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The Dynamical Systems Method (DSM) is justified for solving operatorequations $F(u)=f$, 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 $F\inC^1_{loc}$, that is, it has a continuous with respect to $u$ Fr\'echetderivative $F'(u)$, 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 $u$. Itis proved under these assumptions that the continuous analog of the Newton'smethod $\dot{u}=-[F'(u)]^{-1}(F(u)-f), \quad u(0)=u_0, \quad (*)$ convergesstrongly 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 Cauchyproblem (*) was not established earlier without assuming that $F'(u)$ isLipschitz-continuous. The case when $F$ is not a global homeomorphism but amonotone operator in $H$ is also considered.

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