Equilibrium existence theorems of generalized games for generalized Lθ,Fc -majorized mapping in topological space

Nonlinear Analysis 67 (2007) 316–326 www.elsevier.com/locate/na

Equilibrium existence theorems of generalized games for generalized Lθ,Fc -majorized mapping in topological space✩ Yan-Mei Du a,∗ , Lei Deng b a Department of Mathematics, Tianjin Polytechnic University, Tianjin 300160, PR China b Department of Mathematics, Southwest China Normal University, Chongqing 400715, PR China

Received 17 February 2006; accepted 31 May 2006

Abstract In this paper, we first introduce generalized Lθ,Fc -correspondence and generalized Lθ,Fc -majorized mappings without open lower sections in the nonconvexity setting of topological space. Some existence theorems of maximal elements for generalized Lθ,Fc -correspondence and generalized Lθ,Fc -majorized mappings are obtained in topological spaces without convexity. As applications, we establish new equilibrium existence theorems for qualitative games and generalized games with infinite sets of players and generalized Lθ,Fc -majorized preference correspondences in topological spaces without convexity. Our results generalize and improve the corresponding results in the recent literature. c 2006 Elsevier Ltd. All rights reserved.  CLC: O177.91 MSC: 47H10; 54A05; 54B10; 54C60 Keywords: Generalized Lθ,Fc -correspondence; Generalized Lθ,Fc -majorized mapping; Maximal elements; Generalized game

1. Introduction and preliminaries In mathematical economics, the existence of equilibrium is the main problem in investigating various kind of economic models. In recent years, many authors have obtained an equilibrium ✩ Supported by the National Natural Science Foundation of China (19871057) and by the Major Project of Science and Technology of MOE, PR China (02060). ∗ Corresponding author. E-mail address: [email protected] (Y.-M. Du).

c 2006 Elsevier Ltd. All rights reserved. 0362-546X/$ - see front matter  doi:10.1016/j.na.2006.05.015

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existence theorem for the generalized game by assuming that the strategy sets of players are topological vector spaces and the preference correspondences have open lower sections or are majorized by the correspondence with open lower sections (see [9–18]). In the setting, convexity assumptions play a crucial role. Since Horvath [1] introduced H -space by replacing convex hulls by contract subsets, many authors have put forward abstract convex spaces without linear structure, for example: G-convex space [2], L-convex space [3]. Inspired by the above research, we first introduce the notions of generalized L Fc -correspondence and generalized L Fc -majorized mappings which include the classes of set-valued mapping defined in [6–12] as special cases. Here the convexity assumption of space is not required and the mappings do not have open lower sections. As a result, we establish some new existence theorems of maximal elements for generalized L Fc -correspondence and generalized L Fc -majorized mappings on topological spaces. Next, we give new notions of abstract economy and obtain some existence theorems of equilibrium points for one-person games, qualitative games and generalized games with infinite (countable or uncountable) sets of players and with generalized L Fc -majorized preference correspondences under nonparacompact and nonconvexity settings of topological spaces. Our results unify and generalize the corresponding notions and results introduced by many authors (see [8–18]). Let X be a nonempty subset of topological space E, 2 X and X denote the family of all subsets and all nonempty finite subsets of X, respectively. For A ∈ X, let |A| denote the cardinality of A. Let n denote the standard n-simplex, that is,   n+1 n+1   n+1 n = u ∈ R :u= λi (u)ei , λi (u) ≥ 0, λi (u) = 1 , i=1

