- Research Article
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Connectedness and Compactness of Weak Efficient Solutions for Set-Valued Vector Equilibrium Problems
Journal of Inequalities and Applications volume 2008, Article number: 581849 (2008)
Abstract
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.
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. [1] and Cheng [2] have 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 Yao [7], Konnov and Yao [8], Fu [9], Hou et al. [10], Tan [11], Peng et al. [12], Ansari and Flores-Bazán [13], Lin et al. [14] and Long et al. [15] have studied the existence of solutions for set-valued vector equilibrium and set-valued vector variational inequalities problems. However, the connectedness and the compactness of the set of solutions to the set-valued vector equilibrium problem remained unstudied. In this paper, we study the existence, 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.
2. Preliminaries
Throughout this paper, let ,
be two normed linear spaces, let
be a nonempty subset of
, let
be a set-valued map, and let
be a closed convex pointed cone in
.
We consider the following set-valued vector equilibrium problem (SVEP): find , such that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2008%2F581849/MediaObjects/13660_2007_Article_1828_Equ1_HTML.gif)
Definition 2.1.
Let A vector
satisfying
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2008%2F581849/MediaObjects/13660_2007_Article_1828_Equ2_HTML.gif)
is called a weak efficient solution to the SVEP. Denote by the set of all weak efficient solutions to the SVEP.
Let be the topological dual space of
. Let
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2008%2F581849/MediaObjects/13660_2007_Article_1828_Equ3_HTML.gif)
be the dual cone of .
Definition 2.2.
Let . A vector
is called an
-efficient solution to the SVEP if
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2008%2F581849/MediaObjects/13660_2007_Article_1828_Equ4_HTML.gif)
where means that
, for all
. Denote by
the set of all
-efficient solutions to the SVEP.
Definition 2.3.
Let be a nonempty convex subset in
. A set-valued map
is called to be
-convex in its second variable if, for each fixed
, for every
,
, the following property holds:
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2008%2F581849/MediaObjects/13660_2007_Article_1828_Equ5_HTML.gif)
Definition 2.4.
Let be a nonempty convex subset in
. A set-valued map
is called to be
-concave in its first variable if, for each fixed
, for every
,
, the following property holds:
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2008%2F581849/MediaObjects/13660_2007_Article_1828_Equ6_HTML.gif)
Definition 2.5.
Let be a nonempty subset of
. Let
be a set-valued map, where
is the space of all bounded linear operators from
into
(let
be equipped with operator norm topology). Set
(i)Let be a convex subset of
.
is said to be
-hemicontinuous if, for every pair of points
, the set-valued map
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2008%2F581849/MediaObjects/13660_2007_Article_1828_Equ7_HTML.gif)
is lower semicontinuous at 0.
(ii)Let .
is said to be
-pseudomonotone on
if, for every pair of points
,
, for all
, then
, for all
.
The definition of -hemicontinuity was introduced by Lin et al. [14].
Definition 2.6.
Let be a Hausdorff topological vector space and let
be a nonempty set.
is called to be a KKM map if for any finite set
the relation
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2008%2F581849/MediaObjects/13660_2007_Article_1828_Equ8_HTML.gif)
holds, where denoted the convex hull of
.
For the definition of the upper semicontinuity and lower semicontinuity, see [16].
The following FKKM theorem plays a crucial role in this paper.
Lemma 2.7.
Let be a Hausdorff topological vector space. Let
be a nonempty convex subset of
, and let
be a KKM map. If for each
,
is closed in
, and if there exists a point
such that
is compact, then
.
By definition, we can get the following lemma.
Lemma 2.8.
Let be a nonempty convex subset of
. Let
be a set-valued map, and let
be a closed convex pointed cone. Moreover, suppose that
is
-convex in its second variable. Then, for each
,
is convex.
3. Scalarization
In this section, we extend a result in [3] to set-valued map.
Theorem 3.1.
Suppose that , and that
is a convex set for each
. Then
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2008%2F581849/MediaObjects/13660_2007_Article_1828_Equ9_HTML.gif)
Proof.
