A GENERAL VECTOR-VALUED VARIATIONAL

Internat. J. Math. & Math. Sci.

VOL. 21 NO. 4 (1998) 637-642

637

A GENERAL VECTOR-VALUED VARIATIONAL

INEQUAUTY

AND

ITS

FUZZY EXTENSION

SEHIEPARK

Department

of Mathematics

SeoulNationalUniversity Seoul 151-742.KOREA

BYUNG-SOO LEE

Department

ofMathematics

Kyungmng

University Pusan608-736.KOREA

GUEMYUNG LEE

Department

^ofAppliedMathematics

PukyongNational University Pusan 608-737,KOREA (ReceivedDecember 12,

1996)

ABSTRACT. A generalvector-valued variational inequality

(GVVO

is considered. Weestablishthe existencetheorem for

(GVVI)

inthe noncompact setting, whichis anoncompactgeneralization of the existencetheorem for

(GVVI)

^obtainedby Leeetal., byusing the generalizedformof

KKM

theorem due toPark.

Moreover,

weobtain the fiu2y extensionofourexistencetheorem.

KEY WORDS AND PHRASES: Variationalinequalities, fuzzyextension,

KKM

theorem.

1991AMSSUBJECTCLASSIFICATION CODES: 47H19.

1. INTRODUCTION

Recently,Giannessi

[1

^imroducedavariational inequality for vector-valuedmappingsinaEuclidean space. Sincethen, Chenetal.

[2-6]

have intensivelystudiedvariationalinequalitiesfor vector-valued mappings in Banach spaces.

Lee

et al.

[7]

have established the existence theorem ofavariational inequalityfor a multifimction withvectorvaluesinaBanachspace.

Onthe otherhand, Changand Zhu

[8]

introduced the conceptofvariationalinequalitiesforfuzzy mappingsinlocallyconvexHausdorff topological vectorspacesand investigatedexistencetheoremsfor some kinds ofvariational inequalities forfazzy mappings, ^{which were} the extensions ofsome theorems in

[9,10,11,12].

Leeetal. 13 obtainedthefazzy generalizations ofnewresults of Kim and Tan

[14],

^andthey

[7]

establishedthefuzzyextensionoftheir existencetheorem. Ourmotivation of this paper is to consider the noncompact cases of the existence theorems of variational inequalities for multifimctionswith vectorvalues orfuzzymappingsinBanachspacesobtainedbyLeeet al.

[7].

Let

X

and

Y

betwonormedspacesand

D

anonempty convex subset ofX. Let

T" X

^-,2^L(x’Y) beamultifunction, where

L(X, Y)

^isthespaceof allcontinuous linearmapsfrom

X

into

Y,

andCa closed pointed andconvexconeof

Y

such thatInt C 0,whereIntdenotes theinterior.

Considerthe following generalizedvector-valued variationalinequality:

(GVVI)

Findz0

e D

suchthatfor eachx

e D,

thereexists ans0

T(x0)

such that

(so,

z

z0)

Int

C,

where

(so, V)

denotes the evaluation of

so

^at

638 S.PARK,B. S.LEEANDG.M.LEE

When

T

is a mapping fxom

X

into

L(X,Y), (GVVD

reduces to the following vector-valued variationalinequality

(VVI)

^consideredbyChenetal.[3,5,6].

(VVI)

^Findz0G

D

such that

(T(z0),z z0) ^IntU

^{for allz}

^D.

The above inequality

(VVI)

is a generalization ofthe following classic scalar-valued variational inequality

(vI).

(VI)

^Find z0

D

such that

f(z0),z zo) ^_>

0 for all z

D,

where

f ^R"

^--,

^R"

^{is a given}

mapping.

Our purpose inthis paperis to establishthe existence theorems for

(GVVI)

in the noncompact setting,whichisthe noncompact case of the existence theoremfor

(GVVI)

obtainedby Leeetal.

[7],

by using aparticularformofthe generalized

KKM

^theoremsduetoPark

[15-17].

Ourexistencetheorem subsumes Theorem 2.1 of Cottle andYao [18], the part (i) of Theorem 2.1 of Chen and

Yang

[6], Theorem2of

Yang [19]

and Theorem 2.1ofLeeetal.

