# Fixed Points, Maximal Elements and Equilibria of Generalized Games in Abstract Convex Spaces

Authors: Yan-Mei Du
Journal: Abstract and Applied Analysis
 (2012)

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## Abstract

We firstly prove some new fixed point theorems for set-valued mappings in noncompact abstract convex space. Next, two existence theorems of maximal elements for class of𝒜C,θ mapping and 𝒜C,θ-majorized mapping are obtained. As in applications, we establish new equilibria existence theorems for qualitative games and generalized games. Our theorems improve and generalize the most known results in recent literature.


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## Body

## 1. Introduction

Since Borglin and Keiding [1] proved a new existence theorem for a compact generalized games (=abstract economy) with KF-majorized preference correspondences. Following their ideas, many authors studied the existence of equilibria for generalized games, for example; see Ding [2], Shen [3], Chowdhury et al. [4], Briec and Horvath [5], Ding and Wang [6], Kim et al. [7], Kim and Tan [8], Lin et al. [9], Lin and Liu [10], Du and Deng [11], and so forth. In the setting, convexity assumptions play a crucial role. Since Horvath [12, 13] 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 [14] and FC-space [15]. As a result, many authors established existence theorems of maximal elements and equilibria of generalized games with majorized correspondences in H-space, G-convex space, and FC-space, respectively. For example, see Hou [16], Ding [17], Ding and Xia [18], Yang and Deng [19], Ding and Feng [20], and others.In 2006, Park [21] introduce the concept of an abstract convex space, which is a topological space without any convexity structure and linear structure. Moreover, abstract convex space include topological vector spaces, H-space, G-convex space, and FC-space as special cases (see Park [21–23]). Abstract convex space will be the framework of this paper.In this paper, we will introduce the new class of𝒜C,θ mapping and 𝒜C,θ-majorized mapping in abstract convex space. Some new fixed point theorems for set-valued mappings are proved under very weak coercive conditions. Next, two existence theorems of maximal elements for class of 𝒜C,θ mapping and 𝒜C,θ-majorized mapping are obtained. As in applications, we establish new equilibria existence theorems for qualitative games and generalized games. Our results generalized and improve the corresponding results due to Ding and Feng [20], Ding and Wang [6], Park [22, 24], Yuan [25], Chowdhury et al. [4], Tan and Yuan [26], Borglin-Keiding [1], Yannelis [27], and so forth.


