A Newton iteration for differentiable set-valued maps

We employ recent developments of generalized differentiation concepts for set-valued mappings and present a Newton-like iteration for solving generalized equations of the form f ( x ) + F ( x ) ∋ 0 where f is a single-valued function while F stands for a set-valued map, both of them being smooth mappings acting between two general Banach spaces  X and Y . The Newton iteration we propose is constructed on the basis of a linearization of both f and F ; we prove that, under suitable assumptions on the “derivatives” of f and F , it converges Q-linearly to a solution to the generalized equation in question. When we strengthen our assumptions, we obtain the Q-quadratic convergence of the method.