HOUSTON JOURNAL OF MATHEMATICS
Volume 22, No. 4, 1996
A GEOMETRICAL COEFFICIENT
IMPLYING THE FIXED POINT PROPERTY
AND STABILITY RESULTS
T. DOMINGUEZ BENAVIDES
Communica ed by Gilles Pisie .
ABSTRACT. In his pape we de ine a new geome ic cons an M(X) in
Banach spaces such ha X has he ixed poin p ope y o nonexpansi e
mappings i M(X) > 1. We p o e ha M(X) •_ WCS(X), he inequali y
being s ic in many impo an classes o Banach spaces and we ob ain
lowe bounds o M(X) based upon ei he he modulus o nea uni o m
smoo hness o he modulus o he Opia] p ope y o he conjuga ed space.
We show ha his new cons an gi es us s abili y esul s o he ixed poin
p ope y wi h espec o œp-spaces which imp o e all p e ious esul s.
Le (M, d) be a me ic space. A mapping T: M -• M is said o be
nonexpansi e i •( •, •) _< •(•,•) o e e y x,y • M. A Banach space X
is said o ha e he ixed poin p ope y ( .p.p.) o nonexpansi e mappings
i o e e y con ex and weakly compac subse C o X, e e y nonexpansi e
mapping T: C -• C has a ixed poin . In 1965 B owde [B] and Ki k
[K], espec i ely, p o ed ha e e y uni o mly con ex Banach space and
any Banach space wi h no mal s uc u e has he .p.p. In 1981 Alspach [A]
showed ha L• ails o ha e he .p.p. O e he las 30 yea s many pape s
ha e appea ed s udying geome ic p ope ies o he Banach spaces (uni o m
1991 Ma hema ics Subjec Classi ica ion. 47H09,47H10.
Key wo ds and ph ases. nonexpansi e mapping, ixed poin , no mal s uc u e,
uni o m smoo hness, nea uni o m smoo hness, Opial p ope y.
This esea ch is pa ially suppo ed by he DGICYT ( esea ch p ojec PB 93-
1177-C01) and he Jun a de Andalucia (p ojec 1241)
835
836 T. DOMINGUEZ BENAVIDES
con exi y, uni o m smoo hness, nea uni o m con exi y, uncondi ional ba-
sis, e c) which assu e ei he no mal s uc u e o he .p.p. (see, o ins ance,
[Ma, GK]). A me hod o assu e he .p.p. o a Banach space X is o use
he "p oximi y" o X o ano he Banach space Y which "s ongly" sa is ies
his p ope y. To use his me hod we need a quan i ica ion o he .p.p.
The i s esul s in his di ec ion we e ob ained by Bynum [By1] de ining
ce ain no mal s uc u e coe icien s. In his pape and in la e pape s
[P 3, DL] se e al lowe bounds o he no mal s uc u e coe icien s we e
ob ained based upon he alue o ce ain geome ic coe icien s (Cla kson
modulus o uni o m con exi y, modulus o uni o m smoo hness, modulus o
nea uni o m con exi y, e c). These bounds can be unde s ood as s abili y
esul s o he .p.p. Recen ly Ga cla-False [Gal] de ined a new geome ic
coe icien R(X) which assu es he .p.p. (in pa icula he p o ed ha nea
uni o mly smoo h spaces ha e he .p.p. in spi e his spaces can ail o ha e
no mal s uc u e) and he ob ained s abili y esul s using his coe icien . In
his pape , ollowing he idea in [Gal], we de ine a new coe icien M(X)
and we p o e ha X has he .p.p. i M(X) > 1. This coe icien is, in gen-
e al, equal o g ea e han Bynum's weakly con e gen sequence coe icien
WCS(X), and s ic ly bigge han WCS(X) in many special spaces (see
Theo em 4.1 and ema k a e Theo em 2.5.). So we can imp o e a classic
esul in me ic ixed poin heo y: E e y Banach space wi h weak uni o m
no mal s uc u e has he .p.p. Ob iously, all lowe bounds o WCS(X)
also hold o M(X) and, in addi ion, we show lowe bounds o M(X) us-
ing ei he he modulus o nea uni o m smoo hness, de ined in [Do], ( ecall
ha WCS(X) can be equal o I in nea uni o mly smoo h spaces) o he
Opial modulus (see [LTX]) o he dual space. In he case o /•p-Spaces we
can di ec ly ob ain he alue o M(X). This alue gi es us s abili y esul s,
which a e s ic ely be e han all p e ious s abili y esul s in hese spaces
[JL, Kh, P 2].
