[Gulevich] Fixed Points of Nonexpansive Mappings(BookFi.org)

7/21/2019 [Gulevich] Fixed Points of Nonexpansive Mappings(BookFi.org)

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P r i n c i p a l N o t a t i o n s

W e n o w i n d i c a t e t h e m o s t f r e q u e n t ly u s e d n o t a t io n s :

O - - t h e e m p t y s e t;

N = { 1, 2 , 3 , . . . } - - t h e s e t o f n a t u r a l n u m b e r s ;

IR t h e s e t o f r e a l n u m b e r s ;

R n - - t h e n - d i m e n s i o n a l r e a l E u c l i d e a n s p a c e ;

( X , 1[. 1[) - - a r e a l B a n a c h s p a c e w i t h a n o r m [ [. [[;

X * - - t h e s p a c e d u a l to X ;

d i m X - - t h e d i m e n s i o n o f X ;

B x , r) - -

t h e c l o s e d b a l l i n X w i t h t h e c e n t e r x E X a n d t h e r a d i u s r ;

B X ) = B ( O , 1 ) - - t h e c l o s e d u n i t b a l l in X ;

S x , r) - -

t h e s p h e r e w i t h t h e c e n t e r x 6 X a n d t h e r a d i u s r ;

S X ) = S ( 0 , 1 ) - - t h e u n i t s p h e r e in X ;

A - - t h e c l o s u r e o f A C X i n t h e n o r m t o p o l o g y ;

~ o ( ~ ) _ _ t h e c lo s u r e o f A i n t h e w e a k (w e a k * ) t o p o l o g y ;

O A = A N X \ A - -

t h e

b o u n d a r y

o f A ;

i n t A = A \ 0 A - - t h e i n t e r i o r o f A ;

(c-6-n-V A) c o n y A - - t h e ( d o s e d ) c o n v e x h u l l o f A ;

d i s t ( x , A ) = i n f { [ [ x - y n : y E A } - - t h e

d i s tance

f r o m x E X t o A ;

d i a m A = s u p { I l x - y [ [ :

x , y

E A } - - t h e

d i a m e t e r

o f A ;

r ( A )

=

in f su p [ [x - Yll

t he Chebyshev rad ius

o f A ;

xEconv y E A

C'( A) = {x E c-5-n 'VA : su p

x

-

y l l

= r ( A ) } - - t h e Ch e b y s h e v c e n t e r o f A ;

y E A

d i s t ( A , E ) =

i n f { l [ x - y l [ : x 6 A , y

E E } - - t h e

d i s tance

b e t w e e n d a n d E ;

T : A - X - - a n o n e ~ p a n s i v e m a p p i n g , i . e . I l T x - T y t l ~

I l x

y I ] f o r a l l x , y 6 A ;

F i x T = { x E A : T x = x } - - t h e s e t o f f ix e d p o i n t s o f a m a p T ;

I P I ) - -

t h e B a n a c h s p a c e o f f a m i l i e s o f n u m b e r s x = { x i } i E I f o r w h i c h llx Ilp < c o ;

{ i ~ 6 i ) l / p

l x l t p = Ix~tp , 1 ~< p < o o , _ t h e n o r m i n t h e s p a c e l P ( I ) ;

s u p { l x i l : i E I } , p = o o

I v = / P ( N ) , I~ = / P ( { 1 , 2 , . . . , n } ) ;

c I ) C o I ) ) - -

t h e B a n a c h s p a c e o f s e q u e n c e s o f n u m b e r s w h i c h a r e c o n v e r g e n t ( t o 0 ) w i t h t h e n o r m II.r I =

s u p { I x , [ : i e I } ;

c = c ( N ) ( Co = c 0 ( N ) ) - - t h e S a n a c h s p a c e o f s e q u e n c e s o f n u m b e r s c o n v e r g i n g ( t o 0 ) w i t h t h e n o r m ]]x [] =

s u p { ] x n l : n e N } ;

C K ) - -

t h e B a n a c h s p a c e o f c o n t i n u o u s f u n c t i o n s o n a c o m p a c t s p a c e K w i t h t h e n o r m [[f]] =

s u p { I f x )[ : x E g } ;

L P ( ) = L P ( f /, E , ) - - t h e B a n a c h s p a c e o f / ; - m e a s u r a b l e f u n c t i o n s f o n f~ w i t h p - t h s u m m a b l e p o w e r ;

1 ~< p <~ e c ; ( f~ , E , / ; ) - - t h e s p a c e w i t h a a - f i n i t e m e a s u r e ;

1 , ,

I t f l l = I f l ' d , 1 ~< p < c ~ _ t h e n o r m o n t h e s p a c e L P ( ) ;

e s s s u p { l f ( z ) ] : x 6 f~ } ,

p = o o

L P [ 0 , 1] = L P ( [ 0 , 1 ] , ~ , A ) ; A - t h e L e b e s g u e m e a s u r e o n [ 0, 11;

\

z e x , ) - th e l V ) -d irec t s u m th e d irec t p rodu ct) o f a fami ly o f Ban ach

{ X i } i 6 I ;

p a c e s

\ i 6 1

/ p

i ~ E i ~ X i ) - - t h e c o I ) - d i r e c t s u m o f B a n a c h s p a e e s X i } i e i ;

o

756



d X,r) = i n f { l l F I I IIF-111 : F i s a n i s o m o r p h i s m f r o m X o n t o Y } - - t h e B a n a c h - M a z u r d i s t a n c e

b e t w e e n t w o s p a c e s X a n d Y ;

i f t h e s p a c e s X a n d Y a r e i s o m e t r i c a l l y i s o m o r p h i c , w e w r i t e X ~ Y ;

t h e c o n v e r g e n c e o f a s e q u e n c e { z,~ } C X t o a n e l e m e n t z E X i n t h e s t r o n g ( i n I 1' l t) , w e a k o r w e a k *

H> w w

t o p o l o g i e s o f a s p a c e X w i l l b e d e n o t e d b y x n ~ x ( o r x ,~ x ) , z n > z , a n d x ,~ > z r e s p e c t i v e l y ;

z , y ) = { x + t y - z ) : 0 < t < 1 } - - t h e in t e r v a l w i t h e n d p o i n t s z , y E X ;

x , y ] = { z + t y - x ) : 0 < t ~< 1 } - - t h e h a l f i n t e r v a l ;

[ x , y l = { x + t ( y - ~ ) : 0 < t < z } - t h e s e g m e n t ( d o s e d i n t e r v a l ) .

[ ] d e n o t e s t h e e n d o f a p r o o f .

T h e n o t a t i o n s l i s t e d a b o v e a r e m o r e o r l e ss s t a n d a r d ( s e e t h e b o o k s [ 9, 1 0, 6 1 , 6 9 , 1 6, 2 0]. P e r h a p s t h e

n o t i o n o f th e l p, c 0 - d i re c t p r o d u c t o f B a n a c h s p a c e s n e e d s a m o r e p r e c i se d e f i n i t i o n .

L e t I b e a n o n e m p t y s e t. B y t h e I P I )- d i r e ct s u m ( o r I P I ) - d i r e c t p r o d u c t ) o f a f a m i l y o f B a n a c h s p a ce s

{ X i } i e z w e s h a ll m e a n t h e B a n a c h s p a c e o f a ll f a m i l ie s x = { z i} i E z, x i E X i f o r w h i c h

I l z l l < o ~

w h e r e

1

I1 ~1 1 = I x i l ) ,

s u p { l l z i l l : i E I } ,

l ~ p < ~

S i m i l a r l y , b y t h e

c o I ) - d ir e c t s u m

o f a f a m i l y o f B a n a c h sp a c e s

{ X i } i E I

w e m e a n t h e B a n a c h s p a ce o f a ll

c o n v e r g e n t t o z e r o f a m i l i e s x = { z i} iE X ,

x i E X i ,

l i m I l z i l l = 0 e n d o w e d w i th t h e n o r m

iEI

I lx l l = s u p { l l m , II : i e Z } .

C h a p t e r

N o r m a l S t r u c t u r e i n B a n a c h

S p a c e s a n d I ts G e n e r a l i z a t i o n s

1 .1 . N o r m a l S t r u c t u r e .

T h e n o t i o n o f n o r m a l s t r u c t u r e , o n e o f t h e m o s t i n t e n s iv e l y s tu d i e d d u r i n g t h e l a s t d e c a d e s , w a s i n t ro -

d u c e d b y M . S . B r o d s k i i a n d D . P . M i l m a n i n 1 94 8 i n th e w e l l - k n o w n p a p e r [3] i n o r d e r t o p r o v e t h e e x i s t e n c e

o f a c o m m o n f i xe d p o i n t f o r a f a m i l y o f i s o m e tr i c m a p p i n g s f r o m a c o n v e x c o m p a c t s u b s e t o f a B a n a c h

s p a c e i n t o i t s e l f .

T h i s n o t i o n i s ef f e c t iv e l y u t i l i z e d in t h e f ix e d p o i n t t h e o r y o f n o n e x p a n s i v e m a p p i n g s .

W e n o w g i v e t h e n e c e s s a r y d e f i n i t i o n s .

L e t X b e a B a n a c h s p a c e , A C X b e a c o n v e x b o u n d e d s e t.

D e f i n i t i o n 1 . 1 . 1 . h p o i n t x 0 E A is c a l le d a d i a m e t r i c a l p o i n t o f t h e s e t A i f

s u p f ix 0 - ~ 11 = d i a m A .

xE

T h e s e t A h a s a n o r m a l s t r u c t u r e i f n o t a l l o f i ts p o i n t s a r e d i a m e t r i c a l a n d i f t h e s a m e p r o p e r t y i s s h a r e d

b y e v e r y c o n v ex s u b s e t o f A h a v i n g m o r e t h a n o n e p o i n t.

T h u s A h a s a n o r m a l s t r u c t u r e i f f o r a n y c o n v e x C C A , d i a m C > 0 , t h e r e e x i s t s a n e l e m e n t x E C s u c h

t h a t

s u p l l x - y l l < d i a m C ,

y E c

i .e ., C i s c o n t a i n e d i n a b a l l w h o s e c e n t e r is a p o i n t o f C a n d w h o s e r a d i u s i s l es s t h a n t h e d i a m e t e r o f C .

757



De fi ni ti on 1.1.2. A bounded sequence {zn} C X is called d i a m e t r i c a l if diam{xn } > 0 and

lim dist(z~+l, conv{zk}~=l) = diam{z~}.

In the paper by Brodskii and M ilman menti oned above there is given the following criterion of presence

of a normal structure for a convex set.

Theorem 1 .1 .3 .

A s e t A C X h a s a n o r m a l s t r u c t u r e i f a n d o n l y i f i t h a s n o d i a m e t r i c a l s e q u e n c e s .

Pr oo f. Suppose that A has a diametrical sequence {xn} and let C be the convex huh of this sequence.

It will be sufficient to show that the set C has no norma l structure. If x E C, t he n x E conv{xk}~=x for

so mer nE Na nd x Econv{xk}~=l forum> m. Hence

d i a m C x . a l t >

dist(xn+l,

C O 1 2 V { Z k } ~ = I .

This proves that for any x E C

s u p { z y l l : y E C } = diam C.

Conversely, assu me that A contains a convex set H, diam H = d > 0, which consists of diamet rical

points. It will be sufficient to show tha t H has a diametr ical sequence. Let 0 < e < d an d the points

Zl, z2 E H be chosen so th at IlXl - x211 > d - e . Assu ming that the points

X l , X 2 , . . . , x n

are alre ady found,

we choose xn+l E H so th at

X 1 + X 2 + ' + X n _ _ X r t - 4 - 1 > d g

n ~

/'l

We now show tha t the sequence {xn} thus co nst ruc ted is diametrical. Let z = ~--~k=l kxk , t k >~ 0,

7l

Y ~ k = l t k = 1, t = max {t a,t 2,. .., t,~ }. Then it is easy to check that

= - - z z n } ,

n t ~ t / z k e cony{z, zl , . . . ,

k-.-~ 1

Hence

d - ~_~ < - ~<

k----1

and, therefore,

- nli > d -

This allows us to conclude

diametrical. []

/ I x -x l lt I I x - x l lt < . 7 1 1 - g 7 d ,

k = l

_et >1 d - e__.

Thus

n n

l im dist(zn+l, co nv {x l, x2 , . . ,z,~}) = d.

that d is the diame ter of the sequence {Zn} and t hat th is sequence is

In particular, Theorem 1.1.3 implies that every convex compact subset of an arbit rary Banac h space has

normal structure because it cannot contain a diametrical sequence.

An example of a set without a normal structure is given by the unit ball of the space co, since the

standard basis {en} C co forms a diametrical sequence.

The following theorem by Kirk [119] is a classical result on fixed points whose proof makes essential use

of the notion of normal structure.

Theorem 1 .1 .4 .

L e t K b e a n o n e m p t y c o n v e x w e a k l y c o m p a c t s u b s e t o f a B a n a c h s p a c e . I f K h a s a

n o r m a l s t r u c t u r e , t h e n e v e r y n o n e x p a n s i v e m a p p i n g T : K - + K h a s a f ix e d p o i n t .

Pr oo f. Using Zorn's lemma we find a nonempt y minimal T- invariant convex closed subset C in K. If C

consists of a single point, th en ever ythin g is proved. Suppose tha t d = di am C > 0. Since K h as a norm al

structure, there exists an r E (0, d) such that

D = { x e C : C c B ( x , r ) } ~ 3.

For an arbitr ary x E D the nonexpansi veness of T implies the inclusion

T ( C ) C B ( T x , r ).

Hence

con-5-fivT(C) C B ( T z , r ) .

Bu t con-6-ffVT(C) = C, since con-5-n-vT(C) is a T- in va ri an t subset of C . Thus C C

B ( T x , r ) .

This shows that

T x E D ,

and hence D is T-invariant. Moreover, the set D is convex and closed

because D = N{ B( y, r) : y E C}. The min imal ity of C yietds D = C. Hence IIx - Yil ~< r for any x, y E D,

and, there fore, d iam D ~< r < diam C. This gives a contra dic tion. Hence dia~mC = 0. []

758



D e f i n i t i o n 1 . 1 . 5 . A B a n a c h s p a c e is s a id t o h a v e a n o r m a l s t r u c t u r e i f e v e ry b o u n d e d c o n v e x c l o s e d

s u b s e t o f i t h as a n o r m a l s t r u c t u r e .

W e n o w p a s s t o c o n s i d e r i n g s o m e c la s s es o f B a n a c h s p a c e s h a v i n g a n o r m a l s t r u c t u r e .

D e f i n i t i o n 1 . 1 . 6 . A f u n c t i o n 8 : [0 , 2 ] - + [ 0, 1 ] s u c h t h a t f o r 0 ~< r ~< 2

6 r

i s c a l l e d a m o d u l u s o f c o n v e x i t y o f t h e B a n a c h s p a c e X .

A s p a c e X i s c a l l e d u n i f o r m l y c o n v e z i f 8 x ~ ) > 0 f o r e v e r y r > 0 .

I t c a n b e s h o w n t h a t a B a n a c h s p a c e X i s u n i f o r m l y c o n v e x i f a n d o n l y if o n e o f t h e f o l l o w i n g c o n d i t i o n s

i s f u l f i l l e d :

1 ) f o r a n y r > 0 t h e r e e x i s t s a 8 r > 0 s u c h t h a t i f x , y E B X ) a n d I I x -Y l l ~ , t h e n II7

<

1 - 8 ~ ) ;

2) i f x ,~ , y ,~ E B X ) a n d I lx n + y ~ l l ~ 2 , th n I Ix ~ - y n l l - + 0 , ~ - + ~ ) .

R e m a r k 1 . 1 . 7 . T h e n o t i o n o f u n i f o r m c o n v e x i t y w a s i n t r o d u c e d b y J . A . C l a r k s o n [5 7] . I- Ii lb e rt s p a c e s ,

s p a c e s

Ip ,

L P 1 < p < c x~ ) a r e e x a m p l e s o f u n i f o r m l y c o n v e x s p a c e s .

E v e r y u n i f o r m l y c o n v e x B a n a c h s p a c e is r e fl e x iv e s e e, e .g . , [1 6] ) a n d , t h e r e f o r e , e v e r y b o u n d e d c o n v e x

a n d d o s e d s u b s e t o f i t is w e a k l y c o m p a c t .

S t a t e m e n t 1 . 1 . 8 M . E d e l s t e i n [ 75 ]) .

E v e r y u n i f o r m l y c o n v e z B a n a c h s p a c e h a s a n o r m a l s t r u c t u r e .

P r o o f . F o r an a r b i t r a r y b o u n d e d c o n v e x s u b s e t A o f a u n i f o rm l y c o n v e x s p a c e w i th d = d i a m A > 0

o n e c a n f i n d t w o e l e m e n t s

u , v E A

s u c h t h a t IIu - vii )

d / 2 .

T h e n f o r e a c h x E A w e h a v e IIx - ull ~< d,

I I x - v l l ~ < d , I I x - u ) - x - v ) l l / > d / 2 He n c e I t x - u ) + x - v ) l l ~ <

2 1 - 8 1 / 2 ) ) d ,

i . e I I x - w l l -<

1 - 8 1 / 2 ) ) d ,

w h e r e w = u + v ) / 2 E A . T h u s A C B w , r ) , w h e r e r = 1 - 8 1 / 2 ) ) d < d . [ ]

D e f i n i t i o n 1 . 1 . 9 . A n o n e m p t y p r o p e r s u b s e t C o f a B a n a c h s p a c e X i s s a i d t o h a v e t h e f i z e d p o i n t

p r o p e r t y F P P ) i f a n y n o n e x p a n s i v e m a p p i n g T : C --+ C h a s a f i x ed p o i n t.

A sp a c e X h a s t h e F P P i f a n y n o n e m p t y b o u n d e d c o n v e x a n d c lo s e d s u b s e t o f i t h a s t h e F P P .

T h e o r e m 1 .1 .4 . i m p l i e s t h a t a n y re f le x i v e B a n a c h s p a c e w i t h a n o r m a l s t r u c t u r e h a s t h e F P P .

A s a c o r o l l a r y o f T h e o r e m 1 .1 .4 . a n d S t a t e m e n t 1 . 1 . 8 , o n e h a s

T h e o r e m 1 . 1 . 1 0 F . E . B r o w d e r [3 9] , D . G 6 h d e [ 94 ]) .

E a c h u n i f o r m l y c o n v e z B a n a c h s p a c e h a s t h e

F P P .

D e f i n i t i o n 1 . 1 . 1 1 . A B a n a c h sp a c e X i s c a ll e d

u n i f o r m l y c o n v e z i n e v e r y d i r e c t i o n

U C E D ) i f f o r a n y

z E X \ { 0 } a n d s > 0 t h e r e e x i st s a 8 = 8 e , z ) > 0 s u c h t h a t i f x , y E

S X ) , x - y

= A z , A E R a n d

]l x + Yll > 2 1 - 8) , th e n IAI < r

R e m a r k 1 . 1 . 1 2 . T h e n o t i o n o f u n i f o r m c o n v e x i ty in a n y d i r ec t i o n w a s i n t r o d u c e d b y A . L . G a r k a v i [5]

i n o r d e r t o c h a ra c t e r i z e t h o s e B a n a c h s p a c es i n w h i c h e v e r y b o u n d e d s e t ha s C h e b y s h e v c e n t e r c o n s i st i n g

o f n o t m o r e t h a n o n e p o i n t t h e C h e b y s h e v c e n t e r o f a s e t i s t h e c o l l e c t i o n o f c e n t e r s o f a ll b a l l s o f t h e

s m a l l e s t r a d i u s c o n t a i n i n g t h e g i v e n se t ) .

S t a t e m e n t 1 . 1 . 1 3 [ 62 , 2 2 8] . F o r a B a n a c h s p a c e X t o b e UC E D ) i t i s n e c e s s a r y a n d S u f f i c i e n t t h a t

a n y o f th e f o l l o w i n g c o n d i t i o n s h o l d :

1 )

f o r a n y E X \

{ 0 }

a n d x n , y n E B X ) s u c h t h a t z ~ - y n = A ,~ z, A ~ E R a n d

] lx ,~ + ynt l - -+ 2 ,

o n e

has An --+ 0 ;

2 ) f o r a n y x n , y ~ E B X ) s u c h t h a t x n - y . ~ z a n d

I l x n Y - I I - ~ 2 o n e h a s z = 0 ;

3 )

f o r a n y z E X \

{ 0 }

t h e r e e z i s t s a A > 0 s u c h t h a t

II x + 2z l I < 1 - A ,

i f

llxlI ~< 1

a n d z

+ zll ~< 1;

4 )

f o r a n y z E X \

{ 0 }

a n d r > O , o n e h a s 8 r > O , w h e r e

8 r ~

: x , y E B x ) , x - - y = A z , I A I > . . .e }

i s t h e mo d u l u s o f c o n v e z i t y o f X a t th e p o i n t z .

I t i s e a s y t o s e e t h a t e v e r y u n i f o r m l y c o n v e x s p a c e is U C E D ) .

7 5 9



T h e o r e m 1 . 1 . 1 4 [ 22 8, 6 2] . E v e r y B a n a c h s p a ce w h i c h i s u n i f o r m l y c o n v e z i n e v e r y d i r e c t i o n h a s a

n o r m a l s t r u c t u r e .

P r o o f . I t w i ll b e s u f fi c ie n t t o p ro v e t h a t a n y b o u n d e d c o n v e x s u b s e t A o f a ( U C E D ) B a n a c h s p a c e w i t h

d = d i a m A > 0 c o n t a i n s a n o n - d i a m e t r i c a l p o i n t.

T a k e z , y E A , x r y . W e s h a l l s h o w t h a t u = ( x + y ) / 2 i s a n o n - d i a m e t r i c a l p o i n t. I f n o t , t h e n t h e r e

e x i s t s { v a } C A s u c h t h a t llu - vn]l --+ d . T h e n IIX -- Vnll < d, lly - v~l l <<-d, II(X -- V,~ + y - - Vn)/21 1 - + d

a n d x - v n - ( y - v n ) = x - y . By v i r t u e o f P r o p o s i t i o n 1 . 1. 1 3, it fo l lo w s t h a t z = y . T h i s g i v e s a

c o n t r a d i c t i o n . [ ]

W e p r e s e n t t w o g e n e r a l i z a t i o n s o f T h e o r e m 1 .1 .1 4 .

T h e o r e m 1 . 1 . 1 5 [ 2 0 2 ] .

L e t X b e a B a n a c h s p a c e , A C X b e a c o u n t a b l e s u b s e t , a n d l e t X b e ( U C E D )

w i t h r e s p e c t t o X \ A ( i . e . , f o r a n y z E X \ A , z 7~ O , a n d s > O, t h e m o d u l u s o f c o n v e z i t y ~ ( e , z ) > 0 ) .

T h e n X h a s a n o r m a l s t r u c t u r e .

T h e o r e m 1 . 1 . 1 6 [117]. L e t X b e B a n a c h s p a c e a n d Y C X b e a s u b s p a c e o f f i n i t e c o d i m e n s i o n . I f X

i s ( U C E D ) w i t h r e sp e c t t o S ( Y ) , t h e n X h a s a n o r m a l s t r u c t u r e .

D e f i n i t i o n 1 .1 . 17 1 2 0] . A B a n a c h ( c o n j u g a te ) s pa c e X is s a i d t o b e w e a k l y ( w e a k * ) u n i f o r m l y c o n v e x i f

f o r a n y x ,~ , y ~ e B ( X ) suc h th a t I lxn +Y nl l + 2 i t fo l lows th a t x~ - y~ ~) 0 (x ,~ - y ,~ ~ '> 0 ) . I t i s c l ear

t h a t e v e r y u n i f o r m l y c o n v e x s p a c e is w e a k l y u n if o r m l y c o n v e x, a n d e v e r y w e a k l y u n i f o r m l y c o n v e x s p a c e is

( U C E D ) . H e n c e w e a k l y u n i f o r m l y c on v e x s p ac e s h a v e a n o r m a l s t r u c t u r e .

R e m a r k 1 . 1 . 1 8 . T h e c la ss o f l o ca l ly u n i f o r m l y c o n v e x s p a c e s ( L U C ) i n t r o d u c e d b y A . R . L o v a g l i a [1 65 ]

i s c l os e t o t h e c l a ss o f u n i f o r m l y c o n v e x s p a c e s ( a B a n a c h s p a c e X i s s a i d t o b e ( L U C ) i f f r o m t h e c o n d i t i o n s

x , x ~ e S ( X ) a n d tl x, ~ + x t i --4 2 it f o ll o w s t h a t x ~ ~ x ) . H o w e v e r , M. A . Sm i t h a n d B . T u r e t t [ 2 0 3 ] h a v e

c o n s t r u c t e d a n e x a m p l e o f a r e fl e xi v e ( L U C ) B a n a c h s p a c e w i t h o u t a n o r m a l s t r u c t u r e .

F o r { x l , z 2 , . . . , x k + l } C X t h e v alu e

1 }

V x l , . . . , x k+ l )

su

d e t f l x a ) . . . f l Z k + l )

= . . : f i e S ( X * ) , i = 1 , . . . , k

\ f k ( 1 ) . . . f k ( x k + l )

is c a ll e d t h e k - v o l u m e o f t h e s e t c o n y {x 1 , x z , . . . , x r + l } .

D e f i n i t i o n 1 . 1 . 1 9 [2 08 ]. A B a n a c h sp a c e X is c a l le d k - u n i f o r m l y c o n v e z ( k - U C) , k E N , i f ~ ( ~ ) ( e ) > 0

f o r e v e r y ~ > 0 , w h e r e

5 ~ ) s ) = i n f 1 k --~ 1 x i :

x i 6 B X ) , i = 1 , . . . , k -~ 1, V x i , . . . ,

X k + l ~ g

i s t he m o d u l u s o f k - c o n v e z i t y o f X .

I t i s e a s i l y s e e n

t h t

( 1 - U C ) s p a c e is j u s t a u n i f o r m l y c o n v e x ( U C ) s p a c e .

F . S u l l i v an [ 20 8] s h o w e d t h a t e v e r y ( k - U C ) s p a c e i s su p e r r e f l e x i v e a n d h a s a n o r m a l s t r u c t u r e .

O b s e r v e t h a t t h e c l as s o f ( k - U C ) s p a c e s ( fo r k / > 2 ) is s u b s t a n t i a l l y l a r g e r t h a n t h e c l as s o f ( U C ) s p a c e s .

I n p a r t i c u l a r , ( k - U C ) ~ ( ( k + 1 ) - U C ) f o r a n y k E N .

W e n o w t u r n t o a n o r m a l s t r u c t u r e i n a sp a c e o f K a d e c - K l e e t y p e .

D e f i n i t i o n 1 . 1 . 2 0 [ 10 4] . A (r e a l ) B a n a c h s p a c e X i s c a l le d :

1) a K a d e c - g I e e ( K K ) s p a c e , i f f o r e v e r y s e q u e n c e { x n } C B ( X ) s u c h t h a t x n w > x a n d i n f { ] ] x - n -

x l l: n m } > 0 w e h a v e I lx l t < 1;

2)

a u n i f o r m K a d e c - K l e e ( U K K )

s p a c e , i f f o r a n y ~ > 0 t h e r e e x i s t s a ~ E ( 0 , 1 ) s u c h t h a t f r o m t h e

c o n d i t i o n s

{ x n } C B ( X ) , X n ~ o x ,

i n f { l l x , - Z m l l : n r m } > / e i t f o l l o w s t h a t l x l l < - ~ ;

3) a n e a r l y u n i f o r m l y c o n v e x ( N U C ) s p a c e i f f o r a n y r > 0 t h e r e e x i s t s a ~ > 0 s u c h t h a t f r o m t h e

c o n d i ti o n s { x , } C B X ) , i n f { l l z ~ - X m l l : n r r n ) i> e i t f ol lo w s t h a t c o n v { x n ) ) n B 0 , 1 - Q r o .

760



T h e ( K K ) s p a c e s a re r e m a r k a b l e b y t h e f a ct t h a t o n t h e i r u n it s p h e r e s t r o n g a n d w e a k c o n v e r g e n c e

c o i n c i d e .

R . H u f f [ 10 4] h a s s h o w n t h a t a B a n a c h s p a c e is ( N U C ) i f a n d o n l y i f i t i s r e fl e xi v e a n d ( U K K ) .

T h e s p a c e l 1 i s a n o n - r e f l e x i v e ( N U C ) s p a c e , s in c e in P w e a k a n d s t r o n g c o n v e r g e n c e c o i n c i d e .

W e r e c a ll t h a t a B a n a c h s p a c e X i s u n i f o r m l y c o n v e x (U C ) i f f o r e v e r y e > 0 t h e r e e x i s t s a 5 > 0 s u c h

t h a t f o r a n y x , y 6 B X ) ,

I1~

- y l l / >

~,

w e h a v e z + y) / 2 E B O, 1 - 5 ) .

I t i s o b v i o u s t h a t ( U C ) C ( N U C ) C ( U K K ) C ( K K ) a n d a l l t h e i n c lu s i o n s a r e s t r i c t [1 0 4] .

I n [2 24 ], i t i s p r o v e d t h a t ( k - U C ) C ( U K K ) .

D e f i n i t i o n 1 . 1 . 2 1 1 7 3 ]. A B a n a c h s p a c e X i s c a l le d a w e a k ly u n i f o r m K a d e c - K l e e ( W U K K ) s p a c e i f t h e re

e x i s t e E ( 0, 1 ) a n d 6 > 0 s u c h t h a t t h e c o n d i t i o n s { z , , } C B X ) , x , w+ x , inf{ l lz ,~ - XmH

: n

# m }

E

i m p l y I l x l l

~ 1 - 5 .

I t i s c l e a r t h t ( U K K ) C ( W U K K ) .

T h e o r e m 1 . 1 . 2 2 [7 2].

Le t X b e a r e fl e x iv e W U K I0 s p ac e . Th e n X h a s a n o r m a l s t r u c t u r e .

I n p a r t ic u l a r , e v e r y ( N U C ) s p a c e h a s a n o r m a l s t r u c t u r e [ 10 5 , 7 3].

I n [1 37 ], t h e r e i s g i v e n t h e f o l lo w i n g s u f fi c ie n t c r i te r i o n f o r u n i f o r m c o n v e x i t y o f a B a n a c h s p a c e .

P r o p o s i t i o n 1 . 1 . 2 3 . Le t X b e a r e fl e x iv e B a n a c h s pa ce w i t h a n u n c o n d i t i o n a l b a s is { e , } ~= l s u c h t h a t

f o r s o m e p 6 ( 1 , e ~ )

I 1 ~

y l /~ > / l t = l i p I l y I F ,

i f t here ex i s t s n 6 N such tha t

s u p p ( x ) C { e i} i% l , s u p p ( y ) C { e i } i ~ , + l .

T h e n X i s N U C ) .

S i n c e t h e w e l l - k n o w n B e r n s t e i n s p a c e , w h i c h d o e s n o t h a v e t h e B a n a c h - S a k s p r o p e r t y ( s ee [ 1 0] ), s at is f ie s

t h e c o n d i t i o n s o f P r o p o s i t i o n 1 . 1. 2 3 [ 13 7] b y v i r tu e o f 1 . 1 .2 2 , it h a s a n o r m a l s t r u c t u r e .

D e f i n i t i o n 1 . 1 . 2 4 . A B a n a c h s p a ce X is s a id to b e u n i f o r m l y s m o o t h , i f l i r a p t ) / t = 0 , w h e r e

t - - O O

p t ) = p x t ) = s u p { I 1 = § t y l l § I 1 = - t y l l _ 1 : = e s x ) , y ~ B X ) }

i s t h e m o d u lu s o f sm o o t h n e s s o f t h e s p a c e X .

E x a m p l e s o f u n i f o r m l y s m o o t h s p a c e s a xe s u p p l i e d b y H i l b e r t s p a c e s , s p a c e s Ip , L P ( 1 < p < o c ) e t c .

I t i s k n o w n t h a t a s p a c e X i s u n i f o r m l y s m o o t h if a n d o n l y i f X * i s u n i f o r m l y c o n v e x . M o r e o v e r , e v e r y

u n i f o r m l y s m o o t h B a n a c h s p a c e i s r e f le x i v e ( se e , e . g . , [1 0] ).

T h e o r e m 1 . 1 . 2 5 [2 14 ]. Le t X be a Ban ach space such tha t l i r a p x t ) / t < 1 / 2 . T h e n X h a s a n o r m a l

t ~O ~O

s t ruc ture .

C o r o l l a r y 1 . 1 . 2 6 [ 2 7 ] .

E v e r y u n i f o r m l y s m o o t h B a n a c h s p a c e h a s t h e F P P .

D e f i n i t i o n 1 . 1 . 2 7 ( E . L a m i D o z o [1 3 9] ). A B a n a c h s p a c e X s a t i s fi e s t h e O p i a l c o n d i t i o n i f t h e f o l lo w i n g

w

h o l d s : f r o m x , > z , y # x i t f o l l o w s t h a t

l im in f l }x ,~

-

x l l

< l i m i n f l l x ~ - Y l I .

I . i . I )

C o n d i t i o n ( 1 . 1 .1 ) a p p e a r s f o r t h e f ir s t t im e i n p a p e r b y Z . O p i a l [ 20 ]. I t i s e q u i v a l e n t t o e a c h o f t h e

f o l l o w i n g c o n d i t i o n s [ 1 40 , 1 8 , 1 7 3 ] :

1 ) f r o m z , ~ > x , y # x i t f o l l o w s t h a t

l i m s u P l [ ~ -

- - ~ 1 1 < l i m s u P l l z ~ - - Y l I ;

~ ,I - -~ O Q n - - o ~

761



w

2 ) f r o m x n > x , y # x i t f o l l o w s t h a t

l i m i n f l l x ~ - x l l < l i m s u p l l x , - Y t l;

3 ) f r o m x n ~ > O , l i m i n f l i x I t = 1 , x # 0 i t f o l l o w s t h a t

n - - + o o

l i m i n f l l x n

~11

> 1 .

E v e r y H i l b e r t s p a c e X s a t is f i es t h e O p i a l c o n d i t i o n s in c e f o r zn , x , y E X , w e h a v e

l l x n - y t l ~ = I I x ~ - ~ 1 1 2 + 2 x , ~ - x z - y ) + I I x - y l 2

T h e s p a c e l 1 i s a n o n - r e f l e x i v e B a n a c h s p a c e s a t i s f y i n g t h e O p i a l c o n d i t i o n s i nc e , i n l 1 , w e a k a n d s t r o n g

c o n v e r g e n c e c o i n c i d e .

W e n o w d e s c r i b e s o m e m o r e c l a s se s o f B a n a c h s p a c e s s a t i s f y i n g t h e O p i a l c o n d i t i o n .

B y a sca le f unc t io n w e s h a l l m e a n a c o n t i n u o u s s t r i c t l y m o n o t o n e f u n c t i o n v : IR+ --+ R + s u c h t h a t

. ( 0 ) = 0 ,

~ + 0 r = + o r

D e f i n i t i o n 1 . 1 . 2 8 ( F . E . B r o w d e r [4 0] ). L e t X b e a ( r e al ) B a n a c h sp a c e . A m a p p i n g r : X --+ X * i s

c a l l e d a dual i ty mapping, i f t h e r e e x i s ts a s c a l e f u n c t i o n v s u c h t h a t f o r e v e r y x E X

( ~ x ) ( x ) = I I ~ 1 1 t 1~ 1 1 , t C x l l = ~ ( l l x l l) .

w *

W e s h a l l s a y t h a t a sp a c e X h a s a weakly continuous d uali ty ma ppin g ~ : X --+ X * , i f ff?xn > r as

It

x n ) x .

E x a m p l e 1 . 1 . 2 9 [ 43 ]. T h e s p a c e s l p , 1 < p < o c h a v e w e a k l y c o n t i n u o u s d u a l i t y m a p p i n g s . I n f a c t , f o r

x = ( X l , X 2 , . . . ) E I p p u t

f i x = ( IX ll p - 1 . s i g n (x 1 ) , Ix 2[ p - 1 . s i g n ( x 2 ) , . . . ) .

i~ = ) l / q i~ = ) l / q

e t v ( t ) =

t p/q,

w h e r e

1 / p + 1 / q

= 1 . S ince

l ~ i l p - x ) q = I x i l p

C x E l q . M o r e o v e r ,

~ l l x l l ) = I tx t l p / q = I 1 r a n d

i i < x ) x ) l l = x i l , t , - a s i g n , ) = I 1 1 1 p = I I x l l I 1 1 1 .

i =

T h u s r is a d u a l i t y m a p p i n g .

T h e w e a k c o n t i n u i t y o f 9 i s o b v i o u s s in c e , i n o u r c a s e , w e a k c o n v e r g e n c e c o i n c i d e s w i t h b o u n d e d n e s s

a n d c o o r d i n a t e - w i s e c o n v e r g e n c e .

T h e o r e m 1 . 1 . 3 0 ( J .- P . G o ss e z a n d E . L a m i D o z o [9 8]) . Every Banach space wi th a weakly cont inuous

duality mapping satisf ies the OpiaI condit ion.

I n p a r t i c u l a r , t h e s p a c e s Ip , 1 < p < o c s a t i s f y t h e O p i a l c o n d i t i o n .

A s w a s s h o w n i n [ 1 79 ], t h e s p a c e s L p fo r e v e r y p E ( 1, o e ) , p # 2 , d o n o t s a t i s f y t h e O p i a l c o n d i t i o n , a n d

h e n c e t h e y d o n o t h a v e w e a k l y c o n t i n u o u s d u a l i t y m a p p i n g s .

L e t X b e a B a n a c h s p a c e a n d x , y E X . W e sa y t h a t z is orthogonal t o y ( n o t a t i o n x _ k y ) , i f

I l x l l ~ < I I x + @ 1 1 f o r a n y ) , E R .

D e f i n i t i o n 1 . 1 . 3 1 ( L . A . K a r l o v i t z [ 1 1 1] ). T h e r e l a t i o n _L i s s a i d t o b e un i formly approx ima te l y sym-

met r i c ( U A S ) , i f f o r a n y y E X a n d ~ > 0 t h e r e e x i s t a c lo s e d l in e a r s u b s p a c e U = U y , a ) C X o f f i n i t e

c o d i m e n s i o n a n d 5 =

5 y ,~)

> 0 s u c h t h a t

I l x l l ~ t ~

762





T h i s i s a c o n t r a d i c t i o n . [ ]

A n o t h e r c l a ss o f s p a c es w i t h a n o r m a l s t r u c t u r e a n d r e l a t e d t o t h e s p a c es f r o m 1 .1 .3 3 w a s i n t r o d u c e d

i n [ 9 7 ] . W e s h a l l n e e d s o me i n f o r ma t i o n a b o u t b a s e s ( s e e [ 1 2 , 1 5 ] ) .

A ( s e p a r a b l e ) B a n a c h s p a c e X i s a s p a c e w i t h a S c h a u d e r b a s is { a n } C X , i f f o r a n y x E X t h e r e e x i s ts

a u n i q u e s eq u e n c e { x , } C R s u c h t h a t x ie i - ~ 0 a s n - ~ c o , i .e . , x = ~ z n e n . W i t h t h e b a s i s

i= n= l

t h e r e i s a s s o c i a t e d t h e c o n j u g a t e s y s t e m o f f u n c t i o n a l s { f n } C X * s u c h t h a t f n ( a : ) = x , , f o r e v e r y x E X .

T h e e x p a n s i o n o f z E X h a s t h e f o r m

n= l

l

T h e l i n e a r c o n t i n u o u s m a p p i n g s Pn : X --+ X s u c h t h a t P o x = ~ z i e i a r e c a l l e d n a t u r a l p r o j e c t i o n s

i=

assoc ia ted wi th the bas i s {e,~}.

