# Existence of a tripled coincidence point in ordered ${G}_{b}$-metric spaces and applications to a system of integral equations

## Abstract

In this paper, tripled coincidence points of mappings satisfying some nonlinear contractive conditions in the framework of partially ordered ${G}_{b}$-metric spaces are obtained. Our results extend the results of Aydi et al. (Fixed Point Theory Appl., 2012:101, 2012, doi:10.1186/1687-1812-2012-101). Moreover, some examples of the main result are given. Finally, some tripled coincidence point results for mappings satisfying some contractive conditions of integral type in complete partially ordered ${G}_{b}$-metric spaces are deduced.

MSC: 47H10, 54H25.

## 1 Introduction and preliminaries

The concepts of mixed monotone mapping and coupled fixed point were introduced in  by Bhaskar and Lakshmikantham. Also, they established some coupled fixed point theorems for a mixed monotone mapping in partially ordered metric spaces. For more details on coupled fixed point theorems and related topics in different metric spaces, we refer the reader to  and .

Also, Berinde and Borcut  introduced a new concept of tripled fixed point and obtained some tripled fixed point theorems for contractive-type mappings in partially ordered metric spaces. For a survey of tripled fixed point theorems and related topics, we refer the reader to .

Definition 1.1 

An element $\left(x,y,z\right)\in {X}^{3}$ is called a tripled fixed point of $F:{X}^{3}\to X$ if $F\left(x,y,z\right)=x$, $F\left(y,x,y\right)=y$ and $F\left(z,y,x\right)=z$.

Definition 1.2 

An element $\left(x,y,z\right)\in {X}^{3}$ is called a tripled coincidence point of the mappings $F:{X}^{3}\to X$ and $g:X\to X$ if $F\left(x,y,z\right)=g\left(x\right)$, $F\left(y,x,y\right)=gy$ and $F\left(z,y,x\right)=gz$.

Definition 1.3 

An element $\left(x,y,z\right)\in {X}^{3}$ is called a tripled common fixed point of $F:{X}^{3}\to X$ and $g:X\to X$ if $x=g\left(x\right)=F\left(x,y,z\right)$, $y=g\left(y\right)=F\left(y,x,y\right)$ and $z=g\left(z\right)=F\left(z,y,x\right)$.

Definition 1.4 

Let X be a nonempty set. We say that the mappings $F:{X}^{3}\to X$ and $g:X\to X$ are commutative if $g\left(F\left(x,y,z\right)\right)=F\left(gx,gy,gz\right)$ for all $x,y,z\in X$.

The notion of altering distance function was introduced by Khan et al.  as follows.

Definition 1.5 The function $\psi :\left[0,\mathrm{\infty }\right)\to \left[0,\mathrm{\infty }\right)$ is called an altering distance function if

1. 1.

ψ is continuous and nondecreasing.

2. 2.

$\psi \left(t\right)=0$ if and only if $t=0$.

The concept of generalized metric space, or G-metric space, was introduced by Mustafa and Sims . Mustafa and others studied several fixed point theorems for mappings satisfying different contractive conditions (see ).

Definition 1.6 (G-metric space, )

Let X be a nonempty set and $G:{X}^{3}\to {R}^{+}$ be a function satisfying the following properties:

• (G1) $G\left(x,y,z\right)=0$ iff $x=y=z$;

• (G2) $0 for all $x,y?X$ with $x?y$;

• (G3) $G\left(x,x,y\right)=G\left(x,y,z\right)$ for all $x,y,z?X$ with $z?y$;

• (G4) $G\left(x,y,z\right)=G\left(x,z,y\right)=G\left(y,z,x\right)=?$ (symmetry in all three variables);

• (G5) $G\left(x,y,z\right)=G\left(x,a,a\right)+G\left(a,y,z\right)$ for all $x,y,z,a?X$ (rectangle inequality).

Then the function G is called a G-metric on X and the pair $\left(X,G\right)$ is called a G-metric space.

Example 1.7 If we think that $G\left(x,y,z\right)$ is measuring the perimeter of the triangle with vertices at x, y and z, then (G5) can be interpreted as

$\left[x,y\right]+\left[x,z\right]+\left[y,z\right]\le 2\left[x,a\right]+\left[a,y\right]+\left[a,z\right]+\left[y,z\right],$

where $\left[x,y\right]$ is the ‘length’ of the side x, y. If we take $y=z$, we have

$2\left[x,y\right]\le 2\left[x,a\right]+2\left[a,y\right].$

Thus, (G5) embodies the triangle inequality. And so (G5) can be sharp.

In , Aydi et al. established some tripled coincidence point results for mappings $F:{X}^{3}\to X$ and $g:X\to X$ involving nonlinear contractions in the setting of ordered G-metric spaces.

Theorem 1.8 

Let $\left(X,⪯\right)$ be a partially ordered set and $\left(X,G\right)$ be a G-metric space such that $\left(X,G\right)$ is G-complete. Let $F:{X}^{3}\to X$ and $g:X\to X$. Assume that there exist $\psi ,\varphi :\left[0,\mathrm{\infty }\right)\to \left[0,\mathrm{\infty }\right)$ such that ψ is an altering distance function and ϕ is a lower-semicontinuous and nondecreasing function with $\varphi \left(t\right)=0$ if and only if $t=0$ and for all $x,y,z,u,v,w,r,s,t\in X$, with $gx⪯gu⪯gr$, $gy⪰gv⪰gs$ and $gz⪯gw⪯gt$, we have

$\begin{array}{r}\psi \left(G\left(F\left(x,y,z\right),F\left(u,v,w\right),F\left(r,s,t\right)\right)\right)\\ \phantom{\rule{1em}{0ex}}\le \psi \left(max\left\{G\left(gx,gu,gr\right),G\left(gy,gv,gs\right),G\left(gz,gw,gt\right)\right\}\right)\\ \phantom{\rule{2em}{0ex}}-\varphi \left(max\left\{G\left(gx,gu,gr\right),G\left(gy,gv,gs\right),G\left(gz,gw,gt\right)\right\}\right).\end{array}$

Assume that F and g satisfy the following conditions:

1. (1)

$F\left({X}^{3}\right)\subseteq g\left(X\right)$,

2. (2)

F has the mixed g-monotone property,

3. (3)

F is continuous,

4. (4)

g is continuous and commutes with F.

Let there exist ${x}_{0},{y}_{0},{z}_{0}\in X$ such that $g{x}_{0}⪯F\left({x}_{0},{y}_{0},{z}_{0}\right)$, $g{y}_{0}⪰F\left({y}_{0},{x}_{0},{y}_{0}\right)$ and $g{z}_{0}⪯F\left({z}_{0},{y}_{0},{x}_{0}\right)$. Then F and g have a tripled coincidence point in X, i.e., there exist $x,y,z\in X$ such that $F\left(x,y,z\right)=gx$, $F\left(y,x,y\right)=gy$ and $F\left(z,y,x\right)=gz$.

Also, they proved that the above theorem is still valid for F not necessarily continuous assuming the following hypothesis (see Theorem 19 of ).

1. (I)

If $\left\{{x}_{n}\right\}$ is a nondecreasing sequence with ${x}_{n}\to x$, then ${x}_{n}⪯x$ for all $n\in \mathbb{N}$.

2. (II)

If $\left\{{y}_{n}\right\}$ is a nonincreasing sequence with ${y}_{n}\to y$, then ${y}_{n}⪰y$ for all $n\in \mathbb{N}$.

A partially ordered G-metric space $\left(X,G\right)$ with the above properties is called regular.

In this paper, we obtain some tripled coincidence point theorems for nonlinear $\left(\psi ,\phi \right)$-weakly contractive mappings in partially ordered ${G}_{b}$-metric spaces. This results generalize and modify several comparable results in the literature. First, we recall the concept of generalized b-metric spaces, or ${G}_{b}$-metric spaces.

Definition 1.9 

Let X be a nonempty set and $s\ge 1$ be a given real number. Suppose that a mapping $G:{X}^{3}\to {\mathbb{R}}^{+}$ satisfies:

• (G b 1) $G\left(x,y,z\right)=0$ if $x=y=z$,

• (G b 2) $0 for all $x,y?X$ with $x?y$,

• (G b 3) $G\left(x,x,y\right)=G\left(x,y,z\right)$ for all $x,y,z?X$ with $y?z$,

• (G b 4) $G\left(x,y,z\right)=G\left(p\left\{x,y,z\right\}\right)$, where p is a permutation of x, y, z (symmetry),

• (G b 5) $G\left(x,y,z\right)=s\left[G\left(x,a,a\right)+G\left(a,y,z\right)\right]$ for all $x,y,z,a?X$ (rectangle inequality).

Then G is called a generalized b-metric and the pair $\left(X,G\right)$ is called a generalized b-metric space or a ${G}_{b}$-metric space.

Obviously, each G-metric space is a ${G}_{b}$-metric space with $s=1$. But the following example shows that a ${G}_{b}$-metric on X need not be a G-metric on X (see also ).

Example 1.10 If we think that ${G}_{b}\left(x,y,z\right)$ is the maximum of the squares of length sides of a triangle with vertices at x, y and z such that:

If $x\ne y\ne z$, then ${G}_{b}\left(x,y,z\right)=max\left\{{\left(\left[x,y\right]\right)}^{2},{\left(\left[y,z\right]\right)}^{2},{\left(\left[z,x\right]\right)}^{2}\right\}$.

If $x\ne y=z$, then ${G}_{b}\left(x,y,y\right)={\left(\left[x,y\right]\right)}^{2}$,

where $\left[x,y\right]$ is the ‘length’ of the side x, y. Then it is easy to see that ${G}_{b}\left(x,y,z\right)$ is a ${G}_{b}$ function with $s=2$.

Since by the triangle inequality we have

$\left[x,y\right]\le \left[x,a\right]+\left[a,y\right],\phantom{\rule{2em}{0ex}}\left[z,x\right]\le \left[z,a\right]+\left[a,x\right],$

hence

$\begin{array}{rcl}{G}_{b}\left(x,y,z\right)& =& max\left\{{\left(\left[x,y\right]\right)}^{2},{\left(\left[y,z\right]\right)}^{2},{\left(\left[z,x\right]\right)}^{2}\right\}\\ \le & max\left\{{\left(\left[x,a\right]+\left[a,y\right]\right)}^{2},{\left(\left[y,z\right]\right)}^{2},{\left(\left[z,a\right]+\left[a,x\right]\right)}^{2}\right\}\\ \le & max\left\{2\left({\left(\left[x,a\right]\right)}^{2}+{\left(\left[a,y\right]\right)}^{2}\right),{\left(\left[y,z\right]\right)}^{2},2\left({\left(\left[z,a\right]\right)}^{2}+{\left(\left[a,x\right]\right)}^{2}\right)\right\}\\ \le & 2{\left(\left[x,a\right]\right)}^{2}+max\left\{2{\left(\left[a,y\right]\right)}^{2},{\left(\left[y,z\right]\right)}^{2},2{\left(\left[z,a\right]\right)}^{2}\right\}\\ \le & 2{\left(\left[x,a\right]\right)}^{2}+max\left\{2{\left(\left[a,y\right]\right)}^{2},2{\left(\left[y,z\right]\right)}^{2},2{\left(\left[z,a\right]\right)}^{2}\right\}\\ =& 2\left({G}_{b}\left(x,a,a\right)+{G}_{b}\left(a,y,z\right)\right).\end{array}$

Example 1.11 

Let $\left(X,G\right)$ be a G-metric space and ${G}_{\ast }\left(x,y,z\right)=G{\left(x,y,z\right)}^{p}$, where $p>1$ is a real number. Then ${G}_{\ast }$ is a ${G}_{b}$-metric with $s={2}^{p-1}$.

