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20 pages, 324 KB

Open AccessArticle

Coupled Fixed Point Theory over Quantale-Valued Quasi-Metric Spaces (QVQMS) with Applications in Generalized Metric Structures

by Irem Eroğlu

Axioms 2026, 15(1), 45; https://doi.org/10.3390/axioms15010045 - 8 Jan 2026

Viewed by 458

Abstract

In this study, we establish several coupled fixed point results in quantale-valued quasi-metric spaces (QVQMSs), which constitutes a generalization of metric and probabilistic metric spaces. The obtained results will be illustrated with concrete examples. Furthermore, we introduce the concept of

θ_{s}

-completeness [...] Read more.

In this study, we establish several coupled fixed point results in quantale-valued quasi-metric spaces (QVQMSs), which constitutes a generalization of metric and probabilistic metric spaces. The obtained results will be illustrated with concrete examples. Furthermore, we introduce the concept of

θ_{s}

-completeness and, as an application of the main theorems, we derive some results in both quantale-valued partial metric spaces and probabilistic metric spaces. Full article

(This article belongs to the Special Issue Fixed-Point Theory and Its Related Topics, 5th Edition)

15 pages, 328 KB

Open AccessFeature PaperArticle

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 1097

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)

22 pages, 315 KB

Open AccessArticle

Fixed Point Dynamics in a New Type of Contraction in b-Metric Spaces

by María A. Navascués and Ram N. Mohapatra

Symmetry 2024, 16(4), 506; https://doi.org/10.3390/sym16040506 - 22 Apr 2024

Cited by 13 | Viewed by 2841

Abstract

Since the topological properties of a b-metric space (which generalizes the concept of a metric space) seem sometimes counterintuitive due to the fact that the “open” balls may not be open sets, we review some aspects of these spaces concerning compactness, metrizability, continuity [...] Read more.

Since the topological properties of a b-metric space (which generalizes the concept of a metric space) seem sometimes counterintuitive due to the fact that the “open” balls may not be open sets, we review some aspects of these spaces concerning compactness, metrizability, continuity and fixed points. After doing so, we introduce new types of contractivities that extend the concept of Banach contraction. We study some of their properties, giving sufficient conditions for the existence of fixed points and common fixed points. Afterwards, we consider some iterative schemes in quasi-normed spaces for the approximation of these critical points, analyzing their convergence and stability. We apply these concepts to the resolution of a model of integral equation of Urysohn type. In the last part of the paper, we refine some results about partial contractivities in the case where the underlying set is a strong b-metric space, and we establish some relations between mutual weak contractions and quasi-contractions and the new type of contractivity. Full article

(This article belongs to the Special Issue Symmetry in Nonlinear Dynamics and Chaos II)

16 pages, 306 KB

Open AccessFeature PaperArticle

On Protected Quasi-Metrics

by Salvador Romaguera

Axioms 2024, 13(3), 158; https://doi.org/10.3390/axioms13030158 - 28 Feb 2024

Cited by 5 | Viewed by 3117

Abstract

In this paper, we introduce and examine the notion of a protected quasi-metric. In particular, we give some of its properties and present several examples of distinguished topological spaces that admit a compatible protected quasi-metric, such as the Alexandroff spaces, the Sorgenfrey line, [...] Read more.

In this paper, we introduce and examine the notion of a protected quasi-metric. In particular, we give some of its properties and present several examples of distinguished topological spaces that admit a compatible protected quasi-metric, such as the Alexandroff spaces, the Sorgenfrey line, the Michael line, and the Khalimsky line, among others. Our motivation is due, in part, to the fact that a successful improvement of the classical Banach fixed-point theorem obtained by Suzuki does not admit a natural and full quasi-metric extension, as we have noted in a recent article. Thus, and with the help of this new structure, we obtained a fixed-point theorem in the framework of Smyth-complete quasi-metric spaces that generalizes Suzuki’s theorem. Combining right completeness with partial ordering properties, we also obtained a variant of Suzuki’s theorem, which was applied to discuss types of difference equations and recurrence equations. Full article

(This article belongs to the Section Geometry and Topology)

19 pages, 318 KB

Open AccessArticle

Double-Controlled Quasi M-Metric Spaces

by Irshad Ayoob, Ng Zhen Chuan and Nabil Mlaiki

Symmetry 2023, 15(4), 893; https://doi.org/10.3390/sym15040893 - 10 Apr 2023

Cited by 1 | Viewed by 2098

Abstract

One of the well-studied generalizations of a metric space is known as a partial metric space. The partial metric space was further generalized to the so-called M-metric space. In this paper, we introduce the Double-Controlled Quasi M-metric space as a new [...] Read more.

