Abstract
In this paper, in fuzzy metric spaces (in the sense of Kramosil and Michalek (Kibernetika 11:336-344, 1957)) we introduce the concept of a generalized fuzzy metric which is the extension of a fuzzy metric. First, inspired by the ideas of Grabiec (Fuzzy Sets Syst. 125:385-389, 1989), we define a new G-contraction of Banach type with respect to this generalized fuzzy metric, which is a generalization of the contraction of Banach type (introduced by M Grabiec). Next, inspired by the ideas of Gregori and Sapena (Fuzzy Sets Syst. 125:245-252, 2002), we define a new GV-contraction of Banach type with respect to this generalized fuzzy metric, which is a generalization of the contraction of Banach type (introduced by V Gregori and A Sapena). Moreover, we provide the condition guaranteeing the existence of a fixed point for these single-valued contractions. Next, we show that the generalized pseudodistance (introduced by Włodarczyk and Plebaniak (Appl. Math. Lett. 24:325-328, 2011)) may generate some generalized fuzzy metric on X. The paper includes also the comparison of our results with those existing in the literature.
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1 Introduction
A number of authors generalize Banach’s [1] and Caccioppoli’s [2] result and introduce the new concepts of contractions of Banach and study the problem concerning the existence of fixed points for such a type of contractions; see e.g. Burton [3], Rakotch [4], Geraghty [5, 6], Matkowski [7–9], Walter [10], Dugundji [11], Tasković [12], Dugundji and Granas [13], Browder [14], Krasnosel’skiĭ et al. [15], Boyd and Wong [16], Mukherjea [17], Meir and Keeler [18], Leader [19], Jachymski [20, 21], Jachymski and Jóźwik [22], and many others not mentioned in this paper.
In 1975, Kramosil and Michalek [23] introduced the concept of fuzzy metric spaces. It is worth noticing that there exist at least five different concepts of a fuzzy metric space (see Artico and Moresco [24], Deng [25], George and Veeramani [26], Erceg [27], Kaleva and Seikkala [28], Kramosil and Michalek [23]).
In 1989, Grabiec [29] proved an analog of the Banach contraction theorem in fuzzy metric spaces (in the sense of Kramosil and Michalek [23]). In his proof, he used a fuzzy version of Cauchy sequence. It is worth noticing that in the literature in order to prove fixed point theorems in fuzzy metric space, authors used two different types of Cauchy sequences. For details see [30]. The existence of fixed points for maps in fuzzy metric spaces was studied by many authors; see e.g. Gregori and Sapena [31], Miheţ [32]. Fixed point theory for contractive mappings in fuzzy metric spaces is closely related to the fixed point theory for the same type of mappings in probabilistic metric spaces of Menger type; see Hadžić [33], Sehgal and Bharucha-Reid [34], Schweizer et al. [35], Tardiff [36], Schweizer and Sklar [37], Qiu and Hong [38], Hong and Peng [39], Mohiuddine and Alotaibi [40], Wang et al. [41], Hong [42], Saadati et al. [43], and many others not mentioned in this paper.
In this paper, in fuzzy metric spaces (in the sense of Kramosil and Michalek [23]), we introduce the concept of a generalized fuzzy metric which is the extension of a fuzzy metric. First, inspired by the ideas of Grabiec [29], we define a new G-contraction of Banach type with respect to this generalized fuzzy metric, which is a generalization of a contraction of Banach type (introduced by M Grabiec). Next, inspired by the ideas of Gregori and Sapena [31], we define a new GV-contraction of Banach type with respect to this generalized fuzzy metric, which is a generalization of a contraction of Banach type (introduced by V Gregori and A Sapena). Moreover, we provide the condition guaranteeing the existence of a fixed point for these single-valued contractions. Next, we show that the generalized pseudodistance (introduced by Włodarczyk and Plebaniak [44]) may generate some generalized fuzzy metric on X. Moreover, if we put , where is the usual metric, then is a fuzzy metric generated by d.
