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

Regularity properties of a cubically convergent scheme for generalized equations

Michel H. Geoffroy 1, and  Alain Piétrus 1,

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 $f+G$.

Keywords: Set-valued mappings, strong subregularity, cubic convergence., metric regularity.
Mathematics Subject Classification: Primary: 49J53, 49J40, 90C4.
Citation: Michel H. Geoffroy, Alain Piétrus. Regularity properties of a cubically convergent scheme for generalized equations. Communications on Pure and Applied Analysis, 2007, 6 (4) : 983-996. doi: 10.3934/cpaa.2007.6.983
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