New conditions on fuzzy coupled coincidence fixed point theorem

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.

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

$$M(F(x,y),F(u,v),kt)\ge M(gx,gu,t)\ast M(gy,gv,t)$$

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

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

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 $g(x)⪯g(u)$ and $g(y)⪰g(v)$, 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 $x_n⪯x$ for all $n\in \mathbb{N}\cup \{0\}$,

• (b) if we have a non-increasing sequence $\{y_n\}\to y$, then $y_n⪰y$ for all $n\in \mathbb{N}\cup \{0\}$.

If there exist $x_0,y_0\in X$ such that $g(x_0)⪯F(x_0,y_0)$, $g(y_0)⪰F(y_0,x_0)$, 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 $t>0$,

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

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]

A binary operation $\ast :[0,1]\times [0,1]\to [0,1]$ is continuous 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 $ϵ\in (0,1)$ and $t>0$ there is $n_0\in \mathbb{N}$ such that $M(x_n,x_m,t)>1-ϵ$ 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 $(X,⪯)$ 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 $n\in \mathbb{N}$, $x_n⪯x_{n+1}$. A mapping $g:X\to X$ is called monotonic increasing if for all $x,y\in X$ with $x⪯y$, $g(x)⪯g(y)$.

Definition 2.4 [21]

Let $(X,⪯)$ 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 $x_1,x_2\in X$, $g(x_1)⪯g(x_2)$ implies $F(x_1,y)⪯F(x_2,y)$ for all $y\in X$, and for all $y_1,y_2\in X$, $g(y_1)⪯g(y_2)$ implies $F(x,y_1)⪰F(x,y_2)$ for all $x\in X$.

Definition 2.5 [14]

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

$$F(x,y)=g(x),F(y,x)=g(y).$$

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

$$\gamma (a_n)\ast \gamma (a_n)>a_n,n\in \mathbb{N}.$$

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

$$\gamma (1)\ast \gamma (1)\ge 1,$$

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 $g(x)⪯g(u)$ and $g(y)⪰g(v)$,

• (iii) if a non-decreasing sequence $\{x_n\}\to x$, then $x_n⪯x$ for all $n\in \mathbb{N}\cup \{0\}$,

• (iv) if a non-increasing sequence $\{y_n\}\to y$, then $y_n⪰y$ for all $n\in \mathbb{N}\cup \{0\}$.

If there exist $x_0,y_0\in X$ such that $g(x_0)⪯F(x_0,y_0)$, $g(y_0)⪰F(y_0,x_0)$ 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 $g(x_0)⪯F(x_0,y_0)$ and $F(y_0,x_0)⪯g(y_0)$. Define the sequences $\{x_n\}$ and $\{y_n\}$ in X by

$$g(x_{n+1})=F(x_n,y_n)\text{and}g(y_{n+1})=F(y_n,x_n),\text{for all }n\in \mathbb{N}\cup \{0\}.$$

Along the lines of the proof of [17], we see that

$$g(x_n)⪯g(x_{n+1})\text{and}g(y_n)⪰g(y_{n+1}),\text{for all }n\in \mathbb{N}\cup \{0\}.$$

(3.2)

By (3.1) and (3.2) we have

$$\begin{array}{rl}M(g(x_1),g(x_2),t)& \ge M(g(x_1),g(x_2),\varphi (t))\\ =M(F(x_0,y_0),F(x_1,y_1),\varphi (t))\\ \ge \gamma (M(g(x_0),g(x_1),t)\ast M(g(y_0),g(y_1),t))\\ >M(g(x_0),g(x_1),t)\ast M(g(y_0),g(y_1),t)>0,\mathrm{\forall }t>0,\end{array}$$

(3.3)

and

$$\begin{array}{rl}M(g(y_1),g(y_2),t)& \ge M(g(y_1),g(y_2),\varphi (t))\\ =M(F(y_0,x_0),F(y_1,x_1),\varphi (t))\\ \ge \gamma (M(g(y_0),g(y_1),t)\ast M(g(x_0),g(x_1),t))\\ >M(g(y_0),g(y_1),t)\ast M(g(x_0),g(x_1),t)>0,\mathrm{\forall }t>0.\end{array}$$