2 On fixed point theory in Kramosil and Michalek’s fuzzy metric spaces and George and Veeramani’s fuzzy metric spaces
To begin with, we recall the concept of a fuzzy metric space, which was introduced by Kramosil and Michalek [23] in 1975.
Definition 2.1 [23]
The 3-tuple is a fuzzy metric space if X is an arbitrary set, ∗ is a continuous t-norm, and M is a fuzzy set in satisfying the following conditions:
(M1) ;
(M2) ;
(M3) ;
(M4) ;
(M5) is left-continuous, for all .
Then M is called a fuzzy metric on X.
Definition 2.2 (I) [29] A sequence in X is Cauchy in Grabiec’s sense (we say G-Cauchy) if
(II) [29] A sequence in X is convergent to if
i.e.,
Of course, since is continuous, by (M4) it follows that the limit is uniquely determined.
-
(III)
[29] A fuzzy metric space in which every G-Cauchy sequence is convergent is called complete in Grabiec’s sense (G-complete for short).
Some interesting observations on these definitions can be found in [45].
In 1989, Grabiec [29] established the following extension of Banach’s result in Kramosil and Michalek’s fuzzy metric space.
Theorem 2.1 (Fuzzy Banach contraction theorem, Grabiec [29])
Let be a G-complete fuzzy metric space such that
Let be a mapping satisfying
(G1) .
Then T has a unique fixed point.
Next, we recall the concept of a fuzzy metric space, which was introduced by George and Veeramani [26] in 1994.
Definition 2.3 [26]
The 3-tuple is a fuzzy metric space if X is an arbitrary set, ∗ is a continuous t-norm, and M is a fuzzy set on satisfying the following conditions:
(M1) ;
(M2) ;
(M3) ;
(M4) ;
(M5) is continuous, for all .
Then M is called a fuzzy metric on X.
Definition 2.4 (I) [26] Let be a fuzzy metric space. The open ball for with center and radius r, , is defined as
The family is a neighborhood’s system for a Hausdorff topology on X, which we call induced by the fuzzy metric M.
-
(II)
[31] A sequence in X is Cauchy in George and Veeramani’s sense (we say GV-Cauchy) if
-
(III)
[31] A fuzzy metric space in which every GV-Cauchy sequence is convergent is called complete in George and Veeramani’s sense (GV-complete for short).
In 2002, Gregori and Sapena [31] established the following extension of Banach’s result in George and Veeramani’s fuzzy metric spaces.
Theorem 2.2 (Fuzzy Banach contraction theorem, Gregori and Sapena [31])
Let be a GV-complete fuzzy metric space in which fuzzy contractive sequences, i.e.,
are GV-Cauchy. Let be a mapping satisfying
(G2) .
Then T has a unique fixed point.
3 On generalized fuzzy metric and fixed point theory in Kramosil and Michalek’s fuzzy metric spaces and George and Veeramani’s fuzzy metric spaces
Now in Kramosil and Michalek’s fuzzy metric space we introduce the concept of a generalized fuzzy metric on X. Next, we define a new kind of completeness of the space.
Definition 3.1 Let be a fuzzy metric space. The map N is said to be a G-generalized fuzzy metric on X if the following three conditions hold:
(N1) ;
(N2) is left-continuous, for all ;
(N3) for any sequences and in X such that
and
we have
Remark 3.1 If is a fuzzy metric space, then the fuzzy metric M is a G-generalized fuzzy metric on X. However, there exists a G-generalized fuzzy metric on X which is not a fuzzy metric on X (for details see Example 4.3).
Definition 3.2 (I) A sequence in X is N-Cauchy in Grabiec’s sense (we say N-G-Cauchy) if
-
(II)
A sequence in X is N-convergent to if
-
(III)
A fuzzy metric space is called N-G-complete if each N-G-Cauchy sequence in X is N-convergent to some and
(NC) .
Now we prove the auxiliary lemma.
