Local convergence of some iterative methods for generalized equations

We study generalized equations of the following form: 0 ∈ f (x) +g(x) + F(x), (∗) where f is Fréchet differentiable in a neighborhood of a solution x∗ of (∗) and g is Fréchet differentiable at x∗ and where F is a set-valued map acting in Banach spaces. We prove the existence of a sequence (xk ) satisfying 0 ∈ f (xk )+ g(xk)+ ∇f (xk) + [xk−1,xk;g] (xk+1 −xk )+ F(xk+1) which is super-linearly convergent to a solution of (∗).We also present other versions of this iterative procedure that have superlinear and quadratic convergence, respectively.  2003 Elsevier Inc. All rights reserved.