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Abstract and Applied Analysis
Volume 2012, Article ID 236413, 14 pages
http://dx.doi.org/10.1155/2012/236413
Research Article

Hölder Continuity of Solutions to Parametric Generalized Vector Quasiequilibrium Problems

1School of Mathematical Sciences, Inner Mongolia University, Hohhot 010021, China
2College of Science, Chongqing Jiao Tong University, Chongqing 400074, China

Received 2 October 2011; Accepted 2 December 2011

Academic Editor: Svatoslav Staněk

Copyright © 2012 Z. Y. Peng. 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

By using a linear scalarization method, we establish sufficient conditions for the Hölder continuity of the solution mappings to a parametric generalized vector quasiequilibrium problem with set-valued mappings. These results extend the recent ones in the recent literature, (e.g., Li et al. (2009), Li et al. (2011)). Furthermore, two examples are given to illustrate the obtained result.

1. Introduction

The vector equilibrium problem has been attracting great interest because it provides a unified model for several important problems such as vector variational inequalities, vector complementarity problems, vector optimization problems, vector min-max inequality, and vector saddle point problems. Many different types of vector equilibrium problems have been intensively studied for the past years; see, for example, [13] and the references therein.

It is important to derive results for parametric vector equilibrium problems concerning the properties of the solution mapping when the problems data vary. Among many desirable properties of vector equilibrium problems, the stability analysis of solutions is an essential topic in vector optimization theory and applications. In general, stability may be understood as lower (upper) semicontinuity, continuity, Lipschitz and Hölder continuity and so on. Recently, semicontinuity, especially lower semicontinuity, of solution mappings to parametric vector variational inequalities and parametric vector equilibrium problems has been intensively studied in the literature; see [412]. On the other hand, Hölder continuity of solutions to parametric vector equilibrium problems has also been discussed recently; see [1322], although there are less works in the literature devoted to this property than to semicontinuity. There have been many papers devoted to discussing the local uniqueness and Hölder continuity of the solutions to parametric variational inequalities and parametric equilibrium problems; see [1420] and the references therein. Yên [14] obtained Hölder continuity of the unique solution of a classic perturbed variational inequality by the metric projection method. Ait Mansour and Riahi [15] proved Hölder continuity of the unique solution for a parametric vector equilibrium problem under the concepts of strong monotonicity. Bianchi and Pini [16] introduced the concept of strong pseudomonotonicity and got the Hölder continuity of the unique solution of a parametric vector equilibrium problem. Bianchi and Pini [17] extend the results of [16] to vector equilibrium problems. Anh and Khanh [18] generalized the main results of [16] to the vector case and obtained Hölder continuity of the unique solutions for two classes of perturbed generalized vector equilibrium problems. Anh and Khanh [19] further discussed uniqueness and Hölder continuity of the solutions for perturbed generalized vector equilibrium problems, which improved remarkably the results in [16, 18]. Anh and Khanh [20] extended the results of [19] to the case of perturbed generalized vector quasiequilibrium problems and obtained Hölder continuity of the unique solutions.

For general perturbed vector quasiequilibrium problems, it is well known that a solution mapping is, in general, a set-valued one, but not a single-valued one. Naturally, there is a need to study Hölder continuous properties of the set-valued solution mappings. Under the Hausdorff distance and the strong quasimonotonicity, Lee et al. [21] first showed that the set-valued solution mapping for a parametric vector variational inequality is Hölder continuous. Recently, by virtue of the strong quasimonotonicity, Ait Mansour and Aussel [22] discussed Hölder continuity of set-valued solution mappings for parametric generalized variational inequalities. Li et al. [23] introduced an assumption, which is weaker than the corresponding ones of [16, 18], and established the Hölder continuity of the set-valued solution mappings for two classes of parametric generalized vector quasiequilibrium problems in general metric spaces. Li et al. [24] extended the results of [23] to perturbed generalized vector quasiequilibrium problems. Later, S. J. Li and X. B. Li [25] use a scalarization technique to obtain the Hölder continuity of the set-valued solution mappings for a parametric vector equilibrium problem in general metric spaces.

