# Generalized metric spaces endowed with vector-valued metrics and matrix equations by tripled fixed point theorems

## Abstract

This research focuses on proving the results of tripled fixed point and coincidence point in generalized metric spaces endowed with vector-valued metrics and matrix equations. The results from this study are illustrated by two applications.

## Introduction

Perov described the Banach contraction principle for contraction mappings on spaces equipped with vector-valued metrics in . Later, by a different method, the results of Perov in  were generalized and their fixed point property of a self-mapping over generalized metric space $$(X,d)$$ was studied.

In this article $$M_{m,m}(\mathbb{R^{+}})$$ represents the set of all $$m\times m$$ matrices with components in $$\mathbb{R^{+}}$$, Θ represents the matrix zero and I represents the identity matrix and $$\mathbb{N}_{0}=\mathbb{N}\cup \{0\}$$.

Let $$A\in M_{m,m}(\mathbb{R^{+}})$$, then A is called convergent to zero, if and only if $$A^{n}\rightarrow 0$$ as $$n\rightarrow \infty$$. We refer to [14, 15] for more details.

Let $$\alpha, \beta \in \mathbb{R}^{m}$$, where $$\alpha =(\alpha _{1}, \ldots, \alpha _{m})$$, $$\beta =(\beta _{1},\ldots, \beta _{m})$$ and $$c\in \mathbb{R}$$. Note that $$\alpha _{i}\leq \beta _{i}$$ (resp. $$\alpha _{i}<\beta _{i}$$) for each $$1\leq i\leq m$$ and also $$\alpha _{i}\leq c$$ (resp. $$\alpha _{i}< c$$) for $$1\leq i\leq m$$, respectively. We define

$$\alpha + \beta:= (\alpha _{1} + \beta _{1},\ldots, \alpha _{m} + \beta _{m})$$

and

$$\alpha \cdot \beta:= (\alpha _{1} \cdot \beta _{1},\ldots, \alpha _{m} \cdot \beta _{m}).$$

These are addition and multiplication on $$\mathbb{R}^{m}$$ (see [5, 7, 8]).

### Definition 1.1

()

Let X be a non-empty set. A mapping $$d:X^{2}\longrightarrow \mathbb{R}^{m}$$ is called a vector-valued metric on X, if the following properties hold:

1. (1)

$$d(x^{1}, x^{2})\geq 0$$ for each $$x^{1}, x^{2}\in X$$, if $$d(x^{1}, x^{2})=0$$, if and only if $$x^{1}=x^{2}$$;

2. (2)

$$d(x^{1},x^{2})=d(x^{2},x^{1})$$ for each $$x^{1},x^{2}\in X$$;

3. (3)

$$d(x^{1},x^{2})\leq d(x^{1}, x^{3})+ d(x^{3}, x^{2})$$ for each $$x^{1}, x^{2}, x^{3}\in X$$.

If $$x^{1}, x^{2} \in \mathbb{R}^{m}, x^{1} = (x^{1}_{1}, \ldots, x^{1}_{m})$$ and $$x^{2} = (x^{2}_{1}, \ldots, x^{2}_{m})$$, then $$x^{1} \leq x^{2}$$ if and only if $$x^{1}_{i} \leq x^{2}_{i}$$ for $$1 \leq i \leq m$$. A set X is called a generalized metric space, equipped with a vector-valued metric d and denoted by $$(X, d)$$.

Now, we need the following equivalent propositions. Their proofs are classic results in matrix analysis (see for more details [1, 12, 13]).

1. (1)

$$A\rightarrow 0$$;

2. (2)

$$A^{n}\rightarrow 0$$ as $$n\rightarrow \infty$$;

3. (3)

$$|\lambda |<1$$, for each $$\lambda \in \mathbb{C}$$ with $$\det (A-\lambda I)=0$$;

4. (4)

the matrix $$I-A$$ is nonsingular and

$$(I-A)^{-1}=I+A+\cdots +A^{n}+\cdots;$$
5. (5)

$$A^{n}q\longrightarrow 0$$ and $$qA^{n}\longrightarrow 0$$ as $$n\longrightarrow \infty$$, for each $$q\in \mathbb{R}^{m}$$.

Denote the set of all matrices $$A\in M_{m,m}(\mathbb{R^{+}})$$ where $$A^{n}\longrightarrow 0$$ by $$\mathcal{Z}M$$. For the sake of simplicity, we identify the row and column vectors in $$\mathbb{R}^{m}$$.

### Definition 1.2

()

An element $$(x^{1}, x^{2})\in X^{2}$$ is called a coupled fixed point of the mapping $$F:X^{2}\longrightarrow X$$ if $$F(x^{1}, x^{2})=x^{1}, F(x^{2}, x^{1})=x^{2}$$.

### Definition 1.3

()

Suppose that $$F:X^{2}\longrightarrow X$$ and $$g:X\longrightarrow X$$ are given. An element $$(x^{1}, x^{2})\in X^{2}$$ is called a coupled coincidence point of the mappings F and g if $$F(x^{1}, x^{2})=gx^{1}$$ and $$F(x^{2}, x^{1}) = gx^{2}$$. Then $$(gx^{1}, gx^{2})$$ is called a coupled coincidence point.

### Definition 1.4

()

Let $$(X, d,\preceq )$$ be a partially ordered complete metric space. We consider partially ordered set X. We define on $$X^{3}$$ the following order, for $$(x^{1}, x^{2}, x^{3}), (u^{1}, u^{2}, u^{3}) \in X^{3}$$,

$$\bigl(u^{1}, u^{2}, u^{3}\bigr) \preceq \bigl(x^{1}, x^{2}, x^{3}\bigr) \quad\Leftrightarrow\quad x^{1} \succeq u^{1}, \qquad x^{2} \preceq u^{2},\qquad x^{3} \succeq u^{3}.$$

### Definition 1.5

()

.Let $$(X,\preceq )$$ be a partially ordered set and $$F:X^{3}\rightarrow X$$. We say that F has the mixed monotone property if for any $$x^{1}, x^{2}, x^{3} \in X$$

\begin{aligned} &x^{1}_{1}, x^{1}_{2} \in X,\qquad x^{1}_{1} \preceq x^{1}_{2}\quad \Rightarrow\quad F \bigl(x^{1}_{1}, x^{2}, x^{3}\bigr) \preceq F\bigl(x^{1}_{2}, x^{2}, x^{3} \bigr), \\ &x^{2}_{1}, x^{2}_{2} \in X,\qquad x^{2}_{1} \preceq x^{2}_{2}\quad \Rightarrow \quad F \bigl(x^{1}, x^{2}_{1}, x^{3}\bigr) \succeq F\bigl(x^{1}, x^{2}_{2}, x^{3} \bigr), \\ &x^{3}_{1}, x^{3}_{2} \in X,\qquad x^{3}_{1} \preceq x^{3}_{2} \quad\Rightarrow \quad F \bigl(x^{1}, x^{2}, x^{3}_{1}\bigr) \preceq F\bigl(x^{1}, x^{2}, x^{3}_{2} \bigr), \end{aligned}

that is, $$F(x^{1}, x^{2}, x^{3})$$ is monotone non-decreasing in $$x^{1}$$ and $$x^{3}$$ and is monotone non-increasing in $$x^{2}$$.

Now, we present a triple fixed point of the second kind that used for mixed monotone mappings (see ).

### Definition 1.6

()

An element $$(x^{1}, x^{2}, x^{3})\in X^{3}$$ is called a triple fixed point of the mapping $$F:X^{3}\longrightarrow X$$ if

$$F\bigl(x^{1}, x^{2}, x^{3}\bigr)=x^{1},\qquad F\bigl(x^{2}, x^{1}, x^{2}\bigr)=x^{2},\qquad F \bigl(x^{3}, x^{2}, x^{1}\bigr)=x^{3}.$$

### Definition 1.7

()

Let $$(X, d)$$ be the complete generalized metric space. The mapping $$\overline{d}: X^{3}\rightarrow \mathbb{R}^{m}$$ with

$$\overline{d} \bigl[ \bigl(x^{1}, x^{2}, x^{3} \bigr), \bigl(u^{1}, u^{2}, u^{3}\bigr) \bigr] = d \bigl(x^{1},u^{1}\bigr) + d\bigl(x^{2}, u^{2}\bigr) + d\bigl(x^{3}, u^{3}\bigr)$$

defines a metric on $$X^{3}$$, which, for convenience, we denote by d, too.

### Definition 1.8

()

Let $$(X,\preceq )$$ be a partially ordered set, $$F:X^{3}\longrightarrow X$$ and $$g:X\longrightarrow X$$ be given. We say F has the g-mixed monotone property if for any $$x^{1}, x^{2}, x^{3} \in X$$,

\begin{aligned} &x^{1}_{1}, x^{1}_{2} \in X,\qquad gx^{1}_{1} \preceq gx^{1}_{2}\quad \Rightarrow\quad F\bigl(x^{1}_{1}, x^{2}, x^{3}\bigr) \preceq F\bigl(x^{1}_{2}, x^{2}, x^{3} \bigr), \\ &x^{2}_{1}, x^{2}_{2} \in X,\qquad gx^{2}_{1} \preceq gx^{2}_{2} \quad\Rightarrow\quad F\bigl(x^{1}, x^{2}_{1}, x^{3}\bigr) \succeq F\bigl(x^{1}, x^{2}_{2}, x^{3} \bigr), \\ &x^{3}_{1}, x^{3}_{2} \in X,\qquad gx^{3}_{1} \preceq gx^{3}_{2}\quad \Rightarrow\quad F\bigl(x^{1}, x^{2}, x^{3}_{1}\bigr) \preceq F\bigl(x^{1}, x^{2}, x^{3}_{2} \bigr), \end{aligned}

that is, $$F(x^{1}, x^{2}, x^{3})$$ is monotone non-decreasing in $$x^{1}$$ and $$x^{3}$$, and monotone non-increasing in $$x^{2}$$.

