Volume 2011, Article ID 936428,15pages doi:10.1155/2011/936428
Research Article
Boundedness and Nonemptiness of Solution Sets
for Set-Valued Vector Equilibrium Problems with
an Application
Ren-You Zhong,
1Nan-Jing Huang,
1and Yeol Je Cho
2 1Department of Mathematics, Sichuan University, Chengdu, Sichuan 610064, China 2Department of Mathematics Education and the RINS, Gyeongsang National University,Chinju 660-701, Republic of Korea
Correspondence should be addressed to Yeol Je Cho,[email protected]
Received 25 October 2010; Accepted 19 January 2011
Academic Editor: K. Teo
Copyrightq2011 Ren-You Zhong 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.
This paper is devoted to the characterizations of the boundedness and nonemptiness of solution sets for set-valued vector equilibrium problems in reflexive Banach spaces, when both the mapping and the constraint set are perturbed by different parameters. By using the properties of recession cones, several equivalent characterizations are given for the set-valued vector equilibrium problems to have nonempty and bounded solution sets. As an application, the stability of solution set for the set-valued vector equilibrium problem in a reflexive Banach space is also given. The results presented in this paper generalize and extend some known results in Fan and Zhong2008, He2007, and Zhong and Huang2010.
1. Introduction
LetX andY be reflexive Banach spaces. Let K be a nonempty closed convex subset ofX. Let F : K×K → 2Y be a set-valued mapping with nonempty values. Let P be a closed
convex pointed cone inY with intP /∅. The coneP induces a partial ordering inY, which was defined by y1≤Py2 if and only ify2 − y1 ∈ P. We consider the following set-valued
vector equilibrium problem, denoted by SVEPF, K, which consists in findingx ∈K such that
It is well known that1.1is closely related to the following dual set-valued vector equilibrium problem, denoted by DSVEPF, K, which consists in findingx∈K such that
Fy, x⊂−P, ∀y∈K. 1.2
We denote the solution sets of SVEPF, Kand DSVEPF, KbySandSD, respectively.
LetZ1, d1andZ2, d2be two metric spaces. Suppose that a nonempty closed convex
setL⊂Xis perturbed by a parameteru, which varies overZ1, d1, that is,L:Z1 → 2X is a
set-valued mapping with nonempty closed convex values. Assume that a set-valued mapping
F : X ×X → 2Y is perturbed by a parameterv, which varies over Z
2, d2, that is, F :
X×X×Z2 → 2Y. We consider a parametric set-valued vector equilibrium problem, denoted
by SVEPF·,·, v, Lu, which consists in findingx∈Lusuch that
Fx, y, v∩−intP ∅, ∀y∈Lu. 1.3
Similarly, we consider the parameterized dual set-valued vector equilibrium problem, denoted by DSVEPF·,·, v, Lu, which consists in findingx∈Lusuch that
Fy, x, v⊂−P, ∀y∈Lu. 1.4
We denote the solution sets of SVEPF·,·, v, Luand DSVEPF·,·, v, LubySu, vand
SDu, v, respectively.
In 1980, Giannessi 1 extended classical variational inequalities to the case of vector-valued functions. Meanwhile, vector variational inequalities have been researched quite extensively see, e.g., 2. Inspired by the study of vector variational inequalities, more general equilibrium problems 3 have been extended to the case of vector-valued bifunctions, known as vector equilibrium problems. It is well known that the vector equilibrium problem provides a unified model of several problems, for example, vector optimization, vector variational inequality, vector complementarity problem, and vector saddle point problemsee4–9. In recent years, the vector equilibrium problem has been intensively studied by many authorssee, e.g.,1–3,10–26and the references therein.
