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

Fixed Point Theory and Applications20122012:159

https://doi.org/10.1186/1687-1812-2012-159

• Accepted: 6 September 2012
• Published:

## 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.

## Keywords

• fuzzy set
• ${D}_{G,\mathrm{\infty }}$-convergent
• ${D}_{G,\mathrm{\infty }}$-Cauchy
• fuzzy self-mapping
• fixed point

## 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 $G:X×X×X\to {\mathbb{R}}^{+}$ be a function satisfying:

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

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

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

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

(G 5) $G\left(x,y,z\right)\le G\left(x,a,a\right)+G\left(a,y,z\right)$ 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 $\left(X,G\right)$ is a G-metric space.

Lemma 1.1 Every G-metric space $\left(X,G\right)$ defines a metric space $\left(X,{d}_{G}\right)$ by
Definition 1.2 Let $\left(X,G\right)$ be a G-metric space. The sequence $\left\{{x}_{n}\right\}$ in X is said to be
1. (i)

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

2. (ii)

G-Cauchy if for any $\epsilon >0$, there exists $N\in \mathbb{N}$ such that $G\left({x}_{n},{x}_{m},{x}_{l}\right)<\epsilon$, for all $n,m,l\ge N$.

Lemma 1.2 Let $\left(X,G\right)$ be a G-metric space, then for a sequence $\left\{{x}_{n}\right\}$ in X and point $x\in X$ the following are equivalent:
1. (i)

$\left\{{x}_{n}\right\}$ is G-convergent to x.

2. (ii)

$G\left({x}_{n},{x}_{n},x\right)\to 0$ as $n\to +\mathrm{\infty }$.

3. (iii)

$G\left({x}_{n},x,x\right)\to 0$ as $n\to +\mathrm{\infty }$.

4. (iv)

$G\left({x}_{m},{x}_{n},x\right)\to 0$ as $m,n\to +\mathrm{\infty }$.

Lemma 1.3 Let $\left(X,G\right)$ be a G-metric space, then for a sequence $\left\{{x}_{n}\right\}$ in X, the following are equivalent:
1. (i)

The sequence $\left\{{x}_{n}\right\}$ is G-Cauchy.

2. (ii)

For any $\epsilon >0$, there exists $N\in \mathbb{N}$ such that $G\left({x}_{n},{x}_{m},{x}_{m}\right)<\epsilon$, for all $n,m\ge N$.

3. (iii)

$\left\{{x}_{n}\right\}$ is a Cauchy sequence in the metric space $\left(X,{d}_{G}\right)$.

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

Lemma 1.4 A G-metric space $\left(X,G\right)$ is G-complete if and only if $\left(X,{d}_{G}\right)$ 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., [27]) and in partially ordered G-metric spaces (see, e.g., [813]). Recently, Kaewcharoen and Kaewkhao [14] introduced the following concepts. Let X be a G-metric space and $CB\left(X\right)$ the family of all nonempty closed bounded subsets of X. Let $H\left(\cdot ,\cdot ,\cdot \right)$ be the Hausdorff G-distance on $CB\left(X\right)$, i.e.,
${H}_{G}\left(A,B,C\right)=max\left\{\underset{x\in A}{sup}G\left(x,B,C\right),\underset{x\in B}{sup}G\left(x,C,A\right),\underset{x\in C}{sup}G\left(x,A,B\right)\right\},$
where
$\begin{array}{c}G\left(x,B,C\right)={d}_{G}\left(x,B\right)+{d}_{G}\left(B,C\right)+{d}_{G}\left(x,C\right),\hfill \\ {d}_{G}\left(x,B\right)=\underset{y\in B}{inf}{d}_{G}\left(x,y\right),\hfill \\ {d}_{G}\left(A,B\right)=\underset{x\in A,y\in B}{inf}{d}_{G}\left(x,y\right).\hfill \end{array}$

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., [1722]). 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 $\left(X,{d}_{G}\right)$ be a metric space, a fuzzy set in X is a function with domain X and values in $I=\left[0,1\right]$. If μ is a fuzzy set and $x\in X$, then the function value $\mu \left(x\right)$ is called the grade of membership of x in μ.

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

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

Let $C\left(X\right)$ be the family of all nonempty compact subsets of X. Denote by $\mathcal{C}\left(X\right)$ the totality of fuzzy sets which satisfy that for each $\alpha \in I$, ${\left[\mu \right]}_{\alpha }\in C\left(X\right)$. Let ${\mu }_{1},{\mu }_{2}\in \mathcal{C}\left(X\right)$, then ${\mu }_{1}$ is said to be more accurate than ${\mu }_{2}$, denoted by ${\mu }_{1}\subset {\mu }_{2}$, if and only if ${\mu }_{1}\left(x\right)\le {\mu }_{2}\left(x\right)$ 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 ${\mu }_{1},{\mu }_{2}\in \mathcal{C}\left(X\right)$, define
$\begin{array}{rcl}{D}_{\mathrm{\infty }}\left({\mu }_{1},{\mu }_{2}\right)& =& \underset{0\le \alpha \le 1}{sup}H\left({\left[{\mu }_{1}\right]}_{\alpha },{\left[{\mu }_{2}\right]}_{\alpha }\right)\\ =& \underset{0\le \alpha \le 1}{sup}max\left\{\underset{x\in {\left[{\mu }_{1}\right]}_{\alpha }}{sup}{d}_{G}\left(x,{\left[{\mu }_{2}\right]}_{\alpha }\right),\underset{y\in {\left[{\mu }_{2}\right]}_{\alpha }}{sup}{d}_{G}\left(y,{\left[{\mu }_{1}\right]}_{\alpha }\right)\right\}.\end{array}$

Lemma 2.1 [23]

The metric space $\left(\mathcal{C}\left(X\right),{D}_{\mathrm{\infty }}\right)$ is complete provided $\left(X,{d}_{G}\right)$ is complete.

For ${\mu }_{1},{\mu }_{2},{\mu }_{3}\in \mathcal{C}\left(X\right)$, $\alpha \in I$, we define:
Proposition 2.1 If $A,B\in C\left(X\right)$ and $x\in A$, then there exists $y\in B$ such that
$2\left[G\left(x,y,y\right)+G\left(y,x,x\right)\right]\le {H}_{G}\left(A,B,B\right).$
Proof For $x\in A$, there exists $y\in B$ such that
${d}_{G}\left(x,y\right)={d}_{G}\left(x,B\right)=\frac{1}{2}G\left(x,B,B\right),$
it follows that
$2\left[G\left(x,y,y\right)+G\left(y,x,x\right)\right]=G\left(x,B,B\right)\le {H}_{G}\left(A,B,B\right).$

