Fixed Point Theorems for Compact Multimaps on Almost ¡Convex Sets in
Generalized Convex Spaces
TianYuan Kuo
Fooyin University
151 ChinHsueh Rd.,TaLiao Hsiang,Kaohsiung Hsien 831
Taiwan
Email:sc038@mail.fy.edu.tw
JyhChung Jeng
NanJeon Institute of Technology
Yanshoei,Tainan Hsien 737
Taiwan
Email:jhychung@pchome.com.tw
and
YoungYe Huang
Center for General Education
Southern Taiwan University of Technology
1 NanTai St.YungKang City
Tainan Hsien 710,Taiwan
Email:yueh@mail.stut.edu.tw
In this paper,we introduce the concept of strict KKMproperty and investigate
the ¯xed point problem for multimaps having this property on almost ¡convex
subsets of locally Gconvex uniform spaces.Our new ¯xed point theorem gener
alizes the wellknown FanGlicksberg ¯xed point theorem and partially extends
the Himmelberg ¯xed point theorem.Its application to minimax theorem is also
given.
2000 AMS Classi¯cation:47H10,54H25.
Key words and phrases:¯xed point,strict KKMproperty,Gconvex space,almost
convex.
1
1.Introduction and Preliminaries
For two topological spaces X and Y,an upper semicontinuous multimap
T:X ( Y is said to be a Kakutani multimap if either T is singlevalued (in
which case,Y is simply assumed to be a topological space),or T(x) is a compact
and convex subset of Y for any x in X (in which case,Y is assumed to be a
subset of a topological vector space).The Himmelberg ¯xed point theorem[5],a
generalization of the famous FanGlicksberg ¯xed point theorem[3,4],says that
a compact Kakutani multimap T:X ( X on a nonempty convex subset of a
locally convex topological vector space E has a ¯xed point.
The main purpose of this paper is to deal with the ¯xed point problem on
almost ¡convex subsets in locally Gconvex uniform spaces instead of convex
subsets in locally convex topological vector spaces.In section 2,we start with
generalized KKM mappings on a Gconvex space E to introduce the concept
of the strict KKM property and then show that a Kakutani multimap has the
strict KKM property.The ¯xed point problem for multimaps having the strict
KKMproperty is investigated in section 3,where we established a new ¯xed point
theorem by showing that for an almost ¡convex subset X of a locally Gconvex
uniform space E,any compact and closed multimap T having the strict KKM
property has a ¯xed point.An application to minimax theorem of our ¯xed
ponint theorem is given in section 4.
We now recall some basic de¯nitions and facts.For a nonempty set Y,2
Y
denotes the class of all subsets of Y and hY i denotes the class of all nonempty
¯nite subsets of Y.A multimap T:X!2
Y
is a function from a set X into the
power set 2
Y
of Y.The notation T:X (Y stands for a multimap T:X!2
Y
having nonempty values.
In the sequel,for n ¸ 0,¢
n
denotes the standard nsimplex of R
n+1
,that is,
¢
n
=
(
® = (®
0
;¢ ¢ ¢;®
n
) 2 R
n+1
:®
i
¸ 0 for all i and
n
X
i=0
®
i
= 1
)
;
and fe
0
;¢ ¢ ¢;e
n
g,the standard basis of R
n+1
,is the set of the vertices of ¢
n
.
If X and Y are two subsets of a linear space E,a multimap F:X ( Y
satisfying co(A) µ F(A) for any A 2 hXi is called a KKMmapping,where co(A)
2
denotes the convex hull of A.The most important result for KKMmapping is the
KKM Lemma published in 1929 due to Knaster,Kuratowski and Mazurkiewicz:
KKM Lemma 1.1.(cf.[1,12]) Suppose F
0
;¢ ¢ ¢;F
n
are closed subsets of the
standard nsimplex ¢
n
in R
n+1
.If for any nonempty subset I of f0;¢ ¢ ¢;ng,
cofe
i
:i 2 Ig µ
S
i2I
F
i
,then
T
n
i=0
F
i
6=?:
For a multimap T:X!2
Y
,A µ X and B µ Y,the image of A under T is
the set T(A) =
S
x2A
T(x);and the inverse image of B under T is T
¡
(B) = fx 2
X:T(x)\B 6=?g.
All topological spaces are supposed to be Hausdor®.The closure of a subset
X of a topological space is denoted by
X.Let X and Y be two topological spaces.
A multimap T:X!2
Y
is said to be
(a) upper semicontinuous(u.s.c.) if T
¡
(B) is closed in X for each closed subset
B of Y;
(b) lower semicontinuous(l.s.c.) if T
¡
(B) is open in X for each open subset B of
Y;
(c) compact if T(X) is contained in a compact subset of Y;
(d) closed if its graph Gr(T) = f(x;y):y 2 T(x);x 2 Xg is a closed subset of
X £Y.
Lemma 1.2.(Lassonde[9,Proposition 1]) Let X,Y be topological spaces and
T:X (Y.
(a) If Y is regular and T is u.s.c.with closed values,then T is closed.
(b) If T:X (Y is u.s.c.with compact values and S:X (Y is closed,then
T\S:X ( Y,de¯ned by (T\S)(x) = T(x)\S(x) for each x 2 X,is
u.s.c.;in particular,if S is compact and closed,then S is u.s.c.
