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Existence Result of Generalized Vector Quasiequilibrium Problems in Locally 
-Convex Spaces
Fixed Point Theory and Applications volume 2011, Article number: 967515 (2011)
Abstract
This paper deals with the generalized strong vector quasiequilibrium problems without convexity in locally 
-convex spaces. Using the Kakutani-Fan-Glicksberg fixed point theorem for upper semicontinuous set-valued mapping with nonempty closed acyclic values, the existence theorems for them are established. Moreover, we also discuss the closedness of strong solution set for the generalized strong vector quasiequilibrium problems.
1. Introduction
Let 
be real topological vector space, and let 
be a nonempty closed convex subset of 
. Let 
be a bifunction, where 
is the set of real numbers. The equilibrium problem for 
is to find 
such that
Problem (1.1) was studied by Blum and Oettli [1]. The set of solution of (1.1) is denoted by 
. The equilibrium problem contains many important problems as special cases, including optimization, Nash equilibrium, complementarity, and fixed point problems (see [1–3] and the references therein). Recently, there has been an increasing interest in the study of vector equilibrium problems. Many results on the existence of solutions for vector variational inequalities and vector equilibrium problems have been established (see, e.g., [4–16]).
Let 
and 
be real topological vector spaces and 
a nonempty subset of 
. Let 
be a closed and convex cone in 
with 
, where 
denotes the topological interior of 
. For a bifunction 
, the vector equilibrium problem (for short, VEP) is to find 
such that
which is a unified model of several known problems, for instance, vector variational and variational-like inequality problems, vector complementarity problem, vector optimization problem, and vector saddle point problem; see, for example, [3, 8, 17, 18] and references therein. In 2003, Ansari and Yao [19] introduced vector quasiequilibrium problem (for short, VQEP) to find 
such that
where 
is a multivalued map with nonempty values.
Recently, Ansari et al. [4] considered a more general problem which contains VEP and generalized vector variational inequality problems as special cases. Let 
and 
be real locally convex Hausdorff space, 
a nonempty subset and 
a closed convex pointed cone. Let 
be a given set-valued mapping. Ansari et al. [4] introduced the following problems, to find 
such that
or to find 
such that
It is called generalized vector equilibrium problem (for short, GVEP), and it has been studied by many authors; see, for example, [20–22] and references therein. For other possible ways to generalize VEP, we refer to [23–25]. If 
is nonempty and 
satisfies (1.4), then we call 
a weak efficient solution for VEP, and if 
satisfies (1.5), then we call 
a strong solution for VEP. Moreover, they also proved an existence theorem for a strong vector equilibrium problem (1.5) (see [4]).
On the other hand, it is well known that a strong solution of vector equilibrium problem is an ideal solution; it is better than other solutions such as efficient solution, weak efficient solution, proper efficient solution, and supper efficient solution (see [12]). Thus, it is important to study the existence of strong solution and properties of the strong solution set. In 2003, Ansari and Flores-Bazán [26] considered the following generalized vector quasiequilibrium problem (for short, GVQEP): to find 
such that
Very recently, the generalized strong vector quasiequilibrium problem (in short, GSVQEP) is introduced by Hou et al. [27] and Long et al. [16]. Let 
, and 
be real locally convex Hausdorff topological vector spaces, 
and 
nonempty compact convex subsets, and 
a nonempty closed convex cone. Let 
, 
and 
be three set-valued mappings. They considered the GSVQEP, finding 
such that 
and
Moreover, they gave an existence theorem for a generalized strong vector quasiequilibrium problem without assuming that the dual of the ordering cone has a weak
compact base.
Throughout this paper, motivated and inspired by Hou et al. [27], Long et al. [16], and Yuan [28], we will introduce and study the generalized vector quasiequilibrium problem on locally 
-convex Hausdorff topological vector spaces. Let 
, 
, and 
be real locally 
-convex Hausdorff topological vector spaces, 
and 
nonempty compact subsets, and 
a nonempty closed convex cone. We also suppose that 
, 
and 
are set-valued mappings.
