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

- Xin-Qi Hu
^{1}Email author and - Qing Luo
^{1}

**2012**:196

https://doi.org/10.1186/1687-1812-2012-196

© Hu and Luo; licensee Springer 2012

**Received:**18 May 2012**Accepted:**12 October 2012**Published:**29 October 2012

## 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.

## Keywords

- coupled coincidence point
- common coupled fixed point
- partially ordered set; mixed monotone mapping
- generalized fuzzy metric space

## 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 [6–10].

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 [17–19] 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 $(\phi ,\psi )$-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 [22–25].

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 [14–16] even in metric spaces. We also give an example to illustrate the theorems.

Recall that if $(X,\le )$ is a partially ordered set and $F:X\to X$ satisfies that for $x,y\in X$, $x\le y$ implies $F(x)\le F(y)$, 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 $G:X\times X\times X\to [0,+\mathrm{\infty})$ be a function satisfying the following properties:

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

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

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

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

(G-5) $G(x,y,z)\le G(x,a,a)+G(a,y,z)$ 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 $(X,G)$ is called a *G*-metric space.

**Definition 2.2** [20]

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

**Definition 2.3** [20]

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

**Definition 2.4** [20]

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

**Proposition 2.1** [20]

*In a*

*G*-

*metric space*$(X,G)$,

*the following are equivalent*:

- (i)
*The sequence*$\{{x}_{n}\}$*is**G*-*Cauchy*. - (ii)
*For every*$\epsilon >0$,*there exists*$N\in \mathbb{N}$*such that*$G({x}_{n},{x}_{m},{x}_{m})<\epsilon $*for all*$n,m\ge N$.

**Proposition 2.2** [20]

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

**Proposition 2.3** [20]

*Let*$(X,G)$

*be a*

*G*-

*metric space*;

*then for any*$x,y,z,a\in X$,

*it follows that*

- (i)
*if*$G(x,y,z)=0$,*then*$x=y=z$, - (ii)
$G(x,y,z)\le G(x,x,y)+G(x,x,z)$,

- (iii)
$G(x,y,y)\le 2G(x,x,y)$,

- (iv)
$G(x,y,z)\le G(x,a,z)+G(a,y,z)$,

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

*G*-metric space, where

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

**Definition 2.5** [26]

*t*-norm if ∗ satisfies the following conditions:

- (i)
∗ is commutative and associative;

- (ii)
∗ is continuous;

- (iii)
$a\ast 1=a$ for all $a\in [0,1]$;

- (iv)
$a\ast b\le c\ast d$ whenever $a\le c$ and $b\le d$ for all $a,b,c,d\in [0,1]$.

**Definition 2.6** [21]

A 3-tuple $(X,G,\ast )$ 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 ${X}^{3}\times (0,+\mathrm{\infty})$ satisfying the following conditions for each $t,s>0$:

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

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

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

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

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

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

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

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

A *GF* space is said to be symmetric if $G(x,x,y,t)=G(x,y,y,t)$ 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 $a\ast b=min\{a,b\}$ and for all $x,y,z\in X$ and $t>0$, $G(x,y,z,t)=\frac{t}{t+G(x,y,z)}$. Then $(X,G,\ast )$ is a *GF* space.

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

*GF*space, where

which implies that (GF-5) holds.

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

*GF*space. For $t>0$, the open ball ${B}_{G}(x,r,t)$ with center $x\in X$ and radius $0<r<1$ is defined by

A subset $A\subset X$ is called an open set if for each $x\in A$, there exist $t>0$ and $0<r<1$ such that ${B}_{G}(x,r,t)\subset A$.

