador Romaguera
Prof. Dr. Manuel Sanchis
Guest Editors

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Keywords

  • fuzzy (quasi-)metric spaces
  • fuzzy (quasi-)uniformities
  • fuzzy normed spaces
  • soft metric spaces
  • fixed point theory
  • fuzzy set theory
  • soft set theory
  • applications

Published Papers (9 papers)

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Research

13 pages, 277 KiB  
Article
The Strong Ekeland Variational Principle in Quasi-Pseudometric Spaces
Mathematics 2024, 12(3), 471; https://doi.org/10.3390/math12030471 - 01 Feb 2024
Viewed by 363
Abstract
Roughly speaking, Ekeland’s Variational Principle (EkVP) (J. Math. Anal. Appl. 47 (1974), 324–353) asserts the existence of strict minima of some perturbed versions of lower semicontinuous functions defined on a complete metric space. Later, Pando Georgiev (J. Math. Anal. Appl. 131 (1988), no. [...] Read more.
Roughly speaking, Ekeland’s Variational Principle (EkVP) (J. Math. Anal. Appl. 47 (1974), 324–353) asserts the existence of strict minima of some perturbed versions of lower semicontinuous functions defined on a complete metric space. Later, Pando Georgiev (J. Math. Anal. Appl. 131 (1988), no. 1, 1–21) and Tomonari Suzuki (J. Math. Anal. Appl. 320 (2006), no. 2, 787–794 and Nonlinear Anal. 72 (2010), no. 5, 2204–2209)) proved a Strong Ekeland Variational Principle, meaning the existence of strong minima for such perturbations. Please note that Suzuki also considered the case of functions defined on Banach spaces, emphasizing the key role played by reflexivity. In recent years, an increasing interest was manifested by many researchers to extend EkVP to the asymmetric case, i.e., to quasi-metric spaces (see references). Applications to optimization, behavioral sciences, and others were obtained. The aim of the present paper is to extend the strong Ekeland principle, both Georgiev’s and Suzuki’s versions, to the quasi-pseudometric case. At the end, we ask for the possibility of extending it to asymmetric normed spaces (i.e., the extension of Suzuki’s results). Full article
16 pages, 313 KiB  
Article
Riemann Integral on Fractal Structures
Mathematics 2024, 12(2), 310; https://doi.org/10.3390/math12020310 - 17 Jan 2024
Viewed by 363
Abstract
In this work we start developing a Riemann-type integration theory on spaces which are equipped with a fractal structure. These topological structures have a recursive nature, which allows us to guarantee a good approximation to the true value of a certain integral with [...] Read more.
In this work we start developing a Riemann-type integration theory on spaces which are equipped with a fractal structure. These topological structures have a recursive nature, which allows us to guarantee a good approximation to the true value of a certain integral with respect to some measure defined on the Borel σ-algebra of the space. We give the notion of Darboux sums and lower and upper Riemann integrals of a bounded function when given a measure and a fractal structure. Furthermore, we give the notion of a Riemann-integrable function in this context and prove that each μ-measurable function is Riemann-integrable with respect to μ. Moreover, if μ is the Lebesgue measure, then the Lebesgue integral on a bounded set of Rn meets the Riemann integral with respect to the Lebesgue measure in the context of measures and fractal structures. Finally, we give some examples showing that we can calculate improper integrals and integrals on fractal sets. Full article
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11 pages, 277 KiB  
Article
Presymmetric w-Distances on Metric Spaces
Mathematics 2024, 12(2), 305; https://doi.org/10.3390/math12020305 - 17 Jan 2024
Viewed by 466
Abstract
In an outstanding article published in 2008, Suzuki obtained a nice generalization of the Banach contraction principle, from which a characterization of metric completeness was derived. Although Suzuki’s theorem has been successfully generalized and extended in several directions and contexts, we here show [...] Read more.
In an outstanding article published in 2008, Suzuki obtained a nice generalization of the Banach contraction principle, from which a characterization of metric completeness was derived. Although Suzuki’s theorem has been successfully generalized and extended in several directions and contexts, we here show by means of a simple example that the problem of achieving, in an obvious way, its full extension to the framework of w-distances does not have an emphatic response. Motivated by this fact, we introduce the concept of presymmetric w-distance on metric spaces. We also give some properties and examples of this new structure and show that it provides a reasonable setting to obtain a real and hardly forced w-distance generalization of Suzuki’s theorem. This is realized in our main result, which consists of a fixed point theorem that involves presymmetric w-distances and certain contractions of Suzuki-type. We also discuss the relationship between our main result and the well-known w-distance full generalization of the Banach contraction principle, due to Suzuki and Takahashi. Connected to this approach, we prove another fixed point result that compares with our main result through some examples. Finally, we state a characterization of metric completeness by using our fixed point results. Full article
7 pages, 241 KiB  
Article
On Completeness and Fixed Point Theorems in Fuzzy Metric Spaces
Mathematics 2024, 12(2), 287; https://doi.org/10.3390/math12020287 - 16 Jan 2024
Viewed by 443
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 [...] Read more.
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. Full article
13 pages, 283 KiB  
Article
A New Notion of Fuzzy Function Ideal Convergence
Mathematics 2024, 12(2), 260; https://doi.org/10.3390/math12020260 - 12 Jan 2024
Viewed by 397
Abstract
