# Remarks on Weakly KKM Maps in Abstract Convex Spaces

Authors: Sehie Park
Journal: International Journal of Mathematics and Mathematical Sciences
 (2008-02-03)

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## 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 inG-convex spaces and ϕA-spaces (X,D;{ϕA}A∈〈D〉). Consequently, results by Balaj 2004, Liu
1991, and Tang et al. 2007 can be properly generalized and
unified.


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## Body

## 1. Introduction

Since the  appearance of generalized convex (simply,G-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, L-spaces
due to Ben-El-Mechaiekh et al. [2], spaces having property (H) 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 ϕA-spaces
and are particular forms of G-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 G-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 G-KKM
mappings for G-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 G-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 onG-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 G-convex
spaces and ϕA-spaces (X,D;{ϕA}A∈〈D〉). 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〈D〉 denote the set
of all nonempty finite subsets of a set D.Definition 2.1.
Anabstract convex space(E,D;Γ) consists of a
set E, a nonempty set D, and a multimap Γ:〈D〉⊸E with nonempty
values. One may denote ΓA:=Γ(A) for A∈〈D〉.
For anyD′⊂D, the Γ-convex hull of D′ is denoted and
defined by(2.1)coΓD′:=⋃{ΓA∣A∈〈D′〉}(co is reserved for the convex
hull in vector spaces).
A subsetX of E is called a Γ-convex subset of (E,D;Γ) relative to D′ if for any N∈〈D′〉, one has ΓN⊂X, that is,  coΓD′⊂X. This means that (X,D′;Γ|〈D′〉) itself is an
abstract convex space called a subspace of (E,D;Γ).
WhenD⊂E, the space is denoted by (E⊃D;Γ). In such case, a subset X of E is said to be Γ-convex if, for any A∈〈X∩D〉, one has ΓA⊂X. In case E=D,  let (E;Γ):=(E,E;Γ).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, aconvexity space(E,𝒞)in the
classical sense consists of a
nonempty set E and a family 𝒞 of subsets of E such that E itself is an
element of 𝒞 and 𝒞 is closed under
arbitrary intersection. For details, see [17], where the bibliography lists 283
papers. For any subset X⊂E, its 𝒞-convex hull is defined and
denoted by  Co𝒞X:=⋂{Y∈𝒞∣X⊂Y}. We say that X is 𝒞-convex if X=Co𝒞X.
Now we can consider the map Γ:〈E〉⊸E given by ΓA:=Co𝒞A. Then (E,𝒞) becomes our
abstract convex space (E;Γ).
(2) Ageneralized convex space or a G-convex space(E,D;Γ) consists of a
topological space E, a nonempty set D, and a multimap Γ:〈D〉⊸E such that for
each A∈〈D〉 with the
cardinality |A|=n+1, there exists a continuous function ϕA:Δn→Γ(A) such that J∈〈A〉 implies ϕA(ΔJ)⊂Γ(J).Here,Δn is a standard n-simplex
with vertices {ei}i=0n, and ΔJ the face of Δn corresponding
to J∈〈A〉; that is, if A={a0,a1,…,an} and J={ai0,ai1,…,aik}⊂A, then ΔJ=co{ei0,ei1,…,eik}.We have
established a large amount of literature onG-convex
spaces; see [1, 16, 18–21] and references therein.Recently, we
are concerned with another variant ofG-convex
spaces as follows [6].Definition 2.3.
AϕA-space(2.2)(X,D;{ϕA}A∈〈D〉)consists of a topological space X, a nonempty set D, and a family of continuous functions ϕA:Δn→X (i.e.,
singular n-simplexes)
for A∈〈D〉 with the
cardinality |A|=n+1.Example 2.4.
The following are typical
examples ofG-convex
spaces and ϕA-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 C-space
(or an H-space)
due to Horvath;(5)
[2] an L-space
due to Ben-El-Mechaiekh et al.;(6)
[3] a
topological space Y is said to have
property (H) if, for each N={y0,…,yn}∈〈Y〉, there exists a continuous mapping φN:Δn→Y;(7)
[4, 5, 26, 27] (Y,{φN}) is said to be an FC-space
if Y is a
topological space and for each N={y0,…,yn}∈〈Y〉, where some
elements in N may be the same,
there exists a continuous mapping φN:Δn→Y;(8)
anyG-convex
space is clearly a ϕA-space.
The converse also holds.Proposition 2.5  (see [6]).
AϕA-space(X,D;{ϕA}A∈〈D〉)can be made
into aG-convex
space(X,D;Γ).Proof.
This can be done in two ways.(1)
