2004 Conference on Diff. Eqns. and Appl. in Math. Biology, Nanaimo, BC, Canada. Electronic Journal of Differential Equations, Conference 12, 2005, pp. 79–85. ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu ftp ejde.math.txstate.edu (login: ftp)

CONTINUOUS NEWTON METHOD FOR STAR-LIKE FUNCTIONS

YAKOV LUTSKY

Abstract. We study a continuous analogue of Newton method for solving the nonlinear equation

φ(z) = 0,

where φ(z) holomorphic function and 0 ∈ \overline{φ(D)}. It is proved that this method converges, to the solution for each initial data z ∈ D, if and only if φ(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.

1. Results

Let D ⊂ ℂ be a domain (that is, an open connected subset of ℂ). The set of all holomorphic functions on D will be denoted by Hol(D, ℂ). We consider a nonlinear equation

φ(z) = 0,  (1.1)

where φ(z) ∈ Hol(D, ℂ) and 0 ∈ \overline{φ(D)}.

In the known Newton’s method [6, 7], the solution of (1.1) can be found as a limit of the sequence {z_{n}}, n = 0, 1, 2 . . .. The first term z_{0} ∈ D is given and other terms are constructed by the iterative process

z_{n+1} = z_{n} − \frac{φ(z_{n})}{φ′(z_{n})}.  (1.2)

It is well known that the convergence of process (1.2) depends on the choice of the initial approximation z_{0} ∈ D. If z_{0} is chosen arbitrarily, then sequence may diverge.

The Continuous Newton Method (CNM) and its modifications [2, 6] are an alternative approach to the solution of (1.1). The CNM has been considered as the solution of the following Cauchy problem (continuous analogue of process (1.2))

\begin{matrix}\frac{∂u(t, z)}{∂t} + \frac{φ(u(t, z))}{φ′(u(t, z))} = 0 \\ u(0, z) = z, \end{matrix}  (1.3)

2000 Mathematics Subject Classification. 49M15, 46T25, 47H25.

Key words and phrases. Newton method; star-like functions; continuous semigroup.

\begin{matrix}c \\ ◯ \end{matrix}2005 Texas State University - San Marcos.

Published April 20, 2005.

where the initial condition z is some point which belongs to the domain D. A solution of the (1.1) was obtained as the limit

lim_{t→∞} u(t, z) = τ ∈ \overline{D},  (1.4)

where u(t, z) is the solution of (1.3).

The continuous Newton Method has several advantages over the iterative method (1.2) because convergence theorems for CNM usually can be obtained easily. However, as well as in the iterative process (1.2), 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 φ(z) for which the solution of (1.1) can be found by CNM under arbitrary initial condition z ∈ 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 [10] 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 (1.1) in \overline{D} is proved. This solution is obtained as limit (1.4). More exact results are obtained when D = Δ is an open unit disk in 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 (1.1). As it was mentioned above, 0 ∈ \overline{φ(D)}. It means that the function φ may have no null point in D. Moreover, φ even may be undefined on the boundary ∂D. Therefore, we consider the solution of (1.1) at the boundary points of domain D in following generalized meaning.

Definition 1.1. A point τ ∈ ∂D is said to be a generalized solution of (1.1) on the set \overline{D} if there is a Jordan curve γ ∈ \overline{D} such that γ ∩ ∂D = τ and

lim_{z→τ, z∈γ} φ(z) = 0.  (1.5)

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

Definition 1.2. We will say that CNM is well defined on the domain D if the Cauchy problem (1.3) has a unique solution for each z ∈ D and

{u(t, z), t ≥ 0, z ∈ D} ⊂ D.

Definition 1.3. We will say that the CNM converges globally in the domain D if the limit (1.4) exists for each solution u(t, z) of Cauchy problem (1.3).

Remark 1.4. The CNM is well defined on the domain D if and only if the function f(z) = \frac{φ(z)}{φ′(z)} is a generator of a one-parameter continuous semigroup of holomorphic self-mappings S_{f} = {F_{t} : D → D, t ≥ 0}, where

F_{t} = φ^{−1} ∘ e^{−t} ∘ φ ,  (1.6)
u(t, z) = F_{t}(z) = φ^{−1}(e^{−t}φ(z)) (t ≥ 0, z ∈ D)  (1.7)

is the unique solution of the Cauchy problem (1.3), [9, 10]. In this case the set γ_{z}(t) = {u(t, z), t ≥ 0} is a Jordan curve for each z ∈ D.

