#### Abstract

The aim of this paper is to study the minimax theorems for set-valued mappings with or without linear structure. We define several kinds of cone-convexities for set-valued mappings, give some examples of such set-valued mappings, and study the relationships among these cone-convexities. By using our minimax theorems, we derive some existence results for saddle points of set-valued mappings. Some examples to illustrate our results are also given.

#### 1. Introduction

The minimax theorems for real-valued functions were introduced by Fan [1, 2] in the early fifties. Since then, these were extended and generalized in many different directions because of their applications in variational analysis, game theory, mathematical economics, fixed-point theory, and so forth (see, for example, [3–11] and the references therein). The minimax theorems for vector-valued functions have been studied in [4, 9, 10] with applications to vector saddle point problems. However, the minimax theorems for set-valued bifunctions have been studied only in few papers, namely, [4–8] and the references therein.

In this paper, we establish some new minimax theorems for set-valued mappings. Section 2 deals with preliminaries which will be used in rest of the paper. Section 3 denotes the cone-convexities of set-valued mappings. In Section 4, we establish some minimax theorems by using separation theorems, Fan-Browder fixed-point theorem. In the last section, we discuss some existence results for different kinds of saddle points for set-valued mappings.

#### 2. Preliminaries

Throughout the paper, unless otherwise specified, we assume that , are two nonempty subsets, and is a real Hausdorff topological vector space, is a closed convex pointed cone in with . Let be the topological dual space of , and let

We present some fundamental concepts which will be used in the sequel.

*Definition 2.1 (see [3, 4, 8]). *Let be a nonempty subset of . A point is called a (a)*minimal point* of if ; denotes the set of all minimal points of ;(b)*maximal point* of if ; denotes the set of all maximal points of ;(c)*weakly minimal point* of if ; denotes the set of all weakly minimal points of ;(d)*weakly maximal point* of if ; denotes the set of all weakly maximal points of .

It can be easily seen that and .

Lemma 2.2 (see [3, 4]). * Let be a nonempty compact subset of . Then, *(a)*; *(b)*; *(c)*; *(d)*. *

Following [6], we denote both and by (both and by ) in since both and (both and ) are the same in .

*Definition 2.3. *Let , be Hausdorff topological spaces. A set-valued map with nonempty values is said to be (a)*upper semicontinuous at * if for every and for every open set containing , there exists a neighborhood of such that ;(b)*lower semi-continuous at * if for any sequence such that and any , there exists a sequence such that ;(c)*continuous at * if is upper semi-continuous as well as lower semi-continuous at .

We present the following fundamental lemmas which will be used in the sequel.

Lemma 2.4 (see [9, Lemma 3.1]). * Let , , and be three topological spaces. Let be compact, a set-valued mapping, and the set-valued mapping defined by
*(a)*If is upper semi-continuous on , then is upper semi-continuous on . *(b)*If is lower semi-continuous on , so is . *

Lemma 2.5 (see [9, Lemma 3.2]). *Let be a Hausdorff topological vector space, a set-valued mapping with nonempty compact values, and the functions , defined by and . *(a)*If is upper semi-continuous, so is . *(b)*If is lower semi-continuous, so is . *(c)*If is continuous, so are and . *

We shall use the following nonlinear scalarization function to establish our results.

*Definition 2.6 (see [6, 10]). *Let and . The * Gerstewitz function * is defined by

We present some fundamental properties of the scalarization function.

Proposition 2.7 (see [6, 10]). *Let and . The Gerstewitz function has the following properties: *(a); (b); (c), where is the topological boundary of ; (d); (e); (f) is a convex function; (g) is an increasing function, that is, ; (h) is a continuous function.

Theorem 2.8 ( Fan-Browder fixed-point theorem (see [12])). *Let be a nonempty compact convex subset of a Hausdorff topological vector space and let be a set-valued mapping with nonempty convex values and open fibers, that is, is open for all . Then, has a fixed point.*

#### 3. Cone-Convexities

In this section, we present different kinds of cone-convexities for set-valued mappings and give some relations among them. Some examples of such set-valued mappings are also given.