2.The Strict KKM Property
De¯nition 2.1.A generalized convex space or a Gconvex space (E;¡) consists
of a topological space E and a map ¡:hEi (E such that
3
(a) for any A;B 2 hEi,A µ B implies ¡(A) µ ¡(B);and
(b) for each A = fa
0
;¢ ¢ ¢;a
n
g 2 hEi with jAj = n +1,there exists a continuous
function'
A
:¢
n
!¡(A) such that if 0 · i
0
< i
1
< ¢ ¢ ¢ < i
k
· n,then
'
A
(cofe
i
0
;¢ ¢ ¢;e
i
k
g) µ ¡(fa
i
0
;¢ ¢ ¢;a
i
k
g).
In this paper,we assume that a Gconvex space (E;¡) always satis¯es the extra
condition:x 2 ¡(fxg) for any x 2 E.
A subset K of a Gconvex space (E;¡) is said to be ¡convex if for any A 2 hKi,
¡(A) µ K.For a nonempty subset Q of E,the ¡convex hull of Q,denoted by
¡co(Q),is de¯ned by
¡co(Q) =\fC:Q ½ C ½ E;C is ¡convexg:
It is easy to see that ¡co(Q) is the smallest ¡convex subset of E containing
Q.
For convenience,we also express ¡(A) by ¡
A
.
A uniformity for a set X is a nonempty family U of subsets of X £X such
that
(a) each member of U contains the diagonal ¢;
(b) if U 2 U,then U
¡1
2 U;
(c) if U 2 U,then V ± V µ U for some V in U;
(d) if U and V are members of U,then U\V 2 U;and
(e) if U 2 U and U µ V µ X £X,then V 2 U:
Every member V in U is called an entourage.An entourage V is said to sym
metric if (x;y) 2 V whenever (y;x) 2 V.
If (X;U) is a uniformspace,then the topology T induced by U is the family of
all subsets W of X such that for each x in W there is U in U such that U[x] µ W,
where U[x] is de¯ned as fy 2 X:(x;y) 2 Ug:If H is a subset of X and U is in
U,then U(H):= [
x2H
U[x].For details of uniform spaces we refer to [8].
De¯nition 2.2.A Gconvex uniform space (E;U;¡) is a Gconvex space so that
4
its topology is induced by a uniformity U.A Gconvex uniform space (E;U;¡) is
said to be a locally Gconvex uniform space if the uniformity U has a base B con
sisting of open symmetric entourages such that for each V 2 B and any x 2 E,
V [x]:= fy 2 X:(x;y) 2 V g is ¡convex.
By the de¯nition of a Gconvex uniform space (E;U;¡),it is easy to check
that A µ ¡
A
for any A 2 hEi.
Motivated by the works of Chang and Zhang[2],we make the following de¯
nition.
De¯nition 2.3.Let X be a nonempty subset of a Gconvex space (E;¡) and F:
X (E.If F satis¯es that for any fx
1
;¢ ¢ ¢;x
n
g 2 hXi,there is fy
1
;¢ ¢ ¢;y
n
g 2
hXi such that
¡
fy
i
:i2Ig
µ
[
i2I
F(x
i
)
for any nonempty subset I of f1;¢ ¢ ¢;ng,then F is called a generalized KKM
mapping.If ¡
A
µ F(A) for each A 2 hEi,then F is called a KKM mapping.
It is easy to see that a KKM mapping is a generalized KKM mapping by
putting y
i
= x
i
(i = 1;¢ ¢ ¢;n) for any fx
1
;¢ ¢ ¢;x
n
g 2 hXi.However,a generalized
KKM mapping may not be a KKM mapping as shown in [7].
Proposition 2.4.Let X be a nonempty subset of a Gconvex space (E;¡).
If F:X (E is a generalized KKM mapping,then f
F(x):x 2 Xg has the ¯nite
intersection property.
Proof.For any fx
0
;x
1
;¢ ¢ ¢;x
n
g 2 hXi,since F is a generalized KKM map
ping,there is fy
0
;y
1
;¢ ¢ ¢;y
n
g 2 hXi such that
¡
fy
i
:i2Ig
µ
[
i2I
F(x
i
)
for any nonempty subset I of f0;1;¢ ¢ ¢;ng:Let B = fy
i
:i = 0;1;¢ ¢ ¢;ng.Since
5
(E;¡) is a Gconvex space,there is a continuous function Á
B
:¢
n
!¡
B
such
that
Á
B
(cofe
i
:i 2 Ig) µ ¡
fy
i
:i2Ig
\Á
B
(¢
n
):
for any I 2 hf0;1;¢ ¢ ¢;ngi.So,
cofe
i
:i 2 Ig µ Á
¡
B
¡
¡
fy
i
:i2Ig
\Á
B
(¢
n
)
¢
µ Á
¡
B
(([
i2I
F(x
i
))\Á
B
(¢
n
))
µ Á
¡
B
³³
[
i2I
F(x
i
)
´
\Á
B
(¢
n
)
´
= [
i2I
Á
¡
B
³
F(x
i
)\Á
B
(¢
n
)
´
:
By KKM lemma,\
n
i=0
Á
¡
B
³
F(x
i
)\Á
B
(¢
n
)
´
6=?,so\
n
i=0
Á
¡
B
(
F(x
i
)) 6=?.Any
z 2\
n
i=0
Á
¡
B
(
F(x
i
)) satis¯es that Á
B
(z) 2\
n
i=0
F(x
i
):¤
Theorem 2.5.Let (E;U;¡) be a locally Gconvex uniform space such that
¡
fxg
= fxg for any x 2 E.If X is a nonempty subset of E and F:X (E is
a multimap with closed values such that F(x
0
) is compact for some x
0
2 X,then
F is a generalized KKM mapping if and only if
T
x2X
F(x) 6=?.
