Positive Definite Solution of System of Matrix Equations with X^−k and Y^−l via Coupled Fixed Point Theorem in Partially Ordered Spaces

Abstract

We establish adequate conditions for the existence and uniqueness of solutions to systems of two matrix equations when the unknown matrices are raised to a power $k\in [-1,1]\setminus \left\{0\right\}$. The findings from coupled fixed points for ordered pairs of maps are used. Numerical examples are provided to illustrate the results shown. Some of the known results are the consequence of our acquisitions.

1. Introduction

It is well known that algebraic discrete-type Riccati equations play a central role in modern control theory and signal processing. These equations arise in many important applications, such as in optimal control theory, dynamic programming, stochastic filtering, statistics, and other fields of pure and applied mathematics [1,2,3]. Nonlinear matrix equations are widely employed in scientific and engineering computing. Research into the existence and properties of matrix equation solutions, as well as the accompanying numerical methods, is both theoretically significant and practical. In recent years, researchers have showed tremendous interest in matrix equations, such as

$$X-\sum _{i=1}^m{\mathcal{A}}_i^{*}{X}^{-p_i}{\mathcal{A}}_i=\mathcal{Q},$$

where $0<p_i\le 1(i=1,2,\dots ,m)$, $\mathcal{Q}$ is a positive definite $N\times N$ matrix, and X is searched in the class of positive definite $N\times N$ matrices, with a primary focus on investigating the conditions for positive definite solutions, perturbation analysis, and developing iterative methods for solving these equations. There are some noteworthy studies on some special cases for $m=1,2$ and $p_i=1$ [4,5].

In a series of articles [6,7], the authors study the system of matrix equation $X\pm {\mathcal{A}}^{*}{X}^{-q}\mathcal{A}=\mathcal{Q}$ and obtain deep necessary conditions for the existence of a solution. In these two works, the authors find a solution to the considered equations by fixed point iteration. Using these techniques, we use a couple of fixed point iterations for solving a system of two matrix equations. A simplified generalization $X+{\mathcal{A}}^{*}{X}^{-q}\mathcal{A}-{\mathcal{B}}^{*}{X}^{-q}\mathcal{B}=\mathcal{Q}$ of the mentioned matrix equations in [6,7] is investigated in [8] with the help of coupled fixed points from [9]. If we simultaneously apply the ideas from [8,9,10,11], it seems possible to investigate equation $X+p{\mathcal{A}}^{*}{X}^k\mathcal{A}+q{\mathcal{B}}^{*}{X}^l\mathcal{B}=\mathcal{Q}$, for $p,q\in \{\pm 1\}$ and $k,l\in [-1,1]\setminus \left\{0\right\}$. More complicated matrix equations are studied in [12]. Related to the generalizations presented by us are also [13,14]. We should not miss mentioning [15], where a nonlinear system of matrix equations is investigated through an approach different from ours. The Hermitian and skew-Hermitian splitting (HSS) iteration method has been used in solving some classes of linear matrix equations $AX+XB=C$ [16] and $AXB=C$ [17].

Let $(X,d)$ be a metric space and $T:X\to X$ be a self-map. The contraction mapping theorem [18] and the abstract monotone iterative approach [10] are well known and applicable in several applications. The first result on fixed points in partially ordered spaces is found in [19], but [10] laid the groundwork for numerous studies in this area. Let us point out that for a monotone function $\mathcal{F}$ the matrix equation $X=Q\pm {\sum }_{i=1}^m{A}_i^{*}\mathcal{F}\left(X\right)A_i$ is investigated and deep results about its solutions are obtained in [10]. We will investigate the equation $X=Q+p{A}^{*}{X}^{k}A+q{B}^{*}{X}^{l}B$ for $p,q\in \{\pm 1\}$ and $k,l\in [-1,1]\setminus \left\{0\right\}$; thus, we will consider two monotone functions $F_1$ and $F_2$ instead of one $\mathcal{F}$ and different signs in the summation.

Recently [20,21,22,23,24], a trend has emerged to relax the contraction condition by requiring it only in a partially ordered metric space $(X,d)$, meaning that instead of requiring $d(Tx,Ty)\le \alpha d(x,y)$ for any $x,y\in X$, the condition is needed only when $x\preccurlyeq y$, provided that ≼ is a partial order on $(X,d)$.

Fixed points have recently been used to solve matrix equations with the inverted matrix $X^{-1}$ [8]. Authors convert the matrix equation $X+{\mathcal{A}}^{*}{X}^{-1}\mathcal{A}-{\mathcal{B}}^{*}{X}^{-1}\mathcal{B}=Q$, for $\mathcal{A}$ and $\mathcal{B}$ as arbitrary $N\times N$ square matrices and Q as an $N\times N$ positive definite matrix, into a two-matrix-equation system:

$$\left|\begin{array}{c}X=\mathcal{Q}-{\mathcal{A}}^{*}{X}^{-1}\mathcal{A}+{\mathcal{B}}^{*}{Y}^{-1}\mathcal{B}\hfill \\ Y=\mathcal{Q}-{\mathcal{A}}^{*}{Y}^{-1}\mathcal{A}+{\mathcal{B}}^{*}{X}^{-1}\mathcal{B}\hfill \end{array}\right.$$

(1)

and apply the concept to a coupled fixed point in metric spaces with partial ordering [9].

The system (1) may be viewed as a coupled fixed point problem $X=F(X,Y)$ and $Y=F(Y,X)$ [25] for $F:\mathcal{X}\left(N\right)\times \mathcal{X}\left(N\right)\to \mathcal{X}\left(N\right)$, where $\mathcal{X}\left(N\right)$ stands for the class of $N\times N$ matrices and $F(X,Y)=\mathcal{Q}-{\mathcal{A}}^{*}{X}^{-1}\mathcal{A}+{\mathcal{B}}^{*}{Y}^{-1}\mathcal{B}$.

If particular assumptions on the map $F:X\times X\to X$ and for the underlying normed space $(X,\parallel ·\parallel )$ hold, then $X=Y$, as in [26]. In [26], two maps $F,G:X\times X\to X$ are introduced, allowing the solutions $X_0$ and $Y_0$ of system of equations $X_0=F(X_0,Y_0)$ and $Y_0=G(X_0,Y_0)$ to satisfy $X_0\ne Y_0$. If $G(X,Y)=F(Y,X)$, we obtain the popular result on coupled fixed points [9].

Following the trend noted above [24,27,28] and the observations in [26], this paper expands the concept of mixed monotone maps to provide a unified framework for a broader class of problems, where existing results become particular cases of the new findings. We establish criteria for guaranteeing the uniqueness and existence of solutions to the matrix equation system:

$$\left|\begin{array}{c}X=Q-{\mathcal{A}}^{*}{X}^{-1}\mathcal{A}+{\mathcal{B}}^{*}{Y}^{-1}\mathcal{B},\hfill \\ Y=\mathcal{R}-{\mathcal{C}}^{*}{X}^{-1}\mathcal{C}+{\mathcal{D}}^{*}{Y}^{-1}\mathcal{D}.\hfill \end{array}\right.$$

(2)

The results from [8] become a special case when $\mathcal{R}=Q$, $\mathcal{B}=\mathcal{C}$, and $\mathcal{D}=\mathcal{A}$.

Our work generalizes the results of [10] on the existence and uniqueness of fixed points in partially ordered metric spaces for monotone mappings by eliminating the requirement of continuity. We adapt the concept of coupled fixed points such that the solution $(X,Y)$ of (2) does not require $X=Y$. The main results are applied to investigate coupled fixed points for ordered pairs of two maps exhibiting different monotonic features, such as mixed monotone or total monotone properties, thus extending existing studies in the field.

We will mention that due to the calculation of not only the inverse matrix $X^{-k}$ but also the raising and the power number k between $(0,1)$, the calculations can be very difficult, and the results we obtain are rather theoretical in nature. We find sufficient conditions for the existence and uniqueness of the solutions for the investigated classes of matrix equations, and we believe that with the use of some refined iteration techniques [16,17,29,30,31] it will be possible to find approximate solutions of more complicated matrix equations.

At first glance, the proposed results are similar to those of [11]. We would like to comment that when we can choose the power of the matrices $k,l\in (-1,1)\setminus \left\{0\right\}$ the generalized result obtained in [11] is better seen. It turns out that the idea proposed by [9,25] and applied in [8] requires a specific definition of the partial order in the Cartesian product $x\times X$. We demonstrate that depending on the sign of the power k and the sign $p\in \{\pm 1\}$, with which the addends $p{A}^{*}{X}^{k}A$ and $p{C}^{*}{X}^{k}C$ participate, they naturally give rise to the partial order in the Cartesian product $x\times X$. The same is true for the addends $q{B}^{*}{Y}^{l}B$ and $q{D}^{*}{Y}^{l}D$, where $q\in \{\pm 1\}$. That is to say, if we consider the system of matrix equations

$$\left|\begin{array}{c}X=Q+p{\mathcal{A}}^{*}{X}^k\mathcal{A}+q{\mathcal{B}}^{*}{Y}^l\mathcal{B},\hfill \\ Y=\mathcal{R}+p{\mathcal{C}}^{*}{X}^k\mathcal{C}+q{\mathcal{D}}^{*}{Y}^l\mathcal{D},\hfill \end{array}\right.$$

for $k,l\in (-1,1)\setminus \left\{0\right\}$ and $p,q\in \{\pm 1\}$, then we can introduce a partial order in $(X\times X,\preccurlyeq )$ as follows:

• If $\mathrm{sign}\left(k\right)\ast p\ast \mathrm{sign}\left(l\right)\ast q>0$, then $(x,y)\preccurlyeq (u,v)$ if $x\preccurlyeq u$ and $y\preccurlyeq v$.
• If $\mathrm{sign}\left(k\right)\ast p>0$ and $\mathrm{sign}\left(l\right)\ast q<0$, then $(x,y)\preccurlyeq (u,v)$ if $x\preccurlyeq u$ and $y\succcurlyeq v$.
• If $\mathrm{sign}\left(k\right)\ast p<0$ and $\mathrm{sign}\left(l\right)\ast q<0$, then $(x,y)\preccurlyeq (u,v)$ if $x\succcurlyeq u$ and $y\succcurlyeq v$.

Therefore, we extend the classes of nonlinear systems of matrix equations that can be investigated for the existence and uniqueness of a solution and their approximation to the findings.

2. Materials and Methods

Matrix Equations

Throughout this paper, we will denote by $\mathcal{X}\left(\mathcal{N}\right)$ the set of all $N\times N$ matrices and by $\mathcal{H}\left(\mathcal{N}\right)$ the set of all $N\times N$ Hermitian matrices, and we will assume all matrices to be square ones of one and the same dimension. The $N\times N$ identity matrix will be written as I. We will denote by ${\mathcal{A}}^{*}$ the conjugated elements. A matrix $\mathcal{A}$ is known as Hermitian if $\mathcal{A}={\mathcal{A}}^{*}$, skew-Hermitian provided that $\mathcal{A}=-{\mathcal{A}}^{*}$, unitary whenever $\mathcal{A}{\mathcal{A}}^{*}=I$ holds, where I is identity matrix, and normal if $\mathcal{A}{\mathcal{A}}^{*}={\mathcal{A}}^{*}\mathcal{A}$. A matrix $\mathcal{A}$ is said to be positive semidefinite if $\langle x,\mathcal{A}x\rangle \ge 0$ for all nonzero vectors x. The notation $\mathcal{A}\succcurlyeq 0$ will be used to express the fact that $\mathcal{A}$ is positive matrix. If $\langle x,\mathcal{A}x\rangle >0$ for all nonzero x, we will say $\mathcal{A}$ is a positive definite. We will then write $\mathcal{A}\succ 0$, and we will denote the class of all $N\times N$ positive definite matrices by $\mathcal{P}\left(\mathcal{N}\right)$. A positive matrix is strictly positive if and only if it is invertible. If $\mathcal{A}$ and $\mathcal{B}$ are Hermitian matrices, then we say $\mathcal{A}\succcurlyeq \mathcal{B}(\mathcal{A}\succ \mathcal{B})$ if the matrix $\mathcal{A}-\mathcal{B}$ is positive semidefinite (positive definite), i.e., $\mathcal{A}-\mathcal{B}\succcurlyeq 0(\mathcal{A}-\mathcal{B}\succ 0)$. If the inequalities $\mathcal{A}\preccurlyeq X\preccurlyeq \mathcal{B}$ hold, we will write for simplicity $X\in [\mathcal{A},\mathcal{B}]$.