□

Proposition 2.2 If $A,B\in C\left(X\right)$ and ${A}_{1}\subseteq A$, then there exists ${B}_{1}\in C\left(X\right)$ such that ${B}_{1}\subseteq B$ and
${H}_{G}\left({A}_{1},{B}_{1},{B}_{1}\right)\le {H}_{G}\left(A,B,B\right).$
Proof Let and let ${B}_{1}=C\cap B$. For any $x\in {A}_{1}\subseteq A$ and $B\in C\left(X\right)$, Proposition 2.1 implies that ${B}_{1}$ is nonempty. Moreover, for any $x\in {A}_{1}$, there exists $y\in {B}_{1}$ such that $2\left[G\left(x,y,y\right)+G\left(y,x,x\right)\right]\le {H}_{G}\left(A,B,B\right)$. It follows that
$\begin{array}{rcl}G\left({A}_{1},{B}_{1},{B}_{1}\right)& =& \underset{x\in {A}_{1}}{sup}G\left(x,{B}_{1},{B}_{1}\right)=\underset{x\in {A}_{1}}{sup}2{d}_{G}\left(x,{B}_{1}\right)\\ =& 2\underset{x\in {A}_{1}}{sup}\underset{y\in {B}_{1}}{inf}\left[G\left(x,y,y\right)+G\left(y,x,x\right)\right]\le {H}_{G}\left(A,B,B\right).\end{array}$
(1)
On the other hand, for any $y\in {B}_{1}$, there exists $x\in {A}_{1}$ such that $2\left[G\left(x,y,y\right)+G\left(y,x,x\right)\right]\le {H}_{G}\left(A,B,B\right)$. Hence,
$\begin{array}{rcl}G\left({B}_{1},{B}_{1},{A}_{1}\right)& =& \underset{y\in {B}_{1}}{sup}G\left(y,{B}_{1},{A}_{1}\right)=\underset{y\in {B}_{1}}{sup}\left[{d}_{G}\left(y,{A}_{1}\right)\right]+\underset{y\in {B}_{1}}{sup}\left[{d}_{G}\left(y,{B}_{1}\right)\right]+{d}_{G}\left({A}_{1},{B}_{1}\right)\\ =& \underset{y\in {B}_{1}}{sup}\underset{x\in {A}_{1}}{inf}\left[G\left(x,y,y\right)+G\left(y,x,x\right)\right]+0+\underset{x\in {A}_{1},y\in {B}_{1}}{inf}\left[G\left(x,y,y\right)+G\left(y,x,x\right)\right]\\ \le & \underset{y\in {B}_{1}}{sup}\underset{x\in {A}_{1}}{inf}\left[G\left(x,y,y\right)+G\left(y,x,x\right)\right]+\underset{y\in {B}_{1}}{sup}\underset{x\in {A}_{1}}{inf}\left[G\left(x,y,y\right)+G\left(y,x,x\right)\right]\\ =& \underset{y\in {B}_{1}}{sup}\underset{x\in {A}_{1}}{inf}2\left[G\left(x,y,y\right)+G\left(y,x,x\right)\right]\le {H}_{G}\left(A,B,B\right).\end{array}$
(2)
From (1) and (2), we have
${H}_{G}\left({A}_{1},{B}_{1},{B}_{1}\right)\le {H}_{G}\left(A,B,B\right).$

Finally, we can conclude that ${B}_{1}\in C\left(X\right)$ from the closeness of C and the compactness of B. □

Proposition 2.3 Let ${\mu }_{1},{\mu }_{2}\in \mathcal{C}\left(X\right)$ and ${\mu }_{3}\subset {\mu }_{1}$, then there exists ${\mu }_{4}\in \mathcal{C}\left(X\right)$ such that ${\mu }_{4}\subset {\mu }_{2}$ and
${D}_{G,\mathrm{\infty }}\left({\mu }_{3},{\mu }_{4},{\mu }_{4}\right)\le {D}_{G,\mathrm{\infty }}\left({\mu }_{1},{\mu }_{2},{\mu }_{2}\right).$
Proof Let $\alpha \in I$, by ${\mu }_{3}\subset {\mu }_{1}$, we have ${\left[{\mu }_{3}\right]}_{\alpha }\subseteq {\left[{\mu }_{1}\right]}_{\alpha }$. Let
we can get that ${C}_{\alpha }={D}_{\alpha }$. Let ${B}_{\alpha }={D}_{\alpha }\cap {\left[{\mu }_{2}\right]}_{\alpha }$, 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
${H}_{G}\left({\left[{\mu }_{3}\right]}_{\alpha },{B}_{\alpha },{B}_{\alpha }\right)\le {D}_{G,\mathrm{\infty }}\left({\mu }_{1},{\mu }_{2},{\mu }_{2}\right).$
Similar to the proof of Theorem 3 in [23], we can conclude that there exists a fuzzy set ${\mu }_{4}$ such that ${\left[{\mu }_{4}\right]}_{\alpha }={B}_{\alpha }$ for $\alpha \in I$. By the compactness of ${B}_{\alpha }$, we have ${\mu }_{4}\in \mathcal{C}\left(X\right)$. Therefore,
${D}_{G,\mathrm{\infty }}\left({\mu }_{3},{\mu }_{4},{\mu }_{4}\right)\le {D}_{G,\mathrm{\infty }}\left({\mu }_{1},{\mu }_{2},{\mu }_{2}\right).$

□

Proposition 2.4 Let X be a nonempty set. For any ${\mu }_{1},{\mu }_{2},{\mu }_{3}\in \mathcal{C}\left(X\right)$, the following properties hold:
1. (i)

${D}_{G,\mathrm{\infty }}\left({\mu }_{1},{\mu }_{2},{\mu }_{3}\right)=0$ if and only if ${\mu }_{1}={\mu }_{2}={\mu }_{3}$,

2. (ii)

$0<{D}_{G,\mathrm{\infty }}\left({\mu }_{1},{\mu }_{1},{\mu }_{2}\right)$ for all ${\mu }_{1},{\mu }_{2}\in \mathcal{C}\left(X\right)$ with ${\mu }_{1}\ne {\mu }_{2}$,

3. (iii)

${D}_{G,\mathrm{\infty }}\left({\mu }_{1},{\mu }_{1},{\mu }_{2}\right)\le {D}_{G,\mathrm{\infty }}\left({\mu }_{1},{\mu }_{2},{\mu }_{3}\right)$ for all ${\mu }_{1},{\mu }_{2},{\mu }_{3}\in \mathcal{C}\left(X\right)$ with ${\mu }_{2}\ne {\mu }_{3}$,

4. (iv)

${D}_{G,\mathrm{\infty }}\left({\mu }_{1},{\mu }_{2},{\mu }_{3}\right)={D}_{G,\mathrm{\infty }}\left({\mu }_{1},{\mu }_{3},{\mu }_{2}\right)={D}_{G,\mathrm{\infty }}\left({\mu }_{2},{\mu }_{1},{\mu }_{3}\right)=\cdots$ (symmetry in all three variables),

5. (v)

${D}_{G,\mathrm{\infty }}\left({\mu }_{1},{\mu }_{2},{\mu }_{3}\right)\le {D}_{G,\mathrm{\infty }}\left({\mu }_{1},\mu ,\mu \right)+{D}_{G,\mathrm{\infty }}\left(\mu ,{\mu }_{2},{\mu }_{3}\right)$.

