# Coupled coincidence point theorems for contractions in generalized fuzzy metric spaces

## Abstract

In this paper, we introduce the concept of a mixed g-monotone mapping and prove coupled coincidence and common coupled fixed point theorems for mappings under ϕ-contractive conditions in partially ordered generalized fuzzy metric spaces. We also give an example to illustrate the theorems.

MSC:47H10, 54H25.

## 1 Introduction

The theory of fuzzy sets has evolved in many directions after investigation of the notion of fuzzy sets by Zadeh [1]. Many authors have introduced the concept of a fuzzy metric space in different ways [2, 3]. George and Veeramani [4, 5] modified the concept of a fuzzy metric space introduced by Kramosil and Michalek [3] and defined a Hausdorff topology on this fuzzy metric space. They showed also that every metric induces a fuzzy metric. Later, many fixed point theorems in fuzzy metric spaces and probabilistic metric spaces have been obtained by [610].

Nieto and Lopez [11], Ran and Reurings [12], Petrusel and Rus [13] presented some new results for contractions in partially ordered metric spaces. The main idea in [11, 12] involves combining the ideas of the iterative technique in the contractive mapping principle with those in the monotone technique, discussing the existence of a solution to first-order ordinary differential equations with periodic boundary conditions and some applications to linear and nonlinear matrix equations.

Bhaskar and Lakshmikantham [14], Lakshmikantham and Ćirić [15] discussed coupled coincidence and coupled fixed point theorems for two mappings F and g, where F has the mixed g-monotone property and F and g commute. The results were used to study the existence of a unique solution to a periodic boundary value problem. In [16], Choudhury and Kundu established a similar result under the condition that F and g are compatible mappings and the function g is monotone increasing. For more details on ordered metric spaces, we refer to [1719] and references mentioned therein.

Alternatively Mustafa and Sims [20] introduced a new notion of a generalized metric space called G-metric space. Mujahid Abbas et al. [23] proved a unique fixed point of four R-weakly commuting maps in G-metric spaces, and Mujahid Abbas et al. [24] obtained some common fixed point results of maps satisfying the generalized $\left(\phi ,\psi \right)$-weak commuting condition in partially ordered G-metric spaces. Rao et al. [22] proved two unique common coupled fixed-point theorems for three mappings in symmetric G-fuzzy metric spaces. Sun and Yang [21] introduced the concept of G-fuzzy metric spaces and proved two common fixed-point theorems for four mappings. Some interesting references on G-metric spaces are [2225].

In this paper, we introduce the concept of a mixed g-monotone mapping, which is a generalization of the mixed monotone mapping, and prove coupled coincidence point and coupled common fixed point theorems for mappings under ϕ-contractive conditions in partially ordered G-fuzzy metric spaces. The work is an extension of the fixed point result in fuzzy metric spaces and the condition is different from [1416] even in metric spaces. We also give an example to illustrate the theorems.

Recall that if $\left(X,\le \right)$ is a partially ordered set and $F:X\to X$ satisfies that for $x,y\in X$, $x\le y$ implies $F\left(x\right)\le F\left(y\right)$, then a mapping F is said to be non-decreasing. Similarly, a non-increasing mapping is defined.

Before giving our main results, we recall some of the basic concepts and results in G-metric spaces and G-fuzzy metric spaces.

## 2 Preliminaries

Definition 2.1 [20]

Let X be a nonempty set, and let $G:X×X×X\to \left[0,+\mathrm{\infty }\right)$ be a function satisfying the following properties:

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

(G-2) $0 for all $x,y\in X$ with $x\ne y$,

(G-3) $G\left(x,x,y\right)\le G\left(x,y,z\right)$ for all $x,y,z\in X$ with $y\ne z$,

(G-4) $G\left(x,y,z\right)=G\left(x,z,y\right)=G\left(y,z,x\right)=\cdots$, symmetry in all three variables,

(G-5) $G\left(x,y,z\right)\le G\left(x,a,a\right)+G\left(a,y,z\right)$ for all $x,y,z,a\in X$.

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

Definition 2.2 [20]

The G-metric space $\left(X,G\right)$ is called symmetric if $G\left(x,x,y\right)=G\left(x,y,y\right)$ for all $x,y\in X$.

Definition 2.3 [20]

Let $\left(X,G\right)$ be a G-metric space, and let $\left\{{x}_{n}\right\}$ be a sequence in X. A point $x\in X$ is said to be the limit of $\left\{{x}_{n}\right\}$ if and only if ${lim}_{n,m\to \mathrm{\infty }}G\left(x,{x}_{n},{x}_{m}\right)=0$. In this case, the sequence $\left\{{x}_{n}\right\}$ is said to be G-convergent to x.

Definition 2.4 [20]

Let $\left(X,G\right)$ be a G-metric space, and let $\left\{{x}_{n}\right\}$ be a sequence in X. $\left\{{x}_{n}\right\}$ is called a G-Cauchy sequence if and only if ${lim}_{n,m,l\to \mathrm{\infty }}G\left({x}_{n},{x}_{m},{x}_{l}\right)=0$. $\left(X,G\right)$ is called G-complete if every G-Cauchy sequence in $\left(X,G\right)$ is G-convergent in $\left(X,G\right)$.

Proposition 2.1 [20]

In a G-metric space $\left(X,G\right)$, the following are equivalent:

1. (i)

The sequence $\left\{{x}_{n}\right\}$ is G-Cauchy.

2. (ii)

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

Proposition 2.2 [20]

Let $\left(X,G\right)$ be a G-metric space; then the function $G\left(x,y,z\right)$ is jointly continuous in all three of its variables.

Proposition 2.3 [20]

Let $\left(X,G\right)$ be a G-metric space; then for any $x,y,z,a\in X$, it follows that

1. (i)

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

2. (ii)

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

3. (iii)

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

4. (iv)

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

5. (v)

$G\left(x,y,z\right)\le \frac{2}{3}\left(G\left(x,a,a\right)+G\left(y,a,a\right)+G\left(z,a,a\right)\right)$.

Let $\left(X,d\right)$ be a metric space. One can verify that $\left(X,G\right)$ is a G-metric space, where

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

or

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

If $\left(X,G\right)$ is a G-metric space, it easy to verify that $\left(X,{d}_{G}\right)$ is a metric space, where ${d}_{G}\left(x,y\right)=\frac{1}{2}\left(G\left(x,x,y\right)+G\left(x,y,y\right)\right)$.

