Universality of Newton's method

TL;DR

This paper proves the convergence of Newton's method for operator equations without requiring smoothness of the derivative, establishing conditions under which solutions can be reliably found.

Contribution

It demonstrates that Newton's method converges for a broad class of operator equations without smoothness assumptions on the derivative.

Findings

Convergence of Newton's method is established without smoothness assumptions.

Every solvable operator equation can be solved if initial guess is close and inverse derivative norm is bounded.

Provides conditions for convergence applicable to a wide range of operator equations.

Abstract

Convergence of the classical Newton's method and its DSM version for solving operator equations $F(u)=h$ is proved without any smoothness assumptions on $F'(u)$. It is proved that every solvable equation $F(u)=f$ can be solved by Newton's method if the initial approximation is sufficiently close to the solution and $||[F'(y)]^{-1}||\leq m$, where $m>0$ is a constant.