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

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

(1.1)

(2)to find a point , a point , and a point such that

(1.2)

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 ,

(1.3)

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

(1.4)

(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

(1.5)

Again, let be a set-valued mapping defined by

(1.6)

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

(1.8)

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

(1.9)

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

(1.10)

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

(1.11)

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 ,

(1.12)

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

(1.13)

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 .

Lemma 1.8 (see [1, 19]).

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.

Theorem 1.9 (see [20, 21]).

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

(1.14)

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

(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 .

Suppose further that there exist a nonempty closed and compact resp., weakly closed and weakly compact subset of and a point such that and

(2.2)

Then there exists a point such that

(1),

(2)there exist a point and a point such that

(2.3)

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

(2.4)

Suppose the contrary. Then, for any , either or there exists such that

(2.5)

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

(2.6)

Let

(2.7)

and, for any , set

(2.8)

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

(2.9)

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

(2.11)

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

(2.12)

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

(2.14)

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

(2.15)

Also, we have

(2.16)

Thus, from (2.15), it follows that

(2.17)

When , we have for all , that is,

(2.18)

Therefore, by (2.18), we have

(2.19)

which implies that

(2.20)

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,

(2.21)

Thus it follows that

(2.22)

Hence, by (2.22), we have

(2.23)

Since , we have

(2.24)

Since for all , it follows that

(2.25)

Since , by (2.24) and (2.25), we have

(2.26)

Since is an -quasi-pseudo-monotone type II (resp., strongly -quasi-pseudo-monotone type (II) operator, we have

(2.27)

Since , we have

(2.28)

and so

(2.29)

When , we have for all , that is,

(2.30)

and so, by (2.29),

(2.31)

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

(2.33)

whenever and whenever for all . Consequently, we have

(2.34)

(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,

(2.35)

Now, the rest of the proof is similar to the proof in Step 1 of Theorem 1 in [24]. Hence we have shown that

(2.36)

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

(2.37)

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

(2.38)

Suppose further that there exist a nonempty closed and compact resp., weakly closed and weakly compact subset of and a point such that and

(2.39)

Then there exists a point such that

(1),

(2)there exist a point and a point with

(2.40)

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.