A Note on Continuity of Solution Set for Parametric Weak Vector Equilibrium Problems

Abstract

We consider the parametric weak vector equilibrium problem. By using a weaker assumption of Peng and Chang (2014), the sufficient conditions for continuity of the solution mappings to a parametric weak vector equilibrium problem are established. Examples are provided to illustrate the essentialness of imposed assumptions. As advantages of the results, we derive the continuity of solution mappings for vector optimization problems.

1. Introduction

It is well known that the vector equilibrium problem provides a unified model of several classes of problems, including vector variational inequality problems, vector complementarity problems, vector optimization problems, and vector saddle point problems. There are many papers which have discussed the existence results for different types of vector equilibrium problems (see [1–3] and references therein).

In 2008 Gong [4] studied parametric vector equilibrium problems. Based on a scalarization representation of the solution mapping and the property involving the union of a family of lower semicontinuous set-valued mappings of Cheng and Zhu [5], they established the sufficient conditions for the continuity of the solution set mapping for the mixed parametric monotone weak vector equilibrium problems in topological vector spaces. In the same year, Gong and Yao [6] discussed the lower semicontinuity of the efficient solution mappings to a parametric strong vector equilibrium problem with -strict monotonicity of a vector-valued function, by using a scalarization method and density result. In 2009, Xu and Li [7] presented a new proof of lower semicontinuity of the set of efficient solutions to a parametric strong vector equilibrium problem, which is different from the one used in [6]. In 2010, Chen and Li [8] discussed and improved the lower semicontinuity and continuity results of efficient solution mappings to a parametric strong vector equilibrium problem in [4, 6], without the uniform compactness assumption. By virtue of the scalarization technique, [4, 6–8] have discussed the lower semicontinuity, in the case that -efficient solution set is a singleton. However, in practical, the -solution set may not be singleton but a general set. Recently, by using a weak assumption, Peng and Chang [9] discussed the lower semicontinuity of solution maps for parametric weak vector equilibrium problem under the case that the -efficient solution mapping may not be single-valued as follows. Unfortunately, the results obtained in the corresponding papers [4, 6–9] cannot be used in the case of vector optimization problems. Hence, in this paper, we study the lower semicontinuity of the set of efficient solutions for parametric weak vector equilibrium problems when the -efficient solution set is a general set. Moreover, our theorems can apply for vector optimization problems.

The structure of the paper is as follows. Section 2 presents the efficient solutions to parametric weak vector equilibrium problems and materials used in the rest of this paper. We establish, in Section 3, a sufficient condition for the continuity of the efficient solution mappings. We give some examples to illustrate that our main results are different from the corresponding ones in the literature. Section 4 is reserved for an application of the main result to a weak vector optimization problem.

2. Preliminaries

Throughout this paper, if not otherwise specified, , will denote two real Hausdorff topological vector spaces, and a real topological space, and a nonempty subset of . Let be the topological dual space of . Let be a pointed, closed, and convex cone with . Let be the dual cone of . Denote the quasi-interior of by ; that is, Since , the dual cone of has a compact base. Let . Then, is a compact base of .

Let be neighborhoods of considered points . Let be a set-valued mapping and let be a vector-valued mapping.

For each , we consider the following parametric weak vector equilibrium problem (PWVEP): find , such that Let be the efficient solution set of (4); that is, For each and , let denote the set of -efficient solution set to (4); that is, Throughout this paper, we always assume for all . Now, we recall the definition of semicontinuity of set-valued mappings. Let and be two topological spaces, a set-valued mapping, and .

Definition 1 (see [10]). (i) is said to be lower semicontinuous (l.s.c.) at if, for any open set satisfying , there exists such that , for all .
(ii) is said to be upper semicontinuous (u.s.c) at if, for any open set satisfying , there exists such that , for all .

Proposition 2 (see [11, 12]). (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 exists and a subnet of such that .