Motivated by the work reported in [21, 23, 25], this paper aims at establishing sufficient conditions for Hölder continuity of the solution sets for a class of parametric generalized vector quasiequilibrium problem (, in short) with set-valued mapping, by using a linear scalarization method. The main results in this paper are different from corresponding results in [23, 24] and overcome the drawback, which requires the knowledge of detailed values of the solution mapping in a neighborhood of the point under consideration. Our main results also extend and improve the corresponding ones in [25].

The rest of the paper is organized as follows. In Section 2, we introduce the () and define the solution and -solution to the (). Then, we recall some notions and definitions which are needed in the sequel. In Section 3, we discuss Hölder continuity of the solution mapping for the () and compare our main results with the corresponding ones in the recent literature. We also give two examples to illustrate that our main results are applicable.

2. Preliminaries

Throughout this paper, if not other specified, and denote the norm and metric in any metric space, respectively. Let denote the closed ball with radius and center 0 in any metric linear spaces. Let , be metric linear spaces. Let be the topological dual space of . Let be a pointed, closed, and convex cone with , where denotes the interior of . Let be the dual cone of . Since , the dual cone of has a compact base. Letting be given, then is a compact base of .

Let and be neighborhoods of considered points and , respectively. Let be a set-valued mapping, and let be a set-valued mapping. For each and , consider the following parameterized generalized vector quasiequilibrium problem of finding such that

For each and , let Let be the solution set of (), that is,

For each , each and , let denote the set of -solution set to (), that is,

Special Case
When , that is, does not depend on , the () reduces to the parametric generalized vector equilibrium problem (PGVEP) considered by Li et al. [23]. If , the () collapses to the quasiequilibrium problem (QEP) considered by Anh and Khanh [26]. If and is a vector-valued mapping, that is, , the () reduce to the parametric Ky Fan inequality (PKI) considered by S. J. Li and X. B. Li [25].

Now we recall some basic definitions and their properties which are needed in this paper.

Definition 2.1 (classical notion). A set-valued mapping is said to be -Hölder continuous at if there is a neighborhood of such that, for all , where and .

Definition 2.2. A set-valued mapping is said to be -Hölder continuous at if and only if there exists neighborhoods of and of such that, , where and .

Definition 2.3 (see [25]). A set-valued mapping is said to be -Hölder continuous with respect to at if and only if there exists neighborhoods of such that, for all , where , and .

Definition 2.4. Let be a set-valued mapping with nonempty values; is called -like convex on if and only if for any and any , there exists such that

Remark 2.5. If for each and each , is -like convex on , then is a convex set.

3. Main Results

In this section, we mainly discuss the Hölder continuity of the solution mappings to ().

Lemma 3.1. Suppose that are the given neighborhoods of , , respectively. If for each , is -Hölder continuous with respect to at , then for any , the function is -Hölder continuous at . If for each and , is -Hölder continuous with respect to on , then for each , is also -Hölder continuous on .

Proof. (a) By assumption, there exists a neighborhood of , such that for all ,
So, for any , there exist and such that
Then, by the linearity of , we have
It follows from , and the structure of that
Therefore, (3.3) and (3.4) together yield that
Since is arbitrary and , we have
Due to the symmetry between and , the same estimate is also valid, that is,
Thus, it follows (3.6) and (3.7) that and the proof is completed.
(b) As the proof of is similar to , we omit it. Then the proof is completed.

Lemma 3.2. If for each and each , is -like convex on , that is, is a convex set, then

Proof. In a similar way to the proof of Lemma  3.1 in [8], with suitable modifications, we can obtain the conclusion.

Theorem 3.3. Assume that for each , the -solution set for () exists in a neighborhood of the considered point . Assume further that the following conditions hold. is -Hölder continuous in . For each , is -Hölder continuous with respect to at . For each and , is -Hölder continuous with respect to on ., there exists two constants and such that and .Then, for any , there exists open neighborhoods of , of and of , such that the -solution set on satisfies the following Hölder condition: , where .