## 2. Preliminaries

LetX be a nonempty subset of topological space E. We shall denote by 2X the family of all subsets of X, by 〈X〉 the family of all nonempty finite subsets of X, by intE(X), the interior of X in E, and by clE(X) the closure of X in E.IfY is a topological space and T,S:X→2Y are two mappings, for any D⊂X and y∈Y, let S(D)=∪x∈DS(x) and S-(y)={x∈X:y∈S(x)}. The dom S denotes the domain of S, that is, dom S={x∈X:S(x)≠∅}, and T∩S:X→2Y is a mapping defined by (T∩S)(x)=T(x)∩S(x) for each x∈X. The graph of T is the set Gr(T)={(x,y)∈X×Y:y∈T(x)}, and the mapping T-:X→2Y is defined by T-(x)={y∈Y:(x,y)∈clX×Y(Gr(T))}. The mapping clT:X→2Y is defined by (clT)(x)=clY(T(x)) for each x∈X.A subsetA of E is said to be compactly open (resp., compactly closed) in E if for each nonempty compact subset C of E, A∩C is open (resp., closed) in E. Ding [28] define the compact interior and the compact closure of A denoted by cint(A) and ccl(A) as
(2.1)cint(A)=⋃‍{B⊂E:B⊂AandBiscompactlyopeninE},ccl(A)=⋂‍{B⊂E:A⊂BandBiscompactlyclosedinE}.
It is easy to see that for any nonempty compact subset K of E, we have cint(A)∩K=intK(A∩K), ccl(A)∩K=clK(A∩K), and hence cint(A) (resp., ccl(A)) is compactly open (resp., compactly closed) in E. By the definitions, a subset A of E is compactly open (resp., compactly closed) in E if and only if cint(A)=A (resp., ccl(A)=A).Definition 2.1 (see [21, 22]).
An abstract convex space(E,D;Γ) consists of a topological space E, a nonempty set D, and a mapping Γ:〈D〉→2E with nonempty values ΓA=Γ(A) for each A∈〈D〉.For anyD′⊂D, the Γ-convex hull of D′ is denoted and defined by
(2.2)coΓD′=∪{ΓA∣A∈〈D′〉}⊂E.A subsetX of E is called a Γ-convex subset of (E,D;Γ) relative to D′ if for any N∈〈D′〉, we have ΓN⊂X, that is, coΓD′⊂X. Then, (X,D′;Γ∣〈D′〉) is called a Γ-convex subspace of (E,D;Γ).WhenD⊂E, the space is denoted by (E⊃D;Γ). In such a case, a subset X of E is said to be Γ-convex if coΓ(X∩D)⊂X; in other words, X is Γ-convex relative to D′=X∩D. If E=D, let (E;Γ)=(E,E;Γ).Definition 2.2 (see [22]).
Let(E,D;Γ) be an abstract convex space. If a mapping G:D→2E satisfies
(2.3)ΓA⊂G(A)=⋃y∈A‍G(y)∀A∈〈D〉,
Then, G is called a KKM mapping.Definition 2.3 (see [22]).
The partial KKM principle for an abstract convex space(E,D;Γ) is the statement that, for any closed-valued KKM mapping G:D→2E, the family {G(y)}y∈D has the finite intersection property.Definition 2.4.
LetX be a topological space, and Y be a nonempty subset of an abstract convex space (E;Γ). Let θ:X→Y be a single-valued mapping. Then, the mappings ψ,ϕ:X→2Y are said to be an 𝒜C,θ-pair if(a)
for eachx∈X, θ(x)∉coΓ(ϕ(x))⊂Y and ψ(x)⊂ϕ(x),(b)
the mappingψ-:Y→2X is compactly open valued on Y.Definition 2.5.
LetX be a topological space, and Y be a nonempty subset of an abstract convex space (E;Γ). Let θ:X→Y be a single-valued mapping, and P:X→2Y be a set-valued mapping. Then,(i)
P is said to be of class 𝒜C,θ if there exists an 𝒜C,θ-pair such that
(a)
domP⊂domψ,(b)
for eachx∈X, P(x)⊂ϕ(x).(ii)
(ψx,ϕx;Nx) is said to be an 𝒜C,θ-majorant of P at x∈X if Nx is an open neighborhood of x in X and the mapping ϕx,ψx:X→2Y such that
(a)
for eachz∈Nx, P(z)⊂ϕx(z), θ(z)∉coΓ(ϕx(z)) and {z∈Nx:P(z)≠∅}⊂{z∈Nx:ψx(z)≠∅},(b)
for eachz∈X, ψx(z)⊂ϕx(z) and coΓ(ϕx(z))⊂Y,(c)
the mappingψx-:Y→2X is compactly open valued on Y.(iii)
P is said to be an 𝒜C,θ-majorized mapping if for each x∈X with P(x)≠∅, there exists an 𝒜C,θ-majorant (ψx,ϕx;Nx) of P at x, and for any nonempty finite subset A∈〈domP〉, the set {z∈∩x∈ANx:P(z)≠∅}⊂{z∈∩x∈ANx:ψx(z)≠∅}.Remark 2.6.
We note that our notions of the mappingP being of class 𝒜C,θ (resp., 𝒜C,θ-majorized) improve notions of mapping of class ℒθ,F* (resp., ℒθ,F*-majorized), respectively introduced by Ding and Feng [20] from FC-space to abstract convex space, which in turn generalize the corresponding notions in Ding and Wang [6], Chowdhury et al. [4], Yuan [25], Ding et al. [29], and Ding [30].In this paper, we shall deal mainly with either the case (I)X=Y, and X is an abstract convex space, and θ=IX, which is the identity mapping on X or the case (II) X=Πi∈IXi, and θ=πi:X→Xi is the projection of X onto Xi and Xi is an abstract convex space. In both cases (I) and (II), we shall write 𝒜C in place of 𝒜C,θ.Lemma 2.7 (see, [24]).
Let(E,D;Γ) be an abstract convex space and (X,D′;Γ′) be a subspace. If (E,D;Γ) satisfies the partial KKM principle, then so does (X,D′;Γ′).Lemma 2.8 (see, [22]).
Let(E,D;Γ) be an abstract convex space satisfying the partial KKM principle, and S:D→2E be a mapping such that:(i)
for eachz∈X, S(z) is open,(ii)
E=⋃z∈M‍S(z) for some M∈〈D〉.then, there exists an N∈〈D〉 such that ΓN∩⋂z∈N‍S(z)≠∅.


