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VFuzzy metric space and related fixed point theorems
Fixed Point Theory and Applications volume 2016, Article number: 51 (2016)
Abstract
The paper concerns our sustained efforts for introduction of Vfuzzy metric spaces and to study their basic topological properties. As an application of this concept, we prove coupled common fixed point theorems for mixed weakly monotone maps in partially ordered Vfuzzy metric spaces. An example quoted in this paper also corroborates fully the main result. Also, here we introduce the concept of a symmetric Vfuzzy metric space. Under the influence of symmetry property of Vfuzzy metric spaces, its conversion materializes to the main result from Vfuzzy metric spaces to fuzzy metric spaces.
Introduction and preliminaries
A metric space is just a nonempty set X associated with a function d of two variables enabling us to measure the distance between points. In advanced mathematics, we need to find the distance not only between numbers and vectors, but also between more complicated objects like sequences, sets, and functions. In order to find an appropriate concept of a metric space, numerous approaches exist in this sphere. Thus, new notions of distance lead to new notions of convergence and continuity. A number of generalizations of a metric space have been discussed by many eminent mathematicians. Mustafa and Sims [1] introduced the notion of a Gmetric space and suggested an important generalization of a metric space as follows.
Definition 1.1
[1]
Let X be a nonempty set, and let \(G:X^{3} \rightarrow[0, + \infty)\) be a function satisfying the following conditions for all \(x,y,z,a \in X\):

(G1)
\(G(x,y,z) = 0\) if \(x=y=z\),

(G2)
\(0 \leq G(x,y,z)\) with \(x \neq y\),

(G3)
\(G(x,x,y) \leq G(x,y,z)\) with \(y \neq z\),

(G4)
\(G(x,y,z)=G(x,z,y)=G(y,x,z)=G(z,x,y)=G(y,z,x)=G(z,y,x)\),

(G5)
\(G(x,y,z) \leq G(x,a,a) + G(a,y,z)\).
The function G is called a generalized metric on X, and the pair \((X, G)\) is called a Gmetric space.
Example 1.1
[1]
Let \((X,d)\) be a metric space. Define \(G:X \times X \times X \rightarrow R^{+}\) by
Then \((X,G)\) is a Gmetric space.
Recently, some authors studied some important fixed point theorems with application in Gmetric spaces [2–4]. In 2012, Sedghi et al. [5] introduced a new generalized metric space called an Smetric space.
Definition 1.2
[5]
Let X be a nonempty set. Suppose that a function \(S:X^{3} \rightarrow [0, + \infty)\) satisfies the following conditions:

(S1)
\(S(x,y,z) \geq0\),

(S2)
\(S(x,y,z) = 0\) if and only if \(x=y=z=0\),

(S3)
\(S(x,y,z)\leq S(x,x,a) + S(y,y,a) + S(z,z,a)\) for any \(x,y,z,a \in X\).
Then the ordered pair \((X,S)\) is called an Smetric space.
Example 1.2
[5]
Let ℜ be the real line. Then \(S(x,y,z)=\vert xy\vert +\vert yz\vert \) for all \(x,y,z \in\Re \) is an Smetric on ℜ. This Smetric is called the usual Smetric on ℜ.
Abbas et al. [6] established the notion of Ametric spaces, a generalization of Smetric spaces.
Definition 1.3
[6]
Let X be a nonempty set. A function \(A:X^{n} \rightarrow[0, + \infty )\) is called an Ametric on X if for any \(x_{i}\), \(a\in X\), \(i=1,2,3,\ldots, n\), the following conditions hold:

(A1)
\(A(x_{1}, x_{2}, x_{3}, \ldots, x_{n}) \geq0\),

(A2)
\(A(x_{1}, x_{2}, x_{3}, \ldots, x_{n}) = 0\) if and only if \(x_{1}= x_{2}= x_{3}= \cdots= x_{n} = 0\),

