Generalized variational inequalities for fuzzy maps

FUZZY

sets and systems ELSEVIER

Fuzzy Sets and Systems 69 (1995) 221 229

Generalized variational inequalities for fuzzy maps Zhu Yuanguo Department o/" Mathematics, Gannan Teacher's College, Ganzhou, Jiangxi 341000, People's Republic of" China Received March 1994; revised August 1994

Abstract The variational inequality problem for fuzzy maps was first studied by Chang and Zhu (1989). Chang made further study for this kind of problem in [2] (1991). This paper deals with the existence of solutions to generalized variational inequalities for fuzzy maps and then obtains some applications to fixed point and inclusion theorems. The work is an extension of corresponding work of Chang and Zhu (1989) and Shi and Tan (1989).

Keywords: Generalized variational inequality; Fuzzy map; Fixed point; Topological vector space

1. Introduction Let q~ be a real (or complex) n u m b e r field, E, F be vector spaces over • and ( .," ) : F x E --* • be a bilinear functional. A fuzzy set A on E is a m a p A : E --* [0, 1]. We denote the collection of all fuzzy sets on E by o~(E). Let X be a n o n e m p t y subset o f E . A m a p S is called a fuzzy m a p if it is from X into ~ ( E ) . F o r a fuzzy m a p S : X ~ ~ ( E ) , we always denote the fuzzy set S(x) by Sx, x e X. Let S : X - ~ ( X ) and M , T : X - . ~ ( F ) be fuzzy maps. Suppose that ~, /? : X - , [0,1] are two real functionals and 7 is a real n u m b e r in [0, 1). We will consider the following generalized variational inequality problem. Find a point 35e X such that

tl) s~(~) >/~(~); (2) inf~.eF, r,~,,,~>/~lvI R e ( f -

w,35 -- x ) ~< 0 for all x ~ X with S~(x) >~ ~(y) and a l l f E F with M ~ ( f ) > 7.

2. Preliminary Let E be a topological vector space, F be a vector space over 4, and ( . , . ) : F functional. F o r Xo E E and arbitrary e > 0, let

W(xo;e) = { y e F : I(y, x o ) [ < e~}. This project was supported by the Natural Science Foundation of Jiangxi Province, China. 0165-0114/95/$09.50 © 1995 Elsevier Science B.V. All rights reserved SSDI 0 1 6 5 - 0 1 1 4 ( 9 4 ) 0 0 2 3 3 - 9

× E ~ q~ be a bilinear

Y. Z h u / F u z O, S e t s a n d S y s t e m s 69 ( 1 9 9 5 ) 221 229

222

Let a ( F , E ) be the t o p o l o g y on F generated by the family { W(x; e): x eE, e > 0} as a subbase for the n e i g h b o r h o o d system at 0. We note that F, when equipped with the t o p o l o g y a(F, E), is a locally convex topological vector space (but not necessarily Hausdorff). Definition 1. A fuzzy m a p S: X --* ~ ( E ) is said to be convex, if for each x e X , the fuzzy set Sx is convex, i.e., for t e l 0 , 1] and y, z e E , we have

S~(ty + (1 - t)z) >~ min{S~(y),S~(z)}. Definition 2. F o r A 6 ~ ( E )

and c~e [0, 1 ] , / / e [0, 1), the sets

[ A ] , = {xmE: A(x) >>.c~},

(A)a = {xEE: A(x) >//}

are said to be a-closed cut set of A (or closed cut set) a n d / / - o p e n cut set of A (or open cut set) respectively. Definition 3. A fuzzy m a p S:X--* ~ ( F ) is said to be m o n o t o n e (with respect to the bilinear functional ( - , . ) ) , if for each x , y ~ X and u , v ~ F with Sx(u) > O, Sy(v) > 0, we have Re(v-

u , y - x ) >>.O.

Definition 4. A fuzzy m a p S : X ~ o~(F) is said to be closed if Sx(y) is upper semicontinuous on X × F as a real ordinary function. S is said to be sequentially open (with respect to the relative t o p o l o g y on X and the t o p o l o g y on F), if for each e > 0, r e [0, 1), when generalized sequence x u --* x in X and S~(y) > r (y ~ F), there exists a generalized sequence y, ~ y in F such that S~,(y,) + ~ > r.

