Fixed Point Theory and Related Topics

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

Deadline for manuscript submissions: closed (31 January 2020) | Viewed by 44144

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Department of Mathematics, National Kaohsiung Normal University, Kaohsiung 802, Taiwan
Interests: fuzzy sets; optimization; nonlinear analysis
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Dear Colleagues,

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

Related Special Issues

Published Papers (16 papers)

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Research

8 pages, 225 KiB  
Article
Viscosity Approximation Methods for * −Nonexpansive Multi-Valued Mappings in Convex Metric Spaces
by Azadeh Ghanifard, Hashem Parvaneh Masiha, Manuel De La Sen and Maryam Ramezani
Axioms 2020, 9(1), 10; https://doi.org/10.3390/axioms9010010 - 17 Jan 2020
Cited by 2 | Viewed by 1892
Abstract
In this paper, we prove convergence theorems for viscosity approximation processes involving * −nonexpansive multi-valued mappings in complete convex metric spaces. We also consider finite and infinite families of such mappings and prove convergence of the proposed iteration schemes to common fixed points [...] Read more.
In this paper, we prove convergence theorems for viscosity approximation processes involving * −nonexpansive multi-valued mappings in complete convex metric spaces. We also consider finite and infinite families of such mappings and prove convergence of the proposed iteration schemes to common fixed points of them. Our results improve and extend some corresponding results. Full article
(This article belongs to the Special Issue Fixed Point Theory and Related Topics)
21 pages, 316 KiB  
Article
Measure of Weak Noncompactness and Fixed Point Theorems in Banach Algebras with Applications
by Mohamed Amine Farid, Karim Chaira, El Miloudi Marhrani and Mohamed Aamri
Axioms 2019, 8(4), 130; https://doi.org/10.3390/axioms8040130 - 14 Nov 2019
Cited by 2 | Viewed by 2631
Abstract
In this paper, we prove some fixed point theorems for the nonlinear operator A · B + C in Banach algebra. Our fixed point results are obtained under a weak topology and measure of weak noncompactness; and we give an example of the [...] Read more.
In this paper, we prove some fixed point theorems for the nonlinear operator A · B + C in Banach algebra. Our fixed point results are obtained under a weak topology and measure of weak noncompactness; and we give an example of the application of our results to a nonlinear integral equation in Banach algebra. Full article
(This article belongs to the Special Issue Fixed Point Theory and Related Topics)
13 pages, 261 KiB  
Article
Existence of Positive Solution for the Eighth-Order Boundary Value Problem Using Classical Version of Leray–Schauder Alternative Fixed Point Theorem
by Thenmozhi Shanmugam, Marudai Muthiah and Stojan Radenović
Axioms 2019, 8(4), 129; https://doi.org/10.3390/axioms8040129 - 14 Nov 2019
Cited by 8 | Viewed by 2213
Abstract
In this work, we investigate the existence of solutions for the particular type of the eighth-order boundary value problem. We prove our results using classical version of Leray–Schauder nonlinear alternative fixed point theorem. Also we produce a few examples to illustrate our results. [...] Read more.
In this work, we investigate the existence of solutions for the particular type of the eighth-order boundary value problem. We prove our results using classical version of Leray–Schauder nonlinear alternative fixed point theorem. Also we produce a few examples to illustrate our results. Full article
(This article belongs to the Special Issue Fixed Point Theory and Related Topics)
14 pages, 263 KiB  
Article
Informal Complete Metric Space and Fixed Point Theorems
by Hsien-Chung Wu
Axioms 2019, 8(4), 126; https://doi.org/10.3390/axioms8040126 - 7 Nov 2019
Viewed by 2112
Abstract
The concept of informal vector space is introduced in this paper. In informal vector space, the additive inverse element does not necessarily exist. The reason is that an element in informal vector space which subtracts itself cannot be a zero element. An informal [...] Read more.
The concept of informal vector space is introduced in this paper. In informal vector space, the additive inverse element does not necessarily exist. The reason is that an element in informal vector space which subtracts itself cannot be a zero element. An informal vector space can also be endowed with a metric to define a so-called informal metric space. The completeness of informal metric space can be defined according to the similar concept of a Cauchy sequence. A new concept of fixed point and the related results are studied in informal complete metric space. Full article
(This article belongs to the Special Issue Fixed Point Theory and Related Topics)
4 pages, 202 KiB  
Article
A New Approach to the Interpolative Contractions
by Yaé Ulrich Gaba and Erdal Karapınar
Axioms 2019, 8(4), 110; https://doi.org/10.3390/axioms8040110 - 10 Oct 2019
Cited by 47 | Viewed by 2709