*Definition 3.1. *Let be a nonempty convex subset of a topological vector space . A set-valued mapping is said to be (a)* above **convex* [4] (resp., *above*--*concave* [5]) on if for all and all ,
(b)*below- **convex* [13] (resp., *below- **-concave* [9, 13]) on if for all and all ,
(c)*above- **quasi-convex* (resp., *below- **-quasiconcave*) [7, Definition 2.3] on if the set
is convex for all ;(d)*above-properly **-quasiconvex* (resp., * above-properly **-quasiconcave* [6]) on if for all and all , either
or

*below-properly*

*-quasiconvex*[7] (resp.,

*below-properly*

*-quasiconcave*) on if for all and all , either or (f)

*above-naturally*

*-quasiconvex*[6] on if for all and all , where denotes the convex hull of a set ;(g)

*above*

*convex-like*(resp.,

*above-*

*-concave-like*) on ( is not necessarily convex) if for all and all , there is an such that (h)

*below*

*convex-like*[13] (resp.,

*below*

*concave-like*) on ( is not necessarily convex) if for all and all , there is an such that

It is obvious that every above--convex set-valued mapping or above-properly -quasi-convex set-valued mapping is an above-naturally -quasi-convex set-valued mapping, and every above--convex (above--concave) set-valued mapping is an above--convex-like (above--concave-like) set-valued mapping. Similar relations hold for cases below.

*Remark 3.2. *The definition of above-properly -quasi-convex (above-properly -quasi-concave) set-valued mapping is different from the one mentioned in [7, Definition 2.3] or [5, 6]. The following Examples 3.3 and 3.4 illustrate the reason why they are different from the one mentioned in [5–7]. However, if is a vector-valued mapping or a single-valued mapping, both mappings reduce to the ordinary definition of a properly -quasi-convex mapping for vector-valued functions [7]. The above--convexity in Definition 3.1 is also different from the below--convexity used in [5, 9].

*Example 3.3. *Consider . Let be a set-valued mapping defined by
and for all ,
Then is an above-properly -quasi-convex set-valued mapping, but it is not below-properly -quasi-convex.

On the other hand, let be a set-valued mapping defined by and for all , Then, is a below-properly -quasi-convex set-valued mapping, but it is not above-properly -quasi-convex.

*Example 3.4. * Let . Define by
Then is continuous, above--quasi-convex, below--quasi-concave, above-properly -quasi-convex, and above-properly -quasi-concave, but it is not below-properly -quasi-conconvex.

Proposition 3.5. * Let be a nonempty set (not necessarily convex) and for a given set-valued mapping with nonempty compact values, define a set-valued mapping as
*(a)If is above--convex-like, then is so. (b)If is a topological space and is a continuous mapping, then is upper semicontinuous with nonempty compact values on .

*Proof. *(a) Let be above--convex-like, and let be arbitrary. Since is above--convex-like, for any , there exists such that
By Lemma 2.2,
Therefore, is above--convex-like.

(b) The upper semicontinuity of was deduced in [4, Lemma 2].

Proposition 3.6. * Let be a nonempty convex set, and let be a set-valued mapping with nonempty compact values. Then, the set-valued mapping defined by
**
is above--quasiconvex if is so.*

The following result can be easily derived, and therefore, we omit the proof.

Proposition 3.7. *Let be a nonempty convex set and be above--concave. Then the set-valued mapping is above--concave and below--quasiconcave. Furthermore, if is above-properly -quasiconcave, then the set-valued mapping is also above-properly -quasiconcave and below--quasiconcave.*

Let and be a set-valued mapping. Then, the composition mapping is defined by Clearly, the composition mapping is also a set-valued mapping.

Proposition 3.8. *Let be a nonempty set, a set-valued mapping, and . *(a)If is above--convex-like, then is above--convex-like. (b)If is below--concave-like, then is below--concave-like. (c)If is a topological space and is upper semi-continuous, then so is .

*Proof. *(a) By the definition of above--convex-like set-valued mapping , for any and all , there exists such that . For any , there exist , such that
For any , we have . Hence, . Thus, is above--convex-like.

The proof of (b) and (c) is easy, and therefore, we omit it.

Proposition 3.9. * Let be a nonempty convex set and . *(a)If is above--concave (above-properly -quasi-concave), then is above--concave (above-properly -quasi-concave).(b)If is above-properly -quasi-convex, then is above--quasi-convex and above-properly -quasi-convex.(c)If is above--convex, then is above--convex and above--quasi-convex.

Lemma 3.10. * Let be a real Hausdorff topological vector space and a closed convex pointed cone in with . Let be a nonempty compact subset of a topological space , and let be an upper semi-continuous set-valued mapping with nonempty compact values. Then, for any , there exists such that .*

*Proof. *For any given , the mapping is upper semi-continuous by Proposition 3.8 (c). By the compactness of , there exist and such that . By Lemma 2.2, there exists such that , and hence . On the other hand, , we know that , and then . Therefore, the conclusion holds.

