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Keywords = Fatou set

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21 pages, 608 KB  
Article
On Extending the Applicability of Iterative Methods for Solving Systems of Nonlinear Equations
by Indra Bate, Muniyasamy Murugan, Santhosh George, Kedarnath Senapati, Ioannis K. Argyros and Samundra Regmi
Axioms 2024, 13(9), 601; https://doi.org/10.3390/axioms13090601 - 4 Sep 2024
Cited by 7 | Viewed by 1073
Abstract
In this paper, we present a technique that improves the applicability of the result obtained by Cordero et al. in 2024 for solving nonlinear equations. Cordero et al. assumed the involved operator to be differentiable at least five times to extend a two-step [...] Read more.
In this paper, we present a technique that improves the applicability of the result obtained by Cordero et al. in 2024 for solving nonlinear equations. Cordero et al. assumed the involved operator to be differentiable at least five times to extend a two-step p-order method to order p+3. We obtained the convergence order of Cordero et al.’s method by assuming only up to the third-order derivative of the operator. Our analysis is in a more general commutative Banach algebra setting and provides a radius of the convergence ball. Finally, we validate our theoretical findings with several numerical examples. Also, the concept of basin of attraction is discussed with examples. Full article
(This article belongs to the Special Issue Differential Equations and Related Topics, 2nd Edition)
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18 pages, 4765 KB  
Article
Choosing the Best Members of the Optimal Eighth-Order Petković’s Family by Its Fractal Behavior
by Xiaofeng Wang and Wenshuo Li
Fractal Fract. 2022, 6(12), 749; https://doi.org/10.3390/fractalfract6120749 - 19 Dec 2022
Cited by 7 | Viewed by 2305
Abstract
In this paper, by applying Petković’s iterative method to the Möbius conjugate mapping of a quadratic polynomial function, we attain an optimal eighth-order rational operator with a single parameter r and research the stability of this method by using complex dynamics tools on [...] Read more.
In this paper, by applying Petković’s iterative method to the Möbius conjugate mapping of a quadratic polynomial function, we attain an optimal eighth-order rational operator with a single parameter r and research the stability of this method by using complex dynamics tools on the basis of fractal theory. Through analyzing the stability of the fixed point and drawing the parameter space related to the critical point, the parameter family which can make the behavior of the corresponding iterative method stable or unstable is obtained. Lastly, the consequence is verified by showing their corresponding dynamical planes. Full article
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15 pages, 333 KB  
Article
Convergence Theorems in Interval-Valued Riemann–Lebesgue Integrability
by Anca Croitoru, Alina Gavriluţ, Alina Iosif and Anna Rita Sambucini
Mathematics 2022, 10(3), 450; https://doi.org/10.3390/math10030450 - 30 Jan 2022
Cited by 11 | Viewed by 2959
Abstract
We provide some limit theorems for sequences of Riemann–Lebesgue integrable functions. More precisely, Lebesgue-type convergence and Fatou theorems are established. Then, these results are extended to the case of Riemann–Lebesgue integrable interval-valued multifunctions. Full article
15 pages, 1432 KB  
Article
Fixed Point Root-Finding Methods of Fourth-Order of Convergence
by Alicia Cordero, Lucía Guasp and Juan R. Torregrosa
Symmetry 2019, 11(6), 769; https://doi.org/10.3390/sym11060769 - 6 Jun 2019
Cited by 6 | Viewed by 3401
Abstract
In this manuscript, by using the weight-function technique, a new class of iterative methods for solving nonlinear problems is constructed, which includes many known schemes that can be obtained by choosing different weight functions. This weight function, depending on two different evaluations of [...] Read more.
In this manuscript, by using the weight-function technique, a new class of iterative methods for solving nonlinear problems is constructed, which includes many known schemes that can be obtained by choosing different weight functions. This weight function, depending on two different evaluations of the derivative, is the unique difference between the two steps of each method, which is unusual. As it is proven that all the members of the class are optimal methods in the sense of Kung-Traub’s conjecture, the dynamical analysis is a good tool to determine the best elements of the family in terms of stability. Therefore, the dynamical behavior of this class on quadratic polynomials is studied in this work. We analyze the stability of the presented family from the multipliers of the fixed points and critical points, along with their associated parameter planes. In addition, this study enables us to select the members of the class with good stability properties. Full article
(This article belongs to the Special Issue Fixed Point Theory and Computational Analysis with Applications)
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