Catherine Cabuzel

• PhD
• University of the French Antilles

This paper deals with variational inclusions of the form \(0 \in K-f(x)\) where \(f : \mathbb{R}^{n} \rightarrow \mathbb{R} ^{m}\) is a semismooth function and \(K\) is a nonempty closed convex cone in \(\mathbb{R}^{m}\). We show that the previous problem can be solved by a Newton-type method using the Clarke generalized Jacobian of \(f\). The resu...

This paper deals with the study of an iterative method for solving a variational inclusion of the form 0 ∈ f (x)+F (x) where f is a locally Lipschitz subanalytic function and F is a set-valued map from R n to the closed subsets of R n. To this inclusion, we firstly associate a Newton then secondly an Inexact Newton type sequence and with some semis...

This paper deals with the study of an iterative method for solving a variational inclusion of the form 0 ∈ f (x)+F (x) where f is a locally Lipschitz subanalytic function and F is a set-valued map from Rn to the closed subsets of Rn. To this inclusion, we firstly associate a Newton then secondly an Inexact Newton type sequence and with some semista...

This chapter deals with variational inclusions of the form 0 ∈ f (x) + g(x) + F(x) where f is a locally Lipschitz and subanalytic function, g is a Lipschitz function, F is a set-valued map, acting all in ℝn and n is a positive integer. The study of the previous variational inclusion depends on the properties of the function g. The behaviour as been...

The midpoint method was previously used in the solving of equations and in this work, the aim is to adapt the theory to the set-valued analysis in order to approach solutions of generalized equations of the form 0_inf(x)+F(x). In particular, we show the existence and convergence fo sequences provided the second order derivative of the function f sa...

The aim of this study is the approximation of a solution x ∗ of the generalized equation 0∈f(x)+F(x) in Banach spaces, where f is a single function whose second order Fréchet derivative ∇2f verifies an Hölder condition, and F stands for a set-valued map with closed graph. Using a fixed point theorem and proceeding by induction under the pseudo-Lips...

This paper deals with variational inclusions of the form: 0∈f(x)+F(x) where f is a single function admitting a second order Fréchet derivative and F is a set-valued map acting in Banach spaces. We prove the existence of a sequence (x k ) satisfying 0∈f(x k )+∑ i=1 M a i ∇f(x k +β i (x k+1 -x k ))(x k+1 -x k )+F(x k+1 ) where the single-valued funct...

This paper concerns variational inclusions of the form 0 Î f(x) + F(x)0 \in f(x) + F(x) where f is a single locally Lipschitz subanalytic function and F is a set-valued map acting in Banach spaces. We prove the existence and the convergence of a sequence (x k ) satisfying 0 Î f(xk)+Df(xk)(xk+1-xk)+F(xk+1)0 \in f(x_k)+\Delta f(x_k)(x_{k+1}-x_k)+F(x...

In this paper, we prove the existence of a sequence (x(k)) verifying 0 is an element of f(x(k))+Sigma(M)(i=1) a(i)del f(x(k)+beta(i)(x(k+1)-x(k)))(x(k+1)-x(k))+F(x(k+1)), where the function f admits a second order Frechet derivative which satisfies a Holder condition and F is a set valued map from a banach space X to the subsets of Y which is also...

In this paper we study an iterative method for solving a perturbed variational inclusion of the form 0 is an element of f (x) + g(x) + F(x), where f is a locally Lipschitz subanalytic function, g is a Lipschitz function and F is a set-valued map. We prove existence and convergence of a sequence (x(k)) satisfying 0 is an element of f(x(k)) + g(x(k))...

We prove the existence of a sequence (x k ) satisfying 0∈f(x k )+∑ i=1 M a i ∇f(x k +β i (x k+1 -x k ))(x k+1 -x k )+F(x k+1 ), where f is a function whose second order Fréchet derivative ∇ 2 f satisfies a center-Hölder condition and F is a set-valued map from a Banach space X to the subsets of a Banach space Y. We show that the convergence of this...