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Journal of Inequalities and Applications Volume 2008, Article ID 581849,15pages doi:10.1155/2008/581849

Research Article

Connectedness and Compactness of

Weak Efficient Solutions for Set-Valued

Vector Equilibrium Problems

Bin Chen, Xun-Hua Gong, and Shu-Min Yuan

Department of Mathematics, Nanchang University, Nanchang 330047, China

Correspondence should be addressed to Xun-Hua Gong,xunhuagong@gmail.com

Received 1 November 2007; Revised 17 July 2008; Accepted 5 September 2008

Recommended by C. E. Chidume

We study the set-valued vector equilibrium problems and the set-valued vector Hartman-Stampacchia variational inequalities. We prove the existence of solutions of the two problems. In addition, we prove the connectedness and the compactness of solutions of the two problems in normed linear space.

Copyrightq2008 Bin Chen 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.

1. Introduction

We know that one of the important problems of vector variational inequalities and vector equilibrium problems is to study the topological properties of the set of solutions. Among its topological properties, the connectedness and the compactness are of interest. Recently, Lee et al.1and Cheng2have studied the connectedness of weak efficient solutions set for single-valued vector variational inequalities in finite dimensional Euclidean space. Gong

3–5 has studied the connectedness of the various solutions set for single-valued vector equilibrium problem in infinite dimension space. The set-valued vector equilibrium problem was introduced by Ansari et al.6. Since then, Ansari and Yao7, Konnov and Yao8, Fu

9, Hou et al.10, Tan11, Peng et al.12, Ansari and Flores-Baz´an13, Lin et al.14

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2. Preliminaries

Throughout this paper, letX,Y be two normed linear spaces, letAbe a nonempty subset of X, letF:A×A→2Y be a set-valued map, and letCbe a closed convex pointed cone inY.

We consider the following set-valued vector equilibrium problemSVEP: findxA, such that

Fx, y∩−intC ∅ ∀y∈A. 2.1

Definition 2.1. Let intC /∅.A vectorxAsatisfying

Fx, y∩−intC ∅ ∀y∈A 2.2

is called a weak efficient solution to the SVEP. Denote byVwA, Fthe set of all weak efficient

solutions to the SVEP.

LetY∗be the topological dual space ofY. Let

C{f Y:fy0 ∀yC} 2.3

be the dual cone ofC.

Definition 2.2. LetfC∗\ {0}. A vectorxAis called anf-efficient solution to the SVEP if fFx, y≥0 ∀y∈A, 2.4

wherefFx, y≥0 means thatfz≥0, for allzFx, y. Denote byVfA, Fthe set of all f-efficient solutions to the SVEP.

Definition 2.3. LetAbe a nonempty convex subset inX. A set-valued mapF : A×A→2Y is called to beC-convex in its second variable if, for each fixedxA, for everyy1, y2 ∈A,

t∈0,1, the following property holds:

tFx, y1 1−tFx, y2⊂Fx, ty1 1−ty2 C. 2.5

Definition 2.4. LetAbe a nonempty convex subset inX. A set-valued mapF : A×A→2Y is called to beC-concave in its first variable if, for each fixedyA, for everyx1, x2 ∈ A,

t∈0,1, the following property holds:

Ftx1 1−tx2, ytFx1, y 1−tFx2, y C. 2.6

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iLetAbe a convex subset ofX.Tis said to bev-hemicontinuous if, for every pair of pointsx, yA, the set-valued map

: Tαy 1−αx, yx, α∈0,1, 2.7

is lower semicontinuous at 0.

iiLetfC∗\ {0}.Tis said to bef-pseudomonotone onAif, for every pair of points x, yA,fs, yx≥0, for allsTx, thenfs, yx≥0, for allsTy. The definition ofv-hemicontinuity was introduced by Lin et al.14.