Proposition 3.11. * Let be a nonempty convex set. If is above-properly -quasi-convex, then it is above--quasi-convex.*

*Proof. * For any , let . Then, and are subsets of . Since is above-properly -quasi-convex, for any , is contained in either or , and hence, in . Thus, the set is convex, and therefore, is above--quasi-convex.

Proposition 3.12. * Let be a nonempty convex set. If is above-naturally -quasi-convex, then it is above--quasi-convex.*

*Proof. *Let , , and be the same as given as in Proposition 3.11. Then, since is convex. By the above-naturally -quasi-convexity, for all . Thus, the set is convex, and therefore, is above--quasi-convex.

Proposition 3.13. * Let be a nonempty convex set. If is above-naturally -quasi-convex, then is above-naturally -quasi-convex for any .*

*Proof. *Let be given. From the above-naturally -quasi-convexity of , for any and any ,
For any , there is a such that . Then there exist and , such that . Hence, , and
Therefore, is a above-naturally -quasi-convex.

Proposition 3.14. *Let be a set-valued mapping with nonempty compact values. For any , *(a)if for some , then ;(b)if for some , then .

* Proof. *Let . Suppose that . Then
Then, there exists and . Therefore, there exists such that and . Since , and . This implies that , which is a contradiction. This proves (a).

Analogously, we can prove (b), so we omit it.

*Remark 3.15. *Propositions 3.8 and 3.9, Lemma 3.10, and Propositions 3.13 and 3.14 are always true except Proposition 3.8 (b) if we replace by any Gerstewitz function.

#### 4. Minimax Theorems for Set-Valued Mappings

In this section, we establish some minimax theorems for set-valued mappings with or without linear structure.

Theorem 4.1. *Let , be two nonempty compact subsets (not necessarily convex) of real Hausdorff topological spaces and , respectively. Let the set-valued mapping be lower semi-continuous on and upper semi-continuous on such that for all , is nonempty compact and satisfies the following conditions: *(i)*for each , is below--concave-like on ; *(ii)*for each , is above--convex-like on . ** Then,
*

*Proof. *Since
it is sufficient to prove that
Choose any such that . For any , let
Then, by the lower semi-continuity of the set-valued mapping , the set is closed, hence it is compact for all . By the choice of , we have
Since is compact and the collection covers , there exist finite number of points in such that
or
This implies that
and therefore,
Following the idea of Borwein and Zhuang [14], let
where . Then the set is convex, so is . We note that the interior of is nonempty since
Since , by separation hyperplane theorem [15, Theorem 14.2], there is a such that
where , that is,
By (4.11), (4.13), and the choice of , we have that . Furthermore, from the fact
we have
Hence, by (4.13), we have
or
Thus, we have . Hence, by (4.17), we have
Since is below--concave-like in , there is such that
Therefore,
and hence,
This completes the proof.

*Remark 4.2. *Theorem 4.1 is a modification of [14, Theorem A]. If is a real-valued function, then Theorem 4.1 reduces to the well-known minimax theorem due to Fan [2].

We next establish a minimax theorem for set-valued mappings defined on the sets with linear structure.

Theorem 4.3. *Let , be two nonempty compact convex subsets of real Hausdorff topological vector spaces and , respectively. Let the set-valued mapping be lower semi-continuous on and upper semi-continuous on such that for all , is nonempty compact, and satisfies the following conditions: *(i)*for each , is above--quasi-convex on ; *(ii)*for each , is above--concave, or above-properly -quasi-concave on ; *(iii)*for each , there is a such that
**Then,
*

*Proof. *We only need to prove that
is impossible, since it is always true that
Suppose that there is an such that
Define by
For each , . Since is compact and the set-valued mapping is upper semi-continuous, there is a such that .

On the other hand, from the condition (iii), for each , there is a such that . Hence, for each , . By (i) and Proposition 3.6, the mapping is above--quasi-convex on . By (ii) and Proposition 3.7, the mapping is below--quasi-concave on . Hence, for each , the set is convex. From the lower semi-continuities on and upper semi-continuity on of , the set
is open in . By Fan-Browder fixed-point Theorem 2.8, there exists such that
that is,
which is a contradiction. This completes the proof.

*Remark 4.4. *[5, Propositions 2.7 and 2.1] can be deduced from Theorem 4.3. Indeed, in [5, Proposition 2.1], the above-naturally -quasi-convexity is used. By Proposition 3.12, the condition (i) of Theorem 4.3 holds. Hence the conclusion of Proposition 2.1 in [5] holds. We also note that, in Theorem 4.3, the mapping need not be continuous on . Hence Theorem 4.3 is a slight generalization of [7, Theorem 3.1].

