# On Abstract Economies and Their Applications

Authors: Chien-Hao Huang; Liang-Ju Chu
Journal: Abstract and Applied Analysis
 (2013)

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## Abstract

We establish a new equilibrium existence theorem of
generalized abstract economies with general preference correspondences.
As an application, we derive an existence theorem of generalized quasi-variational
inequalities in the general setting ofl.c.-spaces without any linear structure.


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## Body

## 1. Introduction and Preliminary

LetI be any (finite or infinite) set of agents. A generalized abstract economy is defined as a family of order quintuples Ω=(Xα,Aα,Bα,Fα,Pα)α∈I with X:=∏α∈I‍Xα such that for each α∈I, Xα is a topological space, Aα,Bα:X→2Xα are constraint correspondences, Fα:X→2Xα is a fuzzy constraint correspondence, and Pα:X×X→2Xα is a preference correspondence. In a real market, any preference of a real agent would be unstable by the fuzziness of consumers’ behavior or market situations. Thus, it is reasonable to introduce fuzzy constraint correspondences in defining an abstract economy. An equilibrium point of Ω is a point (x^,y^)∈X×X such that for each α∈I, x^α∈clBα(x^), y^α∈Fα(x^), and Aα(x^)∩Pα(x^,y^)=∅, where x^α and y^α denote the projections of x^ and y^ from X to Xα, respectively.In caseFα(x)=Xα for each x∈X and Pα is independent of the second variable, that is, Pα:X→2Xα, the above generalized abstract economy reduces to the standard abstract economy Ωs∶=(Xα,Aα,Bα,Pα)α∈I, in which an equilibrium point of Ωs is a point x^∈X such that for each α∈I, x^α∈clBα(x^) and Aα(x^)∩Pα(x^)=∅. When Aα=Bα and each Xα is a topological vector space, the standard abstract economy Ωs coincides with the classical definition of Shafer and Sonnenschein [1]. For more details on abstract economies, see, for example, [2–14] and the references therein.Throughout this paper, all topological spaces are assumed to be Hausdorff. In order to establish our main results, we first give some basic notations. For a nonempty setC of a topological space X, we denote the set of all subsets of C by 2C, the set of all nonempty finite subsets of C by 〈X〉, the interior of C by intC, and the closure of C by clC.Let{ΓD} be a family of some nonempty contractible subsets of a topological space X indexed by D∈〈X〉 such that ΓD⊂ΓD′ whenever D⊂D′. The pair (X,{ΓD}) is called an H-space. Given an H-space (X,{ΓD}), a nonempty subset C of X is said to be H-convex if ΓD⊆C for all D∈〈C〉. For a nonempty subset C of X, we define the H-convex hull of C as
(1)H-coC:=⋂‍{K∣KisH-convexinXandC⊆K}.It is known that ifx∈H-coC, then there exists a finit subset D of C such that x∈H-coD. Moreover, for any D∈〈X〉, H-coD is called a polytope. We will say that X is an H-space with precompact polytopes if any polytope of X is precompact. For example, a locally convex topological vector space X is an H-space with precompact polytopes, by setting ΓD=coD for all D∈〈X〉.AnH-space (X,{ΓD}) is called an l.c.-space if X is a uniform space whose topology is induced by its uniformity 𝒰, and there is a base ℬ consisting of symmetric entourages in 𝒰 such that for each V∈ℬ, the set V(E):={y∈X∣(x,y)∈V for some x∈E} is H-convex whenever E is H-convex. We will use the notation (X,𝒰,ℬ) to stand for an l.c.-space. For details of uniform spaces, we refer to [15]. In a recent paper [16], we introduce a new measure of precompactness of a subset A in an l.c.-space (X,𝒰,ℬ) by
