On Completeness and Fixed Point Theorems in Fuzzy Metric Spaces

Abstract

This paper is devoted to showing the relevance of the notion of completeness used to establish a fixed point theorem in fuzzy metric spaces introduced by Kramosil and Michalek. Specifically, we show that demanding a stronger notion of completeness, called p-completeness, it is possible to relax some extra conditions on the space to obtain a fixed point theorem in this framework. To this end, we focus on a fixed point result, proved by Mihet for complete non-Archimedean fuzzy metric spaces (Theorem 1). So, we define a weaker concept than the non-Archimedean fuzzy metric, called t-strong, and we establish an alternative version of Miheţ’s theorem for p-complete t-strong fuzzy metrics (Theorem 2). In addition, an example of t-strong fuzzy metric spaces that are not non-Archimedean is provided.

1. Introduction

In 1975, Kramosil and Michalek introduced the concept of fuzzy metric space in [1], which is commonly used in its reformulation provided by Grabiec in [2]. Later on, George and Veeramani slightly modified this concept in [3] with the aim of defining the notion of fuzzy metrics, which fitted more faithfully with the axioms of a classical metric.

It is worth mentioning that this last concept of fuzzy metric space can be considered, in some sense, a particular case of fuzzy metric space in the context of Kramosil and Michalek. Both notions of fuzzy metric spaces have been studied widely from the theoretical point of view and are still being studied today. Indeed, we can find recent references devoted to the study of different topics in this kind of fuzzy metrics such as fixed point theory, convergence and Cauchyness, the Wijsman topology or the asymptotic dimension (see for instance [4,5,6,7,8,9,10,11]).

Many results proved in fuzzy metric spaces, as defined by George and Veeramani’s, can be extended to the context of Kramosil and Michalek. Even more, from the topological point of view, both aforesaid notions of fuzzy metrics and classical metrics are the same, since it was proved in the literature that fuzzy metrics are metrizable.

However, as for purely metric topics such as fixed point theory, classical and fuzzy metrics show some differences. For instance, the first approach to establish a version of the celebrated fixed point theorem of Banach for fuzzy metrics, as defined by Kramosil and Michalek, was made by Grabiec in [2]. However, the notion of completeness used to prove such a theorem is too strong as has been showed in the literature (see, for instance, [3,12,13]) in which a strong notion of completeness was used to ensure the existence of a fixed point. Later on, several authors adapted some classical fixed point results to the fuzzy context and, in most of them, extra conditions in addition to completeness are required (see, for instance [14,15,16,17,18,19]).

In this direction, Miheţ proved a fixed point result in [20] using the notion of completeness introduced by George an Veeramani in [3], but limited it to non-Archimedean fuzzy metric spaces (see Definition 2).

Moreover, a significant difference between classical and fuzzy metrics arises due to the inclusion of a t parameter in their definition. This fact has allowed for the introduction in the literature of some notions in fuzzy metrics that lack sense in the classical context. In this direction, some new notions of convergence, Cauchyness and completeness have been defined in the literature (see [21] and references therein).

The aim of this paper is to study in detail fuzzy metric spaces in both aforementioned senses, which show differences compared with classical metrics. Concretely, we focus on establishing a discussion of the conditions required in order to prove a fixed point theorem in fuzzy metric spaces. In this direction, we explore the applicability of the fixed point theory of a new notion of completeness of fuzzy metric spaces introduced in the literature. In this sense, we have focused on (Theorem 3.1, [20]), a result established by Miheţ. As was mentioned above, such a theorem was proved for non-Archimedean fuzzy metrics by using the completion introduced by George and Veeramani. In this paper, we propose an alternative formulation of this theorem by relaxing the non-Archimedean condition and strengthening the notion of completeness.

It is worth mentioning that the purpose of the paper is not to obtain a generalization of the result established by Miheţ but rather to discuss the requirements demanded to prove a fixed point theorem in fuzzy metric spaces. So, we show that an alternative notion of completeness can be useful to obtain fixed point results in fuzzy metric spaces without demanding overly stringent conditions.

Therefore, the main contribution of this paper is to lay on the table the study of different notions of completeness of fuzzy metric spaces in order to obtain fixed point results. In this direction, we first introduce the notion of t-strong fuzzy metric spaces and we show an example of fuzzy metric that is t-strong, but not non-Archimedean. Then, we prove a fixed point theorem for contractive mappings in the sense of Miheţ for p-complete t-strong fuzzy metric spaces, a notion of completeness formerly introduced in the literature. Finally, we illustrate the use of such a theorem by means of an example of fuzzy metric spaces that are not non-Archimedean.