Also, in the above example, $\left(X,{G}_{\ast }\right)$ is not necessarily a G-metric space. For example, let $X=\mathbb{R}$ and G-metric G be defined by

$G\left(x,y,z\right)=\frac{1}{3}\left(|x-y|+|y-z|+|x-z|\right)$

for all $x,y,z\in \mathbb{R}$ (see ). Then ${G}_{\ast }\left(x,y,z\right)=G{\left(x,y,z\right)}^{2}=\frac{1}{9}{\left(|x-y|+|y-z|+|x-z|\right)}^{2}$ is a ${G}_{b}$-metric on with $s={2}^{2-1}=2$, but it is not a G-metric on .

Example 1.12 

Let $X=\mathbb{R}$ and $d\left(x,y\right)=|x-y{|}^{2}$. We know that $\left(X,d\right)$ is a b-metric space with $s=2$. Let $G\left(x,y,z\right)=d\left(x,y\right)+d\left(y,z\right)+d\left(z,x\right)$, then $\left(X,G\right)$ is not a ${G}_{b}$-metric space. Indeed, (G b 3) is not true for $x=0$, $y=2$ and $z=1$. To see this, we have

$G\left(0,0,2\right)=d\left(0,0\right)+d\left(0,2\right)+d\left(2,0\right)=2d\left(0,2\right)=8$

and

$G\left(0,2,1\right)=d\left(0,2\right)+d\left(2,1\right)+d\left(1,0\right)=4+1+1=6.$

So, $G\left(0,0,2\right)>G\left(0,2,1\right)$.

However, $G\left(x,y,z\right)=max\left\{d\left(x,y\right),d\left(y,z\right),d\left(z,x\right)\right\}$ is a ${G}_{b}$-metric on with $s=2$. Similarly, if $d\left(x,y\right)=|x-y{|}^{p}$ is selected with $p\ge 1$, then $G\left(x,y,z\right)=max\left\{d\left(x,y\right),d\left(y,z\right),d\left(z,x\right)\right\}$ is a ${G}_{b}$-metric on with $s={2}^{p-1}$.

Now we present some definitions and propositions in a ${G}_{b}$-metric space.

Definition 1.13 

A ${G}_{b}$-metric G is said to be symmetric if $G\left(x,y,y\right)=G\left(y,x,x\right)$ for all $x,y\in X$.

Definition 1.14 Let $\left(X,G\right)$ be a ${G}_{b}$-metric space. Then, for ${x}_{0}\in X$ and $r>0$, the ${G}_{b}$-ball with center ${x}_{0}$ and radius r is

${B}_{G}\left({x}_{0},r\right)=\left\{y\in X\mid G\left({x}_{0},y,y\right)

By some straight forward calculations, we can establish the following.

Proposition 1.15 

Let X be a ${G}_{b}$-metric space. Then, for each $x,y,z,a\in X$, it follows that:

1. (1)

if $G\left(x,y,z\right)=0$, then $x=y=z$,

2. (2)

$G\left(x,y,z\right)\le s\left(G\left(x,x,y\right)+G\left(x,x,z\right)\right)$,

3. (3)

$G\left(x,y,y\right)\le 2sG\left(y,x,x\right)$,

4. (4)

$G\left(x,y,z\right)\le s\left(G\left(x,a,z\right)+G\left(a,y,z\right)\right)$.

Definition 1.16 

Let X be a ${G}_{b}$-metric space. We define ${d}_{G}\left(x,y\right)=G\left(x,y,y\right)+G\left(x,x,y\right)$ for all $x,y\in X$. It is easy to see that ${d}_{G}$ defines a b-metric d on X, which we call the b-metric associated with G.

Proposition 1.17 

Let X be a ${G}_{b}$-metric space. Then, for any ${x}_{0}\in X$ and $r>0$, if $y\in {B}_{G}\left({x}_{0},r\right)$, then there exists $\delta >0$ such that ${B}_{G}\left(y,\delta \right)\subseteq {B}_{G}\left({x}_{0},r\right)$.

From the above proposition, the family of all ${G}_{b}$-balls

$Ϝ=\left\{{B}_{G}\left(x,r\right)\mid x\in X,\phantom{\rule{0.25em}{0ex}}r>0\right\}$

is a base of a topology $\tau \left(G\right)$ on X, which we call the ${G}_{b}$-metric topology.

Now, we generalize Proposition 5 in  for a ${G}_{b}$-metric space as follows.

Proposition 1.18 

Let X be a ${G}_{b}$-metric space. Then, for any ${x}_{0}\in X$ and $r>0$, we have

${B}_{G}\left({x}_{0},\frac{r}{2s+1}\right)\subseteq {B}_{{d}_{G}}\left({x}_{0},r\right)\subseteq {B}_{G}\left({x}_{0},r\right).$

Thus every ${G}_{b}$-metric space is topologically equivalent to a b-metric space. This allows us to readily transport many concepts and results from b-metric spaces into ${G}_{b}$-metric space setting.

Definition 1.19 

Let X be a ${G}_{b}$-metric space. A sequence $\left\{{x}_{n}\right\}$ in X is said to be:

1. (1)

${G}_{b}$-Cauchy if for each $\epsilon >0$, there exists a positive integer ${n}_{0}$ such that, for all $m,n,l\ge {n}_{0}$, $G\left({x}_{n},{x}_{m},{x}_{l}\right)<\epsilon$;

2. (2)

${G}_{b}$-convergent to a point $x\in X$ if for each $\epsilon >0$, there exists a positive integer ${n}_{0}$ such that, for all $m,n\ge {n}_{0}$, $G\left({x}_{n},{x}_{m},x\right)<\epsilon$.

Proposition 1.20 

Let X be a ${G}_{b}$-metric space. Then the following are equivalent:

1. (1)

the sequence $\left\{{x}_{n}\right\}$ is ${G}_{b}$-Cauchy;

2. (2)

for any $\epsilon >0$, there exists ${n}_{0}\in \mathbb{N}$ such that $G\left({x}_{n},{x}_{m},{x}_{m}\right)<\epsilon$ for all $m,n\ge {n}_{0}$.

Proposition 1.21 

Let X be a ${G}_{b}$-metric space. The following are equivalent:

1. (1)

$\left\{{x}_{n}\right\}$ is ${G}_{b}$-convergent to x;

2. (2)

$G\left({x}_{n},{x}_{n},x\right)\to 0$ as $n\to +\mathrm{\infty }$;

3. (3)

$G\left({x}_{n},x,x\right)\to 0$ as $n\to +\mathrm{\infty }$.

Definition 1.22 

A ${G}_{b}$-metric space X is called complete if every ${G}_{b}$-Cauchy sequence is ${G}_{b}$-convergent in X.

Definition 1.23 

Let $\left(X,G\right)$ and $\left({X}^{\prime },{G}^{\prime }\right)$ be two ${G}_{b}$-metric spaces. Then a function $f:X\to {X}^{\prime }$ is ${G}_{b}$-continuous at a point $x\in X$ if and only if it is ${G}_{b}$-sequentially continuous at x, that is, whenever $\left\{{x}_{n}\right\}$ is ${G}_{b}$-convergent to x, $\left\{f\left({x}_{n}\right)\right\}$ is ${G}_{b}^{\prime }$-convergent to $f\left(x\right)$.

Mustafa and Sims proved that each G-metric function $G\left(x,y,z\right)$ is jointly continuous in all three of its variables (see Proposition 8 in ). But, in general, a ${G}_{b}$-metric function $G\left(x,y,z\right)$ for $s>1$ is not jointly continuous in all its variables. Now, we recall an example of a discontinuous ${G}_{b}$-metric.

Example 1.24 

Let $X=\mathbb{N}\cup \left\{\mathrm{\infty }\right\}$ and let $D:X×X\to \mathbb{R}$ be defined by

Then it is easy to see that for all $m,n,p\in X$, we have

$D\left(m,p\right)\le \frac{5}{2}\left(D\left(m,n\right)+D\left(n,p\right)\right).$

Thus, $\left(X,D\right)$ is a b-metric space with $s=\frac{5}{2}$ (see corrected Example 3 in ).

Let $G\left(x,y,z\right)=max\left\{D\left(x,y\right),D\left(y,z\right),D\left(z,x\right)\right\}$. It is easy to see that G is a ${G}_{b}$-metric with $s=5/2$. In , it is proved that $G\left(x,y,z\right)$ is not a continuous function.

So, from the above discussion, we need the following simple lemma about the ${G}_{b}$-convergent sequences in the proof of our main result.

Lemma 1.25 

Let $\left(X,G\right)$ be a ${G}_{b}$-metric space with $s>1$ and suppose that $\left\{{x}_{n}\right\}$, $\left\{{y}_{n}\right\}$ and $\left\{{z}_{n}\right\}$ are ${G}_{b}$-convergent to x, y and z, respectively. Then we have

$\begin{array}{rl}\frac{1}{{s}^{3}}G\left(x,y,z\right)& \le \underset{n⟶\mathrm{\infty }}{lim inf}G\left({x}_{n},{y}_{n},{z}_{n}\right)\\ \le \underset{n⟶\mathrm{\infty }}{lim sup}G\left({x}_{n},{y}_{n},{z}_{n}\right)\le {s}^{3}G\left(x,y,z\right).\end{array}$

In particular, if $x=y=z$, then we have ${lim}_{n⟶\mathrm{\infty }}G\left({x}_{n},{y}_{n},{z}_{n}\right)=0$.

In this paper, we present some tripled coincidence point results in ordered ${G}_{b}$-metric spaces. Our results extend and generalize the results in .

## 2 Main results

Let $\left(X,⪯,G\right)$ be an ordered ${G}_{b}$-metric space and $F:{X}^{3}\to X$ and $g:X\to X$. In the rest of this paper, unless otherwise stated, for all $x,y,z,u,v,w,r,s,t\in X$, let

$\begin{array}{rl}{M}_{F}\left(x,y,z,u,v,w,r,s,t\right)=& max\left\{G\left(F\left(x,y,z\right),F\left(u,v,w\right),F\left(r,s,t\right)\right),\\ G\left(F\left(y,x,y\right),F\left(v,u,v\right),F\left(s,r,s\right)\right),\\ G\left(F\left(z,y,x\right),F\left(w,v,u\right),F\left(t,s,r\right)\right)\right\}\end{array}$

and

${M}_{g}\left(x,y,z,u,v,w,r,s,t\right)=max\left\{G\left(gx,gu,gr\right),G\left(gy,gv,gs\right),G\left(gz,gw,gt\right)\right\}.$

Now, the main result is presented as follows.

Theorem 2.1 Let $\left(X,⪯,G\right)$ be a partially ordered ${G}_{b}$-metric space and $F:{X}^{3}\to X$ and $g:X\to X$ be such that $F\left({X}^{3}\right)\subseteq g\left(X\right)$. Assume that

$\begin{array}{r}\psi \left(s{M}_{F}\left(x,y,z,u,v,w,r,s,t\right)\right)\\ \phantom{\rule{1em}{0ex}}\le \psi \left({M}_{g}\left(x,y,z,u,v,w,r,s,t\right)\right)-\phi \left({M}_{g}\left(x,y,z,u,v,w,r,s,t\right)\right)\end{array}$
(2.1)

for every $x,y,z,u,v,w,r,s,t\in X$ with $gx⪯gu⪯gr$, $gy⪰gv⪰gs$ and $gz⪯gw⪯gt$, or $gr⪯gu⪯gx$, $gs⪰gv⪰gy$ and $gt⪯gw⪯gz$, where $\psi ,\phi :\left[0,\mathrm{\infty }\right)\to \left[0,\mathrm{\infty }\right)$ are altering distance functions.

