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Fixed Point Theory and Its Applications Dedicated to the Memory of Professor William Arthur Kirk

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A special issue of Symmetry (ISSN 2073-8994). This special issue belongs to the section "Mathematics".

Deadline for manuscript submissions: closed (30 September 2024) | Viewed by 26078

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Dr. Alexander Zaslavski

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Department of Mathematics, The Technion—Israel Institute of Technology, Haifa 32000, Israel
Interests: variational and optimal control problems on unbounded domains; optimization theory and related topics; infinite products of operators and their applications
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Special Issue Information

Dear Colleagues,

This Special Issue on fixed point theory and its applications is dedicated to the memory of Professor William Arthur Kirk, who passed away on October 20, 2022.

Professor Kirk received his Bachelor’s degree from DePauw University in 1958 and his Ph.D. from the University of Missouri in 1962. From 1962 to 1967, he was an Assistant Professor at the University of California at Riverside. In 1967, he began to work as an Associate Professor at the University of Iowa, where he became a full professor in 1970.

Professor Kirk was an outstanding and internationally famous mathematician who made significant contributions to Nonlinear Functional Analysis, especially fixed-point theory. He is one of the founders of the modern theory of metric fixed-points, and his works deeply influenced the development of the field. He is known for Kirk’s Fixed-Point Theorem of 1964/1965 and for the Caristi-Kirk Fixed-Point Theorem of 1976. His achievements and leadership in the field were recognized by the title of Doctor Honoris Causa in 2004 from Maria Curie-Sklodowska University, Poland. Professor Kirk published three books and 177 journal articles. His book Topics in Metric Fixed Point Theory, written jointly with K. Goebel and published by Cambridge University Press in 1990, is now a classic in the area of fixed-point theory. Professor Kirk was the advisor of thirteen PhD students.

This Special Issue aims to seek out high-quality articles from researchers in the fixed point theory and other areas of nonlinear functional analysis close to the research activities of Professor Kirk.

Dr. Alexander Zaslavski
Guest Editor

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. Symmetry 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

symmetry analysis
symmetry/asymmetry phenomena
symmetry nonlinear system
complete metric space
contractive mapping
symmetry and fixed point
iterate
nonexpansive mapping
set-valued mapping

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Published Papers (17 papers)

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Editorial

Jump to: Research

6 pages, 228 KiB

Open AccessEditorial

Special Issue: Fixed-Point Theory and Its Applications, Dedicated to the Memory of Professor William Arthur Kirk

by Alexander J. Zaslavski

Symmetry 2024, 16(11), 1408; https://doi.org/10.3390/sym16111408 - 22 Oct 2024

Viewed by 1396

Abstract

Fixed-point theory is a rapidly growing area of research [...] Full article

(This article belongs to the Special Issue Fixed Point Theory and Its Applications Dedicated to the Memory of Professor William Arthur Kirk)

Research

Jump to: Editorial

20 pages, 631 KiB

Open AccessArticle

Enhanced Double Inertial Forward–Backward Splitting Algorithm for Variational Inclusion Problems: Applications in Mathematical Integrated Skill Prediction

by Nipa Jun-On and Watcharaporn Cholamjiak

Symmetry 2024, 16(8), 1091; https://doi.org/10.3390/sym16081091 - 22 Aug 2024

Cited by 3 | Viewed by 1568

Abstract

This paper introduces a new algorithm that combines the forward–backward splitting algorithms with a double inertial technique, utilizing the previous three iterations. The weak convergence theorem is established under certain mild conditions in a Hilbert space, including a relaxed inertial method in real numbers. An example of infinite dimension space is given with numerical results to support our proposed algorithm. The algorithm is applied to an asymmetrical educational dataset of students from 109 schools, utilizing asymmetric inputs as nine attributes to predict the output as students’ mathematical integrated skills. The algorithm’s performance is compared with other algorithms in the literature to demonstrate its effectiveness. The proposed algorithm demonstrates comparable precision, recall, accuracy, and F1 score but performs a relatively lower number of iterations. The contributions of each performance aspect to the mathematical integration skill of students are discussed to improve students’ mathematical learning. Full article

(This article belongs to the Special Issue Fixed Point Theory and Its Applications Dedicated to the Memory of Professor William Arthur Kirk)

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13 pages, 267 KiB

Open AccessArticle

New Fixed Point Theorems for Generalized Meir–Keeler Type Nonlinear Mappings with Applications to Fixed Point Theory

by Shin-Yi Huang and Wei-Shih Du

Symmetry 2024, 16(8), 1088; https://doi.org/10.3390/sym16081088 - 22 Aug 2024

Cited by 2 | Viewed by 1211

Abstract

In this paper, we investigate new fixed point theorems for generalized Meir–Keeler type nonlinear mappings satisfying the condition (DH). As applications, we obtain many new fixed point theorems which generalize and improve several results available in the corresponding literature. An example is [...] Read more.

