Lin, Park. Yu, 1999

8/13/2019 Lin, Park. Yu, 1999

http://slidepdf.com/reader/full/lin-park-yu-1999 1/16

Ž .Journal of Mathematical Analysis and Applications 233, 581596 1999

Article ID jmaa .1999.6311, a vailable online at http: www.idealibrary.com on

R emarks on Fixed P oints, M aximal E lements, a nd

E quilibria of G eneralized G ames

Lai-Jiu Lin

Department of Mathematics, National Changhua Uni ersity of Education, Changhua,

Taiwan, Republic of China 50058

E -mail: maljlin@ mat h.ncue.edu.tw

Sehie Pa rk

Department of Mathematics, Seoul National Uni ersity, Seoul,

Republic of Korea 151-742

E -mail: shpark@ math.snu.edu.kr

and

Zenn-Tsuen Yu

Department of Electrical Engineering, Nan-Kai College of Technology and Commerce,

Nantour, Taiwan, Republic of China 542

Submitted by Arrigo Cellina

R eceived Februa ry 10, 1998

1. INTRODUCTION

Recently, Mehta et al. 12 gave fixed points theorems for -condensingmultimaps in locally convex topological vector spaces, and applied them toexistence theorems of maximal elements of -condensing maps which areeither upper semicontinuous or L -majorized. Moreover they deducedC

existence theorems of equilibria in locally convex topological vector spacesfor the one-person game and generalized game which may have anyŽ .countable or uncountable set of players.

* This research was supported by Mathematical Center of the National Science Council,Republic of China, while the second author was visiting at National Changhua University ofEducation. The second author was supported in part by the Non-directed Research Fund,

Korea Research Foundation, 1997.

5810022-247X 99 $30.00

Copyright 1999 by Academic Press

All rights of reproduction in any form reserved.

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LIN, PARK, AND Y U582

In this paper, using a new fixed point theorem for acyclic maps due to P ark 13 , we obta in collectively fixed point theorems for noncompact maps

defined on a noncompact convex subset of a topological vector space in a

larger class than locally convex ones. Our new results are applied toexistence theorems of maximal elements of -condensing maps and equi-libria for one-person or generalized games. We deal with multimapsdefined on an admissible convex and non-compact subset of a topologicalvector space, not-necessarily locally convex topological vector space. Con-sequently our results generalize and improve the corresponding results

given by Yannelis et al. 21 , Mehta et al. 12 , a nd many other results.This paper is organized as follows:

In Sect. 2, we give some necessary terminology and facts. Our main toolis a particular form of the fixed point theorem due to P ark 13, Theorem

1 .Section 3 deals with collectively fixed point theorems for later use. In

Sect. 4, some of the fixed point results in the previous sections are restatedin the form of existence theorems for maximal elements.

Section 5 deals with various existence theorems for equilibria of general-

ized games. Those results are given in the forms of full generality.

2. PRELIMINARIES

Through this paper, tvs means Hausdorff topological vector spaces. A

multimap or a map T : X

Y is a function from X into the power set 2

Y

Ž . Ž .of Y with values T x Y for x X and fibers T y for y Y . NoteŽ . Ž .that x T y if and only if y T x . A nonempty topological space is

˘ acyclic if all of its reduced Cech homology groups over rationals vanish.For topological spaces X and Y , a m a p T : X Y is said to be upper

Ž . Ž . semicontinuous usc if for each closed set B Y , The set T B x X :Ž . 4T x B is a closed subset of X ; and acyclic if it is usc with acyclic

compact values. A maximal element of T : X Y is a point x X such

Ž . Ž . Ž .that T x . The graph G r T of T is the set x, y X Y :Ž .4 y T x . The interior a nd closure are denoted by int a nd cl, respectively.

Ž . Ž . Ž .4The map T : X Y is defined by T x y Y : x, y cl G r T . X Y

Ž . Ž Ž ..The map cl T : X Y is defined by cl T x cl T x for each x X . ForŽ .Ž .two maps T , G: X Y , the map T G: X Y is defined by T G x

Ž . Ž . T x G x for each x X . I f X is a set, Y is a subset of a vectorŽ .space E and T : X Y is a map such that co T x Y for each x X ,

Ž .Ž . Ž .then the map co T : X Y is defined by co T x co T x for each x X .

