Applications of N-Tupled Fixed Points in Partially Ordered Metric Spaces for Solving Systems of Nonlinear Matrix Equations

Abstract

We unify a known technique used for fixed points and coupled, tripled and N-tupled fixed points for weak monotone maps, i.e., maps that exhibit monotone properties for each of their variables. We weaken the classical contractive condition in partially ordered metric spaces by requiring it to hold only for a sequence of successive iterations, generated by the considered map, provided that it is a monotone one. We show that some known results are a direct consequence of the main result. The introduced technique shows that the partial order in the constructed Cartesian space is induced by both the partial order in the considered metric space and by the monotone properties of the investigated maps. We illustrate the main result, which is applied to solve a nonlinear matrix equation, following key ideas from Berzig, Duan & Samet. We present an illustrative example. We comment that a similar approach can be used to solve systems of nonlinear matrix equations.

1. Introduction and Preliminaries

Matrix equations play a crucial role in modeling across various scientific disciplines. We would like to mention just a few recent non-classical applications of matrix equations: in physics [1,2,3,4]; in computer science [5,6]; in economics [7,8]; and in chemistry [9,10,11]. We should not overlook algebraic discrete-type Riccati equations, which play a crucial role in modern control theory. These equations appear in various applications [12,13,14].

Sometimes, it is not possible to find the exact solution to a matrix equation or a system of matrix equations under consideration. Despite this drawback, often proving the existence and uniqueness of the solution, as well as providing a procedure for an approximate solution, is sufficient to analyze the models under study.

The in-depth results in [15] concerning the investigation of the matrix equation

$$X\pm \sum _{i=1}^m{\mathcal{A}}_i^{*}F\left(X\right){\mathcal{A}}_i=\mathcal{Q},$$

where F is a continuous and monotone map from a set of positive definite matrices to itself, lead to increasing interest in the class of matrix equations

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

where $0<p_i\le 1(i=1,2,\dots ,m)$.

Some special cases were studied in [16,17,18,19]. More complex variations were investigated in [20,21,22,23].

Our investigation will be based on generalizations of the Banach fixed point theorem. Assume $(X,d)$ is a metric space and $T:X\to X$ is a self-map. The contraction mapping theorem [24] and its generalization to partially ordered metric spaces [15] are widely used approaches today. The concept of obtaining fixed point results in partially ordered spaces originated in [25], while [15] laid the groundwork for further development in this field. Following [15,25,26,27,28,29,30], it appears reasonable to reduce the Banach contraction condition by insisting that $d(Tx,Ty)\le \alpha d(x,y)$ holds only for those elements that fulfill $x\preccurlyeq y$, where ≼ is a partial order on $(X,d)$. The concept of considering a partial order was first discussed in [25,31]. It is worth noting that [31] considered a partially ordered cone normed space and developed the concept of connected fixed points for the first time. In [32], coupled fixed points were studied using a partially ordered metric space instead of cone-based ordering. An application of coupled fixed points from [32] has been proposed in [33] for solving the matrix equation $X+{\mathcal{A}}^{*}{X}^{-1}\mathcal{A}-{\mathcal{B}}^{*}{X}^{-1}\mathcal{B}=Q$, where $\mathcal{A}$ and $\mathcal{B}$ are arbitrary $N\times N$ square matrices and Q is a positive definite $N\times N$ matrix.

A drawback of the notion of a coupled fixed point $(x,y)$ is that it usually satisfies $x=y$. Natural models from economics lead to equilibrium ordered pairs $(x,y)$, such that $x\ne y$ [34]. Thus, the notion of coupled fixed points for a map F have been generalized to coupled fixed points for an ordered pair of maps $(F,G)$.

The equation

$$X=\mathcal{Q}-{\mathcal{A}}^{*}{X}^{-1}\mathcal{A}+{\mathcal{B}}^{*}{X}^{-1}\mathcal{B}$$

(1)

can be viewed as a map $F:\mathcal{X}\left(N\right)\times \mathcal{X}\left(N\right)\to \mathcal{X}\left(N\right)$, where $\mathcal{X}\left(N\right)$ represents the space of $N\times N$ matrices, defined by $F(X,Y)=\mathcal{Q}-{\mathcal{A}}^{*}{X}^{-1}\mathcal{A}+{\mathcal{B}}^{*}{Y}^{-1}\mathcal{B}$. Following [31], we seek the coupled fixed point $(X,Y)$ for F, i.e., $X=F(X,Y)$ and $Y=F(Y,X)$ and $X=Y$, under the specific assumption that on map F and underlying partially ordered metric space $\left(\mathcal{X}\right(N),\rho ,\preccurlyeq )$, we get the solution of (1).

The notion of a coupled fixed point for an ordered pair of maps $(F,G)$ was introduced in [35], which allows us to solve problems with coupled fixed points [36], without leading to $X=Y$, i.e., the system

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

The idea of tripled fixed points was introduced in various aspects in [37,38,39]. Later, a more sophisticated notion of n-tuples of fixed points was proposed in [40].

It is assumed that a partially ordered metric space $(X,\rho ,\preccurlyeq )$ generates a partial order in the Cartesian product space $X\times X$ by $(x,y)⪯(u,v)$, provided that $x\preccurlyeq y$ and $u\succcurlyeq v$ [32], and in the Cartesian product space $X\times X\times X$ by $(x,y,z)⪯(u,v,w)$, provided that $x\preccurlyeq y$, $u\succcurlyeq v$ and $z\preccurlyeq w$ [37,39].

A recent observation [41] for coupled fixed points and [42] for tripled fixed points shows that the partial ordering the Cartesian product space may be defined in different ways, and its definition is generated by the monotone properties of the ordered pair or tripled fixed points of the maps involved.

We will try to generalize the ideas from [40] for n-tuples of fixed points in the context of ordered n-tuples of maps $(F_1,\dots ,F_n)$. The proposed notion of n-tuples of fixed points has been used to solve systems of integral equations [43,44] and fractional nonlinear dynamical systems [45]. The first observation that results on n-tupled fixed points can be obtained from those on fixed points was made in [46,47,48,49] and was further investigated in [50,51].

Our goal is to present a weaker notion for the monotone properties of a map and, under these weaker conditions, to obtain results about the existence, uniqueness, and error estimates for the fixed points, which will be a natural generalization of the results from [15]. Further, we aim to generalize the notion about weak monotonicity to an ordered n-tuple of maps $(F_1,F_2,\dots ,F_n)$ and to introduce the notion of an n-tupled fixed point for an ordered n-tuple of maps, which is a generalization of the ideas from [40,42]. Finally, we search sufficient conditions for the existence, uniqueness, and error estimates for the introduced n-tupled fixed points for an ordered n-tuple of maps and apply them in solving nonlinear matrix equations. We intend to show that the partial order in a constructed Cartesian space is induced both by the partial order in the metric space and monotone properties of the considered maps. We would like to say that, whenever the classical mixed monotone property is assumed, we can prove the n-tupled fixed point results by the related fixed point ones as suggested in [46,47,48,49]. Unfortunately, under the weaker monotone properties and contractive conditions that we will assume, it seems that the mentioned technique to obtain n-tupled fixed points by the related fixed points results will not be applicable.

2. Results

2.1. Fixed Points for Weak Monotone Maps

Definition 1 

([24]). Let X be a set and $f:X\to X$. For any $x_0\in X$, we will define inductively the iterated sequence ${\left\{x_n\right\}}_{n=0}^{\infty }$ by $x_1=f{x}_0$, and if we define $x_n$, then $x_{n+1}=f{x}_n$ and will sometimes be denoted by $x_{n+1}=f^{n+1}{x}_0$.

Definition 2. 

Let $(X,\preccurlyeq )$ be a partially ordered set. We say that map $f:X\to X$ is a map with a weak monotone property if, from $x\preccurlyeq fx$, it follows $fx\preccurlyeq f^{2}x$, and if $x\succcurlyeq fx$, it follows $fx\succcurlyeq f^{2}x$.

Proposition 1. 

Let $(X,\preccurlyeq )$ be a partially ordered set. Let $f:X\to X$ be a map possessing a weak monotone property. Then, for any $x_0\in X$, the elements $x_0$ and $f{x}_0$ are comparable, then the sequence of successive iterations $x_n=f{x}_{n-1}$ is monotone.

Proof. 

Let $x_0\preccurlyeq f{x}_0$. Let us consider the iterated sequence ${\left\{x_n\right\}}_{n=0}^{\infty }$, i.e., $x_n=f{x}_{n-1}$ for $n\in \mathbb{N}$. From the weak monotone property, it follows that $x_1=f{x}_0\preccurlyeq f^2{x}_0=f{x}_1=x_2$. Therefore, from the weak monotone property, we get that $x_2=f{x}_1\preccurlyeq f^2{x}_1=f{x}_2=x_3$. Continuing by induction on the index n, let us assume that $x_n\preccurlyeq x_{n+1}$, i.e., $x_n\preccurlyeq x_{n+1}=f{x}_n$ holds. From the weak monotone property, we get $x_{n+1}=f{x}_n\preccurlyeq f^2{x}_n=f{x}_{n+1}=x_{n+2}$, and thereafter, ${\left\{x_n\right\}}_{n=0}^{\infty }$ is a monotone increasing sequence.

By the same argument, we get that if $x_0\succcurlyeq f{x}_0$, then ${\left\{x_n\right\}}_{n=0}^{\infty }$ is a monotone decreasing sequence. □

Proposition 2 

([15,52]). Let $(X,\preccurlyeq )$ be a partially ordered set. Let $f:X\to X$ be a map with a weak monotone property. Let $\xi \in X$ be a fixed point for f and $z_0$ be comparable with ξ, satisfying

• if $\xi \preccurlyeq z_0$ then $z_0\preccurlyeq f{z}_0$
• if $\xi \succcurlyeq z_0$ then $z_0\succcurlyeq f{z}_0$.

Then, each element from the iterated sequence ${\left\{z_n\right\}}_{n=0}^{\infty }$ is comparable with ξ.

Proof. 

Let $\xi$ be a fixed point for f, $\xi \preccurlyeq z_0$, and $z_0\preccurlyeq f{z}_0$. From the assumption that f is a weak monotone map, it follows that $z_0\preccurlyeq f{z}_0$. From the weak monotone property, it follows that ${\left\{z_n\right\}}_{n=0}^{\infty }$ is a monotone sequence, and there holds either $\xi \preccurlyeq z_0\preccurlyeq z_n$ for all $n\in \mathbb{N}$.

The proof for the case $\xi \succcurlyeq z_0$, and $z_0\succcurlyeq f{z}_0$ can be carried out in a similar fashion. □

For completeness, we will mention that the idea of imposing contractive conditions only on the sequence of successive iterations dates back to [53]. Following [53] to obtain a result about the existence of fixed point, it is necessary to add either an additional condition for the map or for the iterative sequence.