Assume that

1. (1)

F has the mixed g-monotone property.

2. (2)

g is ${G}_{b}$-continuous and commutes with F.

Also suppose that

1. (a)

either F is ${G}_{b}$-continuous and $\left(X,G\right)$ is ${G}_{b}$-complete, or

2. (b)

$\left(X,G\right)$ is regular and $\left(g\left(X\right),G\right)$ is ${G}_{b}$-complete.

If there exist ${x}_{0},{y}_{0},{z}_{0}\in X$ such that $g{x}_{0}⪯F\left({x}_{0},{y}_{0},{z}_{0}\right)$, $g{y}_{0}⪰F\left({y}_{0},{x}_{0},{y}_{0}\right)$ and $g{z}_{0}⪯F\left({z}_{0},{y}_{0},{x}_{0}\right)$, then F and g have a tripled coincidence point in X.

Proof Let ${x}_{0},{y}_{0},{z}_{0}\in X$ be such that $g{x}_{0}⪯F\left({x}_{0},{y}_{0},{z}_{0}\right)$, $g{y}_{0}⪰F\left({y}_{0},{x}_{0},{y}_{0}\right)$ and $g{z}_{0}⪯F\left({z}_{0},{y}_{0},{x}_{0}\right)$. Define ${x}_{1},{y}_{1},{z}_{1}\in X$ such that $g{x}_{1}=F\left({x}_{0},{y}_{0},{z}_{0}\right)$, $g{y}_{1}=F\left({y}_{0},{x}_{0},{y}_{0}\right)$ and $g{z}_{1}=F\left({z}_{0},{y}_{0},{x}_{0}\right)$. Then $g{x}_{0}⪯g{x}_{1}$, $g{y}_{0}⪰g{y}_{1}$ and $g{z}_{0}⪯g{z}_{1}$. Similarly, define $g{x}_{2}=F\left({x}_{1},{y}_{1},{z}_{1}\right)$, $g{y}_{2}=F\left({y}_{1},{x}_{1},{y}_{1}\right)$ and $g{z}_{2}=F\left({z}_{1},{y}_{1},{x}_{1}\right)$. Since F has the mixed g-monotone property, we have $g{x}_{0}⪯g{x}_{1}⪯g{x}_{2}$, $g{y}_{0}⪰g{y}_{1}⪰g{y}_{2}$ and $g{z}_{0}⪯g{z}_{1}⪯g{z}_{2}$.

In this way, we construct the sequences $\left\{{a}_{n}\right\}$, $\left\{{b}_{n}\right\}$ and $\left\{{c}_{n}\right\}$ as

$\begin{array}{c}{a}_{n}=g{x}_{n}=F\left({x}_{n-1},{y}_{n-1},{z}_{n-1}\right),\hfill \\ {b}_{n}=g{y}_{n}=F\left({y}_{n-1},{x}_{n-1},{y}_{n-1}\right)\hfill \end{array}$

and

${c}_{n}=g{z}_{n}=F\left({z}_{n-1},{y}_{n-1},{x}_{n-1}\right)$

for all $n\ge 1$.

We will finish the proof in two steps.

Step I. We shall show that $\left\{{a}_{n}\right\}$, $\left\{{b}_{n}\right\}$ and $\left\{{c}_{n}\right\}$ are ${G}_{b}$-Cauchy.

Let

${\delta }_{n}=max\left\{G\left({a}_{n-1},{a}_{n},{a}_{n}\right),G\left({b}_{n-1},{b}_{n},{b}_{n}\right),G\left({c}_{n-1},{c}_{n},{c}_{n}\right)\right\}.$

So, we have

${\delta }_{n}={M}_{F}\left({x}_{n-2},{y}_{n-2},{z}_{n-2},{x}_{n-1},{y}_{n-1},{z}_{n-1},{x}_{n-1},{y}_{n-1},{z}_{n-1}\right)$

and

${\delta }_{n}={M}_{g}\left({x}_{n-1},{y}_{n-1},{z}_{n-1},{x}_{n},{y}_{n},{z}_{n},{x}_{n},{y}_{n},{z}_{n}\right).$

As $g{x}_{n-1}⪯g{x}_{n}$, $g{y}_{n-1}⪰g{y}_{n}$ and $g{z}_{n-1}⪯g{z}_{n}$, using (2.1) we obtain that

$\begin{array}{rcl}\psi \left(s{\delta }_{n+1}\right)& =& \psi \left(s{M}_{F}\left({x}_{n-1},{y}_{n-1},{z}_{n-1},{x}_{n},{y}_{n},{z}_{n},{x}_{n},{y}_{n},{z}_{n}\right)\right)\\ \le & \psi \left({M}_{g}\left({x}_{n-1},{y}_{n-1},{z}_{n-1},{x}_{n},{y}_{n},{z}_{n},{x}_{n},{y}_{n},{z}_{n}\right)\right)\\ -\phi \left({M}_{g}\left({x}_{n-1},{y}_{n-1},{z}_{n-1},{x}_{n},{y}_{n},{z}_{n},{x}_{n},{y}_{n},{z}_{n}\right)\right)\\ =& \psi \left({\delta }_{n}\right)-\phi \left({\delta }_{n}\right)\\ \le & \psi \left(s{\delta }_{n}\right)-\phi \left({\delta }_{n}\right).\end{array}$
(2.2)

Since ψ is an altering distance function, by (2.2) we deduce that

${\delta }_{n+1}\le {\delta }_{n},$

that is, $\left\{{\delta }_{n}\right\}$ is a nonincreasing sequence of nonnegative real numbers. Thus, there is $r\ge 0$ such that

$\underset{n\to \mathrm{\infty }}{lim}{\delta }_{n}=r.$

Letting $n\to \mathrm{\infty }$ in (2.2), from the continuity of ψ and φ, we obtain that

$\psi \left(sr\right)\le \psi \left(sr\right)-\phi \left(r\right),$

which implies that $\phi \left(r\right)=0$ and hence $r=0$.

Next, we claim that $\left\{{a}_{n}\right\}$, $\left\{{b}_{n}\right\}$ and $\left\{{c}_{n}\right\}$ are ${G}_{b}$-Cauchy.

We shall show that for every $\epsilon >0$, there exists $k\in \mathbb{N}$ such that if $m,n\ge k$,

$max\left\{G\left({a}_{m},{a}_{n},{a}_{n}\right),G\left({b}_{m},{b}_{n},{b}_{n}\right),G\left({c}_{m},{c}_{n},{c}_{n}\right)\right\}<\epsilon .$

Suppose that the above statement is false. Then there exists $\epsilon >0$ for which we can find subsequences $\left\{{a}_{m\left(k\right)}\right\}$ and $\left\{{a}_{n\left(k\right)}\right\}$ of $\left\{{a}_{n}\right\}$, $\left\{{b}_{m\left(k\right)}\right\}$ and $\left\{{b}_{n\left(k\right)}\right\}$ of $\left\{{b}_{n}\right\}$ and $\left\{{c}_{m\left(k\right)}\right\}$ and $\left\{{c}_{n\left(k\right)}\right\}$ of $\left\{{c}_{n}\right\}$ such that $n\left(k\right)>m\left(k\right)>k$ and

$max\left\{G\left({a}_{m\left(k\right)},{a}_{n\left(k\right)},{a}_{n\left(k\right)}\right),G\left({b}_{m\left(k\right)},{b}_{n\left(k\right)},{b}_{n\left(k\right)}\right),G\left({c}_{m\left(k\right)},{c}_{n\left(k\right)},{c}_{n\left(k\right)}\right)\right\}\ge \epsilon ,$
(2.3)

where $n\left(k\right)$ is the smallest index with this property, i.e.,

$max\left\{G\left({a}_{m\left(k\right)},{a}_{n\left(k\right)-1},{a}_{n\left(k\right)-1}\right),G\left({b}_{m\left(k\right)},{b}_{n\left(k\right)-1},{b}_{n\left(k\right)-1}\right),G\left({c}_{m\left(k\right)},{c}_{n\left(k\right)-1},{c}_{n\left(k\right)-1}\right)\right\}<\epsilon .$
(2.4)

From (2.4), we have

$\begin{array}{r}\underset{k⟶\mathrm{\infty }}{lim sup}max\left\{G\left({a}_{m\left(k\right)},{a}_{n\left(k\right)-1},{a}_{n\left(k\right)-1}\right),G\left({b}_{m\left(k\right)},{b}_{n\left(k\right)-1},{b}_{n\left(k\right)-1}\right),\\ \phantom{\rule{1em}{0ex}}G\left({c}_{m\left(k\right)},{c}_{n\left(k\right)-1},{c}_{n\left(k\right)-1}\right)\right\}\le \epsilon .\end{array}$
(2.5)

From the rectangle inequality,

$G\left({a}_{m\left(k\right)},{a}_{n\left(k\right)},{a}_{n\left(k\right)}\right)\le s\left[G\left({a}_{m\left(k\right)},{a}_{n\left(k\right)-1},{a}_{n\left(k\right)-1}\right)+G\left({a}_{n\left(k\right)-1},{a}_{n\left(k\right)},{a}_{n\left(k\right)}\right)\right].$
(2.6)

Similarly,

$G\left({b}_{m\left(k\right)},{b}_{n\left(k\right)},{b}_{n\left(k\right)}\right)\le s\left[G\left({b}_{m\left(k\right)},{b}_{n\left(k\right)-1},{b}_{n\left(k\right)-1}\right)+G\left({b}_{n\left(k\right)-1},{b}_{n\left(k\right)},{b}_{n\left(k\right)}\right)\right]$
(2.7)

and

$G\left({c}_{m\left(k\right)},{c}_{n\left(k\right)},{c}_{n\left(k\right)}\right)\le s\left[G\left({c}_{m\left(k\right)},{c}_{n\left(k\right)-1},{c}_{n\left(k\right)-1}\right)+G\left({c}_{n\left(k\right)-1},{c}_{n\left(k\right)},{c}_{n\left(k\right)}\right)\right].$
(2.8)

So,

$\begin{array}{r}max\left\{G\left({a}_{m\left(k\right)},{a}_{n\left(k\right)},{a}_{n\left(k\right)}\right),G\left({b}_{m\left(k\right)},{b}_{n\left(k\right)},{b}_{n\left(k\right)}\right),G\left({c}_{m\left(k\right)},{c}_{n\left(k\right)},{c}_{n\left(k\right)}\right)\right\}\\ \phantom{\rule{1em}{0ex}}\le smax\left\{G\left({a}_{m\left(k\right)},{a}_{n\left(k\right)-1},{a}_{n\left(k\right)-1}\right),G\left({b}_{m\left(k\right)},{b}_{n\left(k\right)-1},{b}_{n\left(k\right)-1}\right),G\left({c}_{m\left(k\right)},{c}_{n\left(k\right)-1},{c}_{n\left(k\right)-1}\right)\right\}\\ \phantom{\rule{2em}{0ex}}+smax\left\{G\left({a}_{n\left(k\right)-1},{a}_{n\left(k\right)},{a}_{n\left(k\right)}\right),G\left({b}_{n\left(k\right)-1},{b}_{n\left(k\right)},{b}_{n\left(k\right)}\right),G\left({c}_{n\left(k\right)-1},{c}_{n\left(k\right)},{c}_{n\left(k\right)}\right)\right\}.\end{array}$
(2.9)

Letting $k\to \mathrm{\infty }$ as ${lim}_{n\to \mathrm{\infty }}{\delta }_{n}=0$, by (2.3) and (2.4), we can conclude that