For any matrix $\mathcal{A}$, the matrix ${\mathcal{A}}^{*}\mathcal{A}$ is always positive, and its unique positive definite square root is denoted by $\left|\mathcal{A}\right|$. The eigenvalues of $\left|\mathcal{A}\right|$ counted with multiplicities are called the singular values of $\mathcal{A}$. We will always enumerate them in a decreasing order and use for them the notation ${\sigma }_1\left(\mathcal{A}\right)\ge {\sigma }_2\left(\mathcal{A}\right)\ge \dots \ge {\sigma }_n\left(\mathcal{A}\right)$. If rank $\mathcal{A}=k$, then ${\sigma }_k\left(\mathcal{A}\right)>0$, but ${\sigma }_{k+1}\left(\mathcal{A}\right)=\dots ={\sigma }_n\left(\mathcal{A}\right)=0$. Every matrix $\mathcal{A}$ is unitarily equivalent (or unitarily similar) to an upper triangular matrix T, i.e., $\mathcal{A}=QT{Q}^{*}$, where Q is unitary.

We write $r\left(\mathcal{A}\right)$ for the spectral radius of $\mathcal{A}$. The spectral norm is denoted by $\parallel ·\parallel$, i.e., $\parallel \mathcal{A}\parallel =\sqrt{{\lambda }^{+}\left({\mathcal{A}}^{*}\mathcal{A}\right)}$, where ${\lambda }^{+}\left({\mathcal{A}}^{*}\mathcal{A}\right)$ is the greatest eigenvalue of ${\mathcal{A}}^{*}\mathcal{A}$. We will denote by ${\lambda }_j\left(\mathcal{A}\right)$, $j=1,2,\dots ,N$, the eigenvalues of $\mathcal{A}\in \mathcal{X}\left(N\right)$. Singular vales ${\sigma }_j\left(\mathcal{A}\right)$ of a matrix $\mathcal{A}$ are the square roots of the eigenvalues of ${\mathcal{A}}^{*}\mathcal{A}$, i.e., ${\sigma }_j\left(\mathcal{A}\right)=\sqrt{{\lambda }_j\left({\mathcal{A}}^{*}\mathcal{A}\right)}$. We consider the space $\mathcal{H}\left(\mathcal{N}\right)$, supplied by the norm ${\parallel ·\parallel }_{\mathrm{tr}}$. Let us recall, for completeness, that ${\parallel \mathcal{A}\parallel }_{\mathrm{tr}}={\sum }_{j=1}^N{\sigma }_j\left(\mathcal{A}\right)$, where $\mathcal{A}\in \mathcal{X}\left(N\right)$. If ${\lambda }_j\left(\mathcal{A}\right)$, $j=1,2,\dots ,N$, are the eigenvalues of $\mathcal{A}$ and $\mathcal{A}\in \mathcal{P}\left(N\right)$, then ${\parallel \mathcal{A}\parallel }_{\mathrm{tr}}={\sum }_{j=1}^p{\sigma }_j\left(\mathcal{A}\right)\ge {\sum }_{j=1}^p{\lambda }_j\left(\mathcal{A}\right)=\mathrm{tr}\left(\mathcal{A}\right)$.

For the illustrative examples, we will denote $\lambda \left(\mathcal{A}\right)={\left\{{\lambda }_j\left(\mathcal{A}\right)\right\}}_{j=1}^N$.

The next lemmas will be useful later.

Lemma 1

([10]). For any two positive semidefinite $N\times N$ matrices $\mathcal{A}$ and $\mathcal{B}$, the following holds:

$$0\le \mathrm{tr}\left(\mathcal{A}\mathcal{B}\right)\le {\parallel \mathcal{A}\parallel }_{\mathrm{tr}}\mathrm{tr}\left(\mathcal{B}\right).$$

Lemma 2

([32]). Let $\mathcal{P},\mathcal{Q}\in \mathcal{P}\left(N\right)$ and $b>0$ such that $\mathcal{P},\mathcal{Q}\ge bI>0$. Then, for any $0<\theta \le 1$ and any unitarily invariant norm $\left|\right|\left|·\right|\left|\right|$, the inequalities

$$\left|\right||{\mathcal{P}}^{\theta }-{\mathcal{Q}}^{\theta }\left|\right||\le \theta b^{\theta -1}\left|\right||\mathcal{P}-\mathcal{Q}\left|\right|\left|and\right|\left|\right|{\mathcal{P}}^{-\theta }-{\mathcal{Q}}^{-\theta }\left|\right||\le \theta b^{-(\theta +1)}\left|\right||\mathcal{P}-\mathcal{Q}\left|\right||$$

hold.

Lemma 3

([32]). Let $\mathcal{A}\in \mathcal{H}\left(\mathcal{N}\right)$ satisfying $-I\prec \mathcal{A}\prec I$. Then, $\parallel \mathcal{A}\parallel <1$.

A classical partial order in a Cartesian product space $(X\times X,\preccurlyeq )$, provided that $(X,\preccurlyeq )$ is partially ordered, can be introduced by $(x,y)\preccurlyeq (u,v)$ if $x\preccurlyeq u$ and $y\succcurlyeq v$ [8,9].

The first appearance of coupled fixed points for ordered pairs of maps $(F,G)$ is in [26], where the author comments on the need to generalize the classical definition of coupled fixed points [9,25] in order to apply the results in solving systems of nonsymmetric equations.

Definition 1

([26]). Let A be a nonempty set and $F,G:A\times A\to A$ be two maps. If $x=F(x,y)$ and $y=G(x,y)$, then $(x,y)\in A\times A$ is considered a coupled fixed point for $(F,G)$ in A.

If $G(x,y)=F(y,x)$, we obtain the classical definition for coupled fixed points from [9,25].

Definition 2

([11,33]). The ordered pair of maps $(F,G)$, $F,G:X\times X\to X$, provided that $(X,\preccurlyeq )$ is a partially ordered set, is said to satisfy the mixed monotone property if for all $x,y\in X$ the following inequalities hold:

$$x_1,x_2\in Xif{x}_1\preccurlyeq x_{2}thenF(x_1,y)\preccurlyeq F(x_2,y),G(x_1,y)\preccurlyeq G(x_2,y)$$

and

$$y_1,y_2\in Xif{y}_1\preccurlyeq y_{2}thenG(x,y_1)\succcurlyeq G(x,y_2),F(x,y_1)\succcurlyeq F(x,y_2)$$

Theorem 1

([11]). Let $F_1,F_2:X\times X\to X$, $(F_1,F_2)$ satisfy the mixed monotone property and $(X,\rho ,\preccurlyeq )$ be a partially ordered metric space, and let there exist $k\in [0,1)$ such that the inequality

$$\rho (F_1(x,y),F_1(u,v))+\rho (F_2(x,y),F_2(u,v))\le k(\rho (x,u)+\rho (y,v))$$

(3)

holds for all $x\succcurlyeq u$, $y\preccurlyeq v$.

Let one of the following hold:

(i) 

$F_1$, $F_2$ are continuous maps.

(ii) 

If ${lim}_{n\to \infty }(x_n,y_n)=(x,y)$, $(x_n,y_n)\in X\times X$, then the following are true:

• $(x_n,y_n)\preccurlyeq (x,y)$, provided that $(x_n,y_n)\preccurlyeq (x_{n+1},y_{n+1})$.
• $(x_n,y_n)\succcurlyeq (x,y)$, provided that $(x_n,y_n)\succcurlyeq (x_{n+1},y_{n+1})$.

Let there exist $x_0,y_0\in X$ so that one of the following holds:

• $x_0\preccurlyeq F_1(x_0,y_0)$ and $y_0\succcurlyeq F_2(x_0,y_0)$.
• $x_0\succcurlyeq F_1(x_0,y_0)$ and $y_0\preccurlyeq F_2(x_0,y_0)$.

Then, there is a coupled fixed point $(\xi ,\eta )\in X\times X$ for $(F_1,F_2)$.

The following estimations of error are valid:

(I) 

A priori error estimate:

$$max\{\rho (x_n,\xi ),\rho (y_n,\eta )\}\le {\displaystyle \frac{k^n}{1-k}}(\rho (x_0,x_1)+\rho (y_0,y_1)).$$

(II) 

A posteriori error estimate:

$$max\{\rho (x_{n+1},\xi ),\rho (y_{n+1},\eta )\}\le {\displaystyle \frac{k}{1-k}}(\rho (x_n,x_{n+1})+\rho (y_n,y_{n+1})).$$

(III) 

The rate of convergence:

$$max\{\rho (x_{n+1},\xi ),\rho (y_{n+1},\eta )\}\le {\displaystyle \frac{k}{1-k}}(\rho (x_n,\xi )+\rho (y_n,\eta )).$$

If there are lower or upper bounds for every pair of elements $x,y\in X$, then the following are true:

• $(\xi ,\eta )$ is a unique coupled fixed point.
• If $F_2(\xi ,\eta )=F_1(\eta ,\xi )$, then $\xi =\eta$.

It is proposed in [11] that a different partial ordering can be defined in $(X\times X,⪯)$ by $(x,y)⪯(u,v)$ if $x\preccurlyeq u$ and $y\preccurlyeq v$, provided that the investigated maps $(F_1,F_2)$ do not satisfy the mixed monotone property. It is shown in [11] that the ordered pair of maps $(F_1,F_2)$ naturally generates a partial order in the Cartesian product space $X\times X$. To not confuse the reader, we will use the notations $(x,y)\preccurlyeq (u,v)$ if $x\preccurlyeq u$ and $y\succcurlyeq v$ and $(x,y)⪯(u,v)$ if $x\preccurlyeq u$ and $y\preccurlyeq v$.

Definition 3

([11]). The ordered pair of maps $(F,G)$, $F,G:X\times X\to X$, provided that $(X,\preccurlyeq )$ is a partially ordered set, is said to satisfy the total decreasing monotone property if for all $x,y\in X$ the following hold:

$$x_1,x_2\in Xif{x}_1\preccurlyeq x_{2}thenF(x_1,y)\succcurlyeq F(x_2,y),G(x_1,y)\succcurlyeq G(x_2,y)$$

and

$$y_1,y_2\in Xif{y}_1\preccurlyeq y_{2}thenG(x,y_1)\succcurlyeq G(x,y_2),F(x,y_1)\succcurlyeq F(x,y_2)$$

Theorem 2

([11]). Let $F_1,F_2:X\times X\to X$, $(F_1,F_2)$ satisfy the mixed monotone property, and $(X,\rho ,\preccurlyeq )$ be a partially ordered metric space, and let there exist $k\in [0,1)$ such that the inequality

$$\rho (F_1(x,y),F_1(u,v))+\rho (F_2(x,y),F_2(u,v))\le k(\rho (x,u)+\rho (y,v))$$

(4)

holds for all $x\preccurlyeq u$, $y\preccurlyeq v$.

Let one of the following hold:

(i) 

$F_1$ and $F_2$ are continuous maps.

(ii) 

If ${lim}_{n\to \infty }(x_n,y_n)=(x,y)$, $(x_n,y_n)\in X\times X$, then the following are true:

• $(x_n,y_n)⪯(x,y)$, provided that $(x_n,y_n)⪯(x_{n+1},y_{n+1})$.
• $(x_n,y_n)⪰(x,y)$, provided that $(x_n,y_n)⪰(x_{n+1},y_{n+1})$.

Let there exist $x_0,y_0\in X$ so that one of the following holds:

• $x_0\succcurlyeq F_1(x_0,y_0)$ and $y_0\succcurlyeq F_2(x_0,y_0)$, i.e., $(x_0,y_0)⪰(F_1(x_0,y_0),F_2(x_0,y_0))$.
• $x_0\preccurlyeq F_1(x_0,y_0)$ and $y_0\preccurlyeq F_2(x_0,y_0)$, i.e., $(x_0,y_0)⪯(F_1(x_0,y_0),F_2(x_0,y_0))$.

Then, there is a coupled fixed point $(\xi ,\eta )\in X\times X$ for $(F_1,F_2)$.

The following estimations of error are valid:

(I) 

A priori error estimate:

$$max\{\rho (x_n,\xi ),\rho (y_n,\eta )\}\le {\displaystyle \frac{k^n}{1-k}}(\rho (x_0,x_1)+\rho (y_0,y_1)).$$

(II) 

A posteriori error estimate:

$$max\{\rho (x_{n+1},\xi ),\rho (y_{n+1},\eta )\}\le {\displaystyle \frac{k}{1-k}}(\rho (x_n,x_{n+1})+\rho (y_n,y_{n+1})).$$

(III) 

The rate of convergence:

$$max\{\rho (x_{n+1},\xi ),\rho (y_{n+1},\eta )\}\le {\displaystyle \frac{k}{1-k}}(\rho (x_n,\xi )+\rho (y_n,\eta )).$$

If there are lower or upper bounds for every pair of elements $x,y\in X$, then the following are true:

• $(\xi ,\eta )$ is a unique coupled fixed point.
• If $F_2(x,y)=F_1(y,x)$, then $\xi =\eta$.

