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

Authors: Haishu Lu; Qingwen Hu
Journal: Journal of Function Spaces and Applications
 (2013)

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## 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.


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## Body

## 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

LetX be a set. We will denote by 2X 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→2Y a set-valued mapping. Then the set-valued mapping T-1:Y→2X 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→2E with nonempty values. One may denote ΓA∶=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
(1)coΓ(D′)∶=⋃‍{ΓA:A∈〈D′〉}⊆E
(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 ΓN⊆X; that is, coΓ(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 coΓ(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)
LetX be a topological space and {FA} be a given family of nonempty contractible subsets of X indexed by A∈〈X〉 such that FA⊆FB whenever A⊆B. The couple (X,FA) is called an H-space (see [14]). A set D⊆X is said H-convex if FA⊆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 aG-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 φA:Δn→Γ(A) such that J∈〈A〉 implies φA(ΔJ)⊆Γ(J). Here, Δn is the standard n-simplex with vertices {ei}i=0n and ΔJ the face of Δn corresponding to J∈〈A〉; that is, if A={a0,a1,…,an} and J={ai0,…,aik}⊆A, then ΔJ=co{ei0,ei1,…,eik}.(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 subsetA 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→2Z with nonempty values, if a set-valued mapping G:D→2Z satisfies F(ΓA)⊆G(A) for  each A∈〈D〉, then G is called a KKM map with respect to F. A KKM map G:D→2E is a KKM map with respect to the identity map 1E.A set-valued mappingF:E→2Z is said to have the KKM property and called a 𝔎-map, if, for any KKM map G:D→2Z with respect to F, the family {G(y):y∈D} has the finite intersection property. We denote
(2)𝔎(E,D,Z)∶={F:E⟶2Z∣Fisa𝔎-map}.LetZ be a topological space. A set-valued mapping F:E→2Z is called a 𝔎ℭ-map (resp., 𝔎𝔒-map) if, for any closed-valued (resp., open-valued) KKM map G:D→2Z 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 1E∈𝔎ℭ(E,D,E); that is, for any closed-valued KKM map G:D→2E, 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., 1E∈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]).
LetX be a topological space, Y a nonempty set, and T:X→2Y 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 eachy∈Y,  T-1(y) contains an open subset Oy of X and X=⋃y∈Y‍Oy.Lemma 7 (see [8]).
Let{(Ei,Di;Γi)}i∈I be a family of abstract convex spaces, where I is an index set. Let E∶=∏i∈I‍Ei be equipped with the product topology and D∶=∏i∈I‍Di. For each i∈I, let πi:D→Di be the projection. Define Γ=∏i∈I‍Γi:〈D〉→2E by Γ(A)∶=∏i∈I‍Γi(πi(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 1X∈𝔎ℭ(X,X) (i.e., (X;Γ) satisfies the partial KKM principle). Let S,T:X→2X be two set-valued mappings satisfying the following conditions:(i)
for eachx∈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{y¯0,…,y¯n}∈〈X〉 such that X=⋃j=0n‍S-1(y¯j).
ThenT has a fixed point x¯∈X; that is, x¯∈T(x¯).By Lemma9, we can obtain the following collectively fixed point theorem which is the main result of our paper.Theorem 10.
LetI be a finite index set; let {(Xi;Γi)}i∈I be a family of abstract convex spaces such that (X;Γ)∶=(∏i∈I‍Xi;Γ) is an abstract convex space defined as in Lemma 7. Let K be nonempty compact subset X. For each i∈I, let Si, Ti:X→2Xi be set-valued mappings such that,(i)
for eachx∈X, coΓiSi(x)⊆Ti(x);(ii)
for eachyi∈Xi, Si-1(yi) is open in X;(iii)
K⊆⋃yi∈Xi‍Si-1(yi);(iv)
for eachNi∈〈Xi〉, there exists a compact Γi-convex subset LNi of (Xi;Γi) containing Ni such that, for L∶=∏i∈I‍LNi, we have
(3)L∖K⊆⋃‍{Si-1(yi):yi∈LNi}.
If(X;Γ) satisfies 1X∈𝔎ℭ(X,X), then there exists x¯=(x¯i)i∈I∈X such that x¯i∈Ti(x¯) for each i∈I.Proof.
SinceK is compact subset of X, by (ii) and (iii), for each i∈I, there exists Ni∈〈Xi〉 such that
(4)K⊆⋃‍{Si-1(yi):yi∈Ni}.