### Definition 1.9

()

Let $$F:X^{3}\longrightarrow X$$ and $$g:X\longrightarrow X$$ be given. F and g are called compatible if

$$\lim_{n\rightarrow +\infty }d\bigl(g(U_{123}), F(V_{123}) \bigr)= 0,$$

where $$U_{123}=F(x^{1}_{n}, x^{2}_{n},x^{3}_{n})$$ and $$V_{123}=(gx^{1}_{n}, gx^{2}_{n}, gx^{3}_{n})$$,

$$\lim_{n\rightarrow +\infty } d\bigl(g(U_{212}), F(V_{212}) \bigr)= 0,$$

where $$U_{212}=F(x^{2}_{n}, x^{1}_{n},x^{2}_{n})$$ and $$V_{212}=(gx^{2}_{n}, gx^{1}_{n}, gx^{2}_{n})$$,

$$\lim_{n\rightarrow +\infty } d\bigl(g(U_{321}), F(V_{321}) \bigr)= 0,$$

where $$U_{321}=F(x^{3}_{n}, x^{2}_{n}, x^{1}_{n})$$ and $$V_{321}=(gx^{3}_{n}, gx^{2}_{n}, gx^{1}_{n})$$,

whenever $$\{x^{1}_{n}\}, \{x^{2}_{n}\}$$, and $$\{x^{3}_{n}\}$$ are sequences in X, such that

\begin{aligned} &\lim_{n\rightarrow +\infty }U_{123}=\lim_{n\rightarrow + \infty }gx^{1}_{n}=x^{1}, \\ &\lim_{n\rightarrow +\infty }U_{212}=\lim_{n\rightarrow + \infty }gx^{2}_{n}= x^{2}, \\ &\lim_{n\rightarrow +\infty }U_{321}=\lim_{n\rightarrow + \infty }gx^{3}_{n}= x^{3}, \end{aligned}

for some $$x^{1}, x^{2}, x^{3} \in X$$.

### Definition 1.10

()

Let $$F:X^{3}\longrightarrow X$$ and $$g:X\longrightarrow X$$. The mappings F and g are called weakly reciprocally continuous if

\begin{aligned} &\lim_{n\rightarrow +\infty }g(U_{123})=gx^{1}\quad \text{or}\quad \lim _{n\rightarrow +\infty }F(V_{123})=F\bigl(x^{1}, x^{2}, x^{3}\bigr), \\ &\lim_{n\rightarrow +\infty }g(U_{212})=gx^{2} \quad\text{or}\quad\lim _{n\rightarrow +\infty }F(V_{212})=F\bigl(x^{2}, x^{1}, x^{2}\bigr), \\ &\lim_{n\rightarrow +\infty }g(U_{321})=gx^{3}\quad \text{or}\quad\lim _{n\rightarrow +\infty }F(V_{321})=F\bigl(x^{3}, x^{2}, x^{1}\bigr), \end{aligned}

whenever $$\{x^{1}_{n}\}, \{x^{2}_{n}\}$$, and $$\{x^{3}_{n}\}$$ are sequences in X, such that

\begin{aligned} &\lim_{n\rightarrow +\infty }U_{123}=\lim_{n \rightarrow +\infty }gx^{1}_{n}=x^{1}, \\ &\lim_{n\rightarrow +\infty }U_{212}=\lim_{n \rightarrow +\infty }gx^{2}_{n}= x^{2}, \\ &\lim_{n\rightarrow +\infty }U_{321}=\lim_{n \rightarrow +\infty }gx^{3}_{n}= x^{3}, \end{aligned}

for some $$x^{1}, x^{2}, x^{3} \in X$$.

### Definition 1.11

()

Let $$F:X^{3}\longrightarrow X$$ and $$g:X\longrightarrow X$$. The mappings F and g are called reciprocally continuous if

\begin{aligned} &\lim_{n\rightarrow +\infty }g(U_{123})=gx^{1}\quad\text{and}\quad\lim _{n \rightarrow +\infty }F(V_{123})=F\bigl(x^{1}, x^{2}, x^{3}\bigr), \\ &\lim_{n\rightarrow +\infty }g(U_{212})=gx^{2} \quad\text{and}\quad\lim _{n \rightarrow +\infty }F(V_{212})=F\bigl(x^{2}, x^{1}, x^{2}\bigr), \\ &\lim_{n\rightarrow +\infty }g(U_{321})=gx^{3} \quad\text{and}\quad\lim _{n \rightarrow +\infty }F(V_{321})=F\bigl(x^{3},x^{2}, x^{1}\bigr), \end{aligned}

whenever $$\{x^{1}_{n}\}, \{x^{2}_{n}\}$$, and $$\{x^{3}_{n}\}$$ are sequences in X, such that

\begin{aligned} &\lim_{n\rightarrow +\infty }U_{123}=\lim_{n\rightarrow + \infty }gx^{1}_{n}=x^{1}, \\ &\lim_{n\rightarrow +\infty }U_{212}=\lim_{n\rightarrow + \infty }gx^{2}_{n}= x^{2}, \\ &\lim_{n\rightarrow +\infty }U_{321}=\lim_{n\rightarrow + \infty }gx^{3}_{n}= x^{3}, \end{aligned}

for some $$x^{1}, x^{2}, x^{3} \in X$$.

### Definition 1.12

()

Let $$(X, d,\preceq )$$ be a partially ordered metric space. We say that X is regular if the following properties hold:

1. (i)

if a non-decreasing sequence $$x^{1}_{n}\rightarrow x^{1}$$, then $$x^{1}_{n} \preceq x^{1}$$ for all $$n \geq 0$$,

2. (ii)

if a non-increasing sequence $$x^{2}_{n}\rightarrow x^{2}$$, then $$x^{2} \preceq x^{2}_{n}$$ for all $$n \geq 0$$.

For the main result of this article, we study existence and uniqueness of triple common fixed point for a sequence of mappings $$T_{n}: X^{3} \rightarrow X$$ and $$g:X\rightarrow X$$, where $$(X, d)$$ is a complete generalized metric space.

First, we have the following two definitions from [6, 15].

### Definition 1.13

()

Let $$(X, d)$$ be a metric space and let $$T_{n}:X^{3}\rightarrow X$$ and $$g:X\longrightarrow X$$ are given. The sequence $$\{T_{n}\}_{n\in \mathbb{N}_{0}}$$ and the mapping g are said to be compatible if

$$\lim_{n\rightarrow +\infty } d\bigl(g\bigl(U'_{123}\bigr), T_{n}\bigl(V'_{123}\bigr)\bigr)= 0,$$

where $$U'_{123}=T_{n}(x^{1}_{n}, x^{2}_{n},x^{3}_{n})$$ and $$V'_{123}=(gx^{1}_{n}, gx^{2}_{n}, gx^{3}_{n})$$

$$\lim_{n\rightarrow +\infty } d\bigl(g\bigl(U'_{212}\bigr), T_{n}\bigl(V'_{212}\bigr)\bigr)= 0,$$

where $$U'_{212}=T_{n}(x^{2}_{n}, x^{1}_{n},x^{2}_{n})$$ and $$V'_{212}=(gx^{2}_{n}, gx^{1}_{n}, gx^{2}_{n})$$

$$\lim_{n\rightarrow +\infty } d\bigl(g\bigl(U'_{321}\bigr), T_{n}\bigl(V'_{321}\bigr)\bigr)= 0,$$

where $$U'_{321}=T_{n}(x^{3}_{n}, x^{2}_{n}, x^{1}_{n})$$ and $$V'_{321}=(gx^{3}_{n}, gx^{2}_{n}, gx^{1}_{n})$$, whenever $$\{x^{1}_{n}\}, \{x^{2}_{n}\}$$, and $$\{x^{3}_{n}\}$$ are sequences in X, such that

\begin{aligned} &\lim_{n\rightarrow +\infty }U'_{123}=\lim _{n \rightarrow +\infty }gx^{1}_{n+1}=x^{1}, \\ &\lim_{n\rightarrow +\infty }U'_{212}=\lim _{n \rightarrow +\infty }gx^{2}_{n+1}= x^{2}, \\ &\lim_{n\rightarrow +\infty }U'_{321}=\lim _{n \rightarrow +\infty }gx^{3}_{n+1}= x^{3} \end{aligned}

for some $$x^{1}, x^{2}, x^{3} \in X$$.

