INTERNATIONAL JOURNAL OF COMPUTERS COMMUNICATIONS & CONTROL
ISSN 1841-9836, 11(2):273-281, April 2016.

Fuzzy b-Metric Spaces

S. Nădăban

Sorin Nădăban
Department of Mathematics and Computer Science
Aurel Vlaicu University of Arad,
Elena Drăgoi 2, RO-310330 Arad, Romania
snadaban@gmail.com

Abstract: Metric spaces and their various generalizations occur frequently in com-
puter science applications. This is the reason why, in this paper, we introduced and
studied the concept of fuzzy b-metric space, generalizing, in this way, both the notion
of fuzzy metric space introduced by I. Kramosil and J. Michálek and the concept
of b-metric space. On the other hand, we introduced the concept of fuzzy quasi-b-
metric space, extending the notion of fuzzy quasi metric space recently introduced by
V. Gregori and S. Romaguera. Finally, a decomposition theorem for a fuzzy quasi-
pseudo-b-metric into an ascending family of quasi-pseudo-b-metrics is established.
The use of fuzzy b-metric spaces and fuzzy quasi-b-metric spaces in the study of de-
notational semantics and their applications in control theory will be an important
next step.
Keywords: Fuzzy b-metric spaces, fuzzy quasi-b-metric, fuzzy quasi-pseudo-b-
metric, b-metric space.

1 Introduction and preliminaries

The concept of b-metric space was introduced by I.A. Bakhtin [5] and exensively used by S.
Czerwic [10, 11].

Definition 1. [10] Let X be a nonempty set and k ≥ 1 be a given real number. A function
d : X ×X → [0,∞) is a b-metric on X if, for all x,y,z ∈ X, the following conditions hold:

(b1) d(x,y) = 0 if and only if x = y;

(b2) d(x,y) = d(y,x);

(b3) d(x,z) ≤ k[d(x,y) + d(y,z)].

The triple (X,d,k) will be called b-metric space.

Some examples of b-metric spaces and some fixed point theorems in b-metric spaces can be
found in [6–8, 21]. We also note that the class of b-metric spaces is larger than that of metric
spaces, since every b-metric is a metric when k = 1. In [22] an example of a b-metric space which
is not a metric space, is given.

Recently, M.A. Alghamdi, N. Hussain, P. Salimi [1] introduced the notion of b-metric-like
space, which is an interesting generalization of metric-like space (introduced by A. Amini-Harandi
[2]) and partial metric space (introduced by S.G. Matthews [17]). In paper [14], N. Hussain and
M.H. Shah introduced the notion of cone b-metric space, generalizing both notions of b-metric
spaces and cone metric spaces.

The concept of quasi-b-metric space was introduced by M.H. Shah and N. Hussain [20] in
2012. In this paper we adopt a slight modification of their definition.

Definition 2. Let X be a nonempty set. A real valued function d : X ×X → [0,∞) is said to be
a quasi-b-metric with constant k ≥ 1 if the following conditions are satisfied:

Copyright © 2006-2016 by CCC Publications



274 S. Nădăban

(qb1) d(x,y) = d(y,x) = 0 if and only if x = y;

(qb3) d(x,z) ≤ k[d(x,y) + d(y,z)],(∀)x,y,z ∈ X.

The triple (X,d,k) will be called quasi-b-metric space.

On the other hand, after L.A. Zadeh has introduced in his famous paper [23] the concept of
fuzzy set, one of the important problems is to obtain an adequate notion of fuzzy metric space. I.
Kramosil and J. Michálek [16] reformulated successfully the notion of probabilistic metric space,
introduced by K. Menger in 1942, in fuzzy context.

Definition 3. [19] A binary operation

∗ : [0,1]× [0,1] → [0,1]

is called triangular norm (t-norm) if it satisfies the following condition:

1. a∗ b = b∗a,(∀)a,b ∈ [0,1];

2. a∗1 = a,(∀)a ∈ [0,1];

3. (a∗ b)∗ c = a∗ (b∗ c),(∀)a,b,c ∈ [0,1];

4. If a ≤ c and b ≤ d, with a,b,c,d ∈ [0,1], then a∗ b ≤ c∗d.

Example 4. Three basic examples of continuous t-norms are ∧, ·,∗L, which are defined by a∧b =
min{a,b}, a·b = ab (usual multiplication in [0,1]) and a∗Lb = max{a+b−1,0} (the Lukasiewicz
t-norm).

