The Hierarchical Minimax Inequalities for Set-Valued Mappings

Abstract

We study the minimax inequalities for set-valued mappings with hierarchical process and propose two versions of minimax inequalities in topological vector spaces settings. As applications, we discuss the existent results of solutions for set equilibrium problems. Some examples are given to illustrate the established results.

1. Introduction and Preliminaries

Let be a nonempty set in a Hausdorff topological vector space, a Hausdorff topological vector space, and a closed convex and pointed cone with apex at the origin with ; that is, is properly closed with and satisfies , for all ; ; and . The scalar hierarchical minimax inequalities are stated as follows: for given mappings , under some suitable conditions, the following inequality holds:

For given mappings , the first version of hierarchical minimax theorems states that under some suitable conditions, the following inequality holds:

The second version of hierarchical minimax theorems states that under some suitable conditions, the following inequality holds:

These versions, and , arise naturally from some minimax theorems in the vector or real-valued settings. We refer to [1–4] and the references therein.

The notations we use in the above relations are as follows.

Definition 1 (see [1, 3]). 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 .

We note that, for a nonempty compact set , both sets and are nonempty. Furthermore, , , , and . Following [3], we denote both and by max (both Min and by min) in since both Max and (both Min and ) are the same in .

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

Definition 2 (see [5, 6]). 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 semicontinuous at if for any net , , implies that there exists net such that ;(c)continuous at if is upper semicontinuous as well as lower semicontinuous at .

We note that if is upper semicontinuous at and is compact, then for any net , , and for any net for each there exists and a subnet such that . We refer to [5, 6] for more details.

Definition 3 (see [3, 7]). Let and . The Gerstewitz function is defined by

Some fundamental properties for the Gerstewitz function are as follows.

Proposition 4 (see [3, 7]). Let and . The Gerstewitz function has the following properties: (a);(b);(c) is a convex, continuous, and increasing function.

We also need the following different kinds of cone-convexities for set-valued mappings.

Definition 5 (see [1]). Let be a nonempty convex subset of a topological vector space. A set-valued mapping is said to be (a)above--convex (resp., above--concave) on if, for all and all , (b)above-naturally -quasiconvex on if, for all and all , where denotes the convex hull of a set ;(c)above--convex-like (resp., above--concave-like) on ( is not necessary convex) if, for all and all , there is an such that

We note that whenever is a scalar function and , the mappings in Definition 5 reduce to the classical ones. The following theorem is a special case of the scalar hierarchical minimax theorem by Lin [8].

Theorem 6. Let be a nonempty compact convex subset of real Hausdorff topological vector space. Let the set-valued mappings such that for all ; and are compact for each and for each and satisfy the following conditions: (i) is lower semicontinuous on for each and is above--concave on for each ;(ii) is above-naturally -quasiconvex for each , and is lower semicontinuous on for each ;(iii) is lower semicontinuous on for each , is above--concave on for each , and is lower semicontinuous for each . Then one has

Lemma 7. Let be such that , , and exist for all . Then

Proof. By using the similar technique of Lemma 3.3 [9], we can show that the conclusion is valid.

2. Scalar Hierarchical Minimax Inequalities

We first state the following scalar hierarchical minimax inequalities.

Theorem 8. Let be a nonempty compact (not necessarily convex) subset of a real Hausdorff topological space. Let the set-valued mappings with nonempty compact values such that (i) and are upper semicontinuous on ;(ii) is convex-like for each , and is concave-like on for each ;(iii)for all , . Then the relation holds.

Proof. From (i), we know that both sides of exist. For any , Define by for all . By (i), the set is closed for all . We claim that the whole intersection is empty. Indeed, if not, there exists such that, for all , . In particular, we choose ; then which, with the aid of condition (iii), contradicts the choice of . Hence, by the compactness of , there exist such that Let for all . Then, by (iii), we have This implies that, for each , there is such that Define two sets as follows: By the concave-like property of , we can see that these two sets are disjoint. For each , by the separation theorem, there exists nonzero vector such that for all . Then, and for all . Let for all . Then we have For each , by taking and noting , we have for all . Since the mapping is convex-like for each , there is such that Since for all , we have for all . By Lemma 7, we know that Therefore, the relation holds.