(This article belongs to the Special Issue Fixed Point Theory and Its Applications Dedicated to the Memory of Professor William Arthur Kirk)

24 pages, 367 KiB

Open AccessArticle

Some Common Fixed Point Results of Tower Mappings in (Pseudo)modular Metric Spaces

by Daniel Francis, Godwin Amechi Okeke and Safeer Hussain Khan

Symmetry 2024, 16(7), 896; https://doi.org/10.3390/sym16070896 - 14 Jul 2024

Cited by 1 | Viewed by 937

Abstract

In this paper, we prove the existence and uniqueness of common fixed point of tower type contractive mappings in complete metric (pseudo)modular spaces involving the theoretic relation. However, the newly introduced contraction in this paper further characterize and includes in their full strength several existing results in metrical fixed point theory. Some nontrivial supportive examples were given to justify our result. Our results generalize, improve, and unify some existing results. Full article

(This article belongs to the Special Issue Fixed Point Theory and Its Applications Dedicated to the Memory of Professor William Arthur Kirk)

10 pages, 268 KiB

Open AccessArticle

Fixed Point of α-Modular Nonexpanive Mappings in Modular Vector Spaces 𝓁_p(·)

by Buthinah A. Bin Dehaish and Mohamed A. Khamsi

Symmetry 2024, 16(7), 799; https://doi.org/10.3390/sym16070799 - 25 Jun 2024

Cited by 2 | Viewed by 1141

Abstract

Let C denote a convex subset within the vector space

𝓁_{p (\cdot)}

, and let T represent a mapping from C onto itself. Assume

α = (α_{1}, \dots, α_{n})

is a multi-index in [...] Read more.

Let C denote a convex subset within the vector space

𝓁_{p (\cdot)}

, and let T represent a mapping from C onto itself. Assume

α = (α_{1}, \dots, α_{n})

is a multi-index in

{[0, 1]}^{n}

such that

\sum_{i = 1}^{n} α_{i} = 1

, where

α_{1} > 0

and

α_{n} > 0

. We define

T_{α} : C \to C

T_{α} = \sum_{i = 1}^{n} α_{i} T^{i}

, known as the mean average of the mapping T. While every fixed point of T remains fixed for

T_{α}

, the reverse is not always true. This paper examines necessary and sufficient conditions for the existence of fixed points for T, relating them to the existence of fixed points for

T_{α}

and the behavior of T-orbits of points in T’s domain. The primary approach involves a detailed analysis of recurrent sequences in

R

. Our focus then shifts to variable exponent modular vector spaces

𝓁_{p (\cdot)}

, where we explore the essential conditions that guarantee the existence of fixed points for these mappings. This investigation marks the first instance of such results in this framework. Full article

(This article belongs to the Special Issue Fixed Point Theory and Its Applications Dedicated to the Memory of Professor William Arthur Kirk)

18 pages, 432 KiB

Open AccessArticle

Fixed Point Method for Nonlinear Fractional Differential Equations with Integral Boundary Conditions on Tetramethyl-Butane Graph

by Juan J. Nieto, Ashish Yadav, Trilok Mathur and Shivi Agarwal

Symmetry 2024, 16(6), 756; https://doi.org/10.3390/sym16060756 - 17 Jun 2024

Cited by 3 | Viewed by 1318

Abstract

Until now, little investigation has been done to examine the existence and uniqueness of solutions for fractional differential equations on star graphs. In the published articles on the subject, the authors used a star graph with one junction node that has edges with the other nodes, although there are no edges between them. These graph structures do not cover more generic non-star graph structures; they are specific examples. The purpose of this study is to prove the existence and uniqueness of solutions to a new family of fractional boundary value problems on the tetramethylbutane graph that have more than one junction node after presenting a labeling mechanism for graph vertices. The chemical compound tetramethylbutane has a highly symmetrical structure, due to which it has a very high melting point and a short liquid range; in fact, it is the smallest saturated acyclic hydrocarbon that appears as a solid at a room temperature of 25 °C. With vertices designated by 0 or 1, we propose a fractional-order differential equation on each edge of tetramethylbutane graph. Employing the fixed-point theorems of Schaefer and Banach, we demonstrate the existence and uniqueness of solutions for the suggested fractional differential equation satisfying the integral boundary conditions. In addition, we examine the stability of the system. Lastly, we present examples that illustrate our findings. Full article