The next result is a slight generalization of the main results from [15,36]. It is shown in [36] that the continuity assumption in [15] can be replaced by (ii) of Theorem 1. We weaken in Theorem 1 the monotonicity assumed in [15,36] by assuming just a weak monotone property.

Theorem 1 

([15,36]). Let $(X,\rho ,\preccurlyeq )$ be a partially ordered complete metric space and $f:X\to X$ be a map with a weak monotone property.

Let one of the following conditions hold:

(i) 

f is continuous.

(ii) 

For any convergent sequence, ${lim}_{n\to \infty }{z}_n=z$.

• If $z_n\preccurlyeq z_{n+1}$, then $z_n\preccurlyeq z$.
• If $z_n\succcurlyeq z_{n+1}$, then $z_n\succcurlyeq z$.

Let there exist $z_0\in X$ such that $z_0$ and $f{z}_0$ are comparable, then there is a fixed point $\xi \in X$ of f, which is a limit of the sequence ${\left\{z_n\right\}}_{n=1}^{\infty }$, defined by $z_n=f{z}_{n-1}=f^n{z}_0$.

Let, for any monotone sequence ${\left\{z_n\right\}}_{n=0}^{\infty }$, the inequality that follows hold

$$\rho (f{z}_{n+1},f{z}_n)\le \alpha \rho (z_n,z_{n-1})$$

(2)

for $n\in \mathbb{N}$.

Then, there exists fixed point ξ for map f.

The error estimations are listed below:

(I) 

A priori error estimate $\rho (z_n,\xi )\le {\displaystyle \frac{{\alpha }^n}{1-\alpha }}\rho (z_0,z_1)$.

(II) 

A posteriori error estimate $\rho (z_{n+1},\xi )\le {\displaystyle \frac{\alpha }{1-\alpha }}\rho (z_n,z_{n+1})$.

(III) 

The rate of convergence $\rho (z_{n+1},\xi )\le {\displaystyle \frac{\alpha }{1-\alpha }}\rho (z_n,z_{n+1})$.

Fixed point ξ is unique if each two elements $x,y\in X$ are not comparable and there is an element, z, that is comparable with both of them, satisfying the following:

• If $x,y\preccurlyeq z$, then $z\preccurlyeq fz$.
• If $x,y\succcurlyeq z$, then $z\succcurlyeq fz$.

and $\rho (f{u}_n,f{v}_n)\le \alpha \rho (u_n,v_n)$ holds for any two sequences that satisfy $u_n\preccurlyeq v_n$.

Proof. 

There exists $x_0$, so either $x_0\preccurlyeq f{x}_0$ or $x_0\succcurlyeq f{x}_0$ holds.

By Proposition 1, it follows that either $x_{n-1}\preccurlyeq x_n$ for all $n\in \mathbb{N}$ or $x_{n-1}\succcurlyeq x_n$ for all $n\in \mathbb{N}$ holds. From the contractive condition (2) for monotone sequences, applying n times, we get

$$\rho (x_{n+1},x_n)=\rho (f{x}_n,f{x}_{n-1})\le \alpha \rho (x_n,x_{n-1})\le \dots \le {\alpha }^n\rho (x_1,x_0).$$

By the classical argument from the proof of the Banach Fixed Point Theorem, it follows that ${\left\{x_n\right\}}_{n=0}^{\infty }$ is a convergent sequence, and by the completeness of the underlying metric space, it follows that there exists $\xi \in X$, so that $\xi ={lim}_{n\to \infty }{x}_n$.

If f is a continuous map, by the classical argument from the proof of the Banach Fixed Point Theorem, it follows that $\xi$ is a fixed point for f.

If (ii) holds, then

$$\rho (\xi ,f\xi )=\underset{n\to \infty }{lim}\rho (x_n,f\xi )=\underset{n\to \infty }{lim}\rho (f{x}_{n-1},f\xi )\le \alpha \underset{n\to \infty }{lim}\rho (x_{n-1},\xi )=0,$$

because $x_n$ and $\xi$ are comparable for all $n\in \mathbb{N}$.

The error estimates can be obtained by the classical argument from the proof of the Banach Fixed Point Theorem.

Let us assume additionally that for two elements $x,y\in X$ that are not comparable, there is an element, z, that is comparable with both of them and the contractive inequality holds for any two comparable elements, and f is a monotone map.

Let us suppose that there are two fixed points $\xi \ne \eta$ that are comparable. Then, $\rho (\xi ,\eta )=\rho (f\xi ,f\eta )\le \alpha \rho (\xi ,\eta )$, a contradiction, and thus, $\xi =\eta$.

Let us suppose that there are two fixed points $\xi \ne \eta$ that are not comparable. There is $z_0$, which is comparable with both $\xi$ and $\eta$. Let us assume that $\xi ,\eta \preccurlyeq z_0$ and $z_0\preccurlyeq f{z}_0$. From Proposition 2, it follows that ${\left\{z_n\right\}}_{n=0}^{\infty }$ is a monotone sequence and either $\xi \preccurlyeq z_0\preccurlyeq z_n$ holds for all $n\in \mathbb{N}$. Then,

$$\rho (\xi ,f{z}_n)=\rho (f\xi ,f{z}_n)\le \alpha \rho (\xi ,z_{n-1})\le {\alpha }^n\rho (\xi ,z_0).$$

We get, in a similar fashion, that $\rho (\eta ,f{z}_n)\le {\alpha }^n\rho (\eta ,z_0)$. There exists $n_0\in \mathbb{N}$ so that ${\alpha }^{n_0}(\rho (\xi ,z_0)+\rho (\xi ,z_0))<\rho (\xi ,\eta )$. Thus,

$$\rho (\xi ,\eta )\le \rho (\xi ,z_{n_0})+\rho (\eta ,z_{n_0})\le {\alpha }^{n_0}(\rho (\xi ,z_0)+\rho (\xi ,z_0))<\rho (\xi ,\eta ),$$

This yields a contradiction, and therefore, $\xi =\eta$.

Let us point out that if $z_0$ exists, so that $\xi \preccurlyeq z_0\eta$ or $\xi \succcurlyeq z_0\succcurlyeq \eta$, then $\xi$ and $\eta$ are comparable. □

2.2. N-Tupled Fixed Points for Maps in Partially Ordered Metric Spaces

We emphasize, once more, that there is a close connection between n-tupled fixed points and fixed points [48,50,51]. This idea was used in the proof of existence, uniqueness and error estimation in [36,41,42]. Unfortunately, when the condition of a contraction type is weakened, by considering maps with a weak monotone property, instead of a mixed monotone one, it seems that it is not possible to perform the proof directly by means of a suitable definition of a new map and passing to fixed points.

To simplify the notations in what follows, let us denote $x\preccurlyeq y$ by $p_1(x,y)$ and $x\succcurlyeq y$ by $p_2(x,y)$.

Definition 3. 

Let $(X,d,\preccurlyeq )$ denote a metric space that is equipped with a partial order, $F_k:X^n\to X$, $k=1,2,\dots ,n$. We say that $(F_1,F_2,\dots ,F_n)$ has a mixed monotone property if, for each $i=1,2,\dots n$, all the maps $F_k(x_1,x_2,\dots ,x_n)$ for $k=1,2,\dots ,n$ are either monotonically increasing for their i-th variable or are monotonely decreasing for its i-th variable.

Example 1. 

Let us consider the maps $F_i:\mathbb{R}\to \mathbb{R}$, defined by

$$F_i(x_1,\dots ,x_n)=\sum _{k=1}^n{(-1)}^{k}k{x}_k$$

for $i=1,2,\dots n$.

Each $F_i$ is an increasing function of its even variables and a decreasing one of its odd variables.

Definition 4. 

Let $(X,d,\preccurlyeq )$ denote a metric space that is equipped with a partial order, $F_k:X^n\to X$, $k=1,2,\dots ,n$, such that $(F_1,F_2,\dots ,F_n)$ has a mixed monotone property. Let $(X^n,⪯)$ be endowed with the partial ordering $(x_1,x_2,\dots x_n)⪯(u_1,u_2,\dots ,u_n)$, provided that $p_1(x_i,u_i)$ if $F_k$ increases its i-th variable and $p_2(x_i,u_i)$ if $F_k$ decreases its i-th variable, where $p_1$ is relation ≼ and $p_2$ is relation ≽.

Example 2. 

Let us consider $(\mathbb{R},\le )$, maps $F_i$, defined in Example 1. Let us introduce a partial ordering ⪯ in ${\mathbb{R}}^n$ by $(x_1,\dots ,x_n)⪯(y_1,\dots ,y_n)$ if $x_k\ge y_k$ for k odd and $x_k\le y_k$, for k even for all $k\in \{1,2,\dots ,n\}$.

For an ordered n-tuple of maps $(F_1,\dots ,F_n)$, we will define the iterated sequence ${\left\{x^{\left(m\right)}\right\}}_{m=1}^{\infty }$. It is a generalization of the iterated sequence defined for maps of two variables [32,35], three variables [37,54], and n variables [40], but for just one map, i.e., $F_j=F_1$ for all $j=2,3,\dots n$.

Definition 5. 

Let X be a set and $F_i:X^n\to X$ for $i=1,2,\dots n$. For any $x^{\left(0\right)}\in X^n$, we define inductively the iterated sequence ${\left\{x^{\left(m\right)}\right\}}_{m=1}^{\infty }$ by $x^{\left(1\right)}=\left(x_1^{\left(1\right)},x_2^{\left(1\right)},\dots ,x_n^{\left(1\right)}\right)$, where $x_i^{\left(1\right)}=F_i\left(x_1^{\left(0\right)},x_2^{\left(0\right)},\dots ,x_n^{\left(0\right)}\right)$ for $i=1,2,\dots n$. If we define $x^{\left(m\right)}$, then $x^{(m+1)}=\left(x_1^{(m+1)},x_2^{(m+1)},\dots ,x_n^{(m+1)}\right)$ is defined by $x_i^{(m+1)}=F_i\left(x_1^{\left(m\right)},x_2^{\left(m\right)},\dots ,x_n^{\left(m\right)}\right)$ for $i=1,2,\dots n$.

Proposition 3. 