The remainder of the paper is structured as follows. The next section is devoted to compiling the notions and results given in the literature necessary to carry out our study. Then, in Section 3 the main results of the paper are presented.

2. Preliminaries

We will denote throughout the paper by ${\mathbb{R}}^{+}$ the set of non-negative real numbers, i.e., ${\mathbb{R}}^{+}$ is the interval $[0,\infty [$.

Definition 1 

([1,2]). A fuzzy metric space is an ordered triple $(\mathcal{X},\mathcal{M},\ast )$ such that $\mathcal{X}$ is a (non-empty) set, ∗ is a continuous t-norm and $\mathcal{M}$ is a fuzzy set on $\mathcal{X}\times \mathcal{X}\times {\mathbb{R}}^{+}$ satisfying the following conditions, for all $x,y,z\in \mathcal{X}$ and $t,s>0$

• (KM1) $\mathcal{M}(x,y,0)=0;$
• (KM2) $\mathcal{M}(x,y,t)=1$, for all $t>0$, if and only if $x=y;$
• (KM3) $\mathcal{M}(x,y,t)=\mathcal{M}(y,x,t);$
• (KM4) $\mathcal{M}(x,y,t)\ast \mathcal{M}(y,z,s)\le \mathcal{M}(x,z,t+s);$
• (KM5) The function ${\mathcal{M}}_{xy}:{\mathbb{R}}^{+}\to [0,1]$ is left-continuous, where ${\mathcal{M}}_{xy}\left(t\right)=\mathcal{M}(x,y,t)$ for each $t\in {\mathbb{R}}^{+}$.

A concept of fuzzy metrics, similar to the previous one, was given by George and Veeramani in [3]. In fact, extending the concept of George and Veeramani, by assuming $\mathcal{M}(x,y,0)=0$ for all $x,y\in \mathcal{X}$, provides a new stronger notion of fuzzy metrics than the one presented in Definition 1. Many concepts and results can be given indistinguishably for both notions of fuzzy metrics, as is the case throughout the remainder of this section.

If $(\mathcal{X},\mathcal{M},\ast )$ is a fuzzy metric space, we say that $(\mathcal{M},\ast )$, or simply $\mathcal{M}$, is a fuzzy metric on $\mathcal{X}$. Also, we say that $(\mathcal{X},\mathcal{M})$, or simply $\mathcal{X}$, is a fuzzy metric space.

Let $(\mathcal{X},d)$ be a metric space. Denote by · the product t-norm and let ${\mathcal{M}}_d$ be the fuzzy set defined on $\mathcal{X}\times \mathcal{X}\times {\mathbb{R}}^{+}$ by

$${\mathcal{M}}_d(x,y,0)=0\mathrm{and}{\mathcal{M}}_d(x,y,t)=\frac{t}{t+d(x,y)},\mathrm{for}\mathrm{all}t>0.$$

(1)

Then, $({\mathcal{M}}_d,·)$ is a fuzzy metric on $\mathcal{X}$ called standard fuzzy metric induced by d [3].

George and Veeramani proved in [3] that every fuzzy metric $\mathcal{M}$ on $\mathcal{X}$ generates a topology ${\tau }_{\mathcal{M}}$ on $\mathcal{X}$, which has as a base the family of open sets of the form $\{{\mathcal{B}}_{\mathcal{M}}(x,r,t):x\in \mathcal{X},r\in ]0,1[,t>0\},$ where ${\mathcal{B}}_{\mathcal{M}}(x,r,t)=\{y\in \mathcal{X}:\mathcal{M}(x,y,t)>1-r\}$ for all $x\in \mathcal{X}$, $r\in ]0,1[$ and $t>0$. In the case of the standard fuzzy metric ${\mathcal{M}}_d$, it is well known that the topology $\tau \left(d\right)$ on $\mathcal{X}$ deduced from d coincides with the topology ${\tau }_{{\mathcal{M}}_d}$ deduced from ${\mathcal{M}}_d$ (see [3]). From now on, we will suppose that $\mathcal{X}$ is endowed with the topology ${\tau }_{\mathcal{M}}$.

