Uniform convexity of the hyperbolic metric and fixed points of holomorphic mappings in the Hilbert ball

036*-546x,*0’0901

Nonlinear Analysrs, Throrr, Merkods & Applicalmns. Vol 4, No. 5, pp IO1I-1021 &I Pcrgamon Press Lid 1980 Prmted in Great Britain

IO1I s02.00/0

UNIFORM CONVEXITY OF THE HYPERBOLIC METRIC AND FIXED POINTS OF HOLOMORPHIC MAPPINGS IN THE HILBERT BALL K. Instytut

G~EBEL, Matematyki

T.

SEKOWSKI

UMCS,

and A.

Nowotki

STACHURA

10, 20-031

Lublin,

Poland

(Received 8 June 1979; revised 27 August 1979) Key words andphrases:

Holomorphic

mappings,

hyperbolic

metric, nonexpansive

mappings,

fixed points

INTRODUCTION AFTER PAPERS by Browder [ 11, GGhde [6] and Kirk [ 151 published in 1965, many authors paid a lot of attention to the problem of existence of fixed points for nonexpansive self-mappings of closed convex sets in Banach spaces. We recall that a mapping 7! D + D, where D is a subset of a Banach space, is said to be nonexpansive iff 11 TX - Ty(j d /Ix - y/j for all x, y ED. Thus nonexpansiveness means simply the Lipschitz condition with constant one. This is a purely metric condition and can be considered as well for mappings defined on arbitrary metric spaces. The reason why nonexpansive mappings are investigated mostly on convex sets is because of the nice metrical structure of these sets. Convexity in the linear sense is important in this theory mainly because it coincides with metrical convexity. However most of the methods used in this field are typically metric. On the other hand, even if D is a closed convex and bounded subset of a Banach space there may exist a nonexpansive mapping T: D + D with no fixed points. Convexity is not enough. If we want to obtain positive results we have to impose some stronger assumptions, e.g., uniform convexity or normal structure. The situation in metric space settings is even more complicated. There are some formal extensions of the results cited above, e.g. [14], but they lack natural realizations other than those on convex sets. There are two aims of this paper. First we want to show that some metric spaces which appear in quite natural ways in the theory of holomorphic mappings in complex Banach spaces are so regular that most of the methods commonly used for convex sets can be successfully applied to these spaces. The space which we discuss in detail is the unit ball in Hilbert space furnished with the so-called hyperbolic metric [7]. Then, as a result of the above we obtain some fixed point theorems for holomorphic selfmappings of the unit ball in Hilbert space, and a characterization of the set of fixed points for such mappings. These results are closely related to and extend earlier work of Hayden and Suffridge [ 10,l l] and Earle and Hamilton [3].

1. HYPERBOLIC

METRIC

IN

BANACH

SPACES

be a complex Banach space. A mapping F defined on an open subset A of E into E is said to be holomorphic if F has a Frtchet derivative at each point of A. Basic theory of holomorphic Let E

1011

1012

K.

GOEBEL,

T. S~KOWSKI

and A. S I ACHURA

mappings may be found in [13]. While some of our considerations may be stated in more general settings, we shall restrict ourself only to the case of mappings transforming the open unit ball B in E into itself. The investigations of hyperbolic metrics in the spaces of more than one dimension were originated by Caratheodory [2]. More information about this notion may be found in [12], [17] and [1X]. We shall cite only some basic facts which are necessary for our purpose. The family .9 of all holomorphic mappings f: B + B forms a semigroup with respect to composition. The derivatives of all functions f E 9 are equibounded at each point of B:

This is a consequence

of the Schwartz-Pick

Lemma

a(z, Y) =,“u: is well-defined.

