Fixed Point Theory and Its Related Topics IV

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

Deadline for manuscript submissions: 31 December 2024 | Viewed by 13999

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

Special Issue Information

Dear Colleagues,

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

The 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 and articles that unify the different types of fixed-point theorems. 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

Published Papers (15 papers)

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Research

19 pages, 325 KiB  
Article
Existence and Uniqueness Results for a Pantograph Boundary Value Problem Involving a Variable-Order Hadamard Fractional Derivative
by Kadda Maazouz, Moussa Daif Allah Zaak and Rosana Rodríguez-López
Axioms 2023, 12(11), 1028; https://doi.org/10.3390/axioms12111028 - 01 Nov 2023
Viewed by 839
Abstract
This paper discusses the problem of the existence and uniqueness of solutions to the boundary value problem for the nonlinear fractional-order pantograph equation, using the fractional derivative of variable order of Hadamard type. The main results are proved through the application of fractional [...] Read more.
This paper discusses the problem of the existence and uniqueness of solutions to the boundary value problem for the nonlinear fractional-order pantograph equation, using the fractional derivative of variable order of Hadamard type. The main results are proved through the application of fractional calculus and Krasnoselskii’s fixed-point theorem. Moreover, the Ulam–Hyers–Rassias stability of the nonlinear fractional pantograph equation is analyzed. To conclude this paper, we provide an example illustrating our findings and approach. Full article
(This article belongs to the Special Issue Fixed Point Theory and Its Related Topics IV)
25 pages, 365 KiB  
Article
F-Contractions Endowed with Mann’s Iterative Scheme in Convex Gb-Metric Spaces
by Amna Naz, Samina Batul, Dur-e-Shehwar Sagheer, Irshad Ayoob and Nabil Mlaiki
Axioms 2023, 12(10), 937; https://doi.org/10.3390/axioms12100937 - 29 Sep 2023
Cited by 1 | Viewed by 1009
Abstract
Recently, Ji et al. established certain fixed-point results using Mann’s iterative scheme tailored to Gb-metric spaces. Stimulated by the notion of the F-contraction introduced by Wardoski, the contraction condition of Ji et al. was generalized in this research. Several fixed-point [...] Read more.
Recently, Ji et al. established certain fixed-point results using Mann’s iterative scheme tailored to Gb-metric spaces. Stimulated by the notion of the F-contraction introduced by Wardoski, the contraction condition of Ji et al. was generalized in this research. Several fixed-point results with Mann’s iterative scheme endowed with F-contractions in Gb-metric spaces were proven. One non-trivial example was elaborated to support the main theorem. Moreover, for application purposes, the existence of the solution to an integral equation is provided by using the axioms of the proven result. The obtained results are generalizations of several existing results in the literature. Furthermore, the results of Ji. et al. are the special case of theorems provided in the present research. Full article
(This article belongs to the Special Issue Fixed Point Theory and Its Related Topics IV)
12 pages, 295 KiB  
Article
Almost Boyd-Wong Type Contractions under Binary Relations with Applications to Boundary Value Problems
by Amal F. Alharbi and Faizan Ahmad Khan
Axioms 2023, 12(9), 896; https://doi.org/10.3390/axioms12090896 - 20 Sep 2023
Cited by 2 | Viewed by 605
Abstract
This article is devoted to investigating the fixed point theorems for a new contracitivity contraction, which combines the idea involved in Boyd-Wong contractions, strict almost contractions and relational contractions. Our results improve and expand existing fixed point theorems of literature. In process, we [...] Read more.
This article is devoted to investigating the fixed point theorems for a new contracitivity contraction, which combines the idea involved in Boyd-Wong contractions, strict almost contractions and relational contractions. Our results improve and expand existing fixed point theorems of literature. In process, we deduce a metrical fixed point theorem for strict almost Boyd-Wong contractions. To demonstrate the credibility of our results, we present a number of a few examples. Applying our findings, we find a unique solution to a particular periodic boundary value problem. Full article
(This article belongs to the Special Issue Fixed Point Theory and Its Related Topics IV)
18 pages, 325 KiB  
Article
Almost Ćirić Type Contractions and Their Applications in Complex Valued b-Metric Spaces
by Muhammad Sarwar, Syed Khayyam Shah, Zoran D. Mitrović, Aiman Mukheimer and Nabil Mlaiki
Axioms 2023, 12(8), 794; https://doi.org/10.3390/axioms12080794 - 16 Aug 2023
Cited by 1 | Viewed by 666
Abstract
In this article, we present the use of a unique and common fixed point for a pair of mappings that satisfy certain rational-type inequalities in complex-valued b-metric spaces. We also provide applications related to authenticity concerns in integral equations. Our results combine well-known [...] Read more.
