Math. Narhr. 199 (1985) 231-245

Quasi-Variational Inequalities in Topological Linear Locally Convex HAUSDORPF Spaces

By NGUYEN XUAN TAN of Hanoi

(Received February 22, 1984)

Abstract

This paper will present some results on quasivariational inequality {C, E , P , 'p}

in topological linear locally convex Hausdorff spaces. We shall be concerning with quasivariatioiial inequalities defined on subsets which are convexe closed, or only closed. The compactness of the subset C is replaced by the condensing pro- perty of the mapping E . Further, we also obtain some results for quasivariational inequality {C, E , P , v}, where the multivalued mapping E maps CI into Zx and satkfiev n general inwnrd boundary condition.

1. Introduction

Quasi-variational inequalities which play an important role on optimal control theory, on fixed point theory, on complementary problems and on mathematical programming e.t.c. have been considered by many authors.

By a constructive method H. TUI [5] proved the following theorem for compu- tation of solutions of quasi-variational inequalities in finite dimensional spaces : Let X be an Euclidean space, C c X a convex compact subset. Let E : C -2' be a continuous multivalued mapping with E(z) , xE C, is a convexe compact nonempty subset and P : C -2" an U.S.C. mapping withP(x), zE C, is a convex compact non- empty subset. Then the quasi-variational inequality, shortly we denote by (C, E, P, o}, has a solution. This means that there exists a point x* E C such that

1. r*EE(x*), 2. there exists y*~P(z*) for which

(x--x*, y*)zO, for all xEE(z*) .

The present paper is devoted to developing a method to extend the above re- sults by imposing a lower semicontinuous proper convex function cp : C -23, and by replacing X with a topological linear locally convex HATJSDORFF space. Further we

232 &lath. Nachr. 1 4 (198.5)

shall be concerned with the quasi-variational inequality { C , E , P, rp}, where C is a convex closed set, or only closed set. The compactness of the set C is replaced By the condensing property of the mapping E. Further we obtain some revults for the existence of quasi-variational inequality (C, E , P , 9) where E : C - z x and satis- fies a general inward boundary condition.

3. .The main results

In the sequel, by S, Y we always clenote real topological linear locally conves Hsusclorff spaces. We write R I for the space of all real numbers. I for a index set. c f o r the closed hull of C in a topological space, co( C) for the closecl convex hull of C'. Further {Vu} - (0) stands for V , s V,, Y z , ~ i and n TT% = {o) where I is a ordered

indes set. By 2' we denote the system of all subsets of the set C in a topological space. Given multivalued mapping F : C +i?'-, C'cY. We recall some well-known following definitions.

Definition 1. a) F is called upper se)),.i-contin.sro.c.s a t a point .q,E C (briefly u . s .~) if for any neighbourhoocl V of the origin in Y there exists il neighbourhood c'of the origin in S such that

5 €1

F(x) c F(x,,) + I', for all N E (x,, + r) n C: . F is u.s.c on C' if F is U.S.C. at every point C'.

closed subset in X x Y . b) F is called closed ifthe qrupli, G , of F (i.e. G = {(x, y) I a.E C' and y ~ F ( x ) ) is a

Renisrk. It can easily be seen that the following assertionx are always true.

i. F is closed if and only if for any net (xJ, z,-z and any net (yJ, ~ J ~ C F ( Z ~ ) , yv+y one has yEF(z).

iii. If F is closed and F ( C ) is a compact subset then F is U.S.C. iii. If F is u.s.c and F(x) , xE C, is a closed subset, then F is closed. iv. If C ia a compact subset of X and if F is u.s.c and P(x) , zE C', is compact, then

F ( C ) is also compact.

Definition 2. F is said to be lower semicontintcous (briefly 1.s.c) at a point x0 if for any open subset G of Y with F(s,) n G * 8, the set U = { X E X I F ( z ) fl G * 0 ) is a neighbourhood of xo. It is a simple matter to show that F is 1.s.c a t x E C if and only if for any Y E F ( z ) and any (x"), x, +x there exists a net (y,) such that y,E F(xl ) and y, -y. If F is 1.s.c at every point of C we shall say that F is 1.s.c on C and if F is a t the same time u.s.c and l a c . on C we shall say that F is continuous.