## 3. Fixed Point Theorems

Theorem 3.1.
Let(E,D;Γ) be an abstract convex space satisfying the partial KKM principle, and K be a nonempty compact subset of E. Suppose that F:E→2D, G:E→2E be mappings such that(i)
F(x)⊂G(x) for each x∈E;(ii)
for eachy∈D, F-(y) is compactly open in E and for each x∈K, F(x)≠∅,(iii)
for eachN∈〈D〉, there exists a compact abstract convex subset LN of E containing N such that
(3.1)LN∖K⊂⋃‍{cint(G-(y)):y∈LN}.Then, there exists a point x^∈E, such that x^∈coΓ(G(x^)).Proof.
Since for eachx∈K, F(x)≠∅, then K⊂∪{F-(y):y∈D}. By (i) and (ii), then K⊂{cint(G-(y)):y∈D}. Since K is a nonempty compact subset of E, there exists a finite set N∈〈D〉 such that
(3.2)K⊂⋃‍{cint(G-(y)):y∈N}.
By (iii), there exists a compact abstract convex subset LN of E containing N such that
(3.3)LN∖K⊂⋃‍{cint(G-(y)):y∈LN}.
By (3.2) and N⊂LN, then
(3.4)LN∩K⊂⋃‍{cint(G-(y)):y∈N}⊂⋃‍{cint(G-(y)):y∈LN}.
By (3.3) and (3.4), we have
(3.5)LN⊂⋃‍{cint(G-(y)):y∈LN}.
Since LN is compact, there exists a finite set B={z1,z2,…,zm}∈〈LN〉, such that
(3.6)LN=⋃i=1m‍cint(G-(zi))∩LN=⋃i=1m‍intLN(G-(zi)∩LN).
Let D′=LN∩D and define Γ′:〈D′〉→2LN by ΓA′=ΓA∩LN for each A∈〈D′〉, then (LN,D′;Γ′) is an abstract convex space. Since (E,D;Γ) satisfies the partial KKM principle, so does (LN,D′;Γ′) by Lemma 2.7.
DefineS:LN→2LN by S(z)=cint(G-(z))∩LN for each z∈LN. It is easy to prove that the all the hypotheses of Lemma 2.8 are satisfied. By Lemma 2.8, there exists a finite set M∈〈LN〉 such that ΓM′∩⋂z∈M‍S(z)≠∅. Let x^∈ΓM′∩⋂z∈M‍S(z), then for each z∈M, x^∈S(z)=cint(G-(z))∩LN⊂G-(z), that is M⊂G(x^) and ΓM⊂coΓ(G(x^)). Since x^∈ΓM′=ΓM∩LN⊂ΓM, thus x^∈coΓ(G(x^)). This completes the proof.Remark 3.2.
Theorem3.1 generalizes Theorem 3.1 of Ding and Feng [20] and Theorem 3.1 of Ding and Wang [6] from FC-space to abstract convex space, and the coercive condition (iii) in Theorem 2.1 is weaker than the condition (3) in Theorem 3.1 of Ding and Wang [6].Corollary 3.3.
Let(E,D;Γ) be an abstract convex space satisfying the partial KKM principle, and K be a nonempty compact subset of E. Suppose that F:E→2D, G:E→2E be mappings such that(i)
F(x)⊂G(x) for each x∈E,(ii)
for eachy∈D, F-(y) is compactly open in E and for each x∈K, F(x)≠∅,(iii)
for eachN∈〈D〉, there exists a compact abstract convex subset LN of E containing N such that for each x∈LN∖K, there exists a point y-∈LN such that x∉clX[X∖G-(y-)].Then, there exists a point x^∈E, such that x^∈coΓ(G(x^)).Proof.
By (iii), for eachN∈〈D〉, there exists a compact abstract convex subset LN of E containing N such that for each x∈LN∖K, there exists y-∈LN such that x∉clX[X∖G-(y-)], then x∈(clX[X∖G-(y-)])c= int(G-(y-)), thus x∈∪{int(G-(y)):y∈LN}⊂{cint(G-(y)):y∈LN}, that is LN∖K⊂{cint(G-(y)):y∈LN}. Hence, all the hypotheses of Theorem 3.1 are satisfied. By Theorem 3.1, there exists a point x^∈E, such that x^∈coΓ(G(x^)). This completes the proof.Remark 3.4.
Corollary3.3 generalizes Theorem 3.2 of Ding and Feng [20] from FC-space to abstract convex space. Moreover, Corollary 3.3 improves the corresponding result of Park [22, 24].


