A modification of the convergence conditions for Picard's iteration.

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

    Universidad de La Rioja

    Logroño, España

    ROR https://ror.org/0553yr311

Revista:
Computational and Applied Mathematics

ISSN: 0101-8205

Año de publicación: 2004

Volumen: 23

Número: 1

Páginas: 55-65

Tipo: Artículo

DOI: 10.1590/S0101-82052004000100003 SCOPUS: 2-s2.0-77049109723 WoS: WOS:000208135000003 GOOGLE SCHOLAR lock_openAcceso abierto editor

Otras publicaciones en: Computational and Applied Mathematics

Repositorio institucional: lock_openAcceso abierto Editor

Resumen

To solve by successive approximation nonlinear equations of the form F(x)=0, where F:Ω⊆X→X, is an operator defined on an open convex domain of a Banach space X with values in X, one uses a fixed point theorem based method which requires the operator G(x)=x−F(x) to be a contraction. This has a very limited scope of applicability. The aim of this paper is to modify the successive approximation method wherein the semilocal convergence of the successive approximation has been studied under the alternate condition: ∥F′(x)−Ix∥≤ω(∥x∥), where ω:R+→R+ is a non-decreasing function. The study presented in this paper belongs to the class of unbounded generalized contraction results, where the main idea is to generalize the fixed point theorem using a nonlinear majorant function in the contraction inequality