Kantorovich’s theorem on Newton’s method for solving generalized equations under the majorant condition

摘要

In this paper we consider a version of the Kantorovich’s theorem for solving the generalized equation F(x)+T(x)∋0, where F is a Fréchet derivative function and T is a set-valued and maximal monotone acting between Hilbert spaces. We show that this method is quadratically convergent to a solution of F(x)+T(x)∋0. We have used the idea of majorant function, which relaxes the Lipschitz continuity of the derivative F′. It allows us to obtain the optimal convergence radius, uniqueness of solution and also to solving generalized equations under Smale’s condition.