814 Sehie Park Hoonjoo Kim equilibrium points of n-person games with constraint preference correspondences on non-compact H-spaces. In 1967, motivated

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1 J. Korean Math. Soc. 36 (1999), No. 4, pp. 813{828 COINCIDENCE THEOREMS ON A PRODUCT OF GENERALIZED CONVEX SPACES AND APPLICATIONS TO EQUILIBRIA Sehie Park Hoonjoo Kim Abstract. In this paper, we give a Peleg type KKM theorem on G-convex spaces using this, we obtain a coincidence theorem. First, these results are applied to a whole intersection property, a section property, an analytic alternative for multimaps. Secondly, these are used to prove existence theorems of equilibrium points in qualitative games with preference correspondences in n-person games with constraint preference correspondences for non-paracompact setting of commodity spaces. 0. Introduction In the last two decades there have appeared many generalizations of the classical Arrow-Debreu result on existence of the Walrasian equilibria in various directions. For compact commodity spaces, Gale Mas-Colell [7] proved existence of a competitive equilibrium, Borglin Keiding [3] proved a new existence theorem for an abstract economy, which was generalized to an abstract economy with innite number of agents by Yennelis Prabhakar [25]. And Tarafdar [24] showed existence of abstract economies whose commodity spaces are H-spaces. Furthermore Tan-Yuan [22], Ding-Tarafdar [6], Ding-Tan [5] proved equilibrium theorems for non-compact generalized games with correspondences dened on a non-compact (but paracompact) commodity space. And Tan-Yu-Yuan [23] obtained existence theorems of Received April 17, Mathematics Subject Classication: Primary 54H25, 90D06, 46N10. Key words phrases: G-convex space, ;-convex set, G-KKM, abstract economy, correspondence, n-person game, equilibrium, qualitative game, ;-quasiconcave. This paper was supported in part by the Non-directed Research Fund, Korea Research Foundation, 1997, by KOSEF

2 814 Sehie Park Hoonjoo Kim equilibrium points of n-person games with constraint preference correspondences on non-compact H-spaces. In 1967, motivated by the search for equilibrium points in noncooperative games, Peleg [21] established an extension of the Knaster- Kuratowski-Mazurkiewicz (KKM) theorem [11]. Since then Peleg's lemma has been widely used in the framework of game theory in order to prove existence results concerning dierent solution concepts, like the bargaining set the kernel. Recently, the authors introduced generalized convex (or G-convex) spaces which are adequate to establish theories on xed points, coincidence points, KKM maps, equilibrium problems, best approximations, many others. For details, see [17-20]. In this paper, we give a Peleg type KKM theorem on G-convex spaces, which was used to derive a coincidence theorem. First, these results are applied to a whole intersection property, a geometric lemma, an analytic alternative for multimaps. Secondly, these are used to prove existence theorems of equilibrium points in qualitative games with preference correspondences in n-person games with constraint preference correspondences for non-paracompact setting of commodity spaces. 1. Preliminaries A multimap (map or correspondence) F : X ( Y is a function from a set X into the power set 2 Y of Y that is, a function with values Fx Y for x 2 X bers F ; y = fx 2 X : y 2 Fxg for y 2 Y. For A X, let F (A) = S ffx : x 2 Ag. As usual, the set f(x y) 2 X Y : y 2 Fxg is called the graph of F denoted by F. A map F : X ( Y is compact provided F (X) is contained in a compact subset of Y. For any B Y, the (lower) inverse (upper) inverse of B under F are dened by F ; (B) =fx 2 X : Fx B 6= g F + (B) =fx 2 X : Fx Bg resp. The (lower) inverse of F : X ( Y is the map F ; : Y ( X dened by x 2 F ; y if only if y 2 Fx. Given two maps F : X ( Y G : Y ( Z, their composite GF : X ( Z is dened by (GF )x = G(Fx) for each x 2 X.