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 $x\in {\left[{\mu }_{1}\right]}_{\alpha }$, $y\in {\left[{\mu }_{2}\right]}_{\alpha }$ and $z\in {\left[{\mu }_{3}\right]}_{\alpha }$, we have
${d}_{G}\left(x,y\right)-{d}_{G}\left(x,z\right)-{d}_{G}\left(z,y\right)\le 0,$
it follows that
$\begin{array}{c}{d}_{G}\left(x,y\right)-{d}_{G}\left(x,{\left[{\mu }_{3}\right]}_{\alpha }\right)-{d}_{G}\left({\left[{\mu }_{2}\right]}_{\alpha },{\left[{\mu }_{3}\right]}_{\alpha }\right)\hfill \\ \phantom{\rule{1em}{0ex}}\le \underset{y\in {\left[{\mu }_{2}\right]}_{\alpha }}{sup}{d}_{G}\left(x,y\right)-\underset{z\in {\left[{\mu }_{3}\right]}_{\alpha }}{inf}{d}_{G}\left(x,z\right)-\underset{y\in {\left[{\mu }_{2}\right]}_{\alpha },z\in {\left[{\mu }_{3}\right]}_{\alpha }}{inf}{d}_{G}\left(z,y\right)\hfill \\ \phantom{\rule{1em}{0ex}}=\underset{y\in {\left[{\mu }_{2}\right]}_{\alpha },z\in {\left[{\mu }_{3}\right]}_{\alpha }}{sup}\left[{d}_{G}\left(x,y\right)-{d}_{G}\left(x,z\right)-{d}_{G}\left(z,y\right)\right]\le 0.\hfill \end{array}$
This implies that
$\underset{x\in {\left[{\mu }_{1}\right]}_{\alpha },y\in {\left[{\mu }_{2}\right]}_{\alpha }}{inf}{d}_{G}\left(x,y\right)-\underset{x\in {\left[{\mu }_{1}\right]}_{\alpha }}{sup}{d}_{G}\left(x,{\left[{\mu }_{3}\right]}_{\alpha }\right)\le {d}_{G}\left({\left[{\mu }_{2}\right]}_{\alpha },{\left[{\mu }_{3}\right]}_{\alpha }\right).$
Then,
${d}_{G}\left({\left[{\mu }_{1}\right]}_{\alpha },{\left[{\mu }_{2}\right]}_{\alpha }\right)\le \underset{x\in {\left[{\mu }_{1}\right]}_{\alpha }}{sup}{d}_{G}\left(x,{\left[{\mu }_{3}\right]}_{\alpha }\right)+{d}_{G}\left({\left[{\mu }_{2}\right]}_{\alpha },{\left[{\mu }_{3}\right]}_{\alpha }\right).$
Hence,
(3)
Similarly, we can prove that
$G\left({\left[{\mu }_{2}\right]}_{\alpha },{\left[{\mu }_{1}\right]}_{\alpha },{\left[{\mu }_{1}\right]}_{\alpha }\right)\le G\left({\left[{\mu }_{2}\right]}_{\alpha },{\left[{\mu }_{1}\right]}_{\alpha },{\left[{\mu }_{3}\right]}_{\alpha }\right).$
(4)
By (3) and (4), we have

Now, we prove the property (v).

For any $\alpha \in I$ and $x\in {\left[{\mu }_{2}\right]}_{\alpha }$, $y\in {\left[\mu \right]}_{\alpha }$, we have
${d}_{G}\left(x,{\left[{\mu }_{1}\right]}_{\alpha }\right)\le {d}_{G}\left(x,y\right)+{d}_{G}\left(y,{\left[{\mu }_{1}\right]}_{\alpha }\right),$
it follows that
$\underset{x\in {\left[{\mu }_{2}\right]}_{\alpha }}{sup}{d}_{G}\left(x,{\left[{\mu }_{1}\right]}_{\alpha }\right)\le \underset{x\in {\left[{\mu }_{2}\right]}_{\alpha }}{sup}{d}_{G}\left(x,{\left[\mu \right]}_{\alpha }\right)+\underset{y\in {\left[\mu \right]}_{\alpha }}{sup}{d}_{G}\left(y,{\left[{\mu }_{1}\right]}_{\alpha }\right).$
(5)
From (5) and
${d}_{G}\left({\left[{\mu }_{1}\right]}_{\alpha },{\left[{\mu }_{3}\right]}_{\alpha }\right)\le {d}_{G}\left({\left[\mu \right]}_{\alpha },{\left[{\mu }_{3}\right]}_{\alpha }\right)+{d}_{G}\left({\left[\mu \right]}_{\alpha },{\left[{\mu }_{1}\right]}_{\alpha }\right),$
we have
$\begin{array}{rcl}{G}_{\alpha }\left({\mu }_{2},{\mu }_{3},{\mu }_{1}\right)& =& G\left({\left[{\mu }_{2}\right]}_{\alpha },{\left[{\mu }_{3}\right]}_{\alpha },{\left[{\mu }_{1}\right]}_{\alpha }\right)\\ =& \underset{x\in {\left[{\mu }_{2}\right]}_{\alpha }}{sup}\left[{d}_{G}\left(x,{\left[{\mu }_{1}\right]}_{\alpha }\right)+{d}_{G}\left(x,{\left[{\mu }_{3}\right]}_{\alpha }\right)+{d}_{G}\left({\left[{\mu }_{1}\right]}_{\alpha },{\left[{\mu }_{3}\right]}_{\alpha }\right)\right]\\ \le & \underset{x\in {\left[{\mu }_{2}\right]}_{\alpha }}{sup}\left[{d}_{G}\left(x,{\left[\mu \right]}_{\alpha }\right)+{d}_{G}\left(x,{\left[{\mu }_{3}\right]}_{\alpha }\right)+{d}_{G}\left({\left[\mu \right]}_{\alpha },{\left[{\mu }_{3}\right]}_{\alpha }\right)\right]\\ +\underset{y\in {\left[\mu \right]}_{\alpha }}{sup}\left[{d}_{G}\left(y,{\left[{\mu }_{1}\right]}_{\alpha }\right)+{d}_{G}\left(y,{\left[\mu \right]}_{\alpha }\right)+{d}_{G}\left({\left[\mu \right]}_{\alpha },{\left[{\mu }_{1}\right]}_{\alpha }\right)\right]\\ =& {G}_{\alpha }\left({\mu }_{2},{\mu }_{3},\mu \right)+{G}_{\alpha }\left(\mu ,\mu ,{\mu }_{1}\right).\end{array}$
(6)
Similarly, we can obtain that
(7)
(8)
By (6), (7) and (8), we have
$\begin{array}{rcl}{D}_{G,\mathrm{\infty }}\left({\mu }_{1},{\mu }_{2},{\mu }_{3}\right)& =& \underset{0\le \alpha \le 1}{sup}max\left\{{G}_{\alpha }\left({\mu }_{2},{\mu }_{3},{\mu }_{1}\right),{G}_{\alpha }\left({\mu }_{3},{\mu }_{2},{\mu }_{1}\right),{G}_{\alpha }\left({\mu }_{1},{\mu }_{2},{\mu }_{3}\right)\right\}\\ \le & \underset{0\le \alpha \le 1}{sup}max\left\{{G}_{\alpha }\left({\mu }_{2},{\mu }_{3},\mu \right),{G}_{\alpha }\left({\mu }_{3},{\mu }_{2},\mu \right),{G}_{\alpha }\left(\mu ,{\mu }_{2},{\mu }_{3}\right)\right\}\\ +\underset{0\le \alpha \le 1}{sup}max\left\{{G}_{\alpha }\left(\mu ,\mu ,{\mu }_{1}\right),{G}_{\alpha }\left(\mu ,\mu ,{\mu }_{1}\right),{G}_{\alpha }\left({\mu }_{1},\mu ,\mu \right)\right\}\\ =& {D}_{G,\mathrm{\infty }}\left(\mu ,{\mu }_{2},{\mu }_{3}\right)+{D}_{G,\mathrm{\infty }}\left({\mu }_{1},\mu ,\mu \right).\end{array}$