Theorem 3

(e.g., [34]). Let $C\subseteq X$ be a nonempty, compact, convex subset of a normed vector space $(X,·\parallel )$. Any $f:C\to C$ self-mapping has a fixed point.

3. Results

Positive Definite Solution of System of Matrix Equations

The notation that we shall employ is ${\Omega }_{aI}=\{X\in \mathcal{H}\left(N\right):X\succcurlyeq aI\}$. We will examine (5).

$$\left|\begin{array}{c}X=\mathcal{Q}+p{\mathcal{A}}^{*}{X}^k\mathcal{A}+q{\mathcal{B}}^{*}{Y}^l\mathcal{B},\hfill \\ Y=\mathcal{R}+p{\mathcal{C}}^{*}{X}^k\mathcal{C}+q{\mathcal{D}}^{*}{Y}^l\mathcal{D},\hfill \end{array}\right.$$

(5)

where $k,l\in [-1,1]\setminus \left\{0\right\}$, $p,q\in \{\pm 1\}$, $Q,\mathcal{R}, {\mathcal{A}}^{*}\mathcal{A}, {\mathcal{B}}^{*}\mathcal{B}, {\mathcal{C}}^{*}\mathcal{C}, {\mathcal{D}}^{*}\mathcal{D}\in \mathcal{P}\left(N\right)$, $\mathcal{A}, \mathcal{B}, \mathcal{C}, \mathcal{D}\in \mathcal{X}\left(N\right)$, and $Q, \mathcal{R}\in \mathcal{H}\left(N\right)$.

Theorem 4.

Let $p=-1$, $q=1$, $k,l\in (0,1]$ in (5) and $0<a<b$, such that the following are fulfilled:

(i) 

$Q-a^{-k}{\mathcal{A}}^{*}\mathcal{A},\mathcal{R}-a^{-k}{\mathcal{C}}^{*}\mathcal{C}\in {\Omega }_{aI}$.

(ii) 

$a^k{b}^l(Q-aI)\succcurlyeq b^l{\mathcal{A}}^{*}\mathcal{A}-a^k{\mathcal{B}}^{*}\mathcal{B}$ and $a^k{b}^l(\mathcal{R}-aI)\preccurlyeq b^l{\mathcal{C}}^{*}\mathcal{C}-a^k{\mathcal{D}}^{*}\mathcal{D}$.

(iii) 

Where ${\delta }_1=max\left\{{\displaystyle \frac{k\parallel {\mathcal{A}}^{*}\mathcal{A}+{\mathcal{C}}^{*}\mathcal{C}\parallel }{a^{k+1}}},{\displaystyle \frac{l\parallel {\mathcal{B}}^{*}\mathcal{B}+{\mathcal{D}}^{*}\mathcal{D}\parallel }{a^{l+1}}}\right\}<1$.

Then, the system (5) has a unique solution $\tilde{X},\tilde{Y}\in {\Omega }_{aI}$. The inductively defined sequences ${\left\{X_n\right\}}_{n=0}^{\infty }$ and ${\left\{Y_n\right\}}_{n=0}^{\infty }$

$$\left\{\begin{array}{c}{X}_0=aI,Y_0=bI,\hfill \\ X_{n+1}=Q-{\mathcal{A}}^{*}{X}_n^{-k}\mathcal{A}+{\mathcal{B}}^{*}{Y}_n^{-l}\mathcal{B},\hfill \\ Y_{n+1}=\mathcal{R}-{\mathcal{C}}^{*}{X}_n^{-k}\mathcal{C}+{\mathcal{D}}^{*}{Y}_n^{-l}\mathcal{D},\hfill \end{array}\right.$$

(6)

are converging to $\tilde{X}$ and $\tilde{Y}$ in $(\mathcal{H}\left(N\right), \parallel ·{\parallel }_{\mathrm{tr}},\preccurlyeq )$, respectively. The following estimations of error are valid:

(I) 

A priori error estimate:

$$max\{\parallel \tilde{X}-X_n{\parallel }_{\mathrm{tr}},\parallel \tilde{Y}-Y_n{\parallel }_{\mathrm{tr}}\}\le {\displaystyle \frac{{\delta }_1^n}{1-{\delta }_1}}(\parallel X_0-X_1{\parallel }_{\mathrm{tr}}+ \parallel Y_0-Y_1{\parallel }_{\mathrm{tr}}).$$

(II) 

A posteriori error estimate:

$$max\{\parallel X_{n+1}-\tilde{X}{\parallel }_{\mathrm{tr}},\parallel Y_{n+1}-\tilde{Y}{\parallel }_{\mathrm{tr}}\}\le {\displaystyle \frac{\delta }{1-\delta }}(\parallel X_{n+1}-X_n{\parallel }_{\mathrm{tr}}+ \parallel Y_{n+1}-Y_n{\parallel }_{\mathrm{tr}}).$$

(III) 

The convergence rate:

$$max\{\parallel \tilde{X}-X_{n+1}{\parallel }_{\mathrm{tr}},\parallel \tilde{Y}-Y_{n+1}{\parallel }_{\mathrm{tr}}\}\le {\displaystyle \frac{{\delta }_1}{1-{\delta }_1}}(\parallel \tilde{X}-X_n{\parallel }_{\mathrm{tr}}+ \parallel \tilde{Y}-Y_n{\parallel }_{\mathrm{tr}}).$$

If, moreover, the inclusions $\mathcal{Q},\mathcal{R},\mathcal{Q}-a^{-1}{\mathcal{A}}^{*}\mathcal{A},\mathcal{R}-a^{-1}{\mathcal{C}}^{*}\mathcal{C}\in \mathcal{P}\left(N\right)$ hold, then there exists $n_0\in \mathbb{N}$ with the property $(\tilde{X},\tilde{Y})\in [aI,n_{0}Q]\times [aI,n_0\mathcal{R}]$.

Proof. 

Let us set $F_1(X,Y)=Q-{\mathcal{A}}^{*}{X}^{-k}\mathcal{A}+{\mathcal{B}}^{*}{Y}^{-l}\mathcal{B}$ and $F_2(X,Y)=\mathcal{R}-{\mathcal{C}}^{*}{X}^{-k}\mathcal{C}+{\mathcal{D}}^{*}{Y}^{-l}\mathcal{D}$. Then, $F_1,F_2:\mathcal{H}\left(N\right)\times \mathcal{H}\left(N\right)\to \mathcal{H}\left(N\right)$

For $X,Y\in {\Omega }_{aI}$, from the inclusion ${\mathcal{B}}^{*}\mathcal{B},{\mathcal{D}}^{*}\mathcal{D}\in \mathcal{P}\left(N\right)$ and (Theorem 4(i)), we conclude

$$F_1(X,Y)=Q-{\mathcal{A}}^{*}{X}^{-k}\mathcal{A}+{\mathcal{B}}^{*}{Y}^{-l}\mathcal{B}\succcurlyeq Q-{\mathcal{A}}^{*}{X}^{-k}\mathcal{A}\succcurlyeq Q-a^{-k}{\mathcal{A}}^{*}\mathcal{A}\succcurlyeq aI,$$

and

$$F_2(X,Y)=\mathcal{R}-{\mathcal{C}}^{*}{X}^{-k}\mathcal{C}+{\mathcal{D}}^{*}{Y}^{-l}\mathcal{D}\succcurlyeq \mathcal{R}-{\mathcal{C}}^{*}{X}^{-k}\mathcal{C}\succcurlyeq \mathcal{R}-a^{-k}{\mathcal{C}}^{*}\mathcal{C}\succcurlyeq aI.$$

Therefore, $F_1,F_2:{\Omega }_{aI}\times {\Omega }_{aI}\to {\Omega }_{aI}$.

In order to prove that $(F_1,F_2)$ has the mixed monotone property, let us choose $X,Y,U,V\in {\Omega }_{aI}$, satisfying $X\succcurlyeq U$ and $Y\preccurlyeq V$. Then, $X^{-k}\preccurlyeq U^{-k}$ and $Y^{-l}\succcurlyeq V^{-l}$ hold. Consequently, from ${\mathcal{A}}^{*}\mathcal{A},{\mathcal{C}}^{*}\mathcal{C}\in \mathcal{P}\left(N\right)$, we obtain

$$F_1(X,Y)-F_1(U,Y)=-{\mathcal{A}}^{*}(X^{-k}-U^{-k})\mathcal{A}={\mathcal{A}}^{*}(U^{-k}-X^{-k})\mathcal{A}\succcurlyeq 0,$$

and

$$F_2(X,Y)-F_2(U,Y)=-{\mathcal{C}}^{*}(X^{-k}-U^{-k})\mathcal{C}={\mathcal{C}}^{*}(U^{-k}-X^{-k})\mathcal{C}\succcurlyeq 0,$$

i.e., each $F_i$, $i=1,2$, is an increasing map on its first variable. In a similar fashion, the inequalities

$$F_1(X,Y)-F_1(X,V)={\mathcal{B}}^{*}(Y^{-l}-V^{-l})\mathcal{B}\succcurlyeq 0,$$

and

$$F_2(X,Y)-F_2(X,V)={\mathcal{D}}^{*}(Y^{-l}-V^{-l})\mathcal{D}\succcurlyeq 0.$$

ensure that each $F_i$, $i=1,2$, is a decreasing map on its second variable. Consequently, $(F_1,F_2)$ has the mixed monotone property.

For an initial guess of $X_0=aI$, $Y_0=bI$ from (Theorem 4(ii)), we obtain that $X_0\preccurlyeq F_1(X_0,Y_0)$ and $Y_0\succcurlyeq F_2(X_0,Y_0)$. Indeed, $F_1(X_0,Y_0)=\mathcal{Q}-a^{-k}{\mathcal{A}}^{*}\mathcal{A}+b^{-l}{\mathcal{B}}^{*}\mathcal{B}\succcurlyeq aI=X_0$ is equivalent to $\mathcal{Q}+-b^{-l}{\mathcal{B}}^{*}\mathcal{B}\succcurlyeq a^{-k}{\mathcal{A}}^{*}\mathcal{A}+aI$, i.e., $a^k{\mathcal{B}}^{*}\mathcal{B}\succcurlyeq b^l{\mathcal{A}}^{*}\mathcal{A}+a^k{b}^l(aI-\mathcal{Q})$. The proof of the inequality $Y_0\succcurlyeq F_2(X_0,Y_0)$ is performed in a similar fashion: $F_2(X_0,Y_0)=\mathcal{R}-a^{-k}{\mathcal{C}}^{*}\mathcal{C}+b^{-l}{\mathcal{D}}^{*}\mathcal{D}\preccurlyeq bI=Y_0$, i.e., $a^k{\mathcal{D}}^{*}\mathcal{D}\preccurlyeq b^l{\mathcal{C}}^{*}\mathcal{C}+a^k{b}^l(aI-\mathcal{R})$.

For any two $X,Y\in \mathcal{H}\left(N\right)$, there is a greatest lower bound and a least upper bound.

$F_1$ and $F_2$ are continuous maps.

