Research Article | Open Access

# Common Fixed Point Theorems in a New Fuzzy Metric Space

**Academic Editor:**Yeong-Cheng Liou

#### Abstract

We generalize the Hausdorff fuzzy metric in the sense of Rodríguez-López and Romaguera, and we introduce a new -fuzzy metric, where -fuzzy metric can be thought of as the degree of nearness between two fuzzy sets with respect to any positive real number. Moreover, under -contraction condition, in the fuzzy metric space, we give some common fixed point theorems for fuzzy mappings.

#### 1. Introduction

The concept of fuzzy sets was introduced initially by Zadeh [1] in 1965. After that, to use this concept in topology and analysis, many authors have expansively developed the theory of fuzzy sets and application [2, 3]. In the theory of fuzzy topological spaces, one of the main problems is to obtain an appropriate and consistent notion of fuzzy metric space. This problem was investigated by many authors [4–13] from different points of view. George and Veeramani’s fuzzy metric space [6] has been widely accepted as an appropriate notion of metric fuzziness in the sense that it provides rich topological structures which can be obtained, in many cases, from classical theorems. Further, it is necessary to mention that this fuzzy metric space has very important application in studying fixed point theorems for contraction-type mappings [7, 14–16]. Besides that, a number of metrics are used on subspaces of fuzzy sets. For example, the sendograph metric [17–19] and the -metric for fuzzy sets [20–25] induced by the Hausdorff-Pompeiu metric have been studied most frequently, where -metric is an ordinary metric between two fuzzy sets. Combining fuzzy metric (in the sense of George and Veeramani) and Hausdorff-Pompeiu metric, Rodríguez-Lópezand Romaguera [26] construct a Hausdorff fuzzy metric, where Hausdorff fuzzy metric can be thought of as the degree of nearness between two crisp nonempty compact sets with respect to any positive real number.

In this present investigation, considering the Hausdorff-Pompeiu metric and theories on fuzzy metric spaces (in the sense of George and Veeramani) together, we study the degree of nearness between two fuzzy sets as a natural generalization of the degree of nearness between two crisp sets, in turn, it helps in studying new problems in fuzzy topology. Based on the Hausdorff fuzzy metric , we introduce a suitable notion for the -fuzzy metric on the fuzzy sets whose -cut are nonempty compact for each . In particular, we explore several properties of -fuzzy metric. Then, under -contraction condition, we give some common fixed point theorems in the fuzzy metric space on fuzzy sets.

#### 2. Preliminaries

According to [27], a binary operation is called a continuous -norm if is an Abelian topological semigroups with unit 1 such that whenever and for all .

*Definition 2.1 (see [6]). *The 3-tuple is said to be a fuzzy metric space if is an arbitrary set, is a continuous -norm, and is a fuzzy set on satisfying the following conditions, for all :(i);(ii) if and only if ;(iii);(iv);(v) is continuous.

If is a fuzzy metric space, it will be said that is a fuzzy metric on .

A simply but useful fact [7] is that is nondecreasing for all . Let be a metric space. Denote by the usual multiplication for all , and let be the fuzzy set defined on by Then, is a fuzzy metric space, and is called the standard fuzzy metric induced by [8].

George and Veeramani [6] proved that every fuzzy metric on generates a topology on which has a base the family of open sets of the form: where for all and . They proved that is a Hausdorff first countable topological space. Moreover, if is a metric space, then the topology generated by coincides with the topology generated by the induced fuzzy metric (see [8]).

Lemma 2.2 (see [6]). *Let be a fuzzy metric space and let be the topology induced by the fuzzy metric. Then, for a sequence in , if and only if as for all .*

*Definition 2.3 (see [6]). *A sequence in a fuzzy metric space is called a Cauchy sequence if and only if for each , there exists such that for all . A fuzzy metric space is said to be complete if and only if every Cauchy sequence is convergent.

*Definition 2.4 (see [13]). *Let be a nonempty subset of a fuzzy metric space . For and , .

Lemma 2.5 (see [28]). *Let be a set and let be a family of subsets of such that*(1)*;*(2)* implies ;*(3)*, implies .**Then, the function defined by has the property that for every .*

Next, we recall some pertinent concepts on Hausdorff fuzzy metric. Denote by the set of nonempty closed and bounded subsets of a metric space . It is well known (see, e.g., [29]) that the function defined on by for all , is a metric on called the Hausdorff-Pompeiu metric. In [30], it is proved that the metric is complete provided is complete.

