# New conditions on fuzzy coupled coincidence fixed point theorem

- Shenghua Wang
^{1}Email author, - Ting Luo
^{1}, - Ljubomir Ćirić
^{2}and - Saud M Alsulami
^{3}

**2014**:153

https://doi.org/10.1186/1687-1812-2014-153

© Wang et al.; licensee Springer. 2014

**Received: **11 February 2014

**Accepted: **13 June 2014

**Published: **22 July 2014

## Abstract

Recently, Choudhury *et al.* proved a coupled coincidence point theorem in a partial order fuzzy metric space. In this paper, we give a new version of the result of Choudhury *et al.* by removing some restrictions. In our result, the mappings are not required to be compatible, continuous or commutable, and the *t*-norm is not required to be of Hadžić-type. Finally, two examples are presented to illustrate the main result of this paper.

**MSC:**54E70, 47H25.

## Keywords

## 1 Introduction

The concept of fuzzy metric spaces was defined in different ways [1–3]. Grabiec [4] presented a fuzzy version of Banach contraction principle in a fuzzy metric space of Kramosi and Michalek’s sense. Fang [5] proved some fixed point theorems in fuzzy metric spaces, which improve, generalize, unify, and extend some main results of Edelstein [6], Istratescu [7], Sehgal and Bharucha-Reid [8].

In order to obtain a Hausdorff topology, George and Veeramani [9, 10] modified the concept of fuzzy metric space due to Kramosil and Michalek [11]. Many fixed point theorems in complete fuzzy metric spaces in the sense of George and Veeramani [9, 10] have been obtained. For example, Singh and Chauhan [12] proved some common fixed point theorems for four mappings in GV fuzzy metric spaces. Gregori and Sapena [13] proved that each fuzzy contractive mapping has a unique fixed point in a complete GV fuzzy metric space in which fuzzy contractive sequences are Cauchy.

The coupled fixed point theorem and its applications in metric spaces are firstly obtained by Bhaskar and Lakshmikantham [14]. Recently, some authors considered coupled fixed point theorems in fuzzy metric spaces; see [15–18].

In [15], the authors gave the following results.

**Theorem 1.1** [[15], Theorem 2.5]

*Let*$a\ast b>ab$

*for all*$a,b\in [0,1]$

*and*$(X,M,\ast )$

*be a complete fuzzy metric space such that*

*M*

*has*

*n*-

*property*.

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

*and*$g:X\to X$

*be two functions such that*

*for all* $x,y,u,v\in X$, *where* $0<k<1$, $F(X\times X)\subseteq g(X)$ *and* *g* *is continuous and commutes with* *F*. *Then there exists a unique* $x\in X$ *such that* $x=gx=F(x,x)$.

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

(*ϕ*-1) *ϕ* is non-decreasing;

(*ϕ*-2) *ϕ* is upper semicontinuous from the right;

(*ϕ*-3) ${\sum}_{n=0}^{\mathrm{\infty}}{\varphi}^{n}(t)<+\mathrm{\infty}$ for all $t>0$ where ${\varphi}^{n+1}(t)=\varphi ({\varphi}^{n}(t))$, $n\in \mathbb{N}$.

In [16], Hu proved the following result.

**Theorem 1.2** [[16], Theorem 1]

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

*be a complete fuzzy metric space*,

*where*∗

*is a continuous*

*t*-

*norm of H*-

*type*.

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

*and*$g:X\to X$

*be two mappings and let there exist*$\varphi \in \mathrm{\Phi}$

*such that*

*for all* $x,y,u,v\in X$, $t>0$. *Suppose that* $F(X\times X)\subseteq g(X)$, *and* *g* *is continuous*; *F* *and* *g* *are compatible*. *Then there exists* $x\in X$ *such that* $x=gx=F(x,x)$, *that is*, *F* *and* *g* *have a unique common fixed point in* *X*.

Choudhury *et al.* [17] gave the following coupled coincidence fixed point result in a partial order fuzzy metric space.

**Theorem 1.3** [[17], Theorem 3.1]

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

*be a complete fuzzy metric space with a Hadžić type*

*t*-

*norm*$M(x,y,t)\to 1$

*as*$t\to \mathrm{\infty}$

*for all*$x,y\in X$.

*Let*⪯

*be a partial order defined on*

*X*.

