On System of Generalized Vector Quasiequilibrium Problems with Applications

Volume 2009, Article ID 430930,10pages doi:10.1155/2009/430930

Research Article

On System of Generalized Vector Quasiequilibrium

Problems with Applications

Jian-Wen Peng

_{and Lun Wan}

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,[email protected]

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.

Copyrightq2009 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

Throughout this paper, for a setAin a topological space, we denote by coA, intA, coAthe convex hull, interior, and the convex closure ofA, respectively.

LetI be an index set. For eachi∈I, letZi,Eiand letFi be topological vector spaces. Consider two family of nonempty convex subsets{Xi}_i∈IwithXi⊆Eiand{Yi}_i∈IwithYi⊆Fi. Let

E

i∈I

Ei, X

i∈I

Xi, F

i∈I

Fi, Y

i∈I

Yi. 1.1

An element of the setXi _j∈I\iX_iwill be denoted byxi, therefore,x∈Xwill be written as

x xi_{, xi∈X}i_{×Xi. Similarly, an element of the setYwill be denoted byy y}i_{, yi∈Y}i_×Yi.

For eachi∈I, letCi :X → 2Zi_,Di _:X×Y → ₂Xi _andTi _:X×Y → ₂Yi _{be set-valued maps}

with nonempty values, and letfi :X ×Y×Xi → Zi be a the vector-valued function. Then

the system of generalized vector quasi-equilibrium problemsin Short, SGVQEPis to find

x, y xi, xi, yi, y_iinX×Y such that for eachi∈I,

xi∈Dix, y, yi∈Tix, y:fix, y, zi/∈ −intCix, ∀zi∈Dix, y. 1.2

Here are some special cases of theSGVQEP.

i For each i ∈ I, let φi : X ×Y → Zi be a vector-valued function. We define a trifunctionfi:X×Y×Xi → Ziasfix, y, ui φixi, y, ui−φix, y,∀x, y, ui∈X×Y×Xi. Then theSGVQEPreduces to the generalized Debreu-type equilibrium problem for vector-valued functionsin short, G-Debreu VEP, which is to findx, y xi, xi, yi, y_iinX×Y such that for eachi∈I,

xi∈Dix, y, y_i ∈Tix, y:φixi, y, zi−φix, y/∈ −intCix, ∀zi∈Dix, y. 1.3

iiWe denote byR andR the set of real numbers and the set of real nonnegative numbers, respectively. For each i ∈ I, if Z_i R, and C_ix R for all x ∈ X, then theSGVQEPreduces to the system of generalized quasi-equilibrium problems in short, SGQEP, which is to findx, y xi, xi, yi, y_iinX×Y such that for eachi∈I,

xi∈Dix, y, yi ∈Tix, y:fix, y, zi≥0, ∀zi∈Dix, y. 1.4

And the G-Debreu VEP reduces to the generalized Debreu-type equilibrium problem for scalar-valued functionsin short, G-Debreu EP, which is to findx, y xi, xi, yi, y_iin

X×Ysuch that for eachi∈I,

xi∈Dix, y, y_i∈Tix, y:φixi, y, zi≥φix, y, ∀zi∈Dix, y. 1.5

iiiLetY {y}. For eachi ∈ I, letDix, y Aix,Tix, y {y_i} for allx ∈ X, where A_i : X → 2Xi _{is a set-valued map. We define a function} ϕ_{i :} X × X_i → Z_i

and a function hi : X ×Y → Zi as ϕix, zi fix, y, zi, for allx, zi ∈ X ×Xi, and

et al.19which contain those mathematical in18,20as special cases. TheSGQEPreduces to the mathematical models in21, page 286and22, pages 152-153and theG-Debreu EP reduces to the abstract economy in23, page 345which contains the noncooperative game in24as a special case.

ivIf the index setIis singleton,Dx, y Dix,Tx, y Tx,andCx C, then

theSGVQEP becomes the implicit vector variational inequality in 9 and the SGQEP reduces to the quasi-equilibrium problem investigated in14–17.