Let $(X,d,\preccurlyeq )$ denote a metric space that is equipped with a partial order, $F_k:X^n\to X$, $k=1,2,\dots ,n$, such that $(F_1,F_2,\dots ,F_n)$ has a mixed monotone property. $(X^n,⪯)$ is endowed with the partial ordering from Definition 4. Let $(p_{j_1},p_{j_2},\dots p_{j_n})$ denote the partial order generated by $(F_1,F_2,\dots ,F_n)$, i.e., $(x_1,x_2,\dots x_n)⪯(y_1,y_2,\dots ,y_n)$ if, and only if, $p_{j_k}(x_k,y_k)$ holds, $j_k\in \{1,2\}$. If there is $x^{\left(0\right)}=\left(x_1^{\left(0\right)},x_2^{\left(0\right)},\dots x_n^{\left(0\right)}\right)\in X^n$, such that

$$\left(x_1^{\left(0\right)},x_2^{\left(0\right)},\dots x_n^{\left(0\right)}\right)⪯\left(F_1\left(x^{\left(0\right)}\right),F_2\left(x^{\left(0\right)}\right),\dots F_n\left(x^{\left(0\right)}\right)\right),$$

i.e, $p_{j_k}\left(x_k^{\left(0\right)},F_k\left(x^{\left(0\right)}\right)\right)$, then $p_{j_k}(x_k^{\left(n\right)},F_k\left(x^{\left(n\right)}\right))$ holds, i.e.,

$$\left(x_1^{\left(n\right)},x_2^{\left(n\right)},\dots x_n^{\left(n\right)}\right)⪯\left(x_1^{(n+1)},x_2^{(n+1)},\dots x_n^{(n+1)}\right).$$

Therefore, the sequence ${\left\{x^{\left(m\right)}\right\}}_{m=0}^{\infty }$ is an increasing sequence for the partial ordering ⪯, and map $F=(F_1,F_2,\dots ,F_n):X^n\to X^n$ has a weak monotone property.

Proof. 

Let $\left(x_1^{\left(0\right)},x_2^{\left(0\right)},\dots x_n^{\left(0\right)}\right)⪯\left(F_1\left(x^{\left(0\right)}\right),F_2\left(x^{\left(0\right)}\right),\dots F_n\left(x^{\left(0\right)}\right)\right)$, i.e., $p_{j_k}\left(x_k^{\left(0\right)},F_k\left(x^{\left(0\right)}\right)\right)$, and thus, $\left(x_1^{\left(0\right)},x_2^{\left(0\right)},\dots x_n^{\left(0\right)}\right)⪯\left(x_1^{\left(1\right)},x_2^{\left(1\right)},\dots x_n^{\left(1\right)}\right)$, i.e., $p_{j_k}\left(x_k^{\left(0\right)},x_k^{\left(1\right)}\right)$.

By the mixed monotone property of $(F_1,F_2,\dots F_n)$, it follows that $p_{j_k}\left(x_k^{\left(1\right)},F_k\left(x^{\left(1\right)}\right)\right)$ for all $k=1,2,\dots ,n$, and consequently, $\left(x_1^{\left(1\right)},x_2^{\left(1\right)},\dots x_n^{\left(1\right)}\right)⪯\left(x_1^{\left(2\right)},x_2^{\left(2\right)},\dots x_n^{\left(2\right)}\right)$.

Continuing by induction on i, let us assume that there holds

$$\left(x_1^{\left(i\right)},x_2^{\left(i\right)},\dots x_n^{\left(i\right)}\right)⪯\left(x_1^{(i+1)},x_2^{(i+1)},\dots x_n^{(i+1)}\right),$$

i.e., $p_{j_k}\left(x_k^{\left(i\right)},x_k^{(i+1)}\right)$ for all $k=1,2,\dots ,n$.

By the mixed monotone property of $(F_1,F_2,\dots F_n)$, it follows that $p_{j_k}\left(x_k^{(i+1)},F_k\left(x^{(i+1)}\right)\right)$ for all $k=1,2,\dots ,n$, and consequently,

$$\left(x_1^{(i+1)},x_2^{(i+1)},\dots x_n^{(i+1)}\right)⪯\left(x_1^{(i+2)},x_2^{(i+2)},\dots x_n^{(i+2)}\right).$$

□

Just for simplicity of the notations, let us put $T=(F_1,F_2,\dots ,F_n):X^n\to X^n$. Using this notation, the iterated sequence from Definition 5 can be written as $x^{(m+1)}=T\left(x^{\left(m\right)}\right)$.

Theorem 2. 

Let $(X,d,\preccurlyeq )$ denote a metric space that is equipped with a partial order, $F_k:X^n\to X$, $k=1,2,\dots ,n$, such that $(F_1,F_2,\dots ,F_n)$ has a mixed monotone property. $(X^n,\preccurlyeq )$ is equipped with the partial order from Definition 4. Let $(p_{j_1},p_{j_2},\dots p_{j_n})$ denote the partial order ⪯ generated by $(F_1,F_2,\dots ,F_n)$, i.e., $(x_1,x_2,\dots x_n)⪯(y_1,y_2,\dots ,y_n)$ if, and only if, $p_{j_k}(x_k,y_k)$ holds.

Let one of the following hold:

(i) 

$F_k$, $k=1,2,\dots ,n$ are continuous maps.

(ii) 

for any convergent sequence ${lim}_{m\to \infty }{x}^{\left(m\right)}=\xi$, $x^{\left(m\right)},\xi \in X^n$.

• If $x^{\left(m\right)}⪯x^{(m+1)}$ then $x^{\left(m\right)}⪯\xi$.
• If $x^{\left(m\right)}⪰x^{(m+1)}$ then $x^{\left(m\right)}⪰\xi$.

Let $\alpha \in [0,1)$ exist, so that for each monotone sequence ${\left\{x^{\left(m\right)}\right\}}_{m=0}^{\infty }\subset X^n$, there holds

$$\sum _{k=1}^n\rho \left(F_k\left(x^{\left(m\right)}\right),F_k\left(x^{(m+1)}\right)\right)\le \alpha \sum _{k=1}^n\rho \left(x_k^{\left(m\right)},x_k^{(m+1)}\right).$$

(3)

If $x^{\left(0\right)}=\left(x_1^{\left(0\right)},x_2^{\left(0\right)},\dots x_1^{\left(0\right)}\right)\in X^n$ exists, one of the following holds:

• $p_{j_k}\left(x_k^{\left(0\right)},F_k\left(x^{\left(0\right)}\right)\right)$, i.e.,

  $$\left(x_1^{\left(0\right)},x_2^{\left(0\right)},\dots x_1^{\left(0\right)}\right)⪯\left(F_1\left(x^{\left(0\right)}\right),F_2\left(x^{\left(0\right)}\right),\dots ,F_n\left(x^{\left(0\right)}\right)\right)$$
• $p_{j_k}\left(F_k\left(x^{\left(0\right)}\right),x_k^{\left(0\right)}\right)$, i.e.,

  $$\left(x_1^{\left(0\right)},x_2^{\left(0\right)},\dots x_1^{\left(0\right)}\right)⪰\left(F_1\left(x^{\left(0\right)}\right),F_2\left(x^{\left(0\right)}\right),\dots ,F_n\left(x^{\left(0\right)}\right)\right).$$

Then, there is an n-tupled fixed point $({\xi }_1,{\xi }_2\dots {\xi }_n)\in X^n$, being a limit of the sequence ${\left\{x^{\left(m\right)}\right\}}_{m=0}^{\infty }$, defined by $x_k^{(m+1)}=F_k\left(x_1^{\left(m\right)},x_2^{\left(m\right)},\dots x_n^{\left(m\right)}\right)$ for $k=1,2,\dots n$ and $m\in \mathbb{N}$.

The following error estimates hold:

(I) 

A priori error estimate

$$max\left\{\rho \left(x_k^{\left(m\right)},{\xi }_k\right):k=1,2,\dots ,n\right\}\le {\displaystyle \frac{{\alpha }^m}{1-\alpha }}\sum _{k=1}^n\rho \left(x_k^{\left(0\right)},x_k^{\left(1\right)}\right)$$

(II) 

A posteriori error estimate

$$max\left\{\rho \left(x_k^{(m+1)},{\xi }_k\right)k=1,2,\dots ,n\right\}\le {\displaystyle \frac{\alpha }{1-\alpha }}\sum _{k=1}^n\rho \left(x_k^{\left(m\right)},x_k^{(m+1)}\right)$$

(III) 

The rate of convergence

$$max\left\{\rho \left(x_k^{(m+1)},{\xi }_k\right)k=1,2,\dots ,n\right\}\le {\displaystyle \frac{\alpha }{1-\alpha }}\sum _{k=1}^n\rho \left(x_k^{\left(m\right)},{\xi }_k\right).$$

The n-tupled fixed point is unique if, for two elements $v,w\in X^n$ that are not comparable, there is an element, z, comparable with both of them, satisfying

• if $x,y⪯z$ then $z⪯Tz$;
• if $x,y⪰z$ then $z⪰Tz$.

where $T=(F_1,F_2,\dots ,F_n)$ and

$$\rho \left(T{u}^{\left(n\right)},T{v}^{\left(n\right)}\right)\le \alpha \rho \left(u^{\left(n\right)},v^{\left(n\right)}\right)$$

holds for any two sequences that satisfy $u^{\left(n\right)}⪯v^{\left(n\right)}$.

Proof. 

By the existence of $x^{\left(0\right)}=\left(x_1^{\left(0\right)},x_2^{\left(0\right)},\dots x_n^{\left(0\right)}\right)\in X^n$, satisfying either

$$\left(x_1^{\left(0\right)},x_2^{\left(0\right)},\dots x_n^{\left(0\right)}\right)\preccurlyeq \left(F_1\left(x^{\left(0\right)}\right),F_2\left(x^{\left(0\right)}\right),\dots ,F_n\left(x^{\left(0\right)}\right)\right)$$

or

$$\left(x_1^{\left(0\right)},x_2^{\left(0\right)},\dots x_n^{\left(0\right)}\right)\succcurlyeq \left(F_1\left(x^{\left(0\right)}\right),F_2\left(x^{\left(0\right)}\right),\dots ,F_n\left(x^{\left(0\right)}\right)\right),$$

the assumption that $(F_1,F_2,\dots ,F_n)$ has a mixed monotone property and Proposition 4, it follows that the sequence of successive iterations ${\left\{x^{\left(m\right)}\right\}}_{m=0}^{\infty }$ is a monotone one. The map

$$T\left(x\right)=(F_1\left(x\right),F_2\left(x\right),\dots ,F_n\left(x\right))$$

satisfies the assumptions in Theorem 1, and this finishes the proof. □

Let us define the permutation ${\pi }_k(1,2,\dots ,n)=(k,k+1,\dots ,n,1,2,\dots ,k-1)$. Then, for $F:X^n\to X$, we get the n-tupled fixed point from [40], i.e.,

$$x_i=F\left({\pi }_i(x_1,x_2,\dots ,x_n)\right)$$

for $i=1,2,\dots ,n$.

Proposition 4. 

Let $(X,d,\preccurlyeq )$ denote a metric space that is equipped with a partial order, $F:X^n\to X$, $k=1,2,\dots ,n$ and $F_k\left(x\right)=F\left({\pi }_k\left(x\right)\right)$, for $k=1,2,\dots ,n$. $(F_1,F_2,\dots ,F_n)$ has a mixed monotone property. $(X^n,⪯)$ is equipped with the partial order from Definition 4. Let $(p_{j_1},p_{j_2},\dots p_{j_n})$ denote the partial order generated by $(F_1,F_2,\dots ,F_n)$, i.e., $(x_1,x_2,\dots x_n)⪯(y_1,y_2,\dots ,y_n)$ if, and only if, $p_{j_k}(x_k,y_k)$ holds. Let $F_k\left(x\right)=F_1\left({\pi }_k\left(x\right)\right)$ for $k=1,2,\dots ,n$. If $\xi \in X^n$ is an n-tupled fixed point for $T=(F_1,F_2,\dots ,F_n)$, then, for any $z^{\left(0\right)}$ comparable with ξ, any element from the iterated sequence ${\left\{z^{\left(m\right)}\right\}}_{m=0}^{\infty }$ is comparable with ξ, too.