$\begin{array}{rl}\frac{\epsilon }{s}\le & \underset{k\to \mathrm{\infty }}{lim inf}max\left\{G\left({a}_{m\left(k\right)},{a}_{n\left(k\right)-1},{a}_{n\left(k\right)-1}\right),G\left({b}_{m\left(k\right)},{b}_{n\left(k\right)-1},{b}_{n\left(k\right)-1}\right),\\ G\left({c}_{m\left(k\right)},{c}_{n\left(k\right)-1},{c}_{n\left(k\right)-1}\right)\right\}.\end{array}$
(2.10)

Since

$G\left({a}_{m\left(k\right)},{a}_{n\left(k\right)},{a}_{n\left(k\right)}\right)\le sG\left({a}_{m\left(k\right)},{a}_{m\left(k\right)+1},{a}_{m\left(k\right)+1}\right)+sG\left({a}_{m\left(k\right)+1},{a}_{n\left(k\right)},{a}_{n\left(k\right)}\right)$
(2.11)

and

$G\left({b}_{m\left(k\right)},{b}_{n\left(k\right)},{b}_{n\left(k\right)}\right)\le sG\left({b}_{m\left(k\right)},{b}_{m\left(k\right)+1},{b}_{m\left(k\right)+1}\right)+sG\left({b}_{m\left(k\right)+1},{b}_{n\left(k\right)},{b}_{n\left(k\right)}\right),$
(2.12)

and

$G\left({c}_{m\left(k\right)},{c}_{n\left(k\right)},{c}_{n\left(k\right)}\right)\le sG\left({c}_{m\left(k\right)},{c}_{m\left(k\right)+1},{c}_{m\left(k\right)+1}\right)+sG\left({c}_{m\left(k\right)+1},{c}_{n\left(k\right)},{c}_{n\left(k\right)}\right),$
(2.13)

we obtain that

$\begin{array}{r}max\left\{G\left({a}_{m\left(k\right)},{a}_{n\left(k\right)},{a}_{n\left(k\right)}\right),G\left({b}_{m\left(k\right)},{b}_{n\left(k\right)},{b}_{n\left(k\right)}\right),G\left({c}_{m\left(k\right)},{c}_{n\left(k\right)},{c}_{n\left(k\right)}\right)\right\}\\ \phantom{\rule{1em}{0ex}}\le smax\left\{G\left({a}_{m\left(k\right)},{a}_{m\left(k\right)+1},{a}_{m\left(k\right)+1}\right),G\left({b}_{m\left(k\right)},{b}_{m\left(k\right)+1},{b}_{m\left(k\right)+1}\right),G\left({c}_{m\left(k\right)},{c}_{m\left(k\right)+1},{c}_{m\left(k\right)+1}\right)\right\}\\ \phantom{\rule{2em}{0ex}}+smax\left\{G\left({a}_{m\left(k\right)+1},{a}_{n\left(k\right)},{a}_{n\left(k\right)}\right),G\left({b}_{m\left(k\right)+1},{b}_{n\left(k\right)},{b}_{n\left(k\right)}\right),\\ \phantom{\rule{2em}{0ex}}G\left({c}_{m\left(k\right)+1},{c}_{n\left(k\right)},{c}_{n\left(k\right)}\right)\right\}.\end{array}$
(2.14)

If in the above inequality $k\to \mathrm{\infty }$ as ${lim}_{n\to \mathrm{\infty }}{\delta }_{n}=0$, from (2.3) we have

$\begin{array}{rl}\frac{\epsilon }{s}\le & \underset{k\to \mathrm{\infty }}{lim sup}max\left\{G\left({a}_{m\left(k\right)+1},{a}_{n\left(k\right)},{a}_{n\left(k\right)}\right),G\left({b}_{m\left(k\right)+1},{b}_{n\left(k\right)},{b}_{n\left(k\right)}\right),\\ G\left({c}_{m\left(k\right)+1},{c}_{n\left(k\right)},{c}_{n\left(k\right)}\right)\right\}.\end{array}$
(2.15)

As $n\left(k\right)>m\left(k\right)$, we have $g{x}_{m\left(k\right)}⪯g{x}_{n\left(k\right)-1}$, $g{y}_{m\left(k\right)}⪰g{y}_{n\left(k\right)-1}$ and $g{z}_{m\left(k\right)}⪯g{z}_{n\left(k\right)-1}$. Putting $x={x}_{m\left(k\right)}$, $y={y}_{m\left(k\right)}$, $z={z}_{m\left(k\right)}$, $u={x}_{n\left(k\right)-1}$, $v={y}_{n\left(k\right)-1}$, $w={z}_{n\left(k\right)-1}$, $r={x}_{n\left(k\right)-1}$, $s={y}_{n\left(k\right)-1}$ and $t={z}_{n\left(k\right)-1}$ in (2.1), we have

$\begin{array}{r}\psi \left(s\cdot max\left\{G\left({a}_{m\left(k\right)+1},{a}_{n\left(k\right)},{a}_{n\left(k\right)}\right),G\left({b}_{m\left(k\right)+1},{b}_{n\left(k\right)},{b}_{n\left(k\right)}\right),G\left({c}_{m\left(k\right)+1},{c}_{n\left(k\right)},{c}_{n\left(k\right)}\right)\right\}\right)\\ \phantom{\rule{1em}{0ex}}=\psi \left(s\cdot {M}_{F}\left({x}_{m\left(k\right)},{y}_{m\left(k\right)},{z}_{m\left(k\right)},{x}_{n\left(k\right)-1},{y}_{n\left(k\right)-1},{z}_{n\left(k\right)-1},{x}_{n\left(k\right)-1},{y}_{n\left(k\right)-1},{z}_{n\left(k\right)-1}\right)\right)\\ \phantom{\rule{1em}{0ex}}\le \psi \left({M}_{g}\left({x}_{m\left(k\right)},{y}_{m\left(k\right)},{z}_{m\left(k\right)},{x}_{n\left(k\right)-1},{y}_{n\left(k\right)-1},{z}_{n\left(k\right)-1},{x}_{n\left(k\right)-1},{y}_{n\left(k\right)-1},{z}_{n\left(k\right)-1}\right)\right)\\ \phantom{\rule{2em}{0ex}}-\phi \left({M}_{g}\left({x}_{m\left(k\right)},{y}_{m\left(k\right)},{z}_{m\left(k\right)},{x}_{n\left(k\right)-1},{y}_{n\left(k\right)-1},{z}_{n\left(k\right)-1},{x}_{n\left(k\right)-1},{y}_{n\left(k\right)-1},{z}_{n\left(k\right)-1}\right)\right)\\ \phantom{\rule{1em}{0ex}}=\psi \left(max\left\{G\left(g{x}_{m\left(k\right)},g{x}_{n\left(k\right)-1},g{x}_{n\left(k\right)-1}\right),G\left(g{y}_{m\left(k\right)},g{y}_{n\left(k\right)-1},g{y}_{n\left(k\right)-1}\right),\\ \phantom{\rule{2em}{0ex}}G\left(g{z}_{m\left(k\right)},g{z}_{n\left(k\right)-1},g{z}_{n\left(k\right)-1}\right)\right\}\right)\\ \phantom{\rule{2em}{0ex}}-\phi \left(max\left\{G\left(g{x}_{m\left(k\right)},g{x}_{n\left(k\right)-1},g{x}_{n\left(k\right)-1}\right),G\left(g{y}_{m\left(k\right)},g{y}_{n\left(k\right)-1},g{y}_{n\left(k\right)-1}\right),\\ \phantom{\rule{2em}{0ex}}G\left(g{z}_{m\left(k\right)},g{z}_{n\left(k\right)-1},g{z}_{n\left(k\right)-1}\right)\right\}\right)\\ \phantom{\rule{1em}{0ex}}=\psi \left(max\left\{G\left({a}_{m\left(k\right)},{a}_{n\left(k\right)-1},{a}_{n\left(k\right)-1}\right),G\left({b}_{m\left(k\right)},{b}_{n\left(k\right)-1},{b}_{n\left(k\right)-1}\right),\\ \phantom{\rule{2em}{0ex}}G\left({c}_{m\left(k\right)},{c}_{n\left(k\right)-1},{c}_{n\left(k\right)-1}\right)\right\}\right)\\ \phantom{\rule{2em}{0ex}}-\phi \left(max\left\{G\left({a}_{m\left(k\right)},{a}_{n\left(k\right)-1},{a}_{n\left(k\right)-1}\right),G\left({b}_{m\left(k\right)},{b}_{n\left(k\right)-1},{b}_{n\left(k\right)-1}\right),\\ \phantom{\rule{2em}{0ex}}G\left({c}_{m\left(k\right)},{c}_{n\left(k\right)-1},{c}_{n\left(k\right)-1}\right)\right\}\right).\end{array}$
(2.16)

Letting $k\to \mathrm{\infty }$ in (2.16),

$\begin{array}{rcl}\psi \left(s\cdot \frac{\epsilon }{s}\right)& \le & \psi \left(s\cdot \underset{k\to \mathrm{\infty }}{lim sup}max\left\{G\left({a}_{m\left(k\right)+1},{a}_{n\left(k\right)},{a}_{n\left(k\right)}\right),G\left({b}_{m\left(k\right)+1},{b}_{n\left(k\right)},{b}_{n\left(k\right)}\right),\\ G\left({c}_{m\left(k\right)+1},{c}_{n\left(k\right)},{c}_{n\left(k\right)}\right)\right\}\right)\\ \le & \psi \left(\underset{k\to \mathrm{\infty }}{lim sup}max\left\{G\left({a}_{m\left(k\right)},{a}_{n\left(k\right)-1},{a}_{n\left(k\right)-1}\right),G\left({b}_{m\left(k\right)},{b}_{n\left(k\right)-1},{b}_{n\left(k\right)-1}\right),\\ G\left({c}_{m\left(k\right)},{c}_{n\left(k\right)-1},{c}_{n\left(k\right)-1}\right)\right\}\right)\\ -\phi \left(\underset{k\to \mathrm{\infty }}{lim inf}max\left\{G\left({a}_{m\left(k\right)},{a}_{n\left(k\right)-1},{a}_{n\left(k\right)-1}\right),G\left({b}_{m\left(k\right)},{b}_{n\left(k\right)-1},{b}_{n\left(k\right)-1}\right),\\ G\left({c}_{m\left(k\right)},{c}_{n\left(k\right)-1},{c}_{n\left(k\right)-1}\right)\right\}\right)\\ \le & \psi \left(\epsilon \right)-\phi \left(\underset{k\to \mathrm{\infty }}{lim inf}max\left\{G\left({a}_{m\left(k\right)},{a}_{n\left(k\right)-1},{a}_{n\left(k\right)-1}\right),G\left({b}_{m\left(k\right)},{b}_{n\left(k\right)-1},{b}_{n\left(k\right)-1}\right),\\ G\left({c}_{m\left(k\right)},{c}_{n\left(k\right)-1},{c}_{n\left(k\right)-1}\right)\right\}\right).\end{array}$
(2.17)

From (2.17), we have

$\begin{array}{r}\phi \left(\underset{k\to \mathrm{\infty }}{lim inf}max\left\{G\left({a}_{m\left(k\right)},{a}_{n\left(k\right)-1},{a}_{n\left(k\right)-1}\right),G\left({b}_{m\left(k\right)},{b}_{n\left(k\right)-1},{b}_{n\left(k\right)-1}\right),\\ \phantom{\rule{1em}{0ex}}G\left({c}_{m\left(k\right)},{c}_{n\left(k\right)-1},{c}_{n\left(k\right)-1}\right)\right\}\right)\le 0.\end{array}$

Therefore,

$\underset{k\to \mathrm{\infty }}{lim inf}max\left\{G\left({a}_{m\left(k\right)},{a}_{n\left(k\right)-1},{a}_{n\left(k\right)-1}\right),G\left({b}_{m\left(k\right)},{b}_{n\left(k\right)-1},{b}_{n\left(k\right)-1}\right),G\left({c}_{m\left(k\right)},{c}_{n\left(k\right)-1},{c}_{n\left(k\right)-1}\right)\right\}=0,$

which is a contradiction to (2.10). Consequently, $\left\{{a}_{n}\right\}$, $\left\{{b}_{n}\right\}$ and $\left\{{c}_{n}\right\}$ are ${G}_{b}$-Cauchy.

