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19 pages, 792 KB

Open AccessArticle

Generalized Ishikawa Iterative Algorithm with Errors and Variable Generalized Ishikawa Iterative Algorithm for Nonexpansive Mappings in Symmetric Banach Spaces

by Liangjuan Yu, Yuhan Zhu and Wenying Zhao

Symmetry 2026, 18(1), 125; https://doi.org/10.3390/sym18010125 - 9 Jan 2026

Abstract

We present a generalized Ishikawa iterative algorithm with an error term and a variable generalized Ishikawa iterative algorithm. Leveraging the geometric symmetry inherent in uniformly convex Banach spaces, we establish their respective weak convergence theorems for nonexpansive mappings. As applications, we extend several [...] Read more.

We present a generalized Ishikawa iterative algorithm with an error term and a variable generalized Ishikawa iterative algorithm. Leveraging the geometric symmetry inherent in uniformly convex Banach spaces, we establish their respective weak convergence theorems for nonexpansive mappings. As applications, we extend several recent results in the literature related to the proximal point algorithm and the split feasibility problem. Consequently, we propose a hyper-generalized proximal point algorithm and a hyper-generalized perturbation CQ algorithm. Our work not only broadens the application scope of these methods but also highlights the foundational role of symmetric space properties in ensuring algorithmic convergence. Full article

(This article belongs to the Special Issue Symmetry in Fixed Point Theory and Optimization: Computations and Applications)

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15 pages, 268 KB

Open AccessArticle

Algorithms for Solving the Resolvent of the Sum of Two Maximal Monotone Operators with a Finite Family of Nonexpansive Operators

by Ali Berrailes and Abdallah Beddani

Axioms 2025, 14(11), 783; https://doi.org/10.3390/axioms14110783 - 25 Oct 2025

Viewed by 329

Abstract

In this paper, we address a variational problem involving the sum of two maximal monotone operators combined with a finite family of nonexpansive operators. To solve this problem, we propose iterative algorithms based on single-valued mappings. First, we examine cases involving two or [...] Read more.

In this paper, we address a variational problem involving the sum of two maximal monotone operators combined with a finite family of nonexpansive operators. To solve this problem, we propose iterative algorithms based on single-valued mappings. First, we examine cases involving two or three maximal monotone operators, introducing novel algorithms to obtain their solutions. Secondly, we extend our analysis by applying the Ishikawa iterative scheme within the framework of fixed-point theory. This allows us to establish strong convergence results. Finally, we provide an illustrative example to demonstrate the effectiveness and applicability of the proposed methods. Full article

(This article belongs to the Special Issue Modern Problems of Analysis, Optimization, Approximation and Their Applications)

25 pages, 2530 KB

Open AccessArticle

Enhancing Production Line Station Efficiency and Performance via Dynamic Modelling Techniques

by Florina Chiscop, Eduard Stefan Jitaru, Carmen-Cristiana Cazacu, Cicerone Laurentiu Popa, Lidia Florentina Parpala and Costel Emil Cotet

Processes 2025, 13(10), 3176; https://doi.org/10.3390/pr13103176 - 6 Oct 2025

Viewed by 901

Abstract

This research investigates the optimization of operational efficiency and cost reduction through the enhancement of material flow management within production line stations. Departing from conventional static analyses, the study employs advanced simulation tools to pinpoint performance bottlenecks and inefficiencies via dynamic modelling techniques. [...] Read more.

This research investigates the optimization of operational efficiency and cost reduction through the enhancement of material flow management within production line stations. Departing from conventional static analyses, the study employs advanced simulation tools to pinpoint performance bottlenecks and inefficiencies via dynamic modelling techniques. The Ishikawa diagram serves as the primary tool for conducting root-cause analysis. Simultaneously, the 5S methodology is implemented to foster workplace organization, standardization, and hygiene practices. In contrast to traditional optimization frameworks, the proposed strategy integrates real-time performance tracking systems, complemented by adaptive feedback mechanisms. This integration permits ongoing assessment of the production process, facilitating iterative improvement cycles. Empirical data gathered from monitored cycle times, equipment utilization rates, and defect frequencies substantiate the validation of implemented changes. The resulting optimized system significantly minimizes downtime and waste, thereby advancing sustainable and scalable operations. Ultimately, this research demonstrates that the fusion of simulation-based insights with lean management principles leads to considerable improvements in manufacturing productivity and overall product quality. Full article

(This article belongs to the Section Manufacturing Processes and Systems)

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16 pages, 673 KB

Open AccessArticle

Recent Developments in Iterative Algorithms for Digital Metrics

by Aasma Shaheen, Afshan Batool, Amjad Ali, Hamed Al Sulami and Aftab Hussain

Symmetry 2024, 16(3), 368; https://doi.org/10.3390/sym16030368 - 18 Mar 2024

Cited by 6 | Viewed by 2152

Abstract

This paper aims to provide a comprehensive analysis of the advancements made in understanding Iterative Fixed-Point Schemes, which builds upon the concept of digital contraction mappings. Additionally, we introduce the notion of an Iterative Fixed-Point Schemes in digital metric spaces. In this study, [...] Read more.

