Generalized Knaster-Kuratowski-Mazurkiewicz type theorems and applications to minimax inequalities

TẠP CHÍ PHÁT TRIỂN KH&CN, TẬP 20, SỐ K2-2017 
131
Abstract— Knaster-Kuratowski-Mazurkiewicz type 
theorems play an important role in nonlinear 
analysis, optimization, and applied mathematics. 
Since the first well-known result, many international 
efforts have been made to develop sufficient 
conditions for the existence of points intersection (and 
their applications) in increasingly general settings: G-
convex spaces [21, 23], L-convex spaces [12], and FC-
spaces [8, 9]. 
Applications of Knaster-Kuratowski-Mazurkiewicz 
type theorems, especially in existence studies for 
variational inequalities, equilibrium problems and 
more general settings have been obtained by many 
authors, see e.g. recent papers [1, 2, 3, 8, 18, 24, 26] 
and the references therein. 
In this paper we propose a definition of generalized 
KnasterKuratowski-Mazurkiewicz mappings to 
encompass R-KKM mappings [5], L-KKM mappings 
[11], T-KKM mappings [18, 19], and many recent 
existing mappings. Knaster-KuratowskiMazurkiewicz 
type theorems are established in general topological 
spaces to generalize known results. As applications, 
we develop in detail general types of minimax 
theorems. Our results are shown to improve or 
include as special cases several recent ones in the 
literature.. 
Index Terms— L - T -KKM mappings, Generalized 
convexity, Transfer compact semicontinuity, Minimax 
theorems, Saddle-points. 
1 INTRODUCTION 
xistence of solutions takes a central place in the 
optimization theory. Studies of the existence of 
solutions of a problem are based on existence 
results for important points in nonlinear analysis 
like fixed points, maximal points, intersection 
points, etc. 
Manuscript Received on July 13th, 2016. Manuscript Revised 
December 06th, 2016. 
This work was supported by University of Information 
Technology, Vietnam National University Hochiminh City 
under grant number D1-2017-07. 
Ha Manh Linh was with the Department of Mathematics, 
Vietnam National University-HoChiMinh City, University of 
Information Technology, Thu Duc district, Saigon, Vietnam e-
mail: linhhm@uit.edu.vn. 
One of the most famous existence theorems in 
nonlinear analysis is the classical KKM theorem, 
which has been generalized by many authors. For 
example see [1, 2, 3, 4, 6, 10, 22, 23, 27]. In early 
forms of this fundamental result, convexity 
assumptions played a crucial role and restricted the 
ranges of applicable areas. After, various 
generalized linear/convex structures have been 
proposed and corresponding types of KKM 
mappings have been defined together with these 
spaces, such as [3, 6, 21] investigated G-convex 
spaces, Ding [7-9] introduced the concept of a FC-
space and then Khanh and Quan [18, 19], Khanh, 
Lin and Long [14], Khanh and Long [15, 16] and, 
Khanh, Long and Quan [17] generalized and 
unified the previous spaces into a notion called a 
GFC-space. 
Applications of KKM-type theorems, especially 
in existence studies for variational inequalities, 
equilibrium problems and more general settings 
have been obtained by many authors, see e.g. recent 
papers [1, 2, 3, 8, 18, 24, 26] and the references 
therein. 
To avoid in a stronger sense convexity structures 
in investigating KKM-type theorems, in this paper 
we propose a definition of a generalized type of 
KKM mappings in terms of a FLS-space and use it 
to establish generalized KKM type theorems. As 
applications we focus only on minimax and saddle-
point problems, which also generalize or improve 
recent results in the literature [3, 5, 6, 10,...]. 
The outline of the paper is as follows. Section 2 
contains definitions and preliminary facts for our 
later use. In Section 3, we give our main results. 
This section contains generalized KKM-type 
theorems, a Ky Fan type matching theorem and 
discuss their consequences in some particular cases. 
In section 4, we obtain the sufficient conditions for 
the solutions existence of minimax and saddle-