Then by (iv), for each i∈I, there exists a compact Γi-convex subset LNi of (Xi;Γi) containing Ni such that, for L∶=∏i∈I‍LNi, we have
(5)L∖K⊆⋃‍{Si-1(yi):yi∈LNi}.
By (4), we have
(6)L⋂‍K⊆⋃‍{Si-1(yi):yi∈LNi}.
Then it follows from (5) and (6) that
(7)L=(L∖K)⋃‍(L⋂‍K)=⋃‍{Si-1(yi)⋂‍L:yi∈LNi},∀i∈I.
For each i∈I, since LNi is Γi-convex subset of (Xi;Γi), it follows from Lemma 1 in Park [9] that (LNi;Γi|〈LNi〉) is an abstract convex space, which is a subspace of (Xi;Γi). 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 Si′,Ti′:L→2LNi by
(8)Si′(x)=Si(x)⋂‍LNi,Ti′(x)=Ti(x)⋂‍LNi,∀x∈L.
Furthermore, we define two set-valued mappings S′,T′:L→2L by
(9)S′(x)=∏i∈I‍Si′(x),∀x∈L,T′(x)=∏i∈I‍Ti′(x),∀x∈L.
Next, we prove that S′ and T′ satisfy all the conditions of Lemma 9 as follows.(a)
For eachx∈L,  M∈〈S′(x)〉 implies that Γ|〈L〉(M)⊆T′(x). In fact, M∈〈S′(x)〉 implies that πi(M)∈〈Si′(x)〉 for each i∈I. Then it follows that πi(M)∈〈Si(x)〉 for each i∈I. By (i) and the fact that (LNi;Γi|〈LNi〉) is a subspace of (Xi;Γi) for each i∈I, we have
(10)Γi(πi(M))=Γi|〈LNi〉(πi(M))⊆Ti(x),∀i∈I,Γi(πi(M))=Γi|〈LNi〉(πi(M))⊆LNi,∀i∈I.Then it follows from (10) that
(11)Γ|〈L〉(M)=∏i∈I‍(Γi|〈LNi〉(πi(M)))⊆∏i∈I‍(Ti(x)⋂‍LNi)=T′(x),which implies thatcoΓ|〈L〉S′(x)⊆T′(x) for each x∈L.(b)
For eachy∈L, (S′)-1(y) is relatively open in L. In fact, by (8), for each i∈I and each yi∈LNi, we have
(12)(Si′)-1(yi)={x∈L:yi∈Si(x)⋂‍LNi}=L⋂‍{x∈X:yi∈Si(x)}=L⋂‍Si-1(yi).Therefore, for eachy∈L, we obtain
(13)(S′)-1(y)={x∈L:y∈S′(x)}={x∈L:yi∈Si′(x),∀i∈I}={x∈L:x∈(Si′)-1(yi),∀i∈I}=⋂i∈I‍(Si′)-1(yi)=L⋂‍(⋂i∈I‍Si-1(yi)).By (ii) and the fact thatI is a finite index set, we know that (S′)-1(y) is relatively open in L for each y∈L.(c)
There exists{y¯0,…,y¯n}∈〈L〉 such that L=⋃j=0n‍(S′)-1(y¯j). In fact, it follows from (7) and (12) that
(14)L=⋃‍{(Si′)-1(yi):yi∈LNi},∀i∈I.Letx∈L be given. Then by (14), for each i∈I, there exists yi∈Si′(x) such that
(15)x∈⋂i∈I‍(Si′)-1(yi)=(S′)-1(y),wherey=(yi)i∈I∈L.Therefore, from the arbitrary ofx, we have that L=⋃y∈L‍(S′)-1(y). Since L is compact, there exists {y¯0,…,y¯n}∈〈L〉 such that L=⋃j=0n‍(S′)-1(y¯j).(d)
Since1X∈𝔎ℭ(X,X), it follows from Lemma 8 that 1L∈𝔎ℭ(L,L).
Hence, by Lemma9, there exists x¯=(x¯i)i∈I∈L⊆X such that x∈T′(x); that is, x¯i∈Ti′(x¯)⊆Ti(x¯) for each i∈I. This completes the proof.Remark 11.
Theorem10 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.
