Systems of Generalized Quasivariational Inclusion Problems with Applications in L Γ -Spaces

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Volume 2011, Article ID 561573,12pages doi:10.1155/2011/561573

Research Article

Systems of Generalized Quasivariational Inclusion Problems with Applications in L Γ -Spaces

Ming-ge Yang,

^{1, 2}

Jiu-ping Xu,

³

and Nan-jing Huang

^{1, 3}

1Department of Mathematics, Sichuan University, Chengdu, Sichuan 610064, China

2Department of Mathematics, Luoyang Normal University, Luoyang, Henan 471022, China

3College of Business and Management, Sichuan University, Chengdu, Sichuan 610064, China

Correspondence should be addressed to Nan-jing Huang,[email protected] Received 27 September 2010; Accepted 22 October 2010

Academic Editor: Yeol J. E. Cho

Copyrightq2011 Ming-ge Yang 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.

We first prove that the product of a family ofLΓ-spaces is also anLΓ-space. Then, by using a Himmelberg type fixed point theorem inLΓ-spaces, we establish existence theorems of solutions for systems of generalized quasivariational inclusion problems, systems of variational equations, and systems of generalized quasiequilibrium problems inLΓ-spaces. Applications of the existence theorem of solutions for systems of generalized quasiequilibrium problems to optimization problems are given inLΓ-spaces.

1. Introduction

In 1979, Robinson1studied the following parametric variational inclusion problem: given x∈Rⁿ, findy∈R^msuch that

0∈g x, y

Q x, y

, 1.1

whereg:Rⁿ×R^m → R^pis a single-valued function andQ:Rⁿ×R^mR^pis a multivalued map. It is known that1.1covers variational inequality problems and a vast of variational system important in applications. Since then, various types of variational inclusion problems have been extended and generalized by many authors see, e.g., 2–7 and the references therein.

On the other hand, Tarafdar 8 generalized the classical Himmelberg fixed point theorem 9 to locally H-convex uniform spaces or LC-spaces. Park 10 generalized the result of Tarafdar 8 to locally G-convex spaces or LG-spaces. Recently, Park 11

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introduced the concept of abstract convex spaces which include H-spaces and G-convex spaces as special cases. With this new concept, he can study the KKM theory and its applications in abstract convex spaces. More recently, Park12 introduced the concept of LΓ-spaces which includeLC-spaces andLG-spaces as special cases. He also established the Himmelberg type fixed point theorem inLΓ-spaces. To see some related works, we refer to 13–21and the references therein. However, to the best of our knowledge, there is no paper dealing with systems of generalized quasivariational inclusion problems inLΓ-spaces.

Motivated and inspired by the works mentioned above, in this paper, we first prove that the product of a family ofLΓ-spaces is also anLΓ-space. Then, by using the Himmelberg type fixed point theorem due to Park12, we establish existence theorems of solutions for systems of generalized quasivariational inclusion problems, systems of variational equations, and systems of generalized quasiequilibrium problems in LΓ-spaces. Applications of the existence theorem of solutions for systems of generalized quasiequilibrium problems to optimization problems are given inLΓ-spaces.

2. Preliminaries

For a setX,Xwill denote the family of all nonempty finite subsets ofX. IfAis a subset of a topological space, we denote by intAandAthe interior and closure ofA, respectively.

A multimapor simply a mapT : X Y is a function from a setXinto the power set 2^Y ofY; that is, a function with the valuesTx⊂Y for allx∈X. Given a mapT :XY, the mapT⁻:Y Xdefined byT⁻y {x∈X:y∈Tx}for ally∈Y, is called thelower inverse ofT. For anyA⊂X,TA:

x∈ATx. For anyB⊂Y,T⁻B:{x∈X:Tx∩B /∅}.