point problems. 
Generalized Knaster-Kuratowski-Mazurkiewicz 
type theorems and applications to minimax 
inequalities 
Ha Manh Linh 
E 
132 SCIENCE & TECHNOLOGY DEVELOPMENT, Vol 20, No.K2- 2017 
2 PRELIMINARIES 
We recall now some definitions for our later use. 
For a set X , by X2 and X we denote the family 
of all nonempty subsets, and the family of 
nonempty finite subsets, respectively. Let Z , X be 
topological spaces and ZBA , , int A , cl A (or A ), 
int AB and cl AB stand for the interior, closure, 
interior in B and closure in B of A . A is called 
compactly open (compactly closed, resp.) if for 
each nonempty compact subset K of Z , KA is 
open (closed, resp.) in K . The compact interior and 
compact closure of A are defined by 
},:{=c inZpactlyopenAandBiscomBZBintA  
}.:{=c edinZpactlyclosAandBiscomBZBclA  
It is clear that cint A (ccl A , resp.) is compactly 
open (compactly closed, resp.) in Z and for each 
nonempty compact subset ZK  with  KA , 
one has K cint A = int )( AKK  and 
K ccl =A cl )( AKK  . It is equally obvious that 
ZA is compactly open (compactly closed, resp.) 
if and only if cint A = A (ccl A = A , resp.). A set-
valued ZXT 2: is said to be upper [lower resp.] 
semicontinuous (usc) [lsc resp.] if for any open 
[closed resp.] subset ZU  , the set 
})(:{:= UxTXxT  is open [closed resp] in X . T 
is said compact if )(XT is compact subset of Z . 
N , Q , and R denote the set of the natural 
numbers, the set of rational numbers, and that of 
the real numbers, respectively, and },{= RR . 
For N n , n stands for the n -simplex with the 
vertices being the unit vectors 1e , 2e , ..., 1 ne of a 
basis of 1 nR . 
Definition 1 Let X be a topological space, Y be 
a nonempty set and  be a family of lower 
semicontinuous mappings ,2: Xn  N n . Then 
a triple ),,( YX is said to be a finitely lower 
semicontinuous topological space ( FLS -space in 
short) if for each finite subset  YyyyN n},...,,{= 10 , 
there is XnN 2:  of the family  . Later we 
also use }){,,( NYX  to denote ),,( YX . 
Remark 1 If N is a continuous single-valued 
mapping, then ),,( YX becomes an GFC -space as 
defitioned in [18-20]. If in addition XY = then 
),,( YX is rewritten as ),( X and becomes an 
FC -space in [7, 8]. The Example 1 below shows 
that in general the inverse is not true. 
Definition 2 (See [18-20]). Let ( X , Y , ) be a 
GFC -space and Z be a topological space. Let 
ZXT 2: , ZYF 2: be two set-valued mappings. 
F is called a generalized KKM mapping with 
respect to T ( T -KKM mapping in short) if for 
each  YyyN n},...,{= 0 and each  Nyy kii
},...,{
0
, 
),())((
0=
ji
k
j
kN yFT  
where  N is corresponding to N and 
}.,...,{=
0 kiik
eeco 
Definition 3 (See [19]). Let ),,( YX be a GFC-
space and Z be a topological space. A set-valued 
mapping ZXT 2: is called better admissible if T 
is usc and compact-valued such that for each 
 YN and each continuous mapping 
nnNT ))((:  , the composition 
n
nNnN
T
 2:| )(  has a fixed point, where 
 N is corresponding to N. 
The class of all such better admissible mapping 
from X to Z is denoted by ),,( ZYXB 
Definition 4 (See [7]). Let Z be a topological 
space and Y be a nonempty set. Let ZYF 2: is a 
set-valued mapping. 
 1. F is called transfer open-valued (transfer 
closed-valued, resp.) if, for each Yy and )(yFz 
( )(yFz , resp.) there exists Yy such that 
 z int )(yF ( z cl )(yF , resp.) 
 2. F is said to be transfer compactly open-
valued (transfer compactly closed-valued, resp.) if 
for each Yy , each nonempty compact subset 
ZK  and each KyFz  )( ( KyFz  )( , resp.), 
there is Yy such that z int ))(( KyFK  
( z cl ))(( KyFK  , resp.) 
We will need the following well-known result. 
Lemma 1 ([7]). Let Y be a set, X be a 
topological space and XYF 2: . The following 
statements are equivalent. 
 1. F is transfer compactly closed-valued 
(transfer compactly open-valued, respectively). 
 2. for each compact subset XK  . 
TẠP CHÍ PHÁT TRIỂN KH&CN, TẬP 20, SỐ K2-2017 
133
),)(cl(=
))(ccl(=))((
KyF
KyFKyF
K
Yy
YyYy




