On Some Generalized Quasi-Equilibrium Problems

JOURNAL OF MATHEMATICAL ANALYSIS AND APPLICATIONS ARTICLE NO.

224, 167]181 Ž1998.

AY985964

On Some Generalized Quasi-Equilibrium Problems* Lai-Jiu Lin Department of Mathematics, National Changhua Uni¨ ersity of Education, Changhua, Taiwan 50058, Republic of China E-mail: [email protected]

and Sehie Park† Department of Mathematics, Seoul National Uni¨ ersity, Seoul 151-742, Republic of Korea E-mail: [email protected] Submitted by Helene ´` Frankowska Received September 11, 1997

0. INTRODUCTION By an equilibrium problem, Blum and Oettli w2x understood the problem of finding ŽEP.

ˆx g X such that f Ž ˆx, y . F 0 for all y g X,

where X is a given set and f : X = X ª R is a given function. We can consider more general problems as follows: A quasi-equilibrium problem is to find ŽQEP.

ˆx g X such that ˆx g SŽ ˆx . and f Ž ˆx, z . F 0 for all z g SŽ ˆx .,

where X and f are as above and S: X -o X is a given multimap. A generalized quasi-equilibrium problem is to find ŽGQEP. ˆ x g X and ˆ y g TŽˆ x . such that ˆ x g SŽ ˆ x . and f Ž ˆ x, ˆ y, z . F 0 for all z g SŽ ˆ x ., * Research supported by Mathematical Center of the National Science Council, Republic of China while the second author was visiting at National Changhua University of Education. † The second author was partially supported by KOSEF-981-0102-013-2. 167 0022-247Xr98 $25.00 Copyright Q 1998 by Academic Press All rights of reproduction in any form reserved.

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where X and S are the same as above, Y is another given set, T : X -o Y is another multimap, and f : X = Y = X ª R is a given function. These problems contain as special cases, for instance, optimization problems, problems of the Nash type equilibrium, complementarity problems, fixed point problems, and variational inequalities, as well as many others. There are many variations or generalizations of these problems Žsee w3, 4, 8]13, 18x.. In this paper, we study some equilibrium problems, quasi-equilibrium problems, and generalized quasi-equilibrium problems in G-convex spaces using a new method of fixed point approach. In fact, the second author w11x obtained an existence theorem of generalized quasi-equilibrium problems in nonnecessarily locally convex spaces. On the other hand, the second author w14, 15x extended the concept of H-spaces to G-convex spaces and established the KKM theory on these spaces. By exploiting these new approaches, we obtain new theorems including the key results in w3x in more exact formulations under much weaker restrictions.

1. PRELIMINARIES Let X and Y be nonempty sets. A multimap or map T : X -o Y is a function from X into the power set of Y with nonempty values. Let x g TyŽ y . if and only y g T Ž x .. For topological spaces X and Y, a map T : X -o Y is said to be upper semicontinuous Žu.s.c.. if, for each closed set B ; Y, the set TyŽ B . s x g X: T Ž x . l B / B4 is a closed subset of X; lower semicontinuous Žl.s.c.. if, for each open set B ; Y, the set TyŽ B . is open; continuous if it is u.s.c. and l.s.c.; closed if its graph GrŽT . s Ž x, y .: x g X, y g T Ž x .4 is closed in X = Y; and compact if the closure T Ž x . of its range T Ž X . is compact in Y. For a set D, ² D : denotes the set of all nonempty finite subsets of D; and let D n be the standard n-simplex with vertices e1 , e2 , . . . , e nq1 , where e i is the ith unit vector in R nq 1. Park and Kim w14, 15x introduced the concept of a generalized con¨ ex space or a G-con¨ ex space Ž X, D; G . consisting of a topological space X, a nonempty subset D of X, and a map G: ² D : -o X such that for each A g ² D : with < A < s n q 1, there exists a continuous function fA : D n ª G Ž A. such that J g ² A: implies fAŽ D J . ; G Ž J ., where D J denotes the face of D n corresponding to J g ² A:. For a G-convex space Ž X, D; G ., a subset C of X is said to be G-con¨ ex if, for each A g ² D :, A ; C implies G Ž A. ; C. We may write G Ž A. s GA for each A g ² D :. If D s X, then Ž X, D; G . will be denoted by Ž X, G ..