3 Coincidence on generalized convex spaces applications 815 For topological spaces X Y, a map F : X ( Y is lower semicontinuous (l.s.c.) if, for each open set B Y, F ; (B) is open in X. For a nonempty set D, let hdi denote the set of all nonempty nite subsets of D. For a set A, let jaj denote the cardinality ofa. Let n denote the stard n-simplex that is, n = ( u 2 R n+1 : u = n+1 X i=1 i (u)e i i (u) 0 n+1 X i=1 i (u) =1 where e i is the i-th unit vector in R n+1. For each u = P n+1 i=1 i(u)e i in n, the (n + 1)-tuple ( 1 (u) n+1 (u)) is called the barycentric coordinate of u 2 n. Let X be a set (in a vector space) D a nonempty subset of X. Then (X D) is called a convex space if convex hulls of any nonempty nite subset of D is contained in X X has a topology that induces the Euclidean topology on such convex hulls. A subset A of X is said to be D-convex if, for each N 2hDi, N A implies co N A, where co denotes the convex hull. If X = D, then X = (X X) becomes a convex space in the sense of Lassonde [12]. Let X be a topological space. A c-structure on X is given by amap F : hxi ( X such that (1) for all A 2hXi, F (A) isnonempty contractible (2) for all A B 2hXi, A B implies F (A) F (B). A pair (X F) is then called a c-space by Horvath [8,9] an H-space by Bardaro Ceppitelli [1]. A generalized convex space or a G-convex space (X D ;) consists of a topological space X a nonempty set D such that for each A 2hDi with jaj = n +1, there exist a subset ;(A) of X a continuous function A : n! ;(A) such that J 2hAi implies A ( J ) ;(J). Here J denotes the face of n corresponding to J 2hAi that is, if A = fa 1 a 2 a n+1 g, then J = fu 2 n : P j j(u) =1 a j 2 Jg. For details on G-convex spaces, see [17-20]. We may write ;(A) = ; A for each A 2 hdi. Note that ; A does not need to contain A for A 2 hdi. If D = X, then (X D ;) will be denoted by (X ;). For simplicity, we assume that D X in this )

4 816 Sehie Park Hoonjoo Kim paper. For an (X D ;), a subset C of X is said to be ;-convex if for each A 2hDi, A C implies ; A C. For a nonempty subset S of X, the ;-convex hull of S, ;-co S, is dened by ;-co S = fy : S Y X Y is ;-convexg: Any convex space (X D) becomes a G-convex space (X D ;) by putting ; A = co A. An H-space (X F) is a G-convex space (X ;). In fact, by putting ; A = F (A) for each A 2 hxi with jaj = n +1, there exists a continuous map A : n! X such that for all J A, A ( J ) F (J) by Horvath [8, Theorem 1]. The other major examples of G-convex spaces are convex subsets of a t.v.s., metric spaces with Michael's convex structure, S-contractible spaces, Horvath's pseudo-convex spaces, Komiya's convex spaces, Bielawski's simplicial convexities, Joo's pseudoconvex spaces for the literature, see [17-20]. Recently, wegave new examples of G-convex spaces, simultaneously, showed that some abstract convexities of other authors are simple particular examples of our G-convexity see [16]. Such examples are L-spaces of Ben-El-Mechaiekh et al., continuous images of G-convex spaces, generalized H-spaces of Verma or Stacho, mc-spaces of Llinares. Moreover, Ben-El-Mechaiekh et al. [2] gave examples of G-convex spaces (X ;) as follows: B 0 -simplicial convexity, hyperconvex metric spaces due to Aronszajn Panitchpakdi, Takahashi's convexity in metric spaces. More recently, it is noted that every almost convex subset of a topological vector space has a G-convexity. For a convex space (X D), a map F : D ( X is called a KKM map if co N F (N) for each N 2hDi. For a G-convex space (X D ;), a map F : D ( X is called a G-KKM map if ; N F (N) for each N 2hDi. 2. Coincidence Theorems In this section, for each i 2 I = f1 ng, let (X i D i ; i ) be a G-convex space, Y a topological space, X = i2i X i,d = i2i D i. We may write ; i (A i )=; Ai for each A i 2hD i i i 2 I. First we need the following generalization of the classical KKM theorem due to Peleg [21]:

5 Coincidence on generalized convex spaces applications 817 Lemma. For each i 2 I, let C j i, j =1 m i +1, be closed subsets of i2i mi such that for each A i f1 m i +1g [ i 2 I, m1 Ai mn C j i j2ai where Ai denotes the face of mi corresponding to A i. Then i2i mi+1 j=1 C j i 6= : Now we generalize Lemma to G-convex spaces: Theorem 1. For each i 2 I, let(x i D i ; i ) be a compact G-convex space, G i : D i ( Y F : X ( Y multimaps such that (1.1) for each i 2 I, x i 2 D i, F + G i x i is closed (1.2) for each A = i2i A i 2hDi, i2i ; Ai i2i [ ff + G i x i : x i 2 A i g: Then there exists an x 2 X such that Fx T i2i T xi2di G ix i. Proof. For any B = i2i B i 2 hdi, there exists a continuous function f i : Bi! ; Bi such that f i ( Ai ) ; Ai for each A i B i i 2 I, where Bi is a (jb i j;1)-simplex Ai denotes the face of jbi j;1 corresponding to A i. Let fz = i2i f i z for each z 2 i2i Bi. For any i 2 I A i B i, the set f ; (F + G i x i ) is closed for all x i 2 A i since f is continuous. Moreover, Y f i ( Ai ) i2i Ai f ; f( Y i2i f ; ( Y i2i Ai )=f ;; Y i2i ; Ai ) f ; ( i2i [ ff + G i x i : x i 2 A i g) = i2i [ ff ; F + G i x i : x i 2 A i g: Hence by Lemma, f ; ( F + G i x i )= i2i xi2bi i2i xi2bi f ; F + G i x i 6=

6 818 Sehie Park Hoonjoo Kim T that is, f T T ; ( i2i xi2di F + G i x i ) 6= since X is compact. Therefore, T we have i2i xi2di T F + G i x i 6= hence, there exists an x 2 X T such that Fx i2i xi2di G ix i. Remark. For an H-space X F = 1 X, Theorem 1 reduces to Marchi Martinez-Legaz [14, Corollary 5]. The following is a generalization of Theorem 1 on non-compact setting: Theorem 2. Let K be a nonempty compact subset of Y. For each i 2 I, suppose G i : D i ( Y F : X ( Y satisfy (1.2) the following: (2.1) for each i 2 I x i 2 D i, F + G i x i G i x i are compactly closed (2.2) for each A = i2i A i 2 hdi, there exists a compact ;-convex subset L = i2i L i of X such that L i contains A i for each i 2 I L i2i ff + G i x i : x i 2 L i D i gf + K: Then we have F (X) K i2i xi2di G i x i 6= : Proof. Suppose the conclusion does not hold. Since F (X) K is compact, there exists an A = i2i A i 2hDi such that F (X) K [ i2i For the set L in (2.2), we have [ xi2ai (Y ng i x i ): L i2i ff + G i x i : x i 2 L i D i gf + K = : But L i2i ff + G i x i : x i 2 L i D i gf + K

7 by (2.2), we have Coincidence on generalized convex spaces applications 819 (2.3) L i2i ff + G i x i : x i 2 L i D i g = : Dene H i x i = F + G i x i L for x i 2 L i D i i 2 I. Note that L i D i 6= for i 2 I. Consider (L i L i D i L H i 1 L ) instead of (X i D i Y G i F) in Theorem 1. Then all of the requirements of Theorem 1 are satised. Hence L F + G i x i = i2i xi2lidi i2i which contradicts (2.3). xi2lidi H i x i 6= Remarks. 1. Condition (2.1) is satised if we assume one of the following: (i) F is l.s.c. G i x i is closed for each x i 2 D i i 2 I. (ii) F is a compact-valued continuous multimap G i x i is compactly closed for each x i 2 D i i 2 I. (iii) F = t : X! Y is a single-valued continuous function G i x i is compactly closed for each x i 2 D i i 2 I. For details, see [15]. 2. For an H-space X, m = 1, a single-valued continuous function F, Theorem 2 reduces to Chang Yang [4, Lemma 1.3]. From Theorem 2, we obtain the following: Theorem 3. Let K be a nonempty compact subset of Y. i 2 I, let S i : D i ( Y, T i : X i ( Y be multimaps, t : X! Y a continuous function satisfying (3.1) for each i 2 I x i 2 D i, S i x i is compactly open in Y (3.2) for each i 2 I y 2 t(x), A i 2hS ; i yi implies ; Ai T ; i y (3.3) for all y 2 t(x) K, y 2 S i (D i ) for some i 2 I (3.4) for each A = i2i A i 2 hdi, there exists a compact ;-convex subset L = i2i L i of X such that L i contains A i for each i 2 I t(l)nk [ i2i S i (L i D i ): For

