Justification of the Dynamical Systems Method (DSM) for global homeomorphisms

TL;DR

This paper justifies the use of the Dynamical Systems Method (DSM) for solving nonlinear operator equations in Hilbert spaces, proving convergence under broad conditions without requiring Lipschitz continuity of the derivative.

Contribution

It establishes the convergence of a continuous Newton-like method for global homeomorphisms in Hilbert spaces without assuming Lipschitz continuity of the derivative.

Findings

Proves strong convergence of the DSM for global homeomorphisms.

Shows existence of solutions without Lipschitz continuity assumption.

Considers cases where the operator is monotone but not a homeomorphism.

Abstract

The Dynamical Systems Method (DSM) is justified for solving operator equations $F(u)=f$, where $F$ is a nonlinear operator in a Hilbert space $H$. It is assumed that $F$ is a global homeomorphism of $H$ onto $H$, that $F\in C^1_{loc}$, that is, it has a continuous with respect to $u$ Fr\'echet derivative $F'(u)$, that the operator $[F'(u)]^{-1}$ exists for all $u\in H$ and is bounded, $||[F'(u)]^{-1}||\leq m(u)$, where $m(u)>0$ is a constant, depending on $u$, and not necessarily uniformly bounded with respect to $u$. It is proved under these assumptions that the continuous analog of the Newton's method $\dot{u}=-[F'(u)]^{-1}(F(u)-f), \quad u(0)=u_0, \quad (*)$ converges strongly to the solution of the equation $F(u)=f$ for any $f\in H$ and any $u_0\in H$. The global (and even local) existence of the solution to the Cauchy problem (*) was not established earlier without assuming that $F'(u)$ is Lipschitz-continuous. The case when $F$ is not a global homeomorphism but a monotone operator in $H$ is also considered.