Let us consider the partially ordered normed space $({\Omega }_{aI}\times {\Omega }_{aI},|\left|\right|·\left|\right||,\preccurlyeq )$, where $\left|\right||x_1,x-2\left)\right|\left|\right|= \parallel x_1{\parallel }_{\mathrm{tr}}+ {\parallel x_2\parallel }_{\mathrm{tr}}$ and $(x_1,x_2)\preccurlyeq (u_1,u-2)$ if $x_1\preccurlyeq u_1$ and $x_2\succcurlyeq u_2$. For the map $T(x,y)=(F_1(x,y),F_2(x,y)):{\Omega }_{aI}\times {\Omega }_{aI}\to {\Omega }_{aI}\times {\Omega }_{aI}$, the following chain of inequalities holds:

$$\begin{array}{ccc}\left|\right|\left|T\right(X,Y)-T(U,V\left)\right|\left|\right|\hfill & =\hfill & \left|\right|\left|\right(F_1(X,Y)-F_1(U,V), F_2(X,Y)-F_2(U,V)\left)\right|\left|\right|\hfill \\ & =\hfill & \parallel (F_1(X,Y)-F_1{(U,V)\parallel }_{\mathrm{tr}} + \parallel F_2(X,Y)-F_2(U,V){)\parallel }_{\mathrm{tr}}\hfill \\ & =\hfill & \parallel {\mathcal{A}}^{*}(U^{-k}-X^{-k})\mathcal{A}+{\mathcal{B}}^{*}(Y^{-l}-V^{-l}){\mathcal{B}\parallel }_{\mathrm{tr}}\hfill \\ & & +\parallel {\mathcal{C}}^{*}(U^{-k}-X^{-k})\mathcal{C}+{\mathcal{D}}^{*}(Y^{-l}-V^{-l}){\mathcal{D}\parallel }_{\mathrm{tr}}\hfill \\ & =\hfill & \mathrm{tr}\left({\mathcal{A}}^{*}\mathcal{A}(U^{-k}-X^{-k})\right)+\mathrm{tr}\left({\mathcal{B}}^{*}\mathcal{B}(Y^{-l}-V^{-l})\right)\hfill \\ & & +\mathrm{tr}\left({\mathcal{C}}^{*}\mathcal{C}(U^{-k}-X^{-k})\right)+\mathrm{tr}\left({\mathcal{D}}^{*}\mathcal{D}(Y^{-l}-V^{-l})\right)\hfill \\ & =\hfill & \mathrm{tr}\left(({\mathcal{A}}^{*}\mathcal{A}+{\mathcal{C}}^{*}\mathcal{C})(U^{-k}-X^{-k})\right)\hfill \\ & & +\mathrm{tr}\left(({\mathcal{B}}^{*}\mathcal{B}+{\mathcal{D}}^{*}\mathcal{D})(Y^{-l}-V^{-l})\right)\hfill \\ & \le \hfill & \parallel {\mathcal{A}}^{*}\mathcal{A}+{\mathcal{C}}^{*}\mathcal{C}\parallel .\mathrm{tr}(U^{-k}-X^{-k})\hfill \\ & & +\parallel {\mathcal{B}}^{*}\mathcal{B}+{\mathcal{D}}^{*}\mathcal{D}\parallel .\mathrm{tr}(Y^{-l}-V^{-l})\hfill \\ & \le \hfill & \parallel {\mathcal{A}}^{*}\mathcal{A}+{\mathcal{C}}^{*}\mathcal{C}\parallel {\displaystyle \frac{k}{a^{k+1}}}\mathrm{tr}(X-U)\hfill \\ & & +\parallel {\mathcal{B}}^{*}\mathcal{B}+{\mathcal{D}}^{*}\mathcal{D}\parallel {\displaystyle \frac{l}{a^{l+1}}}\mathrm{tr}(V-Y)\hfill \\ & \le \hfill & max\left\{{\displaystyle \frac{k\parallel {\mathcal{A}}^{*}\mathcal{A}+{\mathcal{C}}^{*}\mathcal{C}\parallel }{a^{k+1}}},{\displaystyle \frac{l\parallel {\mathcal{B}}^{*}\mathcal{B}+{\mathcal{D}}^{*}\mathcal{D}\parallel }{a^{l+1}}}\right\}(\mathrm{tr}(X-U)+\mathrm{tr}(V-Y))\hfill \\ & \le \hfill & {\delta }_1{(\parallel X-U)\parallel }_{\mathrm{tr}}{+\parallel V-Y\parallel }_{\mathrm{tr}})\hfill \\ & =\hfill & {\delta }_1\left|\right||(X,Y)-(U,V)\left|\right||.\hfill \end{array}$$

For the last inequality, we use the fact that $X-U,V-Y\in \mathcal{P}\left(N\right)$, and therefore, $\mathrm{tr}(X-U)\le {\parallel X-U\parallel }_{\mathrm{tr}}$ and $\mathrm{tr}(V-Y)\le {\parallel V-Y\parallel }_{\mathrm{tr}}$.

A unique ordered pair $(\tilde{X},\tilde{Y})\in {\Omega }_{aI}\times {\Omega }_{aI}$ exists, which is a solution to system (5), since $(F_1,F_2)$ meets the assumption in Theorem 1.

The existence of unique solutions $\tilde{X},\tilde{Y}\succcurlyeq aI$ has been shown. The interval where the solutions are located will be shortened. Assume that $\mathcal{Q}-a^{-k}{\mathcal{A}}^{*}\mathcal{A},\mathcal{R}-a^{-k}{\mathcal{C}}^{*}\mathcal{C},\mathcal{Q},\mathcal{P}\in \mathcal{P}\left(N\right)$ and $F_1$, $F_2$ are continuous.

We will use the fact that, if $\mathcal{W}\in \mathcal{P}\left(N\right)$, then for any $\alpha \in (0,1)$ $W\succcurlyeq \alpha W$ holds, i.e., $W-\alpha W\succcurlyeq 0$, or equivalently, $(1-\alpha )W\in \mathcal{P}\left(N\right)$. Let us mention that if ${lim}_{n\to \infty }{\alpha }_n=0$, ${\alpha }_n>0$, and $\mathcal{W},\mathcal{V}\in \mathcal{P}\left(N\right)$, then there is $n_0\in \mathbb{N}$ such that ${\alpha }_n\mathcal{W}\preccurlyeq \mathcal{V}$ holds for all $n\ge n_0$.

Therefore, there is $n_0\in \mathbb{N}$ big enough such that the following holds:

$$(n_0-1)\mathcal{Q}\succcurlyeq a^{-l}{\mathcal{B}}^{*}\mathcal{B}\succcurlyeq a^{-l}{\mathcal{B}}^{*}\mathcal{B}-n_0^{-Nk}{\mathcal{A}}^{*}{Q}^{-k}\mathcal{A}.$$

Consequently, for all $\mathcal{Q},\mathcal{R}\in \mathcal{P}\left(N\right)$, there exists $n_0\in \mathbb{N}$ satisfying

$$-n_0^{-Nk}{\mathcal{A}}^{*}{Q}^{-k}\mathcal{A}+a^{-l}{\mathcal{B}}^{*}\mathcal{B}\preccurlyeq (n_0-1)Q$$

and

$$-n_0^{-Nk}{\mathcal{C}}^{*}{Q}^{-k}\mathcal{C}+a^{-l}{\mathcal{D}}^{*}\mathcal{D}\preccurlyeq (n_0-1)\mathcal{R}.$$

The last two inequalities are equivalent to

$$\begin{array}{ccc}{F}_1(n_{0}Q,aI)\hfill & =\hfill & Q-{\mathcal{A}}^{*}{\left(n_{0}Q\right)}^{-k}\mathcal{A}+{\mathcal{B}}^{*}{\left(aI\right)}^{-l}\mathcal{B}\hfill \\ \hfill & =\hfill & Q-n_0^{-Nk}{\mathcal{A}}^{*}{Q}^{-k}\mathcal{A}+a^{-l}{\mathcal{B}}^{*}\mathcal{B}\preccurlyeq n_{0}Q,\hfill \end{array}$$

(7)

and

$$\begin{array}{ccc}{F}_2(n_{0}Q,aI)\hfill & =\hfill & \mathcal{R}-{\mathcal{C}}^{*}{\left(n_{0}Q\right)}^{-k}\mathcal{C}+{\mathcal{D}}^{*}{\left(aI\right)}^{-l}\mathcal{D}\hfill \\ \hfill & =\hfill & \mathcal{R}-n_0^{-Nk}{\mathcal{C}}^{*}{Q}^{-k}\mathcal{C}+a^{-l}{\mathcal{D}}^{*}\mathcal{D}\preccurlyeq n_0\mathcal{R},\hfill \end{array}$$

(8)

With the extra condition that $\mathcal{R}-a^{-k}{\mathcal{C}}^{*}\mathcal{C},\mathcal{Q}-a^{-k}{\mathcal{A}}^{*}\mathcal{A}\in \mathcal{P}\left(N\right)$, we obtain

$$\begin{array}{ccc}{F}_1(aI,n_0\mathcal{R})\hfill & =\hfill & Q-{\mathcal{A}}^{*}{\left(aI\right)}^{-k}\mathcal{A}+{\mathcal{B}}^{*}{\left(n_0\mathcal{R}\right)}^{-l}\mathcal{B}\hfill \\ \hfill & =\hfill & Q-a^{-k}{\mathcal{A}}^{*}\mathcal{A}+n_0^{-Nl}{\mathcal{B}}^{*}{\mathcal{R}}^{-l}\mathcal{B}\succcurlyeq aI,\hfill \end{array}$$

(9)

and

$$\begin{array}{ccc}{F}_2(aI,n_0\mathcal{R})\hfill & =\hfill & \mathcal{R}-{\mathcal{C}}^{*}{\left(aI\right)}^{-k}\mathcal{C}+{\mathcal{D}}^{*}{\left(n_0\mathcal{R}\right)}^{-l}\mathcal{D}\hfill \\ \hfill & =\hfill & \mathcal{R}-a^{-k}{\mathcal{C}}^{*}\mathcal{C}+n_0^{-Nl}{\mathcal{D}}^{*}{\mathcal{R}}^{-l}\mathcal{D}\succcurlyeq aI.\hfill \end{array}$$

(10)

Let $X\in [aI,n_{0}Q]$ and $Y\in [aI,n_0\mathcal{R}]$, i.e.,

$$aI\preccurlyeq X\preccurlyeq n_{0}QandaI\preccurlyeq Y\preccurlyeq n_0\mathcal{R}.$$

Using the mixed monotone property of $(F_1,F_2)$ and (7)–(10), we obtain

$$aI\preccurlyeq F_1(aI,n_0\mathcal{R})\preccurlyeq F_1(X,Y)\preccurlyeq F_1(n_{0}Q,aI)\preccurlyeq n_{0}Q,$$

(11)

and

$$aI\preccurlyeq F_2(aI,n_0\mathcal{R})\preccurlyeq F_2(X,Y)\preccurlyeq F_2(n_{0}Q,aI)\preccurlyeq n_0\mathcal{R}.$$

(12)

Consequently, from (11) and (12), it follows that $T:C\to C$, where $C=[aI,n_{0}Q]\times [aI,n_0\mathcal{R}]$ is a compact and convex set, $T(X,Y)=(F_1(X,Y), F_2(X,Y))$ is a continuous map, and according to Theorem 3 the map T has at least one fixed point in C. □

Let us consider the equation

$$X=Q-{\mathcal{A}}^{*}{X}^{-k}\mathcal{A}+{\mathcal{B}}^{*}{X}^{-l}\mathcal{B}$$

(13)

for $\mathcal{Q}\in \mathcal{H}\left(N\right)$, $\mathcal{A},\mathcal{B},\in \mathcal{X}\left(N\right)$, and ${\mathcal{A}}^{*}\mathcal{A},{\mathcal{B}}^{*}\mathcal{B},\mathcal{Q}\in \mathcal{P}\left(N\right)$.

We will apply Theorem 4 for the system of Equation (5) with $\mathcal{C}=\mathcal{A}$, $\mathcal{D}=\mathcal{B}$, and $\mathcal{R}=Q$, i.e.,

$$\begin{array}{c}X=Q+p{\mathcal{A}}^{*}{X}^k\mathcal{A}+q{\mathcal{B}}^{*}{Y}^l\mathcal{B}\\ Y=Q+p{\mathcal{A}}^{*}{Y}^k\mathcal{A}+q{\mathcal{B}}^{*}{X}^l\mathcal{B}\end{array}$$

(14)

Corollary 1.

Let $p=-1$, $q=1$, $k,l\in (0,1]$ in (14) and $0<a<b$, such that the following are fulfilled:

(i) 

$Q\succcurlyeq aI+a^{-k}{\mathcal{A}}^{*}\mathcal{A}$.

(ii) 

$a^k{b}^l(Q-aI)+a^k{\mathcal{B}}^{*}\mathcal{B}\succcurlyeq b^l{\mathcal{A}}^{*}\mathcal{A}$.

(iii) 

Where ${\delta }_2=max\left\{{\displaystyle \frac{k\parallel {\mathcal{A}}^{*}\mathcal{A}\parallel }{a^{k+1}}},{\displaystyle \frac{l\parallel {\mathcal{B}}^{*}\mathcal{B}\parallel }{a^{l+1}}}\right\}<1/2$.