LetI be a finite index set; let {(Xi;Γi)}i∈I be a family of abstract convex spaces such that (X;Γ)∶=(∏i∈I‍Xi;Γ) is an abstract convex space defined as in Lemma 7. Let K be nonempty compact subset X. For each i∈I, let Si,Ti:X→2Xi be set-valued mappings such that,(i)
for eachx∈X, coΓiSi(x)⊆Ti(x);(ii)
K⊆⋃yi∈Xi‍intSi-1(yi);(iii)
for eachNi∈〈Xi〉, there exists a compact Γi-convex subset LNi of (Xi;Γi) containing Ni such that, for L∶=∏i∈I‍LNi, we have
(16)L∖K⊆⋃‍{intSi-1(yi):yi∈LNi}.
If(X;Γ) satisfies 1X∈𝔎ℭ(X,X), then there exists x¯=(x¯i)i∈I∈X such that x¯i∈Ti(x¯) for each i∈I.Proof.
For eachi∈I, define a set-valued mapping Si~:X→2Xi as follows:
(17)Si~(x)=(intSi-1)-1(x),∀x∈X.
Then by (i), we have coΓiSi~(x)⊆Ti(x) for each i∈I and each x∈X. By (17), for each i∈I and each yi∈Xi, we have
(18)Si~-1(yi)=intSi-1(yi),
which is open in X. By (ii) and (18), we have K⊆⋃yi∈Xi‍Si~-1(yi) 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 Ni∈〈Xi〉, there exists a compact Γi-convex subset LNi of (Xi;Γi) containing Ni such that, for L∶=∏i∈I‍LNi, we have
(19)L∖K⊆⋃‍{Si~-1(yi):yi∈LNi}.
Since (X;Γ) satisfies 1X∈𝔎ℭ(X,X), it follows that all the conditions of Theorem 10 for Si~ and Ti hold. Therefore, by Theorem 10, there exists x¯=(x¯i)i∈I∈X such that x¯i∈Ti(x¯) for each i∈I.Remark 13.
We have shown that Theorem10 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 Theorems10 and 12 and their applications obtained by other authors. We omit their proofs.Proposition 14 (see [21]).
LetX1,…,Xn (n≥1) be nonempty compact convex sets, each in a topological vector space (not necessary Hausdorff). Let each Mi:X→2Xi have the local intersection property and nonempty convex values. Then there exists x¯∈X such that x¯i∈Mi(x¯) for each i∈{1,…,n}.Remark 15.
By Remarks6 and 13, we know that Proposition 14 is a particular form of Theorems 10 and 12.By using Proposition14, Prokopovych [21] proved a theorem on the existence of a pure strategy ɛ-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 Proposition14, we can obtain the following famous Fan-Browder fixed theorem.The Fan-Browder Fixed Point Theorem (see [22, 23]).  LetXbe a nonempty compact convex subset of a topological vector space. LetM:X→2Xbe a set-valued mapping such that it has nonempty convex values and open inverse values (i.e., eachM-1(y)is open inX). ThenMhas 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]).
LetX be a nonempty convex subset of a topological vector space, A⊆X nonempty convex, and D⊆A a nonempty compact subset. Let S:A→2A and L:A→2A be two set-valued mappings. Assume that,(a)
for eachx∈A, L(x)is convex andS(x)⊆L(x);(b)
for eachx∈D, S(x)≠∅;(c)
for eachy∈A, S-1(y)is open inA;(d)
for each finite subsetNofA, there is a compact convex subsetLNsuch thatN⊆LN⊆AandS(x)∩LN≠∅for allx∈LN∖D.
ThenL has a fixed point.Remark 17.
The coercivity condition (d) in Proposition16 can be replaced by the following condition:(d)′
for each finite subsetN of A, there is a compact convex subset LN such that N⊆LN⊆A and LN∖D⊆⋃y∈LN‍S-1(y).Many authors applied Proposition16 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 Proposition16 can be found in Ding and Tan [30].Proposition 18 (see [30]).
LetX be a nonempty convex subset of a topological vector space. Let S:X→2X be a set-valued mapping. Assume that,(a)
for eachx∈X,S(x)≠∅;(b)
for eachy∈X,S-1(y)is compactly open; that is, for each nonempty compact subsetCofX, S-1(y)⋂‍Cis open inC;(c)
there exist a nonempty compact convex subsetX0ofXand a nonempty compact subsetKofX(not necessarily convex) such that, for eachx∈X∖K, we haveco(X0⋃‍{x})⋂‍coS(x)≠∅.