Lemma 3.1 Let be a fuzzy metric space and let the map N be a G-generalized fuzzy metric on X. Then for each the following property holds:
Proof Let such that
be arbitrary and fixed. By (N1) and (3.4), we get
Defining the sequences and , from (3.5) and (3.4) we have
and
Hence, the properties (3.1) and (3.2) hold. Therefore, by (N3), we see that
which, by the definition of the sequences and , gives
Hence, by (M2), we conclude that . □
The main result of the paper is the following.
Theorem 3.1 Let be a fuzzy metric space, and let N be a G-generalized fuzzy metric on X such that
Let be an N-G-contraction of Banach type, i.e., T is a mapping satisfying
(B1) .
We assume that a fuzzy metric space is N-G-complete. Then T has a unique fixed point , and for each , the sequence is convergent to w. Moreover, , for all .
Proof The proof will be divided into four steps.
Step I. We see that for each the sequence satisfies
Indeed, let be arbitrary and fixed and let . Let be as in (B1), and let and be arbitrary and fixed. From (B1) we obtain
Consequently, the property (3.7) holds.
Step II. We see that for each the sequence is N-G-Cauchy, i.e., it satisfies
Indeed, let be arbitrary and fixed and let . Let and be arbitrary and fixed. Then by (N1) and (3.7) we calculate
Now, using (3.6) we obtain
Thus (3.8) holds.
Step III. Next we see that for each the sequence is convergent to a fixed point of T.
Indeed, let be arbitrary and fixed and let . By Step II the sequence is N-G-Cauchy in X. By the N-G-completeness of X (Definition 3.2(III)), there exists such that is N-convergent to w (i.e., ). Moreover, by (NC), we get
Next, using (N1) and (B1) we calculate
which, by (3.9), gives
Similarly, using (N1) and (B1) we calculate
which, by (3.9), gives
Now, from (3.10), (3.11), and Lemma 3.1 we obtain , i.e., w is a fixed point of T in X. Moreover, by (N1), (3.10), and (3.11), we obtain
Now, if we define the sequence , then by (3.8) and (3.9) we have
and
Therefore (3.1) and (3.2) hold, so by (N3) we have , which gives
Step IV. Finally we see that w is a unique fixed point of T in X and , for all .
Indeed, assume that for some . Then using (B1) we obtain
which, by (N2) and (3.6), gives
Similarly, using (B1), (N3), and (3.6) we calculate . Hence,
Next, applying Lemma 3.1, we get , thus the fixed point of T is unique. Moreover, by (3.12) we get . □
Remark 3.2 It is worth noticing that in George and Veeramani’s fuzzy metric space we may introduce the concept of a generalized fuzzy metric (in the sense of George-Veeramani) on X (for short, GV-generalized fuzzy metric). Let be a fuzzy metric space. The map N is said to be a GV-generalized fuzzy metric on X if the following three conditions hold:
() ;
() is continuous, for all ;
() for any sequences and in X such that
and
we have
Remark 3.3 Using similar considerations, we may introduce the concepts of N-Cauchy sequences in George and Veeramani’s sense and N-GV-completeness. Precisely: (I) A sequence in X is N-Cauchy in George and Veeramani’s sense (we say N-GV-Cauchy) if
-
(II)
A fuzzy metric space is called N-GV-complete, if each N-GV-Cauchy sequence in X is N-convergent to some and .
Now using similar arguments to the corresponding ones appearing in Section 3 and in the paper of Gregori and Sapena [31] we may conclude the following fixed point theorem in George and Veeramani’s fuzzy metric space.
Theorem 3.2 Let be a fuzzy metric space, and let N be a GV-generalized fuzzy metric on such that N-fuzzy contractive sequences, i.e.,
are N-GV-Cauchy. Let be an N-GS-contraction of Banach type (in the sense of Gregori and Sapena), i.e., a mapping satisfying
(B2) .
We assume that a fuzzy metric space is N-GV-complete. Then T has a unique fixed point , and for each , the sequence , is convergent to w. Moreover, , for all .
