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International Journal of Mathematics and Mathematical Sciences
Volume 2008 (2008), Article ID 423596, 10 pages
http://dx.doi.org/10.1155/2008/423596
Research Article

Remarks on Weakly KKM Maps in Abstract Convex Spaces

1Natural Sciences Division, The National Academy of Sciences, Seoul 137-044, South Korea
2Department of Mathematics, Seoul National University, Seoul 151-747, South Korea

Received 26 September 2007; Accepted 7 January 2008

Academic Editor: Petru Jebelean

Copyright © 2008 Sehie Park. 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

A KKM space is an abstract convex space satisfying the KKM principle. We obtain variants of the KKM principle for KKM spaces related to weakly KKM maps and indicate some applications of them. These results properly generalize the corresponding ones in -convex spaces and -spaces . Consequently, results by Balaj 2004, Liu 1991, and Tang et al. 2007 can be properly generalized and unified.

1. Introduction

Since the appearance of generalized convex (simply, -convex) spaces in 1993 [1], the concept has been challenged by several authors who aimed to obtain more general concepts. In fact, a number of modifications or imitations of the concept have followed, for example, -spaces due to Ben-El-Mechaiekh et al. [2], spaces having property () due to Huang [3], FC-spaces due to Ding [4, 5], and others. It is known that all of such examples belong to the class of -spaces and are particular forms of -convex spaces; see [6]. Some authors also tried to generalize the Knaster-Kuratowski-Mazurkiewicz theorem (simply, the KKM principle) [7] for their own setting. They introduced various types of generalized KKM maps and obtained modifications of known results. Recently, we proposed new concepts of abstract convex spaces and KKM spaces [811] which are proper generalizations of -convex spaces.

In 1991, Liu [12] obtained a form of the KKM principle and applied it to SupInfSup inequalities of von Neumann type and of Ky Fan type. Motivated by this work, Balaj [13] introduced the concept of weakly -KKM mappings for -convex spaces and obtained related results on intersections and the Fan type or the Sion type minimax inequalities. Moreover, based on the misconception that FC-spaces generalize -convex spaces, Tang et al. [14] introduced the so-called W-G-F-KKM mapping and claimed to obtain similar results for the so-called FC-spaces.

In the present paper, our aim is to show that such basic results for weakly KKM maps on -convex spaces can be extended on the more general KKM spaces [6, 10, 11], a particular type of abstract convex spaces satisfying the general KKM principle. These results properly generalize the corresponding ones in -convex spaces and -spaces . We note that some applications of them in [1214] can also be generalized to KKM spaces. Consequently, most results in [1214] can be properly generalized and unified.

2. Abstract convex spaces

In this section, we follow mainly [6, 8, 9, 15].

Let denote the set of all nonempty finite subsets of a set

Definition 2.1. An abstract convex space consists of a set , a nonempty set , and a multimap with nonempty values. One may denote for
For any , the -convex hull of is denoted and defined by(co is reserved for the convex hull in vector spaces).
A subset of is called a -convex subset of relative to if for any , one has , that is, This means that itself is an abstract convex space called a subspace of .
When , the space is denoted by . In such case, a subset of is said to be -convex if, for any , one has . In case let .

Example 2.2. In [6, 16], we gave plenty of examples of abstract convex spaces. Here we give only two of them as follows.
(1) Usually, a convexity spacein the classical sense consists of a nonempty set and a family of subsets of such that itself is an element of and is closed under arbitrary intersection. For details, see [17], where the bibliography lists 283 papers. For any subset , its -convex hull is defined and denoted by . We say that is -convex if . Now we can consider the map given by . Then becomes our abstract convex space .
(2) A generalized convex space or a -convex space consists of a topological space , a nonempty set , and a multimap such that for each with the cardinality , there exists a continuous function such that implies .

Here, is a standard -simplex with vertices , and the face of corresponding to ; that is, if and , then .

We have established a large amount of literature on -convex spaces; see [1, 16, 1821] and references therein.

Recently, we are concerned with another variant of -convex spaces as follows [6].

Definition 2.3. A -spaceconsists of a topological space , a nonempty set , and a family of continuous functions (i.e., singular -simplexes) for with the cardinality .

Example 2.4. The following are typical examples of -convex spaces and -spaces: (1) any nonempty convex subset of a topological vector space (t.v.s.);(2) [22] a convex space due to Komiya;(3) [23] a convex space due to Lassonde; (4)[24, 25] a -space (or an -space) due to Horvath;(5)[2] an -space due to Ben-El-Mechaiekh et al.;(6)[3] a topological space is said to have property () if, for each , there exists a continuous mapping ;(7) [4, 5, 26, 27] is said to be an FC-space if is a topological space and for each , where some elements in may be the same, there exists a continuous mapping ;(8)any -convex space is clearly a -space. The converse also holds.

Proposition 2.5 (see [6]). A-spacecan be made into a-convex space.

