Common fixed points for nonexpansive type fuzzy mappings

J.

VOL. 18 NO. 4 (1995) 813-818

COMMON FIXED POINTS

FOR

NONEXPANSIVE

AND

NONEXPANSIVE

TYPE FUZZY MAPPINGS

BYUNG SOO LEE

Department

of Mathematics,

Kyungsung

University Pusan, 608-736,KOREA

DO SANG KIM

Department

ofAppliedMathematics NationalFisheriesUniversityofPusan

608-737,KOREA

GUE MYUNG LEEandSUNG JIN CHO

Department

ofNationalSciences,PusanNationalUniversityofTechnology Pusan, 608-739, KOREA

813

(ReceivedMarch23,1993and in revised formJuly29,1994)

ABSTRACT.

In

thispaperwe defineg-nonexpansiveandg-nonexpansive type fuzzy mappings and provecommon fixedpointtheorems forsequencesoffuzzy mappings satisfyingcertain conditions on a Banachspace. Thus we obtain fixedpointtheorems fornonexpansive typemulti-valuedmappings.

KEY

WORDS AND PHRASES: Star-shapedset,Opial’scondition, weakconvergence,Hausdorff

metric,nonexpansivefuzzymapping, nonexpansive type fuzzy mapping,fixedpoint,common fixed

point.

1991AMS

SUBJECT

CLASSIFICATION: 47H10, 54H25.

1. INTRODUCTION

Fixedpointtheorems forfuzzy mappingswere obtainedby Chang, Heilpernand others 1-5,7, 9-13,16]. Especially,LeeandCho 10]showed that asequenceoffuzzymappingswiththecondition

(*)satisfies the condition(**), thatasequencewiththecondition(**)hasacommonfixedpointand consequentlythat asequenceoffuzzymappingswiththecondition(*)has a common fixedpoint. These

results arefuzzy analogues ofcommon fixed theoremsforsequencesofg-contractiveandg-contractive typemulti-valuedmappings [8].In 11 and 13] Leeetal.also obtained a common fixedpointtheorem forsequencesoffuzzymappingswhichgeneralizetheresultsin and 10] respectively.

Inthispaperwe defineg-nonexpansiveandg-nonexpansive typefuzzymappingsand show that a

sequenceoffuzzy mappingswith the condition(****),which are definedon anonemptyweaklycompact star-shapedsubset ofaBanachspace Xsatisfying Opial’scondition,hasacommonfixedpoint. As

corollaries,firstlyweshow that similar results are obtained for the conditions *), **)or ***). Secondly

weobtain fixedpointtheorems for nonexpansivetype fuzzy [respectively, compact-valued] mappings F[resp.,f]fromK(cX)to W(K) [resp.,

2K].

Thirdlyweshowthat similar results are obtainedfor nonexpansivefuzzy [resp.,compact-valued] mappings.

2.

PRELIMINARIES

Wereviewbrieflysome definitions andterminologiesneeded.

AfuzzysetAin a metricspaceXis a function with domainXandvaluesin[0,1 ]. (Inparticular,

ifAisanordinary (crisp)subset ofX,its characteristic function

Za

is afuzzysetwith domainXand values {0,1}). Especially {x} is afuzzy setwith amembership functionequal toa characteristic functionof theset{x}. The c-levelsetofA,denotedby

A,

is definedby

A,={x’A(x)>o} if e (0,1],

A0=

{x "A(x) > 0}

W(X)denotes the collection of allfuzzysetsAinXsuch that (i)

A,

iscompactinXforeachoe

[0,1 and(ii)

A

is anonempty subset ofX. For A,Be W(X), A cBmeansA (x)<B (x)foreach x eX. Let A andBbetwo nonempty bounded subsets ofa Banachspace X. The Hausdorff distance betweenAandBis

dn(A,B) max sup inf a bl] sup

]nf

All

a bl]

LaeA bB beB

DEFINITION2.1. Let A,B W(X)andot [0,1]. Thenwe define

D(A B sup

dn(Aa

B,)

WenotethatDis ametriconW(X)such thatm({x},{y})=llx-Yll,wherex,y

_

X.

DEFINITION2.2.Let Xbeanarbitrarysetand Ybeanymetricspace. Fiscalled afuzzy mapping

iffFis amappingfrom thesetXintoW(Y).

