On the Dsm Version of Newton’s Method

Abstract. The DSM (dynamical systems method) version of the Newton’s method is for solving operator equation F (u) = f in Banach spaces is discussed. If F is a global homeomorphism of a Banach space X onto X, that is continuously Fréchet differentiable, and the DSM version of the Newton’s method is u̇ = −[F ′(u)]−1(F (u)− f), u(0) = u0, then it is proved that u(t) exists for all t ≥ 0 and is unique, that there exists u(∞) := limt→∞ u(t), and that F (u(∞)) = f . These results are obtained for an arbitrary initial approximation u0. This means that convergence of the DSM version of the Newton’s method is global. The proof is simple, short, and is based on a new idea. If F is not a global homeomorphism, then a similar result is obtained for u0 sufficiently close to y, where F (y) = f and F is a local homeomorphism of a neighborhood of y onto a neighborhood of f . These neighborhoods are specified.