Abstract and Applied Analysis

VolumeΒ 2011Β (2011), Article IDΒ 683140, 17 pages

http://dx.doi.org/10.1155/2011/683140

## -Well-Posedness for Mixed Quasi Variational-Like Inequality Problems

School of Mathematics, Chongqing Normal University, Chongqing 400047, China

Received 15 January 2011; Revised 18 March 2011; Accepted 14 April 2011

Academic Editor: V.Β Zeidan

Copyright Β© 2011 Jian-Wen Peng and Jing Tang. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

The concepts of -well-posedness, -well-posedness in the generalized sense, L--well-posedness and L--well-posedness in the generalized sense for mixed quasi variational-like inequality problems are investigated. We present some metric characterizations for these well-posednesses.

#### 1. Introduction

Well-posedness plays a crucial role in the stability theory for optimization problems, which guarantees that, for an approximating solution sequence, there exists a subsequence which converges to a solution. The study of well-posedness for scalar minimization problems started from Tykhonov [1] and Levitin and Polyak [2]. Since then, various notions of well-posedness for scalar minimization problems have been defined and studied in [3β8] and the references therein. It is worth noting that the recent study for various types of well-posedness has been generalized to variational inequality problems [9β13], generalized variational inequality problems [14, 15], quasi variational inequality problems [16], generalized quasi variational inequality problems [17], generalized vector variational inequality problems [18], vector quasi variational inequality problems [19], mixed quasi variational-like inequality problems [20], and many other problems.

In this paper, we are interested in investigating four classes of well-posednesses for a mixed quasi variational-like inequality problem. The paper is organized as follows. In Section 2, we introduce the definitions of -well-posedness, -well-posedness in the generalized sense, L--well-posedness and L--well-posedness in the generalized sense for a mixed quasi variational-like inequality problem. In Section 3, some characterizations of -well-posedness, and L--well-posedness for a mixed quasi variational-like inequality problem are obtained. In Section 4, some characterizations of -well-posedness in the generalized sense and L--well-posedness in the generalized sense for a mixed quasi variational-like inequality problem are presented.

#### 2. Preliminaries

Throughout this paper, without other specification, let be a real Banach space with the dual , let be a nonempty closed convex subset of , and let be a set-valued map. Let be a set-valued map with nonempty values, let be a single-valued map, and let be a real-valued function. Ceng et al. [20] introduced the following mixed quasi variational-like inequality problem, which is to find a point such that, for some ,

Denote by the solution set of (MQVLI). Let ; we introduce the notions of several classes of -well-posednesses for (MQVLI).

*Definition 2.1. *A sequence in is an -approximating sequence for (MQVLI) if(i)there exists a sequence in , with ;(ii)there exists a sequence , such that

*Definition 2.2. *(MQVLI) is said to be -well-posed (resp., -well-posed in the generalized sense) if it has a unique solution and every -approximating sequence strongly converges to (resp., if the solution set of (MQVLI) is nonempty and for every -approximating sequence has a subsequence which strongly converges to a point of ).

*Definition 2.3. *A sequence is an L--approximating sequence for (MQVLI) if there exists a real number sequence , such that

*Definition 2.4. *(MQVLI) is said to be L--well-posed (resp., L--well-posed in the generalized sense) if it has a unique solution and every L--approximating sequence strongly converges to (resp., if the solution set of (MQVLI) is nonempty and for every L--approximating sequence has a subsequence which strongly converges to a point of ).

It is worth noting that if , then the definitions of -well-posedness, -well-posedness in the generalized sense, L--well-posedness, and L--well-posedness in the generalized sense for (MQVLI), respectively, reduce to those of the well-posedness, well-posedness in the generalized sense, L-well-posedness, and L-well-posedness in the generalized sense for (MQVLI) in [20]. We also note that Definition 2.2 generalizes and extends -well-posedness and -well-posedness in the generalized sense of variational inequalities in [10] which are related to the continuously differentiable gap function of variational inequalities introduced by Fukushima [21].

In order to investigate the -well-posedness for (MQVLI), we need the following definitions.

We recall the notion of Mosco convergence [22]. A sequence of subsets of Mosco converges to a set if where and are, respectively, the PainlevΓ©-Kuratowski strong limit inferior and weak limit superior of a sequence , that is, where ββ means weak convergence, and ββ means strong convergence.

If , we call the sequence of subsets of Lower Semi-Mosco convergent to a set .

