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Symmetry
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Elementary Fixed Point Theory and Common Fixed Points
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Elementary Fixed Point Theory and Common Fixed Points

A special issue of Symmetry (ISSN 2073-8994). This special issue belongs to the section "Mathematics".

Deadline for manuscript submissions: closed (31 December 2022) | Viewed by 23946

Special Issue Editors

Dr. Salvatore Sessa

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Guest Editor
Dipartimento di Architettura, Università degli Studi di Napoli Federico II, Via Toledo 402, 80134 Napoli, Italy
Interests: fuzzy relation equations; fixed point theory; geographical information systems
Special Issues, Collections and Topics in MDPI journals
Prof. Dr. Mohammad Imdad

E-Mail Website
Guest Editor
Department of Mathematics, Aligarh Muslim University, Aligarh, India
Interests: fixed; point; theory
Special Issues, Collections and Topics in MDPI journals
Dr. Waleed Mohammad Alfaqih

E-Mail Website
Guest Editor
Department of Mathematics, Hajjah University, Hajjah, Yemen
Interests: fixed; point; theory
Special Issues, Collections and Topics in MDPI journals

Special Issue Information

Dear Colleagues,

 

In this Special Issue, we solicit contributions dealing with fixed-point theorems from a general point of view regarding original concepts or ideas into recent advances of this theory, avoiding any trivial extension of results already consolidated, possibly accompanied by concrete examples. We recommend to the authors to consider in the theorems symmetric inequalities or symmetric conditions with respect to the metric (or hyper-metric if multifunctions are involved) used, i.e., anyone can obtain a dual result by considering dual definitions based simply on the change of variables, inequality sign, etc.


For example, this happens mainly in common fixed-point theorems where the use of two (multi-) functions must be interchangeable, i.e., a concept depending on one of two functions must hold with respect to the other. Very often, looking at some of the papers published, if f and g are self-maps of a metric space, some definitions seem only dependent on f and do not hold if g should assume the same role as f. Our aim is to emphasize this aspect mainly in this journal whose title obliges us to keep in mind the above recommendation. Following this line of thought, common fixed-point theorems (which motivates the second part of the title of this Special Issue) are welcome in the context of general environments, but this aspect is left to the discretion of the authors.


Our aim is for this Special Issue to collect meaningful papers which give a jolt to this theory.

 

Prof. Dr. Salvatore Sessa

Prof. Dr. Mohammad Imdad

Dr. Waleed Mohammad Alfaqih

Guest Editors

Prof. Dr. Salvatore Sessa
Prof. Dr. Mohammad Imdad
Dr. Waleed Mohammad Alfaqih
Guest Editors

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. Symmetry is an international peer-reviewed open access monthly 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 2400 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.

Published Papers (20 papers)