The function

[8, lo]. Thus for any z E B and y E E the value O~f(Z)Y/I

p: B x B -+ (0, + co) defined by 1 p(x, y) = inf a(?@), $0)) dr, s 0

where inlimum is taken over the class of all piece-wise differential curves y: (0, 1) -+ B joining x and Y, is a metric on B and is called the hyperbolic metric. In case E = C and B is the open unit disc, this is simply the Poincare metric. Generally, the metric space (B, p) has the following properties : (1) (B, p) is unbounded; (2) (B, p) is complete; (3) the p-topology is equivalent to the original norm topology on any ball B, = [x: l/x/l d r] withr < 1; (4) ~(0, x) = tanh-’ llxll; and the most important to us (5) p(f(x), f(y)) d p(x, y) for any x, y E B and all f E 9; (6) if sup [Ilf(x)II :x E B] < 1, then there exists k < 1 such that p(f(x), f(y)) d kp(x, y). According to (6) all holomorphic mappings which send B ‘strictly inside’ B are p-contractions, and in view of (2) and Banach’s contraction principle each such f has exactly one fixed point. This result in a more general setting has been proved in [lo]. In consequence of (6) if h E F is a biholomorphic mapping of Bonto B, which means that h and h- ’ are holomorphic, then we have (7) &r(x), NY)) = Pb? Y)

for all x, y E B.

Thus all mappings f~ 9 are p-nonexpansive and all biholomorphic ones are p-isometries. The whole class of p-nonexpensive self-mappings of B is much wider than 9 and it contains also some nonholomorphic isometries. For example the function f(x) = Z on the unit disc is such an isometry. 2.

THE

CASE

OF

HILBERT

SPACE

From now on we shall work only with the Hilbert space case of E = Il. Much more can be said about the space (B, p) in this setting. Hayden and Suffridge [ 1 l] gave complete characterization

Uniform convexity of the hyperbolic metric and fixed points of holomorphic mappings in the Hilbert ball 1013

of the group of biholomorphic mappings. These are mappings of the form h = U o M, o V, where U, I/ are unitary transformations and for any a E B, M, is the so-called Mobius transformation: M,(x) = (1 - (Ia()2)l’2(1- &-l’*

1 :TXaa), 7

where a* denotes the linear functional a*(u) = (u, a). An equivalent formula for M, looks as follows:

where P, is the orthogonal projection in the direction a and P,’ = I - P,. Let us notice that all Mobius transforms and thus all biholomorphic mappings are weakly continuous and moreover map afline sets onto affrne sets [ 111. (Affine set means here the intersection of B with a complex affine submanifold of H). Moreover for any a, b E B, M&O) = b, M_,,(a) = 0 and in view of this, M,O M_,(a) = b. Thus B is homogeneous in the sense that any point a E B may be mapped on any b E B via a biholomorphic mapping. In consequence of the above facts and (4): &,Y)

= tanh-‘[(M-,b)l(.

Straight forward calculations (cf. [7]) show that (1 JIM-,(x)ll

=

(1

-

-

l[xlj’,<1 p

_

tx

llYl12)

42

li2 )

.

If we put a(x

,

y)

=

(1 - 11x11’)(1 - IIYII”) 11- (4 Y)J2

then p(x, y) = tanh- ’ (1 - 0(x, Y))“~ and for any x, y, u, u E B the inequality p(u. u) < p(x, y) is equivalent to a(x, y) < a(u, u). Using the above iormulas we can check that for any x # 0 the segment [tx: t E (0, l)] is the unique geodesic segment joining 0 and x and is isometric with the interval [0, ~(0, x)]. Consequently, any two points x, y E B may be joined by the unique geodesic line isometric to the interval [0, p(x, y)]. This line is the image under M, of the segment [tM_,(x): t E (0, l)]. Summarizing, the space (B, p) is convex in sense of Menger and metric segments are uniquely determined.

3. UNIFORM

CONVEXITY

OF (B, p)

In this section we shall prove that the metric convexity of (B, p) is in some sense uniform. Let us denote by $[x, y] the unique metric midpoint between x and y. Thus p(x, y) = 2p(x, +[x, y]) = 2P(Y, 3% Yl). 1. There exists a continuous properties: (i) 6(r, 0) = 0; THEOREM

function 6:(0, co) x (0,2) -+ (0, 1) with the following

1014

K.

GOEBEL,

T. SEKOWSKI AND A. STACHURA

(ii) 6(r, E) increases with respect to E; (iii) for any a, x, y E B and any I E (0, co), E E (0. 2) the following implication holds: P(U, 4 G r ~(a, Y) dx,

G r

*

da, +[x, Y])

G (1 -

&r, d)r.

