Local convergence of Newton's method for solving generalized equations with monotone operator

Abstract

In this paper we study Newton's method for solving the generalized equation $F(x)+T(x)\ni 0$ in Hilbert spaces, where $F$ is a Fréchet differentiable function and $T$ is set-valued and maximal monotone. We show that this method is local quadratically convergent to a solution. Using the idea of majorant condition on the nonlinear function which is associated to the generalized equation, the convergence of the method, the optimal convergence radius and results on the convergence rate are established. The advantage of working with a majorant condition rests in the fact that it allow to unify several convergence results pertaining to Newton's method.