On Abstract Economies and Their Applications

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 of 𝑙.𝑐.-spaces without any linear structure.

1. Introduction and Preliminary

Let 𝐼 be any (finite or infinite) set of agents. A generalized abstract economy is defined as a family of order quintuples 𝛀=(𝑋_𝛼,𝐴_𝛼,𝐵_𝛼,𝐹_𝛼,𝑃_𝛼)_𝛼∈𝐼 with 𝑋∶=∏_𝛼∈𝐼𝑋_𝛼 such that for each 𝛼∈𝐼, 𝑋_𝛼 is a topological space, 𝐴_𝛼,𝐵_𝛼∶𝑋→2^𝑋𝛼 are constraint correspondences, 𝐹_𝛼∶𝑋→2^𝑋𝛼 is a fuzzy constraint correspondence, and 𝑃_𝛼∶𝑋×𝑋→2^𝑋𝛼 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∈cl𝐵_𝛼(𝑥), 𝑦_𝛼∈𝐹_𝛼(𝑥), and 𝐴_𝛼(𝑥)∩𝑃_𝛼(𝑥,𝑦)=∅, where 𝑥_𝛼 and 𝑦_𝛼 denote the projections of 𝑥 and 𝑦 from 𝑋 to 𝑋_𝛼, respectively.

In case 𝐹_𝛼(𝑥)=𝑋_𝛼 for each 𝑥∈𝑋 and 𝑃_𝛼 is independent of the second variable, that is, 𝑃_𝛼∶𝑋→2^𝑋𝛼, the above generalized abstract economy reduces to the standard abstract economy 𝛀_𝐬∶=(𝑋_𝛼,𝐴_𝛼,𝐵_𝛼,𝑃_𝛼)_𝛼∈𝐼, in which an equilibrium point of 𝛀_𝐬 is a point 𝑥∈𝑋 such that for each 𝛼∈𝐼, 𝑥_𝛼∈cl𝐵_𝛼(𝑥) and 𝐴_𝛼(𝑥)∩𝑃_𝛼(𝑥)=∅. When 𝐴_𝛼=𝐵_𝛼 and each 𝑋_𝛼 is a topological vector space, the standard abstract economy 𝛀_𝐬 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 set 𝐶 of a topological space 𝑋, we denote the set of all subsets of 𝐶 by 2^𝐶, 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 {Γ_𝐷} be a family of some nonempty contractible subsets of a topological space 𝑋 indexed by 𝐷∈⟨𝑋⟩ such that Γ_𝐷⊂Γ_𝐷′ whenever 𝐷⊂𝐷^′. The pair (𝑋,{Γ_𝐷}) is called an 𝐻-space. Given an 𝐻-space (𝑋,{Γ_𝐷}), a nonempty subset 𝐶 of 𝑋 is said to be 𝐻-convex if Γ_𝐷⊆𝐶 for all 𝐷∈⟨𝐶⟩. For a nonempty subset 𝐶 of 𝑋, we define the 𝐻-convex hull of 𝐶 as 𝐻-co𝐶∶={𝐾∣𝐾is𝐻-convexin𝑋and𝐶⊆𝐾}. (1)

It is known that if 𝑥∈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 Γ_𝐷=co𝐷 for all 𝐷∈⟨𝑋⟩.

An 𝐻-space (𝑋,{Γ_𝐷}) 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 𝑄(𝐴){(𝑉(𝐾.∶=𝑉∈ℬ∣𝐴⊆cl))forsomeprecompactset𝐾of𝑋 (2)

Let (𝑋_𝛼,𝒰_𝛼,ℬ_𝛼)_𝛼∈𝐼 be a family of 𝑙.𝑐.-spaces with precompact polytopes, where 𝐼 is a finite or infinite index set and 𝑋=∏_𝛼∈𝐼𝑋_𝛼. For each 𝛼∈𝐼, let 𝜋_𝛼 be the projection of 𝑋 onto 𝑋_𝛼 and 𝑄_𝛼 a measure of precompactness in 𝑋_𝛼. We say that a set-valued mapping 𝑇_𝛼∶𝑋→2^𝑋𝛼 is 𝑄_𝛼-condensing if 𝑄_𝛼(𝜋_𝛼(𝐶))⊊𝑄_𝛼(𝑇_𝛼(𝐶)) for every 𝐶 satisfying 𝜋_𝛼(𝐶) is a nonprecompact subset of 𝑋_𝛼. It is clear that for any set-valued mapping 𝑇∶𝑋→2^𝑌 and any measure 𝑄 in 𝑌, 𝑇 is 𝑄-condensing whenever 𝑌 is compact.