Then there existsx¯ 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 Proposition18, “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,FA) be a compact H-space and T:X→2X a set-valued mapping such that,(a)
for eachx∈X,T(x)is a nonemptyH-convex subset ofX;(b)
for eachy∈X,T-1(y)contains an open setOy(Oymay be empty for somey);(c)
X=⋃y∈X‍Oy.
ThenT has a fixed point.By using Proposition20, 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 Proposition20.Proposition 21 (see [35]).
Let(X,FA) be an H-space, D⊆X an H-compact set, and T:X→2X a set-valued mapping such that,(a)
for eachx∈X,T(x)is a nonemptyH-convex subset ofX;(b)
for eachy∈X,T-1(y)is open inX;(c)
T(x)⋂‍D≠∅for  allx∈X.
ThenT has a fixed point.By using Proposition21, 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]).
LetX be a nonempty compact Δ-convex subset of a topological semilattice with path-connected intervals and T:X→2X a set-valued mapping such that(a)
for eachx∈X,T(x)is nonempty andΔ-convex;(b)
for eachy∈X,T-1(y)is open inX.
ThenT 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→2X be two set-valued mappings such that,(a)
for eachx∈X,N∈〈S(x)〉impliesΓ(A)⊆T(x);(b)
X=⋃y∈X‍intS-1(y);(c)
there exists a nonempty compact subsetKofXsuch that, for eachN∈〈X〉, there exists a compactG-convex subsetLNof(X;Γ)containingNsuch that(20)L∖K⊆⋃‍{intS-1(y):y∈LN}.
Then there existsx¯∈X such that x¯∈T(x¯).Remark 24.
The coercivity condition (c) in Proposition23 can be replaced by the following equivalent condition:(c)′
there exists a nonempty compact subsetK of X such that, for each N∈〈X〉, there exists a compact G-convex subset LN of (X;Γ) containing N such that
(21)L⋂‍(⋂‍{X∖intS-1(y):y∈LN})⊆K.
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]).
LetI be a finite index set; let {(Xi;Γi)}i∈I be a family of compact abstract convex spaces such that (X;Γ)∶=(∏i∈I‍Xi;Γ) is an abstract convex space as defined as in Lemma 7, and satisfies the partial KKM principle. For each i∈I, let Ti:X→2Xi be a set-valued mapping such that,(a)
for eachx∈X,coΓiSi(x)⊆Ti(x);(b)
X=⋃yi∈Xi‍intSi-1(yi).
Then there existsx¯=(x¯i)i∈I∈X such that x¯i∈Ti(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.LetI be any set of agents. For each i∈I, let Xi be the strategy set or commodity space of the agent i, and let X=∏i∈I‍Xi. Following the method of Kim and Tan [44], let ɛ=(Xi,Ai,Bi,Fi,Pi)i∈I be a generalized abstract economy, where Ai,Bi:X=∏i∈I‍Xi→2Xi are two constrained correspondences such that Ai(x) and Bi(x) are the states attainable for the agent i at x; Fi:X→2Xi is a fuzzy constrained correspondence such that Fi(x) is the unstable state for the agent i at x, and Pi:X×X→2Xi is a preference correspondence such that Pi(x,y) is the state preference by the agent i at (x,y). An equilibrium for ɛ is a point (x¯,y¯)∈X×X such that for each i∈I,  x¯i∈Bi(x¯),  y¯i∈Fi(x¯), and Pi(x¯,y¯)∩Ai(x¯)=∅.As an application of Theorem10, we derive the following equilibrium existence theorem for generalized abstract economies in noncompact abstract convex spaces.Theorem 26.
LetI be a finite index set, {(Xi;Γi)}i∈I a family of abstract convex spaces such that (X;Γ)∶=(∏i∈I‍Xi;Γ) and (X×X;Γ×Γ) are two abstract convex spaces as defined in Lemma 7. Let ɛ=((Xi;Γi),Ai,Bi,Fi,Pi)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 eachx∈X, Ai(x)≠∅ and coΓiAi(x)⊆Bi(x);(ii)
for eachx∈X, Fi(x) is nonempty Γi-convex;(iii)
for each(x,y)∈Wi={(x,y)∈X×X:Ai(x)⋂‍Pi(x,y)≠∅}, xi∉coΓiPi(x,y);(iv)
for each(ui,vi)∈Xi×Xi, [(Ai-1(ui)⋂‍Fi-1(vi))×X]⋂‍[Pi-1(ui)⋃‍(X×X∖Wi)] is open in X×X;(v)
for eachN0i×N1i∈〈Xi×Xi〉, there exist compact Γi-convex subsets LN0i, LN1i of (Xi;Γi) containing N0i,N1i, respectively, such that, for L∶=∏i∈I‍LN0i×∏i∈I‍LN1i, we have
(22)L∖K⊆⋃(ui,vi)∈LN0i×LN1i‍{[(Ai-1(ui)⋂‍Fi-1(vi))×X]⋃(ui,vi)∈LN0i×LN1i‍⋂‍[Pi-1(ui)⋃‍(X×X∖Wi)][Ai-1(ui)}.