(3.4)

Since ∗ is positive, we have

$$M(g(x_1),g(x_2),t)\ast M(g(y_1),g(y_2),t)>0,\mathrm{\forall }t>0.$$

Repeating the process (3.3) and (3.4), we get

$$M(g(x_2),g(x_3),t)>0\text{and}M(g(y_2),g(y_3),t)>0,\mathrm{\forall }t>0,$$

and further we have

$$M(g(x_2),g(x_3),t)\ast M(g(y_2),g(y_3),t)>0,\mathrm{\forall }t>0.$$

Continuing the above process, we get, for each $n\in \mathbb{N}$,

$$M(g(x_n),g(x_{n+1}),t)>0,\mathrm{\forall }t>0,$$

and

$$M(g(y_n),g(y_{n+1}),t)>0,\mathrm{\forall }t>0.$$

Since ∗ is positive, one has

$$M(g(x_n),g(x_{n+1}),t)\ast M(g(y_n),g(y_{n+1}),t)>0,\mathrm{\forall }n\in \mathbb{N},\mathrm{\forall }t>0.$$

Now we prove by induction that, for each $n\in \mathbb{N}$ and $k\in \mathbb{N}$ with $k\ge n$, one has

$$M(g(x_n),g(x_k),t)\ast M(g(y_n),g(y_k),t)>0,\mathrm{\forall }t>0.$$

(3.5)

Obviously (3.5) holds for $k=n$. Assume that (3.5) holds for some $k\in \mathbb{N}$ with $k>n$. Then we have

$$M(g(x_n),g(x_{k+1}),t)\ge M(g(x_n),g(x_k),t/2)\ast M(g(x_k),g(x_{k+1}),t/2).$$

Since $M(g(x_n),g(x_k),t/2)>0$, $M(g(x_k),g(x_{k+1}),t/2)>0$, and ∗ is positive, we have

$$M(g(x_n),g(x_{k+1}),t)>0,\mathrm{\forall }t>0.$$

Similarly, we have

$$M(g(y_n),g(y_{k+1}),t)>0,\mathrm{\forall }t>0.$$

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

Now we use the method of Wang [22] to show that both $\{g(x_n)\}$ and $\{g(y_n)\}$ are Cauchy sequences. Fix $t>0$. Let

$$a_n=\underset{k\ge n}{inf}M(g(x_n),g(x_k),t)\ast M(g(y_n),g(y_k),t).$$

For $k\ge n+1$, by (3.1) and (3.2) we have

$$\begin{array}{rl}M(g(x_{n+1}),g(x_k),t)& \ge M(g(x_{n+1}),g(x_k),\varphi (t))\\ \ge \gamma (M(g(x_n),g(x_{k-1}),t)\ast M(g(y_n),g(y_{k-1}),t)).\end{array}$$

Similarly,

$$M(g(y_{n+1}),g(y_k),t)\ge \gamma (M(g(x_n),g(x_{k-1}),t)\ast M(g(y_n),g(y_{k-1}),t)).$$

So, by (3.5) and the hypothesis we have

$$\begin{array}{r}M(g(x_{n+1}),g(x_k),t)\ast M(g(y_{n+1}),g(y_k),t)\\ \ge {\ast }^2\left(\gamma (M(g(x_n),g(x_{k-1}),t)\ast M(g(y_n),g(y_{k-1}),t))\right)\\ \ge M(g(x_n),g(x_{k-1}),t)\ast M(g(y_n),g(y_{k-1}),t)>0,\end{array}$$