T h e b a s i s {e , ~} i s s a i d t o b e :

1) m o n o t o n e , i f sup [IP~][ = 1;

n

o o

2) u n c o n d i t i o n a l , i f e v e r y c o n v e r g e n t s e r i e s ) - ] . x n e n c o n v e r g e s u n c o n d i t i o n a l l y , i . e. , t h e s e r i e s ~ e n

n = l n : l

x n e n c o n v e r g e s f o r a n y c o l l e c t i o n o f c o e f f i c i e n t s e n = =El;

rn

3) b o u n d e d l y c o m p l e t e , i f f r o m t h e b o u n d e d n e s s o f t h e p a r t i a l s u m s s u P l I E x n e . l l < c o i t f o l l o w s t h e

n n=l

c o n v e r g e n c e o f t h e s e ri e s ~ z n e , .

n= l

D e f i n i t i o n 1 . 1 . 3 6 [2 09 ]. A B a n a c h s pa c e X w i t h a S c h a u d e r b a s is { e , } s a t i s f i e s t h e G o s s e z - La m i D o z o

c o n d i t i o n ( G L D ) , i f t h e r e e x i s ts a s t r i c t l y i n c r e a s i n g s e q u e n c e { k ,~ } C N s u c h t h a t f o r e v e r y ~ > 0 t h e r e

i s a a = a ( r > 0 s u c h t h a t f o r a n y x E X a n d n E N f o r w h i c h I I P k . I I = 1 a n d f i x - P k . ~> r we hav e

I t f l / > 1

I t i s o b v i o u s t h a t t h e s p a c e s I p , 1 <~ p < c o , w i t h t h e s t a n d a r d b a s i s { e ~ } , s a t i s f y t h e c o n d i t i o n ( G L D )

a s w e ll a s t h e s p a c e s D 1 a n d D ~ b y D a y ( s e e [ 89 ]) .

R e m a r k 1 . 1 . 3 7 . T h e c o n d i ti o n (G L D ) g e n e r a l i z es t h e c o n d i t io n (A ) f r o m th e p a p e r b y A . R . L o-

v a g l i a [ 16 5] , i n w h i c h k n = n f o r e v e r y n E N . I t w a s p r o v e d i n [ 16 5 ] t h a t a Ba n a c h s p a c e s a t i s f y i n g t h e

c o n d i t i o n ( A ) is i s o m o r p h ic t o a l o c a ll y u n i f o r m l y c o n v e x sp a c e. E v e r y u n i f o r m l y c o n v e x B a n a c h s p a ce

w i t h a m o n o t o n e b a s i s s at is f ie s t h e c o n d i t i o n ( A ) ( s e e [5 3] ). O t h e r c o n d i t i o n s w h i c h a r e e q u i v a l e n t t o t h e

c o n d i t i o n ( A ) c a n b e f o u n d i n t h e b o o k b y I . S i n g e r [ 1 9 9 , T h e o r e m I I . 6 . 2 ] .

T h e o r e m 1 . 1 . 3 8

[ 2 0 7 ]

Le t X e a Banach space wi th a bas i s {e~} s a t is f y in g t h e c o n d it i o n G L D ) .

Th e n t h e b a si s {a n } i s b o u n d e d ly c o m p l e t e a n d X i s i s o m o r p h i c t o th e c o n j ug a t e B a n a c h s p a c e g e n e r a te d

b y th e c o n j u g a te s y s t e m o f li n e a r f u n c t i o n a l s { f n } .

T h e o r e m 1 . 1 .3 9 [97]. E v e r y r e fl e x iv e B a n a c h s p a c e w i th a b as is s a ti s fg i n g t h e c o n d i t i o n G LD ) h a s a


W . L . B y n u m [ 47 ] h a s e x t e n d e d T h e o r e m 1 .1 .3 9.

T h e o r e m 1 . 1 . 4 0 . Let X be a re f l ex ive Bana ch space and l e t t here ex i s t e E (0, 1) a n d ~ > 0 s u c h t h a t

f o r a n y a > 0 a n d x E X t h e re a re a we a k l y c o n t i n u o u s s e m i n o r m p o n X a n d a s u b a d d i t i v e f u n c t i o n ~r o n

X wi th the fo l lowin g proper t i es :

1) l Y l l

1 e , t hen l y t l > t (1 + g )p (y ) .

Th e n X h a s a n o r m a l s tr u c t u r e .

I f a s p a c e X w i t h a S c h a u d e r b a s is s a ti s fi e s t h e c o n d i t i o n ( G L D ) , x E X a n d a > 0 , t h e n t h e r e e x i s t s a n

n E N s u c h t h a t t h e s e m i n o r m s p y ) = I I Pny H a n d a ( y ) = I ly - Pnyl[ s a t i s f y c o n d i t i o n s ( 1 ) , (2 ) a n d ( 3 ) o f

T h e o r e m 1 . 1 . 4 0 .

764



D e f i n i t i o n 1 . 1 .4 1 [ 8 5 ] L e t X b e a B a n a c h s p a c e . T h e n u m b e r

~ o x ) = s u p { ~ t > 0 : ~ x s ) = 0 } ,

i s c a l l e d t h e

c h a r a c t e r i s t i c

( o r t h e

c o e f f i c ie n t ) o f c o n v e x i t y

o f X , w h e r e 8 x ( ' ) is t h e m o d u l u s o f c o n v e x i t y

o f X .

T h e o r e m 1 . 1 . 4 2 [8 5] . I f s o ( X ) < 1, th e n t h e B a n a c h s p ac e X h a s a n o r m a l s t r u c t u r e .

O b s e r v e t h a t a s p a c e X i s u n i f o r m l y c o n v e x i f a n d o n l y i f r = 0 . B y n u m [4 6] g a v e a n e x a m p l e o f

a r e f l e x iv e s p a c e X w i t h a n o r m a l s t r u c t u r e f o r w h i c h s 0 ( X ) > 1, a n d c o n s t r u c t e d a s p a c e Y w i t h o u t a

n o r m a l s t r u c t u r e s u c h t h a t ~ 0 ( Y ) = 1 .

E x a m p l e 1 . 1 . 4 3 [ 46 ]. F o r x = ( x n ) E I p , 1 < p < e e , w e d e f i n e s e q u e n c e s z + a n d x - :

x + ) . = m a x { x . , 0 } , ~ - ) . = m a x { - x . , O } .

T h e n z = x + - z - F o r 1 ~ q < c o , l e t

1 ,q

= ( / P , l I ), w h e r e t h e n o r m I zl =

l lx + l l ~ + I l x - I I ~ ) l / q ,

a n d

[ l l i p

i s t h e

s t a n d a r d n o r m o n l ' . F o r l ' ' ~ = ( I P , I l )

t h e n o r m i s

I x [ = m a x { l l x + l l p , I 1 ~ - I I , } .

T h e s p a c e s

Ip'q

( 1 < p < e c , 1 ~< q ~< c ~ ) a r e r e f l e x i v e s i n c e t h e n o r m s [ - [ a n d [ [.

I I ,

a r e e q u i v a l e n t .

S t a t e m e n t 1 . 1 . 4 4 [4 6] . Fo r 1 

1 ; t h e sp a c e Y = l p ' ~ h a s n o n o r m a l s t r u c t u r e a n d s o ( Y ) = 1 .

O b s e r v e t h a t

l v,1

h a s a S c h a u d e r b a s i s { e ~} a n d s a ti s f i e s t h e c o n d i t i o n ( G L D ) . S i n c e

( Ip ,q)* = l / ,q

(1 <

p < c ~ , 1 ~< q ~ c r w h e r e p * a n d q * a r e t h e e x p o n e n t s c o n j u g a t e t o p a n d q , t h e s p a c e s

Ip,1

a n d

l / ' ~

a r e

m u t u a l l y d u a l . T h u s t h e r e i s c o n s t r u c t e d a c l as s o f r e fl e x iv e s p a c e s s u c h t h a t , b y S t a t e m e n t 1 . 1 .4 4 , e v e r y

s p a c e f r o m t h i s c l as s h a s ( d o e s n o t h a v e ) a n o r m a l s t r u c t u r e , w h e r e a s i ts d u a l d o e s n o t h a v e ( h a s ) n o r m a l

s t r u c t u r e s . T h i s s h o w s t h a t a n o r m a l s t r u c t u r e i s n o t a p r o p e r t y t h a t i s p r e s e r v e d u n d e r p a s s a g e t o t h e

d u a l s p a c e .

W e o b s e r v e i n t h i s c o n n e c t i o n t h a t t h e p a s s a g e t o a q u o t i e n t s p a c e d o e s n o t g u a r a n t e e t h e p r e s e r v a t i o n

o f a n o r m a l s t r u c t u r e . I n f a c t , M . A . S m i t h a n d B . T u r e t t [2 05 ] h a v e s h o w n t h a t w i t h t h e ( U C E D ) , a n d ,

h e n c e , fo r t h e B a n a c h s p a c e

X ( l 1 ,

I I I I) w i t h t h e e q u i v a l e n t n o r m Il zl l = l l x l l ~ ]l~:]12)1 / 2 , w h e r e I I ]li i s

t h e s t a n d a r d

l i - n o r m

( i = 1 , 2 ) , z = ( x ~ ) 6 l ~ , a n d 5 = ( z ,~ . 2 - n / 2 ) ~ = ~ , w h i c h h a s a n o r m a l s t r u c t u r e , t h e r e

e x i s t s a s u b s p a c e Y s u c h t h a t t h e q u o t i e n t s p a c e

X ~

Y h a s n o n o r m a l s t r u c t u r e .

T h e f o l lo w i n g re s u l t c o m p l e m e n t s T h e o r e m 1 . 1. 42 a n d i s d u e B . T u r e t t [ 21 4] .

T h e o r e m 1 . 1 . 4 5 .

L e t X b e a B a n a c h s p a c e s u c h t h a t

s 0 ( X * ) < 1 .

T h e n X a n d X * a r e s u p e r r e f l e z i v e

a n d h a v e n o r m a l s t r u c t u r e s .

I n p a r t i c u l a r , a u n i f o r m l y s m o o t h s p a c e X i s s u p e r r e f le x i v e a n d h a s a n o r m a l s t r u c t u r e s i n c e X * i s

u n i f o r m l y c o n v e x a n d s 0 ( X * ) = 0 .

S . S w a m i n a t h a n [2 09 ] h a s d e f i n e d th e c o e f f ic i e n t o f c o n v e x i t y o f a B a n a c h s p a ce a t a p o i n t z E X \ { 0 }

a s a n u m b e r r s u c h t h a t

s z = s u p { s / > 0 : ~ e , z ) = 0 } ,

w h e r e

6 ( e , z )

i s t h e m o d u l u s o f c o n v e x i t y o f X a t z (s e e 1. 1. 1 3) . T h e n i t i s o b v i o u s t h a t X i s ( U C E D ) i f

a n d o n l y i f e z = 0 f o r e v e r y z 6 X \ { 0 } .

T h e o r e m 1 . 1 . 4 6 [2 09 ]. / f s , < 1

f o r e v e r y z 6 X \

{ 0 } ,

t h e n Z h a s a n o r m a l s t r u c t u r e .

R . C . J a m e s [1 08 ] i n t r o d u c e d t h e c l as s o f u n i f o r m l y n o n - s q u a r e s p a c e s .

D e f i n i t i o n 1 . 1 . 4 7 . A B a n a c h s p a c e X i s s a i d t o b e

u n i f o r m l y n o n - s q u a r e ,

i f t h e r e e x i s t s a n e > 0 s u c h

t h a t f o r a n y x , y fi S ( X ) t h e i n e q u a l i t y

I 1 ~ + y l l + I I ~ - u l l ~ < 4 -

h o l d s .

I t i s c l e a r t h a t X i s u n i f o r m l y n o n - s q u a r e i f a n d o n l y i f t h e r e e x i s t s a ~ > 0 s u c h t h a t o n

S ( X )

t h e r e a r e

n o p o i n t s x a n d y f o r w h i c h I I( z + y ) / 2 1 l > 1 - ~ a n d I I( z - y ) /2 1 l > 1 - 5 . H e n c e a u n i f o r m l y c o n v e x s p a c e

i s u n i f o r m l y n o n - s q u a r e .

E v e r y u n i f o r m l y n o n - s q u a r e B a n a c h s p a c e i s r e f le x i v e [1 08 ].

765



T h e o r e m 1 . 1 . 4 8 [85]. A B a n a c h s pa ce i s u n i f o r m l y n o n - s q u a r e i f a n d o n l y i f e o ( X ) < 2 .

K . G o e b e l a n d T . S e k o w s k i [ 93 ] i n t r o d u c e d t h e n o t i o n o f m o d u lu s o f n o n c o m p a c t c o n v e x i t y o f a B a n a c h

s p a c e X a s a fu n c t i o n A x ( e ) :

[ 0 2 1

--+ [0, suc h th a t

A x ( e ) = i n f { 1 - - i n f IIxil : A = c o n v A C B ( X ) , o ~ ( A ) ) e }

xE

w h e r e a ( A ) = i n f { t > 0 : t h e s e t A a d m i t s a p a r t i t i o n i n t o a f i n it e f a m i l y o f s u b s e t s w h o s e d i a m e t e r s a r e

s m a l l e r t h a n t } is K u r a t o w s k i ' s m e a s u r e o f n o n c o m p a c t n e s s o f A ( a b o u t m e a s u r e s o f n o n c o m p a c t n e s s o f

s e ts s e e [1 ]) . T h e n u m b e r e l ( X ) = s u p { e ) 0 : A x ( e ) = 0 } i s c a l le d t h e c o e f f ic i e n t o f n o n c o m p a c t c o n v e x i t y

o f X . C l e a r l y , e ~ ( X ) <~ c o ( X ) .

T h e o r e m

1 1 4 9 [9 31 . f f ~ l X ) < 1 , th e n X i s r e f l e x i v e a n d h a s a ~o rm a l

s t r u c t u r e .

9 = m a x I x . ( i ) l , x n = ( X n ( 1 ) , X n ( 2 ) ,

x a m p l e 1 . 1 . 5 0 [9 3]. L e t X , = ( R " , l I n) , w h e r e t h e n o r m I x , ~ t l < ~ i ~n

. . . , x , ( n ) ) E R " . C o n s i d e r t h e I X - p ro d u c t o f t h e s e s p a c e s :

D =

oo

x = { x . } . = ~ : x . e R ~ , I x . l ~ < + ~ ,

n m l

e q u i p p e d w i t h t h e n o r m I lx l l = [ x nl 2 I t i s k n o w n t h a t D i s r e f l e x i v e [ 59 1. S i n c e 5 D ( e ) = 0 fo r

nm.~l

a n y e > 0 , w e h a v e c o ( D ) = 2 . F u r t h e r m o r e , e l ( D ) = 0 si n c e A o ( e ) = 1 - - V /1 - e ~ / 4 . B y T h e o r e m 1 . 1 .4 9 ,

w e c o n c l u d e t h a t t h e s p a c e D ( t h e D a y s p ac e ) h a s a n o r m a l s t r u c t u r e .

O b s e r v e t h a t t h e r e f l e x i v i t y o f a B a n a c h s p a c e X d o e s n o t i m p l y t h a t e l ( X ) < 1 . F o r e x a m p l e , f o r t h e

r e f l e x i v e s p a c e X = ( l 2 , I " I ~) , w h e r e t h e n o r m

I x l~ = I I z l , x z , . . - ) 1 1 ~ = m a x { ~ l x l l , I lx l1 2 }, ,X / > 1 , w e h a v e

( [9 3] ) e l ( X ) = 2 ( 1 - / ~ - 2 ) 1 / 2 _.+ 2 a s ,~ --+ + o o ( i n p a r t i c u l a r , f o r A = v ~ w e g e t e l ( X ) = v ~ > 1 ).

I n [ 2 9] t h e r e w a s i n t r o d u c e d t h e m o d u l u s o f n o n c o m p a c t c o n v e x i t y A x ( e ) o f t h e s p a c e X , w h i c h i s d e f i n e d

s i m i l a r l y t o A x ( e ) w i t h t h e o n l y d if f er e nc e , t h a t i n s t e a d o f t h e K u r a t o w s k i m e a s u r e o f n o n c o m p a c t n e s s o n e

u s e s t h e H a u s d o r f f m e a s u r e o f n o n c o m p a c t n e s s x ( A ) - - in f { e > 0 : t h e s e t A h a s a f in i t e e - n e t } i n X . L e t

s 0 ) = l im ~ x ( e ) .

e--- 1 -- 0

T h e o r e m 1 . 1 . 5 1 [1 85 ]. Le t X be a Ba nach space . Th en the fo l low ing asser t io ns are va l id :

1) i f 5 x ( 1 - 0 ) > 0, t hen X i s re f l ex ive;

2

i f T X x 1 - O )

= 1 , t h e n X s a t i 4 e s t h e O p i a l c o n d i t i o n ( i n p a r t i c u la r , X h a s a n o r m a l s tr u c t u r e ) ;

3 ) i f 5 x - ( 1 - 0 ) = 1 , t h e n X h a s t he F e e .

O b s e r v e t h a t t h e p r o o f o f t h is t h e o r e m i s c a r r i e d o u t b y n o n - s t a n d a r d m e a n s w i t h t h e u s e o f t h e n o t i o n

o f u l t r a p o w e r o f a B a n a c h s p a c e.

S o m e c o n d i ti o n s , im p l y i n g t h e p r e s e n c e o f n o r m a l s t r u c t u r e i n t h e L e b e s g u e - B o c h n e r s p a c e s w e r e f o u n d

b y D . J . D o w n i n g a n d B . T u r e t t [ 6 8] , a n d b y M . A . S m i t h a n d B . T u r e t t [2 94 ].

L e t ( f~ , E , # ) b e a m e a s u r e s p a c e , X b e a B a n a c h s p a c e , 1 < p < c o , L P ( , X ) b e t h e L e b e s g u e - B o c h n e r

space ( see [10 l ) .

T h e o r e m 1 . 1 . 5 2 . Th e f o l l o w i n g s t a t e m e n t s a r e t r u e :

1) i f X i s a Ba na ch space wi th co (X ) < 1 , an arb i t rary measure , and 1 < p < ec , then eo

L . . , X ) ) < 1 ( i n p a r t i c u l a r , L ( . , X ) h a s a ~ o r m a l s tr u c t u r e ) ;

2) i f ( f~ , E , ) i s an arb i t rary measure space, X i s a Ba nac h space and 1 < p < ~ , then the Lebesg ue-

B o c h n e r s p a c e L ~ ' ( # , X ) h a s a n o r m a l s t r u c t u r e i f a n d o n l y i f X h a s a n o r m a l s t r u c t u r e .

S t a t e m e n t s ( 1 ) a n d ( 2) o f T h e o r e m 1 .1 .5 2 w e r e p r o v e d i n t h e p a p e r s [ 6 8] a n d [ 2 04 ] r e s p e c t i v e l y .

A n o t h e r c la ss o f B a n a c h s p a c es w i t h a n o r m a l s t r u c t u r e is d e s c r i b e d in t h e f o ll o w in g t h e o r e m .

T h e o r e m 1 . 1 . 5 3 [48 ]. Le t X be a Ba nac h space i som orph ic to l p , 1 < p < ec , and such tha t the

B a n a c h - M a z u r d i s t a n c e d ( X , i p ) < 2 1/ p. Th e n X h a s a n o r m a l s t r u c t u r e .

766



I f d ( X , l p ) < . 2 1 /p , t h e n X h a s t h e F P P .

A s a c o n s e q u e n c e , w e o b t a i n t h a t t h e s p a c e l P ' % 1 < p < c o , h a s t h e F P P , a l t h o u g h i t h a s n o n o r m a l

s t r u c t u r e s ( s e e 1 . 1 .4 4 ) . I n d e e d ,

d ( l p ,~ , Ip) = 21/p .

L . A . K a r l o v i t z w a s o n e o f t h e f i r st w h o g i v e a n e x a m p l e o f a r e fl e x iv e B a n a c h s p a c e w i t h t h e F P P , b u t

w i t h o u t a n o r m a l s t r u c t u r e .

L e t

X j = ( l 2 ,

l l II), w h e r e t h e n o r m

I lxll

= m a ~ { l l x l l ~ ,

I lx l l2 /v }

a n d H I1 ~ i s t h e l ~ - n o r m , I I II i s

t h e / 2 - n o r m . T h i s s p a c e w a s i n t r o d u c e d b y R . C . J a m e s . K a r l o v i t z [1 12 ] s h o w e d t h a t t h e r e f le x i v e s p a c e

X j h a s n o n o r m a l s t r u c t u r e s , b u t h a s t h e F P P . T h e k e y ro l e i n t h e p r o o f w a s p la y e d b y t h e l e m m a o f

G o e b e l - K a r l o v i t z .

L e m m a 1 . 1 . 5 4 ( G o e b e l [ 87 ], K a r l o v i t z [ 11 0 ]) . L e t X b e a B a n a c h s p a ce , C b e a c o n v e z w e a k l y c o mp a c t

s u b s e t o f X , T : C --+ C b e a n o n e z p a n s i v e ma p p i n g . S u p p o s e t h a t C i s m i n i m a l i n t h e s e n s e t h a t i t c o n t a i n s

n o p r o p e r c o n v e x c lo s e d s u b s e t i n v a r i a n t u n d e r T . L e t { x ~ } b e a se q u e n c e o f a l m o s t f i z e d p o i n t s , i . e ., z ~ E C

a n d I I x . -

Tx . I I - O . Th e n f o r e v e r y C

l im I Ix -

x n l l

= d i a m C .

n - - +~

T h e r e s u l t b y K a r l o v i t z s t a t i n g t h a t X j h a s th e F P P a l so c a n be d e d u c e d a s a c o n s e q u e n c e o f T h e o r e m

1 . 1 . 5 3 s i n c e

( se e [4 8] ). T . L a n d e s [1 45 ] w a s o n e o f t h e f i rs t w h o s t u d i e d t h e c o n d i t i o n s u n d e r w h i c h a n O r l i c z s e q u e n c e

s p a c e h a s a n o r m a l s t r u c t u r e .

A f u n c t i o n ~ : [ 0, r ~ [ 0, c o ) i s c a l l e d a n O r I i c z f u n c t i o n i f i t i s a c o n t i n u o u s n o n d e c r e a s i n g a n d c o n v e x

f u n c t i o n s u c h t h a t r = 0 a n d q o(c m) = c ~ . T h e c o r r e s p o n d i n g Orl icz sequence space l~ i s d e f i n e d a s

f o l l o w s :

= x : N - R : < o o f o r s o m e t > 0 ,

t l = l

a n d

I lxll

= i n f t > 0 : ~ h a ( I x ( n ) l / t ) 1 i s a n o r m o n t h e s p a c e l ~ . N o t e t h a t ( l ~~ I [ ID i s

a g a n a c h

n= l

s p a c e .

W e s h a l l s a y t h a t a n O r l i c z f u n c t i o n tp :

I ) s a t i s f ie s t h e

A 2 - c o n d i t i o n

a t z e ro , i f T ( 2 t ) )

M ~ p ( t )

f o r s o m e M > 0 a n d f o r a l l t t h a t a r e s u f f i c i e n t l y

c l o s e t o z e r o ;

2 ) i s l i n e a r a t z e r o , i f ~ o(~ ) = c o n s t > 0 f o r a ll s u f f i c i e n t l y s m a l l s .

T h e o r e m 1 . 1 . 5 5 [1 45 ].

F o r t h e Or l i c z s e q u e n c e s p a c e l ~ t o h a v e a n o r m a l s t r u c t u r e i t i s n e c e s s a r y a n d

s u f f i c i e n t t h a t ha s a t i s f y t h e f o l l o w i n g t h r e e c o n d i t i o n s :

1) r

s a t i s f ie s t h e A 2 -c o n d i t i o n a t z e r o ,

2) r i s n o t l i n e a r a t z e r o ,

3 ) s u p { t / > 0 : h a( t) < 1 } > I / 2 .

F u r t h e r m o r e , L a n d e s [1 45 ] e s t a b l i s h e d t h a t c o n v e x s u b s e t s o f 1~~ w h i c h a r e c o m p a c t i n t h e t o p o l o g y o f

p o i n t w i s e c o n v e r g e n c e h a v e a n o r m a l s t r u c t u r e i f a n d o n l y i f ha s a ti s fi e s t h e A 2 - c o n d i t i o n a t 0 a n d i s s t r i c t l y

onvex t

0 ( i . e . , q o ( At ) ~< 3 , ha ( t) f o r t > 0 a n d 0 < A < 1 ) .

R e s u l t s c o n c e r n i n g N S a n d F P P o f O r l i c z s p a c es a l so c a n b e fo u n d i n t h e p a p e r s [ 74 , 14 2 , 5 6 ], a n d o t h e r s .

H . F u j i h a r a a n d T . K a n e k o [8 1] s t u d i e d t h e r e a l B a n a c h s p a c e s X s a t i s f y in g t h e c o n d i t i o n : f o r a n y

f E X * , t > O

d i a m ( { x E B ( X ) :

f ( z ) > i

t } ) < 2 . ( 1 . 1 . 2 )

I n [ 81 ], i t i s s h o w n t h a t t h e c o n d i t i o n ( 1 .1 : 2 ) is w e a k e r t h a n t h e f o l l o w i n g o n e :

f o r a n y x n , y ~ 6 B ( X ) , n = 1 , 2 , . . . s u c h t h a t I I x , - Y n ll --~ 2 , z n + y n w > 0 . H e n c e u n i f o r m l y c o n v e x

s p a c e s s a t i s f y c o n d i t i o n ( 1 . 1 .2 ) .

7 6 7



T h e o r e m 1 . 1 . 5 6 [8 1] .

Le t X be a re f l ex ive Ban ach space sa t i s f y ing con d i t ion

( 1 . 1 . 2 ) .

Th e n X h a s a


W e n o w g i v e e x a m p l e s o f s p a c e s w i t h o u t a n o r m a l s t r u c t u r e , a n d a l s o s o m e r e s u l t s c o n c e r n i n g t h e

r e n o r m a l i z a t i o n o f s p a c e s.

E x a m p l e 1 . 1 . 5 7 . T h e s p a c e s c , co , 11, l ~ 1 76 [ 0 , 1] h a v e n o n o r m a l s t r u c t u r e s . I n d e e d , t h e s t a n d a r d b a s i s

{ e n } f o r m s a d i a m e t r i c a l s e q u e n c e i n e a c h o n e o f t h e s p a c e s c , C o, l 1 , l ~ 1 7 6 o n s i d e r i n C [ 0 , 1] t h e b o u n d e d

c o n v e x a n d c l o s e d s u b s e t _h. = { x =

x ( t )

E C [0 , 1] : 0 ~<

x ( t )

~< 1 , x ( 0 ) = 0 a n d x ( 1 ) = 1 } . O b v i o u s l y ,

d i a m K = 1. I f x E K , t h e n f o r a g i v e n e > 0 t h e r e i s ~ > 0 s u ch t h a t z ( t ) < ~ f o r 0 < t < ~ . S e l e c t a

f u n c t i o n y E K s u c h t h a t

y ( t )

= 1 f o r t /> ~ / 2 . T h e n I Ix - y ] ] ~> 1 - r T h i s p r o v e s t h a t t h e p o i n t x i s

d i a m e t r i c a l . H e n c e K h a s n o n o r m a l s t r u c t u r e s .

F u r t h e r m o r e , C [ 0 , 1] h a s n o F P P , s i n c e t h e m a p p i n g T : K --+ K s u c h t h a t

( T x ) ( t ) = t x ( t )

i s n o n e x p a n -

s i v e a n d F i x T = O ( s e e [ 11 9 ]) .

O n e o f th e f i r st e x a m p l e s o f a r e f le x i v e s p a c e w i t h o u t a n o r m a l s t r u c t u r e w a s f o u n d b y J a m e s ( s ee [ 2 09 ]) .

E x a m p l e 1 . 1 . 5 8 [ 33 ]. I n t r o d u c e o n t h e s p a c e 12 t h e e q u i v a l e n t n o r m N x ][ = m a x ~ l ix ] ]2 , tlx ]]o o , w h e r e

I I 112 i s t h e / 2 - n o r m , a n d ] I ] ]o o i s t h e / ~ 1 7 6 O b v i o u s l y , 2 ] ] z i[ 2 ~< I l x iI ~< ] ]x il 2. T h e s p a c e X = ( l 2 , I I ' II )

i s r e f l e x i v e ( i n f a c t , s u p e r r e f l e x i v e ) s i n c e i t is i s o m o r p h i c t o l 2 . L e t K = { x E 1 2 : I ]x l l ~< 1 ,

xi >t 0

f o r

i = 1 , 2 , . . . } . T h e s e t K i s b o u n d e d , c o n v e x a n d c l o s e d , a n d h e n c e it is w e a k l y c o m p a c t i n l 2. W i t h r e s p e c t

t o t h e n e w n o r m t h e d i a m e t e r o f K i s e q u a l t o 1 . F o r g i v e n x E K a n d ~ > 0 w e s e l e c t a b a s i s v e c t o r e n s o

t h a t l x n ] < e . T h e n IIx - e ,~ I[ /> ] x ~ - 1 ] > > 1 - - e . T h u s e v e r y p o i n t o f K i s d i a m e t r i c a l , i . e ., K h a s n o

n o r m a l s t r u c tu r e s .

T h e n e x t r e s u l t o n r e n o r m a l i z a t i o n i s d u e t o U . Z i z l e r [2 28 ].

T h e o r e m 1 . 1 . 5 9 .

Ev ery separab le Ban ach space X adm i t s an equ iva len t no rm I I such tha t the space

( Z , I ] )

i s ( U C E D ) .

T h i s i m p l i e s, i n p a r t i c u l a r , t h a t e v e r y se p a r a b l e B a n a c h s p a c e is i s o m o r p h i c t o a s p a c e w i t h a n o r m a l

s t r u c t u r e ( s e e a l s o [6 2 ]) . F o r n o n s e p a r a b l e s p a c e s t h i s is n o l o n g e r t r u e . A s h a s b e e n s h o w n b y L a n d e s [ 14 6] ,

t h e s p a c e

c o ( I )

w i t h a n u n c o u n t a b l e I i s n o t is o m o r p h i c t o a n y s p a c e w i t h a n o r m a l s t r u c t u r e . D . v a n

D u l s t [7 1] s u p p l e m e n t e d t h e s e r e s u l ts .

T h e o r e m 1 . 1 .6 0 .

E v e r y i n f i n i t e d i m e n s i o n a l B a n a c h s p a c e c a n b e e q u i v a l e n t l y r e n o r m a l i z e d i n s u c h a

wa y t h a t i t w i l l n o t h a v e a n o r m a l s t r u c t u r e .

H e n c e a s u p e r r e f l e x i v e s p a c e d o e s n o t n e e d t o h a v e a n o r m a l s t r u c t u r e , a s , f o r e x a m p l e , t h e s p a c e ( l 2 , I ' ]),

w h e r e I I i s t h e e q u i v a l e n t n o r m s p o k e n a b o u t i n T h e o r e m 1 .1 .6 0.

R e m a r k 1 . 1 . 6 1 . R . V . R a k h m a n k u l o v f o r m u l a t e d a n d p ro v e d T h e o r e m 1 .1 .6 0 l o n g b e f o r e t h e a p p e a r -

a n c e o f [7 1] i n h i s u n p u b l i s h e d w o r k ( O n n o r m a l s t r u c t u r e i n B a n a c h s p a c e s , L e n i n g r a d , 1 9 76 ).

N o t e o n e m o r e o f t h e e a r l ie s t r e s u l t s c o n c e r n i n g r e n o r m a l i z a t i o n s .

T h e o r e m 1 . 1 . 6 2 [3 3].

Th e r e e x i s t s a r e f l e x i v e s t r i c t l y c o n v e x B a n a c h s p a c e w i t h a n o r m a l s t r u c t u r e

wh i c h i s n o t i s o m o r p h i c t o a n y u n i f o r m l y c o n v e x s p a c e .

1 .2 . U n i f o r m l y N o r m a l S t r u c t u r e .

D e f i n i t i o n 1 . 2 . 1 ( A . A . G i ll e s p ie a n d W . W . W i l li a m s [ 84 ]) . A B a n a c h s p a c e X i s s a i d t o h a v e a

u n i f o r m l y

n o r m a l s t r u c t u r e

( U N S ) , i f f o r s o m e h E ( 0 , 1) a n d f o r e v e r y b o u n d e d s u b s e t

A C X , r ( A ) <~ h .

d i a m A ,

w h e r e r ( A ) = i n f s u p ]Ix - Y I[ i s t h e C h e b y s h e v r a d i u s o f A .

xEconv yE

T h e v a l u e

J ( X )

= s u p { r ( A ) : A C X , d i a m A = 1 ) i s c a l le d t h e

J u n g c o n s t a n t

o f t h e s p a c e X . T h u s a

s p a c e X h a s a u n i f o r m l y n o r m a l s t r u c t u r e i f a n d o n l y i f J ( X ) < 1.

O b s e r v e t h a t 1 / 2 ~ J ( X ) ~< 1 f o r a n y B a n a c h s p a c e X . I t i s c l e a r t h a t a s p a c e w i t h U N S a l s o h a s a

n o r m a l s t r u c t u r e . T h e c o n v e r s e is n o t t r u e [8 4].

E x a m p l e 1 . 2 . 2 [ 4 8] . L e t X b e t h e / 2 - p r o d u c t o f t h e s p ac e s

I n,

n > / 2 . T h e s p a c e X i s r e f l e x i v e a n d

( U C E D ) ( s ee [6 2] ) a n d , t h e r e f o r e , i t h a s a n o r m a l s t r u c t u r e . F o r e v e r y p E (1 , ~ ) a n d f o r t h e s t a n d a r d

768



b a s i s { e i } C I p w e h a v e d i a m { e i } = 2 l ip . T h e n J ( X ) ) J ( l ) >, 2 - U n f o r e v e r y n ) 2 . H e n c e J ( X ) = 1,

i .e ., X h a s n o n o r m a l s t r u c t u r e s .

T h e o r e m 1 . 2 . 3 [8 4]. E v e r y B a n a c h s p ac e w i th U N S h a s th e F P P .

T h i s t h e o r e m c a n b e d e r i v e d a s a c o n s e q u e n c e o f T h e o r e m s 1 .1 . 4 a n d 1 .2 .4 .

T h e o r e m 1 . 2 . 4 ( E . M a l u t a [ 16 7] a n d B a e J o n g S o o k [2 5]) . E v e r y S a n a c h s p a c e w i t h U N S i s r e f l e x i v e .

P r o o f . L e t X b e a B a n a c h s p a c e w i t h U N S , i. e. ,

J ( X )

< 1 . S u p p o s e t h a t X i s n o n r e f l ex i v e . T h e n ,

b y T h e o r e m 2 . 2 f r o m [ 16 ], f o r a n y ~ > 0 t h e r e i s a s e q u e n c e { X n} C X s u c h t h a t f o r a n y m E N , y m E

c o n v { x l , x ~ , . . . , X m } a n d z , , e e o n v { x k } ~ = m + l w e h a v e 1 - ~ 1 - ~ . B y t h e a r b i t r a r i n e s s o f ~ > 0 , w e c o n c l u d e t h a t

J ( X ) --= 1 . T h i s i s a c o n t r a d i c t i o n . [ ]

R e m a r k 1 . 2 . 5 . I n fa c t , e v e r y B a n a c h s p a c e w i t h U N S i s s u p e r re f le x i v e ( se e T h e o r e m 3 .2 .8 ) .

W e n o w i n d i c a t e s o m e c la s se s o f B a n a c h s p a c e s w i t h U N S .

S t a t e m e n t 1 . 2 . 6 . A B a n a c h s p ac e X h a s U N S , i f i t i s o n e o f t h e f o l l o w i n g s p a c e s :

n \ / n s e e [23]);

) a n n - d i m e n s i o n a l s p a ce ; a n d , i n t h is c a se , J ( X ) <~ ~ ( s e e [23]) ; J (R '~) = __ 2( n + 1 )

2 ) a u n i f o r m l y c o n v e x s p a c e, a n d in th i s ca se J ( X ) ~< 1 - 5x (1 ) ( s e e [48]);

3 )

a u n i f o r m l y s m o o t h s p a c e , a n d i n t h is c as e J ( X ) ~ 1 - 5 x .

(1 )

( s ee

[184]);

4 )

a s p a c e w i t h t h e c h a r a c t e r i s t i c o f c o n v e x i t y

c o ( X ) < 1 ( s e e [ 54 ])

s ince

5 x ( 1 ) > O ,

a n d J ( X )

1 - ~ x I ) ;

5) a k - u n i f o r m l y c o n v e x s p a c e ( f o r s o m e k = 2 , 3 ; . . . ) ; and , in th i s case , J ( X ) <~ m a x { 1 - 5 } ~ ) ( e ) , 1 -

1 - e }

kT.s fo r an y

e > 0 ( [23]) ,

w h e r e 5 ~ ) ( . ) i s t h e m o d u l u s o f k - c o n v e x i t y ;

6) a n i n fi n i t e d i m e n s i o n a l m l b e r t sp a ce , a n d , i n t h i s c a s e , J ( z ) = 1 / v ' ~ ( s e e [ 4 8 ] ) ;

7 ) a s p a ce h a v i n g a k - u n i f o r m l y c o n v e x d u a l s pa ce X * f o r s o m e k E N ( [215] ) ;

8 ) a sp a c e w h o s e k - c h a r a c t e r i s t i c o f co n v e x i ty ~ ( k ) ( Z ) = s u p { r > 0 : 5 ( X k ) ( r = 0 } i s s m a l l e r t h a n 1

[26 ;

9) a L e b e s g u e - B o c h n e r s p a ce L P ( # , Y ) w i t h a n a r b i tr a r y m e a s u r e # , 1 1/4 ( [82]) .

D e f i n i t i o n 1 . 2 . 7 ( s ee [ 82 ]) . A B a n a c h s p a c e X i s c a l l e d a n U - s p a c e i f f o r e v e r y e > 0 t h e r e i s 5 > 0

s u c h t h a t , f o r a n y ~ , y ~ s g ) , I Iz + Y ll > 2 1 - ~ ) , t h e i n e q u a l i t y

f ~ y )

> 1 e i s v a l i d f o r e v e r y

f ~ y ) ~ S X * ) ,

f ~ ~ ) = I lx l l-

I n [8 2], i t i s s h o w n t h a t t h e u n i f o r m l y c o n v e x s p a c e s a n d t h e u n i f o r m l y s m o o t h s p a c e s a r e U - s p a c e s .

E v e r y U - s p a c e i s u n i f o r m l y n o n - s q u a r e . A B a n a c h s p a c e X i s a U - s p a c e i f a n d o n l y i f X * i s a U - s p a c e .

T h e o r e m 1 . 2 . 8 ( G a o J i a n d L a u K a - S i n g [8 2]). E v e r y U - s p a c e h a s U N S .

T h e p r o o f o f t h i s t h e o r e m u s e s th e u l t r a p r o d u c t s t e c h n i q u e s .

S . P r u s [1 82 ] p r o v e d u s i n g u l t r a p r o d u c t s t h a t n o t o n l y a u n i f o r m l y s m o o t h b u t a n a r b i t r a r y B a n a c h

s p a c e X h a s U N S , if l i m p x ( t ) / t < 1 / 2 , w h e r e p z ( ' ) is th e m o d u l u s o f s m o o t h n e s s o f X . E a r l ie r T u r e t t

t-- 0 O

p r o v e d [2 14 ] t h a t t h e c o n d i t i o n l i m p x ( t ) / t < 1 / 2 is e q u i v a l e n t t o t h e c o n d i t i o n c o ( X * ) < 1 , w h o s e

t-- 0 0

f u l fi l lm e n t i m p l i e s t h a t X * h a s U N S . N o t e t h a t f o r a n a r b i t r a r y B a n a c h s p a c e X t h e c o n d i t i o n e 0 ( X ) < 1 i s

s a t i s f ie d if a n d o n l y i f i t s L i f s c h i t z c h a r a c t e r i s t i c ~ 0 ( X ) > 1 ( [6 8 ]) ( s e e S e c . 2 . 3 f o r t h e d e f i n i t i o n o f r ( X ) ) .

T h e r e e x is t s a B a n a c h s p a c e X w i t h J ( X ) < 1 a n d c o ( X ) > / 1 .

E x a m p l e 1 . 2 . 9 [ 5 4] . C o n s i d e r t h e J a m e s s p a c e Z ;~ = ( l 2 , 1.1 ~) w i t h t h e n o r m [ x tz = m a x { l l x ] [ 2 , t 3 1 [z ][ ~ }.