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

Definition 1.5. The set Ω ⊂ ℂ is called star-shaped if for any ω ∈ Ω, the point tω belongs to Ω for every t ∈ (0, 1].

Definition 1.6. A univalent holomorphic function f : D → ℂ is said to be star-like if the set f(D) is star-shaped .

Remark 1.7. It follows from [1], that univalent function φ(z) (∈ Hol(D, ℂ)) is a star-like function, if and only if the mapping f(z) = \frac{φ(z)}{φ′(z)} ∈ G(D).

Now we will formulate the main result of this paper.

Theorem 1.8. Let φ(z) (∈ Hol Δ, ℂ)) be a univalent function such that \overline{φ(Δ)} ∋ 0. Then continuous Newton Methos is well defined in Δ if and only if the following inequality holds:

Re\{\overline{z} \frac{φ(z)}{φ′(z)}\} ≥ (1 − |z|^{2}) · Re\{\overline{z} \frac{φ(0)}{φ′(0)}\}, z ∈ Δ.  (1.8)

Moreover, in this case CNM converges globally to a unique point τ ∈ \overline{Δ}. In addition, (i) If τ ∈ Δ is a solution of (1.1) and |τ| ≤ ρ < 1, then

|τ − u(t, z)| ≤ δ^{−1} exp\{−\frac{1 − |z|}{1 + |z|}δt\}|τ − z|, z ∈ Δ, t ≥ 0,  (1.9)

where δ = \frac{1−ρ}{1+ρ} and u(t, z) is a solution of Cauchy problem (1.3). (ii) If τ ∈ ∂Δ is a generalized solution of (1.1), then the limit

β = lim_{r→1^{−}} \frac{φ(rτ)}{φ′(rτ)(r − 1)τ} > 0  (1.10)

exist and

|τ − u(t, z)| ≤ \frac{\sqrt{2}e^{− \frac{β}{2} t}}{\sqrt{1 − z^{2}}} |τ − z|, z ∈ Δ, t ≥ 0,  (1.11)

where u(t, z) is a solution of the Cauchy problem (1.3).

Remark 1.9. As a matter of fact [8], if τ ∈ ∂Δ is a generalized solution of (1.1), then

lim_{z→τ,τ∈γ} φ(z) = 0

along each non-tangential curve γ (i.e. there exists a non-tangential limit at point τ).

The proof of Theorem 1.8 is based on the following result.

Theorem 1.10. Let D be a bounded domain with Jordan boundary ∂D and φ(z) (∈ Hol(D, ℂ)) be a univalent function such that \overline{φ(D)} ∋ 0. Then following two conditions are equivalent:

(i) φ(z) is a star-like function.

(ii) The continuous Newton method is well defined in the domain D. Moreover, if it is this case, CNM globally converges to a unique point τ ∈ \overline{Δ}.

Proof. The equivalence (i) ⟺ (ii) follows from Remarks 1.4 and 1.7. Therefore, it is sufficient to prove the latter assertion of this theorem on the global convergence.

Let Ω = φ(D). We will consider following two cases separately: 0 ∈ Ω and 0 ∈ ∂Ω. If 0 ∈ Ω, then τ = φ^{−1}(0) ∈ D is a unique solution of (1.1). Since φ^{−1}(z) is a continuous function at point 0, we obtain by (1.7) that

lim_{t→∞} u(t, z) = lim_{t→∞} φ^{−1}(e^{−t}φ(z)) = φ^{−1}( lim_{t→∞} e^{−t}φ(z)) = φ^{−1}(0) = τ

for each z ∈ D. So in this case CNM converges globally.

Suppose now, that 0 ∈ ∂Ω. Let h : D → Δ be any conformal mapping of D onto unit open disk Δ = {z ∈ ℂ : |z| < 1}. Then the linear invertible operator T : Hol(Δ, ℂ) → Hol(D, ℂ), defined by

T (f) = [(h^{−1})′]^{−1}f ∘ h^{−1}  (1.12)

is invertible and maps G(Δ) onto G(D) (see [3, 10]). Moreover, if

{F_{t} : D → D, t ≥ 0} and {Ψ_{t} : Δ → Δ, t ≥ 0}

are semigroups of holomorphic self-mappings, generated by f and ψ = T (f), respectively (see [10]), then

F_{t} = h^{−1} ∘ Ψ_{t} ∘ h .  (1.13)

In the considered case f(z) = \frac{φ(z)}{φ′(z)} ∈ G(D) has no null point in D. It follows from (1.12), that the function ψ(z) has no null point in Δ. Therefore, for each point z ∈ Δ there exists a unique limit

e = lim_{t→∞} Ψ_{t} (z) ∈ ∂Δ .

