\documentclass[reqno]{amsart} \usepackage{graphicx} \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 $$\label{e1} \varphi ( z) =0,$$ 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 $$\label{e2} z_{n+1}=z_{n}-\frac{\varphi ( z_{n}) }{\varphi '(z_{n}) }.$$ 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}) $$\label{e3} \begin{gathered} \frac{\partial u(t,z)}{\partial t}+\frac{\varphi (u(t,z))}{\varphi '(u(t,z))}=0 \\ u(0,z)=z, \end{gathered}$$ 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 $$\label{e4} \lim_{t\to \infty }\ u( t,z) =\tau \in \overline{D},$$ 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 $$\lim_{z\to \tau,\ z\in \gamma}\varphi(z)=0.$$ \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: $$\label{e8} \mathop{\rm Re}\{ \overline{z}\frac{\varphi ( z) }{\varphi'( z) }\} \geq (1-\vert z\vert ^{2}) \cdot \mathop{\rm Re}\{ \overline{z}\frac{\varphi ( 0) }{\varphi'(0)}\} , z\in \Delta .$$ 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 $$\label{e9} \vert \tau -u( t,z) \vert \leq \delta ^{-1}\exp \{ -\frac{1-\vert z\vert }{1+\vert z\vert }\delta t\} \vert \tau -z\vert ,\quad z\in \Delta ,\; t\geq 0,$$ 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 $$\label{e10} \beta =\lim_{r\to 1^{-}\ }\frac{\varphi ( r\tau ) }{\varphi'( r\tau ) ( r-1) \tau }>0$/extract_tex] exist and $$\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,$$ 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 $$\label{e12} T( f) =[ ( h^{-1})'] ^{-1}f\circ h^{-1}$$ 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 $$\label{e13} F_{t}=h^{-1}\circ \Psi _{t}\circ h\,.$$ 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 $\gamma _{z_0}( t) =\{ u( t,z_0) ,t\geq 0\}$, $$\label{e14} \lim_{z\to \tau }\varphi ( z) =\lim_{t\to \infty } \varphi ( F_{t}( z_0) ) =\lim_{t\to \infty } \varphi ( \varphi ^{-1}( e^{-t}( z_0) )) =0\,.$$ 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 $$\label{e15} \lim_{z\to \tau _{1,}z\in \gamma _{1}}\varphi ( z) =0.$$ 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 $$\label{e16} \big\vert \frac{\tau -u( t,z) }{1-u( t,z) \tau } \big\vert \leq \vert M_{\tau }( z) \vert \cdot \exp \{ -\frac{1-M_{\tau }( z) }{1+M_{\tau }( z) } t\}, \quad z\in \Delta$$ where $$\label{e17} M_{\tau }( z) =\frac{\tau -z}{1-\tau \overline{z}}$$ 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. 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