[7]. Moreover,

weobtain thefitzzy extension of ourexistencetheorem. Our fuzzyextensionisageneralizationof Theorem3.1ofLeeetal.

[7].

^Now

wegive the definition ofa

KKM

map.

DEFINITION 1.1. Let

D

be a subsetofaconvexspace X. Thenamultifimction

G D

2x_is called

KKM

iffor each nonempty finite subset

N

of

D,

coN C

G(N),

^wherecodenotes the convex hull and

G(N) U ^{Gx

Aconvespace

X

isa nonempty convex (ina vctorspace)withany topologythatinduceathe Euclideantopologyon the convexhulls ofitsfinitesubsets. Thus,aconvexsubset

X

of a topological vector space

E

with therelativetopologyisautomatically a convexspace. Fordetailsof the convex space,seeLassonde

[9].

We say^that asubset

A

ofatopological space

X

is

compact&

^closedin

X

iffor every compact subset

K c X

theset

A K

isclosedin

K.

Weneedthefollowing particular formofthe generalized KKMtheoremsduetoPark

[16-18],

^whichwillbe usedintheproof ofour Theorem2.

THEOREM I. Let

X

beaconvexspace,

K

a nonempty compact subsetof

X,

and

X

2x_a

KKM

multifimction.

Suppose

that

(1)

for each

X, G()

^iscompactly closed;and

(2)

for eachfinitesubsetNof

X,

thereexistsa compact convex subset

Lv

^of

^X

^{such thatNC}

LN

and

s fl ^{S()

^/

^{s) c K.}

Then we have

2. ExistenceTheorems

First,wegive the followingdefinitionsfor theexistencetheorems for

(GVVD.

DEFINITION2.1. Let

X

beanormedspacewithdualspaceX* and

T X

^--,X* amapping.

1.

T

is saidtobemonotoneif forany z,/

X, (T(z) T(),

^z

t) _>

^0.

2.

T

is said to be pseudomonotone if for any z,t

X, (T(z),/-z)_>

0 implies that

(T(I/),

^3.

- ^{T z) >}

^{is said toO. be} hemicontinuous ifforany z,tt,z6

X,

the mapping ^a

- ^{(T(z +}

^cry),

^z)

^is

continuousat0

+.

DEFINITION2.2. Let

X, Y

betwonormedspaces,

T X

^--,

L(X, Y)

amappingand

U

aclosed, pointed and convex cone of

Y

such thatInt

U .

1.

T

is said to be

U-monotone

ifforany z, F

X, (T(z) T(/),

^z

F) U.

2.

T

is saidtobeU-pseudomonotoneifforany z,/

X, (T(z),

F

z) ^{Int U}

implies that

((), .

^,t

^c.

GENERAL VECTOR-VALUED VARIATIONAL INEQUALITY 639 3.

T

is said tobeV-hemicontinuous if forany x,y,z E

X,

the mappingee^-,

(T(x + eel/), z)

is continuousat0

+.

REMARK. When

Y R

^andC

R+,

Definition 2.2becomes Definition 2.1.

DEFINITION2.3. Let

X

and

Y"

betwonormedspaces,

T" X

2^L(X,Y)aset-valuedmapandC aclosed,pointed and convex cone of

Y

such that

Int

C

.

1.

T

is said tobeC-monotoneiffor anyx,yE

X,

s

T(x)

and t

T(y), (s

t,x

F)

EC 2.

T

is said to be C-pseudomonotone if for any x,y

X, (s,y-x)

-IntC for some s

T(x)

impliesthat

It,

y

) Int

Cfor some t

3.

T

is said tobeV-hemicontmuousifforany z,y

X,

ee

>

0and

ta T(x + ^eey),

^thereexists

to T(x)

^{such that}for anyz

X, (ta, z/ (t0, z)

as ee 0

+.

REMARK.

^1. Definition 2.3is ageneralization ofDefinition2.2.

2. Wecaneasilyprovethat theC-monotonicity impliestheC-pseudomonotonicity.

Nowweprovethe following existence theorem forthenoncompact case of

(GVVI).