Let 𝑋 be a topological space, let 𝑌 be an 𝐻-space, and let 𝑆,𝑇∶𝑋→2^𝑌 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 𝑁_𝑥 of 𝑥 such that 𝑇(𝑧)⊆𝑉 for all 𝑧∈𝑁_𝑥. (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→2^𝑋 is defined by 𝑇⁻¹𝑦∶=𝑥∈𝑋∣𝑦∈𝑇(𝑥)∀𝑦∈𝑌. (3)(4) The set-valued mappings 𝑆∩𝑇∶𝑋→2^𝑌 and cl𝑇∶𝑋→2^𝑌 are defined by (𝑥)(𝑥)(𝑥),(𝑥)(𝑇(𝑥𝑆∩𝑇)(∶=𝑆∩𝑇cl𝑇∶=cl)),∀𝑥∈𝑋. (4)

Further, we denote by ℍ(𝑋,𝑌) the class of all 𝑢.𝑠.𝑐. set-valued mappings 𝑇∶𝑋→2^𝑌 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 (𝑋_𝛼,𝒰_𝛼,ℬ_𝛼)_𝛼∈𝐼 be a family of 𝑙.𝑐.-spaces with precompact polytopes, 𝑋∶=∏_𝛼∈𝐼𝑋_𝛼, and let 𝑇_𝛼∶𝑋→2^𝑋𝛼 be 𝑄_𝛼-condensing. Then there exists a nonempty compact 𝐻-convex subset 𝐾∶=∏_𝛼∈𝐼𝐾_𝛼 of  𝑋 such that 𝑇_𝛼(𝐾)⊆𝐾_𝛼.

Theorem B (see [16]). Let (𝑋_𝛼,𝒰_𝛼,ℬ_𝛼)_𝛼∈𝐼 be a family of 𝑙.𝑐.-spaces with precompact polytopes and 𝑋∶=∏_𝛼∈𝐼𝑋_𝛼. If 𝑇_𝛼∶𝑋→2^𝑋𝛼 is an 𝑢.𝑠.𝑐.𝑄_𝛼-condensing mapping with closed 𝐻-convex values for each 𝛼∈𝐼, then 𝑇∶=∏_𝛼∈𝐼𝑇_𝛼 has a fixed point.

Next, we list and establish some essential lemmas as follows.

Lemma 1 (see [12]). If 𝑋 is an 𝑙.𝑐.-space and 𝐸 is an 𝐻-convex subset of 𝑋, then cl𝐸 is also 𝐻-convex.

Lemma 2 (see [12]). Let 𝑋 be a topological space and let (𝑌,{Γ_𝐷}) be a compact 𝑙.𝑐.-space. If 𝑇∶𝑋→2^𝑌 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]). Let 𝑋 and 𝑌 be topological spaces and let 𝑆∶𝑋→2^𝑌 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. Let 𝑋 be paracompact, (𝑌,{Γ_𝐷}) an 𝐻-space, and 𝑆,𝑇∶𝑋→2^𝑌 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 each 𝑥∈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 𝑦_𝑥∈𝑌 such that 𝑥∈int𝑆⁻¹(𝑦_𝑥). 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 {𝑈_𝑦∣𝑦∈𝑌} such that 𝑈_𝑦⊆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∈𝑈_𝑦⊆int𝑆⁻¹(𝑦)⊆𝑆⁻¹(𝑦). This implies 𝑦∈𝑆(𝑥)⊆𝐻-co𝑆(𝑥), and hence {𝑦∈𝑌∣𝑥∈𝑈_𝑦}∈⟨𝐻-co𝑆(𝑥)⟩. It follows that for each 𝑥∈𝑋, we get 𝑓(𝑥)∈Γ_{{𝑦∈𝑌∣𝑥∈𝑈𝑦}}⊆𝐻-co𝑆(𝑥)⊆𝑇(𝑥). (5) Thus, the proof is complete.

