On the DSM version of Newton's method

A. G. Ramm

Eurasian mathematical journal, Tome 2 (2011) no. 3, pp. 89-97

Résumé

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 $t\ge0$ 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.