Proposition 1 

([3]). A sequence ${\left\{x_n\right\}}_n$ in a fuzzy metric space $(\mathcal{X},\mathcal{M},\ast )$ converges to $x_0$ if and only if $\underset{n}{lim}\mathcal{M}(x_0,x_n,t)=1$, for all $t>0$.

Definition 2 

([20]). A fuzzy metric space $(\mathcal{X},\mathcal{M},\ast )$ is said to be non-Archimedean (strong) if for all $x,y,z\in \mathcal{X}$ and all, $t>0$ satisfies

$$\mathcal{M}(x,z,t)\ge \mathcal{M}(x,y,t)\ast \mathcal{M}(y,z,t).$$

Definition 3 

([3]). A sequence ${\left\{x_n\right\}}_n$ in a fuzzy metric space $(\mathcal{X},\mathcal{M},\ast )$ is called Cauchy if for each $\epsilon \in ]0,1[$ and each $t>0$ there exists $n_0\in \mathbb{N}$ such that $\mathcal{M}(x_n,x_m,t)>1-\epsilon$ for all $n,m\ge n_0$ or equivalently $\underset{m,n}{lim}\mathcal{M}(x_n,x_m,t)=1$ for all $t>0$.

A fuzzy metric space $(\mathcal{X},\mathcal{M},\ast )$ is called complete if every Cauchy sequence in $\mathcal{X}$ is convergent with respect to ${\tau }_{\mathcal{M}}$.

Definition 4 

([21]). A sequence ${\left\{x_n\right\}}_n$ in a fuzzy metric space $(\mathcal{X},\mathcal{M},\ast )$ is called p-Cauchy if there exists $t_0>0$ such that for each $ϵ\in ]0,1[$, there exists $n_0\in \mathbb{N}$ such that $\mathcal{M}(x_n,x_m,t_0)>1-ϵ$ for all $n,m\ge n_0$, or equivalently, $\underset{n,m}{lim}\mathcal{M}(x_n,x_m,t_0)=1$. In such a case, we will say that ${\left\{x_n\right\}}_n$ is p-Cauchy for $t_0$.

Obviously, Cauchy sequences are p-Cauchy; however, the converse is not true, as has been deomnstrated in the literature.

Definition 5 

([21]). A fuzzy metric space $(\mathcal{X},\mathcal{M},\ast )$ is called p-complete if every p-Cauchy sequence converges (in ${\tau }_M$).

Definition 6 

([20]). Let Ψ be the class of all mappings $\psi :[0,1]\to [0,1]$ such that ψ is continuous, non-decreasing and $\psi \left(a\right)>a$ for all $a\in ]0,1[$. Let $(\mathcal{X},\mathcal{M},\ast )$ be a fuzzy metric space and let $\psi \in \Psi$. A mapping $T:\mathcal{X}\to \mathcal{X}$ is said to be fuzzy ψ-contractive if

$$\mathcal{M}\left(T\right(x),T(y),t)\ge \psi \left(\mathcal{M}\right(x,y,t\left)\right)forallx,y\in \mathcal{X}andt>0.$$

(2)

Lemma 1 

([22]). If $\psi \in \Psi$, then $\underset{n}{lim}{\psi }^n\left(a\right)=1$ for all $a\in ]0,1]$.

Following the terminology introduced in this section, D. Miheţ established in [20] the following fixed point theorem for complete fuzzy metric spaces for the class of fuzzy $\psi$-contractive mappings.

Theorem 1. 

Let $(\mathcal{X},\mathcal{M},\ast )$ be a complete and strong fuzzy metric space and let $T:\mathcal{X}\to \mathcal{X}$ be a fuzzy ψ-contractive mapping. If there exists $x_0\in \mathcal{X}$ such that $\mathcal{M}(x_0,T\left(x_0\right),t)>0$, for all $t>0$, the T has a fixed point.

3. The Results

We begin this section by introducing a notion that is weaker than a strong fuzzy metric.

Definition 7. 

A fuzzy metric space $(\mathcal{X},\mathcal{M},\ast )$ is said to be t-strong if there exists $t_0>0$ such that

$$\mathcal{M}(x,z,t_0)\ge \mathcal{M}(x,y,t_0)\ast \mathcal{M}(y,z,t_0),forallx,y,z\in \mathcal{X}.$$

In such a case, we will say that $\mathcal{M}$ is t-strong for $t_0$, or simply, t-strong.

