Hindawi Publishing Corporation

International Journal of Mathematics and Mathematical Sciences Volume 2009, Article ID 430930,10 pages doi:10.1155/2009/430930

Research Article

On System of Generalized Vector Quasiequilibrium Problems with Applications

Jian-Wen Peng1 and Lun Wan2

1 College of Mathematics and Computer Science, Chongqing Normal University, Chongqing 400047, China

2 Scientific Research Office, Chongqing Normal University, Chongqing 400047, China

Correspondence should be addressed to Jian-Wen Peng, jwpeng6@yahoo.com.cn Received 3 July 2009; Accepted 19 August 2009 Recommended by Marco Squassina

We introduce a new system of generalized vector quasiequilibrium problems which includes system of vector quasiequilibrium problems, system of vector equilibrium problems, and vector equilibrium problems, and so forth in literature as special cases. We prove the existence of solutions for this system of generalized vector quasi-equilibrium problems. Consequently, we derive some existence results of a solution for the system of generalized quasi-equilibrium problems and the generalized Debreu-type equilibrium problem for both vector-valued functions and scalar-valued functions.

Copyright © 2009 J.-W. Peng and L. Wan. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

1. Introduction and Formulations

In the resent years, the vector equilibrium problems have been studied in [1-7] and the references therein which is a unified model of several problems, for instance, vector varia-tional inequality, vector variational-like inequality, vector complementarity problems, vector optimization problems. A comprehensive bibliography on vector equilibrium problems, vector variational inequalities, vector variational-like inequalities and their generalizations can be found in a recent volume [1]. Ansari and Yao [8] and Chiang et al. [9] introduced and studied some vector quasi-equilibrium problems which generalized those quasi-equilibrium problems in [10-17] to the case of vector-valued function. Very recently, the system of vector equilibrium problems was introduced by Ansari et al. [18] with applications in Nash-type equilibrium problem for vector-valued functions. The system of vector quasi-equilibrium problems was introduced by Ansari et al. [19] with applications in Debreu-type equilibrium problem for vector-valued functions. As a generalization of the above models, we introduce a new system of generalized vector quasi-equilibrium problems, that is, a family of generalized quasi-equilibrium problems for vector-valued maps defined on a product set.

Throughout this paper, for a set A in a topological space, we denote by co A, int A, co A the convex hull, interior, and the convex closure of A, respectively.

Let I be an index set. For each i e I, let Zi, Ei and let Fi be topological vector spaces. Consider two family of nonempty convex subsets {Xi}ieI with Xi c Ei and {Yi}ieI with Yi c Fi. Let

E = nEi, X = nXi, F = nFi, Y = nYi- (1.1)

ieI ieI ieI ieI

An element of the set Xi = ri/e^X will be denoted by xi, therefore, x e X will be written as x = (xi,xi) e Xi x Xi. Similarly, an element of the set Y will be denoted by y = (yi,yi) e Yi x Yi. For each i e I, let Q : X ^ 2Zi, Di : X x Y ^ 2Xi and Ti : X x Y ^ 2Yi be set-valued maps with nonempty values, and let fi : X x Y x Xi ^ Zi be a the vector-valued function. Then the system of generalized vector quasi-equilibrium problems (in Short, SGVQEP) is to find (x,y) = (xi,xi,yi,yi) in X x Y such that for each i e I,

xi e D^(x,y), yi e T^(x,y) : f^(x,y,z^ e - int Ci(x), Vzi e D^(x,y). (1.2)

Here are some special cases of the (SGVQEP).

