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Well-posedness for parametric generalized vector quasivariational inequality problems of the Minty type
Journal of Inequalities and Applications volume 2014, Article number: 178 (2014)
Abstract
In this paper, we introduce the concepts of well-posedness, and of well-posedness in the generalized sense for parametric generalized vector quasivariational inequality problems of the Minty type. The necessary and sufficient conditions for the various kinds of well-posedness of these problems are obtained. Our results are different from some main results in the literature and extend them.
MSC:90C31, 49J53, 49J40, 49J45.
1 Introduction and preliminaries
A vector variational inequality in a finite-dimensional Euclidean space was introduced first by Giannessi [1]. Later, this problem has been extended and studied by many authors in abstract spaces; see [2–6]. Moreover, vector variational inequality problems have many important applications in vector optimization problems [7–9], vector equilibria problems [10, 11], and variational relation problems [12, 13].
The concept of well-posedness for unconstrained scalar optimization problems was first introduced and studied by Tykhonov [8], which has become known as Tykhonov well-posedness. In 1966, Levitin and Polyak [14] introduced the concept of well-posedness for constrained scalar optimization problems. With the development of the theory about optimization problems, the concept of well-posedness has been generalized to several related problems, as vector optimization problems, see [15–20], variational inequality problems, see [15, 21–23], equilibria problems, see [24–33] and the references therein. Recently, Fang and Huang [22] studied the well-posedness for a vector variational inequality of the Minty type and the Stampacchia type. Very recently, Lalitha and Bhatia [23] also studied a quasivariational inequality problem of the Minty type, and the well-posedness for this problem was obtained.
Motivated and inspired by the work mentioned, in this paper, we also study the parametric generalized vector quasivariational inequality problems. However, we only study the well-posedness for generalized vector quasivariational inequality problems of the Minty type. The well-posedness for generalized vector quasivariational inequality problems of the Stampacchia type is the same as the Minty type. Let X, Y, Γ, Λ be metric spaces and be a closed, convex, and pointed cone with . The cone C induces a partial ordering in Y defined by
where intC denotes the interior of C.
Let be the space of all linear continuous operators from X into Y, and be a nonempty subset. Let , , and be set-valued mappings. Let , be continuous single-valued mappings. We denote by the value of a linear operator at , and we always assume that is continuous.
Now we adopt the following notations (see [10, 12, 13]). For subsets M and N under consideration we adopt the notations
where w, m, and s are used for weak, middle, and strong, respectively, kinds of considered problems. Let , , and, for , . We consider the following parametric generalized vector quasivariational inequality problems of the Minty type (in short: (MQVIPγλ)).
(MQVIPγλ) Find such that satisfies
Denote by (MQVIP) the family . For each , , and let . We denote by the solution sets of (MQVIPγλ).
Throughout the article, we assume that for each in the neighborhoods .
Next, we recall some basic definitions and some of their properties.
Let X and Z be two topological vector spaces and let be a multifunction.
-
(i)
G is said to be lower semicontinuous (lsc) at if for each open set implies the existence of a neighborhood V of such that , .
-
(ii)
G is said to be upper semicontinuous (usc) at if for each open set , there is a neighborhood V of such that , .
-
(iii)
G is said to be closed at if for each net , , it follows that .
Let X and Z be two topological vector spaces and be a multifunction.
-
(i)
If Z is compact and G is closed at , then G is usc at .
-
(ii)
If G is usc at and is closed, then G is closed at .
The structure of this article is as follows. In the remaining part of this section, we recall definitions for later use. In Section 2, we introduce concepts of well-posedness, and well-posedness in the generalized sense for parametric generalized vector quasivariational inequality problems of the Minty type. Moreover, the necessary and sufficient conditions for the various kinds of well-posedness of these problems are obtained.
2 Main results
Definition 2.1 Let converges to . A sequence is said to be an approximating sequence for (MQVIP) corresponding to , if
-
(i)
, ∀n;
-
(ii)
there exists a sequence that converges to 0 such that
Definition 2.2 The problem (MQVIP) is said to be well-posed at if
-
(i)
the problem (MQVIP) has a unique solution , i.e., ;
-
(ii)
for any sequence converges to , every approximating sequence for (MQVIP) corresponding to converges to .