Obviously, strong fuzzy metrics are t-strong. However, we can find t-strong fuzzy metrics that are not strong. Below, we recall an example of a non-strong fuzzy metric introduced in [23], which turns out to be t-strong.

Example 1. 

Let d be the usual metric on $\mathbb{R}$ restricted to $]0,1]$ and consider the standard fuzzy metric ${\mathcal{M}}_d$ induced by d. Define the function $\mathcal{M}$ on $]0,1]\times ]0,1]\times {\mathbb{R}}^{+}$ by $\mathcal{M}(x,y,0)=0$ and

$$\mathcal{M}(x,y,t)=\left\{\begin{array}{cc}{\mathcal{M}}_d(x,y,t),\hfill & 0<t\le d(x,y)\hfill \\ {\mathcal{M}}_d(x,y,2t)·\frac{t-d(x,y)}{1-d(x,y)}+{\mathcal{M}}_d(x,y,t)·\frac{1-t}{1-d(x,y)},\hfill & d(x,y)<t\le 1\hfill \\ {\mathcal{M}}_d(x,y,2t),\hfill & t>1\hfill \end{array}\right..$$

Following the same arguments to those used in the proof of (Proposition 9 [23]), we conclude that $\left(\right]0,1],\mathcal{M},·)$ is a fuzzy metric space. Moreover, by (Remark 14 [23], we have that $\mathcal{M}$ is not strong. We will see that $\mathcal{M}$ is a t-strong fuzzy metric.

Fix $t_0=1$, so for each $u,v\in ]0,1]$, we get $\mathcal{M}(u,v,1)={\mathcal{M}}_d(u,v,2)=\frac{2}{2+d(u,v)}$. Therefore, given $x,y,z\in ]0,1]$ arbitrary, we have

$$\begin{gathered} \mathcal{M}(x,z,1)=\frac{2}{2+d(x,z)}\ge \frac{2}{2+d(x,y)+d(y,z)}\ge \\ \ge \frac{2}{2+d(x,y)}·\frac{2}{2+d(y,z)}=\mathcal{M}(x,y,1)·\mathcal{M}(y,z,1). \end{gathered}$$

Hence, $\mathcal{M}$ is a t-strong fuzzy metric on $]0,1]$ for $t_0=1$.

Now, we are able to state our main theorem in which the non-Archimedean condition is relaxed compared with Theorem 2. In compensation, a stronger notion of completeness is required to obtain the fixed point result.

Theorem 2. 

Let $(\mathcal{X},\mathcal{M},\ast )$ be a p-complete and t-strong fuzzy metric space for some $t_0>0$ and let $T:\mathcal{X}\to \mathcal{X}$ be a fuzzy ψ-contractive mapping. If there exists $x_0\in \mathcal{X}$ such that $\mathcal{M}(x_0,T\left(x_0\right),t_0)>0$, then T has a unique fixed point $x^{\ast }\in \mathcal{X}$ in the set $D_{x^{\ast }}=\{y\in \mathcal{X}:\mathcal{M}(x^{\ast },y,t)>0forallt>0\}$.

Proof. 

Let $x_0\in \mathcal{X}$ such that $\mathcal{M}(x_0,T\left(x_0\right),t_0)>0$. Define the sequence ${\left\{x_n\right\}}_n$ by $x_{n+1}=T\left(x_n\right)$ for all $n\in \mathbb{N}$, where $x_1=T\left(x_0\right)$. Taking into account that $\mathcal{M}(x_0,x_1,t_0)=\mathcal{M}(x_0,T\left(x_0\right),t_0)>0$, we conclude, by Lemma 1, that $\underset{n}{lim}{\psi }^n\left(\mathcal{M}(x_0,x_1,t_0)\right)=1$. On the other hand, by definition of ${\left\{x_n\right\}}_n$ and applying n-times the contractive condition we get $\mathcal{M}(x_n,x_{n+1},t_0)\ge {\psi }^n\left(\mathcal{M}(x_0,x_1,t_0)\right)$ for all $n\in \mathbb{N}$. Therefore, $\underset{n}{lim}\mathcal{M}(x_n,x_{n+1},t_0)=1$.