It is easy to see that a sequence of subsets of Mosco converges to a set implies that the sequence also Lower Semi-Mosco converges to the set , but the converse is not true in general.

We will use the usual abbreviations usc and lsc for βupper semicontinuousβ and βlower semicontinuousβ, respectively. For any , will denote the line segment , while and are defined analogously. We will frequently use , and to denote, respectively, the norm topology on , the weak topology on , and the weak* topology on . Given a convex set , a multivalued map will be called upper hemicontinuous, if its restriction on any line segment is usc with respect to the topology on . will be called -monotone if, for any , for all , . We refer the reader to [23, 24] for basic facts about multivalued maps.

Lemma 2.5 (see [25]). *Let be a sequence of nonempty subsets of a Banach space such that *(i)* is convex for every ;*(ii)*;*(iii)*there exists such that . **Then, for every , there exists a positive real number such that
**
where is a closed ball with a center and radius . If is a finite dimensional space, then assumption (iii) can be replaced by *(iii)β²*.*

The following lemmas play important role in this paper.

Lemma 2.6. *Let be a real separable Banach space with the dual , let be a nonempty convex subset of , and let be a set-valued map with nonempty, weakly* compact convex valued, -monotone, and upper hemicontinuous. Let be a single-valued map with , and let be a convex lsc function. Assume that the map is concave for each and usc. If is a convex subset of with the property that, for each and each , then for each , the following conditions are equivalent: *(i)*, such that ,*(ii)*, such that .*

*Proof. *According to the -monotonicity of , (i) (ii) is obvious.

Next prove (ii) (i). Suppose that (ii) holds. Given any , let , for . By the assumptions of for each . It follows from the condition (ii) that for each , there exists such that
Then
which implies that
It follows that for each ,
Since is a weak* compact valued and -usc on the line segment , is )-closed, and )-subcontinuous on , it follows from and that has a subsequence weak* converging to some . By taking the limit of subsequence in (2.10) we get
Define the bifunction on by
For each is weakly* lsc and quasiconvex on the weakly* compact convex set while for each is usc and quasiconcave on the convex set S_{1}. Hence, according to the Sion Minimax Theorem [26],
By (2.11), we have ; hence, , which implies that there exists , such that

Finally, for each , choose , and a sequence in converging to . The function is usc and concave on ; hence its restriction on any line segment is continuous [27, Theoremββ2.35]. Accordingly, (2.14) implies ,
Hence, (i) holds.

Lemma 2.7. *Let be a real Banach space with the dual , let be a nonempty convex subset of , and let be a convex-valued set-valued map from to . Let be a set-valued map with nonempty values, let be a single-valued map with , and let be a convex function. Assume that the function is concave, for each . Then if and only if the following condition holds:
*

*Proof. *The necessity is easy to get; next we start to prove the sufficiency. Let . Since , and is convex-valued, , it follows that
Thus,
which implies that
The above inequality implies, for converging to zero, that is a solution of (MQVLI). This completes the proof.

#### 3. The Characterizations of Well-Posedness for (MQVLI)

In this section, we investigate some metric characterizations of -well-posedness and L--well-posedness for (MQVLI).

For any , we consider the sets

Theorem 3.1. *Let the same assumptions be as in Lemma 2.7. Then, one has the following. *(a)*(MQVLI) is -well-posed if and only if the solution set of (MQVLI) is nonempty and .*(b)*Moreover, if is -monotone, then (MQVLI) is L--well-posed if and only if the solution set of (MQVLI) is nonempty and .*

*Proof. *We only prove (a). The proof of (b) is similar and is omitted here. Suppose that (MQVLI) is -well-posed; then . It follows from Lemma 2.7 that . Suppose by contradiction that exists a real number , such that ; then there exists , with , and , such that . Since the sequences , and are both -approximating sequences for (MQVLI), and strongly converge to the unique solution , and this gives a contradiction. Therefore, .

Conversely, let be an -approximating sequence for (MQVLI). Then there exists a sequence in with and a sequence in with , such that
That is, . It is easy to see and imply that is a singleton point set. Indeed, if there exist two different solutions , then from Lemma 2.7, we know that . Thus, , a contradiction. Let be the unique solution of (MQVLI). It follows from Lemma 2.7 that . Thus, . So strongly converges to . Therefore, (MQVLI) is -well-posed.