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 theSGVQEPand theSGQEPwith or without involvingΦ-condensing maps by using similar techniques in19. InSection 4, as applications of the results ofSection 3, we derive some existence results of a solution for the

G-Debreu VEPand the theG-Debreu EP.

2. Preliminaries

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

Definition 2.119,25. LetMbe a nonempty convex subset of a topological vector spaceE andZa real topological space with a closed and convex coneP with apex at the origin. A vector-valued functionϕ:M → Zis called

iP-quasifunction if and only if, for allz∈Z, the set{x∈M:ϕx∈z−P}is convex,

iinaturalP-quasifunction if and only if,∀x, y∈M, andλ∈0,1,ϕλx 1−λy∈ co{ϕx, ϕy} −P.

Definition 2.213. LetXandYbe two topological spaces.T :X → 2Ybe a set-valued map. ThenTis said to be upper semicontinuous if the set{x∈X:Tx⊆V}is open inXfor every open subsetV ofY. AlsoT is said to be lower semicontinuous if the set{x∈X :Tx∩V} is open inX for every open subsetV ofY ·T is said to have open lower sections if the set

T−1_{y {x∈X:y∈Tx}is open inXfor eachy∈Y.}

Definition 2.326. LetEbe a Hausdorfftopological space andLa lattice with least element, denoted by 0. A mapΦ: 2E → Lis a measure of noncompactness provided that the following conditions hold∀M, N ∈2E:

i ΦM 0 iffMis precompacti.e., it is relatively compact,

ii ΦcoM ΦM,

iii ΦM∪N max{ΦM,ΦN}.

Definition 2.426. LetΦ : 2E → Lbe a measure of noncompactness onEandX ⊆ E. A set-valued mapT :X → 2Eis calledΦ-condensing provided that, ifM⊆XwithΦTM≥

ΦM, thenMis relatively compact.

Remark 2.5. Note that every set-valued map defined on a compact set isΦ-condensing for any measure of noncompactnessΦ. IfEis locally convex andT :X → 2Eis a compact set-valued mapi.e.,TXis precompact, thenTisΦ-condensing for any measure of noncompactness

Φ. It is clear that ifT :X → 2EisΦ-condensing andT∗:X → 2EsatisfiesT∗x⊆Tx∀x∈

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 7and Chebbi and Florenzano

28, Corollary 4.

Lemma 2.619,27. Let{Xi}_i∈Ibe a family of nonempty convex subsets where eachXiis contained in a Hausdorfftopological vector spaceE_i, For eachi∈I, letS_i :X → 2Xi _{be a set-valued map such}

that

ifor eachi∈I,Sixis convex,

iifor eachx∈X,x_i/∈Six,

iiifor eachyi∈Xi,Si−1yiis open inX.

ivthere exist a nonempty compact subsetNofXand a nonempty compact convex subsetB_i ofXifor eachi∈Isuch that for eachx∈X\Nthere existsi∈IsatisfyingSix∩Bi/∅. Then there existsx∈Xsuch thatS_ix ∅for alli∈I.

Lemma 2.719,28. LetIbe any index set and{X_i}_i∈Ibe a family of nonempty, closed and convex subsets where eachXiis contained in a locally convex Hausdorfftopological vector spaceEi. For each

i∈I, letS_i :X → 2Xi _{be a set-valued map. Assume that the set-valued map}S_:X → ₂X _{defined as}

Sx _i∈ISix,∀x∈X, isΦ-condensing and the conditions (i), (ii), (iii) ofLemma 2.6hold. Then

there existsx∈Xsuch thatSix ∅for alli∈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. LetIbe any index set. For eachi∈I, letZibe a topological vector space, letEiandFi be two Hausdorfftopological vector spaces, letX_i ⊆E_iandY_i ⊆F_ibe nonempty and convex subsets, letDi :X×Y → 2Xi _andTi _:X×Y → ₂Yi _{be set-valued maps with nonempty convex values and}

open lower sections, and the setWi {x, y∈X×Y :xi∈Dix, yandyi ∈Tix, y}be closed inX×Y and letf_i:X×Y×X_i → Z_ibe a vector-valued function. For eachi∈I, letC_i:X → 2Zi