T h e s e s p a c e s h a v e n o r m a l s t r u c t u r e i f a n d o n l y i f / 3 ~< x / ~ [2 8] . F u r t h e r m o r e , f o r 1 ~< /3 ~< v ~ , e 0 ( Z ) =

2 ~ / ' ~ 2 - 1 a n d J ( X a ) = / 3 . 2 - x / 2 . T h u s , f o r v / ' 5 /2 <~ f l < x / ~ w e h a v e J ( X a ) < 1 an d r > / 1 .

T h e o r e m 1 . 2 . 1 0 ( se e [ 22 5] ). L e t X ~ ( i = 1 , 2 , . . . , n ) b e B a n a c h s p a ce s w i th U N S . T h e n t h e d ir e ct

p r o d u c t

( X 1

~ (9 X n ) o o h a s U N S a n d

J [ ( g l G ' ( ~ g n ) o j <~ 1 - ( 1 - 1 r ~ i ~ x J ( g i ) ) / 2 n - l

769



T h i s r e s u l t w a s s u p p l e m e n t e d in [2 15 ], w h e r e i t w a s p r o v e d t h a t , i f a B a n a c h s p a c e X h a s U N S , a n d

d i m Y < c ~, t h e n t h e d ir e c t s u m

X 3 Y

h a s U N S , w h e r e

X 3 Y

is e q u i p p e d w i t h a n a r b i t r a r y d i r e c t p r o d u c t

n o r m . I n p a r t i c u l a r , t h e s p a c e Z = 12 (3 l~~ w i t h t h e n o r m I [ ( z, y ) l[ = m a x { ] I x [ J 2 , ] [Y I [~ } f o r x 6 1 2 a n d

y E l ~ h a s U N S . M o r e o v e r , e 0 ( Z ) = 2 s i n c e Z c o n t a i n s a n i s o m e t r i c c o p y o f l ~ . T h u s t h e s p a c e Z h a s

U N S a n d i s n o t u n i f o r m l y n o n - s q u a r e [5 4].

M A S m i t h a n d B . T u r e t t [ 2 05 ] p r o v e d t h a t t h e s p a c e 1 2 1 ( se e 1 .1 .4 3 ) is ( 2 - U C ) a n d h e n c e h a s U N S ,

a l t h o u g h i ts d u a l s p a c e l 2 ~ 1 7 6a s n o n o r m a l s t ru c t u r e s . T h u s U N S i s n o t a p r o p e r t y t h a t i s p r e s e r v e d u n d e r

t h e p a s s a g e t o t h e d u a l s p a c e .

W e n o w p r e s e n t s o m e m o r e f ac ts c o n c e r n i n g t h e J u n g c o n s t a n t.

S . P r u s [ 1 83 ] c o m p u t e d t h e e x a c t v a l u e s o f t h e J u n g c o n s t a n t f o r t h e s p a c es Ip a n d L p ( s e e a l s o [ 6 5 ]) :

J L p) = J IP) = m a x { 2 - l / p , 2 0 - p ) / p }

f o r 1 < p < c ~ .

I f X a n d Y a r e t w o i s o m o r p h i c B a n a c h s p a ce s , t h e n

J Y ) <~ d Z , Y ) . J X )

[ 4 8 ] , w h e r e

d X , Y )

i s t h e

B a n a c h - M a z u r d i s ta n c e . A s a c o n s e q u e n c e w e o b t a i n t h a t i f d X , Y ) < 1 / J X ) , t h e n Y h a s U N S , a n d

h e n c e , t h e F P P . I n p a r t i c u l a r , i f X = I P , 1 < p < c o , t h e n Y h a s t h e F P P ,

i f d I P , Y )

< m i n { 2 l / p , 2 1 - 1 / P } .

P r u s [ 18 3] s t r e n g t h e n e d t h i s r e s u l t b y s h o w i n g t h a t Y h a s t h e F P P i f d IP, Y ) < Cp f o r s o m e 1 m i n { 2 1 / ,

2 1 - - 1 / P } f o r

e x a m p l e C 2 = ( v ~

N o t e t h a t i n t h e p a p e r s [ 16 7 , 2 31 ] lo w e r a n d

D e f i n i t i o n 1 . 2 . 1 1 [7 8]. B y t h e

measure of

w e m e a n t h e v a l u e

+ 1)/2 = 1 . 6 1 8 . . . > v ~

u p p e r b o u n d s f o r t h e v a l u e J I~) a r e g i v e n .

nonconvexity o f a b o u n d e d s u b s e t A o f a B a n a c h s p a c e X

A ( A ) = s u p i n f I I x - y H .

x E c o n v

yE

W i t h t h e a i d o f t h e p r i n c ip l e o f l oc a l r e fl e x iv i ty o f L i n d e n s t r a u s s - R o s e n t h a l t h e f o l l o w i n g r e l a t io n b e t w e e n

t h e J u n g c o n s t a n t a n d t h e m e a s u r e o f n o n c o n v e x i t y is e s t a b l i s h e d i n [6 ].

T h e o r e m 1 . 2 . 1 2 [6]. For any Banach space X the following identity

J Z )

= s u p { ) ~ ( A ) : A C X , d i a m A = 1 } .

holds.

C o r o l l a r y 1 . 2 . 1 3 [ 60]. For any Banach space X the identi ty J Z) = s u p { J ( Y ) : Y is a finite-

dimensional subspace of X } is true; in particular, J X ) = s u p { r ( A ) : A C Z is finite, d i a m A = 1 },

where r A) is the Chebyshev radius of A see 1 .2 .1 ) .

N o t e t h a t C o r o l l a r y 1 . 2. 13 s t r e n g t h e n s t h e a n a l o g o u s r e s u l t b y D . A m i r [ 23 ] o b t a i n e d f o r t h e c a s e o f

r e f l e x i v e s p a c e s .

W e n o w t u r n t o s o m e q u a n t i t a t i v e c h a r a c te r i s ti c s c o n n e c t e d w i t h t h e n o t io n o f n o r m a l s t r u c t u r e .

B y n u m [ 4 8] i n t r o d u c e d t h r e e c o e f fi c ie n t s c h a r a c t e r i z i n g t h e n o r m a l s t r u c t u r e o f a r e f le x i v e B a n a c h s p a -

c e

X :

N X ) = i n f - - r A )

: A c X , 0 < d i a m A < o o

is the normal structure coefficient f o r X ;

. f d i a m . {x__ .~ ( { x . } ) }

S X )

= i n f | : { z , } is a n o n c o n v e r g e n t b o u n d e d s e q u e n c e i n X , t h e

bounde d sequences

coefficient

f o r X ;

d i a m ~ { a n }

w c s x )

= i n f [ : { z . }

s

a w e a k l y ( b u t n o t s t ro n g l y ) c o n v e r g e n t s e q u e n c e i n X . , t h e

weakly convergent sequences coefficient

f o r X . H e r e d i a m , { a n } = n li m ( s u p { ] ] x i - x j I l : i i> n , j ~> n } ) a n d

r a ( {X n } ) = i n f { l i m s u p t ] x n - - x l l : x 6 c o n v x n } a r e t h e

asymptotic diameter

a n d t h e

asymptotic radius

o f

n --r oo

t h e s e q u e n c e { x n } C X .

C l e a r l y ,

Y Z ) = 1 / J X ) ,

w h e r e

J X )

i s t h e J u n g c o n s t a n t .

770



T h e o r e m 1 . 2 . 1 4 [48]. For any reflexive Banach space X

1 <. N (X ) <<.B S ( X ) <. W C S ( X ) <. 2 .

a n d N ( X ) = B S ( X ) g X i s 8 ep ar aU e.

T . - C h . L i r a [ 1 58 ] p r o v e d t h a t N ( X ) = B S ( X ) f o r e v e r y B a n a c h s p a c e X .

H o w e v e r , N ( X ) W C S ( X ) .

E x a m p l e 1 . 2 . 1 5 [4 8]. L e t Z b e t h e / 2 - p r o d u c t o f t h e s p a ce s I ~ = (IRn , I[" I [ ~ ) , n = 1 , 2 , . . . . T h e n

X

i s r e f le x i v e a n d

N ( X ) = 1 < W C S ( X ) = v /2 .

I f

W C S ( X )

> 1 , t h e n X h a s a n o r m a l s t r u c t u r e [ 4 8] . I f

N ( X )

> 1, t h e n X h a s U N S . B u t t h e r e a r e

s p a ce s w i t h a n o r m a l s t r u c t u r e s u c h t h a t

N ( X ) = W C S ( X ) = 1.

E x a m p l e 1 . 2 . 1 6 [4 8]. L e t X b e t h e / 2 - p r o d u c t o f t h e sp a c e s U , n / > 2 . T h e n X i s r e fl e xi v e a n d ( U C E D )

( s ee [6 2 ]) . H e n c e ~ r h a s a n o r m a l s t r u c t u r e . F u r t h e r m o r e ,

W C S ( X ) ~ W C S ( l ~)

= 2 1 / f o r a n y n ~ N .

For 1 < p < cxD W C S ( I p ) = 2 x/p, W C S( l p 'I ) = 21 /p and W C S ( L p) ~ m i n { 2 U p , 2 l l / p } [481.

S . P r u s [ 1 8 3 ] o b t a i n e d f o r L p = L P[ 0, 1 ], 1 < p < e c , t h e e x a c t v a l u e s o f t h e B y n u m c o e f f i c i e n t s : N ( L P ) =

W C S ( L p )

= mi n {2 1 / v , 2 1 - 1 / p } .

T h e o r e m 1 . 2 . 1 7 [48].

If X is a uniformly convez Banac h space and Y is a Banach space such tha t the

Banach -Mazur dis tance d(X , Y ) <. W C S (X ), then Y has the FPP .

I n p a r t i c u l a r , t h e s p a c e X = ( l 2 , I1"

11)

w i t h t h e n o r m

I lx l l = m a ~ { l l x l l 2 , v ~ l l ~ l l o o }

h a s t h e F P P .

F u r t h e r m o r e , l p'~176a s t h e F P P s in c e d(IP,Oo, Ip) = 21/p = W C S( IP ).

P r u s [ 1 83 ] h a s s t r e n g t h e n e d T h e o r e m 1 .2 .1 7 b y s h o w i n g t h a t i ts c o n c l u s i o n i s v a l i d i f

d ( X ,

Y ) < a ( X ) 9

W C S ( X ) , w h e r e

F o r e x a m p l e , a ( / 2 ) = 1 .0 1 38 . . . .

E . M a l u t a [ 16 7] h a s i n t r o d u c e d t w o c o n s t a n t s , c o n c e r n i n g t h e n o r m a l s t r u c t u r e o f B a n a c h s p a c es :

[ l im s u p d i s t ( z . + l , c o n v { x i} n = l )

D ( X )

= s u p / . -- ,c r d i a r n { x . } : { x n } is a b o u n d e d n o n c o n s t a n t s e q u e n c e i n X / ;

t

}

( X ) = su p c t i a m { x , } : A C X , 0 < d i a m A < o o i s t h e constant of uniformity of the normal struc-

ture.

I t is o b v i ou s t h a t N ( X ) =

1 / N ( X )

= J (X ) ; 0 ~<

D (X ) <~ 1.

T h e f o l l o w i n g f a c t s a r e e s t a b l i s h e d i n [ 16 7] :

1 1

1) D ( X ) = 0 if a n d o n l y i f d i m X < c ~ ; f o r a n i n f i n i t e d i m e n s i o n a l s p a c e , d(X) >>.2 ( 1 - 5 x ( 1 ) ) ) 2 ;

2 ) i f X i s a n o n r e f l e x i v e B a n a c h s p a c e , th e n

D ( X )

= 1 ; i f D ( X ) < 1 , t h e n X i s r e f l e x iv e a n d h a s n o r m a l

s t r u c t u r e ( h o w e v e r X = ( ~ 0 I n ) is r e fl e xi v e a n d h a s a n o r m a l st r u c t u r e , a l t h o u g h D ( X ) = 1 );

n 1 2

i n p a r t i c u l a r , i f D ( X ) < 1 , t h e n X h a s t h e F P P ;

3) D ( X ) = s u p { D ( Y ) : Y is a s e p a r a b l e s u b s p a c e o f X } ;

4 )

D(X) <~

J ( X ) ; a n d i f X i s i n f in i t e d i m e n s i o n a l a n d r e f l ex i v e , t h e n

D ( X ) <. 1 / W C S ( X ) ;

5 ) t h e c o n d i t i o n

D ( X )

< 1 d o e s n o t i m p l y t h a t X h a s U N S ; in f a c t, fo r t h e s p a c e X o f E x a r n p l e 1 .2 .1 5

w e h a v e

D ( X )

= 1 / x / ~ a n d

J ( X )

= 1;

6 ) D(IP) = 2 -1 /p fo r 1 < p < oo ; hen ce , D(IP) < J( lp) , 1 < p < 2;

7)

D ( X )

= 1, w he re X = ( l 2 , II"

If)

w i t h t h e n o r m I l x l l

= m a x { l l ~ l l 2 , v ~ l l x l l ~ } .

D . A m i r [2 31 o b s e r v e d t h a t

D ( l a )

= 2 - 1 / ' . F u r t h e r m o r e , h e p r o v e d t h a t

D ( X ) = 1 / W C S ( X )

p r o v i d e d

t h e r e f le x i v e B a n a c h s p a c e X s a t is f ie s t h e O p i a l c o n d i t i o n .

771



T h e o r e m 1 . 2 . 1 8 ( S. P r u s [1 83 ]). F o r a n y i n f in i t e d i m e n s i o n a l B a n a c h s p a ce X

D ( X ) = l / W C S ( X ) = i n f { d i a m { x n } : { xn } c S ( X ) , x , ~ '> 0 } .

I n t h e p a p e r [ 21 6] , t h e r e w a s i n t r o d u c e d t h e f o l l o w i n g c o n s t a n t f o r a re f le x i ve B a n a c h s p a c e X :

W X ) =

i n {

d i a m { x n }

i n f { l i m s u p l l x n -

z l l x E x }

n---szOO

{ x n } i s a w e a k l y , b u t n o t s t r o n g l y , c o n v e r g e n t s e q u e n c e i n X } .

O b v i o u s l y , W C S ( X ) <<.W ( X ) . F u r t h e r m o r e W ( l p ) = 21 /p fo r 1 < p < oo [216] ; W C S ( L p ) = W ( L p ) =

r a i n { 2 l / p , 2 l - l / p } [ 6 5 ]) .

T h e o r e m 1 . 2 . 1 9 [6 5].

L e t ( ~ 2 , ~ , # ) b e a a - f i n i t e m e a s u r e s p a c e , 1 <~ p < o o , a n d L P ( # ) b e i n f i n i t e -

d im e n si on a l. T h e n N ( L ' ( # ) ) = r a i n { 2 ~ /p , 2 1 - l / P } .

Moreover

W C S ( L V ( # ) ) = 1 / D ( L P ( # ) ) = N ( L V ( # ) ) ,

i f p >>.2 , or , i f 1 < p < 2 , an d there ex i s t s a sub se t f~ l C ~2 o f fu l l an d f in i t e me asu re suc h tha t e very

1

m e a s u r a b l e s u b s e t F 1 C f~ l c o n t a i n s a m e a s u r a b l e s u b s e t F 2 w i t h m e a s u r e

# ( F 2 ) = ~ # ( F 1 ) .

H o n g K u n X u [2 20 ] h a s s h o w n t h a t D ( X ) < 1 i f X i s a r e f l ex i v e ( W U K K ) s p a c e ( s e e 1 . 1 . 2 1 ). I n

p a r t i c u l a r ,

D ( X )

< 1 i f X i s a ( N U C ) - s p a c e .

T h e o r e m 1 . 2 . 2 0 [2 20 ]. O n a B a n a c h s p a c e X l e t a n o r m I 11 a n d a s e m i n o r m l 12 b e g i v e n s u c h t h a t

I I~ <~ C t I1 w i t h s o m e c o n s t a n t C . T h e n , i f

D ( ( X , I 1 .1 )) < 1 , w e

h a v e

D ( ( X , I I )) < 1 ,

w h e r e t h e n o r m

1.1=1.11 1-12.

h e B r o w n s p a c e i s d e f i n e d a s t h e s p a c e ( /2 , I I, I 1 ~ w i t h a n e q u i v a le n t n o r m I I . l i b s u c h t h a t i t s r e s t r i c t i o n

t o t h e h y p e r p l a n e { ( x i ) E 12 : X l = 0 } c o i n c id e s w i t h t h e / 2 - n o r m , a n d i ts r e s t r i c t i o n t o t h e s u b s p a c e

s p a n { e l , e n } , n = 3 , 4 , . . . , i s t h e l P ( 0 - n o r m , w h e r e { e , } i s t h e

s t a n d a r d b a s i s o f 12 a n d p ( n ) ~ + o o a s

n ~ o o ; p ( 3 ) = 1 6 ( s e e [8 3] ). I n [ 83 ], i t is s h o w n tb . a t ( l 2 , [ [. t [ s ) h a s a n o r m a l s t r u c t u r e s i n c e t h i s s p a c e

h a s a s u b s p a c e o f f i n i te c o d i m e n s i o n is o m e t r i c to F . F u r t h e r m o r e , i f X = ( l 2 , I1 I I s , t h e n O X) < 1 [220].

F i n a l ly , i t i s s h o w n i n [2 15 ] t h a t t h e B r o w n s p a c e h a s U N S .

X i n T a i Y u [ 22 5] h a s c o n s t r u c t e d a s p a c e ( m o d e l l e d o n t h e B r o w n s p a c e ) w i t h U N S , t h a t i s n e i t h e r

( U C E D ) , n o r ( L U C ) , n o r ( W U K K ) , n o r u n i f o r m l y n o n - s q u a r e , n o r ( k - U C ) .

N o t e t h a t i n [ 22 6] t h e B y n u m c o e ff ic i en t W C S ( X ) i s c o n s i d e r e d f o r t h e / P - p r o d u c t o f a f i n i te n u m b e r

o f s p a c e s X i ( 1 ~< i ~ < n ) , i n w h i c h w e a k c o n v e r g e n c e d o e s n o t c o i n c i d e w i t h s t r o n g ; i t is s h o w n t h a t

W C S ( ( i ~ = l O X i l ) = m i n { W C S ( X i ) : l <. i ~ n } .

z p

J i G a o a n d K . - S . L a u [ 82 ] c o n s i d e r ed t h e m a g n i t u d e

G X)

= s u p { m i n { t l z + Y i t , 1 t x - Y l I }: ~ , Y ~ S ( X ) } ,

f o r w h i c h t h e y e s t a b l i s h e d t h e f o l l o w i n g :

1 ) v ~ <~ G ( X ) ~< 2 f o r a n y B a n a c h s p a c e X ;

2 ) G ( X ) < 2 i f a n d o n l y i f X i s u n i f o r m l y n o n - s q u a r e ; i n p a r t i c u l a r , i f G ( X ) < 2 , t h e n X i s r e f l e x i v e ;

3 )

G ( l P ) = G ( L p )

= m a x { 2 l / p , 2 1 - 1 / P } , 1 ~ O : ~ 5 x ( r ~< 1 - r

T h e o r e m 1 . 2 . 2 1 [8 2]. L e t X b e a B a n a c h s p a ce w i th G ( X ) < 3 / 2 . T h e n X h a s U N S .

S o m e a d d i t i o n a l i n f o r m a t i o n a b o u t t h e c o e ff ic ie n t G ( X ) c a n b e f o u n d i n [ 1 8 4 ] .

1 .3 . C l o s e - t o - N o r m a l S t r u c t u r e s .

C h i S o n g W o n g [2 17 ] in t r o d u c e d t h e n o t io n o f a d o s e - t o - n o r m a l s t r u c t u r e .

772



D e f i n i t i o n 1 . 3 . 1 . A b o u n d e d c o n v e x s u b s e t K o f a B a n a c h s p a c e X i s s a i d t o h a v e a c l o s e - t o - n o r m a l

s t r u c t u r e ( C N S ) i f f o r e v e r y c o n v e x c l o se d s u b s e t H C K w i t h d i a m H > 0 t h e r e is a n x E H s u c h t h a t

I I x - y l l < d i a m H f o r e v e r y y 9

I t is c l e a r t h a t C N S i s a g e n e r a l i z a t i o n o f N S .

T h e o r e m 1 . 3 . 2 [ 21 7]. L e t a B a n a c h s p a c e X p o s s e s s a t l e as t o n e o f th e f o l l o w i n g p r o p e r t i e s :

1) X i s separab le;

2) X i s s t r i c t l y c o n v e x i . e . ,

f i x + y l l < 2

for any x, y 9 S(X), x #

y ) ;

w H

3) f o r a n y s e q u e n c e { x n } C X s u c h t h a t x , > x , a n d x ~ l l --+ Ilxll, i t f o l l o w s t h a t x ~ x .

T h e n e v e r y w e a k ly c o m p a c t c o n v e x s u b s e t o f t h e s p ac e X h a s C N S .

T h e K a d e c - K l e e s p a c e s , a n d , i n p a r t i c u l a r , ( U C ) - s p a c e s , p o s s e s s p r o p e r t y ( 3) s in c e o n t h e u n i t s p h e r e s

o f t h e s e s p a c e s w e a k a n d s t r o n g c o n v e r g e n c e s co i n ci d e .

L e t X a n d K C X b e t h e s a m e a s i n E x a m p l e 1 . 1. 5 8. T h e n X i s a r e f l ex i v e s e p a r a b l e s p a c e , a n d K i s a

c o n v e x w e a k l y c o m p a c t s e t w i t h o u t a n o r m a l s t r u c t u r e . B y T h e o r e m 1 .3 .2 , t h e s e t / ( h a s C N S .

R e m a r k 1 . 3 . 3 . U s i ng th e t e c h n iq u e s o f u l t r a p r o d u c t s K u o k F o n t V u [4] h a s c o n s t r u c t e d a n e x a m p l e of

a B a n a c h s p a c e i s o m o r p h i c t o a H i l b e r t s p a c e a n d a c o nv e x w e a k l y c o m p a c t s e t i n i t w i t h o u t C N S . T h u s ,

n o t e v e r y co n v e x w e a k l y c o m p a c t s e t in a n a r b i t r a r y B a n a c h s p a c e h a s C N S .

C h i S o n g W o n g [2 18 ] e s t a b l i s h e d a c o n n e c t i o n b e t w e e n t h e p r e s e n c e o f C N S a n d t h e e x i s t e n c e o f f ix e d

p o i n t s o f K a n n a n m a p s ( i. e ., T : D C X - -+ X s u c h t h a t

I I T x

- Tyl l

2 t l x

- Tx l l +

I ly - T y l I ) ) .

I

T h e o r e m 1 . 3 . 4 [ 21 8]. L e t X b e a B a n a c h s p a c e. T h e f o l l o w i n g t w o a s s e r t i o n s a r e e q u i v a l e n t :

1) e v e r y c o n v e x w e a k ly c o m p a c t s u b s e t o f X h a s C N S ;

2) e v e r y K a n n a n m a p a c t i n g o n a n o n e m p t y c o n v e x w e a k l y c o m p a c t s u b s e t o f X h a s a u n i q u e ) f ix e d

p o i n t .

D e f i n i t i o n 1 . 3 . 5 [ 10 3, 1 6 6] . A n o n - c o n s t a n t b o u n d e d s e q u e n c e { x ~ } C X i s s a i d t o b e s t r i c t l y d i a m e t -

r ical , i f t h e r e e x i s t s a n m 9 N s u c h t h a t d i s t x n + l , c o n v { x i } n = l ) = d i a m {x i }n = x fo r a n y n > rn .

T h e o r e m 1 . 3 . 6 [ 10 3] . T h e f o l l o w i n g a s s e r t i o n s a r e e q u iv a l e n t :

1) a c o n v e x su b s e t K o f a B a n a c h s p ac e X h a s C N S ;

2) K d o e s n o t c o n t a i n a d i a m e t r i c a l s e q u e n c e ;

3 ) K d o e s n o t c o n t a i n a s e q u e n c e { x ~ } s u c h t h a t f o r s o m e C > 0 a n d e v e r y n , m 9 N, n ~ rn ,

X n l - - --1 ~ X i = C

I I x . - X r . l l = n / = 1

I n a d d i t i o n t o 1 . 3 .3 , w e i n d i c a t e t h e f o l l o w i n g

E x a m p l e 1 . 3 . 7 [ 10 3] . L e t X b e a n u n c o u n t a b l e s et . C o n s i d e r i n t h e s p a c e c o I ) t h e b o u n d e d c o n v e x

s u b s e t K = { x = { x i } i s z E c o I ) : x i >>.0 f o r e v e r y i E I , ~ x i <<. 1 } . I f { y ( n ) } ~ = 1 C K , t h e n , u s i n g

i E I

t h e d i a g o n a l p r o c e d u r e , w e e x t r a c t a s u b s e q u e n c e {y (n k) }~ ~ c o n v e r g i n g c o o r d i n a t e - w i s e to s o m e e l e m e n t

y E K . S i n c e ( c 0 ( I ) ) * ~ I t I) , y , ,k) w> Y as k --+ oo. T h u s K is a c o n v e x w e a k c o m p a c t s e t . F o r e v e r y

x = { x i } i E I E K s e l e c t z = { z i } i E I E K s o t h a t

0 , i T~io

Zi -~-

1, i = :o,

w h e r e i0 E I i s a n i n d e x s u c h t h a t Xio = 0. T h e n n x - z l l = 1 = d i a m K . H e n c e K h a s n o C N S a n d ,

m o r e o v e r , i t h a s n o N S .

T h e o r e m 1 . 3 . 8 [ 10 3, 1 66 ]. E v e r y b o u n d e d cl o se d s u b se t o f a s t r i c t ly c o n v e x B a n a c h s p a c e h a s C N S .

D a y [ 6 0 ] h a s p r o v e d t h a t o n c 0 ( I ) o n e c a n d e f i n e a n e q u i v a l e n t s t r i c t l y c o n v e x n o r m l " I. T h e n t h e b a l l

B X ) o f t h e s p a c e X = ( c 0 ( I ) , l " I) w i ll h a v e C N S b y T h e o r e m 1 .3 .8 . F u r t h e r m o r e , B X ) i s n o t w e a k l y

c o m p a c t s i n c e t h e s p a c e c 0 ( I ) i s n o t r e f l e x i v e .

773



S t a t e m e n t 1 . 3 . 9 [1 43 ]. L e t ( X , I I

I I )

be

an inJ~nite

d i m e n s i o n l

B a n a c h s p ac e . T h e n t h er e e x i s ts an

e q u i v a l e n t n o r m I 1

I I '

o n X

n d

s e q u e n c e { x , } C X s u c h t h a t

=

X i = I l x . - - x m l l ' = 1

I I x - I I = I I x - I I ' X . + X ~ i = 1

f o r a n y n , m 6 N , n ~ m .

P r o o f . T h e r e e x i s t t w o se q u e n ce s { x , } C X a n d { f ~ } C X * s u c h t h a t

I I x ~ l t

=

I t f ~ l l = x ,

j 1 , ~ = m ,

f ~ ( x m )

0 , n m .

D e f i n e

F r o m S t a t e m e n t 1 .3 .9 a n d T h e o r e m 1 .3 .6 w e o b t a i n t h a t e v e r y i n fi n it e d i m e n s i o n a l B a n a c h s p a c e c a n b e

r e n o r m a l i z e d s o t h a t i n t h e n e w n o r m t h e u n i t b a l l w i l l n o t h a v e C N S . F u r t h e r m o r e , i n [1 43 ] i t i s e s t a b l i s h e d

t h a t i f a c o n v e x w e a k l y c o m p a c t s e t K o f a B a n a c h s p a c e X i s n o t c o m p a c t , t h e n K h a s n o C N S w i t h

r e s p e c t t o t h i s n o r m .

A n o t h e r g e n e r a l i z a t i o n o f a n o r m a l s t r u c t u r e w a s g iv e n b y J . B . B a i l lo n a n d R . S c h 6 n b e r g .

D e f i n i t i o n 1 . 3 . 1 0 [2 8] . A B a n a c h s p a c e X h a s a n a s y m p t o t i c n o r m a l s t r u c tu r e ( A N S ) i f f o r an y b o u n d e d

c o n v e x c l o s e d s u b s e t A C X w i t h d i a m A > 0 a n d f o r a n y s e q u e n c e { z ,~ } C A s u c h t h a t l[ x,~ - x ~ + l l l --> 0

a s n ~ o o , t h e r e e x i s t s a n x E A f o r w h i c h l i m i n f m ix , - x lt < d i a m A .

n -~ . o o

I t i s o b v i o u s t h a t e v e r y s u b s p a c e w i t h a n o r m a l s t r u c t u r e h a s a n A N S .

T h e o r e m 1 . 3 . 1 1 [28]. E v e r y r e f le x iv e B a n a c h s p a c e w i t h a n A N S h a s th e F P P .

T h e o r e m 1 . 3 . 1 2 [28]. Le t /3 >>.1 and X # = (/2 , I I#) w i t h t h e n o r m x l ~ = m a x { l l x ] [2 , f l l l x l i ~ } T h e n

t h e f o l l o w i n g a s s e r t i o n s a r e t ru e :

1) X h a s a N S i f a n d o n ly if / 3 < v ~ ;

2) X ~ h a s a n A N S i f a n d o n l y i f t3 < 2.

T h u s t h e s p a c e s X # f o r x / ~ < ~/~ < 2 h a v e a n A N S , b u t h a v e n o N S , a n d t h e s p a c e X 2 h a s n o A N S .

T h e o r e m 1 . 3 . 1 3 [28]. T h e s p a ce X 2 h a s th e F P P .

R e m a r k 1 . 3 . 1 4 . T h e s p a c e s X # h a v e t h e F P P f o r a l l / 3 / > 1 ( [7 9, 3 6] ).

S t u d y i n g r e l a t i o n s b e t w e e n t h e C N S a n d A N S , K . - K . T a n [ 21 1] o b t a i n e d t h e f o l lo w i n g r e s u lt .

T h e o r e m 1 . 3 . 1 5 . L e t I b e a n u n c o u n t a b l e s e t a n d Z A = (12 (I) , I1 ][), w h e r e t h e n o r m ] lxH = ma x{l lx l l2 .

)~ l lx l l~} . T h e n t h e f o l l o w i n g a s s e r t i o n s a r e t r u e :

1) i f 1 <<.~ < v / 2 , t h e n X x h a s a N S , a n d h e n c e a C N S a n d a n A N S ;

2) if 1 ~< A < 2, t h e n X ~ h a s a n A N S ;

3) / f v /2 ~< A < 2 , t h e n X ~ h a s n o C N S .

D . v a n D u l s t [ 70 ] h a s s h o w n t h a t e v e r y in f in i te d i m e n s i o n a l B a n a c h s p a c e c a n b e s o r e n o r m a l i z e d t h a t i n

t h e n e w n o r m i t w i lt h a v e n o A N S . I t is k n o w n t h a t e v e r y B a n a c h s p a c e c a n b e e q u i v a l e n t l y r e n o r m a l i z e d

s o t h a t i t w il l h a v e a N S , a n d h e n c e a n A N S ( se e 1 .1 .5 9 ). H o w e v e r , t h e r e a r e n o n s e p a r a b i e s p a c e s t h a t

c a n n o t b e r e n o r m a l i z e d s o a s t o h a v e a n o r m a l s t r u c t u r e o r a n A N S ( s ee [ 18 0] ).

T h e f o l l o w i n g c r i te r i o n o f h a v i n g a n A N S f o r a B a n a c h s p a c e b e l o n g s t o F . E . L l o r e n s [ 16 4] .

T h e o r e m 1 . 3 . 1 6 . A B a n a c h s p a c e X h a s a n A N S i f a n d o n l y i f f o r a n y b o u n d e d c o n v e x c l o s e d s u b s e t

1

A C X w i t h d i a m A > 0 a n d f o r a n y { x , } C A , { a , } C R , a n >~ n , ~ - - - = c x ~, t h e c o n d i t i o n a n ( x ,

n = l an

X , - -1 ) --> 0 i mp l i e s t h e e x i s t e n c e o f x 6 A f o r w h i c h

l i m i n f l l x - x n l l < d i a m A .

n--- ~

N o t e t h at t h e B y n u m s p a c e Ip ' ~ , 1 < p < ~ , i s a r e fl e xi v e s p a c e w i t h o u t a n A N S b u t w i t h t h e F P P

[48 ]. T h e fo l l o wi n g n o t i o n i s d u e t o W .A. K i rk [12 0 ].

774



D e f i n i t i o n 1 . 3 . 1 7 . A b o u n d e d c o n v e x s u b s e t IX" o f a B a n a c h s p a c e X h a s a n o r m a l s t r u c t u r e w i th

respec t to a sub se t F C_ X i f f o r e v e r y b o u n d e d c o n v e x s u b s e t H C IX" w i t h d i a m H > 0 t h e r e i s x 9 F s u c h

t h a t s u p {] [x - y [ [ : y 9 H } < d i a m H .

T a k i n g i n 1 . 3 .1 7 t h e s e t F _D K ( f o r e x a m p l e F = X ) , w e o b t a i n a g e n e r a l i z a t i o n o f t h e n o t i o n o f n o r m a l

s t r u c t u r e .

E x a m p l e 1 . 3 . 1 8 [ 1 20 ] . I n t h e s p a c e b [0 , 1] o f b o u n d e d r e a l - v a l u e d f u n c t i o n s o n t h e c l o s e d i n t e r v a l [ 0 ,1 ]

w i t h t h e s u p - n o r m , c o n s i d e r t h e c o n v e x b o u n d e d a n d c l o s e d s u b s e t

Ix" = {x = x( t ) 9 C[0 , 1 ] : x (0 ) = 0 , x (1 ) = 1 ,

~<

z(t) <~ 1 } .

T h e n , a s i t i s k n o w n , t h e s e t K h a s n o N S , s i n c e

s u p { l ] x Y l] :

Y 9

K }

=

1

= d i a m i x " f o r a n y z 9 K .

N e v e r t h e l e s s , K h a s a n o r m a l s t r u c t u r e w i t h r e s p e c t t o b [0 , 1]. I n d e e d , l e t H b e a n a r b i t r a r y c o n v e x

s u b s e t o f K w i t h d i a m H > 0 , a n d r e ( t ) = i n f { x ( t ) : z 9 H } , M ( t ) = s u p { z ( t ) : x 9 H } , a n d N ( t ) =

1 1

~ ( m ( t ) + M ( t ) ) . T h e n N ( t ) 9 b [0 , 1] a n d [ I N - VII ~< ~ d i a m H fo r a n y y 9 H .

O b s e r v e t h a t i n t h i s e x a m p l e i n s t e a d o f b [0 , 1] o n e c o u l d t a k e t h e s p a c e L ~ [ 0 , 1 ].

I n T h e o r e m 1 .1 .4 , i t i s p o s s i b l e t o w e a k e n t h e a s s u m p t i o n a b o u t t h e p r e s e n c e o f a n o r m a l s t r u c t u r e i f o n e

c o n f i n e s o n e s e l f t o c o n t r a c t i n g m a p p i n g s ( i. e ., m a p p i n g s T : D C X - + X s u c h t h a t I ITx - Tyl l <

x

- ull

fo r a ll z , y 9 D, z # y ) .

T h e o r e m 1 . 3 . 1 9 [1 20 ]. L e t K b e a n o n e m p t y c o n v e x w e a k ly c o m p a c t s u b s e t o f a B a n a c h s p ac e X , a n d

l e t K h a v e a n o r m a l s t r u c t u r e w i t h r e s pe c t t o K . T h e n e v e r y c o n t r a c t i n g m a p p i n g T : K --+ K h a s a f i x e d

p o i n t .

T h i s t h e o r e m i s n o t t r u e i n t h e c a s e o f a n a r b i t r a r y b o u n d e d c o n v e x c l o s e d s e t.

E x a m p l e 1 . 3 . 2 0 ( s e e [ 1 3 1] ) . L e t K b e t h e u n i t b a l l o f t h e s p a c e C o. D e f i n e T : K - + K s o t h a t

f o r e v e r y x = ( x / ) 9 K , w h e r e t = = 1 - 1 / 2 f o r e v e r y n 9

•

C l e a r l y , T i s a c o n t r a c t i n g m a p p i n g o f t h e b a l l K i n t o it s e lf . M o r e o v e r , T i s a n a f f in e m a p p i n g , i .e . ,

T ( t x + (1 - t ) y ) = t T x + ( 1 - t ) T y f o r a n y t 9 [ 0, 1]. A s s u m e t h a t T h a s a f i x e d p o i n t x = ( x x , x 2 , . . . ) 9 K .

T h e n X 1 = 1, X 2 = t l X l , x a = t 2 x 2 : t l t 2 , . . . , X n q - 1 = t l 9 t 2 " . . . " t n . S i nc e x n + l = f i ( 1 - 1 / 2 n + l >

i = l

1 - 2 - ~ > 2 ' t h e n Xr, -~ 0 a s n --- r 0 . He n c e x ~ c o . T h i s is a c o n t r a d i c t i o n .

i----1

T h e e s s e n t i a l p o i n t h e r e i s t h a t c o i s n o t r e f le x i v e, a n d h e n c e t h e b a l l o f t h i s s p a c e i s n o t w e a k l y c o m p a c t .

N o t e t h a t b y t h e K a k u t a n i t h e o r e m ( se e , f o r e x a m p l e , [9 , T h e o r e m V . 1 0 .8 ] ), e v e r y n o n e x p a n s i v e a f fi ne

m a p p i n g f r o m a c o n v e x w e a k l y c o m p a c t s u b s e t o f a B a n a c h s p a c e i n t o i t s e lf h a s a f i xe d p o i n t .

I n t h i s c o n n e c t i o n , w e s h a l l p o i n t o u t a n o t h e r i n t e r e s t i n g f a c t .

T h e o r e m 1 . 3 . 2 1 [17]. F o r a B a n a c h s p a c e t o b e r e f l e x i v e i t i s n e c e s s a r y a n d s u f f i c i e n t t h a t e v e r y a f f i n e

c o n t i n u o u s m a p p i n g f r o m a n a r b i t r a r y c o n v e x c l os e d s u b s e t i n t o i t s e l f h a v e a f i x ed p o i n t .

I n v e s t i g a ti n g t h e c o n d i t i o n s o f e x i s te n c e o f f i xe d p o i n t s o f n o n e x p a n s i v e m a p p i n g s i n B a n a c h l a t ti c e s ,

P . S o a r d i [ 2 06 ] i n t r o d u c e d t h e f o l lo w i n g n o ti o n .

D e f i n i t i o n 1 . 3 . 2 2 . A c lo s e d s u b s e t K o f a B a n a c h s p a c e X h a s a u n i f o r m r e la t iv e n o r m a l s t r u c t u re

( U R N S ) i f t h e r e e x i s ts a c o n s t a n t c < 1 s u c h t h a t f o r e v e r y n o n e m p t y b o u n d e d c l o s ed s e t M C K t h e r e i s

XM 9 K w i t h t h e f o l l o w i n g p r o p e r t i e s :

1 ) M C B ( X M , C " d i a m M ) ;

2 ) i f y 9 K i s s u c h t h a t M C B ( y , c d i a m M ) , t h e n y 9 B ( X M , c d i a m M ) .

A r e a l p a r t i a l l y o r d e r e d B a n a c h s p a c e X i s c a l l e d a B a n a c h l a t t i c e , i f t h e o r d e r a n d t h e n o r m a r e

c o n n e c t e d b y t h e f o l lo w i n g a x i o m s :

1) x ~ < y i m p l i e s x + z ~ < y + z f o r a l l z , y , z 9

2 ) ax >/ O for e ve ry x 9 X , x >l O and eve ry a 9 lR , a l> O,

3) fo r a l l x , y 9 X t h e r e e x i st t h e l e a s t u p p e r b o u n d x V y a n d t h e g r e a t e s t l o w e r b o u n d x h y ;

4 ) [ [x [ [ ~< [ [y [ [, i f Ix [ <~ [y [, wh er e th e ab so lu t e va lue ]z[ fo r z 9 X i s def in ed by [z[ = z V ( - z ) .

775



A B a n a c h l a t t i c e X i s c a l l e d a n a b s tr a c t M - s p a c e A M - s p a c e ) , if [Ix +Y ii = m a x t l x l l , I l y l l) fo r a l l x , y E X

s uc h t h a t z ~ > 0 , y ) 0 a n d x A y = 0 .