Theorem 3.2. *Let be a real separable Banach space with the dual , let be a nonempty closed convex subset of , and let be a single-valued map with , which is -continuous in each of its variables separately. And let be a convex lsc function; let and be two set-valued maps. Assume the following conditions hold: *(i)* is nonempty convex-valued and, for each sequence in converging to , the sequence Lower Semi-Mosco converging to ;*(ii)*for every converging sequence , there exists , such that ;*(iii)* is nonempty, weak* compact convex valued, -monotone, and upper hemicontinuous;*(iv)*the map is concave for each .**Then, (MQVLI) is -well-posed if and only if
*

The proof of the above theorem relies on the following lemma.

Lemma 3.3. *Let the same assumptions be made as in Theorem 3.2. Let in be an -approximating sequence. If converges to some , then is a solution of (MQVLI).*

*Proof. *Since is an -approximating sequence for (MQVLI), there exists a sequence in with and a sequence in with , such that

For each , choose , such that . It follows from and that . It follows from the assumption (i) that . Thus, .

Assumption (ii) applied to the constant sequence , implies that . For every , it follows from assumptions (i) and (ii) and Lemma 2.5 that there exist and such that . Therefore, for sufficiently large, we have . Notice that is -continuous, is lsc, is -monotone, and is an approximating sequence; we have, for every

Thus, for every and every , we get . Let and ; it follows from Lemma 2.6 that there exists such that for all . According to Lemma 2.7, is a solution of (MQVLI).

*Proof of Theorem 3.2. *The necessity follows from Theorem 3.1 and Lemma 2.7. Now we prove the sufficiency. Suppose that (3.3) holds. Let us show that there exists at most one solution of (MQVLI). Indeed, if there exist two different solutions , then from Lemma 2.7, we know that . Thus, , a contradiction. Note also that there exist -approximate sequences for (MQVLI); indeed, for any sequence in with , and any choice of (which is nonempty by assumption), is an -approximate sequence.

Let be an -approximating sequence for (MQVLI); then . In light of (3.3), is a Cauchy sequence and strongly converging to a point . Applying Lemma 3.3, we get that is a solution of (MQVLI) and so (MQVLI) is -well-posed.

Now, we present a result in which assumption (ii) and the monotonicity of are dropped, while the continuity requirements are strengthened.

Theorem 3.4. *Let be a real separable Banach space with the dual , let be a nonempty closed convex subset of , and let be a single-valued map with , which is -continuous. And let be a convex and continuous function, let and be two set-valued maps. Assume the following assumptions hold: *(i)*the multifunction is nonempty convex-valued and for each sequence in converging to , the sequence Lower Semi-Mosco converging to ;*(ii)* is nonempty, weak* compact, and convex valued, -usc;*(iii)*the map is concave for each .**Then, (MQVLI) is -well-posed if and only if (3.3) holds.*

The proof of the above theorem relies on the following lemma.

Lemma 3.5. *Let the assumptions be as in Theorem 3.4. Let in be an -approximating sequence. If converges to some , then is a solution of (MQVLI).*

*Proof. *Since is an -approximating sequence for (MQVLI), there exist a sequence in with and a sequence in , such that

As in Lemma 3.3, we infer . Since Lower Semi-Mosco converges to , for every , there exists a sequence , such that in the strongly topology. Since is -continuous, the sequence converges strongly to . It follows from (ii) and Propositionββ2.19 in [24] that there exists a subsequence of weak* converging to some . It follows from (ii) and Propositionββ2.17 in [24] that is )-closed, and so . Thus, we have
Hence, and so
Applying Lemma 2.7, is a solution of (MQVLI).

*Proof of Theorem 3.4. *The necessity follows from Theorem 3.1 and Lemma 2.7. Now we prove the sufficiency. Suppose that (3.3) holds. It follows from the proof of Theorem 3.2 that there exists at most one solution of (MQVLI) and there exist -approximate sequences for (MQVLI). Let be an -approximating sequence for (MQVLI); then . In light of (3.3), is a Cauchy sequence and strongly converging to a point . Applying Lemma 3.5, we get that is a solution of (MQVLI) and so (MQVLI) is -well-posed.

We have analogous results for L--well-posedness.