Let be the set of all nonempty compact subsets of a fuzzy metric space , , , according to [26], the Hausdorff fuzzy metric on is defined as where , and is a fuzzy metric on . It is shown that if and only if , and if and only if .

Lemma 2.6 (see [26]). *Let be a fuzzy metric space. Then, is complete if and only if is complete.*

Lemma 2.7 (see [26]). *Let be a metric space. Then, the Hausdorff fuzzy metric of the standard fuzzy metric coincides with standard fuzzy metric of the Hausdorff metric on .*

#### 3. On -Fuzzy Metric

Let be a fuzzy metric space. Denote by the totality of fuzzy sets: which satisfy that, for each , the -cut of , is nonempty compact in .

*Definition 3.1. *Let be a fuzzy metric space. The -fuzzy metric between two fuzzy sets is induced by the Hausdorff fuzzy metric as
where , and
is the fuzzy separation of from .

Lemma 3.2. *Let be a fuzzy metric space, , . Then one has*(1)*,*(2)*,*(3)* if and only if ,*(4)* if and only if ,*(5)*if , then ,*(6)*,*(7)*,
*(8)* is continuous.*

*Proof. *For (1), by the definition of the -cut , for every , is nonempty compact in . By the theorem of nested intervals, there exists a point in for every , likewise, there exists a points in for every . Thus, . Moreover, it is clear that .

For (2), it is clear that .

For (3), since if and only if for all , which implies for all , we have that if and only if .

For (4), it follows from (3).

For (5), for every , any , and , by the proof of Theorem 1 in [26], we have
with all , which implies
for all and all . Since , then . By (iv) of Definition 2.1 and the arbitrariness of and , we have
which implies
Consequently, .

For (6), for every , by the proof of (5) and (iv) of Definition 2.1, we have
Consequently, .

For (7), for every , by the proof of (6), we have
Similarly, it can be shown that
Hence, .

For (8), by the continuity on of the function , it is clear that is continuous.

Theorem 3.3. *Let be a fuzzy metric space. Then, is a fuzzy metric space, where is a fuzzy set on the .*

*Proof. * It is easily proved by Lemma 3.2.

*Example 3.4. *Let be the Euclidean metric on , and let and let be two compact intervals. Then, . Let be a fuzzy metric space, where the usual multiplication for all , and is defined on by
Denote by the totality of fuzzy sets which satisfy that for each , the -cut of is a nonempty compact interval. For any -cuts of fuzzy sets and for all , by a simple calculation, we have
So by Definition 3.1, we get

#### 4. Properties of the -Fuzzy Metric

*Definition 4.1. *Let be a fuzzy metric space. For , define with center a fuzzy set and radius as

Proposition 4.2. *Every is an open set.*

*Proof. * It is identical with the proof in [6].

Proposition 4.3. *Let be a fuzzy metric space. Define .**Then, is a topology on .*

*Proof. *
It is identical with the proof in [6].

*Definition 4.4. *A sequence in a fuzzy metric space is a Cauchy sequence if and only if for each , , there exists such that for all .

Lemma 4.5. *Let be a fuzzy metric space on fuzzy metric and let be the topology induced by the fuzzy metric . Then, for a sequence in , if and only if as .*

*Proof. * It is identical with the proof of Theorem 3.11 in [6].

Theorem 4.6. *The fuzzy metric space is complete provided is complete.*

*Proof. *Let be a complete fuzzy metric space and let a sequence be a Cauchy sequence in . Consider a fixed . Then, is a Cauchy sequence in , where denotes all nonempty compact subsets of .

Since is complete by Lemma 2.6, it follows that . Actually, from the definition of and the continuity of , it is easy to see that , uniformly in .

Now, consider the family , where . Take , we have
Since , it follows that . Thus, for each , if is large enough. Hence, , and by Lemma 3.2, we have .

Now, take and . We have to show that . It is clear that
On the other hand, we have
for fixed . However,
Consequently, for every , there exists such that . For given , since , there exists such that
for . Now,
for any . Since , we obtain
Now, for and all . Note that (since the convergence is uniform in ) does not depend on . Since decreases to , if follows that for some (depending on ).

Thus, , if is large.

Finally, by taking large enough, we obtain
that is,
From (4.3) and (4.9), it yields . Thus, Lemma 2.5 is applicable and there exists for every such that . It remains to show that in .

Let . Then, since is a Cauchy sequence, there exists such that implies .

Let be fixed. Then,
Thus, in the -fuzzy metric. The proof is completed.

