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Symmetry with Operator Theory and Equations Symmetry with Operator Theory and Equations Edited by Ioannis Argyros Printed Edition of the Special Issue Published in Symmetry www.mdpi.com/journal/symmetry Symmetry with Operator Theory and Equations Symmetry with Operator Theory and Equations Special Issue Editor Ioannis Argyros MDPI • Basel • Beijing • Wuhan • Barcelona • Belgrade Special Issue Editor Ioannis Argyros Cameron University USA Editorial Office MDPI St. Alban-Anlage 66 4052 Basel, Switzerland This is a reprint of articles from the Special Issue published online in the open access journal Symmetry (ISSN 2073-8994) in 2019 (available at: https://www.mdpi.com/journal/symmetry/special issues/ Symmetry Operator Theory Equations). For citation purposes, cite each article independently as indicated on the article page online and as indicated below: LastName, A.A.; LastName, B.B.; LastName, C.C. Article Title. Journal Name Year, Article Number, Page Range. ISBN 978-3-03921-666-6 (Pbk) ISBN 978-3-03921-667-3 (PDF) c 2019 by the authors. Articles in this book are Open Access and distributed under the Creative Commons Attribution (CC BY) license, which allows users to download, copy and build upon published articles, as long as the author and publisher are properly credited, which ensures maximum dissemination and a wider impact of our publications. The book as a whole is distributed by MDPI under the terms and conditions of the Creative Commons license CC BY-NC-ND. Contents About the Special Issue Editor ...................................... vii Preface to ”Symmetry with Operator Theory and Equations” .................... ix Alicia Cordero, Jonathan Franceschi, Juan R. Torregrosa and Anna Chiara Zagati A Convex Combination Approach for Mean-Based Variants of Newton’s Method Reprinted from: Symmetry 2019, 11, 1106, doi:10.3390/sym11091106 ................ 1 Alicia Cordero, Ivan Girona and Juan R. Torregrosa A Variant of Chebyshev’s Method with 3αth-Order of Convergence by Using FractionalDerivatives Reprinted from: Symmetry 2019, 11, 1017, doi:10.3390/sym11081017 ................. 17 R.A. Alharbey, Ioannis K. Argyros and Ramandeep Behl Ball Convergence for Combined Three-Step Methods Under Generalized Conditions in Banach Space Reprinted from: Symmetry 2019, 11, 1002, doi:10.3390/sym11081002 ................ 28 Ioannis K. Argyros, Santhosh George, Chandhini Godavarma and Alberto A. Magren´ ˜ an Extended Convergence Analysis of the Newton–Hermitian and Skew–Hermitian Splitting Method Reprinted from: Symmetry 2019, 11, 981, doi:10.3390/sym11080981 ................. 39 R. A. Alharbey, Munish Kansal, Ramandeep Behl and J.A. Tenreiro Machado Efficient Three-Step Class of Eighth-Order Multiple Root Solvers and Their Dynamics Reprinted from: Symmetry 2019, 11, 837, doi:10.3390/sym11070837 ................. 54 Janak Raj Sharma, Sunil Kumar and Ioannis K. Argyros Development of Optimal Eighth Order Derivative-Free Methods for Multiple Roots of Nonlinear Equations Reprinted from: Symmetry , 11, 766, doi:10.3390/sym11060766 .................... 84 Mujahid Abbas, Yusuf Ibrahim, Abdul Rahim Khan and Manuel de la Sen Strong Convergence of a System of Generalized Mixed Equilibrium Problem, Split Variational Inclusion Problem and Fixed Point Problem in Banach Spaces Reprinted from: Symmetry 2019, 11, 722, doi:10.3390/sym11050722 .................101 Mehdi Salimi and Ramandeep Behl Sixteenth-Order Optimal Iterative Scheme Based on Inverse Interpolatory Rational Function for Nonlinear Equations Reprinted from: Symmetry 2019, 11, 691, doi:10.3390/sym11050691 .................122 Munish Kansal, Ramandeep Behl, Mohammed Ali A. Mahnashi and Fouad Othman Mallawi Modified Optimal Class of Newton-Like Fourth-Order Methods for Multiple Roots Reprinted from: Symmetry 2019, 11, 526, doi:10.3390/sym11040526 .................133 Janak Raj Sharma, Deepak Kumar and Ioannis K. Argyros An Efficient Class of Traub-Steffensen-Like Seventh Order Multiple-Root Solvers with Applications Reprinted from: Symmetry , 11, 518, doi:10.3390/sym11040518 ....................144 v Ramandeep Behl, M. Salimi, M. Ferrara, S. Sharifi and Samaher Khalaf Alharbi Some Real-Life Applications of a Newly Constructed DerivativeFree Iterative Scheme Reprinted from: Symmetry 2019, 11, 239, doi:10.3390/sym11020239 .................161 Ioannis K. Argyros, Stepan Shakhno, Halyna Yarmola Two-Step Solver for Nonlinear Equations Reprinted from: Symmetry 2019, 11, 128, doi:10.3390/sym11020128 .................175 Ramandeep Behl, Ioannis K. Argyros, J.A. Tenreiro Machado and Ali Saleh Alshomrani Local Convergence of a Family of Weighted-Newton Methods Reprinted from: Symmetry 2019, 11, 103, doi:10.3390/sym11010103 .................184 vi About the Special Issue Editor Ioannis Argyros, he is a professor at the Department of Mathematics Sciences, Cameron University. His research interests include: Applied mathematics, Operator theory, Computational mathematics and iterative methods especially on Banach spaces. He has published more than a thousand peer reviewed papers, thirty