A note on solution lower semicontinuity for parametric generalized vector equilibrium problems
© Peng et al.; licensee Springer. 2014
Received: 15 February 2014
Accepted: 5 August 2014
Published: 26 August 2014
By using a density technique, sufficient conditions for lower semicontinuity of strong solutions to a parametric generalized vector equilibrium problem are established, where the monotonicity is not necessary. The obtained results are different from the corresponding ones in the literature (Gong and Yao in J. Optim. Theory Appl. 138:197-205, 2008; Gong in J. Optim. Theory Appl. 139:35-46, 2008; Chen and Li in Pac. J. Optim. 6:141-151, 2010; Li and Fang in J. Optim. Theory Appl. 147:507-515, 2010; Gong and Yao in J. Optim. Theory Appl. 138:189-196, 2008). Some examples are given to illustrate the results.
MSC:49K40, 90C29, 90C31.
It is well known that the vector equilibrium problem (VEP, in short) is a very general mathematical model, which embraces the formats of several disciplines, as those for Nash equilibria, those from Game Theory, those from (Vector) Optimization and (Vector) Variational Inequalities, and so on (see [1–4]).
The stability analysis of solution maps for parametric vector equilibrium problems (PVEP, in short) is an important topic in optimization theory and applications. There are some papers discussing the upper and/or lower semicontinuity of solution maps. Cheng and Zhu  obtained a result on the lower semicontinuity of the solution set map to a PVEP in finite-dimensional spaces by using a scalarization method. Huang et al.  used local existence results of the models considered and additional assumptions to establish the lower semicontinuity of solution mappings for parametric implicit vector equilibrium problems. Recently, by virtue of a density result and scalarization technique, Gong and Yao  have first discussed the lower semicontinuity of efficient solutions to parametric vector equilibrium problems, which are called generalized systems in their paper. By using the idea of Cheng and Zhu , Gong  has discussed the continuity of the solution maps to a weak PVEP in Hausdorff topological vector spaces. Kimura and Yao  discussed the semicontinuity of solution maps for parametric vector quasi-equilibrium problems by virtue of the closedness or openness assumptions for some certain sets. Xu and Li  proved the lower semicontinuity for PVEP by using a new proof method, which is different from the one used in Gong and Yao . Chen and Li  studied the continuity of solution sets for parametric generalized systems without the uniform compactness assumption, which improves the corresponding results in . We observe that the semicontinuity of solution maps for the PVEP has been discussed under the assumption of C-strict (strong) monotonicity, which implies that the f-solution set of the PVEP is a singleton for a linear continuous functional f (see [5, 7, 8, 10, 11]). However, it is well known that the f-solution set of (weak) PVEPs should be general, but not a singleton. Moreover, to the best of our knowledge, there are few results of semicontinuity have been established for strong solution maps of PVEP in the literature. So, in this paper, by using a density skill, we aim at studying the lower semicontinuity of the strong solution map for a class of parametric generalized vector equilibrium problems (PGVEPs), when the f-solution set is a general set by removing the assumption of C-strict monotonicity.
The rest of the paper is organized as follows. In Section 2, we introduce a class of parametric generalized vector equilibrium problem, and recall some concepts and their properties. In Section 3, by the density and scalarization technique, we discuss the lower semicontinuity of strong solution mappings to the PGVEP, and compare our main results with the corresponding ones in the recent literature [7, 8, 10, 11]. We also give some examples to illustrate our results.
Throughout this paper, unless specified otherwise, let X, Y be normed spaces and Z be Banach space. Let be the topological dual space of Y, and C be a closed convex pointed cone in Y with nonempty topological interior intC.
where is a convex cone in Y.
where is a set-valued mapping, is a vector-valued mapping with .
Definition 2.1 
The set of the weak solutions to the (PGVEP) is denoted by .
The set of the strong solutions to the (PGVEP) is denoted by .
Definition 2.3 
The set of the f-solution to the (PGVEP) is denoted by .
is called C-monotone on , iff for any given , for each , .
is called C-strictly monotone (i.e., C-strongly monotone in ) on , iff F is a C-monotone on , and for any given , for each with , .
is called C-convex if, for each and , .
is called C-like-convex on , iff for any and any , there exists such that .
A set is called a C-convex set, iff is a convex set in Y.
Throughout this paper, we always assume and for all . This paper aims at investigating the semicontinuity of the strong solution mappings to (PGVEP).
Lemma 2.1 
Let be a C-convex set for each and . If , then .
The notation denotes the open ball with center and radius . Let be a set-valued mapping, and let there be given .
