Journal of Applied Mathematics

Volume 2012 (2012), Article ID 183040, 12 pages

doi:10.1155/2012/183040

Research Article

The Convergent Behavior for Parametric Generalized Vector Equilibrium Problems

Yen-Cherng Lin,¹ Po-Jen Cheng,² and Su-Ling Lee3,4

¹Department of Occupational Safety and Health, China Medical University, Taichung 404, Taiwan

²Department of Applied Mathematics, National Chiayi University, Chiayi 60004, Taiwan

³Szu Chen Junior High School, Taichung 43441, Taiwan

⁴Department of Applied Statistics, Chung Hua University, Hsinchu 300, Taiwan

Received 14 September 2012; Accepted 24 October 2012

Academic Editor: Jen Chih Yao

Copyright © 2012 Yen-Cherng Lin et al. 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 some properties for parametric generalized vector equilibrium problems and the convergent behavior for the correspondent solution sets of this problem under some suitable conditions. Several existence results and the topological structures of the efficient solutions set are established. Some new results of existence for weak solutions and strong solutions are derived. Finally, we give some examples to illustrate our theory including the example studied by Fang (1992), who established the perturbed nonlinear program and described successfully that the optimal solution of will approach the optimal solution of linear program (P).

1. Introduction and Preliminaries

In recent years, the topological structures of the set of efficient solutions for vector equilibrium problems or generalized systems or variational inequality problems have been discussed in several aspects, as we show in [1–29]. More precisely, we divide this subject into several topics as following. First, the closedness of the set of efficient solutions are studied in [1, 4, 6, 13–16, 27]. Second, the lower semicontinuity of the set of efficient solutions are studied in [1, 9, 10, 19, 21, 23–26, 30]. Third, the upper semicontinuity of the set of efficient solutions are studied in [1, 4, 7, 8, 16, 21, 23–26, 30]. Fourth, the connectedness of the set of efficient solutions are studied in [2, 3, 17, 20, 27, 29]. Fifth, the existence of efficient solutions are studied in [5, 6, 8–12, 16–18, 22, 27, 29, 31].

Gong and Yao [19] establish the lower semicontinuity of the set of efficient solutions for parametric generalized systems with monotone bifunctions in real locally convex Hausdorff topological vector spaces. They also discuss the connectedness of the efficient solutions for generalized systems, we refer to [20]. Luc [27, Chapter 6] investigates the structures of efficient point sets of linear, convex, and quasiconvex problems and also points out that the closedness and connectedness of the efficient solutions sets are important in mathematical programming. Huang et al. [8] discuss a class of parametric implicit vector equilibrium problems in Hausdorff topological vector spaces, where the mappings and are perturbed by parameters, say and , respectively. They establish the upper semicontinuity and lower semicontinuity of the solution mapping for such problems and derive the closedness of the set of efficient solutions. Li et al. [1] discuss the generalized vector quasivariational inequality problem and obtain both upper semicontinuous and lower semicontinuous properties of the set of efficient solutions for parametric generalized vector quasivariational inequality problems. The closedness of the set of efficient solutions is also derived. Cheng [2] discusses the connectedness of the set of weakly efficient solutions for vector variational inequalities in . In 1992, Fang [32] established the perturbed nonlinear program and described successfully that the optimal solution of will approach the optimal solution of linear program . We will state the result in Example 3.7 below. We further point out that, in some suitable conditions, such convergent behavior will display continuity. Furthermore, the correspondent solution sets will preserve some kinds of topological properties under the convergent process. These results will show the convergent behavior about the sets of solutions by two kinds of parameters. As mentioned in [20], for the connectedness, “there are few papers which deal with this subject.” But from above descriptions, we can understand and the topological structures of the sets of efficient solutions for some problems are more and more popular and interesting subjects. On the other hand, for our recent result [15], we study the generalized vector equilibrium problems in real Hausdorff topological vector space settings. The concepts of weak solutions and strong solutions are introduced. Several new results of existence for weak solutions and strong solutions of the generalized vector equilibrium problems are derived. These inspired us to discuss the parametric generalized vector equilibrium problems (PGVEPs). Let us introduce some notations as follows. We will use these notations through all this paper.

