Generalized vector-valued variational inequalities and fuzzy extensions

GENERALIZED VECTOR-VALUED VARIATIONAL INEQUALITIES AND FUZZY EXTENSIONS

BVUNG SOO LEE, GUE MVUNG LEE AND Do SANG KIM

1. Introduction and Preliminaries

Recently, Giannessi [9] firstly introduced the vector-valued varia- tional inequalities in a real Euclidean space. Later Chen et al. [5]

intensively discussed vector-valued variational inequalities and vector- val1,led quasi variational inequalities in Banach spaces. They [4-8]

proved some existence theorems for the solutions of vector-valued varia- tional inequalities and vector-valued quasi-variational inequalities. Lee et al. [14] established the existence theorem for the solutions of vector- valued variational inequalities for multifunctions in reflexive Bana~

spaces.

On the other hand, Chang and Zhu [3] investigated the existence theorems of vector-valued variational inequalities for fuzzy mappings in locally convex Hausdorff topological vector spaces, which were the fuzzy extensions of some theorems in [12, 20, 22, 24]. Lee et al. [13]

obtained the fuzzy generalizations of new results of Kim and Tan [11], and they [14] established the fuzzy extensions of their existence theo- rems. The noncompact cases of the existence theorems of vector-valued variational inequalities for multifunctions or fuzzy mappings in Banach spaces obtained by Lee et al. [14] was considerd by Park et al. [19].

In this paper, we establish the existence theorems of the following more generalized vector-valued variational inequalities (GVVI) for mul- tifunctions than inequalities in [14] by using variable dominated cones,

Received September 28, 1995. Revised April3,1996.

1991 AMS Subject Classification: 47H19.

Key words: Variational inequalities, existence theorems, fuzzy extensions, con- vexity, coercivity.

The first author was supported in part by the Basic Science Research Institute program, Ministry of Education, 1996, Project No. BSRI-96-1405. The second and third authors were supported in part by the Basic Science Research Institute program, Ministry of Education, 1996, Project No. BSRI-96-1440.

and obtain the fuzzy extensions of our existence theorems. In section 2, we establish the existence theorems for (GVVI) under the upper- semicontinuity of the multifunction T, and obtain the existence theo- rems for (GVVI), by using theP -monotonicity and V-hemicontinuity of T, under the coercivity condition in Banach spaces. Also we ob- tain the existence theorems for (GVVI) in reflexive Banach spaces. In section 3, the fuzzy analogues of our results in section 2 are dealt with.

Let X and Y be two Banach spaces and D a nonempty convex subset ofX. Let T : X ~ 2L (X,Y) be a multifunction, where L(X, Y) is the space of all continuous linear operators from X into Y. Let {C(x)lx E D}be a family of convex cones inY such that Int C(x)

f=

0,

\Ix E D, where Int denotes the interior.

Consider the following generalized vector-valued variational inequal- ity :

(GVVI) Find Xo E D such that for each x E D, there exists an So ET( xo) such that

(So, x - xo) _~ -Int C(xo),

where (so,y) denotes the evaluation ofSo at y.

When T is an operator from X into L(X,Y), (GVVI) reduces to the following vector-valued variational inequality (VVI) considered by Chen [4].

(VVI) Find Xo ED such that (T(xo), x - xo) _~ -Int C(xo) for all xED.

When for every x E D, C(x)

=

C, where C is a convex cone inY with Int C

f= 0,

(GVVI) [respectively, (VVI)] reduces to the following vector-valued variational inequalities (GVVIy [resp., (VVIy] investi- gated by Park et al. [19] and Lee et al. [14] [resp., Chen et al. [5, 7, 8], Yang [23]].

(GVVI)' Find Xo E D such that for each x E D there exists an So E T(xo) such:that

(So, x - xo)

rt.

-Int C.

(WI)' Find Xo E D such that (T(xo),x - xo)

rt.

-Int C for all x ED.

The above inequality(VVI)'is a generalization of the following clas- sic scalar-valued variational inequality (VI). When Y

=

JR, X

=

JRn, C(x) = R+, Vx E D

c

JRn, then the(VVI)' collapses to the (VI).

(VI) FindXo E D such that (J(xo), x - xo) ~ 0for all x ED eRn, where

f :

^{D - tRn} is a given operator.

Now we give the definition of a KKM multifunction.

DEFINITION 1.1. Let D be a subset of a topological vector spaceX.