Definition 2.5 
F is called lower semicontinuous (l.s.c., for short) at , iff for any open set V satisfying , there exists , such that for every , .
F is called upper semicontinuous (u.s.c., for short) at , iff for any open set V satisfying , there exists , such that for every , .
We say F is l.s.c. (resp. u.s.c.) on Λ, iff it is l.s.c. (resp. u.s.c.) at each . F is said to be continuous on Λ, iff it is both l.s.c. and u.s.c. on Λ.
Proposition 2.1 
F is l.s.c. at if and only if for any sequence with and any , there exists , such that .
If F has compact values (i.e., is a compact set for each ), then F is u.s.c. at if and only if for any sequences with and with , there exist and a subsequence of , such that .
The following lemma plays an important role in the proof of the lower semicontinuity of the solution map .
Lemma 2.2 [, Theorem 2, p.114]
The union of a family of l.s.c. set-valued mappings from a topological space X into a topological space Y is also an l.s.c. set-valued mapping from X into Y, where I is an index set.
Proposition 2.2 
is a closed set.
Q is l.s.c. at if and only if .
3 Lower semicontinuity of strong solution sets to (PGVEP)
In this section, we discuss the lower semicontinuity of the strong solutions for (PVEP). Firstly, we obtain two important lemmas relevant to the (PVEP) as follows.
is continuous with compact convex values on Λ.
For each , is continuous on .
- (iii)For each , , there exist and , such that
Then is l.s.c. on Λ.
Proof Suppose to the contrary that there exists , such that is not l.s.c. at . Then there exist a sequence with and , such that for any , .
which contradicts (3.6). Therefore , which leads to a contradiction. Hence, for each , is l.s.c. on Λ. □
A is nonempty compact set.
is continuous on .
- (iii)For each there exist and , such that
then we see that is l.s.c. on .
Proof By using Lemma 3.1 and following a similar way to the proof to Lemma 3.1 of , we can obtain the conclusion. □
Now, we discuss the lower semicontinuity of strong solution mappings to (PGVEP).
is continuous with compact convex values on Λ.
For each and , is C-like-convex on .
For each , is continuous on .
- (iv)For each , , there exist and , such that
are bounded subsets of Y for each .
Then is lower semicontinuous on Λ.
Proof We prove the result in the following three steps.
By Lemma 3.2, we know that is lower semicontinuous on .
Then . We show that converges to with respect to the topology .
Hence , that is, . This means that converges to with respect to .
Then we can obtain the result that for each (3.11) is valid, and the validity of (3.7) follows readily from (3.10) and (3.11).
Step 2. For each , let . By a similar argument to the proof of Lemma 3.1, we find for each , that is l.s.c. on Λ. It follows from Lemma 2.2 that is l.s.c. on Λ.
Hence, is l.s.c. at . By the arbitrariness of , is l.s.c. on Λ. This completes the proof. □
Remark 3.2 In Theorem 3.1, by using the density technique, we obtain a sufficient condition for the lower semicontinuity of strong solutions to (PGVEP). Our approach is different from the corresponding ones in  (see Theorem 2.1 of ). Furthermore, the condition of C-strong (strict) monotonicity is not required, and the C-convexity of F is generalized to the C-like-convexity. Thus, Theorem 3.1 improves and extends the corresponding results in the literature [5, 7, 8, 10, 11]. The following example is given to illustrate the case.
Thus, for any given and for any given , is -like-convex on , that is, condition (iii) of Theorem 3.1 is satisfied.
Thus, the condition (iv) of Theorem 3.1 is satisfied. It is clear that are bounded subsets of Y for each , , and . Consequently, by Theorem 3.1, is lower semicontinuous on Λ.
Obviously, is not C-strongly monotone on . Then Theorem 2.1 in  and Theorem 3.2 in  are not applicable, and the corresponding results in references [, Theorem 4.2], [, Theorem 3.1] are also not applicable.
Finally, we give an example to illustrate that the assumption (iv) of Theorem 3.1 is essential.
Obviously, we know that conditions (ii), (iii), (v), and (vi) are satisfied.
Using a similar method, there exists , for , we have . Thus, condition (iv) of Theorem 3.1 is not satisfied.
By virtue of Definition 2.5, we know that is not lower semicontinuous at .
This work was supported by the Natural Science Foundation of China (Nos. 11301571, 11271389, 71201126), the science foundation of Ministry of Education of China (No. 12XJC910001), the Natural Science Foundation Project of ChongQing (No. 2012jjA00016) and the Education Committee Research Foundation of Chongqing (KJ130428).
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