Meir–Keeler Fixed-Point Theorems in Tripled Fuzzy Metric Spaces

Abstract

In this paper, we first propose the concept of a family of quasi-G-metric spaces corresponding to the tripled fuzzy metric spaces (or G-fuzzy metric spaces). Using their properties, we give the characterization of tripled fuzzy metrics. Second, we introduce the notion of generalized fuzzy Meir–Keeler-type contractions in G-fuzzy metric spaces. With the aid of the proposed notion, we show that every orbitally continuous generalized fuzzy Meir–Keeler-type contraction has a unique fixed point in complete G-fuzzy metric spaces. In the end, an example illustrates the validity of our results.

1. Introduction

Fixed-point theory is an active research field with a wide range of applications in applied mathematics, engineering, economics, and computer science. As is well known, one of the most famous results in the theory of complete metric spaces is the Banach contraction principle [1]. This principle is used to study the existence and uniqueness of solutions for a wide class of linear and nonlinear functional equations arising in pure and applied mathematics. For example, in fuzzy game theory, fuzzy fixed-point results are related to the existence of equilibrium solution [2]. In a dynamical system, fuzzy fixed-point theorems can be applied in existence, uniqueness, and continuity of solution with some vague parameters [3,4]. As a generalization, the Meir–Keeler contraction principle plays a fundamental role in fixed-point theory [5,6,7]. In 2006, Mustafa and Sims introduced the concept of G-metric spaces as a three-variable viewpoint to extend metric spaces [8]. Based on the notion of G-metric spaces, Mustafa et al. obtained some fixed-point results for mappings satisfying different contractive conditions [9,10,11].

The concept of fuzzy metric spaces was initiated by Kramosil and Michálek [12] in 1975, which is now called KM-fuzzy metric spaces and could be considered to be modifications of the concept of Menger probabilistic metric spaces [13]. To obtain a Hausdorff topology for a KM-fuzzy metric space, George and Veeramani [14] in 1994 reintroduced the concept of fuzzy metric spaces (called GV-fuzzy metric space) by modifying the definition of KM-fuzzy metric spaces. Later, George and Veeramani [14] gave a necessary and sufficient condition for the completeness of fuzzy metric space. Since then, various fixed-point results for mappings satisfying different contractive conditions were established by many researchers [15,16,17,18,19,20,21]. Moreover, in 2019, Zheng and Wang [22] proposed the concept of fuzzy Meir–Keeler contractive mappings in fuzzy metric spaces, which covers fuzzy $\psi$-contractive mappings and fuzzy $\mathcal{H}$-contractive mappings in [23,24] as special cases, and obtained some Meir–Keeler-type fixed-point theorems.

Recently, Tian et al. [25] generalized the concept of G-metric spaces to fuzzy setting, which is called tripled fuzzy metric spaces (or G-fuzzy metric spaces), and it is also a generalization of fuzzy metric spaces in the sense of George and Veeramani. Also, they introduced two kinds of notions of generalized fuzzy contractive mappings and obtained a fixed-point theorem on the mappings in the space. Based on the above analysis, although fixed-point theory in fuzzy metric spaces is studied from various aspects at present, it remains to be studied in G-fuzzy metric spaces. To enrich the fixed-point theory in G-fuzzy metric spaces and apply it to other theories more widely, it is necessary to work on the theoretical framework in G-fuzzy metric spaces.

The structure of the paper is as follows. In Section 2, some necessary definitions and results are applied. In Section 3, we propose the concept of a family of quasi G-metric spaces. Using the properties of quasi G-metric families, we give the characterization of tripled fuzzy metrics. In Section 4, we introduce the concept of generalized Meir–Keeler-type contractions in the context of G-fuzzy metric spaces and present some fixed-point theorems. Then, we also give an example to illustrate a generalized fuzzy Meir–Keeler-type contraction. Finally, a summary is given in Section 5.

2. Preliminaries

In this section, we recall some basic concepts and results which will be used. In the sequel, the letter $\mathbb{N}$ denotes the set of natural numbers.

Definition 1 

([26]). A t-norm ∗ on $[0,1]$ is a binary operation on $[0,1]$ that is commutative (i.e., $a\ast b=b\ast a$, for any $a,b\in [0,1]$), associative (i.e., $a\ast (b\ast c)=(a\ast b)\ast c$, for any $a,b,c\in [0,1]$), increasing (i.e., $a\ast c\le b\ast c$ whenever $a\le b$ with $a,b,c\in [0,1]$) and has neutral element 1 (i.e., $b\ast 1=b$, for any $b\in [0,1]$). A t-norm is said to be continuous if $a\ast b$ is continuous at each point $(a,b)\in [0,1]\times [0,1]$.

A t-norm ∗ is said to be strictly increasing if $a<b\Rightarrow a\ast a<b\ast b$. For $a\in [0,1]$, the sequence ${\left\{{\ast }^{n}a\right\}}_{n=1}^{\infty }$ is defined by ${\ast }^{1}a=a$ and ${\ast }^{n}a=\left({\ast }^{n-1}a\right)\ast a$. A t-norm ∗ is said to be of H-type if the sequence of functions ${\left\{{\ast }^{n}a\right\}}_{n=1}^{\infty }$ is equicontinous at $a=1$.

The t-norm ∧ defined by $a\wedge b=min\{a,b\}$ is a trivial example of a t-norm of H-type. It is known that $a\ast b\le a\wedge b$ for all $a,b\in [0,1]$. The following results present a wide range of t-norms of H-type.

Lemma 1 

([27]). Let $\delta \in (0,1]$ be a real number and let ∗ be a t-norm. Define $x{\ast }_{\delta }y=x\ast y$, if $max\{x,y\}\le 1-\delta$, and $x{\ast }_{\delta }y=min\{x,y\}$, if $max\{x,y\}>1-\delta$. Then, ${\ast }_{\delta }$ is a t-norm of H-type.

Lemma 2 

([28,29,30]). Let ∗ be a t-norm.

(1) 

Suppose that there exists a strictly increasing sequence $\left\{b_n\right\}\subseteq [0,1)$ such that ${lim}_{n\to \infty }{b}_n=1$ and $b_n\ast b_n=b_n$. Then, ∗ is of H-type.

(2) 

Conversely, if ∗ is continuous and of H-type, then there exists a strictly increasing sequence $\left\{b_n\right\}\subseteq [0,1)$ such that ${lim}_{n\to \infty }{b}_n=1$ and $b_n\ast b_n=b_n$.

Example 1 

([26]). The three basic continuous t-norms are defined as follows:

(i) 

the minimum t-norm ${\ast }_m:x{\ast }_{m}y=x\wedge y$;

(ii) 

the product t-norm ${\ast }_p:x{\ast }_{p}y=x·y$;

(iii) 

the ukasiewicz t-norm ${\ast }_L:x{\ast }_{L}y=0\vee (x+y-1)$.

Definition 2 

([8]). Let X be a nonempty set and let $G:X^3⟶[0,\infty )$ be a map satisfying the following conditions:

• (TM-1) $G(x,y,z)=0$ whenever $x=y=z$;
• (TM-2) $G(x,x,y)>0$ for all $x,y\in X$ with $x\ne y$;
• (TM-3) $G(x,x,y)\le G(x,y,z)$ for all $x,y,z\in X$ with $z\ne y$;
• (TM-4) $G(x,y,z)=G(x,z,y)=G(y,z,x)=\cdots ,$ (symmetry in all three variables);
• (TM-5) $G(x,y,z)\le G(x,a,a)+G(a,y,z)$ for all $x,y,z,a\in X$.

Then G is called a G-metric, and the pair $(X,G)$ is called a G-metric space.

Definition 3 

([14]). A triplet $(X,M,\ast )$ is called a fuzzy metric space if X is a nonempty set, ∗ is a continuous t-norm and M is a fuzzy set on $X\times X\times (0,\infty )$ such that the following conditions are valid:

• (FM-1) $M(x,y,t)>0$ for all $x,y\in X$;
• (FM-2) $M(x,y,t)=1$ for all $t>0$ if and only if $x=y$;
• (FM-1) $M(x,y,t)=M(y,x,t)$ for all $x,y\in X$ and $t>0$;
• (FM-3) $M(x,z,r+s)\ge M(x,y,r)\ast M(y,z,s)$ for all $x,y,z\in X$ and $r,s>0$;
• (FM-4) $M(x,y,·):(0,\infty )⟶[0,1]$ is continuous.

Definition 4 

([25]). A triplet $(X,F,\ast )$ is called a tripled fuzzy metric space if X is a nonempty set, ∗ is a continuous t-norm and F is a fuzzy set on $X\times X\times X\times (0,\infty )$ such that the following conditions are valid:

• (TFM-1) $F_{x,y,z}\left(t\right)>0$ for all $x,y,z\in X$ and $t>0$;
• (TFM-2) $F_{x,y,z}\left(t\right)=1$ for all $t>0$ if and only if $x=y=z$;
• (TFM-3) $F_{x,x,y}\left(t\right)\ge F_{x,y,z}\left(t\right)$ for $y\ne z$ and $t>0$;
• (TFM-4) F is invariant under all permutations of $(x,y,z)$, i.e., $F_{x,y,z}\left(t\right)=F_{x,z,y}\left(t\right)=F_{y,x,z}\left(t\right)=\cdots$;
• (TFM-5) $F_{x,z,y}(r+s)\ge F_{x,a,a}\left(r\right)\ast F_{a,y,z}\left(s\right)$ for all $a,x,y,z\in X$ and $r,s>0$;
• (TFM-6) $F_{x,y,z}(·):(0,\infty )\to [0,1]$ is continuous.

If $(X,F,\ast )$ is a G-fuzzy metric space (or G-metric space), we will say that $(F,\ast )$$\left(orsimplyF\right)$ is a tripled fuzzy metric (or G-fuzzy metric) on X.

A tripled fuzzy metric F on X is said to be stationary if F does not depend on t, i.e., if for any $x,y,z\in X$, the function $F_{x,y,z}\left(t\right)$ is constant. In this case, we write $F_{x,y,z}$ instead of $F_{x,y,z}\left(t\right).$

Proposition 1 

([25]). Let $(X,F,\ast )$ be a G-fuzzy metric space. Then for all $x,y\in X$, $F_{x,y,y}(·)$ is nondecreasing.

Next, we will present the topological properties, convergence of sequences, Cauchy sequences, and completeness of G-fuzzy metric spaces.

