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Axioms
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Research on Fixed Point Theory and Application
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Research on Fixed Point Theory and Application

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

Deadline for manuscript submissions: closed (25 June 2024) | Viewed by 18887

Special Issue Editors

Prof. Dr. Shuechin Huang

E-Mail Website
Guest Editor
Department of Applied Mathematics, National Dong Hwa University, Hualien 97401, Taiwan
Interests: complex analysis; fixed point theory
Prof. Dr. Yasunori Kimura

E-Mail Website
Guest Editor
Department of Information Science, Toho University, Miyama, Funabashi 274-8510, Chiba, Japan
Interests: set-valued analysis; nonlinear functional analysis

Special Issue Information

Dear Colleagues,

Fixed point theory is one of the most powerful techniques of modern mathematics, especially in pure and applied analysis, topology, and the lattice, set, and category theories. The fixed point method was first introduced by Poincare in 1866 for the study of differential equations of celestial mechanics. Among the most original and far-reaching results in fixed point theory, the Banach and Brouwer theorems are two classical fixed point theorems that illustrate the difference between the two main branches of the theory: the metric fixed point and topological fixed point theories. For more than a century, fixed point theory has played an important role in nonlinear analysis and has been used an enormous number of applications in various areas, such as control theory, optimization, game theory, and economics.

The object of this Special Issue is to provide a platform for researchers to share and discuss the recent advancements and challenges in fixed point theory. We aim to collate recent, high-quality works in this area. We solicit contributions to fixed point theorems, possibly accompanied by concrete examples, that apply original and novel ideas to recent developments in theory and avoid any trivial extensions to already solidified results.

Potential topics include, but are not limited to, the following:

  • Fixed point theory in various abstract spaces;
  • Best proximity point theory in various abstract spaces;
  • Existence of solutions of differential and integral equations via fixed point results;
  • Equilibrium problems;
  • Variational inequality problems;
  • Saddle point problems;
  • Minimax problems;
  • Numerical methods for obtaining the approximated fixed points.

Prof. Dr. Shuechin Huang
Prof. Dr. Yasunori Kimura
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. Axioms 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.

Keywords

  • fixed point
  • saddle point
  • best proximity point
  • equilibrium
  • Hadamard manifold
 

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Published Papers (15 papers)