One of the well-studied generalizations of a metric space is known as a partial metric space. The partial metric space was further generalized to the so-called M-metric space. In this paper, we introduce the Double-Controlled Quasi M-metric space as a new generalization of the M-metric space. In our new generalization of the M-metric space, the symmetry condition is not necessarily satisfied and the triangle inequality is controlled by two binary functions. We establish some fixed point results, along with the examples and applications to illustrate our results. Full article

(This article belongs to the Special Issue Symmetry Application in Fixed Point Theory)

21 pages, 362 KB

Open AccessArticle

Some Characterizations of Complete Hausdorff KM-Fuzzy Quasi-Metric Spaces

by Salvador Romaguera

Mathematics 2023, 11(2), 381; https://doi.org/10.3390/math11020381 - 11 Jan 2023

Cited by 2 | Viewed by 1980

Abstract

Gregori and Romaguera introduced, in 2004, the notion of a KM-fuzzy quasi-metric space as a natural asymmetric generalization of the concept of fuzzy metric space in the sense of Kramosil and Michalek. Ever since, various authors have discussed several aspects of such spaces, [...] Read more.

Gregori and Romaguera introduced, in 2004, the notion of a KM-fuzzy quasi-metric space as a natural asymmetric generalization of the concept of fuzzy metric space in the sense of Kramosil and Michalek. Ever since, various authors have discussed several aspects of such spaces, including their topological and (quasi-)metric properties as well as their connections with domain theory and their relationship with other fuzzy structures. In particular, the development of the fixed point theory for these spaces and other related ones, such as fuzzy partial metric spaces, has received remarkable attention in the last 15 years. Continuing this line of research, we here establish general fixed point theorems for left and right complete Hausdorff KM-fuzzy quasi-metric spaces, which are applied to deduce characterizations of these distinguished kinds of fuzzy quasi-metric completeness. Our approach, which mixes conditions of Suzuki-type with contractions of

α - ϕ

-type in the well-known proposal of Samet et al., allows us to extend and improve some recent theorems on complete fuzzy metric spaces. The obtained results are accompanied by illustrative and clarifying examples. Full article

(This article belongs to the Special Issue Aggregation Functions and Indistinguishability Operators: Applications)

8 pages, 250 KB

Open AccessArticle

Ordered Vectorial Quasi and Almost Contractions on Ordered Vector Metric Spaces

by Çetin Cemal Özeken and Cüneyt Çevik

Mathematics 2021, 9(19), 2443; https://doi.org/10.3390/math9192443 - 1 Oct 2021

Cited by 3 | Viewed by 1704

Abstract

In this paper, we define ordered vectorial quasi contractions. We show that ordered quasi contractions are ordered vectorial quasi contractions, but the reverse is not true. We also define ordered vectorial almost contractions and present fixed point theorems for this type of contractions. [...] Read more.

In this paper, we define ordered vectorial quasi contractions. We show that ordered quasi contractions are ordered vectorial quasi contractions, but the reverse is not true. We also define ordered vectorial almost contractions and present fixed point theorems for this type of contractions. Hence, we disclose many results in the literature. With the help of examples, we illustrate the relationship between these two types of contractions and some others in the literature. Full article

16 pages, 558 KB

Open AccessArticle

Some Fixed Point Results on Relational Quasi Partial Metric Spaces and Application to Non-Linear Matrix Equations

by Reena Jain, Hemant Kumar Nashine and Zoran Kadelburg

Symmetry 2021, 13(6), 993; https://doi.org/10.3390/sym13060993 - 2 Jun 2021

Cited by 10 | Viewed by 3144

Abstract

We introduce a

q_{ϱ}

-implicit contractive condition by an implicit relation on relational quasi partial metric spaces and establish new (unique) fixed point results and periodic point results based on it. We justify the results by two suitable examples and compare with [...] Read more.

We introduce a

q_{ϱ}

-implicit contractive condition by an implicit relation on relational quasi partial metric spaces and establish new (unique) fixed point results and periodic point results based on it. We justify the results by two suitable examples and compare with them related work. We discuss sufficient conditions ensuring the existence of a unique positive definite solution of the non-linear matrix equation

U = B + \sum_{i = 1}^{m} A_{i}^{*} G (U) A_{i}

, where

B

is an

n \times n

Hermitian positive definite matrix,

A_{1}

,

A_{2}

, …

A_{m}

are

n \times n

matrices, and

G

is a non-linear self-mapping of the set of all Hermitian matrices which is continuous in the trace norm. Two examples (with randomly generated matrices and complex matrices, respectively) are given, together with convergence and error analysis, as well as average CPU time analysis and visualization of solution in surface plot. Full article

(This article belongs to the Section Mathematics)

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8 pages, 278 KB

Open AccessArticle

Fixed Point of Interpolative Rus–Reich–Ćirić Contraction Mapping on Rectangular Quasi-Partial b-Metric Space

by Pragati Gautam, Luis Manuel Sánchez Ruiz and Swapnil Verma

Symmetry 2021, 13(1), 32; https://doi.org/10.3390/sym13010032 - 28 Dec 2020

Cited by 20 | Viewed by 3677

Abstract

The purpose of this study is to introduce a new type of extended metric space, i.e., the rectangular quasi-partial b-metric space, which means a relaxation of the symmetry requirement of metric spaces, by including a real number s in the definition of the [...] Read more.