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

*and*$g:X\to X$

*be two mappings such that*

*F*

*has mixed*

*g*-

*monotone property and satisfies the following conditions*:

- (i)
$F(X\times X)\subseteq g(X)$,

- (ii)
*g**is continuous and monotonic increasing*, - (iii)
$(g,F)$

*is a compatible pair*, - (iv)
$M(F(x,y),F(u,v),kt)\ge \gamma (M(g(x),g(u),t)\ast M(g(y),g(v),t))$

*for all*$x,y,u,v\in X$, $t>0$*with*$g(x)\u2aafg(u)$*and*$g(y)\u2ab0g(v)$,*where*$k\in (0,1)$, $\gamma :[0,1]\to [0,1]$*is a continuous function such that*$\gamma (a)\ast \gamma (a)\ge a$*for each*$0\le a\le 1$.

*Also suppose that*

*X*

*has the following properties*:

- (a)
*if we have a non*-*decreasing sequence*$\{{x}_{n}\}\to x$,*then*${x}_{n}\u2aafx$*for all*$n\in \mathbb{N}\cup \{0\}$, - (b)
*if we have a non*-*increasing sequence*$\{{y}_{n}\}\to y$,*then*${y}_{n}\u2ab0y$*for all*$n\in \mathbb{N}\cup \{0\}$.

*If there exist* ${x}_{0},{y}_{0}\in X$ *such that* $g({x}_{0})\u2aafF({x}_{0},{y}_{0})$, $g({y}_{0})\u2ab0F({y}_{0},{x}_{0})$, *and* $M(g({x}_{0}),F({x}_{0},{y}_{0}),t)\ast M(g({y}_{0}),F({y}_{0},{x}_{0}),t)>0$ *for all* $t>0$, *then there exist* $x,y\in X$ *such that* $g(x)=F(x,y)$ *and* $g(y)=F(y,x)$, *that is*, *g* *and* *F* *have a coupled coincidence point in* *X*.

Wang *et al.* [18] proved the following coupled fixed point result in a fuzzy metric space.

**Theorem 1.4** [[18], Theorem 3.1]

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

*be a fuzzy metric space under a continuous*

*t*-

*norm*∗

*of H*-

*type*.

*Let*$\varphi :(0,\mathrm{\infty})\to (0,\mathrm{\infty})$

*be a function satisfying*${lim}_{n\to \mathrm{\infty}}{\varphi}^{n}(t)=0$

*for any*$t>0$.

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

*and*$g:X\to X$

*be two mappings with*$F(X\times X)\subseteq g(X)$

*and assume that for any*$t>0$,

*for all* $x,y,u,v\in X$. *Suppose that* $F(X\times X)$ *is complete and* *g* *and* *F* *are* *w*-*compatible*, *then* *g* *and* *F* *have a unique common fixed point* ${x}^{\ast}\in X$, *that is*, ${x}^{\ast}=g({x}^{\ast})=F({x}^{\ast},{x}^{\ast})$.

In this paper, by modifying the conditions on the result of Choudhury *et al.* [17], we give a new coupled coincidence fixed point theorem in partial order fuzzy metric spaces. In our result, we do not require that the *t*-norm is of Hadžić-type [19], the mappings are compatible [16], commutable, continuous or monotonic increasing. Our proof method is different from the one of Choudhury *et al.* Finally, some examples are presented to illustrate our result.

## 2 Preliminaries

**Definition 2.1** [9]

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

- (1)
∗ is associative and commutative,

- (2)
∗ is continuous,

- (3)
$a\ast 1=a$ for all $a\in [0,1]$,

- (4)
$a\ast b\le c\ast d$ whenever $a\le c$ and $b\le d$ for all $a,b,c,d\in [0,1]$.

Typical examples of the continuous *t*-norm are $a{\ast}_{1}b=ab$ and $a{\ast}_{2}b=min\{a,b\}$ for all $a,b\in [0,1]$.

A *t*-norm ∗ is said to be positive if $a\ast b>0$ for all $a,b\in (0,1]$. Obviously, ∗_{1} and ∗_{2} are positive *t*-norms.

**Definition 2.2** [9]

The 3-tuple $(X,M,\ast )$ is called a fuzzy metric space if *X* is an arbitrary non-empty set, ∗ is a continuous *t*-norm and *M* is a fuzzy set on ${X}^{2}\times (0,\mathrm{\infty})$ satisfying the following conditions for each $x,y,z\in X$ and $t,s>0$:

(GV-1) $M(x,y,t)>0$,

(GV-2) $M(x,y,t)=1$ if and only if $x=y$,

(GV-3) $M(x,y,t)=M(y,x,t)$,

(KM-4) $M(x,y,\cdot ):(0,\mathrm{\infty})\to [0,1]$ is continuous,

(KM-5) $M(x,y,t+s)\ge M(x,z,t)\ast M(y,z,s)$.

