An Existence Theorem of Solutions for the System of Generalized Vector Quasi-Variational-Like Inequalities

In this paper, we introduce and study the system of generalized vector quasi-variational-like inequalities in Hausdorff topological vector spaces, which include the system of vector quasi-variational-like inequalities, the system of vector variational-like inequalities, the system of vector quasi-variational inequalities, and several other systems as special cases. Moreover, a number of C-diagonal quasiconvexity properties are proposed for set-valued maps, which are natural generalizations of the g-diagonal quasiconvexity for real functions. Together with an application of continuous selection and fixed-point theorems, these conditions enable us to prove unified existence results of solutions for the system of generalized vector quasi-variational-like inequalities. The results of this paper can be seen as extensions and generalizations of several known results in the literature.

• The System of Generalized Vector Quasi-Variational-Like Inequalities; Fixed Point Theorem; Open Lower Section; Upper Semicontinuous; C-Diagonal Quasiconvexity

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

2) Strong type II system of generalized vector quasivariational-like inequalities which is to find such that, and

3) Weak type I system of generalized vector quasivariational-like inequalities which is to find such that, and

4) Weak type II system of generalized vector quasivariational-like inequalities which is to find such that, and

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

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

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

4) If, problem (1.4) reduces to generalized vector quasi-variational-like inequalities of finding such that and

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

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.

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