□

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

Definition 2.1 Let $\left(\mathcal{C}\left(X\right),{D}_{G,\mathrm{\infty }}\right)$ be a metric space. The sequence $\left\{{\mu }_{n}\right\}$ in $\mathcal{C}\left(X\right)$ is said to be
1. (i)

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

2. (ii)

${D}_{G,\mathrm{\infty }}$-Cauchy if for every $\epsilon >0$, there exists $N\in \mathbb{N}$ such that ${D}_{G,\mathrm{\infty }}\left({\mu }_{n},{\mu }_{m},{\mu }_{l}\right)<\epsilon$ for all $n,m,l\ge N$.

Proposition 2.5 Let $\left(\mathcal{C}\left(X\right),{D}_{G,\mathrm{\infty }}\right)$ be a metric space, then for a sequence $\left\{{\mu }_{n}\right\}\subset \mathcal{C}\left(X\right)$ and $\mu \in \mathcal{C}\left(X\right)$, the following are equivalent:
1. (i)

$\left\{{\mu }_{n}\right\}$ is ${D}_{G,\mathrm{\infty }}$-convergent to μ.

2. (ii)

${D}_{\mathrm{\infty }}\left(\mu ,{\mu }_{n}\right)\to 0$ as $n\to +\mathrm{\infty }$.

3. (iii)

${D}_{G,\mathrm{\infty }}\left({\mu }_{n},{\mu }_{n},\mu \right)\to 0$ as $n\to +\mathrm{\infty }$.

4. (iv)

${D}_{G,\mathrm{\infty }}\left(\mu ,\mu ,{\mu }_{n}\right)\to 0$ 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 ${D}_{G,\mathrm{\infty }}\left(\mu ,{\mu }_{n},{\mu }_{m}\right)\to 0$ as $n,m\to +\mathrm{\infty }$, then
${G}_{\mathrm{\infty }}\left(\mu ,{\mu }_{n},{\mu }_{m}\right)=\underset{0\le \alpha \le 1}{sup}\underset{x\in {\left[\mu \right]}_{\alpha }}{sup}\left[{d}_{G}\left(x,{\left[{\mu }_{n}\right]}_{\alpha }\right)+{d}_{G}\left(x,{\left[{\mu }_{m}\right]}_{\alpha }\right)+{d}_{G}\left({\left[{\mu }_{n}\right]}_{\alpha },{\left[{\mu }_{m}\right]}_{\alpha }\right)\right]\to 0$
and
${G}_{\mathrm{\infty }}\left({\mu }_{n},\mu ,{\mu }_{m}\right)=\underset{0\le \alpha \le 1}{sup}\underset{x\in {\left[{\mu }_{n}\right]}_{\alpha }}{sup}\left[{d}_{G}\left(x,{\left[\mu \right]}_{\alpha }\right)+{d}_{G}\left(x,{\left[{\mu }_{m}\right]}_{\alpha }\right)+{d}_{G}\left({\left[\mu \right]}_{\alpha },{\left[{\mu }_{m}\right]}_{\alpha }\right)\right]\to 0.$
Thus, for any $\alpha \in I$,
(9)
and
(10)
It follows that

“(ii) (i)” Suppose ${D}_{\mathrm{\infty }}\left(\mu ,{\mu }_{n}\right)\to 0$ as $n\to +\mathrm{\infty }$, then (9) and (10) hold.

Moreover, $0\le {d}_{G}\left({\left[{\mu }_{n}\right]}_{\alpha },{\left[\mu \right]}_{\alpha }\right)\le {sup}_{x\in {\left[{\mu }_{n}\right]}_{\alpha }}\left[{d}_{G}\left(x,{\left[\mu \right]}_{\alpha }\right)\right]$ implies that as $n\to +\mathrm{\infty }$,
${d}_{G}\left({\left[{\mu }_{n}\right]}_{\alpha },{\left[\mu \right]}_{\alpha }\right)\to 0.$
(11)
From (9), (10) and (11), we have as $n\to +\mathrm{\infty }$,
$\begin{array}{c}{D}_{G,\mathrm{\infty }}\left({\mu }_{n},{\mu }_{n},\mu \right)\hfill \\ \phantom{\rule{1em}{0ex}}=\underset{0\le \alpha \le 1}{sup}max\left\{G\left({\left[{\mu }_{n}\right]}_{\alpha },{\left[{\mu }_{n}\right]}_{\alpha },{\left[\mu \right]}_{\alpha }\right),G\left({\left[\mu \right]}_{\alpha },{\left[{\mu }_{n}\right]}_{\alpha },{\left[{\mu }_{n}\right]}_{\alpha }\right)\right\}\hfill \\ \phantom{\rule{1em}{0ex}}=\underset{0\le \alpha \le 1}{sup}max\left\{\underset{x\in {\left[{\mu }_{n}\right]}_{\alpha }}{sup}\left[{d}_{G}\left(x,{\left[\mu \right]}_{\alpha }\right)+{d}_{G}\left({\left[{\mu }_{n}\right]}_{\alpha },{\left[\mu \right]}_{\alpha }\right)\right],\underset{x\in {\left[\mu \right]}_{\alpha }}{sup}2{d}_{G}\left(x,{\left[{\mu }_{n}\right]}_{\alpha }\right)\right\}\to 0.\hfill \end{array}$
Thus, from
$\begin{array}{rcl}0& \le & {D}_{G,\mathrm{\infty }}\left(\mu ,{\mu }_{n},{\mu }_{m}\right)={D}_{G,\mathrm{\infty }}\left({\mu }_{n},\mu ,{\mu }_{m}\right)\le {D}_{G,\mathrm{\infty }}\left({\mu }_{n},\mu ,\mu \right)+{D}_{G,\mathrm{\infty }}\left(\mu ,\mu ,{\mu }_{m}\right)\\ \le & {D}_{G,\mathrm{\infty }}\left(\mu ,{\mu }_{n},{\mu }_{n}\right)+{D}_{G,\mathrm{\infty }}\left({\mu }_{n},\mu ,{\mu }_{n}\right)+{D}_{G,\mathrm{\infty }}\left(\mu ,{\mu }_{m},{\mu }_{m}\right)+{D}_{G,\mathrm{\infty }}\left({\mu }_{m},\mu ,{\mu }_{m}\right)\\ =& 2{D}_{G,\mathrm{\infty }}\left({\mu }_{n},{\mu }_{n},\mu \right)+2{D}_{G,\mathrm{\infty }}\left({\mu }_{m},{\mu }_{m},\mu \right),\end{array}$
we can conclude that