As usual, the set{x, y∈X×Y :y∈Tx} ⊂X×Y is called the graph ofT. For topological spacesXandY, a mapT :XYis called

iclosed if its graph GraphTis a closed subset ofX×Y,

iiupper semicontinuousin short, u.s.c.if for any x ∈ X and any open setV inY withTx⊂V, there exists a neighborhoodUofxsuch thatTx⊂Vfor allx ∈U, iiilower semicontinuousin short, l.s.c.if for anyx∈Xand any open setV inYwith Tx∩V /∅, there exists a neighborhoodUofxsuch thatTx∩V /∅for allx ∈U, ivcontinuous ifTis both u.s.c. and l.s.c.,

vcompact ifTXis contained in a compact subset ofY.

Lemma 2.1see22. LetXandY be topological spaces,T :X Y be a map. Then,T is l.s.c. at x∈Xif and only if for anyy∈Txand for any net{xα}inXconverging tox, there exists a net {yα}inY such thaty_α∈Txαfor eachαandy_αconverges toy.

Lemma 2.2see23. LetXandY be Hausdorﬀtopological spaces andT :X Y be a map.

iIfT is an u.s.c. map with closed values, thenT is closed.

iiIfY is a compact space andTis closed, thenTis u.s.c.

iiiIfXis compact andTis an u.s.c. map with compact values, thenTXis compact.

In what follows, we introduce the concept of abstract convex spaces and map classes R,RCand ROhaving certain KKM properties. For more details and discussions, we refer the reader to11,12,24.

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Definition 2.3see11. An abstract convex spaceE, D;Γconsists of a topological spaceE, a nonempty setD, and a mapΓ:DEwith nonempty values. We denoteΓA : ΓAfor A∈ D.

In the caseE D, let E;Γ : E, E;Γ. It is obvious that any vector spaceE is an abstract convex space with Γ co, where co denotes the convex hull in vector spaces. In particular,R; cois an abstract convex space.

LetE, D;Γbe an abstract convex space. For anyD ⊂ D, theΓ-convex hull ofD is denoted and defined by

co_ΓD :ΓA |A∈

D ⊂E, 2.1

co is reserved for the convex hull in vector spaces. A subsetX ofEis called a Γ-convex subset ofE, D;Γrelative to D if for anyN ∈ D, we haveΓN ⊂ X; that is, co_ΓD ⊂ X.

This means thatX, D;Γ|Ditself is an abstract convex space called a subspace ofE, D;Γ.

WhenD ⊂ E, the space is denoted byE ⊃D;Γ. In such case, a subsetX ofEis said to be Γ-convex if co_ΓX ∩D ⊂ X; in other words,X isΓ-convex relative toD X∩D. When E;Γ R; co,Γ-convex subsets reduce to ordinary convex subsets.

Let E, D;Γ be an abstract convex space andZ a set. For a map F : E Z with nonempty values, if a mapG:DZsatisfies

FΓA⊂GA, ∀A∈ D, 2.2

thenGis called a KKM map with respect toF. A KKM mapG:D Eis a KKM map with respect to the identity map 1_E. A mapF:EZis said to have the KKM property and called aR-map if, for any KKM mapG : D Zwith respect toF, the family{Gy}_y∈Dhas the finite intersection property. We denote

RE, Z:

F:EZ|F is aR-map . 2.3

Similarly, whenZis a topological space, aRC-map is defined for closed-valued maps G, and aRO-map is defined for open-valued mapsG. In this case, we have

RE, Z⊂RCE, Z∩ROE, Z. 2.4

Note that ifZis discrete, then three classesR,RCandROare identical. Some authors use the notation KKME, Zinstead ofRCE, Z.

Definition 2.4 see24. For an abstract convex space E, D;Γ, the KKM principle is the statement 1E∈RCE, E∩ROE, E.

A KKM space is an abstract convex space satisfying the KKM principle.

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Definition 2.5. LetY;Γbe an abstract convex space,Zbe a real t.v.s., andF:Y Za map.

Then,

iF is {0}-quasiconvex-like if for any {y1, y2, . . . , yn} ∈ Y and any y ∈ Γ{y1, y₂, . . . , y_n}there existsj∈ {1,2, . . . , n}such thatFy⊂Fyj,

iiFis{0}-quasiconvex if for any{y1, y2, . . . , yn} ∈ Yand anyy∈Γ{y1, y2, . . . , yn} there existsj∈ {1,2, . . . , n}such thatFyj⊂Fy.