### Definition 1.14

()

Let $$(X, d)$$ be a metric space and let $$T_{n}:X^{3}\rightarrow X$$ and $$g:X\longrightarrow X$$ are given. $$\{T_{n}\}_{n\in \mathbb{N}_{0}}$$ and g are called weakly reciprocally continuous if

\begin{aligned} &\lim_{n\rightarrow +\infty }g\bigl(U'_{123} \bigr)=gx^{1}, \\ &\lim_{n\rightarrow +\infty }g\bigl(U'_{212} \bigr)=gx^{2}, \\ &\lim_{n\rightarrow +\infty }g\bigl(U'_{321} \bigr)=gx^{3}, \end{aligned}

whenever $$\{x^{1}_{n}\}, \{x^{2}_{n}\}$$, and $$\{x^{3}_{n}\}$$ are sequences in X, such that

\begin{aligned} \lim_{n\rightarrow +\infty }U'_{123}=\lim _{n \rightarrow +\infty }gx^{1}_{n+1}=x^{1}, \\ \lim_{n\rightarrow +\infty }U'_{212}=\lim _{n \rightarrow +\infty }gx^{2}_{n+1}= x^{2}, \\ \lim_{n\rightarrow +\infty }U'_{321}=\lim _{n \rightarrow +\infty }gx^{3}_{n+1}= x^{3} \end{aligned}

for some $$x^{1}, x^{2}, x^{3} \in X$$.

## Main results

We start with the following statement, which we will use in the main theorem. Inspired by Definition 1.8 we have the following definition.

### Definition 2.1

Let $$(X,\preceq )$$ be a partially ordered set, $$T_{n}:X^{3}\rightarrow X, n\in \mathbb{N}_{0}$$, and $$g:X\rightarrow X$$. We say that $$\{T_{n}\}_{n\in \mathbb{N}_{0}}$$ has the g-mixed monotone property if for any $$x^{1}, x^{2}, x^{3}, x^{\prime 1}, x^{\prime 2}, x^{\prime 3}\in X$$,

\begin{aligned} gx^{1}\preceq gx^{\prime 1},\qquad gx^{\prime 2} \preceq gx^{2} \quad\textit{and}\quad gx^{3} \preceq gx^{\prime 3}, \end{aligned}
(2.1)

imply that

\begin{aligned} \begin{aligned} &T_{n}\bigl(x^{1}, x^{2}, x^{3}\bigr) \preceq T_{n+1}\bigl(x^{\prime 1}, x^{\prime 2}, x^{\prime 3}\bigr),\qquad T_{n+1}\bigl(x^{\prime 2}, x^{\prime 1}, x^{\prime 2}\bigr) \preceq T_{n} \bigl(x^{2}, x^{1}, x^{2}\bigr) \quad\textit{and} \\ &T_{n+1}\bigl(x^{3}, x^{2}, x^{1}\bigr) \preceq T_{n}\bigl(x^{\prime 3}, x^{\prime 2}, x^{\prime 1} \bigr). \end{aligned} \end{aligned}
(2.2)

### Definition 2.2

Suppose that $$T_{i}:X^{3}\rightarrow X$$ and $$g:X\rightarrow X$$ are given. We say $$\{T_{i}\}_{i\in \mathbb{N}_{0}}$$ and g satisfy the $$( K )$$ property if

\begin{aligned} d(T_{i}\bigl(x^{1}, x^{2}, x^{3}\bigr),T_{j}\bigl(u^{1}, u^{2}, u^{3}\bigr)\leq {}&A\bigl[ d\bigl( g \bigl(x^{1}\bigr),T_{i}\bigl(x^{1}, x^{2}, x^{3}\bigr)\bigr) \\ &{}+ d\bigl(gu^{1},T_{j}\bigl(u^{1}, u^{2}, u^{3}\bigr)\bigr)\bigr] \\ &{}+ B\bigl( d \bigl(gu^{1}, gx^{1}\bigr)\bigr) \end{aligned}
(2.3)

for $$x^{1}, x^{2}, x^{3}, u^{1}, u^{2}, u^{3} \in X$$ with $$gx^{1} \succeq gu^{1}, gu^{2} \succeq gx^{2}, gx^{3} \succeq gu^{3}$$ or $$gx^{1} \preceq gu^{1}, gu^{2} \preceq gx^{2}, gx^{3} \preceq gu^{3}$$, $$I\neq A = (a_{ij}), I\neq B = (b_{ij}) \in M_{m, m}(\mathbb{R}^{+})$$, $$(A + B)(I-A)^{-1} \in \mathcal{Z}M$$.

### Definition 2.3

If $$T_{0}$$ and g have a non-decreasing transcendence point in $$x^{1}_{0}, x^{3}_{0}$$ and a non-increasing transcendence point in $$x^{2}_{0}$$, then we say $$T_{0}$$ and g have a mixed triple transcendence point, if there exist $$x^{1}_{0}, x^{2}_{0}, x^{3}_{0} \in X$$ such that

\begin{aligned} T_{0}\bigl(x^{1}_{0}, x^{2}_{0}, x^{3}_{0}\bigr)\succeq gx^{1}_{0},\qquad T_{0}\bigl(x^{2}_{0}, x^{1}_{0}, x^{2}_{0}\bigr)\preceq g x^{2}_{0}\quad \textit{and}\quad T_{0}\bigl(x^{3}_{0}, x^{1}_{0}, x^{2}_{0}\bigr)\succeq g x^{3}_{0}. \end{aligned}
(2.4)

### Lemma 2.4

Let$$(X, d, \preceq )$$be a partially ordered complete generalized metric space. Letgbe a self-mapping onXand$$\{T_{i}\}_{i\in \mathbb{N}_{0}}$$be a sequence of mappings from$$X^{3}$$intoXand having ag-mixed monotone property with$$T_{i}(X^{3}) \subseteq g(X)$$. If$$T_{0}$$andghave a mixed triple transcendence point, then

1. (a)

there are sequences$$\{x^{1}_{n}\}, \{x^{2}_{n}\}$$and$$\{x^{3}_{n}\}$$inXsuch that

\begin{aligned} &gx^{1}_{n}= T_{n-1} \bigl(x^{1}_{n-1}, x^{2}_{n-1}, x^{3}_{n-1}\bigr),\qquad gx^{2}_{n} = T_{n-1} \bigl(x^{2}_{n-1}, x^{1}_{n-1}, x^{2}_{n-1}\bigr)\quad\textit{and} \\ & gx^{3}_{n} = T_{n-1} \bigl(x^{3}_{n-1}, x^{1}_{n-1}, x^{2}_{n-1}\bigr); \end{aligned}
2. (b)

$$\{gx^{1}_{n}\}, \{gx^{3}_{n}\}$$are non-decreasing sequences and$$\{gx^{2}_{n}\}$$is a non-increasing sequence.

### Proof

(a) By hypothesis, let for $$x^{1}_{0}, x^{2}_{0}, x^{3}_{0} \in X$$ the condition (2.4) hold. Since $$T_{0}(X^{3}) \subseteq g(X)$$, we can define $$x^{1}_{1}, x^{2}_{1}, x^{3}_{1}\in X$$ such that

\begin{aligned} &gx^{1}_{1} = T_{0}\bigl(x^{1}_{0}, x^{2}_{0}, x^{3}_{0}\bigr),\qquad gx^{2}_{1} = T_{0}\bigl(x^{2}_{0}, x^{1}_{0}, x^{2}_{0}\bigr)\quad \textit{and} \\ &gx^{3}_{1} = T_{0}\bigl(x^{3}_{0}, x^{2}_{0}, x^{1}_{0}\bigr). \end{aligned}

Again since $$T_{0}(X^{3}) \subseteq g(X)$$, there exist $$x^{1}_{2}, x^{2}_{2}, x^{3}_{2} \in X$$ such that

\begin{aligned} &gx^{1}_{2}= T_{1}\bigl(x^{1}_{1}, x^{2}_{1}, x^{3}_{1}\bigr),\qquad gx^{2}_{2} = T_{1}\bigl(x^{2}_{1}, x^{1}_{1}, x^{2}_{1}\bigr) \quad\textit{and} \\ &gx^{3}_{2}= T_{1}\bigl(x^{3}_{1}, x^{2}_{1}, x^{1}_{1}\bigr). \end{aligned}

Continuing this technique, we get

\begin{aligned} \begin{aligned} &gx^{1}_{n} = T_{n-1} \bigl(x^{1}_{n-1}, x^{2}_{n-1}, x^{3}_{n-1}\bigr),\qquad gx^{2}_{n} = T_{n}\bigl(x^{2}_{n-1}, x^{1}_{n-1}, x^{2}_{n-1}\bigr)\quad \textit{and} \\ &gx^{3}_{n} = T_{n}\bigl(x^{3}_{n-1}, x^{2}_{n-1}, x^{1}_{n-1}\bigr),\quad \textit{for all } n \geq 0. \end{aligned} \end{aligned}
(2.5)

(b) Now, by mathematical induction, we show that

\begin{aligned} gx^{1}_{n} \preceq gx^{1}_{n+1},\qquad gx^{2}_{n}\succeq gx^{2}_{n+1}\quad \textit{and}\quad gx^{3}_{n} \preceq gx^{3}_{n+1}, \end{aligned}
(2.6)

for all $$n \geq 0$$. To this end, since (2.4) holds, in the light of

$$gx^{1}_{1} = T_{0}\bigl(x^{1}_{0}, x^{2}_{0}, x^{3}_{0}\bigr),\qquad gx^{2}_{1}= T_{0}\bigl(x^{2}_{0}, x^{1}_{0}, x^{3}_{0}\bigr) \quad\textit{and}\quad gx^{3}_{1}= T_{0}\bigl(x^{3}_{0}, x^{2}_{0}, x^{1}_{0}\bigr),$$

we have

$$gx^{1}_{0}\preceq gx^{1}_{1},\qquad gx^{2}_{0}\succeq gx^{2}_{1},\qquad gx^{3}_{0} \preceq gx^{3}_{1},$$

that is, (2.6) holds for $$n = 0$$. We assume that (2.6) holds for some $$n > 0$$. Now, by (2.5) and (2.6), the result is achieved. Thus, we are done. □

Before expressing the main theorems, we first give the following examples.