For eachA∈〈D〉, by putting ΓA:=X, we obtain a trivial G-convex
space (X,D;Γ).(2)
Let{Γα}α be the family
of maps Γα:〈D〉⊸X giving a G-convex
space (X,D;Γα) such that ϕA(Δn)⊂ΓAα for each A∈〈D〉 with |A|=n+1. Note that, by (1), this family is not empty. Then,
for each α and each A∈〈D〉 with |A|=n+1, we have(2.3)ϕA(Δn)⊂ΓAα,ϕA(ΔJ)⊂ΓJαforJ⊂A.
Let Γ:=⋂αΓα, that is, ΓA=⋂αΓAα. Then(2.4)ϕA(Δn)⊂ΓA,ϕA(ΔJ)⊂ΓJforJ⊂A.Therefore, (X,D;Γ) is a G-convex
space.Therefore,G-convex
spaces and ϕA-spaces
are essentially the same.For aG-convex
space (X,D;Γ), a multimap G:D⊸X is called a KKM map if ΓA⊂G(A) for each A∈〈D〉.Proposition 2.6  (see [6]).
For aϕA-space(X,D;{ϕA}A∈〈D〉),any mapT:D⊸Xsatisfying(2.5)ϕA(ΔJ)⊂T(J)foreachA∈〈D〉,J∈〈A〉,is a KKM map on a G-convex
space (X,D;Γ).Proof.
DefineΓ:〈D〉⊸X by ΓA:=T(A) for each A∈〈D〉. Then (X,D;Γ) becomes a G-convex
space. In fact, for each A with |A|=n+1, we have a
continuous function ϕA:Δn→T(A)=:Γ(A) such that J∈〈A〉 implies ϕA(ΔJ)⊂T(J)=:Γ(J). Moreover, note that ΓA⊂T(A) for each A∈〈D〉 and hence T:D⊸X is a KKM map on
a G-convex
space (X,D;Γ).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 G-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 (X,{φA}) could extend
the triple (X,D;Γ).The concept of
KKM maps forG-convex
spaces is refined as follows.Definition 2.8.
Let(E,D;Γ) be an abstract
convex space and Z a set. For a
multimap F:E⊸Z with nonempty
values, if a multimap G:D⊸Z satisfies(2.6)F(ΓA)⊂G(A):=⋃y∈AG(y)∀A∈〈D〉,then G is called a KKM map with respect to F. A KKM mapG:D⊸E is a KKM map
with respect to the identity map 1E.A multimapF:E⊸Z is said to have
the KKM property and called a𝔎-map if, for any KKM map G:D⊸Z with respect to F, the family {G(y)}y∈D has the finite
intersection property. We denote(2.7)𝔎(E,Z):={F:E⊸Z∣Fisa𝔎−map}.Similarly, whenZ is a
topological space, a ℜℭ-map is defined for closed-valued maps G, and a ℜ𝔒-map for open-valued maps G. Note that if Z is discrete,
then three classes 𝔎, ℜℭ, and ℜ𝔒 are identical.
Some authors use the notation KKM(E,Z) instead of ℜℭ(E,Z).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 mapF∈𝔎(E,Z) for any
nonempty set Z. In fact, for each x∈E, choose F(x):=Z or F(x):={z0} for some z0∈Z.
If1E∈𝔎(E,E), then f∈𝔎(E,Z) for any
function f:E→Z. If E and Z have any
topology, this holds for ℜℭ or ℜ𝔒 for any
continuous f.
(2)  For aG-convex
space (X,D;Γ) and a
topological space Z, we defined the classes 𝔎,ℜℭ,ℜ𝔒 of multimaps F:X⊸Z [16]. It is
known that for a G-convex
space (X,D;Γ), we have the identity map 1X∈ℜℭ(X,X)∩ℜ𝔒(X,X); see [19–21]. Moreover, for any topological space Y, if F:X→Y is a continuous
single-valued map or if F:X⊸Y has a
continuous selection, then F∈ℜℭ(X,Y)∩ℜ𝔒(X,Y).
(3)  Let(X,D;Γ) be a G-convex
space, Y a nonempty set,
and T:X→2Y, S:D→2Y two mappings. We say that S is a generalized G-KKM mapping [13] with respect to T if for each A∈〈D〉,  T(Γ(A))⊂S(A). If Y is a
topological space, T:X→2Y is said to have
the G-KKM property if for any map S:D→2Y generalized G-KKM
with respect to T, the family {S(z)¯∣z∈D} has the finite
intersection property.
This simply tells thatS is a KKM map
with respect to T and T∈ℜℭ(X,Y).
(4)  Let(X,φA) be an FC-space, Y a nonempty set,
and T,S:X→2Y two mappings. We say that S is a generalized KKM mapping [14] with respect to T if for each A∈〈X〉, each B∈〈A〉, T(φA(ΔB))⊂S(B). If Y is a
topological space, T:X→2Y is said to have
the F-KKM property if for any map S:X→2Y generalized KKM
with respect to T, the family {S(z)¯∣z∈X} has the finite
intersection property.
Note that(X,φA) becomes a G-convex
space (X,A;Γ) for each A∈〈X〉 with Γ(B):=φA(ΔB) for each B∈〈A〉. Then S|A:A→2Y is a KKM map
with respect to T:X→2Y and T∈ℜℭ(X,Y).
(5)  Let(X,φA) be an FC-space, Y a nonempty
subset, and S:Y→2X. Then S is a generalized F-KKM mapping [14] if for each
finite subset A˜={y0,…,yn} of Y, there exists a finite subset A={x0,…,xn} of X such that for
any subset B={xi0,…,xik} of A,(2.8)φA(co{ei0,…,eik})⊂⋃j=0kS(yij).A generalizedF-KKM map in the above sense can be made
into a KKM map on a G-convex
space (X⊃Y;Γ) where Γ(A˜):=S(A˜) and ϕA˜:=φA as above.