(i) For each i e I, let $i : X x Y ^ Zi be a vector-valued function. We define a trifunction fi : X x Y x Xi ^ Zi as fi(x, y, ui) = $i(xi, y, ui) - $i(x, y), V(x, y,ui) e X x Y x Xi. Then the (SGVQEP) reduces to the generalized Debreu-type equilibrium problem for vector-valued functions (in short, G-Debreu VEP), which is to find (x,y) = (x\xi,y\yi) in X x Y such that for each i e I,

xi e Di(x,y), y e T^x,y) : - $^(x,y) / - intCi(x), Vzi e D^x,y). (1.3)

(ii) We denote by R and R+ the set of real numbers and the set of real nonnegative numbers, respectively. For each i e I, if Zi = R, and Ci (x) = R+ for all x e X, then the (SGVQEP) reduces to the system of generalized quasi-equilibrium problems (in short, SGQEP), which is to find (x,y) = (xi,xi,yi,yi) in X x Y such that for each i e I,

xi e D^(x,y), y e Ti(x,y) : f^x,y,z^ > 0, Vzi e D^(x,y). (1.4)

And the G-Debreu VEP reduces to the generalized Debreu-type equilibrium problem for scalar-valued functions (in short, G-Debreu EP), which is to find (x,y) = (xi,xi,yi,yi) in X x Y such that for each i e I,

xi e Di(x,y), yi e Ti(x,y) : > $^(x,y), Vzi e D^x,y). (1.5)

(iii) Let Y = {y}. For each i e I, let Di(x,y) = Ai(x), Ti(x,y) = {yi} for all x e X, where Ai : X ^ 2Xi is a set-valued map. We define a function yi : X x Xi ^ Zi and a function hi : X x Y ^ Zi as yi(x,zi) = fi(x,y,zi), for all (x,zi) e X x Xi, and hi(x) = $i(x,y), for all x e X, then (SGVQEP) and (G-Debreu VEP), respectively, reduce to the system of vector quasi-equilibrium problems and the (Debreu VEP) introduced by Ansari

et al. [19] which contain those mathematical in [18,20] as special cases. The (SGQEP) reduces to the mathematical models in [21, page 286] and [22, pages 152-153] and the (G-Debreu EP) reduces to the abstract economy in [23, page 345] which contains the noncooperative game in [24] as a special case.

(iv) If the index set I is singleton, D(x,y) = Di(x), T(x,y) = T(x), and C(x) = C, then the (SGVQEP) becomes the implicit vector variational inequality in [9] and the (SGQEP) reduces to the quasi-equilibrium problem investigated in [14-17].

The rest of this paper is arranged in the following manner. The following section deals with some preliminary definitions, notations and results which will be used in the sequel. In Section 3, we establish existence results for a solution to the (SGVQEP) and the (SGQEP) with or without involving O-condensing maps by using similar techniques in [19]. In Section 4, as applications of the results of Section 3, we derive some existence results of a solution for the (G-Debreu VEP) and the the (G-Debreu EP).

2. Preliminaries

In order to prove the main results, we need the following definitions.

Definition 2.1 ([19, 25]). Let M be a nonempty convex subset of a topological vector space E and Z a real topological space with a closed and convex cone P with apex at the origin. A vector-valued function y : M ^ Z is called

(i) P-quasifunction if and only if, for all z e Z, the set {x e M : y(x) e z - P} is convex,

(ii) natural P-quasifunction if and only if, Vx,y e M, and X e [0,1], y(Xx + (1 - X)y) e c°{y(x),y(y)}-p.

Definition 2.2 ([13]). Let X and Y be two topological spaces. T : X ^ 2Y be a set-valued map. Then T is said to be upper semicontinuous if the set {x e X : T(x) c V} is open in X for every open subset V of Y. Also T is said to be lower semicontinuous if the set {x e X : T(x) n V} is open in X for every open subset V of Y ■ T is said to have open lower sections if the set T-1(y) = {x e X : y e T(x)} is open in X for each y e Y.

Definition 2.3 ([26]). Let E be a Hausdorff topological space and L a lattice with least element, denoted by 0. A map O : 2E ^ L is a measure of noncompactness provided that the following conditions hold VM, N e 2E:

(i) O(M) = 0 iff M is precompact (i.e., it is relatively compact),

(ii) O(coM) = O(M),

(iii) O(M U N) =max{O(M), O(N)}.

