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Abstract and Applied Analysis
Volume 2013 (2013), Article ID 419053, 7 pages
http://dx.doi.org/10.1155/2013/419053
Research Article

Well-Posedness for Generalized Set Equilibrium Problems

Department of Occupational Safety and Health, College of Public Health, China Medical University, Taichung 40421, Taiwan

Received 14 July 2013; Accepted 13 September 2013

Academic Editor: Jen-Chih Yao

Copyright © 2013 Yen-Cherng Lin. 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

We study the well-posedness for generalized set equilibrium problems (GSEP) and propose two types of the well-posed concepts for these problems in topological vector space settings. These kinds of well-posedness arise from some well-posedness in the vector settings. We also study the relationship between these well-posedness concepts and present several criteria for the well-posedness of GSEP. Our results are new or include as special cases recent existing results.

1. Introduction and Preliminaries

Let , , be three topological vector spaces, a nonempty closed convex subset of a closed convex and pointed cone with apex at the origin, and ; that is, is properly closed with nonempty interior and satisfies , for all ; ; and .

The set-valued mapping satisfies for all and for all , and a set-valued mapping is given. The generalized set equilibrium problem (GSEP) is to find an with some such that We denote the set of all solutions for (GSEP) by .

The concept of well-posedness is inspired by numerical methods producing optimizing sequences for optimization problems [1]. There are many cases so that the solutions may not be unique for a minimization problem. A naturally generalized concept of well-posedness which permits the existence but not uniqueness of minimizers and the convergence of some subsequence of every minimizing sequence toward a minimizer. Other more general notions of well-posedness have been introduced in [2] and there are many others in the literature; see, for example, [315]. Our main purpose is to derive some properties of well-posedness for the generalized set equilibrium problems. We also study the relations between these properties.

A minimizing mapping is defined by for all , where and for all . Assume that . We note that for all since for all and for all . denotes the collection of neighborhoods around in , similar notations for and . For any mapping , denotes the union .

We propose some properties that can be easily derived from the definition. For the sake of clarity, we give the following proof.

Lemma 1. (i) for all and for all .
(ii) with if and only if .
(iii) with if and only if .

Proof. (i) If not, there exists for some and . Then and . Hence, which contradicts the fact that for all and for all .
(ii) if and only if if and only if .
(iii) By (i) and (ii), we have with if and only if if and only if .

Definition 2. A sequence is a minimizing sequence for if for every neighborhood of 0, there is , such that for all .

Definition 3. (GSEP) is -well-posed if it satisfies the following conditions: (i) there exists at least one solution, that is, the set ; (ii) for every minimizing sequence and for every , there is such that for all .

Definition 4. For , the -approximate solution set of (GSEP) is defined by for some .

We can easily see that in Definition 4. Indeed, with some if and only if if and only if from Lemma 1(iii).

Definition 5. (GSEP) is -well-posed if it satisfies the following conditions: (i) there exists at least one solution, that is, the set ; (ii) the mapping is upper Hausdorff continuous at ; that is, for every , there exists such that for every .

Definition 3 arises from [8], and Definition 5 is originally proposed by [9].

Definition 6 (see [16, 17]). A set-valued mapping is (i) upper semicontinuous if for every and every open set in with , there exists a neighborhood of such that ; (ii) lower semicontinuous if for every and every open neighborhood of every , there exists a neighborhood of such that for all ; (iii) continuous if it is both upper semicontinuous and lower semicontinuous.

We note that is upper semicontinuous at and is compact; then for any net , , and for any net for each , there exists and a subnet such that . We can refer to [18] for more details. We also note that is lower semicontinuous at if for any net , , implies that there exists net such that . For more details, we refer the reader to [16] or [17]. Another more weaker upper semi-continuity is said above -upper Hausdorff semicontinuous [9]. A mapping is above -upper Hausdorff semicontinuous if for every and every open set , there exists a neighborhood such that . Obviously, the upper Hausdorff continuity is weaker than the upper semi-continuity, and an upper Hausdorff continuous mapping is an above -upper Hausdorff semi-continuous mapping.

2. -Well-Posed and -Well-Posed

In this section, we will discuss the relationship between these two kinds of well-posedness. The first one is given as follows.

Example 7. (i) There is an example that satisfies -well-posed, but not -well-posed. (ii) There is an example that satisfies both -well-posed and -well-posed.
Solution. (i) The first one is inspired by the example of [10]. Let , , , , satisfy for all . The set-valued mapping is defined by for all with .
Then the set of all solutions for (GSEP) is . For any and any , the minimizing mapping is
If we choose , a neighborhood of 0, and , for all , we can easily see that as , and for all . Thus, (GSEP) is not -well-posed.
Nevertheless, for any minimizing sequence for with for all and for every , there exists such that for . This will force that as , and hence for all . Therefore, (GSEP) is -well-posed.
(ii) We modify the above example as follows. Let , , , , , be given the same as in (i). The set-valued mapping is defined by for all with .
Then the set of all solutions for (GSEP) is . For any and any , the minimizing mapping is
Since for any minimizing sequence , and for every , we always have . Thus, (GSEP) is -well-posed. Furthermore, since , for all , we always have . Hence, (GSEP) is -well-posed.