## 3. The KKM
spaces

We introduced the following in [6].Definition 3.1.
For an abstract convex
topological space(E,D;Γ), the KKM
principle is the statement 1E∈ℜℭ(E,E)∩ℜ𝔒(E,E).
AKKM 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)
EveryG-convex
space is a KKM space; see [19–21].(2)
A connected
ordered space(X,≤) can be made
into an abstract convex topological space (X⊃D;Γ) for any
nonempty D⊂X by defining ΓA:=[minA,maxA]={x∈X∣minA≤x≤maxA} for each A∈〈D〉. Further, it is a KKM space; see [15, Theorem 5(i)].(3)
The
extended long lineL* can be made
into a KKM space (L*⊃D;Γ); see [15]. In fact, L* is constructed
from the ordinal space D:=[0,Ω] consisting of
all ordinal numbers less than or equal to the first uncountable ordinal Ω, together with the order topology. Recall that L* is a
generalized arc obtained from [0,Ω] by placing a
copy of the interval (0,1) between each
ordinal α and its
successor α+1 and we give L* the order
topology. Now let Γ:〈D〉⊸L* be the one as
in (2).
ButL* is not a G-convex
space. In fact, since Γ{0,Ω}=L* is not path
connected, for A:={0,Ω}∈〈L*〉 and Δ1:=[0,1], there does not exist a continuous function ϕA:[0,1]→ΓA such that ϕA{0}⊂Γ{0}={0} and ϕA{1}⊂Γ{Ω}={Ω}. Therefore, (L*⊃D;Γ) is not G-convex.Therefore, the
concepts of KKM spaces properly generalize those ofG-convex
spaces and ϕA-spaces.From the
definition of the KKM map, we have the following form of Fan's matching
theorem.Theorem 3.3.
Let(E,D;Γ)
be a KKM space
and  S:D⊸E
a map
satisfying what follows:
(1)
S(z) is open (resp.,
closed) for each z∈D;(2)
X=⋃z∈MS(z) for some M∈〈D〉.
Then there exists an N∈〈D〉 such that(3.1)ΓN∩⋂z∈NS(z)≠∅.Proof.
LetG:D⊸E be a map given
by G(z):=E∖S(z) for z∈D. Then G has closed
(resp., open) values. Suppose, on the contrary to the conclusion, that for any N∈〈D〉, we have ΓN∩⋂z∈NS(z)=∅, that is, ΓN⊂E∖⋂z∈NS(z)=⋃z∈N(E∖S(z))=G(N). Therefore, G is a KKM map.
Since (E,D;Γ) is a KKM space,
there exists a y^∈⋂z∈MG(z)=⋂z∈M(X∖S(z)). Hence, y^∉S(z) for all z∈M. This violates condition (2).Corollary 3.4.
Let(E,D;Γ)be a KKM space,A∈〈D〉,{Mz|z∈A}an open or
closed cover ofE.Then there exists a nonempty subsetBofAsuch thatΓ(B)∩⋂{Mz∣z∈B}≠∅.Corollary 3.5  (see [13, Lemma 1]).
Let(X,D;Γ)be aG-convex
space,  A∈〈D〉,{Mz∣z∈A}an open or
closed cover ofX.Then there exists a nonempty subsetBofAsuch thatΓ(B)∩⋂{Mz∣z∈B}≠∅.Balaj [13]
deduced Corollary 3.5 from a previous result of the present author.Corollary 3.6  (see [14, Theorem 3.2]).
Let(X,φA)be anFC-space,  A∈〈X〉,{Mx∣x∈A}an open or
closed cover ofX.Then there exists a nonempty subsetBofAsuch thatφA(ΔB)∩⋂{Mx∣x∈B}≠∅.This is a very
particular form of Corollary3.5 with Γ(B):=φA(ΔB). In fact, (X,φA) becomes a G-convex
space (X,A;Γ) with Γ(B):=φA(ΔB) for each A∈〈X〉 and each B∈〈A〉.Note also that our proof of Theorem3.3 is much more
simple than that of [14, Theorem 3.2].


