Abstract
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 Zhong (2008), He (2007), and Zhong and Huang (2010).
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1. Introduction
Let 
and 
be reflexive Banach spaces. Let 
be a nonempty closed convex subset of 
. Let 
be a set-valued mapping with nonempty values. Let 
be a closed convex pointed cone in 
with 
. The cone 
induces a partial ordering in 
, which was defined by 
. We consider the following set-valued vector equilibrium problem, denoted by SVEP
, which consists in finding 
such that
It is well known that (1.1) is closely related to the following dual set-valued vector equilibrium problem, denoted by DSVEP
, which consists in finding 
such that
We denote the solution sets of SVEP
and DSVEP
by 
and 
, respectively.
Let 
and 
be two metric spaces. Suppose that a nonempty closed convex set 
is perturbed by a parameter 
, which varies over 
, that is, 
is a set-valued mapping with nonempty closed convex values. Assume that a set-valued mapping 
is perturbed by a parameter 
, which varies over 
, that is, 
. We consider a parametric set-valued vector equilibrium problem, denoted by SVEP
, which consists in finding 
such that
Similarly, we consider the parameterized dual set-valued vector equilibrium problem, denoted by DSVEP
, which consists in finding 
such that
We denote the solution sets of SVEP
and DSVEP
by 
and 
, 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 intensively studied by many authors (see, e.g., [1–3, 10–26] and the references therein).
Among many desirable properties of the solution sets for vector equilibrium problems, stability analysis of solution set is of considerable interest (see, e.g, [27–33] and the references therein). Assuming that the barrier cone of 
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. [37] discussed 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 Zhong [39] extended the corresponding results of He [38] to the case that the perturbation was imposed on the mapping and the constraint set simultaneously. Very recently, Zhong and Huang [40] studied 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 
-pseudomonotone mapping in a reflexive Banach space, when both the mapping and the constraint set are perturbed by different parameters, which generalized and extended some known results in [38, 39].
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 Zhong [39], He [38], Zhong and Huang [40] from the variational inequality to the vector equilibrium problem.
The rest of the paper is organized as follows. In Section 2, we recall some concepts in convex analysis and present some basic results. In Section 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.
Let 
be a reflexive Banach space and 
be a nonempty closed convex subset of 
. The symbols "
" and "
" are used to denote strong and weak convergence, respectively.
The barrier cone of 
, denoted by 
, is defined by
The recession cone of 
, denoted by 
, is defined by
It is known that for any given 
,
We give some basic properties of recession cones in the following result which will be used in the sequel. Let 
be any family of nonempty sets in 
. Then
If, in addition, 
and each set 
is closed and convex, then we obtain an equality in the previous inclusion, that is,
Let 
be a proper convex and lower semicontinuous function. The recession function 
of 
is defined by
where 
is any point in 
. Then it follows that
The function 
turns out to be proper convex, lower semicontinuous and so weakly lower semicontinuous with the property that
Definition 2.1.
A set-valued mapping 
is said to be
-
(i)
upper semicontinuous at
if, for any neighborhood 
of 
, there exists a neighborhood 
of 
such that
(2.9) -
(ii)
lower semicontinuous at
if, for any 
and any neighborhood 
of 
, there exists a neighborhood 
of 
such that
(2.10)
if, for any neighborhood 
of 
, there exists a neighborhood 
of 
such that
if, for any 
and any neighborhood 
of 
, there exists a neighborhood 
of 
such that
We say 
is continuous at 
if it is both upper and lower semicontinuous at 
, and we say 
is continuous on 
if it is both upper and lower semicontinuous at every point of 
.
It is evident that 
is lower semicontinuous at 
if and only if, for any sequence 
with 
and 
, there exists a sequence 
with 
such that 
.
Definition 2.2.
A set-valued mapping 
is said to be weakly lower semicontinuous at 
if, for any 
and for any sequence 
with 
, there exists a sequence 
such that 
.
We say 
is weakly lower semicontinuous on 
if it is weakly lower semicontinuous at every point of 
. By Definition 2.2, we know that a weakly lower semicontinuous mapping is lower semicontinuous.