(A3)
\(A(x_{1}, x_{2}, x_{3}, \ldots, x_{n}) \leq A(x_{1}, x_{1}, x_{1}, \ldots, (x_{1})_{n1}, a) + A(x_{2}, x_{2}, x_{2}, \ldots, (x_{2})_{n1}, a)+ \cdots+A(x_{n}, x_{n}, x_{n}, \ldots, (x_{n})_{n1}, a)\).
The pair \((X,A)\) is called an Ametric space.
Example 1.3
[6]
Let \(X=\Re\). Define the function \(A:X^{n} \rightarrow[0, + \infty)\) by
Then \((X,A)\) is called the usual Ametric space.
Fixed point theorems have been studied in many contexts, one of which is the fuzzy setting. The concept of fuzzy sets was initially introduced by Zadeh [7] in 1965. To use this concept in topology and analysis, the theory of fuzzy sets and its applications have been developed by many eminent authors. It is well known that a fuzzy metric space is an important generalization of a metric space.
Many authors have introduced fuzzy metric spaces in different ways. For instance, George and Veeramani [8] modified the concept of a fuzzy metric space introduced by Kramosil and Michalek [9] and defined the Hausdorff topology of a fuzzy metric space.
Definition 1.4
[10]
A binary operation \(\ast: [0,1] \times[0,1] \rightarrow[0,1]\) is called a continuous tnorm if ∗ satisfies following conditions:

(i)
∗ is commutative and associative,

(ii)
∗ is continuous,

(iii)
\(a \ast1 = a\), ∀ \(a \in[0,1]\),

(iv)
\(a\ast b \leq c \ast d\) whenever \(a \leq c\) and \(b \leq d\) for all \(a,b,c,d\in[0,1]\).
Definition 1.5
[9]
The triplet \((X, M, \ast)\) is called a fuzzy metric space (shortly, FMspace) if X is an arbitrary set, ∗ is a continuous tnorm, and M is a fuzzy set in \(X \times X \times[0, \infty)\) satisfying the following conditions for all \(x,y,z \in X\) and \(s,t > 0\):

(FM1)
\(M(x, y , 0) = 0\),

(FM2)
\(M(x, y ,t) = 1\) if and only if \(x = y\),

(FM3)
\(M(x, y ,t) = M(y, x, t)\),

(FM4)
\(M(x, y ,t) \ast M(y, z, s) \leq M(x, z ,t+s)\),

(FM5)
\(M(x, y, \cdot) : [0 ,\infty) \rightarrow[0, 1]\) is left continuous.
Note that \(M(x, y ,t)\) can be thought of as a degree of nearness between x and y with respect to t.
Example 1.4
[8]
Let \((X,d)\) be a metric space. Define the tnorm \(a\ast b = ab\) or \(a \ast b = \min \{a,b \}\). For all \(x,y \in X\), \(t>0\), let
Then \((X,M,\ast)\) is a fuzzy metric space.
Lemma 1.1
[11]
Let \((X,M,\ast)\) be a fuzzy metric space. If there exists \(k \in(0,1)\) such that, for all \(x,y \in X\) and \(t > 0\),
for all \(x,y \in X\), \(t > 0\), then \(x=y\).
In the process of generalization of fuzzy metric spaces, Sun and Yang [12] coined the notion of Gfuzzy metric spaces and established common fixedpoint theorems for four mappings.
Definition 1.6
[12]
A triplet \((X, V, \ast)\) is said to be a Gfuzzy metric space (shortly, GFspace) if X is an arbitrary nonempty set, ∗ is a continuous tnorm, and G is a fuzzy set on \(X \times X \times X \times[0, \infty)\) satisfying the following conditions for all \(x,y,z \in X\) and \(s,t > 0\):

