Justification of the Dynamical Systems Method (DSM) for global homeomorphisms
Provides a theoretical justification for a dynamical systems method in nonlinear functional analysis, addressing a gap in existing convergence proofs.
The paper proves global convergence of the continuous analog of Newton's method for solving nonlinear operator equations under relaxed smoothness conditions, removing the Lipschitz continuity requirement on the derivative.
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 $u$ Fréchet 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.