Fixed Point Theory and Its Related Topics II

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

Deadline for manuscript submissions: closed (31 January 2021) | Viewed by 23372

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Special Issue 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 “Fixed Point Theory and Related Topics”.

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, 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 (11 papers)

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Research

7 pages, 247 KiB  
Article
(ρ,η,μ)-Interpolative Kannan Contractions I
Axioms 2021, 10(3), 212; https://doi.org/10.3390/axioms10030212 - 03 Sep 2021
Cited by 6 | Viewed by 1631
Abstract
We point out a vital error in the paper of Gaba et al. (2019), showing that a (ρ,η,μ) interpolative Kannan contraction in a complete metric space need not have a fixed point. Then we give an appropriate [...] Read more.
We point out a vital error in the paper of Gaba et al. (2019), showing that a (ρ,η,μ) interpolative Kannan contraction in a complete metric space need not have a fixed point. Then we give an appropriate restriction on a (ρ,η,μ)-interpolative Kannan contraction that guarantees the existence of a fixed point and provide an equivalent formulation. Moreover, we show that this formulation can be extended to the interpolative Reich-Rus-Ćirić type contraction. Full article
(This article belongs to the Special Issue Fixed Point Theory and Its Related Topics II)
10 pages, 283 KiB  
Article
Cascading Operators in CAT(0) Spaces
Axioms 2021, 10(1), 20; https://doi.org/10.3390/axioms10010020 - 09 Feb 2021
Viewed by 1516
Abstract
In this work, we introduce the notion of cascading non-expansive mappings in the setting of CAT(0) spaces. This family of mappings properly contains the non-expansive maps, but it differs from other generalizations of this class of maps. Considering the concept [...] Read more.
In this work, we introduce the notion of cascading non-expansive mappings in the setting of CAT(0) spaces. This family of mappings properly contains the non-expansive maps, but it differs from other generalizations of this class of maps. Considering the concept of Δ-convergence in metric spaces, we prove a principle of demiclosedness for this type of mappings and a Δ-convergence theorem for a Mann iteration process defined using cascading operators. Full article
(This article belongs to the Special Issue Fixed Point Theory and Its Related Topics II)
31 pages, 896 KiB  
Article
Using the Supremum Form of Auxiliary Functions to Study the Common Coupled Coincidence Points in Fuzzy Semi-Metric Spaces
Axioms 2021, 10(1), 5; https://doi.org/10.3390/axioms10010005 - 05 Jan 2021
Cited by 1 | Viewed by 1316
Abstract
This paper investigates the common coupled coincidence points and common coupled fixed points in fuzzy semi-metric spaces. The symmetric condition is not necessarily satisfied in fuzzy semi-metric space. Therefore, four kinds of triangle inequalities are taken into account in order to study the [...] Read more.
This paper investigates the common coupled coincidence points and common coupled fixed points in fuzzy semi-metric spaces. The symmetric condition is not necessarily satisfied in fuzzy semi-metric space. Therefore, four kinds of triangle inequalities are taken into account in order to study the Cauchy sequences. Inspired by the intuitive observations, the concepts of rational condition and distance condition are proposed for the purpose of simplifying the discussions. Full article
(This article belongs to the Special Issue Fixed Point Theory and Its Related Topics II)
8 pages, 240 KiB  
Article
e-Distance in Menger PGM Spaces with an Application
Axioms 2021, 10(1), 3; https://doi.org/10.3390/axioms10010003 - 30 Dec 2020
Cited by 5 | Viewed by 1342
Abstract
The main purpose of the present paper is to define the concept of an e-distance (as a generalization of r-distance) on a Menger PGM space and to introduce some of its properties. Moreover, some coupled fixed point results, in terms of [...] Read more.
The main purpose of the present paper is to define the concept of an e-distance (as a generalization of r-distance) on a Menger PGM space and to introduce some of its properties. Moreover, some coupled fixed point results, in terms of this distance on a complete PGM space, are proved. To support our definitions and main results, several examples and an application are considered. Full article