It is clear that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2008%2F581849/MediaObjects/13660_2007_Article_1828_Equ10_HTML.gif)
Now we prove that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2008%2F581849/MediaObjects/13660_2007_Article_1828_Equ11_HTML.gif)
Let . By definition,
, for all
. Thus
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2008%2F581849/MediaObjects/13660_2007_Article_1828_Equ12_HTML.gif)
As is a convex pointed cone, we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2008%2F581849/MediaObjects/13660_2007_Article_1828_Equ13_HTML.gif)
By assumption, is a convex set. By the separation theorem of convex sets, there exist some
, such that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2008%2F581849/MediaObjects/13660_2007_Article_1828_Equ14_HTML.gif)
By (3.6), we obtain that and
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2008%2F581849/MediaObjects/13660_2007_Article_1828_Equ15_HTML.gif)
Therefore, . Hence
. Thus we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2008%2F581849/MediaObjects/13660_2007_Article_1828_Equ16_HTML.gif)
4. Existence of The Weak Efficient Solutions
Theorem 4.1.
Let be a nonempty closed convex subset of
and let
be a closed convex pointed cone with
. Let
be a set-valued map with
for all
. Suppose that for each
,
is lower semicontinuous on
, and that
is
-convex in its second variable. If there exists a nonempty compact subset
of
, and
, such that
, for all
, then, for any
,
,
,
, and
.
Proof.
Let . Define the set-valued map
by
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2008%2F581849/MediaObjects/13660_2007_Article_1828_Equ17_HTML.gif)
By assumption, , for all
, so
. We claim that
is a KKM map. Suppose to the contrary that there exists a finite subset
of
, and there exists
such that
. Then
for some
,
, with
, and
, for all
. Then there exist
, such that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2008%2F581849/MediaObjects/13660_2007_Article_1828_Equ18_HTML.gif)
As is
-convex in its second invariable, we can get that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2008%2F581849/MediaObjects/13660_2007_Article_1828_Equ19_HTML.gif)
By (4.3), we know that there exist ,
, such that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2008%2F581849/MediaObjects/13660_2007_Article_1828_Equ20_HTML.gif)
Hence . By assumption, we have
. By (4.2), however, we have
. This is a contradiction. Thus
is a KKM map. Now we show that for each
,
is closed. For any sequence,
and
. Because
is a closed set, we have
. By assumption, for each
,
is lower semicontinuous on
, then by [16], for each fixed
, and for each
, there exist
, such that
. Because
, we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2008%2F581849/MediaObjects/13660_2007_Article_1828_Equ21_HTML.gif)
Thus . By the continuity of
and
, we have
. By the arbitrariness of
, we have
, that is,
. Hence
is closed. By the assumption, we have
, and
is closed. Since
is compact,
is compact. By Lemma 2.7, we have
. Thus there exists
. This means that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2008%2F581849/MediaObjects/13660_2007_Article_1828_Equ22_HTML.gif)
Therefore, . Next we show that
. If
, then
. It follows from
that
, and by Theorem 3.1, we have
.
Theorem 4.2.
Let be a nonempty closed convex subset of
and let
be a closed convex pointed cone with
. Let
. Assume that
is a
-hemicontinuous,
-pseudomonotone mapping. Moreover, assume that the set-valued map
defined by
is
-convex in its second variable. If there exists a nonempty compact subset
of
, and
, such that
, for all
, then
and
.
Proof.
Let . Define the set-valued maps
,
by
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2008%2F581849/MediaObjects/13660_2007_Article_1828_Equ23_HTML.gif)
respectively. As for each , we have
, then
. The proof of the theorem is divided into four steps.
-
(I)
is a KKM map on
.
Suppose to the contrary that there exists a finite subset of
, and there exists
such that
. Then
for some
,
, with
, and
, for all
. Then there exist 
such that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2008%2F581849/MediaObjects/13660_2007_Article_1828_Equ24_HTML.gif)
Since is
-convex in its second variable, we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2008%2F581849/MediaObjects/13660_2007_Article_1828_Equ25_HTML.gif)
Let , for each
. By (4.9), we know there exists
, such that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2008%2F581849/MediaObjects/13660_2007_Article_1828_Equ26_HTML.gif)
As , we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2008%2F581849/MediaObjects/13660_2007_Article_1828_Equ27_HTML.gif)
While by (4.8), we have . This is a contraction. Hence
is a KKM map on
.
-
(II)
for all
and
is a KKM map.
By the -pseudomonotonicity of
, for each
, we have
. Since
is a KKM map, so is
.
