- Research Article
- Open access
- Published:
Solving the Set Equilibrium Problems
Fixed Point Theory and Applications volume 2011, Article number: 945413 (2011)
Abstract
We study the weak solutions and strong solutions of set equilibrium problems in real Hausdorff topological vector space settings. Several new results of existence for the weak solutions and strong solutions of set equilibrium problems are derived. The new results extend and modify various existence theorems for similar problems.
1. Introduction and Preliminaries
Let ,
,
be arbitrary real Hausdorff topological vector spaces, let
be a nonempty closed convex set of
, and let
be a proper closed convex and pointed cone with apex at the origin and
, that is,
is proper closed with
and satisfies the following conditions:
(1), for all
;
(2);
(3).
Letting ,
be two sets of
, we can define relations "
" and "
" as follows:
(1);
(2).
Similarly, we can define the relations "" and "
" if we replace the set
by
.
The trimapping and mapping
are given. The set equilibrium problem (SEP)
I
is to find an
such that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F945413/MediaObjects/13663_2010_Article_1445_Equ1_HTML.gif)
for all and for some
. Such solution is called a weak solution for (SEP)
I
. We note that (1.1) is equivalent to the following one:
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F945413/MediaObjects/13663_2010_Article_1445_Equ2_HTML.gif)
for all and for some
.
For the case when does not depend on
, that is, to find an
with some
such that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F945413/MediaObjects/13663_2010_Article_1445_Equ3_HTML.gif)
for all , we will call this solution a strong solution of (SEP)
I
. We also note that (1.3) is equivalent to the following one:
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F945413/MediaObjects/13663_2010_Article_1445_Equ4_HTML.gif)
for all .
We note that if is a vector-valued function and the mapping
is constant for each
, then (SEP)
I
reduces to the vector equilibrium problem (VEP), which is to find
such that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F945413/MediaObjects/13663_2010_Article_1445_Equ5_HTML.gif)
for all . Existence of a solution of this problem is investigated by Ansari et al. [1, 2].
If is a vector-valued function and
which is denoted the space of all continuous linear mappings from
to
and
, where
denotes the evaluation of the linear mapping
at
, then (SEP)
I
reduces to (GVVIP): to find
and
such that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F945413/MediaObjects/13663_2010_Article_1445_Equ6_HTML.gif)
for all . It has been studied by Chen and Craven [3].
If we consider ,
,
, and
, where
denotes the evaluation of the linear mapping
at
, then (SEP)
I
reduces to the (GVVIP) which is discussed by Huang and Fang [4] and Zeng and Yao [5]: to find a vector
and
such that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F945413/MediaObjects/13663_2010_Article_1445_Equ7_HTML.gif)
If ,
is a single-valued mapping,
, then (SEP)
I
reduces to the (weak) vector variational inequalities problem which is considered by Fang and Huang [6], Chiang and Yao [7], and Chiang [8] as follows: to find a vector
such that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F945413/MediaObjects/13663_2010_Article_1445_Equ8_HTML.gif)
for all . The vector variational inequalities problem was first introduced by Giannessi [9] in finite-dimensional Euclidean space.
Summing up the above arguments, they show that for a suitable choice of the mapping and the spaces
,
, and
, we can obtain a number of known classes of vector equilibrium problems, vector variational inequalities, and implicit generalized variational inequalities. It is also well known that variational inequality and its variants enable us to study many important problems arising in mathematical, mechanics, operations research, engineering sciences, and so forth.
In this paper we aim to derive some solvabilities for the set equilibrium problems. We also study some results of existence for the weak solutions and strong solutions of set equilibrium problems. Let be a nonempty subset of a topological vector space
. A set-valued function
from
into the family of subsets of
is a KKM mapping if for any nonempty finite set
, the convex hull of
is contained in
. Let us first recall the following results.
Fan's Lemma (see [10]).
Let be a nonempty subset of Hausdorff topological vector space
. Let
be a KKM mapping such that for any
,
is closed and
is compact for some
. Then there exists
such that
for all
.
Definition 1.1 (see [11]).
Let be a vector space, let
be a topological vector space, let
be a nonempty convex subset of
, and let
be a proper closed convex and pointed cone with apex at the origin and
, and
is said to be
(1)-convex if
for every
and
;
(2)naturally quasi-convex if
for every
and
.
The following definition can also be found in [11].
Definition 1.2.