Definition 5. [16] The triple (X,M,∗) is said to be a fuzzy metric space if X is an arbitrary set,
∗ is a continuous t-norm and M is a fuzzy metric, i.e. a fuzzy set in X ×X × [0,∞) such that
for all x,y,z ∈ X we have:

(M1) M(x,y,0) = 0;

(M2) [M(x,y,t) = 1,(∀)t > 0] if and only if x = y;

(M3) M(x,y,t) = M(y,x,t),(∀)t ≥ 0;

(M4) M(x,z,t + s) ≥ M(x,y,t)∗M(y,z,s),(∀)t,s ≥ 0;

(M5) M(x,y, ·) : [0,∞) → [0,1] is left continuous and lim
t→∞

M(x,y,t) = 1.

We note that A. George and P. Veeramani [12] modified the concept of fuzzy metric space
introduced by I. Kramosil and J. Michálek and defined a Hausdorff topology on this fuzzy space.
Another approach for fuzzy metric spaces was introduced by O. Kaleva and S. Seikkala in paper
[15], by setting the distance between two points to be a non-negative, upper semicontinuous,
normal and convex fuzzy number.

In recent years, different types of fuzzy generalized metric spaces was considered by different
authors in different approaches. Thus, V. Gregori and S. Romaguera introduced in paper [13]
the concept of fuzzy quasi-metric space, generalizing in this way the notions of fuzzy metric
introduced by I. Kramosil and J. Michálek and by A. George and P. Veeramani to the quasi-
metric setting.

On the other hand, the idea of fuzzy cone metric space has been introduced in [3] and some
basic properties and fixed point theorems for different types of contraction mappings have been
developed in fuzzy cone metric spaces. In paper [4], T. Bag introduced the concept of fuzzy



Fuzzy b-Metric Spaces 275

cone b-metric space and some fixed point theorems are established in such spaces for contraction
mappings. We must note that Bag’s definitions for fuzzy cone metric space and for fuzzy cone
b-metric spaces generalized the notion of fuzzy metric introduced by Kaleva and Seikkala.

In this paper we introduced and studied the concept of fuzzy b-metric space, generalizing, in
this way, both the notion of fuzzy metric space introduced by I. Kramosil and J. Michálek and
the concept of b-metric space. On the other hand, we introduced the concept of fuzzy quasi-b-
metric space, extending the notion of fuzzy quasi-metric space recently introduced by V. Gregori
and S. Romaguera. Finally, a decomposition theorem for a fuzzy quasi-pseudo-b-metric into an
ascending family of quasi-pseudo-b-metrics is established.

2 Fuzzy b-metric spaces

Definition 6. Let X be a nonempty set, let k ≥ 1 be a given real number and ∗ be a continuous
t-norm. A fuzzy set M in X × X × [0,∞) is called fuzzy b-metric if, for all x,y,z ∈ X, the
following conditions hold:

(bM1) M(x,y,0) = 0;

(bM2) [M(x,y,t) = 1,(∀)t > 0] if and only if x = y;

(bM3) M(x,y,t) = M(y,x,t),(∀)t ≥ 0;

(bM4) M(x,z,k(t + s)) ≥ M(x,y,t)∗M(y,z,s),(∀)t,s ≥ 0;

(bM5) M(x,y, ·) : [0,∞) → [0,1] is left continuous and lim
t→∞

M(x,y,t) = 1.

The quadruple (X,M,∗,k) is said to be a fuzzy b-metric space.
Remark 7. The class of fuzzy b-metric spaces is larger than the class of fuzzy metric spaces,
since a fuzzy b-metric space is a fuzzy metric space when k = 1.
Example 8. Let (X,d,k) be a b-metric space. Let

Md : X ×X × [0,∞) → [0,1],Md(x,y,t) =

{
t

t+d(x,y)
if t > 0

0 if t = 0
.

Then (X,Md,∧,k) is a fuzzy b-metric space. Md will be called standard fuzzy b-metric.

Proof: We check only (bM4), because verifying the other conditions is standard.
Let x,y,z ∈ X and t,s > 0. Without restraining the generality we assume that Md(x,y,t) ≤

Md(y,z,s). Thus tt+d(x,y) ≤
s

s+d(y,z)
, i.e. td(y,z) ≤ sd(x,y).

