#### Abstract

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

#### 1. Introduction

Since Borglin and Keiding [1] proved a new existence theorem for a compact generalized games (=abstract economy) with -majorized preference correspondences. Following their ideas, many authors studied the existence of equilibria for generalized games, for example; see Ding [2], Shen [3], Chowdhury et al. [4], Briec and Horvath [5], Ding and Wang [6], Kim et al. [7], Kim and Tan [8], Lin et al. [9], Lin and Liu [10], Du and Deng [11], and so forth. In the setting, convexity assumptions play a crucial role. Since Horvath [12, 13] introduced -space by replacing convex hulls by contract subsets, many authors have put forward abstract convex spaces without linear structure, for example: -convex space [14] and -space [15]. As a result, many authors established existence theorems of maximal elements and equilibria of generalized games with majorized correspondences in -space, -convex space, and -space, respectively. For example, see Hou [16], Ding [17], Ding and Xia [18], Yang and Deng [19], Ding and Feng [20], and others.

In 2006, Park [21] introduce the concept of an abstract convex space, which is a topological space without any convexity structure and linear structure. Moreover, abstract convex space include topological vector spaces, -space, -convex space, and -space as special cases (see Park [21–23]). Abstract convex space will be the framework of this paper.

In this paper, we will introduce the new class of mapping and -majorized mapping in abstract convex space. Some new fixed point theorems for set-valued mappings are proved under very weak coercive conditions. Next, two existence theorems of maximal elements for class of mapping and -majorized mapping are obtained. As in applications, we establish new equilibria existence theorems for qualitative games and generalized games. Our results generalized and improve the corresponding results due to Ding and Feng [20], Ding and Wang [6], Park [22, 24], Yuan [25], Chowdhury et al. [4], Tan and Yuan [26], Borglin-Keiding [1], Yannelis [27], and so forth.

#### 2. Preliminaries

Let be a nonempty subset of topological space . We shall denote by the family of all subsets of , by the family of all nonempty finite subsets of , by , the interior of in , and by the closure of in .

If is a topological space and are two mappings, for any and , let and . The dom denotes the domain of , that is, dom , and is a mapping defined by for each . The graph of is the set Gr, and the mapping is defined by . The mapping is defined by ( for each .

A subset of is said to be compactly open (resp., compactly closed) in if for each nonempty compact subset of , is open (resp., closed) in . Ding [28] define the compact interior and the compact closure of denoted by and as It is easy to see that for any nonempty compact subset of , we have , , and hence cint() (resp., ) is compactly open (resp., compactly closed) in . By the definitions, a subset of is compactly open (resp., compactly closed) in if and only if (resp., ).

*Definition 2.1 (see [21, 22]). *An abstract convex space consists of a topological space , a nonempty set , and a mapping with nonempty values for each .

For any , the -convex hull of is denoted and defined by

A subset of is called a -convex subset of relative to if for any , we have , that is, . Then, is called a -convex subspace of .

When , the space is denoted by . In such a case, a subset of is said to be -convex if ; in other words, is -convex relative to . If , let .

*Definition 2.2 (see [22]). *Let be an abstract convex space. If a mapping satisfies
Then, is called a KKM mapping.

*Definition 2.3 (see [22]). *The partial KKM principle for an abstract convex space is the statement that, for any closed-valued KKM mapping , the family has the finite intersection property.

*Definition 2.4. *Let be a topological space, and be a nonempty subset of an abstract convex space . Let be a single-valued mapping. Then, the mappings are said to be an -pair if(a) for each , and ,(b) the mapping is compactly open valued on .

*Definition 2.5. *Let be a topological space, and be a nonempty subset of an abstract convex space . Let be a single-valued mapping, and be a set-valued mapping. Then,(i) is said to be of class if there exists an -pair such that(a),(b) for each , .(ii) is said to be an -majorant of at if is an open neighborhood of in and the mapping such that(a) for each , , and ,(b) for each , and ,(c) the mapping is compactly open valued on .(iii) is said to be an -majorized mapping if for each with , there exists an -majorant of at , and for any nonempty finite subset , the set .

*Remark 2.6. *We note that our notions of the mapping being of class (resp., -majorized) improve notions of mapping of class (resp., -majorized), respectively introduced by Ding and Feng [20] from -space to abstract convex space, which in turn generalize the corresponding notions in Ding and Wang [6], Chowdhury et al. [4], Yuan [25], Ding et al. [29], and Ding [30].