Step II. We shall show that F and g have a tripled coincidence point.

First, let (a) hold, that is, F is ${G}_{b}$-continuous and $\left(X,G\right)$ is ${G}_{b}$-complete.

Since X is ${G}_{b}$-complete and $\left\{{a}_{n}\right\}$ is ${G}_{b}$-Cauchy, there exists $a\in X$ such that

$\underset{n\to \mathrm{\infty }}{lim}G\left({a}_{n},{a}_{n},a\right)=\underset{n\to \mathrm{\infty }}{lim}G\left(g{x}_{n},g{x}_{n},a\right)=0.$
(2.18)

Similarly, there exist $b,c\in X$ such that

$\underset{n\to \mathrm{\infty }}{lim}G\left({b}_{n},{b}_{n},b\right)=\underset{n\to \mathrm{\infty }}{lim}G\left(g{y}_{n},g{y}_{n},b\right)=0$
(2.19)

and

$\underset{n\to \mathrm{\infty }}{lim}G\left({c}_{n},{c}_{n},c\right)=\underset{n\to \mathrm{\infty }}{lim}G\left(g{z}_{n},g{z}_{n},c\right)=0.$
(2.20)

Now, we prove that $\left(a,b,c\right)$ is a tripled coincidence point of F and g.

Continuity of g and Lemma 1.25 yields that

$\begin{array}{rcl}0& =& \frac{1}{{s}^{3}}G\left(ga,ga,ga\right)\le lim\underset{n\to \mathrm{\infty }}{inf}G\left(g\left(g{x}_{n}\right),g\left(g{x}_{n}\right),ga\right)\\ \le & lim\underset{n\to \mathrm{\infty }}{sup}G\left(g\left(g{x}_{n}\right),g\left(g{x}_{n}\right),ga\right)\le {s}^{3}G\left(ga,ga,ga\right)=0.\end{array}$

Hence,

$\underset{n\to \mathrm{\infty }}{lim}G\left(g\left(g{x}_{n}\right),g\left(g{x}_{n}\right),ga\right)=0$
(2.21)

and similarly,

$\underset{n\to \mathrm{\infty }}{lim}G\left(g\left(g{y}_{n}\right),g\left(g{y}_{n}\right),gb\right)=0$
(2.22)

and

$\underset{n\to \mathrm{\infty }}{lim}G\left(g\left(g{z}_{n}\right),g\left(g{z}_{n}\right),gc\right)=0.$
(2.23)

Since $g{x}_{n+1}=F\left({x}_{n},{y}_{n},{z}_{n}\right)$, $g{y}_{n+1}=F\left({y}_{n},{x}_{n},{y}_{n}\right)$ and $g{z}_{n+1}=F\left({z}_{n},{y}_{n},{x}_{n}\right)$, the commutativity of F and g yields that

$g\left(g{x}_{n+1}\right)=g\left(F\left({x}_{n},{y}_{n},{z}_{n}\right)\right)=F\left(g{x}_{n},g{y}_{n},g{z}_{n}\right),$
(2.24)
$g\left(g{y}_{n+1}\right)=g\left(F\left({y}_{n},{x}_{n},{y}_{n}\right)\right)=F\left(g{y}_{n},g{x}_{n},g{y}_{n}\right)$
(2.25)

and

$g\left(g{z}_{n+1}\right)=g\left(F\left({z}_{n},{y}_{n},{x}_{n}\right)\right)=F\left(g{z}_{n},g{y}_{n},g{x}_{n}\right).$
(2.26)

From the continuity of F and (2.24), (2.25) and (2.26) and Lemma 1.25, $\left\{g\left(g{x}_{n+1}\right)\right\}$ is ${G}_{b}$-convergent to $F\left(a,b,c\right)$, $\left\{g\left(g{y}_{n+1}\right)\right\}$ is ${G}_{b}$-convergent to $F\left(b,a,b\right)$ and $\left\{g\left(g{z}_{n+1}\right)\right\}$ is ${G}_{b}$-convergent to $F\left(c,b,a\right)$. From (2.21), (2.22) and (2.23) and uniqueness of the limit, we have $F\left(a,b,c\right)=ga$, $F\left(b,a,b\right)=gb$ and $F\left(c,b,a\right)=gc$, that is, g and F have a tripled coincidence point.

In what follows, suppose that assumption (b) holds.

Following the proof of the previous step, there exist $u,v,w\in X$ such that

$\underset{n\to \mathrm{\infty }}{lim}G\left(g{x}_{n},g{x}_{n},gu\right)=0,$
(2.27)
$\underset{n\to \mathrm{\infty }}{lim}G\left(g{y}_{n},g{y}_{n},gv\right)=0$
(2.28)

and

$\underset{n\to \mathrm{\infty }}{lim}G\left(g{z}_{n},g{z}_{n},gw\right)=0,$
(2.29)

as $\left(g\left(X\right),G\right)$ is ${G}_{b}$-complete.

Now, we prove that $F\left(u,v,w\right)=gu$, $F\left(v,u,v\right)=gv$ and $F\left(w,v,u\right)=gw$. From regularity of X and using (2.1), we have

$\begin{array}{r}\psi \left(s{M}_{F}\left({x}_{n},{y}_{n},{z}_{n},u,v,w,u,v,w\right)\right)\\ \phantom{\rule{1em}{0ex}}\le \psi \left(max\left\{G\left(g{x}_{n},gu,gu\right),G\left(g{y}_{n},gv,gv\right),G\left(g{z}_{n},gw,gw\right)\right\}\right)\\ \phantom{\rule{2em}{0ex}}-\phi \left(max\left\{G\left(g{x}_{n},gu,gu\right),G\left(g{y}_{n},gv,gv\right),G\left(g{z}_{n},gw,gw\right)\right\}\right).\end{array}$
(2.30)

As $\left\{g{x}_{n}\right\}$ is ${G}_{b}$-convergent to gu, from Lemma 1.25, we have ${lim}_{n\to \mathrm{\infty }}G\left(g{x}_{n},gu,gu\right)=0$. Analogously, ${lim}_{n\to \mathrm{\infty }}G\left(g{y}_{n},gv,gv\right)={lim}_{n\to \mathrm{\infty }}G\left(g{z}_{n},gw,gw\right)=0$.

As ψ and φ are continuous, from (2.30) we have

$\underset{n\to \mathrm{\infty }}{lim}{M}_{F}\left({x}_{n},{y}_{n},{z}_{n},u,v,w,u,v,w\right)=0,$

or, equivalently,

$\underset{n\to \mathrm{\infty }}{lim}G\left(g{x}_{n+1},F\left(u,v,w\right),F\left(u,v,w\right)\right)=0.$
(2.31)

Similarly,

$\underset{n\to \mathrm{\infty }}{lim}G\left(g{y}_{n+1},F\left(v,u,v\right),F\left(v,u,v\right)\right)=\underset{n\to \mathrm{\infty }}{lim}G\left(g{z}_{n+1},F\left(w,v,u\right),F\left(w,v,u\right)\right)=0.$
(2.32)

On the other hand,

$\begin{array}{r}G\left(gu,F\left(u,v,w\right),F\left(u,v,w\right)\\ \phantom{\rule{1em}{0ex}}\le sG\left(gu,g{x}_{n+1},g{x}_{n+1}\right)+sG\left(g{x}_{n+1},F\left(u,v,w\right),F\left(u,v,w\right).\end{array}$
(2.33)

Taking the limit when $n\to \mathrm{\infty }$ and using (2.27) and (2.31), we get

$\begin{array}{rl}G\left(gu,F\left(u,v,w\right),F\left(u,v,w\right)\right)\le & s\underset{n\to \mathrm{\infty }}{lim}G\left(gu,g{x}_{n+1},g{x}_{n+1}\right)\\ +s\underset{n\to \mathrm{\infty }}{lim}G\left(g{x}_{n+1},F\left(u,v,w\right),F\left(u,v,w\right)=0,\end{array}$
(2.34)

that is, $gu=F\left(u,v,w\right)$.

Analogously, we can show that $gv=F\left(v,u,v\right)$ and $gw=F\left(w,v,u\right)$.

Thus, we have proved that g and F have a tripled coincidence point. This completes the proof of the theorem. □

Let

$M\left(x,y,z,u,v,w,r,s,t\right)=max\left\{G\left(x,u,r\right),G\left(y,v,s\right),G\left(z,w,t\right)\right\}.$

Taking $g={I}_{X}$ (the identity mapping on X) in Theorem 2.1, we obtain the following tripled fixed point result.

Corollary 2.2 Let $\left(X,⪯,G\right)$ be a ${G}_{b}$-complete partially ordered ${G}_{b}$-metric space, and let $F:{X}^{3}\to X$ be a mapping with the mixed monotone property. Assume that

$\begin{array}{r}\psi \left(s{M}_{F}\left(x,y,z,u,v,w,r,s,t\right)\right)\\ \phantom{\rule{1em}{0ex}}\le \psi \left(M\left(x,y,z,u,v,w,r,s,t\right)\right)-\phi \left(M\left(x,y,z,u,v,w,r,s,t\right)\right)\end{array}$
(2.35)

for every $x,y,z,u,v,w,r,s,t\in X$ with $x⪯u⪯r$, $y⪰v⪰s$ and $z⪯w⪯t$, or $r⪯u⪯x$, $s⪰v⪰y$ and $t⪯w⪯z$, where $\psi ,\phi :\left[0,\mathrm{\infty }\right)\to \left[0,\mathrm{\infty }\right)$ are altering distance functions.

Also suppose that

1. (a)

either F is ${G}_{b}$-continuous, or

2. (b)

$\left(X,G\right)$ is regular.

If there exist ${x}_{0},{y}_{0},{z}_{0}\in X$ such that ${x}_{0}⪯F\left({x}_{0},{y}_{0},{z}_{0}\right)$, ${y}_{0}⪰F\left({y}_{0},{x}_{0},{y}_{0}\right)$ and ${z}_{0}⪯F\left({z}_{0},{y}_{0},{x}_{0}\right)$, then F has a tripled fixed point in X.

Taking $\psi \left(t\right)=t$ and $\phi \left(t\right)=\frac{{t}^{2}}{1+t}$ for all $t\in \left[0,\mathrm{\infty }\right)$ in Corollary 2.2, we obtain the following tripled fixed point result.

Corollary 2.3 Let $\left(X,⪯,G\right)$ be a ${G}_{b}$-complete partially ordered ${G}_{b}$-metric space and $F:{X}^{3}\to X$ with the mixed monotone property. Assume that

$s{M}_{F}\left(x,y,z,u,v,w,r,s,t\right)\le \frac{M\left(x,y,z,u,v,w,r,s,t\right)}{1+M\left(x,y,z,u,v,w,r,s,t\right)}$
(2.36)

for every $x,y,z,u,v,w,r,s,t\in X$ with $x⪯u⪯r$, $y⪰v⪰s$ and $z⪯w⪯t$, or $r⪯u⪯x$, $s⪰v⪰y$ and $t⪯w⪯z$.

Also suppose that

1. (a)

either F is ${G}_{b}$-continuous, or

2. (b)

$\left(X,G\right)$ is regular.