8 820 Sehie Park Hoonjoo Kim Then there exists an ~x = i2i ~x i 2 X such that t~x 2 T i ~x i for some i 2 I. Proof. For each i 2 I, let G i : D i ( Y be a multimap dened by G i x i = Y ns i x i for each x i 2 D i. Clearly G i has compactly closed values. By (3.3), [ t(x) K G i x i = t(x) K fyn i2i xi2di i2i S i (D i )g = : T There exists an A = i2i A i 2hDisuch that t( i2i ; Ai ) 6 i2i G i(a i ) by Theorem 2. Take ~x 2 i2i ; T T Ai such thatt~x = ~y =2 i2i G i(a i ). For some i 2 I, ~y =2 G i (A i ) or ~y 2 xi2ai S ix i that is, A i S ; i ~y. Hence, there exists an ~x i 2 ; Ai T i ; ~y by (3.2). Therefore ~y 2 T i ~x i for some i 2 I. Remark. For a compact H-space X = Y = D t =1 X, Theorem 3 reduces to Marchi Martinez-Legaz [14, Theorem 6]. From Theorem 2, we have another whole intersection property as follows: Theorem 4. Let K be a nonempty compact subset of Y. For each i 2 I, let G i : D i ( Y, H i : X i ( Y be multimaps, t : X! Y a continuous function such that (4:1) for each i 2 I x i 2 D i, G i x T i is compactly closed (4:2) for each x = i2i x i 2 X, tx 2 i2i H ix i (4:3) for each y 2 t(x) i 2 I, A i 2 hd i ng ; i yi implies ; Ai X i nh i ; y (4.4) for each A = i2i A i 2 hdi, there exists a compact ;-convex subset L = i2i L i of X such that L i contains A i for each i 2 I t(l) i2i fgi x i : x i 2 L i D i gk: Then t(x) K i2i xi2di G i x i 6= :

9 Coincidence on generalized convex spaces applications 821 Proof. Suppose T that there exists an A = i2i A i 2 hdi such that t( i2i ; Ai ) 6 i2i G i(a i ) that T is, there exists an ~x = i2i ~x i 2 i2i ; Ai such that ~y = t~x =2 i2i G i(a i ). In other words, A i 2 hd i ng ; i ~yi for some i 2 I. By (4.3), ; Ai X i nh i ; ~y, since ~x i 2 ; Ai, we have ~x i =2 H i ; ~y or ~y =2 H i~x i, which contradicts (4.2). So all of the requirements of Theorem 2 are satised for F = t, hence the conclusion holds. 3. Section Properties For each i 2 I = f1 ng, let (X i D i ; i ), Y, X, D be the same as in Section 2. We now deduce a section property or a geometric form of Theorem 4. Theorem 5. Let K be a nonempty compact subset of Y, t : X! Y a continuous function, N i X i Y M i D i Y for each i 2 I. Suppose that (5.1) for each i 2 I x i 2 D i, fy 2 Y :(x i y) 2 M i g is compactly closed in Y (5.2) for each x = i2i x i 2 X, (x i tx) 2 N i for all i 2 I (5.3) for each y 2 t(x) i 2 I, A i 2 hfx i 2 D i : (x i y) =2 M i gi implies ; Ai fx i 2 X i :(x i y) =2 N i g (5.4) for each A = i2i A i 2 hdi, there exists a compact ;-convex subset L = i2i L i of X such that L i contains A i for each i 2 I t(l) i2i xi2lidi fy 2 Y :(x i y) 2 M i gk: Then there exists a ~y 2 t(x) such that i2i ; Di f~yg i2i M i. Proof. For each i 2 I x i 2 D i, let G i x i = fy 2 Y :(x i y) 2 M i g which is compactly closed by (5.1). Moreover, for each i 2 I x i 2 X i, let H i x i = fy 2 Y : (x i y) 2 N i g. Then (5.2){(5.4) imply