-

Now we introduce the following definitions :

Definition 3. a) A family of multivalued mappings {P,/rcI} P, : C -2' is cal- led (*)-comergent to the multivalued mapping P : C -2' if for any convergent net

Sguyen Siian Tan, Quasi-Vsriationel Inequalities 233

( z , ) , ~ ~ in C and for any net ( Y , , ) , , ~ ~ in Y, y,EP,(z,) there esists a net ( z , ) , ~ ~ in Y, z,EP(z,) such that z,-y,--O.

b) A family {P,, r E I } of multivaluecl mappings P, : C -2' is called (:::>)-COW

vergent to the multivalued mapping P : C! -2' if i t is (:::)-convergent to P and if for any convergent net (zJrcl in C and any net ( Y , ) , , ~ ~ , y,EP(x,) there exists a net ( z ~ ) ~ ~ ~ , z,EP,(z,) such that z, -yJYdO. If (P,, v E Z } is (:%)-convergent to P ((::::)-con- vergent to P ) we shall denote it in symbols P, 2 P (P, (- P respect.).

Next we recall the definition of almost conves subset.

Definition 4. A subset C of a topological locally convex Hausdorff space X is called almost convex if for any neighbourhood V of the origin in S and any finite subset {v,, v,, ..., v n } c C there exists a finite subset {zI, z2 , ..., z,)cC such that

E&z1,z4, ..., z J c C ancl s i - z i i ~ l ; for i = 1 , 2, ..., 'IE.

Now, let X be a topological linear locally conves Hausclorff space aiicl let S+ be the space of all linear functionals on S. We define the strong topology on S+ i.e. the topology of uniformly convergens on the system of bounded subsets of X. Thus S+ becomes a topological linear locally convex HAVSDORFB space ancl we denote i t by X". It is easy to show that (S, S*) is a, pair of dual space with the cludity defined by

(x, y )=y( z ) , X E S , ycX* I

Let 11s consider the following mappings

E ; : C - P , v E I

P, : C - P * , V E I

fp,:C--R~, Y E I

and suppose that E,- E : C-zx

p+p: c - p -----t y v -rP: C-RI

(= means cp, converges uniformly to cp on C). We have:

Theorem 1. Suppose that

i. The multivalued mapping P is u.s.c and P(z) , xEC, is compact, ii. The multivalued mapping E is closed and lower semicontinuous iii. For any net (z,) cC, z, +z, any net (y,) cX*, y y -y one has (z, y)=lim (z,, yy) . Then every limit point in C of the net (x,),~~ of solutions of the problem {C, E,, P,, 9,) is a solution of the problem {C, E , P, y ) .

U

234 Math. Nechr. 193 (1985)

Proof. Assume that x: is a solution of the problem {C, E,, P,, rp?} and the net ( z & ~ converges to z*. We shall show that x*EE(x*) and there exists y * ~ P ( r * ) such that (1) (x - z* , y*)+(p(x)-rp(z*) no, for all xEE(x*) . Indeed, Since E'""! E , x: e x * , x:EE,(xT), one has a net ( z , ) ,~~ , z, €E(z:) such that z, -x: -0. It follows that z, -x*. Therefore from the closedness of E we draw z * ~ E ( z * ) . Now, since P,%P andx:+x*, yiKEP,(x:) one then has a net ( v , ) , ~ ~ , v,E Y(x'E) with v, -y: -0. Further, take a system {V,) ,uE I of neighbourhoods of the origin in X such that {V,,} --{O}. From the upper semicontinuitp of P and zz +z* one claims that for any V,, these exists Y , ) = Y & ~ ) such that P(x:) c P ( x * ) + V,, for all YZY,,. Consequently, v,EP(r:) can be written as v ,=av+u, with u,EP(x*) and zt,EV,,. But, from the Compactness of P(x*) and {V,,} -{O), without loss of gen- erality, we map assume that a, &y*. Thisimpliesv, -y*, andso y, -. y*.Therefore one has y*EP(x*).Lastlp, since E is 1.s.c. and z: +x*, for any x ~ E ( r t * ) there exists a net ( z , )v~I , z ,~E(z~) such that z y - x . Fromthefact thatE,'-E.r~~+x*,z,EE(z~) one has a net (w,) Y E I , zv ,EE,(x~) with zv, - z, -0, and so zc, -5. R u t xp is a solu- tion of the problem {C, E,, P,, (p,} we therefore have