Among many desirable properties of the solution sets for vector equilibrium problems, stability analysis of solution set is of considerable interestsee, e.g,27–33and the references therein. Assuming that the barrier cone of K has nonempty interior, McLinden 34
presented a comprehensive study of the stability of the solution set of the variational inequality, when the mapping is a maximal monotone set-valued mapping. Adly 35, Adly et al.36, and Addi et al.37discussed the stability of the solution set of a so-called semicoercive variational inequality. He 38 studied the stability of variational inequality problem with either the mapping or the constraint set perturbed in reflexive Banach spaces. Recently, Fan and Zhong39extended the corresponding results of He38to the case that the perturbation was imposed on the mapping and the constraint set simultaneously. Very recently, Zhong and Huang 40studied the stability analysis for a class of Minty mixed variational inequalities in reflexive Banach spaces, when both the mapping and the constraint set are perturbed. They got a stability result for the Minty mixed variational inequality with
Inspired and motivated by the works mentioned above, in this paper, we further study the characterizations of the boundedness and nonemptiness of solution sets for set-valued vector equilibrium problems in reflexive Banach spaces, when both the mapping and the constraint set are perturbed. We present several equivalent characterizations for the vector equilibrium problem to have nonempty and bounded solution set by using the properties of recession cones. As an application, we show the stability of the solution set for the set-valued vector equilibrium problem in a reflexive Banach space, when both the mapping and the constraint set are perturbed by different parameters. The results presented in this paper extend some corresponding results of Fan and Zhong39, He38, Zhong and Huang40
from the variational inequality to the vector equilibrium problem.
The rest of the paper is organized as follows. InSection 2, we recall some concepts in convex analysis and present some basic results. InSection 3, we present several equivalent characterizations for the set-valued vector equilibrium problems to have nonempty and bounded solution sets. In Section 4, we give an application to the stability of the solution sets for the set-valued vector equilibrium problem.
2. Preliminaries
In this section, we introduce some basic notations and preliminary results.
LetX be a reflexive Banach space andK be a nonempty closed convex subset ofX. The symbols “→” and “” are used to denote strong and weak convergence, respectively.
The barrier cone ofK, denoted by barrK, is defined by
barrK:
x∗∈X∗: sup
x∈K
x∗, x<∞
. 2.1
The recession cone ofK, denoted byK∞, is defined by
K∞:{d∈X:∃tn−→0,∃xn∈K, tnxn d}. 2.2
It is known that for any givenx0∈K,
K∞{d∈X:x0 λd∈K,∀λ >0}. 2.3
We give some basic properties of recession cones in the following result which will be used in the sequel. Let{Ki}i∈Ibe any family of nonempty sets inX. Then
i∈I
Ki
∞
⊂
i∈I
If, in addition, i∈IKi/∅and each setKiis closed and convex, then we obtain an equality in
the previous inclusion, that is,
i∈I
Ki
∞
i∈I
Ki∞. 2.5
LetΦ:K → R∪ { ∞}be a proper convex and lower semicontinuous function. The recession functionΦ∞ofΦis defined by
Φ∞x:tlim→ ∞Φx0 tx−Φx0
t , 2.6
wherex0is any point in DomΦ. Then it follows that
Φ∞x:t→lim∞Φtx
t . 2.7
The function Φ∞· turns out to be proper convex, lower semicontinuous and so weakly lower semicontinuous with the property that
Φu v≤Φu Φ∞v, ∀u∈DomΦ, v∈X. 2.8
Definition 2.1. A set-valued mappingG:K → 2Y is said to be
iupper semicontinuous atx0∈Kif, for any neighborhoodNGx0ofGx0, there
exists a neighborhoodNx0ofx0such that
Gx⊂ NGx0, ∀x∈ Nx0; 2.9
iilower semicontinuous atx0∈Kif, for anyy0∈Gx0and any neighborhoodNy0
ofy0, there exists a neighborhoodNx0ofx0such that
GxNy0
/
∅, ∀x∈ Nx0. 2.10
We sayGis continuous atx0 if it is both upper and lower semicontinuous atx0, and
we sayGis continuous onKif it is both upper and lower semicontinuous at every point of
K.
It is evident thatGis lower semicontinuous atx0 ∈Kif and only if, for any sequence {xn}withxn → x0andy0 ∈Gx0, there exists a sequence{yn}withyn ∈Gxnsuch that
yn → y0.
Definition 2.2. A set-valued mappingG:K → 2Y is said to be weakly lower semicontinuous
atx0 ∈K if, for anyy0 ∈Gx0and for any sequence{xn} ∈Kwithxn x0, there exists a
We sayGis weakly lower semicontinuous onKif it is weakly lower semicontinuous at every point ofK. ByDefinition 2.2, we know that a weakly lower semicontinuous mapping is lower semicontinuous.
Definition 2.3. A set-valued mappingG:K → 2Y is said to be
iupperP-convex onKif for anyx1andx2∈K,t∈0,1,
tGx1 1−tGx2⊂Gtx1 1−tx2 P; 2.11
iilowerP-convex onKif for anyx1andx2∈K,t∈0,1,
Gtx1 1−tx2⊂tGx1 1−tGx2−P. 2.12
We say thatGisP-convex ifGis both upperP-convex and lowerP-convex.
