A Maximal Element Theorem in FWC-Spaces and Its Applications

Abstract

A maximal element theorem is proved in finite weakly convex spaces (FWC-spaces, in short) which have no linear, convex, and topological structure. Using the maximal element theorem, we develop new existence theorems of solutions to variational relation problem, generalized equilibrium problem, equilibrium problem with lower and upper bounds, and minimax problem in FWC-spaces. The results represented in this paper unify and extend some known results in the literature.

1. Introduction and Preliminaries

In 1983, by using fixed point theorems for set-valued mappings, Yannelis and Prabhakar [1] proved three existence theorems of maximal elements under the setting of locally convex topological vector spaces. In 1985, Yannelis [2] improved the Fan-Browder-type fixed point theorem and obtained an existence result of maximal elements by using this fixed point theorem. Since then, many maximal element theorems and their applications have been established in the setting of topological vector spaces; see, for example, [3–8] and the references therein.

It is well known that the linearity and convexity assumptions play crucial roles in most of the known existence results of maximal elements, which strictly restrict the applicable range of these maximal element theorems. Considering this fact, Zhang and Wu [9] proved an existence theorem of maximal elements in noncompact -spaces and obtained some minimax inequalities, variational inequalities, and quasivariational inequalities by using this maximal element theorem. Subsequently, Wu [10] used existence theorems of maximal elements to prove equilibrium existence theorems for qualitative games and abstract economies in noncompact -spaces. Recently, by using a generalization of the Fan-Browder fixed point theorem, Balaj and Lin [11] proved a new fixed point theorem for set-valued mappings in -convex spaces from which they derived several coincidence theorems and existence theorems for maximal elements. As applications, they obtained some existence theorems of solutions to the generalized equilibrium problem and minimax problem.

Motivated and inspired by the work mentioned above, in this paper, we prove a new maximal element theorem in -spaces (see Definition 1) without any linear, convex, and topological structure. As applications of this theorem, we obtain some new existence theorems of solutions to variational relation problem, generalized equilibrium problem, equilibrium problem with lower and upper bounds, and minimax problem in -spaces.

Now, we introduce some notation and definitions. For a nonempty set , and denote the family of all subsets of and the family of nonempty finite subsets of , respectively. For every , denotes the cardinality of . If is a topological space, then denotes the closure of . If is a vector space, then we denote by the convex hull of . Let be a set-valued mapping with being a nonempty set. We define the mapping by for each . If is a topological space, we say that is compact if is compact. If and are both topological spaces, we say that is upper semicontinuous (resp., lower semicontinuous) if for every closed subset of , the set (resp., ) is closed. Let denote the standard -dimensional simplex with vertices . For a nonempty subset , let denote the convex hull of the vertices .

A nonempty topological space is contractible if the identity mapping on is homotopic to a constant mapping. Every nonempty convex subset of a topological vector space is contractible, but the converse is not true in general. A subset of a topological space is called to be compactly closed (resp., compactly open) in if for each nonempty compact subset of , is closed (resp., open) in . The notions of compactly closed (resp., compactly open) sets are true generalizations of closed (resp., open) sets. Note that there exists a nonempty subset of the topological vector space such that for each nonempty compact subset of , is closed in , but is not closed. For details, see Kelley [12, page 240] or Wilansky [13, page 143].

Let be a topological vector space. For every , let us define a continuous mapping by for each . This mapping motivates us to introduce an abstract convex space which does not possess any linear, convex, and topological structure and is described in the following definition.

Definition 1 (see [14]). A triple is said to be a finite weakly convex space (-space, in short) if , are two nonempty sets and for each where some elements in may be same, there exists a set-valued mapping with nonempty values. When , the space is denoted by . In case , let . Let and . is said to be an -subspace of relative to if for each and for each , we have , where . We note that if is nonempty and is an -subspace of relative to , then is automatically nonempty. When , is said to be an -subspace of .
It is worthwhile noticing that and in Definition 1 do not possess any linear, convex, and topological structure. Major examples of -spaces are convex subsets of topological vector spaces, hyperconvex metric spaces introduced by Aronszajn and Panitchpakdi [15], Lassonde’s convex spaces in [16], -spaces introduced by Horvath [17], -convex spaces introduced by Park and Kim [18], -convex spaces introduced by Ben-El-Mechaiekh et al. [19], --spaces introduced by Verma [20–22], pseudo--spaces introduced by Lai et al. [23], -spaces due to Khanh et al. [24], -spaces due to Ding [25], and many other topological spaces with abstract convex structure (see, e.g., [26] and the references therein).
Taking Lassonde’s convex space, -space, and hyperconvex metric space as examples, we show that these three spaces are particular forms of -spaces. Let be a convex space in [16]; that is, a nonempty convex set in a vector space with any topology that induces the Euclidean topology on the convex hulls of its finite subsets. Then for every , define a continuous mapping as follows: Therefore, forms an -space. Let be an -space in [17], where is a family of nonempty contractible subsets of indexed by such that whenever . Then by Theorem 1 of Horvath [17], for every , there exists a continuous mapping and thus, is an -space. Let be a hyperconvex metric space in [15]. Then by Lemma 2.3 of Yuan [27], we know that is an -space and thus, is an -space.