Definition 2.5 [26]

A binary operation $\ast :\left[0,1\right]×\left[0,1\right]\to \left[0,1\right]$ is a continuous t-norm if satisfies the following conditions:

1. (i)

is commutative and associative;

2. (ii)

is continuous;

3. (iii)

$a\ast 1=a$ for all $a\in \left[0,1\right]$;

4. (iv)

$a\ast b\le c\ast d$ whenever $a\le c$ and $b\le d$ for all $a,b,c,d\in \left[0,1\right]$.

Definition 2.6 [21]

A 3-tuple $\left(X,G,\ast \right)$ is said to be a G-fuzzy metric space (denoted by GF space) if X is an arbitrary nonempty set, is a continuous t-norm and G is a fuzzy set on ${X}^{3}×\left(0,+\mathrm{\infty }\right)$ satisfying the following conditions for each $t,s>0$:

(GF-1) $G\left(x,x,y,t\right)>0$ for all $x,y\in X$ with $x\ne y$;

(GF-2) $G\left(x,x,y,t\right)\ge G\left(x,y,z,t\right)$ for all $x,y,z\in X$ with $y\ne z$;

(GF-3) $G\left(x,y,z,t\right)=1$ if and only if $x=y=z$;

(GF-4) $G\left(x,y,z,t\right)=G\left(p\left(x,y,z\right),t\right)$, where p is a permutation function;

(GF-5) $G\left(x,a,a,t\right)\ast G\left(a,y,z,s\right)\le G\left(x,y,z,t+s\right)$ (the triangle inequality);

(GF-6) $G\left(x,y,z,\cdot \right):\left(0,\mathrm{\infty }\right)\to \left[0,1\right]$ is continuous.

Remark 2.1 Let $x=w$, $y=u$, $z=u$, $a=v$ in (GF-5), we have

$G\left(w,u,u,t+s\right)\ge G\left(w,v,v,t\right)\ast G\left(v,u,u,s\right),$

which implies that

$G\left(u,u,w,s+t\right)\ge G\left(u,u,v,s\right)\ast G\left(v,v,w,t\right),$

for all $u,v,w\in X$ and $s,t>0$.

A GF space is said to be symmetric if $G\left(x,x,y,t\right)=G\left(x,y,y,t\right)$ for all $x,y\in X$ and for each $t>0$.

Example 2.1 Let X be a nonempty set, and let G be a G-metric on X. Define the t-norm $a\ast b=min\left\{a,b\right\}$ and for all $x,y,z\in X$ and $t>0$, $G\left(x,y,z,t\right)=\frac{t}{t+G\left(x,y,z\right)}$. Then $\left(X,G,\ast \right)$ is a GF space.

Remark 2.2 If $\left(X,M,\ast \right)$ is a fuzzy metric space [4], then $\left(X,G,\ast \right)$ is a GF space, where

$G\left(x,y,z,t\right)=min\left\{M\left(x,y,t\right),M\left(y,z,t\right),M\left(z,x,t\right)\right\}.$

In fact, we only need to verify (GF-5). Since

we have

which implies that (GF-5) holds.

Remark 2.3 If $\left(X,G,\ast \right)$ is a symmetric GF space, let $M\left(x,y,t\right)=G\left(x,y,y,t\right)$, then $\left(X,M,\ast \right)$ is a fuzzy metric space [4].

Let $\left(X,G,\ast \right)$ be a GF space. For $t>0$, the open ball ${B}_{G}\left(x,r,t\right)$ with center $x\in X$ and radius $0 is defined by

${B}_{G}\left(x,r,t\right)=\left\{y\in X:G\left(x,y,y,t\right)>1-r\right\}.$

A subset $A\subset X$ is called an open set if for each $x\in A$, there exist $t>0$ and $0 such that ${B}_{G}\left(x,r,t\right)\subset A$.

Definition 2.7 [21]

Let $\left(X,G,\ast \right)$ be a GF space, then

1. (1)

a sequence $\left\{{x}_{n}\right\}$ in X is said to be convergent to x (denoted by ${lim}_{n\to \mathrm{\infty }}{x}_{n}=x$) if

$\underset{n\to \mathrm{\infty }}{lim}G\left({x}_{n},{x}_{n},x,t\right)=1$

for all $t>0$.

1. (2)

a sequence $\left\{{x}_{n}\right\}$ in X is said to be a Cauchy sequence if

$\underset{n,m\to \mathrm{\infty }}{lim}G\left({x}_{n},{x}_{n},{x}_{m},t\right)=1,$

that is, for any $\epsilon >0$ and for each $t>0$, there exists ${n}_{0}\in \mathbb{N}$ such that

$G\left({x}_{n},{x}_{n},{x}_{m},t\right)>1-\epsilon ,$

for $n,m\ge {n}_{0}$.

1. (3)

A GF space $\left(X,G,\ast \right)$ is said to be complete if every Cauchy sequence in X is convergent.

Lemma 2.1 [21]

Let $\left(X,G,\ast \right)$ be a GF space. Then $G\left(x,y,z,t\right)$ is non-decreasing with respect to t for all $x,y,z\in X$.

Lemma 2.2 [21]

Let $\left(X,G,\ast \right)$ be a GF space. Then G is a continuous function on ${X}^{3}×\left(0,+\mathrm{\infty }\right)$.

In the rest of the paper, $\left(X,G,\ast \right)$ will denote a GF space with a continuous t-norm defined as $a\ast b=min\left\{a,b\right\}$ for all $a,b\in \left[0,1\right]$, and we assume that

$\underset{t\to \mathrm{\infty }}{lim}G\left(x,y,z,t\right)=1,\phantom{\rule{1em}{0ex}}\mathrm{\forall }x,y,z\in X.$
(P)

Define $\mathrm{\Phi }=\left\{\varphi :{R}^{+}\to {R}^{+}\right\}$, where ${R}^{+}=\left[0,+\mathrm{\infty }\right)$ and each $\varphi \in \mathrm{\Phi }$ satisfies the following conditions:

(Φ-1) ϕ is strict increasing;

(Φ-2) ϕ is upper semi-continuous from the right;

(Φ-3) ${\sum }_{n=0}^{\mathrm{\infty }}{\varphi }^{n}\left(t\right)<+\mathrm{\infty }$ for all $t>0$, where ${\varphi }^{n+1}\left(t\right)=\varphi \left({\varphi }^{n}\left(t\right)\right)$.