Definition 2.4. Let{An}be a sequence of sets inX. We define
ω-lim sup
n→ ∞
An:{x∈X:∃{nk}, xnk ∈Ank such thatxnk x}. 2.13
Lemma 2.5see36. LetKbe a nonempty closed convex subset of X withintbarrK/∅. Then there exists no sequence{xn} ⊂Ksuch thatxn → ∞andxn/xn 0.
Lemma 2.6see39. LetKbe a nonempty closed convex subset ofXwithintbarrK/∅. Then there exists no sequence{dn} ⊂K∞with eachdn1such thatdn0.
Lemma 2.7see39. LetZ, dbe a metric space andu0 ∈Zbe a given point. LetL: Z→ 2X
be a set-valued mapping with nonempty values and letLbe upper semicontinuous atu0. Then there
exists a neighborhoodUofu0such thatLu∞⊂Lu0∞for allu∈U.
Lemma 2.8see41. LetKbe a nonempty convex subset of a Hausdorfftopological vector spaceE
andG:K → 2Ebe a set-valued mapping fromKintoEsatisfying the following properties:
iGis aKKMmapping, that is, for every finite subsetAofK,coA⊂x∈AGx;
iiGxis closed inEfor everyx∈K;
iiiGx0is compact in E for somex0∈K.
Then x∈KGx/∅.
3. Boundedness and Nonemptiness of Solution Sets
LetKbe a nonempty convex and closed subset ofX. Assume thatF :K×K → 2Y is
a set-valued mapping satisfying the following conditions:
f0for eachx∈K,Fx, x 0;
f1for eachx, y∈K,Fx, y∩−intP ∅implies thatFy, x⊂−P;
f2for eachx∈K,Fx,·isP-convex onK;
f3for eachx∈K,Fx,·is weakly lower semicontinuous onK;
f4for eachx, y ∈K, the set{ξ ∈x, y :Fξ, y −intP ∅}is closed, herex, y
stands for the closed line segment joiningxandy.
Remark 3.1. If
Fx, yAx, y−x Φy−Φx, ∀x, y∈K, 3.1
where A:K → 2X∗ is a set-valued mapping, Φ : K → R{ ∞} is a proper, convex, lower semicontinuous function andP R , then condition f1 reduces to the following
Φ-pseudomonotonicity assumption which was used in40.See40, Definition 2.2iiiof
40: for allx, x∗,y, y∗in the graphA,
x∗, y−x Φy−Φx≥0⇒y∗, y−x Φy−Φx≥0. 3.2
Remark 3.2. If, for each y ∈ K, the mapping F·, y is lower semicontinuous in K, then
conditionf4is fulfilled. Indeed, for eachx, y∈Kand for any sequence{ξn} ⊂ {ξ ∈x, y:
Fξ, y −intP ∅}with ξn → ξ0, we have ξ0 ∈ x, yand Fξ0, y −intP ∅. By
the lower semicontinuity ofF·, y, for anyz ∈Fξ0, y, there existszn ∈Fξn, ysuch that
zn → z. SinceFξn, y −intP ∅, we havezn ∈ Y \−intPand soz ∈ Y \−intP
by the closedness of Y \ −intP. This implies that Fξ0, y −intP ∅ and the set {ξ∈x, y:Fξ, y −intP ∅}is closed.
The following example shows that conditionsf0–f4can be satisfied.
Example 3.3. LetXR,Y R2,P R2 andK 1,2. Let
Fx, yy−x,1,1 xy−x, ∀x, y∈K. 3.3
It is obvious that f0 holds. Since for each x, y ∈ K, Fx,· and F·, y are lower semicontinuous onK, byRemark 3.2, we known that conditionsf3andf4hold. For each
x, y∈K, ifFx, y∩−R2 ∅, then we havey−x≥0. This implies that
Fy, xx−y,1,1 yx−y⊂−R2 3.4
and sof1holds. Moreover, for eachx∈K,y1, y2 ∈Kandt1, t2 ∈0,1witht1 t21, it is
easy to verify that
Fx, t1y1 t2y2
t1F
x, y1
t2F
x, y2
which shows thatFx,·isR2-convex onKand sof2holds. Thus,Fsatisfies all conditions f0–f4.
