An Existence Theorem of Solutions for the System of Generalized Vector Quasi-Variational-Like Inequalities ()
1. Introduction and Formulation
In recent years, the system of generalized vector quasivariational-like inequality, which is a unified model for the system of vector quasi-variational-like inequalities, the system of vector variational-like inequalities, the system of vector variational inequalities, the system of vector equilibrium problems and the system of variational inequalities etc., has been studied (see [1-18] and references therein).
In this paper, we consider the systems of four kinds of generalized vector quasi-variational-like inequalities with set-valued mappings and discuss the existence of its solutions in locally convex topological vector space (l.c.s. in short), motivated and inspired by the recent works of Peng [1] and Ansari et al. [2].
Throughout this paper, unless otherwise specified, assume that 
be an index set. For each
, let 
be a locally convex topological vector space (l.c.s., in short) and 
be a nonempty convex subset of Hausdorff topological vector space (t.v.s., in short)
. Let 
be a subset of continuous function space 
from 
into
, where 
is equipped with a 
- topology. Let 
and 
denote the interior and convex hull of a set 
respectively. Let 
be a set-valued mapping such that 
for each
. Denote that 
and 
.
For each
, let 
be a vectorvalued mapping, 
,
, 
and 
be four set-valued mappings. Then1) Strong type I system of generalized vector quasivariational-like inequalities which is to find 
such that
, 
and
(1.1)
2) Strong type II system of generalized vector quasivariational-like inequalities which is to find 
such that
, 
and
(1.2)
3) Weak type I system of generalized vector quasivariational-like inequalities which is to find 
such that
, 
and
(1.3)
4) Weak type II system of generalized vector quasivariational-like inequalities which is to find 
such that
, 
and
(1.4)
where 
denotes the evaluation of 
at
. By the corollary of the Schaefer [3], 
becomes a l.c.s.. By Ding and Tarafdar [4], the bilinear map 
is continuous.
The following problems are the special cases of above four kinds of systems of generalized vector quasi-variational-like inequalities.
The above system of generalized vector quasi-variational-like inequalities encompass many models of system of variational inequalities. The following problems are the special cases of problem (1.4).
1) If for each 
let 
be an identity mapping, 
, problem (1.4) reduces to the system of generalized quasi-variational-like inequalities of finding 
such that for each
, 
and
(1.5)
which was introduced and studied by Peng [1].
2) If for each 
let 
be an identity mapping, 
and
, problem (1.5) reduces to the system of generalized variational-like inequalities of finding 
such that for each
, 
and
(1.6)
In addition, let 
and let 
for all
, then problem (1.5) reduces to the system of generalized vector quasi-variational inequalities studied by Ansari and Yao [5].
3) If for each 
be an identity mapping, 
, 
and 
then problem (1.5) reduces to the system of generalized vector variational inequalities of finding 
such that for each
, 
and
(1.7)
4) If
, problem (1.4) reduces to generalized vector quasi-variational-like inequalities of finding 
such that 
and
(1.8)
such type of problem studied in [6-10].
5) If 
and 
is single valued mapping, 
be an identity mapping, 
, and 
for all 
then problem (1.4) reduces to classical variational inequality problem of finding 
such that 
and
(1.9)
which was introduced and studied by Hartman and Stampacchia [11].
2. Preliminaries
Definition 2.1. [12] Let 
and 
be two t.v.s. and 
be a convex subset of t.v.s.
. Let 
and
be two set-valued mappings. Assume given any finite subset 
in
, any
, with 
for
, and
.
Then, 1) 
is said to be strong Type I C-diagonally quasiconvex (SIC-DQC, in short) in the second argument if for some
,
2) 
is said to be strong Type II C-diagonally quasiconvex (SIIC-DQC, in short) in the second argument if for some
,
3) 
is said to be weak Type I C-diagonally quasiconvex (WIC-DQC, in short) in the second argument if for some
,
4) 
is said to be weak Type II C-diagonally quasiconvex (WIIC-DQC, in short) in the second argument if for some
,
It is easy to verify that the following proposition, 1) SIC-DQC implies SIIC-DQC; 2) SIIC-DQC implies WIC-DQC; 3) WIC-DQC implies WIIC-DQC. The converse is not true. Following example shows that the con0 verse is not true.
