Lin, Chen, 2003

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COINCIDENCE THEOREMS FOR FAMILIESOF MULTIMAPS AND THEIR APPLICATIONSTO EQUILIBRIUM PROBLEMS

LAI-JIU LIN AND HSIN I CHEN

Received 30 October 2001

We apply some continuous selection theorems to establish coincidence theoremsfor a family of multimaps under various conditions. Then we apply these coin-

cidence theorems to study the equilibrium problem with m families of players

and 2m families of constraints on strategy sets. We establish the existence theo-

rems of equilibria of this problem and existence theorem of equilibria of abstract

economics with two families of players.

1. Introduction

For multimaps F : X Y and S : Y X , a point (x, y )∈ X ×Y is called a coin-

cidence point of F and S if y ∈ F (x ) and x ∈ S( y ). In 1937, Neumann [19] and in1966 Fan [8] established the well-known coincidence theorems. In 1984, Brow-

der [4] combined Kakutani-Fan fixed-point theorem and Fan-Browder fixed-

point theorem to obtain a coincidence theorem. So many authors gave some

coincidence theorems and applied them in various fields as equilibrium prob-

lem, minimax theorem, quasi-variational inequalities, game theory, mathemat-

ical economics, and so on, see [1, 5, 6, 9, 11, 18] and references therein. In 1991,

Horvath [10, Theorem 3.2] obtained a continuous selection theorem. In [21,

Theorem 1], Wu and Shen established another continuous selection theorem.

Let I and J be any index sets. For each i ∈ I and j ∈ J , let X i and Y j benonempty sets and H j : X =

i∈I X i Y j ; T i : Y =

j∈ J Y j X i be multimaps.

A point (x, ¯ y )∈ X ×Y , where x = (x i)i∈I and ¯ y = ( ¯ y j ) j∈ J

is called a coincidence

point of two families of multimaps, if ¯ y j ∈ H j (x ) and x i ∈ T i( ¯ y ) for each i ∈ I

and j ∈ J . In this paper, we apply the continuous selection theorem of Horvath

[10] and the fixed-point theorem of Park [17] to derive the coincidence theo-

rems for two families of multimaps. Our coincidence theorems for two families

of multimaps include Fan-Browder fixed-point theorem [3] and Browder coin-

cidence theorem [4] as special cases.

Copyright © 2003 Hindawi Publishing CorporationAbstract and Applied Analysis 2003:5 (2003) 295–309

2000 Mathematics Subject Classification: 54H25, 46N10, 91A13, 91B50

URL: http://dx.doi.org/10.1155/S1085337503210034

http://dx.doi.org/10.1155/S1085337503210034

http://dx.doi.org/10.1155/S1085337503210034

8/13/2019 Lin, Chen, 2003


296 Coincidence theorems with applications

We will employ our results on coincidence theorems for two families of mul-

timaps to consider the equilibrium problem with m families of players and 2mfamilies of constraints on the strategy sets introduced by Lin et al. [ 14]. We con-

sider the following problem: let I be any index set and for each k ∈ I , let J k be

a finite index set, X k j denote the strategy set of jth player in kth family, Y k = j∈ J k X k j , Y =

k∈I Y k , Y k =

l ∈I,l =k Y l , and Y = Y k ×Y k. Let F k j : X k j ×Y k

Rl k j be the payoff of the j th player in the kth family, let Ak j : Y k X k j be the con-

straint which restricts the strategy of the jth player in the kth family to the subset

Ak j (Y k) of X k j when all players in other families have chosen their strategies x i j ,

i ∈ I , i = k, and j ∈ J i, and let Bk : Y k Y k be the constraint which restricts the

strategies of all the families except kth family to the subset Bk(Y k) of Y k when

all the players in the k family have chosen their strategies y k = (x k j ) j∈ J k , k ∈ I .

Our problem is to find a strategies combination ¯ y = ( ¯ y k)k∈I ∈

k∈I Y k = Y ,u = (uk)k∈I ∈ Y , ¯ y k = (x k j ) j∈ J k , ¯ y k ∈ Ak( ¯ y k), uk ∈ Bk( ¯ y k), and z k j ∈ F k j (x k j , uk)

such that

z k j − z k j ∈ − intRl k j

+ , (1.1)

for all z k j ∈ F k j (x k j , uk), x k j ∈ Ak j ( ¯ y k), and for all k ∈ I and j ∈ J k. In the Nash

equilibrium problem, the strategy of each player is subject to no constraint. In

the Debreu equilibrium problem, the strategy of each player is subject to a con-straint which is a function of the strategies of the other players. For the special

case of our problem, if each of the families contain one player, we find that the

strategy of each player is subject a constraint which is a function of strategies

of the other players, and for each k ∈ I , where I is the index set of players, the

strategies combination of the players other than the kth player is a function of

strategy of kth player. Therefore, our problem is diff erent from the Nash equilib-

rium problem and their generalizations. As we note from Remark 4.8, for each

k ∈ I , if J k = {k} be a singleton set, then the above problem reduces to the prob-

lem which is diff erent from the Debreu social equilibrium problem [6] and theNash equilibrium problem [19]. Lin et al. [14] demonstrate the following ex-

ample of this kind of equilibrium problem in our real life. Let I = {1 , 2 , . . . ,m}denote the index set of the companies. For each k ∈ I , let J k = {1 , 2 , . . . , nk} de-

note the index set of factories in the kth company, F k j denote the payoff function

of the jth factory in the kth company. We assume that the products between the

factories in the same company are diff erent, and the financial systems and man-

agement systems are independent between the factories in the same company,

while some collections of products are the same and some collections of prod-

ucts are diff erent between diff erent factories in diff erent companies. Therefore,

the strategy of the jth factory in the kth company depends on the strategies of

all factories in diff erent companies. The payoff function F k j of the jth factory in

the kth company depends on its strategy and the strategies of factories in other

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L.-J. Lin and H. I Chen 297

companies. We also assume that for each k ∈ I , the strategies of the k company

influence the strategies of all other companies. With this strategies combination,

each factory can choose a collection of products, and from these collection of

products, there exists a product that minimizes the loss of each factory. In this

type of abstract economic problem with two families of players, the strategy and

the preference correspondence of each player in family A depend on the strate-

gies combination of all players in family B, but does not depend on the strategies

combination of the players in family A. The same situation occurs to each player

of family B. The abstract economic problem studied in the literature, the strategy

and preference correspondence of each player depend on the strategies combi-

nation of all the players. Therefore, the abstract economic problem, we studied

in this paper, is diff erent from the abstract economic studied in the literature.

