A Collectively Fixed Point Theorem in Abstract Convex Spaces and Its Applications

Abstract

The main purpose of this paper is to establish a new collectively fixed point theorem in noncompact abstract convex spaces. As applications of this theorem, we obtain some new existence theorems of equilibria for generalized abstract economies in noncompact abstract convex spaces.

1. Introduction

Collectively fixed point theorems for a family of set-valued mappings play a vital role in studying various nonlinear problems. In 1991, Tarafdar [1] established a collectively fixed point theorem in topological vector spaces and gave applications to mathematical economies, game theory, and problems of social sciences. Since then, a lot of generalizations and applications of collectively fixed point theorem under different assumptions and different underlying spaces have been studied by many authors (see [2–7] and the references therein).

Inspired and motivated by the above results, in this paper, we establish a new collectively fixed point theorem in noncompact abstract convex spaces. As applications of this fixed point theorem, some new existence theorems of equilibria for generalized abstract economies are proved under the setting of noncompact abstract convex spaces.

2. Preliminaries

Let 𝑋 be a set. We will denote by 2^𝑋 the family of all subsets of 𝑋, by ⟨𝑋⟩ the family of nonempty finite subsets of 𝑋. Let 𝐴 be a subset of a topological space 𝑋; we will denote the interior and the closure of 𝐴 by int𝐴 and cl𝐴, respectively. Let 𝑋 and 𝑌 be two nonempty sets and 𝑇∶𝑋→2^𝑌 a set-valued mapping. Then the set-valued mapping 𝑇⁻¹∶𝑌→2^𝑋 is defined by 𝑇⁻¹(𝑦)={𝑥∈𝑋∶𝑦∈𝑇(𝑥)} for each 𝑦∈𝑌.

Definition 1 (see [8]). An abstract convex space (𝐸,𝐷;Γ) consists of a topological space 𝐸, a nonempty set 𝐷, and a set-valued mapping Γ∶𝐷→2^𝐸 with nonempty values. One may denote Γ_𝐴∶=Γ(𝐴) for each 𝐴∈⟨𝐷⟩.

Let (𝐸,𝐷;Γ) be an abstract convex space. For any 𝐷^′⊂𝐷, the Γ-convex hull of 𝐷^′ is denoted and defined by co_Γ𝐷^′∶=Γ_𝐴∶𝐴∈𝐷^′⊆𝐸 (1) (co is reserved for the convex hull in vector spaces). A subset 𝑋 of 𝐸 is called a Γ-convex subset of (𝐸,𝐷;Γ) relative to 𝐷^′ if, for each 𝑁∈⟨𝐷^′⟩, we have Γ_𝑁⊆𝑋; that is, co_Γ(𝐷^′)⊆𝑋. This means that (𝑋,𝐷^′;Γ|_{⟨𝐷′⟩}) itself is an abstract convex space called a subspace of (𝐸,𝐷;Γ). When 𝐷⊆𝐸, the space is denoted by (𝐸⊇𝐷;Γ). In such case, a subset 𝑋 of 𝐸 is said to be Γ-convex if co_Γ(𝑋⋂𝐷)⊆𝑋; in other words, 𝑋 is Γ-convex relative to 𝐷^′∶=𝐷⋂𝑋. In case 𝐸=𝐷, let (𝐸;Γ)∶=(𝐸,𝐸;Γ).

Remark 2. There are a lot of examples of abstract convex spaces; see [7–13] and references therein. Here, for convenience, we give the following three examples of abstract convex spaces which are cited in this paper.(a)Let 𝑋 be a topological space and {𝐹_𝐴} be a given family of nonempty contractible subsets of 𝑋 indexed by 𝐴∈⟨𝑋⟩ such that 𝐹_𝐴⊆𝐹_𝐵 whenever 𝐴⊆𝐵. The couple (𝑋,𝐹_𝐴) is called an 𝐻-space (see [14]). A set 𝐷⊆𝑋 is said 𝐻-convex if 𝐹_𝐴⊆𝐷 for each 𝐴∈⟨𝐷⟩. A set 𝐾⊆𝑋 is said 𝐻-compact if, for each 𝐴∈⟨𝑋⟩, there is a compact 𝐻-convex set 𝐷⊆𝑋 such that 𝐾⋃𝐴⊆𝐷.(b)A generalized convex space or a 𝐺-convex space (𝑋,𝐷;Γ) (see [15]) consists of a topological space 𝑋 and a nonempty set 𝐷 such that, for each 𝐴∈⟨𝐷⟩ with the cardinality |𝐴|=𝑛+1, there exist a subset Γ(𝐴) of 𝑋 and a continuous function 𝜑_𝐴∶Δ_𝑛→Γ(𝐴) such that 𝐽∈⟨𝐴⟩ implies 𝜑_𝐴(Δ_𝐽)⊆Γ(𝐽). Here, Δ_𝑛 is the standard 𝑛-simplex with vertices {𝑒_𝑖}^𝑛_𝑖=0 and Δ_𝐽 the face of Δ_𝑛 corresponding to 𝐽∈⟨𝐴⟩; that is, if 𝐴={𝑎₀,𝑎₁,…,𝑎_𝑛} and 𝐽={𝑎_𝑖0,…,𝑎_𝑖𝑘}⊆𝐴, then Δ_𝐽=co{𝑒_𝑖0,𝑒_𝑖1,…,𝑒_𝑖𝑘}.(c)A semilattice (see [16]) is a partially ordered set 𝑋, with the partial ordering denoted by ≤, for which, any pair (𝑥,𝑥^′) of elements has a least upper bound, denoted by 𝑥∨𝑥^′. A topological semilattice is a topological space 𝑋 with a partial ordering ≤ for which it is a semilattice with a continuous sup operation; that is, the function 𝑋×𝑋→𝑋, (𝑥,𝑥^′)→𝑥∨𝑥^′ is continuous.

It is evident that each nonempty finite subset 𝐴 of a semilattice has a least upper bound, denoted by sup𝐴. In a partially ordered set (𝑋,≤), two arbitrary elements 𝑥 and 𝑥^′ do not have to be comparable. In case 𝑥≤𝑥^′, the set [𝑥,𝑥^′]={𝑦∈𝑋∶𝑥≤𝑦≤𝑥^′} is called an order interval. Now assume that (𝑋,≤) is a semilattice and 𝐴 is a nonempty finite subset of 𝑋. Then the set Δ(𝐴)∶=⋃_𝑎∈𝐴[𝑎,sup𝐴] is well defined. A subset 𝐷 of 𝑋 is called Δ-convex if, for any 𝑁∈⟨𝐷⟩, we have Δ(𝑁)⊆𝐷.

Definition 3 (see [13]). Let (𝐸,𝐷;Γ) be an abstract convex space and 𝑍 a set. For a set-valued mapping 𝐹∶𝐸→2^𝑍 with nonempty values, if a set-valued mapping 𝐺∶𝐷→2^𝑍 satisfies 𝐹(Γ_𝐴)⊆𝐺(𝐴) for  each 𝐴∈⟨𝐷⟩, then 𝐺 is called a KKM map with respect to 𝐹. A KKM map 𝐺∶𝐷→2^𝐸 is a KKM map with respect to the identity map 1_𝐸.

A set-valued mapping 𝐹∶𝐸→2^𝑍 is said to have the KKM property and called a 𝔎-map, if, for any KKM map 𝐺∶𝐷→2^𝑍 with respect to 𝐹, the family {𝐺(𝑦)∶𝑦∈𝐷} has the finite intersection property. We denote 𝔎(𝐸,𝐷,𝑍)∶=𝐹∶𝐸⟶2^𝑍∣𝐹isa𝔎-map. (2)

Let 𝑍 be a topological space. A set-valued mapping 𝐹∶𝐸→2^𝑍 is called a 𝔎ℭ-map (resp., 𝔎𝔒-map) if, for any closed-valued (resp., open-valued) KKM map 𝐺∶𝐷→2^𝑍 with respect to 𝐹, the family {𝐺(𝑦)∶𝑦∈𝐷} has the finite intersection property. In this case, we denote 𝐹∈𝔎ℭ(𝐸,𝐷,𝑍) (resp., 𝐹∈𝔎𝔒(𝐸,𝐷,𝑍)). When 𝐸=𝐷, we will write 𝔎ℭ(𝐸,𝑍) (resp., 𝔎𝔒(𝐸,𝑍)) instead of 𝔎ℭ(𝐸,𝐷,𝑍) (resp., 𝔎𝔒(𝐸,𝐷,𝑍)). Note that if 𝑍 is a discrete space, then three classes 𝔎, 𝔎ℭ, and 𝔎𝔒 are identical. For more details, we refer to [12, 13, 17, 18] and the references therein.

Definition 4 (see [17, 18]). The partial KKM principle for an abstract convex space (𝐸,𝐷;Γ) is the statement 1_𝐸∈𝔎ℭ(𝐸,𝐷,𝐸); that is, for any closed-valued KKM map 𝐺∶𝐷→2^𝐸, the family {𝐺(𝑦)∶𝑦∈𝐷} has the finite intersection property. The KKM principle is the statement that the same property also holds for any open-valued KKM map (i.e., 1_𝐸∈𝔎ℭ(𝐸,𝐷,𝐸)⋂𝔎𝔒(𝐸,𝐷,𝐸)).