We remark that Lemma 4 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. Let 𝑋 be a compact 𝐻-space, and let 𝑃∶𝑋×𝑋→2^𝑋 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 each 𝑧₀∈𝑋, we fix an (𝑥₀,𝑦₀)∈(𝐻-co𝑃)⁻¹(𝑧₀). Since 𝑧₀∈𝐻-co𝑃(𝑥₀,𝑦₀), there is a finite set {𝑧₁,𝑧₂,…,𝑧_𝑛} in 𝑃(𝑥₀,𝑦₀) such that 𝑧₀∈𝐻-co{𝑧₁,𝑧₂,…,𝑧_𝑛}. Since each 𝑃⁻¹(𝑧_𝑖) is open, it follows that the set 𝑈∶=⋂^𝑛_𝑖=1𝑃⁻¹(𝑧_𝑖) 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(𝑧_𝑖) for all 𝑖=1,2,…,𝑛. Accordingly, 𝑧_𝑖∈𝑃(𝑥,𝑦) for all 𝑖=1,2,…,𝑛. Hence, 𝑧₀∈𝐻-co𝑧₁,𝑧₂,…,𝑧_𝑛⊆𝐻-co𝑃𝑥,𝑦. (6)
That is, (𝑥,𝑦)∈(𝐻-co𝑃)⁻¹(𝑧₀). Consequently, 𝑈⊆(𝐻-co𝑃)⁻¹(𝑧₀).

Theorem 6. Let 𝛀=(𝑋_𝛼,𝐴_𝛼,𝐵_𝛼,𝐹_𝛼,𝑃_𝛼)_𝛼∈𝐼 be a generalized abstract economy, where 𝐼 is a set of agents and 𝑋=∏_𝛼∈𝐼𝑋_𝛼 such that for each 𝛼∈𝐼, (1)𝑋_𝛼 is an 𝑙.𝑐.-space with precompact polytopes, (2)𝐴_𝛼(𝑥)⊆cl𝐵_𝛼(𝑥) for each 𝑥∈𝑋, (3) both cl𝐵_𝛼 and 𝐹_𝛼 are 𝑄_𝛼-condensing mappings in ℍ(𝑋,𝑋_𝛼), (4)𝑥_𝛼∉cl(𝐻-co𝑃_𝛼(𝑥,𝑦)) for each 𝑥,𝑦∈𝑋, (5)𝐴_𝛼∩(𝐻-co𝑃_𝛼) is transfer open inverse valued in 𝑋_𝛼, (6)𝑊_𝛼∶={(𝑥,𝑦)∈𝑋×𝑋∣𝐴_𝛼(𝑥)∩(𝐻-co𝑃_𝛼(𝑥,𝑦))≠∅} is paracompact. Then 𝛀 has an equilibrium point (𝑥,𝑦)∈𝑋×𝑋.

