On a Newton Type Iterative Method for Solving Inclusions

We introduce a notion of strict differentiability for multifunctions by means of a notion of tangency based on a uniform property of Clarke's tangent cone. Given a multifunction G and a point (a, b) ∈ G and assuming that the derivative DG(a, b) is surjective and has a bounded inverse, we build a sequence ((x_n, y_n)) ⊂ G, such that d(x_n+1 − x_n, DG(a, b)⁻¹(−y_n)) converges to 0. The sequence ((x_n, y_n)) is shown to converge to (x, 0) where x is a solution of 0 ∈ G(x) provided the norm of ‖y₀‖ is small enough. As a consequence we obtain an open mapping theorem for multifunctions whose proof is constructive.