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Generalized BiQuasivariational Inequalities for QuasiPseudomonotone Type II Operators on Noncompact Sets
Journal of Inequalities and Applications volumeÂ 2010, ArticleÂ number:Â 237191 (2010)
Abstract
We prove some existence results of solutions for a new class of generalized biquasivariational inequalities (GBQVI) for quasipseudomonotone type II and strongly quasipseudomonotone type II operators defined on noncompact sets in locally convex Hausdorff topological vector spaces. To obtain these results on GBQVI for quasipseudomonotone type II and strongly quasipseudomonotone type II operators, we use Chowdhury and Tan's generalized version (1996) of Ky Fan's minimax inequality (1972) as the main tool.
1. Introduction and Preliminaries
In this paper, we obtain some results on generalized biquasivariational inequalities for quasipseudomonotone type II and strongly quasipseudomonotone type II operators defined on noncompact sets in locally convex Hausdorff topological vector spaces. Thus we begin this section by defining the generalized biquasivariational 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 biquasivariational 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)variationallike inequalities and generalized biquasivariational inequalities (see [2â€“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 biquasi 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 biquasivariational 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 biquasivariational inequality (GBQVI) problem for the triple (, , ) is:
(1)to find a point and a point such that
(2)to find a point , a point , and a point such that
Let be a nonempty subset of and let be a setvalued mapping. Then is said to be monotone on if, for any , , and .
Let and be topological spaces and let be a setvalued 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 semicontinuous on .
Let be a convex set in a topological vector space . Then is said to be lower semicontinuous if, for all , is closed in .
If is a convex set in a vector space , then is said to be concave if, for all and ,
Our main results in this paper are to obtain some existence results of solutions of the generalized biquasivariational inequalities using Chowdhury and Tan's following definition of quasipseudomonotone type II and strongly quasipseudomonotone 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 setvalued mappings and .
(1) is called an quasipseudomonotone (resp., stronglyquasipseudomonotone) type II operator if, for any and every net in converging to (resp., weakly to ) with
(2) is said to be a quasipseudomonotone (resp., strongly quasipseudomonotone) type II operator if is an quasipseudomonotone (resp., strongly quasipseudomonotone) type II operator with .
The following is an example on quasipseudomonotone type II operators given in [3].
Example 1.4.
Consider and . Then . Let be a setvalued mapping defined by
Again, let be a setvalued mapping defined by
Then is lower semicontinuous and is upper semicontinuous. It can be shown that becomes a quasipseudomonotone type II operator on .

(i)
To show that is lower semicontinuous, 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 .
Finally, if , then . We take for some so that and . Thus, for all , we have . Hence , where for . Consequently, is lower semicontinuous on .

(ii)
To show that is upper semicontinuous, 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 semicontinuous on .

(iii)
Finally, we will show that is also a quasipseudomonotone type II operator. To show this, let us assume first that is a net in such that in . We now show that
(1.7)
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 quasipseudomonotone type II operator.
The above example is a particular case of a more general result on quasipseudomonotone 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 setvalued mappings such that is lower semicontinuous and is upper semicontinuous. Suppose further that, for any , and are weak*compact sets in . Then is both a quasipseudomonotone type II and a strongly quasipseudomonotone type II operator.
Proof.
Suppose that is a net in and with (resp., weakly) and . Then it follows that, for any ,
To obtain the above inequalities, we use the following facts. For any , and . Since is compact and and are weak*compact valued for any , using the lower semicontinuity of and the upper semicontinuity of it can be shown that (details can be verified by the reader easily) and . Thus we obtain
in the last inequality above. Consequently, is both a quasipseudomonotone type II and a strongly quasipseudomonotone type II operator.
In Sectionâ€‰3 of this paper, we obtain some general theorems on solutions for a new class of generalized biquasivariational inequalities for quasipseudomonotone type II and strongly quasipseudomonotone 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 semicontinuous 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 semicontinuous mapping such that is a bounded subset of for any . Then, for any continuous linear functional on , the mapping defined by is upper semicontinuous; that is, for any , the set is open in .
Let and be topological spaces, let be nonnegative and continuous and let be lower semicontinuous. Then the mapping defined by for all is lower semicontinuous.
Let be a nonempty convex subset of a vector space and let be a nonempty compact convex subset of a Hausdorff topological vector space. Suppose that is a realvalued function on such that, for each fixed , the mapping , that is, is lower semicontinuous and convex on and, for each fixed , the mapping , that is, is concave on . Then
2. Existence Results
In this section, we will obtain and prove some existence theorems for the solutions to the generalized biquasivariational inequalities of quasipseudomonotone type II and strongly quasipseudomonotone 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 Lemmaâ€‰3.1 in [3].
Lemma 2.1.
Let be a Hausdorff topological vector space over , let be a vector space over , and let be a nonempty compact subset of . Let be a bilinear functional such that separates points in . Suppose that the equips with the topology; for any , is continuous on and , are upper semicontinuous maps such that and are compact for any . Let and be continuous. Define a mapping by
Suppose that is continuous over the compact subset of . Then is lower semicontinuous 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 semicontinuous such that each is compact and convex,
(b) is convex and is bounded,
(c) is an quasipseudomonotone type II resp., strongly quasipseudomonotone type II operator and is upper semicontinuous such that each is compact resp., weakly compact and convex and is strongly bounded,
(d) is an upper semicontinuous mapping such that each is weakly compact and convex,
(e)the set is open in .
Suppose further that there exist a nonempty closed and compact resp., weakly closed and weakly compact subset of and a point such that and
Then there exists a point such that
(1),
(2)there exist a point and a point such that
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.
We need to show that there exists a point such that and
Suppose the contrary. Then, for any , either or there exists such that
that is, for any , either or . If , then, by a separation theorem for convex sets in locally convex Hausdorff topological vector spaces, there exists such that
Let
and, for any , set
Then Since each is open in by Lemma 1.7 and is open in by hypothesis, is an open covering for . Since is paracompact, there exists a continuous partition of unity for subordinated to the covering (see Dugundji [22, Theoremâ€‰VIII, 4.2]); that is, for any , and are continuous functions such that, for any , for all and for all and , is locally finite and for any . Note that, for any , is continuous on (see [23, Corollaryâ€‰10.1.1]). Define a mapping by
Then we have the following.

