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On Generalized Implicit Vector Equilibrium Problems in Topological Ordered Spaces
Fixed Point Theory and Applications volume 2009, Article number: 513408 (2009)
Abstract
We discuss three classes of generalized implicit vector equilibrium problems in topological ordered spaces. Under some conditions, we prove three new existence theorems of solutions for the generalized implicit vector equilibrium problems in topological ordered spaces by using the Fan-Browder fixed point theorem.
1. Introduction and Preliminaries
It is well known that the vector equilibrium problem is closely related to vector variational inequality, vector optimization problem, and many others (see, e.g., [1–6] and the references therein).
Recently, a large of generalized vector equilibrium problems have been studied in different conditions by many authors and a lot of results concerned with the existence of solutions and properties of solutions have been given in finite and infinite dimensional spaces (see [7] and the references therein).
The main purpose of this paper is to extend some known results for vector equilibrium problems to topological ordered spaces (see [8]). We discuss three classes of generalized implicit vector equilibrium problems in topological ordered spaces. Under some conditions, we prove three new existence theorems of solutions for the generalized implicit vector equilibrium problems in topological ordered spaces by using the Fan-Browder fixed point theorem.
A semilattice is a partially ordered set , with the partial ordering denoted by
, for which any pair
of elements has a least upper bound, denoted by
. It is easy to see that any nonempty finite subset
of
has a least upper bound, denoted by sup 
. In the case
, the set
is called an order interval. Now assume that
is a semilattice and
is a nonempty finite subset. Thus, the set
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F513408/MediaObjects/13663_2009_Article_1149_Equ1_HTML.gif)
is well defined and it has the following properties:
(a),
(b)if , then
.
A subset is said to be
-convex if, for any nonempty finite subset
, we have
.
For any denotes the family of all finite subsets of
and
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F513408/MediaObjects/13663_2009_Article_1149_Equ2_HTML.gif)
Let be a topological semilattice,
a nonempty
-convex subset,
a Hausdorff topological vector space. Assume that
,
,
, and
such that, for any
,
is a closed, pointed, and convex cone in
and
.
In this paper, we consider the following three classes generalized implicit vector equilibrium problems:
weak generalized implicit vector equilibrium problem (WGIVEP): find
, such that
and for any
, there exists an
such that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F513408/MediaObjects/13663_2009_Article_1149_Equ3_HTML.gif)
strong generalized implicit vector equilibrium problem (SGIVEP): find
such that
and
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F513408/MediaObjects/13663_2009_Article_1149_Equ4_HTML.gif)
uniform generalized implicit vector equilibrium problem (UGIVEP): find
such that
and there exists
such that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F513408/MediaObjects/13663_2009_Article_1149_Equ5_HTML.gif)
Definition 1.1.
Let and
be two topological spaces.
A mapping
is called upper semicontinuous (usc) at
if, for any neighborhood
of
, there exists a neighborhood
of
such that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F513408/MediaObjects/13663_2009_Article_1149_Equ6_HTML.gif)
F is called usc on if it is usc at each point of
.
A mapping
is called lower semicontinuous (lsc) at
if, for any net
in
such that
and for any
, there exists
such that
.
is called lsc on
if it is lsc at each point of
.
A mapping
is called complement pseudo-upper semicontinuous (c p-usc) at
if, for
with
, we have
.
is called c p-usc on
if
is c p-usc at each point
of
.
Remark 1.2.
By [9], if
is usc with closed values, then for any net
in
such that
and for any net
in
with
such that
in
, we have
.
If
is usc with closed values, then
is c p-usc, where
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F513408/MediaObjects/13663_2009_Article_1149_Equ7_HTML.gif)
Lemma 1.3 (see [9]).
Let and
be two topological spaces. Let
be compact and
be usc such that
is compact for each
. Then
is compact.
Definition 1.4.
Let be a topological semilattice or a
-convex subset of a topological semilattice, let
be a Hausdorff topological vector space, and let
be a closed, pointed, and convex cone with
.
A mapping
is called a
-convex mapping (or a
-concave mapping) with respect to
if, for any nonempty finite subset
,
,
,
with
and
, there exists
such that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F513408/MediaObjects/13663_2009_Article_1149_Equ8_HTML.gif)
A mapping
is called to have
-inheritance if, for any nonempty finite subset
of
,
,
with
and
, we have
.
Let
. A mapping
is called
-
-convex (or
-
-concave) with respect to
in second argument if, for any nonempty finite subset
,
,
,
,
with
and
, there exists
such that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F513408/MediaObjects/13663_2009_Article_1149_Equ9_HTML.gif)
Remark 1.5.
