Hindawi Publishing Corporation
Journal of Inequalities and Applications
Volume 2011, Article ID 936428, 15 pages
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,
1
Nan-Jing Huang,
1
andYeolJeCho
2
1
Department of Mathematics, Sichuan University, Chengdu, Sichuan 610064, China
2
Department 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
Copyright q 2011 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
∅, ∀y ∈ K. 1.1
2 Journal of Inequalities and Applications
It is well known that 1.1 is closely related to the following dual set-valued vector
equilibrium problem, denoted by DSVEPF, K, which consists in finding x ∈ K such that
F
y, x
⊂
−P
, ∀y ∈ K. 1.2
We denote the solution sets of SVEPF, K and DSVEPF, K by S and S
D
, respectively.
Let Z
1
,d
1
and Z
2
,d
2
be two metric spaces. Suppose that a nonempty closed convex
set L ⊂ X is perturbed by a parameter u, which varies over Z
1
,d
1
. 1.3
Similarly, we consider the parameterized dual set-valued vector equilibrium problem,
denoted by DSVEPF·, ·,v,Lu, which consists in finding x ∈ Lu such that
F
y, x, v
⊂
−P
, ∀y ∈ L
u
. 1.4
We denote the solution sets of SVEPF·, ·,v,Lu
and DSVEPF·, ·,v,Lu by Su, v and
S
D
u, 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 problem see 4–9. In recent years, the vector equilibrium problem has been
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.
Let X be a reflexive Banach space and K be a nonempty closed convex subset of X.
The symbols “ → ”and“” are used to denote strong and weak convergence, respectively.
The barrier cone of K, denoted by barrK, is defined by
barr
K
:
x
∗
∈ X
∗
:sup
x∈K
x
∗
,x
< ∞
. 2.1
The recession cone of K, denoted by K
∞
i
}
i∈I
be any family of nonempty sets in X. Then
i∈I
K
i
∞
⊂
i∈I
K
i
∞
.
2.4
4 Journal of Inequalities and Applications
If, in addition,
i∈I
K
i
/
∅ and each set K
i
Φ
x
0
tx
− Φ
x
0
t
,
2.6
where x
0
is any point in Dom Φ. Then it follows that
Φ
∞
x
: lim
t → ∞
Φ
tx
t
.
0
of x
0
such that
G
x
⊂N
G
x
0
, ∀x ∈N
x
0
; 2.9
ii lower semicontinuous at x
0
∈ K if, for any y
0
∈ Gx
0
and any neighborhood Ny
0
K.
It is evident that G is lower semicontinuous at x
0
∈ K if and only if, for any sequence
{x
n
} with x
n
→ x
0
and y
0
∈ Gx
0
, there exists a sequence {y
n
} with y
n
∈ Gx
n
such that
y
n
→ y
0
.
Definition 2.2. A set-valued mapping G : K → 2
Y
is said to be weakly lower semicontinuous
at x
and x
2
∈ K, t ∈ 0, 1,
tG
x
1
1 − t
G
x
2
⊂ G
tx
1
1 − t
x
2
P; 2.11
ii lower P-convex on K if for any x
We say that G is P-convex if G is both upper P-convex and lower P-convex.
Definition 2.4. Let {A
n
} be a sequence of sets in X. We define
ω-lim sup
n →∞
A
n
:
{
x ∈ X : ∃
{
n
k
}
,x
n
k
∈ A
n
k
such that x
n
k
x
}
.
2.13
Lemma 2.5 see 36. Let K be a nonempty closed convex subset of X with intbarrK
/
0
such that Lu
∞
⊂ Lu
0
∞
for all u ∈ U.
Lemma 2.8 see 41. Let K be a nonempty convex subset of a Hausdorff topological vector space E
and G : K → 2
E
be a set-valued mapping from K into E satisfying the following properties:
i G is a KKM mapping, that is, for every finite subset A of K, coA ⊂
x∈A
Gx;
ii Gx is closed in E for every x ∈ K;
iii Gx
0
is compact in E for some x
0
∈ K.
Then
x∈K
Gx
/
∅.
