1. Introduction
Variational inequality theory has appeared as an effective and powerful tool to study and investigate a wide class of problems arising in pure and applied sciences including elasticity, optimization, economics, transportation, and structural analysis, see for instance [1,2]. In 1966, Browdev [3] first formulated and proved the basic existence theorems of solutions to a class of nonlinear variational inequalities. In 1980, Giannessi [1] introduced the vector variational inequality in a finite dimensional Euclidean space. Since then Chen et al. [4] have intensively studied vector variational inequalities in abstract spaces and have obtained existence theorems for their inequalities.
The pseudo-monotone type operators was first introduced in [5] with a slight variation in the name of this operator. Later these operators were renamed as pseudomonotone operators in [6]. The pseudomonotone operators are set-valued generalization of the classical pseudomonotone operator with slight variations. The classical definition of a single-valued pseudo-monotone operator was introduced by Brezis, Nirenberg and Stampacchia [7].
In this paper we obtained some general theorems on solutions for a new class of generalized quasi variational type inequalities for (η,h)-quasi-pseudo-monotone operators defined as compact sets in topological vector spaces. We have used the generalized version of Ky Fan’s minimax inequality [8] due to Chowdhury and Tan [9].
Let 
and 
be the topological spaces, 
be the mapping and the graph of 
is the set
. In this paper, 
denotes either the real field 
or the complex field
. Let 
be a topological vector space over
, 
be a vector space over 
and 
be a bilinear functional.
For each nonempty subset 
of 
and 
let 
and
for
. Let 
be the (weak) topology on 
generated by the family 
as a subbase for the neighbourhood system at 0 and 
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. The bilinear functional 
separates points in
, i.e., for each
, there exists 
such that
, then 
also becomes Hausdorff. Furthermore, for a net 
in 
and for
1) 
in 
if and only if 
for each 
and 2) 
in 
if and only if 
uniformaly for 
for each nonempty bounded subset 
of
.
Given a set-valued map 
and two set valued maps 
the generalized quasi variational type inequality (GQVTI) problem is to find 
and 
such that 
and
where
.
If
, then generalized quasi variational type inequality (GQVTI) is equivalent to generalized quasi variational inequality (GQVI).
Find 
and 
such that 
and
and 
was introduced by Shih and Tan [10] in 1989 and later was stated by Chowdhury and Tan in [11].
Definition 1. Let 
be a nonempty subset of a topological vector space 
over 
and 
be a topological vector space over
, which is equipped with the 
Let
be a bilinear functional. Suppose we have the following four maps.
1) 
2) 
3) 
4)
.
1) Then 
is said to be an (η,h)-quasi pseudo-monotone type operator if for each 
and every net 
in 
converging to y (or weakly to y) with
We have
2) 
is said to be h-quasi-pseudomonotone operator if 
is (η,h)-quasi-pseudomonotone operator with 
and for some
,
3) a quasi-pseudo monotone operator if 
is an h-quasi pseudo-monotone operator with
.
Remark 1. If 
and 
is replaced by
, then h-quasi-pseudo monotone operator reduces to the h-pseudo monotone operator, see for example [5]. The h-pseudo monotone operator defined in [5] is slightly more general than the definition of h-pseudo monotone operator given in [12]. Also we can find the generalization of quasi-pseudo monotone operator in [11] and for more detail see [13].
Theorem 1. [8] Let 
be a topological vector space, 
be a nonempty convex subset of 
and 
be such that 1) For each 
and each fixed
, 
is lower semicontinuous on
;
2) For each 
and each
, 
;
3) For each 
and each
, every net 
in 
converging to 
with
for all 
and all 
we have
;
4) There exist a nonempty closed compact subset 
of 
and 
such that
Then there exists 
such that
2. Preliminaries
In this section, we shall mainly state some earlier work which will be needed in proving our main results.
Lemma 1. [14] Let 
be a nonempty subset of a Hausdorff topological vector space 
and 
be an upper semicontinuous map such that 
is a bounded subset of 
for each
. Then for each continuous linear functional 
on
, the map 
defined by
the set 
is open in 
.
Lemma 2. [15] Let 
be topological spaces, 
be non-negative and continuous and 
be lower semicontinuous. Then the map
, defined by 
for all
, is lower semicontinuous.
Lemma 3. [11] Let 
be a topological vector space over
, 
be a nonempty compact subset of 
and 
be a Hausdorff topological vector space over
. Let 
be a bilinear functional and 
be an upper semicontinuous map such that each 
is compact. Let 
be a nonempty compact subset of
, 
and 
be continuous. Define 
by
Suppose that 
is continuous on the (compact)
subset 
of
. Then 
is lower semicontinuous on
.