10 822 Sehie Park Hoonjoo Kim (4.2){(4.4), resp. Therefore, we have t(x) K i2i xi2di G i x i 6= : Hence there exists a ~y 2 t(x) K such that ~y 2 T i2i that is, i2i ; Di f~yg i2i M i. Remark. Theorem 5 generalizes [14, Corollary 8]. The following is a reformulation of Theorem 3. T xi2di G ix i Theorem 6. Let K be a nonempty compact subset of Y. For each i 2 I, let N i M i Z i be sets, t : X! Y a continuous function, g i : D i Y! Z i h i : X i Y! Z i functions. Suppose that (6.1) for each i 2 I x i 2 D i, fy 2 Y : g i (x i y) 2 N i g is compactly open in Y (6.2) for each y 2 t(x) i 2 I, A i 2 hfx i 2 D i : g i (x i y) 2 N i gi implies ; Ai fx i 2 X i : h i (x i y) 2 M i g (6.3) for each A = i2i A i 2 hdi, there exists a compact ;-convex subset L = i2i L i of X such that L i contains A i for each i 2 I Then either t(l)nk [ i2i [ xi2lidi fy 2 Y : g i (x i y) 2 N i g: (a) there exists a ~y 2 t(x)k such that g i (x i ~y) =2 N i for all i 2 I x i 2 D i or (b) there exists an ~x = i2i ~x i 2 X such that h i (~x i t~x) 2 M i for some i 2 I. Proof. For each i 2 I, consider the multimaps S i : D i ( Y T i : X i ( Y given by S i x i = fy 2 Y : g i (x i y) 2 N i g for x i 2 D i T i x i = fy 2 Y : h i (x i y) 2 M i g for x i 2 X i :

11 Coincidence on generalized convex spaces applications 823 Then (6.1){(6.3) imply (3.1), (3.2), (3.4), resp. Suppose that (a) does not hold. Then for each y 2 t(x) K, there exists an S i 2 I an x i 2 D i such that g i (x i y) 2 N i that is, t(x) K i2i S i(d i ). Hence (3.3) holds. Therefore, by Theorem 3, there exists an ~x = i2i ~x i 2 X an i 2 I such that t~x 2 T i ~x i thatis, (b) holds. From Theorem 6, we have the following analytic alternative, which is a basis of various minimax inequalities: Theorem 7. Let K be a nonempty compact subset of Y. For each i 2 I, let i i 2 R, t : X! Y a continuous function, g i : D i Y! R, h i : X i Y! R extended real-valued functions. Suppose that (7.1) for each i 2 I x i 2 D i, fy 2 Y : g i (x i y) > i g is compactly open (7.2) for each y 2 t(x) i 2 I, A i 2 hfx i 2 D i : g i (x i y) > i gi implies ; Ai fx i 2 X i : h i (x i y) > i g (7.3) for each A = i2i A i 2 hdi, there exists a compact ;-convex subset L = i2i L i of X such that L i contains A i for each i 2 I Then either t(l)nk [ i2i [ xi2lidi fy 2 Y : g i (x i y) > i g: (a) there exists a ~y 2 t(x) K such that g i (x i ~y) i for all i 2 I x i 2 D i or (b) there exists an ~x = i2i ~x i 2 X such that h i (~x i t~x) > i for some i 2 I. Proof. Put Z i = R, N i = ( i 1] M i = ( i 1] for each i 2 I in Theorem Equilibrium Existence Theorems If X is a topological space, (Y D ;) is a G-convex space, F : X ( Y is a multimap, then ;-co F F : X ( Y are multimaps dened by (;-co F )x = ;-co(fx), Fx = fy 2 Y : (x y) 2 cl XY F g for each x 2 X, resp.

12 824 Sehie Park Hoonjoo Kim An abstract economy (or generalized game) is a family of quadruples (X i A i, B i P i ) i2i where I is a (nite or innite) set of agents (players) such that for each i 2 I, X i is a nonempty subset of a topological space, X = i2i X i A i B i : X ( X i are constraint correspondences, P i : X ( X i is a preference correspondence. When I = f1 ::: ng, (X i A i B i P i ) i2i is also called an n-person game. An equilibrium of (X i A i B i P i ) i2i is a point x = (x i ) i2i 2 X such that for each i 2 I x i 2 B i x A i x P i x =. From now on we only consider I = f1 ::: ng. Denote X i = j2infig X i, x =(x 1 ::: x n ) 2 X, x i =(x 1 ::: x i;1 x i+1 ::: x n ) 2 X i, (x i y i )=(x 1 :::, x i;1 y i x i+1 ::: x n ) 2 X. Following the notion of Gale Mas-Colell [7], the collection (X i P i ) i2i will be called a qualitative game. A point x 2 X is said to be an equilibrium of the game (X i P i ) i2i if P i x = for all i 2 I. See also [13, 23]. Theorem 8. For each i 2 I, let (X i P i ) i2i be a qualitative game such that (X i ; i ) is a G-convex space, P i : X ( X i a correspondence. Suppose that (8:1) for each i 2 I x 2 X, x i =2 ;-cop i x (8:2) for each i 2 I y i 2 X i, P i ; y i is compactly open (8:3) for each A = i2i A i 2 hxi, there exists a compact ;-convex subset L = i2i L i of X such that L i contains A i for each i 2 I LnK [ i2i P i ; (L i ): Then (X i P i ) i2i has an equilibrium point ~x 2 K that is, P i ~x = for all i 2 I. Proof. Suppose that (X i P i ) i2i has no equilibrium point ink that is, for each x 2 K, P i x 6= for some i 2 I. For i 2 I, put S i T i : X i ( X by S i x i = P i ; x i S T i x i = (;-cop i ) ; x i, resp. By the assumption, K i2i P i ;1 (X i )= i2i S i(x i ) that is, (3.3) holds for t =1 X D i = X i for i 2 I. Since other conditions of Theorem 3 hold also, there exists an ~x 2 X such that ~x 2 T i ~x i for some i 2 I which implies ~x i 2 T i ; ~x = ;-cop i ~x. This contradicts (8.1).