).)(int(=
))(cint(=))((
KyF
KyFKyF
K
Yy
YyYy




Definition 5 Let ),,( YX be a FLS -space and Z 
be a topological space. Let ZYF 2: and 
ZXT 2: be set-valued mappings 
 1. F is said to be a generalized L -KKM 
mapping wrt T ( L - T -KKM mapping in short) if, 
for each  YyyyN n},...,,{= 10 and each 
Nyyy
kiii
},...,,{
10
, one has ),())((
0= j
i
k
j
kN yFT   
where  N is corresponding to N and 
},...,,{=
10 kiiik
eeeco . 
 2. We say that a set-valued mapping 
ZXT 2: has the generalized L -KKM property if, 
for each L - T -KKM mapping ZYF 2: , the 
family }:)({ YyyF has the finite intersection 
property, i.e. all finite intersections of sets of this 
family are nonempty. The class of all mappings 
ZXT 2: which have the generalized L -KKM 
property is denoted by L -KKM(X,Y,Z). 
 3. Let XYS 2: be a set-valued mapping. A 
subset D of Y is called an L - S -subset of Y if, for 
each  YyyN n},...,{= 0 and each ,},...,{ 0
DNyy
kii
 
one has XnN 2:  of  such that 
),()( DSkN   where k is face of n 
corresponding to },...,{
0 kii
yy . 
Remark 2 Note that the Definition 5 (i) is a 
generalization of the Definition 2.1 of [11]. We 
also see that every L - T -KKM mapping is a T -
KKM mapping when N is a continuous single-
valued mapping. If in addition XY = and T is the 
identity map then L - T -KKM mapping becomes an 
R -KKM mapping of [5] and thw Definition 2.2 of 
[7]. 
The following example shows that the Definition 
5 (i) contains the Definition 2. 
Example 1 Suppose that )[0,== ZX and 
N=Y . For each  YN , we define XnN 2: by 

.,
,},...,{{0},
=)( 0
otherwise
eeeif
e nN 
We see that N is lower semicontinuous but not 
continuous. Hence ),,( YX is a FLS-space. 
Let ZYF 2: and ZXT 2: be defined as 
follows 2)[0,=)( yyF for each Yy and 
.[0,1]=)( XforeachxxT Then F is not a T -KKM 
mapping. However F is the generalized L - T -
KKM mapping. Also, the class }:({ YyyF has the 
finite intersection property. 
Lemma 2 (Classical) Let ZXT 2: be upper 
semicontinuous with compact valued from a 
compact space X to Y. Then T(X) is compact. 
Lemma 3 Let ),,( YX be a GFC-space and Z be 
a topological space. Then ),,( ZYXB L-
KKM(X,Y,Z). 
 Proof. For each ),,( ZYXT B , let F is a 
generalized L - T -KKM. Suppose to the contrary 
that  YyyN n},...,{= 0 exists such that 
.=)(
0=
i
n
i
yF 
It follows that 
 =)())((
0=
i
n
i
nN yFT  
and 
].))(())(\[(=))((
0=
nNi
n
i
nN TyFZT   
Then ninNi TyFZ 0=}))(())(\{(  is an open 
covering of the compact set ))(( nNT . Let 
n
ii 0=}{ 
be a continuous partition of unit associated with 
this covering and nnNT ))((:  be defined by 
.)(=)(
0=
ii
n
i
ett   Then  is continuous. Since 
),,( ZYXT B , the composition NnN
T  )(| has 
a fixed point. Hence, there is ))((0 nNTz such 
that )))((( 00 zTz N  . Where 
)0(0(0)
0 )(=)( zJij
Jj
ezz   with 
0})(:}{0,1,...,{=) 00( znjzJ j 
134 SCIENCE & TECHNOLOGY DEVELOPMENT, Vol 20, No.K2- 2017 
On the other hand, as F is L - T -KKM ( T -
KKM), one has 
,)(=
)(
))(()))(((
)0(
)0(
)0(00
j
zJj
j
zJj
zJNN
yF
yF
TzTz