Remark 2.6. IfY is a nonempty convex subset of a t.v.s. withΓ co, thenDefinition 2.5iand iireduce toDefinition 2.4iiiandviin Lin5, respectively.

Definition 2.7see25. A uniformity for a setXis a nonempty familyUof subsets ofX×X satisfying the following conditions:

ieach member ofUcontains the diagonalΔ, iifor eachU∈ U,U⁻¹∈ U,

iiifor eachU∈ U, there existsV ∈ Usuch thatV ◦V ⊂U, ivifU∈ U,V ∈ U, thenU∩V ∈ U,

vifU∈ UandU⊂V ⊂X×X, thenV ∈ U.

The pairX,Uis called a uniform space. Every member inUis called an entourage.

For anyx∈Xand anyU∈ U, we defineUx:{y∈X :x, y∈U}. The uniformityUis called separating if

{U⊂X×X:U∈ U} Δ. The uniform spaceX,Uis Hausdorﬀif and only ifUis separating. For more details about uniform spaces, we refer the reader to Kelley 25.

Definition 2.8see12. An abstract convex uniform spaceE, D;Γ;Bis an abstract convex space with a basisBof a uniformity ofE.

Definition 2.9see12. An abstract convex uniform spaceE⊃D;Γ;Bis called anLΓ-space if

iDis dense inE, and

iifor eachU ∈ Band eachΓ-convex subsetA ⊂E, the set{x∈ E:A∩Ux/∅}is Γ-convex.

Lemma 2.10see12, Corollary 4.5. LetE ⊃ D;Γ;Bbe a HausdorﬀKKMLΓ-space andT : EEa compact u.s.c. map with nonempty closedΓ-convex values. Then,Thas a fixed point.

Lemma 2.11see24, Lemma 8.1. Let{Ei, D_i;Γi}_i∈I be any family of abstract convex spaces.

LetE :

i∈IEi andD :

i∈IDi. For eachi ∈ I, letπi : D → Di be the projection. For each A∈ D, defineΓA:

i∈IΓiπiA. Then,E, D;Γis an abstract convex space.

Lemma 2.12. LetI be any index set. For eachi ∈ I, letXi;Γi;Bibe anLΓ-space. If one defines

X :

i∈IXi,ΓA :

i∈IΓiπiA for eachA ∈ X and B : {_n

j1U^j : U^j ∈ S, j 1,2, . . . , n and n ∈ N}, whereS : {{x, y ∈ X×X : xi, yi ∈ Ui} :i ∈ I, Ui ∈ Bi}. Then, X;Γ;Bis also anLΓ-space.

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Proof. ByLemma 2.11,X;Γis an abstract convex space. It is easy to check thatSis a subbase of the product uniformity ofX. SinceBis the basis generated byS, we obtain thatBis a basis of the product uniformity, and the associated uniform topology onX.

Now, we prove that for eachU∈ Band eachΓ-convex subsetA⊂X, the set{x∈X : A∩Ux/∅}isΓ-convex. Firstly, we show that for eachi ∈ I,π_iAis aΓi-convex subset ofX_i. For anyN_i ∈ πiA, we can find someN ∈ Awithπ_iN N_i. SinceAis aΓ- convex subset ofX, we haveΓN⊂A. It follows thatΓiπiN ΓiNi⊂πiA. Thus, we have shown thatπ_iAis aΓi-convex subset ofX_i. Secondly, we show that the set{x ∈X : A∩Ux/∅}isΓ-convex. Since eachU^j ∈ Shas the formU^j {x, y∈X×X:xij, y_i_j∈U_i_j} for someij∈IandUij∈ Bij, we have that

