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
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: findx ∈ A, such that
Fx, y∩−intC ∅ ∀y∈A. 2.1
Definition 2.1. Let intC /∅.A vectorx∈Asatisfying
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∗:fy≥0 ∀y∈C} 2.3
be the dual cone ofC.
Definition 2.2. Letf∈C∗\ {0}. A vectorx∈Ais called anf-efficient solution to the SVEP if fFx, y≥0 ∀y∈A, 2.4
wherefFx, y≥0 means thatfz≥0, for allz∈Fx, 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 fixedx ∈ A, 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 fixedy ∈ A, for everyx1, x2 ∈ A,
t∈0,1, the following property holds:
Ftx1 1−tx2, y⊂tFx1, y 1−tFx2, y C. 2.6
iLetAbe a convex subset ofX.Tis said to bev-hemicontinuous if, for every pair of pointsx, y∈A, the set-valued map
Jα: Tαy 1−αx, y−x, α∈0,1, 2.7
is lower semicontinuous at 0.
iiLetf ∈C∗\ {0}.Tis said to bef-pseudomonotone onAif, for every pair of points x, y∈A,fs, y−x≥0, for alls∈Tx, thenfs, y−x≥0, for alls ∈Ty. The definition ofv-hemicontinuity was introduced by Lin et al.14.
Definition 2.6. LetX be a Hausdorfftopological vector space and letK ⊂ Xbe 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 eachx∈K,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
letC⊂Y be a closed convex pointed cone. Moreover, suppose thatFx, yisC-convex in its second variable. Then, for eachx∈A,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 eachx∈A. Then
VwA, F
f∈C∗\{0}
VfA, F. 3.1
Proof. It is clear that
VwA, F⊃
f∈C∗\{0}
Now we prove that
VwA, F⊂
f∈C∗\{0}
VfA, F. 3.3
Letx∈VwA, F. By definition,Fx, y∩−intC ∅, for ally∈A. 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 somef∈Y∗\ {0}, such that
inf{fFx, y c:y∈A, c∈C} ≥sup{f−c:c∈C}. 3.6
By3.6, we obtain thatf∈C∗\ {0}and
fFx, y≥0 ∀y∈A. 3.7
Therefore,x∈VfA, F. HenceVwA, F⊂f∈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 letC ⊂ Y be a closed convex
pointed cone withintC /∅. LetF:A×A→2Y be a set-valued map withFx, x⊂Cfor allx∈A. Suppose that for eachy ∈ A,F·, yis lower semicontinuous onA, and that Fx, yisC-convex in its second variable. If there exists a nonempty compact subset D of A, andy ∈ D, such that Fx, y∩−intC/∅, for allx∈A\D, then, for anyf∈C∗\ {0},VfA, F/∅,VfA, F⊂D, VwA, F/∅, andVwA, F⊂D.
Proof. Letf∈C∗\ {0}. Define the set-valued mapG:A→2Aby
Gy {x∈A:fFx, y≥0}, y∈A. 4.1
x∈co{y1, . . . , yn}such thatx /∈ni1Gyi. Thenx
n
i1tiyifor someti ≥0, 1≤i≤n, with
n
i1ti1, andx /∈Gyi, for alli1, . . . , n. Then there existzi∈Fx, 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, yn⊂Fx, x C. 4.3
By4.3, we know that there existz∈Fx, x,c∈C, 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 eachy∈A,Gyis closed. For any sequence,{xn} ⊂Gyandxn→x0.
BecauseAis a closed set, we have x0 ∈ A. By assumption, for eachy ∈ A,F·, yis lower
semicontinuous onA, then by16, for each fixedy ∈ A, and for eachz0 ∈ Fx0, y, there
existzn∈Fxn, y, such thatzn→z0. Because{xn} ⊂Gy, we have
fFxn, y≥0. 4.5
Thusfzn ≥0. By the continuity off andzn→z0, we havefz0 ≥ 0. By the arbitrariness
ofz0 ∈ Fx0, y, we havefFx0, y ≥ 0, that is,x0 ∈ Gy. HenceGyis closed. By the
assumption, we haveGy ⊂D, andGyis closed. SinceDis compact,Gyis compact. ByLemma 2.7, we havey∈AGy/∅. Thus there existsx∈y∈AGy. This means that
fFx, y≥0 ∀y∈A. 4.6
Therefore, x ∈ VfA, F. Next we show that VfA, F ⊂ D. If x ∈ VfA, F, then x ∈
y∈AGy ⊂ Gy ⊂ D. It follows from VfA, F ⊂ VwA, F that VwA, F/∅, and by
Theorem 3.1, we haveVwA, F⊂D.