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Research

15 pages, 304 KiB  
Article
On General Class of Z-Contractions with Applications to Spring Mass Problem
by Monairah Alansari and Mohammed Shehu Shagari
Symmetry 2023, 15(2), 302; https://doi.org/10.3390/sym15020302 - 21 Jan 2023
Viewed by 830
Abstract
One of the latest techniques in metric fixed point theory is the interpolation approach. This notion has so far been examined using standard functional equations. A hybrid form of this concept is yet to be uncovered by observing the available literature. With this [...] Read more.
One of the latest techniques in metric fixed point theory is the interpolation approach. This notion has so far been examined using standard functional equations. A hybrid form of this concept is yet to be uncovered by observing the available literature. With this background information, and based on the symmetry and rectangular properties of generalized metric spaces, this paper introduces a novel and unified hybrid concept under the name interpolative Y-Hardy–Rogers–Suzuki-type Z-contraction and establishes sufficient conditions for the existence of fixed points for such contractions. As an application, one of the obtained results was employed to examine new criteria for the existence of a solution to a boundary valued problem arising in the oscillation of a spring. The ideas proposed herein advance some recently announced important results in the corresponding literature. A comparative example was constructed to justify the abstractions and pre-eminence of our obtained results. Full article
(This article belongs to the Special Issue Elementary Fixed Point Theory and Common Fixed Points)
17 pages, 326 KiB  
Article
Fixed Points of (α, β, F*) and (α, β, F**)-Weak Geraghty Contractions with an Application
by Khairul Habib Alam, Yumnam Rohen and Naeem Saleem
Symmetry 2023, 15(1), 243; https://doi.org/10.3390/sym15010243 - 16 Jan 2023
Cited by 4 | Viewed by 1685
Abstract
This study aims to provide some new classes of (α,β,F*)-weak Geraghty contraction and (α,β,F**)-weak Geraghty contraction, which are self-generalized contractions on any metric space. Furthermore, we [...] Read more.
This study aims to provide some new classes of (α,β,F*)-weak Geraghty contraction and (α,β,F**)-weak Geraghty contraction, which are self-generalized contractions on any metric space. Furthermore, we find that the mappings satisfying the definition of such contractions have a unique fixed point if the underlying space is complete. In addition, we provide an application showing the uniqueness of the solution of the two-point boundary value problem. Full article
(This article belongs to the Special Issue Elementary Fixed Point Theory and Common Fixed Points)
24 pages, 373 KiB  
Article
Control Functions in G-Metric Spaces: Novel Methods for θ-Fixed Points and θ-Fixed Circles with an Application
by Hasanen A. Hammad, Maryam G. Alshehri and Ayman Shehata
Symmetry 2023, 15(1), 164; https://doi.org/10.3390/sym15010164 - 05 Jan 2023
Cited by 2 | Viewed by 922
Abstract
The purpose of this paper is to present some new contraction mappings via control functions. In addition, some fixed point results for Θ,α,θ,Ψ contraction, rational Θ,α,θ,Ψ contraction and almost [...] Read more.
The purpose of this paper is to present some new contraction mappings via control functions. In addition, some fixed point results for Θ,α,θ,Ψ contraction, rational Θ,α,θ,Ψ contraction and almost Θ,α,θ,Ψ contraction mappings are obtained. Moreover, under contraction mappings of types (I), (II), and (III) of Θ,θ,Ψυ0, several fixed circle solutions are provided in the setting of a G-Metric space. Our results extend, unify, and generalize many previously published papers in this direction. In addition, some examples to show the reliability of our results are presented. Finally, a supporting application that discusses the possibility of a solution to a nonlinear integral equation is incorporated. Full article
(This article belongs to the Special Issue Elementary Fixed Point Theory and Common Fixed Points)
17 pages, 345 KiB  
Article
Symmetrical Hybrid Coupled Fuzzy Fixed-Point Results on Closed Ball in Fuzzy Metric Space with Applications
by Tahair Rasham, Farhan Saeed, Ravi P. Agarwal, Aftab Hussain and Abdelbsset Felhi