(i)
Since is Hausdorff, for any and fixed , the mapping
(2.10)
is lower semicontinuous (resp., weakly lower semicontinuous) on by Lemma 2.1 and so the mapping
is lower semicontinuous (resp., weakly lower semicontinuous) on by Lemma 1.8. Also, for any fixed ,
is continuous on . Hence, for any and fixed , the mapping is lower semicontinuous (resp., weakly lower semicontinuous) on .

(ii)
For any and , . Indeed, if this is false, then, for some and (say where with , we have . Then, for any ,
(2.13)
which implies that
which is a contradiction.

(iii)
Suppose that , , and is a net in converging to (resp., weakly to ) with for all and .
Case 1 ().
Note that for any and . Since is strongly bounded and is a bounded net, it follows that
Also, we have
Thus, from (2.15), it follows that
When , we have for all , that is,
Therefore, by (2.18), we have
which implies that
Hence, by (2.17) and (2.20), we have .
Case 2 ().
Since , there exists such that for any . When , we have for all , that is,
Thus it follows that
Hence, by (2.22), we have
Since , we have
Since for all , it follows that
Since , by (2.24) and (2.25), we have
Since is an quasipseudomonotone type II (resp., strongly quasipseudomonotone type (II) operator, we have
Since , we have
and so
When , we have for all , that is,
and so, by (2.29),
Hence we have .

(iv)
By the hypothesis, there exists a nonempty compact and so a closed (resp., weakly closed and weakly compact) subset of and a point such that and
(2.32)
Thus it follows that
whenever and whenever for all . Consequently, we have
(If is a strongly quasipseudomonotone type II operator, then we equip with the weak topology.) Thus satisfies all the hypotheses of Theorem 1.6 and so, by Theorem 1.6, there exists a point such that for all , that is,
Now, the rest of the proof is similar to the proof in Stepâ€‰1 of Theoremâ€‰1 in [24]. Hence we have shown that
Then, by applying Theorem 1.9 as we proved in Stepâ€‰3 of Theoremâ€‰1 in [24], we can show that there exist a point and a point such that
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 semicontinuous (resp., weakly lower semicontinuous), 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 quasipseudomonotone type II resp., strongly quasipseudomonotone type II operator and is an upper semicontinuous mapping such that each is strongly, that is, compact and convex resp., weakly, i.e., compact and convex,
(d) is an upper semicontinuous mapping such that each is compact convex and, for any , is upper semicontinuous at some point in with , where
Suppose further that there exist a nonempty closed and compact resp., weakly closed and weakly compact subset of and a point such that and
Then there exists a point such that
(1),
(2)there exist a point and a point with
Moreover, if for all , then is not required to be locally convex.
Proof.
The proof is similar to the proof of Theoremâ€‰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 Theoremsâ€‰3.2 andâ€‰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].
() The first paper on generalized biquasivariational inequalities was written by Shih and Tan in 1989 in [1] and the results were obtained on compact sets where the setvalued mappings were either lower semicontinuous or upper semicontinuous. Our present paper is another extension of the original work in [1] using quasipseudomonotone type II operators on noncompact sets.

(3)
The results in [4] were obtained on compact sets where one of the setvalued mappings is a quasipseudomonotone type I operators which were defined first in [4] and extends the results in [1]. The quasipseudomonotone type I operators are generalizations of pseudomonotone type I operators introduced first in [17]. In all our results on generalized biquasivariational inequalities, if the operators and the operators are replaced by , then we obtain results on generalized quasivariational inequalities which generalize the corresponding results in the literature (see [18]).
() The results on generalized biquasivariational inequalities given in [5] were obtained for setvalued quasisemimonotone and biquasisemimonotone operators and the corresponding results in [2] were obtained for setvalued upperhemicontinuous operators introduced in [6]. Our results in this paper are also further extensions of the corresponding results in [2, 5] using setvalued quasipseudomonotone type II operators on noncompact sets.
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Acknowledgment
This work was supported by the Korea Research Foundation Grant funded by the Korean Government (KRF2008313C00050).
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Chowdhury, M., Cho, Y. Generalized BiQuasivariational Inequalities for QuasiPseudomonotone Type II Operators on Noncompact Sets. J Inequal Appl 2010, 237191 (2010). https://doi.org/10.1155/2010/237191
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DOI: https://doi.org/10.1155/2010/237191