Fixed Point Theory and Its Related Topics III

A special issue of Axioms (ISSN 2075-1680). This special issue belongs to the section "Logic".

Deadline for manuscript submissions: closed (31 December 2022) | Viewed by 21664

Special Issue Editor


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Guest Editor
Department of Mathematics, National Kaohsiung Normal University, Kaohsiung 802, Taiwan
Interests: fuzzy optimization; fuzzy real analysis; fuzzy statistical analysis; operations research; computational intelligence; soft computing; fixed point theory; applied functional analysis
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Special Issue Information

Dear Colleagues,

This issue is a continuation of the previous successful Special Issue series.

Fixed point theory arose from the Banach contraction principle and has been studied for a long time. Its application mostly relies on the existence of solutions to mathematical problems that are formulated from economics and engineering. After the existence of the solutions is guaranteed, the numerical methodology will be established to obtain the approximated solution. Fixed points of function depend heavily on the considered spaces that are defined using the intuitive axioms. In particular, variant metrics spaces are proposed, like a partial metric space, b-metric space, fuzzy metric space and probabilistic metric space, etc. Different spaces will result in different types of fixed point theorems. In other words, there are a lot of different types of fixed point theorems in the literature. Therefore, this Special Issue welcomes survey articles. Articles that unify the different types of fixed point theorems are also very welcome. The topics of this Special Issue include:

  • fixed point theorems in metric space
  • fixed point theorems in fuzzy metric space
  • fixed point theorems in probabilistic metric space
  • fixed point theorems of set-valued functions in various spaces
  • the existence of solutions in game theory
  • the existence of solutions for equilibrium problems
  • the existence of solutions of differential equations
  • the existence of solutions of integral equations
  • numerical methods for obtaining the approximated fixed points

Prof. Dr. Hsien-Chung Wu
Guest Editor

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Keywords

  • Fixed Point
  • Best Proximity Point
  • Equilibrium
  • Cauchy Sequences
  • Completeness
  • Game Theory

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Published Papers (17 papers)