Theorem 3.6. *Let be a real separable Banach space with the dual , let be a nonempty closed convex subset of , and let be a single-valued map with , which is -continuous in each of its variables separately. And let be a convex lsc function; let and be two set-valued maps. Assume the following assumptions hold: *(i)*the multifunction is nonempty convex-valued and for each sequence in converging to , the sequence Lower Semi-Mosco converging to ;*(ii)*for every converging sequence , there exists , such that ;*(iii)* is a set-valued map with nonempty, weak* compact convex valued, -monotone and upper hemicontinuous;*(iv)*the map is concave for each .**Then (MQVLI) is L--well-posed if and only if
*

Lemma 3.7. *Let the same assumptions be as in Theorem 3.6. Let in be an L--approximating sequence. If converges to some , then is a solution of (MQVLI).*

*Proof. *Since is an L--approximating sequence for (MQVLI), there exists a sequence in , , such that , and
From the proof of Lemma 3.3, (i) and (ii), we can obtain , , and for each , one has for sufficiently large. It follows from (iii) that for every and every , we have

Let and ; it follows from Lemma 2.6 that there exists such that for all . According to Lemma 2.7, is a solution of (MQVLI).

*Proof of Theorem 3.6. *Assume that (3.9) holds. Let in be an L--approximating sequence for (MQVLI); then there exists a sequence in , such that . It is easy to see that and imply that is a singleton point set. Indeed, if there exist two different solutions , then from Lemma 2.7 and the -monotonicity of , we know that . Thus, , a contradiction. Let be the unique solution of (MQVLI). It follows from Lemma 2.7 and the -monotonicity of that . Thus, . So strongly converge to . It follows from Lemma 3.7 that . Therefore, (MQVLI) is L--well-posed.

Conversely, assume that the problem is L--well-posed, It follows from the -monotonicity of that . Suppose by contradiction that a real number exists, such that ; then there exists , with , and , such that . Since the sequences and are both L--approximating sequences for (MQVLI), and strongly converge to the unique solution , and this gives a contradiction. Therefore, .

Theorem 3.8. *Let be a real separable Banach space, let be a nonempty closed convex subset of , and let be a single-valued map with , which is -continuous. And let be a convex continuous function; be a set-valued map from to . Assume the following assumptions hold: *(i)*the multifunction is nonempty convex-valued and for each sequence in converging to , the sequence Lower Semi-Mosco converging to ;*(ii)* is a set-valued map with nonempty, weak* compact convex-valued, -usc, and -monotone;*(iii)*the map is concave for each .**
Then (MQVLI) is L--well-posed if and only if (3.9) holds.*

Lemma 3.9. *Let the same assumptions be as in Theorem 3.8. Let in be an L--approximating sequence. If converges to some , then is a solution of (MQVLI).*

*Proof. *Since is an L--approximating sequence for (MQVLI), there exists a sequence in , , such that , and

It follows from the Lower Semi-Mosco convergence of and the proof of Lemma 3.3 that . Since Lower Semi-Mosco converges to , for every , there exists a sequence , strongly converging to . For each select . It follows from (ii) and Propositionββ2.19 in [24] that there exists a subsequence of weak* converging to some . It follows from (ii) and Propositionββ2.17 in [24] that is )-closed, and so . By the continuity of and similar argument with the proof of Lemma 3.5, we know that
It follows from (3.12) that
We deduce from the above inequality that
Let ; by Lemma 2.6 we know that there exists , such that
Then using Lemma 2.7, is a solution of (MQVLI).

*Proof of Theorem 3.8. *Assume that (3.9) holds. If in is an L--approximating sequence, then from the proof of Theorem 3.6, we know that converges to some . By Lemma 3.9, is a solution of (MQVLI) and so (MQVLI) is L--well-posed. The converse is exactly same as that in the proof of Theorem 3.6.

#### 4. The Characterizations of -Well-Posed in the Generalized Sense for (MQVLI)

In this section, we investigate some metric characterizations of -well-posedness in the generalized sense for (MQVLI).

*Definition 4.1 (see [8]). *Let be a nonempty subset of . The measure of noncompactness of the set is defined by

*Definition 4.2 (see [8]). *Let be a metric space and let and be nonempty subsets of . The Hausdorff distance between and is defined by
where with .

