A Newton–Secant method for differentiable set-valued maps in Nachi–Penot sense

Abstract

This paper is concerned with the solving of variational inclusions of the form \(0\in f(x) + g(x) + F(x) – K\), where \(g\) is a function which is differentiable at a solution \(x^{*}\) of the inclusion but may be not differentiable in a neighborhood of \(x^{*}\). The function \(f\) and the set-valued mapping \(F\) are differentiable in the sense of Nachi–Penot [14] and \(K\) is a nonempty closed convex cone.
We introduce a Newton-Secant method to solve our problem and the sequence associated is semilocally convergent to \(x^{*}\) with an order equal to \(\frac{1 +\sqrt{5}}{2}\). Finally, some numerical results are also given to illustrate the convergence of the proposed method.