ABSTRACT

In this paper, we define the concepts of (η,h)-quasi pseudo-monotone operators on compact set in locally convex Hausdorff topological vector spaces and prove the existence results of solutions for a class of generalized quasi variational type inequalities in locally convex Hausdorff topological vector spaces.

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 for1) 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.

REFERENCES