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Novel Approaches in Fuzzy Sets and Metric Spaces
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Novel Approaches in Fuzzy Sets and Metric Spaces

A special issue of Mathematics (ISSN 2227-7390). This special issue belongs to the section "D2: Operations Research and Fuzzy Decision Making".

Deadline for manuscript submissions: closed (30 November 2024) | Viewed by 10925

Special Issue Editors

Prof. Dr. Vishal Gupta

E-Mail Website
Guest Editor
Department of Mathematics, Maharishi Markandeshwar (Deemed to be University), Mullana 133207, Haryana, India
Interests: fuzzy set theory; fuzzy mappings; fixed point theory; topology; differential and integral equations
Dr. Pooja Dhawan

E-Mail Website
Guest Editor
Department of Mathematics, Maharishi Markandeshwar (Deemed to be University), Mullana 133207, Haryana, India
Interests: fuzzy set theory; fuzzy mappings; fixed point theory; topology; differential and integral equations

Special Issue Information

Dear Colleagues,

The Special Issue aims to cumulate novel advancements in the field of fuzzy sets and related metric spaces. The concept of fuzzy set theory was originally introduced by Zadeh (1965) and it has been developed by a huge circle of researchers over the last two decades. This theory has turned out to be one of the most efficient decision aid techniques providing the ability to deal with uncertainty and vagueness. Till today, several papers are published in prominent journals along with various fuzzy set applications in the fields of mathematics, decision making, information technology, automatic control and artificial intelligence and much more because of its flexibility and tolerance in dealing with imprecise data.

The purpose of this Special Issue is to further investigate modern trends of fuzzy metric spaces in new frontiers that promote a comprehensive and state-of-the-art understanding of fuzzy metric theory-based extensions and relevant applications.

Prof. Dr. Vishal Gupta
Dr. Pooja Dhawan
Guest Editors

Manuscript Submission Information

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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 set
  • fuzzy mappings
  • fuzzy metric space
  • metric space
  • contractions

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Published Papers (9 papers)