Proof. Let be arbitrarily given. For all , , and , we set for the sake of convenient statement in the sequel. We prove that (3.11) holds by the following three steps.Step 1. We first show that, for all , for all , Obviously, if , we have that (3.12) holds. So we suppose . Since , and by the Hölder continuity of , there exists and such that
Since are -solutions to () at parameters , respectively, we obtain By virtue of (iv), we get that which together with (3.14) yields that Then, from Lemma 3.1, (3.13), we have The assumption (v) yields that Hence, we have that (3.12) holds.
Step 2. Now we show that, for all , for all , Obviously, we only need to prove that (3.19) holds when . By virtue of assumption (i), there exists and such that
By the Hölder continuity of , there exists and such that From the definition of -solution for (), we have From assumptions (ii)–(iv), (3.22), and Lemma 3.1, we have By virtue of (3.20)–(3.21) and (3.23), we can get Therefore, it follows from (v) that and the conclusion (3.19) holds.
Step 3. Finally, by the arbitrariness of , (3.12) and (3.19), we can easily get that and the conclusion (3.11) holds. This completes the proof.

Remark 3.4. Theorem 3.3 generalizes Lemma 3.3 in S. J. Li and X. B. Li [25] from vector-valued version to set valued version. Moreover, the assumption () of Lemma  3.3 in [25] is removed.

Now, we give an example to illustrate that Theorem 3.3 is applicable under the case that the mapping is set valued.

Example 3.5. Let and . Let be defined by and a set-valued mapping defined by Then, . Consider that and . Direct computation shows that .

It can be checked that is -Hölder continuous in ; for all , is -Hölder continuous with respect to at ; for each and , is 3.1-Hölder continuous with respect to on . Here . Take and , for any and for all , we have and also have and . Hence, all assumptions of Theorem 3.3 hold, and thus it is valid.

Theorem 3.6. Assume that for each , the -solution set for () exists in a neighborhood of the considered point . Assume further that the following conditions hold: is -Hölder continuous in ;for each , is -Hölder continuous with respect to at ;for each and , is -Hölder continuous with respect to on ;, there exist two constants and such that , is -like convex on ; and .Then there exist neighborhoods of and of , such that the solution set on is nonempty and satisfies the following Hölder continuous condition, for all :

Proof. Since the system of , which are given by Theorem 3.3, is an open covering of the weak* compact set , there exist a finite number of points from such that Hence, let and . Then and are desired neighborhoods of and , respectively. Indeed, let be given arbitrarily. For any , by virtue of (3.32), there exists such that . From the construction of the neighborhoods and , one has Then, from the assumption of existence for -solution set and Lemma 3.2, is nonempty.
Now, we show that (3.31) holds. Indeed, taking any , we need to show that for any , there exists satisfying
Since , there exists such that It follows from (3.32) that there exists such that . Thus, by the construction of the neighborhoods and , we have Obviously, thanks to Theorem 3.3, we have Let . Then, (3.34) holds, and the proof is complete.

Remark 3.7. Theorem 3.6 generalizes, and improves the corresponding results of S. J. Li and X. B. Li [25] in the following three aspects. The vector-valued mapping is extended to set-valued, and the parametric vector equilibrium problem is extended to the parametric vector quasiequilibrium problem.The assumption () of Theorem 3.1 in [25] is removed.The -convexity of (see Theorem 3.1 in [25]) is extended to -convexlikeness.

In addition, it is easy to see that the assumption (iv) of Theorem 3.6 is different form the assumption () of Theorem 3.1 in S. J. Li and X. B. Li [25].

Moreover, we also can see that the obtained result extends the ones of [23]. Now, we give the following example to illustrate the case.

Example 3.8. Let , and . Let be defined by , and let be a set-valued mapping defined by Consider that and . Then, and .

Obviously, is -Hölder continuous in ; for all , is -Hölder continuous with respect to at ; for each and , is -Hölder continuous with respect to on . Here . Take and , for any and for all , we have and also have and . Therefore, all assumptions of Theorem 3.3 hold, and thus it is applicable.

However, the assumption (ii) of Theorem 3.1 (or (ii) of Theorem 4.1) in [23] does not hold. In fact, for any , for any and , there exists such that for all . Thus, Theorems 3.1 and 4.1 in Li et al. [23] are not applicable.

Acknowledgments

The author would like to thank the anonymous referees for valuable comments and suggestions, which helped to improve the paper. This work was supported by the Natural Science Foundation of China (no. 10831009. 11001287), the Natural Science Foundation Project of ChongQing (no. CSTC, 2010BB9254. 2011AC6104), and the Research Grant of Chongqing Key Laboratory of Operations and System Engineering.

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