'I

(u'"-5:, ?J~)+rp,(zo,) --rp"($) so . Hence

n o , for all x ~ E ( z * ) . This means that z* is a solution of the problem {C, E , P , tp} and hence the proof is complete.

Theorem 2. Let X be u n-dimensional Euclidean space, C cz convex compact non- empty subset of X , and let P : C-2" be an z6.s.c. niultivalzied niupping zcrith P(x), xEC, convex compact nonempty. Further, let E : C -2c be a closed and 1.s.c. mzsltiva- heel mapping with E(x), X E C , convex conipnct nonempty and p; C -R, CL proper convex lozcer semicontinuous fzrnction. Then the problem {C, E , P, rp} has a solw tion.

Proof. Let us consider two following cases : a) 'p = 0 on C. (see the proof in H. TUI [ 5 ] ) b) r p + O on C .

Without loss of generality se can assume that int C c 0 (in the contrary case we take the affine space geiieratedc insteadof X). Let (U,} YE I b e a system of convex closed neighbourhoods of the origin in X such that { U,} --{O}. From the convexity and the compactness of C, for any U. one can find a finite subset {a,, a?, ..., a,,,} c c i n t C such that

Cc{a,, a?, ..., n,)+U; .

Sguyen Xiinn Tan, Quasi-Variational Inequalities 2 35

P u t Cv=co {a1, a?_, ..., an}

E,(x)=(E(x) + Uv)i7Cv, xCC,

P V ( X ) = P(x) + arp(x) 7 5 E cv t

where arp(x) is the subgradient of IF a t x. One can easily verifyg, is an U.S.C. multi- valued mapping from C, into 2“ with Qv(x), xEC,, convex compact nonempty and E, is a closed and 1.s.c. multivalued mapping withE,(x), xFC,, convex compact nonempty (see in the proof of Theorem 3 below).

Applying the case a ) yields that the problem {C, E,, Q,., 0 ) has a solution, e.g. there exists x:EC‘., x;~EE,(x,*) and u ~ ~ Q , ( x f ) such that

Since uTCQv(x:) i t follows that

This yields

and so the problem {Cv, E,, P, cp} has n solution. Now, setting

(x-x:, u:)zO, for all zEE,,(z:) .

2L; = y; f where yp E P(xi:) and 2:: E +(.z;?) .

(x-x:, yp)++) -e(x:’) r O , for all x~E,(x1:’)

E’(x)={E(z), if -. xCC\C, . E,(x), if SCC,

W e deduce the problem {C, E,, P, cp> has it solution. fur ther , i t can easily be seen

The compactness of C and Theorem 1 imply that the problem {C, E , Y, rp} has cz solution. This completes the proof of theorem.

Theorem 3. Let C‘ be a compact subset of the topological linear locully convex HAUSDORFF space X, P: C -Bs* an U.S.C. mz~ltivaluecl mapping with P(x), X E C , compact. Let E: C -2‘ be a closed and Z.S.C. multivnlued mapping with E(x), xEC, compact and tp : C -. RI a proper lower sem.icontinuous function. Further suppose that there exists an almost convex subset A dense in C such that P(x), E(x), xE A, are nom- empty convex subsets and cp is convex 0% any convex subset of A. The38 the problem.

Proof. Take a system {Vv} Y E I of convex closed and symetric neighbourhoods of the origin in X such tha t {V,} +{U}. From the compactness of C and the fact that A is dense in C, for any V , one can find a finite subset {vl, v2, ..., v,}cAsuch that

that i ,(*,:L+ - E (see also in the proof of Theorem 3 below).

{C, E, P, q) hCt8 S O h ! i W .