Proof.If F is a generalized KKM mapping,then,in view of F(x
0
) being com
pact and Proposition 2.4,
T
x2X
F(x) 6=?.Conversely,if
T
x2X
F(x) 6=?,then
fF(x):x 2 Xg has the ¯nite intersection property.So for any fx
1
;¢ ¢ ¢;x
n
g 2
hXi,there is y 2
T
n
i=1
F(x
i
).Putting y
i
= y for any i = 1;¢ ¢ ¢;n,we con
clude that ¡
fy
i
:i2Ig
= ¡
fyg
= fyg µ
S
i2I
F(x
i
),for any nonempty subset I of
f1;¢ ¢ ¢;ng.Hence F is a generalized KKM mapping.¤
The concept of strict KKM property on a topological vector space in [7] can
be extended to that on Gconvex spaces..
De¯nition 2.6.Suppose X and Y are two nonempty subsets of a Gconvex space
(E;¡),and T,F:X (Y.We say that F is a generalized KKM mapping with
respect to T if for any A = fx
1
;¢ ¢ ¢;x
n
g 2 hXi there is B = fy
1
;¢ ¢ ¢;y
n
g 2 hXi
6
satisfying
(a) ¡
B
µ X,and
(b) T
¡
¡
fy
i
:i2Ig
¢
µ
S
i2I
F(x
i
) for any nonempty subset I of f1;¢ ¢ ¢;ng.
If a multimap T:X ( Y satis¯es that for any generalized KKM mapping
F:X (Y with respect to T,the family f
F(x):x 2 Xg has the ¯nite intersec
tion property,then T is said to have the strict KKM property.
Checking the proof of Watson [11,Lemma 1],the following lemma holds.
Lemma 2.7.(Watson[11]) Suppose X is a nonempty compact subset of a locally
Gconvex uniform space (E;U;¡),p:X!¢
n
is continuous and T:¢
n
(X
is u.s.c.with closed ¡convex values.Then p ± T:¢
n
(¢
n
has a ¯xed point.
Proposition 2.8.Suppose X is a nonempty subset of a locally Gconvex uni
form space (E;U;¡) and T:X (E is compact and u.s.c.with closed ¡convex
values.Then T has the strict KKM property.
Proof.Assume that T does not have the strict KKM property.Then there is a
closedvalued generalized KKMmapping F:X (E with respect to T such that
T
n
i=0
F(x
i
) =?for some fx
0
;¢ ¢ ¢;x
n
g 2 hXi.Choose B = fy
0
;¢ ¢ ¢;y
n
g 2 hXi
such that ¡
B
µ X and T(¡
fy
i
:i2Ig
) µ
S
i2I
F(x
i
) for any nonempty subset I of
f0;1;¢ ¢ ¢;ng.Since T is compact,the set K:=
T(X) is a compact subset of E.
By the de¯nition of a Gconvex space,there is a continuous function
'
B
:¢
n
!¡
B
µ X
such that'
B
(¢
J
) µ ¡
fy
i
:i2Jg
for any J 2 hf0;1;¢ ¢ ¢;ngi.Noting that K µ E =
S
n
i=0
F(x
i
)
c
,there is a partition of unity f®
i
g
n
i=0
subordinated to fF(x
i
)
c
g
n
i=0
.
De¯ne p:K!¢
n
by p(x) =
P
n
i=0
®
i
(x)e
i
.It is clear that p±T ±'
B
:¢
n
(¢
n
is u.s.c.By Watson Lemma,p±T±'
B
has a ¯xed point ^x,that is ^x 2 (p±T±'
B
)(^x).
Choose ^y 2 (T ±'
B
)(^x) so that ^x = p(^y).Put J = fi 2 f0;1;¢ ¢ ¢;ng:®
i
(^y) > 0g.
It is easy to see that i 2 J if and only if ^y =2 F(x
i
).So ^y =2
S
i2J
F(x
i
),which in
7
view of
^y 2 T('
B
(^x)) µ T ('
B
(¢
J
)) µ T
¡
¡
fy
i
:i2Jg
¢
implies that T
¡
¡
fy
i
:i2Jg
¢
*
S
i2J
F(x
i
),a contradiction to the fact that F is a
generalized KKM mapping with respect to T.Hence,T has the strict KKM
property,completing the proof.¤
3.Fixed Point Theorems
In this section we at ¯rst extend the concept of almost convex sets in topo
logical vector spaces to locally Gconvex uniform spaces and then investigate the
¯xed point problem on such sets.
De¯nition 3.1.A nonempty subset X of a Gconvex uniform space (E;U;¡) is
said to be almost ¡convex if for any fx
1
;¢ ¢ ¢;x
n
g 2 hXi and for any entourage
U 2 U there is fy
1
;¢ ¢ ¢;y
n
g 2 hXi such that y
i
2 U[x
i
] for each i 2 f1;¢ ¢ ¢;ng
and ¡co(fy
1
;¢ ¢ ¢;y
n
g) µ X.