### Example 2.5

1. 1.

$A= 1 4 ( 1 1 1 1 )$ and $B= 1 6 ( 1 1 1 1 )$ are matrices in $$\mathcal{Z}M$$. It is easy to see that $$(A+B)(I-A)^{-1}\in \mathcal{Z}M$$.

2. 2.

$A= ( 1 3 0 0 1 3 )$ and $B= ( 0 1 3 1 3 0 )$ are matrices in $$\mathcal{Z}M$$. It is easy to see that $$(A+B)(I-A)^{-1}\in \mathcal{Z}M$$.

3. 3.

Let $$A=\alpha I$$ and $$B=((I-\alpha )^{3}-\alpha )I$$ be matrices in $$\mathcal{Z}M$$. Then for $$\alpha = \frac{1}{4}, \frac{1}{5}, \frac{1}{7}, \frac{1}{8}$$ it is clear that $$(A+B)(I-A)^{-1}\in \mathcal{Z}M$$.

### Theorem 2.6

In addition to the conditions of Lemma 2.4, let$$g(X)\subseteq X$$be complete, $$\{T_{i}\}_{i\in \mathbb{N}_{0}}$$andgbe compatible, weakly reciprocally continuous, wheregis monotonic non-decreasing, continuous, and satisfies the condition (K). If$$g(X)$$is regular and$$A, B$$are nonzero matrices in$$\mathcal{Z}M$$, then$$\{T_{i}\}_{i\in \mathbb{N}_{0}}$$andghave a triple coincidence point.

### Proof

Let $$\{x^{1}_{n}\}, \{x^{2}_{n}\}\textit{ and }\{x^{3}_{n}\}$$ be the same sequences which are constructed in Lemma 2.4. By (2.3), we get

\begin{aligned} d\bigl(gx^{1}_{n}, gx^{1}_{n+1}\bigr) ={}& d\bigl(T_{n-1}\bigl(x^{1}_{n-1}, x^{2}_{n-1}, x^{3}_{n-1}\bigr),T_{n}\bigl(x^{1}_{n}, x^{2}_{n},x^{3}_{n}\bigr)\bigr) \\ \leq{}& A\bigl[d\bigl(gx^{1}_{n-1},T_{n-1} \bigl(x^{1}_{n-1}, x^{2}_{n-1}, x^{3}_{n-1}\bigr)\bigr) \\ &{}+ d\bigl(gx^{1}_{n},T_{n}\bigl(x^{1}_{n}, x^{2}_{n},x^{3}_{n}\bigr)\bigr)\bigr]+ B(d\bigl(gx^{1}_{n}, gx^{1}_{n-1}\bigr) \\ ={}& A\bigl[d\bigl(gx^{1}_{n-1}, gx^{1}_{n} \bigr)+ d\bigl(gx^{1}_{n}, gx^{1}_{n+1} \bigr)\bigr] \\ &{}+ B(d\bigl(gx^{1}_{n}, gx^{1}_{n-1} \bigr). \end{aligned}

It follows that

\begin{aligned} d\bigl(gx^{1}_{n}, gx^{1}_{n+1} \bigr) \leq (A + B) (I-A)^{-1}d\bigl(gx^{1}_{n-1}, gx^{1}_{n}\bigr) \end{aligned}
(2.7)

and similarly

\begin{aligned} d\bigl(gx^{2}_{n}, gx^{2}_{n+1} \bigr) \leq (A + B) (I-A)^{-1}d\bigl(gx^{2}_{n-1}, gx^{2}_{n}\bigr) \end{aligned}
(2.8)

and

\begin{aligned} d\bigl(gx^{3}_{n}, gx^{3}_{n+1} \bigr) \leq (A + B) (I-A)^{-1}d\bigl(gx^{3}_{n-1}, gx^{3}_{n}\bigr). \end{aligned}
(2.9)

\begin{aligned} \delta _{n}:={}&d\bigl(gx^{1}_{n}, gx^{1}_{n+1}\bigr)+ d\bigl(gx^{2}_{n}, gx^{2}_{n+1}\bigr) + d\bigl(gx^{3}_{n}, gx^{3}_{n+1}\bigr) \\ \leq {}&(A + B) (I-A)^{-1}\bigl[d\bigl(gx^{1}_{n-1}, gx^{1}_{n}\bigr)+d\bigl(gx^{2}_{n-1}, gx^{2}_{n}\bigr) \\ &{}+d\bigl(gx^{3}_{n-1}, gx^{3}_{n}\bigr) \bigr] \\ ={}&\bigl((A + B) (I-A)^{-1}\bigr)\delta _{n-1}. \end{aligned}

We set $$C = (A + B)(I-A)^{-1}$$, for all $$n \in \mathbb{N}$$, then

$$\varTheta \leq \delta _{n} \leq C\delta _{n-1} \leq C^{2}\delta _{n-2} \leq \cdots \leq C^{n}\delta _{0}.$$

Moreover, with repeated use of the triangle inequality and for $$p > \varTheta$$, we get

\begin{aligned} &d\bigl(gx^{1}_{n}, gx^{1}_{n+p}\bigr) + d\bigl(gx^{2}_{n}, gx^{2}_{n+p}\bigr) + d \bigl(gx^{3}_{n}, gx^{3}_{n+p}\bigr) \\ &\quad\leq d\bigl(gx^{1}_{n}, gx^{1}_{n+1} \bigr) + d\bigl(gx^{2}_{n}, gx^{2}_{n+1} \bigr) + d\bigl(gx^{3}_{n}, gx^{3}_{n+1} \bigr) \\ &\qquad{}+d\bigl(gx^{1}_{n+1}, gx^{1}_{n+2}\bigr) + d\bigl(gx^{2}_{n+1}, gx^{2}_{n+2}\bigr) + d\bigl(gx^{3}_{n+1}, gx^{3}_{n+2}\bigr) \\ &\qquad{}+\cdots +d\bigl(gx^{1}_{n+p-1}, gx^{1}_{n+p} \bigr) + d\bigl(gx^{2}_{n+p-1}, gx^{2}_{n+p} \bigr) \\ &\qquad{}+ d\bigl(gx^{3}_{n+p-1}, gx^{3}_{n+p}\bigr) \\ &\quad=\delta _{n}+\delta _{n+1}+\cdots +\delta _{n+p-1} \\ &\quad\leq \bigl(C^{n}+C^{n+1}+\cdots +C^{n+p-1}\bigr) \delta _{0} \\ &\quad\leq C^{n}\bigl(I+C+\cdots +C^{p-1}+\cdots \bigr)\delta _{0} \\ &\quad=C^{n}(I-C)^{-1}\delta _{0}. \end{aligned}

We have

\begin{aligned} &d\bigl(gx^{1}_{n}, gx^{1}_{n+p}\bigr) + d\bigl(gx^{2}_{n}, gx^{2}_{n+p}\bigr) + d \bigl(gx^{3}_{n}, gx^{3}_{n+p}\bigr) \\ &\quad\leq \bigl((A + B) (I-A)^{-1}\bigr)^{n}\bigl(I-(A+B) (I-A)^{-1}\bigr)^{-1}\delta _{0}. \end{aligned}

Now, taking the limit as $$n\rightarrow +\infty$$, we conclude

$$\lim_{n\rightarrow +\infty }d\bigl(gx^{1}_{n}, gx^{1}_{n+p}\bigr) + d\bigl(gx^{2}_{n}, gx^{2}_{n+p}\bigr) + d\bigl(gx^{3}_{n}, gx^{3}_{n+p}\bigr)=0.$$

This implies that

\begin{aligned} \lim_{n\rightarrow +\infty }d\bigl(gx^{1}_{n}, gx^{1}_{n+p}\bigr) = \lim_{n\rightarrow +\infty } d \bigl(gx^{2}_{n}, gx^{2}_{n+p}\bigr)= \lim _{n\rightarrow +\infty } d\bigl(gx^{3}_{n}, gx^{3}_{n+p}\bigr)=0. \end{aligned}

Thus, $$\{gx^{1}_{n}\}, \{gx^{2}_{n}\}$$ and $$\{gx^{3}_{n}\}$$ are Cauchy sequences in X. Since $$g(X)$$ is complete, there exists $$(x^{\prime 1}, x^{\prime 2}, x^{\prime 3}) \in X ^{3}$$, with

\begin{aligned} &\lim_{n\rightarrow +\infty }\bigl\{ gx^{1}_{n}\bigr\} =gx^{\prime 1}:= x^{1},\qquad\lim_{n \rightarrow +\infty } \{gy_{n}\}=gx^{\prime 2}:= x^{2}\quad\textit{and} \\ &\lim_{n\rightarrow +\infty }\{gz_{n}\}= gx^{\prime 3}:= x^{3}. \end{aligned}