Definition 2.4 ([26]). Let O : 2E ^ L be a measure of noncompactness on E and X c E. A set-valued map T : X ^ 2E is called O-condensing provided that, if M c X with O(T(M)) > O(M), then M is relatively compact.

Remark 2.5. Note that every set-valued map defined on a compact set is O-condensing for any measure of noncompactness O. If E is locally convex and T : X ^ 2E is a compact set-valued map (i.e., T(X) is precompact), then T is O-condensing for any measure of noncompactness O. It is clear that if T : X ^ 2E is O-condensing and T* : X ^ 2E satisfies T*(x) c T(x) Vx e X, then T* is also O-condensing.

We will use the following particular forms of two maximal element theorems for a family of set-valued maps due to Deguire et al. [27, Theorem 7] and Chebbi and Florenzano [28, Corollary 4].

Lemma 2.6 ([19,27]). Let {Xi}ieI be a family of nonempty convex subsets where each Xi is contained in a Hausdorff topological vector space Ei, For each i e I, let Si : X ^ 2Xi be a set-valued map such that

(i) for each i e I, Si(x) is convex,

(ii) for each x e X, xi / Si(x),

(iii) for each yi e Xi, Sf1 (yi) is open in X.

(iv) there exist a nonempty compact subset N of X and a nonempty compact convex subset Bi of Xi for each i e I such that for each x e X \ N there exists i e I satisfying Si(x) n Bi = 0. Then there exists x e X such that Si(x) = 0 for all i e I.

Lemma 2.7 ([19,28]). Let I be any index set and {Xi}ieI be a family of nonempty, closed and convex subsets where each Xi is contained in a locally convex Hausdorff topological vector space Ei. For each i e I, let Si : X ^ 2Xi be a set-valued map. Assume that the set-valued map S : X ^ 2X defined as S(x) = n ieISi(x), Vx e X, is O-condensing and the conditions (i), (ii), (iii) of Lemma 2.6 hold. Then there exists x e X such that Si(x) = 0 for all i e I.

3. Existence Results

An existence result of a solution for the system of generalized vector quasi-equilibrium problems with or without O-condensing maps are will shown in this section.

Theorem 3.1. Let I be any index set. For each i e I, let Zi be a topological vector space, let Ei and Fi be two Hausdorff topological vector spaces, let Xi c Ei and Yi c Fi be nonempty and convex subsets, let Di : X x Y ^ 2Xi and Ti : X x Y ^ 2Yi be set-valued maps with nonempty convex values and open lower sections, and the set Wi = {(x,y) e X x Y : xi e Di(x,y) and yi e Ti(x,y)} be closed in X x Y and let fi : X x Y x Xi ^ Zi be a vector-valued function. For each i e I, let Ci : X ^ 2Zi be a set-valued map such that Ci(x) be a proper closed and convex cone with apex at the origin and int Ci(x) = 0 for all x e X and Pi = nxeXCi(x). Assume that

(i) for all x = (xi, xi) e X,for all y e Y, fi(x, y, xi) e - int Q(x);

(ii) for each (x,y) e X x Y, zi ^ fi(x,y,zi) is natural Pi-quasifunction;

(iii) for all zi e Xi, the set {(x, y) e X x Y : fi(x, y, zi) e - int Ci(x)} is closed in X x Y;

(iv) there exist nonempty and compact subsets N c X and K c Y and nonempty, compact and convex subsets Bi c Xi, Ai c Yi for each i e I such that V(x,y) = (xi,xi,y) e X x Y \ N x K 3i e I and 3ui e Bi, Vi e Ai satisfying ui e Di(x,y), Vi e Ti(x,y) and fi(x,y,Ui) e -intCi(x).

Then, there exists (x,y) = (xi,xi,yl,yi) in X x Y such that for each i e I,

xi e D^(x,y), y{ e T-i(x,y) : f^(x,y,z^ e - int Ci(x), Vzi e D^(x,y). (3.1) That is, the solution set of the (SGVQEP) is nonempty.