Theorem 9. Let be a nonempty compact convex subset of a real Hausdorff topological vector space. Let the set-valued mappings with nonempty compact values such that (i) and are upper semicontinuous on ;(ii) is quasiconcave for each ; that is, for each , the set is convex in ;(iii)for all , . Then the relation holds.

Proof. By (i), we know that both sides of exist. Choose any satisfies Define by for all . By (ii), the set is convex for all . By (iii), we have Hence, for all . By the upper semicontinuity of , we know that the mapping is upper semicontinuous for each . Thus, for each , is closed; hence it is compact. In order to claim that the mapping is upper semicontinuous on , we only need to show that the mapping has a closed graph. Since, for any net we have the net . converges to some point . Then, for each , . Since the mapping is upper semicontinuous, we have Thus, . Suppose that for all . Then, by Kakutani fixed point theorem, the mapping has a fixed point which is a contradiction to (24). Hence, there is an such that . From this, we know that This implies that the relation holds.

The following examples illustrate Theorems 8 and 9.

Example 10. Let and for all . Define by , , , and . Obviously, all conditions of Theorem 8 hold. Hence the relation holds. Indeed, by simple calculation, we can see that

Example 11. Let . The mappings , and are the same as in Example 10. Then, all conditions of Theorem 9 hold. We can see that both values of and are the same as those in Example 10. Hence the relation holds.

3. Hierarchical Minimax Inequalities

In this section, we will present two versions of hierarchical minimax inequalities. The following theorem is the first result satisfies the relation .

Theorem 12. Let be a nonempty compact convex subset of a real Hausdorff topological vector space. Let the set-valued mappings with nonempty compact values such that for all satisfy the following conditions: (i) is upper semicontinuous, is above--concave on for each , and is lower semicontinuous on for each ;(ii) is above-naturally -quasiconvex for each , and is lower semicontinuous on for each ;(iii) is lower semicontinuous and above--concave on for each , and is lower semicontinuous on for each ;(iv)for each , Then the relation is valid.

Proof. Let for all . From Lemma 2.4 and Proposition  3.5 in [1], the mapping is upper semicontinuous with nonempty compact values on . Hence is compact and so is . Then is a closed convex set with nonempty interior. Suppose that . By separation theorem, there is a , , and a nonzero continuous linear functional such that for all and . From this we can see that , where , and for all . By Proposition 3.14 of [1], for any , there is a and with such that Let us choose and in (29); we have for all . Therefore, From conditions (i)–(iii), by applying Proposition 3.9 and Proposition  3.13 in [1], all conditions of Theorem 6 hold. Hence, we have Since is compact, there is such that Thus, and, hence, Therefore, By taking into account condition (iv), we know that Hence, the relation is valid.

The following example illustrates that Theorem 12 is valid.

Example 13. Let , , and define and define for all .

We can easily see that for all and conditions (i)–(iii) of Theorem 12 are valid. Now we claim that condition (iv) holds. Indeed,

Hence, .

On the other hand, . Hence, Thus, condition (iv) of Theorem 12 holds. By Theorem 12, the relation is valid. Indeed, Hence, Thus, and hence the conclusion of Theorem 12 is valid.

Theorem 14. Let be a nonempty compact convex subset of real Hausdorff topological vector space. Let the set-valued mappings such that for all and satisfy the following conditions: (i) is continuous with nonempty compact values, and is above--concave on for each and any Gerstewitz function ;(ii) is above-naturally -quasiconvex for each , and is lower semicontinuous on for each ;(iii) is upper semicontinuous with nonempty compact values, is above--concave on for each , and is lower semicontinuous on for each and any Gerstewitz function ;(iv)for each , Then the relation is valid.