**Definition 2.7** [21]

*GF*space, then

- (1)a sequence $\{{x}_{n}\}$ in
*X*is said to be convergent to*x*(denoted by ${lim}_{n\to \mathrm{\infty}}{x}_{n}=x$) if$\underset{n\to \mathrm{\infty}}{lim}G({x}_{n},{x}_{n},x,t)=1$

- (2)a sequence $\{{x}_{n}\}$ in
*X*is said to be a Cauchy sequence if$\underset{n,m\to \mathrm{\infty}}{lim}G({x}_{n},{x}_{n},{x}_{m},t)=1,$

- (3)
A

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

**Lemma 2.1** [21]

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

**Lemma 2.2** [21]

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

*GF*space with a continuous

*t*-norm ∗ defined as $a\ast b=min\{a,b\}$ for all $a,b\in [0,1]$, and we assume that

Define $\mathrm{\Phi}=\{\varphi :{R}^{+}\to {R}^{+}\}$, where ${R}^{+}=[0,+\mathrm{\infty})$ 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) ${\sum}_{n=0}^{\mathrm{\infty}}{\varphi}^{n}(t)<+\mathrm{\infty}$ for all $t>0$, where ${\varphi}^{n+1}(t)=\varphi ({\varphi}^{n}(t))$.

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

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

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

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

**Lemma 2.4**

*Let*$(X,G,\ast )$

*be a*

*GF*

*space*.

*If we define*${E}_{\lambda}:X\times X\times X\to [0,\mathrm{\infty})$

*by*

*for all*$\lambda \in (0,1]$

*and*$x,y,z\in X$,

*then we have*:

- (1)
*for each*$\lambda \in (0,1]$,*there exists*$\mu \in (0,1]$*such that*${E}_{\lambda}({x}_{1},{x}_{1},{x}_{n})\le \sum _{i=1}^{n-1}{E}_{\mu}({x}_{i},{x}_{i},{x}_{i+1}),\phantom{\rule{1em}{0ex}}\mathrm{\forall}{x}_{1},\dots ,{x}_{n}\in X.$ - (2)
*The sequence*${\{{x}_{n}\}}_{n\in \mathbb{N}}$*in**X**is convergent if and only if*${E}_{\lambda}({x}_{n},{x}_{n},x)\to 0$*as*$n\to \mathrm{\infty}$*for all*$\lambda \in (0,1]$.

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

- (2)Since
*G*is continuous in its fourth argument, by Definition 2.1 of ${E}_{\mu}$, we have$G({x}_{n},{x}_{n},x,\eta )>1-\lambda \phantom{\rule{1em}{0ex}}\text{for all}\eta 0.$

This proved the lemma. □

**Lemma 2.5**

*Let*$(X,G,\ast )$

*be a*

*GF*

*space and*$\{{y}_{n}\}$

*be a sequence in*

*X*.

*If there exists*$\varphi \in \mathrm{\Phi}$

*such that*

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

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

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 ({p}_{n})<\varphi ({a}_{n})+\epsilon $. From the definition of ${E}_{\lambda}$ by (2.1), it follows from ${p}_{n}>{a}_{n}={E}_{\lambda}({y}_{n-1},{y}_{n-1},{y}_{n})$ that $G({y}_{n-1},{y}_{n-1},{y}_{n},{p}_{n})>1-\lambda $ for all $n\in \mathbb{N}$.

*ε*, we have

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

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

## 3 Main results

**Definition 3.1** [14]

*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 $x,y\in X$,

**Definition 3.2** [14]

**Definition 3.3** [15]

*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 $x,y\in X$,

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

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

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

**Definition 3.4** [15]

*X*be a nonempty set, $F:X\times X\to X$ and $g:X\to X$, then

- (1)An element $(x,y)\in X\times X$ is called a coupled coincidence point of the mappings
*F*and*g*if$F(x,y)=g(x),\phantom{\rule{2em}{0ex}}F(y,x)=g(y).$ - (2)An element $(x,y)\in X\times X$ is called a common coupled fixed point of the mappings
*F*and*g*if$F(x,y)=g(x)=x,\phantom{\rule{2em}{0ex}}F(y,x)=g(y)=y.$

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

*X*such that

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

**Definition 3.6** [16]

*w*-compatible if

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

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

**Theorem 3.1**

*Let*$(X,\le )$

*be a partially ordered set and*$(X,G,\ast )$

*be a complete*

*GF*

*space*.

