#### Abstract

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 where satisfies certain monotonicity conditions with respect to the given ordering. As an application of our obtained results, we present some iterative algorithms to solve a class of matrix equations. A numerical example is also presented to test the validity of the algorithms.

#### 1. Introduction

The following global attractivity result from [1] (see also [2]) is very useful in establishing convergence results in many situations.

Theorem 1 (see [1]). *Let be a closed and bounded interval of real numbers and let satisfy the following conditions.*(i)*The function is monotonic in each of its arguments.*(ii)*For each and for each , one defines
**and assume that if is a solution of the system
**then . **Then there exists exactly one equilibrium of the equation
**
and every solution of the above equation converges to . *

The above result in Theorem 1 attracted considerable attention of the leading specialists in difference equations and discrete dynamic systems and has been generalized and extended to the case of maps in , see [3], and maps in Banach space with cone, see [4–6].

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., [7–19]). These fixed-point results have been applied mainly to the existence of solutions of boundary value problems for differential equations, namely [18], has been applied for solving a class of matrix equations.

In [20], Burgić et al. obtained the following global attractivity result for mixed monotone mappings in partially ordered complete metric spaces (see also Gnana Bhaskar and Lakshmikantham [10]).

Theorem 2 (see Burgić et al. [20]). *Let be a partially ordered set and suppose there is a metric on such that is a complete metric space. Let be a map such that is nonincreasing in for all and nondecreasing in for all . Suppose that the following conditions hold.*(i)*There exists with
*(ii)*There exists such that the following condition holds:
*(iii)*If is a nondecreasing convergent sequence such that , then , for all and if is a nonincreasing convergent sequence such that , then , for all ; if for every , then . **Then one has the following.*(a)*For every initial point such that condition (ii) holds, , , , where satisfy
*(b)*If in condition (ii), then . If in addition , then , converge to the equilibrium of the equation
*(c)*In particular, every solution of
**such that , converges to the equilibrium of (7). *

In this paper, motivated by the results and ideas in a recent work of Berinde and Borcut [9], we extend Theorem 2 to mappings . Such extension allows us to study the third-order difference equation The presented theorems also extend and generalize the work in [9]. We use our obtained results to build some iterative algorithms to solve a class of matrix equations. A numerical example is also presented to test the validity of the algorithms.

Now we introduce the following concepts.

*Definition 3. *Let be a nonempty set and a given mapping. One says that is a fixed-point of the third order if

*Definition 4. *Let be a partially ordered set and a given mapping. one says that has the mixed monotone property if is monotonously increasing in and and is monotonously decreasing in ; that is, for any ,(i),(ii), and(iii).

Through this paper we will use the following notations.

Let be a partially ordered set. (i)For the notation means that and .(ii)We endow with the partial order that we denote also by , defined by (iii)Let be a given mapping. For all , we denote

#### 2. Main Result

Our first result is the following.

Theorem 5. *Let be a partially ordered set and suppose there is a metric on such that is a complete metric space. Let be a mapping having the mixed monotone property on . Suppose that the following conditions hold.*(i)* There exists with
* *for all . *(ii)* There exist such that
*(iii)* If is a nondecreasing convergent sequence such that , then for all , if is a nonincreasing convergent sequence such that , then for every , and if for every , then . **Then one has the following.*(a)* For every initial point such that condition (14) holds:
* *where satisfy
* *If and in condition (14), then and , , and converge to the equilibrium of the equation
*(b)* In particular, every solution of
* *such that (or ) converges to the equilibrium of (18).*(c)* The following estimates hold:
*

*Proof. *Let such that condition (14) is satisfied. Denote , , and . Since , , and , from the mixed monotone property of , we get that
Consider the sequences , , and defined by (17). By induction and using the mixed monotone property of , we obtain easily that
For the sake of clarity, for all , denote
We claim that, for all , we have
By (13) and (21), we obtain
Thus we get that
Then our claim holds for . Suppose now that (23) holds for some fixed . Similarly, by (13) and (21), we obtain
Similarly, one can show that
Then by the induction principle, (23) holds for all .

Now we will prove that , , and are Cauchy sequences in the metric space . Using (23) and the triangular inequality, for , we have
This implies that is a Cauchy sequence. Similarly, one can prove that and are Cauchy sequences.

Since is complete, there exist such that
From condition (iii) and (21), we get that
We claim that . Indeed, we have

Then, our claim holds; that is, . Similarly, one can show that and . Thus we proved (16).

On the other hand, from (28), for with being fixed, we have
Letting in the above inequality, we obtain
that is,
Similarly, one can show that
Thus we proved (19).

Now, if and , we claim that, for all and . Indeed, by the mixed monotone property of ,
Assume that and for some . Then,
and similarly for and . Thus we proved that

Next, from (38), we have
which implies that . Similarly, we obtain that , that is, . Then .

