Generalized Quasi-Variational Inequalities for Pseudo-Monotone Type III and Strongly Pseudo-Monotone Type III Operators on Non-Compact Sets
Mohammad S. R. Chowdhury^{1, *}, Yeol Je Cho^{2, 3}
^{1}Department of Mathematics, University of Management and Technology, Lahore, Pakistan
^{2}Department of Mathematics Education and RINS, Gyeongsang National University, Jinju, Korea
^{3}Department of Mathematics, King Abdulaziz University, Jeddah, Saudi Arabia
Email address:
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
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