#### Abstract

Under the -contraction conditions, we prove common fixed point theorems for self-mappings in the space of the bounded closed sets in the complete stationary fuzzy metric space with the -fuzzy metric for the bounded closed sets.

#### 1. Introduction

It is well known that not only the Hausdorff metric is very important concept in general topology and analysis, but also many authors have expansively developed the theory of fuzzy sets and application (see [1–11]). As a natural generalization of the concept of set, fuzzy sets were introduced initially by Zadeh [12] in 1965. Various concepts of the fuzzy metrics on ordinary set were considered in [13–19].

In [20], Rodríguez-López and Romaguera introduced and discussed a suitable notion for the Hausdorff fuzzy metric of a given fuzzy metric space (in the sense of George and Veeramani) on the set of its nonempty compact subsets. It is necessary to note that such fuzzy metric space has very important application in studying fixed point theorems for contraction-type mappings [21–30]. In fuzzy functional analysis, many researches have been done on the fixed point theory in the space of compact fuzzy sets equipped with the supremum metric [1, 16, 31–38].

In this paper, we will establish the completeness of with respect to the completeness of the stationary fuzzy metric space , where is the class of sets with nonempty bounded closed subsets of , and is the stationary Hausdorff fuzzy metric on induced by . Finally, we will prove some common fixed point theorems for self-mappings in the space .

#### 2. Preliminaries

We start this section by recalling some pertinent concepts.

*Definition 1 (see [39]). *A triangular norm (or -norm for short) is a binary operation on the unit interval , that is, a function , such that for all the following four axioms are satisfied:(1) (boundary condition);(2) whenever and (monotonicity);(3) (commutativity);(4) (associativity). A -norm is said to be continuous if it is a continuous function in ; a -norm is said to be positive if whenever . The following are examples of -norms: ; , where denotes the usual multiplication for all .

*Definition 2 (see [40]). *A stationary fuzzy metric space is an ordered triple such that is an arbitrary nonempty set, is a continuous -norm, and is a fuzzy set of satisfying the following conditions, for all :(1),(2) if and only if ,(3),(4).

If is a stationary fuzzy metric space, we will say that is a stationary fuzzy metric on . Since a stationary fuzzy metric is a special fuzzy metric, just like fuzzy metrics in [14], we can prove that every stationary fuzzy metric on generates a topology on which has as a base the family of sets of the form , where for all . A sequence in a stationary fuzzy metric space is said to be Cauchy if ; a sequence in converges to if [40].

*Example 3 (see [14]). *Let be a metric space. Denote by the usual multiplication for all , and define on by
for all . Then is a stationary fuzzy metric on which will be called a standard stationary fuzzy metric.

*Definition 4 (see [41]). *A stationary fuzzy pseudometric space is an ordered triple such that is an arbitrary nonempty set, is a continuous -norm, and is a fuzzy set of satisfying the following conditions, for all :(1) if and only if ,(2),(3).

If is a stationary fuzzy pseudometric space, we will say that is a stationary fuzzy pseudometric on . In the following we always suppose that the -norm is positive.

*Definition 5 (see [42]). *Let be a stationary fuzzy metric space and . If, for all , , then is an accumulation point of ; the set of all accumulation points of is called the derived set of , denoted by ; the union of and is called the closure of , denoted by . If , then is a closed set of .

*Definition 6 (see [42]). *Let be a stationary fuzzy metric space, and . If there exists such that for all we have , then we say that is a bounded subset of ; if itself is a bounded set we will say that is a bounded stationary fuzzy metric space.

Given a stationary fuzzy metric space , we will denote by , , and , the powerset, the set of nonempty subsets, and the set of nonempty bounded closed subsets of , respectively.

Let be a nonempty subset of a stationary fuzzy metric space . For all , let

For the empty index set , we will make the convention that, for , It follows that .

*Definition 7 (see [43]). *Let be a stationary fuzzy metric space. For all , we define a function on by
where .

*Definition 8. *Let be any stationary metric space. is said to be a fixed point of a self-mapping of if and only if .

#### 3. Main Results

Now we will establish our main theorems.