## 4. Existence of Maximal Elements

LetX be a topological space, and P:X→2X be a mapping. A point x^∈X is called a maximal element of T if T(x^)=∅.In the section, We firstly prove a selection theorem for𝒜C-majorized mapping. Next, we will establish some new existence theorems of maximal elements for class 𝒜C mapping and 𝒜C-majorized mapping defined on noncompact abstract convex space.Lemma 4.1.
LetX be a regular topological space, and Y be a nonempty subset of an abstract convex space (E;Γ). Let θ:X→E, and P:X→2Y be an 𝒜C,θ-majorized mapping. If each open subset of X containing the set B=domP is paracompact, then there exists a 𝒜C,θ-pair ψ,ϕ:X→2Y such that P(x)⊂ϕ(x) for each x∈X and domP⊂domψ.Proof.
SinceP is an 𝒜C,θ-majorized mapping, for each x∈B, let Nx be an open neighborhood of x in X, and ψx,ϕx:X→2Y be mappings such that(1)
for eachz∈Nx, P(z)⊂ϕx(z), θ(z)∉coΓ(ϕx(z)) and {z∈Nx:P(z)≠∅}⊂{z∈Nx:ψx(z)≠∅},(2)
for eachz∈X, ψx(z)⊂ϕx(z) and coΓ(ϕx(z))⊂Y,(3)
the mappingψx-:Y→2X is compactly open valued on Y,(4)
for eachA∈〈domP〉, {z∈∩x∈ANx:P(z)≠∅}⊂{z∈∩x∈ANx:∩x∈Aψx(z)≠∅}.
SinceX is regular, for each x∈B, there exists an open neighborhood Gx of x in X such that clXGx⊂Nx. Let G=∪x∈BGx, then G is an open subset of X containing B, so that G is paracompact by assumption. By Theorem VIII.1.4 of Dugundji [31], the open covering {Gx} of G has an open precise neighborhood finite refinement Gx′. Given any x∈B, we define the mappings ψx′,ϕx′:G→2Y by
(4.1)ψx′(z)={ψx(z)ifz∈G∩clXGx′,Yifz∈G∖clXGx′,ϕx′(z)={ϕx(z)ifz∈G∩clXGx′,Yifz∈G∖clXGx′,
then we have(i)
by (2), for eachz∈G, ψx′(z)⊂ϕx′(z),(ii)
by (1),domP⊂domψx′, and P(z)⊂ϕx′(z) for each z∈G,(iii)
for eachy∈Y, the set
(4.2)(ψx′)-(y)={z∈G∩clXGx′:y∈ψx(z)}∪{z∈G∖clXGx′:y∈Y}=[(G∩clXGx′)∩ψx-(y)]∪(G∖clXGx′)=[G∩ψx-(y)]∪(G∖clXGx′).It follows that for each nonempty compact subset C of X, the set
(4.3)(ψx′)-(y)∩C=[G∩ψx-(y)∩C]∪[(G∖clXGx′)∩C]
is open in C by (3). Thus, the mapping (ψx′)-:Y→2G is compactly open valued on Y. Now, define ψ,ϕ:X→2Y by
(4.4)ψ(z)={∩x∈Bψx′(z)ifz∈G,∅ifz∈X∖G,ϕ(z)={∩x∈Bϕx′(z)ifz∈G,∅ifz∈X∖G.(a)
For eachz∈X, by (i), ψ(z)⊂ϕ(z) and coΓ(ϕ(z))⊂Y. If z∈X∖G, then ϕ(z)=∅, so that θ(z)∉coΓ(ϕ(z)); if z∈G, then z∈G∩clXGx′ for some x∈B, so that ϕx′(z)=ϕx(z), and hence ϕ(z)⊂ϕx(z), by (1), we have θ(z)∉coΓ(ϕ(z)). Therefore, θ(z)∉coΓ(ϕ(z)) for all z∈X.(b)
Now, we show that the mappingψ-:Y→2X is compactly open valued on Y. Indeed, let y∈Y be such that ψ-(y)≠∅ and C be a nonempty compact subset of X. Given a point u,
(4.5)u∈ψ-(y)∩C={z∈X:y∈ψ(z)}∩C={z∈G:y∈ψ(z)}∩C.Since {Gx′} is a neighborhood finite refinement, there exists an open neighborhood Mu of u in G such that {x∈B:Mu∩Gx′≠∅}={x1,x2,…,xn}. Note that for each x∈B with x∉{x1,x2,…,xn}, ∅=Mu∩Gx′=Mu∩clXGx′, so that ψx′(z)=Y for z∈Mu. Then, we have ψ(z)=∩x∈Bψx′(z)=∩i=1nψxi′(z) for all z∈Mu. It follows that
(4.6)ψ-(y)={z∈X:y∈ψ(z)}={z∈G:y∈∩x∈Bψx′(z)}⊃{z∈Mu:y∈∩x∈Bψx′(z)}={z∈Mu:y∈∩i=1nψxi′(z)}=Mu∩{z∈G:y∈∩i=1nψxi′(z)}=Mu∩[∩i=1n(ψxi′)-(y)].
Since (ψxi′)-(y) is compactly open in X by (iii), then Mu′=Mu∩[∩i=1n(ψxi′)-(y)]∩C is an open neighborhood of u in C such that Mu′⊂ψ-(y)∩C. This shows that ψ-:Y→2X is compactly open valued on Y.
By (a) and (b), thus(ψ,ϕ) is an 𝒜C,θ-pair.