The generalized vector quasiequilibrium problem of type (I) (GSVQEP I) is to find 
such that
The generalized vector quasiequilibrium problem of type (II) (GSVQEP II) is to find 
such that
We denote the set of all solution to the (GSVQEP I) and (GSVQEP II) by 
and 
, respectively. The main motivation of this paper is to prove the existence theorems of the generalized strong vector quasiequilibrium problems in locally 
-convex spaces, by using Kakutani-Fan-Glicksberg fixed point theorem for upper semicontinuous set-valued mapping with nonempty closed acyclic values, and the closedness of 
and 
. The results in this paper generalize, extend, and unify some well-known some existence theorems in the literature.
2. Preliminaries
Let 
be the standard 
-dimensional simplex in 
with vertices 
. For any nonempty subset 
of 
, we denote 
by the convex hull of the vertices 
. The following definition was essentially given by Park and Kim [29].
Definition 2.1.
A generalised convex space, or say, a 
-convex space 
consists of a topological space 
, a nonempty subset 
of 
and a function 
such that
(i)for each 
if 
,
(ii)for each 
with 
, there exists a continuous function 
such that 
for each 
, where 
and 
denotes the face of 
corresponding to the subindex of 
in 
.
A subset 
of the 
-convex space 
is said to be 
-convex if for each 
for all 
. For the convenience of our discussion, we also denote 
by 
or 
if there is no confusion for 
, where 
is the set of all indices for the set 
; that is, 
. A space 
is said to have a 
-convex structure if and only if 
is a 
-convex space.
In order to cover general economic models without linear convex structures, Park and Kim [29] introduced another abstract convexity notion called a 
-convex space, which includes many abstract convexity notions such as 
-convex spaces as special cases. For the details on G-convex spaces, see [30–34], where basic theory was extensively developed.
Definition 2.2.
A 
-convex 
is said to be a locally 
-convex space if 
is a uniform topological space with uniformity 
, which has an open base 
of symmetric entourages such that for each 
, the set 
is a 
-convex set for each 
.
We recall that a nonempty space is said to be acyclic if all of its reduced ÄŒech homology groups over the rationals vanish.
Definition 2.3 (see [35]).
Let 
be a topological space. A subset 
of 
is called contractible at 
, if there is a continuous mapping 
such that 
for all 
and 
for all 
.
In particular, each contractible space is acyclic and thus any nonempty convex or star-shaped set is acyclic. Moreover, by the definition of contractible set, we see that each convex space is contractible.
Definition 2.4.
Let 
and 
be two topological vector spaces and 
a nonempty subset of 
, and let 
be a set-valued mapping.
(i)
is called upper 
-continuous at 
if, for any neighbourhood 
of the origin in 
, there is a neighbourhood 
of 
such that, for all 
,
(ii)
is called lower 
-continuous at 
if, for any neighbourhood 
of the origin in 
, there is a neighbourhood 
of 
such that for all 
,
Definition 2.5.
Let 
and 
be two topological vector spaces and 
a nonempty convex subset of 
. A set-valued mapping 
is said to be properly 
-quasiconvex if, for any 
and 
, we have
Definition 2.6.
Let 
and 
be two topological vector spaces and 
a set-valued mapping.
(i)
is said to be upper semicontinuous at 
if, for any open set 
containing 
, there exists an open set 
containing 
such that for all 
, 
; 
is said to be upper semicontinuous on 
if it is upper semicontinuous at all 
.
(ii)
is said to be lower semicontinuous at 
if, for any open set 
with 
, there exists an open set 
containing 
such that for all 
, 
; 
is said to be lower semicontinuous on 
if it is lower semicontinuous at all 
.
(iii)
is said to be continuous on 
if it is at the same time upper semicontinuous and lower semicontinuous on 
.
(iv)
is said to be closed if the graph, 
, of 
, that is, 
, is a closed set in 
.