Theorem 4.3. *Let the same assumptions be as in Lemma 2.7. Then, one has the following. *(a)*(MQVLI) is -well-posed in the generalized sense if and only if the solution set of (MQVLI) is nonempty compact and .*(b)*Moreover, if is -monotone, then (MQVLI) is --well-posed in the generalized sense if and only if the solution set of (MQVLI) is nonempty compact and .*

*Proof. *We only prove (a). The proof of (b) is similar and is omitted here. Assume that (MQVLI) is -well-posed in the generalized sense; then is nonempty and compact. It follows from Lemma 2.7 that . Now we show that . Suppose by contradiction that there exists and , such that . It follows from that is an -approximating sequence for (MQVLI). Since (MQVLI) is -well-posedness in the generalized sense, there exists a subsequence of strongly converging to a point of . This contradicts . Thus .

For the converse, let be an -approximating sequence for (MQVLI); then . It follows from that there exists a sequence , such that . Since is compact, there exists a subsequence of strongly converging to . Thus the corresponding subsequence of is strongly converging to . Therefore, (MQVLI) is -well-posed in the generalized sense.

Theorem 4.4. *Let the same assumptions be as in Theorem 3.4. Then (MQVLI) is -well-posed in the generalized sense if and only if
*

*Proof. *Assume that (MQVLI) is -well-posed in the generalized sense; so . By Theorem 4.3(a), is nonempty compact and . For any , we have
since is compact, . For every , the following relation holds (see, e.g., [13])
It follows from that .

Conversely, assume that (4.3) holds. Then, for any , cl is nonempty closed and increasing with . By (4.3), , where cl is the closure of . By the generalized Cantor theorem [23, page 412], we know that
where is nonempty compact.

Now we show that
It follows from Lemma 2.7 that . So we need to prove that . Indeed, let . Then for every . Given , , for every there exists such that . Hence, and

It follows from (4.8), , and the proof of Lemma 3.3 that .

Since Lower Semi-Mosco converges to , for every , there exists a sequence , such that in the strong topology.

Since is -continuous, the sequence converges strongly to . It follows from (ii) and Propositionββ2.19 in [24] that there exists a subsequence of weak* converging to some . It follows from (ii) and Propositionββ2.17 in [24] that is )-closed, and so . It follows from the proof of Lemma 3.5 that
Hence,
that is,
By Lemma 2.7, we know that . Thus, . It follows from (4.6) and (4.7) that . It follows from the compactness of and Theorem 4.3(a) that (MQVLI) is -well-posed in the generalized sense. The proof is completed.

Theorem 4.5. *Let the same assumptions be as in Theorem 3.8. Then (MQVLI) is L--well-posed in the generalized sense if and only if
*

*Proof. *Assume that (MQVLI) is L--well-posed in the generalized sense. It follows from Lemma 2.7 and the -monotonicity of that . And so , for each . By similar argument with that in the proof of Theorem 4.3(a), we can get as . From the proof of Theorem 4.4, we also obtain
Thus, .

Conversely, assume that (4.13) holds. Then, for any , cl is nonempty closed and increasing with . By (4.13), , where cl is the closure of . By the generalized Cantor theorem [23, Page 412], we know that
where is nonempty compact.

Now we show that
It follows from Lemma 2.7 and the monotonicity of that . So we need to prove that . Indeed, let . Then for every . Given , , for every there exists such that . Hence, and

It follows from (4.17), , and the proof of Lemma 3.3 that .

Since Lower Semi-Mosco converges to , for every , there exists a sequence , such that in the strong topology.

For each select . Since is -usc with weak* compact convex values, we can find a subsequence of weak* converging to some . By the continuity of and similar argument with the proof of Lemma 3.5, we know that
Hence,
We deduce from the above inequality that
By Lemma 2.6 we know that there exist , such that
It follows from Lemma 2.7 that . Thus, . It follows from (4.15) and (4.16) that . It follows from the compactness of and Theorem 4.3(b) that (MQVLI) is L--well-posed in the generalized sense. The problem is completed.

*Remark 4.6. *(i) It is easy to see that if , then by the main results in our paper, we can recover the corresponding results in [20] with the weaker condition Lower Semi-Mosco converging to instead of the condition Mosco converging to .

(ii) The proof methods of Theorems 4.4 and 4.5 are different from those ofTheoremsββ4.1 andββ4.2in [20].

#### Acknowledgments

The authors would like to express their thanks to the referee for helpful suggestions. This research was supported by the National Natural Science Foundation of China (Grants 10771228 and 10831009), the Natural Science Foundation of Chongqing (Grant no. CSTC, 2009BB8240), and the Research Project of Chongqing Normal University (Grant 08XLZ05).

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