Let be a Hausdorff topological vector space, let
be a proper closed convex and pointed cone with apex at the origin and
, and let
be a nonempty subset of
. Then
(1)a point is called a minimal point of
if
;
is the set of all minimal points of
;
(2)a point is called a maximal point of
if
;
is the set of all maximal points of
;
(3)a point is called a weakly minimal point of
if
;
is the set of all weakly minimal points of
;
(4)a point is called a weakly maximal point of
if
;
is the set of all weakly maximal points of
.
Definition 1.3.
Let ,
be two topological spaces. A mapping
is said to be
(1)upper semicontinuous if for every and every open set
in
with
, there exists a neighborhood
of
such that
;
(2)lower semicontinuous if for every and every open neighborhood
of every
, there exists a neighborhood
of
such that
for all
;
(3)continuous if it is both upper semicontinuous and lower semicontinuous.
We note that is lower semicontinuous at
if for any net
,
,
implies that there exists net
such that
. For other net-terminology properties about these two mappings, one can refer to [12].
Lemma 1.4 (see [13]).
Let ,
, and
be real topological vector spaces, and let
and
be nonempty subsets of
and
, respectively. Let
,
be set-valued mappings. If both
and
are upper semicontinuous with nonempty compact values, then the set-valued mapping
defined by
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F945413/MediaObjects/13663_2010_Article_1445_Equ9_HTML.gif)
is upper semicontinuous with nonempty compact values.
By using similar technique of [11, Proposition  2.1], we can deduce the following lemma that slight-generalized the original one.
Lemma 1.5.
Let ,
be two Hausdorff topological vector spaces, and let
,
be nonempty compact convex subsets of
and
, respectively. Let
be continuous mapping with nonempty compact valued on
; the mapping
is naturally quasi
-convex on
for each
, and the mapping
is
-convex on
for each
. Assume that for each
, there exists
such that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F945413/MediaObjects/13663_2010_Article_1445_Equ10_HTML.gif)
Then, one has
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F945413/MediaObjects/13663_2010_Article_1445_Equ11_HTML.gif)
2. Existence Theorems for Set Equilibrium Problems
Now, we state and show our main results of solvabilities for set equilibrium problems.
Theorem 2.1.
Let ,
,
be real Hausdorff topological vector spaces, let
be a nonempty closed convex subset of
, and let
be a proper closed convex and pointed cone with apex at the origin and
. Given mappings
,
, and
, suppose that
(1) for all
;
(2)for each , there is an
such that for all
,
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F945413/MediaObjects/13663_2010_Article_1445_Equ12_HTML.gif)
(3)for each , the set
is convex;
(4)there is a nonempty compact convex subset of
, such that for every
, there is a
such that for all
,
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F945413/MediaObjects/13663_2010_Article_1445_Equ13_HTML.gif)
-
(5)
for each
, the set
is open in
.
Then there exists an which is a weak solution of (SEP)I. That is, there is an
such that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F945413/MediaObjects/13663_2010_Article_1445_Equ14_HTML.gif)
for all and for some
.
Proof.
Define by
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F945413/MediaObjects/13663_2010_Article_1445_Equ15_HTML.gif)
for all . From condition (5) we know that for each
, the set
is closed in
, and hence it is compact in
because of the compactness of
.
Next, we claim that the family has the finite intersection property, and then the whole intersection
is nonempty and any element in the intersection
is a solution of (SEP)
I
, for any given nonempty finite subset
of
. Let
, the convex hull of
. Then
is a compact convex subset of
. Define the mappings
, respectively, by
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F945413/MediaObjects/13663_2010_Article_1445_Equ16_HTML.gif)
for each . From conditions (1) and (2), we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F945413/MediaObjects/13663_2010_Article_1445_Equ17_HTML.gif)
and for each , there is an
such that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F945413/MediaObjects/13663_2010_Article_1445_Equ18_HTML.gif)
Hence , and then
for all
.
We can easily see that has closed values in
. Since, for each
,
, if we prove that the whole intersection of the family
is nonempty, we can deduce that the family
has finite intersection property because
and due to condition (4). In order to deduce the conclusion of our theorem, we can apply Fan's lemma if we claim that
is a KKM mapping. Indeed, if
is not a KKM mapping, neither is
since
for each
. Then there is a nonempty finite subset
of
such that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F945413/MediaObjects/13663_2010_Article_1445_Equ19_HTML.gif)
Thus there is an element such that
for all
, that is,
for all
. By (3), we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F945413/MediaObjects/13663_2010_Article_1445_Equ20_HTML.gif)
and hence which contradicts (2.6). Hence
is a KKM mapping, and so is
. Therefore, there exists an
which is a solution of (SEP)
I
. This completes the proof.
