On some known fixed point results in the complex domain: survey Текст научной статьи по специальности «Математика»

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ON SOME KNOWN FIXED POINT %

RESULTS IN THE COMPLEX DOMAIN: SURVEY I

Tatjana M. Dosenovic3, Henk Koppelaar^, Stojan N. Radenovicc

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University of Novi Sad, Faculty of Technology, E

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Novi Sad, Republic of Serbia, o

e-mail: [email protected], ORCID ID: http://orcid.org/0000-0002-3236-4410 b Delft University of Technology, Faculty of Electrical Engineering, Mathematics and Computer Science, Delft, Netherlands, e-mail: [email protected], ORCID ID: https://orcid.org/0000-0001-7487-6564

' University of Belgrade, Faculty of Mechanical Engineering, °

Belgrade, Republic of Serbia, -a

e-mail: [email protected], x

ORCID ID: ©http://orcid.org/0000-0001-8254-6688 ^

DOI: 10.5937/vojtehg66-17103; https://doi.org/10.5937/vojtehg66-17103

FIELD: Mathematics ^

ARTICLE TYPE: Review Paper E

ARTICLE LANGUAGE: English

Abstract: O

In this survey paper, we consider some known results from the fixed ^

point theory with complex domain. The year 1926 is very significant for 0 this subject. This is the beginning of the research and application of the

fixed point theory in complex analysis. The Denjoy-Wolf theorem, |

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together with the Banach contraction principle, is one of the main tools in the mathematical analysis. o

Keywords: fixed point, Jordan curve, analytic function, complex D Banach space.

Introduction and preliminaries

Let f be an analytic map of the unit disk D = {z e C: |z| < l} into

ACKNOWLEDGMENT: The first author is grateful for the financial support from the Ministry of Education and Science and Technological Development of the Republic of Serbia (Matematicki modeli nelinearnosti, neodredenosti i odlucivanja, 174009) and from the Provincial Secretariat for Higher Education and Scientific Research, Province of Vojvodina, Republic of Serbia, project no. 142-451-2838/2017-01.

itself. Now, we want to consider the fixed point of f and especially a lemma implies that f has at most one fixed point in D and some § maps have no fixed point. If f : D ^ D isa continuous map, then f

> must have a fixed point in D. From now on, we make the assumption that f is an analytic function on the open disk D .

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w Definition 1.1 Suppose that f is an analytic map of the unit disk D

^ into itself. We say that a e D is a fixed point of f if f (a) = a. Also, a e 3D is a fixed point of f if lim^r f (ra) = a.

<

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The Julia-Caratheodory theorem implies that if a gSD is a fixed

w point of f, then limr^r f (ra) exists (we call it f '(a)) and 0< f '(a)<œ. >-

a:

¡< Theorem 1.2 ( The Denjoy-Wolf theorem from 1926. ) If f is an

analytic map of D into itself, but not the identity map, there is a unique fixed point a of f in D such that |f ' (a) < 1.

<

The point a in the above theorem is called the Denjoy-Wolff point of f. g The Schwarz-Pick Lemma implies that f has at most one fixed

point in D and if f has a fixed point in D, it must be the Denjoy-Wolf point.

1

z + —

Example 1.3 1) The mapping f (z) =-2 is an automorphism of

1 + -2

D with the fixed points 1 and -1, but the Denjoy-Wolff point is a = 1 because f '() = 1 (f (-1) = 3)

2) The mapping f : D ^ D given by f (z) = —^^ has three fixed

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2 - z

points: 0, 1, and -1. The Denjoy-Wolff point is a = 0 since f'(o) = 2 and f ' ( 1)=3.

/ x 2 z3 +1

3) The mapping f (z ) =-— is an inner function fixing 1 and -1,

2 + z

with the Denjoy-Wolff point a = 1 because f '(l) = 1 ( note that

f '(-1) = 9).

4) The inner function f(z) = expjj^—-jj has a fixed point in D

which is the Denjoy-Wolff point (a « .21365), and infinitely many fixed points on ÔD.

Definition 1.4 Let f : D ^ D be an analytic map. The set F = j z g D : lim f (rz )= z J is called the fixed point set of f .

Theorem 1.5 If f is an analytic function that maps the unit disk into itself, then there exists E c F which has linear measure zero.

