# On System of Generalized Vector Quasiequilibrium Problems with Applications

Authors: Jian-Wen Peng; Lun Wan
Journal: International Journal of Mathematics and Mathematical Sciences
 (2009)

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## Abstract

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.


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## Body

## 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 variational 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 setA in a topological space, we denote by coA, intA, co¯A the convex hull, interior, and the convex closure of A, respectively.LetI be an index set. For each i∈I, let Zi, Ei and let Fi be topological vector spaces. Consider two family of nonempty convex subsets {Xi}i∈I with Xi⊆Ei and {Yi}i∈I with Yi⊆Fi. Let(1.1)E=∏i∈IEi,X=∏i∈IXi,F=∏i∈IFi,Y=∏i∈IYi.
An element of the set Xi=∏j∈I∖iXi will be denoted by xi, therefore, x∈X will be written as x=(xi,xi)∈Xi×Xi. Similarly, an element of the set Y will be denoted by y=(yi,yi)∈Yi×Yi. For each i∈I, let Ci:X→2Zi,  Di:X×Y→2Xi and Ti:X×Y→2Yi be set-valued maps with nonempty values, and let fi:X×Y×Xi→Zi be a the vector-valued function. Then the system of generalized vector quasi-equilibrium problems (in Short, SGVQEP) is to find (x̅,y̅)=(x̅i,x̅i,y̅i,y̅i) in X×Y such that for each i∈I,(1.2)x̅i∈Di(x̅,y̅),y̅i∈Ti(x̅,y̅):fi(x̅,y̅,zi)∉-intCi(x̅),∀zi∈Di(x̅,y̅).Here  are some special cases of the (SGVQEP).(i) For eachi∈I, let ϕi:X×Y→Zi be a vector-valued function. We define a trifunction fi:X×Y×Xi→Zi as fi(x,y,ui)=ϕi(xi,y,ui)-ϕi(x,y), ∀(x,y,ui)∈X×Y×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 (x̅,y̅)=(x̅i,x̅i,y̅i,y̅i) in X×Y such that for each i∈I,(1.3)x̅i∈Di(x̅,y̅),y̅i∈Ti(x̅,y̅):ϕi(x̅i,y̅,zi)-ϕi(x̅,y̅)∉-intCi(x̅),∀zi∈Di(x̅,y̅).(ii) We denote byR and R+ the set of real numbers and the set of real nonnegative numbers, respectively. For each i∈I, if Zi=R, and Ci(x)=R+ for all x∈X, then the (SGVQEP) reduces to the system of generalized quasi-equilibrium problems (in short, SGQEP), which is to find (x̅,y̅)=(x̅i,x̅i,y̅i,y̅i) in X×Y such that for each i∈I,(1.4)x̅i∈Di(x̅,y̅),y̅i∈Ti(x̅,y̅):fi(x̅,y̅,zi)≥0,∀zi∈Di(x̅,y̅).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(x̅,y̅)=(x̅i,x̅i,y̅i,y̅i) in X×Y such that for each i∈I,(1.5)x̅i∈Di(x̅,y̅),y̅i∈Ti(x̅,y̅):ϕi(x̅i,y̅,zi)≥ϕi(x̅,y̅),∀zi∈Di(x̅,y̅).(iii) LetY={y̅}. For each i∈I, let Di(x,y̅)=Ai(x), Ti(x,y̅)={y̅i} for all x∈X, where Ai:X→2Xi is a set-valued map. We define a function φi:X×Xi→Zi and a function hi:X×Y→Zi as φi(x,zi)=fi(x,y̅,zi), forall(x,zi)∈X×Xi, and hi(x)=ϕi(x,y̅),forallx∈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 setI 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 Section3, we establish existence results for a solution to the (SGVQEP) and the (SGQEP) with or without involving Φ-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]).