Theorem 2.2.
Let ,
,
be real Hausdorff topological vector spaces, let
be a nonempty closed convex subset of
, and let
be a proper closed convex and pointed cone with apex at the origin and
. Let the mapping
be such that for each
, the mappings
and
are upper semicontinuous with nonempty compact values and
. Suppose that conditions (1)–(4) of Theorem 2.1 hold. Then there exists an
which is a solution of (SEP)I. That is, there is an
such that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F945413/MediaObjects/13663_2010_Article_1445_Equ21_HTML.gif)
for all and for some
.
Proof.
For any fixed , we define the mapping
by
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F945413/MediaObjects/13663_2010_Article_1445_Equ22_HTML.gif)
for all and
. Since the mappings
and
are upper semicontinuous with nonempty compact values, by Lemma 1.4, we know that
is upper semicontinuous on
with nonempty compact values. Hence, for each
, the set
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F945413/MediaObjects/13663_2010_Article_1445_Equ23_HTML.gif)
is open in . Then all conditions of Theorem 2.1 hold. From Theorem 2.1, (SEP)
I
has a solution.
In order to discuss the results of existence for the strong solution of (SEP)
I
, we introduce the condition (). It is obviously fulfilled that if ,
is single-valued function.
Theorem 2.3.
Under the framework of Theorem 2.2, one has a weak solution of (SEP)I with
. In addition, if
,
, and
is compact,
is convex, the mapping
is continuous with nonempty compact valued on
, the mapping
is naturally quasi
-convex on
for each
, and the mapping
is
-convex on
for each
. Assuming that for each
, there exists
such that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F945413/MediaObjects/13663_2010_Article_1445_Equ24_HTML.gif)
then is a strong solution of (SEP)I; that is, there exists
such that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F945413/MediaObjects/13663_2010_Article_1445_Equ25_HTML.gif)
for all . Furthermore, the set of all strong solutions of (SEP)I is compact.
Proof.
From Theorem 2.2, we know that such that (1.1) holds for all
and for some
. Then we have
.
From condition () and the convexity of , Lemma 1.5 tells us that
. Then there is an
such that
. Thus for all
, we have
. Hence there exists
such that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F945413/MediaObjects/13663_2010_Article_1445_Equ26_HTML.gif)
for all . Such an
is a strong solution of (SEP)
I
.
Finally, to see that the solution set of (SEP)
I
is compact, it is sufficient to show that the solution set is closed due to the coercivity condition (4) of Theorem 2.2. To this end, let denote the solution set of (SEP)
I
. Suppose that net
which converges to some
. Fix any
. For each
, there is an
such that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F945413/MediaObjects/13663_2010_Article_1445_Equ27_HTML.gif)
Since is upper semicontinuous with compact values and the set
is compact, it follows that
is compact. Therefore without loss of generality, we may assume that the sequence
converges to some
. Then
and
. Let
. Since the mapping
is upper semicontinuous with nonempty compact values, the set
is open in
. Hence
is closed in
. By the facts
and
, we have
. This implies that
. We then obtain
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F945413/MediaObjects/13663_2010_Article_1445_Equ28_HTML.gif)
Hence and
is closed.
We would like to point out that condition () is fulfilled if we take and
is a single-valued function. The following is a concrete example for both Theorems 2.1 and 2.3.
Example 2.4.
Let ,
,
,
, and
. Choose
to be defined by
for every
and
is defined by
, where
,
with
, for some
,
, and
is defined by
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F945413/MediaObjects/13663_2010_Article_1445_Equ29_HTML.gif)
Then all conditions of Theorems 2.1 and 2.3 are satisfied. By Theorems 2.1 and 2.3, respectively, the (SEP)
I
not only has a weak solution, but also has a strong solution. A simple geometric discussion tells us that is a strong solution for (SEP)
I
.
Corollary 2.5.
Under the framework of Theorem 2.1, one has a weak solution of (SEP)I with
. In addition, if
and
,
is compact,
is convex,
-convex on
for each
and the mapping
is
-convex on
for each
,
such that
is continuous with nonempty compact values for each
, and
is upper semicontinuous with nonempty compact values. Assume that condition () holds, then
is a strong solution of (SEP)I; that is, there exists
such that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F945413/MediaObjects/13663_2010_Article_1445_Equ30_HTML.gif)
for all . Furthermore, the set of all strong solutions of (SEP)I is compact.