(This article belongs to the Special Issue Fixed Point Theory and Its Applications Dedicated to the Memory of Professor William Arthur Kirk)

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10 pages, 213 KiB

Open AccessArticle

Existence of a Fixed Point and Convergence of Iterates for Self-Mappings of Metric Spaces with Graphs

by Alexander J. Zaslavski

Symmetry 2024, 16(6), 705; https://doi.org/10.3390/sym16060705 - 6 Jun 2024

Cited by 1 | Viewed by 780

Abstract

In the present paper, under certain assumptions, we establish the convergence of iterates for self-mappings of complete metric spaces with graphs which are of a contractive type. The class of mappings considered in the paper contains the so-called cyclical mappings introduced by W. A. Kirk, P. S. Srinivasan and P. Veeramani in 2003 and the class of monotone nonexpansive operators. Our results hold in the case of a symmetric graph. Full article

(This article belongs to the Special Issue Fixed Point Theory and Its Applications Dedicated to the Memory of Professor William Arthur Kirk)

23 pages, 341 KiB

Open AccessArticle

Best Proximity Point Results for Multi-Valued Mappings in Generalized Metric Structure

by Asad Ullah Khan, Maria Samreen, Aftab Hussain and Hamed Al Sulami

Symmetry 2024, 16(4), 502; https://doi.org/10.3390/sym16040502 - 21 Apr 2024

Cited by 2 | Viewed by 1666

Abstract

In this paper, we introduce the novel concept of generalized distance denoted as

J_{θ}

and call it an extended b-generalized pseudo-distance. With the help of this generalized distance, we define a generalized point to set distance [...] Read more.

In this paper, we introduce the novel concept of generalized distance denoted as

J_{θ}

and call it an extended b-generalized pseudo-distance. With the help of this generalized distance, we define a generalized point to set distance

J_{θ} (u, H^{★}),

a generalized Hausdorff type distance and a

P^{J_{θ}}

-property of a pair

(H^{★}, K^{★})

of nonempty subsets of extended b-metric space

(U^{★}, ρ_{θ}) .

Additionally, we establish several best proximity point theorems for multi-valued contraction mappings of Nadler type defined on b-metric spaces and extended b-metric spaces. Our findings generalize numerous existing results found in the literature. To substantiate the introduced notion and validate our main results, we provide some concrete examples. Full article

(This article belongs to the Special Issue Fixed Point Theory and Its Applications Dedicated to the Memory of Professor William Arthur Kirk)

10 pages, 281 KiB

Open AccessArticle

Common Fixed-Point Theorem and Projection Method on a Hadamard Space

by Yasunori Kimura

Symmetry 2024, 16(4), 483; https://doi.org/10.3390/sym16040483 - 16 Apr 2024

Cited by 1 | Viewed by 1562

Abstract

In this paper, we obtain an equivalent condition to the existence of a common fixed point of a given family of nonexpansive mappings defined on a Hadamard space. Moreover, if the space is bounded, we show that the generating process of the approximate sequence by a specific projection method will stop in finite steps if there is no common fixed point. It is a significant advantage to reveal the nonexistence of a common fixed point in a finite time. Full article

(This article belongs to the Special Issue Fixed Point Theory and Its Applications Dedicated to the Memory of Professor William Arthur Kirk)

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20 pages, 325 KiB

Open AccessArticle

On Normed Algebras and the Generalized Maligranda–Orlicz Lemma

by Mieczysław Cichoń and Kinga Cichoń

Symmetry 2024, 16(1), 56; https://doi.org/10.3390/sym16010056 - 31 Dec 2023

Cited by 2 | Viewed by 1376

Abstract

In this paper, we discuss some extensions of the Maligranda–Orlicz lemma. It deals with the problem of constructing a norm in a subspace of the space of bounded functions, for which it becomes a normed algebra so that the norm introduced is equivalent to the initial norm of the subspace. This is done by satisfying some inequality between these norms. We show in this paper how this inequality is relevant to the study of operator equations in Banach algebras. In fact, we study how to equip a subspace of the space of bounded functions with a norm equivalent to a given one so that it is a normed algebra. We give a general condition for the construction of such norms, which allows us to easily check whether a space with a given norm is an algebra with a pointwise product and the consequences of such a choice for measures of noncompactness in such spaces. We also study quasi-normed spaces. We introduce a general property of measures of noncompactness that allows the study of quadratic operator equations, prove a fixed-point theorem suitable for such problems, and complete the whole with examples and applications. Full article