If there exist ${x}_{0},{y}_{0},{z}_{0}\in X$ such that ${x}_{0}⪯F\left({x}_{0},{y}_{0},{z}_{0}\right)$, ${y}_{0}⪰F\left({y}_{0},{x}_{0},{y}_{0}\right)$ and ${z}_{0}⪯F\left({z}_{0},{y}_{0},{x}_{0}\right)$, then F has a tripled fixed point in X.

Remark 2.4 Theorem 1.8 is a special case of Theorem 2.1.

Remark 2.5 Theorem 2.1 part (a) holds if we replace the commutativity assumption of F and g by compatibility assumption (also see Remark 2.2 of ).

The following corollary can be deduced from our previously obtained results.

Corollary 2.6 Let $\left(X,⪯\right)$ be a partially ordered set and $\left(X,G\right)$ be a ${G}_{b}$-complete ${G}_{b}$-metric space. Let $F:{X}^{3}\to X$ be a mapping with the mixed monotone property such that

$\begin{array}{rl}\psi \left(s{M}_{F}\left(x,y,z,u,v,w,r,s,t\right)\right)\le & \psi \left(\frac{G\left(x,u,r\right)+G\left(y,v,s\right)+G\left(z,w,t\right)}{3}\right)\\ -\phi \left(max\left\{G\left(x,u,r\right),G\left(y,v,s\right),G\left(z,w,t\right)\right\}\right)\end{array}$
(2.37)

for every $x,y,z,u,v,w,r,s,t\in X$ with $x⪯u⪯r$, $y⪰v⪰s$ and $z⪯w⪯t$, or $r⪯u⪯x$, $s⪰v⪰y$ and $t⪯w⪯z$.

Also suppose that

1. (a)

either F is ${G}_{b}$-continuous, or

2. (b)

$\left(X,G\right)$ is regular.

If there exist ${x}_{0},{y}_{0},{z}_{0}\in X$ such that ${x}_{0}⪯F\left({x}_{0},{y}_{0},{z}_{0}\right)$, ${y}_{0}⪰F\left({y}_{0},{x}_{0},{y}_{0}\right)$ and ${z}_{0}⪯F\left({z}_{0},{y}_{0},{x}_{0}\right)$, then F has a tripled fixed point in X.

Proof If F satisfies (2.37), then F satisfies (2.35). So, the result follows from Theorem 2.1. □

In Theorem 2.1, if we take $\psi \left(t\right)=t$ and $\phi \left(t\right)=\left(1-k\right)t$ for all $t\in \left[0,\mathrm{\infty }\right)$, where $k\in \left[0,1\right)$, we obtain the following result.

Corollary 2.7 Let $\left(X,⪯\right)$ be a partially ordered set and $\left(X,G\right)$ be a ${G}_{b}$-complete ${G}_{b}$-metric space. Let $F:{X}^{3}\to X$ be a mapping having the mixed monotone property and

${M}_{F}\left(x,y,z,u,v,w,r,s,t\right)\le \frac{k}{s}M\left(x,y,z,u,v,w,r,s,t\right)$
(2.38)

for every $x,y,z,u,v,w,r,s,t\in X$ with $x⪯u⪯r$, $y⪰v⪰s$ and $z⪯w⪯t$, or $r⪯u⪯x$, $s⪰v⪰y$ and $t⪯w⪯z$.

Also suppose that

1. (a)

either F is ${G}_{b}$-continuous, or

2. (b)

$\left(X,G\right)$ is regular.

If there exist ${x}_{0},{y}_{0},{z}_{0}\in X$ such that ${x}_{0}⪯F\left({x}_{0},{y}_{0},{z}_{0}\right)$, ${y}_{0}⪰F\left({y}_{0},{x}_{0},{y}_{0}\right)$ and ${z}_{0}⪯F\left({z}_{0},{y}_{0},{x}_{0}\right)$, then F has a tripled fixed point in X.

Corollary 2.8 Let $\left(X,⪯\right)$ be a partially ordered set and $\left(X,G\right)$ be a ${G}_{b}$-complete ${G}_{b}$-metric space. Let $F:{X}^{3}\to X$ be a mapping with the mixed monotone property such that

${M}_{F}\left(x,y,z,u,v,w,r,s,t\right)\le \frac{k}{3s}\left[G\left(x,u,r\right)+G\left(y,v,s\right)+G\left(z,w,t\right)\right]$
(2.39)

for every $x,y,z,u,v,w,r,s,t\in X$ with $x⪯u⪯r$, $y⪰v⪰s$ and $z⪯w⪯t$, or $r⪯u⪯x$, $s⪰v⪰y$ and $t⪯w⪯z$.

Also suppose that

1. (a)

either F is ${G}_{b}$-continuous, or

2. (b)

$\left(X,G\right)$ is regular.

If there exist ${x}_{0},{y}_{0},{z}_{0}\in X$ such that ${x}_{0}⪯F\left({x}_{0},{y}_{0},{z}_{0}\right)$, ${y}_{0}⪰F\left({y}_{0},{x}_{0},{y}_{0}\right)$ and ${z}_{0}⪯F\left({z}_{0},{y}_{0},{x}_{0}\right)$, then F has a tripled fixed point in X.

Proof If F satisfies (2.39), then F satisfies (2.38). □

Note that if $\left(X,⪯\right)$ is a partially ordered set, then we can endow ${X}^{3}$ with the following partial order relation:

$\left(x,y,z\right)⪯\left(u,v,w\right)\phantom{\rule{1em}{0ex}}⟺\phantom{\rule{1em}{0ex}}x⪯u,\phantom{\rule{2em}{0ex}}y⪰v,\phantom{\rule{2em}{0ex}}z⪯w$

for all $\left(x,y,z\right),\left(u,v,w\right)\in {X}^{3}$ (see ).

In the following theorem, we give a sufficient condition for the uniqueness of the common tripled fixed point (also see, e.g., [4, 46, 50] and ).

Theorem 2.9 In addition to the hypotheses of Theorem 2.1, suppose that for every $\left(x,y,z\right)$ and $\left({x}^{\ast },{y}^{\ast },{z}^{\ast }\right)\in X×X×X$, there exists $\left(u,v,w\right)\in {X}^{3}$ such that $\left(F\left(u,v,w\right),F\left(v,u,v\right),F\left(w,v,u\right)\right)$ is comparable with $\left(F\left(x,y,z\right),F\left(y,x,y\right),F\left(z,y,x\right)\right)$ and $\left(F\left({x}^{\ast },{y}^{\ast },{z}^{\ast }\right),F\left({y}^{\ast },{x}^{\ast },{y}^{\ast }\right),F\left({z}^{\ast },{y}^{\ast },{x}^{\ast }\right)\right)$. Then F and g have a unique common tripled fixed point.

Proof From Theorem 2.1 the set of tripled coincidence points of F and g is nonempty. We shall show that if $\left(x,y,z\right)$ and $\left({x}^{\ast },{y}^{\ast },{z}^{\ast }\right)$ are tripled coincidence points, that is,

$g\left(x\right)=F\left(x,y,z\right),\phantom{\rule{2em}{0ex}}g\left(y\right)=F\left(y,x,y\right),\phantom{\rule{2em}{0ex}}g\left(z\right)=F\left(z,y,x\right)$

and

$g\left({x}^{\ast }\right)=F\left({x}^{\ast },{y}^{\ast },{z}^{\ast }\right),\phantom{\rule{2em}{0ex}}g\left({y}^{\ast }\right)=F\left({y}^{\ast },{x}^{\ast },{y}^{\ast }\right),\phantom{\rule{2em}{0ex}}g\left({z}^{\ast }\right)=F\left({z}^{\ast },{y}^{\ast },{x}^{\ast }\right),$

then $gx=g{x}^{\ast }$ and $gy=g{y}^{\ast }$ and $gz=g{z}^{\ast }$.

Choose an element $\left(u,v,w\right)\in {X}^{3}$ such that $\left(F\left(u,v,w\right),F\left(v,u,v\right),F\left(w,v,u\right)\right)$ is comparable with

$\left(F\left(x,y,z\right),F\left(y,x,y\right),F\left(z,y,x\right)\right)$

and

$\left(F\left({x}^{\ast },{y}^{\ast },{z}^{\ast }\right),F\left({y}^{\ast },{x}^{\ast },{y}^{\ast }\right),F\left({z}^{\ast },{y}^{\ast },{x}^{\ast }\right)\right).$

Let ${u}_{0}=u$, ${v}_{0}=v$ and ${w}_{0}=w$ and choose ${u}_{1}$, ${v}_{1}$ and ${w}_{1}\in X$ so that $g{u}_{1}=F\left({u}_{0},{v}_{0},{w}_{0}\right)$, $g{v}_{1}=F\left({v}_{0},{u}_{0},{v}_{0}\right)$ and $g{w}_{1}=F\left({w}_{0},{v}_{0},{u}_{0}\right)$. Then, similarly as in the proof of Theorem 2.1, we can inductively define sequences $\left\{g{u}_{n}\right\}$, $\left\{g{v}_{n}\right\}$ and $\left\{g{w}_{n}\right\}$ such that $g{u}_{n+1}=F\left({u}_{n},{v}_{n},{w}_{n}\right)$, $g{v}_{n+1}=F\left({v}_{n},{u}_{n},{v}_{n}\right)$ and $g{w}_{n+1}=F\left({w}_{n},{v}_{n},{u}_{n}\right)$. Since $\left(gx,gy,gz\right)=\left(F\left(x,y,z\right),F\left(y,x,y\right),F\left(w,y,x\right)\right)$ and $\left(F\left(u,v,w\right),F\left(v,u,v\right),F\left(w,v,u\right)\right)=\left(g{u}_{1},g{v}_{1},g{w}_{1}\right)$ are comparable, we may assume that $\left(gx,gy,gz\right)⪯\left(g{u}_{1},g{v}_{1},g{w}_{1}\right)$. Then $gx⪯g{u}_{1}$, $gy⪰g{v}_{1}$ and $gz⪯g{w}_{1}$. Using the mathematical induction, it is easy to prove that $gx⪯g{u}_{n}$, $gy⪰g{v}_{n}$ and $gz⪯g{w}_{n}$ for all $n\ge 0$.

Applying (2.1), as $gx⪯g{u}_{n}$, $gy⪰g{v}_{n}$ and $gz⪯g{w}_{n}$, one obtains that

$\begin{array}{r}\psi \left(smax\left\{G\left(gx,g{u}_{n+1},g{u}_{n+1}\right),G\left(gy,g{v}_{n+1},g{v}_{n+1}\right),G\left(gz,g{w}_{n+1},g{w}_{n+1}\right)\right\}\right)\\ \phantom{\rule{1em}{0ex}}=\psi \left(s{M}_{F}\left(x,y,z,{u}_{n},{v}_{n},{w}_{n},{u}_{n},{v}_{n},{w}_{n}\right)\right)\\ \phantom{\rule{1em}{0ex}}\le \psi \left({M}_{g}\left(x,y,z,{u}_{n},{v}_{n},{w}_{n},{u}_{n},{v}_{n},{w}_{n}\right)\right)-\phi \left({M}_{g}\left(x,y,z,{u}_{n},{v}_{n},{w}_{n},{u}_{n},{v}_{n},{w}_{n}\right)\right)\\ \phantom{\rule{1em}{0ex}}=\psi \left(max\left\{G\left(gx,g{u}_{n},g{u}_{n}\right),G\left(gy,g{v}_{n},g{v}_{n}\right),G\left(gz,g{w}_{n},g{w}_{n}\right)\right\}\right)\\ \phantom{\rule{2em}{0ex}}-\phi \left(max\left\{G\left(gx,g{u}_{n},g{u}_{n}\right),G\left(gy,g{v}_{n},g{v}_{n}\right),G\left(gz,g{w}_{n},g{w}_{n}\right)\right\}\right).\end{array}$
(2.40)

From the properties of ψ, we deduce that

$\left\{max\left\{G\left(gx,g{u}_{n},g{u}_{n}\right),G\left(gy,g{v}_{n},g{v}_{n}\right),G\left(gz,g{w}_{n},g{w}_{n}\right)\right\}\right\}$

is nonincreasing.

