Special Issue "Abstract Metric Spaces: Usefulness in Statistics and Fixed Point Theory"

A special issue of Mathematics (ISSN 2227-7390). This special issue belongs to the section "Fuzzy Set Theory".

Deadline for manuscript submissions: closed (31 March 2021).

Special Issue Editor

Prof. Dr. Antonio Francisco Roldán López de Hierro
E-Mail Website
Guest Editor
Department of Didactics of Mathematics, University of Granada, Avda. del Hospicio, 18071 Granada, Spain
Interests: fuzzy numbers; fuzzy decision making; fuzzy regression; aggregation functions; fixed point theory
Special Issues and Collections in MDPI journals

Special Issue Information

Dear Colleagues,

Four properties characterize the notion of “metric”: non-negativity, identity of indiscernibles, symmetry, and triangle inequality (which is, maybe, the most important inequality in Mathematics). By combining the power of these four axioms, many possible applications have been created in several fields of study, especially in Mathematics and Physics. After the pioneering works of Fréchet (1906) and Hausdorff (1914), a great variety of modifications of the original notion have been introduced in order to extend and adapt this definition to each respective researcher’s interest area: semimetrics, quasimetrics, pseudometrics, partial metric spaces, Branciari distances, RS-distances, G-metric spaces, etc.

Taking into account convergent and Cauchy sequences in metric spaces, it is well known that the notion of “metric” plays a key role in fixed point theory. However, such a concept is also essential in Statistics, because it can model natural random phenomena better than other algebraic tools. Thus, statistical metric spaces, Menger spaces, fuzzy metric spaces, intuitionistic metric spaces, etc. have been introduced. Furthermore, in regression analysis, errors are computed using some generalized metrics in order to compare observed and theoretical points.

This Special Issue is devoted (but not limited) to the wide range of applications of the notion of “metric” in all fields of study, including topics such as:

  • Fixed point theory;
  • Fuzzy distances;
  • Generalized distances in abstract metrics spaces;
  • Applications of metrics in Statistics;
  • Use of metrics in regression analysis;
  • Convergence analysis;
  • Metrics in Computer Science.

Prof. Dr. Antonio Francisco Roldán López de Hierro
Guest Editor

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Published Papers (18 papers)