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Research

20 pages, 338 KiB  
Article
Best Proximity Point for ΓτF-Fuzzy Proximal Contraction
by Uma Devi Patel, Vesna Todorcevic, Slobodan Radojevic and Stojan Radenović
Axioms 2023, 12(2), 165; https://doi.org/10.3390/axioms12020165 - 06 Feb 2023
Cited by 3 | Viewed by 857
Abstract
In this writing, first, we disclose the first and second category of a ΓτF-fuzzy proximal contraction for a mapping O:UV which is nonself and also declare a fuzzy q-property to confirm the existence of the [...] Read more.
In this writing, first, we disclose the first and second category of a ΓτF-fuzzy proximal contraction for a mapping O:UV which is nonself and also declare a fuzzy q-property to confirm the existence of the best proximity point for nonself function O. Then, we discover a few results using the ΓτF-fuzzy proximal contraction of the first category for a continuous and discontinuous nonself function O in a non-Archimedean fuzzy metric space. Later, we discuss another result for the ΓτF-fuzzy proximal contraction of the second category as well. In between the fuzzy proximal theorems, many examples are presented in support of the definitions and theorems proved in this writing. Full article
(This article belongs to the Special Issue Fixed Point Theory and Its Related Topics III)
13 pages, 300 KiB  
Article
Some Extended Results for Multivalued F-Contraction Mappings
by Hakan Sahin, Mustafa Aslantas and Ali Abdulkareem Nasir Nasir
Axioms 2023, 12(2), 116; https://doi.org/10.3390/axioms12020116 - 22 Jan 2023
Cited by 3 | Viewed by 803
Abstract
In this study, we introduce a new notion, so-called KW-type F-contraction mapping, inspired by the ideas of Wardowski and Klim–Wardowski. Then, we investigate the existence of a best proximity point for such mappings by considering a new family, which is [...] Read more.
In this study, we introduce a new notion, so-called KW-type F-contraction mapping, inspired by the ideas of Wardowski and Klim–Wardowski. Then, we investigate the existence of a best proximity point for such mappings by considering a new family, which is larger than the family of functions that is often used in fixed-point results for multivalued mappings. To demonstrate the effectiveness of our result, we also give a comparative example to which similar results in the literature cannot be applied. Moreover, we present an application of our main result to homotopic mappings. Full article
(This article belongs to the Special Issue Fixed Point Theory and Its Related Topics III)
16 pages, 310 KiB  
Article
Fixed Point Theorems for Mann’s Iteration Scheme in Convex Gb-Metric Spaces with an Application
by Dong Ji, Chaobo Li and Yunan Cui
Axioms 2023, 12(2), 108; https://doi.org/10.3390/axioms12020108 - 19 Jan 2023
Cited by 3 | Viewed by 1092
Abstract
In this paper, we present a series of fixed point results for Mann’s iteration scheme in the framework of Gb-metric spaces. First, we introduce the concept of convex Gb-metric space by means of a convex structure and Mann’s iteration [...] Read more.
In this paper, we present a series of fixed point results for Mann’s iteration scheme in the framework of Gb-metric spaces. First, we introduce the concept of convex Gb-metric space by means of a convex structure and Mann’s iteration algorithm is extended to this space. Furthermore, using Mann’s iteration scheme, we prove some fixed point results for several mappings satisfying various suitable conditions on complete convex Gb-metric spaces. Some examples supporting our main results are also presented. We also discuss the well-posedness of the fixed point problems and the P property for given mappings. Moreover, as an application, we apply our main result to prove the existence of the solutions to integral equations. Full article
(This article belongs to the Special Issue Fixed Point Theory and Its Related Topics III)
17 pages, 334 KiB  
Article
Solving an Integral Equation via Tricomplex-Valued Controlled Metric Spaces
by Rajagopalan Ramaswamy, Gunaseelan Mani, Arul Joseph Gnanaprakasam, Ola A. Ashour Abdelnaby and Stojan Radenović
Axioms 2023, 12(1), 56; https://doi.org/10.3390/axioms12010056 - 03 Jan 2023
Cited by 1 | Viewed by 887
Abstract
In this present study, we propose the concept of tricomplex-controlled metric spaces as a generalization of both controlled metric-type spaces and tricomplex metric-type spaces. In this work, we establish fixed point results using Banach, Kannan and Fisher-type contractions supported with nontrivial examples in [...] Read more.
In this present study, we propose the concept of tricomplex-controlled metric spaces as a generalization of both controlled metric-type spaces and tricomplex metric-type spaces. In this work, we establish fixed point results using Banach, Kannan and Fisher-type contractions supported with nontrivial examples in the setting of the proposed space. We apply the derived result to find the analytical solution of an integral equation using the fixed point technique under the same metric. Full article
