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Abstract and Applied Analysis

Volume 2014 (2014), Article ID 368098, 6 pages

http://dx.doi.org/10.1155/2014/368098
Research Article

Levitin-Polyak Well-Posedness of an Equilibrium-Like Problem in Banach Spaces

College of Economic Mathematics, Southwestern University of Finance and Economics, Chengdu, Sichuan 611130, China

Received 8 February 2014; Accepted 26 April 2014; Published 21 May 2014

Academic Editor: Qing-bang Zhang

Copyright © 2014 Ru-liang Deng. 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 concept of Levitin-Polyak well-posedness of an equilibrium-like problem in Banach spaces is introduced. Under suitable conditions, some characterizations of its Levitin-Polyak well-posedness are established. Some conditions under which an equilibrium-like problem in Banach spaces is Levitin-Polyak well-posed are also derived.

1. Introduction

In 1966, Tykhonov [1] first established the well-posedness of a minimization problem, which has been known as Tykhonov well-posedness. Since it is important in optimization problems, various concepts of well-posedness have been introduced and studied in past decades. For more about the well-posedness, we refer to [24] and the references therein.

The Tykhonov well-posedness of a constrained minimization problem requires that every minimizing sequence should lie in the constraint set. In many situations, the minimizing sequence produced by a numerical optimization method usually fails to be feasible but gets closer and closer to the constraint set. Levitin and Polyak [5] generalized the concept of Tykhnov well-posedness by requiring the existence and uniqueness of minimizer and the convergence of every generalized minimizing sequence toward the unique minimizer, which has been known as Levitin and Polyak well-posedness. There are a lot of results concerned with Tykhonov well-posedness, LP well-posedness, and their generalizations for minimization problems. For details, we refer to [13, 57].

Recently, the concept of well-posedness has been extended to many other fields, including Nash equilibrium [8], inclusion problems, and fixed point problems [913]. Lemaire [12, 13] studied the relations between the well-posedness of minimization problems, inclusion problems, and fixed point problems. Fang et al. [11] proved that the well-posedness of a general mixed variational inequality is equivalent to the existence and the uniqueness of its solution in the Hilbert space. Recently, Ceng and Yao [9] got some results for the well-posedness of the generalized mixed variational inequality, the corresponding inclusion problem, and the corresponding fixed point problem. On the other hand, Li and Xia [14] considered the Levitin-Polyak well-posedness of a generalized variational inequality in Banach space. And they showed that the Levitin-Polyak well-posedness of a generalized variational inequality is equivalent to the uniqueness and existence of its solutions. However, there has been no result for the Levitin-Polyak well-posedness of an equilibrium-like problem.

Motivated and inspired by the research work going on in this field, in this paper, we extend the notion of Levitin-Polyak well-posedness to an equilibrium-like problem in Banach spaces and give some metric characterizations of its Levitin-Polyak well-posedness. Finally, we derive some conditions under which an equilibrium-like problem is Levitin-Polyak well-posed.

2. Preliminaries

Let be a real reflexive Banach space with its dual and let be a nonempty, closed, and convex subset of . Let be a set-valued mapping, and let be a functional. In this paper, we consider the following equilibrium-like problem associated with :

Definition 1. Let be nonempty subsets of . The Hausdorff metric between and is defined by where with .

Lemma 2 (Nadler’s theorem [7]). Let be a normed vector space and let be the Hausdorff metric on the collection of all nonempty, closed, and bounded subsets of , induced by a metric in terms of , which is defined by , for and in , where with . If and lie in , then, for any and any , there exists such that . In particular, whenever and are compact subsets in , one has .

Definition 3 (see [9]). A nonempty set-valued mapping is said to be(i) -hemicontinuous if, for any , the function from into is continuous at , where is the Hausdorff metric defined on ;(ii) -uniformly continuous if, for all , there exists such that for all with , one has , where is the Hausdorff metric defined on .