Y) 2 Er

Remark. This function is a counterpart of the modulus of convexity of Banach spaces. Its properties make our space similar to the uniformly convex Banach spaces. However in Banach spaces, the modulus of convexity does not depend on r. Proof. In view of homogeneity we may assume a = 0. Suppose ~(0, x) < r, ~(0, y) < r, p(x, y) 3 er and put z = $[x, y]. The mapping M _-zmaps z into 0 and the geodesic passing through x, y onto a straight line passing through 0. Thus if w = M_.(x) then M_.(y) = - w and ~(0, w) = ~(0, - w) 2 +/2). Put c = tanh r, d = tanh (q/2). We have

r 3 ~(0, x) = ~(0, M,(w)) = tanh-’

(1 - a(-~, w))~‘~,

r 3 ~(0, y) = ~(0, M,( - w)) = tanh- ’ (1 - C(- z, - w))~‘~.

Thus a(-z,w)

=

(1 -

Ilzll')(l-

G(_z

3_w)

=

llwl12)>

1 _

C2 9

11 + (1 -

(z, 41"

lzl12)(1 - ll+d2) 2 1 _ c2 p - (z,41’ .

Notice that at least one of the denominators exceed 1. Moreover (1w (( > tl, so we get

of the left sides of the above inequalities must

1 - ((z((2 a

s

which implies ~(0, z) = tanh-‘llzll

< tanh-’

J($$).

In view of this we can put 6(r, E) = 1 - L tanh- 1 J(G), r

r#O,

6(0, E) = 0,

and check that 6 satisfies our assumptions. Uniform convexity has some interesting consequences. Let us start with the definition: Definition 1: A set X c B is said to be p-convex if for any two points x, y E X the geodesic segment

[x, y] joining x and y is contained in X. X is said to be geodesically closed if for all x, y E X, x # y the whole geodesic line passing by x and y is contained in X.

Uniform convexity of the hyperbolic metric and fixed points of holomorphic mappings in the Hilbert ball

1015

The following properties of p-convex sets are easy to observe: (i) (ii) (iii) (iv) (v)

Each ball is p-convex; The intersection of any family of p-convex sets is p-convex; The union of any lineary ordered by inclusion family of p-convex sets is p-convex; The closure of a p-convex set is p-convex; The biholomorphic image of a p-convex set is p-convex.

THEOREM 2. The intersection of a decreasing sequence of nonempty bounded (in the p-topology) closed, p-convex sets is nonempty.

Proof.LetX,,n= 1,2,3 ,..., X,,, c X, be a sequence with the required property. We can assume that 0 4 X,. Put rn = dist (0, X,), r = lili-r I”. Take a decreasing sequence E, converging to zero and define the sets x = [x E X,, ~(0, x) d r + EJ. Obviously Y,, n = 1,2,3,. . . , are nonempty, p-bounded, p-convex and closed. For any x, y E Y, we have p(0, +[x, y]) > r” and in view of Theorem 1 we get

and consequently

(

r, G 1 - 6 r [

+

E,,

)I

(I + CJ, diamx

I+E

n

which in view of continuity of 6 implies lim diam Y, = 0. n-rm Thus

n X,= n r, z 4.

After a slight modification of the proof we can get the same theorem for an arbitrary decreasing (not necessarily denumerable) family of sets. Remark.

This theorem should be compared with the well-known fact that any decreasing family of nonempty, bounded, closed and convex sets in a reflexive Banach space has nonempty intersection. This is a consequence of weak compactness. Our proof above carries over to the case of uniformly convex Banach spaces. 3. Let X be a nonempty, closed, p-convex subset of B. Then for any x E B there exists exactly one point y E X such that p(x, y) = dist (x, X).

THEOREM

Proof. Apply the method used in the proof of Theorem 2 to the sequence of sets X, = [z E X: p(x, z) < dist (x, X) + l/n].

K. GOEBEL, T. SFKOWSKI and A. STACHURA

1016

We shall call such point y the metrical projection of x onto X and denote it y = P,x or simply Px if X is fixed. Further properties of metrical projections follow from the following: Let x, y, z E B and let [x, z], [y, z] d enote the geodesic segments joining x, z and y, z respectively. If p(y, z) = inf [p(y, u): u E [x, z]] then p(x, z) = inf [p(x, u): u E [y, z]].