be a set-valued map such thatCixbe a proper closed and convex cone with apex at the origin and

intCix/∅for allx∈XandPi∩x∈XCix. Assume that

ifor allx xi, x_i∈X, for ally∈Y,f_ix, y, x_i/∈ −intCix;

iifor eachx, y∈X×Y,zi→fix, y, ziis naturalPi-quasifunction;

iiifor allz_i∈X_i, the set{x, y∈X×Y :f_ix, y, z_i/∈ −intC_ix}is closed inX×Y;

ivthere exist nonempty and compact subsetsN ⊆ X and K ⊆ Y and nonempty, compact and convex subsetsB_i ⊆ X_i,A_i ⊆ Y_i for eachi ∈ I such that ∀x, y xi, x_i, y ∈

X×Y\N×K ∃i∈I and∃u_i ∈B_i,v_i ∈A_isatisfyingu_i ∈D_ix, y,v_i ∈T_ix, yand

fix, y, ui∈ −intCix.

Then, there existsx, y xi, xi, yi, y_iinX×Ysuch that for eachi∈I,

xi∈Dix, y, yi∈Tix, y:fix, y, zi/∈ −intCix, ∀zi∈Dix, y. 3.1

Proof. For eachi∈I, let us define a set-valued mapQi:X×Y → 2Xi_by

Qix, yzi∈Xi:fix, y, zi∈ −intCix , ∀x, y∈X×Y. 3.2

Then,∀i∈Iand∀x, y∈X×Y,Q_ix, yis a convex set.

To prove it, let us fix arbitraryi ∈ I andx, y ∈ X ×Y. Let zi1, zi2 ∈ Qix, yand

λ∈0,1, then we have

fix, y, zij

∈ −intCix, forj 1,2. 3.3

Sincefix, y,·is naturalPi-quasifunction,∃μ∈0,1such that

fix, y, λzi1 1−λzi2

∈μfix, y, zi1

1−μfix, y, zi2

−Pi. 3.4

From3.3and3.4, we get

fix, y, λzi1 1−λzi2

∈ −intC_ix−intC_ix−P_i⊆ −intC_ix. 3.5

Henceλz_i1 1−λz_i2 ∈Q_ix, yand, therefore,Q_ix, yis convex.

It follows from conditionithat, for eachi∈Iand for allx, y xi, xi, y∈X×Y,

xi/∈Qix, y. 3.6

It follows from conditioniiithat for eachi∈Iand eachzi∈Xi, the set

Qi−1_{zi x, y∈X×Y :fix, y, zi∈ −intCix 3.7}

is open inXi. That is,Qi has open lower sections onX×Y. For eachi ∈ I, we also define another set-valued mapSi:X×Y → 2Xi×Yi _by

Six, y

⎧ ⎨ ⎩

Dix, y∩Qix, y×Tix, y, if x, y∈Wi,

Dix, y×Tix, y, if x, y/∈Wi.

3.8

Then, it is clear that∀i∈Iand∀x, y∈X×Y,S_ix, yis convex, andx_i, y_i/∈Six, y. Since

∀i∈Iand∀ui, vi∈Xi×Yi,

Si−1_{ui, vi Q}−1

i ui∩

Di−1_ui∩Ti−1_vi

∪X×Y \Wi∩

Di−1ui

∩Ti−1vi

,

3.9

From conditioniv, there exist a nonempty and compact subsetN×K ⊆ X×Y and a nonempty, compact, and convex subsetB_i×A_i ⊆X_i×Y_ifor eachi∈Isuch that∀x, y

xi_{, xi, y∈X×Y\N×K ∃i∈I and ∃ui, vi∈Six, y∩Bi×Ai. Hence, byLemma 2.6,}

∃x, y∈X×Y such thatSix, y ∅,∀i∈I. Since∀i∈Iand∀x, y∈X×X,Dix, yand

Tix, yare nonempty, we havex, y ∈WiandDix, y∩Qix, y ∅,∀i∈ I. This implies

x, y∈WiandDix, y∩Qix, y ∅,∀i∈I. Therefore,∀i∈I,

xi∈Dix, y, y_i∈Tix, y, fix, y, zi/∈ −intCix, ∀zi∈Dix, y. 3.10

That is, the solution set of theSGVQEPis nonempty.