Let ${\varphi }_{1}\left(t\right)=\frac{t}{t+1}$, ${\varphi }_{2}\left(t\right)=kt$, where $0, then ${\varphi }_{1},{\varphi }_{2}\in \mathrm{\Phi }$.

It is easy to prove that if $\varphi \in \mathrm{\Phi }$, then $\varphi \left(t\right) for all $t>0$.

Using (P), one can prove the following lemma.

Lemma 2.3 Let $\left(X,G,\ast \right)$ be a GF space. If there exists $\varphi \in \mathrm{\Phi }$ such that if $G\left(x,y,z,\varphi \left(t\right)\right)\ge G\left(x,y,z,t\right)$ for all $t>0$, then $x=y=z$.

Lemma 2.4 Let $\left(X,G,\ast \right)$ be a GF space. If we define ${E}_{\lambda }:X×X×X\to \left[0,\mathrm{\infty }\right)$ by

${E}_{\lambda }\left(x,y,z\right)=inf\left\{t>0,G\left(x,y,z,t\right)>1-\lambda \right\}$
(2.1)

for all $\lambda \in \left(0,1\right]$ and $x,y,z\in X$, then we have:

1. (1)

for each $\lambda \in \left(0,1\right]$, there exists $\mu \in \left(0,1\right]$ such that

${E}_{\lambda }\left({x}_{1},{x}_{1},{x}_{n}\right)\le \sum _{i=1}^{n-1}{E}_{\mu }\left({x}_{i},{x}_{i},{x}_{i+1}\right),\phantom{\rule{1em}{0ex}}\mathrm{\forall }{x}_{1},\dots ,{x}_{n}\in X.$
2. (2)

The sequence ${\left\{{x}_{n}\right\}}_{n\in \mathbb{N}}$ in X is convergent if and only if ${E}_{\lambda }\left({x}_{n},{x}_{n},x\right)\to 0$ as $n\to \mathrm{\infty }$ for all $\lambda \in \left(0,1\right]$.

Proof (1) For any $\lambda \in \left(0,1\right]$, let $\mu \in \left(0,1\right]$ and $\mu <\lambda$, and so, by the triangular inequality (GF-5) and Remark 2.1, for any $\delta >0$, we have

which implies, by Definition 2.1 of ${E}_{\mu }$, that

${E}_{\lambda }\left({x}_{1},{x}_{1},{x}_{n}\right)\le {E}_{\mu }\left({x}_{1},{x}_{1},{x}_{2}\right)+{E}_{\mu }\left({x}_{2},{x}_{2},{x}_{3}\right)+\cdot \cdot \cdot +{E}_{\mu }\left({x}_{n-1},{x}_{n-1},{x}_{n}\right)+\left(n-1\right)\delta .$

Since $\delta >0$ is arbitrary, we have

${E}_{\lambda }\left({x}_{1},{x}_{1},{x}_{n}\right)\le {E}_{\mu }\left({x}_{1},{x}_{1},{x}_{2}\right)+{E}_{\mu }\left({x}_{2},{x}_{2},{x}_{3}\right)+\cdot \cdot \cdot +{E}_{\mu }\left({x}_{n-1},{x}_{n-1},{x}_{n}\right).$
1. (2)

Since G is continuous in its fourth argument, by Definition 2.1 of ${E}_{\mu }$, we have

This proved the lemma. □

Lemma 2.5 Let $\left(X,G,\ast \right)$ be a GF space and $\left\{{y}_{n}\right\}$ be a sequence in X. If there exists $\varphi \in \mathrm{\Phi }$ such that

$G\left({y}_{n},{y}_{n},{y}_{n+1},\varphi \left(t\right)\right)\ge G\left({y}_{n-1},{y}_{n-1},{y}_{n},t\right)\ast G\left({y}_{n},{y}_{n},{y}_{n+1},t\right)$
(2.2)

for all $t>0$ and $n=1,2,\dots$, then $\left\{{y}_{n}\right\}$ is a Cauchy sequence in X.

Proof Let ${\left\{{E}_{\lambda }\left(x,y,z\right)\right\}}_{\lambda \in \left(0,1\right]}$ be defined by (2.1). For each $\lambda \in \left(0,1\right]$ and $n\in \mathbb{N}$, putting ${a}_{n}={E}_{\lambda }\left({y}_{n-1},{y}_{n-1},{y}_{n}\right)$, we will prove that

${a}_{n+1}\le \varphi \left({a}_{n}\right),\phantom{\rule{1em}{0ex}}\mathrm{\forall }n\in \mathbb{N}.$
(2.3)

Since ϕ is upper semi-continuous from right, for given $\epsilon >0$ and each ${a}_{n}$, there exists ${p}_{n}>{a}_{n}$ such that $\varphi \left({p}_{n}\right)<\varphi \left({a}_{n}\right)+\epsilon$. From the definition of ${E}_{\lambda }$ by (2.1), it follows from ${p}_{n}>{a}_{n}={E}_{\lambda }\left({y}_{n-1},{y}_{n-1},{y}_{n}\right)$ that $G\left({y}_{n-1},{y}_{n-1},{y}_{n},{p}_{n}\right)>1-\lambda$ for all $n\in \mathbb{N}$.

Thus, by (2.2), (2.3) and Lemma 2.1, we get

Again by Definition 2.1, we get

$\begin{array}{rcl}{E}_{\lambda }\left({y}_{n},{y}_{n},{y}_{n+1}\right)& \le & \varphi \left(max\left\{{p}_{n},{p}_{n+1}\right\}\right)=max\left\{\varphi \left({p}_{n}\right),\varphi \left({p}_{n+1}\right)\right\}\\ \le & max\left\{\varphi \left({a}_{n}\right),\varphi \left({a}_{n+1}\right)\right\}+\epsilon .\end{array}$

By the arbitrariness of ε, we have

${a}_{n+1}={E}_{\lambda }\left({y}_{n},{y}_{n},{y}_{n+1}\right)\le max\left\{\varphi \left({a}_{n}\right),\varphi \left({a}_{n+1}\right)\right\}.$
(2.4)

So, we can infer that ${a}_{n+1}\le \varphi \left({a}_{n}\right)$. If not, then by (2.4), we have ${a}_{n+1}\le \varphi \left({a}_{n+1}\right)<{a}_{n+1}$, which is a contradiction. Hence, (2.4) implies that ${a}_{n+1}\le \varphi \left({a}_{n}\right)$, and (2.3) is proved.