3. Main result In this and the following sections, except where otherwise noted, we always let E be a locally convex H a u s d o r f f t o p o l o g i c a l vector space over ~b, X c E be a n o n e m p t y c o m p a c t convex subset, F be a topological vector space over 4~ and ( -,- ) : F x E ~ q~ be a bilinear functional which is continuous on c o m p a c t subsets of F x X. Besides, 2 v will denote the family of all subsets of F. Let ~,//: X ~ [-0, 1] be two lower semicontinuous functions and 7 E [0, 1) be a real number. We first give the following assumptions. (a) F u z z y m a p S : X ~ ~ ( X ) is convex and closed. The closed cut set [Sx]~x~ is n o n e m p t y for each x e X . (b) Fuzzy m a p M : X ~ ~ ( F ) is m o n o t o n e (with respect to ( ',. )). The open cut set (Mx) 7 is n o n e m p t y for each x ~ X. (c) F u z z y m a p T : X ~ o~(F) is closed. The closed cut set [Tx]~x~ is n o n e m p t y closed for each x e X and Ux~x [Tx]~x~ is compact. (d) F u z z y m a p M : X ---, ~ ( F ) is sequentially open (with respect to the line segments in X and the t o p o l o g y or(F, E) on F). Theorem 1. Suppose that assumptions (a) (c) are satisfied and that the set

= {yeX"

sup

sup

inf

~:~ X .['e F w~F S~,(x) ) ct{y) M , ( . f ) > )' ~,(w) ~ [~(y)

Re(f--

1

w,y -- x ) > O~ 3

223

Y. Z h u / F u z z y Sets and S y s t e m s 69 (1995) 221 229

is open in X. Then there exists a point ~ • X such that (1) S~(37)/> ~(37); (2) infw~F: r~(,.)>~l~(~.)R e ( f -- w,~ -- x ) <. Ofor all x • X with Sy(x) >1 ~(~) and all f • F with M ~ ( f ) > 7. In addition, if assumption (d) is also satisfied, then (3) infw~r,T, iw)>~l~l;iR e ( f - - w,~ -- x ) <~Ofor all x • X with S~(x) >1 ~(~) and all f • F with M~(f) > 7. Furthermore, if et(x) - Ofor all x • X, E is not required to be locally convex. If[ T~]ptx) =-- {O}for all x • X, the continuity assumption on ( . , . ~ can be weakened to: for each f • F , x ~ ( f x ) is continuous on X. Proof. Let

g:X--,2 x, x~[Sx],(x);

~ 7 I : X ~ 2 v,

x ~-*(Mx)~;

T : X ~ 2 r,

x~--,[T~]p(~).

(i) We will show that S(x) is closed convex for each x • X and g is upper semicontinuous. Firstly, for y, z • i(x), t • [0, 1], i.e., Sx(y) >~~(x), Sx(z) >1 ct(x), by the convexity of S in assumption (a), we have Sx(ty + (1 - t)z) >1 min{Sx(y),S~(z)} >1 ~(x). Consequently, ty + (1 - t ) z • [Sx],(x) = i(x), i.e., i(x) is convex. Secondly, let y, • i(x), Yu --* Yo. Since S is closed, S~(y) is upper semicontinuous on X x X and then we have S~(yo) >1 lim sup Sx(y,) >~ e(x). Thus Yo • i(x) and then i(x) is closed. Finally, we verify that the graph of i, graph i = { (x, y): x • X, y • if(x) }, is closed in X x X. Let (x,, y,) ~ (Xo, Yo) and y, • i(x,). In view of the closedness of S and lower semicontinuity of ct, we have Sxo(yo) >~ lim sup Sx,(y,) >>-lim sup e(xu) >/lim inf ~(x,) ~> e(Xo). Hence Yo • [Sxo]~tXo) and then (xo,Y0)•graph ~ This proves that graph i is closed. Therefore, 5~ is upper semicontinuous [1, Corollary III.1.9] (ii) Now we verify that M is monotone. For x, y • X, u • M(x), w • ~l(y), or Mx(u) > 7 >~O, My(w) > 7 >~O, by the m o n o t o n y of M (assumption (b)), we obtain the proof of m o n o t o n y of M. (iii) From assumption (c), we know that i~(x)= [T~]a(x) is nonempty compact for each x • X and T ( X ) = U ~ x [T~],tx) is compact. According to (i), it is easy to see that T is upper semicontinuous. (iv) The set Z={y•X:

sup

sup

inf R e ( f - w , y - x ) > O }

x e S(y) ,fe fit (x) w e T(y)

: {y•X:

sup

sup

inf

Re(f-w,y-x)>O}

xeX feF weF S~,(x) >~:((y) M~(f) > 7 T~,(w)>>.fl(y)

is open by the assumption of the theorem. Therefore, by [5, Theorem 1], there exists 37•X such that 3~•i(37) and inf

Re(f-

w,)~ - x ) ~< 0

for all x e i ( f )

w~(f)

This proves parts (1) and (2) of the theorem.

andf•A~r(x).