LEMMA 1.

Proof. We can assume that z = 0. Our assumption holds iff a(y, tx) < a(y, 0) for all t E (0,l). Check that this inequality is equivalent to Re(x, y) d 0 and consequently to a(x, ty) < 0.(x, 0). THEOREM 4. Let X be a closed p-convex subset of B and let x E B. For any y E X and any u E [x, Pxl, P(YVPx) G P(Ylu).

Proof. Notice that Pu = Px and the result thus follows from the fact that

p(x, Px) = inf [p(x, u): u E [Px, y]] and Lemma 1. Remark. The analogous theorem for the norm metric projection on a convex set in Banach spaces

is not valid even under assumption of uniform convexity. However it is true in Hilbert spaces. 4. ASYMPTOTIC The notion

of asymptotic

CENTER

center of a bounded

and became a very useful tool in investigations defined in purely metrical

OF A SEQUENCE

sequence has been introduced

of fixed points of nonexpansive

by Edelstein

mappings.

[4]

It can be

terms.

Let (M, d) be a metric space and let {x,>

be a bounded

sequence

of elements

of

M. For any

x EM denote sup &,

$6 (x.}) = lim

x,),

r({x,H

= inf [r(y, (x,}):

A({&))

=

1x: T({x,))

Y E Ml,

= T(x,

and call them respectively,

the asymptotic

asymptotic

center of ix”}.

Obviously

without

set A({x,})

may be empty

or contain

a lot of points.

{-491,

radius of (x.1 at x, the asymptotic any further assumptions

radius of {x,> and the

about the space

(M, d) the

Suppose T: M + M is a nonexpansive mapping, which means that d(Tx, Ty) d d(x, y) for all x, y E M. The following two facts show the usefulness of the asymptotic center concept. LEMMA 2. If for some

x E M the sequence

{x,}

=

(T” x 1 is bounded

pnd its asymptotic center

consists of exactly one point z, then z = Tz. LEMMA 3. If (y,}

is a bounded

sequence of points

such that ?i_ma d(y,, y,) = 0 and if A({y,}) consist

of exactly one point z, then z = Tz. Proof. To show the first fact it is enough to notice that r( Tz, (x,})

= lim sup d( Tz, TX,_ J G lim sup d(z, x,_ 1) = r({x,}),

Uniform convexity of the hyperbolic metric and fixed points of holomorphic mappings in the Hilbert ball

1017

and the uniqueness of z completes the proof. Similarly r(7z, {y,)) = lim sup d(E, y,) d lim sup (d(% and

TY,)

+ WY,,

Y,)) G h

SUP 4z,

y,) = r({y,l),

the same argument works.

Remark. The question of whether the asymptotic center is a singleton depends strongly on the regularity of the space (M, d). For example for Banach spaces, uniform convexity is a sufficient condition to assure that.

5. ASYMPTOTIC

CENTER

OF SEQUENCES

IN (B, p)

In our space (B, p) the sequence {xv} is bounded if and only if lim sup (/x,/I < 1. The main result of this section is the following. THEOREM 5.

Any p-bounded sequence in B has asymptotic center consisting of one point.

Proof. Let {xn} be a bounded sequence and let r > 0 be its asymptotic radius (if I = 0 then A( {x,}) = 1im XJ Take E > 0 and put A, = fi

fi

k=l

i=k

E(xi, r + E),

where K(x, r + E) is the closed p-ball centered at xi of radius r + E. Notice that A({x,}) may be equivalently defined as A (Ix,)) = n AC. Hence in view of basic properties of p-convex sets and Theorem 2, A({x,}) # 4. To prove uniqueness suppose that x, y E A({x,}). Thus we have lim sup p(x,, x) < r, Km sup P&

Y) < I,

implying r d *imsupp(x,,+[x,y])

and we get a contradiction

6 [I - 6(r,q)]r

unless p(x, y) = 0.

The next question is how the asymptotic center is situated with respect to the sequence.