LetM 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 φ:M→Z is called(i)
P-quasifunction if and only if, for all z∈Z, the set {x∈M:φ(x)∈z-P} is convex,(ii)
naturalP-quasifunction if and only if, ∀x,y∈M, and λ∈[0,1], φ(λx+(1-λ)y)∈co{φ(x),φ(y)}-P.Definition 2.2 ([13]).
LetX 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∈X:T(x)⊆V} is open in X for every open subset V of Y. Also T is said to be lower semicontinuous if the set {x∈X:T(x)∩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∈X:y∈T(x)} is open in X for each y∈Y.Definition 2.3 ([26]).
LetE be a Hausdorff topological space and L a lattice with least element, denoted by 0. A map Φ:2E→L is a measure of noncompactness provided that the following conditions hold ∀M,N∈2E:(i)
Φ(M)=0 iff M is precompact ( i.e., it is relatively compact),(ii)
Φ(co¯M)=Φ(M),(iii)
Φ(M∪N)=max{Φ(M),Φ(N)}.Definition 2.4 ([26]).
LetΦ:2E→L be a measure of noncompactness on E and X⊆E. A set-valued map T:X→2E is called Φ-condensing provided that, if M⊆X with Φ(T(M))≥Φ(M), then M is relatively compact.Remark 2.5.
Note that every set-valued map defined on a compact set isΦ-condensing for any measure of noncompactness Φ. If E is locally convex and T:X→2E is a compact set-valued map (i.e., T(X) is precompact), then T is Φ-condensing for any measure of noncompactness Φ. It is clear that if T:X→2E is Φ-condensing and T*:X→2E satisfies T*(x)⊆T(x)∀x∈X, then T* is also Φ-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}i∈I be a family of nonempty convex subsets where each Xi is contained in a Hausdorff topological vector space Ei, For each i∈I, let Si:X→2Xi be a set-valued map such that (i)
for eachi∈I, Si(x) is convex,(ii)
for eachx∈X, xi∉Si(x),(iii)
for eachyi∈Xi, Si-1(yi) is open in X.(iv)
there exist a nonempty compact subsetN of X and a nonempty compact convex subset Bi of Xi for each i∈I such that for each x∈X∖N there exists i∈I satisfying Si(x)∩Bi≠∅. Then there exists x̅∈X such that Si(x̅)=∅ for all i∈I.Lemma 2.7 ([19, 28]).
LetI be any index set and {Xi}i∈I 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∈I, let Si:X→2Xi be a set-valued map. Assume that the set-valued map S:X→2X defined as S(x)=∏i∈ISi(x), ∀x∈X, is Φ-condensing and the conditions (i), (ii), (iii) of Lemma 2.6 hold. Then there exists x̅∈X such that Si(x̅)=∅ for all i∈I.


## 3. Existence Results

An existence result of a solution for the system of generalized vector quasi-equilibrium problems with or withoutΦ-condensing maps are will shown in this section.Theorem 3.1.
LetI be any index set. For each i∈I, let Zi be a topological vector space, let Ei and Fi be two Hausdorff topological vector spaces, let Xi⊆Ei and Yi⊆Fi be nonempty and convex subsets, let Di:X×Y→2Xi and Ti:X×Y→2Yi be set-valued maps with nonempty convex values and open lower sections, and the set Wi={(x,y)∈X×Y:xi∈Di(x,y)andyi∈Ti(x,y)} be closed in X×Y and let  fi:X×Y×Xi→Zi be a vector-valued function. For each i∈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 intCi(x)≠∅ for all x∈X and Pi=∩x∈XCi(x). Assume that(i)
for allx=(xi,xi)∈X, for all y∈Y, fi(x,y,xi)∉-intCi(x);(ii)
for each(x,y)∈X×Y, zi↦fi(x,y,zi) is natural Pi-quasifunction;(iii)
for allzi∈Xi, the set {(x,y)∈X×Y:fi(x,y,zi)∉-intCi(x)} is closed in X×Y;(iv)
there exist nonempty and compact subsetsN⊆X and K⊆Y and nonempty, compact and convex subsets Bi⊆Xi, Ai⊆Yi for each i∈I such that ∀(x,y)=(xi,xi,y)∈X×Y∖N×K∃i∈I and  ∃u̅i∈Bi, v̅i∈Ai satisfying u̅i∈Di(x,y), v̅i∈Ti(x,y) and fi(x,y,u̅i)∈-intCi(x).Then, there exists (x̅,y̅)=(x̅i,x̅i,y̅i,y̅i) in X×Y such that for each i∈I,
(3.1)x̅i∈Di(x̅,y̅),y̅i∈Ti(x̅,y̅):fi(x̅,y̅,zi)∉-intCi(x̅),∀zi∈Di(x̅,y̅).
