Hindawi Publishing Corporation Abstract and Applied Analysis Volume 2014, Article ID 307903, 4 pages http://dx.doi.org/10.1155/2014/307903

Research Article

Constrained Weak Nash-Type Equilibrium Problems

W. C. Shuai,1,2 K. L. Xiang,2 and W. Y. Zhang2

1 Department of Mathematics, Sichuan University for Nationalities, Kangding 626000, China

2 Department of Economic Mathematics, Southwestern University of Finance and Economics, Chengdu 610000, China Correspondence should be addressed to W. Y. Zhang; zhangwy@swufe.edu.cn

Received 28 January 2014; Accepted 20 March 2014; Published 14 April 2014 Academic Editor: Sheng-Jie Li

Copyright © 2014 W. C. Shuai et al. 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.

A constrained weak Nash-type equilibrium problem with multivalued payoff functions is introduced. By virtue of a nonlinear scalarization function, some existence results are established. The results extend the corresponding one of Fu (2003). In particular, if the payoff functions are singlevalued, our existence theorem extends the main results of Fu (2003) by relaxing the assumption of convexity.

1. Introduction

For a long time, real valued functions have played a central role in game theory. More recently, motivated by applications to real-world situations, many authors have studied the existence of solutions of Pareto equilibria of multiobjective game with vector payoff functions; for example, see [1-4] and the references therein. Notice that most payoffs may be one collection of things from many collections of things in the real world; reference [5] studied the constrained Nash-type equilibrium problem with multivalued payoff functions and proved the existence results.

In the paper, let I be an index set, Z* a real topological vector space, and X* (i el) a Hausdorff topological space. Let X = n^i^t and X* = UjeijtiXj. For each x e X, let x* and x* denote the ith coordinate of x and the projection of x on X', respectively. Inthesequel, wemaywrite x = (x*)*ei = (x*,x'). For all i e I, let C* be a convex, closed, and pointed cone of Z*, with apex at the origin and with nonempty interior; let F* : X* x X* ^ 2Z> and S* : X ^ 2X>. We consider a class of constrained weak Nash-type equilibrium problems with multivalued payoff functions.

(CWNEP) Finding an x = (x)*ei e X such that, for each i el, u* e S* (x), and z* e F*(x*,x'), there exists z* e F*(u*,x') satisfying

z* -z* i- int C*. (1)

Then, x is a solution of (CWNEP).

The following problems are special cases of (CWNEP).

(i) If, for each i e I, F* is a singlevalued function, Z* = R, and St(X) = X*, (CWNEP) reduces to the Nash equilibrium problem [6].

(ii) Let X, Y, and Z be real Hausdorff topological vector spaces, and let C and D be two nonempty subsets of X and Y, respectively. Let P c Z be a closed convex and pointed cone with int P = 0, let S : C x D ^ 2C and T : C x D ^ 2D be two set-valued mappings, and let f,g:CxD ^ Z be two vector-valued mappings. The problem (CWNEP) reduces to a class of symmetric vector quasiequilibrium problems (for short, SVQEP) that consists in finding (x,y) e Cx D such that x e S(x, y), ~y e T(x, y), and

f (x, y) - f (x, y) i - int P, Vx e S (x, y),

_ _ _ _ _ (2) g (x, y) - g (x, y) i - int P, Vy eT (x, y),

which was considered by Fu [7].

In this paper, we obtain the existence result for (CWNEP). Our existence theorem extends the main result of [6] from singlevalued case to multivalued case. In particular, if the payoff functions are singlevalued, our existence theorem extends the corresponding result in [7] by relaxing the assumption of convexity.

The rest of the paper is organized as follows. In Section 2, we state some notations and preliminary results for multivalued mappings. We recall the nonlinear scalarization function and its properties. In Section 3, we show existence result for (CWNEP).

