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We establish fixed-point theorems for mixed monotone mappings in the setting of ordered metric spaces which satisfy a contractive condition for all points that are related by a given ordering. We also give a global attractivity result for all solutions of the difference equation

The following global attractivity result from [

Let

The function

For each

and assume that if

then

Then there exists exactly one equilibrium

The above result in Theorem

Moreover, there has been recent interest in establishing fixed-point theorems in partially ordered complete metric spaces with a contractivity condition which holds for all points that are related by partial ordering, (see, e.g., [

In [

Let

There exists

There exists

If

Then one has the following.

For every initial point

If

In particular, every solution

such that

In this paper, motivated by the results and ideas in a recent work of Berinde and Borcut [

Now we introduce the following concepts.

Let

Let

Through this paper we will use the following notations.

Let

For

We endow

Let

Our first result is the following.

Let

There exists

for all

There exist

If

Then one has the following.

For every initial point

where

If

In particular, every solution

such that

The following estimates hold:

Let

Now we will prove that

Since

Then, our claim holds; that is,

On the other hand, from (

Now, if

Next, from (

Now, assume that

If we replace the condition, if

In addition to the hypotheses of Theorem

From (a) of Theorem

Let

There exists

for all

There exist

If

Then one has the following.

For every initial point

where

If

In particular, every solution

such that

Note that (

In addition to the hypotheses of Corollary

Let

There exists

There exist

If

Then one has the following.

For every initial point

where

If

In particular, every solution

such that

The following estimates hold:

In addition to the hypotheses of Corollary

In this section, we apply our main results to the study of a class of third-order difference matrix equations. At first, we start by fixing some notations and recalling some preliminaries.

We will use the symbol

The following lemmas will be useful later.

Let

Let

Finally, we recall the well-known Schauder fixed-point theorem.

Let

Now, we consider the class of third-order difference matrix equations:

Suppose that

Equation (

The sequences

converge to

for all

For every

In order to make the proof easy, we divide it into several steps.

Now, all the hypotheses of Theorem

Since

From Steps 1 and 2, we know that (

From (

Consider a third-order difference matrix equation

We are interested to approximate the unique positive equilibrium to (

We use Algorithm (iii) of Theorem

After

The residual errors are

The convergence history is given by Figure

Convergence history for (