(GF1)
\(G(x,x,y,t) > 0\) with \(x \neq y\),

(GF2)
\(G(x,x,y,t) \geq G(x,y,z,t)\) with \(y \neq z\),

(GF3)
\(G(x,y,z,t) = 1 \) if and only if \(x=y=z\),

(GF4)
\(G(x,y,z,t)= G(x,z,y,t) =G(y,x,z,t)= G(z,x,y,t)= G(y,z,x,t)=G(z,y,x,t)\),

(GF5)
\(G(x,y,z,t+s) \geq G(x,a,a,t) \ast G(a,y,z,s)\),

(GF6)
\(G(x,y,z,\cdot): [0 ,\infty) \rightarrow[0, 1]\) is left continuous.
Example 1.5
[12]
Let G be a Gmetric on a nonempty set X. Define the tnorm \(a \ast b = \min \{a,b \}\). For all \(x,y \in X\) and \(t>0\), denote
Then \((X,G,\ast)\) is a GFspace.
Lemma 1.2
[12]
Let \((X,G, \ast)\) be a GFspace. Then \(G(x,y,z,t)\) is nondecreasing with respect to t for all \(x,y,z \in X\).
On the other hand, the concepts of coupled fixed points and mixed monotone property of a fuzzy metric space are established by Bhaskar and Lakshmikantham [13]. Lakshmikantham and Ciric [14] discussed the mixed monotone mappings and gave some coupled fixed point theorems, which can be used to discuss the existence and uniqueness of a solution for a periodic boundary value problem.
Definition 1.7
[14]
An element \((x,y) \in X \times X\) is called a coupled fixed point of a mapping \(P:X \times X \rightarrow X\) if \(P(x,y) =x\) and \(P(y,x) =y\).
Definition 1.8
[14]
An element \((x,y) \in X \times X\) is called a coupled coincidence point of the mappings \(P:X \times X \rightarrow X\) and \(Q:X \rightarrow X\) if \(P(x,y) =Q(x)\) and \(P(y,x) =Q(y)\).
Definition 1.9
[14]
An element \((x,y) \in X \times X\) is called a common coupled fixed point of the mappings \(P:X \times X \rightarrow X\) and \(Q:X \rightarrow X\) if \(x= P(x,y) =Q(x)\) and \(y= P(y,x) =Q(y)\).
Example 1.6
Let \(X=[0,1]\). Define \(P:X \times X \rightarrow X\) and \(Q:X \rightarrow X\) as
For \(x=0\) and \(y=\frac{1}{2}\), we have
and
So, \((x,y)=(0,\frac{1}{2})\) is a common coupled fixed point of the mappings \(P:X \times X \rightarrow X\) and \(Q:X \rightarrow X\).
Definition 1.10
[14]
An element \(x \in X \) is called a common fixed point of the mappings \(P:X \times X \rightarrow X\) and \(Q:X \rightarrow X\) if \(x= P(x,x) =Q(x)\).
Example 1.7
Define \(P:X \times X \rightarrow X\) and \(Q:X \rightarrow X\) where \(X=[0,1]\) as
Then we have
For \(x=1\), we get \(P(1,1) =1= Q(1)\). So, \((x,y)=(1,1)\) is a common fixed point of the pair \((P,Q)\).
Definition 1.11
[13]
Let \((X,\preceq)\) be a partially ordered set. The mapping \((x,y)\) is said to have the mixed monotone property if P is monotone nondecreasing in its first argument and is monotone nonincreasing in its second argument; that is, for any \(x,y \in X\),
and
Example 1.8
Let \(P:X \times X \rightarrow X\) where \(X=[0,1]\) be defined by
For \(x_{1},x_{2}\in X\), \(x_{1} \preceq x_{2}\), we have
Therefore, the mapping P has the mixed monotone property.
Definition 1.12
[13]
Let \((X,\preceq)\) be a partially ordered set, and \(P:X \times X \rightarrow X\) and \(Q:X \rightarrow X\). We say that P has the mixed Qmonotone property if P is monotone Qnondecreasing in its first argument and is monotone Qnonincreasing in its second argument, that is, for any \(x,y \in X\),
and
Example 1.9
Let \(P:X \times X \rightarrow X\) and \(Q:X \rightarrow X\) where \(X=[1,1]\) be two functions given by
Therefore, the map P has the mixed Qmonotone property.
Definition 1.13
[15]
The mappings \(P:X \times X \rightarrow X\) and \(Q:X \rightarrow X\) are said to be Wcompatible if
and
whenever \(P(x,y) = Q(x)\) and \(P(y,x) = Q(y)\) for some \((x,y) \in X \times X\).
Example 1.10
Define \(P:X \times X \rightarrow X\) and \(Q:X \rightarrow X\) where \(X=[1,1]\) as \(P(x,y) =\frac{x^{2}+y^{2}}{2}\), \(Q(x) =x\), which satisfies \(Q (P(x,y) ) =P ((Q x,Q y) )\) and \(Q (P(y,x) ) =P ((Q y,Q x) )\). For \(x=1\) and \(y=1\), we get \(P(x,y)= Q(x)\) and \(P(y,x)= Q(y)\). This implies that the mappings \(P:X \times X \rightarrow X\) and \(Q:X \rightarrow X\) are Wcompatible mappings.
Many others eminent authors proved significant results which contributed in the arena of fixed point theory (see [16–21]).
VFuzzy metric space
After the exhaustive review of the previous literature, we introduce Vfuzzy metric spaces and discuss their properties given below.
Definition 2.1
Let X be a nonempty set. A triple \((X, V, \ast)\) is said to be a Vfuzzy metric space (denoted by VFspace), where ∗ is a continuous tnorm, and V is a fuzzy set on \(X^{n} \times(0, \infty )\) satisfying the following conditions for all \(t, s > 0\):