   
 so there is )( 0zJj such that )(0 jyFz 
However, in view of the definitions of )( 0zJ and 
of the partition nii 0=}{ 
,)(\
))(())(\(
0})(:))(({0
j
nNj
jnN
yFZ
TyFZ
zTzz

 
 
a contradiction. 
3 GENERALIZED L - T -KKM TYPE THEOREMS 
Theorem 1 Let ),,( YX be a FLS -space and Z 
be topological spaces. Let ZYF 2: and ZXT 2: 
be set-valued mappings. Assume that the following 
conditions hold 
 1. F is L-T-KKM; 
 2. T L-KKM(X,Y,Z) and )(XT is a compact 
subset of Z; 
 3. there are  YA and a nonempty compact 
subset K of Z such that 
;)(ccl KyF
Ay

 
 4. F is transfer compactly closed-valued. 
 Then 
.))(()(  
yFXTK
Yy
 
 Proof. Define a new set-valued mapping 
)(2: XTYH by 
 )(=)( XTyH ccl )(yF , for each Yy . 
 Then H has closed-values in )(XT . We show 
that H is L - T -KKM. Indeed, since F is L - T -
KKM, for each  YyyN n},...,{= 0 and each 
Nyy
kii
},...,{
0
 one has 
).(
])()([=
)()(
)())((=))((
0=
0=
0=
ji
k
j
ji
k
j
ji
k
j
kNkN
yH
XTyF
XTyF
XTTT






 
Therefore, H is the L - T -KKM mapping. 
Moreover, since LT -KKM(X,Y,Z) it follows 
that the family 
}:)({=}:)({ YyyHYyyH 
has the finite intersection property. Since )(XT is 
compact and }:)({ YyyH is a family of closed 
subsets in )(XT , one has 
)).(c)((=)( yclFXTyH
YyYy
 
 
Hence, there exists  
)((ˆ XTz
Yy ccl ))( yF , 
i.e., zˆ ccl )),(yF for each Yy . By (iii), there is 
 YA and a compact subset K of Z such that 
.)(cˆ KyclFz
Ay
 
 
By (iv) and Lemma 1, we have 
).(
)()(=
)()(c
yF
XTzF
XTzclFz
Yz
Yz


 