From the above observation, the -well-posed is weaker than -well-posed for (GSEP). What conditions need to be added so that the converse statement can be valid? The following results will be one of the answers.

Proposition 8. (a) If (GSEP) is -well-posed, then it is -well-posed. (b) If (GSEP) is -well-posed, and for every , there exists such that where for all , then (GSEP) is -well-posed.

Proof. For the idea of the proof, we can use the similar direction of [10, Propositions 3 and 4]. For the sake of clarity, we give the proof of (b) as follows. Suppose that (GSEP) is not -well-posed. Then there is a neighborhood of 0, and sequences with and such that This means Since , there exists such that Now, we separate into two cases.
Case 1. If the sequence is a minimizing sequence, then by -well-posedness, for this , there is a such that , for all , which contradicts (8).
Case 2. If the sequence is not a minimizing sequence, then there is a and a subsequence of with a corresponding subsequence such that By relation (10), we have , for all . For this and condition (7), there is a symmetric neighborhood such that , for all . For large enough, . Taking for all . This implies that, for large enough, which contradicts (11). This completes the proof.

We need the following lemma for the next criterion for -well-posedness of (GSEP).

Lemma 9. Let be a regular topological vector spaces, and let be a nonempty compact subset of . Suppose that ; then there is a neighborhood of such that . In particular, .

Proof. Suppose that . For all , we have . Since is regular, there is a neighborhood of such that Since is a nonempty compact subset, the set There exist , such that Let . Then Since , we have

We note that, although every compact regular space is a normal space, Lemma 9 is not so intuitive. Furthermore, if the set is not compact, the conclusion may not hold. For example, we choose and .

Now, we present first criterion of -well-posedness for (GSEP).

Theorem 10. Let , , be three Hausdorff topological vector spaces where is a finite dimensional space and is regular, let be a nonempty closed convex subset of , and let be a closed convex and pointed cone with apex at the origin and . The mapping is upper semi-continuous with nonempty compact values and satisfies for all and for all , and the mapping is upper semi-continuous with nonempty compact values, such that (i) the solution set of (GSEP) is nonempty and bounded; (ii) the minimizing mapping is upper Hausdorff continuous on ; (iii) for all , for all and for all ; (iv) the mapping is above-C-upper Hausdorff continuous on for every , and the mapping is above--concave [19] on for every and ; (v) for every minimizing sequence , and for each , there is a sequence with for each is a bounded sequence in . Then (GSEP) is -well-posedness.

Proof. We prove it by contradiction. Suppose that (GSEP) is not -well-posedness. Then there exists a minimizing sequence and such that for infinitely many , where denotes the unit open ball in . Let us choose a subsequence from so that the relation (17) holds for all elements of the subsequence. Such a subsequence is still a minimizing sequence, and we still denote it by if there is no any confusion. Now, we separate our discussion into two cases.
Case 1. If the sequence is bounded, then it has a convergent subsequence that converges to some point with a corresponding subsequence with for every . By the upper semi-continuity of , there exists a convergent subsequence of (without any confuse, we still denote it by ) converges to some point . From (17), for every . Hence, , and by Lemma 1, we have . By Lemma 9, there is a neighborhood such that Since is a minimizing sequence, for each , we can choose such that . Since is upper Hausdorff continuity of at , we have Hence, for large enough, which contradicts (18).
Case 2. If the sequence is unbounded. Since is bounded, so are and . Then the set is compact. We denote that , where means the boundary of . Since the sequence is unbounded, there is a subsequence with as . Without loss of generality, we may assume that for all . Fix any and let for any , where . After a simple calculation, we can see that as . Hence, has a subsequence that converges to some point . For the similar process in Case 1, we have a subsequence of the corresponding sequence with converges to some point . By the above -concavity of in , we have By condition (v), there is a bounded sequence with for each in . Hence, by (20), we have Next, we claim that Indeed, if . By Lemma 9, there is a neighborhood such that For this , by the above -upper Hausdorff continuity of and the fact that , we have for large enough. Since the sequence is bounded, the left-hand side of (21) will fell into for large enough. But in this situation, we can see that which contradicts (23). Thus, the relation (22) holds. Since and , by condition (iii) we have which contradicts the fact that .
From the discussions of above two cases, (GSEP) is -well-posedness.

Remark 11. Theorem 10 generalize the Theorem 1 of [10] to (GSEP).

Lemma 12. Suppose that and satisfy for all , then for all .

Proof. For any fixed . Choose any ; we have and . There exist and such that and . That is, . Since and , we know that . Thus, , and hence, . Therefore,

Lemma 13. Let , , , be given the same as in Theorem 10, let be as given in Lemma 12, and let be upper semi-continuous with nonempty compact values such that the set is bounded, where . Any sequence satisfies for every neighborhood of 0; there is , such that for all . Then there exists a bounded sequence in with , for all .