(2)Q(A)∶={V∈ℬ∣A⊆cl(V(K))0forsomeprecompactsetKofX}.Let(Xα,𝒰α,ℬα)α∈I be a family of l.c.-spaces with precompact polytopes, where I is a finite or infinite index set and  X=∏α∈I‍Xα. For each α∈I, let πα be the projection of X onto Xα and Qα a measure of precompactness in Xα. We say that a set-valued mapping Tα:X→2Xα is Qα-condensing if Qα(πα(C))⊊Qα(Tα(C)) for every C satisfying πα(C) is a nonprecompact subset of Xα. It is clear that for any set-valued mapping T:X→2Y and any measure Q in Y, T is Q-condensing whenever Y is compact.LetX be a topological space, let Y be an H-space, and let S,T:X→2Y be two set-valued mappings.(1)
T is said to be upper semicontinuous(u.s.c.) if for each x∈X and each open subset V of Y with T(x)⊆V, there exists a neighborhood Nx of x such that T(z)⊆V for all z∈Nx.(2)
T is said to be transfer open valued on X if for each x∈X, for each y∈T(x), there exists some x′∈X such that y∈intT(x′).(3)
T is said to be transfer open inverse valued in Y if T-1 is transfer open valued on Y, where T-1:Y→2X is defined by
(3)T-1(y)∶={x∈X∣y∈T(x)}∀y∈Y.(4)
The set-valued mappingsS∩T:X→2Y and clT:X→2Y are defined by
(4)0000(S∩T)(x)∶=S(x)∩T(x),clT(x)∶=cl(T(x)),∀x∈X.Further, we denote byℍ(X,Y) the class of all u.s.c. set-valued mappings T:X→2Y with nonempty closed H-convex values.


## 2. Main Results

The following fundamental theorems will play an important role in proving our main theorem.Theorem A (see [16]).
Let(Xα,𝒰α,ℬα)α∈I be a family of l.c.-spaces with precompact polytopes, X∶=∏α∈I‍Xα, and let Tα:X→2Xα be Qα-condensing. Then there exists a nonempty compact H-convex subset K∶=∏α∈I‍Kα of  X such that Tα(K)⊆Kα.Theorem B (see [16]).
Let(Xα,𝒰α,ℬα)α∈I be a family of l.c.-spaces with precompact polytopes and X∶=∏α∈I‍Xα. If Tα:X→2Xα is an u.s.c.Qα-condensing mapping with closed H-convex values for each α∈I, then T∶=∏α∈I‍Tα has a fixed point.Next, we list and establish some essential lemmas as follows.Lemma 1 (see [12]).
IfX is an l.c.-space and E is an H-convex subset of X, then  clE is also H-convex.Lemma 2 (see [12]).
LetX be a topological space and let (Y,{ΓD}) be a compact l.c.-space. If T:X→2Y is an u.s.c. set-valued mapping, then the mapping x↦cl[H-coT(x)] is also u.s.c. with compact H-convex values.Lemma 3 (see [7]).
LetX and Y be topological spaces and let S:X→2Y be a transfer open valued mapping. Then ⋃x∈X‍S(x)=Y∖(∩x∈Xcl(Y∖S(x))) and hence ⋃x∈X‍S(x) is open in Y.Lemma 4.
LetX be paracompact, (Y,{ΓD}) an H-space, and S,T:X→2Y be two set-valued mappings such that (1)
S(x)≠∅ and H-coS(x)⊆T(x) for each x∈X,(2)
S is transfer open inverse valued in Y. Then T has a continuous selection; that is, there exists a continuous function f:X→Y such that f(x)∈T(x) for each x∈X.Proof.
Since for eachx∈X, S(x)≠∅, it follows that x∈S-1(y) for some y∈Y. Since S is transfer open inverse valued in Y, there exists some yx∈Y such that x∈intS-1(yx). This yields that {intS-1(y)∣y∈Y} forms an open cover of X. Since X is paracompact, there exists a locally finite open cover {Uy∣y∈Y} such that Uy⊆intS-1(y) for each y∈Y. By [17, Theorem 3.1], there exists a continuous function f:X→Y such that f(x)∈Γ{y∈Y∣x∈Uy} for all x∈X. Note that for any x∈X, there exist finitely many y∈Y such that x∈Uy⊆intS-1(y)⊆S-1(y). This implies y∈S(x)⊆H-coS(x), and hence {y∈Y∣x∈Uy}∈〈H-coS(x)〉. It follows that for each x∈X, we get
(5)f(x)∈Γ{y∈Y∣x∈Uy}⊆H-coS(x)⊆T(x).
