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\AtBeginDocument{{\noindent\small
2004 Conference on Diff. Eqns. and Appl. in Math. Biology,  Nanaimo, BC, Canada.\newline
{\em Electronic Journal of Differential Equations},
Conference 12, 2005, pp. 79--85.\newline
ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or
http://ejde.math.unt.edu
\newline ftp ejde.math.txstate.edu  (login: ftp)}
\thanks{\copyright 2005 Texas State University - San Marcos.}
\vspace{9mm}}
\setcounter{page}{79}

\begin{document}

\title[\hfilneg EJDE/Conf/12 \hfil Continuous Newton method]
{Continuous Newton method for star-like functions}

\author[Y. Lutsky \hfil EJDE/Conf/12 \hfilneg]
{Yakov Lutsky}

\address{Yakov Lutsky \hfill\break
Department of  Mathematics, Ort Braude College,
Karmiel 21982, Israel}
\email{yalutsky@yahoo.com}

\date{}
\thanks{Published April 20, 2005.}
\subjclass[2000]{49M15, 46T25, 47H25}
\keywords{Newton method; star-like functions; continuous semigroup}

\begin{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.
\end{abstract}

\maketitle
\numberwithin{equation}{section}
\newtheorem{theorem}{Theorem}[section]
\newtheorem{definition}[theorem]{Definition}
\newtheorem{remark}[theorem]{Remark}

\section{Results}

Let $D\subset\mathbb{C}$ be a domain
(that is, an open connected subset of $\mathbb{C}$).
The set of all holomorphic functions on $D$ will be denoted by
$\mathop{\rm Hol}\nolimits(D,\mathbb{C})$.
We consider a nonlinear equation
\begin{equation} \label{e1}
\varphi ( z) =0,
\end{equation}
where $\varphi(z)\in\mathop{\rm Hol}\nolimits( D,\mathbb{C})$ and
$0\in\overline{\varphi ( D)}$.

In the known Newton's method \cite{AK, OP}, the
solution of \eqref{e1} can be found as a limit of the sequence
$\{z_{n}\}$, $n=0,1,2\dots $. The first term $z_{0}\in D$ is given and
other terms are constructed by the iterative process
\begin{equation} \label{e2}
z_{n+1}=z_{n}-\frac{\varphi ( z_{n}) }{\varphi '(z_{n}) }.
\end{equation}
It is well known that the convergence of process \eqref{e2} depends on the
choice of the initial approximation $z_{0}\in D$. If $z_{0}$ is chosen
arbitrarily, then sequence may diverge.

The Continuous Newton Method (CNM) and its modifications \cite{AGS, AK} are an
alternative approach to the solution of \eqref{e1}.
The CNM has been considered as the solution of the following Cauchy
problem (continuous analogue of process \eqref{e2})
\begin{equation} \label{e3}
\begin{gathered}
\frac{\partial u(t,z)}{\partial t}+\frac{\varphi (u(t,z))}{\varphi '(u(t,z))}=0 \\
u(0,z)=z,
\end{gathered}
\end{equation}
where the initial condition $z$ is some point which belongs to the domain
$D$.
A solution of the \eqref{e1} was obtained as the limit

\begin{equation} \label{e4}
\lim_{t\to \infty }\ u( t,z) =\tau \in \overline{D},
\end{equation}
where $u( t,z) $ is the solution of \eqref{e3}.

The continuous Newton Method has several advantages over the iterative method \eqref{e2} because
convergence theorems for CNM usually can be obtained easily.
 However, as well as in the iterative process \eqref{e2}, for its
realization in a general case it is necessary
to choose an initial condition by a special way.

This problem leads us to the following question: Are there  functions
$\varphi(z)$ for which the solution of \eqref{e1} can be found by  CNM
under arbitrary initial condition $z\in D$ in the domain $D$?

In this article the question is answered. It is proved that
star-like functions and only they satisfy this requirement. Our
study of CNM is based on results of the theory of one-parameter continuous
semigroups (see \cite{SD} and the references given there). It has
permitted us to consider the convergence of CNM both to an interior point
and to a boundary point by the same approach.