Remark 3.2. 1The conditioniiiofTheorem 3.1is satisfied if the following conditions hold

∀i∈I:

aCi : X → 2Zi _{is a set-valued map such that int}Cix_/∅ _{for each}x ∈ X _{and the}

set-valued mapMiZi\−intCi:X → 2Zi is upper semicontinuous;

bfor allzi∈Xi, the mapx, y→fix, y, ziis continuous onX×Y;

2If∀i∈I, and∀x ∈X,C_ix C_i, afixedproper, closed and convex cone inZ_i, then the conditioniiandiiiofTheorem 3.1can be replaced, respectively, by the following conditions:

c∀i ∈ I, the vector-valued function ∀x, y ∈ X × Y, z_i → f_ix, y, z_i is C_i -quasifunction;

d∀i∈I,∀z_i∈X_i, the mapx, y→f_ix, y, z_iisC_i-upper semicontinuous onX×Y;

3Theorem 3.1extends and generalizes in19, Theorem 2,20, Theorem 2.1and

18, Theorem 2.1in several ways.

4If∀i∈I,Xiis a nonempty, compact and convex subset of a Hausdorfftopological vector spaceEi, then the conclusion ofTheorem 3.1holds without conditioniv.

Theorem 3.3. LetIbe any index set. For eachi∈I, letZ_ibe a topological vector space, letE_iandF_ibe two locally convex Hausdorfftopological vector spaces, letXi⊆EiandYi⊆Fibe nonempty, closed and convex subsets, letDi:X×Y → 2Xi_andTi_:X×Y → ₂Yi_{be set-valued maps with nonempty convex}

values and open lower sections, the setW_i {x, y ∈ X×Y : x_i ∈D_ix, yandy_i ∈ T_ix, 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}Cix_{be a proper closed and convex cone with apex}

at the origin and intC_ix_/∅for allx ∈ Xand P_i ∩_x∈XC_ix. Assume that the set-valued map

D×T _i∈IDi×_i∈ITi:X×Y → 2X×Ydefined asD×Tx, y _i∈IDix, y×_i∈ITix, y, ∀x, y∈X×Y, isΦ-condensing and for eachi∈I, the conditions (i), (ii) and (iii) ofTheorem 3.1 hold. Then the solution set of the (SGVQEP) is nonempty.

Proof. In view ofLemma 2.7and the proof ofTheorem 3.1, it is sufficient to show that the set-valued mapS :X×Y → 2X×Y defined asSx, y _i∈ISix, y,for allx, y ∈X×Y, isΦ-condensing, whereS_i’s are the same as in the proof ofTheorem 3.1. By the definition ofSi,Six, y ⊆ Dix, y×Tix, yfor allx, y ∈ X ×Y and for eachi ∈ I, and therefore

ByTheorem 3.1andRemark 3.2, we can easily get the following result.

Corollary 3.4. Let I be any index set. For each i ∈ I, let E_i and F_i be two Hausdorff topological vector spaces, letXi ⊆ Ei andYi ⊆ Fibe nonempty and convex subsets, letDi :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∈Dix, yandyi∈Tix, y}be closed inX×Y,andfi:X×Y×Xi → R

be a function. Assume that

ifor allx xi, xi∈X, for ally∈Y,fix, y, xi≥0;

iifor eachx, y∈X×Y,z_i→f_ix, y, z_iis quasiconvex;

iiifor allzi∈Xi, the set{x, y∈X×Y :fix, y, zi≥0}is closed inX×Y;

ivthere exist nonempty and compact subsetsN ⊆ X and K ⊆ Y and nonempty, compact and convex subsetsBi ⊆ Xi,Ai ⊆ Yi for eachi ∈ I such that ∀x, y xi, xi, y ∈

X×Y \N×K ∃i∈Iand ∃ui ∈Bi,vi ∈Aisatisfyingui∈Dix, y,vi∈Tix, yand

fix, y, ui<0.