Then a multifunction G :D ^{- t} 2x _{is called KKM}if for each nonempty finite subset N of D, co N

c

G(N), where codenotes the convex hull and G(N)

=

U{Gx : x EN}.

A convex spaceX is a nonempty convex set (in a vector space) with any topology that induces the Euclidean topology on the convex hulls of its finite subsets. Thus, a convex subset D of a topological vector space X with the relative topology is automatically a convex space.

For details of the convex space, _~ee Lassonde [12].

We need the following particular form of the generalized KKM the- orems due to Park [16-18], which will be used inthe proof of our main results.

THEOREM 1. Let X be a convex space, K a nonempty compact subset ofX, andG : X ^{- t} 2^x a KKM multifunction. Suppose that

(1) for each yE X, G(y) is closed; and

(2) for each nonempty finite subset N of X, there exists a compact convex subset LN ofX such that N C LN and LN

n

^{n{G(y) : y E}

LN}

c

K.

Then we have

Knn{G(y): y E X} =1=

0.

Inparticular, ifX = K, that is, X is a compact convex space, then the condition (2)is obviously held, and henceinthis case, without the condition (2), it is true that

n{

G(y) : yEX} =1=

0.

2. Existence Theorems

First, we give the following definitions for the existence theorems for (GVV!).

DEFINITION 2.1. Let F be a multifunetion from a topological space X into a topological spaceY.

1. F is said to be closed at x E X if for each sequences {xn }~=1 converging to x and{Yn}_~=1 converging to y such thatYn E F(x_n)for alln, we have y E F( x). F is said to be closed if it is closed at every

x EX.

2. F is said to be upper semi-continuous at x E X if for every open setV inY containingF(x), there exists a neighborhood N(x) ofx such that F(z) C V for allz E N(x). F is said to be upper semz-continuous if it is upper semi-continuous at every x E X.

DEFINITION 2.2. Let X and Y be two normed spaces, T : X ~

L(X,Y) an operator, andP a nonempty closed convex cone inY with Int P =1=

0.

1. T is said to be P -monotone if for any x, y E X, (T(x) - T(y), x-

0E~

,

2. T is said to be P -ps eudomonotone iffor any x, y EX, (T( x ), y - x)

tf.

-Int P implies that (T(y),y - x)

tf.

^-IntP.

3. T is said to be V-hemicontinuous if for any x, y, z EX, the mapping Cl! f-+ (T(x

+

ay),z) is continuous at 0+.

DEFINITION 2.3. Let X and Y be two normed spaces, T : X ~

2L (X,Y) a multifunetion, andP a nonempty convex cone inY withInt P =1=

0.

1. T is said to be P -monotone if for any x, y E X, s E T( x) and tET(y), (s-t,x-y) EP.

2. T is said to be P -pseudomonotone iffor any x, y E X, (s, y - x)

tf.

-IntP for some s E T(x) implies that (t, y - x)

tf.

-Int P for some t E T(y).

3. T is said to be V-hemicontinuous if for any x, y E X,a > 0 and ta E T(x

+

ay), there exists a to ^E T(x) such that for any z E

X,(ta, z) ^I--t (to, z) as a -+0+.

REMARK. 1. Definition 2.3 is a generalization of Definition 2.2.

2. We can easily prove that the P -monotonicity implies the P -pseudomonotonicity.

Now we prove the following existence theorems for (GVVI) in Ba- nach spaces.

THEOREM 2. Let X and Y be Banach spaces, D a nonempty closed convex subset ofX, and C : D -+ 2Y a multifunction such that for each x E D, C(x) is a convex cone in Y with Int C(x) =F

0

and C(x) =F Y. Let W : D -+ 2^Y be a closed multifunction defined by W(x) = Y\(-Int C(x)) for any x E D.

Let T : X -+2L (X,Y) is uppersemi-continuous and compact-valued.

Suppose that T(D) is containedin a compact subset of L(X, Y).

Then (GVVI) is solvable.

Furthermore, the solution set of (GVVI) is a compact subset of D.

Proof. Define a multifunction F_{I :}D -+ 2^D by

FI(y) = {x E D: (s,y - x) <I. -Int C(x) for some s E T(x)}

for y E D. ThenFI is a KKM multifunction onD.

In fact, suppose that N = {Xl'·· ·,xn } C D, _I:~=IG:i

=

^1,ai ~ O,i

=

1,···,n and x

=

I:~l aiXi

rt.