Theorem 4.5. *Let and be nonempty compact (not necessarily convex) subsets of real Hausdorff topological vector spaces and , respectively. Let the mapping be upper semi-continuous with nonempty compact values and lower semi-continuous on such that *(i)*for each , is below--concave-like on ; *(ii)*for each , is above--convex-like on ;*(iii)*for every ,
**Then for any
**
there is a
**
such that
**
that is,
*

*Proof. *Let for all . From Lemma 2.4 and Proposition 3.5, the set-valued mapping is upper semi-continuous with nonempty compact values on . Hence the set is compact, and so is . Then is a closed convex set with nonempty interior. Suppose that . By separation hyperplane theorem [15, Theorem 14.2], there exist , and a nonzero continuous linear functional such that
Therefore,
This implies that and for all .

Let . From Lemma 3.10, for each fixed , there exist and with such that . Choosing and in (4.36), we have
Therefore,

By the conditions (i), (ii) and Proposition 3.8, the set-valued mapping is below--concave-like on for all , and the set-valued mapping is above--convex-like on for all. From Theorem 4.1, we have
Since is compact, there is an such that . For any and all , we have
that is,
Thus, , and hence,
If , by the condition (iii), which contradicts (4.43). Hence, for every ,
that is,
or

The following examples illustrate Theorem 4.5.

*Example 4.6. *Let , and
It is obviously that is below--concave-like on and above--convex-like on . We now verify the condition (iii) of Theorem 4.5. Indeed, for any ,
Then,
Thus, for every ,
and the condition (iii) of Theorem 4.5 holds.

Furthermore, for any ,
Then,
Thus,
Hence, the conclusion of Theorem 4.5 holds.

*Example 4.7. *Let , , , and be defined by
Let for all . Then is upper semi-continuous, but not lower semi-continuous on , and is not continuous but is upper semi-continuous on . Moreover, has nonempty compact values and is lower semi-continuous on . It is easy to see that is below--concave-like on and is above--convex-like on . We verify the condition (iii) of Theorem 4.5. Indeed, for all , .
Then,
Therefore, the condition (iii) of Theorem 4.5 holds.

Since
for all , and , for each , we can choose such that
Furthermore,
Therefore,
Hence, the conclusion of Theorem 4.5 holds.

*Remark 4.8. *Theorem 3.1 in [5] Theorem 3.1 in [6], or Theorem 4.2 in [7] cannot be applied to Examples 4.6 and 4.7 because of the following reasons: (i)the two sets and are not convex in Example 4.6; (ii) is not continuous on in Examples 4.6 and 4.7.

Theorem 4.9. * Let , be two nonempty compact convex subsets of real Hausdorff topological vector spaces and , respectively. Suppose that the set-valued mapping has nonempty compact values, and it is continuous on and lower semi-continuous on such that *(i)*for each , is above-naturally -quasi-convex on ;*(ii)*for each , is above--concave or above-properly -quasi-concave on ;*(iii)*for every ,
**(iv) for any continuous increasing function and for each , there exists such that
**
Then, for any , there is a
**
such that , that is,
*

*Proof. *Let be defined as the same as in the proof of Theorem 4.5. Following the same perspective as in the proof of Theorem 4.5, suppose that . For any and Gerstewitz function . By Proposition 2.7(d), we have
Let . From Lemma 3.10, for the mapping and Remark 3.15, for each *,* there exist and with such that . Choosing in (4.65), we have
Therefore,

By conditions (i), (ii) and Remark 3.15, the set-valued mapping is upper semi-continuous, and either above--concave or above-properly -quasi-concave on , and the set-valued mapping is lower semi-continuous and above--quasi-convex on . From Theorem 4.3, we have
Since the set-valued mapping is lower semi-continuous on , by Lemma 2.4 (b) and Lemma 2.5 (b), the set-valued mapping is upper semi-continuous on . By the compactness of , there exists such that . For all and all , we have . Thus, , and hence,
If , by the condition (iii), which contradicts (4.69). Hence, for every ,
that is,
This completes the proof.

The following example illustrates Theorem 4.9.

*Example 4.10. *Let , and be a set-valued mapping defined as
Let for all . Then is lower semi-continuous, but not upper semi-continuous on , and is continuous on , and has nonempty compact values and is lower semi-continuous on . It is easy to see that is above--concave or above-properly -quasi-concave on and is above-naturally -quasi-convex on.

We verify the condition (iii) of Theorem 4.9. Indeed, for all , and . Hence,