An abstract convex space is called a (partial) KKM space if it satisfies the (partial) KKM principle. A lot of examples of (partial) KKM spaces can be found in [9, 17, 18] and the references therein.

Definition 5 (see [19]). Let 𝑋 be a topological space, 𝑌 a nonempty set, and 𝑇∶𝑋→2^𝑌 a set-valued mapping. 𝑇 is said to have local intersection property if, for each 𝑥∈𝑋 with 𝑇(𝑥)≠∅, there exists an open neighborhood 𝑁(𝑥) of 𝑥 such that ⋂_{𝑧∈𝑁(𝑥)}𝑇(𝑧)≠∅.

Remark 6. By Proposition 1 in Lin [19] and Lemma 3.1 in Llinares [20], we can see that the following conditions are equivalent.(i)𝑇 has the local intersection property and, for all 𝑥∈𝑋, 𝑇(𝑥) is nonempty;(ii)𝑋=⋃_𝑦∈𝑌int𝑇⁻¹(𝑦);(iii)for each 𝑦∈𝑌,  𝑇⁻¹(𝑦) contains an open subset 𝑂_𝑦 of 𝑋 and 𝑋=⋃_𝑦∈𝑌𝑂_𝑦.

Lemma 7 (see [8]). Let {(𝐸_𝑖,𝐷_𝑖;Γ_𝑖)}_𝑖∈𝐼 be a family of abstract convex spaces, where 𝐼 is an index set. Let 𝐸∶=∏_𝑖∈𝐼𝐸_𝑖 be equipped with the product topology and 𝐷∶=∏_𝑖∈𝐼𝐷_𝑖. For each 𝑖∈𝐼, let 𝜋_𝑖∶𝐷→𝐷_𝑖 be the projection. Define Γ=∏_𝑖∈𝐼Γ_𝑖∶⟨𝐷⟩→2^𝐸 by Γ(𝐴)∶=∏_𝑖∈𝐼Γ_𝑖(𝜋_𝑖(𝐴)) for each 𝐴∈⟨𝐷⟩. Then (𝐸,𝐷;Γ) is an abstract convex space.

Lemma 8 (see [13]). Let (𝐸,𝐷;Γ) be an abstract convex space, (𝑋,𝐷^′;Γ|_{⟨𝐷′⟩}) a subspace of (𝐸,𝐷;Γ), and 𝑍 a topological space. If 𝐹∈𝔎ℭ(𝐸,𝐷,𝑍), then 𝐹|_𝑋∈𝔎ℭ(𝑋,𝐷^′,cl(𝐹(𝑋))).

3. A Collectively Fixed Point Theorem

The following lemma is a special case of Theorem 11 in Park [12].

Lemma 9. Let (𝑋;Γ) be an abstract convex space with 1_𝑋∈𝔎ℭ(𝑋,𝑋) (i.e., (𝑋;Γ) satisfies the partial KKM principle). Let 𝑆,𝑇∶𝑋→2^𝑋 be two set-valued mappings satisfying the following conditions:(i)for each 𝑥∈𝑋, co_Γ𝑆(𝑥)⊆𝑇(𝑥) (i.e., for each 𝑀∈⟨𝑆(𝑥)⟩, Γ(𝑀)⊆𝑇(𝑥));(ii)𝑆⁻¹ has open values;(iii)there exists {𝑦₀,…,𝑦_𝑛}∈⟨𝑋⟩ such that 𝑋=⋃^𝑛_𝑗=0𝑆⁻¹(𝑦_𝑗).
Then 𝑇 has a fixed point 𝑥∈𝑋; that is, 𝑥∈𝑇(𝑥).

By Lemma 9, we can obtain the following collectively fixed point theorem which is the main result of our paper.

Theorem 10. Let 𝐼 be a finite index set; let {(𝑋_𝑖;Γ_𝑖)}_𝑖∈𝐼 be a family of abstract convex spaces such that (𝑋;Γ)∶=(∏_𝑖∈𝐼𝑋_𝑖;Γ) is an abstract convex space defined as in Lemma 7. Let 𝐾 be nonempty compact subset 𝑋. For each 𝑖∈𝐼, let 𝑆_𝑖, 𝑇_𝑖∶𝑋→2^𝑋𝑖 be set-valued mappings such that,(i)for each 𝑥∈𝑋, co_Γ𝑖𝑆_𝑖(𝑥)⊆𝑇_𝑖(𝑥);(ii)for each 𝑦_𝑖∈𝑋_𝑖, 𝑆_𝑖⁻¹(𝑦_𝑖) is open in 𝑋;(iii)𝐾⊆⋃_{𝑦𝑖∈𝑋𝑖}𝑆_𝑖⁻¹(𝑦_𝑖);(iv)for each 𝑁_𝑖∈⟨𝑋_𝑖⟩, there exists a compact Γ_𝑖-convex subset 𝐿_𝑁𝑖 of (𝑋_𝑖;Γ_𝑖) containing 𝑁_𝑖 such that, for 𝐿∶=∏_𝑖∈𝐼𝐿_𝑁𝑖, we have 𝐿⧵𝐾⊆𝑆_𝑖⁻¹𝑦_𝑖∶𝑦_𝑖∈𝐿_𝑁𝑖. (3)
If (𝑋;Γ) satisfies 1_𝑋∈𝔎ℭ(𝑋,𝑋), then there exists 𝑥=(𝑥_𝑖)_𝑖∈𝐼∈𝑋 such that 𝑥_𝑖∈𝑇_𝑖(𝑥) for each 𝑖∈𝐼.