If(X×X;Γ×Γ) satisfies 1X×X∈𝔎ℭ(X×X,X×X), then there exists (x¯,y¯)∈X×X such that, for each i∈I, x¯i∈Bi(x¯), y¯i∈Fi(x¯) and Pi(x¯,y¯)∩Ai(x¯)=∅.Proof.
By Lemma7, for each i∈I, (Xi×Xi;Γi×Γi) is an abstract convex space. For each i∈I, define two set-valued mappings Si, Ti:X×X→2Xi×Xi by
(23)Si(x,y)={[Pi(x,y)⋂‍Ai(x)]×Fi(x),if(x,y)∈Wi,Ai(x)×Fi(x),if(x,y)∈X×X∖Wi,Ti(x,y)={[coΓiPi(x,y)⋂‍Bi(x)]×Fi(x),if(x,y)∈Wi,Bi(x)×Fi(x),if(x,y)∈X×X∖Wi.
By (i), (ii), and the definition of Wi, we have Si(x,y)≠∅ and coΓi×Γi(Si(x,y))⊆Ti(x,y) for each i∈I and for all (x,y)∈X×X. For each i∈I and for all (ui,vi)∈Xi×Xi, we see from
(24)Si-1(ui,vi)=[Pi-1(ui)⋂‍(Ai-1(ui)×X)⋂‍(Fi-1(vi)×X)]⋃‍[(X×X∖Wi)⋂‍(Ai-1(ui)×X)⋃‍⋂‍(Fi-1(vi)×X)]=[(Ai-1(ui)⋂‍Fi-1(vi))×X]⋂‍[Pi-1(ui)⋃‍(X×X∖Wi)]
and (iv) that Si-1(ui,vi) is open in X×X. Since Si(x,y)≠∅ for each i∈I and for all (x,y)∈X×X, we have the following:
(25)X×X=⋃(ui,vi)∈Xi×Xi‍Si-1(ui,vi),∀i∈I,
and so, we have
(26)K⊆X×X=⋃(ui,vi)∈Xi×Xi‍Si-1(ui,vi),∀i∈I.
By (v), for each i∈I and each Ni=N0i×N1i∈〈Xi×Xi〉, there exists a compact Γi×Γi-convex subset LN0i×LN1i of (Xi×Xi;Γi×Γi) containing Ni, such that, for L∶=∏i∈I‍LN0i×∏i∈I‍LN1i, we have
(27)L∖K⊆⋃‍{Si-1(ui,vi):(ui,vi)∈LN0i×LN1i}.
Since (X×X;Γ×Γ) satisfies 1X×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)∈Ti(x¯,y¯) for each i∈I. If (x¯,y¯)∈Wi for some i, then we have
(28)(x¯i,y¯i)∈[coΓiPi(x¯,y¯)⋂‍Bi(x¯)]×Fi(x¯).
And, hence, x¯i∈coΓiPi(x¯,y¯)∩Bi(x¯) and so x¯i∈coΓiPi(x¯,y¯), which contradicts (iii). Therefore, we must have (x¯,y¯)∈X×X∖Wi for all i∈I. It follows from the definitions of Ti and Wi that, for each i∈I, x¯i∈Bi(x¯), y¯i∈Fi(x¯), and Pi(x¯,y¯)∩Ai(x¯)=∅.Remark 27.
Theorem26 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.