(This article belongs to the Special Issue Fixed Point Theory and Its Related Topics II)
13 pages, 749 KiB  
Article
Fixed Points of g-Interpolative Ćirić–Reich–Rus-Type Contractions in b-Metric Spaces
Axioms 2020, 9(4), 132; https://doi.org/10.3390/axioms9040132 - 16 Nov 2020
Cited by 7 | Viewed by 1809
Abstract
We use interpolation to obtain a common fixed point result for a new type of Ćirić–Reich–Rus-type contraction mappings in metric space. We also introduce a new concept of g-interpolative Ćirić–Reich–Rus-type contractions in b-metric spaces, and we prove some fixed point results [...] Read more.
We use interpolation to obtain a common fixed point result for a new type of Ćirić–Reich–Rus-type contraction mappings in metric space. We also introduce a new concept of g-interpolative Ćirić–Reich–Rus-type contractions in b-metric spaces, and we prove some fixed point results for such mappings. Our results extend and improve some results on the fixed point theory in the literature. We also give some examples to illustrate the given results. Full article
(This article belongs to the Special Issue Fixed Point Theory and Its Related Topics II)
16 pages, 312 KiB  
Article
Strong Convergence of Extragradient-Type Method to Solve Pseudomonotone Variational Inequalities Problems
Axioms 2020, 9(4), 115; https://doi.org/10.3390/axioms9040115 - 13 Oct 2020
Cited by 6 | Viewed by 2163
Abstract
A number of applications from mathematical programmings, such as minimax problems, penalization methods and fixed-point problems can be formulated as a variational inequality model. Most of the techniques used to solve such problems involve iterative algorithms, and that is why, in this paper, [...] Read more.
A number of applications from mathematical programmings, such as minimax problems, penalization methods and fixed-point problems can be formulated as a variational inequality model. Most of the techniques used to solve such problems involve iterative algorithms, and that is why, in this paper, we introduce a new extragradient-like method to solve the problems of variational inequalities in real Hilbert space involving pseudomonotone operators. The method has a clear advantage because of a variable stepsize formula that is revised on each iteration based on the previous iterations. The key advantage of the method is that it works without the prior knowledge of the Lipschitz constant. Strong convergence of the method is proved under mild conditions. Several numerical experiments are reported to show the numerical behaviour of the method. Full article
(This article belongs to the Special Issue Fixed Point Theory and Its Related Topics II)
15 pages, 300 KiB  
Article
The Split Various Variational Inequalities Problems for Three Hilbert Spaces
Axioms 2020, 9(3), 103; https://doi.org/10.3390/axioms9030103 - 07 Sep 2020
Viewed by 2145
Abstract
There are many methods for finding a common solution of a system of variational inequalities, a split equilibrium problem, and a hierarchical fixed-point problem in the setting of real Hilbert spaces. They proved the strong convergence theorem. Many split feasibility problems are generated [...] Read more.
There are many methods for finding a common solution of a system of variational inequalities, a split equilibrium problem, and a hierarchical fixed-point problem in the setting of real Hilbert spaces. They proved the strong convergence theorem. Many split feasibility problems are generated in real Hillbert spaces. The open problem is proving a strong convergence theorem of three Hilbert spaces with different methods from the lasted method. In this research, a new split variational inequality in three Hilbert spaces is proposed. Important tools which are used to solve classical problems will be developed. The convergence theorem for finding a common element of the set of solution of such problems and the sets of fixed-points of discontinuous mappings has been proved. Full article
(This article belongs to the Special Issue Fixed Point Theory and Its Related Topics II)
24 pages, 1384 KiB  
Article
A General Inertial Projection-Type Algorithm for Solving Equilibrium Problem in Hilbert Spaces with Applications in Fixed-Point Problems
Axioms 2020, 9(3), 101; https://doi.org/10.3390/axioms9030101 - 31 Aug 2020
Cited by 10 | Viewed by 2571
Abstract