Proof. For each 𝛼∈𝐼, we define 𝜙_𝛼∶𝑋×𝑋→2^𝑋𝛼 by 𝜙_𝛼𝑥,𝑦∶=𝐴_𝛼(𝑥)∩𝐻-co𝑃_𝛼𝑥,𝑦,∀𝑥,𝑦∈𝑋×𝑋. (7)
Assume that 𝑊_𝛼≠∅. Then for each (𝑥,𝑦)∈𝑊_𝛼, we have some 𝑦_𝛼∈𝜙_𝛼(𝑥,𝑦). Equivalently, (𝑥,𝑦)∈𝜙_𝛼⁻¹(𝑦_𝛼). It follows that 𝑊_𝛼=⋃_{𝑦𝛼∈𝑋𝛼}𝜙_𝛼⁻¹(𝑦_𝛼). Since each 𝜙_𝛼 is transfer open inverse valued in 𝑋_𝛼 by (5), it follows from Lemma 3 that 𝑊_𝛼 is open in 𝑋×𝑋.
For 𝑧_𝛼∈𝑋_𝛼, if (𝑥,𝑦)∈𝜙_𝛼⁻¹(𝑧_𝛼), by using (5), we have some 𝑧^′_𝛼∈𝑋_𝛼 such that (𝑥,𝑦)∈int𝜙_𝛼⁻¹(𝑧^′_𝛼)⊆𝑊_𝛼. Thus, the restriction 𝜙_𝛼|_𝑊𝛼∶𝑊_𝛼→2^𝑋𝛼 is transfer open inverse valued in 𝑋_𝛼. Moreover, by (3), each 𝜙_𝛼∣_𝑊𝛼(𝑥,𝑦) is nonempty and 𝐻-convex. Therefore, by Lemma 4, there exists a continuous function 𝑓_𝛼∶𝑊_𝛼→𝑋_𝛼 such that 𝑓_𝛼(𝑥,𝑦)∈𝜙_𝛼|_𝑊𝛼(𝑥,𝑦) for each (𝑥,𝑦)∈𝑊_𝛼.
Since cl𝐵_𝛼 and 𝐹_𝛼 are 𝑄_𝛼-condensing, applying Theorem A, we have two nonempty compact 𝐻-convex subsets 𝐾∶=∏_𝛼∈𝐼𝐾_𝛼 and 𝐾^′∶=∏_𝛼∈𝐼𝐾^′_𝛼 of 𝑋 such that cl𝐵_𝛼(𝐾)⊆𝐾_𝛼 and 𝐹_𝛼(𝐾^′)⊆𝐾^′_𝛼. Using these notations, we define a set-valued mapping 𝑆_𝛼∶𝐾×𝐾^′→2^{𝐾𝛼×𝐾′𝛼} by 𝑆_𝛼𝑥,𝑦=cl𝐻-co𝑓_𝛼𝑥,𝑦×𝐹_𝛼(𝑥),if𝑥,𝑦∈𝐾×𝐾^′∩𝑊_𝛼,cl𝐵_𝛼(𝑥)×𝐹_𝛼(𝑥),if𝑥,𝑦∈𝐾×𝐾^′⧵𝑊_𝛼. (8)
We will show that 𝑆_𝛼∈ℍ(𝐾×𝐾^′,𝐾_𝛼×𝐾^′_𝛼). Let 𝑉_𝛼 be an open subset of 𝐾_𝛼×𝐾^′_𝛼. Since cl[𝐻-co𝑓_𝛼(𝑥,𝑦)]⊆cl[𝐻-co𝜙_𝛼(𝑥,𝑦)]⊆cl𝐵_𝛼(𝑥) for each (𝑥,𝑦)∈𝑊_𝛼, we have 𝑈_𝛼=𝑥,𝑦∈𝐾×𝐾^′∣𝑆_𝛼𝑥,𝑦⊆𝑉_𝛼=𝑥,𝑦∈𝐾×𝐾^′∩𝑊_𝛼∣cl𝐻-co𝑓_𝛼𝑥,𝑦×𝐹_𝛼(𝑥)⊆𝑉_𝛼∪𝑥,𝑦∈𝐾×𝐾^′⧵𝑊_𝛼∣cl𝐵_𝛼(𝑥)×𝐹_𝛼(𝑥)⊆𝑉_𝛼=𝑥,𝑦∈𝐾×𝐾^′∩𝑊_𝛼∣cl𝐻-co𝑓_𝛼𝑥,𝑦×𝐹_𝛼(𝑥)⊆𝑉_𝛼∪𝑥,𝑦∈𝐾×𝐾^′∣cl𝐵_𝛼(𝑥)×𝐹_𝛼(𝑥)⊆𝑉_𝛼. (9)
It follows from Lemma 2 and the upper semicontinuity of cl𝐵_𝛼×𝐹_𝛼 that 𝑈_𝛼 is open in 𝐾×𝐾^′. Hence, 𝑆_𝛼 is 𝑢.𝑠.𝑐. Further, by (3) and Lemma 1, each 𝑆_𝛼(𝑥,𝑦) is nonempty, closed, and 𝐻-convex. Therefore, 𝑆_𝛼∈ℍ(𝐾×𝐾^′,𝐾_𝛼×𝐾^′_𝛼).
Next, we define a set-valued mapping 𝑇_𝛼∶𝐾×𝐾^′→2^{𝐾𝛼×𝐾′𝛼} by 𝑇_𝛼=𝑆𝑥,𝑦_𝛼𝑥,𝑦,if𝑥,𝑦∈𝐾×𝐾^′∩𝑊_𝛼,cl𝐵_𝛼(𝑥)×𝐹_𝛼(𝑥),if𝑥,𝑦∈𝐾×𝐾^′⧵𝑊_𝛼. (10)
Since 𝐾_𝛼×𝐾^′_𝛼 is compact, each 𝑇_𝛼 is 𝑄_𝛼-condensing in ℍ(𝐾×𝐾^′,𝐾_𝛼×𝐾^′_𝛼). Hence, by Theorem B, the set-valued mapping ∏_𝛼∈𝐼𝑇_𝛼 has a fixed point (𝑥,𝑦)∈𝐾×𝐾^′; that is, (𝑥_𝛼,𝑦_𝛼)∈𝑇_𝛼(𝑥,𝑦) for each 𝛼∈𝐼. If (𝑥,𝑦)∈𝑊_𝛼, then 𝑥_𝛼,𝑦_𝛼∈cl𝑆_𝛼𝑦𝐵𝑥,⊆cl_𝛼(𝑥)∩𝐻-co𝑃_𝛼𝑥,𝑦×𝐹_𝛼(𝑥). (11)
Thus, 𝑥_𝛼∈cl(𝐻-co𝑃_𝛼(𝑥,𝑦)), which contradicts with (4). Therefore, (𝑥,𝑦)∉𝑊_𝛼 and hence 𝑥_𝛼∈cl𝐵_𝛼(𝑥), 𝑦_𝛼∈𝐹_𝛼(𝑥), and 𝐴_𝛼(𝑥)∩𝑃_𝛼(𝑥,𝑦)=∅ for each 𝛼∈𝐼. That is, (𝑥,𝑦) is an equilibrium of 𝛀.

