American Journal of Applied Mathematics
Volume 3, Issue 3-1, June 2015, Pages: 46-53

Generalized Quasi-Variational Inequalities for Pseudo-Monotone Type III and Strongly Pseudo-Monotone Type III Operators on Non-Compact Sets

Mohammad S. R. Chowdhury1, *, Yeol Je Cho2, 3

1Department of Mathematics, University of Management and Technology, Lahore, Pakistan

2Department of Mathematics Education and RINS, Gyeongsang National University, Jinju, Korea

3Department of Mathematics, King Abdulaziz University, Jeddah, Saudi Arabia

Email address:

(M. S. R. Chowdhury)
(M. S. R. Chowdhury)
(Y. J. Cho)

To cite this article:

Mohammad S. R. Chowdhury, Yeol Je Cho. Generalized Quasi-Variational Inequalities for Pseudo-Monotone Type III and Strongly Pseudo-Monotone Type III Operators on Non-Compact Sets. American Journal of Applied Mathematics. Special Issue: Proceedings of the 1st UMT National Conference on Pure and Applied Mathematics (1st UNCPAM 2015). Vol. 3, No. 3-1, 2015, pp. 46-53. doi: 10.11648/j.ajam.s.2015030301.18


Abstract: In this paper, the authors prove some existence results of solutions for a new class of generalized quasi-variational inequalities (GQVI) for pseudo-monotone type III operators and strongly pseudo-monotone type III operators defined on non-compact sets in locally convex Hausdorff topological vector spaces. In obtaining these results on GQVI for pseudo-monotone type III operators, we shall use Chowdhury and Tan’s generalized version [1] of Ky Fan’s minimax inequality [2] as the main tool.

Keywords: Generalized Quasi-Variational Inequalities, Pseudo-Monotone Type III Operators, Locally Convex Topological Vector Spaces


1. Introduction

Let  be a non-empty set, and  be the family of all non-empty subsets of . Let  be a topological vector space. We shall denote by  the continuous dual of , by  the pairing between  and  for  and  and by  the real part of . Given the maps  and , the generalized quasi-variational inequality problem (GQVI) is to find a point  and a point  such that  for all . The GQVI was introduced by Chan and Pang [3] in 1982 when  is finite dimensional and by Shih and Tan [4] in 1985 when  is infinite dimensional.

In [5] we established some existence theorems of generalized variational inequalities and generalized complementarity problems in topological vector spaces for pseudo-monotone type III operators defined as follows:

Definition 1.1. Let  be a topological vector space,  a non-empty subset of  and a map. If , then  is said to be an -pseudo-monotone (respectively, a strongly -pseudo-monotone) type III operator if for each  and every net  in  converging to  (respectively, weakly to ) with

,

we have

 is said to be a pseudo-monotone (respectively, a strongly pseudo-monotone) type III operator if T is an h-pseudo-monotone type III (respectively, a strongly h-pseudo-monotone type III) operator with

The above operators were originally named -hemi-continuous (respectively, strong -hemi-continuous) operators in [5]. Later, in [6], we re-named these operators pseudo-monotone type III operators.

The following result in [5] justified the validity of a set-valued pseudo-monotone (respectively, strongly pseudo-monotone) type III operator.

Proposition 1.1. Let  be a non-empty compact subset of a topological vector space  and  an upper semi-continuous mapping from the relative weak topology on  to the strong topology on , such that each  is a strongly compact subset of . Then  is both a pseudo-monotone and a strongly pseudo-monotone type III operator.

If  is single-valued and continuous, the compactness of  is not required and the following result was obtained in [5]:

Proposition 1.2. Let  be a non-empty bounded subset of a topological vector space  and  a continuous mapping from the relative weak topology on  to the strong topology on . Then  is both a pseudo-monotone and a strongly pseudo-monotone type III operator.