(This article belongs to the Special Issue Fixed Point Theory and Its Related Topics III)
13 pages, 286 KiB  
Article
Existence Results for Generalized Vector Quasi-Equilibrium Problems in Hadamard Manifolds
by Shuechin Huang
Axioms 2023, 12(1), 40; https://doi.org/10.3390/axioms12010040 - 29 Dec 2022
Viewed by 975
Abstract
The purpose of this article was to establish manifold versions of existence theorems for generalized vector quasi-equilibrium problems in locally compact and σ-compact spaces without any continuity assumption. The fixed-point theorem in a product Hadamard manifold is the key focus of our [...] Read more.
The purpose of this article was to establish manifold versions of existence theorems for generalized vector quasi-equilibrium problems in locally compact and σ-compact spaces without any continuity assumption. The fixed-point theorem in a product Hadamard manifold is the key focus of our discussion. We further applied our theorems to saddle point and minimax problems. Full article
(This article belongs to the Special Issue Fixed Point Theory and Its Related Topics III)
14 pages, 336 KiB  
Article
Solution to Integral Equation in an O-Complete Branciari b-Metric Spaces
by Menaha Dhanraj, Arul Joseph Gnanaprakasam, Gunaseelan Mani, Ozgur Ege and Manuel De la Sen
Axioms 2022, 11(12), 728; https://doi.org/10.3390/axioms11120728 - 13 Dec 2022
Cited by 2 | Viewed by 968
Abstract
In this paper, we prove fixed point theorem via orthogonal Geraghty type α-admissible contraction map in an orthogonal complete Branciari b-metric spaces context. An example is presented to strengthen our main result. We provided an application to find the existence and [...] Read more.
In this paper, we prove fixed point theorem via orthogonal Geraghty type α-admissible contraction map in an orthogonal complete Branciari b-metric spaces context. An example is presented to strengthen our main result. We provided an application to find the existence and uniqueness of a solution to the Volterra integral equation. We have compared the approximate solution and exact solution numerically. Full article
(This article belongs to the Special Issue Fixed Point Theory and Its Related Topics III)
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14 pages, 294 KiB  
Article
New Results for Multivalued Mappings in Hausdorff Neutrosophic Metric Spaces
by Mathuraiveeran Jeyaraman, Hassen Aydi and Manuel De La Sen
Axioms 2022, 11(12), 724; https://doi.org/10.3390/axioms11120724 - 13 Dec 2022
Viewed by 971
Abstract
The fundamental goal of this paper is to derive common fixed-point results for a sequence of multivalued mappings contained in a closed ball over a complete neutrosophic metric space. A basic and distinctive procedure has been used to prove the proposed results. Full article
(This article belongs to the Special Issue Fixed Point Theory and Its Related Topics III)
28 pages, 361 KiB  
Article
Asymptotically Coupled Coincidence Points and Asymptotically Coupled Fixed Points in Fuzzy Semi-Metric Spaces
by Hsien-Chung Wu
Axioms 2022, 11(12), 688; https://doi.org/10.3390/axioms11120688 - 30 Nov 2022
Viewed by 689
Abstract
Asymptotically coupled coincidence points and asymptotically coupled fixed points in fuzzy semi-metric spaces are studied in this paper. The fuzzy semi-metric space is taken into account, which lacks symmetric conditions. In this case, the desired results are separately investigated based on four different [...] Read more.
Asymptotically coupled coincidence points and asymptotically coupled fixed points in fuzzy semi-metric spaces are studied in this paper. The fuzzy semi-metric space is taken into account, which lacks symmetric conditions. In this case, the desired results are separately investigated based on four different types of triangle inequalities. The uniqueness of asymptotically coupled coincidence points cannot be guaranteed, and it can only be addressed in a weak sense of uniqueness. However, the uniqueness of asymptotically coupled fixed points can be guaranteed using different arguments. Full article
(This article belongs to the Special Issue Fixed Point Theory and Its Related Topics III)
23 pages, 404 KiB  
Article
Hybrid Fuzzy Contraction Theorems with Their Role in Integral Inclusions
by Faryad Ali, Mohammed Shehu Shagari and Akbar Azam
Axioms 2022, 11(11), 580; https://doi.org/10.3390/axioms11110580 - 22 Oct 2022
Viewed by 932
Abstract
The focus of this paper is to establish a new concept of b-hybrid fuzzy contraction regarding the study of fuzzy fixed-point theorems in the setting of b-metric spaces. This idea harmonizes and refines several well-known results in the direction of point-valued, [...] Read more.
