### 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 2 C1
loc, that is, it has
the Fr´echet derivative F0(u) continuous with respect to u, that the operator [F0(u)]−1
exists for all u 2 H and is bounded, ||[F0(u)]−1|| m(u), where m(u) > 0 depends on
u, and is not necessarily uniformly bounded with respect to u. It is proved under these
assumptions that the continuous analogue of the Newton’s method
u˙ = −[F0(u)]−1(F(u) − f), u(0) = u0, ( )
converges strongly to the solution of the equation F(u) = f for any f 2 H and any
u0 2 H. The global (and even local) existence of the solution to the Cauchy problem
( ) was not established earlier without assuming that F0(u) is Lipschitz-continuous.
The case when F is not a global homeomorphism but a monotone operator in H is also
considered.

### Description:

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 2 C1
loc, that is, it has
the Fr´echet derivative F0(u) continuous with respect to u, that the operator [F0(u)]−1
exists for all u 2 H and is bounded, ||[F0(u)]−1|| m(u), where m(u) > 0 depends on
u, and is not necessarily uniformly bounded with respect to u. It is proved under these
assumptions that the continuous analogue of the Newton’s method
u˙ = −[F0(u)]−1(F(u) − f), u(0) = u0, ( )
converges strongly to the solution of the equation F(u) = f for any f 2 H and any
u0 2 H. The global (and even local) existence of the solution to the Cauchy problem
( ) was not established earlier without assuming that F0(u) is Lipschitz-continuous.
The case when F is not a global homeomorphism but a monotone operator in H is also
considered.