In this paper, we shall first obtain some general theorems on solutions for a new class of generalized quasi-variational inequalities for pseudo-monotone type III operators and strongly pseudo-monotone type III operators defined on non-compact sets in topological vector spaces. In obtaining these results, we shall mainly use the following generalized version of Ky Fan’s minimax inequality [2] due to M.S.R. Chowdhury and K.-K Tan [1].

Theorem 1.3. Let  be a topological vector space,  be a non-empty convex subset of ,  be lower semi-continuous on  for each , and  be such that

(a)       for each  and each fixed ,  is lower semi-continuous on ;

(b)       for each  and each , ;

(c)       for each  and each , every net  in  converging to  with  for all  and all , we have ;

(d)       there exist a non-empty closed and compact subset  of  and  such that  for all .

Then there exists  such that  for all .

2. Preliminaries

Let  be a topological vector space over . Then, for each , each non-empty subset  of  and each , let  and .

Let  be the topology on  generated by the family  as a subbase for the neighborhood system at  and  be the topology on  generated by the family  is a non-empty bounded subset of  and  as a base for the neighborhood system at 0. We note that , when equipped with the topology  or the topology , becomes a locally convex Hausdorff topological vector space. Furthermore, for a net  in  and for , (i)  in  if and only if  for each  and (ii)  in  if and only if  uniformly for  for each non-empty bounded subset  of . The topology  (respectively, ) is called the weak*-topology (respectively, the strong topology) on .

If  is a topological space and  is an open cover for , then a partition of unity subordinated to the open cover  is a family  of continuous real-valued functions  such that

(a)        for all ,

(b)       support  is locally finite and

(c)        for each .

We shall first state the following result which is Lemma 1 of Shih and Tan in [4, pp.334-335]:

Lemma 2.1. Let  be a non-empty subset of a Hausdorff topological vector space  and  be an upper semi-continuous map such that  is a bounded subset of  for each . Then for each continuous linear functional  on , the map  defined by  is upper semi-continuous; i.e. for each , the set  is open in .

The following result is Lemma 3 of Takahashi in [7, p.177] (see also Lemma 3 in [8, pp.68-85]):

Lemma 2.2. Let  and  be topological spaces,  be non-negative and continuous and  be lower semi-continuous. Then the map , defined by  for all , is lower semi-continuous.

We shall need the following Kneser’s minimax theorem in [9, pp.2418-2420] (see also [10, pp.40-41]):

Theorem 2.3. Let  be a non-empty convex subset of a vector space and  be a non-empty 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 semi-continuous and convex on  and for each fixed , the map , i.e.  is concave on . Then

.

The following result is Lemma 3 in [1]:

Lemma 2.4. Let  be a Hausdorff topological vector space, , , and  be upper semi-continuous from  to the weak*-topology on  such that  is weak*-compact. Let  be defined by  for all . Then for each fixed ,  is lower semi-continuous on .

3. Generalized Quasi-Variational Inequalities of Pseudo-Monotone Type III and Strongly Pseudo-Monotone Type III Operators

In this section, we shall obtain some general existence theorems for the solutions to the generalized quasi-variational inequalities for pseudo-monotone type III operators and strongly pseudo-monotone type III operators on non-compact sets.

We shall first establish the following result:

Theorem 3.1. Let  be a locally convex Hausdorff topological vector space,  be a non-empty paracompact convex and bounded subset of  and  be convex with  bounded. Let  be upper semi-continuous such that each  is compact convex and  be an -pseudo-monotone type III (respectively, strongly h-pseudo-monotone type III) operator and be upper semi-continuous from  to the weak*-topology on  for each  and  is strongly bounded. Also, for each ,  is weak*-compact convex. Suppose that the set

is open in  and the following conditions are satisfied:

(a)       for each  and each  and any net  in  converging to , we have whenever , and

(b)       whenever .

Suppose further that there exists a non-empty compact subset  of  and a point  such that  and  for all . Then there exists a point  such that

(i)          and

(ii)       there exists a point  with  for all .

Proof. We shall complete the proof in three steps as follows:

Step 1. There exists a point  such that  and

.