Again and again using (2.3), we get

By Lemma 2.4, for each $\lambda \in \left(0,1\right]$, there exists $\mu \in \left(0,\lambda \right]$ such that

(2.5)

Since $\varphi \in \mathrm{\Phi }$, by condition (Φ-3) we have ${\sum }_{n=0}^{\mathrm{\infty }}{\varphi }^{n}\left({E}_{\mu }\left({y}_{0},{y}_{0},{y}_{1}\right)\right)<+\mathrm{\infty }$. So, for given $\epsilon >0$, there exists ${n}_{0}\in \mathbb{N}$ such that ${\sum }_{i={n}_{0}}^{\mathrm{\infty }}{\varphi }^{i}\left({E}_{\mu }\left({y}_{0},{y}_{0},{y}_{1}\right)\right)<\epsilon$. Thus, it follows from (2.5) that

${E}_{\lambda }\left({y}_{n},{y}_{n},{y}_{m}\right)\le \sum _{i=n}^{\mathrm{\infty }}{\varphi }^{i}\left({E}_{\mu }\left({y}_{0},{y}_{0},{y}_{1}\right)\right)<\epsilon ,\phantom{\rule{1em}{0ex}}\mathrm{\forall }\phantom{\rule{0.2em}{0ex}}n\ge {n}_{0},$

which implies that $G\left({y}_{n},{y}_{n},{y}_{m},\epsilon \right)>1-\lambda$ for all $m,n\in \mathbb{N}$ with $m>n\ge {n}_{0}$. Therefore, $\left\{{y}_{n}\right\}$ is a Cauchy sequence in X. □

## 3 Main results

Definition 3.1 [14]

Let $\left(X,\le \right)$ be a partially ordered set. The mapping F is said to have the mixed monotone property if F is monotone non-decreasing in its first argument and is monotone non-increasing in its second argument; that is, for any $x,y\in X$,

${x}_{1},{x}_{2}\in X,\phantom{\rule{2em}{0ex}}{x}_{1}\le {x}_{2}\phantom{\rule{1em}{0ex}}⇒\phantom{\rule{1em}{0ex}}F\left({x}_{1},y\right)\le F\left({x}_{2},y\right),$
(3.1)

and

${y}_{1},{y}_{2}\in X,\phantom{\rule{2em}{0ex}}{y}_{1}\le {y}_{2}\phantom{\rule{1em}{0ex}}⇒\phantom{\rule{1em}{0ex}}F\left(x,{y}_{1}\right)\ge F\left(x,{y}_{2}\right).$
(3.2)

Definition 3.2 [14]

An element $\left(x,y\right)\in X×X$ is called a coupled fixed point of the mapping $F:X×X$ if

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

Definition 3.3 [15]

Let $\left(X,\le \right)$ be a partially ordered set and $F:X×X\to X$ and $g:X\to X$. We say F has the mixed g-monotone property if F is monotone g-non-decreasing in its first argument and is monotone g-non-increasing in its second argument; that is, for any $x,y\in X$,

(3.3)

and

(3.4)

Note that if g is the identity mapping, then Definition 3.3 reduces to Definition 3.1.

Example 3.1 Let $X=\left[-1,1\right]$ with the natural ordering of real numbers. Let $g:X\to X$ and $F:X×X\to X$ be defined as

$g\left(x\right)={x}^{4},\phantom{\rule{2em}{0ex}}F\left(x,y\right)={x}^{2}-{y}^{2}.$

Then F is not mixed monotone but mixed g-monotone.

Definition 3.4 [15]

Let X be a nonempty set, $F:X×X\to X$ and $g:X\to X$, then

1. (1)

An element $\left(x,y\right)\in X×X$ is called a coupled coincidence point of the mappings F and g if

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

An element $\left(x,y\right)\in X×X$ is called a common coupled fixed point of the mappings F and g if

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

Definition 3.5 The mappings $F:X×X\to X$ and $g:X\to X$ are said to be compatible if

$\underset{n\to \mathrm{\infty }}{lim}G\left(gF\left({x}_{n},{y}_{n}\right),gF\left({x}_{n},{y}_{n}\right),F\left(g\left({x}_{n}\right),g\left({y}_{n}\right)\right),t\right)=1$

and

$\underset{n\to \mathrm{\infty }}{lim}G\left(gF\left({y}_{n},{x}_{n}\right),gF\left({y}_{n},{x}_{n}\right),F\left(g\left({y}_{n}\right),g\left({x}_{n}\right)\right),t\right)=1$

for all $t>0$ whenever $\left\{{x}_{n}\right\}$ and $\left\{{y}_{n}\right\}$ are sequences in X such that

$\underset{n\to \mathrm{\infty }}{lim}F\left({x}_{n},{y}_{n}\right)=\underset{n\to \mathrm{\infty }}{lim}g\left({x}_{n}\right)=x,\phantom{\rule{2em}{0ex}}\underset{n\to \mathrm{\infty }}{lim}F\left({y}_{n},{x}_{n}\right)=\underset{n\to \mathrm{\infty }}{lim}g\left({y}_{n}\right)=y$

for all $x,y\in X$ are satisfied.

Definition 3.6 [16]

The mappings $F:X×X\to X$ and $g:X\to X$ are called w-compatible if

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

whenever $g\left(x\right)=F\left(x,y\right)$ and $g\left(y\right)=F\left(y,x\right)$ for some $\left(x,y\right)\in X×X$.

Remark 3.1 It is easy to prove that if F and g are compatible then they are w-compatible.

Theorem 3.1 Let $\left(X,\le \right)$ be a partially ordered set and $\left(X,G,\ast \right)$ be a complete GF space. Let $F:X×X\to X$ and $g:X\to X$ be two mappings such that F has the mixed g-monotone property and there exists $\varphi \in \mathrm{\Phi }$ such that

(3.5)

for all $x,y,u,v\in X$, $t>0$ for which $g\left(x\right)\le g\left(u\right)$ and $g\left(y\right)\ge g\left(v\right)$, or $g\left(x\right)\ge g\left(u\right)$ and $g\left(y\right)\le g\left(v\right)$.

