In their relevant article [1
], Kada, Suzuki and Takahashi gave and discussed a concept of w
-distance for metric spaces. In fact, they generalized several important theorems including, among others, Caristi’s fixed point theorem and Ekeland’s Variational Principle, with the help of this concept. Almost simultaneously, Suzuki and Takahashi [2
] obtained a characterization of metric completeness through a generalization of the Banach Contraction Principle that uses w
-distances. Since then, many authors have used and extended w
-distances, mainly in the context of the fixed point theory and from different views (see, e.g., [3
] and their bibliographies).
We remind that a w-distance on a metric space is a function (the set of non-negative real numbers) such that for every the following conditions hold:
is a lower semicontinuous function;
for each there is such that and imply
Obviously, every metric
on a set S
is a w
-distance on the metric space
], Example 1). Other interesting instances of w
-distances on a metric space can be seen in [1
In accordance with [2
] a weakly contraction on a metric space
is a self map f
for which there are a w
and a constant
verifying, for any
Then, Suzuki and Takahashi proved their renowned and aforementioned theorem which should be stated in the following way.
], Theorem 4). A necessary and sufficient condition for a metric space to be complete is that every weakly contraction on it has a fixed point.
In this paper, we propose a notion of w
-distance for fuzzy metric spaces, in the sense of Kramosil and Michalek [11
], which allows us to obtain a fuzzy counterpart of Suzuki and Takahashi theorem (Theorem 1 above). For our approach, (fuzzy) contractions in the sense of Hicks [12
] will play a fundamental role. Thus, in Section 2
we remind some meaningfull notions and properties that we shall use throughout this note. In Section 3
we introduce our notion of w
-distance in the setting of fuzzy metric spaces and present several pertinent examples. Section 4
is devoted to prove a fixed point theorem, with the help of w
-distances, which will provides the “only if” part for the characterization of fuzzy metric completeness that will be obtained in Section 5
Our study is mainly motivated by the recent articles [13
] where fuzzy extensions of the famous characterizations of metric completeness due to Kirk [16
], Hu [17
] and Subrahmanyam [18
], respectively, were obtained.
we denote the set of non-negative integer numbers and by * any continuous t-norm (a deep and extensive study of continuous t-norms may be found in [19
]). Many results and examples on fuzzy metric spaces and related structures are given in [20
As we have point out we shall consider fuzzy metric spaces in the sense of Kramosil and Michalek.
Thus, and following the current terminology (see, e.g., [14
]), by a fuzzy metric space we mean a triple
is a set, m is a function from
, i.e., a fuzzy set in
and * is a continuous t-norm, such that for every
the next conditions are satisfied:
is a left continuous function.
In this case we say that (or simply m) is a fuzzy metric on
It is well known (see ([21
], Lemma 4)) that for each
is a non-decreasing function on
. We shall use this property without explicit reference.
Given a fuzzy metric on a set S put whenever and . Then, the collection is a base of open sets for a metrizable topology on called the topology generated by
Note that a sequence converges to an with respect to if and only if for each m eventually.
If a sequence converges to an with respect to we simply write if no confusion arises.
A fuzzy metric space is called complete if every Cauchy sequence converges with respect to the topology where a sequence in S is a Cauchy sequence assuming that for each and there is an such that whenever
At the end of this section we remind the following well-known and paradigmatic example of a fuzzy metric space (see, e.g., ([14
], Example 1)).
Given a metric space denote by the fuzzy set in defined as if and if Then is a fuzzy metric on S for any continuous t-norm *, called the induced fuzzy metric. Furthermore, the topology generated by ρ agrees with the topology generated by We also have that is complete if and only if is complete.
3. Fuzzy w-Distances on Fuzzy Metric Spaces
We begin this section by introducing our notion of w-distance for fuzzy metric spaces, after which we will give some remarks and examples to support it.