A B a n a c h l a t t ic e X i s s a i d to b e o r d e r - c o m p l e t e i f e v e r y s e t A C X w i t h a n u p p e r b o u n d h a s a l e a st

u p p e r b o u n d .

L e t X b e a n o r d e r - c o m p l e t e A M - s p a c e w i t h u n i t y ( i .e ., w i t h a n e l e m e n t e E X s u c h t h a t t h e b a l l B X )

i s t h e o r d e r i n t e r v a l I - e , e] = { x E X : - e ~< x ~< e } ) . T h e n X i s i s o m e t r i c a l l y l a t t i c e i s o m o r p h i c t o t h e

s p a c e C S ) o f c o n t i n u o u s r e a l - v a l u e d f u n c t i o n s d e f i n ed o n so m e e x t r e m a l l y d i s c o n n e c t e d c o m p a c t H a u s d o r f f

s p a c e S .

T h i s a n d o t h e r n e c e s s a r y i n f o r m a t i o n a b o u t B a n a c h l a t ti c e s m a y b e f o u n d i n [ 13 8, 1 9 2] .

S t a t e m e n t 1 . 3 . 2 3 [ 2 0 6 ] . E v e r y o r d e r- c o m p l et e A M - s p a c e w i t h u n i t y h as U R N S .

I t is e a s i l y s e e n t h a t e v e r y c l o s e d b a l l a n d o r d e r i n t e r v a l { z X : a ~< z ~< b } h a s U R N S i n a n

o r d e r - c o m p l e t e A M - s p a c e X w i t h u n i ty .

T h e o r e m 1 . 3 . 2 4 [ 2 0 6 ] . L e t X b e a c o n j u g a t e B a n a c h s p a c e a n d K C X b e a w e a k * c l o s e d s e t w i t h

U R N S . L e t T : K --+ K b e a n o n e x p a n s i v e m a p p i n g h a v i n g a n in v a r i a n t w e ak * c o m p a c t s u b s e t M C K .

T h e n T h a s a f i x e d p o i n t i n K .

I t is k n o w n ( s e e [1 92 ]) t h a t a n A M - s p a c e is d u a l t o a n A L - s p a c e .

T h e o r e m

1 . 3 . 2 5 [ 2 0 6 ] . L e t X b e t h e d u a l t o a n A L - s p a c e . T h e n t h e f o l l o w i n g a r e t r u e :

1) i f A C X i s a c l o s e d b a ll o r a cl o s e d o r d e r i n t e r v a l a n d T : A - -+ A i s a n o n e x p a n s i v e ma p p i n g , t h e n

T h a s a f i x e d p o i n t ;

2) i f T : X -+ X i s a n o n e x p a n s i v e ma p p i n g h a v i n g a n i n v a r i a n t w e ak * c o mp a c t s u b s e t , t h e n T h a s a

f i x e d p o i n t i n X .

L e t ( Y , E , # ) b e a m e a s u r e s p a c e w i t h a a - f in i t e m e a s u r e . T h e n t h e s p a c e L ~ Y , E , # ) i s t h e d u a l t o t h e

A L - s p a c e L 1 ( IF , E , # ) .

C o r o l l a r y

1 .3 .2 6 [1 9 7 , 2 0 6 ] . T h e f o l lo w i n g a s s e r t i o n s a re t r u e :

1) A c l o s e d b a ll a n d a c l o s e d o r d e r i n t e r v a l i n L ~ o r C S ) h a v e t h e F P P ;

2) I f T : L ~ - + L ~ i s a n o n e x p a n s i v e m a p p i n g t r a n s f o r m i n g i n t o i t s e lf a w e a k * c o m p a c t s u b s e t , t h e n

T h a s a f i x e d p o i n t i n L ~ .

W e n o w g i v e a n e x a m p l e o f a s p a c e w i t h a U R N S , w h i c h is n o t a n A M - s p a c e .

E x a m p l e 1 . 3 . 2 7 [ 20 6 ]. L e t X b e a u n i f o r m l y c o n v e x S a n a c h s p a c e a n d II It, 1 1- II * b e t h e n o r m s o n

X a n d o n X * r e s p e ct i v el y . D e n o t e b y Z t h e s p a c e o f s e q u e n c e s z = ( Zl ; z 2 , . . . , z n , . . . ) , z,~ E X , s u c h t h a t

s u p l lz n ll = l l zt [ ~ < c o . T h e s p a c e Z i s n o t a n A M - s p a c e , s i n c e X i s n o t a n A M - s p a c e . M o r e o v e r , Z i s t h e

n

o o

d u a l t o t h e s p a c e o f a ll s e q u e n c e s y = ( y i , y 2 , . . . ) , y n E X * , s u c h t h a t ~ I l y n l l ~ . T h e n Z a n d e v e r y

r ~

c l o s e d b a l l B C Z h a v e U R N S . T h e r e f o r e , a n a n a l o g o f T h e o r e m 1 . 3. 2 5 i s v a l i d f o r Z .

R e m a r k

1 . 3 . 2 8 . R e s u l t s 1 .3 .2 5 1 ) a n d 1 .3 .2 6 ( 1 ) a r e a l so c o n s e q u e n c e s o f t h e p r e s e n c e o f t h e F P P

i n b o u n d e d h y p e r c o n v e x m e t r i c s p a c e s [ 12 4] (a m e t r i c s p a c e ( M , p ) is c a l le d h y p e r c o n v e x i f e v e r y f a m i l y o f

c l o s e d b a l l s { B x i , r i ) } i n M s a t i s f y i n g t h e c o n d i t i o n p xi , x j ) <~ ri Jr r j h a s n o n e m p t y in t e r s e c t i o n ) . C l o s e d

b a l l s a n d o r d e r i n t e r v a l s i n L ~ o r C S ) a r e e x a m p l e s o f b o u n d e d h y p e r c o n v e x m e t r i c s p a c e s.

W e n o w c o n s i d e r s o m e r e s u l t s o n f i x ed p o i n t s o f is o m e t r i c m a p p i n g s .

D e f i n i t i o n 1 . 3 . 2 9 [ 14 7] . A s u b s e t K o f a B a n a c h s p a c e X h a s a f ix e d p o in t p r o p e r t y f o r i s o m e t r i c

m a p p i n g s ( F P P I ) i f t h e r e e x i s t s a n x E K s u c h t h a t T x = x f o r a n y i s o m e t r i c m a p p i n g T o f K i n t o i ts e l f,

W e n o w g i v e s o m e e x a m p l e s o f s e t s w i t h t h e F P P I :

1 ) e v e r y c o n v e x w e a k l y c o m p a c t s e t i n X w i t h a n o r m a l s t r u c t u r e [3 ];

2 ) e v e r y c o n v e x w e a k l y c o m p a c t s e t i n a s t r i c t l y c o n v e x X [ 19 0] ;

3 ) e v e r y b o u n d e d c o n v e x c l o se d s u b s e t o f a r ef l ex i v e B a n a c h s p a c e w i t h a n o n e m p t y i n t e r i o r [1 31 ].

T h e o r e m 1 . 3 . 3 0 ( A . T . - M . L a u [ 14 7 ]) . L e t X b e a B a n a c h s p a c e a n d K b e a n o n e m p t y b o u n d e d c l o se d

b u t n o t n e c e s sa r i ly c o n v e x ) s u b s e t o f X w i t h a U R N S . T h e n K h a s t h e F P P L

1 4 W e a k l y N o r m a l S t ru c t u r e s

776



D e f i n i t i o n 1 . 4 . 1 . A B a n a c h s p a ce X h a s a we a k l y n o r m a l s t r u c t u r e ( w -N S ) i f a n y w e a k l y c o m p a c t

c o n v e x s u b s e t A o f i t w i t h d i a m A > 0 h a s a n o r m a l s t r u c t u r e .

D e f i n i t i o n 1 . 4 . 2 . A w e a k * cl os e d c o n v ex s u b s e t A o f a c o n j u g a t e B a n a c h s p a c e h a s a w e a k * n o r m a l

s t r u c t u r e ( w * - N S ) , i f e v e r y w e a k * c o m p a c t c o n v e x s u b s e t K C A , d i a m K > 0 , c o n t a i n s a p o i n t x 0 s u c h

t h a t

s u p { I l l - z 0 [ [ : z E K } <

d i a m

K .

A c o n j u g a t e B a n a c h s p a c e h a s a we ak * n o r m a l s t r u c t u r e ( w * - N S) i f e v e r y w e a k * c l o s e d c o n v e x n o n t r i v i a l

s u b s e t o f i t h a s w e a k * n o r m a l s t r u c t u r e .

T h e n o t i o n o f w - N S w a s i n t r o d u c e d b y L a m i D o z o ( s e e [2 09 ]) a n d t h a t o f w * - N S b y T . - C . L i m [ 15 5] .

I t is o b v i o u s t h a t e v e r y ( c o n j u g a t e ) B a n a c h s p a c e w i t h a N S h a s a ( w * - N S ) w - N S . I n a r e f le x i v e B a n a c h

s p a c e , t h e n o t i o n s o f N S , w - N S a n d w * - N S , c o i n c id e . F o r t h e n o n - r e f le x i v e s p a c e s th i s i s n o l o n g e r t r u e . F o r

e x a m p l e , t h e s p a c e l 1 h a s a w - N S s i nc e e v e r y w e a k l y c o m p a c t s u b s e t i n l 1 i s a ( s t r o n g l y ) c o m p a c t s u b s e t ,

b u t h a s n o N S .

T h e p r e s e n c e o f a w * - N S i n a c o n j u g a t e s p a c e i m p l i e s t h e p r e s e n c e o f a w - N S s i n c e t h e w * - t o p o l o g y i s

w e a k e r t h a n t h e w - t op o l o g y .

W e n o w i n d i c a t e s o m e c l a s se s o f n o n - r e fl e x iv e B a n a c h s p a c e s w i t h a w - N S :

1 ) t h e ( U CE D ) s p a c e s [ 2 2 8 ] ;

2 ) t h e s p a c e s s a t i s f y i n g t h e O p i a l c o n d i t i o n [ 9 8 ] ;

3 ) t h e s p a c e s w i t h a S c h a u d e r b a s i s s a t i s f y i n g t h e c o n d i t i o n ( G L D ) ( s e e 1 . 1. 3 6) [ 9 7] ;

4 ) t h e ( U K K ) s p a c e s ( s e e 1 .1 .2 1) [ 73 ].

T h e o r e m 1 . 4 . 3 [1 55 ].

Let K be a weak* comp act con vez subse t o f a con juga te Ba nac h space , wi th

w * - N S . T h e n K h a s t h e F P P .

T h e f o l l o w i n g e x a m p l e s h o w s t h a t t h e p r e s e n c e o f a w * - N S i n a w e a k * c o m p a c t s e t i n T h e o r e m 1 .4 .2 is

e s s e n t i a l .

E x a m p l e 1 . 4 . 4 [ 15 5] . O n t h e s p a c e s (c o , [ l' [[ oo ) a n d ( l 1 , ] [ ' n l ) , w e i n t r o d u c e , r e s p e c t i v e l y , t h e e q u i v a l e n t

n o r ms [ x l =

I I x + l l ~ +

I Ix - I loo and

t l x l l = m a x { l l z l [ 1 , ! t ~ - I I 1 } ,

w h e r e x + ( x - ) i s t h e p o si ti v e ( n e g a t i v e ) p a r t

o f t h e s e q u e n c e x . T h e n o r m s I I a n d 11 I[ ~ a r e e q u i v a l e n t s in c e I 1~ t1 ~ - < t x l ~ < 2 } i ~ I I ~ , a n d t h e s p a c e s

( l l ,

I 1" I I) ~ d ( c o , I " I )*

a r e i s o m e t r i c a l l y i s o m o r p h i c . M o r e o v e r ,

8 9 ~ 1 0, E xi <~ 1 } . T h e s e t K i s w e a k * c o m p a c t a n d c o n v e x in ( t l , l l . I I ) a s t h e

i=

i n t e r s e c t i o n o f t h e u n i t b a l l a n d t h e w e a k * c l o se d s et

{z E 11 : :ci /> O, i = 1 ,2 , . . . }.

L e t T : K - -+ K b e a m a p p i n g s u c h t h a t T z = ( 1 - E x i , X l , X 2 , . . - , x n , . . . ) f or e v e r y x E K . T h e n

i=

] [ T x - T y[ [ = [ I x - y [[ f o r a ll x , y E K . H e n c e T is a n a f f i n e i s o m e t r y . Bu t i t i s e a s i l y v e r i f i e d t h a t F i x T = ~ .

S t a t e m e n t 1 . 4 . 5 [1 45 ]. The space 11 has a w*-NS.

C o r o l l a r y 1 . 4 . 6 [ 1 1 1 ] . Ev ery no ne mp ty weak* com pact subse t in 11 for ezample , the un i t ba ll ) has the

F P P .

N o t e t h a t t h e s p a c e l I s a t is f i e s t h e w e a k * O p i a l c o n d i t i o n [ 11 1] , i .e . , f o r e v e r y s e q u e n c e {x ~} C l ~ , x , ~ ')

z , a n d f o r e v e r y y r x w e h a v e

l i m i n f [ ] z , - x l l < l i m i n f l l x n - - Y l[.

I n f a c t , h e r e w e h a v e ( s e e [ 1 5 5 ] )

l i m i n f [ I x . ~ l l ~ l i m i n f I [ z y l l . [ 1 ~ y l l .

K . G o e b e l a n d T . K u c z u m o w [ 91 ] h a v e c o n s t r u c t e d a d e c r e a s i n g s e q u e n ce o f b o u n d e d c l o se d ( b u t n o t

w e a k * c o m p a c t ) s e t s in t h e s p a c e 11 s u c h t h a t t h e s e t s w i t h o d d i n d i c e s h a v e t h e F P P , a n d t h e s e t s w i t h

777



e v e n i n d ic e s d o n o t . M o r e o v e r , t h e s e se t s c a n b e s o c h o s e n t h a t t h e i r in t e r s e c t i o n i s n o n e m p t y a n d h a s

( d o e s n o t h a v e ) t h e F P P a t o u r c ho i c e. T h e r e i s g i v e n a n e x a m p l e o f t w o b o u n d e d c o n v e x c lo s e d s e t s in 11

h a v i n g t h e F P P , b u t w h o s e i n te r s e c t io n d o e s n o t .

P . S o a r d i [ 2 07 ] h a s s h o w n t h a t e v e r y w e a k* c o m p a c t c o n v e x su b s e t o f a B a n a c h s p a c e X h a s t h e F P P i f

t h e B a n a c h - M a z u r d i s ta n c e

d X , l 1)

< 2 . H e re t h e b o u n d 2 i s s h a rp s i n ce fo r t h e s p a ce X = ( l 1 , [1 ]]) f r o m

E x a m p l e 1 .4 .4 , i n w h i c h t h e r e e x is t s a w e a k* c o m p a c t c o n v e x s u b s e t w i t h o u t F P P , w e h a v e d X , 1 1 ) = 2 .

A c t u a l l y , i f Y = ( l 1 , ] . ] ), w h e re t h e n o r m ] . ] i s eq u i v a l en t t o t h e n o rm I ] ]11 an d

d Y , l 1)

< 2 , t h en Y

has a w*-NS (see [115]) .

T h e q u e s t i o n o f e x i s t e n c e o f a w e a k l y n o r m a l s t r u c t u r e i n s p a c e s o f t h e K a d e c - K l e e t y p e ( s e e 1 .1 .2 0 ,

1 . 1 . 2 1 ) i s s o l v ed b y t h e fo l l o w i n g t h eo rem.

T h e o r e m 1 . 4 . 7 [72]. L e t a B a n a c h s pa ce X be W U K t O . T h e n X h a s a w - N S .

I n p a r t i c u l a r , e v e r y ( U K K ) - s p a c e h a s a w - N S . H e n c e , e v e r y w e a k l y c o m p a c t c o n v e x su b s e t i n a ( W U K K ) -

s p a c e h a s t h e F P P .

D e f i n i t i o n 1 . 4 . 8 [7 3] . A c o n j u g a t e B a n a c h s p a c e X i s c a l le d ( W U K K * ) i f t h e r e e x is t a E (0 , 1 ) a n d ~ > 0

s u c h t h a t f o r a n y s e q u e n c e { x n } C B X ) w i t h i n f { [IXn--Xr~ ]l: n r m } >i e w e h a v e { x n }w A B ( 0 , 1 - ~ ) r O .

I f f o r ev e r y e e ( 0 , 1 ) t h e r e i s a $ = 6 ( e ) > 0 fo r w h i ch 1 . 4. 8 h o l d s , t h e n X i s c a l l ed (U K K * ) .

T h e o r e m 1 . 4 . 9 [73 ]. E v e r y W U g K * ) - s p a c e h a s a w * - Y S .

F u r t h e r m o r e , i f a s p a c e i s ( U K K * ) , t h e n t h e C h e b y s h e v c e n t e r o f e v e r y w e a k* c o m p a c t c o n v e x su b s e t i s

a c o m p a c t s e t ( t h e s e t r = { z e A : s u p H z - VII = r ( A ) } , w h e r e

r A )

is t h e C h e b y s h e v r a d i u s o f a

b o u n d e d c o n v e x s e t A , i s c a l l e d t h e C h e b y s h e v c e n t e r o f A ) .

S i n c e I 1 i s ( U K K * ) - s p a c e [ 73 ], a s a c o n s e q u e n c e w e g e t t h a t e v e r y w e a k * c o m p a c t c o n v e x s u b s e t o f t 1

h a s a c o m p a c t C h e b y s h e v c e n t e r [ 1 5 5 ] .

D e f i n i t i o n 1 . 4 . 1 0 [7 1]. A ( c o n j u g a te ) B a n a c h s p a c e X is s a id t o h a v e t h e w e a k * ) w e a k f i z e d p o i n t

p r o p e r t y ( ( w * - F P P ) w - F P P ) , i f e v e r y ( w e a k *) w e a k ly c o m p a c t c o n v e x s u b s e t A C X h a s t h e F P P , t h a t is ,

e v e r y n o n e x p a n s i v e m a p p i n g T : A -+ A h a s a fi x e d p o i n t .

T h e o r e m 1 . 4 . 1 1 [71]. I f a B a n a c h s p a ce X i s s e pa r a b le , t h e n t h e s p a ce s X a n d X * a d m i t e q u i v a l e n t

n o r m s w i t h r e s p ec t t o w h i c h t h e y w i ll h a v e t h e w - F P P a n d t h e w * - F P P r e s p ec t iv e l y.

D e f i n i t i o n 1 . 4 . 1 2 [7 1]. A c o n j u g a t e B a n a e h s p a c e X i s s a i d t o p o s s e ss t h e

w * - O p i a l p r o p e r t y

i f f o r

lo

e v e r y s e q u e n c e { x n } C X , x > x0 a n d e v e r y x # x 0 o n e h a s

i i m i n f

I l x . x o l l <

i m i n f

l x . x l I .

Fo r e x am p l e ( s ee [11 1] ), l I s a t i s f ie s t h e w * -O p i a l p ro p e r t y .

T h e o r e m 1 . 4 . 1 3 [11 1].

L e t X b e a s ep a ra b le B a n a c h s p a c e a n d a s s u m e t h a t X * s a t i sf i e s t h e w * - O p i a I

p r o p e rt y. T h e n X * h a s t h e w * - F P P .

I n p a r t ic u l a r , f r o m t h i s w e g et t h a t t h e s p a c es 11 a n d t h e J T ( t h e J a m e s t r e e s p a ce ) h a v e t h e w * - F P P

(see [111]).

T h e p r e s e n c e o f a w e a k * n o r m a l s t r u c t u r e i n t h e L o r e n t z s p a c e s w a s e s t a b l i s h e d i n [ 50 ].

L e t ( ~ , ~ , # ) b e a m e a s u r e s p a c e w i t h a a - f i n i te m e a s u r e . F o r a ~ u - m e a su r a bl e f u n c t i o n f w e s h a l l

p u t d l t ) = / ~ ({x : [ f ( x ) l > t ) , f * ( t ) = i n f {s > 0 : d r ( s ) ~< t } , 0 < t < o o . Fo r 1 < p < o o , t h e

L o r e n t z s p a c e L p,a ( /~ ) i s t h e B a n a c h s p a c e o f e q u i v a l e n c e c l a ss e s o f y - m e a s u r a b l e f u n c t i o n s f w i t h t h e n o r m

c o

I[fll = f f * t ) d t l / P ) . N o t e t h a t L p , l ( # ) i s a c o n j u g a t e s p a c e .

S t a t e m e n t 1 . 4 . 1 4 [50]. L p , a ( # ) i s

a W U g K * ) - s p a c e .

W e o b t a i n a s a c o n s e q u e n c e o f T h e o r e m 1 .4 .9 . t h a t L p , I ( # ) h a s t h e w * - F P P .

I t is k n o w n ( B . T u r e t t [ 21 4]) t h a t i f X is a B a n a c h s p a c e s u c h t h a t t h e c h a r a c t e r i s t i c o f c o n v e x i t y

G 0 ( X *) < 1 , t h e n b o t h s p a c e s X a n d X * a r e su p e r r e f l e x i v e a n d h a v e n o r m a l s t r u c t u r e s . T o s t u d y t h e w *

- N S , K . G o e b e l a n d T . S e k ow s k i [9 3] i n t r o d u c e d f o r th e c o n j u g a t e s p a c e s d u a l a n a l o g s o f th e m o d u l u s a n d

778



c o e f fi c ie n t o f n o n c o m p a c t c o n v e x i t y :

{ 9 }

~ ( e ) = i n f 1 - d i s t ( 0 , A ) : A =c--6-fiVw A c B ( X ) , a ( A ) > / e

;

= s u p { 9 [ 0 , 2 ] : x ;c (e ) = 0 } ,

w h e r e a ( . ) i s t h e K u r a t o w s k i m e a s u r e o f n o n c o m p a c t n e s s . I t is o bv i o u s t h a t A 3 r1 62 >~ A x ( e ) fo r e v e r y

c o n j u g a t e B a n a c h s p a c e . I n p a r t i c u l a r , A ~ I ( e ) = ~ > A t l ( e ) . S e k o w s ki [1 9 4] h a s p r o v e d t h a t A 3 c ( e ) ~<

f o r a n y B a n a c h s p a c e X .

T h e o r e m 1 . 4 . 1 5 [ 9 3 ] .

I f X i s a con juga t e Banach s pace and e ; X ) < 1, t hen X has a w *-N S .

T h i s t h e o r e m a l s o im p l i e s t h a t 11 h a s a w * - N S . A . T . - M . L a u a n d P . F . M a h [1 48 ], d e v e l o p i n g t h e c o n c e p t

o f n e a r l y n o r m a l s t r u c t u r e , g a v e t h e f o l lo w i n g

D e f i n i t i o n 1 . 4 . 1 6 . A ( c o n ju g a t e ) B a n a c h sp a c e X is s a id t o h a v e

a quas i- w eak* - ) w ea k l y -nor m a l

s t ructure quas i -w*

- N S )

q u a s i - w - N S )

i f e v e r y ( w e a k * ) w e a k l y c o m p a c t c o n v e x s u b s e t K C X , d i a m Ix > 0 ,

c o n t a i n s a p o i n t z s u c h t h a t [Ix - y ][ < d i a m K f o r a n y y E K .

I n [ 14 8] i t is s h o w n t h a t e v e r y c o n j u g a t e B a n a c h s p a c e X h a s a q u a s i- w * - N S i f X s a t is f ie s t h e L i r a

c o n d i t i o n :

l iml Ix~

y l l - - l im I I z ~ l l

+ I ly ll fo r any y E _32 i f t he n e t z a w 0 , an d i f t he re ex i s t s l iml l z~ l l .

L e t H b e a ( r e a l ) H i l b e r t s p a c e , B H ) t h e B a n a c h a l g e b r a o f b o u n d e d l i n e a r o p e r a t o r s o n H , g H ) b e

t h e i d e a l o f c o m p a c t o p e r a t o r s , T H ) b e t h e i d e a l o f o p e r a t o r s w i t h t r a c e . T h e f o l lo w i n g r e s u l t s a r e p r o v e d

i n [ 1 4 8] a m o n g o t h e r s :

1)

B H )

h a s a q u a s i - w * - N S , : ',, d i m H < c o ;

2 ) C H ) ha s a qu asi - w -N S -', ',- H is sep ara ble ;

3 )

T H )

h a s a q u a s i - w * - N S;

4) T H ) sa t i s f i es the L i m c on d i t io n -', ,-' d im H < oo ;

5 ) i f M i s a lo c a l l y c o m p a c t H a u s d o r f f s p a c e , t h e n [ c 0 ( M ) ] * s a t i s fi e s t h e L i m c o n d i t i o n 4 ---4 - [ c 0 ( M ) ] * ----

l i I )

f o r s o m e n o n e m p t y s et I , w h e r e

c o M )

is t h e B a n a c h s p a c e o f c o n t i n u o u s f u n c t i o n s o n M

v a n i s h i n g a t i n f i n i t y .

A s a m a t t e r o f f a c t ( K . L e n n a r d [ 15 0] )

T H )

h a s a w * - N S , s i n c e i t is ( U K K * ) - s p a c e .

A f u r t h e r d e v e l o p m e n t o f t h e s e r e s u l t s c a n b e f o u n d i n [ 14 9].

O b s e r v e , in c o n n e c t i o n w i t h t h e s p a c e s o f l i n e a r o p e r a t o r s , t h e f a c t t h a t t h e s p a c e

C1 Ip Iq)

o f o p e r a t o r s

o f t r a c e c l a s s f r o m l ' i n t o l q ( 1 <

p , q

< oc - + - = 1 ) has a w*-N S (M. Bes bes [35]) .

P q

C o n s i d e r t h e

class ical quas i -re f lexive James space J ,

t h a t i s, t h e s p a c e o f a l l r e a l s e q u e n c e s x = {x , } E c o

s u c h t h a t

I lxl I j

= s u p ( ( x m - - x p = ) 2 + ( x p 2 - - x . 3 ) 2 ' + x . . _ , - x . . ) 2 + x . . - - x m ) 2) ~ < o %

w h e r e t h e s u p r e m u m i s t a k e n o v e r a l l n 6 N a n d a l l f i n i te i n c r e a s i n g s e q u e n c e s p l < p 2 < ' < p n i n N .

T h e s p a c e J h a s t h e f o l l o w i n g p r o p e r t i e s ( J a m e s [ 1 0 7] ):

i J ~ J ;

2 ) T ( J ) i s a c l o s e d s u b s p a c e i n J * * o f c o d i m e n s i o n 1 , t h a t i s,

d i m [ J * * / ~ J ) ]

= 1 (her e ~2 i s t he

c a r io n i c a l e m b e d d i n g f r o m J i n t o J * * ) ;

3 ) t h e s e q u e n c e { an } f o r m s a m o n o t o n e s h r in k i n g , b u t n o t b o u n d e d l y c o m p l e t e b a s is i n J ;

4 ) J = 3 1 + 3 2 , w he re J1 an d 3 2 a re so m e c losed subs pa ces in J suc h tha t J1 N 3 2 = {0} an d J1 a nd

J 2 a r e i s o m o r p h i c t o l 2 .

T h u s J is a n o n r e f le x i v e s p a c e i s o m e t r i c t o i ts s e c o n d c o n j u g a t e .

T h e o r e m 1 . 4 . 1 7 ( D . T i n g l e y [ 21 3] ).

T he Jam es quas i -r e f lex i ve s pace J has a w -N S .

C o r o l l a r y 1 . 4 . 1 8 ( M . A . K h a m s i ) .

T he James s pace J has t he w -FPP.

T h e f o l lo w i n g n o t i o n i s a n a t u r a l g e n e r a l i z a t i o n o f t h e S c h a u d e r b a s is .

779



D e f i n i t i o n 1 . 4 . 1 9 ( s e e [ 16 2 ]) . A s e q u e n c e o f f i n i te d i m e n s i o n a l su b s p a c e s { X , } i s c a l l e d a f i n i t e d i m e n -

s i o n a l S c h a u d e r d e c o m p o s i t i o n ( F D S D ) o f a s p a c e X i f e v e r y x E X i s u n i q u e l y r e p r e s e n t a b l e i n t h e f o r m

oo

z = ~ x i , w h e r e x i E X i f o r i = l , 2 , . . . .

i = l

oo

F o r x = ~ x , p u t s u p p ( x ) = { i E N : z i 0 } . L et A , B C N a n d k E N . W r i t e A < B ( A < B + k ) i f

i =

f o r a l l (a, b) E a • B , a 0 : ( l l x l l ' + I l y l l' ) 1 / ' ~< ) q lx + Y ]I f o r al l x , y E X s u c h t h a t s u p p ( z ) < s u p p ( y ) } .

I f l '1 i s t h e B y n u m s p a c e , t h e n ~1(1 '1) = 2 1 - 1 / ' f o r a n y p E [ 1, o o] . Fo r t h e J a m e s s p a c e J 1 = {x =

E c o : I l x l l < o o } w h e r e t h e n o r m I1 I I is c o n s t r u c t e d f r o m I 1 I l J b y d i s c a r d i n g t h e l a s t s u m m a n d ,

w e h a v e P 2 ( J 1 ) = 1 . N o t e t h a t t h e s p a c e J1 h a s n o N S , a l t h o u g h i t is i s o m o r p h i c t o t h e s p a c e J . O n e c a n

s h o w t h a t ~ , ( l ' ) = 1.

S t a t e m e n t 1 . 4 . 2 0 [115]. L e t X a n d Y be t w o i s o m o r p h i c B a n a c h s p ac e s, a n d l e t H a d m i t a n F D S D .

T h e n Y a d m i t s a n F D S D a n d / 3 p ( Y ) <~ d ( X , Y ) / 3 p ( X ) .

T h e o r e m 1 . 4 . 2 1 [ 1 1 5 ] . L e t X b e a B a n a c h s p a c e h a v i n g a s u b s p a c e Y o f f i n i t e c o d i me n s i o n s u c h t h a t

Y a d m i ts a n F D S D a n d / 3 , ( Y ) < 2 1/ f o r s o m e p E [1, co]. T h e n Z h a s a w - N S .

C o r o l l a r y 1 . 4 . 2 2 [ 1 1 5 ] .

T h e J a m e s s p a c e J 1 h a s a w - N S , a n d h e n c e t h e w - F P P .

C o r o l l a r y 1 . 4 . 2 3 [ 1 1 5 ] .

I f a B a n a c h s p ac e X i s i s o m o r p h i c to l a n d d ( X , l ' )

< 2 l /p ,

t h e n X h a s a

N S .

T h e c o n s t a n t 2 1 / ' i n 1 . 4 .2 3 i s s h a r p , s i n c e f o r e v e r y p E ( 1, e o ) t h e r e i s a s p a c e X , s u c h t h a t d ( X , , 1 1 / ' ) =

2 1 /p a n d X , h a s n o N S [ 11 5] . O n e c a n t a k e f o r X p t h e s p a c e ( l p , 1 . 1 , ) , w h e r e ] xlp = ma x ( ] l x H p , 21 /p nx H 1 ) .

T h e n t h e c a n o n i c a l b a si s i n Ip w i l l b e a d i a m e t r i c a l s e q u e n c e i n t h e s p a c e X p .

T h e o r e m 1 . 4 . 2 4 [115]. L e t X b e a B a n a c h s p ac e w~ th a s h r i n k i n g b a sis , a n d / 3 p ( X * ) < 2 1 / ' . T h e n X *

h a s a w * - N S , a n d h e n c e t h e w * - F P P .

T h e f o ll o w in g t h e o r e m g i ve s a p a r t i a l a n s w e r to t h e q u e s t io n o f T . L a n d e s [1 44 ]: D o e s t h e / l - d i r e c t s u m

o f s p a c e s w i t h a N S ( w - N S ) h a v e a N S ( w - N S ) ?

T h e o r e m 1 . 4 . 2 5 [115].

L e t

{ X i }

b e a s e q u e nc e o f B a n a c h s p a ce s a d m i t t i n g a n F D S D , s u c h t h a t

sup/31

i EN

(Xi ) < 2 . T h e n t h e l l- d i r e c t s u m o f t h e s e sp a c e s h a s a w - N S .

M . A . K h a m s i [1 17 ] h a s i n t r o d u c e d t h e n o ti o n s o f m o d e l - n o r m a l s t r u c t u r e a n d s u p e r - n o r m a l s t r u c t u r e a n d

h a s s h o w n t h a t i f s o m e s u b s p a c e o f a B a n a c h s p a c e p o s se s se s o n e o f t h e s e s t r u c t u r e s , t h e n t h e s p a c e i ts e lf

h a s a N S o r a w - N S r e s p e c t i v e l y .

J . - P . P e n o t [ 1 81 ] g a v e a n a b s t r a c t v e r s i o n o f T h e o r e m 1 .1 .4 . W e p r e s e n t t h e r e s u l t s o f P e n o t f o l l o w in g

p a p e r s [ 12 2, 1 2 4 ] b y K i r k .

L e t ( M , p ) b e a m e t r i c s p a c e . F o r D C M p u t

r ( D )

= i n f s u p

p ( u , v ) .

A s s u m e t h a t M h a s a

c o n v e x

uE rE

s t r u c t u r e , i .e . , a f a m i l y o f s u b s e t s T C 2 M s u c h t h a t N {F : F E ~- } E T f o r e a c h ~ - C T . A f a m i l y T i s s a i d

t o b e n o r m a l i f r ( D ) < d i a m D f o r a n y D E T w i t h d i a m D > 0 , a n d T is s a i d t o b e ( c o u n t a b I y ) c o m p a c t i f

e v e r y ( c o u n t a b l e ) s u b f a m i l y o f s e ts f r o m T w i t h t h e f i n i te i n t e rs e c t io n p r o p e r t y h a s n o n e m p t y i n t e rs e c t io n .

T h e o r e m 1 . 4 . 2 6 [181]. L e t ( M , p ) b e a b o u n de d m e t r i c s p a c e a n d a s s u m e t h a t M h a s a n o r m a l c o u n t a b l y

c o m p a c t c o n v e x s t r u c t u r e 7 t h a t c o n t a i n s cl o se d b alls o f M . T h e n e v e r y n o n e z p a n s i v e m a p p i n g T : M --+ M

h a s a f ix e d p o i n t .

O b s e r v e th a t b y v i r t u e o f t h e a s s u m p t i o n s o f t h is t h e o r e m , M E T a n d M is a c o m p l e t e m e t r i c s p ac e .

L e t X b e a B a n a c h s p a c e a n d l e t r b e a t o p o l o g y o n X f o r w h i c h t h e b a ll s c l o s e d i n t h e n o r m t o p o l o g y

a r e r - c l o s e d .

D e f i n i t i o n 1 . 4 . 2 7 [1 22 ]. A c o n ve x r - c l o s e d s u b s e t K C X i s s a i d t o h a v e a r - n o r m a l s t r u c t u r e ( r - N S )

i f f o r a n y b o u n d e d v - c l o s ed s u b s e t D C K , d i a m D > 0 , w e h a v e r ( D ) < d i a m D .

T h e o r e m 1 . 4 . 2 8 [122]. L e t X be a B a n a c h s pa c e, a n d K a n o n e m p t y b o u n d e d r - c l o s e d c o n v e x s u b s e t

w i t h a r - N S , a n d t h a t is c o u n t a b l y c o m p a c t i n t h e t o po l og y r . T h e n K h a s t h e F P P .

780



T h e f r u i t f u l n e s s o f s u c h a n a b s t r a c t a p p r o a c h i s d e m o n s t r a t e d in [ 1 5 0 ] b y C . L e n n a r d .

L e t ( f~ , E , ) b e a m e a s u r e s p a c e w i t h a p o s it i v e a - f in i t e a n d c o u n t a b l y a d d i t i v e m e a s u r e ;~ . L e t L ~

b e t h e s e t o f a l l E - m e a s u r a b l e f u n c t i o n s o n f l. T h e topology o f local convergence in me asure ( lcm-top ology)

o n L ~ i s t h e t o p o l o g y t h a t i s g e n e r a t e d b y t h e f o ll o w i n g m e t r i c p w h i c h i s i n v a r i a n t u n d e r t r a n s l a t i o n s :

l e t {E ,}n ~_ _i b e a E - p a r t i t i o n o f f ~ , w h e r e E = {E E E : ( E ) E ( 0 , o ~ ) } , f o r a l l

f , g E L ~

~ f

l f - g l

P ( f ' g ) = ~ - ~ 2 n # ( E ~ ) j

1 - t f - gl

d .

n = l

En

I f ( f l ) < o o , t h e n t h e m e t r i c p ( f , g ) = f I /- 91 d g e n e r a t e s t h e topology o f convergen ce in measure

l l / - g l

I2

( cm- t opo l ogy) .

N o t e t h a t e v e r y a l m o s t e v e r y w h e r e c o n v e r g e n t s e q u e n c e i n L ~ i s l o c a l ly c o n v e r g e n t i n

m e a s u r e . C o n v e r s e l y , e v e r y s e q u e n c e i n L ~ w h i c h i s l o c a l ly c o n v e r g e n t i n m e a s u r e c o n t a i n s a s u b s e q u e n c e

w h i c h is a l m o s t e v e r y w h e r e c o n v e r g e n t t o t h e s a m e l im i t .

Z ~ , ) d e n o t e s , a s u s u a l , t h e B a n a c h s p a c e o f a ll (e q u i v a l e n c e c l as s e s) o f m e a s u r a b l e f u n c t i o n s f : f~ ~ R ,

f o r w h i c h

I If H ~ = f n

I f ( w ) l

d # ( w )

< c o . T h e l c m - t o p o l o g y a n d t h e c m - t o p o l o g y o n L I ( ) a r e in d u c e d b y

t h e c o r r e s p o n d i n g to p o l o g i e s f r o m L ~ t o L 1 ( ) .

D e f i n i t i o n 1 . 4 . 2 9 [ 15 0] . A B a n a c h s p a c e X is s a i d t o b e ( U K K ( r ) ) , w h e r e 7- i s a t o p o l o g y o n X , i f f o r

e a c h e > 0 t h e r e i s a 6 > 0 s u ch t h a t f or a n y s eq u en c e { xn }

C B (X ) , x ,~ ~> z

a n d i n f { l l x n - X m l l :

n #

m } > e , w e h a v e I l x ll

~< ~

T h e o r e m 1 . 4 . 3 0 [ 1 5 0 ] .

The space

L x ( )

is ( UK K(7 .)), whe re 7. is the Icm-topology.

I n p a r t i c u l a r , L ~ , ) h a s a 7 . -N S s e e a l so [ 3 4 ] )

C o r o l l a r y 1 . 4 . 3 1 [ 1 4 2 ] .

Ev er y none mp t y bounded 7 .- compac t conve z s ubs e t o f t he s pace L 1 (# ) has t he

FPP, w her e 7 . i s t he l cm- t opo l ogy on L I (# ) .

1 5 F i x e d P o i n t s a n d P r o d u c t S p a c e s .

B e l l uc e , K i r k a n d S t e i n e r [ 3 3 ] o b t a i n e d o n e o f t h e f i r s t r e s u l t s c o n c e r n i n g t h e p r e s e n c e o f a n o r m a l

s t r u c t u r e i n t h e p r o d u c t o f sp a c e s h a v in g a n o r m a l s t r u c t u r e i n e v e r y fa c t o r .

\

L e t ( ~ ]

@ X i ~

b e t h e l P ( I ) - d i r e c t s u m ( d i r e c t p r o d u c t ) o f a f a m i l y o f B a n a c h s p a c e s { X , } i E z , t h a t i s,

\i l ] p

t h e B a n a c h s p a c e o f a ll x

=

{:r.i}iEi, zi ~ Xi f o r w h i c h l z l l < o o , w h e r e

I l x l l = I lx il l p , l ~ < p <

s u p { l l z i l l : i E 1 } , p =

~ .

T h e o r e m 1 . 5 . 1 [33 ].

L e t X I , X z , . . . , X , be B a n a c h sp a ce s w i t h a n o r m a l s t r u c tu r e . T h e n t h e l ~ -

d i r ect s um ( ~-]~ ~ X i ) has a nor m a l s t r uc t u r e , and , i n par t i cu la r , t he w -F PP .

