Generalized Minimax Theorems On Nonconvex Domains

Generalized Minimax Theorems On Nonconvex Domains

Byung-Soo Lee

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,

min_Xsup

Y

f(x; y) = sup

Y

min_Xf(x; y)

for a convex-concave like functionf : X Y !Rinvolving 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 [

x2X

Max_wF(x; X) Max [

x2X

F(x; x) S

and

Max [

x2X

Min_wF(X; x) Min [

x2X

F(x; x) +S

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 ofx₀2X₀ such that Min_wF(x₀; Y) Min[y2Y co(Max_wF(x₀; y)) +S

Mathematics Sub ject Classi…cations: 49J40, 49K35, 65K10.

yDepartment of Mathematics, Kyungsung University, Busan 608-736, Korea

46

on convex domainsXandY by using more generalized Ky Fan’s section theorem. Very recently, Zhang et al. [10] investigated the existences of

z12Max [

x2X

MinwF(x; Y)andz22Min [

y2Y

MaxwF(X; y)

such thatz12z2+S on compact convex subsets ofX Y by using Fan-Browder …xed point theorem.

In 2010, Yang et al. [8] considered

; 6=Min_w [

x2X

Max_wF(x; X) Max [

x2X

F(x; x) +Znint(S);

; 6=Maxw

[

x2X

F(x; x) Min [

x2X

MaxwF(x; X) +Zn( int(S)) and

Maxw

[

x2X

F(x; x) Min [

x2X

MaxwF(x; X) +S

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 Min_lex [

y2Y0

Max_lex [

x2X0

f(x; y) =Max_lex [

x2X0

Min_lex [

y2Y0

f(x; y)

for a vector-valued mappingf :X₀ Y₀!Z;whereX₀andY₀are nonempty compact subsets of metric spaces X and Y, respectively and Z is a nonempty subset of the n-dimensional Euclidean spaceRⁿ with a lexicographic coneC_lex:

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 rea- sonable 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 [

x2X;y2Y

G(x; y) Max [

x2X

MinwG(x; Y) S

and

Max [

x2X

MinwG(x; Y) Min [

x2X;y2Y

G(x; y) +S:

for a vector set-valued mapping Gde…ned on X Y using Fan-KKM theorem on a nonconvex domain X and a convex domain Y of Hausdor¤ topological vector spaces E andF, respectively.

2 Preliminaries

Throughout this paper, let E; F and V be real Hausdor¤ topological vector spaces and X andY be subsets ofE andF, respectively. Assume thatS is a pointed closed convex cone inV with the nonempty interior intS. We recall the following de…nitions and lemmas in [3–9].

DEFINITION 2.1. LetA V be a nonempty set. Then

(i) A pointz 2Ais a minimal point of A ifA\(z S) =fzg and MinA denotes the set of all minimal points of A;

(ii) A pointz2Ais a weakly minimal point ofAifA\(z intS) =; and MinwA denotes the set of all weakly minimal points ofA;

(iii) A point z2A is a maximal point ofAifA\(z+S) =fzg and MaxA denotes the set of all maximal points ofA;

(iv) A pointz2Ais a weakly maximal point ofAifA\(z+intS) =;and MaxwA denotes the set of all weakly maximal points ofA.

REMARK 2.1. It is known that

MinA Min_wA and MaxA Max_wA:

LEMMA 2.1 Let A V be a nonempty compact subset. Then MinA 6= ;, A MinA+S, MaxA6=; andA MaxA S:

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 allx2N(x):

(ii) Gis said to be upper semicontinuous (u.s.c.) at xif for every open setW E withG(x) W;there exists a neighborhood N(x)ofxsuch thatG(x) W for allx2N(x):

(iii) Gis said to be l.s.c. (resp., u.s.c.) onX E if it is l.s.c. (resp., u.s.c.) at every pointx2X:

PROPOSITION 2.1 ([4, 7]). The following statements (i) and (ii) hold:

(i) Gis l.s.c. at x2X if and only if for any netfx g X with x !xand any y2G(x), there existsy 2G(x )such thaty !y:

(ii) If Ghas compact set-values (i.e., G(x) is a compact set for each x2X), then the following (a) and (b) are equivalent

(a) Gis u.s.c. at x2X.