In this article, we present the use of a unique and common fixed point for a pair of mappings that satisfy certain rational-type inequalities in complex-valued b-metric spaces. We also provide applications related to authenticity concerns in integral equations. Our results combine well-known contractions, such as the Ćirić contraction and almost contractions. Full article
(This article belongs to the Special Issue Fixed Point Theory and Its Related Topics IV)
19 pages, 330 KiB  
Article
A Study on Existence and Controllability of Conformable Impulsive Equations
by Nazim I. Mahmudov and Gülbahar Akgün
Axioms 2023, 12(8), 787; https://doi.org/10.3390/axioms12080787 - 14 Aug 2023
Viewed by 715
Abstract
We study the existence/uniqueness of conformable fractional type impulsive nonlinear systems as well as the controllability of linear/semilinear conformable fractional type impulsive controlled systems. Using the conformable fractional derivative approach, we introduce the conformable controllability operator and the conformable controllability Gramian matrix in [...] Read more.
We study the existence/uniqueness of conformable fractional type impulsive nonlinear systems as well as the controllability of linear/semilinear conformable fractional type impulsive controlled systems. Using the conformable fractional derivative approach, we introduce the conformable controllability operator and the conformable controllability Gramian matrix in order to obtain the necessary and sufficient conditions for the complete controllability of linear impulsive conformable systems. We present a set of sufficient conditions for the controllability of the conformable semilinear impulsive systems. Full article
(This article belongs to the Special Issue Fixed Point Theory and Its Related Topics IV)
30 pages, 355 KiB  
Article
Solving Some Integral and Fractional Differential Equations via Neutrosophic Pentagonal Metric Space
by Gunaseelan Mani, Poornavel Subbarayan, Zoran D. Mitrović, Ahmad Aloqaily and Nabil Mlaiki
Axioms 2023, 12(8), 758; https://doi.org/10.3390/axioms12080758 - 01 Aug 2023
Viewed by 562
Abstract
In this paper, we first introduce the notion of neutrosophic pentagonal metric space. We prove several interesting results for some classes contraction mappings and prove some fixed point theorems in neutrosophic pentagonal metric space. Finally, we prove the uniqueness and existence of the [...] Read more.
In this paper, we first introduce the notion of neutrosophic pentagonal metric space. We prove several interesting results for some classes contraction mappings and prove some fixed point theorems in neutrosophic pentagonal metric space. Finally, we prove the uniqueness and existence of the integral equation and fractional differential equation to support our main result. Full article
(This article belongs to the Special Issue Fixed Point Theory and Its Related Topics IV)
21 pages, 619 KiB  
Article
Stability Results and Reckoning Fixed Point Approaches by a Faster Iterative Method with an Application
by Hasanen A. Hammad and Doha A. Kattan
Axioms 2023, 12(7), 715; https://doi.org/10.3390/axioms12070715 - 23 Jul 2023
Viewed by 771
Abstract
In this manuscript, we investigate some convergence and stability results for reckoning fixed points using a faster iterative scheme in a Banach space. Also, weak and strong convergence are discussed for close contraction mappings in a Banach space and for Suzuki generalized nonexpansive [...] Read more.
In this manuscript, we investigate some convergence and stability results for reckoning fixed points using a faster iterative scheme in a Banach space. Also, weak and strong convergence are discussed for close contraction mappings in a Banach space and for Suzuki generalized nonexpansive mapping in a uniformly convex Banach space. Our method opens the door to many expansions in the problems of monotone variational inequalities, image restoration, convex optimization, and split convex feasibility. Moreover, some experimental examples were conducted to gauge the usefulness and efficiency of the technique compared with the iterative methods in the literature. Finally, the proposed approach is applied to solve the nonlinear Volterra integral equation with a delay. Full article
(This article belongs to the Special Issue Fixed Point Theory and Its Related Topics IV)
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20 pages, 318 KiB  
Article
Solving Integral Equations via Fixed Point Results Involving Rational-Type Inequalities
by Syed Shah Khayyam, Muhammad Sarwar, Asad Khan, Nabil Mlaiki and Fatima M. Azmi
Axioms 2023, 12(7), 685; https://doi.org/10.3390/axioms12070685 - 12 Jul 2023
Viewed by 879
Abstract
In this study, we establish unique and common fixed point results in the context of a complete complex-valued b-metric space using rational-type inequalities. The presented work generalizes some well-known results from the existing literature. Furthermore, to ensure the validity of the findings, we [...] Read more.
In this study, we establish unique and common fixed point results in the context of a complete complex-valued b-metric space using rational-type inequalities. The presented work generalizes some well-known results from the existing literature. Furthermore, to ensure the validity of the findings, we have included some examples and a section on the existence of solutions for the systems of Volterra–Hammerstein integral equations and Urysohn integral equations, respectively. Full article