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FIXED POINTS, MAXIMAL ELEMENTS AND EQUILIBRIA OF GAMES 583

Let X be a topological space, Y a nonempty subset of a vector space EŽ .and : X E a function. A map : X Y is said to be of class if a

, C

Ž . Ž . Ž . Ž .for each x X , co x Y and x co x ; b there exists a mapŽ . Ž . Ž .T : X Y such that for each x X , T x x and T y is compactly

Ž . Ž . 4 Ž .open in X for each y Y ; a nd c x X : x x X : T x 4 ; see Tan and Y uan 15 and Mehta et al. 12 .We now introduce a different class as follows:

Ž .A m ap : X Y is said to be of class if a for ea ch x X ,

Ž . Ž . Ž . Ž .co x Y and x co x ; b there exists a map T : X Y suchŽ . Ž . Ž . 4 Ž .that for each x X , T x x and x X : T x int T y :

4 Ž . Ž . 4 Ž . 4 y Y ; a nd c x X : x x X : T x .

I f X Y and 1 , the identity map on X , we shall denote and X C

in place of and , respectively. Note that if X is compact, then we , C

have . , C

Ž .Let I be a possibly infinite set of players. Suppose tha t for each i I ,the ith player’s strategy set X is a nonempty subset of a tvs E andi i

4 X Ł X denote the product space of X . I f P : X X is ai I i i i I i i

Ž .preference map for each i I , the collection X , P is called ai i i I

qualitati e game

. An equilibrium point x X

of the game is a pointˆŽ .satisfying P x for all i I . A generalized game is a family ofˆi

Ž .quadruples X ; A ; B ; P such that for each i I , X is ai i i i i I i

nonempty subset of a tvs E and A , B : X Ł X X are maps. Ani i i i I i i

equilibrium of a generalized game is a point x X such that, for eachˆŽ . Ž . Ž .i I , x B x and A x P x .ˆ ˆ ˆ ˆi i i i

Let E be a tvs and C a lattice with a least element 0. A function : E 2 C is called a measure of noncompactness 7 if t he following condi-

tions hold for any A, B 2 E:

Ž . Ž .1 A 0 if and only if A is relative compact;

Ž . Ž . Ž .2 co A A , where co A denotes the closed convex hull of A; and

Ž . Ž . Ž . Ž .43 A B ma x A , B .

Ž . Ž .It follows that A B

implies

A

B . The above notion is a

generalization of the set-measure and the ball-measure of noncom-pactness defined in terms of a family of seminorms or a norm.

Let D be a subset of E. A map T : D E is said to be - condensing ifŽ Ž .. Ž . Dwhenever T for 2 , then is relatively compact.

From now on, we a ssume that is a measure of noncompactness on t hegiven tvs E if necessary.

Note that each map defined on a compact set is -condensing. If E is

locally convex, then a compact map T : X E is - or -condensingwhenever X E is complete or E is quasi-complete.

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LEMMA 1 12 . Let X be a nonempty closed con ex subset of a tvs E andT : X X a - condensing map. Then there exists a nonempty compact con ex

Ž . subset K of X such that T K K .

In 12 , E is assumed to be a locally convex tvs, but Lemma 1 is true forany tvs as we can see in its proof.

The following is well-known:

LEMMA 2 9 . Let X be a paracompact space Y , a con ex subset of a tvs Ž . 4 E, and S: X Y is a map such that X int S y : y Y . Then co S

has a continuous selection; that is, there is a continuous map f : X Y suchŽ . Ž .that f x co S x for each x X .

Moreo er , if X itself is compact, then f co A for some finite subset A of Y .

ŽLEMMA 3 12 . Let X and Y be topological spaces and A a closed resp.. Ž . open subset of X . Suppose F : X Y and F : A Y are lower resp. upper 1 2

Ž . Ž . semicontinuous such that F x F x for all x A. Then the map F :2 1

X Y defined by

F x if x A;Ž .1

F x Ž . ½ F x if x AŽ .2

Ž .is also lower resp. upper semicontinuous.

