On Some Generalized Ky Fan Minimax Inequalities

Abstract

Some generalized Ky Fan minimax inequalities for vector-valued mappings are established by applying the classical Browder fixed point theorem and the Kakutani-Fan-Glicksberg fixed point theorem.

1. Introduction

It is well known that Ky Fan minimax inequality [1] plays a very important role in various fields of mathematics, such as variational inequality, game theory, mathematical economics, fixed point theory, control theory. Many authors have got some interesting achievements in generalization of the inequality in various ways. For example, Ferro [2] obtained a minimax inequality by a separation theorem of convex sets. Tanaka [3] introduced some quasiconvex vector-valued mappings to discuss minimax inequality. Li and Wang [4] obtained a minimax inequality by using some scalarization functions. Tan [5] obtained a minimax inequality by the generalized G-KKM mapping. Verma [6] obtained a minimax inequality by an R-KKM mapping. Li and Chen [7] obtained a set-valued minimax inequality by a nonlinear separation function . Ding [8, 9] obtained a minimax inequality by a generalized R-KKM mapping. Some other results can be found in [10–16].

In this paper, we will establish some generalized Ky Fan minimax inequalities forvector-valued mappings by the classical Browder fixed point theorem and the Kakutani-Fan-Glicksberg fixed point theorem.

2. Preliminaries

Now, we recall some definitions and preliminaries needed. Let and be two nonempty sets, and let be a nonempty set-valued mapping, if and only if , . Throughout this paper, assume that every space is Hausdorff.

Definition 2.1 (see [10]).

For topological spaces and , a mapping is said to be

(i)upper semicontinuous (usc), if for each open set , the set is open subset of ;

(ii)lower semicontinuous (lsc), if for each closed set , the set is closed subset of ;

(iii)continuous, if it is both (usc) and (lsc);

(iv)compact-valued, if is compact in for any .

Definition 2.2 (see [11]).

Let be a topological vector space and be a pointed convex cone with a nonempty interior , and let be a nonempty subset of . A point is said to be

(i)a minimal point of if ;

(ii)a weakly minimal point of if ;

(iii)a maximal point of if ;

(iv)a weakly maximal point of if .

By , , , , we denote, respectively, the set of all minimal points, the set of all weakly minimal points, the set of all maximal points, the set of all weakly maximal points of .

Lemma 2.3 (see [11]).

Let be a nonempty compact subset of a topological vector space with a closed pointed convex cone . Then

(i);

(ii);

(iii);

(iv).

Lemma 2.4 (see [11]).

Let and be two topological vector spaces, , and let be a set-valued mapping. If is compact, and is upper semicontinuous and compact-valued, then is compact set.

Lemma 2.5 (see [2, Theorem  3.1]).

Let be a topological vector space, let be a topological vector space with a closed pointed convex cone , , let and be two nonempty compact subsets of , and let be a continuous mapping. Then both defined by and defined by are upper semicontinuous and compact-valued.

Definition 2.6.

Let be a topological vector space and let be a closed pointed convex cone in , . Given and , the function and are, respectively, defined by , and .

We quote some of their properties as follows (see [12]):

(i); ;

(ii); ;

(iii); ;

(iv); ;

(v) is a continuous and convex function; is a continuous and concave function;

(vi) and are strictly monotonically increasing (monotonically increasing), that is, if (), where denotes or .

Definition 2.7 (see [3]).

Let be a topological vector space, let be a nonempty convex subsets of , and let be a topological vector space with a pointed convex cone , . A vector-valued mapping is said to be

(i)-quasiconcave if for each , the set is convex;

(ii)properly -quasiconcave if for any and , either or .

The following two propositions are very important in proving Proposition 2.10.

Proposition 2.8 (see [4]).

Let be a topological vector space and let be a closed pointed convex cone in , , :

(i) is -quasiconcave if and only if for all  and for all , is quasiconcave;

(ii)if is properly -quasiconcave.

Then is quasiconcave.

Proposition 2.9.

Let be a topological vector space and let be a nonempty convex subset of , . Then the following two statements are equivalent:

(i)for any , is convex;

(ii)for any , is convex.

Proof.

(i)(ii) For any , . Let , then . By (i), we have is convex, then . Thus, is convex.

(ii)(i) For any , , then for all , . By (ii), we have is convex, that is, . Since is arbitrary, then is convex.

Proposition 2.10.

Let be a topological vector space, let be a topological vector space with a closed pointed convex cone , , and let be a nonempty compact convex subset of , be a vector mapping. Then the following two statements are equivalent:

(i)for any , is convex, that is, is -quasiconcave;

(ii)for any , is convex.

Proof.

(i)(ii) for all and for all , let . By Proposition 2.8, we have is quasiconcave, that is, for any , is convex, then by Proposition 2.9, we have for any , is convex. Thus, is convex. Therefore, we have is convex since by property (i) of .

(ii)(i) By Proposition 2.8, we need only prove for all and for all , is quasiconcave, that is, for any , is convex.