QUASI-EQUILIBRIUM PROBLEMS

169

An extended real-valued function g: X ª R on a topological space X is lower Žresp. upper . semicontinuous Žl.s.c.. Žresp. u.s.c.. if x g X: g Ž x . ) r 4 Žresp. x g X: g Ž x . - r 4. is open for each r g R. If X is a G-convex space, then g: X ª R is G-quasi-con¨ ex wresp. G-quasi-conca¨ e x if x g X: g Ž x . - r 4 Žresp. x g X: g Ž x . ) r 4. is G-convex for each r g R. It is easy to see from Park and Kim w14x that any H-space Ž X, D ., Lassonde’s convex space, and a convex subset of a topological vector space are G-convex spaces. For example, any convex space Ž X, D . becomes a G-convex space Ž X, D . by putting GA s co A, where co A denotes the convex hull of A. Throughout this paper, we assume that every space is Hausdorff, and t.v.s. means topological vector spaces. A nonempty subset X of a t.v.s. E is said to be admissible Žin the sense of Klee w7x. provided that, for every compact subset K of X and every neighborhood V of the origin 0 of E, there exists a continuous map h: K ª X such that x y hŽ x . g V for all x g K and hŽ K . is contained in a finite dimensional subspace L of E. It is well known that every nonempty convex subset of a locally convex t.v.s. is admissible. Other examples of admissible t.v.s. are l p , L p Ž0, 1. for 0 - p - 1, the space SŽ0, 1. of equivalence classes of measurable functions on w0, 1x, the Hardy space H p for 0 - p - 1, certain Orlicz spaces, and ultrabarrelled t.v.s. admitting Schauder basis. Moreover, a locally convex subset of an F-normable t.v.s. and every compact convex locally convex subset of a t.v.s. are admissible. For details, see Hadzic ˇ ´ w5x, Weber w17x, and references therein. ˇ A nonempty topological space is acyclic if all of its reduced Cech o homology groups over rationals vanish. A map T : X - Y is said to be acyclic if it is u.s.c. with acyclic compact values. The following generalized quasi-equilibrium existence theorem is needed in this paper: THEOREM 0 w11x. Let X and Y be admissible con¨ ex subset of t.¨ .s. E and F, respecti¨ ely, let S: X -o X be a compact closed map, let T : X -o Y be a compact acyclic map, and let f : X = Y = X ª R be an u.s.c. function. Suppose that Ži.

the function m: X = Y ª R defined by m Ž x, y . s max f Ž x, y, u . ugS Ž x .

for Ž x, y . g X = Y

is l.s.c.; and Žii. for each Ž x, y . g X = Y, the set M Ž x, y . s u g S Ž x . : f Ž x, y, u . s m Ž x, y . 4

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is acyclic. Then there exists an Ž x, y . g X = Y such that x g SŽ x. , y g T Ž x. ,

f Ž x, y, x . G f Ž x, y, u .

for all u g S Ž x . .

The following is well known w1x: BERGE’S THEOREM. Let X and Y be topological spaces, f : X = Y ª R a real function, F: X -o Y a multimap, and fˆŽ x . s sup f Ž x, y . ,

G Ž x . s y g F Ž x . : f Ž x, y . s fˆŽ x . 4

ygF Ž x .

for xg X . Ža. If f is u.s.c. and F is u.s.c. with compact ¨ alues, then fîs u.s.c. Žb. If f is l.s.c. and F is l.s.c., then fîs l.s.c. Žc. If f is continuous and F is continuous with compact ¨ alues, then fîs continuous and G is u.s.c.

2. A SELECTION THEOREM AND THE FAN]BROWDER TYPE FIXED POINT THEOREM We begin with the following selection theorem: THEOREM 1. Let X be a compact space, Ž Y, G . a G-con¨ ex space, and F: X -o Y. Suppose that

w3x..

Ži. Žii.

for each x g X, F Ž x . is G-con¨ ex; and X s D y g Y Int FyŽ y . Ž that is, Fy has transfer open ¨ alues; see

Then there is a continuous function f : X ª Y such that f Ž x . g F Ž x . for all x g X; that is, F has a continuous selection. Proof. Since X is compact, there exists a finite subset B s nq 1 y 1 , y 2 , . . . , ynq14 of Y such that X s Dis1 Int Fy Ž yi .. Since Ž Y, G . is a G-convex space, there exists a continuous map f B : D n ª Y such that nq 1 f B Ž D n . g GB and f B Ž D J . ; GJ for each J g ² B :. Let l i 4is1 be the y nq 1 partition of unity subordinated to the cover Int F Ž yi .4is1 of X. Define a continuous map p: X ª D n by nq1

pŽ x . s

Ý li Ž x . ei s Ý li Ž x . ei is1

igN x

for x g X ,


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where i g Nx m l i Ž x . / 0 « x g FyŽ yi . m yi g F Ž x .. By Ži., we have

