We introduce the concepts of generalized relaxed monotonicity and generalized relaxed semimonotonicity. We consider a class of generalized vector variationa-llike inequality problem involving generalized relaxed semimonotone mapping. By using Kakutani-Fan-Glicksberg’s fixed-point theorem, we prove the solvability for this class of vector variational-like inequality with relaxed monotonicity assumptions. The results presented in this paper generalize some known results for vector variational inequality in recent years.

Vector variational inequalities were initially introduced and considered by Giannessi [

The concept monotonicity and the compactness operators are very useful in nonlinear functional analysis and its applications. In 1968, Browder [

In this paper, we pose two new concepts of generalized relaxed monotonicity and generalized relaxed semimonotonicity as well as two classes of generalized vector variational-like inequalities with generalized relaxed monotone mappings and generalized relaxed semimonotone mappings. We investigate the solvability of vector variational-like inequalities with generalized relaxed semimonotone mappings by means of the Kakutani-Fan-Glicksberg fixed-point theorem. The results presented in this paper generalize the results of Chen [

Throughout the paper unless otherwise specified, let

The partial order

First, we give the concept of generalized relaxed monotone mapping. In order to do so, the definition of a vector monotone mapping is needed, which was posed by Chen [

Let

We now give the concept of generalized relaxed monotone mapping.

Let

(i) If

(ii) In the above inclusion (

(iii) In the above inclusion (

Then

We recall the following concepts and results which are needed in the sequel.

A mapping

A mapping

Let

Let

A mapping

Now, we have the following Minty’s type Lemma.

Let

the set-valued mapping

Then the following two problems are equivalent:

Following the lines of proof given by Chen [

Let

Define two set-valued mappings

Now we prove

Indeed suppose

Throughout this section, let

Some nonlinear mapping consisting with two variables, may be monotone with respect to the first variable and compact with respect to the second one. However, we cannot always expect for them to be monotone or compact with respect to the two variables simultaneously. Keeping this complexity in mind we are interested in the so called semimonotone mapping. We now give the concept of a generalized relaxed semimonotone mapping.

Let

for every

for every

When

Let

Let us suppose

The norm of

Now, for fixed

Therefore, the mapping defined as above is a generalized relaxed semimonotone mapping.

Now we will pose the main problem of our study. In this paper, we investigate the following generalized vector variational-like inequality problem (for short, GVVLIP) is to find a vector

The GVVLIP (

Some special cases of GVVLIP (

If

If

We recall the following fixed-point theorem, by Zeidle [

The set-valued mapping

Now, we have the following existence results for GVVLIP (

Let

The set-valued mapping

For each fixed

Let

Define a set-valued mapping

Let

From the above we know that

If the boundedness of

Let

For each

Theorems