Next, we claim that domP⊂ dom ψ, indeed, for each w∈domP, we must have w∈G. Since {Gx′} is neighborhood finite, then the set {x∈B:w∈clXGx′}={x1′,x2′,…,xm′}. If x∉{x1′,x2′,…,xm′}, w∈G∖clXGx′, and ψx′(w)=Y, thus we have ψ(w)=∩x∈Bψx′(w)=∩i=1mψxi′′(w). Since w∈∩i=1mclXGxi′′⊂∩i=1mNxi′, by (4), ψ(w)≠∅. Hence, domP⊂domψ.
Finally, we prove thatP(z)⊂ϕ(z) for each z∈X. Indeed, let z∈X with P(z)≠∅, then z∈G. For each x∈B, if z∈G∖clXGx′, then ϕx′(z)=Y, and so P(z)⊂ϕx′(z), and if z∈G∩clXGx′, we have z∈clXGx′⊂clXGx⊂Nx, so that by (1), P(z)⊂ϕx(z)⊂ϕx′(z). It follows that P(z)⊂ϕx′(z) for all x∈B so that P(z)⊂∩x∈Bϕx′(z)=ϕ(z). This completes the proof.Remark 4.2.
Lemma4.1 generalizes Lemma 4.1 of Ding and Feng [20], Lemma  5.1 of Ding and Wang [15], Theorem  3.1 of Yuan [25], Lemma 3.1 of Chowdhury et al. [4], and Lemma 3.1 of Tan and Yuan [26].Theorem 4.3.
Let(X;Γ) be an abstract convex space satisfying the partial KKM principle, and let K be a nonempty compact subset of X. Suppose that the mapping P:X→2X is of class 𝒜C and satisfy(i)
for eachN∈〈X〉, there exists a compact abstract convex subset LN of X containing N such that
(4.7)LN∖K⊂⋃‍{cint(P-(y)):y∈LN}.Then, there exists a point x^∈K such that P(x^)=∅.Proof.
SinceP is of class 𝒜C, then there exists an 𝒜C-pair (ϕ,ψ) such that(a)
domP⊂domψ, P(x)⊂ϕ(x) for each x∈X,(b)
for eachx∈X, x∉coΓ(ϕ(x))⊂X and ψ(x)⊂ϕ(x),(c)
the mappingψ-:X→2X is compactly open valued on X.
Suppose that  for eachx∈K, P(x)≠∅. By (a), ψ(x)≠∅ for each x∈K. By (i) and (a), for each N∈〈X〉, there exists a compact abstract convex subset LN of X containing N such that LN∖K⊂∪{cint(P-(y)):y∈LN}⊂∪{cint(ϕ-(y)):y∈LN}. Therefore, ψ and ϕ satisfy all the hypotheses of Theorem 3.1. By Theorem 3.1, there exists a point x-∈K such that x-∈coΓ(ϕ(x-)). Which contradicts with condition (b). Hence, there exists a point x^∈K such that P(x^)=∅. This completes the proof.Remark 4.4.
Theorem4.3 generalizes most existence theorems of maximal elements in the literature, for example; see Theorem  4.1 of Ding and Feng [20], Theorem  3.2 of Yuan [25], Theorem  3.1 of Chowdhury et al. [4], Theorem  3.2 of Tan and Yuan [26], Theorem  5.2 of Ding and Wang [6], and so on.As an application of Lemma4.1 and Theorem 4.3, we have the following existence theorem maximal elements.Theorem 4.5.
Let(X;Γ) be a paracompact abstract convex space satisfying the partial KKM principle, and let K be a nonempty compact subset of X. Let P:X→2X be an 𝒜C-majorized mapping and satisfy(i)
for eachN∈〈X〉, there exists a compact abstract convex subset LN of X containing N such that
(4.8)LN∖K⊂⋃‍{cint(P-(y)):y∈LN}.Then, there exists a point x^∈K such that P(x^)=∅.Proof.
Suppose thatP(x)≠∅ for all x∈X, then domP=X is paracompact. By Lemma 4.1   P is of class 𝒜C. Therefore, all the hypotheses of Theorem 4.3 are satisfied. By Theorem 4.3, there exists a point x^∈K such that P(x^)=∅. Which is a contradiction. Thus, there exists a point x^∈X such that P(x^)=∅. By the assumptions, x^ must in K. This completes the proof.Remark 4.6.
Theorem4.5 generalizes Theorem  5.3 of Ding and Wang [6], Theorem  3.2 of Chowdhury et al. [4], Theorem  3.3 of Tan and Yuan [26], Corollary 1 of Borglin-Keiding [1], and Theorem  2 of Yannelis [27].