Definition 2.3 The problem (MQVIP) is said to be well-posed in the generalized sense at if
-
(i)
the solution set of (MQVIP) is nonempty;
-
(ii)
for any sequence that converges to , every approximating sequence for (MQVIP) corresponding to has a subsequence which converges to some point of .
For , , and , we denote the approximate solution set of (MQVIP) by :
Remark 2.4
-
(i)
In the special case, where , , , , , and Q is an identity map, let be a single-valued mapping, then the problem (MQVIPγλ) reduces to the problem (MVVIλ) studied in [22].
-
(ii)
In the special case as in Remark 2.4(i), then Definitions 2.1, 2.2, and 2.3 reduce to Definitions 2.2, 2.5, and 2.6, respectively, of Fang and Huang in [22].
-
(iii)
Well-posedness for vector problems has been defined in different ways. In this paper, we denote instead of ϵe, with ϵ being positive numbers and , i.e., only a fixed direction e is allowed (see [24, 28]).
Remark 2.5 ([36])
Let X and Z be two metric spaces and be a multifunction. If is compact, then G is usc at if and only if for any sequence that converges to and for any sequence , there is a subsequence of converging to some . If, in addition, is a singleton, then the above limit point y must be and the whole converges to .
The following theorem gives sufficient conditions for the well-posedness and the well-posedness in the generalized sense for (MQVIP).
Theorem 2.6 Assume for problem (MQVIP) that
-
(i)
E is usc at and is a compact set;
-
(ii)
in , is lsc;
-
(iii)
in , T is usc and compact-valued if (or ), and lsc if .
Then (MQVIP) is well-posed in the generalized sense at . Moreover, if is a singleton, then this problem is well-posed at .
Proof Since , we have in fact three cases. However, the proof techniques are similar. We consider only the case . We first prove that is upper semicontinuous at . Indeed, we suppose to the contrary the existence of an open subset V of such that for all it converges to , that is, , , for all n. Since E is usc and is compact-valued at , we can assume that tends to for some . If , , such that
By the lower semicontinuity of , T at and , , there exists , such that , . Since , we have
Let be an identity map, by the continuity of η, Q, and , it follows that is continuous (where id is continuous). So (2.1) implies
which is impossible. Hence, belongs to , which is again a contradiction, since , for all n. Therefore, is usc at .
Now we prove that is compact, by checking its closedness. Indeed, let , . This proof is similar to above and so we have and hence is compact. By Remark 2.5, we complete the proof. □
The following example shows that the upper semicontinuity and compactness of E are essential.
Example 2.7 Let , , , , Q be an identity map, , , and be defined by
Then we have and , . We show that assumptions (ii) and (iii) of Theorem 2.6 are fulfilled. But the family is not well-posed in the generalized sense at . The reason is that E is not usc at 0 and is not compact. In fact
The following example shows that the lower semicontinuity of is essential.
Example 2.8 Let , , , , , Q be an identity map, , , and be defined by
We have , . Hence E is usc at 0 and is compact and the conditions (ii) and (iii) of Theorem 2.6 are easily seen to be fulfilled. But the family is not well-posed in the generalized sense at . The reason is that is not lower semicontinuous at . In fact
Theorem 2.9 Assume for problem (MQVIP) the assumptions (ii) and (iii) as in Theorem 2.6 and replace (i) by (i′):
(i′) A is compact, is closed in .
Then (MQVIP) is well-posed in the generalized sense at . Moreover, if is a singleton, then this problem is well-posed at .
Proof We omit the proof since the technique is similar to that for Theorem 2.6 with suitable modifications. □
The following example shows that the compactness of A cannot be dropped.
Example 2.10 Let , , , , H be the identity map, , , and be defined by
We see that is closed at , the assumptions (ii) and (iii) of Theorem 2.9 are satisfied. But the family is not well-posed in the generalized sense at . The reason is that A is not compact. In fact, and , .
