Existence of Solutions for Open image in new window -Generalized Vector Variational-Like Inequalities

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  1. Selected Papers from the 10th International Conference 2009 on Nonlinear Functional Analysis and Applications

Abstract

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

Keywords

Convex Subset Vector Variational Inequality Vector Equilibrium Problem Hausdorff Topological Vector Space Strong Vector 

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, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 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 Open image in new window -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 Open image in new window -generalized vector variational-like inequalities and a class of Open image in new window -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 Open image in new window -generalized vector variational-like inequalities under suitable conditions. By using FKKM theorem, we prove some new existence results of solutions for the Open image in new window -generalized vector variational-like inequalities and Open image in new window -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 Open image in new window and Open image in new window be two real Hausdorff topological vector spaces, Open image in new window a nonempty, closed, and convex subset, and Open image in new window a closed, convex, and pointed cone with apex at the origin. Recall that the Hausdorff topological vector space Open image in new window is said to an ordered Hausdorff topological vector space denoted by Open image in new window if ordering relations are defined in Open image in new window as follows:
If the interior Open image in new window is nonempty, then the weak ordering relations in Open image in new window are defined as follows:

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

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

-Generalized Vector Variational-Like Inequality (for short, Open image in new window -GVVLI): for each Open image in new window and Open image in new window , find Open image in new window such that
-Generalized Strong Vector Variational-Like Inequality (for short, Open image in new window -GSVVLI): for each Open image in new window and Open image in new window , find Open image in new window such that
-GVVLI and Open image in new window -GSVVLI encompass many models of variational inequalities. For example, the following problems are the special cases of Open image in new window -GVVLI and Open image in new window -GSVVLI.

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

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

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

Definition 2.1.

Definition 2.2.

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

Definition 2.3.

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

Lemma 2.4 (see [21]).

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

(1) Open image in new window and Open image in new window imply Open image in new window ;

(2) Open image in new window and Open image in new window imply Open image in new window ;

(3) Open image in new window and Open image in new window imply Open image in new window ;

(4) Open image in new window and Open image in new window imply Open image in new window .

Lemma 2.5 (see [22]).

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

Lemma 2.6 (see [23]).

Let Open image in new window be a Hausdorff topological space, Open image in new window be nonempty compact convex subsets of Open image in new window . Then Open image in new window is compact.

Lemma 2.7 (see [24]).

Let Open image in new window and Open image in new window be two topological spaces. If Open image in new window is upper semicontinuous with closed values, then Open image in new window is closed.

3. Main Results

Theorem 3.1.

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

(i)find Open image in new window , such that Open image in new window , for all Open image in new window

(ii)find Open image in new window , such that Open image in new window , for all Open image in new window

where Open image in new window is defined by Open image in new window for all Open image in new window .

Proof.

Suppose that (i) holds. We can find Open image in new window , such that
On the other hand, we know Open image in new window is affine and Open image in new window . It follows that
By Lemma 2.4,

and so Open image in new window is a solution of (ii).

Conversely, suppose that (ii) holds. Then there exists Open image in new window such that
Considering the Open image in new window -hemicontinuity of Open image in new window and letting Open image in new window , we have

This completes the proof.

Remark 3.2.

If Open image in new window and Open image in new window for all Open image in new window , then Theorem 3.1 is reduced to Lemma Open image in new window of [17].

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

Theorem 3.3.

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

(i) Open image in new window and Open image in new window is Open image in new window -monotone in Open image in new window ;

(ii) Open image in new window is an upper semicontinuous set-valued mapping.

Proof.