Proof. For each i e I, let us define a set-valued map Qi : X x Y ^ 2Xi by

Qi(x,y) = {zi e Xi: fi(x,y,zi) e-intCi(x)}, V(x,y) e X x Y. (3.2)

Then, Vi e I and V(x, y) e X x Y, Qi(x, y) is a convex set.

To prove it, let us fix arbitrary i e I and (x,y) e X x Y. Let zi1/zi2 e Qi(x,y) and X e [0,1], then we have

fi(x,y, zij^ e-int Ci(x), for j = 1,2. (3.3)

Since fi(x, y, ■) is natural Prquasifunction, 3^ e [0,1] such that

fi(x,y,Xzh +(1 - X)zh) e tfi(x,y,zh) + (1 - p)fi(x,y,zi2) - Pi. (3.4) From (3.3) and (3.4), we get

f^x,y,Xzi1 + (1 - X)zi2) e - int Ci(x) - int Ci(x) - Pi c - int Ci(x). (3.5)

Hence Xzi1 +(1 - X)zi2 e Qi(x,y) and, therefore, Qi(x,y) is convex.

It follows from condition (i) that, for each i e I and for all (x, y) = (xi, xi, y) e X x Y,

xi /Qi (x,y). (3.6)

It follows from condition (iii) that for each i e I and each zi e Xi, the set

Qi-1 (zi) = {(;x,y) e X x Y : fl(x,y,zl) e- int Ci(x)} (3.7)

is open in Xi. That is, Qi has open lower sections on X x Y. For each i e I, we also define another set-valued map Si : X x Y ^ 2XixYi by

f[Di(x,y) n Qi(x,y)\ x Ti(x,y), if (x,y) e Wu Si{x,y) = \ (3.8)

[D^x,^ x T^x,^, if (x,y) / Wi.

Then, it is clear that Vi e I and V(x,y) e X x Y, Si(x,y) is convex, and (xi,yi) / Si(x,y). Since Vi e I and V(ui, vi) e Xi x Yi,

Si~1(ui,vi) = \Q-1(Ui) n fa-\ui)) n (TrVi))]

u [(X x Y \ Wi) n (Df\u)) n (V1^))],

and Di^1(ui), Ti^1(vi), Qi^1(ui), and X x Y \ Wi are open in X x Y, we have Si-1(ui,vi) is open in X Y.

From condition (iv), there exist a nonempty and compact subset N x K c X x Y and a nonempty, compact, and convex subset Bi x Ai c Xi x Yi for each i e I such that V(x, y) = (xi,xi,y) e X x Y \ N x K 3i e I and 3(ui,vi) e Si(x,y) n (Bi x Ai). Hence, by Lemma 2.6, 3(x,y) e X x Y such that Si(x,y) = 0,Vi e I. Since Vi e I and V(x,y) e X x X, Di(x,y) and Ti(x,y) are nonempty, we have (x,y) e Wi and Di(x,y) n Qi(x,y) = 0, Vi e I. This implies (x,y) e Wi and Di(x,y) n Qi(x,y) = 0, Vi e I. Therefore, Vi e I,

xi e Di(x,y), yi e Ti(x,y), f^x,y,z^ / - intCi(x), Vzi e D^(x,y). (3.10)

That is, the solution set of the (SGVQEP) is nonempty. □

Remark 3.2. (1) The condition (iii) of Theorem 3.1 is satisfied if the following conditions hold Vi e I:

(a) Ci : X ^ 2Zi is a set-valued map such that int Ci(x) / 0 for each x e X and the set-valued map Mi = Zi \ (-int Ci) : X ^ 2Zi is upper semicontinuous;

(b) for all zi e Xi, the map (x, y) ^ fi(x,y, zi) is continuous on X x Y;

(2) If Vi e I, and Vx e X, Ci(x) = Ci, a (fixed) proper, closed and convex cone in Zi, then the condition (ii) and (iii) of Theorem 3.1 can be replaced, respectively, by the following conditions:

(c) Vi e I, the vector-valued function V(x,y) e X x Y, zi ^ fi(x,y,zi) is Ci-quasifunction;

(d) Vi e I, Vzi e Xi, the map (x,y) ^ fi(x,y,zi) is Ci-upper semicontinuous on X x Y;

(3) Theorem 3.1 extends and generalizes in [19, Theorem 2], [20, Theorem 2.1] and [18, Theorem 2.1] in several ways.