Definition 5 

([25]). Let $(X,F,\ast )$ be a G-fuzzy metric space, $\lambda \in (0,1)$, $t>0$, and $x\in X$. Then the set

$$U(x,\lambda ,t)=\{y\in X:F_{x,y,y}\left(t\right)>1-\lambda ,F_{x,x,y}\left(t\right)>1-\lambda \}$$

is called a neighborhood with center x and radius λ with respect to t.

Theorem 1 

([25]). Let $(X,F,\ast )$ be a G-fuzzy metric space. Define

$${\tau }_F=\{A\subseteq X:\mathit{for}\mathit{any}x\in A,\mathit{there}\mathit{exists}\lambda \in (0,1)\mathit{and}t>0\mathit{such}\mathit{that}U(x,\lambda ,t)\subseteq A\}.$$

Then ${\tau }_F$ is a topology on X.

Example 2 

([25]). Let $(X,d)$ be a metric space and $\ast ={\ast }_p$. Define a fuzzy set F on $X\times X\times X\times (0,\infty )$ by

$$F_{u,v,w}^d\left(t\right)=\frac{t}{t+d(u,v)}\ast \frac{t}{t+d(v,w)}\ast \frac{t}{t+d(w,u)}.$$

Then $(X,F^d,{\ast }_p)$ is a G-fuzzy metric space, and $F^d$ is usually called the standard G-fuzzy metric induced by d. The topology ${\tau }_{F^d}$ coincides with the topology ${\tau }^d$ on X deduced from d.

Definition 6 

([25]). Let $(X,F,\ast )$ be a G-fuzzy metric space. Then

(1) 

A sequence $\left\{x_n\right\}$ in X is said to be F-convergent (or simply convergent) to a point $x\in X$, denoted by $x_n\to x$, if for every $t>0$ and $\lambda \in (0,1)$, there exists an integer $N_{\lambda ,t}>0$ such that $x_n\in U(x,\lambda ,t)$ for all $n>N_{\lambda ,t}$.

(2) 

A sequence $\left\{x_n\right\}$ in X is said to be Cauchy if for any $t>0$ and $\lambda \in (0,1)$, there exists an integer $N_{\lambda ,t}>0$ such that $F_{x_n,x_m,x_l,}\left(t\right)>1-\lambda$ for all $n,m,l>N_{\lambda ,t}$.

Proposition 2 

([25]). Let $(X,F,\ast )$ be a G-fuzzy metric space. Then, the following statements are equivalent.

(1) 

The sequence $\left\{x_n\right\}$ is Cauchy.

(2) 

For any $t>0$ and $\lambda \in (0,1)$, there exists an integer $N_{\lambda ,t}>0$ such that $F_{x_n,x_m,x_m}\left(t\right)>1-\lambda$ for all $m,n\ge N_{\lambda ,t}$.

Proposition 3 

([25]). Let $(X,F,\ast )$ be a G-fuzzy metric space. Then, the following statements are equivalent.

(1) 

The sequence $\left\{x_n\right\}$ in X is F-convergent to a point $x\in X$.

(2) 

For all $t>0$, $F_{x_n,x_n,x}\left(t\right)\to 1$ as $n\to \infty$.

(3) 

For all $t>0$, $F_{x_n,x,x}\left(t\right)\to 1$ as $n\to \infty$.

Definition 7 

([25]). Let $(X,F,\ast )$ be a G-fuzzy metric space. The triple $(X,F,\ast )$ is called F-complete, or simply complete, if every Cauchy sequence is convergent in $(X,F,\ast )$.

3. Some Properties of the Quasi G-Metric Families

In this section, we mainly introduce the following quasi G-metric families corresponding to the tripled fuzzy metric spaces.

Definition 8. 

Let $(X,F,\ast )$ be a G-fuzzy metric space. For each $\lambda \in (0,1]$, define a mapping $G_{\lambda }:X^3⟶[0,\infty )$ by

$$G_{\lambda }(x,y,z)=inf\{t:F_{x,y,z}\left(t\right)>1-\lambda \}(\forall x,y,z\in X).$$

Then $\{G_{\lambda }(·,·,·):\lambda \in (0,1]\}$ is called a quasi G-metric family with respect to the tripled fuzzy metric F on X.

The following lemma follows from Definition 8.

Lemma 3. 

Let $(X,F,\ast )$ be a G-fuzzy metric space and let $\{G_{\lambda }:\lambda \in (0,1]\}$ be the quasi G-metric family, $\epsilon >0$ and $x,y,z\in X$. Then

(1) 

$F_{x,y,z}\left(\epsilon \right)>1-\lambda ⟺G_{\lambda }(x,y,z)<\epsilon$;

(2) 

$G_{\lambda }(x,y,z)=0$ for all $\lambda \in (0,1]⟺x=y=z$;

(3) 

$G_{\lambda }(x,x,y)\le G_{\lambda }(x,y,z)$ whenever $z\ne y$;

(4) 

$G_{\lambda }(x,y,z)=G_{\lambda }(x,z,y)=G_{\lambda }(y,z,x)=\cdots ,$ (symmetry in all three variables);

(5) 

if $F_{x,y,z}(·)$ is continuous, then $F_{x,y,z}\left(G_{\lambda }(x,y,z)\right)=1-\lambda$ and

$$G_{\lambda }(x,y,z)\le \epsilon ⟹F_{x,y,z}\left(\epsilon \right)\ge 1-\lambda ;$$

(6) 

if F is strictly increasing, then $F_{x,y,z}\left(\epsilon \right)\ge 1-\lambda ⟹G_{\lambda }(x,y,z)\le \epsilon .$

Proof. 

The verification of (1)–(4) is obvious by Definition 8.

(5) Since F is continuous by (TFM-6), we have

$$\begin{array}{cc}\hfill F_{x,y,z}\left(G_{\lambda }(x,y,z)\right)& =F_{x,y,z}(inf\{t:F_{x,y,z}\left(t\right)>1-\lambda \})\hfill \\ & =inf\{F_{x,y,z}\left(t\right):F_{x,y,z}\left(t\right)>1-\lambda \}=1-\lambda .\hfill \end{array}$$

Moreover, since $G_{\lambda }(x,y,z)\le ϵ$, we have $F_{x,y,z}\left(\epsilon \right)\ge F_{x,y,z}\left(G_{\lambda }(x,y,z)\right)=1-\lambda ,$ as desired.

(6) Let $F_{x,y,z}\left(\epsilon \right)\ge 1-\lambda$ and suppose $G_{\lambda }(x,y,z)>ϵ$. Since F is strictly increasing, then

$$F_{x,y,z}\left(\epsilon \right)<F_{x,y,z}\left(G_{\lambda }(x,y,z)\right)=1-\lambda ,$$

a contradiction. □

Example 3. 

Let $(X,G)$ be a G-metric space and let $\ast ={\ast }_m=\wedge$. Define

$$F_{x,y,z}^G\left(t\right)=\frac{t}{t+G(x,y,z)}(\forall x,y,z\in X,\forall t>0).$$

Then $(X,F^G,\wedge )$ is a triple fuzzy metric space [25]. For the space $(X,F_G,\wedge )$, we have

$$G_{\lambda }(x,y,z)=G(x,y,z)=\frac{G(x,y,z)(1-\lambda )}{\lambda }$$

for all $\lambda \in (0,1]$, i.e., $\{G_{\lambda }(x,y,z):\lambda \in (0,1]\}$ is a quasi G-metric family with respect to $(X,F^G,\wedge )$.

Theorem 2. 

Let $(X,F,\ast )$ be a G-fuzzy metric space and $x,y,z\in X$. Then, $G_{\lambda }(x,y,z)$ is a nonincreasing left continuous function on $\lambda \in (0,1]$.

Proof. 

Let ${\lambda }_1,{\lambda }_2\in (0,1]$ and ${\lambda }_1<{\lambda }_2$. From

$$\{t:F_{x,y,z}\left(t\right)>1-{\lambda }_1\}\subseteq \{t:F_{x,y,z}\left(t\right)>1-{\lambda }_2\},$$

it is easy to see that $G_{\lambda }(x,y,z)$ is nonincreasing. Now, we show that ${lim}_{\alpha \to 0^{+}}{G}_{\lambda -\alpha }(x,y,z)=G_{\lambda }(x,y,z)$. Assume that ${lim}_{\alpha \to 0^{+}}{G}_{\lambda -\alpha }(x,y,z)>G_{\lambda }(x,y,z)$ for $\lambda \in (0,1]$ and $x,y,z\in X$. Then, there exists $r>0$ such that ${lim}_{\alpha \to 0^{+}}{G}_{\lambda -\alpha }(x,y,z)>r>G_{\lambda }(x,y,z)$. This means that $G_{\lambda -\alpha }(x,y,z)>r>G_{\lambda }(x,y,z)$ for all $\alpha \in (0,\lambda )$. From $r>G_{\lambda }(x,y,z)$, we have $F_{x,y,z}\left(r\right)>1-\lambda$. However, from $G_{\lambda -\alpha }(x,y,z)>r$, it implies that $F_{x,y,z}\left(r\right)\le 1-\lambda +\alpha$, and so $F_{x,y,z}\left(r\right)\le 1-\lambda$ by the arbitrariness of $\alpha$, a contradiction. □

Theorem 3. 

Let $(X,F,\ast )$ be a G-fuzzy metric space and let $x,y,z\in X$ be distinct. Then,

(1) 

$G_{\lambda }(x,y,z)$ is continuous on $\lambda \in (0,1]$ if and only if $F_{x,y,z}(·)$ is strictly increasing;

(2) 

$G_{\lambda }(x,y,z)$ is strictly decreasing on $\lambda \in (0,1]$ if and only if $F_{x,y,z}(·)$ is continuous.

Proof. 