By construction, we have

\begin{aligned} &\lim_{n\rightarrow +\infty }gx^{1}_{n+1}= \lim _{n\rightarrow + \infty }T_{n}\bigl(x^{1}_{n}, x^{2}_{n},x^{3}_{n}\bigr)=x^{1}, \\ &\lim_{n\rightarrow +\infty }gx^{2}_{n+1} = \lim _{n\rightarrow + \infty }T_{n}\bigl(x^{2}_{n}, x^{1}_{n},x^{2}_{n}\bigr)=x^{2}, \end{aligned}

and

\begin{aligned} \lim_{n\rightarrow +\infty }gx^{3}_{n+1}=\lim _{n\rightarrow +\infty }T_{n}\bigl(x^{3}_{n}, x^{2}_{n},x^{1}_{n}\bigr)=x^{3}. \end{aligned}

Since $$\{T_{i}\}_{i\in \mathbb{N}_{0}}$$ and g are weakly reciprocally continuous and compatible, we have

\begin{aligned} &\lim_{n\rightarrow +\infty }T_{n}\bigl(gx^{1}_{n}, gx^{2}_{n}, gx^{3}_{n}\bigr)= gx^{1}, \\ &\lim_{n\rightarrow +\infty }T_{n}\bigl(gx^{2}_{n}, gx^{1}_{n}, gx^{2}_{n}\bigr)= gx^{2}, \end{aligned}

and

\begin{aligned} \lim_{n\rightarrow +\infty } T_{n}\bigl(gx^{3}_{n}, gx^{2}_{n}, gx^{1}_{n}\bigr)= gx^{3}. \end{aligned}

Since $$\{gx^{1}_{n}\}$$ and $$\{gx^{3}_{n}\}$$ are non-decreasing and $$\{gx^{2}_{n}\}$$ is non-increasing, using the regularity of X, we have $$gx^{1}_{n} \preceq x^{1}, x^{2} \preceq gx^{2}_{n}$$ and $$gx^{3}_{n} \preceq x^{3}$$ for all $$n \geq 0$$. So by (2.3), we get

\begin{aligned} d(T_{i}\bigl(x^{1}, x^{2}, x^{3} \bigr),T_{n}\bigl(gx^{1}_{n}, gx^{2}_{n}, gx^{3}_{n}\bigr) \leq{}& A\bigl[ d\bigl( gx^{1},T_{i} \bigl(x^{1}, x^{2}, x^{3}\bigr)\bigr) \\ &{}+ d(g\bigl(gx^{1}_{n},T_{n}\bigl(gx^{1}_{n}, gx^{2}_{n}, gx^{3}_{n}\bigr)\bigr)\bigr] \\ &{}+ B( d \bigl(g\bigl(gx^{1}_{n}, gx^{1}\bigr) \bigr). \end{aligned}

Taking the limit as $$n\rightarrow +\infty$$, we obtain $$gx^{1}=T_{i}(x^{1}, x^{2}, x^{3})$$. Similarly, it can be proved that $$gx^{2}=T_{i}(x^{2}, x^{1}, x^{2})$$ and $$gx^{3} =T_{i}(x^{3}, x^{2}, x^{1})$$. Thus, $$(x^{1}, x^{2}, x^{3})$$ is a triple coincidence point of $$\{T_{i}\}_{i \in \mathbb{N}}$$ and g. □

If in Theorem 2.6g is the identity mapping, then we have the following corollary.

### Corollary 2.7

Let$$(X, d, \preceq )$$be a partially ordered complete generalized metric space. Let$$\{T_{i}\}_{i\in \mathbb{N}\cup \{0\}}$$be a mixed monotone sequence of mappings from$$X^{3}$$intoX, where$$\{T_{m}\}$$and$$\mathrm{Id}: X \rightarrow X$$satisfy the$$(K)$$property. Also$$T_{0}$$andIdhave a mixed transcendence point. If$$g(X)$$is regular, then there exists$$(x^{1}, x^{2}, x^{3}) \in X^{3}$$, such that$$x^{1} = T_{i}(x^{1}, x^{2}, x^{3}), x^{2}= T_{i}(x^{2}, x^{1}, x^{2})$$, and$$x^{3}= T_{i}(x^{3}, x^{2}, x^{1})$$for$$i \in \mathbb{N}_{0}$$.

### Definition 2.8

We say that $$(x^{1}, x^{2}, x^{3})$$ is a triple comparable with $$(u^{1}, u^{2}, u^{3})$$ if and only if

\begin{aligned} &x^{1}\succeq u^{1},\qquad x^{2}\preceq u^{2},\qquad x^{3}\succeq u^{3} \quad\textit{or} \\ &x^{1}\preceq u^{1},\qquad x^{2}\succeq u^{2},\qquad x^{3}\preceq u^{3} \quad\textit{or} \\ &x^{1}\succeq u^{2},\qquad x^{2}\preceq u^{3},\qquad x^{3}\succeq u^{1} \quad\textit{or} \\ &x^{1}\preceq u^{2},\qquad x^{2}\succeq u^{3},\qquad x^{3}\preceq u^{1}\quad \textit{or} \\ &x^{1}\succeq u^{3},\qquad x^{2}\preceq u^{1},\qquad x^{3}\succeq u^{2} \quad\textit{or} \\ & x^{1}\preceq u^{3},\qquad x^{2}\succeq u^{1},\qquad x^{3}\preceq u^{2}. \end{aligned}

If in the above definition we replace $$(x^{1}, x^{2}, x^{3} )$$ and $$(u^{1}, u^{2}, u^{3})$$ with $$( gx^{1}, gx^{2}, gx^{3})$$ and $$(gu^{1}, gu^{2}, gu^{3})$$, we call $$(x^{1}, x^{2}, x^{3})$$ a triple comparable with $$(u^{1}, u^{2}, u^{3})$$ with respect to g.

### Theorem 2.9

Let$$(X, d, \preceq )$$be a partially ordered complete generalized metric space. Letgbe a self-mapping onXand$$\{T_{i}\}_{i\in \mathbb{N}_{0}}$$be a sequence of mappings from$$X^{3}$$intoX. Let$$\{T_{i}\}_{i\in \mathbb{N}_{0}}$$andgsatisfy the condition (K) and have triple coincidence points comparable with respect tog, then$$\{T_{i}\}_{i \in \mathbb{N}_{0}}$$andghave a unique triple common fixed point.

### Proof

According to Theorem 2.6, the set of tripled coincidence points is non-empty. First, we show that, if $$(x^{1}, x^{2}, x^{3})$$ and $$(x^{\prime 1}, x^{\prime 2}, x^{\prime 3})$$ are triple coincidence points, that is, if

\begin{aligned} &gx^{1} = T_{i}\bigl(x^{1}, x^{2}, x^{3}\bigr),\qquad gx^{2} = T_{i}\bigl(x^{2}, x^{1}, x^{2}\bigr),\qquad gx^{3}= T_{i} \bigl(x^{3}, x^{2}, x^{1}\bigr), \\ &gx^{\prime 1}= T_{i}\bigl(x^{\prime 1}, x^{\prime 2}, x^{\prime 3}\bigr),\qquad gx^{\prime 2} = T_{i}\bigl(x^{\prime 2}, x^{\prime 1}, x^{\prime 2}\bigr),\qquad gx^{\prime 3}= T_{i} \bigl(x^{\prime 3}, x^{\prime 2}, x^{\prime 1}\bigr), \end{aligned}

then $$gx^{1} = gx^{\prime 1}, gx^{2} = gx^{\prime 2}$$ and $$gx^{3}=gx^{\prime 3}$$. Since the set of triple coincidence points is a triple comparable, applying condition (2.3) implies

\begin{aligned} d\bigl(gx^{1}, gx^{\prime 1}\bigr)={}& d\bigl(T_{i} \bigl(x^{1}, x^{2}, x^{3}\bigr),T_{j} \bigl(x^{\prime 1}, x^{\prime 2}, x^{\prime 3}\bigr)\bigr) \\ \leq{}& A\bigl[ d\bigl( gx^{1},T_{i}\bigl(x^{1}, x^{2}, x^{3}\bigr)\bigr)+ d\bigl(gx^{\prime 1},T_{j} \bigl(x^{\prime 1}, x^{\prime 2}, x^{\prime 3}\bigr)\bigr)\bigr] \\ &{}+ Bd \bigl(gx^{\prime 1}, gx^{1}\bigr). \end{aligned}

Therefore, as $$I\neq B\in \mathcal{Z}M$$, $$d(gx^{1}, gx^{\prime 1}) = \varTheta$$, that is, $$gx^{1} = gx^{\prime 1}$$. Similarly, it can be proved that $$gx^{2} = gx^{\prime 2}$$ and $$gx^{3} = gx^{\prime 3}$$. So $$gx^{1} =gx^{2} = gx^{3} =gx^{\prime 1}= gx^{\prime 2}=gx^{\prime 3}$$.

