Applied Mathematics E-Notes, 15(2015), 46-53 c
Available free at mirror sites of http://www.math.nthu.edu.tw/ amen/
ISSN 1607-2510
Generalized Minimax Theorems On Nonconvex
Domains
Byung-Soo Leey
Received 25 September 2014
Abstract
In this paper, the author considers generalized minimax theorems for vector
set-valued mappings using Fan-KKM theorem on nonconvex domains of Hausdor¤
topological vector spaces.
1
Introduction
In 1953, K. Fan [2] considered the following minimax equality which is called Ky Fan
minimax inequality,
minX sup f (x; y) = sup minX f (x; y)
Y
Y
for a convex-concave like function f : X Y ! R involving no linear structures. Since
then, there have been many generalized results on Ky Fan minimax theorems due to
the important roles of the theorems to many …elds, such as variational inequalities,
game theory, mathematical economics, control theory, equilibrium problem and …xed
point theory. In generalizing Ky Fan minimax theorems, authors have used famous
known theorems. For examples, Zhang and Li [9] considered
[
[
Min
Maxw F (x; X) Max
F (x; x) S
x2X
and
Max
[
x2X
Minw F (X; x)
x2X
Min
[
F (x; x) + S
x2X
for a set-valued mapping F de…ned on X X using Kakutani-Fan-Glicksberg …xed
point theorem, where X is a convex domain and S is a pointed closed convex cone.
Chang et al. [1] obtained a Ky Fan minimax inequality for vector-valued mappings
on W -spaces by applying a generalized section theorem and a generalized …xed point
theorem.
Li et al. [5] considered the existence of x0 2 X0 such that
Minw F (x0 ; Y )
Min [y2Y co(Maxw F (x0 ; y)) + S
Mathematics Subject Classi…cations: 49J40, 49K35, 65K10.
of Mathematics, Kyungsung University, Busan 608-736, Korea
y Department
46
B. S. Lee
47
on convex domains X and Y by using more generalized Ky Fan’s section theorem. Very
recently, Zhang et al. [10] investigated the existences of
z1 2 Max
[
x2X
Minw F (x; Y ) and z2 2 Min
[
such that z1 2 z2 + S on compact convex subsets of X
point theorem.
In 2010, Yang et al. [8] considered
[
;=
6 Minw
;=
6 Maxw
Maxw F (x; X)
Max
x2X
[
x2X
F (x; x)
[
Min
x2X
x2X
and
[
Maxw
[
F (x; x)
Min
x2X
Maxw F (X; y)
y2Y
Y by using Fan-Browder …xed
F (x; x) + Z n int(S);
Maxw F (x; X) + Z n ( int(S))
[
Maxw F (x; X) + S
x2X
for a vector-valued mapping F : X X ! Z using Fan-Browder type …xed point
theorem and maximal element theorem in abstract convex spaces.
In 2010, Li et al. [6] considered
Minlex
[
Maxlex
y2Y0
[
f (x; y) = Maxlex
x2X0
[
Minlex
x2X0
[
f (x; y)
y2Y0
for a vector-valued mapping f : X0 Y0 ! Z; where X0 and Y0 are nonempty compact
subsets of metric spaces X and Y , respectively and Z is a nonempty subset of the
n-dimensional Euclidean space Rn with a lexicographic cone Clex :
In the previous cited works, almost all the results on generalized Ky Fan minimax
theorem were considered on convex domains. Hence it would be desirable and reasonable to consider generalized Ky Fan minimax theorems on nonconvex domains for
many further applications.
In this paper, the author considers the following generalized minimax theorems
Min
[
G(x; y)
x2X;y2Y
and
Max
[
x2X
Minw G(x; Y )
Max
[
Minw G(x; Y )
S
x2X
Min
[
G(x; y) + S:
x2X;y2Y
for a vector set-valued mapping G de…ned on X Y using Fan-KKM theorem on a
nonconvex domain X and a convex domain Y of Hausdor¤ topological vector spaces
E and F , respectively.
48
2
Generalized Minimax Theorems
Preliminaries
Throughout this paper, let E; F and V be real Hausdor¤ topological vector spaces
and X and Y be subsets of E and F , respectively. Assume that S is a pointed closed
convex cone in V with the nonempty interior intS. We recall the following de…nitions
and lemmas in [3–9].
DEFINITION 2.1. Let A
V be a nonempty set. Then
(i) A point z 2 A is a minimal point of A if A \ (z
the set of all minimal points of A;
S) = fzg and MinA denotes
(ii) A point z 2 A is a weakly minimal point of A if A \ (z
denotes the set of all weakly minimal points of A;
intS) = ; and Minw A
(iii) A point z 2 A is a maximal point of A if A \ (z + S) = fzg and MaxA denotes
the set of all maximal points of A;
(iv) A point z 2 A is a weakly maximal point of A if A \ (z + intS) = ; and Maxw A
denotes the set of all weakly maximal points of A.
