V-Fuzzy metric space and related fixed point theorems
- Vishal Gupta^{1}Email author and
- Ashima Kanwar^{1}
https://doi.org/10.1186/s13663-016-0536-1
© Gupta and Kanwar 2016
Received: 10 December 2015
Accepted: 29 March 2016
Published: 12 April 2016
Abstract
The paper concerns our sustained efforts for introduction of V-fuzzy 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 V-fuzzy metric spaces. An example quoted in this paper also corroborates fully the main result. Also, here we introduce the concept of a symmetric V-fuzzy metric space. Under the influence of symmetry property of V-fuzzy metric spaces, its conversion materializes to the main result from V-fuzzy metric spaces to fuzzy metric spaces.
Keywords
MSC
1 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 G-metric space and suggested an important generalization of a metric space as follows.
Definition 1.1
[1]
- (G-1)
\(G(x,y,z) = 0\) if \(x=y=z\),
- (G-2)
\(0 \leq G(x,y,z)\) with \(x \neq y\),
- (G-3)
\(G(x,x,y) \leq G(x,y,z)\) with \(y \neq z\),
- (G-4)
\(G(x,y,z)=G(x,z,y)=G(y,x,z)=G(z,x,y)=G(y,z,x)=G(z,y,x)\),
- (G-5)
\(G(x,y,z) \leq G(x,a,a) + G(a,y,z)\).
The function G is called a generalized metric on X, and the pair \((X, G)\) is called a G-metric space.
Example 1.1
[1]
Recently, some authors studied some important fixed point theorems with application in G-metric spaces [2–4]. In 2012, Sedghi et al. [5] introduced a new generalized metric space called an S-metric space.
Definition 1.2
[5]
- (S-1)
\(S(x,y,z) \geq0\),
- (S-2)
\(S(x,y,z) = 0\) if and only if \(x=y=z=0\),
- (S-3)
\(S(x,y,z)\leq S(x,x,a) + S(y,y,a) + S(z,z,a)\) for any \(x,y,z,a \in X\).
Example 1.2
[5]
Let ℜ be the real line. Then \(S(x,y,z)=\vert x-y\vert +\vert y-z\vert \) for all \(x,y,z \in\Re \) is an S-metric on ℜ. This S-metric is called the usual S-metric on ℜ.
Abbas et al. [6] established the notion of A-metric spaces, a generalization of S-metric spaces.
Definition 1.3
[6]
- (A-1)
\(A(x_{1}, x_{2}, x_{3}, \ldots, x_{n}) \geq0\),
- (A-2)
\(A(x_{1}, x_{2}, x_{3}, \ldots, x_{n}) = 0\) if and only if \(x_{1}= x_{2}= x_{3}= \cdots= x_{n} = 0\),
- (A-3)
\(A(x_{1}, x_{2}, x_{3}, \ldots, x_{n}) \leq A(x_{1}, x_{1}, x_{1}, \ldots, (x_{1})_{n-1}, a) + A(x_{2}, x_{2}, x_{2}, \ldots, (x_{2})_{n-1}, a)+ \cdots+A(x_{n}, x_{n}, x_{n}, \ldots, (x_{n})_{n-1}, a)\).
Example 1.3
[6]
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]
- (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]
- (FM-1)
\(M(x, y , 0) = 0\),
- (FM-2)
\(M(x, y ,t) = 1\) if and only if \(x = y\),
- (FM-3)
\(M(x, y ,t) = M(y, x, t)\),
- (FM-4)
\(M(x, y ,t) \ast M(y, z, s) \leq M(x, z ,t+s)\),
- (FM-5)
\(M(x, y, \cdot) : [0 ,\infty) \rightarrow[0, 1]\) 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]
Lemma 1.1
[11]
In the process of generalization of fuzzy metric spaces, Sun and Yang [12] coined the notion of G-fuzzy metric spaces and established common fixed-point theorems for four mappings.
Definition 1.6
[12]
- (GF-1)
\(G(x,x,y,t) > 0\) with \(x \neq y\),
- (GF-2)
\(G(x,x,y,t) \geq G(x,y,z,t)\) with \(y \neq z\),
- (GF-3)
\(G(x,y,z,t) = 1 \) if and only if \(x=y=z\),
- (GF-4)
\(G(x,y,z,t)= G(x,z,y,t) =G(y,x,z,t)= G(z,x,y,t)= G(y,z,x,t)=G(z,y,x,t)\),
- (GF-5)
\(G(x,y,z,t+s) \geq G(x,a,a,t) \ast G(a,y,z,s)\),
- (GF-6)
\(G(x,y,z,\cdot): [0 ,\infty) \rightarrow[0, 1]\) is left continuous.
Example 1.5
[12]
Lemma 1.2
[12]
Let \((X,G, \ast)\) be a GF-space. 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
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
Definition 1.11
[13]
Example 1.8
Definition 1.12
[13]
Example 1.9
Definition 1.13
[15]
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 W-compatible mappings.