i=1

where ei (i = 1, . . . , n + 1) is i th unit vector in Rn+1 the . Let N = {x 0 , x 1 , . . . , x k } ∈ X(k < k n+1 n),  N = {u ∈ R : u = i=1 λi (u)ei , λi (u) ≥ 0, ki=1 λi (u) = 1} denote the face of n corresponding to N. For A ⊂ X, cl X (A) and int X (A) denote the closure and the interior of A in X, respectively. If Y is a topological space and F : X → 2Y is a mapping, for any D ⊂ X and y ∈ Y , let F(D) = ∪x∈D F(x) and F − (y) = {x ∈ X : y ∈ F(x)}. Let S, T : X → 2Y be two mappings; then T ∩ S : X → 2Y is a mapping defined by (T ∩ S)(x) = T (x) ∩ S(x) for each x ∈ X. T is said to have open lower sections if for each y ∈ Y , T − (y) is open in X. T is said to have compactly open lower sections if for each y ∈ Y and for any nonempty compact subset K of X, T − (y) ∩ K is open in K . The graph of T is the set Gr (T ) = {(x, y) ∈ X × Y : y ∈ T (x)}. The correspondence T¯ : X → 2Y is defined by T¯ (x) = {y ∈ Y : (x, y) ∈ cl X ×Y (Gr (T ))} and the correspondence cl T : X → 2Y is defined by (cl T )(x) = clY (T (x)) for each x ∈ X. The following notions were introduced by Ding [4]. Let E be a topological space, For any given subset A of E, the compact interior and the compact closure of A are denoted by cint(A) and ccl(A) as  cint(A) = {B ⊂ E : B ⊂ A and B is compactly open in E};  ccl(A) = {B ⊂ E : A ⊂ B and B is compactly closed in E}. It is easy to see that cint(A) ∩ K = int K (A ∩ K ), ccl(A) ∩ K = cl K (A ∩ K ), then cint(A) (resp., ccl(A)) is compactly open (resp., compactly closed) in E. Hence, a subset A of E is compactly open (resp., compactly closed) in E if and only if cint(A) = A (resp., ccl(A) = A).

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Definition 1.1 ([4]). Let X be a nonempty set and Y be a topological space. A correspondence F : X → 2Y is said to be transfer compactly closed valued (resp., transfer compactly open valued) on X if for each x ∈ X and for any nonempty compact subset K of Y , y ∈ F(x) ∩ K (resp., y ∈ F(x) ∩ K ) implies that there exists an x  ∈ X such that y ∈ ccl(F(x  )) ∩ K (resp., y ∈ cint(F(x  )) ∩ K ). Definition 1.2 ([5]). Let X be a nonempty set and Y be a topological space. T : X → 2Y is said to be a generalized relatively KKM (R-KKM) mapping if for any N = {x 0 , x 1 , . . . , x n } ∈ X, there exists a continuous mapping ϕ N : n → Y such that, for each ei0 , ei1 , . . . , eik , ϕ N (k ) ⊂ k j =0 T (x i j ), where k is a standard k-subsimplex of n with vertices ei0 , ei1 , . . . , eik . Definition 1.3. Let X and Y be topological spaces. Let θ : X → Y be a single-valued mapping and A : X → 2Y be a set-valued mapping. Then (i) A is said to be a generalized Lθ,Fc -correspondence if (a) A− : Y → 2 X is transfer compactly open valued on Y ; (b) for each N = {y0 , y1 , . . . , yn } ∈ Y , there exists a continuous mapping ϕ N : n → Y such that for each x ∈ X, N ∩ A(x) = {yi0 , yi1 , . . . , yik } = ∅ implies θ (x) ∈ ϕ N ( N∩A(x) ); (ii) (Bx ; Nx ) is said to be a generalized Lθ,Fc -majorant of A at x ∈ X if Nx is an open neighborhood of x in X and the mapping Bx : X → 2Y is such that (a) A(z) ⊂ Bx (z) for each z ∈ X; (b) for each N = {y0 , y1 , . . . , yn } ∈ Y , there exists a continuous mapping ϕ N : n → Y such that for each z ∈ Nx , N ∩ Bx (z) = {yi0 , yi1 , . . . , yik } = ∅ implies θ (z) ∈ ϕ N ( N∩Bx (z) ); (c) Bx− : Y → 2 X is transfer compactly open valued on Y ; (iii) A is said to be generalized Lθ,Fc -majorized if for each x ∈ X with A(x) = ∅, there exists a generalized Lθ,Fc -majorant (Bx ; Nx ) of A at x, and for any N ∈ {x ∈ X : A(x) = ∅}, the mapping x∈N Bx− is transfer compactly open valued in Y . Remark 1.1. Definition 1.3 unifies Definition 2.1 of Ding [6] from G-convex to topological space. Moreover, Definition 1.3 improves correspondence notions from the literature [8–10]. In