Then, (13) has a unique solution $\tilde{X}\in {\Omega }_{aI}$. The inductively defined sequences ${\left\{X_n\right\}}_{n=0}^{\infty }$ and ${\left\{Y_n\right\}}_{n=0}^{\infty }$

$$\left\{\begin{array}{c}{X}_0=aI,Y_0=bI,\hfill \\ X_{n+1}=Q-{\mathcal{A}}^{*}{X}_n^{-k}\mathcal{A}+{\mathcal{B}}^{*}{Y}_n^{-l}\mathcal{B},\hfill \\ Y_{n+1}=Q-{\mathcal{A}}^{*}{Y}_n^{-k}\mathcal{A}+{\mathcal{B}}^{*}{X}_n^{-l}\mathcal{B},\hfill \end{array}\right.$$

(15)

are both converging to $\tilde{X}$ in $(\mathcal{H}\left(N\right), \parallel ·{\parallel }_{\mathrm{tr}},\preccurlyeq )$. The following estimations of error are valid:

(I) 

A priori error estimate:

$$max\{\parallel \tilde{X}-X_n{\parallel }_{\mathrm{tr}},\parallel \tilde{X}-Y_n{\parallel }_{\mathrm{tr}}\}\le {\displaystyle \frac{{\delta }_2^n}{1-{\delta }_2}}(\parallel X_0-X_1{\parallel }_{\mathrm{tr}}+ \parallel Y_0-Y_1{\parallel }_{\mathrm{tr}}).$$

(II) 

A posteriori error estimate:

$$max\{\parallel \tilde{X}-X_{n+1}{\parallel }_{\mathrm{tr}},\parallel \tilde{X}-Y_{n+1}{\parallel }_{\mathrm{tr}}\} \le {\displaystyle \frac{{\delta }_2}{1-{\delta }_2}}(\parallel X_n-X_{n+1}{\parallel }_{\mathrm{tr}}+ \parallel Y_n-Y_{n+1}{\parallel }_{\mathrm{tr}}).$$

(III) 

The convergence rate:

$$max\{\parallel \tilde{X}-X_{n+1}{\parallel }_{\mathrm{tr}},\parallel \tilde{X}-Y_{n+1}{\parallel }_{\mathrm{tr}}\} \le {\displaystyle \frac{{\delta }_2}{1-{\delta }_2}}(\parallel \tilde{X}-X_n{\parallel }_{\mathrm{tr}}+ \parallel \tilde{X}-Y_n{\parallel }_{\mathrm{tr}}).$$

If, moreover, the inclusions $\mathcal{Q},\mathcal{Q}-a^{-1}{\mathcal{A}}^{*}\mathcal{A}\in \mathcal{P}\left(N\right)$ hold, then there exists $n_0\in \mathbb{N}$ with the property $\tilde{X}\in [aI,n_{0}Q]$.

For the proof of the corollary, we need to comment that by Theorem 1, it follows that $\tilde{X}=\tilde{Y}$, because $F_2(X,Y)=F_1(Y,X)$.

It was commented in [11] that we can change the signs in (5) in a way to keep the assumption for the ordered pair of maps $(F_1,F_2)$ to be with the mixed monotone property.

Let us consider (5) with $p=1$, $q=-1$, $k\in (0,1]$, and $l\in [-1,0)$, i.e.,

$$\left|\begin{array}{c}X=Q+{\mathcal{A}}^{*}{X}^k\mathcal{A}+{\mathcal{B}}^{*}{Y}^{-l}\mathcal{B},\hfill \\ Y=\mathcal{R}+{\mathcal{C}}^{*}{X}^k\mathcal{C}+{\mathcal{D}}^{*}{Y}^{-l}\mathcal{D}.\hfill \end{array}\right.$$

(16)

Theorem 5.

For some $0<a<b$, let the following be fulfilled:

(i) 

$Q\succcurlyeq aI-a^k{\mathcal{A}}^{*}\mathcal{A}$ and $\mathcal{R}\succcurlyeq aI-a^k{\mathcal{C}}^{*}\mathcal{C}$.

(ii) 

$b^l{a}^k{\mathcal{A}}^{*}\mathcal{A}\succcurlyeq b^l(aI-Q)-{\mathcal{B}}^{*}\mathcal{B}$ and $b^l{a}^k{\mathcal{C}}^{*}\mathcal{C}\succcurlyeq b^l(aI-\mathcal{R})-{\mathcal{D}}^{*}\mathcal{D}$.

(iii) 

Where ${\delta }_3=max\left\{{\displaystyle \frac{k\parallel {\mathcal{A}}^{*}\mathcal{A}+{\mathcal{C}}^{*}\mathcal{C}\parallel }{a^{k-1}}},{\displaystyle \frac{l\parallel {\mathcal{B}}^{*}\mathcal{B}+{\mathcal{D}}^{*}\mathcal{D}\parallel }{a^{l+1}}}\right\}<1$.

Then, (16) has a unique solution $\tilde{X},\tilde{Y}\in {\Omega }_{aI}$. The inductively defined sequences ${\left\{X_n\right\}}_{n=0}^{\infty }$ and ${\left\{Y_n\right\}}_{n=0}^{\infty }$

$$\left\{\begin{array}{c}{X}_0=aI,Y_0=bI,\hfill \\ X_{n+1}=Q+{\mathcal{A}}^{*}{X}_n^k\mathcal{A}+{\mathcal{B}}^{*}{Y}_n^{-l}\mathcal{B},\hfill \\ Y_{n+1}=\mathcal{R}+{\mathcal{C}}^{*}{X}_n^k\mathcal{C}+{\mathcal{D}}^{*}{Y}_n^{-l}\mathcal{D},\hfill \end{array}\right.$$

(17)

are converging to $\tilde{X}$ and $\tilde{Y}$ in $(\mathcal{H}\left(N\right),\parallel ·{\parallel }_{\mathrm{tr}},\preccurlyeq )$, respectively. The following estimations of error are valid:

(I) 

A priori error estimate:

$$max\{\parallel \tilde{X}-X_n{\parallel }_{\mathrm{tr}},\parallel \tilde{Y}-Y_n{\parallel }_{\mathrm{tr}}\}\le {\displaystyle \frac{{\delta }_3^n}{1-{\delta }_3}}(\parallel X_0-X_1{\parallel }_{\mathrm{tr}}+ \parallel Y_0-Y_1{\parallel }_{\mathrm{tr}}).$$

(II) 

A posteriori error estimate:

$$max\{\parallel \tilde{X}-X_{n+1}{\parallel }_{\mathrm{tr}},\parallel \tilde{Y}-Y_{n+1}{\parallel }_{\mathrm{tr}}\}\le {\displaystyle \frac{{\delta }_3}{1-{\delta }_3}}(\parallel X_n-X_{n+1}{\parallel }_{\mathrm{tr}}+ \parallel Y_n-Y_{n+1}{\parallel }_{\mathrm{tr}}).$$

(III) 

The convergence rate:

$$max\{\parallel \tilde{X}-X_{n+1}{\parallel }_{\mathrm{tr}},\parallel \tilde{Y}-Y_{n+1}{\parallel }_{\mathrm{tr}}\}\le {\displaystyle \frac{{\delta }_3}{1-{\delta }_3}}(\parallel \tilde{X}-X_n{\parallel }_{\mathrm{tr}}+ \parallel \tilde{Y}-Y_n{\parallel }_{\mathrm{tr}}).$$

If, moreover, the inclusions $\mathcal{Q}-a^{-1}{\mathcal{A}}^{*}\mathcal{A},\mathcal{R}-a^{-1}{\mathcal{C}}^{*}\mathcal{C},\mathcal{Q},\mathcal{R}\in \mathcal{P}\left(N\right)$, then there exists $n_0\in \mathbb{N}$ with the property $(\tilde{X},\tilde{Y})\in [aI,n_{0}Q]\times [aI,n_0\mathcal{R}]$.

Proof. 

Let us set $F_1(X,Y)=\mathcal{Q}+{\mathcal{A}}^{*}{X}^k\mathcal{A}+{\mathcal{B}}^{*}{Y}^{-l}\mathcal{B}$ and $F_2(X,Y)=\mathcal{R}+{\mathcal{C}}^{*}{X}^k\mathcal{C}+{\mathcal{D}}^{*}{Y}^{-l}\mathcal{D}$. Then, $F_1,F_2:\mathcal{H}\left(N\right)\times \mathcal{H}\left(N\right)\to \mathcal{H}\left(N\right)$.

For $X,Y\in {\Omega }_{aI}$, from the inclusion ${\mathcal{B}}^{*}\mathcal{B},{\mathcal{D}}^{*}\mathcal{D}\in \mathcal{P}\left(N\right)$ and (Theorem 5(i)), we conclude

$$F_1(X,Y)=Q+{\mathcal{A}}^{*}{X}^k\mathcal{A}+{\mathcal{B}}^{*}{Y}^{-l}\mathcal{B}\succcurlyeq Q+{\mathcal{A}}^{*}{X}^k\mathcal{A}\succcurlyeq Q+a^k{\mathcal{A}}^{*}\mathcal{A}\succcurlyeq aI,$$

and

$$F_2(X,Y)=\mathcal{R}+{\mathcal{C}}^{*}{X}^k\mathcal{C}+{\mathcal{D}}^{*}{Y}^{-l}\mathcal{D}\succcurlyeq \mathcal{R}+{\mathcal{C}}^{*}{X}^k\mathcal{C}\succcurlyeq \mathcal{R}+a^k{\mathcal{C}}^{*}\mathcal{C}\succcurlyeq aI.$$

Therefore, $F_1,F_2:{\Omega }_{aI}\times {\Omega }_{aI}\to {\Omega }_{aI}$.

In order to prove that $(F_1,F_2)$ has the mixed monotone property, let us choose $X,Y,U,V\in {\Omega }_{aI}$, satisfying $X\succcurlyeq U$ and $Y\preccurlyeq V$. Then, $X^{-k}\preccurlyeq U^{-k}$ and $Y^{-l}\succcurlyeq V^{-l}$ hold. Consequently, from ${\mathcal{A}}^{*}\mathcal{A},{\mathcal{C}}^{*}\mathcal{C}\in \mathcal{P}\left(N\right)$, we obtain

$$F_1(X,Y)-F_1(U,Y)={\mathcal{A}}^{*}(X^k-U^k)\mathcal{A}\succcurlyeq 0,$$

and

$$F_2(X,Y)-F_2(U,Y)={\mathcal{C}}^{*}(X^k-U^k)\mathcal{C}\succcurlyeq 0,$$

i.e., each $F_i$, $i=1,2$, is an increasing map on its first variable. In a similar fashion, the inequalities

$$F_1(X,Y)-F_1(X,V)={\mathcal{B}}^{*}(Y^{-l}-V^{-l})\mathcal{B}\succcurlyeq 0,$$

and

$$F_2(X,Y)-F_2(X,V)={\mathcal{D}}^{*}(Y^{-l}-V^{-l})\mathcal{D}\succcurlyeq 0.$$

ensure that each $F_i$, $i=1,2$, is a decreasing map on its second variable. Consequently, $(F_1,F_2)$ has the mixed monotone property.

For an initial guess of $X_0=aI$, $Y_0=bI$ from (Theorem 5(ii)), we obtain that $X_0\preccurlyeq F_1(X_0,Y_0)$ and $Y_0\succcurlyeq F_2(X_0,Y_0)$. Indeed, $F_1(X_0,Y_0)=Q+a^k{\mathcal{A}}^{*}\mathcal{A}+b^{-l}{\mathcal{B}}^{*}\mathcal{B}\succcurlyeq aI=X_0$ is equivalent to $a^k{\mathcal{A}}^{*}\mathcal{A}+b^{-l}{\mathcal{B}}^{*}\mathcal{B}\succcurlyeq aI-Q$, i.e., $b^l{a}^k{\mathcal{A}}^{*}\mathcal{A}+{\mathcal{B}}^{*}\mathcal{B}\succcurlyeq b^l(aI-Q)$. The proof that $Y_0\succcurlyeq F_2(X_0,Y_0)$ is conducted in a similar fashion: $F_2(X_0,Y_0)=\mathcal{R}+a^k{\mathcal{C}}^{*}\mathcal{C}+b^{-l}{\mathcal{D}}^{*}\mathcal{D}\preccurlyeq aI=Y_0$ is equivalent to $a^k{\mathcal{C}}^{*}\mathcal{C}+b^{-l}{\mathcal{D}}^{*}\mathcal{D}\succcurlyeq aI-\mathcal{R}$, i.e., $b^l{a}^k{\mathcal{C}}^{*}\mathcal{C}+{\mathcal{D}}^{*}\mathcal{D}\succcurlyeq b^l(aI-\mathcal{R})$.

There exists a greatest lower bound and a least upper bound for each $X,Y\in \mathcal{H}\left(N\right)$.

The maps $F_1$ and $F_2$ are continuous.