(VF1)
\(V(x, x, x,\ldots, x, y, t) > 0\) for all \(x, y \in X\) with \(x \neq y\);

(VF2)
\(V(x_{1}, x_{1}, x_{1}, \ldots, x_{1}, x_{2}, t) \geq V(x_{1}, x_{2}, x_{3}, \ldots, x_{n}, t)\) for all \(x_{1}, x_{2}, x_{3}, \ldots, x_{n} \in X\) with \(x_{2} \neq x_{3} \neq \cdots\neq x_{n}\);

(VF3)
\(V(x_{1}, x_{2}, x_{3}, \ldots, x_{n}, t) = 1\) if and only if \(x_{1} = x_{2} = x_{3} = \cdots= x_{n}\);

(VF4)
\(V(x_{1}, x_{2}, x_{3}, \ldots, x_{n}, t) = V(p(x_{1}, x_{2}, x_{3}, \ldots, x_{n}), t)\), where p is a permutation function;

(VF5)
\(V(x_{1}, x_{2}, x_{3},\ldots, x_{n1}, t+s) \geq V(x_{1}, x_{2}, x_{3}, \ldots,x_{n1}, l, t) \ast V(l,l,l, \ldots, l,x_{n}, s)\);

(VF6)
\(\lim_{t \rightarrow\infty}V(x_{1}, x_{2}, x_{3}, \ldots , x_{n}, t) = 1\);