(This article belongs to the Special Issue Fixed Point Theory and Its Applications Dedicated to the Memory of Professor William Arthur Kirk)

15 pages, 333 KiB

Open AccessArticle

A Fixed Point Theorem in the Lebesgue Spaces of Variable Integrability L^p(·)

by Amnay El Amri, Mohamed Amine Khamsi and Osvaldo D. Méndez

Symmetry 2023, 15(11), 1999; https://doi.org/10.3390/sym15111999 - 30 Oct 2023

Cited by 3 | Viewed by 1294

Abstract

We establish a fixed point property for the Lebesgue spaces with variable exponents

L^{p (\cdot)}

, focusing on the scenario where the exponent closely approaches 1. The proof does not impose any additional conditions. In particular, our investigation centers on [...] Read more.

We establish a fixed point property for the Lebesgue spaces with variable exponents

L^{p (\cdot)}

, focusing on the scenario where the exponent closely approaches 1. The proof does not impose any additional conditions. In particular, our investigation centers on

ρ

-non-expansive mappings defined on convex subsets of

L^{p (\cdot)}

, satisfying the “condition of uniform decrease” that we define subsequently. Full article

(This article belongs to the Special Issue Fixed Point Theory and Its Applications Dedicated to the Memory of Professor William Arthur Kirk)

11 pages, 288 KiB

Open AccessArticle

Uniform Convexity in Variable Exponent Sobolev Spaces

by Mostafa Bachar, Mohamed A. Khamsi and Osvaldo Méndez

Symmetry 2023, 15(11), 1988; https://doi.org/10.3390/sym15111988 - 27 Oct 2023

Cited by 3 | Viewed by 1216

Abstract

We prove the modular convexity of the mixed norm

L^{p} (ℓ^{2})

on the Sobolev space

W^{1, p} (Ω)

in a domain

Ω \subset R^{n}

under the sole assumption that the exponent [...] Read more.

We prove the modular convexity of the mixed norm

L^{p} (ℓ^{2})

on the Sobolev space

W^{1, p} (Ω)

in a domain

Ω \subset R^{n}

under the sole assumption that the exponent

p (x)

is bounded away from 1, i.e., we include the case

\sup_{x \in Ω} p (x) = \infty

. In particular, the mixed Sobolev norm is uniformly convex if

1 < \inf_{x \in Ω} p (x) \leq \sup_{x \in Ω} p (x) < \infty

and

W_{0}^{1, p} (Ω)

is uniformly convex. Full article

(This article belongs to the Special Issue Fixed Point Theory and Its Applications Dedicated to the Memory of Professor William Arthur Kirk)

9 pages, 231 KiB

Open AccessArticle

Three Convergence Results for Iterates of Nonlinear Mappings in Metric Spaces with Graphs

by Alexander J. Zaslavski

Symmetry 2023, 15(9), 1756; https://doi.org/10.3390/sym15091756 - 13 Sep 2023

Cited by 1 | Viewed by 911

Abstract

In 2007, in our joint work with D. Butnariu and S. Reich, we proved that if for a self-mapping of a complete metric that is uniformly continuous on bounded sets all its iterates converge uniformly on bounded sets, then this convergence is stable under the presence of small errors. In the present paper, we obtain an extension of this result for self-mappings of a metric space with a graph. Full article

(This article belongs to the Special Issue Fixed Point Theory and Its Applications Dedicated to the Memory of Professor William Arthur Kirk)

17 pages, 319 KiB

Open AccessArticle

Recent Advancements in KRH-Interpolative-Type Contractions

by Ansar Abbas, Amjad Ali, Hamed Al Sulami and Aftab Hussain

Symmetry 2023, 15(8), 1515; https://doi.org/10.3390/sym15081515 - 1 Aug 2023

Cited by 1 | Viewed by 1567

Abstract

The focus of this paper is to conduct a comprehensive analysis of the advancements made in the understanding of Interpolative contraction, building upon the ideas initially introduced by Karapinar in 2018. In this paper, we develop the notion of Interpolative contraction mappings to the case of non-linear Kannan Interpolative, Riech Rus Ćirić interpolative and Hardy–Roger Interpolative contraction mappings based on controlled function, and prove some fixed point results in the context of controlled metric space, thereby enhancing the current understanding of this particular analysis. Furthermore, we provide a concrete example that illustrates the underlying drive for the investigations presented in this context. An application of the proposed non-linear Interpolative-contractions to the Liouville–Caputo fractional derivatives and fractional differential equations is provided in this paper. Full article