3. Boundedness and Nonemptiness of Solution Sets
In this section, we present several equivalent characterizations for the set-valued vector
Ax, y − x
Φ
y
− Φ
x
, ∀x, y ∈ K, 3.1
where A:K → 2
X
∗
is a set-valued mapping, Φ : K → R
{∞} is a proper, convex,
lower semicontinuous function and P R
, then condition f
1
reduces to the following
Φ-pseudomonotonicity assumption which was used in 40. See 40 , Definition 2.2iii of
40: for all x, x
∗
, y,y
∗
in the graphA,
4
is fulfilled. Indeed, for each x, y ∈ K and for any sequence {ξ
n
}⊂{ξ ∈ x, y :
Fξ, y
− int P∅} with ξ
n
→ ξ
0
, we have ξ
0
∈ x, y and Fξ
0
,y
− int P∅.By
the lower semicontinuity of F·,y, for any z ∈ Fξ
0
,y, there exists z
n
∈ Fξ
n
,y such that
z
n
→ z. Since Fξ
n
,y
1, 1 x
y − x
, ∀x, y ∈ K.
3.3
It is obvious that f
0
holds. Since for each x, y ∈ K, Fx, · and F·,y are lower
semicontinuous on K,byRemark 3.2, we known that conditions f
3
and f
4
hold. For each
x, y ∈ K,ifFx, y ∩ −R
2
∅, then we have y − x ≥ 0. This implies that
F
y, x
x − y,
1, 1 y
y
1
t
2
y
2
t
1
F
x, y
1
t
2
F
x, y
2
3.5
Journal of Inequalities and Applications 7
which shows that Fx, · is R
2
-convex on K and so f
2
holds. Thus, F satisfies all conditions
f
1 − t
F
x
t
,x
tF
x
t
,y
⊂ F
x
t
,x
t
P. 3.6
Since Fx
t
,x ⊂ −P,weobtain
tF
x
t
,y
4
. If the solution set S is nonempty, then
S
∞
S
D
∞
R
1
:
y∈K
d ∈ K
∞
: F
y, y λd
⊂
−P
, ∀λ>0
.
3.8
Proof. From the proof of Theorem 3.4, we know that
S S
D
D
y∈K
K ∩ S
y
. By the assumptions f
2
and f
3
, we know that the set S
y
is nonempty closed and convex. It follows from 2.5 and
Theorem 3.4 that
S
∞
S
D
∞
⎛
⎝
y∈K
K ∩ S
y
⎞
⎠
∞
y
∞
y∈K
d ∈ K
∞
: y λd ∈ S
y
, ∀λ>0
y∈K
d ∈ K
∞
: F
y, y λd
⊂−P, ∀λ>0
.
y∈K
d ∈ K
∞
: F
y, y λd
⊂
−P
, ∀λ>0
K
∞
∩
d ∈ X : y
∗
,y λd − y Φ
y λd
− Φ
y
≤ 0, ∀y ∈ K, y
Y
be a set-
valued mapping satisfying assumptions f
0
–f
4
. Suppose that intbarrK
/
∅. Then the following
statements are equivalent:
i the solution set of SVEPF, K is nonempty and bounded;
ii the solution set of DSVEPF, K is nonempty and bounded;
iii R
1
y∈K
{d ∈ K
∞
: Fy, y λd ⊂ −P, ∀λ>0} {0};
iv there exists a bounded set C ⊂ K such that for every x ∈ K \ C, there exists some y ∈ C
such that Fy, x
/
⊂−P.
Proof. The implications i⇔ii and ii⇒iii follow immediately from Theorems 3.4 and 3.5
and the definition of recession cone.
Now we prove that iii implies iv.Ifiv does not hold, then there exists a sequence
{x
n
}⊂K such that for each n, x
y
λ
x
n
x
n
⊂
1 −
λ
x
n
F
y, y
λ
x
n
F
{0}.Thusiv holds.
Journal of Inequalities and Applications 9
Since i and ii are equivalent, it remains to prove that iv implies ii.LetG : K →
2
K
be a set-valued mapping defined by
G
y
:
x ∈ K : F
y, x
⊂
−P
, ∀y ∈ K. 3.15
We first prove that Gy is a closed subset of K. Indeed, for any x
n
∈ Gy with x
n
→ x
0
,
we have Fy, x
t
2
y
2
··· t
n
y
n
∈ co{y
1
,y
2
, ,y
n
} such that y/∈∪
i∈{1,2, ,n}
Gy
i
. Then
F
y
i
, y
/
⊂
−P
2
··· t
n
F
y, y
n
⊂ F
y, y
P ⊂ P, 3.18
which is a contradiction with 3.17. Thus we know that G is a KKM mapping.