Symmetry 2023, 15(1), 30; https://doi.org/10.3390/sym15010030 - 22 Dec 2022
Cited by 4 | Viewed by 1142
Abstract
In this research, we establish some new fixed-point results for a symmetric coupled dominated fuzzy mapping satisfying a new advanced contraction on a closed ball in the setting of complete fuzzy metric spaces. In addition, the new notion of hybrid fuzzy-graph-dominated mappings introduced [...] Read more.
In this research, we establish some new fixed-point results for a symmetric coupled dominated fuzzy mapping satisfying a new advanced contraction on a closed ball in the setting of complete fuzzy metric spaces. In addition, the new notion of hybrid fuzzy-graph-dominated mappings introduced in fuzzy metric spaces achieves some new advanced fuzzy fixed-point problems. Some new definitions and illustrative examples are given to validate our new findings. Lastly, to demonstrate the originality of our new results, we present an application to the Fredholm-type integral equation. Full article
(This article belongs to the Special Issue Elementary Fixed Point Theory and Common Fixed Points)
16 pages, 1807 KiB  
Article
New Stability Results of an ABC Fractional Differential Equation in the Symmetric Matrix-Valued FBS
by Zahra Eidinejad, Reza Saadati, Radko Mesiar and Chenkuan Li
Symmetry 2022, 14(12), 2667; https://doi.org/10.3390/sym14122667 - 16 Dec 2022
Cited by 1 | Viewed by 1068
Abstract
By using a class of aggregation control functions, we introduce the concept of multiple-HU-OS1-stability and get an optimum approximation for a nonlinear single fractional differential equation (NS-ABC-FDE) with a Mittag–Leffler kernel. We apply an alternative fixed-point theorem to prove [...] Read more.
By using a class of aggregation control functions, we introduce the concept of multiple-HU-OS1-stability and get an optimum approximation for a nonlinear single fractional differential equation (NS-ABC-FDE) with a Mittag–Leffler kernel. We apply an alternative fixed-point theorem to prove the existence of a unique solution and the multiple-HU-OS1-stability for the NS-ABC-FDE in the symmetric matrix-valued FBS. Finally, with an example, we show the application of the obtained results. Full article
(This article belongs to the Special Issue Elementary Fixed Point Theory and Common Fixed Points)
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11 pages, 286 KiB  
Article
Solving a Boundary Value Problem via Fixed-Point Theorem on ®-Metric Space
by Gunaseelan Mani, Gopinath Janardhanan, Ozgur Ege, Arul Joseph Gnanaprakasam and Manuel De la Sen
Symmetry 2022, 14(12), 2518; https://doi.org/10.3390/sym14122518 - 29 Nov 2022
Cited by 1 | Viewed by 1093
Abstract
In this paper, we prove the fixed-point theorem for rational contractive mapping on ®-metric space. Additionally, an Euclidean metric space with a binary relation example and an application to the first-order boundary value problem are given. Moreover, the obtained results generalize and extend [...] Read more.
In this paper, we prove the fixed-point theorem for rational contractive mapping on ®-metric space. Additionally, an Euclidean metric space with a binary relation example and an application to the first-order boundary value problem are given. Moreover, the obtained results generalize and extend some of the well-known results in the literature. Full article
(This article belongs to the Special Issue Elementary Fixed Point Theory and Common Fixed Points)
24 pages, 453 KiB  
Article
Survivability of AIDS Patients via Fractional Differential Equations with Fuzzy Rectangular and Fuzzy b-Rectangular Metric like Spaces
by Naeem Saleem, Salman Furqan, Hossam A. Nabwey and Reny George
Symmetry 2022, 14(11), 2450; https://doi.org/10.3390/sym14112450 - 18 Nov 2022
Cited by 1 | Viewed by 1115
Abstract
As it is not always true that the distance between the points in fuzzy rectangular metric spaces is one, so we introduce the notions of rectangular and b-rectangular metric-like spaces in fuzzy set theory that generalize many existing results, which can be [...] Read more.
