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This article is cited in 1 scientific paper (total in 1 paper)
On the DSM version of Newton's method
A. G. Ramm Mathematics Department, Kansas State University, Manhattan, KS, USA
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 $t\ge0$ 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.
Keywords and phrases:
nonlinear equations, homeomorphism, surjectivity, dynamical systems method (DSM).
Received: 22.01.2011
Citation:
A. G. Ramm, “On the DSM version of Newton's method”, Eurasian Math. J., 2:3 (2011), 89–97
Linking options:
https://www.mathnet.ru/eng/emj64 https://www.mathnet.ru/eng/emj/v2/i3/p89
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