Abstract and Applied AnalysisVolume 2011 (2011), Article ID 367476, 14 pagesdoi:10.1155/2011/367476

Research Article

A Note on Hölder Continuity of Solution Set for Parametric Vector Quasiequilibrium Problems

Xian-Fu Hu

School of Information Engineering, Chongqing City Management Vocational College, Chongqing 401331, China

Received 20 April 2011; Accepted 23 June 2011

Academic Editor: W. A. Kirk

Copyright © 2011 Xian-Fu Hu. 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 scalarization technique, we extend and sharpen the results in S. Li and X. Li (2011) on the Hölder continuity of the solution sets of parametric vector equilibrium problems to the case of parametric vector quasiequilibrium problems in metric spaces. Furthermore, we also give an example to illustrate that our main results are applicable.

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. For the details, we can refer to [1–7] and the reference therein.

It is well known that stability analysis of solution mapping for parametric vector equilibrium problems or variational inequalities is another important topic in optimization theory and applications. Stability may be understood as lower or upper semicontinuity, continuity, and Lipschitz or Hölder continuity. Recently, the semicontinuity, especially the lower semicontinuity, of solution mappings to parametric vector variational inequalities and parametric vector equilibrium problems has been intensively studied in the literature, such as [8–16]. We observe that rather few works in the literature on the Hölder continuity for parametric vector equilibrium problems, and for this direction one can only refer to [17–29]. Most of the research in the area of stability analysis for parametric variational inequalities and parametric equilibrium problems has been performed under assumptions which implied the local uniqueness of perturbed solutions so that the solution mapping was single valued, see [17–23, 29] and the references therein. Yen [17] obtained the Hölder continuity of the unique solution of a classic perturbed variational inequality by the metric projection method. Mansour and Riahi [18] proved the Hölder continuity of the unique solution for a parametric vector equilibrium problem under the concepts of strong monotonicity and Hölder continuity. Bianchi and Pini [19] introduced the concept of strong pseudomonotonicity and got the Hölder continuity of the unique solution of a parametric vector equilibrium problem. Bianchi and Pini [20] extend the results of [19] to vector equilibrium problems. Anh and Khanh [21] generalized the main results of [19] to the vector case and obtained Hölder continuity of the unique solutions for two classes of perturbed generalized vector equilibrium problem. Anh and Khanh [22] further discussed uniqueness and Hölder continuity of the solutions for perturbed generalized vector equilibrium problems, which improve remarkably the results in [19, 21], and which become properly stronger than the result of [17], when applied to variational inequalities. Anh and Khanh [23] extended the results of [22] to the case of perturbed generalized vector quasiequilibrium problems and obtained the 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. There have also been a few papers to study more general situations where the solution sets of variational inequalities or equilibrium problems may be set-valued. Under the Hausdorff distance and the strong quasimonotonicity, Lee et al. [24] 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 [25] have discussed the Hölder continuity of set-valued solution mappings for a parametric generalized variational inequalities. Li et al. [26] introduced an assumption, which is weaker than the corresponding ones in the literature, 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. [27] extended the results of [26] to perturbed generalized vector quasiequilibrium problems, and improved the main results in [27]. Later, S. Li and X. Li [28] use a scalarization technique to obtain the Hölder continuity of the set-valued solution mappings for a parametric weak vector equilibrium problems with set-valued mappings in general metric spaces.

Motivated by the work reported in [24, 26–28], this paper aims at establishing the Hölder continuity of a solution mapping, which is set valued in general, to a parametric weak vector quasiequilibrium problems, by using a scalarization technique. Of course, the main consequences of our results are different from corresponding results in [26, 27] 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 [28]. However, in this paper, the main conditions are quite explicit under which we can verify directly. Moreover, we compare our results with corresponding ones in [22, 23] and give an example to illustrate the application of our results.

The rest of the paper is organized as follows. In Section 2, we introduce the parametric vector quasiequilibrium problem and define the solution and -solution to parametric vector quasiequilibrium problem. Then, we recall some definitions and their properties which are needed in the sequel. In Section 3, we discuss the Hölder continuity of the solution mapping for the parametric vector quasiequilibrium problem and compare our main results with the corresponding ones in the recent literature. We also give an example 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 three metric linear spaces. Let be the topological dual space of . For any , we introduce the norm , where denotes the value of at . Let be a pointed, closed and convex cone with , where stands for the interior of a subset . Let , for all be the dual cone of . Since , the dual cone of has a weak* compact base. Let , which is a weak* compact base of .

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

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

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

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

Definition 2.1. A vector-valued function is said to be convex on if and only if for any and .