Definition 3. Let and be two vector spaces. Let be a nonempty subset of . A vector-valued function is said to be(a)strictly convex on a convex subset of , if (b)-convex on a convex subset of , if (c)-convexlike on convex subset of , if, for any , and any , there exist such that .

Obviously, we get that

Next, we recall the definitions of monotonicity which are in common use in review literature.

Definition 4. Let and be two vector spaces. Let be a nonempty subset of . A bifunction is said to be(i)monotone on subset of , if (ii)strictly monotone on subset of , if is monotone and

Remark 5. It is clear that (ii) implies (i) but the converse is not true. An easy example is that for all where ; we see that for all .

Now, we collect two vital lemmas.

Lemma 6 (see [13]). Suppose that for each and , is a convex set; then

Lemma 7 (see [14, Theorem  2, p. 114]). The union of a family of l.s.c. set-valued mappings from a topological space to a topological space is also l.s.c. set-valued mapping from to is also l.s.c. set-valued mapping from to , where is an index set.

3. Main Results

In this section, we present the continuity of the efficient solution mapping to PWVEP.

Theorem 8. Let be a considered point for (PWVEP). Suppose that the following conditions are satisfied: (i) is continuous with nonempty compact convex values at ;(ii) is continuous on ;(iii) is monotone on ;(iv) is -strictly convex on .
Then, for each , is continuous on .

Proof. We first prove that is lower semicontinuous at . Suppose the contrary that there exists a such that is not l.s.c. at . Then there exists a net with and such that for any , . Since , we have and By the lower semicontinuity of at , there exists a net such that .
For any , by the upper semicontinuity and compactness of at , we get that there exists and a subsequence of such that , denoted by . We have By continuity of and on , we get that We want to show that . Assume that , then by strict convexity of and linearity of imply that Monotonicity assumption of implies that This implies that Adding (18) and (19), it follows from linearity of and monotonicity of that This is impossible by the contradiction assumption. This proof is complete.

Before comparing our result with the result of [9], we first recall that result as follows.

Theorem 9 (see [9, Theorem  3.1]). Let be a considered point for (PWVEP). Suppose that the following conditions are satisfied: (i) is a mapping with nonempty compact convex valued and continuous at ;(ii)for each , is continuous on ;(iii)for any , if , then .
Then, for each , is l.s.c. at .

Remark 10. In [9], they assumed the condition of -strict monotonicity (or called -strongly monotone in [6, 7]) at the considered point . In the case, the -solution set may be a general set, but not a singleton. Unfortunately, that result of [9] cannot be used in the case of vector optimization problems. Theorem 8 discusses the lower semicontinuity of the -solution mappings. Compared with Theorem  3.1 of [9], assumption (iii) of Theorem 8 is relaxed from assumption (iii) in Theorem  3.1 in [9]. An advantage Theorem 8 is that it works for vector optimization problems. However, in some situations Theorem 8 is applicable while Theorem  3.1 in [9] is not, as shown by the following example.

Example 11. Let , , , be a subset of . Let be a considered point for (PVEP). Let be a mapping defined by and let be a mapping defined by It is clear that is monotone on , but not satisfied condition (iii) in Theorem  3.1 of [9]. Indeed, for each , we have Also, satisfy -strictly convex on . Indeed, for any and , we have Let . We directly compute that , for each . Thus, we can easily get that is a general set-valued one for each (where is any neighborhood of ), but not a singleton. Moreover, by Theorem 8, we can get that is l.s.c. at .

However, to relax the condition (iii) in [9], we add the condition of strict convexity of . The following example illustrates that the strict convexity of is needed.

Example 12. Let , , , be a subset of . Let be a considered point for (PVEP). Let be a mapping defined by and let be a mapping defined by It is clear that is monotone on and also does not satisfy -strictly convex on . Let . It follows from direct computation that Clearly, we see that is not l.s.c. at . Hence, the assumed strict convexity of is essential.