Definition 2. Let be an -space and a topological space. The class of better admissible mappings is defined as follows: a set-valued mapping belongs to if and only if for every and for every continuous mapping , the composition has a fixed point. When , we will write instead of .

Remark 3. Since and in Definition 2 are nonempty sets which do not possess any linear, convex, and topological structure, the class includes many important classes of mappings as special cases, for example, the class of Kakutani's mappings (i.e., the upper semicontinuous set-valued mappings with nonempty compact convex values and codomain being convex set in a topological vector space), the class in Park and Kim [18], the class in Ben-El-Mechaiekh et al. [19], and the class in Ding [25].

Example 4. Let and , where is a family of subsets of . We can verify that is not a topological space. For simplicity, we write instead of . Let with the Euclidean metric topology. For each , define a set-valued mapping by for each . It is easy to see that forms an -space. Now we define a set-valued mapping by Then for each , we have . Therefore, the composition is an upper semicontinuous set-valued mapping with nonempty compact contractible values. By Lemma 1 of [28], for every continuous function , the composition has a fixed point. Therefore, .

Lemma 5. Let be an index set. For each , let be an -space. Let , , and . Then is also an -space.

Proof. For each , let be the projection of onto . For every , let . Since each is an -space, it follows that there exists a set-valued mapping with nonempty values. Define a set-valued mapping by It is clear that has nonempty values. Therefore, is also an -space.

2. A Maximal Element Theorem

Our first result is the following maximal element theorem.

Theorem 6. Let be an FWC-space, a Hausdorff topological space, and a nonempty compact subset of . Let , , , and be set-valued mappings such that(i)for each , ;(ii)for each , is compactly open;(iii)for each and each , (iv)one of the following conditions holds:(iv₁)there exists such that and for each , is a compact subset of ;(iv₂)for each , there exists a subset of containing such that is an -subspace of relative to , is a compact subset of , and
Then there exists such that .

Proof. We prove Theorem 6 distinguishing the following two cases.
Case (iv₁). Assume (iv₁) holds. Suppose that the conclusion of Theorem 6 does not hold. Then for each , and hence, there exists ; that is, . Therefore, we have which implies that . Since is compact and each is compactly open by (ii), it follows that there exists such that By the first part of (iv₁), we have Then it follows from (7) and (8) that where . By the definition of -spaces, there exists a set-valued mapping with nonempty values. By the second part of (iv₁), is compact subset of . By (9), we have and thus, ; that is, is an open cover of the compact set . Let be the partition of unity subordinated to this cover and then define a mapping by for each . Clearly, is continuous and for each , we have where is defined by . Then we have Define a set-valued mapping as follows: Now, we show that for each , is an -subspace of relative to . In fact, let and . Then . By (iii), ; that is, . Therefore, we have . By (i), we know that , which implies that for each , is an -subspace of relative to . Hence, by (12) and (13), we have This shows that
On the other hand, since , it follows that the composition mapping has a fixed point ; that is, . Let such that . Choose such that Then by (13) and (16), we have which contradicts (15). Thus, there must exist a point such that .
Case (iv₂). Assume (iv₂) holds. Suppose that the conclusion of Theorem 6 is not true. Then by using the same method as in Case (iv₁), we have which implies that . By (ii), we know that is a family of closed sets in . Thus, there exists such that that is, . By (iv₂), there exists a subset of containing such that is an -subspace of relative to and Therefore, we have
Since , it follows that . Therefore, we have Since is a compact subset of , it follows from (ii) and (22) that there exists such that By the fact that is an -subspace of relative to , we can see that the triple is also an -space. Hence, there exists a set-valued mapping with nonempty values. Assume that is the partition of unity subordinated to the open cover . Then for every , we have Furthermore, we define a continuous mapping by for each . Let . Since , it follows that . Then the composition has a fixed point ; that is, . Let such that . Then we have where . Let . By (i) and (iii), for each and each , we have thus, we have the following: Hence, there exists such that . On the other hand, by the definitions of and of the partition , we have which is a contradiction. Therefore, the conclusion of Theorem 6 holds.

Remark 7. (iii) of Theorem 6 can be replaced by the following equivalent condition:
for each , is an -subspace of relative to .

Proof. (iii) : let and . Then . Thus, . Since by (iii) of Theorem 6, it follows that ; that is, . Therefore, we have , which implies that holds.
(iii): let , , and . Then there exists such that . Therefore, we get This means that . By , we have and hence, . Let . Then , which implies that

Example 8. Let be endowed with the Euclidean topology. Let and , where is a family of subsets of . It is easy to check that is not a topological space. For simplicity, we will write instead of . Define a set-valued mappings such that which is open in . Therefore, is compactly open-valued, and hence, (ii) of Theorem 6 is satisfied. Furthermore, define a set-valued mapping such that Now, let be defined by Then we have , which is compact subset of . Let . Then (iv₁) of Theorem 6 is satisfied automatically. In order to check (iii) of Theorem 6, for every , we define a set-valued mapping by for all . Then forms an -space. For each , we have We can see that for each , is an -subspace of relative to . Therefore, by Remark 7, we know that (iii) of Theorem 6 holds. Finally, we show that, for each , and . By the definition of , we can obtain as follows: Thus, we can easily see that for each . Hence, for each , . By using the same method as in Example 4, we can prove that . Therefore, all the hypotheses of Theorem 6 are satisfied. We can see that there exists a point such that .