Ž fB p . Ž x . g f B Ž D N x . ; GN x ; F Ž x . . Let f [ f B p. Then f : X ª Y is continuous and f Ž x . g F Ž x . for all x g X. Note that Theorem 1 reduces to w3, Lemma 2.1Ž4.x whenever Y is an H-space. By the same argument, we have the following fixed point theorem: THEOREM 2. Let Ž X, G . be a compact G-con¨ ex space and let F: X -o X be a map such that Ži. for all x g X, F Ž x . is G-con¨ ex; and Žii. X s D y g X Int FyŽ y .. Then F has a fixed point. Proof. As in the proof of Theorem 1, we see that pf B : D n ª D n is continuous, hence pf B has a fixed point z g D n ; that is, z s Ž pf B .Ž z .. Let x s f B Ž z .. Then x s f B Ž z . s f B Ž Ž pf B . Ž z . . s Ž f B p . Ž x . s f Ž x . g F Ž x . , where f s f B p. Therefore F has a fixed point. As a consequence of Theorem 2, we obtain a Fan]Browder type fixed point theorem. COROLLARY 1. Let Ž X, G . be a compact G-con¨ ex space and let F: X -o X be a map such that Ži. Žii.

for all x g X, F Ž x . is nonempty G-con¨ ex; and for all y g X, FyŽ y . is open.

Then F has a fixed point. Proof. Since F Ž x . is nonempty for all x g X, there exists y g F Ž x .. This shows that x g FyŽ y . and X s D y g X FyŽ y . s D y g X Int FyŽ y .. Then all of the requirements of Theorem 2 are satisfied, and the conclusion follows. Remark. H-space versions of Theorems 1]2 and Corollary 1 were due to Horvath in his earlier works; for the references, see w6x. Moreover, generalized forms of Theorem 2 and Corollary 1 were given by Park and Kim w14x.

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From Corollary 1, we have the following equilibrium existence results: COROLLARY 2. Let Ž X, G . be a compact G-con¨ ex space, and let c : X = X ª R be a function such that Ži. Žii. Žiii.

for each x g X, y g X: c Ž x, y . - 04 is G-con¨ ex; for each y g X, x g X: c Ž x, y . - 04 is open; and c Ž x, x . G 0 for all x g X.

Then there exists an ˆ x g X such that

cŽˆ x, y . G 0

for all y g X .

Proof. Let F: X -o X be defined by F Ž x . s y g X : c Ž x, y . - 0 4

for x g X .

Then Fy Ž y . s x g X : c Ž x, y . - 0 4

for z g X .

Suppose F Ž x . / B for all x g X. Then, by Corollary 1, there exists an x g X such that x g F Ž x ., which violates condition Žiii.. Therefore, there exists an ˆ x g X such that F Ž ˆ x . s B; that is c Ž ˆ x, y . G 0 for all y g X. COROLLARY 3. Let Ž X, G . be a compact G-con¨ ex space, Y a topological space, T : X -o Y a map ha¨ ing a continuous selection f, and f : X = Y = X ª R a function such that Ži. Žii. Žiii.

f Ž x, y, z . is G-quasi-con¨ ex in z; f Ž x, y, z . is u.s.c. in Ž x, y .; and f Ž x, f Ž x ., x . G 0 for all x g X.

Then there exist an ˆ x g X and a ˆ y g TŽˆ x . such that

fŽ ˆ x, ˆ y, z . G 0

for all z g X .

Proof. Put c Ž x, z . s f Ž x, f Ž x ., z . for Ž x, z . g X = X. Then c satisfies all of the requirements of Corollary 2. Therefore, we have the conclusion. Remark. Chang et al. w3, Corollary 3.2x obtained a particular form of Corollary 3 under the assumption that f is continuous instead of Žii..