Proof. Since 𝐾 is compact subset of 𝑋, by (ii) and (iii), for each 𝑖∈𝐼, there exists 𝑁_𝑖∈⟨𝑋_𝑖⟩ such that 𝐾⊆𝑆_𝑖⁻¹𝑦_𝑖∶𝑦_𝑖∈𝑁_𝑖. (4) Then by (iv), for each 𝑖∈𝐼, there exists a compact Γ_𝑖-convex subset 𝐿_𝑁𝑖 of (𝑋_𝑖;Γ_𝑖) containing 𝑁_𝑖 such that, for 𝐿∶=∏_𝑖∈𝐼𝐿_𝑁𝑖, we have 𝐿⧵𝐾⊆𝑆_𝑖⁻¹𝑦_𝑖∶𝑦_𝑖∈𝐿_𝑁𝑖. (5) By (4), we have 𝐿𝐾⊆𝑆_𝑖⁻¹𝑦_𝑖∶𝑦_𝑖∈𝐿_𝑁𝑖. (6) Then it follows from (5) and (6) that ()𝐿𝐾=𝑆𝐿=𝐿⧵𝐾_𝑖−1𝑦_𝑖𝐿∶𝑦_𝑖∈𝐿_𝑁𝑖,∀𝑖∈𝐼. (7) For each 𝑖∈𝐼, since 𝐿_𝑁𝑖 is Γ_𝑖-convex subset of (𝑋_𝑖;Γ_𝑖), it follows from Lemma 1 in Park [9] that (𝐿_𝑁𝑖;Γ_𝑖|_{⟨𝐿𝑁𝑖⟩}) is an abstract convex space, which is a subspace of (𝑋_𝑖;Γ_𝑖). Then by Lemma 7, (𝐿;Γ|_⟨𝐿⟩) is an abstract convex space, which is a subspace of (𝑋;Γ). Now, for each 𝑖∈𝐼, define two set-valued mappings 𝑆^′_𝑖,𝑇^′_𝑖∶𝐿→2^𝐿𝑁𝑖 by 𝑆^′_𝑖(𝑥)=𝑆_𝑖(𝑥)𝐿_𝑁𝑖,𝑇^′_𝑖(𝑥)=𝑇_𝑖(𝑥)𝐿_𝑁𝑖,∀𝑥∈𝐿. (8) Furthermore, we define two set-valued mappings 𝑆^′,𝑇^′∶𝐿→2^𝐿 by 𝑆^′(𝑥)=_𝑖∈𝐼𝑆^′_𝑖(𝑥)𝑇,∀𝑥∈𝐿,_′(𝑥)=_𝑖∈𝐼𝑇^′_𝑖(𝑥),∀𝑥∈𝐿. (9) Next, we prove that 𝑆^′ and 𝑇^′ satisfy all the conditions of Lemma 9 as follows.(a) For each 𝑥∈𝐿,  𝑀∈⟨𝑆^′(𝑥)⟩ implies that Γ|_⟨𝐿⟩(𝑀)⊆𝑇^′(𝑥). In fact, 𝑀∈⟨𝑆^′(𝑥)⟩ implies that 𝜋_𝑖(𝑀)∈⟨𝑆^′_𝑖(𝑥)⟩ for each 𝑖∈𝐼. Then it follows that 𝜋_𝑖(𝑀)∈⟨𝑆_𝑖(𝑥)⟩ for each 𝑖∈𝐼. By (i) and the fact that (𝐿_𝑁𝑖;Γ_𝑖|_{⟨𝐿𝑁𝑖⟩}) is a subspace of (𝑋_𝑖;Γ_𝑖) for each 𝑖∈𝐼, we have Γ_𝑖𝜋_𝑖(𝑀)=Γ_{𝑖⟨𝐿𝑁𝑖⟩}𝜋_𝑖(𝑀)⊆𝑇_𝑖(𝑥)Γ,∀𝑖∈𝐼,_𝑖𝜋_𝑖(𝑀)=Γ_{𝑖⟨𝐿𝑁𝑖⟩}𝜋_𝑖(𝑀)⊆𝐿_𝑁𝑖,∀𝑖∈𝐼. (10) Then it follows from (10) that Γ|_⟨𝐿⟩(𝑀)=_𝑖∈𝐼Γ_{𝑖⟨𝐿𝑁𝑖⟩}𝜋_𝑖(𝑀)⊆_𝑖∈𝐼𝑇_𝑖(𝑥)𝐿_𝑁𝑖=𝑇^′(𝑥), (11) which implies that co_{Γ|⟨𝐿⟩}𝑆^′(𝑥)⊆𝑇^′(𝑥) for each 𝑥∈𝐿.(b) For each 𝑦∈𝐿, (𝑆^′)⁻¹(𝑦) is relatively open in 𝐿. In fact, by (8), for each 𝑖∈𝐼 and each 𝑦_𝑖∈𝐿_𝑁𝑖, we have 𝑆^′_𝑖⁻¹𝑦_𝑖=𝑥∈𝐿∶𝑦_𝑖∈𝑆_𝑖(𝑥)𝐿_𝑁𝑖=𝐿𝑥∈𝑋∶𝑦_𝑖∈𝑆_𝑖(𝑥𝑆)=𝐿_𝑖⁻¹𝑦_𝑖. (12) Therefore, for each 𝑦∈𝐿, we obtain 𝑆^′−1𝑦=𝑥∈𝐿∶𝑦∈𝑆^′(𝑥=)𝑥∈𝐿∶𝑦_𝑖∈𝑆^′_𝑖(𝑥)=𝑆,∀𝑖∈𝐼𝑥∈𝐿∶𝑥∈^′_𝑖⁻¹𝑦_𝑖=,∀𝑖∈𝐼_𝑖∈𝐼𝑆^′_𝑖⁻¹𝑦_𝑖=𝐿_𝑖∈𝐼𝑆_𝑖⁻¹𝑦_𝑖. (13) By (ii) and the fact that 𝐼 is a finite index set, we know that (𝑆^′)⁻¹(𝑦) is relatively open in 𝐿 for each 𝑦∈𝐿.(c) There exists {𝑦₀,…,𝑦_𝑛}∈⟨𝐿⟩ such that 𝐿=⋃^𝑛_𝑗=0(𝑆^′)⁻¹(𝑦_𝑗). In fact, it follows from (7) and (12) that 𝐿=𝑆^′_𝑖⁻¹𝑦_𝑖∶𝑦_𝑖∈𝐿_𝑁𝑖,∀𝑖∈𝐼. (14) Let 𝑥∈𝐿 be given. Then by (14), for each 𝑖∈𝐼, there exists 𝑦_𝑖∈𝑆^′_𝑖(𝑥) such that 𝑥∈_𝑖∈𝐼𝑆^′_𝑖⁻¹𝑦_𝑖=𝑆^′−1𝑦,where𝑦=𝑦_{𝑖𝑖∈𝐼}∈𝐿. (15) Therefore, from the arbitrary of 𝑥, we have that 𝐿=⋃_𝑦∈𝐿(𝑆^′)⁻¹(𝑦). Since 𝐿 is compact, there exists {𝑦₀,…,𝑦_𝑛}∈⟨𝐿⟩ such that 𝐿=⋃^𝑛_𝑗=0(𝑆^′)⁻¹(𝑦_𝑗).(d) Since 1_𝑋∈𝔎ℭ(𝑋,𝑋), it follows from Lemma 8 that 1_𝐿∈𝔎ℭ(𝐿,𝐿).
Hence, by Lemma 9, there exists 𝑥=(𝑥_𝑖)_𝑖∈𝐼∈𝐿⊆𝑋 such that 𝑥∈𝑇^′(𝑥); that is, 𝑥_𝑖∈𝑇^′_𝑖(𝑥)⊆𝑇_𝑖(𝑥) for each 𝑖∈𝐼. This completes the proof.

Remark 11. Theorem 10 is a new result and completely different from the corresponding collectively fixed point theorems in [1–6], the proofs of which are mainly based on the unity partition theorem. Therefore, the topological spaces in these fixed point theorems satisfy Hausdorff property.

Theorem 12. Let 𝐼 be a finite index set; let {(𝑋_𝑖;Γ_𝑖)}_𝑖∈𝐼 be a family of abstract convex spaces such that (𝑋;Γ)∶=(∏_𝑖∈𝐼𝑋_𝑖;Γ) is an abstract convex space defined as in Lemma 7. Let 𝐾 be nonempty compact subset 𝑋. For each 𝑖∈𝐼, let 𝑆_𝑖,𝑇_𝑖∶𝑋→2^𝑋𝑖 be set-valued mappings such that,(i)for each 𝑥∈𝑋, co_Γ𝑖𝑆_𝑖(𝑥)⊆𝑇_𝑖(𝑥);(ii)𝐾⊆⋃_{𝑦𝑖∈𝑋𝑖}int𝑆_𝑖⁻¹(𝑦_𝑖);(iii)for each 𝑁_𝑖∈⟨𝑋_𝑖⟩, there exists a compact Γ_𝑖-convex subset 𝐿_𝑁𝑖 of (𝑋_𝑖;Γ_𝑖) containing 𝑁_𝑖 such that, for 𝐿∶=∏_𝑖∈𝐼𝐿_𝑁𝑖, we have 𝐿⧵𝐾⊆int𝑆_𝑖⁻¹𝑦_𝑖∶𝑦_𝑖∈𝐿_𝑁𝑖. (16)
If (𝑋;Γ) satisfies 1_𝑋∈𝔎ℭ(𝑋,𝑋), then there exists 𝑥=(𝑥_𝑖)_𝑖∈𝐼∈𝑋 such that 𝑥_𝑖∈𝑇_𝑖(𝑥) for each 𝑖∈𝐼.

Proof. For each 𝑖∈𝐼, define a set-valued mapping 𝑆_𝑖∶𝑋→2^𝑋𝑖 as follows: 𝑆_𝑖(𝑥)=int𝑆_𝑖⁻¹⁻¹(𝑥),∀𝑥∈𝑋. (17) Then by (i), we have co_Γ𝑖𝑆_𝑖(𝑥)⊆𝑇_𝑖(𝑥) for each 𝑖∈𝐼 and each 𝑥∈X. By (17), for each 𝑖∈𝐼 and each 𝑦_𝑖∈𝑋_𝑖, we have 𝑆_𝑖⁻¹𝑦_𝑖=int𝑆_𝑖⁻¹𝑦_𝑖, (18) which is open in 𝑋. By (ii) and (18), we have 𝐾⊆⋃_{𝑦𝑖∈𝑋𝑖}𝑆_𝑖⁻¹(𝑦_𝑖) for each 𝑖∈𝐼. By (iii) and (18), there exists a nonempty compact subset 𝐾 of 𝑋 such that, for each 𝑖∈𝐼 and each 𝑁_𝑖∈⟨𝑋_𝑖⟩, there exists a compact Γ_𝑖-convex subset 𝐿_𝑁𝑖 of (𝑋_𝑖;Γ_𝑖) containing 𝑁_𝑖 such that, for 𝐿∶=∏_𝑖∈𝐼𝐿_𝑁𝑖, we have 𝐿⧵𝐾⊆𝑆_𝑖⁻¹𝑦_𝑖∶𝑦_𝑖∈𝐿_𝑁𝑖. (19) Since (𝑋;Γ) satisfies 1_𝑋∈𝔎ℭ(𝑋,𝑋), it follows that all the conditions of Theorem 10 for 𝑆_𝑖 and 𝑇_𝑖 hold. Therefore, by Theorem 10, there exists 𝑥=(𝑥_𝑖)_𝑖∈𝐼∈𝑋 such that 𝑥_𝑖∈𝑇_𝑖(𝑥) for each 𝑖∈𝐼.

Remark 13. We have shown that Theorem 10 implies Theorem 12. It is clear that Theorem 12 implies Theorem 10. Therefore, Theorem 10 is equivalent to Theorem 12.

4. Particular Fixed Point Theorems

In this section, we give simple consequences of Theorems 10 and 12 and their applications obtained by other authors. We omit their proofs.

Proposition 14 (see [21]). Let 𝑋₁,…,𝑋_𝑛 (𝑛≥1) be nonempty compact convex sets, each in a topological vector space (not necessary Hausdorff). Let each 𝑀_𝑖∶𝑋→2^𝑋𝑖 have the local intersection property and nonempty convex values. Then there exists 𝑥∈𝑋 such that 𝑥_𝑖∈𝑀_𝑖(𝑥) for each 𝑖∈{1,…,𝑛}.

Remark 15. By Remarks 6 and 13, we know that Proposition 14 is a particular form of Theorems 10 and 12.

By using Proposition 14, Prokopovych [21] proved a theorem on the existence of a pure strategy 𝜀-Nash equilibrium in every compact, quasiconcave, and payoff secure game; meanwhile, by means of Proposition 14, he also proved an approximate equilibrium existence theorem that covers a number of known game models.

By Proposition 14, we can obtain the following famous Fan-Browder fixed theorem.

The Fan-Browder Fixed Point Theorem (see [22, 23]).  Let 𝑋 be a nonempty compact convex subset of a topological vector space. Let 𝑀∶𝑋→2^𝑋 be a set-valued mapping such that it has nonempty convex values and open inverse values (i.e., each 𝑀⁻¹(𝑦) is open in 𝑋). Then 𝑀 has a fixed point.