**Lemma 2.1** [4]

*Let* $(X,M,\ast )$ *be a fuzzy metric space*. *Then* $M(x,y,\ast )$ *is non*-*decreasing for all* $x,y\in X$.

**Lemma 2.2** [20]

*Let* $(X,M,\ast )$ *be a fuzzy metric space*. *Then* *M* *is a continuous function on* ${X}^{2}\times (0,\mathrm{\infty})$.

**Definition 2.3** [9]

Let $(X,M,\ast )$ be a fuzzy metric space. A sequence $\{{x}_{n}\}$ in *X* is called an *M*-Cauchy sequence, if for each $\u03f5\in (0,1)$ and $t>0$ there is ${n}_{0}\in \mathbb{N}$ such that $M({x}_{n},{x}_{m},t)>1-\u03f5$ for all $m,n\ge {n}_{0}$. The fuzzy metric space $(X,M,\ast )$ is called *M*-complete if every *M*-Cauchy sequence is convergent.

Let $(X,\u2aaf)$ be a partially ordered set and *F* be a mapping from *X* to itself. A sequence $\{{x}_{n}\}$ in *X* is said to be non-decreasing if for each $n\in \mathbb{N}$, ${x}_{n}\u2aaf{x}_{n+1}$. A mapping $g:X\to X$ is called monotonic increasing if for all $x,y\in X$ with $x\u2aafy$, $g(x)\u2aafg(y)$.

**Definition 2.4** [21]

Let $(X,\u2aaf)$ be a partially ordered set and $F:X\times X\to X$ and $g:X\to X$ be two mappings. The mapping *F* is said to have the mixed *g*-monotone property if for all ${x}_{1},{x}_{2}\in X$, $g({x}_{1})\u2aafg({x}_{2})$ implies $F({x}_{1},y)\u2aafF({x}_{2},y)$ for all $y\in X$, and for all ${y}_{1},{y}_{2}\in X$, $g({y}_{1})\u2aafg({y}_{2})$ implies $F(x,{y}_{1})\u2ab0F(x,{y}_{2})$ for all $x\in X$.

**Definition 2.5** [14]

Here $(gx,gy)$ is called a coupled point of coincidence.

## 3 Main results

**Lemma 3.1** *Let* $\gamma :[0,1]\to [0,1]$ *be a left continuous function and* ∗ *be a continuous* *t*-*norm*. *Assume that* $\gamma (a)\ast \gamma (a)>a$ *for all* $a\in (0,1)$. *Then* $\gamma (1)=1$.

*Proof*Let $\{{a}_{n}\}\subseteq (0,1)$ be a non-decreasing sequence with ${lim}_{n\to \mathrm{\infty}}{a}_{n}=1$. By hypothesis we have

*γ*is left continuous and ∗ is continuous, we get

which implies that $\gamma (1)\ast \gamma (1)=1$. Since $\gamma (1)\ge \gamma (1)\ast \gamma (1)$, one has $\gamma (1)=1$. This completes the proof. □

**Theorem 3.1**

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

*be a fuzzy metric space with a continuous and positive*

*t*-

*norm*.

*Let*⪯

*be a partial order defined on*

*X*.

*Let*$\varphi :(0,\mathrm{\infty})\to (0,\mathrm{\infty})$

*be a function satisfying*$\varphi (t)\le t$

*for all*$t>0$

*and let*$\gamma :[0,1]\to [0,1]$

*be a left continuous and increasing function satisfying*$\gamma (a)\ast \gamma (a)>a$

*for all*$a\in (0,1)$.

*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 assume that*$g(X)$

*is complete*.

*Suppose that the following conditions hold*:

- (i)
$F(X\times X)\subseteq g(X)$,

- (ii)
*we have*$M(F(x,y),F(u,v),\varphi (t))\ge \gamma (M(g(x),g(u),t)\ast M(g(y),g(v),t)),$(3.1)

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

*with*$g(x)\u2aafg(u)$

*and*$g(y)\u2ab0g(v)$,

- (iii)
*if a non*-*decreasing sequence*$\{{x}_{n}\}\to x$,*then*${x}_{n}\u2aafx$*for all*$n\in \mathbb{N}\cup \{0\}$, - (iv)
*if a non*-*increasing sequence*$\{{y}_{n}\}\to y$,*then*${y}_{n}\u2ab0y$*for all*$n\in \mathbb{N}\cup \{0\}$.

