Hindawi Publishing Co po a ion
Fixed Poin Theo y and Applica ions
Volume 2010, A icle ID 546761, 20 pages
doi:10.1155/2010/546761
Resea ch A icle
Does Ki k’s Theo em Hold o Mul i alued
Nonexpansi e Mappings?
T. Dom´
ınguez Bena ides and B. Ga i a
Facul ad de Ma em´
a icas, Uni e sidad de Se illa, P.O. Box 1160, 41080 Se illa, Spain
Co espondence should be add essed o T. Dom´
ınguez Bena ides, [email p o ec ed]
Recei ed 25 Sep embe 2009; Accep ed 29 Decembe 2009
Academic Edi o : Mohamed A. Khamsi
Copy igh q2010 T. Dom´
ınguez Bena ides and B. Ga i a. This is an open access a icle
dis ibu ed unde he C ea i e Commons A ibu ion License, which pe mi s un es ic ed
use, dis ibu ion, and ep oduc ion in any medium, p o ided he o iginal wo k is p ope ly ci ed.
Fixed Poin Theo y o mul i alued mappings has many use ul applica ions in Applied Sciences,
in pa icula , in Game Theo y and Ma hema ical Economics. Thus, i is na u al o y o ex ending
he known ixed poin esul s o single- alued mappings o he se ing o mul i alued mappings.
Some heo ems o exis ence o ixed poin s o single- alued mappings ha e al eady been ex ended
o he mul i alued case. Howe e , many o he ques ions emain s ill open, o ins ance, he
possibili y o ex ending he well-known Ki k’s Theo em, ha is: do Banach spaces wi h weak
no mal s uc u e ha e he ixed poin p ope y FPP o mul i alued nonexpansi e mappings?
The e a e many p ope ies o Banach spaces which imply weak no mal s uc u e and consequen ly
he FPP o single- alued mappings  o example, uni o m con exi y, nea ly uni o m con exi y,
uni o m smoo hness,.... Thus, i is na u al o conside he ollowing p oblem: do hese p ope ies
also imply he FPP o mul i alued mappings? In his way, some pa ial answe s o he p oblem
o ex ending Ki k’s Theo em ha e appea ed, p o ing ha hose p ope ies imply he exis ence
o ixed poin o mul i alued nonexpansi e mappings. He e we p esen he main known esul s
and cu en esea ch di ec ions in his subjec . This pape can be conside ed as a su ey, bu some
new esul s a e also shown.
1. In oduc ion
The p esence o absence o a ixed poin i.e., a poin which emains in a ian unde a map
is an in insic p ope y o a map. Howe e , many necessa y o sufficien condi ions o he
exis ence o such poin s in ol e a mix u e o algeb aic, opological, o me ic p ope ies o
he mapping o i s domain. By Me ic Fixed Poin Theo y, we unde s and he b anch o
Fixed Poin Theo y conce ning hose esul s which depend on a me ic and which a e no
p ese ed when his me ic is eplaced by ano he equi alen me ic. The i s me ic ixed
poin heo em was gi en by Banach in 1922.
2 Fixed Poin Theo y and Applica ions
Theo em 1.1 Banach Con ac ion P inciple, 1.Le Xbe a comple e me ic space and T:X→
Xa con ac i e mapping, ha is, he e exis s k∈0,1such ha dTx,Ty≤kdx,y o e e y
x,y ∈X.ThenThas a (unique) ixed poin x0. Mo eo e , x0limnTnx o e e y x∈X.
Banach Theo em is a basic ool in Func ional Analysis, Nonlinea Analysis and
Diffe en ial Equa ions. Thus, i is na u al o look o some gene aliza ions unde weake
assump ions.
Fo many yea s Me ic Fixed Poin Theo y jus s udied some ex ensions o Banach
Theo em elaxing he con ac i eness condi ion, and he ex ension o his esul o
mul i alued mappings. In he 1960s, Me ic Fixed Poin Theo y ecei ed a s ong boos when
Ki k 2p o ed ha e e y single aluednonexpansi e mapping T:C→C, de ined om
a con ex closed bounded subse Co a e lexi e Banach space wi h no mal s uc u e, has a
ixed poin .