In supposition of the theorem the boundary ∂D is a Jordan curve, thus, applying Caratheodory Theorem, we conclude, that the function h(z) has a continuous extension to D ∪ ∂D, [8]. Therefore τ = h^{−1}(e) ∈ ∂D and for any z ∈ D by (1.13), we have

lim_{t→∞} u(t, z) = lim_{t→∞} F_{t}(z)

= lim_{t→∞} h^{−1}(Ψ_{t}(h(z)))

= h^{−1}( lim_{t→∞} Ψ_{t}(h(z)))

= h^{−1}(e) = τ .

Further, it follows from (1.7), that for each point z_{0} ∈ D along the curve γ_{z_{0}}(t) = {u(t, z_{0}), t ≥ 0},

lim_{z→τ} φ(z) = lim_{t→∞} φ(F_{t}(z_{0})) = lim_{t→∞} φ(φ^{−1}(e^{−t}(z_{0}))) = 0 .  (1.14)

Thus τ is a generalized solution of equation (1.1) in the set \overline{D}. Therefore, to complete our proof in the case 0 ∈ ∂Ω, we need to show the uniqueness of generalized solution τ. Assume, that there exist another generalized solution τ_{1} ∈ ∂D of (1.1). Then there is a Jordan curve γ_{1} ⊂ \overline{D} which begins at some point z_{0} ∈ D such that γ_{1} ∩ ∂D = τ_{1} and

lim_{z→τ_{1,}z∈γ_{1}} φ(z) = 0.  (1.15)

Since γ_{z_{0}}(t) = {u(t, z_{0}), t ≥ 0} is a Jordan curve, then the curve γ = γ_{1} ∪ γ_{z_{0}}(t) is Jordan too (see Fig .1).

Let the points τ̂, τ̂_{1} ∈ ∂D be different from τ, τ_{1}. We will use the following notation:

γ̂ is some curve which connects the points τ̂ and τ̂_{1}, such that γ̂ ∈ \overline{D}, γ ∩ γ̂ = ∅ and γ̂ ∩ ∂D = {τ̂, τ̂_{1}};

λ (respectively λ_{1}) is the part of boundary ∂D which connects the points τ and τ̂ (respectively τ_{1} and τ̂_{1}).

Let D_{1} be a domain which boundary is ∂D_{1} = γ ∪ λ ∪ γ̂ ∪ λ_{1} and Ω_{1} = φ(D_{1}). Then it follows from (1.14) and (1.15), that the curve φ(γ) (⊂ ∂Ω_{1}) is closed and 0 ∈ φ(γ) (see Fig. 2).

Moreover, the domain Ω_{1} is placed in the external part of the complex plane ℂ with respect to the curve φ(γ). Therefore, there are ω ∈ Ω_{1} and t ∈ (0, 1] such that tω ∉ Ω_{1}. It means, that Ω_{1} is not star-shaped. This contradicts the star-likeness of the function φ(z). Thus uniqueness of the generalized solution τ is proved. The Theorem 1.10 is proved. □

Proof of Theorem 1.8. It is proved in [10] that the function f(z) ∈ G(Δ) if and only if the following inequality holds.

Re\{\overline{z}f(z)\} ≤ (1 − |z|^{2}) · Re\{\overline{z}f(0)\}, z ∈ Δ.

Hence, by Theorem 1.10 and Remark 1.7 we obtain, that inequality (1.8) is equivalent to the assertions (i), (ii) of the Theorem 1.10. Thus the CNM converges globally to a unique solution τ ∈ \overline{Δ} of (1.1). Now we will show that the estimate (1.9) holds.