-
(III)
.
Now we show that for each ,
is closed. Let
be a sequence in
such that
converges to
. By the closedness of
, we have
. Since
, then for each
, we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2008%2F581849/MediaObjects/13660_2007_Article_1828_Equ28_HTML.gif)
As , and the continuity of
, then for each
, we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2008%2F581849/MediaObjects/13660_2007_Article_1828_Equ29_HTML.gif)
Consequently, . Hence
is closed. By the assumption, we have
. Then
is compact since
is compact. By step (II), we know
is a KKM map. By Lemma 2.7,
.
-
(IV)
.
Because , we have
. Now let us show that
. Let
. For each
, and each
, we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2008%2F581849/MediaObjects/13660_2007_Article_1828_Equ30_HTML.gif)
For any and for each fixed
, define the set-valued mapping
by
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2008%2F581849/MediaObjects/13660_2007_Article_1828_Equ31_HTML.gif)
We pick a sequence such that
and set
. Since
is a convex set,
for each
. It is clear that
. Let
. We have
Since
is
-hemicontinuous,
is lower semicontinuous at 0. By [16], there exist
, such that
. As
, there exist
such that
. By
, we have
. By (4.14), we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2008%2F581849/MediaObjects/13660_2007_Article_1828_Equ32_HTML.gif)
Since ,
. Hence
since
is continuous and
. Therefore, for any
and for each
, we have
. Hence
. Thus
. This means that there exists
, for each
, we have
, for all
. It follows that
, thus
. By the proof of Theorem 4.1, we know
. Since
, we have
. 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.
Let be a nonempty closed convex subset of
, let
be a closed convex pointed cone with
, and let
be a set-valued map. Assume that the following conditions are satisfied:
(i)for each ,
is lower semicontinuous on
;
(ii) is
-concave in its first variable and
-convex in its second variable;
(iii), for all
;
(iv) is a bounded subset in
;
(v)there exists a nonempty compact convex subset of
, and
, such that
, for all
.
Then is a nonempty connected compact set.
Proof.
We define the set-valued map by
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2008%2F581849/MediaObjects/13660_2007_Article_1828_Equ33_HTML.gif)
By Theorem 4.1, for each , we have
, hence
and
. It is clear that
is convex, so it is a connected set. Now we prove that, for each
,
is a connected set. Let
, we have
and
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2008%2F581849/MediaObjects/13660_2007_Article_1828_Equ34_HTML.gif)
Because is
-concave in its first variable, for each fixed
, and for above
, and
, we have
since
is convex, and
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2008%2F581849/MediaObjects/13660_2007_Article_1828_Equ35_HTML.gif)
Hence for each ,
, there exist
,
, and
, such that
. As
and by (5.2), we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2008%2F581849/MediaObjects/13660_2007_Article_1828_Equ36_HTML.gif)
Thus
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2008%2F581849/MediaObjects/13660_2007_Article_1828_Equ37_HTML.gif)
that is . So
is convex, therefore it is a connected set.
Now we show that is upper semicontinuous on
. Since
is a nonempty compact set, by [16], we only need to prove that
is closed. Let the sequence
and
, where
converge to
with respect to the norm topology. As
, we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2008%2F581849/MediaObjects/13660_2007_Article_1828_Equ38_HTML.gif)
that is, , for all
. As
and
is compact, we have
. Since for each
,
is lower semicontinuous on
, for each fixed
, and each
, there exist
, such that
. From
, we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2008%2F581849/MediaObjects/13660_2007_Article_1828_Equ39_HTML.gif)
By the continuity of and
, we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2008%2F581849/MediaObjects/13660_2007_Article_1828_Equ40_HTML.gif)
Let . By assumption,
is a bounded set in
, then there exist some
, such that for each
, we have
. For any
, because
with respect to norm topology, there exists
, and when 
, we have
. Therefore, there exists
, and when 
, we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2008%2F581849/MediaObjects/13660_2007_Article_1828_Equ41_HTML.gif)
Hence
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2008%2F581849/MediaObjects/13660_2007_Article_1828_Equ42_HTML.gif)
Consequently, by (5.8), (5.10), we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2008%2F581849/MediaObjects/13660_2007_Article_1828_Equ43_HTML.gif)
By (5.7), we have . So for any
and for each
, we have
. Hence
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2008%2F581849/MediaObjects/13660_2007_Article_1828_Equ44_HTML.gif)
This means that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2008%2F581849/MediaObjects/13660_2007_Article_1828_Equ45_HTML.gif)
Hence the graph of is closed. Therefore,
is a closed map. By [16],
is upper semicontinuous on
. Because
is
-convex in its second variable, by Lemma 2.8, for each
,
is convex. It follows from Theorem 3.1 that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2008%2F581849/MediaObjects/13660_2007_Article_1828_Equ46_HTML.gif)
Thus by [17, Theorem 3.1] is a connected set.
