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

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 ∈ C 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 ∈ H and is bounded, ||[F ′(u)]−1|| ≤ 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 u̇ = −[F ′(u)]−1(F (u) − f), u(0) = u0, (∗) converges strongly to the solution of the equation F (u) = f for any f ∈ H and any u0 ∈ 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. MSC: 4705; 4706; 47J35