Definition 2.6. LetX be a Hausdorfftopological vector space and letKXbe a nonempty set.G:K→2Xis called to be a KKM map if for any finite set{x1, . . . , xn} ⊂Kthe relation

co{x1, . . . , xn} ⊂ n

i1

Gxi 2.8

holds, where co{x1, . . . , xn}denoted the convex hull of{x1, . . . , xn}.

For the definition of the upper semicontinuity and lower semicontinuity, see16. The following FKKM theorem plays a crucial role in this paper.

Lemma 2.7. LetXbe a Hausdorfftopological vector space. LetKbe a nonempty convex subset ofX,

and letG:K→2Kbe a KKM map. If for eachxK,Gxis closed inX, and if there exists a point x0∈Ksuch thatGx0is compact, thenx∈KGx/.

By definition, we can get the following lemma.

Lemma 2.8. LetAbe a nonempty convex subset ofX. LetF :A×A→2Y be a set-valued map, and

letCY be a closed convex pointed cone. Moreover, suppose thatFx, yisC-convex in its second variable. Then, for eachxA,Fx, A Cis convex.

3. Scalarization

In this section, we extend a result in3to set-valued map.

Theorem 3.1. Suppose thatintC /, and thatFx, A Cis a convex set for eachxA. Then

VwA, F

f∈C∗\{0}

VfA, F. 3.1

Proof. It is clear that

VwA, F

f∈C∗\{0}

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Now we prove that

VwA, F

f∈C∗\{0}

VfA, F. 3.3

LetxVwA, F. By definition,Fx, y∩−intC ∅, for allyA. Thus

Fx, A∩−intC ∅. 3.4

AsCis a convex pointed cone, we have

Fx, A C∩−intC ∅. 3.5

By assumption,Fx, A Cis a convex set. By the separation theorem of convex sets, there exist somefY∗\ {0}, such that

inf{fFx, y c:yA, cC} ≥sup{f−c:cC}. 3.6

By3.6, we obtain thatfC∗\ {0}and

fFx, y≥0 ∀y∈A. 3.7

Therefore,xVfA, F. HenceVwA, Ff∈C\{0}VfA, F. Thus we have

VwA, F

f∈C∗\{0}

VfA, F. 3.8

4. Existence of the weak efficient solutions

Theorem 4.1. LetA be a nonempty closed convex subset ofX and letCY be a closed convex

pointed cone withintC /. LetF:A×A→2Y be a set-valued map withFx, xCfor allxA. Suppose that for eachyA,F·, yis lower semicontinuous onA, and that Fx, yisC-convex in its second variable. If there exists a nonempty compact subset D of A, andyD, such that Fx, y∩−intC/, for allxA\D, then, for anyfC∗\ {0},VfA, F/,VfA, FD, VwA, F/, andVwA, FD.

Proof. LetfC∗\ {0}. Define the set-valued mapG:A→2Aby

Gy {x∈A:fFx, y≥0}, y∈A. 4.1

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x∈co{y1, . . . , yn}such thatx /ni1Gyi. Thenx

n

i1tiyifor someti ≥0, 1≤in, with

n

i1ti1, andx /Gyi, for alli1, . . . , n. Then there existziFx, yi, such that

fzi<0 ∀i1, . . .,n. 4.2

AsFx, yisC-convex in its second invariable, we can get that

t1Fx, y1 t2Fx, y2 · · ·tnFx, ynFx, x C. 4.3

By4.3, we know that there existzFx, x,cC, such that

t1z1t2z2· · ·tnznzc. 4.4

Hencefzc ft1z1 t2z2· · ·tnzn. By assumption, we havefzc ≥ 0. By4.2,

however, we haveft1z1t2z2· · ·tnzn<0. This is a contradiction. ThusGis a KKM map.

Now we show that for eachyA,Gyis closed. For any sequence,{xn} ⊂Gyandxnx0.

