On the Local convergence of two-step Newton type Method in Banach Spaces under generalized Lipschitz Conditions

TL;DR

This paper analyzes the local convergence of a two-step Newton method with cubic convergence rate for solving nonlinear equations in Banach spaces, under generalized Lipschitz conditions on derivatives.

Contribution

It establishes convergence results for the method assuming the derivative satisfies generalized Lipschitz conditions, including radius or center Lipschitz conditions with weak L-average.

Findings

Convergence proven under generalized Lipschitz conditions.

Method achieves cubic convergence rate.

Results applicable to derivatives with weak L-average.

Abstract

The motive of this paper is to discuss the local convergence of a two-step Newton type method of convergence rate three for solving nonlinear equations in Banach spaces. It is assumed that the first order derivative of nonlinear operator satisfies the generalized Lipschitz i.e. $L$-average condition. Also, some results on convergence of the same method in Banach spaces are established under the assumption that the derivative of the operators satisfies the radius or center Lipschitz condition with a weak $L$-average particularly it is assumed that $L$ is positive integrable function but not necessarily non-decreasing.