## 5. Existence of Equilibrium of Points

LetI be a (finite or infinite) set of players. For each i∈I, let its strategy set X and Yi(i∈I) be nonempty set, and let Y=∏i∈I‍Yi. Pi:X→2Yi be the preference correspondence of i-th player. The collection Λ=(X;Yi;Pi)i∈I will be called a qualitative game. A point x^∈X is said to be an equilibrium of the qualitative game, if Pi(x^)=∅ for each i∈I.A generalized game is a quintuple familyΛ=(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∈I‍Yi. 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 x^∈X such that for each i∈I,θi(x^)∈B-i(x^) and Ai(x^)∩Pi(x^)=∅. If θi=πi:X→Xi is the projection of X onto Xi, then our definition of an equilibrium point coincides with the standard definition given by Chowdhury et al. [4], and if, in addition, Ai=Bi for each i∈I, our definition of an equilibrium point generalizes the standard definition, for example, Borglin and Keiding [1] and Gale and Mas-Colell [32].Lemma 5.1 (see [33]).
Let{(Xi,Di;Γi)}i∈I be a family of abstract convex spaces. Let X:=Πi∈IXi be equipped with the product topology, and let D:=Πi∈IDi. For each i∈I, let πi:D→Di be the projection. For each A∈〈D〉, define Γ(A):=Πi∈IΓi(πi(A)). Then, (X,D;Γ) is an abstract convex space.As an application of Theorem4.3, we prove the following existence theorem of equilibrium points for one person game in abstract convex space.Theorem 5.2.
Let(X;Γ) be an abstract convex space satisfying the partial KKM principle and K be a closed and compact subset of X, and let K=coΓK. Suppose that the mappings A,B,andP:X→2X satisfy(i)
for eachx∈X, coΓ(A(x))⊂B¯(x) and A(x)∩P(x)⊂K,(ii)
the mappingA-:X→2X is compactly open valued on X,(iii)
the mappingA∩P:X→2X is of class 𝒜C and A(x)≠∅ for each x∈K,(iv)
for eachN∈〈X〉, there exists a compact abstract convex subset LN of X containing N such that
(5.1)LN∖K⊂⋃‍{cint((A∩P)-(y)):y∈LN}.Then, there exists a point x^∈K such that x^∈B¯(x^) and A(x^)∩P(x^)=∅.Proof.
LetW={x∈X:x∉B¯(x)}, then W is open in X. Define Q:X→2X by
(5.2)Q(x)={Kifx∈X∖K,A(x)ifx∈K∩W,A(x)∩P(x)ifx∈K∖W.
By (iii), A∩P is of class 𝒜C, then there exist two mappings ψ,ϕ:X→2X such that(a)
for eachx∈X, A(x)∩P(x)⊂ϕ(x) and dom (A∩P)⊂domψ,(b)
for eachx∈X, x∉coΓ(ϕ(x)) and ψ(x)⊂ϕ(x),(c)
the mappingψ-:X→2X is compactly open valued on X.Define Ψ,Φ:X→2X by
(5.3)Ψ(x)={Kifx∈X∖K,A(x)ifx∈K∩W,ψ(x)ifx∈K∖W,Φ(x)={Kifx∈X∖K,A(x)ifx∈K∩W,ϕ(x)ifx∈K∖W.
Then, we have(a′)
for eachx∈X, Q(x)⊂Φ(x) by (a) and dom Q⊂domΨ,(b′)
for eachx∈X, Ψ(x)⊂Φ(x) by (b), if x∈X∖K, x∉K=coΓK=coΓΦ(x) and if x∈K∩W, x∉B¯(x) by (i), and x∉coΓ(A(x))=coΓ(Φ(x)) and if x∈K∖W, x∉coΓ(ϕ(x))=coΓ(Φ(x)) by (b), thus x∉coΓ(Φ(x)) for each x∈X,(c′)
for eachy∈X, then it is easy to verify that the set
(5.4)Ψ-(y)={x∈X:y∈Ψ(x)}=(X∖K)∪[A-(y)∩K∩W]∪[ψ-(y)∩(K∖W)]is compactly open in X by (ii) and (c). This shows that Q is of class 𝒜C. By the definition of Q, condition (i) and (iv), for each N∈〈X〉, there exists a compact abstract convex subset LN of X containing N such that
(5.5)LN∖K⊂⋃‍{cint((A∩P)-(y)):y∈LN}⊂⋃‍{cint(Q-(y)):y∈LN}.
Hence, all the hypotheses of Theorem 4.3 are satisfied. By Theorem 4.3, there exists a point x^∈K such that Q(x^)=∅. By the definition of Q and condition (iii), x^∈K∖W⊂K, that is, x^∈B¯(x^) and A(x^)∩P(x^)=∅. This completes the proof.Remark 5.3.