That is, the solution set of the (SGVQEP) is nonempty.Proof.
For eachi∈I, let us define a set-valued map Qi:X×Y→2Xi by
(3.2)Qi(x,y)={zi∈Xi:fi(x,y,zi)∈-intCi(x)},∀(x,y)∈X×Y.
Then, ∀i∈I and ∀(x,y)∈X×Y, Qi(x,y) is a convex set.
To prove it, let us fix arbitraryi∈I and (x,y)∈X×Y. Let zi1,zi2∈Qi(x,y) and λ∈[0,1], then we have
(3.3)fi(x,y,zij)∈-intCi(x),forj=1,2.
Since fi(x,y,·) is natural Pi-quasifunction,  ∃μ∈[0,1] such that
(3.4)fi(x,y,λzi1+(1-λ)zi2)∈μfi(x,y,zi1)+(1-μ)fi(x,y,zi2)-Pi.
From (3.3) and (3.4), we get
(3.5)fi(x,y,λzi1+(1-λ)zi2)∈-intCi(x)-intCi(x)-Pi⊆-intCi(x).
Henceλzi1+(1-λ)zi2∈Qi(x,y) and, therefore, Qi(x,y) is convex.
It follows from condition (i) that, for eachi∈I and for all (x,y)=(xi,xi,y)∈X×Y,
(3.6)xi∉Qi(x,y).
It follows from condition (iii) that for eachi∈I and each zi∈Xi, the set
(3.7)Qi-1(zi)={(x,y)∈X×Y:fi(x,y,zi)∈-intCi(x)}
is open in Xi. That is, Qi has open lower sections on X×Y. For each i∈I, we also define another set-valued map Si:X×Y→2Xi×Yi by
(3.8)Si(x,y)={[Di(x,y)∩Qi(x,y)]×Ti(x,y),if(x,y)∈Wi,Di(x,y)×Ti(x,y),if(x,y)∉Wi.
Then, it is clear that ∀i∈I and ∀(x,y)∈X×Y, Si(x,y) is convex, and (xi,yi)∉Si(x,y). Since ∀i∈I and ∀(ui,vi)∈Xi×Yi,
(3.9)Si-1(ui,vi)=[Qi-1(ui)∩(Di-1(ui))∩(Ti-1(vi))]∪[(X×Y∖Wi)∩(Di-1(ui))∩(Ti-1(vi))],
and Di-1(ui), Ti-1(vi), Qi-1(ui), and X×Y∖Wi are open in X×Y, we have Si-1(ui,vi) is open in X×Y.
From condition (iv), there exist a nonempty and compact subsetN×K⊆X×Y and a nonempty, compact, and convex subset Bi×Ai⊆Xi×Yi for each i∈I such that ∀(x,y)=(xi,xi,y)∈X×Y∖N×K∃i∈I and ∃(u̅i,v̅i)∈Si(x,y)∩(Bi×Ai). Hence, by Lemma 2.6, ∃(x̅,y̅)∈X×Y such that Si(x̅,y̅)=∅,∀i∈I. Since ∀i∈I and ∀(x,y)∈X×X, Di(x,y) and Ti(x,y) are nonempty, we have (x̅,y̅)∈Wi and Di(x̅,y̅)∩Qi(x̅,y̅)=∅, ∀i∈I. This implies (x̅,y̅)∈Wi and Di(x̅,y̅)∩Qi(x̅,y̅)=∅, ∀i∈I. Therefore, ∀i∈I,
(3.10)x̅i∈Di(x̅,y̅),y̅i∈Ti(x̅,y̅),fi(x̅,y̅,zi)∉-intCi(x̅),∀zi∈Di(x̅,y̅).