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Research

15 pages, 286 KiB  
Article
Integral Equations: New Solutions via Generalized Best Proximity Methods
by Amer Hassan Albargi and Jamshaid Ahmad
Axioms 2024, 13(7), 467; https://doi.org/10.3390/axioms13070467 - 11 Jul 2024
Cited by 1 | Viewed by 627
Abstract
This paper introduces the concept of proximal (α,F)-contractions in F-metric spaces. We establish novel results concerning the existence and uniqueness of best proximity points for such mappings. The validity of our findings is corroborated through a non-trivial [...] Read more.
This paper introduces the concept of proximal (α,F)-contractions in F-metric spaces. We establish novel results concerning the existence and uniqueness of best proximity points for such mappings. The validity of our findings is corroborated through a non-trivial example. Furthermore, we demonstrate the applicability of these results by proving the existence of solutions for Volterra integral equations related to population growth models. This approach not only extends best proximity theory, but also paves the way for further research in applied mathematics and beyond. Full article
(This article belongs to the Special Issue Research on Fixed Point Theory and Application)
12 pages, 258 KiB  
Article
On a Version of Dontchev and Hager’s Inverse Mapping Theorem
by Thanaa A. Alarfaj and Saud M. Alsulami
Axioms 2024, 13(7), 445; https://doi.org/10.3390/axioms13070445 - 30 Jun 2024
Viewed by 881
Abstract
By revisiting an open question raised by Kirk and Shahzad, we are able to prove a generalized version of Nadler’s fixed-point theorem in the context of strong b-metric space. Such a result leads us to prove a new version of Dontchev and [...] Read more.
By revisiting an open question raised by Kirk and Shahzad, we are able to prove a generalized version of Nadler’s fixed-point theorem in the context of strong b-metric space. Such a result leads us to prove a new version of Dontchev and Hager’s inverse mapping theorem. Some examples are provided to illustrate the results. Full article
(This article belongs to the Special Issue Research on Fixed Point Theory and Application)
15 pages, 319 KiB  
Article
Recent Advances in Proximity Point Theory Applied to Fractional Differential Equations
by Nabil Mlaiki, Dur-e-Shehwar Sagheer, Sana Noreen, Samina Batul and Ahmad Aloqaily
Axioms 2024, 13(6), 395; https://doi.org/10.3390/axioms13060395 - 13 Jun 2024
Viewed by 728
Abstract
This article introduces the concept of generalized (ffF,b,ϕ˘) contraction in the context of b-metric spaces by utilizing the idea of F contraction introduced by Dariusz Wardowski. The main findings of the research focus on [...] Read more.
This article introduces the concept of generalized (ffF,b,ϕ˘) contraction in the context of b-metric spaces by utilizing the idea of F contraction introduced by Dariusz Wardowski. The main findings of the research focus on the existence of best proximity points for multi-valued (ffF,b,ϕ˘) contractions in partially ordered b-metric spaces. The article provides examples to illustrate the main results and demonstrates the existence of solutions to a second-order differential equation and a fractional differential equation using the established theorems. Additionally, several corollaries are presented to show that the results generalize many existing fixed-point and best proximity point theorems. Full article
(This article belongs to the Special Issue Research on Fixed Point Theory and Application)
22 pages, 352 KiB  
Article
On Approximate Variational Inequalities and Bilevel Programming Problems
by Balendu Bhooshan Upadhyay, Ioan Stancu-Minasian, Subham Poddar and Priyanka Mishra
Axioms 2024, 13(6), 371; https://doi.org/10.3390/axioms13060371 - 30 May 2024
Viewed by 886
Abstract
In this paper, we investigate a class of bilevel programming problems (BLPP) in the framework of Euclidean space. We derive relationships among the solutions of approximate Minty-type variational inequalities (AMTVI), approximate Stampacchia-type variational inequalities (ASTVI), and local ϵ-quasi solutions of the BLPP, [...] Read more.
In this paper, we investigate a class of bilevel programming problems (BLPP) in the framework of Euclidean space. We derive relationships among the solutions of approximate Minty-type variational inequalities (AMTVI), approximate Stampacchia-type variational inequalities (ASTVI), and local ϵ-quasi solutions of the BLPP, under generalized approximate convexity assumptions, via limiting subdifferentials. Moreover, by employing the generalized Knaster–Kuratowski–Mazurkiewicz (KKM)-Fan’s lemma, we derive some existence results for the solutions of AMTVI and ASTVI. We have furnished suitable, non-trivial, illustrative examples to demonstrate the importance of the established results. To the best of our knowledge, there is no research paper available in the literature that explores relationships between the approximate variational inequalities and BLPP under the assumptions of generalized approximate convexity by employing the powerful tool of limiting subdifferentials. Full article
