General study of iterative processes of R-order at least three under weak convergence conditions.

  1. Hernández, M.A. 1
  2. Romero, N. 1
  1. 1 Universidad de La Rioja
    info

    Universidad de La Rioja

    Logroño, España

    ROR https://ror.org/0553yr311

Revista:
Journal of Optimization Theory and Applications

ISSN: 0022-3239

Año de publicación: 2007

Volumen: 133

Número: 2

Páginas: 163-177

Tipo: Artículo

DOI: 10.1007/S10957-007-9197-X SCOPUS: 2-s2.0-34547341209 WoS: WOS:000248206000003 GOOGLE SCHOLAR

Otras publicaciones en: Journal of Optimization Theory and Applications

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2005/00064/001

Resumen

We consider a family of Newton-type iterative processes solving nonlinear equations in Banach spaces, that generalizes the usually iterative methods of R-order at least three. The convergence of this family in Banach spaces is usually studied when the second derivative of the operator involved is Lipschitz continuous and bounded. In this paper, we relax the first condition, assuming that F (x)-F (y) ω(x-y), where ω is a nondecreasing continuous real function. We prove that the different R-orders of convergence that we can obtain depend on the quasihomogeneity of the function ω. We end the paper by applying the study to some nonlinear integral equations. © 2007 Springer Science+Business Media, LLC.