That is, the solution set of the (SGVQEP) is nonempty.Remark 3.2.
(1) The condition (iii) of Theorem3.1 is satisfied if the following conditions hold ∀i∈I:
(a)
Ci:X→2Zi is a set-valued map such that intCi(x)≠∅ for each x∈X and the set-valued map Mi=Zi∖(-intCi):X→2Zi is upper semicontinuous;(b)
for allzi∈Xi, the map (x,y)↦fi(x,y,zi) is continuous on X×Y;
(2) If∀i∈I, and ∀x∈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)
∀i∈I, the vector-valued function ∀(x,y)∈X×Y, zi↦fi(x,y,zi) is Ci-quasifunction;(d)
∀i∈I, ∀zi∈Xi, the map (x,y)↦fi(x,y,zi) is Ci-upper semicontinuous on X×Y;
(3) Theorem3.1 extends and generalizes in [19, Theorem  2], [20, Theorem  2.1] and [18, Theorem  2.1] in several ways.
(4) If∀i∈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.
LetI be any index set. For each i∈I, let Zi be a topological vector space, let  Ei and Fi be two locally convex Hausdorff topological vector spaces, let  Xi⊆Ei and Yi⊆Fi be nonempty, closed and convex subsets, let Di:X×Y→2Xi and Ti:X×Y→2Yi be set-valued maps with nonempty convex values and open lower sections, the set Wi={(x,y)∈X×Y:xi∈Di(x,y)andyi∈Ti(x,y)} be closed in X×Y and fi:X×Y×Xi→Zi be a vector-valued function. For each i∈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 intCi(x)≠∅ for all x∈X and Pi=∩x∈XCi(x). Assume that the set-valued map D×T=(∏i∈IDi×∏i∈ITi):X×Y→2X×Y defined as (D×T)(x,y)=∏i∈IDi(x,y)×∏i∈ITi(x,y), ∀(x,y)∈X×Y, is Φ-condensing and for each i∈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 Lemma2.7 and the proof of Theorem 3.1, it is sufficient to show that the set-valued map S:X×Y→2X×Y defined as S(x,y)=∏i∈ISi(x,y),forall(x,y)∈X×Y, is Φ-condensing, where Si's are the same as in the proof of Theorem 3.1. By the definition of Si, Si(x,y)⊆Di(x,y)×Ti(x,y) for all (x,y)∈X×Y and for each i∈I, and therefore S(x,y)⊆D(x,y)×T(x,y) for all (x,y)∈X×Y. Since D×T is Φ-condensing, by Remark 2.5, we have S is also Φ-condensing.By Theorem3.1 and Remark 3.2, we can easily get the following result.Corollary 3.4.
LetI be any index set. For each i∈I, let Ei and Fi be two Hausdorff topological vector spaces,  let Xi⊆Ei and Yi⊆Fi be nonempty and convex subsets, let Di:X×Y→2Xi and Ti:X×Y→2Yi be set-valued maps with nonempty convex values and open lower sections, let  the set Wi={(x,y)∈X×Y:xi∈Di(x,y)andyi∈Ti(x,y)} be closed in X×Y, and fi:X×Y×Xi→R be a function. Assume that(i)
for allx=(xi,xi)∈X, for all y∈Y, fi(x,y,xi)≥0;(ii)
for each(x,y)∈X×Y, zi↦fi(x,y,zi) is quasiconvex;(iii)
for allzi∈Xi, the set {(x,y)∈X×Y:fi(x,y,zi)≥0} is closed in X×Y;(iv)
there exist nonempty and compact subsetsN⊆X and K⊆Y and nonempty, compact and convex subsets Bi⊆Xi, Ai⊆Yi for each i∈I such that ∀(x,y)=(xi,xi,y)∈X×Y∖N×K∃i∈I and ∃u̅i∈Bi, v̅i∈Ai satisfying u̅i∈Di(x,y), v̅i∈Ti(x,y) and fi(x,y,u̅i)<0.Then, there exists (x̅,y̅)=(x̅i,x̅i,y̅i,y̅i) in X×Y such that for each i∈I,
(3.11)x̅i∈Di(x̅,y̅),y̅i∈Ti(x̅,y̅):fi(x̅,y̅,zi)≥0,∀zi∈Di(x̅,y̅).
