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.