FI(N). Then for any sE T(x), we have (s, Xi - x) E -Int C(x),i = 1,···,n. Thus we have

n n

(s, x)

=

^{(s, L aixi)}

=

^{L ai(S, Xi)}

i=l i=l

n

E LG:i(S,X) - Int C(x) i=l

= (s,x) - Int C(x).

Hence 0 E Int C(x), which contradicts the assumption C(x) =F Y.

Therefore, FI is a KKM multifunction on D. We claim that FI is

closed-valued. In fact, let _{Xn}~=l be a sequence in F1(y) ^converging to x* ^E D for any fixed y^E D. Since Xn E F1(y) for all n, there exists an Sn ET(x_{n )} such that

(2.1) for all n.

On the other hand, by assumptionT(D) is compact.

Hence without loss of generality, we can assume that there exists a S* E L(X, Y) such that Sn converges to S*. Since T is upper semi- continuous and compact-valued, T is closed[l]' so S* E T(x*). More- over we have

lI(sm y-x n) - (s*, y - x*)¹¹

~ lI(sn, x* -

xn)11 + II(sn -

S*,Y - x*)1I

~ IIsnll· I\x*- xnl\

+

I\sn -

s*I\·lly -

x*l\.

Since{Sn} is bounded in L(X, Y), (sn,Y - x n)converges to (s*, y- x*). By (2.1) and the closedness ofW, we have (s*, y - x*) EW(x*).

Consequently, there exists an s* E T(x*) such that (s*, y - x*)

tI.

-Int C(x).

HenceF1(y) is closed. Therefore, by Theorem 1 there exists anXo E n{F1(y) : y E D}. Thus there exists an Xo E D such that for each x E

D, there exists an So ET(xo) such that (so, x - xo)

tI.

-Int C(xo). It is clear that the solution set of (GVVI), n{F1(y) :yE D} is compact.

The following theorem shows that (GVVI) is solvable under a coer- civity condition in Banach spaces.

THEOREM 3. Let X .and Y be Banach spaces, D a nonempty convex subset of X and K a nonempty compact subset of X. Let C : D ^--t2Y

be a multifunction such that for eachx ED, C(x) is a convex cone in Y with lnt C(x) =I=- 0andC(x) =I=-Y, andP:= nXEDC(x) a nonempty convex cone in Y with Int P =I=- 0. Let W : D --t 2^Y be a closed multifunction defined byW(x) = Y\(-Int C(x)) for any x E D, and T : X ^--t 2L (X,Y) a multifunetion.

Suppose that

(1) T is P -monotone, compact-valued and V-hemicontinuous.

(2) for each nonempty finite subset N of D, there exists a nonempty compact convex subset LN of D such that N

c

LN and for each

x E LN\K there exists ayE LN such that (t, y - x) E -lnt C(x) for allt ET(y).

Then (GVVI) issolvable.

Proof. Define a multifunction F_{I :}D ~ 2^D by

FI(y) = {x E D: (s,y-x)

rt

-lnt C(x) for some sET(x)} for yE D.

Then by the same argument as the proof in Theorem 2, FI is a KKM multifunction on D. Define a multifunction F_{2 :}D _~ 2^D by

F2(y) = {x E D :(t, y - x)

rt

-lnt C(x) for somet ET(y)}

for yE D. ThenF2 is also a KKM multifunction onD. In fact, for any x E FI(y), there exists an s E T(x) such that (s,y - x) ~ -lnt C(x).

By the P -monotonicity of T,

(s-t,y-x) E -Pc -C(x)

for any t E T(y). Hence for any t E T(y) (t,y - x)

rt

-lnt C(x) and thus x E F2(y). Hence FI(y) C F2(y) for any y ED. Therefore F2 is also a KKM multifunction on D. We claim that F₂ is closed-valued.

Indeed, for any fixed yE D, let {Xn}~=lbe a sequence in F2(y) which converges to x* E D. Since Xn E F₂(y) for each n, there exists a t_n E T(y) such that

(2.2)

Since T(y) is compact, we may assume that {tn}~=l converges to some t* ET(y). Note that

II(t

^{n ,}Y - x n) - (t*, y - x*)1I = lI(t n, x* - x n)

+

(t n - t*, y - x*)

11

::; lI(t n,x* - x n)1I

+

lI(t n - t*,y - x*)1I ::; IIt nll ·lI x * - xnll

+

^{IIt n -}

t*1I

·lIy -

x*ll·

Since {tn}~=l is bounded in L(X, Y), (t n,^{Y -} x n) converges to (t*, y - x*). By (2.2) and the closedness ofW we have (t*, y - x*) E W(x*). Hence (t*, y - x*)

rt

-lnt C(x*), whence we have x* E F2(y).

