- Research Article
- Open access
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Existence Results for System of Variational Inequality Problems with Semimonotone Operators
Journal of Inequalities and Applications volume 2010, Article number: 251510 (2010)
Abstract
We introduce the system of variational inequality problems for semimonotone operators in reflexive Banach space. Using the Kakutani-Fan-Glicksberg fixed point theorem, we obtain some existence results for system of variational inequality problems for semimonotone with finite-dimensional continuous operators in real reflexive Banach spaces. The results presented in this paper extend and improve the corresponding results for variational inequality problems studied in recent years.
1. Introduction
Let be a Banach space, let
be the dual space of
, and let
denote the duality pairing of
and
. If
is a Hilbert space and
is a nonempty, closed, and convex subset of
, then let,
be a nonempty, closed, and convex subset of a Hilbert space
and let
be a mapping. The classical variational inequality problem, denoted by
, is to find
such that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F251510/MediaObjects/13660_2010_Article_2096_Equ1_HTML.gif)
for all .
The variational inequality problem (VIP) has been recognized as suitable mathematical models for dealing with many problems arising in different fields, such as optimization theory, game theory, economic equilibrium, mechanics. In the last four decades, since the time of the celebrated Hartman Stampacchia theorem (see [1, 2]), solution existence of variational inequality and other related problems has become a basic research topic which continues to attract attention of researchers in applied mathematics (see, e.g., [3–14] and the references therein). Related to the variational inequalities, we have the problem of finding the fixed points of the nonexpansive mappings, which is the current interest in functional analysis. It is natural to consider a unified approach to these different problems; see, for example, [10, 11].
Let be a nonempty, closed, and convex subset of
and let
be single valued. For the system of generalized variational inequality problem (SGVIP), find
such that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F251510/MediaObjects/13660_2010_Article_2096_Equ2_HTML.gif)
There are many kinds of mappings in the literature of recent years; see, for example, [12, 13, 15–18]. In 1999, Chen [19] introduced the concept of semimonotonicity for a single valued mapping, which occurred in the study of nonlinear partial differential equations of divergence type. Recently, Fang and Huang [20] introduced two classes of variational-like inequalities with generalized monotone mappings in Banach spaces. Using the KKM technique, they obtained the existence of solutions for variational-like inequalities with relaxed monotone mappings in reflexive Banach spaces. Moreover, they present the solvability of variational-like inequalities with relaxed semimonotone mappings in arbitrary Banach spaces by means of the Kakutani-Fan-Glicksberg fixed point theorem.
On the other hand, some interesting and important problems related to variational inequalities and complementarity problems were considered in recent papers. In 2004, Cho et al. [21], introduced, and studied a system of nonlinear variational inequalities. They proved the existence and uniqueness of solution for this problem and constructed an iterative algorithm for approximating the solution of system of nonlinear variational inequalities. In 2000, [22], systems of variational inequalities were introduced and an existence theorem was obtained by Ky Fan lemma. In 2002, Kassay et al. [23] introduced and studied Minty and Stampacchia variational inequality systems by the Kakutani-Fan-Glicksberg fixed point theorem. Very recently, Fang and Huang [20] introduced and studied systems of strong implicit vector variational inequalities by the same fixed point theorem. Zhao and Xia [24] introduced and established some existence results for systems of vector variational-like inequalities in Banach spaces by also using the Kakutani-Fan-Glicksberg fixed point theorem.
Let be a semimonotone mapping and
a closed convex set. The variational inequality problem (VIP) is to find
such that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F251510/MediaObjects/13660_2010_Article_2096_Equ3_HTML.gif)
Let be a nonempty closed convex subset of a real reflexive Banach space
with dual space
, and let
be two semimonotone mappings. We consider the following as the system of variational inequality problem (SVIP). Find
such that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F251510/MediaObjects/13660_2010_Article_2096_Equ4_HTML.gif)
In particular, if setting by
for all
, then the system of variational inequality problem reduces to the variational inequality problem (VIP).
In this paper, we introduce the system of generalized variational inequality in real reflexive Banach space. By using the Kakutani-Fan-Glicksberg fixed point theorem, we obtain some existence results for system of generalized variational inequality for semimonotone and finite dimensional continuous in real reflexive Banach spaces. The results presented in this paper extend and improve the corresponding results of Chen [19] and many others.
