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Fixed points of fuzzy contractive set-valued mappings and fuzzy metric completeness
Fixed Point Theory and Applications volume 2013, Article number: 276 (2013)
Abstract
In this paper, we introduce a new fuzzy contraction via a new concept of the fuzzy sets called fw-distances initiated in the paper, which is a generalization of a fuzzy contractive mapping initiated in the article (Fuzzy Sets Syst. 159:739-744, 2008). A fixed point theorem is established by using this type of contraction of set-valued mappings in fuzzy metric spaces which are complete in the sense of George and Veeramani. As an application of our results, we give characterizations of fuzzy metric completeness. The results are supported by examples.
1 Introduction
In Fuzzy metric spaces we refer to as KM-spaces were initiated by Kramosil and Michálek [1]. The conditions which they formulated were modified later by George and Veeramani [2] via proposing new fuzzy metric spaces called GV-spaces in this paper, with the help of continuous t-norms (see [3]) in order to obtain a Hausdorff topology in fuzzy metrics paces. The paper of Grabiec [4] started the investigations concerning a fixed point theory in fuzzy metric spaces by extending the well-known Banach contraction principle to KM-spaces. Many authors followed this concept by introducing and investigating the different types of fuzzy contractive mappings. Some instances of these works are in [5–19]. For instance, in 2002, Gregori and Sapena [5] have introduced a kind of contractive mappings and proved fuzzy fixed point theorems in GV-spaces and KM-spaces by using a strong condition for completeness, now called the completeness in the sense of Grabiec or G-completeness, which can be considered a fuzzy version of the Banach contraction theorem. These results have become recently of interest for many authors.
However, as a complete fuzzy metric space in the usual sense, that is, M-complete, i.e., the Cauchy sequence in the usual George and Veeramani’s sense is convergent (defined, for short, M-Cauchy), needs not be G-complete (see [2, 6]). Being aware of this problem, Gregori and Sapena in [5] raised the question whether the fuzzy contractive sequences are M-Cauchy. Very recently, many papers have appeared concerning this subject (see, for example, [7–10]). In particular, in [10], Wardowski considered a generalization of a fuzzy contractive mapping of Gregori and Sapena in M-complete GV-spaces, also in [7], Mihet defined a new fuzzy contraction called fuzzy ψ-contraction which enlarges the class of fuzzy contractive mappings of Gregori and Sapena and considered these mappings in KM-spaces. They have shown that every generalized fuzzy contractive sequence is M-Cauchy in respective fuzzy metric spaces and proved fuzzy contraction fixed point theorems under different hypotheses. For instance, Mihet assumed that the space under consideration is an M-complete non-Archimedean KM-space. Moreover, he posed an open question whether this fixed point theorem holds if the non-Archimedean fuzzy metric space is replaced by a fuzzy metric space. Vetro [8] introduced a notion of weak non-Archimedean fuzzy metric space and proved common fixed point results for a pair of generalized contractive-type mappings. Wang [9] gave a positive answer for the open question.
Motivated by the works mentioned above, in this paper, we will establish fixed point theorems for weakly fuzzy contractive set-valued mappings on M-complete GV-spaces. To this end, we first introduce a new concept called fw-distance here. Next, using this fw-distance, we introduce a fuzzy ψ-p-contractive set-valued mapping and formulate the conditions guaranteeing the convergence of a fuzzy ψ-p-contractive sequence and the existence of fixed points of a fuzzy ψ-p-contractive set-valued mapping in M-complete GV-spaces and KM-spaces. The established notion of contraction turns out to be a generalization of the fuzzy contractive condition of Gregori and Sapena. Moreover, the paper includes a comprehensive set of examples showing that a fuzzy ψ-p-contractive mapping is fuzzy ψ-contractive and the converse is false. So our results and demonstration are also a generalization of those of [7, 9]. To further illustrate the applicability of the fw-distance, we give characterizations of fuzzy metric completeness, that is, a GV-space X is M-complete if and only if every fuzzy ψ-p-contractive mapping from X into itself has a fixed point in X.
Finally, the idea of the present paper has originated from the study of an analogous problem examined by Suzuki [20] for set-valued contractive mappings and [21] for single-valued contractive mappings on complete determinacy metric spaces.
2 Preliminaries
Let us recall [3] that a continuous t-norm is a binary operation such that is an ordered Abelian topological monoid with unit 1. In this sequel, we always assume that ∗ is positive, i.e., for all .
As examples of t-norm satisfying the conditions above, we enumerate , and for , respectively.