ŽA nonempty subset X of a tvs E is said to be admissible in the sense of .Klee 11 provided that, for every compact subset K of X and every

neighborhood V of the origin 0 of E, there exists a continuous map h:Ž . Ž . K X such that x h x V for all x K and h K is contained in a

finite dimensional subspace L of E.It is well known that every nonempty convex subset of a locally convex

p pŽ .tvs is admissible. Other examples of admissible tvs are l , L 0, 1 forŽ .0 p 1, the space S 0, 1 of eq uivalence classes of measurable functions

pon 0, 1 , the H ardy space H for 0 p 1 every compact convex locally convex subset of tvs, and many others. For details, see Pa rk 13 and

references therein. We need the following particular form of P ark 13, Theorem 1 :

TH E O R E M 0. Let X be an admissible con ex subset of a tvs E and T : X X a compact acyclic map. Then T has a fixed point x X ; that is,

Ž . x T x .

3. C OLLE CTIVELY FIXE D PO INTS

We begin with the following collectively fixed point theorem which isessential in this paper.

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TH E O R E M 1. Let I be an index set. For each i I , let X be a con exi

subset of a t s E . Suppose that X Ł X is a closed con ex locally con exi i I i

subset of the t s E Ł E and S , T : X X are maps satisfying thei I i i i i

following conditions:

Ž . Ž . Ž .1 for each x X and i I , co S x T x ;i i

Ž . Ž . 42 X int S y : y X for each i I ; andi i

Ž .3 the map T : X X defined by

T x T x for x X Ž . Ž .Ł ii I

is - condensing .

Ž . Ž .Then there exists an x x X such that x T x for each i I .i i I i i

Proof. Since T is -condensing, it follows from Lemma 1 that there isŽ .a nonempty compact convex subset K of X such that T K K . Since K

is compact, it follows from Lemma 2 that co S has a continuous K i

selection for each i I . Hence T has a continuous selection f : K K , K i i i

where K K and is the projection of E Ł E onto E . Let f :i i i i I i i

K K be defined by

f x f x for x K .Ž . Ž .Ł ii I

Then f is continuous. Since K is a compact convex locally convex subset of

X , K

is an admissible subset of X

. Then by Theorem 0, there is anŽ . Ž . Ž . x x K such that x f x T x . Hence for each i I , x i i I i

Ž . Ž . f x T x .i i

Ž .C OROLLARY 1. In Theorem 1, if the condition 2 is replaced by

Ž . Ž .2 for each i I , S y is compactly open in X for all y X andi i

Ž .S x for all x X .i

Ž . Ž .Then there exists an x x X such that x T x for each i I

.i i I i i

Ž . Ž . Proof. Let S: X X be defined by S x Ł S x for x X . Sincei I i

Ž . Ž .T is -condensing and S x T x for all x X , it follows that S is also-condensing. By Lemma 1, there exists a compact convex subset K of X

Ž .such that S K K . Le t K be the projection of K into E . Theni i

Ž . Ž .S K K . Let S S : K K . Since for each y X and i I , S y K i i i i i i i

is compactly open in X , it follows that for each y K ,i

1 S y x K : y S x x K : y S x K S y 4 4Ž . Ž . Ž . Ž . Ž .i i i i

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Ž .is open in K . B y assumption, we see that for each x K with S x ,i Ž . Ž .Ž .if y S x , then y K and x S y . Consequently, K i i i

Ž .Ž . 4 Ž .Ž . 4 S y : y K int S y : y K . Let T T . Then for K i i i i Ž . Ž .each x K and each i I , co S x T x . Since K is a compact convexi ilocally convex subset of X , K is an admissible closed set. Then the

conclusion follows immediately from Theorem 1.When I is a singleton, we do not need the admissibility.

TH E O R E M 2. Let X be a nonempty closed con ex set of a t s E and S, T : X X maps such that

Ž . Ž . Ž .1 for each x X , co S x T x ; and

Ž . Ž . 42 X int S y : y X .

If T is - condensing , then T has a fixed point.