THEOREM2. Let

X

and

Y

be Banachspaces,

C

aclosed, pointedand convex cone in

Y

with C

:

^0,

^D

^{anonemptyconvexsubset of}

^{X, K}

^anonempty compact subset of

X,

and

T X

^2L(X’Y) Supposethat

(1)

T

isC-pseudomonotone,compact-valued, and V-hemicontinuous; and

(2)

for each nonempty finite subset

N

of

D,

there exists a nonempty compact convex subset

LN

^of

D

^{such that}

N

C

Lv

and for eachz

LN\K

^thereexistsa y E

LN

^suchthat

^t,

^y

xl ^Int

^Cfor

all t

T(F).

^Then(GVVI)issolvable.

PROOF. Define a multifunction

F1 ^D

^2zby

Ft(y) {z D"

{s,y-

x) q IntC

forsome s

T(x)}

for y⁽

D.

Then

Ft

^{is a}

^KKM

multifunctionon

D.

In fact, suppose that

N {x,...,xn}

C

D, V.=eei

^{1, eei}

_

^{O, 1,...,n and}

x

E=lee,

^x

FI (N).

^{Thenfor anys}

T(x),

wehave

(s,

Xi-

X . ^{Int C,}

^{1,...,n Thus}

wehave

($,X) S, OtiXi eei8,

Xi) e Oi(8, X ^{IntC (s,z)}

^intC.

i=1 i=1 i=1

Hence0/nt

C,

which contradictsthe poimedness ofC. Therefore,

F

^isa

^KKM

multifunctionon

D

Definea multifimetion

F2 ^D

^--,2z)by

F2(y) {z D"

(t,y

z)

lntCfor some E

for y

D.

For anyz

Ft ^(y)

^thereexists an s

T(z)

^{such that}

(s,

y-

z) f ^Int

^{C. BytheC-}

pseudomonotonieity of

T,

there existsa t

T(y)

such that

(t,y- z)

^{lntC. Thusz E}

F2(y)

Henceforanyy

D, Ft (y) c F2(y).

Therefore

F2

^isalsoa

^KKM

multifunction on

D.

We claim that

F2

^is closed-valued. In fact, for any y

D,

let

{z,}

^{be a} sequence in

F2()

which converges to

x. D.

Since z,,

F2(y)

for each n, there exists a t, E

T(y)

such that

(tn,y-z,) Y\(-/ntC).

^Since

T(y)

iscompact, we mayassume that

{t,}

convergesto some t.

T(y).

Notethat

Since

{t, }

^isbounded in

L(X, Y), (tn -

^convergesto

^(t.,

^Z/-

^z.).

^Hence

^t.

^z.

^{Int C,}

whencewehave

z. F2 ^(y).

6Z,0 S. PARK, B. S.LEEAND G. M.LEE

Further,notethat assumption

(2)

implies that,for eachz6

LN\K

there exists a y6

LN

^{such that}

:

F2(t/).

^Hence

Lv

^f3

["I{F2(/):

Z/E

Lv}

^C

^K.

^{Therefore,condition}

(2)ofTheorem

holds.

Therefore,byTheorem 1, there exists^{an z}

K n f’l ^{{F2 (/)}

^Z/

^D}.

^{Thenforany /E}

^D,

^there

exists a

t

^E

^T/such

^that

^(tu,

^Z/-

^z}

^Int

^{C. By}

the convexity of

D,

for anyzE

(0,1),

there exists a

to ET(c=i/+(1-c=)z)

^such that

{ta,c(Z/-z)} -IntC.

^{Dividing by c=, we have}

(to,

_Z

z} f

^{IntC. By the} V-hemicontinuity of

T,

there exists

to

E

T(z)

such that

(to,

Z/-

z) f

^{IntC. HencezE}

f’l ^{{F1 (/)}

^/E

^{D} #}

^(Z). Consequently,there existsanz0

K

such thatforeach z E

D,

there existsan₈₀E

T(zo)

^suchthat

(80,z zo) f ^IntC.

COROLLARY2.1.

In

Theorem 2, if

D

isclosed,then the coercivity

(2)

canbereplaced bythe followingwithoutaffectingitsconclusion:

(2’)

^{thereexists a}nonempty compact subset

K

of

D

anda_Z/0E

K

such that

(t,I/o Z)

^{E IntC for z E}

^{D- K}

^and

tET(!/0).