Thus, the proof is complete.We remark that Lemma4 extends [7, Theorem 2] from topological vector spaces to general H-spaces. When S=T and S has open lower sections, Lemma 4 reduces to [18, Theorem 3.1].Lemma 5.
LetX be a compact H-space, and let P:X×X→2X be a set-valued mapping such that for each x∈X, P-1(x) is open: then so is (H-coP)-1(x).Proof.
For eachz0∈X, we fix an (x0,y0)∈(H-coP)-1(z0). Since z0∈H-coP(x0,y0), there is a finite set {z1,z2,…,zn} in P(x0,y0) such that z0∈H-co{z1,z2,…,zn}. Since each P-1(zi) is open, it follows that the set U:=⋂i=1n‍P-1(zi) is also open and (x0,y0)∈U. To complete the proof, we will show that U⊆(H-coP)-1(z0). For any (x,y)∈U, we have (x,y)∈P-1(zi) for all i=1,2,…,n. Accordingly, zi∈P(x,y) for all i=1,2,…,n. Hence,
(6)z0∈H-co{z1,z2,…,zn}⊆H-coP(x,y).
That is,(x,y)∈(H-coP)-1(z0). Consequently, U⊆(H-coP)-1(z0).Theorem 6.
LetΩ=(Xα,Aα,Bα,Fα,Pα)α∈I be a generalized abstract economy, where I is a set of agents and X=∏α∈I‍Xα such that for each α∈I, (1)
Xα is an l.c.-space with precompact polytopes,(2)
Aα(x)⊆clBα(x) for each x∈X,(3)
bothclBα and Fα are Qα-condensing mappings in ℍ(X,Xα),(4)
xα∉cl(H-coPα(x,y)) for each x,y∈X,(5)
Aα∩(H-coPα) is transfer open inverse valued in Xα,(6)
Wα∶={(x,y)∈X×X∣Aα(x)∩(H-coPα(x,y))≠∅} is paracompact.Then Ω has an equilibrium point (x^,y^)∈X×X.Proof.
For eachα∈I, we define ϕα:X×X→2Xα by
(7)ϕα(x,y)∶=Aα(x)∩(H-coPα(x,y)),∀(x,y)∈X×X.
Assume thatWα≠∅. Then for each (x,y)∈Wα, we have some yα∈ϕα(x,y). Equivalently, (x,y)∈ϕα-1(yα). It follows that Wα=⋃yα∈Xα‍ϕα-1(yα). Since each ϕα is transfer open inverse valued in Xα by (5), it follows from Lemma 3 that Wα is open in X×X.
Forzα∈Xα, if (x,y)∈ϕα-1(zα), by using (5), we have some zα′∈Xα such that (x,y)∈intϕα-1(zα′)⊆Wα. Thus, the restriction ϕα|Wα:Wα→2Xα is transfer open inverse valued in Xα. Moreover, by (3), each ϕα∣Wα(x,y) is nonempty and H-convex. Therefore, by Lemma 4, there exists a continuous function fα:Wα→Xα such that fα(x,y)∈ϕα|Wα(x,y) for each (x,y)∈Wα.
SinceclBα and Fα are Qα-condensing, applying Theorem A, we have two nonempty compact H-convex subsets K∶=∏α∈I‍Kα and K′:=∏α∈I‍Kα′ of X such that clBα(K)⊆Kα and Fα(K′)⊆Kα′. Using these notations, we define a set-valued mapping Sα:K×K′→2Kα×Kα′ by
(8)Sα(x,y)={cl(H-cofα(x,y))×Fα(x),if(x,y)∈(K×K′)∩Wα,clBα(x)×Fα(x),if(x,y)∈(K×K′)∖Wα.