The celeb a ed Ki k’s heo em had a p o ound impac in he de elopmen o Fixed
Poin Theo y and inicia ed he sea ch o mo e gene al condi ions o a Banach space and o
asubse Cwhich assu e he exis ence o ixed poin s.
The esul ob ained by Ki k is, in some sense, su p ising because i uses geome ic
p ope ies o Banach spaces commonly used in Linea Func ional Analysis, bu a ely
conside ed in Nonlinea Analysis un il hen. Thus, i is he s a ing poin o a new
ma hema ical ield: he applica ion o he Geome ic Theo y o Banach Spaces o Fixed
Poin Theo y. F om ha momen on, many esea che s ha e ied o exploi his connec ion,
essen ially conside ing some o he geome ic p ope ies o Banach spaces which can be
applied o p o e he exis ence o ixed poin s o diffe en ypes o nonlinea ope a o s e.g.,
uni o m smoo hness, Opial p ope y, nea ly uni o m con exi y, nea ly uni o m smoo hness,
e c..
Fixed Poin Theo y o mul i alued mappings has use ul applica ions in Applied
Sciences, in pa icula , in Game Theo y and Ma hema ical Economics. Thus, i is na u al
o s udy he p oblem o he ex ension o he known ixed poin esul s o single alued
mappings o he se ing o mul i alued mappings.
Some heo ems o exis ence o ixed poin s o single- alued mappings ha e al eady
been ex ended o he mul i alued case. Fo example, in 1969 Nadle 3ex ended he Banach
Con ac ion P inciple o mul i alued con ac i e mappings in comple e me ic spaces.
Howe e , many o he ques ions emain open, o ins ance, he possibili y o ex ending he
well-known Ki k’s Theo em 2, ha is, do Banach spaces wi h weak no mal s uc u e ha e
he ixed poin p ope y FPP o mul i alued nonexpansi e mappings?
The e a e many p ope ies o Banach spaces which imply weak no mal s uc u e and
consequen ly he FPP o single alued mappings e.g., uni o m con exi y, nea ly uni o m
con exi y, uni o m smoo hness, .... Thus, i is na u al o conside he ollowing p oblem:
Do hese p ope ies also imply he FPP o mul i alued mappings? As a consequence, some
pa ial answe s o he p oblem o ex ending Ki k’s Theo em ha e appea ed, which a e
di ec ed o p o e ha hose p ope ies imply he exis ence o ixed poin o mul i alued
nonexpansi e mappings.
He e we p esen he main known esul s and cu en esea ch di ec ions in his subjec .
This pape can be conside ed as a su ey, bu some new esul s a e also included.
2. P elimina ies
In his sec ion we ecall he no ion o no mal s uc u e and some p ope ies o Banach spaces
which imply no mal s uc u e.
Fixed Poin Theo y and Applica ions 3
No mal s uc u e plays an essen ial ole in some p oblems o Me ic Fixed Poin
Theo y, especially hose conce ning nonexpansi e mappings. The no ion o no mal s uc u e
was in oduced by B odski˘
ıand Mil’man 4in 1948 in o de o s udy ixed poin s o
isome ies. La e , he no ion o no mal s uc u e was gene alized o he weak opology.
De ini ion 2.1. A Banach space Xis said o ha e no mal s uc u e NS esp., weak no mal
s uc u e w-NS i o e e y bounded closed  esp., weakly compac con ex subse Co X
wi h diamC:sup{x−y:x, y ∈C}>0, he e exis s x∈Csuch ha sup{x−y:y∈
C}<diamC.
In 1965 Ki k 2ob ained a s ong connec ion be ween no mal s uc u e and he FPP
o nonexpansi e mappings.
Theo em 2.2. Le Cbe a bounded closed ( esp., weakly compac ) con ex subse o a Banach space X
and le T:C→Cbe a nonexpansi e mapping (i.e., Tx−Ty≤x−y o e e y x, y ∈C). I Xis
a e lexi e Banach space wi h no mal s uc u e ( esp., a Banach space wi h w-NS), hen Thas a ixed
poin .
Bynum 5de ined wo coefficien s ela ed o no mal s uc u e and weak no mal
s uc u e.
De ini ion 2.3. The no mal s uc u e coefficien o a Banach space Xis de ined by
NXin diamA
A:A⊂Xcon ex closed and bounded wi h diamA>0,2.1
whe e diamAdeno es he diame e o Ade ined by diamAsup{x−y:x, y ∈A}and
Adeno es he Chebyshe adius o Ade ined by Ain {sup{x−y:y∈A}:x∈A}.