the paper, we will deal with (1) X = Y and θ = I X is the identity mapping on X, and (2) X = i∈I X i , Y = X i and θi = πi : X → X i , where πi is the projection of X onto X i and each X i is a topological space. In both cases, we shall write L Fc in place of Lθ,Fc . Lemma 1.1 ([5]). Let X be a nonempty set and Y be a topological space. Let T : X → 2Y be a generalized R-KKM mapping with is transfer compactly closed valued. If some M ∈ X is such that x∈M ccl(T (x)) is compact, then x∈X T (x) = ∅. 2. Existence theorem of maximal elements Let X and Y be topological spaces and A : X → 2Y be a correspondence. A point xˆ ∈ X is said to be a maximal element of A if A(x) ˆ = ∅. Theorem 2.1. Let X be a topological space. A : X → 2 X is a generalized L Fc -correspondence such that a nonempty compact subset K of X and M ∈ X such that X \ K ⊂  (i) There exists − (y)). Then there exists a point xˆ ∈ X such that A( x) cint(A ˆ = ∅. y∈M

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Proof. Suppose for each x ∈ X, A(x) = ∅. Since A is a generalized L Fc -correspondence then (a) A− : X → 2 X is transfer compactly open valued on X; (b) for each N = {x 0 , x 1 , . . . , x n } ∈ X, there exists a continuous mapping ϕ N : n → X such that for each x ∈ X, N ∩ A(x) = {x i0 , x i1 , . . . , x ik } = ∅ implies x ∈ ϕ N ( N∩A(x) ). Define B : X → 2 X by B(x) = X \ A− (x),

for each x ∈ X.

Then B is transfer compactly closed valued on X by (a). We claim that B is a generalized and for any continuous mapping R-KKM mapping. If not, there exists N = {x 0 , x 1 , . . . , x n },  ϕ N : n → X, there is ei0 , ei1 , . . . , eik such that ϕ N (k ) ⊂ kj =0 B(x i j ). That is, there exists   x ∈ ϕ N (k ) such that x ∈ kj =0 B(x i j ) = kj =0 X \ A− (x i j ); then {x i0 , x i1 , . . . , x ik } ⊂ A(x). In particular, for ϕ N in (b), we have x ∈ ϕ N (k ) = ϕ N ( N∩A(x) ). This contradicts (a). By (i), for some M ∈ X,    ccl(B(y)) = ccl(X \ A− (y)) = X \ cint(A− (y)) ⊂ K y∈M

and thus

y∈M



y∈M

ccl(B(y)) is compact. By Lemma 1.1,   ∅ = B(x) = (X \ A− (x)). y∈M

x∈X

x∈X

B(x) = ∅. Thus we get

x∈X

Then there exists xˆ ∈ X such that xˆ ∈ A− (x) for each x ∈ X. Thus A(x) ˆ = ∅. This contradicts the assumption. Hence by (i), there exists xˆ ∈ K such that A(x) ˆ = ∅.  Theorem 2.2. Let X be a topological space. A : X → 2 X is a generalized L Fc -majorized mapping such that a nonempty compact subset K of X and M ∈ X such that X \ K ⊂  (i) There exists − ˆ = ∅. y∈M cint(A (y)). Then there exists a point xˆ ∈ X such that A( x) Proof. Suppose that A(x) = ∅ for each x ∈ X. Since A is a generalized L Fc -majorized mapping, for each x ∈ X, there exist an open neighborhood Nx of x in X and a mapping Bx : X → 2 X such that (a) A(z) ⊂ Bx (z) for each z ∈ X; (b) for each N = {x 0 , x 1 , . . . , x n } ∈ X, there exists a continuous mapping ϕ N : n → X such that for each z ∈ Nx , N ∩ Bx (z) = {x i0 , x i1 , . . . , x ik }

= ∅ implies θ (z) ∈ ϕ N ( N∩Bx (z) ); (c) for any N ∈ {x ∈ X : A(x) = ∅}, the mapping x∈N Bx− is transfer compactly open valued in X. It is clear that the family {Nx : x ∈ K } is an open covering ofK . By the compactness of K , there exists a finite subset {x 1 , x 2 , . . . , x n } ⊂ K such that K ⊂ ni=1 Nxi = U ; then U is open in X. Define B : X → 2 X by ⎧ n ⎪ ⎨ B (z) if z ∈ U ; x B(z) = i=1 i ⎪ ⎩∅ if z ∈ U. Then by (b), for each N ∈ X with |N| = n + 1, there exists a continuous mapping ϕ N : n → X such that for each x ∈ X, N ∩ B(x) = ∅ implies x ∈ ϕ N ( N∩B(x) ).