Theorem 2.6.
Let ,
,
,
,
,
,
be as in Theorem 2.1. Assume that the mapping
is
-convex on
for each
and
such that
(1)for each , there is an
such that
;
(2)there is a nonempty compact convex subset of
, such that for every
, there is a
such that for all
,
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F945413/MediaObjects/13663_2010_Article_1445_Equ31_HTML.gif)
(3)for each , the set
is open in
.
Then there is an which is a weak solution of (SEP)I.
Proof.
For any given nonempty finite subset of
. Letting
, then
is a nonempty compact convex subset of
. Define
as in the proof of Theorem 2.1, and for each
, let
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F945413/MediaObjects/13663_2010_Article_1445_Equ32_HTML.gif)
We note that for each ,
is nonempty and closed since
by conditions (1) and (3). For each
,
is compact in
. Next, we claim that the mapping
is a KKM mapping. Indeed, if not, there is a nonempty finite subset
of
, such that
. Then there is an
such that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F945413/MediaObjects/13663_2010_Article_1445_Equ33_HTML.gif)
for all and
. Since the mapping
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F945413/MediaObjects/13663_2010_Article_1445_Equ34_HTML.gif)
is -convex on
, we can deduce that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F945413/MediaObjects/13663_2010_Article_1445_Equ35_HTML.gif)
for all . This contradicts condition (1). Therefore,
is a KKM mapping, and by Fan's lemma, we have
. Note that for any
, we have
by condition (2). Hence, we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F945413/MediaObjects/13663_2010_Article_1445_Equ36_HTML.gif)
for each nonempty finite subset of
. Therefore, the whole intersection
is nonempty. Let
. Then
is a solution of (SEP)
I
.
Corollary 2.7.
Let ,
,
,
,
,
,
be as in Theorem 2.1. Assume that the mapping
is
-convex on
for each
and
,
such that
is continuous with nonempty compact values for each
, and
is upper semicontinuous with nonempty compact values. Suppose that
(1)for each , there is an
such that
;
(2)there is a nonempty compact convex subset of
, such that for every
, there is a
such that for all
,
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F945413/MediaObjects/13663_2010_Article_1445_Equ37_HTML.gif)
Then there is an which is a weak solution of (SEP)I.
Proof.
Using the technique of the proof in Theorem 2.2 and applying Theorem 2.6, we have the conclusion.
The following result is another existence theorem for the strong solutions of (SEP). We need to combine Theorem 2.6 and use the technique of the proof in Theorem 2.3.
Theorem 2.8.
Under the framework of Theorem 2.6, on has a weak solution of (SEP)I with
. In addition, if
and
,
is compact,
is convex and the mapping
is naturally quasi
-convex on
for each
,
such that
is continuous with nonempty compact values for each
, and
is upper semicontinuous with nonempty compact values. Assuming that condition () holds, then
is a strong solution of (SEP)I; that is, there exists
such that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F945413/MediaObjects/13663_2010_Article_1445_Equ38_HTML.gif)
for all . Furthermore, the set of all strong solutions of (SEP)I is compact.
Using the technique of the proof in Theorem 2.3, we have the following result.
Corollary 2.9.
Under the framework of Corollary 2.7, one has a weak solution of (SEP)I with
. In addition, if
and
,
is compact,
is convex, and the mapping
is naturally quasi
-convex on
for each
. Assuming that condition () holds, then
is a strong solution of (SEP)I; that is, there exists
such that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F945413/MediaObjects/13663_2010_Article_1445_Equ39_HTML.gif)
for all . Furthermore, the set of all strong solutions of (SEP)I is compact.
Next, we discuss the existence results of the strong solutions for (SEP)
I
with the set without compactness setting from Theorems 2.10 to 2.14 below.
Theorem 2.10.
Letting be a finite-dimensional real Banach space, under the framework of Theorem 2.1, one has a weak solution
of (SEP)I with
. In addition, if
and
,
is convex,
for all
and for all
, the mapping
is
-convex on
for each
and
and the mapping
is naturally quasi
-convex on
for each
,
such that
is continuous for each
, and
is upper semicontinuous with nonempty compact values. Assume that for some
, such that for each
, there is a
such that the condition
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F945413/MediaObjects/13663_2010_Article_1445_Equ40_HTML.gif)
is satisfied, where . Then
is a strong solution of (SEP)I; that is, there exists
such that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F945413/MediaObjects/13663_2010_Article_1445_Equ41_HTML.gif)
for all . Furthermore, the set of all strong solutions of (SEP)I is compact.
