Ball Convergence of a Parametric Efficient Family of Iterative Methods for Solving Nonlinear Equations

Samundra Regmi,Christopher Argyros,Santhosh George,Ioannis Argyros

doi:10.3390/foundations1010004

Abstract

The goal is to extend the applicability of Newton-Traub-like methods in cases not covered in earlier articles requiring the usage of derivatives up to order seven that do not appear in the methods. The price we pay by using conditions on the first derivative that actually appear in the method is that we show only linear convergence. To find the convergence order is not our intention, however, since this is already known in the case where the spaces coincide with the multidimensional Euclidean space. Note that the order is rediscovered by using ACOC or COC, which require only the first derivative. Moreover, in earlier studies using Taylor series, no computable error distances were available based on generalized Lipschitz conditions. Therefore, we do not know, for example, in advance, how many iterates are needed to achieve a predetermined error tolerance. Furthermore, no uniqueness of the solution results is available in the aforementioned studies, but we also provide such results. Our technique can be used to extend the applicability of other methods in an analogous way, since it is so general. Finally note that local results of this type are important, since they demonstrate the difficulty in choosing initial points. Our approach also extends the applicability of this family of methods from the multi-dimensional Euclidean to the more general Banach space case. Numerical examples complement the theoretical results.

Highlights

Let B1, B2 denote Banach spaces and T ⊆ B1 be a nonempty convex and open set
In this article we extended the applicability of Newton-Traub-like methods in cases not covered before requiring the usage of derivatives up to order seven that do not appear in the methods
To find the convergence order is not, our intention, since this is already known in the case of spaces that coincide with the multidimensional Euclidean space

Summary

Introduction

Let B1, B2 denote Banach spaces and T ⊆ B1 be a nonempty convex and open set. Set L B(B1, B2) = {V : B1 → B2 is by a bounded linear operator}. Another problem is that there are no error bounds on xn − ξ or results on uniqueness of ξ or how close to ξ we should start, and the selection of x0 is really “ a shot in the dark” To address all these concerns about this very efficient and useful method, we only use conditions on the first derivative. Assume there exists a function φ : D0 → R that is continuous and increasing satisfying φ(0) = 0. Assume there exist functions φ1 : D1 → R, φ2 : D1 × D1 → R, and φ3 : D1 → R that are continuous and increasing. 2 2K0 +K was obtained by Argyros in [1] as the convergence radius for Newton’s method under conditions (17)–(19). Notice that the convergence radius for Newton’s method given independently by Rheinboldt [23] and.

Numerical Examples

Conclusions