*If there exist* ${x}_{0},{y}_{0}\in X$ *such that* $g({x}_{0})\u2aafF({x}_{0},{y}_{0})$, $g({y}_{0})\u2ab0F({y}_{0},{x}_{0})$ *and* $M(g({x}_{0}),F({x}_{0},{y}_{0}),t)\ast M(g({y}_{0}),F({y}_{0},{x}_{0}),t)>0$ *for all* $t>0$, *then there exist* ${x}^{\ast},{y}^{\ast}\in X$ *such that* $g({x}^{\ast})=F({x}^{\ast},{y}^{\ast})$ *and* $g({y}^{\ast})=F({y}^{\ast},{x}^{\ast})$.

*Proof*Let ${x}_{0},{y}_{0}\in X$ such that $g({x}_{0})\u2aafF({x}_{0},{y}_{0})$ and $F({y}_{0},{x}_{0})\u2aafg({y}_{0})$. Define the sequences $\{{x}_{n}\}$ and $\{{y}_{n}\}$ in

*X*by

Therefore, (3.5) holds for all $k\in \mathbb{N}$ with $k\ge n$.

*γ*is increasing, we have

*i.e.*,

*γ*is left continuous, by hypothesis we get

This is a contradiction. So $a=1$.

It follows that both $\{g({x}_{n})\}$ and $\{g({y}_{n})\}$ are Cauchy sequences. Since $g(X)$ is complete, there exist ${x}^{\ast},{y}^{\ast}\in X$ such that $g({x}_{n})\to g({x}^{\ast})$ and $g({y}_{n})\to g({y}^{\ast})$ as $n\to \mathrm{\infty}$.

*γ*is left continuous and ∗ is continuous, letting $n\to \mathrm{\infty}$ in (3.9), we get

It follows that $F({x}^{\ast},{y}^{\ast})=g({x}^{\ast})$. Similarly, we can prove that $F({y}^{\ast},{x}^{\ast})=g({y}^{\ast})$. This completes the proof. □

If $\varphi (t)=t$ for all $t>0$ in Theorem 3.1, we get the following corollary.

**Corollary 3.1**

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

*be a fuzzy metric space with a positive*

*t*-

*norm*.

*Let*⪯

*be a partial order defined on*

*X*.

*Let*$\gamma :[0,1]\to [0,1]$

*be a left continuous and increasing function satisfying*$\gamma (a)\ast \gamma (a)>a$

*for all*$a\in (0,1)$.

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

*and*$g:X\to X$

*be two mappings such that*

*F*

*has mixed*

*g*-

*monotone property and assume that*$g(X)$

*is complete*.

*Suppose that the following conditions hold*:

- (i)
$F(X\times X)\subseteq g(X)$.

- (ii)
*We have*$M(F(x,y),F(u,v),t)\ge \gamma (M(g(x),g(u),t)\ast M(g(y),g(v),t)),$

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

*with*$g(x)\u2aafg(u)$

*and*$g(y)\u2ab0g(v)$.

- (iii)
*If we have a non*-*decreasing sequence*$\{{x}_{n}\}\to x$,*then*${x}_{n}\u2aafx$*for all*$n\in \mathbb{N}\cup \{0\}$. - (iv)
*If we have a non*-*increasing sequence*$\{{y}_{n}\}\to y$,*then*${y}_{n}\u2ab0y$*for all*$n\in \mathbb{N}\cup \{0\}$.

*If there exist* ${x}_{0},{y}_{0}\in X$ *such that* $g({x}_{0})\u2aafF({x}_{0},{y}_{0})$, $g({y}_{0})\u2ab0F({y}_{0},{x}_{0})$ *and* $M(g({x}_{0}),F({x}_{0},{y}_{0}),t)\ast M(g({y}_{0}),F({y}_{0},{x}_{0}),t)>0$ *for all* $t>0$, *then there exist* ${x}^{\ast},{y}^{\ast}\in X$ *such that* $g({x}^{\ast})=F({x}^{\ast},{y}^{\ast})$ *and* $g({y}^{\ast})=F({y}^{\ast},{x}^{\ast})$.

Letting $g(x)=x$ for all $x\in X$ in Theorem 3.1 and Corollary 3.1, we get the following corollaries.

**Corollary 3.2**

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

*be a complete fuzzy metric space with a positive*

*t*-

*norm*.

*Let*⪯

*be a partial order defined on*

*X*.