□

Proposition 2.6 Let $\left(\mathcal{C}\left(X\right),{D}_{G,\mathrm{\infty }}\right)$ be a metric space and $\left\{{\mu }_{n}\right\}$ a sequence in $\mathcal{C}\left(X\right)$, then the following are equivalent:
1. (i)

The sequence $\left\{{\mu }_{n}\right\}$ is ${D}_{G,\mathrm{\infty }}$-Cauchy.

2. (ii)

For every $\epsilon >0$, there exists $N\in \mathbb{N}$ such that ${D}_{G,\mathrm{\infty }}\left({\mu }_{n},{\mu }_{m},{\mu }_{m}\right)<\epsilon$ for all $n,m>N$.

3. (iii)

$\left\{{\mu }_{n}\right\}$ is a Cauchy sequence in the metric space $\left(\mathcal{C}\left(X\right),{D}_{\mathrm{\infty }}\right)$.

Proof “(i) (ii)” is evidence.

“(ii) (iii)” Suppose that for every $\epsilon >0$, there exists $N\in \mathbb{N}$ such that ${D}_{G,\mathrm{\infty }}\left({\mu }_{n},{\mu }_{m},{\mu }_{m}\right)<\epsilon$, for all $n,m>N$, then as $n,m\to +\mathrm{\infty }$,
${G}_{\mathrm{\infty }}\left({\mu }_{n},{\mu }_{m},{\mu }_{m}\right)\to 0$
(12)
and
${G}_{\mathrm{\infty }}\left({\mu }_{m},{\mu }_{m},{\mu }_{n}\right)\to 0.$
(13)
It follows that
$\underset{0\le \alpha \le 1}{sup}\underset{x\in {\left[{\mu }_{n}\right]}_{\alpha }}{sup}{d}_{G}\left(x,{\left[{\mu }_{m}\right]}_{\alpha }\right)=\frac{1}{2}{G}_{\mathrm{\infty }}\left({\mu }_{n},{\mu }_{m},{\mu }_{m}\right)\to 0.$
(14)
From (13) and
$0\le \underset{0\le \alpha \le 1}{sup}\underset{x\in {\left[{\mu }_{m}\right]}_{\alpha }}{sup}{d}_{G}\left(x,{\left[{\mu }_{n}\right]}_{\alpha }\right)\le {G}_{\mathrm{\infty }}\left({\mu }_{m},{\mu }_{m},{\mu }_{n}\right),$
we have
(15)
By (14) and (15), we have

that is, $\left\{{\mu }_{n}\right\}$ is a Cauchy sequence in the metric space $\left(\mathcal{C}\left(X\right),{D}_{\mathrm{\infty }}\right)$.

“(iii) (ii)” Suppose ${D}_{\mathrm{\infty }}\left({\mu }_{n},{\mu }_{m}\right)\to 0$ as $n,m\to +\mathrm{\infty }$, then (14) and (15) hold. Moreover,
$0\le \underset{0\le \alpha \le 1}{sup}{d}_{G}\left({\left[{\mu }_{m}\right]}_{\alpha },{\left[{\mu }_{n}\right]}_{\alpha }\right)\le \underset{0\le \alpha \le 1}{sup}\underset{x\in {\left[{\mu }_{m}\right]}_{\alpha }}{sup}{d}_{G}\left(x,{\left[{\mu }_{n}\right]}_{\alpha }\right)$
and (15) imply that
(16)
From (14), (15) and (16), we have as $n,m\to +\mathrm{\infty }$,
$\begin{array}{rcl}{G}_{\mathrm{\infty }}\left({\mu }_{n},{\mu }_{m},{\mu }_{m}\right)& =& \underset{0\le \alpha \le 1}{sup}G\left({\left[{\mu }_{n}\right]}_{\alpha },{\left[{\mu }_{m}\right]}_{\alpha },{\left[{\mu }_{m}\right]}_{\alpha }\right)\\ =& \underset{0\le \alpha \le 1}{sup}\underset{x\in {\left[{\mu }_{n}\right]}_{\alpha }}{sup}2{d}_{G}\left(x,{\left[{\mu }_{m}\right]}_{\alpha }\right)\to 0\end{array}$
(17)
and
$\begin{array}{rcl}{G}_{\mathrm{\infty }}\left({\mu }_{m},{\mu }_{m},{\mu }_{n}\right)& =& \underset{0\le \alpha \le 1}{sup}G\left({\left[{\mu }_{m}\right]}_{\alpha },{\left[{\mu }_{m}\right]}_{\alpha },{\left[{\mu }_{n}\right]}_{\alpha }\right)\\ =& \underset{0\le \alpha \le 1}{sup}\underset{x\in {\left[{\mu }_{m}\right]}_{\alpha }}{sup}\left[{d}_{G}\left(x,{\left[{\mu }_{n}\right]}_{\alpha }\right)+{d}_{G}\left({\left[{\mu }_{m}\right]}_{\alpha },{\left[{\mu }_{n}\right]}_{\alpha }\right)\right]\to 0.\end{array}$
(18)
We can get from (17) and (18) that

□

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 $\left(\mathcal{C}\left(X\right),{D}_{G,\mathrm{\infty }}\right)$ is complete provided $\left(X,G\right)$ 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 $\mu ,{\mu }_{1},{\mu }_{2}\in \mathcal{C}\left(X\right)$ and ${\mu }_{1}\subset {\mu }_{2}$, then
1. (i)

${G}_{\mathrm{\infty }}\left({\mu }_{1},\mu ,\mu \right)\le {G}_{\mathrm{\infty }}\left({\mu }_{2},\mu ,\mu \right)$,

2. (ii)

${G}_{\mathrm{\infty }}\left(\mu ,{\mu }_{2},{\mu }_{2}\right)\le {G}_{\mathrm{\infty }}\left(\mu ,{\mu }_{1},{\mu }_{1}\right)\le {D}_{G,\mathrm{\infty }}\left(\mu ,{\mu }_{1},{\mu }_{1}\right)$,

3. (iii)

${G}_{\mathrm{\infty }}\left({\mu }_{1},{\mu }_{2},{\mu }_{2}\right)=0$.