*Let*$F:X\times X\to X$

*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*

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

*Suppose*$F(X\times X)\subseteq g(X)$,

*g*

*is continuous and*

*F*

*and*

*g*

*are compatible*.

*Also suppose*

- (a)
*F**is continuous or* - (b)

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

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

*X*such that

for all $n\ge 0$.

*F*has the mixed

*g*-monotone property, from (3.8) and (3.3),

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

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

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

*F*and

*g*are compatible, we have by (3.15)

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

*F*and

*g*are continuous, by Lemma 2.2, taking limits as $n\to \mathrm{\infty}$ in (3.16) and (3.17), we get

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

*F*and

*g*are compatible and

*g*is continuous, by (3.16) and (3.17), we have

*G*, using (3.8) and (3.19), we have

Letting $k\to 1$, which implies that $gx=F(x,y)$ by Lemma 2.3, and similarly, by the virtue of (3.8), (3.15) and (3.20), we get $gy=F(y,x)$. 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*$(X,\le )$

*be a partially ordered set and*$(X,G,\ast )$

*be a complete*

*GF*

*space*.

*Let*$F:X\times X\to X$

*be a mapping such that*

*F*

*has the mixed monotone property and there exists*$\varphi \in \mathrm{\Phi}$

*such that*

*for all*$x,y,u,v\in X$, $t>0$

*for which*$x\le u$

*and*$y\ge v$.

*Suppose*

- (a)
*F**is continuous or* - (b)
*X**has the following properties*: - (i)
*if a non*-*decreasing sequence*${x}_{n}\to x$,*then*${x}_{n}\le x$*for all**n*, - (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* ${x}_{0}\le F({x}_{0},{y}_{0})$ *and* ${y}_{0}\ge F({y}_{0},{x}_{0})$, *then there exist* $x,y\in X$ *such that* $x=F(x,y)$ *and* $y=F(y,x)$; *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*$(S,\le )$

*is a partially ordered set*,

*then we endow the product*$S\times S$

*with the following partial order*:

**Theorem 3.2**

*In addition to the hypotheses of Theorem*3.1,

*suppose that for every*$(x,y),({x}^{\star},{y}^{\star})\in X\times X$,

*there exists a*$(u,v)\in X\times X$

*satisfying*$g(u)\le g(v)$

*or*$g(v)\le g(u)$

*such that*$(F(u,v),F(v,u))\in X\times X$

*is comparable to*$(F(x,y),F(y,x))$, $(F({x}^{\star},{y}^{\star}),F({y}^{\star},{x}^{\star}))$.

*Then*

*F*

*and*

*g*

*have a unique common coupled fixed point*;

*that is*,

*there exists a unique*$(x,y)\in X\times X$

*such that*

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

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

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

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

*F*and

*g*, we can get the

*w*-compatibility of

*F*and

*g*, which implies

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

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

**Corollary 3.2**

*Let*$(X,\le )$

*be a partially ordered set and*$(X,F,\ast )$

*be a complete fuzzy metric space*.

*Let*$F:X\times X\to X$

*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*

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

*Suppose*$F(X\times X)\subseteq g(X)$,

*g*

*is continuous and*

*F*

*and*

*g*

*are compatible*.

*Also suppose*

- (a)
*F**is continuous or* - (b)
*X**has the following properties*: - (i)
*if a non*-*decreasing sequence*${x}_{n}\to x$,*then*${x}_{n}\le x$*for all**n*, - (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* $g({x}_{0})\le F({x}_{0},{y}_{0})$ *and* $g({y}_{0})\ge F({y}_{0},{x}_{0})$, *then there exist* $x,y\in X$ *such that* $g(x)=F(x,y)$ *and* $g(y)=F(y,x)$, *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:

Next, we give an example to demonstrate Theorem 3.1.

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

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

*F* obeys the mixed *g*-monotone property.

*X*such that

*F*and

*g*are compatible in

*X*. Also, ${x}_{0}=0$ and ${y}_{0}=c$ are two points in

*X*such that

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

We consider the following cases:

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

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

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

## Declarations

### 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).

## Authors’ Affiliations

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