BecauseAis a closed set, we have x0 ∈ A. By assumption, for eachyA,F·, yis lower

semicontinuous onA, then by16, for each fixedyA, and for eachz0 ∈ Fx0, y, there

existznFxn, y, such thatznz0. Because{xn} ⊂Gy, we have

fFxn, y≥0. 4.5

Thusfzn ≥0. By the continuity off andznz0, we havefz0 ≥ 0. By the arbitrariness

ofz0 ∈ Fx0, y, we havefFx0, y ≥ 0, that is,x0 ∈ Gy. HenceGyis closed. By the

assumption, we haveGyD, andGyis closed. SinceDis compact,Gyis compact. ByLemma 2.7, we havey∈AGy/∅. Thus there existsxy∈AGy. This means that

fFx, y≥0 ∀y∈A. 4.6

Therefore, xVfA, F. Next we show that VfA, FD. If xVfA, F, then x

y∈AGyGyD. It follows from VfA, FVwA, F that VwA, F/∅, and by

Theorem 3.1, we haveVwA, FD.

Theorem 4.2. LetA be a nonempty closed convex subset ofX and letCY be a closed convex

pointed cone withintC /. LetfC∗\ {0}. Assume thatT :A→2LX,Yis av-hemicontinuous, f-pseudomonotone mapping. Moreover, assume that the set-valued mapF :A×A→2Y defined by Fx, y Tx, yxisC-convex in its second variable. If there exists a nonempty compact subsetD ofA, andyD, such thatTx, yx∩−intC/, for allxA\D, thenVfA, F/and VfA, FD.

Proof. LetfC∗\ {0}. Define the set-valued mapsE,G:A→2Aby Ey {x∈A:fs, yx≥0 ∀s∈Tx}, yA,

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respectively. As for eachyA, we haveyEy, thenEy/∅. The proof of the theorem is divided into four steps.

IEis a KKM map onA.

Suppose to the contrary that there exists a finite subset{y1, . . . , yn} ofA, and there

existsx ∈co{y1, . . . , yn}such thatx /in1Eyi. ThenxA, x

n

i1tiyi for someti ≥ 0,

1≤in, withni1ti1, andx /Eyi, for alli1, . . . , n. Then there existsiT xsuch that fsi, yix<0 for each i1,2, . . . , n. 4.8

SinceFx, yisC-convex in its second variable, we have

t1T x, y1−x · · ·tnT x, ynxT x, xx CC. 4.9

Letzi si, yix, for eachi1,2, . . . , n. By4.9, we know there existscC, such that t1z1t2z2· · ·tnznc. 4.10

AsfC∗\ {0}, we have

fc≥0. 4.11

While by4.8, we haveft1z1t2z2· · ·tnzn<0. This is a contraction. HenceEis a KKM

map onA.

IIEyGyfor allyAandGis a KKM map.

By thef-pseudomonotonicity ofT, for eachyA, we haveEyGy. SinceEis a KKM map, so isG.

IIIy∈AGy/∅.

Now we show that for eachyA,Gyis closed. Let{xn}be a sequence inGysuch thatxn converges tox. By the closedness ofA, we havexA. Since{xn} ⊂ Gy, then for

eachsTy, we have

fs, yxn≥0. 4.12

Asxnx, and the continuity off, then for eachsTy, we have

fs, yx≥0. 4.13

Consequently,xGy. HenceGyis closed. By the assumption, we haveGyD. Then Gyis compact sinceDis compact. By stepII, we knowGis a KKM map. ByLemma 2.7,

y∈AGy/∅.

IVy∈AGy y∈AEy.