Now, we will see by contradiction that ${\left\{x_n\right\}}_n$ is a p-Cauchy sequence for $t_0$. So assume that ${\left\{x_n\right\}}_n$ is not p-Cauchy for $t_0$. By definition of the p-Cauchy sequence, there exists ${\epsilon }_0\in ]0,1[$ such that for each $k\in \mathbb{N}$, we can find $m\left(k\right)>n\left(k\right)\ge k$, satisfying $\mathcal{M}(x_{m\left(k\right)},x_{n\left(k\right)},t_0)\le 1-{\epsilon }_0$. Observe that we are able to pick the aforementioned $m\left(k\right)>n\left(k\right)\ge k$ fulfilling, in addition, the condition $\mathcal{M}(x_{m\left(k\right)-1},x_{n\left(k\right)},t_0)>1-{\epsilon }_0$. Then, taking into account that $\mathcal{M}$ is t-strong for $t_0$, we get, for all $k\in \mathbb{N}$, the following inequality

$$\begin{gathered} 1-{\epsilon }_0\ge \mathcal{M}(x_{m\left(k\right)},x_{n\left(k\right)},t_0)\ge \mathcal{M}(x_{m\left(k\right)},x_{m\left(k\right)-1},t_0)\ast \mathcal{M}(x_{m\left(k\right)-1},x_{n\left(k\right)},t_0)\ge \\ \mathcal{M}(x_{m\left(k\right)},x_{m\left(k\right)-1},t_0)\ast (1-{\epsilon }_0). \end{gathered}$$

Therefore, since $\underset{k}{lim}\mathcal{M}(x_{m\left(k\right)},x_{m\left(k\right)-1},t_0)=1$, taking limits in the preceding inequality, by continuity of ∗, we obtain $1-{\epsilon }_0\ge \mathcal{M}(x_{m\left(k\right)},x_{n\left(k\right)},t_0)\ge 1-{\epsilon }_0$ and so $\underset{k}{lim}\mathcal{M}(x_{m\left(k\right)},x_{n\left(k\right)},t_0)=1-{\epsilon }_0$. Nevertheless, keeping in mind again that $\mathcal{M}$ is t-strong for $t_0$ and using the contractive condition, we have the next inequality

$$\begin{gathered} \mathcal{M}(x_{m\left(k\right)},x_{n\left(k\right)},t_0)\ge \\ \ge \mathcal{M}(x_{m\left(k\right)},x_{m\left(k\right)+1},t_0)\ast \mathcal{M}(x_{m\left(k\right)+1},x_{n\left(k\right)+1},t_0)\ast \mathcal{M}(x_{n\left(k\right)+1},x_{n\left(k\right)},t_0)\ge \\ \ge \mathcal{M}(x_{m\left(k\right)},x_{m\left(k\right)+1},t_0)\ast \psi \left(\mathcal{M}(x_{m\left(k\right)},x_{n\left(k\right)},t_0)\right)\ast \mathcal{M}(x_{n\left(k\right)+1},x_{n\left(k\right)},t_0). \end{gathered}$$

Taking limits in the above inequality, we conclude, by continuity of $\psi$ and ∗ and since $\underset{k}{lim}\mathcal{M}(x_{m\left(k\right)},x_{m\left(k\right)+1},t_0)=\underset{k}{lim}\mathcal{M}(x_{n\left(k\right)+1},x_{n\left(k\right)},t_0)=1$, that

$$1-{\epsilon }_0=\underset{k}{lim}\mathcal{M}(x_{m\left(k\right)},x_{n\left(k\right)},t_0)\ge \underset{k}{lim}\psi \left(\mathcal{M}(x_{m\left(k\right)},x_{n\left(k\right)},t_0)\right)=\psi (1-{\epsilon }_0).$$

Due to $\psi \left(a\right)>a$ for all $a\in ]0,1[$, we have concluded that $1-{\epsilon }_0\ge \psi (1-{\epsilon }_0)>1-{\epsilon }_0$, which is a contradiction. Thus, ${\left\{x_n\right\}}_n$ is a p-Cauchy sequence for $t_0$ and, since $(\mathcal{X},\mathcal{M},\ast )$ is p-complete, we have that ${\left\{x_n\right\}}_n$ converges to some $x^{\ast }\in \mathcal{X}$.