That is, the solution set of the (SGQEP) is nonempty.By Theorem3.3, we can easily get the following result.Corollary 3.5.
LetI be any index set. For each i∈I, let Zi be a topological vector space,  let Ei and Fi be two locally convex Hausdorff topological vector spaces,  let Xi⊆Ei and Yi⊆Fi be nonempty, closed and convex subsets, let Di:X×Y→2Xi and Ti:X×Y→2Yi be set-valued maps with nonempty convex values and open lower sections, the set Wi={(x,y)∈X×Y:xi∈Di(x,y)andyi∈Ti(x,y)} be closed in X×Y and fi:X×Y×Xi→R be a function. Assume that the set-valued map D×T=(∏i∈IDi×∏i∈ITi):X×Y→2X×Y defined as (D×T)(x,y)=∏i∈IDi(x,y)×∏i∈ITi(x,y), ∀(x,y)∈X×Y, is Φ-condensing and for each i∈I, the conditions (i), (ii) and (iii) of Corollary 3.4 hold. Then the solution set of the (SGQEP) is nonempty.Remark 3.6.
Theorem3.3 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.
LetI be any index set. For each i∈I, let Zi be a topological vector space, let  Ei and Fi be two Hausdorff topological vector spaces,  let Xi⊆Ei and Yi⊆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 intCi(x)≠∅ for each x∈X and Pi=∩x∈XCi(x), Di:X×Y→2Xi and Ti:X×Y→2Yi be set-valued maps with nonempty convex values and open lower sections, the set Wi={(x,y)∈X×Y:xi∈Di(x,y)andyi∈Ti(x,y)} be closed in X×Y and ϕi be a bifunction from X×Y into Zi. For each i∈I, assume that(i)
Mi=Zi∖(-intCi):X→2Zi is upper semicontinuous;(ii)
For allxi∈Xi and y∈Y, zi↦ϕi(xi,y,zi) is natural Pi-quasifunction, where Pi=∩x∈XCi(x);(iii)
ϕi is continuous on X×Y;(iv)
there exist nonempty and compact subsetsN⊆X and K⊆Y and nonempty, compact and convex subsets Bi⊆Xi, Ai⊆Yi for each i∈I such that ∀(x,y)=(xi,xi,y)∈X×Y∖N×K∃i∈I and ∃u̅i∈Bi, v̅i∈Ai satisfying u̅i∈Di(x,y), v̅i∈Ti(x,y) and ϕi(xi,y,u̅i)-ϕi(x,y)∈-intCi(x).Then, there exists (x̅,y̅)=(x̅i,x̅i,y̅i,y̅i) in X×Y such that for each i∈I,
(4.1)x̅i∈Di(x̅,y̅),y̅i∈Ti(x̅,y̅):ϕi(x̅i,y̅,zi)-ϕi(x̅,y̅)∉-intCi(x̅),∀zi∈Di(x̅,y̅).
That is, the solution set of the (G-Debreu VEP) is nonempty.Proof.
For eachi∈I, we define a trifunction fi:X×Y×Xi as
(4.2)fi(x,y,ui)=ϕi(xi,y,ui)-ϕi(x,y),∀(x,y,ui)∈X×Y×Xi.