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Now we show that B − : X → 2 X is transfer compactly open valued in X. Indeed, for each y ∈ X, we have   n n   − Bxi (z) = U ∩ Bx−i (y). (1) B (y) = z ∈ U : y ∈ i=1

i=1

− ∩C = For each  n y ∈− X and for any nonempty compact subset C of X, if x ∈ B (y)  ∈ X such U∩ B (y) ∩ C. By condition (c) and U being open in X, there exists y i=1 x i that     n n   −  −  Bxi (y ) ∩ C = cint U ∩ Bxi (y ) ∩ C. x ∈ U ∩ cint i=1

B−

i=1

This proves that :X→ is transfer compactly open valued in X. Thus B is a generalized L Fc -correspondence. Finally, by (i) and condition (a), we have   cint(A− (y)) ⊂ cint(Bx−i (y)) i = 0, 1, . . . , n X\K ⊂ y∈M

2X

y∈M

and then X\K ⊂

n  

cint(Bx−i (y)) =

i=1 y∈M

 y∈M

 cint

n 

i=1

 Bx−i (y) =



cint(B − (y)).

y∈M

The condition of Theorem 2.1 is satisfied. By Theorem 2.1, there exists a point xˆ ∈ K such that B(x) ˆ = ∅. By the definition of B, we have A(z) ⊂ B(z) for each z ∈ K ⊂ U , and thus A(x) ˆ = ∅. This is a contradiction. Hence there exists a point xˆ ∈ X such that A(x) ˆ = ∅. By condition (i), xˆ must be in K .  Remark 2.1. Theorem 2.2 generalized Theorem 3.3 of Ding [6] in the following ways: (1) from G-convex space to a topological space without convexity; (2) L Fc -correspondence is replaced by generalized L Fc -correspondence. Theorem 2.2 also unifies correspondence results in [9,10]. 3. Existence of equilibrium points Let I be a (finite or infinite)

set of players. For each i ∈ I , let its strategy set X and Yi (i ∈ I ) be a nonempty set and Y = i∈I Yi . Let Pi : X → 2Yi be the preference correspondence of the i th player. The collection Λ = (X; Yi ; Pi )i∈I will be called a qualitative game. A point xˆ ∈ X is said to be an equilibrium of the qualitative game if Pi (x) ˆ = ∅ for each i ∈ I . A generalized game (=abstract economy) is a quintuple family Λ = (X; Yi ; Ai ; Bi ; Pi ; θi )i∈I where X is a nonempty set, I

is a (finite or infinite) set of players such that for each i ∈ I , Yi is a nonempty set and Y = i∈I Yi . Ai , Bi : X → 2Yi , θi : X → Yi are the constraint correspondences and Pi : X → 2Yi is the preference correspondence. An equilibrium of the generalized game Λ is a point xˆ ∈ X such that for each i ∈ I, θi (x) ˆ ∈ B¯ i (x) ˆ and Ai (x) ˆ ∩ Pi (x) ˆ = ∅. Theorem 3.1. Let X be a topological space. A, B, P : X → 2 X is such that (i) Z = {x ∈ X : A(x) = ∅} is a nonempty subset of X and A is a generalized L Fc correspondence on X;