Y. Zhu / Fuzzy Sets and Systems 69 (1995) 221-229

224

If (d) is also satisfied, we will show that M is lower semicontinuous along line segments in X to the t o p o l o g y cr(F,E) on F. F o r xu ~ x in a line segment of X and yeMI(x), we have M~(y) > 7. Let ~/be a real n u m b e r such that M x ( y ) > ~/ > 7. Denote e. = r / - 7. By Definition 4, there exists a generalized sequence y, in F such that Yu--* Y and M~,(yu) + e > q. Thus, Mx,(y~) > q - e = 7- This means that y~effl(xu). H e n c e , / ~ is lower semicontinuous by [-1, Definition III.1.2]. Part (3) of the t h e o r e m follows immediately from [5]. If ~(x) - 0 for all x e X , [Sx]~(~) = X, i.e., ~q(x) = X for all x e X . By virtue of [5], E is not required to be locally convex. By the same reason, if [-T~]~(x~ = {0} for all x E X , T - 0 which indicates that the continuity a s s u m p t i o n on ( . , . ) can be weakened to: for f e F , x ~ ( f , x ) is continuous on X. This completes the proof. []

4. Consequences of Theorem 1 Theorem 2. Let E be a Hausdorff topological vector space (not necessarily locally convex) and assumptions (b) and (c) hold. Then there exists ~ X such that (1) infw~V, T,.(w)>~/~(hR e ( . f -- w, # -- x ) <~ O for all x ~ X and f ~ F , m~(.f) > 7. In addition,/f(d) also holds, then (2) infw~r, T,I,,)>~I~(~)Re(f -- w, ~) -- x ) <~ OJor all x e X and f ~ F , m x ( f ) > 7. Proof. Let Z'=

y E X : sup xeX

sup

inf

Re(f-

w,y-

x ) > 0}.

.feF weF Mx(.f) > 7 T~(w)>~fl(y)

I f Z ' = X, by applying T h e o r e m 1 with ~(x) - 0 for all x e X , we obtain a contradiction. Thus 2:" ¢ X which proves part (1). Part (2) follows from the p r o o f of T h e o r e m 1. []

Theorem 3. Suppose that assumptions (a) and (b) are satisfied and the continuity assumption on ( . , ' ) weakened to:for each f e F , x ~ ( f x ) is continuous on X. I f the set Z =

yeX:

sup

sup

Re(fy-

is

x) >0

x~X .feF S~(x) >1~(y) M~(f) > 7

is open in X , then there exists y ~ X such that (1) S~,(f) ~> ~(3~); (2) supfeF, M,(f)>;, R e ( f 37 -- x ) <~ O for all x 6 X with Sy.(x) >~ et(~). In addition, (f(d) is also satisfied, then (3) supteF, i,.(tl>. ' R e ( f , 33 -- x ) <~ O for all x e X with S~,(x) >~ ~(~). Proof. Let fuzzy m a p T: X -* ~ ( F ) x~

T x ( y ) = 1 if y = 0

Let f i ( x ) = 1 for all x m X . T h e o r e m 1. []

and

as follows: Tx(y)=O ify¢0.

Then [Tx]/~lx/= 0 for all x e X .

The conclusion follows immediately from

Y. Zhu / Fuzz), Sets and Systems 69 (1995) 221-229

225

Theorem 4. Let E be a Hausdorff topological vector space, ( .,. ) be the same as in Theorem 3 and assumptions (b) and (d) hold. Then there exists ~e X such that sup

Re(,f,y-x)<~0

for all x e X .

fe F M.~. t ) > 7

Proof. By applying Theorem 3 and the proof of Theorem 2, the assertion follows.

