1 Introduction

Perov described the Banach contraction principle for contraction mappings on spaces equipped with vector-valued metrics in [11]. Later, by a different method, the results of Perov in [5] 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

([5])

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

([3])

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

([10])

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

([15])

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

([2])

.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 [9]).

Definition 1.6

([9])

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

([2])

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

([4])

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

([15])

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

([15])

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

([15])

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

([15])

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

([15])

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

([15])

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 \(x^{1}, x^{2}, x^{3} \in X\).

2 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)

Adding (2.7)–(2.9), we have

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

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 \(\max \lbrace A,B \rbrace =B\).

3 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 \(i \in \mathbb{N}\). □

4 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

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