Let , and be arbitrary real Hausdorff topological vector spaces, where and are finite dimensional. Let , and be two parametric sets, be a mapping with nonempty values, , a set-valued mapping such that for each , is a proper closed convex and pointed cone with apex at the origin and . For each , we can define relations “” and “” as follows: (1) and (2) . Furthermore, we use the following notations:

Similarly, we can define the relations “” and “” if we replace the set by . If the mapping is constant, then we denote it by . The mappings and are given. The parametric generalized vector equilibrium problem (PGVEP, for short) is as follows: For every , we will like to find an such that

for all and for some . Such set of weak efficient solutions for (PGVEP) is denoted by . If we find and some such that

for all . Such set of efficient solutions for (PGVEP) is denoted by . Our main purpose is to find some topological structures for these two sets, and , of efficient solutions of the parametric generalized vector equilibrium problem. Furthermore, we try to find some sufficient conditions lead them to be nonempty or closed or connected or even compact sets.

2. Some Properties for

Theorem 2.1. Let , and be given as in Section 1, the parametric spaces be two Hausdorff topological vector spaces. Let the mapping be such that is continuous and is -convex for every , the mapping be an upper semicontinuous with nonempty compact values, and the mapping is continuous with nonempty compact and convex values. Suppose that the following conditions hold the following:

(a)for any , , there is an , such that ;(b)the mapping is closed [33] on .

Then, we have

(1)for every , the weak efficient solutions for (PGVEP) exist, that is, the set is nonempty, where for some for all .(2) is upper semicontinuous on with nonempty compact values.

Proof. (1) For any fixed , we can easy check that the mappings , satisfy all conditions of Corollary 2.2 in [15] with and . Hence, from this corollary, we know that is nonempty.

(2) For any fixed , we first claim that is closed in , hence it is compact. Indeed, let a net and for some . Then, and for all and for some . Since is compact, . For each and for each , there exists an such that . Since is upper semicontinuous with nonempty compact values, and the set is compact, is compact. Therefore, without loss of generality, we may assume that the net converges to some . Then . Since the mapping is continuous, we have

Since , and the mapping is closed, we have

This proves that , and hence is closed. Since is compact, so is .

We next prove that the mapping is upper semicontinuous. That is, for any , if there is a net converges to and some , we need to claim that there is a and a subnet of such that . Indeed, since and are upper semicontinuous with nonempty compact values, there is a and a subnet of such that .

If we can claim that , then we can see that is upper semicontinuous on , and complete our proof. Indeed, if not, there is a such that for every we have

Since is lower semicontinuous, there is a net with and . Since , we have and, for each ,

for some .

Since is upper semicontinuous and the net , without loss of generality, we may assume that for some . Since the mapping is continuous, we have

From (2.4) and the closedness of the mapping , we have

which contradicts (2.3). Hence, we have .

3. Some Properties for

In the section, we discuss the set of the efficient solutions for (PGVEP), where there is an , such that for all . The sets of minimal points, maximum points, weak minimal points, and weak maximum points for some set with respect to the cone are denoted by , , , and , respectively. For more detail, we refer the reader to Definition 1.2 of [28].

Theorem 3.1. Under the framework of Theorem 2.1, for each , there is an with . In addition, if is convex, the mapping is properly quasi -convex (Definition 1.1 of [28]) on for each . Assume that the mapping satisfies the following conditions:

(i)

for every ;(ii) for any fixed , if and cannot be comparable with which does not equal to , then ; (iii) if , there exists an such that .

Then, we have

(a)for every , the efficient solutions exists, that is, the set is nonempty, furthermore, it is compact; (b)the mapping is upper semicontinuous on with nonempty compact values; (c)for each , the set is connected if is constant, and for any , and , is convex.

Proof. (a) Fixed any , we can easy see that all conditions of Theorem 2.3 of [15] hold, hence from Theorem 2.3 of [15], we know that is nonempty and compact.

(b) Let be a net such that and be a net with . Since and are upper semicontinuous with nonempty compact values, there are an and a subnet of such that . Since is upper semicontinuous with nonempty compact values, is compact. Since , there is an such that a subnet of converges to . Without loss of generality, we still denote the subnet by , and hence .

If , then there is a such that

Since is compact, there is a net, say , in converges to . Since the mapping is continuous, and the mapping is closed, we have

which contracts (3.2). Thus, .

In order to prove (c), we introduce Lemmas 3.2–3.4 as follows.

Let be the topological dual space of . For each ,

Let , then is nonempty and connected. If is a constant mapping, then for all . In the sequel, we suppose that is not a singleton. That is, , and hence it is connected. For each , let us denote the set of -efficient solutions to (PGVEP) by

Lemma  3.2.   Under the framework of Theorem 3.1,

for every .

Proof. From (a) of Theorem 3.1, we know that, for each , there is an with such that

for all and for all . Thus, for every . Hence, for every .

Lemma  3.3.  Suppose that for any and , are bounded. Then, the mapping is upper semicontinuous with compact values.

Proof. Fixed any . We first claim that the mapping is closed. Let , and with respect to the strong topology in .