Suppose the contrary. Then for each , either  or there exists  such that ; that is, for each , either  or . If , then by a separation theorem for convex sets in locally convex Hausdorff topological vector spaces, there exists  such that . For each , set

.

Let  and for each , set

.

Then . Since each  is open in  by Lemma 2.1 and  is open in  by hypothesis, { is an open covering for . Since  is paracompact, there is a continuous partition of unity { for  subordinated to the open cover { (see, for example, Theorem VIII.4.2 of Dugundji in [11]), i.e. for each ,  and  are continuous functions such that for each ,  for all  and  for all  and  is locally finite and  for each . Note that for each ,  is continuous on  (see e.g. [12, Corollary 10.1.1, p.83]). Define  by

for each . Then we have the following:

(i)         Since  is Hausdorff, for each  and each fixed , the map

is continuous on  by Lemma 2.3 and the fact that  is continuous on  and therefore the map

 

is lower semi-continuous on  by Lemma 2.2. Also, for each fixed ,

 

is continuous on . Hence, for each  and each fixed , the map  is lower semi-continuous on .

(ii)       For each  and for each , . If this were false, then there exists some  and some , say  with , such that . Then for each ,

,

so that

which is a contradiction.

(iii)      Suppose that , and  is a net in  converging to  with  for all  and all .

Case 1: .

Since  is continuous and , we have . Note that  for each  and . Since  is strongly bounded and  is a bounded set, it follows that

(2.1)

Also, we have

.

Thus it follows that

(2.2)

When , we have  for all , i.e.

(2.3)

for all . Therefore, by (2.3), we have

and so

(2.4)

Hence, by (2.2) and (2.4), we have .

Case 2. .

Since , there exists  such that  for all .

When , we have  for all , i.e.

for all .

Thus

(2.5)

Hence

 

Since , we have

(2.6)

Since  for all , it follows that

(2.7)

Since , by (2.6) and (2.7) we have

Then, by hypothesis (a), we have

Since  is a pseudo-monotone type III operator, we have

.

Then, by hypothesis (b), we have

.

Since , we have

(2.8)

Thus,

Again, when  we have  for all , i.e.

for all .

Thus

(2.9)

Hence, we have .

(iv)     By hypothesis, there exists a non-empty compact (and therefore closed) subset  of  and a point  such that  and  for all .

Thus, for each , . Hence,  and  for all ; also,  whenever  for .

Consequently,

for all .

Thus, the hypothesis of (d) of Theorem 1.3 is satisfied trivially. (If  is a strongly -quasi-pseudo-monotone type III operator, we equip  with the weak topology.) Thus  satisfies all the hypotheses of Theorem 1.3. Hence, by Theorem 1.3, there exists a point  such that  for all , i.e.

(2.10)

for all .

If , then , so that . Choose  such that

.

Then it follows that

.

If  for some , then  and hence

and so . Then we see that  whenever  for . Since  or  for some , it follows that

,

which contradicts (2.10). This contradiction proves Step 1. Hence we have shown that there exists a point  such that  and

.

Step 2. We need to show that there exists a point  such that  for all .

From Step 1, we have

(2.11)

where  is a weak*-compact convex subset of the Hausdorff topological vector space  and  is a convex subset of .

Now, we define  by  for each  and . Then, for each fixed , the mapping  is convex and continuous on  and, for each fixed , the mapping  is concave on . So, we can apply Kneser’s Minimax Theorem (Theorem 2.3) and obtain the following:

.

Hence, by (2.11), we obtain

.

Since  is compact, there exists  such that

for all . This completes the proof.

When  is compact, we obtain the following immediate consequence of Theorem 3.1:

Theorem 3.2. Let  be a locally convex Hausdorff topological vector space,  be a non-empty compact convex subset of  and  be convex with  bounded. Let  be upper semi-continuous such that each  is closed convex and  be an -pseudo-monotone type III (respectively, a strongly -pseudo-monotone type III) operator and be upper semi-continuous from  to the weak*-topology on  for each  and  is strongly bounded. Also, for each ,  is weak*-compact convex. Suppose that the set

is open in  and the following conditions are satisfied:

(a)       For each , each , and any net  in  converging to , we have

(b)       , whenever  and

(c)       , whenever .

