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Generalized Bi-Quasivariational Inequalities for Quasi-Pseudomonotone Type II Operators on Noncompact Sets
Journal of Inequalities and Applications volume 2010, Article number: 237191 (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.
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:
(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 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
.
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 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
.
(1) is called an
-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.
Consider and
. Then
. Let
be a set-valued mapping defined by

Again, let be a set-valued mapping defined by

Then is lower semi-continuous and
is upper semi-continuous. It can be shown that
becomes a quasi-pseudo-monotone type II operator on
.
-
(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
.
Finally, if , then
. We take
for some
so that
and
. Thus, for all
, we have
. Hence
, where
for
. Consequently,
is lower semi-continuous on
.
-
(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)
Finally, we will show that
is also a quasi-pseudo-monotone 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 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.
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 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.
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 real-valued function on
such that, for each fixed
, the mapping
, that is,
is lower semi-continuous 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 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.
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 semi-continuous 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 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
.
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, Theorem 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, Corollary 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 semi-continuous (resp., weakly lower semi-continuous) on by Lemma 2.1 and so the mapping

is lower semi-continuous (resp., weakly lower semi-continuous) on by Lemma 1.8. Also, for any fixed
,

is continuous on . Hence, for any
and fixed
, the mapping
is lower semi-continuous (resp., weakly lower semi-continuous) 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
-quasi-pseudo-monotone type II (resp., strongly
-quasi-pseudo-monotone 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
-quasi-pseudo-monotone 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 Step 1 of Theorem 1 in [24]. Hence we have shown that

Then, by applying Theorem 1.9 as we proved in Step 3 of Theorem 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 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
,
(d) is an upper semi-continuous mapping such that each
is
-compact convex and, for any
,
is upper semi-continuous 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 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].
() The first paper on generalized bi-quasi-variational inequalities was written by Shih and Tan in 1989 in [1] and the results were obtained on compact sets where the set-valued mappings were either lower semi-continuous or upper semi-continuous. Our present paper is another extension of the original work in [1] using quasi-pseudo-monotone type II operators on noncompact sets.
-
(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.
References
Shih M-H, Tan K-K: Generalized bi-quasi-variational inequalities. Journal of Mathematical Analysis and Applications 1989, 143(1):66–85. 10.1016/0022-247X(89)90029-2
Chowdhury MSR: Generalized bi-quasi-variational inequalities for upper hemi-continuous operators in non-compact settings. Acta Mathematica Hungarica 2001, 92(1–2):111–120.
Chowdhury MSR, Tan K-K: Generalized bi-quasi-variational inequalities for quasi-pseudo-monotone type II operators on compact sets. Central European Journal of Mathematics 2010, 8(1):158–169. 10.2478/s11533-009-0066-8
Chowdhury MSR, Tan K-K: Generalized bi-quasi-variational inequalities for quasi-pseudo-monotone type I operators on compact sets. Positivity 2008, 12(3):511–523. 10.1007/s11117-007-2141-3
Chowdhury MSR, Tarafdar E: Generalized bi-quasi-variational inequalities for quasi-semi-monotone and bi-quasi-monotone operators with applications in non-compact settings and minimization problems. Journal of Inequalities and Applications 2000, 5(1):63–89. 10.1155/S1025583400000059
Chowdhury MSR, Tan K-K: Generalized variational inequalities for quasi-monotone operators and applications. Bulletin of the Polish Academy of Sciences 1997, 45(1):25–54.
Chowdhury MSR, Tan K-K: Generalized variational-like inequalities for pseudo-monotone type III operators. Central European Journal of Mathematics 2008, 6(4):526–536. 10.2478/s11533-008-0049-1
Chowdhury MSR, Tan K-K: Study of generalized quasi-variational inequalities for lower and upper hemi-continuous operators on non-compact sets. Mathematical Inequalities & Applications 1999, 2(1):121–134.
Chowdhury MSR, Tan K-K: Applications of pseudo-monotone operators with some kind of upper semicontinuity in generalized quasi-variational inequalities on non-compact sets. Proceedings of the American Mathematical Society 1998, 126(10):2957–2968. 10.1090/S0002-9939-98-04436-0
Chowdhury MSR, Tan K-K: Generalized quasi-variational inequalities for upper semi-continuous operators on non-compact sets. Nonlinear Analysis: Theory, Methods & Applications 1997, 30(8):5389–5394. 10.1016/S0362-546X(97)00387-8
Chowdhury MSR, Tan K-K: Note on generalized bi-quasi-variational inequalities. Applied Mathematics Letters 1996, 9(3):97–102. 10.1016/0893-9659(96)00039-0
Tarafdar EU, Chowdhury MSR: Topological Methods for Set-Valued Nonlinear Analysis. World Scientific, Hackensack, NJ, USA; 2008:xiv+612.
Chowdhury MSR, Tarafdar E: Existence theorems of generalized quasi-variational inequalities with upper hemi-continuous and demi operators on non-compact sets. Mathematical Inequalities & Applications 1999, 2(4):585–597.
Chowdhury MSR, Tarafdar E, Thompson HB: Non-compact generalized variational inequalities for quasi-monotone and hemi-continuous operators with applications. Acta Mathematica Hungarica 2003, 99(1–2):105–122.
Chowdhury MSR, Thompson HB: Generalized variational-like inequalities for pseudomonotone type II operators. Nonlinear Analysis: Theory, Methods & Applications 2005, 63(5–7):e321-e330.
Fan K: A minimax inequality and applications. In Inequalities, III. Edited by: Shisha O. Academic Press, New York, NY, USA; 1972:103–113.
Chowdhury MSR, Tan K-K: Generalization of Ky Fan's minimax inequality with applications to generalized variational inequalities for pseudo-monotone operators and fixed point theorems. Journal of Mathematical Analysis and Applications 1996, 204(3):910–929. 10.1006/jmaa.1996.0476
Shih MH, Tan K-K: Generalized quasivariational inequalities in locally convex topological vector spaces. Journal of Mathematical Analysis and Applications 1985, 108(2):333–343. 10.1016/0022-247X(85)90029-0
Takahashi W: Nonlinear variational inequalities and fixed point theorems. Journal of the Mathematical Society of Japan 1976, 28(1):168–181. 10.2969/jmsj/02810168
Aubin J-P: Applied Functional Analysis. John Wiley & Sons, New York, NY, USA; 1979:xv+423.
Kneser H: Sur un théorème fondamental de la théorie des jeux. Comptes Rendus de l'Académie des Sciences 1952, 234: 2418–2420.
Dugundji J: Topology. Allyn and Bacon, Inc., Boston, Mass, USA; 1966:xvi+447.
Rockafellar RT: Convex Analysis, Princeton Mathematical Series, no. 28. Princeton University Press, Princeton, NJ, USA; 1970:xviii+451.
Chowdhury MSR, Tan K-K: Application of upper hemi-continuous operators on generalized bi-quasi-variational inequalities in locally convex topological vector spaces. Positivity 1999, 3(4):333–344. 10.1023/A:1009849400516
Acknowledgment
This work was supported by the Korea Research Foundation Grant funded by the Korean Government (KRF-2008-313-C00050).
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Chowdhury, M., Cho, Y. Generalized Bi-Quasivariational Inequalities for Quasi-Pseudomonotone 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
Keywords
- Compact Subset
- Convex Subset
- Compact Convex
- Inequality Problem
- Compact Convex Subset