(This article belongs to the Special Issue Fixed Point Theory and Its Applications Dedicated to the Memory of Professor William Arthur Kirk)

19 pages, 1202 KiB

Open AccessArticle

Hybrid Functions Approach via Nonlinear Integral Equations with Symmetric and Nonsymmetrical Kernel in Two Dimensions

by Sahar M. Abusalim, Mohamed A. Abdou, Mohamed A. Abdel-Aty and Mohamed E. Nasr

Symmetry 2023, 15(7), 1408; https://doi.org/10.3390/sym15071408 - 13 Jul 2023

Cited by 5 | Viewed by 1486

Abstract

The second kind of two-dimensional nonlinear integral equation (NIE) with symmetric and nonsymmetrical kernel is solved in the Banach space

L_{2} [0, 1] \times L_{2} [0, 1]

. Here, the NIE’s existence and singular solution [...] Read more.

The second kind of two-dimensional nonlinear integral equation (NIE) with symmetric and nonsymmetrical kernel is solved in the Banach space

L_{2} [0, 1] \times L_{2} [0, 1]

. Here, the NIE’s existence and singular solution are described in this passage. Additionally, we use a numerical strategy that uses hybrid and block-pulse functions to obtain the approximate solution of the NIE in a two-dimensional problem. For this aim, the two-dimensional NIE will be reduced to a system of nonlinear algebraic equations (SNAEs). Then, the SNAEs can be solved numerically. This study focuses on showing the convergence analysis for the numerical approach and generating an estimate of the error. Examples are presented to prove the efficiency of the approach. Full article

(This article belongs to the Special Issue Fixed Point Theory and Its Applications Dedicated to the Memory of Professor William Arthur Kirk)

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13 pages, 274 KiB

Open AccessArticle

Banach Fixed Point Theorems in Generalized Metric Space Endowed with the Hadamard Product

by Saleh Omran, Ibtisam Masmali and Ghaliah Alhamzi

Symmetry 2023, 15(7), 1325; https://doi.org/10.3390/sym15071325 - 28 Jun 2023

Cited by 4 | Viewed by 1944

Abstract

In this paper, we prove some Banach fixed point theorems in generalized metric space where the contractive conditions are endowed with the Hadamard product of real symmetric positive definite matrices. Since the condition that a matrix A converges to zero is not needed, this produces stronger results than those of Perov. As an application of our results, we study the existence and uniqueness of the solution for a system of matrix equations. Full article

(This article belongs to the Special Issue Fixed Point Theory and Its Applications Dedicated to the Memory of Professor William Arthur Kirk)

27 pages, 491 KiB

Open AccessArticle

Computational Techniques for Solving Mixed (1 + 1) Dimensional Integral Equations with Strongly Symmetric Singular Kernel

by Sharifah E. Alhazmi, Amr M. S. Mahdy, Mohamed A. Abdou and Doaa Sh. Mohamed

Symmetry 2023, 15(6), 1284; https://doi.org/10.3390/sym15061284 - 19 Jun 2023

Cited by 14 | Viewed by 1703

Abstract

This paper describes an effective strategy based on Lerch polynomial method for solving mixed integral equations (MIE) in position and time with a strongly symmetric singular kernel in the space [...] Read more.

This paper describes an effective strategy based on Lerch polynomial method for solving mixed integral equations (MIE) in position and time with a strongly symmetric singular kernel in the space

L_{2} (- 1, 1) \times C [0, T], (T < 1) .

The Quadratic numerical method (QNM) was applied to obtain a system of Fredholm integral equations (SFIE), then the Lerch polynomials method (LPM) was applied to transform SFIE into a system of linear algebraic equations (SLAE). The existence and uniqueness of the integral equation’s solution are discussed using Banach’s fixed point theory. Also, the convergence and stability of the solution and the stability of the error are discussed. Several examples are given to illustrate the applicability of the presented method. The Maple program obtains all the results. A numerical simulation is carried out to determine the efficacy of the methodology, and the results are given in symmetrical forms. From the numerical results, it is noted that there is a symmetry utterly identical to the kernel used when replacing each x with y. Full article

(This article belongs to the Special Issue Fixed Point Theory and Its Applications Dedicated to the Memory of Professor William Arthur Kirk)

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