ARCHIVUMMATHEMATICUM(BRNO)Tomus40(2004),
345353FIXEDPOINTTHEOREMSFORNONEXPANSIVEMAPPINGSINMODULARSPACESPOOMKUMAMAbstract.
Inthispaper, weextendseveral
conceptsfromgeometryofBa-nachspacestomodularspaces. Withacareful
generalization, wecancoverall corresponding results in the former
setting. Main result we prove says thatifisaconvex,
-completemodularspacesatisfyingtheFatoupropertyandr-uniformlyconvexforallr>
0,Caconvex, -closed, -boundedsubsetofX,T:C
Ca-nonexpansivemapping,thenThasaxedpoint.1. IntroductionThe theory
of modular spaces was initiated by Nakano [15] in 1950 in
connectionwiththetheoryof
orderspacesandredenedandgeneralizedbyMusielakandOrlicz[14] in1959.
Itiswell knownthatoneof thestandardproofof Banachsxed point theorem
is based on Cantors theorem in complete metric spaces [5,
6].Tothisend,
usingsomeconvenientconstantsinthecontractionassumption, wepresent a
generalization of Banachs xed point theorem in some classes of
modularspaces.In this paper, we extend many concepts and results in
normed spaces to modularspaces.2. PreliminariesWestart byreviewing
somebasic factsabout modular spaces as formulated
byMusielakandOrlicz[14]. Formoredetailsthereaderisreferedto[7, 9,
10] and[13].Denition2.1(cf. [7]). LetXbeanarbitrary vectorspace.(a)
Afunction: X[0, ] iscalledamodularonXif forarbitraryx, yinX,(i)(x)
= 0ifandonlyifx = 0,(ii)(x) = (x)foreveryscalarwith| |=
1,and2000MathematicsSubjectClassication: 46B20, 46E30,47H10.Key
words and phrases: xed point, modular spaces, -nonexpansive
mapping, -normalstructure, -uniformnormalstructure,
r-uniformlyconvex.ReceivedNovember7,2002.346 P. KUMAM(iii)(x +y)
(x) +(y)if += 1and, 0.(b) If (iii) is replacedby(iii)(x+y) (x) +(y)
if +=1and, 0,esaythatisaconvexmodular.(c) A modular denes a
corresponding modular space, i.e. the vector space XgivenbyX= {x X:
(x) 0 as 0} .XisalinearsubspaceofX.Ingeneral themodularisnot
subadditiveand therefore doesnot behave asanormoradistance.
ButonecanassociatetoamodularanF-norm
(see[13]).ThemodularspaceXcanbeequippedwithanF-norm
(see[13])denedbyx= inf_ > 0; _x_ _.Namely, if isconvex,
thenthefunctional |x|=inf_ > 0; _x_
1_isanorminXwhichisequivalenttotheF-norm..Denition2.2(cf. [7,8]).
LetXbeamodularspace.(a) Asequence(xn) Xis saidtobe -convergent
toxXandwritexnx,if(xnx) 0asn .(b) A sequence (xn) is called -Cauchy
whenever (xnxm) 0 as n, m .(c) The modular is called -complete if
any -Cauchy sequence is -convergent.(d) Asubset BXis called -closed
if for any sequence (xn) B-convergent tox X,wehavex B.(e)
A-closedsubsetB Xiscalled-compactifanysequence(xn) Bhasa-convergent
subsequence.(f) issaidtosatisfythe2-condition if(2xn) 0whenever(xn)
0asn .(g) Wesaythat has theFatouproperty if (x) liminfn(xn)
wheneverxnx.(h) AsubsetB Xissaidtobe-boundedifdiam(B) <
,wherediam(B) = sup{(x y); x, y B}iscalledthe-diameterofB.(i)
Denethe-distancebetweenx XandB Xasdis(x, B) = inf{(x y); y B} .(j)
Denethe-Ball, B(x, r),centeredatx XwithradiusrasB(x, r) = {y X; (x
y) r} .Let(X, .)beanormedspace.