Now, we show that is a compact set. We first show that
is a closed set. Let
with
. Since
is compact,
We claim that
Suppose to the contrary that
, then there exist some
such that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2008%2F581849/MediaObjects/13660_2007_Article_1828_Equ47_HTML.gif)
Thus there exists such that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2008%2F581849/MediaObjects/13660_2007_Article_1828_Equ48_HTML.gif)
Hence is a neighborhood of
. Since
is lower semicontinuous at
, there exists some neighborhood
of
such that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2008%2F581849/MediaObjects/13660_2007_Article_1828_Equ49_HTML.gif)
Since , there exist some
, and when 
, we have
. By (5.17),
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2008%2F581849/MediaObjects/13660_2007_Article_1828_Equ50_HTML.gif)
This contradicts . Thus
This means that
is a closed set. Since
is compact and
,
is compact.
Theorem 5.2.
Let be a nonempty closed convex subset of
, and let
be a closed convex pointed cone with
. Assume that for each
,
is a
-hemicontinuous,
-pseudomonotone mapping. Moreover, assume that the set-valued map
defined by
is
-convex in its second variable, and the set
is a bounded set in
. If there exists a nonempty compact convex subset
of
, and
, such that
, for all
, then
is a nonempty connected set.
Proof.
We define the set-valued map by
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2008%2F581849/MediaObjects/13660_2007_Article_1828_Equ51_HTML.gif)
By Theorem 4.2, for each , we have
and
. Hence
and
. Clearly,
is a convex set, hence it is a connected set. Define the set-valued maps
,
by
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2008%2F581849/MediaObjects/13660_2007_Article_1828_Equ52_HTML.gif)
respectively. Now we prove that for each ,
is a connected set. Let
, then
. By the proof of Theorem 4.2, we have
, so
. Hence for
and for each
,
, we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2008%2F581849/MediaObjects/13660_2007_Article_1828_Equ53_HTML.gif)
Then, for each ,
, and
, we have
since
is convex and
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2008%2F581849/MediaObjects/13660_2007_Article_1828_Equ54_HTML.gif)
Hence . Thus
. Consequently, for each
,
is a convex set. Therefore, it is a connected set. The following is to prove that
is upper semicontinuous on
. Since
is a nonempty compact set, by [16] we only need to show that
is a closed map. Let sequence
and
, where
converges to
with respect to the norm topology of
. As
, we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2008%2F581849/MediaObjects/13660_2007_Article_1828_Equ55_HTML.gif)
Then, for each , we have that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2008%2F581849/MediaObjects/13660_2007_Article_1828_Equ56_HTML.gif)
By assumption, for each ,
is
-pseudomonotone, and by (5.24), for each
, for the above
, and for each
, we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2008%2F581849/MediaObjects/13660_2007_Article_1828_Equ57_HTML.gif)
As , we have
, and
. As
, and
is compact, we have
. Let
. By assumption,
is a bounded set in
. Then, there exists
, such that for each
, we have
. For any
, because
with respect to the norm topology, there exists
, and when 
, we have
. Therefore, there exists
, and when 
, we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2008%2F581849/MediaObjects/13660_2007_Article_1828_Equ58_HTML.gif)
Hence
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2008%2F581849/MediaObjects/13660_2007_Article_1828_Equ59_HTML.gif)
Then
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2008%2F581849/MediaObjects/13660_2007_Article_1828_Equ60_HTML.gif)
Then, by (5.25), (5.28), we have . Hence for each
, and for each
, we have
. Since
is
-pseudomonotone, for each
, and for each
, we have
. Hence
. Therefore, the graph of
is closed, and
is a closed map. By [16], we know that
is upper semicontinuous on
. Because
is
-convex in its second variable, for each
,
is convex. It follows from Theorem 3.1 that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2008%2F581849/MediaObjects/13660_2007_Article_1828_Equ61_HTML.gif)
Then, by [17, Theorem 3.1], we know that is a connected set. The proof of the theorem is completed.