Ux

y∈X: x, y

∈U

⎧⎨

⎩y∈X : x, y

∈ⁿ

j1

U^j

⎫⎬

⎭

y∈X: xij, yij

∈Uij ∀j1,2, . . . , n

y∈X:yij ∈Uij

xij

∀ j1,2, . . . , n

i∈I\{ⁱ^j:j1,2,...,n}

X_i×ⁿ

j1

U_i_j x_i_j

,

2.5

{x∈X:A∩Ux/∅}

⎧⎨

⎩x∈X:A∩

⎛

⎝

i∈I\{ij:j1,2,...,n}

Xi×ⁿ

j1

Uij

xij

⎞

⎠/∅

⎫⎬

⎭

⎧⎨

⎩x∈X:

i∈I\{ij:j1,2,...,n}

πiA∩Xi×ⁿ

j1

πijA∩Uij

xij

/∅

⎫⎬

⎭

⎧⎨

⎩x∈X: n

j1

πijA∩Uij

xij

/∅

⎫⎬

⎭

ⁿ

j1

x∈X:πijA∩Uij

xij

/∅

ⁿ

j1

⎛

⎝

i∈I\{ij}

Xi×

xij ∈Xij :πijA∩Uij

xij

/∅⎞

⎠.

2.6 By the definition ofLΓ-spaces, we obtain that for eachj ∈ {1,2, . . . , n}, the set{xij ∈ Xij : π_i_jA∩U_i_jxij/∅}isΓij-convex. It follows from2.6that the set{x∈X :A∩Ux/∅}is aΓ-convex subset ofX. ThereforeX;Γ;Bis anLΓ-space. This completes the proof.

Remark 2.13. Lemma 2.12 generalizes26, Theorem 2.2from locallyFC-uniform spaces to LΓ-spaces. The proof ofLemma 2.12is diﬀerent with the proof of26, Theorem 2.2.

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3. Existence Theorems of Solutions for Systems of Generalized Quasivariational Inclusion Problems

LetIbe any index set. For eachi∈I, letZibe a topological vector space,Xi;Γ¹_i;B¹_ibe anLΓ- space, andYi;Γ²_i;B²_ibe anLΓ-space with 1Yi ∈RCYi, Y_i. LetX

i∈IX_i,Y

i∈IY_iand X×Y;Γ;Bbe the productLΓ-space as defined inLemma 2.12. Furthermore, we assume that X×Y;Γ;Bis a KKM space. Throughout this paper, we use these notations unless otherwise specified, and assume that all topological spaces are Hausdorﬀ.

The following theorem is the main result of this paper.

Theorem 3.1. For eachi∈I, suppose that

iA_i:X×Y X_iis a compact u.s.c. map with nonempty closedΓ¹_i-convex values, iiT_i :X Y_iis a compact continuous map with nonempty closedΓ²_i-convex values, iiiG_i:X×Y_i×Y_iZ_iis a closed map with nonempty values,

ivfor eachx, vi∈X×Yi,yiGix, yi, viis{0}-quasiconvex; for eachx, yi∈X×Yi, viGix, yi, viis{0}-quasiconvex-like and 0∈Gix, yi, yi.

Then, there existsx, y ∈ X ×Y withx xi_i∈I andy y_i_i∈I such that for eachi ∈ I,x_i ∈ A_ix, y,y_i∈T_ixand 0∈G_ix, y_i, v_ifor allv_i∈T_ix.

Proof. For eachi∈I, defineHi:X TiXby Hix

yi∈Tix: 0∈Gi

x, yi, vi

∀vi∈Tix , ∀x∈X. 3.1 Then,Hixis nonempty for eachx∈X. Indeed, fix anyi∈Iandx∈X, defineQ^x_i :Tix T_ixby

Q_i^xvi

yi∈Tix: 0∈Gi

x, yi, vi , ∀vi∈Tix. 3.2 First, we show thatQ_i^xis a KKM map w.r.t. 1Tix. Suppose to the contrary that there exists a finite subset{v_i¹, v_i², . . . , v_iⁿ} ⊂T_ixsuch thatΓ²_i{v_i¹, v_i², . . . , v_iⁿ}/⊂_n

k1Q_i^xv^k_i. Hence, there existsvi ∈ Γ²_i{v¹_i, v²_i, . . . , vⁿ_i}satisfyingvi/∈Q_i^xv^k_ifor allk 1,2, . . . , n. SinceTixisΓ²_i- convex, we have v_i ∈ Γ²_i{v¹_i, v²_i, . . . , vⁿ_i} ⊂ T_ix. By v_i/∈Q^x_iv^k_ifor all k 1,2, . . . , n, we know that 0/∈G_ix, vi, v^k_ifor allk 1,2, . . . , n. Sincev_i G_ix, vi, v_iis{0}-quasiconvex- like, there exists 1≤j ≤nsuch that