Proof. 

Let us assume $\xi \preccurlyeq z^{\left(0\right)}$ and $p_{j_k}\left({\xi }_k,z_k^{\left(0\right)}\right)$. From the assumption that F satisfies the mixed monotone property and $\xi$ is an n-tupled fixed point for map T, it follows that ${\xi }_i=F\left({\pi }_i\left(\xi \right)\right)$ for $i=1,2,\dots ,n$ and $p_{j_k}\left({\xi }_k,F\left({\pi }_k\left(z_k^{\left(0\right)}\right)\right)\right)$, i.e., $p_{j_k}\left({\xi }_k,z_k^{\left(1\right)}\right)$. Therefore, $\xi ⪯z^{\left(m\right)}$ for every $m\ge 0$. By similar arguments, we get that if $\xi ⪰z^{\left(0\right)}$, then $\xi ⪰z^{\left(m\right)}$ for every $m\ge 0$. □

Theorem 3. 

We incorporate the following additional assumptions into the hypotheses of Theorem 2:

(i) 

$F_k\left(x\right)=F_1\left({\pi }_k\left(x\right)\right)$, for $k=2,3,\dots ,n$;

(ii) 

If for each to of components $x,y\in X^n$, that are not comparable, there is $z\in X^n$, comparable with both of them;

We get that the components of the n-tupled fixed point $({\xi }_1,{\xi }_2,\dots ,{\xi }_n)$ are equal, i.e., ${\xi }_1={\xi }_k$ for $k=2,3,\dots ,n$.

Proof. 

Let us put $x=(x_1,x_2,\dots x_n)$ and $F\left(x\right)=F_1\left(x\right)$, just to simplify the notations and to fit the next formulas into the text field. Let there be $z^{\left(0\right)}\in X^n$, which is comparable with both fixed points $\xi$ and ${\pi }_1\left(\xi \right)=({\xi }_2,{\xi }_3,\dots ,{\xi }_n,{\xi }_1)$. By Proposition 4, it follows that each element from the sequence ${\left\{z^{\left(m\right)}\right\}}_{m=0}^{\infty }$ is simultaneously comparable with both $\xi$ and ${\pi }_1\left(\xi \right)$, too. Let us say that

$${\pi }_1\left(\xi \right)=({\xi }_2,{\xi }_3,\dots ,{\xi }_n,{\xi }_1)=(F\left({\pi }_2\left(\xi \right)\right),F\left({\pi }_3\left(\xi \right)\right),\dots ,F\left({\pi }_n\left(\xi \right)\right),F\left({\pi }_1\left(\xi \right)\right))$$

is an n-tupled fixed points for the map

$$(F\left({\pi }_2\left(x\right)\right),F\left({\pi }_3\left(x\right)\right),\dots ,F\left({\pi }_n\left(x\right)\right),F\left({\pi }_1\left(x\right)\right)):X^n\to X^n.$$

By Proposition 4, it follows that $z^{\left(m\right)}$ is comparable to ${\pi }_2\left(\xi \right)$ for all $m=0,1,2,\dots$.

Then, if we choose $m\in \mathbb{N}$ so that ${\alpha }^m\left({\rho }_1\left(\xi ,z^{\left(0\right)}\right)+{\rho }_1\left(z^{\left(0\right)},{\pi }_2\left(\xi \right)\right)\right)<{\rho }_1(\xi ,{\pi }_2\left(\xi \right))$, forming a chain of inequalities

$$\begin{array}{ccc}{\rho }_1(\xi ,{\pi }_2\left(\xi \right))\hfill & =\hfill & {\displaystyle \sum _{i=1}^n\rho ({\xi }_i,{\xi }_{i+1})}\hfill \\ & \le \hfill & {\displaystyle \sum _{i=1}^n\rho \left({\xi }_i,{\pi }_i\left(z^{\left(m\right)}\right)\right)+\sum _{i=1}^n\rho \left({\pi }_i\left(z^{\left(m\right)}\right),{\xi }_{i+1}\right)}\hfill \\ & =\hfill & {\displaystyle \sum _{i=1}^n\rho \left(F\left({\pi }_i\left(\xi \right)\right),F\left({\pi }_i\left(z^{(m-1)}\right)\right)\right)+\sum _{i=1}^n\rho \left(F\left({\pi }_i\left(z^{(m-1)}\right)\right),F\left({\pi }_i\left({\pi }_2\left(\xi \right)\right)\right)\right)}\hfill \\ & \le \hfill & {\displaystyle {\alpha }^m\sum _{k=1}^n\rho \left({\xi }_k,z_k^{\left(0\right)}\right)+{\alpha }^m\sum _{k=1}^n\rho \left(z_k^{\left(0\right)},{\xi }_{k+1}\right)}\hfill \\ & =\hfill & {\alpha }^m\left({\rho }_1\left(\xi ,z^{\left(0\right)}\right)+{\rho }_1\left(z^{\left(0\right)},{\pi }_2\left(\xi \right)\right)\right)\hfill \end{array}$$

we reach a contradiction. Thus, ${\xi }_i={\xi }_{i+1}$ for all $i=1,2,\dots ,n$, and therefore,

$${\xi }_1={\xi }_2=\dots ,={\xi }_n.$$

□

3. Applications

3.1. Applications of Theorems 2 and 3 in the Investigations of Coupled Fixed Points

If $n=2$, we get the results from [36].

Definition 6 

([36,55]). Given partially ordered set $(X,\preccurlyeq )$, we say that an ordered pair of mappings $(F,G)$, where $F,G:X\times X\to X$, possesses the mixed monotone property if, for all $x,y\in X$, the following inequalities are satisfied:

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

and

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

The next result is a slight generalization of the result in [36].

Theorem 4. 

Let $(X,d,\preccurlyeq )$ be a metric space with a partial ordering, $F,G:X^2\to X$, such that $(F,G)$ has the mixed monotone property. $(X^2,⪯)$ is endowed with the partial ordering from Definition 4, i.e., $(x_1,x_2)⪯(y_1,y_2)$ if, and only if, $x_1\preccurlyeq y_1$ and $x_2\succcurlyeq y_2$ hold.

Let one of the following hold:

(i) 

F and G are continuous maps.

(ii) 

For any convergent sequence ${lim}_{m\to \infty }{x}^{\left(m\right)}=\xi$, $x^{\left(m\right)},\xi \in X^2$.

• If $x^{\left(m\right)}⪯x^{(m+1)}$, then $x^{\left(m\right)}⪯\xi$.
• If $x^{\left(m\right)}⪰x^{(m+1)}$, then $x^{\left(m\right)}⪰\xi$.

Let $\alpha \in [0,1)$ exist, so that, for each monotone sequence ${\left\{x^{\left(m\right)}\right\}}_{m=0}^{\infty }\subset X^2$, there holds

$$\rho (F\left(x_1^{\left(m\right)}\right),F\left(x_2^{(m+1)}\right)+\rho (G\left(x_1^{\left(m\right)}\right),G\left(x_2^{(m+1)}\right)\le \alpha \left(\rho (x_1^{\left(m\right)},x_1^{(m+1)})+\rho (x_2^{\left(m\right)},x_2^{(m+1)})\right).$$

(4)

We consider if there is $x^{\left(0\right)}=\left(x_1^{\left(0\right)},x_2^{\left(0\right)}\right)\in X^2$ so that one of the following holds:

• $x_1^{\left(0\right)}\preccurlyeq F\left(x^{\left(0\right)}\right)$ and $x_2^{\left(0\right)}\succcurlyeq G\left(x^{\left(0\right)}\right)$.
• $x_1^{\left(0\right)}\succcurlyeq F\left(x^{\left(0\right)}\right)$ and $x_2^{\left(0\right)}\preccurlyeq G\left(x^{\left(0\right)}\right)$.

Then, an coupled fixed point $({\xi }_1,{\xi }_2)\in X^2$ exists, which is a limit of the sequence ${\left\{x^{\left(m\right)}\right\}}_{m=0}^{\infty }$, defined by $x_1^{(m+1)}=F\left(x_1^{\left(m\right)},x_2^{\left(m\right)}\right)$ and $x_2^{(m+1)}=G\left(x_1^{\left(m\right)},x_2^{\left(m\right)}\right)$.

The error estimates from Theorem 2 hold.

Coupled fixed point ξ is unique if, for two elements $x,y\in X^2$ that are not comparable, there is an element, z, comparable with both of them, satisfying the following:

• If $x,y⪯z$, then $z⪯(Fz,Gz)$;
• If $x,y⪰z$, then $z⪰(Fz,Gz)$.

And $\rho ((F{u}_n,G{u}_n),(F{v}_n,F{v}_n))\le \alpha \rho (u_n,v_n)$ holds for any two sequences that satisfy $u_n⪯v_n$.

If map G satisfies $G(x,y)=F(y,x)$, then coupled fixed point $\xi =({\xi }_1,{\xi }_2)$ satisfies ${\xi }_1={\xi }_2$.

Proof. 

It is sufficient to check that map $Tz=T(x,y)=\left(F\right(x,y),G(x,y\left)\right)=(Fz,GZ)$, where $z=(x,y)$, has the weak monotone property, provided that ordered pair $(F,G)$ has the mixed monotone property. Let there exist $x^{\left(0\right)}=\left(x_1^{\left(0\right)},x_2^{\left(0\right)}\right)$ such that $x_1^{\left(0\right)}\preccurlyeq F{x}^{\left(0\right)}$ and $x_2^{\left(0\right)}\succcurlyeq G{x}^{\left(0\right)}$. Thus,

$$x^{\left(0\right)}=\left(x_1^{\left(0\right)},x_1^{\left(0\right)}\right)⪯\left(F{x}^{\left(0\right)},G{x}^{\left(0\right)}\right)=T{x}^{\left(0\right)}$$

and

$$T^2{x}^{\left(0\right)}=T\left(T{x}^{\left(0\right)}\right)=\left(F\left(F{x}^{\left(0\right)},G{x}^{\left(0\right)}\right),G\left(F{x}^{\left(0\right)},G{x}^{\left(0\right)}\right)\right)⪰\left(F{x}^{\left(0\right)},G{x}^{\left(0\right)}\right)=T{x}^{\left(0\right)}$$

If $x_1^{\left(0\right)}\succcurlyeq F{x}^{\left(0\right)}$ and $x_2^{\left(0\right)}\preccurlyeq G{x}^{\left(0\right)}$ hold, then we get $x^{\left(0\right)}⪰T{x}^{\left(0\right)}$ and $T^2{x}^{\left(0\right)}⪯T{x}^{\left(0\right)}$. Consequently, we can apply Theorem 1. □

Definition 7 

([36]). Given partially ordered set $(X,\preccurlyeq )$, we say that an ordered pair of mappings $(F,G)$, where $F,G:X\times X\to X$, possesses the total monotone property if, for all $x,y\in X$, the following inequalities are satisfied:

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

and

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

The following result extends the main theorem presented in [36].