Example 1.6 Let K be a compact set of measure zero in ÔD. There is a function f analytic in D and continuous on D such that f (D)c D and the fixed point set of f is {0}o> K.

Theorem 1.7 Let f be a univalent analytic function that maps the unit disk into itself. Then the set F has capacity zero.

Example 1.8 For a given compact set K in ÔD of capacity zero and a point a gôD \ K, there exists an analytic and univalent function in D ( say f ) such that f (D)c D and F = {a}^ K.

In the next theorem, it is required that the Denjoy-Wolff point a = b0 be normalized, i.e., b0 = 0 or b0 = 1.

Theorem 1.9 Let f be an analytic function with f (D)c D and suppose that b0,b^...,bn are fixed points of f.

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• If b0 = 0, then

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• If b0 =1 and f' (l) = 1, then

V 1 < f (1) j=1 f (bj)-1"1 - f (1)'

If b0 =1 and 0< f (1) <1, then

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1 -b,\ / \ ( 1 >

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(bj )-1 < 2 Re

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f (0)

Moreover, the equality holds if and only if f is the Blaschke product of order n +1 in case 1) or of order n in cases 2) and 3).

If f has infinitely many fixed points, then the appropriate inequality holds for any choice of finitely many fixed points. In particular, only countably many fixed points of f can have a finite angular derivative.

If b0, b1, b2,... are countably many fixed points for which f (bj) <

then the corresponding infinite sum converges and the appropriate inequality holds.

With the assumption that an analytic function f is univalent and that the Denjoy-Wolff point is b0 = 0 or b0 = 1, we have the next assertion.

Theorem 1.10 Let f be a univalent analytic function with f (D) tz D and suppose b0, b1,..., bn are fixed points of f.

If b0 = 0, then

V(log f (bj)) < 2 Re B 1

j=1

f

where B = limr log

A

f (b )

J' (°)rbiy

< f'(1) <

and lim z ^logf^l = log f (0).

If bo =0 and 0 < f (1) <1, then ¿(log f ()) <-(log f (l)) .

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Moreover, the equality holds if and only if f is embeddable in a semigroup and f (D) = D with n analytic arcs removed in case 1) or n -1 in case 2).

• If ¿0=1 and f (1) = 1, then ¿cj(log f () < 2log1 -f-f(°)) ,

where

ci = limr Im

( (

log

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1 f K)- f (o) 1 - f (0)fr

bj 1 - f (0)f (rbj ) f (r )- f(0)

Remark 1.11 In (Anderson & Vasil'ev, 2008, pp.101-110), the authors proved for b0 = 0, case 1), the inequality

n/ (bj )

1

j >

j=1

f (0)

where ccj > 0 and ¿c = 1. The equality holds only for the unique

j=1

solution of a given complex differential equation with a given initial condition.

Considering b0 = 0, case 3) and the assumption that f is

embeddable in a continuous semigroup, in (Contreras et al, 2006, pp.125-142) was proved that

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^ 1 - Re b 1 ,, v

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where G is the infinitesimal generator of the semi-group and a is the map from the linear fractional model.

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In case 3), an equality condition is not included and maybe this inequality is not the best possible. Also, a sharp inequality to describe the general case has not yet been obtained.

Main results

It is known that a continuous mapping of a simply connected, closed, bounded set of the Euclidean plane into itself has at least one fixed point. w Let F be an analytic function in some domain S of the complex

=3 plane. Standard results in real numerical analysis show that the equation o z = F(z) has at least one solution, called a fixed point of F. If S is

y bounded and simply connected, F is continuous on the closure S of S,

and F(s)e S. If the mapping F is a contraction, then there is a unique

fixed point, and the iteration sequence defined by zn+1 = F(zn),

ct —

¡< n = 0,1,2,... converges to the fixed point for every choice of z0 eS. If S

is convex, then a necessary and sufficient condition for the mapping to

be a contraction is that the derivative F' of F satisfies |f'(z)< k,

Q z e S, where k < 1.

d The purpose of the next theorem is to show that the conclusion is

* not affected if we replace the assumption that F is a contraction by the condition that F is an analytic function.