4 Examples illustrating Theorem 3.2 and some comparisons
Now, we will present some examples illustrating the concepts that have been introduced so far. We will show a fundamental difference between Theorem 2.2 and Theorem 3.2. Examples will show that Theorem 3.2 is the essential generalization of Theorem 2.2. First, we recall an example of the standard fuzzy metric induced by the metric d.
Example 4.1 [[31], Definition 2.5]
Let X be a metric space. Let ∗ be the usual product on . Then the 3-tuple where
(MD) , ,
is a George and Veeramani fuzzy metric space (standard fuzzy metric space), and is fuzzy metric on X.
Recently, in 2011, Włodarczyk and Plebaniak introduced the concept of generalized pseudodistances which, in a natural way, are extensions of metrics. For details see [44]. We recall the concept of a generalized pseudodistance.
Definition 4.1 Let X be a metric space with a metric . The map is said to be a generalized pseudodistance on X if the following two conditions hold:
(J1) ;
(J2) for any sequences and in X such that
and
we have
We recall also the following remark.
Remark 4.1 (A) If is a metric space, then the metric is a generalized pseudodistance on X. However, there exists a generalized pseudodistance on X which is not a metric (see Example 4.2).
(B) From (J1) and (J2) it follows that if , , then
Indeed, if and , then , since, by (J1), we get . Now, defining and for , we conclude that (4.1) and (4.2) hold. Consequently, by (J2), we get (4.3), which implies . However, since , we have . Contradiction.
-
(C)
From (B) it follows that if , then
Now we introduce and use some particular kind of generalized pseudodistance to construct the generalized fuzzy metrics.
Example 4.2 Let X be a metric space with metric . Let be a bounded and closed set, containing at least two different points, be arbitrary and fixed. Let be such that , where are arbitrary and fixed. Define the map as follows:
We can show that the map J is a generalized pseudodistance on X. Indeed, let be arbitrary and fixed. We consider the following four cases:
Case 1. If , then by (4.4) we obtain , so by the triangle inequality for d, we get (if ), and (if , since ). In consequence, in both situations
Case 2. If , then by (4.4) we obtain and , so by (4.4) (if ) and (if ). In consequence, in both situations
Case 3. If , then by (4.4) we obtain and , so by (4.4), (if ) and (if ). In consequence, in both situations
Case 4. If , then by (4.4) we obtain and , so by (4.4), , (if ) and (if ). In consequence, in both situations
Therefore, , i.e., the condition (J1) holds.
For proving that (J2) holds we assume that the sequences and in X satisfy (4.1) and (4.2). Then, in particular, (4.2) yields
By (4.5) and (4.4), since , we conclude that
From (4.6), (4.4), and (4.5), we get . Therefore, the sequences and satisfy (4.3). Consequently, the property (J2) holds.
In the remaining part of the work, the generalized pseudodistance defined by (4.4) will be called a generalized pseudodistance generated by d.
Example 4.3 Let be a standard metric space. Let be a generalized pseudodistance on X generated by d (i.e., defined in Example 4.2). Let ∗ be a continuous t-norm given by . Then the where
, is a GV-generalized fuzzy metric on X.
Part I. We prove ().
Let be arbitrary and fixed. By (J1) we get
Assume that there exist and such that . This, by (4.7), gives
Hence by a simple calculation we obtain a contradiction. In consequence (N1) and () hold.
Part II. We prove ().
Let be arbitrary and fixed. Then for we have
Thus, is continuous, for each . In consequence (N2) and () hold.
Part III. Next we prove ().
We assume that the sequences and in X satisfy (3.13) and (3.14). Then, in particular, (3.14) yields
Since , by a simple calculation we have
Next, from (4.9) and (4.10) we obtain
Now, let . We obtain , next, by (4.7) we have , so and finally , which, by (4.4), gives . Therefore .
Hence, using (4.9) we obtain
Consequently (3.15) holds. Hence () holds.