Thus we arrive at the conclusion 
.))(()(  
yFXTK
Yy
 
Remark 3 Theorem 1 unifies and generalizes 
Theorem 3.2 of [5], Theorem 3.2 of [11] and 
Theorem 3.2 of [21] under much weaker 
assumptions. By Lemma 3, Theorem 1 improves the 
assertion (iii1 ) of Theorem 2.2 of [19]. 
The following example shows that we cannot use 
of known results in FC -spaces of [7] or GFC -
convex spaces of [18-20], but is easily investigated 
by FLS -spaces. 
Example 2 Let {0}= NY and )[0;== ZX . For 
each  YyyyN n},...,,{= 10 , we define ,: XnN 
by ii
n
i
N ye   0==)( , where nii
n
i
ee  0== and 
TẠP CHÍ PHÁT TRIỂN KH&CN, TẬP 20, SỐ K2-2017 
135
1=
0=
i
n
i
 . Then }){,,( NYX is the GFC -space. Let 
ZYF 2: and ZXT 2: are defined as follows 
 )(yF = ,0=yif{0}{ [ ..if0,0.5] otherwise 
 )(xT = {0}{ [0, 1) , [0, 1] ..if otherwise 
 We can see that F is not T -KKM. Indeed, we 
choose  YyN 1}={= 0* , one has 
 1=)( 0*
 N and 
(1).=[0,0.5][0,1]=))(( 0*
FT N Ú 
 Hence the results in [18-20] are out of use for 
this case. 
To apply our Theorem 1, we now define a FLS -
space by {0}= NY , )[0;= X and the 
corresponding ,2: XnN  by 
 )(eN = },,...,{eif{0}{ 0 nee [0; 
0.5] ..if otherwise 
 We see that N is lower semicontinuous 
mapping, so }){,( NYX  is a FLS -space. 
Furthermore, for each  YyyyN n},...,,{= 10 we have 
 )({0}=))(( yFT nN   for each Yy 
 Therefore F is a L - T -KKM mapping, so (i) of 
Theorem 1 is satisfied. Clearly [0,1]=)(XT is the 
compact subset of Z and the class }:({ YyyF has 
the finite intersection property,i.e., (ii)of Theorem 1 
is fulfilled. If we choose {0,1}=A and [0,1]=K then 
assumptions (iii) of Theorem 1 are satisfied. 
Moreover it is easy to see that F is transfer 
compactly closed-valued. By Theorem 1, one 
concludes that 
.{0}=))(()(  
yFXTK
Yy
 
Theorem 2 Let (X,Y,  ) be a FLS -space and Z 
be topological spaces. Let XYS 2: , ZYF 2: 
and ZXT 2: be s ... },...,{
0
 such that 
.)())((
0=
  
ji
k
j
kM yFT  
 Proof. Suppose that the conclusion is not true. 
Then for any  YyyN n},...,{= 0 and any 
Nyy
kii
},...,{
0
,  =)())((
0=
j
k
j
kN yFT  . 
Therefore )())((
0=
j
k
j
kN yHT   , where 
)(\=)( yFZyH . It follows that H is L - T -KKM. By 
(i), H is transfer compactly closed-valued. Clearly, 
all conditions of Theorem 2 are satisfied. It follows 
from Theorem 2 that 
  
)())(( yHYST
Yy . 
 Hence, )())(( YFYST Ö , but this contradictions 
(iii). Thus there exist  YyyM m},...,{= 0 and 
Myy
kii
},...,{
0
 such that 
 .)())((
0=
  j
k
j
kM yFT  
Remark 6 Theorem 3 generalizes Theorem 8 of 
[21] and Theorem 3.1 of [12] since being G -KKM 
mapping and R -KKM mapping are special cases of 
L - T -KKM mapping. 
Theorem 5 Theorem 2 and 4 are equivalent. 
 Proof. We saw that Theorem 4 can be proved 
by using Theorem 2. Now we derive Theorem 2 
from Theorem 4. Suppose that 
.=)())(( 
yFYST
Yy
 
Let )(\=)( yFZyH . Then )(yH is transfer 
compactly open-valued and )())(( YHYST  . It 
follows from Theorem 4 that there exist  YM 
and Myy
kii
},...,{
0
 such that 
,)())((
0=
  
