Local convergence of Newton's method under majorant condition

TL;DR

This paper provides a comprehensive local convergence analysis of Newton's method for nonlinear equations under a majorant condition, relaxing traditional assumptions and establishing convergence properties, uniqueness range, and convergence rate.

Contribution

It introduces a new convergence analysis framework for Newton's method without requiring convexity of the majorant function's derivative.

Findings

Convergence is guaranteed under a majorant condition without convexity assumptions.

The largest range for the solution's uniqueness is determined.

Optimal convergence radius and rate are established.

Abstract

A local convergence analysis of Newton's method for solving nonlinear equations, under a majorant condition, is presented in this paper. Without assuming convexity of the derivative of the majorant function, which relaxes the Lipschitz condition on the operator under consideration, convergence, the biggest range for uniqueness of the solution, the optimal convergence radius and results on the convergence rate are established. Besides, two special cases of the general theory are presented as an application.