2. Preliminaries

Let us first recall some definitions of continuity for set-valued mappings. Let X and Y be two topological spaces. T : X ^ 2y is a set-valued mapping. T is said to be upper semicontinuous at x0 e X if, for each open set V containing T(x0), there is an open set U containing x0 such that, for each t e U, T(t) c V. It is said to be upper semicontinuous if it is upper semicontinuous at every point x e X. T is said to be lower semicontinuous at x0 e X if, for each open set V with T(x0) nV =0, there is an open set U containing x0 such that, for each t e U, T(t) n V = 0. It is said to be lower semicontinuous on X if it is lower semicontinuous at every point x e X. T is said to be continuous at x0 if it is both upper semicontinuous and lower semicontinuous at x0. It is said to be continuous on X if it is continuous at every point x e X.

From [7, Lemma 2], T is l.s.c. at x e X if and only if, for any y e T(x) and any net [xn], xn ^ x, there is a net [yn] such that yn e T(xn) and yn ^ y. T is closed if and only if, for any net [xn], xn ^ x, and any net [yn], yn e T(xn), yn ^ y, one has y e T(x).

Definition 1. Assume that X isa Hausdorff topological space and Z is a real topological vector space. Let E be a nonempty convex subset of X, let H : E ^ 2Z be a set-valued mapping, and let P c Z be a closed convex and pointed cone with int P = 0. H is said to be generalized Luc's quasi-P-convex on E if, for every x1, x2 e E, X e [0,1], and y e H(Xx1 + (1-X)x2), there exist zx e H(x1) and z2 e H(x2) such that

yez-C, zeC(z1,z2 ),

where C(z1,z2) is the set of all upper bounds of z1 and z2; that

C (z1, z2 ) = {z eZ\z1 ez-P,z2 e z-P}. (4)

Remark 2. Definition 1 is a generalization of the concept of Luc's quasi-P-convexity in [8].

Now we recall the definition of the nonlinear scalarization function [9,10] as follows.

Definition 3. Let Z be a real topological vector space, and let P c Z be a closed convex and pointed cone with e e int P. The nonlinear scalarization function %e : Z ^ R is defined by

(y) = min [t eR \ y e te-P}.

Lemma 4 (see [9]). The nonlinear scalarization function has the following main properties:

(i) $e(■) is continuous and convex;

(ii) $e() is subadditive; that is, £>e(y1 +y2) < ^e(y1)+^e(y2);

(iii) $e(■) is strictly monotone; that is, ify1 -y2 e int P, then

> ^2).

3. Existence for the Solution of (CWNEP)

Throughout this section, let Et (i e I) be a locally convex Hausdorff topological vector space, and let Zi be a real Hausdorff topological vector space. Let Xt be a nonempty, compact convex subset of Zi, respectively. Let Ci c Zi be a closed convex and pointed cone with ei e int Ci. Suppose that Sj : X ^ 2X' is a continuous set-valued mapping with compact convex values and Fj : Xj xX' ^ 2 ' is a continuous set-valued mapping with compact values. For every i e I, set ^ (F'(X,y)) = \JUi€F(X,y) te, (u').

Lemma 5 (see [11]). Let E be a nonempty compact convex subset of a locally convex Hausdorff topological space X. If G : E ^ 2e is upper semicontinuous and, for each x e E, G(x) is a nonempty, closed, and convex subset, then there exists an x e E such that x e G(x).

Theorem 6. Suppose that the following conditions hold:

(i) S' : X ^ 2X is continuous with compact convex values;

(ii) Fj : Xj x X' ^ 2Zi are continuous with compact values;

(iii) for each fixed x{ e X', F'(^x') is generalized Luc's quasi-C'-convex.

Then, there exists an x e X' x X' such that, for each i el, u{ e S'(x), andZ' e Fj(x',x'), there exists Zj e Fi(ui,'xt) satisfying

Zj - Zj i - int Cj.

