Common fixed point theorems for fuzzy mappings in G-metric spaces

Abstract

In this paper, we introduce the concept of Hausdorff G-metric in the space of fuzzy sets induced by the metric $d_G$ and obtain some results on Hausdorff G-metric. We also prove common fixed point theorems for a family of fuzzy self-mappings in the space of fuzzy sets on a complete G-metric space.

MSC:47H10, 54H25.

1 Introduction and preliminaries

Fixed point theory is very important in mathematics and has applications in many fields. A number of authors established fixed point theorems for various mappings in different metric spaces. In 2006, Mustafa and Sims [1] introduced the G-metric space as a generalization of metric spaces. We now recall some definitions and results in G-metric spaces in [1].

Definition 1.1 Let X be a nonempty set, and let $G:X\times X\times X\to {\mathbb{R}}^{+}$ be a function satisfying:

(G 1) $G(x,y,z)=0$ if $x=y=z$,

(G 2) $0<G(x,x,y)$ for all $x,y\in X$ with $x\ne y$,

(G 3) $G(x,x,y)\le G(x,y,z)$ for all $x,y,z\in X$, with $y\ne z$,

(G 4) $G(x,y,z)=G(x,z,y)=G(y,z,x)=\cdots$ (symmetry in all three variables),

(G 5) $G(x,y,z)\le G(x,a,a)+G(a,y,z)$ for all $x,y,z,a\in X$ (rectangle inequality).

Then the function G is called a generalized metric or, more specifically, a G-metric on X, and the pair $(X,G)$ is a G-metric space.

Lemma 1.1 Every G-metric space $(X,G)$ defines a metric space $(X,d_G)$ by

$$d_G(x,y)=G(x,y,y)+G(x,x,y),\mathit{\text{for all }}x,y\in X.$$

Definition 1.2 Let $(X,G)$ be a G-metric space. The sequence $\{x_n\}$ in X is said to be

• (i) G-convergent to x if for any $\epsilon >0$, there exists $x\in X$ and $N\in \mathbb{N}$ such that $G(x,x_n,x_m)<\epsilon$, for all $n,m\ge N$.

• (ii) G-Cauchy if for any $\epsilon >0$, there exists $N\in \mathbb{N}$ such that $G(x_n,x_m,x_l)<\epsilon$, for all $n,m,l\ge N$.

Lemma 1.2 Let $(X,G)$ be a G-metric space, then for a sequence $\{x_n\}$ in X and point $x\in X$ the following are equivalent:

• (i) $\{x_n\}$ is G-convergent to x.

• (ii) $G(x_n,x_n,x)\to 0$ as $n\to +\mathrm{\infty }$.

• (iii) $G(x_n,x,x)\to 0$ as $n\to +\mathrm{\infty }$.

• (iv) $G(x_m,x_n,x)\to 0$ as $m,n\to +\mathrm{\infty }$.

Lemma 1.3 Let $(X,G)$ be a G-metric space, then for a sequence $\{x_n\}$ in X, the following are equivalent:

• (i) The sequence $\{x_n\}$ is G-Cauchy.

• (ii) For any $\epsilon >0$, there exists $N\in \mathbb{N}$ such that $G(x_n,x_m,x_m)<\epsilon$, for all $n,m\ge N$.

• (iii) $\{x_n\}$ is a Cauchy sequence in the metric space $(X,d_G)$.

Definition 1.3 A G-metric space $(X,G)$ is said to be G-complete if every G-Cauchy sequence in $(X,G)$ is G-convergent in $(X,G)$.

Lemma 1.4 A G-metric space $(X,G)$ is G-complete if and only if $(X,d_G)$ is a complete metric space.

Based on the notion of G-metric spaces, many authors obtained fixed point theorems for mappings satisfying different contractive-type conditions in G-metric spaces (see, e.g., [2–7]) and in partially ordered G-metric spaces (see, e.g., [8–13]). Recently, Kaewcharoen and Kaewkhao [14] introduced the following concepts. Let X be a G-metric space and $CB(X)$ the family of all nonempty closed bounded subsets of X. Let $H(\cdot ,\cdot ,\cdot )$ be the Hausdorff G-distance on $CB(X)$, i.e.,

$$H_G(A,B,C)=max\{\underset{x\in A}{sup}G(x,B,C),\underset{x\in B}{sup}G(x,C,A),\underset{x\in C}{sup}G(x,A,B)\},$$

where

$$\begin{array}{c}G(x,B,C)=d_G(x,B)+d_G(B,C)+d_G(x,C),\hfill \\ d_G(x,B)=\underset{y\in B}{inf}{d}_G(x,y),\hfill \\ d_G(A,B)=\underset{x\in A,y\in B}{inf}{d}_G(x,y).\hfill \end{array}$$

Kaewcharoen and Kaewkhao [14] and Tahat et al. [15] obtained some common fixed point theorems for single-valued and multi-valued mappings in G-metric spaces.

The existence of fixed points of fuzzy mappings has been an active area of research interest since Heilpern [16] introduced the concept of fuzzy mappings in 1981. Many results have appeared related to fixed points for fuzzy mappings in ordinary metric spaces (see, e.g., [17–22]). Qiu and Shu [23, 24] proved some fixed point theorems for fuzzy self-mappings in ordinary metric spaces. However, there are very few results on fuzzy self-mappings in G-metric spaces. The purpose of this paper is to introduce the notion of Hausdorff G-metric in the space of fuzzy sets which extends the Hausdorff G-distance in [14]. We also establish common fixed point theorems for a family of fuzzy self-mappings in the space of fuzzy sets on a complete G-metric space.

2 A Hausdorff G-metric in the space of fuzzy sets

Let $(X,d_G)$ be a metric space, a fuzzy set in X is a function with domain X and values in $I=[0,1]$. If μ is a fuzzy set and $x\in X$, then the function value $\mu (x)$ is called the grade of membership of x in μ.

The α-level set of μ, denoted by ${[\mu ]}_{\alpha }$, is defined as

$$\begin{array}{c}{[\mu ]}_{\alpha }=\{x:\mu (x)\ge \alpha \},\text{if }\alpha \in (0,1],\hfill \\ {}_0=\overline{\{x:\mu (x)>0\}},\hfill \end{array}$$

where $\overline{B}$ is the closure of the non-fuzzy set B.

