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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.

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 [

In 2008 Gong [

The structure of the paper is as follows. Section

Throughout this paper, if not otherwise specified,

Let

For each

(i)

(ii)

(i)

(ii) If

Let

Obviously, we get that

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

Let

It is clear that (ii) implies (i) but the converse is not true. An easy example is that

Now, we collect two vital lemmas.

Suppose that for each

The union

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

Let

Then, for each

We first prove that

For any

Before comparing our result with the result of [

Let

for each

for any

Then, for each

In [

Let

However, to relax the condition (iii) in [

Let

Let

for each

for each

Then,

Since, for each

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

Let

Let

Then,

Suppose the contrary that

We will show that

Since

The following theorem is directly obtained from Theorems

Let

for each

for each

Then,

Since the parametric weak vector equilibrium problem (PWVEP) contains the parametric weak vector optimization problems, we can derive from Theorem

For each

We directly obtain the following theorem from Theorem

Let

for each

for each

Then,

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

Let

In this paper, we study the lower semicontinuity of the set of efficient solutions for parametric weak vector equilibrium problems when the

The authors declare that they have no competing interests.

All authors read and approved the final paper.

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.