Theorem 5. 

Let $(X,d,\preccurlyeq )$ be a metric space with a partial ordering, $F,G:X^n\to X$, such that $(F,G)$ has the total monotone property. $(X^2,⪯)$ is endowed with the partial ordering from Definition 4, i.e., $(x_1,x_2)⪯(y_1,y_2)$ if, and only if, $x_1\preccurlyeq y_1$ and $x_2\preccurlyeq y_2$ hold.

Let one of the following hold:

(i) 

F and G are continuous maps.

(ii) 

For any convergent sequence ${lim}_{m\to \infty }{x}^{\left(m\right)}=\xi$, $x^{\left(m\right)},\xi \in X^2$.

• If $x^{\left(m\right)}⪯x^{(m+1)}$, then $x^{\left(m\right)}⪯\xi$.
• If $x^{\left(m\right)}⪰x^{(m+1)}$, then $x^{\left(m\right)}⪰\xi$.

Let $\alpha \in [0,1)$ exists, so that for each monotone sequence ${\left\{x^{\left(m\right)}\right\}}_{m=0}^{\infty }\subset X^2$, there holds

$$\rho (F\left(x_1^{\left(m\right)}\right),F\left(x_2^{(m+1)}\right)+\rho (G\left(x_1^{\left(m\right)}\right),G\left(x_2^{(m+1)}\right)\le \alpha \left(\rho (x_1^{\left(m\right)},x_1^{(m+1)})+\rho (x_2^{\left(m\right)},x_2^{(m+1)})\right).$$

(5)

We consider if there is $x^{\left(0\right)}=(x_1^{\left(0\right)},x_2^{\left(0\right)})\in X^2$ so that one of the following holds:

• $x_1^{\left(0\right)}\preccurlyeq F\left(x_1^{\left(0\right)}\right)$ and $x_2^{\left(0\right)}\preccurlyeq G\left(x_2^{\left(0\right)}\right)$.
• $x_1^{\left(0\right)}\succcurlyeq F\left(x_1^{\left(0\right)}\right)$ and $x_2^{\left(0\right)}\succcurlyeq G\left(x_2^{\left(0\right)}\right)$.

Then, coupled fixed point $({\xi }_1,{\xi }_2)\in X^2$ exists, which is a limit of the sequence ${\left\{x^{\left(m\right)}\right\}}_{m=0}^{\infty }$, defined by $x_1^{(m+1)}=F\left(x_1^{\left(m\right)},x_2^{\left(m\right)}\right)$ and $x_2^{(m+1)}=G\left(x_1^{\left(m\right)},x_2^{\left(m\right)}\right)$.

The error estimates from Theorem 2 hold.

Coupled fixed point ξ is unique if, for two elements $x,y\in X^2$ that are not comparable, there is an element, z, comparable with both of them, satisfying the following:

• If $x,y⪯z$, then $z⪯(Fz,Gz)$.
• If $x,y⪰z$, then $z⪰(Fz,Gz)$.

And $\rho ((F{u}_n,G{u}_n),(F{v}_n,G{v}_n))\le \alpha \rho (u_n,v_n)$ holds for any two sequences that satisfy $u_n⪯v_n$.

If map G satisfies $G(x,y)=F(y,x)$, then coupled fixed point $\xi =({\xi }_1,{\xi }_2)$ satisfies ${\xi }_1={\xi }_2$.

Proof. 

It is sufficient to check that map $Tz=T(x,y)=\left(F\right(x,y),G(x,y\left)\right)=(Fz,GZ)$, where $z=(x,y)$, has the weak monotone property, provided that ordered pair $(F,G)$ has the total monotone property. Let $x^{\left(0\right)}=\left(x_1^{\left(0\right)},x_2^{\left(0\right)}\right)$ exist, such that $x_1^{\left(0\right)}\preccurlyeq F{x}^{\left(0\right)}$ and $x_2^{\left(0\right)}\preccurlyeq G{x}^{\left(0\right)}$. Thus, $x^{\left(0\right)}=\left(x_1^{\left(0\right)},x_1^{\left(0\right)}\right)⪯\left(F{x}^{\left(0\right)},G{x}^{\left(0\right)}\right)=T{x}^{\left(0\right)}$ and

$$T^2{x}^{\left(0\right)}=T\left(T{x}^{\left(0\right)}\right)=\left(F\left(F{x}^{\left(0\right)},G{x}^{\left(0\right)}\right),G\left(F{x}^{\left(0\right)},G{x}^{\left(0\right)}\right)\right)⪰\left(F{x}^{\left(0\right)},G{x}^{\left(0\right)}\right)=T{x}^{\left(0\right)}$$

If $x_1^{\left(0\right)}\succcurlyeq F{x}^{\left(0\right)}$ and $x_2^{\left(0\right)}\succcurlyeq G{x}^{\left(0\right)}$ hold, we get, in a similar fashion, $x^{\left(0\right)}⪰T{x}^{\left(0\right)}$ and $T^2{x}^{\left(0\right)}⪯T{x}^{\left(0\right)}$. Consequently, we can apply Theorem 1. □

3.2. Applications of Theorems 2 and 3 in the Investigations of Tripled Fixed Points

If $n=3$, we get the results from [36].

Definition 8 

([37]). Let $(X,\preccurlyeq )$ be a partially ordered set and consider mapping $F:X\times X\times X\to X$. We say that F possesses the mixed monotone property if it is monotone non-decreasing in its first (x) and third (z) arguments, and monotone non-increasing in its second argument (y). That is, for any $x,y,z\in X$, the following inequalities are satisfied:

$$for{x}_1,x_2\in X,x_1\preccurlyeq x_{2}there holdsF(x_1,y,z)\preccurlyeq F(x_2,y,z),$$

$$for{y}_1,y_2\in X,y_1\preccurlyeq y_{2}there holdsF(x,y_1,z)\succcurlyeq F(x,y_2,z)$$

and

$$for{z}_1,z_2\in X,z_1\preccurlyeq z_{2}there holdsF(x,y,z_1)\preccurlyeq F(x,y,z_2).$$

The following result extends the main theorem presented in [36].

Theorem 6. 

Let $(X,\rho ,\preccurlyeq )$ be a complete metric space equipped with a partial order, and let $(F_1,F_2,F_3)$ be a triple of continuous mappings that possess the mixed monotone property. Suppose there exists constant $\alpha \in [0,1)$ such that the inequality

$$\sum _{k=1}^3\rho (F_k(x,y,z),F_k(u,v,w))\le \alpha (\rho (x,u)+\rho (y,v)+\rho (z,w))$$

(6)

holds true for all $x\succcurlyeq u$, $y\preccurlyeq v$ and $z\succcurlyeq w$.

Let one of the following hold:

(i) 

$F_k$, $k=1,2,3$ are continuous maps.

(ii) 

For any convergent sequence ${\displaystyle \underset{n\to \infty }{lim}(x_n,y_n,z_n)=(x,y,z)}$, $(x_n,y_n,z_n)\in X^3$.

• If $(x_n,y_n,z_n)\preccurlyeq (x_{n+1},y_{n+1},z_{n+1})$, then $(x_n,y_n,z_n)\preccurlyeq (x,y,z)$.
• If $(x_n,y_n,z_n)\succcurlyeq (x_{n+1},y_{n+1},z_{n+1})$, then $(x_n,y_n,z_n)\succcurlyeq (x,y,z)$.

We consider if there are $x_0,y_0,z_0\in X$ so that one of the following conditions holds:

• $x_1^{\left(0\right)}\preccurlyeq F_1\left(x_1^{\left(0\right)},x_2^{\left(0\right)},x_3^{\left(0\right)}\right)$, $x_2^{\left(0\right)}\succcurlyeq F_2\left(x_1^{\left(0\right)},x_2^{\left(0\right)},x_3^{\left(0\right)}\right)$, and $x_3^{\left(0\right)}\preccurlyeq F_3\left(x_1^{\left(0\right)},x_2^{\left(0\right)},x_3^{\left(0\right)}\right)$.
• $x_1^{\left(0\right)}\succcurlyeq F_1\left(x_1^{\left(0\right)},x_2^{\left(0\right)},x_3^{\left(0\right)}\right)$, $x_2^{\left(0\right)}\preccurlyeq F_2\left(x_1^{\left(0\right)},x_2^{\left(0\right)},x_3^{\left(0\right)}\right)$, and $x_3^{\left(0\right)}\succcurlyeq F_3\left(x_1^{\left(0\right)},x_2^{\left(0\right)},x_3^{\left(0\right)}\right)$.

Then, a tripled fixed point $({\xi }_1,{\xi }_2,{\xi }_3)\in X\times X\times X$ exists, which is a limit of the sequence ${\left\{\left(x_1^{\left(n\right)},x_2^{\left(n\right)},x_3^{\left(n\right)}\right)\right\}}_{n=0}^{\infty }$, defined by $x_k^{\left(n\right)}=F_k\left(x_1^{\left(n\right)},x_2^{\left(n\right)},x_3^{\left(n\right)}\right)$ for $k=1,2,3$ and $n\in \mathbb{N}$.

The error estimates hold.

If any pair of components $(x,y,z),(u,v,w)\in (X\times X\times X,\preccurlyeq )$, where $(x_1,x_2,x_3)\preccurlyeq (y_1,y_2,y_3)$ if $x_1\preccurlyeq y_1$, $x_2\succcurlyeq y_2$, and $x_3\preccurlyeq y_3$, has either a lower bound or an upper bound, then $({\xi }_1,{\xi }_2,{\xi }_3)$ is a unique coupled fixed point.

A similar generalization holds if we consider an ordered triple of maps, where each of the maps monotonely increases all of its variables.

3.3. Application of Theorem 3

We recall commonly known concepts from linear algebra, related to matrix equations. In this exposition, we follow [56]. The presented notions and notations can also be found in [33,42]. Only for the sake of the completeness of the text, will we briefly recall these well-known concepts. In what follows, let $\mathcal{H}\left(M\right)$ represent the collection of all $M\times M$, Hermitian matrices, with $M>1$ being a natural number. Following conventions from [56], the symbol ${\mathcal{V}}^{*}$ denotes the conjugate transpose of a matrix $\mathcal{V}$. A matrix is classified as follows:

• Hermitian if it satisfies $\mathcal{V}={\mathcal{V}}^{*}$;
• Skew–Hermitian if $\mathcal{V}=-{\mathcal{V}}^{*}$;
• Unitary if $\mathcal{V}{\mathcal{V}}^{*}=I$ (where I is the identity matrix);
• Normal if $\mathcal{V}{\mathcal{V}}^{*}={\mathcal{V}}^{*}\mathcal{V}$.