## 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 $\mathcal{C}\left(X\right)$ into $\mathcal{C}\left(X\right)$, i.e., $F\left(\mu \right)\in \mathcal{C}\left(X\right)$ for each $\mu \in \mathcal{C}\left(X\right)$. ${\mu }_{0}\in \mathcal{C}\left(X\right)$ is said to be a fixed point of a fuzzy self-mapping F of $\mathcal{C}\left(X\right)$ if and only if ${\mu }_{0}\subset F\left({\mu }_{0}\right)$.

Let Φ denote all functions $\varphi :\left[0,+\mathrm{\infty }\right)\to \left[0,+\mathrm{\infty }\right)$ satisfying:
1. (i)

ϕ is non-decreasing and continuous from the right,

2. (ii)

${\sum }_{n=1}^{\mathrm{\infty }}{\varphi }^{n}\left(t\right)<+\mathrm{\infty }$, 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 \left(t\right).

Theorem 3.1 Let $\left(X,G\right)$ be a G-complete metric space and ${\left\{{T}_{i}\right\}}_{i=1}^{\mathrm{\infty }}$ a sequence of fuzzy self-mappings of $\mathcal{C}\left(X\right)$. Suppose that for each ${\mu }_{1},{\mu }_{2}\in \mathcal{C}\left(X\right)$ and for arbitrary positive integers i and j, $i\ne j$,
(19)

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

Proof Let ${\mu }_{0}\in \mathcal{C}\left(X\right)$ and ${\mu }_{1}\subset {T}_{1}{\mu }_{0}$, by Proposition 2.3, there exists ${\mu }_{2}\in \mathcal{C}\left(X\right)$ such that ${\mu }_{2}\subset {T}_{2}{\mu }_{1}$ and
${D}_{G,\mathrm{\infty }}\left({\mu }_{1},{\mu }_{2},{\mu }_{2}\right)\le {D}_{G,\mathrm{\infty }}\left({T}_{1}{\mu }_{0},{T}_{2}{\mu }_{1},{T}_{2}{\mu }_{1}\right).$
Again by Proposition 2.3, we can find ${\mu }_{3}\in \mathcal{C}\left(X\right)$ such that ${\mu }_{3}\subset {T}_{3}{\mu }_{2}$ and
${D}_{G,\mathrm{\infty }}\left({\mu }_{2},{\mu }_{3},{\mu }_{3}\right)\le {D}_{G,\mathrm{\infty }}\left({T}_{2}{\mu }_{1},{T}_{3}{\mu }_{2},{T}_{3}{\mu }_{2}\right).$
Continuing this process, we can construct a sequence $\left\{{\mu }_{n}\right\}$ in $\mathcal{C}\left(X\right)$ such that
${\mu }_{n+1}\subset {T}_{n+1}{\mu }_{n},\phantom{\rule{1em}{0ex}}n=0,1,2,\dots$
and
${D}_{G,\mathrm{\infty }}\left({\mu }_{n},{\mu }_{n+1},{\mu }_{n+1}\right)\le {D}_{G,\mathrm{\infty }}\left({T}_{n}{\mu }_{n-1},{T}_{n+1}{\mu }_{n},{T}_{n+1}{\mu }_{n}\right),\phantom{\rule{1em}{0ex}}n=1,2,\dots .$
(20)
By (19), (20), Proposition 2.8 and (v) in Proposition 2.4, we have
(21)
Suppose that $0\le {D}_{G,\mathrm{\infty }}\left({\mu }_{n-1},{\mu }_{n},{\mu }_{n}\right)<{D}_{G,\mathrm{\infty }}\left({\mu }_{n},{\mu }_{n+1},{\mu }_{n+1}\right)$, then
${D}_{G,\mathrm{\infty }}\left({\mu }_{n},{\mu }_{n+1},{\mu }_{n+1}\right)\le \varphi \left({D}_{G,\mathrm{\infty }}\left({\mu }_{n},{\mu }_{n+1},{\mu }_{n+1}\right)\right)<{D}_{G,\mathrm{\infty }}\left({\mu }_{n},{\mu }_{n+1},{\mu }_{n+1}\right),$

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

Hence,
${D}_{G,\mathrm{\infty }}\left({\mu }_{n},{\mu }_{n+1},{\mu }_{n+1}\right)\le {D}_{G,\mathrm{\infty }}\left({\mu }_{n-1},{\mu }_{n},{\mu }_{n}\right)$
(22)
and
${D}_{G,\mathrm{\infty }}\left({\mu }_{n},{\mu }_{n+1},{\mu }_{n+1}\right)\le \varphi \left({D}_{G,\mathrm{\infty }}\left({\mu }_{n-1},{\mu }_{n},{\mu }_{n}\right)\right)\le \cdots \le {\varphi }^{n}\left({D}_{G,\mathrm{\infty }}\left({\mu }_{0},{\mu }_{1},{\mu }_{1}\right)\right).$
(23)

Now, we prove that $\left\{{\mu }_{n}\right\}$ 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 ${D}_{G,\mathrm{\infty }}\left({\mu }_{0},{\mu }_{1},{\mu }_{1}\right)=0$, then ${\mu }_{0}={\mu }_{1}$. Inequality (21) implies that
$\begin{array}{rcl}{D}_{G,\mathrm{\infty }}\left({\mu }_{1},{\mu }_{2},{\mu }_{2}\right)& \le & \varphi \left(max\left\{{D}_{G,\mathrm{\infty }}\left({\mu }_{0},{\mu }_{1},{\mu }_{1}\right),{D}_{G,\mathrm{\infty }}\left({\mu }_{1},{\mu }_{2},{\mu }_{2}\right)\right\}\right)\\ =& \varphi \left({D}_{G,\mathrm{\infty }}\left({\mu }_{1},{\mu }_{2},{\mu }_{2}\right)\right).\end{array}$

It follows from $\varphi \left(t\right) that ${D}_{G,\mathrm{\infty }}\left({\mu }_{1},{\mu }_{2},{\mu }_{2}\right)=0$, 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 ${D}_{G,\mathrm{\infty }}\left({\mu }_{0},{\mu }_{1},{\mu }_{1}\right)>0$. ${\sum }_{i=1}^{\mathrm{\infty }}{\varphi }^{i}\left(t\right)<\mathrm{\infty }$ and (24) yield that
(25)
Case 2. If $m, from (25) and
$\begin{array}{rcl}0& \le & {D}_{G,\mathrm{\infty }}\left({\mu }_{n},{\mu }_{m},{\mu }_{m}\right)={D}_{G,\mathrm{\infty }}\left({\mu }_{m},{\mu }_{m},{\mu }_{n}\right)\\ \le & {D}_{G,\mathrm{\infty }}\left({\mu }_{m},{\mu }_{n},{\mu }_{n}\right)+{D}_{G,\mathrm{\infty }}\left({\mu }_{n},{\mu }_{m},{\mu }_{n}\right)=2{D}_{G,\mathrm{\infty }}\left({\mu }_{m},{\mu }_{n},{\mu }_{n}\right),\end{array}$
we can get that
(26)