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3. COLLECTIVELY FIXED POINT THEOREMS We need the following version of the KKM theorem for G-KKM map: LEMMA 1. Let Ž X, G . be a G-con¨ ex space, let K be a nonempty compact subset of X, and F: X -o X be a G-KKM map Ž that is, for each N g ² X :, GN g F Ž N .. such that Ži. F x g X F Ž x . s F x g X F Ž x . Ž that is, F is transfer closed-¨ alued; see w3x.; and Žii. for each N g ² X :, there exists a compact G-con¨ ex subset L N of X containing N such that L N l F F Ž x . : x g L N 4 ; K. Then K l F x g X F Ž x . / B. Proof. Define F: X -o X by F Ž x . s F Ž x . for each x g X. Then F is a G-KKM map with closed values. Therefore, by Park and Kim w14, Theorem 3x, we have K l F x g X F Ž x . / B. Hence, by Ži., we have the conclusion. Note that Lemma 1 is a G-convex version of Chang et al. w3, Lemma 2.2x and that the coercivity condition Žii. is more general than theirs. LEMMA 2. Let Ž X, G . be a G-con¨ ex space, let K be a nonempty compact subset of X, let B / A ; B be any sets, and let f : X = X ª B be a function such that Ži. the map y ¬ x g X: f Ž x, y . g A4 is transfer closed-¨ alued; Žii. for each x g X, the set y g X: f Ž x, y . f A4 is G-con¨ ex; and Žiii. for each N g ² X :, there exists a compact G-con¨ ex subset L N of X such that LN l

F x g X : f Ž x, y . g A4 ; K .

ygL N

Then one of the following holds: Ž1. Ž2.

there exists a y g X such that f Ž y, y . f A; or there exists an x g X such that f Ž x, y . g A for all y g X.

Proof. Define F: X -o X by F Ž y . s x g X: f Ž x, y . g A4 for y g X. Then Ži. and Žiii. imply condition Ži. and Žii. in Lemma 1. Suppose that Ž1. does not hold. Then f Ž y, y . g A

for all y g X .

We claim that F is a G-KKM map. Otherwise, there exists an N g ² X : such that GN o F Ž N .; that is, z f F Ž N . for some z g GN . Hence f Ž z, y .

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f A for all y g N and so N ; y g X : f Ž z, y . f A4 . By Žii., we have z g GN ; y g X : f Ž z, y . f A4 and hence f Ž z, z . f A, which is a contradiction. Therefore F is a G-KKM map. Now, by Lemma 1, we have

F F Ž y . / B.

ygX

Therefore, there exists an x g X such that x g F Ž y . for all y g X; that is, f Ž x, y . g A for all y g X. This completes our proof. Note that Lemma 2 extends Chang et al. w3, Lemma 2.3x. From Lemmas 1 and 2, we deduce the following collectively fixed point theorem for later use: THEOREM 3. Let Ž X i , Gi . i g I be a family of G-con¨ ex spaces, let X s Ł i g I X i , let K be a nonempty compact subset of X, and let Ti : X -o X i be multimaps. Suppose that, for each i g I, Ži. Ti Ž x . is nonempty G-con¨ ex for each x g X; Žii. Ža. X s D x g X Int TiyŽ x i . for each i g I whene¨ er I is finite; or i i Žb. X s D y g X Int TyŽ y . whene¨ er I is infinite, where TyŽ y . s F i g I TiyŽ yi . for y s Ž yi . i g I g X; and Žiii. for each N g ² X :, there exists a compact G-con¨ ex subset L N of X containing N such that LN l

F x g X:

y j f Tj Ž x . for some j g I 4 ; K .

ygL N


ˆx g Ł Ti Ž ˆx . igI

or ˆ x i g Ti Ž ˆ x.

for all i g I.

Proof. Suppose the contrary. Then, for any x g X, we have xf

Ł Ti Ž x . .

Ž 1.

igI

Hence, for any x g X, there exists a j g I such that x j f Tj Ž x .. We define G s Ž x, y . g X = X: y f Ł i g I Ti Ž x .4 . Then, by Ž1., Ž x, x . g G for all


175

x g X and G is nonempty. Since Ž X i , Gi . i g I is a family of G-convex spaces, if we define G: X -o X by G Ž A. s Ł i g I Gi Žp i A. for each A g ² X :, it is known that Ž X, G . is a G-convex space, where p i : X -o X i is the projection for each i g I Žsee w16, Theorem 4.1x.. By Ži., for any x g X, the set

y g X : Ž x, y . f G4 s y g X : y g Ł Ti Ž x . s Ł Ti Ž x .