*Let*$\varphi :(0,\mathrm{\infty})\to (0,\mathrm{\infty})$

*be a function satisfying*$\varphi (t)\le t$

*for all*$t>0$

*and let*$\gamma :[0,1]\to [0,1]$

*be a left continuous and increasing function satisfying*$\gamma (a)\ast \gamma (a)>a$

*for all*$a\in (0,1)$.

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

*and assume*

*F*

*has mixed monotone property*.

*Suppose that the following conditions hold*:

- (i)
*We have*$M(F(x,y),F(u,v),\varphi (t))\ge \gamma (M(x,u,t)\ast M(y,v,t)),$*for all*$x,y,u,v\in X$, $t>0$*with*$x\u2aafu$*and*$y\u2ab0v$. - (ii)
*If we have a non*-*decreasing sequence*$\{{x}_{n}\}\to x$,*then*${x}_{n}\u2aafx$*for all*$n\in \mathbb{N}\cup \{0\}$. - (iii)
*If we have a non*-*increasing sequence*$\{{y}_{n}\}\to y$,*then*${y}_{n}\u2ab0y$*for all*$n\in \mathbb{N}\cup \{0\}$.

*If there exist* ${x}_{0},{y}_{0}\in X$ *such that* ${x}_{0}\u2aafF({x}_{0},{y}_{0})$, ${y}_{0}\u2ab0F({y}_{0},{x}_{0})$ *and* $M({x}_{0},F({x}_{0},{y}_{0}),t)\ast M({y}_{0},F({y}_{0},{x}_{0}),t)>0$ *for all* $t>0$, *then there exist* ${x}^{\ast},{y}^{\ast}\in X$ *such that* ${x}^{\ast}=F({x}^{\ast},{y}^{\ast})$ *and* ${y}^{\ast}=F({y}^{\ast},{x}^{\ast})$.

**Corollary 3.3**

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

*be a complete fuzzy metric space with a positive*

*t*-

*norm*.

*Let*⪯

*be a partial order defined on*

*X*.

*Let*$\gamma :[0,1]\to [0,1]$

*be a left continuous and increasing function satisfying*$\gamma (a)\ast \gamma (a)>a$

*for all*$a\in (0,1)$.

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

*and assume*

*F*

*has mixed monotone property*.

*Suppose that the following conditions hold*:

- (i)
*We have*$M(F(x,y),F(u,v),t)\ge \gamma (M(x,u,t)\ast M(y,v,t)),$*for all*$x,y,u,v\in X$, $t>0$*with*$x\u2aafu$*and*$y\u2ab0v$. - (ii)
*If we have a non*-*decreasing sequence*$\{{x}_{n}\}\to x$,*then*${x}_{n}\u2aafx$*for all*$n\in \mathbb{N}\cup \{0\}$. - (iii)
*If we have a non*-*increasing sequence*$\{{y}_{n}\}\to y$,*then*${y}_{n}\u2ab0y$*for all*$n\in \mathbb{N}\cup \{0\}$.

*If there exist* ${x}_{0},{y}_{0}\in X$ *such that* ${x}_{0}\u2aafF({x}_{0},{y}_{0})$, ${y}_{0}\u2ab0F({y}_{0},{x}_{0})$, *and* $M({x}_{0},F({x}_{0},{y}_{0}),t)\ast M({y}_{0},F({y}_{0},{x}_{0}),t)>0$ *for all* $t>0$, *then there exist* ${x}^{\ast},{y}^{\ast}\in X$ *such that* ${x}^{\ast}=F({x}^{\ast},{y}^{\ast})$ *and* ${y}^{\ast}=F({y}^{\ast},{x}^{\ast})$.

First, we illustrate Theorem 3.1 by modifying [[17], Example 3.4] as follows.

**Example 3.1**Let $(X,\u2aaf)$ is the partially ordered set with $X=[0,1]$ and the natural ordering ≤ of the real numbers as the partial ordering ⪯. Define $M:{X}^{2}\times (0,\mathrm{\infty})$ by

Let $a\ast b=ab$ for all $a,b\in [0,1]$. Then $(X,M,\ast )$ is a (complete) fuzzy metric space.

Let $\psi (t)=t$ for all $t>0$ and $\gamma (s)={s}^{\frac{1}{3}}$ for all $s\in [0,1]$. It is easy to see that $\gamma (s)\ast \gamma (s)>s$ for all $s\in (0,1)$.