Since ϕi(xi,y,·) is natural Pi quasi-function, by [19, Remark  2], for all ui1,ui2∈Xi   and for all λ∈[0,1],∃α∈[0,1] such that
(4.3)ϕi(xi,y,λui1+(1-λ)ui2)∈αϕi(xi,y,ui1)+(1-α)ϕi(xi,y,ui2)-Pi,
Hence
(4.4)fi(x,y,λui1+(1-λ)ui2)∈αfi(x,y,ui1)+(1-α)fi(x,y,ui2)-Pi.
Hence, for all (x,y)∈X×Y, fi(x,y,·) is natural Pi quasifunction.
By condition (iii), we know that for allzi∈Xi, the map (x,y)↦fi(x,y,zi) is continuous on 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 Theorem3.3, Corollaries 3.4 and 3.5, respectively, we have the following results.Theorem 4.2.
LetI be any index set. For each i∈I, let Zi be a topological vector space, let Ei and Fi be two locally convex Hausdorff topological vector spaces, let  Xi⊆Ei and Yi⊆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)≠∅ for each x∈X and Pi=∩x∈XCi(x), Di:X×Y→2Xi and Ti:X×Y→2Yi be set-valued maps with nonempty convex values and open lower sections, the set Wi={(x,y)∈X×Y:xi∈Di(x,y)andyi∈Ti(x,y)} be closed in X×Y and φi:X×Y→Zi be a vector-valued function. Assume that the set-valued map D×T=(∏i∈IDi×∏i∈ITi):X×Y→2X×Y defined as (D×T)(x,y)=∏i∈IDi(x,y)×∏i∈ITi(x,y), ∀(x,y)∈X×Y, is Φ-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.
LetI be any index set. For each i∈I, let Xi⊆Ei and Yi⊆Fi be nonempty and convex subsets, let Di:X×Y→2Xi and Ti:X×Y→2Yi be set-valued maps with nonempty convex values and open lower sections, the set Wi={(x,y)∈X×Y:xi∈Di(x,y)andyi∈Ti(x,y)} be closed in X×Y and ϕi be a bifunction from X×Y into R. For each i∈I, assume that(i)
for allxi∈Xi and y∈Y, zi↦ϕi(xi,y,zi) is quasiconvex;(ii)
ϕi is continuous on X×Y;(iii)
there exist nonempty and compact subsetsN⊆X and K⊆Y and nonempty, compact and convex subsets Bi⊆Xi, Ai⊆Yi for each i∈I such that ∀(x,y)=(xi,xi,y)∈X×Y∖N×K∃i∈I and ∃u̅i∈Bi, v̅i∈Ai satisfying u̅i∈Di(x,y), v̅i∈Ti(x,y) and ϕi(xi,y,u̅i)<ϕi(x,y).Then, there exists (x̅,y̅)=(x̅i,x̅i,y̅i,y̅i) in X×Y such that for each i∈I,
(4.5)x̅i∈Di(x̅,y̅),y̅i∈Ti(x̅,y̅):ϕi(x̅i,y̅,zi)≥ϕi(x̅,y̅),∀zi∈Di(x̅,y̅).
That is, the solution set of the (G-Debreu EP) is nonempty.Theorem 4.4.
LetI be any index set. For each i∈I, let Ei and Fi be two locally convex Hausdorff topological vector spaces, Xi⊆Ei and Yi⊆Fi be nonempty, closed and convex subsets, let Di:X×Y→2Xi and Ti:X×Y→2Yi be set-valued maps with nonempty convex values and open lower sections, the set Wi={(x,y)∈X×Y:xi∈Di(x,y)andyi∈Ti(x,y)} be closed in X×Y and φi:X×Y→R be a function. Assume that the set-valued map D×T=(∏i∈IDi×∏i∈ITi):X×Y→2X×Y defined as (D×T)(x,y)=∏i∈IDi(x,y)×∏i∈ITi(x,y), ∀(x,y)∈X×Y, is Φ-condensing and (i), and (ii) of Theorem 4.3 hold. Then, the solution set of the (G-Debreu EP) is nonempty.Remark 4.5.
Theorem4.1 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].



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*Source: 430930-2009-09-06.xml*