PROOF. Itsufficestoshow that

(2’)

implies

(2). In

fact,foranynonempty finite subset

N

of we let

Lv o({/0}

^UNU

K)

^C

D. By (2),

for any zE

Lr ^K

^C

^{D K,}

there exists a Z/0E

K

C

Lv

^suchthat

(,

Z/0

z)

E IntCfor all t E

T(Z/0).

Hence

(2)

^holds.

REMARK. Evenforasingle-valued

T,

Corollary2.1ismoregeneralthan

Yang

19, Theorem2].

For

D K,

Theorem 2reducestothefollowing

COROLLARY2.2 Let

X

and

Y

beBanachspaces,

C

aclosed pointed and convex conein

Y

with Int C (Z),

D

anonempty compact and convex subsetof

X

and

T X

^--,2^L(x’’)C-pseudomonotone, compact-valued,and V-hemicontinuous. Then

(GVVI)

is solvable.

REMARK.

Corollary2.2extends Chen and

Yang [6,

Theorem 2.1,Part

(i)].

COROLLARY2.3

[7].

Let

X

be a reflexive Banachspace,

Y

a Banachspace,

C

aclosed pointed and convexconein

Y

with

Int

C

-(Z), D

anonempty bounded closed and convex subset of

X,

and

T

:X^--,2^L(x’’) C-pseudomonotone, compact-valued and V-hemicontinuous. Then

(GVVI)

is solvable.

PROOF. Switch tothe weaktopologyonX.

COROLLARY 2.4. Let

X

be a Banach spacewithdual space

X’, D

anonempty compact and convex subset of

X

and

T :X

--. X* pseudomonotone and hemicontinuous. Then there exists an z0E

D

such that

(T(=:0),

^=:

=:0) ^>_

⁰for all =: ED.

REMARK. Corollary2.4generalizesCottle andYao [11,Theorem

2.1].

Notethat for

Y R

and C

R+,

corollaries extend or reduce to well-known scalar valued variational inequalities due to HartmanandStampacchia, Browder,Stampacchia,

Mosco,

Dungundjii andGranasandmanyothers.

3. FUZZY

EXTENSION

Let

X

and

Y

betwonormedspacesand

;T(L(X, Y))

the collection of allfitzzysetson

L(X, Y).

^A

mapping

F

from

X

into

.TC(L(X, Y))

^iscalledafuzzy mapping.

If

F

:X^--,

.TC(L(X,Y))

^isafuzzy mapping,then

F(z),z

^E

X (denoted

by

F=),

isafttzzyset in

(L(X,Y))

^and

F=(8),8

E

L(X,Y),

^isthedegreeof membership of⁸in

F=.

^Let

^A

^E

and/ [0,1].

^{Then theset}

(A) ^{ L(X, Y) A() _>/}

is said tobean a-cut setof

A.

DEFINITION 3.1

[20]. A

fuzzy set

A

in

L(X, Y)

is compact if for

each/

E

(0,1], (A)#

^is

compactin

L(X, Y).

DEFINITION

^3.2. Let

X

and

Y

betwonormedspaces,

F X

^--,

1:(L(X, Y))

^afuzzymapping and

C

aclosed, pointedand convex coneof

Y

suchthat

Int C .

1.

F

is said to be C-monotone iffor any z,/E

X

and 8,tE

L(X,Y)

^with

F=(8)>

O and

’() >

^0,

⁽

^t,z

) e c.

OENERAL VECTOR-VALUEDVAR/ATIONALINEQUALITY 641 2.

F

is said tobeC-pseudomonotoneiffor any z,9E

X and/3

E

(0,1], (s,

9

z> ^Ir

^Cfor

some s

L(X,Y)

with

Fx(8)>/3

implies that

<t,9-z>

-InC for some t

L(X,Y)

with

F()

_3.