Now, since A is almost convex one then has a finite subset {z~, z2, ..., z,}cC 1

with co {zl, z2 , ..., z , } cA and vj-ziE- V” for i = l , 2, ..., n. Let us define 4

C,=CO { ~ l j 22. ..., 2,)

236 3fat.h. Nachr. PlS (1985)

and

~ , = ( ~ ( ~ ) + q n c , , ~ E C ,

One can easily SO that E,(z ) , z E C,, is convex and compact. Now, for any z E C, we have

and hence

E ~ ( ~ ) = ( E ( ~ ) + V , ) ~ C ~ M . Further w e claim that E , is a closed ancl 1.s.c. multivaluecl mapping from C, into itself.

Indeed, suppose that zc CC,, xc +x, yP EE,(z,) and yr -y. We have y,,E E,(.v,) = = (E(x , ) + V,)n Cv. The compactness of C, gives y e C,. Furthermore w e can write yP=+ fv,, with z a ~ E ( x J and vJIE V,. Since C is compact and E is U.S.C. it follows that for any system {C?} of neighbourhoods of the origin in X with {G,,} -(O) one can fintl. for any iYq, po=pn(q) such that

i. The closedness of E,:

E, , t (x)cE( . r )+ LT, for all p s p 0 .

This implies :/, c l i ( , r ) + L‘, for all p spa. By the compactness E(N) nncl { V,,} - (0) \vc! niay nssume that z,, -z ancl so zE U(s) . Further we have ye -zj,=vp€V, theruforo y,, - z ~ , + g - - z ~ V , , and hence yEz+V,,cE(z)+V,;, . Consequently

P E ( q x ) + rqnc, = E ” ( Z ) . This shows E is closed.

Let N,) be an arbitrary poiiit in Cl, and yo€E,(zo). We claim that for any neighbour- hood U of the origin in X there exists n, neighbourhood11.I of the origin such that

ii. The lower semicontinuity of E,:

(2) E,(x)n(y,,+ U ) + a for any zE ( .co+M)nC,. Indeed, take z small enough, 0 <a< 1, such that z (C,-yo) c U and put

Since y o ~ E v ( z o ) = ( E ( z 0 ) +V,)nCu, it follows that yo can be mit ten as yo=zo+w with zoEE(zo) and vE V,. From the lower semicontinuity ofE andz,EE(x,) one can find a neighbourhood M of the origin in X such that

E(z)n(zo+Li’)*O for all zc(zo+i%I)nC.

We no claim that

(3) Ev(z)n(yo+ U)*0 for all zE(zo+M)flC‘, .

Sgtiyeii Xuan Tan, Quasi-Variational Inequulities 237

Indeed, take zEE(x;)n(zo+ 77) we draw z = z O + u with uE C'. On the other hand 1 1 2 2

Z E E ( ~ ) C C . ' , , + - V - ~ , thereforez canbe writt,en asz=n+zu with nEC',anclwE-Vv.

This yields C C = Z ~ + U - W . Put

Y = y o + ~ @-Yo) *

From the convexity of C, we draw y e C,. Now we have

y - z =yo +a (a -yo) - (z , , +u) =zo + u + a (zo + u - zu -20 + v) - (zo + Z L )

and so YE:+ t ; cE(z ) + F-,. Consequently Y E ( E ( z ) + vv)n C', =Ev(x;) for any X E

E ( z O + N l ) f l C,. This (3) is proved and hence E, is I.s.c. Further, let X,=sp C, (the smallest linear space containg Cu). It is a simple

matter to show that dim X, < +a and X,, become a, topological linear space with the topology induced by the topology of X. Put

P,(z) = {ye .X:/y is a retriction of some z ~ P ( z ) on X.} where is the space of all linear functionals on Xu with the topology induced by the topology of X*. It is easily verified that (X,,, .x*) is a pair of dual spaces. Let cpy be the restriction of cp on C,. It can easily be seen that cpv is a proper convex lower semicontinuous function on C,. We have

E, : c, --Sc1.

x; P, : c,-2 cp,: C,- -RI .

The above proofs show that E,, Pi and q ~ , satisfy all the assumptions of Theorem 2 on C, (here we can identify X: with X,). Therefore the problem (C,,E,, P,, cpy) has a solution for any V E I .