If in Theorem 6, then Theorem 6 deduces the following result.

Corollary 9. Let be an FWC-space, a Hausdorff topological space, and a nonempty compact subset of . Let , , and be set-valued mappings such that(i)for each , is compactly open;(ii)for each and each , (iii)one of the following conditions holds:(iii₁)there exists such that and for each , is a compact subset of ;(iii₂)for each , there exists a subset of containing such that is an -subspace of relative to , is a compact subset of , and
Then there exists such that .

Taking and in Corollary 9, we can obtain the following result.

Corollary 10. Let be an FWC-space, a Hausdorff topological space, and a nonempty compact subset of . Let and be set-valued mappings such that(i)for each , is compactly open;(ii)for each and each , (iii)one of the following conditions holds:(iii₁)there exists such that and for each , is a compact subset of ;(iii₂)for each , there exists a subset of containing such that is an -subspace of , is a compact subset of , and
Then there exists such that .

3. Existence of Solutions to Variational Relation Problem

In 2008, Luc [29] introduced a variational relation problem which unifies many equilibrium problems, optimization problems, and variational or differential inclusion problems. Since then, further studies on variational relation problems were investigated by many authors; see, for example, [30–32] and the references therein.

Let be an -space, a Hausdorff topological space, a nonempty compact subset of , and a set-valued mapping. In this section, we will study the following variational relation problems in -spaces.(1)Let be a relation linking and . Find such that holds for each .(2)Let be a nonempty set, a set-valued mapping, and a relation linking and . Find such that holds for each and each .

By applying Corollary 10, we have the following existence theorem of solutions to the variational relation problem in -spaces.

Theorem 11. Let be an FWC-space, a Hausdorff topological space, and a nonempty compact subset of . Let be a set-valued mapping and let be a relation linking elements , such that(i)for each , the set is compactly open;(ii)for each , each and each , there exists such that holds;(iii)one of the following conditions holds:(iii₁)there exists such that and for each , is a compact subset of ;(iii₂)for each , there exists a subset of containing such that is an -subspace of , is a compact subset of , and
Then there exists such that holds for each .

Proof. Define a set-valued mapping by By (i), for each , is compactly open. By (ii), for each and each , we have which implies that By (iii), we know that one of the following conditions holds:(a)there exists such that and for each , is a compact subset of ;(b)for each , there exists a subset of containing such that is an -subspace of , is a compact subset of , and Therefore, by Corollary 10, there exists such that ; that is, holds for each .

By taking and for every in Theorem 11, we can obtain the following result.

Corollary 12. Let be an FWC-space and a nonempty compact subset of , where is a Hausdorff topological space. Let , where is the identity mapping on . Let be a relation linking elements , such that(i)for each , the set is compactly open;(ii)for each , each and each , there exists such that holds;(iii)one of the following conditions holds:(iii₁)there exists such that and is a compact set-valued mapping for each ;(iii₂)for each , there exists a compact subset of containing such that is an -subspace of and
Then there exists such that holds for each .

Corollary 13. Let be an FWC-space, where is a Hausdorff compact topological space. Let , where is the identity mapping on . Let be a relation linking elements , such that the following conditions hold:(i)for each , the set is compactly open;(ii)for each , each and each , there exists such that holds.
Then there exists such that holds for each .

Proof. Let . Then (iii₁) of Corollary 12 is satisfied automatically. Hence, the conclusion of Corollary 13 follows from Corollary 12.

Remark 14. It is interesting to compare Corollary 13 with Theorem 2.1 of Pu and Yang [32] in the following aspects: (i) of Corollary 13 is weaker than (i) of Theorem 2.1 of Pu and Yang [32], which can be stated as follows: for each , the set is closed; (ii) of Theorem 2.1 of Pu and Yang [32] can be stated as follows: for each , there exists a continuous mapping such that, for each , there exists such that holds, where . By (ii) of Theorem 2.1 of Pu and Yang [32], we know that in Theorem 2.1 of Pu and Yang [32] forms an -space; in Corollary 13, the topological space needs not to have the fixed point property, but in Theorem 2.1 of Pu and Yang [32] needs to possess the fixed point property; for the identity mapping on in Theorem 2.1 of Pu and Yang [32], we must have . In fact, for every and for every continuous mapping , the composition is continuous, where coincides with the one in (ii) of Theorem 2.1 of Pu and Yang [32]. Then by Brouwer fixed point theorem, there exists such that , which implies that .

