- Research Article
- Open Access

# Generalized Bi-Quasivariational Inequalities for Quasi-Pseudomonotone Type II Operators on Noncompact Sets

- MohammadSR Chowdhury
^{1}and - YeolJe Cho
^{2}Email author

**2010**:237191

https://doi.org/10.1155/2010/237191

© M. S. R. Chowdhury and Y. J. Cho. 2010

**Received:**3 November 2009**Accepted:**18 January 2010**Published:**1 February 2010

## Abstract

We prove some existence results of solutions for a new class of generalized bi-quasivariational inequalities (GBQVI) for quasi-pseudomonotone type II and strongly quasi-pseudomonotone type II operators defined on noncompact sets in locally convex Hausdorff topological vector spaces. To obtain these results on GBQVI for quasi-pseudomonotone type II and strongly quasi-pseudomonotone type II operators, we use Chowdhury and Tan's generalized version (1996) of Ky Fan's minimax inequality (1972) as the main tool.

## Keywords

- Compact Subset
- Convex Subset
- Compact Convex
- Inequality Problem
- Compact Convex Subset

## 1. Introduction and Preliminaries

In this paper, we obtain some results on generalized bi-quasi-variational inequalities for quasi-pseudo-monotone type II and strongly quasi-pseudo-monotone type II operators defined on noncompact sets in locally convex Hausdorff topological vector spaces. Thus we begin this section by defining the generalized bi-quasi-variational inequalities. For this, we need to introduce some notations which will be used throughout this paper.

Let be a nonempty set and let be the family of all nonempty subsets of . If and are topological spaces and , then the graph of is the set . Throughout this paper, denotes either the real field or the complex field .

Let be a topological vector space over , let be a vector space over and let be a bilinear functional.

For any , any nonempty subset of , and any , let and . Let be the (weak) topology on generated by the family as a subbase for the neighbourhood system at 0 and let be the (strong) topology on generated by the family is a nonempty bounded subset of and as a base for the neighbourhood system at 0. We note then that , when equipped with the (weak) topology or the (strong) topology , becomes a locally convex topological vector space which is not necessarily Hausdorff. But, if the bilinear functional separates points in , that is, for any with , there exists such that , then also becomes Hausdorff. Furthermore, for any net in and ,

(1) in if and only if for any ,

(2) in if and only if uniformly for any , where a nonempty bounded subset of .

The generalized bi-quasi-variational inequality problem was first introduced by Shih and Tan [1] in 1989. Since Shih and Tan, some authors have obtained many results on generalized (quasi)variational inequalities, generalized (quasi)variational-like inequalities and generalized bi-quasi-variational inequalities (see [2–15]).

The following is the definition due to Shih and Tan [1].

Definition 1.1 ..

Let
and
be a vector spaces over
, let
be a bilinear functional, and let
be a nonempty subset of
. If
and
, the *generalized bi-quasi variational inequality problem* (GBQVI) for the triple (
,
,
) is to find
satisfying the following properties:

(1) ,

(2) for any and .

The following definition of the generalized bi-quasi-variational inequality problem is a slight modification of Definition 1.1.

Definition 1.2.

Let
and
be vector spaces over
, let
be a bilinear functional, and let
be a nonempty subset of
. If
and
, then the *generalized bi-quasivariational inequality* (GBQVI) problem for the triple (
,
,
) is:

Let
be a nonempty subset of
and let
be a set-valued mapping. Then
is said to be *monotone* on
if, for any
,
, and
.

Let and be topological spaces and let be a set-valued mapping. Then is said to be:

(1)*upper* (resp., *lower*) *semicontinuous* at
if, for each open set
in
with
(resp.,
), there exists an open neighbourhood
of
in
such that
(resp.,
) for all
,

(2)*upper* (resp., *lower*) *semicontinuous* on
if
is upper (resp., lower) semicontinuous at each point of
,

(3)*continuous* on
if
is both lower and upper semi-continuous on
.

Let
be a convex set in a topological vector space
. Then
is said to be *lower semi-continuous* if, for all
,
is closed in
.

