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 ![](https://www.scirp.org/html/1-1040077\99d40c89-ec6a-43fc-b2e5-40cad4a0f3d5.jpg)
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
,
![](https://www.scirp.org/html/1-1040077\4b10fc9e-198c-41f6-927e-68f267e056ad.jpg)
2)
is said to be strong Type II C-diagonally quasiconvex (SIIC-DQC, in short) in the second argument if for some
,
![](https://www.scirp.org/html/1-1040077\ef12dfd5-a6ef-41aa-94bf-e247563540ec.jpg)
3)
is said to be weak Type I C-diagonally quasiconvex (WIC-DQC, in short) in the second argument if for some
,
![](https://www.scirp.org/html/1-1040077\e9310b42-ffc5-4614-ad35-5c5eb03be527.jpg)
4)
is said to be weak Type II C-diagonally quasiconvex (WIIC-DQC, in short) in the second argument if for some
,
![](https://www.scirp.org/html/1-1040077\99a68bed-b3be-4b0f-9ffd-07d12f59ba1a.jpg)
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
,
![](https://www.scirp.org/html/1-1040077\9c0a4906-6122-49d0-92ab-2288e6e3372a.jpg)
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) ![](https://www.scirp.org/html/1-1040077\a7b1cf1c-73a9-4bfd-9539-131085f5afba.jpg)
Then there exist
such that
, that is,
for each
, where
is the projection of
onto ![](https://www.scirp.org/html/1-1040077\7dfd0066-c346-466a-b2d2-57bc5d2e574e.jpg)
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
![](https://www.scirp.org/html/1-1040077\dd38cd53-c207-4d62-a595-882a5c72e27d.jpg)
is open.
Then there exist
and
such that
![](https://www.scirp.org/html/1-1040077\3ae66cb3-fc7b-4436-b44c-8ac97b1df577.jpg)
Proof. Define a set-valued mapping
by
![](https://www.scirp.org/html/1-1040077\81eaa03a-7498-4876-a011-961b8f852de2.jpg)
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 ![](https://www.scirp.org/html/1-1040077\5f8d2cc1-21e8-412d-abda-6dfbb500dabf.jpg)
with
such that
and
for all
such that
![](https://www.scirp.org/html/1-1040077\8c3b6cb3-3a06-458d-b6bf-241b6b3f33fb.jpg)
which contradicts the hypothesis 2). Hence, ![](https://www.scirp.org/html/1-1040077\26200456-10b4-4ff0-a7bb-84f92aa74249.jpg)
By hypothesis 3), for each
and each
, we known that
![](https://www.scirp.org/html/1-1040077\feed77b7-c099-46d9-b7ac-e0a859ab7da2.jpg)
is open and so
has open lower sections.
For each
, consider a set-valued mapping
defind by
![](https://www.scirp.org/html/1-1040077\0ab6c1f0-168e-4734-8c89-0c9570d99f2d.jpg)
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
![](https://www.scirp.org/html/1-1040077\b13f6893-c674-4546-a4ed-3ce8d9c7e33f.jpg)
There are two cases to consider. In the case
, we have
![](https://www.scirp.org/html/1-1040077\e72e554f-e186-4a1f-927d-9d72a4c72c76.jpg)
This implies that,
,
![](https://www.scirp.org/html/1-1040077\07fdb54c-8c95-4c6f-907a-7a40e53c645c.jpg)
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
![](https://www.scirp.org/html/1-1040077\58e6153e-c478-49d7-8859-948be84e5c37.jpg)
This implies
. Hence, in this particular case, the assertion of the theorem holds.
We now consider the case
. Define a setvalued mapping
by
![](https://www.scirp.org/html/1-1040077\4491128b-ed63-4737-a93e-d7da49b02d7b.jpg)
Then,
is a convex set-valued mapping and for each
,
is open. For each
, consider the set-valued mapping
defined by
![](https://www.scirp.org/html/1-1040077\c1c56d70-e4de-437e-bd7f-f59db60641c7.jpg)
By condition 1) and the properties of
,
satisfies all the conditions of Lemma 2.4. Thereforethere exists
such that
. Suppose that
, then
![](https://www.scirp.org/html/1-1040077\49cefb5d-a7f0-4949-8daf-f50e3122bc2d.jpg)
so that
. This is a contradiction.