Theorem 15. Let be an FWC-space, a Hausdorff topological space, a nonempty compact subset of , and a nonempty set. Let , be set-valued mappings and a relation linking elements , . Assume that(i)for each , the set is compactly open, where is defined by for each ;(ii)for each , each and each , there exists such that holds for each ;(iii)one of the following conditions holds:(iii₁)there exists such that and for each , is a compact subset of ;(iii₂)for each , there exists a subset of containing such that is an -subspace of , is a compact subset of , and
Then there exists such that holds for each and each .

Proof. Let the relation on and be defined by holds if and only if . Then by (i), for each , the set is compactly open. By (ii), for each , each and each , there exists such that that is, . Thus, holds. By (iii), we know that one of the following conditions holds:(a)there exists such that and for each , is compact subset of ;(b)for each , there exists a subset of containing such that is an -subspace of and where is a compact subset of . Therefore, by Theorem 11, there exists such that holds for each ; that is, holds for each and each .

Remark 16. Theorem 15 generalizes Theorem 3.1 of Balaj and Lin [30] in the following aspects: (a) The underlying spaces of Theorem 15 and Theorem 3.1 of Balaj and Lin [30] are -spaces and convex spaces, respectively. It follows from the previous analysis that -spaces include convex spaces as special cases; (b) The class of better admissible mappings in Theorem 15 and Theorem 3.1 of Balaj and Lin [30] are and , respectively. By Remark 3, we know that is contained in ; (c) in Theorem 15 does not possess any topological structure; (d) in Theorem 15 needs not to be compact. In fact, if in Theorem 15 is compact, then we know that (iii₁) of Theorem 15 is satisfied by taking ; (e) (ii) of Theorem 3.1 of Balaj and Lin [30] can be stated as follows: for each , each and each , there exists such that for all . (ii) of Theorem 15 is weaker than (ii) of Theorem 3.1 of Balaj and Lin [30]. In fact, for every , we can define a continuous mapping by Therefore, forms an -space. On the basis of this fact, we can see that (ii) of Theorem 3.1 of Balaj and Lin [30] implies (ii) of Theorem 15.
(2) Theorem 15 is equivalent to Theorem 11. In fact, we have shown that Theorem 11 implies Theorem 15. Conversely, if and for each in Theorem 15, then Theorem 15 becomes Theorem 11.

Corollary 17. Let be defined by for each . Theorem 15 is true if (i) of Theorem 15 is replaced by one of the following conditions: for each , the set is compactly closed; is a topological space, the set-valued mapping is lower semicontinuous, and has open values.

Proof. Suppose that is satisfied. Then is compactly closed for each . Thus, is compactly open. If holds, then by the definition of a lower semicontinuous set-valued mapping, for each , the set is open and thus, compactly open.

4. Generalized Equilibrium Theorems

In recent years, many authors (see, e.g., [33–35] and the references therein) studied one or more of the following generalized equilibrium problems.

Let and be nonempty sets and a topological space. Let and be set-valued mappings. Find such that one of the following situations occurs:

Let be another nonempty set, a set-valued mapping, and a single-valued mapping. The generalized implicit vector equilibrium problem is to find such that, for each , there exists satisfying . For more details, the reader may consult [35] and the references therein.

In this section, as applications of Theorem 6, we will prove new existence theorems of solutions to generalized equilibrium problems in -spaces without any linear, convex, and topological structure.

Theorem 18. Let be an -space, a Hausdorff topological space, a nonempty compact subset of , and a nonempty set. Let , , , , and be set-valued mappings such that(i)for each and each , ;(ii)for each , the set is compactly closed;(iii)for each , the set is an -subspace of relative to the set ;(iv)one of the following conditions holds:(iv₁) for some and for each , is a compact subset of ;(iv₂)for each , there exists a subset of containing such that is an -subspace of relative to , is a compact subset of , and
Then there exists such that for each .

Proof. Define and by By (i), we have for each . By (ii), for each , is compactly open. Now, we show that (iii) of Theorem 6 is satisfied. Suppose the contrary. Then there exist and such that which implies that there exists such that for each ; that is, . By (iii), we have which implies that Since , it follows that there exists such that ; that is, , which contradicts (55). Therefore, (iii) of Theorem 6 holds. Suppose that (iv₁) of Theorem 18 is fulfilled. Then by (iv₁) and the definition of , we know that there exists such that and for each , is a compact subset of . Therefore, (iv₁) of Theorem 6 is satisfied. If (iv₂) of Theorem 18 holds, then by (iv₂) and the definition of again, we know that for each , there exists a subset of containing such that is an -subspace of relative to and where is a compact subset of . Therefore, (iv₂) of Theorem 6 is satisfied. Thus, all conditions of Theorem 6 are fulfilled. By Theorem 6, there exists a point such that ; that is, there exists a point such that for each . This completes the proof.