(This article belongs to the Special Issue Research on Fixed Point Theory and Application)
15 pages, 267 KiB  
Article
Quasi-Contraction Maps in Subordinate Semimetric Spaces
by Areej Alharbi, Hamed Alsulami and Maha Noorwali
Axioms 2024, 13(5), 318; https://doi.org/10.3390/axioms13050318 - 10 May 2024
Viewed by 765
Abstract
Throughout this study, we discuss the subordinate Pompeiu–Hausdorff metric (SPHM) in subordinate semimetric spaces. Moreover, we present a well-behaved quasi-contraction (WBQC) to solve quasi-contraction (QC) problems in subordinate semimetric spaces under some local constraints. Furthermore, we provide examples to support our conclusion. Full article
(This article belongs to the Special Issue Research on Fixed Point Theory and Application)
12 pages, 369 KiB  
Article
Asymptotic Behavior of Some Differential Inequalities with Mixed Delays and Their Applications
by Axiu Shu, Xiaoliang Li and Bo Du
Axioms 2024, 13(5), 302; https://doi.org/10.3390/axioms13050302 - 2 May 2024
Viewed by 982
Abstract
In this paper, we focus on the asymptotic stability of the trajectories governed by the differential inequalities with mixed delays using the fixed-point theorem. It is interesting that the Halanay inequality is a special case of the differential inequality studied in this paper. [...] Read more.
In this paper, we focus on the asymptotic stability of the trajectories governed by the differential inequalities with mixed delays using the fixed-point theorem. It is interesting that the Halanay inequality is a special case of the differential inequality studied in this paper. Our results generalize and improve the existing results on Halanay inequality. Finally, three numerical examples are utilized to illustrate the effectiveness of the obtained results. Full article
(This article belongs to the Special Issue Research on Fixed Point Theory and Application)
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15 pages, 400 KiB  
Article
Iterative Stability Analysis for Generalized α-Nonexpensive Mappings with Fixed Points
by Maryam Iqbal, Amjad Ali, Hamid Al Sulami and Aftab Hussain
Axioms 2024, 13(3), 156; https://doi.org/10.3390/axioms13030156 - 27 Feb 2024
Cited by 2 | Viewed by 1390
Abstract
This article introduces a novel iterative process, denoted as F★, designed for the class of generalized α-Nonexpensive mappings. The study establishes strong and weak convergence theorems within the context of Banach spaces, supported by carefully chosen assumptions. The convergence results [...] Read more.
This article introduces a novel iterative process, denoted as F★, designed for the class of generalized α-Nonexpensive mappings. The study establishes strong and weak convergence theorems within the context of Banach spaces, supported by carefully chosen assumptions. The convergence results contribute to the theoretical foundation of iterative processes in functional analysis. The presented framework is applied to address nonlinear integral equations, showcasing the versatility and applicability of the proposed F* for the class of generalized iteration process. Additionally, the article includes numerical examples that not only validate the theoretical findings but also provide insights into the practical utility of the developed methodology. Full article
(This article belongs to the Special Issue Research on Fixed Point Theory and Application)
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13 pages, 211 KiB  
Article
Convergence Results for Contractive Type Set-Valued Mappings
by Alexander J. Zaslavski
Axioms 2024, 13(2), 112; https://doi.org/10.3390/axioms13020112 - 7 Feb 2024
Viewed by 1209
Abstract
In this work, we study an iterative process induced by a contractive type set-valued mapping in a complete metric space and show its convergence, taking into account computational errors. Full article
(This article belongs to the Special Issue Research on Fixed Point Theory and Application)
11 pages, 260 KiB  
Article
On Some Properties of Multi-Valued Feng–Liu-Type Operators in Metric Spaces
by Adrian Petruşel, Gabriela Petruşel and Lijun Zhu
Axioms 2024, 13(1), 24; https://doi.org/10.3390/axioms13010024 - 29 Dec 2023
Cited by 1 | Viewed by 1130
Abstract
In the context of a complete metric space, the most important fixed-point result for multi-valued operators was given in 1969. Many extensions of this fixed-point principle for multi-valued operators were proved by different authors. Based on some of the above-mentioned results, we introduce [...] Read more.