(VF7)
\(V(x_{1}, x_{2}, x_{3}, \ldots, x_{n}, \cdot): (0, \infty) \rightarrow[0, 1]\) is continuous.
Example 2.1
Let \((X, A)\) be an Ametric space. Define the tnorm \(a \ast b = ab\) or \(a \ast b = \min \{a, b \}\). For all \(x_{1}, x_{2}, x_{3}, \ldots, x_{n} \in X\), \(t > 0\), denote
Then \((X, V, \ast)\) is a Vfuzzy metric space.
Lemma 2.1
Let \((X, V, \ast)\) be a Vfuzzy metric space. Then \(V(x_{1}, x_{2}, x_{3}, \ldots, x_{n}, t)\) is nondecreasing with respect to t.
Proof
Since \(t > 0\) and \(t + s > t\) for \(s > 0\), by letting \(l = x_{n}\) in condition (VF5) of a Vfuzzy metric space, we get
This implies that \(V(x_{1}, x_{2}, x_{3}, \ldots, x_{n1}, x_{n}, t+s) \geq V(x_{1}, x_{2}, x_{3}, \ldots, x_{n1}, x_{n}, t)\).
So, \(V(x_{1}, x_{2}, x_{3}, \ldots, x_{n1}, x_{n}, t)\) is nondecreasing with respect to t. □
Lemma 2.2
Let \((X, V, \ast)\) be a Vfuzzy metric space such that
with \(k \in(0, 1)\). Then \(x_{1} = x_{2} = x_{3} = \cdots= x_{n}\).
Proof
By assumption
for \(t > 0\). Since \(kt < t\), by Lemma 2.1 we have
From (1), (2), and the definition of a Vfuzzy metric space we get \(x_{1} = x_{2} = x_{3} = \cdots= x_{n}\). □
Definition 2.2
Let \((X, V, \ast)\) be a Vfuzzy metric space. A sequence \(\{ x_{r} \}\) is said to converge to a point \(x \in X\) if \(V(x_{r}, x_{r}, x_{r}, \ldots, x_{r}, x, t) \rightarrow1\) as \(r \rightarrow\infty\) for all \(t > 0\), that is, for each \(\epsilon> 0\), there exists \(n \in N\) such that for all \(r \geq N\), we have \(V(x_{r}, x_{r}, x_{r}, \ldots,x_{r}, x, t) > 1  \epsilon\), and we write \(\lim_{r \rightarrow\infty}x_{r} = x\).
Definition 2.3
Let \((X, V, \ast)\) be a Vfuzzy metric space. A sequence \(\{ x_{r} \}\) is said to be a Cauchy sequence if \(V(x_{r}, x_{r}, x_{r}, \ldots, x_{r}, x_{q}, t) \rightarrow1\) as \(r, q \rightarrow\infty\) for all \(t > 0\), that is, for each \(\epsilon> 0\), there exists \(n_{0} \in N\) such that for all \(r, q \geq n_{0}\), we have \(V(x_{r}, x_{r}, x_{r}, \ldots,x_{r}, x_{q}, t) > 1  \epsilon\).
Definition 2.4
The Vfuzzy metric space \((X, V, \ast)\) is said to be complete if every Cauchy sequence in X is convergent.
Definition 2.5
The mappings \(P: X \times X \rightarrow X\) and \(Q: X \rightarrow X\) are said to be compatible on Vfuzzy metric spaces if
and
whenever \(\{x_{r} \}\) and \(\{y_{r} \}\) are sequences in X such that \(\lim_{r \rightarrow\infty}Q(x_{r}) = \lim_{r \rightarrow\infty}P(x_{r}, y_{r}) = x\) and \(\lim_{r \rightarrow \infty}Q(y_{r}) = \lim_{r \rightarrow\infty}P(y_{r}, x_{r}) = y\) for all \(x, y \in X\) and \(t >0\).
Main results
In this section, we explicitly prove fixed point theorems for coupled maps on partially ordered Vfuzzy metric spaces.
Theorem 3.1
Let \((X, V, \ast)\) be a complete Vfuzzy metric space, and \((X,\preceq )\) be a partially ordered set. Let \(P: X \times X \rightarrow X\) and \(Q : X \times X \rightarrow X\) be two mappings such that

(T1)
\(P(X \times X) \subseteq Q(X)\);

(T2)
P has the mixed Qmonotone property;

(T3)
there exists \(k \in(0, 1)\) such that
$$\begin{aligned}& V\bigl(P(x, y), P(x, y),\ldots,P(x, y), P(u, v), kt\bigr) \\& \quad \geq V(Qx, Qx, \ldots, Qx, Qu, t) \\& \qquad {}\ast V\bigl(Qx, Qx,\ldots, Qx, P(x, y), t\bigr) \\& \qquad {} \ast V\bigl(Qu, Qu,\ldots, Qu, P(u,v), t\bigr) \end{aligned}$$for all \(x, y, u, v \in X\) and \(t > 0\) for which \(Q(x) \leq Q(u)\) and \(Q(y) \geq Q(v)\) or \(Q(x) \geq Q(u)\) and \(Q(y) \leq Q(v)\);