The weakly con e gen sequence coefficien o Xis de ined by
WCSXin diama{xn}
a{xn},2.2
whe e he in imum is aken o e all weakly con e gen sequences {xn}which a e no no m
con e gen , whe e,
diama{xn}lim
k→∞sup{xn−xm:n, m ≥k},
a{xn}in lim sup
nxn−x:x∈co{xn}2.3
deno e he asymp o ic diame e and adius o {xn}, espec i ely.
We ecall ha Xis said o ha e uni o m no mal s uc u e UNS esp., weak uni o m
no mal s uc u e w-UNS i NX>1 esp., WCSX>1. No ice ha his is no
he common de ini ion o weak uni o m no mal s uc u e and is o en known as Bynum’s
condi ion. I is known ha i Xhas uni o m no mal s uc u e, hen Xis e lexi e 6.
4 Fixed Poin Theo y and Applica ions
In he la es i y yea s, some geome ical p ope ies implying no mal s uc u e ha e
been s udied. He e we a e going o ecall some o hese p ope ies and some esul s which
p o e ha hese p ope ies imply he exis ence o ixed poin o mul i alued mappings.
Fi s we conside he Opial p ope y. Opial 7was he i s who s udied such a
p ope y gi ing applica ions o Fixed Poin Theo y. The uni o m Opial p ope y was de ined
in 8by P us, and he Opial modulus was in oduced in 9by Lin e al.
De ini ion 2.4. We will say ha a Banach space Xsa is ies he Opial p ope y i o e e y
weakly null sequence {xn}and e e y x/
0inX, we ha e
lim in
n→∞ xn<lim in
n→∞ xnx.2.4
We will say ha Xsa is ies he nons ic Opial p ope y i
lim in
n→∞ xn≤lim in
n→∞ xnx2.5
unde he same condi ions.
The Opial modulus o Xis de ined o c≥0as
Xcin lim in
nxnx−1,2.6
whe e he in imum is aken o e all x∈Xwi h x≥cand all weakly null sequences {xn}
in Xwi h lim in nxn≥1.
We will say ha Xsa is ies he uni o m Opial p ope y i Xc>0 o all c>0.
The e a e some ela ionships be ween he no ions o Opial p ope y and no mal
s uc u e. I Xis a Banach space which sa is ies he Opial p ope y, hen Xhas w-NS 10.
On he o he hand, WCSX≥1 X19, Theo em 3.2. Consequen ly, Xhas w-UNS i
X1>0.
Nex we s udy he uni o m con exi y o he space, which is ano he geome ical
p ope y ela ed wi h no mal s uc u e. We ecall ha a Banach space Xis uni o mly con ex
UCi and only i
δX:in 1−



xy
2


:x,y ∈Bx,
x−y
≥>02.7
o each ∈0,2, o equi alen ly
ε0X:sup{≥0:δX0}0.2.8
The Cla kson modulus δXand he coefficien o no mal s uc u e NXa e ela ed
by he ollowing inequali y: NX≥1−δX1−1. Consequen ly, he condi ion δX1>0
implies ha Xis e lexi e and has uni o m no mal s uc u e. In pa icula , no ice ha no
only do uni o mly con ex spaces ha e no mal s uc u e, bu so do all hose spaces which do
no ha e segmen s o leng h g ea e han o equal o 1 nea he uni sphe e.
Fixed Poin Theo y and Applica ions 5
In 1980 Huff11ini ia ed he s udy o nea ly uni o m con exi y which is an
in ini e-dimensional gene aliza ion o uni o m con exi y. Independen ly o Huff,Goebeland
Se¸kowski 12also in oduced a p ope y which is equi alen o nea ly uni o m con exi y
unde he name o noncompac uni o m con exi y. I is known ha a Banach space Xis
nea ly uni o mly con ex NUCi and only i
ΔX,φ:in 1−d0,A
:A⊂BXcon ex,φ
A>
>02.9
o each >0, o equi alen ly
εφX:sup≥0:ΔX,φ00,2.10
whe e φis a measu e o noncompac ness. Also we a e going o use he ollowing equi alen
de ini ion: Xis NUC i and only i Xis e lexi e and
ΔX:in 1−x:{xn}⊂BX,x
nx,lim in
nxn−x≥>02.11
o each >0, o equi alen ly
Δ0X:sup{>0:ΔX0}0.2.12
When Xis a e lexi e Banach space, βis he sepa a ion measu e and χis he Hausdo ff
measu e  o de ini ions see, o ins ance, 13o 14, we ha e he ollowing ela ionships
among he diffe en moduli:
ΔX,β≤ΔX≤ΔX,χ,2.13
and consequen ly,
εβX≥Δ0X≥εχX.2.14
I he space Xsa is ies he nons ic Opial p ope y, hen Δ0Xcoincides wi h εχX.