Now, we are focused on proving that $x^{\ast }$ is a fixed point of T. Considering an arbitrary $t>0$, then, by axiom (KM4) and the contractive condition, we have that

$$\mathcal{M}(x^{\ast },T\left(x^{\ast }\right),t)\ge \mathcal{M}(x^{\ast },x_n,t/2)\ast \mathcal{M}(x_n,T\left(x^{\ast }\right),t/2)\ge \mathcal{M}(x^{\ast },x_n,t/2)\ast \psi \left(\mathcal{M}(x_{n-1},x^{\ast },t/2)\right),$$

for all $n\in \mathbb{N}$. Taking limits in the above inequality, by continuity of $\psi$ and ∗, and due to ${\left\{x_n\right\}}_n$ converges to $x^{\ast }$, we get $\mathcal{M}(x^{\ast },T\left(x^{\ast }\right),t)=1$. On account of t is arbitrary, we conclude that $T\left(x^{\ast }\right)=x^{\ast }$.

Finally, we will show by contradiction that $x^{\ast }$ is the unique fixed point of T in the set $D_{x^{\ast }}=\{y\in \mathcal{X}:\mathcal{M}(x^{\ast },y,t)>0\mathrm{for}\mathrm{all}t>0\}$. Suppose that $y^{\ast }\in D_{x^{\ast }}$ is a fixed point of T. Then, for all $t>0$, we have, by the contractive condition, the next inequality

$$\mathcal{M}(x^{\ast },y^{\ast },t)=\mathcal{M}(T\left(x^{\ast }\right),T\left(y^{\ast }\right),t)\ge \psi \left(\mathcal{M}(x^{\ast },y^{\ast },t)\right).$$

Since $\mathcal{M}(x^{\ast },y^{\ast },t)>0$ for all $t>0$ and due to $\psi \left(a\right)>a$ for all $a\in ]0,1[$, we conclude that $\mathcal{M}(x^{\ast },y^{\ast },t)=1$ for all $t>0$ and so $x^{\ast }=y^{\ast }$.   □

Below, we provide an example of a p-complete t-strong fuzzy metric that is not strong. Such an example illustrates the applicability of the preceding result compared with previous fixed point results concerning fuzzy $\psi$-contractive mappings. The interest lies not so much in finding a fixed point of the mapping considered as in the conditions satisfied by the considered fuzzy metric.

Example 2. 

Let d be the usual metric on $\mathbb{R}$. Define the fuzzy set $\mathcal{M}$ on $\mathbb{R}\times \mathbb{R}\times {\mathbb{R}}^{+}$ as follows:

$$\mathcal{M}(x,y,t)=\left\{\begin{array}{cc}0,\hfill & 0\le t\le d(x,y)\hfill \\ {\mathcal{M}}_d(x,y,t),\hfill & d(x,y)<t\le 1\hfill \\ {\mathcal{M}}_d(x,y,2t),\hfill & t>1\hfill \end{array}\right..$$

We will see that $(\mathbb{R},\mathcal{M},·)$ is a fuzzy metric space.

By definition of $\mathcal{M}$, axioms (KM1), (KM3) and (KM5) are satisfied. So, our focus is on demonstrating that the remaining axioms are also satisfied.

• (KM2) Let $x,y\in \mathbb{R}$ and assume that $\mathcal{M}(x,y,t)=1$, for all $t>0$. Then, $d(x,y)<t$ for all $t>0$ and so $d(x,y)=0$. Thus, we conclude that $x=y$. Obviously, if $x=y$, we have $\mathcal{M}(x,y,t)=1$ for all $t>0$.
• (KM4) Let $x,y,z\in \mathbb{R}$ and $t,s>0$. Note that if $t+s>1$, then $\mathcal{M}(x,z,t+s)={\mathcal{M}}_d(x,z,2(t+s))\ge {\mathcal{M}}_d(x,y,2t)·{\mathcal{M}}_d(y,z,2s)\ge \mathcal{M}(x,y,t)·\mathcal{M}(y,z,s)$. So, we will show the triangle inequality for the case $0\le t+s\le 1$. For this case, we distinguish two possibilities:

  1. 

  Suppose that $0\le t+s<d(x,z)$. Then, $0\le t<d(x,y)$ or $0\le s<d(y,z)$ or, on the contrary, $d(x,y)+d(y,z)\le t+s<d(x,z)$ which is not possible. So, $\mathcal{M}(x,z,t+s)=0=\mathcal{M}(x,y,t)·\mathcal{M}(y,z,s)$.