(This article belongs to the Special Issue Fixed Point Theory and Its Related Topics IV)
15 pages, 358 KiB  
Article
General New Results on (ϕ,F)Contractions in bMetric-like-Spaces
by Kastriot Zoto, Milanka Gardašević-Filipović, Ilir Vardhami, Zoran Mitrović and Stojan Radenović
Axioms 2023, 12(7), 672; https://doi.org/10.3390/axioms12070672 - 07 Jul 2023
Viewed by 697
Abstract
Thispaper recognizes a general approach related to recent fixed point results about the classes of interpolative and hybrid contractions in metric space and general metric spaces. Considering auxiliary functions, so called Wardowski functions, and a rich set of implicit relations, we introduce types [...] Read more.
Thispaper recognizes a general approach related to recent fixed point results about the classes of interpolative and hybrid contractions in metric space and general metric spaces. Considering auxiliary functions, so called Wardowski functions, and a rich set of implicit relations, we introduce types of (αvq,ϕ,F)contractions and rorder hybrid (αvq,ϕ,F)contractions in the setting of bmetric-like spaces. They generate and simplify many forms of contractions widely used in the literature. The resulting theorems significantly extend, generalize, and unify an excellent work on fixed point theory. Full article
(This article belongs to the Special Issue Fixed Point Theory and Its Related Topics IV)
23 pages, 334 KiB  
Article
Existence, Approximation and Stability of Fixed Point for Cirić Contraction in Convex b-Metric Spaces
by Gazal, Savita Rathee, Mahima, Anshuka Kadyan, Minakshi and Anil Kumar
Axioms 2023, 12(7), 646; https://doi.org/10.3390/axioms12070646 - 28 Jun 2023
Viewed by 723
Abstract
We establish a new fixed point theorem in the setting of convex b-metric spaces that ensures the existence of fixed point for Cirić contraction with the assumption k<1s2. Also, the fixed point is approximated by Krasnoselskij iterative [...] Read more.
We establish a new fixed point theorem in the setting of convex b-metric spaces that ensures the existence of fixed point for Cirić contraction with the assumption k<1s2. Also, the fixed point is approximated by Krasnoselskij iterative procedure. Moreover, we discuss the stability of fixed point for the aforesaid contraction. As a consequence, we develop a common fixed point and coincidence point result. Finally, we provide a number of examples to illustrate the findings presented here and incorporate these findings to solve an initial value problem. Full article
(This article belongs to the Special Issue Fixed Point Theory and Its Related Topics IV)
24 pages, 509 KiB  
Article
Best Proximity Points for p–Cyclic Infimum Summing Contractions
by Miroslav Hristov, Atanas Ilchev, Petar Kopanov, Vasil Zhelinski and Boyan Zlatanov
Axioms 2023, 12(7), 628; https://doi.org/10.3390/axioms12070628 - 25 Jun 2023
Viewed by 663
Abstract
We investigate fixed points for p cyclic maps by introducing a new notion of p–cyclic infimum summing maps and a generalized best proximity point for p–cyclic maps. The idea generalizes some results about best proximity points in order to widen the [...] Read more.
We investigate fixed points for p cyclic maps by introducing a new notion of p–cyclic infimum summing maps and a generalized best proximity point for p–cyclic maps. The idea generalizes some results about best proximity points in order to widen the class of sets and maps for which we can ensure the existence and uniqueness of best proximity points. The replacement of the classical notions of best proximity points and distance between the consecutive set arises from the well-known group traveling salesman problem and presents a different approach to solving it. We illustrate the new result with a map that does not satisfy the known results about best proximity maps for p–cyclic maps. Full article
(This article belongs to the Special Issue Fixed Point Theory and Its Related Topics IV)
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12 pages, 270 KiB  
Article
Well-Posedness Scheme for Coupled Fixed-Point Problems Using Generalized Contractions
by Dur-e-Shehwar Sagheer, Isma Urooj, Samina Batul, Ahmad Aloqaily and Nabil Mlaiki
Axioms 2023, 12(6), 523; https://doi.org/10.3390/axioms12060523 - 26 May 2023
Viewed by 724
Abstract
In this study, we present a more general class of rational-type contractions in the domain of Hilbert spaces, along with a novel coupled implicit relation. We develop several intriguing results and consequences for the existence of unique coupled fixed points. Further, we investigate [...] Read more.
In this study, we present a more general class of rational-type contractions in the domain of Hilbert spaces, along with a novel coupled implicit relation. We develop several intriguing results and consequences for the existence of unique coupled fixed points. Further, we investigate a necessary condition that guarantees the well-posedness of a coupled fixed-point problem of self-mappings in Hilbert spaces. Some new observations proposed in this research broaden and extend previously published results in the literature. Full article
(This article belongs to the Special Issue Fixed Point Theory and Its Related Topics IV)