As it is not always true that the distance between the points in fuzzy rectangular metric spaces is one, so we introduce the notions of rectangular and b-rectangular metric-like spaces in fuzzy set theory that generalize many existing results, which can be regarded as the main advantage of this paper. In these spaces, the symmetry property is preserved, but the self distance may not be equal to one. We discuss topological properties and demonstrate that neither of these spaces is Hausdorff. Using α−ψ-contraction and Geraghty contractions, respectively, in our main results, we establish fixed point results in these spaces. We present examples that justify our definitions and results. We use our main results to demonstrate that the solution of a nonlinear fractional differential equation for HIV is unique. Full article
(This article belongs to the Special Issue Elementary Fixed Point Theory and Common Fixed Points)
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18 pages, 3035 KiB  
Article
Graphical Views of Intuitionistic Fuzzy Double-Controlled Metric-Like Spaces and Certain Fixed-Point Results with Application
by Umar Ishtiaq, Naeem Saleem, Fahim Uddin, Salvatore Sessa, Khaleel Ahmad and Ferdinando di Martino
Symmetry 2022, 14(11), 2364; https://doi.org/10.3390/sym14112364 - 09 Nov 2022
Cited by 6 | Viewed by 1101
Abstract
In this article, we establish the concept of intuitionistic fuzzy double-controlled metric-like spaces by “assuming that the self-distance may not be zero”; if the value of the metric is zero, then it has to be “a self-distance”. We derive numerous fixed-point results for [...] Read more.
In this article, we establish the concept of intuitionistic fuzzy double-controlled metric-like spaces by “assuming that the self-distance may not be zero”; if the value of the metric is zero, then it has to be “a self-distance”. We derive numerous fixed-point results for contraction mappings. In addition, we provide several non-trivial examples with their graphical views and an application of integral equations to show the validity of the proposed results. Full article
(This article belongs to the Special Issue Elementary Fixed Point Theory and Common Fixed Points)
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28 pages, 6300 KiB  
Article
Generalized Cyclic p-Contractions and p-Contraction Pairs Some Properties of Asymptotic Regularity Best Proximity Points, Fixed Points
by Manuel De la Sen and Asier Ibeas
Symmetry 2022, 14(11), 2247; https://doi.org/10.3390/sym14112247 - 26 Oct 2022
Cited by 1 | Viewed by 999
Abstract
This paper studies a general p-contractive condition of a self-mapping T on X, where X ,d is either a metric space or a dislocated metric space, which combines the contribution to the upper-bound of [...] Read more.
This paper studies a general p-contractive condition of a self-mapping T on X, where X ,d is either a metric space or a dislocated metric space, which combines the contribution to the upper-bound of dTx , Ty, where x and y are arbitrary elements in X of a weighted combination of the distances dx,y , dx,Tx,dy,Ty,dx,Ty,dy,Tx, dx,Tx−dy,Ty and dx,Ty−dy,Tx. The asymptotic regularity of the self-mapping T on X and the convergence of Cauchy sequences to a unique fixed point are also discussed if X,d is complete. Subsequently, T, S generalized cyclic p-contraction pairs are discussed on a pair of non-empty, in general, disjoint subsets of X. The proposed contraction involves a combination of several distances associated with the T, S-pair. Some properties demonstrated are: (a) the asymptotic convergence of the relevant sequences to best proximity points of both sets is proved; (b) the best proximity points are unique if the involved subsets are closed and convex, the metric is norm induced, or the metric space is a uniformly convex Banach space. It can be pointed out that both metric and a metric-like (or dislocated metric) possess the symmetry property since their respective distance values for any given pair of elements of the corresponding space are identical after exchanging the roles of both elements. Full article
(This article belongs to the Special Issue Elementary Fixed Point Theory and Common Fixed Points)
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20 pages, 1016 KiB  
Article
Minimum Superstability of Stochastic Ternary Antiderivations in Symmetric Matrix-Valued FB-Algebras and Symmetric Matrix-Valued FC-⋄-Algebras