Example 2.1. Let 
and 
.
1) If
. Then 
is SIIC-DQC, but it is not SIC-DQC.
2) If
. Then 
is WIICDQC, but it is not WIC-DQC.
Definition 2.2. [13] Let 
and 
be two t.v.s. and 
be a convex subset of t.v.s.
. A mapping 
is called (generalized) vector 0- diagonally convex if for any finite subset
of 
and any 
with 
for
, and
,
Definition 2.3. [14] Let 
and 
be two topological spaces and 
be a set-valued mapping. Then1) 
is said to have open lower sections if the set 
is open in 
for every
;
2) 
is said to be upper semicontinuous (u.s.c., in short) if for each 
and each open set 
in 
with
, there exists an open neighborhood 
of 
in 
such that 
for each
;
3) 
is said to be lower semicontinuous (l.s.c., in short) if for each 
and each open set 
in 
with
, there exists an open neighborhood 
of 
in 
such that 
for each
;
4) 
is said to be continuous if it is both upper and lower semicontinuous;
5) 
is said to be closed if for any net 
in 
such that 
and any net 
in 
such that 
and 
for any
, we have 
.
Lemma 2.1. [15] Let 
and 
be two topological spaces. If 
is u.s.c. set-valued mapping with closed values, then 
is closed.
Lemma 2.2. [16] Let 
and 
be two topological spaces and 
is u.s.c. mapping with compact values. Suppose 
is a net in 
such that
. If 
for each
, then there are a 
and a subnet 
of 
such that
.
Lemma 2.3. [17] Let 
and 
be two topological spaces. Suppose that 
and 
are set-valued mappings having open lower sections, then 1) A set-valued mapping 
defined by, for each
, 
has open lower sections;
2) A set-valued mapping 
defined by, for each
, 
has open lower sections.
For each
, 
a Hausdorff t.v.s. Let 
be a family of nonempty compact convex subsets with each 
in
. Let 
and
. The following system of fixed-point theorem is needed in this paper.
Lemma 2.4. [18] For each
, let 
be a set-valued mapping. Assume that the following conditions hold.
1) For each
, 
is convex set-valued mapping;
2) 
Then there exist 
such that
, that is, 
for each
, where 
is the projection of 
onto 
3. Main Results
Theorem 3.1. For each
, let 
be a l.c.s., 
a nonempty compact convex subset of Hausdorff t.v.s.
, 
a nonempty compact convex subset of
, which is equipped with a 
-topology. For each
, assume that the following conditions are satisfied.
1) 
and 
are two nonempty convex set-valued mappings and have open lower sections;
2) For each 
and
, the mapping
is WIIC-DQC;
3) For each
, the set
is open.
Then there exist 
and 
such that
Proof. Define a set-valued mapping 
by
We first prove that 
for all
. To see this, suppose, by way of contradiction, that there exist some 
and some point 
such that
. Then, there exist finite points 
in 
and 
with 
such that 
and
for all 
such that
which contradicts the hypothesis 2). Hence, 
By hypothesis 3), for each 
and each
, we known that
is open and so 
has open lower sections.
For each
, consider a set-valued mapping 
defind by
Since 
has open lower sections by hypothesis 1), we may apply Lemma 2.3 to assert that the set-valued mapping 
has also open lower sections. Let
There are two cases to consider. In the case
, we have
This implies that, 
,
On the other hand, by condition 1), and the fact 
is a compact convex subset of
, we can apply Lemma 2.4 to assert the existence of a fixed point
. Since
, picking
, we have
This implies
. Hence, in this particular case, the assertion of the theorem holds.
We now consider the case
. Define a setvalued mapping 
by
Then, 
is a convex set-valued mapping and for each
, 
is open. For each
, consider the set-valued mapping 
defined by
By condition 1) and the properties of
, 
satisfies all the conditions of Lemma 2.4. Thereforethere exists 
such that
. Suppose that
, then
so that
. This is a contradiction.