Theorem 4.2. LetA be a nonempty closed convex subset ofX and letC ⊂ Y be a closed convex
pointed cone withintC /∅. Letf ∈C∗\ {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, y−xisC-convex in its second variable. If there exists a nonempty compact subsetD ofA, andy ∈D, such thatTx, y −x∩−intC/∅, for allx∈ A\D, thenVfA, F/∅and VfA, F⊂D.
Proof. Letf∈C∗\ {0}. Define the set-valued mapsE,G:A→2Aby Ey {x∈A:fs, y−x≥0 ∀s∈Tx}, y∈A,
respectively. As for eachy∈A, we havey∈Ey, 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. Thenx ∈ A, x
n
i1tiyi for someti ≥ 0,
1≤i≤n, withni1ti1, andx /∈Eyi, for alli1, . . . , n. Then there existsi ∈T xsuch that fsi, yi−x<0 for each i1,2, . . . , n. 4.8
SinceFx, yisC-convex in its second variable, we have
t1T x, y1−x · · ·tnT x, yn−x⊂T x, x−x CC. 4.9
Letzi si, yi−x, for eachi1,2, . . . , n. By4.9, we know there existsc∈C, such that t1z1t2z2· · ·tnznc. 4.10
Asf∈C∗\ {0}, we have
fc≥0. 4.11
While by4.8, we haveft1z1t2z2· · ·tnzn<0. This is a contraction. HenceEis a KKM
map onA.
IIEy⊂Gyfor ally∈AandGis a KKM map.
By thef-pseudomonotonicity ofT, for eachy∈A, we haveEy⊂Gy. SinceEis a KKM map, so isG.
IIIy∈AGy/∅.
Now we show that for eachy∈A,Gyis closed. Let{xn}be a sequence inGysuch thatxn converges tox. By the closedness ofA, we havex ∈A. Since{xn} ⊂ Gy, then for
eachs∈Ty, we have
fs, y−xn≥0. 4.12
Asxn→x, and the continuity off, then for eachs∈Ty, we have
fs, y−x≥0. 4.13
Consequently,x∈Gy. HenceGyis closed. By the assumption, we haveGy⊂D. Then Gyis compact sinceDis compact. By stepII, we knowGis a KKM map. ByLemma 2.7,
y∈AGy/∅.
IVy∈AGy y∈AEy.
Because Ey ⊂ Gy, we have y∈AGy ⊃ y∈AEy. Now let us show that
y∈AGy⊂y∈AEy. Letx∈y∈AGy. For eachy∈A, and eachs∈Ty, we have
For anys∈T xand for each fixedy∈A, define the set-valued mappingJ:0,1→2Y by Jα Tαy 1−αx, y−x α∈0,1. 4.15
We pick a sequence{αn} ⊂0,1such thatαn→0 and setxn αny 1−αnx. SinceAis a
convex set,xn∈Afor eachn. It is clear thatxn→x. Letw s, y−x. We havew∈J0.Since
T isv-hemicontinuous,Jαis lower semicontinuous at 0. By16, there existwn ∈ Jαn
Tαny 1−αnx, y−x, such thatwn→w. Aswn ∈Jαn, there existsn∈Txnsuch that
wn sn, y−x. Bywn→w, we havesn, y−x→s, y−x. By4.14, we have
αnfsn, y−x fsn, αny 1−αnx−x≥0. 4.16
Sinceαn > 0,fsn, y−x≥ 0. Hencefs, y−x ≥ 0 sincef is continuous andwn→w. Therefore, for anys∈T xand for eachy∈A, we havefs, y−x≥0. Hencex∈y∈AEy. Thusy∈AEy y∈AGy/∅. This means that there existsx∈A, for eachs∈Tx, we have fs, y−x≥0, for ally∈A. It follows thatx∈VfA, F, thusVfA, F/∅. By the proof of
Theorem 4.1, we knowVfA, F⊂D. SinceVfA, F⊂VwA, 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, letC⊂Y be a closed convex pointed
cone withintC /∅, and letF:A×A→2Ybe a set-valued map. Assume that the following conditions are satisfied:
ifor eachy∈A,F·, yis lower semicontinuous onA;
iiFx, yisC-concave in its first variable andC-convex in its second variable;
iiiFx, x⊂C, for allx∈A;
iv{Fx, y:x, y∈A}is a bounded subset inY;
vthere exists a nonempty compact convex subsetDofA, andy ∈D, such thatFx, y∩
−intC/∅, for allx∈A\D.
