A general vector valued variational inequality and its fuzzy extension

(1)

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

-,₂L(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,

(2)

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

2xis

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 saythat 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

2xa

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* a}_mapping. 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/),

-

z) >

O.

3.

T

is said to 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)

_a_mapping_and

U

_a_closed,

pointed and convex cone of

Y

such thatInt

U

.

1.

T

is said to be

U-monotone

iffor_{any z, F}

X,

(T(z)

T(/),

z

F)

U.

2.

T

is saidtobeU-pseudomonotoneifforany z,/

X,

(T(z),

F

z)

Int

U

implies that

(3)

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

2L(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 thatfor 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. Bythe

C-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,

(4)

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 existsan 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 anonempty compact subset

K

of

D

andaZ/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

--,2L(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--,₂L(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))

_is_a_{fuzzy 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

.

(5)

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

---

0

+"

Nowweobtain 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 a9

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’o_for_any_z

X,

’(z)

F(z).

_{It follows}_from

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 isC-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

Theorem2.1thereexistsz0

D

such that for each z

D,

there exists

so

E

(z0)

such that

(so,

z

z0>

IntC. Hencethere exists anx0

D

such thatfor eachz

D,

there existss0

L(X,Y)

with

F,o

(s0)

>/3

such that

<8o,

z

zo>

.

IrtC.

COROLLARY3.1. InTheorem 3, if

D

isclosed,then the coercivity

(2)

canbe replaced bythe followingwithoutaffectingits conclusion:

(2’)

there existsanonempty compactsubset

K

of

D

anda 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 afuzzy 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

(6)

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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