Journal of Applied Mathematics

Volume 2012 (2012), Article ID 691401, 11 pages

doi:10.1155/2012/691401

Research Article

Continuity of the Solution Maps for Generalized Parametric Set-Valued Ky Fan Inequality Problems

Z. Y. Peng1,2 and X. B. Li1,3

¹College of Science, Chongqing Jiaotong University, Chongqing 400074, China

²Department of Mathematics, Inner Mongolia University, Hohhot 010021, China

³College of Mathematics and Statistics, Chongqing University, Chongqing 401331, China

Received 24 January 2012; Revised 27 March 2012; Accepted 10 April 2012

Academic Editor: Ya Ping Fang

Copyright © 2012 Z. Y. Peng and X. B. Li. 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.

Under new assumptions, we provide suffcient conditions for the (upper and lower) semicontinuity and continuity of the solution mappings to a class of generalized parametric set-valued Ky Fan inequality problems in linear metric space. These results extend and improve some known results in the literature (e.g., Gong, 2008; Gong and Yoa, 2008; Chen and Gong, 2010; Li and Fang, 2010). Some examples are given to illustrate our results.

1. Introduction

The Ky Fan inequality is a very general mathematical format, which embraces the formats of several disciplines, as those for equilibrium problems of mathematical physics, those from game theory, those from (vector) optimization and (vector) variational inequalities, and so on (see [1, 2]). Since Ky Fan inequality was introduced in [1, 2], it has been extended and generalized to vector or set-valued mappings. The Ky Fan Inequality for a set-/vector-valued mapping is known as the (weak) generalized Ky Fan inequality ((W)GKFI, in short). In the literature, existing results for various types of (generalized) Ky Fan inequalities have been investigated intensively, see [3–5] and the references therein.

It is well known that the stability analysis of solution maps for parametric Ky Fan inequality (PKFI, in short) is an important topic in optimization theory and applications. There are some papers to discuss the upper and/or lower semicontinuity of solution maps. Cheng and Zhu [6] discussed the upper semicontinuity and the lower semicontinuity of the solution map for a PKFI in finite-dimensional spaces. Anh and Khanh [7, 8] studied the stability of solution sets for two classes of parametric quasi-KFIs. Huang et al. [9] discussed the upper semicontinuity and lower semicontinuity of the solution map for a parametric implicit KFI. By virtue of a density result and scalarization technique, Gong [10] first discussed the lower semicontinuity of the set of efficient solutions for a parametric KFI with vector-valued maps. By using the ideas of Cheng and Zhu [6], Gong and Yao [11] studied the continuity of the solution map for a class of weak parametric KFI in topological vector spaces. Then, Kimura and Yao [12] discussed the semicontinuity of solution maps for parametric quasi-KFIs. Based on the work of [6, 10], the continuity of solution sets for PKFIs was discussed in [13] without the uniform compactness assumption. Recently, Li and Fang [14] obtained a new sufficient condition for the lower semicontinuity of the solution maps to a generalized PKFI with vector-valued mappings, where their key assumption is different from the ones in [11, 13].

Motivated by the work reported in [10, 11, 14, 15], this paper aims at studying the stability of the solution maps for a class of generalized PKFI with set-valued mappings. We obtain some new sufficient conditions for the semicontinuity of the solution sets to the generalized PKFI. Our results are new and different from the corresponding ones in [6, 10, 11, 13–17].

The rest of the paper is organized as follows. In Section 2, we introduce a class of generalized set-valued PKFI and recall some concepts and their properties which are needed in the sequel. In Section 3, we discuss the upper semicontinuity and lower semicontinuity of the solution mappings for the class of generalized PKFI and compare our main results with the corresponding ones in the recent literature ([10, 11, 13–15]). We also give two examples to illustrate that our main results are applicable.

2. Preliminaries

Throughout this paper, if not, otherwise, specified, denotes the metric in any metric space. Let denote the closed ball with radius and center 0 in any metric linear spaces. Let and be two real linear metric spaces. Let be a linear metric space and let be a nonempty subset of . Let be the topological dual space of , and let be a closed, convex, and pointed cone in with , where denotes the interior of . Let

be the dual cone of .

Let be a nonempty subset of , and let be a set-valued mapping. We consider the following generalized KFI which consist in finding such that

When the set and the function are perturbed by a parameter which varies over a set of , we consider the following weak generalized PKFI which consist in finding such that

where is a set-valued mapping and is a set-valued mapping with .

For each , the solution set of (PKFI) is defined by

For each and for each , the -solution set of (PKFI) is defined by

Special Case

(i)If for any , , where is a set-valued mapping and is a single-valued mapping, the (PKFI) reduces to the weak parametric vector equilibrium problem ((W)PVEP) considered in [15].(ii)When is a vector-valued mapping, that is, , the (PKFI) reduces to the parametric Ky Fan inequality in [14].(iii)If for any , , where and are two vector-valued maps, the (PKFI) reduces to the parametric (weak) vector equilibrium problem (PVEP) considered in [10, 11, 13, 16].