Because EyGy, we have y∈AGyy∈AEy. Now let us show that

y∈AGyy∈AEy. Letxy∈AGy. For eachyA, and eachsTy, we have

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For anysT xand for each fixedyA, define the set-valued mappingJ:0,1→2Y by Tαy 1−αx, yx α∈0,1. 4.15

We pick a sequence{αn} ⊂0,1such thatαn→0 and setxn αny 1−αnx. SinceAis a

convex set,xnAfor eachn. It is clear thatxnx. Letw s, y−x. We havewJ0.Since

T isv-hemicontinuous,Jαis lower semicontinuous at 0. By16, there existwnJαn

Tαny 1−αnx, yx, such thatwnw. AswnJαn, there existsnTxnsuch that

wn sn, yx. Bywnw, we havesn, yxs, yx. By4.14, we have

αnfsn, yx fsn, αny 1−αnxx≥0. 4.16

Sinceαn > 0,fsn, yx≥ 0. Hencefs, yx ≥ 0 sincef is continuous andwnw. Therefore, for anysT xand for eachyA, we havefs, yx≥0. Hencexy∈AEy. Thusy∈AEy y∈AGy/∅. This means that there existsxA, for eachsTx, we have fs, yx≥0, for allyA. It follows thatxVfA, F, thusVfA, F/∅. By the proof of

Theorem 4.1, we knowVfA, FD. SinceVfA, FVwA, F, we haveVwA, F/∅. The

proof of the theorem is completed.

5. Connectedness and compactness of the solutions set

In this section, we discuss the connectedness and the compactness of the weak efficient solutions set for set-valued vector equilibrium problems and the set-valued vector Hartman-Stampacchia variational inequalities in normed linear space.

Theorem 5.1. LetAbe a nonempty closed convex subset ofX, letCY be a closed convex pointed

cone withintC /, and letF:A×A→2Ybe a set-valued map. Assume that the following conditions are satisfied:

ifor eachyA,F·, yis lower semicontinuous onA;

iiFx, yisC-concave in its first variable andC-convex in its second variable;

iiiFx, xC, for allxA;

iv{Fx, y:x, yA}is a bounded subset inY;

vthere exists a nonempty compact convex subsetDofA, andyD, such thatFx, y

−intC/, for allxA\D.

ThenVwA, Fis a nonempty connected compact set.

Proof. We define the set-valued mapH:C∗\ {0} →2Dby

Hf VfA, F, fC∗\ {0}. 5.1

ByTheorem 4.1, for eachfC∗\ {0}, we haveHf/∅, henceVwA, F/∅andVwA, F

D. It is clear thatC\ {0}is convex, so it is a connected set. Now we prove that, for each

fC\ {0},Hfis a connected set. Letx

1, x2∈Hf, we havex1, x2∈Dand

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BecauseFx, yisC-concave in its first variable, for each fixedyA, and for abovex1, x2 ∈

D, andt∈0,1, we havetx1 1−tx2∈DsinceDis convex, and

Ftx1 1−tx2, ytFx1, y 1−tFx2, y C. 5.3

Hence for eachyA,zFtx1 1−tx2, y, there existz1 ∈Fx1, y,z2 ∈ Fx2, y, and

cC, such thatztz1 1−tz2c. AsfC∗\ {0}and by5.2, we have

fz tfz1 1−tfz2 fc≥0. 5.4

Thus

fFtx1 1−tx2, y≥0 ∀y∈A, 5.5

that istx1 1−tx2∈Hf. SoHfis convex, therefore it is a connected set.

Now we show thatHfis upper semicontinuous onC∗\ {0}. SinceDis a nonempty compact set, by16, we only need to prove thatHis closed. Let the sequence{fn, xn} ⊂

GraphH and fn, xnf0, x0, where {fn} converge to f0 with respect to the norm

topology. Asfn, xn∈GraphH, we have

xnHfn VfnA, F, 5.6

that is,fnFxn, y≥0, for allyA. Asxnx0andDis compact, we havex0∈D. Since for

eachyA,F·, yis lower semicontinuous onA, for each fixedyA, and eachz0∈Fx0, y,

there existznFxn, y, such thatznz0. FromfnFxn, y≥0, we have

fnzn≥0. 5.7

By the continuity off0andznz0, we have

f0zn−→f0z0. 5.8

LetQ {Fx, y:x, yA}. By assumption,Qis a bounded set inY, then there exist some M >0, such that for eachzQ, we havez ≤ M. For anyε > 0, becausefnf0→0 with

respect to norm topology, there existsn0 ∈ N, and when nn0, we havefnf0 < ε.

