Symmetry

Research

15 pages, 2229 KiB

Open AccessArticle

Convergence on Kirk Iteration of Cesàro Means for Asymptotically Nonexpansive Mappings

by Lale Cona and Deniz Şimşek

Symmetry 2025, 17(3), 393; https://doi.org/10.3390/sym17030393 - 5 Mar 2025

Viewed by 461

This article addresses the convergence of iteration sequences in Cesàro means for asymptotically nonexpansive mappings. Specifically, this study explores the behavior of Kirk iteration in the Cesàro means in the context of uniformly convex and reflexive Banach spaces equipped with uniformly Gâteaux differentiable [...] Read more.

This article addresses the convergence of iteration sequences in Cesàro means for asymptotically nonexpansive mappings. Specifically, this study explores the behavior of Kirk iteration in the Cesàro means in the context of uniformly convex and reflexive Banach spaces equipped with uniformly Gâteaux differentiable norms. The focus is to determine the conditions under which the Kirk iteration sequence converges strongly or weakly to a fixed point. Finally, some examples are given in this article to demonstrate the advantages of the preferred iteration method and to verify the results obtained. Full article

(This article belongs to the Special Issue Elementary Fixed Point Theory and Common Fixed Points II)

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16 pages, 290 KiB

Open AccessArticle

Common Fixed Point Theorems in Complex-Valued Controlled Metric Spaces with Application

by Amnah Essa Shammaky and Jamshaid Ahmad

Symmetry 2024, 16(11), 1442; https://doi.org/10.3390/sym16111442 - 31 Oct 2024

Viewed by 1237

Abstract

The objective of this article is to establish common fixed point results in the background of complex-valued controlled metric spaces for generalized rational contractions. Our findings generalize a number of well-established results in the literature. To highlight the uniqueness of our key finding, [...] Read more.

The objective of this article is to establish common fixed point results in the background of complex-valued controlled metric spaces for generalized rational contractions. Our findings generalize a number of well-established results in the literature. To highlight the uniqueness of our key finding, we present an example. As a demonstration of the applicability of our principal theorem, we solve the Volterra integral equation. Full article

(This article belongs to the Special Issue Elementary Fixed Point Theory and Common Fixed Points II)

20 pages, 317 KiB

Open AccessArticle

Fixed Point Results for New Classes of k-Strictly Asymptotically Demicontractive and Hemicontractive Type Multivalued Mappings in Symmetric Spaces

by Imo Kalu Agwu, Faeem Ali, Donatus Ikechi Igbokwe and Iqbal Ahmad

Symmetry 2024, 16(9), 1104; https://doi.org/10.3390/sym16091104 - 24 Aug 2024

Viewed by 1254

Abstract

Fixed point theory is a significant area of mathematical analysis with applications across various fields such as differential equations, optimization, and dynamical systems. Recently, multivalued mappings have gained attention due to their ability to model more complex and realistic problems. ln this work, [...] Read more.

Fixed point theory is a significant area of mathematical analysis with applications across various fields such as differential equations, optimization, and dynamical systems. Recently, multivalued mappings have gained attention due to their ability to model more complex and realistic problems. ln this work, novel classes of nonlinear mappings called k-strictly asymptotically demicontractive-type and asymptotically hemicontractive-type multivalued mappings are introduced in real Hilbert spaces that are symmetric spaces. In addition, we discuss the weak and strong convergence results by considered modified algorithms, and a demiclosedness property, for these classes of mappings are proved. Several non-trivial examples are demonstrated to validate the newly defined mappings. Consequently, the results and iterative methods obtained in this study improve and extend several known outcomes in the literature. Full article

(This article belongs to the Special Issue Elementary Fixed Point Theory and Common Fixed Points II)

14 pages, 272 KiB

Open AccessArticle

A Novel Fixed-Point Iteration Approach for Solving Troesch’s Problem

by Doaa Filali, Faeem Ali, Mohammad Akram and Mohammad Dilshad

Symmetry 2024, 16(7), 856; https://doi.org/10.3390/sym16070856 - 6 Jul 2024

Cited by 3 | Viewed by 1991

Abstract

This paper introduces a novel F fixed-point iteration method that leverages Green’s function for solving the nonlinear Troesch problem in Banach spaces, which are symmetric spaces. The Troesch problem, characterized by its challenging boundary conditions and nonlinear nature, is significant in various physical [...] Read more.