Afuzzy mappingFisafuzzysubset onXxYwith amembershipfunctionF(x) (y). The function valueF(x) (y)isthegradeofmembershipofyinF(x). IncaseX Y, F(x)isa functionfromXinto

[0,1]. Especiallyfor a multi-valuedmappingf:X 2

x,

)(;-x)is

afunctionfromXto{0,1}. Hencea

fuzzymappingF X

.-

W(X)isanotherextensionofamulti-valuedmapping

f:

X 2

x.

DEFINITION2.3. Letg beamappingfromaBanachspace (X, toitself. Afuzzy mapping F:X W(X)is g-contractive[respectively, g-nonexpansive] ifD(F(x),F(y)) <k I[g(x)-g(y)ll for all x,y X,for some fixed k, 0<k < [resp.,k ].

PROPOSITION2.4[9]. Let (X, 11)beaBanachspace, F :X

.--

W(X)afuzzy mappingand x eX,then there exists

ux

Xsuch that{u}cF(x).

DEFINITION2.5. Let gbeamappingfrom a Banachspace (X, to itself. Wecall afuzzy mapping F:X W(X) g-contractive type [respectively, g-nonexpansive type] if for all x eX,{ux} cF(x)there exists{vv}cF(y)forally Xsuch thatD({ux},{v,,})<k [[g(x)-g(y)[[

for somefixedk, 0<k< [resp.,k ].

REMARK. When g is an identity, a g-contractive [respectively, g-contractive type, g-nonexpansive, g-nonexpansive type] fuzzy mapping Fissaid tobe contractive[resp., contractive-type, nonexpansive, nonexpansive type].

LEMMA

2.6. Let A,Be W(X). Then for each {x}cA, there exists {y} B such that

D({x},{y})< O(A,B).

PROOF. If{x }cA,then x e

A.

By

compactnessof

B,

wecanchoose aye

B,

i.e.,{y}cB,

such that[[x-yl[ <dn(A,B). Bythe factsD({x},{y})=llx-yl[ anddn(A,B)<D(A,B),wehave

O({x},{y}) < O(A,B).

PROPOSITION2.7.Letg beamappingfromaBanachspace(X, toitself. IfF X W(X)

isag-nonexpansive [respectively, g-contractive] fuzzy mapping,thenFisg-nonexpansive type[resp., g-contractive type].

PROOF.Itcanbeeasily proved by Lemma 2.6.

3.

COMMON

FIXED

POINTS FOR FUZZY MAPPINGS

Foramapping gof a BanachspaceXintoitself and asequence

(F,)*__

offuzzy mappings ofXinto

W(X)we consider thefollowingconditions(*), (**), (***)and(****).

(**) there exists a constant k with 0<k<1 such that for each pair of fuzzy mappings

F,,F:

:X ---> W(X)and foranyx X,{u} F,(x)impliesthat there is{v,} cFj(y)for ally Xwith

D({u},{v,})< k IIg(x)-g(y)ll.

(***) for eachpairoffuzzymappings

F,,F:

:X --> W(X),

D(F,(x),Fj(y))

<llg(x)-g(y)ll for all

x,yX.

(****) for each pair offuzzymappings

F,,F

X ---> W(X),andforanyx X,{u}cF,(x)implies

that thereis{v,}

cF(y)

for ally XwithO({u},{v,})<[[g(x)-g(y)ll.

Itiseasilyprovedthat the condition (*) [respectively,(***)] implies thecondition (**) [resp., (****)] byLemma2.6, but thefollowing exampleshows that the converses donotholdingeneral.

EXAMPLE

3.1. Letg be anidentity mappingfrom a Euclidean metricspace([0, "),

"1

toitself.

Let

(F,)=

beasequenceoffuzzy mappingsfrom[0, oo)intoW([0,)),whereF,(x) [0,,,,,) --> [0,1]is definedasfollows;

1, z=O,

if x=O, F,(x)(z)=

0 z:O,

O<z<x/2,

otherwise, _F,(x)(z)= 1/2, x/2

<z

<-

ix,

[0,

z>ix

Thenthesequence

(F,)*__

satisfies the condition(****),but doesnotsatisfy thecondition(***). Inthis sectionweshowthat asequenceoffuzzy mappingswiththe condition(****),which are defined on anonempty weakly compactstar-shapedsubsetKof a BanachspaceXwhich satisfiesOpial’s

condition, has a common fixedpoint usinga common fixedpointtheorem duetoLeeand Cho 10],and

consequentlyasequenceoffuzzymappingswiththe condition(*),(**)or(***)has a common fixed

point. As corollaries we obtain fixed point theorems for nonexpansive type fuzzy [respectively, compact-valued] mappingsF[resp.,f]from anonempty weaklycompactandstar-shapedsubsetKof aBanachspace Xwhich satisfiesOpial’sconditiontoW(X)[resp.,

2x].