Let $C(X)$ be the family of all nonempty compact subsets of X. Denote by $\mathcal{C}(X)$ the totality of fuzzy sets which satisfy that for each $\alpha \in I$, ${[\mu ]}_{\alpha }\in C(X)$. Let ${\mu }_1,{\mu }_2\in \mathcal{C}(X)$, then ${\mu }_1$ is said to be more accurate than ${\mu }_2$, denoted by ${\mu }_1\subset {\mu }_2$, if and only if ${\mu }_1(x)\le {\mu }_2(x)$ for each $x\in X$. ${\mu }_1={\mu }_2$ if and only if ${\mu }_1\subset {\mu }_2$ and ${\mu }_2\subset {\mu }_1$.

Let ${\mu }_1,{\mu }_2\in \mathcal{C}(X)$, define

$$\begin{array}{rcl}{D}_{\mathrm{\infty }}({\mu }_1,{\mu }_2)& =& \underset{0\le \alpha \le 1}{sup}H({[{\mu }_1]}_{\alpha },{[{\mu }_2]}_{\alpha })\\ =& \underset{0\le \alpha \le 1}{sup}max\{\underset{x\in {[{\mu }_1]}_{\alpha }}{sup}{d}_G(x,{[{\mu }_2]}_{\alpha }),\underset{y\in {[{\mu }_2]}_{\alpha }}{sup}{d}_G(y,{[{\mu }_1]}_{\alpha })\}.\end{array}$$

Lemma 2.1 [23]

The metric space $(\mathcal{C}(X),D_{\mathrm{\infty }})$ is complete provided $(X,d_G)$ is complete.

For ${\mu }_1,{\mu }_2,{\mu }_3\in \mathcal{C}(X)$, $\alpha \in I$, we define:

Proposition 2.1 If $A,B\in C(X)$ and $x\in A$, then there exists $y\in B$ such that

$$2[G(x,y,y)+G(y,x,x)]\le H_G(A,B,B).$$

Proof For $x\in A$, there exists $y\in B$ such that

$$d_G(x,y)=d_G(x,B)=\frac{1}{2}G(x,B,B),$$

it follows that

$$2[G(x,y,y)+G(y,x,x)]=G(x,B,B)\le H_G(A,B,B).$$

 □

Proposition 2.2 If $A,B\in C(X)$ and $A_1\subseteq A$, then there exists $B_1\in C(X)$ such that $B_1\subseteq B$ and

$$H_G(A_1,B_1,B_1)\le H_G(A,B,B).$$

Proof Let $C=\{y:\text{ there exists }x\in A_1,\text{ such that }2(G(x,y,y)+G(y,x,x))\le H_G(A,B,B)\}$ and let $B_1=C\cap B$. For any $x\in A_1\subseteq A$ and $B\in C(X)$, Proposition 2.1 implies that $B_1$ is nonempty. Moreover, for any $x\in A_1$, there exists $y\in B_1$ such that $2[G(x,y,y)+G(y,x,x)]\le H_G(A,B,B)$. It follows that

$$\begin{array}{rcl}G(A_1,B_1,B_1)& =& \underset{x\in A_1}{sup}G(x,B_1,B_1)=\underset{x\in A_1}{sup}2d_G(x,B_1)\\ =& 2\underset{x\in A_1}{sup}\underset{y\in B_1}{inf}[G(x,y,y)+G(y,x,x)]\le H_G(A,B,B).\end{array}$$

(1)

On the other hand, for any $y\in B_1$, there exists $x\in A_1$ such that $2[G(x,y,y)+G(y,x,x)]\le H_G(A,B,B)$. Hence,

$$\begin{array}{rcl}G(B_1,B_1,A_1)& =& \underset{y\in B_1}{sup}G(y,B_1,A_1)=\underset{y\in B_1}{sup}[d_G(y,A_1)]+\underset{y\in B_1}{sup}[d_G(y,B_1)]+d_G(A_1,B_1)\\ =& \underset{y\in B_1}{sup}\underset{x\in A_1}{inf}[G(x,y,y)+G(y,x,x)]+0+\underset{x\in A_1,y\in B_1}{inf}[G(x,y,y)+G(y,x,x)]\\ \le & \underset{y\in B_1}{sup}\underset{x\in A_1}{inf}[G(x,y,y)+G(y,x,x)]+\underset{y\in B_1}{sup}\underset{x\in A_1}{inf}[G(x,y,y)+G(y,x,x)]\\ =& \underset{y\in B_1}{sup}\underset{x\in A_1}{inf}2[G(x,y,y)+G(y,x,x)]\le H_G(A,B,B).\end{array}$$

(2)

From (1) and (2), we have

$$H_G(A_1,B_1,B_1)\le H_G(A,B,B).$$

Finally, we can conclude that $B_1\in C(X)$ from the closeness of C and the compactness of B. □

Proposition 2.3 Let ${\mu }_1,{\mu }_2\in \mathcal{C}(X)$ and ${\mu }_3\subset {\mu }_1$, then there exists ${\mu }_4\in \mathcal{C}(X)$ such that ${\mu }_4\subset {\mu }_2$ and

$$D_{G,\mathrm{\infty }}({\mu }_3,{\mu }_4,{\mu }_4)\le D_{G,\mathrm{\infty }}({\mu }_1,{\mu }_2,{\mu }_2).$$

Proof Let $\alpha \in I$, by ${\mu }_3\subset {\mu }_1$, we have ${[{\mu }_3]}_{\alpha }\subseteq {[{\mu }_1]}_{\alpha }$. Let

we can get that $C_{\alpha }=D_{\alpha }$. Let $B_{\alpha }=D_{\alpha }\cap {[{\mu }_2]}_{\alpha }$, then $B_{\alpha }$ is nonempty compact and $B_{\alpha }\subseteq B_{\beta }$, for $0\le \beta \le \alpha \le 1$. From the proof of Proposition 2.2, we have

$$H_G({[{\mu }_3]}_{\alpha },B_{\alpha },B_{\alpha })\le D_{G,\mathrm{\infty }}({\mu }_1,{\mu }_2,{\mu }_2).$$