Hence,
. Therefore,
Thus
This implies
Consequently, the assertion of the theorem holds in this case.
Corollary 3.2. For each
, let 
be a l.c.s., 
a nonempty compact convex subset of Hausdorff t.v.s.
, 
a nonempty compact convex subset of
, which is equipped with a 
-topology. For each
, assume that the following conditions are satisfied.
1) 
and 
are two nonempty convex set-valued mappings and have open lower sections;
2) For all
, the mapping
is an u.s.c. setvalued mapping;
3) 
is a convex set-valued mapping with 
for all
;
4) 
is affine in the first argument and for all
,
;
5) 
is a generalized vector 0-diagonally convex set-valued mapping;
6) For a given
, and a neighborhood 
of
, for all 
Then there exists 
and 
such that
Proof. Define a set-valued mapping 
by
We first prove that 
for all 
. By contradiction, for each
, suppose there exists some point 
such that 
. Then, there exist finite points 
in
, such that
Since 
is affine and 
is convexfor 
with 
such that 
and 
for all 
such that
Since 
for all 
which contradicts the hypothesis 5). Therefore 
We now prove that for each
is open. Indeed, let
, that is
. Since
is an u.s.c. setvalued mapping, there exists a neighborhood 
of 
such that
By 6),
Hence, 
This implies, 
is open for each 
and so 
have open lower sections. For the remainder of the proof, we can just follow that of Theorem 3.1. This completes the proof.
Corollary 3.3. For each
, let 
be a l.c.s., 
a nonempty compact convex subset of Hausdorff t.v.s.
, 
a nonempty compact convex subset of
, which is equipped with a 
-topology. For each
, assume that the following conditions are satisfied.
1) 
and 
are two nonempty convex set-valued mappings and have open lower sections;
2) For all
, the mapping 
is an u.s.c. setvalued mapping;
3) 
is a convex set-valued mapping such that for each
, 
is a convex cone with
;
4) 
is affine in the first argument and for all
,
;
5) 
is a generalized vector 0-diagonally convex set-valued mapping;
6) For a given
, and a neighborhood 
of
, for all 
Then there exist 
and 
such that
Proof. By hypothesis 3), the condition 4) in Corollary 3.2 is satisfied. Hence, all the conditions are satisfied as in Corollary 3.2.
Corollary 3.4. For each
, let 
be a l.c.s., 
a nonempty compact convex subset of Hausdorff t.v.s.
, 
a nonempty compact convex subset of
, which is equipped with a 
-topology. For each
, assume that 
and 
are single valued mappings and the following conditions are satisfied.
1) 
and 
are two nonempty convex set-valued mappings and have open lower sections;
2) For all
, the mapping
is continuous;
3) 
is a convex set-valued mapping with 
for all
;
4) 
is affine in the first argument and for all
,
;
5) 
is a vector 0-diagonally convex mapping;
6) 
is an u.s.c. set-valued mapping.
Then there exist 
and 
such that
Proof. Define a set-valued mapping 
by
We now prove that for each
is open, that is, the set
is closed. Indeed, let 
be a net in 
such that 
and
Since 
is continuous, hence
Since 
is an u.s.c. set-valued mapping with closed values, by Lemma 2.1, we have
and hence 
in the set
This implies 
is open for each 
and so 
has open lower sections. For the remainder of the proof, we can just follow that of Theorem 3.1 and Corollary 3.2. This completes the proof.
Theorem 3.5. For each
, let 
be a l.c.s., 
a nonempty compact convex subset of Hausdorff t.v.s.
, 
a nonempty compact convex subset of
, which is equipped with a 
-topology. For each
, assume that the following conditions are satisfied.
1) 
and 
are two nonempty convex set-valued mappings and have open lower sections;
2) For each 
and
, the mapping 
is WIC-DQC;
3) for each
, the set
is open.
Then there exist 
and 
such that
Proof. Define a set-valued mapping 
by
For the remainder proof, we just follow that of Theorem 3.1.
