Justification of the Dynamical Systems Method for Global Homeomorphism

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 ∈ C loc, that is, it has the Fréchet derivative F ′(u) continuous with respect to u, that the operator [F ′(u)]−1 exists for all u ∈ H and is bounded, ||[F ′(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