This paper aims to provide a comprehensive analysis of the advancements made in understanding Iterative Fixed-Point Schemes, which builds upon the concept of digital contraction mappings. Additionally, we introduce the notion of an Iterative Fixed-Point Schemes in digital metric spaces. In this study, we extend the idea of Iteration process Mann, Ishikawa, Agarwal, and Thakur based on the ϝ-Stable Iterative Scheme in digital metric space. We also design some fractal images, which frame the compression of Fixed-Point Iterative Schemes and contractive mappings. Furthermore, we present a concrete example that exemplifies the motivation behind our investigations. Moreover, we provide an application of the proposed Fractal image and Sierpinski triangle that compress the works by storing images as a collection of digital contractions, which addresses the issue of storing images with less storage memory in this paper. Full article

(This article belongs to the Special Issue New Trends in Fixed Point Theory with Emphasis on Symmetry)

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16 pages, 323 KB

Open AccessArticle

Approximating Fixed Points of Relatively Nonexpansive Mappings via Thakur Iteration

by V. Pragadeeswarar, R. Gopi and M. De la Sen

Symmetry 2022, 14(6), 1107; https://doi.org/10.3390/sym14061107 - 27 May 2022

Viewed by 1997

Abstract

The study of symmetry is a major tool in the nonlinear analysis. The symmetricity of distance function in a metric space plays important role in proving the existence of a fixed point for a self mapping. In this work, we approximate a fixed [...] Read more.

The study of symmetry is a major tool in the nonlinear analysis. The symmetricity of distance function in a metric space plays important role in proving the existence of a fixed point for a self mapping. In this work, we approximate a fixed point of noncyclic relatively nonexpansive mappings by using a three-step Thakur iterative scheme in uniformly convex Banach spaces. We also provide a numerical example where the Thakur iterative scheme is faster than some well known iterative schemes such as Picard, Mann, and Ishikawa iteration. Finally, we provide a stronger version of our proposed theorem via von Neumann sequences. Full article

(This article belongs to the Special Issue Nonlinear Analysis and Applications, Geometry of Banach Spaces and Symmetry)

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18 pages, 325 KB

Open AccessArticle

Strong Convergence Theorems for a Finite Family of Enriched Strictly Pseudocontractive Mappings and Φ_T-Enriched Lipschitizian Mappings Using a New Modified Mixed-Type Ishikawa Iteration Scheme with Error

by Naeem Saleem, Imo Kalu Agwu, Umar Ishtiaq and Stojan Radenović

Symmetry 2022, 14(5), 1032; https://doi.org/10.3390/sym14051032 - 18 May 2022

Cited by 23 | Viewed by 2374

Abstract

In this paper, we introduce two new classes of mappings known as

λ

-enriched strictly pseudocontractive mappings and

Φ_{T}

-enriched Lipshitizian mappings in the setup of a real Banach space. In addition, a new modified mixed-type Ishikawa iteration scheme was constructed, and [...] Read more.

In this paper, we introduce two new classes of mappings known as

λ

-enriched strictly pseudocontractive mappings and

Φ_{T}

-enriched Lipshitizian mappings in the setup of a real Banach space. In addition, a new modified mixed-type Ishikawa iteration scheme was constructed, and it was proved that our iteration method converges strongly to the common fixed points of finite families of the above mappings in the framework of a real uniformly convex Banach space. Moreover, we provided a non-trivial example to support our main result. Our results extend and generalize several results existing in the literature. Full article

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

15 pages, 2256 KB

Open AccessArticle

An Accelerated Fixed-Point Algorithm with an Inertial Technique for a Countable Family of G-Nonexpansive Mappings Applied to Image Recovery

by Kobkoon Janngam and Rattanakorn Wattanataweekul

Symmetry 2022, 14(4), 662; https://doi.org/10.3390/sym14040662 - 24 Mar 2022

Cited by 3 | Viewed by 2319

Abstract

Many authors have proposed fixed-point algorithms for obtaining a fixed point of G-nonexpansive mappings without using inertial techniques. To improve convergence behavior, some accelerated fixed-point methods have been introduced. The main aim of this paper is to use a coordinate affine structure [...] Read more.