Similar to the proof of Theorem 3 in [23], we can conclude that there exists a fuzzy set ${\mu }_4$ such that ${[{\mu }_4]}_{\alpha }=B_{\alpha }$ for $\alpha \in I$. By the compactness of $B_{\alpha }$, we have ${\mu }_4\in \mathcal{C}(X)$. Therefore,

$$D_{G,\mathrm{\infty }}({\mu }_3,{\mu }_4,{\mu }_4)\le D_{G,\mathrm{\infty }}({\mu }_1,{\mu }_2,{\mu }_2).$$

 □

Proposition 2.4 Let X be a nonempty set. For any ${\mu }_1,{\mu }_2,{\mu }_3\in \mathcal{C}(X)$, the following properties hold:

• (i) $D_{G,\mathrm{\infty }}({\mu }_1,{\mu }_2,{\mu }_3)=0$ if and only if ${\mu }_1={\mu }_2={\mu }_3$,

• (ii) $0<D_{G,\mathrm{\infty }}({\mu }_1,{\mu }_1,{\mu }_2)$ for all ${\mu }_1,{\mu }_2\in \mathcal{C}(X)$ with ${\mu }_1\ne {\mu }_2$,

• (iii) $D_{G,\mathrm{\infty }}({\mu }_1,{\mu }_1,{\mu }_2)\le D_{G,\mathrm{\infty }}({\mu }_1,{\mu }_2,{\mu }_3)$ for all ${\mu }_1,{\mu }_2,{\mu }_3\in \mathcal{C}(X)$ with ${\mu }_2\ne {\mu }_3$,

• (iv) $D_{G,\mathrm{\infty }}({\mu }_1,{\mu }_2,{\mu }_3)=D_{G,\mathrm{\infty }}({\mu }_1,{\mu }_3,{\mu }_2)=D_{G,\mathrm{\infty }}({\mu }_2,{\mu }_1,{\mu }_3)=\cdots$ (symmetry in all three variables),

• (v) $D_{G,\mathrm{\infty }}({\mu }_1,{\mu }_2,{\mu }_3)\le D_{G,\mathrm{\infty }}({\mu }_1,\mu ,\mu )+D_{G,\mathrm{\infty }}(\mu ,{\mu }_2,{\mu }_3)$.

Proof The properties (i), (ii) and (iv) are readily derived from the definition of $D_{G,\mathrm{\infty }}$.

First, we prove the property (iii).

For any $\alpha \in I$ and $x\in {[{\mu }_1]}_{\alpha }$, $y\in {[{\mu }_2]}_{\alpha }$ and $z\in {[{\mu }_3]}_{\alpha }$, we have

$$d_G(x,y)-d_G(x,z)-d_G(z,y)\le 0,$$

it follows that

$$\begin{array}{c}{d}_G(x,y)-d_G(x,{[{\mu }_3]}_{\alpha })-d_G({[{\mu }_2]}_{\alpha },{[{\mu }_3]}_{\alpha })\hfill \\ \le \underset{y\in {[{\mu }_2]}_{\alpha }}{sup}{d}_G(x,y)-\underset{z\in {[{\mu }_3]}_{\alpha }}{inf}{d}_G(x,z)-\underset{y\in {[{\mu }_2]}_{\alpha },z\in {[{\mu }_3]}_{\alpha }}{inf}{d}_G(z,y)\hfill \\ =\underset{y\in {[{\mu }_2]}_{\alpha },z\in {[{\mu }_3]}_{\alpha }}{sup}[d_G(x,y)-d_G(x,z)-d_G(z,y)]\le 0.\hfill \end{array}$$

This implies that

$$\underset{x\in {[{\mu }_1]}_{\alpha },y\in {[{\mu }_2]}_{\alpha }}{inf}{d}_G(x,y)-\underset{x\in {[{\mu }_1]}_{\alpha }}{sup}{d}_G(x,{[{\mu }_3]}_{\alpha })\le d_G({[{\mu }_2]}_{\alpha },{[{\mu }_3]}_{\alpha }).$$

Then,

$$d_G({[{\mu }_1]}_{\alpha },{[{\mu }_2]}_{\alpha })\le \underset{x\in {[{\mu }_1]}_{\alpha }}{sup}{d}_G(x,{[{\mu }_3]}_{\alpha })+d_G({[{\mu }_2]}_{\alpha },{[{\mu }_3]}_{\alpha }).$$

Hence,

(3)

Similarly, we can prove that

$$G({[{\mu }_2]}_{\alpha },{[{\mu }_1]}_{\alpha },{[{\mu }_1]}_{\alpha })\le G({[{\mu }_2]}_{\alpha },{[{\mu }_1]}_{\alpha },{[{\mu }_3]}_{\alpha }).$$

(4)

By (3) and (4), we have

Now, we prove the property (v).

For any $\alpha \in I$ and $x\in {[{\mu }_2]}_{\alpha }$, $y\in {[\mu ]}_{\alpha }$, we have

$$d_G(x,{[{\mu }_1]}_{\alpha })\le d_G(x,y)+d_G(y,{[{\mu }_1]}_{\alpha }),$$

it follows that

$$\underset{x\in {[{\mu }_2]}_{\alpha }}{sup}{d}_G(x,{[{\mu }_1]}_{\alpha })\le \underset{x\in {[{\mu }_2]}_{\alpha }}{sup}{d}_G(x,{[\mu ]}_{\alpha })+\underset{y\in {[\mu ]}_{\alpha }}{sup}{d}_G(y,{[{\mu }_1]}_{\alpha }).$$

(5)

From (5) and

$$d_G({[{\mu }_1]}_{\alpha },{[{\mu }_3]}_{\alpha })\le d_G({[\mu ]}_{\alpha },{[{\mu }_3]}_{\alpha })+d_G({[\mu ]}_{\alpha },{[{\mu }_1]}_{\alpha }),$$

we have

$$\begin{array}{rcl}{G}_{\alpha }({\mu }_2,{\mu }_3,{\mu }_1)& =& G({[{\mu }_2]}_{\alpha },{[{\mu }_3]}_{\alpha },{[{\mu }_1]}_{\alpha })\\ =& \underset{x\in {[{\mu }_2]}_{\alpha }}{sup}[d_G(x,{[{\mu }_1]}_{\alpha })+d_G(x,{[{\mu }_3]}_{\alpha })+d_G({[{\mu }_1]}_{\alpha },{[{\mu }_3]}_{\alpha })]\\ \le & \underset{x\in {[{\mu }_2]}_{\alpha }}{sup}[d_G(x,{[\mu ]}_{\alpha })+d_G(x,{[{\mu }_3]}_{\alpha })+d_G({[\mu ]}_{\alpha },{[{\mu }_3]}_{\alpha })]\\ +\underset{y\in {[\mu ]}_{\alpha }}{sup}[d_G(y,{[{\mu }_1]}_{\alpha })+d_G(y,{[\mu ]}_{\alpha })+d_G({[\mu ]}_{\alpha },{[{\mu }_1]}_{\alpha })]\\ =& G_{\alpha }({\mu }_2,{\mu }_3,\mu )+G_{\alpha }(\mu ,\mu ,{\mu }_1).\end{array}$$