Our main results in this paper are to obtain some existence results of solutions of the generalized bi-quasi-variational inequalities using Chowdhury and Tan's following definition of quasi-pseudo-monotone type II and strongly quasi-pseudo-monotone type II operators given in [3].

Definition 1.3.

Let be a topological vector space, let be a nonempty subset of , and let be a topological vector space over . Let be a bilinear functional. Consider a mapping and two set-valued mappings and .

*quasi-pseudo-monotone*(resp.,

*strongly*

*-quasi-pseudo-monotone*)

*type*II

*operator*if, for any and every net in converging to (resp., weakly to ) with

(2)
is said to be a *quasi-pseudo-monotone* (resp., *strongly quasi-pseudo-monotone*) *type* II *operator* if
is an
-quasi-pseudo-monotone (resp., strongly
-quasi-pseudo-monotone) type II operator with
.

The following is an example on quasi-pseudo-monotone type II operators given in [3].

Example 1.4.

- (i)
To show that is lower semi-continuous, consider . Then . Let be given. Then, if , then . Let be so chosen that . Now, if we take , then, for all , we have . Thus . Hence .

If , . Then, for , we can take . Thus for all , because implies .

- (ii)
To show that is upper semi-continuous, let be such that . Then . Let be an open set in such that . Let be such that . Consider . Then, for all , since . Again, if , then . Let be an open set in such that . Let be such that . Let which is an open neighbourhood of in . Then for all , we have if and if . Now, . Also, for all with , we have . Hence is upper semi-continuous on .

- (iii)

We have

(considering , the value will be also if we consider ). So, it follows that, for all ,

The values can be obtained similarly for the cases where and . Also, it follows that, for all ,

The values can be obtained similarly for the cases when . Therefore, in all the cases, we have shown that

Hence is a quasi-pseudo-monotone type II operator.

The above example is a particular case of a more general result on quasi-pseudo-monotone type II operators. We will establish this result in the following proposition.

Proposition 1.5.

Let be a nonempty compact subset of a topological vector space . Suppose that and are two set-valued mappings such that is lower semi-continuous and is upper semi-continuous. Suppose further that, for any , and are weak*-compact sets in . Then is both a quasi-pseudo-monotone type II and a strongly quasi-pseudo-monotone type II operator.

Proof.

in the last inequality above. Consequently, is both a quasi-pseudo-monotone type II and a strongly quasi-pseudo-monotone type II operator.

In Section 3 of this paper, we obtain some general theorems on solutions for a new class of generalized bi-quasi-variational inequalities for quasi-pseudo-monotone type II and strongly quasi-pseudo-monotone type II operators defined on noncompact sets in topological vector spaces. To obtain these results, we mainly use the following generalized version of Ky Fan's minimax inequality [16] due to Chowdhury and Tan [17].

Theorem 1.6.

Let be a topological vector space, let be a nonempty convex subset of , and let be such that

(a)for any and fixed , is lower semi-continuous on ,

(b)for any and , ,

(c)for any and , every net in converging to with for all and , one has ,

(d)there exist a nonempty closed and compact subset of and such that for all .

Then there exists such that for all .

Now, we use the following lemmas for our main results in this paper.

Lemma 1.7 (see [18]).

Let be a nonempty subset of a Hausdorff topological vector space and let be an upper semi-continuous mapping such that is a bounded subset of for any . Then, for any continuous linear functional on , the mapping defined by is upper semi-continuous; that is, for any , the set is open in .

Let and be topological spaces, let be nonnegative and continuous and let be lower semi-continuous. Then the mapping defined by for all is lower semi-continuous.

## 2. Existence Results

In this section, we will obtain and prove some existence theorems for the solutions to the generalized bi-quasi-variational inequalities of quasi-pseudo-monotone type II and strongly quasi-pseudo-monotone type II operators with noncompact domain in locally convex Hausdorff topological vector spaces. Our results extend and/or generalize the corresponding results in [1].

Before we establish our main results, we state the following result which is Lemma 3.1 in [3].

