2004 Conference on Diff. Eqns. and Appl. in Math. Biology, Nanaimo, BC, Canada.
Electron. J. Diff. Eqns., Conference 12, 2005, pp. 79-85.

Continuous Newton method for star-like functions

Yakov Lutsky

Abstract:
We study a continuous analogue of Newton method for solving the nonlinear equation
$$ \varphi (z) =0, $$
where $\varphi(z)$ holomorphic function and $0\in\overline{\varphi ( D)}$. It is proved that this method converges, to the solution for each initial data $z\in D$, if and only if $\varphi(z)$ is a star-like function with respect to either an interior or a boundary point. Our study is based on the theory of one parameter continuous semigroups. It enables us to consider convergence in the case of an interior as well as a boundary location of the solution by the same approach.

Published April 20, 2005.
Math Subject Classifications: 49M15, 46T25, 47H25.
Key Words: Newton method; star-like functions; continuous semigroup.

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Yakov Lutsky
Department of Mathematics
Ort Braude College
Karmiel 21982, Israel
email: yalutsky@yahoo.com

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