The problem of characterizing complete metric spaces by means of fixed point properties has its origin, in great part, in a work of Connell published in 1959, where the author constructed an example of a non-complete metric space for which every Banach contraction has fixed point [1
] (second part of Example 4). In connection to Connell’s example, Hu proved in [2
] that a metric space is complete if and only if any Banach contraction on closed subsets thereof has a fixed point. Later, Subrahmanyam [3
] and Kirk [4
], respectively proved that both the famous Kannan fixed point theorem [5
] and the famous Caristi fixed point theorem [6
] characterize the metric completeness. Remarkable is also the contribution of Suzuki and Takahashi who gave in [7
] (Theorem 4) a necessary and sufficient condition for a metric space to be complete by means of weakly contractive mappings having a fixed point. In a paper published in 2008, Suzuki [8
] obtained a nice and elegant characterization of the metric completeness with the help of a weak form of the Banach contraction principle, and, very recently, it was proved in [9
] that an important fixed point theorem by Samet, Vetro and Vetro [10
] (Theorem 2.2) is also able of characterizing complete metric spaces.
In contrast to the situation described in the preceding paragraph, the natural question of characterizing complete fuzzy metric spaces via fixed point properties has received little attention. Indeed, despite the large backlog of published fixed point theorems, only appear in the literature a few efforts, with positive partial results, to obtain a suitable version of Caristi’s theorem that allows us to characterize the completeness of fuzzy metric spaces [11
This paper deals with giving an impulse to the study of characterizing complete fuzzy metric spaces by means of fixed point results. In this direction, our main result extends to the framework of fuzzy metric spaces, in the sense of Kramosil and Michalek [13
], Hu’s characterization cited above. In fact, we will show that Hu’s theorem can be recovered from our main result, and, as an application, we will deduce that a well-known fixed point theorem of Mihet [14
] (Theorem 2.2) also allows us to obtain a characterization of fuzzy metric completeness. Since the main ingredient in our approach is the celebrated Hicks fixed point theorem [15
], we conclude the paper showing that another fundamental and distinguished the fuzzy version of the Banach contraction principle due to Sehgal and Bharucha-Reid [17
] is not suitable to characterize complete fuzzy metric spaces in this setting.
In order to help the reader, we start this section by collecting some concepts and properties which will be useful in the rest of the paper.
The sets of real numbers and positive integer numbers will be denoted by
, respectively. Our basic reference for general topology is [18
A binary operation is said to be a continuous t-norm if the following conditions hold: (i) The pair is an Abelian semigroup with neutral element (ii) * is continuous on ; (iii) if , where .
The books [19
] provide excellent references in the study of continuous t-norms. In particular, the following are basic but crucial examples of continuous t-norms.
Minimum ∧ given by
Product given by
The ukasiewicz t-norm given by
We will also consider continuous t-norms of H-type (see e.g., [20
] (Chapter 1, Section 1.6). Recall that ∧ is of H-type but
It is well known that In fact, for any continuous t-norm
Definition 1. (Kramosil and Michalek ). A fuzzy metric on a set is a pair such that * is a continuous t-norm and is a function from to such that for all
(km2) if and only if for all
(km4) for all
(km5) is left continuous.
If is a fuzzy metric on a set the triple is said to be a fuzzy metric space.
is a fuzzy metric on a set
, is a base of open sets for a metrizable topology
, namely, the topology induced by
A sequence in a fuzzy metric space is called a Cauchy sequence if for any and there exists a such that for all
A fuzzy metric space is said to be complete if every Cauchy sequence converges with respect to the topology
The following is an easy but very useful example of a fuzzy metric space.
Let be a metric space. Define a function as if and if Then, is a fuzzy metric on for any continuous t-norm Moreover, the topologies induced by σ and coincide, and we also have that is complete if and only if is complete.
The concepts and results from Hicks, Radu, and Miheţ, cited in the sequel was originally established by these authors in the slightly more general framework of Menger spaces.