(6)

Similarly, we can obtain that

(7)

(8)

By (6), (7) and (8), we have

$$\begin{array}{rcl}{D}_{G,\mathrm{\infty }}({\mu }_1,{\mu }_2,{\mu }_3)& =& \underset{0\le \alpha \le 1}{sup}max\{G_{\alpha }({\mu }_2,{\mu }_3,{\mu }_1),G_{\alpha }({\mu }_3,{\mu }_2,{\mu }_1),G_{\alpha }({\mu }_1,{\mu }_2,{\mu }_3)\}\\ \le & \underset{0\le \alpha \le 1}{sup}max\{G_{\alpha }({\mu }_2,{\mu }_3,\mu ),G_{\alpha }({\mu }_3,{\mu }_2,\mu ),G_{\alpha }(\mu ,{\mu }_2,{\mu }_3)\}\\ +\underset{0\le \alpha \le 1}{sup}max\{G_{\alpha }(\mu ,\mu ,{\mu }_1),G_{\alpha }(\mu ,\mu ,{\mu }_1),G_{\alpha }({\mu }_1,\mu ,\mu )\}\\ =& D_{G,\mathrm{\infty }}(\mu ,{\mu }_2,{\mu }_3)+D_{G,\mathrm{\infty }}({\mu }_1,\mu ,\mu ).\end{array}$$

 □

Remark 2.1 Proposition 2.4 implies that $D_{G,\mathrm{\infty }}$ is a G-metric in $\mathcal{C}(X)$, or more specially a Hausdorff G-metric in $\mathcal{C}(X)$.

Definition 2.1 Let $(\mathcal{C}(X),D_{G,\mathrm{\infty }})$ be a metric space. The sequence $\{{\mu }_n\}$ in $\mathcal{C}(X)$ is said to be

• (i) $D_{G,\mathrm{\infty }}$-convergent to μ if for every $\epsilon >0$, there exists $\mu \in \mathcal{C}(X)$ and $N\in \mathbb{N}$ such that $D_{G,\mathrm{\infty }}(\mu ,{\mu }_n,{\mu }_m)<\epsilon$ for all $n,m\ge N$,

• (ii) $D_{G,\mathrm{\infty }}$-Cauchy if for every $\epsilon >0$, there exists $N\in \mathbb{N}$ such that $D_{G,\mathrm{\infty }}({\mu }_n,{\mu }_m,{\mu }_l)<\epsilon$ for all $n,m,l\ge N$.

Proposition 2.5 Let $(\mathcal{C}(X),D_{G,\mathrm{\infty }})$ be a metric space, then for a sequence $\{{\mu }_n\}\subset \mathcal{C}(X)$ and $\mu \in \mathcal{C}(X)$, the following are equivalent:

• (i) $\{{\mu }_n\}$ is $D_{G,\mathrm{\infty }}$-convergent to μ.

• (ii) $D_{\mathrm{\infty }}(\mu ,{\mu }_n)\to 0$ as $n\to +\mathrm{\infty }$.

• (iii) $D_{G,\mathrm{\infty }}({\mu }_n,{\mu }_n,\mu )\to 0$ as $n\to +\mathrm{\infty }$.

• (iv) $D_{G,\mathrm{\infty }}(\mu ,\mu ,{\mu }_n)\to 0$ as $n\to +\mathrm{\infty }$.

Proof Since $D_{G,\mathrm{\infty }}$ is a G-metric, Lemma 1.2 implies that (i), (iii) and (iv) are equivalent. Now, we prove that (ii) is also an equivalent condition.

“(i) ⟹ (ii)” Suppose $D_{G,\mathrm{\infty }}(\mu ,{\mu }_n,{\mu }_m)\to 0$ as $n,m\to +\mathrm{\infty }$, then

$$G_{\mathrm{\infty }}(\mu ,{\mu }_n,{\mu }_m)=\underset{0\le \alpha \le 1}{sup}\underset{x\in {[\mu ]}_{\alpha }}{sup}[d_G(x,{[{\mu }_n]}_{\alpha })+d_G(x,{[{\mu }_m]}_{\alpha })+d_G({[{\mu }_n]}_{\alpha },{[{\mu }_m]}_{\alpha })]\to 0$$

and

$$G_{\mathrm{\infty }}({\mu }_n,\mu ,{\mu }_m)=\underset{0\le \alpha \le 1}{sup}\underset{x\in {[{\mu }_n]}_{\alpha }}{sup}[d_G(x,{[\mu ]}_{\alpha })+d_G(x,{[{\mu }_m]}_{\alpha })+d_G({[\mu ]}_{\alpha },{[{\mu }_m]}_{\alpha })]\to 0.$$

Thus, for any $\alpha \in I$,

$$\underset{x\in {[\mu ]}_{\alpha }}{sup}\left[d_G(x,{[{\mu }_n]}_{\alpha })\right]\to 0\text{as }n\to +\mathrm{\infty },$$

(9)

and

$$\underset{x\in {[{\mu }_n]}_{\alpha }}{sup}\left[d_G(x,{[\mu ]}_{\alpha })\right]\to 0\text{as }n\to +\mathrm{\infty }.$$

(10)

It follows that

$$D_{\mathrm{\infty }}(\mu ,{\mu }_n)=\underset{0\le \alpha \le 1}{sup}H({[\mu ]}_{\alpha },{[{\mu }_n]}_{\alpha })\to 0\text{as }n\to +\mathrm{\infty }.$$

“(ii) ⟹ (i)” Suppose $D_{\mathrm{\infty }}(\mu ,{\mu }_n)\to 0$ as $n\to +\mathrm{\infty }$, then (9) and (10) hold.