Definition 4. Let and be two topological spaces and . A set-valued mapping is said to be upper semicontinuous (u.s.c. in short) at , if for any neighbourhood of , there exists a neighbourhood of such that , for all . If is u.s.c. at each point of , we say that is u.s.c. on .

Definition 5 (see [15]). Let be a nonempty subset of . The measure of noncompactness of the set is defined by where denotes the diameter of the set , for .

Definition 6. Let be a real reflexive Banach space with its dual and let be a set-valued mapping. A functional is said to be monotone with respect to , if for any and , .

Remark 7. If , for all and , it is easy to know that is monotone with respect to which reduces to   being monotone.

We first prove the following proposition.

Proposition 8. Let be a nonempty, closed, and convex subset of   and let be a nonempty compact-valued mapping which is -hemicontinuous. Let be monotone with respect to , continuous in first argument, and concave in third argument. Moreover, , for all , . Then, for a given , the following statements are equivalent:(i)there exists such that , for all ;(ii) , for all , .

Proof. First, we assume that for some , , for all . Because is monotone with respect to , we have

Conversely, suppose that for all , , we obtain

For any given , we define for all . Replacing by in the left-hand side of the last inequality, we have that, for each , This implies that Since is a nonempty compact-valued mapping, and are nonempty compact and hence lie in . From Lemma 2, we get that, for each and for each fixed , there exists a such that Since is compact, without loss of generality, we assume that as . Since is -hemicontinuous, we get that as , This implies that as . Since is continuous in first argument, by we obtain that there exists an such that This completes the proof.

3. Levitin-Polyak Well-Posedness of

In this section, we extend the concepts of Levitin-Poylak well-posedness to the equilibrium-like problem and establish its metric characterizations. Let be a given number, and let , , , and be defined as the previous section.

Definition 9. A sequence is called an LP -approximating sequence for , if there exist with and such that for all and there exists such that

If , then every LP -approximating sequence is LP -approximating. When , we say that is an LP approximating sequence for .

Definition 10. is strongly LP -well-posed if has an unique solution and every LP -approximating sequence converges strongly to the unique solution. In the sequel, strong LP 0-well-posedness is always called as strong LP well-posedness. If , then strong LP -well-posedness implies strong LP -well-posedness.

Definition 11. is strongly LP -well-posed in the generalized sense if has nonempty solution set and every LP -approximating sequence has a subsequence which converges strongly to some point of . In the sequel, strong LP 0-well-posedness in the generalized sense is always called as strong LP well-posedness in the generalized sense. If , then strong LP -well-posedness in the generalized sense implies strong LP -well-posedness in the generalized sense.

Remark 12. If , for all , , then Definitions 10 and 11 reduce to Definitions 3.3 and 3.4 of [14], respectively. Moreover, when is a Hilbert space, , and , Definitions 10 and 11 reduce to Definitions 3.2 and 3.3 of [11], respectively.

To obtain the metric characterizations of LP -well-posedness, we consider the following LP -approximating solution set of :

Theorem 13. Let be a nonempty, closed, and convex subset of and let be a -hemicontinuous and nonempty compact-valued mapping. Let be monotone with respect to , lower semicontinuous in second argument, and concave in third argument. Moreover, , for all , . Then, is strongly LP -well-posed if and only if

Proof. First, we assume that is strongly LP -well-posed and is the unique solution of . It is easy to see that . If as , then there exist constant and sequences with and with such that Because of , by the definition of , for , we obtain and there exists such that Since is closed and convex, then there exists such that . Let ; we get and . This implies that . Thus, is an LP approximating sequence for . By the similar argument, we obtain that is an LP approximating sequence for . So they have to converge strongly to the unique solution of , which contradicts condition (13).