Proposition 9. *Let be a stationary fuzzy metric space. Then, for all ,*(1)* if and only if ** if and only if ** if and only if *,
(2)*for all **, **,*(3),
(4),
(5)*if **, then **,*(6),
(7)* if and only if **.*

*Proof. *In fact, we can prove this proposition by a similar proof of Proposition 1 in [42].

Theorem 10. *Let be a stationary fuzzy metric space, and then is a stationary fuzzy pseudometric space. *

*Proof. *Let ; by the definition of , (1) of Proposition 9, and the commutativity of , it is clear that if and only if and .

In addition, by (4) of Proposition 9 and the commutativity of , we obtain
Consequently, by the definition of , we get
We conclude that is a stationary fuzzy pseudometric space.

Proposition 11 (see [42]). *Let be a stationary fuzzy metric space. If are any two bounded subsets of , then is a bounded subset of . *

Theorem 12. *Let be a stationary fuzzy metric space. Then is a stationary fuzzy metric space. *

*Proof. *Let . By Proposition 11, we have , which means there exists such that , for all , . Hence, for any , we can get that
Thus we obtain
Similarly, we can get
Consequently, by the positivity of , we have .

By the definition of , (7) of Proposition 9, and the commutativity of , it is clear that if and only if and . In addition, by (4) of Proposition 9 and the commutativity of , we obtain
Consequently, by the definition of , we get
We conclude that is a stationary fuzzy metric space.

*Example 13. *Let be a metric space. Then the Hausdorff stationary fuzzy metric of the standard fuzzy metric coincides with the standard fuzzy metric of the Hausdorff fuzzy metric on .

In fact, let ; for each , we have
Consequently, we obtain
We conclude that on .

Let us recall that if is a uniform space, then the Hausdorff-Bourbaki uniformity (of ) on has as a base the family of sets of the form where [44].

The restriction of to will be also denoted by . In addition, if is a stationary fuzzy metric space, then is a (countable) base for the uniformity on compatible with , where for all . is called the uniformity induced by . In particular, is the uniformity induced by the Hausdorff stationary fuzzy metric of . We have the following useful result.

Theorem 14. *Let be a stationary fuzzy metric space. Then the Hausdorff-Bourbaki uniformity coincides with the uniformity on .*

*Proof. *If , then, for any , there exists , such that . Thus we obtain
for all . Consequently we have
Similarly, we can get
for all . Thus we conclude that

If , then, for each , we have
Let , and then, for each , there exists such that
Thus we obtain .

Similarly, by
we can get . It follows that
Hence we obtain the following relations:

We conclude that on .

Theorem 15. *Let be a stationary fuzzy metric space. Then is complete if and only if is complete. *

*Proof. *In fact, we can prove it by a similar proof of Theorem 3 in [20].

Another type of convergence for a sequence of sets was defined by Kuratowski.

We say that a sequence of sets , , converges to a set , denoted by , if where , .

We mention that, for sequence of closed sets, convergence in Hausdorff metric implies convergence in the sense of Kuratoski. But for sequence of bounded closed sets, both types of convergence are equivalent provided that the limit set is nonempty [37].

Lemma 16. *Let be a stationary fuzzy metric space and . Then*(1)*for arbitrarily and any , there exists such that ;*(2)*for any and any , there exists such that .*

*Proof. *We only prove (1) since it is equivalent to (2).

For each , there exists such that, for any ,
This completes the proof.

In fact, we can get a more general result.

Lemma 17. *Let be a stationary fuzzy metric space and . Then*(1)*for arbitrarily and any closed subset , there exists closed subset such that ;*(2)*for any closed subset and any , there exists closed subset such that .*

*Proof. *We only prove (1) since it is equivalent to (2). Let and let
Assume . For any , there exists a such that . By (2) of Proposition 9, we have
By the arbitrariness of , we have . Then we get

Conversely, suppose . Take a descending positive number sequence such that as . For each , there exists a such that

Let . Then we have , and . Hence we can get
Thus, we obtain . Let . For each , by Lemma 16, there exists a such that
Consequently, is a nonempty closed subset of .