Then, there existsx, y xi, xi, yi, y_iinX×Ysuch that for eachi∈I,

xi∈Dix, y, y_i ∈Tix, y:fix, y, zi≥0, ∀zi∈Dix, y. 3.11

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

ByTheorem 3.3, we can easily get the following result.

Corollary 3.5. LetIbe any index set. For eachi∈I, letZibe a topological vector space, letEiandFi be two locally convex Hausdorfftopological vector spaces, letX_i⊆E_iandY_i⊆F_ibe nonempty, closed and convex subsets, letDi :X×Y → 2Xi _andTi _:X×Y → ₂Yi_{be set-valued maps with nonempty}

convex values and open lower sections, the setWi {x, y ∈ X ×Y : xi ∈ Dix, yand yi ∈

Tix, y}be closed inX×Y andfi:X×Y×Xi → Rbe a function. Assume that the set-valued map D×T _i∈IDi×_i∈ITi:X×Y → 2X×Ydefined asD×Tx, y _i∈IDix, y×_i∈ITix, y, ∀x, y∈X×Y, isΦ-condensing and for eachi∈I, the conditions (i), (ii) and (iii) ofCorollary 3.4 hold. Then the solution set of the (SGQEP) is nonempty.

Remark 3.6. Theorem 3.3is a generalization of19, Theorem 3. Corollaries3.4and3.5extend and generalize the main results in10–17.

4. Applications

In this section, we present some existence of a solution for theG-Debreu VEPand the G-Debreu EP.

Theorem 4.1. LetIbe any index set. For eachi∈I, letZibe a topological vector space, letEiandFi be two Hausdorfftopological vector spaces, letXi ⊆EiandYi ⊆Fibe nonempty and convex subsets, let C_i : X → 2Zi _{be a set-valued map such that} C_ix _{is a proper, closed and convex cone with}

apex at the origin and intCix_/∅for eachx ∈ X and Pi ∩x∈XCix,Di : X×Y → 2Xi and Ti :X×Y → 2Yi _{be set-valued maps with nonempty convex values and open lower sections, the set}

fromX×Y intoZi. For eachi∈I, assume that

iM_iZ_i\−intC_i:X → 2Zi _{is upper semicontinuous;}

iiFor all xi ∈ Xi andy ∈ Y,z_i → φ_ixi, y, z_iis natural P_i-quasifunction, where P_i ∩x∈XCix;

iiiφiis continuous onX×Y;

ivthere exist nonempty and compact subsetsN ⊆ X and K ⊆ Y and nonempty, compact and convex subsetsB_i ⊆ X_i,A_i ⊆ Y_i for eachi ∈ I such that ∀x, y xi, x_i, y ∈

X×Y \N×K ∃i∈Iand ∃ui ∈Bi,vi ∈Aisatisfyingui∈Dix, y,vi∈Tix, yand

φixi_{, y, ui−φix, y∈ −intCix.}

Then, there existsx, y xi, x_i, yi, y_iinX×Ysuch that for eachi∈I,

xi∈Dix, y, y_i∈Tix, y:φixi, y, zi−φix, y/∈ −intCix, ∀zi ∈Dix, y.

4.1

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

Proof. For eachi∈I, we define a trifunctionfi:X×Y×Xias

fix, y, uiφixi_{, y, ui−φix, y, ∀x, y, ui∈X×Y×Xi. 4.2}

Sinceφ_ixi, y,·is naturalP_iquasi-function, by19, Remark 2, for allu_i1, u_i2 ∈X_iand for all

λ∈0,1, ∃α∈0,1such that

φi

xi_{, y, λu}

i1 1−λui2

∈αφi

xi_{, y, u}

i1

1−αφ_ixi, y, u_i2−P_i, 4.3

Hence

fix, y, λui1 1−λui2

∈αfix, y, ui1

1−αf_ix, y, u_i2

−Pi. 4.4

Hence, for allx, y∈X×Y,f_ix, y,·is naturalP_iquasifunction.