Let denote the base of neighborhoods of 0 of
. By [18], for each bounded subset
and for each neighborhood
of 0 in
, we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2008%2F581849/MediaObjects/13660_2007_Article_1828_Equ62_HTML.gif)
Lemma 5.3.
Let be a nonempty convex subset of
, and let
If
is lower semicontinuous on
, then
is
-hemicontinuous on
.
Proof.
For any fixed , we need to show that the set-valued mapping
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2008%2F581849/MediaObjects/13660_2007_Article_1828_Equ63_HTML.gif)
is lower semicontinuous at 0. For any and for any neighborhood
of
, there exists
such that
and there exists a neighborhood
of
in
, such that
. Since
is lower semicontinuous at
, for the neighborhood
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2008%2F581849/MediaObjects/13660_2007_Article_1828_Equ64_HTML.gif)
of , there exists a neighborhood
of
such that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2008%2F581849/MediaObjects/13660_2007_Article_1828_Equ65_HTML.gif)
For the above , since
, when
, there exists
, and when 
, we have
By (5.33), we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2008%2F581849/MediaObjects/13660_2007_Article_1828_Equ66_HTML.gif)
Thus there exist such that
, for all
. By (5.32), we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2008%2F581849/MediaObjects/13660_2007_Article_1828_Equ67_HTML.gif)
that is
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2008%2F581849/MediaObjects/13660_2007_Article_1828_Equ68_HTML.gif)
This means that is lower semicontinuous at 0. By definition,
is
-hemicontinuous on
.
Theorem 5.4.
Let be a nonempty closed bounded convex subset of
, and let
be a closed convex pointed cone with
. Assume that for each
,
is a
-pseudomonotone, lower semicontinuous mapping. Moreover, assume that the set-valued map
defined by
is
-convex in its second variable, and the set
is a bounded set in
. If there exists a nonempty compact convex subset
of
, and
, such that
, for all
, then
is a nonempty connected compact set.
Proof.
By Lemma 5.3 and Theorem 5.2, is a nonempty connected set. Since
is a compact set, we need only to show that
is closed. Let
It is clear that
and
. We claim that
Suppose to the contrary that
, then there exists
such that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2008%2F581849/MediaObjects/13660_2007_Article_1828_Equ69_HTML.gif)
that is
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2008%2F581849/MediaObjects/13660_2007_Article_1828_Equ70_HTML.gif)
Thus there exists such that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2008%2F581849/MediaObjects/13660_2007_Article_1828_Equ71_HTML.gif)
Hence there exists such that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2008%2F581849/MediaObjects/13660_2007_Article_1828_Equ72_HTML.gif)
where is the unit ball of
. Set
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2008%2F581849/MediaObjects/13660_2007_Article_1828_Equ73_HTML.gif)
By definition, is a neighborhood of zero of
, since
is bounded. Since
is lower semicontinuous at
, for the above
and the above
, there exists a neighborhood
of
such that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2008%2F581849/MediaObjects/13660_2007_Article_1828_Equ74_HTML.gif)
Since there exists
such that when
, we have
Thus by (5.42) we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2008%2F581849/MediaObjects/13660_2007_Article_1828_Equ75_HTML.gif)
Thus there exist for all
. We have
, and hence
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2008%2F581849/MediaObjects/13660_2007_Article_1828_Equ76_HTML.gif)
Therefore,
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2008%2F581849/MediaObjects/13660_2007_Article_1828_Equ77_HTML.gif)
Since and
, we have
. Hence there exists
, and when 
we have
. This combining (5.45) implies that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2008%2F581849/MediaObjects/13660_2007_Article_1828_Equ78_HTML.gif)
Hence
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2008%2F581849/MediaObjects/13660_2007_Article_1828_Equ79_HTML.gif)
By (5.40) and (5.47), we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2008%2F581849/MediaObjects/13660_2007_Article_1828_Equ80_HTML.gif)
On the other hand, since , we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2008%2F581849/MediaObjects/13660_2007_Article_1828_Equ81_HTML.gif)
This contradicts (5.48), because Thus
. This means that
is a closed subset of
.
