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Mathematics 2019, 7(2), 207; https://doi.org/10.3390/math7020207

Extended Local Convergence for the Combined Newton-Kurchatov Method Under the Generalized Lipschitz Conditions

1
Department of Mathematical Sciences, Cameron University, Lawton, OK 73505, USA
2
Department of Theory of Optimal Processes, Ivan Franko National University of Lviv, 79000 Lviv, Ukraine
*
Author to whom correspondence should be addressed.
Received: 4 February 2019 / Revised: 19 February 2019 / Accepted: 20 February 2019 / Published: 23 February 2019
(This article belongs to the Special Issue Iterative Methods for Solving Nonlinear Equations and Systems)
Full-Text   |   PDF [249 KB, uploaded 23 February 2019]

Abstract

We present a local convergence of the combined Newton-Kurchatov method for solving Banach space valued equations. The convergence criteria involve derivatives until the second and Lipschitz-type conditions are satisfied, as well as a new center-Lipschitz-type condition and the notion of the restricted convergence region. These modifications of earlier conditions result in a tighter convergence analysis and more precise information on the location of the solution. These advantages are obtained under the same computational effort. Using illuminating examples, we further justify the superiority of our new results over earlier ones. View Full-Text
Keywords: nonlinear equation; iterative process; non-differentiable operator; Lipschitz condition nonlinear equation; iterative process; non-differentiable operator; Lipschitz condition
This is an open access article distributed under the Creative Commons Attribution License which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited (CC BY 4.0).
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Argyros, I.K.; Shakhno, S. Extended Local Convergence for the Combined Newton-Kurchatov Method Under the Generalized Lipschitz Conditions. Mathematics 2019, 7, 207.

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