Abstract
In this paper, we obtain sufficient conditions for the lower semicontinuity of an approximate solution mapping for a parametric generalized vector equilibrium problem involving set-valued mappings. By using a scalarization method, we obtain the lower semicontinuity of an approximate solution mapping for such a problem without the assumptions of monotonicity and compactness.
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1 Introduction
The vector equilibrium problem is a unified model of several problems, for example, the vector optimization problem, the vector variational inequality problem, the vector complementarity problem and the vector saddle point problem. In the literature, existence results for various types of vector equilibrium problems have been investigated intensively, e.g., see [1–4] and the references therein. The stability analysis of the solution mappings for VEP is an important topic in vector equilibrium theory. Recently, the semicontinuity, especially the lower semicontinuity, of solution mappings to parametric vector equilibrium problems has been studied in the literature, see [5–16]. In the mentioned results, the lower semicontinuity of solution mappings to parametric generalized strong vector equilibrium problems is established under the assumptions of monotonicity and compactness. Very recently, Han and Gong [17] studied the lower semicontinuity of solution mappings to parametric generalized strong vector equilibrium problems without the assumptions of monotonicity and compactness.
On the other hand, exact solutions of the problems may not exist in many practical problems because the data of the problems are not sufficiently ‘regular’. Moreover, these mathematical models are solved usually by numerical methods which produce approximations to the exact solutions. So it is impossible to obtain an exact solution of many practical problems. Naturally, investigating approximate solutions of parametric equilibrium problems is of interest in both practical applications and computations. Anh and Khanh [18] considered two kinds of approximate solution mappings to parametric generalized vector quasiequilibrium problems and established the sufficient conditions for their Hausdorff semicontinuity (or Berge semicontinuity). Among many approaches for dealing with the lower semicontinuity and continuity of solution mappings for parametric vector variational inequalities and parametric vector equilibrium problems, the scalarization method is of considerable interest. By using a scalarization method, Li and Li [19] discussed the Berge lower semicontinuity and Berge continuity of an approximate solution mapping for a parametric vector equilibrium problem.
Motivated by the work reported in [17–19], in this paper we aim to establish efficient conditions for the lower semicontinuity of an approximate solution mapping for a parametric generalized vector equilibrium problem involving set-valued mappings. By using a scalarization method, we obtain the lower semicontinuity of an approximate solution mapping for such a problem without the assumptions of monotonicity and compactness.
2 Preliminaries
Throughout this paper, let X and Y be real Hausdorff topological vector spaces, and let Z be a real topological space. We also assume that C is a pointed closed convex cone in Y with its interior . Let be the topological dual space of Y. Let be the dual cone of C, where denotes the value of ξ at y. Since , the dual cone of C has a weak∗ compact base. Let . Then is a weak∗ compact base of .
Suppose that K is a nonempty subset of X and is a set-valued mapping. We consider the following generalized vector equilibrium problem (GVEP) of finding such that
When the set K and the mapping F are perturbed by a parameter μ which varies over a set M of Z, we consider the following parametric generalized vector equilibrium problem (PGVEP) of finding such that
where is a set-valued mapping, is a set-valued mapping with . For each and , the approximate solution set of (PGVEP) is defined by
where . For each and , by we denote the ξ-approximate solution set of (PGVEP), i.e.,
Definition 2.1 Let D be a nonempty convex subset of X. A set-valued mapping is said to be:
-
(i)
C-convex on D if, for any and for any , we have
-
(ii)
C-concave on D if, for any and for any , we have
Definition 2.2 [17]
Let M and be topological vector spaces. Let D be a nonempty subset of M. A set-valued mapping is said to be uniformly continuous on D if, for any neighborhood V of , there exists a neighborhood of such that for any with .
Definition 2.3 [20]
Let M and be topological vector spaces. A set-valued mapping is said to be:
-
(i)
Hausdorff upper semicontinuous (H-u.s.c.) at if, for any neighborhood V of , there exists a neighborhood of such that
-
(ii)
Lower semicontinuous (l.s.c.) at if, for any and any neighborhood V of x, there exists a neighborhood of such that
The following lemma plays an important role in the proof of the lower semicontinuity of the solution mapping .
Lemma 2.4 [[21], Theorem 2]
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.
3 Lower semicontinuity of the approximate solution mapping for (PGVEP)
In this section, we establish the lower semicontinuity of the approximate solution mapping for (PGVEP) at the considered point with .
Firstly, using the same argument as in the proof given in [[22], Lemma 3.1], we can prove the following useful result.