Therefore, there existsn0∈N, and whennn0, we have

|fnznf0zn||fnf0zn| ≤ fnf0znMε. 5.9

Hence

lim

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Consequently, by5.8,5.10, we have

lim

n→ ∞fnzn nlim→ ∞fnznf0zn f0zn

lim

n→ ∞fnznf0zn n→ ∞limf0zn

f0z0.

5.11

By5.7, we havef0z0≥0. So for anyyAand for eachz0∈Fx0, y, we havef0z0≥0.

Hence

f0Fx0, y≥0 ∀y∈A. 5.12

This means that

x0∈Vf0A, F Hf0. 5.13

Hence the graph of H is closed. Therefore, H is a closed map. By 16, H is upper semicontinuous onC∗\{0}. BecauseFx, yisC-convex in its second variable, byLemma 2.8, for eachxA,Fx, A Cis convex. It follows fromTheorem 3.1that

VwA, F

f∈C∗\{0}

VfA, F. 5.14

Thus by17, Theorem 3.1VwA, Fis a connected set.

Now, we show that VwA, F is a compact set. We first show that VwA, F is a

closed set. Let {xn} ⊂ VwA, F with xnx0. Since D is compact, x0 ∈ D. We claim that

x0∈VwA, F.Suppose to the contrary thatx0/VwA, F, then there exist somey0 ∈Asuch

that

Fx0, y0∩−intC/∅. 5.15

Thus there existsz0∈Fx0, y0such that

z0∈ −intC. 5.16

Hence−intCis a neighborhood ofz0. SinceF·, y0is lower semicontinuous atx0, there exists

some neighborhoodUx0ofx0such that

Fx, y0∩−intC/∅ ∀x∈Ux0∩A. 5.17

Sincexnx0, there exist somen0, and when nn0, we havexnUx0∩A. By5.17,

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This contradicts{xn} ⊂VwA, F. Thusx0 ∈VwA, F.This means thatVwA, Fis a closed

set. SinceDis compact andVwA, FD,VwA, Fis compact.

Theorem 5.2. LetAbe a nonempty closed convex subset ofX, and let CY be a closed convex

pointed cone withintC /. Assume that for eachfC∗\{0},T :A→2LX,Yis av-hemicontinuous, f-pseudomonotone mapping. Moreover, assume that the set-valued mapF :A×A→2Y defined by Fx, y Tx, yxis C-convex in its second variable, and the set {Fx, y : x, yA} is a bounded set inY. If there exists a nonempty compact convex subsetDofA, andyD, such that

Tx, yx∩−intC/, for allxA\D, thenVwA, Fis a nonempty connected set. Proof. We define the set-valued mapH:C∗\ {0} →2Dby

Hf VfA, F for eachfC∗\ {0}. 5.19

ByTheorem 4.2, for eachfC∗\{0}, we haveHf VfA, F/∅andVfA, FD. Hence

VwA, F/∅andVwA, FD. Clearly,C∗\ {0}is a convex set, hence it is a connected set.

Define the set-valued mapsE,G:A→2Aby

Ey {x∈A:fs, yx≥0, ∀s∈Tx}, yA, Gy {x∈A:fs, yx≥0, ∀s∈Ty}, yA,

5.20

respectively. Now we prove that for eachfC∗\ {0},Hfis a connected set. Letx1, x2 ∈