The focus of this paper is to establish a new concept of b-hybrid fuzzy contraction regarding the study of fuzzy fixed-point theorems in the setting of b-metric spaces. This idea harmonizes and refines several well-known results in the direction of point-valued, multivalued, and fuzzy-set-valued maps in the comparable literature. To attract new researchers to this field, some important results are shown to be corollaries. Moreover, a result is presented to establish sufficient conditions for the existence of solutions of integral inclusion of Fredholm type. Lastly, illustrations are presented to validate the suppositions of the given theorems. Full article
(This article belongs to the Special Issue Fixed Point Theory and Its Related Topics III)
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17 pages, 331 KiB  
Article
Reich-Type and (α, F)-Contractions in Partially Ordered Double-Controlled Metric-Type Spaces with Applications to Non-Linear Fractional Differential Equations and Monotonic Iterative Method
by Muhammad Farhan, Umar Ishtiaq, Muhammad Saeed, Aftab Hussain and Hamed Al Sulami
Axioms 2022, 11(10), 573; https://doi.org/10.3390/axioms11100573 - 20 Oct 2022
Cited by 4 | Viewed by 1187
Abstract
In this manuscript, we defined (α, F)-contractions in the context of double-controlled metric spaces and partially ordered double-controlled metric spaces. We established new fixed-point results and defined the notion of double-controlled metric space on a Reich-type contraction. Our findings are [...] Read more.
In this manuscript, we defined (α, F)-contractions in the context of double-controlled metric spaces and partially ordered double-controlled metric spaces. We established new fixed-point results and defined the notion of double-controlled metric space on a Reich-type contraction. Our findings are generalizations of a few well-known findings in the literature. Some non-trivial examples and certain consequences are also provided to illustrate the significance of the presented results. The existence and uniqueness of the solution of non-linear fractional differential equations and the monotone iterative method are also determined using the fixed-point method. Full article
(This article belongs to the Special Issue Fixed Point Theory and Its Related Topics III)
6 pages, 234 KiB  
Article
Equivalence between the Existence of Best Proximity Points and Fixed Points for Some Classes of Proximal Contractions
by Sumit Som, Moosa Gabeleh and Manuel De la Sen
Axioms 2022, 11(9), 468; https://doi.org/10.3390/axioms11090468 - 13 Sep 2022
Viewed by 1150
Abstract
In the year 2014, Almeida et al. introduced a new class of mappings, namely, contractions of Geraghty type. Additionally, in the year 2021, Beg et al. introduced the concept of generalized F-proximal contraction of the first kind and generalized F-proximal contraction [...] Read more.
In the year 2014, Almeida et al. introduced a new class of mappings, namely, contractions of Geraghty type. Additionally, in the year 2021, Beg et al. introduced the concept of generalized F-proximal contraction of the first kind and generalized F-proximal contraction of the second kind, respectively. After developing these concepts, authors mainly studied the best proximity points for these classes of mappings. In this short note, we prove that the problem of the existence of the best proximity points for the said classes of proximal contractions is equivalent to the corresponding fixed points problems. Full article
(This article belongs to the Special Issue Fixed Point Theory and Its Related Topics III)
11 pages, 286 KiB  
Article
Fixed Point of Modified F-Contraction with an Application
by Min Wang, Naeem Saleem, Shahid Bashir and Mi Zhou
Axioms 2022, 11(8), 413; https://doi.org/10.3390/axioms11080413 - 18 Aug 2022
Cited by 2 | Viewed by 1062
Abstract
In this paper, we prove the existence of fixed points for every 2-rotative continuous mapping in Banach spaces to answer an open question raised by Goebel and Koter. Further, we modify F-contraction by developing F-rotative mapping and establish some fixed-point theorems. [...] Read more.
In this paper, we prove the existence of fixed points for every 2-rotative continuous mapping in Banach spaces to answer an open question raised by Goebel and Koter. Further, we modify F-contraction by developing F-rotative mapping and establish some fixed-point theorems. Finally, we apply our results to prove the existence of a solution of a non-linear fractional differential equation. Full article
(This article belongs to the Special Issue Fixed Point Theory and Its Related Topics III)
16 pages, 287 KiB  
Article
Solving Integral Equations Using Weakly Compatible Mappings via D*-Metric Spaces
by Roqia Butush, Zead Mustafa and M. M. M. Jaradat
Axioms 2022, 11(5), 189; https://doi.org/10.3390/axioms11050189 - 20 Apr 2022
Viewed by 1365
Abstract
We introduce a new pair of mappings (S,T) on D*-metric spaces called DS*-W.C. and DRS*-W.C. Many examples are presented to show the difference between these mappings and other types of [...] Read more.