Moreover, $0\le d_G({[{\mu }_n]}_{\alpha },{[\mu ]}_{\alpha })\le {sup}_{x\in {[{\mu }_n]}_{\alpha }}[d_G(x,{[\mu ]}_{\alpha })]$ implies that as $n\to +\mathrm{\infty }$,

$$d_G({[{\mu }_n]}_{\alpha },{[\mu ]}_{\alpha })\to 0.$$

(11)

From (9), (10) and (11), we have as $n\to +\mathrm{\infty }$,

$$\begin{array}{c}{D}_{G,\mathrm{\infty }}({\mu }_n,{\mu }_n,\mu )\hfill \\ =\underset{0\le \alpha \le 1}{sup}max\{G({[{\mu }_n]}_{\alpha },{[{\mu }_n]}_{\alpha },{[\mu ]}_{\alpha }),G({[\mu ]}_{\alpha },{[{\mu }_n]}_{\alpha },{[{\mu }_n]}_{\alpha })\}\hfill \\ =\underset{0\le \alpha \le 1}{sup}max\{\underset{x\in {[{\mu }_n]}_{\alpha }}{sup}[d_G(x,{[\mu ]}_{\alpha })+d_G({[{\mu }_n]}_{\alpha },{[\mu ]}_{\alpha })],\underset{x\in {[\mu ]}_{\alpha }}{sup}2d_G(x,{[{\mu }_n]}_{\alpha })\}\to 0.\hfill \end{array}$$

Thus, from

$$\begin{array}{rcl}0& \le & D_{G,\mathrm{\infty }}(\mu ,{\mu }_n,{\mu }_m)=D_{G,\mathrm{\infty }}({\mu }_n,\mu ,{\mu }_m)\le D_{G,\mathrm{\infty }}({\mu }_n,\mu ,\mu )+D_{G,\mathrm{\infty }}(\mu ,\mu ,{\mu }_m)\\ \le & D_{G,\mathrm{\infty }}(\mu ,{\mu }_n,{\mu }_n)+D_{G,\mathrm{\infty }}({\mu }_n,\mu ,{\mu }_n)+D_{G,\mathrm{\infty }}(\mu ,{\mu }_m,{\mu }_m)+D_{G,\mathrm{\infty }}({\mu }_m,\mu ,{\mu }_m)\\ =& 2D_{G,\mathrm{\infty }}({\mu }_n,{\mu }_n,\mu )+2D_{G,\mathrm{\infty }}({\mu }_m,{\mu }_m,\mu ),\end{array}$$

we can conclude that

$$D_{G,\mathrm{\infty }}(\mu ,{\mu }_n,{\mu }_m)\to 0\text{as }n,m\to +\mathrm{\infty }.$$

 □

Proposition 2.6 Let $(\mathcal{C}(X),D_{G,\mathrm{\infty }})$ be a metric space and $\{{\mu }_n\}$ a sequence in $\mathcal{C}(X)$, then the following are equivalent:

• (i) The sequence $\{{\mu }_n\}$ is $D_{G,\mathrm{\infty }}$-Cauchy.

• (ii) For every $\epsilon >0$, there exists $N\in \mathbb{N}$ such that $D_{G,\mathrm{\infty }}({\mu }_n,{\mu }_m,{\mu }_m)<\epsilon$ for all $n,m>N$.

• (iii) $\{{\mu }_n\}$ is a Cauchy sequence in the metric space $(\mathcal{C}(X),D_{\mathrm{\infty }})$.

Proof “(i) ⟺ (ii)” is evidence.

“(ii) ⟹ (iii)” Suppose that for every $\epsilon >0$, there exists $N\in \mathbb{N}$ such that $D_{G,\mathrm{\infty }}({\mu }_n,{\mu }_m,{\mu }_m)<\epsilon$, for all $n,m>N$, then as $n,m\to +\mathrm{\infty }$,

$$G_{\mathrm{\infty }}({\mu }_n,{\mu }_m,{\mu }_m)\to 0$$

(12)

and

$$G_{\mathrm{\infty }}({\mu }_m,{\mu }_m,{\mu }_n)\to 0.$$

(13)

It follows that

$$\underset{0\le \alpha \le 1}{sup}\underset{x\in {[{\mu }_n]}_{\alpha }}{sup}{d}_G(x,{[{\mu }_m]}_{\alpha })=\frac{1}{2}{G}_{\mathrm{\infty }}({\mu }_n,{\mu }_m,{\mu }_m)\to 0.$$

(14)

From (13) and

$$0\le \underset{0\le \alpha \le 1}{sup}\underset{x\in {[{\mu }_m]}_{\alpha }}{sup}{d}_G(x,{[{\mu }_n]}_{\alpha })\le G_{\mathrm{\infty }}({\mu }_m,{\mu }_m,{\mu }_n),$$

we have

$$\underset{0\le \alpha \le 1}{sup}\underset{x\in {[{\mu }_m]}_{\alpha }}{sup}{d}_G(x,{[{\mu }_n]}_{\alpha })\to 0\text{as }n,m\to +\mathrm{\infty }.$$

(15)

By (14) and (15), we have

$$D_{\mathrm{\infty }}({\mu }_n,{\mu }_m)\to 0\text{as }n,m\to +\mathrm{\infty },$$

that is, $\{{\mu }_n\}$ is a Cauchy sequence in the metric space $(\mathcal{C}(X),D_{\mathrm{\infty }})$.