Then there exists a point  such that

(i)          and

(ii)       there exists a point  with  for all .

Note that if the map  is, in addition, lower semi-continuous and for each ,  is upper semi-continuous at  in , then the set  in Theorem 3.1 is always open in  and we obtain the following theorem:

Theorem 3.3. Let  be a locally convex Hausdorff topological vector space,  be a non-empty paracompact convex and bounded subset of  and  be convex with  bounded. Let  be continuous such that each  is compact convex,  be an -pseudo-monotone type III (respectively, strongly -pseudo-monotone type III) operator which is upper semi-continuous from  to the weak*-topology on  for each , with  strongly bounded. Also, for each ,  is weak*-compact convex. Suppose that for each ,  is upper semi-continuous at  from the relative topology on  to the strong topology on  and the following conditions are satisfied:

(a)       For each , each , and any net  in  converging to , we have , whenever , and

(b)       , whenever .

Suppose further that there exists a non-empty compact subset  of  and a point  such that  and  for all .

Then there exists a point  such that

(i)          and

(ii)       there exists a point  with  for all .

The proof is similar to the proof of Theorem 3.1 in [13]. But for completeness, we shall include the detailed proof here.

Proof. The proof will follow from Theorem 3.1 if we can show that the set

is open in . To show that  is open in , we start as follows:

Let  be an arbitrary point. We show that there exists an open neighborhood  of  in  such that . Now, by definition of , there exists a point  in  with

.

Let

.

Thus, . Again, let

.

Then  is a strongly open neighborhood of  in  and so  is an open neighborhood of  in . Since  is upper semi-continuous at , there exists an open neighborhood  of  in  such that  for all . Since the mapping  is continuous at , there exists an open neighborhood  of  in  such that

for all .

Since  and  is lower semi-continuous at , there exists an open neighborhood  of  in  such that  for all . Since the mapping  is continuous at , there exists an open neighborhood  of  in  such that

for all .

Let . Then  is an open neighborhood of  in  such that for each , we have the following:

(a)        as ; so, we can choose any .

(b)        as .

(c)        as .

(d)        as .

Hence, we can obtain the following by omitting the details:







Consequently, we have

since . Hence,  for all . Therefore, . But  was arbitrary. Consequently,  is open in . Thus, all the hypotheses of Theorem 3.1 are satisfied. Hence, the conclusion follows from Theorem 3.1. This completes the proof.

When  is compact, we obtain the following theorem:

Theorem 3.4. Let  be a locally convex Hausdorff topological vector space,  be a non-empty compact convex subset of  and  be convex with  bounded. Let  be continuous such that each  is closed convex,  be an -pseudo-monotone type III (respectively, strongly -pseudo-monotone type III) operator which is upper semi-continuous from  to the weak*-topology on  for each , with  strongly bounded. Also, for each ,  is weak*-compact convex. Suppose that for each ,  is upper semi-continuous at  from the relative topology on  to the strong topology on  and the following conditions are satisfied:

(a)       For each , each , and any net  in  converging to , we have  whenever , and

(b)       whenever .

Then there exists a point  such that

(i)          and

(ii)       there exists a point  with  for all .

Remark 3.5. (1) Theorems 3.1, 3.2, 3.3 and 3.4 of this paper are further extensions of the results obtained in [4] on generalized quasi-variational inequalities of pseudo-monotone type III and strongly pseudo-monotone type III operators.

(2) In 1985, Shih and Tan ([4]) obtained results on generalized quasi-variational inequalities in locally convex topological vector spaces and their 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 [4] using pseudo-monotone type III and strongly pseudo-monotone type III operators on non-compact sets.

Acknowledgement

The second author was supported by the Basic Science Research Program through the National Research Foundation of Korea (NRF) funded by the Ministry of Science, ICT and Future Planning (2014R1A2A2A01002100).


References

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