A fuzzy w-distance on a fuzzy metric space is a fuzzy setwin that satisfies the next conditions for every
if and then for all and
for each there is such that and imply
Remark 1. By similarity with the concept of w-distance for metric spaces, we could attempt to reformulate condition (fw2) above as follows: “for each and the function is lower semicontinuous”. However, Example 4 below provides a useful instance (see the proof of Theorem 4) of a fuzzy w-distance such that is not lower semicontinuous for an Notice also that condition (fw2) is neither strange nor artificial; in fact, Grabiec proved in (, Lemma 6) that it is satisfied by every fuzzy metric. Remark 2. An antecedent of Definition 1 is in the article , where the authors introduced an discussed the concept of r-distance. Hence, and following (, Definition 2.1), by an r-distance on a fuzzy metric space we mean a fuzzy setrin satisfying conditions (fw1) and (fw3) above and the following one instead of (fw2): “for each and the function is continuous”. Example 4 (see also Remark 1)shows that condition (fw2) is more appropriate in our framework. We stress that actually the authors defined the concept of r-distance in the slight more general context of Menger probabilistic metric spaces. Example 2.
Let be a fuzzy metric space and let T be a closed subset of Define a fuzzy setwin byw if and , andw otherwise. We show thatwverifies the conditions of Definition 1, so it is a fuzzy w-distance on
We first assume that Hence
Otherwise, we havew orw so
If or the sequence is eventually in we get eventually, so
Otherwise, and the sequence has a subsequence such that for all Since T is closed we deduce that and hence
Given by the continuity of *, there is for which If and
we get , and Therefore
Let be a fuzzy metric space. Then is a fuzzy w-distance on Indeed, this fact is an immediate consequence of Example 2 when
Denote by the restriction to of the Euclidean metric. Let ( be the induced fuzzy metric as constructed in Example 1, and let Since T is closed we can apply the construction done in Example 2 and thus the fuzzy setwin given byw if and , andw otherwise, is a fuzzy w-distance on
We show that, for and any the function is not lower semicontinuous. Obviously with respect to the topology generated by ( Nevertheless, for any and because for all Conistenly, for any the function is not lower semincontinuous, and thus not continuous.
The following example provides an efficient method to generate fuzzy w-distances from w-distances on metric spaces.
Let w be a w-distance on a metric space and let be the fuzzy metric space associated to as constructed in Example 1. Define a fuzzy setwin by
w if and and
We show thatwis a fuzzy w-distance on the fuzzy metric space Indeed,
If orw thenw.
Otherwise we havew so and Hence by condition (w1), and also Therefore
Ifw the conclusion is obvious.
Ifw we get or
In the first case, by condition (w2), there is such that for all Hence for all so
In the second one, the same argument, changing w with also shows that
Given there is for which condition (w3) is fulfilled. Put Then and hence fromw andw it followsw i.e., and We get so
Example 6. Let and letmbe the fuzzy set in given by and for all and It is well known that m is a fuzzy metric space where * is the usual product on [0, 1] (see, e.g., (, Example 10.1.3)). Furthermore, the topology m) agrees with the usual topology on
We show that the fuzzy setwin given byw andw for all is a fuzzy w-distance on m Indeed,
We get, for
Let and Notice that for the usual topology on S and thus for the usual topology on Therefore
Given suppose and Then and Consequently
4. A Fixed Point Theorem in Terms of Fuzzy w-Distances
In his article [12
], Hicks introduced a relevant notion of contraction under the name of C
-contraction. Hicks’ notion was called Hicks contraction by several authors (see, e.g., [14
A self map f
of a fuzzy metric space
is a Hicks contraction provided that there is
such that, for each
We generalize this notion to our setting in a natural fashion as follows.
Definition 2. A self map f of a fuzzy metric space is aw-Hicks contraction provided that there are a fuzzy w-distancewon and an such that for each and
Next we prove a fixed point result in terms of w-Hicks contractions.
Everyw-Hicks contraction on a complete fuzzy metric space has a fixed point.