(T4)
Q is continuous, and P and Q are compatible.
Also suppose that

(a)
P is continuous or

(b)
X has the following properties:

(i)
if \(\{x_{r} \}\) is a nondecreasing sequence such that \(x_{r}\rightarrow x\), then \(x_{r} \leq x\) for all \(r \in N\);

(ii)
if \(\{y_{r} \}\) is a nonincreasing sequence \(y_{r}\rightarrow y\), then \(y_{r} \geq y\) for all \(r \in N\).

(i)

(a)
If there exist \(x_{0}, y_{0} \in X\) such that \(Q(x_{0}) \leq P(x_{0},y_{0})\) and \(Q(y_{0}) \geq P(y_{0},x_{0})\), then P and Q have a coupled coincidence point in X.
Proof
Let \((x_{0}, y_{0})\) be a given point in \(X \times X\) such that \(Q(x_{0}) \leq P(x_{0}, y_{0})\) and \(Q(y_{0}) \geq P(y_{0}, x_{0})\). Using (T1), choose \(x_{1}\), \(y_{1}\) such that
Construct two sequences \(\{x_{r} \}\) and \(\{y_{r} \} \) in X such that \(P(x_{r}, y_{r}) = Q(x_{r+1})\) and
Now we shall prove that
We use mathematical induction.
Step 1. Let \(r = 0\). Since \(Q(x_{0}) \leq P(x_{0}, y_{0})\) and \(Q(y_{0}) \geq P(y_{0}, x_{0})\), using condition (3), we have
So inequalities (5) hold for \(r = 0\).
Step 2. Now suppose that (5) hold for some fixed \(s \geq0\). So we get
Step 3. Since P has the mixed Qmonotone property, using (4), we have
and
Also, \(Q(x_{r+2}) = P(x_{r+1}, y_{r+1})\geq P(x_{r+1}, y_{r})\) and
From (T3) and (4) we get
Now, two cases arise.
Case 1. If \(V(Qx_{r1},Qx_{r1},\ldots,Qx_{r1}, Qx_{r},t) < V(Qx_{r},Qx_{r},\ldots, Qx_{r}, Qx_{r+1}, t)\), then
Then by simple induction we have that, for all \(t > 0\) and \(r = 1,2, \ldots,\infty\),
Thus, by condition (VF5) of the definition of a Vfuzzy metric space, for any positive integer p and real number \(t > 0\), we have
Therefore, taking \(r \rightarrow\infty\), by definition (VF6) we get
which implies that \(\{Qx_{n} \}\) is a Cauchy sequence in X.
Case 2. If \(V(Qx_{r1},Qx_{r1},\ldots,Qx_{r1}, Qx_{r},t) > V(Qx_{r},Qx_{r},\ldots, Qx_{r}, Qx_{r+1}, t)\), then
By Lemma 2.2 we get \(Qx_{r} = Qx_{r+1}\).
Thus, there exists a positive integer m such that \(r \geq m\) implies \(Qx_{r} = Qx_{m}\), ∀r, which shows that \(\{Qx_{n} \} \) is a convergent sequence and so a Cauchy sequence in X.
Taking \(x = y_{r}\), \(y = x_{r}\), \(u = y_{r1}\), \(v = x_{r1}\) in (T3), we get
So, from equation (4) we have
In the same way (discussed before), \(\{Qy_{n} \}\) is a Cauchy sequence in X.
Since X is a complete space, there exist \(x,y \in X\) such that
By considering condition (T4) and \(r \rightarrow\infty\) we have
and
as \(r \rightarrow\infty\).
By conditions (T4) and (a), since P and Q are continuous, from (10) we have
and
This implies that \(P(x, y) = Q(x)\) and \(P(y, x) = Q(y)\), and thus, we have proved that P and Q have a coupled coincidence point in X.
Now, suppose that conditions (T4) and (b) hold. Since Q is continuous and P, Q are compatible mappings, we have
and
By condition (VF5) of a Vfuzzy metric space, as \(r \rightarrow \infty\), we get
We get
Using condition (T3) and equations (11)(12), we get
By Lemma 2.2 we have \(P(x, y) = Q(x)\). Similarly, we get \(P(y, x) = Q(y)\). Hence, we proved that P and Q have a coupled coincidence point in X. □
Theorem 3.2