In the context of a complete metric space, the most important fixed-point result for multi-valued operators was given in 1969. Many extensions of this fixed-point principle for multi-valued operators were proved by different authors. Based on some of the above-mentioned results, we introduce the notion of the multi-valued Feng–Liu-type operator and we construct an abstract fixed-point theory for this general class of multi-valued operators. Our results extend and complement some theorems in metric fixed-point theory for multi-valued operators. An application to a Cauchy problem related to a first-order differential inclusion is also given. In this case, our theorem improves several previous theorems on this subject by relaxing the contraction-type condition (with respect to its second argument) on the multi-valued right-hand side. Full article
(This article belongs to the Special Issue Research on Fixed Point Theory and Application)
14 pages, 626 KiB  
Article
Fixed-Point Convergence of Multi-Valued Non-Expansive Mappings with Applications
by Akbar Azam, Maliha Rashid, Amna Kalsoom and Faryad Ali
Axioms 2023, 12(11), 1020; https://doi.org/10.3390/axioms12111020 - 29 Oct 2023
Viewed by 1363
Abstract
This paper is dedicated to the advancement of fixed-point results for multi-valued asymptotically non-expansive maps regarding convergence criteria in complete uniformly convex hyperbolic metric spaces that are endowed with a graph. The famous fixed-point theorems of Goebel and Kirk, Khamsi and Khan, along [...] Read more.
This paper is dedicated to the advancement of fixed-point results for multi-valued asymptotically non-expansive maps regarding convergence criteria in complete uniformly convex hyperbolic metric spaces that are endowed with a graph. The famous fixed-point theorems of Goebel and Kirk, Khamsi and Khan, along with other recent results in the literature can be obtained as corollaries of these main results. A nice graph and an interesting example are also provided in support of the hypothesis of the main results. Full article
(This article belongs to the Special Issue Research on Fixed Point Theory and Application)
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21 pages, 1237 KiB  
Article
Inertial Method for Solving Pseudomonotone Variational Inequality and Fixed Point Problems in Banach Spaces
by Rose Maluleka, Godwin Chidi Ugwunnadi and Maggie Aphane
Axioms 2023, 12(10), 960; https://doi.org/10.3390/axioms12100960 - 11 Oct 2023
Viewed by 1207
Abstract
In this paper, we introduce a new iterative method that combines the inertial subgradient extragradient method and the modified Mann method for solving the pseudomonotone variational inequality problem and the fixed point of quasi-Bregman nonexpansive mapping in p-uniformly convex and uniformly smooth [...] Read more.
In this paper, we introduce a new iterative method that combines the inertial subgradient extragradient method and the modified Mann method for solving the pseudomonotone variational inequality problem and the fixed point of quasi-Bregman nonexpansive mapping in p-uniformly convex and uniformly smooth real Banach spaces. Under some standard assumptions imposed on cost operators, we prove a strong convergence theorem for our proposed method. Finally, we perform numerical experiments to validate the efficiency of our proposed method. Full article
(This article belongs to the Special Issue Research on Fixed Point Theory and Application)
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16 pages, 324 KiB  
Article
Common Fixed Point Results on a Double-Controlled Metric Space for Generalized Rational-Type Contractions with Application
by Khaleel Ahmad, Ghulam Murtaza, Salha Alshaikey, Umar Ishtiaq and Ioannis K. Argyros
Axioms 2023, 12(10), 941; https://doi.org/10.3390/axioms12100941 - 30 Sep 2023
Viewed by 1097
Abstract
In this manuscript, we prove several common fixed point theorems for generalized rational-type contraction mappings under several conditions in the context of double-controlled metric spaces. Further, we utilize a double-controlled metric space equipped with a graph to prove rational-type common fixed point theorems. [...] Read more.
In this manuscript, we prove several common fixed point theorems for generalized rational-type contraction mappings under several conditions in the context of double-controlled metric spaces. Further, we utilize a double-controlled metric space equipped with a graph to prove rational-type common fixed point theorems. Furthermore, we establish non-trivial examples to show the validity of the main results. These results improve and generalize already known results. At the end, we solve the Fredholm-type integral equation by utilizing the main results. Full article
(This article belongs to the Special Issue Research on Fixed Point Theory and Application)
27 pages, 401 KiB  
Article