In addition the uniqueness of solution of \eqref{e1} in $\overline{D}$ is
proved. This solution is obtained as limit \eqref{e4}. More exact results are
obtained when $ D=\Delta $  is an open unit disk in $\mathbb{C}$. In
particular, in this case the exponential convergence of CNM is established.

It is important to note one more problem in realization of CNM for the
solution of \eqref{e1}. As it was mentioned above,
$0\in\overline{\varphi ( D) }$. It means that the function $\varphi$ may
have no null point in $D$. Moreover, $\varphi$ even may be undefined on
the boundary $\partial D$. Therefore, we  consider the solution
of \eqref{e1} at the boundary points of domain $D$ in following
generalized meaning.

\begin{definition} \rm
 A point $\tau\in \partial D$ is said to be a generalized solution of
\eqref{e1} on the set $\overline{D}$ if there is a Jordan curve $\gamma
\in \overline{D}$ such that $\gamma \cap \partial D=\tau$ and
\begin{equation}
\lim_{z\to \tau,\ z\in \gamma}\varphi(z)=0.
\end{equation}
\end{definition}

Here we give some definitions which will be used in the sequel.

\begin{definition} \rm
We will say that CNM is well defined on the domain $D$ if the Cauchy
problem \eqref{e3}  has a unique solution for each $z\in D$ and
\[
\{ u (t,z),\ t\geq 0,\ z\in D\} \subset D.
\]
\end{definition}

\begin{definition} \rm
We will say that the CNM  converges globally  in the domain $D$ if the
limit \eqref{e4} exists for each solution $u( t,z) $ of Cauchy problem
\eqref{e3}.
\end{definition}

\begin{remark} \label{rmk1} \rm
The CNM is well defined on the domain D if and only if the function
$f( z) =\frac{\varphi ( z) }{\varphi '(z) }$ is a generator of a
one-parameter continuous semigroup of
holomorphic self-mappings
$S_{f}=\{ F_{t}:D\to D,\ t\geq 0\} $, where
\begin{gather} \label{e6}
F_{t}=\varphi ^{-1}\circ e^{-t}\circ \varphi\,,\\
u( t,z) =F_{t}( z) =\varphi ^{-1}( e^{-t}\varphi
( z) ) \quad (t\geq 0,\ z\in D) \label{e7}
\end{gather}
is the unique solution of the Cauchy problem \eqref{e3},
\cite{RS,SD}. In this case the set
 $\gamma_{z}( t) =\{ u( t,z) ,t\geq 0\} $
is a Jordan curve for each $z\in D$.
\end{remark}

The set of generators of one-parameter semigroups of  holomorphic
self-mappings in $D$ will be denoted by $\mathcal{G}( D)$.

\begin{definition} \rm
 The set $\Omega \subset \mathbb{C}$ is called star-shaped if for any
 $\omega \in \Omega $, the point $t\omega $ belongs to $\Omega $
 for every $t\in ( 0,1]$.
\end{definition}

\begin{definition}
 A univalent holomorphic function $f: D\to \mathbb{C}$ is said to
be star-like if the set $f( D) $ is star-shaped .
\end{definition}

\begin{remark} \label{rmk2} \rm
It follows from \cite{AR}, that univalent function 
$\varphi ( z)\; (\in \mathop{\rm Hol}( D,\mathbb{C})) $ is a star-like
function, if and only if the mapping
$f( z) =\frac{\varphi (z) }{\varphi '( z) }\in \mathcal{G}( D)$.
\end{remark}

Now we will formulate the main result of this paper.

