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The aim of this paper is to study generalized vector quasi-equilibrium problems (GVQEPs) by scalarization method in locally convex topological vector spaces. A general nonlinear scalarization function for set-valued mappings is introduced, its main properties are established, and some results on the existence of solutions of the GVQEPs are shown by utilizing the introduced scalarization function. Finally, a vector variational inclusion problem is discussed as an application of the results of GVQEPs.

Recently, various vector equilibrium problems were investigated by adopting many different methods, such as the scalarization method (e.g., [

The scalarization method is an important and efficacious tool of translating the vector problems into the scalar problems. In 1992, Chen [

In this paper, we will discuss the GVQEPs by utilizing scalarization method. The essential preliminaries are listed in Section

Suppose that

Let

Let

Let

The generalized

Let

Let

If

If

Let

Let

From now on, unless otherwise specified, let

In this section, suppose that

The general nonlinear scalarization function

If

Let

Let

Obviously, the strict monotonicity wrt

Let

Let

Now some main properties of the general nonlinear scalarization function are established. First, according to [

For each

Letting

Suppose that

If

then

If

(1) It is sufficient to attest the fact that for each

(2) It's enough to argue that for each

If

Let

(1) Define

(2) Consider the following mapping:

In this section, further suppose that

GVQEP1: Seek

where

GVQEP2: Find

where

It's worth noting that the GVQEP considered in [

Let

It's sufficient to testify that

Now a result on existence of solutions of the GVQEP1 is verified by making use of the general nonlinear scalarization function defined in Section

Let

both

for each

for each

Denote

Define

For each

Now consider a set-valued mapping

If mapping

Let

both

for each

for each

Let

both

for each

for each

Denoting

The result below follows from Theorem

Let

both

for each

for each

Let

Assume that

Let

(a) and (b) Since

(c) Obviously,

In fact, for each fixed

(d) For each

(e) The assertion that

Thus there exist

Incidentally, the set-valued mapping

The work was supported by both the Doctoral Fund of innovation of Beijing University of Technology (2012) and the 11th graduate students Technology Fund of Beijing University of Technology (No. ykj-2012-8236).