LetI be a finite index set; let {(Xi;Γi)}i∈I be a family of abstract convex spaces such that (X;Γ)∶=(∏i∈I‍Xi;Γ) and (X×X;Γ×Γ) are two abstract convex spaces as defined in Lemma 7. Let ɛ=((Xi;Γi),Ai,Bi,Fi,Pi)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 eachx∈X, Ai(x)≠∅ and coΓiAi(x)⊆Bi(x);(ii)
for eachx∈X, Fi(x) is nonempty Γi-convex;(iii)
for each(x,y)∈Wi={(x,y)∈X×X:Ai(x)⋂‍Pi(x,y)≠∅}, xi∉coΓiPi(x,y);(iv)
for eachui∈Xi, Ai-1(ui), Fi-1(ui), and Pi-1(ui) are open sets;(v)
the setWi is closed in X×X;(vi)
for eachN0i×N1i∈〈Xi×Xi〉, there exist compact Γi-convex subsets LN0i,LN1i of (Xi;Γi) containing N0i,N1i, respectively, such that, for L∶=∏i∈I‍LN0i×∏i∈I‍LN1i, we have
(29)L∖K⊆⋃(ui,vi)∈LN0i×LN1i‍{[(Ai-1(ui)⋂‍Fi-1(vi))×X]⋃(ui,vi)∈LN0i×LN1i‍⋂‍[Pi-1(ui)⋃‍(X×X∖Wi)]}.
If(X×X;Γ×Γ) satisfies 1X×X∈𝔎ℭ(X×X,X×X), then there exists (x¯,y¯)∈X×X such that, for each i∈I, x¯i∈Bi(x¯), y¯i∈Fi(x¯), and Pi(x¯,y¯)∩Ai(x¯)=∅.Proof.
By (iv) and (v), for each(ui,vi)∈Xi×Xi, the set [(Ai-1(ui)∩Fi-1(vi))×X]∩[Pi-1(ui)∪(X×X∖Wi)] is open in X×X. Hence, the conclusion of Corollary 28 follows from Theorem 26.Corollary 29.
LetI be a finite index set; let {(Xi;Γi)}i∈I be a family of abstract convex spaces such that (X;Γ)∶=(∏i∈I‍Xi;Γ) and (X×X;Γ×Γ) are two abstract convex spaces as defined in Lemma 7. Let ɛ=((Xi;Γi),Ai,Bi,Fi,Pi)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 eachx∈X, Ai(x)≠∅ and coΓiAi(x)⊆Bi(x);(ii)
for eachx∈X, Fi(x) is nonempty Γi-convex;(iii)
for each(x,y)∈Wi={(x,y)∈X×X:Ai(x)∩Pi(x,y)≠∅}, xi∉cl(coΓiPi(x,y));(iv)
for eachui∈Xi, Ai-1(ui), Fi-1(ui), and Pi-1(ui) are open sets;(v)
the setWi is closed in X×X;(vi)
for eachN0i×N1i∈〈Xi×Xi〉, there exist compact Γi-convex subsets LN0i,LN1i of (Xi;Γi) containing N0i,N1i, respectively, such that, for L∶=∏i∈I‍LN0i×∏i∈I‍LN1i, we have
(30)L∖K⊆⋃(ui,vi)∈LN0i×LN1i‍{[(Ai-1(ui)⋂‍Fi-1(vi))×X]⋃(ui,vi)∈LN0i×LN1i‍⋂‍[Pi-1(ui)⋃‍(X×X∖Wi)]Ai-1(ui)}.
If(X×X;Γ×Γ) satisfies 1X×X∈𝔎ℭ(X×X,X×X), then there exists (x¯,y¯)∈X×X such that, for each i∈I, x¯i∈Bi(x¯), y¯i∈Fi(x¯), and Pi(x¯,y¯)∩Ai(x¯)=∅.Proof.
By (iii), for each(x,y)∈Wi={(x,y)∈X×X:Ai(x)∩Pi(x,y)≠∅}, xi∉coΓiPi(x,y). Hence, the conclusion of Corollary 29 follows from Corollary 28.IfAi(x)=Bi(x)=Xi for each i∈I and each x∈X in Theorem 26, then we have the following theorem.Theorem 30.