A plethora of applications from mathematical programming, such as minimax, and mathematical programming, penalization, fixed point to mention a few can be framed as equilibrium problems. Most of the techniques for solving such problems involve iterative methods that is why, in this paper, [...] Read more.
A plethora of applications from mathematical programming, such as minimax, and mathematical programming, penalization, fixed point to mention a few can be framed as equilibrium problems. Most of the techniques for solving such problems involve iterative methods that is why, in this paper, we introduced a new extragradient-like method to solve equilibrium problems in real Hilbert spaces with a Lipschitz-type condition on a bifunction. The advantage of a method is a variable stepsize formula that is updated on each iteration based on the previous iterations. The method also operates without the previous information of the Lipschitz-type constants. The weak convergence of the method is established by taking mild conditions on a bifunction. For application, fixed-point theorems that involve strict pseudocontraction and results for pseudomonotone variational inequalities are studied. We have reported various numerical results to show the numerical behaviour of the proposed method and correlate it with existing ones. Full article
(This article belongs to the Special Issue Fixed Point Theory and Its Related Topics II)
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24 pages, 411 KiB  
Article
Inertial Subgradient Extragradient Methods for Solving Variational Inequality Problems and Fixed Point Problems
Axioms 2020, 9(2), 51; https://doi.org/10.3390/axioms9020051 - 11 May 2020
Cited by 3 | Viewed by 2374
Abstract
We propose two new iterative algorithms for solving K-pseudomonotone variational inequality problems in the framework of real Hilbert spaces. These newly proposed methods are obtained by combining the viscosity approximation algorithm, the Picard Mann algorithm and the inertial subgradient extragradient method. We establish [...] Read more.
We propose two new iterative algorithms for solving K-pseudomonotone variational inequality problems in the framework of real Hilbert spaces. These newly proposed methods are obtained by combining the viscosity approximation algorithm, the Picard Mann algorithm and the inertial subgradient extragradient method. We establish some strong convergence theorems for our newly developed methods under certain restriction. Our results extend and improve several recently announced results. Furthermore, we give several numerical experiments to show that our proposed algorithms performs better in comparison with several existing methods. Full article
(This article belongs to the Special Issue Fixed Point Theory and Its Related Topics II)
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12 pages, 799 KiB  
Article
Fuzzy b-Metric Spaces: Fixed Point Results for ψ-Contraction Correspondences and Their Application
Axioms 2020, 9(2), 36; https://doi.org/10.3390/axioms9020036 - 31 Mar 2020
Cited by 16 | Viewed by 3169
Abstract
In this paper we introduce the concepts of ψ -contraction and monotone ψ -contraction correspondence in “fuzzy b -metric spaces” and obtain fixed point results for these contractive mappings. The obtained results generalize some existing ones in fuzzy metric spaces and “fuzzy [...] Read more.
In this paper we introduce the concepts of ψ -contraction and monotone ψ -contraction correspondence in “fuzzy b -metric spaces” and obtain fixed point results for these contractive mappings. The obtained results generalize some existing ones in fuzzy metric spaces and “fuzzy b -metric spaces”. Further we address an open problem in b -metric and “fuzzy b -metric spaces”. To elaborate the results obtained herein we provide an example that shows the usability of the obtained results. Full article
(This article belongs to the Special Issue Fixed Point Theory and Its Related Topics II)
6 pages, 224 KiB  
Article
On a Common Jungck Type Fixed Point Result in Extended Rectangular b-Metric Spaces
Axioms 2020, 9(1), 4; https://doi.org/10.3390/axioms9010004 - 27 Dec 2019
Cited by 5 | Viewed by 2200
Abstract
In this paper, we present a Jungck type common fixed point result in extended rectangular b-metric spaces. We also give some examples and a known common fixed point theorem in extended b-metric spaces. Full article
(This article belongs to the Special Issue Fixed Point Theory and Its Related Topics II)
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