ji
k
j
kM yHT  (where  M ). 
Hence )())((
0= j
i
k
j
kM yFT Ö . This contradicts the 
fact that F is L - T -KKM. Thus the conclusion of 
Theorem 2 follows Theorem 4. 
4 KY FAN TYPE MINIMAX INEQUALITIES 
In this section, by applying L - T -KKM 
theorems, we shall establish some new Ky Fan type 
minimax inequalities and saddle point problems. 
Definition 6 Let ),,( YX be a FLS -space and 
Z be a topological space. Let 
}{:,2:  RZYgXT Z and R  . g is called 
 - L -quasiconvex (  - L -quasiconcave, resp.) wrt 
T in y if,   YN and 
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137
)),((,},...,{
0 kNkii
TzNyy   one has 
max  ),(0 zyg jikj
 (min  ),(0 zyg jikj
, resp.). 
Remark 7 Definition 6 generalizes Definition 
4.1 of [9], Definition 4.1 of [20] and Definition 1.7 
of [25] 
Definition 7 Let ),,( YX be a FLS -space and 
Z be a topological space. Let 
R ZYgXT Z :,2: and R  , with  . g 
is called -  - L -quasiconcave wrt T in y if, 
  YN , )),((,},...,{
0 kNkii
TzNyy   there is an 
}{0,..., kr satisfying  ),( zyg
ri
. If = , then 
the notion in Definition 7 reduces to the 
corresponding notion in Definition 6. 
We need also the following notion of Definition 
2.6 in [6]. 
Definition 8 Let Y be a nonempty set and Z be 
a topological space. Let R ZYf : and R  . f 
is called  -transfer compactly lower (upper, resp.) 
semicontinuous in z if for each compact subset K 
of Z and for each Kz , there exists a Yy such 
that >),( zyf ( <),( zyf , resp.,) implies that there 
exists an open neighborhood )(zU of z and a point 
Yy 0 such that >),( 0 zyf ( <),( 0 zyf , resp.,) for 
all )(zUz . 
Let ZYF 2: by }),(:{=)(  zyfZzyF 
(  ),( zyf , resp.). Then f is  -transfer compactly 
lower (upper, resp.) semicontinuous in z if and 
only if F is transfer compactly closed-valued 
(open-valued resp.). 
Theorem 6 Let ),,( YX be a FLS -space and Z 
be a topological space. Let 
}{:,,2:  RZYgfXT Z and R  be such 
that 
 1. for each ),(),(,),( zygzyfZYzy ; 
 2. g is generalized  -L-quasiconcave wrt T 
in y; 
 3. f is  transfer compactly in z; 
 4. T L-KKM(X,Y,Z) and )(XT is a 
compact subset of Z; 
 5. there exist  YA and a nonempty compact 
subset K of Z such that the set 
KzyfZzcl
Ay
 