0∈Gix, vi, vi⊂Gi

x, vi, v_i^j

. 3.3

This leads to a contradiction. Therefore,Q^x_i is a KKM map w.r.t. 1_T_i_x. Next, we show that Q^x_iviis closed for eachvi ∈ Tix. Indeed, ifyi ∈ Q^x_ivi, then there exists a net{y_i^α}_α∈Λ inQ^x_ivisuch thaty_i^α → y_i. For eachα ∈Λ, we havey_i^α ∈ T_ixand 0 ∈Gix, y^α_i, v_i. By conditionii,T_ixis closed, and hencey_i ∈T_ix. By conditioniii,G_iis closed, and hence 0∈Gix, yi, vi. It follows thatyi∈Q^x_ivi. Therefore,Q^x_iviis closed. Since 1Yi ∈RCYi, Yi andT_ixisΓ²_i-convex, we have that 1Tix ∈RCTix, Tix. Having thatT_iis compact, we can deduce that

vi∈TixQ^x_ivi/∅. That isH_ixis nonempty.

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Hi is closed for each i ∈ I. Indeed, if x, yi ∈ GraphHi, then there exists a net {x^α, y^α_i}_α∈Λ in GraphHi such that x^α, y^α_i → x, yi. One has y^α_i ∈ T_ix^α and 0 ∈ G_ix^α, y^α_i, v_ifor all v_i ∈ T_ix^α. By condition ii, T_i is closed, and hence y_i ∈ T_ix.

Letvi∈Tix, sinceTiis l.s.c., there exists a net{v_i^α}satisfyingv^α_i ∈Tix^αandv_i^α → vi. We have 0∈ Gix^α, y^α_i, v^α_i. SinceGiis closed, we obtain 0 ∈Gix, yi, vi. Thus, we have shown thatx, yi∈GraphHi. Hence,H_iis closed.

Hix isΓ²_i-convex for eachi ∈ I andx ∈ X. Indeed, if{y_i¹, y²_i, . . . , y_iⁿ} ∈ Hix, then we have that {y_i¹, y²_i, . . . , yⁿ_i} ⊂ Tix and 0 ∈ Gix, y_i^k, vi for all vi ∈ Tixand all k1,2, . . . , n. For any giveny_i∈Γ²_i{y¹_i, y²_i, . . . , yⁿ_i}, we havey_i∈T_ixbecauseT_ixisΓ²_i- convex. For eachvi∈Tix, sinceyi Gix, yi, viis{0}-quasiconvex, there exists 1≤j ≤n such that

G_i

x, y_i^j, v_i

⊂G_i

x, y_i, v_i

. 3.4

Hence, 0∈G_ix, y_i, v_ifor allv_i∈T_ix. It follows thaty_i∈H_ixandH_ixisΓ²_i-convex.

SinceHiX ⊂TiXandTiXis compact. It follows fromLemma 2.2iithatHiis a compact u.s.c. map for eachi∈I. DefineQ:X×Y X×Yby

Q x, y

i∈I

A_i x, y

×

i∈I

H_ix

, ∀ x, y

∈X×Y. 3.5

It follows from the above discussions that for eachi ∈ I,Hi is a compact u.s.c. map with nonempty closedΓ²_i-convex values. Thus,Qis a compact u.s.c. map with nonempty closed Γ-convex values. ByLemma 2.10, there existsx, y∈X×Ysuch thatx, y∈Qx, y. That is there existsx, y∈X×Ywithx xi_i∈Iandy y_i_i∈Isuch that for eachi∈I,xi∈Aix, y, y_i∈Tixand 0∈Gix, y_i, vifor allvi∈Tix. This completes the proof.