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(ii) for each x ∈ Z , there exist an open neighborhood Mx of x in Z and φx : Z → 2 X such that (1) for each z ∈ Z , (A ∩ P)(z) ⊂ φx (z) and the correspondence φx− : X → 2 Z is transfer compactly open valued on X; (2) for each N ∈ X with |N| = n + 1, there exists a continuous mapping ϕ N : n → X z ∈ ϕ N ( N∩φx (z) ); such that for each z ∈ Nx , N ∩ φx (z) = ∅ implies (3) for each N ∈ {x ∈ Z : (A ∩ P)(x) = ∅}, A− ∩ x∈N φx− is transfer compactly open valued in X; (iii) there exists a nonempty compact subset K of Z and M ∈ Z  such that Z \ K ⊂  − y∈M cint((A ∩ P) (y)). ¯ x) Then there exists a point xˆ ∈ X such that xˆ ∈ B( ˆ and A(x) ˆ ∩ P(xˆ ) = ∅. Proof. Let W = {x ∈ Z : x ∈  A(x) ∩ P(x) Q(x) = A(x)

¯ B(x)}, then W is open in Z . Define a mapping Q : Z → 2 X by if z ∈

W; if z ∈ W.

Suppose x ∈ Z with Q(x) = ∅. Case I: If x ∈ W , then W is an open neighborhood of x in Z . Now define φx : Z → 2 X by φx (z) = A(z) for each z ∈ Z . Let Nx = W ; then by (i) we have (a) Q(z) ⊂ A(z) = φx (z) for each z ∈ Z ; (b) the correspondence φx− : X → 2 Z is transfer compactly open valued in X; (c) for each N = {x 0 , x 1 , . . . , x n } ∈ X with |N| = n + 1, there exists a continuous mapping ϕ N : n → X such that for each z ∈ Nx , N ∩ φx (z) = ∅ implies z ∈ ϕ N ( N∩φx (z) ). Then (φx ; Nx ) is a generalized L Fc -majorant of Q at x. Case II: If x ∈ W , then Q(x) = A(x) ∩ P(x) = ∅. By condition (ii), there exists an open neighborhood Mx of x in Z and φx : Z → 2 X such that (a ) for each z ∈ Z , (A ∩ P)(z) ⊂ φx (z) and the correspondence φx− : X → 2 Z is transfer compact open valued in X; (b ) for each N = {x 0 , x 1 , . . . , x n } ∈ X with |N| = n + 1, there exists a continuous mapping ϕ N : n → X such that for each z ∈ Nx , N ∩ φx (z) = ∅ implies z ∈ ϕ N ( N∩φx (z) ); (c ) for each N ∈ {x ∈ Z : (A ∩ P)(x) = ∅}, A− ∩ x∈N φx− is transfer compactly open valued in X. Now for each x ∈ Z with Q(x) = ∅, let  Mx if x ∈ W ; Nx = W if x ∈ W and define the correspondence Φx : Z → 2 X by  A(z) ∩ φx (z) if z ∈ W ; Φx (z) = A(z) if z ∈ W. Then for each x ∈ Z with Q(x) = ∅, Nx is an open neighborhood of x in Z such that (a ) for each z ∈ Z , Q(z) ⊂ Φx (z) by (a) and (a ); (b ) for each N = {x 0 , x 1 , . . . , x n } ∈ X with |N| = n + 1, there exists a continuous mapping ϕ N : n → X such that for each z ∈ Nx , N ∩ φx (z) = ∅ implies z ∈ ϕ N ( N∩φx (z) ) by (b) and (b );

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(c ) for each y ∈ X, Φx− (y) = {z ∈ Z \ W : y ∈ Φx (z)} ∪ {z ∈ W : y ∈ Φx (z)} = {z ∈ Z \ W : y ∈ A(z) ∩ φx (z)} ∪ {z ∈ W : y ∈ A(z)} = [(Z \ W ) ∩ A− (y) ∩ φx− (y)] ∪ [W ∩ A− (y)] = [W ∪ φx− (y)] ∩ [ A− (y) ∩ Z ] = [W ∩ A− (y)] ∪ [φx− (y) ∩ A− (y)].