Let us consider the partially ordered normed space $({\Omega }_{aI}\times {\Omega }_{aI},|\left|\right|·\left|\right||,\preccurlyeq )$, where $\left|\right||x_1,x-2\left)\right|\left|\right|= \parallel x_1{\parallel }_{\mathrm{tr}}+ {\parallel x_2\parallel }_{\mathrm{tr}}$ and $(x_1,x_2)\preccurlyeq (u_1,u-2)$ if $x_1\preccurlyeq u_1$ and $x_2\succcurlyeq u_2$. For the map $T(x,y)=(F_1(x,y),F_2(x,y)):{\Omega }_{aI}\times {\Omega }_{aI}\to {\Omega }_{aI}\times {\Omega }_{aI}$, the following chain of inequalities holds:

$$\begin{array}{ccc}\left|\right|\left|T\right(X,Y)-T(U,V\left)\right|\left|\right|\hfill & =\hfill & \left|\right|\left|\right(F_1(X,Y)-F_1(U,V),F_2(X,Y)-F_2(U,V)\left)\right|\left|\right|\hfill \\ & =\hfill & \parallel (F_1(X,Y)-F_1{(U,V)\parallel }_{\mathrm{tr}} + \parallel F_2(X,Y)-F_2(U,V){)\parallel }_{\mathrm{tr}}\hfill \\ & =\hfill & \parallel {\mathcal{A}}^{*}(X^k-U^k)\mathcal{A}+{\mathcal{B}}^{*}(Y^{-l}-V^{-l}){\mathcal{B}\parallel }_{\mathrm{tr}}\hfill \\ & & +\parallel {\mathcal{C}}^{*}(X^k-U^k)\mathcal{C}+{\mathcal{D}}^{*}(Y^{-l}-V^{-l}){\mathcal{D}\parallel }_{\mathrm{tr}}\hfill \\ & =\hfill & \mathrm{tr}\left({\mathcal{A}}^{*}\mathcal{A}(X^k-U^k)\right)+\mathrm{tr}\left({\mathcal{B}}^{*}\mathcal{B}(Y^{-l}-V^{-l})\right)\hfill \\ & & +\mathrm{tr}\left({\mathcal{C}}^{*}\mathcal{C}(X^k-U^k)\right)+\mathrm{tr}\left({\mathcal{D}}^{*}\mathcal{D}(Y^{-l}-V^{-l})\right)\hfill \\ & =\hfill & \mathrm{tr}\left(({\mathcal{A}}^{*}\mathcal{A}+{\mathcal{C}}^{*}\mathcal{C})(X^k-U^k)\right)\hfill \\ & & +\mathrm{tr}\left(({\mathcal{B}}^{*}\mathcal{B}+{\mathcal{D}}^{*}\mathcal{D})(Y^{-l}-V^{-l})\right)\hfill \\ & \le \hfill & \parallel {\mathcal{A}}^{*}\mathcal{A}+{\mathcal{C}}^{*}\mathcal{C}\parallel .\mathrm{tr}(X^k-U^k)\hfill \\ & & +\parallel {\mathcal{B}}^{*}\mathcal{B}+{\mathcal{D}}^{*}\mathcal{D}\parallel .\mathrm{tr}(Y^{-l}-V^{-l})\hfill \\ & \le \hfill & \parallel {\mathcal{A}}^{*}\mathcal{A}+{\mathcal{C}}^{*}\mathcal{C}\parallel {\displaystyle \frac{k}{a^{k-1}}}\mathrm{tr}(X-U)\hfill \\ & & +\parallel {\mathcal{B}}^{*}\mathcal{B}+{\mathcal{D}}^{*}\mathcal{D}\parallel {\displaystyle \frac{l}{a^{l+1}}}\mathrm{tr}(V-Y)\hfill \\ & \le \hfill & max\left\{{\displaystyle \frac{k\parallel {\mathcal{A}}^{*}\mathcal{A}+{\mathcal{C}}^{*}\mathcal{C}\parallel }{a^{k-1}}},{\displaystyle \frac{l\parallel {\mathcal{B}}^{*}\mathcal{B}+{\mathcal{D}}^{*}\mathcal{D}\parallel }{a^{l+1}}}\right\}(\mathrm{tr}(X-U)+\mathrm{tr}(V-Y))\hfill \\ & \le \hfill & {\delta }_3{(\parallel X-U)\parallel }_{\mathrm{tr}}{+\parallel V-Y\parallel }_{\mathrm{tr}})\hfill \\ & =\hfill & {\delta }_3\left|\right||(X,Y)-(U,V)\left|\right||.\hfill \end{array}$$

A unique ordered pair $(\tilde{X},\tilde{Y})\in {\Omega }_{aI}\times {\Omega }_{aI}$ exists, which is a solution to system (16), since $(F_1,F_2)$ meets the assumption in Theorem 1. □

The matrix equation $X=Q\pm {\sum }_{k=1}^n{\mathcal{A}}_k^{*}{\mathcal{F}}_{\infty }\left(X\right){\mathcal{A}}_k$, where ${\mathcal{F}}_{\infty }$ is a monotone map, was considered in [10]. If we simplify it by letting $n=2$, we obtain $X=Q\pm {\sum }_{k=1}^2{\mathcal{A}}_k^{*}{\mathcal{F}}_{\infty }\left(X\right){\mathcal{A}}_k$. We cannot apply the technique from [8] as the map $F_1(X,Y)=Q\pm ({\mathcal{A}}_1^{*}{\mathcal{F}}_{\infty }\left(X\right){\mathcal{A}}_1+{\mathcal{A}}_2^{*}{\mathcal{F}}_{\infty }\left(X\right){\mathcal{A}}_2)$ does not meet the mixed monotone property. Fortunately, it will be either a totally monotone increasing or totally monotone decreasing map, and consequently, we can rely on Theorem 2.

Let us consider (5) with $p,q=1$, $k,l\in (0,1]$, i.e.,

$$\left|\begin{array}{c}X=Q+{\mathcal{A}}^{*}{X}^{-k}\mathcal{A}+{\mathcal{B}}^{*}{Y}^{-l}\mathcal{B}\hfill \\ Y=\mathcal{R}+{\mathcal{C}}^{*}{X}^{-k}\mathcal{C}+{\mathcal{D}}^{*}{Y}^{-l}\mathcal{D}\hfill \end{array}\right.$$

(18)

and ${\mathcal{A}}^{*}\mathcal{A},{\mathcal{B}}^{*}\mathcal{B},{\mathcal{C}}^{*}\mathcal{C},{\mathcal{D}}^{*}\mathcal{D},\mathcal{Q},\mathcal{R},\in P\left(N\right)$.

Theorem 6.

Let $0<b<a$, such that the following conditions are fulfilled:

(i) 

$Q,\mathcal{R}\in {\Omega }_{aI}$.

(ii) 

Where ${\delta }_4=max\left\{\parallel {\mathcal{A}}^{*}\mathcal{A}+{\mathcal{C}}^{*}\mathcal{C}\parallel {\displaystyle \frac{k}{a^{k+1}}},\parallel {\mathcal{B}}^{*}\mathcal{B}+{\mathcal{D}}^{*}\mathcal{D}\parallel {\displaystyle \frac{l}{a^{l+1}}}\right\}<1$.

Then, the following are true:

(1) 

The system (18) has a solution $(\tilde{X},\tilde{Y})\in {\Omega }_Q\times {\Omega }_{\mathcal{R}}$.

(2) 

The sequences defined by

$$\left\{\begin{array}{c}{X}_0=aI,Y_0=bI\hfill \\ X_{n+1}=Q+{\mathcal{A}}^{*}{X}_n^{-k}\mathcal{A}+{\mathcal{B}}^{*}{Y}_n^{-l}\mathcal{B},\hfill \\ Y_{n+1}=\mathcal{R}+{\mathcal{C}}^{*}{X}_n^{-k}\mathcal{C}+{\mathcal{D}}^{*}{Y}_n^{-l}\mathcal{D},\hfill \end{array}\right.$$

(19)

are converging to $\tilde{X}$ and $\tilde{Y}$ in $(\mathcal{H}\left(N\right),\parallel ·{\parallel }_{\mathrm{tr}},\preccurlyeq )$.

The following estimations of error are valid:

(I) 

A priori error estimate:

$$max\{\parallel \tilde{X}-X_n{\parallel }_{\mathrm{tr}},\parallel \tilde{Y}-Y_n{\parallel }_{\mathrm{tr}}\}\le {\displaystyle \frac{{\delta }_4^n}{1-{\delta }_4}}(\parallel X_0-X_1{\parallel }_{\mathrm{tr}}+ \parallel Y_0-Y_1{\parallel }_{\mathrm{tr}}),$$

(II) 

A posteriori error estimate:

$$max\{\parallel \tilde{X}-X_{n+1}{\parallel }_{\mathrm{tr}},\parallel \tilde{Y}-Y_{n+1}{\parallel }_{\mathrm{tr}}\}\le {\displaystyle \frac{{\delta }_4}{1-{\delta }_4}}(\parallel X_n-X_{n+1}{\parallel }_{\mathrm{tr}}+ \parallel Y_n-Y_{n+1}{\parallel }_{\mathrm{tr}})$$

(III) 

The rate of convergence:

$$max\{\parallel \tilde{X}-X_{n+1}{\parallel }_{\mathrm{tr}},\parallel \tilde{Y}-Y_{n+1}{\parallel }_{\mathrm{tr}}\}\le {\displaystyle \frac{{\delta }_4}{1-{\delta }_4}}(\parallel \tilde{X}-X_n{\parallel }_{\mathrm{tr}}+ \parallel \tilde{Y}-Y_n{\parallel }_{\mathrm{tr}}).$$

The inclusion $(\tilde{X},\tilde{Y})\in [Q,F_1(Q,\mathcal{R})]\times [\mathcal{R},F_2(Q,\mathcal{R})]$ holds.

Proof. 

Let us set $F_1(X,Y)=\mathcal{Q}+{\mathcal{A}}^{*}{X}^{-k}\mathcal{A}+{\mathcal{B}}^{*}{Y}^{-l}\mathcal{B}$ and $F_2(X,Y)=\mathcal{R}+{\mathcal{C}}^{*}{X}^{-k}\mathcal{C}+{\mathcal{D}}^{*}{Y}^{-l}\mathcal{D}$. Then, $F_1,F_2:\mathcal{H}\left(N\right)\times \mathcal{H}\left(N\right)\to \mathcal{H}\left(N\right)$

From ${\mathcal{A}}^{*}\mathcal{A}$,${\mathcal{B}}^{*}\mathcal{B}$,${\mathcal{C}}^{*}\mathcal{C}$, ${\mathcal{D}}^{*}\mathcal{D}\in P\left(N\right)$, for all $X\in {\Omega }_Q$ and $X\in {\Omega }_{\mathcal{R}}$, the inequalities $F_1(X,Y)\succcurlyeq Q$ and $F_2(X,Y)\succcurlyeq \mathcal{R}$ hold, as far as ${\mathcal{A}}^{*}{X}^{-k}\mathcal{A}+{\mathcal{B}}^{*}{Y}^{-l}\mathcal{B}\succcurlyeq 0$ and ${\mathcal{C}}^{*}{X}^{-k}\mathcal{C}+{\mathcal{D}}^{*}{Y}^{-l}\mathcal{D}\succcurlyeq 0$. Consequently, $F_1:{\Omega }_Q\times {\Omega }_{\mathcal{R}}\to {\Omega }_Q$ and $F_2:{\Omega }_Q\times {\Omega }_{\mathcal{R}}\to {\Omega }_{\mathcal{R}}$.

We will show that the ordered pair of maps $(F_1,F_2)$ satisfies the total decreasing monotone property. For all $X,U\in {\Omega }_Q$ and $Y,V\in {\Omega }_{\mathcal{R}}$ chosen to satisfy $X\preccurlyeq U$ and $Y\preccurlyeq V$, $X^{-k}\succcurlyeq U^{-k}$ and $Y^{-l}\succcurlyeq V^{-l}$ hold. Thus,

$$F_1(X,Y)-F_1(U,Y)={\mathcal{A}}^{*}(X^{-k}-U^{-k})\mathcal{A}\succcurlyeq 0,$$

and

$$F_2(X,Y)-F_2(U,Y)={\mathcal{C}}^{*}(X^{-k}-U^{-k})\mathcal{C}\succcurlyeq 0,$$

i.e., each $F_i$, $i=1,2$, is a decreasing map on its first variable. In a similar fashion, from

$$F_1(X,Y)-F_1(X,V)={\mathcal{B}}^{*}(Y^{-l}-V^{-l})\mathcal{B}\succcurlyeq 0,$$

and

$$F_2(X,Y)-F_2(X,V)={\mathcal{D}}^{*}(Y^{-l}-V^{-l})\mathcal{D}\succcurlyeq 0$$

we conclude that each $F_i$, $i=1,2$, is a decreasing map on its second variable. Consequently, $(F_1,F_2)$ has the total decreasing monotone property.