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Research

Article
A Perov Version of Fuzzy Metric Spaces and Common Fixed Points for Compatible Mappings
Mathematics 2021, 9(11), 1290; https://doi.org/10.3390/math9111290 - 04 Jun 2021
Viewed by 384
Abstract
In this paper, we define and study the Perov fuzzy metric space and the topology induced by this space. We prove Banach contraction theorems. Moreover, we devised new results for Kramosil and Michálek fuzzy metric spaces. In the process, some results about multidimensional [...] Read more.
In this paper, we define and study the Perov fuzzy metric space and the topology induced by this space. We prove Banach contraction theorems. Moreover, we devised new results for Kramosil and Michálek fuzzy metric spaces. In the process, some results about multidimensional common fixed points as coupled/tripled common fixed point results are derived from our main results. Full article
Article
New Fixed Point Results via a Graph Structure
Mathematics 2021, 9(9), 1013; https://doi.org/10.3390/math9091013 - 29 Apr 2021
Viewed by 295
Abstract
The main aim of this paper is to introduce and study some fixed point results for rational multivalued G-contraction and F-Khan-type multivalued contraction mappings on a metric space with a graph. At the end, we give an illustrative example. Full article
Article
Common α-Fuzzy Fixed Point Results for F-Contractions with Applications
Mathematics 2021, 9(3), 277; https://doi.org/10.3390/math9030277 - 30 Jan 2021
Cited by 1 | Viewed by 371
Abstract
F-contractions have inspired a branch of metric fixed point theory committed to the generalization of the classical Banach contraction principle. The study of these contractions and α-fuzzy mappings in b-metric spaces was attempted timidly and was not successful. In this article, the main objective is to obtain common α-fuzzy fixed point results for F-contractions in b-metric spaces. Some multivalued fixed point results in the literature are derived as consequences of our main results. We also provide a non-trivial example to show the validity of our results. As applications, we investigate the solution for fuzzy initial value problems in the context of a generalized Hukuhara derivative. Our results generalize, improve and complement several developments from the existing literature. Full article
Article
On Some New Contractive Conditions in Complete Metric Spaces
Mathematics 2021, 9(2), 118; https://doi.org/10.3390/math9020118 - 07 Jan 2021
Viewed by 363
Abstract
One of the main goals of this paper is to obtain new contractive conditions using the method of a strictly increasing mapping F:(0,+)(,+). According to the recently obtained results, this was possible (Wardowski’s method) only if two more properties (F2) and (F3) were used instead of the aforementioned strictly increasing (F1). Using only the fact that the function F is strictly increasing, we came to new families of contractive conditions that have not been found in the existing literature so far. Assuming that α(u,v)=1 for every u and v from metric space Ξ, we obtain some contractive conditions that can be found in the research of Rhoades (Trans. Amer. Math. Soc. 1977, 222) and Collaco and Silva (Nonlinear Anal. TMA 1997). Results of the paper significantly improve, complement, unify, generalize and enrich several results known in the current literature. In addition, we give examples with results in line with the ones we obtained. Full article
Article
Fixed Point Theory Using ψ Contractive Mapping in C -Algebra Valued B-Metric Space
Mathematics 2021, 9(1), 92; https://doi.org/10.3390/math9010092 - 04 Jan 2021
Viewed by 715
Abstract
In this paper, fixed point theorems using ψ contractive mapping in C-algebra valued b-metric space are introduced. By stating multiple scenarios that illustrate the application domains, we demonstrate several applications from the obtained results. In particular, we begin with the definition of the positive function and then recall some properties of the function that lay the fundamental basis for the research. We then study some fixed point theorems in the C-algebra valued b-metric space using a positive function. Full article
Article
The Technique of Quadruple Fixed Points for Solving Functional Integral Equations under a Measure of Noncompactness
Mathematics 2020, 8(12), 2130; https://doi.org/10.3390/math8122130 - 28 Nov 2020
Viewed by 587
Abstract
Under the idea of a measure of noncompactness, some fixed point results are proposed and a generalization of Darbo’s fixed point theorem is given in this manuscript. Furthermore, some novel quadruple fixed points results via a measure of noncompactness for a general class [...] Read more.
Under the idea of a measure of noncompactness, some fixed point results are proposed and a generalization of Darbo’s fixed point theorem is given in this manuscript. Furthermore, some novel quadruple fixed points results via a measure of noncompactness for a general class of functions are presented. Ultimately, the solutions to a system of non-linear functional integral equations by the fixed point results obtained are discussed, and non-trivial examples to illustrate the validity of our study are derived. Full article
Article
Some Remarks on Fuzzy sb-Metric Spaces
Mathematics 2020, 8(12), 2123; https://doi.org/10.3390/math8122123 - 27 Nov 2020
Viewed by 415
Abstract
Fuzzy strong b-metrics called here by fuzzy sb-metrics, were introduced recently as a fuzzy version of strong b-metrics. It was shown that open balls in fuzzy sb-metric spaces are open in the induced topology (as different from the case of fuzzy b-metric spaces) and thanks to this fact fuzzy sb-metrics have many useful properties common with fuzzy metric spaces which generally may fail to be in the case of fuzzy b-metric spaces. In the present paper, we go further in the research of fuzzy sb-metric spaces. It is shown that the class of fuzzy sb-metric spaces lies strictly between the classes of fuzzy metric and fuzzy b-metric spaces. We prove that the topology induced by a fuzzy sb-metric is metrizable. A characterization of completeness in terms of diameter zero sets in these structures is given. We investigate products and coproducts in the naturally defined category of fuzzy sb-metric spaces. Full article
Article
On \({\mathcal{F}}\)-Contractions for Weak α-Admissible Mappings in Metric-Like Spaces
Mathematics 2020, 8(9), 1629; https://doi.org/10.3390/math8091629 - 21 Sep 2020
Cited by 5 | Viewed by 562
Abstract
In the paper, we consider some fixed point results of F-contractions for triangular α-admissible and triangular weak α-admissible mappings in metric-like spaces. The results on F-contraction type mappings in the context of metric-like spaces are generalized, improved, unified, and enriched. We prove the main result but using only the property (F1) of the strictly increasing mapping F:0,+,+. Our approach gives a proper generalization of several results given in current literature. Full article
Article
Common Attractive Points of Generalized Hybrid Multi-Valued Mappings and Applications
Mathematics 2020, 8(8), 1307; https://doi.org/10.3390/math8081307 - 06 Aug 2020
Cited by 1 | Viewed by 448
Abstract