Lemma 4. [11] Let 
be a topological vector space over
, 
be a vector space over 
and 
be a nonempty convex subset of
. Let 
be a bilinear functional, equip 
with the 
topology. Let 
be convex with second argument and 
for all
. Let 
be lower semicontinuous along line segments in 
to the 
-topology on
. Let 
and 
be two maps. Let the continuous map 
be convex with second argument, 
for every
. Suppose that there exists 
such that
, 
is convex and
Then
Theorem 2. [16] Let 
be a nonempty convex subset of a vector space and 
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 map
, i.e., 
is lower semicontinuous and convex on Y and for each fixed
, the map
, i.e., 
is concave on
. Then
3. Existence Result
In this section, we prove the existence theorem for the solutions to the generalized quasi variational type inequalities for (η,h)-quasi-pseudo monotone operator with compact domain in locally convex Hausdorff topological vector spaces.
Theorem 3. Let 
be a locally convex Hausdorff topological vector space over
, 
be a nonempty compact convex subset of 
and 
a Hausdorff topological vector space over
. Let 
be a bilinear continuous functional on compact subset of
. Suppose that 1) 
is upper semicontinuous such that each 
is closed and convex;
2) 
is convex with second argument, 
is lower semicontinuous and 
for
;
3) 
is convex with second argument, 
is continuous and 
for all
;
4) 
is an (η,h)-quasi-pseudo-monotone operator and is upper semicontinuous such that each 
is compact, convex and 
is strongly bounded;
5) 
is a linear and upper semicontinuous map in 
such that each 
is (weakly) compact convex;
6) the set
is open in
.
Then there exists 
such that a)
and b) there exists 
with
Moreover if 
for all
, 
is not required to be locally convex and if
, the continuity assumption on 
can be weakened to the assumption that for each
, the map 
is continuous on
.
Proof. We divide the proof into three steps.
Step 1. There exists 
such that 
and
Contrary suppose that for each
, either 
or there exists 
such that
that is for each 
either 
or
. If
, then by a Hahn-Banach separation theorem for convex sets is locally convex Hausdorff topological vector spaces, there exists 
such that
.
For each
, set
.
Then 
is open in 
by Lemma 1 and 
is open in
by hypothesis. Now 
and
is an open covering for
. Since 
is compact subset of
, there exists 
such that 
for
. Let
for 
and 
be a continuous partition of unity on 
subordinated to the covering
. Then 
are continuous non-negative real valued functions on 
such that 
vanishes on 
for each 
and 
for all 
(see [17] p. 83).
Define 
by
for each
. Then we have 1) 
is Hausdorff for each 
and each fixed 
the map
is lower semicontinuous on 
by Lemma 3 and the fact that 
is continuous on
, therefore the map
is lower semicontinuous on 
by Lemma 2. Also for each fixed
,
is continuous on
. Hence for each 
and each fixed
, the map 
is lower semicontinuous on
.
2) for each 
and each
,
. Indeed, if these were false then for some
and some 
(say
, where 
with
), we have
. Then for each
,
So that
which is a contradiction.
Thus we have 
for each 
and each
.
3) Suppose that
, 
and 
is a net in 
converging to 
with 
for all
,
.
Case 1.
.
Note that 
for each 
and 
. Since 
is strongly bounded and 
is a bounded net, therefore
(1)
Also
Thus
(2)
When
, we have 
for all 
i.e.,
(3)
for all
.
Therefore by (3), we have
Thus
(4)
Hence by (2) and (4), we have
.
Case 2.
.
Since
, there exists 
such that 
for all
. When
, we have 
for all
, i.e.,
for all
.
Thus
(5)
Hence
Since
we have
(6)
Since 
for all
. It follows that
(7)
Since 
by (6) and (7), we have
Since 
is (η,h)-quasi pseudomonotone operator, we have
Since
, we have
Thus
(8)
When
, we have 
for all
, i.e.,
for all
.
Thus
(9)
Hence, we have
.
Since 
is a compact subset of the Hausdorff topological vector space
, it is also closed. Now if we take
, then for any
, we have
Thus 
satisfies all the hypothesis of Theorem 1. Hence by Theorem 1, there exists 
such that
(10)
Now the rest of the proof of Step 1 is similar to the proof in Step 1 of Theorem 1 in [11]. Hence Step 1 is proved.
Step 2.
From Step 1, we have 
and
Since 
is a convex subset of 
and 
is linear, continuous along line segments in
, by Lemma 4 we have
Step 3. There exists 
with
By Step 2 and applying Theorem 2 as proved in Step 3 of Theorem 1 in [11], we can show that there exists 
such that
We observe from the above proof that the requirement that 
be locally convex is needed when and only when the separation theorem is applied to the case
. Thus if 
is the constant map 
for all
, 
is not required to be locally convex.
Finally, if
, in order to show that for each
, 
is lower semicontinuous, Lemma 3 is no longer needed and the weaker continuity assumption as 
that for each
, the map 
is continuous on 
is sufficient. This completes the proof.