Many others eminent authors proved significant results which contributed in the arena of fixed point theory (see [16–21]).
2 V-Fuzzy metric space
After the exhaustive review of the previous literature, we introduce V-fuzzy metric spaces and discuss their properties given below.
Definition 2.1
- (VF-1)
\(V(x, x, x,\ldots, x, y, t) > 0\) for all \(x, y \in X\) with \(x \neq y\);
- (VF-2)
\(V(x_{1}, x_{1}, x_{1}, \ldots, x_{1}, x_{2}, t) \geq V(x_{1}, x_{2}, x_{3}, \ldots, x_{n}, t)\) for all \(x_{1}, x_{2}, x_{3}, \ldots, x_{n} \in X\) with \(x_{2} \neq x_{3} \neq \cdots\neq x_{n}\);
- (VF-3)
\(V(x_{1}, x_{2}, x_{3}, \ldots, x_{n}, t) = 1\) if and only if \(x_{1} = x_{2} = x_{3} = \cdots= x_{n}\);
- (VF-4)
\(V(x_{1}, x_{2}, x_{3}, \ldots, x_{n}, t) = V(p(x_{1}, x_{2}, x_{3}, \ldots, x_{n}), t)\), where p is a permutation function;
- (VF-5)
\(V(x_{1}, x_{2}, x_{3},\ldots, x_{n-1}, t+s) \geq V(x_{1}, x_{2}, x_{3}, \ldots,x_{n-1}, l, t) \ast V(l,l,l, \ldots, l,x_{n}, s)\);
- (VF-6)
\(\lim_{t \rightarrow\infty}V(x_{1}, x_{2}, x_{3}, \ldots , x_{n}, t) = 1\);
- (VF-7)
\(V(x_{1}, x_{2}, x_{3}, \ldots, x_{n}, \cdot): (0, \infty) \rightarrow[0, 1]\) is continuous.
Example 2.1
Lemma 2.1
Let \((X, V, \ast)\) be a V-fuzzy metric space. Then \(V(x_{1}, x_{2}, x_{3}, \ldots, x_{n}, t)\) is nondecreasing with respect to t.
Proof
So, \(V(x_{1}, x_{2}, x_{3}, \ldots, x_{n-1}, x_{n}, t)\) is nondecreasing with respect to t. □
Lemma 2.2
Proof
Definition 2.2
Let \((X, V, \ast)\) be a V-fuzzy 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 V-fuzzy 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 V-fuzzy metric space \((X, V, \ast)\) is said to be complete if every Cauchy sequence in X is convergent.
Definition 2.5
3 Main results
In this section, we explicitly prove fixed point theorems for coupled maps on partially ordered V-fuzzy metric spaces.
Theorem 3.1
- (T1)
\(P(X \times X) \subseteq Q(X)\);
- (T2)
P has the mixed Q-monotone property;
- (T3)there exists \(k \in(0, 1)\) such thatfor 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 \(Q(y) \leq Q(v)\);$$\begin{aligned}& V\bigl(P(x, y), P(x, y),\ldots,P(x, y), P(u, v), kt\bigr) \\& \quad \geq V(Qx, Qx, \ldots, Qx, Qu, t) \\& \qquad {}\ast V\bigl(Qx, Qx,\ldots, Qx, P(x, y), t\bigr) \\& \qquad {} \ast V\bigl(Qu, Qu,\ldots, Qu, P(u,v), t\bigr) \end{aligned}$$
- (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)
Proof
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.
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
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 \((P(x,y),P(y,x))\), \((P(l,m),P(m,l))\).
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.
Here, we furnish an example to demonstrate the validity of the hypothesis of the above results.
Example 3.1
Then \((X, V, \ast)\) is a complete V-fuzzy metric space.
We take \(k = \frac{1}{2}\) and consider the sequences \(\{x_{r} \}\), \(\{y_{r} \}\) in X defined by \(x_{r} = \frac{1}{2r}\), \(y_{r} = \frac{1}{3r}\).
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
- (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)
Proof
By assuming \(Q = I\) (the identity mapping) in Theorem 3.1 we get the result. □
4 Symmetric V-fuzzy metric space
Definition 4.1
Remark 4.1
Remark 4.2
If \((X,V, \ast)\) is a symmetric V-fuzzy 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
- (T1)
\(P(X \times X) \subseteq Q(X)\);
- (T2)
P has the mixed Q-monotone property;
- (T3)there exists \(k \in(0, 1)\) such thatfor 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 \(Q(y) \leq Q(v)\);$$ M\bigl(P(x, y), P(u,v),kt\bigr) \geq M(Qx, Qu, t) \ast M\bigl(Qx, P(x, y),t \bigr)\ast M\bigl(Qu, P(u,v), t\bigr) $$
- (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)
Declarations
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Authors’ Affiliations
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