In some economic model with two companies (say A and B), the strategy of each factory of company A depend on the strategies combination of factories in

company B. The same case occurs in company B. We can use this example to

explain the abstract economic problem we study in this paper. We also apply the

coincidence theorems for a family of multimaps to consider the abstract eco-

nomic problem with two families of players. In this paper, we want to establish

the existence theorems of equilibria of constrained equilibrium problems with

m families of players and 2m families of constraints and existence theorem of

equilibria of abstract economic with two families of players.

2. Preliminaries

In order to establish our main results, we first give some concepts and notations.

Throughout this paper, all topological spaces are assumed to be Hausdorff .

Let A be a nonempty subset of topological vector space (t.v.s.) X , we denote by

int A the interior of A, by ¯ A the closure of A in X , by co A the convex hull of

A, and by co A the closed convex hull of A. Let X , Y , and Z be nonempty sets.

A multimap (or map) T : X Y is a function from X into the power set of Y

and T − : Y X is defined by x ∈ T −( y ) if and only if y ∈ T (x ). Let B ⊂ Y , we

define T −

(B) = {x ∈ X : T (x )∩B =∅}. Given two multimaps F : X

Y andG : Y Z , the composite GF : X Z is defined by GF (x ) = G(F (x )) for all

x ∈ X .

Let X and Y be two topological spaces, a multimap T : X Y is said to be

compact if there exists a compact subset K ⊂ Y such that T ( X )⊂K ; to be closed

if its graph Gr (T ) = {(x, y ) | x ∈ X, y ∈ T (x )} is closed in X × Y ; to have lo-

cal intersection property if, for each x ∈ X with T (x ) =∅, there exists an open

neighborhood N (x ) of x such that

z ∈N (x ) T (z ) =∅; to be upper semicontin-

uous (u.s.c.) if T −( A) is closed in X for each closed subset A of Y ; to be lower

semicontinuous (l.s.c.) if T −(G) is open in X for each open subset G of Y , and

to be continuous if it is both u.s.c. and l.s.c.

A topological space is said to be acyclic if all of its reduced Cech homology

groups vanish. In particular, any convex set is acyclic .

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Definition 2.1 (see [15]). Let X be a nonempty convex subset of a t.v.s. E. A

multimap G : X R is said to be R+-quasiconvex if, for any α ∈R, the set

x ∈ X : there is a y ∈G(x ) such that α− y ≥ 0

(2.1)

is convex.

Definition 2.2 (see [15]). Let Z be a real t.v.s. with a convex solid cone C and Abe a nonempty subset of Z . A point ¯ y ∈ A is called a weak vector minimal point

of A if, for any y ∈ A, y − ¯ y ∈− int C . Moreover, the set of weak vector minimal

points of A is denoted by wMinC A.

Lemma 2.3. Let I be any index set and {Ei}i∈I be a family of locally convex t.v.s.

For each i ∈ I , let X i be a nonempty convex subset of Ei , F i , H i such that X :=i∈I X i X i be multimaps satisfying the following conditions:

(i) for each x ∈ X , co F i(x )⊂H i(x ) ;(ii) X =

{int F −i (x i) : x i ∈ X i} ;

(iii) H i is compact.

Then there exists a point x = (x i)i∈I ∈ X such that x ∈H (x ) :=

i∈I H i(x ) ; that is,

x i ∈H i(x ) for each i ∈ I .

Proof. Lemma 2.3 follows immediately from [12, Proposition 1] and [21, Theo-

rem 2].

3. Coincidence theorems for families of multivalued maps

Theorem 3.1. Let I and J be any index sets, and let {U i}i∈I and {V j} j∈ J be families

of locally convex t.v.s. For each i ∈ I and j ∈ J , let X i and Y j be nonempty convex

subsets, each in U i and V j , respectively. Let F j , H j : X :=

i∈I X i Y j ; Si , T i :

Y :=

j∈ J Y j X i be multimaps satisfying the following conditions:

(i) for each x ∈ X , co F j (x )⊂H j (x ) ;

(ii) X ={int F j

−( y j ) : y j ∈ Y j} ;

(iii) for each y ∈ Y , co Si( y )⊂ T i( y ) ;

(iv) Y ={int Si−(x i) : x i ∈ X i} ;

(v) T i is compact.

Then there exist x = (x i)i∈I ∈ X and ¯ y = ( ¯ y j ) j∈ J ∈ Y such that ¯ y j ∈ H j (x ) and

x i ∈ T i( ¯ y ) for each i ∈ I and j ∈ J .

Proof. Since for each i ∈ I , T i is compact, there exists a compact subset Di ⊂ X isuch that T i(Y ) ⊂ Di for each i ∈ I . Let D =

i∈I Di and K = coD, it follows

from [7, Lemma 1] that K is a nonempty paracompact convex subset in X . For

each i ∈ I , let K i be the ith projection of K . By assumption (iii), Si, T i : Y K iand co Si( y ) ⊂ T i( y ) for each y ∈ Y . By (i), for each x ∈ K , coF j |K (x ) ⊂ H j |K

(x ). By (ii), K ={intK F j−( y j ) : y j ∈ Y j}.