By using the Fan-Browder fixed point theorem, Yu and Yuan [24] obtained the existence results of weight Nash-equilibria and Pareto equilibria for multiobjective games. Kim and Yuan [25] applied the Fan-Browder fixed point theorem to prove a maximal element theorem for 𝐿-majorized mappings in topological vector spaces from which they obtained an existence theorem of maximal elements for the family of 𝐿-majorized mappings in which domains are not compact. Balaj and Muresan [26] obtained two minimax inequalities by using the Fan-Browder fixed point theorem. Recently, Luo [27] applied the Fan-Browder fixed point theorem to establish some generalized Ky Fan minimax inequalities for vector-valued mappings.

In 1992, Park [28] generalized the Fan-Browder fixed point theorem to noncompact setting and obtained the following result, which is a particular case of Theorems 10 and 12 with 𝐼={1}.

Proposition 16 (see [28]). Let 𝑋 be a nonempty convex subset of a topological vector space, 𝐴⊆𝑋 nonempty convex, and 𝐷⊆𝐴 a nonempty compact subset. Let 𝑆∶𝐴→2^𝐴 and 𝐿∶𝐴→2^𝐴 be two set-valued mappings. Assume that,(a)for each 𝑥∈𝐴, 𝐿(𝑥) is convex and 𝑆(𝑥)⊆𝐿(𝑥);(b)for each 𝑥∈𝐷, 𝑆(𝑥)≠∅;(c)for each 𝑦∈𝐴, 𝑆⁻¹(𝑦) is open in 𝐴;(d)for each finite subset 𝑁 of 𝐴, there is a compact convex subset 𝐿_𝑁 such that 𝑁⊆𝐿_𝑁⊆𝐴 and 𝑆(𝑥)∩𝐿_𝑁≠∅ for all 𝑥∈𝐿_𝑁⧵𝐷.
Then 𝐿 has a fixed point.

Remark 17. The coercivity condition (d) in Proposition 16 can be replaced by the following condition:(d)′ for each finite subset 𝑁 of 𝐴, there is a compact convex subset 𝐿_𝑁 such that 𝑁⊆𝐿_𝑁⊆𝐴 and 𝐿_𝑁⧵𝐷⊆⋃_{𝑦∈𝐿𝑁}𝑆⁻¹(𝑦).

Many authors applied Proposition 16 or particular forms of Proposition 16 to study various nonlinear problems in topological vector spaces, for example, generalized Minty vector variational inequality problems, generalized variational inequality problems, generalized vector equilibrium problems, and the constrained or the competitive Nash type equilibrium problems. For more details, see Park [29] and the references therein.

The following particular form of Proposition 16 can be found in Ding and Tan [30].

Proposition 18 (see [30]). Let 𝑋 be a nonempty convex subset of a topological vector space. Let 𝑆∶𝑋→2^𝑋 be a set-valued mapping. Assume that,(a)for each 𝑥∈𝑋,  𝑆(𝑥)≠∅;(b)for each 𝑦∈𝑋, 𝑆⁻¹(𝑦) is compactly open; that is, for each nonempty compact subset 𝐶 of 𝑋, 𝑆⁻¹(𝑦)⋂𝐶 is open in 𝐶; (c)there exist a nonempty compact convex subset 𝑋₀ of 𝑋 and a nonempty compact subset 𝐾 of 𝑋 (not necessarily convex) such that, for each 𝑥∈𝑋⧵𝐾, we have co(𝑋₀⋃{𝑥})⋂co𝑆(𝑥)≠∅.
Then there exists 𝑥 such that 𝑥∈co𝑆(𝑥).

Ding and Tan [30] applied Proposition 18 to obtain an existence theorem of equilibria for one person games. By using Proposition 18, Ding and Yuan [31] proved a maximal element theorem from which they obtained some existence theorems of equilibria for generalized games without lower semicontinuity for both constraint and preference correspondences.

Remark 19. In condition (b) of Proposition 18, “compactly” can be removed; see Park [11].

The following result (see Corollary 2.3 in Tarafdar [2]) is known in the setting of 𝐻-spaces without linear structure.

Proposition 20 (see [2]). Let (𝑋,𝐹_𝐴) be a compact 𝐻-space and 𝑇∶𝑋→2^𝑋 a set-valued mapping such that,(a)for each 𝑥∈𝑋, 𝑇(𝑥) is a nonempty 𝐻-convex subset of 𝑋;(b)for each 𝑦∈𝑋, 𝑇⁻¹(𝑦) contains an open set 𝑂_𝑦 (𝑂_𝑦 may be empty for some 𝑦);(c)𝑋=⋃_𝑦∈𝑋𝑂_𝑦.
Then 𝑇 has a fixed point.

By using Proposition 20, Chang et al. [32] proved an existence theorem of solutions for the quasi-variational inequality problem in the setting of 𝐻-spaces. After that, Chang et al. [33] applied Proposition 20 with each 𝑇⁻¹(𝑦) being open to prove some existence theorems of loose saddle point, saddle point, and minimax problems for vector-valued multifunctions in the framework of 𝐻-spaces. On the basis of an equivalent form of Proposition 20, Wu [34] obtained two existence theorems for maximal elements in 𝐻-spaces from which he proved the existence of solutions of Fan-Yen minimax inequalities, qualitative games, and abstract economies.

The following result is a noncompact generalization of Proposition 20.

Proposition 21 (see [35]). Let (𝑋,𝐹_𝐴) be an 𝐻-space, 𝐷⊆𝑋 an 𝐻-compact set, and 𝑇∶𝑋→2^𝑋 a set-valued mapping such that,(a)for each 𝑥∈𝑋, 𝑇(𝑥) is a nonempty 𝐻-convex subset of 𝑋;(b)for each 𝑦∈𝑋, 𝑇⁻¹(𝑦) is open in 𝑋;(c)𝑇(𝑥)⋂𝐷≠∅ for  all 𝑥∈𝑋.
Then 𝑇 has a fixed point.

By using Proposition 21, Cubiotti and Nordo [36] obtained an existence result for the Nash equilibria of generalized games with strategy sets in 𝐻-spaces.

Luo [37] proved the following fixed point theorem in topological ordered spaces.

Proposition 22 (see [37]). Let 𝑋 be a nonempty compact Δ-convex subset of a topological semilattice with path-connected intervals and 𝑇∶𝑋→2^𝑋 a set-valued mapping such that(a)for each 𝑥∈𝑋, 𝑇(𝑥) is nonempty and Δ-convex;(b)for each 𝑦∈𝑋, 𝑇⁻¹(𝑦) is open in 𝑋.
Then 𝑇 has a fixed point.

Luo [37, 38] applied Proposition 22 to prove a saddle-point theorem, existence theorems of solutions for some generalized quasi-Ky Fan inequalities, and Nash equilibrium points for a game system in the setting of topological ordered spaces. By using Proposition 22, Vinh [39] proved a coincidence theorem from which he obtained a Sion-Neumann type minimax theorem.

As it is well known, 𝐺-convex spaces are typical example of abstract convex spaces. The following extension of the Fan-Browder fixed point theorem to 𝐺-convex spaces is a particular form of Theorem 3.3 in Park [29], and it includes the fixed point theorems mentioned previously as special cases.

Proposition 23. Let (𝑋;Γ) be a 𝐺-convex space, and let 𝑆, 𝑇∶𝑋→2^𝑋 be two set-valued mappings such that,(a)for each 𝑥∈𝑋, 𝑁∈⟨𝑆(𝑥)⟩ implies Γ(𝐴)⊆𝑇(𝑥);(b)𝑋=⋃_𝑦∈𝑋int𝑆⁻¹(𝑦);(c)there exists a nonempty compact subset 𝐾 of 𝑋 such that, for each 𝑁∈⟨𝑋⟩, there exists a compact 𝐺-convex subset 𝐿_𝑁 of (𝑋;Γ) containing 𝑁 such that𝐿⧵𝐾⊆int𝑆⁻¹𝑦∶𝑦∈𝐿_𝑁. (20)
Then there exists 𝑥∈𝑋 such that 𝑥∈𝑇(𝑥).

Remark 24. The coercivity condition (c) in Proposition 23 can be replaced by the following equivalent condition:(c)′ there exists a nonempty compact subset 𝐾 of 𝑋 such that, for each 𝑁∈⟨𝑋⟩, there exists a compact 𝐺-convex subset 𝐿_𝑁 of (𝑋;Γ) containing 𝑁 such that 𝐿𝑋⧵int𝑆⁻¹𝑦∶𝑦∈𝐿_𝑁⊆𝐾. (21)
Lin [19] applied the equivalent form of Proposition 23 to obtain some minimax inequalities, existence of maximal element, intersection theorems, and KKM type theorems. At the same year, Lin and Yu [40] applied special cases of Proposition 23 to study scalar equilibrium problems and vectorial equilibrium problems in the setting of 𝐺-convex spaces. Ding and Park [41] applied Proposition 23 to a class of abstract generalized vector equilibrium problems in 𝐺-convex spaces. Recently, by using Proposition 23, Balaj and Lin [42] proved a new fixed point theorem for set-valued mappings in 𝐺-convex spaces from which they obtained some coincidence theorems and existence theorems for maximal elements. Applications of these results to generalized equilibrium and minimax theory were also given.

