Title: Convergence of an iterative method in Banach spaces with Lipschitz continuous first derivative

Authors: P.K. Parida; D.K. Gupta

Addresses: Center for Applied Mathematics, Central University of Jharkhand, Ranchi 835205, India ' Department of Mathematics, Indian Institute of Technology, Kharagpur 721 302, West Bengal, India

Abstract: In this paper, the convergence of a third order Newton-like method used for solving F(x) = 0 in Banach spaces is established by using recurrence relations, when the first Fréchet derivative of F satisfies the Lipschitz condition. Here, we relaxed the necessary conditions on F in order to study the convergence. This work is useful when either second derivative of F may not exist or may not satisfy Lipschitz condition. An existence and uniqueness theorem is derived for the root x* of F(x) = 0. A numerical example is given to demonstrate the applicability of the method.

Keywords: nonlinear operator equations; Lipschitz continuous first derivative; recurrence relations; cubic convergence; a priori error bounds; Banach spaces.

DOI: 10.1504/IJANS.2014.068254

International Journal of Applied Nonlinear Science, 2014 Vol.1 No.4, pp.289 - 299

Received: 03 Mar 2014
Accepted: 30 Jun 2014
Published online: 08 Apr 2015 *

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