For any , let . By property (i) of , we have

(2.1)

Thus, for any , is convex since is convex by (ii). Therefore, by Proposition 2.9, we have for any , is convex.

3. Generalized Ky Fan Minimax Inequalities

In this section, we will establish some generalized Ky Fan minimax inequalities and a corollary by Propositions 1.1, 1.3 and Lemmas 3.1, 3.2.

Lemma 3.1 (see [13]).

Let be a topological vector space, let be a nonempty compact and convex set, and let , such that

(i)for each , is nonempty and convex;

(ii)for each , is open.

Then has a fixed point.

Lemma 3.2 (see [11], Kakutani-Fan-Glicksberg fixed point theorem).

Let be a locally convex topological vector space and let be a nonempty compact and convex set. If is upper semicontinuous, and for any , is a nonempty, closed and convex subset, then has a fixed point.

Theorem 3.3.

Let be a topological vector space, let be a topological vector space with a closed pointed convex cone , , let be a nonempty compact convex subset of , and let be a continuous mapping, such that

(i)for all , for any , is convex.

Then

(3.1)

Proof.

Let , then by the definition of the weakly maximal point, we have

(*)

For each , let

(3.2)

Now, we prove that there exists , such that .

Supposed for each , , then by condition (i), we have for each , is nonempty and convex. In addition, we have for each , is open since is continuous.

Thus, by Lemma 3.1, there exists , such that , that is, , which contradicts (*).

Therefore, there exists , such that , that is, for any ,

(3.3)

Since , then (because of , and Lemma 2.3).

Remark 3.4.

By Proposition 2.10, in the above Theorem 3.3, the condition (i) can be replaced by "for each , is -quasiconcave in ".

Theorem 3.5.

Let be a topological vector space, let be a topological vector space with a closed convex pointed cone , , let be a nonempty compact convex subset of , and let be a continuous mapping, such that

(i)for each , is properly -quasiconcave in .

Then

(3.4)

Proof.

Since is compact, and is continuous, then by Lemma 2.3, we have for any , and .

For any , there exists , such that . Let , by the definition of the weakly minimal point, we have . Thus, for each , let

(3.5)

Now, we prove that there exists , such that .

For all , let , the function is defined by

(3.6)

Let , then is continuous since both and are continuous. By property (iv) of , we have

(**)

For any , let , then it satisfies the all conditions of Lemma 3.1.

In fact, firstly, by , we have , and for each , is open since is continuous. Secondly, by condition (i) and Proposition 2.8, we have is quasiconcave in , that is, for any , is convex. Thus, by Proposition 2.9, is convex.

By Lemma 3.1, there exists , such that , that is,

(3.7)

Since is compact, then has a subnet converging to . Let in the above expression, together with (**), yields

(3.8)

Thus,

(3.9)

Therefore, for all , we have

(3.10)

Theorem 3.6.

Let be a locally convex topological vector space, let be a topological vector space with a closed convex pointed cone , , let be a nonempty compact and convex subset of , let be a continuous mapping, and let such that

(i)for each , is nonempty convex.

Then

(3.11)

Proof.

For each , we define by

(3.12)

Now, we prove that has a fixed point.

(1)By the condition (i), we have for each , is closed and convex since is continuous and is closed.

(2) is upper semicontinuous mapping.

For each , is compact since is compact and is closed. We only need to prove has a closed graph.

In fact, Let , and a net in converging to .

Since is continuous and is closed, then

(3.13)

Thus,

(3.14)

Therefore, by Lemma 3.2 (KFG fixed point theorem), has a fixed point such that

(3.15)

Then

(3.16)

Remark 3.7.

If for each , is -quasiconcave in and , then the condition (i) holds. Thus, we can obtain the following corollary.

Corollary 3.8.

Let be a locally convex topological vector space, let be a topological vector space with a closed convex pointed cone , , let be a nonempty compact and convex subset of , and let be a continuous mapping such that

(i) is -quasiconcave in for each ;

(ii) for each .

Then

(3.17)

Proof.

Let , and for each , let . By condition (ii), is nonempty. And by condition (i), is convex. Thus, by Theorem 3.6, the conclusion holds.

Remark 3.9.

By Definition 2.7, the condition (i) can be replaced by "(i) is properly -quasiconcave in for each ."

Example 3.10.

Let , , , . Given a fixed , for each , we define by

(3.18)

In Figure 1, the red line denotes the graph of for each .

Figure 1

The function's graph.

Now we prove satisfies the conditions of Corollary 3.8:

(i) is a continuous.

Let is closed, let , and . Then by the definition of , we have

(3.19)

Thus there exists a subnet yet denoted by , and , such that since is closed. Hence, , and . Therefore, is closed.

(ii)From Figure 1, we can check that is properly -quasiconcave in for each .

(iii)From Figure 1, we can check that for each . Thus, for each .

Finally, from Figure 1, we can check that , that is, Corollary 3.8 holds.