Remark that condition (4) of Theorem 6 can be replaced by a milder condition 𝑥_𝛼∉cl(𝐵_𝛼(𝑥)∩𝐻-co𝑃_𝛼(𝑥,𝑦)) for each (𝑥,𝑦)∈𝑊_𝛼. Further, when each 𝑙.𝑐.-space (𝑋_𝛼,Γ^𝛼) satisfies Γ^𝛼_{𝑥𝛼}={𝑥_𝛼}, condition (4) can be modified by 𝑥_𝛼∉𝐻-co𝑃_𝛼(𝑥,𝑦) without affecting the conclusion.

Corollary 7. Let 𝛀=(𝑋_𝛼,𝐴_𝛼,𝐵_𝛼,𝐹_𝛼,𝑃_𝛼)_𝛼∈𝐼 be a generalized abstract economy, where 𝐼 is a set of agents and 𝑋=∏_𝛼∈𝐼𝑋_𝛼 such that for each 𝛼∈𝐼, (1)(𝑋_𝛼,Γ^𝛼) is an 𝑙.𝑐.-space with precompact polytopes, and Γ^𝛼_{𝑥𝛼}={𝑥_𝛼} for each 𝑥_𝛼∈𝑋_𝛼, (2)𝐴_𝛼(𝑥)⊆cl𝐵_𝛼(𝑥) for each 𝑥∈𝑋, (3) both cl𝐵_𝛼 and 𝐹_𝛼 are 𝑄_𝛼-condensing mappings in ℍ(𝑋,𝑋_𝛼), (4)𝑥_𝛼∉𝐻-co𝑃_𝛼(𝑥,𝑦) for each 𝑥,𝑦∈𝑋, (5)𝐴_𝛼∩(𝐻-co𝑃_𝛼) is transfer open inverse valued in 𝑋_𝛼, (6)𝑊_𝛼∶={(𝑥,𝑦)∈𝑋×𝑋∣𝐴_𝛼(𝑥)∩(𝐻-co𝑃_𝛼(𝑥,𝑦))≠∅} is paracompact. Then 𝛀 has an equilibrium point (𝑥,𝑦)∈𝑋×𝑋.

Proof. According to the proof of Theorem 6 and by virtue of the condition Γ^𝛼_{𝑥𝛼}={𝑥_𝛼} for each 𝑥_𝛼∈𝑋_𝛼, we obtain cl(𝐻-co𝑓_𝛼(𝑥,𝑦))=𝑓_𝛼(𝑥,𝑦). It follows that the set-valued mapping 𝑆_𝛼 can be defined by 𝑆_𝛼=𝑓𝑥,𝑦^𝛼𝑥,𝑦×𝐹_𝛼(𝑥),if𝑥,𝑦∈𝐾×𝐾^′∩𝑊_𝛼,cl𝐵_𝛼(𝑥)×𝐹_𝛼(𝑥),if𝑥,𝑦∈𝐾×𝐾^′⧵𝑊_𝛼. (12)
Thus, by an analogue proof to Theorem 6, we may conclude that 𝛀 has an equilibrium point.