Hf VfA, F, thenx1, x2∈y∈AEy. By the proof ofTheorem 4.2, we havey∈AGy

y∈AEy, sox1, x2 ∈

y∈AGy. Hence fori1,2,and for eachyA,sTy, we have

fs, yxi≥0. 5.21

Then, for eachyA,sTy, andt∈0,1, we havetx1 1−tx2∈DsinceDis convex and

fs, ytx1 1−tx2≥0. 5.22

Hencetx1 1−tx2 ∈ y∈AGy y∈AEy. Thustx1 1−tx2 ∈ Hf. Consequently,

for eachfC∗\ {0},Hfis a convex set. Therefore, it is a connected set. The following is to prove thatHis upper semicontinuous onC∗\ {0}. SinceDis a nonempty compact set, by

16we only need to show thatHis a closed map. Let sequence{fn, xn} ⊂GraphHand

fn, xnf0, x0, where{fn}converges tof0 with respect to the norm topology ofY∗. As

fn, xn∈GraphH, we have

xnHfn VfnA, F. 5.23

Then, for eachsTxn, we have that

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By assumption, for each n, T : A→2LX,Y isfn-pseudomonotone, and by5.24, for each

yA, for the abovexn, and for eachsTy, we have

fns, yxn≥0 ∀y∈A. 5.25

Asxnx0, we haves, yxns, yx0, andf0s, yxnf0s, yx0. Asxnx0,

and D is compact, we havex0 ∈ D. LetQ {Fx, y : x, yA}. By assumption, Qis a

bounded set inY. Then, there existsM >0, such that for eachzQ, we havez ≤M. For anyε > 0, becausefnf0→0 with respect to the norm topology, there existsn0 ∈ N, and

when nn0, we havefnf0 < ε. Therefore, there existsn0 ∈N, and when nn0, we

have

|fns, yxnf0s, yxn||fnf0s, yxn| ≤Mε. 5.26

Hence

lim

n→ ∞fns, yxnf0s, yxn 0. 5.27

Then

lim

n→ ∞fns, yxn nlim→ ∞fns, yxnf0s, yxn f0s, yxn

f0s, yx0.

5.28

Then, by5.25,5.28, we havef0s, yx0≥0. Hence for eachyA, and for eachsTy,

we havef0s, yx0 ≥ 0. Since T isf0-pseudomonotone, for each yA, and for each

sTx

0, we havef0s, yx0≥ 0. Hencex0 ∈ Hf0 Vf0A, F. Therefore, the graph ofHis closed, andHis a closed map. By16, we know thatHis upper semicontinuous on C\ {0}. BecauseFx, yisC-convex in its second variable, for eachx A,Fx, A Cis

convex. It follows fromTheorem 3.1that

VwA, F

f∈C∗\{0}

VfA, F. 5.29

Then, by 17, Theorem 3.1, we know that VwA, F is a connected set. The proof of the

theorem is completed.

LetNb0denote the base of neighborhoods of 0 ofLX, Y. By18, for each bounded

subsetQX,and for each neighborhoodVof 0 inY, we have

WQ, V:{s∈LX, Y:sQV} ∈Nb0. 5.30

Lemma 5.3. Let A be a nonempty convex subset of X, and let T : A→2LX,Y. If T is lower

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Proof. For any fixedx, yA, we need to show that the set-valued mapping

: Tαy 1−αx, yx, α∈0,1, 5.31

is lower semicontinuous at 0. For anyzJ0and for any neighborhoodUzofz, there existss0∈Txsuch thatz s0, yx,and there exists a neighborhoodV of 0 inY, such that

Uz zV. SinceTis lower semicontinuous atx, for the neighborhood

Us0:s0Wyx, V s0{s∈LX, Y: s, yxV} 5.32

ofs0, there exists a neighborhoodUxofxsuch that

TxUs0/∅ ∀x ∈UxA. 5.33

For the abovey, since αy 1−αxx, whenα→0, there exists 0 < α0 < 1, and when

0< α < α0, we haveαy 1−αxUx.By5.33, we have

Tαy 1−αxUs0/∅ ∀0< α < α0. 5.34

Thus there existTαy 1−αxsuch thatUs0, for all 0< α < α0. By5.32, we

have

sα, yxUz ∀0< α < α0, 5.35

that is

Uz/∅ ∀0≤α < α0. 5.36

This means that J is lower semicontinuous at 0. By definition, T is v-hemicontinuous on A.