Theorem5.2 improves and generalizes Theorem 5.1 of Ding and Feng [20], Theorem 6.1 of Ding and Wang [6], Theorem 4.1 of Yuan [25], Theorem 4.1 of Chowdhury et al. [4], and Theorem 4.1 of Tan and Yuan [26].As another application of Theorem4.5, we can obtain the following existence of equilibria for qualitative games.Theorem 5.4.
LetΛ=(X;Xi;Pi)i∈I be a qualitative game, For each i∈I, suppose that  the following conditions are satisfied(i)
(Xi;Γi)i∈I is a family of paracompact abstract convex space such that (X;Γ) satisfies the partial KKM principle, and K is a nonempty compact subset of X,(ii)
Pi:X→2Xi is an 𝒜C-majorized mapping,(iii)
Wi={x∈X:Pi(x)≠∅} is open in X,(iv)
for eachN∈〈X〉, there exists a compact abstract convex subset LN of X containing N such that
(5.6)LN∖K⊂⋃‍{cint(Pi-(πi(y))):y∈LN}.Then, Λ has an equilibrium point in K.Proof.
For eachx∈X, let I(x)={i∈I:Pi(x)≠∅}. Define a mapping Pi′:X→2X by Pi′(x)=πi-(Pi(x))=Πj∈I,j≠iXi⊗Pi(x) for each x∈X, where the mapping πi:X→Xi is projection of X onto Xi. Furthermore, define the mapping P:X→2X by
(5.7)P(x)={∩i∈I(x)Pi′(x)ifI(x)≠∅,∅ifI(x)=∅.
Then, for each x∈X, P(x)≠∅ if and only if I(x)≠∅. We will show that P is an 𝒜C-majorized mapping. For each x∈X with P(x)≠∅, let i∈I(x) with Pi(x)≠∅, by (ii), and let Nx be an open neighborhood of x in X, and ψi,x,ϕi,x:X→2Xi be mappings such that(a)
for eachz∈Nx, Pi(z)⊂ϕi,x(z), zi∉coΓ(ϕi,x(z)), and {z∈Nx:Pi(z)≠∅}⊂{z∈Nx:ψi,x(z)≠∅},(b)
for eachz∈X, ψi,x(z)⊂ϕi,x(z) and coΓ(ϕi,x(z))⊂Xi,(c)
the mappingψi,x-:Xi→2X is compactly open valued on Xi,(d)
for each finite subsetA∈〈domPi〉, {z∈∩x∈ANx:Pi(z)≠∅}⊂{z∈∩x∈ANx:∩x∈Aψi,x(z)≠∅}.By (iii), we may assume that Nx⊂Wi, hence Pi(z)≠∅ and i∈I(z) for all z∈Nx. Now, define two mappings ψx,ϕx:X→2X by
(5.8)ψx(z)=πi-(ψi,x(z)),ϕx(z)=πi-(ϕi,x(z))foreachz∈X.
Then, we have(a')
for eachz∈Nx, by (a), P(z)=∩i∈I(z)Pi′(z)⊂Pi′(z)=πi-(Pi(z))⊂πi-(ϕi,x(z))=ϕx(z) and z∉coΓ(ϕx(z)),(b')
for eachz∈X, ψx(z)⊂ϕx(z) by (b),(c')
for eachy∈X, ψx-(y)={z∈X:y∈ψx(z)}={z∈X:yi∈ψi,x(z)}=ψi,x-(y) is compactly open in X by (c),(d')
for each finite subsetA={x1,x2,…,xn}∈〈domP〉, put ∪{B:B⊂Aand∩x∈BI(x)≠∅}={xi1,xi2,…,xim}. For each i∈∩k=1mI(xik), then
(5.9)∩x∈Aψx(z)=∩x∈Aπi-(ψi,x(z))=Πj≠ik,k=1,…,nXj⊗∩k=1mψi,xik(z)⊗Πk=m+1nψik,xik(z).For each z∈∩x∈ANx, if ∩x∈Aψx(z)=∅, then there exists a set A1={xi1,xi2,…,xim}⊂A, such that ∩x∈A1ψi,x(z)=∅, by (d), Pi(z)=∅. Thus, {z∈∩x∈ANx:P(z)≠∅}⊂{z∈∩x∈ANx:∩x∈Aψx(z)≠∅}. This shows that P is an 𝒜C-majorized mapping. By P-(y)=Pi-(πi(y)) and condition (iv), for each N∈〈X〉, there exists a compact abstract convex subset LN of X containing N such that
(5.10)LN∖K⊂⋃‍{cint(Pi-(πi(y))):y∈LN}=⋃‍{cint(P-(y)):y∈LN}.
Hence, all the hypotheses of Theorem 4.5 are satisfied. By Theorem 4.5, there exists a point x^∈K such that P(x^)=∅. This implies that I(x^)=∅ and therefore Pi(x^)=∅ for each i∈I, that is, x^ is an equilibrium point of Λ.Remark 5.5.
Theorem5.4 improves and generalizes Theorem 5.2 of Ding and Feng [20], Theorem 6.2 of Ding and Wang [6], Theorem 4.2 of Yuan [25], Theorem 4.2 of Chowdhury et al. [4], and Theorem 4.2 of Tan and Yuan [26].Applying Theorem5.4, we prove that the following equilibria existence theorem for a noncompact generalized games.Theorem 5.6.
LetΛ=(X;Xi;Ai;Bi;Pi,π)i∈I be a generalized game. Let K be a compact and closed subset of X and coΓ(πi(K))=πi(K). Suppose that for each i∈I, (i)
(Xi,Γi) is a paracompact abstract convex space such that (X;Γ) satisfies the partial KKM principle,(ii)
for eachx∈X, coΓ(Ai(x))⊂B¯i(x), Ai(x)∩Pi(x)⊂πi(K) and domAi=X,(iii)