THEOREM 6. Let X be a p-convex, closed subset of B and let {x,> be a p-bounded sequence of elements of X. Then A({x,)) E X.

Proof. Let z = A((x,}) and let Pz be the metric projection of z onto X. Then in view of Theorem 4 we have r(Pz, {x,>) d r(z, {x,)) = r({x,j), implying z = Pz.

K. GOEBEL,

1018

T. SFKOWSKI

and

A. STACHURA

If we denote p-conv X the smallest p-convex, closed set containing X, then in view of Theorem 6 we get A({x,})E fi p-conv[x,:

k = n, n + 1,. . .].

?!=I Remark. The fact above makes the asymptotic center similar to the notion of the weak limit. However it should be mentioned that in a Banach space setting, the weak limit of a weakly convergent sequence usually does not coincide with its asymptotic center. Both notions coincide for the class of weakly convergent sequences in Hilbert space and all Zpspaces, but not in Lp spaces [163. THEOREM

7. If

{x”} is a p-bounded sequence which converges weakly to x, then x = A({x,}).

Proof. Because of weak continuity x = 0. For any y # 0 we have

of all Mijbius transforms we can consider only the case - llYl12) 11- (X”7Y)12

lim sup 0(x,, y) = lim sup (I - llxJ)(I

= (1 - 11 y 11') lim sup(1 - 11 x, /j‘) < lim sup fr(xn, 0) and thus r(y, {x,>) 2 ~(0, {x,>), which completes the proof. This shows that p-bounded, p-convex sets are weakly compact. It is also worth noticing that generally the asymptotic center of a sequence {x”} with respect to the metric p does not coincide with the asymptotic center with respect to norm. For example if {x”} is a periodic sequence taking only two values x, y then the respective asymptotic centers are $[x, y] and (x + y)/2.

6. NONEXPANSIVE

MAPPINGS

ON

BAND

FIXED

POINTS

Suppose C = B is a nonempty closed and p-convex set. In view of Lemma 2, Lemma 3 and the results of previous sections we immediately get the following: 8. If

THEOREM

9. If T: C -+ C is nonexpansive then the following statements are equivalent:

(i) (ii) (iii) (iv)

C is bounded then any p-nonexpansive

mapping T: C -+ C has a fixed point.

THEOREM

T has a fixed point in C; There exists x E C such that the sequence {x,,} = {T”x} is p-bounded; For any x E C the sequence {xJ = {T”xj is p-bounded; There exists a p-bounded sequence {y,} of elements of C such that p(y,, Ty,) --f 0.

The above theorems are exact copies of well-known results holding for norm nonexpansive mappings defined on closed convex sets in uniformly convex Banach spaces ([l], C517C61). Remark:

THEOREM

p-convex.

10. If T:

C -+ C is a p-nonexpansive

mapping then the set fix T = [x: x = TX] is

Uniform

convexity

of the hyperbolic

metric and fixed points of holomorphic

mappings

in the Hilbert ball

1019

Proof. It is enough to show that if x, y E fix T then z = $[x, y] E fix T. We have p( Tz, x) 6 p(z, x) y) implying Tz = i[x, y].

= $.+x, y) and similarly p(Tz, y) < $(x,

7. FIXED

POINTS

OF HOLOMORPHIC

MAPPINGS

So far we have not made use of analyticity, but only of p-nonexpansiveness. Theorem 9 we have:

According to

THEOREM 11. A holomorphic mapping T : B -+ B has a fixed point if and only if for some x E B the sequence {x,,> = (T”x} satisfies lim sup/Ix, I( -c 1.

This extends the result of Earle, Hamilton [3]. Analyticity allow us to give some more precise characterization THEOREM

12. If T: B + B is holomorphic

of the set fix 7:

then the set fix T is affine.