This paper introduces a novel F fixed-point iteration method that leverages Green’s function for solving the nonlinear Troesch problem in Banach spaces, which are symmetric spaces. The Troesch problem, characterized by its challenging boundary conditions and nonlinear nature, is significant in various physical and engineering applications. The proposed method integrates fixed-point theory with Green’s function techniques to develop an iteration process that ensures convergence, stability, and accuracy. The numerical experiments demonstrate the method’s efficiency and robustness, highlighting its potential for broader applications in solving nonlinear differential equations in Banach spaces. Full article

(This article belongs to the Special Issue Elementary Fixed Point Theory and Common Fixed Points II)

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13 pages, 328 KiB

Open AccessArticle

Relation-Preserving Functional Contractions Involving a Triplet of Auxiliary Functions with an Application to Integral Equations

by Doaa Filali and Faizan Ahmad Khan

Symmetry 2024, 16(6), 691; https://doi.org/10.3390/sym16060691 - 4 Jun 2024

Cited by 1 | Viewed by 683

Abstract

This article addresses certain fixed-point results in a metric space equipped with a locally transitive binary relation under a functional contraction containing three auxiliary functions. The findings proved herein enrich and improve a number of existing results. In order to prove the credibility [...] Read more.

This article addresses certain fixed-point results in a metric space equipped with a locally transitive binary relation under a functional contraction containing three auxiliary functions. The findings proved herein enrich and improve a number of existing results. In order to prove the credibility of our findings, an illustrative example is provided. Making use of our findings, we study the genuineness of the unique solution to a Fredholm integral equation. Full article

(This article belongs to the Special Issue Elementary Fixed Point Theory and Common Fixed Points II)

18 pages, 326 KiB

Open AccessArticle

Strong and Weak Convergence Theorems for the Split Feasibility Problem of (β,k)-Enriched Strict Pseudocontractive Mappings with an Application in Hilbert Spaces

by Asima Razzaque, Naeem Saleem, Imo Kalu Agwu, Umar Ishtiaq and Maggie Aphane

Symmetry 2024, 16(5), 546; https://doi.org/10.3390/sym16050546 - 2 May 2024

Viewed by 952

Abstract

The concept of symmetry has played a major role in Hilbert space setting owing to the structure of a complete inner product space. Subsequently, different studies pertaining to symmetry, including symmetric operators, have investigated real Hilbert spaces. In this paper, we study the [...] Read more.

The concept of symmetry has played a major role in Hilbert space setting owing to the structure of a complete inner product space. Subsequently, different studies pertaining to symmetry, including symmetric operators, have investigated real Hilbert spaces. In this paper, we study the solutions to multiple-set split feasibility problems for a pair of finite families of

β

-enriched, strictly pseudocontractive mappings in the setup of a real Hilbert space. In view of this, we constructed an iterative scheme that properly included these two mappings into the formula. Under this iterative scheme, an appropriate condition for the existence of solutions and strong and weak convergent results are presented. No sum condition is imposed on the countably finite family of the iteration parameters in obtaining our results unlike for several other results in this direction. In addition, we prove that a slight modification of our iterative scheme could be applied in studying hierarchical variational inequality problems in a real Hilbert space. Our results improve, extend and generalize several results currently existing in the literature. Full article

(This article belongs to the Special Issue Elementary Fixed Point Theory and Common Fixed Points II)

12 pages, 263 KiB

Open AccessArticle

An Existence Result for Second-Order Boundary-Value Problems via New Fixed-Point Theorems on Quasi-Metric Space

by Gonca Durmaz Güngör and Ishak Altun

Symmetry 2024, 16(1), 99; https://doi.org/10.3390/sym16010099 - 13 Jan 2024

Cited by 2 | Viewed by 1172

Abstract

We introduce the new idea of

(α - θ_{σ})

-contraction in quasi-metric spaces in this paper. For these kinds of mappings, we then prove new fixed-point theorems on left K, left M, and left [...] Read more.

We introduce the new idea of

(α - θ_{σ})

-contraction in quasi-metric spaces in this paper. For these kinds of mappings, we then prove new fixed-point theorems on left K, left M, and left

S m y t h

-complete quasi-metric spaces. We also apply our results to infer the existence of a solution to a second-order boundary-value problem. Full article

(This article belongs to the Special Issue Elementary Fixed Point Theory and Common Fixed Points II)

23 pages, 330 KiB

Open AccessArticle

Existence and Uniqueness of Solutions of Hammerstein-Type Functional Integral Equations

by Cemil Tunç, Fehaid Salem Alshammari and Fahir Talay Akyildiz

Symmetry 2023, 15(12), 2205; https://doi.org/10.3390/sym15122205 - 15 Dec 2023

Cited by 8 | Viewed by 1362

Abstract

The authors deal with nonlinear and general Hammerstein-type functional integral equations (HTFIEs). The first objective of this work is to apply and extend Burton’s method to general and nonlinear HTFIEs in a Banach space via the Chebyshev norm and complete metric. The second [...] Read more.