References
Lee GM, Kim DS, Lee BS, Yen ND: Vector variational inequality as a tool for studying vector optimization problems. Nonlinear Analysis: Theory, Methods & Applications 1998,34(5):745–765. 10.1016/S0362-546X(97)00578-6
Cheng Y: On the connectedness of the solution set for the weak vector variational inequality. Journal of Mathematical Analysis and Applications 2001,260(1):1–5. 10.1006/jmaa.2000.7389
Gong X-H: Efficiency and Henig efficiency for vector equilibrium problems. Journal of Optimization Theory and Applications 2001,108(1):139–154. 10.1023/A:1026418122905
Gong X-H, Fu WT, Liu W: Super efficiency for a vector equilibrium in locally convex topological vector spaces. In Vector Variational Inequalities and Vector Equilibria, Nonconvex Optimization and Its Applications. Volume 38. Edited by: Giannessi F. Kluwer Academic Publishers, Dordrecht, The Netherlands; 2000:233–252. 10.1007/978-1-4613-0299-5_13
Gong X-H: Connectedness of the solution sets and scalarization for vector equilibrium problems. Journal of Optimization Theory and Applications 2007,133(2):151–161. 10.1007/s10957-007-9196-y
Ansari QH, Oettli W, Schläger D: A generalization of vectorial equilibria. Mathematical Methods of Operations Research 1997,46(2):147–152. 10.1007/BF01217687
Ansari QH, Yao J-C: An existence result for the generalized vector equilibrium problem. Applied Mathematics Letters 1999,12(8):53–56. 10.1016/S0893-9659(99)00121-4
Konnov IV, Yao JC: Existence of solutions for generalized vector equilibrium problems. Journal of Mathematical Analysis and Applications 1999,233(1):328–335. 10.1006/jmaa.1999.6312
Fu J-Y: Generalized vector quasi-equilibrium problems. Mathematical Methods of Operations Research 2000,52(1):57–64. 10.1007/s001860000058
Hou SH, Yu H, Chen GY: On vector quasi-equilibrium problems with set-valued maps. Journal of Optimization Theory and Applications 2003,119(3):485–498.
Tan NX: On the existence of solutions of quasivariational inclusion problems. Journal of Optimization Theory and Applications 2004,123(3):619–638. 10.1007/s10957-004-5726-z
Peng J-W, Lee H-WJ, Yang X-M: On system of generalized vector quasi-equilibrium problems with set-valued maps. Journal of Global Optimization 2006,36(1):139–158. 10.1007/s10898-006-9004-5
Ansari QH, Flores-Bazán F: Recession methods for generalized vector equilibrium problems. Journal of Mathematical Analysis and Applications 2006,321(1):132–146. 10.1016/j.jmaa.2005.07.059
Lin L-J, Ansari QH, Huang Y-J: Some existence results for solutions of generalized vector quasi-equilibrium problems. Mathematical Methods of Operations Research 2007,65(1):85–98. 10.1007/s00186-006-0102-4
Long X-J, Huang N-J, Teo K-L: Existence and stability of solutions for generalized strong vector quasi-equilibrium problem. Mathematical and Computer Modelling 2008,47(3–4):445–451. 10.1016/j.mcm.2007.04.013
Aubin J-P, Ekeland I: Applied Nonlinear Analysis, Pure and Applied Mathematics. John Wiley & Sons, New York, NY, USA; 1984:xi+518.
Warburton AR: Quasiconcave vector maximization: connectedness of the sets of Pareto-optimal and weak Pareto-optimal alternatives. Journal of Optimization Theory and Applications 1983,40(4):537–557. 10.1007/BF00933970
Robertson AP, Robertson W: Topological Vector Spaces, Cambridge Tracts in Mathematics and Mathematical Physics, no. 53. Cambridge University Press, New York, NY, USA; 1964:viii+158.
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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Chen, B., Gong, XH. & Yuan, SM. Connectedness and Compactness of Weak Efficient Solutions for Set-Valued Vector Equilibrium Problems. J Inequal Appl 2008, 581849 (2008). https://doi.org/10.1155/2008/581849
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DOI: https://doi.org/10.1155/2008/581849