Modified Inertial-Type Subgradient Extragradient Methods for Variational Inequalities and Fixed Points of Finite Bregman Relatively Nonexpansive and Demicontractive Mappings
by Cong-Shan Wang, Lu-Chuan Ceng, Bing Li, Sheng-Long Cao, Hui-Ying Hu and Yun-Shui Liang
Axioms 2023, 12(9), 832; https://doi.org/10.3390/axioms12090832 - 28 Aug 2023
Cited by 3 | Viewed by 998
Abstract
In this paper, we design two inertial-type subgradient extragradient algorithms with the linear-search process for resolving the two pseudomonotone variational inequality problems (VIPs) of and the common fixed point problem (CFPP) of finite Bregman relatively nonexpansive operators and Bregman relatively demicontractive operators in [...] Read more.
In this paper, we design two inertial-type subgradient extragradient algorithms with the linear-search process for resolving the two pseudomonotone variational inequality problems (VIPs) of and the common fixed point problem (CFPP) of finite Bregman relatively nonexpansive operators and Bregman relatively demicontractive operators in Banach spaces of both p-uniform convexity and uniform smoothness, which are more general than Hilbert ones. By the aid of suitable restrictions, it is shown that the sequences fabricated by the suggested schemes converge weakly and strongly to a solution of a pair of VIPs with a CFPP constraint, respectively. Additionally, the illustrative instance is furnished to back up the practicability and implementability of the suggested methods. This paper reveals the competitive advantage of the proposed algorithms over the existing algorithms; that is, the existing hybrid projection method for a single VIP with an FPP constraint is extended to develop the modified inertial-type subgradient extragradient method for a pair of VIPs with an CFPP constraint. Full article
(This article belongs to the Special Issue Research on Fixed Point Theory and Application)
18 pages, 346 KiB  
Article
Positive Solutions for a System of Hadamard Fractional Boundary Value Problems on an Infinite Interval
by Alexandru Tudorache and Rodica Luca
Axioms 2023, 12(8), 793; https://doi.org/10.3390/axioms12080793 - 16 Aug 2023
Cited by 5 | Viewed by 1445
Abstract
Our investigation is devoted to examining the existence, uniqueness, and multiplicity of positive solutions for a system of Hadamard fractional differential equations. This system is defined on an infinite interval and is subject to coupled nonlocal boundary conditions. These boundary conditions encompass both [...] Read more.
Our investigation is devoted to examining the existence, uniqueness, and multiplicity of positive solutions for a system of Hadamard fractional differential equations. This system is defined on an infinite interval and is subject to coupled nonlocal boundary conditions. These boundary conditions encompass both Hadamard fractional derivatives and Riemann–Stieltjes integrals, and the nonlinearities within the system are non-negative functions that may not be bounded. To establish the main results, we rely on the utilization of mathematical theorems such as the Schauder fixed-point theorem, the Banach contraction mapping principle, and the Avery–Peterson fixed-point theorem. Full article
(This article belongs to the Special Issue Research on Fixed Point Theory and Application)
11 pages, 277 KiB  
Article
Banach Contraction Principle-Type Results for Some Enriched Mappings in Modular Function Spaces
by Safeer Hussain Khan, Abdullah Eqal Al-Mazrooei and Abdul Latif
Axioms 2023, 12(6), 549; https://doi.org/10.3390/axioms12060549 - 2 Jun 2023
Cited by 2 | Viewed by 1741
Abstract
The idea of enriched mappings in normed spaces is relatively a newer idea. In this paper, we initiate the study of enriched mappings in modular function spaces. We first introduce the concepts of enriched ρ-contractions and enriched ρ-Kannan mappings in modular [...] Read more.
The idea of enriched mappings in normed spaces is relatively a newer idea. In this paper, we initiate the study of enriched mappings in modular function spaces. We first introduce the concepts of enriched ρ-contractions and enriched ρ-Kannan mappings in modular function spaces. We then establish some Banach Contraction Principle type theorems for the existence of fixed points of such mappings in this setting. Our results for enriched ρ-contractions are generalizations of the corresponding results from Banach spaces to modular function spaces and those from contractions to enriched ρ-contractions. We make a first ever attempt to prove existence results for enriched ρ-Kannan mappings and deduce the result for ρ-Kannan mappings. Note that even ρ-Kannan mappings in modular function spaces have not been considered yet. We validate our main results by examples. Full article
(This article belongs to the Special Issue Research on Fixed Point Theory and Application)
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