In this paper, we shall deal mainly with either the case (I) , and is an abstract convex space, and , which is the identity mapping on or the case (II) , and is the projection of onto and is an abstract convex space. In both cases (I) and (II), we shall write in place of .

Lemma 2.7 (see, [24]). *Let be an abstract convex space and be a subspace. If satisfies the partial KKM principle, then so does . *

Lemma 2.8 (see, [22]). *Let be an abstract convex space satisfying the partial KKM principle, and be a mapping such that:*(i)* for each , is open,*(ii)* for some .**then, there exists an such that .*

#### 3. Fixed Point Theorems

Theorem 3.1. *Let be an abstract convex space satisfying the partial KKM principle, and be a nonempty compact subset of . Suppose that , be mappings such that*(i)* for each ;*(ii)* for each , is compactly open in and for each , ,*(iii)* for each , there exists a compact abstract convex subset of containing such that
**Then, there exists a point , such that . *

*Proof. *Since for each , , then . By (i) and (ii), then . Since is a nonempty compact subset of , there exists a finite set such that
By (iii), there exists a compact abstract convex subset of containing such that
By (3.2) and , then
By (3.3) and (3.4), we have
Since is compact, there exists a finite set , such that
Let and define by for each , then is an abstract convex space. Since satisfies the partial KKM principle, so does by Lemma 2.7.

Define by for each . It is easy to prove that the all the hypotheses of Lemma 2.8 are satisfied. By Lemma 2.8, there exists a finite set such that . Let , then for each , , that is and . Since , thus . This completes the proof.

*Remark 3.2. *Theorem 3.1 generalizes Theorem 3.1 of Ding and Feng [20] and Theorem 3.1 of Ding and Wang [6] from -space to abstract convex space, and the coercive condition (iii) in Theorem 2.1 is weaker than the condition (3) in Theorem 3.1 of Ding and Wang [6].

Corollary 3.3. * Let be an abstract convex space satisfying the partial KKM principle, and be a nonempty compact subset of . Suppose that , be mappings such that*(i)* for each ,*(ii)* for each , is compactly open in and for each , ,*(iii)* for each , there exists a compact abstract convex subset of containing such that for each , there exists a point such that .**Then, there exists a point , such that . *

*Proof. *By (iii), for each , there exists a compact abstract convex subset of containing such that for each , there exists such that , then int, thus , that is . Hence, all the hypotheses of Theorem 3.1 are satisfied. By Theorem 3.1, there exists a point , such that . This completes the proof.

*Remark 3.4. *Corollary 3.3 generalizes Theorem 3.2 of Ding and Feng [20] from -space to abstract convex space. Moreover, Corollary 3.3 improves the corresponding result of Park [22, 24].

#### 4. Existence of Maximal Elements

Let be a topological space, and be a mapping. A point is called a maximal element of if .

In the section, We firstly prove a selection theorem for -majorized mapping. Next, we will establish some new existence theorems of maximal elements for class mapping and -majorized mapping defined on noncompact abstract convex space.

Lemma 4.1. *Let be a regular topological space, and be a nonempty subset of an abstract convex space . Let , and be an -majorized mapping. If each open subset of X containing the set is paracompact, then there exists a -pair such that for each and . *

*Proof. *Since is an -majorized mapping, for each , let be an open neighborhood of in , and be mappings such that(1) for each , , and ,(2) for each , and , (3) the mapping is compactly open valued on , (4) for each , .

Since is regular, for each , there exists an open neighborhood of in such that . Let , then is an open subset of containing , so that is paracompact by assumption. By Theorem VIII.1.4 of Dugundji [31], the open covering of has an open precise neighborhood finite refinement . Given any , we define the mappings by
then we have(i) by (2), for each , ,(ii) by (1), , and for each ,(iii) for each , the set
It follows that for each nonempty compact subset of , the set
is open in by (3). Thus, the mapping is compactly open valued on . Now, define by
(a) For each , by (i), and . If , then , so that ; if , then for some , so that , and hence , by (1), we have . Therefore, for all .(b) Now, we show that the mapping is compactly open valued on . Indeed, let be such that and be a nonempty compact subset of . Given a point ,
Since is a neighborhood finite refinement, there exists an open neighborhood of in such that . Note that for each with , , so that for . Then, we have for all . It follows that
Since is compactly open in by (iii), then is an open neighborhood of in such that . This shows that is compactly open valued on .