(2)

Hence for any compact subset C of Z and for each z ∈ Φx− (y) ∩ C, we have z ∈ [W ∩ A− (y) ∩ C] ∪ [φx− (y) ∩ A− (y) ∩ C]. If z ∈ W ∩ A− (y) ∩ C, by condition (i), there exists y  ∈ X such that z ∈ cint(A− (y  )) ∩ C. Then z ∈ cint(A− (y  )) ∩ W ∩ C = cint(A− (y  ) ∩ W ) ∩ C ⊂ cint{[ A− (y  ) ∩ W ] ∪ [φx− (y  ) ∩ A− (y  )]} ∩ C = cint(Φx− (y  )) ∩ C. If z ∈ φx− (y) ∩ A− (y) ∩ C, by condition (c ), there exists y  ∈ X such that z ∈ cint(φx− (y  ) ∩ A− (y  )) ∩ C

⊂ cint{[ A− (y  ) ∩ W ] ∪ [φx− (y  ) ∩ A− (y  )]} ∩ C = cint(Φx− (y  )) ∩ C.

Then Φx− : X → 2 Z is transfer compactly open valued in X. Hence (Φx , Nx ) is a generalized L FC -majorant of Q at x. For any x ∈ Z , if x ∈ W , by Case I, Φx (z) = A(z). If x ∈ W , by Case II,  A(z) ∩ φx (z) if z ∈ W ; Φx (z) = A(z) if z ∈ W. So for each N ∈ {x ∈ X : Q(x) = ∅}, if N  ⊂ N such that N \ N  ⊂ W , then by (2) we have    Φx− (y) = Φx− (y) ∩ Φx− (y) x∈N 

x∈N

=



x∈N 

x∈N\N 

{[W ∪ φx− (y)] ∩ A− (y)} ∩



= W∪

 x∈N  −





(A− (y) ∩ W )

x∈N\N 

φx− (y) ∩ [ A− (y) ∩ W ]  −

= [W ∩ A (y)] ∪ W ∩ A (y) ∩

 x∈N 

 φx− (y)

.

By conditions (i) and (c ), we have that ni=1 Φx−i is transfer compactly open valued on Y . Therefore Q is a generalized L Fc -majorized mapping. From the definition of Q, it follows that (A ∩ P)(z) ⊂ Q(z) for each z ∈ Z . Now, by condition compact subset K of Z and M ∈ Z  such that   (iii), there exists a nonempty Z \ K ⊂ y∈M cint((A ∩ P)− (y)) ⊂ y∈M cint(Q − (y)). By Theorem 2.2, there exists a point ¯ x) xˆ ∈ K such that Q(x) ˆ = ∅. By the definition of Q, we must have xˆ ∈ B( ˆ and A(x)∩ ˆ P(xˆ ) = ∅. 

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Remark 3.1. Theorem 3.1 does not require X to possess convexity; thus Theorem 3.1 generalized Theorem 4.1 of Ding [5], Theorem 4.1 of Chowdhury [9] and Theorem 4.1 of Yuan [10]. Theorem 3.2. Let

Λ = (X; X i ; Pi )i∈I be a qualitative game. For each i ∈ I , X i is topological space with X = i∈I X i . Let Pi : X → 2 X i be such that (i) (ii) (iii)

Wi = {x ∈ X : Pi (x) = ∅} is open in X; Pi : X → 2 X i is a generalized L Fc -majorized mapping; there exists a nonempty compact subset K of X and Mi ∈ X i  such that X \ K ⊂  − y∈Mi cint(Pi (y)).

Then Λ has an equilibrium point in xˆ ∈ K . Proof. For each x ∈ X, let I (x) = {i ∈ I : Pi (x) = ∅}. For each i ∈ I , define the correspondence Pi : X → 2 X by Pi (x) = πi− (Pi (x)) where πi : X → X i is the projection of X onto X i . Furthermore, define the correspondence P : X → 2 X by ⎧  ⎨ Pi (x) if I (x) = ∅; P(x) = i∈I (x) ⎩ ∅ if I (x) = ∅. Clearly, for each x ∈ X, P(x) = ∅ if and only if I (x) = ∅. Let x ∈ X with P(x) = ∅. For any fixed i ∈ I (x), by condition (ii), there exists an open neighborhood Nx of x in X and φi, x : X → 2 X i such that (a) for each z ∈ X, Pi (z) ⊂ φi,x (z); (b) for each N = {x i0 , x i1 , . . . , x in } ∈ X i , there exists a continuous mapping ϕi,N : n → X i such that for each z ∈ Nx , N ∩ φi,x (z) = ∅ implies z i ∈ ϕi,N ( N∩φi,x (z) ); − is transfer compactly open valued in X. (c) for each N ∈ {x ∈ X : Pi (x) = ∅}, x∈N φi,x By condition (i), we may assume that Nx ⊂ Wi so that Pi (z) = ∅ for all z ∈ Nx . For each x ∈ X with P(x) = ∅, define the correspondence Φx : X → 2 X by Φx (z) = πi− (φi, x (z)). Then we have (a ) By (a), for each z ∈ X,  P(z) = Pi (z) ⊂ Pi (z) = πi− (Pi (z)) ⊂ πi− (φi, x (z)) = Φx (z). i∈I (z)