Lemma 3.1 For each , , if for each , is a convex set, then
Proof For any , there exists such that . Thus, we can obtain that and , . Then, for each and , , which arrives at . It then follows that, for each ,
which gives that . Hence, . Conversely, let be arbitrary. Then and , . Thus, we have
and hence
Because is a convex set, by the well-known Edidelheit separation theorem (see [23], Theorem 3.16), there exist a continuous linear functional and a real number γ such that
for all , and . Since C is a cone, we have for all . Thus, for all , that is, . Moreover, it follows from , and the continuity of ξ that for all . Thus, for all , we have , i.e., . □
Theorem 3.2 We assume that for any given , there exists such that the ξ-approximate solution set exists in , where is a neighborhood of . Assume further that the following conditions are satisfied:
-
(i)
is nonempty convex;
-
(ii)
K is H-u.s.c. at and l.s.c. at ;
-
(iii)
for any , is C-concave on ;
-
(iv)
is uniformly continuous on .
Then the ξ-approximate solution mapping is l.s.c. at .
Proof Suppose to the contrary that is not l.s.c. at , then there exist and a neighborhood of . For any neighborhoods and of and , respectively, there exist and such that . In particular, there exist sequences and such that
For the above , there exists a neighborhood of such that
We define a ξ-set-valued mapping by
Notice that . Next, we claim that is l.s.c. at 0. Suppose to the contrary that is not l.s.c. at 0, then there exist and a neighborhood of . For any neighborhood U of 0, there exists such that . In particular, there exists a nonnegative sequence such that
Since , we choose . Since , there exists such that
We claim that . In fact, since and , for any , we have and . Then, for any ,
and for any ,
By the C-concavity of , we have that
It follows that, for any , there exist , and such that . It follows from the linearity of ξ that , which gives that . For all , by (3.5) and (3.6), we have
This implies that , that is, . By (3.4), we get that , which contradicts (3.3). Therefore, is l.s.c. at 0. Since is l.s.c. at 0, for above and for above , there exists a balanced neighborhood of 0 such that , . In particular, from , there exits such that . Let .
For any , since , there exists such that
Since is uniformly continuous on , for above , there exists a neighborhood of , a neighborhood of and a neighborhood of , for any with , and , we have
Since K is H-u.s.c. at , for above , there exists a neighborhood of such that
We see that . Since K is l.s.c. at , for , there exists a neighborhood of such that
It follows from that there exists a positive integer such that . Noting that (3.9) and (3.10), we obtain
and
By (3.12), we choose
Next, we prove that . For any , by (3.11), there exists such that . It follows from (3.13) that . Noting that and (3.8), we have
By (3.7), we have
Hence, for any and , there exist and such that
It follows from the linearity of ξ that for all . This leads to . Thus
Hence . Also, since and by (3.2) and (3.13), we have
This means that , which contradicts (3.1). This completes the proof. □
Theorem 3.3 We assume that for any given , there exists such that the approximate solution set exists in . Suppose that conditions (i)-(iv) as in Theorem 3.2 are satisfied. Assume further that for each , is a convex set. Then the approximate solution mapping is l.s.c. at .
Proof Since is a convex set for each , by virtue of Lemma 3.1, it holds that . It follows from Theorem 3.2 that for each , is l.s.c. at . Thus, in view of Lemma 2.4, we obtain that is l.s.c. at . □
The following example illustrates all of the assumptions in Theorem 3.3.
Example 3.4 Let , and . Let be the closed ball of radius in . Let , and the set-valued mapping be defined by
where and . Define a set-valued mapping for all , by . We choose , , and . We can see that and . Further, for any , there exists such that . Hence, exists in . It is easy to observe that for any , is C-concave on . Clearly, condition (ii) is true. It is obvious that . Let , we can see that is uniformly continuous on . Finally, we can check that for each , is a convex set. Applying Theorem 3.3, we obtain that is l.s.c. at .
The following example illustrates that the concavity of F cannot be dropped.
Example 3.5 Let , and . Let , and the set-valued mapping be defined by
Define a set-valued mapping for all , by . We choose , , . Then, all the assumptions of Theorem 3.3 are satisfied except (iii). Indeed, taking , , and , we have
but . The direct computation shows that
Clearly, we see that is even not l.s.c. at since is not C-concave on .
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Acknowledgements
The authors were partially supported by the Thailand Research Fund, Grant No. PHD/0078/2554 and Grant No. RSA5780003. The authors would like to thank the referees for their remarks and suggestions, which helped to prove the paper.
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Wangkeeree, R., Boonman, P. & Preechasilp, P. Lower semicontinuity of approximate solution mappings for parametric generalized vector equilibrium problems. J Inequal Appl 2014, 421 (2014). https://doi.org/10.1186/1029-242X-2014-421
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DOI: https://doi.org/10.1186/1029-242X-2014-421