We introduce a new pair of mappings (S,T) on D*-metric spaces called DS*-W.C. and DRS*-W.C. Many examples are presented to show the difference between these mappings and other types of mappings in the literature. Moreover, we obtain several common fixed point results by using these types of mappings and the (E.A) property. We then employ the fixed point results to establish the existence and uniqueness of a solution for a class of nonlinear integral equations. Full article
(This article belongs to the Special Issue Fixed Point Theory and Its Related Topics III)
21 pages, 319 KiB  
Article
Rectangular Gb-Metric Spaces and Some Fixed Point Theorems
by Chaobo Li and Yunan Cui
Axioms 2022, 11(3), 108; https://doi.org/10.3390/axioms11030108 - 01 Mar 2022
Viewed by 2093
Abstract
In this paper, we first introduce the concept of rectangular Gb-metric space which generalizes the notion of rectangular metric space and Gb-metric space. Then, some fixed point results connected with certain contractions are obtained in the setting of rectangular [...] Read more.
In this paper, we first introduce the concept of rectangular Gb-metric space which generalizes the notion of rectangular metric space and Gb-metric space. Then, some fixed point results connected with certain contractions are obtained in the setting of rectangular Gb-metric spaces. Additionally, we also introduce the concept of convex rectangular Gb-metric space by means of the convex structure and study the fixed points of enriched type contractions in this space. Full article
(This article belongs to the Special Issue Fixed Point Theory and Its Related Topics III)
17 pages, 318 KiB  
Article
Fixed Point Results for New Types of Fuzzy Contractions via Admissible Functions and FZ -Simulation Functions
by Abdelhamid Moussaoui, Naeem Saleem, Said Melliani and Mi Zhou
Axioms 2022, 11(3), 87; https://doi.org/10.3390/axioms11030087 - 23 Feb 2022
Cited by 8 | Viewed by 1546
Abstract
In this paper, we introduce two new concepts, generalized α-η-FZ-contraction and modified α-η-FZ-contraction, which unify several types of contractions in the context of fuzzy metric spaces. We discuss the existence and uniqueness results [...] Read more.
In this paper, we introduce two new concepts, generalized α-η-FZ-contraction and modified α-η-FZ-contraction, which unify several types of contractions in the context of fuzzy metric spaces. We discuss the existence and uniqueness results of such mappings in the setting of a complete fuzzy metric space in the sense of George and Veeramani and present several significant consequences of our obtained results by using variant examples for FZ-simulation functions and admissible mappings. Some examples are provided to illustrate the usability of our main results. Full article
(This article belongs to the Special Issue Fixed Point Theory and Its Related Topics III)
12 pages, 229 KiB  
Article
Stable Convergence Theorems for Products of Uniformly Continuous Mappings in Metric Spaces
by Alexander J. Zaslavski
Axioms 2021, 10(3), 156; https://doi.org/10.3390/axioms10030156 - 19 Jul 2021
Viewed by 1255
Abstract
We study the behavior of inexact products of uniformly continuous self-mappings of a complete metric space that is uniformly continuous and bounded on bounded sets. It is shown that previously established convergence theorems for products of non-expansive mappings continue to hold even under [...] Read more.
We study the behavior of inexact products of uniformly continuous self-mappings of a complete metric space that is uniformly continuous and bounded on bounded sets. It is shown that previously established convergence theorems for products of non-expansive mappings continue to hold even under the presence of computational errors. Full article
(This article belongs to the Special Issue Fixed Point Theory and Its Related Topics III)
23 pages, 331 KiB  
Article
Coupled Optimal Results with an Application to Nonlinear Integral Equations
by Pulak Konar, Sumit Chandok, Shrutinil Dutta and Manuel De la Sen
Axioms 2021, 10(2), 73; https://doi.org/10.3390/axioms10020073 - 22 Apr 2021
Cited by 2 | Viewed by 1625
Abstract
In the present work, we consider the best proximal problem related to a coupled mapping, which we define using control functions and weak inequalities. As a consequence, we obtain some results on coupled fixed points. Our results generalize some recent results in the [...] Read more.
In the present work, we consider the best proximal problem related to a coupled mapping, which we define using control functions and weak inequalities. As a consequence, we obtain some results on coupled fixed points. Our results generalize some recent results in the literature. Also, as an application of the results obtained, we present the solution to a system of a coupled Fredholm nonlinear integral equation. Our work is supported by several illustrations. Full article
(This article belongs to the Special Issue Fixed Point Theory and Its Related Topics III)
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