In 2010, Park [7] established the following generalized Fan-Browder fixed point theorem in abstract convex spaces.

Proposition 25 (see [7]). Let 𝐼 be a finite index set; let {(𝑋_𝑖;Γ_𝑖)}_𝑖∈𝐼 be a family of compact abstract convex spaces such that (𝑋;Γ)∶=(∏_𝑖∈𝐼𝑋_𝑖;Γ) is an abstract convex space as defined as in Lemma 7, and satisfies the partial KKM principle. For each 𝑖∈𝐼, let 𝑇_𝑖∶𝑋→2^𝑋𝑖 be a set-valued mapping such that,(a)for each 𝑥∈𝑋, co_Γ𝑖𝑆_𝑖(𝑥)⊆𝑇_𝑖(𝑥);(b)𝑋=⋃_{𝑦𝑖∈𝑋𝑖}int𝑆_𝑖⁻¹(𝑦_𝑖).
Then there exists 𝑥=(𝑥_𝑖)_𝑖∈𝐼∈𝑋 such that 𝑥_𝑖∈𝑇_𝑖(𝑥) for each 𝑖∈𝐼.

Park [7] applied Proposition 25 to establish the von Neumann-Fan type intersection theorem under the setting of abstract convex spaces satisfying the partial KKM principle. By using Proposition 25 with 𝐼 being a singleton and 𝑆=𝑇, Yang et al. [43] established some minimax theorems for vector-valued mappings in abstract convex spaces. They also gave some examples to illustrate their results.

5. Equilibria for Generalized Abstract Economies

Considering any preference of a real agent could be unstable because of the fuzziness of consumers’ behavior or market situations, Kim and Tan [44] introduced the fuzzy constraint correspondences in defining the following generalized abstract economy.

Let 𝐼 be any set of agents. For each 𝑖∈𝐼, let 𝑋_𝑖 be the strategy set or commodity space of the agent 𝑖, and let 𝑋=∏_𝑖∈𝐼𝑋_𝑖. Following the method of Kim and Tan [44], let 𝜀=(𝑋_𝑖,𝐴_𝑖,𝐵_𝑖,𝐹_i,𝑃_𝑖)_𝑖∈𝐼 be a generalized abstract economy, where 𝐴_𝑖,𝐵_𝑖∶𝑋=∏_𝑖∈𝐼𝑋_𝑖→2^𝑋𝑖 are two constrained correspondences such that 𝐴_𝑖(𝑥) and 𝐵_𝑖(𝑥) are the states attainable for the agent 𝑖 at 𝑥; 𝐹_𝑖∶𝑋→2^𝑋𝑖 is a fuzzy constrained correspondence such that 𝐹_𝑖(𝑥) is the unstable state for the agent 𝑖 at 𝑥, and 𝑃_𝑖∶𝑋×𝑋→2^𝑋𝑖 is a preference correspondence such that 𝑃_𝑖(𝑥,𝑦) is the state preference by the agent 𝑖 at (𝑥,𝑦). An equilibrium for 𝜀 is a point (𝑥,𝑦)∈𝑋×𝑋 such that for each 𝑖∈𝐼,  𝑥_𝑖∈𝐵_𝑖(𝑥),  𝑦_𝑖∈𝐹_𝑖(𝑥), and 𝑃_𝑖(𝑥,𝑦)∩𝐴_𝑖(𝑥)=∅.

As an application of Theorem 10, we derive the following equilibrium existence theorem for generalized abstract economies in noncompact abstract convex spaces.

Theorem 26. Let 𝐼 be a finite index set, {(𝑋_𝑖;Γ_𝑖)}_𝑖∈𝐼 a family of abstract convex spaces such that (𝑋;Γ)∶=(∏_𝑖∈𝐼𝑋_𝑖;Γ) and (𝑋×𝑋;Γ×Γ) are two abstract convex spaces as defined in Lemma 7. Let 𝜀=((𝑋_𝑖;Γ_𝑖),𝐴_𝑖,𝐵_𝑖,𝐹_𝑖,𝑃_𝑖)_𝑖∈𝐼 be a generalized abstract economy, and let 𝐾 be a nonempty compact subset of 𝑋×𝑋. For each 𝑖∈𝐼, assume that,(i)for each 𝑥∈𝑋, 𝐴_𝑖(𝑥)≠∅ and co_Γ𝑖𝐴_𝑖(𝑥)⊆𝐵_𝑖(𝑥);(ii)for each 𝑥∈𝑋, 𝐹_𝑖(𝑥) is nonempty Γ_𝑖-convex;(iii)for each (𝑥,𝑦)∈𝑊_𝑖={(𝑥,𝑦)∈𝑋×𝑋∶𝐴_𝑖(𝑥)⋂𝑃_𝑖(𝑥,𝑦)≠∅}, 𝑥_𝑖∉co_Γ𝑖𝑃_𝑖(𝑥,𝑦);(iv)for each (𝑢_𝑖,𝑣_𝑖)∈𝑋_𝑖×𝑋_𝑖, [(𝐴_𝑖⁻¹(𝑢_𝑖)⋂𝐹_𝑖⁻¹(𝑣_𝑖))×𝑋]⋂[𝑃_𝑖⁻¹(𝑢_𝑖)⋃(𝑋×𝑋⧵𝑊_𝑖)] is open in 𝑋×𝑋;(v)for each 𝑁_0𝑖×𝑁_1𝑖∈⟨𝑋_𝑖×𝑋_𝑖⟩, there exist compact Γ_𝑖-convex subsets 𝐿_𝑁0𝑖, 𝐿_𝑁1𝑖 of (𝑋_𝑖;Γ_𝑖) containing 𝑁_0𝑖,𝑁_1𝑖, respectively, such that, for 𝐿∶=∏_𝑖∈𝐼𝐿_𝑁0𝑖×∏_𝑖∈𝐼𝐿_𝑁1𝑖, we have 𝐿⧵𝐾⊆_{𝑢𝑖,𝑣𝑖∈𝐿𝑁0𝑖×𝐿𝑁1𝑖}𝐴_𝑖⁻¹𝑢_𝑖𝐹_𝑖⁻¹𝑣_𝑖𝑃×𝑋_𝑖−1𝑢_𝑖𝑋×𝑋⧵𝑊_𝑖. (22)
If (𝑋×𝑋;Γ×Γ) satisfies 1_𝑋×𝑋∈𝔎ℭ(𝑋×𝑋,𝑋×𝑋), then there exists (𝑥,𝑦)∈𝑋×𝑋 such that, for each 𝑖∈𝐼, 𝑥_𝑖∈𝐵_𝑖(𝑥), 𝑦_𝑖∈𝐹_𝑖(𝑥) and 𝑃_𝑖(𝑥,𝑦)∩𝐴_𝑖(𝑥)=∅.