Remark 19. Theorem 18 generalizes Theorem 4.1 of Fang and Huang [34] in the following aspects: (a) The underlying spaces of Theorem 18 and Theorem 4.1 of Fang and Huang [34] are -spaces and -spaces, respectively. By the previous analysis, we know that -spaces include -spaces as special cases; (b) The class of better admissible mappings in Theorem 18 and Theorem 4.1 of Fang and Huang [34] are and , respectively. By Remark 3, we know that is contained in ; (c) (ii) of Theorem 18 is weaker than (i) of Theorem 4.1 of Fang and Huang [34]; (d) (iii) of Theorem 18 is weaker than (iii) of Theorem 4.1 of Fang and Huang [34]; (e) (iv₂) of Theorem 18 is weaker than (iv) of Theorem 4.1 of Fang and Huang [34]. It should be emphasized that the proof of Theorem 18 is different from that of Theorem 4.1 of Fang and Huang [34].
By using the same argument as in Theorem 18, we can obtain Theorems 20, 22, and 23. We omit their proofs.

Theorem 20. Let be an -space, a Hausdorff topological space, a nonempty compact subset of , and a nonempty set. Let , , , , and be set-valued mappings. Assume that(i)for each and each , ;(ii)for each , the set is compactly closed;(iii)for each , the set is an -subspace of relative to the set ;(iv)one of the following conditions holds:(iv₁) for some and for each , is a compact subset of ;(iv₂)for each , there exists a subset of containing such that is an -subspace of relative to , is a compact subset of , and
Then there exists such that for each .

Remark 21. Theorem 20 generalizes Theorem 4.3 of Fang and Huang [34] in the following aspects: (a) The underlying spaces of Theorem 18 and Theorem 4.3 of Fang and Huang [34] are -spaces and -spaces, respectively. It follows from the previous analysis that -spaces include -spaces as special cases; (b) The class of better admissible mappings in Theorem 18 and Theorem 4.3 of Fang and Huang [34] are and , respectively. It follows from Remark 3 that is contained in ; (c) (ii) of Theorem 18 is weaker than (i) and (ii) of Theorem 4.3 of Fang and Huang [34]; (d) (iii) of Theorem 18 is weaker than (v) of Theorem 4.3 of Fang and Huang [34]; (e) (iv₂) of Theorem 18 is weaker than (vi) of Theorem 4.3 of Fang and Huang [34].

Theorem 22. Let be an -space, a Hausdorff topological space, a nonempty compact subset of , and a nonempty set. Let , , , , and be set-valued mappings. Assume that(i)for each and each , ;(ii)for each , the set is compactly closed;(iii)for each , the set is an -subspace of relative to the set ;(iv)one of the following conditions holds:(iv₁) for some and for each , is a compact subset of ;(iv₂)for each , there exists a subset of containing such that is an -subspace of relative to , is a compact subset of , and
Then there exists such that for each .

Theorem 23. Let be an -space, a Hausdorff topological space, a nonempty compact subset of , and a nonempty set. Let , , , , and be set-valued mappings. Assume that(i)for each and each , ;(ii)for each , the set is compactly closed;(iii)for each , the set is an -subspace of relative to the set ;(iv)one of the following conditions holds:(iv₁) for some and for each , is a compact subset of ;(iv₂)for each , there exists a subset of containing such that is an -subspace of relative to , is a compact subset of , and
Then there exists such that for each .

By Theorem 20, we can obtain the following existence theorem of solutions to the generalized implicit vector equilibrium problem.

Theorem 24. Let be an -space, a Hausdorff topological space, a nonempty compact subset of , and a nonempty set. Let , , and be set-valued mappings. Let be a nonempty set and a set-valued mapping. Let and be two single-valued mappings. Assume that(i)for each and each , ;(ii)for each , the set is compactly closed;(iii)for each , the set is an -subspace of relative to the set ;(iv)one of the following conditions holds:(iv₁) for some and for each , is a compact subset of ;(iv₂)for each , there exists a subset of containing such that is an -subspace of relative to , is a compact subset of , and
Then there exists such that, for each , there exists satisfying .

Proof. Define two set-valued mappings and by It is clear that all conditions of Theorem 20 are satisfied. By Theorem 20, there exists a point such that for each . Hence, for each , there exists satisfying . This completes the proof.

Remark 25. Theorem 24 generalizes Theorem 4.4 of Fang and Huang [34] in the following aspects: (a) The underlying spaces of Theorem 18 and Theorem 4.4 of Fang and Huang [34] are -spaces and -spaces, respectively. It follows from the previous analysis that -spaces include -spaces as special cases; (b) the class in Theorem 24 includes the class in Theorem 4.4 of Fang and Huang [33] as a special case; (c) (ii) Theorem 24 is weaker than (i) and (ii) of Theorem 4.4 of Fang and Huang [34]. In fact, by the proof of Theorem 4.4 of Fang and Huang [34], we can see that (i) and (ii) of Theorem 4.4 of Fang and Huang [34] imply (ii) of Theorem 24; (d) (iv₂) of Theorem 24 is weaker than (vi) of Theorem 4.4 of Fang and Huang [34]. We point out that Theorem 24 also generalizes Theorem 3.2 and Corollary 3.2 of Lee and Kum [35] from topological vector spaces to -spaces. We emphasis that , , , and in Theorem 24 do not possess any linear, convex, and topological structure.
(2) Let and in Theorem 24 be two topological spaces. Then by Theorem 4.4 of Fang and Huang [33], we can replace (ii) of Theorem 24 by the following conditions:the graph of is open in ; is upper semicontinuous on each compact subset of with nonempty compact values and for each , is continuous on each compact subset of .