(4) If Vi e I, Xi is a nonempty, compact and convex subset of a Hausdorff topological vector space Ei, then the conclusion of Theorem 3.1 holds without condition (iv).

Theorem 3.3. Let I be any index set. For each i e I, let Zi be a topological vector space, let Ei and Fi be two locally convex Hausdorff topological vector spaces, let Xi c Ei and Yi c Fi be nonempty, closed and convex subsets, let Di: X x Y ^ 2Xi and Ti : X x Y ^ 2Yi be set-valued maps with nonempty convex values and open lower sections, the set Wi = {(x,y) e X x Y : xi e Di(x,y) and yi e Ti(x,y)} be closed in X x Y and fi : X x Y x Xi ^ Zi be a vector-valued function. For each i e I, let Ci : X ^ 2Zi be a set-valued map such that Ci(x) be a proper closed and convex cone with apex at the origin and int Ci(x)/ 0 for all x e X and Pi = nxeXCi(x). Assume that the set-valued map D x T = (nieIDi x nieITi) : Xx Y ^ 2XxY defined as (D x T)(x,y) = nieIDi(x,y)xnieITi(x,y), V(x,y) e X x Y, is O-condensing and for each i e I, the conditions (i), (ii) and (iii) of Theorem 3.1 hold. Then the solution set of the (SGVQEP) is nonempty.

Proof. In view of Lemma 2.7 and the proof of Theorem 3.1, it is sufficient to show that the set-valued map S : X x Y ^ 2XxY defined as S(x,y) = nieISi(x,y),for all (x,y) e X x Y, is O-condensing, where Si's are the same as in the proof of Theorem 3.1. By the definition of Si, Si(x,y) c Di(x,y) x Ti(x,y) for all (x,y) e X x Y and for each i e I, and therefore S(x, y) c D(x, y) x T(x, y) for all (x, y) e X x Y. Since D x T is O-condensing, by Remark 2.5, we have S is also O-condensing. □

By Theorem 3.1 and Remark 3.2, we can easily get the following result.

Corollary 3.4. Let I be any index set. For each i e I, let Ei and Fi be two Hausdorff topological vector spaces, let Xi c Ei and Yi c Fi be nonempty and convex subsets, let Di : X x Y ^ 2Xi and Ti : X x Y ^ 2Yi be set-valued maps with nonempty convex values and open lower sections, let the set Wi = {(x,y) e Xx Y : xi e Di(x,y) and yi e Ti(x, y)} be closed in X x Y, and fi : Xx YxXi ^ R be a function. Assume that

(i) for all x = (xi, xi) e X,for all y e Y, fi(x, y, xi) > 0;

(ii) for each (x,y) e X x Y, zi ^ fi(x,y, zi) is quasiconvex;

(iii) for all zi e Xi, the set {(x,y) e X x Y : fi(x,y, zi) > 0} is closed in X x Y;

(iv) there exist nonempty and compact subsets N c X and K c Y and nonempty, compact and convex subsets Bi c Xi, Ai c Yi for each i e I such that V(x,y) = (xi,xi,y) e X x Y \ N x K 3i e I and 3ui e Bi, vi e Ai satisfying ui e Di(x,y), vi e Ti(x, y) and fi(x,y,ui) < 0.

Then, there exists (x,y) = (xl,xi,yl,yi) in X x Y such that for each i e I,

xi e D^(x,y), y e T^x,y) : f^x,y,z^ > 0, Vzi e D^x,y). (3.11)

That is, the solution set of the (SGQEP) is nonempty.

By Theorem 3.3, we can easily get the following result.