Therefore, $$\{T_{i}\}_{i \in \mathbb{N}}$$ and g have a unique triple coincidence point $$(gx^{1}, gx^{1})$$. Since two compatible mappings commute at their coincidence points, thus, clearly, $$\{T_{i}\}_{i \in \mathbb{N}}$$ and g have a unique tripled common fixed point whenever $$\{T_{i}\}_{i \in \mathbb{N}}$$ and g are weakly compatible. □

### Example 2.10

Let $$X = [0, 1]$$. Define

$$d\bigl(x^{1}, x^{2}\bigr)= \begin{pmatrix} \vert x^{1}-x^{2} \vert \\ \vert x^{1}-x^{2} \vert \end{pmatrix}.$$

Then $$(X, d)$$ is a partially ordered complete generalized metric space. Define

$$A= \begin{pmatrix} \frac{1}{3} & 0 \\ 0 & \frac{1}{3} \end{pmatrix} \quad\textit{and}\quad B= \begin{pmatrix} 0 & \frac{1}{3} \\ \frac{1}{3} & 0 \end{pmatrix}.$$

Because A and B are nonzero matrices in $$\mathcal{Z}M$$ and considering the mapping $$T_{i}: X^{3} \rightarrow X$$ and $$g:X\rightarrow X$$ with

$$T_{i}\bigl(x^{1},x^{2}, x^{3}\bigr) = \frac{x^{1}+x^{2}+ x^{3}}{3^{i}},\qquad g\bigl(x^{1}\bigr)=9x^{1},$$

it can be easily verified by mathematical induction that the inequality (2.3) holds for all $$x^{1}, x^{2}, x^{3} \in X$$, that is, we see that the greatest value of the first side happens when $$i=1, j\rightarrow \infty$$, in this case for $$i=1$$ we have

\begin{aligned} & \begin{pmatrix} \vert \frac{x^{1}+x^{2}+x^{3}}{3}- \frac{u^{1}+u^{2}+ u^{3}}{3^{j}} \vert \\ \vert \frac{x^{1}+x^{2}+x^{3}}{3}- \frac{u^{1}+u^{2}+ u^{3}}{3^{j}} \vert \end{pmatrix} \\ &\quad \leq \begin{pmatrix} \frac{1}{3}& 0 \\ 0 & \frac{1}{3} \end{pmatrix} \begin{pmatrix} \vert 9x^{1}-\frac{x^{1}+x^{2}+x^{3}}{3} \vert + \vert 9u^{1}- \frac{u^{1}+u^{2}+ u^{3}}{3^{j}} \vert \\ \vert 9x^{1}-\frac{x^{1}+x^{2}+x^{3}}{3} \vert + \vert 9u^{1}- \frac{u^{1}+u^{2}+ u^{3}}{3^{j}} \vert \end{pmatrix} \\ & \qquad{}+ \begin{pmatrix} 0 & \frac{1}{3} \\ \frac{1}{3}& 0 \end{pmatrix} \begin{pmatrix} \vert 9(u^{1}-x^{1}) \vert \\ \vert 9(u^{1}-x^{1}) \vert \end{pmatrix}. \end{aligned}

Now for $$j=j+1$$ we have

\begin{aligned} \alpha:={}& \begin{pmatrix} \vert \frac{x^{1}+x^{2}+x^{3}}{3}- \frac{1}{3} \frac{u^{1}+u^{2}+ u^{3}}{3^{j}} \vert \\ \vert \frac{x^{1}+x^{2}+x^{3}}{3}- \frac{1}{3} \frac{u^{1}+u^{2}+ u^{3}}{3^{j}} \vert \end{pmatrix} \\ \leq{}& \begin{pmatrix} \frac{1}{3}& 0 \\ 0 & \frac{1}{3} \end{pmatrix} \begin{pmatrix} \vert 9x^{1}-\frac{x^{1}+x^{2}+x^{3}}{3} \vert + \vert 3u^{1}- \frac{1}{3}\frac{u^{1}+u^{2}+ u^{3}}{3^{j}} \vert \\ \vert 9x^{1}-\frac{x^{1}+x^{2}+x^{3}}{3} \vert + \vert 3u^{1}- \frac{1}{3}\frac{u^{1}+u^{2}+ u^{3}}{3^{j}} \vert \end{pmatrix} \\ &{}+ \begin{pmatrix} 0 & 3 \\ 3 & 0 \end{pmatrix} \begin{pmatrix} \vert (\frac{u^{1}}{3}-x^{1}) \vert \\ \vert (\frac{u^{1}}{3}-x^{1}) \vert \end{pmatrix}:=\beta. \end{aligned}

So

\begin{aligned} \alpha \leq{}& \frac{1}{3} \begin{pmatrix} \vert \frac{x^{1}+x^{2}+x^{3}}{3}- \frac{u^{1}+u^{2}+ u^{3}}{3^{j}} \vert \\ \vert \frac{x^{1}+x^{2}+x^{3}}{3}- \frac{u^{1}+u^{2}+ u^{3}}{3^{j}} \vert \end{pmatrix}+\frac{2}{3} \begin{pmatrix} \vert \frac{x^{1}+x^{2}+x^{3}}{3} \vert \\ \vert \frac{x^{1}+x^{2}+x^{3}}{3} \vert \end{pmatrix} \\ \leq{}& \frac{1}{3} \begin{pmatrix} \frac{1}{3}& 0 \\ 0 & \frac{1}{3} \end{pmatrix} \begin{pmatrix} \vert 9x^{1}-\frac{x^{1}+x^{2}+x^{3}}{3} \vert + \vert 9u- \frac{u^{1}+u^{2}+ u^{3}}{3^{j}} \vert \\ \vert 9x^{1}-\frac{x^{1}+x^{2}+x^{3}}{3} \vert + \vert 9u- \frac{u^{1}+u^{2}+ u^{3}}{3^{j}} \vert \end{pmatrix} \\ &{}+\frac{1}{3} \begin{pmatrix} 0 & 3 \\ 3 & 0 \end{pmatrix} \begin{pmatrix} \vert u^{1}-x^{1} \vert \\ \vert u^{1}-x^{1} \vert \end{pmatrix}+ \frac{2}{3} \begin{pmatrix} \vert \frac{x^{1}+x^{2}+x^{3}}{3} \vert \\ \vert \frac{x^{1}+x^{2}+x^{3}}{3} \vert \end{pmatrix}\leq \beta. \end{aligned}

Thus all the hypotheses of Theorem 2.6 are satisfied and $$(0,0, 0)$$ is the triple coincident point of g and $$\{T_{i}\}_{i \in \mathbb{N}_{0}}$$. Moreover, using the same $$\{T_{i}\}_{i \in \mathbb{N}_{0}}$$ and g in Theorem 2.9, $$(0,0, 0)$$ is the unique triple common fixed point of g and $$\{T_{i}\}_{i \in \mathbb{N}_{0}}$$.

Before explaining the application, it is necessary to provide the following definition, which we will use in Theorem 3.1.

### Definition 2.11

Let $$A=( a_{ij})$$ and $$B=( b_{ij})$$ be two matrices in $$\mathcal{Z}M$$. Then

\begin{aligned} &A\leq B\quad\Leftrightarrow\quad a_{ij}\leq b_{ij},\quad 1\leq i,j \leq m \\ &\max \lbrace A,B \rbrace =C=( c_{ij}) \quad\textit{where } c_{ij}= \max \lbrace a_{ij}, b_{ij} \rbrace. \end{aligned}

Clearly if $$A\leq B$$ then $$\max \lbrace A,B \rbrace =B$$.

## Application 1

In this part, we will use the results of Sect. 2 to extract some results for the existence and uniqueness of solutions of the integral equations system. Consider the following integral equations system:

\begin{aligned} \begin{aligned} &x^{1}(t)= \int ^{T}_{0} (f \bigl(t, s, x^{1} (s) \bigr) + g \bigl(t, s, x^{2} (s)\bigr)+ h \bigl(t, s, x^{3} (s)\bigr)\, ds+v(t), \\ &x^{2}(t)= \int ^{T}_{0} (f \bigl(t, s, x^{2} (s) \bigr) + g \bigl(t, s, x^{3} (s)\bigr)+ h \bigl(t, s, x^{1} (s)\bigr)\, ds+v(t), \\ &x^{3}(t)= \int ^{T}_{0} (f \bigl(t, s, x^{3} (s) \bigr) + g \bigl(t, s, x^{1} (s)\bigr)+ h \bigl(t, s, x^{2} (s)\bigr)\, ds+v(t), \end{aligned} \end{aligned}
(3.1)

for all $$t, s \in [0, T]$$, for some $$T > 0$$.

Let $$X=C([0, T], \mathbb{R})$$ be continuous real functions, defined on the interval $$[0, T]$$, endowed with a metric

$$d\bigl(x^{1}, x^{2}\bigr)= \begin{pmatrix} \max_{0\leq t \leq T} \vert x^{1}(t)-x^{2}(t) \vert \\ \max_{0\leq t \leq T} \vert x^{1}(t)-x^{2}(t) \vert \end{pmatrix}.$$

We define the partial order “” on X as follows:

for $$x^{1}, x^{2}\in X, x^{1} \preceq x^{2} \Leftrightarrow x^{1}(t) \preceq x^{2}(t)$$ for any $$t\in [0, T]$$.

