We prove a generalized result on the existence of equilibria for a monotone set-valued map defined on noncompact domain and take its values in an order of topological vector space. As consequence, we give a new variational inequality.

In the literature, the notion of an equilibrium point (or equilibrium problem) has been firstly introduced by Karamardian in [

The classical hypothesis used to prove this type of equilibrium result concerns the convexity and compactness of the domain

In this paper, we investigate the extension of equilibrium points to set-valued maps

We extend the notions of convexity, monotonicity, and continuity given previously to set-valued maps. If

We firstly need to define an order on the codomain of set-valued maps as it has done for single valued maps. If

By using this order, we naturally extend the notion of convexity for set-valued maps as follows.

Given a set-valued map

Note that in particular, if

As in the case of single valued maps, we can find many kinds of monotonicity for set-valued maps in the literature. We will use the notion of pseudomonotonicity defined in [

Let

We recall the classical notions of continuity for set-valued maps as follows.

Given a set-valued map

A set-valued map which is both lower and upper semicontinuous is called continuous.

In this paper, we will use the definition of coercing family borrowed from [

Consider a subset

for each

for each

for each

Definition

Note that in the case where the family is reduced to one element, condition (iii) of Definition

The following generalization of KKM principle obtained in [

Let

As it is mentioned in the introduction, at an abstract level all possible extension of equilibria can be handled equally well. But there are great practical differences if we try to replace the resulting abstract conditions by simpler, verifiable hypotheses like convexity or semicontinuity. This is even more so if we admit a “moving” ordering cone

Let

Let

For all

For all

For all

For all

There exists a family

Let us consider a set-valued map

Then we can see firstly that

In fact, let

It is also clear from condition

In addition, we can verify that condition

We deduce that

Let

For all

For all

Then there exists

By Theorem

The following result, which corresponds to Theorem 1 in [

Let

There exists a nonempty compact set

Then there exists

By taking for all

Then by Theorem

but this contradicts hypothesis (5′).

Let

For all

For all

For all

The map

For all

For all

There exists a family

Following [

Now let

Let a map

For all

The map

For all

There exists a family

Take

To verify hypothesis

Note that Corollaries

The authors declare that there is no conflict of interests regarding the publication of this paper.

The authors extend their appreciation to the Deanship of Scientific Research at King Saud University for funding the work through the research group Project (RGP-VPP 237).