Hicks introduced in [15
] the following notion, under the name of a C
-contraction, in the study of the fixed point theory for Menger spaces and fuzzy metric spaces.
Definition 2. (). Let be a fuzzy metric space. We say that a mapping is a Hicks contraction on (with constant if it satisfies the following condition: There exists such that for any and
In fact, Hicks proved that if is a complete fuzzy metric space, then every Hicks contraction on has a unique fixed point.
], Radu proved the following improvement of Hicks’ fixed point theorem, which can be also deduced as an immediate consequence of [21
] (Theorem 2.2).
Let be a complete fuzzy metric space. Then, every Hicks contraction on has a unique fixed point.
Later, Miheţ’s introduced in [14
] the notion of a weak-Hicks contraction and obtained, among other results, a fixed point theorem for this class of contractions that strictly contains the class of Hicks contractions [14
] (Theorem 2.2, Corollary 2.2.1 and Proposition 2.1).
Miheţ’s approach motivates the following notion.
Let be a fuzzy metric space. We say that a mapping is a Miheţ contraction on if it satisfies the following two conditions:
] (Theorem 2.2) and its proof we deduce the following restatement of [14
] (Corollary 2.2.1).
Theorem 2. (Miheţ ) Let be a complete fuzzy metric space such that or * is of H-type. Then, every Miheţ contraction on has a fixed point.
According to [14
] (Definition 2.2), a mapping
satisfying condition (mi2) in Definition 3 is said to be a weak-Hicks contraction. It is well-known [14
] (Lemma 2.1) that every weak-Hicks contraction is a continuous mapping. Obviously, every Hicks contraction is a weak Hicks contraction and hence a continuous mapping. In fact, we have the following better conclusion.
Every Hicks contraction is a Miheţ contraction.
Proof of Proposition 1.
Let be a Hicks contraction (with constant c) on a fuzzy metric space It suffices to show that satisfies condition (mi1) in Definition 3. Indeed, suppose that for all Since we deduce that so Putting we deduce that a contradiction. □
3. Results and Examples
As we recalled in Section 1
, the Banach contraction principle does not characterize the metric completeness. The following example (based on the example given in [7
]) shows that, similarly to the metric case, there exist non-complete fuzzy metric spaces for which every Hicks contraction has fixed points.
Example 2. (compare , Example on pages 377-378). Let
and let σ be the restriction of the Euclidean metric on to Clearly is not complete and thus the fuzzy metric space is not complete (see Example 1). However, every continuous mapping has at least a fixed point, as Suzuki and Takahashi proved in . Therefore, every Hicks contraction on has at least a fixed point because Hicks contractions are continuous mappings.
Nevertheless, we can obtain a characterization of fuzzy metric completeness similar to the one given by Hu for metric spaces with the help of Hicks contractions having a fixed point. To this end, the notion of a semi-metric and an important theorem due to Radu (Theorem 3 below) will be fundamental to our approach.
A semi-metric (compare e.g., [22
]) for a topological space
is a function
satisfying the following conditions for every
a subset of
(sm3) (As usual, denotes the closure of in
As in the metric case, if is a semi-metric for a topological space a sequence in is called a Cauchy sequence if for each there exists such that for all
Theorem 3. (Radu  Proposition 2.1.1) Let be a fuzzy metric space. For each put
Then is a semi-metric for satisfying and
for all Therefore, a sequence in is a Cauchy sequence for σ if and only if it is a Cauchy sequence in
Moreover, if then σ is a metric on X whose induced topology coincides with
Our main result is the following.
A fuzzy metric space is complete if and only if every Hicks contraction on any closed subset of has a fixed point.
Proof of Theorem 4.
Suppose that is a complete fuzzy metric space. Let be a closed subset of Then is a complete fuzzy metric space, where by we denote the restriction of to Therefore, every Hicks contraction on has a (unique) fixed point by Theorem 1.
