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 ∈ C¹_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:

&udot; = -[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.