Proof. By Lemma 7, for each 𝑖∈𝐼, (𝑋_𝑖×𝑋_𝑖;Γ_𝑖×Γ_𝑖) is an abstract convex space. For each 𝑖∈𝐼, define two set-valued mappings 𝑆_𝑖, 𝑇_𝑖∶𝑋×𝑋→2^{X𝑖×𝑋𝑖} by 𝑆_𝑖=𝑃𝑥,𝑦^𝑖𝑥,𝑦⋂𝐴_𝑖(𝑥)×𝐹_𝑖(𝑥),if𝑥,𝑦∈𝑊_𝑖,𝐴_𝑖(𝑥)×𝐹_𝑖(𝑥),if𝑥,𝑦∈𝑋×𝑋⧵𝑊_𝑖,𝑇_𝑖𝑥,𝑦=co_Γ𝑖𝑃_𝑖𝑥,𝑦⋂𝐵_𝑖(𝑥)×𝐹_𝑖(𝑥),if𝑥,𝑦∈𝑊_𝑖,𝐵_𝑖(𝑥)×𝐹_𝑖(𝑥),if𝑥,𝑦∈𝑋×𝑋⧵𝑊_𝑖. (23) By (i), (ii), and the definition of 𝑊_𝑖, we have 𝑆_𝑖(𝑥,𝑦)≠∅ and co_{Γ𝑖×Γ𝑖}(𝑆_𝑖(𝑥,𝑦))⊆𝑇_𝑖(𝑥,𝑦) for each 𝑖∈𝐼 and for all (𝑥,𝑦)∈𝑋×𝑋. For each 𝑖∈𝐼 and for all (𝑢_𝑖,𝑣_𝑖)∈𝑋_𝑖×𝑋_𝑖, we see from 𝑆_𝑖⁻¹𝑢_𝑖,𝑣_𝑖=𝑃_𝑖⁻¹𝑢_𝑖𝐴_𝑖⁻¹𝑢_𝑖×𝑋𝐹_𝑖⁻¹𝑣_𝑖×𝑋𝑋×𝑋⧵𝑊_𝑖𝐴_𝑖⁻¹𝑢_𝑖𝐹×𝑋_𝑖−1𝑣_𝑖=𝐴×𝑋_𝑖−1𝑢_𝑖𝐹_𝑖⁻¹𝑣_𝑖𝑃×𝑋_𝑖−1𝑢_𝑖𝑋×𝑋⧵𝑊_𝑖 (24) and (iv) that 𝑆_𝑖⁻¹(𝑢_𝑖,𝑣_𝑖) is open in 𝑋×𝑋. Since 𝑆_𝑖(𝑥,𝑦)≠∅ for each 𝑖∈𝐼 and for all (𝑥,𝑦)∈𝑋×𝑋, we have the following: 𝑋×𝑋=_{(𝑢𝑖,𝑣𝑖)∈𝑋𝑖×𝑋𝑖}𝑆_𝑖⁻¹𝑢_𝑖,𝑣_𝑖,∀𝑖∈𝐼, (25) and so, we have 𝐾⊆𝑋×𝑋=_{(𝑢𝑖,𝑣𝑖)∈𝑋𝑖×𝑋𝑖}𝑆_𝑖⁻¹𝑢_𝑖,𝑣_𝑖,∀𝑖∈𝐼. (26) By (v), for each 𝑖∈𝐼 and each 𝑁_𝑖=𝑁_0𝑖×𝑁_1𝑖∈⟨𝑋_𝑖×𝑋_𝑖⟩, there exists a compact Γ_𝑖×Γ_𝑖-convex subset 𝐿_𝑁0𝑖×𝐿_𝑁1𝑖 of (𝑋_𝑖×𝑋_𝑖;Γ_𝑖×Γ_𝑖) containing 𝑁_𝑖, such that, for 𝐿∶=∏_𝑖∈𝐼𝐿_𝑁0𝑖×∏_𝑖∈𝐼𝐿_𝑁1𝑖, we have 𝐿⧵𝐾⊆𝑆_𝑖⁻¹𝑢_𝑖,𝑣_𝑖∶𝑢_𝑖,𝑣_𝑖∈𝐿_𝑁0𝑖×𝐿_𝑁1𝑖. (27) Since (𝑋×𝑋;Γ×Γ) satisfies 1_𝑋×𝑋∈𝔎ℭ(𝑋×𝑋,𝑋×𝑋), we can see that all the conditions of Theorem 10 are satisfied. So, by Theorem 10, there exists (𝑥,𝑦)∈𝑋×𝑋 such that (𝑥_𝑖,𝑦_𝑖)∈𝑇_𝑖(𝑥,𝑦) for each 𝑖∈𝐼. If (𝑥,𝑦)∈𝑊_𝑖 for some 𝑖, then we have 𝑥_𝑖,𝑦_𝑖∈co_Γ𝑖𝑃_𝑖𝑥,𝑦𝐵_𝑖(𝑥)×𝐹_𝑖(𝑥). (28) And, hence, 𝑥_𝑖∈co_Γ𝑖𝑃_i(𝑥,𝑦)∩𝐵_𝑖(𝑥) and so 𝑥_𝑖∈co_Γ𝑖𝑃_𝑖(𝑥,𝑦), which contradicts (iii). Therefore, we must have (𝑥,𝑦)∈𝑋×𝑋⧵𝑊_𝑖 for all 𝑖∈𝐼. It follows from the definitions of 𝑇_𝑖 and 𝑊_𝑖 that, for each 𝑖∈𝐼, 𝑥_𝑖∈𝐵_𝑖(𝑥), 𝑦_𝑖∈𝐹_𝑖(𝑥), and 𝑃_𝑖(𝑥,𝑦)∩𝐴_𝑖(𝑥)=∅.

Remark 27. Theorem 26 is a new result, which can be compared with Theorem 3.2.1 in [45], Theorems 4.1-4.2 in [46], and Theorem 3.1 in [47] in several aspects.

Corollary 28. Let 𝐼 be a finite index set; let {(𝑋_𝑖;Γ_𝑖)}_𝑖∈𝐼 be a family of abstract convex spaces such that (𝑋;Γ)∶=(∏_𝑖∈𝐼𝑋_𝑖;Γ) and (𝑋×𝑋;Γ×Γ) are two abstract convex spaces as defined in Lemma 7. Let 𝜀=((𝑋_𝑖;Γ_𝑖),𝐴_𝑖,𝐵_𝑖,𝐹_𝑖,𝑃_𝑖)_𝑖∈𝐼 be a generalized abstract economy, and let 𝐾 be a nonempty compact subset of 𝑋×𝑋. For each 𝑖∈𝐼, assume that,(i)for each 𝑥∈𝑋, 𝐴_𝑖(𝑥)≠∅ and co_Γ𝑖𝐴_𝑖(𝑥)⊆𝐵_𝑖(𝑥);(ii)for each 𝑥∈𝑋, 𝐹_𝑖(𝑥) is nonempty Γ_𝑖-convex;(iii) for each (𝑥,𝑦)∈𝑊_𝑖={(𝑥,𝑦)∈𝑋×𝑋∶𝐴_𝑖(𝑥)⋂𝑃_𝑖(𝑥,𝑦)≠∅}, 𝑥_𝑖∉co_Γ𝑖𝑃_𝑖(𝑥,𝑦);(iv)for each 𝑢_𝑖∈𝑋_𝑖, 𝐴_𝑖⁻¹(𝑢_𝑖), 𝐹_𝑖⁻¹(𝑢_𝑖), and 𝑃_𝑖⁻¹(𝑢_𝑖) are open sets;(v)the set 𝑊_𝑖 is closed in 𝑋×𝑋;(vi)for each 𝑁_0𝑖×𝑁_1𝑖∈⟨𝑋_𝑖×𝑋_𝑖⟩, there exist compact Γ_𝑖-convex subsets 𝐿_𝑁0𝑖,𝐿_𝑁1𝑖 of (𝑋_𝑖;Γ_𝑖) containing 𝑁_0𝑖,𝑁_1𝑖, respectively, such that, for 𝐿∶=∏_𝑖∈𝐼𝐿_𝑁0𝑖×∏_𝑖∈𝐼𝐿_𝑁1𝑖, we have 𝐿⧵𝐾⊆_{𝑢𝑖,𝑣𝑖∈𝐿𝑁0𝑖×𝐿𝑁1𝑖}A_𝑖⁻¹𝑢_𝑖𝐹_𝑖⁻¹𝑣_𝑖𝑃×𝑋_𝑖−1𝑢_𝑖𝑋×𝑋⧵𝑊_𝑖. (29)
If (𝑋×𝑋;Γ×Γ) satisfies 1_𝑋×𝑋∈𝔎ℭ(𝑋×𝑋,𝑋×𝑋), then there exists (𝑥,𝑦)∈𝑋×𝑋 such that, for each 𝑖∈𝐼, 𝑥_𝑖∈𝐵_𝑖(𝑥), 𝑦_𝑖∈𝐹_𝑖(𝑥), and 𝑃_𝑖(𝑥,𝑦)∩𝐴_𝑖(𝑥)=∅.

Proof. By (iv) and (v), for each (𝑢_𝑖,𝑣_𝑖)∈𝑋_𝑖×𝑋_𝑖, the set [(𝐴_𝑖⁻¹(𝑢_𝑖)∩𝐹_𝑖⁻¹(𝑣_𝑖))×𝑋]∩[𝑃_𝑖⁻¹(𝑢_𝑖)∪(𝑋×𝑋⧵𝑊_𝑖)] is open in 𝑋×𝑋. Hence, the conclusion of Corollary 28 follows from Theorem 26.

Corollary 29. Let 𝐼 be a finite index set; let {(𝑋_𝑖;Γ_𝑖)}_𝑖∈𝐼 be a family of abstract convex spaces such that (𝑋;Γ)∶=(∏_𝑖∈𝐼𝑋_𝑖;Γ) and (𝑋×𝑋;Γ×Γ) are two abstract convex spaces as defined in Lemma 7. Let 𝜀=((𝑋_𝑖;Γ_𝑖),𝐴_𝑖,𝐵_𝑖,𝐹_𝑖,𝑃_𝑖)_𝑖∈𝐼 be a generalized abstract economy, and let 𝐾 be a nonempty compact subset of 𝑋×𝑋. For each 𝑖∈𝐼, assume that,(i)for each 𝑥∈𝑋, 𝐴_𝑖(𝑥)≠∅ and co_Γ𝑖𝐴_𝑖(𝑥)⊆𝐵_𝑖(𝑥);(ii)for each 𝑥∈𝑋, 𝐹_𝑖(𝑥) is nonempty Γ_𝑖-convex;(iii)for each (𝑥,𝑦)∈𝑊_𝑖={(𝑥,𝑦)∈𝑋×𝑋∶𝐴_𝑖(𝑥)∩𝑃_𝑖(𝑥,𝑦)≠∅}, 𝑥_𝑖∉cl(co_Γ𝑖𝑃_𝑖(𝑥,𝑦));(iv)for each 𝑢_𝑖∈𝑋_𝑖, 𝐴_𝑖⁻¹(𝑢_𝑖), 𝐹_𝑖⁻¹(𝑢_𝑖), and 𝑃_𝑖⁻¹(𝑢_𝑖) are open sets;(v)the set 𝑊_𝑖 is closed in 𝑋×𝑋;(vi)for each 𝑁_0𝑖×𝑁_1𝑖∈⟨𝑋_𝑖×𝑋_𝑖⟩, there exist compact Γ_𝑖-convex subsets 𝐿_𝑁0𝑖,𝐿_𝑁1𝑖 of (𝑋_𝑖;Γ_𝑖) containing 𝑁_0𝑖,𝑁_1𝑖, respectively, such that, for 𝐿∶=∏_𝑖∈𝐼𝐿_𝑁0𝑖×∏_𝑖∈𝐼𝐿_𝑁1𝑖, we have 𝐿⧵𝐾⊆_{(𝑢𝑖,𝑣𝑖)∈𝐿𝑁0𝑖×𝐿𝑁1𝑖}𝐴_𝑖⁻¹𝑢_𝑖𝐹_𝑖⁻¹𝑣_𝑖𝑃×𝑋_𝑖−1𝑢_𝑖𝑋×𝑋⧵𝑊_𝑖. (30)
If (𝑋×𝑋;Γ×Γ) satisfies 1_𝑋×𝑋∈𝔎ℭ(𝑋×𝑋,𝑋×𝑋), then there exists (𝑥,𝑦)∈𝑋×𝑋 such that, for each 𝑖∈𝐼, 𝑥_𝑖∈𝐵_𝑖(𝑥), 𝑦_𝑖∈𝐹_𝑖(𝑥), and 𝑃_𝑖(𝑥,𝑦)∩𝐴_𝑖(𝑥)=∅.