Thus, $$(X, d, \preceq )$$ is a partially ordered complete generalized metric space. For (3.1) we consider the following hypotheses:

1. (i)

$$f, g, h \in [0, T]\times [0, T] \times \mathbb{R}\longrightarrow \mathbb{R}^{2}$$ are continuous;

2. (ii)

$$v:[0, T]\longrightarrow \mathbb{R}$$ is continuous;

3. (iii)

there exists $$\rho:[0, T]\longrightarrow M_{2\times 2}([0, T])$$, such that, for all $$x^{1}, x^{2}\in X$$,

\begin{aligned} \begin{aligned} &0\leq \bigl\vert f \bigl(t, s, x^{1}(s) \bigr) - f \bigl(t, s, x^{2}(s) \bigr) \bigr\vert \leq \rho _{1}(t)d\bigl(x^{1}, x^{2}\bigr), \\ &0\leq \bigl\vert g\bigl(t, s, x^{2}(s)\bigr) - g\bigl(t, s, x^{1}(s) \bigr) \bigr\vert \leq \rho _{2}(t)d \bigl(x^{1}, x^{2}\bigr), \\ &0\leq \bigl\vert h\bigl(t, s, x^{1}(s) \bigr) - h \bigl(t, s, x^{2}(s) \bigr) \bigr\vert \leq \rho _{3}(t)d \bigl(x^{1}, x^{2}\bigr), \end{aligned} \end{aligned}
(3.2)

for all $$s, t\in [0, T]$$ with $ρ(t)≤A= ( 1 3 0 0 1 3 )$ and $ρ(t)≤B= ( 0 1 3 1 3 0 )$. Because A and B are nonzero matrices in $$\mathcal{Z}M$$;

4. (iv)

we suppose that $$\rho _{1}(t)+\rho _{2}(t)+\rho _{3}(t)<1$$ and

$$\rho (t) = \max \bigl\{ \rho _{1}(t), \rho _{2}(t), \rho _{3}(t)\bigr\} ;$$
5. (v)

there are functions $$\alpha, \beta, \gamma: [0, T] \longrightarrow \mathbb{R}$$ which are continuous, such that

\begin{aligned} &\alpha \leq \int ^{T}_{0} (f \bigl(t, s, \alpha (s)\bigr) + g \bigl(t, s, \beta (s)\bigr)+ h \bigl(t, s, \gamma (s)\bigr)\, ds+v(t), \\ &\beta \geq \int ^{T}_{0} (f \bigl(t, s, \beta (s)\bigr) + g \bigl(t, s, \alpha (s)\bigr)+ h \bigl(t, s, \beta (s)\bigr)\, ds+v(t), \\ &\gamma \leq \int ^{T}_{0} (f \bigl(t, s, \gamma (s)\bigr) + g \bigl(t, s, \beta (s)\bigr)+ h \bigl(t, s, \alpha (s)\bigr)\, ds+v(t). \end{aligned}

### Theorem 3.1

Under hypotheses (i)–(v), (3.1) has a unique solution inX.

### Proof

We consider the operator defined by $$T_{i}: X ^{3}\longrightarrow X$$, with

\begin{aligned} T\bigl(x^{1}, x^{2}, x^{3} \bigr) &=T_{i}\bigl(x^{1}, x^{2}, x^{3} \bigr) \\ &= \int ^{T}_{0} (f \bigl(t, s, x^{1}(s) \bigr) + g \bigl(t, s, x^{2}(s)\bigr)+ h \bigl(t, s, x^{3}(s) \bigr)\, ds+v(t), \end{aligned}

for any $$x^{1}, x^{2}, x^{3}\in X$$ and $$t, s \in [0, T]$$.

We prove that the operator $$\{T_{i}\}_{i \in \mathbb{N}}$$ fulfills the conditions of Corollary 2.7. First, we show that $$\{T_{i}\}_{i \in \mathbb{N}}$$ has the mixed monotone property. Let $$x^{1}, u^{1}\in X$$ with $$x^{1}\leq u^{1}$$ and $$t, s \in [0, T]$$, then we have

\begin{aligned} T_{i}\bigl(u^{1}, x^{2}, x^{3} \bigr) (t)-T_{i}\bigl(x^{1}, x^{2}, x^{3} \bigr) (t)= \int ^{T}_{0} (f \bigl(t, s, u^{1}(s) \bigr) -f \bigl(t, s, x^{1}(s)\bigr)\, ds. \end{aligned}

Given that $$x^{1}(t)\leq u^{1}(t)$$ for all $$t\in [0, T]$$ and based on our assumption (3.2), we have

$$T_{i}\bigl(u^{1}, x^{2}, x^{3} \bigr) (t)-T_{i}\bigl(x^{1}, x^{2}, x^{3} \bigr) (t)\geq 0,$$

that is, $$T_{i}(u^{1}, x^{2}, x^{3} )(t)\geq T_{i}(x^{1}, x^{2}, x^{3} )(t)$$. For $$x^{2}, u^{2}\in X$$ with $$x^{2}\leq u^{2}$$ and $$t, s \in [0, T]$$, then we have

\begin{aligned} T_{i}\bigl(x^{1}, x^{2}, x^{3} \bigr) (t)-T_{i}\bigl(x^{1}, u^{2}, x^{3} \bigr) (t)= \int ^{T}_{0} (f \bigl(t, s, x^{2}(s) \bigr) -f \bigl(t, s, u^{2}(s)\bigr)\, ds. \end{aligned}

Given that $$x^{2}(t)\leq u^{2}(t)$$ for all $$t\in [0, T]$$ and based on our assumption (3.2), we have

$$T_{i}\bigl(x^{1}, x^{2}, x^{3} \bigr) (t)-T_{i}\bigl(x^{1}, u^{2}, x^{3} \bigr) (t)\leq 0,$$

that is, $$T_{i}(x^{1}, x^{2}, x^{3} )(t)\geq T_{i}(x^{1}, u^{2}, x^{3} )(t)$$. Similarly, we have

$$T_{i}\bigl(x^{1}, x^{2}, u^{3} \bigr) (t)-T_{i}\bigl(x^{1}, u^{2}, x^{3} \bigr) (t)\geq 0,$$

that is, $$T_{i}(x^{1}, x^{2}, x^{3} )(t)\leq T_{i}(x^{1}, x^{2}, u^{3} )(t)$$. So, $$\{T_{i}\}_{i \in \mathbb{N}}$$ has the mixed monotone property. Now, we estimate $$d(T_{i}(x^{1}, x^{2}, x^{3} ), T_{j}(u^{1}, u^{2}, u^{3} ))$$ for $$x^{1} \preceq u^{1}, u^{2} \preceq x^{2}, x^{3} \preceq u^{3}$$ or $$x^{1} \succeq u^{1}, u^{2} \succeq x^{2}, x^{3} \succeq u^{3}$$ and with $$\{T_{i}\}_{i \in \mathbb{N}}$$ having the mixed monotone property, we get

\begin{aligned} &d\bigl(T_{i}\bigl(x^{1}, x^{2}, x^{3} \bigr), T_{j}\bigl(u^{1}, u^{2}, u^{3} \bigr)\bigr) \\ &\quad= \begin{pmatrix} \max_{0\leq t \leq T} \vert T_{i}(x^{1}, x^{2}, x^{3} )(t)-T_{j}(u^{1}, u^{2}, u^{3} )(t) \vert \\ \max_{0\leq t \leq T} \vert T_{i}(x^{1}, x^{2}, x^{3} )(t)-T_{j}(u^{1}, u^{2}, u^{3} )(t) \vert \end{pmatrix}. \end{aligned}

Now, for all $$t\in [0, T]$$ by using (3.2), we have

\begin{aligned} &\bigl\vert T_{i}\bigl(x^{1}, x^{2}, x^{3} \bigr) (t)- T_{j}\bigl(u^{1}, u^{2}, u^{3} \bigr) (t) \bigr\vert \\ &\quad= \biggl\vert \int ^{T}_{0} (f \bigl(t, s, x^{1}(s) \bigr) + g \bigl(t, s, x^{2}(s)\bigr)+h \bigl(t, s, x^{3}(s) \bigr) ds \\ &\qquad{}- \int ^{T}_{0} (f \bigl(t, s, u^{1}(s) \bigr) + g \bigl(t, s, u^{2}(s)\bigr)+ h \bigl(t, s, u^{3}(s) \bigr) \,ds \biggr\vert \\ &\quad\leq \int ^{T}_{0} \bigl\vert (f \bigl(t, s, x^{1}(s)\bigr) - f \bigl(t, s, u^{1}(s)\bigr) \bigr\vert \,ds \\ &\qquad{}+ \int ^{T}_{0} \bigl\vert (g\bigl(t, s, x^{2}(s)\bigr) - g \bigl(t, s, u^{2}(s)\bigr) \bigr\vert \,ds \\ &\qquad{}+ \int ^{T}_{0} \bigl\vert (h\bigl(t, s, x^{3}(s)\bigr) - h\bigl(t, s, u^{3}(s)\bigr) \bigr\vert \,ds \\ &\quad\leq \rho _{1}(t)d\bigl(x^{1}, u^{1}\bigr)+\rho _{2}(t)d\bigl(x^{2}, u^{2}\bigr)+\rho _{3}(t)d\bigl(x^{3}, u^{3}\bigr) \\ &\quad\leq \rho (t) \bigl(d\bigl(x^{1}, u^{1}\bigr)+d \bigl(x^{2}, u^{2}\bigr)+d\bigl(x^{3}, u^{3}\bigr)\bigr) \\ &\quad\leq B \bigl(d\bigl(x^{1}, u^{1}\bigr)+d \bigl(x^{2}, u^{2}\bigr)+d\bigl(x^{3}, u^{3}\bigr)\bigr). \end{aligned}

