APPLICATIONES MATHEMATICAE 34,4 (2007), pp. 493–503 Catherine Cabuzel and Alain Pietrus (Pointe-`a-Pitre) SOLVING VARIATIONAL INCLUSIONS BY A MULTIPOINT ITERATION METHOD UNDER CENTER-H ¨ OLDER CONTINUITY CONDITIONS Abstract. We prove the existence of a sequence (x k ) satisfying 0 ∈ f (x k ) + ∑ M i=1 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 Fr´echet derivative ∇ 2 f satifies a center-H¨older 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. 1. Introduction. In a previous paper [5], we used a multipoint iteration formula to solve the “abstract” generalized equation (1) 0 ∈ f (x)+ F (x) where f is a function from X into Y which possesses a second order Fr´echet derivative, F is a set-valued map from X to the subsets of Y with closed graph, and X , Y are two Banach spaces. We obtained cubic convergence when the second order Fr´echet derivative is Lipschitz. Recall that equation (1) is an abstract model for various problems: • When F = {0}, (1) is an equation. • When F is the positive orthant in R m , (1) is a system of inequalities. • When F is the normal cone to a convex and closed set in X , (1) may represent variational inequalities. Now, we recall some results obtained for the original problem. When the Fr´echet derivative ∇f is locally Lipschitz, Dontchev [6, 7] associated to (1) a Newton-type method based on a partial linearization which provides local quadratic convergence. Following his work, Pietrus [19] obtained a 2000 Mathematics Subject Classification : 49J53, 47H04, 65K10. Key words and phrases : set-valued mapping, generalized equations, Aubin continuity, pseudo-Lipschitz map, multipoint iteration formula, center-H¨older continuity. [493] c  Instytut Matematyczny PAN, 2007