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 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.