Assume that X is a totally ordered set in addition to the hypotheses of Theorem 3.1. Then P and Q have a unique common coupled fixed point.
Proof
Suppose that \((x, y)\) and \((l, m)\) are a coupled coincidence point of P and Q, that is,
and
Let us show that \(Q(x) = Q(l)\), \(Q(y) = Q(m)\).
If X is a totally ordered set, then for all \((x,y), (l,m) \in X \times X\), there exists \((\alpha, \beta) \in X \times X\) such that \((P(\alpha, \beta), P(\beta, \alpha))\) is comparable with \((P(x,y),P(y,x))\), \((P(l,m),P(m,l))\).
The sequences \(\{Q(\alpha_{r}) \}\), \(\{Q(\beta_{r}) \} \) and their limits are defined similarly as in Theorem 3.1, so that
By condition (T3) we have
We obtained that
Again by (T3),
Letting \(r \rightarrow\infty\), we have
Following the above steps with the help of condition (T3), we obtain
From (13), (14), (15), and the definition of a Vfuzzy metric space we have
This implies that
Now, we can easily prove that
So, we have \(P(x, y) = Q(x)\), \(P(y, x) = Q(y)\), and the compatibility of P and Q implies the wcompatibility of P and Q given by
This implies that \((Q(x),Q(y))\) is a coupled coincidence point.
Assuming that \(l = Q(x)\), \(m = Q(y)\), by (16)(17) we have
and
So, \((Q(x), Q(y))\) is a common coupled fixed point of P and Q.
We can easily prove the uniqueness of common coupled fixed point under the assumption that \((x^{\ast}, y^{\ast})\) is another common coupled fixed point of P and Q.
From (16)(19) we can show that
and
This implies that P and Q have a unique common coupled fixed point. □
Here, we furnish an example to demonstrate the validity of the hypothesis of the above results.
Example 3.1
Let \((X, \leq)\) be a partially ordered set with \(X = [0,1]\), \(a \ast b = \min \{a, b \}\). Let \(P: X \times X \rightarrow X\) and \(Q: X \rightarrow X\) be two mappings defined as
This implies that P satisfies the definition of the mixed Qmonotone property.
Let
where \(A(x_{1}, x_{2},\ldots, x_{n})\) is the Ametric space defined as
for all \(x_{1}, x_{2},\ldots,x_{n} \in X\), \(t > 0\).
Then \((X, V, \ast)\) is a complete Vfuzzy metric space.
We take \(k = \frac{1}{2}\) and consider the sequences \(\{x_{r} \}\), \(\{y_{r} \}\) in X defined by \(x_{r} = \frac{1}{2r}\), \(y_{r} = \frac{1}{3r}\).
Since
Also, \(P: X \times X \rightarrow X\) and \(Q: X \rightarrow X\) are compatible mappings in X. From Theorem 3.1 we have that \(Q(x) \leq Q(u)\) and \(Q(y) \geq Q(v)\). This implies \(x \leq u\), \(y \geq v\). If we consider \(x \geq y\), \(u \geq v\), then we have
If we consider \(x < y\), \(u \geq v\), then we have
If we consider \(x < y\), \(u < v\), then we get directly condition (T3) of Theorem 3.1.
Therefore, all hypotheses of Theorem 3.1 hold. So we conclude that \((w, w')\) is a common coupled fixed point of P and Q.
Theorem 3.3
Let \((X, V, \ast)\) be a complete Vfuzzy metric space, and \((X, \leq )\) be a partially ordered set. Let \(P: X \times X \rightarrow X\) be a mapping such that P has the mixed monotone property and there exists \(k \in(0,1)\) such that
for all \(x, y, u, v \in X\), \(t > 0\) such that \(x \leq u\) and \(y \geq v\).
Also suppose that