(1) Suppose $F_{x,y,z}(·)$ is strictly increasing. From Theorem 2, we see that $G_{\lambda }(x,y,z)$ is left continuous and ${lim}_{\alpha \to 0^{+}}{G}_{\lambda +\alpha }(x,y,z)\le G_{\lambda }(x,y,z)$ for $\lambda \in (0,1)$. We will prove that $G_{\lambda }(x,y,z)$ is right continuous for $\lambda \in (0,1)$. Assume that $g_{\lambda }(x,y,z)={lim}_{\alpha \to 0^{+}}{G}_{\lambda +\alpha }(x,y,z)$$<G_{\lambda }(x,y,z)$ and $t\in (g_{\lambda }(x,y,z),G_{\lambda }(x,y,z))$. Then, from $t<G_{\lambda }(x,y,z)$, it implies that $F_{x,y,z}\left(t\right)\le 1-\lambda$. In addition, from $g_{\lambda }(x,y,z)<t$, it implies that $G_{\lambda +\alpha }(x,y,z)<t$ for all $\alpha \in (0,1-\lambda )$, i.e., $F_{x,y,z}\left(t\right)>1-\lambda -\alpha$. Letting $\alpha \to 0^{+}$, we have $F_{x,y,z}\left(t\right)\ge 1-\lambda$. Thus, for each $t\in (g_{\lambda }(x,y,z),G_{\lambda }(x,y,z))$, we have $F_{x,y,z}\left(t\right)=1-\lambda$. Since $F_{x,y,z}(·)$ is strictly increasing, it is a contradiction. Conversely, suppose $G_{\lambda }(x,y,z)$ is continuous on $\lambda \in (0,1]$. We will prove that $F_{x,y,z}(·)$ is strictly increasing. Assume that there exist $t_1,t_2\in [0,\infty )$ with $t_1<t_2$ such that $0<F_{x,y,z}\left(t_1\right)=F_{x,y,z}\left(t_2\right)=1-\lambda <1$. Then, $F_{x,y,z}\left(t_1\right)>1-\lambda -\epsilon$ and $F_{x,y,z}\left(t_2\right)<1-\lambda +\epsilon$ for $\epsilon >0$. It follows that $G_{\lambda +\epsilon }(x,y,z)<t_1<t_2\le G_{\lambda -\epsilon }(x,y,z)$. Letting $\epsilon \to 0^{+}$, by the continuity of $G_{\lambda }(x,y,z)$ on $\lambda$, we have $G_{\lambda }(x,y,z)\le t_1<t_2\le G_{\lambda }(x,y,z)$, a contradiction.

(2) It is similar to (1). □

Theorem 4. 

Let $(X,F,\ast )$ be a G-fuzzy metric space. Then,

(1) 

if $\ast =\wedge$, then

$$G_{\lambda }(x,y,z)\le G_{\lambda }(x,a,a)+G_{\lambda }(a,y,z)(\forall \lambda \in (0,1],\forall x,y,z,a\in X);$$

(2) 

If F is strictly increasing and ∗ is of H-type, then there exists a strictly decreasing sequence $\left\{{\lambda }_n\right\}\subseteq (0,1]$ such that ${lim}_{n\to \infty }{\lambda }_n=0$ and

$$G_{{\lambda }_n}(x,y,z)\le G_{{\lambda }_n}(x,a,a)+G_{{\lambda }_n}(a,y,z)(\forall n\in {\mathbb{Z}}^{+},\forall x,y,z,a\in X).$$

Proof. 

(1) Let $\ast =\wedge$. Then, for each $\epsilon >0$, from (TFM-5), we have

$$F_{x,y,z}(G_{\lambda }(x,a,a)+G_{\lambda }(a,y,z)+2\epsilon )\ge F_{x,a,a}(G_{\lambda }(x,a,a)+\epsilon )\wedge F_{a,y,z}(G_{\lambda }(a,y,z)+\epsilon )>1-\lambda ,$$

which shows that $G_{\lambda }(x,y,z)\le G_{\lambda }(x,a,a)+G_{\lambda }(a,y,z)+2\epsilon$. By the arbitrariness of $\epsilon$, we obtain $G_{\lambda }(x,y,z)\le G_{\lambda }(x,a,a)+G_{\lambda }(a,y,z)$, as desired.

(2) Since ∗ is of H-type, it follows from Lemma 2, there exists a strictly increasing sequence $\left\{b_n\right\}$ in $[0,1)$ such that ${lim}_{n\to \infty }{b}_n=1$ and $b_n\ast b_n=b_n$. Putting ${\lambda }_n=1-b_n$, we have $\left\{{\lambda }_n\right\}$ in $(0,1]$ and ${lim}_{n\to \infty }{\lambda }_n=0$. Then, for each $\epsilon >0$, from (TFM-5), we have

$$\begin{array}{cc}\hfill & F_{x,y,z}(G_{{\lambda }_n}(x,a,a)+G_{{\lambda }_n}(a,y,z)+2\epsilon )\hfill \\ \hfill & \ge F_{x,a,a}(G_{{\lambda }_n}(x,a,a)+\epsilon )\wedge F_{a,y,z}(G_{{\lambda }_n}(a,y,z)+\epsilon )\hfill \\ \hfill & \ge (1-{\lambda }_n)\ast (1-{\lambda }_n)\hfill \\ \hfill & =1-{\lambda }_n.\hfill \end{array}$$

Since F is strictly increasing, we have that $G_{{\lambda }_n}(x,y,z)\le G_{{\lambda }_n}(x,a,a)+G_{{\lambda }_n}(a,y,z)+2\epsilon$. By the arbitrariness of $\epsilon$, we obtain $G_{{\lambda }_n}(x,y,z)\le G_{{\lambda }_n}(x,a,a)+G_{{\lambda }_n}(a,y,z)$, as desired. □

Theorem 5. 

Let $(X,F,\ast )$ be a G-fuzzy metric space. Then, for each $\lambda \in (0,1]$, there exists $\mu \in (0,\lambda ]$ such that

$$G_{\lambda }(x,y,z)\le G_{\lambda }(x,a,a)+G_{\mu }(a,y,z)(\forall x,y,z,a\in X).$$

Proof. 

Assume that there is $\lambda \in (0,1]$, for each $n\in {\mathbb{Z}}^{+}$, ${\mu }_n=\lambda /n$, there exist $x_n,y_n,z_n,a_n\in X$ such that $G_{\lambda }(x_n,y_n,z_n)>G_{\lambda }(x_n,a_n,a_n)+G_{\lambda }(a_n,y_n,z_n).$ Then, there exist $s_n,t_n\in [0,+\infty )$ such that

$$G_{\lambda }(x_n,y_n,z_n)=s_n+t_n,s_n>G_{\lambda }(x_n,a_n,a_n)\mathrm{and}{t}_n>G_{\lambda }(a_n,y_n,z_n).$$

Thus, $F_{x_n,y_n,z_n}(s_n+t_n)\le 1-\lambda ,F_{x_n,a_n,a_n}\left(s_n\right)>1-\lambda$, and $F_{a_n,y_n,z_n}\left(t_n\right)>1-{\mu }_n$. Putting $b_n=F_{x_n,a_n,a_n}\left(s_n\right)$, ${\delta }_n=F_{x_n,a_n,a_n}\left(s_n\right)-(1-\lambda )$, we have $b_n,{\delta }_n\in (0,1]$. Since ∗ is continuous, ${lim}_{a\to 1^{-}}{b}_n\ast a=b_n>b_n-{\delta }_n$, there exists ${\mu }_0\in (0,\lambda ]$ such that $b_n\ast (1-\mu )>b_n-{\delta }_n=1-\lambda$ for all $\mu \in (0,{\mu }_0]$. Hence, for ${\mu }_n\in (0,{\mu }_0]$, from (FTM-5), we obtain

$$1-\lambda \ge F_{x_n,y_n,z_n}(s_n+t_n)\ge F_{x_n,a_n,a_n}\left(s_n\right)\ast F_{a_n,y_n,z_n}\left(t_n\right)\ge b_n\ast (1-{\mu }_n)>1-\lambda ,$$

a contradiction. □

Theorem 6. 

Let $(X,F,\ast )$ be a G-fuzzy metric space such that F is strictly increasing (or ∗ is strictly increasing), $\lambda \in (0,1]$, $\nu =1-(1-\lambda )\ast (1-\lambda )$. Then

$$G_{\nu }(x,y,z)\le G_{\lambda }(x,a,a)+G_{\lambda }(a,y,z)(\forall x,y,z,a\in X).$$

Proof. 

Let $\epsilon >0$. Then $F_{x,a,a}\left(G_{\lambda +\epsilon }(x,a,a)\right)>1-\lambda$ and $F_{a,y,z}\left(G_{\lambda +\epsilon }(a,y,z)\right)>1-\lambda$. If ∗ is strictly increasing, then from (FN-4) we have

$$\begin{array}{cc}\hfill & F_{x,y,z}(G_{\lambda }(x,a,a)+G_{\lambda }(a,y,z)+2\epsilon )\hfill \\ \hfill & \ge F_{x,a,a}(G_{\lambda }(x,a,a)+\epsilon )\ast F_{a,y,z}(G_{\lambda }(a,y,z)+\epsilon )\hfill \\ \hfill & >(1-\lambda )\ast (1-\lambda )=1-\nu ,\hfill \end{array}$$

and so $G_{\nu }(x,y,z)\le G_{\lambda }(x,a,a)+G_{\lambda }(a,y,z)+2\epsilon$; if N is strictly increasing, then from

$$\begin{array}{cc}\hfill & F_{x,y,z}(G_{\lambda }(x,a,a)+G_{\lambda }(a,y,z)+2\epsilon )\hfill \\ \hfill & \ge F_{x,a,a}(G_{\lambda }(x,a,a)+\epsilon )\ast F_{a,y,z}(G_{\lambda }(a,y,z)+\epsilon )\ge 1-\nu ,\hfill \end{array}$$

we also have $G_{\nu }(x,y,z)\le G_{\lambda }(x,a,a)+G_{\lambda }(a,y,z)+2\epsilon$. Letting $\epsilon \to 0^{+}$, we obtain the desired inequality. □

Theorem 7. 

Let $(X,F,\ast )$ be a G-fuzzy metric space, $x\in X$ and $\left\{x_n\right\}\subseteq X$. Then,

(1) 

${lim}_{n\to \infty }{x}_n=x⟺{lim}_{n\to \infty }{G}_{\lambda }(x_n,x,x)=0⟺{lim}_{n\to \infty }{G}_{\lambda }(x_n,x_n,x)=0$ for each $\lambda \in (0,1]$;

(2) 

for each $\lambda \in (0,1]$, $G_{\lambda }(·,·,·):X^3⟶[0,\infty )$ is continuous;

(3) 

$\beta \left(A\right)=1⟺{sup}_{x,y,z\in A}{G}_{\lambda }(x,y,z)<+\infty$ for each $\lambda \in (0,1]$, where

$$\beta \left(A\right)=\underset{t>0}{sup}\underset{x,y,z\in A}{inf}{F}_{x,y,z}\left(t\right).$$

Proof. 