Theorem 3.4. LetKbe a nonempty closed convex subset ofXandF:K×K → 2Y be a set-valued mapping satisfying assumptionsf0-f4. ThenSSD.
Proof. From the assumptionf1, it is easy to see thatS⊂SD. We now prove thatSD⊂S. Let
x ∈ SD. Then for ally ∈ K,Fy, x ⊂ −P. Set x
t x ty−x, wheret ∈ 0,1. Clearly,
xt∈K. From the upperP-convexity ofFx,·, we have
1−tFxt, x tF
xt, y
⊂Fxt, xt P. 3.6
SinceFxt, x⊂−P, we obtain
tFxt, y
⊂ −1−tFxt, x 0 P ⊂P P ⊂P. 3.7
This implies thatFxt, y⊂P and soFxt, y∩−intP ∅. Lettingt → 0 , by assumption
f4, we haveFx, y∩−intP ∅. Thus,x∈SandSD⊂S. This completes the proof.
Theorem 3.5. LetKbe a nonempty closed convex subset ofXandF:K×K → 2Y be a set-valued mapping satisfying assumptionsf0–f4. If the solution setSis nonempty, then
S∞SD∞R1:
y∈K
d∈K∞:Fy, y λd⊂−P, ∀λ >0. 3.8
Proof. From the proof ofTheorem 3.4, we know that
SSDx∈K:Fy, x⊂−P,∀y∈K
y∈K
x∈K:Fy, x⊂−P. 3.9
LetSy {x ∈X :Fy, x⊂ −P}. ThenS SD y∈KK∩Sy. By the assumptionsf2
andf3, we know that the setSy is nonempty closed and convex. It follows from2.5and
Theorem 3.4that
S∞SD∞
⎛
⎝
y∈K
K∩Sy
⎞ ⎠
∞
y∈K
K∩Sy∞
y∈K
K∞∩Sy∞
y∈K
d∈K∞:d∈Sy∞
y∈K
d∈K∞:y λd∈Sy, ∀λ >0
y∈K
d∈K∞:Fy, y λd⊂ −P, ∀λ >0.
3.10
Remark 3.6. If
Fy, xAy, x−y Φx−Φy, ∀x, y∈K, 3.11
whereA:K → 2X∗ is a set-valued mapping,Φ:K → R{ ∞}is a proper, convex, lower
semicontinuous function andPR , then it follows from3.8and2.8that
SD∞
y∈K
d∈K∞:Fy, y λd⊂−P, ∀λ >0
K∞∩d∈X:y∗, y λd−y Φy λd−Φy≤0,∀y∈K, y∗∈Ay, ∀λ >0
K∞∩d∈X:y∗, d Φ∞d≤0,∀y∗∈AK.
3.12
Thus, we know that Theorem 3.5 is a generalization of 40, Theorem 3.1. Moreover, by
40, Remark 3.1,Theorem 3.5is also a generalization of38, Lemma 3.1.
Theorem 3.7. Let K be a nonempty closed convex subset of X and F : K ×K → 2Y be a set-valued mapping satisfying assumptionsf0–f4. Suppose thatintbarrK/∅. Then the following statements are equivalent:
ithe solution set ofSVEPF, Kis nonempty and bounded;
iithe solution set ofDSVEPF, Kis nonempty and bounded;
iiiR1 y∈K{d∈K∞:Fy, y λd⊂−P, ∀λ >0}{0};
ivthere exists a bounded setC⊂Ksuch that for everyx∈K\C, there exists somey ∈C
such thatFy, x/⊂−P.
Proof. The implicationsi⇔iiandii⇒iiifollow immediately from Theorems3.4and3.5
and the definition of recession cone.
Now we prove thatiiiimpliesiv. Ifivdoes not hold, then there exists a sequence
{xn} ⊂Ksuch that for eachn,xn ≥nandFy, xn⊂−Pfor everyy ∈Kwithy ≤n.
Without loss of generality, we may assume thatdn xn/xnweakly converges tod. Then
d∈K∞by the definition of the recession cone. Since intbarrK/∅, byLemma 2.5, we know thatd /0. Lety ∈ K and λ > 0 be any fixed points. For nsufficiently large, by the lower
P-convexity ofFy,·,
F
y,
1− λ
xn
y λ xnxn
⊂
1− λ
xn
Fy, y λ xnF
y, xn
−P⊂0−P−P⊂ −P.