A matrix $\mathcal{V}$ is termed positive semidefinite when $0\le \langle v,\mathcal{V}v\rangle$ for every non-zero vector v, denoted as $\mathcal{V}\succcurlyeq 0$. If $0<\langle v,\mathcal{V}v\rangle$ for all non-zero u, the matrix is positive definite, written as $\mathcal{V}\succ 0$. A positive definite matrix is invertible, and this property is both necessary and sufficient. For Hermitian matrices $\mathcal{A}$ and $\mathcal{B}$, the notation $\mathcal{V}\succcurlyeq \mathcal{W}$ (or $\mathcal{V}\succ \mathcal{W}$) indicates that $\mathcal{V}-\mathcal{W}$ is positive semidefinite (or positive definite). If X satisfies $\mathcal{V}\preccurlyeq X\preccurlyeq \mathcal{W}$, we denote this by $X\in [\mathcal{V},\mathcal{W}]$.

For any matrix $\mathcal{V}$, the product $\mathcal{V}{\mathcal{V}}^{*}$ is always positive semidefinite, and its unique positive definite square root is denoted as $\left|\mathcal{V}\right|$. The singular values of $\mathcal{V}$, labeled ${\sigma }_1\left(\mathcal{V}\right)\ge {\sigma }_2\left(\mathcal{V}\right)\ge \dots \ge {\sigma }_m\left(\mathcal{V}\right)$, correspond to the eigenvalues of $\left|\mathcal{V}\right|$ arranged in descending order. If $\mathrm{rank}\left(\right|\mathcal{V})=s$, then ${\sigma }_s\left(\mathcal{V}\right)>0$, while ${\sigma }_{s+1}\left(\mathcal{V}\right)=\dots {\sigma }_m\left(\mathcal{V}\right)=0$.

Lastly, every matrix $\mathcal{V}$ can be transformed via unitary equivalence into an upper triangular matrix $\mathcal{T}$, meaning $\mathcal{V}={\mathcal{UTU}}^{*}$ for unitary matrix $\mathcal{U}$.

By $r\left(\mathcal{V}\right)$, we denote the spectral radius of $\mathcal{V}$. The spectral norm is written as $\parallel \mathcal{V}\parallel$, defined by $\parallel \mathcal{V}\parallel =\sqrt{{\lambda }^{+}\left({\mathcal{V}}^{*}\mathcal{V}\right)}$, where ${\lambda }^{+}$ is the largest eigenvalue of ${\mathcal{V}}^{*}\mathcal{V}$.

Furthermore, we equip $\mathcal{H}\left(N\right)$ with the trace norm ${\left|·\right|}_{\mathrm{tr}}$, given by ${\parallel \mathcal{U}\parallel }_{\mathrm{tr}}={\sum }_{j=1}^M{\sigma }_j\left(\mathcal{U}\right)$, where ${\sigma }_j\left(\mathcal{U}\right)$, for $j=1,2,\dots ,M$, are the singular values of $\mathcal{U}$.

The following lemmas will be instrumental in our subsequent analysis.

Lemma 1 

([15]). Suppose $\mathcal{U}$ and $\mathcal{W}$ are $M\times M$ positive semidefinite matrices. Then, the trace satisfies $0\le \mathrm{tr}\left(\mathcal{U}\mathcal{W}\right)\le \parallel \mathcal{U}\parallel \mathrm{tr}\left(\mathcal{W}\right)$.

Lemma 2 

([56]). Let $0<\theta \le 1$, and let U and W be positive definite matrices of the same size such that $U,W\succcurlyeq bI>0$. Then, for any unitarily invariant norm $\left|\right|\left|·\right|\left|\right|$, the following inequalities hold: $\left|\right||U^{\theta }-V^{\theta }\left|\right||\le \theta b^{\theta -1}\left|\right||U-V\left|\right||$ and $\left|\right||U^{-\theta }-V-\theta \left|\right||\le \theta b^{-(\theta +1)}\left|\right||U-V\left|\right||$.

Lemma 3 

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

Theorem 7 

([57]). Let S be a nonempty, compact, convex subset of a normed vector space. Every continuous function $f:S\to S$ mapping S into itself has a fixed point.

Let us put ${\Omega }_{\alpha I}=\{U\in \mathcal{H}\left(N\right):X\succcurlyeq \alpha I\}$. Let us consider the matrix Equation (7)

$$X=\mathcal{Q}+\sum _{i=1}^s{p}_i{\mathcal{A}}_i^{*}{X}^{{\theta }_i}{\mathcal{A}}_i+\sum _{i=s+1}^n{p}_i{\mathcal{A}}_i^{*}{X}^{{\theta }_i}{\mathcal{A}}_i,$$

(7)

where $p_k\in \{-1,1\}$, ${\mathcal{A}}_i\in \mathcal{X}\left(N\right)$, $\mathcal{Q}\in \mathcal{H}\left(N\right)$, $\mathcal{Q}\in \mathcal{P}\left(N\right)$ for $i,k=1,2,\dots ,n$, and $p_i{\theta }_i>0$ for $i=1,2,\dots s$, $p_i{\theta }_i<0$ for $i=s+1,\dots n$.

Theorem 8. 

Let there be $0<a<b$, so that

(i) 

${\displaystyle \mathcal{Q}+\sum _{k=s+1}^n{p}_k{a}^{{\theta }_k}{\mathcal{A}}_i^{*}{\mathcal{A}}_i\succcurlyeq aI}$, for $i=s+1,\dots n$

(ii) 

there exists $X_0\in {\left({\Omega }_{aI}\right)}^n$, so that $X^{\left(0\right)}⪯F\left(X^{\left(0\right)}\right)$

(iii) 

$\delta =\left({\displaystyle \sum _{i\in {\Delta }^{+}}\parallel A_i^{*}{A}_i\parallel {\theta }_i{a}^{{\theta }_i-1}+\sum _{i\in {\Delta }^{-}}\parallel A_i^{*}{A}_i\parallel {\theta }_i{a}^{-({\theta }_i+1)}}\right)<1$

are satisfied.

Then, there is a unique solution, $\tilde{X}\in {\Omega }_{aI}$, for $i=1,2,\dots n$ for the system (7).

Proof. 

Consider the maps $F:{\left(\mathcal{H}\left(N\right)\right)}^n\to \mathcal{H}\left(N\right)$ defined by

$$F(X_1,X_2,\dots ,X_n)=\mathcal{Q}+\sum _{k=1}^n{p}_i{\mathcal{A}}_i^{*}{X}_k^{{\theta }_k}{\mathcal{A}}_i.$$

Let us put $X=(X_1,\dots ,X_n)$ and $F_i\left(X\right)=F\left({\pi }_i\left(X\right)\right)$ for $i=1,\dots ,n$. Just to simplify the notations, we will identify $X_{n+p}=X_p$ for $p\in \mathbb{N}$. Let $X_i\in {\Omega }_{aI}$, i.e., $X_i\succcurlyeq aI$, $i=1,2,\dots n$. Then, from $\mathcal{Q},{\mathcal{A}}_i^{*}{\mathcal{A}}_i\in \mathcal{P}\left(N\right)$ for $i=1,2,\dots n$ and (i) of Theorem 8 we get

$$\begin{array}{ccc}{F}_i\left(X\right)\hfill & =\hfill & {\displaystyle F\left({\pi }_i\left(X\right)\right)=Q+\sum _{k=1}^n{p}_i{\mathcal{A}}_i^{*}{X}_{k+i-1}^{{\theta }_k}{\mathcal{A}}_i\succcurlyeq Q+\sum _{k=s+1}^n{p}_i{\mathcal{A}}_i^{*}{X}_{k+i-1}^{{\theta }_k}{\mathcal{A}}_i}\hfill \\ & \succcurlyeq \hfill & {\displaystyle Q+\sum _{k=s+1}^n{p}_i{a}^{{\theta }_k}{\mathcal{A}}_i^{*}{\mathcal{A}}_i\succcurlyeq aI.}\hfill \end{array}$$

Thus, $F_i:{\left({\Omega }_{aI}\right)}^n\to {\Omega }_{aI}$.

Let us put

$$Z_i=(X_1,\dots ,X_{i-1},X,X_{i+1},\dots ,X_n),$$

$$W_i=(X_1,\dots ,X_{i-1},U,X_{i+1},\dots ,X_n)$$

for $i=1,2,\dots n$. Let $X,U\in {\Omega }_{aI}$, such that $X\succcurlyeq U$. Then, $p_i{X}^{{\theta }_i}\succcurlyeq p_i{U}^{{\theta }_i}$ for $p_i{\theta }_i>0$, i.e., $i=1,\dots s$, and $p_i{X}^{{\theta }_i}\preccurlyeq p_i{U}^{{\theta }_i}$ for $p_i{\theta }_i<0$, i.e., $i=s+1,\dots n$.

Therefore, by ${\mathcal{A}}_i^{*}{\mathcal{A}}_i\in \mathcal{P}\left(N\right)$, we get

$$F_i\left(Z_i\right)-F_i\left(W_i\right)=p_i{\mathcal{A}}_i^{*}(X^{\theta }-U^{\theta }){\mathcal{A}}_i={\mathcal{A}}_i^{*}(p_i{X}^{\theta }-p_i{U}^{\theta }){\mathcal{A}}_i\succcurlyeq 0,$$

for $i=1,2\dots ,s$ and

$$F_i\left(Z_i\right)-F_i\left(W_i\right)=p_i{\mathcal{A}}_i^{*}(X^{\theta }-U^{\theta }){\mathcal{A}}_i={\mathcal{A}}_i^{*}(p_i{X}^{\theta }-p_i{U}^{\theta }){\mathcal{A}}_i\preccurlyeq 0$$

for $i=s+1,\dots ,n$.

Let us put

$$T\left(X\right)=(F_1\left(X\right),F_2\left(X\right),\dots ,F_n\left(X\right)):{\left({\Omega }_{aI}\right)}^n\to {\left({\Omega }_{aI}\right)}^n$$

Consequently, the n-tuple of maps $T=(F_1,F_2,\dots ,F_n)$ has the mixed monotone property and, consequently, the weak mixed monotone property.

Let us introduce a partial ordering in $\left({\left({\Omega }_{aI}\right)}^n,\preccurlyeq \right)$, generated by the mixed monotone property of map T.