(b ) By (b), for each

N ∈ X with |N| = n+1, there exists a continuous mapping ϕ N : n → X by ϕ N (u) = i∈I (x) ϕi,N (u) ⊗ i∈I \I (x) X i for each u ∈ n satisfying for each z ∈ Nx , N ∩ Φx (z) = ∅ implies z ∈ ϕ N ( N∩Φx (z) ). By the definition of Φx , for each y ∈ X, Φx− (y) = {z ∈ X : y ∈ Φx (z)} = {z ∈ X : yi ∈ φi, x (z)} = φi,− x (yi ).

(3)

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(c ) By (3), for each N ∈ {x ∈ X : P(x) = ∅},   Φx− (y) = ϕi,− x (yi ). x∈N

By (c),



x∈N − x∈N Φx

is transfer compactly open valued in Y .

Hence P is generalized L Fc -majorized mapping. each y ∈ X. By condition By the definitions of P and Pi , we have P − (y) = Pi− (πi (y)) for

(iii), thereexists a nonempty compact subset K of X and M = i∈I Mi ∈ X such that X \ K ⊂ y∈M cint(P − (y)). By Theorem 2.2, there exists a point xˆ ∈ K such that P(xˆ ) = ∅ which implies I (x) ˆ = ∅. Therefore Pi (x) ˆ = ∅ for all i ∈ I .  Remark 3.2. Theorem 3.2 generalized Theorem 4.2 of Ding [5], Theorem 2.2 of Yuan [10]. Theorem 3.3. Let Λ = (X; X i ; Ai ; Bi ; Pi ; π i )i∈I be a generalized game. Suppose that for each i ∈ I , X i is topological space with X = i∈I X i and πi is projection from X onto X i . Let Ai , Bi , Pi : X → 2 X i be such that (i) Z = i∈I {x ∈ X : Ai (x) = ∅} is a nonempty subset of X and Ai is a generalized L Fc -correspondence; (ii) Wi = {x ∈ Z : (Ai ∩ Pi )(x) = ∅} is open in Z ; (iii) for each x ∈ Wi , there exist an open neighborhood Mx of x in Z and φi, x : Z → 2 X i such that − (1) for each z ∈ Z , (Ai ∩ Pi )(z) ⊂ φi,x (z) and the correspondence φi,x : X i → 2 Z is transfer compactly open valued on X i ; (2) for each N ∈ X i  with |N| = n +1, there exists a continuous mapping ϕi,N : n → X i such that, for each z ∈ Nx , N ∩ φi,x (z) = ∅ implies z i ∈ ϕi,N ( N∩φi,x (z) ); − (3) for each N ∈ {x ∈ Z : (Ai ∩ Pi )(x) = ∅}, A− x∈N φi,x is transfer compactly i ∩ open valued on Z ; (iv) there exists a nonempty compact subset K of Z and M ∈ Z  such that Z \ K ⊂  − y∈M cint((A i ∩ Pi ) (y)). Then Λ has an equilibrium point in xˆ ∈ K . Proof. For each i ∈ I , let Ui = {x ∈ Z : x i ∈ B¯ i (x)}; then Wi is open in Z . Define a mapping Q i : Z → 2 X i by  Ai (x) ∩ Pi (x) if x ∈ Ui ; Q i (x) = if x ∈ Ui . Ai (x) Now we will show that the qualitative game Λ = (Z ; Yi ; Q i )i∈I satisfies all assumptions of Theorem 3.2. For each i ∈ I , the set {x ∈ Z : Q i (x) = ∅} = {x ∈ Ui : Q i (x) = ∅} ∪ {x ∈ Z \ Ui : Q i (x) = ∅} = Ui ∪ [(Z \ Ui ) ∩ Wi ] = Ui ∪ Wi is open in Z by (ii) and hence condition (i) of Theorem 3.2 is satisfied. Now for each x ∈ Z with Q i (x) = ∅, let  Mx if x ∈ Ui ; Nx = Ui if x ∈ Ui .