Now, assume that . Then, in view of the monotonicity of , we have
Continuing this process, one can show that
If we assume that , in view of the monotonicity of , we have
Continuing in a similar way, we can prove that
Letting and using (iii) and (29), we get that as , where is the equilibrium of (17). Similarly, if and , we obtain also the same result.

*Remark 6. *If we replace the condition, if is a nondecreasing convergent sequence such that , then for all and if is a nonincreasing convergent sequence such that , then for every , by the continuity of , we can check easily that the result of Theorem 5 holds also in this case.

Theorem 7. * In addition to the hypotheses of Theorem 5, suppose that for every , , there exists such that and . Then one obtains the uniqueness of the fixed point of the third order. *

*Proof. *From (a) of Theorem 5, we know that admits a fixed point of the third order ; that is,
where
Suppose that is another fixed point of the third order of ; that is,
We will prove that
where
From the hypothesis of Theorem 7, there exists such that
Since is a mixed monotone operator, we have
We have
which implies that
that is,
*Example *8. Let be an ordered set with the natural ordering of real numbers and a usual metric on . Let be defined by
It is easy to check that has the mixed monotone property on . For with , we have
Thus (13) is satisfied for . Thus all the conditions of Theorems 5 and 7 are satisfied. Moreover, there exists a unique in such that

Corollary 9. *Let be a partially ordered set and suppose there is a metric on such that is a complete metric space. Let be a mapping having the mixed monotone property on . Suppose that the following conditions hold. *(1)* There exists for and with
* *for all . *(2)* There exist such that
*(3)* If is a nondecreasing convergent sequence such that , then for all , if is a nonincreasing convergent sequence such that , then for every , and if for every , then .**Then one has the following.*(a)* For every initial point such that condition (14) holds,
* *where satisfy
* *If and in condition (14), then and , , and converge to the equilibrium of the equation
*(b)* In particular, every solution of
* *such that (or ) converges to the equilibrium of (18).*

*Proof. *Note that (14) implies that
for all , where and the result follows from Theorem 5.

Corollary 10. *In addition to the hypotheses of Corollary 9, suppose that for every , , there exists such that and . Then one obtains the uniqueness of the fixed point of the third order. *

Corollary 11. *Let be a partially ordered set and suppose there is a metric on such that is a complete metric space. Let be a mapping having the mixed monotone property on . Suppose that the following conditions hold. *(1)*There exists with
*(2)*There exist such that
*(3)*If is a nondecreasing convergent sequence such that , then for all ; if is a nonincreasing convergent sequence such that , then for every , and if for every , then . **Then one has the following.*(a)* For every initial point such that condition (14) holds,
* *where satisfy
* *If and in condition (14), then and , , and converge to the equilibrium of the equation
*(b)* In particular, every solution of
* *such that (or ) converges to the equilibrium of (18).*(c)* The following estimates hold:
*

Corollary 12. * In addition to the hypotheses of Corollary 11, suppose that for every , , there exists such that and . Then one obtains the uniqueness of the fixed point of the third order. *

#### 3. Application: Solving a Class of Third-Order Difference Matrix Equations

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 for the set of all Hermitian matrices. We denote by the set of all Hermitian positive definite matrices. Instead of we will also write . Furthermore, means that is positive semidefinite. As a different notation for and we will use, respectively, , and . The symbol denotes the spectral norm, that is, , the largest eigenvalue of . We denote by the Ky Fan norm defined by , where are the singular values of . For a given , we define the modified norm given by . The set endowed with this norm is a complete metric space for any positive definite matrix . For any matrix , we denote by tr the trace of the matrix .

The following lemmas will be useful later.

Lemma 13 (see [18]). *Let and be matrices, then
*

Lemma 14 (See [21]). *Let satisfy , then . *

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

Theorem 15 (The Schauder fixed-point theorem). *Let be a nonempty, compact, and convex subset of a normed vector space. Every continuous function mapping into itself has a fixed point. *

Now, we consider the class of third-order difference matrix equations: for given , where and , , and are Hermitian matrices. This type of difference equations often arises from many areas such as ladder networks [22, 23], dynamic programming [24, 25], and control theory [26, 27].

##### 3.1. A Convergence Result

Theorem 16. *Suppose that
**
Then, one has the following.*(i)* Equation (72) has one and only one equilibrium point .*(ii)*, where
*(iii)* The sequences and defined by , , and
* *converge to , and the error estimation is given by
* *for all , where is a certain constant in .*(iv)* For every , every solution of (72) converges to .*

*Proof. *In order to make the proof easy, we divide it into several steps.*Step *1. We claim that there exists a unique solution to the system
Consider the mapping defined by
for all . It is clear that is a mapping having the mixed monotone property with respect to the partial order . Let such that , , and . Using Lemma 13, we have