Proof. Since T is -condensing, by Lemma 1, there exists a nonemptyŽ .compact convex subset K of X such that T K K . For any x K , there

Ž . Ž . Ž .exists a y S x such that x S y ; that is, K is covered by S y :

4

Ž . 4 y K . Moreover, we have K K X K int S y : y X Ž . 4 int S y : y X . F o r a n y x K , there exists y X such that K

Ž . Ž . Ž . x int S y S y , or y S x K . Therefore, we have K K

Ž . 4 int S y : y K . B y applying Lemma 2, there exists a continuous K

Ž .selection f : K K of T such that f K co A K for some finite K

subset A of K . Then f has a fixed point by the B rouwer fixed pointco A

theorem, which is a particular form of Theorem 0. Therefore T has a fixedpoint in K . This completes our proof.

Remark. I f X itself is compact, T is not-necessarily -condensing inTheorem 2. This is known as the Fan Browder fixed point theorem.

C OROLLARY 2. Let X be a nonempty closed con ex subset of a t s E.Suppose that T : X X is a - condensing map such that

Ž . Ž .1 T x is nonempty con ex for each x X ;

Ž . Ž .2 for each y X , T y is compactly open in X .

Then T has a fixed point.

Remark. For a locally convex tvs E, C orollary 2 reduces to 12, Theo-rem 2.2 . Note also tha t Theorem 2 properly generalized C orollary 2.

Ž Ž . Ž E XAMPLE . Let E , X , 0 and T x , x . Then forŽ . Ž .each y X , T y y, 0 . I t is easy to see that X int T y : X

4 Ž . y X . B ut for each y X , T y is not compactly open.

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We have one more collectively fixed point theorem:

TH E O R E M 3. Let I be an index set and for each i I , X a con ex subseti

of a t s E . Suppose that the set X Ł X is a con ex locally con exi i I i

closed subset of E Ł E and for each i I , T : X X is usc and for i I i i i

Ž . Ž . each x X , T x is con ex except for a finite number of i’ s for which T x isi i

Ž . Ž . acyclic. Suppose that T : X X defined by T x Ł T x for each x X i I i

Ž . Ž .is - condensing . Then there exists an x x X such that x T xi i I i i

for each i I .

Proof. Since T is -condensing, it follows from Lemma 1 that there isŽ .a compact convex subset K of X such that T K K . By assumption, for

each i I , T : X X is usc. Let T T for each i, and T T K K i i i i

Ł T . Since the product of a finite number of acyclic sets is acyclic, T

isi I i

usc with compact acyclic values. Thus T : K K is a compact acyclic map.

Since K is a compact convex locally convex subset of a tvs X , K isadmissible, it follows from Theorem 0 that T

has a fixed point; that is, Ž . Ž . Ž .there exists an x x K X such that x T x T x for alli i I i i i

i I .For the case I is a singleton, we have the following particular form of

13, Theorem 2 :

TH E O R E M 4. Let X be a closed con ex locally con ex subset of a t s E and T : X X an acyclic map. If T is - condensing , then T has a fixed point.

Remarks. 1. Theorem 4 can be deduced from Theorem 0 and Lemma 1.

2. I f T is convex-valued and E is locally convex, then Theorem 4 reduces to 12, Theorem 2.3 . Note t hat Theorem 0 and 4 include most of

Ž .well-known fixed point theorems in locally convex topological vector spaces; see 13 .

4. MAXIMAL ELEMENTS

Some of the fixed point results in the previous sections can be restated

to existence theorems for maximal elements.TH E O R E M 5. Let X be a closed con ex locally con ex subset of a t s E.

Suppose that T : X X is an usc - condensing map with closed con exŽ . alues. If x T x for each x X , then T has a maximal element; that is,

Ž .there exists an x X such that T x .

Ž . Proof. Suppose that T x for all x X . Since every nonemptyconvex set is acyclic, T : X X becomes an acyclic map. Since T is

-condensing, by Theorem 4, T has a fixed point. This contradicts thehypothesis.

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Remark. I f E is locally convex, then Theorem 5 reduces t o 12, Theo-rem 3.2 .

TH E O R E M 6. Let X be a nonempty closed con ex subset of a t s E.