By (a) and (b), thus is an -pair.

Next, we claim that dom dom , indeed, for each , we must have . Since is neighborhood finite, then the set . If , , and , thus we have . Since , by (4), . Hence, .

Finally, we prove that for each . Indeed, let with , then . For each , if , then , and so , and if , we have , so that by , . It follows that for all so that . This completes the proof.

*Remark 4.2. *Lemma 4.1 generalizes Lemma 4.1 of Ding and Feng [20], Lemma 5.1 of Ding and Wang [15], Theorem 3.1 of Yuan [25], Lemma 3.1 of Chowdhury et al. [4], and Lemma 3.1 of Tan and Yuan [26].

Theorem 4.3. *Let be an abstract convex space satisfying the partial KKM principle, and let be a nonempty compact subset of . Suppose that the mapping is of class and satisfy*(i)* for each , there exists a compact abstract convex subset of containing such that
**Then, there exists a point such that . *

*Proof. *Since is of class , then there exists an -pair such that(a) dom , for each ,(b) for each , and ,(c) the mapping is compactly open valued on .

Suppose that for each , . By (a), for each . By (i) and (a), for each , there exists a compact abstract convex subset of containing such that . Therefore, and satisfy all the hypotheses of Theorem 3.1. By Theorem 3.1, there exists a point such that . Which contradicts with condition (b). Hence, there exists a point such that . This completes the proof.

*Remark 4.4. *Theorem 4.3 generalizes most existence theorems of maximal elements in the literature, for example; see Theorem 4.1 of Ding and Feng [20], Theorem 3.2 of Yuan [25], Theorem 3.1 of Chowdhury et al. [4], Theorem 3.2 of Tan and Yuan [26], Theorem 5.2 of Ding and Wang [6], and so on.

As an application of Lemma 4.1 and Theorem 4.3, we have the following existence theorem maximal elements.

Theorem 4.5. *Let be a paracompact abstract convex space satisfying the partial KKM principle, and let be a nonempty compact subset of . Let be an -majorized mapping and satisfy*(i)* for each , there exists a compact abstract convex subset of containing such that
**Then, there exists a point such that .*

*Proof. *Suppose that for all , then is paracompact. By Lemma 4.1 is of class . Therefore, all the hypotheses of Theorem 4.3 are satisfied. By Theorem 4.3, there exists a point such that . Which is a contradiction. Thus, there exists a point such that . By the assumptions, must in . This completes the proof.

*Remark 4.6. *Theorem 4.5 generalizes Theorem 5.3 of Ding and Wang [6], Theorem 3.2 of Chowdhury et al. [4], Theorem 3.3 of Tan and Yuan [26], Corollary 1 of Borglin-Keiding [1], and Theorem 2 of Yannelis [27].

#### 5. Existence of Equilibrium of Points

Let be a (finite or infinite) set of players. For each , let its strategy set and be nonempty set, and let . be the preference correspondence of -th player. The collection will be called a qualitative game. A point is said to be an equilibrium of the qualitative game, if for each .

A generalized game is a quintuple family , where is a nonempty set, is a (finite or infinite) set of players such that for each , is a nonempty set and . , are the constraint correspondences, and is the preference correspondence. An equilibrium of the generalized game is a point such that for each and . If is the projection of onto , then our definition of an equilibrium point coincides with the standard definition given by Chowdhury et al. [4], and if, in addition, for each , our definition of an equilibrium point generalizes the standard definition, for example, Borglin and Keiding [1] and Gale and Mas-Colell [32].

Lemma 5.1 (see [33]). *Let be a family of abstract convex spaces. Let be equipped with the product topology, and let . For each , let be the projection. For each , define . Then, is an abstract convex space. *

As an application of Theorem 4.3, we prove the following existence theorem of equilibrium points for one person game in abstract convex space.

Theorem 5.2. *Let be an abstract convex space satisfying the partial KKM principle and be a closed and compact subset of , and let . Suppose that the mappings satisfy*(i)* for each , and ,*(ii)* the mapping is compactly open valued on ,*(iii)* the mapping is of class and for each , *(iv)* for each , there exists a compact abstract convex subset of containing such that
**Then, there exists a point such that and . *

*Proof. *Let , then is open in . Define by
By (iii), is of class , then there exist two mappings such that(a) for each , and dom ,(b) for each , and , (c) the mapping is compactly open valued on .Define by
Then, we have for each , by (a) and dom , for each , by (b), if , and if , by (i), and and if , by (b), thus for each , for each , then it is easy to verify that the set
is compactly open in by (ii) and (c). This shows that is of class . By the definition of , condition (i) and (iv), for each , there exists a compact abstract convex subset of containing such that
Hence, all the hypotheses of Theorem 4.3 are satisfied. By Theorem 4.3, there exists a point such that . By the definition of and condition (iii), , that is, and . This completes the proof.

