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Existence of Solutions to the System of Generalized Implicit Vector Quasivariational Inequality Problems
Journal of Inequalities and Applications volume 2009, Article number: 654370 (2009)
Abstract
We study the system of generalized implicit vector quasivariational inequality problems and prove a new existence result of its solutions by Kakutani-Fan-Glicksberg's fixed points theorem. As a special case, we also derive a new existence result of solutions to the generalized implicit vector quasivariational inequality problems.
1. Introduction
The system of generalized implicit vector quasivariational inequality problems generalizes the generalized implicit vector quasivariational inequality problems, and the latter had been studied in [1–3]. In this paper, we study the system of generalized implicit vector quasivariational inequality problems and prove a new existence result of its solutions by Kakutani-Fan-Glicksberg's fixed points theorem. For other existence results with respect to the system of generalized implicit vector quasivariational inequality problems, we refer the reader to [4–6] and references therein.
Let be an index set (finite or infinite). For each
, let
and
be two Hausdorff topological vector spaces,
a nonempty subset of
, and
a closed, convex and pointed cone of
with
, where
denotes the interior of
. Denote that
,
. For each
, we can write
. For each
, let
be a nonempty subset of the continuous linear operators space
from
into
and let
,
,
be three set-valued maps, where
and
denote the family of all nonempty subsets of
and
, respectively. The system of generalized implicit vector quasivariational inequality problems (briefly, SGIVQIP) is as follows: find
such that for each
and
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F654370/MediaObjects/13660_2009_Article_1987_Equ1_HTML.gif)
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F654370/MediaObjects/13660_2009_Article_1987_IEq30_HTML.gif)
is said to be a solution of the SGIVQIP. An SGIVQIP is usually denoted by .
If is a singleton, then the SGIVQIP coincides with the generalized implicit vector quasivariational inequality problems (briefly, GIVQIP). A GIVQIP is usually denoted by
.
Throughout this paper, unless otherwise specified, assume that for each ,
is a nonempty convex compact subset of a Banach space
is a Hausdorff topological vector space, and
is a closed, convex, and pointed cone of
with
, where
denotes the interior of
.
2. Preliminaries
In this section, we introduce some useful notations and results.
Definition 2.1.
Let and
be two topological spaces and
a nonempty convex subset of
.
is a set-valued map.
(1) is called upper semicontinuous at
if, for any open set
, there exists an open neighborhood
of
in
such that for all
,
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F654370/MediaObjects/13660_2009_Article_1987_Equ2_HTML.gif)
and upper semicontinuous on if it is upper semicontinuous at every point of
(2) is called lower semicontinuous at
if, for any open set
, there exists an open neighborhood
of
in
such that for all
,
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F654370/MediaObjects/13660_2009_Article_1987_Equ3_HTML.gif)
and lower semicontinuous on if it is lower semicontinuous at every point of
(3) is called continuous at
if, it is both upper semicontinuous and lower semicontinuous at
; and continuous on
if it is continuous at every point of
.
Definition 2.2.
Let and
be two topological vector spaces and
a nonempty convex subset of
.Also
is a set-valued map.
(1) is called upper
semicontinuous at
if, for any open neighborhood
of the zero element
in
, there exists an open neighborhood
of
in
such that, for all
,
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F654370/MediaObjects/13660_2009_Article_1987_Equ4_HTML.gif)
and upper semicontinuous on
if it is upper
semicontinuous at every point of
.
(2) is called lower
semicontinuous at
if, for any open neighborhood V of the zero element
in
, there exists an open neighborhood
of
in
such that, for all
,
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F654370/MediaObjects/13660_2009_Article_1987_Equ5_HTML.gif)
and lower semicontinuous on
if it is lower
semicontinuous at every point of
.
(3) is called
continuous at
if it is upper
semicontinuous and lower
semicontinuous at
; and
continuous on
if it is
continuous at every point of
.
Definition 2.3.
Let and
be two topological vector spaces and
a nonempty convex subset of
. Let
be a set-valued map.
(1) is called
convex if, for each
,
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F654370/MediaObjects/13660_2009_Article_1987_Equ6_HTML.gif)
and concave if
is
convex.
(2) is called
quasiconvex-like if, for each
,
,
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F654370/MediaObjects/13660_2009_Article_1987_Equ7_HTML.gif)
and quasiconcave-like if
is
quasiconvex-like.
Lemma 2.4 ([7, Theorem 1]).
Let be a nonempty paracompact subset of a Hausdorff topological space
and,
be a nonempty subset of a Hausdorff topological vector space
. Suppose that
be two set-valued maps with following conditions:
(1)for each ,
;
-
(2)
for each
,
is open.
Then has a continuous selection, that is, there is a continuous map
such that
for each
.
3. Existence of Solutions to the SGIVQIP
Lemma 3.1.
Let be three Hausdorff topological spaces,
a topological vector space, and
a closed, convex, and pointed cone of
. Let
and
be two set-valued maps. Assume that
and
(1) is upper semicontinuous on
with nonempty and compact values;
(2) is upper
semicontinuous on
with nonempty and compact values;
-
(3)
for each
.