Corollary 3.5. Let I be any index set. For each i e I, let Zi be a topological vector space, let Ei and Fi be two locally convex Hausdorff topological vector spaces, let Xi c Ei and Yi c Fi be nonempty, closed and convex subsets, let Di : X x Y ^ 2Xi and Ti : X x Y ^ 2Yi be set-valued maps with nonempty convex values and open lower sections, the set Wi = {(x,y) e X x Y : xi e Di(x,y) and yi e Ti(x, y)} be closed in X x Y and fi : X x Y x Xi ^ R be a function. Assume that the set-valued map DxT = (HieiDi xU.ieiTi) : XxY ^ 2XxY defined as (DxT)(x,y) = H.ieIDi(x,y)xHieITi(x,y), V(x,y) e X x Y, is O-condensing and for each i e I, the conditions (i), (ii) and (iii) of Corollary 3.4 hold. Then the solution set of the (SGQEP) is nonempty.

Remark 3.6. Theorem 3.3 is a generalization of [19, Theorem 3]. Corollaries 3.4 and 3.5 extend and generalize the main results in [10-17].

4. Applications

In this section, we present some existence of a solution for the (G-Debreu VEP) and the (G-Debreu EP).

Theorem 4.1. Let I be any index set. For each i e I, let Zi be a topological vector space, let Ei and Fi be two Hausdorff topological vector spaces, let Xi c Ei and Yi c Fi be nonempty and convex subsets, let Ci : X ^ 2Zi be a set-valued map such that Ci(x) is a proper, closed and convex cone with apex at the origin and int Ci(x) = 0 for each x e X and Pi = nxeXCi(x), Di : X x Y ^ 2Xi and Ti : X x Y ^ 2Yi be set-valued maps with nonempty convex values and open lower sections, the set Wi = {(x,y) e X x Y : xi e Di(x,y) and yi e Ti(x,y)} be closed in X x Y and $i be a bifunction

from X x Y into Zi. For each i e I, assume that

(i) Mi = Zi \ (- int Ci) : X ^ 2Zi is upper semicontinuous;

(ii) For all xi e Xi and y e Y, zi ^ $i(xi,y,zi) is natural Pi-quasifunction, where Pi =

nxeXCi(x);

(iii) is continuous on X x Y;

(iv) there exist nonempty and compact subsets N c X and K c Y and nonempty, compact and convex subsets Bi c Xi, Ai c Yi for each i e I such that V(x,y) = (xi,xi,y) e X x Y \ N x K 3i e I and 3ui e Bi, Vi e Ai satisfying ui e Di(x,y), vi e Ti(x, y) and $i(xi,y,ui) - $i(x,y) e - int Ci(x).

Then, there exists (x,y) = (xi,xi,yl,yi) in X x Y such that for each i e I,

xi e D^x,y), yi e T^(x,y) : y,z^ - &(x,y) e - intCi(x), Vzi e D^x,y).

That is, the solution set of the (G-Debreu VEP) is nonempty. Proof. For each i e I, we define a trifunction fi : X x Y x Xi as

f^x,y,u^ = $^xi,y,ui^ - x,y), V(x,y,u) e X x Y x Xi. (4.2)

Since $i(xi,y, ■) is natural Pi quasi-function, by [19, Remark 2], for all ui1 ,ui2 e Xi and for all 1 e [0,1], 3a e [0,1] such that

$^xi,y,Xui1 + (1 - 1)ui2^ e a$^xi,y,ui1^ + (1 - a)$^xi,y,ui2^ - Pi, (4.3)

f^x,y,1ui1 + (1 - 1)ui2) e af^x,y,ui1) + (1 - a)fi(x,y,ui2) - Pi. (4.4)

Hence, for all (x, y) e X x Y, fi(x, y, ■) is natural Pi quasifunction.