> . _F

_{is said tobe} hemicontinuous ifforany x,9

where /3E

(0,1],

there exists

to L(X,Y)

with

Fz(to)

^>_/3 for any z

X, <ta, z> ^<to, zl

^as

C

---

^Now0

^+"

^{weobtain afuzzyextensionofTheorem 2.}

THEOREM3. Let

X

and

Y

beBanachspaces, Caclosed,pointed and convex conein

Y

with Int

C #

0,

D

a nonempty convex subset of

X, K

a nonempty compact subset of

X,

and

F X Y(L(X, Y))

^afimzymappingsuch that thereexists areal

number/3 (0,1]

suchthatfor each x

X, (Fx)

is a nonempty subsetof

L(X, Y). Suppose

that

(1)

^FisC-pseudomonotoneand hemicontinuous, and for eachz E

X, F,

isacompact fuzzysetin

L(X,r),

(2)

for each nonempty finite subsetNof

D,

thereexistsa compact convex subset

Lv

^of

^D

^such

that N

c Lv

and for each z

Lc\K

^{there exists a}9

Lv

^{such that}

<t,

9-

z>

^{Int Cfor all}

L(X, r) .am _F.(t) >/.

Then there exists an _z0

D

such that for each z E

D,

there exists an ₈₀

L(X,Y)

^with

F.0 (0) _>

^{such that}

<o,

*0 lt

C.

PROOF. Defineamultifunction

" ^X

^{--,2L(X’oforanyz}

^{X, ’(z) F(z).}

^{It followsfrom}

the C-pseudomonotonicity of

F

thatfor any z,9

X, <8,

9

z>

^{Int Cfor some8}

^(z)

^implies

that

<t,

9-

z>

^{Int Cforsome tE}

^’(9).

^{Thisimplieshat is}C-pseudomonotone. Furthermore, the hemicontinuity of

F

implies the V-hemicontinuity of

.

^{Sincefor eachz}

^{X, F,}

^isacompact

fuzzysetin

L(X, Y),

^thenfor eachz

K, (z)

^{iscompact. Condition}

(2)

implies that assumption

(2)

inTheorem 2issatisfiedfor the multifimction

’. By

Theorem2.1thereexists_z0

D

such that for each z

D,

there exists

so

^E

(z0)

^{such that}

(so,

z

z0>

IntC. Hencethere exists anx0

D

such thatfor each_COROLLARYz

D,

there exists_{3.1. In}Theorem 3, if_s0

L(X,Y) D

withisclosed,

F,o (s0)

then the coercivity

>/3

^{such that}

<8o, (2)

zcan

zo>

be

.

replaced by^IrtC. the followingwithoutaffectingits conclusion:

(2’)

there existsanonempty compactsubset

K

of

D

and_{a 90}

K

such that

<t, g0-z>E

Int C for all x

D K

and all For O

K,

Theorem3reducestothe following

COROLLARY

:.:.

^Let

X

and

Y

be Banachspace,

C

aclosedpointedand convex cone in

Y

with Int C

#

0,

D

bea nonempty compact and convex subset of

X

and

F X ’(L(X, Y))

a fuzzy mapping such that there existsarealnumber/3

(0,1]

^{suchthatfor eachz}

X, (F,)

is a nonempty subset of

L(X, Y). Suppose

that

F

is C-pseudomonotone and hemicontinuous, and that for each z E

F, F=

^isacompact fimzyset in

L(X, Y’).

Then there exists anz0

there exists an₈₀

L(X,Y)

^with

Fzo(.0)

^>_/3suchthat

COROLLARY3.3

[12].

Let

X

bea reflexiveBanach space and

Y

aBanachspace. Let

D

bea nonempty,bounded,closed and convex subset of

X

andCaclosed, pointed andconvexconein

Y

with

IntC # .

^Let

^{F X}

^--.

^(L(X,Y))

^{be a}fuzzy mapping such that

F

is C-pseudomonotone and hemicontinuous and that for eachz

X, F,

isa compactfuzzysetin

L(X, Y). Suppose

further that there existsareal

number/3

E

(0,1]

^suchthatforeachz

X, (F=)

is a nonempty subset of

L(X, it).

Then there exists anz0

O

such thatforeach x

D,

there existsan such that

<s0,

z _z0 Irtt

C.

62 S.PARK, B. S.LEE ANDG.M.LEE

ACKNOWLEDGEMENT. The first author was supported in

pan

by the Basic Science Research Institute

Program,

ProjectNo.BSRI-97-1413, the second BSRI-97-1405 and the third BSRI-97-1440.

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