(b) for any net fx g X with x ! x and any y 2 G(x ); there exists y2G(x)and a subnetfy g offy gsuch that y !y:

LEMMA 2.2. LetX0be a nonempty subset ofE, andG:X!2^V be a set-valued mapping. If X is compact and G is upper semicontinuous and compact set-valued, thenG(X) =S

x2XG(x)is compact.

Now we de…ne ag-KKM mappingG:X!2^F:

DEFINITION 2.3. Let X and Y be nonempty subsets ofE and F, respectively, and g : X ! Y be a mapping. A set-valued mapping H : X ! 2^F is said to be a g-KKM mapping ifco(g(A)) S

x2AH(x)for every …nite subsetAofX, whereco(A) is the convexhull of A.

REMARK 2.2. Ifg is the identity whenX =Y, theng-KKM mapping reduces to the usual KKM mapping.

THEOREM 2.1 (Fan-KKM Theorem). LetX andY be nonempty subsets ofEand F, respectively, andg :X !Y be a mapping. If H :X !2^F is ag-KKM mapping with closed set-values and there exists x02X such thatH(x0)is compact, then

\

x2X

H(x)6=;:

DEFINITION 2.4 ([5]). A set-valued mappingG:K!2^Y is said to beS-concave if for all x; y2K andt2[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 spaceX andS is a pointed convex cone in an ordered vector spaceY. GisS-convex if GisS-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 ofF. LetG:X Y !2^V be a upper semicontinuous set-valued mapping with compact set-values andg:X!Y be a mapping.

IfGisS-concave in the second variable, then

Min [

x2X;y2Y

G(x; y) Max [

x2X

Min_wG(x; Y) S

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. Letv2Min[^x2X;y2YG(x; y);thenv2MinG(x; y)for each x2 X and eachy2Y:Since

G(x; y) MinG(x; y) +S by Lemma 2.1, we have

v2 G(x; y) +S forx2X and y2Y: (1) De…ne a set-valued mapping H :X !2^F by

H(x) =fy2Y :G(x; y) v+Sg forx2X;

then H is a g-KKM mapping. If not, there exists a …nite subset fxi ; i= 1; ngofX such that

Xn i=1

tig(xi)2= [n i=1

H(xi)forti2[0;1] (i= 1; n)with Xn i=1

ti= 1:

By the de…nition ofH,

G 0

@x_i; Xn j=1

t_jg(x_j) 1

A6 v+S fori= 1; n: (2)

Hence by the condition thatGisS-concave in the second variable,

G 0

@x_i; Xn j=1

tjg(xj) 1 A

Xn j=1

tjG(xi; g(xj)) +S: (3)

Thus by (2) and (3),

Xn j=1

t_jG(x_i; g(x_j))6 v+S;

which shows that

G(xi; g(xj))6 v+S(i; j= 1; n)

leads a contradiction to (1). On the other hand, H(x) 6= ; for allx 2 X by (1).

Moreover, for all x2X; H(x)is closed. In fact, for any netfy g inH(x) converging toy and anyz 2G(x; y )withz 2v+S;there existz2G(x; y)withz2v+S and a subnet fz g of fz g such thatz ! z by Proposition 2.1. Hencey 2 H(x); which says thatH(x)is closed. Further,F(x)is compact for allx2X;from the fact thatY is compact. Hence by Fan-KMM theorem, we have

\

x2X

H(x)6=;:

Thus there exists y2Y such thatG(x; y) v+S for allx2X;which implies that MinwG(x; y) v+S:

Hence we have

v2MinwG(x; Y) S [

x2X

MinwG(x; Y) S Max [

x2X

MinwG(x; Y) S

by Lemma 2.1.

EXAMPLE 3.1. LetG:X Y !2^V be a set-valued mapping de…ned by G(x; y) = [x+y; x y] [x²; x²+y²]

for (x; y) 2 X Y, where X = [0;¹₂][[²₃;1]; Y = [ 1;0], V =R² and S = R² :=

f(x; y) :x 0; y 0g:ThenGis upper semicontinuous andG(x; y)is compact for all (x; y)2X Y:Moreover,GisS-concave in the second variable. In fact,

G(x; ty₁+ (1 t)y₂) tG(x; y₁) + (1 t)G(x; y₂)

= [x²; x²+ (ty1+ (1 t)y2)²] t[x²; x²+y₁²] + (1 t)[x²; x²+y²₂]

