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Solving variational inclusions by a multipoint iteration method under center-Hölder continuity conditions

Catherine Cabuzel;  Alain Pietrus — 2007

Applicationes Mathematicae

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 ∇²f satifies 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 method is superquadratic.

Local analysis of a cubically convergent method for variational inclusions

Steeve Burnet;  Alain Pietrus — 2011

Applicationes Mathematicae

This paper deals with variational inclusions of the form 0 ∈ φ(x) + F(x) where φ is a single-valued function admitting a second order Fréchet derivative and F is a set-valued map from ℝ q to the closed subsets of ℝ q . When a solution z̅ of the previous inclusion satisfies some semistability properties, we obtain local superquadratic or cubic convergent sequences.

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