By condition iii, we know that for all zi ∈ Xi, the map x, y → fix, y, zi is continuous onX×Y. So it follows fromRemark 3.2that conditioniiiofTheorem 3.1holds. It is easy to verify that the other conditions ofTheorem 3.1are satisfied. ByTheorem 3.1, we know that the conclusion holds.

Similarly, byTheorem 3.3, Corollaries3.4and3.5, respectively, we have the following results.

Theorem 4.2. LetIbe any index set. For eachi∈ I, letZi be a topological vector space, letEiand

Fi be two locally convex Hausdorfftopological vector spaces, letXi ⊆ Ei andYi ⊆ Fibe nonempty, closed and convex subsets, letC_i :X → 2Zi _{be a set-valued map such that}C_ix_{is a proper, closed}

and convex cone with apex at the origin and intCix_/∅ for each x ∈ X and Pi ∩x∈XCix,

Di : X ×Y → 2Xi _and Ti _: X ×Y → ₂Yi _{be set-valued maps with nonempty convex values}

closed inX×Y andϕi :X×Y → Zibe a vector-valued function. Assume that the set-valued map

D×T _i∈IDi×i∈ITi:X×Y → 2X×Ydefined asD×Tx, y i∈IDix, y×i∈ITix, y,

∀x, y∈X×Y, isΦ-condensing and (i), (ii), and (iii) ofTheorem 4.1hold. Then, the solution set of the (G-Debreu VEP) is nonempty.

Theorem 4.3. LetIbe any index set. For eachi∈I, letX_i⊆E_iandY_i⊆F_ibe nonempty and convex subsets, letDi:X×Y → 2Xi_andTi_:X×Y → ₂Yi_{be set-valued maps with nonempty convex values}

and open lower sections, the setWi{x, y∈X×Y :xi∈Dix, yandyi∈Tix, y}be closed in

X×Yandφ_ibe a bifunction fromX×YintoR. For eachi∈I, assume that

ifor allxi∈Xiandy∈Y,zi→φixi, y, ziis quasiconvex;

iiφ_iis continuous onX×Y;

iiithere exist nonempty and compact subsetsN ⊆ X and K ⊆ Y and nonempty, compact and convex subsetsBi ⊆ Xi,Ai ⊆ Yi for eachi ∈ I such that ∀x, y xi, xi, y ∈

X×Y \N×K ∃i∈Iand ∃ui ∈Bi,vi ∈Aisatisfyingui∈Dix, y,vi∈Tix, yand

φixi, y, ui< φix, y.

Then, there existsx, y xi, xi, yi, y_iinX×Ysuch that for eachi∈I,

xi∈Dix, y, yi∈Tix, y:φi

xi, y, zi

≥φix, y, ∀zi∈Dix, y. 4.5

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

Theorem 4.4. LetIbe any index set. For eachi∈I, letE_i andF_ibe two locally convex Hausdorff topological vector spaces, X_i ⊆ E_i and Y_i ⊆ F_i be nonempty, closed and convex subsets, letD_i :

X×Y → 2Xi_andTi _:X×Y → ₂Yi_{be set-valued maps with nonempty convex values and open lower}

sections, the setW_i {x, y∈ X×Y :xi ∈Dix, yandyi ∈ Tix, y}be closed inX×Y and ϕi:X×Y → Rbe a function. Assume that the set-valued mapD×T i∈IDi×i∈ITi:X×Y →

2X×Y defined asD×Tx, y _i∈IDix, y×_i∈ITix, y,∀x, y∈X×Y, isΦ-condensing and (i), and (ii) ofTheorem 4.3hold. Then, the solution set of the (G-Debreu EP) is nonempty.

Remark 4.5. Theorem 4.1extends and generalizes 19, Theorem 5 and20, Theorems 3.1, 3.6 and Corollaries 3.2, 3.3, and 3.5. Theorem 4.2extends 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 in21–24.

Acknowledgment

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

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