- Open Access
Fixed points of fuzzy contractive set-valued mappings and fuzzy metric completeness
© Hong and Peng; licensee Springer. 2013
- Received: 24 June 2013
- Accepted: 10 September 2013
- Published: 8 November 2013
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.
In Fuzzy metric spaces we refer to as KM-spaces were initiated by Kramosil and Michálek . The conditions which they formulated were modified later by George and Veeramani  via proposing new fuzzy metric spaces called GV-spaces in this paper, with the help of continuous t-norms (see ) in order to obtain a Hausdorff topology in fuzzy metrics paces. The paper of Grabiec  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  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  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 , Wardowski considered a generalization of a fuzzy contractive mapping of Gregori and Sapena in M-complete GV-spaces, also in , 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  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  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  for set-valued contractive mappings and  for single-valued contractive mappings on complete determinacy metric spaces.
Let us recall  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 
for all ,
for all and for some implies ,
for all ,
for all and
for all if and only if ;
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.
is a continuous function on for and
is nondecreasing for all .
The first fact for the proof we refer to [, Proposition 1]. To prove the second fact, by (F4), we notice that for with .
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 ). 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 
A sequence in X is said to be convergent to a point , denoted by , if for any .
A sequence in X is called Cauchy sequence if for each and , there exists such that for any .
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 . In , 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.
for all and all .
For any , , is upper semicontinuous, and is left continuous for .
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).
This implies that (w3) holds. However, some other following examples of fw-distances show that the converse is false.
Then M is a fuzzy metric, and is a GV-space (see ), is an fw-distance but not a fuzzy metric on X.
holds, that is, (w3) is met. This reduces that is an fw-distance.
However, is not a fuzzy metric since it is nonsymmetric. □
with and . Then is an fw-distance but not a fuzzy metric on .
Some properties for the fw-distance are useful in this sequel.
if, for , and for any , then ; in particular, if and , then ;
if, for , and for any , then converges to z;
if, for , for any with , then is a Cauchy sequence;
if, for , for any , then is a Cauchy sequence;
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 , 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 [, 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 .
for all , (1) is valid.
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.
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.
By (4) and Proposition 2.6(1), we have which implies that . This proof is complete. □
has a fixed point in X.
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.
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 . So Corollary 3.4 is a positive answer for the open question of , but also an essential extension and improvement of Theorem 3.1 in  (see ) and Theorem 3 in , also, the corresponding results of [5, 26].
4 Characterizations of completeness
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. □
is an fw-distance on , where is a nondecreasing continuous function with .
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.
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.
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 .
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.
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.
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. □
Supported by the Natural Science Foundation of Zhejiang Province (LY12A01002).
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