(3.6)

which implies that

$$a_{n+1}\ge a_n>0.$$

Since $\{a_n\}$ is bounded, there exists $a\in (0,1]$ such that ${lim}_{n\to \mathrm{\infty }}{a}_n=a$. Assume that $a<1$. Since γ is increasing, we have

$$\begin{array}{r}{\ast }^2\left(\gamma (M(g(x_n),g(x_{k-1}),t)\ast M(g(y_n),g(y_{k-1}),t))\right)\\ \ge {\ast }^2\left(\gamma \left(\underset{k\ge n+1}{inf}(M(g(x_n),g(x_{k-1}),t)\ast M(g(y_n),g(y_{k-1}),t))\right)\right)\end{array}$$

and further

$$\begin{array}{r}\underset{k\ge n+1}{inf}{\ast }^2\left(\gamma \left((M(g(x_n),g(x_{k-1}),t)\ast M(g(y_n),g(y_{k-1}),t))\right)\right)\\ \ge {\ast }^2\left(\gamma \left(\underset{k\ge n+1}{inf}(M(g(x_n),g(x_{k-1}),t)\ast M(g(y_n),g(y_{k-1}),t))\right)\right).\end{array}$$

(3.7)

From (3.6) and (3.7) it follows that

$$\begin{array}{r}\underset{k\ge n+1}{inf}(M(g(x_{n+1}),g(x_k),t)\ast M(g(y_{n+1}),g(y_k),t))\\ \ge {\ast }^2\left(\gamma \left(\underset{k\ge n+1}{inf}(M(g(x_n),g(x_{k-1}),t)\ast M(g(y_n),g(y_{k-1}),t))\right)\right),\end{array}$$

i.e.,

$$a_{n+1}\ge \gamma (a_n)\ast \gamma (a_n),\mathrm{\forall }n\in \mathbb{N}.$$

Since γ is left continuous, by hypothesis we get

$$a\ge \gamma (a)\ast \gamma (a)>a.$$

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

For any given $ϵ>0$, there exists $n_0\in \mathbb{N}$ such that

$$1-a_n<ϵ\text{for all }n\ge n_0.$$

Thus for each $k\ge n\ge n_0$,

$$M(g(x_n),g(x_k),t)\ast M(g(y_n),g(y_k),t)>1-ϵ,$$

which implies that

$$min\{M(g(x_n),g(x_k),t),M(g(y_n),g(y_k),t)\}>1-ϵ.$$

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 $n\to \mathrm{\infty }$.

By hypothesis, we have

$$g(x_n)⪯g\left(x^{\ast }\right)\text{and}g(y_n)⪰g\left(y^{\ast }\right),n\in \mathbb{N}.$$

(3.8)

Now, for all $t>0$, by (3.1) and (3.8) we have

$$\begin{array}{rl}M(F(x^{\ast },y^{\ast }),g\left(x^{\ast }\right),t)\ge & M(F(x^{\ast },y^{\ast }),F(x_n,y_n),t/2)\ast M(F(x_n,y_n),g\left(x^{\ast }\right),t/2)\\ \ge & M(F(x^{\ast },y^{\ast }),F(x_n,y_n),\varphi (t/2))\ast M(F(x_n,y_n),g\left(x^{\ast }\right),\varphi (t/2))\\ \ge & \gamma (M(g\left(x^{\ast }\right),g(x_n),t/2)\ast M(g\left(y^{\ast }\right),g(y_n),t/2))\\ \ast M(F(x_n,y_n),g\left(x^{\ast }\right),\varphi (t/2)).\end{array}$$

(3.9)

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

$$\begin{array}{rl}M(F(x^{\ast },y^{\ast }),g\left(x^{\ast }\right),t)\ge & \underset{n\to \mathrm{\infty }}{lim}[\gamma (M(g\left(x^{\ast }\right),g(x_n),t/2)\ast M(g\left(y^{\ast }\right),g(y_n),t/2))\\ \ast M(F(x_n,y_n),g\left(x^{\ast }\right),\varphi (t/2))]\\ =& \gamma (1\ast 1)\ast 1=1,\mathrm{\forall }t>0.\end{array}$$

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 $g(x)⪯g(u)$ and $g(y)⪰g(v)$.

• (iii) If we have a non-decreasing sequence $\{x_n\}\to x$, then $x_n⪯x$ for all $n\in \mathbb{N}\cup \{0\}$.

• (iv) If we have a non-increasing sequence $\{y_n\}\to y$, then $y_n⪰y$ for all $n\in \mathbb{N}\cup \{0\}$.