For the special case ofTheorem 3.1, we have the following corollary which is actually an existence theorem of solutions for variational equations.

Corollary 3.2. For eachi∈I, suppose that conditions (i) and (ii) inTheorem 3.1hold. Moreover, iii1G_i:X×Y_i×Y_i → Z_iis a continuous mapping;

iv1for eachx, vi∈X×Yi,yi → Gix, yi, viis{0}-quasiconvex; for eachx, yi∈X×Yi, v_i → G_ix, yi, v_iis also{0}-quasiconvex andG_ix, yi, y_i 0.

Then, there existsx, y ∈ X ×Y withx xi_i∈I andy y_i_i∈I such that for eachi ∈ I,x_i ∈ Aix, y,y_i∈TixandGix, y_i, vi 0 for allvi ∈Tix.

Theorem 3.3. For eachi∈I, suppose that conditions (i) and (ii) inTheorem 3.1hold. Moreover, iii2H_i:XZ_iis a closed map with nonempty values andQ_i:X×Y_i×Y_iZ_iis an u.s.c.

map with nonempty compact values;

iv2for eachx, vi∈X×Y_i,y_iQ_ix, yi, v_iis{0}-quasiconvex; for eachx, yi∈X×Y_i, viQix, yi, viis{0}-quasiconvex-like and 0∈Hix Qix, yi, yi.

Then, there existsx, y ∈ X ×Y withx xi_i∈I andy y_i_i∈I such that for eachi ∈ I,xi ∈ A_ix, y,y_i∈T_ixand 0∈H_ix Q_ix, y_i, v_ifor allv_i∈T_ix.

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Proof. For eachi∈I, defineGi:X×Yi×YiZiby G_i

x, y_i, v_i

H_ix Q_i

x, y_i, v_i , ∀

x, y_i, v_i

∈X×Y_i×Y_i. 3.6 Obviously, Gi has nonempty values. Now, we show that Gi is closed. Indeed, if x, yi, vi, zi∈GraphGi, then there exists a net{x^α, y_i^α, v_i^α, z^α_i}_α∈Λin GraphGisuch that x^α, y^α_i, v^α_i, z^α_i → x, yi, v_i, z_i. Since

z^α_i ∈Gi

x^α, y^α_i, v^α_i

Hix^α Qi

x^α, y^α_i, v^α_i

, 3.7

there existu^α_i ∈Hix^αandw_i^α∈Qix^α, y^α_i, v^α_isuch thatz^α_i u^α_i w^α_i. Let K{x^α:α∈Λ} ∪ {x}, L_i

y^α_i :α∈Λ ∪

y_i , M_i

v_i^α:α∈Λ ∪ {v_i}. 3.8

ThenKis a compact subset ofX,L_iandM_iare compact subsets ofY_i. By conditioniii2and Lemma 2.2iii,QiK×Li×Miis a compact subset ofZi. Thus, we can assume thatw_i^α → wi. By conditioniii2,Qiis closed, and hencewi ∈Qix, yi, vi. Sincez^α_i −w^α_i u^α_i ∈Hix^αand H_iis closed, we havez_i−w_i∈H_ix. Lettingu_iz_i−w_i, it follows that

ziuiwi∈Hix Qi

x, yi, vi Gi

x, yi, vi

, 3.9

and soGiis closed.

By the above discussions, we know that conditioniiiofTheorem 3.1is satisfied. It is easy to check that conditionivofTheorem 3.1is also satisfied. ByTheorem 3.1, there exists x, y∈X×Y withx xi_i∈Iandy y_i_i∈I such that for eachi∈I,xi ∈Aix, y,y_i∈Tix and

0∈Gi

x, y_i, vi

Hix Qi

x, y_i, vi

, 3.10

for allvi∈Tix. This completes the proof.

For the special case ofTheorem 3.3, we have the following corollary which is actually an existence theorem of solutions for variational equations.