Lemma 2.7 (see [36]).
Let 
and 
be two Hausdorff topological vector spaces and 
a set-valued mapping. Then, the following properties hold:
(i)if 
is closed and 
is compact, then 
is upper semicontinuous, where 
and 
denotes the closure of the set 
,
(ii)if 
is upper semicontinuous and for any 
is closed, then 
is closed,
(iii)
is lower semicontinuous at 
if and only if for any 
and any net 
, there exists a net 
such that 
and 
.
We now have the following fixed point theorem in locally 
-convex spaces given by Yuan [28] which is a generalization of the Fan-Glickberg-type fixed point theorems for upper semicontinuous set-valued mapping with nonempty closed acyclic values given in several places (e.g., see Kirk and Shin [37], Park and Kim [29], and others in locally convex spaces).
Lemma 2.8 (see [28]).
Let 
be a compact locally 
-convex space and 
an upper semicontinuous set-valued mappings with nonempty closed acyclic values. Then, 
has a fixed point; that is, there exists an 
such that 
.
3. Main Results
In this section, we apply the Kakutani-Fan-Glicksberg fixed point theorem for upper semicontinuous set-valued mapping with nonempty closed acyclic values to establish two existence theorems of strong solutions and obtain the closedness of the strong solutions set for generalized strong vector quasiequilibrium problem.
Theorem 3.1.
Let 
, 
, and 
be real locally 
-convex topological vector spaces, 
and 
nonempty compact subsets, and 
a nonempty closed convex cone. Let 
be a continuous set-valued mapping such that for any 
, the set 
is a nonempty closed contractible subset of 
. Let 
be an upper semicontinuous set-valued mapping with nonempty closed acyclic values and 
a set-valued mapping satisfy the following conditions:
(i)for all 
,
(ii)for all 
are properly 
-quasiconvex,
(iii)
are upper 
-continuous,
(iv)for all 
are lower 
-continuous.
Then, the solutions set 
is nonempty and closed subset of 
.
Proof.
For any 
, we define a set-valued mapping 
by
Since for any 
is nonempty. So, by assumption (i), we have that 
is nonempty. Next, we divide the proof into five steps.
Step 1 (to show that 
is acyclic).
Since every contractible set is acyclic, it is enough to show that 
is contractible. Let 
, thus 
and 
. Since 
is contractible, there exists a continuous mapping 
such that 
and 
. Now, we set 
for all 
. Then, 
is a continuous mapping, and we see that 
and 
for all 
. Let 
. We claim that 
. In fact, if 
, then there exists 
such that
Since 
is properly 
-quasiconvex, we can assume that
It follows that
which contradicts 
. Therefore, 
, and hence 
is contractible.
Step 2 (to show that 
is a closed subset of 
).
Let 
be a sequence in 
such that 
. Then, 
. Since 
is a closed subset of 
, 
. Since 
is a lower semicontinuous, it follows by Lemma 2.7(iii) that for any 
and any net 
, there exists a net 
such that 
and 
. This implies that
Since 
are lower 
-continuous, we note that for any neighbourhood 
of the origin in 
, there exists a subnet 
of 
such that
From (3.5) and (3.6), we have
We claim that 
. Assume that there exists 
and 
. Then, we note that 
, and the set 
is closed. Thus, 
is open, and 
. Since 
is a locally 
-convex space, there exists a neighbourhood 
of the origin such that 
and 
. Thus, we note that 
, and hence 
, which contradicts to (3.7). Hence, 
, and therefore, 
. Then, 
is a closed subset of 
.
Step 3 (to show that 
is upper semicontinuous).