Here,we like to give a concrete example of an almost ¡convex subset of a
Gconvex uniform space.Let E be the closed triangle with vertices (0;0),(1;0)
and (0;1) in the Euclidean space R
2
and let U be the usual Euclidean uniformity
restricted to E.De¯ne ¡:hEi ( E by ¡(fxg) = fxg for each x 2 E and
¡(fx
1
;¢ ¢ ¢;x
n
g) = [
n
i=1
[0;x
i
] for any ¯nite subset fx
1
;¢ ¢ ¢;x
n
g of E,where [0;x]
the closed line segment joining 0 with x.Put A to be the subset of E with the
open line segment ((0;0);(1;0)) deleted.By the concept of cstructure of Horvath
[6],we see that (E;U;¡) is a Gconvex uniform space.Moreover,it is easy to
check that A is almost ¡convex but not ¡convex in E.
Proposition 3.2.Suppose the locally Gconvex uniform space (E;U;¡) sat
is¯es the property that V [K] is ¡convex whenever K is a ¡convex subset of E
and V is any member in some base B consisting of open symmetric entourages of
8
U.Then the closure
X of an almost ¡convex subset X of E is ¡convex.
Proof.Let A = fx
1
;¢ ¢ ¢;x
n
g 2
X
®
.We have to show that ¡
A
µ
X.For
any W 2 B,choose a symmetric entourage V so that V ± V µ W;and for any
i 2 f1;¢ ¢ ¢;ng,choose a y
i
2 X\V [x
i
].Since X is almost ¡convex,there is
fz
1
;¢ ¢ ¢ z
n
g 2 hXi such that for any i = 1;¢ ¢ ¢;n,
z
i
2 V [y
i
] and ¡co(fz
1
;¢ ¢ ¢;z
n
g) µ X:
It follows from(x
i
;y
i
) 2 V and (y
i
;z
i
) 2 V that (x
i
;z
i
) 2 V ±V,so z
i
2 (V ±V )[x
i
].
By symmetry,we have
x
i
2 (V ± V )[z
i
] µ (V ± V )[fz
1
;¢ ¢ ¢;z
n
g]
µ (V ± V )[¡cofz
1
;¢ ¢ ¢;z
n
g]
µ W[¡cofz
1
;¢ ¢ ¢;z
n
g]:
Since W[¡cofz
1
;¢ ¢ ¢;z
n
g] is ¡convex by assumption,we infer that
¡
fx
1
;¢¢¢;x
n
g
µ W[¡cofz
1
;¢ ¢ ¢;z
n
g] µ W[X]:
Therefore,¡
fx
1
;¢¢¢;x
n
g
µ\
W2U
W[X] =
X.¤
We now take up the ¯xed point problem on an almost ¡convex subset of a
locally Gconvex uniform space.To begin with,a key lemma will be established.
Lemma 3.3.Let X be an almost ¡convex subset of a locally Gconvex uniform
space (E;U;¡).If T:X (X is compact and has the strict KKM property,then
for any U 2 U,there is x
U
2 X such that U[x
U
]\T(x
U
) 6=?.
Proof.On the contrary,assume there is a U 2 U such that
U[x]\T(x) =?(1)
for any x 2 X.Let K =
T(X) and choose a symmetric entourage V so that
V µ V ± V µ U.By assumption,K is compact.De¯ne F:X!2
X
by
9
F(x) = K n V [x] for each x 2 X.Since U[x]\T(x) =?,we have that
?6= T(x) µ K n U[x]
µ K n V [x] = F(x);
so F(x) 6=?for each x 2 X,that is F:X ( X.We now show that F
is a generalized KKM mapping with respect to
T
.If not,there exists
A
=
fx
1
;¢ ¢ ¢;x
n
g 2 hXi such that for any B = fy
1
;¢ ¢ ¢;y
n
g 2 hXi with ¡
B
µ X,one
has T(¡
fy
i
:i2Ig
) *
S
i2I
F(x
i
) for some nonempty subset I of f1;¢ ¢ ¢;ng.Since X
is almost ¡convex,there is fz
1
;¢ ¢ ¢;z
n
g 2 hXi such that ¡cofz
1
;¢ ¢ ¢;z
n
g µ X
and
x
i
2 V [z
i
] (2)
for any i = 1;¢ ¢ ¢;n.Choose a nonempty subset I of f1;¢ ¢ ¢;ng such that
T
¡
¡
fz
i
:i2Ig
¢
*
[
i2I
F(x
i
);
and then choose ¹ 2 ¡
fz
i
:i2Ig
and ³ 2 T(¹) so that ³ =2
S
i2I
F(x
i
).Then
³ 2 V [x
i
] for any i 2 I,and so,in view of (2),³ 2 (V ± V )[z
i
] for any i 2 I,that
is z
i
2 (V ± V )[³] µ U[³] for any i 2 I.So,by noting that U[³] is ¡convex,we
infer that ¡
fz
i
:i2Ig
µ U[³],and hence ³ 2 U[p] for any p 2 ¡
fz
i
:i2Ig
.In particular,
³ 2 U[¹].But then ³ 2 U[¹]\T(¹),a contradiction to (1).Therefore,we
conclude that F is a generalized KKM mapping with respect to T.
Finally,since T has the strict KKM property and F is compactvalued,the
family fF(x):x 2 Xg has the ¯nite intersection property,so
T
x2X
F(x) 6=?.