ThenVwA, Fis a nonempty connected compact set.
Proof. We define the set-valued mapH:C∗\ {0} →2Dby
Hf VfA, F, f ∈C∗\ {0}. 5.1
ByTheorem 4.1, for eachf∈C∗\ {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
f∈C∗\ {0},Hfis a connected set. Letx
1, x2∈Hf, we havex1, x2∈Dand
BecauseFx, yisC-concave in its first variable, for each fixedy∈A, and for abovex1, x2 ∈
D, andt∈0,1, we havetx1 1−tx2∈DsinceDis convex, and
Ftx1 1−tx2, y⊂tFx1, y 1−tFx2, y C. 5.3
Hence for eachy ∈A,z ∈Ftx1 1−tx2, y, there existz1 ∈Fx1, y,z2 ∈ Fx2, y, and
c∈C, such thatztz1 1−tz2c. Asf ∈C∗\ {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, xn→f0, x0, where {fn} converge to f0 with respect to the norm
topology. Asfn, xn∈GraphH, we have
xn∈Hfn VfnA, F, 5.6
that is,fnFxn, y≥0, for ally∈A. Asxn→x0andDis compact, we havex0∈D. Since for
eachy∈A,F·, yis lower semicontinuous onA, for each fixedy∈A, and eachz0∈Fx0, y,
there existzn∈Fxn, y, such thatzn→z0. FromfnFxn, y≥0, we have
fnzn≥0. 5.7
By the continuity off0andzn→z0, we have
f0zn−→f0z0. 5.8
LetQ {Fx, y:x, y∈A}. By assumption,Qis a bounded set inY, then there exist some M >0, such that for eachz∈Q, we havez ≤ M. For anyε > 0, becausefn−f0→0 with
respect to norm topology, there existsn0 ∈ N, and when n ≥ n0, we havefn−f0 < ε.
Therefore, there existsn0∈N, and whenn≥n0, we have
|fnzn−f0zn||fn−f0zn| ≤ fn−f0zn ≤Mε. 5.9
Hence
lim
Consequently, by5.8,5.10, we have
lim
n→ ∞fnzn nlim→ ∞fnzn−f0zn f0zn
lim
n→ ∞fnzn−f0zn n→ ∞limf0zn
f0z0.
5.11
By5.7, we havef0z0≥0. So for anyy∈Aand 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 eachx∈A,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 xn→x0. 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
Sincexn→x0, there exist somen0, and when n≥n0, we havexn∈Ux0∩A. By5.17,
This contradicts{xn} ⊂VwA, F. Thusx0 ∈VwA, F.This means thatVwA, Fis a closed
set. SinceDis compact andVwA, F⊂D,VwA, Fis compact.
Theorem 5.2. LetAbe a nonempty closed convex subset ofX, and let C ⊂ Y be a closed convex
pointed cone withintC /∅. Assume that for eachf ∈C∗\{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, y−xis C-convex in its second variable, and the set {Fx, y : x, y ∈ A} is a bounded set inY. If there exists a nonempty compact convex subsetDofA, andy ∈D, such that
Tx, y −x∩−intC/∅, for allx∈A\D, thenVwA, Fis a nonempty connected set. Proof. We define the set-valued mapH:C∗\ {0} →2Dby
Hf VfA, F for eachf∈C∗\ {0}. 5.19
ByTheorem 4.2, for eachf∈C∗\{0}, we haveHf VfA, F/∅andVfA, F⊂D. Hence
VwA, F/∅andVwA, F⊂D. 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, y−x≥0, ∀s∈Tx}, y∈A, Gy {x∈A:fs, y−x≥0, ∀s∈Ty}, y∈A,
5.20
respectively. Now we prove that for eachf ∈C∗\ {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 eachy∈A,s∈Ty, we have
fs, y−xi≥0. 5.21
Then, for eachy∈A,s∈Ty, andt∈0,1, we havetx1 1−tx2∈DsinceDis convex and
fs, y−tx1 1−tx2≥0. 5.22
Hencetx1 1−tx2 ∈ y∈AGy y∈AEy. Thustx1 1−tx2 ∈ Hf. Consequently,
for eachf ∈C∗\ {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, xn→f0, x0, where{fn}converges tof0 with respect to the norm topology ofY∗. As
fn, xn∈GraphH, we have
xn∈Hfn VfnA, F. 5.23
Then, for eachs ∈Txn, we have that
By assumption, for each n, T : A→2LX,Y isfn-pseudomonotone, and by5.24, for each
y∈A, for the abovexn, and for eachs∈Ty, we have
fns, y−xn≥0 ∀y∈A. 5.25
Asxn→x0, we haves, y−xn→s, y−x0, andf0s, y−xn→f0s, y−x0. Asxn→x0,
and D is compact, we havex0 ∈ D. LetQ {Fx, y : x, y ∈ A}. By assumption, Qis a
bounded set inY. Then, there existsM >0, such that for eachz∈Q, we havez ≤M. For anyε > 0, becausefn−f0→0 with respect to the norm topology, there existsn0 ∈ N, and
when n ≥ n0, we havefn−f0 < ε. Therefore, there existsn0 ∈N, and when n ≥ n0, we
have
|fns, y−xn−f0s, y−xn||fn−f0s, y−xn| ≤Mε. 5.26
Hence
lim
n→ ∞fns, y−xn−f0s, y−xn 0. 5.27
Then
lim
n→ ∞fns, y−xn nlim→ ∞fns, y−xn−f0s, y−xn f0s, y−xn
f0s, y−x0.
