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 \dot{u} = -[F'(u)]^{-1}(F(u) - f), u(0) = u_0, then it is proved that u(t) exists for all t >= 0 and is unique, that there exists u(\infty) := \lim_{t \to \infty} u(t), and that F(u(\infty)) = f. These results are obtained for an arbitrary initial approximation u_0. 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 u_0 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.