The following example shows that the closedness of is essential.
Example 2.11 Let , , , , H be an identity map, , , and be defined by
We show that A is compact and the conditions (ii), (iii) of Theorem 2.9 are easily seen to be fulfilled. But the family is not well-posed in the generalized sense at . The reason is that is not closed at . In fact,
The following example shows that all assumptions of Theorem 2.6 are satisfied.
Example 2.12 Let , , , , , H be an identity map, and let , , and be defined by
Then , . We see that all assumptions of Theorem 2.9 are satisfied. So, the family is well-posed in the generalized sense at . In fact, , .
For , , and positive ξ, we define the following sets of approximate solutions of the family :
where and are the closed balls centered at and with radius ξ.
Observe that, for every ,
-
(i)
;
-
(ii)
.
Theorem 2.13 Assume X is complete and the following conditions hold:
-
(i)
is closed in , and in , is lsc;
-
(ii)
in , T is usc and compact-valued if (or ), and lsc if .
Then (MQVIP) is well-posed at if and only if
Proof Similar arguments can be applied to the three cases. We present only the proof for the case where . If (MQVIP) is well-posed at , then (MQVIP) has a unique solution and hence , , as . If as , then there exist and , , such that , , and
Then there exist such that . Hence there exist , and such that , satisfy
and , satisfy
i.e., and are approximating sequences for (MQVIP) corresponding to and , respectively. Hence, the sequences and converges to the unique solution of (), contradicting the fact that , .
Conversely, let and , and be approximating sequences for (MQVIP) corresponding to and . Then there is such that , satisfying
This yields with , as . Since as , it follows that is Cauchy and converges to a point . By the closedness of at , .
Next, we verify that . Using the same argument as for Theorem 2.6, we deduce that .
Now we prove that () has a unique solution. If has two distinct solutions and , it is not hard to see that , , . It follows that
which is impossible. Hence, (MQVIP) is well-posed at . □
The following example shows that the uniqueness of well-posed is essential.
Example 2.14 Let , , , , , H be an identity map, and let , , and be defined by
We show that the conditions (i) and (ii) of Theorem 2.13 are easily seen to be fulfilled and the family is well-posed at . But as .
The following example shows that all assumptions of Theorem 2.13 are satisfied.
Example 2.15 Let , , , , H be an identity map, and let , , and be defined by
We show that the conditions (i) and (ii) of Theorem 2.13 are easily seen to be fulfilled and and
and and the family is well-posed at , and as .
Next, we consider the following notions of measures of noncompactness.
Let X is complete. The Kuratowski measure of the set is defined by
A, B be nonempty subsets of X. The Hausdorff metric between A and B is defined by
where with .
By the definitions of ζ and H, we have
for every all bounded sets A and B.
The function ζ is a regular measure of noncompactness defined by that satisfies the following conditions:
-
(i)
if and only if the set D is unbounded;
-
(ii)
;
-
(iii)
from it follows that D is a totally bounded set;
-
(iv)
from it follows that ;
-
(v)
if X is a complete space, and if is a sequence of closed subsets of X such that for each and , then is a nonempty compact set and , where H is a Hausdorff metric.
Lemma 2.19 Assume we have problem (MQVIP). Let Γ, Λ be finite dimensional and the following conditions hold:
-
(i)
is closed in , and in , is lsc;
-
(ii)
in , T is usc and compact-valued if (or ), and lsc if .
Then is closed, for all , .
Proof Similar arguments can be applied in the three cases. We present only the proof for the case where . We let such that . Hence, for all , there exist and and , such that
Since and are compact, we can assume that and . By the closedness of at , we find that . We show that , such that
i.e., . Indeed, if , then , such that
By the lower semicontinuity of and T, there exist , such that , , for all n. As we have
By the continuity of Q, η, and id, it follows that is continuous. So we have
and we see a contradiction. Hence . Thus, is closed. □
Next, we provide sufficient conditions for the two sets to coincide.
Lemma 2.20 Assume for problem (MQVIP) the following conditions to hold:
-
(i)
is closed at , and is lsc at ;
-
(ii)
in , T is usc and compact-valued if (or ), and lsc if .