We will show thatSα∈ℍ(K×K′,Kα×Kα′). Let Vα be an open subset of Kα×Kα′. Since cl[H-cofα(x,y)]⊆cl[H-coϕα(x,y)]⊆clBα(x) for each (x,y)∈Wα, we have
(9)Uα={(x,y)∈K×K′∣Sα(x,y)⊆Vα}={(x,y)∈(K×K′)∩Wα∣cl[H-cofα(x,y)]0×Fα(x)⊆Vα∈(K×K′)∩Wα}∪{(x,y)∈(K×K′)∖Wα∣clBα(x)×Fα(x)⊆Vα}={(x,y)∈(K×K′)∩Wα∣cl[H-cofα(x,y)]0×Fα(x)⊆Vα(K×K′)∩Wα}∪{(x,y)∈K×K′∣clBα(x)×Fα(x)⊆Vα}.
It follows from Lemma2 and the upper semicontinuity of clBα×Fα that Uα is open in K×K′. Hence, Sα is u.s.c. Further, by (3) and Lemma 1, each Sα(x,y) is nonempty, closed, and H-convex. Therefore, Sα∈ℍ(K×K′,Kα×Kα′).
Next, we define a set-valued mappingTα:K×K′→2Kα×Kα′ by
(10)Tα(x,y)00={Sα(x,y),if(x,y)∈(K×K′)∩Wα,clBα(x)×Fα(x),if(x,y)∈(K×K′)∖Wα.
SinceKα×Kα′ is compact, each Tα is Qα-condensing in ℍ(K×K′,Kα×Kα′). Hence, by Theorem B, the set-valued mapping ∏α∈I‍Tα has a fixed point (x^,y^)∈K×K′; that is, (x^α,y^α)∈Tα(x^,y^) for each α∈I. If (x^,y^)∈Wα, then
(11)(x^α,y^α)∈cl(Sα(x^,y^))⊆cl(Bα(x)∩(H-coPα(x^,y^)))×Fα(x^).
Thus,x^α∈cl(H-coPα(x^,y^)), which contradicts with (4). Therefore, (x^,y^)∉Wα and hence x^α∈clBα(x^), y^α∈Fα(x^), and Aα(x^)∩Pα(x^,y^)=∅ for each α∈I. That is, (x^,y^) is an equilibrium of Ω.Remark that condition (4) of Theorem6 can be replaced by a milder condition xα∉cl(Bα(x)∩H-coPα(x,y)) for each (x,y)∈Wα. Further, when each l.c.-space (Xα,Γα) satisfies Γ{xα}α={xα}, condition (4) can be modified by xα∉H-coPα(x,y) without affecting the conclusion.Corollary 7.
LetΩ=(Xα,Aα,Bα,Fα,Pα)α∈I be a generalized abstract economy, where I is a set of agents and X=∏α∈I‍Xα such that for each α∈I, (1)
(Xα,Γα) is an l.c.-space with precompact polytopes, and Γ{xα}α={xα} for each xα∈Xα,(2)
Aα(x)⊆clBα(x) for each x∈X,(3)
bothclBα and Fα are Qα-condensing mappings in ℍ(X,Xα),(4)
xα∉H-coPα(x,y) for each x,y∈X,(5)
Aα∩(H-coPα) is transfer open inverse valued in Xα,(6)
Wα:={(x,y)∈X×X∣Aα(x)∩(H-coPα(x,y))≠∅} is paracompact. Then Ω has an equilibrium point (x^,y^)∈X×X.Proof.
According to the proof of Theorem6 and by virtue of the condition Γ{xα}α={xα} for each xα∈Xα, we obtain cl(H-cofα(x,y))=fα(x,y). It follows that the set-valued mapping Sα can be defined by
(12)Sα(x,y)={fα(x,y)×Fα(x),if(x,y)∈(K×K′)∩Wα,clBα(x)×Fα(x),if(x,y)∈(K×K′)∖Wα.