FUrther, note that assumption(2) implies that, for each x ELN\K there exists ayE LN such that x

tI.

^F2(y). Hence LN

n

^n{F2(y) : y E LN}

c

K. Therefore, the condition (2) of Theorem 1 holds. Thus, by Theorem 1, there exists an x E K

n

n{F2(y) : y E D}. Then for any y E D, there exists a t y E T(y) such that (ty,y - x)

tI.

-lnt C(x).

By the convexity ofD, for any a E (0,1), there exists a t a E T(ay

+

(l-a)x) such that (ta,a(y-x))

tI.

-lnt C(x). Dividing by a, we have (ta,y - x)

tI.

-lnt C(x). By the V-hemicontinuity of T, there exists a to E T(x) such that (to,y - x)

tI.

-lnt C(x). Hence x E n{F1(y) : y E D}. Thus n{F1(y) : y E D}

1= 0.

Consequently, there exists an Xo ED such that for each x ED, there exists anSo E T(xo) such that (so,x - xo)

f/.

-lnt C(xo).

COROLLARY 2.1. In Theorem 3, if D is closed, then the coercivity (2) can bereplaced by the following without affecting its conclusion:

(2)' there exist a nonempty compact subset K ofD and ayE K such that (t,y - x) E -lnt C(x) for x ED\K andt E T(y).

Proof. It sufficies to show that (2)' implies (2). In fact, for any nonempty finite subset N of D, we let LN = co (N U(K nD)) c D.

By (2)',for any x ELN\K c D\K, there exists ayE (KnD) c LN such that (t, y - x) E -lnt C(x) for allt E T(y). Hence (2) holds.

For D = K, Theorem 3 reduces to the following corollary ;

COROLLARY 2.2. Let X and Y be Banach spaces, D a nonempty compact convex subset ofX, C :D --t 2^Y a multifunction such that for each x E D, C(x) is a convex cone in Y with 1nt C(x)

1= 0

and C(x)

=J

Y, and P := nXEDC(x) a nonempty convex conein Y with 1nt P

1= 0.

Let W : D ~ 2^Y be a closed multifunction defined by W(x)

=

Y\( -lnt C(x)) for any x E X, and T : X ^{--t 2.z;,(X,Y)} a multifunction. HT: X --t 2L (X,Y) is P -monotone, compact-valued and V -hemicontinuous, then (GVVI) is solvable.

Now we prove the following existence theorem for (GVVI) in reflex- ive Banach spaces.

THEOREM 4. Let X be a reflexive Banach space, Y a Banach space, and D a nonempty closed, bounded and convex subset of X. Let C : D ^--t 2Y be amultifunction such that for each x ED, C (x) is a convex

cone in Y: with Int C(x) =1= 0 and C(x) =1= Y, and P := nXEDC(x) a nonempty mnvex cone in Y with Int P =1= 0. Let W : D ^{- t} 2^Y be a weakly closed multifunction defined by W(x) = Y\( -lnt C(x)) for any x E D, and T : X ^{- t} 2L (X,Y) a multifunction.

HT is P -monotone, compact-valued and V -hemicontinuous, then (GVVI) is solvable.

Proof. Define a multifunction F1 : D ^{- t} 2^D by

F1(y)

=

{x E D: (s,y-x) ~ -lnt C(x) for somesE T(x)} for yE D.

Then by the same argument as the proof in Theorem 2, F1is a KKM multifunction on D. Define a multifunction F2 : D ^{- t} 2^D by

F2(y)

=

{x E D :(t,y - x) ~ -lnt C(x) for somet ET(y)}

for y E D. Then by the same argument as the proof in Theorem 3, F1(y) C F2(y)for anyyE D,and henceF2is also aKKMmultifunction on D. Now we claim that F₂ is weakly closed-valued. Indeed, for any fixedyE D,let {xn}~=1be a sequence inF2(y) which converges weakly to x* ED. Since ^Xn^{E F}2(y) for eachn, there exists a t n^ET(y) such that

(2.3)

Since T(y) is compact, we may assume that {tn} ;::'=1 converges to somet* E T(y). Note that for any qE Y*, where y* is the topological dual ofY,

Iq((tn,y - x n) - (t*,y - x*))1

~Iq((t n - t*,^{y -} xn))1

+ Iq(

(t*, x* - xn))1

~llqlllltn -

t*lllIy - xnll + I(q

⁰ t*)(x* - x n

)!

