On the Local Convergence of Two-Step Newton Type Method in Banach Spaces under Generalized Lipschitz Conditions

Akanksha Saxena,Jai P Jaiswal,Ioannis K Argyros,Christopher Argyros,Kamal R Pardasani

doi:10.3390/math9060669

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. Our new idea gives a tighter convergence analysis without new conditions. The proposed technique is useful in expanding the applicability of iterative methods. Useful examples justify the theoretical conclusions.

Highlights

Consider a nonlinear operator t : Ω ⊆ X → Y such that X and Y are two Banach spaces, Ω is a non-empty open convex subset and t is Fréchet differentiable nonlinear operator
For re-investigating the local convergence of Newton’s method, generalized Lipschitz conditions was constructed by Wang [16], in which a non-decreasing positive integrable function was used instead of usual Lipschitz constant
A new technique is developed in view of which we achieve a tighter local convergence analysis compared with earlier studies, without additional hypothesis

Summary

Introduction

Consider a nonlinear operator t : Ω ⊆ X → Y such that X and Y are two Banach spaces, Ω is a non-empty open convex subset and t is Fréchet differentiable nonlinear operator. For re-investigating the local convergence of Newton’s method, generalized Lipschitz conditions was constructed by Wang [16], in which a non-decreasing positive integrable function was used instead of usual Lipschitz constant. Wang and Li [17] derived some results on convergence of Newton’s method in Banach spaces when derivative of the operators satisfies the radius or center Lipschitz condition but with a weak L-average. Shakhno [18] have studied the local convergence of the two step Secant-type method [2], when the first-order divided differences satisfy the generalized Lipschitz conditions. Numerical examples are presented to justify the significance of the results

Generalized Lipschitz Conditions

The Uniqueness Ball for the Solution of Equations

Convergence under Weak L-Average

Numerical Examples

Conclusions