Then(x)=xisaconvexmodularonX.Onecancheckthat -ballsare-closedif
andonlyif
hastheFatouproperty(cf.[8]).FIXEDPOINTTHEOREMSINMODULARSPACES
347Example2.3.(1) TheOrlicz modular is denedfor everymeasurablereal
functionf bytheformula(f) =_R(|f(t)|) dm(t)
,wheremdenotestheLebesguemeasureinRand: R[0,
)iscontinuous.Wealsoassumethat(u)=0iu=0and(t) asn .
ThemodularspaceinducedbytheOrliczmodulariscalledtheOrliczspaceL.(2)TheMusielak-Orliczmodularspaces(see.[17]).
Let(f) =_(, f())d() ,whereisa-nitemeasureon,and : R [0,
)satisfythefollowing:(i)(, u)isacontinuousevenfunctionof
uwhichisnondecreasingforu>0,suchthat(, 0) = 0, (, u) > 0foru
= 0,and(, u) asn .(ii)(, u)isameasurablefunctionofforeachu
R.ThecorrespondingmodularspaceiscalledtheMusielak-Orliczspaces,
andisdenotedbyL.Denition2.4(cf. [8]).
AmodularspaceXissaidtohave-normal
structureifforanynonempty-bounded-closedconvexsubsetCofXnotreducedtoaonepoint,thereexistsapointx
Csuchthatr(x, C) := sup{(x y); y C} < diam(C) .A modular space
Xis said to have -uniformlynormalstructure if there
existsaconstantc (0, 1)suchthatforanysubsetCasabove, thereexistsx
Csuchthatr(x, C) < c diam(C) .Clearly-uniformlynormal
structureis-normal
structure.LetXbeamodularspaceandletCbeanonempty-boundedand-closedconvex
subset C of X. We will say that C has the xed point property (fpp)
if every-nonexpansiveselfmapdenedonC(i.e.,T:CC, (T(x) T(y)) (x
y)for every x, y C) has a xed point, that is, there eistsx Csuch
that T(x) = x.Also, amodularspaceXissaidtohavethexedpoint
property(fpp)if everynonempty-bounded-closed
convexsubsetofXhasthexedpointproperty.In Banach spaces, when we
think about reexivity automatically the dual spaceis present in our
taught. But in modular spaces, it is very hard to conceive the
dualspace. Tocircumventtheproblem,weusesomecharacterization
ofreexivity.Theorem2.5(Smulian1939, cf. [12]). A normed spaceXis
reexive if and onlyif
nCn= whenever (Cn) is a sequence of nonempty, closed bounded and
convexsubsetsofXsuchthatCn Cn+1foreachn N.Denition2.6(cf. [8]).
LetXbeamodularspace. Wewill saythatXorsatisestheproperty(R)if
everydecreasingsequenceof
nonempty-closedand-boundedconvexsubsetsofX,hasanonemptyintersection.Thefollowingtheoremisknown.348
P. KUMAMTheorem2.7(cf. [8]). Let Xbea-completemodular space.
Assumethat isconvexandsatisestheFatouproperty. Moreover, assumethat
Xhasthe-normal structure and hasthe property (R)
andCisany-closed-bounded convexnonempty subsetofX.
Thenany-nonexpansive mappingT: CChasaxedpointinC.3. ResultsWe start
this chapter with generalizations as well as their corresponding
resultsofuniformconvexityandnormalstructurecoecientsinmodularspaces.Denition3.1.
Forr> 0, a modular spaceXis said to ber-uniformlyconvexif
foreach>0, thereexists>0suchthatforanyx, yX, theconditions(x)
r, (y) rand(x y) rimply_x +y2_ (1 )r .Denition3.2. Let Xbe aModular
space. For any 0andr >0, themodulusof r-uniformconvexity
ofXisdenedby(r, ) = inf_1 1r_x +y2_ : (x) r, (y) r, (x y)
r_.Denition3.3. ThenormalstructurecoecientofXisthenumberN(X) =
inf_diam(C)R(C):C XCis-closedconvex,-boundedanddiam(C) >
0_,whereR(C):=inf{r(x, C): xC}whichiscalledthe-Chebyshevradius
ofC(cf.[7]).Remark3.4.(1) ItisnothardtoshowthatR(C) = 0.