(3) Proof. We define a set-valued mapping A{:X ^ 2Xi by

Ai (x) =

Uj e Sj (x) \ max (Fj (u„x))

= min ^ max %e, (Fj (xj,x1))

Xi^Si(x)

It follows from [12, pages 110-119, Propositions 6 and 21] that max(Fj(-, x')) is upper semicontinuous for each fixed x* e X*. By [12, page 112, Proposition 11], the set

y max ^ (F(d,y)) ^

QtS(x)

is compact. Therefore, A¡(x) is nonempty for every x e X. Let

{xn} e X, xn x,

h,n e Aj (xn)

Ui,n "j,Q.

We must show that uio e A*(xo).First,notethatuin e Ai(xn) and then uin e Si(xn). As St(-) is upper semicontinuous and the set S*(xo) is compact, it follows that uio e S¡(x0). Suppose that uio i A*(xo). Then, there exists a vector wio e S*(xo) satisfying

max ^ {pi {wi,O,X0)) < max ^ {pi {ui,O'X0 )) • (10)

As S*(■) is lower semicontinuous, there exists win e Si(xn), such that win ^ wio. It follows from compactness of F*(win, x'n) that there exists zin e F*(win, x'n) such that

^ (z',n) = max^ {F' {W'n X'n)) •

It follows from the upper semicontinuity of Fi(■, ■) and the compactness of X* x Xi that F(x\ xi) is compact. Hence, for the net {zin}, there exists a subnet of [zi n] converging to zifl. Without loss of generality, assume zin ^ zi o. Now we prove that

^ (z',o) = max ^ {F' {W',0'X0 )) •

Since the mapping F(, ■) is upper semicontinuous and the set Fi(wi,0, x'0) is compact, we have (zifi) e (Fi(wifi, x*0)).

Now, suppose that (zio) = max (Fi(wio,x'o)). Namely, there exists vi0 e Fi(wi0,x'0) such that

L (v i,0) > $e, (zi,0). As Fi(■'■) is lower semicontinuous,

there exists vin e Fi(win,x'n) such that vin ^ vio. Since (■) is continuous, for n large enough,

^ {V',n) > ^ (z',n) '

which is a contradiction to (11).

From the compactness of Fi(uin,x'n), we take zin e F(uini, x'n) such that

^ (z',n) = max ^ {F' iui,n' X'n)) •

By the compactness of f*(x*, x'), we can choose a converging subnet of {zin}, which is denoted without loss of generality by the original net {zin}. Assume zi n ^ zi0. Similar to the preceding proof, we have

^ (z',o) = max(F{u',o,x'o)) •

Then, by (10), ^(zho)<Ze, (zuol

It follows from the continuity of %e (■) that %e (zi n) ^ (z*fi) and ^(Z*,J ^ (z*,'). Therefore, ^(Z*,J < ^(z*,n), when n is large enough. It is said that

max^ {Fi {wi,n' Xn)) < max^ {Fi {ui,n' x'n)) • (16)

By the definition of A *(■) and ui n e A*(xn), we have

max^ {F* {"in x'n)) = min U max^ {F* {Xi,n> X'n)) •

This, however, contradicts the fact uin e A*(xn). Therefore, the mapping A *(■) is closed.

Let ui1,ui2 e A*(x), X e (0,1), and

ao = min U max^ {f* {0*, x')) • (18)

di^Siix)

From the definition of A(-), we have u^, ui 2 e S*(x) and max^ {f{u*i,x)) = max^ {F{uia,x')) = a(19)

As S*(x) is convex-valued, Xu^ + (1 - X)ui2 e S*(x).

According to the generalized Luc's quasi-C*-convexity of F*(^, x*),wegetthat, forall z't e F*(Xui1 + (1 - X)ui 2, x'),there exist z*i e F*(ui1,xi) and z^2 e F*(ui 2, x') such that

z'i ez' -C', VZ' eCtz'^Z'J.