“(iii) ⟹ (ii)” Suppose $D_{\mathrm{\infty }}({\mu }_n,{\mu }_m)\to 0$ as $n,m\to +\mathrm{\infty }$, then (14) and (15) hold. Moreover,

$$0\le \underset{0\le \alpha \le 1}{sup}{d}_G({[{\mu }_m]}_{\alpha },{[{\mu }_n]}_{\alpha })\le \underset{0\le \alpha \le 1}{sup}\underset{x\in {[{\mu }_m]}_{\alpha }}{sup}{d}_G(x,{[{\mu }_n]}_{\alpha })$$

and (15) imply that

$$\underset{0\le \alpha \le 1}{sup}{d}_G({[{\mu }_m]}_{\alpha },{[{\mu }_n]}_{\alpha })\to 0\text{as }n,m\to +\mathrm{\infty }.$$

(16)

From (14), (15) and (16), we have as $n,m\to +\mathrm{\infty }$,

$$\begin{array}{rcl}{G}_{\mathrm{\infty }}({\mu }_n,{\mu }_m,{\mu }_m)& =& \underset{0\le \alpha \le 1}{sup}G({[{\mu }_n]}_{\alpha },{[{\mu }_m]}_{\alpha },{[{\mu }_m]}_{\alpha })\\ =& \underset{0\le \alpha \le 1}{sup}\underset{x\in {[{\mu }_n]}_{\alpha }}{sup}2d_G(x,{[{\mu }_m]}_{\alpha })\to 0\end{array}$$

(17)

and

$$\begin{array}{rcl}{G}_{\mathrm{\infty }}({\mu }_m,{\mu }_m,{\mu }_n)& =& \underset{0\le \alpha \le 1}{sup}G({[{\mu }_m]}_{\alpha },{[{\mu }_m]}_{\alpha },{[{\mu }_n]}_{\alpha })\\ =& \underset{0\le \alpha \le 1}{sup}\underset{x\in {[{\mu }_m]}_{\alpha }}{sup}[d_G(x,{[{\mu }_n]}_{\alpha })+d_G({[{\mu }_m]}_{\alpha },{[{\mu }_n]}_{\alpha })]\to 0.\end{array}$$

(18)

We can get from (17) and (18) that

$$D_{G,\mathrm{\infty }}({\mu }_n,{\mu }_m,{\mu }_m)\to 0\text{as }n,m\to +\mathrm{\infty }.$$

 □

The next proposition follows directly from Lemma 1.4, Lemma 2.1, Proposition 2.5 and Proposition 2.6.

Proposition 2.7 The metric space $(\mathcal{C}(X),D_{G,\mathrm{\infty }})$ is complete provided $(X,G)$ is G-complete.

From the definitions of $G_{\mathrm{\infty }}$ and $D_{G,\mathrm{\infty }}$, we can get the next proposition readily.

Proposition 2.8 If $\mu ,{\mu }_1,{\mu }_2\in \mathcal{C}(X)$ and ${\mu }_1\subset {\mu }_2$, then

• (i) $G_{\mathrm{\infty }}({\mu }_1,\mu ,\mu )\le G_{\mathrm{\infty }}({\mu }_2,\mu ,\mu )$,

• (ii) $G_{\mathrm{\infty }}(\mu ,{\mu }_2,{\mu }_2)\le G_{\mathrm{\infty }}(\mu ,{\mu }_1,{\mu }_1)\le D_{G,\mathrm{\infty }}(\mu ,{\mu }_1,{\mu }_1)$,

• (iii) $G_{\mathrm{\infty }}({\mu }_1,{\mu }_2,{\mu }_2)=0$.

3 Fixed point theorems for fuzzy self-mappings

In this section, we establish two fixed point theorems for fuzzy self-mappings. First, we recall the concept of a fuzzy self-mapping in [23].

Definition 3.1 [23]

Let X be a metric space. A mapping F is said to be a fuzzy self-mapping if and only if F is a mapping from the space $\mathcal{C}(X)$ into $\mathcal{C}(X)$, i.e., $F(\mu )\in \mathcal{C}(X)$ for each $\mu \in \mathcal{C}(X)$. ${\mu }_0\in \mathcal{C}(X)$ is said to be a fixed point of a fuzzy self-mapping F of $\mathcal{C}(X)$ if and only if ${\mu }_0\subset F({\mu }_0)$.

Let Φ denote all functions $\varphi :[0,+\mathrm{\infty })\to [0,+\mathrm{\infty })$ satisfying:

• (i) ϕ is non-decreasing and continuous from the right,

• (ii) ${\sum }_{n=1}^{\mathrm{\infty }}{\varphi }^n(t)<+\mathrm{\infty }$, for all $t>0$, where ${\varphi }^n$ denotes the n th iterative function of ϕ.

Remark 3.1 It can be directly verified that for any $\varphi \in \mathrm{\Phi }$ and all $t>0$, $\varphi (t)<t$.

Theorem 3.1 Let $(X,G)$ be a G-complete metric space and ${\{T_i\}}_{i=1}^{\mathrm{\infty }}$ a sequence of fuzzy self-mappings of $\mathcal{C}(X)$. Suppose that for each ${\mu }_1,{\mu }_2\in \mathcal{C}(X)$ and for arbitrary positive integers i and j, $i\ne j$,

(19)

where $\varphi \in \mathrm{\Phi }$. Then there exists at least one ${\mu }_{\ast }\in \mathcal{C}(X)$ such that ${\mu }_{\ast }\subset T_i{\mu }_{\ast }$ for all $i\in {\mathbb{Z}}^{+}$.

Proof Let ${\mu }_0\in \mathcal{C}(X)$ and ${\mu }_1\subset T_1{\mu }_0$, by Proposition 2.3, there exists ${\mu }_2\in \mathcal{C}(X)$ such that ${\mu }_2\subset T_2{\mu }_1$ and

$$D_{G,\mathrm{\infty }}({\mu }_1,{\mu }_2,{\mu }_2)\le D_{G,\mathrm{\infty }}(T_1{\mu }_0,T_2{\mu }_1,T_2{\mu }_1).$$

Again by Proposition 2.3, we can find ${\mu }_3\in \mathcal{C}(X)$ such that ${\mu }_3\subset T_3{\mu }_2$ and

$$D_{G,\mathrm{\infty }}({\mu }_2,{\mu }_3,{\mu }_3)\le D_{G,\mathrm{\infty }}(T_2{\mu }_1,T_3{\mu }_2,T_3{\mu }_2).$$

Continuing this process, we can construct a sequence $\{{\mu }_n\}$ in $\mathcal{C}(X)$ such that

$${\mu }_{n+1}\subset T_{n+1}{\mu }_n,n=0,1,2,\dots$$

and

$$D_{G,\mathrm{\infty }}({\mu }_n,{\mu }_{n+1},{\mu }_{n+1})\le D_{G,\mathrm{\infty }}(T_n{\mu }_{n-1},T_{n+1}{\mu }_n,T_{n+1}{\mu }_n),n=1,2,\dots .$$

(20)

By (19), (20), Proposition 2.8 and (v) in Proposition 2.4, we have

(21)

Suppose that $0\le D_{G,\mathrm{\infty }}({\mu }_{n-1},{\mu }_n,{\mu }_n)<D_{G,\mathrm{\infty }}({\mu }_n,{\mu }_{n+1},{\mu }_{n+1})$, then

$$D_{G,\mathrm{\infty }}({\mu }_n,{\mu }_{n+1},{\mu }_{n+1})\le \varphi (D_{G,\mathrm{\infty }}({\mu }_n,{\mu }_{n+1},{\mu }_{n+1}))<D_{G,\mathrm{\infty }}({\mu }_n,{\mu }_{n+1},{\mu }_{n+1}),$$

which is a contradiction since $D_{G,\mathrm{\infty }}({\mu }_n,{\mu }_{n+1},{\mu }_{n+1})>0$.