by Zahra Eidinejad, Reza Saadati, Donal O’Regan and Fehaid Salem Alshammari
Symmetry 2022, 14(10), 2064; https://doi.org/10.3390/sym14102064 - 03 Oct 2022
Cited by 2 | Viewed by 977
Abstract
Our main goal in this paper is to investigate stochastic ternary antiderivatives (STAD). First, we will introduce the random ternary antiderivative operator. Then, by introducing the aggregation function using special functions such as the Mittag-Leffler function (MLF), the Wright function (WF), the H [...] Read more.
Our main goal in this paper is to investigate stochastic ternary antiderivatives (STAD). First, we will introduce the random ternary antiderivative operator. Then, by introducing the aggregation function using special functions such as the Mittag-Leffler function (MLF), the Wright function (WF), the H-Fox function (HFF), the Gauss hypergeometric function (GHF), and the exponential function (EXP-F), we will select the optimal control function by performing the necessary calculations. Next, by considering the symmetric matrix-valued FB-algebra (SMV-FB-A) and the symmetric matrix-valued FC-⋄-algebra (SMV-FC-⋄-A), we check the superstability of the desired operator. After stating each result, the superstability of the minimum is obtained by applying the optimal control function. Full article
(This article belongs to the Special Issue Elementary Fixed Point Theory and Common Fixed Points)
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21 pages, 385 KiB  
Article
Certain New Development to the Orthogonal Binary Relations
by Amjad Ali, Aftab Hussain, Muhammad Arshad, Hamed Al Sulami and Muhammad Tariq
Symmetry 2022, 14(10), 1954; https://doi.org/10.3390/sym14101954 - 20 Sep 2022
Cited by 9 | Viewed by 1145
Abstract
In this study, inspired by the concept of B-metric-like space (BMLS), we introduce the concept of orthogonal B-metric-like space (OBMLS) via a hybrid pair of operators. Additionally, we establish the concept of orthogonal dynamic system (ODS) as a generalization of the [...] Read more.
In this study, inspired by the concept of B-metric-like space (BMLS), we introduce the concept of orthogonal B-metric-like space (OBMLS) via a hybrid pair of operators. Additionally, we establish the concept of orthogonal dynamic system (ODS) as a generalization of the dynamic system (DS), which improves the existing results for analysies such as those presented here. By applying this, some new refinements of the F⊥-Suzuki-type (F⊥-ST) fixed-point results are presented. These include some tangible instances, and applications in the field of nonlinear analysis are given to highlight the usability and validity of the theoretical results. Full article
(This article belongs to the Special Issue Elementary Fixed Point Theory and Common Fixed Points)
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14 pages, 297 KiB  
Article
Meir–Keeler Type Contraction in Orthogonal M-Metric Spaces
by Ateq Alsaadi, Bijender Singh, Vizender Singh and Izhar Uddin
Symmetry 2022, 14(9), 1856; https://doi.org/10.3390/sym14091856 - 06 Sep 2022
Cited by 4 | Viewed by 1041
Abstract
In this article, we prove fixed point results for a Meir–Keeler type contraction due to orthogonal M-metric spaces. The results of the paper improve and extend some recent developments in fixed point theory. The extension is assured by the concluding remarks and [...] Read more.
In this article, we prove fixed point results for a Meir–Keeler type contraction due to orthogonal M-metric spaces. The results of the paper improve and extend some recent developments in fixed point theory. The extension is assured by the concluding remarks and followed by the main theorem. Finally, an application of the main theorem is established in proving theorems for some integral equations and integral-type contractive conditions. Full article
(This article belongs to the Special Issue Elementary Fixed Point Theory and Common Fixed Points)
11 pages, 307 KiB  
Article
φ-ψ-Contractions under W-Distances Employing Symmetric Locally T-Transitive Binary Relation
by Mohammad Arif, Mohammad Imdad and Salvatore Sessa
Symmetry 2022, 14(7), 1456; https://doi.org/10.3390/sym14071456 - 16 Jul 2022
Viewed by 862
Abstract