By condition (iii), we know that for all zi e Xi, the map (x,y) ^ fi(x,y,zi) is continuous on X x Y. So it follows from Remark 3.2 that condition (iii) of Theorem 3.1 holds. It is easy to verify that the other conditions of Theorem 3.1 are satisfied. By Theorem 3.1, we know that the conclusion holds. □

Similarly, by Theorem 3.3, Corollaries 3.4 and 3.5, respectively, we have the following results.

Theorem 4.2. Let I be any index set. For each i e I, let Zi be a topological vector space, let Ei and Fi be two locally convex Hausdorff topological vector spaces, let Xi c Ei and Yi c Fi be nonempty, closed and convex subsets, let Ci : X ^ 2Zi be a set-valued map such that Ci(x) is a proper, closed and convex cone with apex at the origin and intCi(x) = 0 for each x e X and Pi = nxeXCi(x), Di : X x Y ^ 2Xi and Ti : X x Y ^ 2Yi be set-valued maps with nonempty convex values and open lower sections, the set Wi = {(x,y) e X x Y : xi e Di(x,y) and yi e Ti(x,y)} be

closed in X x Y and yi : X x Y ^ Zi be a vector-valued function. Assume that the set-valued map DxT = (HieIDi xHeITi) : XxY ^ 2XxY defined as (DxT)(x,y) = UieIDi(x,y)xHieITi(x,y), V(x,y) e X x Y, is O-condensing and (i), (ii), and (iii) of Theorem 4.1 hold. Then, the solution set of the (G-Debreu VEP) is nonempty.

Theorem 4.3. Let I be any index set. For each i e I, let Xi c Ei and Yi c Fi be nonempty and convex subsets, let Di: X x Y ^ 2Xi and Ti: X x Y ^ 2Yi be set-valued maps with nonempty convex values and open lower sections, the set Wi = {(x,y) e X x Y : xi e Di(x,y) and yi e Ti(x, y)} be closed in X x Y and be a bifunction from X x Y into R. For each i e I, assume that

(i) for all xi e Xi and y e Y, zi ^ $i(xi, y, zi) is quasiconvex;

(ii) is continuous on X x Y;

(iii) there exist nonempty and compact subsets N c X and K c Y and nonempty, compact and convex subsets Bi c Xi, Ai c Yi for each i e I such that V(x,y) = (xi,xi,y) e X x Y \ N x K 3i e I and 3ui e Bi, vt e Ai satisfying ui e Di(x,y), vi e Ti(x, y) and $i(xi,y,ui) < $i(x,y).

Then, there exists (x,y) = (xl ,xi,yl ,yi) in X x Y such that for each i e I,

xi e D^x,y), yi e T^x,y) : > $^x,y), Vzi e D^x,y). (4.5)

That is, the solution set of the (G-Debreu EP) is nonempty.

Theorem 4.4. Let I be any index set. For each i e I, let Ei and Fi be two locally convex Hausdorff topological vector spaces, Xi c Ei and Yi c Fi be nonempty, closed and convex subsets, let Di : X x Y ^ 2Xi and Ti : X x Y ^ 2Yi be set-valued maps with nonempty convex values and open lower sections, the set Wi = {(x,y) e X x Y : xi e Di(x,y) and yi e Ti(x,y)} be closed in X x Y and : X x Y ^ R be a function. Assume that the set-valued map D x T = (n ieIDi xJl ieITi) : X x Y ^ 2XxY defined as (D x T)(x,y) = nieIDi(x,y) ^nieITi(x,y), V(x,y) e X x Y,is O-condensing and (i), and (ii) of Theorem 4.3 hold. Then, the solution set of the (G-Debreu EP) is nonempty.

Remark 4.5. Theorem 4.1 extends and generalizes [19, Theorem 5] and [20, Theorems 3.1, 3.6 and Corollaries 3.2, 3.3, and 3.5]. Theorem 4.2 extends and generalizes [19, Theorem 6]. Theorems 4.3 and 4.4 are generalizations of [20, Corollaries 3.5 and 3.7] and the corresponding results in [21-24].

Acknowledgment

This research was supported by the National Natural Science Foundation of China (Grant no. 10771228 and Grant no. 10831009).

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