Suppose $F\left(X×X\right)\subseteq g\left(X\right)$, g is continuous and F and g are compatible. Also suppose

1. (a)

F is continuous or

2. (b)

X has the following properties:

(3.6)
(3.7)

If there exists ${x}_{0},{y}_{0}\in X$ such that $g\left({x}_{0}\right)\le F\left({x}_{0},{y}_{0}\right)$ and $g\left({y}_{0}\right)\ge F\left({y}_{0},{x}_{0}\right)$, then there exist $x,y\in X$ such that $g\left(x\right)=F\left(x,y\right)$ and $g\left(y\right)=F\left(y,x\right)$; that is, F and g have a coupled coincidence point in X.

Proof Let ${x}_{0},{y}_{0}\in X$ be such that $g\left({x}_{0}\right)\le F\left({x}_{0},{y}_{0}\right)$ and $g\left({y}_{0}\right)\ge F\left({y}_{0},{x}_{0}\right)$. Since $F\left(X×X\right)\subseteq g\left(X\right)$, we can choose ${x}_{1},{y}_{1}\in X$ such that $g\left({x}_{1}\right)=F\left({x}_{0},{y}_{0}\right)$ and $g\left({y}_{1}\right)=F\left({y}_{0},{x}_{0}\right)$. Continuing in this way, we construct two sequences $\left\{{x}_{n}\right\}$ and $\left\{{y}_{n}\right\}$ in X such that

(3.8)

We shall show that

(3.9)
(3.10)

for all $n\ge 0$.

We shall use the mathematical induction. Let $n=0$. Since $g\left({x}_{0}\right)\le F\left({x}_{0},{y}_{0}\right)$ and $g\left({y}_{0}\right)\ge F\left({y}_{0},{x}_{0}\right)$, and as $g\left({x}_{1}\right)=F\left({x}_{0},{y}_{0}\right)$ and $g\left({y}_{1}\right)=F\left({y}_{0},{x}_{0}\right)$, we have $g\left({x}_{0}\right)\le g\left({x}_{1}\right)$ and $g\left({y}_{0}\right)\ge g\left({y}_{1}\right)$. Thus, (3.9) and (3.10) hold for $n=0$. Suppose now that (3.9) and (3.10) hold for some fixed $n\ge 0$. Then since $g\left({x}_{n}\right)\le g\left({x}_{n+1}\right)$ and $g\left({y}_{n}\right)\ge g\left({y}_{n+1}\right)$, and as F has the mixed g-monotone property, from (3.8) and (3.3),

$\begin{array}{l}g\left({x}_{n+1}\right)=F\left({x}_{n},{y}_{n}\right)\le F\left({x}_{n+1},{y}_{n}\right),\\ F\left({y}_{n+1},{x}_{n}\right)\le F\left({y}_{n},{x}_{n}\right)=g\left({y}_{n+1}\right),\end{array}\right\}$
(3.11)

and from (3.8) and (3.4),

$\begin{array}{l}g\left({x}_{n+2}\right)=F\left({x}_{n+1},{y}_{n+1}\right)\ge F\left({x}_{n+1},{y}_{n}\right),\\ F\left({y}_{n+1},{x}_{n}\right)\ge F\left({y}_{n+1},{x}_{n+1}\right)=g\left({y}_{n+2}\right).\end{array}\right\}$
(3.12)

Now from (3.11) and (3.12), we get $g\left({x}_{n+1}\right)\le g\left({x}_{n+2}\right)$ and $g\left({y}_{n+1}\right)\ge g\left({y}_{n+2}\right)$. Thus, by mathematical induction, we conclude that (3.9) and (3.10) hold for all $n\ge 0$. Therefore,

$g\left({x}_{0}\right)\le g\left({x}_{1}\right)\le g\left({x}_{2}\right)\le \cdots \le g\left({x}_{n}\right)\le g\left({x}_{n+1}\right)\le \cdots$
(3.13)

and

$g\left({y}_{0}\right)\ge g\left({y}_{1}\right)\ge g\left({y}_{2}\right)\ge \cdots \ge g\left({y}_{n}\right)\ge g\left({y}_{n+1}\right)\ge \cdots .$
(3.14)

By putting ($x={x}_{n-1}$, $y={y}_{n-1}$, $u={x}_{n}$, $v={y}_{n}$) in (3.5), we get

So, by (3.8), we have

Now, by Lemma 2.5, $\left\{g\left({x}_{n}\right)\right\}$ is a Cauchy sequence.

By putting ($x={y}_{n}$, $y={x}_{n}$, $u={y}_{n-1}$, $v={x}_{n-1}$) in (3.5), we get

So, by (3.8), we have

Now, by Lemma 2.5, $\left\{g\left({y}_{n}\right)\right\}$ is also a Cauchy sequence.

Since X is complete, there exist $x,y\in X$ such that

$\underset{n\to \mathrm{\infty }}{lim}F\left({x}_{n},{y}_{n}\right)=\underset{n\to \mathrm{\infty }}{lim}g\left({x}_{n}\right)=x,\phantom{\rule{2em}{0ex}}\underset{n\to \mathrm{\infty }}{lim}F\left({y}_{n},{x}_{n}\right)=\underset{n\to \mathrm{\infty }}{lim}g\left({y}_{n}\right)=y.$
(3.15)

Since F and g are compatible, we have by (3.15)

$\underset{n\to \mathrm{\infty }}{lim}G\left(g\left(F\left({x}_{n},{y}_{n}\right)\right),g\left(F\left({x}_{n},{y}_{n}\right)\right),F\left(g\left({x}_{n}\right),g\left({y}_{n}\right)\right),t\right)=1$
(3.16)

and

$\underset{n\to \mathrm{\infty }}{lim}G\left(g\left(F\left({y}_{n},{x}_{n}\right)\right),g\left(F\left({y}_{n},{x}_{n}\right)\right),F\left(g\left({y}_{n}\right),g\left({x}_{n}\right)\right),t\right)=1$
(3.17)

for all $t>0$. Next, we prove that $g\left(x\right)=F\left(x,y\right)$ and $g\left(y\right)=F\left(y,x\right)$.

Let (a) hold. Since F and g are continuous, by Lemma 2.2, taking limits as $n\to \mathrm{\infty }$ in (3.16) and (3.17), we get

$G\left(g\left(x\right),g\left(x\right),F\left(x,y\right),t\right)=1,\phantom{\rule{2em}{0ex}}G\left(g\left(y\right),g\left(y\right),F\left(y,x\right),t\right)=1$

for all $t>0$. We have $g\left(x\right)=F\left(x,y\right)$, $g\left(y\right)=F\left(y,x\right)$.

