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December 2009 Secant-like Method for Solving Generalized Equations
Ioannis K. Argyros, Saïd Hilout
Methods Appl. Anal. 16(4): 469-478 (December 2009).


In A Kantorovich–type analysis for a fast iterative method for solving nonlinear equations, and Convergence and applications of Newton–type iterations, Argyros introduced a new derivative–free quadratically convergent method for solving a nonlinear equation in Banach space. In this paper, we extend this method to generalized equations in order to approximate a locally unique solution. The method uses only divided differences operators of order one. Under some Lipschitz–type conditions on the first and second order divided differences operators and Lipschitz–like property of set–valued maps, an existence–convergence theorem and a radius of convergence are obtained. Our method has the following advantages: we extend the applicability of this method than all the previous ones, and we do not need to evaluate any Fréchet derivative. We provide also an improvement on the radius of convergence for our algorithm, under some center–condition and less computational cost. Numerical examples are also provided.


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Ioannis K. Argyros. Saïd Hilout . "Secant-like Method for Solving Generalized Equations." Methods Appl. Anal. 16 (4) 469 - 478, December 2009.


Published: December 2009
First available in Project Euclid: 12 October 2010

zbMATH: 1206.65167
MathSciNet: MR2734496

Primary: 47H04 , 47H17 , 49M15 , 65B05 , 65G99 , 65H10 , 65K10

Keywords: Aubin’s continuity , Banach space , divided differences , Fréchet derivative , generalized equation , Radius of convergence , secant method , set–valued map

Rights: Copyright © 2009 International Press of Boston


Vol.16 • No. 4 • December 2009
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