If is a
-
-convex mapping (or a
-
-concave mapping) with respect to
in second argument, then for any
,
is a
-convex mapping (or a
-concave mapping) with respect to
.
Lemma 1.6 (see [10]).
Let be a nonempty compact
-convex subset of a topological semilattice with path-connected intervals
, let
be a mapping with nonempty
-convex values such that, for each
,
is an open set in
. Then
has a fixed point.
2. Existence Theorems
Theorem 2.1.
Let be a nonempty compact
-convex subset of a topological semilattice with path-connected intervals
, let
be a Hausdorff topological vector space. Let
be a mapping with nonempty
-convex values, and let
and
be mappings and
be a mapping such that, for each
,
is a closed, pointed, and convex cone in
with
. Assume that
(1)For any ,
is open;
(2) is closed;
(3) is usc with compact values;
(4) for any
and
;
(5)for any and
,
is
-concave with respect to
;
(6)for any ,
is lsc;
(7) is usc, where
for each
.
Then there exists an such that
and for any
, there exists an
such that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F513408/MediaObjects/13663_2009_Article_1149_Equ10_HTML.gif)
Furthermore, the solution set of (WGIVEP) is closed, and hence is compact.
Proof.
Define by
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F513408/MediaObjects/13663_2009_Article_1149_Equ11_HTML.gif)
We first prove that for any ,
is open, that is,
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F513408/MediaObjects/13663_2009_Article_1149_Equ12_HTML.gif)
is closed. Let a net and
. Then there exists
such that
,
, for any
. By
and Lemma 1.3, we know that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F513408/MediaObjects/13663_2009_Article_1149_Equ13_HTML.gif)
is compact and so has a cluster point
. We may assume that
and thus,
. For any
, by
, there exists
such that
and so
. It follows from
that
and hence
. Thus,
is closed and so
is open.
Suppose that there exists an such that
is not
-convex, that is, there exist
such that
. Hence, there exists
,
, that is, there exists
such that
. For each
,
, take
,
. Let
,
. For any
,
and
, we have
. By
, there exists
such that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F513408/MediaObjects/13663_2009_Article_1149_Equ14_HTML.gif)
Since , we know that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F513408/MediaObjects/13663_2009_Article_1149_Equ15_HTML.gif)
which is a contradiction. Therefore, for any ,
is
-convex.
By and Lemma 1.6,
. Define
by
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F513408/MediaObjects/13663_2009_Article_1149_Equ16_HTML.gif)
Then is
-convex for each
. It follows from
and
that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F513408/MediaObjects/13663_2009_Article_1149_Equ17_HTML.gif)
is open.
Suppose that for all ,
is nonempty. Then, by Lemma 1.6  
has a fixed point, that is, there exists
, such that
. If
, then
, hence
, for all
,
which contradicts to assumption
; If
, then
, hence
which contradicts with
. Therefore, there exists
, such that
. Since
is nonempty for any
, then
,
, that is,
and for any
,
. Therefore,
and for any
, there exists an
such that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F513408/MediaObjects/13663_2009_Article_1149_Equ18_HTML.gif)
Let denote the solution set of (WGIVEP) and
with
. We show that
, that is,
, and for all
, there exists
such that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F513408/MediaObjects/13663_2009_Article_1149_Equ19_HTML.gif)
In fact, it follows from that
. For any
,
. By
, there exists an open neighborhood
of
such that
. Since
, there exists
such that for any
,
. Thus,
and so there exists
such that
, that is,
. Since
is compact,
has a cluster point
. We may assume that
. From
, we have
. By
, for any
, there exists
such that
. It follows from
that
, that is,
. Thus,
is closed, and hence is compact. This completes the proof.
Theorem 2.2.
Let be a nonempty compact
-convex subset of a topological semilattice with path-connected intervals
, let
be a Hausdorff topological vector space. Let
be with nonempty
-convex values, and let
and
be mappings and
be a mapping such that, for each
,
is a closed, pointed, and convex cone in
with
. Assume that
(1)For any ,
is open;
(2) is closed;
(3) is lsc;
(4)for all and
,
;
(5)for all ,
is
-concave with respect to
in second argument;
(6)for all ,
is lsc;
(7) is usc, where
for all
.
Then there exists an such that
and
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F513408/MediaObjects/13663_2009_Article_1149_Equ20_HTML.gif)
Furthermore, the solution set of (SGIVEP) is closed, and hence is compact.
