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Common fixed point theorems for fuzzy mappings in G-metric spaces
Fixed Point Theory and Applications volume 2012, Article number: 159 (2012)
Abstract
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.
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 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.
Lemma 1.1 Every G-metric space defines a metric space by
Definition 1.2 Let be a G-metric space. The sequence in X is said to be
-
(i)
G-convergent to x if for any , there exists and such that , for all .
-
(ii)
G-Cauchy if for any , there exists such that , for all .
Lemma 1.2 Let be a G-metric space, then for a sequence in X and point the following are equivalent:
-
(i)
is G-convergent to x.
-
(ii)
as .
-
(iii)
as .
-
(iv)
as .
Lemma 1.3 Let be a G-metric space, then for a sequence in X, the following are equivalent:
-
(i)
The sequence is G-Cauchy.
-
(ii)
For any , there exists such that , for all .
-
(iii)
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.
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 the family of all nonempty closed bounded subsets of X. Let be the Hausdorff G-distance on , i.e.,
where
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 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.
The α-level set of μ, denoted by , is defined as
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 .
Let , define
Lemma 2.1 [23]
The metric space is complete provided is complete.
For , , we define:
Proposition 2.1 If and , then there exists such that
Proof For , there exists such that
it follows that
□
Proposition 2.2 If and , then there exists such that and
Proof Let and let . For any and , Proposition 2.1 implies that is nonempty. Moreover, for any , there exists such that . It follows that
On the other hand, for any , there exists such that . Hence,
From (1) and (2), we have
Finally, we can conclude that from the closeness of C and the compactness of B. □
Proposition 2.3 Let and , then there exists such that and
Proof Let , by , we have . Let
we can get that . Let , then is nonempty compact and , for . From the proof of Proposition 2.2, we have
Similar to the proof of Theorem 3 in [23], we can conclude that there exists a fuzzy set such that for . By the compactness of , we have . Therefore,
□
Proposition 2.4 Let X be a nonempty set. For any , the following properties hold:
-
(i)
if and only if ,
-
(ii)
for all with ,
-
(iii)
for all with ,
-
(iv)
(symmetry in all three variables),
-
(v)
.
Proof The properties (i), (ii) and (iv) are readily derived from the definition of .
First, we prove the property (iii).
For any and , and , we have
it follows that
This implies that
Then,
Hence,
Similarly, we can prove that
By (3) and (4), we have
Now, we prove the property (v).
For any and , , we have
it follows that
From (5) and
we have
Similarly, we can obtain that
By (6), (7) and (8), we have
□
Remark 2.1 Proposition 2.4 implies that is a G-metric in , or more specially a Hausdorff G-metric in .
Definition 2.1 Let be a metric space. The sequence in is said to be
-
(i)
-convergent to μ if for every , there exists and such that for all ,
-
(ii)
-Cauchy if for every , there exists such that for all .
Proposition 2.5 Let be a metric space, then for a sequence and , the following are equivalent:
-
(i)
is -convergent to μ.
-
(ii)
as .
-
(iii)
as .
-
(iv)
as .
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.
“(i) ⟹ (ii)” Suppose as , then
and
Thus, for any ,
and
It follows that
“(ii) ⟹ (i)” Suppose as , then (9) and (10) hold.
Moreover, implies that as ,
From (9), (10) and (11), we have as ,
Thus, from
we can conclude that
□
Proposition 2.6 Let be a metric space and a sequence in , then the following are equivalent:
-
(i)
The sequence is -Cauchy.
-
(ii)
For every , there exists such that for all .
-
(iii)
is a Cauchy sequence in the metric space .
Proof “(i) ⟺ (ii)” is evidence.
“(ii) ⟹ (iii)” Suppose that for every , there exists such that , for all , then as ,
and
It follows that
From (13) and
we have
By (14) and (15), we have
that is, is a Cauchy sequence in the metric space .
“(iii) ⟹ (ii)” Suppose as , then (14) and (15) hold. Moreover,
and (15) imply that
From (14), (15) and (16), we have as ,
and
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 is complete provided is G-complete.
From the definitions of and , we can get the next proposition readily.
Proposition 2.8 If and , then
-
(i)
,
-
(ii)
,
-
(iii)
.
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 into , i.e., for each . is said to be a fixed point of a fuzzy self-mapping F of if and only if .
Let Φ denote all functions satisfying:
-
(i)
ϕ is non-decreasing and continuous from the right,
-
(ii)
, for all , where denotes the n th iterative function of ϕ.
Remark 3.1 It can be directly verified that for any and all , .
Theorem 3.1 Let be a G-complete metric space and a sequence of fuzzy self-mappings of . Suppose that for each and for arbitrary positive integers i and j, ,
where . Then there exists at least one such that for all .
Proof Let and , by Proposition 2.3, there exists such that and
Again by Proposition 2.3, we can find such that and
Continuing this process, we can construct a sequence in such that
and
By (19), (20), Proposition 2.8 and (v) in Proposition 2.4, we have
Suppose that , then
which is a contradiction since .
Hence,
and
Now, we prove that is a -Cauchy sequence. For positive integers m, n, we distinguish the following two cases.
Case 1. If , then
Assume that , then . Inequality (21) implies that
It follows from that , that is, . By induction, we have . Thus, , .
Suppose that . and (24) yield that
Case 2. If , from (25) and
we can get that
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 .
Now, Proposition 2.8 and (19) imply that
Letting , we can see from (27) and Proposition 2.5 that
It implies that , that is, .
If in Theorem 3.1 we choose , where is a constant, we obtain the following corollary. □
Corollary 3.1 Let be a G-complete metric space and a sequence of fuzzy self-mappings of . Suppose that for each and for arbitrary positive integers i and j, ,
where . Then there exists at least one such that for all .
The following example illustrates Theorem 3.1.
Example 3.1 Let . Define by
Then X is a complete nonsymmetric G-metric space [5].
For , and , owing to Zadeh’s extension principle [25], scalar multiplication and addition are defined by
and
For any and , we can get easily from the definition of that
and
Now, suppose , define by
Suppose , define a sequence of fuzzy self-mappings of as
For any , without loss of generality, suppose . For each , by (28), (29) and the definition of α-level set, we have
Therefore, satisfy the conditions of Theorem 3.1 with . Moreover, for each ,
is a common fixed point of .
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, 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.
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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).
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Zhu, L., Zhu, CX. & Chen, CF. Common fixed point theorems for fuzzy mappings in G-metric spaces. Fixed Point Theory Appl 2012, 159 (2012). https://doi.org/10.1186/1687-1812-2012-159
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DOI: https://doi.org/10.1186/1687-1812-2012-159