3.13
Since
1− λ
xn
y λ
xnxn y λd
3.14
andFy,·is weakly lower semicontinuous, we know thatFy, y λd⊂ −Pand sod∈R1.
Sinceiandiiare equivalent, it remains to prove thativimpliesii. LetG:K →
2Kbe a set-valued mapping defined by
Gy:x∈K:Fy, x⊂−P, ∀y∈K. 3.15
We first prove thatGyis a closed subset ofK. Indeed, for anyxn ∈ Gywith xn → x0,
we have Fy, xn ⊂ −P. It follows from the weakly lower semicontinuity of Fy,· that
Fy, x0⊂−P. This shows thatx0∈Gyand soGyis closed.
We next prove thatGis a KKM mapping fromK toK. Suppose to the contrary that there existt1, t2, . . . , tn∈0,1witht1 t2 · · · tn 1,y1, y2, . . . , yn∈Kandyt1y1 t2y2 · · · tnyn∈co{y1, y2, . . . , yn}such thaty /∈ ∪i∈{1,2,...,n}Gyi. Then
Fyi, y
/
⊂−P, i1,2, . . . , n. 3.16
By assumptionf1, we have
Fy, yi
∩−intP/∅, i1,2, . . . , n. 3.17
It follows from the upperP-convexity ofFy,·that
t1F
y, y1
t2F
y, y2
· · · tnF
y, yn
⊂Fy, y P ⊂P, 3.18
which is a contradiction with3.17. Thus we know thatGis a KKM mapping.
We may assume thatCis a bounded closed convex setotherwise, consider the closed convex hull ofCinstead ofC. Let{y1, . . . , ym}be finite number of points inKand letM:
coC∪ {y1, . . . , ym}. Then the reflexivity of the spaceX yields thatMis weakly compact
convex. Consider the set-valued mappingGdefined byGy : Gy∩Mfor ally ∈ M. Then eachGy is a weakly compact convex subset ofMand G is a KKM mapping. We claim that
∅/
y∈M
Gy⊂C. 3.19
Indeed, byLemma 2.8, intersection in3.19is nonempty. Moreover, if there exists somex0∈
y∈MGybutx0∈/C, then byiv, we haveFy, x0/⊂−Pfor somey∈C. Thus,x0∈/Gy
and sox0∈/Gy, which is a contradiction to the choice ofx0.
Letz∈ y∈MGy. Thenz∈Cby3.19and soz∈ mi1Gyi∩C. This shows that
the collection{Gy∩C:y∈K}has finite intersection property. For eachy ∈K, it follows from the weak compactness ofGy∩Cthat y∈KGy∩Cis nonempty, which coincides with the solution set of DSVEPF, K.
Remark 3.8. Theorem 3.7establishes the necessary and sufficient conditions for the vector
equilibrium problem to have nonempty and bounded solution sets. If
whereA:K → 2X∗ is a set-valued mapping,Φ:K → R{ ∞}is a proper, convex, lower
semicontinuous function and P R , then problem 1.2 reduces to the following Minty mixed variational inequality: findingx∈Ksuch that
y∗, y−x Φy−Φx≥0, ∀y∈K, y∗∈Ay, 3.21
which was considered by Zhong and Huang40. Therefore,Theorem 3.7is a generalization of40, Theorem 3.2. Moreover, by40, Remark 3.2,Theorem 3.7is also a generalization of
Theorem 3.4due to He38.
Remark 3.9. By using a asymptotic analysis methods, many authors studied the necessary
and sufficient conditions for the nonemptiness and boundedness of the solution sets to variational inequalities, optimization problems, and equilibrium problems, we refer the reader to references42–49for more details.
4. An Application
As an application, in this section, we will establish the stability of solution set for the set-valued vector equilibrium problem when the mapping and the constraint set are perturbed by different parameters.
LetZ1, d1and Z2, d2be two metric spaces.F : X ×X×Z2 → 2Y is a set-valued
mapping satisfying the following assumptions:
f0for eachu∈Z1,v∈Z2,x∈Lu,Fx, x, v 0;
f1for eachu∈Z1,v∈Z2,x, y∈Lu,Fx, y, v∩−intP ∅implies thatFy, x, v⊂ −P;
f2for eachu∈Z1,v∈Z2,x∈Lu,Fx,·, visP-convex onLu;
f3for eachu∈Z1, v∈Z2,x, y∈Luandz∈Fx, y, v, for any sequences{xn},{yn}
and {vn} withxn → x,yn yand vn → v, there exists a sequence{zn}with
zn∈Fxn, yn, vnsuch thatzn → z.