LetI be a finite index set; let {(Xi;Γi)}i∈I be a family of abstract convex spaces such that (X;Γ)∶=(∏i∈I‍Xi;Γ) and (X×X;Γ×Γ) are two abstract convex spaces as defined as in Lemma 7. Let ɛ=((Xi;Γi),Fi,Pi)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)∈Wi={(x,y)∈X×X:Pi(x,y)≠∅}, xi∉coΓiPi(x,y);(ii)
for each(ui,vi)∈Xi×Xi, [Fi-1(vi)×X]⋂‍[Pi-1(ui)⋃‍(X×X∖Wi)] is open in X×X;(iii)
for eachN0i×N1i∈〈Xi×Xi〉, there exist compact Γi-convex subsets LN0i,LN1i of (Xi;Γi) containing N0i,N1i, respectively, such that, for L∶=∏i∈I‍LN0i×∏i∈I‍LN1i, we have
(31)L∖K⊆⋃(ui,vi)∈LN0i×LN1i‍{∫[Fi-1(vi)×X]⋃(ui,vi)∈LN0i×LN1i‍⋂‍[Pi-1(ui)⋃‍(X×X∖Wi)]∫}.
If(X×X;Γ×Γ) satisfies 1X×X∈𝔎ℭ(X×X,X×X), then the generalized qualitative game ɛ has a constrained maximal element; that is, there exists (x¯,y¯)∈X×X such that, for each i∈I, y¯i∈Fi(x¯) and Pi(x¯,y¯)=∅.By Theorem30, we can obtain the following equilibrium existence theorem for generalized abstract economies.Theorem 31.
LetI be a finite index set; let {(Xi;Γi)}i∈I be a family of abstract convex spaces such that (X;Γ)∶=(∏i∈I‍Xi;Γ) and (X×X;Γ×Γ) are two abstract convex spaces as defined in Lemma 7. Let ɛ=((Xi;Γi),Ai,Bi,Fi,Pi)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 setVi={x∈X:xi∉Bi(x)} is open in X;(ii)
if(x,y)∈X×X with x∈Vi, we have Ai(x)≠∅ and xi∉coΓiAi(x);  if (x,y)∈X×X with x∉Vi and Pi(x,y)∩Ai(x)≠∅, we have xi∉coΓi(Pi(x,y)∩Ai(x));(iii)
for eachui∈Xi, Ai-1(ui), Fi-1(ui), and Pi-1(ui) are open sets;(iv)
the setWi = {(x,y)∈X×X:x∈ViandAi(x)≠∅}⋃‍{(x,y)∈X×X:x∉ViandPi(x,y)∩Ai(x)≠∅} is closed in X×X;(v)
for eachN0i×N1i∈〈Xi×Xi〉, there exist compact Γi-convex subsets LN0i,LN1i of (Xi;Γi) containing N0i,N1i, respectively, such that, for each (x,y)∈L∖K, there exists (ui,vi)∈N0i×N1i satisfying
(32)(x,y)∈Fi-1(vi)×X,(x,y)∈((Pi-1(ui)⋃‍(Vi×X))⋂‍(Ai-1(ui)×X))⋃‍(X×X∖Wi),whereL∶=∏i∈I‍LN0i×∏i∈I‍LN1i.
If(X×X;Γ×Γ) satisfies 1X×X∈𝔎ℭ(X×X,X×X), then there exists (x¯,y¯)∈X×X such that, for each i∈I, x¯i∈Bi(x¯), y¯i∈Fi(x¯), and Pi(x¯,y¯)∩Ai(x¯)=∅.Proof.
For eachi∈I, define a set-valued mapping Hi:X×X→2Xi by
(33)Hi(x,y)={Pi(x,y)⋂‍Ai(x),if(x,y)∈X×Xwithx∉Vi,Ai(x),if(x,y)∈X×Xwithx∈Vi.
Then for each i∈I and each ui∈Xi, we have
(34)Hi-1(ui)={(x,y)∈X×X:ui∈Hi(x,y)}={∫(x,y)∈X×X:x∉Viandui∈Pi(x,y)⋂‍Ai(x)∫}⋃‍{(x,y)∈X×X:x∈Viandui∈Ai(x)}=[Pi-1(ui)⋂‍((Ai-1(ui)⋂‍(X∖Vi))×X)]⋃‍[(Vi⋂‍Ai-1(ui))×X]=[Pi-1(ui)⋃‍(Vi×X)]⋂‍[Ai-1(ui)×X].
By (i), (iii), and (iv), for each i∈I and each (ui,vi)∈Xi×Xi, the set [Fi-1(vi)×X]∩[Hi-1(ui)∪(X×X∖Wi)] is open in X×X. By (ii), for each i∈I and each (x,y)∈X×X with Hi(x,y)≠∅, we have xi∉coΓiHi(x,y). By (v), for each i∈I and each N0i×N1i∈〈Xi×Xi〉, there exist compact Γi-convex subsets LN0i,LN1i of (Xi;Γi) containing N0i,N1i, respectively, such that, for L∶=∏i∈I‍LN0i×∏i∈I‍LN1i, we have
(35)L∖K⊆⋃(ui,vi)∈LN0i×LN1i‍{∫[Fi-1(vi)×X]⋃(ui,vi)∈LN0i×LN1i‍⋂‍[Hi-1(ui)⋃‍(X×X∖Wi)]∫}.