Then there exist open neighborhood of
and open neighborhood
of
, and open neighborhood
of
such that
whenever
,
,
Proof.
By (3) and compactness of , there exists an open neighborhood
of the zero element
of
such that
. By (2), there exist open neighborhood
of
and open neighborhood
of
, open neighborhood
of
such that
whenever
. Since
is compact and
, there exist finite
such that
. Taking
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F654370/MediaObjects/13660_2009_Article_1987_Equ8_HTML.gif)
Clearly, and
are open neighborhood of
and
, respectively. Thus for each
, we have
whenever
,
By (1), there exist open neighborhood
of
with
and open neighborhood
of
such that
whenever
, which implies that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F654370/MediaObjects/13660_2009_Article_1987_Equ9_HTML.gif)
whenever ,
,
The proof is finished.
By Lemma 3.1, we obtain the following result.
Theorem 3.2.
Consider an SGIVQIP . For each
, assume that
(1) is continuous on
with convex compact values and for each
;
(2) is upper semicontinuous on
with nonempty and compact values;
(3) is upper
semicontinuous on
with nonempty and compact values;
-
(4)
for each
and each
,
is
convex or
quasiconvex-like;
-
(5)
for each
and each
, if
, then
, where
is the
component of
.
Then the SGIVQIP has a solution, that is, there exists such that for each
and
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F654370/MediaObjects/13660_2009_Article_1987_Equ10_HTML.gif)
Proof.
For each , define a set-valued map
by
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F654370/MediaObjects/13660_2009_Article_1987_Equ11_HTML.gif)
Step 1.
We prove that the set is closed. For any sequence
with
, we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F654370/MediaObjects/13660_2009_Article_1987_Equ12_HTML.gif)
If , then there exists
such that for each
. By Lemma 3.1, there exist open neighborhood
of
and open neighborhood
of
, such that
whenever
. By (1), there exist
such that
, which implies that there exists a positive integer
such that
whenever
. Thus we have
whenever
, a contradiction. This shows that
is closed,that is,
is open.
Without loss of generality, assume that .
Define a set-valued map by
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F654370/MediaObjects/13660_2009_Article_1987_Equ13_HTML.gif)
Step 2.
We prove that for each is nonempty and convex.
For each , we have
. By Lemma 3.1, there exists an open neighborhood
of
such that
whenever
, which implies that
, that is,
is open. By (4), it is easy to verify that
is convex.
Since is convex and
, then for each
is nonempty and convex.
Step 3.
We prove that has a continuous selection
.
For each , we have
and
. By
, there exists
such that
, where
. Since
is continuous with convex compact values, then there exists an open neighborhood
of
such that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F654370/MediaObjects/13660_2009_Article_1987_Equ14_HTML.gif)
whenever , where
. Thus
whenever
, which implies that
whenever
, that is,
whenever
. This shows that the set
is open. By
, we have
. By Lemma 3.1, there exists an open neighborhood
of
such that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F654370/MediaObjects/13660_2009_Article_1987_Equ15_HTML.gif)
whenever , which implies that
, that is,
is open. Hence, for each
, the set
is open.
By Lemma 2.4, has a continuous selection
.
Step 4.
We prove that the SGIVQIP has a solution.
For each , define the set-valued map
by
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F654370/MediaObjects/13660_2009_Article_1987_Equ16_HTML.gif)
Note that is upper semicontinuous when
and
is upper semicontinuous when
, and it is easy to verify that
is also upper semicontinuous when
, where
denotes the boundary of
. Thus,
is upper semicontinuous with nonempty convex compact values. By [8, Theorem  7.1.15], the set-valued map
defined by
is closed with nonempty convex values. By Kakutani-Fan-Glicksberg's fixed points theorem (see [9, pages 550]),
has a fixed point, that is, there exists
. The condition (5) implies that for each
,
, that is,
for each
. Thus we have that for each
,
and
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F654370/MediaObjects/13660_2009_Article_1987_Equ17_HTML.gif)
The proof is finished.
If is a singleton, we obtain the following existence result of solutions to the GIVQIP by Theorem 3.2.
Corollary 3.3.
Consider a GIVQIP . Assume that
(1) is continuous on
with convex compact values and for each
;
(2) is upper semicontinuous on
with nonempty and compact values;
(3) is upper
semicontinuous on
with nonempty and compact values;
-
(4)
for each
and each
,
is
convex or
quasiconvex-like;
-
(5)
for each
and each
, if
, then
.
Then the GIVQIP has a solution, that is, there exists such that
,
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F654370/MediaObjects/13660_2009_Article_1987_Equ18_HTML.gif)
Remark 3.4.
Theorem 3.2, Corollary 3.3, and each corresponding result in literatures [1–6] do not include each other as a special case.
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Acknowledgments
The research was supported by the Natural Science Foundation of CQ CSTC.
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Lin, Z. Existence of Solutions to the System of Generalized Implicit Vector Quasivariational Inequality Problems. J Inequal Appl 2009, 654370 (2009). https://doi.org/10.1155/2009/654370
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DOI: https://doi.org/10.1155/2009/654370