Proof. By (iii), for each (𝑥,𝑦)∈𝑊_𝑖={(𝑥,𝑦)∈𝑋×𝑋∶𝐴_𝑖(𝑥)∩𝑃_𝑖(𝑥,𝑦)≠∅}, 𝑥_𝑖∉co_Γ𝑖𝑃_𝑖(𝑥,𝑦). Hence, the conclusion of Corollary 29 follows from Corollary 28.

If 𝐴_𝑖(𝑥)=𝐵_𝑖(𝑥)=𝑋_𝑖 for each 𝑖∈𝐼 and each 𝑥∈𝑋 in Theorem 26, then we have the following theorem.

Theorem 30. Let 𝐼 be a finite index set; let {(𝑋_𝑖;Γ_𝑖)}_𝑖∈𝐼 be a family of abstract convex spaces such that (𝑋;Γ)∶=(∏_𝑖∈𝐼𝑋_𝑖;Γ) and (𝑋×𝑋;Γ×Γ) are two abstract convex spaces as defined as in Lemma 7. Let 𝜀=((𝑋_𝑖;Γ_𝑖),𝐹_𝑖,𝑃_𝑖)_𝑖∈𝐼 be a generalized qualitative game, and let 𝐾 be a nonempty compact subset of 𝑋×𝑋. For each 𝑖∈𝐼, assume that,(i)for each (𝑥,𝑦)∈𝑊_𝑖={(𝑥,𝑦)∈𝑋×𝑋∶𝑃_𝑖(𝑥,𝑦)≠∅}, 𝑥_𝑖∉co_Γ𝑖𝑃_𝑖(𝑥,𝑦);(ii)for each (𝑢_𝑖,𝑣_𝑖)∈𝑋_𝑖×𝑋_𝑖, [𝐹_𝑖⁻¹(𝑣_𝑖)×𝑋]⋂[𝑃_𝑖⁻¹(𝑢_𝑖)⋃(𝑋×𝑋⧵𝑊_𝑖)] is open in 𝑋×𝑋;(iii)for each 𝑁_0𝑖×𝑁_1𝑖∈⟨𝑋_𝑖×𝑋_𝑖⟩, there exist compact Γ_𝑖-convex subsets 𝐿_𝑁0𝑖,𝐿_𝑁1𝑖 of (𝑋_𝑖;Γ_𝑖) containing 𝑁_0𝑖,𝑁_1𝑖, respectively, such that, for 𝐿∶=∏_𝑖∈𝐼𝐿_𝑁0𝑖×∏_𝑖∈𝐼𝐿_𝑁1𝑖, we have 𝐿⧵𝐾⊆_{(𝑢𝑖,𝑣𝑖)∈𝐿𝑁0𝑖×𝐿𝑁1𝑖}𝐹_𝑖⁻¹𝑣_𝑖𝑃×𝑋_𝑖−1𝑢_𝑖𝑋×𝑋⧵𝑊_𝑖. (31)
If (𝑋×𝑋;Γ×Γ) satisfies 1_𝑋×𝑋∈𝔎ℭ(𝑋×𝑋,𝑋×𝑋), then the generalized qualitative game 𝜀 has a constrained maximal element; that is, there exists (𝑥,𝑦)∈𝑋×𝑋 such that, for each 𝑖∈𝐼, 𝑦_𝑖∈𝐹_𝑖(𝑥) and 𝑃_𝑖(𝑥,𝑦)=∅.

By Theorem 30, we can obtain the following equilibrium existence theorem for generalized abstract economies.

Theorem 31. Let 𝐼 be a finite index set; let {(𝑋_𝑖;Γ_𝑖)}_𝑖∈𝐼 be a family of abstract convex spaces such that (𝑋;Γ)∶=(∏_𝑖∈𝐼𝑋_𝑖;Γ) and (𝑋×𝑋;Γ×Γ) are two abstract convex spaces as defined in Lemma 7. Let 𝜀=((𝑋_𝑖;Γ_𝑖),𝐴_𝑖,𝐵_𝑖,𝐹_𝑖,𝑃_𝑖)_𝑖∈𝐼 be a generalized abstract economy, and let 𝐾 be a nonempty compact subset of 𝑋×𝑋. For each 𝑖∈𝐼, assume that,(i) the set 𝑉_𝑖={𝑥∈𝑋∶𝑥_𝑖∉𝐵_𝑖(𝑥)} is open in 𝑋;(ii)if (𝑥,𝑦)∈𝑋×𝑋 with 𝑥∈𝑉_𝑖, we have 𝐴_𝑖(𝑥)≠∅ and 𝑥_𝑖∉co_Γ𝑖𝐴_𝑖(𝑥); if (𝑥,𝑦)∈𝑋×𝑋 with 𝑥∉𝑉_𝑖 and 𝑃_𝑖(𝑥,𝑦)∩𝐴_𝑖(𝑥)≠∅, we have 𝑥_𝑖∉co_Γ𝑖(𝑃_𝑖(𝑥,𝑦)∩𝐴_𝑖(𝑥));(iii)for each 𝑢_𝑖∈𝑋_𝑖, 𝐴_𝑖⁻¹(𝑢_𝑖), 𝐹_𝑖⁻¹(𝑢_𝑖), and 𝑃_𝑖⁻¹(𝑢_𝑖) are open sets;(iv)the set 𝑊_𝑖 = {(𝑥,𝑦)∈𝑋×𝑋∶𝑥∈𝑉_𝑖𝑎𝑛𝑑𝐴_𝑖(𝑥)≠∅}⋃{(𝑥,𝑦)∈𝑋×𝑋∶𝑥∉𝑉_𝑖𝑎𝑛𝑑𝑃_𝑖(𝑥,𝑦)∩𝐴_𝑖(𝑥)≠∅} is closed in 𝑋×𝑋;(v)for each 𝑁_0𝑖×𝑁_1𝑖∈⟨𝑋_𝑖×𝑋_𝑖⟩, there exist compact Γ_𝑖-convex subsets 𝐿_𝑁0𝑖,𝐿_𝑁1𝑖 of (𝑋_𝑖;Γ_𝑖) containing 𝑁_0𝑖,𝑁_1𝑖, respectively, such that, for each (𝑥,𝑦)∈𝐿⧵𝐾, there exists (𝑢_𝑖,𝑣_𝑖)∈𝑁_0𝑖×𝑁_1𝑖 satisfying 𝑥,𝑦∈𝐹_𝑖⁻¹𝑣_𝑖∈𝑃×𝑋,𝑥,𝑦_𝑖⁻¹𝑢_𝑖𝑉_𝑖×𝑋𝐴_𝑖⁻¹𝑢_𝑖×𝑋𝑋×𝑋⧵𝑊_𝑖, (32) where 𝐿∶=∏_𝑖∈𝐼𝐿_𝑁0𝑖×∏_𝑖∈𝐼𝐿_𝑁1𝑖.
If (𝑋×𝑋;Γ×Γ) satisfies 1_𝑋×𝑋∈𝔎ℭ(𝑋×𝑋,𝑋×𝑋), then there exists (𝑥,𝑦)∈𝑋×𝑋 such that, for each 𝑖∈𝐼, 𝑥_𝑖∈𝐵_𝑖(𝑥), 𝑦_𝑖∈𝐹_𝑖(𝑥), and 𝑃_𝑖(𝑥,𝑦)∩𝐴_𝑖(𝑥)=∅.