Indeed,supposeR(C) = 0andlet,x0, y0Cbesuchthat x0=y0. SinceR(C)
=infyC r(x, C) =0, sothereexistsasequence(xn)inCsuchthatlimnr(xn,
C) = 0. Thus_x0y02_ = _(x0xn) + (xny0)2_ (x0xn) +(xny0) 0asn .
Thereforex0= y0,acontradiction.(2) Foranyx CwehaveR(C) r(x, C)
diam(C).(3)
ItisobviousformthedenitionthatXhas-uniformnormalstructureifandonlyifN(X)
> 1(see[11]).Lemma3.5. Letr> 0.
AmodularspaceXisr-uniformlyconvexifandonlyif(r, ) > 0forall >
0.Proof. Let>0. If Xis r-uniformlyconvex, thenthereexists
>0suchthat for anyx, y Xwith(x) r, (y) r, and(x y) r.
wehave_x+y2_(1 )r. Thus, forthesexandy, 1 1r_x+y2_. Hence(r,
)FIXEDPOINTTHEOREMSINMODULARSPACES 349>0. Conversely,suppose(r,
)>0forsome>0and>0. Takeanyx, y Xsuchthat(x) r, (y) rand(x
y) r. Bydenitionof,weget(r, ) 1 1r_x+y2_. Hence1r_x +y2_ 1 (r, ) 1
.ThereforeXisr-uniformlyconvex.Lemma3.6. The modulus (r, .) of
uniformconvexityof Xis increasingon[0, ).Proof.
Letr>0and1>20. Letx, yXbesuchthat(x)rand(y) r. If(x y)
1r,then(x y) 2r. Thisshowthat_1 1r_x +y2_ : (x) r, (y) r, (x y)
r1__1 1r_x +y2_ : (x) r, (y) r, (x y) r2_.Thisimpliesthat(r, 1) (r,
2).Theorem3.7. If the modulus of convexityof amodular space
Xsatises(d, ) > 0forall d, > 0,thenXhas-normal
structure.Proof. Let Cbe a nonempty -bounded-closedconvexsubset of
Xwithdiam(C) = d > 0. Let (0, 1)thereexistx, y Csuchthat(x y) d
.Let z =x+y2andwC. Thus, z C, (wx) d, (wy) dand((w x) (w y)) = (x
y) d.Consequently,_w _x +y2__ = _(w x) + (w y)2_ (1 (d, ))d
.HencesupwC(w z) _1 (d, )_d .Since(d, ) > 0,wegetsupwC(w z) <
d = diam(C) .SincethisistrueforanyC,thisprovesthatXhas-normal
structure.Lemma3.5andTheorem3.7giveusimmediatelyCorollary3.8. For
amodular space X, if Xis r-uniformly convex for allr>
0,thenXhas-normallystructure.Corollary3.9(cf. [4]).
Closedboundedconvexsubsetsof uniformlyconvexBa-nachspaceshavenormal
structure.350 P. KUMAMTheorem3.10. Let X be a -complete modular
space. If is convex and
satisestheFatoupropertyandXisr-uniformlyconvexforall
r>0,thenXhastheproperty(R).Proof. Let (Cn) be a decreasing
sequence of -bounded, -closed nonempty convexsubsetsofX, z
XwhichdoesnotbelongtoC1andr =limndis(z, Cn) .Dene Dn=CnB(z, r)
andlet dnbe the diameter of Dn. Bythe Fatoupropertyof , (Dn) is
adecreasingsequenceof nonempty-bounded,
-closedconvexsubsetsofXbecauseB(z,
r)isthena-closedset(see[8]).Letrnbe a sequence of positive number
that decreases to zero anddnrn> 0for all n. Thereexist x,
yDnsuchthat (x y) dn rn. Thus, bythedenitionof(r,dnrnr),wehave(z x
+y2) = _(z x) + (z y)2__1 _r, dnrnr__r .Hence1rdis(z, Cn) 1r_z x
+y2_ 1 _r, dnrnr_. ()Putd = limndnandan=
dn1n,andconsidertwocases.Case1(an d,for alln large enough).