Thus, by an analogue proof to Theorem6, we may conclude that Ω has an equilibrium point.Following the proof of Theorem6 by taking ϕα(x,y)∶=clBα(x)∩(H-coPα(x,y)), we may obtain a new version of equilibrium existence theorem as follows.Corollary 8.
LetΩ=(Xα,Aα,Bα,Fα,Pα)α∈I be a generalized abstract economy, where I is a set of agents and X=∏α∈I‍Xα such that for each α∈I, (1)
Xα is an l.c.-space with precompact polytopes,(2)
Aα(x)⊆clBα(x) for each x∈X,(3)
bothclBα and Fα are Qα-condensing mappings in ℍ(X,Xα),(4)
xα∉cl(H-coPα(x,y)) for each x,y∈X,(5)
clBα∩(H-coPα) is transfer open inverse valued in Xα,(6)
Wα∶={(x,y)∈X×X∣clBα(x)∩(H-coPα(x,y))≠∅} is paracompact. Then Ω has an equilibrium point (x^,y^)∈X×X.Notice that Theorem6 generalizes [7, Kim-Tan, Theorem 2], in which they deal with the case of locally convex topological vector spaces under some compactness conditions, and it also improves [19, Wu-Yuan, Theorem 3] in the setting of locally H-convex spaces. We also note that if X is metrizable, the set Wα is also metrizable and hence is paracompact. Therefore, the assumption (6) of Theorem 6 is automatically satisfied. Furthermore, if each Xα is compact, then both clBα and Fα are obviously Qα-condensing. Thus, we have an immediate consequence, which is a generalization of [7, Kim-Tan, Corollary 1] to H-spaces.Corollary 9.
LetΩ=(Xα,Aα,Bα,Fα,Pα)α∈I be a generalized abstract economy, where I is a set of agents such that for each α∈I, (1)
(Xα,Γα) is a metrizable compact l.c.-space, and Γ{xα}α={xα} for each xα∈Xα,(2)
Aα(x)⊆clBα(x) for each x∈X,(3)
clBα∈ℍ(X,Xα), and Fα∈ℍ(X,Xα),(4)
xα∉H-coPα(x,y) for each x,y∈X,(5)
Aα∩(H-coPα) is transfer open inverse valued in Xα. Then Ω has an equilibrium point (x^,y^)∈X×X.We note that our main results focus on the setting of generall.c.-spaces without any linear structure; further, the correspondences are not necessarily lower semicontinuous and do not require the usual open lower section assumption, such as the earlier works [3, Theorem 4], [13, Theorem 3 and its Corollary], [19, Theorems 1 and 3], and [18, Theorem 6.1]. In fact, we can give a simple example applicable for Corollary 9, while previous results do not.Example 10.
Consider the setI of agents is singleton. Let X=[0,1] and the correspondences A,B,F:X→2X be defined by A(x)=B(x)=[0,1], and F(x)={x} for each x∈X. The preference correspondence P:X×X→2X is defined as follows:
(13)P(x1,x2)={(x1+x22,1],ifx1<x2,x1,x2∈Q,(x1+2x23,1],ifx1<x2,x1∉Q,orx2∉Q,∅,ifx1=x2,{0},ifx1>x2.
