Semilocal Convergence of the Extension of Chun’s Method

Alicia Cordero,Eulalia Martínez,Javier G Maimó,María P Vassileva,Juan R Torregrosa

doi:10.3390/axioms10030161

Abstract

In this work, we use the technique of recurrence relations to prove the semilocal convergence in Banach spaces of the multidimensional extension of Chun’s iterative method. This is an iterative method of fourth order, that can be transferred to the multivariable case by using the divided difference operator. We obtain the domain of existence and uniqueness by taking a suitable starting point and imposing a Lipschitz condition to the first Fréchet derivative in the whole domain. Moreover, we apply the theoretical results obtained to a nonlinear integral equation of Hammerstein type, showing the applicability of our results.

Highlights

IntroductionThe simplicity of its iterative expression and second order of convergence confers to Newton’s method a very useful efficiency in many applied problems
We focus on solving nonlinear systems of equations, that is F(x) = 0 where F : Ω ⊂ X −→ Y, is a nonlinear continuous and twice differentiable Fréchet operator in an open convex set Ω, and X and Y Banach spaces
The divided difference operator can be expressed in an integral way by means of the Genocchi-Hermite formula [x, y; F] =

Summary

Introduction

The simplicity of its iterative expression and second order of convergence confers to Newton’s method a very useful efficiency in many applied problems. In the recent years, one can find in the literature a great variety of iterative methods that can reach higher convergence order and better efficiency than Newton’s method, see [2,3] and the references therein. In these texts, we can see the study about the convergence order of the methods always related with the computational efficiency reached

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Results

Conclusion