Let us put $X^{\left(0\right)}=(\underset{s}{\underbrace{aI,\dots ,aI}},\underset{n-s}{\underbrace{bI,\dots ,bI}})$. The inequalities

$$F_i\left(X\right)\succcurlyeq \mathcal{Q}+\sum _{i=s+1}^n{p}_i{b}^{{\theta }_i}{\mathcal{A}}_i^{*}{\mathcal{A}}_i\succcurlyeq aI$$

for $i=1,2,\dots ,s$ and

$$F_i\left(X\right)\preccurlyeq \mathcal{Q}+\sum _{i=s+1}^n{p}_i{a}^{{\theta }_i}{\mathcal{A}}_i^{*}{\mathcal{A}}_i\preccurlyeq bI$$

for $i=s+1,\dots n$ coincide with (ii) of Theorem 8. Consequently, the inequality $X^{\left(0\right)}⪯F\left(X^{\left(0\right)}\right)$ holds.

There exists the greatest lower bound and lowest upper bound for each $V,W\in {\left(\mathcal{H}\left(N\right)\right)}^n$, and thus, for any two matrices $V,W\in {\left(\mathcal{H}\left(N\right)\right)}^n$, there exists a matrix comparable with both of them.

Maps $F_i$ and $i=1,2,\dots n$ are continuous.

Let us equip $\left({\left({\Omega }_{aI}\right)}^n,\left|\right|\left|·\right|\left|\right|,\preccurlyeq \right)$ with the norm $\left|\right||(X_1,\dots ,X_n)\left|\right||={\sum }_{k=1}^n{\parallel X_k\parallel }_{\mathrm{tr}}$.

There holds the following chain of equalities:

$$\begin{array}{ccc}{S}_1\hfill & =\hfill & \left|\right|\left|T\right(X)-T(U\left)\right|\left|\right|\hfill \\ & =\hfill & \left|\right|\left|\right(F_1\left(X\right)-F_1\left(U\right),F_2\left(X\right)-F_2\left(U\right),\dots ,F_n\left(X\right)-F_n\left(U\right)\left)\right|\left|\right|\hfill \\ & =\hfill & \left|\right|\left|\right(F\left({\pi }_1\left(X\right)\right)-F\left({\pi }_1\left(U\right)\right),\dots ,F\left({\pi }_n\left(X\right)\right)-F\left({\pi }_n\left(U\right)\right)\left)\right|\left|\right|\hfill \\ & =\hfill & {\displaystyle \sum _{i=1}^n{\parallel F\left({\pi }_i\left(X\right)\right)-F\left({\pi }_i\left(U\right)\right)\parallel }_{\mathrm{tr}}.}\hfill \end{array}$$

Let us denote by ${\Delta }^{+}=\{j:{\theta }_j>0\}$ and ${\Delta }^{-}=\{j:{\theta }_j<0\}$.

$$\begin{array}{ccc}{S}_2\hfill & =\hfill & {\displaystyle \parallel F\left({\pi }_i\left(X\right)\right)-F\left({\pi }_i\left(U\right)\right){\parallel }_{\mathrm{tr}}}\hfill \\ & =\hfill & {\displaystyle {∥\sum _{j=1}^s{p}_j{\mathcal{A}}_j^{*}\left(X_{j+i-1}^{{\theta }_j}-U_{j+i-1}^{{\theta }_j}\right){\mathcal{A}}_j+\sum _{j=s+1}^n{p}_j{\mathcal{A}}_j^{*}\left(U_{j+i-1}^{{\theta }_j}-X_{j+i-1}^{{\theta }_j}\right){\mathcal{A}}_j∥}_{\mathrm{tr}}}\hfill \\ & =\hfill & {\displaystyle \mathrm{tr}\left(\sum _{i\in {\Delta }^{+}}{A}_i^{*}{A}_i{p}_i\left(X_{j+i-1}^{{\theta }_i}-U_{j+i-1}^{{\theta }_i}\right)+\sum _{i\in {\Delta }^{-}}{A}_i^{*}{A}_i{p}_i\left(U_{j+i-1}^{{\theta }_i}-X_{j+i-1}^{{\theta }_i}\right)\right)}\hfill \end{array}$$

$$\begin{array}{ccc}{S}_3\hfill & =\hfill & {\displaystyle \sum _{i=1}^n{\parallel F\left({\pi }_i\left(X\right)\right)-F\left({\pi }_i\left(U\right)\right)\parallel }_{\mathrm{tr}}}\hfill \\ & \le \hfill & {\displaystyle \sum _{j=1}^n\mathrm{tr}\left(\sum _{i\in {\Delta }^{+}}{p}_i{A}_i^{*}{A}_i\left(X_{j+i-1}^{{\theta }_i}-U_{j+i-1}^{{\theta }_i}\right)+\sum _{i\in {\Delta }^{-}}{p}_i{A}_i^{*}{A}_i\left(U_{j+i-1}^{{\theta }_i}-X_{j+i-1}^{{\theta }_i}\right)\right)}\hfill \\ & =\hfill & {\displaystyle \sum _{j=1}^n\mathrm{tr}\left(\sum _{i\in {\Delta }^{+}}{p}_i{A}_i^{*}{A}_i\left(X_{j+i-1}^{{\theta }_i}-U_{j+i-1}^{{\theta }_i}\right)\right)}\hfill \\ & & {\displaystyle +\sum _{j=1}^n\mathrm{tr}\left(\sum _{i\in {\Delta }^{-}}{p}_i{A}_i^{*}{A}_i\left(U_{j+i-1}^{{\theta }_i}-X_{j+i-1}^{{\theta }_i}\right)\right)}\hfill \\ & =\hfill & {\displaystyle \sum _{i\in {\Delta }^{+}}\sum _{j=1}^n\mathrm{tr}\left(\left(p_i{A}_i^{*}{A}_i\left(X_{j+i-1}^{{\theta }_i}-U_{j+i-1}^{{\theta }_i}\right)\right)\right)}\hfill \\ & & {\displaystyle +\sum _{i\in {\Delta }^{-}}\sum _{j=1}^n\mathrm{tr}\left(\left(p_i{A}_i^{*}{A}_i\left(U_{j+i-1}^{{\theta }_i}-X_{j+i-1}^{{\theta }_i}\right)\right)\right)}\hfill \\ & =\hfill & {\displaystyle \sum _{i\in {\Delta }^{+}}\sum _{j=1}^n\parallel A_i^{*}{A}_i\parallel \mathrm{tr}\left(X_{j+i-1}^{{\theta }_i}-U_{j+i-1}^{{\theta }_i}\right)+\sum _{i\in {\Delta }^{-}}\sum _{j=1}^n\parallel A_i^{*}{A}_i\parallel \mathrm{tr}\left(U_{j+i-1}^{{\theta }_i}-X_{j+i-1}^{{\theta }_i}\right)}\hfill \\ & \le \hfill & {\displaystyle \sum _{i\in {\Delta }^{+}}\sum _{j=1}^n\parallel A_i^{*}{A}_i\parallel {\theta }_i{a}^{{\theta }_i-1}\mathrm{tr}(X_{j+i-1}-U_{j+i-1})}\hfill \\ & & {\displaystyle +\sum _{i\in {\Delta }^{-}}\sum _{j=1}^n\parallel A_i^{*}{A}_i\parallel {\theta }_i{a}^{-({\theta }_i+1)}\mathrm{tr}(X_{j+i-1}-U_{j+i-1})}\hfill \\ & \le \hfill & {\displaystyle \sum _{j=1}^n\left(\sum _{i\in {\Delta }^{+}}\parallel A_i^{*}{A}_i\parallel {\theta }_i{a}^{{\theta }_i-1}+\sum _{i\in {\Delta }^{-}}\parallel A_i^{*}{A}_i\parallel {\theta }_i{a}^{-({\theta }_i+1)}\right)\mathrm{tr}(X_j-U_j)}\hfill \\ & \le \hfill & {\displaystyle \left(\sum _{i\in {\Delta }^{+}}\parallel A_i^{*}{A}_i\parallel {\theta }_i{a}^{{\theta }_i-1}+\sum _{i\in {\Delta }^{-}}\parallel A_i^{*}{A}_i\parallel {\theta }_i{a}^{-({\theta }_i+1)}\right)\sum _{j=1}^n\mathrm{tr}(X_j-U_j)}\hfill \\ & =\hfill & \delta \sum _{j=1}^n\parallel X_j-U_j{\parallel }_{\mathrm{tr}}=\delta \left|\right||X-U\left|\right||,\hfill \end{array}$$

where $\delta =\left({\sum }_{i\in {\Delta }^{+}}\parallel A_i^{*}{A}_i\parallel {\theta }_i{a}^{{\theta }_i-1}+{\sum }_{i\in {\Delta }^{-}}\parallel A_i^{*}{A}_i\parallel {\theta }_i{a}^{-({\theta }_i+1)}\right)<1$.

Consequently, the ordered n-tuple of map T satisfies Theorem 3’s assumptions, and therefore, a unique solution $\left(\tilde{X}\right)\in {\Omega }_{aI}$ to (7) exists. □

Moreover, in the iterative sequences ${\left\{X^{\left(m\right)}\right\}}_{m=0}^{\infty }$, where

$$X^{\left(m\right)}=\left(X_1^{\left(m\right)},\dots ,X_n^{\left(m\right)}\right),$$

$X^{\left(m\right)}=T\left(X^{(m-1)}\right)$ each ${\left\{X_i^{\left(m\right)}\right\}}_{m=0}^{\infty }$ converges to $\tilde{X}$.

The following error estimations are listed below:

(I) 

A priori error estimate

$$max\left\{{∥X_i^{\left(m\right)}-\tilde{X}∥}_{\mathrm{tr}}:i=1,2,\dots n\right\}\le {\displaystyle \frac{{\delta }^m}{1-\delta }}M\left(X^{\left(1\right)},X^{\left(1\right)}\right)$$

(II) 

A posteriori error estimate

$$max\left\{{∥X_i^{(m+1)}-\tilde{X}∥}_{\mathrm{tr}}:i=1,2,\dots n\right\}\le {\displaystyle \frac{\delta }{1-\delta }}M\left(X^{(m+1)},X^{\left(m\right)}\right)$$

(8)

(III) 

The rate of convergence

$$max\left\{{∥X_i^{(m+1)}-\tilde{X}∥}_{\mathrm{tr}}:i=1,2,\dots n\right\}\le {\displaystyle \frac{\delta }{1-\delta }}M\left(X^{\left(m\right)},\tilde{X}\right),$$

where we use the notation $M(X,Y)={\sum }_{i=1}^n{\parallel X_i-Y_i\parallel }_{\mathrm{tr}}$.

4. Illustrative Examples

Application of Theorem 8

We illustrate example with $n=5$.

Example 3. 