Definition 2.1 [2]
A fuzzy metric space is an ordered triple such that X is a (non- empty) set, ∗ is a continuous t-norm, and M is a fuzzy set on that satisfies the following conditions for all :
-
(F1)
for all ,
-
(F2)
for all and for some implies ,
-
(F3)
for all ,
-
(F4)
for all and
-
(F5)
is continuous.
In the definition of Kramosil and Michalek [1], M is a fuzzy set on that satisfies (F3) and (F4), while (F1), (F2), (F5) are replaced by (K1), (K2), (K5), respectively, as follows:
-
(K1)
;
-
(K2)
for all if and only if ;
-
(K5)
is left continuous.
As we have mentioned, we refer to these spaces as KM-spaces and refer to the spaces given in Definition 2.1 as GV-spaces. In addition, when X is called a fuzzy metric space means, it may be a GV-space or a KM-space.
In this sense, M is called a fuzzy metric on X. Some simple but useful facts are that
-
(I)
is a continuous function on for and
-
(II)
is nondecreasing for all .
The first fact for the proof we refer to [[22], Proposition 1]. To prove the second fact, by (F4), we notice that for with .
Let be a GV-space. For and , the open ball with center is defined by
A subset is called open if for each , there exist and such that . Let denote the family of all open subsets of X. Then is a topology on X induced by the fuzzy metric M. This topology is metrizable (see [23]). Therefore, A closed subset B of X is equivalent to if and only if there exists a sequence such that topologically converges to x. In fact, the topological convergence of sequences can be indicated by the fuzzy metric as follows.
Definition 2.2 [2]
Let be a fuzzy metric space.
-
(i)
A sequence in X is said to be convergent to a point , denoted by , if for any .
-
(ii)
A sequence in X is called Cauchy sequence if for each and , there exists such that for any .
-
(iii)
A fuzzy metric space , in which every Cauchy sequence is convergent, is said to be complete.
There exist two fuzzy versions of Cauchy sequences and completeness, i.e., besides called M-Cauchy sequence and M-completeness in the sense of Definition 2.2, G-Cauchy sequence defined by for all and corresponding G-completeness introduced by [4]. In [6], the authors have pointed out that a G-Cauchy sequence is not an M-Cauchy in general. It is clear that an M-Cauchy sequence is G-Cauchy, and hence, a fuzzy metric space is M-complete if it is G-complete. From now on, by Cauchy sequence and completeness we mean an M-Cauchy sequence and M-completeness.
We now introduce a new notion as follows.
Definition 2.3 Let be a fuzzy metric space. A fuzzy set on is said to be an fw-distance if the following hypotheses are satisfied:
-
(w1)
for all and all .
-
(w2)
For any , , is upper semicontinuous, and is left continuous for .
-
(w3)
Let . For any and , there exists and such that and imply .
Note that neither of the implications (namely (F2)) necessarily hold, and is nonsymmetric, i.e., in general, does not satisfy (F3).
The fuzzy metric M is an fw-distance on X. In fact, (F4) implies that (w1) holds. The properties I and II of M combining conditions (F5) or (K5) guarantee that (w2) is valid. Finally, for any and , from the properties of ∗, we can take a small enough such that . Now, putting and , by means of (F4), we have
This implies that (w3) holds. However, some other following examples of fw-distances show that the converse is false.
Example 2.4 Let be a one-to-one continuous function, and let be an increasing continuous function. Define for all . Fixed , define M and by, respectively,
Then M is a fuzzy metric, and is a GV-space (see [24]), is an fw-distance but not a fuzzy metric on X.
Proof We observe that
Hence, , i.e., (w1) holds. (w2) is valid. Trivial. For any and , set and such that and and , we can distinguish two cases:
Now, we have, respectively,
and
It is easy to verify that in the two cases, the inequality
holds, that is, (w3) is met. This reduces that is an fw-distance.
However, is not a fuzzy metric since it is nonsymmetric. □
Example 2.5 Let with the fuzzy metric with g as in Example 2.4. Fixed . Define by
with and . Then is an fw-distance but not a fuzzy metric on .
Some properties for the fw-distance are useful in this sequel.
Proposition 2.6 Let ’ be a fuzzy metric space, and let be an fw-distance on X. Then for sequences and in X, the function sequences and with converging to 0 for , and we have the following:
-
(1)
if, for , and for any , then ; in particular, if and , then ;
-
(2)
if, for , and for any , then converges to z;
-
(3)
if, for , for any with , then is a Cauchy sequence;
-
(4)
if, for , for any , then is a Cauchy sequence;
-
(5)
if and in X with and for some , then .