2. Preliminaries
In this section, let be a real Banach space, and let
be the unit sphere of
. A Banach space
is said to be strictly convex if, for any
,
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F251510/MediaObjects/13660_2010_Article_2096_Equ5_HTML.gif)
It is also said to be uniformly convex if, for each , there exists
such that, for any
,
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F251510/MediaObjects/13660_2010_Article_2096_Equ6_HTML.gif)
It is known that a uniformly convex Banach space is reflexive and strictly convex, and we define a function called the modulus of convexity of
as follows:
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F251510/MediaObjects/13660_2010_Article_2096_Equ7_HTML.gif)
Then is uniformly convex if and only if
for all
. A Banach space
is said to be smooth if the limit
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F251510/MediaObjects/13660_2010_Article_2096_Equ8_HTML.gif)
exists for all . It is also said to be uniformly smooth if the limit (2.4) is attained uniformly for
. We recall that
is uniformly convex if and only if
is uniformly smooth. It is well known that
is smooth if and only if
is strictly convex.
Definition 2.1 (KKM mapping).
Let be a nonempty subset of a linear space
. A set-valued mapping
is said to be a KKM mapping if, for any finite subset
of
, we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F251510/MediaObjects/13660_2010_Article_2096_Equ9_HTML.gif)
where denotes the convex hull of
.
Lemma 2.2 (Fan-KKM Theorem).
Let be a nonempty convex subset of a Hausdorff topological vector space
, and let
be a KKM mapping with closed values. If there exists a point
such that
is a compact subset of
, then
.
Definition 2.3.
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 either
or
.
Definition 2.4.
Let and
be two topological vector spaces, and
be 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 2.5 (see [25]).
Let and
be two Hausdorff topological vector spaces, and let
be 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
.
Lemma 2.6 (see Kakutani-Fan-Glicksberg [26]).
Let be a nonempty compact subset of locally convex Hausdorff vector topology space
. If
is upper semicontinuous and, for any
,
is nonempty convex and closed, then there exists an
such that
.
Lemma 2.7 (see [27]).
For each ,
is a Fréchet differentiable convex function on
, and the Fréchet derivative
of
is equal to Yosida approximation
of
. More precisely,
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F251510/MediaObjects/13660_2010_Article_2096_Equ10_HTML.gif)
holds for and
.
Let be a nonempty closed convex subset of a real reflexive Banach space
with dual space
. A mapping
is said to be monotone if it satisfies
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F251510/MediaObjects/13660_2010_Article_2096_Equ11_HTML.gif)
Definition 2.8 (see [19]).
A mapping is said to be semimonotone if it satisfies the following:
(a)for each ,
is monotone; that is,
, for all
;
(b)For each fixed ,
is completely continuous; that is, if
in weak topology of
, then
has a subsequence
in norm topology of
.
An operator is said to be hemicontinuous at
, if, for any
,
with
, we have
for all
at
.
Lemma 2.9.
Let be a hemicontinuous monotone operator, let
be a convex subset, and let
be a convex function and
a given point. Then
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F251510/MediaObjects/13660_2010_Article_2096_Equ12_HTML.gif)
if and only if
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F251510/MediaObjects/13660_2010_Article_2096_Equ13_HTML.gif)
Proof.
Let such that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F251510/MediaObjects/13660_2010_Article_2096_Equ14_HTML.gif)
By the monotonicity of , we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F251510/MediaObjects/13660_2010_Article_2096_Equ15_HTML.gif)
On the other hand, suppose that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F251510/MediaObjects/13660_2010_Article_2096_Equ16_HTML.gif)
For any given and any
, taking
since
is convex and Replacing
by
into the above inequality, one has
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F251510/MediaObjects/13660_2010_Article_2096_Equ17_HTML.gif)
It follows from the convexity of on
that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F251510/MediaObjects/13660_2010_Article_2096_Equ18_HTML.gif)
Letting and using the hemicontinuity of
, we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F251510/MediaObjects/13660_2010_Article_2096_Equ19_HTML.gif)
This completes the proof.
3. The Existence of the System of Generalized Variational Inequality
In this section, we prove two existence theorems for system of variational inequality problems for semimonotone with finite dimensional continuous operators in real reflexive Banach spaces. First, we prove an existence theorem for system of variational inequality problems for continuous mappings as follows.
Theorem 3.1.
Let be a reflexive Banach spac, let
be a compact convex subset of
, and let
be two continuous mappings. Then the problem (1.2) has a solution and the set of solutions of (1.2) is closed.
Proof.
Fix , for each
, the sets
and
are defined as follows:
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F251510/MediaObjects/13660_2010_Article_2096_Equ20_HTML.gif)
Step 1 (Show that and
are nonempty compact convex subsets of
).
For any , we note that
and
. Thus,
and
are nonempty subsets of
. Moreover, it follows from the definitions of
and
that both of them are compact convex subsets of
.
Step 2 (Show that and
are KKM mappings).
For any finite set , we claim that
and
. Let
. Then
, where
and
. We observe that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F251510/MediaObjects/13660_2010_Article_2096_Equ21_HTML.gif)
So, there is at least one number such that
. Therefore,
. Similarly, we obtain that
. Hence, we have
and
. This implies that
and
are KKM mappings.
Step 3 (Show that and
are closed for all
).
