On Some Generalized Ky Fan Minimax Inequalities

Volume 2009, Article ID 194671,9pages doi:10.1155/2009/194671

Research Article

On Some Generalized Ky Fan Minimax Inequalities

Xianqiang Luo

Department of Mathematics, Wuyi University, Jangmen, 529020, China

Correspondence should be addressed to Xianqiang Luo,[email protected] Received 31 October 2008; Revised 26 March 2009; Accepted 21 April 2009 Recommended by Naseer Shahzad

Some generalized Ky Fan minimax inequalities for vector-valued mappings are established by applying the classical Browder fixed point theorem and the Kakutani-Fan-Glicksberg fixed point theorem.

Copyrightq2009 Xianqiang Luo. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

1. Introduction

It is well known that Ky Fan minimax inequality1plays a very important role in various fields of mathematics, such as variational inequality, game theory, mathematical economics, fixed point theory, control theory. Many authors have got some interesting achievements in generalization of the inequality in various ways. For example, Ferro2obtained a minimax inequality by a separation theorem of convex sets. Tanaka3introduced some quasiconvex vector-valued mappings to discuss minimax inequality. Li and Wang4obtained a minimax inequality by using some scalarization functions. Tan 5 obtained a minimax inequality by the generalized G-KKM mapping. Verma 6 obtained a minimax inequality by an R- KKM mapping. Li and Chen7 obtained a set-valued minimax inequality by a nonlinear separation functionξ_k,a. Ding8,9obtained a minimax inequality by a generalized R-KKM mapping. Some other results can be found in10–16.

In this paper, we will establish some generalized Ky Fan minimax inequalities forvector-valued mappings by the classical Browder fixed point theorem and the Kakutani- Fan-Glicksberg fixed point theorem.

2. Preliminaries

Now, we recall some definitions and preliminaries needed. LetX andY be two nonempty sets, and letT : X → 2^Y be a nonempty set-valued mapping, x ∈ T⁻¹y if and only if y∈Tx,TX

x∈XTx. Throughout this paper, assume that every space is Hausdorff.

Definition 2.1see10. For topological spacesXandY, a mappingT :X → 2^Y is said to be iupper semicontinuoususc, if for each open setB⊂ Y, the setT⁻¹B {x∈X :

Tx⊂B}is open subset ofX;

iilower semicontinuouslsc, if for each closed setB ⊂Y, the setT⁻¹B {x ∈X : Tx⊂B}is closed subset ofX;

iiicontinuous, if it is bothuscandlsc;

ivcompact-valued, ifTxis compact inY for anyx∈X.

Definition 2.2see11. LetZbe a topological vector space andC⊂Zbe a pointed convex cone with a nonempty interior intC, and letBbe a nonempty subset ofZ. A pointz∈Bis said to be

ia minimal point ofBifB∩z−C {z};

iia weakly minimal point ofBifB∩z−intC ∅;

iiia maximal point ofBifB∩zC {z};

iva weakly maximal point ofBifB∩zintC ∅.

By minB, min_wB, maxB, max_wB, we denote, respectively, the set of all minimal points, the set of all weakly minimal points, the set of all maximal points, the set of all weakly maximal points ofB.

Lemma 2.3see11. LetBbe a nonempty compact subset of a topological vector spaceZwith a closed pointed convex coneC. Then

iminB /∅;

iiB⊂minBC⊂minwBC;

iiimaxB /∅;

ivB⊂maxB−C⊂max_wB−C.

Lemma 2.4see11. LetEandZbe two topological vector spaces,∅/X⊂E, and letF:X → 2^Z be a set-valued mapping. IfXis compact, andF is upper semicontinuous and compact-valued, then

FX

x∈XFxis compact set.

Lemma 2.5see2, Theorem 3.1. LetEbe a topological vector space, letZbe a topological vector space with a closed pointed convex coneC, intC /∅, letXandY be two nonempty compact subsets of E, and letf : X ×Y → Z be a continuous mapping. Then both F₁ : X → 2^Z defined by F₁x max_wfx, YandF₂:X → 2^Zdefined byF₂x min_wfx, Yare upper semicontinuous and compact-valued.

Definition 2.6. LetZbe a topological vector space and letCbe a closed pointed convex cone inZ, intC /∅. Givene∈intCanda∈Z, the functionh_e,aandg_e,a:Z → Rare, respectively, defined byhe,az min{t∈R:z∈ate−C}, andge,az max{t∈R:z∈ateC}.