The intent of this paper is to prove the relation-theoretic fixed point results under φ-ψ-contractions involving W-distance on a metric space and equipped with a symmetric locally T-transitive binary relation (not necessarily transitive relation). Our results enrich and [...] Read more.
The intent of this paper is to prove the relation-theoretic fixed point results under φ-ψ-contractions involving W-distance on a metric space and equipped with a symmetric locally T-transitive binary relation (not necessarily transitive relation). Our results enrich and improve several fixed point results of the existing literature. Full article
(This article belongs to the Special Issue Elementary Fixed Point Theory and Common Fixed Points)
13 pages, 295 KiB  
Article
Order-Theoretic Common Fixed Point Results in Rmb-Metric Spaces
by Mohammad Asim, Mohammad Imdad and Salvatore Sessa
Symmetry 2022, 14(7), 1376; https://doi.org/10.3390/sym14071376 - 04 Jul 2022
Viewed by 1029
Abstract
In this paper, we prove common fixed point theorems for a pair of self-mappings in the framework of Rmb-metric spaces. We also deduce some corollaries of our main result. In order to support our main result, we also set an [...] Read more.
In this paper, we prove common fixed point theorems for a pair of self-mappings in the framework of Rmb-metric spaces. We also deduce some corollaries of our main result. In order to support our main result, we also set an example. Full article
(This article belongs to the Special Issue Elementary Fixed Point Theory and Common Fixed Points)
13 pages, 281 KiB  
Article
On Prešić–Ćirić-Type α-ψ Contractions with an Application
by Faruk Sk, Mohammed Ahmed Osman Tom, Qamrul Haq Khan and Faizan Ahmad Khan
Symmetry 2022, 14(6), 1166; https://doi.org/10.3390/sym14061166 - 06 Jun 2022
Cited by 3 | Viewed by 1103
Abstract
In this paper, we extend the idea of α-ψ contraction mapping to the product spaces by introducing Prešić–Ćirić-type α-ψ contractions and utilize them to prove some coincidence and common fixed-point theorems in the context of ordered metric spaces using [...] Read more.
In this paper, we extend the idea of α-ψ contraction mapping to the product spaces by introducing Prešić–Ćirić-type α-ψ contractions and utilize them to prove some coincidence and common fixed-point theorems in the context of ordered metric spaces using α-admissibility of the mapping. Our newly established results generalize a number of well-known fixed-point theorems from the literature. Moreover, we give some examples that attest to the credibility of our results. Further, we give an application to the nonlinear integral equations, which can be employed to study the existence and uniqueness of solutions to the integral equations. Full article
(This article belongs to the Special Issue Elementary Fixed Point Theory and Common Fixed Points)
18 pages, 325 KiB  
Article
Strong Convergence Theorems for a Finite Family of Enriched Strictly Pseudocontractive Mappings and ΦT-Enriched Lipschitizian Mappings Using a New Modified Mixed-Type Ishikawa Iteration Scheme with Error
by Naeem Saleem, Imo Kalu Agwu, Umar Ishtiaq and Stojan Radenović
Symmetry 2022, 14(5), 1032; https://doi.org/10.3390/sym14051032 - 18 May 2022
Cited by 11 | Viewed by 1349
Abstract
In this paper, we introduce two new classes of mappings known as λ-enriched strictly pseudocontractive mappings and ΦT-enriched Lipshitizian mappings in the setup of a real Banach space. In addition, a new modified mixed-type Ishikawa iteration scheme was constructed, and [...] Read more.
In this paper, we introduce two new classes of mappings known as λ-enriched strictly pseudocontractive mappings and ΦT-enriched Lipshitizian mappings in the setup of a real Banach space. In addition, a new modified mixed-type Ishikawa iteration scheme was constructed, and it was proved that our iteration method converges strongly to the common fixed points of finite families of the above mappings in the framework of a real uniformly convex Banach space. Moreover, we provided a non-trivial example to support our main result. Our results extend and generalize several results existing in the literature. Full article
(This article belongs to the Special Issue Elementary Fixed Point Theory and Common Fixed Points)
22 pages, 1744 KiB  
Article
Fixed Point of (α,β)-Admissible Generalized Geraghty F-Contraction with Application
