General study of iterative processes of R-order at least three under weak convergence conditions.
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Universidad de La Rioja
info
ISSN: 0022-3239
Year of publication: 2007
Volume: 133
Issue: 2
Pages: 163-177
Type: Article
More publications in: Journal of Optimization Theory and Applications
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Abstract
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.