A modification of the convergence conditions for Picard's iteration.
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Universidad de La Rioja
info
ISSN: 0101-8205
Ano de publicación: 2004
Volume: 23
Número: 1
Páxinas: 55-65
Tipo: Artigo
Outras publicacións en: Computational and Applied Mathematics
Resumo
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