Research

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

Viewed by 206

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 [...] Read more.

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

Viewed by 690

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 [...] Read more.

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

Viewed by 639

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

Viewed by 657

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

Viewed by 498

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 [...] Read more.

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 - 01 Aug 2023

Viewed by 994

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 [...] Read more.

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 1 | Viewed by 840

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 2 | Viewed by 1152

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, [...] Read more.

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 4 | Viewed by 972

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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