13 Coincidence on generalized convex spaces applications 825 Theorem 9. Let (X i A i B i P i ) i2i be an n-person game such that for each i 2 I, (X i ; i ) is a G-convex space, A i B i P i : X ( X i correspondences, K a nonempty compact subset of X. Suppose that (9:1) for each i 2 I x 2 X, A i x is nonempty ;-coa i x B i x (9:2) for each x 2 X, B i x =cl Xi B i x for each i 2 I (9:3) for each i 2 I x i 2 X i, A ; i x i P ; i x i is compactly open (9:4) x i =2 ;-cop i x for all x 2 X i 2 I (9:5) for each N = i2i N i 2 hxi, there exists a compact ;-convex subset L = i2i L i of X such that L i contains N i for each i 2 I LnK [ i2i(a i P i ) ; (L i ): Then (X i A i B i P i ) i2i has an equilibrium point ~x 2 X that is, for each i 2 I, ~x i 2 cl Xi B i ~x A i ~x P i ~x =. Proof. For each i 2 I, let F i = fx 2 X : x i =2 cl Xi B i xg. Then F i is open in X by (9.2). Dene ' i : X ( X i by Ai x P i x if x =2 F i ' i x = A i x if x 2 F i : Fix any i 2 I. If x 2 F i, ;-co ' i x = G-coA i x B i x, since x i =2 cl Xi B i x = B i x, x i =2 ;-co ' i x. If x =2 F i, x i =2 ;-cop i x implies that x i =2 ;-co (A i x P i x) = ;-co ' i x. Further, ' ; i y = A; i y (F i [ P ; i y)is compactly open by (9.3). By (9.5) the denition of ' i, ' i satises (8.3). By Theorem 8, there exists an ~x 2 K such that ' i ~x = for all i 2 I. Since A i ~x 6= for all i 2 I, ~x i 2 cl Xi B i ~x A i ~x P i ~x = for all i 2 I. Let (X ;) be a G-convex space. A function f : X! R is said to be ;-quasiconcave if for each A 2hXi x 2 ; A, fx min y2a fy. Theorem 10. For each i 2 I, let (X i ; i ) be a G-convex space, F i = X i ( X i a correspondence, u i : X ( R a function. Suppose that (10:1) for each i 2 I x i 2 X i, F i x i is nonempty ;-convex in X i