Suppose that T : X X is a map such that

Ž . Ž .1 T is - condensing and T x is con ex for all x X ;

Ž . Ž .2 for each y X , T y is compactly open; and

Ž . Ž .3 for each x X , x T x .

Then T has a maximal element.

Ž . Proof. Suppose that for a ll x X , T x . Then by Corollary 2,Ž . Ž .there exists an x X such that x T x . This contradicts 3 . Then the

conclusion follows.

Ž . Remarks. 1. In Theorem 5, if the condition 2 is replaced by X Ž . 4 int T y : y X , then the conclusion is not true. In fact, for any

Ž . Ž . x X , there exists a y X such that x int T y T y . HenceŽ . y T x .

2. If E is locally convex, then Theorem 6 reduces to 12, Theorem3.1 .

TH E O R E M 7. Let X be a nonempty closed con ex subset of a t s E and T : X X be of class . If T is - condensing , then there exists a point x X

Ž . such that T x .

Ž . Proof. Suppose tha t T x is nonempty for each x X . Since T : X

X is of class , we have

Ž . Ž . Ž .a for ea ch x X , co T x X and x co T x ;

Ž . Ž .b there exists a map G: X X such that for each x X , G x Ž . Ž . 4 Ž .T x and x X : G x int G y ; and y X

Ž . Ž . 4 Ž . 4c x X : G x x X : T x .

Ž . Ž . 4Since T x for all x X , we have X x X : T x . Thus Ž . 4 Ž . Ž . X int G y : y X by b a nd c . Since T is -condensing, it

Ž . Ž . Ž .follows that co T is -condensing. B y b , we see that co G x co T xfor all x X . Then by Theorem 2, there exists x

X such that x

Ž . Ž .co T x . This contradicts a . H ence there exists an x X such thatŽ .T x .

C OROLLARY 3. Let X be a nonempty closed con ex subset of a t s E and

T : X X be of class . If T is - condensing , then there exists a point x X C

Ž . such that T x .

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Ž . Proof. Suppose tha t T x is nonempty for each x X . Since T : X X is of class , it follows thatC

Ž . Ž . Ž .a for ea ch x X , co T x X and x co T x ;

Ž . Ž .b there exists a map G: X X such that for each x X , G x Ž . Ž .T x and G y is compactly open for each y X ;

Ž . Ž . 4 Ž . 4c x X : T x x X : G x .

Since T is -condensing, there exists a compact convex subset K of X Ž . Ž .such that T K K . Le t G G . Since for each y K , G y is K

Ž .Ž .compactly open, it is easy to see G y is open in K and K

Ž .Ž .

4

int G y : y K . It is easy to check that T : K

K is of class

. K K Since K is a nonempty compact convex subset of E, by Theorem 7, there

Ž . Ž .exists an x K such that T x . This contradicts T x for all x X .

Remark. I f E is locally convex, then Corollary 3 reduces to 12, Theorem 3.4 . Similarly, 12, Theorem 3.5 can be improved without

assuming the local convexity as follows:

C OROLLARY 4. Let X be a nonempty closed con ex subset of a t s E and P : X X an - majorized - condensing map. Then P has a maximalC

element.

Remarks. 1. For the definition of -majorized ma p, see 12 ; and theC

proof of 12, Theorem 3.4 works for C orollary 4.

2. In a ll of the results in Sects. 3 and 4, the -condensing maps can

be replaced by compact maps, for which the closedness of X is redundant, see P ark 13 . For example, C orollary 4 can be stated for compact maps. Then we obtain a generalization of B orglin and Keiding 2, Corollary 1 ,

Toussaint 16, Theorem 2.2 , Tulcea 18, Theorem 2 , and Y annelis and P rabhakas 21, C orollary 5.1 .