Next, we suppose that (b) holds. By (3.9), (3.10), (3.15), we have for all $n\ge 0$

$g\left({x}_{n}\right)\le x,\phantom{\rule{2em}{0ex}}g\left({y}_{n}\right)\ge y.$
(3.18)

Since F and g are compatible and g is continuous, by (3.16) and (3.17), we have

$\underset{n\to \mathrm{\infty }}{lim}g\left(g{x}_{n}\right)=gx=\underset{n\to \mathrm{\infty }}{lim}g\left(F\left({x}_{n},{y}_{n}\right)\right)=\underset{n\to \mathrm{\infty }}{lim}F\left(g\left({x}_{n}\right),g\left({y}_{n}\right)\right)$
(3.19)

and

$\underset{n\to \mathrm{\infty }}{lim}g\left(g{y}_{n}\right)=gy=\underset{n\to \mathrm{\infty }}{lim}g\left(F\left({y}_{n},{x}_{n}\right)\right)=\underset{n\to \mathrm{\infty }}{lim}F\left(g\left({y}_{n}\right),g\left({x}_{n}\right)\right).$
(3.20)

Now, we have

$\begin{array}{rcl}G\left(gx,gx,F\left(x,y\right),\varphi \left(t\right)\right)& \ge & G\left(gx,gx,g\left(g{x}_{n+1}\right),\varphi \left(t\right)-\varphi \left(kt\right)\right)\\ \ast G\left(g\left(g{x}_{n+1}\right),g\left(g{x}_{n+1}\right),F\left(x,y\right),\varphi \left(kt\right)\right)\end{array}$

for all $0\le k<1$. Taking the limit as $n\to \mathrm{\infty }$ in the above inequality, by continuity of G, using (3.8) and (3.19), we have

By (3.5), (3.19) and the above inequality, we have that

Letting $k\to 1$, which implies that $gx=F\left(x,y\right)$ by Lemma 2.3, and similarly, by the virtue of (3.8), (3.15) and (3.20), we get $gy=F\left(y,x\right)$. Thus, we have proved that F and g have a coupled coincidence point in X.

This completes the proof of Theorem 3.1. □

Taking $g=I$ (the identity mapping) in Theorem 3.1, we get the following consequence.

Corollary 3.1 Let $\left(X,\le \right)$ be a partially ordered set and $\left(X,G,\ast \right)$ be a complete GF space. Let $F:X×X\to X$ be a mapping such that F has the mixed monotone property and there exists $\varphi \in \mathrm{\Phi }$ such that

$G\left(F\left(x,y\right),F\left(x,y\right),F\left(u,v\right),\varphi \left(t\right)\right)\ge G\left(x,x,u,t\right)\ast G\left(x,x,F\left(x,y\right),t\right)\ast G\left(u,u,F\left(u,v\right),t\right)$

for all $x,y,u,v\in X$, $t>0$ for which $x\le u$ and $y\ge v$. Suppose

1. (a)

F is continuous or

2. (b)

X has the following properties:

3. (i)

if a non-decreasing sequence ${x}_{n}\to x$, then ${x}_{n}\le x$ for all n,

4. (ii)

if a non-increasing sequence ${y}_{n}\to y$, then ${y}_{n}\ge y$ for all n.

If there exists ${x}_{0},{y}_{0}\in X$ such that ${x}_{0}\le F\left({x}_{0},{y}_{0}\right)$ and ${y}_{0}\ge F\left({y}_{0},{x}_{0}\right)$, then there exist $x,y\in X$ such that $x=F\left(x,y\right)$ and $y=F\left(y,x\right)$; that is, F has a coupled fixed point in X.

Now, we shall prove the existence and uniqueness theorem of a coupled common fixed point. Note that if $\left(S,\le \right)$ is a partially ordered set, then we endow the product $S×S$ with the following partial order:

$\mathit{\text{for}}\left(x,y\right),\left(u,v\right)\in S×S,\phantom{\rule{1em}{0ex}}\left(x,y\right)\le \left(u,v\right)\phantom{\rule{1em}{0ex}}⇔\phantom{\rule{1em}{0ex}}x\le u,\phantom{\rule{2em}{0ex}}y\ge v.$

Theorem 3.2 In addition to the hypotheses of Theorem  3.1, suppose that for every $\left(x,y\right),\left({x}^{\star },{y}^{\star }\right)\in X×X$, there exists a $\left(u,v\right)\in X×X$ satisfying $g\left(u\right)\le g\left(v\right)$ or $g\left(v\right)\le g\left(u\right)$ such that $\left(F\left(u,v\right),F\left(v,u\right)\right)\in X×X$ is comparable to $\left(F\left(x,y\right),F\left(y,x\right)\right)$, $\left(F\left({x}^{\star },{y}^{\star }\right),F\left({y}^{\star },{x}^{\star }\right)\right)$. Then F and g have a unique common coupled fixed point; that is, there exists a unique $\left(x,y\right)\in X×X$ such that

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

Proof From Theorem 3.1, the set of coupled coincidence points is nonempty. We shall show that if $\left(x,y\right)$ and $\left({x}^{\star },{y}^{\star }\right)$ are coupled coincidence points, that is, if $g\left(x\right)=F\left(x,y\right)$, $g\left(y\right)=F\left(y,x\right)$ and $g\left({x}^{\star }\right)=F\left({x}^{\star },{y}^{\star }\right)$, $g\left({y}^{\star }\right)=F\left({y}^{\star },{x}^{\star }\right)$, then

$g\left(x\right)=g\left({x}^{\star }\right),\phantom{\rule{2em}{0ex}}g\left(y\right)=g\left({y}^{\star }\right).$
(3.21)

By assumption, there is $\left(u,v\right)\in X×X$ such that $\left(F\left(u,v\right),F\left(v,u\right)\right)$ is comparable with $\left(F\left(x,y\right),F\left(y,x\right)\right)$, $\left(F\left({x}^{\star },{y}^{\star }\right),F\left({y}^{\star },{x}^{\star }\right)\right)$. Put ${u}_{0}=u$, ${v}_{0}=v$ and choose ${u}_{1},{v}_{1}\in X$ so that $g\left({u}_{1}\right)=F\left({u}_{0},{v}_{0}\right)$ and $g\left({v}_{1}\right)=F\left({v}_{0},{u}_{0}\right)$. Then, similarly as in the proof of Theorem 3.1, we can inductively define sequences $\left\{g\left({u}_{n}\right)\right\}$ and $\left\{g\left({v}_{n}\right)\right\}$ such that

$g\left({u}_{n+1}\right)=F\left({u}_{n},{v}_{n}\right),\phantom{\rule{2em}{0ex}}g\left({v}_{n+1}\right)=F\left({v}_{n},{u}_{n}\right).$

With the similar proof as in Theorem 3.1, we can prove that the limits of $\left\{g\left({u}_{n}\right)\right\}$ and $\left\{g\left({v}_{n}\right)\right\}$ exist.