½

5

igI

igI

is G-convex. Moreover, for any y g X, we have

x g X : Ž x, y . f G4 s x g X : y g Ł Ti Ž x .

½

igI

5

s x g X : yi g Ti Ž x . for each i g I 4 s

F x g X:

x g Tiy Ž yi . 4 s

igI

F Tiy Ž yi . .

Ž 2.

igI

If I is infinite, by Žii.Žb. and Ž2., the multimap y ¬ x g X: Ž x, y . f G4 is transfer open-valued. If I is finite, by Žii.Ža., Tiy is transfer open-valued for each i g I. Hence if

F Tiy Ž yi . ,

xg

igI

then x g TiyŽ yi . and there exists yXi g X i such that x g Int Tiy Ž yiX .

for all i g I.

Since I is finite, we have xg

F Int Tiy Ž yiX . s Int F Int Tiy Ž yiX . igI

; Int

igI

ž FT

y i

Ž yiX . sInt x g X : Ž x, y9 . f G4 .

igI

/

This shows that the multimap y ¬ x g X: Ž x, y . f G4 is transfer openvalued if I is finite. By Žiii., we see that LN l

F x g X : Ž x, y . g G4

ygL N

s LN l

F

ygL N

s LN l

½ x g X: y f Ł T Ž x. 5

F x g X:

ygL N

igI

i

y j f Tj Ž x . for some j g I 4 ; K .

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Then it follows from Lemma 2 that there exists an x g X such that Ž x, y . g G for all y g X; that is, y f Ł i g I Ti Ž x . for all y g X. This implies Ł i g I Ti Ž x . is empty. Therefore there exists j g I such that Tj Ž x . is empty; this contradicts condition Ž1.. Hence there exists an ˆ x g X such that ˆ x g Ł i g I Ti Ž ˆ x .. If Ž X i , Gi . i g I is a family of compact G-convex spaces, then condition Žiii. of Theorem 3 holds automatically. Therefore, we have the following generalization of Theorem 2. COROLLARY 4. Let Ž X i , Gi . i g I be a family of compact G-con¨ ex spaces, let X s Ł i g I X i , and let Ti : X -o X i be a multimap for all i g I. Suppose that Ži. for any x g X and i g I, Ti Ž x . is a nonempty G-con¨ ex set; Žii. one of the following conditions holds: Ža. for any i g , Tiy : X i -o X is transfer open-¨ alued if I is finite; Žb. the multimap Ty: X -o X defined by TyŽ y . s F i g I TiyŽ yi . for each y s Ž yi . i g I g X is transfer open-¨ alued if I is infinite. Then there exists an ˆ x g X such that

ˆx g Ł Ti Ž ˆx . . igI

Remark. If I is a singleton, then Corollary 4 reduces to Theorem 2. Note that Theorem 3 and Corollary 4 are G-convex space versions of w3, Theorems 2.4 and 2.5x, respectively.

4. QUASI-EQUILIBRIUM PROBLEMS In this section, we deal with existence of solutions of certain quasiequilibrium problems in G-convex spaces without any linear structure. We begin with the following. THEOREM 4. Let Ž X, G . be a compact G-con¨ ex space, and let S: X -o X be a map with nonempty G-con¨ ex ¨ alues and open fibers Ž that is, SyŽ z . is open for each z g X . such that S: X -o X is u.s.c. Suppose that c : X = X ª R is a continuous function such that c Ž x, ? . is G-quasi-con¨ ex and

c Ž x, x . G 0

for all x g X .


ˆx g S Ž ˆx .

and

cŽˆ x, x . G 0

for all x g S Ž ˆ x. .


177

Proof. For each natural number n, define a map Fn : X -o X by

½

Fn Ž x . s z g S Ž x . : c Ž x, z . - min c Ž x, u . q ugS Ž x .

1 n

5

for x g X. Since c Ž x, ? . is l.s.c. and SŽ x . is compact as closed subset of X, it attains minimum. Moreover, c Ž x, ? . is G-quasi-convex and SŽ x . is G-convex, and hence each FnŽ x . is nonempty and G-convex. Furthermore, for any z g X, we have Fny Ž z . s Sy Ž z . l x g X : c Ž x, z . q max

½

ugS Ž x .

yc Ž x, u . -

1 n

5

.