On the other hand
Md(x,z,k(t + s)) =

k(t + s)

k(t + s) + d(x,z)
≥

≥
k(t + s)

k(t + s) + k[d(x,y) + d(y,z)]
=

t + s

t + s + d(x,y) + d(y,z)
.

We will prove that
t + s

t + s + d(x,y) + d(y,z)
≥

t

t + d(x,y)
.

Hence we will obtain that Md(x,z,k(t + s)) ≥ Md(x,y,t) = Md(x,y,t) ∧Md(y,z,s), what had
to be verified. We remark that

t + s

t + s + d(x,y) + d(y,z)
≥

t

t + d(x,y)
⇔



276 S. Nădăban

t2 + st + td(x,y) + sd(x,y) ≥ t2 + st + td(x,y) + td(y,z) ⇔ sd(x,y) ≥ td(y,z) ,

which is true. 2

Definition 9. Let k ≥ 1 be a real given number. A function f : R → R will be called k-
nondecreasing if for t < s we will have that f(t) ≤ f(ks).
Proposition 10. For all x,y ∈ X the mapping M(x,y, ·) : [0,∞) → [0,1] is k-nondecreasing.

Proof: For 0 < t < s we have

M(x,y,ks) ≥ M(x,x,s− t)∗M(x,y,t) = 1∗M(x,y,t) = M(x,y,t) .

2

Theorem 2.1. Let (X,M,∗,k) be a fuzzy b-metric space. For x ∈ X,r ∈ (0,1),
t > 0 we define the open ball

B(x,r,t) := {y ∈ X : M(x,y,t) > 1−r} .

Then
TM := {T ⊂ X : x ∈ T iff (∃)t > 0,r ∈ (0,1) : B(x,r,t) ⊆ T}

is a topology on X.

Proof: It is obvious that ∅ and X belong to TM .
Let {Ti}i∈I ⊆TM and T =

∪
i∈I

Ti. We will show that T ∈TM . Let x ∈ T . Then there exists

i0 ∈ I such that x ∈ Ti0 . As Ti0 ∈ TM , there exist t > 0, r ∈ (0,1) such that B(x,r,t) ⊆ Ti0 .
Thus B(x,r,t) ⊆

∪
i∈I

Ti = T .

Let now {Ti}ni=1 ⊆ TM and T =
n∩

i=1
Ti. We will show that T ∈ TM . Let x ∈ T . We obtain

that x ∈ Ti,(∀)i = 1,n. Thus

(∃)ti > 0,ri ∈ (0,1) : B(x,ri, ti) ⊆ Ti,(∀)i = 1,n .

Let
r = min{ri, i = 1,n}, t = min

{
ti
k
,i = 1,n

}
.

We have that B(x,r,t) ⊆ B(x,ri, ti),(∀)i = 1,n. Indeed, for y ∈ B(x,r,t), we have
M(x,y,t) > 1 − r ≥ 1 − ri,(∀)i = 1,n. As t ≤ tik ,(∀)i = 1,n, we obtain that
M(x,y,t) ≤ M(x,y,ti). Thus M(x,y,ti) > 1−ri,(∀)i = 1,n. Hence y ∈ B(x,ri, ti),(∀)i = 1,n.
Therefore B(x,r,t) ⊆ Ti,(∀)i = 1,n. Thus B(x,r,t) ⊆

n∩
i=1

Ti = T . 2

Remark 11. Previous theorem extends to fuzzy b-metric space a similar result obtained by A.
George and P. Veeramani [12] in the context of fuzzy metric space. The definitions for convergent
sequence and Cauchy sequence given by A. George and P. Veeramani [12] in the context of fuzzy
metric space can be translated in the context of fuzzy b-metric space, as follows.

Definition 12. Let (X,M,∗,k) be a fuzzy b-metric space and (xn) be a sequence in X. The
sequence (xn) is said to be convergent if there exists x ∈ X such that M(xn,x,t) = 1,(∀)t > 0.
In this case, x is called the limit of the sequence (xn) and we note lim

n→∞
xn = x, or xn → x.

Remark 13. Let (X,M,∗,k) be a fuzzy b-metric space. A sequence (xn) is convergent to x if
and only if (xn) is convergent to x in topology TM .