Abstract
We propose a refinement in the interpolative approach in fixed-point theory. In particular, using this method, we prove the existence of fixed points and common fixed points for Kannan-type contractions and provide examples to support our results. Full article
(This article belongs to the Special Issue Fixed Point Theory and Related Topics)
18 pages, 345 KiB  
Article
C*-Algebra Valued Fuzzy Soft Metric Spaces and Results for Hybrid Pair of Mappings
by Daripally Ram Prasad, Gajula Naveen Venkata Kishore, Hüseyin Işık, Bagathi Srinuvasa Rao and Gorantla Adi Lakshmi
Axioms 2019, 8(3), 99; https://doi.org/10.3390/axioms8030099 - 16 Aug 2019
Cited by 6 | Viewed by 2744
Abstract
In this paper, we establish some results on coincidence point and common fixed point theorems for a hybrid pair of single valued and multivalued mappings in complete C * -algebra valued fuzzy soft metric spaces. In addition, we provided some coupled fixed point [...] Read more.
In this paper, we establish some results on coincidence point and common fixed point theorems for a hybrid pair of single valued and multivalued mappings in complete C * -algebra valued fuzzy soft metric spaces. In addition, we provided some coupled fixed point theorems. Finally, we have given examples which support our main results. Full article
(This article belongs to the Special Issue Fixed Point Theory and Related Topics)
8 pages, 240 KiB  
Article
Relatively Cyclic and Noncyclic P-Contractions in Locally K-Convex Space
by Edraoui Mohamed, Aamri Mohamed and Lazaiz Samih
Axioms 2019, 8(3), 96; https://doi.org/10.3390/axioms8030096 - 6 Aug 2019
Cited by 5 | Viewed by 2555
Abstract
Our main goal of this research is to present the theory of points for relatively cyclic and relatively relatively noncyclic p-contractions in complete locally K -convex spaces by providing basic conditions to ensure the existence and uniqueness of fixed points and best [...] Read more.
Our main goal of this research is to present the theory of points for relatively cyclic and relatively relatively noncyclic p-contractions in complete locally K -convex spaces by providing basic conditions to ensure the existence and uniqueness of fixed points and best proximity points of the relatively cyclic and relatively noncyclic p-contractions map in locally K -convex spaces. The result of this paper is the extension and generalization of the main results of Kirk and A. Abkar. Full article
(This article belongs to the Special Issue Fixed Point Theory and Related Topics)
13 pages, 256 KiB  
Article
On Fixed Point Results for Modified JS-Contractions with Applications
by Vahid Parvaneh, Nawab Hussain, Aiman Mukheimer and Hassen Aydi
Axioms 2019, 8(3), 84; https://doi.org/10.3390/axioms8030084 - 24 Jul 2019
Cited by 14 | Viewed by 2784
Abstract
In [Fixed Point Theory Appl., 2015 (2015):185], the authors introduced a new concept of modified contractive mappings, generalizing Ćirić, Chatterjea, Kannan, and Reich type contractions. They applied the condition ( θ 4 ) (see page 3, Section 2 of the above paper). Later, [...] Read more.
In [Fixed Point Theory Appl., 2015 (2015):185], the authors introduced a new concept of modified contractive mappings, generalizing Ćirić, Chatterjea, Kannan, and Reich type contractions. They applied the condition ( θ 4 ) (see page 3, Section 2 of the above paper). Later, in [Fixed Point Theory Appl., 2016 (2016):62], Jiang et al. claimed that the results in [Fixed Point Theory Appl., 2015 (2015):185] are not real generalizations. In this paper, by restricting the conditions of the control functions, we obtain a real generalization of the Banach contraction principle (BCP). At the end, we introduce a weakly JS-contractive condition generalizing the JS-contractive condition. Full article
(This article belongs to the Special Issue Fixed Point Theory and Related Topics)
12 pages, 287 KiB  
Article
Best Proximity Point Results for Geraghty Type Ƶ-Proximal Contractions with an Application
by Hüseyin Işık, Hassen Aydi, Nabil Mlaiki and Stojan Radenović
Axioms 2019, 8(3), 81; https://doi.org/10.3390/axioms8030081 - 18 Jul 2019
Cited by 6 | Viewed by 3091
Abstract
In this study, we establish the existence and uniqueness theorems of the best proximity points for Geraghty type Ƶ-proximal contractions defined on a complete metric space. The presented results improve and generalize some recent results in the literature. An example, as well as [...] Read more.
In this study, we establish the existence and uniqueness theorems of the best proximity points for Geraghty type Ƶ-proximal contractions defined on a complete metric space. The presented results improve and generalize some recent results in the literature. An example, as well as an application to a variational inequality problem are also given in order to illustrate the effectiveness of our generalizations. Full article
(This article belongs to the Special Issue Fixed Point Theory and Related Topics)
36 pages, 432 KiB  
Article
Recent Advances on the Results for Nonunique Fixed in Various Spaces
by Erdal Karapınar