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o Theorem 2.1 (Henrici, 1969)

q We first prove a reduced form of the theorem. Let S denote the

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interior of a Jordan curve r, let F be analytic in S and continuous on S ur, and let F(S u r) e S. Then F has exactly one fixed point, and the iteration sequence defined by zn+1 = F(zn), n = 0,1,2,... converges to the fixed point for arbitrary z0 e S u r.

Note that the function F (z ) = 2 z100 in |z| < 1 satisfies the hypothesis and |f' (z) is arbitrary large.

Proof. First let S be the unit disk. By the hypothesis, we have

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t (z ) =

Let w = t (zn). Since

= t (zn+1 ) = f (F (zn )) = t (F (t-1 (wn))) = G (wn), we conclude from (2) that |wn+1| < k|wn hence that |wn| < kn|w0|, and

finally that wn ^ 0.

From the above, it follows that the iteration sequence converges to the fixed point, i.e., zn = t -1 (wn t -1 (0) = s.

We now turn to the case where S is an arbitrary Jordan domain. By the Osgood-Caratheodory theorem (Caratheodory, 1960), there exists a function q that maps S conformally onto Izl <1 and S uT continuously

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To prove the existence of a zero of z = F (z), we apply the Rouche

theorem (Ahlfors, 1953) with z in the role of the 'big' function and F(z)

in the role of the 'small' function. Then, using (1), we can conclude that

z - F (z) and z have the same number of zeros inside |z| = 1, namely

exactly one. d

With s denote the unique fixed point and let d

z - S j|

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1 - zs

This is a linear transformation which maps |z| < 1 onto itself and

sends s into 0. Hence, the function G = t o F o t-1 has the fixed point 0 and it is continuous mapping of |z| < 1 onto a closed subset of |z| <1. Hence, p

k := sup |G (z) <1.

|z| <1

We can certainly assume that k > 0 , since otherwise, G and F are constant and the proof is straightforward. The function k -1G vanishes at 0 and is bounded by 1, hence by the Schwarz lemma (Ahlfors, 1953) we |

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G(z )< k|z| (2)

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for all z such that |z| < 1.

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and one-to-one onto |z| < 1. It is easily seen that the function H = g o F o g-1 satisfies the hypotheses of the theorem for the unit disk. Furthermore, if the points zn are defined by zn+1 = F(zn), n = 0,1,2,... and wn = g(zn), then wn+1 = H(wn). Consequently, the validity of the theorem for the unit disk implies the validity for the general case.

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— We add some problems related to the previous theorem.

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o • It can be shown that k <-■„

° 1 + r2

o • If S is the unit disk, than |zn - s| <(1 + r)kn, n = 0,1,2

5 • Let F'(s) = F''(s) = - = F(m-1)(s) = 0, F(m)(sW 0, for some

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integer m >1. If S is the unit disk, show that

£ |zn -s <(1 + r))1 + k2 +••• + mn-1, n e N.

• A research problem is whether similar results can be established for systems of analytic equations.

^^ A set S is said to lie strictly inside a subset D of a Banach space if

d there is some s>0 such that D whenever xeS. The following

>c3 theorem may be viewed as a holomorphic version of the Banach's

contraction mapping theorem.

° Theorem 2.2 (Earle & Hamilton, 1970, pp.61-65)

o Let D be a nonempty domain in a complex Banach space X and

h: D ^ D a bounded holomorphic function. If h(D) lies strictly inside D, then h has a unique fixed point in D.

Proof. Let us construct a metric p, called the CFR-pseudometric, with the contraction h. Define

a(x, v) = sup|g'(.x)v|: g: D ^ Aholomorphic} for x e D and v e X and set

i

L(y) = \a(y(t ),/(t))

for y in the set r of all curves in D with piecewise continuous ^ derivative. Clearly a specifies a seminorm at each point of D. We view L(y) as the length of the curve y measured with respect to a. Define

p(x, y) = inf (L(y): y e T, y(o) = x, y(1) = y} for x, y e D. It is easy to verify that p is a pseudometric on D. |

Let x e D and v e X. By the chain rule, we have

(g o h)(x) = g'(h(x))h'(x)v |

for any holomorphic function g: D ^ A. Hence, x

a(h(x), h' (x)v) < a(x, v). (3) |

By integrating this and applying the chain rule, we obtain L(hoy)<L(y) for all yeT and thus the Schwarz-Pick inequality p(h(x), h(y)) < p(x, y) holds for all x, y e D.