/=1

T . L a n d e s [ 14 4 ] g a v e a g e n e r a l i z a t i o n o f t h i s r e s u lt . A B a n a c h s p a c e Z i s c a l le d

p e r m u t a t i o n a l s pa c e

( w i t h

t h e in d e x se t I 0 , w h e r e I m a y h a v e a n y c a r d i n a l i ty ) if Z h a s a S c h a u d e r b a s i s { e i} iE Z ( u n c o n d i t i o n a l ,

i f I i s u n c o u n t a b l e ) , a n d t h e

n o r m

on Z i s

m o n o t o n e ,

t ha t i s I l z l l

~<

I lzql, pr ov id ed 0 ~<

z( i) <~ E(i)

fo r a l l

i E I ( z , E E Z ) .

T h e s p a c e s IP (I ), 1 <~ p <~ oo, o r Co(I) f o r a n y s e t I , i n p a r t i c u l a r , Ip = lP(N ), Co -- c0(N ), l~ =

Ip ({ 1, 2 , . . . , n } ) a r e e x a m p l e s o f p e r m u t a t i o n a l s p a c es .

L e t Z b e a p e r m u t a t i o n a l s p a ce w i th t h e i n de x se t I , a n d { X i } i~ z a f a m i l y o f B a n a c h s p a c e s . T h e n

t h e o p e r a t o r U

d e f i n e d o n t h e s e t o f a l l

z E

1-I x ~ f o r w h i c h ~ I lz i )] le ~ E Z , a n d

a c t i n g b y t h e r u l e

i I i I

z = ~ IIz(i)l lei ,

i s ca l l ed a

permutat ional operator .

T h e Z - d i r e c t s u m (~ -~

@ X i ) z

o f t h e f a m i l y { X i } iE z

iEI iEl

is t h e d o m a i n o f d ef i n it i on o f U e q u i p p e d w i t h t h e n o r m

I lx l l = I I U z l l z

W e s h a ll s a y t h a t a n o r m a l s t r u c t u r e i s p r e s e rv e d u n d e r t h e Z - d i r e c t s u m i f (~ -]

@ X i ) z

h a s a n o r m a l

iEI

s t r u c t u r e w h e n a l l X i h a v e i t .

781



T h e o r e m 1 . 5 . 2 [1 44 ].

Le t Z be a per mu ta t ion a l space w i th the index se t I -~-

{ 1 , 2 , . . . , n }

s u c h t h a t

I I z < 2 i f I l z l l = I I l l = 1 ,

z ( i ) >>. o , i ( i ) > i o f o r a l l i E I , a n d = S (i ) o n l y ] o r t h o s e i I f o r w h i c h

z ( i ) = " ~( i) = O. Th e n a n o r m a l s t r u c t u r e i s p r e s e r v e d u n d e r t h e Z - d i r e c t s u m .

I n p a r t i c u l a x , T h e o r e m 1 . 5 .2 i s v a l i d if Z i s a s t r i c t l y c o n v e x s p a c e , f o r e x a m p l e l ~ ( 1 < p < o z ) , o r

Z = I n 1 7 6 u r t h e r m o r e , L a n d e s h a s p r o v e d t h a t t h e c o n c l u s i o n o f t h is t h e o r e m r e m a i n s t r u e a ls o i n t h e c a se

w h e r e Z i s a n a r b i t r a r y i n f i n i t e - d i m e n s i o n a l ( U C ) p e r m u t a t i o n a l s p a c e , f o r e x a m p l e , Z = IP ( 1 < p < o o ) .

A Schur space i s a B a n a c h s p a c e i n w h i c h w e a k a n d s t r o n g c o n v e r g e n c e c o i n c i d e .

S t a t e m e n t 1 . 5 . 3 [1 44 ]. Le t Z be a per mu ta t ion a l space sa t i s f y ing the ( GL D) con d i t ion ( see 1 .1 .36) .

Th e n t h e Z - d i r e c t s u m o f a f a m i l y o f S c h u r s p ac e s h a s a w- N S .

I n p a r t i c u l a r , o n e c a n t a k e f o r Z t h e s p a c e 11 ( I ) .

W e n o w t u r n t o t h e r e s u lt s c o n c e r n i n g t h e e x i s te n c e o f f ix e d p o in t s o f n o n e x p a n s i v e m a p p i n g s i n a

p r o d u c t s p a c e .

D e f i n i t i o n 1 . 5 . 4 [ 10 1] . A c o n v e x s u b s e t K o f a B a n a c h s p a ce X is s a id t o h a v e t h e

e f f e c t i v e F P P

i f

f o r e v e r y n o n e x p a n s i v e m a p p i n g T : K ~ K t h e r e e x is t s x 0 E K s u c h t h a t t h e s e t { x ~ E K : x t =

t T x t + (1 - t ) x o , 0 < t < 1 } is p r e c o m p a c t .

O b v i o u s l y , a c l o s e d c o n v e x s e t w i t h t h e e f fe c ti v e F P P h a s t h e F P P t o o .

R . H a y d o n , E . O d e l l a n d Y . S t e r n f e l d ( s e e [1 30 ]) s h o w e d t h a t i f K 1 a n d / x ' 2 a r e c l o s e d s u b s e t s o f B a n a c h

s p a c e s X 1 , X 2 r e s p e c t iv e l y , a n d K x i s s e p a r a b l e a n d h a s t h e F P P , a n d K 2 h a s t h e e f f e c t iv e F P P , t h e n t h e

se t K1 E9/x '2 C (X92 @ X 2 ) o ~ h a s t h e F P P .

K i r k a n d S t e r n f e l d [ 13 0] h a v i n g i m p r o v e d t h e m e t h o d o f [ 10 1] h a v e p r o v e d a m o r e g e n e r a l r e s u l t .

T h e o r e m 1 . 5 . 5 [1 30 ]. Le t K i be c losed subse t s o f Ba nac h spaces X i

i = 1 , 2 , . . . , n )

and l e t K1 have

t he F P P , a n d K 2 , K 3 , . . . , K n be b o u nd e d a n d co n v ex . T h e n t h e s e t K 1 @ ' . " @ K n C ( X 1 @ " '" G X n ) h a s

t h e F P P f o r m a p p i n g s t h a t a r e n o n e x p a n s i v e w i t h r e s pe c t to th e n o r m o f ( X 1 q ~ . . . ~ X n ) o o , i f f o r e ac h

j >1 2 one o f the fo l low ing cond i t ion s i s sa ti s f ied :

1) K i is separable for i <<.j - 1 and h ~ has the ef fec t ive FP P ;

2 ) X j i s un i for m ly convex , o r X j i s re f l ex ive and sa t is f i es the Opia l cond i t ion;

3 ) X j i s u n i f o r m l y s m o o t h ;

4 )

X j i s a u n i f o r m l y c o n v e x B a n a c h s pa ce w i t h t h e n o r m d i ff e r e n ti a b le i n t h e s e n s e o f F r d c h e t .

W e s h a ll s a y t h a t a s u b s e t K o f a B a n a c h s p a c e X h a s t h e B r o w d e r - G h h d e p r o p e r t y ( s e e [ 42 ]) i f f o r a n y

n o n e x p a n s i v e m a p p i n g T : K ~ K t h e m a p p i n g I - T i s d e m i c l o s e d , i .e . , i f, g i v e n a s e q u e n c e { x n } C K

s u c h t h a t x ~ w ) x a n d ( I - T ) x , ~ !!'11) y , t h e n x E K a n d ( I - T ) x = y ( h e r e I : X - + X i s t h e i d e n t i ty

m a p p i n g ) . A B a n a c h s p a c e i s s a id t o h a v e t h e B r o w d e r - G h h d e p r o p e r t y i f e v er y n o n e m p t y b o u n d e d c o n ve x

a n d c l o s ed s u b s e t o f i t h a s t h i s p r o p e r t y .

C l e ar ly , e v e r y n o n e m p t y c o n v e x w e a k l y c o m p a c t s u b s e t w i t h th e B r o w d e r - G h h d e p r o p e r t y h a s t h e F P P .

B r o w d e r [42 ], g e n e ra l iz i n g t h e m e t h o d o f G h h d e [9 4], p r o v e d t h a t e v e ry n o n e m p t y c o n v e x a n d c lo s e d

s u b s e t o f a u n i f o r m l y c o n v e x B a n a c h s p a c e h a s t h e B r o w d e r - G 6 h d e p r o p e r ty . A n a n a l o g o u s f a c t i s e a si ly

e s t a b l i s h e d f o r a s p a c e s a t i s f y i n g t h e O p i a l c o n d i t i o n ( s ee [ 1 40 ]) , i n p a r t ic u l a r , f o r s p a c e s h a v i n g a w e a k l y

c lo s ed s e q u e n t i al l y c o n t i n u o u s d u a l i t y m a p p i n g .

T h e o r e m 1 . 5 . 6 ( K i r k [ 12 5] ). Let Ki be a bounded convex c losed subse t o f a Banach space X i ( i = 1 , 2 ) .

A s s u m e t h a t K1 h a s t h e F P P a n d X 2 i s r e f le x i v e a n d h as th e B r o w d e r - G h h d e p r o p e r t y . Th e n K1 9 K2 h a s

t h e F P P f o r t h e m a p p i n g s t h a t a r e n o n e x p a n s i v e w i t h r e s pe c t to ( X 1 G X 2 ) ~ .

P . -K . L i n [ t6 1 ], s t u d y i n g t h e q u e s t i o n o f p r e s e r v a ti o n o f t h e B r o w d e r - G h h d e p r o p e r t y i n p r o d u c t s p a ce s ,

s u p p l e m e n t e d t h e r e s u l t s o f K i rk .

L e t I I" I Iz b e a n o r m o n t h e p l a n e R 2 s u c h t h a t f o r a l l

s , t

E IR I t ( s ,O) l [ z = I I (O, s ) l l z

= Is l , I I ( s , t ) l l z =

I I( Is l, I t l )I I z . G i v e n B a n a c h s p a c e s ( X i , I[ 9 t[ i) ( i = 1 , 2 ) , t h e n o r m o n ( X 1 9 X 2 , 1t " I I z ) i s d e f i n e d a s

T h e o r e m 1 . 5 . 7 [1 61 ]. Th e f o l l o w i n g a s s e r t i o n s a r e t r u e :

1) i f t here ex i s t s , t > 0 such tha t I I ( s , t ) l l z = I s l , the n ( l 2 G R , II " t l z ) d o e s n o t h a v e t h e B r o w d e r - G h h d e

proper ty ;

782



2 ) i f X i s a u n i f o r m l y c o n v e x B a n a c h s p ac e a n d Y i s a 5 c h u r s p a c e a n d I [" ] [z is a n o r m s u c h t h a t

[[ ( s , t ) l [ z > Is[ f o r a n y t > O , t h e n t h e s p a ce ( X 9 Y , [[" I[z) h a s t h e B r o w d e r - G h h d e p r o p e r t y .

L i n h a s s h o w n t h a t t h e r e a r e ( 2 - U C ) B a n a c h s p a c e s t h a t d o n o t h a v e th e B r o w d e r - G h h d e p r o p e r t y .

K i r k a n d M a r t i n e z - Y a f i e z [1 27 ], a n d K i r k [ I 2 6 ], u s i n g u l t r a n e t s , g e n e r a l i z e d t h e r e s u l t s o f p a p e r s [1 30 ]

a nd [125] .

W e s h a l l s a y t h a t t h e s e t ( K 1 G K 2 ) p h a s t h e F P P i f t h e s u b s e t K 1 @ K 2 C ( X 1 | X 2 ) p h a s t h e F P P f o r

m a p p i n g s t h a t a r e n o n e x p a n s i v e w i t h r e s p e c t t o t h e n o r m o f / P - d i r e c t s u m ( X 1 9 X 2 ) p , w h e r e 1 ~< p ~< o o .

T h e o r e m 1 . 5 . 8 [1 27 ]. G i v e n s u b s e t s K i o f B a n a c h s p a c e s X i ( i = 1 , 2 ) . T h e f o l l o w i n g a s s e r t i o n s a r e

t r u e :

1) i l K 1 h a s th e F P P , a n d K 2 i s a c o n ve x w e ak ly c o m p a c t s e t h a v i n g th e s tr i c t B r o w d e r - G 6 h d e p r o p e r t y

( i . e . , f o r e v e r y n e t { x ~ } C K 2 s u c h t h a t f r o m x a w> x a n d x ~ T x ~ ] i'l l

y i t f o l l o w s t h a t x E K 2

a n d x - T x = y ) , t h e n ( K 1 G K 2 ) p has t he E P P fo r 1 <~ p <~ oo;

2) i f K 1 i s c l o s ed a n d h a s t h e F P P , a n d K 2 i s c o n v e x a n d c l os e d, a n d h a s t h e e ff e c t iv e F P P , t h e n

( K 1

@ I s h a s t h e F P P ;

3 ) i f I~ 1 a n d K 2 h a v e E R R , t h e n ( h'~ @ K 2 ) p has t he F P F fo r 1 <<.p < oo .

K . - K . T a n a n d H . - K . X u [2 12 ] h a v e s h o w n u s i n g e l e m e n t a r y t o o l s ( n o t u s i n g u l t r a n e t s ) t h a t i n a s s e r t i o n ( 1 )

f r o m 1 . 5 . 8 t h e s t r i c t B r o w d e r - G h h d e p r o p e r t y m a y b e r e p l a c e d b y t h e o r d i n a r y B r o w d e r - G h h d e p r o p e r t y .

K i r k [1 26 ] h a s s h o w n t h a t i f K ~ , Ii'2 h a v e th e F P P , K 2 i s c o n v e x a n d w e a k l y c o m p a c t , a n d X 2 i s a

( K K ) - s p a c e , t h e n ( K 1 @ K 2 )o ~ h a s t h e F P P . T . K u c z u m o w [1 34 ] h a s s h o w n t h a t h e r e X ~ m a y b e a n

a r b i t r a r y B a n a c h s p a c e i f o n e r e q u i re s t h a t K 2 h a v e t h e c o m m o n F P P ( t h a t i s , e a c h n o n e x p a n s i v e m a p p i n g

T : K 2 - + K 2 h a s a fi x e d p o i n t i n e v e r y n o n e m p t y c o n v e x c l o s ed T - i n v a r i a n t s u b s e t o f K 2 ) . I n p a r t i c u l a r ,

t h e f o l l o w i n g re s u l t h o l d s .

T h e o r e m 1 . 5 . 9 [1 34 ]. L e t K 1 a n d K 2 b e n o n e m p t y c o n v ex w e a k l y c o m p a c t s u b s e ts o f B a n a c h s p ac es

w i th t he c o m m o n F P P . T h e n ( K 1 9 K 2 ) ~ h a s t h e E P P .

A s i m i l a r r e s u l t f o r a s p e c i a l c la s s o f n o r m s w a s o b t a i n e d b y K h a m s i [1 17 ].

D e f i n i t i o n 1 . 5 . 1 0 [ 11 7] . A n o r m N o n t h e d ir e c t s u m X ~ Y o f B a n a c h s p a c e s X a n d Y i s s a id t o h a v e

t y p e ( L ) if:

1 ) t h e r e s t r i c t i o n s o f N t o X a n d Y c o i n c i d e w i t h t h e n o r m s o f X a n d Y ;

2 ) t h e n a t u r a l p r o j e c t i o n s c o n n e c t e d w i t h X G Y h a v e n o r m 1 ;

3 ) f o r a n y x , z ' e Z , y E Y w e h a v e : i f Y ( x + y ) <~ N ( x ' + y ) , th e n [[xI[ ~< ]Ix' l[ .

T h e o r e m 1 . 5 . 1 1 [1 17 ]. L e t X a n d Y b e B a n a c h s p a c e s w i t h t h e w - E R R , a n d ][. I[ b e a n o r m o f ty p e

( L ) o n X ~ Y . T h e n f o r a l l w e a k l y c o m p a c t c o n v e x s u b s e t s K I a n d K 2 o f X a n d Y r e s p e c ti v e ly , t h e s e t

K x @ K 2 h a s t h e E R R f o r m a p p i n g s t h a t a re n o n e z p a n s i v e f o r t h e n o r m I[ " I[ o n X @ Y .

P r o o f . L e t T : K 1 ~ K 2 - - > / s 9 I ~2 b e a n o n e x p a n s i v e m a p p i n g . T h e n T - - T 1 + T 2 a n d T ( x + y ) =

T l ( X + y ) + T 2 ( x + y ) f o r al l (x , y ) E K ~ x / s L e t u s f i x x i n K ~ a n d c o n s i d e r t h e m a p p i n g T ~ : K 2 - -+ I x'2

s u c h t h a t T ~ ( y ) = T 2 ( x + y ) f o r a ll y e K 2 . B y ( 2 ) f r o m 1 . 5. 1 0, T~ is a n o n e x p a n s i v e m a p p i n g . S i n c e

Y h a s t h e w - F P P , f o r e v e r y X l E K t h e r e i s a f ix e d p o i n t y ~ f o r th e m a p p i n g T ~ . D e f i n e To : K 1 - + K 1

s o t h a t T o = T ~ ( y z + x ) . T h e n T o i s a n o n e x p a n s i v e m a p p i n g . I n d e e d , [ [T ( x + y x ) - T ( x ' + y==,)][ ~<

IIx + y== - x ' - y~, II. B u t T ( x + y~ ) = T o ( z ) + y~ a n d T ( x ' + y ~ , ) : T 0 ( z ' ) + y ~ , . U s i n g (3 ) a n d 1 . 5 . 1 0 w e

o b t a i n HTo(x ) - T 0 (x ')l I ~< IIx - x ' ll . S i n c e X h a s t h e w - F P P , F i x T 0 # ~ . L e t x 0 b e a f i x e d p o i n t o f T o .

T h e n T ( x o + Y ~o ) = T o ( x o ) + Y ~ o , a n d T ( x o + Y ~ o ) = x o + Y ~o . [ ]

K h a m s i [1 17 ] r a i s e d t h e f o l l o w i n g q u e s t i o n : I s t h e c o n c l u s i o n o f T h e o r e m 1 .5 .1 1 v a l id f o r e v e r y c o n v e x

w e a k l y c o m p a c t s u b s e t i n X 9 Y , i f Y i s a f i n i t e - d i m e n s i o n a l s p a c e ?

T a n a n d X u [ 2 1 2 ] g a v e a p a r t i a l a n s w e r t o t h i s q u e s t i o n .

D e f i n i t i o n 1 . 5 . 1 2 [2 12 ]. A B a n a c h s p a c e X is s a id t o h a v e p r o p e r t y ( P ) i f f o r a n y n o n c o n s t a n t s e q u e n c e

{ x ~} C X w e a k l y c o n v e r g i n g t o z

l i m i n f H x . - x II < d i a n a { x . ) .

R at OO

I t is ea s i ly s e e n t h a t t h e f o l l o w i n g s p a c e s h a v e p r o p e r t y ( P ) :

1 ) t h o s e w i t h a u n i f o r m l y n o r m a l s t r u c t u r e ;

783



2 ) t h o s e s a t i s f y i n g th e O p i a l c o n d i t i o n ;

3 ) t h o s e f o r w h i c h M a l u t a ' s c o n s t a n t

D ( X )

< 1 , f o r e x a m p l e , t h e ( N U C ) s p a c e s .

T h e o r e m 1 . 5 . 1 3 [2 12 ].

Let X be a Ba nac h space wi th prop erty (P) , Y be a f ini t e dim en sio na l space ,

and X @ Y be prov ide d w i th a norm sa t i s f y ing the f o l l owing c ond i t ions :

( a )

t he re s t r i c t i ons o f t he no rm o f X ~ Y to X and Y c o inc ide wi th t he nor ms o f X and Y ,

( b )

t he na tura l pro j e c t ions P 1 an d P 2 c onne c t ed w i th X @ Y hav e norm 1 .

The n X G Y has t he w- F P P .

P r o o f . L e t C b e a n a r b i t r a r y w e a k l y c o m p a c t c o n v e x s u b s e t o f X E3 Y , a n d l e t T : C --4 C b e a

n o n e x p a n s i v e m a p p i n g . C o n s t r u c t a m i n i m a l n o n e m p t y c o n v e x c lo s ed T - i n v a r i an t s u b s e t K C C . S e l ec t a

s e q u e n c e { z n} C g s u c h t h a t l i m

]lTz,, -

z ,~ ]l = 0 . T h e n b y t h e G o e b e l - K a r l o v i t z l e m m a , l i m ] ] zn - z l[ =

n --') OD I'L --) OO

d i ax n K f o r e v e r y z E K . S u p p o s e t h a t d i a m K > 0 . W i t h o u t l o s s o f g e n e r a l i ty , w e c a n a s s u m e t h a t

d i a m K = l . L e t z , ~ = x , , + y n , w h e r e x n 6 X , y n 6 Y . W e a l s o c a n a s s u m e t h a t z n ~ x 0 a n d y n I I. II y 0.

T h e n z , ~ > z 0, w h e r e z0 = x 0 + Y0 6 K . O b s e r v e t h a t d i a m { x , } <~ d i am { z , ~ } s i n c e t h e p r o j e c t i o n

P x : X 9 Y --+ X h a s n o r m 1 . S i n c e X h a s p r o p e r t y ( P ) , w e o b t a i n

1 = l i r a I l z n -z 0 ] l ~ < l i m i n f l ] x , - x 0 1 1 z + l i m H y , - y 0 I I Y =

n--~t OO n---+ ~ n---+ ~

= li m i n f ll x , ~ x 0 11 x < d i a m { x , } ~< d i a m { z , } = 1 .

T h e c o n t r a d i c t i o n d e r i v e d s h o w s t h a t d i a m K = 0 , i .e ., K c o n s i s ts o f a s in g l e p o i n t w h i c h i s a f ix e d p o i n t

o f t h e m a p p i n g T . [ ]

S i n c e f o r 1 ~< p ~ < o o , t h e l P - n o r m I1" lip s a t i s fi e s c o n d i t i o n s ( a ) a n d ( b ) f r o m 1 . 5 .1 3 , w e c o n c l u d e f o r X

a n d Y f r o m 1 . 5. 13 t h a t ( X ~ Y ) p h a s t h e w - F P P ( [ 2 1 2 ]) .

T o c o n c l u d e C h a p t e r 1 w e l i st s o m e s u r v e y s a n d b o o k s t h a t a r e e n t ir e ly o r p a rt i a ll y d e v o t e d t o t h e n o r m a l

s t r u c t u r e i n B a n a c h s p a c e s : W . A . K i r k [ 89 , 1 2 3, 1 2 4] , S . S w a m i n a t h a n [2 09 ], J . L . N e l s o n , K . L . S i n g h a n d

J . H . M . W h i t f i e l d [ 1 7 7 ] , K . G o e b e l a n d o t h e r s .

C h a p t e r 2

T h e M e t h o d o f A s y m p t o t i c C e n t e r s

2 .1 . A s y m p t o t i c C e n t e r s o f B o u n d e d S e q u e n c e s .

D e f i n i t i o n 2 . 1 . 1 . F o r a s u b s e t K o f a B a n a c h s p a c e X a n d f o r a b o u n d e d s e q u e n c e { x n } in X t h e

n u m b e r

A R ( K , { x n } )

= i n f { l i m s u p l l x - Y lI: Y E K }

~- ' -+ Oo

a n d t h e s e t

A C ( K ,

{ x,~ }) = { z E K : l i m s u p l [ z

- x n[ I = A R ( K ,

{ x n } ) }

n- - + oo

a r e c a l l e d , r e s p e c t i v e l y , t h e asy mpto t i c rad ius a n d t h e asy mpto t i c c e n t e r o f t h e s e q u e n c e { x n } w i t h r e s p e c t

t o K .

T h e n o t i o n o f a s y m p t o t i c c e n t e r w a s i n t r o d u c e d b y M . E d e l s t e in [7 6, 77 ] t o p r o v e t h e o r e m s o n f ix e d

p o i n t s o f n o n e x p a n s i v e m a p p i n g s a c t i n g o n u n i f o r m l y c o n v e x B a n a c h s p a ce s .

O b v i o u s l y , t h e a s y m p t o t i c c e n t e r

A C ( K ,

{ x n } ) i s a b o u n d e d c o n v e x a n d c lo s e d s e t , p r o v i d e d t h a t K is

c o n v e x a n d c l o se d . F u r t h e r m o r e ,

A C ( K ,

{ x n } ) y6 @ i f K i s w e a k l y c o m p a c t .

W e n o w i n d i c a t e s o m e p r o p e r t i e s o f a s y m p t o t i c c e n t e r s ( s e e [1 31 , 1 54 , 1 5 5, 2 2 3 , 8 9 ]) .

S t a t e m e n t 2 . 1 . 2 .

Le t X be a B an ac h space , K be a non e mp ty c onv e x c lose d subse t o f X , and

{ x n }

be

a bounde d se quenc e in X . The n the f o l l owing are true :

1)

A C ( K ,

{ x n } )

i s a non e m pty c onv e x we ak ly compac t se t , i~ K i s a bounde d we ak ly c ompa c t se t ;

2 )

A C ( g , { x n } ) = A C ( K , { y n } ) , i f i i x , ,- y n ] ] --+ 0;

3 )

there ex is ts a subsequence

{ z n , } C { x n }

s uc h th at g R ( K , { x n o } ) = d R ( K , { x n , } ) a nd

A C ( K ,

{ x n , } )

C A C ( K ,

{ x n 0 } ) f o r

every subsequence

{ X n 0 } C { x n , }

( a n d w it h t hi s A C ( K ,

{ x , , , } )

= d e ( K , { z~ 0 }) i f Z i s (UC EO ) , and Ix i s a bounded weakly comp act se t) ;

784





T h e o r e m 2 . 2 . 1 . L e t X , K , T b e a s a bo ve , a n d T ( O K ) C K . T h e n T h a s a f i x e d p o i n t i n th e f o l l o w i n g

cases :

1) X i s a Hi lb e r t space , a nd Ix i s a c losed ba l l ( Br o w d e r [ 3 7 ] ) ;

2 ) X i s u n i f o r m l y c o n v e x , a n d T i s d e f in e d i n a n e i g h b o r h o o d o f K ( Br o w d e r [ 4 1 ] ) ;

3 ) X i s r e f le x i ve , a n d K h a s a N S (Kirk [121]) .

I n t h e i r g e n e r a l i z a t io n o f t h e c la s si ca l S c h a u d e r - T y c h o n o v t h e o r e m o n t h e f i x e d p o in t o f a c o n t i n u o u s

m a p p i n g f r o m a c o n v e x c o m p a c t s e t in t o it se lf , B . R . H a l p e r n a n d G . M . B e r g m a n [ 10 0 ] u s e d t h e n o t i o n o f

i n w a r d m a p . L e t X , K C X , T : K --+ X b e as a b o v e . L e t x E K . T h e n t h e s e t I h ' ( x ) = { ( 1 - t ) x + t y : y E

K , t / > 0 } i s c a l l e d t h e

i n n e r s e t

f o r x w i t h r e s p e c t to K . T h e s e t

I g ( x )

is t h e u n i o n o f al l r a y s e m a n a t i n g

f r o m x a n d p a s s i n g t h r o u g h s o m e o t h e r p o in t y E K . S i n c e K is c on v e x , Iho(x ) = {(1 - t ) x + t y : y E

K , t > . 1}.

D e f i n i t i o n 2 . 2 . 2 [100]. A ma pp ing T : K --+ X i s ca l l ed :

1) i n w a r d o n K , i f f o r e v e r y x E K , T x E I h ' ( x ) ;

2 )

w e a k l y i n w a r d

o n K , i f f o r e v e r y x E K ,

T x E I K ( z ) .

I t is o b vi o u s, t h a t t h e c o n d i t i o n o f w e a k i n w a r d n e s s is m o r e g e n e r a l t h a n t h e R o t h e c o n d i t i o n o r t h e

c o n d i t io n t h a t ( x , T x ] M g r 0 f o r e v e r y z E O h ' , w h e r e ( x , T x ] = { x + t ( T x - x ) : 0 < t <~ 1}.

S . R e i c h [ 1 8 7] e x t e n d e d t h e r e s u l t o f K i r k ( s ee 2. 2 .1 . (3 ) ) t o t h e c a s e o f n o n e x p a n s i v e m a p p i n g T : K --+ X

w h i c h i s i n w a r d o n K . K . C h r i s ti [ 52 ] , g e n e r a l i z in g t h e r e s u l t o f R e i c h , p r o v e d t h e f o l l o w i n g t h e o r e m .

T h e o r e m 2 . 2 . 3 . L e t X b e a B a n a c h s p a ce a n d K b e a n o n e m p t y c o n v e x c l o s ed ( b u t n o t n e c e s s a r i l y

b o u nd e d) s u b s e t o f X w i t h th e F P P . T h e n e v e ry n o n e x p a n s i v e m a p p i n g T : K - ~ X , w h i c h is w e a k l y i n w a r d

o n K , h a s a f i x e d p o i n t .

R e m a r k 2 . 2 . 4 . A l t h o u g h in T h e o r e m 2 .2 .3 t h e s e t K is n o t a s s u m e d t o b e b o u n d e d , t h e p r e s e n c e o f

t h e F P P f o r K c a n l e a d to t h e b o u n d e d n e s s o f K . T h u s , f o r e x a m p l e , i n o r d e r t h a t a n o n e m p t y c o n v e x a n d

c l o s e d s u b s e t o f a H i l b e r t s p a c e h a v e t h e F P P , i t is n e c e s s a r y a n d s u f f ic i e n t t h a t i t b e b o u n d e d ( R a y [1 86 ]) .

W e n o w i n d i c a t e s o m e w o r k s i n w h i c h t h e r e a r e g i v e n e x i s t e n c e c o n d i t i o n s fo r f i x e d p o i n t s o f n o n e x p a n s i v e

m a p p i n g s d e f i n e d o n u n b o u n d e d s e t s: [ 12 9, 5 1 ,1 6 9 ] .

W e n o w p r o c e e d t o t h e e x p o s i ti o n o f r e s u lt s c o n c e r n i n g m u l t i - v a l u e d m a p p i n g s .

L e t ( X , If I I) b e n B a n a c h s p a c e a n d P ( X ) b e t h e c o l le c t io n o f a ll n o n e m p t y b o u n d e d c l o se d s u b s e ts o f

X . A m a p p i n g T : X -+ P ( X ) i s ca l l ed a m u l t i - v a l u e d m a p p i n g f r o m X i n t o X . T h e H a u s d o r f f m e t r i c o n

P ( X )

is d e f i n e d b y t h e f o r m u l a

H ( A 1 , A 2 ) = m a x ~ s u p i nf ] ] x - y l I , s up i n f I I x - y i ] }

kxE A1 yEAs xEA2 yEA1

w h e r e Aa A 2 E P ( X ) . I n t h e m e t r i c s p a c e ( P ( X ) , H ) l im A , = A m e a n s t h a t l im H ( A , , A ) = O. A

c o n n e c t i o n b e t w e e n t h e n o r m 11. ]1 o n X a n d t h e m e t r i c H o n P ( X ) i s e s t a b l i s h e d b y t h e f o l l o w i n g s i m p l e

f a c t .

S t a t e m e n t 2 . 2 . 5 .

T h e f o l l o w i n g a r e t r u e :

1) L e t A 1 , A 2 e P ( X ) . T h e n f o r e a c h x E A 1 a n d ~ > 0 th e r e i s y E A 2 s u c h t h a t f ix - Y ll <<-

H ( A 1 , A 2 ) + e .

2) L e t z n I1 11z i n ( X , ] ] - I I ) , A n H A in ( P ( Z ) , g ) , a n d z n E A n f o r n = l , 2 , . . . . T h e n z E d .

I n p a r t i c u l a r , i t f o ll o w s f r o m 2 . 2 .5 t h a t f o r a c o m p a c t s e t A 2 a n d f o r a n y z E A 1 t h e r e i s y E A 2 s u c h

that IIx - y]l ~< H ( A 1 , A 2 ) . L e t D C X . A p o i n t z E D i s c a l l e d a f i x e d p o i n t o f a m u l t i - v a l u e d m a p p i n g

T : D --+ P ( X ) i f x E T x . Fi x T i s t h e s e t o f al l f i x e d p o i n t s o f T .

D e f i n i t i o n 2 2 6 A m u l t i - v a l u e d m a p p i n g T : D - + P ( X ) i s s a i d t o b e n o n e z p a n s i v e i f H ( T x , T y ) <~

[Ix - y l l for al l x , y E D.

J . T . M a r k i n o b t a i n e d t h e f i r st r e s u lt o n f i x ed p o i n ts o f n o n e x p a n s i v e m u l t i - v a l u e d m a p p i n g s , g e n e r a li z in g

a t h e o r e m o f B r o w d e r [ 38 ].

786



T h e o r e m 2 . 2 . 7 [ 17 0] .

Let X be a real Hilbert space, B be a closed bali in X, Kc (B) be the fami ly of

nonemp ty convex compact subsets of B, and T : B --~ ICe(B) be a nonexpansive multi-va lued mapping.

Then

F i x T # ~ .

M a r k i n u s e d a m u l t i - v a l u e d a n a l o g o f t h e c o n t r a c t i o n m a p p i n g p r i n c i p l e d u e t o B a n a c h .

T h e o r e m 2 . 2 . 8 ( J . T . M a r k i n [ 17 0], S . B . N a d l e r [1 7 6] ).

Let (M,p) be a complete metric space and

T : M --+ P ( M ) be a contractive multi -valued mapping, that is, for a fixed

k G (0 , 1)

and x, y E M we have

H( Tx , T y) <~ kp(x , y).

Then

F i x T # 0 .

g

H e r e

H ( A 1 , A 2 ) =

m a x ~ s u p i n f

p(x,y),

sup in:[

p(x ,y )~

fo r

At ,A2 E P(M).

k x 6 A y E A = z 6 A = y 6 A I

R e m a r k 2 . 2 . 9 . T h e r e s u l t s o n f ix e d p o i n t s o f c o n t r a c t i v e m u l t i -v a l u e d m a p p i n g s , w i t h p r o o f s , c a n b e

f o u n d i n t h e b r i l l i a n t s u r v e y [ 1 1 ] b y A . A . I v a n o v .

B r o w d e r ( s ee [4 3]) h a s g e n e r a li z e d T h e o r e m 2 .2 . 7 t o t h e c a s e w h e r e B is a n a r b i t r a r y n o n e m p t y b o u n d e d

c o n v e x c l o se d s u b s e t o f a r e f le x i ve B a n a c h s p a c e w i t h a w e a k l y c o n t in u o u s d u a l i t y m a p p i n g . T h i s r e s u l t

w a s , i n i t s t u r n , g e n e r a l i z e d b y L a m i D o z o [ 1 4 0 ] .

T h e o r e m 2 . 2 . 1 0 .

Let X be a Banach space satisfying the Opial condition, M be a nonempty convex

weakly compact subset of X , and T : M --+ K( M ) be a nonexpansive mapping, where K ( M ) is the fami ly

of nonempty compact subsets of M provided with the Hausdor ff metric. Then

F i x T 7~ ~ .

T h e p r o o f o f t h e t h e o r e m i s b a s e d o n t h e i m p o r t a n t p r o p e r t y o f b e i n g d e m i c l o s e d f o r t h e m a p p i n g I - T ,

w h e r e I : X - + X i s t h e i d e n t i t y m a p p i n g .

S t a t e m e n t 2 . 2 . 1 1 [1 40 ].

Let X and M be the same as in

2 .2 .10 ,

and T : M -+ K ( X ) be a nonexpanslve

multi-valued mapping. Then [ - T is demicIosed, i.e., for every sequence

{ x , }

C M and any y,~ E (I -

T)x, , n = 1 , 2 , . . . , such that x , ~ ~ x, yn I 1 1 1 ~ Y, we have y E ( I - T)x .

P r o o f o f t h e s t a t e m e n t . L e t { x,~ } C M and yn E (I -- T)x,~ b e s u c h t h a t x ,~ w z a n d y = --+ y . I t i s

o b v i o u s t h a t x E M . S in c. e

y E x, - Tx ,

w e o b t a i n

y , = x , - v , , v ~ 6 T x n . ( 2 . 2 .1 )

t

B y v i r t u e o f S t a t e m e n t 2 . 2. 5 , t h e r e is a v, , 6 T z s u c h t h a t

I I v . - v ll • H T x . ,T x ) < . I t x . - x l l .

(2 .2 .2 )

F r o m ( 2 . 2. 1 ) a n d ( 2 . 2 .2 ) , p a s s i n g t o t h e l i m i t w i t h r e s p e c t t o n , w e o b t a i n

l i m i n f l l : r . - x l l / > l i m i n f l l v . - v ~ [ I = l i m i n f l l x . - y . - v ~ l l -

n } o o

n,---} oo n ---}13o

(2 .2 .3 )

S i n c e t h e s e t Tx i s c o m p a c t w e c a n a s s u m e p a s s i n g t o a s u b s e q u e n c e , i f n e c e s s a ry , t h a t v, -+ v E Tx. T h e n

(2 .2 .3 ) y i e l d s

l i m i n f l l x , - x l l / > l i m i n f l l z , - Y - v ii -

t l

S i n c e X s a t i sf i e s t h e O p i a l c o n d i t i o n a n d x ~ ~ x th i s y ie l d s x = y + v . T h u s

y = x - v E x - Tx. []

P r o o f o f T h e o r e m 2 . 2 . 1 0 . L e t x 0 6 M b e a f i x e d e l e m e n t , a n d f o r m = 1 , 2 , . . . , 0 < t i n , t ,n [ 1 , x E M

T,,x = tmTx +

(1 -

tm)zo.

(2 .2 .4 )

T h e n

T,n : M -+ K ( M )

i s a c o n t r a c ti v e m u l t i - v a lu e d m a p p i n g . B y T h e o r e m 2 . 2. 8 , t h e r e i s a n z m E

w

T ~ x ~ , m = 1 , 2 , . . . . S i n c e m i s w e a k l y c o m p a c t , t h e r e i s a s u b s e q u e n c e { x ' } C { a m } s u c h t h a t x ,

z E M . F r o m ( 2 . 2. 4 ) w e d e r i v e

x . = t . v . + ( 1 - t . > o , v . e

787



T h u s

l l x

-

v . l l

=

( 1

-

t . ) l l x 0

- v , ll. I n par t i cu la r , d i s t (x ~ , T x~ ) --+ 0 as n --+ ~ . T h e n y ,~ = x ,~ - v , E

( I - T ) x n a n d y n - ~ 0. B y v i r t u e o f S t a t e m e n t 2 . 2 .1 1 , w e h a v e 0 E ( I - T ) z , i .e . , x E T x . [ ]

A s s a d a n d K i r k [ 2 4 ] s h o w e d t h a t t h e c o n c lu s io n o f T h e o r e m 2 .2 .1 0 r e m a i n s v a l i d i f w e a s s u m e t h a t

T : M -+ K ( X ) a n d T x C M f o r e a c h x E O M . F u r t h e r m o r e , it is s u f fi c ie n t t h a t M b e a w e a k l y c o m p a c t

s t a r - s h a p e d s e t h e r e ( [ 1 0 6 ] ) .

I n t h e c a s e w h e r e X i s a s e p a r a b l e r e f le x i ve s t r i c t l y c o n v e x B a r t a c h s p a c e w i t h a w e a k l y c o n t i n u o u s

d u a l i t y m a p p i n g ( fo r e x a m p l e , Ip , 1 < p < ~ ) o n e c a n r e p l a c e t h e f a m i l y K ( M ) i n T h e o r e m 2 . 2 . 1 0 b y

P c ( M ) , w h e r e P c ( M ) i s t h e f a m i l y o f n o n e m p t y b o u n d e d c o n v e x c l o s e d s u b s e ts o f M ( s e e [ 17 1] ).

