On semilocal convergence of three-step Kurchatov method under weak condition

Himanshu Kumar

doi:10.1007/s40065-020-00308-8

Abstract

The purpose of this paper to establish the semilocal convergence analysis of three-step Kurchatov method under weaker conditions in Banach spaces. We construct the recurrence relations under the assumption that involved first-order divided difference operators satisfy the omega condition. Theorems are given for the existence-uniqueness balls enclosing the unique solution. The application of the iterative method is shown by solving nonlinear system of equations and nonlinear Hammerstein-type integral equations. It illustrates the theoretical development of this study.

Highlights

In this article, we consider the problem of approximating a unique solution ∗ of a nonlinear equation B( ) = 0, (1)where B : C ⊂ X → Y is continuous but non-differentiable nonlinear operator defined on a nonempty open convex subset C of Banach space X with values in Banach space Y
The semilocal convergence is based on the information around an initial point, to give criteria ensuring the convergence of the iterative method; while the local one is, based on the information around a solution, to find estimates of the radii of convergence balls
The article was devoted to study the semilocal convergence of three-step Kurchatov-type method under ωcondition on the first-order divided difference using recurrence relations

Summary

Introduction

We consider the problem of approximating a unique solution ∗ of a nonlinear equation. Kumar and Parida [29] gave semilocal convergence of this method under Lipschitz continuity conditions on first-order divided difference operator by using recurrence relations. Argyros et al [8] introduced the Chebyshev–Secant-type method and gave the semilocal convergence by the use of recurrence relations under Lipschitz-type conditions on the first-order divided difference operator. Authors established the semilocal convergence of method using recurrence relations under Lipschitz continuity condition In all these above-mentioned methods, divided difference operator was frozen. Ezquerro et al [21] worked on semilocal convergence of Kurchatov two-step method (4) by the help of recurrence relations under condition (5) They established the R-order of convergence with efficiency index and computational efficiency index. R is the least positive real root and B( 0, R), B( 0, R) ⊆ C

Semilocal convergence of Kurchatov three-step method

Optimal computational efficiency

Application of two-dimensional problem

Application of Higher-dimensional problem

Conclusions