Now, by hypothesis, there exists an — >0 such that B—(x)cD whenever xe D. We may assume that D is bounded by replacing D by

{BE(h(x)): x e D}

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Fix t with 0< t < —, where 8 denotes the diameter of h(D). Given ^

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xe D, define

h (y ) = h(y) + t[h(y)- h(x)] and note that h x: D ^ D is holomorphic. Given x e D and v e X, it follows from h 1 (x)v = (1 +1)h'(x)v and (3) with h replaced by h 1 that

a(h(x), h' (x )v) < a(x, v). Integrating this as before, we obtain

p(h(x ^ h(y)) < P(x, y)

for all x, y e D.

Now pick a point x0 e D and let {xn} be the sequence of iterates given by xn = hn(x0). Then {xn} is a p -Cauchy sequence by the proof of the contraction mapping theorem.

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^ Since X is complete in the norm metric, it suffices to show that

there exists a constant m >0 such that

<§ p(x, y )> m\\x- y|| (4)

o 1

for all x, y e D. Since D is bounded, we may take m = —, where d is

o d

the diameter of D. Given x e D and v e X, define p(y) = ml(y - x)

LU * II II

E where l e X with ||l| = 1. Then g: D ^A is holomorphic and o Dg(x) = ml(v). Hence a(x,v)> m||v|| by the Hahn-Banach theorem.

< Integrating as before, we obtain (4). The previous result, so called the Earle-Hamilton theorem, is still

CH applied in cases where the holomorphic function does not necessarily E map its domain strictly inside itself. The following fixed point theorem is a

£ consequence of two applications of the Earle-Hamilton theorem.

<

Theorem 2.3 (Khatskevich et al, 1995, pp.305-316, S. Reic et al, 1996, pp.1-44)

Let D be a nonempty bounded convex domain in a Banach space < and h: D ^ D a holomorphic function having a uniformly continuous

extension to D. If there exists an s > 0 such that ||h(x)- x|| > s whenever

x e 5D, then h has a unique fixed point in D .

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° The hypothesis that ||h(x)—x|| > s for all x e SD is satisfied when D

" IM <,.

contains the origin and supxeSD" <1. Considerably stronger results

have been obtained for the case where D is the open unit ball of a Hilbert space.

Theorem 2.4 (Goebel et al, 1980, pp.1011-1021) Let D be the open unit ball of a Hilbert space and h: B ^ B a holomorphic function. If there is a point x0 in B such that the sequence

|hn (x0)} of iterates lies strictly inside B, then h has a fixed point in B . If x e SD, then h has a unique fixed point in D .

It is more complicated to obtain fixed points for nonexpansive mappings which are not contractive. One important result is that a nonexpansive self-mapping of a closed bounded convex set in a uniformly convex Banach space has a fixed point. In (Goebel et al, 1980, pp.1011—1021), it was shown that the CRF-metric p in the open unit ball B of a Hilbert space is uniformly convex and the fixed point theorem for holomorphic self-mappings of B was obtained.

Theorem 2.5 Let B be the open unit ball of a Hilbert space and h : B ^ B an arbitrary function satisfying the Schwarz-Pick inequality:

p(h(4 h(y ))<p(x, y ) for all x, y e B. If h has a continuous extension to B, then h has a fixed point in B.

Corollary 2.6 If h : B ^ B isa holomorphic function that has a continuous extension to B, then h has a fixed point in B.

For a treatment of the Cartesian products of the Hilbert balls, we refer the reader to ( Kuczumow et al, 2001, pp.437-515).