Proof (1) For any and , let . By our assumptions, there exists such that and which implies that and for large enough n. In view of (w3), one has . Now, the arbitrariness of ε implies that , i.e., .
(2) Similarly to the argument of (1), for any and , we can find such that for each , that is, .
(3) For any and , there exists such that for . Let with . Then, by means of the assumption of (3), we have and . (w3) guarantees that . From Definition 2.2(ii) is a Cauchy sequence.
As an analogous argument in (3), we can verify that (4) is valid.
(5) If and for some , by (w2) . Therefore, (5) holds. □
3 Fixed point theorems
In the sequel, by , we denote the collection consisting of all nonempty closed subsets of X (obviously, every closed subset of X is bounded in the sense of fuzzy metric spaces). Let be a fuzzy metric space and be a set-valued mapping. An element is called a fixed point of T if .
The following collection Ψ of functions is given in [7], that is, implies that ψ from into itself is continuous, nondecreasing and for each .
Let and be an fw-distance. The set-valued mapping T is called a fuzzy ψ-p-contractive mapping if the following implication takes place: for any and , there exists such that for each . In particular, the fuzzy ψ-M-contraction corresponds to the fuzzy ψ-contraction according to [[7], Definition 3.1]. A fuzzy ψ-p-contractive sequence in X is any sequence in X such that for all and .
Theorem 3.1 Let be a complete fuzzy metric space, and let T be a fuzzy ψ-p-contractive set-valued mapping from X into . If there exists such that for some and any , then T has at least a fixed point . Moreover, if , then for all .
Proof From our assumption, there exists such that for some and any . For fixed , by the contractive condition, there exists such that
for all . Applying again the contractive condition for , , we can choose such that
Thus, by induction, we obtain a sequence in X such that and
for every and . Next, for each , we prove by induction that, for all ,
We have shown that the claim is true for . Assume that for all and with . Then, by virtue of (w1), we have
Since , , from the fact that ∗ is positive we have
for all , (1) is valid.
Now, any fix . Let for . Then is a function from into and
for any with . We will prove that converges to 0. To this end, it is sufficient to verify that
with . For , by (1) we have
This yields that for all , i.e., is a decreasing sequence. So is convergent. Let . By virtue of the continuity of ψ, we have which yields that , and hence (2) is valid. Moreover, by virtue of Proposition 2.6(3), we see that is a Cauchy sequence. Hence converges to a point by the completeness of X.
Fix a large enough . Since converges to and is upper semicontinuous, we have
This implies that
Again, the contractive hypothesis reduces that there exists such that . Consequently, we have a sequence such that , for all and . Fix , let
In view of (4), we obtain that converges to 0. By this, combining , we have converging to 1, which implies that converges to 0. By Proposition 2.6(2), converges to . Since is closed, , i.e., is a fixed point of T.
Finally, for such , if , there exists such that . Thus, we also have a sequence in X such that and for every . Let for fixed . Repeating the proof process of (2), we can infer that as . By Proposition 2.6(4), is a Cauchy sequence. Hence converges to a point . Since is upper semicontinuous, , and hence for all . For any ,
By (4) and Proposition 2.6(1), we have which implies that . This proof is complete. □
Example 3.2 Let , for any and be given as in Example 2.4 with , and . For given , the set-valued mapping as follows
has a fixed point in X.
Proof Using similar arguments as the ones in [[12], Theorem 16], one can show that is a complete GV-space. Let for . Then . Let
It is not hard to verify that is an fw-distance. For any and , if , then and choose , we have
If , choose , we have
if and
if . Consequently, T is ψ-p-contractive and all conditions of Theorem 3.1 are satisfied. Hence, T has a fixed point (in fact, ). □
Remark 3.3 We observe that T in Example 3.2 is not fuzzy ψ-contractive. Hence, there exists a mapping which is fuzzy ψ-p-contractive but not fuzzy ψ-contractive. However, every fuzzy ψ-contractive mapping is obviously fuzzy ψ-p-contractive.
In fact, set and take , in Example 3.2, we have , . Note that
Let and . Then . Take , we have
that is, . Consequently, T is not fuzzy ψ-M-contractive.
Let be a fuzzy metric space and T a single-valued mapping from X into itself. T is said to satisfy nonzero property if there exists such that for all .
Corollary 3.4 Let be a complete fuzzy metric space, and the mapping T from X into itself is fuzzy ψ-p-contractive with the fw-distance satisfying for any . If T satisfies the nonzero property, then T has a unique fixed point . Further, satisfies for all .