By [12, Proposition 1] and [21, Theorem 1], H j |K (x ) has a continuous selec-

tion, that is, for each j ∈ J , there exists a continuous function f j : K → Y j such

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that f j (x ) ∈H j (x ) for all x ∈ K . Let f : K → Y be defined by f (x ) =

j∈ J f j (x )

and P i, W i : K K i be defined by W i(x ) = Si( f (x )) and P i(x ) = T i( f (x )) for

all x ∈ K . It is easy to see that W i−(x i) = f −1(Si

−(x i)) for all x i ∈ K i and for all

i ∈ I . By assumption (iii), for all i ∈ I and for all x ∈ X , co W i(x )= coSi( f (x ))⊂

T i( f (x )) = P i(x ). Since Si(Y )⊂ T i(Y ) ⊂K i, it follows from assumption (iv) and

the continuity of f that

K = f −1(Y )= f −1

int Si−

x i

: x i ∈ X i

= f −1

int Si−

x i

: x i ∈K i

⊂

int f −1

Si−

x i

: x i ∈K i

= intW i

−x i :

x i∈

K i⊂

K.

(3.1)

Hence,

K =

int W i−

x i

: x i ∈K i

. (3.2)

Then by Lemma 2.3, there exists a point x = (x i)i∈I ∈ K ⊂ X such that x i ∈

P i(x ) = T i( f (x )) for each i ∈ I . Let ¯ y = ( ¯ y j ) j∈ J ∈ Y such that ¯ y = f (x ), then,

for each i ∈ I and j ∈ J , ¯ y j = f j (x ) ∈ H j (x ) and x i ∈ T i( ¯ y ). The proof is com-

plete.

Corollary 3.2 [22, Theorem 8]. In Theorem 3.1 , if the condition (v) is replaced by (v ), then the conclusion is still true, where (v ) X i is compact.

Proof. Since X i is compact, T i is compact and the conclusion of Corollary 3.2

follows from Theorem 3.1.

Remark 3.3. Theorem 3.1 improves [15, Theorem 8].

Theorem 3.4. Let I and J be any index sets, let {U i}i∈I and {V j} j∈ J be families

of locally convex t.v.s. For each i ∈ I and j ∈ J , let X i and Y j be nonempty convex

subsets of U i and V j , respectively, and let D j be a nonempty compact metrizable

subset of Y j . For each i ∈ I and j ∈ J , let S j , T j : X :=

i∈I X i D j ; F i , H i : Y := j∈ J Y j X i be multimaps satisfying the following conditions:

(i) for each x ∈ X , coS j (x )⊂ T j (x ) and S j (x ) =∅ ;

(ii) S j is l.s.c.;

(iii) for each y ∈ Y , co F i( y )⊂H i( y ) ;

(iv) Y ={int F i

−(x i) : x i ∈ X i} ;

(v) H i is compact.

Then there exist x = (x i)i∈I ∈ X and ¯ y = ( ¯ y j ) j∈ J ∈ D :=

j∈ J D j such that ¯ y j ∈

T j (x ) and x i ∈H i( ¯ y ) for each i ∈ I and j ∈ J .

Proof. Since for each i ∈ I , H i is compact, there exists a compact subset C i ⊂ X isuch that H i(Y ) ⊂ C i for each i ∈ I . Let C =

i∈I C i and D :=

j∈ J D j , then

coC and K = coD are nonempty paracompact convex subsets each in X and Y ,

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respectively by [7, Lemma 1]. By assumptions (i), (ii), and following the same

argument as in the proof of [20, Theorem 1], there exists an u.s.c. multimap P j :

coC D j with nonempty compact convex values such that P j (x ) ⊂ T j (x ) for

all x ∈ coC . Define P : co C D by P (x ) =

j∈ J P j (x ) for all x ∈ coC . Then it

follows from [8, Lemma 3] that P is an u.s.c. multimap with nonempty compact

convex values.

By [12, Proposition 1] and [21, Theorem 1], for each i ∈ I , H i |K ( y ) has a

continuous selection f i : K → C i such that f i( y ) ∈ H i( y ) for all y ∈ K . Let f :

K → C be defined by f ( y ) =

i∈I f i( y ) for all y ∈ K , and let W : K D be

defined by W ( y ) = P |C ( f ( y )) for all y ∈ K . It is easy to see that W is an u.s.c.

multimap with nonempty closed convex values. Then, by [17, Theorem 7], there

exists ¯ y ∈D such that ¯ y ∈W ( ¯ y )= P |C ( f ( ¯ y )). Let x ∈ C such that x = f ( ¯ y ) and

¯ y ∈ P |C (x ), then for each i ∈ I and j ∈ J , ¯ y j ∈ P j (x ) ⊂ T j (x ) and x i = f i( ¯ y ) ∈H i( ¯ y ).

Theorem 3.5. Let I and J be any index sets. For each i ∈ I and j ∈ J , let X i and

Y j be nonempty convex subsets in locally convex t.v.s. U i and V j , respectively, let D j

be a nonempty compact metrizable subset of Y j , and let C i be a nonempty compact

metrizable subset of X i. For each i ∈ I and j ∈ J , let S j , T j : X :=

i∈I X i D j ; F i ,

H i : Y :=

j∈ J Y j C i be multimaps satisfying the following conditions:

(i) for each x ∈ X , coS j (x )⊂ T j (x ) and S j (x ) =∅ ;

(ii) S j is l.s.c.;

(iii) for each y ∈ Y , coF i( y )⊂H i( y ) and F i( y ) =∅ ;

(iv) F i is l.s.c.

Then there exist x = (x i)i∈I ∈ coC := co

i∈I C i and ¯ y = ( ¯ y j ) j∈ J ∈ coD :=

co

j∈ J D j such that ¯ y j ∈ T j (x ) and x i ∈H i( ¯ y ) for each i ∈ I and j ∈ J .