(1) ${lim}_{n\to \infty }{x}_n=x$ ⇔ for $\lambda \in (0,1]$ and $\epsilon >0$, there exists $N\in {\mathbb{Z}}^{+}$ such that $F_{x_n,x,x}\left(\epsilon \right)$$>1-\lambda$ and $F_{x_n,x_n,x}\left(\epsilon \right)>1-\lambda$, i.e., $G_{\lambda }(x_n,x,x)<\epsilon \mathrm{and}{G}_{\lambda }(x_n,x_n,x)<\epsilon \iff {lim}_{n\to \infty }$$G_{\lambda }(x_n,x,x)=0\mathrm{and}{lim}_{n\to \infty }{G}_{\lambda }(x_n,x_n,x)=0$ for each $\lambda \in (0,1]$.

(2) From Theorem 5, for each $\lambda \in (0,1]$, there exists $\mu \in (0,\lambda ]$ such that

$$G_{\lambda }(x,y,z)\le G_{\mu }(x,x_n,x_n)+G_{\lambda }(x_n,y,z)\mathrm{and}{G}_{\lambda }(x_n,y,z)\le G_{\mu }(x_n,x,x)+G_{\lambda }(x,y,z),$$

which implies $|G_{\lambda }(x_n,y,z)-G_{\lambda }(x,y,z)|\le G_{\mu }(x_n,x,x)$. Hence, by (1) and the symmetry, we obtain the desired conclusion.

(3) If $\beta \left(A\right)=1$, then for each $\lambda \in (0,1]$, there exists ${\delta }_{\lambda }>0$ such that for all $x,y,z\in A,F(x,y,z,{\delta }_{\lambda })>1-\lambda$, and so $G_{\lambda }(x,y,z)<{\delta }_{\lambda }$. Thus, for a fixed point $y_0\in A$, by Theorem 5, there is $\mu \in (0,\lambda ]$ such that

$$G_{\lambda }(x,y,z)\le G_{\mu }(x,y_0,y_0)+G_{\lambda }(y_0,y,z)\le {\delta }_{\mu }+G_{\lambda }(y_0,y,z)(\forall x,y,z\in A).$$

Hence, ${sup}_{x,y,z\in A}{G}_{\lambda }(x,y,z)\le {\delta }_{\mu }+G_{\lambda }(y_0,y,z)<+\infty$.

Conversely, for each $\lambda \in (0,1]$, suppose ${sup}_{x,y,z\in A}{G}_{\lambda }(x,y,z)<+\infty$ and set ${\delta }_{\lambda }={sup}_{x,y,z\in A}{G}_{\lambda }(x,y,z)$. Then, for each $x,y,z\in A$, we have $G_{\lambda }(x,y,z)\le {\delta }_{\lambda }$. By Theorem 5, there is $\mu =\mu \left(\lambda \right)\in (0,\lambda ]$ such that for all $x,y,z,a\in A$,

$$G_{\lambda }(x,y,z)\le G_{\mu }(x,a,a)+G_{\lambda }(a,y,z)\le {\delta }_{\mu }+{\delta }_{\lambda }<+\infty ,$$

i.e., $F_{x,y,z}({\delta }_{\mu }+{\delta }_{\lambda })\ge 1-\lambda$, which implies that $\beta \left(A\right)={sup}_{t>0}{inf}_{x,y,z\in A}{F}_{x,y,z}\left(t\right)=1$. □

From Lemma 3 and Theorems 3 and 6, we obtain the following consequence.

Corollary 1. 

Let $(X,F,\ast )$ be a G-fuzzy metric space and $\{G_{\lambda }:\lambda \in (0,1]\}$ a quasi G-metric family corresponding to F on X. Then,

(1) 

$G_1(x,y,z)=0$ for all $x,y,z\in X$;

(2) 

$G_{\lambda }(x,y,z)=0$ for all $\lambda \in (0,1]$⟺$x=y$;

(3) 

$G_{\lambda }(x,x,y)\le G_{\lambda }(x,y,z)$ for all $x,y,z\in X$ with $z\ne y$;

(4) 

$G_{\lambda }(x,y,z)=G_{\lambda }(x,z,y)=G_{\lambda }(y,z,x)=\cdots ,$ (symmetry in all three variables), for all $\lambda \in (0,1]$ and $x,y,z\in X$;

(5) 

for each $\lambda \in (0,1]$, there exists $\mu \in (0,\lambda ]$ such that

$$G_{\lambda }(x,y,z)\le G_{\mu }(x,a,a)+G_{\lambda }(a,y,z)(\forall x,y,z,a\in X);$$

Theorem 8. 

Let $(X,F,\ast )$ be a G-fuzzy metric space and $\{G_{\lambda }:\lambda \in (0,1]\}$ a quasi G-metric family corresponding to F on X. Then,

$$F_{x,y,z}\left(t\right)=1-sup\{\lambda \in (0,1]:G_{\lambda }(x,y,z)\ge t\}=1-inf\{\lambda \in (0,1]:G_{\lambda }(x,y,z)<t\}.$$

Proof. 

Let ${\mu }_t=sup\{\lambda \in (0,1]:G_{\lambda }(x,y,z)\ge t\}$ and ${\nu }_t=inf\{\lambda \in (0,1]:G_{\lambda }(x,y,z)<t\}$. Since $G_{\lambda }(x,y,z)$ is nonincreasing, ${\mu }_t\le {\nu }_t$. For each $\epsilon >0$, by the definition of ${\mu }_t$, we have $G_{{\mu }_t+\epsilon }(x,y,z)<t$. By the definition of ${\nu }_t$, this means that ${\mu }_t+\epsilon \ge {\nu }_t$, i.e., ${\mu }_t\ge {\nu }_t$. Hence, ${\mu }_t={\nu }_t$. For each $\epsilon >0$, since $G_{{\mu }_t-\epsilon }(x,y,z)\ge t$ and $G_{{\nu }_t+\epsilon }(x,y,z)<t$, we have $F_{x,y,z}\left(t\right)\le 1-({\mu }_t-\epsilon )$ and $F_{x,y,z}\left(t\right)>1-({\nu }_t+\epsilon )$. By the arbitrariness of $\epsilon$, the assertion holds. □

Theorem 9. 

Let X be a nonempty set and let $\{G_{\lambda }(·,·,·):\lambda \in (0,1]\}$ be a quasi G-metric family satisfying the following conditions:

• (G-0) $G_1(x,y,z)=0$ for all $x,y,z\in X$;
• (G-1) $G_{\lambda }(x,y,z)=0$ for all $\lambda \in (0,1]$⟺$x=y=z$;
• (G-2) $G_{\lambda }(x,x,y)\le G_{\lambda }(x,y,z)$ for all $x,y,z\in X$ with $z\ne y$;
• (G-3) $G_{\lambda }(x,y,z)=G_{\lambda }(x,z,y)=G_{\lambda }(y,z,x)=\cdots ,$ (symmetry in all three variables);
• (G-4) $G_{\lambda }(x,y,z)\le G_{\lambda }(x,a,a)+G_{\lambda }(a,y,z)$ for all $\lambda \in (0,1]$ and $x,y,z,a\in X;$
• (G-5) $G_{\lambda }(·,·,·)$ is strictly decreasing left continuous on $\lambda \in (0,1]$;

Let

$$\begin{array}{cc}\hfill F_{x,y,z}\left(t\right)& =inf\{\alpha \in [0,1):G_{1-\alpha }(x,y,z)\ge t\}\hfill \\ \hfill & =sup\{\alpha \in [0,1):G_{1-\alpha }(x,y,z)<t\}(\forall x,y,z,\in X,\forall t>0).\hfill \end{array}$$

Then, $(X,F,\wedge )$ is a G-fuzzy metric space. If ∗ is continuous, then $(X,F,\ast )$ is a G-fuzzy metric space.

Proof. 

It is evident that F is a fuzzy set on $X\times [0,+\infty )$. By (G-5), it holds $inf\{\alpha \in [0,1):G_{1-\alpha }(x,y,z)\ge t\}=sup\{\alpha \in [0,1):G_{1-\alpha }(x,y,z)<t\}$.

(TFM-1) From the definition of $F_{x,y,z}\left(t\right)$, we see that $F_{x,y,z}\left(t\right)>1-\lambda ⟺G_{\lambda }(x,y,z)<t$. Thus, from (G-0), we have $G_1(x,y,z)=0<t$ for all $t>0$. This implies that $F_{x,y,z}\left(t\right)>0$ for all $t>0$, as desired.

(TFM-2) Let $x=y=z$ and $t>0$. From (G1), we see that $G_{1-\alpha }(x,y,z)=0$ for all $\alpha \in [0,1)$. It follows that $F_{x,y,z}\left(t\right)=1$. Conversely, suppose that $F_{x,y,z}\left(t\right)=1$ but $x,y,z$ are distinct. For each $t>0$, from (G-5), there exists $\lambda \in (0,1]$ such that $G_{\lambda }(x,y,z)>t$, and so $F_{x,y,z}\left(t\right)\le 1-\lambda <1$, a contradiction.

(TFM-3) For $x,y,z\in X$ with $y\ne z$ and $t>0$, by (G-2) and (G-5), we have

$$F_{x,x,y}\left(t\right)=inf\{\alpha \in [0,1):G_{1-\alpha }(x,x,y)\ge t\}\ge inf\{\alpha \in [0,1):G_{1-\alpha }(x,y,z)\ge t\}=F_{x,y,z}\left(t\right).$$

(TFM-4) It is straightforward by (G-3).

(TFM-5) Let $x,y,z,a\in X$ and $t,s>0,F_{x,y,z}(t+s)=1-\lambda$, $\lambda =1-\alpha \in (0,1]$. Then, for each $\epsilon >0$,

$$t+s\le G_{1-(\alpha +\lambda )}(x,y,z)=G_{\lambda -\epsilon }(x,y,z).$$

Letting $\epsilon \to 0^{+}$, by (G-4), we have $t+s\le G_{\lambda }(x,y,z)$. Thus, by (N3), we have

$$t+s\le G_{\lambda }(x,y,z)\le G_{\lambda }(x,a,a)+G_{\lambda }(a,y,z),$$

which implies that $t\le G_{\lambda }(x,a,a)$ or $s\le G_{\lambda }(a,y,z)$. Hence, $F_{x,a,a}\left(t\right)\le 1-\lambda$ or $F_{a,y,z}\left(s\right)\le 1-\lambda$. Therefore $F_{x,a,a}\left(t\right)\wedge F_{a,y,z}\left(s\right)\le 1-\lambda =F_{x,y,z}(t+s).$ Since $a\ast b\le a\wedge b$, (TFM-5) holds for a general t-norm ∗.