[~

Theorem 5. Suppose that assumptions (a) and (c) are satisfied and T, in addition, is convex. Let the set Z' = { y e X :

~xSUp

~.~Finf R e ( w , y -

x ) >O}

S,.(x) ~> ~(~.) T,.(w)/>/~()')

is open in X. Then there exists ~ ~ X such that (1) Sv()7) > ~(37); (2) there exists a point ~ F with T~(~) >~fl(f) and Re(2,37 - x ) <~ 0 for all x e X with Sy(x) >1 ~(9). l f a(x) - Ofor all x e X, E is not required to be locally convex. Proof. Let fuzzy map M : X ~ ~ ( F ) as follows: x ~ - * M x ( y ) = 1 if y = 0

and

M~(y)=O ify:~0.

It is clear that M satisfies assumptions (b) and (d). In the proof of Theorem 1, replacing T by - T, we easily know that there exists a point p~ X such that

s~.(y) > ~(y), sup

(,)

inf

Re(w,f--x)<~O.

(**)

-'¢e[S,]~l,, ~ v.,~[T~]l. ,)

As T is convex, we know that [T~.]I~U,) is convex. In addition, [S;]:(f)is also convex. So by Kneser's minimax theorem [4], we have inf

sup

R e ( w , ) ~ - x) =

sup

inf

Re(w, 9 - x) ~< 0.

x~[S,.]~,~ w~[T~.]/.,. ~

'*E[T,]/~O, ~ xE[S,,J~(y~

Since [T~]~(~I is compact, there exists Ye [~]/;(;) such that sup

R e ( L ) 7 - x) ~ 0.

x e [S~]~ ,,~

This means that ~ e F with T~(Z) >~ fi()~) and Re(~,)~ - x) ~< 0 for all x e X with Sf(x) >~ ~(~). Theorem 6. Let E be a Hausdorff topological vector space, let assumption (c) hold and T be convex. Then there exist ~ e X , 5 6 F such that T;,(y) >>.fl(y) and Re(&)7 - x ) <<.Ofor all x 6 X . Proof. Let

Z" = { y ~ X : sup x~X

inf T,.(w) ~ l~(_v)

Re(w,y--x)>O}.

226

Y. Zhu / Fuzzy Sets and Systems 69 (1995) 221-229

If Z" = X, by a p p l y i n g T h e o r e m 1 as seen in the p r o o f of T h e o r e m 5 with ~(x) - 0 for all x ~ X, we o b t a i n a c o n t r a d i c t i o n . T h u s v , # X a n d t h e n there exists 37eX such t h a t sup

inf

x~X

w~F Tv(w) >1[~(.~)

Re(w,37 - x ) ~< 0.

By K n e s e r ' s m i n i m a x t h e o r e m as seen in the p r o o f of T h e o r e m 5, we c o m p l e t e the proof.

[~

5. Application Let H be a H i l b e r t space, ( .," ), Jl " IJ be i n n e r a n d n o r m o n H, respectively, a n d X be a n o n e m p t y s u b s e t in H. F o r 2 e(0, 1], a fuzzy m a p T: X ~ ~ ( H ) is said to be a 2 - p s e u d o c o n t r a c t i o n , if for x, y e X, u, v e H with Tx(u) >~ 2, Ty(v) ~> 2 a n d for every r > 0, we h a v e Ilx -- Yll ~< I1(1 + r)(x -- y) -- r(u -- v)ll T h e o r e m 7. L e t H be a Hilbert space and X be a n o n e m p t y bounded convex subset in H. F u z z y map T: X ~ ~ ( H ) is a fl-pseudocontraction (fl 6 (0, 1)) and sequentially open (with respect to the line segments in X and the weak topology on H). In addition, (Tx)# is n o n e m p t y f o r all x ~ X . Then f o r a given z ~ H, there exists # ~ X such that f o r all u e H with T;,(u) > fl, we have Re(z + )7 - u, )7) = rain Re(z + )7 - u, x). x~X

I f in addition, f o r each x ~ X ,

there exists y s X

such that Tx(y + z) > fl, then Ty(~ + z) > ft.