For any , there exists such that , which implies that
Thus we obtain

By the definition of , we can get
for all . Thus we obtain
Consequently, we easily obtain the following inequality:

This completes the proof.

Lemma 18 (see [29]). *Let be a nondecreasing function satisfying the following conditions:*(i)* is continuous from the left,*(ii)*,**where denote the th iterative function of . Then*(1)*for each , such that ,*(2)*. *

Theorem 19. *Let be a complete stationary fuzzy metric space and let be a sequence of self-mappings of . If there exists a constant , such that, for each , and for arbitrary positive integers and , ,
**
where satisfies the conditions of Lemma 18. Then there exists an such that , for all . *

*Proof. *Let and , and . By Lemma 17, there exists , such that and
Again by Lemma 17, 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 inequality (38) and formula (41), we have
where , which implies that .

If , then
From , it follows that . Hence, there are two cases.*Case **1*. If , by (2) of Lemma 18 we can get
that is, .*Case **2*. If , by (1) of Lemma 18, we can get
Obviously, (43) and (45) are contradictory. Hence, we have
that is, .

Consequently, we easily obtain the following inequalities:
Thus, for arbitrary positive integer , we have

Since , for all , we get
which implies that . Hence, is a Cauchy sequence. In addition, since is a complete stationary fuzzy metric space, by Theorem 15, we get that is complete. Thus there exists an such that as ; that is, .

Next, we show that , that is, , for all . In fact, for arbitrary positive integers and , , by Proposition 9, we have
Moreover, we have
Since is continuous from the left and is a continuous positive -norm, we can obtain
Consequently, we conclude that
that is, . By (1) of Proposition 9, we obtain , for all .

Corollary 20. *Let be a complete stationary fuzzy metric space and let be a self-mapping of . If there exists a constant , such that, for each ,
**
where satisfies the conditions of Lemma 18, then there exists an such that . *

*Proof. *In fact, we can define a sequence of fuzzy self-mappings of as , for . Thus, this result is a special case of Theorem 19.

Theorem 21. *Let be a complete stationary fuzzy metric space and let be a sequence of self-mappings of . If there exists a constant , such that, for each , and for arbitrary positive integers and , ,
**
where is nondecreasing and continuous from the left for each variable, let , where . If
**
where denotes the th iterative function of , then there exists an such that , for all . *

*Proof. *Let and , and . By Lemma 17, there exists , such that and
Again by Lemma 17, 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 inequality (55) and formula (59), we have
where , which implies that .

If , then
From
we get . Hence, there are two cases.*Case **1*. If , by (2) of Lemma 18, we can get
that is, .*Case **2*. If , by (1) of Lemma 18, we can get
Obviously, (61) and (64) are contradictory. Hence, we have
that is, .

Consequently, we easily obtain the following relations:
Thus, for arbitrary positive integer , we have

By (, for all , and continuity of , we can get
which implies that . Hence, we get that is a Cauchy sequence. In addition, since is a complete stationary fuzzy metric space, by Theorem 15, we have that is complete. Thus there exists an such that as ; that is, .

Next, we show that , that is, , for all . In fact, for arbitrary positive integers and , , by Proposition 9, we have
Moreover, we have
Since is continuous from the left and is a continuous positive -norm, hence, we can obtain
Consequently, we conclude that
that is, . By (1) of Proposition 9, we obtain , for all .

Corollary 22. *Let be a complete stationary fuzzy metric space and let be a self-mapping of . If there exists a constant , such that, for each ,
**
where is nondecreasing and continuous from the left for each variable, let , where . If
**
where denotes the th iterative function of , then there exists an such that . *

*Proof. *In fact, we can define a sequence of fuzzy self-mappings of as , for . Thus, this result is a special case of Theorem 21.

Theorem 23. *Let be a complete stationary fuzzy metric space and let be a sequence of self-mappings of . If there exists a constant , such that, for each , and, for arbitrary positive integers and , ,
**
where satisfies the conditions of Lemma 18, then there exists an such that , for all . *

*Proof. *Let and , and . By Lemma 17, there exists , such that and
Again by Lemma 17, we can find such that and
By induction, we produce a sequence of points of such that