Theorem 13. Let be a considered point. Suppose that the following conditions are satisfied: (i) is continuous with nonempty compact convex values at ;(ii) is continuous on ;(iii) is monotone on ;(iv)for each , is -strictly convex on ;(v)for each and , is -convexlike on .
Then, is l.s.c. at .

Proof. Since, for each and for each , is -convexlike on , is convex. It follows from Lemma 6 that By Theorem 8, for each , is l.s.c. at . Therefore, by Lemma 7 it implies that is l.s.c. at . This completes the proof.

Now, we give an example to illustrate that our result improves that of [9].

Example 14. Let , , , be a subset of . Let be a considered point for (PVEP). Let be a mapping defined by and let be a mapping defined by It is clear that is monotone on , but not satisfied -strict monotone on . Also, satisfy -strictly convex on . It follows from direct computation that , for each . Thus, we can easily get that is a general set-valued one for each (where is any neighborhood of ), but not a singleton. Moreover, by Theorem 13, we can get that is l.s.c. at .

Theorem 15. Let be a considered point. Suppose that the following conditions are satisfied: (i) is continuous with nonempty compact convex values at ;(ii) is continuous on .
Then, is u.s.c. at .

Proof. Suppose the contrary that is not upper semicontinuous at . Then, there exist an open neighborhood of and a net converging to such that Then there exists some such that Since , we have . By the assumption, is u.s.c. with compact valued at , then we have that there exists subnet such that .
We will show that ; suppose the contrary that . Then there exists such that Since is l.s.c. at and and , we have that there exists such that . It follows from that By (ii) it implies that , which leads to a contradiction with (30). Hence, we have .
Since and is an open set, there exists some such that which leads to contradiction with (29). Thus is u.s.c. at .

The following theorem is directly obtained from Theorems 13 and 15.

Theorem 16. Let be a considered point. Suppose that the following conditions are satisfied: (i) is continuous with nonempty compact convex values at ;(ii) is continuous on ;(iii)for each , is -strictly convex on ;(iv)for each and , is -convexlike on ;(v) is monotone on .
Then, is continuous at .

4. Vector Optimization Problem

Since the parametric weak vector equilibrium problem (PWVEP) contains the parametric weak vector optimization problems, we can derive from Theorem 17 direct consequences. We denote the ordering induced by as follows: The ordering and the ordering are defined similarly. Let be a vector-valued mapping. For each , consider the problem of parametric weak optimization problem (PWVOP) finding such that Setting , PWVEP becomes a special case of PWVOP.

For each , the efficient solution set of (34) is denoted by The -efficient solution set of (34) is

We directly obtain the following theorem from Theorem 16.

Theorem 17. Let be a considered point. Suppose that the following conditions are satisfied: (i) is continuous with nonempty compact convex values at ;(ii) is continuous on ;(iii)for each , is -strictly convex on ;(iv)for each and , is -convexlike on .
Then, is continuous at .

The following example illustrates that the strict convexity cannot be dropped.

Example 18. Let , , , be a subset of . Let be a considered point for PWVOP. Let be a mapping defined by and let be a mapping defined by It is clear that does not satisfy -strictly convex on . It follows from direct computation that Clearly, we see that is not l.s.c. at . Hence, the assumed strict convexity of is essential.

5. Conclusions

In this paper, we study the lower semicontinuity of the set of efficient solutions for parametric weak vector equilibrium problems when the -efficient solution set is a general set. Moreover, our theorems can apply for vector optimization problems.

Conflict of Interests

The authors declare that they have no competing interests.

Authors’ Contribution

All authors read and approved the final paper.

Acknowledgments

Pakkapon Preechasilp was supported by Pibulsongkram Rajabhat University. Rabian Wangkeeree was partially supported by the Thailand Research Fund, Grant no. RSA5780003. The authors would like to thank the referees for reading this paper carefully, providing valuable suggestions and comments on the original version of this paper.