Following the proof of Theorem 6 by taking 𝜙_𝛼(𝑥,𝑦)∶=cl𝐵_𝛼(𝑥)∩(𝐻-co𝑃_𝛼(𝑥,𝑦)), we may obtain a new version of equilibrium existence theorem as follows.

Corollary 8. Let 𝛀=(𝑋_𝛼,𝐴_𝛼,𝐵_𝛼,𝐹_𝛼,𝑃_𝛼)_𝛼∈𝐼 be a generalized abstract economy, where 𝐼 is a set of agents and 𝑋=∏_𝛼∈𝐼𝑋_𝛼 such that for each 𝛼∈𝐼, (1)𝑋_𝛼 is an 𝑙.𝑐.-space with precompact polytopes, (2)𝐴_𝛼(𝑥)⊆cl𝐵_𝛼(𝑥) for each 𝑥∈𝑋, (3) both cl𝐵_𝛼 and 𝐹_𝛼 are 𝑄_𝛼-condensing mappings in ℍ(𝑋,𝑋_𝛼), (4)𝑥_𝛼∉cl(𝐻-co𝑃_𝛼(𝑥,𝑦)) for each 𝑥,𝑦∈𝑋, (5)cl𝐵_𝛼∩(𝐻-co𝑃_𝛼) is transfer open inverse valued in 𝑋_𝛼, (6)𝑊_𝛼∶={(𝑥,𝑦)∈𝑋×𝑋∣cl𝐵_𝛼(𝑥)∩(𝐻-co𝑃_𝛼(𝑥,𝑦))≠∅} is paracompact. Then 𝛀 has an equilibrium point (𝑥,𝑦)∈𝑋×𝑋.

Notice that Theorem 6 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 𝑊_𝛼 is also metrizable and hence is paracompact. Therefore, the assumption (6) of Theorem 6 is automatically satisfied. Furthermore, if each 𝑋_𝛼 is compact, then both cl𝐵_𝛼 and 𝐹_𝛼 are obviously 𝑄_𝛼-condensing. Thus, we have an immediate consequence, which is a generalization of [7, Kim-Tan, Corollary 1] to 𝐻-spaces.

Corollary 9. Let 𝛀=(𝑋_𝛼,𝐴_𝛼,𝐵_𝛼,𝐹_𝛼,𝑃_𝛼)_𝛼∈𝐼 be a generalized abstract economy, where 𝐼 is a set of agents such that for each 𝛼∈𝐼, (1)(𝑋_𝛼,Γ^𝛼) is a metrizable compact 𝑙.𝑐.-space, and Γ^𝛼_{𝑥𝛼}={𝑥_𝛼} for each 𝑥_𝛼∈𝑋_𝛼, (2)𝐴_𝛼(𝑥)⊆cl𝐵_𝛼(𝑥) for each 𝑥∈𝑋, (3)cl𝐵_𝛼∈ℍ(𝑋,𝑋_𝛼), and 𝐹_𝛼∈ℍ(𝑋,𝑋_𝛼), (4)𝑥_𝛼∉𝐻-co𝑃_𝛼(𝑥,𝑦) for each 𝑥,𝑦∈𝑋, (5)𝐴_𝛼∩(𝐻-co𝑃_𝛼) is transfer open inverse valued in 𝑋_𝛼. Then 𝛀 has an equilibrium point (𝑥,𝑦)∈𝑋×𝑋.

We note that our main results focus on the setting of general 𝑙.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 set 𝐼 of agents is singleton. Let 𝑋=[0,1] and the correspondences 𝐴,𝐵,𝐹∶𝑋→2^𝑋 be defined by 𝐴(𝑥)=𝐵(𝑥)=[0,1], and 𝐹(𝑥)={𝑥} for each 𝑥∈𝑋. The preference correspondence 𝑃∶𝑋×𝑋→2^𝑋 is defined as follows: 𝑃𝑥₁,𝑥₂=𝑥₁+𝑥₂2,1,if𝑥₁<𝑥₂,𝑥₁,𝑥₂𝑥∈𝑄,₁+2𝑥₂3,1,if𝑥₁<𝑥₂,𝑥₁∉𝑄,or𝑥₂∉𝑄,∅,if𝑥₁=𝑥₂,{0},if𝑥₁>𝑥₂. (13) 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 𝑎_𝑛=(1/2)−(1/𝑛√2) and let 𝑏_𝑛=9/10; then (𝑎_𝑛,𝑏_𝑛)∈𝑋×𝑋⧵(𝐴∩𝑃)⁻¹(1/2) and (𝑎_𝑛,𝑏_𝑛) 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].