Without loss of generality, suppose l1 = %e.(zi x) and l2 = Ze,(z*a), h ^ l2> we have z^ e lie* - C* and z^ e ^e* -C* c l1e* - C*. From (20), z't e l1e* - C*. By the monotonicity of

He,(■),

Ze, {4) * Ze, (lie,) = ll• (21)

li < max {max{f{uiA,x )), max{F {u*a, x*))) = ao•

therefore, (z[) < ao. Since

z.1 e F* {Xu*i + (1 - A) ui22, x*) (23)

is arbitrary, we have

max{f* {Xu*i + (1 - X) ui 2, x')) < a(24)

By the fact that F*(Xui1 + (1 - X)ui 2, x') is compact and %e (■) is continuous, there exists

zt e F* {XuiA + (1-X)ui1,xi) (25)

such that

^ (z*) = max^ {F* {Xu*1 + (1 - A) u*a, x')) • (26)

Thus, %e (z*) < ao. It follows from the definition of ao that

max{f* {Xu*1 + (1 - X) ui 2, x')) = a(27)

Thus, Xu'1 + (1-X)ui 2 e A*(x);namely, A*(x) is a convex set.

Define A : X ^ 2X by A(x) = ni€lA'(x),Vx e X. Therefore, A(x) is a nonempty, convex, and closed subset of X for each x e X. Since A*(-) is closed, so is A(-), and since A(x) c X, X is compact, by [12, page 111, Corollary 9], A(^) is upper semicontinuous. By Lemma 5, there exists a point x e X such that x e A(x).

By the definition of A(-), we have x, e S, (x),

max(f; (xi,xi)) > max(F (x,,x')) (28) e Si (%), i el.

From (28), Vzi e Fi(xi,xi),

max^ (Fi (Xi,*)) > ^ (z,). (29)

By the compactness of f,(x,,x') and the continuity of (•), there exists z, e Fi(xi,xi), such that (z,) = max(Fi(xi,xi)). Thus, for all z, e FCx^x'), there exists z, e Fi(xi,x') such that (z,) < (z,). Then, it follows from the subadditivity of (•) that

By Lemma 4, we get

(z, - z¡) > 0.

Z: - Z¡<t- int P.

So x is a solution of (CWNEP) and this completes the proof.

Let X, Y, and Z be real Hausdorff topological vector spaces, and let C and D be two compact subsets of X and Y, respectively.

Corollary 7. Let X, Y, and Z be real Hausdorff topological vector spaces, and let C and D be two nonempty subsets of X and Y, respectively. Let P c Z be a closed convex and pointed cone with int P = 0. Assume that

(1) S : C xD ^ 2C and T : C xD ^ 2D are continuous and compact, and for each (x, y) e C x D, S(x, y) and T(x, y) are nonempty, closed convex subsets;

(2) f,g:CxD ^ Z are continuous;

(3) for any fixed y e D, /(•, y) is Luc's quasi-P-convex; for any fixed x e C, g(x, •) is Luc's quasi-P-convex.

Then there exists (x,~y) e C x D such that x e S(x, y), ~y e T(x, y), and

f (x, y) - f (x, y) Í - int P, Vx e S (x, y), g (x, y) - g (x, y) Í - int P, Vy e T (x, ~y).

Remark 8. Since both the class of properly quasi-P-convex functions and the class of P-convex functions (see [7]) are larger than the class of Luc's quasi-P-convex functions, Corollary 7 improves [7, Theorem].

Example 9. Suppose that X = Y = R, C = D = [0,1], and P = R+ and let S : C x D ^ 2C and T : C x D ^ 2D be defined as S(x, y) = C and T(x, y) = D, respectively For all (x, y) e R2, let

f (x, y) = (x2,1 - X2, y) , g(x, y) = (x, y2,!- y2) .

It is clear that the mappings f and g are not properly quasi-P-convex (see [7]), but all the conditions of Corollary 7 hold. It is easy to see from [7] that both the class of properly quasi-P-convex functions and the class of P-convex functions (see [7]) are larger than the class of Luc's quasi-P-convex functions, and then Corollary 7 improves [7, Theorem].

Conflict of Interests

The authors declare that there is no conflict of interests regarding the publication of this paper.

Acknowledgment

This paper is supported by the Fundamental Research Funds for the Central Universities (JBK130401 and JBK140924).

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