Bybeingincreasing and (), wehaveforallnlargeenough,1rdis(z, Cn) 1
_r, anr_ 1 _r, dr_.Lettingn ,weget1 1 _r,
dr_,whichimpliesthat(r,dr) = 0.
Byr-uniformconvexityofXandLemma3.1.6wehave(r, ) > 0forall >
0,whenced = 0.Case2 (0 < an< d, for innitely many n). There
exists a subsequence (an ) suchthatan d,whencethelimitlimn
(r,an
r)existsandby(),wehave1 1 limn
_r, an
r_.Consequently, limn
(r,an
r) = 0. Since an d and (r, ) > 0 for all >
0,wehaved=limndn=0as well. Thus, thereexists a-Cauchysequence(xn),
wherexn Dnforeachn. SinceXis-complete, (xn)-converges tosomex0X.
Usingthe-closenessof Dn, wededucethat x0Dnforall
n1.ThisimpliesthatnNDn= and so nNCn= aswell.
Theproofisthereforecomplete.FIXEDPOINTTHEOREMSINMODULARSPACES
351Corollary3.11(cf. [4]). Let Xbea-complete modular
spacewithconvexandsatisfyingtheFatouproperty.
IfXisr-uniformlyconvexforallr>
0,thenXhasthexedpointproperty.Proof. By Corollary 3.8 and Theorem
3.10,Xhas-normal structure and prop-erty(R).
Consequently,Theorem2.7canbeappliedtoconcludethatXhasthexedpointproperty.Corollary3.12(cf.
[4]). If Cisanonemptyclosedboundedconvexsubset of
auniformlyconvexBanachspace,theneverynonexpansivemappingT:
CChasaxedpointinC.Theorem3.13. LetXbe a modular space with modulus
of convexity(1, ) = 1forsome (0, 1). Ifweassumethat(x) = (x)forall
> 0,thenN(X) 11 (1, ).Proof. LetCbea-closed,
-boundedconvexsubsetofXwith diam(C)=d > 0.Since (0,
1),thereexistx, y Csuchthat(x y) d .Let z =x+y2CandwC. Then(wxd)
=1d(wx) 1, (wyd) =1d(w y) 1,and__w xd__w yd__ =1d(x y)
.Bythedenitionof(1, ),weobtain1d_w x +y2_ =1d_(w x) + (w y)2_ 1 (1,
) .HenceitfollowsthatR(C) supwK(z w) d_1 (1,
)_.Consequently,diam(C)R(C)11 (1, ) .ThereforeN(X) 11 (1, )
.Remark3.14. If weassumethat inColloray3.8(x) =(x) forall
>0,thenXwillhave-uniformlynormalstructure.Corollary3.15. If Xis
a modular space with the modulus of convexity (1, ) (0, 1)forsome
(0, 1),thenXhas-uniformlynormal structure.352 P. KUMAMProof.
ByTheorem 3.13wehaveN(X) > 1. Thus,byRemarks
3.4(3),Xhas-uniformlynormal structure.Corollary3.16. IfXisa
Banachspace spacewithmodulus ofconvexityX() (0, 1) forsome (0, 1)
andwe put(x) = x, thenwe getthatXhasuniformlynormal
structure.Corollary3.16stronglyimproves [1] whichstates that
anyuniformlyconvexBanach
spacehasuniformlynormalstructure.NotethataBanachspaceXisuniformlyconvexifandonlyifitsmodulusofconvexitysatisesX()
> 0forall > 0(see[5]).Acknowlegement. Theauthorisgrateful
toProf. SompongDhompongsaformanyhelpful comments andsuggestions.
Theauthoralsowishes thanks
toananonymousrefereeforhis/hersuggestionswhichledtosubstantialimprovementsofthispaper.
Lastbutnotleast, IwouldliketothankProf.
HenrykHudziktohisEnglishsuggestionsandcorrection.References[1]
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[email protected]