=f0g 0; t(t 1)(y1 y2)² 2S

forx2X andy1; y22Y:On the other hand,

Min [

x2X;y2Y

G(x; y) =f( 1;2)g

and [

x2X

Min_wG(x; Y) f(a; b) : 1 a 0and1 b 2g: Hence

Min [

x2X;y2Y

G(x; y) Max [

x2X

MinwG(x; Y)2 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 ! 2^V be a upper semicontinuous set- valued mapping with compact set-values andg:X !Y be a mapping. If the following conditions hold:

(i) GisS-convex in the second variable, and (ii) for eachx2X,

Max [

x2X

MinwG(x; Y) G(x; Y) +S:

Then

Max [

x2X

Min_wG(x; Y) Min [

x2X;y2Y

G(x; y) +S:

PROOF. Letv2MaxS

x2XMin_wG(x; Y). Then

v2G(x; Y) +S for each x2X: (4) Thus, for eachx2X;there existsy2Y such thatv2G(x; y) +S. De…ne a set-valued mapping

H :X !2^F

byH(x) =fy 2Y :G(x; y) v Sg;then by the above argument, it is easily shown that H(x)6=; for allx2X:Moreover, for all x2X; H(x)is closed. In fact, for any net fy g inH(x) converging toy and anyz 2G(x; y )withz 2v S; there exist z2G(x; y)withz2v Sand a subnetfz goffz gsuch thatz !zby Proposition 2.1. Hence y2H(x);which says thatH(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 ofX such that

Xn i=1

tig(xi)2= [n i=1

H(xi)forti2[0;1] (i= 1; n)with Xn i=1

ti= 1:

By the de…nition ofH,

G 0

@x_i; Xn j=1

tjg(xj) 1

A6 v S fori= 1; n:

SinceGisS-convex in the second variable, Xn

j=1

tjG(xi; g(xj))6 v S;

which shows that

G(x_i; g(x_j))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 y2Y such thatG(x; y) v S for allx2X;which implies that v2G(x; y) +S [

x2X;y2Y

G(x; y) +S Min [

x2X;y2Y

G(x; y) +S

by Lemma 2.1.

REMARK 3.1. The same results can be obtained for a set-valued mapping F : X Y !2^Ron a nonconvex domain X and a convex domainY ofR, as corollaries.

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 domainsX X;

Max [

x2X

MinwG(x; X) Min [

x2X

G(x; x) +S

and

Min [

x2X

G(x; x) Max [

x2X

MinwG(x; X) S:

Acknowledgements. The author thanks the referee for his valuable comments and suggestions.

References

[1] S. S. Chang, G. M. Lee and B. S. Lee, Minimax inequalities for vector-valued mappings on W-spaces. J. Math. Anal. Appl., 198(1996), 371–380.

[2] K. Fan, Minimax theorems, National Academy of Sciences, Washington, DC, Pro- ceedings USA 39(1953), 42–47.

[3] A. P. Farajzadeh, B. Hatamnejad and B. S. Lee, On generalized Ky Fan’s minimax theorems, to appear.

[4] F. Ferro, A minimax theorem for vector-valued functions, J. Optim. Theory Appl., 60(1989), 19–31.

[5] S. J. Li, G. Y. Chen and G. M. Lee, Minimax theorems for set-valued mappings, J. Optim. Theory Appl., 106(2000), 183–200.

[6] X. B. Li, S. J. Li and Z. M. Fang, A minimax theorem for vector-valued functions in lexicographic order, Nonlinear Anal., 73(2000), 1101–1108.

[7] Y. D. Xu and S. J. Li, On the lower semicontinuity of the solution mappings to a parametric generalized strong vector equilibrium problem, 17(2013), 341–353.

[8] M. G. Yang, J. P. Xu, N. J. Huang and S. J. Yu, Minimax theorems for vector- valued mappings in abstract convex spaces, Taiwanese J. Math., 14(2010), 719–

732.

[9] Y. Zhang and S. J. Li, Ky Fan minimax inequalities for set-valued mappings., Fixed Point Theory Appl., 2012, 2012:64, 12 pp.

[10] Y. Zhang, S. J. Li and S. K. Zhu, Minimax problems for set-valued mappings, Numer. Funct. Anal. and Optim., 33(2012), 239–253.