Kantorovich's theorem on Newton's method for solving strongly regular generalized equation

TL;DR

This paper proves that under strong regularity conditions, Newton's method for solving generalized equations converges quadratically to a unique solution, extending and unifying existing theories using Banach perturbation and majorant techniques.

Contribution

It establishes quadratic convergence of Newton's method for generalized equations under strong regularity, unifying previous results with new analytical techniques.

Findings

Quadratic convergence of Newton's method under strong regularity.

Uniqueness of the solution in a neighborhood of the starting point.

Unified framework using Banach perturbation and majorant techniques.

Abstract

In this paper we consider the Newton's method for solving the generalized equation of the form $ f(x) +F(x) \ni 0, $ where $f:{\Omega}\to Y$ is a continuously differentiable mapping, $X$ and $Y$ are Banach spaces, $\Omega\subseteq X$ an open set and $F:X \rightrightarrows Y$ be a set-valued mapping with nonempty closed graph. We show that, under strong regularity of the generalized equation, concept introduced by S.M.Robinson in [27], and starting point satisfying the Kantorovich's assumptions, the Newton's method is quadratically convergent to a solution, which is unique in a suitable neighborhood of the starting point. The analysis presented based on Banach Perturbation Lemma for generalized equation and the majorant technique, allow to unify some results pertaining the Newton's method theory.