Remark 26. The solution sets of generalized equilibrium problems considered in Theorems 18, 20, and 22–24 are compact subsets of . Indeed, by the proof of Theorems 6, 18, 20, and 22–24, we can see that these solution sets can be represented by , where is a set-valued mapping with compactly open values. Thus, these solution sets are compactly closed subsets of the compact set . Therefore, the solution sets of generalized equilibrium problems considered in Theorems 18, 20, and 22–24 are compact subsets of .

5. Applications

Let be a nonempty closed subset of a locally convex semireflexive topological vector space , and let be a real-valued function on . In 1999, Isac et al. [36] first raised the open problem of finding such that for each , where , are two real numbers with . Later, Li [37] introduced the concept of extremal subsets and then, by using the Fan-KKM theorem in topological vector spaces, he gave some positive answers to this open problem mentioned above. Recently, Fakhar and Zafarani [38] obtained an existence theorem of solutions to the equilibrium problems with lower and upper bounds under the setting of -convex spaces.

In this section, we apply Theorem 18 to obtain existence results of solutions to the equilibrium problem with lower and upper bounds and minimax problem in -spaces.

Theorem 27. Let be an -space and a nonempty compact subset of a Hausdorff topological space . Let and be set-valued mappings. Let and be real-valued functions on and , respectively. Let and be real-valued functions on such that for each . Assume that(i)for each and each , ;(ii)for each , the set is compactly closed;(iii)for each , the set is an -subspace of relative to the set ;(iv)one of the following conditions holds:(iv₁) for some and for each , is a compact subset of ;(iv₂)for each , there exists a subset of containing such that is an -subspace of relative to , is a compact subset of , and
Then there exists such that for each .

Proof. Let . Define three set-valued mappings , , , and as follows: It is clear that all conditions of Theorem 18 with are satisfied. Therefore, by Theorem 18 with , there exists a point such that for each ; that is, , for each . This completes the proof.

Remark 28. Theorem 27 generalizes Corollary 3.2 of Mitrović and Merkle [39] in the following aspects: The underlying spaces of Theorem 27 and Corollary 3.2 in [39] are -spaces and Hausdorff compact topological vector spaces, respectively. By the previous analysis, we know that -spaces include Hausdorff compact topological vector spaces as special cases; The condition that there are four functions in Theorem 27 is more general than the condition that there are three functions in Corollary 3.2 in [39];    (ii) of Theorem 27 is weaker than of Corollary of Mitrović and Merkle [39]; (iii) of Theorem 27 is weaker than of of Corollary 3.2 of Mitrović and Merkle [39]. We point out that the proof of Theorem 27 is different from that of Corollary 3.2 of Mitrović and Merkle [39].

Let and for all , where and are real numbers such that . In this case, Theorem 27 deduces the following corollary.

Corollary 29. Let be an -space and a nonempty compact subset of a Hausdorff topological space . Let and be set-valued mappings. Let and be real-valued functions on and , respectively. Let and be two real numbers such that . Assume that(i)for each and each , ;(ii)for each , the set is compactly closed;(iii)for each , the set is an -subspace of relative to the set ;(iv)one of the following conditions holds:(iv₁) for some and for each , is a compact subset of ;(iv₂)for each , there exists a subset of containing such that is an -subspace of relative to , is a compact subset of , and
Then there exists such that for each .

Another special case of Corollary 29, stated below, is the case where .

Corollary 30. Let be an -space and a nonempty compact subset of a Hausdorff topological space . Let and be set-valued mappings. Let and be real-valued functions on and , respectively. Let be a real number. Assume that(i)for each and each , ;(ii)for each , the set is compactly closed;(iii)for each , the set is an -subspace of relative to the set ;(iv)one of the following conditions holds:(iv₁) for some and for each , is a compact subset of ;(iv₂)for each , there exists a subset of containing such that is an -subspace of relative to , is a compact subset of , and
Then there exists such that for each .

Remark 31. It is interesting to compare Corollary 30 with Corollary 3.1 of Li [37] in the following aspects: the underlying spaces in Corollary 30 are -spaces without any linear, convex, and topological structure, which include the corresponding underlying spaces (i.e, Hausdorff topological vector spaces) in Corollary 3.1 of Li [37] as special cases; (i) of Corollary 30 is weaker than (i) of Corollary 3.1 of Li [37]; (ii) of Corollary 30 is weaker than (iv) of Corollary 3.1 of Li [37]; (iii) of Corollary 30 is weaker than (ii) of Corollary 3.1 of Li [37]; (iv) of Corollary 30 is neither stronger nor weaker than (iii) of Corollary 3.1 of Li [37].