Hence, if we proceed as in Theorem 2.1, we can show that

$\underset{n\to \mathrm{\infty }}{lim}max\left\{G\left(gx,g{u}_{n},g{u}_{n}\right),G\left(gy,g{v}_{n},g{v}_{n}\right),G\left(gz,g{w}_{n},g{w}_{n}\right)\right\}=0,$

that is, $\left\{g{u}_{n}\right\}$, $\left\{g{v}_{n}\right\}$ and $\left\{g{w}_{n}\right\}$ are ${G}_{b}$-convergent to gx, gy and gz, respectively.

Similarly, we can show that

$\underset{n\to \mathrm{\infty }}{lim}max\left\{G\left(g{x}^{\ast },g{u}_{n},g{u}_{n}\right),G\left(g{y}^{\ast },g{v}_{n},g{v}_{n}\right),G\left(g{z}^{\ast },g{w}_{n},g{w}_{n}\right)\right\}=0,$

that is, $\left\{g{u}_{n}\right\}$, $\left\{g{v}_{n}\right\}$ and $\left\{g{w}_{n}\right\}$ are ${G}_{b}$-convergent to $g{x}^{\ast }$, $g{y}^{\ast }$ and $g{z}^{\ast }$, respectively. Finally, since the limit is unique, $gx=g{x}^{\ast }$, $gy=g{y}^{\ast }$ and $gz=g{z}^{\ast }$.

Since $gx=F\left(x,y,z\right)$, $gy=F\left(y,x,y\right)$ and $gz=F\left(z,y,x\right)$, by commutativity of F and g, we have $g\left(gx\right)=g\left(F\left(x,y,z\right)\right)=F\left(gx,gy,gz\right)$, $g\left(gy\right)=g\left(F\left(y,x,y\right)\right)=F\left(gy,gx,gy\right)$ and $g\left(gz\right)=g\left(F\left(z,y,x\right)\right)=F\left(gz,gy,gx\right)$. Let $gx=a$, $gy=b$ and $g\left(z\right)=c$. Then $ga=F\left(a,b,c\right)$, $gb=F\left(b,a,b\right)$ and $gc=F\left(c,b,a\right)$. Thus, $\left(a,b,c\right)$ is another tripled coincidence point of F and g. Then $a=gx=ga$, $b=gy=gb$ and $c=gz=gc$. Therefore, $\left(a,b,c\right)$ is a tripled common fixed point of F and g.

To prove the uniqueness, assume that $\left(p,q,r\right)$ is another tripled common fixed point of F and g. Then $p=gp=F\left(p,q,r\right)$, $q=gq=F\left(q,p,q\right)$ and $r=gr=F\left(r,p,q\right)$. Since $\left(p,q,r\right)$ is a tripled coincidence point of F and g, we have $gp=gx$, $gq=gy$ and $gr=gz$. Thus, $p=gp=ga=a$, $q=gq=gb=b$ and $r=gr=gc=c$. Hence, the tripled common fixed point is unique. □

## 3 Examples

The following examples support our results.

Example 3.1 Let $X=\left(-\mathrm{\infty },\mathrm{\infty }\right)$ be endowed with the usual ordering and the ${G}_{b}$-complete ${G}_{b}$-metric

$G\left(x,y,z\right)={\left(|x-y|+|y-z|+|z-x|\right)}^{2},$

where $s=2$.

Define $F:{X}^{3}\to X$ as

$F\left(x,y,z\right)=\frac{x-2y+4z}{96}$

for all $x,y,z\in X$ and $g:X\to X$ with $g\left(x\right)=\frac{x}{2}$ for all $x\in X$.

Let $\phi :\left[0,\mathrm{\infty }\right)\to \left[0,\mathrm{\infty }\right)$ be defined by $\phi \left(t\right)=ln\left(t+1\right)$, and let $\psi :\left[0,\mathrm{\infty }\right)\to \left[0,\mathrm{\infty }\right)$ be defined by $\psi \left(t\right)=ln\left(\frac{4t+4}{t+4}\right)$.

Now, from the fact that for $\alpha ,\beta ,\gamma \ge 0$, ${\left(\alpha +\beta +\gamma \right)}^{p}\le {2}^{2p-2}{\alpha }^{p}+{2}^{2p-2}{\beta }^{p}+{2}^{p-1}{\gamma }^{p}$, we have

$\begin{array}{r}\psi \left(sG\left(F\left(x,y,z\right),F\left(u,v,w\right),F\left(r,s,t\right)\right)\right)\\ \phantom{\rule{1em}{0ex}}=ln\left(\begin{array}{c}2\left(\frac{1}{96}\left[|\left(x-2y+4z\right)-\left(u-2v+4w\right)|\right]+\frac{1}{96}\left[|\left(u-2v+4w\right)-\left(r-2s+4t\right)|\right]\\ {+\frac{1}{96}\left[|\left(r-2s+4t\right)-\left(x-2y+4z\right)|\right]\right)}^{2}+1\end{array}\right)\\ \phantom{\rule{1em}{0ex}}\le ln\left(\begin{array}{c}2\left(\frac{1}{48}|\frac{x}{2}-\frac{u}{2}|+\frac{1}{24}|\frac{y}{2}-\frac{v}{2}|+\frac{1}{12}|\frac{z}{2}-\frac{w}{2}|+\frac{1}{48}|\frac{u}{2}-\frac{r}{2}|+\frac{1}{24}|\frac{v}{2}-\frac{s}{2}|\\ {+\frac{1}{12}|\frac{w}{2}-\frac{t}{2}|+\frac{1}{48}|\frac{r}{2}-\frac{x}{2}|+\frac{1}{24}|\frac{s}{2}-\frac{y}{2}|+\frac{1}{12}|\frac{t}{2}-\frac{z}{2}|\right)}^{2}+1\end{array}\right)\\ \phantom{\rule{1em}{0ex}}=ln\left(\begin{array}{c}2\left(\frac{1}{48}\left[|\frac{x}{2}-\frac{u}{2}|+|\frac{u}{2}-\frac{r}{2}|+|\frac{r}{2}-\frac{x}{2}|\right]+\frac{1}{24}\left[|\frac{y}{2}-\frac{v}{2}|+|\frac{v}{2}-\frac{s}{2}|+|\frac{s}{2}-\frac{y}{2}|\right]\\ {+\frac{1}{12}\left[|\frac{z}{2}-\frac{w}{2}|+|\frac{w}{2}-\frac{t}{2}|+|\frac{t}{2}-\frac{z}{2}|\right]\right)}^{2}+1\end{array}\right)\\ \phantom{\rule{1em}{0ex}}\le ln\left(\begin{array}{c}\frac{8}{{48}^{2}}\left({\left[|\frac{x}{2}-\frac{u}{2}|+|\frac{u}{2}-\frac{r}{2}|+|\frac{r}{2}-\frac{x}{2}|\right]}^{2}+\frac{8}{{24}^{2}}{\left[|\frac{y}{2}-\frac{v}{2}|+|\frac{v}{2}-\frac{s}{2}|+|\frac{s}{2}-\frac{y}{2}|\right]}^{2}\\ +\frac{4}{{12}^{2}}{\left[|\frac{z}{2}-\frac{w}{2}|+|\frac{w}{2}-\frac{t}{2}|+|\frac{t}{2}-\frac{z}{2}|\right]}^{2}\right)+1\end{array}\right)\\ \phantom{\rule{1em}{0ex}}\le ln\left(\begin{array}{c}\frac{1}{12}\left({\left[|\frac{x}{2}-\frac{u}{2}|+|\frac{u}{2}-\frac{r}{2}|+|\frac{r}{2}-\frac{x}{2}|\right]}^{2}+\frac{1}{12}{\left[|\frac{y}{2}-\frac{v}{2}|+|\frac{v}{2}-\frac{s}{2}|+|\frac{s}{2}-\frac{y}{2}|\right]}^{2}\\ +\frac{1}{12}{\left[|\frac{z}{2}-\frac{w}{2}|+|\frac{w}{2}-\frac{t}{2}|+|\frac{t}{2}-\frac{z}{2}|\right]}^{2}\right)+1\end{array}\right)\\ \phantom{\rule{1em}{0ex}}\le ln\left(\begin{array}{c}\frac{1}{4}max\left\{{\left[|\frac{x}{2}-\frac{u}{2}|+|\frac{u}{2}-\frac{r}{2}|+|\frac{r}{2}-\frac{x}{2}|\right]}^{2},{\left[|\frac{y}{2}-\frac{v}{2}|+|\frac{v}{2}-\frac{s}{2}|+|\frac{s}{2}-\frac{y}{2}|\right]}^{2},\\ {\left[|\frac{z}{2}-\frac{w}{2}|+|\frac{w}{2}-\frac{t}{2}|+|\frac{t}{2}-\frac{z}{2}|\right]}^{2}\right\}+1\end{array}\right)\\ \phantom{\rule{1em}{0ex}}\le ln\left(\frac{1}{4}max\left\{G\left(gx,gu,gr\right),G\left(gy,gv,gs\right),G\left(gz,gw,gt\right)\right\}+1\right)\\ \phantom{\rule{1em}{0ex}}=ln\left(max\left\{G\left(gx,gu,gr\right),G\left(gy,gv,gs\right),G\left(gz,gw,gt\right)\right\}+1\right)\\ \phantom{\rule{2em}{0ex}}-ln\left(\frac{4max\left\{G\left(gx,gu,gr\right),G\left(gy,gv,gs\right),G\left(gz,gw,gt\right)\right\}+4}{max\left\{G\left(gx,gu,gr\right),G\left(gy,gv,gs\right),G\left(gz,gw,gt\right)\right\}+4}\right)\\ \phantom{\rule{1em}{0ex}}=\psi \left(max\left\{G\left(gx,gu,gr\right),G\left(gy,gv,gs\right),G\left(gz,gw,gt\right)\right\}\right)\\ \phantom{\rule{2em}{0ex}}-\phi \left(max\left\{G\left(gx,gu,gr\right),G\left(gy,gv,gs\right),G\left(gz,gw,gt\right)\right\}\right).\end{array}$

Analogously, we can show that

$\begin{array}{r}\psi \left(G\left(F\left(y,x,y\right),F\left(v,u,v\right),F\left(s,r,s\right)\right)\right)\\ \phantom{\rule{1em}{0ex}}\le \psi \left(max\left\{G\left(gx,gu,gr\right),G\left(gy,gv,gs\right),G\left(gz,gw,gt\right)\right\}\right)\\ \phantom{\rule{2em}{0ex}}-\phi \left(max\left\{G\left(gx,gu,gr\right),G\left(gy,gv,gs\right),G\left(gz,gw,gt\right)\right\}\right)\end{array}$

and

$\begin{array}{r}\psi \left(G\left(F\left(z,y,x\right),F\left(w,v,u\right),F\left(t,s,r\right)\right)\right)\\ \phantom{\rule{1em}{0ex}}\le \psi \left(max\left\{G\left(gx,gu,gr\right),G\left(gy,gv,gs\right),G\left(gz,gw,gt\right)\right\}\right)\\ \phantom{\rule{2em}{0ex}}-\phi \left(max\left\{G\left(gx,gu,gr\right),G\left(gy,gv,gs\right),G\left(gz,gw,gt\right)\right\}\right).\end{array}$

Thus,

$\begin{array}{rl}\psi \left(s{M}_{F}\left(x,y,z,u,v,w,r,s,t\right)\right)\le & \psi \left({M}_{g}\left(x,y,z,u,v,w,r,s,t\right)\right)\\ -\phi \left({M}_{g}\left(x,y,z,u,v,w,r,s,t\right)\right).\end{array}$

Hence, all of the conditions of Theorem 2.1 are satisfied. Moreover, $\left(0,0,0\right)$ is the unique common tripled fixed point of F and g.