Theresultsfor thenonexpansivecompact-Valued mappingsarethe caseofreplacingconvexity

withstar-shapednessinTheorem3.5due to Husain and Latif[8].

Following

Nguyen

14]wedefine:LetX,Yand Zbeany nonemptysets, andA if(X)andB .q(Y)

where.q(X)isthe collection of allfuzzysetsinX. If

f:

X---)Y,thenthefuzzysetf(A)is defined viathe extensionprinciple byf(A) J(Y)andf(A)(y) sup A (x).

/-I(y)

Iff:X

Y --> Z,then thefuzzy setf(A,B)is defined via the extensionprinciplebyf(A,B) .Z)

andf(A,B)(z)= sup [min{A(x),B(y)}].

(x, v) f-I(z)

PROPOSITION (NGUYEN). Let

f:

X Y --->ZandA .q(X)andB .9(Y). Then anecessary

and sufficient condition for the equality [f(A,B)],=f(A,B) for all ct [0,1] is that for all

z Z, sup [min{A (x), B(y)}]isattained.

(x,v) f-(z)

AsubsetKof aBanachspaceXissaid tobestar-shapedif there exists apointv Ksuch that tv+(1 t)x Kfor all x Kand 0< <1. The pointviscalled thestar centerofK.

THEOREM3.2 10]. Letg bea nonexpansivemappingfrom acompletemetric linearspace (X,d)

KIM, LEE AND S. J. CHO

PROPOSITION

3.3. Let Kbe a nonempty boundedstar-shapedsubset ofaBanachspace Xand ganonexpansivemappingfromXinto itself. If

(F,)=

is asequenceoffuzzymappingsofKintoW(X) satisfyingthe condition(****),then there existasequence

(x,,),__

inKandasequence

(u,,),__

inX satisfying{u,,}cF,(x,,)for all e Nsuch thatIlx,,-u,,ll

PROOF. Let_x0be thestar-centerofK. Choosearealsequence

(k,,),__,

such that 0 <k,,< and

k,,--0 as n --)oo. Then for each x e K,

k.x

+ (1-k,,)x K. Define a fuzzy mapping F[’ ofK

intoW(X) by setting

F;’

(x) k,,{x0} + k,,)F, (x)for all

N.

thenby Proposition3.3 in 14]itfollows that

[F’(x)],

k,.o

+(1 k,,)

[F,(x)],

for all Nand eacho: [0,1 ].Nowwe show that for eachn N,

is asequenceoffuzzy mappings satisfyingthe condition(**). If we let{u,}c

F’(x)

for each

(F,),__,

x K,we get

ux

k,

xo

+(1-k,,)vxfor somev K such that{v} F,(x). Since

(F,)7=,

satisfiesthe condition(** **),there existsa{ v, }cFj(y)for ally Ksuch that

vx

v, < g(x) g (y)l] < x y

II.

Put u,

=k,,xo+(1-k,,)v,

clearly by definition of _Fj(y) we get {u,.}

cFj’(y)

and

u-y,l[

=ll(1-k.)(v,,-v,,)ll

_<(1-k,,)llg(x)-g(y)ll

-<

(1-k.)llx

Yll

whichprovesthat

(FI"),__,’

isa

sequenceoffuzzymappings satisfying the condition(**). Thecommonfixed pointtheorem for a

sequenceoffuzzy mappingsduetoLeeandCho 10]i.e.,Theorem3.2guarantees that for each fixed n N,

(FI"),__,**

has a common fixedpointinK, say {x,,}

F,"(x,,)

W(K)for all N. Fromthe definition of

F’,’(x,,)

thereexistsa{u,,}cF,(x,,)such thatx,, k,,x0+(1-k,,)u,, for all Nandeach fixed n N. Thus

x.

u.II

kxo(1 k)u,, u,,ll

k.II

x0-

u.ll.