Hence,
. Therefore,
![](https://www.scirp.org/html/1-1040077\19397687-5f1f-41af-a70f-408b70575817.jpg)
Thus
![](https://www.scirp.org/html/1-1040077\400e6d18-1ae8-4adf-b38c-77200aefb424.jpg)
This implies
![](https://www.scirp.org/html/1-1040077\55176250-4880-43a5-b843-93e3a4c66c23.jpg)
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
![](https://www.scirp.org/html/1-1040077\5f54855f-3c62-40ec-9aed-eda2cbcb108e.jpg)
Then there exists
and
such that
![](https://www.scirp.org/html/1-1040077\221e3ddd-8875-496e-b90c-ab485ad6cbb2.jpg)
Proof. Define a set-valued mapping
by
![](https://www.scirp.org/html/1-1040077\54a15153-1289-4ab9-b648-c43643f7882e.jpg)
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
![](https://www.scirp.org/html/1-1040077\ef90be16-a3d5-443f-92ad-c4b8411b53f1.jpg)
Since
is affine and
is convexfor
with
such that ![](https://www.scirp.org/html/1-1040077\4ad5065c-af80-47d0-8b08-5b83217582ab.jpg)
and
for all
such that
![](https://www.scirp.org/html/1-1040077\6d3325bb-bf2a-4cd3-9482-ab6b9bef455b.jpg)
Since
for all ![](https://www.scirp.org/html/1-1040077\01573adc-fd00-4875-a48e-d4bb8ddf01d8.jpg)
![](https://www.scirp.org/html/1-1040077\2ae27487-42fd-4a53-94d3-b086152ba09b.jpg)
which contradicts the hypothesis 5). Therefore ![](https://www.scirp.org/html/1-1040077\e71caa28-cd96-45bb-aeda-d7df63dbdf83.jpg)
We now prove that for each
![](https://www.scirp.org/html/1-1040077\3092feba-4b8c-4111-bac8-040aafd5e1b6.jpg)
is open. Indeed, let
, that is
. Since
is an u.s.c. setvalued mapping, there exists a neighborhood
of
such that
![](https://www.scirp.org/html/1-1040077\464839e4-f511-40e2-8f52-161b184e0f8d.jpg)
By 6),
![](https://www.scirp.org/html/1-1040077\86379eb1-553c-4f47-ae1b-f8426ef319b2.jpg)
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
![](https://www.scirp.org/html/1-1040077\ae8aa84f-37ea-4308-819e-3ab2104cb5f2.jpg)
Then there exist
and
such that
![](https://www.scirp.org/html/1-1040077\dbea6cc5-50b8-468d-ba6b-03d4073ebbe9.jpg)
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
![](https://www.scirp.org/html/1-1040077\634e96eb-292e-452c-b0c4-b56558473408.jpg)
Proof. Define a set-valued mapping
by
![](https://www.scirp.org/html/1-1040077\85cedb4c-dd34-46b7-abae-5da13385ea2d.jpg)
We now prove that for each
![](https://www.scirp.org/html/1-1040077\af6fe9d1-1544-4f8e-a78c-a2daaa172e50.jpg)
is open, that is, the set
![](https://www.scirp.org/html/1-1040077\1daaf826-7699-4ef5-9117-2ce0e6ef59c4.jpg)
is closed. Indeed, let
be a net in
such that
and
![](https://www.scirp.org/html/1-1040077\a6dbf853-cc55-4b5b-b3de-9384a81516d0.jpg)
Since
is continuous, hence
![](https://www.scirp.org/html/1-1040077\4ef6b944-bf9c-45c1-b0f4-af7a8e5359c5.jpg)
Since
is an u.s.c. set-valued mapping with closed values, by Lemma 2.1, we have
![](https://www.scirp.org/html/1-1040077\a90b5e43-7eed-45e9-a4a2-1539b6f9c467.jpg)
and hence
in the set
![](https://www.scirp.org/html/1-1040077\9e60b411-fb7d-4991-8474-a7d7cbb7574c.jpg)
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
![](https://www.scirp.org/html/1-1040077\b8042992-d929-4a2d-9e72-20c4dc6bc4c4.jpg)
is open.