Then , for every .
Proof We present only the proof for the case where . We first prove that . It is easy to see that . Thus, we only need to show that . Indeed, let , there are and such that , satisfying
Since , and is closed, we have . Now we verify that . Indeed, for each , by the semicontinuity of at and the semicontinuity of T at , there exist and such that , . As , we have
By the continuity of Q, η, , , and . We have
i.e.,
Hence
It is clear that
□
The following theorem shows the well-posedness in the generalized sense at for (MQVIP) by using the Kuratowski measure ζ.
Theorem 2.21 Let X be complete, Γ, Λ be finite dimensional and the following conditions hold:
-
(i)
is closed in , and in , is lsc;
-
(ii)
in , T is usc and compact-valued if (or ), and lsc if .
Then (MQVIP) is well-posed in the generalized sense at if and only if
Proof Similar arguments can be applied in the three cases. We present only the proof for the case where . Now we suppose that (MQVIP) is well-posed in the generalized sense at . Let be a solution set of (MQVIPγλ) for all . Then, from Theorem 2.6, we see that is a nonempty compact. Clearly , , . Now we show that
Indeed, since , , . Using the concept of Hausdorff metric, we have
Suppose that , , , for some .
We set .
We claim that . Indeed, let . Then . Since , we see that
Hence, there is k such that , i.e., . So
Note further that
Hence,
Since is compact, , so we have
Now we prove that
Suppose to the contrary that
There are , , and such that
is an approximating sequence of (MQVIP). By the well-posedness in the generalized sense of (MQVIP) at , there is a subsequence of converging to some point of , which is impossible as , . Hence
Conversely, as . By Lemma 2.19, we see that is closed, for all , . By Lemma 2.20, we have
Since as , the regular measure properties of ζ imply that is compact and
Let be an approximating sequence for (MQVIP) corresponding to , where and . There is such that , satisfying
This means that with . We see that
Hence, there is such that
By the compactness of , there is a subsequence of convergent to some point of . Therefore, the corresponding subsequence of tends to . Hence, (MQVIP) is well-posed in the generalized sense at . □
Remark 2.22 In cases as in Remark 2.4(i), Theorems 3.3, 3.4, and 3.5-3.6 in [22] are particular cases of Theorems 2.13, 2.21, and 2.9, respectively. However, the assumptions and our proof methods are very different from Theorems 3.3, 3.4, and 3.5-3.6 in [22].
The following example shows that the closedness of in Theorem 2.21 cannot be dropped.
Example 2.23 Let , , , , , H be an identity map, and let , , and be defined by
We show that is lsc in and the condition (ii) of Theorem 2.21 is easily seen to be fulfilled and . Hence, as . But the family is not well-posed in the generalized sense at . The reason is that is not closed at . Indeed, we let , as and , . It is clear that is convergent to . In fact, .
The following example shows that the lower semicontinuity of in Theorem 2.21 is essential.
Example 2.24 Let , , , , , , , H be an identity map, and let , , and be defined by
We show that is lsc in and the condition (ii) of Theorem 2.21 is easily seen to be fulfilled and . Hence, as . But the family is not well-posed in the generalized sense at . The reason is that is not lower semicontinuous. In fact
The following example shows that all assumptions of Theorem 2.21 are fulfilled.
Example 2.25 Let , , , , , , H be an identity map, and let , , and be defined by
We show that the assumptions (i) and (ii) of Theorem 2.21 are easily seen to be fulfilled and
and . Hence, as , and the family is well-posed in the generalized sense at .
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Acknowledgements
The author is grateful to Prof. Phan Quoc Khanh and Prof. Lam Quoc Anh for their help in the research process. The author also thanks the three anonymous referees for their valuable remarks and suggestions, which helped them to improve considerably the article.
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Hung, N.V. Well-posedness for parametric generalized vector quasivariational inequality problems of the Minty type. J Inequal Appl 2014, 178 (2014). https://doi.org/10.1186/1029-242X-2014-178
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DOI: https://doi.org/10.1186/1029-242X-2014-178