  2. 

  Now, assume $d(x,z)\le t+s<1$. Then, $0\le t,s<1$ and so $\mathcal{M}(x,y,t)\le {\mathcal{M}}_d(x,y,t)$ and $\mathcal{M}(y,z,s)\le {\mathcal{M}}_d(y,z,s)$. Therefore,

  $$\mathcal{M}(x,z,t+s)={\mathcal{M}}_d(x,z,t+s)\ge {\mathcal{M}}_d(x,y,t)·{\mathcal{M}}_d(y,z,s)\ge \mathcal{M}(x,y,t)·\mathcal{M}(y,z,s).$$

  Hence, (KM4) is fulfilled for all possible cases.

Following similar arguments to those used in Example 1, we conclude that $(\mathcal{X},\mathcal{M},\ast )$ is t-strong for $t_0=2$. Moreover, $(\mathcal{X},\mathcal{M},\ast )$ is p-complete. Indeed, observe that for each $t>1$, the family of p-Cauchy sequences coincides with the family of Cauchy sequences in $\mathbb{R}$ for d. Therefore, each p-Cauchy sequence in $(\mathcal{X},\mathcal{M},\ast )$ is a Cauchy sequence of $\mathbb{R}$ for d, since p-Cauchy sequences for some $0<t\le 1$ are also p-Cauchy for each $t>1$. Taking into account that $\mathbb{R}$ is complete for d, we have that $(\mathcal{X},\mathcal{M},\ast )$ is p-complete (indeed, it is complete).

Now, consider the self-mapping $T:\mathbb{R}\to \mathbb{R}$ given by $T\left(x\right)=\frac{x}{2}$ for all $x\in \mathbb{R}$, which, obviously has a unique fixed point $x^{\ast }=0$. We will see that T is a fuzzy ψ-contractive mapping for $\psi :[0,1]\to [0,1]$ given by $\psi \left(a\right)=\frac{a}{a+\frac{1}{2}(1-a)}$ for $a\in [0,1]$.

First, observe that $d(T\left(x\right),T\left(y\right))=\frac{1}{2}d(x,y)$ for all $x,y\in \mathbb{R}$. So, given $x,y\in \mathbb{R}$ and $t>0$, we distinguish three cases:

1. 

Suppose that $0<t\le d\left(T\right(x),T(y\left)\right)$. Then, $0<t\le d(x,y)$ and so

$$\mathcal{M}\left(T\right(x),T(y),t)=0=\psi \left(0\right)=\psi \left(\mathcal{M}\right(x,y,t\left)\right).$$

2. 

Assume $d\left(T\right(x),T(y\left)\right)<t\le 1$. Then,

$$\mathcal{M}(T\left(x\right),T\left(y\right),t)=\frac{t}{t+\frac{1}{2}d(x,y)}=\psi \left(\frac{t}{t+d(x,y)}\right)\ge \psi \left(\mathcal{M}(x,y,t)\right).$$

3. 

Now, consider $t>1$. Then,

$$\mathcal{M}(T\left(x\right),T\left(y\right),t)=\frac{2t}{2t+\frac{1}{2}d(x,y)}=\psi \left(\frac{2t}{2t+d(x,y)}\right)=\psi \left(\mathcal{M}(x,y,t)\right).$$

Thus, T is a fuzzy ψ-contractive mapping. Besides, note that for each $x_0\in \mathbb{R}$, we have that $\mathcal{M}(x_0,T\left(x_0\right),t_0)>0$ for all $t_0>1$. Hence, all the hypotheses of Theorem 2 are satisfied, which ensures the existence of the aforementioned fixed point $x^{\ast }=0$. However, Theorem 1 cannot be applied to this mapping in the fuzzy metric considered since some conditions imposed in it are not fulfilled. Concretely, such a fuzzy metric is not strong since if we take $x=0$, $y=\frac{1}{4}$ and $z=\frac{1}{2}$, we have, for $t=\frac{1}{2}$, the following:

$$\mathcal{M}(x,z,t)=0<\frac{\frac{1}{2}}{\frac{1}{2}+\frac{1}{4}}·\frac{\frac{1}{2}}{\frac{1}{2}+\frac{1}{4}}=\mathcal{M}(x,y,t)·\mathcal{M}(y,z,t).$$