We now extend E,, P, on C by the following ways:

E(z ) if ~ E C \ C , , if xCC,

This shows that the problem {C, &, g,, 'p} has a solution.

235 M&th. Nachr. 122 (1985)

.. (44 Finally we claim that k,* E and P, -P. Indeed, P, -P is obvious,

Now, let ( x , ) , ~ ~ c C , x,-x and ( y Y ) Y E I ~ C , ~ , € E , ( Z , ) . We set i t remains to be shown 5, (- E.

(4) z v = y r for vEI for which z,EC\C,. For Y E I , I, E C,, we have Z?,(x,) = ( E ( x Y ) + TJY) n C,, hence y. EE,(x,) can be written as yy =t , +v, with z , ~ E(z,) and v,E V,. W e deduce ( 5 ) y,-s,=vu.

Combining (1) and (5) givesz,<E(z,)and y, -2°C V, for all Y E I . Since {V,)-{O], we then have

-

- (:!9 yy-zy-O and so 23- E .

Further. let ( X , ) , ~ ~ C C', x,+z and (yJYEIc C', y,EE(z,). We set

(6) z,=y, for any YE I such that z,E C\C,.

For ~ , E I s u c h t h a t ~ ~ , E C ' , w e havey,EE(z,)cC',+ +t',, thciiyl,=c,+v,with c,EC, and v, C T/',. We set

(7)

t~nt l hence z , ~ ( E ( z , ) + V,,) n C,= E,(x,) = &(.c,). Combining ((5) and (7) yields z,, E El,(x,) and z, - yv E V, for all YE I . Since { V,) - (0) i t follows that IZ, - y, -c 0. This shows E , I...\ E.

The compactness of C' and the upper semicontinuity of Y yield the compactness of P ( C ) . Then for any nets ( x , ) c C , xu -x., ( ~ J , ) , ~ , c P ( C ) , y,-y, one obtains

1

d

: ,, = c, = y , - VI , - -

- (,.,.I%)

(2, y) = lim (xu , y ") . V

To complete the proof, i t remains to apply Theorem 1. From this theorem we draw immediately the following corollary.

Corollary. Let X be a topological linear locally convex HAUSDORFF space, C c X a convex compact nonemnpty subset. Let P : C -2"' be an U.S.C. multivalued mapping with P ( x ) , x c C', convex compact nonempty and E : C 42' a closed and 1.s.c. multi- valued mapping with E ( x ) , X E C, convex compact nonempty and let q~ : C-Rl be a proper convex lower semicontinuous function. Then the problem {C, E , P, 'p) has a soliltion.

In the follows we shall prove some extentioils of the above Corollary t o the case where C is not compact. The compactness of C is replaced by the condensing property of the multivalued mapping E. First of all we introduce some following definitions.

Definition 5. A function r. : 2"-[0, +-3 is called a measure of noncoinpactness if i t satisfies the following conditions :

Sguyen Sum Tan, Quasi-Variational Ineqrialities 239

i,

iii, a(A) su(B) for A S B , iv, r*(A U B) I; niax {u(A), a(R)} . This notion is a, generalization of the notion of a measure of noncompactness in- troduced hy C. KURATOVSKI in [2].

a(B) =0 if and only if 13 is totally bounded, .. 11. rX(E0 (B) )=r (B) ,

Definition 6. Let S he a, space with a measure of noncompactness z, C c X a subset. A multivalued mapping E': C-TY is called condensing i f it satisfies the following conditions :

i, ~ ( P ( B ) ) L ~ ( B ) for 811 BEP, ij , if a(B) > O one has u(k"(23)) <x(B). We have :

Theorem 4. Let ,Y be a topological liuecw locally convex HAU~UORFF space with a mensure of noncompactness u and C c X n nonenipty conzex qzrasi-conzplete subset. Let P : C -?"* be an U . S . C . .multivaEued Iiiopping with P ( T ) , z~ C, convex compact nonempty, B : C + 2' a closed and 1.s.c. rind contleminy rrtdtivnlzrerl tiicipping with E(x) , I ~ E (', co9ire.r closed nolaempty and q? : 6'- R, rr proper lower semicontinwm9 concex fimction. Then the pTob1enz { ( ', E , Y, y} hn.v (I .wlutioii.