Consequently,

\begin{aligned} d\bigl(T_{i}\bigl(x^{1}, x^{2}, x^{3} \bigr), T_{j}\bigl(u^{1}, u^{2}, u^{3} \bigr)\bigr)\leq{}& B \begin{pmatrix} d(x^{1}, u^{1})+d(x^{2}, u^{2})+d(x^{3}, u^{3}) \\ d(x^{1}, u^{1})+d(x^{2}, u^{2})+d(x^{3}, u^{3}) \end{pmatrix} \\ \leq{}& A\bigl[ d\bigl( x^{1},T_{i}\bigl(x^{1}, x^{2}, x^{3}\bigr)\bigr)+ d\bigl(u^{1},T_{j} \bigl(u^{1}, u^{2}, u^{3}\bigr)\bigr)\bigr] \\ &{}+B d \bigl(u^{1}, x^{1}\bigr). \end{aligned}

Let $$\alpha, \beta, \gamma$$ be the same as (v); then we have

$$\alpha \leq {T}_{i}(\alpha, \beta, \gamma ),\qquad \beta \geq {T}_{i}( \beta, \alpha, \beta ),\qquad \gamma \leq {T}_{i}( \gamma, \beta, \alpha ).$$

If $$x^{1}_{0}= \alpha, x^{2}_{0}=\beta, x^{3}_{0}=\gamma$$, then all assumptions of Corollary 2.7 are fulfilled. So, there exists a triple fixed point $$(x^{1}, x^{2}, x^{3})$$ for the operator $$\{T_{i}\}_{i \in \mathbb{N}}$$; that is, $$T_{i}(x^{1}, x^{2}, x^{3})=x^{1},T_{i}(x^{2}, x^{1}, x^{2}) = x^{2}$$, and $$T_{i}(x^{3}, x^{2}, x^{1}) =x^{3}$$ for $$i \in \mathbb{N}$$. □

## Application 2

Now if we consider the sequence of the integral equations system below, in which

\begin{aligned} \begin{aligned} &x^{1}(t)= \int ^{T}_{0} (f_{i} \bigl(t, s, x^{1} (s)\bigr) + g_{i} \bigl(t, s, x^{2} (s) \bigr)+ h_{i} \bigl(t, s, x^{3} (s)\bigr) \,ds+v(t), \\ &x^{2}(t)= \int ^{T}_{0} (f_{i} \bigl(t, s, x^{2} (s)\bigr) + g_{i} \bigl(t, s, x^{3} (s) \bigr)+ h_{i} \bigl(t, s, x^{1} (s)\bigr)\, ds+v(t), \\ &x^{3}(t)= \int ^{T}_{0} (f_{i} \bigl(t, s, x^{3} (s)\bigr) + g_{i} \bigl(t, s, x^{1} (s) \bigr)+ h_{i} \bigl(t, s, x^{2} (s)\bigr)\, ds+v(t), \end{aligned} \end{aligned}
(4.1)

for all $$t, s \in [0, T]$$, for some $$T > 0$$, then, similar to Theorem 3.1, this sequence of the integral equations system with the conditions given below will have a simultaneous solution.

Let $$X=C([0, T], \mathbb{R})$$ be equipped with metric defined in Sect. 3 and “” be the partial order on X. Thus, $$(X, d, \preceq )$$ is a partially ordered complete generalized metric space. For (4.1) we consider the following hypotheses:

1. (i)

$$f_{i}, g_{i}, h_{i} \in [0, T]\times [0, T] \times \mathbb{R} \longrightarrow \mathbb{R}^{2}$$ are continuous;

2. (ii)

$$v:[0, T]\longrightarrow \mathbb{R}$$ is continuous;

3. (iii)

there exists $$\rho:[0, T]\longrightarrow M_{2\times 2}([0, T])$$, such that, for all $$x^{1}, x^{2}\in X$$, we have

\begin{aligned} \begin{aligned} &0\leq \bigl\vert f_{i} \bigl(t, s, x^{1}(s) \bigr) - f_{i} \bigl(t, s, x^{2}(s) \bigr) \bigr\vert \leq \rho _{1}(t)d\bigl(x^{1}, x^{2}\bigr), \\ &0\leq \bigl\vert g_{i}\bigl(t, s, x^{2}(s) \bigr) - g_{i}\bigl(t, s, x^{1}(s) \bigr) \bigr\vert \leq \rho _{2}(t)d\bigl(x^{1}, x^{2}\bigr), \\ &0\leq \bigl\vert h_{i}\bigl(t, s, x^{1}(s) \bigr) - h_{i} \bigl(t, s, x^{2}(s) \bigr) \bigr\vert \leq \rho _{3}(t)d\bigl(x^{1}, x^{2}\bigr), \end{aligned} \end{aligned}
(4.2)

for all $$s, t\in [0, T]$$ with $ρ(t)≤A= ( 1 3 0 0 1 3 )$ and $ρ(t)≤B= ( 0 1 3 1 3 0 )$;

4. (iv)

we suppose that $$\rho _{1}(t)+\rho _{2}(t)+\rho _{3}(t)<1$$ and

$$\rho (t) = \max \bigl\{ \rho _{1}(t), \rho _{2}(t), \rho _{3}(t)\bigr\} ;$$
5. (v)

there are functions $$\alpha, \beta, \gamma: [0, T] \longrightarrow \mathbb{R}$$ which are continuous, such that

\begin{aligned} &\alpha \leq \int ^{T}_{0} (f_{i} \bigl(t, s, \alpha (s)\bigr) + g_{i} \bigl(t, s, \beta (s)\bigr)+ h_{i} \bigl(t, s, \gamma (s)\bigr) \,ds+v(t), \\ &\beta \geq \int ^{T}_{0} (f_{i} \bigl(t, s, \beta (s)\bigr) + g_{i} \bigl(t, s, \alpha (s)\bigr)+ h_{i} \bigl(t, s, \beta (s)\bigr)\, ds+v(t), \\ &\gamma \leq \int ^{T}_{0} (f_{i} \bigl(t, s, \gamma (s)\bigr) + g_{i} \bigl(t, s, \beta (s)\bigr)+ h_{i} \bigl(t, s, \alpha (s)\bigr)\, ds+v(t). \end{aligned}

## References

1. 1.

Allaire, G., Kaber, S.M.: Numerical Linear Algebra, Applied Mathematics, vol. 55. Springer, New York (2008)

2. 2.

Berinde, V., Borcut, M.: Tripled fixed point theorems for contractive type mappings in partially ordered metric spaces. Nonlinear Anal. 74(15), 4889–4897 (2011)

3. 3.

Bhaskar, T.G., Lakshmikantham, V.: Fixed point theorems in partially ordered metric space and applications. Nonlinear Anal. 65, 1379–1393 (2006)

4. 4.

Borcut, M., Berinde, V.: Tripled coincidence theorems for contractive type mappings in partially ordered metric spaces. Appl. Math. Comput. 218(10), 5929–5936 (2012)

5. 5.

Filip, A.D., Petruşel, A.: Fixed point theorems on spaces endowed with vector-valued metrics. Fixed Point Theory Appl. 2010, 281381 (2010)

6. 6.

Hadi Bonab, S., Abazari, R., Bagheri Vakilabad, A.: Partially ordered cone metric spaces and coupled fixed point theorems via α-series. Math. Anal. Contemp. Appl. 1(1), 50–61 (2019)

7. 7.

Hosseinzadeh, H.: Some fixed point theorems in generalized metric spaces endowed with vector-valued metrics and application in nonlinear matrix equations. Sahand Commun. Math. Anal. 17(2), 37–53 (2020)

8. 8.

Hosseinzadeh, H., Jabbari, A., Razani, A.: Fixed point theorems and common fixed point theorems on spaces equipped with vector-valued metrics. Ukr. Math. J. 65(5), 814–822 (2013)

9. 9.

Kadelburg, Z., Radenović, S.: Fixed point and tripled fixed point theorems under Pata-type conditions in ordered metric spaces. Int. J. Anal. Appl. 6(1), 113–122 (2014)

10. 10.

Lakshmikantham, V., Ciric, L.: Coupled fixed point theorems for nonlinear contractions in partially ordered metric spaces. Nonlinear Anal. 70, 4341–4349 (2009)

11. 11.

Perov, A.I.: On the Cauchy problem for a system of ordinary differential equations. Pviblizhen. Met. Reshen. Differ. Uvavn. 2, 115–134 (1964)

12. 12.

Precup, R.: The role of matrices that are convergent to zero in the study of semilinear operator systems. Math. Comput. Model. 49(3–4), 703–708 (2009)

13. 13.

Rus, I.A.: Principles and Applications of the Fixed Point Theory. Dacia, Cluj-Napoca (1979)

14. 14.

Varga, R.S.: Matrix Iterative Analysis. Computational Mathematics, vol. 27. Springer, Berlin (2000)

15. 15.

Vats, R.K., Tas, K., Sihag, V., Kumar, A.: Triple fixed point theorems via α-series in partially ordered metric spaces. J. Inequal. Appl. 2014, 176 (2014)

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The authors thank the referee for useful proposals to improve the paper.

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