Proof From Theorem 3.1, there exists with and for all . Let . If then . This contradiction implies that . So, by and Proposition 2.6(1), we have . □
Remark 3.5 In the case of , T is exactly fuzzy ψ-contractive initiated by Mihet [7]. So Corollary 3.4 is a positive answer for the open question of [7], but also an essential extension and improvement of Theorem 3.1 in [7] (see [25]) and Theorem 3 in [9], also, the corresponding results of [5, 26].
4 Characterizations of completeness
As an application of Corollary 3.4 and the fw-distance, we propose the following profounder result of fuzzy fixed point theory which gives characterizations of the fuzzy metric completeness. We need the following assumption:
We first list the following lemmas regarding the fw-distance which plays a key role in this section.
Lemma 4.1 Let X be a GV-space with the fuzzy metric M, let be an fw-distance on X, and let be a function from into satisfying (w1), (w2) in Definition 2.3. Suppose that for every , . Then is also an fw-distance on X. In particular, if satisfies (w1), (w2) in Definition 2.3 and for every , , then is an fw-distance on X.
Proof We show that satisfies (w3). Let and . Since is an fw-distance, there exists a positive number and such that and imply that . Then and imply that . This proof is complete. □
Lemma 4.2 Let be a GV-space and contain at least two points, and . Then the fuzzy subset defined by
is an fw-distance on , where is a nondecreasing continuous function with .
Proof It is clear that for since A contains at least two points. If , we have
In the other case, without loss of generality, we suppose that , then and . Hence,
Let . If for some , we have . Let for all . If , then implies that . So, we have . If , we have . In each case, the set is closed. Therefore, is upper semicontinuous. is obviously continuous.
Let and . Then there exists a positive number such that . Let . Then and imply that . So, we have
This proof is complete. □
Remark 4.3 If A is a compact subset of X, then for all . Indeed, suppose that this is not true, then there exists such that . Thus, for each , there exist such that . Since A is compact, there exist the subsequences , of , , respectively, such that and with . Note that M is continuous, we have , a contradiction. In addition, we observe that is nondecreasing for any given . Therefore, is nondecreasing, and this guarantees the existence of , say, .
Theorem 4.4 Let be a fuzzy metric space. Then X is complete if and only if every fuzzy ψ-p-contractive mapping from X into itself satisfying the nonzero property has a fixed point in X.
Proof Since the ‘only if’ part is proved in Corollary 3.4, we only need to prove the ‘if’ part. Assume that X is not complete. Then there exists a sequence in X which is Cauchy and does not converge. So, there exists such that for any . Let for any . Moreover, we have also for all . Thus, any fix , for some and each , we can choose such that . We now obtain a subsequence such that
Let
for . Then we may assume that there exists a sequence in X satisfying the following conditions:
-
(i)
is Cauchy;
-
(ii)
does not converge;
-
(iii)
for any , , where
Put . Then A is bounded and closed. We next prove that given as in Lemma 4.2 is positive. In fact, if for some , then for every , there exists such that which implies that . On the other hand, since is a Cauchy sequence, we have , a contradiction. Hence, for all .
Let us define the fuzzy set on by
where is an increasing continuous function with . It is clear that is an fw-distance on X by Lemmas 4.1 and 4.2. Further, for any and , i.e., satisfies the symmetry.
Define a mapping as follows:
Then it is easy to see that T has no fixed point in X. Moreover, from (F1), it follows that , that is, T satisfies the nonzero property. To complete the proof, it is sufficient to show that T is fuzzy ψ--contractive with . If or , then
Let us assume that . Then, without loss of generality, we may assume that , and . For any , from (iii), combining the monotonicity of , it follows that
This implies that
On the other hand, by (iii), combining (4) and the symmetry of , we have
This shows that T is fuzzy ψ--contractive. □
Example 4.5 For any nonempty set X, let us consider the fuzzy metric space with as in Example 3.2. Let for any fixed . Then the fuzzy metric space is not complete.
Proof Consider the mapping with and , we have . Let , we have for any and . From Lemma 4.1, it follows that is an fw-distance in X. Now, we have for all and
for . This implies that T is fuzzy ψ-p-contractive. We assert that T has no fixed point in . Indeed, if , then . Consequently, Theorem 4.4 guarantees that is not complete. □
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Supported by the Natural Science Foundation of Zhejiang Province (LY12A01002).
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Hong, S., Peng, Y. Fixed points of fuzzy contractive set-valued mappings and fuzzy metric completeness. Fixed Point Theory Appl 2013, 276 (2013). https://doi.org/10.1186/1687-1812-2013-276
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DOI: https://doi.org/10.1186/1687-1812-2013-276