(TFM-6) Since $G_{\lambda }(x,y,z)$ is strictly decreasing on $\lambda \in (0,1]$ for distinct $x,y,z\in X$, it follows from Theorem 3(2) that $F_{x,y,z}(·)$ is continuous. □

4. Generalized Fuzzy Meir–Keeler Type Contractions

We start with the definition of orbital continuity in the context of G-fuzzy metric spaces.

Definition 9. 

Let $(X,F,\ast )$ be a G-fuzzy metric space and $f:X⟶X$ be a self-map. We say that f is orbitally F-continuous (or simply orbitally continuous) whenever ${lim}_{i\to \infty }{F}_{f^{n_i}x,w,w}\left(t\right)=1$ implies that ${lim}_{i\to \infty }{F}_{f{f}^{n_i}x,fw,fw}\left(t\right)=1$ for each $x,w\in X$, $t>0$ and $n_i\in \mathbb{N}$.

Recall that in a G-fuzzy metric space $(X,F,\ast )$, a map $f:X⟶X$ is said to be F-continuous (or simply continuous) if for any sequence $\left\{x_n\right\}$ convergent to x with respect to ${\tau }_F$, $\left\{f\left(x_n\right)\right\}$ is F-convergent to $f\left(x\right)$. Obviously, every F-continuous map is orbitally F-continuous.

An interesting and general contraction condition for self-maps on fuzzy metric spaces was considered by Zheng and Wang in a recent paper [22]:

Denote $\Delta =\{\delta :(0,1]⟶(0,1]\mid \delta \mathrm{is}\mathrm{right}-\mathrm{continuous}\}.$

Lemma 4 

([22]). If $\delta \in \Delta$, then for $t\in (0,1)$, there exists an integer $k=k\left(t\right)>0$ such that

$$t+\frac{\delta \left(t\right)}{k}<1\mathrm{and}\delta \left(t+\frac{\delta \left(t\right)}{k}\right)-\frac{\delta \left(t\right)}{k}>0.$$

Definition 10. 

Let $(X,M,\ast )$ be a fuzzy metric space. A map $f:X⟶X$ is said to be fuzzy Meir–Keeler contraction with respect to $\delta \in \Delta$ if the following condition holds:

$$\forall ϵ\in (0,1),ϵ-\delta \left(ϵ\right)<M(x,y,t)\le ϵ\Rightarrow M(fx,fy,t)>ϵ,$$

(1)

for all $x,y\in X$ and $t>0$.

It is pointed out that fuzzy Meir–Keeler contractions cover fuzzy $\psi$-contractions and fuzzy $\mathcal{H}$-contractions as special cases [22]. Following this line of thought, we shall introduce a notion of generalized fuzzy Meir–Keeler-type contractions on G-fuzzy metric spaces.

Definition 11. 

Let $(X,F,\ast )$ be a G-fuzzy metric space. A map $f:X⟶X$ is called a generalized fuzzy Meir–Keeler-type contraction with respect to $\delta \in \Delta$ if the following condition holds:

$$\forall ϵ\in (0,1),ϵ-\delta \left(ϵ\right)<G_{x,y,z}\left(t\right)\le ϵ\Rightarrow F(fx,fy,fz,t)>ϵ,$$

(2)

where $G_{x,y,z}\left(t\right)=min\{F_{x,y,z}\left(t\right),F_{fx,x,x}\left(t\right),F_{fy,y,y}\left(t\right),F_{fz,z,z}\left(t\right)\}$ for all $x,y,z\in X$ and $t>0$.

Remark 1. 

If f is a generalized fuzzy Meir–Keeler-type contraction with respect to $\delta \in \Delta$, then

$$F_{fx,fy,fz}\left(t\right)>G_{x,y,z}\left(t\right)$$

(3)

for all $x,y,z\in X$ and $t>0$.

We now present our main results.

Proposition 4. 

Let $(X,F,\ast )$ be a G-fuzzy metric space and $f:X⟶X$ be a generalized fuzzy Meir–Keeler-type contraction with respect to $\delta \in \Delta$. Then ${lim}_{n\to \infty }{F}_{f^{n+1}x,f^{n}y,f^{n}z}\left(t\right)=1$ for all $x\in X$ and $t>0$.

Proof. 

Let $x_0\in X$. We define an iterative sequence $\left\{x_n\right\}$ as follows:

$$x_{n+1}=f{x}_n=f^{n+1}{x}_0.$$

(4)

for all integers $n\ge 0$. If $x_{n_0+1}=x_{n_0}$ for some $n_0\ge 0$, then $x_{n_0}$ is the desired fixed point of f. Indeed, $f{x}_{n_0}=x_{n_0+1}=x_{n_0}$. In this case, the proposition follows. Throughout the proof, we assume that $x_{k+1}\ne x_k$ for all $k\ge n_0$. Consequently, we have $G_{x_{n+1},x_n,x_n}\left(t\right)<1$ for all $n\ge 0$. By Remark 1, we obtain

$$\begin{array}{cc}\hfill & F_{x_{n+2},x_{n+1},x_{n+1}}\left(t\right)\hfill \\ \hfill & =F_{f{x}_{n+1},f{x}_n,f{x}_n}\left(t\right)\hfill \\ \hfill & >G_{x_{n+1},x_n,x_n}\left(t\right)\hfill \\ \hfill & =min\{F_{x_{n+1},x_n,x_n}\left(t\right),F_{f{x}_{n+1},x_{n+1},x_{n+1}}\left(t\right),F_{f{x}_n,x_n,x_n}\left(t\right),F_{f{x}_n,x_n,x_n}\left(t\right)\}\hfill \\ \hfill & =min\{F_{x_{n+1},x_n,x_n}\left(t\right),F_{x_{n+2},x_{n+1},x_{n+1}}\left(t\right)\}.\hfill \end{array}$$

Since $0<G_{x_{n+1},x_n,x_n}\left(t\right)<1$ for each n, we find that

$$F_{x_{n+2},x_{n+1},x_{n+1}}\left(t\right)>G_{x_{n+1},x_n,x_n}\left(t\right)\ge min\{F_{x_{n+1},x_n,x_n}\left(t\right),F_{x_{n+2},x_{n+1},x_{n+1}}\left(t\right)\}$$

using Remark 1 again. Notice that the case where

$$min\{F_{x_{n+1},x_n,x_n}\left(t\right),F_{x_{n+2},x_{n+1},x_{n+1}}\left(t\right)\}=F_{x_{n+2},x_{n+1},x_{n+1}}\left(t\right)$$

is impossible. Hence, we derive that

$$F_{x_{n+2},x_{n+1},x_{n+1}}\left(t\right)>G_{x_{n+1},x_n,x_n}\left(t\right)=F_{x_{n+1},x_n,x_n}\left(t\right)$$

(5)

for every n.

Thus, $\{F_{x_{n+2},x_{n+1},x_{n+1}}\left(t\right):n\in \mathbb{N}\}$ is an increasing sequence which is upper bounded by 1. Hence, it converges to some $ϵ\in (0,1]$ such that

$$\underset{n\to \infty }{lim}{F}_{x_{n+1},x_n,x_n}\left(t\right)=ϵ.$$

(6)

In particular, we have

$$\underset{n\to \infty }{lim}{G}_{x_{n+1},x_n,x_n}\left(t\right)=ϵ.$$

(7)

We claim that $ϵ=1$. Suppose, on the contrary, that $ϵ<1$. Since $\delta \in \Delta$, there exists $k\in \mathbb{N}$ such that

$$ϵ+\frac{\delta \left(ϵ\right)}{k}<1\mathrm{and}\delta \left(ϵ+\frac{\delta \left(ϵ\right)}{k}\right)-\frac{\delta \left(ϵ\right)}{k}>0.$$

Since ${lim}_{n\to \infty }{G}_{x_{n+1},x_n,x_n}\left(t\right)=ϵ$, there exists $n_0$ such that when $n>n_0$,

$$G_{x_{n+1},x_n,x_n}\left(t\right)>ϵ-\delta \left(ϵ+\frac{\delta \left(ϵ\right)}{k}\right)+\frac{\delta \left(ϵ\right)}{k}=\left(ϵ+\frac{\delta \left(ϵ\right)}{k}\right)-\delta \left(ϵ+\frac{\delta \left(ϵ\right)}{k}\right).$$

(8)

On the other hand, there also exists $n_1$ such that when $n>n_1$,

$$G_{x_{n+1},x_n,x_n}\left(t\right)\le ϵ+\frac{\delta \left(ϵ\right)}{k}.$$

(9)

Inequalities (8) and (9) can be satisfied whenever $n>max\{n_0,n_1\}$. It follows from Definition 11 that

$$F_{f{x}_{n+1},f{x}_n,f{x}_n}\left(t\right)=F_{x_{k+2},x_{k+1},x_{k+1}}\left(t\right)>ϵ+\frac{\delta \left(ϵ\right)}{k},$$

(10)

which contradicts to (6). Therefore, $ϵ=1$. □

Lemma 5. 

If f is a generalized fuzzy Meir–Keeler-type contraction with respect to $\delta \in \Delta$, and $\left\{x_n\right\},\left\{y_n\right\}\subseteq X$ such that ${lim}_{n\to \infty }{F}_{x_n,y_n,y_n}\left(t\right)={lim}_{n\to \infty }{F}_{f{x}_n,f{y}_n,f{y}_n}\left(t\right)=\lambda >0$, then $\lambda =1$.

Proof. 

Suppose, on the contrary, that $\lambda <1$. Since $\delta \in \Delta$, there exists $k\in \mathbb{N}$ such that

$$\lambda +\frac{\delta \left(\lambda \right)}{k}<1\mathrm{and}\delta \left(\lambda +\frac{\delta \left(\lambda \right)}{k}\right)-\frac{\delta \left(\lambda \right)}{k}>0.$$

Since ${lim}_{n\to \infty }{F}_{x_n,y_n,y_n}\left(t\right)=\lambda$, there exists $n_0\in \mathbb{N}$ such that

$$F_{x_n,y_n,y_n}\left(t\right)>\lambda -\delta \left(\lambda +\frac{\delta \left(\lambda \right)}{k}\right)+\frac{\delta \left(\lambda \right)}{k}(\forall n>n_0).$$

(11)

On the other hand, there exists $n_1\in \mathbb{N}$ such that

$$F_{x_n,y_n,y_n}\left(t\right)\le \lambda +\frac{\delta \left(\lambda \right)}{k}(\forall n>n_1).$$

(12)

Please note that $G_{x_n,y_n,y_n}\left(t\right)=min\{F_{x_n,y_n,y_n}\left(t\right),F_{x_{n+1},x_n,x_n}\left(t\right),F_{y_{n+1},y_n,y_n}\left(t\right)\}$. We distinguish three cases.