Let 
be given such that 
, and let 
such that 
. Since 
and 
is upper semicontinuous, it follows by Lemma 2.7(ii) that 
. We claim that 
. Assume that 
. Then, there exists 
such that
which implies that there is a neighbourhood 
of the origin in 
such that
Since 
is upper 
-continuous, it follows that for any neighbourhood 
of the origin in 
, there exists a neighbourhood 
of 
such that
Without loss of generality, we can assume that 
. This implies that
Thus, there is 
such that
it is a contradiction to 
. Hence, 
, and therefore, 
is a closed mapping. Since 
is a compact set and 
is a closed subset of 
, 
is compact. This implies that 
is compact. Then, by Lemma 2.7(i), we have 
is upper semicontinuous.
Step 4 (to show that the solutions set 
is nonempty).
Define the set-valued mapping 
by
Then, 
is an upper semicontinuous mpping. Moreover, we note that 
is a nonempty closed acyclic subset of 
for all 
. By Lemma 2.8, there exists a point 
such that 
. Thus, we have 
, 
. It follows that there exists 
and 
such that 
and
Hence, the solutions set 
.
Step 5 (to show that the solutions set 
is closed).
Let 
be a net in 
such that 
. By definition of the solutions set 
, we note that 
, and there exist 
satisfying
Since 
is a continuous closed valued mapping, 
. From the compactness of 
, we can assume that 
. Since 
is an upper semicontinuous closed valued mapping, it follows by Lemma 2.7(ii) that 
is closed. Thus, we have 
. Since 
is a lower 
-continuous, we have
This means that 
belongs to 
. Therefore, the solutions set 
is closed. This completes the proof.
Theorem 3.1 extends Theorem 3.1 of Long et al. [16] to locally 
-convex which includes locally convex Hausdorff topological vector spaces.
Corollary 3.2.
Let 
, 
and 
be real locally convex Hausdorff topological vector spaces, 
and 
two nonempty compact convex subsets, and 
a nonempty closed convex cone. Let 
be a continuous set-valued mapping such that for any 
, 
is a nonempty closed convex subset of 
. Let 
be an upper semicontinuous set-valued mapping such that for any 
, 
is a nonempty closed convex subset of 
. Let 
be a set-valued mapping satisfying the following conditions:
(i)for all 
,
(ii)for all 
are properly 
-quasiconvex,
(iii)
are upper 
-continuous,
(iv)for all 
are lower 
-continuous.
Then, the solutions set 
is nonempty and closed subset of 
.
Theorem 3.3.
Let 
, 
and 
be real locally 
-convex topological vector spaces, 
and 
nonempty compact subsets, and 
a nonempty closed convex cone. Let 
be a continuous set-valued mapping such that for any 
, the set 
is a nonempty closed contractible subset of 
. Let 
be an upper semicontinuous set-valued mapping with nonempty closed acyclic values and 
a set-valued mapping satisfying the following conditions:
(i)for all 
,
(ii)for all 
are properly 
-quasiconvex,
(iii)
are upper 
-continuous,
(iv)for all 
are lower 
-continuous.
Then, the solutions set 
is nonempty and closed subset of 
.
Proof.
For any 
, define a set-valued mapping 
by
Proceeding as in the proof of Theorem 3.1, we need to prove that 
is closed acyclic subset of 
for all 
. We divide the remainder of the proof into three steps.
Step 1 (to show that 
is a closed subset of 
).
Let 
be a sequence in 
such that 
. Then, 
and 
. Since 
is a closed subset of 
, we have 
. By the lower semicontinuity of 
and Lemma 2.7(iii), we note that for any 
and any net 
, there exists a net 
such that 
and 
. Thus, we have
which implies that there exists a neighbourhood 
of the origin in 
such that
Since 
are lower 
-continuous, it follows that for any neighbourhood 
of the origin in 
, there exists a subnet 
of 
such that
Without loss of generality, we can assume that 
. Then, by (3.18), (3.19), and (3.20), we have
This means that 
and so 
is a closed subset of 
.
Step 2 (to show that 
is upper semicontinuous).