Conversely, suppose that condition (12) holds. Let be an LP -approximating sequence for . Then, there exists with such that , and there exist and such that Since , then there exists such that . It is obvious that . Suppose that ; we get that . From (12), we have that is a Cauchy sequence and converges strongly to a point . Since is monotone with respect to and lower semicontinuous in second argument, it follows from (16) that, for any , , For any , let , for all . Since is a nonempty, closed, and convex subset, we have that . Then, (17) implies that Since is concave in third argument and , for all , , Since is a nonempty compact-valued mapping and -hemicontinuous, by Lemma 2, for each fixed and each , there exists a such that . Since is -hemicontinuous, we get that as . Since is compact, without loss of generality, we assume that as . Thus, we obtain that This implies that as . It follows from (19) that Therefore, solves .

To complete the proof, we only need to prove that has a unique solution. Suppose that has two distinct solutions and . Then, it is obvious that for all and a contradiction to (12). This completes the proof.

Theorem 14. Let be a nonempty, closed, and convex subset of and let be a -hemicontinuous and nonempty compact-valued mapping. Let be monotone with respect to and lower semicontinuous in second argument. Moreover, , for all , . Then, is strongly LP -well-posed in the generalized sense if and only if

Proof. Assume that is strongly LP -well-posed in the generalized sense. Let be the solution set of . Then, is nonempty and compact. Indeed, let be any sequence in . Then, is an LP -approximating sequence for . Since is strongly -well-posed in the generalized sense, has a subsequence which converges strongly to some point of . Thus, is compact. It is easy to see that for all . Now we show that It is easy to see that, for every , Taking into account the compactness of , we obtain To prove (23), it is sufficient to show that Indeed, if as , then there exist and with , and such that where is the closed ball centered at 0 with radius . By the definition of , we know that , and there exists such that Thus, there exists such that . Let ; then, we have with . So is an LP -approximating sequence for . Since is strongly LP -well-posed in the generalized sense, there exists a subsequence of which converges strongly to some point of . This contradicts (28) and so

Conversely, suppose that (23) holds. We first show that is closed for all . Let with ; then, there exists such that and Since is an upper semicontinuous and nonempty compact-valued mapping, there exist a sequence of and some such that . Therefore, it follows from (31) and the lower semicontinuity of that It is obvious that . This implies that and so is nonempty closed for all . Observe that Since , the theorem in page 412 of [15] can be applied and one concludes that is nonempty and compact with Let be an LP -approximating sequence for . Then, there exists with such that , and there exist and such that Since , then there exists such that . It follows that Set ; we get . From (23) and the definition of , we obtain Since is compact, there exists such that From the compactness of , there exists a subsequence of which converges strongly to . Hence, the corresponding subsequence of converges strongly to . Thus, is strongly LP -well-posed in the generalized sense. The proof is complete.

4. Conditions for Levitin-Polyak Well-Posedness

In this section, we get some conditions under which the in Banach spaces is Levitin-Polyak well-posed.

For any , we denote . We have the following result.

Theorem 15. Let be a nonempty, closed, and convex subset of and let be a -hemicontinuous and nonempty compact-valued mapping. Let be monotone with respect to , lower semicontinuous in first and second arguments, and concave in third argument. Moreover, , for all , . If there exists some with such that is compact, then is strongly LP -well-posed in the generalized sense.

Proof. Let be an LP approximating sequence for . Then, there exist and with such that and there exists satisfying Since , then there exists such that . Thus, Let ; we can get . Without loss of generality, suppose that for is sufficiently large. By the compactness of , there exist a subsequence of and such that . It is easy to see that . Furthermore, by the u.s.c. of at and compactness of , there exist a subsequence of and some such that . Since is lower semicontinuous in first and second arguments, it follows from (40) that For any , let , for all ; it is obvious that . Now, from (42), we have By the convexity of , it follows that, for each , we obtain Let in the last inequality; then, we have This shows that solves . Thus, is strongly LP -well-posed in the generalized sense.

Conflict of Interests

The author declares that there is no conflict of interests regarding the publication of this paper.

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