Proof. We 1rove this theorem by using the mcthotl develol)ed by S. R I ~ ) [:3]. Let us take a fixed point W E c' and define

3l'i={DcC'/DaQ, convex closed, WED and E ( l > ) c D f . . One has X + D (because C ' E X ) . For each Dc3K we put

We have y(D) = E o ( E ( D ) IJ ( w } ) .

E(g(D)) , ,cE(D) S E O ( E ( D ) u {w})=g(D) . This means g ( l ) ) ~ 3 & . We define on 3&, the partial order I by D i s D 1 - if and only if D IsD,. Thus, $li becomes n partially ordered set which is inductive. Let {D,} be a chain in 911, xve set D = n 0,. We then have WED and D is a convex closed subset.

Since E ( D ) c E ( D , ) c D , for any Y i t follows that E ( D ) c n D , = D and so

D E 3K. This shows that every chain in 3lZ has a lower bound. Using ZORX'S Lemma yields that all has the minimal element, say D,. But as was to be shown g(Do) E X , therefore

Y

V

g(D,) =Do = 50 (E(D,) u (w}) .

.*(Do) =a(eo (E(Du) U {w}) s m a i {a(E(Do), .({w>))!

Since E is condensing we then have

=a(E(Do)) - This implies u(Do) = 0 and hence Do is a compact subset.

240 Ninth. Xinchr. l?? (1985)

Now we are in the position: E : Do--2% y : D"+s"*

q? : DO-R,. Applying the above Corollary gives that {DO, E , P , 9) has a solution, so the prob- 1t.m (C, E, P, rp} has also solution. This completes the proof of Theorem 4.

Theorem 6. Let X be a topological linear locally convex HAUSDOBFF space with a nieusure of noncompactnesa u, G c X a n open convex subset containg the Magin and with 6 qzmsicomplete. Let P : &-aS' be an U.S.C. multivalued mapping with P(x) , re G, convex closed nonempty and E : G --2" n closed, lower sernicontinz.mus condensing multivalued mapping with E(x) , Z E 6, convez closed nonempty and with E ( 6 ) bound- ed. Suppose that for all x ~ a G (the frontier of G) and /?€ R1, B-1 it holds /?z@E(x) . Further let q~ be a proper convex lower semicontinuow function from. X into RI. T?ben the problem {G, E, P, 9) hus a solution.

Proof. Let p be a Minkovski function of G. Since o E G and G is open it follows t h a t p is continuous. From the boundenless of E(G) one can find a Aw2 such that

Set

- -

E ( C ) ~ B = { & K I p ( x ) S A ] .

- It is easily verified that 8 is il continuous function and e(x) = 0 for all zE G. Fur- ther, let us define the multivalued mappingk : X--2" by

~ ( x ) = ( l - - e ( x ) ) E ( x ( z ) ) , X E X , where

x if X E G if x @ G .

n(x)= x I,, - It is a simple matter to show that the multivalued mapping E is closed and 1.s.c.

We now claim that g i n condensing. Indeed, let DEZX. We have .

~(E(D)) sa(co ((01 u E ( ~ ( D ) ) ) sa(E(n(D)) ) s u ( n ( ~ ) ) s a ( D ) . Suppose that D < 2" and a(D) -0 :

i, if a(n(l)))=O, then a(E(n(D)) ) ia(n(D))=O<a(D) ,

Xguyen Xunn Tan, Quasi-Variational Inequalities 241

and

ii, if a(z(D))=-O then

u(E(n(D1)) -=u(z(D)) a ( D ) .

a(E(D)) -=@) ,

P(.) =P(n(x) ) , 2EX.

In any case we have

- and so E is a condensing multivaluecl mapping. Lastly, put

Since x is a continuous mapping and P is an u.s.c inultivalued mapping, it follows that 1; is U.S.C. Thus, the problem {S, E , P, cp}satisfies all assumptions of Theorem 4.