Since , there is an such that for all . Since is upper semicontinuous with nonempty compact values, by a similar argument in the proof of Theorem 3.1(b), there is an such that a subnet of converges to . Without loss of generality, we still denote the subnet by .

For each , we define for all . We note that the set is bounded by assumption, hence is well defined and is a seminorm of . For any , let be a neighborhood of 0 with respect to . Since , there is a such that for every . That is, for every . This implies that

for all . Since the mapping is continuous and , we have

By the continuity of , we have

for some and all . Let us choose . Combining (3.8) and (3.10), we know that, for all ,

That is . Since , there is an such that , which proves that . Therefore, the mapping is closed. By the compactness and Corollary 7 in [33, page 112], the mapping is upper semicontinuous with compact values.

Lemma  3.4.  Suppose that for any , and , is convex. Then

Furthermore, if is constant, then we have

Proof. We first claim that .

If , there is a such that . Then, there is a such that

for all . This implies that for all . Indeed, if there is a such that . Since , we have which contracts (3.14). Thus, . This proves (3.12) holds.

Second, if is constant, we claim that .

If , then with and for all , that is, . Hence,

Since is convex, by Eidelheit separation theorem, there is a and such that

for all , , . Then,

for all , , .

Without loss of generality, we denote by , then

for all , , . By the left-hand side inequality of (3.18) and the linearity of , we have for all . Since is closed, for any in the boundary of , there is a net such that . By the continuity of , . Hence, for all , , that is .

By the right-hand side inequality of (3.18), for all , there is an such that for all . This implies that for all if we choose . Hence, for all . Thus, . Therefore, , and hence

Combining this with (3.12), we have

Now, we go back to prove Theorem 3.1(c).

Proof of Theorem 3.1(c). From Lemmas 3.2 and 3.3, the mapping is upper semicontinuous with nonempty compact values. From Lemma 3.4 and Theorem 3.1 [29], we know that for each , the set is connected.

Modifying the Example 3.1 [8], we give the following examples to illustrate Theorems 2.1 and 3.1 as follows.

Example 3.5   3.5. Let , for all , , for all . Choose by for all . Define for all . Then, all the conditions of Theorem 2.1 hold, and for all . Indeed, since there are two choices for , one is , and the other is . If the nonnegative number is less than 1, for any in , and we always choose , then for this case, the set will contain all elements of the set . Furthermore, if we always choose , then the set will contain all elements of the set . If the nonnegative number is greater than or equal to 1, then the set will contain all elements of the set . Hence,

Here, we note that we cannot apply Theorem 3.1 since is not convex.

Example 3.6. Following Example 3.5, let for all . By Theorem 2.1, the set . We choose any , and we can see the mapping is properly quasi -convex on for any . Since for all . So, condition (i) of Theorem 3.1 holds. Obviously, the condition (ii) also holds, since no such exists in this example. Now, we can see condition (iii) holds. Indeed, from the facts

we know that if , then we can choose such that . Hence, we can apply Theorem 3.1, and we know that is nonempty compact and connected. Let us compute the set for any . If we choose any for some , we can see that all the points in the set are efficient solutions for (PGVEP). Hence,

for any .

Example 3.7 (see [32]). The perturbed nonlinear program described successfully that the optimal solutions set of will approach the optimal solutions set of linear program , where and are as follows

where .

We further note that, such convergent behavior will be described by upper semicontinuity by Theorems 2.1 and 3.1. That is,

Furthermore, the correspondent solution sets will preserve some kinds of topological properties, such as compactness and connectedness, under the convergent process.

We would like to point out an open question that naturally raises from Theorems 2.1 and 3.1. Under what conditions the mappings and will be lower semicontinuous?

Acknowledgments

This research of the first author was supported by Grant NSC99-2115-M-039-001- and NSC100-2115-M-039-001- from the National Science Council of Taiwan. The authors would like to thank the reviewers for their valuable comments and suggestions to improve the paper.

References

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.

Y. H. Cheng, “On the connectedness of the solution set for the weak vector variational inequality,” Journal of Mathematical Analysis and Applications, vol. 260, no. 1, pp. 1–5, 2001.

X. H. Gong, “Connectedness of the solution sets and scalarization for vector equilibrium problems,” Journal of Optimization Theory and Applications, vol. 133, no. 2, pp. 151–161, 2007.

L. Gong, “Global stability result for the generalized quasivariational inequality problem,” Journal of Optimization Theory and Applications, vol. 70, no. 2, pp. 365–375, 1991.

R. L. Tobin, “Sensitivity analysis for variational inequalities,” Journal of Optimization Theory and Applications, vol. 48, no. 1, pp. 191–204, 1986.