Since $\left(F\left(x,y\right),F\left(y,x\right)\right)=\left(g\left({x}_{1}\right),g\left({y}_{1}\right)\right)=\left(g\left(x\right),g\left(y\right)\right)$ and $\left(F\left(u,v\right),F\left(v,u\right)\right)=\left(g\left({u}_{1}\right),g\left({v}_{1}\right)\right)$ are comparable, it is easy to show that $\left(g\left(x\right),g\left(y\right)\right)$ and $\left(g\left({u}_{n}\right),g\left({v}_{n}\right)\right)$ are comparable for all $n\ge 1$. Thus, from (3.5),

for each $n\ge 1$. Letting $n\to \mathrm{\infty }$, we get

$\underset{n\to \mathrm{\infty }}{lim}g\left({u}_{n}\right)=g\left(x\right),\phantom{\rule{2em}{0ex}}\underset{n\to \mathrm{\infty }}{lim}g\left({v}_{n}\right)=g\left(y\right).$
(3.22)

Similarly, one can prove that

$\underset{n\to \mathrm{\infty }}{lim}g\left({u}_{n}\right)=g\left({x}^{\star }\right),\phantom{\rule{2em}{0ex}}\underset{n\to \mathrm{\infty }}{lim}g\left({v}_{n}\right)=g\left({y}^{\star }\right).$
(3.23)

By (3.22) and (3.23), we have

$G\left(gx,gx,g{x}^{\star },t\right)\ge G\left(gx,gx,g{u}_{n+1},\frac{t}{2}\right)\ast G\left(g{u}_{n+1},g{u}_{n+1},g{x}^{\star },\frac{t}{2}\right)\to 1\phantom{\rule{1em}{0ex}}\left(n\to \mathrm{\infty }\right),$

which shows that $g\left(x\right)=g\left({x}^{\star }\right)$.

Similarly, one can prove that $g\left(y\right)=g\left({y}^{\star }\right)$. Thus, we proved (3.21).

Since $g\left(x\right)=F\left(x,y\right)$ and $g\left(y\right)=F\left(y,x\right)$, by the compatibility of F and g, we can get the w-compatibility of F and g, which implies

$g\left(g\left(x\right)\right)=g\left(F\left(x,y\right)\right)=F\left(g\left(x\right),g\left(y\right)\right),$
(3.24)

and

$g\left(g\left(y\right)\right)=g\left(F\left(y,x\right)\right)=F\left(g\left(y\right),g\left(x\right)\right).$
(3.25)

Denote $g\left(x\right)=z$, $g\left(y\right)=w$. Then from (3.24) and (3.25),

$g\left(z\right)=F\left(z,w\right),\phantom{\rule{2em}{0ex}}g\left(w\right)=F\left(w,z\right).$
(3.26)

Thus, $\left(z,w\right)$ is a coupled coincidence point. From (3.21) with ${x}^{\star }=z$, ${y}^{\star }=w$, it also follows $g\left(z\right)=g\left(x\right)$, $g\left(w\right)=g\left(y\right)$, that is,

$g\left(z\right)=z,\phantom{\rule{2em}{0ex}}g\left(w\right)=w.$
(3.27)

From (3.26) and (3.27), we get

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

Therefore, $\left(z,w\right)$ is a common coupled fixed point of F and g. To prove the uniqueness, assume that $\left(p,q\right)$ is another coupled common fixed point. Then by (3.21) we have $p=g\left(p\right)=g\left(z\right)=z$ and $q=g\left(q\right)=g\left(w\right)=w$. □

From Remark 2.3, let $\left(X,G,\ast \right)$ be a symmetric GF space. From Theorem 3.1, we get the following

Corollary 3.2 Let $\left(X,\le \right)$ be a partially ordered set and $\left(X,F,\ast \right)$ be a complete fuzzy metric space. Let $F:X×X\to X$ and $g:X\to X$ be two mappings such that F has the mixed g-monotone property and there exists $\varphi \in \mathrm{\Phi }$ such that

$M\left(F\left(x,y\right),F\left(u,v\right),\varphi \left(t\right)\right)\ge M\left(gx,gu,t\right)\ast M\left(gx,F\left(x,y\right),t\right)\ast M\left(gu,F\left(u,v\right),t\right)$

for all $x,y,u,v\in X$, $t>0$, for which $g\left(x\right)\le g\left(u\right)$ and $g\left(y\right)\ge g\left(v\right)$, or $g\left(x\right)\ge g\left(u\right)$ and $g\left(y\right)\le g\left(v\right)$.

Suppose $F\left(X×X\right)\subseteq g\left(X\right)$, g is continuous and F and g are compatible. Also suppose

1. (a)

F is continuous or

2. (b)

X has the following properties:

3. (i)

if a non-decreasing sequence ${x}_{n}\to x$, then ${x}_{n}\le x$ for all n,

4. (ii)

if a non-increasing sequence ${y}_{n}\to y$, then ${y}_{n}\ge y$ for all n.

If there exist ${x}_{0},{y}_{0}\in X$ such that $g\left({x}_{0}\right)\le F\left({x}_{0},{y}_{0}\right)$ and $g\left({y}_{0}\right)\ge F\left({y}_{0},{x}_{0}\right)$, then there exist $x,y\in X$ such that $g\left(x\right)=F\left(x,y\right)$ and $g\left(y\right)=F\left(y,x\right)$, that is, F and g have a coupled coincidence point in X.