Let us set $T(x,y)=(F_1(x,y),F_2(x,y)):{\Omega }_Q\times {\Omega }_{\mathcal{R}}\to {\Omega }_Q\times {\Omega }_{\mathcal{R}}$, $({\Omega }_{\mathcal{R}}\times {\Omega }_{\mathcal{R}},|\left|\right|·\left|\right|\left|\right)$ and $\left|\right|\left|(x,y)\right|\left|\right|={\parallel x\parallel }_{\mathrm{tr}}+{\parallel y\parallel }_{\mathrm{tr}}$.

$$\begin{array}{ccc}\left|\right|\left|T\right(X,Y)-T(U,V\left)\right|\left|\right|\hfill & =\hfill & \left|\right|\left|\right(F_1(X,Y)-F_1(U,V),F_2(X,Y)-F_2(U,V)\left)\right|\left|\right|\hfill \\ & =\hfill & \parallel (F_1(X,Y)-F_1{(U,V)\parallel }_{\mathrm{tr}} + \parallel F_2(X,Y)-F_2(U,V){)\parallel }_{\mathrm{tr}}\hfill \\ & =\hfill & \parallel {\mathcal{A}}^{*}(X^{-k}-U^{-k})\mathcal{A}+{\mathcal{B}}^{*}(Y^{-l}-V^{-l}){\mathcal{B}\parallel }_{\mathrm{tr}}\hfill \\ & & +\parallel {\mathcal{C}}^{*}(X^{-k}-U^{-k})\mathcal{C}+{\mathcal{D}}^{*}(Y^{-l}-V^{-l}){\mathcal{D}\parallel }_{\mathrm{tr}}\hfill \\ & =\hfill & \mathrm{tr}\left({\mathcal{A}}^{*}\mathcal{A}(X^{-k}-U^{-k})\right)+\mathrm{tr}\left({\mathcal{B}}^{*}\mathcal{B}(Y^{-l}-V^{-l})\right)\hfill \\ & & +\mathrm{tr}\left({\mathcal{C}}^{*}\mathcal{C}(X^{-k}-U^{-k})\right)+\mathrm{tr}\left({\mathcal{D}}^{*}\mathcal{D}(Y^{-l}-V^{-l})\right)\hfill \\ & =\hfill & \mathrm{tr}\left(({\mathcal{A}}^{*}\mathcal{A}+{\mathcal{C}}^{*}\mathcal{C})(X^{-k}-U^{-k})\right)\hfill \\ & & +\mathrm{tr}\left(({\mathcal{B}}^{*}\mathcal{B}+{\mathcal{D}}^{*}\mathcal{D})(Y^{-l}-V^{-l})\right)\hfill \\ & \le \hfill & \parallel {\mathcal{A}}^{*}\mathcal{A}+{\mathcal{C}}^{*}\mathcal{C}\parallel \mathrm{tr}(X^{-k}-U^{-k})+ \parallel {\mathcal{B}}^{*}\mathcal{B}\hfill \\ & & +{\mathcal{D}}^{*}\mathcal{D}\parallel \mathrm{tr}(Y^{-l}-V^{-l})\hfill \\ & \le \hfill & \parallel {\mathcal{A}}^{*}\mathcal{A}+{\mathcal{C}}^{*}\mathcal{C}\parallel {\displaystyle \frac{k}{a^{k+1}}}\mathrm{tr}(X-U)\hfill \\ & & +\parallel {\mathcal{B}}^{*}\mathcal{B}+{\mathcal{D}}^{*}\mathcal{D}\parallel {\displaystyle \frac{l}{a^{l+1}}}\mathrm{tr}(V-Y)\hfill \\ & \le \hfill & {\delta }_4{(\parallel X-U)\parallel }_{\mathrm{tr}}{ +\parallel V-Y\parallel }_{\mathrm{tr}})\hfill \\ & =\hfill & {\delta }_4\left|\right||(X,Y)-(U,V)\left|\right||.\hfill \end{array}$$

If we set $Q_0=Q$ and ${\mathcal{R}}_0=\mathcal{R}$, we obtain $Q_0\preccurlyeq F_1(Q_0,{\mathcal{R}}_0)=Q_1$ and ${\mathcal{R}}_0\preccurlyeq F_2(Q_0,{\mathcal{R}}_0)={\mathcal{R}}_1$. Therefore, $Q_0\preccurlyeq F_1(Q_0,{\mathcal{R}}_0)$ and ${\mathcal{R}}_0\preccurlyeq F_2(Q_0,{\mathcal{R}}_0)$ hold. For any $X,Y\in \mathcal{H}\left(\mathcal{N}\right)$, there is a least upper bound and a greatest lower one. The maps $F_1$ and $F_2$ are continuous.

From Theorem 2 follows the existence of a unique $(\tilde{X},\tilde{Y})\in {\Omega }_Q\times {\Omega }_{\mathcal{R}}$ solution of (18).

For all $(X,Y)\in {\Omega }_Q\times {\Omega }_{\mathcal{R}}$, by the total decreasing property of $(F_1,F_2)$, we obtain $F_1(X,Y)\preccurlyeq F_1(Q,\mathcal{R})$ and $F_2(X,Y)\preccurlyeq F_2(Q,\mathcal{R})$. Consequently, for all $(X,Y)\in [Q,F_1(Q,\mathcal{R})]\times [\mathcal{R},F_2(Q,\mathcal{R})]$, the inclusion $T:[Q,F_1(Q,\mathcal{R})]\times [\mathcal{R},F_2(Q,\mathcal{R})]\to [Q,F_1(Q,\mathcal{R})]\times [\mathcal{R},F_2(Q,\mathcal{R})]$ holds, where $T(X,Y)=(F_1(X,Y),F_2(X,Y))$.

As far as T is a continuous map and $[Q,F_1(Q,\mathcal{R})]\times [\mathcal{R},F_2(Q,\mathcal{R})]$ is a convex and compact set, according to Theorem 3, it follows that at least one fixed point in $[Q,F_1(Q,\mathcal{R})]\times [\mathcal{R},F_2(Q,\mathcal{R})]$ for T exists. □

4. Illustrative Examples

We will illustrate the results with matrices from $\mathcal{X}\left(3\right)$. Calculations can be performed with matrices with higher dimensions, but the matrices will be difficult to fit in the text field.

Let us set

$$\mathcal{A}=\left(\begin{array}{ccc}0.021\hfill & \hfill 0.01& 0.1\hfill \\ 0.1\hfill & \hfill 0.03& 0.1\hfill \\ 1\hfill & \hfill 0.01& 0.2\hfill \end{array}\right),\mathcal{B}=\left(\begin{array}{ccc}0.0833\hfill & \hfill -0.0167& -0.0167\hfill \\ 0.025\hfill & \hfill 0.0833& -0.025\hfill \\ 0.0333\hfill & \hfill 0.0333& \hfill 0.0833\end{array}\right),$$

$$\mathcal{C}=\left(\begin{array}{ccc}0.0187\hfill & \hfill 0.0044& 0.125\hfill \\ 0.125\hfill & 0.05\hfill & 0.0625\hfill \\ 0.0025\hfill & 0.125\hfill & 0.0625\hfill \end{array}\right),\mathcal{D}=\left(\begin{array}{ccc}0.0714\hfill & \hfill 0.0214& -0.0214\hfill \\ 0.0286\hfill & \hfill 0.0714& \hfill 0.0286\\ 0.0357\hfill & \hfill 0.0357& \hfill 0.0714\end{array}\right),$$

$$\mathcal{Q}=\left(\begin{array}{ccc}3.5\hfill & \hfill 0.5& 0.5\hfill \\ 0.5\hfill & \hfill 3.5& 1\hfill \\ 0.5\hfill & 1\hfill & 4\hfill \end{array}\right),\mathcal{R}=\left(\begin{array}{ccc}15\hfill & \hfill 7.5& \hfill 5.1\hfill \\ \hfill 7.5& 15\hfill & 10.05\hfill \\ \hfill 5.1& 10.05\hfill & 15\hfill \end{array}\right)$$

and $a=2$, $b=9$, $k={\displaystyle \frac{1}{5}}$, $l={\displaystyle \frac{1}{13}}$.

Example 1.

Let consider the system of matrix equations

$$\left|\begin{array}{c}X=\mathcal{Q}-{\mathcal{A}}^{*}{X}^{-k}\mathcal{A}+{\mathcal{B}}^{*}{Y}^{-l}\mathcal{B}\hfill \\ Y=\mathcal{R}-{\mathcal{C}}^{*}{X}^{-k}\mathcal{C}+{\mathcal{D}}^{*}{Y}^{-l}\mathcal{D}.\hfill \end{array}\right.$$

(20)

To solve (20), we will apply Theorem 4.

We calculate $\lambda \left({\mathcal{A}}^{*}\mathcal{A}\right)=\{1.0558,0.0154,0.0002\}$, $\lambda \left({\mathcal{B}}^{*}\mathcal{B}\right)=\{0.0108,0.0066,0.0074\}$, $\lambda \left({\mathcal{C}}^{*}\mathcal{C}\right)=\{0.0391,0.0104,0.0081\}$, $\lambda \left({\mathcal{D}}^{*}\mathcal{D}\right)=\{0.0142,0.0049,0.0013\}$. Therefore, ${\mathcal{A}}^{*}\mathcal{A}\succ 0$, ${\mathcal{B}}^{*}\mathcal{B}\succ 0$, ${\mathcal{C}}^{*}\mathcal{C}\succ 0$, ${\mathcal{D}}^{*}\mathcal{D}\succ 0$, and thus $\mathcal{Q},\mathcal{R},{\mathcal{A}}^{*}\mathcal{A},{\mathcal{B}}^{*}\mathcal{B},{\mathcal{C}}^{*}\mathcal{C}\in \mathcal{P}\left(N\right)$.

$$\mathcal{Q}-a^{-k}{\mathcal{A}}^{*}\mathcal{A}-aI\approx \left(\begin{array}{ccc}0.62036\hfill & 0.48851\hfill & 0.31536\hfill \\ 0.4885\hfill & 1.49904\hfill & 0.99478\hfill \\ 0.315356\hfill & 0.99478\hfill & 1.9478\hfill \end{array}\right)\succ 0$$

and

$$\mathcal{R}-a^{-k}{\mathcal{C}}^{*}\mathcal{C}-aI\approx \left(\begin{array}{ccc}12.9861\hfill & \hfill 7.4942& \hfill 5.091\hfill \\ \hfill 7.4942& \hfill 12.9842& 10.04\hfill \\ \hfill 5.091\hfill & 10.04\hfill & 12.9796\hfill \end{array}\right)\succ 0.$$

The inequalities ${\lambda }^{+}({\mathcal{A}}^{*}\mathcal{A}+{\mathcal{C}}^{*}\mathcal{C})<1.077$ and ${\lambda }^{+}({\mathcal{B}}^{*}\mathcal{B}+{\mathcal{D}}^{*}\mathcal{D})<0.025$ hold, where ${\lambda }^{+}$ is the largest eigenvalue, and consequently we obtain

$${\displaystyle \frac{k\parallel {\mathcal{A}}^{*}\mathcal{A}+{\mathcal{C}}^{*}\mathcal{C}\parallel }{a^{k+1}}}<0.09033\mathrm{a}\mathrm{n}\mathrm{d}{\displaystyle \frac{l\parallel {\mathcal{B}}^{*}\mathcal{B}+{\mathcal{D}}^{*}\mathcal{D}\parallel }{a^{l+1}}}<0.00576.$$

For a stop criterion, we use

$$max\{\parallel \tilde{X}-X_{n+1}{\parallel }_{\mathrm{tr}},\parallel \tilde{Y}-Y_{n+1}{\parallel }_{\mathrm{tr}}\}\le {\displaystyle \frac{{\delta }_1}{1-{\delta }_1}}(\parallel X_n-X_{n+1}{\parallel }_{\mathrm{tr}}+ \parallel Y_n-Y_{n+1}{\parallel }_{\mathrm{tr}})\le {10}^{-10},$$

where

$$\delta =max\left\{{\displaystyle \frac{k\parallel {\mathcal{A}}^{*}\mathcal{A}+{\mathcal{C}}^{*}\mathcal{C}\parallel }{a^{k+1}}},{\displaystyle \frac{l\parallel {\mathcal{B}}^{*}\mathcal{B}+{\mathcal{D}}^{*}\mathcal{D}\parallel }{a^{l+1}}}\right\}<1.$$