On he o he hand, i εβX<1in pa icula , i Xis NUC, hen Xis e lexi e and has
weak uni o m no mal s uc u e see 13, page 125.
The dual concep o uni o m con exi y is uni o m smoo hness which is also ela ed o
no mal s uc u e. A Banach space Xis said o be uni o mly smoo h USi
ρ
X0lim
→0
ρX 
0,2.15

6 Fixed Poin Theo y and Applica ions
whe e ρXis he modulus o smoo hness o X, de ined by
ρX sup1
2
x y

x− y
−1:x≤1,
y
≤12.16
o ≥0.
I is known ha ρ
X0<1/2 implies ha Xis e lexi e and has uni o m no mal
s uc u e 15–17. Howe e , he in ini e-dimensional gene aliza ion o uni o m smoo hness,
nea ly uni o m smoo hness, does no imply no mal s uc u e 13, Example VI.2.
3. Some P ope ies Implying Weak No mal S uc u e and
he FPP o Mul i alued Mappings
In his sec ion we a e going o show some esul s which p o e ha some p ope ies implying
weak no mal s uc u e also imply he exis ence o ixed poin o mul i alued nonexpansi e
mappings. As a consequence hese esul s gi e some pa ial answe s o he p oblem o
ex ending Ki k’s Theo em.
Th oughou his sec ion KX esp., KCX will deno e he amily o all nonemp y
compac  esp., compac con exsubse s o X. We ecall ha a mul i alued mapping T:
X→KXis said o be nonexpansi e i HTx,Ty≤x−y o e e y x, y ∈X, whe e
H·,·deno es he Hausdo ffme ic gi en by
HA, B:maxsup
a∈A
da, B,sup
b∈B
db,A3.1
o e e y bounded subse s Aand Bo X.
In 1973 Lami Dozo ga e he ollowing esul o exis ence o ixed poin o hose spaces
which sa is y he Opial p ope y.
Theo em 3.1 Lami Dozo 18, Theo em 3.2.Le Xbe a Banach space which sa is ies he Opial
p ope y, le Cbe a weakly compac con ex subse o X, and le T:C→KCbe a nonexpansi e
mapping. Then Thas a ixed poin , ha is, he e exis s x∈Csuch ha x∈Tx.
In 1974 Lim 19ga e a simila esul o uni o mly con ex spaces using Edels ein’s
me hod o asymp o ic cen e s 20.
Theo em 3.2 Lim 19.Le Xbe a uni o mly con ex Banach space, le Cbe a closed bounded
con ex subse o Xand T:C→KCbe a nonexpansi e mapping. Then Thas a ixed poin .
In 1990 Ki k and Massa p o ed he ollowing pa ial gene aliza ion o Lim’s Theo em
using asymp o ic cen e s o sequences and ne s. We ecall ha , gi en a bounded sequence
{xn}in a Banach space Xand a subse Co X, he asymp o ic cen e o {xn}wi h espec o
Cis de ined by
AC, {xn}:x∈C: lim sup
nxn−x C, {xn},3.2
Fixed Poin Theo y and Applica ions 7
whe e C, {xn}deno es he asymp o ic adius o {xn}wi h espec o Cde ined by
C, {xn}:in lim sup
nxn−x:x∈C.3.3
Theo em 3.3 Ki k and Massa 21.Le Cbe a closed bounded con ex subse o a Banach space
Xand T:C→KCCa nonexpansi e mapping. I he asymp o ic cen e in Co each bounded
sequence o Xis nonemp y and compac , hen Thas a ixed poin .