As a consequence of Corollary 29, we can obtain the following corollary, which improves and generalizes Theorems of Verma [20], Theorem 2.6 of Verma [21], and Corollary 3.4 of Fakhar and Zafarani [40].

Corollary 32. Let be an -space and a nonempty compact subset of a Hausdorff topological space . Let and be set-valued mappings. Let and be real-valued functions on and , respectively. Let be a real number. Assume that(i)for each and each , ;(ii)for each , the set is compactly closed;(iii)for each , the set is an -subspace of relative to the set ;(iv)one of the following conditions holds:(iv₁) for some and for each , is a compact subset of ;(iv₂)for each , there exists a subset of containing such that is an -subspace of relative to , is a compact subset of , and
Then there exists such that for each .

Proof. Define real-valued functions and by Taking and in Corollary 29, we can see that all conditions of Corollary 29 are satisfied for and . Therefore, by Corollary 29, there exists a point such that for each ; that is, for each . This completes the proof.

Remark 33. (ii)-(iii) of Corollary 32 can be replaced by the following conditions, respectively.For every , is lower semicontmuous on each nonempty compact subset of .For every , every , and every , we have for all .
It is clear that implies (ii) of Corollary 32. Now, we show that implies (iii) of Corollary 32. In fact, if (iii) of Corollary 32 does not hold, then there exist , , and such that . Hence, there exists such that . Since , we have for each . By , we obtain the following contradiction: Therefore, (iii) of Corollary 32 must hold.

Remark 34. Theorem 6 is equivalent to Corollary 32. We first show that Theorem 6 implies that Corollary 32. Define and by By using the same method as in Theorem 18, we know that (iii) of Theorem 6 holds. We can see that the other conditions of Theorem 6 are satisfied. Therefore, by Theorem 6, there exists a point such that , which implies that for each .
Conversely, let be given. Let us define two real-valued functions and by We can see that and satisfy all conditions of Corollary 32. Therefore, by Corollary 32, there exists a point such that for each ; that is, for each , which implies that .

Corollary 35. Let , be as in Theorem 27, and let be a compact set-valued mapping. Let and be real-valued functions on and , respectively. Let be a real number. Assume that(i)for each and each , ;(ii)for each , the set is compactly closed;(iii)for each , the set is an -subspace of relative to the set .
Then there exists such that for each .

Proof. Define a set-valued mapping by for each . For each , let . Let . Then (iv₂) of Corollary 32 is satisfied automatically. Therefore, the conclusion of Corollary 35 follows from Corollary 32.

Remark 36. Corollary 35 generalizes Theorem 3.3 of Tan [41] in the following aspects: The underlying spaces of Corollary 35 are -spaces which contain G-convex spaces adopted in Theorem 3.3 of Tan [41]; There are two functions in Corollary 35, but there is only one function in Theorem 3.3 of Tan [41]; the condition that each -co is compact in Theorem 3.3 of Tan [41] is dropped; (ii) of Corollary 35 is weaker than of Theorem 3.3 of Tan [41]. In fact, the lower semicontinuity of a function implies (ii) of Corollary 35; (5) combining Proposition of Tan [41] and Remark , we can see that (iii) of Corollary 35 is weaker than of Theorem 3.3 of Tan [41]. Corollary 35 also generalizes Theorem 3.1 of Zeng et al. [33] from topological vector spaces to -spaces without any linear, convex, and topological structure. The comparison details between Corollary 35 and Theorem 3.1 of Zeng et al. [33] are left up to the reader to finish.

Theorem 37. Let be an -space and a nonempty compact subset of a Hausdorff topological space . Let and be set-valued mappings. Let and be real-valued functions on and , respectively. Let be a real number. Assume that(i)for each , the set is compactly closed;(ii)for each , the set is an -subspace of relative to the set ;(iii)one of the following conditions holds:(iii₁) for some and for each , is a compact subset of ;(iii₂)for each , there exists a subset of containing such that is an -subspace of relative to , is a compact subset of , and
Then we have the following alternatives:(a)there exist and such that ;(b)there exists such that for each .

Proof. If (a) is false, then it follows that for each and each , . Hence, by Corollary 32, there exists a point such that for each . This completes the proof.

Theorem 38. Let be an -space and a nonempty compact subset of a Hausdorff topological space . Let and be set-valued mappings. Let and be real-valued functions on and , respectively. Assume that and the following conditions hold:(i)for each , the set is compactly closed;(ii)for each , the set is an -subspace of relative to the set ;(iii)either(iii₁) for some and for each , is a compact subset of or(iii₂)for each , there exists a subset of containing such that is an -subspace of relative to , is a compact subset of , and
Then there exists such that for each . In particular, we have .

Proof. Let . By the definition of , (a) of Theorem 37 does not hold. Hence, (b) of Theorem 37 is satisfied. So, there exists a point such that for each . In particular, we have . This completes the proof.

Remark 39. By setting and and adjusting the corresponding conditions of Theorem 38, we know that Theorem 38 can be restated with the conclusion that there exists a point such that for each . In particular, we have . Thus, Theorem 38 generalizes Theorem 3 of Yuan [28] from Hausdorff topological vector spaces to -spaces.