Axioms 2019, 8(2), 72; https://doi.org/10.3390/axioms8020072 - 5 Jun 2019
Cited by 9 | Viewed by 2778
Abstract
In this short survey, we aim to underline the importance of the non-unique fixed point results in various abstract spaces. We recall a brief background on the topic and we combine, collect and unify several existing non-unique fixed points in the literature. Some [...] Read more.
In this short survey, we aim to underline the importance of the non-unique fixed point results in various abstract spaces. We recall a brief background on the topic and we combine, collect and unify several existing non-unique fixed points in the literature. Some interesting examples are considered. Full article
(This article belongs to the Special Issue Fixed Point Theory and Related Topics)
9 pages, 264 KiB  
Article
Fixed Point Theorems via α-ϱ-Fuzzy Contraction
by Badshah-e- Rome, Muhammad Sarwar and Poom Kumam
Axioms 2019, 8(2), 69; https://doi.org/10.3390/axioms8020069 - 31 May 2019
Cited by 1 | Viewed by 2837
Abstract
Some well known results from the existing literature are extended and generalized via new contractive type mappings in fuzzy metric spaces. A non trivial supporting example is also provided to demonstrate the validity of the obtained results. Full article
(This article belongs to the Special Issue Fixed Point Theory and Related Topics)
10 pages, 247 KiB  
Article
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 2325
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)
20 pages, 330 KiB  
Article
Relation Theoretic Common Fixed Point Results for Generalized Weak Nonlinear Contractions with an Application
by Atiya Perveen, Idrees A. Khan and Mohammad Imdad
Axioms 2019, 8(2), 49; https://doi.org/10.3390/axioms8020049 - 24 Apr 2019
Cited by 3 | Viewed by 2894
Abstract
In this paper, by introducing the concept of generalized Ćirić-type weak ( ϕ g , R ) -contraction, we prove some common fixed point results in partial metric spaces endowed with binary relation R . We also deduce some useful consequences showing the [...] Read more.
In this paper, by introducing the concept of generalized Ćirić-type weak ( ϕ g , R ) -contraction, we prove some common fixed point results in partial metric spaces endowed with binary relation R . We also deduce some useful consequences showing the usability of our results. Finally, we present an application to establish the solution of a system of integral equations. Full article
(This article belongs to the Special Issue Fixed Point Theory and Related Topics)
8 pages, 220 KiB  
Article
Applications of Square Roots of Diffeomorphisms
by Yoshihiro Sugimoto
Axioms 2019, 8(2), 43; https://doi.org/10.3390/axioms8020043 - 11 Apr 2019
Viewed by 1986
Abstract
In this paper, we prove that on any contact manifold ( M , ξ ) there exists an arbitrary C -small contactomorphism which does not admit a square root. In particular, there exists an arbitrary C -small contactomorphism which is not [...] Read more.
In this paper, we prove that on any contact manifold ( M , ξ ) there exists an arbitrary C -small contactomorphism which does not admit a square root. In particular, there exists an arbitrary C -small contactomorphism which is not “autonomous”. This paper is the first step to study the topology of C o n t 0 ( M , ξ ) Aut ( M , ξ ) . As an application, we also prove a similar result for the diffeomorphism group Diff ( M ) for any smooth manifold M. Full article
(This article belongs to the Special Issue Fixed Point Theory and Related Topics)
12 pages, 234 KiB  
Article
Fixed Point Theorems for Geraghty Contraction Type Mappings in b-Metric Spaces and Applications
by Hamid Faraji, Dragana Savić and Stojan Radenović
Axioms 2019, 8(1), 34; https://doi.org/10.3390/axioms8010034 - 14 Mar 2019
Cited by 35 | Viewed by 3870
Abstract
In this paper, some new results are given on fixed and common fixed points of Geraghty type contractive mappings defined in b-complete b-metric spaces. Moreover, two examples are represented to show the compatibility of our results. Some applications for nonlinear integral [...] Read more.
In this paper, some new results are given on fixed and common fixed points of Geraghty type contractive mappings defined in b-complete b-metric spaces. Moreover, two examples are represented to show the compatibility of our results. Some applications for nonlinear integral equations are also given. Full article
(This article belongs to the Special Issue Fixed Point Theory and Related Topics)
13 pages, 652 KiB  
Article
Best Proximity Point Theorems on Rectangular Metric Spaces Endowed with a Graph
by Nizar Souayah, Hassen Aydi, Thabet Abdeljawad and Nabil Mlaiki
Axioms 2019, 8(1), 17; https://doi.org/10.3390/axioms8010017 - 1 Feb 2019
Cited by 16 | Viewed by 3568
Abstract
In this paper, we ensure the existence and uniqueness of a best proximity point in rectangular metric spaces endowed with a graph structure. Full article
(This article belongs to the Special Issue Fixed Point Theory and Related Topics)
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