Then it has a solution. Let us denote i t by x*. Suppose now that x * ( G . Since z* B. 8B (because if x*E 6B then O(x*) = 1 and so 3?(x*) = (01. This ilnplies z* = = O E G ) . We then have 0 < e(z*) < 1. Therefore we deduce

This shows

- -

-e(x*)) ~ ( ~ ( : r * ) ) .

2* -- ----EE(n(z*)). I - e(z*)

But z*=p(.r*) z(z*). Consequently

P@*) i - e(x*) Since - =- 1 and n(z*) E 86 it follows a contradiction. Therefore we ob-

tain X * E G and so

Z*cE(z*) = ( I -e(z*)) E ( ~ ( Z * ) ) = ( I -0) z(c*) = ~ ( z * ) - and there exists y * ~ P ( s * ) =P(n(x*)) =P(J*) such that

(z-x”, y*)+p(x) -cp(z*)zO, for all .rEE(x*) . This means that the problem {G, E , P, cp} has a solution. This completes the proof of Theorem 5.

Lastly, we prove one theorem on the existence of solutions of the problem {C, E, P, a}, where E : C -2“ and satisfies a general inward boundary condition. First of all we introduce the following definitions

Definition 7. Let x be a topo~ogical linear hCally convex HAUSDORFF space with a measure of noncompactness a and C c X a subset. A singlevalued mapping F : X-C is said to be condensing outside the subset C if i, a ( F ( B ) ) s a ( B ) for all BE2X; ii, If B c n C with a(B) > O then a(F(B))<cc(B). 1 G Math. Naclu., Bd. I??

242 Math. Sachr. 1252 (198.5)

Defiiiitiou 6. Let X be a topological linear locally convex HAUSDORFF space with a measure of noncompactness a and C c X a subset. A multivalued mapping T : CI --3x is said to satisfy a general inward boudary condition if there exists a continuous retraction x : X - C which is condensing outside C such that for any zEC, BER,, 8 > 1 one has

X(.) + e (z-n(+~ T ( ~ ( z ) ) . We have:

Leninia 1. Let X be a topological linear locally convex HAUSDORFF space with a measzwe of noncompactness a, C c X a subset. Let 8 : X-[0, 11 be a fzmction with O(x) = O for all XE C and n : X- C be a retraction condensing outside C. Then the multivaluecl mapping F : X - -2x defined by

F ( 4 = (1 - W,)f(n(.)) + W ) 4%) is conrlensing for any condensing multivaluecl mapping f : C! - F.

Proof . Indeed, for any BETr we have

Nguyen Suan Tan, Qunsi-Variational Inequalities 243

It follows : If r(B2) = O then

(13) If z(B2)>0 and a(n(B2))=0 then

( 14 a ~ F ( B . ) ) ~ a ( n ( B " ) ) = O < a ( B ) . If z(B1)>O but a(n(B?))>O then

(15)

Combining (13), ( la) and (15) gives

(16) a(F(B. ) )<a(B) . Since B=B,UB,! it follows that F ( B ) = F ( B , ) U F ( B , ) and hence (12) and (16) yield

Which shows that F is condensing. This completes the proof of Lemma 1.

Lemma 2. Let S be a topological linear loccdly convex HACSDORFF space, C ' c S a subset. Suppoee that 0 : X-[0, 11 is a continuous function and z : X - C i s a conti- nuous retraction. Then the ?n,ultivalued mapping F : X - 2 x defined by

is closed and E.s.c.for any closed and 1.s.c. multivalued snapping f : C -5" with f( C ) is bounded.

z( F ( B 2 ) ) s a( z( EL)) z a(&) = 0 < a(B) .

a(F(B2)) ~ a ( n ( B 2 ) ) <a(&) a a ( B ) .