I n T h e o r e m 2 . 2 . 1 0 , it is s u f f i c i e n t t o r e q u i r e (s e e [ 17 3] ) t h a t T : M --+ K ( X ) a n d T x C I ~M (X ) =

{ x + t ( y - - x ) : y E M , T >>. 1 } f o r e v e r y x E M , a n d M b e a w e a k l y c o m p a c t s t a r - s h a p e d s e t (i .e .,

t h e r e e x is t s x 0 E M s u c h t h a t [ x 0 ,x ] C M f o r a n y x E M ) . I n t h e p r o o f o f t h i s re s u l t , t h e r e w a s

u s e d t h e f a ct t h a t f o r a n y a r b i t r a r y n o n e m p t y c lo s e d s u b s e t A o f a B a n a c h s p a c e Y a n y m u l t i - v a l u e d

c o n t r a c ti o n m a p p i n g T : A --+ K ( Y ) s u c h t h a t T z C I 'A(X ) f o r e v e r y x E A , h a s a f ix e d p o i n t . A n d

w i t h t h i s , t h e c o n d i t i o n T x C IrA(X) ' c a n n o t b e r e p l a c e d b y t h e c o n d i t i o n T x C I A ( x ) ( e x a m p l e :

X = R 2 , A = { x E X : Ilxi] = l } , T x - { 0 } ) .

L i m [ 1 5 1 ] , u s i n g t h e a s y m p t o t i c c e n t e r s t e c h n i q u e s a n d t r a n s f i n i t e i n d u c t i o n , o b t a i n e d t h e f o l l o w i n g

r e s u l t .

T h e o r e m 2 . 2 . 1 2 . Le t X b e a ( U C E D ) B a n a c h s pa ce , M a n o n e m p t y c o n v e x we a k l y c o m p a c t s u b s e t o f

X , a n d T : M --+ K ( M ) a n o n e x p a n s i v e m u l t iv a l u e d m a p p i n g . Th e n F i x T r ~ .

I n p a r t i c u l a r , t h i s t h e o r e m i s v a li d f o r e v e r y u n i f o r m l y c o n v e x B a n a c h s p a c e .

I n [ 15 3] , L i m o b s e r v e d t h a t i n T h e o r e m 2 . 2 .1 2 it is s u f f i c i e n t t o r e q u i r e t h a t T : M --+ K ( X ) a n d T x C M

f o r e v e r y x E O M .

G o e b e l [ 8 6] a n d L i r a [ 1 52 ] h a v e s i m p l i fi e d t h e p r o o f o f T h e o r e m 2 .2 . 12 .

L e t M b e a n o n e m p t y w e a k l y c o m p a c t c o n v e x s u b s e t o f a B a n a c h s p a c e X a n d x i a b o u n d e d s e q u e n c e i n

Z . L e t r { x i} = A R ( M , { x / } ) , d { x i } = A C ( M , { z i } ) b e t h e a s y m p t o t i c r a d i u s a n d t h e a s y m p t o t i c c e n t e r

o f t h e s e q u e n c e { x i} w i t h r e s p e c t t o M . T h e a s y m p t o t i c c e n te r i n a ( U C E D ) B a n a c h s p a c e c o n si s ts e x a c t ly

of a s ing le po in t ( see [62] ).

I t is o b v io u s , t h a t i f t w o s e q u e n c e s d i f f e r o n l y b y a f i n it e n u m b e r o f t e r m s , t h e n t h e i r a s y m p t o t i c r a d i i

a n d t h e a s y m p t o t i c c e n t e r s c o i n c id e . M o r e o v e r , f o r e v e r y s u b s e q u e n c e { y i } o f a s e q u e n c e { x i } w e h a v e

r { y }

D e f i n i t i o n 2 . 2 . 1 3 [8 6]. A s e q u e n c e { x i } i s sa id to be :

1) regular , i f e v e r y s u b s e q u e n c e o f it h a s t h e s a m e a s y m p t o t i c r a d i u s a s { x i } ;

2) a l m o s t c o n v e r g e n t , i f e v e r y s u b s e q u e n c e o f i t h a s t h e s a m e a s y m p t o t i c c e n t e r a s { x i } .

S t a t e m e n t 2 . 2 . 1 4 [ 86 , 1 52 ]. The fo l lowing are t rue:

1) every bounded sequence {x i } in a Banach space con ta ins a regu lar subsequence;

2) i f X i s a ( UC ED ) B an ach space , the n every regu lar sequence i s a lmo s t co nvergen t .

P r o o f . ( 1) . F o r a n a r b i t r a r y b o u n d e d s e q u e n ce { z i } p u t r 0 { z i } = i n f { r { v i } : { v i } i s a s u b s e q u e n c e o f

{z i }} . O b v i o u s l y , r 0 {z i } ~< r {z i } , a n d , i f { w i } i s a s u b s e q u e n c e o f { z i } , t h e n r o { z i } <~ r o { w i } .

L e t { x i } b e a b o u n d e d s e q u e n ce . W e s h a ll c o n s t r u c t a f a m i l y o f s e q u e n c e s { x ~ } , n = 1 , 2 , . . . , s a ti s fy i n g

1

t h e f o l l o w i n g c o n d i t i o n s : {x ~} = {z i } ; {x ~ + 1 } is a s u b s e q u e n c e o f {z ~ } f o r w h i c h r {z ~ + 1 } ~< r 0 {x ~ } + - .n

I t i s e a s y t o s e e t h a t t h e d i a g o n a l s e q u e n c e { x l} is re g u l a r .

( 2 ) L e t { x i } b e a r e g u l a r s e q u e n c e a n d z = A { z i } . D e n o t e b y r t h e c o m m o n v a lu e o f t h e a s y m p t o t i c

r a d i i o f t h e s u b s e q u e n c e s o f t h e { z i } . S u p p o s e t h a t t h e r e e x i st s a s u b s e q u e n c e { y i} C { z i } f o r w h i c h

A { y i } = y , y r z . T h e n l i ms u p l l y i - z ll ~< l i ms u P l l x i - z ll = r , l i ms u p l lY i - z l l = r . F r o m t h e u n i f o r m

i-- oo i- - ~ i--~eo

c o n v e x i t y i n e v e r y d i r e c t i o n o f t h e s p a c e X w e h a v e

l i m s u p yi Y + z

i--+oo 2 < r = r{ y i} .

T h i s p r o v i d e s a c o n t r a d i c t i o n . [ ]

788



P r o o f o f T h e o r e m 2 . 2. 1 2 . U s i n g T h e o r e m 2 .2 .8 , w e s e le c t a s u b s e q u e n c e { z i} C M f o r w h i c h d is t ( x i,

T x i ) - + 0 a s i --+ ~ . B y v i r t u e o f S t a t e m e n t 2 .2 .1 4 , w e m a y a s s u m e t h a t { z i } i s r e g u l a r a n d h e n c e a l m o s t

c o n v e rg e n t . L e t z = A { z ~ } , r = r { z i } . F o r e a c h z , , i = 1 , 2 , . . . , find yi E T z i s u c h t h a t Ilz~ - yjl[ ~ 0 as

i ~ ~ . F o r e a c h y i , i = 1 , 2 , . . . f i n d z i E T z f o r w h i c h Ilyi - z ~ l l ~< H ( T z i , T z ) <~ [ Ix i - z l l . T h i s i s p o s s i b l e

b e c a u s e t h e s e t s T z i a n d T z a r e c o m p a c t . hoose a s u b s e q u e n c e {zi~ } s o t h a t z i k -+ v E T z . S i n c e { z i }

i s r e g u l a r , r { x i , } = r { z i } = r , a n d A { z i k } = A { z i } = z . T h e i n e q u a l i t i e s

Ilzi~ - vii ~< IIv - z ,~ II §

I l zi ~ - y i ~

II § Ily i~ - z , . I1 ,

I l z i k

y i ~

II ~< l ie - z ~ II

i m p l y t h a t

l i m s u p l l z i k - v i i ~ < l i m s u p l l z i ~ - z l l r .

k-- co k-- oo

He n c e , v = A { x i k } = z , a n d z E T z . [ ]

R e m a r k 2 . 2 . 1 5 . I n T h e o r e m s 2 . 2. 10 a n d 2 .2 .1 2 , t h e c o n d i t io n T : M - + I f ( M ) c a n n o t b e r e p l a c e d

b y t h e c o n d i t i o n T : M --+ I f ( X ) a n d M f - I T z ~ 0 fo r e v e ry z E M . T o s e e t h i s , l e t X = R 2 t i le

E u c l i d e a n p la n e , M = { x E X :

l z l l ~ <

1 } , a n d T : M ~ K ( X ) b e s u c h

t h t

T z = { y E X :

I l y

-

z l l = ~ }

f o r e v e r y x E M , w h e r e e E ( 0 , 1 ) i s a f i x e d n u m b e r . T h e n H ( T z , T y ) =

I I z - y l l

fo r a l l z , y E X . B u t

F i x T = ~ , a lt h o u g h M N T z # 0 f o r e a c h z E M . O b s e r v e t h a t h e r e T z r I M ( Z ) . T h i s e x a m p l e i s t a k e n

f r o m [ 8 9 1

T h e n e x t r e s u l t s t i m u l a t e d t h e d e v e l o p m e n t o f i n v e s t ig a t i o n s o n t h e f i xe d p o i n t s o f i n w a r d n o n e x p a n s i v e

m u l t i - v a l u e d m a p p i n g s .

T h e o r e m 2 . 2 . 1 6 [6 6, I 8 9] . L e t C b e a c o n v e z c l o s e d s u b s e t o f a B a n a c h s p a c e X , a n d T : C - -+ K ( X )

be a m u l t i - v a l u e d c o n t r a c t i o n m a p p i n g w h i c h i s w e a k l y in w a r d o n C ( i .e . , T x C I c ( x ) f o r e v e r y x E C ,

w he re I v ( z ) = { x + t ( y - x ) : y E C , t ) 0 } ) . T h e n F i x T # 0 .

A s w a s o b s e r v e d in [ 67 ], in T h e o r e m 2 . 2. 1 6 , i n s t e a d o f t h e c o n d i t i o n T : C - + K ( X ) i t i s su f f ic ien t

t o t a k e t h e c o n d i t i o n T : C --+ E ( X ) , w h e r e E ( X ) = { A C X : A i s a n e x i s t e n c e s e t} ( a n o n e m p t y s e t

A C X i s c a l l e d a n e x i s t e n c e s e t i f f o r e v e r y z E X t h e r e i s a n a E A s u c h t h a t I I z - a l l - - d a s t x ,

A ) ) .

n p a r t i c u l a r , i n s t e a d o f T : C --+ K ( X ) i t i s s u f f ic i e n t t o t a k e T : C - + K w ( X ) , w h e r e K ~ ( X ) is

t h e t o t a l i t y o f a ll n o n e m p t y w e a k l y c o m p a c t s u b s e t s o f X , s i nc e e v e r y n o n e m p t y w e a k l y c o m p a c t s e t i s a n

e x i s t e n c e s e t .

S . Z h a n g [ 2 2 7 ] h a s g e n e r a l i z e d T h e o r e m 2 . 2 . 1 6 .

T h e o r e m 2 . 2 . 1 7 . L e t X b e a B a n a c h s pa c e, C a n o n e m p t y c l o s e d (b u t n o t n e c e s s a r i l y c o n v e x ) s u b s e t

a n d T : C - + K ( X ) b e a m u l t i - v a l u e d c o n t r a c ti o n m a p p i n g s u c h t h a t T z M I v ( z ) # 0 f o r e a c h z E C , w h e r e

I b ( z ) = { z + t ( y - x ) : y E C , t ) 1 } . T h e n F i x T # O .

N o t e t h a t i f C i s c o n v e x , t h e n I ~ c ( Z ) = I t ( z ) f o r e v e r y z E C . M o r e o v e r , f o r t h e v a l i d i t y o f t h e

T h e o r e m i t i s s u f f ic i e n t t o p u t

{ z ~ T x :

I I z - z ll = d i s t ( z , T z ) } n _ r b( z ) # e i n s t e a d o f t h e

c o n d i t i o n

T z n I b ( z ) # e

( s e e [ 2 2 7 1 )

G e n e r a l i zi n g T h e o r e m 2 . 2. 1 2 , L i m [ t 54 ] a n d K u c z u m o w [1 33 ] p r o v e d t h e f o l l o w i n g r e s u l t.

T h e o r e m 2 .2 . 1 8 . L e t X b e a ( U C E D ) B a n a c h s p a c e, C b e a w e a k l y c o m p a c t s u b s e t o f X , T : C - + K ( X )

b e a n o n e z p a n s i v e m u l t i - v a l u e d m a p p i n g w h i c h is w e a k l y i n w a r d o n C . T h e n F i x T r 0 .

E a r li e r, t h is t h e o r e m w a s p r o v e d b y R e i c h [ 1 89 ] f o r th e c a s e o f a n i n w a r d m a p p i n g , g e n e r a l i z i n g t h e

c o r r e s p o n d i n g r e s u l t o f D o w n i n g a n d K i r k [6 6], o b t a i n e d f o r t h e u n i f o r m l y c o n v e x s p a c e s .

S t a t e m e n t

2 . 2 . 1 9 [ 1 5 4 ] . L e t X a n d C b e t h e s a m e a s i n 2 .2 .18 , a n d { z n } b e a b o u n d ed s e q ue n c e i n X .

I f z i s th e a s y m p t o t i c c e n t e r o f { Z n } w i t h re s p ec t t o C , t h e n z i s a l so t h e a s y m p t o t i c c e n t e r o f { z n } w i t h

r e s p e c t t o I v ( z ) .

P r o o f . O b s e r v e t h a t t h e a s y m p t o t i c c e n t e r in a ( U C E D ) s p a c e c o n s i st s o f a s in g l e p o i n t i f i t i s n o t

e m p t y . L e t y b e t h e a s y m p t o t i c c e n t e r o f { z , } w i t h r e s p e c t t o l c ( z ) a n d y ~ z . S i n c e C C I v ( z ) , w e h a v e

y ~ z c ( z ) \ c a n d r ( y ) < r ( z ) ( h e r e r ( z ) = l i m s u p l l z n - z l l, z ~ X ) . T h e f u n c t i o n r ( . ) is c o n t i n u o u s s i n c e

11---+OO

789



I r )

- d v ) l - v ii fo r a l l u , v e X . H e n c e t h e r e i s z E I t x ) \ C SU Ch t h a t r z ) < r x ) . T h e r e f o r e ,

z = 1 - a ) x + a w f or s o m e w e C , a > l , a n d b y t h e c o n v e x i t y o f r ( - ) , r w ) = r l z + 1 - 1 ) x ) <<.

~ r z ) + 1 - ~ ) r x ) < r x ) . T h i s c o n t r a d i c t s t h e d e f i n i ti o n o f t h e e l e m e n t x . T h u s y = x . [ ]

P r o o f o f T h e o r e m 2 .2 .1 8 . L e t x0 b e a f ix e d e l e m e n t o f C . B y T h e o r e m 2 .2 .1 6 , th e c o n t r a c t i o n m a p p i n g

T , : C - -4 K X ) , w h e r e T , x = (1 - t , ) x o + t n T x fo r a l l x e C, 0 < t~ , tn 1 1 , ha s a f ix ed po in t x ,~ , i . e .,

x n E T ~ x , , n = 1 , 2 , . . . . B y v i r t u e o f S t a t e m e n t 2 . 2 .1 4 , o n e c a n a s s u m e t h a t { x,~ } i s a r e g u l a r a n d a l m o s t

c o n v e r g e n t s e q u e n c e . L e t z b e t h e a s y m p t o t i c c e n t e r o f { x , } w i t h re s p e c t t o C . F o l lo w i n g t h e p r o o f of

T h e o r e m 2 . 2 . 1 2 , w e f i n d a n e l e m e n t v E T z s u c h t h a t f o r s o m e s u b s e q u e n c e { x ~ k } C { x n } w e h a v e

l im su pl l{ x, ~ } - v ii ~< l i ra sup l l{x,~ } - zl l.

k-- ~ k--- ~

S i n c e, b y 2 . 2 .1 9 , t h e p o i n t z i s t h e a s y m p t o t i c c e n t e r o f { x , } w i t h r e s p e c t t o I v z ) , a n d v E I v z ) w e

o b t a i n v = z ( s i n c e t h e a s y m p t o t i c c e n t e r i n a ( U C E D ) s p a c e c o n t ai n s n o m o r e t h a n o n e e l e m e n t ) . H e n c e

z E T z . [ ]

R e t u r n i n g t o t h e s p a c e s s a t i s f y i n g t h e O p i a l c o n d i t i o n , n o t e t h a t K . Y a n a g i [ 2 2 3 ] h a s e x t e n d e d T h e o r e m

2 .2 .1 0 t o t h e c a s e o f a w e a k l y i n w a r d m a p p i n g .

T h e o r e m 2 . 2 . 2 0 . Le t X be a Ba na ch space sa t i s f y ing the Opia l cond i t ion , M be a no ne m pty weakly

c o m p a c t c o n v e x s u b s e t o f X , a n d T : M --+ K X ) b e a we a k l y i n wa r d n o n e x p a n s i v e m u l t i - v a l u e d m a p p i n g ,

t h a t is , T x C I M x ) f o r e v e r y x E M . T h e n F i x T 7~ 0 .

T h e p r o o f o f t h i s t h e o r e m r e p e a t s t h e p r o o f o f T h e o r e m 2 .2 .1 0 w o r d f o r w o r d w i t h t h e o n l y d i f fe r e nc e

t h a t i n s t e a d o f T h e o r e m 2 .2 .8 o n e h a s t o u s e T h e o r e m 2 .2 .1 6.

S i n c e f o r a c o m p a c t s e t M i n a B a n a c h s p a c e , o bv i o u sl y ,

l i m i n f l l x . - x < l i m i n f l l x . - y l l

I1--~ oo 7 l- . oo

i f { x n } C M , x n ~ > x a n d y 7~ x , f r o m T h e o r e m 2 . 2. 2 0 w e o b t a i n

S t a t e m e n t 2 . 2 . 2 1 [ 2 2 3 ] . Le t M be a no ne mp ty co mpac t subse t in a Ba na ch space X and T : M --+

K X ) b e a n o n e x p a n s i v e m u l t i- v a l u e d m a p p i n g w h i c h i s . T h e n F i x T 7~ O.

I f, in T h e o r e m 2 .2 .2 0 , o n e a s s u m e s t h a t i n t M ~ 0 , t h e n t h e c o n d i t io n o f w e a k i n w a r d n e s s m a y b e

r e p l a c e d b y a w e a k e r b o u n d a r y c o n d i t i o n o f t h e L e r a y - S c h a u d e r t y p e .

T h e o r e m 2 . 2 . 2 2 ( C . M o r a l e s [ 17 5] ). Given a Ba nac h space X sa t i s f y in g the Opia l cond i t ion , M a

we a k l y c o m p a c t s u b s e t n o t n e c e s s a r i l y c o n ve x ) o f X a n d 0 E i n t M , T : M --+ K X ) a n o n e x p a n s i v e

m u l t i - v a l u e d m a p p i n g . I f T s a t i s fi e s t h e Le r a y - S c h a u d e r c o n d i ti o n :

A x ~ T x f or e a c h x E O M a n d A > 1,

t h e n Fi x T # O .

E a r l i e r , T h e o r e m 2 . 2. 2 2 w a s p r o v e d b y A s s a d a n d K i r k [ 2 4 ] i n t h e c a s e w h e r e X i s a H i l b e r t s p a c e a n d

M i s t h e c l o s e d u n i t b a l l .

S . Z h a n g [ 22 7], w i t h t h e a i d o f T h e o r e m 2 . 2 .1 7 , p r o v e d t h a t , i n T h e o r e m 2 . 2 . 2 0, i t is s u f f i c ie n t t o r e q u i r e ,

i n s t e a d o f t h e c o n v e x i t y o f X , t h a t M b e a s t a r - s h a p e d s e t.

W e n o w p r e s e n t t w o r e s u lt s a b o u t m a p p i n g s d e f i n e d o n u n b o u n d e d s e ts .

T h e o r e m 2 . 2 . 2 3 [175]. Le t X b e a U C ) B a n a c h s pa ce , M b e a c o n v e x c lo s e d s u b s e t o f X , 0 E M , T :

M --+ K M ) b e a n o n e x p a n s i v e m u l t i - v a l u e d m a p p i n g . I f t h e s e t o f e i g e n v e c t o r s { x 6 M : A x 6 T x f o r

s o m e A > 1} i s bounded, the n F i x T # O .

T h e o r e m 2 . 2 . 2 4 [67]. Le t X be a UC) Bana ch space, M a convex c losed subse t o f X , T : M --+ Kc M )

a n o n e x p a n si v e m u l t i- v a l u e d m a p p in g . I f / o r s o m e x o E M t he s e t { x e M :

I I x

- x 0 1 t

> / d i s t x , T x o ) }

is

bounded, then F i x T # ; 3.

T h e d e v e l o p m e n t o f r e s u l t s c o n n e c t e d w i t h T h e o r e m 2 . 2. 24 c a n b e f o u n d i n [ 49 , 16 8] .

790



W e n o w p r o c e e d t o t h e r e s u l ts o f K i r k a n d M a s s a o n f i x e d p oi n t s o f m u l t i - v a lu e d m a p p i n g s o f s pa c e s

w i t h c o m p a c t a s y m p t o t i c c e n t e r s .

L e t X b e a B a n a c h s p a c e , M b e a s u b s e t o f X , { x n } C X b e a b o u n d e d s e q u e n c e . R e c a l l t h a t

t h e s e q u e n c e { X n } i s s a id t o b e regular, i f r { x , ~ } = r{x,} f o r e v e r y s u b s e q u e n c e { x ,~ } C { x , } ( h e r e

r{z,} = AR(M, { z,~ }) is t h e a s y m p t o t i c r a d i u s o f t h e s e q u e n c e { z n } C X w i t h r e s p e c t t o M ) .

D e f i n i t i o n 2 . 2 . 2 5 [1 25 ]. A s e q u e n c e { x , } C X i s s a i d t o b e asymptotically uniform with respect to M, i f

t h e a s y m p t o t i c c e n t e r

A{x,k} = A{xn}

f o r e v e r y s u b s e q u e n c e

{znk} C

{x ,~} ( h e r e

A{z ,} = AC(M,

{z,~})

is t h e a s y m p t o t i c c e n t e r o f t h e s e q u e n c e { zn } C X w i t h r e s p e c t t o M C X ) . I f, m o r e o v e r , A { x , } c o n t a i n s

e x a c t l y o n e p o i n t , t h e n { X n} is c a l le d almost convergent.

S t a t e m e n t 2 . 2 . 2 6 [125]. Let M be a separable subset of a Banach space X . Then every bounded

sequence {xn} C X contains an asymptotically uni form subsequence with respect to M.

P r o o f . B y v i r t u e o f S t a t e m e n t 2 .2 .1 4 , w e m a y s u p p o s e t h a t { x} is r e g u l a r . S i n c e M is s e p a r a b l e , w i t h

t h e h e l p o f t h e d i a g o n a l p r o c e d u r e w e c h o o se a s u b s e q u e n c e { x,~ k} C { x,~ }, w h i c h w e a g a i n d e n o t e b y

{ x ~ } , s u c h t h a t l i m [[y - x ni[ e x is t s f o r e v e r y y e M . L e t { v , } b e a n a r b i t r a r y s u b s e q u e n c e o f {x ~ } a n d

n - -b o o

~ ( y ) = l i m s u p i ] y - V n [I. T h e m i n i m u m o f t h e f u n c t i o n ~ ( . ) i s r{v~} = r{x,}, a n d i t i s a t t a i n e d o n t h e s e t

n - - o o

A{v,} = AC(M, { v ~ } ) . T h u s f o r e v e r y y 9 A{v~} w e h a v e ~ ( y ) = lirn oo n y - x ~ l [ = r { x , ) = d R(M , { x ~ } ) .

H e n c e

d{v,} C A{x~}.

Si n c e {v , } C {x ~} f o r

y 9 A{x,}

w e h a v e r { v ~ } ~< l i m s u p i i Y - v ~ [] ~< l i m s u p i i Y -

x~l[ = r{xn} = r { v , } , i . e . , A{x~} C A{v~}. F r o m t h is w e c o n c l u d e t h a t d{v~ } = A{x n}. []

W e s h a l l n e e d t h e f o l lo w i n g r e su l t :

S t a t e m e n t 2 . 2 . 2 7 [99 ]. Let V be a nonempty convex compact set in a Banach space X, and T : V -+

(P c( X), H) be a continuous multi-valued mapping such that Tx M Iv(z) r ~ fo r every x E V. Then

F i x T # ~ .

I n p a r t i c u l a r , s i n c e V C Iv(x) , t h e c o n c l u s i o n o f t h e t h e o r e m i s v a l id f o r a n y n o n e x p a n s i v e m u l t i - v a l u e d

m a p p i n g T : V --+ Kc (X) s u c h t h a t Tx M V # ~ f o r e a c h x 9 V .

T h e o r e m 2 . 2 . 2 8 ( K i r k a n d M a s s a [1 28 ]). Given a Banach space X and a bounded convex closed subset

M of X such that every sequence in M has a nonempty asymptotic c e n t e r with respect to M. Then e v e r y

nonexpansive multi-valued mapping T : M --~ K c(M ) has a fixed point.

P r o o f . [ 1 3 5 ] ( s e e a l so [8 9]) . A s i n t h e p r o o f o f T h e o r e m 2 . 2. 1 0, c h o o s e a s e q u e n c e { x , } C M s u c h t h a t

d i s t ( x n ,

T x , ) -'+ 0 as n --~ oo.

B y v i r t u e o f S t a t e m e n t 2 . 2. 14 , w e c a n a s s u m e t h a t

{xn}

i s r e g u l a r . M o r e o v e r ,

i f M i s s e p a r a b le , w e c a n a s s u m e t h a t { xn } is a s y m p t o t i c a l l y u n i f o r m . W e n o w s h o w t h a t f o r a n o n s e p a r a b l e

M t h e r e i s a s e p a r a b l e s u b s e t M f o r w h i c h t h e c o n d i t i o n s o f t h e t h e o r e m h o l d . L e t V A(M, { X n } ) , w h e r e

A(M, { x , } ) = AC(M, { x n } ) , a n d d e f i n e t h e s e q u e n c e { M n } o f s u b s e t s o f M a s fo l lo w s :

M1 = c--0-fiV(VU { x , } ) ,

Mn+a = c-6-ffC(Mn U TMn) , n = 1 , 2 , . . . .

O0 N

P u t M = U M n . T h e n M i s a s e p a r a b le T - i n v a r i a n t s u b s e t o f M w h i c h c o n t a i n s { x ~ ) a n d V . T h u s

, l = 1

Y = A(M, {X n ) ) = A(M, { X n } ) . S i n c e { a n ) is r e g u l a r f o r e v e r y s u b s e q u e n c e { Y , } o f { x ,, }

v c { y n =

A M ,

y n .

H e n c e e v e r y s u b s e q u e n c e {Yn} h a s a n o n e m p t y c o m p a c t a s y m p t o t i c c e n t e r w i t h r e s p ec t to M . B y S t a t e m e n t

2 .2 .2 6 , t h e r e is a s u b s e q u e n c e { y n } C { x n } w h i c h i s a s y m p t o t i c a l l y u n i f o r m . T a k e a n a r b i t r a r y z E V =

A(M, { x , } ) a n d c o n s t r u c t s e q u e n c e s { y ,~ } , { z n }, { z ~ , } a n d a p o i n t v E Tz j u s t a s in t h e p r o o f o f T h e o r e m

2 .2 .1 2 . S i n c e t h e s e t V m a y c o n t a i n m o r e t h a n o n e p o in t , w e c a n n o t p r o v e , a s i n T h e o r e m 2 . 2 .1 2 , t h a t

z 6 F i x T . H o w e v e r , j u s t a s i n T h e o r e m 2 . 2 .1 2 , it is n o t h a r d t o v e r i f y t h a t v 6 A(M,{xnh}) = V. T h u s ,

f o r e v e r y e l e m e n t z o f t h e c o n v e x c o m p a c t s e t V w e h a v e

Tz M V ~ ~.

B y v i r t u e o f S t a t e m e n t 2 . 2 .2 7 ,

F i x T ~ . [ ]

791



C o r o l l a r y 2 . 2 . 2 9 [ 1 2 5 ] . I f M is a w* - c o m p a c t c o n v e x s u b s e t

o f l

and ?t: : M --+ K c M ) i s a nonex pan -

s i r e m u l t i - v a l u e d m a p p i n g , t h e n F i x T 0 .

P r o o f . L i ra [ 15 5] h a s s h o w n t h a t t h e a s y m p t o t i c c e n t e r w i th r e s p e c t to s u c h M o f e v e r y s e q u e n c e in M

i s a c o m p a c t s e t . [ ]

L i m ( M R 8 7 i: 4 7 0 6 8 ) o b s e r v e d t h a t i n 2 . 2 .2 9 i t i s s u f f i c i e n t t h a t T : M --+ K M ) s ince 11 sa t i s f i es the

w * - O p i a l c o n d i t i o n .

K i r k [ 1 25 ] h a s s h o w n t h a t e v e r y n o n e m p t y b o u n d e d c o n v e x c l os e d s u b s e t M o f a ( k - U C ) B a n a c h s p a c e

X p o s s e s se s t h e p r o p e r t y i n d i c a t e d i n T h e o r e m 2 . 2 .2 8 . M o r e e x a c tl y , f o r e v e r y s e q u e n c e { x , } C M t h e s e t

A M , { x n } ) i s n o n e m p t y , c o n v e x , c l o se d , a n d h a s d i m e n s i o n n o t g r e a t e r t h a n k - 1 . H e n c e w e h a v e

T h e o r e m 2 . 2 . 3 0 [1 25 ]. Le t X b e a k - U C) B a n a c h s p a c e f o r s o m e k E N , M b e a n o n e m p t y b o u n d e d

c o n v e x s u b s e t o f X , a n d T : M --+ Kc M ) b e a n o n e x p a n s i v e m u l t i - v a l u e d m a p p i n g . Th e n F i x T r 0 .

N o t e o n e r e s u l t fo r t h e O r l i c z s p a c e l ~ . L a m i D o z o [ 1 41 ] h a s s h o w n t h a t a n o n e x p a n s i v e m u l t i - v a l u e d

m a p p i n g T : M --+ K M ) h a s a f i x e d p o i n t i f M i s a b o u n d e d s t a r - s h a p e d s u b s e t o f l ~ w h i c h i s c o m p a c t i n

t h e t o p o l o g y o f p o i n t w i s e c o n v e r g e n c e a n d t h e f u n c t i o n ~ s a ti s fi e s a n a d d i t i o n a l c o n d i t i o n .

S o m e s u p p l e m e n t a r y i n f o r m a t i o n a b o u t n o n e x p a n s i v e m u l t i - v a lu e d m a p p i n g s c a n b e f o u n d i n [2 , 8 9,

222, 2001

2 . 3 U n i f o r m l y L i p s c h i t z i a n M a p p i n g s . L e t X b e a B a n a c h s p ac e , D b e a s u b s et o f X . A m a p p i n g

T : D --+ X is sa id to b e Li p s c h i t z w i t h c o n s t a n t k , i f fo r a l l z , y E D

I I T x - T y l l < k l l x - Y l I .

T h e n e x t e x a m p l e s h o w s t h a t t h e B r o w d e r - G h d e - K i r k t h e o r e m o n f ix e d p o i nt s o f a n o n e x p a n s i v e m a p -

p i n g ( i. e ., a L i p s c h i t z i a n m a p p i n g w i t h c o n s t a n t k = 1 ) i s n o t v a l i d f o r L i p s c h i tz m a p p i n g s w i t h c o n s t a n t

k > 1 a n d a r b i t r a r i l y c l o s e t o 1 .

E x a m p l e 2 . 3 . 1 [1 19 ]. L e t B b e t h e u n i t b a l l in l 2 . T h e s p a c e l ~ i s r e f l e x iv e a n d B h a s a n o r m a l

s t r u c t u r e . C o n s i d e r t h e m a p p i n g T : B --+ B d e f i n e d b y

T x l , X 2 , . . . ) = ( t( 1 - I l x l l ) , x l , x = , . - ) ,

w h e r e t h e c o n s t a n t t E ( 0 , 1 ). T h e n T i s L i p s c h i tz m a p p i n g w i t h c o n s t a n t k = x / 1 + t 2 a n d F i x T = 0 .

G o e b e l a n d K i r k [ 8 8] i n t r o d u c e d a c la s s o f m a p p i n g s t h a t is i n t e r m e d i a t e b e t w e e n n o n e x p a n s i v e a n d

L i p s c h i tz w i t h c o n s t a n t k > 1 , f o r w h i c h t h e f i x e d p o i n t t h e o r e m s ( o f B r o w d e r - G h h d e - - K i r k t y p e ) a x e s ti ll

va l id .

D e f i n i t i o n 2 . 3 . 2 [ 88 ]. A m a p p i n g T : D C X --+ X i s s a i d t o b e u n i f o r m l y k - L i p s c h i t z i f fo r a l l x , y E D

a n d n = - 1 , 2 , . . .

I I T x -

T Y l l

< k l l x - Y l l -

O b v i o u s l y , e v e r y n o n e x p a n s i v e m a p p i n g i s u n i f o r m l y 1 - L i p s c h i t z .

T h e f o ll o w in g t h e o r e m w a s o n e o f th e f ir s t fi x e d p o i n t r e s u lt s f o r a u n i f o r m l y k - L i p s c h i tz m a p p i n g .

T h e o r e m 2 . 3 . 3 [88]. G i v e n a U C) B a n a c h s pa ce X w i t h m o d u l u s o f c o n v e x i t y 6 . ) a n d A a n o n e m p t y

b o u n d ed c o n v e x a n d c l os e d su b s e t o f X . Le t T : A --+ A b e a u n i f o r m l y k - L i p s c h i t z m a p p i n g w i t h c o n s t a n t

k sa t i s f y ing the cond i t ion

k . l - 6 i / k ) ) i . 2 . 3 . 1 )

T h e n F i x T O .

N o t e t h a t ( 2 . 3 . 1 ) i m p l i e s t h a t

1

k < ~ , . , . ( 2 . 3. 2 )

1

H e r e i s a g e n e r a l i z a t i o n o f T h e o r e m 2 .3 .3 .

792



T h e o r e m 2 . 3 . 4 [90].

G i v e n a B a n a c h s p a c e X w i t h c h a r a c t e r i s t ic o f c o n v e x i t y ~ o ( X ) < 1 , a n d l e t

1 < 7 b e a s o l u t i o n o f t h e e q u a t i o n

3'(1 -

5 x ( 1 / 7 ) ) = 1 . I f A i s a n o n e m p t y b o u n d e d a n d c l o s e d s u b s e t o f

X , T : A ~ A i s a u n i f o r m l y k - L i p s c h i t z m a p p i n g , a n d k < 7 , t h e n

F ix T ~ O .

A s t r o n g e r r e s u l t i s d u e t o E . A . L i f s c h i t z [ 1 4 ] .

L e t ( M , p ) b e a B a n a c h s p a c e . I f a > 0 is a s u f f ic i e n t l y s ma l l n u m b e r ( f o r e x a m p le , a ~< 1 ), t h e n t h e r e i s

a n u m b e r b =

b(a)

> 1 s u c h t h a t f o r al l x , y E M a n d r > 0 f o r w h ic h

p( x, y) >1 r

t h e r e i s a z E M f o r w h ic h

B ( x , d r ) N B ( y , b r ) C B ( z , r ) .

D e f i n i t i o n 2 . 3 . 5 . T h e l e a s t u p p e r b o u n d o f t h e n u m b e r s a sa t is f y in g th e p r o p e r t y i n d i c a t e d a b o v e is

c a l l e d t h e

L i f s c h i t z c h a r a c t er i s ti c x ( M )

o f t h e m e t r i c s p a c e ( U , p ).

O b v i o u s l y , x ( M ) /> 1 .

I f X i s a n o n r e f l e x iv e B a n a c h s p a c e , t h e n t h e L i f s c h i t z c h a r a c t e r i s t i c o f a n y b a l l i n X i s e q u a l t o 1 . I f X

i s a H i lb e r t s p a c e , t h e n f o r a n y n o n e m p ty c o n v e x s e t M C X w e h a v e x ( iV / ) ) V ~ ( s e e [1 3 , 1 4 ]) .

S t a t e m e n t 2 . 3 . 6 [14].

L e t X b e B a n a c h s p a c e w i t h c h a r a c t e r i s t ic o f c o n v e x i ty ~ o ( X ) < 1 . T h e n f o r

a n y n o n e m p t y c o n v e x s u b s e t M C X

1

x ( M ) /> 1 - 5 ( 1 ) ' ( 2 . 3 .4 )

T h e o r e m 2 . 3 . 7 [1 4].

G i v e n a c o m p l e te m e t r i c s p a c e ( M , p ) a n d T : M --+ M a u n i f o r m l y k - L i p s c h i t z

m a p p i n g w i t h c o n s t a n t k < x ( M ) ( t h a t is , p ( T ' x , T " y ) <~ k p ( x , y ) f o r a l l x , y E M , n E N ) . A s s u m e t h a t

t h e t r a j e c t o r y

{Y0,

T y o , T 2 y o , . . . } o f s o m e p o i n t Y o E M i s b o u nd e d. T h e n

F i x T # O .

P r o o f . A b a ll

B ( x , r ) = { y E M : p ( y , x ) <<.r }

i s c a l l e d

r e g u l a r

i f i t c o n t a i n s t h e t r a j e c t o r y o f s o m e p o i n t .

A s e q u e n c e o f b a l l s B ~ = B ( x n , r n ) , n = 1 , 2 , . . . , i s c a l l e d

f u n d a m e n t a l

i f t h e i n t e r s e c t i o n s B , N B , ~ +I , n =

1 , 2 , . . . , a r e n o n e m p t y a n d i f

r n + l ~ # 9 r n ,

w h e r e # < 1 . O b v i o u s l y , t h e c e n t e rs x n , n = 1 , 2 , . . . o f a

f u n d a m e n t a l s e q u e n c e o f b a l l s f o r m a c o n v e r g e n t s e q u e n c e .

I f t h e r e i s a f u n d a m e n t a l s e q u e n c e o f r e g u l a r b a l ls , t h e n e v e r y n e i g h b o r h o o d o f t h e l i m i t x * o f i t s c e n t e r s

c o n t a in s t h e t r a j e c t o r y o f s o m e p o i n t . T h e r e fo r e ,

T x * = x * .

L e t k < a < x ( M ) a n d # E (0 , 1 ) b e a n u m b e r s u c h t h a t "7 = m i n { # 9

b(a) ,

# a k - 1 } > i . W e w i l l s h o w

t h a t f o r e v e r y re g u l a r b a l l

B x , r )

o n e c an c o n s t r u c t a b a l l B ( x l , # r ) c o n t a in i n g t h e t r a j e c t o r y o f s o m e p o i n t

f r o m B z , r) c o n t a i n e d i n e z , r ) . T h i s w i ll c o m p l e te t h e p r o o f.

L e t

B z o , r o )

b e a r e g u l a r b a ll . P u t 5 = i n f { r > 0 :

B xo , r)

i s r e g u l a r } . W e n e e d to c o n s i d e r

o n l y

t h e

c a s e w h e r e 5 > 0 . D e n o t e b y m a n u m b e r s u c h t h a t

p ( T m x o , x o )

> /.t 9 5 . S ince t he ba l l

B 0 =

B xo,-~5)

i s r e g u l a r i t c o n t a i n s s o m e t r a j e c t o r y

u o , T u o , T 2 u o , . . . .

T h e m a p p i n g T is u n i f o r m l y k - L i p s c h i tz , t h e

t r a j e c t o r y o f t h e p o i n t u l =

T m u o

l i e s i n t h e b a l l B 1 =

B ( T ' ~ x o ,

k 7 5 ) . T h e n t h i s t r a j e c t o r y l ie s i n t h e

in t e r s e c t i o n B 0 (3 B 1 w h ic h is c o n t a in e d i n a b a l l o f r a d iu s # 6 . [ ]

o r o l l a r y 2 . 3 . 8

[14]. L e t M b e a n o n e m p t y b o u n d e d c o n v e x a n d c l os e d s u b s e t o f a H i l b e r t sp a c e , a n d

T : M --+ M b e a u n i f o r m l y k - L i p s c h i t z m a p p i n g w i t h c o n s t a n t k < x / ~ . T h e n

F ix T # O .

I t is u n k n o w n w h e t h e r t h e c o n s t a n t v f 2 is s h a r p . T h e n e x t e x a m p l e s h o w s th a t t h e c o n s t a n t k i n 2. 3. 8

m u s t b e s t r i c t l y s m a l l e r t h a t 7 r / 2 .