Definition 2.2 (see [11]). A set-valued map is said to be -Hölder continuous at if and only if there exist neighborhoods of and of such that, for all , for all , where and .

Definition 2.3. A set-valued mapping is said to be -Hölder continuous at , -relative to if and only if there exists a neighborhood of such that, for all , for all , where , and .

From of [15, Lemma 3.1], we have

Lemma 2.4. If for each and each , is a -convex set, that is, is a convex set, then

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

In this paper, we use the following notation, for any , If or is unbounded, then . It is known [30] that solution sets of quasicomplementarity problems are in general unbounded. Hence, so are solution sets of quasiequilibrium problems, which are more general problems.

3. Main Results

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

Lemma 3.1. Suppose that and are the given neighborhoods of and , respectively. (a)If is -Hölder continuous at , θ-relative to , then for each , is also -Hölder continuous at , θ-relative to .(b)If for each and , is -Hölder continuous in , then for each , is also -Hölder continuous in .

Proof. (a)By assumption, there exits a neighborhood of , denoted without loss of generality by , such that, for all , for all , Then, for each , we obtain that (b)As the proof of (b) is similar to (a), we omit it. Then the proof is complete.

Theorem 3.2. Assume that for each , the -solution set for (PVQEP) exists in a neighborhood of the considered point . Assume further that the following conditions hold (i) is -Hölder continuous in ;(ii)for all , for all , for all , there exist two constants and such that (iii) is -Hölder continuous at , θ-relative to , and for all , for all , is -Hölder continuous in ;(iv) and Then, for any , there exist open neighborhoods of , of and of , such that, the -solution set on is a singleton and satisfies the following Hölder condition: for all , for all , where .

Proof. For any , let be open (where depend on ). Obviously, is nonempty for each . Let and be fixed. For any , , and , we set for the sake of convenient statement in the sequel. We shall divide the proof of (3.4) into three steps.Step 1. We first show that, for all , for all , Obviously, the conclusion (3.5) is trivial when . So we suppose that . Since and by the Hölder continuity of , there exist and such that Noting that are the -solution to (PVQEP) at parameters , respectively, we obtain By (ii), we have which together with (3.7) yield that Therefore, it follows from Lemma 3.1, (3.6) that This, by (iv), implies that and hence (3.5) holds.Step 2. Now we show that, for all , for all , Without loss of generality, we assume that . Thanks to (i), there exist and such that By the Hölder continuity of , there exist and such that From the definition of -solution for (PVQEP), we have It follows from (ii), (3.15) that which together with (ii) and Lemma 3.1 yields that From (3.13), (3.14), and (3.17), we have Therefore, it follows from (iv) and (3.18) that and the conclusion (3.12) holds.Step 3. Finally, we easily see from (3.5) and (3.12) that for all , for all , and the conclusion (3.4) holds. This implies that Taking and in (3.21), we can get the diameter of is 0, that is, this set is singleton . is similar. Therefore, the uniqueness of -solution for (PVQEP) has been demonstrated, and the proof is complete.

Remark 3.3. If in (iii) of Theorem 3.2 is bounded, then we can take in (iii), since for some , for all . Thus, the condition in Theorem 3.2 can be omitted.

Assumption (ii) of Theorem 3.2 look seemingly complicated. Now, we give a sufficient condition for (ii) of Theorem 3.2. Namely, we have the following result.

Corollary 3.4. Theorem 3.2 is still valid if assumption (ii) is replaced with (ii’) for all is -Hölder strongly monotone in , that is, there exist two constants and such that, for all ,

Proof. We only need to prove that assumption (ii) of Theorem 3.2 holds. Indeed, it follows from (ii') that for all , Noting that linearity of , we have which together with implies that Therefore, and the proof is complete.

Remark 3.5. When , (PVQEP) collapses to the quasiequilibrium problem (QEP) considered by Anh and Khanh [29]. In this case, Corollary 3.4 is same as the Theorem 2.1 of [29]. From Proposition 1.1 of [22] or Corollary 3.4, we can easily see that Theorem 3.2 improves the Theorem 2.1 of [29].

Theorem 3.6. Assume that for each , the -solution set for the problem (PVQEP) exists in a neighborhood of the considered point . Assume further that the following conditions hold: (i) is -Hölder continuous in ;(ii) for all , for all , for all , there exist two constants and such that (iii) is -Hölder continuous at , θ-relative to , and for all , for all , is -Hölder continuous in ;(iv) and ;(v) for all , for all is convex on . 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 3.2, 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 (3.29), there exists such that . From the construction of the neighborhoods and , one has Then, according to Theorem 3.2, the -solution is a nonempty singleton. Hence, in view of Lemma 2.4, is nonempty.Now, we show that (3.28) holds. Indeed, taking any , we need to show that for any , there exists satisfying Since , there exists such that Thanks to (3.29), there exists such that . Thus, by the construction of the neighborhoods and , we have Then, it follows from Theorem 3.2 that Let . Then, (3.31) holds and the proof is complete.