If in Theorem 38 is a single-valued mapping, then we have the following corollary.

Corollary 40. Let , , , and be as in Theorem 27. Let be a single-valued mapping. Let and be real-valued functions on and , respectively. Assume that and the following conditions hold:(i)for each , the set is compactly closed;(ii)for each , the set is an -subspace of relative to the set ;(iii)either(iii₁) for some and for each , is a compact subset of or(iii₂)for each , there exists a subset of containing such that is an -subspace of relative to , is a compact subset of , and
Then there exists such that for each . In particular, we have .

By taking and for all , we can obtain the following result from Corollary 40.

Corollary 41. Let be an -space, where is a Hausdorff topological space. Let be a nonempty compact subset of . Let be a set-valued mapping and let , where is the identity mapping on . Let and be real-valued functions on and , respectively. Assume that and the following conditions hold:(i)for each , the set is compactly closed;(ii)for each , the set is an -subspace of relative to the set ;(iii)either(iii₁) for some and is a compact set-valued mapping for each or(iii₂)for each , there exists a subset of containing such that is a compact -subspace of relative to and
Then there exists such that for each . In particular, we have .

Remark 42. Corollary 41 generalizes Corollary 5 of Jin and Cheng [42] in the following aspects: The underlying spaces of Corollary 41 are -spaces which include L-convex spaces adopted in Corollary 5 of Jin and Cheng [42] as special cases; the condition that each in Corollary 5 of Jin and Cheng [42] is compact is dropped; (i) of Corollary 41 is weaker than (i) of Corollary 5 of Jin and Cheng [42]. In fact, the condition that a function is lower semicontinuous on compact subset of implies (i) of Corollary 41; it is easy to verify that (ii) of Corollary 41 is weaker than (ii) of Corollary 5 of Jin and Cheng [42]; (iii) of Corollary 41 is weaker than (iii) of Corollary 5 of Jin and Cheng [42]. Additionally, Corollary 41 is quite differen7t from Theorem 1 of Kim [43] because the underlying spaces of Theorem 1 of Kim [43] are Hausdorff topological vector spaces and the conditions of Theorem 1 of Kim [43] are different from that of Corollary 41.

Corollary 43. Let be an -space, where is a Hausdorff topological space. Let be a nonempty compact subset of . Let be a set-valued mapping and let , where is the identity mapping on . Let and be real-valued functions on and , respectively. Assume that and the following conditions hold:(i)for each , the set is compactly closed;(ii)for each , the set is an -subspace of relative to the set ;(iii)either(iii₁) for some and is a compact set-valued mapping for each or(iii₂)for each , there exists a subset of containing such that is a compact -subspace of relative to and
Then there exists such that for each . In particular, we have .

Proof. By setting and , we can see that the conclusion of Corollary 43 holds from Corollary 41.

Theorem 44. Let and be two -spaces, where and are two Hausdorff topological spaces. Let be a compact set-valued mapping, where is an -space defined as in Lemma 5. Let be a real-valued function on . Assume that(i)for each , each , and each , or ;(ii)for each and each , the sets and are compactly closed;(iii)for each and each , the set is an -subspace of ;(iv)for each and each , the set is an -subspace of .
Then .

Proof. It is clear that the following inequality is always true. In order to prove that the equality holds, it suffices to show the following inequality: Suppose the contrary. Then we have It follows that there exists such that which shows that for each , there exists such that Following the method in the proof of Theorem 4.4 of Tan [41], we define the real-valued function by By (i) and the definition of , for each , each , we have . For each , we have Then by (ii), we know that for each , the set is compactly closed.
Now, we show that for each , the set is an -space of . In fact, for each , we have the following: Then by (iii) and (iv), for each and each , we have Since , it follows from (84) that Therefore, we have , which implies that for each , the set is an -subspace of . Thus, by Corollary 35 with and , there exists such that for each . Hence, for each , either or , which contradicts (80). Therefore, we have . This completes the proof.

Remark 45. Theorem 44 generalizes Theorem 4.4 of Tan [41] in the following aspects: (a) The underlying spaces of Theorem 44 are -spaces which contain G-convex spaces adopted in Theorem 4.4 of Tan [41]; (b) the assumption that each - and each - in Theorem 4.4 of Tan [41] are compact is dropped; (c) (ii) of Theorem 44 is weaker than and of Theorem 4.4 of Tan [41]; (d) (iii) and (iv) of Theorem 44 are weaker than of Theorem 4.4 of Tan [41].

Conflict of Interests

The authors declare that there is no conflict of interests regarding the publication of this paper.

Acknowledgments

The authors thank the referees for their many valuable suggestions and helpful comments which improved the exposition of this paper. This work was supported by the Planning Foundation for Humanities and Social Sciences of Ministry of Education of China (“Research on utilizing conflict of water resources and initial water rights allocation in a river basin based on game theory,” no. 12YJAZH084). At the same time, this work was also supported by Jiangsu Overseas Research Training Program for University Prominent Young Middle-Aged Teachers and Presidents.