Conversely, suppose that
is not complete. Then, we can find a Cauchy sequence
, which does not converge for
Therefore, the set
is closed and also are closed all sets of the form
So, by condition (sm3) in the definition of a semi-metric, we have that for each
Since, by Theorem 3,
is also a Cauchy sequence for the semi-metric
, there is a strictly increasing function
Now let be the mapping given by for all Since so has no fixed points. We shall show that, nevertheless, is a Hicks contraction on the fuzzy metric space with constant 1/2.
By Theorem 3,
Assume, without loss of generality, that
Consequently by Theorem 3, which implies that is a Hicks contraction on . This concludes the proof. □
Now we apply Theorem 4 to deduce that Theorem 2 also allows us to characterize fuzzy completeness for two large classes of fuzzy metric spaces.
Let be a fuzzy metric space such that or * is of H-type. Then is complete if and only if every Miheţ contraction on any closed subset of has a fixed point.
Proof of Theorem 5.
Suppose that is complete and let be a closed subset of Then is a complete fuzzy metric space. It follows from Theorem 2 that every Miheţ contraction on has a fixed point.
Conversely, let be a closed subset of and suppose that every Miheţ contraction on has a fixed point. Since, by Proposition 1, every Hicks contraction is a Miheţ contraction, we deduce that every Hicks contraction on has a fixed point. Hence is complete by Theorem 4. □
Next, we shall deduce the classical Hu theorem from our main result. Two simple auxiliary results will be useful to this end.
If is a metric space, we denote by the metric defined on by for all Obviously, the topologies induced by and coincide, and is complete if and only if is complete. Therefore is complete if and only if is complete.
Let be a metric space. Then, every Banach contraction on is a Banach contraction on
Proof of Proposition 2.
Let be a Banach contraction on Then, there exists such that for all
we deduce that
which is not possible. Therefore
We conclude that is a Banach contraction on □
Let be a metric space with . Then, every Hicks contraction on is a Banach contraction on .
Proof of Proposition 3.
Let be a Hicks contraction on with constant c. We want to show that for all
Indeed, assume the contrary. Then, there exist such that Choose an for which and Since and is a Hicks contraction we deduce that On the other hand, from it follows that This contradiction concludes the proof. □
Theorem 6. (Hu ). A metric space is complete if and only if every Banach contraction on any closed subset of has a fixed point. Proof of Theorem 6.
Since the proof of the “only if” part is obvious, we only show the “if” part. Suppose that every Banach contraction on any closed subset of has a fixed point. Let be a closed subset of the fuzzy metric space and let be a Hicks contraction on Since it follows from Proposition 3 that is a Banach contraction on where denotes the restriction of the metric to Therefore is a Banach contraction on by Proposition 2. Hence has a fixed point in . Consequently is complete by Theorem 4. Thus is complete and, hence, is complete. □
Sehgal and Baharucha-Reid proved in [17
] the first fixed point theorem for fuzzy metric spaces (actually, they obtained their result in the realm of Menger spaces).
Theorem 7. (Sehgal and Baharucha-Reid ). Let be a complete fuzzy metric space. If is a mapping such that there exists satisfying
for all and then has a unique fixed point.
We finish the paper showing that, contrarily to the case of Hicks contractions, this fundamental theorem is not suitable to obtain a characterization of fuzzy metric completeness. More precisely, we present an example of a non-complete fuzzy metric space such that for any closed subset every mapping satisfying the conditions of Theorem 7 is constant and, hence, has a unique fixed point.
Let and let given as if and if for all Then is a fuzzy metric space (see  (Example on page 2016)). Clearly is a non convergent Cauchy sequence in so it is not complete. Now let be any subset of (note that all subsets of are closed because is the discrete topology on and be a mapping such that there exists for which for all and Take Then so
Therefore and, thus, for all We conclude that is constant and, hence, it has a unique fixed point in . Note that, by Theorem 4, there exists a (closed) subset of and a Hicks contraction without fixed points.