~lIqlllltn

- t*II(llyll + II

x nll)

+ I(q

⁰ t*)(x* - xn)l·

SincexnE D for all nandD is bounded, {xn} ;::'=1 is bounded. Since Ilt n - t*1I ^{- t 0,}

On the other hand, sinceq0 t* is continuous and linear from X to lR, we have

Consequently,

{(t n, Y - xn)}~=l converges weakly to (t*, y - x*). By (2.3) and the weak closedness ofW we have (t*, y-x*) E W(x*). Hence (t*, y-x*) tJ.

-lnt C(x*), whence we have x* E F2(y).

SinceD is a closed, bounded and convex subset of a reflexive Banach space X, D is weakly compact. Thus, by Theorem 1, there exists an x E n{F2(y) : y E D}. It follows from the V-hemicontinuity of T that there exists a to E T(x) such that (to,Y - x) tJ. -lnt C(x). Hence x E n{F1(y) : y E D}. Thus n{F1(y) : y E D} =1=

0.

Consequently, there exists an ^Xo E D such that for each x E D, there exists an So E T(xo) such that (so, x - xo) tJ. -lnt C(xo).

REMARK. WhenC(x) is a constant cone in Theorem 4, we can show that (GVVI)' is solvable under the P-pseudomonotonicity ofT [14].

When T is a single-valued mapping, we can obtain the following corollary from Theorem 4.

COROLLARY2.3 [4]. LetX bearefIexiveBanach space, Y aBanach space, D a nonempty bounded, closed and convex subset ofX, C : D --+ 2^Y a multifunetion such that for each x E D, C(x) is a convex cone in Y with Int C(x) =1=

0

and C(x) =1= Y, and P := nXEDC(x) a nonempty convex cone in Y with Int P =1=

0.

Let W be a weakly closed multifunetion defined by W(x) = Y\( -lnt C(x)) for any x E X, and T : X --+ L(X, Y) an operator. HT: X --+ L(X, Y) is P-monotone, and V -hemicontinuous, then (VVI) is solvable.

REMARK. Note that for Y

=

lR and C

=

lR+, corollaries extend or reduce to the well-known scalar valued variational inequalities due to Hartman and Stampacchia [10], Browder [2], Stampacchia [21], Mosco [15] and many others.

3. Fuzzy Extensions

Let X andY be two nonned spaces and F(L(X,Y)) the collection ofallfuzzy sets on L(X, Y). A mapping F from X intoF(L(X, Y)) is called a fuzzy mapping.

If F : X ~ F(L(X, Y)) is a fuzzy mapping, then F(x), x E X (denoted by Fx), is a fuzzy set in F(L(X, Y)) and Fx(s),sE L(X, Y), is the degree of membership of s in Fx. ^Let A E F(L(X, Y)) and (3 E [0,1]. Then the set (A)p = {s EL(X,Y) : A(s) ~ (3} is said to be a (3-cut ofA.

DEFINITION 3.1 [25]. A fuzzy set A on L(X, Y) is compact if for each (3 E (0,1]' (A)p is compact in L(X,Y).

DEFINITION 3.2. Let X and Y be two normed spaces, F : X ~

F(L(X, Y)) a fuzzy mapping, and P a nonempty convex cone in Y with 1nt P

#0.

1. F is said to beP-monotoneif for any x, y E X ands, t E L(X, Y) with Fx(s) >

°

^{and Fy(t) > 0,}(s - t,x - y) EP.

2. F is said to be P-pseudomonotone if for any x, y E X and (3 E (0,1], (s,y-x)

rt.

-lnt P for somesE L(X, Y) withFx(s) ~ (3 implies that (t, y - x)

rt.

-lnt P for some t E L(X, Y) with Fy(t) ~ (3.

3. F is said to be hemicontinuous if for any x, y E X and to! E

L(X, Y) with Fx+O!y(tO!) ~ (3 where et,(3 E (0,1], there exists to E L(X, Y) with Fx(to) ~ (3 such that for any zE X, (to!,z) -+ (to,z) as

a~O+.

4. F is said to be closed at xo. E X if for each open subset V of L(X, Y) such that if Fxo(s) ~ (3 where (3 E (0,1]' then s E V, there exists a neighborhood N(xo) of Xo such that ifx E N(xo) and Fx(s) ~ (3, then s EV. F is called closedif it is closed at each point ofX.