~ ( F ( B ) ) s m a x { a ( P ( B i ) ) , a ( F ( B ? ) ) } c a ( B ) ,

qz) = ( 1 - e(z) f(+)) - e(z) z(z)

Proof. a) The closedness of 3' Indeed, let z, d x , y . ~ F(z , ) and y, --y. We have

Consider the case e(z) * 1. Then for infinite numbers of Y one has e (x , ) =k 1. Take a subnet

t ~ ~ = ( i - e ( ~ J ) z,.+O(z,) I+,.), with z,Ef(+,)).

and so

Now, consider the case e(z) = 1. Then me have e(z,) - 1. Since (z,) E C and f (C) is bounded, it follows that

which implies that

Thus F is closed.

y E (1 - 4%)) f(W) + w4 4 x 1 = F(4 -

Y" = (1 - 2, + e ( ~ , ) +,) -(z)

Y = z ( z ) € ( ~ - I ) f ( ~ ( ~ ) ) + n ( z ) = F ( z ) .

lti*

244 Math. Nschr. In? (1985)

b) The lower semicontinuity of F. Let x E X , yEF(z) and x, -5. From the conti- nuity of 3c and 8 one deduces n(x,) -n(x) and 8(x,) --8(x). Since yfF(x) one then has

Since f is 1.s.c. and zE f (n (x ) ) , n(x,,) -n(z), it follows that there exists a net z , z,,Ef(n(z,)) such that z, -2. Setting

one can see y , ~ ( 1 -8(z , ) ) f (n(xv) )+8(x , ) n(z,) =F(z , ) and yv -y which shows that P is 1.s.c. This completes the proof of Lemma 2.

y = ( I - O(z)) x + O(z) n(x) with Z E f ( ~ ( z ) ) .

yV = (1 - w,)) Z, + e ( x v ) ~ ( X J

Now we can conclude our paper by the following theorem. Theorem 6. Let X be a quasicomplete topological linear locally convex HAUSDORFF

spcice with a mensure of noncompactness a , C C S a closed subset. Let P : C -PY* be an U.S.C. mz~ltivalued mapping with P ( x ) , x E C, convex compact nonentpty, and E : C-2" n closed and 1.s.c. and condensing multivnlued mctpping satisfying the general inward boundary condition and having E ( x ) , xE C, convex closed nonempty and E ( C ) bozindecl. Zzirthermore suppose thnt y : C - R i s a proper convex lower semi - continuozu function. Then the problem {C, E , P, q} has a sohition.

P r o of. From the fact that every topological linear locally convex HAUSDOREP space is n complete regular topological space and C is n closed subset of ,Y we draw that there exists n continuous function 8 from X into [0, 11 with O - l ( O ) = C. Define -

E ( X ) = ( 1 - e ( x ) ) + e (n) . where n is a continuous retraction taking from the general inward boundary con- dition of E. Applying Lemma 1 and Lemma 2 yields that E is a closed, 1.s.c. and condensing multivalued mapping from X into 2.'. It is clear that E(z) , x E C', is convex closed and nonempty. Further, set

I t can easily be seen that @ is an U.S.C. multivalued mapping and&), x E S , is convex compact and nonempty. Using Theorem 4 gives that the problem {XI 2, P, 9) has a solution, say, x*. Suppose now that x* 4 C. This follows 8(z*) > O and

- -

P ( x ) = P ( z ( x ) ) , X E X .

(17) $*EI$X*)=(I - o ( ~ * ) ) ~ ( ~ ( X * ) ) + e ( x * ) n(z*) . If O(z*) = 1 then (17) yields z* =n(x*) < C. It is not possible, therefore we have 0 c O(z*) < 1. This deduces

1 Since -1. So, it contradicts the general inward boundary condition

of E. Thus we draw z* E C and the problem {C, E, P, 'p} has a solution. This com- pletes the proof of Theorem 6.

1 - e(x*)

Nguyen Xuan Tan, Quasi-Variational Inequalities 24.5

References

[l] C. BERGE, Topological spaces, New York 1963 [2] C. KURATOWSXI, Topologie 11. Wersawa 1950 [3] S. REID, Fixed point Theorems in locally convex spaces, Math. Zeitschrift (1973) 17-31 [4] H. H. SCEAEFER, Topological vector spaces, New York 1966 151 H. TUI, Combinatoriel method for solving nonlinear equations. Preprint, communicated by

R. Kluge, Berlin 1980

Institut of Natheniatics Hanoi - Vieham