Hence,

$$D_{G,\mathrm{\infty }}({\mu }_n,{\mu }_{n+1},{\mu }_{n+1})\le D_{G,\mathrm{\infty }}({\mu }_{n-1},{\mu }_n,{\mu }_n)$$

(22)

and

$$D_{G,\mathrm{\infty }}({\mu }_n,{\mu }_{n+1},{\mu }_{n+1})\le \varphi (D_{G,\mathrm{\infty }}({\mu }_{n-1},{\mu }_n,{\mu }_n))\le \cdots \le {\varphi }^n(D_{G,\mathrm{\infty }}({\mu }_0,{\mu }_1,{\mu }_1)).$$

(23)

Now, we prove that $\{{\mu }_n\}$ is a $D_{G,\mathrm{\infty }}$-Cauchy sequence. For positive integers m, n, we distinguish the following two cases.

Case 1. If $m>n$, then

(24)

Assume that $D_{G,\mathrm{\infty }}({\mu }_0,{\mu }_1,{\mu }_1)=0$, then ${\mu }_0={\mu }_1$. Inequality (21) implies that

$$\begin{array}{rcl}{D}_{G,\mathrm{\infty }}({\mu }_1,{\mu }_2,{\mu }_2)& \le & \varphi (max\{D_{G,\mathrm{\infty }}({\mu }_0,{\mu }_1,{\mu }_1),D_{G,\mathrm{\infty }}({\mu }_1,{\mu }_2,{\mu }_2)\})\\ =& \varphi (D_{G,\mathrm{\infty }}({\mu }_1,{\mu }_2,{\mu }_2)).\end{array}$$

It follows from $\varphi (t)<t$ that $D_{G,\mathrm{\infty }}({\mu }_1,{\mu }_2,{\mu }_2)=0$, that is, ${\mu }_1={\mu }_2$. By induction, we have ${\mu }_0={\mu }_1=\cdots ={\mu }_k=\cdots$. Thus, ${\mu }_0={\mu }_k\subset T_k{\mu }_{k-1}=T_k{\mu }_0$, $k=1,2,\dots$ .

Suppose that $D_{G,\mathrm{\infty }}({\mu }_0,{\mu }_1,{\mu }_1)>0$. ${\sum }_{i=1}^{\mathrm{\infty }}{\varphi }^i(t)<\mathrm{\infty }$ and (24) yield that

$$D_{G,\mathrm{\infty }}({\mu }_n,{\mu }_m,{\mu }_m)\to 0,\text{as }n,m\to +\mathrm{\infty }.$$

(25)

Case 2. If $m<n$, from (25) and

$$\begin{array}{rcl}0& \le & D_{G,\mathrm{\infty }}({\mu }_n,{\mu }_m,{\mu }_m)=D_{G,\mathrm{\infty }}({\mu }_m,{\mu }_m,{\mu }_n)\\ \le & D_{G,\mathrm{\infty }}({\mu }_m,{\mu }_n,{\mu }_n)+D_{G,\mathrm{\infty }}({\mu }_n,{\mu }_m,{\mu }_n)=2D_{G,\mathrm{\infty }}({\mu }_m,{\mu }_n,{\mu }_n),\end{array}$$

we can get that

$$D_{G,\mathrm{\infty }}({\mu }_n,{\mu }_m,{\mu }_m)\to 0\text{as }n,m\to +\mathrm{\infty }.$$

(26)

Thus, (25) and (26) imply that $\{{\mu }_n\}$ is a $D_{G,\mathrm{\infty }}$-Cauchy sequence. As $(X,G)$ is G-complete, by Proposition 2.7, we conclude that $(\mathcal{C}(X),D_{G,\mathrm{\infty }})$ is complete. There exists ${\mu }_{\ast }\in \mathcal{C}(X)$ such that $D_{G,\mathrm{\infty }}({\mu }_{\ast },{\mu }_m,{\mu }_m)\to 0$ as $m\to +\mathrm{\infty }$.

Now, Proposition 2.8 and (19) imply that

(27)

Letting $j\to +\mathrm{\infty }$, we can see from (27) and Proposition 2.5 that

$$G_{\mathrm{\infty }}({\mu }_{\ast },T_i{\mu }_{\ast },T_i{\mu }_{\ast })\le \varphi (G_{\mathrm{\infty }}({\mu }_{\ast },T_i{\mu }_{\ast },T_i{\mu }_{\ast })).$$

It implies that $G_{\mathrm{\infty }}({\mu }_{\ast },T_i{\mu }_{\ast },T_i{\mu }_{\ast })=0$, that is, ${\mu }_{\ast }\subset T_i{\mu }_{\ast }$.

If in Theorem 3.1 we choose $\varphi (t)=kt$, where $k\in (0,1)$ is a constant, we obtain the following corollary. □

Corollary 3.1 Let $(X,G)$ be a G-complete metric space and ${\{T_i\}}_{i=1}^{\mathrm{\infty }}$ a sequence of fuzzy self-mappings of $\mathcal{C}(X)$. Suppose that for each ${\mu }_1,{\mu }_2\in \mathcal{C}(X)$ and for arbitrary positive integers i and j, $i\ne j$,

$$\begin{array}{c}{D}_{G,\mathrm{\infty }}(T_i{\mu }_1,T_j{\mu }_2,T_j{\mu }_2)\hfill \\ \le k(max\{D_{G,\mathrm{\infty }}({\mu }_1,{\mu }_2,{\mu }_2),G_{\mathrm{\infty }}({\mu }_1,T_i{\mu }_1,T_i{\mu }_1),G_{\mathrm{\infty }}({\mu }_2,T_j{\mu }_2,T_j{\mu }_2),\hfill \\ \frac{1}{2}[G_{\mathrm{\infty }}({\mu }_1,T_j{\mu }_2,T_j{\mu }_2)+G_{\mathrm{\infty }}({\mu }_2,T_i{\mu }_1,T_i{\mu }_1)]\left\}\right),\hfill \end{array}$$

where $k\in (0,1)$. Then there exists at least one ${\mu }_{\ast }\in \mathcal{C}(X)$ such that ${\mu }_{\ast }\subset T_i{\mu }_{\ast }$ for all $i\in {\mathbb{Z}}^{+}$.