Choosing ´ 2
T
x2X
F(x) and noting that
T
x2X
F(x) = K n
S
x2X
V [x],we see
that ´ =2 V [´],a contradiction.Thus there is x
U
2 X such that U[x
U
]\T(x
U
) 6=
?.This completes the proof.¤
Theorem 3.4.Let X be an almost ¡convex subset of a locally Gconvex uni
form space (E;U;¡).If T:X (X is compact,closed and has the strict KKM
property,then T has a ¯xed point.
10
Proof.Let B be a base of U consisting of open symmetric entourages.By
Lemma 3.3,for any V 2 B there is x
V
2 X such that V [x
V
]\T(x
V
) 6=?.
Choose y
V
2 V [x
V
]\T(x
V
).Since T is compact,we may assume that fy
V
g
V 2B
converges to y
0
.For any W 2 B,choose U 2 B such that U ± U µ W.Since
fy
V
g
V 2B
converges to y
0
,there is V
0
2 B such that V
0
µ U and
y
V
2 U[y
0
];8V 2 B with V µ V
0
;
that is,
(y
V
;y
0
) 2 U;8V 2 B with V µ V
0
:
Thus,for V 2 B with V µ V
0
,it follows from
(x
V
;y
V
) 2 V µ U and (y
V
;y
0
) 2 U
that (x
V
;y
0
) 2 U ± U µ W.Hence x
V
2 W[y
0
].This shows that fx
V
g
V 2B
converges to y
0
.Since T is closed,we conclude that y
0
2 T(y
0
),completing the
proof.¤
The above theorem partially extends Himmelberg [5,Theorem 2] as the following
corollary shows.
Corollary 3.5.Suppose X is an almost ¡convex subset of a locally Gconvex
uniform space (E;U;¡) and T:X ( E is compact and u.s.c.with closed ¡
convex values.If T(X) µ X,then it has a ¯xed point.
Proof.This follows immediately from Proposition 2.8 and Theorem 3.4.¤
As an application of Theorem 3.4,we establish a quasiequlibrium theorem.
Theorem3.6.Let X be an almost ¡convex subset of a locally Gconvex uniform
space (E;U;¡),f:X £X!R be upper semicontinuous and H:X (X be
compact and closed.Assume that
(a) the function M de¯ned on X by
M(x) = max
y2H(x)
f(x;y)
11
for x 2 X is lower semicontinuous,and
(b) the multimap T:X (X de¯ned by
T(x) = fy 2 H(x):f(x;y) = M(x)g
has the strict KKM property.
Then there exists an ^x 2 X such that ^x 2 H(^x) and f(^x;^x) = M(^x).
Proof.Since f is upper semicontinuous and H(x) is compact,T(x) is nonempty.
Also,let K be a compact subset of X such that H(X) µ K.We show that
Gr(T) is closed in X £ K.To see this,let f(x
®
;y
®
)g
®
be a net in Gr(T) and
(x
®
;y
®
)!(x;y).We have
f(x;y) ¸
lim
®
f(x
®
;y
®
) =
lim
®
M(x
®
)
¸ lim
®
M(x
®
)
¸ M(x);
where the last inequality follows from (a).Since H is closed and y
®
2 H(x
®
) for
any ®,we see that (x;y) 2 Gr(H),that is,y 2 H(x).Thus,y 2 fz 2 H(x):
f(x;z) = M(x)g,which shows that T is closed.Moreover,since H is compact,
so is T.Therefore T has a ¯xed point ^x in X by Theorem 3.4,that is,^x 2 H(^x)
and f(^x;^x) = M(^x).¤
By means of Corollary 3.5,we can extend the Theorem 1 of Himmelberg [5]
in the following manner.
Theorem 3.7.Suppose the locally Gconvex uniform space (E;U;¡) satis¯es
the conditions:
(a) ¡
C
is closed and ¡convex for any C 2 hEi,
(b) there is a base B of U consisting of closed symmetric entourages so that for
any V 2 B,V [K] is ¡convex whenever K is a ¡convex subset of E.
If X is a nonempty compact subset of E and T:X (X is an u.s.c.multimap
such that T(x) is closed for all x 2 X and ¡convex for all x in some dense
12
almost ¡convex subset A of X,then T has a ¯xed point.
Proof.For each V 2 B,let F
V
= fx 2 X:x 2 V [T(x)]g.In view of
x 2
\
V 2B
F
V
,x 2
\
V 2B
V [T(x)]
,x 2
T(x) = T(x);since
\
V 2B
V [T(x)] =
T(x);
to ¯nd a ¯xed point of T it su±ces to show that
\
V 2B
F
V
6=?:(1)
As for (1),noting that F
U
\F
V
¶ F
U\V
and X is compact,we need only to show
each F
V
is closed and nonempty.
For each V 2 B,de¯ne T
V
:X (X by
T
V
(x) = V [T(x)]\X
for each x 2 X.We claim that T
V
is closed,that is,Gr(T
V
) is closed in X £X.