The purpose of this study is to introduce a new type of extended metric space, i.e., the rectangular quasi-partial b-metric space, which means a relaxation of the symmetry requirement of metric spaces, by including a real number s in the definition of the rectangular metric space defined by Branciari. Here, we obtain a fixed point theorem for interpolative Rus–Reich–Ćirić contraction mappings in the realm of rectangular quasi-partial b-metric spaces. Furthermore, an example is also illustrated to present the applicability of our result. Full article

(This article belongs to the Special Issue Advanced Calculus in Problems with Symmetry)

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11 pages, 302 KB

Open AccessArticle

Interpolative Reich–Rus–Ćirić and Hardy–Rogers Contraction on Quasi-Partial b-Metric Space and Related Fixed Point Results

by Vishnu Narayan Mishra, Luis Manuel Sánchez Ruiz, Pragati Gautam and Swapnil Verma

Mathematics 2020, 8(9), 1598; https://doi.org/10.3390/math8091598 - 17 Sep 2020

Cited by 33 | Viewed by 3186

Abstract

The aim of this paper was to obtain common fixed point results by using an interpolative contraction condition given by Karapinar in the setting of complete metric space. Here in this paper, we have redefined the Reich–Rus–Ćirić type contraction and Hardy–Rogers type contraction [...] Read more.

The aim of this paper was to obtain common fixed point results by using an interpolative contraction condition given by Karapinar in the setting of complete metric space. Here in this paper, we have redefined the Reich–Rus–Ćirić type contraction and Hardy–Rogers type contraction in the framework of quasi-partial b-metric space and proved the corresponding common fixed point theorem by adopting the notion of interpolation. The results are further validated with the application based on them. Full article

(This article belongs to the Special Issue Variational Inequality)

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10 pages, 247 KB

Open AccessArticle

Best Approximation Results in Various Frameworks

by Taoufik Sabar, Abdelhafid Bassou and Mohamed Aamri

Axioms 2019, 8(2), 67; https://doi.org/10.3390/axioms8020067 - 27 May 2019

Cited by 1 | Viewed by 3091

Abstract

We first provide a best proximity point result for quasi-noncyclic relatively nonexpansive mappings in the setting of dualistic partial metric spaces. Then, those spaces will be endowed with convexity and a result for a cyclic mapping will be obtained. Afterwards, we prove a [...] Read more.

We first provide a best proximity point result for quasi-noncyclic relatively nonexpansive mappings in the setting of dualistic partial metric spaces. Then, those spaces will be endowed with convexity and a result for a cyclic mapping will be obtained. Afterwards, we prove a best proximity point result for tricyclic mappings in the framework of the newly introduced extended partial

S_{b}

-metric spaces. In this way, we obtain extensions of some results in the literature. Full article

(This article belongs to the Special Issue Fixed Point Theory and Related Topics)

17 pages, 789 KB

Open AccessArticle

Common Fixed Point Theorems for Generalized Geraghty (α,ψ,ϕ)-Quasi Contraction Type Mapping in Partially Ordered Metric-Like Spaces

by Haitham Qawaqneh, Mohd Noorani, Wasfi Shatanawi and Habes Alsamir

Axioms 2018, 7(4), 74; https://doi.org/10.3390/axioms7040074 - 25 Oct 2018

Cited by 18 | Viewed by 4523

Abstract

The aim of this paper is to establish the existence of some common fixed point results for generalized Geraghty

(α, ψ, ϕ)

-quasi contraction self-mapping in partially ordered metric-like spaces. We display an example and an application to show [...] Read more.

The aim of this paper is to establish the existence of some common fixed point results for generalized Geraghty

(α, ψ, ϕ)

-quasi contraction self-mapping in partially ordered metric-like spaces. We display an example and an application to show the superiority of our results. The obtained results progress some well-known fixed (common fixed) point results in the literature. Our main results cannot be specifically attained from the corresponding metric space versions. This paper is scientifically novel because we take Geraghty contraction self-mapping in partially ordered metric-like spaces via

α -

admissible mapping. This opens the door to other possible fixed (common fixed) point results for non-self-mapping and in other generalizing metric spaces. Full article

(This article belongs to the Special Issue Mathematical Analysis and Applications II)

16 pages, 264 KB

Open AccessArticle

Best Proximity Point Theorems in Partially Ordered b-Quasi Metric Spaces

by Ali Abkar, Narges Moezzifar and Azizollah Azizi

Mathematics 2016, 4(4), 66; https://doi.org/10.3390/math4040066 - 26 Nov 2016

Viewed by 4459

Abstract

In this paper, we introduce the notion of an ordered rational proximal contraction in partially ordered b-quasi metric spaces. We shall then prove some best proximity point theorems in partially ordered b-quasi metric spaces. Full article

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