14 pages, 313 KiB  
Article
On the Fixed Circle Problem on Metric Spaces and Related Results
by Nabil Mlaiki, Nihal Özgür, Nihal Taş and Dania Santina
Axioms 2023, 12(4), 401; https://doi.org/10.3390/axioms12040401 - 20 Apr 2023
Cited by 2 | Viewed by 1119
Abstract
The fixed-circle issue is a geometric technique that is connected to the study of geometric characteristics of certain points, and that are fixed by the self-mapping of either the metric space or of the generalized space. The fixed-disc problem is a natural resultant [...] Read more.
The fixed-circle issue is a geometric technique that is connected to the study of geometric characteristics of certain points, and that are fixed by the self-mapping of either the metric space or of the generalized space. The fixed-disc problem is a natural resultant that arises as a direct outcome of this problem. In this study, our goal is to examine new classes of self-mappings that meet a new particular sort of contraction in a metric space. The common geometrical characteristic of the set of fixed points of any element of these classes is that a circle or even a disc, that is either termed the fixed circle or even the fixed disc of the appropriate self-map, is included within that set. In order to accomplish this, we establish two new classifications of contraction mapping: Fc-contractive mapping and Fc-expanding mapping. In the investigation of neural networks, activation functions with either fixed circles (or even fixed discs) are observed frequently. This demonstrates how successful our results with the fixed-circle (respectively, the fixed-disc) model were. Full article
(This article belongs to the Special Issue Fixed Point Theory and Its Related Topics IV)
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16 pages, 365 KiB  
Article
On the Existence of Solutions to a Boundary Value Problem via New Weakly Contractive Operator
by Rhoda Chiroma, Trad Alotaibi, Mohammed Shehu Shagari, Awad A. Bakery, OM Kalthum S. K. Mohammed and Arafa O. Mustafa
Axioms 2023, 12(4), 397; https://doi.org/10.3390/axioms12040397 - 20 Apr 2023
Cited by 1 | Viewed by 965
Abstract
In this paper, the notion of generalized quasi-weakly contractive operators in metric-like spaces is introduced, and new conditions for the existence of fixed points for such mappings are investigated. A non-trivial example which highlights the novelty of our principal idea is constructed. It [...] Read more.
In this paper, the notion of generalized quasi-weakly contractive operators in metric-like spaces is introduced, and new conditions for the existence of fixed points for such mappings are investigated. A non-trivial example which highlights the novelty of our principal idea is constructed. It is observed comparatively that the proposed concepts herein subsume some important results in the corresponding literature. As an application, one of our obtained findings is utilized to setup novel criteria for the existence of solutions to two-point boundary value problems of a second order differential equation. To attract new researchers in the directions examined in this article, a significant number of corollaries are pointed out and discussed. Full article
(This article belongs to the Special Issue Fixed Point Theory and Its Related Topics IV)
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11 pages, 257 KiB  
Article
New Iterative Scheme Involving Self-Adaptive Method for Solving Mixed Variational Inequalities
by Aiman Mukheimer, Saleem Ullah, Muhammad Bux, Muhammad Shoaib Arif and Kamaleldin Abodayeh
Axioms 2023, 12(3), 310; https://doi.org/10.3390/axioms12030310 - 20 Mar 2023
Cited by 1 | Viewed by 893
Abstract
Variational inequalities (VI) problems have been generalized and expanded in various ways. The VI principle has become a remarkable study area combining pure and applied research. The study of variational inequality in mathematics is significantly aided by providing an important framework by fixed-point [...] Read more.
Variational inequalities (VI) problems have been generalized and expanded in various ways. The VI principle has become a remarkable study area combining pure and applied research. The study of variational inequality in mathematics is significantly aided by providing an important framework by fixed-point theory. The concept of fixed-point theory can be considered an inherent component of the VI. We consider a mixed variational inequality (MVI) a useful generalization of a classical variational inequality. The projection method is not applicable to solve MVI due to the involvement of the nonlinear term ϕ. MVI is equivalent to fixed-point problems and the resolvent equation techniques. This technique is commonly used in the research on the existence of a solution to the MVI. This paper uses a new self-adaptive method using step size to modify the fixed-point formulation for solving the MVI. We will also provide the convergence of the proposed scheme. Our output could be seen as a significant refinement of the previously known results for MVI. A numerical example is also provided for the implementation of the generated algorithm. Full article
(This article belongs to the Special Issue Fixed Point Theory and Its Related Topics IV)
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