Let (x;y) 2
Gr(T
V
) and choose a net f(x
®
;y
®
)g
®
in Gr(T
V
) so that lim
®
(x
®
;y
®
) =
(x;y).By the de¯nition of T
V
,for each ®,there is z
®
2 T(x
®
) such that y
®
2
V [z
®
]\X.Since X is compact,there is a subnet fz
®
j
g of fz
®
g such that z
®
j
!z
for some z 2 X,and so,(y
®
j
;z
®
j
)!(y;z).In light of the closedness of V in
E £ E,it follows that (y;z) 2 V,that is,y 2 V [z].Moreover,since x
®
j
!x,
z
®
j
!z and T is closed,we have z 2 T(x),and hence,y 2 V [T(x)],which shows
that (x;y) 2 Gr(T
V
).Now let ¢ be the diagonal in X £X.Since
x 2 F
V
,x 2 V [T(x)]\X
,x 2 T
V
(x)
,(x;x) 2 Gr(T
V
)\¢
and since Gr(T
V
)\¢ is closed in X £X,we infer that F
V
is closed.
It remains to show that F
V
is nonempty.Choose W 2 B so that
W ± W ± W ± W µ V:(2)
13
Since X µ [
x2X
W[x] and X is compact,there is a ¯nite subset fx
1
;¢ ¢ ¢;x
m
g of
X such that
X µ
m
[
i=1
W[x
i
]:(3)
By the denseness of A in X,there is a ¯nite subset fy
1
;¢ ¢ ¢;y
m
g of A such that
y
i
2 W[x
i
];8i = 1;¢ ¢ ¢;m:(4)
Since A is almost ¡convex,there is fz
1
;¢ ¢ ¢;z
m
g 2 hAi such that
¡cofz
1
;¢ ¢ ¢;z
m
g µ A and z
i
2 W[y
i
];8i = 1;¢ ¢ ¢;m:(5)
Then (4) and (5) imply that
(x
i
;z
i
) 2 W ± W;8i = 1;¢ ¢ ¢;m:(6)
Let C = ¡cofz
1
;¢ ¢ ¢;z
m
g.Since fz
1
;¢ ¢ ¢;z
n
g µ ¡
fz
1
;¢¢¢;z
n
g
and ¡cofz
1
;¢ ¢ ¢;z
m
g
is the smallest ¡convex subset containing fz
1
;¢ ¢ ¢;z
n
g,we have
¡
fz
1
;¢¢¢;z
n
g
= ¡cofz
1
;¢ ¢ ¢;z
m
g = C;
and so it follows from hypothesis (a) that C is closed.Also,we have C µ A by
(5).De¯ne H
V
:C!2
C
by H
V
(x) = T
V
(x)\C.In view of Lemma 1.2,H
V
is
u.s.c.Since both of T
V
(x) and C are ¡convex and closed,so is H
V
(x).Moreover,
H
V
(x) 6=?.To see this,for any y 2 W[T(x)]\X,choose k 2 T(x) such that
y 2 W[k]\X.By (3),there is x
j
such that y 2 W[x
j
],and so (x
j
;k) 2 W ± W.
Meanwhile,(x
j
;z
j
) 2 W ± W by (6).Hence,(z
j
;k) 2 W ± W ± W ± W µ V,and
so
z
j
2 V [k]\C µ V [T(x)]\C
= T
V
(x)\C = H
V
(x);
which shows that H
V
(x) 6=?for each x 2 C.Consequently,H
V
:C (C has a
¯xed point ^x by Corollary 3.5.Then
^x 2 H
V
(^x) = T
V
(^x)\C
= V [T(^x)]\C;
14
that is,^x 2 F
V
,so F
V
is nonempty.¤
4.Application to Minimax Theorem
Tan and Zhang[10] showed that the product of an arbitrary family of Gconvex
spaces is a Gconvex space:Suppose f(E
i
;¡
i
)g
i2I
is any family of Gconvex
spaces.Let E = ¦
i2I
E
i
be equipped with product topology.For each i 2 I,
let ¼
i
:E!E
i
be the projection.De¯ne ¡ = ¦
i2I
¡
i
:hEi (E by
¡(A) = ¦
i2I
¡
i
(¼
i
(A))
for each A 2 hEi.Then (E;¡) is a Gconvex space.
Lemma 4.1.Let (E;¡) = (¦
i2I
E
i
;¦
i2I
¡
i
) be the product Gconvex space of
a family of Gconvex spaces (E
i
;¡
i
),i 2 I.Then K:= ¦
i2I
K
i
is ¡convex in E
provided for each i 2 I,K
i
is a ¡
i
convex subset of E
i
.
Proof.Let ¼
i
be the projection from E to E
i
> For each A 2 hKi,it is clear
that ¼
i
(A) 2 hK
i
i.Since K
i
is ¡
i
convex,we have ¡
i
(¼
i
(A)) µ K
i
,and so
¡
A
= ¦
i2I
¡
i
(¼
i
(A)) µ ¦
i2I
K
i
;
which shows that ¦
i2I
K
i
is ¡convex.¤
Since the topology of the product uniformity is the product topology,the fol
lowing proposition holds.
Proposition 4.2.Suppose f(E
i
;U
i
;¡
i
)g
i2I
is any family of locally Gconvex uni
form spaces.Let E = ¦
i2I
E
i
be equipped with product topology and U = ¦
i2I
U
i
be the product uniformity on E.Then (E;U;¡) is a locally Gconvex uniform
space.
Proof.It su±ces to show that the product uniformity U has a base B consisting
15
of open symmetric entourages such that for each V 2 B and for each x 2 E,V [x]
is ¡convex.For any i 2 I,let B
i
be a base of U
i
consisting of open symmetric
entourages such that for each V
i
2 B
i
and for each x
i
2 E
i
,V
i
[x
i
] is ¡
i
convex.
