A study of convergence for a fourth-order two-point iteration in Banach spaces.

  1. Ezquerro, J.A. 1
  2. Hernandez, M.A. 1
  1. 1 Universidad de La Rioja
    info

    Universidad de La Rioja

    Logroño, España

    ROR https://ror.org/0553yr311

Revista:
Kodai mathematical journal

ISSN: 0386-5991

Año de publicación: 1999

Volumen: 22

Número: 3

Páginas: 373-383

Tipo: Artículo

SCOPUS: 2-s2.0-85024742577 GOOGLE SCHOLAR lock_openAcceso abierto editor

Otras publicaciones en: Kodai mathematical journal

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Resumen

The authors consider the method (1)x2n+1:=x2n−f′(x2n)−1f(x2n),x2n+2:=x2n+1−34H(I−32H)(x2n+1−x2n),H:=H(x2n,x2n+1):=f′(x2n)−1[f′(x2n+23(x2n+1−x2n))−f′(x2n)],for solving operator equations f(x)=0 in Banach spaces. A Kantorovich-type convergence analysis is carried out under the assumption that the third derivative of the operator f satisfies the Lipschitz condition. The theorem proved establishes a convergence condition, existence and uniqueness radii, and a priori error bounds. As an application, these bounds are calculated for the integral equation x(s)=s[1−12∫10cos(x(t))dt] with trivial (linear) solution. {Reviewer's remark: The authors' claim that the method (1) exhibits the convergence rate of order four contradicts Maĭstrovskiĭ's classical result on the optimality of Newton's method