Proof. For each 𝑖∈𝐼, define a set-valued mapping 𝐻_𝑖∶𝑋×𝑋→2^𝑋𝑖 by 𝐻_𝑖=𝑃𝑥,𝑦^𝑖𝑥,𝑦⋂𝐴_𝑖(𝑥),if𝑥,𝑦∈𝑋×𝑋with𝑥∉𝑉_𝑖,𝐴_𝑖(𝑥),if𝑥,𝑦∈𝑋×𝑋with𝑥∈𝑉_𝑖. (33) Then for each 𝑖∈𝐼 and each 𝑢_𝑖∈𝑋_𝑖, we have 𝐻_𝑖⁻¹𝑢_𝑖=𝑥,𝑦∈𝑋×𝑋∶𝑢_𝑖∈𝐻_𝑖=𝑥,𝑦𝑥,𝑦∈𝑋×𝑋∶𝑥∉𝑉_𝑖and𝑢_𝑖∈𝑃_𝑖𝑥,𝑦𝐴_𝑖(𝑥)𝑥,𝑦∈𝑋×𝑋∶𝑥∈𝑉_𝑖and𝑢_𝑖∈𝐴_𝑖(𝑥=𝑃_𝑖−1)𝑢_𝑖𝐴_𝑖⁻¹𝑢_𝑖𝑋⧵𝑉_𝑖𝑉×𝑋_𝑖𝐴_𝑖⁻¹𝑢_𝑖=𝑃×𝑋_𝑖−1𝑢_𝑖𝑉_𝑖×𝑋𝐴_𝑖⁻¹𝑢_𝑖×𝑋. (34) By (i), (iii), and (iv), for each 𝑖∈𝐼 and each (𝑢_𝑖,𝑣_𝑖)∈𝑋_𝑖×𝑋_𝑖, the set [𝐹_𝑖⁻¹(𝑣_𝑖)×𝑋]∩[𝐻_𝑖⁻¹(𝑢_𝑖)∪(𝑋×𝑋⧵𝑊_𝑖)] is open in 𝑋×𝑋. By (ii), for each 𝑖∈𝐼 and each (𝑥,𝑦)∈𝑋×𝑋 with 𝐻_𝑖(𝑥,𝑦)≠∅, we have 𝑥_𝑖∉co_Γ𝑖𝐻_𝑖(𝑥,𝑦). By (v), for each 𝑖∈𝐼 and each 𝑁_0𝑖×𝑁_1𝑖∈⟨𝑋_𝑖×𝑋_𝑖⟩, there exist compact Γ_𝑖-convex subsets 𝐿_𝑁0𝑖,𝐿_𝑁1𝑖 of (𝑋_𝑖;Γ_𝑖) containing 𝑁_0𝑖,𝑁_1𝑖, respectively, such that, for 𝐿∶=∏_𝑖∈𝐼𝐿_𝑁0𝑖×∏_𝑖∈𝐼𝐿_𝑁1𝑖, we have 𝐿⧵𝐾⊆_{(𝑢𝑖,𝑣𝑖)∈𝐿𝑁0𝑖×𝐿𝑁1𝑖}𝐹_𝑖⁻¹𝑣_𝑖𝐻×𝑋_𝑖−1𝑢_𝑖𝑋×𝑋⧵𝑊_𝑖. (35) So, all the conditions of Theorem 30 are satisfied. Hence, by Theorem 30, there exists (𝑥,𝑦)∈𝑋×𝑋 such that 𝑦_𝑖∈𝐹_𝑖(𝑥),𝐻_𝑖𝑥,𝑦=∅,∀𝑖∈𝐼. (36) If 𝑥∈𝑉_𝑗 for some 𝑗∈𝐼, then, by the definition of 𝐻_𝑖, we have 𝐻_𝑖𝑥,𝑦=𝐴_𝑖(𝑥)=∅, (37) which contradicts the first part of (ii). Thus, we have 𝑥,𝑦∈𝑋×𝑋with𝑥∉𝑉_𝑖,∀𝑖∈𝐼, (38) which implies that, for each 𝑖∈𝐼, 𝑥_𝑖∈𝐵_𝑖(𝑥), 𝑦_𝑖∈𝐹_𝑖(𝑥), and 𝑃_𝑖(𝑥,𝑦)∩𝐴_𝑖(𝑥)=∅; that is, (𝑥,𝑦) is an equilibrium point of generalized abstract economy 𝜀.

In Theorems 26 and 31, when 𝑃_𝑖(𝑥,𝑦)=𝑃_𝑖(𝑥) and 𝐹_𝑖(𝑥)=𝑋_𝑖 for each 𝑖∈𝐼 and for all (𝑥,𝑦)∈𝑋×𝑋, we can derive the following equilibrium existence results for abstract economies.

Corollary 32. Let 𝐼 be a finite index set; let {(𝑋_𝑖;Γ_𝑖)}_𝑖∈𝐼 be a family of abstract convex spaces such that (𝑋;Γ)∶=(∏_𝑖∈𝐼𝑋_𝑖;Γ) is an abstract convex space as defined in Lemma 7. Let 𝜀=((𝑋_𝑖;Γ_𝑖),𝐴_𝑖,𝐵_𝑖,𝑃_𝑖)_𝑖∈𝐼 be an abstract economy, and let 𝐾 be a nonempty compact subset of 𝑋. For each 𝑖∈𝐼, assume that,(i)for each 𝑥∈𝑋, 𝐴_𝑖(𝑥)≠∅ and co_Γ𝑖𝐴_𝑖(𝑥)⊆𝐵_𝑖(𝑥);(ii)for each 𝑥∈𝑊_𝑖={𝑥∈𝑋∶𝐴_𝑖(𝑥)⋂𝑃_𝑖(𝑥)≠∅}, 𝑥_𝑖∉co_Γ𝑖𝑃_𝑖(𝑥);(iii)for each 𝑢_𝑖∈𝑋_𝑖, 𝐴_𝑖⁻¹(𝑢_𝑖)⋂(𝑃_𝑖⁻¹(𝑢_𝑖)⋃(𝑋⧵𝑊_𝑖)) is open in 𝑋;(iv)for each 𝑁_𝑖∈⟨𝑋_𝑖⟩, there exists a compact Γ_𝑖-convex subset 𝐿_𝑁𝑖 of (𝑋_𝑖;Γ_𝑖) containing 𝑁_𝑖, such that, for 𝐿∶=∏_𝑖∈𝐼𝐿_𝑁𝑖, we have 𝐿⧵𝐾⊆_{𝑢𝑖∈𝐿𝑁𝑖}𝐴_𝑖⁻¹𝑢_𝑖𝑃_𝑖⁻¹𝑢_𝑖𝑋⧵𝑊_𝑖. (39)
If (𝑋;Γ) satisfies 1_𝑋∈𝔎ℭ(𝑋,𝑋), then there exists 𝑥∈𝑋 such that, for each 𝑖∈𝐼, 𝑥_𝑖∈𝐵_𝑖(𝑥) and 𝑃_𝑖(𝑥)∩𝐴_𝑖(𝑥)=∅.

Corollary 33. Let 𝐼 be a finite index set; let {(𝑋_𝑖;Γ_𝑖)}_𝑖∈𝐼 be a family of abstract convex spaces such that (𝑋;Γ)∶=(∏_𝑖∈𝐼𝑋_𝑖;Γ) is an abstract convex space as defined in Lemma 7. Let 𝜀=((𝑋_𝑖;Γ_𝑖),𝐴_𝑖,𝐵_𝑖,𝑃_𝑖)_𝑖∈𝐼 be an abstract economy, and let 𝐾 be a nonempty compact subset of 𝑋. For each 𝑖∈𝐼, assume that,(i)the set 𝑉_𝑖={𝑥∈𝑋∶𝑥_𝑖∉𝐵_𝑖(𝑥)} is open in 𝑋;(ii)if 𝑥∈𝑉_𝑖, we have 𝐴_𝑖(𝑥)≠∅ and 𝑥_𝑖∉co_Γ𝑖𝐴_𝑖(𝑥); if 𝑥∉𝑉_𝑖 with 𝑃_𝑖(𝑥)⋂𝐴_𝑖(𝑥)≠∅, then we have 𝑥_𝑖∉co_Γ𝑖(𝑃_𝑖(𝑥)⋂𝐴_𝑖(𝑥));(iii)for each 𝑢_𝑖∈𝑋_𝑖, the sets 𝐴_𝑖⁻¹(𝑢_𝑖) and 𝑃_𝑖⁻¹(𝑢_𝑖) are open in 𝑋;(iv)the set 𝑊_𝑖={𝑥∈𝑉_𝑖∶𝐴_𝑖(𝑥)≠∅}⋃{𝑥∉𝑉_𝑖∶𝑃_𝑖(𝑥)⋂𝐴_𝑖(𝑥)≠∅} is closed in 𝑋;(v)for each 𝑁_𝑖∈⟨𝑋_𝑖⟩, there exists a compact Γ_𝑖-convex subset 𝐿_𝑁𝑖 of (𝑋_𝑖;Γ_𝑖) containing 𝑁_𝑖 such that, for each 𝑥∈𝐿⧵𝐾, there exists 𝑢_𝑖∈𝑁_𝑖 satisfying 𝑥∈𝑃_𝑖⁻¹𝑢_𝑖𝑉_𝑖𝐴_𝑖⁻¹𝑢_𝑖𝑋⧵𝑊_𝑖, (40) where 𝐿∶=∏_𝑖∈𝐼𝐿_𝑁𝑖.
If (𝑋;Γ) satisfies 1_𝑋∈𝔎ℭ(𝑋,𝑋), then there exists 𝑥∈𝑋 such that, for each 𝑖∈𝐼, 𝑥_𝑖∈𝐵_𝑖(𝑥) and 𝑃_𝑖(𝑥)∩𝐴_𝑖(𝑥)=∅.

Acknowledgments

The authors thank the referee for his many valuable suggestions and helpful comments which improve the exposition of this paper. This work was partially supported by the Planning Foundation for Humanities and Social Sciences of Ministry of Education of China (Research on utilizing conflict of water resource and initial water right allocation in a river basin—on the basis of game theory, no. 12YJAZH084) and Jiangsu Overseas Research & Training Program for University Prominent Young & Middle-aged Teachers and Presidents.