We quote some of their properties as followssee12:

ihe,az< r⇔z∈are−intC;ge,az> r ⇔z∈areintC;

iih_e,az≤r⇔z∈are−C;g_e,az≥r⇔z∈areC;

iiih_e,az> r⇔z /∈are−C;g_e,az< r⇔z /∈areC;

ivhe,az≥r⇔z /∈are−intC;ge,az≤r⇔z /∈areintC;

vh_e,ais a continuous and convex function;g_e,ais a continuous and concave function;

vihe,aandge,aare strictly monotonically increasingmonotonically increasing, that is, ifz₁−z₂∈intC⇒fz1> fz2 z1−z₂∈C⇒fz1≥fz2, wherefdenotes h_e,aorg_e,a.

Definition 2.7 see3. LetE be a topological vector space, let X be a nonempty convex subsets ofE, and letZbe a topological vector space with a pointed convex coneC, intC /∅.

A vector-valued mappingf :X → Zis said to be

iC-quasiconcave if for eachz∈Z, the set{x∈X:fx∈zC}is convex;

iiproperlyC-quasiconcave if for anyx, y∈Xandt∈0,1, eitherftx 1−ty∈ fx Corftx 1−ty∈fy C.

The following two propositions are very important in provingProposition 2.10.

Proposition 2.8see4. LetZbe a topological vector space and letCbe a closed pointed convex cone inZ, intC /∅,f :X → Z:

ifisC-quasiconcave if and only if for alle∈intCand for alla∈Z,g_e,afis quasiconcave;

iiiffis properlyC-quasiconcave.

Thenh_e,afis quasiconcave.

Proposition 2.9. LetEbe a topological vector space and letX be a nonempty convex subset ofE, f:X → R. Then the following two statements are equivalent:

ifor anyr∈R,{x∈X:fx≥r}is convex;

iifor anyt∈R,{x∈X:fx> t}is convex.

Proof. i⇒iiFor anyt ∈ R,x1, x2 ∈ {x ∈ X : fx > t}. Letr min{fx1, fx2} > t, then x1, x2 ∈ {x ∈ X : fx ≥ r}. By i, we have{x ∈ X : fx ≥ r} is convex, then co{x1, x₂}⊂ {x∈X:fx≥r > t}. Thus, co{x1, x₂}⊂ {x∈X:fx> t}is convex.

ii⇒iFor anyr ∈R,x1, x2 ∈ {x∈X :fx≥r}, then for allε >0,x1, x2 ∈ {x∈X : fx> r−ε}. Byii, we have{x∈X :fx> r−ε}is convex, that is, co{x1, x2}⊂ {x∈X: fx> r−ε}. Sinceεis arbitrary, then co{x1, x₂}⊂ {x∈X :fx≥r}is convex.

Proposition 2.10. LetEbe a topological vector space, letZbe a topological vector space with a closed pointed convex coneC, intC /∅, and letXbe a nonempty compact convex subset ofE,f :X → Z be a vector mapping. Then the following two statements are equivalent:

ifor anyz∈Z,{x∈X:fx∈zC}is convex, that is,fxisC-quasiconcave;

iifor anyz∈Z,{x∈X:fx∈zintC}is convex.

Proof. i⇒iifor allz ∈ Zand for alle ∈ intC, leta z−e. ByProposition 2.8, we have g_e,afxis quasiconcave, that is, for anyr ∈ R,{x ∈ X : g_e,afx ≥ r}is convex, then byProposition 2.9, we have for anyt ∈ R,{x ∈ X : g_e,afx > t}is convex. Thus, {x ∈ X :ge,afx >1}is convex. Therefore, we have{x∈X :fx∈zintC}is convex since {x∈X:fx∈zintC}{x∈X:g_e,afx>1}by propertyiofg_e,a.

ii⇒iBy Proposition 2.8, we need only prove for alle ∈ intCand for all a ∈ Z, g_e,afxis quasiconcave, that is, for anyr∈R,{x∈X :g_e,afx≥r}is convex.

For anyt∈R, letzate. By propertyiofg_e,a, we have x∈X :fx∈zintC

x∈X:g_e,a fx

> t

. 2.1

Thus, for anyt∈R,{x∈X :g_e,afx> t}is convex since{x∈X :fx∈zintC}is convex byii. Therefore, byProposition 2.9, we have for anyr∈R,{x∈X :ge,afx≥r} is convex.

3. Generalized Ky Fan Minimax Inequalities

In this section, we will establish some generalized Ky Fan minimax inequalities and a corollary by Propositions1.1,1.3and Lemmas3.1,3.2.

Lemma 3.1see13. LetEbe a topological vector space, letX ⊂ Ebe a nonempty compact and convex set, and letT :X → 2^X, such that

ifor eachx∈X,Txis nonempty and convex;

iifor eachx∈X,T⁻¹xis open.

ThenT has a fixed point.