E x a m p l e 2 . 3 . 9 ( L i fs c h i tz [1 4 ]) . L e t B b e t h e u n i t b a l l i n t h e s p a c e 12. C o n s i d e r t h e c o n t i n u o u s m a p p i n g

T : B --+ B for wh ich

T x

= c o s e l + ~ s i n

w h e r e e , = ( 1 , 0 , 0 , . . . ) a n d

P ( x l , X 2 , . . .

) = ( 0 , X l , X 2 , . . . ) ( i. e. , P i s t h e sh i f t o p e r a t o r ) . I t i s i m m e d i a t e

r

t h a t

I I T x - T y l l

~ ~ I I x - y H f o r a l l x , y E B . S i n ce

T n = p n - 1 o T a n d P i s

a n i s o m e t r y w e c o n c l u d e

r

t h a t T is u n i f o r m l y ~ - L i p s c h i tz . B u t F i x T = O , b e c a u s e , i f x =

T x ,

t h e n I l x l l = 1 a n d x =

P x ,

w h i c h i s

i m p o s s i b l e .

C o m p a r i n g ( 2 .3 . 4 ) a n d ( 2 .3 . 2 ) , w e s e e t h a t T h e o r e m 2 . 3 .3 i s a c o n s e q u e n c e o f T h e o r e m 2 . 3 .7 .

D o w n i n g a n d T u r e t t [ 68 ] i n t r o d u c e d t h e n o t i o n o f

L i f s c h i t z c h a r a c t e r i s t i c > c o ( X )

o f a B a n a c h s p a c e X :

x 0 x ) = i n f i x M ) : M is a n o n e m p t y b o u n d e d co n v ex cl os ed su b s et o f X } .

793



O b v i o u s l y , > c0 (X ) ) 1 f o r e v e r y X . F r o m t h e r e s u l t s o f L i f sc h i t z [ 1 4 ] i t ~o ll ow s t h a t

1 ) i f X i s non ref l e x ive , th en >c0(X) = 1 ;

2 ) i f x 0 ( X ) > 1 , t h e n X i s r e f l e x iv e ;

3 ) i f X i s a Hi lb er t spa ce , th en >c0(X) = v /'2;

4 ) i f

c o ( X )

< 1 , t h e n x 0 ( X ) > 1 ;

5 ) x 0 ( X ) / > % w h e r e 1 < 7 is a s o l u t i o n o f t h e e q u a t i o n

. 1 - = 1 .

D e f i n i t i o n 2 . 3 . 1 0 . A B a n a c h s p ac e X is s a id t o h a v e t h e

f i xed po in t p r oper ty fo r un i fo r m ly k -L ips ch i t z

m app ings

( F P P ( k ) ) i f f o r a n y n o n e m p t y b o u n d e d c o n v e x c l os e d s u b s et M C X a n d e v e r y u n i f o r m l y k-

L i p s c h i t z m a p p i n g T : M ~ M , F i x T r ;~ .

T h e o r e m 2 . 3 .7 c a n b e r e f o r m u l a t e d fo r a B a n a c h s p a c e X a s fo ll o w s.

T h e o r e m 2 . 3 . 1 1 . Let X be a Ban ach space such that the L i fschi tz cha racter is t ic ;4o(X) > 1 . The n X

has the ERR(k ) w i th cons tan t k < xo (X) .

T h e n e x t t h e o r e m s h o w s t h a t T h e o r e m s 2 .3 .4 a n d 2 .3 .1 1 a r e c o m p a x a b le q u a li t at i v e ly , a l t h o u g h q u a n t i -

t a t i v e l y t h e r e s u l t o f L i fs c h i tz i s m o r e e x a c t .

T h e o r e m 2 . 3 . 1 2 ( D o w n i n g a n d T u r e t t [6 8]).

Let X be a Ban ach sp ac e. The n >co(X) > 1 < :.

o X ) < 1 .

T h e n e x t t h e o r e m

T h e o r e m 2 . 3 . 1 3

w a s p r o v e d w i t h t h e a i d o f t h e a s y m p t o t i c c e n t e r s t e c h n iq u e s .

( E . C a s i n i a n d E . M a l u t a [ 5 4 ] ) .

L e t X be a Banach s pace w i th a un i fo r m ly nor m a l

s t r uc tur e ( i . e . , J (Z ) < 1). T hen Z has the ER R(k ) w i th constan t k < ( J (X ) ) -1 /2

I t f o ll o w s f r o m T h e o r e m s 2 .3 .1 2 , 2 .3 .1 3 a n d E x a m p l e 1 .2 .9 t h a t t h e s p a c e X ~ = ( l 2 , I IX ) w i t h t h e

n o r m I x] ~ = m a x { n x n 2 , f l l lx l l ~ } h a s t h e F P P ( k ) w i t h k < 2 1 / 4 . f l - ~ / 2 . S i n c e f o r f l = v ~ / 2 w e h a v e

e 0 ( X ~ ) = x 0 ( X ~ ) = 1 t h e n T h e o r e m 2 . 3. 13 is n o t a c o n s e q u e n c e o f T h e o r e m s 2. 3 .1 1 a n d 2 . 3 .4 .

A r e l a t i o n s h i p b e t w e e n x 0 ( X ) a n d

J ( X )

i s e s t a b l i s h e d b y t h e f o l l o w i n g t h e o r e m .

T h e o r e m 2 . 3 . 1 4 ( J .R . L . W e b b a n d W . Z h a o [ 21 6] ).

For every Banach space X

x o ( X ) ~ ( J ( X ) ) - 1 .

N o t e t h a t , f o r a n o n r e f l e x i v e X , x o ( X ) : J ( X ) = 1 . I f X i s a H i l b e r t s p a c e , t h e n > c o ( X ) = ( J ( X ) ) - I :

x /~ . I n [ 21 6] , t h e r e a x e a l s o g i v e n s o m e lo w e r b o u n d s f o r x 0 ( L P) , 1 

1 a n d g 0 ( X ~ ) = 1 . M o r e o v e r ,

J ( X ~ )

< 1 . H e n c e ,

x o ( X z ) < ( J ( X z ) ) - 1 .

W i t h t h e a i d o f t h e L P - i n e q u al it ie s a n d t h e a s y m p t o t i c c e n t e r s t e c h n iq u e s q u i t e a n u m b e r o f r e s u l ts o n

f i x e d p o i n t s o f u n i f o r m l y L i p s c h i t z m a p p i n g s a c t i n g o n t h e s p a c e s L p h a s b e e n d e r i v e d ( [ 1 56 , 1 5 9 , 2 0 1 , 2 2 1 ] ) .

W e p r e s e n t s o m e o f t h e s e r e s u l t s .

( l + a ' - l ) 1 /p

T h e o r e m 2 . 3 . 1 5 ( L i ra [ 15 6] ).

The space L p , 2 < p < 0 has the F P P( k) w i th

k < 1 + (1 + a ) p -1 '

where a is a unique solutio n of the equation (p - 2)z p -1 + (p - 1 )x p -2 = 1 on the in terval (0 , 1) .

L e t X b e a n a b s t r a c t L ' - s p a c e , t h a t i s , X i s a B a n a c h l a t t i c e s u c h t h a t IIx §

y p

= Ilxl l ' + I lyl l ' for

a l l x , y

E X, x A y = 0 . T he n , as i t i s sho wn in [201] , t he fo l lowin g ine qu al i ty i s va l id fo r 1 < p ~< 2 :

I 1 1

- t )~ +

t y l l .< 1

- t)l lxll 2 +

t l l y l l ~ -

-

p

- 1)t (1 - t) t lx - yl l 2 for al l x , y E X an d 0 < t < 1 . T h i s

i n e q u a l i t y i s u s e d i n t h e p r o o f o f t h e f o l lo w i n g

T h e o r e m 2 . 3 . 1 6 ( R . S m a r z e w s k i [2 01 ]). Let X be an abstract LP-space, 1 < p <. 2. Th en X has the

F P P ( k ) i f k < v ~ .

G . K u h n [1 36 ], u s i n g t h e f a c t o f p r e s e n c e o f a u n i f o r m r e l a ti v e n o r m a l s t r u c t u r e ( U R N S ) i n a n o r d e r

c o m p l e t e A M - s p a c e , o b t a i n e d t h e f o l lo w i n g r e s u l t.

794



T h e o r e m 2 . 3 .1 7 . L e t X b e a n o r d e r c o m p l e t e A M - s p a c e w i t h u n i t y a n d I C X a n o r d e r i n t e r v a l . T h e n

e v e r y u n i f o r m l y k - L i p s c h i t z m a p p i n g T : I -+ I w i t h k < ~ h a s a f i x e d p o i n t .

I f Y i s a c o m p l e x i f i c a t i o n o f X a n d B C Y i s a c l o s ed b al l, t h e n e v e r y u n i f o r m l y k - L i p s c h i t z m a p p i n g

T : B --+ B w i t h k < r h a s a f i x e d p o i n t .

N o t e t h a t t h i s t h e o r e m g e n e r a li z e s t h e c o r r e s p o n d i n g r e s u l t s o f P . S o a r d i [ 2 0 6 ] a n d R . S i n e [1 97 ].

F u r t h e r m o r e , K u h n [1 36 ] p r o v e d t h a t e v e ry u n i f o r m l y k - L ip s c h it z m a p p i n g f r o m a c l o se d b a ll o f t h e

s p a c e ( ~ ( g / 2 ) i n t o i t s e lf h a s a f i x ed p o i n t i f k < C / 2 ( t h is i s d u e t o t h e f a c t t h a t t h e s p a c e ( ~ | 2 )

n=l / r n=l c r

h a s t h e U R N S ) .

S i n c e x ( [ ) = x ( B ) = 1 , T h e o r e m 2 . 3 .1 7 i s n o t a c o n s e q u e n c e o f T h e o r e m 2 . 3. 1 1 .

A r t i n t e r e s t i n g g e n e r a l i z a t i o n o f T h e o r e m s 2 . 3. 11 a n d 2 . 3 . 14 w a s f o u n d b y J . G o r n i c k i a n d K r ~ t p p e l.

L e t r n b e t h e s p a c e o f r e a l b o u n d e d s e q u e n c e s . F o r { s n } E r n d e n o t e b y L i m s , ~ t h e B a n a c h l i m i t ( s e e,

n ---->oo

e . g. , [9 ]). L e t A C N a n d IA I b e t h e c a r d i n a l i t y o f A . C o n s i d e r t h e b o u n d e d s e q u e n c e

I A n { 1 2 . .. n } I

a n- -= ~ n = 1 , 2 , . . . .

n

D e f i n i t i o n 2 . 3 . 1 8 [1 32 ]. T h e n u m b e r ( A ) = L i m s , i s c a ll e d t h e B a n a c h d e n s i t y o f a s e t A C N .

n - - o o

S t a t e m e n t 2 . 3 . 1 9 [96]. T h e f o l l o w i n g a s s e r t i o n s a r e tr u e fo r A , B C N :

i 0 ~ < ( A ) ~ < 1 ;

2 ) = 1 ;

3) i f A C B , t h e n ( A ) <~ I ~ ( B );

4 ) i f A N B = C), t hen ( A U B ) = ( A ) + ( B ) ;

5 ) i f ( A ) = 1 t h e n / z ( A N B ) = ( B ) ;

6) / . t (A + i = ( A ) , w h e r e A + 1 = { a + 1 : a E A } ;

1

7) ( s A ) = ~ ( A ) , w h e r e s A = { s a : a e A } , s e

N ;

n

8 ) i r A = { e l , a 2 , . . . } a n d l i m - - = r , t h e n ( A ) = r .

n - - o o a n

D e f i n i t i o n 2 . 3 . 2 0 [9 6]. L e t A C N b e s u c h t h a t ( A ) > 1 / 2 . A m a p p i n g T : D C X --+ X i s s a i d t o b e

p - u n i f o r m l y k - L i p s c h i t z i f f o r a l l z , y E D , n E A ,

I I T x - T y l [ < . k l l = - Y l I .

A n e x a m p l e o f s u c h m a p p i n g i s g iv e n i n [ 96 ].

T h e o r e m 2 . 3 . 2 1 [ 96 , 9 5 ]. L e t X b e a B a n a c h s p a ce . T h e n X h a s t he F P P f o r - u n i f o r m l y k - L i p s c h i t z

m a p p i n g s , i l k < m a x { x 0 ( X ) , ( J ( X ) ) - l / 2 } .

U s i n g L P - i n e q u al it ie s a n d t h e a s y m p t o t i c c e n t e rs t e c h n iq u e s , H o n g - K u n X u [2 21] h a s i m p r o v e d T h e o r e m s

2 . 3 . 1 5 a n d 2 . 3 . 1 6 .

T h e o r e m 2 . 3 . 2 2 . T h e s p a c e L p , 1 2 ,

i s t h e u n i q u e s o l u t i o n o f t h e e q u a t i o n

p - 2 ) z p - 1 + p - 1 ) z p - 2 = 1

o n t h e i n t e r v a l 0 < x < 1 .

795



C h a p t e r 3

N o n s t a n d a r d M e t h o d s i n t h e

F i x e d P o i n t T h e o r y

3 1 T h e E x a m p l e o f A l s p a c h

F o r a l o n g ti m e o n e o f th e m a i n p r o b l e m s i n t h e f i x ed p o i n t t h e o r y o f n o n e x p a n s i v e m a p p i n g s w a s t h e

fo l low ing (S . Re ich [188]) :

D o e s a n y n o n e m p t y c o n v e x w e a k l y c o m p a c t s u b s e t o f a B a n a c h s p a c e h a v e t h e F P P ?

D . A l s p a c h g a v e a n e g a t i v e a n s w e r t o t h i s q u e s t i o n .

E x a m p l e 3 . 1 . 1 ( A l s p a c h [2 2]) . I n t h e s p a c e L I [ 0 , 1 ], c o n s i d e r t h e s e t

K =

1

{ s L [ o , . . s ( o

= 1 , 0 ~< f ~< 2 a l m o s t e v e r y w h e r e } .

O b v i o u s ly , K i s a c o n v e x c l o se d a n d h e n c e w e a k l y cl o s e d s e t. M o r e o v e r , K i s w e a k l y c o m p a c t s i n c e t h e

o r d e r i n t e r v a l { f E L i [ 0 , 1 ] : 0 ~< f ~< 2 ) i s w e a k l y c o m p a c t i n L l [ 0 , 1] ( s e e , f o r e x a m p l e , [ 9] ). D e f i n e t h e

m a p p i n g T : K ~ K i n t h e fo l lo w i n g w a y :

2 f ( 2 t ) A

2

0 ~< t ~< 1/2,

( T y ) ( t ) = 2 f 2 t - 1 ) - 2 ) 0 , 1 / 2 < t ~< 1 ,

w h e r e x ( t) V y ( t) = m i n { x ( t ) , y ( t ) } a n d x ( t) A y ( t) = m a x { x ( t ) , y ( t ) } f o r x , y E L ' [ 0 , 1 ] . N o t e t h a t a ll

e q u a l i ti e s a n d i n e q u a l i t i e s i n 3 .1 .1 a r e v a l i d u p t o s e t s o f m e a s u r e z e r o .

W i t h t h e a i d o f t h e r e l a t i o n

I lx h z - y A z [ I + n z v z - y v z l l = I I x - y l l,

w hic h i s va l i d fo r a l l x , y , z E L ' [0 , 1 ]

( s e e [ 16 3] ), i t i s e a s y t o e s t a b l i s h t h a t T : K ~ g i s a n i s o me t r y , i .e . ,

I I T X - T y l l = I Ix - y l l

fo r a l l

x , y E K . A s s u m e t h a t t h e m a p p i n g T h a s a f i x e d p o i n t g E K . L e t A = { t : g ( t ) = 2}. S ince

A = { t : T g t ) = 2 } = ~ : g t ) = 2 T : g t ) = 2 : l < g t ) < 2 and the m easure of the

s e t { 2 : g ( t ) = 2 } U { l + t }~ : g ( t )

= 2 is e q u a l t o t h e m e a s u r e o f

A ,

w e o b t ai n t h a t t h e m e a s u r e o f t h e

set { t : 1 ~< g ( t ) < 2 } is z e r o . Co n s i d e r i n g t h e s e t { t : g ( t ) = 0 } a n a l o g o u s l y w e o b t a i n t h a t t h e m e a s u r e o f

t h e s e t { t : 0 <

g ( t )

~< 1 } i s z e r o . H e n c e t h e m e a s u r e o f t h e s e t { t : 0 <

g ( t )

< 2 } i s e q u a l t o z e r o . T h e n t h e

m e a s u r e o f A is 1 / 2 s i n c e f o g g A = 1 . T a k i n g i n t o a c c o u n t t h a t T g = g , w e o b t a i n A = A U A + 5 =

I A U i 1

~ A u f l A I

i , i i I i

=

A

= =

T h e s e e q u a l i t ie s a l l o w u s t o p r o v e e a s i l y ( b y i n d u c t i o n o n n E

N

t h a t t h e i n t e r s e c t i o n o f A w i t h a n y i n t e r v a l

o f t h e f o r m

( k / 2 n , ( k +

1 ) / 2 n ) , w h e r e n E 1~ a n d k E { 0, 1 , . . . , 2 n - 1 }, a n d h e n c e w i t h a n y i n t e r v a l f r o m

[0, 1] h a s t h e m e a s u r e e q u a l t o t h e h a l f -l e n g t h o f t h is i n t e r w l . T h i s c o n t r a d i c t s t h e L e b e s g u e t h e o r e m o n

t h e d e n s i t y p o i n ts o f a m e a s u r a b l e s e t. T h u s t h e m a p p i n g T h a s n o f i x e d p o i n ts .

G . S c h e c h t m a n [ 19 3 ] i m p r o v e d a n d g e n e r a li z e d th e i d e a o f A ls p a c h .

L e t (f ~, E , ) b e a m e a s u r e s p a c e a n d

r - i

: f~ --+ [0, 1] x f l a m e a s u r e p r e s e r v i n g t r a n s f o r m a t i o n . D e f i ne

t h e m a p p i n g T ~ o n t h e s e t { f E L i ( ) : 0 ~< f ~< 1 } in t h e f o l l o w i n g w a y :

T~f( s ) = X~({ ( t ,~ ) :o ~<t4

f(s)}),

w h e r e X i s t h e c h a r a c t e r i s t i c f u n c t i o n o f a s e t. I t i s e a s y t o c h e c k t h a t

T,.

i s a n i s o m e t r y . I n p a r t i c u l a r ,

f T . : f d = f f d p

fo r ev e ry f E LI(/~ ), 0 ~< f ~< 1. F or 0 < a ~< ̀ 8 < 1 let W~ ,~ = { f E LI (/~ ) : 0 ~< f ~<

1 , o~ <~ f f d ~ <~ , 8 } . W ~ , a

i s a c o n v e x w e a k l y c o m p a c t s u b s e t o f L a ( ) a n d

T, . (W, : , , a )

C W , , , a . I t is ea sy

t o c h e c k t h a t T ~ h a s a f i x e d p o i n t i n W ~, a i f a n d o n l y i f t h e r e i s a s e t A E E s u c h t h a t a ~< ( A ) ~< / 3 a n d

r ( [ 0 , 1] x A ) = A a l m o s t e v e r y w h e r e . I f s u c h a n A e x i st s , t h e n

X A

is

a fi x e d p o i n t o f T r

796



S t a t e m e n t 3 . 1 . 2 [19 3]. L e t ~ = [0, 1] ~, a n d # b e t h e m u l t i p l i c a t iv e L e b e s g u e m e a s u r e e q u a l t o th e

p r o d u c t o f th e L e b e s g u e m e a s u r e s o f a c o u n ta b l e n u m b e r o f f a c t o r s [0, 1]. T h e n t h e r e e x i s t s a c o n v e x w e a k l y

c o m p a c t s u b s e t W C L 1 ( # ) a n d a n o n e x p a n s i v e m a p p i n g T : W --+ W w i t h o u t f i x e d p o i n t s .

P r o o f . L e t W = W 1/ 2,1 /2

a n d T t , S l , S 2 , . . . ) ) - - t ,

Sl , S2 , . . . ) . T h e n ~ -( [0 , 1] A ) = A a l m o s t e v e r y -

w h e r e i f a n d o n l y i f p ( A ) = 0 o r / ~ ( A ) = 1 . T h u s T = T~ h a s n o f i x e d p o i n t s . [ ]

S t a t e m e n t 3 . 1 . 3 [193]. L e t 12 = [0, t], # b e t h e g e b e s g u e m e a s u r e o n [0, 1], a n d

r e ~ 2 - 6 ,~2 -n e l e2 52

, = 7 + N + . . . ,

r t . = l r t = l

w h e r e ~ , , 5n = 0 , 1 , n = 1 , 2 , . . . . T h e n t h e i s o m e t r y T r : W 1 / 2 , 1 /2 --+ W 1 / 2 , 1 /2 h a s n o f ix e d p o i n t s .

T h e o r e m 3 . 1 . 4 [ 1 9 3 ] . F o r e v e r y n = 1 , 2 , . . . , t h e re i s a c o n v e x w e a k l y c o m p a c t s u b s e t W C L 1 a n d a

f a m i l y T 1 , T 2 , . . . ,T ,~+ l o f c o m m u t i n g n o n e x p a n s i v e m a p p i n g s f r o m W i n t o i t s el f s u c h t h a t a n y n o f t h e m

h a v e a c o m m o n f i x e d p o i n t , b u t t h e c o m p o s i t i o n T 1 o T 2 o . . . o T n + l h a s n o f i x e d p o i n t s . I n p a r ti c u l a r ,

1 1, T ~ , . . . , T , + I h a v e n o c o m m o n f i x e d p o i n t .

T h e f o l lo w i n g m o d i f i c a t i o n o f t h e e x a m p l e o f A l s p a c h , d u e t o R . S i n e [ 19 8] , i s a n o t h e r c o n c r e t e r e a l -

i z a ti o n o f t h e g e n e r a l i d e a o f S c h e c h t m a n c o n s i de r e d a bo v e . S i ne u s es t h e e r g o d i c i t y p r o p e r t y o f b a k e r ' s

t r a n s f o r m a t i o n r ~ - -+ f l (s e e [ 19 ]) , w h e r e 12 = { ( x , y ) E ] R 2 : 0 < z , y <<.1 } a n d

1

r v =

( 2 1 -( x + 1 ) , 2 y ) m o d 1,

0 ~< y < 1/ 2,

1/ 2 ~< y < 1.

L e t I = {0 ~< f ~< 1 } b e t h e o r d e r i n t e r v a l i n L I [ 0 , 1] a n d A S = { ( x , y ) E • : y ~< f ( z ) } b e t h e s u b g r a p h

o f a f u n c t i o n f E I . T h e s et r i s t h e s u b g r a p h o f s o m e f u n c t i o n g E I . P u t T f = g . T h e n T : I --+ I i s

a n i s o m e t r y . S i n c e b a k e r ' s t r a n s f o r m a t i o n r is e r g o d i c ( s ee [ 19 ]) ( i. e ., e v e r y r m e a s u r a b l e s u b s e t

o f f l h a s m e a s u r e 0 o r 1 ), w e se e t h a t T - i n v a r i a n t i n I a r e o n l y t h o s e f u n c t i o n s f o r w h i c h t h e i n t e g r a l i s e q u a l

t o 0 o r 1 . T h u s F i x T = { 0 , 1 } , t h a t i s, t h e s e t o f f i x e d p o i n t s o f t h e i s o m e t r y T i s n o t c o n n e c t e d ( c o m p a r e

t h i s, f o r e x a m p l e , w i t h t h e f a c t ( s e e [6 3] ) t h a t t h e s e t o f f i x e d p o i n t s o f a n y n o n e x p a n s i v e m a p p i n g d e f i n e d

o n a c o n v e x s u b s e t o f a s t r i c t l y c o n v e x B a n a c h s p a c e m u s t b e c o n v e x ) . A s F i x T c o n s i s t s o f o n l y t w o p o i n t s ,

t h e s e t o f p o i n t s l y i n g m i d w a y b e t w e e n t h e m w i l l b e c o n v e x in v a r i a n t a n d f i x e d p o i n t f r e e . P r e c i s e l y th i s

s e t a p p e a r s i n t h e e x a m p l e o f A l s p a c h ( th i s s e t is t h e C h e b y s h e v c e n t e r C ( I ) o f th e o r d e r i n t e r v a l I ) .

N o t e t h a t , f o r t h e s e t K f r o m t h e e x a m p l e o f A l s pa c h , d i a m K - - 2 ,

r ( K ) = 1, a n d f - 1 E C ( K ) .

3 2 U l t r a p o w e r a n d S u p e r r e f l e x i v i ty

I n t h is s e c t io n , w e s h a ll g i v e t h e d e f i n i t i o n o f u l t r a p r o d u c t o f B a n a c h s p a c e s a n d f o r m u l a t e s o m e u s e f u l

f a c t s c o n n e c t e d w i t h t h i s n o t i o n . I n o u r e x p o s i t i o n w e s h a l l f o ll o w S . H e i n r i c h [ 1 02 ], a n d K . G o e b e l a n d

W . A . K i r k [ 89 ]. F u r t h e r m o r e , w e p r o v e h e r e w i t h t h e a i d o f t h e u l t r a p o w e r t h e t h e o r e m o n s u p e r r e f l e x i v it y

o f B a n a c h s p a ce s w i t h a u n i f o r m n o r m a l s t r u c t u r e , w h i c h g iv e s a p o s i t i v e a n s w e r t o a n o l d q u e s t i o n o f

D . A m i r [ 23 ] t h a t w a s r e c e n t l y r a i s e d a g a i n b y J . W a n g a n d X . Y u [ 2 15 ].

L e t I b e a n o n e m p t y s e t . A f a m i l y ~ o f n o n e m p t y s u b s e t s o f I i s s a i d to b e a f i l t e r on I i f i t s a t i s f i es

t h e f o l l o w i n g c o n d i t i o n s :

1 ) A , B E ~ i mp l i e s A A B E U ,

2 ) A E 9 v , B C I a n d A C B i m p l i e s B E ~ ' .

G i v e n t w o f i l te r s f 'x a n d 9v2 o n I . W e s a y t h a t U 2 m a j o r i z e s J = l i f e v e r y e l e me n t o f 5 v l b e l o n g s t o 5 v s .

A f i l t e r / 2 i s c a l l e d a n u l t r a f i l t e r i f e v e r y f i lt e r w h i c h m a j o r i z e s / g c o i n c i d e s w i t h L /. Z o r n ' s l e m m a i m p l i e s

t h a t e v e r y fi lt e r o n I i s m a j o r i z e d b y s o m e u l t r a f il t e r . F o r e v e r y u l t r a f i l t e r / 2 o n I t h e f o l l o w i n g is va l id : i f

A C_ I , t h e n e i t h e r A E / 2 , o r I \ A E / / . A n u l t r a f i l t e r b / i s c a l l e d t r i v i a l i f i t is g e n e r a t e d b y s o m e e l e m e n t

i o E I , t h a t i s , / 2 = {A C I : i o E A } . N o n t r i v i a l u l t r a f il t e r s a r e s a i d t o b e f r e e . A n u l tr a s i s s a i d t o b e

c o u n t a b I y i n c o m p l e t e

i f i t c o n t a i n s a d e c r e a s i n g s e q u e n c e o f s e t s A 1 D A2 D w i t h t h e e m p t y i n t e r s e c ti o n :

797



oo

A A n = ~ . N o t e t h a t , i n t h e c a s e w h e r e I = N , a n u l t r a f i l t e r is c o u n t a b l y i n c o m p l e t e i f a n d o n l y i f i t i s

n l

f ree .

G i v e n a n u l t r a f i l t e r / 2 o n I , a t o p o l o g ic a l s p a c e W , a n d x : I ~ W . D e n o t e t h e v a l u e x i ) , i E I , b y

x i . W e s h al l s a y t h a t a f a m i l y { x i } iE z = { x i : i E I } c o n v e r g e s o v e r / 2 t o a n e l e m e n t y E W , i f f o r e v e r y

n e i g h b o r h o o d V o f y th e s e t s { i E I : z i E V } b e l o n g t o / 2 . I n th i s c a s e , w e w r i t e l i m x i = y , o r l i m z i = y .

i- /.d 12

N o t e t h a t i f a f a m i l y

{ X i } i I

h a s a l i m i t p o i n t z , t h e n t h e r e e x i s t s l i m z i = z .

U

S t a t e m e n t 3 . 2 . 1 .

G i v e n a c o m p a c t H a u s d o r f f s pa c e K , a n d a n u l t r a f il t e r/ 2 o n I . T h e n f o r e v e r y f a m i l y

{ x i } i E l C K t h e re i s a u n i q u e y E K s u c h t h a t

l i m

z i = y .

U

G i v e n a n u l t r a f i lt e r / 2 o n I , a n d a f a m i l y o f B a n a c h s p a c e s { X i } i e i . D e n o t e b y l ~ I , X , ) t h e s p a c e o f

a l l f a m i l i e s { x i } i E l , x / E X i s u c h t h a t

[ [ { x I H = s u p { l t x i l l : I } < o o . 3 . 2 . 1 )

l ~ 1 7 6 X i ) is a s u b s p a c e o f th e p r o d u c t s p a c e I-I x i e q u i p p e d w i t h n o r m ( 3 .2 . 1) .

i I

Le t .Af(N) = {{ x i} E

l ~ 1 76 X i ) :

l i ~ n n x in = 0 } . T h e n

l ~ 1 7 6 X i )

w i t h n o r m ( 3 . 2 . 1 ) i s a B a n a c h s p a c e ,

a n d A / ( /2 ) i s a c l o s e d s u b s p a c e o f l ~ 1 7 6 X i ) .

D e f i n i t i o n 3 . 2 . 2 [58]. B y t h e

u l t r a p r o d u c t X ~ ) u

o f t h e B a n a c h s p a c e s

{ X i } i E I w i t h r e s p e c t t o t h e

u l t r a f i l t e r / 2

o n I w e m e a n t h e q u o t i e n t s p a c e

x i ) u = l I ,

e q u i p p e d w i t h t h e n o r m

I I , ) u l l = lium Ilxin, (3. 2.2 )

w h e r e

x ~ ) u

i s a n e l e m e n t o f t h e u l t r a p r o d u c t

X ~ ) u ,

t h a t i s, t h e e q u i v a le n c e c l a s s c o r r e s p o n d i n g t o a f a m i l y

{ z i } iE I E l ~ 1 7 6 X i ) .

I f X i = X f o r a l l i E I , t h e n t h e u l t r a p r o d u c t

X ) u

i s c a l l e d t h e

u l t r a p o w e r

o f X w i t h r e s p e c t t o / 2 . F o r

a n u l t r a p o w e r o f X w e s h a ll u s e th e n o t a t i o n X u ( o r ) ~ , w h e n i t d o e s n o t l e a d to a m b i g u i t y ) .

B y S t a t e m e n t 3 . 2 .1 , t h e l i m i t ( 3 .2 . 2 ) a l w a y s e x i s ts . M o r e o v e r , t h e n o r m d e f i n e d b y ( 3 .2 . 2 ) is a q u o t i e n t

n o r m , a n d X u w i t h t h i s n o r m i s a B a n a c h s p a c e .

I n w h a t f o l l o w s , w e s h a l l m a i n l y c o n f i n e o u r s e l v e s t o f r e e u l t r a f i l t e r s / 4 o n t h e i n d e x s e t I = N . F u r t h e r -

m o r e , t h e s y m b o l { x i } w i l l d e n o t e b o t h a n e l e m e n t o f t h e s p a c e l ~ 1 7 6 = l ~ 1 7 6 X ) a n d t h e e q u i v a l e n c e

c l a s s {x i } + A / ( /2 ) E X , w h e re )~ = l ~ Z ) / ~ / 2 ) i s t h e u l t r a p o w e r o f Z . T h e s p a c e Z w i ll b e i d e n ti f ie d

w i t h t h e s u b s p a c e

{ { x , x , . . . } : x E X } C X

w h i c h i s i s o m e t r i c t o X ( t h i s s u b s p a c e o f ) ~ c o n s i s t s o f a l l

e l e m e n t s { z i } + A / ( /2 } s u c h t h a t

z i = x

fo r a l l i E I ) .

W e n o w t u r n t o t h e n o t i o n o f s u p e r r e f le x i v i ty o f a B a n a c h s p a c e a n d t o i ts r e l a t i o n s w i t h t h e n o t i o n o f

u l t r a p o w e r .

D e f i n i t i o n 3 . 2 . 3 [ 10 9] . A B a n a c h s p a c e Y is s a i d t o b e

f i n i t e l y r e p r e s e n t a b l e

i n a B a n a c h s p a c e X i f f o r

a n y f i n it e d i m e n s i o n a l s u b s p a c e F C Y a n d e v e r y e > 0 t h e r e is a s u b s p a c e G C X o f t h e s a m e d i m e n s i o n ,

s u ch t h a t t h e B a n a c h - M a z u r d i s ta n c e

d F , G) < . 1 + e .

W e n o w f o r m u l a t e s o m e k n o w n f a c t s o f th e t h e o r y o f B a n a c h s p a c e s in t e r m s o f f in i te r e p r e s e n t a b i l i t y :

1 ) t h e s p a c e 12 i s f i n i t e l y r e p r e s e n t a b l e i n e v e r y i n f i n i t e - d im e n s i o n a l B a n a c h s p a c e ( a t h e o r e m o f D v o r e t -

z k y ) ;

2 ) f o r a n y B a n a c h s p a c e X t h e s e c o n d d u a l s p a c e X * * is f i n it e ly r e p r e s e n t a b l e i n X ( a c o n s e q u e n c e o f

t h e L i n d e n s t r a u s s - R o s e n t h a l p r i n c ip l e o f l o ca l r ef l ex i v it y );

3 ) e v e r y B a n a c h s p a c e i s f i n i t e ly r e p r e s e n t a b l e i n C o.

T h i s a n d o t h e r f a c t s f r o m t h e g e o m e t r y o f B a n a c h s p a c e s c a n b e f o u n d i n t h e s u r v e y b y M . I . K a d e c [1 2]

w h i c h i s r i c h i n c o n t e n t .

798



T h e o r e m 3 . 2 . 4 . Le t X be a Ban ach space . The n the u l t rapower X i s f i n i t e l y represen tab le in X .

R e f e r e n c e s f o r t h i s t h e o r e m c a n b e f o u n d i n [ 1 0 2 ] .

D e f i n i t i o n 3 . 2 . 5 ( R . C . J a m e s [ 10 9] ). A B a n a c h s p ac e X is s a id to b e superre f l ex ive i f e v e r y B a n a c h

s p a c e Y t h a t i s f i n i t e l y r e p r e s e n t a b l e i n X i s r e f le x i v e .

O b v i o u s l y , a s u p e r r e f i e x i v e s p a c e is r ef l e x iv e ( t h e c o n v e r s e is n o t t r u e ) .

I n t h e n e x t t h e o r e m , w e h a v e c o l le c te d f u n d a m e n t a l r e s u lt s e s t a b li s h in g p r o f o u n d r e l a t io n s b e t w e e n

s u p e r r e f l e x i v i t y , u n i f o r m c o n v e x i t y , a n d o t h e r n o t i o n s .

T h e o r e m 3 . 2 . 6 . The fo l lowing asser t ions are equ iva len t :

1) a Ba nac h space X i s superre f l ex ive;

2 ) X i s i som orph ic to a un i fo rm ly convex space;

3 ) X i s i som orph ic to a un i fo rm ly smooth space;

4 ) i n X o n e c a n i n tr o d u c e a n e q u i v a le n t n o r m wh i c h i s u n i f o r m l y c o n v e x a n d u n i f o r m l y s m o o t h a t t h e

s a m e t i m e ;

5 ) X i s i som orp h ic to a un i fo rm ly non-square space;

6 ) X * i s s u p e r r e f t e x iv e .

T h e r e s u l t s c o l l e c t e d i n 3 . 2 . 6 a r e d u e t o J a m e s a n d E n f l o ( s e e t h e L i t e r a t u r e C i t e d i n [ 1 2 ] ) .

T h e o r e m 3 . 2 . 7 ( se e [ t0 2 ]) . A Ban ach space X i s superre f l ez i ve i f and on ly i f every u l t rapower X u i s

ref lexive.

F r o m 3 .2 .7 w e , i n p a r t i c u l a r , o b t a i n t h a t t h e local propert ies o f a s p a c e X , t h a t i s, t h o s e p r o p e r t i e s t h a t

a r e d e s c ri b e d i n t e r m s o f f i n it e - d im e n s i o n a l s u b s p ac e s o f X , a r e i n h e r i t e d b y t h e u l t r a p o w e r X .

T h u s t h e f o l l o w i n g p r i n c i p l e h o l d s ( s ee [ 10 2]): X h a s a l o c a l p r o p e r t y ( P ) i f a n d o n l y i f f o r e v e r y u lt r a s

5 ( t h e u l t r a p o w e r X u h a s t h e p r o p e r t y ( P ) . F o r e x a m p l e , a s t h e m o d u l i o f c o n v e x i ty a n d s m o o t h n e s s 6 x

a n d p x o b v i o u s l y h a v e t w o - d i m e n s i o n a l c h a r a c t e r, w e g e t

6 x . ) = @ . ) , p x . ) = p x . ) ,

S i n ce t h e J u n g c o n s t a n t J X ) h a s f i n i t e - d im e n s i o n a l c h a r a c t e r ( s ee 1 .2 . 13 ) w e o b t a i n J X ) = J . ~ ) .

T h e q u e s t i o n : I s a n y B a n a c h s p a c e w i t h a u n i f o r m l y n o r m a l s tr u c t u r e ( U N S ) s u p e r re f ie x i v e ? - - w a s

r a i s e d b y D . A m i r [2 3] a n d b y E . C a s i n i a n d E . M a l u t a [5 5].

W e g i v e a p o s i t i v e a n s w e r t o t h i s q u e s t i o n :

T h e o r e m 3 . 2 . 8 . E v e r y B a n a c h s pa ce X w i t h U N S is su p e r r e fl e x iv e .

P r o o f . B y v i r t u e o f S t a t e m e n t 1 .2 .1 3 , w e h a v e J ( )C ) - - s u p { J ( Y ) : Y i s a f i n i te - d i m e n s i o n a l s u b s p a c e

o f ) ~ } , w h e r e ) ( i s t h e u l t r a p o w e r o f X w i t h r e s p e c t t o a n a r b i t r a r y f i x e d u l t r a f i l te r b /. S i n c e X i s f i n it e l y

r e p r e s e n t a b l e i n X , f o r a n y f i n i t e - d i m e n s i o n a i s u b s p a c e ~ C . ~ a n d f o r e a c h c > 0 , th e r e i s a s u b s p a c e

Y C X s u c h t h a t d ( F ' , Y ) ~< 1 + ~ . H e n c e

J Y ) <<.d Y , Y ) . J Y ) <~ (1 + e ) J Y ) .

T h i s i m p l i e s t h a t J ( X ) ~< J X ) . T h e o p p o s i t e i n e q u a l i t y is o b v i o u s , s i n c e X e m b e d s i s o m e t r i c a l l y i n t o

X . T h u s J ( . ~ ) = J X ) < 1 . B y T h e o r e m 1 . 2 .4 , t h e s p a c e f f i s r e fl ex i v e. T h e n , b y T h e o r e m 3 . 2. 7 , w e

o b t a i n t h a t X is s u p e r r e f l e x iv e . [ ]

3 .3 . U l t r a p o w e r a n d F i x e d P o i n t s .

B y t h e e a r ly 8 0 s t h e r e b e g a n t o b e fe lt t h e n e ce s s it y o f a t t r a c t i n g n e w m e t h o d s t o e s t a b l is h t h e F P P

f o r B a n a c h s p a ce s w i t h o u t n o r m a l s t r u c tu r e . T h e f a m i l i ar e x a m p l e o f A l s p a c h [ 2 2 ] s t i m u l a t e d e f f o rt s in

t h i s d i r e c t io n . I t w a s t h e n t h a t t h e p a p e r b y B . M a u r e y [1 74 ], w h i c h i n i t ia t e d t h e n e w p e r i o d i n t h e

d e v e l o p m e n t o f t h e f ix e d p o i n t t h e o r y f o r n o n e x p a n s i v e m a p p i n g s a p p e a r e d . T h e p a p e r [1 74 ] w a s t h e fi rs t

o n e to e x p l oi t th e n e w m e t h o d b a s e d o n t h e u s e o f t h e n o t i o n o f u l t r a p o w e r t h a t a p p e a r e d i n th e f r a m e w o r k

o f n o n s t a n d a r d a n a ly s i s. T h i s e n a b l e d M a u r e y t o d e r iv e a n u m b e r o f s t r o n g r e s u l ts .