}),(:{c  . 
 Then there exists a point Zz ˆ such that 
Yyzyf  ,)ˆ,(  . 
 Proof. First, we define two set-valued mappings 
ZYGF 2:, by 
 }),(:{=)(  zyfZzyF 
and .},),(:{=)( YyzygZzyG   
 By (i), we have that YyyFyG  ),()( . By (ii) 
and Definition 6, for each  YyyN n},...,{= 0 , each 
 Nyy
nii
},...,{
0
 and each ))(( kNTz  , 
min  ),(0 zyg jikj
. Hence there exists }{0,..., kr 
such that  )( ,zri
yg , i.e., 
)()()(
0=0= j
i
k
jj
i
k
jr
i yFyGyGz   . Since 
))(( kNTz  is arbitrary, we have 
 )()((
0= j
i
k
j
kN yFT   . 
 Hence, F is a generalized L - T -KKM mapping. 
The condition (iii) implies that F is transfer 
compactly closed-valued. The condition (v) implies 
that there exists  YA and a nonempty compact 
subset K of Z such that 
.)(c KyclFAy  
Add the condition (iv), all conditions of Theorem 
1 are satisfied. By Theorem 1 we have, 
.)(   yFYy 
Taking any )(ˆ yFz Yy  , we obtain 
.,)ˆ,( Yyzyf   W 
Remark 8 Theorem 6 generalize Theorem 2.1-
2.4 of [26]. 
Theorem 7 Let ),,( YX be a FLS -space and Z 
be a topological space. Let 
}{:,,2:  RZYgfXT Z and R  be such 
that 
 1. for each ),(),(,),( zygzyfZYzy ; 
 2. g is generalized  -L-quasiconcave wrt T 
in y; 
 3. f is  -transfer compactly lower 
semicontinuous in z; 
 4. T L-KKM(X,Y,Z); There is XYS 2: 
such that Y is an L-S-subset of itself and ))(( YST is 
compact. 
 Then there exists a point Zz ˆ such that 
Yyzyf  ,)ˆ,(  . 
 Proof. Define two set-valued mappings 
ZYGF 2:, by 
 }),(:{=)(  zyfZzyF and 
.},),(:{=)( YyzygZzyG   
 By (i), we have that YyyFyG  ),()( . By (ii) 
and Definition 6, for each  YyyN n},...,{= 0 , each 
 Nyy
nii
},...,{
0
 and each ))(( kNTz , 
138 SCIENCE & TECHNOLOGY DEVELOPMENT, Vol 20, No.K2- 2017 
min  ),(0 zyg jikj
. Hence there exists }{0,..., kr 
such that  )( ,zri
yg , i.e., 
)()()(
0=0= j
i
k
jj
i
k
jr
i yFyGyGz   . Since 
))(( kNTz  is arbitrary, we have 
 )()((
0= j
i
k
j
kN yFT   . 
 Hence, F is a generalized L - T -KKM mapping. 
The condition (iii) implies that F is transfer 
compactly closed-valued. All conditions of 
Theorem 2 are satisfied. By Theorem 2 we have 
.)(   yFYy Then, there is )(ˆ yFz Yy  such that 
.,)ˆ,( Yyzyf   
Theorem 8 Let ),,( YX , ),,(  XY be two FLS -
spaces and Z be a topological space. Let 
ZXT 2: , YXH 2: , }{:  RZYg . 
Assumption that 
 1. g is generalized 0-L-quasiconcave wrt T in 
y and generalized 0-L-quasiconvex wrt H in z; 
 2. g is 0 -transfer compactly lower 
semicontinuous in z and 0 -transfer compactly 
upper semicontinuous in y; 
 3. T L-KKM(X,Y,Z); there is XYS 2:1 
such that Y is an L- 1S -subset of itself and ))(( 1 YST 
is compact; 
 4. H L-KKM(X,Z,Y); there is XZS 2:2 
such that Z is an L- 2S -subset of itself and ))(( 2 ZST 
is compact. 
 Then, g has a saddle point ZYzy )ˆ,ˆ( , i.e., 
.),(),,ˆ()ˆ,ˆ()ˆ,( ZYzyzygzygzyg  
In particular, we have 
0.=),(is=),(si zygnfupzygupnf ZzYyYyZz 
Proof. Applying Theorem 7 with 0= and 
gf  , there exists a point Zz ˆ such that 0)ˆ,( zyg 
for all Yy . Let ),(=),( zygyzf for all .),( YZyz 
We apply Theorem 7 with 0= again, there is a 
point Yy ˆ such that 0)ˆ,( yzf for all Zz . Then 
we have .),(),,ˆ(0)ˆ,( ZYzyzygzyg  Thus, 
0=)ˆ,ˆ( zyg and 
 ZYzyzygzygzyg  ),(),,ˆ()ˆ,ˆ()ˆ,( , 
 which implies 
),(is)ˆ,ˆ(),(si zygnfupzygzygupnf ZzYyYyZz 
Since ),(is),(si zygnfupzygupnf ZzYyYyZz is 
always hold, we get 
0.=),(is=),(si zygnfupzygupnf ZzYyYyZz 
 W 
Remark 9 Theorem 8 contains Theorem 4.2 of 
[25]. 
Theorem 9 Let ),,( YX be a FLS -space and Z 
be a topological space. Let 
}{:,,2:  RZYgfXT Z and R  , with 
 be such that 
 1. for each  ),(,),( zygZYzy implies 
 ),( zyf ; 
 2. g is generalized -  -L-quasiconcave wrt 
T in y; 
 3. f is  -transfer compactly lower 
semicontinuous in z and -transfer compactly 
upper semicontinuous in z; 
 4. T L-KKM(X,Y,Z); There is XYS 2: 
such that Y is an L-S-subset of itself and ))(( YST is 
compact. 
 Then, there exists a point Zz ˆ such that 
.,)ˆ,( Yyzyf   
 Proof. We also define two mappings 
ZYGF 2:, by 
 }),(:{=)(  zyfZzyF and 
YyzygZzyG  },),(:{=)(  
 Then, by (i), we have )()( yFyG  for all Yy . 
By (ii), for each  YyyN n},...,{= 0 , each 
Nyy
kii
},...,{
0
 and each )),(( kNTz there is an 
}{0,..., kr satisfying .),(  zyg
ri
 It follows that 
).()()(=}),(:{
0= k
i
k
jr
iriri
yFyFyGzygZzz   
 Since ))(( kNTz  is arbitrary, we have 
)())((
0= j
i
k
j
kN yFT   . Hence F is a L - T -KKM 
mapping. 
We set 
 }),(:{:=)(  zyfZzyA 
}),(:{:=)( zyfZzyB 
 Then one has ).()(=)( yByAyF  The condition 
(iii) implies that A and B are transfer compactly 
closed-valued. We need show that F is transfer 
compactly closed-valued. For each compact subset 
K of Z , by Lemma 1, we have 
))(c(=))(( KyAlKyA K
YyYy