Proof. Following the same argument as in the proof of [20, Theorem 1], for

each i ∈ I and j ∈ J , there are two u.s.c. multimaps P j : co C D j and Qi :

coD C i with nonempty closed convex values such that P j (x ) ⊂ T j (x ) for all

x ∈ coC , and Qi( y ) ⊂H i( y ) for all y ∈ coD. Define P : co C D, Q : coD C

by P (x )=

j∈ J P j (x ) for all x ∈ coC , and Q( y )=

i∈I Qi( y ) for all y ∈ coD. By [8, Lemma 3], P and Q both are u.s.c. multimaps with nonempty closed con-

vex values. Let W : co C × coD C ×D be defined by W (x, y ) = (Q( y ) , P (x ))

for (x, y ) ∈ (coC )× (coD). It is easy to see that W is an u.s.c. multimap with

nonempty closed convex values. Therefore, by [17, Theorem 7], there exist x =(x i)i∈I ∈ coC and ¯ y = ( ¯ y j ) j∈ J ∈ coD such that (x, ¯ y ) ∈ W (x, ¯ y ). Then for each

i ∈ I and j ∈ J , ¯ y j ∈ P j (x )⊂ T j (x ) and x i ∈Qi( ¯ y )⊂H i( ¯ y ).

Remark 3.6. In Theorem 3.1, we do not assume that { X i} and {Y j} are metriz-

able for each i ∈ I and j ∈ J .

Theorem 3.7. Let I and J be finite index sets, let {U i}i∈I be a family of t.v.s., and

let {V j} j∈ J be a family of locally convex t.v.s. For each i ∈ I and j ∈ J , let X i be a

nonempty convex subset of U i , let Y j be a nonempty compact convex subset of V j ,

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and let F j : X :=

i∈I X i Y j and let Gi : Y :=

j∈ J Y j X i be two multimaps

satisfying the following conditions:

(i) F j is an u.s.c. multimap with nonempty closed acyclic values;(ii) Y =

{int Gi

−(x i) : x i ∈ X i} and Gi( y ) is convex for all y ∈ Y .

Then there exist x = (x i)i∈I ∈ X and ¯ y = ( ¯ y j ) j∈ J ∈ Y such that ¯ y j ∈ F j (x ) and

x i ∈Gi( ¯ y ) for each i ∈ I and j ∈ J .

Proof. By [12, Proposition 1] and [21, Theorem 1], Gi has a continuous selection

g i : Y → X i such that g i( y ) ∈ Gi( y ) for all y ∈ Y . Let g : Y → X be defined by

g ( y )=

i∈I g i( y ) for all y ∈ Y , then g is also continuous. Define P j : Y Y j and

P : Y Y by P j ( y ) = F j ( g ( y )), and P ( y ) =

j∈ J P j ( y ) for all y ∈ Y , then P j :

Y Y j is an u.s.c. multimap with nonempty closed acyclic values. By Kunneth

formula (see [16] and [8, Lemma 3]), P : Y Y is also an u.s.c. multimap withnonempty closed acyclic values. Therefore, by [17, Theorem 7], there exists ¯ y ∈

Y such that ¯ y ∈ P ( ¯ y ) =

j∈ J P j ( ¯ y )=

j∈ J F j ( g ( ¯ y )). Let x = g ( ¯ y ) such that ¯ y i ∈

F i(x ), then x i = g i( ¯ y )∈Gi( ¯ y ) and ¯ y j ∈ F j (x ) for all i ∈ I and j ∈ J .

Remark 3.8. (i) In particular, if I = J is a singleton, X = Y , E = V , and F = I X ,

the identity mapping on X in the above theorem, then we can obtain well-known

Browder fixed-point theorem [3].

(ii) If I = J is a singleton, F is an u.s.c. multimap with nonempty closed con-

vex values, G( y ) is nonempty for all y ∈ Y , and G−(x ) is open for all x ∈ X , then

Theorem 3.7 reduces to Browder coincidence theorem [4].

Theorem 3.9. Let I and J be finite index sets, {U i}i∈I and {V j} j∈ J be families of

locally convex t.v.s. For each i ∈ I and j ∈ J , let X i be a nonempty convex metrizable

compact subsets of U i and let Y j be a nonempty compact convex subsets of V j . For

each i ∈ I and j ∈ J , let F j : X :=

i∈I X i Y j and Gi : Y :=

j∈ J Y j X i be two

multimaps satisfying the following conditions:

(i) F j is an u.s.c. multimap with nonempty closed acyclic values;

(ii) Gi is a l.s.c. multimap with nonempty closed convex values.

Then there exist x = (x i)i∈I ∈ X and ¯ y = ( ¯ y j ) j∈ J ∈ Y such that ¯ y j ∈ F j (x ) and x i ∈Gi( ¯ y ) for each i ∈ I and j ∈ J .

Proof. Since Y j is compact for each j ∈ J , Y =

j∈ J Y j is compact. By assump-

tion (ii) and following the same argument as in the proof of [ 20, Theorem 1],

for each i ∈ I , there exists an u.s.c. multimap ϕi : Y X i with nonempty closed

convex values such that ϕi( y ) ⊂ Gi( y ) for all y ∈ Y . Let ϕ : Y X defined by

ϕ( y )=

i∈I ϕi( y ) for all y ∈ Y , then ϕ is u.s.c. with nonempty convex compact

values. Define a multimap F : X Y by F (x ) =

j∈ J F j (x ) for all x ∈ X , then

F : X Y is also an u.s.c. multimap with nonempty closed acyclic values.

Define a multimap W : X ×Y X ×Y by W (x, y )= (ϕ( y ) , F (x )) for all x ∈

X and for all y ∈ Y . It is easy to see that W is a compact u.s.c. multimap with

nonempty closed acyclic values. It follows from [17, Theorem 7] that there exists

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(x, ¯ y ) ∈ X ×Y such that x ∈ ϕ( ¯ y ) and ¯ y ∈ F (x ). Therefore, x i ∈ Gi( ¯ y ) and ¯ y j ∈

F j (x ) for each i ∈ I and j ∈ J .