We may assume that C is a bounded closed convex set otherwise, consider the closed
convex hull of C instead of C.Let{y
1
, ,y
m
} be finite number of points in K and let M :
coC ∪{y
1
, ,y
m
}. Then the reflexivity of the space X yields that M is weakly compact
convex. Consider the set-valued mapping G
defined by G
/
⊂−P for some y ∈ C.Thus,x
0
/∈ Gy
and so x
0
/∈ G
y, which is a contradiction to the choice of x
0
.
Let z ∈
y∈M
G
y. Then z ∈ C by 3.19 and so z ∈
m
i1
Gy
i
∩ C. This shows that
the collection {Gy ∩ C : y ∈ K} has finite intersection property. For each y ∈ K, it follows
from the weak compactness of Gy ∩ C that
y∈K
Gy ∩ C is nonempty, which coincides
with the solution set of DSVEPF, K.
Φ
y
− Φ
x
≥ 0, ∀y ∈ K, y
∗
∈ A
y
, 3.21
which was considered by Zhong and Huang 40. Therefore, Theorem 3.7 is a generalization
of 40, Theorem 3.2. Moreover, by 40, Remark 3.2, Theorem 3.7 is also a generalization of
Theorem 3.4 due to He 38.
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 references 42–49 for 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.
Let Z
1
,d
2
for each u ∈ Z
1
, v ∈ Z
2
, x ∈ Lu, Fx, ·,v is P-convex on Lu;
f
3
for each u ∈ Z
1
,v ∈ Z
2
, x, y ∈ Lu and z ∈ Fx, y, v, for any sequences {x
n
}, {y
n
}
and {v
n
} with x
n
→ x, y
n
yand v
n
→ v, there exists a sequence {z
n
} with
z
→ 2
X
be a continuous set-valued mapping with nonempty closed convex values and
intbarrLu
0
/
∅. Suppose that F : X × X × Z
2
→ 2
Y
is a set-valued mapping satisfying the
assumptions f
0
–f
3
.If
R
1
u
0
,v
0
y∈L
,v
0
such that
R
1
u, v
y∈L
u
d ∈ L
u
∞
: F
y, y λd, v
⊂
−P
, ∀λ>0
u
n
,v
n
/
{0}.
Since R
1
u
n
,v
n
is cone, we can select a sequence {d
n
} with d
n
∈ R
1
u
n
,v
n
such that
d
n
1 for every n 1, 2, AsX is reflexive, without loss of generality, we can assume
that d
n
d
∞
. Moreover, it follows from Lemma 2.6 that d
0
/
0.
For any λ>0, y ∈ Lu
0
and y
∗
∈ Fy, y λd
0
,v
0
, from the lower semicontinuity of
L, there exists y
n
∈ Lu
n
such that y
n
→ y. Since d
n
d
0
, it follows that y
n
λd
n
y d
0
n
,v
n
, we have Fy
n
,y
n
λd
n
,v
n
⊂ −P and y
∗
n
∈−P. Letting
n →∞,weobtainthaty
∗
∈ −P. Since y ∈ Lu
0
and y
∗
∈ Fy, yλd
0
,v
0
are arbitrary, from
the above discussion, we obtain d
0
∈ R
1
x
− Φ
y
, ∀x, y ∈ L
u
, 4.3
where A : X × Z
2
→ 2
X
∗
is a set-valued mapping, Φ : X → R
{∞} is a proper, convex,
lower semicontinuous function and P R
,fromRemark 3.6, we know that 4.1 and
4.2 in Theorem 4.1 reduce to 4.1 and 4.2 in 40, Theorem 4.1, respectively. Therefore,
Theorem 4.1 is a generalization of 40, Theorem 4.1. Moreover, by 40, Remark 4.1,
Theorem 4.1 is also a generalization of 39, Theorem 3.1.
From Theorem 4.1, we derive the following stability result of the solution set for the
vector equilibrium problem.
Theorem 4.3. Let Z
1
,d
0
-f
3
.IfSu
0
,v
0
is nonempty and bounded, then
i there exists a neighborhood U × V of u
0
,v
0
such that for every u, v ∈ U × V , Su, v is
nonempty and bounded;
ii ω-lim sup
u,v → u
0
,v
0
Su, v ⊂ Su
0
,v
0
.