Many authors have proposed fixed-point algorithms for obtaining a fixed point of G-nonexpansive mappings without using inertial techniques. To improve convergence behavior, some accelerated fixed-point methods have been introduced. The main aim of this paper is to use a coordinate affine structure to create an accelerated fixed-point algorithm with an inertial technique for a countable family of G-nonexpansive mappings in a Hilbert space with a symmetric directed graph G and prove the weak convergence theorem of the proposed algorithm. As an application, we apply our proposed algorithm to solve image restoration and convex minimization problems. The numerical experiments show that our algorithm is more efficient than FBA, FISTA, Ishikawa iteration, S-iteration, Noor iteration and SP-iteration. Full article

(This article belongs to the Special Issue New Directions in Theory of Approximation and Related Problems)

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19 pages, 310 KB

Open AccessArticle

A Krasnoselskii–Ishikawa Iterative Algorithm for Monotone Reich Contractions in Partially Ordered Banach Spaces with an Application

by Nawab Hussain, Saud M. Alsulami and Hind Alamri

Mathematics 2022, 10(1), 76; https://doi.org/10.3390/math10010076 - 27 Dec 2021

Cited by 4 | Viewed by 2682

Abstract

Iterative algorithms have been utilized for the computation of approximate solutions of stationary and evolutionary problems associated with differential equations. The aim of this article is to introduce concepts of monotone Reich and Chatterjea nonexpansive mappings on partially ordered Banach spaces. We describe [...] Read more.

Iterative algorithms have been utilized for the computation of approximate solutions of stationary and evolutionary problems associated with differential equations. The aim of this article is to introduce concepts of monotone Reich and Chatterjea nonexpansive mappings on partially ordered Banach spaces. We describe sufficient conditions for the existence of an approximate fixed-point sequence (AFPS) and prove certain fixed-point results using the Krasnoselskii–Ishikawa iterative algorithm. Moreover, we present some interesting examples to highlight the superiority of our results. Lastly, we provide both weak and strong convergence results for such mappings and consider an application of our results to prove the existence of a solution to an initial value problem. Full article

(This article belongs to the Special Issue Advances in Abstract Inequalities, Partial Functional Differential Equations and Their Applications)

15 pages, 305 KB

Open AccessArticle

Equivalence of Certain Iteration Processes Obtained by Two New Classes of Operators

by Mujahid Abbas, Rizwan Anjum and Vasile Berinde

Mathematics 2021, 9(18), 2292; https://doi.org/10.3390/math9182292 - 17 Sep 2021

Cited by 19 | Viewed by 2093

Abstract

The aim of this paper is two fold: the first is to define two new classes of mappings and show the existence and iterative approximation of their fixed points; the second is to show that the Ishikawa, Mann, and Krasnoselskij iteration methods defined [...] Read more.

The aim of this paper is two fold: the first is to define two new classes of mappings and show the existence and iterative approximation of their fixed points; the second is to show that the Ishikawa, Mann, and Krasnoselskij iteration methods defined for such classes of mappings are equivalent. An application of the main results to solve split feasibility and variational inequality problems are also given. Full article

(This article belongs to the Special Issue General Metric Spaces: Usage and Applications in Statistics and Fixed Point Theory)

15 pages, 781 KB

Open AccessArticle

Common Attractive Points of Generalized Hybrid Multi-Valued Mappings and Applications

by Lili Chen, Ni Yang and Jing Zhou

Mathematics 2020, 8(8), 1307; https://doi.org/10.3390/math8081307 - 6 Aug 2020

Cited by 4 | Viewed by 2097

Abstract

In this paper, we first propose the concepts of

(ζ, η, λ, π)

-generalized hybrid multi-valued mappings, the set of all the common attractive points (

C A_{f, g}

) and the set of all the [...] Read more.

In this paper, we first propose the concepts of

(ζ, η, λ, π)

-generalized hybrid multi-valued mappings, the set of all the common attractive points (

C A_{f, g}

) and the set of all the common strongly attractive points (

C_{s} A_{f, g}

), respectively for the multi-valued mappings f and g in a CAT(0) space. Moreover, we give some elementary properties in regard to the sets

A_{f}

,

F_{f}

and

C A_{f, g}

for the multi-valued mappings f and g in a complete CAT(0) space. Furthermore, we present a weak convergence theorem of common attractive points for two

(ζ, η, λ, π)

-generalized hybrid multi-valued mappings in the above space by virtue of Banach limits technique and Ishikawa iteration respectively. Finally, we prove strong convergence of a new viscosity approximation method for two

(ζ, η, λ, π)

-generalized hybrid multi-valued mappings in CAT(0) spaces, which also solves a kind of variational inequality problem. Full article