Now we can easily obtain fuzzy analogues of Theorem 2 and Theo- rem 3 respectively.

THEOREM 5. Let X and Y be Banach spaces, D a nonempty closed convex subset of X. Let C : D -+ 2^Y be a multifunction such that for each x E D, C(x) is a convex cone in Y with 1nt C(x)

#

0and C(x)

#

^{Y. Let W : D} ~ 2^Y be a closed multifunetion defined by W(x)

=

Y\(-lnt C(x)) for any x E D, and F: X ^-+ F(L(X,Y)) a

fuzzy mapping such that there exists a real number (3 E (0,1] such that for each x E X,(Fx){3 is a nonempty subset of L(X,V). Suppose that F is closed, and for eachx EX,F_x is a compact fuzzy set on L(X,V).

IfUXED(Fx){3 is contained in a compact subset of L(X,V), then there exists an Xo E D such that for each x E D, there exists an So EL(X, Y) with Fxo(so) ~ (3 such that (so, x - xo) t/:. -lnt C(xo).

Proof. Define a multifunction F : X - t 2L (X,Y) by for any x E X,F(x)

=

(Fx){3.Let Xl EX andV be any open set such that F(XI) C V, then s EV for anys EL(X,Y) withF_X1(s) ~ (3. By the closedness ofF, there exists a neighborhood N(xI) of^Xl such that ifx EN(XI) andFx(s) ~ (3, then sEV, that is, there exists a neighborhoodN(XI) of Xl such that x E N(xI) implies F(x) C V. Hence F is upper semi-continuous. Since for each x EX, F_x is a compact fuzzy set on L(X,V), then F(x) is compact. By Theorem 2, we know that there exists an Xo ED such that for each x ED, there exists an So E F(xo) such that (so, x -xo) _~ -lntC(xo). Hence there exists anXo E D such that for each x E D, there exists an So E L(X, Y) with Fxo(so) ~ (3 such that (so,x - xo) ~ -lntC(xo).

THEOREM 6. Let X and Y be Banach spaces, D a nonempty convex subset of X, and K a nonempty compact subset of X. Let C : D ^{- t}2Y

be a multifunction such that for each x ED, C(x) is a convex cone in Y with Int C(x) =I=-

0

andC(x) =I=-Y, and P:= nXEDC(x) anonempty convex cone in Y with Int P =I=-

0.

Let W : D ---+ 2^Y be a closed multifunction defined byW(x)

=

Y\( -lntC(x)) for anyx E X, and F : X ^{- t} F(L(X, V)) a fuzzy mapping such that there exists a real munber(3 E (0,1] such that for each x EX,(Fx){3is a nonempty subset ofL(X, V). Suppose that

(1) F is P -monotone, hemicontinuous, and for each x EX,Fx is a compact fuzzy set on L(X, V).

(2) foreach nonempty finite subset N ofD, there exists a nonempty compact convex subset LN of D such that N C LN and for each x E LN\K there exists ayE L N such that (t,y - x) E -lnt C(x) for allt E L(X, Y) with Fy(t) ~

13.

Then there exists an Xo ED such that for each x E D, there exists anSo EL(X, Y) withFxo(so) 2:: (3such that (so, x -xo) ~ -lnt C(xo).

Proof. Define a multifunction F :X ---+ 2L (X,Y) by F(x)

=

_(Fx){3

for any x EX. It follows from the P -monotonicity of F that for any x, y E X, for anys E F(x) and t E F(y), (s-t, x-y) E P. This implies that

F

is P -monotone. The V-hemicontinuity of

F

is easily proved and the compactness of F(x) for each i E X is proved similarly as in the proof of Theorem 5. Condition (2)implies that assumption (2) in Theorem 3 is satisfied for the multifunction

F.

By Theorem 3 there exists an Xo E D such that for each x E D, there exists an So E F(xo) such that (so,x-xo)

fI.

-1nt C(xo). Hence there exists an Xo E D such that for each x E D, there exists an So E L(X,Y) with Fxo(so) _~

f3

such that (so,x - xo)

fI.

-1nt C(xo).

COROLLARY 3.1. In Theorem 6, ifD is closed, then the coercivity (2) can bereplaced by the following without affecting its conclusion:

(2

t

there exist anonempty compact subset K of X andayE K nD such that (t, y - x) E -1nt C(x) for all t E L(X,Y) with Fy(t) ~

f3

and x ED\K.