Throughout this paper, we always assume for all . This paper aims at investigating the semicontinuity and continuity of the solution mapping as set-valued map from the set into . Now, we recall some basic definitions and their properties which are needed in this paper.

Definition 2.1. Let be a trifunction.

(i) is called -convex function on , if and only if for every and , .(ii) is called -like-convex function on , if and only if for any and any , there exists such that .(iii) is called -monotone on , if and only if for any and , . The mapping is called -strictly monotone (or called -strongly monotone in [10]) on if is -monotone and if for any given , for all and , s.t. .

Definition 2.2 (see [18]). Let and be topological spaces, be a set-valued mapping.

(i) is said to be upper semicontinuous (u.s.c., for short) at if and only if for any open set containing , there exists an open set containing such that for all .(ii) is said to be lower semicontinuous (l.s.c., for short) at if and only if for any open set with , there exists an open set containing such that for all .(iii) is said to be continuous at , if it is both l.s.c. and u.s.c. at . is said to be l.s.c. (resp. u.s.c.) on , if and only if it is l.s.c. (resp., u.s.c.) at each .

From [19, 20], we have the following properties for Definition 2.2.

Proposition 2.3. Let and be topological spaces, let be a set-valued mapping.

(i) is l.s.c. at if and only if for any net with and any , there exists such that .(ii)If has compact values (i.e., is a compact set for each ), then is u.s.c. at if and only if for any net with and for any , there exist and a subnet of , such that .

3. Semicontinuity and Continuity of the Solution Map for (PKFI)

In this section, we obtain some new sufficient conditions for the semicontinuity and continuity of the solution maps to the (PKFI).

Firstly, we provide a new result of sufficient condition for the upper semicontinuity and closeness of the solution mapping to the (PKFI).

Theorem 3.1. For the problem (PKFI), suppose that the following conditions are satisfied:

(i) is continuous with nonempty compact value on ;(ii) is . on .

Then, is . and closed on .

Proof. (i) Firstly, we prove is . on . Suppose to the contrary, there exists some such that is not u.s.c. at . Then, there exist an open set satisfying and sequences and , such that

Since and are u.s.c. at with compact values by Proposition 2.3, there is an such that (here, we can take a subsequence of if necessary).

Now, we need to show that . By contradiction, assume that . Then, there exists such that

that is,

By the lower semicontinuity of at , for , there exists such that .

It follows from and that

Since is l.s.c. at , for , there exists such that

From (3.3), (3.5), and the openness of , there exists a positive integer sufficiently large such that for all ,

which contradicts (3.4). So, we have

Since (here we can take a subsequence of if necessary), we can find (3.7) contradicts (3.1). Consequently, is . on .

(ii) In a similar way to the proof of (i), we can easily obtain the closeness of on . This completes the proof.

Remark 3.2. Theorem 3.1 generalizes and improves the corresponding results of Gong [10, Theorem 3.1] in the following four aspects:

(i)the condition that is convex values is removed;(ii)the vector-valued mapping is extended to set-valued mapping, and the condition that -monotone of mapping is removed;(iii)the assumption (iii) of Theorem  3.1 in [10] is removed;(iv)the condition that is uniformly compact near is not required.

Moreover, we also can see that the obtained result extends Theorem 2.1 of [15].

Now, we give an example to illustrate that Theorem 3.1 is applicable.

Example 3.3. Let , , be a subset of . Let be a set-valued mapping defined by and let defined by .

It follows from direct computation that

Then, we can verify that all assumptions of Theorem 3.1 are satisfied. By Theorem 3.1, is u.s.c. and closed on . Therefore, Theorem 3.1 is applicable.

When is a vector-valued mapping, one can get the following corollary.

Corollary 3.4. For the problem (PKFI), suppose that is a vector-valued mapping and the following conditions are satisfied:

(i) is continuous with nonempty compact value on ;(ii) is continuous on .

Then, is . and closed on .

Now, we give a sufficient condition for the lower semicontinuity of the solution maps to the (PKFI).

Theorem 3.5. Let . Suppose that the following conditions are satisfied:

(i) is continuous with nonempty compact value on ;(ii) is . with nonempty compact values on ;(iii) for each , , there exists , such that

where is a positive constant.

Then, is . on .

Proof. By the contrary, assume that there exists , such that is not l.s.c. at . Then, there exist with and , such that for any with .

Since and are l.s.c. at , there exists such that . We claim that . If not, for , it follows from above-mentioned assumption that , which is a contradiction.

By (iii), there exists , such that

For , because is u.s.c. at with compact values by Proposition 2.3, there exist and a subsequence of such that . In particular, for (3.10), we have

Then, there exist and such that

Since is u.s.c. with compact values on by Proposition 2.3, there exist and such that , . From , , the continuity of , and the closedness of , we have

It follows from and that . Particularly, we have

On the other hand, since and , we have . Also, we have . It follows from the continuity of that we have

By (3.14), (3.15), and the linearity of , we get

For the above and , we consider two cases:

Case i. If , by (3.13), we can obtain that

Then, it follows from that

which is a contradiction to (3.16).