\begin{theorem} \label{thm1}
Let $\varphi ( z)\;(\in \mathop{\rm Hol}\Delta ,\mathbb{C}))$
be a univalent function such that $\overline{\varphi ( \Delta ) }\ni 0$.
Then continuous Newton Methos is well defined in $\Delta $ if and only if
the following  inequality holds:
\begin{equation} \label{e8}
\mathop{\rm Re}\{ \overline{z}\frac{\varphi ( z) }{\varphi'( z) }\}
 \geq (1-\vert z\vert ^{2}) \cdot
\mathop{\rm Re}\{ \overline{z}\frac{\varphi ( 0) }{\varphi'(0)}\} , z\in \Delta .
\end{equation}
Moreover, in this case CNM  converges globally to a
unique point $\tau \in \overline{\Delta }$. In addition,
\begin{itemize}
\item[(i)] If $\tau \in \Delta $ is a solution of
\eqref{e1} and $\vert \tau\vert \leq \rho <1$, then
\begin{equation} \label{e9}
\vert \tau -u( t,z) \vert \leq \delta ^{-1}\exp
\{ -\frac{1-\vert z\vert }{1+\vert z\vert }\delta
t\} \vert \tau -z\vert ,\quad  z\in \Delta ,\; t\geq 0,
\end{equation}
where $\delta =\frac{1-\rho }{1+\rho }$ and
$u(t,z) $ is a solution of Cauchy  problem \eqref{e3}.

\item[(ii)] If $\tau \in \partial \Delta $ is a generalized
solution of \eqref{e1}, then the limit
\begin{equation} \label{e10}
\beta =\lim_{r\to 1^{-}\ }\frac{\varphi ( r\tau )
}{\varphi'( r\tau ) ( r-1) \tau }>0\
\end{equation}
exist and
\begin{equation} \label{e11}
\vert \tau -u( t,z) \vert \leq \frac{\sqrt{2}e^{-\frac{ \beta
}{2}t}}{\sqrt{1-z^{2}}}\vert \tau -z\vert ,\quad z\in \Delta ,\;
t\geq 0,
\end{equation}
where $u(t,z)$ is a solution of the Cauchy problem \eqref{e3}.
\end{itemize}
\end{theorem}


\begin{remark} \label{rmk3} \rm
As a matter of fact \cite{POM}, if $\tau \in \partial \Delta $ is a generalized
solution of \eqref{e1}, then
\[
\lim_{z\to \tau ,\tau \in \gamma }\varphi (z) =0
\]
along each non-tangential curve $\gamma $ (i.e. there exists a
non-tangential limit at point $\tau $).
\end{remark}

The proof of Theorem \ref{thm1} is based on the following result.


\begin{theorem} \label{thm2}
Let $D$ be a bounded domain with Jordan boundary
$\partial D$ and $\varphi ( z)\; ( \in \mathop{\rm Hol}( D,\mathbb{C}) ) $
be a univalent function such that $\overline{\varphi ( D) }\ni 0$.
Then following two conditions are equivalent:
\begin{itemize}
\item[(i)] $\varphi ( z) $ is a star-like function.

\item[(ii)] The continuous Newton method is well defined in the domain
$D$.
\end{itemize}
Moreover, if it is this case, CNM  globally converges
to a unique point $\tau \in \overline{\Delta }$.
\end{theorem}

\begin{proof}
The equivalence $(i)\Longleftrightarrow (ii)$ follows from
Remarks \ref{rmk1} and \ref{rmk2}. Therefore, it is sufficient to prove the
latter assertion of this theorem on the global convergence.

Let  $\Omega= \varphi (D)$. We will consider following two
cases separately:  $0\in \Omega $ and $0\in \partial \Omega $.
If $0\in \Omega $, then $\tau =\varphi ^{-1}( 0) \in D$ is a unique
solution of \eqref{e1}. Since $\varphi ^{-1}( z) $ is a
continuous function at point $0$, we obtain by \eqref{e7} that
\[
\lim_{t\to \infty }u( t,z) \ =\lim_{t\to \infty
}\varphi ^{-1}( e^{-t}\varphi ( z) ) =\varphi
^{-1}( \lim_{t\to \infty }e^{-t}\varphi ( z) )
=\varphi ^{-1}( 0) =\tau
\]
for each $z\in D$. So in this case CNM converges globally.