The followingTheorem 4.1plays an important role in proving our results.
Theorem 4.1. LetZ1, d1andZ2, d2be two metric spaces,u0∈Z1andv0 ∈Z2be given points.
Let L : Z1 → 2X be a continuous set-valued mapping with nonempty closed convex values and
intbarrLu0/∅. Suppose thatF : X×X ×Z2 → 2Y is a set-valued mapping satisfying the
assumptionsf0–f3. If
R1u0, v0
y∈Lu0
d∈Lu0∞:Fy, y λd, v0
⊂−P,∀λ >0{0}, 4.1
then there exists a neighborhoodU×Vofu0, v0such that
R1u, v
y∈Lu
Proof. Assume that the conclusion does not hold, then there exist a sequence{un, vn} in
Z1×Z2withun, vn → u0, v0such thatR1un, vn/{0}.
SinceR1un, vnis cone, we can select a sequence{dn}withdn ∈R1un, vnsuch that
dn 1 for everyn 1,2, . . .. AsX is reflexive, without loss of generality, we can assume
thatdn d0, asn → ∞. SinceL is a continuous set-valued mapping, hence,Lis upper
semicontinuous and lower semicontinuous at u0. From the upper semicontinuity ofL, by
Lemma 2.7, we haveLun∞ ⊂ Lu0∞asnlarge enough and hencedn ∈ Lu0∞asn
large enough. SinceLu0∞is a closed convex cone and hence weakly closed. This implies thatd0 ∈Lu0∞. Moreover, it follows fromLemma 2.6thatd0/0.
For anyλ > 0,y∈Lu0andy∗ ∈Fy, y λd0, v0, from the lower semicontinuity of
L, there existsyn∈Lunsuch thatyn → y. Sincedn d0, it follows thatyn λdn y d0.
Together with vn → v0, from assumptionf3, there exists y∗n ∈ Fyn, yn λdn, vn such
thaty∗n → y∗. Sincedn∈R1un, vn, we haveFyn, yn λdn, vn⊂−Pandyn∗∈ −P. Letting
n → ∞, we obtain thaty∗∈−P. Sincey∈Lu0andy∗∈Fy, y λd0, v0are arbitrary, from
the above discussion, we obtaind0 ∈R1u0, v0withd0/0. This contradicts our assumption
thatR1u0, v0 {0}. This completes the proof.
Remark 4.2. If
Fy, x, vAy, v, x−y Φx−Φy, ∀x, y∈Lu, 4.3
whereA: X×Z2 → 2X
∗
is a set-valued mapping,Φ :X → R{ ∞}is a proper, convex, lower semicontinuous function and P R , from Remark 3.6, we know that 4.1 and
4.2inTheorem 4.1reduce to4.1and4.2in40, Theorem 4.1, respectively. Therefore,
Theorem 4.1 is a generalization of 40, Theorem 4.1. Moreover, by 40, Remark 4.1,
Theorem 4.1is also a generalization of39, Theorem 3.1.
FromTheorem 4.1, we derive the following stability result of the solution set for the vector equilibrium problem.
Theorem 4.3. LetZ1, d1andZ2, d2be two metric spaces,u0∈Z1andv0 ∈Z2be given points.
Let L : Z1 → 2X be a continuous set-valued mapping with nonempty closed convex values and
intbarrLu0/∅. Suppose thatF : X×X ×Z2 → 2Y is a set-valued mapping satisfying the
assumptionsf0-f3. IfSu0, v0is nonempty and bounded, then
ithere exists a neighborhoodU×V ofu0, v0such that for everyu, v∈U×V,Su, vis
nonempty and bounded;
iiω-lim supu,v→u
0,v0Su, v⊂Su0, v0.
Proof. IfSu0, v0is nonempty and bounded, then byTheorem 3.7we haveR1u0, v0 {0}.
It follows from Theorem 4.1 that there exists a neighborhood U×V of u0, v0, such that
R1u, v {0} for every u, v ∈ U×V. By using Theorem 3.7again, we have Su, v is nonempty and bounded for everyu, v∈U×V. This verifies the first assertion.