Let be a sequence in
such that
. Then
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F251510/MediaObjects/13660_2010_Article_2096_Equ22_HTML.gif)
Since and
are continuous, we have
for all
. Thus, we see that
. This implies that
is closed for all
. Similarly, we note that
is closed for all
.
Step 4 (Show that ).
Since and
are closed subsets of
and
is compact, it follows that
and
are compact subsets of
. It follows from Lemma 2.2 that
. Moreover, we note that
and
are closed and convex.
Step 5 (Show that the problem (1.2) has a solution).
Define the set-valued mapping by
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F251510/MediaObjects/13660_2010_Article_2096_Equ23_HTML.gif)
From Step 4, we note that is nonempty closed convex subset of
for all
. Since
, and
is compact,
and
are compact. It follows from Lemma 2.5(i) that
is upper semicontinuous. Hence, by the Kakutani-Fan-Glicksberg theorem, there exists a point
; that is,
and
for all
. By definition of
and
, we get
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F251510/MediaObjects/13660_2010_Article_2096_Equ24_HTML.gif)
Hence, are the solutions of problem (1.2).
Step 6 (Show that the set of solutions of problem (1.2) is closed).
Let be a net in the set of solutions of problem (1.2) such that
. By definition of the set of solutions of problem (1.2) we obtain that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F251510/MediaObjects/13660_2010_Article_2096_Equ25_HTML.gif)
Since are continuous and
is compact, it follows that
and
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F251510/MediaObjects/13660_2010_Article_2096_Equ26_HTML.gif)
This mean that belongs to the set of solution of problem (1.2). Hence, the set of solution of problem (1.2) is closed set. This completes the proof.
Theorem 3.2.
Let be a nonempty bounded closed convex subset of a real reflexive Banach space
with dual space
. Suppose that
is a lower semicontinuous convex function with
. Let
and
be two mappings satisfying the following conditions:
(i)for each ,
and
are monotone;
(ii)for each ,
and
are completely continuous;
(iii)for any given ,
and
are finite dimensional continuous; that is, for any finite dimensional subspace
,
and
are continuous.
Then, there exists such that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F251510/MediaObjects/13660_2010_Article_2096_Equ27_HTML.gif)
Proof.
Let be the subdifferential of
and
the Yosida approximation of
. Let
be a finite dimensional subspace of
with
. For any given
, we consider the following systems of variational inequalities
. Find
such that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F251510/MediaObjects/13660_2010_Article_2096_Equ28_HTML.gif)
Since is nonempty bounded closed convex and
and
are continuous on
for each fixed
, it follows from Theorem 3.1 that
has a solution
. By Lemma 2.9 and the Yosida approximation of
, we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F251510/MediaObjects/13660_2010_Article_2096_Equ29_HTML.gif)
Now, we defined a mapping by the following:
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F251510/MediaObjects/13660_2010_Article_2096_Equ30_HTML.gif)
Next, we will show that this mapping has at least one fixed point in . To prove this, we need the following conditions.
(1)For all ,
as
has a solution.
(2) is convex for all
.
Let and
. Thus, we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F251510/MediaObjects/13660_2010_Article_2096_Equ31_HTML.gif)
Adding (3.12), we get
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F251510/MediaObjects/13660_2010_Article_2096_Equ32_HTML.gif)
It implies that is convex.
(3) is closed for all
.
In fact, leting such that
, we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F251510/MediaObjects/13660_2010_Article_2096_Equ33_HTML.gif)
Since , and
are continuous and
is lower semicontinuous, we get
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F251510/MediaObjects/13660_2010_Article_2096_Equ34_HTML.gif)
Thus implies that
is closed.
(4) is bounded for
.
Since is bounded, it is obvious that
is bounded.
(5)The mapping is upper semicontinuous.
By the completely continuity of ,
and
being semicontinuous, we note that
is upper semicontinuous.
Hence, by the Kakutani-Fan-Glicksberg fixed point theorem, has a fixed point; that is, there exists
such that
. Thus, we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F251510/MediaObjects/13660_2010_Article_2096_Equ35_HTML.gif)
Let is finite dimensional and
. For each
, we let
be the set of all solutions of the following problem. Find
such that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F251510/MediaObjects/13660_2010_Article_2096_Equ36_HTML.gif)
By (3.10), we know that is nonempty bounded. We observe that
, where
is the weak* closure of
in
. Since
is reflexive, it follows that
is weak* compact. For any
, we note that
. So
has the finite intersection property. Therefore, it follows that
. Let
, then we have
.