Let 
be given such that 
, and let 
such that 
. Then, 
and 
. Since 
is upper semicontinuous closed valued mapping, it follows by Lemma 2.7(ii) that 
. We claim that 
. Indeed, if 
, then there exists a 
such that
Since 
is upper 
-continuous, we note that for any neighbourhood 
of the origin in 
, there exists a neighbourhood 
of 
such that
From (3.22) and (3.23), we obtain
As in the proof of Step 2 in Theorem 3.1, we can show that 
for all 
. Hence, there is 
such that
it is a contradiction to 
. Hence, 
, and therefore, 
is a closed mapping. Since 
is a compact set and 
is a closed subset of 
, 
is compact. This implies that 
is compact. Then, by Lemma 2.7(i), we have that 
is upper semicontinuous.
Step 3 (to show that the solutions set 
is nonempty and closed).
Define the set-valued mapping 
by
Then, 
is an upper semicontinuous mapping. Moreover, we note that 
is a nonempty closed acyclic subset of 
for all 
. Hence, by Lemma 2.8, there exists a point 
such that 
. Thus, we have 
and 
. This implies that there exists 
and 
such that 
and
Hence, 
. Similarly, by the proof of Step 5 in Theorem 3.1, we have 
is closed. This completes the proof.
4. Stability
In this section, we discuss the stability of the solutions for the generalized strong vector quasiequilibrium problem (GSVQEP II).
Throughout this section, let 
, 
be Banach spaces, and let 
be a real locally 
-convex Hausdorff topological vector space. Let 
and 
be nonempty compact subsets, and let 
be a nonempty closed convex cone. Let 
is a continuous set-valued mapping with nonempty closed contractible values, and 
is an upper semicontinuous set-valued mapping with nonempty closed acyclic values}.
Let 
be compact sets in a normed space. Recall that the Hausdorff metric is defined by
where 
.
For 
, we define
where 
being the appropriate Hausdorff metrics. Obviously, 
is a metric space. Now, we assume that 
satisfies the assumptions of Theorem 3.3. Then, for each 
, (GSVQEP II) has a solution 
. Let
Thus, 
, which conclude that 
defines a set-valued mapping from 
into 
.
We also need the following lemma in the sequel.
Let 
be a metric space, and let 
be compact sets in 
. Suppose that for any open set 
, there exists 
such that 
for all 
. Then, any sequence 
satisfying 
has a convergent subsequence with limit in 
.
In the following theorem, we replaced the convex set by the contractible set and acyclic set in Theorem 4.1 in [16]. The following theorem can acquire the same result appearing on the Theorem 4.1 by utilized Lemma 4.1. Now, we need only to present stability theorem for the solution set mapping 
for (GSVQEP II).
Theorem 4.2.
is an upper semicontinuous mapping with compact values.
Proof.
Since 
is compact, we need only to show that 
is a closed mapping. In fact, let 
be such that 
. Since 
, we have 
, and there exists 
such that
By the same argument as in the proof of Theorem 4.1 in [16], we can show that 
and 
.
Since 
is lower semicontinuous at 
and 
, it follows by Lemma 2.7(iii) that for any 
, there exists 
such that 
. To finish the proof of the theorem, we need to show that 
for all 
. Since 
, it follows by the same argument as in the proof of Theorem 4.1 in [16] that there exists a subsequence 
of 
such that 
, 
, 
, and
From the upper 
-continuous of 
, we have
Then, 
, and so 
is closed. The theorem is proved.
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S. Plubtieng would like to thank the Thailand Research Fund for financial support under Grant no. BRG5280016. Moreover, K. Sitthithakerngkiet would like to thank the Office of the Higher Education Commission, Thailand, for supporting by grant fund under Grant no. CHE-Ph.D-SW-RG/41/2550, Thailand.
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Plubtieng, S., Sitthithakerngkiet, K. Existence Result of Generalized Vector Quasiequilibrium Problems in Locally 
-Convex Spaces.
Fixed Point Theory Appl 2011, 967515 (2011). https://doi.org/10.1155/2011/967515
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DOI: https://doi.org/10.1155/2011/967515