Case I. If $G_{x_n,y_n,y_n}\left(t\right)=F_{x_n,y_n,y_n}\left(t\right)$, then from Definition 11 with $ϵ=\lambda +\frac{\delta \left(\lambda \right)}{k}$, we have

$$F_{f{x}_n,f{y}_n,f{y}_n}\left(t\right)>\lambda +\frac{\delta \left(\lambda \right)}{k}(\forall n>max\{n_0,n_1\}),$$

which contradicts to $li{m}_{n\to \infty }{F}_{f{x}_n,f{y}_n,f{y}_n}\left(t\right)=\lambda$. Thus, $\lambda =1$.

Case II. If $G_{x_n,y_n,y_n}\left(t\right)=F_{x_{n+1},x_n,x_n}\left(t\right)$, then by Proposition 4, we have

$$\underset{n\to \infty }{lim}{G}_{x_n,y_n,y_n}\left(t\right)=\underset{n\to \infty }{lim}{F}_{x_{n+1},x_n,x_n}\left(t\right)=1.$$

By Remark 1, we obtain $li{m}_{n\to \infty }{F}_{f{x}_n,f{y}_n,f{y}_n}\left(t\right)\ge {lim}_{n\to \infty }{G}_{x_n,y_n,y_n}\left(t\right)=1$.

Case III. If $G_{x_n,y_n,y_n}\left(t\right)=F_{y_{n+1},y_n,y_n}\left(t\right)$, we also have that

$$\underset{n\to \infty }{lim}{F}_{f{x}_n,f{y}_n,f{y}_n}\left(t\right)\ge \underset{n\to \infty }{lim}{G}_{x_n,y_n,y_n}\left(t\right)=\underset{n\to \infty }{lim}{F}_{y_{n+1},y_n,y_n}\left(t\right)=1.$$

This completes the proof. □

Theorem 10. 

Let $(X,F,\ast )$ be a complete G-fuzzy metric space and $f:X⟶X$ be an orbitally continuous generalized fuzzy Meir–Keeler-type contraction with respect to $\delta \in \Delta$. Then f has a unique fixed point if and only if there is $x_0\in X$ such that ${\bigwedge }_{t>0}{F}_{f{x}_0,x_0,x_0}\left(t\right)>0$ and ${\bigwedge }_{t>0}{F}_{x_0,f{x}_0,f{x}_0}\left(t\right)>0$.

Proof. 

Suppose that f has a unique fixed point, then there is $x_0\in X$ such that $f{x}_0=x_0$. Thus, $F_{x_0,f{x}_0,f{x}_0}\left(t\right)=F_{x_0,x_0,x_0}\left(t\right)=1$ and $F_{f{x}_0,x_0,x_0}\left(t\right)=F_{x_0,x_0,x_0}\left(t\right)=1$ for each $t>0$. So ${\bigwedge }_{t>0}{F}_{x_0,f{x}_0,f{x}_0}\left(t\right)>0$ and ${\bigwedge }_{t>0}{F}_{f{x}_0,x_0,x_0}\left(t\right)>0$.

Conversely, suppose that f is an orbitally continuous generalized fuzzy Meir–Keeler contractive map with respect to $\delta \in \Delta$, and that there exists $x_0\in X$ such that ${\bigwedge }_{t>0}{F}_{f{x}_0,x_0,x_0}\left(t\right)$$>0$. We define an iterative sequence $\left\{x_n\right\}$ as follows:

$$x_n=f^n{x}_0(\forall n\ge 1).$$

Then it follows from Proposition 4 that

$$\underset{n\to \infty }{lim}{F}_{x_{n+1},x_n,x_n}\left(t\right)=1(\forall t>0).$$

(13)

From Remark 1, putting $x=x_{n+1}$, $y=z=x_n$, we have

$$\begin{array}{cc}\hfill F_{x_{n+1},x_n,x_n}\left(t\right)& >G_{x_n,x_{n-1},x_{n-1}}\left(t\right)\hfill \\ \hfill & =min\{F_{x_n,x_{n-1},x_{n-1}}\left(t\right),F_{x_{n+1},x_n,x_n}\left(t\right)\}\hfill \end{array}$$

(14)

Hence, $\{F_{x_{n+1},x_n,x_n}\left(t\right):n\in \mathbb{N}\}$ is an increasing sequence. Thus,

$$\underset{t>0}{\bigwedge }{F}_{x_{n+1},x_n,x_n}\left(t\right)\ge \underset{t>0}{\bigwedge }{F}_{x_n,x_{n-1},x_{n-1}}\left(t\right).$$

Let $b_n={\bigwedge }_{t>0}{F}_{x_{n+1},x_n,x_n}\left(t\right)$. Then $\left\{b_n\right\}$ is an increasing sequence and $b_n\le 1$. So there exists $q\in (0,1]$ such that

$$\underset{n\to \infty }{lim}{b}_n=q.$$

Suppose $q<1$. Since $\delta \in \Delta$, there exists $k\in \mathbb{N}$ such that

$$q+\frac{\delta \left(t\right)}{k}<1\mathrm{and}\delta \left(q+\frac{\delta \left(q\right)}{k}\right)-\frac{\delta \left(q\right)}{k}>0.$$

Since $\{b_n:n\in \mathbb{N}\}$ is increasing, there exists $n_0$ such that

$$q-\delta \left(q+\frac{\delta \left(q\right)}{k}\right)+\frac{\delta \left(q\right)}{k}<b_n\le q(\forall n\ge n_0).$$

In particular, pick $n_1=n_0+1$, we have

$$q-\delta \left(q+\frac{\delta \left(q\right)}{k}\right)+\frac{\delta \left(q\right)}{k}<b_{n_1}\le q(\forall n\ge n_0),$$

i.e., $F_{x_{n_1+1},x_{n_1},x_{n_1}}\left(t\right)>q-\delta \left(q+\frac{\delta \left(q\right)}{k}\right)+\frac{\delta \left(q\right)}{k}$ for all $t>0$.

On the other hand,

$$b_{n_1+1}=\underset{t>0}{\bigwedge }{F}_{x_{n_1+1},x_{n_1},x_{n_1}}\left(t\right)\le q.$$

By Proposition 1, there exists $t_0>0$ such that

$$F_{x_{n_1+1},x_{n_1},x_{n_1}}\left(t\right)\le q(\forall 0<t\le t_0).$$

Thus,

$$q-\delta \left(q+\frac{\delta \left(q\right)}{k}\right)+\frac{\delta \left(q\right)}{k}<F_{x_{n_1+1},x_{n_1},x_{n_1}}\left(t\right)\le q\le q+\frac{\delta \left(q\right)}{k}(\forall 0<t\le t_0).$$

Since

$$\begin{array}{cc}\hfill & G_{x_{n_1+1},x_{n_1},x_{n_1}}\left(t\right)\hfill \\ \hfill & =min\{F_{x_{n_1+1},x_{n_1},x_{n_1}}\left(t\right),F_{x_{n_1+2},x_{n_1+1},x_{n_1+1}}\left(t\right)\}\hfill \\ \hfill & =F_{x_{n_1+1},x_{n_1},x_{n_1}}\left(t\right),\hfill & \hfill (\mathrm{by}\mathrm{increasing}\mathrm{property})\end{array}$$

it follows that

$$q-\delta \left(q+\frac{\delta \left(q\right)}{k}\right)+\frac{\delta \left(q\right)}{k}<G_{x_{n_1+1},x_{n_1},x_{n_1}}\left(t\right)\le q\le q+\frac{\delta \left(q\right)}{k}(\forall 0<t\le t_0).$$

By Definition 11, we have $F_{x_{n_1+2},x_{n_1+1},x_{n_1+1}}\left(t\right)>q+\frac{\delta \left(q\right)}{k}$ for all $0<t\le t_0$. Therefore,

$$b_{n_1+1}=\underset{t>0}{\bigwedge }{F}_{x_{n_1+2},x_{n_1+1},x_{n_1+1}}\left(t\right)\ge q+\frac{\delta \left(q\right)}{k},$$

a contradiction. Hence $q=1$. That is to say,

$$\underset{n\to \infty }{lim}{b}_n=\underset{n\to \infty }{lim}\left(\underset{t>0}{\bigwedge }{F}_{x_{n+1},x_n,x_n}\left(t\right)\right)=1.$$

(15)

Making similar technique as in the proof of (15) with the condition ${\bigwedge }_{t>0}{F}_{x_0,f{x}_0,f{x}_0}\left(t\right)>0$, we have

$$\underset{n\to \infty }{lim}{c}_n=\underset{n\to \infty }{lim}\left(\underset{t>0}{\bigwedge }{F}_{x_n,x_{n+1},x_{n+1}}\left(t\right)\right)=1.$$

(16)

Next, we shall prove that $\left\{x_n\right\}$ is a Cauchy sequence in X.