Fuzzy b-Metric Spaces 277

Indeed,
xn → x in topology TM ⇔

⇔ (∀)r ∈ (0,1),(∀)t > 0,(∃)n0 ∈ N : xn ∈ B(x,r,t),(∀)n ≥ n0 ⇔

⇔ (∀)r ∈ (0,1),(∀)t > 0,(∃)n0 ∈ N : M(xn,x,t) > 1−r,(∀)n ≥ n0 ⇔

⇔ lim
n→∞

M(xn,x,t) = 1,(∀)t > 0 .

Definition 14. Let (X,M,∗,k) be a fuzzy b-metric space and (xn) be a sequence in X. The
sequence (xn) is said to be a Cauchy sequence if

(∀)r ∈ (0,1),(∀)t > 0,(∃)n0 ∈ N : M(xn,xm, t) > 1−r,(∀)n,m ≥ n0 .

A fuzzy b-metric space in which every Cauchy sequence is convergent is called complete fuzzy
b-metric space.

3 Fuzzy quasi-b-metric spaces

Definition 15. A fuzzy quasi-b-metric space is a quadruple (X,M,∗,k), where X is a nonempty
set, ∗ is a continuous t-norm, k ≥ 1 is a given real number and M is a fuzzy set in X×X×[0,∞)
such that for all x,y,z ∈ X we have:

(qbM1) M(x,y,0) = 0;

(qbM2) [M(x,y,t) = M(y,x,t) = 1,(∀)t > 0] if and only if x = y;

(qbM3) M(x,z,k(t + s)) ≥ M(x,y,t)∗M(y,z,s),(∀)t,s ≥ 0;

(qbM4) M(x,y, ·) : [0,∞) → [0,1] is left continuous and lim
t→∞

M(x,y,t) = 1.

Remark 16. V. Gregori and S. Romaguera [13] also gave this definition in the particular case
k = 1 and the triple (X,M,∗) is called fuzzy quasi-metric space.
Proposition 17. If Q is a fuzzy quasi-b-metric, then Q−1 defined by Q−1(x,y,t) = Q(y,x,t) is
also a fuzzy quasi-b-metric (called the conjugate of Q).

Proof: We have to check only (qbM3).

Q−1(x,z,k(t + s)) = Q(z,x,k(s + t)) ≥ Q(z,y,s)∗Q(y,x,t) = Q−1(x,y,t)∗Q−1(y,z,s) .

2

Definition 18. [18]. Let ∗,◦ be two t-norms. We say that ◦ dominates ∗ and we denote ◦≫∗ if

(x1 ◦x2)∗ (y1 ◦y2) ≤ (x1 ∗y1)◦ (x2 ∗y2),(∀)x1,x2,y1,y2 ∈ [0,1].

Remark 19. [18]. For any t-norm ∗ we have ∧≫∗.
Proposition 20. Let (X,Q,∗,k) be a fuzzy quasi-b-metric space and ◦ be a continuous t-norm
such that ◦≫∗. Let M be a fuzzy set in X ×X × [0,∞) defined by

M(x,y,t) = Q(x,y,t)◦Q−1(x,y,t) .

Then (X,M,∗,k) is a fuzzy b-metric space.



278 S. Nădăban

Proof: It is easy to check (bM1)− (bM3) and (bM5). We prove (bM4).

M(x,z,k(t + s)) = Q(x,z,k(t + s))◦Q−1(x,z,k(t + s)) ≥

≥ [Q(x,y,t)∗Q(y,z,s)]◦ [Q−1(x,y,t)∗Q−1(y,z,s)] ≥

≥ [Q(x,y,t)◦Q−1(x,y,t)]∗ [Q(y,z,s)◦Q−1(y,z,s)] = M(x,y,t)∗M(y,z,s) .

2

Corollary 21. Let (X,Q,∗,k) be a fuzzy quasi-b-metric space and

M(x,y,t) = min{Q(x,y,t),Q(y,x,t)} .

Then (X,M,∗,k) is a fuzzy b-metric space.

Proof: We apply previous proposition for ◦ = ∧≫∗. 2

Example 22. Let (X,d,k) be a quasi-b-metric space. Let

Md : X ×X × [0,∞) → [0,1],Md(x,y,t) =

{
t

t+d(x,y)
if t > 0

0 if t = 0
.