Szhwarz lemma and its application in the fixed point theory

In this section, we will restrict our attention to the paper (Xu et al, 2016, 2016:84) where the sharp estimates of a boundary fixed point is obtained using the Schwarz lemma. This lemma provides a very powerful tool for studying several research fields in complex analysis. For example, almost all results in the geometric function theory rely heavily on the Schwarz lemma (Ahlfors, 1953), (Anderson & Vasil'ev, 2008, pp.101-110), (Beardon, 1990, pp.41-150), (Beardon, 1997, pp.12571266), (Budzynska et al, 2012, pp. 504-512), (Budzynska et al, 2013a, 621-648), (Budzynska et al, 2013b, pp.747-756), (Burckel, 1981, pp.396-407), (Caratheodory, 1960), (Contreras et al, 2006, pp.125-142), (Cowen, 2010), (Cowen, 1981, pp.69-95), (Cowen & Pommerenke, 1982, pp.271-289), (Denjoy, 1926, pp. 255-257), (Earle & Hamilton, 1970, pp.61-65), (Goebel et al, 1980, pp.1011-1021), (Goebel, 1982, pp.1327-1334), (Harris, 2003, pp.261-274), (Hayden & Suffridge, 1971, pp.419-422), (Hayden & Suffridge, 1976, pp.95-105), (Henrici, 1969), (Julia, 1918, pp.47-295), (Khatskevich et al, 1995, pp.305-316),

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(Kuczumow, 1984, pp.417-419), (Kuczumow et al, 2001, pp.437-515), (Lemmens et al, 2016), (Mateljevic, 1998, pp. 1-4), (Reich & Shoikhet, 1996, pp.1-44), (Rudin, 1978, pp.25-28), (Suffridge, 1974, pp.309-314), 8 (Wolff, 1926), (Xu et al, 2016, 2016:84). On the other hand, the Schwarz ° lemma at the boundary is also useful in complex analysis, and various interesting results have been obtained (Ahlfors, 1953), (Anderson & Vasil'ev, 2008, pp.101-110), (Beardon, 1990, pp.41-150), (Beardon, dc 1997, pp.1257-1266), (Budzynska et al, 2012, pp. 504-512), (Budzynska et al, 2013a, pp.621-648), (Budzynska et al, 2013b, pp.747-g 756), (Burckel, 1981, pp.396-407), (Caratheodory, 1960), (Contreras et o al, 2006, pp.125-142), (Cowen, 2010), (Cowen, 1981, pp.69-95), (Cowen

< & Pommerenke, 1982, pp.271-289), (Denjoy, 1926, pp.255-257). ° We will summarize without proofs the relevant material on (Xu et al, g 2016, 2016:84). First we set up the notation and the terminology. h Let D denote the unit disk in C. With the notion H(D, D), we have oc the class of holomorphic self-mappings of D. Here, N stands for the set

of all positive integers. The boundary point E e 8D is called a fixed point

of f e H (D, D) if

f (E) = lim f (rE) = E

<

^ The classification of the boundary fixed points is given at the

2 begging of this survey. This classification can be done via the value of the angular derivative

. f E)= Z lim^

O z^E z - E

o which belongs to (0, œ) due to the Julia-Caratheodory theorem (see Julia, 1918, pp.47-295). This theorem also asserts that the finite angular derivative at the boundary fixed point E exists if and only if the

holomorphic function f (z) has the finite angular limit Zlim^ f (z). For a boundary fixed point E of f, if f (E)e(0, œ), then E is called a regular fixed point. The regular fixed point is attractive if f (E)e(0,1),

neutral if f (E) = 1, or repulsive if f (E)e(1, œ).

By the Julia-Caratheodory theorem (Julia, 1918, pp.47-295) and the Wolf lemma (Wolff, 1926), if f e H (D, D) with no interior fixed point, then there exists a unique regular boundary fixed point E such that

>o

f (g) e (0,1] and if f e H(D, D) with an interior fixed point, then ^ f (g) > 1 for any boundary fixed point g e 3D. S

The following known results are very significant.

Theorem 3.3 If f e H (d, d) with % = 1 as its regular boundary fixed point, then

2

f' (1) >

R f 1 - f2 (0)+f (0)1 Re V (1 -f (0))2 y

For a fuller treatment and a deeper discussion of fixed point results in complex domain, we refer the reader to (Xu et al, 2016, 2016:84) and the references given there.

Theorem 3.1 Assume that f e H(D, D) has a regular boundary | fixed point 1 and f (0) = 0. Then

o

+ 1 f (0 ) J

a

'»f 1

Moreover, the equality holds if and only if f is of the form

f(z) = -z-pL, z e D, 1 - az

for some constant a e (-1,0].

The next theorem is the improvement of the previous ones. It was announced 60 years later and showed how to dispense with the assumption f (0) = 0. J

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Theorem 3.2 If f e H (d, d) with % = 1 as its regular boundary fixed point, then

/(,)> 2(1 -f(°ff " f W"1 -If(0)2 +|f (0).