Proof.
Define by
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F513408/MediaObjects/13663_2009_Article_1149_Equ21_HTML.gif)
Then the proof is similar to that of Theorem 2.1 and so we omit it.
Theorem 2.3.
Let be a nonempty compact
-convex subset of a topological semilattice with path-connected intervals
and let
be a Hausdorff topological vector space. Let
be with nonempty
-convex values, let
and
be mappings, and let
be a mapping such that, for each
,
is a closed, pointed, and convex cone in
with
. Assume that
(1) is usc with compact values;
(2)For any ,
is nonempty
-convex;
(3) is c p-usc on
;
(4) is usc with nonempty compact values;
(5)for all ,
;
-
(6)
for all
,
is
-concave with respect to
;
(7) has
-inheritance;
(8) is c p-usc on
.
Then there exists an such that
and there exists
such that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F513408/MediaObjects/13663_2009_Article_1149_Equ22_HTML.gif)
Proof.
Define by
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F513408/MediaObjects/13663_2009_Article_1149_Equ23_HTML.gif)
where
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F513408/MediaObjects/13663_2009_Article_1149_Equ24_HTML.gif)
The proof is divided into the following five steps.
-
(I)
For any
,
is nonempty.
If it is false, then there exists such that
, that is, for any
, there exists
such that
. Let
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F513408/MediaObjects/13663_2009_Article_1149_Equ25_HTML.gif)
Then is nonempty values. If there exists
such that
is not
-convex, then there exist
such that
, that is, there exists
with
. Thus,
. For each
,
, take
. For any
,
and
, we have
. By
, there exists
such that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F513408/MediaObjects/13663_2009_Article_1149_Equ26_HTML.gif)
Since ,
. Hence,
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F513408/MediaObjects/13663_2009_Article_1149_Equ27_HTML.gif)
which is a contradiction. Thus, for any ,
is nonempty
-convex.
For any ,
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F513408/MediaObjects/13663_2009_Article_1149_Equ28_HTML.gif)
It follows from that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F513408/MediaObjects/13663_2009_Article_1149_Equ29_HTML.gif)
is closed and so is open. Since
is nonempty compact and
-convex, by Lemma 1.6  
has a fixed point. Thus, there exists
such that
, that is,
which contradicts with Assumption
. Hence
for any
.
-
(II)
For any
,
is
-convex. If it is false, then there exists
such that
is not
-convex, that is, there exist
such that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F513408/MediaObjects/13663_2009_Article_1149_Equ30_HTML.gif)
Thus, there exists such that
. Then
and for all
,
,
. Since
is
-convex,
. By
,
. Since
, there exists
such that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F513408/MediaObjects/13663_2009_Article_1149_Equ31_HTML.gif)
which is a contradiction. Therefore, for any ,
is
-convex.
-
(III)
is nonempty
-convex for any
. By steps (I) and (II), the conclusion follows directly from
.
-
(IV)
For any
,
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F513408/MediaObjects/13663_2009_Article_1149_Equ32_HTML.gif)
is open. In fact, we only need to show that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F513408/MediaObjects/13663_2009_Article_1149_Equ33_HTML.gif)
is closed. Let a net and
. If
, then
and hence
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F513408/MediaObjects/13663_2009_Article_1149_Equ34_HTML.gif)
If and there exists
such that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F513408/MediaObjects/13663_2009_Article_1149_Equ35_HTML.gif)
Take . By
and Lemma 1.3,
is compact and hence
has a cluster point
. We may assume that
and so
. Similarly, by
,
has a cluster point
. We assume that
and hence
. Since
closed,
. Thus,
,
and
. Hence,
is closed. Let a net
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F513408/MediaObjects/13663_2009_Article_1149_Equ36_HTML.gif)
and , then
. By
, we have
, and hence
is closed. Thus,
is closed and so
is open.
-
(V)
The UGIVEP has a solution. By Lemma 1.6,
has a fixed point. Thus, there exists
such that
, that is,
and
such that
and
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F513408/MediaObjects/13663_2009_Article_1149_Equ37_HTML.gif)
This completes the proof.
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Acknowledgments
This research was supported by the Natural Science Foundations of Guangdong Province (9251064101000015). The author is grateful to the referees for the valuable comments and suggestions.
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Luo, Q. On Generalized Implicit Vector Equilibrium Problems in Topological Ordered Spaces. Fixed Point Theory Appl 2009, 513408 (2009). https://doi.org/10.1155/2009/513408
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DOI: https://doi.org/10.1155/2009/513408