Remark 3.2 Compared with the results in [15, 16], we can find that Theorem 3.1 is different in the following aspects:

1. (1)

We assume that F and g are compatible, which is weaker than the conditions in [15, 16], where Theorem 2.1 in [15] assumes commutation for F and g, and Theorem 3.1 in [16] requires g to be a monotone function.

(2) We have a different contractive condition from [15, 16] even in a metric space.

1. (3)

In our paper, we assume that $\varphi \in \mathrm{\Phi }$, which is a stronger condition than that in [15, 16]. But we would like to point out that in the case of $\varphi \left(t\right)=kt$, where $0, the two conditions are equivalent.

Next, we give an example to demonstrate Theorem 3.1.

Example 3.2 Let $X=\left[0,1\right]$, $a\ast b=min\left\{a,b\right\}$. Then $\left(X,\le \right)$ is a partially ordered set with the natural ordering of real numbers. Let

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

for all $x,y,z\in \left[0,1\right]$. Then $\left(X,G,\ast \right)$ is a complete GF space.

Let $g:X\to X$ and $F:X×X\to X$ be defined as

F obeys the mixed g-monotone property.

Let $\varphi \left(t\right)=\frac{t}{3}$ for $t\in \left[0,\mathrm{\infty }\right)$. Let $\left\{{x}_{n}\right\}$ and $\left\{{y}_{n}\right\}$ be two sequences in X such that

$\underset{n\to \mathrm{\infty }}{lim}F\left({x}_{n},{y}_{n}\right)=a,\phantom{\rule{2em}{0ex}}\underset{n\to \mathrm{\infty }}{lim}g\left({x}_{n}\right)=a,\phantom{\rule{2em}{0ex}}\underset{n\to \mathrm{\infty }}{lim}F\left({y}_{n},{x}_{n}\right)=b,\phantom{\rule{2em}{0ex}}\underset{n\to \mathrm{\infty }}{lim}g\left({y}_{n}\right)=b,$

then $a=0$, $b=0$. Now, for all $n\ge 0$,

and

Then it follows that

Hence, the mappings F and g are compatible in X. Also, ${x}_{0}=0$ and ${y}_{0}=c$ are two points in X such that

$g\left({x}_{0}\right)=g\left(0\right)=F\left(0,c\right)=F\left({x}_{0},{y}_{0}\right)$

and

$g\left({y}_{0}\right)=g\left(c\right)={c}^{2}\ge \frac{{c}^{2}}{3}=F\left(c,0\right)=F\left({y}_{0},{x}_{0}\right).$

We next verify the inequality of Theorem 3.1. We take $x,y,u,v\in X$ such that $g\left(x\right)\le g\left(u\right)$ and $g\left(y\right)\ge g\left(v\right)$, that is, ${x}^{2}\le {u}^{2}$, ${y}^{2}\ge {v}^{2}$.

We consider the following cases:

Case 1: $x\ge y$ and $u\ge v$, then

$\begin{array}{rcl}G\left(F\left(x,y\right),F\left(x,y\right),F\left(u,v\right),\varphi \left(t\right)\right)& =& G\left(\frac{{x}^{2}-{y}^{2}}{3},\frac{{x}^{2}-{y}^{2}}{3},\frac{{u}^{2}-{v}^{2}}{3},\varphi \left(t\right)\right)\\ =& \frac{\frac{t}{3}}{\frac{t}{3}+|\frac{\left({x}^{2}-{u}^{2}\right)-\left({y}^{2}-{v}^{2}\right)}{3}|}\\ =& \frac{t}{t+|\left({x}^{2}-{u}^{2}\right)-\left({y}^{2}-{v}^{2}\right)|}\\ \ge & \frac{t}{t+|{u}^{2}-\frac{{u}^{2}-{v}^{2}}{3}|}\\ =& G\left(g\left(u\right),g\left(u\right),F\left(u,v\right),t\right)\\ \ge & G\left(g\left(x\right),g\left(x\right),g\left(u\right),t\right)\ast G\left(g\left(x\right),g\left(x\right),F\left(x,y\right),t\right)\\ \ast G\left(g\left(u\right),g\left(u\right),F\left(u,v\right),t\right).\end{array}$

Case 2: $x\ge y$, $u. Since $x\le u$, then $u cannot happen.

Case 3: $x and $u\ge v$, then

$\begin{array}{rcl}G\left(F\left(x,y\right),F\left(x,y\right),F\left(u,v\right),\varphi \left(t\right)\right)& =& G\left(0,0,\frac{{u}^{2}-{v}^{2}}{3},\varphi \left(t\right)\right)\\ =& \frac{\frac{t}{3}}{\frac{t}{3}+|\frac{\left({u}^{2}-{v}^{2}\right)}{3}|}\\ =& \frac{t}{t+|{u}^{2}-{v}^{2}|}\\ \ge & \frac{t}{t+2|{u}^{2}-{x}^{2}|}\\ =& G\left(g\left(x\right),g\left(x\right),g\left(u\right),t\right)\\ \ge & G\left(g\left(x\right),g\left(x\right),g\left(u\right),t\right)\ast G\left(g\left(x\right),g\left(x\right),F\left(x,y\right),t\right)\\ \ast G\left(g\left(u\right),g\left(u\right),F\left(u,v\right),t\right).\end{array}$

Case 4: $x and $u with ${x}^{2}\le {u}^{2}$ and ${y}^{2}\ge {v}^{2}$, then $F\left(x,y\right)=0$ and $F\left(u,v\right)=0$, that is, $G\left(F\left(x,y\right),F\left(x,y\right),F\left(u,v\right),\varphi \left(t\right)\right)=0$. Obviously, (3.5) is satisfied.

Thus, it is verified that the functions F, g, ϕ satisfy all the conditions of Theorem 3.1. Here $\left(0,0\right)$ is the coupled coincidence point of F and g in X, which is also their common coupled fixed point.

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## Acknowledgements

The authors thank the referees for useful comments and suggestions for the improvement of the paper. This work was supported by the National Natural Science Foundation of China (71171150).

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Correspondence to Xin-Qi Hu.

### Competing interests

The authors declare that they have no competing interests.

### Authors’ contributions

All authors contributed equally and significantly in writing this paper. All authors read and approved the final manuscript.

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Hu, XQ., Luo, Q. Coupled coincidence point theorems for contractions in generalized fuzzy metric spaces. Fixed Point Theory Appl 2012, 196 (2012). https://doi.org/10.1186/1687-1812-2012-196