Proof. If Su
0
,v
0
n
}∈U × V with u
n
,v
n
→ u
0
,v
0
, we need to prove that
ω-lim sup
n →∞
Su
n
,v
n
⊂ Su
0
,v
0
.Letx ∈ ω-lim sup
n →∞
Su
n
,v
n
. Then there exists a
sequence {x
n
j
j
k
} of {x
n
j
} and some ε
0
> 0, such that dx
n
j
k
,Lu
0
≥ ε
0
,
for all k 1, 2, This implies that x
n
j
k
/∈ Lu
0
ε
0
B0, 1 and so Lu
n
j
k
/
∈ Fy, x, v
0
, from the lower semicontinuity of L, there exist
y
n
j
∈ Lu
n
j
such that lim
j →∞
y
n
j
y. Moreover, from assumption f
3
, there exists a
sequence of elements y
∗
n
j
∈ Fy
n
j
,x
n
j
,v
n
∈−P. Letting j →∞,weobtainthaty
∗
∈ −P. Since
y
∗
∈ Fy, x, v
0
is arbitrary, we have Fy, x, v
0
⊂ −P. This yields that x ∈ S
D
u
0
,v
0
Su
0
,v
0
. Thus, have the second assertion. This completes the proof.
Remark 4.4. If
F
y, x, v
A
y
∗
,y− x
Φ
y
− Φ
x
≥ 0, ∀y ∈ L
u
,y
∗
∈ A
y, v
, 4.5
which was considered by Zhong and Huang 40. Therefore, Theorem 4.3 is a generalization
of 40, Theorem 4.2. Moreover, by 40, Remark 4.2, Theorem 4.3 ia also a generalizationof
Theorems 4.1 and 4.4 due to He 38 and Theorem 3.5 due to Fan and Zhong 39.
The following examples show the necessity of the conditions of Theorem 4.3.
Example 4.5. Let X Y R, P R
, Z
y
2
− x
2
,v 0.
4.6
Note that L· is continuous on Z
1
. However, F·, ·, · is not lower semicontinuous at 1/2,
1/4, 0 ∈ X × X × Z
2
. Clearly, we have S0, 0{0} and S0,v0, 1 for any v
/
0. Thus,
lim sup
v → 0
S
0,v
0, 1
/
⊂S
0, 0
.
F
x, y, v
y
2
− x
2
, for any x, y ∈ L
u
,v ∈ Z
2
. 4.8
Journal of Inequalities and Applications 13
Note that F satisfies the assumptions f
0
–f
3
,andLu is upper semicontinuous. However,
Lu is not lower semicontinuous at u 0. Clearly, we have S0, 0{1} and Su, 0{2}
for any u
/
0. Thus,
lim sup
u → 0
S
⎧
⎨
⎩
2, 3
,u 0,
1, 3
,u
/
0,
F
x, y, v
y
2
− x
2
, for any x, y ∈ L
u
,v ∈ Z
2
. 4.10
Note that F satisfies the assumptions f
Foundation Grant funded by the Korean Government KRF-2008-313-C00050.
References
1 F. Giannessi, “Theorems of alternative, quadratic programs and complementarity problems,” in
Variational Inequalities and Complementarity Problems, R. W. Cottle, F. Giannessi, and J. L. Lions, Eds.,
pp. 151–186, John Wiley & Sons, Chichester, UK, 1980.
2 F. Giannessi, Ed., Vector Variational Inequalities and Vector Equilibrium, vol. 38, Kluwer Academic
Publishers, Dordrecht, The Netherlands, 2000.
3 E. Blum and W. Oettli, “From optimization and variational inequalities to equilibrium problems,” The
Mathematics Student, vol. 63, no. 1–4, pp. 123–145, 1994.
4 G. Chen, X. Huang, and X. Yang, Vector Optimization: Set-Valued and Variational Analysis, vol. 541 of
Lecture Notes in Economics and Mathematical Systems, Springer, Berlin, Germany, 2005.
5 G. Y. Chen, X. Q. Yang, and H. Yu, “A nonlinear scalarization function and generalized quasi-vector
equilibrium problems,” Journal of Global Optimization, vol. 32, no. 4, pp. 451–466, 2005.
6 N J. Huang and Y P. Fang, “Strong vector F-complementary problem and least element problem of
feasible set,” Nonlinear Analysis. Theory, Methods & Applications, vol. 61, no. 6, pp. 901–918, 2005.
7 N. J. Huang and Y. P. Fang, “On vector variational inequalities in reflexive Banach spaces,” Journal of
Global Optimization, vol. 32, no. 4, pp. 495–505, 2005.