Definition 2.3.
A set-valued mapping 
is said to be
-
(i)
upper
-convex on 
if for any 
and 
, 
,
(2.11) -
(ii)
lower
-convex on 
if for any 
and 
, 
,
(2.12)
-convex on 
if for any 
and 
, 
,
-convex on 
if for any 
and 
, 
,
We say that 
is 
-convex if 
is both upper 
-convex and lower 
-convex.
Definition 2.4.
Let 
be a sequence of sets in 
. We define
Lemma 2.5 (see [36]).
Let 
be a nonempty closed convex subset of 
with 
. Then there exists no sequence 
such that 
and 
.
Lemma 2.6 (see [39]).
Let 
be a nonempty closed convex subset of 
with 
. Then there exists no sequence 
with each 
such that 
.
Lemma 2.7 (see [39]).
Let 
be a metric space and 
be a given point. Let 
be a set-valued mapping with nonempty values and let 
be upper semicontinuous at 
. Then there exists a neighborhood 
of 
such that 
for all 
.
Lemma 2.8 (see [41]).
Let 
be a nonempty convex subset of a Hausdorff topological vector space 
and 
be a set-valued mapping from 
into 
satisfying the following properties:
-
(i)
is a KKM mapping, that is, for every finite subset 
of 
, 
; -
(ii)
is closed in 
for every 
; -
(iii)
is compact in E for some 
.
is a KKM mapping, that is, for every finite subset 
of 
, 
;
is closed in 
for every 
;
is compact in E for some 
.Then 
.
3. Boundedness and Nonemptiness of Solution Sets
In this section, we present several equivalent characterizations for the set-valued vector equilibrium problem to have nonempty and bounded solution set. First of all, we give some assumptions which will be used for next theorems.
Let 
be a nonempty convex and closed subset of 
. Assume that 
is a set-valued mapping satisfying the following conditions:
(
)for each 
, 
;
(
)for each 
, 
implies that 
;
(
)for each 
, 
is 
-convex on 
;
(
)for each 
, 
is weakly lower semicontinuous on 
;
(
)for each 
, the set 
is closed, here 
stands for the closed line segment joining 
and 
.
Remark 3.1.
If
where 
is a set-valued mapping, 
is a proper, convex, lower semicontinuous function and 
, then condition 
reduces to the following 
-pseudomonotonicity assumption which was used in [40]. (See [40, Definition 2.2(iii)] of [40]): for all 
in the graph
,
Remark 3.2.
If, for each 
, the mapping 
is lower semicontinuous in 
, then condition 
is fulfilled. Indeed, for each 
and for any sequence 
with 
, we have 
and 
. By the lower semicontinuity of 
, for any 
, there exists 
such that 
. Since 
, we have 
and so 
by the closedness of 
. This implies that 
and the set 
is closed.
The following example shows that conditions 
can be satisfied.
Example 3.3.
Let 
, 
, 
and 
. Let
It is obvious that 
holds. Since for each 
, 
and 
are lower semicontinuous on 
, by Remark 3.2, we known that conditions 
and 
hold. For each 
, if 
, then we have 
. This implies that
and so 
holds. Moreover, for each 
, 
and 
with 
, it is easy to verify that
which shows that 
is 
-convex on 
and so 
holds. Thus, 
satisfies all conditions 
.
Theorem 3.4.
Let 
be a nonempty closed convex subset of 
and 
be a set-valued mapping satisfying assumptions 
-
. Then 
.
Proof.
From the assumption 
, it is easy to see that 
. We now prove that 
. Let 
. Then for all 
, 
. Set 
, where 
. Clearly, 
. From the upper 
-convexity of 
, we have
Since 
, we obtain
This implies that 
and so 
. Letting 
, by assumption 
, we have 
. Thus, 
and 
. This completes the proof.
Theorem 3.5.
Let 
be a nonempty closed convex subset of 
and 
be a set-valued mapping satisfying assumptions 
. If the solution set 
is nonempty, then
Proof.