Thus, (25) and (26) imply that $\left\{{\mu }_{n}\right\}$ is a ${D}_{G,\mathrm{\infty }}$-Cauchy sequence. As $\left(X,G\right)$ is G-complete, by Proposition 2.7, we conclude that $\left(\mathcal{C}\left(X\right),{D}_{G,\mathrm{\infty }}\right)$ is complete. There exists ${\mu }_{\ast }\in \mathcal{C}\left(X\right)$ such that ${D}_{G,\mathrm{\infty }}\left({\mu }_{\ast },{\mu }_{m},{\mu }_{m}\right)\to 0$ 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
${G}_{\mathrm{\infty }}\left({\mu }_{\ast },{T}_{i}{\mu }_{\ast },{T}_{i}{\mu }_{\ast }\right)\le \varphi \left({G}_{\mathrm{\infty }}\left({\mu }_{\ast },{T}_{i}{\mu }_{\ast },{T}_{i}{\mu }_{\ast }\right)\right).$

It implies that ${G}_{\mathrm{\infty }}\left({\mu }_{\ast },{T}_{i}{\mu }_{\ast },{T}_{i}{\mu }_{\ast }\right)=0$, that is, ${\mu }_{\ast }\subset {T}_{i}{\mu }_{\ast }$.

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

Corollary 3.1 Let $\left(X,G\right)$ be a G-complete metric space and ${\left\{{T}_{i}\right\}}_{i=1}^{\mathrm{\infty }}$ a sequence of fuzzy self-mappings of $\mathcal{C}\left(X\right)$. Suppose that for each ${\mu }_{1},{\mu }_{2}\in \mathcal{C}\left(X\right)$ and for arbitrary positive integers i and j, $i\ne j$,
$\begin{array}{c}{D}_{G,\mathrm{\infty }}\left({T}_{i}{\mu }_{1},{T}_{j}{\mu }_{2},{T}_{j}{\mu }_{2}\right)\hfill \\ \phantom{\rule{1em}{0ex}}\le k\left(max\left\{{D}_{G,\mathrm{\infty }}\left({\mu }_{1},{\mu }_{2},{\mu }_{2}\right),{G}_{\mathrm{\infty }}\left({\mu }_{1},{T}_{i}{\mu }_{1},{T}_{i}{\mu }_{1}\right),{G}_{\mathrm{\infty }}\left({\mu }_{2},{T}_{j}{\mu }_{2},{T}_{j}{\mu }_{2}\right),\hfill \\ \phantom{\rule{2em}{0ex}}\frac{1}{2}\left[{G}_{\mathrm{\infty }}\left({\mu }_{1},{T}_{j}{\mu }_{2},{T}_{j}{\mu }_{2}\right)+{G}_{\mathrm{\infty }}\left({\mu }_{2},{T}_{i}{\mu }_{1},{T}_{i}{\mu }_{1}\right)\right]\right\}\right),\hfill \end{array}$

where $k\in \left(0,1\right)$. Then there exists at least one ${\mu }_{\ast }\in \mathcal{C}\left(X\right)$ 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 $X=\left\{0,1,2,3,\dots \right\}$. Define $G:X×X×X\to X$ by

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

For $\mu ,\nu \in \mathcal{C}\left(X\right)$, $y\in X$ and $\lambda >0$, owing to Zadeh’s extension principle [25], scalar multiplication and addition are defined by
$\left(\lambda \mu \right)\left(y\right)=\mu \left(\frac{y}{\lambda }\right)$
and
$\left(\mu +\nu \right)\left(x\right)=\underset{{x}_{1},{x}_{2}:{x}_{1}+{x}_{2}=x}{sup}min\left\{\mu \left({x}_{1}\right),\nu \left({x}_{2}\right)\right\}.$
For any $0 and $\mu ,\nu ,\omega \in \mathcal{C}\left(X\right)$, we can get easily from the definition of $G\left(x,y,z\right)$ that
${D}_{G,\mathrm{\infty }}\left(a\mu ,a\nu ,a\omega \right)=a{D}_{G,\mathrm{\infty }}\left(\mu ,\nu ,\omega \right)$
(28)
and
${D}_{G,\mathrm{\infty }}\left(\mu ,a\nu ,a\nu \right)\le {D}_{G,\mathrm{\infty }}\left(\mu ,\nu ,\nu \right).$
(29)
Now, suppose $0, define ${\mu }_{0}:X\to \mathcal{C}\left(X\right)$ by
Suppose $0, define ${\left\{{T}_{i}\right\}}_{i=1}^{\mathrm{\infty }}$ a sequence of fuzzy self-mappings of $\mathcal{C}\left(X\right)$ as
For any $i,j\in {Z}^{+}$, without loss of generality, suppose $i. For each ${\mu }_{1},{\mu }_{2}\in \mathcal{C}\left(X\right)$, by (28), (29) and the definition of α-level set, we have
$\begin{array}{c}{D}_{G,\mathrm{\infty }}\left({T}_{i}{\mu }_{1},{T}_{j}{\mu }_{2},{T}_{j}{\mu }_{2}\right)\hfill \\ \phantom{\rule{1em}{0ex}}={D}_{G,\mathrm{\infty }}\left({q}^{i}{\mu }_{1}+{\mu }_{0},{q}^{j}{\mu }_{2}+{\mu }_{0},{q}^{j}{\mu }_{2}+{\mu }_{0}\right)\hfill \\ \phantom{\rule{1em}{0ex}}\le {q}^{i}{D}_{G,\mathrm{\infty }}\left({\mu }_{1},{q}^{j-i}{\mu }_{2},{q}^{j-i}{\mu }_{2}\right)\le {q}^{i}{D}_{G,\mathrm{\infty }}\left({\mu }_{1},{\mu }_{2},{\mu }_{2}\right)\hfill \\ \phantom{\rule{1em}{0ex}}\le {q}^{i}\left(max\left\{{D}_{G,\mathrm{\infty }}\left({\mu }_{1},{\mu }_{2},{\mu }_{2}\right),{G}_{\mathrm{\infty }}\left({\mu }_{1},{T}_{i}{\mu }_{1},{T}_{i}{\mu }_{1}\right),{G}_{\mathrm{\infty }}\left({\mu }_{2},{T}_{j}{\mu }_{2},{T}_{j}{\mu }_{2}\right),\hfill \\ \phantom{\rule{2em}{0ex}}\frac{1}{2}\left[{G}_{\mathrm{\infty }}\left({\mu }_{1},{T}_{j}{\mu }_{2},{T}_{j}{\mu }_{2}\right)+{G}_{\mathrm{\infty }}\left({\mu }_{2},{T}_{i}{\mu }_{1},{T}_{i}{\mu }_{1}\right)\right]\right\}\right).\hfill \end{array}$
Therefore, ${\left\{{T}_{i}\right\}}_{i=1}^{\mathrm{\infty }}$ satisfy the conditions of Theorem 3.1 with $\varphi \left(t\right)={q}^{i}t$. Moreover, for each $0,

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

## 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, ${\left\{{T}_{i}\right\}}_{i=1}^{n}$ in Example 3.1 is an IFFS and $\left\{{\mu }_{b}:0, where A is the set of attractors of IFFS. Moreover, we can estimate the area of attractors basing on the fixed points of ${\left\{{T}_{i}\right\}}_{i=1}^{n}$. 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 [2628]). Therefore, our results provide a new method for studying the fuzzy differential equation in G-metric spaces.