Lemma 4.7. *Let be a compact fuzzy metric space and compact subsets . Then, for each and , there exists a such that .*

*Proof. *Suppose there exists a such that for any and . Then,
that is,
So,
This is a contradiction with .

Lemma 4.8. *Let be a compact fuzzy metric space, and . Then, for any compact set , there exists a compact set such that .*

*Proof. *Let there exists a such that and let . For any , by Lemma 4.7, there exists a such that
Thus, , moreover, is compact since it is closed in and .

Now, for any , , there exists a such that
Thus, we have , which implies that
Similarly, it can be shown that .

Hence, . This completes the proof.

Theorem 4.9. *Let be a compact fuzzy metric space and , . Then, for any satisfying , there exists a such that and
*

*Proof. *Since , and are normal, we have and for all . Let
and let . For any , by Lemma 4.7, there exists a such that
Thus, is nonempty compact in , moreover, if .

From the proof of Lemma 4.8, we have
By Lemma 3.1 in [28], there exists a fuzzy set with the property that for . Since are nonempty compact for all , we have . Consequently,
This completes the proof.

*Definition 4.10 (see [24]). *Let be any fuzzy metric space. is said to be a fuzzy mapping if and only if is a mapping from the space into , that is, for each .

#### 5. Common Fixed Point Theorems in the Fuzzy Metric Space on Fuzzy Sets

Theorem 5.1. *Let be a compact fuzzy metric space and let be a sequence of fuzzy self-mappings of . Let be a nondecreasing function satisfying the following condition: is continuous from the left and
**
where denote the th iterative function of . Suppose that for each , and for arbitrary positive integers and ,
**
then there exists such that for all .*

*Proof. * Let and . By Theorem 4.9, for any , there exists such that and
Again by Theorem 4.9, for any , we can find such that and
By induction, we produce a sequence of points of such that
Now, we prove that is a Cauchy sequence in . In fact, for arbitrary positive integer , by the inequality (5.2), Lemma 3.2, and the formula (5.5), we have
where implies , by (3) of Lemma 3.2. In addition, it is easy to get that for all . In fact, suppose that there exists some such that . Since is nondecreasing, we have
Since as , for all , then we have as . From the inequality (5.7), we have . This is a contradiction which implies for all . We can prove that . In fact, if , then from the inequality (5.6), we get
which is a contradiction. Thus, from the inequality (5.6), we have
Furthermore, for arbitrary positive integers and , we have
and as , for all , it follows that
is convergent, which implies that is a Cauchy sequence in . Since is a compact fuzzy metric space, it follows is complete. By Theorem 4.6, is complete. Let . Next, we show that for all . In fact, for arbitrary positive integers and , , by Theorem 4.9, we have
where implies . Letting , and using the left continuity of , we have
which implies . Hence, by Lemma 3.2, it follows that . Then, the proof is completed.

Theorem 5.2. *Let be a compact fuzzy metric space and let be a sequence of fuzzy self-mappings of . Suppose that for each , and for arbitrary positive integers and , , ,
**
where is nondecreasing and continuous from the left for each variable. Denote , where . If
**
where denote the th iterative function of , then there exists such that for all .*

*Proof. * Let and . By Theorem 4.9, for any , there exists such that and
Again by Theorem 4.9, for any , we can find such that and
By induction, we produce a sequence of points of such that
Now, we prove that is a Cauchy sequence in . In fact, for arbitrary positive integer , by the inequality (5.14), Lemma 3.2, and the formula (5.18), we have
where implies by (3) in Lemma 3.2 Likewise, we have for all , . If , then from the inequality (5.19), we obtain
which is a contradiction. Thus, from the inequality (5.19), we have
Furthermore, for arbitrary positive integers and , we have
Furthermore, for arbitrary positive integers and , we have
Since as , for all , it follows that
is convergent, this implies that is a Cauchy sequence in . Since is a compact fuzzy metric space, it follows that is complete. By Theorem 4.6, is complete. Let . Now, we show that for all . In fact, for arbitrary positive integers and , , by Theorem 4.9, we have
where implies . Letting , and using the left continuity of , we have
which implies . Hence, by Lemma 3.2, it follows that , then the proof is completed.

Now, we give an example to illustrate the validity of the results in fixed point theory. For simplicity, we only exemplify Theorem 5.1, while the example may be similarly constructed for Theorem 5.2.

*Example 5.3. *Let be a fuzzy metric space, where , , and are the same as in Example 3.4. Then, is a compact metric sp