Lemma 3.2see11, Kakutani-Fan-Glicksberg fixed point theorem. LetEbe a locally convex topological vector space and letX⊂Ebe a nonempty compact and convex set. IfT :X → 2^Xis upper semicontinuous, and for anyx∈X,Txis a nonempty, closed and convex subset, thenThas a fixed point.

Theorem 3.3. LetEbe a topological vector space, letZbe a topological vector space with a closed pointed convex coneC, intC /∅, letXbe a nonempty compact convex subset ofE, and letf :X×X → Zbe a continuous mapping, such that

ifor allz∈maxw_t∈Xft, t, for anyx∈X,{y∈X:fx, y∈zintC}is convex.

Then

maxw

t∈X ft, t⊂min

x∈X maxw y∈X f

x, y

Z\−intC. 3.1

Proof. Letz∈maxw_t∈Xft, t, then by the definition of the weakly maximal point, we have

for any x∈X, fx, x/∈zintC. ∗

For eachx∈X, let

Tx

y∈X:f x, y

∈zintC

. 3.2

Now, we prove that there existsx₀∈X, such thatTx0 ∅.

Supposed for eachx ∈ X, Tx/∅, then by conditioni, we have for each x ∈ X, Txis nonempty and convex. In addition, we have for eachy∈X,T⁻¹yis open sincefis continuous.

Thus, byLemma 3.1, there existsx∈X, such thatx∈Tx, that is,fx, x∈zintC, which contradicts∗.

Therefore, there existsx₀ ∈X, such thatTx0 ∅, that is, for anyy∈X, z /∈f

x₀, y

−intC. 3.3

Since max_wfx0, X/∅, thenz∈max_wfx0, XZ\−intC⊂

x∈Xmax_wfx, XZ\

−intC min_x∈Xmaxw_y∈Xfx, y Z\−intC because ofZ\−intC Z\−intC C, andLemma 2.3.

Remark 3.4. ByProposition 2.10, in the aboveTheorem 3.3, the conditionican be replaced by “for eachx∈X,fx, yisC-quasiconcave iny”.

Theorem 3.5. LetEbe a topological vector space, letZbe a topological vector space with a closed convex pointed coneC, intC /∅, letXbe a nonempty compact convex subset ofE, and letf :X×X → Zbe a continuous mapping, such that

ifor eachx∈X,fx, yis properlyC-quasiconcave iny.

Then

min_w

x∈X max_w

y∈X f x, y

⊂max

t∈X ft, t Z\intC. 3.4

Proof. SinceXis compact, andf is continuous, then byLemma 2.3, we have for anyx ∈X, max_wfx, X/∅andminw_x∈Xmaxw_y∈Xfx, y/∅.

For any x ∈ X, there exists yx ∈ X, such that fx, yx ∈ maxwfx, X. Let z ∈ minw_x∈Xmaxw_y∈Xfx, y, by the definition of the weakly minimal point, we have fx, yx/∈z−intC. Thus, for eachx∈X, let

Tx

y∈X:f x, y

/∈z−intC

/∅. 3.5

Now, we prove that there existsx₀∈X, such thatx₀∈Tx0.

For alle∈intC, letaz−e∈Z, the functionh_e,a:Z → Ris defined by

h_e,az min{t∈R:z∈ate−C}. 3.6

Let gx, y h_e,afx, y, then gx, y is continuous since both h_e,a and f are continuous. By propertyivofhe,a, we have

Tx

y∈X :f x, y

/∈z−intC

y∈X:g x, y

≥1

. ∗∗

For anyn∈N, letTnx {y∈X:gx, y>1−1/n}, then it satisfies the all conditions ofLemma 3.1.

In fact, firstly, byTx⊂Tnx, we haveTnx/∅, and for eachy ∈X,T_n⁻¹yis open sincegx, yis continuous. Secondly, by conditioniandProposition 2.8, we havegx, y is quasiconcave in y, that is, for any r ∈ R, {y ∈ X : gx, y ≥ r} is convex. Thus, by Proposition 2.9,Tnx {y∈X :gx, y>1−1/n}is convex.

ByLemma 3.1, there existsxn∈X, such thatxn∈Tnx, that is, gxn, xn>1− 1

n. 3.7

SinceX is compact, then{xn}has a subnet converging tox0 ∈ X. Letn → ∞in the above expression, together with∗∗, yields

gx0, x₀≥1⇐⇒x₀∈Tx0. 3.8

Thus,

z /∈fx0, x₀ intC. 3.9

Therefore, for allz∈minw_x∈Xmaxw_y∈Xfx, y, we have z∈fx0, x₀ Z\intC⊂max

t∈X ft, t−CZ\intCmax

t∈X ft, t Z\intC. 3.10 Theorem 3.6. LetEbe a locally convex topological vector space, letZbe a topological vector space with a closed convex pointed coneC, intC /∅, letX be a nonempty compact and convex subset ofE, letf:X×X → Zbe a continuous mapping, and letz0∈Zsuch that

ifor eachx∈X,Tx {y∈X :fx, y∈z0C}is nonempty convex.

