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(**)hasacommonfixedpoint^and consequentlythat asequenceoffuzzymappingswiththecondition(*)has a common fixedpoint. These results arefuzzy analogues ofcommon fixed theoremsforsequencesofg-contractiveandg-contractive typemulti-valuedmappings [8].In 11 and 13] Lee^etal.also obtained a common fixedpointtheorem forsequences^offuzzymappingswhichgeneralizetheresultsin 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.

Afuzzy^setAin a metricspaceXis a function with domainXandvaluesin[0,1 ]. (Inparticular, ifAisanordinary (crisp)subset ofX,its characteristic function

Za

^{is afuzzyset}with 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}

whereBdenotes the closure of the(nonfuzzy)^setB.

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

A,

^iscompactinXforeachoe [0,1 and(ii)

A

^{is a}nonempty 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) maxLaeAsup ^bBinf a bl] ^beBsup

]nf All

^{a bl]}

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)isthegrade^ofmembershipofyinF(x). IncaseX Y, F(x)isa functionfromXinto [0,1]. Especiallyfor a multi-valuedmappingf:X 2

x, _)(;-x)is

afunctionfromXto{0,1}. Hence^a 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< such that for each pair of fuzzy mappings F,,Fj:X W(X),D(F,(x),Fj(y))<k IIg(x)-g(Y)ll ^{for all}x,ye X.

(**) 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 all}y 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,)=

^{beasequenceof}fuzzy 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 mappings^withthe 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 consequently^asequenceoffuzzymappingswiththe 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 point^viscalled thestar centerofK.

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

toitself. If

(F,)*__

^{isasequenceof}fuzzy mappingsofXintoW(X)satisfyingthe condition(**),then there exists apointx Xsuch that{x } c7’_- F,(x).

KIM, LEE AND S. J. CHO

PROPOSITION

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

(F,)=

^{is asequenceof}fuzzymappingsofKintoW(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

--

^We

^,

^{weuse thehavefollowing}

^{x. -u.II}

^{notion0 asdue ton}

--

^{Opial 5]. ABanachspace Xis saidto}satisfy 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

Thus we have shown that ’’-2-m x,,

xll -> ^’T--

x,, (y

+

v)ll.

817

Since

(x,)’=

^{convergestoxweakly,Opial’ condition}implies 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 asequenceof}fuzzy mappingsofKintoW(K)satisfying thecondition (****),then

(F,)’__

has a common fixedpoint.

PROOF.

SinceKisweakly compact,it isaboundedsubset ofX.

By

theProposition^3.3there exist asequence

(x,,)*=

^{inKand asequence}

(u,)’=

ⁱⁿ

^X

^satisfying{u,}^cF,(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__

^isasequence^offuzzy 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 2^x has a fixedpoint.

PROOF.DefineF K^.--)W(K)by F(x) %-tx)^thenFisanonexpansive-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^PROOF.F:K^If

--

^{aBanachW(K)hasspaceafixedXadmitspoint.aweaklycontinuous}duality mapping,then it satisfiesOpial’s condition[6].

ACKNOWLEDGEMENT. Wewould like to express ourthanks to the referee forhis valuable commentsandsuggestions.

818

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