Finally, the previous result has been improved and the better estimate has been obtained.

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References

Ahlfors, L. 1953. Complex Analysis. New York: McGraw-Hill, 1st ed.

Anderson, J.M., & Vasil'ev, A. 2008. Lower Schwarz-Pick Estimates and Angular Derivatives. Ann. Acad. Sci. Fennicae Math., 33, pp.101-110. Available at: http://www.acadsci.fi/mathematica/Vol33/AndersonVasilev.html. Accessed: 10.03.2018.

Beardon, A.F. 1990. Iteration of contractions and analytic maps. J. Lond. Math. Soc., s2-41(1), pp.141-150. Available at: https://doi.org/10.1112/jlms/s2-41.1.141.

Beardon, A.F. 1997. The dynamics of contractions. Ergodic Theory and Dynamical Systems, 17(6), pp.1257-1266. Available at: https://doi.org/10.1017/s0143385797086434.

Budzynska, M., Kuczumow, T., & Reich, S. 2012. A Denjoy-Wolff theorem for compact holomorphic mappings in reflexive Banach spaces. J. Math. Anal. Appl. 396(2), pp.504-512. Available at:

https://doi.org/10.1016/jjmaa.2012.06.044.

Budzynska, M., Kuczumow, T., & Reich, S. 2013a. Theorems of Denjoy-Wolff type. Annali di Matematica Pura ed Applicata, 192(4), pp.621-648. Available at: https://doi.org/10.1007/s10231-011-0240-z.

Budzynska, M., Kuczumow, T., & Reich, S. 2013b. A Denjoy-Wolff theorem for compact holomorphic mappings in complex Banach spaces. Annales Academiae Scientiarum Fennicae Mathematica, 38, pp.747-756. Available at: https://doi.org/10.5186/aasfm.2013.3846.

Burckel, R.B. 1981. Iterating analytic self-maps of discs. Am. Math. Monthly, 88(6), pp.396-407. Available at: https://doi.org/10.2307/2321822.

Caratheodory, M. 1960. Theory of functions of a complex variable. Chelsea, New York. Vol. 2, English edition.

Contreras, M.D., Diaz-Madrigal, S., & Pommerenke, C. 2006. On boundary critical points for semigroups of analytic functions. Mathematica Scandinavica, 98(1). Available at: https://doi.org/10.7146/math.scand.a-14987.

Cowen, C.C. 1981. Iteration and the Solution of Functional Equations for Functions Analytic in the Unit Disk. Trans. Amer. Math. Soc., 265, pp.69-95. Available at: https://doi.org/10.1090/S0002-9947-1981 -0607108-9.

Cowen, C.C. 2010. Fixed points of functions analytic in the unit disk. In: Conference on complex analysis, University of Illinois, May 22. University of Illinois.

Cowen, C.C., & Pommerenke, Ch. 1982. Inequalities for the Angular Derivative of an Analytic Function in the Unit Disk. J. London Math. Soc., s2-26(2), pp.271-289. Available at: https://doi.org/10.1112/jlms/s2-26.2.271.

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Ключевые слова: неподвижная точка, кривая Жордана, аналитические функции, полное Банахово пространство.

НЕКИ ПОЗНАТИ РЕЗУЛТАТИ ИЗ НЕПОКРЕТНЕ ТАЧКЕ У КОМПЛЕКСНОМ ДОМЕНУ: ИСТРАЖИВА^Е

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Paper accepted for publishing on / Дата окончательного согласования работы / Датум коначног прихвата^а чланка за об]ав^ива^е: 12.04.2018.

© 2018 The Authors. Published by Vojnotehnicki glasnik / Military Technical Courier o^

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© 2018 Авторы. Опубликовано в «Военно-технический вестник / Vojnotehnicki glasnik / Military Technical Courier» (www.vtg.mod.gov.rs, втг.мо.упр.срб). Данная статья в открытом доступе и распространяется в соответствии с лицензией «Creative Commons» oi

(http://creativecommons.org/licenses/by/3.0/rs/). ^

© 2018 Аутори. Обjавио [^нотехнички гласник / Vojnotehnicki glasnik / Military Technical Courier (www.vtg.mod.gov.rs, втг.мо.упр.срб). Ово jе чланак отвореног приступа и дистрибуира се у О

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