 
and 
).)(c(=))(( KyBlKyB K
YyYy

 
It follows that 
TẠP CHÍ PHÁT TRIỂN KH&CN, TẬP 20, SỐ K2-2017 
139
).)](c)(c([=))]()(([ KyBlyAlKyByA KK
YyYy

 
On the other hand, 
).)](c)(c([
))]()([c())]()(([
KyBlyAl
KyByAlKyByA
KK
Yy
K
YyYy




Therefore F is transfer compactly closed-
valued. Clearly, all conditions of Theorem 2 are 
satisfied. Applying theorem 2  
)(yF
Yy . 
Taking ),(ˆ yFz
Yy we obtain Zz ˆ such that 
 .,)ˆ,( Yyzyf   
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Ha Manh Linh was with the Department of 
Mathematics, Vietnam National University-
HoChiMinh City, University of Information 
Technology, Thu Duc district, Saigon, Vietnam e-
mail: linhhm@uit.edu.vn. 
140 SCIENCE & TECHNOLOGY DEVELOPMENT, Vol 20, No.K2- 2017 
Tóm tắt - Các định lý loại Kanaster-
Kuratowski-Mazurkiewicz đóng một vai trò 
quan trọng trong các lĩnh vực giải tích phi 
tuyến, tối ưu và toán ứng dụng. Kể từ khi xuất 
hiện, nhiều nhà nghiên cứu đã nỗ lực phát triển 
các điều kiện đủ cho sự tồn tại các điểm giao (và 
các áp dụng của chúng) trong các không gian 
tổng quát như: Các không gian G-Lồi [21,23], 
không gian L-lồi [12], và FC-không gian [8,9] 
Các áp dụng của các định lý loại Kanaster-
Kuratowski-Mazurkiewicz, đặc biệt là nghiên 
cứu sự tôn tại cho các bất đẳng thức biến phân, 
các bài toán cân bằng và các bài toán tổng quát 
khác đã được thu được bởi nhiều tác giả, xem 
các bài báo gần đây [1, 2, 3, 8, 18, 24, 26] và 
trong các tài liệu tham khảo của các bài báo này. 
Trong bài báo này, chúng tôi đề xuất khái 
niệm ánh xạ L-T-KKM nhằm bao hàm các định 
nghĩa ánh xạ R-KKM [5], ánh xạ L-KKM [11], 
ánh xạ T-KKM ơ18,19], và các khái niệm đã có 
gần đây. Các định lý KKM suy rộng là được 
thiết lập để mở rộng các kết quả trước đó. 
Trong phần áp dụng, chúng tôi phát triển các 
định lý minimax ở dạng tổng quát. Các kết quả 
chúng tôi được chỉ ra là cải tiến hoặc chứa các 
kết quả khác như trường hợp đặc biệt. 
Từ khóa - Các ánh xạ L-T-KKM; Lồi suy rộng; 
Truyền compact nữa liên tục dưới, Các định lý 
minimax, Các điểm yên ngựa vô hạn. 
Các định lý loại Knaster-Kuratowski-
Mazurkiewicz và các áp dụng cho các bất đẳng 
thức minimax 
Hà Mạnh Linh