On the Kantorovich's theorem for Newton's method for solving generalized equations under the majorant condition

TL;DR

This paper extends Kantorovich's theorem to generalized equations involving set-valued and monotone operators, demonstrating quadratic convergence under relaxed conditions using a majorant function approach.

Contribution

It introduces a Kantorovich-type theorem for generalized equations with set-valued operators, relaxing Lipschitz conditions via a majorant function, and establishes quadratic convergence and solution uniqueness.

Findings

Quadratic convergence of the method to a solution.

Optimal convergence radius and solution uniqueness.

Applicability under Smale's condition.

Abstract

In this paper we consider a version of the Kantorovich's theorem for solving the generalized equation $F(x)+T(x)\ni 0$, where $F$ is a Fr\'echet 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 $F(x)+T(x)\ni 0$. 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.