We do no know a comple e cha ac e iza ion o hose spaces in which asymp o ic
cen e s o bounded sequences a e compac . Ne e heless, he e a e some pa ial answe s,
o example, k-uni o mly con ex Banach spaces sa is y ha condi ion 22. Howe e , an
example gi en by Kuczumo and P us 23shows ha in nea ly uni o mly con ex spaces,
he asymp o ic cen e o a bounded sequence wi h espec o a closed bounded con ex subse
is no necessa ily compac . The e o e, he p oblem o ob aining ixed poin esul s in nea ly
uni o mly con ex spaces emained open. This ques ion  oge he wi h he same ques ion o
uni o mly smoo h spacesexplici ly appea ed in a su ey abou Me ic Fixed Poin Theo y
o mul i alued mappings published by Xu 24in 2000.
The analysis o he impo ance o he asymp o ic cen e in Ki k-Massa Theo em led
Dom´
ınguez Bena ides and Lo enzo o s udy some connec ions be ween asymp o ic cen e s
and he geome y o ce ain spaces, including nea ly uni o mly con ex spaces. Thus, in 25
Dom´
ınguez and Lo enzo ob ained he ollowing ela ionship be ween he Chebyshe adius
o he asymp o ic cen e o a bounded sequence and he modulus o noncompac con exi y
wi h espec o he measu es βand χ.
Theo em 3.4 see 25, Theo em 3.4.Le Cbe a closed con ex subse o a e lexi e Banach space
Xand {xn}a bounded sequence in Cwhich is egula wi h espec o C(i.e., he asymp o ic adius is
in a ian o all subsequences o {xn}). Then
CAC, {xn} ≤1−ΔX,β1− C, {xn},3.4
whe e he Chebyshe adius o a bounded subse Do X ela i e o Cis de ined by
CD:in sup
x−y
:y∈D:x∈C.3.5
Mo eo e , i Xsa is ies he nons ic Opial p ope y, hen
CAC, {xn} ≤1−ΔX,χ1− C, {xn}.3.6
The p e ious inequali ies gi e an i e a i e me hod which educes a each s ep he
alue o he Chebyshe adius o a chain o asymp o ic cen e s. Consequen ly, Dom´
ınguez
and Lo enzo deduced in 26 he ollowing pa ial ex ension o Ki k’s Theo em which, in
pa icula , assu es ha nea ly uni o mly con ex spaces ha e he ixed poin p ope y o
mul i alued nonexpansi e mappings.
8 Fixed Poin Theo y and Applica ions
Theo em 3.5 see 26, Theo em 3.5.Le Cbe a nonemp y closed bounded con ex subse o a
Banach space Xsuch ha εβX<1.Le T:C→KCCbe a nonexpansi e mapping. Then Thas
a ixed poin .
This esul gua an ees, in pa icula , he exis ence o ixed poin s in nea ly uni o mly
con ex spaces because εβX0i Xis NUC, gi ing a posi i e answe o one o he
p e ious open p oblems p oposed by Xu.
Dhompongsa e al. 27obse ed ha he main ool used in he p oo s in 25,26,in
o de o ob ain ixed poin esul s o mul i alued nonexpansi e mappings, is a ela ionship
be ween he Chebyshe adius o he asymp o ic cen e o a bounded sequence and
he asymp o ic adius o he sequence. This ela ionship also gi es an i e a i e me hod
which educes a each s ep he alue o he Chebyshe adius o a chain o asymp o ic
cen e s. Consequen ly, in 27,28 hey in oduced he Dom´
ınguez-Lo enzo condi ion DL-
condi ion, in sho and p ope y Din he ollowing way.
We ecall ha a sequence {xn}is egula wi h espec o Ci C, {xn} C, {xni}
o all subsequences {xni}o {xn},and{xn}is asymp o ically uni o m wi h espec o Ci
AC, {xn}AC, {xni} o all subsequences {xni}o {xn}.