In this paper, we first propose the concepts of (ζ,η,λ,π)-generalized hybrid multi-valued mappings, the set of all the common attractive points (CAf,g) and the set of all the common strongly attractive points (CsAf,g), respectively for the multi-valued mappings f and g in a CAT(0) space. Moreover, we give some elementary properties in regard to the sets Af, Ff and CAf,g for the multi-valued mappings f and g in a complete CAT(0) space. Furthermore, we present a weak convergence theorem of common attractive points for two (ζ,η,λ,π)-generalized hybrid multi-valued mappings in the above space by virtue of Banach limits technique and Ishikawa iteration respectively. Finally, we prove strong convergence of a new viscosity approximation method for two (ζ,η,λ,π)-generalized hybrid multi-valued mappings in CAT(0) spaces, which also solves a kind of variational inequality problem. Full article
Article
Some Remarks on Reich and Chatterjea Type Nonexpansive Mappings
Mathematics 2020, 8(8), 1270; https://doi.org/10.3390/math8081270 - 03 Aug 2020
Cited by 1 | Viewed by 445
Abstract
In the paper, we show that some results related to Reich and Chatterjea type nonexpansive mappings are still valid if we relax or remove some hypotheses. Full article
Article
Common Fixed Point Theorems in Intuitionistic Generalized Fuzzy Cone Metric Spaces
Mathematics 2020, 8(8), 1212; https://doi.org/10.3390/math8081212 - 23 Jul 2020
Viewed by 474
Abstract
In the present work, we study many fixed point results in intuitionistic generalized fuzzy cone metric space. Precisely, we prove new common fixed point theorems for three self mappings that do not require any commutativity or continuity but a generalized contractive condition. Our [...] Read more.
In the present work, we study many fixed point results in intuitionistic generalized fuzzy cone metric space. Precisely, we prove new common fixed point theorems for three self mappings that do not require any commutativity or continuity but a generalized contractive condition. Our results are generalizations for many results in the literature. Some examples are given to support these results. Full article
Article
An Application of the Fixed Point Theory to the Study of Monotonic Solutions for Systems of Differential Equations
Mathematics 2020, 8(7), 1183; https://doi.org/10.3390/math8071183 - 18 Jul 2020
Viewed by 503
Abstract
In this paper, we establish some conditions for the existence and uniqueness of the monotonic solutions for nonhomogeneous systems of first-order linear differential equations, by using a result of the fixed points theory for sequentially complete gauge spaces. Full article
Article
On the Topology Induced by C*-Algebra-Valued Fuzzy Metric Spaces
Mathematics 2020, 8(6), 905; https://doi.org/10.3390/math8060905 - 03 Jun 2020
Viewed by 516
Abstract
We define the notion of C * -algebra-valued fuzzy metric spaces and we study the topology induced by these spaces. Full article
Article
On ℋ-Simulation Functions and Fixed Point Results in the Setting of ωt-Distance Mappings with Application on Matrix Equations
Mathematics 2020, 8(5), 837; https://doi.org/10.3390/math8050837 - 21 May 2020
Cited by 1 | Viewed by 701
Abstract
The concepts of b-metric spaces and ω t -distance mappings play a key role in solving various kinds of equations through fixed point theory in mathematics and other science. In this article, we study some fixed point results through these concepts. We introduce a new kind of function namely, H -simulation function which is used in this manuscript together with the notion of ω t -distance mappings to furnish for new contractions. Many fixed point results are proved based on these new contractions as well as some examples are introduced. Moreover, we introduce an application on matrix equations to focus on the importance of our work. Full article
Article
Cone Metric Spaces over Topological Modules and Fixed Point Theorems for Lipschitz Mappings
Mathematics 2020, 8(5), 724; https://doi.org/10.3390/math8050724 - 04 May 2020
Viewed by 709
Abstract
In this paper, we introduce the concept of cone metric space over a topological left module and we establish some coincidence and common fixed point theorems for self-mappings satisfying a condition of Lipschitz type. The main results of this paper provide extensions as [...] Read more.
In this paper, we introduce the concept of cone metric space over a topological left module and we establish some coincidence and common fixed point theorems for self-mappings satisfying a condition of Lipschitz type. The main results of this paper provide extensions as well as substantial generalizations and improvements of several well known results in the recent literature. In addition, the paper contains an example which shows that our main results are applicable on a non-metrizable cone metric space over a topological left module. The article proves that fixed point theorems in the framework of cone metric spaces over a topological left module are more effective and more fertile than standard results presented in cone metric spaces over a Banach algebra. Full article
Article
On Some New Multivalued Results in the Metric Spaces of Perov’s Type
Mathematics 2020, 8(3), 438; https://doi.org/10.3390/math8030438 - 17 Mar 2020
Viewed by 778
Abstract
The purpose of this paper is to present some new fixed point results in the generalized metric spaces of Perov’s sense under a contractive condition of Hardy–Rogers type. The data dependence of the fixed point set, the well-posedness of the fixed point problem [...] Read more.
The purpose of this paper is to present some new fixed point results in the generalized metric spaces of Perov’s sense under a contractive condition of Hardy–Rogers type. The data dependence of the fixed point set, the well-posedness of the fixed point problem and the Ulam–Hyers stability are also studied. Full article
Article
Coupled Fixed Point Theorems Employing CLR-Property on V -Fuzzy Metric Spaces
Mathematics 2020, 8(3), 404; https://doi.org/10.3390/math8030404 - 12 Mar 2020
Cited by 5 | Viewed by 701
Abstract
The introduction of the common limit range property on V -fuzzy metric spaces is the foremost aim of this paper. Furthermore, significant results for coupled maps are proven by employing this property on V -fuzzy metric spaces. More precisely, we introduce the notion of C L R Ω -property for the mappings Θ : M × M M and Ω : M M . We utilize our new notion to present and prove our new fixed point results. Full article
Article
A Result on a Pata-Ćirić Type Contraction at a Point
Mathematics 2020, 8(3), 393; https://doi.org/10.3390/math8030393 - 10 Mar 2020
Viewed by 604
Abstract
In this manuscript, we define a new contraction mapping, Pata-Ćirić type contraction at a point, that merges distinct contractions defined by Seghal, Pata and Ćirić. We proved that in a complete space, each Pata-Ćirić type contraction at a point possesses a fixed point. [...] Read more.
In this manuscript, we define a new contraction mapping, Pata-Ćirić type contraction at a point, that merges distinct contractions defined by Seghal, Pata and Ćirić. We proved that in a complete space, each Pata-Ćirić type contraction at a point possesses a fixed point. We express an example to illustrate the observed result. Full article
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