Remark 3.10. In Theorem 3.5, if F j is an u.s.c. multimap with nonempty closedconvex values for each j ∈ J , then J may be any index set.

Let I be any index set and, for each i ∈ I , let {U i}i∈I be a family of locally

convex t.v.s. For each i ∈ I , let X i be a nonempty convex subset in t.v.s. Ei for

each i ∈ I . Let X =

i∈I X i, X i =

j∈I, j=i X j and we write X = X i × X i. For each

x ∈ X , x i ∈ X i denotes the ith coordinate and x i ∈ X i the projection of x onto X i,

and we also write x = (x i , x i).

Theorem 3.11. Let I be a finite index set and let {U i}i∈I be a family of locally

convex t.v.s. For each i∈

I , let

X i be a nonempty compact convex subset of

U i and

let F i : X i X i and Gi : X i X i be multimaps satisfying the following conditions:

(i) F i is an u.s.c. multimap with nonempty closed acyclic values;

(ii) X i ={int X i G−

i (x i) : x i ∈ X i} and Gi(x i) is convex for all x i ∈ X i.

Then there exist x = (x i)i∈I ∈ X and u = (ui)i∈I ∈ X such that x i ∈ F i(ui) and

ui ∈Gi(x i) for all i ∈ I .

Proof. Since X i ={int X i G−

i (x i) : x i ∈ X i}, by [12, Proposition 1] and [21, The-

orem 1], Gi has a continuous selection g i : X i → X i.

Define multimaps P i : X X i and P : X X by P i(x )= F i( g i(x i)) and P (x )=i∈I P i(x ) for all x = (x i)i∈I ∈ X , then P i is an u.s.c. multimap with nonempty

closed acyclic values for all i∈ I . Therefore, P : X X is also an u.s.c. multimap

with nonempty closed acyclic values. Since X is a nonempty compact convex

subset in a locally convex t.v.s. E =

i∈I Ei, it follows from [17, Theorem 7] that

there exists x = (x i)i∈I ∈ X such that x ∈ P (x ) =

i∈I P i(x ) =

i∈I F i( g i(x i)),

that is, for all i ∈ I , x i ∈ F i( g i(x i)). For all i ∈ I , let ui = g i(x i), then ui ∈ X i.

Hence, ui = g i(x i)∈Gi(x i) and x i ∈ F i(ui) for all i ∈ I .

Remarks 3.12. (i) In Theorem 3.11, if F i is an u.s.c. multimap with nonempty

closed convex values for each i ∈ I , then I may be any index set.(ii) The proofs and conditions between Theorem 3.7 and Theorem 3.11 are

somewhat diff erent.

4. Applications of coincidence theorem for families of multimaps

to equilibrium problems

In this section, we establish the existence theorem of equilibrium problem with

m families of players and 2m families of constraints on strategy sets which has

been introduced by Lin et al. [14].

Let I be a finite index set and for each k ∈ I and j ∈ J k, let X k j , Y k, Y k , Y ,F k j , and Ak j be the same as in introduction. For each k ∈ I and j ∈ J k, let W k j ∈

Rl k j

+ \{0} and W k =

j∈ J k W k j . For each k ∈ I , let Ak : Y k Y k be defined by

8/13/2019 Lin, Chen, 2003



Ak( y k) =

j∈ J k Ak j ( y k) where y k = (x k j ) j∈ J k ∈ Y k and y k = ( y l )l ∈I,l =k ∈ Y k. Let

SW k : Y k ×Y k R be defined by

SW k y k , y k

=

j∈ J k

W k j ·F k j

x k j , y k

=

u : u=

j∈ J k

W k j · z k j , for z k j ∈ F k j

x k j , y k

.

(4.1)

For each k ∈ I , let M W k : Y k Y k be defined by

M W k y k= y k ∈ Ak

y k

: inf SW k y k , y k

= inf SW k

Ak

y k , y k

,

M ( y )=k∈I M

W k y

k.

(4.2)

Throughout the paper, we will use the above mentioned notations, unless

otherwise specified.

Theorem 4.1. For each k ∈ I and j ∈ J k , let X k j be a nonempty compact convex

subset of a locally convex t.v.s. Ek j satisfying the following conditions:

(i) the multimap SW k is continuous with compact values;

(ii) Ak j : Y k X k j is a continuous multimap with nonempty closed convex val-

ues;(iii) for all y k ∈ Y k , M W k ( y k) is an acyclic set; and

(iv) for each k ∈ I , let Bk : Y k Y k be a multimap with convex values and

Y k ={intY k B−1

k ( y k) : y k ∈ Y k}.

Then there exist ¯ y = ( ¯ y k)k∈I ∈ Y , ¯ y k = (x k j ) j∈ J k , u = (uk)k∈I ∈ Y , ¯ y k ∈ Ak(uk) ,

uk ∈ Bk( ¯ y k) , and z k j ∈ F k j (x k j , uk) such that


+ , (4.3)

for all z k j ∈ F k j (x k j , uk

) , x k j ∈ Ak j (uk

) and for all k ∈ I , j ∈ J k.Proof. By assumptions (i), (ii), (iii), and following the same argument as in the

proof of [13, Theorem 3.2], we show that M W k : Y k Y k is an u.s.c. multimap

with nonempty compact acyclic values. Since Bk( y k) is convex for each k ∈ I ,

y k ∈ Y k, and Y k ={intY k B−1

k ( y k) : y k ∈ Y k}, it follows from Theorem 3.11

that there exist (uk)k∈I ∈ Y and ( ¯ y k)k∈I ∈ Y such that ¯ y k ∈ M W k (uk) and uk ∈

Bk( ¯ y k) for all k ∈ I . Therefore, for each k ∈ I , ¯ y k ∈ Ak(uk), and minSW k ( ¯ y k , uk)=

min SW k ( Ak(uk) , uk) ≤ min SW k ( pk , uk) for all pk ∈ Ak(uk). Let ¯ y k = (x k j ) j∈ J k ∈

Ak(uk). Following the same argument as in the proof of [13, Theorem 3.1], then

we show Theorem 4.1.