Proof. It is proved similarly as in the proof of Corollary 2.1.

For D

=

^K, Theorem 6 reduces to the following corollary ;

COROLLARY 3.2. Let X and Y be Banach spaces, D a nonempty compact convex subset ofX, C : D _~ 2Y be a multifunction such that for each x E D, C(x) is a convex cone in Y with 1nt C(x) =I- 0

and C(x) =I- Y, and P := n xED C(x) ^a nonempty convex cone in Y with 1nt P =I-

0.

Let W : D _~ 2^Y be a closed multifunction defined by W(x)

=

Y\(-1nt C(x)) for any x E D, and F :X ~ F(L(X, Y)) a fuzzy mapping such that there exists a real number

f3

E (0,1] such that for each x E X, _(Fx)f3 is a nonempty subset ofL(X,Y). IfF is P -monotone, hemicontinuous, and for each x E X,Fx is a compact fuzzy set on L(X,Y), then there exists an Xo E D such that for each x E D there exists an So E L(X,Y) with Fxo(s'o) ~

f3

such that (so, x - xo)

fI.

-1nt C(xo).

Now we obtain the following fuzzy extension of Theorem 4.

THEOREM 7. Let X be a reflexive Banach space, Y a Banach space, and D a nonempty closed, bounded and convex subset of X. Let C : D _~ 2^Y be amultifunction such that for each x ED, C (x) is a convex cone in Y with Int C(x) =I-

0

and C(x) =I- Y, and P := nxED C(x)

a nonempty convex cone in Y witb 1nt P =1=

0.

Let W : D --+ 2Y be a weakly closed multifunction defined by W(x)

=

Y\(-lnt C(x)) for any x E X, and F : X --+ F(L(X,Y)) a fuzzy mapping such tbat tbere exists a real munber

f3

E (0,1] sucb tbat for eacb x EX,(Fx){3is a nonempty subset of L(X,Y). IT F is P -monotone, bemicontinuous, and for eacb x EX,Fx is a compact fuzzy set on L(X,Y), tben tbere exists an Xo ED such tbat for each x ED, tbere exists anSo EL(X,Y) witb Fxo(so) _~

f3

sucb tbat (so,x - xo) tJ. -lnt C(xo).

Proof. Define a multifunction

P :

X --+ 2^{L (X,Y)} by p(x)

=

(Fx){3 for any x EX. It follows from the P -monotonicity of F that for any x,y E X, for any s E F(x) and t E F(y), (s - t,x - y) E P. This implies that

F

is P -monotone. The V-hemicontinuity of

F

is easily proved and the compactness ofF(x)for eachx E X is proved similarly as in the proof of Theorem5. Censequently by Theorem 4 there exists an Xo E D such that for each x E D, there exists an So E F(xo) such that (so,x-xo) tJ. -lnt C(xo). Hence there exists an Xo E D such that for each x E D, there exists an So E L(X,Y) with Fxo(so) _~

f3

such that (so,x - xo) tJ. -lnt C(xo).

COROLLARY 3.3. Let X be a reflexive Banach space and Y a Ba- nach space. Let D be a nonempty bounded, closed and convex sub- set of X, and C : D --+ 2Y be a multifunction sucb tbat for eacb x E D, C(x) is a convex cone in Y witb 1nt C(x) =1=

0

andC(x) =1= Y, and P := nXEDC(x) a nonempty convex cone in Y witb 1nt P =1=

0. Let W : D --+ 2Y be a weakly closed multifunction defined by W(x)

=

Y\(-lnt C(x)) for any x E D, and F : X --+ F(L(X,Y)) a fuzzy mapping such tbat tbere exists a real number

f3

E (0,1] sucb tbat for eacb x E X, (Fx){3 is a nonempty subset of L(X,Y). IT F is P -monotone, hemicontinuous, and for each x E X,F_x is a com- pact fuzzy set on L(X,Y), tben tbere exists an Xo E D sucb tbat for eacb x E D tbere exists an So E L(X,Y) witb Fxo(so) ~

f3

sucb tbat (so,x - xo) tJ. -lnt C(xo).

REMARK. When C(x) is a constant cone in Corollary 3.3, we can show that the result of Corollary3.3 holds under P-pseudomonotonicity ofF [14].