(This article belongs to the Special Issue Abstract Metric Spaces: Usefulness in Statistics and Fixed Point Theory)

9 pages, 256 KB

Open AccessArticle

Approximation of Fixed Points of C*-Algebra-Multi-Valued Contractive Mappings by the Mann and Ishikawa Processes in Convex C*-Algebra-Valued Metric Spaces

by Azadeh Ghanifard, Hashem Parvaneh Masiha and Manuel De La Sen

Mathematics 2020, 8(3), 392; https://doi.org/10.3390/math8030392 - 10 Mar 2020

Cited by 5 | Viewed by 2017

Abstract

The aim of the present paper is to state and prove some convergence theorems for the Mann and Ishikawa iteration schemes involving

C^{*}

-algebra-multi-valued contractive mappings in the setting of convex

C^{*}

-algebra-valued metric spaces. The convergence theorems of the proposed [...] Read more.

The aim of the present paper is to state and prove some convergence theorems for the Mann and Ishikawa iteration schemes involving

C^{*}

-algebra-multi-valued contractive mappings in the setting of convex

C^{*}

-algebra-valued metric spaces. The convergence theorems of the proposed iterations to a common fixed point of finite and infinite family of such mappings are also established. Full article

19 pages, 2938 KB

Open AccessArticle

Generation of Julia and Mandelbrot Sets via Fixed Points

by Mujahid Abbas, Hira Iqbal and Manuel De la Sen

Symmetry 2020, 12(1), 86; https://doi.org/10.3390/sym12010086 - 2 Jan 2020

Cited by 28 | Viewed by 5266

Abstract

The aim of this paper is to present an application of a fixed point iterative process in generation of fractals namely Julia and Mandelbrot sets for the complex polynomials of the form [...] Read more.

The aim of this paper is to present an application of a fixed point iterative process in generation of fractals namely Julia and Mandelbrot sets for the complex polynomials of the form

T (x) = x^{n} + m x + r

where

m, r \in C

and

n \geq 2

. Fractals represent the phenomena of expanding or unfolding symmetries which exhibit similar patterns displayed at every scale. We prove some escape time results for the generation of Julia and Mandelbrot sets using a Picard Ishikawa type iterative process. A visualization of the Julia and Mandelbrot sets for certain complex polynomials is presented and their graphical behaviour is examined. We also discuss the effects of parameters on the color variation and shape of fractals. Full article

(This article belongs to the Special Issue Symmetry in Applied Mathematics)

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28 pages, 1613 KB

Open AccessArticle

Approximating Fixed Points of Bregman Generalized α-Nonexpansive Mappings

by Kanikar Muangchoo, Poom Kumam, Yeol Je Cho, Sompong Dhompongsa and Sakulbuth Ekvittayaniphon

Mathematics 2019, 7(8), 709; https://doi.org/10.3390/math7080709 - 6 Aug 2019

Cited by 2 | Viewed by 3729

Abstract

In this paper, we introduce a new class of Bregman generalized

α

-nonexpansive mappings in terms of the Bregman distance. We establish several weak and strong convergence theorems of the Ishikawa and Noor iterative schemes for Bregman generalized

α

-nonexpansive mappings in Banach [...] Read more.

In this paper, we introduce a new class of Bregman generalized

α

-nonexpansive mappings in terms of the Bregman distance. We establish several weak and strong convergence theorems of the Ishikawa and Noor iterative schemes for Bregman generalized

α

-nonexpansive mappings in Banach spaces. A numerical example is given to illustrate the main results of fixed point approximation using Halpern’s algorithm. Full article

(This article belongs to the Special Issue Computational Methods in Analysis and Applications)

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6 pages, 354 KB

Open AccessArticle

Mann and Ishikawa Type Perturbed Iterative Algorithms for Generalized Nonlinear Variational Inclusions

by Hanif Salahuddin, R. Ahmad and M. F. Khan

Math. Comput. Appl. 2001, 6(1), 47-52; https://doi.org/10.3390/mca6010047 - 1 Apr 2001

Cited by 2 | Viewed by 1654

Abstract

In this paper, we consider the generalized nonlinear variational inclusions and develop Mann and Ishikawa type perturbed iterative algorithms for finding the approximate solution of this problem. By using the definition of multivalued relaxed Lipschitz operators, we discuss the convergence criteria for the [...] Read more.

In this paper, we consider the generalized nonlinear variational inclusions and develop Mann and Ishikawa type perturbed iterative algorithms for finding the approximate solution of this problem. By using the definition of multivalued relaxed Lipschitz operators, we discuss the convergence criteria for the perturbed algorithms. Full article

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