Regularity properties of a cubically convergent scheme for generalized equations

Michel H. Geoffroy; Alain Piétrus

doi:10.3934/cpaa.2007.6.983

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2007, Volume 6, Issue 4: 983-996. Doi: 10.3934/cpaa.2007.6.983

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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

Received: November 30, 2006

Revised: February 28, 2007

Published: November 2007

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Abstract

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 $f+G$.

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Mathematics Subject Classification: Primary: 49J53, 49J40, 90C48.

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