By

the definition of W(K), {u,,} F,(x,,)e W(K)implies u,, K. Thus

{llu.-x011

} isbounded. So bythe fact thatk,,-0 as n

--

,

wehave

x.

-u.II

0 asn

--Weuse thefollowingnotiondue toOpial 5]. ABanachspace Xis saidtosatisfy Opial’s condition

5]if for each x Xand eachsequence

(x,,)’__,

weaklyconvergenttox,

lim lim

x.

y >

x,,

x

n

---

n

-for all y

::

x.

PROPOSITION3.4.LetKbeanonemptysubset of a BanachspaceXwhich satisfiesOpial’s

conditionandFa g-nonexpansive typefuzzy mappingofKintoW(K). Let

(x,,)__,

be asequenceinK

whichconverges weaklytoanelement x K. If

(y,,)’=,

isasequenceinXsuch that{x. y.} F(x.)

andconvergestoy X,then{x-y} F(x).

PROOF. SinceFisag-nonexpansive type fuzzy mapping, thereexistsa{v,,}c:F(x)suchthat

IIx,,

-y,, v,,ll < g(x,,) g(x) <_IIx,,-xll. Itfollows that

"----m

X,, y,,

--V.II

<

"-’-r--"

X,,

--xlI.

Sinceevery weaklyconvergentsequenceisnecessarily bounded,limits in theproceeding expressionarefinite. Since

(v.)’__

isasequencein acompactsubset[F(x)],ofXfor eachct [0,1], thereisasubsequenceof

(v.)’=,

also denoted by

(v.)’.,,

convergingtov [F(x)],foreache [0,1]. Hence {v} c:F(x), therefore

lim

(!1

x.

(y+ v)ll

-II

(Y,, +

v,,)

(y+v)ll

lim lim

x,,

(y

+

v)ll

+

(-IIY,,

+ v,,

y v

noo

lim

817

Since

(x,)’=

convergestoxweakly,Opial’ conditionimplies thatx y

+

v, sox y v [F(x)],

foreach o [0,1 ]. Hence {x y}cF(x)andthepropositionisproved.

REMARK.

From the above proof it follows that the weak limit of fixed points of a

nonexpansive-typefuzzy mappingFdefinedonanonempty subsetKofaBanach

space X

satisfying Opial’scondition, inparticularfor a Hilbertspaceisalso a fixedpointofF.

THEOREM

3.5. Let Kbe anonemptyweakly compact star-shaped subsetof aBanachspace X

which satisfiesOpial’scondition. If

(F,)7__t

is asequenceoffuzzy mappingsofKintoW(K)satisfying

thecondition (****),then

(F,)’__

has a common fixedpoint.

PROOF.

SinceKisweakly compact,it isaboundedsubset ofX.

By

theProposition3.3there exist asequence

(x,,)*=

inKand asequence

(u,)’=

in

X

satisfying{u,}c_F,(x,)for all Nsuchthat

x, u, ---->0 as n--->*,,. Puty, x, u,. Kbeingweaklycompact,we can find aweakly convergent

subsequence

(x,,,),

_-,of

(x,),=

t. Let

xo

betheweak limit of thesequence

(x,.).

v Clearly Xo Kand we havey,,, x,,, u,,,{u,,}cF,(x,,)for all N. Then itfollows that y,,, --->0 andby Proposition 3.4

there exists a fixedpoint

xo

Xsuch that{x0} F,(xo)for all N.

THEOREM3.ti. LetKbe anonemptyweakly compactstar-shapedsubset of a Banachspace X

which satisfiesOpial’scondition. If

(F,)7__

isasequenceoffuzzy mappingsofKintoW(K) satisfying

the condition(*),(**)or(***).then

(F,)*=

has a common fixedpoint.

PROOF. Itisprovedbythefact thatthe condition(***)[respectively,(*)]impliesthe condition

(****) [resp., (**)].

If weput

F,

Ffor all Nin Proposition 3.3, then thesequenceoffuzzy mappings

(F,)’=

(F)

in the condition(****)is asequence ofg-nonexpansive typefuzzy mappings. Thus we obtain the

followingcorollaryforg-nonexpansive typefuzzy mappings.

COROLLARY3.7. LetKbe anonemptyweakly compactstar-shapedsubset of a Banachspace Xwhich satisfiesOpial’scondition. Then eachg-nonexpansive type fuzzy mappingF K --> W(K)has a fixedpoint.