From the proof of Theorem 3.4, we know that
Let 
. Then 
. By the assumptions 
and 
, we know that the set 
is nonempty closed and convex. It follows from (2.5) and Theorem 3.4 that
Then this completes the proof.
Remark 3.6.
If
where 
is a set-valued mapping, 
is a proper, convex, lower semicontinuous function and 
, then it follows from (3.8) and (2.8) that
Thus, we know that Theorem 3.5 is a generalization of [40, Theorem 3.1]. Moreover, by [40, Remark 3.1], Theorem 3.5 is also a generalization of [38, Lemma 3.1].
Theorem 3.7.
Let 
be a nonempty closed convex subset of 
and 
be a set-valued mapping satisfying assumptions 
. Suppose that 
. Then the following statements are equivalent:
-
(i)
the solution set of SVEP
is nonempty and bounded; -
(ii)
the solution set of DSVEP
is nonempty and bounded; -
(iii)
; -
(iv)
there exists a bounded set
such that for every 
, there exists some 
such that 
.
is nonempty and bounded;
is nonempty and bounded;
;
such that for every 
, there exists some 
such that 
.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). If (iv) does not hold, then there exists a sequence 
such that for each 
, 
and 
for every 
with 
. Without loss of generality, we may assume that 
weakly converges to 
. Then 
by the definition of the recession cone. Since 
, by Lemma 2.5, we know that 
. Let 
and 
be any fixed points. For 
sufficiently large, by the lower 
-convexity of 
,
Since
and 
is weakly lower semicontinuous, we know that 
and so 
. However, it contradicts the assumption that 
. Thus (iv) holds.
Since (i) and (ii) are equivalent, it remains to prove that (iv) implies (ii). Let 
be a set-valued mapping defined by
We first prove that 
is a closed subset of 
. Indeed, for any 
with 
, we have 
. It follows from the weakly lower semicontinuity of 
that 
. This shows that 
and so 
is closed.
We next prove that 
is a KKM mapping from 
to 
. Suppose to the contrary that there exist 
with 
, 
and 
such that 
. Then
By assumption 
, we have
It follows from the upper 
-convexity of 
that
which is a contradiction with (3.17). Thus we know that 
is a KKM mapping.
We may assume that 
is a bounded closed convex set (otherwise, consider the closed convex hull of 
instead of 
). Let 
be finite number of points in 
and let 
. Then the reflexivity of the space 
yields that 
is weakly compact convex. Consider the set-valued mapping 
defined by 
for all 
. Then each 
is a weakly compact convex subset of 
and 
is a KKM mapping. We claim that
Indeed, by Lemma 2.8, intersection in (3.19) is nonempty. Moreover, if there exists some 
but 
, then by (iv), we have 
for some 
. Thus, 
and so 
, which is a contradiction to the choice of 
.
Let 
. Then 
by (3.19) and so 
. This shows that the collection 
has finite intersection property. For each 
, it follows from the weak compactness of 
that 
is nonempty, which coincides with the solution set of DSVEP
.
Remark 3.8.
Theorem 3.7 establishes the necessary and sufficient conditions for the vector equilibrium problem to have nonempty and bounded solution sets. If
where 
is a set-valued mapping, 
is a proper, convex, lower semicontinuous function and 
, then problem (1.2) reduces to the following Minty mixed variational inequality: finding 
such that
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 
and 
be two metric spaces. 
is a set-valued mapping satisfying the following assumptions:
(
)for each 
, 
, 
, 
;
(
)for each 
, 
, 
, 
implies that 
;
(
)for each 
, 
, 
, 
is 
-convex on 
;
(
)for each 
, 
and 
, for any sequences 
, 
and 
with 
, 
and 
, there exists a sequence 
with 
such that 
.
The following Theorem 4.1 plays an important role in proving our results.
Theorem 4.1.
Let 
and 
be two metric spaces, 
and 
be given points. Let 
be a continuous set-valued mapping with nonempty closed convex values and 
. Suppose that 
is a set-valued mapping satisfying the assumptions 
. If
then there exists a neighborhood 
of 
such that
Proof.
Assume that the conclusion does not hold, then there exist a sequence 
in 
with 
such that 
.
