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Open AccessArticle

Unified Local Convergence for Newton’s Method and Uniqueness of the Solution of Equations under Generalized Conditions in a Banach Space

1
Department of Mathematics Sciences Lawton, Cameron University, Lawton, OK 73505, USA
2
Departamento de Matemáticas y Computación, Universidad de La Rioja, 26006 Logroño, Spain
3
Facultad de Educación, Universidad Internacional de La Rioja, 26006 Logroño, Spain
4
Escuela Superior de Ingeniería y Tecnología, Universidad Internacional de La Rioja, 26006 Logroño, Spain
*
Author to whom correspondence should be addressed.
Mathematics 2019, 7(5), 463; https://doi.org/10.3390/math7050463
Received: 5 March 2019 / Revised: 7 May 2019 / Accepted: 13 May 2019 / Published: 23 May 2019
(This article belongs to the Special Issue Computational Methods in Analysis and Applications)
Under the hypotheses that a function and its Fréchet derivative satisfy some generalized Newton–Mysovskii conditions, precise estimates on the radii of the convergence balls of Newton’s method, and of the uniqueness ball for the solution of the equations, are given for Banach space-valued operators. Some of the existing results are improved with the advantages of larger convergence region, tighter error estimates on the distances involved, and at-least-as-precise information on the location of the solution. These advantages are obtained using the same functions and Lipschitz constants as in earlier studies. Numerical examples are used to test the theoretical results. View Full-Text
Keywords: Newton’s method; local convergence; Newton-Mysovskii conditions; Banach space Newton’s method; local convergence; Newton-Mysovskii conditions; Banach space
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MDPI and ACS Style

Argyros, I.K.; Magreñán, Á.A.; Orcos, L.; Sarría, Í. Unified Local Convergence for Newton’s Method and Uniqueness of the Solution of Equations under Generalized Conditions in a Banach Space. Mathematics 2019, 7, 463.

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