Then (X,Md,∧,k) is a fuzzy quasi-b-metric space. Md will be called standard fuzzy quasi-b-
metric.

Proof: The proof is standard. 2

Proposition 23. If (X,M,∗,k) is a fuzzy quasi-b-metric space, then the relation ≤M on X defined
by

x ≤M y if and only if M(x,y,t) = 1,(∀)t > 0

is a partial ordering.

Proof. It is easy to check.

4 Fuzzy quasi-pseudo-b-metric spaces

Definition 24. Let X be a nonempty set. A function d : X ×X → [0,∞) is called quasi-pseudo-
b-metric with constant k ≥ 1 if the following conditions are satisfied:

(qpb1) d(x,x) = 0;

(qpb3) d(x,z) ≤ k[d(x,y) + d(y,z)],(∀)x,y,z ∈ X.

The triple (X,d,k) will be called quasi-pseudo-b-metric space.

Definition 25. A fuzzy quasi-pseudo-b-metric space is a quadruple (X,M,∗,k), where X is a
nonempty set, ∗ is a continuous t-norm, k ≥ 1 is a given real number and M is a fuzzy set in
X ×X × [0,∞) such that for all x,y,z ∈ X we have:

(qpbM1) M(x,y,0) = 0;

(qpbM2) [M(x,x,t) = 1,(∀)t > 0];

(qpbM3) M(x,z,k(t + s)) ≥ M(x,y,t)∗M(y,z,s),(∀)t,s ≥ 0;



Fuzzy b-Metric Spaces 279

(qpbM4) M(x,y, ·) : [0,∞) → [0,1] is left continuous and lim
t→∞

M(x,y,t) = 1.

Theorem 4.1. Let (X,M,∧,k) be a fuzzy quasi-pseudo-b-metric space and

dα(x,y) := inf{t > 0 : M(x,y,t) > α},α ∈ (0,1) .

Then D = {dα}α∈(0,1) is an ascending family of quasi-pseudo-b-metrics on X.

Proof: (qp1) dα(x,x) = inf{t > 0 : M(x,x,t) > α} = 0.
(qp2)

k[dα(x,y) + dα(y,z)] = k[inf{t > 0 : M(x,y,t) > α}+ inf{s > 0 : M(y,z,s) > α}] =

= k[inf{t + s > 0 : M(x,y,t) > α,M(y,z,s) > α}] =

= inf{k(t + s) > 0 : M(x,y,t)∧M(y,z,s) > α}≥

≥ inf{k(t + s) > 0 : M(x,z,k(t + s)) > α} = dα(x,z) .

It remains to prove that D = {dα}α∈(0,1) is an ascending family. Let α1 ≤ α2. Then

{t > 0 : M(x,y,t) > α2}⊆{t > 0 : M(x,y,t) > α1} .

Thus
inf{t > 0 : M(x,y,t) > α2}≥ inf{t > 0 : M(x,y,t) > α1} ,

namely dα2(x,y) ≥ dα1(x,y),(∀)(x,y) ∈ X ×X. 2

5 Conclusions and further works

In this paper we introduce the notions of fuzzy b-metric space and fuzzy quasi-b-metric space.
Thus, we have built a fertile ground to study, in further papers, some fixed point theorems in these
spaces. The first problem is to established fuzzy versions of Banach contraction mapping principle
in fuzzy b-metric spaces. From here we will obtain a lot of applications both in Mathematics as
well as in Engineering and Computer Science. The second issue is to study set-valued contractions
in fuzzy b-metric spaces and their applications in control theory and convex optimization. A real
challenge is to extend the results of C. Chifu and G. Petruşel [9] in fuzzy b-metric spaces. We
intend to obtain some fixed point theorems for multivalued operators in fuzzy b-metric spaces
endowed with a graph. This paper may be of interest for researchers working in the following
fields belonging to Computer Science and Information Technology:

(i) Integrated solution in computer-based control and communications

(ii) Computational intelligence methods

(iii) Advanced decision support systems

where fuzzy metric spaces will be applied in dealing with the problems such as: fixed point
theorems and their applications in the semantics of programs; distance measurement between
programs with important results to measure the complexity of programs and algorithms; color
image processing and image denoising; the use of some types of fuzzy metrics in cognitive infor-
mation, in time series and in bioinformatics; the appplications in neural networks; data mining
and web mining applications.



280 S. Nădăban

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