1. No a ions and p elimina ies.
In he ollowing, X will be a Banach space, Bx he closed uni ball,
o X, and $x he uni sphe e. We shall o en use Bynum's weakly con e -
gen sequence coe icien WCS(X). Be o e in oducing i , we ecall some
de ini ions.
The asymp o ic diame e and adius o a sequence {x•} in a Banach
FIXED POINT PROPERTY 837
space X will be de ined by:
diama({x,,}) = limsupsup{llx,, - xmll ß n, • • k},
k
a{Xn} = in {lim sup IIx• - YlI'Y E {•} },
n
The weakly con e gen sequence coe icien o a Banach space X is de ined
by
WCS(X) =in { diama({Xn}) ß {Xn} is a weakly con e gen sequence
which is no no m con e gen }.
I is known [Byl] ha X has weak no mal s uc u e, ha is, e e y weakly
compac con ex subse o X wi h mo e han one membe is no diame al,
when WCS(X) > 1.
The ollowing esul shows how he coe icien WCS(X) can be use ul
o p o e he s abili y o he ixed poin p ope y.
Theo em 1.1 [Byl]. Le X and Y be iso no phic Banach spaces, hen
wcs(x) _< d(x, )wcs( ).
Se e al imp o emen s o his esul can be ound in [P 2]. The ol-
lowing o m o WCS(X) [DL, DLX1] will be e y impo an in his pape
Theo em 1.2. Le X be a Banach space wi hou he $chu p ope y. Then:
WeS(X) = in { limn'm;•7•m II• - Xml[ . {•} con e ges weakly
lim sup II•ll
o ze o and lim I1• - xmllexis s}
n,m;n•m
We ecall ha he mapping Px ( ) de ined by
{1 }
px( ) = sup •(ll• •- Yll •-IIx - YI[) - I ß II•ll • 1, Ilyll •
838 T. DOMINGUEZ BENAVIDES
is called modulus o uni o m smoo hness o X. A Banach space X is said
o be uni o mly smoo h i lim -•0 px( ) = O.
A mo e gene al concep is he nea uni o m smoo hness, he dual no ion o
he nea uni o m con exi y (see [P 1]). A Banach space X is said o be
nea uni o mly smoo h i o all s > 0 he e exis s q > 0 such ha o each
, 0 < < q, and o each basic sequence {xn} in Bx he e exis s k ) 1
such ha
IIx• + x•11 < 1 + s .
In [Do] a modulus o nea uni o m smoo hness is de ined in e lexi e Banach
spaces by
F( )=sup{in { Xl + Xn + Xl - xn , -1 'n>l
2
I is easy o check ha 0 < F( ) < o e e y > 0.
A Banach space X is said o sa is y he Opial condi ion [Op] i
lim in IIx• - xll < lim in IIx• - yll
o e e y sequence {x•} in X weakly con e gen o x and e e y poin y • x.
We say ha X sa is ies he uni o m Opial condi ion [P 4] i o e e y c > 0,
he e exis s an = (c) > 0 such ha
I + < limin IIx + x•ll
o all x e X wi h IIxl] _> c and all weakly null sequences {xn} in X such
ha limin •_• IIx•ll _> 1.
In [LTX] he ollowing modulus associa ed o he Opial condi ion has
been de ined:
De ini ion 1.3. Le X be a Banach space. The modulus o Opial o X is
de ined as
x(c) := in {limin llx + xnll- 1}, c >_ o,
whe e he in imum is aken o e all x • X wi h x I >- c and all weakly null
sequences {x•} in X wi h limin IIx•ll ) 1.
I is easily seen ha he uni o m Opial condi ion implies he Opial
condi ion and ha X sa is ies he uni o m Opial condi ion i and only i
x(c) > 0 o all c > O.
' {xn}weakly null in Bx}
FIXED POINT PROPERTY 839
The ollowing cons an o a Banach space X is de ined in [Ga2]:
R(X) = sup{limin IIx +
whe e he sup emum is aken o e all weakly null sequences in Bx and
o e all ec o s x in Bx. In [Gall he ollowing esul o exis ence o ixed
poin s and s abili y o he .p.p. is p o ed
Theo em 1.4. Le X and Y isomo phic Banach spaces. I d(X, Y)R(X) <
2 hen Y has he .p.p.
Finally, o a Banach space X, [X] will deno e, as usual, he quo ien
space eoo(X)/co(X) endowed wi h he no m II[z]ll- limsup IIz•ll, whe e
[z•] deno es he equi alen class o {z•} • eo•(X). By iden i ying x • X
wi h he class [x,x,...] we can conside X as a subse o [X]. I K is a
subse o X we can conside he se [K] = {[z•] • [X]: z• • K o e e y
n • iN}. I T is a mapping om K in o K we de ine [ ]: -. [•:] by
= I aqi.