If $\left\{x_n\right\}$ is not a Cauchy sequence, then there are $\eta \in (0,1)$, $t_0>0$ and sequences $\left\{p\right(n\left)\right\}$ and $\left\{q\right(n\left)\right\}$ such that for all $n\in \mathbb{N}$,

$$p\left(n\right)>q\left(n\right)>n,F_{x_{p\left(n\right)},x_{q\left(n\right)},x_{q\left(n\right)}}\left(t_0\right)\le 1-\eta ,F_{x_{p\left(n\right)-1},x_{q\left(n\right)},x_{q\left(n\right)}}\left(t_0\right)>1-\eta .$$

From (TFM-5), for $s\in (0,t_0)$, we have

$$\begin{array}{cc}\hfill 1-\eta & >F_{x_{p\left(n\right)},x_{q\left(n\right)},x_{q\left(n\right)}}\left(t_0\right)\hfill \\ \hfill & \ge F_{x_{p\left(n\right)},x_{p\left(n\right)-1},x_{p\left(n\right)-1}}\left(s\right)\ast F_{x_{p\left(n\right)-1},x_{q\left(n\right)},x_{q\left(n\right)}}(t_0-s).\hfill \end{array}$$

Taking limit as $s\to 0$, by the continuity of ∗ and by (TFM-6), it holds that

$$\begin{array}{cc}\hfill 1-\eta & \ge F_{x_{p\left(n\right)},x_{q\left(n\right)},x_{q\left(n\right)}}\left(t_0\right)\hfill \\ \hfill & \ge \underset{s\to 0}{lim}{F}_{x_{p\left(n\right)},x_{p\left(n\right)-1},x_{p\left(n\right)-1}}\left(s\right)\ast F_{x_{p\left(n\right)-1},x_{q\left(n\right)},x_{q\left(n\right)}}(t_0-s)\hfill \\ \hfill & =\underset{s\to 0}{lim}{F}_{x_{p\left(n\right)},x_{p\left(n\right)-1},x_{p\left(n\right)-1}}\left(s\right)\ast \underset{s\to 0}{lim}{F}_{x_{p\left(n\right)-1},x_{q\left(n\right)},x_{q\left(n\right)}}(t_0-s)\hfill \\ \hfill & =\left(\underset{t>0}{\bigwedge }{F}_{x_{p\left(n\right)},x_{p\left(n\right)-1},x_{p\left(n\right)-1}}\left(t\right)\right)\ast F_{x_{p\left(n\right)-1},x_{q\left(n\right)},x_{q\left(n\right)}}\left(t_0\right)\hfill \\ \hfill & \ge \left(\underset{t>0}{\bigwedge }{F}_{x_{p\left(n\right)},x_{p\left(n\right)-1},x_{p\left(n\right)-1}}\left(t\right)\right)\ast (1-\eta )\hfill \\ \hfill & =b_{p\left(n\right)-1}\ast (1-\eta ),\hfill \end{array}$$

i.e., $1-\eta \ge F_{x_{p\left(n\right)},x_{q\left(n\right)},x_{q\left(n\right)}}\left(t_0\right)\ge b_{p\left(n\right)-1}\ast (1-\eta )$. Since ${lim}_{n\to \infty }{b}_n=1$, we have

$$\underset{n\to \infty }{lim}{F}_{x_{p\left(n\right)},x_{q\left(n\right)},x_{q\left(n\right)}}\left(t_0\right)=1-\eta .$$

(17)

On the other hand, by Remark 1, we obtain

$$\begin{array}{cc}\hfill & F_{x_{p\left(n\right)+1},x_{q\left(n\right)+1},x_{q\left(n\right)+1}}\left(t_0\right)\hfill \\ \hfill & >G_{x_{p\left(n\right)},x_{q\left(n\right)},x_{q\left(n\right)}}\left(t_0\right)\hfill \\ \hfill & =min\{F_{x_{p\left(n\right)},x_{q\left(n\right)},x_{q\left(n\right)}}\left(t_0\right),F_{x_{p\left(n\right)+1},x_{p\left(n\right)},x_{p\left(n\right)}}\left(t_0\right),F_{x_{q\left(n\right)+1},x_{q\left(n\right)},x_{q\left(n\right)}}\left(t_0\right)\}\hfill \\ \hfill & =min\{F_{x_{p\left(n\right)},x_{q\left(n\right)},x_{q\left(n\right)}}\left(t_0\right),F_{x_{q\left(n\right)+1},x_{q\left(n\right)},x_{q\left(n\right)}}\left(t_0\right)\}.\hfill \end{array}$$

If $p\left(n\right)=q\left(n\right)+1$, then ${lim\; inf}_{n\to \infty }{F}_{x_{p\left(n\right)+1},x_{q\left(n\right)+1},x_{q\left(n\right)+1}}\left(t_0\right)\ge {lim}_{n\to \infty }{F}_{x_{p\left(n\right)},x_{q\left(n\right)},x_{q\left(n\right)}}\left(t_0\right)=1-\eta .$ If $p\left(n\right)>q\left(n\right)+1$, i.e., $p\left(n\right)-1\ge q\left(n\right)+1$, then

$$\begin{array}{cc}\hfill & \underset{n\to \infty }{lim\; inf}{F}_{x_{p\left(n\right)+1},x_{q\left(n\right)+1},x_{q\left(n\right)+1}}\left(t_0\right)\hfill \\ \hfill & \ge \underset{n\to \infty }{lim}{F}_{x_{p\left(n\right)},x_{q\left(n\right)},x_{q\left(n\right)}}\left(t_0\right)\wedge \underset{n\to \infty }{lim}{F}_{x_{q\left(n\right)+1},x_{q\left(n\right)},x_{q\left(n\right)}}\left(t_0\right)\hfill \\ \hfill & \ge (1-\eta )\wedge (1-\eta )=1-\eta .\hfill \end{array}$$

Thus, ${lim\; inf}_{n\to \infty }{F}_{x_{p\left(n\right)+1},x_{q\left(n\right)+1},x_{q\left(n\right)+1}}\left(t_0\right)\ge =1-\eta$ for all $p\left(n\right)>q\left(n\right)>n$. Again by (TFM-5), for $s\in (0,t_0/2)$, we have

$$\begin{array}{cc}\hfill & F_{x_{p\left(n\right)},x_{q\left(n\right)},x_{q\left(n\right)}}\left(t_0\right)\hfill \\ \hfill & \ge F_{x_{p\left(n\right)},x_{p\left(n\right)+1},x_{p\left(n\right)+1}}\left(s\right)\ast F_{x_{p\left(n\right)+1},x_{q\left(n\right)},x_{q\left(n\right)}}(t_0-s)\hfill \\ \hfill & \ge F_{x_{p\left(n\right)},x_{p\left(n\right)+1},x_{p\left(n\right)+1}}\left(s\right)\ast F_{x_{p\left(n\right)+1},x_{q\left(n\right)+1},x_{q\left(n\right)+1}}(t_0-2s)\ast F_{x_{q\left(n\right)+1},x_{q\left(n\right)},x_{q\left(n\right)}}\left(s\right).\hfill \end{array}$$

Taking limit as $s\to 0$, by the continuity of ∗ and by (TFM-6), it holds that

$$\begin{array}{cc}\hfill & F_{x_{p\left(n\right)},x_{q\left(n\right)},x_{q\left(n\right)}}\left(t_0\right)\hfill \\ \hfill & \ge \underset{s\to 0}{lim}{F}_{x_{p\left(n\right)},x_{p\left(n\right)+1},x_{p\left(n\right)+1}}\left(s\right)\ast F_{x_{p\left(n\right)+1},x_{q\left(n\right)+1},x_{q\left(n\right)+1}}(t_0-2s)\ast F_{x_{q\left(n\right)+1},x_{q\left(n\right)},x_{q\left(n\right)}}\left(s\right)\hfill \\ \hfill & =\underset{s\to 0}{lim}{F}_{x_{p\left(n\right)},x_{p\left(n\right)+1},x_{p\left(n\right)+1}}\left(s\right)\ast \underset{s\to 0}{lim}{F}_{x_{p\left(n\right)+1},x_{q\left(n\right)+1},x_{q\left(n\right)+1}}(t_0-2s)\ast \underset{s\to 0}{lim}{F}_{x_{q\left(n\right)+1},x_{q\left(n\right)},x_{q\left(n\right)}}\left(s\right)\hfill \\ \hfill & =\left(\underset{t>0}{\bigwedge }{F}_{x_{p\left(n\right)},x_{p\left(n\right)+1},x_{p\left(n\right)+1}}\left(t\right)\right)\ast F_{x_{p\left(n\right)+1},x_{q\left(n\right)+1},x_{q\left(n\right)+1}}\left(t_0\right)\ast \left(\underset{t>0}{\bigwedge }{F}_{x_{q\left(n\right)+1},x_{q\left(n\right)},x_{q\left(n\right)}}\left(t\right)\right)\hfill \\ \hfill & =c_{p\left(n\right)}\ast F_{x_{p\left(n\right)+1},x_{q\left(n\right)+1},x_{q\left(n\right)+1}}\left(t_0\right)\ast b_{q\left(n\right)}\hfill \\ \hfill & =F_{x_{p\left(n\right)+1},x_{q\left(n\right)+1},x_{q\left(n\right)+1}}\left(t_0\right)\ast (b_{q\left(n\right)}\ast c_{p\left(n\right)}).\hfill \end{array}$$

Since ${lim}_{n\to \infty }{b}_n=1$, ${lim}_{n\to \infty }{c}_n=1$ and ${lim}_{n\to \infty }{F}_{x_{p\left(n\right)},x_{q\left(n\right)},x_{q\left(n\right)}}\left(t_0\right)=1-\eta$, we have

$$\underset{n\to \infty }{lim\; sup}{F}_{x_{p\left(n\right)+1},x_{q\left(n\right)+1},x_{q\left(n\right)+1}}\left(t_0\right)\le \underset{n\to \infty }{lim}{F}_{x_{p\left(n\right)},x_{q\left(n\right)},x_{q\left(n\right)}}\left(t_0\right)=1-\eta .$$

Hence,

$$\underset{n\to \infty }{lim}{F}_{x_{p\left(n\right)+1},x_{q\left(n\right)+1},x_{q\left(n\right)+1}}\left(t_0\right)=1-\eta .$$

(18)

By (17), (18) and Lemma 5, we obtain $1-\eta =1$, which contradicts to $\eta >0$. Hence,

$$\underset{n,m\to \infty }{lim}{F}_{x_n,x_m,x_m}\left(t\right)=1(\forall t>0).$$

Therefore, it follows from Proposition 2 that $\left\{x_n\right\}$ is a Cauchy sequence in X.

Since $(X,F,\ast )$ is complete and $\left\{x_n\right\}$ is a Cauchy sequence in X, there exists $\overline{x}\in X$ such that

$$\underset{n\to \infty }{lim}{F}_{x_n,\overline{x},\overline{x}}\left(t\right)=\underset{n\to \infty }{lim}{F}_{x_n,x_n,\overline{x}}\left(t\right)=1(\forall t>0).$$

Next, we shall show that $\overline{x}$ is a fixed point of f.

Since f is orbitally continuous and ${lim}_{n\to \infty }{F}_{x_n,\overline{x},\overline{x}}\left(t\right)=1$ for all $t>0$, we obtain

$$li{m}_{n\to \infty }{F}_{x_{n+1},f\overline{x},f\overline{x}}\left(t\right)=1.$$

Thus, $\left\{x_{n+1}\right\}$ converges to $f\overline{x}$ with respect to ${\tau }_F$. By the uniqueness of the limit, we obtain $f\overline{x}=\overline{x}$.

