<abstract xmlns="http://www.w3.org/1999/xhtml"><p>This paper is devoted to the study of a multi-step method with divided differences for solving nonlinear equations in Banach spaces. In earlier studies, hypotheses on the Fréchet derivative up to the sixth order of the operator under consideration is used to prove the convergence of the method. That restricts the applicability of the method. In this paper we extended the applicability of the sixth-order multi-step method by using only hypotheses on the first derivative of the operator involved. Our convergence conditions are weaker than the conditions used in earlier studies. Numerical examples where earlier results cannot be applied to solve equations but our results can be applied are also given in this study.</p></abstract>

This paper is devoted to the study of a multi-step method with divided differences for solving nonlinear equations in Banach spaces. In earlier studies, hypotheses on the Fréchet derivative up to the sixth order of the operator under consideration is used to prove the convergence of the method. That restricts the applicability of the method. In this paper we extended the applicability of the sixth-order multi-step method by using only hypotheses on the first derivative of the operator involved. Our convergence conditions are weaker than the conditions used in earlier studies. Numerical examples where earlier results cannot be applied to solve equations but our results can be applied are also given in this study.

Local convergence of a multi-step high order method with divided differences under hypotheses on the first derivative

Department of Mathematical and Computational Sciences, NIT Karnataka

Annales Universitatis Paedagogicae Cracoviensis. Studia Mathematica

This work is licensed under a Creative Commons Attribution Share-Alike 4.0 License.

This paper is devoted to the study of a multi-step method with divided differences for solving nonlinear equations in Banach spaces. In earlier studies,...

Local convergence of a multi-step high order method with divided differences under hypotheses on the first derivative

Published Online: Jan 27, 2018

Page range: 41 - 50

Received: Feb 02, 2017

Accepted: Jun 08, 2017

DOI: https://doi.org/10.1515/aupcsm-2017-0003

Keywords
Multi-step method, restricted convergence domain, radius of convergence, local convergence

© 2017 Ioannis K. Argyros et al., published by De Gruyter Open

This work is licensed under a Creative Commons Attribution Share-Alike 4.0 License.

Local convergence of a multi-step high order method with divided differences under hypotheses on the first derivative

Published Online: Jan 27, 2018

Page range: 41 - 50

Received: Feb 02, 2017

Accepted: Jun 08, 2017

DOI: https://doi.org/10.1515/aupcsm-2017-0003

KeywordsMulti-step method, restricted convergence domain, radius of convergence, local convergence

© 2017 Ioannis K. Argyros et al., published by De Gruyter Open

This work is licensed under a Creative Commons Attribution Share-Alike 4.0 License.

Keywords
Multi-step method, restricted convergence domain, radius of convergence, local convergence