Proof. Since is bounded, its closure is compact. By the upper semi-continuity of , is compact. Hence it is bounded, so is . Fixed , a symmetric neighborhood of 0. Since the sequence satisfies for every neighborhood of 0, there is , such that for all . From Lemma 12, we have for all . We can pick some points , and such that for all . Since is symmetric, we have for all . Since is bounded, so is . Therefore, the sequence is bounded.

Now, let us present another criterion for -well-posedness of (GSEP).

Theorem 14. Let , , , , , , be given the same as in Theorem 10. Suppose that (i) the solution set of (GSEP) is nonempty and compact; (ii) for every and , if , then ; (iii) for all , for all and for all ; (iv) the mapping is above -upper Hausdorff continuous on for every , and the mapping is above -concave on for every and ; (v) for each minimizing sequence , and for each , there is a sequence with for each is a bounded sequence in ; (vi) for every and for every , . Then (GSEP) is -well-posedness.

Proof. We prove it by contradiction. Suppose that (GSEP) is not -well-posedness. Hence there exists a minimizing sequence and such that the relation (17) holds. If the sequence is unbounded, by a similar process in Case 2 of Theorem 10 we know that (GSEP) is -well-posedness. If the sequence is bounded, then it has a convergent subsequence that converges to some point with a corresponding subsequence with for every . We still denote it by if there is no confusion. The relation (17) tells us that Since is a minimizing sequence, we can choose a sequence with , where for all . For the same process as in Case 1 of Theorem 10, by the upper semi-continuity of , there exists a convergent subsequence of (without any confusion, we still denote it by ) that converges to some point . Since , by condition (vi), . Then, for each , there is a such that . Since is compact, there is a subsequence of that converges to some point . Now we claim that . Indeed, suppose that . By Lemma 9, there is such that or By the above -upper Hausdorff continuity of , for this , there is such that for all . Thus, from (32), which contradicts the fact that . Hence, . By condition (ii), . Since , by condition (iii), which contradicts (30). Hence (GSEP) is -well-posedness.

Let us present the third criterion for -well-posedness of (GSEP) as follows.

Theorem 15. Let , , , , , , be given the same as in Theorem 10. Suppose that (i) the solution set of (GSEP) is nonempty and bounded; (ii) and for all , for all and for all ; (iii) the mapping is above -concave on for every , and the mapping is lower semi-continuous on , for every ; (iv) for every and for every , . Then (GSEP) is -well-posedness.

Proof. Suppose that (GSEP) is not -well-posedness. Then we have a minimizing sequence and such that the relation (17) holds for all . We can use the same process as in the proof of Theorem 10 under the situation when we replace by . If the sequence is unbounded, combining Lemmas 12 and 13, we have the sequences , , , , , , , and points , , with the same properties as in the proof (Case 2) of Theorem 10. By condition (iii), we have the mapping which is above -concave on for every and the relations (20) and (21) hold. Since the mapping is lower semi-continuous on for every and fix any , there exists such that . For this , by (21), there exists such that Since the mapping is upper semi-continuous, hence it is upper Hausdorff continuous on for every , and , for any given , for large enough. From (34) and (35) and the fact that , we have Next, we claim that Indeed, if not, . By Lemma 9, there exists such that . But this contradicts (36), and hence (37) holds. Since , by condition (ii), which contradicts the fact . Hence, (GSEP) is -well-posed. On the other hand, if the sequence is bounded, the sequences , , , the points , , and the number are the same as in the Case 1 of Theorem 10, so that for all . Hence . Since is minimizing, we can choose a sequence with . By condition (v), . For each , there is such that . Since is compact, there is a subsequence of that converges to some point . Since is upper semi-continuous with nonempty compact values, we have From condition (ii) and the fact that , we have which contradicts the fact . Hence, (GSEP) is -well-posed.

Example 7 tells us that if (GSEP) is -well-posed, then it is -well-posed. But the converse is not true. Proposition 8 proposes a possible condition so that the converse holds. To the end, we state this result as follows.

Corollary 16. Under the framework of Theorem 14 (resp., Theorem 15) the following condition (A) holds: (A) for every , there is a such that where and is given in (7).
Then (GSEP) is -well-posed.

Proof. From Theorem 14 (resp., Theorem 15), (GSEP) is -well-posed. By condition (vi) of Theorem 14 (resp., condition (iv) of Theorem 15), we have for every and . Hence, Combining this with condition (A), the condition (7) holds. Hence, by Proposition 8, (GSEP) is -well-posed.

Acknowledgments

This work was supported by Grant no. NSC101-2115-M-039-001 of the National Science Council of Taiwan which is gratefully acknowledged. The author would like to thank the referees for their useful comments which have helped to improve some results and the presentation of the paper.

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