AAA Abstract and Applied Analysis 1687-0409 1085-3375 Hindawi Publishing Corporation 307903 10.1155/2014/307903 307903 Research Article Constrained Weak Nash-Type Equilibrium Problems Shuai W. C. 1, 2 Xiang K. L. 2 Zhang W. Y. 2 Li Sheng-Jie 1 Department of Mathematics Sichuan University for Nationalities Kangding 626000 China swun.edu.cn 2 Department of Economic Mathematics Southwestern University of Finance and Economics Chengdu 610000 China swufe.edu.cn 2014 1442014 2014 28 01 2014 20 03 2014 14 4 2014 2014 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  and the references therein. Notice that most payoffs may be one collection of things from many collections of things in the real world; reference  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, Zi a real topological vector space, and Xi(iI) a Hausdorff topological space. Let X=iIXi and Xi=jI,jiXj. For each xX, let xi and xi denote the ith coordinate of x and the projection of x on Xi, respectively. In the sequel, we may write x=(xi)iI=(xi,xi). For all iI, let Ci be a convex, closed, and pointed cone of Zi, with apex at the origin and with nonempty interior; let Fi:Xi×Xi2Zi and Si:X2Xi. We consider a class of constrained weak Nash-type equilibrium problems with multivalued payoff functions.

( C W N E P ) Finding an x¯=(x¯)iIX such that, for each iI, uiSi(x¯), and z¯iFi(x¯i,x¯i), there exists ziFi(ui,x¯i) satisfying (1)zi-z¯i-intCi. Then, x¯ is a solution of (CWNEP).

The following problems are special cases of (CWNEP).

If, for each iI, Fi is a singlevalued function, Zi=R, and Si(X)=Xi, (CWNEP) reduces to the Nash equilibrium problem .

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 PZ be a closed convex and pointed cone with intP, let S:C×D2C and T:C×D2D be two set-valued mappings, and let f,g:C×DZ 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¯)C×D such that x¯S(x¯,y¯), y¯T(x¯,y¯), and (2)f(x,y¯)-f(x¯,y¯)-intP,xS(x¯,y¯),g(x¯,y)-g(x¯,y¯)-intP,yT(x¯,y¯),

which was considered by Fu .

In this paper, we obtain the existence result for (CWNEP). Our existence theorem extends the main result of  from singlevalued case to multivalued case. In particular, if the payoff functions are singlevalued, our existence theorem extends the corresponding result in  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:X2Y is a set-valued mapping. T is said to be upper semicontinuous at x0X if, for each open set V containing T(x0), there is an open set U containing x0 such that, for each tU, T(t)V. It is said to be upper semicontinuous if it is upper semicontinuous at every point xX. T is said to be lower semicontinuous at x0X if, for each open set V with T(x0)V, there is an open set U containing x0 such that, for each tU, T(t)V. It is said to be lower semicontinuous on X if it is lower semicontinuous at every point xX. 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 xX.

From [7, Lemma 2], T is l.s.c. at xX if and only if, for any yT(x) and any net {xn}, xnx, there is a net {yn} such that ynT(xn) and yny. T is closed if and only if, for any net {xn}, xnx, and any net {yn}, ynT(xn), yny, one has yT(x).

Definition 1.

Assume that X is a Hausdorff topological space and Z is a real topological vector space. Let E be a nonempty convex subset of X, let H:E2Z be a set-valued mapping, and let PZ be a closed convex and pointed cone with intP. H is said to be generalized Luc's quasi-P-convex on E if, for every x1, x2E, λ[0,1], and yH(λx1+(1-λ)x2), there exist z1H(x1) and z2H(x2) such that (3)yz-C,zC(z1,z2), where C(z1,z2) is the set of all upper bounds of z1 and z2; that is, (4)C(z1,z2)={zZz1z-P,z2z-P}.

Remark 2.

Definition 1 is a generalization of the concept of Luc's quasi-P-convexity in .

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 PZ be a closed convex and pointed cone with eintP. The nonlinear scalarization function ξe:ZR is defined by (5)ξe(y)=min{tRyte-P}.

Lemma 4 (see [<xref ref-type="bibr" rid="B9">9</xref>]).

The nonlinear scalarization function has the following main properties:

ξe(·) is continuous and convex;

ξe(·) is subadditive; that is, ξe(y1+y2)ξe(y1)+ξe(y2);

ξe(·) is strictly monotone; that is, if y1-y2intP, then ξe(y1)>ξe(y2).