The ollowing lemma is a basic ool in his pape :
Lin's lemma 1.5 [L]. Le X be a Banach space and K be a minimal
weakly compac con ex subse o X which is in a ian unde a nonexpansi e
mapping T. I [W] is a nonemp y closed con ex subse o [K] which is
in a ian unde [T] hen
sup{ll[w]- xll: [w]}- gla e(K)
o e e y x • K.
2. The coe icien M(X) and he .p.p.
In his sec ion we a e going o in oduce a new coe icien in Banach
spaces which yields a new ixed poin heo em. As we shall see, his heo em
enables us o p o e he exis ence o a ixed poin in Banach spaces wi hou
no mal s uc u e. P e iously we need o de ine a unipa ame e amily o
coe icien s.
De ini ion 2.1. Le X be a Banach space. Fo any nonnega i e numbe
a we de ine he coe icien
R(a,X) - sup{limin IIx• + xll}
whe e he sup emum is aken o e all x E X wi h Ilxll _< a and all weakly
null sequences in Bx such ha lim•,,•;•,• IIx - x•11 _< 1.

840 T. DOMINGUEZ BENAVIDES
Theo em 2.2. Le X be a Banach space and assume ha o some a _• 0
we ha e R(a, X) • 1 + a. Then X has he ixed poin p ope y.
P oo . We ollow an a gumen simila o ha in [Gal]. Assume ha X
ails o ha e he .p.p. Then we can ind a weakly compac and con ex
subse K o X such ha diam (K) -- I and K is minimal in a ian o a
nonexpansi e mapping T which has no ixed poin and we can also ind a
weakly null app oxima ed ixed poin sequence {x•} o T in K. We conside
he se
[W]- {[z•] e [K]' I[[zd-[xdl[ <_ 1- and limsuplimsup Ilz•-z,•l[ < }
• m
whe e = 1/(1 + a). I is easy o check ha [W] is a closed,con ex and
[T]-in a ian se . Fu he mo e [W] is non-emp y because i con ains [ x•].
The e o e, om Lemma 1.5 we know ha
sup{[[[wn]- x][ ' [w•] e [W]} -- 1
o e e y x e K. We ake [z•] e [W] and choose a weakly con e -
gen subsequence {y•} o {z•} such ha limsup[z•[[ - lim[[y• [and
lim•,,•;,•m []y• -y,•[] exis s. In his way we ha e
lim ly•-y-•l[ = limsuplimsup ][y•-y,,][ <_ limsuplimsup [[z•-z,•[] _< .
n, n ;n• n n n n n
We deno e he weak limi o {y• } as y. Fo e e y n e N we ha e Ily•- yll <
lim in ,• [[y,• - •1[. Hence
lim sup I[• - •11 = lim supli a sup [ly• - •ml[ < .
[ [ m
A posi i e ] can be chosen such ha lR(a,X) < 1 - R(a,X)/(1 + a).
Fo a la ge enough n we ha e 11• -•ll < + .. Fu he mo e 11•1 <
lim in ][y• - x• [[ _< 1 - . Hence
•-• = +. -•-S <R -W,x =aa, X).
Thus limsup [Iz•l[- limlly•[[ < R(a,X)( + V) < 1 which is con adic ion
wi h Lemma 1.5. []
The ollowing s abili y esul , simila o hose in Theo ems 1.1 and
1.4, can be p o ed by a s aigh o wa d a gumen :
FIXED POINT PROPERTY 841
Theo em 2.3. Le X and Y be isomo phic Banach spaces. Then
R(a,Y) < d(x,Y)R(a,x)
o e e y nonnega i e numbe a.
De ini ion 2.4. Le X be a Banach space. We de ine he coe icien M(X)
as { l+a }
sup R(a,X) ' a _> 0 .
The ollowing esul is a di ec consequence o Theo ems 2.2 and 2.3:
Theo em 2.5. Le X be a Banach space. I M(X) > i hen X has he
.p.p. I Y is ano he Banach space which is isomo phic o X and d(X, Y) <
M(X) hen Y has he .p.p.
Rema ks. (a) F om Theo em 1.2 i is clea ha R(0, X) -- 1/WCS(X).