Remark 4.2. It is easy to see that the conclusion of Theorem 4.1 remains true if

condition (iii) is replaced by

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(iii) for each k ∈ I and for each fixed y k ∈ Y k , the multimap uk SW k (uk , y k)

is R+-quasiconvex.

As a simple consequence of Theorem 4.1, we give a simple proof of the fol-lowing corollaries.

Corollary 4.3 ([14]). For each k ∈ I and j ∈ J k , let X k j be a nonempty compact

convex subset of a locally convex t.v.s. Ek j , let Ak j : Y k X k j be a multimap with

nonempty convex values such that for each x k j ∈ X k j , A−1k j

(x k j ) is open in Y k and

¯ Ak j is an u.s.c. multimap. For each k ∈ I , let Bk : Y k Y k be a multimap with

convex values satisfying the following conditions:

(i) Y k =

{intY k B−1

k ( y k) : y k ∈ Y k} ;

(ii) SW k

: Y k×

Y k R

is a continuous multimap with compact values;(iii) for any y k ∈ Y k , uk SW k (uk , y k) is R+-quasiconvex.

Then there exist ¯ y = ( ¯ y k)k∈I ∈ Y , u = (uk)k∈I ∈ Y , ¯ y k ∈ ¯ Ak(uk) , ¯ y k = (x k j ) j∈ J k ,

uk ∈ Bk( ¯ y k) , and z k j ∈ F k j (x k j , uk) such that


+ , (4.4)

for all z k j ∈ F k j (x k j , uk) , x k j ∈ ¯ Ak j (uk) and for all k ∈ I , j ∈ J k.

Proof. Since A

−1

k j (x k j ) is open in Y k

for each x k j ∈

X k j , Ak j is a l.s.c. multimap. Itis easy to see that ¯ Ak j is also a l.s.c. multimap. By assumption, ¯ Ak j is an u.s.c. mul-

timap. Therefore, ¯ Ak j is a continuous multimap with nonempty convex closed

values. Let ¯ Ak =

j∈ J k¯ Ak j . Applying assumption (iii) and following the same ar-

gument as in [13, Theorem 3.3], it is easy to see that H W k ( y k) is an convex set

for all y k ∈ Y k, where

H W k y k= y k ∈ ¯ Ak

y k

: inf SW k y k , y k

= inf SW k

¯ Ak

y k , y k

. (4.5)

By Theorem 4.1, there exist ¯ y =( ¯ y k)k∈I ∈Y , u=(uk)k∈I ∈ Y with ¯ y k = (x k j ) j∈ J k

∈ ¯ Ak(uk), and uk ∈ Bk( ¯ y k), z k j ∈ F k j (x k j , uk) such that


+ (4.6)

for all z k j ∈ F k j (x k j , uk), x k j ∈ ¯ Ak j (uk) and for all k ∈ I , j ∈ J k.

Corollary 4.4. For each k ∈ I and j ∈ J k , let X k j be a nonempty compact convex


(i) SW k is a continuous multimap with compact values;

(ii) for each j ∈ J k , Ak j : Y k X k j is a continuous multimap with nonempty

closed convex values;

(iii) for all y k ∈ Y k , M W k ( y k) is an acyclic set.

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Then there exist ¯ y = ( ¯ y k)k∈I ∈ Y , u = (uk)k∈I ∈ Y , ¯ y k = (x k j ) j∈ J k ∈ Ak(uk) , and

z k j ∈ F k j (x k j , uk) such that


+ , (4.7)


Proof. For each k ∈ I , let Bk : Y k Y k be defined by Bk( y k) = Y k , then all the

conditions of Theorem 4.1 are satisfied. Therefore, there exist ¯ y = ( ¯ y k)k∈I ∈ Y ,

u= (uk)k∈I ∈ Y , ¯ y k = (x k j ) j∈ J k ∈ ¯ Ak(uk), uk ∈ Bk( ¯ y k), and z k j ∈ F k j (x k j , uk) such

that


+ , (4.8)

for all z k j ∈ F k j (x k j , uk), x k j ∈ ¯ Ak j (uk) and for all k ∈ I , j ∈ J k.

Theorem 4.5. For each k ∈ I and j ∈ J k , let X k j be a nonempty compact convex


(i) F k j is a continuous multimap with nonempty closed values;

(ii) Ak j

: Y k X k j

is a continuous multimap with nonempty closed values;

(iii) for all y k ∈ Y k , {x k j ∈ Ak j ( y k) : F k j (x k j , y k)∩wMinF k j ( Ak j ( y k) , y k) =∅}

is an acyclic set;

(iv) for each k ∈ I , Bk : Y k Y k is a multimap with convex values and Y k ={intY k B−1

k ( y k) : y k ∈ Y k}.

Then there exist ¯ y = ( ¯ y k)k∈I ∈ Y , u = (uk)k∈I ∈ Y , ¯ y k = (x k j ) j∈ J k ∈ ¯ Ak(uk) , uk ∈

Bk( ¯ y k) , and z k j ∈ F k j (x k j , uk) such that

z k j − z k j ∈ − intR

l k j

+ , (4.9)

for all z k j ∈ F k j (x k j , uk) , x k j ∈ Ak j (uk) and for all k ∈ I , j ∈ J k.