Let 𝑋 and 𝑌 be two topological spaces. Given three set-valued mappings 𝑇∶𝑋→2^𝑌, 𝐹∶𝑋→2^𝑋, 𝐴∶𝑋→2^𝑋, and a function 𝜙∶𝑋×𝑋×𝑌→ℝ, a generalized quasi-variational inequality is defined as follows: ()𝑦(𝑥),(𝑥),(𝑥),𝜙𝑦(𝑥)∩𝐹𝐆𝐐𝐕𝐈Find𝑥,𝑤,∈𝑋×𝑋×𝑌suchthat𝑥∈cl𝐴𝑤∈𝐹𝑦∈𝑇𝑧,𝑥,≥0,∀𝑧∈𝐴⁻¹(𝑤)^𝐶. (14)

In particular, if 𝐹(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: ()𝑦(𝑥),(𝑥),𝜙𝑦(𝑥)⧵{𝑥}.𝐐𝐕𝐈Find𝑥,∈𝑋×𝑌suchthat𝑥∈cl𝐴𝑦∈𝑇𝑧,𝑥,≥0,∀𝑧∈𝐴 (15)

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→2^𝑌 and 𝐹,𝐴∶𝑋→2^𝑋 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)𝜙_𝛼(𝑥,𝑥,𝑦)≥0 for all 𝑥∈𝑋 and 𝑦∈𝑇(𝑥), (2) for each fixed 𝑧∈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, 𝜙𝑧,𝑥,𝑦≤max_𝑢∈𝐷𝜙𝑢,𝑥,𝑦,∀𝑧∈𝐻-co𝐷. (16)Then there is a solution to (𝐆𝐐𝐕𝐈).

Proof. Define a set-valued mapping 𝑃∶𝑋×𝑋→2^𝑋 by 𝑃(𝑥,𝑤)∶=𝑧∈𝑋∣inf_{(𝑥)𝑦∈𝑇}𝜙𝑧,𝑥,𝑦<0∩𝐹⁻¹(𝑤)^𝐶,∀(𝑥,𝑤)∈𝑋×𝑋. (17)
By [20, Proposition 23, page 121], for each fixed 𝑧∈𝑋, the mapping 𝑥↦inf_{𝑦∈𝑇(𝑥)}𝜙(𝑧,𝑥,𝑦) is lower semicontinuous. Thus, the set {𝑥∈𝑋∣inf_{𝑦∈𝑇(𝑥)}𝜙(𝑧,𝑥,𝑦)>0} is open for each 𝑧∈𝑋. It follows that 𝑃⁻¹(𝑧)=𝑥∈𝑋∣inf_{(𝑥)𝑦∈𝑇}𝜙∩(𝑇(𝑧𝑧,𝑥,𝑦>0×𝑋𝑋×))^𝐶 (18) 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 0≤inf_{𝑦∈𝑇(𝑥0)}𝜙𝑥₀,𝑥₀,𝑦≤inf_{𝑦∈𝑇(𝑥0)}max_𝑧∈𝐷𝜙𝑧,𝑥₀,𝑦. (19)
By Kneser’s minimax theorem [21], together with 𝑧∈𝑃(𝑥₀,𝑤₀) for all 𝑧∈𝐷, we have inf_{𝑦∈𝑇(𝑥0)}max_𝑧∈𝐷𝜙𝑧,𝑥₀,𝑦=max_𝑧∈𝐷inf_{𝑦∈𝑇(𝑥0)}𝜙𝑧,𝑥₀,𝑦<0. (20)
This is a contradiction. Thus, all hypotheses of Corollary 7 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 inf_𝑦∈𝑇𝑥𝜙𝑧,𝑥,𝑦≥0,∀𝑧∈𝐴(𝑥)∩𝐹⁻¹(𝑤)^𝐶. (21)
Since 𝑇(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).