14 826 Sehie Park Hoonjoo Kim (10:2) for each i 2 I x i 2 X i, F i x i =cl Xi F i x i (10:3) for each y i 2 X i, F ; i y i is compactly open in X i (10:4) u i is continuous in x ;-quasiconcave in x i (10:5) there exist a nonempty compact subset K of X an M 2hXi such that for each x 2 XnK, there exists a y 2 M satisfying y i 2 F i x i for all i 2 I u i (x i x i ) < u i (x i y i ) for some i 2 I. Then the generalized game (X i F i u i ) i2i has an equilibrium point ~x 2 X that is, ~x i 2 F i ~x i u i (~x i ~x i ) = sup yi 2Fi ~x i u i(~x i y i ). Proof. For i 2 I, dene A i P i : X ( X i by A i x = F i x i P i x = fy i 2 X i : u i (x i y i ) > u i xg for all x = (x i x i ) 2 X. Clearly (9.1) holds with A i = B i for i 2 I. Fix an x 2 X i 2 I. It is clear that cl Xi A i x A i x. And if y i 2 A i x, then (x i y i ) 2 F i y i 2 F i x i =cl Xi F i x i =cl Xi A i x: Hence (10.2) implies (9.2). For i 2 I x 2 X, x i =2 P i x. Now we show that P i x is ;-convex for i 2 I x 2 X. Indeed, for each N i 2hP i xi z i 2 ; Ni, u i (x i z i ) min u i (x i y i ) >u i (x i x i )=u i x yi2ni that is, z i 2 P i x, hence ; Ni P i x. For each y i 2 X i i 2 I, A ; i y i = fx 2 X : x i 2 F i ; y ig = F i ; y i X i P i ; y i = fx 2 X : u i x < u i (x i y i )g are compactly open in X by (10.3), (10.4). And (10.5) implies (9.5). Hence, there exists an ~x 2 X such that ~x i 2 cl Xi A i ~x = F i ~x i A i ~x P i ~x = that is, ~x i 2 F i ~x i u i ~x = sup yi 2Fi ~x i u i(~x i y i ). Remark. Theorem 10 is a correct generalized form of Huang [10, Theorem 2]. In particular, (10.2) is essential as the following example shows: Example. Let X 1 = [0 1], X 2 = [1 2], F 1 : [1 2] ( [0 1] F 2 :[0 1] ( [1 2] be dened by F 1 (x 2 )= 8 >< >: [ 1 2 1] x 2 =1 f 1 2 g x 2 = 3 2 [0 1] x 2 6=1 3 2

15 Coincidence on generalized convex spaces applications 827 F 2 (x 1 )= 8 >< >: f 3 2 g x 1 =0 [ 7 4 2] x 1 = 1 2 [1 2] x 1 6=0 1 2 for each x 1 2 [0 1] x 2 2 [1 2], resp. And let u 1 u 2 :[0 1] [1 2]! R be dened by u 1 (x 1 x 2 )=;x 2 1 u 2 (x 1 x 2 )=;x 2 2 for each (x 1 x 2 ) 2 [0 1] [1 2], resp. Then F 1 F 2 u 1 u 2 satisfy (10.1), (10.3) (10.4). But F 1 (1) = [0 1] 6= cl X1 F 1 (1) = [ 1 1]. And 2 (X i F i u i ) i=1 2 has no equilibrium point. References [1] C. Bardaro R. Ceppitelli, Some further generalizations of Knaster-Kuratowski-Mazurkiewicz theorem minimax inequalities, J. Math. Anal. Appl. 132 (1988), 484{490. [2] H. Ben-El-Mechaiekh, S. Chebbi, M. Florenzano, J. V. Llinares, Abstract convexity xed points, J. Math. Anal. Appl. 222 (1998), 138{151. [3] A. Borglin H. Keiding, Existence of equilibrium action of equilibrium: A note on the `new' existence theorems, J. Math. Econom. 3 (1976), 313{316. [4] S. S. Chang L. Yang, Section theorems on H-spaces with applications, J. Math. Anal. Appl. 179 (1993) 214{231. [5] X. P. DingK.K.Tan, On equilibria of noncompact generalized games, J. Math. Anal. Appl. 177 (1993), 226{238. [6] X. P. Ding E. Tarafdar, Fixed point theorems equilibria on noncompact generalized games, Fixed Point Theory Applications, (K.-K. Tan, ed.), World Scientic, River Edge, NJ, 1992, pp. 80{96. [7] D. Gale A. Mas-Colell, On the role of complete, transitive preferences in equilibrium theory, Equilibrium Disequilibrium in Economic Theory (G. Schwodiauer, ed.), Reider, Rordrecht, 1978, pp. 7{14. [8] C. D. Horvath, Contractibility generalized convexity, J. Math. Anal. Appl. 156 (1991), 341{357. [9], Extension selection theorems in topological spaces with a generalized convexity structure, Ann. Fac. Sci. Toulouse 2 (1993), 253{269. [10] Y. Huang, Fixed point theorems with an application in generalized games, J. Math. Anal. Appl. 186 (1994), 634{642. [11] B. Knaster, K. Kuratowski, S. Mazurkiewicz, Ein Beweis des Fixpunktsatzes fur n-dimensionale Simplexe, Fund. Math.14 (1929), 132{137.

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