Proof. D e f i n e a fuzzy m a p M as follows: M : X - - , o~(H),

x ~--~Zz +.,-iT,)~,

w h e r e )~o d e n o t e s the c h a r a c t e r i s t i c f u n c t i o n o n the set D. A p p l y T h e o r e m 4 with E = F = H where ( - ,. ) is the i n n e r (- ,. ) a n d the t o p o l o g y or(F, E) o n H is the w e a k t o p o l o g y o n H. W h e n we e q u i p H with the w e a k t o p o l o g y , X is a c o m p a c t c o n v e x s u b s e t in H. N o t e t h a t (Mx)~, ¢ 0 for all x e X with y = 0. W e n o w s h o w t h a t M is m o n o t o n e . I n fact, for x, y E M, u, v e H with Mx(u) > 0 a n d My(v) > 0, we have u e z + x - (Tx)~ a n d v ~ z + y - (Ty)p. T h u s , there exist P~(Tx)t~ a n d q~(Ty)#, i.e., T~(p) > iS, Ty(q) > fl such t h a t u = z + x - p, v = z + y - q. As T is a / ~ - p s e u d o c o n t r a c t i o n , for every r > 0, we have ]Ix - YN ~< I1(1 + r)(x -- y) -- r(p -- q)Jl. F r o m a b o v e i n e q u a l i t y we c a n o b t a i n r l l x - p - y + q[]2 + 2 R e ( x - y - p + q , x - y) >~ O. By l e t t i n g r ~ 0, we h a v e Re(x - y - p + q , x - y ) / > 0. T h a t is Re(u - v,x - y ) / > 0, w h i c h m e a n s t h a t M is m o n o t o n e .

Y. Zhu ,/Fuzzy Sets and Srstems 69 (1995) 221 229

227

N e x t we p r o v e t h a t M is s e q u e n t i a l l y o p e n (with respect to the line s e g m e n t s in X a n d the w e a k t o p o l o g y o n H). F o r e > 0, if x , --* x (in a line s e g m e n t of X ) a n d M x ( y ) > q (y e H, r/e [0, 1)), i.e., ;(: + x Irwin(Y) > q o r yez+x--(Tx)B, then t h e r e exists w e H with Tx(w)>/3 such t h a t y = z + x - w . Let ~ ( 0 , 1 ) with T~(w) > ~ > / 3 a n d d e n o t e 6 = ~ - / 3 > 0. Since T is s e q u e n t i a l l y o p e n , there exist w, ~ H with w, ~ w such t h a t T~.(w.) + / ) > ~. H e n c e T~.(w.) > ~ - ~ = / 3 or wf~(T~°)~. Let y . = z + x . - w.. T h e n y . --* y in H a n d M~,(yu) = 1. It is o b v i o u s t h a t M x . ( y . ) + e > r/, w h i c h is sufficient. By T h e o r e m 4, we can find a p o i n t )7 e X such t h a t sup

Re(f)5-

x) ~< 0

for all x E X

./'e H M e 11 > 0 or

sup

for all x e X.

Re(z + ) 5 - u , ) 5 - x) ~< 0

uEH

H e n c e , for all u e H with T~.(u) > fi, we h a v e Re(z + )5 - u,)5) = m i n Re(z + )5 - u,x). xEX

If in a d d i t i o n , for x e X, there exists y ~ X such t h a t Tx(y + z) > fl, t h e n for )5 e X, there exists x ~ X such t h a t 7~r(x + z) > ft. Let u = x + z. W e h a v e Re(z + )5 - u,)5) 4 Re(z + 27 - u, u - z). C o n s e q u e n t l y , IIz + 9 - u II = 0. T h e r e f o r e , u = )5 + z a n d t h e n Tf()5 + z) > ft.

[]

C o r o l l a r y 8. Let H be a Hilbert space and X be a nonempty bounded convex-closed set in H. Fuzzy map T: X --* o~ (X ) is a 1-pseudocontraction and [Tx] 1 is nonempty for all x ~ X. Let T be closed (in the sense of the weak topology on H) and sequentially open (with respect to the line segments in X and the weak relative topology on X ). Then T has a fixed point. Proof. N o t e t h a t T is also f l - p s e u d o c o n t r a c t i o n a n d (Tx)p is n o n e m p t y for f i e ( 0 , 1) a n d x e X . A p p l y i n g T h e o r e m 7 with fl = 1 - (I/n) (n = 1,2 . . . . ) a n d z = 0, we h a v e p o i n t s y, c X such t h a t T~',(Y,) > 1 - 1In. E q u i p p i n g H with the w e a k t o p o l o g y , we k n o w t h a t {y,} has a c o n v e r g e n t s u b s e q u e n c e . Let s i m p l y y, ~ 37eX. Since T is closed, we h a v e

T~()5)>f lim T , . ( y , ) > j l i m ( 1

!)=1.

This m e a n s t h a t Tv()5) = 1, i.e., )5 is a fixed p o i n t of T.

