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Abstract and Applied Analysis
Volume 2012 (2012), Article ID 310818, 26 pages
http://dx.doi.org/10.1155/2012/310818
Research Article

Minimax Theorems for Set-Valued Mappings under Cone-Convexities

1Department of Occupational Safety and Health, College of Public Health, China Medical University, Taichung 404, Taiwan
2Department of Mathematics, Aligarh Muslim University, Aligarh 202 002, India
3Department of Mathematics and Statistics, King Fahd University of Petroleum and Minerals, Dhahran 31261, Saudi Arabia
4Department of Mathematics, National Tsing Hua University, Hsinchu 300, Taiwan

Received 7 September 2012; Accepted 27 October 2012

Academic Editor: Ondrej Dosly

Copyright © 2012 Yen-Cherng Lin et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

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, [311] 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, [48] 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

(e)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 [57]. 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, Therefore, the condition (iii) of Theorem 4.9 holds.
Since for any , we can choose such that For any continuous increasing function , the condition (iv) of Theorem 4.9 holds.
Furthermore, since for each , we have Thus, Therefore, the conclusion of Theorem 4.9 holds.

Remark 4.11. Theorem 3.1 in [5], Theorem 3.1 in [6], or Theorem 4.2 in [7] cannot be applied to Example 4.10 as is not continuous on .
If we choose and in Theorems 4.5 and 4.9, we always have