Since c is l.s.c. and S is u.s.c. with compact values, by Berge’s theorem, x ¬ max u g S Ž x . wy c Ž x, u .x is u.s.c. Theorefore, x ¬ c Ž x, z . q max u g SŽ x .wyc Ž x, u.x is u.s.c. Note that Sy Ž z . is open, and hence Fny Ž z . is open. Therefore, by Corollary 1, Fn has a fixed point x n g FnŽ x n . for each n. Since X is compact, we may assume x n ª ˆ x g X. Since x n g S Ž x n . ; SŽ x n . and the graph of S is closed, we have ˆ x g SŽ ˆ x .. On the other hand, xn g SŽ xn .

c Ž x n , x n . - min c Ž x n , u . q

and

ugS Ž x n .

1 n

.

Since S is l.s.c. and c is u.s.c., the function x¬

inf c Ž x, u . q

ugS Ž x .

1 n

yc Ž x, u . q

s y sup ugS Ž x .

1 n

is u.s.c. Therefore, we have

cŽˆ x, ˆ x . F lim

nª`

inf ugS Ž x n .

c Ž x n , u. q

1 n

F

inf c Ž ˆ x, u . , ugS Ž ˆ x.

and hence

cŽˆ x, x . G

inf c Ž ˆ x, u . G c Ž ˆ x, ˆ x. G 0

ugS Ž ˆ x.

for all x g SŽ ˆ x . by Žii.. This completes our proof. From Theorem 4, we obtain the following GQEP, which is a correct version of Chang et al. w3, Theorem 3.1x. COROLLARY 5. Let X and S be the same as in Theorem 4, let Y be any topological space, let T : X -o Y be a map ha¨ ing continuous selection f :

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X ª Y, let and f : X = Y = X ª R be a continuous function such that Ži. Žii.

f Ž x, y, ? . is G-quasi-con¨ ex for each Ž x, y . g X = Y; f Ž x, f Ž x ., x . G 0 for all x g X.

Then there exist an ˆ x g SŽ ˆ x . and a ˆ y s fŽ ˆ x. g T Ž ˆ x . such that

fŽ ˆ x, ˆ y, x . G 0

for all x g S Ž ˆ x. .

Proof. Put c Ž x, z . s f Ž x, f Ž x ., z . in Theorem 4. Remarks 1. Corollary 5 improves Chang et al. w3, Corollary 3.3x even for convex spaces. 2. As we have seen in the above, certain GQEPs are simple consequences of corresponding QEPs whenever the map T has a continuous selection. 3. In w3, Theorem 3.1x, the authors assumed that Žii. S: X -o X is a continuous multimap with nonempty compact H-convex values and SyŽ x . is open for any x g X. This assumption has several defects. First, since SyŽ z . is open for any z g X, S is already l.s.c. Second, since S is u.s.c. with closed values and X is compact, the graph of S is closed in X = X and hence each SyŽ z . is closed for any z g X. This implies that, for important examples of H-spaces such as convex spaces or contractible spaces, S becomes a constant map; that is SŽ x . s X for all x g X. Even in this case, Corollary 5 is much more general than w3, Corollary 3.2x. 4. Similarly, w3, Theorems 3.4 and 3.5, Corollaries 3.5 and 3.6x could be obtained for a more simple and general situation. For example, condition Živ. of w3, Theorem 3.4x simply says that X can be covered by a finite number of sets of the form

½xgS

y

Ž z . : c Ž x, f Ž x . , z . s min c Ž x, f Ž x . , u . ugS Ž x .

5

. zgX

5. GENERALIZED QUASI-EQUILIBRIUM PROBLEMS In this section, we deduce the following GQE existence theorem from Theorem 0: THEOREM 5. Let X and Y be admissible con¨ ex subsets of t.¨ .s. E and F, respecti¨ ely, let S: X -o X be a compact continuous map with closed con¨ ex ¨ alues, let T : X -o Y be a compact acyclic map, and let f : X = Y = X ª R


179

be a continuous function such that Ži. Žii.

f Ž x, y, x . G 0 for all x g X and y g T Ž x .; and u ¬ f Ž x, y, u. is quasi-con¨ ex.

Then there exist an x g SŽ x . and a y g T Ž x . such that

f Ž x, y, x . G 0

for all x g S Ž x .