From Corollary 3.4, Theorems 3.2 and 3.6, we can easily obtain the following result.

Corollary 3.7. Assume that for each , the -solution set for the problem (PVQEP) exists in a neighborhood of the considered point . Assume further that the conditions (i), (iii), (iv), and (v) in Theorem 3.6 hold and the condition (ii) in Theorem 3.6 is replaced with (ii’) for all , is -Hölder strongly monotone in , that is, there exist two constants and such that, for all , Then, there exist neighborhoods of and of , such that, the weak solution set on is nonempty and satisfies the following Hölder continuous condition: for all ,

Remark 3.8. Theorem 3.6 and Corollary 3.7 generalize and improve the corresponding results of S. Li and X. Li [28] in the following three aspects. (i)The assumption (H₄) of Theorem 3.1 in S. Li and X. Li [28] is removed.(ii)The Hölder degree of the solution set is remarkably sharpened since the assumption (H₃) of Theorem 3.1 in S. Li and X. Li [28] is replaced with (iii) of Theorem 3.6 and Corollary 3.7.(iii)We extend the result of S. Li and X. Li [28] on the Hölder continuity of the solution set of parametric vector equilibrium problems to parametric vector quasiequilibrium problems in metric spaces. Moreover, it is easy to see that the assumption (H₁) of Theorem 3.1 in S. Li and X. Li [28] implies the assumption (ii) of Theorem 3.6. However, the converse may not hold. Therefore, Theorem 3.2 generalizes and improves Theorem 3.1 of S. Li and X. Li [28] by weakening the corresponding Hölder-related assumptions.

Now, we give an example to illustrate Theorem 3.6, or Corollary 3.7 is applicable when the solution mapping is set valued.

Example 3.9. Let , and . Let be defined by and defined by Consider that and . Obviously, ; is -Hölder continuous in ; is -Hölder continuous 1-relative to ; for all and , is -Hölder continuous; for all , is 2.2-Hölder strongly monotone in . Here, is arbitrary, and . Hence, we take to see (iv) is satisfied. Therefore, Theorem 3.6 (or Corollary 3.7) derives the Holder continuity of the solution around (in fact, for all).

Acknowledgment

This research was partially supported by the National Natural Science Foundation of China (Grant number: 11001287).

References

Q. H. Ansari, “Vector equilibrium problems and vector variational inequalities,” in Vector Variational Inequalities and Vector Equilibria. Mathematical Theories, F. Giannessi, Ed., vol. 38, pp. 1–16, Kluwer Academic Publishers, Boston, Mass, USA, 2000.

Q. H. Ansari and F. Flores-Bazan, “Generalized vector quasi-equilibrium problems with applications,” Journal of Mathematical Analysis and Applications, vol. 277, no. 1, pp. 246–256, 2003.

Q. H. Ansari and J. C. Yao, “An existence result for the generalized vector equilibrium problem,” Applied Mathematics Letters, vol. 12, no. 8, pp. 53–56, 1999.

M. Bianchi, N. Hadjisavvas, and S. Schaible, “Vector equilibrium problems with generalized monotone bifunctions,” Journal of Optimization Theory and Applications, vol. 92, no. 3, pp. 527–542, 1997.

G. Y. Chen, X. Q. Yang, and H. Yu, “A nonlinear scalarization function and generalized quasi-vector equilibrium problems,” Journal of Global Optimization, vol. 32, no. 4, pp. 451–466, 2005.

S. J. Li, K. L. Teo, and X. Q. Yang, “Generalized vector quasi-equilibrium problems,” Mathematical Methods of Operations Research, vol. 61, no. 3, pp. 385–397, 2005.

X. B. Li and S. J. Li, “Existence of solutions for generalized vector quasi-equilibrium problems,” Optimization Letters, vol. 4, no. 1, pp. 17–28, 2010.

S. J. Li, G. Y. Chen, and K. L. Teo, “On the stability of generalized vector quasivariational inequality problems,” Journal of Optimization Theory and Applications, vol. 113, no. 2, pp. 283–295, 2002.

L. Q. Anh and P. Q. Khanh, “Semicontinuity of the solution set of parametric multivalued vector quasiequilibrium problems,” Journal of Mathematical Analysis and Applications, vol. 294, no. 2, pp. 699–711, 2004.