Then there exist
and
such that
![](https://www.scirp.org/html/1-1040077\df6e158c-a789-4ab8-b0fc-77425141a0aa.jpg)
Proof. Define a set-valued mapping
by
![](https://www.scirp.org/html/1-1040077\423d9a4d-0575-4ecf-9b47-54cbfcb20e59.jpg)
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
![](https://www.scirp.org/html/1-1040077\7f23e900-b550-450c-84c3-ee79cb4e61e5.jpg)
Proof. Let
be a set-valued mapping define in Theorem 3.5. We just prove that for each
![](https://www.scirp.org/html/1-1040077\f2a34165-2d3a-435e-8630-385c08724311.jpg)
is open, that is, the set
![](https://www.scirp.org/html/1-1040077\7e5f012a-a25f-410c-8dff-08ecc961f7c7.jpg)
is closed. Indeed, let
be a net in
such that
and
![](https://www.scirp.org/html/1-1040077\8fd968da-6de0-4a18-ab8f-c59425723636.jpg)
This implies
![](https://www.scirp.org/html/1-1040077\ca8af999-6ff5-43a7-ac66-0aaf707ec7ad.jpg)
We now prove that
![](https://www.scirp.org/html/1-1040077\3a98aef4-7bb1-4b96-9fd8-990f83835bc8.jpg)
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
![](https://www.scirp.org/html/1-1040077\9242e383-e5c2-41fb-afa5-58ad956ca4aa.jpg)
Since
is an l.s.c.
set-valued mapping and
is an u.s.c. set-valued mapping, there exists a neighborhood
such that
![](https://www.scirp.org/html/1-1040077\d69dd52e-0526-49a5-b61b-d6a2ab53c3eb.jpg)
and a neighborhood
of
such that
![](https://www.scirp.org/html/1-1040077\c7032848-ba17-4369-a5f0-6432bd265012.jpg)
Hence, for all
there exists
such that
, which is contradiction.
Therefore, the set
![](https://www.scirp.org/html/1-1040077\a0612e3f-d6a6-494c-ad6b-468b0b79edb4.jpg)
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
![](https://www.scirp.org/html/1-1040077\712b7d4c-2d03-4ae8-877a-2320dc1eabcd.jpg)
is open.
Then there exist
and
such that
![](https://www.scirp.org/html/1-1040077\4c7ad6bb-d07e-4ea1-b57d-50f9e71d5c01.jpg)
Proof. Define a set-valued mapping
by
![](https://www.scirp.org/html/1-1040077\20697b2a-ba26-4a0e-8e3a-ba43aca6b078.jpg)
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
![](https://www.scirp.org/html/1-1040077\49ae8355-67c1-4f4c-beb5-ef176630b992.jpg)
Proof. Let
be a set-valued mapping defined in Theorem 3.7. We prove that for each
![](https://www.scirp.org/html/1-1040077\1734ba2c-d019-41f3-ac39-f41a0cb1c9c8.jpg)
is open, that is, the set
![](https://www.scirp.org/html/1-1040077\ba6d511d-d637-4d38-be42-e7a02fd40cb9.jpg)
is open. If
, since
is open set and for all
, an u.s.c. set-valued mapping, there exists a neighborhood
of
, for all ![](https://www.scirp.org/html/1-1040077\fa9f37ef-531b-4127-83a8-d1edf93268f4.jpg)
![](https://www.scirp.org/html/1-1040077\77d1ad71-2097-4890-a4e7-6c809d3bacd0.jpg)
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
![](https://www.scirp.org/html/1-1040077\73dc5b98-35fb-44a6-b24f-366839dffd9d.jpg)
Proof. Define a set-valued mapping
by
![](https://www.scirp.org/html/1-1040077\18b67f6e-c09e-4a42-9d94-a6a4f83f0b1a.jpg)
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
![](https://www.scirp.org/html/1-1040077\71f0b566-62e6-4a2a-bb97-b440a11c9265.jpg)
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
![](https://www.scirp.org/html/1-1040077\c0b8c273-39c8-49b4-8f8f-8ed633608b34.jpg)
We claim that
![](https://www.scirp.org/html/1-1040077\bfd1d61f-51f7-48f0-8659-cc46f4db6144.jpg)
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