Proof. Since f : X = Y = X ª R is u.s.c. and S is l.s.c., by Berge’s theorem,

Ž x, y . ª max yf Ž x, y, u . ugS Ž x .

is l.s.c. on Ž x, y . g X = Y. Moreover, since SŽ x . is convex and u ¬ f Ž x, y, u. is quasi-convex for each Ž x, y . g X = Y, the set M Ž x, y . s u g SŽ x .: f Ž x, Y, u. s max z g SŽ x .wyf Ž x, y, z .x4 is convex. Therefore, by Theorem 0, there exists an Ž x, y . g X = Y such that x g SŽ x . ,

y g T Ž x. ,

yf Ž x, y, x . G yf Ž x, y, u . for all ugS Ž x . .

Since f Ž x, y, x . G 0 for all x g X and y g T Ž x ., we have

f Ž x, y, x . G 0

for all x g S Ž x . .

This completes our proof. Remarks. 1. Note that Theorem 5 extends Chang et al. w3, Theorem 3.8x in several aspects. 2. As in Chen and Park w4x, Theorem 5 can be applied to obtain generalized-forms of variational inequalities due to Hartman and Stampacchia, Browder, Lions and Stampacchia, Saigal, Chan and Pang, Shih and Tan, Kim, Chang, Parida and Sen, Yao, and others. As applications of Theorem 5, we obtain the following general variational inequality: THEOREM 6. Let e be a real reflexi¨ e Banach space, let X ; E be a nonempty closed con¨ ex subset, let Y be an admissible con¨ ex subset of a t.¨ .s. F, and let T : X -o Y be an acyclic map. Let M: X = Y ª E* be a continuous map with respect to the weak topology on X, the topology on Y, and the norm topology of EU . Let h : X = X ª E be a weakly continuous mapping Ž that is, a continuous mapping with respect to the weak topology on

180

LIN AND PARK

X and the weak topology on E .. Suppose further that Ži. h Ž x, x . s 0 for all x g X; Žii. the function u ¬ ² M Ž x, y ., h Ž u, x .: is con¨ ex; and Žiii. there exists a u g X with 5 u 5 - r such that max ygT Ž x .

² M Ž x, y . , h Ž u, x .: F 0

for all x g X with 5 x 5 s r .

Then there exist an x g X and a y g T Ž x . such that

² M Ž x, y . , h Ž x, x .: G 0

for all x g X .

Proof. We let c Ž x, y, u. s ² M Ž x, y ., h Ž u, x .: and X r s X l B Ž0, r ., where B Ž0, r . s x g E: 5 x 5 F r 4 . Since E is a reflexive Banach space, X r is a weakly compact convex subset of X. By assumption, we see that c : X r = Y = X r ª R is continuous. Then it follows from Theorem 5 that there exist an x g X r and a y g T Ž x . such that

² M Ž x, y . , h Ž x, x .: G 0

for all x g X r .

Then we follow the argument of Chang et al. w3, Theorem 3.9x and get the conclusion. Remark. Chang et al. w3, Theorem 3.9x assumed that F is a Frechet ´ space, Y ; F is a nonempty closed convex set, and T : X -o Y is an u.s.c. multimap with compact convex values with respect to the weak topology on X and the topology on Y. Let E be a real t.v.s., let X ; E, and let h : X = X ª E be a function with h Ž x, x . s 0 for all x g X. A multimap G: X -o E* is said to be h-monotone w3x if for each x, u g X and any y g GŽ x ., ¨ g GŽ u., we have

² y, h Ž u, x .: q ² ¨ , h Ž x, u .: F 0. THEOREM 7. Let E, X, F, Y, T, and M be the same as in Theorem 6. Let h : X = X ª E be a weakly continuous function such that Ži. h Ž x, x . s 0 for all x g X; Žii. the function u ª ² M Ž x, y ., h Ž u, x .: is con¨ ex and the multimap G: X ª 2 E* defined by G Ž x . s M Ž x, y . : y g T Ž x . 4 is h-monotone; and Žiii. there exist a u g X and a ¨ g T Ž u. such that lim

5 x 5ª` xgX

² M Ž u, ¨ . , h Ž x, u .: ) 0.


181

Then there exist an x g X and a y g T Ž x . such that

² M Ž x, y . , h Ž x, x .: G 0

for all x g X .

Proof. Applying Theorem 6 and by following the argument of Chang et al. w3, Theorem 3.10x, we can obtain the conclusion. Remark. The results in Section 4 and 5 of w3x are consequences of our results. For example, it is easy to see that w3, Theorem 4.1x can be deduced from Theorem 2 and w3, Theorem 5.4x from Theorem 7.

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