P.M. Pu and Y.M. Liu extended Moore-Smith’s convergence of nets to fuzzy topology and Y.M. Liu provided analogous results to J. Kelley’s classical characterization theorem of net convergence by introducing the notion of fuzzy convergence classes. In a previous paper, the authors of [...] Read more.
P.M. Pu and Y.M. Liu extended Moore-Smith’s convergence of nets to fuzzy topology and Y.M. Liu provided analogous results to J. Kelley’s classical characterization theorem of net convergence by introducing the notion of fuzzy convergence classes. In a previous paper, the authors of this study provided modified versions of this characterization by using an alternative notion of convergence of fuzzy nets, introduced by B.M.U. Afsan, named fuzzy net ideal convergence. Our main scope here is to generalize and simplify the preceding results. Specifically, we insert the concept of a fuzzy function ideal convergence class, L, on a non-empty set, X, consisting of triads (f,e,I), where f is a function from a non-empty set, D, to the set FP(X) of fuzzy points in X, which we call fuzzy function, eFP(X), and I is a proper ideal on D, and we provide necessary and sufficient conditions to establish the existence of a unique fuzzy topology, δ, on X, such that (f,e,I)L iff fI-converges to e, relative to the fuzzy topology δ. Full article
11 pages, 287 KiB  
Article
Applications to Solving Variational Inequality Problems via MR-Kannan Type Interpolative Contractions
Mathematics 2023, 11(22), 4694; https://doi.org/10.3390/math11224694 - 19 Nov 2023
Viewed by 519
Abstract
The aim of this paper is manifold. We first define the new class of operators called MR-Kannan interpolative type contractions, which includes the Kannan, enriched Kannan, interpolative Kannan type, and enriched interpolative Kannan type operators. Secondly, we prove the existence of a unique [...] Read more.
The aim of this paper is manifold. We first define the new class of operators called MR-Kannan interpolative type contractions, which includes the Kannan, enriched Kannan, interpolative Kannan type, and enriched interpolative Kannan type operators. Secondly, we prove the existence of a unique fixed point for this class of operators. Thirdly, we study Ulam-Hyers stability, well-posedness, and periodic point properties. Finally, an application of the main results to the variational inequality problem is given. Full article
16 pages, 311 KiB  
Article
Existence of Fuzzy Fixed Points and Common Fuzzy Fixed Points for FG-Contractions with Applications
Mathematics 2023, 11(18), 3981; https://doi.org/10.3390/math11183981 - 19 Sep 2023
Viewed by 492
Abstract
This article contains results of the existence of fuzzy fixed points of fuzzy mappings that satisfy certain contraction conditions using the platform of partial b-metric spaces. Some non-trivial examples are provided to authenticate the main results. The constructed results in this work [...] Read more.
This article contains results of the existence of fuzzy fixed points of fuzzy mappings that satisfy certain contraction conditions using the platform of partial b-metric spaces. Some non-trivial examples are provided to authenticate the main results. The constructed results in this work will likely stimulate new research directions in fuzzy fixed-point theory and related hybrid models. Eventually, some fixed-point results on multivalued mappings are established. These theorems provide an excellent application of main theorems on fuzzy mappings. The results of this article are extensions of many already existing results in the literature. Full article
13 pages, 295 KiB  
Article
Fuzzy Metrics in Terms of Fuzzy Relations
Mathematics 2023, 11(16), 3528; https://doi.org/10.3390/math11163528 - 15 Aug 2023
Cited by 1 | Viewed by 632
Abstract
In this paper, we study the concept of fuzzy metrics from the perspective of fuzzy relations. Specifically, we analyze the commonly used definitions of fuzzy metrics. We begin by noting that crisp metrics can be uniquely characterized by linear order relations. Further, we [...] Read more.
In this paper, we study the concept of fuzzy metrics from the perspective of fuzzy relations. Specifically, we analyze the commonly used definitions of fuzzy metrics. We begin by noting that crisp metrics can be uniquely characterized by linear order relations. Further, we explore the criteria that crisp relations must satisfy in order to determine a crisp metric. Subsequently, we extend these conditions to obtain a fuzzy metric and investigate the additional axioms involved. Additionally, we introduce the definition of an extensional fuzzy metric or E-d-metric, which is a fuzzification of the expression d(x,y)=t. Thus, we examine fuzzy metrics from both the linear order and from the equivalence relation perspectives, where one argument is a value d(x,y) and the other is a number within the range [0,+). Full article
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14 pages, 321 KiB  
Article
Ulam–Hyers Stability and Well-Posedness of Fixed Point Problems in C*-Algebra Valued Bipolar b-Metric Spaces
Mathematics 2023, 11(10), 2323; https://doi.org/10.3390/math11102323 - 16 May 2023
Cited by 1 | Viewed by 780
Abstract
Here, we shall introduce the new notion of C*-algebra valued bipolar b-metric spaces as a generalization of usual metric spaces, C*-algebra valued metric space, b-metric spaces. In the above-mentioned spaces, we shall define [...] Read more.
Here, we shall introduce the new notion of C*-algebra valued bipolar b-metric spaces as a generalization of usual metric spaces, C*-algebra valued metric space, b-metric spaces. In the above-mentioned spaces, we shall define (αAψA) contractions and prove some fixed point theorems for these contractions. Some existing results from the literature are also proved by using our main results. As an application Ulam–Hyers stability and well-posedness of fixed point problems are also discussed. Some examples are also given to illustrate our results. Full article
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