^{1}

^{2}

^{1}

^{2}

We introduce the concepts of ordered variational inequalities and ordered complementarity problems with both domain and range in Banach lattices. Then we apply the Fan-KKM theorem and KKM mappings to study the solvability of these problems.

Let

Since most classical Banach spaces are Banach lattices equipped with some lattice orders on which the positive operators appear naturally, the domain of an ordinal variational inequality defined in (

The ranges of the pairing in the variational inequality (

For any positive integer

Based on this motivation, we consider two Banach lattices

This paper is organized as follows. In Section

In this section, we recall some definitions of Banach lattices and provide some properties that are useful in this paper. Here, we adopt the notions from [

Let

Let

The proofs of Parts 1, 2, and 3 are straightforward (e.g., see page 4 in [

A net

The positive cone of a Banach lattice

Let

Let

Let

It is clear that the positive cone

Let

A linear operator

Let

A linear operator

If

Every Banach lattice with order-continuous norm is Dedekind complete.

Every reflexive Banach lattice has order-continuous norm (Nakano theorem); therefore, every reflexive Banach lattice is Dedekind complete.

The class of Banach lattices with order-continuous norms is pretty large and includes many useful Banach spaces. For example, the classical

If the norm of a Banach lattice

Suppose that

In this section, we introduce the concepts of ordered variational inequalities and ordered complementarity problems on suitable Banach lattices. Then we extend some already known solvability results about variational inequalities and complementarity problems (see [

Let

Let

For a given Banach lattice

In Definitions

There are close connections between variational inequality problems and complementarity problems in Banach spaces (e.g., see [

Let

It can be seen that

Let

Now we prove the main theorem of this paper.

Let

That a mapping

In the proof and the following contents, not causing confusion, we drop the foot marks for the norms of the Banach spaces

Now we show that

In particular, if

Let

In the following result, we apply Theorem

Let

From the properties of Banach lattices with order-continuous norms,

It is well known that every reflexive Banach lattice has order-continuous norm. As reflexive Banach lattices have been widely used in many mathematics fields, we list the following result as a special case of Corollary

Let

Taking into account Lemma

Let

Recall that for a given Banach lattice

Let

As mentioned in the introduction, Li and Yao [

A linear operator

a positive operator whenever it maps positive element to positive element, that is, whenever

a negative operator whenever it maps positive element to negative element, that is, whenever

The collection of all positive (negative) operators between

The following results are easy consequences of positive and negative operators. We state them as a lemma without proof.

Let

If

If

Let

totally order increasing on

totally order decreasing on

Noticing that

Let

If

If

At first we prove Part 1. Suppose that

As an immediate consequence, we have the following result.

Let

If

If

Since

Lemma

Let

order monotone if

order pseudomonotone if

From the above definition, it is clear that every order-monotone mapping is order-pseudomonotone. It is well-known that, in the special case

Let

Let

If

Let

From Theorem

In this section, we give an example of ordered variational inequality problem in finite-dimensional cases as an application of Theorem

In an economy, we consider two finite-dimensional Hilbert lattices

The capitals

At first, we show that