Common fixed point theorems for fuzzy mappings in G-metric spaces
© Zhu et al.; licensee Springer 2012
Received: 9 June 2012
Accepted: 6 September 2012
Published: 19 September 2012
In this paper, we introduce the concept of Hausdorff G-metric in the space of fuzzy sets induced by the metric 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.
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  introduced the G-metric space as a generalization of metric spaces. We now recall some definitions and results in G-metric spaces in .
Definition 1.1 Let X be a nonempty set, and let be a function satisfying:
(G 1) if ,
(G 2) for all with ,
(G 3) for all , with ,
(G 4) (symmetry in all three variables),
(G 5) for all (rectangle inequality).
Then the function G is called a generalized metric or, more specifically, a G-metric on X, and the pair is a G-metric space.
G-convergent to x if for any , there exists and such that , for all .
G-Cauchy if for any , there exists such that , for all .
is G-convergent to x.
The sequence is G-Cauchy.
For any , there exists such that , for all .
is a Cauchy sequence in the metric space .
Definition 1.3 A G-metric space is said to be G-complete if every G-Cauchy sequence in is G-convergent in .
Lemma 1.4 A G-metric space is G-complete if and only if is a complete metric space.
The existence of fixed points of fuzzy mappings has been an active area of research interest since Heilpern  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 . 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 be a metric space, a fuzzy set in X is a function with domain X and values in . If μ is a fuzzy set and , then the function value is called the grade of membership of x in μ.
where is the closure of the non-fuzzy set B.
Let be the family of all nonempty compact subsets of X. Denote by the totality of fuzzy sets which satisfy that for each , . Let , then is said to be more accurate than , denoted by , if and only if for each . if and only if and .
Lemma 2.1 
The metric space is complete provided is complete.
Finally, we can conclude that from the closeness of C and the compactness of B. □
if and only if ,
for all with ,
for all with ,
(symmetry in all three variables),
Proof The properties (i), (ii) and (iv) are readily derived from the definition of .
First, we prove the property (iii).
Now, we prove the property (v).
Remark 2.1 Proposition 2.4 implies that is a G-metric in , or more specially a Hausdorff G-metric in .
-convergent to μ if for every , there exists and such that for all ,
-Cauchy if for every , there exists such that for all .
is -convergent to μ.
Proof Since 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.
“(ii) ⟹ (i)” Suppose as , then (9) and (10) hold.
The sequence is -Cauchy.
For every , there exists such that for all .
is a Cauchy sequence in the metric space .
Proof “(i) ⟺ (ii)” is evidence.
that is, is a Cauchy sequence in the metric space .
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 is complete provided is G-complete.
From the definitions of and , we can get the next proposition readily.
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 .
Definition 3.1 
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 into , i.e., for each . is said to be a fixed point of a fuzzy self-mapping F of if and only if .
ϕ is non-decreasing and continuous from the right,
, for all , where denotes the n th iterative function of ϕ.
Remark 3.1 It can be directly verified that for any and all , .
where . Then there exists at least one such that for all .
which is a contradiction since .
Now, we prove that is a -Cauchy sequence. For positive integers m, n, we distinguish the following two cases.
It follows from that , that is, . By induction, we have . Thus, , .
Thus, (25) and (26) imply that is a -Cauchy sequence. As is G-complete, by Proposition 2.7, we conclude that is complete. There exists such that as .
It implies that , that is, .
If in Theorem 3.1 we choose , where is a constant, we obtain the following corollary. □
where . Then there exists at least one such that for all .
The following example illustrates Theorem 3.1.
Then X is a complete nonsymmetric G-metric space .
is a common fixed point of .
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 , we can get an iterated fuzzy function system (i.e., IFFS) corresponding the IFS . For example, in Example 3.1 is an IFFS and , where A is the set of attractors of IFFS. Moreover, we can estimate the area of attractors basing on the fixed points of . 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.
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).
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