*Remark 5.3. *Theorem 5.2 improves and generalizes Theorem 5.1 of Ding and Feng [20], Theorem 6.1 of Ding and Wang [6], Theorem 4.1 of Yuan [25], Theorem 4.1 of Chowdhury et al. [4], and Theorem 4.1 of Tan and Yuan [26].

As another application of Theorem 4.5, we can obtain the following existence of equilibria for qualitative games.

Theorem 5.4. *Let be a qualitative game, For each , suppose that the following conditions are satisfied*(i)* is a family of paracompact abstract convex space such that satisfies the partial KKM principle, and is a nonempty compact subset of , *(ii)* is an -majorized mapping,*(iii)* is open in ,*(iv)* for each , there exists a compact abstract convex subset of containing such that
**Then, has an equilibrium point in . *

*Proof. *For each , let . Define a mapping by for each , where the mapping is projection of onto . Furthermore, define the mapping by
Then, for each , if and only if . We will show that is an -majorized mapping. For each with , let with , by (ii), and let be an open neighborhood of in , and be mappings such that(a) for each , , , and ,(b) for each , and ,(c) the mapping is compactly open valued on ,(d) for each finite subset , .By (iii), we may assume that , hence and for all . Now, define two mappings by
Then, we have for each , by (a), and , for each , by (b), for each , is compactly open in by (c), for each finite subset , put . For each , then
For each , if , then there exists a set , such that , by (d), . Thus, . This shows that is an -majorized mapping. By and condition (iv), for each , there exists a compact abstract convex subset of containing such that
Hence, all the hypotheses of Theorem 4.5 are satisfied. By Theorem 4.5, there exists a point such that . This implies that and therefore for each , that is, is an equilibrium point of .

*Remark 5.5. *Theorem 5.4 improves and generalizes Theorem 5.2 of Ding and Feng [20], Theorem 6.2 of Ding and Wang [6], Theorem 4.2 of Yuan [25], Theorem 4.2 of Chowdhury et al. [4], and Theorem 4.2 of Tan and Yuan [26].

Applying Theorem 5.4, we prove that the following equilibria existence theorem for a noncompact generalized games.

Theorem 5.6. *Let be a generalized game. Let be a compact and closed subset of and . Suppose that for each , *(i)* is a paracompact abstract convex space such that satisfies the partial KKM principle,*(ii)* for each , , and ,*(iii)* for each , is compactly open in ,*(iv)* is open in , *(v)* is an -majorized mapping,*(vi)* for each , there exists a compact abstract convex subset of containing such that
**Then, has an equilibrium point in . *

*Proof. *For each , let , then is open in . Define by
We will prove that the qualitative game satisfies all the hypotheses of Theorem 5.4. For each , we have that the set
is open in and hence the condition (iii) of Theorem 5.4 is satisfied. By (v), for each , there exist an open neighborhood of in and two mappings such that(a) for each , , and ,(b) for each , and ,(c) the mapping is compactly open valued on ,(d) for each nonempty finite subset ,
Define by
Now for each with , the set
is open in . Then, for each , by (a) if , and if , by (a) if , by (ii), that is, for each , and by (a), for each , by (b) and , for each ,
is compactly open by (c) and (iii), for each , put
*Case I*. If , , then
*Case II*. If , then (1) if , then , and
(2) If , , that is similar to the condition (1).(3) If , then
Thus, is an -majorized mapping. By the definition of and condition (vi) for each , there exists a compact abstract convex subset of containing such that
Hence, all the hypotheses of Theorem 5.4 are satisfied. By Theorem 5.4, there exists a point such that . By the definition of , , that is for all , , and . This completes the proof.

*Remark 5.7. *Theorem 5.6 generalized Theorem 5.3 of Ding and Feng [20], Theorem 6.3 of Ding and Wang [6], Theorem 4.4 of Chowdhury et al. [4], and Theorem 4.3 of Tan and Yuan [26].

#### Acknowledgment

This work is partially supported by the National Youth Fund of China (51203112).