After calculations, we obtain

$$\mathcal{Q}-a^{-k}{\mathcal{A}}^{*}\mathcal{A}\approx \left(\begin{array}{ccc}2.62036\hfill & 0.4885\hfill & 0.31536\hfill \\ 0.4885\hfill & 3.49904\hfill & 0.99478\hfill \\ 0.31536\hfill & 0.99478\hfill & 3.94777\hfill \end{array}\right)\succ 0$$

and

$$\mathcal{R}-a^{-k}{\mathcal{C}}^{*}\mathcal{C}\approx \left(\begin{array}{ccc}14.98609\hfill & 7.49422\hfill & 5.091\hfill \\ 7.49422\hfill & 14.9842\hfill & 10.04\hfill \\ 5.09102\hfill & 10.04\hfill & 14.9796\hfill \end{array}\right)\succ 0.$$

If we take $n_0=5$, we obtain

$$-{\displaystyle \frac{{\mathcal{A}}^{*}{\mathcal{Q}}^{-k}\mathcal{A}}{n_0^{9k}}}+{\displaystyle \frac{{\mathcal{B}}^{*}\mathcal{B}}{a^l}}-(n_0-1)\mathcal{Q}\approx \left(\begin{array}{ccc}\hfill -14.03397& \hfill -1.99877& \hfill -2.00786\\ \hfill -1.99877& \hfill -13.99214& \hfill -3.99931\\ \hfill -2.00786& \hfill -3.99931& \hfill -15.99496\end{array}\right)\prec 0$$

and

$$-{\displaystyle \frac{{\mathcal{C}}^{*}{\mathcal{Q}}^{-k}\mathcal{C}}{n_0^9}}+{\displaystyle \frac{{\mathcal{D}}^{*}\mathcal{D}}{a^l}}-(n_0-1)\mathcal{R}\approx \left(\begin{array}{ccc}\hfill -59.99387& \hfill -29.99565& \hfill -20.39867\\ \hfill -29.99565& \hfill -59.99426& \hfill -40.19652\\ \hfill -20.39867& \hfill -40.19652& \hfill -59.99491\end{array}\right)\prec 0.$$

Thus, the assumptions of Theorem 4 are satisfied. It remains to find the interval, where the coupled fixed point is situated. From

$$aI\preccurlyeq \mathcal{Q}-a^{-k}{\mathcal{A}}^{*}\mathcal{A}+b^{-l}{\mathcal{B}}^{*}\mathcal{B}=\left(\begin{array}{ccc}2.62769\hfill & 0.49\hfill & 0.316\hfill \\ 0.49\hfill & 3.50608\hfill & 0.9956\hfill \\ 0.316\hfill & 0.9956\hfill & 3.95439\hfill \end{array}\right),$$

$$n_{0}Q\succcurlyeq n_0(\mathcal{Q}-b^{-k}{\mathcal{A}}^{*}\mathcal{A}+a^{-l}{\mathcal{A}}^{*}\mathcal{A})=\left(\begin{array}{ccc}14.28554\hfill & 2.466\hfill & 1.82024\hfill \\ 2.466\hfill & 17.53596\hfill & 4.98528\hfill \\ 1.82024\hfill & 4.98528\hfill & 19.84388\hfill \end{array}\right)$$

and

$$aI\preccurlyeq \mathcal{R}-a^{-1}{\mathcal{C}}^{*}\mathcal{C}+b^{-1}{\mathcal{D}}^{*}\mathcal{D}=\left(\begin{array}{ccc}\hfill 14.99216& \hfill 7.49831& \hfill 5.09257\\ \hfill 7.49831& \hfill 14.98998& \hfill 10.04349\\ \hfill 5.09257& \hfill 10.04349& \hfill 14.98498\end{array}\right),$$

$$n_{0}R\succcurlyeq n_0(\mathcal{R}-b^{-k}{\mathcal{C}}^{*}\mathcal{C}+a^{-l}{\mathcal{D}}^{*}\mathcal{D}=\left(\begin{array}{ccc}74.91440\hfill & 37.45562\hfill & 25.45807\hfill \\ 37.45562\hfill & 74.90913\hfill & 50.19341\hfill \\ 25.45807\hfill & 50.19341\hfill & 74.89425\hfill \end{array}\right)$$

it follows that

$$\tilde{X}\in \left[\left(\begin{array}{ccc}2.62769\hfill & 0.49\hfill & 0.316\hfill \\ 0.49\hfill & 3.50608\hfill & 0.9956\hfill \\ 0.316\hfill & 0.9956\hfill & 3.95439\hfill \end{array}\right),\left(\begin{array}{ccc}14.28554& 2.466& 1.82024\\ 2.466& 17.53596& 4.98528\\ 1.82024& 4.98528& 19.84388\end{array}\right)\right]$$

and

$$\tilde{Y}\in \left[\left(\begin{array}{ccc}14.99216\hfill & 7.49831\hfill & 5.09257\hfill \\ 7.49831\hfill & 14.98998\hfill & 10.04349\hfill \\ 5.09257\hfill & 10.04349\hfill & 14.98498\hfill \end{array}\right),\left(\begin{array}{ccc}74.91440\hfill & 37.45562\hfill & 25.45807\hfill \\ 37.45562\hfill & 74.90913\hfill & 50.19341\hfill \\ 25.45807\hfill & 50.19341\hfill & 74.89425\hfill \end{array}\right)\right].$$

We calculate the first iteration by setting $X_0=aI$ and $Y_0=bI$.

From $\delta =0.09033<1$, the approximate solutions with $\epsilon ={10}^{-10}$, using the a posteriori error estimate, will be

$$X_6\approx \left(\begin{array}{ccc}2.7412382983\hfill & 0.4923801869\hfill & 0.3443255326\hfill \\ 0.4923801869\hfill & 3.505990543\hfill & 0.9962501337\hfill \\ 0.3443255326\hfill & 0.9962501337\hfill & 3.9626655625\hfill \end{array}\right),$$

$$Y_6\approx \left(\begin{array}{ccc}\hfill 14.9931997842& \hfill 7.4991315725& \hfill 5.0938562578\\ \hfill 7.4991315725& \hfill 14.9919453526& \hfill 10.0450946927\\ \hfill 5.0938562578& \hfill 10.0450946927& \hfill 14.9871884554\end{array}\right),$$

and the a posteriori error is $0.196\ast {10}^{-12}$.

Example 2.

Let consider the system of matrix equations

$$\left|\begin{array}{c}X=\mathcal{Q}-{\mathcal{A}}^{*}{X}^{-k}\mathcal{A}+{\mathcal{B}}^{*}{Y}^{-l}\mathcal{B}\hfill \\ \\ Y=\mathcal{Q}-{\mathcal{A}}^{*}{Y}^{-k}\mathcal{A}+{\mathcal{B}}^{*}{X}^{-l}\mathcal{B}.\hfill \end{array}\right.$$

(21)

We will apply Corollary 1. By similar calculations, we obtain

$$\mathcal{Q}-a^{-k}{\mathcal{A}}^{*}\mathcal{A}-aI\succ 0,\mathcal{Q}-a^{-k}{\mathcal{A}}^{*}\mathcal{A}+b^{-l}{\mathcal{B}}^{*}\mathcal{B}\succ 0$$

and

$$\delta =max\left\{{\displaystyle \frac{k\parallel {\mathcal{A}}^{*}\mathcal{A}\parallel }{a^{k+1}}},{\displaystyle \frac{l\parallel {\mathcal{B}}^{*}\mathcal{B}\parallel }{a^{l+1}}}\right\}\approx 0.08945.$$

The approximate solutions are with $\epsilon ={10}^{-10}$, using the a posteriori error estimate. Since X and Y are very close, we will write them down to the fourteenth digit, and they will be

$$X_6\approx \left(\begin{array}{ccc}2.74227175472527\hfill & 0.49272443691449\hfill & 0.34435524116285\hfill \\ 0.49272443691449\hfill & 3.50669694897017\hfill & 0.99648041181033\hfill \\ 0.34435524116285\hfill & 0.99648041181033\hfill & 3.96309236125448\hfill \end{array}\right),$$

$$Y_6\approx \left(\begin{array}{ccc}2.74227175472691\hfill & 0.49272443691453\hfill & 0.34435524116361\hfill \\ 0.49272443691453\hfill & 3.50669694897017\hfill & 0.99648041181035\hfill \\ 0.34435524116361\hfill & 0.99648041181035\hfill & 3.96309236125484\hfill \end{array}\right).$$

The solution to the matrix equation $X=Q-A^{*}{X}^{-k}A+B^{*}{X}^{-l}B$ is

$$\tilde{X}\approx \left(\begin{array}{ccc}2.74227175472621\hfill & 0.49272443691452\hfill & 0.34435524116329\hfill \\ 0.49272443691452\hfill & 3.50669694897017\hfill & 0.99648041181034\hfill \\ 0.34435524116329\hfill & 0.99648041181034\hfill & 3.96309236125468\hfill \end{array}\right)\in [X_6,Y_6]$$

and the a posteriori error is $0.321\ast {10}^{-10}$.

Example 3.

Let consider the system of matrix equations

$$\left|\begin{array}{c}X=\mathcal{Q}+{\mathcal{A}}^{*}{X}^k\mathcal{A}+{\mathcal{B}}^{*}{Y}^{-l}\mathcal{B}\hfill \\ Y=\mathcal{R}+{\mathcal{C}}^{*}{X}^k\mathcal{C}+{\mathcal{D}}^{*}{Y}^{-l}\mathcal{D}.\hfill \end{array}\right.$$

(22)

We will apply Theorem 5. By similar calculations, we obtain

$$\mathcal{Q}+a^k{\mathcal{A}}^{*}\mathcal{A}-aI\succ 0,\mathcal{R}+a^k{\mathcal{C}}^{*}\mathcal{C}-aI\succ 0,$$

$$\mathcal{Q}+a^k{\mathcal{A}}^{*}\mathcal{A}+b^l{\mathcal{B}}^{*}\mathcal{B}-aI\succ 0,\mathcal{R}+a^k{\mathcal{C}}^{*}\mathcal{C}+b^l{\mathcal{D}}^{*}\mathcal{D}-aI\succ 0,$$

and

$$\delta =max\left\{{\displaystyle \frac{k\parallel {\mathcal{A}}^{*}\mathcal{A}+{\mathcal{B}}^{*}\mathcal{B}\parallel }{a^{k-1}}},{\displaystyle \frac{l\parallel {\mathcal{B}}^{*}\mathcal{B}+{\mathcal{D}}^{*}\mathcal{D}\parallel }{a^{l+1}}}\right\}\approx 0.361328<1.$$

The approximate solutions with $\epsilon ={10}^{-10}$, using the a posteriori error estimate, will be

$$X_7\approx \left(\begin{array}{ccc}2.1718757189\hfill & 0.4816073647\hfill & 0.2137022751\hfill \\ 0.4816073647\hfill & 3.5053525133\hfill & 0.9922434765\hfill \\ 0.2137022751\hfill & 0.9922434765\hfill & 3.9259235039\hfill \end{array}\right),$$

$$Y_7\approx \left(\begin{array}{ccc}\hfill 14.9850796906& \hfill 7.4939715330& \hfill 5.0871396270\\ \hfill 7.4939715330& \hfill 14.9808205788& \hfill 10.0371005411\\ \hfill 5.0871396270& \hfill 10.0371005411& \hfill 14.9755119200\end{array}\right).$$

and the a posteriori error is $0.337\ast {10}^{-12}$.

5. Conclusions

In this paper, we have created a generalization of the coupled fixed points described in [9,25], using the deep results in [10,11,35,36]. We have shown that the definition of partial ordering in the suggested concept in [25] of investigating coupled fixed points for maps with the mixed monotone property in partially ordered metric spaces is closely related to the type of monotonicity of the maps being studied. This enables us to solve different kinds of matrix equation systems. The method suggested by [8] for solving only symmetric systems of matrix equations is generalized by the concept that an ordered pair of maps should be considered rather than just one map. We want to ask some open questions. In [37], deep observation in solving matrix inequalities systems is offered. Is it feasible to solve the system of matrix inequalities

$$\left|\begin{array}{c}X+{\mathcal{A}}^{*}{X}^{-k}\mathcal{A}-{\mathcal{B}}^{*}{Y}^{-l}\mathcal{B}\preccurlyeq \mathcal{Q},\hfill \\ Y+{\mathcal{C}}^{*}{X}^{-k}\mathcal{C}-{\mathcal{D}}^{*}{Y}^{-l}\mathcal{D}\preccurlyeq \mathcal{R},\hfill \\ 0\prec X\preccurlyeq Y,\hfill \end{array}\right.$$

using the method from [37], even for some particular cases $k=l=1$ and $\mathcal{Q},\mathcal{R}=I$?