## 4. Weakly KKM
maps

Definition 4.1.
Let(E,D;Γ) be an abstract
convex space and Z a set. For a
multimap F:E⊸Z with nonempty
values, if a multimap G:D⊸Z satisfies(4.1)F(x)∩G(A)≠∅∀A∈〈D〉andallx∈Γ(A),then G is called a weakly KKM map with respect to F.Clearly, each
KKM map with respect toF is weakly KKM,
and a weakly KKM map G:D⊸E with respect to
the identity map 1E is simply a KKM
map.Example 4.2.
(1) WhenX:=E is a nonempty
subset of a vector space, F is said to be trappable by Gc iff G is not weakly
KKM with respect to F, where Gc(a):=Z∖G(a) for each a∈D [12].
(2) Let(X,D;Γ) be a G-convex
space, Y a nonempty set,
and T:X→2Y,S:D→2Y two mappings
[13]. We say that S is weakly G-KKM mapping with respect to T if for each A∈〈D〉 and any x∈Γ(A), T(x)∩S(A)≠∅.
(3) Let(X,φA) be an FC-space, Y a nonempty set,
and T,S:X→2Y two mappings
[14]. We say that S is weakly generalized F-KKM mapping with
respect to T (for short, W-G-F-KKM mapping with respect to T) if for
each A∈〈X〉, each B⊂A and any x∈φA(ΔB), T(x)∩S(B)≠∅.In 1991, Liu
[12] obtained a form of the KKM principle. Motivated by the form, we deduce the
following generalization.Theorem 4.3.
Let(X,D;Γ)be a compact
KKM space,Ya nonempty set,
andF:X⊸YandG:D⊸Ymaps such that(1)
G is weakly KKM
map with respect to F(2)
for eachz∈D, the set {x∈X∣F(x)∩G(z)≠∅} is closed.
Then there
exists an x0∈X such that F(x0)∩G(z)≠∅ for each z∈D.Proof.
Suppose that the conclusion does not hold. Then for eachx∈X, there exists a z∈D such that F(x)∩G(z)=∅. Define an open set(4.2)Mz:={x∈X∣F(x)∩G(z)=∅}forz∈D.Since X is compact,
there is an A∈〈D〉 such that {Mz∣z∈A} is an open
cover of X. Then, by Corollary 3.4, there exist a subset B of A and a point(4.3)x0∈Γ(B)∩⋂{Mz∣z∈B}≠∅.Since G is weakly KKM
with respect to F and x0∈Γ(B), we have F(x0)∩G(B)≠∅.
On the other hand,x0∈⋂{Mz∣z∈B} implies F(x0)∩G(z)=∅ for all z∈B and hence, F(x0)∩G(B)=∅. This is a
contradiction.Corollary 4.4  (see [13, Theorem 2]).
Let(X,D;Γ)be a compactG-convex
space,  Ya nonempty set,
andT:X⊸YandS:D⊸Ytwo maps such
that(i)
S is weakly G-KKM map
with respect to T;(ii)
for eachz∈D, the set {x∈X∣T(x)∩S(z)≠∅} is closed.
Then there
exists an x0∈X such that T(x0)∩S(z)≠∅ for each z∈D.The following
is an immediate consequence of Corollary4.4.Corollary 4.5  (see [14, Theorem 3.3]).
Let(X,φA)be a compactFC-space,  Ya nonempty set,
andT,S:X⊸Ytwo maps such
that(i)
S is W-G-F-KKM map
with respect to T;(ii)
for eachz∈D, the set {x∈X∣T(x)∩S(z)≠∅} is closed.
Then there
exists an x0∈X such that T(x0)∩S(z)≠∅ for each z∈D.Example 4.6.
In [13, Remark 1], it is noted
that condition (2) in Theorem 4.3 is satisfied if Y is a
topological space, F is upper
semicontinuous, and G has closed
values. In this case, [12, Theorem 2.1] is for a compact convex subset X=D of a
topological vector space, and [14, Theorem 3.4] for a compact FC-space (X,φA).We have the
following variant of Theorem4.3.Theorem 4.7.
Let(X,D;Γ)be a KKM space,Ya  nonempty set,