De ini ion 3.6. A Banach space Xis said o sa is y he DL-condi ion i he e exis s λ∈0,1
such ha o e e y weakly compac con ex subse Co Xand o e e y bounded sequence
{xn}in Cwhich is egula wi h espec o C
CAC, {xn} ≤λ C, {xn}.3.7
A Banach space Xis said o sa is y p ope y Di he e exis s λ∈0,1such ha o
any nonemp y weakly compac con ex subse Co X, any bounded sequence {xn}in Cwhich
is egula and asymp o ically uni o m wi h espec o C, and any sequence {yn}⊂AC, {xn}
which is egula and asymp o ically uni o m wi h espec o C, we ha e
C, yn≤λ C, {xn}.3.8
F om he de ini ion i is easy o deduce ha p ope y Dis weake han he DL-
condi ion. Dhompongsa e al. p o ed in 28, Theo em 3.2and 28, Theo em 3.5 ha
p ope y Dimplies w-NS and he FPP o mul i alued nonexpansi e mappings.
Theo em 3.7 see 28, Theo em 3.3.Le Xbe a Banach space sa is ying p ope y (D). Then Xhas
w-NS.
Theo em 3.8 see 28, Theo em 3.6.Le Cbe a nonemp y weakly compac con ex subse o a
Banach space Xwhich sa is ies p ope y (D). Le T:C→KCCbe a nonexpansi e mapping. Then
Thas a ixed poin .
F om Theo em 3.5 e e y Banach space wi h εβX<1sa is ies heDL-condi ion.
In his pape we p esen some o he p ope ies conce ning geome ical cons an s o Banach
spaces which also imply he DL-condi ion o p ope y D.
Since ou goal is o s udy i p ope ies implying w-NS also imply he FPP o
mul i alued mappings, a possible app oach o ha p oblem is o s udy i hese p ope ies
imply ei he he DL-condi ion o p ope y D. These esul s will gi e only pa ial answe s
Fixed Poin Theo y and Applica ions 9
o he p oblem o ex ending Ki k’s Theo em o mul i alued mappings because we know
ha uni o m no mal s uc u e does no imply p ope y D29,P oposi ion5; he e o e,
he p oblem o ex ending Ki k’s Theo em canno be ully sol ed by his app oach. In his
se ing he ollowing esul s ha e been ob ained.
Theo em 3.9 Dhompongsa e al. 27, Theo em 3.4.Le Xbe a uni o mly nonsqua e Banach
space wi h p ope y WORTH. Then Xsa is ies he (DL)-condi ion.
We ecall ha a Banach space Xis uni o mly nonsqua e i he e exis s δ>0such ha x
y∧x−y≤2−δ o e e y x, y ∈BXo equi alen ly JX<2,whe e JXdeno es he James
cons an o Xde ined by
JXsup
xy
∧
x−y
:x,y ∈BX.3.9
Xis said o sa is y p ope y WORTH i
lim sup
nxnxlim sup
nxn−x3.10
o any x∈Xand any weakly null sequence {xn}in X.
Theo em 3.10 Dhompongsa e al. 28, Theo em 3.7.Le Xbe Banach space such ha
CNJX<1WCSX2
4,3.11
whe e CNJXdeno es he Jo dan- on Neumann cons an o Xde ined by
CNJXsup
xy
2
x−y
2
2x22
y
2:x,y ∈Xno bo h ze o.3.12
Then Xsa is ies p ope y (D).
Theo em 3.11 Dom´
ınguez Bena ides and Ga i a 29, Co olla y 1.Le Xbe a Banach space
such ha
ρ
X0<1
2.3.13
Then Xsa is ies he (DL)-condi ion. In pa icula , uni o mly smoo h Banach spaces (ρ
X00)
sa is y he (DL)-condi ion.
Theo em 3.12 Dom´
ınguez Bena ides and Ga i a 29, Co olla y 2.Le Xbe a Banach space
such ha one o he ollowing wo equi alen condi ions is sa is ied:
1 X1>0,
2Δ
0X<1.
Then Xsa is ies he (DL)-condi ion.
16 Fixed Poin Theo y and Applica ions
Theo em 4.7 see 40, Theo em 3.1.Le ρbe a con ex unc ion modula sa is ying he Δ2- ype
condi ion, Ca nonemp y ρ-bounded ρ-closed subse o Lρ, and T:C→FρCaρ-con ac ion
mapping, ha is, he e exis s a cons an k∈0,1such ha
HρT ,Tg≤kρ −g, ,g∈C. 4.10
Then Thas a ixed poin .
By using ha esul , hey p o ed he exis ence o ixed poin s o mul i alued ρ-
nonexpansi e mappings.