5 . E Q U I L I B R I A O F G E N E R A L I Z E D G A M E S

Applying Theorem 4 a nd following the argument in the proof of 12,Theorem 4.3 , we have the following theorem:

Ž .TH E O R E M 8. Let X , A , P be a generalized game, where I is ai i i i I

Ž . countable or uncountable set of players. Suppose that the following condi-tions are satisfied for each i I :

Ž .1 X is a con ex subset of a t s E such that X Ł X is a closedi i i I i

con ex locally con ex subset of the t s E Ł E ;i I i

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Ž . Ž .2 A : X X is an acyclic map such that for each x X , A x isi i i

con ex except a finite number of i’ s;

Ž . Ž . Ž .3 the map A: X X defined by A x Ł A x for each x X i I i

is - condensing ;Ž . Ž . Ž . Ž .4 for each x X , x A x P x ;i i i

Ž . Ž . Ž . 45 the set U x X : A x P x is open in X ; andi i i

Ž . Ž .Ž .6 A P is usc on U such that for each x U , A P x isi i i i i i

Ž .Ž . closed con ex except a finite number of i’ s for which A P x is acyclic.i i

Ž . Ž .Then there exists an x X such that, for each i I , x A x andi i

Ž . Ž . A x P x

.i i

Remark. For the case E is locally convex, Theorem 8 reduces to 12,1

Theorem 4.3 . Note that in Theorem 8, the involving maps have acyclicvalues on a finite number of points instead of convex values.

The following is a variant of 12, Theorem 4.1 :

TH E O R E M 9. Let X be a nonempty closed con ex subset of t s E. Suppose A, B, P : X X are maps such that

Ž . Ž . Ž . Ž .1 for each x X , A x is nonempty and co A x B x ;

Ž . Ž .2 for each y X , A y is compactly open in X ;

Ž .3 A P is of class ; and

Ž .4 the map A is - condensing .

Ž . Ž . Ž .Then there exists a point x X such that x B x and P x B x

. Proof. Since A is -condensing, by Lemma, 1 there exists a compact

Ž . Ž .4convex subset K of X such that A K K . Let M x K : x B x .Then M is open in K . D efine S: K K by

A x P x if x K M ,Ž . Ž .S x Ž .

½ A x if x M ,Ž .

where A A . K

Ž Ž . Ž ..Since A P is of class , for each x X , x co A x P x andthere exists a map : X X such that

Ž . Ž . Ž . Ž .a for ea ch x X , x A x P x ;

Ž . Ž . 4 Ž . Ž . 4b x X : x x X : A x P x ; andŽ . Ž . 4 Ž . 4c x X : x int y : y Y .

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Now we define T : X X by

x if x K M ,Ž .T x Ž .

½ A x if x M ,Ž .

where . K

Ž . 4 Ž .Ž .We want to show that x K : x int y . In y K K

fact,

x K : x x K : x K x X : x 4 4 4Ž . Ž . Ž .

K int yŽ .Ž . y X

K int y : y X 4Ž .

int y : y X .Ž . Ž . 4 K

Ž . Ž .Ž .I t is easy t o see that f or each y K , T y M y Ž .Ž . Ž .Ž . Ž . A y and A y K A y is open in K . L e t x u K :

Ž . 4T u . Then either x M or x K M .

Ž . Ž . Ž .Case 1. I f x M , then T x A x . Let y A x K , thenŽ .Ž . Ž .Ž . Ž .Ž . Ž . x A y . H ence x M int y A y T y . B ut

Ž . Ž . Ž . Ž . M int y A y is open in K , hence x M

Ž .Ž . Ž .Ž . Ž . Ž . 4int y A y int T y int T y : y K . K K

Ž . Ž . Case 2. If x K M , then T x x . Thus x u K : Ž . 4 Ž . 4 x int y : y X . There exists y X such that x K

Ž .Ž . Ž . Ž . Ž . Ž . Ž .int y . This shows that y x A x P x A x A x Ž .Ž . Ž .Ž . Ž .Ž . K . H ence x A y and x M int y A y

Ž . 4 int T y : y K . K

Ž . 4 Ž . 4In any case, we see that x K : T x

int T y : y K . K Ž . 4 Ž . 4 Ž .B ut int T y : y K x K : T x . Hence x K : T x K

. Ž . 4 Ž . Ž . int T y : y K . Then clearly for each x X , T x S x K

Ž . 4 Ž . 4and x X : S x x X : T x . Moreover, if x M , thenŽ . Ž . Ž . Ž . x B x , it follows from 1 that x co A x co S x , if x M , thenŽ Ž . Ž .. Ž . x co A x P x co S x . This shows that S is of class . Since A I K

Ž . Ž .is -condensing and S x A x for all x X , S is -condensing. ThenŽ . Ž .by Theorem 2, there exists x K such that S x . As A x for

Ž . Ž . Ž .all x X , we must have x M and hence x B x and A x P x . Now the proof is completed.

