# Justification of the Dynamical Systems Method DSM for global homeomorphisms

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The Dynamical Systems Method DSM is justified for solving operator equations $Fu=f$, 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 $Fu$, that the operator $Fu^{-1}$ exists for all $u\in H$ and is bounded, $||Fu^{-1}||\leq mu$, where $mu0$ 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 Newtons method $\dot{u}=-Fu^{-1}Fu-f, \quad u0=u 0, \quad *$ converges strongly to the solution of the equation $Fu=f$ 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 $Fu$ is Lipschitz-continuous. The case when $F$ is not a global homeomorphism but a monotone operator in $H$ is also considered.

Autor: A. G. Ramm

Fuente: https://archive.org/