Example 4.4 Let be a standard fuzzy metric space, where , ∗ be a continuous t-norm given by . Let the closed set and let be given by
Let be defined by
Let be a single-valued map given by
-
(A)
By Example 4.2, J is a generalized pseudodistance on X. Next, by Example 4.3, is a GV-generalized fuzzy metric on X.
-
(B)
We observe that T is -GS-contraction of Banach type, i.e., T satisfies the condition (B2). The proof will be divided into two steps.
Step I. First, we show that T satisfies the following conditions:
Indeed, let and let be arbitrary and fixed. We consider the following two cases:
Case 1. If then by (4.12), . Moreover, since thus by (4.14), . Hence, by (4.12), we obtain
Case 2. If then by (4.12), . Moreover, since
and by (4.12), . Hence we obtain
Concluding, from (4.16) and (4.17), we obtain (4.15).
Step II. We show that T satisfies the following conditions:
Let . Let , be arbitrary and fixed. By (4.15) we know that . Hence, we obtain the following chain of equivalences:
Hence, the condition (4.18) is true, and the map T is -GS-contraction of Banach type.
-
(C)
Observe that T is not contraction of Banach type (in the sense of Gregori and Sapena), i.e., T does not satisfy the condition (G2). Indeed, suppose that T is contraction of Banach type (in the sense of Gregori and Sapena). Then there exists such that
(4.19)
In particular, for and , by (4.14), we have , . Hence . Moreover, and consequently, for each , by (MD) and (4.19), we have
Hence , which is impossible (recall ).
-
(D)
Now we see that is GV-complete standard fuzzy metric space.
Indeed, we see that is complete metric space, thus by [[31], Result 4.3] we conclude that the standard fuzzy metric space is GV-complete.
-
(E)
Next, we observe that the fuzzy metric space is N-GV-complete.
Indeed, let be a sequence such that is -GV-Cauchy, i.e.,
Now, by (4.13) and (4.20) we have
Hence, in particular, (4.21) yields
Hence by (4.12) we get
which gives . Moreover, by (4.20), after simple calculations we see that the sequence is GV-Cauchy. Now from (D) we obtain the result that there exists such that
Now, from (4.22) and (MD) we know that . Moreover, since E is a closed set, we obtain . Hence
which, by (4.12), gives
Finally, by (4.13) and (4.23) we have . Hence, by (4.23) we obtain
Hence we find that is N-GV-complete.
-
(F)
Now we see that each N-fuzzy contractive sequence is N-GV-Cauchy.
Indeed, let be an N-fuzzy contractive sequence, i.e.,
Hence,
which gives
Now, by (4.12),
and
Hence, the sequence is contractive in , thus (by the completeness of ) convergent. Consequently, is Cauchy in X. Therefore is GV-Cauchy in , i.e.,
Now let and be arbitrary and fixed. Then there exists such that, by (4.24) and (4.25), we obtain
Hence the sequence is N-GV-Cauchy.
-
(G)
Finally, we observe that all assumptions of Theorem 3.2 are satisfied. The point is a fixed point of T in X. Moreover, for each , the sequence satisfies condition . Hence, by (MD), we obtain . In consequence, for each , the sequence is convergent (in the standard fuzzy metric space ) to w.
Remark 4.2 (I) We observe that if we put in Theorem 3.2, then we find that Theorems 3.2 and 2.2 are identical.
-
(II)
The introduction of the concept of a generalized fuzzy metric is essential. If X and T are such as in Example 4.4, then we can show that T is an -GS-contraction of Banach type, but it is not a contraction of Banach type with respect to (see Example 4.4(B), (C)). Hence, we see that our theorem is a generalization of Theorem 2.2 (Gregori and Sapena [31]).
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Plebaniak, R. New generalized fuzzy metrics and fixed point theorem in fuzzy metric space. Fixed Point Theory Appl 2014, 241 (2014). https://doi.org/10.1186/1687-1812-2014-241
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DOI: https://doi.org/10.1186/1687-1812-2014-241