Next, we prove the second assertion ω-lim supu,v→u
0,v0Su, v ⊂ Su0, v0. For
any given sequence {un, vn} ∈ U×V with un, vn → u0, v0, we need to prove that
ω-lim supn→ ∞Sun, vn ⊂ Su0, v0. Let x ∈ ω-lim supn→ ∞Sun, vn. Then there exists a
sequence{xnj}with eachxnj ∈Sunj, vnjsuch thatxnjweakly converges tox. We claim that
there exist that a subsequence{xnjk}of{xnj}and someε0 > 0, such thatdxnjk, Lu0 ≥ε0,
for allk1,2, . . .. This implies thatxnjk ∈/Lu0 ε0B0,1and soLunjk/⊂Lu0 ε0B0,1,
which contradicts with the upper semicontinuity ofL·. Thus, we have the claim. Moreover, we obtainx∈Lu0asLu0is a closed convex subset ofXand hence weakly closed.
Now we proveFy, x, v0⊂−Pfor ally∈Lu0and hencex∈SDu0, v0 Su0, v0.
For any y ∈ Lu0 and y∗ ∈ Fy, x, v0, from the lower semicontinuity of L, there exist
ynj ∈ Lunj such that limj→ ∞ynj y. Moreover, from assumption f3, there exists a
sequence of elementsynj∗ ∈ Fynj, xnj, vnjsuch thaty∗nj → y∗. Sincexnj ∈ Sunj, vnj, we
haveFynj, xnj, vnj⊂−Pand soynj∗ ∈ −P. Lettingj → ∞, we obtain thaty∗∈−P. Since
y∗ ∈ Fy, x, v0is arbitrary, we haveFy, x, v0 ⊂ −P. This yields thatx ∈ SDu
0, v0
Su0, v0. Thus, have the second assertion. This completes the proof.
Remark 4.4. If
Fy, x, vAy, v, x−y Φx−Φy, ∀x, y∈Lu, 4.4
whereA: X×Z2 → 2X
∗
is a set-valued mapping,Φ :X → R{ ∞}is a proper, convex, lower semicontinuous function and P R , then problem1.4 reduces to the following parametric Minty mixed variational inequality: findingx∈Lusuch that
y∗, y−x Φy−Φx≥0, ∀y∈Lu, y∗∈Ay, v, 4.5
which was considered by Zhong and Huang40. Therefore,Theorem 4.3is a generalization of40, Theorem 4.2. Moreover, by40, Remark 4.2,Theorem 4.3ia also a generalizationof Theorems 4.1 and 4.4 due to He38andTheorem 3.5due to Fan and Zhong39.
The following examples show the necessity of the conditions ofTheorem 4.3.
Example 4.5. LetXY R,P R ,Z1Z2 −1,1andu0 v0 0,
Lu≡0,1, Fx, y, v
⎧ ⎨ ⎩
{0}, v /0,
y2−x2, v0. 4.6
Note thatL·is continuous onZ1. However,F·,·,·is not lower semicontinuous at 1/2,
1/4,0∈X×X×Z2. Clearly, we haveS0,0 {0}andS0, v 0,1for anyv /0. Thus,
lim sup
v→0
S0, v 0,1⊂/S0,0. 4.7
Example 4.6. LetXY R,P R ,Z1Z2 −1,1andu0 v0 0,
Lu
⎧ ⎨ ⎩
2,3, u0,
1,3, u /0, F
Note thatFsatisfies the assumptionsf0–f3, andLuis upper semicontinuous. However,
Luis not lower semicontinuous atu 0. Clearly, we haveS0,0 {1}andSu,0 {2} for anyu /0. Thus,
lim sup
u→0
Su,0 {2}/⊂S0,0. 4.9
Example 4.7. LetXY R,P R ,Z1Z2 −1,1,u0v00,
Lu
⎧ ⎨ ⎩
2,3, u0,
1,3, u /0, F
x, y, vy2−x2, for any x, y∈Lu, v∈Z2. 4.10
Note thatFsatisfies the assumptionsf0–f3andLuis lower semicontinuous. However,
Luis not upper semicontinuous atu 0. Clearly, we haveS0,0 {2}andSu,0 {1} for anyu /0. Thus,
lim sup
u→0
Su,0 {1}/⊂S0,0. 4.11
Acknowledgments
The authors are grateful to the editor and reviewers for their valuable comments and suggestions. This work was supported by the Key Program of NSFCGrant no. 70831005, the National Natural Science Foundation of China 10671135 and the Korea Research Foundation Grant funded by the Korean GovernmentKRF-2008-313-C00050.
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