Corollary 3.4. For eachi∈I, suppose that conditions (i) and (ii) inTheorem 3.1hold. Moreover, iii3Hi:X → Ziis a continuous map andQi:X×Yi×Yi → Ziis a continuous map;

iv3for eachx, vi∈X×Yi,yi → Qix, yi, viis{0}-quasiconvex; for eachx, yi∈X×Yi, v_i → Q_ix, yi, v_iis also{0}-quasiconvex andH_ix Q_ix, yi, y_i 0.

Then, there existsx, y ∈ X ×Y withx xi_i∈I andy y_i_i∈I such that for eachi ∈ I,x_i ∈ A_ix, y,y_i∈T_ixandH_ix Q_ix, y_i, v_i 0 for allv_i∈T_ix.

FromTheorem 3.3, we establish the following corollary which is actually an existence theorem of solutions for systems of generalized vector quasiequilibrium problems.

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Corollary 3.5. For eachi∈I, suppose that conditions (i) and (ii) inTheorem 3.1hold. Moreover, iii4Ci:XZiis a closed map with nonempty values andQi:X×Yi×YiZiis an u.s.c.

map with nonempty compact values;

iv4for eachx, vi∈X×Y_i,y_iQ_ix, yi, v_iis{0}-quasiconvex; for eachx, yi∈X×Y_i, viQix, yi, viis{0}-quasiconvex-like andQix, yi, yi∩Cix/∅.

Then, there existsx, y ∈ X ×Y withx xi_i∈I andy y_i_i∈I such that for eachi ∈ I,xi ∈ A_ix, y,y_i∈T_ix, andQ_ix, y_i, v_i∩C_ix/∅for allv_i∈T_ix.

Proof. DefineHi : X Zi byHix −Cixfor allx ∈ X. SinceCiis a closed map with nonempty values, we have thatHiis a closed map with nonempty values. All the conditions ofTheorem 3.3are satisfied. The conclusion ofCorollary 3.5follows fromTheorem 3.3. This completes the proof.

4. Applications to Optimization Problems

LetZbe a real topological vector space,Da proper convex cone inZ. A pointy∈Ais called a vector minimal point ofAif for anyy∈A,y−y /∈ −D\ {0}. The set of vector minimal point ofAis denoted by MinDA.

Lemma 4.1see27. LetZbe a Hausdorﬀt.v.s.,Dbe a closed convex cone inZ. IfAis a nonempty compact subset ofZ, then Min_DA /∅.

Theorem 4.2. For eachi∈ I, suppose that conditions (i), (ii) inTheorem 3.1and conditions (iii)4, (iv)₄inCorollary 3.5hold. Furthermore, leth:X×Y Zbe an u.s.c. map with nonempty compact values, whereZis a real t.v.s. ordered by a proper closed convex cone inZ. Then, there exists a solution to:

Min_x,yh x, y

, 4.1

wherex xi_i∈Iandy yi_i∈Isuch that for eachi∈I,xi∈Aix, y,yi∈Tix, andQix, yi, vi∩ C_ix/∅for allv_i∈T_ix.

Proof. ByCorollary 3.5, there existsx, y∈ X×Y withx xi_i∈I andy y_i_i∈I such that for eachi∈I,xi ∈Aix, y,y_i ∈TixandQix, y_i, vi∩Cix/∅for allvi ∈Tix. For each i∈I, let

M_i x, y

∈X×Y :x_i∈A_i x, y

, y_i∈T_ix, Q_i

x, y_i, v_i

∩C_ix/∅ ∀vi∈T_ix , 4.2

andM

i∈IMi. Thenx, y ∈ MandM /∅. We show thatMi is closed for eachi ∈ I.

Indeed, ifx, y∈M_i, then there exists a net{x^α, y^α}_α∈ΛinM_isuch thatx^α, y^α → x, y.

