Methods and Applications of Analysis

Volume 16 (2009)

Number 4

Secant-like Method for Solving Generalized Equations

Pages: 469 – 478



Ioannis K. Argyros

Saïd Hilout


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.


Fréchet derivative, Banach space, divided differences, Secant method, generalized equation, set-valued map, Aubin's continuity, radius of convergence

2010 Mathematics Subject Classification

47H04, 49M15, 65B05, 65G99, 65H10, 65K10

Published 1 January 2009