5.28
Then, by5.25,5.28, we havef0s, y−x0≥0. Hence for eachy∈A, and for eachs∈Ty,
we havef0s, y−x0 ≥ 0. Since T isf0-pseudomonotone, for each y ∈ A, and for each
s∗ ∈Tx
0, we havef0s∗, y−x0≥ 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
subsetQ⊂X,and for each neighborhoodVof 0 inY, we have
WQ, V:{s∈LX, Y:sQ⊂V} ∈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
Proof. For any fixedx, y∈A, we need to show that the set-valued mapping
Jα: Tαy 1−αx, y−x, α∈0,1, 5.31
is lower semicontinuous at 0. For anyz ∈ J0and for any neighborhoodUzofz, there existss0∈Txsuch thatz s0, y−x,and there exists a neighborhoodV of 0 inY, such that
Uz zV. SinceTis lower semicontinuous atx, for the neighborhood
Us0:s0Wy−x, V s0{s∈LX, Y: s, y−x∈V} 5.32
ofs0, there exists a neighborhoodUxofxsuch that
Tx ∩Us0/∅ ∀x ∈Ux∩A. 5.33
For the abovey, since αy 1−αx→x, whenα→0, there exists 0 < α0 < 1, and when
0< α < α0, we haveαy 1−αx∈Ux.By5.33, we have
Tαy 1−αx∩Us0/∅ ∀0< α < α0. 5.34
Thus there existsα ∈Tαy 1−αxsuch thatsα ∈Us0, for all 0< α < α0. By5.32, we
have
sα, y−x∈Uz ∀0< α < α0, 5.35
that is
Jα∩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 C⊂ Y be a closed
convex pointed cone with intC /∅. Assume that for eachf ∈ C∗\ {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, y−xisC-convex in its second variable, and the set{Fx, y : x, y ∈ A} is a bounded set in Y. If there exists a nonempty compact convex subset D ofA, and y ∈ D, such that Tx, y −x∩−intC/∅, for allx ∈ A\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, xn→x0.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
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
s∈LX, 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
Tx∩s0W
y0−A,
ε
2
B/∅ ∀x∈Ux0∩A. 5.42
Sincexn→x0,there existsNsuch that whenn≥N, we havexn ∈Ux0.Thus by5.42we
have
Txn∩
s0W
y0−A,
ε 2 B /
∅. 5.43
Thus there exist sn ∈ Txn ∩s0 Wy0 −A,ε/2B,for all n ≥ N. We have sn −s0 ∈
Wy0−A,ε/2B, and hence
sn−s0, y0−A⊂
ε
2
B. 5.44
Therefore,
sn−s0, y0−xn∈
ε
2
Sincexn→x0 ands0 ∈ LX, Y, we haves0, x0−xn→0. Hence there existsN1 ≥ N, and
when n≥N1,we haves0, x0−xn∈ε/2B. This combining5.45implies that
sn, y0−xn−s0, y0−x0
sn, y0−xn−s0, y0−xn s0, y0−xn−s0, y0−x0
sn−s0, y0−xn s0, x0−xn∈
ε
2
B ε
2
BεB ∀n≥N1.
5.46
Hence
sn, y0−xn∈s0, y0−x0 εB ∀n≥N1. 5.47
By5.40and5.47, we have
sn, y0−xn∈ −intC ∀n≥N1. 5.48
On the other hand, since{xn} ⊂VwA, F, we have
Txn, y−xn∩−intC ∅ ∀y∈A, ∀n. 5.49
This contradicts5.48, becausesn∈Txn.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.
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