On a Newton Type Iterative Method for Solving Inclusions
D. Azé, Chin-Chen Chou · Mathematics of Operations Research · 1995
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 ‖y 0 ‖ is small enough. As a consequence we obtain an open mapping theorem for multifunctions whose proof is constructive.