# American Institute of Mathematical Sciences

December  2007, 6(4): 983-996. doi: 10.3934/cpaa.2007.6.983

## Regularity properties of a cubically convergent scheme for generalized equations

 1 Laboratoire Analyse Optimisation Contrôle, Dept. de Mathématiques, Université des Antilles et de la Guyane, B.P. 250, 97157 Pointe à Pitre, Guadeloupe, France, France

Received  December 2006 Revised  March 2007 Published  September 2007

We consider the perturbed generalized equation $v \in f(x) +G(x)$ where $v$ is a perturbation parameter, $f$ is a function acting from a Banach space $X$ to a Banach space $Y$ while $G: X \rightarrow Y$ is a set-valued mapping. We associate to this generalized equation the following iterative procedure:

$v \in f(x_n)+ \nabla f(x_n)(x_{n+1}-x_n) +\frac{1}{2}\nabla^2 f(x_n) (x_{n+1}-x_n)^2 +G(x_{n+1}).$ $\quad$ (*)

We investigate some stability properties of the method (*) and we study the behavior of the sequences that it generates, more precisely, we show that they inherit some regularity properties from the mapping $f+G$.

Citation: Michel H. Geoffroy, Alain Piétrus. Regularity properties of a cubically convergent scheme for generalized equations. Communications on Pure & Applied Analysis, 2007, 6 (4) : 983-996. doi: 10.3934/cpaa.2007.6.983
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