S. J. Yu and J. C. Yao, “On vector variational inequalities,” Journal of Optimization Theory and Applications, vol. 89, no. 3, pp. 749–769, 1996.

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.

N. J. Huang, J. Li, and H. B. Thompson, “Stability for parametric implicit vector equilibrium problems,” Mathematical and Computer Modelling, vol. 43, no. 11-12, pp. 1267–1274, 2006.

O. Chadli, N. C. Wong, and J. C. Yao, “Equilibrium problems with applications to eigenvalue problems,” Journal of Optimization Theory and Applications, vol. 117, no. 2, pp. 245–266, 2003.

Q. H. Ansari, S. Schaible, and J.-C. Yao, “The system of generalized vector equilibrium problems with applications,” Journal of Global Optimization, vol. 22, no. 1–4, pp. 3–16, 2002.

X.-P. Ding, J.-C. Yao, and L.-J. Lin, “Solutions of system of generalized vector quasi-equilibrium problems in locally -convex uniform spaces,” Journal of Mathematical Analysis and Applications, vol. 298, no. 2, pp. 398–410, 2004.

L. J. Lin, Z. T. Yu, and G. Kassay, “Existence of equilibria for multivalued mappings and its application to vectorial equilibria,” Journal of Optimization Theory and Applications, vol. 114, no. 1, pp. 189–208, 2002.

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

F. Ferro, “A minimax theorem for vector-valued functions,” Journal of Optimization Theory and Applications, vol. 60, no. 1, pp. 19–31, 1989.

Y.-C. Lin, “On generalized vector equilibrium problems,” Nonlinear Analysis, vol. 70, no. 2, pp. 1040–1048, 2009.

P. Q. Khanh and L. M. Luu, “Upper semicontinuity of the solution set to parametric vector quasivariational inequalities,” Journal of Global Optimization, vol. 32, no. 4, pp. 569–580, 2005.

X. H. Gong, “Efficiency and Henig efficiency for vector equilibrium problems,” Journal of Optimization Theory and Applications, vol. 108, no. 1, pp. 139–154, 2001.

R. N. Mukherjee and H. L. Verma, “Sensitivity analysis of generalized variational inequalities,” Journal of Mathematical Analysis and Applications, vol. 167, no. 2, pp. 299–304, 1992.

X. H. Gong and J. C. Yao, “Lower semicontinuity of the set of efficient solutions for generalized systems,” Journal of Optimization Theory and Applications, vol. 138, no. 2, pp. 197–205, 2008.

X. H. Gong and J. C. Yao, “Connectedness of the set of efficient solutions for generalized systems,” Journal of Optimization Theory and Applications, vol. 138, no. 2, pp. 189–196, 2008.

X. H. Gong, “Continuity of the solution set to parametric weak vector equilibrium problems,” Journal of Optimization Theory and Applications, vol. 139, no. 1, pp. 35–46, 2008.

I. V. Konnov and J. C. Yao, “On the generalized vector variational inequality problem,” Journal of Mathematical Analysis and Applications, vol. 206, no. 1, pp. 42–58, 1997.

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.

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.

P. Q. Khanh and L. M. Luu, “Lower semicontinuity and upper semicontinuity of the solution sets and approximate solution sets of parametric multivalued quasivariational inequalities,” Journal of Optimization Theory and Applications, vol. 133, no. 3, pp. 329–339, 2007.

D. T. Luc, Theory of Vector Optimization, vol. 319 of Lecture Notes in Economics and Mathematical Systems, Springer, Berlin, Germany, 1989.

Y.-C. Lin and M.-M. Wong, “Note on -implicit generalized vector variational inequalities,” Taiwanese Journal of Mathematics, vol. 14, no. 2, pp. 707–718, 2010.

A. R. Warburton, “Quasiconcave vector maximization: connectedness of the sets of Pareto-optimal and weak Pareto-optimal alternatives,” Journal of Optimization Theory and Applications, vol. 40, no. 4, pp. 537–557, 1983.

K. Kimura and J.-C. Yao, “Sensitivity analysis of vector equilibrium problems,” Taiwanese Journal of Mathematics, vol. 12, no. 3, pp. 649–669, 2008.

Q. H. Ansari, S. Schaible, and J.-C. Yao, “System of vector equilibrium problems and its applications,” Journal of Optimization Theory and Applications, vol. 107, no. 3, pp. 547–557, 2000.

S.-C. Fang, “An unconstrained convex programming view of linear programming,” Zeitschrift für Operations Research, vol. 36, no. 2, pp. 149–161, 1992.

C. Berge, Espaces Topologiques, Dunod, Paris, France, 1959.