The following example has been constructed according to Example 2.12 of .

Example 3.2 Let $X=\left\{\left(x,0,x\right)\right\}\cup \left\{\left(0,x,0\right)\right\}\subset {R}^{3}$, where $x\in \left[0,\mathrm{\infty }\right]$ with the order defined as

$\left({x}_{1},{y}_{1},{z}_{1}\right)⪯\left({x}_{2},{y}_{2},{z}_{2}\right)\phantom{\rule{1em}{0ex}}⟺\phantom{\rule{1em}{0ex}}{x}_{1}\le {x}_{2},\phantom{\rule{2em}{0ex}}{y}_{1}\le {y}_{2},\phantom{\rule{2em}{0ex}}{z}_{1}\le {z}_{2}.$

Let d be given as

$d\left(x,y\right)=max\left\{{|{x}_{1}-{x}_{2}|}^{2},{|{y}_{1}-{y}_{2}|}^{2},{|{z}_{1}-{z}_{2}|}^{2}\right\}$

and

$G\left(x,y,z\right)=max\left\{d\left(x,y\right),d\left(y,z\right),d\left(z,x\right)\right\},$

where $x=\left({x}_{1},{y}_{1},{z}_{1}\right)$ and $y=\left({x}_{2},{y}_{2},{z}_{2}\right)$. $\left(X,G\right)$ is, clearly, a ${G}_{b}$-complete ${G}_{b}$-metric space.

Let $g:X\to X$ and $F:{X}^{3}\to X$ be defined as follows:

$F\left(x,y,z\right)=x$

and

$g\left(\left(x,0,x\right)\right)=\left(0,x,0\right)\phantom{\rule{1em}{0ex}}\text{and}\phantom{\rule{1em}{0ex}}g\left(\left(0,x,0\right)\right)=\left(x,0,x\right).$

Let $\psi ,\phi :\left[0,\mathrm{\infty }\right)\to \left[0,\mathrm{\infty }\right)$ be as in the above example.

According to the order on X and the definition of g, we see that for any element $x\in X$, $g\left(x\right)$ is comparable only with itself.

By a careful computation, it is easy to see that all of the conditions of Theorem 2.1 are satisfied. Finally, Theorem 2.1 guarantees the existence of a unique common tripled fixed point for F and g, i.e., the point $\left(\left(0,0,0\right),\left(0,0,0\right),\left(0,0,0\right)\right)$.

## 4 Applications

In this section, we obtain some tripled coincidence point theorems for a mapping satisfying a contractive condition of integral type in a complete ordered ${G}_{b}$-metric space.

We denote by Λ the set of all functions $\mu :\left[0,+\mathrm{\infty }\right)\to \left[0,+\mathrm{\infty }\right)$ verifying the following conditions:

1. (I)

μ is a positive Lebesgue integrable mapping on each compact subset of $\left[0,+\mathrm{\infty }\right)$.

2. (II)

For all $\epsilon >0$, ${\int }_{0}^{\epsilon }\mu \left(t\right)\phantom{\rule{0.2em}{0ex}}dt>0$.

Corollary 4.1 Replace the contractive condition (2.1) of Theorem 2.1 by the following condition:

There exists $\mu \in \mathrm{\Lambda }$ such that

$\begin{array}{r}{\int }_{0}^{\psi \left(s{M}_{F}\left(x,y,z,u,v,w,r,s,t\right)\right)}\mu \left(t\right)\phantom{\rule{0.2em}{0ex}}dt\\ \phantom{\rule{1em}{0ex}}\le {\int }_{0}^{\psi \left({M}_{g}\left(x,y,z,u,v,w,r,s,t\right)\right)}\mu \left(t\right)\phantom{\rule{0.2em}{0ex}}dt-{\int }_{0}^{\phi \left({M}_{g}\left(x,y,z,u,v,w,r,s,t\right)\right)}\mu \left(t\right)\phantom{\rule{0.2em}{0ex}}dt.\end{array}$
(4.1)

If the other conditions of Theorem 2.1 are satisfied, then F and g have a tripled coincidence point.

Proof Consider the function $\mathrm{\Gamma }\left(x\right)={\int }_{0}^{x}\mu \left(t\right)\phantom{\rule{0.2em}{0ex}}dt$. Then (4.1) becomes

$\begin{array}{r}\mathrm{\Gamma }\left(\psi \left(s{M}_{F}\left(x,y,z,u,v,w,r,s,t\right)\right)\right)\\ \phantom{\rule{1em}{0ex}}\le \mathrm{\Gamma }\left(\psi \left({M}_{g}\left(x,y,z,u,v,w,r,s,t\right)\right)\right)-\mathrm{\Gamma }\left(\phi \left({M}_{g}\left(x,y,z,u,v,w,r,s,t\right)\right)\right).\end{array}$

Taking ${\psi }_{1}=\mathrm{\Gamma }o\psi$ and ${\phi }_{1}=\mathrm{\Gamma }o\phi$ and applying Theorem 2.1, we obtain the proof (it is easy to verify that ${\psi }_{1}$ and ${\phi }_{1}$ are altering distance functions). □

Corollary 4.2 Substitute the contractive condition (2.1) of Theorem 2.1 by the following condition:

There exists $\mu \in \mathrm{\Lambda }$ such that

$\begin{array}{r}\psi \left({\int }_{0}^{s{M}_{F}\left(x,y,z,u,v,w,r,s,t\right)}\mu \left(t\right)\phantom{\rule{0.2em}{0ex}}dt\right)\\ \phantom{\rule{1em}{0ex}}\le \psi \left({\int }_{0}^{{M}_{g}\left(x,y,z,u,v,w,r,s,t\right)}\mu \left(t\right)\phantom{\rule{0.2em}{0ex}}dt\right)-\phi \left({\int }_{0}^{{M}_{g}\left(x,y,z,u,v,w,r,s,t\right)}\mu \left(t\right)\phantom{\rule{0.2em}{0ex}}dt\right).\end{array}$
(4.2)

If the other conditions of Theorem 2.1 are satisfied, then F and g have a tripled coincidence point.

Proof Again, as in Corollary 4.1, define the function $\mathrm{\Gamma }\left(x\right)={\int }_{0}^{x}\mu \left(t\right)\phantom{\rule{0.2em}{0ex}}dt$. Then (4.2) changes to

$\begin{array}{rl}\psi \left(\mathrm{\Gamma }\left(s{M}_{F}\left(x,y,z,u,v,w,r,s,t\right)\right)\right)\le & \psi \left(\mathrm{\Gamma }\left({M}_{g}\left(x,y,z,u,v,w,r,s,t\right)\right)\right)\\ -\phi \left(\mathrm{\Gamma }\left({M}_{g}\left(x,y,z,u,v,w,r,s,t\right)\right)\right).\end{array}$

Now, if we define ${\psi }_{1}=\psi o\mathrm{\Gamma }$ and ${\phi }_{1}=\phi o\mathrm{\Gamma }$ and apply Theorem 2.1, then the proof is completed. □

Corollary 4.3 Replace the contractive condition (2.1) of Theorem 2.1 by the following condition:

There exists $\mu \in \mathrm{\Lambda }$ such that

$\begin{array}{r}{\psi }_{1}\left({\int }_{0}^{{\psi }_{2}\left(s{M}_{F}\left(x,y,z,u,v,w,r,s,t\right)\right)}\mu \left(t\right)\phantom{\rule{0.2em}{0ex}}dt\right)\\ \phantom{\rule{1em}{0ex}}\le {\psi }_{1}\left({\int }_{0}^{{\psi }_{2}\left({M}_{g}\left(x,y,z,u,v,w,r,s,t\right)\right)}\mu \left(t\right)\phantom{\rule{0.2em}{0ex}}dt\right)-{\phi }_{1}\left({\int }_{0}^{{\phi }_{2}\left({M}_{g}\left(x,y,z,u,v,w,r,s,t\right)\right)}\mu \left(t\right)\phantom{\rule{0.2em}{0ex}}dt\right)\end{array}$
(4.3)

for altering distance functions ${\psi }_{1}$, ${\psi }_{2}$, ${\phi }_{1}$ and ${\phi }_{2}$. If the other conditions of Theorem 2.1 are satisfied, then F and g have a tripled coincidence point.

Similar to , let $N\in \mathbb{N}$ be fixed. Let ${\left\{{\mu }_{i}\right\}}_{1\le i\le N}$ be a family of N functions which belong to Λ. For all $t\ge 0$, we define

$\begin{array}{r}{I}_{1}\left(t\right)={\int }_{0}^{t}{\mu }_{1}\left(s\right)\phantom{\rule{0.2em}{0ex}}ds,\\ {I}_{2}\left(t\right)={\int }_{0}^{{I}_{1}t}{\mu }_{2}\left(s\right)\phantom{\rule{0.2em}{0ex}}ds={\int }_{0}^{{\int }_{0}^{t}{\mu }_{1}\left(s\right)\phantom{\rule{0.2em}{0ex}}ds}{\mu }_{2}\left(s\right)\phantom{\rule{0.2em}{0ex}}ds,\\ {I}_{3}\left(t\right)={\int }_{0}^{{I}_{2}t}{\mu }_{3}\left(s\right)\phantom{\rule{0.2em}{0ex}}ds={\int }_{0}^{{\int }_{0}^{{\int }_{0}^{t}{\mu }_{1}\left(s\right)\phantom{\rule{0.2em}{0ex}}ds}{\mu }_{2}\left(s\right)\phantom{\rule{0.2em}{0ex}}ds}{\mu }_{3}\left(s\right)\phantom{\rule{0.2em}{0ex}}ds,\\ \cdots ,\\ {I}_{N}\left(t\right)={\int }_{0}^{{I}_{\left(N-1\right)}t}{\mu }_{N}\left(s\right)\phantom{\rule{0.2em}{0ex}}ds.\end{array}$

We have the following result.

Corollary 4.4 Replace inequality (2.1) of Theorem 2.1 by the following condition:

$\begin{array}{rl}\psi \left({I}_{N}\left(s{M}_{F}\left(x,y,z,u,v,w,r,s,t\right)\right)\right)\le & \psi \left({I}_{N}\left({M}_{g}\left(x,y,z,u,v,w,r,s,t\right)\right)\right)\\ -\phi \left({I}_{N}\left({M}_{g}\left(x,y,z,u,v,w,r,s,t\right)\right)\right).\end{array}$
(4.4)

If the other conditions of Theorem 2.1 are satisfied, then F and g have a tripled coincidence point.

Proof Consider $\stackrel{ˆ}{\mathrm{\Psi }}=\psi o{I}_{N}$ and $\stackrel{ˆ}{\mathrm{\Phi }}=\phi o{I}_{N}$. Then the above inequality becomes

$\begin{array}{rl}\stackrel{ˆ}{\mathrm{\Psi }}\left(s{M}_{F}\left(x,y,z,u,v,w,r,s,t\right)\right)\le & \stackrel{ˆ}{\mathrm{\Psi }}\left({M}_{g}\left(x,y,z,u,v,w,r,s,t\right)\right)\\ -\stackrel{ˆ}{\mathrm{\Phi }}\left({M}_{g}\left(x,y,z,u,v,\end{array}$