2004 Conference on Diff. Eqns. and Appl. in Math. Biology, Nanaimo, 
BC, Canada.
Electronic Journal of Differential Equations,
Conference 12, 2005, pp. 79-85. 

Title: Continuous Newton method for star-like functions

Author: Yakov Lutsky (Ort Braude College, Karmiel, Israel)
 
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.