The Australian Journal of Mathematical Analysis and Applications


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ISSN 1449-5910  

 

Paper Information

Paper Title:

Traub-Potra-Type Method for Set-Valued Maps

Author(s):

Ioannis K. Argyros and Sad Hilout

Cameron University,
Department of Mathematics Sciences,
Lawton, OK 73505,
USA

iargyros@cameron.edu

URL: http://www.cameron.edu/~ioannisa/

Poitiers University,
Laboratoire de Mathematiques et Applications,
Bd. Pierre et Marie Curie, Teleport 2, B.P. 30179,
86962 Futuroscope Chasseneuil Cedex,
France

said.hilout@math.univ-poitiers.fr

http://www-math.univ-poitiers.fr/~hilout/

Abstract:

We introduce a new iterative method for approximating a locally unique solution of variational inclusions in Banach spaces by using generalized divided differences of the first order. This method extends a method considered by Traub  (in the scalar case) and by Potra  (in the Banach spaces case) for solving nonlinear equations to variational inclusions. An existence-convergence theorem and a radius of convergence are given under some conditions on divided differences operator and Lipschitz-like continuity property of set-valued mappings. The R-order of the method is equal to the unique positive root of a certain cubic equation, which is $1.839..., and as such it compares favorably to related methods such as the Secant method which is only of order $1.618....

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