Then

z₀∈max

x∈X fx, x−C. 3.11

Proof. For eachx∈X, we defineT :X → 2^Xby

Tx

yx∈X:f x, yx

∈z0C

. 3.12

Now, we prove thatT has a fixed point.

1By the conditioni, we have for eachx∈X,Tx/∅is closed and convex sincef is continuous andCis closed.

2T is upper semicontinuous mapping.

For eachx∈X,Txis compact sinceX is compact andTx⊂ Xis closed. We only need to proveThas a closed graph.

In fact, Letx, y∈GrT, and a netxα, y_αin GrTconverging tox, y. Sincefis continuous andz0Cis closed, then

f x_α, y_α

−→f x, y

∈z₀C. 3.13

Thus,

y∈T x

⇒ x, y

∈GrT. 3.14

Therefore, byLemma 3.2KFG fixed point theorem,Thas a fixed pointx3such that

x3∈Tx3. 3.15

Then

z0∈fx3, x3−C⊂

x∈X

fx, x−C⊂max

x∈X fx, x−C. 3.16

Remark 3.7. If for eachx ∈X,fx, yisC-quasiconcave inyandz₀ ⊂ fx, X−C, then the conditioniholds. Thus, we can obtain the following corollary.

Corollary 3.8. LetEbe a locally convex topological vector space, letZbe a topological vector space with a closed convex pointed coneC, intC /∅, letX be a nonempty compact and convex subset ofE, and letf:X×X → Zbe a continuous mapping such that

ifx, yisC-quasiconcave inyfor eachx∈X;

ii minw_x∈Xmaxw_y∈Xfx, y⊂fx, X−Cfor eachx∈X.

Then

minw x∈X maxw

y∈X f x, y

⊂max

x∈Xfx, x−C. 3.17

Proof. Let z₀ ∈ minw_x∈Xmaxw_y∈Xfx, y, and for each x ∈ X, let Tx {yx ∈ X : fx, yx∈z₀C}. By conditionii,Txis nonempty. And by conditioni,Txis convex.

Thus, byTheorem 3.6, the conclusion holds.

Remark 3.9. By Definition 2.7, the condition i can be replaced by “ifx, y is properly C-quasiconcave inyfor eachx∈X.”

Example 3.10. LetER,X 0,1,ZR²,C{x, y∈R×R:|x| ≤y}. Given a fixedx∈X, for eachy∈X, we definef:X×X → Zby

f x, y

⎧⎨

⎩ x, y

, ify≤x y, y

, ify≥x. 3.18

InFigure 1, the red line denotes the graph offx, yfor eachx∈X.

y

C

1,1 X 1 X

Figure 1: The function’s graph.

Now we provefsatisfies the conditions ofCorollary 3.8:

ifis a continuous.

LetB ⊂ Zis closed, let xα, yα ⊂ f⁻¹B {x, y : fx, y ∈ B}, and xα, yα → x, y. Then by the definition off, we have

f xα, yα

⎧⎪

⎨

⎪⎩ xα, yα

, ifyα≤xα

yα, yα

, ifyα≥xα.

3.19

Thus there exists a subnet yet denoted by xα, yα, and yα ≤ xα, such that fxα, yα

xα, yα → x, y ∈ B since B is closed. Hence,y ≤ x, and fx, y x, y ∈ B ⇒ x, y∈f⁻¹B. Therefore,f⁻¹Bis closed.

iiFromFigure 1, we can check thatfx, yis properlyC-quasiconcave inyfor each x∈X.

iiiFrom Figure 1, we can check that minw_x∈Xmaxw_y∈Xfx, y {x, x : x ∈ 0,1} ⊂ 1,1−C ⊂ maxwfx, X {y, y : y ∈ x,1} −Cfor each x ∈ X.

Thus,minw_x∈Xmaxw_y∈Xfx, y⊂max_wfx, X−Cfor eachx∈X.

Finally, fromFigure 1, we can check thatminw_x∈Xmaxw_y∈Xfx, y {x, x :x ∈ 0,1} ⊂1,1−Cmax_x∈Xfx, x−C, that is,Corollary 3.8holds.

Acknowledgments

The author gratefully acknowledges the referee for his/her ardent corrections and valuable suggestions, and is thankful to Professor Junyi Fu and Professor Xunhua Gong for their help.

This work was supported by the Young Foundation of Wuyi University.

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