COROLLARY3.8. Let Kbe anonemptyweakly compactstar-shapedsubset of a Banachspace Xwhich satisfiesOpial’ condition.Then eachnonexpansive type,compact-valued mappingf:K 2x

has a fixedpoint.

PROOF.DefineF K.--)_{W(K)by F(x) %-tx)thenF}isanonexpansive-type fuzzy mapping. By Corollary3.7 there exists apointx eXsuch that{x}

cF(x)=%itx)i.e.,x

_

f(x).

Corollary3.8 is a generalization of thefollowingtheorem duetoHusainand Latif[8].

THEOREM3.9. LetKbe anonemptyweakly compactconvexsubset of a Banachspace Xwhich satisfiesOpial’scondition. Then eachnonexpansive type,compact-valued mapping

f:

K 2xhas a fixedpoint.

COROLLARY3.10. LetKbe anonemptyweakly compactstar-shapedsubset of aBanachspace XwhichsatisfiesOpial’scondition. Then eachnonexpansivefuzzy mappingF K W(K)has a fixed

point.

COROLLARY3.11. LetKbe a nonemptyweaklycompactstar-shapedsubset ofaBanachspace X having a weakly continuous duality mapping. Then each nonexpansive-type fuzzy mapping F:K

--

W(K)hasafixedpoint.

PROOF.IfaBanachspaceXadmitsaweaklycontinuousduality mapping,then it satisfiesOpial’s

condition[6].

818

REFERENCES

[1] BOSE, R. K. andSAHANI, D.,

Fuzzy

mappingsand fixed point theorems,FuzzySetsand

Systems21(1987),53-58.

[2] CHANG, S.S.andHUANG, N. J.,Fixed pointtheorems forgeneralizedfuzzymappings, Acta

of

EngineeringMath.No.2(1984), 135-137(inChinese).

[3] CHANG, S.S.,Fixed pointtheoremsforfuzzymappings,KexueTongbao 14 (1984), 833-836 (in Chinese).

[41

[5]

[61

[7]

is]

[9l

[10]

[11]

[12]

[13]

[14]

[15]

[16]

CHANG, S. S.,Fixeddegreeforfuzzymappingsand a generalizationof

Ky

Fan’stheorem, FuzzySets andSystems24(1987), 103-112.

CHITRA,A.,Anote on the fixedpointsoffuzzymapsonpartiallyorderedtopologicalspaces, FuzzySets andSystems19(1986), 305-308.

GOSSEZ, J.P.andLAMI DOZO, E., Somegeometric propertiesrelatedtothe fixed pointtheory

ofnon-expansive mappings,

Pacific

J.Math.40(1972), 565-573.

HEILPERN, S.,

Fuzzy

mappings and fixedpoint theorem, J. Math. Anal. Appl. 83 (1981), 566-569.

HUSAIN,T.andLATIF,A.,Fixedpointsof multivalued nonexpansivemaps, Internat. J.Math.

&

Math. Sci., Vol. 14,No. 3 (1991),421-430.

LEE,B.S.andCHO,S.J., Afixedpointtheorem forcontractive-type fuzzy mappings,Fuzzy SetsandSystems61 (1994), 309-312.

LEE,B.S. andCHO, S.J.,Commonfixedpointtheoremsforsequencesoffuzzymappings,to

appearinInternat.J.Math.

&

Math. Sci.

LEE,B.S.,CHO, S. J.,LEE,G.M.andKIM,D. S., Onthecommonfixedpointtheoremsfor asequenceoffuzzy mappings,submitted.

LEE,B. S., LEE, G. M., CHO, S.J.andKIM,D. S.,Acommon fixedpoint theoremfor apair

offuzzymappings,submitted.

LEE,B.S.,LEE, G.M., CHO,S.J.andK/M,D. S.,Generalizedcommon fixedpointtheorems for asequenceoffuzzy mappings,toappearinInternat. J.Math.&Math.Sci.

NGUYEN, H. T.,

A

note on the extensionprinciple forfuzzyset,J.Math. Anal.Appl. 64 (1978), 369-380.

OPIAL, Z.,Weakconvergenceof thesequenceof successiveapproximations fornonexpansive mappings, Bull.Amer.Math.Soc.73(1967), 591-597.