The authors deal with nonlinear and general Hammerstein-type functional integral equations (HTFIEs). The first objective of this work is to apply and extend Burton’s method to general and nonlinear HTFIEs in a Banach space via the Chebyshev norm and complete metric. The second objective of the paper is to extend and improve some earlier results to nonlinear HTFIEs. The authors prove two new theorems with regard to the existence and uniqueness of solutions (EUSs) of HTFIEs via a technique called progressive contractions, which belongs to T. A. Burton, and the Chebyshev norm and complete metric. Full article

(This article belongs to the Special Issue Elementary Fixed Point Theory and Common Fixed Points II)

14 pages, 390 KiB

Open AccessArticle

Exploring Fuzzy Triple Controlled Metric Spaces: Applications in Integral Equations

by Fatima M. Azmi

Symmetry 2023, 15(10), 1943; https://doi.org/10.3390/sym15101943 - 20 Oct 2023

Cited by 2 | Viewed by 1278

Abstract

In this article, we delve into the study of fuzzy triple controlled metric spaces, investigating their properties and presenting a range of illustrative examples. We emphasize the broader applicability of this concept in comparison to fuzzy rectangular metric spaces and fuzzy rectangular b [...] Read more.

In this article, we delve into the study of fuzzy triple controlled metric spaces, investigating their properties and presenting a range of illustrative examples. We emphasize the broader applicability of this concept in comparison to fuzzy rectangular metric spaces and fuzzy rectangular b-metric spaces. By introducing the novel concept of (

α

-

ψ

)-fuzzy contractive mappings, we derive fixed point results specifically designed for complete fuzzy triple controlled metric spaces. Our theorems extend and enrich previous findings in this field. Additionally, we demonstrate the practical significance of our study by applying our findings to the solution of an integral equation and providing an example of its application. Furthermore, we propose potential avenues for future research endeavors. Full article

(This article belongs to the Special Issue Elementary Fixed Point Theory and Common Fixed Points II)

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17 pages, 303 KiB

Open AccessArticle

Fixed-Point Theorems on Fuzzy Bipolar

b

-Metric Spaces

by Balaji Ramalingam, Ozgur Ege, Ahmad Aloqaily and Nabil Mlaiki

Symmetry 2023, 15(10), 1831; https://doi.org/10.3390/sym15101831 - 27 Sep 2023

Cited by 7 | Viewed by 1149

Abstract

In this manuscript, we establish some fixed-point theorems without continuity by using the triangular property on a fuzzy bipolar

b

-metric space as a generalized version and expansion of the well-known results. We also provide some examples and applications of the integral equation [...] Read more.

In this manuscript, we establish some fixed-point theorems without continuity by using the triangular property on a fuzzy bipolar

b

-metric space as a generalized version and expansion of the well-known results. We also provide some examples and applications of the integral equation to the solution for our main results. Full article

(This article belongs to the Special Issue Elementary Fixed Point Theory and Common Fixed Points II)

16 pages, 328 KiB

Open AccessArticle

Development of Fixed Point Results for α_Γ-F-Fuzzy Contraction Mappings with Applications

by Salvatore Sessa, Fahad Jahangeer, Doha A. Kattan and Umar Ishtiaq

Symmetry 2023, 15(7), 1300; https://doi.org/10.3390/sym15071300 - 22 Jun 2023

Cited by 4 | Viewed by 1239

Abstract

This manuscript contains several fixed point results for

α_{Γ}

-F-fuzzy contractive mappings in the framework of orthogonal fuzzy metric spaces. The symmetric property guarantees that the distance function is consistent and does not favour any one direction in orthogonal fuzzy [...] Read more.

This manuscript contains several fixed point results for

α_{Γ}

-F-fuzzy contractive mappings in the framework of orthogonal fuzzy metric spaces. The symmetric property guarantees that the distance function is consistent and does not favour any one direction in orthogonal fuzzy metric spaces. No matter how the points are arranged, it enables a fair assessment of the separations between all of them. In fixed point results, the symmetry condition is preserved for several types of contractive self-mappings. Moreover, we provide several non-trivial examples to show the validity of our main results. Furthermore, we solve non-linear fractional differential equations, the Atangana–Baleanu fractional integral operator and Fredholm integral equations by utilizing our main results. Full article

(This article belongs to the Special Issue Elementary Fixed Point Theory and Common Fixed Points II)

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