Y.-M. Du, L. Deng / Nonlinear Analysis 67 (2007) 316–326

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By (iii) define the correspondence Φi, x : Z → 2 X i by  Ai (z) ∩ φi, x (z) if z ∈ Ui ; Φi, x (z) = if z ∈ Ui . Ai (z) Then for each x ∈ X with Q i (x) = ∅, Nx is an open neighborhood of x in Z . By (i) and (iii), we have (a) For each z ∈ Z , Q i (z) ⊂ Φi,x (z) by (1) of (iii); (b) For each N ∈ X i  with |N| = n + 1, there exists a continuous mapping ϕi,N : n → X i such that for each z ∈ Nx , N ∩ Φi,x (z) = ∅ implies z ∈ ϕi,N ( N∩Φi,x (z) ), by (i) and (2) of (iii); (c) For each y ∈ X i , we have Φi,− x (y) = {z ∈ Z \ Ui : y ∈ Φi, x (z)} ∪ {z ∈ Ui : y ∈ Φi, x (z)} = {z ∈ Z \ Ui : y ∈ Ai (z) ∩ φi, x (z)} ∪ {z ∈ Ui : y ∈ Ai (z)} − − = [(Z \ Ui ) ∩ A− i (y) ∩ φi, x (y)] ∪ [Ui ∩ A i (y)] = [Ui ∪ φi,− x (y)] ∩ [ A− i (y) ∩ Z ] − − = [Ui ∩ A− i (y)] ∪ [φi, x (y) ∩ A i (y)].

(4)

Φi,− x

is transfer compactly open valued in X i (this is By (i) and (1) of (iii), we can show that the same as in Theorem 3.1); (d) For each N ∈ {x ∈ Z : Q i (x) = ∅}, by (4),    − − [Ui ∩ A− Φi,− x (y) = (y)] ∪ [ϕ (y) ∩ A (y)] i i, x i x∈N

x∈N

= [Ui ∩

 A− i (y)] ∪

A− i (y) ∩

Hence, by (3) of (iii) and (i), we have that Xi .



 ϕi,− x (y)

.

x∈N

x∈N

Φi, x is transfer compactly open valued in

Therefore for each i ∈ I , Q i is a generalized L Fc -majorized mapping. compact subset K of Z and By condition (iv) and the definition of Q i , there exists a nonempty   M ∈ Z  such that Z \ K ⊂ y∈M cint((Ai ∩ Pi )− (y)) ⊂ y∈M cint(Q − i (y)). All assumptions ˆ = ∅ for all i ∈ I . of Theorem 3.2 are satisfied. Hence, there exists a point xˆ ∈ K such that Q i (x) By the definition of Q i , we must have that for each i ∈ I , xˆi ∈ B¯ i (x) ˆ and Ai (x) ˆ ∩ Pi (x) ˆ = ∅. 

Remark 3.3. Theorem 3.3 generalized Theorem 4.3 of Ding [5], Theorem 4.3 of Chowdhury [9] and Theorem 2.3 of Yuan [10]. Acknowledgements The first author was supported by the National Natural Science Foundation of China (10471113) and by Natural Science Foundation Project of CQ CSTC (2005BB2097). References [1] C. Horvath, Some results on multivalued mappings and inequalities without convexity, in: B.L. Lin, S. Simons (Eds.), Nonlinear and Convex Analysis, in: Lecture Notes in Pure and Appl. Math. Series, vol. 107, Marcel Dekker, New York, 1987, pp. 99–106.

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