Then A∩(H-coP)=A∩P is transfer open inverse valued in X. Indeed, (A∩P)-1(0)=P-1(0)={(x1,x2)∣x1>x2} is open in X×X, and for any t∈(0,1] and (x1,x2)∈(A∩P)-1(t)=P-1(t), we always have (x1,x2)∈intP-1(1)=int(A∩P)-1(1). However, the lower section (A∩P)-1(1/2) is not open. Indeed, let an=(1/2)-(1/n2) and let bn=9/10; then (an,bn)∈X×X∖(A∩P)-1(1/2) and (an,bn) converges to (1/2,9/10), which does not belong to X×X∖(A∩P)-1(1/2). This means that the set X×X∖(A∩P)-1(1/2) is not closed, and hence (A∩P)-1(1/2) is not open. Further, for each x1,x2∈X, x1∉P(x1,x2)=H-coP(x1,x2). Thus, all hypotheses of Corollary 9 are satisfied so that the generalized abstract economy Ω has an equilibrium point in X×X. In fact, all the equilibria of Ω are the points (a,a), where a∈[0,1].LetX and Y be two topological spaces. Given three set-valued mappings T:X→2Y, F:X→2X, A:X→2X, and a function ϕ:X×X×Y→ℝ, a generalized quasi-variational inequality is defined as follows:
(14)(GQVI){Find(x^,w^,y^)∈X×X×Ysuchthatx^∈clA(x^),w^∈F(x^),y^∈T(x^),ϕ(z,x^,y^)≥0,∀z∈A(x^)∩(F-1(w^))C.In particular, ifF(x)={x} for each x∈X, then (F-1(w))C={w}C=X∖{w}. Therefore, the  (GQVI) reduces to the usual quasi-variational inequality as follows:
(15)(QVI){Find(x^,y^)∈X×Ysuchthatx^∈clA(x^),y^∈T(x^),ϕ(z,x^,y^)≥0,∀z∈A(x^)∖{x^}.Theorem 11.
Let(X,Γ) be an l.c.-space with precompact polytopes, Γ{x}={x} for each x∈X, and let Y be a topological space. The set-valued mappings T:X→2Y and F,A:X→2X satisfy T∈H(X,Y), F∈H(X,X), and clA∈H(X,X), and A-1(x) is open for all x∈X. Suppose that ϕ:X×X×Y→ℝ is a function such that (1)
ϕα(x,x,y)≥0 for all x∈X and y∈T(x),(2)
for each fixedz∈X, the mapping (x,y)↦ϕ(z,x,y) is lower semicontinuous,(3)
for each fixed(x,y)∈X×Y, the mapping z↦ϕ(z,x,y) is H-quasiconvex in the following sense that for any finite set D in X,
(16)ϕ(z,x,y)≤maxu∈Dϕ(u,x,y),∀z∈H-coD.Then there is a solution to  (GQVI).Proof.
Define a set-valued mappingP:X×X→2X by
(17)P(x,w)≔{z∈X∣infy∈T(x)ϕ(z,x,y)<0}∩(F-1(w))C,∀(x,w)∈X×X.
By [20, Proposition 23, page 121], for each fixed z∈X, the mapping x↦infy∈T(x)ϕ(z,x,y) is lower semicontinuous. Thus, the set {x∈X∣infy∈T(x)ϕ(z,x,y)>0} is open for each z∈X. It follows that
(18)P-1(z)=({x∈X∣infy∈T(x)ϕ(z,x,y)>0}×X)∩(X×(T(z))C)
is open. By Lemma 5, (H-coP)-1(z) is also open. Next, we show that x∉H-coP(x,w) for all x,w∈X. Assume that there are x0 and w0 satisfying x0∈H-coP(x0,w0). Then there is a finite subset D of P(x0,w0) such that x0∈H-coD. For each fixed y∈T(x0), since the mapping z↦ϕ(z,x0,y) is H-quasiconvex, it follows that
(19)0≤infy∈T(x0)ϕ(x0,x0,y)≤infy∈T(x0)maxz∈Dϕ(z,x0,y).
By Kneser’s minimax theorem [21], together with z∈P(x0,w0) for all z∈D, we have
(20)infy∈T(x0)maxz∈Dϕ(z,x0,y)=maxz∈Dinfy∈T(x0)ϕ(z,x0,y)<0.
This is a contradiction. Thus, all hypotheses of Corollary7 are satisfied. Therefore, there exist x^,w^∈X such that x^∈clA(x^), w^∈F(x^), and A(x^)∩P(x^,w^)=∅, It follows that
(21)infy∈T(x^)ϕ(z,x^,y)≥0,∀z∈A(x^)∩(F-1(w^))C.
SinceT(x^) is compact, there is y^∈T(x^) such that ϕ(z,x^,y^)≥0 for all z∈A(x^)∩(F-1(w^))C. That is, (x^,w^,y^) is a solution to (GQVI).



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