Theo em 4.8 see 40, Theo em 3.4.Le ρbe a con ex unc ion modula sa is ying he Δ2- ype
condi ion, Ca nonemp y ρ-a.e. compac ρ-bounded con ex subse o Lρ, and T:C→KρCa
ρ-nonexpansi e mapping. Then Thas a ixed poin .
They also applied he abo e heo em o ob ain ixed poin esul s in he Banach space
L1 esp., 1 o mul i alued mappings whose domains a e compac in he opology o he
con e gence locally in measu e  esp., w∗- opology.
Conside he space LpΩ,μ o a σ- ini e measu e μwi h he usual no m. Le Cbe
a bounded closed con ex subse o Lp o 1 <p<∞and T:C→KCa mul i alued
nonexpansi e mapping. Because o uni o m con exi y o Lp, i is known ha Thas a
ixed poin . Fo p1, Tcan ail o ha e a ixed poin e en in he single alued case o
a weakly compac con ex se Csee 43. Howe e , since L1is a modula space whe e
ρ Ω| |dμ   o all ∈L1,Theo em 4.8 implies he exis ence o a ixed poin when
we de ine mappings on a ρ-a.e. compac ρ-bounded con ex subse o L1. Thus he ollowing
can be s a ed.
Co olla y 4.9 see 40, Co olla y 3.5.Le Ω,μbe as abo e, C⊂L1Ω,μa nonemp y bounded
con ex se which is compac o he opology o he con e gence locally in measu e, and T:C→KC
a nonexpansi e mapping. Then Thas a ixed poin .
In he case o he space 1,we also can ob ain a bounded closed con ex se Cand a
nonexpansi e mapping T:C→Cwhich is ixed poin ee. Indeed, conside he ollowing
easy and well-known example.
Le
C{xn}∈1:0≤xn≤1,∞

n1
xn1.4.11
De ine a nonexpansi e mapping T:C→Cby
Tx0,x
1,x
2,x
3,...
,whe e x{xn},4.12
hen Tis a ixed poin ee map. Howe e , i we conside Lρ1,whe e ρxx, o all
x∈1, hen ρ-a.e. con e gence and ω∗-con e gence a e iden ical on bounded subse s o 1
see 36. This ac leads o he ollowing co olla y.

Fixed Poin Theo y and Applica ions 17
Co olla y 4.10 see 40, Co olla y 3.6.Le Cbe a nonemp y ω∗-compac con ex subse o 1and
T:C→KCa nonexpansi e mapping. Then Thas a ixed poin .
Nex we will gi e a p ope y o closed con ex bounded subse s o 1mo e gene al han
weak s a compac ness which implies he ixed poin p ope y o nonexpansi e mappings.
Dom´
ınguez e al. in oduced in 44some compac ness condi ions conce ning
p oximinal subse s called P ope y P. Following his idea we will use he ollowing simila
no ion o modula unc ion spaces.
De ini ion 4.11. Le Cbe a nonemp y ρ-closed con ex ρ-bounded subse o Lρ.I issaid ha C
has P ope y Pρi o e e y ∈Lρ,which is he ρ-a.e. limi o a sequence in C, hese Pρ,C 
is a nonemp y and ρ-compac subse o C, whe e Pρ,C {g∈C:ρg− dis ρ ,C}.
Using ha no ion and he ollowing wo lemmas, we ob ain a new ixed poin esul
o mul i alued ρ-nonexpansi e mappings.
Lemma 4.12 see 40, Lemma 3.3.Le ρbe a con ex unc ion modula sa is ying he Δ2- ype
condi ion, ∈Lρ, and Ka nonemp y ρ-compac subse o Lρ. Then he e exis s g0∈Ksuch ha
ρ −g0dis ρ ,K.4.13
Lemma 4.13 see 37, Lemma 1.3.Le ρbe a unc ion modula sa is ying he Δ2- ype condi ion,
and { n}nbe a sequence in Lρsuch ha n
ρ-a.e.
→ ∈Lρand he e exis s k>1such ha supnρk n−
 <∞. Then,
lim in
n→∞ ρ n−glim in
n→∞ ρ n− ρ −g∀g∈Lρ.4.14
Theo em 4.14. Le ρbe a con ex unc ion modula sa is ying he Δ2- ype condi ion, Ca nonemp y
ρ-closed ρ-bounded con ex subse o Lρsa is ying P ope y Pρsuch ha e e y sequence in Chas a
ρ-a.e. con e gen subsequence in Lρ, and T:C→KρCCaρ-nonexpansi e mapping. Then Thas
a ixed poin .
