Journal of Mathematical Chemistry

, Volume 55, Issue 7, pp 1427–1442

# Local convergence of a relaxed two-step Newton like method with applications

• I. K. Argyros
• Á. A. Magreñán
• L. Orcos
• J. A. Sicilia
Original Paper

## Abstract

We present a local convergence analysis for a relaxed two-step Newton-like method. We use this method to approximate a solution of a nonlinear equation in a Banach space setting. Hypotheses on the first Fréchet derivative and on the center divided-difference of order one are used. In earlier studies such as Amat et al. (Numer Linear Algebra Appl 17:639–653, 2010, Appl Math Lett 25(12):2209–2217, 2012, Appl Math Comput 219(24):11341–11347, 2013, Appl Math Comput 219(15):7954–7963, 2013, Reducing Chaos and bifurcations in Newton-type methods. Abstract and applied analysis. Hindawi Publishing Corporation, Cairo, 2013) these methods are analyzed under hypotheses up to the second Fréchet derivative and divided differences of order one. Numerical examples are also provided in this work.

### Keywords

Two-step Newton’s method Banach space Fréchet derivative Divided difference of first order Local–semilocal convergence

65D10 65D99

## Notes

### Acknowledgements

This research was supported by Universidad Internacional de La Rioja (UNIR, http://www.unir.net), under the Plan Propio de Investigación, Desarrollo e Innovación 3 [2015–2017]. Research group: Modelación matemática aplicada a la ingeniería(MOMAIN), by the the grant SENECA 19374/PI/14 and by Ministerio de Ciencia y Tecnología MTM2014-52016-C2-$$\{01\}$$-P.

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© Springer International Publishing Switzerland 2017

## Authors and Affiliations

• I. K. Argyros
• 1
• Á. A. Magreñán
• 2
• L. Orcos
• 2
• J. A. Sicilia
• 2
1. 1.Department of Mathematical SciencesCameron UniversityLawtonUSA
2. 2.Universidad Internacional de La Rioja (UNIR)Logroño, La RiojaSpain