Since τ ∈ Δ is a solution of (1.1) and (\frac{φ}{φ′})′(τ) = 1, we obtain by [10] that

|\frac{τ − u(t, z)}{1 − u(t, z)τ} | ≤ |M_{τ} (z)| · exp\{−\frac{1 − M_{τ} (z)}{1 + M_{τ} (z)}t\}, z ∈ Δ  (1.16)

where

M_{τ} (z) = \frac{τ − z}{1 − τ\overline{z}}  (1.17)

is the Möbius transform of the unit open disk and all values of M_{τ} (z) are found in the open disk centered at

c = −\frac{1 − ρ^{2}}{1 − ρ^{2}|z|^{2}} · z with radius r = \frac{1 − ρ^{2}}{1 − ρ^{2}|z|^{2}} · ρ

Therefore,

|M_{τ} (z)| ≤ |c| + ρ ≤ \frac{|z| + ρ}{1 + ρ|z|},

and from (1.17), we obtain

\frac{1 − |M_{τ} (z)|}{1 + |M_{τ} (z)|} ≥ [1 − \frac{|z| + ρ}{1 + ρ|z|}] · [1 + \frac{|z| + ρ}{1 + ρ|z|}]^{−1} = \frac{(1 − ρ)(1 − |z|)}{(1 + ρ)(1 + |z|)}.

Now, it follows by (1.16) , that

|τ − u(t, z)| ≤ \frac{|1 − u(t, z)\overline{τ}|}{|1 − τ\overline{z}|} · |τ − z| exp\{−\frac{(1 − ρ)(1 − |z|)}{(1 + ρ)(1 + |z|)}t\}.

Since

\frac{|1 − u(t, z)\overline{τ}|}{|1 − τ\overline{z}|} ≤ \frac{(1 + ρ)}{(1 − ρ)},

then we obtain that estimate (1.9) holds.

Now we will prove assertion (ii). It is known that φ(z) is a star-like function, therefore the function φ_{τ} (z) = φ(τz) is star-like too and

f_{τ} (z) = \frac{φ_{τ} (z)}{(φ_{τ} (z))′} = \frac{φ(τz)}{τφ′(τz)} ∈ G(Δ).

Then, it follows by [5], that

lim_{r→ 1^{−}} \frac{f_{τ} (z)}{r − 1} = β > 0,

(i.e. (1.10) holds), and there exist following representation of function φ_{τ} (z):

φ_{τ} (z) = \frac{(1 − z)^{2}}{z} · q_{τ} (z),

where q_{τ} (z) is a star-like function, such that q_{τ} (0) = 0. Now, by using (1.15) and the dynamical extension of the Julia-Wolf-Caratheodory Theorem given in [5], we obtain (1.11). Then Theorem 1.8 is proved. □

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.

References

[1] R. G. Airapetyan, Continuous Newton Method And its Modification, Applicable Analysis 1 (1999), 463–484.

[2] R. G. Airapetyan, A. G. Ramm and A. B. Smirnova, Continuous Analog of the Gauss-Newton Method, Mathematical Models and Methods in Applied Sciences 9 (1999), 1–13.

[3] M. Elin, S. Reich and D. Shoikhet; Dynamics of the Inequalitiesin Geometric Function Theory, J. of Inequalities and Applications 6 (2001), 651-654.

[4] M. Elin, S. Reich and D. Shoikhet; Asymptotic behavior of Semigroups of Holomorphic Mappings, Progress in Nonlinear Differential Equations and Their Applications, 42, 449– 458. Birrchauser Verlag, Basel, (2000).

[5] M. Elin and D. Shoikhet; Dynamic Extension of the Julia-Wolf-Caratheodory Theorem, Dynamics Systems and Applications 10 (2001), 421-438.

[6] L. Kantorovich and G. Akilov; Functional Analysis in Normed Spaces, Pergamon Press, Oxford1964

[7] J. M. Ortega, W. G. Poole; An Introduction to Numerical Methods for Differential Equations, Pitman Publishing Inc.1981.

[8] Sh. Pommerenke; Boundary Behavior of Conformal Maps, Springer, Berlin, 1992.

[9] S. Reich and D. Shoikhet; Semigroups and Generators on Convex Domains with the Hyperbolic Metric, Atti. Appl. Anal. 1, , (1996), 1-44.

[10] D. Shoikhet, Semigroups in geometrical Function Theory, Kluwer, Dordrecht, 2001.

Yakov Lutsky Department of Mathematics, Ort Braude College, Karmiel 21982, Israel

  E-mail address: yalutsky@yahoo.com