for eachy∈Xi, Ai-(y) is compactly open in X,(iv)
Wi={x∈X:(Ai∩Pi)(x)≠∅} is open in X,(v)
Ai∩Pi:X→2Xi is an 𝒜C-majorized mapping,(vi)
for eachN∈〈X〉, there exists a compact abstract convex subset LN of X containing N such that
(5.11)LN∖K⊂⋃‍{cint((Ai∩Pi)-(y)):y∈LN}.Then, Λ has an equilibrium point x^ in K.Proof.
For eachi∈I, let Fi={x∈X:xi∉B¯i(x)}, then Fi is open in X. Define Qi:X→2Xi by
(5.12)Qi(x)={πi(K)ifx∈X∖K,(Ai∩Pi)(x)ifx∈K∖Fi,Ai(x)ifx∈Fi∩K.
We will prove that the qualitative game Λ′=(X;Xi,Qi)i∈I satisfies all the hypotheses of Theorem 5.4. For each i∈I, we have that the set
(5.13){x∈X:Qi(x)≠∅}={x∈X∖K:πi(K)≠∅}∪{x∈K∖Fi:(Ai∩Pi)(x)≠∅}∪{x∈K∩Fi:Ai(x)≠∅}=(X∖K)∪[(K∖Fi)∩Wi]∪(Ki∩Fi)=(X∖K)∪[K∩(Wi∪Fi)]=(X∖K)∪Wi∪Fi
is open in X and hence the condition (iii) of Theorem 5.4 is satisfied. By (v), for each x∈Wi, there exist an open neighborhood Nx of x in X and two mappings ψi,x,ϕi,x:X→2Xi such that(a)
for eachz∈Nx, (Ai∩Pi)(z)⊂ϕi,x(z), zi∉coΓ(ϕi,x(z)) and {z∈Nx:(Ai∩Pi)(z)≠∅}⊂{z∈Nx:ψi,x(z)≠∅},(b)
for eachz∈X, ψi,x(z)⊂ϕi,x(z) and coΓ(ϕi,x(z))⊂Xi,(c)
the mappingψi,x-:Xi→2X is compactly open valued on Xi,(d)
for each nonempty finite subsetA∈〈dom(Ai∩Pi)〉,
(5.14){z∈∩x∈ANx:(Ai∩Pi)(z)≠∅}⊂{z∈∩x∈ANx:∩x∈Aψi,x(z)≠∅}.Define Ψi,x,Φi,x:X→2Xi by
(5.15)Ψi,x(z)={πi(K)ifz∈X∖K,ψi,x(z)ifz∈K∖Fi,Ai(z)ifz∈Fi∩K,Φi,x(z)={πi(K)ifz∈X∖K,ϕi,x(z)ifz∈K∖Fi,Ai(z)ifz∈Fi∩K.
Now for each x∈X with Qi(x)≠∅, the set
(5.16)Ux={X∖Kifx∈X∖K,Nxifx∈K∖Fi,Fiifx∈Fi∩K
is open in X. Then,(a')
for eachz∈Ux, Qi(z)⊂Φi,x(z) by (a) if z∈X∖K, zi∉coΓ(πi(K))=coΓ(Φi,x(z)) and if z∈Nx, z∉coΓ(ϕi,x(z))=coΓ(Φi,x(z)) by (a) if z∈Fi, z∉coΓ(Ai(z))=coΓ(Φi,x(z)) by (ii), that is, z∉coΓ(Φi,x(z)) for each z∈X, and {z∈Ux:Qi(z)≠∅}⊂{z∈Ux:Ψi,x(z)≠∅} by (a),(b')
for eachz∈X, Ψi,x(z)⊂Φi,x(z) by (b) and coΓ(Ψi,x(z))⊂Xi,(c')
for eachy∈Xi,
(5.17)Ψi,x-(y)={z∈X:y∈Ψi,x(z)}={z∈X∖K:y∈πi(K)}∪{z∈K∖Fi:y∈ψi,x(z)}∪{z∈K∩Fi:y∈Ai(z)}=(X∖K)∪[ψi,x-(y)∩(K∩Fi)]∪[Ai-(y)∩K∩Fi]is compactly open by (c) and (iii),(d')
for eachA∈〈domQi〉, put
(5.18)A=[A∩(X∖K)]∪[A∩(K∖Fi)]∪[A∩K∩Fi]=A1∪A2∪A3.
Case I. If A3≠∅, ∩x∈AUx⊂Fi∩K, then
(5.19){z∈∩x∈AUx:∩x∈AΨi,x(z)≠∅}={z∈∩x∈AUx:Ai(z)≠∅}={z∈∩x∈AUx:Qi(z)≠∅}.
Case II. If A3=∅, then (1)
ifA1≠∅,A2≠∅, then ∩x∈AUx=(∩x∈A2Nx)∩(X∖K)⊂X∖K, and
(5.20){z∈∩x∈AUx:∩x∈AΨi,x(z)≠∅}={z∈∩x∈AUx:πi(K)≠∅}={z∈∩x∈AUx:Qi(z)≠∅}.(2)
IfA2=∅,A1≠∅, ∩x∈AUx=X∖K, that is similar to the condition (1).(3)
IfA1=∅,A2≠∅, then
(5.21){z∈∩x∈AUx:∩x∈AΨi,x(z)≠∅}={z∈(∩x∈A2Nx)∩(X∖K):Ψi,x(z)≠∅}∪{z∈(∩x∈A2Nx)∩(K∖Fi):Ψi,x(z)≠∅}∪{z∈(∩x∈A2Nx)∩(K∩Fi):Ψi,x(z)≠∅}={z∈(∩x∈A2Nx)∩(X∖K):πi(K)≠∅}∪{z∈(∩x∈A2Nx)∩(K∖Fi):∩x∈Aψi,x(z)≠∅}∪{z∈(∩x∈A2Nx)∩(K∩Fi):Ai(z)≠∅}⊃{z∈(∩x∈A2Nx)∩(X∖K):πi(K)≠∅}∪{z∈(∩x∈A2Nx)∩(K∖Fi):(Ai∩Pi)(z)≠∅}∪{z∈(∩x∈A2Nx)∩(K∩Fi):Ai(z)≠∅}={z∈∩x∈A2Nx:Qi(z)≠∅}.Thus, Qi is an 𝒜C-majorized mapping. By the definition of Q and condition (vi) for each N∈〈X〉, there exists a compact abstract convex subset LN of X containing N such that
(5.22)LN∖K⊂⋃‍{cint((Ai∩Pi)-(y)):y∈LN}⊂⋃‍{cint(Qi-(y)):y∈LN}.
Hence, all the hypotheses of Theorem 5.4 are satisfied. By Theorem 5.4, there exists a point x^∈K such that Qi(x^)=∅(i∈I). By the definition of Qi, x^∈K∖Fi⊂K, that is for all i∈I, x^i∈B¯i(x^), and Ai(x^)∩Pi(x^)=∅. This completes the proof.Remark 5.7.
Theorem5.6 generalized Theorem  5.3 of Ding and Feng [20], Theorem 6.3 of Ding and Wang [6], Theorem 4.4 of Chowdhury et al. [4], and Theorem  4.3 of Tan and Yuan [26].



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*Source: 842515-2012-12-24.xml*