Proof. Let be defined the same as that in Theorem 12 for all . From the process in the proof of Theorem 12, we know that the set is nonempty compact. Suppose that . For any , there is a Gerstewitz function such that for all . Then, for each , there is and with such that Choosing in (47), we have for all . Therefore,
By conditions (i)–(iii), we know that all conditions of Theorem 6 hold for the mappings , , and , and, hence, we have Since is compact, there is a such that Thus, and, hence, If , then, by (iv), we have which contradicts (54). From this, we can deduce that the relation is valid.

4. Strong and Weak Solutions for SEP

In our previous work [10], we establish existence of solutions for set equilibrium problems (SEP, for short). Let be a Hausdorff topological vector space, and let be a nonempty compact convex subset of a Hausdorff topological vector space. For a given mapping and a trimapping , a weak solution for is a point such that for all and for some . A strong solution for is a point with some such that for all . A strong solution is obviously a weak solution for (SEP) for the same mapping.

We recall that a set-valued mapping is called a KKM mapping if for each finite subset .

Fan Lemma (see [11]). Let be a KKM mapping with nonempty closed values. If there exists an such that is a compact set of , then .

We first state that the existent result of weak solution for (SEP) is as follows.

Theorem 15. Let be a finite dimensional space and the set-valued mappings and are two upper semicontinuous mappings with nonempty compact values such that, (i)for each , there is such that ;(ii)for each , the sets and are convex. Then (SEP) has a weak solution.

Proof. Define by for all . By (i), for all . Hence the set is nonempty for all . Next, we claim that the set is closed for all . Let a net converge to some point . Then there are and such that . By the upper semicontinuities of and , the sets and are compact. Hence, there is a convergent subnet of that converges to some point . Furthermore, the net has a convergent subnet which converges to some point . Again, by the upper semicontinuities of and , we have and . Since the set is closed, . Hence, , and, thus, is closed for all . We next claim that the mapping is a KKM mapping. Indeed, if not, there exist and such that Then there is where and for all .
Since , for all , choose any ; we have By (ii), This implies that which contradicts (i). Thus, the mapping is a KKM mapping. By the Fan lemma, the whole intersection is nonempty. Any point in the whole intersection is a weak solution for .

For the existence of strong solution for (SEP), we propose the following results.

Theorem 16. Under the framework of Theorem 15, in addition, the mappings with nonempty compact values such that (i)the mapping is upper semicontinuous mappings for each ;(ii)both sets and are compact for ;(iii)the mapping is concave-like for each , and the mapping is convex-like for each ;(iv)for each ;(v)for each , there is an with such that Then (SEP) has a strong solution.

Proof. According to Theorem 15, we know that (SEP) has a weak solution. That is, there is an such that for all and for some . For any , from Proposition 4, the Gerstewitz function satisfies Hence, there is such that, for each , Thus, we have By conditions (i)–(v), all conditions of Theorem 6 hold; hence we have Since is compact, there is such that This implies that or for all . Therefore, has a strong solution.

Finally, we give the following example to illustrate that Theorems 15 and 16 are valid.

Example 17. Let , , , and be defined by for all . Then we define which are defined by for all .

Then, the set-valued mappings and are two upper semicontinuous mappings with nonempty compact. We can easily see that for all and if we choose any . So, condition (i) of Theorem 15 holds. It is obvious that condition (ii) of Theorem 15 holds since the mapping is linear. Hence has a weak solution by Theorem 15. Indeed, is a weak solution for where we can choose .

Next, we claim that has a strong solution. We can easily deduce that conditions (i), (iii), and (iv) hold. The condition (ii) is valid since is compact for and so is for . Finally, condition (v) of Theorem 16 is valid, since, for each , we can choose an with and such that Indeed, with is a strong solution for .

Conflict of Interests

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

Acknowledgments

In this research, the first author was supported by Grant no. NSC102-2115-M-039-001- of the National Science Council of Taiwan (Taiwan). The second author was supported partly by National Science Council of the Republic of China.