Proof: It is enough to prove this under the assumption 0 E fix T, (T(0) = 0). If this is not the case and z E fix T then we can consider the mapping S = M _-z0 TO M, and notice that 0 E fix S and fuc T = Mz(fix s). Because Mobius transforms map affine sets onto afline sets our restriction does not lead to a loss of generality. Suppose now that for z # 0 we have z = Tz. Thus in view of Theorem 10 tz = T(tz) for all t E (0,l) and consequently in view of analyticity, u - Tu = 0 for all u belonging to the intersection of B with a one-dimensional space spanned by z. Thus the set fix T contains an affine subset. The same reasoning shows that for any two points x, y E fix T the whole geodesic line (not only the segment [x, y]) passing through x, y is contained in fix T. Let U be a maximal afline subset of fix T which contains 0. We shall prove that actually U = fix T. Assume the contrary. Thus there exists w # U such that w = Tw. Obviously U is closed and w can be represented as w = z + u where z E U, (z, u) = 0. Take the geodesic line passing through 0 and u. It is a linear segment consisting of all points of the form tu, t E ( - l/ (1u (I, l/ ((u (I).Notice that

M&v) = (JU - I(z/l‘Pi + p,)

1

r ;vzz, =tJ(l -

11 z )12b + z.

7

Thus the linear segment consisting of all points z + t,/(l - l/z I12)u,t E ( - l/I)v II, (l/II u (I) is also geodesic. This line contains two fixed points z (t = 0) and z + u (t = I/,,/(1 - \\z/“)) and in view of that is totally contained in fix lYThus (z - v) E fu T(t = - l,/(l - 11~11”)). Consequently -(z-v)=(-z+v)~fixTandfinally~[-z+u,z+u]=v~fixT.LetVbeana~neset spanned by u. It is just a technicality to prove that the smallest geodesically closed set containing U and V coincides with the smallest afhne set spanned by II and I/: This set is contained in fix T, which contradicts maximality of U. Thus U = fix T. The fact described in Theorem 12 has been proved for biholomorphic mappings in [ 111. To obtain further results let us start with the following: LEMMA 4.

Let T: B -+ B be a holomorphic /yll > llzl/, Ty # Ayforall1 > 1.

mapping and let z E fix T. For any y E B with

1020

K. GOEBEL, T. S~KOWSKI and A. STACHURA

Proof: The point y is the metrical projection of ly onto the ball centered at zero of radius M&Thus P@Y, z) ’ P(Y7z) (recall Theorem 4) which contradicts Ty = ily and completes the

Let us associate with a given holomorphic T: B -+ B the family of mappings T, = tT, t E (0, 1). Each mapping T, maps B strictly inside itself and thus has exactly one fixed point. Let us denote such point by z(t). It is easy to notice that z(t) depends continuously on t (even analytically [lo]) and that IIz(t) - Tz(t)I/ = (1 - t))j E(t)/1 < (1 - t) + 0 as t -+ 1. THEOREM

norm.

13. If fix T # 4, then lim z(t) = z where z is the fixed point of T with the smallest r-1

Proof. Let z be as required. Then 11 z(t) 11G IIzll for all t E (0, 1) because otherwise it would contradict Lemma 4. Take any sequence t, -+ 1 such that {z(t,)> converges weakly. Let u = w-lim z(t,) = A({z(f)}). Th us u EON T and llull < llzll which implies u = z. But now the inequality IIz(t,) II d IIz II implies that actually (z(t,)} converges strongly to z and we get our conclusion. THEOREM

14. A holomorphic mapping T: B + B has a fixed point if and only if there exists r < 1 such that TX # Ax, A > 1 for all x such that JJx1J= r.

ProoJ: The ‘only if part is a consequence of Lemma 4. If there exists such r, then ((z(t)// f r for all t E (0, 1) and in view of Theorem 9 (Statement (iv)), fix T # I$.

In consequence of Theorems 13 and 14 fix T is nonempty if and only if !\y /)z(t) 1)< 1. THEOREM 15. Suppose T: I? -+ B is holomorphic ball B. Then T has a fixed point in B.

and has a continuous extension to the closed

Let z(t) be as above. If lim /lz(t) (( < 1 then T has a fixed point in B. Assume that 1-1 lim ((z(t)(( = 1. Take a sequence tn --+ 1 such that {z(tJ> converges weakly and let u = w - lim z(tJ. Proof

1-1

We shall prove that l/u (/ = 1. Suppose the contrary:

l/u (1< 1. Then we have

c( i%(Q), T(u)) 2 +@,), u) and more precisely

Consequently 1 > 1 - (l/t.zJJz(&)/2 B (I - )lu))211 - (ll~“)M~,)~WI2

’