Suppose now, that $0\in \partial \Omega $. Let $h:D\to \Delta $ be
any conformal mapping of  $D$ onto unit open disk
$\Delta =\{ z\in \mathbb{C}:\vert z\vert  < 1\} $. Then the linear
invertible operator
$T:\mathop{\rm Hol}( \Delta ,\mathbb{C}) \to \mathop{\rm Hol}( D,\mathbb{C})$,
defined by
\begin{equation} \label{e12}
T( f) =[ ( h^{-1})'] ^{-1}f\circ h^{-1}
\end{equation}
is invertible and maps $\mathcal{G}( \Delta ) $ onto $\mathcal{G}( D) $
(see \cite{ER,SD}). Moreover, if
\[
\{ F_{t}:D\to D,\; t\geq 0\}
\quad\mbox{and}\quad
\{ \Psi _{t} :\Delta \to \Delta ,\; t\geq 0\}
\]
are semigroups of holomorphic self-mappings, generated by $f$ and
$\psi=T( f) $, respectively (see \cite{SD}), then
\begin{equation} \label{e13}
F_{t}=h^{-1}\circ \Psi _{t}\circ h\,.
\end{equation}
In the considered case $f( z) =\frac{\varphi ( z) }{\varphi '( z) }
\in \mathcal{G}( D) $ has no null point in $D$. It follows from
\eqref{e12}, that the function $\psi (z) $ has no null point in $\Delta $.
 Therefore, for each point $z\in \Delta $ there exists a unique limit
\[
e=\lim_{t\to \infty }\Psi _{t}\ ( z) \in \partial \Delta\,.
\]
In supposition of the theorem the
boundary $\partial D$ is a Jordan curve, thus, applying Caratheodory
Theorem, we conclude, that the function $h( z) $ has a continuous
extension  to $D\cup \partial D$, \cite{POM}.
Therefore $\tau =h^{-1}( e) \in \partial D$ and for any $z\in D$
by \eqref{e13}, we have
\begin{align*}
\lim_{t\to \infty }\ u( t,z)
&=\lim_{t\to \infty }\ F_{t}( z) \\
&=\lim_{t\to \infty }\ h^{-1}( \Psi _{t}( h(z) ) ) \\
&= h^{-1}( \lim_{t\to \infty}\Psi _{t}( h( z) ) )\\
&=h^{-1}( e)=\tau\,.
\end{align*}
Further, it follows from \eqref{e7}, that for each point $z_0\in D$ 
along the curve
$\gamma _{z_0}( t) =\{ u( t,z_0) ,t\geq 0\} $,
\begin{equation} \label{e14}
\lim_{z\to \tau }\varphi ( z) =\lim_{t\to
\infty } \varphi ( F_{t}( z_0) ) =\lim_{t\to
\infty } \varphi ( \varphi ^{-1}( e^{-t}( z_0) )) =0\,.
\end{equation}
Thus $\tau $ is a generalized solution of equation \eqref{e1} in the
set $\overline{D}$. Therefore, to complete our proof in the case
$0\in \partial \Omega $, we need to show the uniqueness of generalized
solution $\tau $. Assume, that there exist another generalized solution
$\tau_{1}\in \partial D$ of \eqref{e1}. Then
there is a Jordan curve $\gamma _{1}\subset \overline{D}$ which begins at
some point $z_{0}\in D$ such that $\gamma _{1}\cap \partial D=\tau _{1}$
and
\begin{equation} \label{e15}
\lim_{z\to \tau _{1,}z\in \gamma _{1}}\varphi ( z) =0.
\end{equation}
Since $\gamma _{z_{0}}(t)=\{ u( t,z_{0}) ,t\geq 0\} $
is a Jordan curve, then the curve $\gamma =\gamma _{1}\cup \gamma_{z_{0}}( t) $
is Jordan too (see Fig .1).

\begin{figure}[htbp]
\begin{center}
\includegraphics[width=0.7\textwidth]{fig1}
\end{center}
\end{figure}

Let the points $\widehat{\tau }$, $\widehat{\tau }_{1}\in \partial D$ be
different from $\tau $, $\tau _{1}$. We will use the following notation:

$\widehat{\gamma }$ is some curve which connects the points $\widehat{\tau }$
and $\widehat{\tau }_{1}$, such that $\widehat{\gamma }\in \overline{D}$,
$\gamma \cap \widehat{\gamma }=\emptyset$ and
$\widehat{\gamma }\cap \partial D=\{\widehat{\tau },\widehat{\tau }_{1}\}$;

$\lambda $ (respectively $\lambda _{1})$ is  the part of boundary $\partial D$
which connects the points $\tau $ and $\widehat{\tau }$ (respectively
$\tau _{1} $ and $\widehat{\tau }_{1})$.