Case ii. If , since , , this contradicts that for any , do not converge to . Thus, is . on . The proof is completed.

Remark 3.6. Theorem 3.5 generalizes and improves the corresponding results of [14, Lemma 3.1] in the following three aspects:

(i)the condition that is convex values is removed;(ii)the vector-valued mapping is extended to set-valued map;(iii)the constant can be any positive constant () in Theorem 3.5, while it should be strictly restricted to in Lemma 3.1 of [14].

Moreover, we also can see that the obtained result extends the ones of Gong and Yao [11, Theorem 2.1], where a strong assumption that -strict/strong monotonicity of the mappings is required.

The following example illustrates that the assumption (iii) of Theorem 3.5 is essential.

Example 3.7. Let . Let be a subset of . For each , let and be a set-valued mapping defined by

Obviously, assumptions (i) and (ii) of Theorem 3.5 are satisfied, and . For any given , let . Then, it follows from a direct computation that

However, is even not l.c.s. at . The reason is that the assumption (iii) is violated. Indeed, if , for and for all , there exist , such that

if , for and for all , there exist , using a similar method, we have . Therefore, (iii) is violated.

Now, we show that is not l.s.c. at . Indeed, there exists and there exists a neighborhood of 0, for any neighborhood of 3, there exists such that and

Thus,

By Definition 2.2 (or page 108 in [18]), we know that is not l.c.s. at . So, the assumption (iii) of Theorem 3.5 is essential.

By virtue of Theorem  1.1 in [15] (or Lemma  2.1 in [16]), we can get the following proposition.

Proposition 3.8. Suppose that for each and , is a convex set, then

Theorem 3.9. For the problem (PKFI), suppose that the following conditions are satisfied:

(i) is continuous with nonempty compact convex value on ;(ii) is continuous with nonempty compact values on ;(iii)for each , , there exists , such that

where is a positive constant.(iv)for each and for each , is -like-convex on .

Then, is closed and continuous (i.e., both . and .) on .

Proof. From Theorem 3.1, it is easy to see that is . and closed on . Now, we will only prove that is . on . For each and for each , since is -like-convex on , is convex. Thus, by virtue of Proposition 3.8, for each , it holds

By Theorem 3.5, for each , is . on . Therefore, in view of Theorem 2 in [20, page 114], we have is l.s.c. on . This completes the proof.

Remark 3.10. Theorem 3.9 generalizes and improves the work in [15, Theorems  3.4-3.5]. Our approach on the (semi)continuity of the solution mapping is totally different from the ones by Chen and Gong [15]. In [15], the is strictly to be a singleton, while it may be a set-valued one in our paper. In addition, the assumption that -strictly monotonicity of the mapping is not required and the -convexity of is generalized to the -like-convexity.

When the mapping is vector-valued, we obtain the following corollary.

Corollary 3.11. For the problem (PKFI), suppose that is a vector-valued mapping and the following conditions are satisfied:

(i) is continuous with nonempty compact convex value on ;(ii) is continuous on ;(iii)for each , , there exists , such that

where is a positive constant.(iv)for each and for each , is -like-convex on .

Then, is closed and continuous (i.e., both . and .) on .

Remark 3.12. Corollary 3.11 generalizes and improves [10, Theorem 4.2] and [13, Theorem 4.2], respectively, because the assumption that -strict monotonicity of the mapping is not required.

Next,we give the following example to illustrate the case.

Example 3.13. Let , be a subset of . Let be a mapping defined by

and define by .

Obviously, is a continuous set-valued mapping from to with nonempty compact convex values, and conditions (ii) and (iv) of Corollary 3.11 are satisfied.

Let , it follows from a direct computation that for any . Hence, for any , there exists , such that,

Thus, the condition (iii) of Corollary 3.11 is also satisfied. By Corollary 3.11, is closed and continuous (i.e., both . and .) on .

However, the condition that is a -strictly monotone mapping is violated. Indeed, for any and , there exist with , such that

which implies that is not -strictly monotone on . Then, Theorem  4.2 in [10] and Theorem  4.2 in [13] are not applicable, and the corresponding results in references (e.g., [11, Lemma 2.2, Theorem 2.1]) are also not applicable.

Acknowledgments

The authors would like to express their deep gratitude to the anonymous referees and the associate editor for their valuable comments and suggestions, which helped to improve the paper. This work was supported by the National Natural Science Foundation of China (10831009, 11001287), the Natural Science Foundation Project of Chongqing (CSTC2011BA0030, CSTC2011AC6104, CSTC2010BB9254), and the Education Committee Project Research Foundation of Chongqing nos. (KJ100711, KJ120401).

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