Proof.
Let us choose such that condition () holds. Letting
, then the set
is nonempty and compact in
. We replace
by
in Theorem 2.3; all conditions of Theorem 2.3 hold. Hence by Theorem 2.3, we have
such that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F945413/MediaObjects/13663_2010_Article_1445_Equ42_HTML.gif)
for all . For any
, choose
small enough such that
. Putting
in (2.29), we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F945413/MediaObjects/13663_2010_Article_1445_Equ43_HTML.gif)
We note that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F945413/MediaObjects/13663_2010_Article_1445_Equ44_HTML.gif)
which implies that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F945413/MediaObjects/13663_2010_Article_1445_Equ45_HTML.gif)
for all . This completely proves the theorem.
Corollary 2.11.
Letting be a finite-dimensional real Banach space, under the framework of Theorem 2.2, one has a weak solution
of (SEP)I with
. In addition, if
and
,
is convex,
for all
and for all
, the mapping
is
-convex on
for each
and
, and the mapping
is naturally quasi
-convex on
for each
. Assume that for some
, condition () holds. Then
is a strong solution of (SEP)I; that is, there exists
such that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F945413/MediaObjects/13663_2010_Article_1445_Equ46_HTML.gif)
for all Furthermore, the set of all strong solutions of (SEP)I is compact.
Using a similar argument to that of the proof in Theorem 2.10 and combining Theorem 2.6 and Corollary 2.7, respectively, we have the following two results of existence for the strong solution of (SEP) I .
Theorem 2.12.
Let be a finite-dimensional real Banach space, under the framework of Theorem 2.6, one has a weak solution
of (SEP)I with
. In addition, if
and
,
is convex,
for all
and for all
, the mapping
is naturally quasi
-convex on
for each
,
such that
is continuous for each
, and
is upper semicontinuous with nonempty compact values. Assume that for some
, condition () holds. Then
is a strong solution of (SEP)I; that is, there exists
such that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F945413/MediaObjects/13663_2010_Article_1445_Equ47_HTML.gif)
for all . Furthermore, the set of all strong solutions of (SEP)I is compact.
In order to illustrate Theorems 2.10 and 2.12 more precisely, we provide the following concrete example.
Example 2.13.
Let ,
,
,
, and
. Choose
to be defined by
for every
and
is defined by
, where
,
, and
is defined by
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F945413/MediaObjects/13663_2010_Article_1445_Equ48_HTML.gif)
We claim that condition () holds. Indeed, We know that the weak solution . For each
, if we choose any
, then
and
. Hence condition () and all other conditions of Theorems 2.10 and 2.12 are satisfied. By Theorems 2.10 and 2.12, respectively, the (SEP)
I
not only has a weak solution, but also has a strong solution. We can see that
is a strong solution for (SEP)
I
.
Theorem 2.14.
Letting be a finite-dimensional real Banach space, under the framework of Corollary 2.7, one has a weak solution
of (SEP)I with
. In addition, if
and
,
is convex,
for all
and for all
, and the mapping
is naturally quasi
-convex on
for each
. Assume that for some
, condition () holds. Then
is a strong solution of (SEP)I; that is, there exists
such that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F945413/MediaObjects/13663_2010_Article_1445_Equ49_HTML.gif)
for all . Furthermore, the set of all strong solutions of (SEP)I is compact.
We would like to point out an open question naturally arising from Theorem 2.3: is Theorem 2.3 extendable to the case of or more general spaces, such as Hausdorff topological vector spaces?
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Acknowledgments
The authors would like to thank the referees whose remarks helped improving the paper. This work was partially supported by Grant no. 98-Edu-Project7-B-55 of Ministry of Education of Taiwan (Republic of China) and Grant no. NSC98-2115-M-039-001- of the National Science Council of Taiwan (Republic of China) that are gratefully acknowledged.
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Lin, YC., Chen, HJ. Solving the Set Equilibrium Problems. Fixed Point Theory Appl 2011, 945413 (2011). https://doi.org/10.1155/2011/945413
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DOI: https://doi.org/10.1155/2011/945413