[]

F i n a l l y , we c o n c l u d e an i n c l u s i o n t h e o r e m a n d a f i x e d - p o i n t t h e o r e m for s e t - v a l u e d m a p s . A m a p G : X - - , 2 x is said to be p s e u d o c o n t r a c t i v e if for x, y ~ X, u ~ G(x), v ~ G(y), we have IIx-yll

~ I](1 + r ) ( x - y ) - r ( u - v ) l l

forallr>0.

Theorem 9. Let H be a Hilbert space and X be a nonempty bounded closed-convex set in H. Set-valued map Q : X --* 2 H is pseudocontractive and lower semicontinuous along the line segments in X to the weak topology on H. I f for given point z E H , ( Q ( x ) - z)c~X--/:0 for all x ~ X , then there exists a point ) 5 6 X such that z~Q(y)

-

)5.

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228

Proof. A p p l y T h e o r e m 7 with the fuzzy m a p follows. [] R e m a r k 1. If Q ( x ) ~ X ~ 0 for all x e X , T h e o r e m 9.

T : X - - , J ~ ( H ) as follows: x ~ Ze(x~- The conclusion

we c o n c l u d e t h a t Q has a fixed p o i n t in X by letting z = 0 in

Theorem 10. Let H be a Hilbert space and X be a nonempty bounded closed-convex set in H. Suppose that single-valued map p : X --+ H is pseudocontractive and lower semicontinuous along the line segments in X to the weak topology on H, set-valued map G : X --+ 2 n is upper semicontinuous (in the sense o f the weak topology on H) and G(x) is bounded closed-convex set f o r all x ~ X. I f in addition, p(x) + G(x) ~ X f o r all x ~ X , then there exists a point y e X such that )7~p()7) + G()7). Proof. Let M:X--+~(H), T: X--* ~ ( H ) ,

x ~Z~

j,(:~,

x ~ Z~c~l.

E q u i p p i n g H with the w e a k t o p o l o g y , we easily k n o w that M is m o n o t o n e , sequentially open (with respect to the line segments in X a n d the weak t o p o l o g y on H) a n d T is closed. In a d d i t i o n , [T:,]I is c o m p a c t convex a n d (Jx~xETx]l = U x ~ x G ( x ) = G ( X ) is c o m p a c t by El, P r o p o s i t i o n I I I . l . l l ] . By T h e o r e m 2 ( / 3 = l, 7 = 03, inf

Re(f-

w,)7 - x) ~< 0

for all x e X and.fe F w i t h M y ( f ) > 0

weH Tdw)

= I

or

sup

inf

xEX

wEH T~4,a l

Re()7--p()7)--w,f--x)~<0. 1

By a p p l y i n g Kneser's m i n i m a x t h e o r e m as seen in the p r o o f of T h e o r e m 5, we have inf

sup R e ( f -

p(f)-

w , ) 7 - x) ~< 0.

w~H xEX T,4w) I

Hence, there exists ~ ' ~ H with 7 } ( ~ ) = 1, i.e., ~ e G ( ) 7 ) such that sup Re()7 - p()7) - w, y - x) <~ O. xEX

Since p()7) + G()7) c X , we can find x = p()7) + ~ X .

Therefore, we have

Re()7 -- p()7) -- ~,, 37 -- p()7) - ~,) ~< 0. Thus, 37 - p()7) - ¢: = 0 a n d then )Tep()7) + G()7). This c o m p l e t e s the proof.

[]

R e m a r k 2. If p is n o n e x p a n s i v e , i.e., ]]p(x) - p(y) l] ~< ]]x - y ]p for all x, y ~ X, then it is easily k n o w n that p is p s e u d o c o n t r a c t i v e a n d c o n t i n u o u s a l o n g the line segments in X to the n o r m t o p o l o g y on H a n d then to the weak t o p o l o g y on H.

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References [1] J.P. Aubin and 1. Ekeland, Applied Nonlinear Analysis (Wiley, New York, 1984). [2] Shih-sen Chang, Variational Inequality and Complementarity Problem Theory with Applications (Shanghai Science and Technology Press, Shanghai, China, 1991). [3] Shih-sen Chang and Yuan-guo Zhu, On variational inequalities for fuzzy mappings, Fuzz), Sets and Systems 32 (1989) 359 368. [-4] H. Kneser, Sur un theoreme fondamental de la theorie des jeux, C.R. Acad. Sci. Paris 234 (1952) 2419 2420. [5] M.H. Shih and K.K. Tan, Generalized bi-quasi-variational inequalities, J. Math. Anal. Appl. 143 (1989) 66 85.