If there exist $x_0,y_0\in X$ such that $g(x_0)⪯F(x_0,y_0)$, $g(y_0)⪰F(y_0,x_0)$ 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 $x⪯u$ and $y⪰v$.

• (ii) If we have a non-decreasing sequence $\{x_n\}\to x$, then $x_n⪯x$ for all $n\in \mathbb{N}\cup \{0\}$.

• (iii) If we have a non-increasing sequence $\{y_n\}\to y$, then $y_n⪰y$ for all $n\in \mathbb{N}\cup \{0\}$.

If there exist $x_0,y_0\in X$ such that $x_0⪯F(x_0,y_0)$, $y_0⪰F(y_0,x_0)$ 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 $x⪯u$ and $y⪰v$.

• (ii) If we have a non-decreasing sequence $\{x_n\}\to x$, then $x_n⪯x$ for all $n\in \mathbb{N}\cup \{0\}$.

• (iii) If we have a non-increasing sequence $\{y_n\}\to y$, then $y_n⪰y$ for all $n\in \mathbb{N}\cup \{0\}$.

If there exist $x_0,y_0\in X$ such that $x_0⪯F(x_0,y_0)$, $y_0⪰F(y_0,x_0)$, 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 $(X,⪯)$ 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

$$M(x,y,t)=e^{-|x-y|/t},\mathrm{\forall }x,y\in X,\mathrm{\forall }t>0.$$

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 $s\in (0,1)$.

Define the mappings $g:X\to X$ by

$$g(x)=x^2,\mathrm{\forall }x\in X,$$

and $F:X\times X\to X$ by

$$F(x,y)=\frac{x^2-y^2}{3}+\frac{2}{3},\mathrm{\forall }x,y\in X.$$

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 $t>0$.

Next we show that for all $t>0$ and all $x,y,u,v\in X$ with $g(x)\le g(u)$ and $g(y)\ge g(v)$, i.e., $x\le u$ and $y\ge v$, one has

$$M(F(x,y),F(u,v),t)\ge {\left(M(g(x),g(u),t)M(g(y),g(v),t)\right)}^{\frac{1}{3}}.$$

(3.10)

We prove the above inequality by a contradiction. Assume

$$M(F(x,y),F(u,v),t)<{\left(M(g(x),g(u),t)M(g(y),g(v),t)\right)}^{\frac{1}{3}}.$$

Then

$$e^{-|(x^2-y^2)/3-(u^2-v^2)/3|/t}<e^{-(|x^2-u^2|+|y^2-v^2|)/3t},$$

i.e.,

$$|(x^2-u^2)-(y^2-v^2)|>|x^2-u^2|+|y^2-v^2|.$$

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 $(X,⪯)$ 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.

Define the mappings $g:X\to X$ and $F:X\times X\to X$ by

$$g(x)=\{\begin{array}{cc}\frac{1}{2}(1-x),\hfill & \text{if }0\le x<1,\hfill \\ 0,\hfill & \text{if }x=2,\hfill \end{array}$$

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 $t>0$.

Let $\varphi (t)=t$ for all $t>0$. Let γ be a function from $[0,1]$ to $[0,1]$ defined by

$$\gamma (s)=\{\begin{array}{cc}\sqrt[3]{s},\hfill & \text{if }0\le s\le \frac{1}{2},\hfill \\ \sqrt[4]{s},\hfill & \text{if }\frac{1}{2}<s\le 1.\hfill \end{array}$$

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

Let $t>0$ and $x,y,u,v\in X$ with $g(x)\le g(u)$ and $g(y)\ge g(v)$, i.e., $u\le x$ and $y\le v$, since

$$\begin{array}{rl}M(F(x,y),F(u,v),\varphi (t))& =e^{-|F(x,y)-F(u,v)|}=e^{-|\frac{x-y}{16}-\frac{u-v}{16}|}\\ \ge max\{e^{-\frac{|x-y|+|y-v|}{6}},e^{-\frac{|x-y|+|y-v|}{8}}\}\\ \ge \gamma (M(g(x),g(u),t)\ast M(g(y),g(v),t)).\end{array}$$

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.