## Declarations

### Acknowledgements

The authors thank the editor and the referees for their useful comments and suggestions. The research was supported by the National Natural Science Foundation of China (11071108) and supported partly by the Provincial Natural Science Foundation of Jiangxi, China (2010GZS0147, 20114BAB201007, 20114BAB201003).

## Authors’ Affiliations

(1)
Department of Mathematics, Nanchang University, Nanchang, 330031, P.R. China
(2)
Department of Mathematics, Jiangxi Agricultural University, Nanchang, 330045, P.R. China

## References

1. Mustafa Z, Sims B: A new approach to generalized metric spaces. J. Nonlinear Convex Anal. 2006, 7: 289–297.
2. Abbas M, Khan AR, Nazir T: Coupled common fixed point results in two generalized metric spaces. Appl. Math. Comput. 2011, 217: 6328–6336. 10.1016/j.amc.2011.01.006
3. Abbas M, Nazir T, Dorić D: Common fixed point of mappings satisfying (E.A) property in generalized metric spaces. Appl. Math. Comput. 2012, 218: 7665–7670. 10.1016/j.amc.2011.11.113
4. Abbas M, Nazir T, Vetro P: Common fixed point results for three maps in G -metric spaces. Filomat 2011, 25: 1–17.
5. Choudhury BS, Maity P: Coupled fixed point results in generalized metric spaces. Math. Comput. Model. 2011, 54: 73–79. 10.1016/j.mcm.2011.01.036
6. Mustafa Z, Sims B: Fixed point theorems for contractive mappings in complete G -metric spaces. Fixed Point Theory Appl. 2009., 2009: Article ID 917175Google Scholar
7. Shatanawi W: Fixed point theory for contractive mappings satisfying Φ-maps in G -metric spaces. Fixed Point Theory Appl. 2010., 2010: Article ID 181650Google Scholar
8. Abbas M, Nazir T, Radenović S: Common fixed point of generalized weakly contractive maps in partially ordered G -metric spaces. Appl. Math. Comput. 2012, 218: 9383–9395. 10.1016/j.amc.2012.03.022
9. Aydi H, Damjanović B, Samet B, Shatanawi W: Coupled fixed point theorems for nonlinear contractions in partially ordered G -metric spaces. Math. Comput. Model. 2011, 54: 2443–2450. 10.1016/j.mcm.2011.05.059
10. Aydi H, Postolache M, Shatanawi W: Coupled fixed point results for $\left(\psi ,\phi \right)$-weakly contractive mappings in ordered G -metric spaces. Comput. Math. Appl. 2012, 63: 298–309. 10.1016/j.camwa.2011.11.022
11. Cho YJ, Rhoades BE, Saadati R, Samet B, Shatanawi W: Nonlinear coupled fixed point theorems in ordered generalized metric spaces with integral type. Fixed Point Theory Appl. 2012., 2012: Article ID 8. doi:10.1186/1687–1812–2012–8Google Scholar
12. Luong NV, Thuan NX: Coupled fixed point theorems in partially ordered G -metric spaces. Math. Comput. Model. 2012, 55: 1601–1609. 10.1016/j.mcm.2011.10.058
13. Saadati R, Vaezpour SM, Vetro P, Rhoades BE: Fixed point theorems in generalized partially ordered G -metric spaces. Math. Comput. Model. 2010, 52: 797–801. 10.1016/j.mcm.2010.05.009
14. Kaewcharoen A, Kaewknao A: Common fixed points for single-valued and multi-valued mappings in G -metric spaces. Int. J. Math. Anal. 2011, 5: 1775–1790.Google Scholar
15. Tahat N, Aydi H, Karapinar E, Shatanawi W: Common fixed points for single-valued and multi-valued maps satisfying a generalized contraction in G -metric spaces. Fixed Point Theory Appl. 2012., 2012: Article ID 48. doi:10.1186/1687–1812–2012–48Google Scholar
16. Heilpern S: Fuzzy mappings and fuzzy fixed point theorems. J. Math. Anal. Appl. 1981, 83: 566–569. 10.1016/0022-247X(81)90141-4
17. Abu-Donia HM: Common fixed point theorems for fuzzy mappings in metric space under φ -contraction condition. Chaos Solitons Fractals 2007, 34: 538–543. 10.1016/j.chaos.2005.03.055
18. Azam A, Arshad M, Beg I: Fixed points of fuzzy contractive and fuzzy locally contractive maps. Chaos Solitons Fractals 2009, 42: 2836–2841. 10.1016/j.chaos.2009.04.026
19. Beg I, Ahmed MA: Common fixed point for generalized fuzzy contraction mappings satisfying an implicit relation. Appl. Math. Lett. 2012. doi:10.1016/j.aml.2011.12.019Google Scholar
20. Ćirić L, Abbas M, Damjanović B: Common fuzzy fixed point theorems in ordered metric spaces. Math. Comput. Model. 2011, 53: 1737–1741. 10.1016/j.mcm.2010.12.050
21. Kamran T: Common fixed points theorems for fuzzy mappings. Chaos Solitons Fractals 2008, 38: 1378–1382. 10.1016/j.chaos.2008.04.031
22. Lee BS, Lee GM, Cho SJ, Kim DS: A common fixed point theorem for a pair of fuzzy mappings. Fuzzy Sets Syst. 1998, 98: 133–136. 10.1016/S0165-0114(96)00345-4
23. Qiu D, Shu L: Supremum metric on the space of fuzzy sets and common fixed point theorems for fuzzy mappings. Inf. Sci. 2008, 178: 3595–3604. 10.1016/j.ins.2008.05.018
24. Qiu D, Shu L, Guan J: Common fixed point theorems for fuzzy mappings under Φ-contraction condition. Chaos Solitons Fractals 2009, 41: 360–367. 10.1016/j.chaos.2008.01.003
25. Zadeh LA: The concept of linguistic variable and its application to approximate seasoning-part I. Inf. Sci. 1975, 8: 199–249. 10.1016/0020-0255(75)90036-5
26. Balasubramaniam P, Muralisankar S: Existence and uniqueness of a fuzzy solution for the nonlinear fuzzy neutral functional differential equation. Comput. Math. Appl. 2001, 42: 961–967. 10.1016/S0898-1221(01)00212-7
27. Balasubramaniam P, Muralisankar S: Existence and uniqueness of a fuzzy solution for semilinear fuzzy integrodifferential equations with nonlocal conditions. Comput. Math. Appl. 2004, 47: 1115–1122. 10.1016/S0898-1221(04)90091-0
28. Nieto JJ, Rodríguez-López R, Georgiou DN: Fuzzy differential systems under generalized metric spaces approach. Dyn. Syst. Appl. 2008, 17: 1–24.Google Scholar