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Research

15 pages, 328 KiB  
Article
Partial Metrics Viewed as w-Distances: Extending Some Powerful Fixed-Point Theorems
by Salvador Romaguera and Pedro Tirado
Mathematics 2024, 12(24), 3991; https://doi.org/10.3390/math12243991 - 18 Dec 2024
Viewed by 547
Abstract
Involving w-distances and hybrid contractions that combine conditions of the Ćirić type and Samet et al. type, we obtain some general fixed-point results for quasi-metric spaces from which powerful and significant fixed-point theorems on partial metric spaces are deduced as special cases. [...] Read more.
Involving w-distances and hybrid contractions that combine conditions of the Ćirić type and Samet et al. type, we obtain some general fixed-point results for quasi-metric spaces from which powerful and significant fixed-point theorems on partial metric spaces are deduced as special cases. We present examples showing that our results are real generalizations of those corresponding to the partial metric case and we give an application to the study of recursive equations where the usual Baire partial metric on a domain of words is replaced with a suitable w-distance. Our approach is inspired on the nice fact, stated by Matthews, that every partial metric induces a weighted quasi-metric. Then, we define the notion of a strong w-distance and deduce that every partial metric is a symmetric strong w-distance for its induced weighted quasi-metric space. Full article
(This article belongs to the Special Issue Novel Approaches in Fuzzy Sets and Metric Spaces)
33 pages, 371 KiB  
Article
Embedding the Different Families of Fuzzy Sets into Banach Spaces by Using Cauchy Sequences
by Hsien-Chung Wu
Mathematics 2024, 12(23), 3660; https://doi.org/10.3390/math12233660 - 22 Nov 2024
Cited by 1 | Viewed by 593
Abstract
The family F(U) of all fuzzy sets in a normed space cannot be a vector space. This deficiency affects its application to practical problems. The topics of functional analysis and nonlinear analysis in mathematics are based on vector spaces. Since [...] Read more.
The family F(U) of all fuzzy sets in a normed space cannot be a vector space. This deficiency affects its application to practical problems. The topics of functional analysis and nonlinear analysis in mathematics are based on vector spaces. Since these two topics have been well developed such that their tools can be used to solve practical economics and engineering problems, lacking a vector structure for the family F(U) diminishes its applications to these kinds of practical problems when fuzzy uncertainty has been detected in a real environment. Embedding the whole family F(U) into a Banach space is still not possible. However, it is possible to embed some interesting and important subfamilies of F(U) into some suitable Banach spaces. This paper presents the concrete and detailed structures of these kinds of Banach spaces such that their mathematical structures can penetrate the core of practical economics and engineering problems in fuzzy environments. The important issue of uniqueness for these Banach spaces is also addressed via the concept of isometry. Full article
(This article belongs to the Special Issue Novel Approaches in Fuzzy Sets and Metric Spaces)
20 pages, 7113 KiB  
Article
A Novel Outlier-Robust Accuracy Measure for Machine Learning Regression Using a Non-Convex Distance Metric
by Ahmad B. Hassanat, Mohammad Khaled Alqaralleh, Ahmad S. Tarawneh, Khalid Almohammadi, Maha Alamri, Abdulkareem Alzahrani, Ghada A. Altarawneh and Rania Alhalaseh
Mathematics 2024, 12(22), 3623; https://doi.org/10.3390/math12223623 - 20 Nov 2024
Viewed by 1219
Abstract
Regression, a supervised machine learning approach, establishes relationships between independent variables and a continuous dependent variable. It is widely applied in areas like price prediction and time series forecasting. The performance of regression models is typically assessed using error metrics such as the [...] Read more.
Regression, a supervised machine learning approach, establishes relationships between independent variables and a continuous dependent variable. It is widely applied in areas like price prediction and time series forecasting. The performance of regression models is typically assessed using error metrics such as the Mean Squared Error (MSE), Mean Absolute Error (MAE), and Root Mean Squared Error (RMSE). However, these metrics present challenges including sensitivity to outliers (notably MSE and RMSE) and scale dependency, which complicates comparisons across different models. Additionally, traditional metrics sometimes yield values that are difficult to interpret across various problems. Consequently, there is a need for a metric that consistently reflects regression model performance, independent of the problem domain, data scale, and outlier presence. To overcome these shortcomings, this paper introduces a new regression accuracy measure based on the Hassanat distance, a non-convex distance metric. This measure is not only invariant to outliers but also easy to interpret as it provides an accuracy-like value that ranges from 0 to 1 (or 0–100%). We validate the proposed metric against traditional measures across multiple benchmarks, demonstrating its robustness under various model scenarios and data types. Hence, we suggest it as a new standard for assessing regression models’ accuracy. Full article
(This article belongs to the Special Issue Novel Approaches in Fuzzy Sets and Metric Spaces)
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29 pages, 608 KiB  
Article
Novel Fuzzy Ostrowski Integral Inequalities for Convex Fuzzy-Valued Mappings over a Harmonic Convex Set: Extending Real-Valued Intervals Without the Sugeno Integrals
by Mesfer H. Alqahtani, Der-Chyuan Lou, Fahad Sikander, Yaser Saber and Cheng-Chi Lee
Mathematics 2024, 12(22), 3495; https://doi.org/10.3390/math12223495 - 8 Nov 2024
Viewed by 790
Abstract
This study presents new fuzzy adaptations of Ostrowski’s integral inequalities through a novel class of convex fuzzy-valued mappings defined over a harmonic convex set, avoiding the use of the Sugeno integral. These innovative inequalities generalize the recently developed interval forms of real-valued Ostrowski [...] Read more.