Let $D_{1}$ be a domain which boundary is
$\partial D_{1}=\gamma \cup \lambda \cup \widehat{\gamma }\cup \lambda _{1}$
and $\Omega _{1}=\varphi ( D_{1}) $. Then it follows from \eqref{e14}
 and \eqref{e15}, that the curve
$\varphi (\gamma )$ ($\subset \partial \Omega _{1}$) is closed and
$0\in \varphi (\gamma )$ (see Fig. 2).

Moreover, the domain $\Omega _{1}$ is placed in the external part of the
complex plane $\mathbb{C}$ with respect to the curve $\varphi (\gamma )$.
Therefore, there are $\omega \in \Omega _{1}$ and $t\in ( 0,1] $
such that $t\omega \notin \Omega _{1}$. It means, that $\Omega _{1}$ is not
star-shaped. This contradicts the star-likeness of the function
$\varphi(z) $. Thus uniqueness of the generalized solution $\tau $ is
proved.
The Theorem \ref{thm2} is proved.
\end{proof}

\begin{figure}[htbp]
\begin{center}
\includegraphics[width=0.7\textwidth]{fig2}
\end{center}
\end{figure}

\begin{proof}[Proof of Theorem \ref{thm1}]
It is proved in \cite{SD} that the function $f( z) \in \mathcal{G}( \Delta ) $
if and only if the following inequality holds.
\[
\mathop{\rm Re}\{ \overline{z}f( z) \} \leq (1-\vert
z\vert ^{2})\cdot \mathop{\rm Re}\{ \overline{z}f(0)\} ,\quad z\in \Delta .
\]
Hence, by Theorem \ref{thm2} and Remark \ref{rmk2} we obtain, that
inequality \eqref{e8} is equivalent to the assertions (i), (ii) of the
Theorem \ref{thm2}. Thus the CNM converges globally
to a unique solution $\tau \in \overline{\Delta }$ of
\eqref{e1}. Now we will show that the estimate \eqref{e9} holds.

Since $\tau \in \Delta $ is a solution of \eqref{e1} and
$(\frac{\varphi}{\varphi'})'( \tau ) =1$, we obtain by \cite{SD} that
\begin{equation} \label{e16}
\big\vert \frac{\tau -u( t,z) }{1-u( t,z) \tau }
\big\vert \leq \vert M_{\tau }( z) \vert \cdot \exp
\{ -\frac{1-M_{\tau }( z) }{1+M_{\tau }( z) }
t\}, \quad z\in \Delta
\end{equation}
where
\begin{equation} \label{e17}
M_{\tau }( z) =\frac{\tau -z}{1-\tau \overline{z}}
\end{equation}
is the M\"{o}bius transform of the unit open disk and all values of
$M_{\tau}( z) $ are found in the open disk centered at
\[
c=-\frac{1-\rho ^{2}}{1-\rho ^{2}\vert z\vert ^{2}}\cdot z
\quad\mbox{with radius}\quad
r=\frac{1-\rho ^{2}}{1-\rho ^{2}\vert z\vert ^{2}}\cdot \rho
\]
Therefore,
\[
\vert M_{\tau }( z) \vert \leq \vert c\vert
+\rho \leq \frac{\vert z\vert +\rho }{1+\rho \vert
z\vert },
\]
and from \eqref{e17}, we obtain
\[
\frac{1-\vert M_{\tau }( z) \vert }{1+\vert
M_{\tau }( z) \vert }\geq [ 1-\frac{\vert
z\vert +\rho }{1+\rho \vert z\vert }] \cdot [ 1+%
\frac{\vert z\vert +\rho }{1+\rho \vert z\vert }]
^{-1}=\frac{(1-\rho )(1-\vert z\vert )}{(1+\rho )(1+\vert
z\vert )}.
\]
Now, it follows by \eqref{e16} , that
\[
\vert \tau -u(t,z)\vert \leq \frac{\vert 1-u(t,z)
\overline{\tau }\vert }{\vert 1-\tau \overline{z}\vert }\cdot
\vert \tau -z\vert \exp \{ -\frac{(1-\rho )(1-\vert
z\vert )}{(1+\rho )(1+\vert z\vert )}t\} .
\]
Since
\[
\frac{\vert 1-u(t,z)\overline{\tau }\vert }{\vert 1-\tau
\overline{z}\vert }\leq \frac{(1+\rho )}{(1-\rho )},
\]
then we obtain that estimate \eqref{e9} holds.