Next, we claim that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F251510/MediaObjects/13660_2010_Article_2096_Equ37_HTML.gif)
Indeed, for each , we choose
such that
. Since
, there exists a sequence
such that
converge weakly to
. This implies that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F251510/MediaObjects/13660_2010_Article_2096_Equ38_HTML.gif)
for all . Since
is lower semicontinuous, it follows that
is weakly lower semicontinuous. Therefore, by using the completely continuity of
and
, respectively, and letting
, we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F251510/MediaObjects/13660_2010_Article_2096_Equ39_HTML.gif)
By Lemma 2.9, we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F251510/MediaObjects/13660_2010_Article_2096_Equ40_HTML.gif)
This completes the proof.
Setting in Theorem 3.2, we have the following result.
Corollary 3.3.
Let be a nonempty bounded closed convex subset of a real reflexive Banach space
with dual space
. Let
be two mappings satisfying the following conditions:
(i)for each ,
and
are monotone;
(ii)for each ,
and
are completely continuous;
(iii)for any given ,
and
are finite dimensional continuous; that is, for any finite dimensional subspace
,
and
are continuous.
Then, there exists such that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F251510/MediaObjects/13660_2010_Article_2096_Equ41_HTML.gif)
Corollary 3.4 (see [19]).
Let be a real reflexive Banach space and
a bounded closed convex subset. Suppose that
is a lower semicontinuous convex function with
is semimonotone, and
is finite dimensional continuous for each
. Then there exists
such that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F251510/MediaObjects/13660_2010_Article_2096_Equ42_HTML.gif)
Proof.
Define a mapping by
for all
. We observe that
is monotone and
is completely continuous for all
. Moreover,
is finite dimensional continuous. Therefore, by Theorem 3.2, there exists
such that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F251510/MediaObjects/13660_2010_Article_2096_Equ43_HTML.gif)
Next, we consider the system of generalized variational inequality in which is unbounded. We have the following result.
Theorem 3.5.
Let be a real reflexive Banach space with dual space
, and let
be a nonempty unbounded closed convex subset with
. Suppose that
is a lower semicontinuous convex function with
. Let
,
be two mappings satisfying the following conditions:
(i)for each ,
and
are monotone;
(ii)for each ,
and
are completely continuous;
(iii)for any given ,
and
are finite dimensional continuous;
(iv) and
.
Then, there exists such that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F251510/MediaObjects/13660_2010_Article_2096_Equ44_HTML.gif)
Proof.
Let be the closed ball in
at center zero with radius
such that
. By Theorem 3.2, there exists
such that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F251510/MediaObjects/13660_2010_Article_2096_Equ45_HTML.gif)
Leting in (3.26), we get
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F251510/MediaObjects/13660_2010_Article_2096_Equ46_HTML.gif)
which implies that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F251510/MediaObjects/13660_2010_Article_2096_Equ47_HTML.gif)
By condition , we know that
is bounded. So, we may assume that
converge weakly to
as
. From (3.26), it follows by Lemma 2.9 that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F251510/MediaObjects/13660_2010_Article_2096_Equ48_HTML.gif)
Since are complete continuous and
is weakly lower semicontinuous, it follows by letting
that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F251510/MediaObjects/13660_2010_Article_2096_Equ49_HTML.gif)
Using Lemma 2.9 again, we obtain
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F251510/MediaObjects/13660_2010_Article_2096_Equ50_HTML.gif)
This completes the proof.
Setting in Theorem 3.5, we have the following result.
Corollary 3.6.
Let be a real reflexive Banach space with dual space
, and let
be a nonempty unbounded closed convex subset with
. Let
be two mappings satisfying the following conditions:
(i)for each ,
and
are monotone;
(ii)for each ,
and
are completely continuous;
(iii)for any given ,
and
are finite dimensional continuous; that is, for any finite dimensional subspace
,
and
are continuous;
(iv) and
Then, there exists such that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F251510/MediaObjects/13660_2010_Article_2096_Equ51_HTML.gif)
Similarly as in the proof of Corollary 3.4, we have the following result.
Corollary 3.7 (see [19]).
Let be a real reflexive Banach space and
an unbounded closed convex subset with
. Suppose that
is a lower semicontinuous convex function with
,
is semimonotone, and
is finite dimensional continuous for each
. Assume that the following condition holds:
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F251510/MediaObjects/13660_2010_Article_2096_Equ52_HTML.gif)
Then there exists , such that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F251510/MediaObjects/13660_2010_Article_2096_Equ53_HTML.gif)
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The first author is thankful to the Thailand Research Fund for financial support under Grant BRG5280016. Moreover, the second author would like to thank the Office of the Higher Education Commission, Thailand for supporting by grant fund under Grant CHE-Ph.D-THA- SUP/86/2550, Thailand.
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Plubtieng, S., Sombut, K. Existence Results for System of Variational Inequality Problems with Semimonotone Operators. J Inequal Appl 2010, 251510 (2010). https://doi.org/10.1155/2010/251510
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DOI: https://doi.org/10.1155/2010/251510