Existence of Solutions for -Generalized Vector Variational-Like Inequalities

Abstract

We introduce and study a class of -generalized vector variational-like inequalities and a class of -generalized strong vector variational-like inequalities in the setting of Hausdorff topological vector spaces. An equivalence result concerned with two classes of -generalized vector variational-like inequalities is proved under suitable conditions. By using FKKM theorem, some new existence results of solutions for the -generalized vector variational-like inequalities and -generalized strong vector variational-like inequalities are obtained under some suitable conditions.

1. Introduction

Vector variational inequality was first introduced and studied by Giannessi [1] in the setting of finite-dimensional Euclidean spaces. Since then, the theory with applications for vector variational inequalities, vector complementarity problems, vector equilibrium problems, and vector optimization problems have been studied and generalized by many authors (see, e.g., [2–15] and the references therein).

Recently, Yu et al. [16] considered a more general form of weak vector variational inequalities and proved some new results on the existence of solutions of the new class of weak vector variational inequalities in the setting of Hausdorff topological vector spaces.

Very recently, Ahmad and Khan [17] introduced and considered weak vector variational-like inequalities with -generally convex mapping and gave some existence results.

On the other hand, Fang and Huang [18] studied some existence results of solutions for a class of strong vector variational inequalities in Banach spaces, which give a positive answer to an open problem proposed by Chen and Hou [19].

In 2008, Lee et al. [20] introduced a new class of strong vector variational-type inequalities in Banach spaces. They obtained the existence theorems of solutions for the inequalities without monotonicity in Banach spaces by using Brouwer fixed point theorem and Browder fixed point theorem.

Motivated and inspired by the work mentioned above, in this paper we introduce and study a class of -generalized vector variational-like inequalities and a class of -generalized strong vector variational-like inequalities in the setting of Hausdorff topological vector spaces. We first show an equivalence theorem concerned with two classes of -generalized vector variational-like inequalities under suitable conditions. By using FKKM theorem, we prove some new existence results of solutions for the -generalized vector variational-like inequalities and -generalized strong vector variational-like inequalities under some suitable conditions. The results presented in this paper improve and generalize some known results due to Ahmad and Khan [17], Lee et al. [20], and Yu et al. [16].

2. Preliminaries

Let and be two real Hausdorff topological vector spaces, a nonempty, closed, and convex subset, and a closed, convex, and pointed cone with apex at the origin. Recall that the Hausdorff topological vector space is said to an ordered Hausdorff topological vector space denoted by if ordering relations are defined in as follows:

(2.1)

If the interior is nonempty, then the weak ordering relations in are defined as follows:

(2.2)

Let be the space of all continuous linear maps from to and . We denote the value of on by . Throughout this paper, we assume that is a family of closed, convex, and pointed cones of such that for all , is a mapping from into , and is a mapping from into .

In this paper, we consider the following two kinds of vector variational inequalities:

-Generalized Vector Variational-Like Inequality (for short, -GVVLI): for each and , find such that

(2.3)

-Generalized Strong Vector Variational-Like Inequality (for short, -GSVVLI): for each and , find such that

(2.4)

-GVVLI and -GSVVLI encompass many models of variational inequalities. For example, the following problems are the special cases of -GVVLI and -GSVVLI.

() If and for all , then -GVVLI reduces to finding , such that for each ,

(2.5)

which is introduced and studied by Ahmad and Khan [17]. In addition, if for each , then -GVVLI reduces to the following model studied by Yu et al. [16].

Find such that for each ,

(2.6)

() If and for all , then -GSVVLI is equivalent to the following vector variational inequality problem introduced and studied by Lee et al. [20].

Find satisfying

(2.7)

For our main results, we need the following definitions and lemmas.

Definition 2.1.

Let and be two mappings and . is said to be -monotone in if and only if

(2.8)

Definition 2.2.

Let and be two mappings. We say that is -hemicontinuous if, for any given and , the mapping is continuous at .