Let consider the matrix Equation (7) with $j=1,2,3,4,5$.

$$X=\mathcal{Q}-{\mathcal{A}}_1^{*}{X}^{-\theta }{\mathcal{A}}_1+{\mathcal{A}}_2^{*}{X}^{-\theta }{\mathcal{A}}_2-{\mathcal{A}}_3^{*}{X}^{-\theta }{\mathcal{A}}_3+{\mathcal{A}}_4^{*}{X}^{-\theta }{\mathcal{A}}_4-{\mathcal{A}}_5^{*}{X}^{-\theta }{\mathcal{A}}_5.$$

(9)

Let us put

$${\mathcal{A}}_1=\left(\begin{array}{ccc}0.021\hfill & \hfill 0.01& 0.1\hfill \\ 0.1\hfill & \hfill 0.03& 0.1\hfill \\ 1\hfill & \hfill 0.01& 0.2\hfill \end{array}\right),{\mathcal{A}}_2=\left(\begin{array}{ccc}0.18\hfill & \hfill 0.54& -0.54\hfill \\ 0.72\hfill & \hfill 1.8& 0.72\hfill \\ 0.9\hfill & \hfill 0.9& 1.8\hfill \end{array}\right),$$

$${\mathcal{A}}_3=\left(\begin{array}{ccc}0.02\hfill & \hfill 0.025& 0.006\hfill \\ 0\hfill & \hfill 0.07& 0.06\hfill \\ 0.005\hfill & \hfill 0.005& -0.16\hfill \end{array}\right),{\mathcal{A}}_4=\left(\begin{array}{ccc}0.1\hfill & \hfill -0.02& -0.02\hfill \\ 0.03\hfill & \hfill 0.1& -0.03\hfill \\ 0.04\hfill & \hfill 0.04& 0.1\hfill \end{array}\right),$$

$${\mathcal{A}}_5=\left(\begin{array}{ccc}0.0033\hfill & \hfill 0.004167& 0.001\hfill \\ 0\hfill & \hfill 0.01167& 0.01\hfill \\ 0.00083\hfill & \hfill 0.00083& -0.0267\hfill \end{array}\right),\mathcal{Q}=\left(\begin{array}{ccc}3.5\hfill & \hfill 0.5& 0.5\hfill \\ 0.5\hfill & \hfill 3.5& 1\hfill \\ 0.5\hfill & \hfill 1& 4\hfill \end{array}\right),$$

and $a=2$, $b=55$, $\theta ={\displaystyle \frac{1}{5}}$.

We provide a numerical example of matrix sequences $X_k^{\left(m\right)}$ created by the relevant systems,

$$\left|\begin{array}{ccc}{X}_1\hfill & =\hfill & \mathcal{Q}-{\mathcal{A}}_1^{*}{X}_1^{-\theta }{\mathcal{A}}_1+{\mathcal{A}}_2^{*}{X}_2^{-\theta }{\mathcal{A}}_2-{\mathcal{A}}_3^{*}{X}_3^{-\theta }{\mathcal{A}}_3+{\mathcal{A}}_4^{*}{X}_4^{-\theta }{\mathcal{A}}_4-{\mathcal{A}}_5^{*}{X}_5^{-\theta }{\mathcal{A}}_5\hfill \\ X_2\hfill & =\hfill & \mathcal{Q}-{\mathcal{A}}_1^{*}{X}_2^{-\theta }{\mathcal{A}}_1+{\mathcal{A}}_2^{*}{X}_3^{-\theta }{\mathcal{A}}_2-{\mathcal{A}}_3^{*}{X}_4^{-\theta }{\mathcal{A}}_3+{\mathcal{A}}_4^{*}{X}_5^{-\theta }{\mathcal{A}}_4-{\mathcal{A}}_5^{*}{X}_1^{-\theta }{\mathcal{A}}_5\hfill \\ X_3\hfill & =\hfill & \mathcal{Q}-{\mathcal{A}}_1^{*}{X}_3^{-\theta }{\mathcal{A}}_1+{\mathcal{A}}_2^{*}{X}_4^{-\theta }{\mathcal{A}}_2-{\mathcal{A}}_3^{*}{X}_5^{-\theta }{\mathcal{A}}_3+{\mathcal{A}}_4^{*}{X}_1^{-\theta }{\mathcal{A}}_4-{\mathcal{A}}_5^{*}{X}_2^{-\theta }{\mathcal{A}}_5\hfill \\ X_4\hfill & =\hfill & \mathcal{Q}-{\mathcal{A}}_1^{*}{X}_4^{-\theta }{\mathcal{A}}_1+{\mathcal{A}}_2^{*}{X}_5^{-\theta }{\mathcal{A}}_2-{\mathcal{A}}_3^{*}{X}_1^{-\theta }{\mathcal{A}}_3+{\mathcal{A}}_4^{*}{X}_2^{-\theta }{\mathcal{A}}_4-{\mathcal{A}}_5^{*}{X}_3^{-\theta }{\mathcal{A}}_5\hfill \\ X_5\hfill & =\hfill & \mathcal{Q}-{\mathcal{A}}_1^{*}{X}_5^{-\theta }{\mathcal{A}}_1+{\mathcal{A}}_2^{*}{X}_1^{-\theta }{\mathcal{A}}_2-{\mathcal{A}}_3^{*}{X}_2^{-\theta }{\mathcal{A}}_3+{\mathcal{A}}_4^{*}{X}_3^{-\theta }{\mathcal{A}}_4-{\mathcal{A}}_5^{*}{X}_4^{-\theta }{\mathcal{A}}_5,\hfill \end{array}\right.$$

which converge to the solutions $\tilde{X}\succ 0$ of (9).

We apply a stopping procedure imposed by the a posteriori error estimate (8) with

$$\underset{i=1,2,3,4,5}{max}\left\{{∥X_i^{(m+1)}-\widetilde{X_i}∥}_{\mathrm{tr}}\right\}\le {10}^{-10}.$$

We apply Theorem 8. It is easy to check that all the conditions of Theorem 8 are satisfied. It is easy to calculate $A_i^{*}{A}_i\succ 0$, $Q,A_i^{*}{A}_i\in \mathcal{P}\left(N\right)$, and

$$Q-a^{\theta }{\mathcal{A}}_1^{*}{\mathcal{A}}_1-a^{\theta }{\mathcal{A}}_3^{*}{\mathcal{A}}_3-a^{\theta }{\mathcal{A}}_5^{*}{\mathcal{A}}_5\succcurlyeq aI.$$

There exists $X_0(aI,bI,aI,bI,aI)$ so that $X_0\preccurlyeq T\left(X_0\right)$.

We put for an initial start $X^{\left(0\right)}=(aI,bI,aI,bI,aI)$.

From

$$\delta ={\displaystyle \frac{\theta \parallel {\mathcal{A}}_1^{*}{\mathcal{A}}_1\parallel }{a^{1-\theta }}}+{\displaystyle \frac{\theta \parallel {\mathcal{A}}_2^{*}{\mathcal{A}}_2\parallel }{a^{1+\theta }}}+{\displaystyle \frac{\theta \parallel {\mathcal{A}}_3^{*}{\mathcal{A}}_3\parallel }{a^{1-\theta }}}+{\displaystyle \frac{\theta \parallel {\mathcal{A}}_4^{*}{\mathcal{A}}_4\parallel }{a^{1+\theta }}}+{\displaystyle \frac{\theta \parallel {\mathcal{A}}_5^{*}{\mathcal{A}}_5\parallel }{a^{1-\theta }}}<0.81<1$$

the approximate solutions for $\epsilon ={10}^{-10}$, given by the a posteriori error estimate, are

$$X_1^{\left(14\right)}\approx \left(\begin{array}{ccc}3.6813816965\hfill & \hfill 1.8805511292& 1.7074607094\hfill \\ 1.8805511292\hfill & \hfill 6.3070644307& 2.6262586938\hfill \\ 1.7074607094\hfill & \hfill 2.6262586938& 6.8008621078\hfill \end{array}\right),$$

$$X_2^{\left(14\right)}\approx \left(\begin{array}{ccc}3.6813816966\hfill & \hfill 1.8805511294& 1.7074607095\hfill \\ 1.8805511294\hfill & \hfill 6.3070644312& 2.6262586939\hfill \\ 1.7074607095\hfill & \hfill 2.6262586939& 6.8008621079\hfill \end{array}\right),$$

$$X_3^{\left(14\right)}\approx \left(\begin{array}{ccc}3.6813816965\hfill & \hfill 1.8805511292& 1.7074607094\hfill \\ 1.8805511292\hfill & \hfill 6.3070644308& 2.6262586938\hfill \\ 1.7074607094\hfill & \hfill 2.6262586938& 6.8008621078\hfill \end{array}\right),$$

$$X_4^{\left(14\right)}\approx \left(\begin{array}{ccc}3.6813816966\hfill & \hfill 1.8805511293& 1.7074607095\hfill \\ 1.8805511293\hfill & \hfill 6.3070644311& 2.6262586939\hfill \\ 1.7074607095\hfill & \hfill 2.6262586939& 6.8008621079\hfill \end{array}\right),$$

$$X_5^{\left(14\right)}\approx \left(\begin{array}{ccc}3.6813816966\hfill & \hfill 1.8805511293& 1.7074607095\hfill \\ 1.8805511293\hfill & \hfill 6.3070644310& 2.6262586938\hfill \\ 1.7074607095\hfill & \hfill 2.6262586938& 6.8008621078\hfill \end{array}\right),$$

and the solution in the matrix equation is

$$\tilde{X}\approx \left(\begin{array}{ccc}3.6813816966\hfill & \hfill 1.8805511293& 1.7074607095\hfill \\ 1.8805511293\hfill & \hfill 6.3070644310& 2.6262586938\hfill \\ 1.7074607095\hfill & \hfill 2.6262586938& 6.8008621078\hfill \end{array}\right),$$

and the a posterior error is $0.39163097399\times {10}^{-10}$.

5. Discussion

We would like to note that we can use Theorem 2 to solve nonlinear systems of matrix equations. Of course, the conditions that need to be specified are quite laborious, so we omit them here. It would be interesting to use the ideas from [46,47,48,49,58] and to investigate the proposed generalization of n-tupled fixed points of ordered n-tuples of maps $(F_1,F_2,\dots ,F_n)$. Furthermore, it would be worthwhile to apply the main result in a system of n different integral [43,44] or ordinary differential [59] equations. Finally, we would like to know if it is possible to obtain similar results to those in [60,61,62] in the proposed notion of n-tupled fixed points.

6. Conclusions

In this study, we link the observations in [15,36,40,46,47,48,49,50,51] to create an extension of the coupled fixed points initiated in [31,32]. The concept of n-tupled fixed points for ordered n-tuples of maps appears to be natural [63,64]. We have shown that the notion of partial ordering in the suggested concept in [31] of investigating coupled fixed points for maps with a mixed monotonicity property in partly ordered metric spaces is closely related to the kind of monotonicity of the mappings being studied. This enables us to solve different kinds of matrix equation problems. Moreover, we suggest a technique that can be applied to whole classes of nonlinear matrix equations. It is worth mentioning the case when the power, $\alpha$, of the matrices in the equations satisfies $\alpha \notin [-1,0)\cup (0,1]$, which is interesting and open to investigation. The method suggested in [33] for solving only symmetric systems of matrix equations is generalized by the concept that an ordered N-tuple of maps should be considered rather than a single map.