Let S be the family of all sets of the form f(x;y) 2 E £ E:(x
i
;y
i
) 2 V
i
g for
i 2 I and V
i
2 B
i
.It is easy to check that S is a subbase of U.Let B be the base
generated by S,that is,
B = fV = V
1
\¢ ¢ ¢\V
n
:V
i
2 S;i = 1;¢ ¢ ¢;n;n 2 Ng:
Since each V
j
is of the form
V
j
= f(x;y) 2 E £E:(x
i
j
;y
i
j
) 2 V
i
j
g
for some i
j
2 I and V
i
j
2 B
i
j
,we see that for each x 2 E,
V
j
[x] = ¦
i2Infi
j
g
E
i
£V
i
j
[¼
i
j
(x)];
so that each V
j
[x] is ¡convex by Lemma 4.1.Therefore,
V [x] = ¦
i2Infi
j
:j=1;¢¢¢;ng
E
i
£¦
n
j=1
V
i
j
[¼
i
j
(x)]
is ¡convex.¤
Lemma 4.3.Let (E;U;¡) = (¦
i2I
E
i
;¦
i2I
U
i
;¦
i2I
¡
i
) be the product locally G
convex uniform space of the family of locally Gconvex uniform spaces (E
i
;U
i
;¡
i
),
i 2 I.Then A:= ¦
i2I
A
i
is almost ¡convex in E provided for each i 2 I,A
i
is
an almost ¡
i
convex subset of E
i
.
Proof.Let fx
1
;¢ ¢ ¢;x
n
g 2 h¦
i2I
A
i
i and U 2 U.We see fromthe proof of the pro
ceeding lemma that there are fi
j
:j = 1;¢ ¢ ¢;mg µ I and V
i
j
2 U
i
j
,j = 1;¢ ¢ ¢;m,
such that V:=\
m
j=1
V
j
µ U,where each V
j
= f(x;y) 2 E £E:(x
i
j
;y
i
j
) 2 V
i
j
g.
Put
U
i
=
(
E
i
£E
i
;if i 2 I n fi
j
:j = 1;¢ ¢ ¢;mg;
V
i
j
;if i 2 fi
j
:j = 1;¢ ¢ ¢;mg:
16
Then each U
i
2 U
i
.Since f¼
i
x
1
;¢ ¢ ¢;¼
i
x
n
g 2 hA
i
i and A
i
is almost ¡
i
convex,
there exists fy
1
i
;¢ ¢ ¢;y
n
i
g 2 hA
i
i such that
y
k
i
2 U
i
[¼
i
x
k
];8k = 1;¢ ¢ ¢;n;and
¡
i
cofy
1
i
;¢ ¢ ¢;y
n
i
g µ A
i
:
For each k = 1;¢ ¢ ¢;n,let y
k
= (y
k
i
)
i2I
2 ¦
i2I
E
i
.It is obvious that
fy
1
;¢ ¢ ¢;y
n
g 2 h¦
i2I
A
i
i;
and y
k
2 V [x
k
] µ U[x
k
] for any k = 1;¢ ¢ ¢;n.Furthermore,since both of
¡cofy
1
;¢ ¢ ¢;y
n
g and ¦
i2I
¡
i
cofy
1
i
;¢ ¢ ¢;y
n
i
g contain fy
1
;¢ ¢ ¢;y
n
g and since ¦
i2I
¡
i

cofy
1
i
;¢ ¢ ¢;y
n
i
g is ¡convex by Lemma 4.1,we infer that
¡cofy
1
;¢ ¢ ¢;y
n
g µ ¦
i2I
¡
i
cofy
1
i
;¢ ¢ ¢;y
n
i
g µ ¦
i2I
A
i
:
This shows that ¦
i2I
A
i
is almost ¡convex.¤
Lemma 4.4.Suppose for each i 2 I the locally Gconvex uniformspace (E
i
;U
i
;¡
i
)
has the properties:
(a) ¡
i
(C
i
) is closed and ¡
i
convex for any C
i
2 hE
i
i,and
(b) there is a base B
i
of U
i
consisting of closed symmetric entourages such that
for any V
i
2 B
i
,V
i
[K
i
] is ¡
i
convex whenever K
i
is a ¡
i
convex subset of E
i
.
Then (E;U;¡) = (¦
i2I
E
i
;¦
i2I
U
i
;¦
i2I
¡
i
) has the properties:
(i) ¡
C
is closed and ¡convex for any C 2 hEi,and
(ii) there is a base B of U consisting of closed symmetric entourages such that
for any V 2 B,V [K] is ¡convex whenever K is a ¡convex subset of E.
Proof.Let C 2 hEi.Noting that ¡co(C) = ¡
C
= ¦
i2I
¡
i
(¼
i
(C)) and for
each i 2 I,¡
i
(¼
i
(C)) is closed and ¡
i
convex,we infer that ¡
C
is closed and
¡convex.This establishes (i).
(ii) can be proved in a like manner as that of Proposition 4.2.¤
Theorem 4.5.Suppose for each i 2 I the locally Gconvex uniform space
17
(E
i
;U
i
;¡
i
) has the properties:
(a) ¡
i
(C
i
) is closed and ¡
i
convex for any C
i
2 hE
i
i,and
(b) there is a base B
i
of U
i
consisting of closed symmetric entourages such that
for any V
i
2 B
i
,V
i
[K
i
] is ¡
i
convex whenever K
i
is a ¡
i
convex subset of E
i
.