Proof. This can be done in two ways. (1)For each , by putting , we obtain a trivial -convex space .(2) Let be the family of maps giving a -convex space such that for each with . Note that, by (1), this family is not empty. Then, for each and each with , we have Let , that is, ThenTherefore, is a -convex space.

Therefore, -convex spaces and -spaces are essentially the same.

For a -convex space , a multimap is called a KKM map if for each .

Proposition 2.6 (see [6]). For a-space any mapsatisfyingis a KKM map on a -convex space .

Proof. Define by for each . Then becomes a -convex space. In fact, for each with we have a continuous function such that implies . Moreover, note that for each and hence is a KKM map on a -convex space .

Remark 2.7. In [14], its authors repeated Ding's false claim in a large number of his own papers as follows. “Recently, Ding [4] introduced FC-space which extended -convex space further and proved the corresponding KKM theorem. From this, many new KKM-type theorems and applications were founded in FC-spaces." For Ding's claim, see [5, 26, 27] and references of [6]. One wonders how a pair could extend the triple .

The concept of KKM maps for -convex spaces is refined as follows.

Definition 2.8. Let be an abstract convex space and a set. For a multimap with nonempty values, if a multimap satisfiesthen is called a KKM map with respect to . A KKM map is a KKM map with respect to the identity map .

A multimap is said to have the KKM property and called a-map if, for any KKM map with respect to , the family has the finite intersection property. We denote

Similarly, when is a topological space, a -map is defined for closed-valued maps , and a -map for open-valued maps . Note that if is discrete, then three classes , , and are identical. Some authors use the notation KKM instead of .

Example 2.9. The above terminology unifies various concepts in other author's usage as follows.
(1) Every abstract convex space in our sense has a map for any nonempty set . In fact, for each , choose or for some .
If , then for any function . If and have any topology, this holds for or for any continuous .
(2) For a -convex space and a topological space , we defined the classes of multimaps [16]. It is known that for a -convex space , we have the identity map ; see [1921]. Moreover, for any topological space , if is a continuous single-valued map or if has a continuous selection, then .
(3) Let be a -convex space, a nonempty set, and , two mappings. We say that is a generalized G-KKM mapping [13] with respect to if for each , . If is a topological space, is said to have the G-KKM property if for any map generalized -KKM with respect to , the family has the finite intersection property.
This simply tells that is a KKM map with respect to and .
(4) Let be an FC-space, a nonempty set, and two mappings. We say that is a generalized KKM mapping [14] with respect to if for each , each , . If is a topological space, is said to have the F-KKM property if for any map generalized KKM with respect to , the family has the finite intersection property.
Note that becomes a -convex space for each with for each . Then is a KKM map with respect to and .
(5) Let be an FC-space, a nonempty subset, and . Then is a generalized F-KKM mapping [14] if for each finite subset of , there exists a finite subset of such that for any subset of ,

A generalized F-KKM map in the above sense can be made into a KKM map on a -convex space where and as above.

3. The KKM spaces

We introduced the following in [6].

Definition 3.1. For an abstract convex topological space , the KKM principle is the statement .
A KKM space is an abstract convex topological space satisfying the KKM principle.

In our recent work [9], we studied elements or foundations of the KKM theory on abstract convex spaces and noticed that many important results therein are related to KKM spaces. Moreover, in [10, 11], a fundamental theory and its applications on KKM spaces are extensively investigated.

Example 3.2. We give examples of KKM spaces. (1)Every -convex space is a KKM space; see [1921].(2)A connected ordered space can be made into an abstract convex topological space for any nonempty by defining for each . Further, it is a KKM space; see [15, Theorem 5(i)].(3)The extended long line can be made into a KKM space ; see [15]. In fact, is constructed from the ordinal space consisting of all ordinal numbers less than or equal to the first uncountable ordinal , together with the order topology. Recall that is a generalized arc obtained from by placing a copy of the interval between each ordinal and its successor and we give the order topology. Now let be the one as in (2).
But is not a -convex space. In fact, since is not path connected, for and , there does not exist a continuous function such that and . Therefore, is not -convex.

Therefore, the concepts of KKM spaces properly generalize those of -convex spaces and -spaces.

From the definition of the KKM map, we have the following form of Fan's matching theorem.

Theorem 3.3. Let be a KKM space and a map satisfying what follows: (1) is open (resp., closed) for each ;(2) for some . Then there exists an such that

Proof. Let be a map given by for . Then has closed (resp., open) values. Suppose, on the contrary to the conclusion, that for any , we have , that is, . Therefore, is a KKM map. Since is a KKM space, there exists a . Hence, for all . This violates condition (2).

Corollary 3.4. Letbe a KKM space,an open or closed cover of Then there exists a nonempty subsetofsuch that

Corollary 3.5 (see [13, Lemma 1]). Letbe a-convex space, an open or closed cover of Then there exists a nonempty subsetofsuch that

Balaj [13] deduced Corollary 3.5 from a previous result of the present author.