Then Open image in new window and Open image in new window are nonempty since Open image in new window and Open image in new window . The proof is divided into the following three steps.
  1. (I)
    First, we prove the following conclusion: Open image in new window is a KKM mapping. Indeed, assume that Open image in new window is not a KKM mapping; then there exist Open image in new window , Open image in new window with Open image in new window and Open image in new window such that
     
That is,
On the other hand, we know Open image in new window . Then we have Open image in new window . It is impossible and so Open image in new window is a KKM mapping.
  1. (II)
    Further, we prove that
     
In fact, if Open image in new window , then Open image in new window From the proof of Theorem 3.1, we know that Open image in new window is Open image in new window -monotone in Open image in new window . It follows that
By Lemma 2.4, we have
Conversely, suppose that Open image in new window Then
It follows from Theorem 3.1 that
which implies that
( Open image in new window ) Last, we prove that Open image in new window Indeed, since Open image in new window is a KKM mapping, we know that, for any finite set Open image in new window one has

This shows that Open image in new window is also a KKM mapping.

Now, we prove that Open image in new window is closed for all Open image in new window . Assume that there exists a net Open image in new window with Open image in new window . Then
Using the definition of Open image in new window , we have
Since Open image in new window and Open image in new window are continuous, it follows that
Since Open image in new window is upper semicontinuous mapping with close values, by Lemma 2.7, we know that Open image in new window is closed, and so
This implies that
and so Open image in new window is closed. Considering the compactness of Open image in new window and closeness of Open image in new window , we know that Open image in new window is compact. By Lemma 2.5, we have Open image in new window and it follows that Open image in new window , that is, for each Open image in new window and Open image in new window there exists Open image in new window such that

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

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

Theorem 3.6.

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

(i) Open image in new window and Open image in new window is Open image in new window -monotone in Open image in new window ;

(ii) Open image in new window is an upper semicontinuous set-valued mapping;

(iii)there exists a nonempty compact and convex subset Open image in new window of Open image in new window and for each Open image in new window , Open image in new window , Open image in new window , there exist Open image in new window such that

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 Open image in new window , such that

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

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

Let Open image in new window be defined as follows:
Using the proof of Theorem 3.3, we obtain

and so there exists Open image in new window

Next we prove that Open image in new window . In fact, if Open image in new window then the assumption implies that there exists Open image in new window such that

which contradicts Open image in new window and so Open image in new window .

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

In the following, we prove the solvability of Open image in new window -GSVVLI under some suitable conditions by using FKKM theorem.

Theorem 3.7.

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

(i)for each Open image in new window , Open image in new window the set Open image in new window is open in Open image in new window

Proof.

Since Open image in new window is a closed subset of Open image in new window , considering the compactness of Open image in new window and closedness of Open image in new window , we know that Open image in new window is compact.

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

Let Open image in new window be defined as follows:
We claim that Open image in new window is a KKM mapping. Indeed, assume that Open image in new window is not a KKM mapping. Then there exist Open image in new window , Open image in new window with Open image in new window and Open image in new window such that
That is,
On the other hand, we know Open image in new window and so

which is impossible. Therefore, Open image in new window is a KKM mapping.

Since Open image in new window is a closed subset of Open image in new window , it follows that Open image in new window is compact. By Lemma 2.5, we have

Thus, there exists Open image in new window

Next we prove that Open image in new window . In fact, if Open image in new window then the condition (ii) implies that there exists Open image in new window such that

which contradicts Open image in new window and so Open image in new window .

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

Thus, Open image in new window -GSVVI is solvable. This completes the proof.

Remark 3.8.

If Open image in new window is compact, Open image in new window , and Open image in new window , then Theorem 3.7 is reduced to Theorem Open image in new window in [20].

Notes

Acknowledgments

The authors greatly appreciate the editor and the anonymous referees for their useful comments and suggestions. This work was supported by the Key Program of NSFC (Grant no. 70831005), the Kyungnam University Research Fund 2009, and the Open Fund (PLN0904) of State Key Laboratory of Oil and Gas Reservoir Geology and Exploitation (Southwest Petroleum University).

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© Xi Li et al. 2010

This article is published under license to BioMed Central Ltd. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Authors and Affiliations

  1. 1.Department of MathematicsSichuan UniversityChengduChina
  2. 2.Department of MathematicsKyungnam UniversityMasanSouth Korea
  3. 3.Science & Technology Finance and Mathematical Finance Key Laboratory of Sichuan ProvinceSichuan UniversitySichuanChina

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