1 - IIzk)l12

(1 - 11T~11~))1- (4&J, u)l’

Passing with n to infinity we obtain 1 Z l/a(Tu, u) implying p(u, Tu) = 0 or u E fix T. This in

Uniform

convexity

of the hyperbolic

metric and fixed points of holomorphic

mappings

in the Hilbert

ball

1021

view of Lemma 4 contradicts the assumption ‘I’r;lt1)z(t) 11= 1. Thus )Iu 11= 1 and {z(t,)> converges strongly to u, and because !iy ((z(t,) - T(z(t,)) // = 0, u must be fixed. This last theorem improves the result of Hayden and Suffridge [lo] in the case of Hilbert space. Also it solves the problem raised by Harris [9]. REFERENCES nonlinear operators in a Banach space, Proc. Nat. Acad. Sci. U.S.A. 54, 10411. BROWDER F. E., Nonexpansive 1044 (1965).,, Funktionen von zwei komplexen Verlnderlichen, 2. CARATHI?ODORTC., tiber das Schwarzsche Lemma bei analytischen Math. Ann. 97,7698 (19261. mappings, Proc. Symposia Pure Math., vol. 16, 3. EARLE C. J., HAMILTON R. S., A fixed point theorem for holomorphic Amer. Math. Sot. Providence, RI. (1970x pp. 61-65. of an asymptotic center with a fixed point property, Bull. Am. mafh. Sot. 78 2OfG-208 4. EDELSTEINM., The construction (1972+ 5. GOEBEL K., An elementary proof of the fixed point theorem of Browder and Kirk, Michigan Math. J. 16, 381-383 (1969). Abbildung, Math. Nachr. 30, 251-258 (1965). 6. G~~HDED., Zum Prinzip der Kontraktiven 7. HAHN K. T., Geometry on the unit ball of a complex Hilbert space, Can. J. Math. XXX, No. 1, 22-31 (1978). functions of vectors, Proc. Coil. Analysis, Rio de Janeiro 8. HARRIS L. A., Bounds on the derivatives of holomorphic (1972), pp. 14+163; Act. Sci. ef Znd., Hermann, Paris (1975). systems of p,seudometrics for domains in a,normed linear space, Advanced in Holo9. HARRIS L. A.,Schwarz-Pick morphy Proc. ‘Seminario de Holomortia,‘Univ. Fed. do Rio de ,Janeiro,_l,977#‘J. A. Bar sso (ed). North Holland, Pup! 1979, pp. 345-406. maps in Banach spaces, Proc. Am. math. ‘so,. 60, 10. HAYDEN T. L. & SUFFRIDCE T. J., Fixed points of holomorphic 95-105 (1976). maps in Hilbert space have a fixed point, PaciJ J. Math. 38, 11. HAYDEN T. L. & SUFFRIDGE T. J., Biholomorphic 419422 (1971). Research 12. HERON M., Several Complex Variables, Local Theory, Oxford Univ. Press and Tata Instituteof Fundamental (1963). analysis and semigroups, Amer. Math. Sot. Coil. Publ. Vol. 3 1, Amer. Math. 13 HILLE E. & PHILLIPS R. S., Functional Sot., Providence, R.1. (1957). mappings in metric space, Kodai Math. Sem. 14. KIIIMA Y. & TAKAHASHI W., A fixed point theorem for non-expansive Rep. 21,326330(1969). 15. KIRK W. A., A fixed point theorem for mappings which do not increase distances, Am. math. Monthly 72, 1004-1006 (1965). of the sequence of successive approximations for nonexpansive mappings, Bull. Am. 16. OPIAL Z., Weak convergence math. Sot. 73,59ll597(1967). Distanz und ihre zugehorige Differentialmetrik, Math. Ann. 161,3 15-324 (1965). 17. REIFFEN H. J., Die Caratheodorysche geometrischen Eigenschaften der invarianten Distanzfunktion von Carathtodory, 18. REIFFEN H. J., Die differential Schr$t Math. Inst. Univ. Miinster 26 (1963).