Proof. Define a multimap M k j : Y k X k j by M k j ( y k) = {x k j ∈ Ak j ( y k) : F k j (x k j ,

y k)∩wMinF k j ( Ak j ( y k) , y k) =∅} for all y k ∈ Y k and for each k ∈ I and j ∈ J k.

Since F k j is a continuous multimap and X k j is compact for each k ∈ I and j ∈

J k, it follows from [2, Proposition 3, page 42] that F k j is a compact continuous

multimap with closed values. By [2, Proposition 2, page 41], M k j is a closed

compact u.s.c. multimap for each k ∈ I and j ∈ J k. Moreover, for each k ∈ I and

j ∈ J k , M k j is an u.s.c. multimap with compact acyclic values. Define a multimap

M k : Y k Y k by M k( y k) =

j∈ J k M k j ( y k) for all y k ∈ Y k and for each k ∈ I .

Then M k is also an u.s.c. multimap with compact acyclic values for each k ∈ I .

8/13/2019 Lin, Chen, 2003



By Theorem 3.11, there exist (uk)k∈I ∈ Y and ( ¯ y k)k∈I ∈ Y such that ¯ y k ∈

M k(uk)and uk ∈ Bk( ¯ y k) for all k ∈ I . Let ¯ y k=(x k j ) j∈ J k ∈ M k(uk)=

j∈ J k M k j (uk),

then x k j ∈ M k j (uk) for each k ∈ I and j ∈ J k. This implies, there exists z k j ∈

F k j (x k j , uk) such that


+ , (4.10)

for all z k j ∈ F k j (x k j , uk), x k j ∈ Ak j (uk) and for all k ∈ I , j ∈ J k.

Applying Theorem 4.5 and following the same argument as in the proof of

Corollary 4.3, we have the following corollary.

Corollary

4.6. For each k∈

I and j∈

J k , let X k j be a nonempty compact convex subset of a locally convex t.v.s. Ek j satisfying the following conditions:

(i) F k j is a continuous multimap with nonempty closed values;

(ii) Ak j : Y k X k j is a multimap with nonempty convex values such that for

each x k j ∈ X k j , A−1k j

(x k j ) is open in Y k and ¯ Ak j is an u.s.c. multimap;

(iii) for all y k ∈ Y k , {x k j ∈ Ak j ( y k) : F k j (x k j , y k)∩wMinF k j ( Ak j ( y k) , y k) =∅}

is an acyclic set; and

(iv) for each k ∈ I , let Bk : Y k Y k be a multimap with convex values and

Y k ={intY k B−1k ( y k) : y k ∈ Y k}.

Then there exist ¯ y = ( ¯ y k)k∈I ∈ Y , u = (uk)k∈I ∈ Y , ¯ y k = (x k j ) j∈ J k ∈ ¯ Ak(uk) , uk ∈Bk( ¯ y k) , and z k j ∈ F k j (x k j , uk) such that


+ , (4.11)


Corollary 4.7. Let I be a finite index set. For each k ∈ I , let X k be a nonempty

compact convex subset of a locally convex t.v.s. and let f k : X → R be a continuous

function, and for each x k

∈ X k

, the function x k → f k(x k , x k

) is quasiconvex. Thenthere exist x = (x k)k∈I ∈ X and ¯ y = ( ¯ y k)k∈I ∈ X ,

f k

x k , ¯ y k≥ f k

x k , ¯ y k

, (4.12)

for all x k ∈ X k and for all k ∈ I .

Proof. For each k ∈ I , J k is a singleton. By assumption, {x k ∈ X k : f k(x k , y k) =

Min f k( X k , y k)} is a convex set. The conclusion of Corollary 4.7 follows from

Corollary 4.6 by taking Ak(x k)= X k and Bk(x k)= X k for all k ∈ I .

Remark 4.8. (i) The index I in Corollary 4.7 can be any index set.(ii) The conclusion between Corollary 4.7 and Nash equilibrium theorem

[16] is somewhat diff erent.

8/13/2019 Lin, Chen, 2003



5. Abstract economics with two families of players

In this section, we consider the following abstract economics with two families

of players.Let I and J be any index sets and let {U i}i∈I and {V j} j∈ J be families of lo-

cally convex t.v.s. For each i ∈ I and j ∈ J , let X i and Y j be nonempty con-

vex subsets each in U i and V j , respectively. Two families of abstract economy

Γ = ( X i , Ai , Bi , P i , Y j , C j , D j , Q j ), where Ai , Bi : Y :=

j∈ J Y j X i, and C j , D j :

X :=

i∈I X i Y j are constraint correspondences, P i : Y X i and Q j : X Y jare preference correspondences. An equilibrium for Γ is to find x = (x i)i∈I ∈ X

and ¯ y = ( ¯ y j ) j∈ J ∈ Y such that for each i ∈ I and j ∈ J , x i ∈ Bi( ¯ y ), ¯ y j ∈ D j (x ),

Ai( ¯ y )∩P i( ¯ y )=∅, and C j (x )∩Q j (x )=∅.

With the above notation, we have the following theorem.

Theorem 5.1. Let Γ= ( X i , Ai , Bi , P i , Y j , C j , D j , Q j )i∈I, j∈ J be two families of abstract

economics satisfying the following conditions:

(i) for each i ∈ I and y ∈ Y , co( Ai( y ))⊂ Bi( y ) and Ai( y ) is nonempty;

(ii) for each j ∈ J and x ∈ X , co(C j (x ))⊂D j (x ) and C j (x ) is nonempty;

(iii) for each i ∈ I , Y =

x i∈ X i intY [{(coP i)−(x i)∪ (Y \H i)}∩ A−i (x i)] , where

H i = { y ∈ Y : Ai( y )∩P i( y ) =∅} ;

(iv) for each j ∈ J , X =

y j∈Y j int X [{(coQ j )−( y j )∪( X \ M j )}∩C − j ( y j )] , where

M j = {x ∈ X : C j (x )∩Q j (x ) =∅} ;

(v) for each i∈I , j ∈ J and each x = (x i)i∈I ∈ X , y =( y j ) j∈ J ∈ Y , x i∈co(P i( y )) ,and y i ∈ coQ j (x ) ;

(vi) D j is compact.