We point out that the starting idea for the construction of a Cauchy sequence of iterates, constructed below, comes from [23
be a complete fuzzy metric space and let f
be a w-Hicks contraction on
Then, there exist a fuzzy w
-distance w on S
and a constant
such that for any
Applying again condition (1), we deduce that
and following this process we obtain
Now fix and put for all
Let We are going to prove the following two assertions (A) and (B):
There existssuch that for all with
Thus, since is arbitrary, the sequence will be a Cauchy sequence in
where is the limit of the sequence
Thus, since is arbitrary, a will be a fixed point of
for which condition (fw3) in Definition 1 holds with respect to
By inequality (2) we have
it follows from condition (fw3), taking
We have shown assertion (A). Consequently is a Cauchy sequence in
Since is complete there exists such that
Next we prove assertion (B).
Since we can choose such that Then, by condition (fw2), in Definition 1,
Consequently, we can find
which implies by (1) that
we deduce from inequalities (3) and (4) and condition (fw3) that
We have proved that a is a fixed point of □
As an immediate consequence of Example 3 and Theorem 2 we get the next enhancement of Hicks’ fixed point theorem (see [24
Every Hicks contraction on a complete fuzzy metric space has a unique fixed point.
Next we present and example where we Theorem 2 works but not Corollary 1. The next result will be of great help for our purposes.
]). Let be a metric space and T be a bounded and closed subset of S with . If K is a positive constant such that then, the function defined by if and otherwise, is a w-distance on
Let be the fuzzy metric space constructed in Example 4. In fact, it is complete because the metric space it is.
Define as if and if
Since f is not continuous (at it is not a Hicks contraction on
Now put and By Lemma 1, the function defined by if and otherwise, is a w-distance on
We show that f is aw-Hicks contraction on (with constant for the fuzzy w-distancewon given by (compare Example 5)
w if and and
We differentiate the three next cases.
If it follows i.e., so Therefore
If it follows Suppose that Then or Since we deduce that a contradiction. Hence
Therefore, Theorem 2 works for this example.
5. A Necessary and Sufficient Condition for Completeness of Fuzzy
In this section we prove our fuzzy counterpart of Theorem 1. In fact, we show that the fixed point theorem obtained in Theorem 2 jointly with the following characterization of fuzzy metric completeness obtained in [14
] yield such a counterpart.
A necessary and sufficient condition for a fuzzy metric space to be complete is that every Hicks contraction on any of its closed subsets has a fixed point.
A necessary and sufficient condition for a fuzzy metric space to be complete is that everyw-Hicks contraction on it has a fixed point.
Necessity: It follows from Theorem 2.
Sufficiency: Let T be a closed subset of a fuzzy metric space and f be a Hicks contraction on T (with constant
Fix and construct a self map g of S as follows:
Following the construction given in Example 2, let be the fuzzy w-distance on given by
if and and
We are going to show that g is a w-Hicks contraction on for this fuzzy w-distance (with constant
Indeed, let and such that
We distinguish three cases:
Then, we get
(we do not need to consider the case
is symmetric). Then w
and thus m
is a Hicks contraction on T
we deduce that m
We conclude that g is a w-Hicks contraction on , so, by assumption it has a fixed point By the definition of and consequently is a fixed point of We have proved that every self map of T has a fixed point. Hence, by Theorem 3, is complete. □
We finish the paper by studying a case that illustrates the preceding theorem.
Let be the fuzzy metric space where * is any continuous t-norm and is the fuzzy metric on S given, for any by if and if
Since is a non convergent Cauchy sequence, it follows that is not complete, so, by Theorem 4, there exist -Hicks contractions on without fixed points. We proceed to explicitly construct one of such contractions.
Let f be the self map of S given by for all Now let be the -distance on given by and for all Suppose with Then so and hence We conclude that f is a -Hicks contraction without fixed point. Observe that f is also a -Hicks contraction with respect to the fuzzy metric .