Definition 2.3.

A multivalued mapping is said to be upper semicontinuous on if, for all and for each open set in with , there exists an open neighbourhood of such that for all .

Lemma 2.4 (see [21]).

Let be an ordered topological vector space with a closed, pointed, and convex cone with . Then for any , we have

(1) and imply ;

(2) and imply ;

(3) and imply ;

(4) and imply .

Lemma 2.5 (see [22]).

Let be a nonempty, closed, and convex subset of a Hausdorff topological space, and a multivalued map. Suppose that for any finite set , one has (i.e., is a KKM mapping) and is closed for each and compact for some , where denotes the convex hull operator. Then .

Lemma 2.6 (see [23]).

Let be a Hausdorff topological space, be nonempty compact convex subsets of . Then is compact.

Lemma 2.7 (see [24]).

Let and be two topological spaces. If is upper semicontinuous with closed values, then is closed.

3. Main Results

Theorem 3.1.

Let be a Hausdorff topological linear space, a nonempty, closed, and convex subset, and an ordered topological vector space with for all . Let and be affine mappings such that for each . Let be an -hemicontinuous mapping. If and is -monotone in then for each , , the following statements are equivalent

(i)find , such that , for all

(ii)find , such that , for all

where is defined by for all .

Proof.

Suppose that (i) holds. We can find , such that

(3.1)

Since is -monotone, for each we have

(3.2)

On the other hand, we know is affine and . It follows that

(3.3)

Hence is also -monotone. That is

(3.4)

Since , for all , we obtain

(3.5)

By Lemma 2.4,

(3.6)

and so is a solution of (ii).

Conversely, suppose that (ii) holds. Then there exists such that

(3.7)

For each , , we let . Obviously, . It follows that

(3.8)

Since and are affine and , we have

(3.9)

That is

(3.10)

Considering the -hemicontinuity of and letting , we have

(3.11)

This completes the proof.

Remark 3.2.

If and for all , then Theorem 3.1 is reduced to Lemma of [17].

Let be a closed convex subset of a topological linear space and a family of closed, convex, and pointed cones of a topological space such that for all . Throughout this paper, we define a set-valued mapping as follows:

(3.12)

Theorem 3.3.

Let be a Hausdorff topological linear space, a nonempty, closed, compact, and convex subset, and an ordered topological vector space with for all . Let and be affine mappings such that for each . Let be an -hemicontinuous mapping. Assume that the following conditions are satisfied

(i) and is -monotone in ;

(ii) is an upper semicontinuous set-valued mapping.

Then for each , , there exist such that

(3.13)

Proof.

For each , we denote and define

(3.14)

Then and are nonempty since and . The proof is divided into the following three steps.

• (I) First, we prove the following conclusion: is a KKM mapping. Indeed, assume that is not a KKM mapping; then there exist , with and such that

  

  (3.15)

That is,

(3.16)

Since and are affine, we have

(3.17)

On the other hand, we know . Then we have . It is impossible and so is a KKM mapping.

• (II) Further, we prove that

  

  (3.18)

In fact, if , then From the proof of Theorem 3.1, we know that is -monotone in . It follows that

(3.19)

and so

(3.20)

By Lemma 2.4, we have

(3.21)

and so for each . That is, and so

(3.22)

Conversely, suppose that Then

(3.23)

It follows from Theorem 3.1 that

(3.24)

That is, and so

(3.25)

which implies that

(3.26)

() Last, we prove that Indeed, since is a KKM mapping, we know that, for any finite set one has

(3.27)

This shows that is also a KKM mapping.

Now, we prove that is closed for all . Assume that there exists a net with . Then

(3.28)

Using the definition of , we have

(3.29)

Since and are continuous, it follows that

(3.30)

Since is upper semicontinuous mapping with close values, by Lemma 2.7, we know that is closed, and so

(3.31)

This implies that

(3.32)

and so is closed. Considering the compactness of and closeness of , we know that is compact. By Lemma 2.5, we have and it follows that , that is, for each and there exists such that

(3.33)

Thus, -GVVLI is solvable. This completes the proof.