The following example illustrates Theorem 3.1.

Example 3.1 Let $X=\{0,1,2,3,\dots \}$. Define $G:X\times X\times X\to X$ by

$$G(x,y,z)=\{\begin{array}{cc}x+y+z,\hfill & \text{if }x,y,z\text{ are all distinct and different from zero};\hfill \\ x+z,\hfill & \text{if }x=y\ne z\text{ and all are different from zero};\hfill \\ y+z+1,\hfill & \text{if }x=0,y\ne z\text{ and }y,z\text{ are different from zero};\hfill \\ y+2,\hfill & \text{if }x=0,y=z\ne 0;\hfill \\ z+1,\hfill & \text{if }x=0,y=0,z\ne 0;\hfill \\ 0,\hfill & \text{if }x=y=z.\hfill \end{array}$$

Then X is a complete nonsymmetric G-metric space [5].

For $\mu ,\nu \in \mathcal{C}(X)$, $y\in X$ and $\lambda >0$, owing to Zadeh’s extension principle [25], scalar multiplication and addition are defined by

$$(\lambda \mu )(y)=\mu \left(\frac{y}{\lambda }\right)$$

and

$$(\mu +\nu )(x)=\underset{x_1,x_2:x_1+x_2=x}{sup}min\{\mu (x_1),\nu (x_2)\}.$$

For any $0<a<1$ and $\mu ,\nu ,\omega \in \mathcal{C}(X)$, we can get easily from the definition of $G(x,y,z)$ that

$$D_{G,\mathrm{\infty }}(a\mu ,a\nu ,a\omega )=a{D}_{G,\mathrm{\infty }}(\mu ,\nu ,\omega )$$

(28)

and

$$D_{G,\mathrm{\infty }}(\mu ,a\nu ,a\nu )\le D_{G,\mathrm{\infty }}(\mu ,\nu ,\nu ).$$

(29)

Now, suppose $0<p\le 1$, define ${\mu }_0:X\to \mathcal{C}(X)$ by

$${\mu }_0(x)=\{\begin{array}{cc}p,\hfill & \text{if }x=0;\hfill \\ 0,\hfill & \text{otherwise}.\hfill \end{array}$$

Suppose $0<q<1$, define ${\{T_i\}}_{i=1}^{\mathrm{\infty }}$ a sequence of fuzzy self-mappings of $\mathcal{C}(X)$ as

$$T_i(\mu )=q^i\mu +{\mu }_0\text{for any }\mu \in \mathcal{C}(X).$$

For any $i,j\in Z^{+}$, without loss of generality, suppose $i<j$. For each ${\mu }_1,{\mu }_2\in \mathcal{C}(X)$, by (28), (29) and the definition of α-level set, we have

$$\begin{array}{c}{D}_{G,\mathrm{\infty }}(T_i{\mu }_1,T_j{\mu }_2,T_j{\mu }_2)\hfill \\ =D_{G,\mathrm{\infty }}(q^i{\mu }_1+{\mu }_0,q^j{\mu }_2+{\mu }_0,q^j{\mu }_2+{\mu }_0)\hfill \\ \le q^i{D}_{G,\mathrm{\infty }}({\mu }_1,q^{j-i}{\mu }_2,q^{j-i}{\mu }_2)\le q^i{D}_{G,\mathrm{\infty }}({\mu }_1,{\mu }_2,{\mu }_2)\hfill \\ \le q^i(max\{D_{G,\mathrm{\infty }}({\mu }_1,{\mu }_2,{\mu }_2),G_{\mathrm{\infty }}({\mu }_1,T_i{\mu }_1,T_i{\mu }_1),G_{\mathrm{\infty }}({\mu }_2,T_j{\mu }_2,T_j{\mu }_2),\hfill \\ \frac{1}{2}[G_{\mathrm{\infty }}({\mu }_1,T_j{\mu }_2,T_j{\mu }_2)+G_{\mathrm{\infty }}({\mu }_2,T_i{\mu }_1,T_i{\mu }_1)]\left\}\right).\hfill \end{array}$$

Therefore, ${\{T_i\}}_{i=1}^{\mathrm{\infty }}$ satisfy the conditions of Theorem 3.1 with $\varphi (t)=q^{i}t$. Moreover, for each $0<b\le p$,

$${\mu }_b(x)=\{\begin{array}{cc}b,\hfill & \text{if }x=0;\hfill \\ 0,\hfill & \text{otherwise}\hfill \end{array}$$

is a common fixed point of ${\{T_i\}}_{i=1}^{\mathrm{\infty }}$.

4 Conclusion

In this work, by using the new concept of Hausdorff G-metric in the space of fuzzy sets, we establish some common fixed point theorems for a family of fuzzy self-mappings in the space of fuzzy sets on a complete G-metric space. These results are useful in fractal. An iterated function system (i.e., IFS) is the significant content in fractal, and the attractor of the IFS plays a very important role in the fractal graphics. On account of the fuzziness of parameters in fractal, by Zadeh’s extension principle [25], we can get an iterated fuzzy function system (i.e., IFFS) corresponding the IFS [24]. For example, ${\{T_i\}}_{i=1}^n$ in Example 3.1 is an IFFS and $\{{\mu }_b:0<b\le p\}\subseteq A$, where A is the set of attractors of IFFS. Moreover, we can estimate the area of attractors basing on the fixed points of ${\{T_i\}}_{i=1}^n$. Our results are also useful in fuzzy differential equation. As we all know, the existence of a solution for a fuzzy differential equation can be established via the fixed point analysis approach (see [26–28]). Therefore, our results provide a new method for studying the fuzzy differential equation in G-metric spaces.