REMARK 2.1. It is known that
MinA
A
Minw A and MaxA
Maxw A:
LEMMA 2.1 Let A
V be a nonempty compact subset.
MinA + S, MaxA 6= ; and A MaxA S:
Then MinA 6= ;,
DEFINITION 2.2.
(i) G is said to be lower semicontinuous (l.s.c.) at a point x if for any open set
W
E with W \ G(x) 6= ;; there exists a neighborhood N (x) of x such that
G(x) \ W 6= ; for all x 2 N (x):
(ii) G is said to be upper semicontinuous (u.s.c.) at x if for every open set W
E
with G(x) W; there exists a neighborhood N (x) of x such that G(x) W for
all x 2 N (x):
(iii) G is said to be l.s.c. (resp., u.s.c.) on X
point x 2 X:
E if it is l.s.c. (resp., u.s.c.) at every
PROPOSITION 2.1 ([4, 7]). The following statements (i) and (ii) hold:
(i) G is l.s.c. at x 2 X if and only if for any net fx g
y 2 G(x), there exists y 2 G(x ) such that y ! y:
X with x ! x and any
(ii) If G has compact set-values (i.e., G(x) is a compact set for each x 2 X), then
the following (a) and (b) are equivalent
B. S. Lee
49
(a) G is u.s.c. at x 2 X.
(b) for any net fx g
X with x ! x and any y 2 G(x ); there exists
y 2 G(x) and a subnet fy g of fy g such that y ! y:
LEMMA 2.2. Let X0 be a nonempty subset of E, and G : X ! 2V be a set-valued
mapping. If X
S is compact and G is upper semicontinuous and compact set-valued,
then G(X) = x2X G(x) is compact.
Now we de…ne a g-KKM mapping G : X ! 2F :
DEFINITION 2.3. Let X and Y be nonempty subsets of E and F , respectively,
and g : X ! Y be a mapping.S A set-valued mapping H : X ! 2F is said to be a
g-KKM mapping if co(g(A))
x2A H(x) for every …nite subset A of X, where co(A)
is the convexhull of A.
REMARK 2.2. If g is the identity when X = Y , then g-KKM mapping reduces to
the usual KKM mapping.
THEOREM 2.1 (Fan-KKM Theorem). Let X and Y be nonempty subsets of E and
F , respectively, and g : X ! Y be a mapping. If H : X ! 2F is a g-KKM mapping
with closed set-values and there exists x0 2 X such that H(x0 ) is compact, then
\
H(x) 6= ;:
x2X
DEFINITION 2.4 ([5]). A set-valued mapping G : K ! 2Y is said to be S-concave
if for all x; y 2 K and t 2 [0; 1]; we have
G((1
t)x + ty)
(1
t)G(x) + tG(y) + S;
where K is a convex subset of a vector space X and S is a pointed convex cone in an
ordered vector space Y . G is S-convex if G is S-concave.
3
Main Result
In this section, we show two generalized minimax theorems on nonconvex domains
using Fan-KKM theorem.
THEOREM 3.1. Let X be a nonempty compact subset of E and Y a nonempty
compact convex subset of F . Let G : X Y ! 2V be a upper semicontinuous set-valued
mapping with compact set-values and g : X ! Y be a mapping.
If G is S-concave in the second variable, then
[
[
Min
G(x; y) Max
Minw G(x; Y ) S
x2X;y2Y
x2X
50
Generalized Minimax Theorems
PROOF. It is easily shown that MinG(x; y) 6= ; and Min [x2X;y2Y G(x; y) 6= ; by
Lemma 2.1 and Lemma 2.2. Let v 2 Min[x2X;y2Y G(x; y); then v 2 MinG(x; y) for each x 2
X and each y 2 Y: Since
G(x; y) MinG(x; y) + S
by Lemma 2.1, we have
v2
G(x; y) + S for x 2 X and y 2 Y:
(1)
De…ne a set-valued mapping H : X ! 2F by
H(x) = fy 2 Y : G(x; y)
v + Sg for x 2 X;
then H is a g-KKM mapping. If not, there exists a …nite subset fxi ; i = 1; ng of X
such that
n
X
i=1
ti g(xi ) 2
=
n
[
i=1
H(xi ) for ti 2 [0; 1] (i = 1; n) with
n
X
ti = 1:
i=1
By the de…nition of H,
0
G @xi ;
n
X
j=1
1
tj g(xj )A 6
v + S for i = 1; n:
Hence by the condition that G is S-concave in the second variable,
0
1
n
n
X
X
G @ xi ;
tj g(xj )A
tj G(xi ; g(xj )) + S:
j=1
(2)
(3)
j=1
Thus by (2) and (3),
n
X
j=1
which shows that
tj G(xi ; g(xj )) 6
G(xi ; g(xj )) 6
v + S;
v + S (i; j = 1; n)
leads a contradiction to (1). On the other hand, H(x) 6= ; for all x 2 X by (1).