P oo . Fix 0∈C. Fo each n∈N, heρ-con ac ion Tn:C→FρCis de ined by
Tn 1
n 01−1
nT , ∈C. 4.15
By Theo em 4.7, we can conclude ha Tnhas a ixed poin , say n.I iseasy osee ha
dis ρ n,T
n≤1
ndiamρC−→ 0.4.16
By ou assump ions, we can assume, by passing h ough a subsequence, ha n
ρ-a.e.
→ o
some ∈Lρ.ByLemma 4.12, o each n∈N he e exis s gn∈T nsuch ha
ρ n−gndis ρ n,T
n.4.17
18 Fixed Poin Theo y and Applica ions
Now we a e going o show ha Pρ,C ∩Th/
∅ o each h∈Pρ,C . Taking any h∈Pρ,C ,
om he ρ-compac ness o Th and Lemma 4.12, we can ind hn∈Th such ha
ρgn−hndis ρgn,Th
≤HρT n,Th
≤ρ n−h,4.18
and we can assume, by passing h ough a subsequence, ha hn
ρ
→h0 o some h0∈Th.F om
abo e and using Lemma 4.13, i ollows ha
lim in
nρ n−h0lim in
nρgn−h0lim in
nρgn−hn≤lim in
nρ n−h
lim in
nρ n− ρ −h.
4.19
On he o he hand, by Lemma 4.13 we also ha e
lim in
nρ n−h0lim in
nρ n− ρ −h0.4.20
Thus, we deduce ρ −h0≤ρ −h, which implies ha h0∈Pρ,C and so Pρ,C ∩Th/
∅.
Now we de ine he mapping 
T:Pρ,C →KCPρ,C  by 
ThPρ,C ∩Th.
F om 45,P oposi ion2.45we know ha he mapping 
Tis uppe semicon inuous. Since
Pρ,C ∩Th is a nonemp y ρ-compac con ex se and he ρ- opology is a no m- opology, we
can apply he Kaku ani-Bohnenblus -Ka lin Theo em see 14 o ob ain a ixed poin o 
T
and hence o T.
I we apply he p e ious heo em in he pa icula case o he space L1Ω,μ o a
σ- ini e measu e μwi h he usual no m, we ob ain he ollowing esul , which can be also
deduced om 44, Theo em 4.9.
Co olla y 4.15. Le Ω,μbe as abo e, C⊂L1Ω,μa nonemp y closed bounded con ex se which
sa is ies P ope y (P). Suppose, in addi ion, ha e e y sequence in Chas a con e gen locally in
measu e subsequence in L1.I T:C→KCCis a nonexpansi e mapping, hen Thas a ixed
poin .
I we conside now he space 1, hen he assump ion o exis ence o a w∗-con e gen
subsequence o e e y sequence in Ccan be emo ed and we can s a e he ollowing esul .
Co olla y 4.16. Le Cbe a nonemp y closed bounded con ex subse o 1which sa is ies P ope y (P).
I T:C→KCCis a nonexpansi e mapping, hen Thas a ixed poin .
No ice ha in 1 he e exis s a subse wi h P ope y Pwhich is no w∗-compac .
Example 4.17 see 44, Example 4.8.Le anbe a bounded sequence o nonnega i e eal
numbe s and le enbe he s anda d Schaude basis o 1. I is clea ha he se C:con xn,
whe e xn:1anen, is ne e weakly s a compac . Ne e heless, by using 46, Example
1i is easy o show ha Chas P ope y Pi and only i N0:{n∈N:anin m∈Nam}is
nonemp y and ini e.
Fixed Poin Theo y and Applica ions 19
Acknowledgmen s
The au ho s a e e y g a e ul o he anonymous e e ee o some use ul sugges ions o
imp o e he p esen a ion o his pape . This esea ch was pa ially suppo ed by DGES
G an no.BFM2006-13997-C02-01 and Jun a de Andaluc´
ıa G an no.FQM-127. This esea ch
is dedica ed o W. A. Ki k celeb a ing his wide and deep con ibu ion in Me ic Fixed Poin
Theo y.
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