This study presents new fuzzy adaptations of Ostrowski’s integral inequalities through a novel class of convex fuzzy-valued mappings defined over a harmonic convex set, avoiding the use of the Sugeno integral. These innovative inequalities generalize the recently developed interval forms of real-valued Ostrowski inequalities. Their formulations incorporate integrability concepts for fuzzy-valued mappings (FVMs), applying the Kaleva integral and a Kulisch–Miranker fuzzy order relation. The fuzzy order relation is constructed via a level-wise approach based on the Kulisch–Miranker order within the fuzzy number space. Additionally, numerical examples illustrate the effectiveness and significance of the proposed theoretical model. Various applications are explored using different means, and some complex cases are derived. Full article
(This article belongs to the Special Issue Novel Approaches in Fuzzy Sets and Metric Spaces)
20 pages, 304 KiB  
Article
Fixed Point Results for Compatible Mappings in Extended Parametric Sb-Metric Spaces
by Sunil Beniwal, Naveen Mani, Rahul Shukla and Amit Sharma
Mathematics 2024, 12(10), 1460; https://doi.org/10.3390/math12101460 - 8 May 2024
Cited by 2 | Viewed by 882
Abstract
This study aims to establish common fixed point theorems for a pair of compatible self-mappings within the framework of extended parametric Sb-metric spaces. To support our assertions, we provide corollaries and examples accompanied with graphical representations. Moreover, we leverage our principal [...] Read more.
This study aims to establish common fixed point theorems for a pair of compatible self-mappings within the framework of extended parametric Sb-metric spaces. To support our assertions, we provide corollaries and examples accompanied with graphical representations. Moreover, we leverage our principal outcome to guarantee the existence of a common solution to a system of integral equations. Full article
(This article belongs to the Special Issue Novel Approaches in Fuzzy Sets and Metric Spaces)
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11 pages, 251 KiB  
Article
Fixed Point Results in Modified Intuitionistic Fuzzy Soft Metric Spaces with Application
by Vishal Gupta, Aanchal Gondhi and Rahul Shukla
Mathematics 2024, 12(8), 1154; https://doi.org/10.3390/math12081154 - 11 Apr 2024
Cited by 2 | Viewed by 1147
Abstract
This paper establishes a new type of space, modified intuitionistic fuzzy soft metric space (MIFSMS). Basic properties and topological structures are defined in the setting of this new notion with valid examples. Moreover, we have given some new results along with suitable examples [...] Read more.
This paper establishes a new type of space, modified intuitionistic fuzzy soft metric space (MIFSMS). Basic properties and topological structures are defined in the setting of this new notion with valid examples. Moreover, we have given some new results along with suitable examples to show their validity. An application for finding the solution of an integral equation is also given by utilizing our newly developed results. Full article
(This article belongs to the Special Issue Novel Approaches in Fuzzy Sets and Metric Spaces)
16 pages, 3274 KiB  
Article
Metric Relations in the Fuzzy Right Triangle
by Ronald Manríquez
Mathematics 2023, 11(19), 4056; https://doi.org/10.3390/math11194056 - 25 Sep 2023
Cited by 2 | Viewed by 1151
Abstract
The study of fuzzy geometry and its different components has grown in recent years, establishing the formal foundations for its development. This paper is devoted to addressing some metric relations in the fuzzy right triangle; in particular, a version of the Pythagorean theorem [...] Read more.
The study of fuzzy geometry and its different components has grown in recent years, establishing the formal foundations for its development. This paper is devoted to addressing some metric relations in the fuzzy right triangle; in particular, a version of the Pythagorean theorem and geometric mean theorem are provided in analytical fuzzy geometry. The main results show, under certain conditions of the fuzzy vertices, a subset relation between the fuzzy distances associated with the fuzzy right triangle, which is very similar to the classical statements of the Pythagorean theorem and the geometric mean in Euclidean geometry. Full article
(This article belongs to the Special Issue Novel Approaches in Fuzzy Sets and Metric Spaces)
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15 pages, 434 KiB  
Article
Intuitionistic Fuzzy Metric-like Spaces and Fixed-Point Results
by Şuara Onbaşıoğlu and Banu Pazar Varol
Mathematics 2023, 11(8), 1902; https://doi.org/10.3390/math11081902 - 17 Apr 2023
Cited by 2 | Viewed by 2489
Abstract
The objective of this paper is to describe the concept of intuitionistic fuzzy metric-like spaces. This space is an extension of metric-like spaces and fuzzy metric spaces, and intuitionistic fuzzy metric spaces. We discuss convergence sequences, contractive mapping and some fixed-point theorems in [...] Read more.
The objective of this paper is to describe the concept of intuitionistic fuzzy metric-like spaces. This space is an extension of metric-like spaces and fuzzy metric spaces, and intuitionistic fuzzy metric spaces. We discuss convergence sequences, contractive mapping and some fixed-point theorems in intuitionistic fuzzy metric-like space. We also give explanations, examples and counterexamples to validate the superiority of these results. Our results provide a substantial extension of several important results from fuzzy metric-like spaces. Full article
(This article belongs to the Special Issue Novel Approaches in Fuzzy Sets and Metric Spaces)
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17 pages, 325 KiB  
Article
An Infinite System of Fractional Sturm–Liouville Operator with Measure of Noncompactness Technique in Banach Space
by Ahmed Salem, Hunida Malaikah and Eid Sayed Kamel
Mathematics 2023, 11(6), 1444; https://doi.org/10.3390/math11061444 - 16 Mar 2023
Cited by 1 | Viewed by 1162
Abstract
In the current contribution, an appropriate quantity connected to the space of all convergent sequences is provided and shown to be a measure of noncompactness in a Banach space. Through the application of the fixed point theorems of Darbo and Meir–Keeler, this amount [...] Read more.
In the current contribution, an appropriate quantity connected to the space of all convergent sequences is provided and shown to be a measure of noncompactness in a Banach space. Through the application of the fixed point theorems of Darbo and Meir–Keeler, this amount is used to discuss whether a solution to an infinite system of fractional Sturm–Liouville operators exists. We offer a numerical example as an application of the key finding in the study. Full article
(This article belongs to the Special Issue Novel Approaches in Fuzzy Sets and Metric Spaces)
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