two books and seventeen chapters in books. He is an active reviewer of a plethora of papers and books, and has received several national and international awards. He has supervised two PhD students, several MSc. and undergraduate students, and has been the external evaluator for many PhD theses, tenure and promotion applicants. vii Preface to ”Symmetry with Operator Theory and Equations” The development of iterative procedures for solving systems or nonlinear equations in abstract spaces is an important and challenging task. Recently such procedures have been extensively used in many diverse disciplines such as Applied Mathematics; Mathematical: Biology; Chemistry; Economics; Physics, and also Engineering to mention a few. The main purpose of this special issue is to present new ideas in this area of research with applications. This issue gives an opportunity to researchers and practitioners to communicate their recent works. Topics included in this issue are: From the 35 articles received, this special issue includes 13 high-quality peer-reviewed papers reflecting recent trends in the aforementioned topics. We hope that the presented results would lead to new ideas in the future. Ioannis Argyros Special Issue Editor ix S S symmetry Article A Convex Combination Approach for Mean-Based Variants of Newton’s Method Alicia Cordero 1, Jonathan Franceschi 1,2,*, Juan R. Torregrosa 1 and Anna C. Zagati 1,2 1 Institute of Multidisciplinary Mathematics, Universitat Politècnica de València, Camino de Vera, s/n, 46022-Valencia, Spain 2 Universitá di Ferrara, via Ludovico Ariosto, 35, 44121 Ferrara, Italy * Correspondence: [email protected] Received: 31 July 2019; Accepted: 18 August 2019; Published: 2 September 2019 Abstract: Several authors have designed variants of Newton’s method for solving nonlinear equations by using different means. This technique involves a symmetry in the corresponding fixed-point operator. In this paper, some known results about mean-based variants of Newton’s method (MBN) are re-analyzed from the point of view of convex combinations. A new test is developed to study the order of convergence of general MBN. Furthermore, a generalization of the Lehmer mean is proposed and discussed. Numerical tests are provided to support the theoretical results obtained and to compare the different methods employed. Some dynamical planes of the analyzed methods on several equations are presented, revealing the great difference between the MBN when it comes to determining the set of starting points that ensure convergence and observing their symmetry in the complex plane. Keywords: nonlinear equations; iterative methods; general means; basin of attraction 1. Introduction We consider the problem of finding a simple zero α of a function f : I ⊂ R → R, defined in an open interval I. This zero can be determined as a fixed point of some function g by means of the one-point iteration method: = ( ) = xn+1 g xn , n 0, 1, . , (1) where x0 is the starting point. The most widely-used example of these kinds of methods is the classical Newton’s method given by: ( ) = − f xn = xn+1 xn , n 0, 1, . (2) f (xn) It is well known that it converges quadratically to simple zeros and linearly to multiple zeros. In the literature, many modifications of Newton’s scheme have been published in order to improve its order of convergence and stability. Interesting overviews about this area of research can be found in [1–3]. The works of Weerakoon and Fernando [4] and, later, Özban [5] have inspired a whole set of variants of Newton’s method, whose main characteristic is the use of different means in the iterative expression. { } α It is known that if a sequence xn n≥0 tends to a limit in such a way that there exist a constant > C 0 and a positive integer n0 such that: | − α|≤ | − α|p ∀ ≥ xn+1 C xn , n n0, (3) Symmetry 2019, 11, 1106; doi:10.3390/sym11091106 www.mdpi.com/journal/symmetry 1 Symmetry 2019, 11, 1106 for p ≥ 1, then p is called the order of convergence of the sequence and C is known as the asymptotic error constant. For p = 1, constant C satisfies 0 < C ≤ 1. If we denote by en = xn − α the exact error of the nth iterate, then the relation: = p + O( p+1) en+1 Cen en (4) is called the error equation for the method and p is the order of convergence. Let us suppose that f : I ⊆ R → R is a sufficiently-differentiable function and α is a simple zero of f . It is plain that: x f (x)= f (xn)+ f (t) dt. (5) xn Weerakoon and Fernando in [4] approximated the definite integral (5) by using the trapezoidal rule and taking x = α, getting: 0 ≈ f (xn)+1/2(α − xn)(f (xn)+ f (α)), (6) α and therefore, a new approximation xn+1 to is given by: ( ) ( ) = − f xn = − f xn = xn+1 xn , zn xn , n 0, 1, . (7) ( f (xn)+ f (zn))/2 f (xn) Thus, this variant of Newton’s scheme can be considered to be obtained by replacing the denominator f (xn) of Newton’s method (2) by the arithmetic mean of f (xn) and f (zn). Therefore, it is known as the arithmetic mean Newton method (AN). In a similar way, the arithmetic mean can be replaced by other means.
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