So, all the conditions of Theorem 30 are satisfied. Hence, by Theorem 30, there exists (x¯,y¯)∈X×X such that
(36)y¯i∈Fi(x¯),Hi(x¯,y¯)=∅,∀i∈I.
If x¯∈Vj for some j∈I, then, by the definition of Hi, we have
(37)Hi(x¯,y¯)=Ai(x¯)=∅,
which contradicts the first part of (ii). Thus, we have
(38)(x¯,y¯)∈X×Xwithx¯∉Vi,∀i∈I,
which implies that, for each i∈I, x¯i∈Bi(x¯), y¯i∈Fi(x¯), and Pi(x¯,y¯)∩Ai(x¯)=∅; that is, (x¯,y¯) is an equilibrium point of generalized abstract economy ɛ.In Theorems26 and 31, when Pi(x,y)=Pi(x) and Fi(x)=Xi for each i∈I and for all (x,y)∈X×X, we can derive the following equilibrium existence results for abstract economies.Corollary 32.
LetI be a finite index set; let {(Xi;Γi)}i∈I be a family of abstract convex spaces such that (X;Γ)∶=(∏i∈I‍Xi;Γ) is an abstract convex space as defined in Lemma 7. Let ɛ=((Xi;Γi),Ai,Bi,Pi)i∈I be an abstract economy, and let K be a nonempty compact subset of X. For each i∈I, assume that,(i)
for eachx∈X, Ai(x)≠∅ and coΓiAi(x)⊆Bi(x);(ii)
for eachx∈Wi={x∈X:Ai(x)⋂‍Pi(x)≠∅}, xi∉coΓiPi(x);(iii)
for eachui∈Xi, Ai-1(ui)⋂‍(Pi-1(ui)⋃‍(X∖Wi)) is open in X;(iv)
for eachNi∈〈Xi〉, there exists a compact Γi-convex subset LNi of (Xi;Γi) containing Ni, such that, for L∶=∏i∈I‍LNi, we have
(39)L∖K⊆⋃ui∈LNi‍[Ai-1(ui)⋂‍(Pi-1(ui)⋃‍(X∖Wi))].
If(X;Γ) satisfies 1X∈𝔎ℭ(X,X), then there exists x¯∈X such that, for each i∈I, x¯i∈Bi(x¯) and Pi(x¯)∩Ai(x¯)=∅.Corollary 33.
LetI be a finite index set; let {(Xi;Γi)}i∈I be a family of abstract convex spaces such that (X;Γ)∶=(∏i∈I‍Xi;Γ) is an abstract convex space as defined in Lemma 7. Let ɛ=((Xi;Γi),Ai,Bi,Pi)i∈I be an abstract economy, and let K be a nonempty compact subset of X. For each i∈I, assume that,(i)
the setVi={x∈X:xi∉Bi(x)} is open in X;(ii)
ifx∈Vi, we have Ai(x)≠∅ and xi∉coΓiAi(x);  if x∉Vi with Pi(x)⋂‍Ai(x)≠∅, then we have xi∉coΓi(Pi(x)⋂‍Ai(x));(iii)
for eachui∈Xi, the sets Ai-1(ui) and Pi-1(ui) are open in X;(iv)
the setWi={x∈Vi:Ai(x)≠∅}⋃‍{x∉Vi:Pi(x)⋂‍Ai(x)≠∅} is closed in X;(v)
for eachNi∈〈Xi〉, there exists a compact Γi-convex subset LNi of (Xi;Γi) containing Ni such that, for each x∈L∖K, there exists ui∈Ni satisfying
(40)x∈((Pi-1(ui)⋃‍Vi)⋂‍Ai-1(ui))⋃‍(X∖Wi),
where L∶=∏i∈I‍LNi.
If(X;Γ) satisfies 1X∈𝔎ℭ(X,X), then there exists x¯∈X such that, for each i∈I, x¯i∈Bi(x¯) and Pi(x¯)∩Ai(x¯)=∅.



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