8 K. K. Tan, J. Yu, and X. Z. Yuan, “Existence theorems for saddle points of vector-valued maps,” Journal
of Optimization Theory and Applications, vol. 89, no. 3, pp. 731–747, 1996.
9 X. Q. Yang, “Vector complementarity and minimal element problems,” Journal of Optimization Theory
and Applications, vol. 77, no. 3, pp. 483–495, 1993.
10 Q. H. Ansari, W. K. Chan, and X. Q. Yang, “The system of vector quasi-equilibrium problems with
applications,” Journal of Global Optimization, vol. 29, no. 1, pp. 45–57, 2004.
14 Journal of Inequalities and Applications
11 Q. H. Ansari, W. Oettli, and D. Schl
¨
ager, “A generalization of vectorial equilibria,” Mathematical
Methods of Operations Research, vol. 46, no. 2, pp. 147–152, 1997.
12 Q. H. Ansari, S. Schaible, and J. C. Yao, “System of vector equilibrium problems and its applications,”
Journal of Optimization Theory and Applications, vol. 107, no. 3, pp. 547–557, 2000.
26 S. J. Li, K. L. Teo, and X. Q. Yang, “Generalized vector quasi-equilibrium problems,” Mathematical
Methods of Operations Research, vol. 61, no. 3, pp. 385–397, 2005.
27 L. Q. Anh and P. Q. Khanh, “On the stability of the solution sets of general multivalued vector
quasiequilibrium problems,” Journal of Optimization Theory and Applications, vol. 135, no. 2, pp. 271–
284, 2007.
28 L. Q. Anh and P. Q. Khanh, “Semicontinuity of solution sets to parametric quasivariational inclusions
with applications to traffic networks—II. Lower semicontinuities applications,” Set-Valued Analysis,
vol. 16, no. 7-8, pp. 943–960, 2008.
29 J. C. Chen and X. H. Gong, “The stability of set of solutions for symmetric vector quasi-equilibrium
problems,” Journal of Optimization Theory and Applications, vol. 136, no. 3, pp. 359–374, 2008.
30 X. H. Gong, “Continuity of the solution set to parametric weak vector equilibrium problems,” Journal
of Optimization Theory and Applications, vol. 139, no. 1, pp. 35–46, 2008.
31 X. H. Gong and J. C. Yao, “Lower semicontinuity of the set of efficient solutions for generalized
systems,” Journal of Optimization Theory and Applications, vol. 138, no. 2, pp. 197–205, 2008.
32 N. J. Huang, J. Li, and H. B. Thompson, “Stability for parametric implicit vector equilibrium
problems,” Mathematical and Computer Modelling, vol. 43, no. 11-12, pp. 1267–1274, 2006.
33 X J. Long, N. J. Huang, and K l. Teo, “Existence and stability of solutions for generalized strong
vector quasi-equilibrium problem,” Mathematical and Computer Modelling, vol. 47, no. 3-4, pp. 445–
451, 2008.
34 L. McLinden, “Stable monotone variational inequalities,” Mathematical Programming,vol.48,no.2,
pp. 303–338, 1990.
35 S. Adly, “Stability of linear semi-coercive variational inequalities in Hilbert spaces: application to the
Signorini-Fichera problem,” Journal of Nonlinear and Convex Analysis, vol. 7, no. 3, pp. 325–334, 2006.
36 S. Adly, M. Th
´
era, and E. Ernst, “Stability of the solution set of non-coercive variational inequalities,”
Communications in Contemporary Mathematics, vol. 4, no. 1, pp. 145–160, 2002.
Journal of Inequalities and Applications 15
37 K. Addi, S. Adly, D. Goeleven, and H. Saoud, “A sensitivity analysis of a class of semi-coercive
variational inequalities using recession tools,” Journal of Global Optimization, vol. 40, no. 1–3, pp. 7–
an and C. Vera, “Characterization of the nonemptiness and compactness of solution sets
in convex and nonconvex vector optimization,” Journal of Optimization Theory and Applications, vol.
130, no. 2, pp. 185–207, 2006.
49
X. X. Huang, X. Q. Yang, and K. L. Teo, “Characterizing nonemptiness and compactness of the
solution set of a convex vector optimization problem with cone constraints and applications,” Journal
of Optimization Theory and Applications, vol. 123, no. 2, pp. 391–407, 2004.
Copyright: Tài liệu đại học ©