For eachα∈Λ,x^α, y^α∈Miimplies that x^α_i ∈Ai

x^α, y^α

, y^α_i ∈Tix^α, Qi

x^α, y_i^α, vi

∩Cix^α/∅ ∀vi∈Tix^α. 4.3

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By the closedness ofAi andTi, we have that xi ∈ Aix, yandyi ∈ Tix. Now, we prove thatQ_ix, yi, v_i∩C_ix/∅for allv_i∈T_ix. For anyv_i ∈T_ix, sinceT_iis l.s.c., there exists a net{v^α_i}_α∈Λsatisfyingv_i^α ∈Tix^αandv_i^α → vi. Letu^α_i ∈Qix^α, y_i^α, v_i^α∩Cix^α. SinceQiis u.s.c. with nonempty compact values, we can assume thatu^α_i → ui ∈Zi. By the closedness ofQ_iandC_i, we have thatu_i ∈ Q_ix, yi, v_i∩C_ix. Thus,Q_ix, yi, v_i∩C_ix/∅. It follows thatM_iis closed. Hence,Mis closed. Note thatM⊂

i∈IA_iX×Y×

i∈IT_iX. We know thatMis a nonempty compact subset ofX×Y. It follows fromLemma 2.2iiithathMis a nonempty compact subset ofZ. ByLemma 4.1, MinDhM/∅. That is there exists a solution of the problem: Min_x,yhx, ywherex, y∈M. This completes the proof.

Theorem 4.3. For eachi ∈ I, suppose thatXi is compact and condition (ii) inTheorem 3.1holds.

Moreover,

iii5Qi:X×Yi×Yi → Ris a continuous function;

iv5for eachx, vi∈X×Yi,yi → Qix, yi, viis{0}-quasiconvex; for eachx, yi∈X×Yi, v_i → Q_ix, yi, v_iis also{0}-quasiconvex andQ_ix, yi, y_i≥0.

Furthermore, leth:X×Y → Ris a l.s.c. function. Then there exists a solution to:

min_x,yh x, y

, 4.4

wherex xi_i∈I andy yi_i∈I such that for eachi∈ I,y_i ∈ T_ixandQ_ix, yi, v_i ≥0 for all v_i∈T_ix.

Proof. For eachi∈I, defineAi:X×Y XiandCi:XRby A_i

x, y

X_i, ∀ x, y

∈X×Y,

C_ix 0,∞, ∀x∈X, 4.5 respectively. It is easy to check that all the conditions ofCorollary 3.5are satisfied. For each i∈I, define

Mi x, y

∈X×Y :yi∈Tix, Qi

x, yi, vi

≥0∀vi∈Tix , 4.6

andM

i∈IMi. Then, byCorollary 3.5, there existsx, y∈Mand henceM /∅. Arguing as Theorem 4.2, we can prove thatMis a nonempty compact subset ofX×Y. Hence there exists a solution to the problem min_x,yhx, ywherex, y∈M. This completes the proof.

Remark 4.4. Theorem 4.3 generalizes 28, Corollary 3.5 from locally convex topological vector spaces toLΓ-spaces.

Theorem 4.5. For eachi ∈ I, suppose thatX_i is compact and condition (ii) inTheorem 3.1holds.

Moreover,

iii6F_i:X×Y_i → Ris a continuous function;

iv6for eachx∈X,y_i → F_ix, yiis{0}-quasiconvex.

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Furthermore, leth:X×Y → Rbe a l.s.c. function. Then, there exists a solution to the problem:

min_x,yh x, y

, 4.7

where x xi_i∈I and y yi_i∈I such that for each i ∈ I, yi is the solution of the problem minvi∈TixFix, vi.

Proof. For eachi∈I, defineQ_i:X×Y_i×Y_i → Rby Qi

x, yi, vi

Fix, vi−Fi x, yi

, ∀

x, yi, vi

∈X×Yi×Yi. 4.8 It is easy to check that all the conditions ofTheorem 4.3are satisfied.Theorem 4.5follows immediately fromTheorem 4.3. This completes the proof.

Acknowledgments

This work was supported by the Key Program of NSFCGrant no. 70831005and the Open FundPLN0904of State Key Laboratory of Oil and Gas Reservoir Geology and Exploitation Southwest Petroleum University.

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