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

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 $F\in C_{loc}^1$, that is, it has a continuous with respect to $u$ Fréchet derivative $F^{\prime }(u)$, that the operator $[F^{\prime }(u){]}^{-1}$ exists for all $u\in H$ and is bounded, $||[F^{\prime }(u){]}^{-1}||\le m(u)$, 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 $\dot{u}=-[F^{\prime }(u){]}^{-1}(F(u)-f),u(0)=u_0,(\ast )$ 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 $F^{\prime }(u)$ is Lipschitz-continuous. The case when $F$ is not a global homeomorphism but a monotone operator in $H$ is also considered.