Now we will prove assertion $(ii)$. It is known that
$\varphi (z) $ is a star-like function, therefore the function
$\varphi _{\tau}( z) =\varphi ( \tau z) $ is star-like too and
\[
f_{\tau }( z) =\frac{\varphi _{\tau }( z) }{(\varphi
_{\tau }( z) )'}
=\frac{\varphi ( \tau z) }{\tau \varphi'( \tau z) }\in \mathcal{G}( \Delta ) .
\]
Then, it follows by \cite{ES}, that
\[
\lim_{r\to \ 1^{-}}\frac{f_{\tau }( z) }{r-1}\
=\beta >0,
\]
(i.e. \eqref{e10} holds), and there exist following representation of
function $\varphi _{\tau }( z) $:
\[
\varphi _{\tau }( z) =\frac{(1-z)^{2}}{z}\cdot q_{\tau }(
z),
\]
where $q_{\tau }( z) $ is a star-like function, such that
$q_{\tau}( 0) =0$. Now, by using \eqref{e15} and the dynamical extension of
the Julia-Wolf-Caratheodory Theorem given in \cite{ES}, we obtain
\eqref{e11}. Then Theorem \ref{thm1} is proved.
\end{proof}

\subsection*{Acknowledgments} The author is very grateful to Professor
David Shoikhet and Dr. Mark Elin for their useful suggestions and for
attention during the preparation of this paper.

\begin{thebibliography}{99}

\bibitem{AR} R. G. Airapetyan, Continuous Newton Method And its Modification,
\textit{Applicable Analysis} \textbf{1} (1999), 463--484.

\bibitem{AGS} R. G. Airapetyan, A. G. Ramm and A. B. Smirnova, Continuous
Analog
of the Gauss-Newton Method, \textit{Mathematical Models and Methods in
Applied Sciences} \textbf{9} (1999), 1--13.

\bibitem{ER} M. Elin, S. Reich and D. Shoikhet;
Dynamics of the Inequalitiesin Geometric Function Theory,
\textit{J. of Inequalities and Applications}
\textbf{6} (2001), 651-654.

\bibitem{ERS} M. Elin, S. Reich and D. Shoikhet; Asymptotic behavior of
Semigroups of Holomorphic Mappings, \textit{Progress in Nonlinear Differential
Equations and Their Applications,} \textbf{42}, 449--458. Birrchauser Verlag,
Basel, (2000).

\bibitem{ES} M. Elin and D. Shoikhet; Dynamic Extension of the
Julia-Wolf-Caratheodory Theorem, \textit{Dynamics Systems and Applications}
\textbf{10} (2001), 421-438.

\bibitem{AK} L. Kantorovich and G. Akilov; \textit{Functional Analysis in
Normed Spaces}, Pergamon Press, Oxford1964

\bibitem{OP} J. M. Ortega, W. G. Poole; \textit{An Introduction to
Numerical Methods for Differential Equations}, Pitman Publishing Inc.1981.

\bibitem{POM} Sh. Pommerenke; \textit{Boundary Behavior of Conformal Maps},
Springer, Berlin, 1992.

\bibitem{RS} S. Reich and D. Shoikhet; \textit{Semigroups and Generators on
Convex Domains with the Hyperbolic Metric}, \textit{Atti. Appl. Anal.}
\textbf{1, }, (1996), 1-44.

\bibitem{SD} D. Shoikhet, \textit{Semigroups in geometrical Function Theory},
Kluwer, Dordrecht, 2001.

\end{thebibliography}

\end{document}