International Journal of Mathematics and Mathematical Sciences

Volume 2008 (2008), Article ID 423596, 10 pages

http://dx.doi.org/10.1155/2008/423596

## Remarks on Weakly KKM Maps in Abstract Convex Spaces

^{1}Natural Sciences Division, The National Academy of Sciences, Seoul 137-044, South Korea^{2}Department 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 [8–11] 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 [12–14] can also be generalized to KKM spaces. Consequently, most results in [12–14] 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 space**in 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, 18–21] and references therein.

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

*Definition 2.3. *A -*space*consists 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*-*space**can 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 map**satisfying**is 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 [19–21]. 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 [19–21].(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. *Let**be a KKM space,**an open or
closed cover of** Then there exists a nonempty subset**of**such that*

Corollary 3.5 (see [13, Lemma 1]). *Let**be a*-convex
space, *an open or
closed cover of** Then there exists a nonempty subset**of**such that*

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

Corollary 3.6 (see [14, Theorem 3.2]). *Let**be an**FC*-space, *an open or
closed cover of** Then there exists a nonempty subset**of**such 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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