Lemma 2.1.

Suppose that is continuous over the compact subset of . Then is lower semi-continuous on .

Now, we establish our first main result as follows.

Theorem 2.2.

Let be a locally convex Hausdorff topological vector space over , let be a nonempty paracompact convex and bounded subset of , and let be a Hausdorff topological vector space over . Let be a bilinear functional which is continuous over compact subsets of . Suppose that

(a) is upper semi-continuous such that each is compact and convex,

(b) is convex and is bounded,

(c) is an -quasi-pseudo-monotone type II resp., strongly -quasi-pseudo-monotone type II operator and is upper semi-continuous such that each is compact resp., weakly compact and convex and is strongly bounded,

(d) is an upper semi-continuous mapping such that each is weakly compact and convex,

(e)the set is open in .

Then there exists a point such that

(1) ,

Moreover, if for all , then is not required to be locally convex, and if , then the continuity assumption on can be weakened to the assumption that, for any , the mapping is continuous resp., weakly continuous on .

Proof.

- (ii)

- (iii)
Suppose that , , and is a net in converging to (resp., weakly to ) with for all and .

Case 1 ( ).

Hence, by (2.17) and (2.20), we have .

Case 2 ( ).

Hence we have .

- (iv)

We observe from the above proof that the requirement that is locally convex is needed if and only if the separation theorem is applied to the case . Thus, if is the constant mapping for all , the is not required to be locally convex.

Finally, if , in order to show that, for any , is lower semi-continuous (resp., weakly lower semi-continuous), Lemma 2.1 is no longer needed and the weaker continuity assumption on that, for any , the mapping is continuous (resp., weakly continuous) on is sufficient. This completes the proof.

We will now establish our last result of this section.

Theorem 2.3.

Let be a locally convex Hausdorff topological vector space over , let be a nonempty paracompact convex and bounded subset of , and let be a vector space over . Let be a bilinear functional such that separates points in , is continuous over compact subsets of , and, for any , the mapping is continuous on . Suppose that equips with the strong topology and

(a) is a continuous mapping such that each is compact and convex,

(b) is convex and is bounded,

(c) is an -quasi-pseudo-monotone type II resp., strongly -quasi-pseudo-monotone type II operator and is an upper semi-continuous mapping such that each is strongly, that is, -compact and convex resp., weakly, i.e., -compact and convex ,

Then there exists a point such that

(1) ,

Moreover, if for all , then is not required to be locally convex.

Proof.

The proof is similar to the proof of Theorem 2 in [24] and so the proof is omitted here.

Remark 2.4.

( ) Theorems 2.2 and 2.3 of this paper are generalizations of Theorems 3.2 and 3.3 in [3], respectively, on noncompact sets. In Theorems 2.2 and 2.3, is considered to be a paracompact convex and bounded subset of locally convex Hausdorff topological vector space whereas, in [3], is just a compact and convex subset of . Hence our results generalize the corresponding results in [3].

- (3)
The results in [4] were obtained on compact sets where one of the set-valued mappings is a quasi-pseudo-monotone type I operators which were defined first in [4] and extends the results in [1]. The quasi-pseudo-monotone type I operators are generalizations of pseudo-monotone type I operators introduced first in [17]. In all our results on generalized bi-quasi-variational inequalities, if the operators and the operators are replaced by , then we obtain results on generalized quasi-variational inequalities which generalize the corresponding results in the literature (see [18]).

( ) The results on generalized bi-quasi-variational inequalities given in [5] were obtained for set-valued quasi-semi-monotone and bi-quasi-semi-monotone operators and the corresponding results in [2] were obtained for set-valued upper-hemi-continuous operators introduced in [6]. Our results in this paper are also further extensions of the corresponding results in [2, 5] using set-valued quasi-pseudo-monotone type II operators on noncompact sets.

## Declarations

### Acknowledgment

This work was supported by the Korea Research Foundation Grant funded by the Korean Government (KRF-2008-313-C00050).

## Authors’ Affiliations

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