Assume for each i 2 I,A
i
is a dense almost ¡
i
convex subset of a compact
subset K
i
of E
i
.Let K = ¦fK
i
:i 2 Ig,A
0
i
= ¦fA
¸
:¸ 2 I;¸ 6= ig and
K
0
i
= ¦fK
¸
:¸ 2 I;¸ 6= ig.If for each i 2 I,F
i
:K
0
i
(K
i
is a closed multimap
so that F
i
(x
0
i
) is ¡
i
convex for each x
0
i
2 A
0
i
.Then\fGr(F
i
):i 2 Ig 6=?.
Proof.For each i 2 I,de¯ne H
i
:K!K
i
by
H
i
(x) = F
i
(x
0
i
);
where x
0
i
is the projection of x on K
0
i
.Since H
i
is the composition of a continuous
function and an u.s.c.multimap,it is u.s.c.,and so it has closed graph.De¯ne
H:K ( K by H(x) = ¦
i2I
H
i
(x) and let A = ¦
i2I
A
i
.We see from Lemma
4.3 that A is almost ¡convex,and H(x) is ¡convex for any x 2 A by Lemma
4.1.Since each F
i
has closed graph,so is each H
i
,and hence H has closed graph.
In addition,the hypotheses (a) and (b) and Lemma 4.4 show that the product
locally Gconvex uniform space (E;U;¡) posseses the properties (a) and (b) of
Theorem 3.7.Consequently,all of the requirements of Theorem 3.7 are satis¯ed,
and hence H has a ¯xed point ^x.Obviously,^x 2\
i2I
Gr(F
i
).¤
A realvalued function':E!R on a Gconvex space (E;¡) is said to
quasiconvex if for each ¸ 2 R,the set fx 2 E:'(x) · ¸g is ¡convex;and
quasiconcave if for each ¸ 2 R,the set fx 2 E:'(x) ¸ ¸g is ¡convex.
Theorem4.6.Suppose for i = 1;2 the locally Gconvex uniformspace (E
i
;U
i
;¡
i
)
has the properties:
(a) ¡
i
(C
i
) is closed and ¡convex for any C
i
2 hE
i
i,and
(b) there is a base B
i
of U
i
consisting of closed symmetric entourages such that
for any V
i
2 B
i
,V
i
[K
i
] is ¡
i
convex whenever K
i
is a ¡
i
convex subset of E
i
.
Let K
i
be a compact subset of the locally Gconvex uniform space (E
i
;U
i
;¡
i
),and
18
A
i
be a dense almost ¡
i
convex subset of K
i
,i = 1;2.If f:K
1
£ K
2
!R is
continuous and satis¯es that
(i) for all x 2 A
1
,f(x;) is quasiconvex on A
2
;
(ii) for all y 2 A
2
,f(;y) is quasiconcave on A
1
,
then max
x2K
1
min
y2K
2
f(x;y) = min
y2K
2
max
x2K
1
f(x;y).
Proof.Since f is continuous on the compact set K
1
£K
2
,both of
max
x2K
1
min
y2K
2
f(x;y) and min
y2K
2
max
x2K
1
f(x;y)
are wellde¯ned.Now,let ² > 0 be given.For any ¹y 2 K
2
,since
max
x2K
1
f(x;¹y) ¡² ¸ min
y2K
2
max
x2K
1
f(x;y) ¡²;
there is ¹x 2 K
1
such that f(¹x;¹y) > min
y2K
2
max
x2K
1
f(x;y)¡²:In a like manner,
for any ¹x 2 K
1
,there is ¹y 2 K
2
such that f(¹x;¹y) < max
x2K
1
min
y2K
2
f(x;y) +²:
So we can de¯ne F
1
:K
2
(K
1
and F
2
:K
1
(K
2
by
F
1
(y) = fx 2 K
1
:f(x;y) ¸ min
y2K
2
max
x2K
1
f(x;y) ¡²g
F
2
(x) = fy 2 K
2
:f(x;y) · max
x2K
1
min
y2K
2
f(x;y) +²g:
By the continuity of f on K
1
£ K
2
,we see that both of F
1
and F
2
are closed
multimaps.Furthermore,by assumptions (i) and (ii),F
1
(y) is ¡
1
convex for all
y 2 A
2
,and F
2
(x) is ¡
2
convex for all x 2 A
1
.Thus,it follows from Theorem 4.5
that Gr(F
1
)\Gr(F
2
) 6=?,that is,there exists (x
²
;y
²
) 2 K
1
£K
2
such that
min
y2K
2
max
x2K
1
f(x;y) ¡² · f(x
²
;y
²
) · max
x2K
1
min
y2K
2
f(x;y) +²:
Then,
liminf
²!0
µ
min
y2K
2
max
x2K
1
f(x;y) ¡²
¶
· liminf
²!0
f(x
²
;y
²
)
· limsup
²!0
f(x
²
;y
²
)
· limsup
²!0
µ
max
x2K
1
min
y2K
2
f(x;y) +²
¶
;
19
and hence min
y2K
2
max
x2K
1
f(x;y) · max
x2K
1
min
y2K
2
f(x;y):But,it is obvious
that max
x2K
1
min
y2K
2
f(x;y) · min
y2K
2
max
x2K
1
f(x;y):Therefore,
max
x2K
1
min
y2K
2
f(x;y) = min
y2K
2
max
x2K
1
f(x;y):
¤
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