Thus M(X) _> WC$(X). This inequali y can be s ic . Fo ins ance, we
conside Bynum's space X = g2,oo, ha is, X is g2 wi h he no m
max{[[x+[[, [[x-[[} whe e x+(n)- max{x(n), 0} and x-(n)- max{-x(n), 0}
a e espec i ely he posi i e and he nega i e pa o x, and [[. [[ is he eu-
clidean no m. Since g•,• ails o ha e no mal s uc u e [By2] we know ha
WC$(g•,•) = 1. Howe e we shall p o e in Sec ion 4 ha M(g2,•) =
(b) Theo em 2.4 is also a s ic imp o emen o he esul in [Gall.
Indeed, conside X = g•,l, ha is, g• wi h he no m
This space has no mal s uc u e [By2, DLX], so M(X) >_ WC'S(X) = x/-•.
Howe e , conside ing he ec o x: el and he sequence xn = -en+l i is
clea ha R(•2,1) = 2.
3. Lowe bounds o M(X).
Since M(X) _> WCS(X), all lowe bounds o WCS(X) based upon
he Cla kson modulus o con exi y, he modulus o nea uni o m con exi y
and he modulus o uni o m smoo hness (see [By1, P 3, DL]) also hold
o M(X). We shall gi e in his sec ion se e al new bounds which do no
longe hold o WC$(X).
842 T. DOMINGUEZ BENAVIDES
Theo em 3.1. Le X be a e iezi e Banach space and deno e
F=in l+F(s)-•'sß[0,1] .
Then R(a,X) _< I + aF i a _< 2 and R(a,X) _< a + 2F - I i a _> 2. In
pa icula , M(X) _> 3/(1 + 2F) and M(X) > I i F'(0) < 1/2.
P oo . The s a emen is ob ious i a = 0. Assume 2 _> a > 0. Le {xn} be
a weakly null sequence in Bx and x ß X be a ec o such ha x I = _<
a. Taking subsequences we can assume ha lim IIxn + xll exis s. Fo an
a bi a y posi i e numbe •7, a numbe ß [0, /2] can be chosen such ha
1--+F <F+o.
Wi h hese assump ions we ha e
[[x + x,l[ = - + -- _< - + -x, +(l- ).
I {x•} --> 0 i is clea ha
limin [lx+x•ll_< (l+F(2•)) +(i- )_< •I+ F+l_<l+aF+a•7.
(No e ha ( ) _> 0 implies F _> 1/2 > 0 and hus F < aF). I {x•}
does no con e ge o ze o, we can assume ha he sequence {yn} de ined
by Yl = x, Yn '- Xn-1 o n > 1 is a basic sequence wi h a bi a y basic
cons an c > I (see [LT, page 5]). Hence, we ha e
1112•+2 •11 < 1 1
IIx+ 11- _ (1111+11+2 x11)< (cll-2 xll+ll+2 nll).
Taking again subsequences we can assume
-+ x• + - x• -I<F -- +7.
FIXED POINT PROPERTY 843
Thus
----x,• + -+--x,• +(l- )_<
IIx+x•ll_•
c 1]
[l+F(2•)+ /]+(l- ) <-c[ (l+F(2•)+ /-•) + +(c-1) _<
c(1 + F + 2 ) + (c- 1)a _< c(1 + aF + 2a ) + (c- 1)a.
Hence
R(a,X) _< c(1 + aF + 2a ) + (c- 1)a.
Since c > 1 and /> 0 a e a bi a y we ob ain R(a,X) _< 1 + aF. I a _> 2
we ha e
Applying he abo e esul o he sequence 2x/ + x,• we ha e R(a, X) _<
(a- 2) + 1 + 2 = a- 1 + 2 . Taking a = 2 we ob ain M(X) _> 3/(1 + 2 ).
Finally, i F'(0) < 1/2 i is clea ha F < 1. []
We ha e no used in he p oo o Theo em 3.1 he condi ion lim I1• -
x,• _< 1. This condi ion le s us imp o e he esul :
Theo em 3.2. Le X be a e lexi e Banach space and deno e
swcs(x) }
F•=in l+F(s)- 2 'sE [0,1] .
Then M(X) _> 3/(1+2F'). In pa icula , M(X) > 1 i F'(O) < WC$(X)/2.
P oo . We use he same a gumen s as hose in he p oo o Theo em 3.1,
no ing ha he condi ion lim•,• ;•#,• x•-x• ] < 1 le s us assume li a I•l <_
1/WC$(X). []
I is an open ques ion o us i 1/F is a lowe bound o M(X) in a
simila way as he lowe bound o WC$(X) ob ained in [P 3], using he
modulus o uni o m smoo hness.