Y. H. Cheng and D. L. Zhu, “Global stability results for the weak vector variational inequality,” Journal of Global Optimization, vol. 32, no. 4, pp. 543–550, 2005.

L. Q. Anh and P. Q. Khanh, “Various kinds of semicontinuity and the solution sets of parametric multivalued symmetric vector quasiequilibrium problems,” Journal of Global Optimization, vol. 41, no. 4, pp. 539–558, 2008.

L. Q. Anh and P. Q. Khanh, “Semicontinuity of the approximate solution sets of multivalued quasiequilibrium problems,” Numerical Functional Analysis and Optimization, vol. 29, no. 1-2, pp. 24–42, 2008.

L. Q. Anh and P. Q. Khanh, “On the stability of the solution sets of general multivalued vector quasiequilibrium problems,” Journal of Optimization Theory and Applications, vol. 135, no. 2, pp. 271–284, 2007.

S. J. Li and C. R. Chen, “Stability of weak vector variational inequality,” Nonlinear Analysis: Theory, Methods & Applications, vol. 70, no. 4, pp. 1528–1535, 2009.

C. R. Chen, S. J. Li, and K. L. Teo, “Solution semicontinuity of parametric generalized vector equilibrium problems,” Journal of Global Optimization, vol. 45, no. 2, pp. 309–318, 2009.

L. Q. Anh and P. Q. Khanh, “Continuity of solution maps of parametric quasiequilibrium problems,” Journal of Global Optimization, vol. 46, no. 2, pp. 247–259, 2010.

N. D. Yen, “Hölder continuity of solutions to a parametric variational inequality,” Applied Mathematics and Optimization, vol. 31, no. 3, pp. 245–255, 1995.

M. A. Mansour and H. Riahi, “Sensitivity analysis for abstract equilibrium problems,” Journal of Mathematical Analysis and Applications, vol. 306, no. 2, pp. 684–691, 2005.

M. Bianchi and R. Pini, “A note on stability for parametric equilibrium problems,” Operations Research Letters, vol. 31, no. 6, pp. 445–450, 2003.

M. Bianchi and R. Pini, “Sensitivity for parametric vector equilibria,” Optimization, vol. 55, no. 3, pp. 221–230, 2006.

L. Q. Anh and P. Q. Khanh, “On the Hölder continuity of solutions to parametric multivalued vector equilibrium problems,” Journal of Mathematical Analysis and Applications, vol. 321, no. 1, pp. 308–315, 2006.

L. Q. Anh and P. Q. Khanh, “Uniqueness and Hölder continuity of the solution to multivalued equilibrium problems in metric spaces,” Journal of Global Optimization, vol. 37, no. 3, pp. 449–465, 2007.

L. Q. Anh and P. Q. Khanh, “Sensitivity analysis for multivalued quasiequilibrium problems in metric spaces: Hölder continuity of solutions,” Journal of Global Optimization, vol. 42, no. 4, pp. 515–531, 2008.

G. M. Lee, D. S. Kim, B. S. Lee, and N. D. Yen, “Vector variational inequality as a tool for studying vector optimization problems,” Nonlinear Analysis: Theory, Methods & Applications, vol. 34, no. 5, pp. 745–765, 1998.

M. A. Mansour and D. Ausssel, “Quasimonotone variational inequalities and quasiconvex programming: quantitative stability,” Pacific Journal of Optimization, vol. 2, no. 3, pp. 611–626, 2006.

S. J. Li, X. B. Li, L. N. Wang, and K. L. Teo, “The Hölder continuity of solutions to generalized vector equilibrium problems,” European Journal of Operational Research, vol. 199, no. 2, pp. 334–338, 2009.

S. J. Li, C. R. Chen, X. B. Li, and K. L. Teo, “Hölder continuity and upper estimates of solutions to vector quasiequilibrium problems,” European Journal of Operational Research, vol. 210, no. 2, pp. 148–157, 2011.

S. J. Li and X. B. Li, “Hölder continuity of solutions to parametric weak generalized Ky fan inequality,” Journal of Optimization Theory and Applications, vol. 149, no. 3, pp. 540–553, 2011.

L. Q. Anh and P. Q. Khanh, “Hölder continuity of the unique solution to quasiequilibrium problems in metric spaces,” Journal of Optimization Theory and Applications, vol. 141, no. 1, pp. 37–54, 2009.

W. Oettli and N. D. Yen, “Quasicomplementarity problems of type ,” Journal of Optimization Theory and Applications, vol. 89, no. 2, pp. 467–474, 1996.