andF:X⊸YandG:D⊸Ymaps such that(1)
G is weakly KKM
with respect to F;(2)
the set{x∈X∣F(x)∩G(z)≠∅} is either all
closed or all open for all z∈D.
Then for each A∈〈D〉 there exists an x0∈Γ(A) such that F(x0)∩G(z)≠∅ for all z∈A.Proof.
LetA∈〈D〉. Then it is easily checked that the subspace (X,A;Γ|〈A〉) 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]).
Let(X,D;Γ)be aG-convex
space,  Ya   nonempty set,
andT:X⊸YandS:D⊸Ymaps such that(i)
S is weakly G-KKM map
with respect to T;(ii)
the set{x∈X∣T(x)∩S(z)≠∅} is either all
closed or all open for all z∈D.
Then for each A∈〈D〉 there exists an x0∈Γ(A) such that T(x0)∩S(z)≠∅ for all z∈A.From Corollary4.8, we have the following.Corollary 4.9  (see [14, Theorem 3.5]).
Let(X,φA)be anFC-space,
φA(Δn)anFC-subspace
for each A∈〈X〉,Ya nonempty set,
andT,S:X⊸Ytwo maps such
that(i)
S is W-G-F-KKM map
with respect to T;(ii)
the set{x∈X∣T(x)∩S(z)≠∅} is either all
closed or all open for all z∈D.
Then for each A∈〈X〉 there exists an x0∈φA(Δn) such that T(x0)∩S(z)≠∅ for each z∈φA(Δn).Remark 4.10.
In [13, Remark 2], it is noted
that condition (3.2) in Theorem 4.7 is satisfied if Y is a
topological space and either F is upper
semicontinuous and G has closed
values or F is lower
semicontinuous and G has open
values. This is exploited in [14, Theorem 3.6].As an example
of applications of Theorem4.3, we give the following.Theorem 4.11.
Let(X,D;Γ)be a compact
KKM space,Ya topological
space. LetT:X⊸Ybe an upper semicontinuous map,ψ:D×Y→ℝ,
φ:X×Y→ℝtwo functions
andβ=infx∈Xsupy∈T(x)φ(x,y).Suppose that(1)
for eachz∈D,             ψ(z,·) is upper semicontinuous on Y;(2)
for anyλ<β and y∈T(x), coΓ{z∈D∣ψ(z,y)<λ}⊂{x∈X∣φ(x,y)<λ}.
(a)
Then the
following holds:(4.4)inf∥∥∥x∈Xsupy∈T(x)φ(x,y)≤supx∈Xinf∥∥∥z∈Dsupy∈T(x)ψ(z,y).(b)
Further, ifT is compact
valued, then there exists an x0∈X such that(4.5)inf∥∥∥x∈Xsupy∈T(x)φ(x,y)≤inf∥∥∥z∈Dsupy∈T(x0)ψ(z,y).Proof.
Just follow that of [13, Theorem 4].Corollary 4.12.
In
Theorem4.11,(X,D;Γ)can be replaced
by a compactG-convex
space without affecting its conclusion.Note that
Corollary4.12 contains some known forms of the Fan type minimax
inequalities; see [13].Corollary 4.13  (see [14, Theorem 4.1]).
Let(X,φA)be a compactFC-space
and  Ya topological
space. LetT:X⊸Ybe a u.s.c.
map,f,g:X×Y→ℝtwo functions,
andβ=infx∈Xsupy∈T(x)f(x,y).Suppose that(i)
for eachz∈X, g(z,·) is u.s.c. on Y;(ii)
for anyλ<β and y∈T(x), if for each A∈〈X〉 and B∈〈A∩{x∈X∣g(x,y)<λ}〉 one has φA(ΔB)⊂{x∈X∣f(x,y)<λ}.
(a)
Then the
following holds:(4.6)inf∥∥∥x∈Xsupy∈T(x)f(x,y)≤supx∈Xinf∥∥∥z∈Xsupy∈T(x)g(z,y).(b)
Moreover, ifT is
compact-valued, then there exists an x0∈X such that(4.7)inf∥∥∥x∈Xsupy∈T(x)f(x,y)≤inf∥∥∥z∈Xsupy∈T(x0)g(z,y).


## 5. Further
remarks

Until now, in this paper, we showed that basic results
in [12] for topological vector spaces, in [13] for G-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 G-convex
spaces.



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*Source: 423596-2008-03-11.xml*