Then $F(X\times X)\subseteq g(X)$, *F* satisfies the mixed *g*-monotone property; see [[17], Example 3.4]. Obviously $g(X)$ is complete.

Let ${x}_{0}=0$ and ${y}_{0}=1$, then $g({x}_{0})\le F({x}_{0},{y}_{0})$ and $g({y}_{0})\ge F({y}_{0},{x}_{0})$; see [[17], Example 3.4]. Moreover, $M(g({x}_{0}),F({x}_{0},{y}_{0}),t)\ast M(g({y}_{0}),F({y}_{0},{x}_{0}),t)>0$ for all $t>0$.

*i.e.*, $x\le u$ and $y\ge v$, one has

*i.e.*,

This is a contradiction. Thus, (3.10) holds. Therefore, all the conditions of Theorem 3.1 are satisfied. Then by Theorem 3.1 we conclude that there exist ${x}^{\ast}$, ${y}^{\ast}$ such that $g({x}^{\ast})=F({x}^{\ast},{y}^{\ast})$ and $g({y}^{\ast})=F({y}^{\ast},{x}^{\ast})$. It is easy to see that $({x}^{\ast},{y}^{\ast})=(\sqrt{\frac{2}{3}},\sqrt{\frac{2}{3}})$, as desired.

**Example 3.2** Let $(X,\u2aaf)$ is the partially ordered set with $X=[0,1)\cup \{2\}$ and the natural ordering ≤ of the real numbers as the partial ordering ⪯. Define a mapping $M:{X}^{2}\times (0,\mathrm{\infty})$ by $M(x,x,t)={e}^{-|x-y|}$ for all $x,y\in X$ and $t>0$. Let $a\ast b=ab$ for all $a,b\in [0,1]$. Then $(X,M,\ast )$ is a fuzzy metric space but not complete.

and $F(x,y)=\frac{y-x}{16}+\frac{1}{8}$ for all $x,y\in X$. Then $F(X\times X)\subseteq g(X)$, *F* satisfies the mixed *g*-monotone property, and $g(X)$ is complete. Take $({x}_{0},{y}_{0})=(\frac{23}{28},\frac{1}{4})$. By a simple calculation we see that $g({x}_{0})\le F({x}_{0},{y}_{0})$ and $g({y}_{0})\ge F({y}_{0},{x}_{0})$. Moreover, $M(g({x}_{0}),F({x}_{0},{y}_{0}),t)\ast M(g({y}_{0}),F({y}_{0},{x}_{0}),t)>0$ for all $t>0$.

*γ*be a function from $[0,1]$ to $[0,1]$ defined by

Obviously, *γ* is left continuous and increasing, and $\gamma (s)\ast \gamma (s)>s$ for all $s\in (0,1)$.

*i.e.*, $u\le x$ and $y\le v$, since

Hence (3.1) is satisfied. Therefore, all the conditions of Theorem 3.1 are satisfied. Then by Theorem 3.1 *F* and *g* have a coincidence point. It is easy to check that $({x}^{\ast},{y}^{\ast})=(\frac{3}{4},\frac{3}{4})$.

The above two examples cannot be applied to [[17], Theorem 3.1], since ∗ is not of Hadžić-type, or *g* is not monotonic increasing or continuous, or $M(x,y,t)\nrightarrow 1$ as $t\to \mathrm{\infty}$ for all $x,y\in X$.

## 4 Conclusion

In this paper, we prove a new coupled coincidence fixed point result in a partial order fuzzy metric space in which some restrictions required in [[17], Theorem 3.1] are removed, such that the conditions required in our result are fewer than the ones required in [[17], Theorem 3.1]. The purpose of this paper is to give some new conditions on the coupled coincidence fixed point theorem. Our result is not an improvement of [[17], Theorem 3.1], since we add some other restrictions such as requiring that the function *γ* is increasing and $M(g({x}_{0}),F({x}_{0},{y}_{0}),t)\ast M(g({y}_{0}),F({y}_{0},{x}_{0}),t)>0$ for all $t>0$. As pointed out in the conclusion part of [17], it still is an interesting open problem to find simpler or fewer conditions on the coupled coincidence fixed point theorem in a fuzzy metric space.

## Declarations

### Acknowledgements

This work is supported by the Fundamental Research Funds for the Central Universities (Grant Numbers: 13MS109, 2014ZD44) and funded by the Deanship of Scientific Research (DSR), King Abdulaziz University, Jeddah. The authors Alsulami and Ćirić, therefore, acknowledge with thanks the DSR financial support.

## Authors’ Affiliations

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