References

1. Berge, C., Topological spaces, Oliver & Boyd Ltd., Edinburgh and London, 1963.

2. Browder, F. R., Existence and approximation of solutions of nonlinear varia- tional inequalities, Proc. Natl. Acad. Sci. USA 56 (1966), 1080-1086.

3. Chang, S. S. and Zhu, Y. G., On variational inequalities for fuzzy mappings, Fuzzy Sets and Systems 32 (1989), 359-367.

4. Chen, G. Y., Existence of solutions for a vector variational inequality: An ex- tension of the Hartmann-Stampacchia theorem,J. Optim. Th. Appl.74(1992), 445-456.

5. Chen, G. Y. and Cheng, G. M., Vector variational inequality and vector opti- mization,Lect. Notes in Econ. & Math. Syst. 285 (1987), 408-416, Springer- Verlag.

6. Chen, G. Y. and Craven, B. D., Approximate dual and approximate vector variational inequality for multiobjective optimization, J. Austral. Math. Soc.

(Series A) 47 (1989), 418-423.

7. Chen, G. Y. and Craven,B. D., A vector variational inequality and optimization over an efficient set, Zeitscriftfur Operations Research 3 (1990),1-12.

8. Chen, G. Y. and Yang, X. Q., The vector complementarity problem and its equivalence with the weak minimal element in ordered sets, J. Math. Anal.

Appl. 153 (1990), 136-158.

9. Giannessi, F., Theorems of alternative, quadratic programs and complementar- ity problems,Variational Inequalities and Complementarity Problems (Edited by Cottle, R. W., Giannessi, F. and Lions, J. L.), John Wiley and Sons, Chich- ester, England (1980), 151-186.

10. Hartmann, P., and Stampacchia, G., On some nonlinear elliptic differential functional equations, Acta Math. 115 (1966),271-310.

11. Kim, W. K. and Tan, K. K., A variational inequality in non-compact sets and its applications,Bull. Ausrtal. Math. Soc. 46 (1992), 139-148.

12. Lassonde, M., On the use of KKM multifunctions in fixed point theory and related topics, J. Math. Anal. Appl. 97 (1983), 151-201.

13. Lee, B. S., Lee, G. M., Cho, S. J. and Kim, D.S., A variational inequality for fuzzy mappings, Proceedings of Fifth International Fuzzy Systems Association World Congress, Seoul (1993), 326-329.

14. Lee, G. M., Kim, D. S., Lee, B. S. and Cho, S. J., Generalized vector variational inequality and fuzzy extension,Appl. Math. Lett. 6 (1993),47-51.

15. Mosco, D., Implicit variational problems and quasi-variational inequalities, Lecture Notes in Mathematics 543 (1976).

16. Park, S., Generalizations of Ky Fan's matching theorems and their applications, II, J. Korean Math. Soc. 28 (1991), 275-283.

17. , Some coincidence theorems on acyclic multifunctions and applications to KKM theory,Fixed Point Theory and Applications, (Edited by Tan, K.-K.), World Scientific, River Edge, NJ (1992), 248-277.

18. , On minimax inequalities on spaces having certain contractible subsets, Bull. Austral. Math. Soc. 47 (1993), 25-40.

19. Park, S., Lee, B. S. & Lee, G. M., A general vector-valued variational inequal- ity and its fuzzy extension, submitted to Applied Mathematics Letters.

20. Shih, M. H. and Tan, K. K., Generalized quasi-variational inequalities in locally convex topological spaces, J. Math. Anal. Appl. 108 (1985), 333-343.

21. Stampacchia, G., Variational Inequalities, In Theory and Applications of Mono- tone Operators (Ghizzetti, A., Ed.) Edizioni Oderisi, Gubbio, Italy, 1968.

22. Takahashi, W., Nonlinear variational inequalities and fixed point theorems, J.

Math. Soc. Japan 28 (1976), 168-181.

23. Yang, X. Q., Vector variational inequality and its duality, Nonlinear Anal. T.

M. A. 21 (1993), 869-877.

24. Yen, C. L., A minimax inequality and its applications to variational inequali- ties, Pacific J. Math. 97 (1981), 142-150.

25. Weiss, M. D., Fixed point, separation and induced topologies for fuzzy sets,J.

Math. Anal. Appl.50 (1975), 142-150.

Byung Soo Lee

Department of Mathematics Kyungsung University Pusan 608-736, Korea

Gue Myung Lee and Do Sang Kim Department of Applied Mathematics Pukyong National University

Pusan 608-737, Korea