(a)
P is continuous or

(b)
X has the following properties:

(i)
if \(\{x_{n} \}\) is a nondecreasing sequence \(x_{r} \rightarrow x\), then \(x_{r} \leq x\) for all \(r \in N\),

(ii)
if \(\{y_{n} \}\) is a nondecreasing sequence \(y_{r} \rightarrow y\), then \(y_{r} \geq x\) for all \(r \in N\).

(i)
If there exist \(x_{0}, y_{0} \in X\) such that \(x_{0} \leq P(x_{0},y_{0})\) and \(y_{0} \geq P(y_{0},x_{0})\), then P has a coupled fixed point in X.
Proof
By assuming \(Q = I\) (the identity mapping) in Theorem 3.1 we get the result. □
Symmetric Vfuzzy metric space
Definition 4.1
A Vfuzzy metric space \((X, V,\ast)\) is said to be symmetric if
for all \(x, y \in X\), \(t > 0\).
Remark 4.1
If \((X, M, \ast)\) is a fuzzy metric space, then \((X, V, \ast)\) is a Vfuzzy metric space, where
Now we have
which implies
So, all conditions of a Vfuzzy metric space are satisfied.
Remark 4.2
If \((X,V, \ast)\) is a symmetric Vfuzzy metric space and \(V(x, x, \ldots, y, t) = M(x, y, t)\) (using Remark 4.1), then \((X, M, \ast )\) is a fuzzy metric space.
By this remark we get the following desired result.
Theorem 4.1
Let \((X, M, \ast)\) be a complete fuzzy metric space, and \((X, \preceq)\) be a partially ordered set. Let \(P: X \times X \rightarrow X\) and \(Q:X \rightarrow X\) be two mappings such that

(T1)
\(P(X \times X) \subseteq Q(X)\);

(T2)
P has the mixed Qmonotone property;

(T3)
there exists \(k \in(0, 1)\) such that
$$ M\bigl(P(x, y), P(u,v),kt\bigr) \geq M(Qx, Qu, t) \ast M\bigl(Qx, P(x, y),t \bigr)\ast M\bigl(Qu, P(u,v), t\bigr) $$for all \(x, y, u, v \in X\), \(t > 0\) such that \(Q(x) \leq Q(u)\) and \(Q(y) \geq Q(v)\) or \(Q(x) \geq Q(u)\) and \(Q(y) \leq Q(v)\);

(T4)
Q is continuous, and P and Q are compatible.
Also suppose that

(a)
P is continuous or

(b)
X has the following properties:

(i)
if \(\{x_{r} \}\) is a nondecreasing sequence \(x_{r} \rightarrow x\), then \(x_{r} \leq x\) for all \(r \in N\);

(ii)
if \(\{y_{r} \}\) is a nondecreasing sequence \(y_{r} \rightarrow y\), then \(y_{r} \leq x\) for all \(r \in N\).

(i)

(a)
If there exist \(x_{0}, y_{0} \in X\) such that \(Q(x_{0}) \leq P(x_{0}, y_{0})\) and \(Q(y_{0}) \geq P(y_{0}, x_{0})\), then P and Q have a coupled coincidence point in X.
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Gupta, V., Kanwar, A. VFuzzy metric space and related fixed point theorems. Fixed Point Theory Appl 2016, 51 (2016). https://doi.org/10.1186/s1366301605361
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MSC
 47H10
 54H25
Keywords
 partial ordered set
 mixed monotone mappings
 common coupled fixed point
 Gfuzzy metric space
 Vfuzzy metric space