Corollary 3.6 (see [14, Theorem 3.2]). Letbe anFC-space, an open or closed cover of Then there exists a nonempty subsetofsuch that

This is a very particular form of Corollary 3.5 with . In fact, becomes a -convex space with for each and each .

Note also that our proof of Theorem 3.3 is much more simple than that of [14, Theorem 3.2].

4. Weakly KKM maps

Definition 4.1. Let be an abstract convex space and a set. For a multimap with nonempty values, if a multimap satisfiesthen is called a weakly KKM map with respect to .

Clearly, each KKM map with respect to is weakly KKM, and a weakly KKM map with respect to the identity map is simply a KKM map.

Example 4.2. (1) When is a nonempty subset of a vector space, is said to be trappable by iff is not weakly KKM with respect to , where for each [12].
(2) Let be a -convex space, a nonempty set, and two mappings [13]. We say that is weakly G-KKM mapping with respect to if for each and any , .
(3) Let be an FC-space, a nonempty set, and two mappings [14]. We say that is weakly generalized F-KKM mapping with respect to (for short, W-G-F-KKM mapping with respect to ) if for each , each and any , .

In 1991, Liu [12] obtained a form of the KKM principle. Motivated by the form, we deduce the following generalization.

Theorem 4.3. Letbe a compact KKM space,a nonempty set, andandmaps such that(1) is weakly KKM map with respect to (2)for each , the set is closed. Then there exists an such that for each .

Proof. Suppose that the conclusion does not hold. Then for each , there exists a such that . Define an open setSince is compact, there is an such that is an open cover of . Then, by Corollary 3.4, there exist a subset of and a pointSince is weakly KKM with respect to and , we have
On the other hand, implies for all and hence, This is a contradiction.

Corollary 4.4 (see [13, Theorem 2]). Letbe a compact-convex space, a nonempty set, andandtwo maps such that(i) is weakly -KKM map with respect to ;(ii)for each , the set is closed. Then there exists an such that for each .

The following is an immediate consequence of Corollary 4.4.

Corollary 4.5 (see [14, Theorem 3.3]). Letbe a compactFC-space, a nonempty set, andtwo maps such that(i) is ---KKM map with respect to ;(ii) for each , the set is closed. Then there exists an such that for each .

Example 4.6. In [13, Remark 1], it is noted that condition (2) in Theorem 4.3 is satisfied if is a topological space, is upper semicontinuous, and has closed values. In this case, [12, Theorem 2.1] is for a compact convex subset of a topological vector space, and [14, Theorem 3.4] for a compact FC-space .

We have the following variant of Theorem 4.3.

Theorem 4.7. Letbe a KKM space,a nonempty set, andandmaps such that(1) is weakly KKM with respect to ;(2)the set is either all closed or all open for all Then for each there exists an such that for all .

Proof. Let . Then it is easily checked that the subspace is also a KKM space. Now the conclusion follows from the same argument in the proof of Theorem 4.3.

Corollary 4.8 (see [13, Theorem 3]). Letbe a-convex space, a nonempty set, andandmaps such that(i) is weakly -KKM map with respect to ;(ii) the set is either all closed or all open for all Then for each there exists an such that for all .

From Corollary 4.8, we have the following.

Corollary 4.9 (see [14, Theorem 3.5]). Letbe anFC-space, anFC-subspace for each a nonempty set, andtwo maps such that(i) is ---KKM map with respect to ;(ii)the set is either all closed or all open for all . Then for each there exists an such that for each .

Remark 4.10. In [13, Remark 2], it is noted that condition (3.2) in Theorem 4.7 is satisfied if is a topological space and either is upper semicontinuous and has closed values or is lower semicontinuous and has open values. This is exploited in [14, Theorem 3.6].

As an example of applications of Theorem 4.3, we give the following.

Theorem 4.11. Letbe a compact KKM space,a topological space. Letbe an upper semicontinuous map,, two functions andSuppose that(1) for each , is upper semicontinuous on ;(2) for any and , .

Proof. Just follow that of [13, Theorem 4].

Corollary 4.12. In Theorem 4.11,can be replaced by a compact -convex space without affecting its conclusion.

Note that Corollary 4.12 contains some known forms of the Fan type minimax inequalities; see [13].

Corollary 4.13 (see [14, Theorem 4.1]). Letbe a compactFC-space and a topological space. Letbe a u.s.c. map,two functions, andSuppose that(i) for each , is u.s.c. on ;(ii)for any and , if for each and one has .

5. Further remarks

Until now, in this paper, we showed that basic results in [12] for topological vector spaces, in [13] for -convex spaces, and in [14] for FC-spaces, are all extended to KKM spaces. Therefore, most of their applications in each paper can be also generalized to KKM spaces. The readers can show this in case they are urgently needed. Finally, note that results in [14] are all particular to corresponding ones for -convex spaces.

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