Theorem 5.4. LetAbe a nonempty closed bounded convex subset ofX, and let CY be a closed

convex pointed cone with intC /. Assume that for eachfC∗\ {0},T : A→2LX,Y is a f -pseudomonotone, lower semicontinuous mapping. Moreover, assume that the set-valued mapF :A× A→2Y defined byFx, y Tx, yxisC-convex in its second variable, and the set{Fx, y : x, yA} is a bounded set in Y. If there exists a nonempty compact convex subset D ofA, and yD, such that Tx, yx∩−intC/, for allxA\D, thenVwA, Fis a nonempty connected compact set.

Proof. ByLemma 5.3and Theorem 5.2,VwA, F is a nonempty connected set. SinceDis a

compact set, we need only to show thatVwA, Fis closed. Let{xn} ⊂VwA, F, xnx0.It

is clear thatx0∈Dand{xn} ⊂D. We claim thatx0∈VwA, F.Suppose to the contrary that

x0/VwA, F, then there existsy0∈Asuch that

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that is

Tx0, y0−x0∩−intC/∅. 5.38

Thus there existss0∈Tx0such that

s0, y0−x0∈ −intC. 5.39

Hence there existsε >0 such that

s0, y0−x0 εB⊂ −intC, 5.40

whereBis the unit ball ofY. Set

W

y0−A,

ε 2 B

sLX, Y: s, y0−A

ε

2

B . 5.41

By definition,Wy0−A,ε/2Bis a neighborhood of zero ofLX, Y, sinceAis bounded.

SinceTis lower semicontinuous atx0, for the aboves0∈Tx0and the aboveWy0−A,ε/2B,

there exists a neighborhoodUx0ofx0such that

Txs0W

y0−A,

ε

2

B/∅ ∀x∈Ux0∩A. 5.42

Sincexnx0,there existsNsuch that whennN, we havexnUx0.Thus by5.42we

have

Txn

s0W

y0−A,

ε 2 B /

∅. 5.43

Thus there exist snTxns0 Wy0 −A,ε/2B,for all nN. We have sns0 ∈

Wy0−A,ε/2B, and hence

sns0, y0−A

ε

2

B. 5.44

Therefore,

sns0, y0−xn

ε

2

(14)

Sincexnx0 ands0 ∈ LX, Y, we haves0, x0−xn→0. Hence there existsN1 ≥ N, and

when nN1,we haves0, x0−xnε/2B. This combining5.45implies that

sn, y0−xns0, y0−x0

sn, y0−xns0, y0−xn s0, y0−xns0, y0−x0

sns0, y0−xn s0, x0−xn

ε

2

B ε

2

BεB ∀n≥N1.

5.46

Hence

sn, y0−xns0, y0−x0 εB ∀n≥N1. 5.47

By5.40and5.47, we have

sn, y0−xn∈ −intCnN1. 5.48

On the other hand, since{xn} ⊂VwA, F, we have

Txn, yxn∩−intC ∅ ∀y∈A, ∀n. 5.49

This contradicts5.48, becausesnTxn.Thusx0 ∈VwA, F. This means thatVwA, Fis a

closed subset ofX.

Acknowledgments

This research was partially supported by the National Natural Science Foundation of China and the Natural Science Foundation of Jiangxi Province, China.

References

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Journal of Optimization Theory and Applications, vol. 133, no. 2, pp. 151–161, 2007.

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9 J.-Y. Fu, “Generalized vector quasi-equilibrium problems,” Mathematical Methods of Operations Research, vol. 52, no. 1, pp. 57–64, 2000.

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Journal of Optimization Theory and Applications, vol. 119, no. 3, pp. 485–498, 2003.

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