Remark 3.4.

The condition (ii) in Theorem 3.3 can be found in several papers (see, e.g., [25, 26]).

Remark 3.5.

If and for all in Theorem 3.3, then condition (ii) holds and condition (i) is equivalent to the -monotonicity of . Thus, it is easy to see that Theorem 3.3 is a generalization of [17, Theorem ].

In the above theorem, is compact. In the following theorem, under some suitable conditions, we prove a new existence result of solutions for -GVVLI without the compactness of .

Theorem 3.6.

Let be a Hausdorff topological linear space, a nonempty, closed, and convex subset, and be an ordered topological vector space with for all . Let and be affine mappings such that for each . Let be an -hemicontinuous mapping. Assume that the following conditions are satisfied:

(i) and is -monotone in ;

(ii) is an upper semicontinuous set-valued mapping;

(iii)there exists a nonempty compact and convex subset of and for each , , , there exist such that

(3.34)

Then for each , , there exist such that

(3.35)

Proof.

By Theorem 3.1, we know that the solution set of the problem (ii) in Theorem 3.1 is equivalent to the solution set of following variational inequality: find , such that

(3.36)

For each and we denote Let be defined as follows:

(3.37)

Obviously, for each ,

(3.38)

Using the proof of Theorem 3.3, we obtain that is a closed subset of . Considering the compactness of and closedness of , we know that is compact.

Now we prove that for any finite set , one has Let Since is a real Hausdorff topological vector space, for each , is compact and convex. Let . By Lemma 2.6, we know that is a compact and convex subset of .

Let be defined as follows:

(3.39)

Using the proof of Theorem 3.3, we obtain

(3.40)

and so there exists

Next we prove that . In fact, if then the assumption implies that there exists such that

(3.41)

which contradicts and so .

Since and for each , it follows that . Thus, for any finite set , we have Considering the compactness of for each , we know that there exists such that Therefore, the solution set of -GVVLI is nonempty. This completes the proof.

In the following, we prove the solvability of -GSVVLI under some suitable conditions by using FKKM theorem.

Theorem 3.7.

Let be a Hausdorff topological linear space, a nonempty, closed, and convex set, and an ordered Hausdorff topological vector space with for all . Assume that for each and are affine, and for all . Let be a mapping such that

(i)for each , the set is open in

(ii)there exists a nonempty compact and convex subset of and for each , , there exists such that

(3.42)

Then for each , there exists such that

(3.43)

Proof.

For each and we denote . Let be defined as follows:

(3.44)

Obviously, for each ,

(3.45)

Since is a closed subset of , considering the compactness of and closedness of , we know that is compact.

Now we prove that for any finite set , one has Let Since is a real Hausdorff topological vector space, for each , is compact and convex. Let . By Lemma 2.6, we know that is a compact and convex subset of .

Let be defined as follows:

(3.46)

We claim that is a KKM mapping. Indeed, assume that is not a KKM mapping. Then there exist , with and such that

(3.47)

That is,

(3.48)

Since and are affine, we have

(3.49)

On the other hand, we know and so

(3.50)

which is impossible. Therefore, is a KKM mapping.

Since is a closed subset of , it follows that is compact. By Lemma 2.5, we have

(3.51)

Thus, there exists

Next we prove that . In fact, if then the condition (ii) implies that there exists such that

(3.52)

which contradicts and so .

Since and for each , it follows that . Thus, for any finite set , we have Considering the compactness of for each , it is easy to know that there exists such that Therefore, for each , there exists such that

(3.53)

Thus, -GSVVI is solvable. This completes the proof.

Remark 3.8.

If is compact, , and , then Theorem 3.7 is reduced to Theorem in [20].