Corollary 3.6. For each
, let 
be a l.c.s., 
a nonempty compact convex subset of Hausdorff t.v.s.
, 
a nonempty compact convex subset of
, which is equipped with a 
-topology. For each
, assume that the following conditions are satisfied.
1) 
and 
are two nonempty convex set-valued mappings and have open lower sections;
2) For each 
and
, the mapping
is WIC-DQC;
3) 
is an u.s.c. set-valued mapping.
Then there exist 
and 
such that
Proof. Let 
be a set-valued mapping define in Theorem 3.5. We just prove that for each
is open, that is, the set
is closed. Indeed, let 
be a net in 
such that 
and
This implies
We now prove that
If it is not true, then there exists a
such that
. Since 
is Hausdorff t.v.s.
(l.c.s. is Hausdorff space) and 
is closed, there exists two open sets 
such that
Since 
is an l.s.c.
set-valued mapping and 
is an u.s.c. set-valued mapping, there exists a neighborhood
such that
and a neighborhood 
of 
such that
Hence, for all 
there exists 
such that
, which is contradiction.
Therefore, the set
is closed. Hence, all the conditions of Theorem 3.5 satisfied. Consequently, the assertion of the theorem holds.
Theorem 3.7. For each
, let 
be a l.c.s., 
a nonempty compact convex subset of Hausdorff t.v.s.
, 
a nonempty compact convex subset of
, which is equipped with a 
-topology. For each
, assume that the following conditions are satisfied.
1) 
and 
are two nonempty convex set-valued mappings and have open lower sections;
2) For each 
and
, the mapping 
is SIIC-DQC;
3) for each
, the set
is open.
Then there exist 
and 
such that
Proof. Define a set-valued mapping 
by
For the remainder proof, we just follow that of Theorem 3.1.
Corollary 3.8. For each
, let 
be a l.c.s., 
a nonempty compact convex subset of Hausdorff t.v.s.
, 
a nonempty compact convex subset of
, which is equipped with a 
-topology. For each
, assume that the following conditions are satisfied.
1) 
and 
are two nonempty convex set-valued mappings and have open lower sections;
2) For each 
and
, the mapping 
is SIIC-DQC;
3) For all 
is closed convex cone
.
Then there exist 
and 
such that
Proof. Let 
be a set-valued mapping defined in Theorem 3.7. We prove that for each
is open, that is, the set
is open. If
, since 
is open set and for all
, an u.s.c. set-valued mapping, there exists a neighborhood 
of
, for all 
This implies 
is open for each 
Therefore, all the conditions of Theorem 3.7 are satisfied. Consequently the assertion of the theorem holds.
Theorem 3.9. For each
, let 
be a l.c.s., 
a nonempty compact convex subset of Hausdorff t.v.s.
, 
a nonempty compact convex subset of
, which is equipped with a 
-topology. For each
, assume that the following conditions are satisfied.
1) 
and 
are two nonempty convex set-valued mappings and have open lower sections;
2) For each 
and
, the mapping 
is SIC-DQC;
3) for each
, the set
is open.
Then there exist 
and 
such that
Proof. Define a set-valued mapping 
by
The rest of the proof is similar to that of Theorem 3.1.
Corollary 3.10. For each
, let 
be a l.c.s., 
a nonempty compact convex subset of Hausdorff t.v.s.
, 
a nonempty compact convex subset of
, which is equipped with a 
-topology. For each
, assume that the following conditions are satisfied.
1) 
and 
are two nonempty convex set-valued mappings and have open lower sections;
2) For each 
and
, the mapping 
is SIC-DQC;
3) 
is an u.s.c. mapping with closed values.
Then there exist 
and 
such that
Proof. Let 
a set-valued mapping defined in Theorem 3.9. We prove that for each
, the set
is open, that is, the set
is closed. Indeed, let 
be a net in 
such that 
and
We claim that
To prove this assertion, we can just follow that of Corollary 3.6. Hence, the set
is open. Therefore, all the conditions of Theorem 3.9 are satisfied. Consequently, the assertion of the corollary hold.
NOTES