Finally, we show that $\overline{x}$ is a unique fixed point of f. If y is another fixed point of f such that $f\overline{x}=\overline{x}\ne fy=y$. Then there exists $t_0>0$ such that $F_{y,\overline{x},\overline{x}}\left(t_0\right)\ne 1$, and so we obtain

$$G_{y,\overline{x},\overline{x}}\left(t_0\right)\le F_{y,\overline{x},\overline{x}}\left(t_0\right)<1.$$

Since f is a generalized Meir–Keeler-type contraction, we have

$$\begin{array}{cc}\hfill 1>F_{y,\overline{x},\overline{x}}\left(t_0\right)& =F_{fy,f\overline{x},f\overline{x}}\left(t_0\right)\hfill \\ \hfill & >G_{y,\overline{x},\overline{x}}\left(t_0\right)\hfill \\ \hfill & =min\{F_{y,\overline{x},\overline{x}}\left(t_0\right),F_{fy,y,y,}\left(t_0\right),F_{f\overline{x},\overline{x},\overline{x}}\left(t_0\right)\}\hfill \\ \hfill & =min\{F_{y,\overline{x},\overline{x}}\left(t_0\right),1,1,1\}\hfill \\ \hfill & =F_{y,\overline{x},\overline{x}}\left(t_0\right)\hfill \end{array}$$

a contraction. Thus, we find that $F_{y,\overline{x},\overline{x}}\left(t\right)=1$ for all $t>0$. So, by (TFM-2) we conclude that $\overline{x}=y$. Therefore, the fixed point of f is unique, and the proof is completed. □

Corollary 2. 

Let $(X,F,\ast )$ be a complete stationary G-fuzzy metric space and $f:X⟶X$ be a generalized fuzzy Meir–Keeler-type contraction with respect to $\delta \in \Delta$. Then f has a unique fixed point $\overline{x}$ and the sequence $\left\{f^{n}x\right\}$ converges to $\overline{x}$ for every $x\in X$.

Definition 12. 

Let $(X,F,\ast )$ be a G-fuzzy metric space and suppose that there exist $k\in \mathbb{N}$ and $\lambda \in [0,1)$ such that $F_{f^{k}x,f^{k}y,f^{k}z}\left(t\right)\ge \lambda$ for all $x,y,z\in X$ and $t>0$. A map $f:X⟶X$ is said to be a generalized fuzzy Meir–Keeler-type contraction with respect to $(k,\lambda ,\delta )$, where $\delta \in \Delta$, if the following condition holds:

$$\forall ϵ\in (\lambda ,1),ϵ-\delta \left(ϵ\right)<G_{x,y,z}\left(t\right)\le ϵ\Rightarrow F_{fx,fy,fz}\left(t\right)>ϵ,$$

(19)

where $G_{x,y,z}\left(t\right)=min\{F_{x,y,z}\left(t\right),F_{fx,x,x}\left(t\right),F_{fy,y,y}\left(t\right),F_{fz,z,z}\left(t\right)\}$ for all $x,y,z\in X$ and $t>0$.

Obviously, a generalized fuzzy Meir–Keeler-type contraction with respect to $\delta \in \Delta$ is just a generalized fuzzy Meir–Keeler-type contraction with respect to $(0,0,\delta )$.

Theorem 11. 

Let $(X,F,\ast )$ be a complete G-fuzzy metric space and $f:X⟶X$ be a generalized fuzzy Meir–Keeler-type contraction with respect to $(k,\lambda ,\delta )$ with $\lambda >0$. Then, f has a unique fixed point.

Proof. 

This proof is similar to that of Theorem 10. □

We finally present an example to illustrate Theorem 10.

Example 4. 

Let $X=[0,4]$ be with the Euclidean metric, and define $f:X⟶X$ by

$$fx=\left\{\begin{array}{cc}\frac{x}{2},\hfill & \mathrm{if}0\le x<2;\hfill \\ 1,\hfill & \mathrm{if}2\le x\le 4.\hfill \end{array}\right.$$

Let $F_{x,y,z}=\frac{1}{1+d(x,y)}\wedge \frac{1}{1+d(y,z)}\wedge \frac{1}{1+d(z,x)}$. Then $(X,F,·)$ is a complete stationary G-fuzzy metric space, where ∧ is the minimum t-norm.

Consider $\delta :(0,1]⟶(0,1]$ as follows:

$$\delta \left(t\right)=\left\{\begin{array}{cc}\frac{1}{6},\hfill & \mathrm{if}0\le t\le \frac{2}{3};\hfill \\ \frac{1}{n(n+2)},\hfill & \mathrm{if}\frac{n-1}{n}\le t<\frac{n}{n+1}\mathrm{and}n\ge 3.\hfill \end{array}\right.$$

Obviously, $\delta \in \Delta$.

Next, we show that f is a generalized fuzzy Meir–Keeler-type contraction with respect to $(2,1/2,\delta )$.

From the definition of f, for all $x,y,z\in X$, we have $f^{2}x,f^{2}y,f^{2}z\in [0,1/2)$, and thus

$$F_{f^{2}x,f^{2}y,f^{2}z}=\frac{1}{1+d(f^{2}x,f^{2}y)}\wedge \frac{1}{1+d(f^{2}y,f^{2}z)}\wedge \frac{1}{1+d(f^{2}z,f^{2}x)}>\frac{1}{1+\frac{1}{2}}=\frac{2}{3}.$$

We want to show that the following condition holds:

$$\forall ϵ\in (2/3,1),ϵ-\delta \left(ϵ\right)<G_{x,y,z}\le ϵ\Rightarrow F_{fx,fy,fz}>ϵ,$$

where $G(x,y,z)=min\{F_{x,y,z},F_{fx,x,x},F_{fy,y,y},F_{fz,z,z}\}$ for all $x,y,z\in X$ and $t>0$.

Now let $ϵ\in (2/3,1)$, then $\frac{n-1}{n}\le ϵ<\frac{n}{n+1}$ for some $n\ge 3$, in this case, $\delta \left(ϵ\right)=\frac{1}{n(n+2)}$.

If $ϵ-\delta \left(ϵ\right)<G_{x,y,z}\le ϵ$, then

$$\frac{n-1}{n}-\frac{1}{n(n+2)}\le ϵ-\delta \left(ϵ\right)<G_{x,y,z}\le ϵ<\frac{n}{n+1}.$$

Without loss of generality and by symmetry, take $x\le y\le z$. We have the following cases:

Case I: $0\le x\le y\le z<2$. Here we have

$$\begin{array}{cc}\hfill G_{x,y,z}& =min\left\{\frac{1}{1+d(z,x)},\frac{1}{1+d(fx,x)},\frac{1}{1+d(fy,y)},\frac{1}{1+d(fz,z)}\right\}\hfill \\ \hfill & =min\left\{\frac{1}{1+d(z,x)},\frac{1}{1+d(fz,z)}\right\},\hfill \end{array}$$

which implies that $d(z,x)<\frac{2}{n}$. Hence

$$F_{fx,fy,fz}=\frac{1}{1+d(fz,fx)}=\frac{1}{1+\frac{1}{2}d(z,x)}>\frac{n}{1+\frac{1}{n}}=\frac{1}{n+1}>ϵ.$$

Case II: $0\le x\le y<2$ and $2\le z\le 4$. Here we have

$$\begin{array}{cc}\hfill G_{x,y,z}& =min\left\{\frac{1}{1+d(z,x)},\frac{1}{1+d(fx,x)},\frac{1}{1+d(fy,y)},\frac{1}{1+d(fz,z)}\right\}\hfill \\ \hfill & =min\left\{\frac{1}{1+d(z,x)},\frac{1}{1+d(fz,z)}\right\},\hfill \end{array}$$

which implies that $d(z,x)<\frac{2}{n}$. Hence

$$F_{fx,fy,fz}=\frac{1}{1+d(fz,fx)}>\frac{1}{1+\frac{1}{2}d(z,x)}>\frac{1}{1+\frac{2}{n}}=\frac{n}{n+2}>ϵ.$$

Case III: $0\le x<2$ and $2\le y\le z\le 4$. Here we have

$$G_{x,y,z}=min\left\{\frac{1}{1+d(z,x)},\frac{1}{1+d(fx,x)},\frac{1}{1+d(fy,y)},\frac{1}{1+d(fz,z)}\right\}=\frac{1}{1+d(fz,z)}=\frac{1}{z},$$

which implies that $z<\frac{n(n+2)}{n^2+n-1}$, but it is impossible whenever $n\ge 3$.

Case IV: $2\le x\le y\le z\le 4$. Here we have

$$G_{(x,y,z}=min\left\{\frac{1}{1+d(z,x)},\frac{1}{1+d(fx,x)},\frac{1}{1+d(fy,y)},\frac{1}{1+d(fz,z)}\right\}=\frac{1}{1+d(fz,z)}=\frac{1}{z},$$

which implies that $z<\frac{n(n+2)}{n^2+n-1}$, but it is impossible whenever $n\ge 3$.

In each case, f is a generalized fuzzy Meir–Keeler-type contraction with respect to $(2,1/2,\delta )$. Also, the mapping f is F-continuous on X, and clearly it is orbitally F-continuous. All the hypotheses of Theorem 11 are satisfied, and so f has a fixed point: $x=0$.

5. Conclusions

In the present work, by means of quasi G-metric families, we give the characterization of G-fuzzy metric space. Also, we introduce the notion of generalized fuzzy Meir–Keeler-type contractive mappings in G-fuzzy metric spaces and present some fixed-point theorems on the mappings. In future work, we will consider the following problems:

• Based on the idea of KM-fuzzy metric spaces, Mardones-Pérez and de Prada [31] study the degree to which some topological-type properties of fuzzy metric spaces are fulfilled. Moreover, Zhong and Šostak [32] introduced the definition of fuzzy k-pseudo metrics and constructed its induced fuzzifying structures, such as fuzzifying topologies, fuzzifying neighborhood systems, fuzzifying uniformity, and fuzzifying closure operators. However, the topology induced by tripled fuzzy metrics by Tian [25] is crisp; thus, it lost the characteristics of fuzzy mathematics. Therefore, it motivates us to consider the fuzzifying structures constructed by tripled fuzzy metrics and explore the relationships.
• As a generalization of GV-fuzzy metrics, KM-fuzzy metrics, Morsi fuzzy metrics, and Shi’s fuzzy metrics, Shi [33] proposed the notion of $(L,M)$-fuzzy metric spaces and induced an $(L,M)$-fuzzy topology. Therefore, it will be interesting to generalize the G-metrics to $(L,M)$-fuzzy case and give research to the characterizations of this type of metrics.