3. Existence for the Solution of (CWNEP)

Throughout this section, let Ei(iI) be a locally convex Hausdorff topological vector space, and let Zi be a real Hausdorff topological vector space. Let Xi be a nonempty, compact convex subset of Zi, respectively. Let CiZi be a closed convex and pointed cone with eiintCi. Suppose that Si:X2Xi is a continuous set-valued mapping with compact convex values and Fi:Xi×Xi2Zi is a continuous set-valued mapping with compact values. For every iI, set ξei(Fi(x,y))=uiFi(x,y)ξei(ui).

Lemma 5 (see [<xref ref-type="bibr" rid="B11">11</xref>]).

Let E be a nonempty compact convex subset of a locally convex Hausdorff topological space X. If G:E2E is upper semicontinuous and, for each xE, G(x) is a nonempty, closed, and convex subset, then there exists an x¯E such that x¯G(x¯).

Theorem 6.

Suppose that the following conditions hold:

Si:X2Xi is continuous with compact convex values;

Fi:Xi×Xi2Zi are continuous with compact values;

for each fixed xiXi, Fi(·,xi) is generalized Luc's quasi-Ci-convex.

Then, there exists an x¯Xi×Xi such that, for each iI, uiSi(x¯), and z¯iFi(x¯i,x¯i), there exists ziFi(ui,x¯i) satisfying (6)zi-z¯i-intCi.

Proof.

We define a set-valued mapping Ai:X2Xi by (7)Ai(x)={xiSi(x)uiSi(x)maxξei(Fi(ui,xi))vvvvvvvvv=minxiSi(x)maxξei(Fi(xi,xi))}.

It follows from [12, pages 110–119, Propositions 6 and 21] that maxξei(Fi(·,xi)) is upper semicontinuous for each fixed xiXi. By [12, page 112, Proposition 11], the set (8)θS(x)maxξei(F(θ,y)) is compact. Therefore, Ai(x) is nonempty for every xX.

Let (9){xn}X,xnx0,ui,nAi(xn),ui,nui,0. We must show that ui,0Ai(x0). First, note that ui,nAi(xn) and then ui,nSi(xn). As Si(·) is upper semicontinuous and the set Si(x0) is compact, it follows that ui,0Si(x0). Suppose that ui,0Ai(x0). Then, there exists a vector wi,0Si(x0) satisfying (10)maxξei(Fi(wi,0,x0i))<maxξei(Fi(ui,0,x0i)). As Si(·) is lower semicontinuous, there exists wi,nSi(xn), such that wi,nwi,0. It follows from compactness of Fi(wi,n,xni) that there exists zi,nFi(wi,n,xni) such that (11)ξei(zi,n)=maxξei(Fi(wi,n,xni)).

It follows from the upper semicontinuity of Fi(·,·) and the compactness of Xi×Xi that Fi(xi,xi) is compact. Hence, for the net {zi,n}, there exists a subnet of {zi,n} converging to zi,0. Without loss of generality, assume zi,nzi,0. Now we prove that (12)ξei(zi,0)=maxξei(Fi(wi,0,x0i)). Since the mapping Fi(·,·) is upper semicontinuous and the set Fi(wi,0,x0i) is compact, we have ξei(zi,0)ξei(Fi(wi,0,x0i)).

Now, suppose that ξei(zi,0)maxξei(Fi(wi,0,x0i)). Namely, there exists vi,0Fi(wi,0,x0i) such that ξei(vi,0)>ξei(zi,0). As Fi(·,·) is lower semicontinuous, there exists vi,nFi(wi,n,xni) such that vi,nvi,0. Since ξei(·) is continuous, for n large enough, (13)ξei(vi,n)>ξei(zi,n), which is a contradiction to (11).

From the compactness of Fi(ui,n,xni), we take z~i,nF(ui,n,xni) such that (14)ξei(z~i,n)=maxξei(Fi(ui,n,xni)). By the compactness of Fi(xi,xi), we can choose a converging subnet of {z~i,n}, which is denoted without loss of generality by the original net {z~i,n}. Assume z~i,nz~i,0. Similar to the preceding proof, we have (15)ξei(z~i,0)=maxξei(F(ui,0,x0i)). Then, by (10), ξei(zi,0)<ξei(z~i,0).

It follows from the continuity of ξei(·) that ξei(zi,n)ξei(zi,0) and ξei(z~i,n)ξei(z~i,0). Therefore, ξei(zi,n)<ξei(z~i,n), when n is large enough. It is said that (16)maxξei(Fi(wi,n,xni))<maxξei(Fi(ui,n,xni)). By the definition of Ai(·) and ui,nAi(xn), we have (17)maxξei(Fi(ui,n,xni))=minxi,nSi(xn)maxξei(Fi(xi,n,xni)). This, however, contradicts the fact ui,nAi(xn). Therefore, the mapping Ai(·) is closed.