Moreover, for all x 2 X; H(x) is closed. In fact, for any net fy g in H(x) converging
to y and any z 2 G(x; y ) with z 2 v + S; there exist z 2 G(x; y) with z 2 v + S and
a subnet fz g of fz g such that z ! z by Proposition 2.1. Hence y 2 H(x); which
says that H(x) is closed. Further, F (x) is compact for all x 2 X; from the fact that Y
is compact. Hence by Fan-KMM theorem, we have
\
H(x) 6= ;:
x2X
Thus there exists y 2 Y such that G(x; y)
Minw G(x; y)
v + S for all x 2 X; which implies that
v + S:
B. S. Lee
51
Hence we have
v 2 Minw G(x; Y )
[
S
Minw G(x; Y )
S
Max
x2X
[
Minw G(x; Y )
S
x2X
by Lemma 2.1.
Y ! 2V be a set-valued mapping de…ned by
EXAMPLE 3.1. Let G : X
G(x; y) = [x + y; x
[x2 ; x2 + y 2 ]
y]
for (x; y) 2 X Y , where X = [0; 21 ] [ [ 23 ; 1]; Y = [ 1; 0], V = R2 and S = R2 :=
f(x; y) : x 0; y 0g: Then G is upper semicontinuous and G(x; y) is compact for all
(x; y) 2 X Y: Moreover, G is S-concave in the second variable. In fact,
G(x; ty1 + (1
2
t)y2 )
2
2
= [x ; x + (ty1 + (1
= f0g
0; t(t
tG(x; y1 ) + (1
2
t)G(x; y2 )
t[x ; x + y12 ] + (1
t)y2 ) ]
2
t)[x2 ; x2 + y22 ]
y2 )2
1)(y1
2S
for x 2 X and y1 ; y2 2 Y: On the other hand,
[
Min
G(x; y) = f( 1; 2)g
x2X;y2Y
and
[
Minw G(x; Y )
f(a; b) :
x2X
Hence
Min
[
G(x; y)
Max
x2X;y2Y
1
a
[
Minw G(x; Y ) 2
x2X
0 and 1
b
2g:
S:
THEOREM 3.2. Let X be a nonempty compact subset of E and Y a nonempty
compact convex subset of F . Let G : X Y ! 2V be a upper semicontinuous setvalued mapping with compact set-values and g : X ! Y be a mapping. If the following
conditions hold:
(i) G is S-convex in the second variable, and
(ii) for each x 2 X,
Max
[
Minw G(x; Y )
G(x; Y ) + S:
x2X
Then
Max
[
x2X
Minw G(x; Y )
Min
[
x2X;y2Y
G(x; y) + S:
52
Generalized Minimax Theorems
PROOF. Let v 2Max
S
x2X Minw G(x; Y
). Then
v 2 G(x; Y ) + S for each x 2 X:
(4)
Thus, for each x 2 X; there exists y 2 Y such that v 2 G(x; y) + S. De…ne a set-valued
mapping
H : X ! 2F
by H(x) = fy 2 Y : G(x; y) v Sg; then by the above argument, it is easily shown
that H(x) 6= ; for all x 2 X: Moreover, for all x 2 X; H(x) is closed. In fact, for any
net fy g in H(x) converging to y and any z 2 G(x; y ) with z 2 v S; there exist
z 2 G(x; y) with z 2 v S and a subnet fz g of fz g such that z ! z by Proposition
2.1. Hence y 2 H(x); which says that H(x) is closed and compact.
Now we show that H is a g-KKM mapping. If not, there exists a …nite subset
fxi ; i = 1; ng of X such that
n
X
i=1
ti g(xi ) 2
=
n
[
i=1
H(xi ) for ti 2 [0; 1] (i = 1; n) with
n
X
ti = 1:
i=1
By the de…nition of H,
0
G @xi ;
n
X
j=1
1
tj g(xj )A 6
v
S for i = 1; n:
Since G is S-convex in the second variable,
n
X
j=1
tj G(xi ; g(xj )) 6
v
S;
which shows that
G(xi ; g(xj )) 6
v
S (i; j = 1; n);
leading a contradiction to (4). Hence by Fan-KMM theorem, we have
\
x2X
H(x) 6= ;:
Thus there exists y 2 Y such that G(x; y)
v 2 G(x; y) + S
[
x2X;y2Y
v
G(x; y) + S
S for all x 2 X; which implies that
Min
[
G(x; y) + S
x2X;y2Y
by Lemma 2.1.
X
REMARK 3.1. The same results can be obtained for a set-valued mapping F :
Y ! 2R on a nonconvex domain X and a convex domain Y of R, as corollaries.
B. S. Lee
53
REMARK 3.2. Putting X = Y and g = I in Theorem 3.1 and Theorem 3.2, we
obtain the following results in [3] on convex domains X X;
[
[
Max
Minw G(x; X) Min
G(x; x) + S
x2X
and
Min
[
x2X
G(x; x)
x2X
Max
[
Minw G(x; X)
S:
x2X
Acknowledgements. The author thanks the referee for his valuable comments
and suggestions.
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