T h e f i rs t re s u l t o f M a u r e y t u r n e d o u t t o b e r a t h e r u n e x p e c t e d a s c o m p a r e d w i t h t h e e x a m p l e o f A l s p a ch .

799



T h e o r e m

3 . 3 . 1 [ 1 7 4 ] .

Le t X be a re f l ex ive su bspace o f L l [O,

1].

T h e n A h a s t h e F P P .

A n o t h e r r e s u l t c o n c e r n s t h e

Ha r d y s p a c e H 1 ,

t h a t i s, t h e s p a c e o f a n a l y t i c f u n c t i o n s i n t h e u n i t d i s k

w i t h t h e n o r m i n d u c e d b y t h a t o f L 1. F o r

f = Y ~ a , e i ~

n =

f~

I l f l lH

= n t

dO

0 - -

T h e o r e m

3 . 3 . 2 [ 1 7 4 ] .

T h e H a r d y s p ac e H 1 h a s t h e w - F P P .

I n p a p e r s [1 01 ] a n d [ 1 7 8 ] ,t h e r e w e r e f o u n d s o m e c la s s es o f w e a k l y c o m p a c t ( b u t n o t n e c e s s a r i l y c o n v e x )

s u b s e t s o f co w i t h t h e F P P . T h e r e i t w a s c o n j e c t u r e d t h a t t h e s p a c e co h a d t h e w - F P P . T h e n e w a p p r o a c h

a l l o w e d M a u r e y t o p r o v e t h is c o n j e c t u r e .

T h e o r e m 3 . 3 . 3 [1 74 ].

T h e s p a c e c o h a s t h e w - F P P .

T h e p r o o f o f t h i s t h e o r e m i s b a s e d o n t h r e e s i m p l e L e m m a s 3 . 3 .4 - 3 .3 . 6 t h a t a r e o f i n d e p e n d e n t i n t er e s t.

T h e f i r s t t w o l e m m a s d e s c r i b e t h e s t r u c t u r e o f m i n i m a l i n v a r i a n t s e ts .

L e m m a 3 . 3 . 4 [1 74 ].

L e t X b e a B a n a c h s p a c e , K b e a c o n v e x w e a k l y c o m p a c t s u b s e t o f X , a n d T :

K --+ K b e a n o n e x p a n s i v e m a p p i n g . L e t C b e a m i n i m a l c o n v e x w e a k ly c o m p a c t T - i n v a r i a n t s u b s e t o f I V.

T h e n f o r a n y x E C

s u p { l l x - Y l l: Y E C } = d i a m C .

P r o o f . B y Z o r n s l e m m a , t h e w e a k c o m p a c t n e s s o f K i m p li e s t h a t t h e r e is a m i n i m a l c o n v e x w e a k l y

c o m p a c t s u b s e t C C K t h a t i s i n v a r i a n t u n d e r T , t h a t i s ,

T C C C .

T h e i n c l u s i o n s

T(c--d- ff~TC) C T C C

con-b gvTC im ply th a t C = con--d-ff~C. Ow ing to th e m ini m a l i t y of

C ,

e v e r y c o n v e x l o w e r s e m i c o n t i n u o u s

f u n c t i o n p : C --+ R s u c h t h a t

p ( T x ) <~ p ( x )

f o r a l l x E C i s c o n s t a n t o n C . I n d e e d , p a t t a i n s i t s l e a s t v a lu e

o n C a t s o m e p o i n t x 0 . T h e n t h e s e t C 1 = { y E C :

p(y) <~

p ( x 0 ) } i s a n o n e m p t y c o n v e x d o s e d T - i n v a r i a n t

s u b s e t o f C . H e n c e C 1 = C a n d

p ( y ) = p ( x o )

f o r a l l y E C . A s s u c h a f u n c t i o n c o n s id e r

p x)

s u p {

I I x y l l :

y E C } . T h e f u n c t i o n

p ( x )

i s co n v e x a n d c o n t i n u o u s , a n d t h u s

p ( x )

= s u p { i t x -

T y l i :

y E C } , b e c a u s e

C = con-WfivTC. T h is im pl ie s

p Tx)

s u p i l T x

t l

<

s u p I [ x Y l l p x ) .

y6C y6C

H e n c e p x) is c o n s t a n t o n C . I t i s e a s i l y s e e n t h a t p ( x ) = d i a m C f o r a ll x e C . [ ]

T h e n e x t l e m m a i s a n u l tr a - a n a lo g o f t h e G o e b e l a n d K a rl o vi t z l e m m a ( s e e 1 . 1 . 54 ) . B u t f ir st w e

i n t r o d u c e a n o t i o n t h a t w i l l b e n e c e s s a r y i n t h e s e q u e l .

A s e q u e n c e i x , } C C i s c a l l e d

q u a s i - s t a b l e

f o r a m a p p in g T : C - -+ C , i f l i~ n t I T x ,, - x n l l = 0 , w h e r e /g

is a f r ee u l t r a f i l te r on N.

L e m m a 3 . 3 . 5 [1 74 ].

L e t C b e a c o n v e x m i n i m a l T - i n v a r i a n t w e a k l y c o m p a c t s e t. T h e n f o r a n y x E C

a n d a n y q u a s i- s ta b l e s e q ue n c e { x , } C C

l i r a I I x x n l l

= d i a m C .

u

P r o o f . F i r s t , v e r if y t h a t C h a s , a t l e a s t o n e , q u a s i - s t a b l e s e q u e n ce . T a k e x0 E C a n d c E (0 , 1 ). C o n s i d e r

t h e m a p p i n g T ~ : C --+ C s u c h t h a t

T ~x = e X o + ( 1 - e ) T x

f o r a l l x E C . T h e n

I I T ~ x l - T ~ x 2

I[ ~< ( 1 -e ) l lx l -x2] [

f o r al l X l , x 2 E C s i n c e T i s a n o n e x p a n s i v e m a p p i n g . T h u s T~ i s a c o n t r a c t i o n o n C a n d , t h e r e f o r e , i t h a s

a u n i q u e f i x e d p o i n t x , 6 C f o r w h i c h

x ~ = T , x ~ = e x o +

(1 -

e ) T x ~ .

H e n c e

IITx~ - x~ll = eltTz~ - xoll <~

d ia m C . ( 3 . 3 .1 )

L e t / J b e a f r ee u l t r a f i lt e r o n N . F r o m ( 3 .3 . 1 ) i t f o l lo w s t h a t a n y b o u n d e d c o n v e x c l o s e d T - i n v a r i a n t s u b s e t

o f t h e S a n a c h s p a c e c o n t a i n s a q u a s i - s t a b l e s e q u e n c e { z , } s u c h t h a t l i ~n

] [ T x , -

xn[[ = 0 . L e t {x n} be an

8 0 0



a r b i t r a r y q u a s i - s t a b l e s e q u e n c e . C o n s i d e r o n C t h e f u n c t i o n r = l ~ n l l x - x , ~ ll . O b v i o u s l y , ~b i s c o n v e x ,

c o n t i n u o u s , a n d r <~ r f o r al l z E C . S i n c e C is a c o n v e x m i n i m a l T - i n v a r i a n t w e a k l y c o m p a c t s e t ,

t h e f u n c t i o n r i s c o n s t a n t o n C . L e t y b e a w e a k l i m i t p o i n t o f t h e s e q u e n c e { z,~ }. T h e n f o r a n y x E C

w e h a v e I l x v i i -<< li~n I l x x n II = r B y L e m m a 3 .3 .4 , w e c o n c l u d e t h a t s u p { f i x y l l : x e c } = d i a m C .

[ ]

W e n o w p r e s e n t s o m e fa c t s w h i c h a r e n e c e ss a r y f o r t h e p r o o f o f t h e n e x t l e m m a . L e t X b e a B a n a c h

s p a c e , C b e a b o u n d e d c o n v e x c l o se d s u b s e t o f X , T : C - + C b e a n o n e x p a n s i v e m a p p i n g . L e t 2 ( =

X u

b e t h e u l t r a p o w e r o f X b y a f r e e u l t r a f i l t e r / t / o n N , t h a t i s, t h e q u o t i e n t s p a c e o f t h e s p a c e

l ~ ( X )

o f

b o u n d e d s e q u e n c e s o f p o i n t s f r o m X b y t h e c lo s ed s u b s p a c e A f = { { x n } E I ~ ( X ) :

J i l l l l = 0 } .

I f

{ x n } is a r e p r e s e n t a t i v e o f a n e q u i v a l e n c e c la s s ~ E ) f t h e n t h e n o r m l l l l = l i m I I x ~ l [. W e s h a l l d e n o t e

n--+/X

b y C = { { y , } E ) ( : y n E C } t h e u l t r a p o w e r o f C , t h a t i s , t h e s e t o f a ll t h o s e p o i n t s ~ E ) ~ w h i c h

h a v e r e p r e s e n t a t i v e s { x n } s u c h t h a t xn E C f o r a l l n E N . C l e a r l y , C i s a c l o s e d c o n v e x s u b s e t o f ~ ' ,

a n d d i a m C = d i a m C . E x t e n d T t o C b y d e f in i ng T ~ a s t h e e q u iv a l e n ce c la ss o f { T x n } , w h e r e { z n } i s a

r e p r e s e n t a t i v e o f ~ .

O b v i o u s l y , t h e e x t e n s i o n o f T i s a n o n e x p a n s i v e m a p p i n g o n C . T h e c o n v e x se t C c a n b e i d e n t i f i e d w i t h

t h e c o n v e x c l o s e d s u b s e t o f C , c o n s i s t in g o f a ll p o i n ts ~ w h o s e r e p r e s e n t a t i v e s a r e c o n s t a n t p o i n t s e q u e n c e s

i n C . O b s e r v e t h a t T a l w a y s h a s a f i x e d p o i n t i n C . I n f a c t , T ~ = ~ f o r a l l ~ E C w h o s e r e p r e s e n t a t i v e s

{ xn } are q u a s i - s t a b l e s e q u e n c e s .

I f C i s a c o n v e x w e a k l y c o m p a c t m i n i m a l T - i n v a r i a n t s e t, t h e n b y L e m m a 3 .3 . 5 f o r a n y ~ E C s u c h t h a t

T ~ = ~ a n d a n y x E C w e h a v e

] ] 5 - x]] = d i a m C = d i a m C . ( 3 .3 . 2)

L e t Y b e a B a n a c h s p a c e a n d x , y E Y . A n e l e m e n t z E Y w i ll b e c a l l e d a

quas i -mi dpo i n t

o f t h e i n t e r v a l

1

[x,y]

i f [Ix - z l[ = ] ]y - z [[ = ~ [ t x - Y [ I . Re m a r k t h a t i f t h e s p a c e Y i s n o t s t r i c t l y c o n v e x , t h e n n o t e v e r y

q u a s i - m i d p o i n t m u s t b e l o n g t o Ix , y ].

L e m m a 3 . 3 . 6 [ 1 7 4 ] . L e t ~ and ~ be f i xed po i n t s o f t he ex t ens i on o f T t o C . T hen T has a f i xed po i n t

~ E 0 tha t is a q ua si-m idp oin t f or [~, y-'].

P r o o f , C o n s i d er t h e s et

K = { t q C :

ll~-tll=ll~-tll=l[l~ -yl[}.

I t is e a s y t o c h e c k t h a t K i s c o n v e x , c l o se d , n o n e m p t y ( 2 (~ + y~ E K ) , a n d T - i n v a r i a n t . B y ( 3 . 3 .1 ) f o r

a n y r n 6 N t h e r e i s a ~ ,,, 6 g s u c h t h a t [[T'trn - ~m[[ ~< - - . 1 L e t {x , , } , {y , } , { t , , ,, , } b e r e p r e s e n t a t i v e s

m

of th e cla sse s ~',~', an d t 'm re sp ec tiv ely . T h e n .-+ulim ]xn - trn,nl[ = .-,ulim [yn -- tm ,.][ = 111~. _ ~.1[, a n d

t oo

l i m

I l T t m , ,

- t i n , , II l / r e . U s i n g t h e d i a g o n a l p r o c e d u r e c h o o s e f r o m { m,,~ }m ,n =l a d i a g o n a l s u b s e q u e n c e

n- -~U

{ t } s uch that

l i m I l z n - t - I I = 1 ~ t lY , ~ - t r i l l = l I l x - y ] l ,

n-- U

l i m I I T T , ~ - ~ n ] l = 0 .

n-~-U

T h e n t h e c l as s F w i t h t h e r e p r e s e n t a t i v e { t . } w i ll b e a f ix e d p o i n t f o r T a n d a q u a s i - m i d p o i n t f o r [ ~', y ~. [ ]

T h u s T h e o r e m 3 . 3 .6 s h o w s t h a t i n t h e l a r g e r u l t r a - s p a c e ) ~ t h e s e t o f f i x e d p o i n t s o f a n o n e x p a n s i v e

m a p p i n g i s n o t o n l y a u t o m a t i c a l l y n o n - e m p t y b u t i t is q u a s i - c o n v e x in t h e s e n s e t h a t f o r a n y t w o f ix e d

p o i n t s t h e r e i s a t h i r d o n e t h a t l ie s h a l f - w a y b e t w e e n t h e m .

P r o o f o f T h e o r e m 3 .3 .3 . L e t C b e a n o n e m p t y c o n ve x w e a k l y c o m p a c t m i n i m a l T - in v a r ia n t s u b se t o f

t h e s p a c e co , d i a m C > 0 a n d l e t

{ xn } C C

b e s o t h a t l i m ][ T x n

- x n

[[ = 0 . U s i n g , i f n e c e s s a r y , a t r a n s l a t i o n ,

n

8 0 1



a n o r m a l i z a t i o n a n d a p a s s a g e t o a s u b s e q u e n c e , w e c a n s u p p o s e t h a t d l a m C = 1 a n d x , ~ 0 . B y

L e m m a 3 .3 .5 w e h a v e l im

I I z ~ l l

= 1. U s i n g t h e d ia g o n a l p r o c e d u r e c h o o s e a s u b s e q u e n c e { z n } f o r w h i c h

n

I I Ix k l A I ~ l l l ~<

1 k

a n d l i m l l z k ~ 1 1 = 1 f o r a = { a , } a n d b =

{ h i }

f r o m

o

d e f i n e l a l , a A b , a V b E C o

k

s o t h a t l a l ) i = a i l , ( a A b i = = m i n { a i , b i } , a V b )i = m a x { a i , b i } ) . T h e p o i n t s ~ " = { z } k a n d f f = { z , k }

a r e f i x e d p o i n t s o f T i n C , a n d

I 1 ~

-

~ 1 1

= 1 . Le t

~

= {zk}

b e

a q u a s i -m i d p o i n t o f t h e i n t e r v a l [~ ', 9"] wh i c h

i s f i x e d u n d e r T . F r o m ( 3 . 3 .2 ) i t f o l lo w s t h a t l i m I l z k t l = 1 a n d

k- - L /

l i m I I ~ k - z k l l = l im t lznk - zkl] = 1 /2 .

k - - e L / k - - L /

H o w e v e r , I zk l ~ I z k

-

x k [ V lz k - x n l + I ~ 1 A I x n ~ l .

ence

l i ra I lz k l l = 1 / 2 . T h e c o n t r a d i c t i o n o b t a i n e d

k - L /

s h o w s t h a t d i a m C = 0 . [ ]

J . E l t o n , P . - K . L i n , E . O d e l l a n d S . S z ar e k [ 7 9 ] e x t e n d e d t h e i d e a s o f M a u r e y a n d g a v e a s i m p l e r p r o o f

o f t h e d i ff i cu l t T h e o r e m 3.3.1. M o r e o v e r , t h e y o b t a i n e d a n u m b e r o f s t r o n g r e s u lt s .

T h e o r e m 3 . 3 . 7 [79]. The follow ing asse rtions are valid:

1) the space C K ) has the w -F P P i l K is a countable compact ordinal less than w~~

2)

the Ts ire lson space Ts and i ts dual T ; have the FP P;

3) the space X ~ see 1 .8 .12) has the FP P for a l l f l ) 1 .

In p a r t i c u l a r , t h e C (~o + 1 ) - B a n a c h s p a c e ( c , I 1" I I ) o f c o n v e r g e n t s e q u e n c e s w i t h t h e s u p r e m u m - n o r m I1" II

h a s t h e w - F P P .

R e m a r k t h a t f o r f i ) v ~ t h e s p a c e X a d o e s n o t h a v e a N S . F o r f i > 2 a s s e r t i o n ( 3 ) o f 3 . 3 . 7 g e n e r a l i z e s

r e s u l t s f ro m [1 1 2 ] a n d [2 8 ] .

I n t h e p r o o f o f T h e o r e m 3 .3 . 7, e s s e n ti a l u s e is m a d e o f t h e p r o p e r t i e s o f B a n a c h s p a c e s w i t h 1 - u n c o n d i t i o n a l

b a s e s .

I n [ 79 ], w i t h t h e a i d o f u l t r a p r o d u c t s , a g e n e r a l i z a t i o n o f T h e o r e m 3 .3 .1 t o t h e c a s e o f B a n a c h l a t t i c e s i s

g i v e n .

D e f i n i t i o n 3 . 3 . 8 [7 9]. A B a n a c h l a t t ic e X h a s a u n i f o r m l y m o n o t o n e n o r m ( U M ) i f f o r a n y e > 0 t h e r e

i s a 6 > 0 s u c h t h a t i f x > y ) 0 a n d

I l x

- y l l / >

~

and IlYll

~ <

1 , t h e n

I l x l l

> Ilyll + ~.

O b s e r v e t h a t t h e s p a c e s L p , 1 ~< p < o o , h a v e a ( U M ) - n o r m . F u r t h e r m o r e , a B a n a c h l a t t i c e X h a s a

( U M ) - n o r m i f a n d o n l y if t h e u l t r a p o w e r . ~ h a s a ( U M ) - n o r m ( [7 9] ).

T h e o r e m 3 . 3 . 9 [79]. Let X be a Ban ach la t tice wi th a UM )-norm a nd suppose that X has a subspace

Y such that 11 is not f in i te ly representable in Y . The n Y has the F P P .

O b s e r v e t h a t a s u b s p a c e Y i n L 1 i s r e f l ex i v e i f a n d o n l y i f 11 i s n o t f i n i te l y r e p r e s e n t a b l e i n Y .

F i n a l l y , i n [ 79 ], th e r e i s g i v e n a p r o o f ( r a t h e r s o p h i s t i c a t e d ) o f t h e f a c t , d u e t o M a u r e y , t h a t e v e r y

i s o m e t r ic m a p p i n g o f a n a r b i t r a r y c o n v e x w e a k l y c o m p a c t s u b s e t o f a s u p e r r e f le x i v e s p a c e i n t o i t s e l f h a s a

f i x e d p o i n t .

J . B o r w e i n a n d B . S i m s [ 3 6 ] u s e d i n s t e a d o f u l t r a p o w e r a c o n s t r u c t i o n c l o s e t o i t . A s a n a n a l o g o f t h e

u l t r a p o w e r o f a B a n a c h s p a c e X , t h e y i n t r o d u c e d t h e q u o t i e n t s p a c e l i m ( X ) = l ~ 1 7 6 w h e r e I ~ 1 7 6

is th e s p a c e o f b o u n d e d p o i n t s e q u e n c e s i n X , a n d c o X ) i s t h e s p a c e o f se q u e n c e s c o n v e r g e n t t o z e r o i n X .

T h e e q u i v a l e n c e c l a s s e s a r e d e n o t e d

by [~

o r N , w h e r e [ ~ = ~ + c o X ) , an d ~" = {x~ } E l ~ 1 7 6 T h e

q u o t i e n t n o r m i s

[l[x~ll[

= l i m s u p l l z n ] [ 9 W i t h t h is n o r m t h e s p a c e l ~ 1 7 6 i s a n o r d e r c o m p l e t e

l

B a n a c h l a t ti c e i f X i s. W i t h t h e h e lp o f t h e n o t i o n l i m ( X ) t h e a n a l o g s o f L e m m a s 3 . 3 . 4 - 3 . 3 . 6 a r e p r o v e d

in [ 3 6 ] I t is s h o w n t h a t l i r a ( X ) h a s m a n y p r o p e r t i e s t h a t a r e a p p r o p r i a t e t o t h e u l t r a p o w e r .

I n [ 3 6] , t w o n o t i o n s u s e f u l fo r fi x e d p o i n t t h e o r y a r e i n t r o d u c e d :

1 ) t h e

Riesz angle

o f a B a n a c h l a t t i c e X i s t h e n u m b e r

802





L e t { e , ~} b e a n u n c o n d i t i o n a l b a s i s i n X , F C N , a n d P F x n e ~ = ~ x ~ e , b e t h e n a t u r a l p r o j e c -

nEF

t i o n a s s o c i a t e d w i t h F . A n u n c o n d i t i o n a l b a s is is c a l l e d a suppress ion bas is i f IIPFII = 1 f o r e v e r y F C N .

E v e r y s u p p r e s s i o n b a s i s is m o n o t o n e . P u t

c = s u p { l l P r l l : F C N } . 3 . 3 .3 )

O b s e r v e t h a t c = 1 i f {e ,~} i s a s u p p r e s s i o n b a s i s . A s A = s u p { I [ 2 P F - II1: F c N } , o b v i o u s l y , 1 ~< c ~< ~ ~<

2c

L i n o b t a i n e d a s t r o n g e r r e s u l t f o r sp a c e s w i t h a s u p p r e s s i o n b a s i s t h a t i s, f o r c = 1 , A ~< 2 ) .

T h e o r e m 3 . 3 . 1 5 [1 60 ].

Let X be a superreflezive spa ce with a suppression basis i .e. ,

c = 1 ) . T h e n X

h a s t h e F P P .

M . A . K h a m s i [ 1 13 ] s u p p l e m e n t e d t h e r e s u l ts o f L i n .

T h e o r e m 3 . 3 . 1 6 .

Let X be a Ban ach space wi th an uncon di t ional bas is sa t is fy ing the condi t ion e . A +

2) < 4.

T hen X has the w-F PP .

I n p a r t i c u l a r , a s p a c e X w i t h a s u p p r e s s i o n b a s i s i .e . , w i t h c = 1 ) h a s t h e w - F P P i f h < 2 .

U s i n g t h e w e a k * - l e m m a o f G o e b e l - K a r l o v i tz , K h a m s i [ 11 4] o b t a i n e d s im i l a r re s u l ts f o r c o n j u g a t e s p a ce s .

D e f i n i t i o n 3 . 3 . 1 7 . A b a si s

{el}

i s s a id to be

shr inking ,

i f f o r e v e r y f E X * , s u p ~ f x ) : x =

x ie i

k

i = n l

llxll <~ ~ > 0 a s n

% ,

_ >

o o

. , ,

O b s e r v e s e e , f o r e x a m p l e , [ 1 9 9 ] ) t h a t a b a s i s { e ~ } C X is s h r i n k i n g i f a n d o n l y i f { f J i s a b a s i s o f t h e

s p a c e X * , w h e r e f i E X * a n d

f i x ) = x i

f o r e v e r y x E X , i = 1 , 2 , . . . .

T h e o r e m 3 . 3 . 1 8 [114]. Let X be a conjugate) Ba nac h space wi th 1-uncondi t ional shr inkin g ba s is . Th en

X h as t he w - F P P t h e w * - F P P ) .

I n p a r t i c u l a r , w e o b t a i n t h a t 11 h a s t h e w * - F P P .

T h e o r e m 3 . 3 . 1 9 [114]. Given a Ban ach space X wi th a suppression shr inking bas is , and le t A < 2 .

T h e n X X * ) h as t h e w - F P P t h e w * - F P P ) .

K h a m s i [ 11 4] h a s e s t a b l i s h e d a c o n n e c t i o n b e t w e e n t h e a l t e r n a t e s i gn s p r o p e r t y o f B a n a c h - S a k s a n d t h e

p r e s en c e o f t h e w - F P P t h e w * - F P P ) .

D e f i n i t i o n 3 . 3 . 2 0 [4 5]. A B a n a c h s p a ce X is s a i d t o h a v e t h e

al ternate s igns prope r ty o f Ban ach-

Saks

A S P B S ) i f e v e r y b o u n d e d s e q u e n c e { z ,, } C X h a s a s u b s e q u e n c e {x ,~k } su c h t h a t t h e s e q u e n c e

{ k - l i ~= l - -1 )i xn , } k l l c o n v e r g e s .

O b s e r v e t h a t c o h a s A S P B S ) . E v e r y B - c o n v e z B a n a c h s p a c e a s p a c e X is B - c o n v e x if P i s n o t f i n i t e ly

r e p r e s e n t a b l e i n X ) , f o r e x a m p l e a s u p e r r e f l e x i v e B a n a c h s p a c e , h a s A S P B S ) s e e , f o r e x a m p l e , [ a0 1) .

T h e o r e m 3 . 3 . 2 1 [1 14 ].

Let X be a Ban ach space wi th a suppress ion shr inking bas is , having A SP B S) .

T h e n X h as t h e w - F P P .

T h e o r e m 3 . 3 . 2 2 [114].

Let X be a Ban ach space wi th a suppress ion shr inking bas is and suppose that

X * h a s A S P B S ) . T h e n X * h a s t he w * - F P P .

B . S i m s [ 19 6], u s i n g o n l y s t a n d a r d t o o ls , g e n e r a l i z e d t h e w e a k * l e m m a o f G o e b e l - K a r l o v i t z [ 1 1 4] t o

t h e c a s e o f a

w*-orthogonal conjugate Banach lattice

t h a t i s, a c o n j u g a t e l a t t i c e X s u c h t h a t f o r e v e r y

{ z , } C X , z , > 0 , a n d f o r e v e r y z E X l i m l l l z I A x l l l = 0 h o l d s ) , a n d o b t a i n e d t h e f o l l o w i n g r e s u l t .

T h e o r e m 3 . 3 . 2 3 . Let X be a w*-orthogonal conjugate Ba nac h la tt ice . The n X has the w* -F PP .

S i m s [ 19 5] , g e n e r a l i z i n g t h e r e s u l t o f 3 .3 .1 1 , e s t a b l i s h e d t h a t e v e r y w - o r t h o g o n a l B a n a c h l a t t i c e X i .e .,

s u c h a l a t t i c e X t h a t f o r {x n } C X , x ,, w > 0 a n d a n y x X h o l d s

l i m l t l x ~ I i z ] l t

= 0 ) h a s t h e w - F P P .

D e f i n i t i o n 3 . 3 . 2 4 [1 95 ]. A B a n a c h s p a c e X is s a i d t o b e w-or thogonaI i f f o r e v e r y s e q u e n c e { x ,, } C

.w >

X , x ,, 0 , a n d e v e r y z E X w e h a v e

~ i m ] l l x . x l l - I I x . - x l l] = 0

8 4



U s i n g t h e n o t i o n o f l i m ( X ) i n t r o d u c e d b y B o r w e i n a n d S i m s i n [ 3 6] , J . G . F a l s e r p r o v e d t h e f o l lo w i n g

t h e o r e m .

T h e o r e m 3 . 3 . 2 5 .

Let X be a w-orthogonal uni formly non-square Banach space. Then X has the FPP.

M . A . K h a m s i a n d P h . T u r p i n [ 11 8 ] r e f i n e d t h e m e t h o d o f L i n [1 60 ], w h i c h a l l o w e d t h e m t o u s e t h e

a s y m p t o t i c c e n t e r s t e c h n i q u e s ( M . E d e l s t e i n [ 76 , 7 7] ) fo r a l a r g e r c l a ss o f s p a c e s t h a n w a s d o n e b e f o r e , a n d

t o g e n e r a l i z e r e s u l t s 3 . 3 .1 3 , 3 . 3 . 1 4 .

T h e o r e m 3 . 3 . 2 6 [1 18 1. Let (X; I1

I1

be a real Banach space endowed with the structure of vector lattice

satisfying the conditions:

(a) for given x, y E X , the conditions z + <~ y+ and x - <~ y - imply

t1~11

~<

Ilyll,

(/3) there is a constant k < 2 such that for x,y E X,

I~I l y t , w e

have

I1~11 kllyll-

Let v be the weakest linear topology on X for which the mapping z --+ I t t I t I from x to R is continuous

at O for each

u E X = { x E X : x = z + } .

Then every nonempty r-compact convex subset of X has the

FPP.

F o r e x a m p l e , a r e a l B a n a c h s p a c e X w i t h a n u n c o n d i t i o n a l S c h a u d e r ba s is { e n } is a v e c t o r l a t t i c e w i t h

t h e c o o r d i n a t e - w i s e o r d e r : ~

z~e~ <~ ~ y~e~

i f x ,~ ~< y ,~ fo r a l l n E N. Th e t op o lo gy v i s , as on e e as i ly

n l n l

s e e s , t h e

topology of coordinate-wise convergence: x (i) ~ x as i -+ oo r x(~) -+ z,~ as i --+ oo (n = 1, 2 , . . . ).

As x = E

xne,~,

w e f i n d t h a t c o n d i t i o n s ( a ) a n d ( [3 ) a r e e q u i v a l e n t, r e s p e c t iv e l y , t o c o n d i t i o n s ( a ' ) a n d

( [ 3 ' ) :

( [ 3 ' )

l l t l, = 0 , 1 , E x ,

.< k l l x l l , = + 1 , 9 E X .

C o r o l l a r y 3 . 3 . 2 7 [ 1 1 8 ] .

Let X be a real Banach space with an uncondit ional basis sat isf ying conditions

(a') and ([3') with k < 2. Then every nonempty convex subset K C X which is compact in the topology of

coordinate-wise convergence, has the FPP.

R e m a r k t h a t 3 .3 .2 7 w a s p r o v e d b y L i n [ 16 0], i n t h e s p e c i a l c a s e w h e r e K i s a c o n v e x w e a k l y c o m p a c t

s e t , a n d k < ( v / ~ - 3 ) / 2 ~ 1, 3 7 25 .

E x a m p l e 3 . 3 . 2 8 [ 11 8]. G i v e n a n o n e m p t y c o n v e x w e a k * c o m p a c t s u b s e t K o f 11. N o t e t h a t l 1 i s a

c o n j u g a t e s p a c e ( l I ~ c ~ ), a n d w * - c o m p a c t n e s s i n l 1 i s e q u i v a l e n t t o c o o r d i n a t e - w i s e c o m p a c t n e s s w i t h

r e s p e c t t o t h e n a t u r a l b a s i s {e n } i n 11 . T h e n , b y v i r t u e o f 3 . 3 .2 7 , t h e m a p p i n g T : K --+ K h a s a f i x e d p o i n t

i f T i s n o n e x p a n s i v e f o r t h e n o r m ] l z l l . / = ll xll l V ( 7 1 1 x l l~ ) , w h e r e 7 E I R+, ] l{x,~}l ll = ~ Ix ,~[, I[{x,~} l l~ =

n l

s u p

lxn

I. O b s e r v e t h a t f o r 3' >~ 2 t h e s e t K n e e d n o t h a v e t h e w * - N S . I n f a ct , c o n s i d e r t h e w * - c o m p a c t c o n v e x

n

se t K = { x E Z : x . > 0 , t lx l la 1 , n = 1 , 2 , . . . } . T h e n {e ,~} C K . I f 7 ) 2 , t h e n l l -e l l -+ d iaml l . l l , I f ,

a s n -+ o o a n d f o r a n y x E K . H e n c e K h a s n o w * - N S . M o r e o v e r , t h e R i e sz a n g l e ( s ee 3 . 3 . 1 0 o f t h e s p a c e

( ll , I1 I I , i s equa l to 2 .

A s a n e x a m p l e o f L i m [ 1 5 5] s h o w s , 3 . 3 . 2 7 d o e s n o t h o l d i f k = 2 .

K h a m s i a n d T u r p i n [ 11 8 ] h a v e e x t e n d e d 3 .3 .2 7 t o s p a c e s o f m e a s u r a b l e f u n c t i o n s , a n d , i n p a r t i c u l a r , t o

O r l i cz f u n c t i o n s p a c e s . G i v e n a m e a s u r e s p a c e ( f / , E , # ) w i t h a a - f i n i t e m e a s u r e , le t M = M ( f / , E , o ') b e t h e

v e c t o r l a t t i c e o f a l l e q u i v a l e n c e c l a ss e s o f # - m e a s u r a b l e f u n c t i o n s o n (~ 2, E , # ) . A n

order ideal

o f t h e s p a c e

M i s a v e c t o r s u b s p a c e X C M s u c h t h a t x E X i f x E M a n d I zl ~< l yl f o r s o m e y E X . A n o r m ] ]. ]l o n X

i s s a i d t o b e

sequentially order-continuous

i f l i ra ] lx n ]l = 0 f o r a n y n o n i n c r e a s i n g s e q u e n c e {z n } C X + s u c h

n

t h a t i n f {x n : n = 1 , 2 , . . . } = 0 .

T h e o r e m 3 . 3 . 2 9 [11 8].

Let a Banach space X be an order ideal in

M ( f ~ , E , # )

with a sequentially

order-continuous n orm satisfying conditions (a) and ([3) from

3.3.26

for some k < 2. Then ever y nonempty

convex v-compact subset X , where v is the topology o f local convergence in measure, has the FP P.

I n t e r e s ti n g a p p l ic a t io n s o f th e u l t r a p r o d u c t s t e c h n i q u e s t o t h e f i xe d p o in t t h e o r y o f n o n e x p a n s i v e m a p -

805



pings c a n be f ou nd in [ 32 , 31 , 82 , 80 , 182, 185 , 4 ], a nd o the r s . T h e boo k [ 21] by A . G . Ak soy a n d

M . A . K h a m s i i s e n t ir e l y d e v o t e d t o n o n s t a n d a r d m e t h o d s i n f ix e d p o i n t t h eo r y . T h e r e m a r k a b l e b o o k [8 9]

b y K . G o e b e l a n d W . A . K i r k i n cl u d e s a l ar g e c h a p t e r d e v o t e d t o t h e n o n s t a n d a r d m e t h o d s .

S o m e U n s o l v e d P r o b l e m s

H e r e w e f o r m u l a t e s o m e f u n d a m e n t a l u n s o l v ed p r o b l e m s c o n n e c t e d t o t h e s u b j e c t s o f o u r s u r v e y .

T h e n e x t p r o b l e m i s co n s i d e r e d t o b e c e n t ra l in t h e f ix e d p o i n t t h e o r y o f n o n e x p a n s i v e m a p p i n g s i n

Sa na c h spa c e s se e , f o r e xa m ple , [79]) .

P r o b l e m 1 . D o e s e v e ry re fl ex i ve B a n a c h s p a ce h a v e t h e fi x ed p o i n t p r o p e r t y ?

I t w o u l d b e

o f

i n t e r e s t t o f i n d a n y p a r t i a l s o l u t i o n t o t h i s p r o b l e m f o r t h e f o l lo w i n g i m p o r t a n t c l a s s e s o f

re f lexive spaces :

1 ) supe r r e f i e x ive ;

2 ) u n i f o r m l y n o n - s q u a r e ;

3 ) s t r i c t l y c onve x ;

4 ) p o s s e s s i n g t h e B a n a c h - S a k s p r o p e r t y ;

5 ) p o s s e s si n g t h e R a d o n - N i k o d y m p r o p e rt y ;

6 ) w i t h a n u n c o n d i t i o n a l b a s is .

T h e f o l l o w i n g p r o b l e m i s a p a r t i a l c a s e o f P r o b l e m 1 .

P r o b l e m 2 . D o e s e v e r y r ef le x iv e B a n a c h s p a ce h a v e t h e f ix e d p o i n t p r o p e r t y w i t h r e s p e c t t o i s o m e t r ic

m a p p i n g s ?

T h e a n s w e r t o t h i s q u e s t i o n i s p o s i t i v e i f t h e s p a c e i s s u p e r r e f le x i v e B . M a u r e y [7 9] ).

P r o b l e m 3 T . L a n d e s [1 44 ]) . L e t X a n d Y b e B a n a c h s p ac e s w i t h n o r m a l s t r u c t u r e . W i l l t h e i r I~

- d i r e c t s u m , t h a t i s , t h e s p a c e

h a v e a n o r m a l s t r u c tu r e ?

P r o b l e m 4 . L e t A b e a n o n e m p t y w e a k l y c o m p a c t c o n v e x su b se t o f a B a n a c h s p ac e X , p o ss e ss i ng t h e

f i xe d p o i n t p r o p e r t y f or s i n gl e - va l u ed ) n o n e x p a n s i v e m a p p i n g s . D o e s e v e r y n o n e x p a n s i v e m u l t i - v al u e d

m a p p i n g T : - + K X ) w h i c h is w e a k l y i n w a r d o n A h a v e a f i xe d p o in t ?

Also of interest is a solution of any particular case of this problem :

1) T i s i nw a r d on A ;

2) T : A --+ K A ) see [188]);

3) T : A --+ K c A );

4 ) T : A --+ K c X ) ;

5 ) A h a s a n o r m a l s t r u c t u r e ;

6 ) X h a s a u n i f o r m l y n o r m a l s t r u c t u r e ;

7 ) X i s a U C E D ) - s p a c e ;

8 ) X is a u n i f o r m l y s m o o t h sp a c e .

H e r e

K M ) K c M ) )

i s t h e f am i l y o f n o n e m p t y c o m p a c t c o n v ex ) s u b s e t s o f M M C X ) p r o v i d e d w i t h

t h e H a u s d o r f f m e t r i c .

J . P . X i e [ 2 19 ] h a s a n n o u n c e d a p o s i ti v e s o l u t i o n to t h i s p r o b l e m , in t h e c a s e w h e r e A h a s a n o r m a l

s t r u c t u r e a n d T : A --+ K~ A ).

P r o b l e m 5 . D o e s H i l b e r t s p a ce h a v e t h e F P P k ) w i t h t h e c o n s t a n t k E , ~ ?

A b o u t t h i s p r o b l e m s e e S u b s ec . 2 .3 . 8, 2 . 3. 9, a n d 2 . 3 . t 0 .

Fo r m u la t i on s a n d o r ) d i sc uss ion s o f t he se p r ob l e m s c a n be f oun d in [ 79, 64 , 123, 208 , 82].

L i t e r a t u r e C i t e d

1 . R . R . A k h m e r o v , M . I . K a m e n s k i i , A . S . P o t a p o v , A . E . R o d k i n a , a n d B . N . S a d o v s k i i , Measure o/

goncompac~ness and Condensing Operators [ in Ru ss i a n] , Na uka , Novo ~ ib i rsk 1986) .

8 6



2 . Y u . G . Bor i s ov ich , B . D . G e lm an , A . D . M ys k i s , and V . V . O bukhovs k i i , O n ne w r es u l t s in the

t h e o r y o f m u l t i - v a l u e d m a p p i n g s . 1 . T o p o l o g ic a l c h a ra c t e r is t ic s a n d s o l v a b il i ty o f o p e r a t o r r e l a ti o n s ,

In : Ma t. Ana liz VoI. 25 I togi Na uki i Tekhn. [Rus s ian ] , A l l -U n ion In s t i tu t e fo r S c ien t i fi c and Techn ica l

In fo rmat ion (V IN ITI ) , A kad . N auk S S S R, M os cow (1987) , pp . 123 -197 .

3 . M . S . B rods k i i and D . P . M i lman , O n the ce n te r o f a convex s e t , Dokl . Akad . Nauk SSSR 59, No. 5 ,

837-840 (1948).

4 . K u o k F o n g V u , O n c o n v e x se t s o f a l m o s t n o r m a l s t ru c t u r e ,

Funk. Anal. i ego Pril .

18, No. 2 , 87-88

(1984).

5 . A . L . G arkav i , O n th e bes t ne t a nd th e bes t s ec t ion o f s e t s in a no r m ed s pace , I zv . Akad. Nauk.

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