Let ui,1,ui,2Ai(x), λ(0,1), and (18)α0=minθiSi(x)maxξei(Fi(θi,xi)). From the definition of Ai(·), we have ui,1,ui,2Si(x) and (19)maxξei(F(ui,1,xi))=maxξei(F(ui,2,xi))=α0. As Si(x) is convex-valued, λui,1+(1-λ)ui,2Si(x).

According to the generalized Luc's quasi-Ci-convexity of Fi(·,xi), we get that, for all ziFi(λui,1+(1-λ)ui,2,xi), there exist zi,1Fi(ui,1,xi) and zi,2Fi(ui,2,xi) such that (20)zizi-Ci,ziC(zi,1,zi,2). Without loss of generality, suppose l1=ξei(zi,1) and l2=ξei(zi,2), l1l2; we have zi,1l1ei-Ci and zi,2l2ei-Cil1ei-Ci. From (20), zil1ei-Ci. By the monotonicity of ξei(·), (21)ξei(zi)ξei(l1ei)=l1. As (22)l1max(maxξei(F(ui,1,xi)),maxξei(F(ui,2,xi)))=α0. therefore, ξei(zi)α0.

Since (23)ziFi(λui,1+(1-λ)ui,2,xi) is arbitrary, we have (24)maxξei(Fi(λui,1+(1-λ)ui,2,xi))α0. By the fact that Fi(λui,1+(1-λ)ui,2,xi) is compact and ξei(·) is continuous, there exists (25)z~iFi(λui,1+(1-λ)ui,1,xi) such that (26)ξei(z~i)=maxξei(Fi(λui,1+(1-λ)ui,2,xi)). Thus, ξei(z~i)α0. It follows from the definition of α0 that (27)maxξei(Fi(λui,1+(1-λ)ui,2,xi))=α0. Thus, λui,1+(1-λ)ui,2Ai(x); namely, Ai(x) is a convex set.

Define A:X2X by A(x)=ΠiIAi(x),xX. Therefore, A(x) is a nonempty, convex, and closed subset of X for each xX. Since Ai(·) is closed, so is A(·), and since A(x)X, X is compact, by [12, page 111, Corollary 9], A(·) is upper semicontinuous. By Lemma 5, there exists a point x¯X such that x¯A(x¯).

By the definition of A(·), we have(28)x¯iSi(x¯),maxξei(Fi(xi,x¯i))maxξei(F(x¯i,x¯i))maxξei(Fi(xi,x¯i))vvxiSi(x¯),iI.

From (28), z¯iFi(x¯i,x¯i), (29)maxξei(Fi(xi,x¯i))ξei(z¯i). By the compactness of Fi(xi,x¯i) and the continuity of ξei(·), there exists ziFi(xi,x¯i), such that ξei(zi)=maxξei(Fi(xi,x¯i)). Thus, for all z¯iF(x¯i,x¯i), there exists ziFi(xi,x¯i) such that ξei(z¯i)ξei(zi). Then, it follows from the subadditivity of ξei(·) that (30)ξei(zi-z¯i)0. By Lemma 4, we get (31)zi-z¯i-intP. 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 PZ be a closed convex and pointed cone with intP. Assume that

S:C×D2C and T:C×D2D are continuous and compact, and for each (x,y)C×D, S(x,y) and T(x,y) are nonempty, closed convex subsets;

f,g:C×DZ are continuous;

for any fixed yD, f(·,y) is Luc's quasi-P-convex; for any fixed xC, g(x,·) is Luc's quasi-P-convex.

Then there exists (x¯,y¯)C×D such that x¯S(x¯,y¯), y¯T(x¯,y¯), and (32)f(x,y¯)-f(x¯,y¯)-intP,xS(x¯,y¯),g(x¯,y)-g(x¯,y¯)-intP,yT(x¯,y¯).

Remark 8.

Since both the class of properly quasi-P-convex functions and the class of P-convex functions (see ) 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+3 and let S:C×D2C and T:C×D2D be defined as S(x,y)=C and T(x,y)=D, respectively. For all (x,y)R2, let (33)f(x,y)=(x2,1-x2,y),g(x,y)=(x,y2,1-y2). It is clear that the mappings f and g are not properly quasi-P-convex (see ), but all the conditions of Corollary 7 hold. It is easy to see from  that both the class of properly quasi-P-convex functions and the class of P-convex functions (see ) 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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