Then there exist x = (x i)i∈I ∈ X and ¯ y = ( ¯ y j ) j∈ J ∈ Y such that x i ∈ Bi( ¯ y ) , ¯ y j ∈

D j (x ) , Ai( ¯ y )∩P i( ¯ y )=∅ , and C j (x )∩Q j (x )=∅ for all i ∈ I and j ∈ J .

Proof. For each i ∈ I and j ∈ J , we define multivalued maps Si, T i : Y X i by

Si( y )=coP i( y )∩ Ai( y ) , for y ∈H i ,

Ai( y ) , for y ∈ Y \H i ,

T i( y )=

coP i( y )∩Bi( y ) , for y ∈H i ,

Bi( y ) , for y ∈ Y \H i ,

(5.1)

and F j , G j : X Y i by

F j (x )=

coQ j (x )∩C j (x ) , for x ∈ M j ,

C j (x ) , for x ∈ X \ M j ,

G j (x )=

coQ j (x )∩D j (x ) , for x ∈ M j ,

D j (x ) , for x ∈ X \ M j .

(5.2)

8/13/2019 Lin, Chen, 2003



Then for each i ∈ I , j ∈ J , x ∈ X , y ∈ Y , co Si( y )⊆ T i( y ), Si( y ) is nonempty, and

coF j (x )⊆G j (x ), F j (x ) is nonempty.

For each i ∈ I , j ∈ J , x ∈ X , and y ∈ Y , it is easy to see that

S−i

x i=

coP i−

x i∪

Y \H i

∩ A−i

x i

(5.3)

and F − j ( y j )= [{(coQ j )−( y j )∪ ( X \ M j )}]∩C − j ( y j ). From (iii),

Y =

x i∈ X i

intY

coP i

−x i∪

Y \H i∩ A−

i

x i=

x i∈ X i

intY S−i

x i

. (5.4)

From (iv),

X =

y j∈Y j

int X

coQ j− y j∪ X \ X j

∩C − j

y j=

y j∈Y j

int X F − j y j

.

(5.5)

Then all conditions of Theorem 3.1 are satisfied. It follows from Theorem 3.1,

there exist x = (x i)i∈I ∈ X and ¯ y = ( ¯ y j ) j∈ J ∈ Y such that x i ∈ T i( ¯ y ) and ¯ y j ∈

G j (x ) for all i ∈ I and j ∈ J . By (v), we have x i ∈ Bi( ¯ y ), ¯ y j ∈ D j (x ), Ai( ¯ y )∩

P i( ¯ y )=∅, and C j (x )∩Q j (x )=∅.

AcknowledgmentThis research was supported by the National Science Council of Taiwan.

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Lai-Jiu Lin: Department of Mathematics, National Changhua University of Education,

Changhua 50058, Taiwan

E-mail address: [email protected]

Hsin I Chen: Department of Mathematics, National Changhua University of Education,

Changhua 50058, Taiwan

E-mail address: [email protected]

mailto:[email protected]




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is devoted to notes and bibliographical and historical remarks.

The book is addressed to a wide audience of specialists such as mathematicians, engineers, biologists,

and physicists. Besides serving as a reference tool for researchers in difference equations, this book can

also be easily used as a textbook for undergraduate or graduate classes. It is written at a level easy to

understand for college students who have had courses in calculus.

8/13/2019 Lin, Chen, 2003


Editorial Board

Topics: The conference will cover all the major

research areas in analysis and dynamics with focuses on

applications, modeling and computations.

Format: There will be one-hour plenary talks, and 30-

minute special session and contributed session talks

The conference will be diversified in subject so as to attract

all researchers in relevant fields; while it is concentrated

enough in each chosen subfield so as to sustain a largenumber of strong special sessions.

Deadlines: March 1, 2006, for registration and abstracts.

A proceedings will be published as a special edition of

DCDS. All papers will be refereed.

Funding from various sources including NSF may beavailable to qualified graduate students and youngresearchers.

Detailed program and updates will be posted athttp://aimSciences.orgFor updated information, contact

Alain Miranville, Chair of Organizing Committee (France)

Tel: (33) (0)5 49 49 68 91, Fax: (33) (0)5 49 49 69 01

[email protected]

Xin Lu, Conference Coordinator (USA)Tel: 910-962-3673, Fax: [email protected]

2006

6th AIMS INTERNATIONAL CONFERENCE ON

Dynamical Systems, DifferentialEquations Applications

Sunday-Wednesday June 25 - 28

Alberto Bressan(USA)

Odo Diekmann(Netherlands)

Pierre-Louis Lions(France)

Alexander Mielke(Germany)

Masayasu Mimura(Japan)

Peter Polacik(USA)

Patrizia Pucci(Italy)

Björn Sandstede(UK)

Lan Wen(China)

I N V I T E D P

L E N A R Y

S P E A K E R S

P r o c e e d i n g s

F u n d i n g

University of Poitiers

Poitiers, France

Organized by:

American Institute of Mathematical Sciences (AIMS)

Department of Mathematics, University of Poitiers.

Organizing Committee:Alain Miranville (Chair)Laurence CherfilsHassan EmamiradMorgan PierrePhilippe RogeonArnaud Rougirel

A I M S

&

Scientific Committee:

Shouchuan Hu (Chair)Antonio AmbrosettiJerry BonaMichel ChipotTom HouTatsien LiCarlangelo LiveraniHiroshi MatanoAlain MiranvilleWei-Ming NiJuergen SprekelsRoger Temam

C O M M I T T E E S

P r o g r a m a n d T o p i c s

C o n t a c t I n f o r m a t i o n

Documents

Lin, Chen, 2003