Since 
is cone, we can select a sequence 
with 
such that 
for every 
. As 
is reflexive, without loss of generality, we can assume that 
, as 
. Since 
is a continuous set-valued mapping, hence, 
is upper semicontinuous and lower semicontinuous at 
. From the upper semicontinuity of 
, by Lemma 2.7, we have 
as 
large enough and hence 
as 
large enough. Since 
is a closed convex cone and hence weakly closed. This implies that 
. Moreover, it follows from Lemma 2.6 that 
.
For any 
, 
and 
, from the lower semicontinuity of 
, there exists 
such that 
. Since 
, it follows that 
. Together with 
, from assumption 
, there exists 
such that 
. Since 
, we have 
and 
. Letting 
, we obtain that 
. Since 
and 
are arbitrary, from the above discussion, we obtain 
with 
. This contradicts our assumption that 
. This completes the proof.
Remark 4.2.
If
where 
is a set-valued mapping, 
is a proper, convex, lower semicontinuous function and 
, from Remark 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 
and 
be two metric spaces, 
and 
be given points. Let 
be a continuous set-valued mapping with nonempty closed convex values and 
. Suppose that 
is a set-valued mapping satisfying the assumptions 
-
. If 
is nonempty and bounded, then
-
(i)
there exists a neighborhood
of 
such that for every 
, 
is nonempty and bounded; -
(ii)
-
.
of 
such that for every 
, 
is nonempty and bounded;
-
.Proof.
If 
is nonempty and bounded, then by Theorem 3.7 we have 
. It follows from Theorem 4.1 that there exists a neighborhood 
of 
, such that 
for every 
. By using Theorem 3.7 again, we have 
is nonempty and bounded for every 
. This verifies the first assertion.
Next, we prove the second assertion 
-
. For any given sequence 
with 
, we need to prove that 
-
. Let 
-
. Then there exists a sequence 
with each 
such that 
weakly converges to 
. We claim that there exists 
such that 
. Indeed, if the claim does hold, then there exist that a subsequence 
of 
and some 
, such that 
, for all 
. This implies that 
and so 
, which contradicts with the upper semicontinuity of 
. Thus, we have the claim. Moreover, we obtain 
as 
is a closed convex subset of 
and hence weakly closed.
Now we prove 
for all 
and hence 
. For any 
and 
, from the lower semicontinuity of 
, there exist 
such that 
. Moreover, from assumption 
, there exists a sequence of elements 
such that 
. Since 
, we have 
and so 
. Letting 
, we obtain that 
. Since 
is arbitrary, we have 
. This yields that 
. Thus, have the second assertion. This completes the proof.
Remark 4.4.
If
where 
is a set-valued mapping, 
is a proper, convex, lower semicontinuous function and 
, then problem (1.4) reduces to the following parametric Minty mixed variational inequality: finding 
such that
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 
, 
, 
and 
,
Note that 
is continuous on 
. However, 
is not lower semicontinuous at 
. Clearly, we have 
and 
for any 
. Thus,
Example 4.6.
Let 
, 
, 
and 
,
Note that 
satisfies the assumptions 
, and 
is upper semicontinuous. However, 
is not lower semicontinuous at 
. Clearly, we have 
and 
for any 
. Thus,
Example 4.7.
Let 
, 
, 
, 
,
Note that 
satisfies the assumptions 
and 
is lower semicontinuous. However, 
is not upper semicontinuous at 
. Clearly, we have 
and 
for any 
. Thus,
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The authors are grateful to the editor and reviewers for their valuable comments and suggestions. This work was supported by the Key Program of NSFC (Grant no. 70831005), the National Natural Science Foundation of China (10671135) and the Korea Research Foundation Grant funded by the Korean Government (KRF-2008-313-C00050).
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Zhong, RY., Huang, NJ. & Cho, Y. Boundedness and Nonemptiness of Solution Sets for Set-Valued Vector Equilibrium Problems with an Application. J Inequal Appl 2011, 936428 (2011). https://doi.org/10.1155/2011/936428
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DOI: https://doi.org/10.1155/2011/936428