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C OROLLARY 5. Let X be a nonempty closed con ex subset of a t s E.Suppose A, B, P : X X are maps such that

Ž . Ž . Ž . Ž .1 for each x X , A x is nonempty and co A x B x ;

Ž . Ž .2 for each y Y , A y is compactly open in X ;Ž .3 A P is of class ;C

Ž .4 the map A is - condensing .

Ž . Ž . Ž .Then there exists a point x X such that x B x and A x P x .

Proof. Since A is -condensing, by Lemma 1 there exists a compactŽ .convex subset K of X such that A K K . By assumption, A P is of

Ž Ž . Ž ..class . Hence for all x X , x co A x P x and there exists aC

ma p : X X such that

Ž . Ž . Ž . Ž .a for ea ch x X , x A x P x ,

Ž . Ž .b for ea ch y X , y is compactly open,

Ž . Ž . 4 Ž . Ž . 4 c x X : x x X : A x P x . Let . K

Ž .Ž .Then : K K and for al l y K , y is open in K . Therefore

Ž . 4 Ž

.

Ž . 4

x K : x int y : y K . Let A A , B B K K K and P P . Then it is easy to see that A P is of class . Then the K I K

conclusion follows from Theorem 9.

Remark. In ca se E is locally convex, Corollary 5 reduces to Mehta et al. 12, Theorem 4.1 .

From now on, we give various solutions of equilibria problems forgeneralized games:

Ž .TH E O R E M 10. Let X , A , P be a generalized game, where I isi i i i I

a set of players. Suppose that the following conditions are satisfied for eachi I :

Ž .1 X is a con ex subset of a t s E and X Ł X is a closedi i i I i

con ex locally con ex subset of E Ł E ;i I i

Ž .2 for each x X , A: X X is usc with nonempty compact con exi

alues;

Ž .3 the mapping A: X X defined by A Ł A for each x X isi I i

- condensing ;

Ž . Ž . Ž . 44 the set U x X : A x P x is paracompact and openi i i

in X ;

Ž . Ž . 4 Ž .5 U int T y : y X and for each x U , T x is con ex,i i i i i

where T A P .i i i

Ž . Ž .Then there exists an x X such that for each i I , either x A xi i

Ž . Ž . Ž . Ž . Ž . P x or x A x and A x P x .i i i i i

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Ž . Ž . Proof. For each i I , by 4 , 5 and Lemma 2, there exists a continu-Ž . Ž . Ž .ous function f : U X such that f x A x P x for each x U .i i i i i i i

Now we define F : X X byi i

f x if x U 4Ž .i i F x Ž .i ½ A x if xU .Ž .i i

Ž . Ž .Then 2 , 4 and Lemma 3 imply that F is usc with nonempty compacti

Ž . Ž .convex values. Let F : X X be a mapping defined by F x Ł F x .i I i

Ž .Then F : X X is usc and -condensing and F x is closed and convexfor each x X . Now, by applying Theorem 4, we have the conclusion.

Ž .TH E O R E M 11. Let X , A , B , P be a generalized economy,i i i i i I

where I is the set of agents such that for each i I ,

Ž .1 X is a con ex subset of a t s E ;i i

Ž .2 X Ł X is a closed con ex locally con ex subset of E i I i

Ł E ;i I i

Ž . Ž . Ž . Ž .3 for each x X , B x is nonempty con ex and A x B x X ;i i i i

Ž . Ž .Ž . Ž Ž ..4 the mapping cl B : X X defined by cl B x cl B x for i i i i

each x X is usc;

Ž . Ž .Ž . Ž .5 the map cl B: X X defined by cl B x Ł cl B x for i I i

each x X is - condensing ;