by Min Wang, Naeem Saleem, Xiaolan Liu, Arslan Hojat Ansari and Mi Zhou
Symmetry 2022, 14(5), 1016; https://doi.org/10.3390/sym14051016 - 17 May 2022
Cited by 2 | Viewed by 1485
Abstract
In this paper, we introduce some new types of extended Geraghty contractions, called (α,β)-admissible generalized Geraghty F-contractions, and prove some fixed point results for such contractions in the setting of partial b-metric spaces. Moreover, based on [...] Read more.
In this paper, we introduce some new types of extended Geraghty contractions, called (α,β)-admissible generalized Geraghty F-contractions, and prove some fixed point results for such contractions in the setting of partial b-metric spaces. Moreover, based on the obtained fixed point results and the property of symmetry, we inaugurate a fixed point result for graphic generalized Geraghty F-contractions defined on partial metric spaces endowed with a directed graph. As an application, we examine the existence of a unique solution to the first-order periodic boundary value by the obtained fixed point result. Moreover, some examples are presented to illustrate the validity of the new results. Full article
(This article belongs to the Special Issue Elementary Fixed Point Theory and Common Fixed Points)
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10 pages, 256 KiB  
Article
Fredholm Type Integral Equation in Controlled Rectangular Metric-like Spaces
by Salma Haque, Fatima Azmi and Nabil Mlaiki
Symmetry 2022, 14(5), 991; https://doi.org/10.3390/sym14050991 - 12 May 2022
Cited by 4 | Viewed by 1178
Abstract
In this article, we present an extension of the controlled rectangular b-metric spaces, so-called controlled rectangular metric-like spaces, where we keep the symmetry condition and we only change the condition [...] Read more.
In this article, we present an extension of the controlled rectangular b-metric spaces, so-called controlled rectangular metric-like spaces, where we keep the symmetry condition and we only change the condition [D(s,r)=0⇔s=r]to[D(s,r)=0⇒s=r], which means we may have a non-zero self distance; also, D(s,s) is not necessarily less than D(s,r). This new type of metric space is a generalization of controlled rectangular b-metric spaces and partial rectangular metric spaces. Full article
(This article belongs to the Special Issue Elementary Fixed Point Theory and Common Fixed Points)
18 pages, 311 KiB  
Article
Existence and Well-Posedness of Tripled Fixed Points with Application to a System of Differential Equations
by Rashwan A. Rashwan, Hasanen A. Hammad, Ahmed Nafea and Fahd Jarad
Symmetry 2022, 14(4), 745; https://doi.org/10.3390/sym14040745 - 05 Apr 2022
Cited by 1 | Viewed by 1062
Abstract
The purpose of this manuscript is to demonstrate the existence and uniqueness of triple fixed-point results for Geraghty-type contractions in ordinary metric spaces with binary relations. Moreover, the well-posedness of the tripled fixed point problem is investigated. Consequently, α-dominated mapping on such [...] Read more.
The purpose of this manuscript is to demonstrate the existence and uniqueness of triple fixed-point results for Geraghty-type contractions in ordinary metric spaces with binary relations. Moreover, the well-posedness of the tripled fixed point problem is investigated. Consequently, α-dominated mapping on such space is discussed. Ultimately, to promote our paper, some illustrative examples are derived, and the existence of the solution to a system of differential equations is obtained as an application. Full article
(This article belongs to the Special Issue Elementary Fixed Point Theory and Common Fixed Points)
7 pages, 220 KiB  
Article
Coincidence Points for Mappings in Metric Spaces Satisfying Weak Commuting Conditions
by Salvatore Sessa and Mohamed Akkouchi
Symmetry 2022, 14(3), 504; https://doi.org/10.3390/sym14030504 - 01 Mar 2022
Cited by 1 | Viewed by 1283
Abstract
In this note, we prove some results of elementary fixed point theory for mappings defined in metric spaces satisfying conditions of weak commutativity. Suitable examples are proven as well. Full article
(This article belongs to the Special Issue Elementary Fixed Point Theory and Common Fixed Points)
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