Ž . Ž . Ž . 46 the set W x X : A x P x is a paracompact openi i i

subset of X ;

Ž . Ž . Ž . Ž .7 for each x x X , x co A x P x andi i I i i i

Ž .8 the map T : X X defined byi i

T x A x P x for x X Ž . Ž . Ž .i i i

Ž . 4 has the property W int T y : y X .i i i

Ž .Then has a equilibrium point x x X ; that is, for each i I ,i i I

Ž . Ž . Ž . x cl B x and A x P x .i i i i

Ž . Ž . Proof. For each i I , let S : X X be defined by S x co T x .i i i i

By Lemma 2, for each i I , S has a continuous selection f : W X .W i i i ii

Define

f x if x W 4Ž .i iG x Ž .i ½ cl B x if xW .Ž .i i

Then it follows from Lemma 3 that G is usc. Let G: X X be defined byi

Ž . Ž .G x Ł G x for x X . Then G is usc with nonempty closed convexi I i

Ž . Ž .values. Since B is -condensing and G x B x for all x X , it is easy

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to see that G is -condensing. B y Theorem 4, there exists x X such thatŽ . Ž . Ž . Ž . x x G x ; that is, x G x for each i I . Hence x cl B xi i I i i i i

Ž . Ž .and A x P x .i i

Remark. A compact map version of Theorem 11 for locally convex tvs was due to Wu a nd Shen 20, Theorem 10 , which includes earlier works of

Y annelis a nd P rabhakar 21, Theorem 6.1 , C hang 3, Theorem 3.1 , and Tian 19 .

Ž .C OROLLARY 6. In Theorem 11, if the condition W int T y : y i i

4 Ž . X is replaced by the condition that for each y X , T y is compactlyi i i

open. Then we ha e the same conclusion.

Proof. Since cl B: X X is -condensing, it follows that there exists acompact convex subset K of X such that cl B: K K . By assumption,

Ž . Ž . Ž . Ž . T x A x P x B x for all x K . Hence T : K K . Let T i i i i i i i

Ž . Ž .T . Since for each y X , T y is compactly open. It follows that T y K i i i i

is open for each y K . Let W

W K . Then W is paracompact. It isi i i i

Ž . 4easy to see that W int T y : y K . Then the conclusion followsi K i i

from Theorem 11.

TH E O R E M 12. Let I be any set of players. For each i I , suppose that X iis a compact con ex subset of a t s E , X Ł X is an admissible subseti i I i

Ž . 4 of E Ł E and P : X X . Let U x X : P x be paracom-i I i I i i i

Ž . pact and open for each i I and P x is con ex for all x U and i I .i i

Ž . 4 Suppose further that U int P y : y X , then there exists an x X i i i

Ž . Ž . Ž . such that for each i I , either x P x or P x .i i i

Proof.

It follows from Lemma 2 that P

: U

X has a continuous

U i i ii

selection f : U X . D efine F : X X byi i i i i

f x if x U ; 4Ž .i i F x Ž .i ½ X if xU ,i i

It follows from Lemma 3 that F is usc with nonempty compact convexi

values such that

F x f x P x for all x U . 4Ž . Ž . Ž .i i i i

Ž . Ž .Now define F : X X by F x Ł F x for each x X . Then F isi I i

usc with nonempty compact convex values. Since X is an admissiblecompact set and F : X X is a compact acyclic map, it follows from

Ž .Theorem 0 that there exists x X such that x F x . It follows thatŽ . Ž . Ž .for each i I , either x P x or P x .i i i

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TH E O R E M 13. In Theorem 12, if we assume that X is a nonemptyi

compact con ex subset of a finite dimensional space E and U is closed, theni i

we ha e the same conclusion.

Proof. Note that X Ł X is automatically admissible. It followsi I i

from Lemma 2 that P has a continuous selection f : U X . D efineU i i i ii

f x if x U ; 4Ž .i iG x Ž .i ½ X if xU .i i

Then, by fo llowing the proof of 12, Theorem 5.3 , we have the conclusion.

Remark. Some results related to this paper can be found in 1, 46, 8,10, 14, 17 .

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