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Probabilistic bmetric spaces and nonlinear contractions
Fixed Point Theory and Applicationsvolume 2017, Article number: 29 (2017)
Abstract
This work is for giving the probabilistic aspect to the known bmetric spaces (Czerwik in Atti Semin. Mat. Fis. Univ. Modena 46(2):263276, 1998), which leads to studying the fixed point property for nonlinear contractions in this new class of spaces.
Introduction
Fixed point theory plays a basic role in applications of many branches of mathematics. Finding a fixed point of contractive mappings has become the center of strong research activity. There are many works about the fixed point of contractive maps (see, for example, [2, 3]). In [3], Polish mathematician Banach proved a very important result regarding a contraction mapping, known as the Banach contraction principle, in 1922.
After that, based on this finding, a large number of fixed point results have appeared in recent years. Generally speaking, there usually are two generalizations on them. One is from mappings. The other is from spaces.
Concretely, for one thing, from mappings, for example, the concept of a φcontraction mapping was introduced in 1968 by Browder [4].
For another thing, from spaces, there are too many generalizations of metric spaces. For instance, recently, Bakhtin [5], introduced bmetric spaces as a generalization of metric spaces. He proved the contraction mapping principle in bmetric spaces that generalized the famous Banach contraction principle in metric spaces. Starting with the paper of Bakhtin, many fixed point results have been established in those interesting spaces (see [1, 6–8]).
Let us recall the notion of a bmetric space.
Definition 1.1
([1])
Let M be a (nonempty) set and \(s \geq 1\) be a given real number. A function \(d : M \times M \to \mathbb{R}^{+}\) is a bmetric iff, for all \(x,y,z \in M \), the following conditions hold:

(1)
\(d(x,y) =0\) iff \(x = y\),

(2)
\(d(x,y) = d(y,x)\),

(3)
\(d(x,z)\leq s[d(x,y) + d(y,z)]\).
In this case, the pair \((M,d)\) is called a bmetric space.
It should be noted that the class of bmetric spaces is effectively larger than the class of metric spaces since a bmetric is a metric when \(s =1\).
This paper is organized as follows. In Section 2, we present some basic concepts and relevant lemmas on probabilistic metric spaces (pms). In Section 3, we generalize the concept of pms by defining a probabilistic (fuzzy) bmetric space and discuss some topological proprieties of these new structures. In Section 4, we prove the main theorem in this paper, i.e., a new fixed point theorem for probabilistic (fuzzy) φcontraction in probabilistic (fuzzy) bmetric spaces. Subsequently, as an application of our results, in Sections 5, we provide an example and prove a fixed point theorem in bmetric spaces. Our results generalize some wellknown results in the literature.
Preliminaries
We begin by briefly recalling some definitions and notions from probabilistic metric spaces theory that we will use in the sequel. For more details, we refer the reader to [9].
A nonnegative real function f defined on \(\mathbb{R}^{+}\cup\{\infty\}\) is called a distance distribution function (briefly, a d.d.f.) if it is nondecreasing, leftcontinuous on \((0,\infty)\), with \(f(0)=0\) and \(f(\infty)=1\). The set of all d.d.f’s will be denoted by \(\Delta^{+}\); and the set of all \(f\in\Delta^{+}\) for which \(\lim_{s\to\infty}f(s)=1\) by \(D^{+}\).
A simple example of distribution function is a Heavyside function in \(D^{+}\)
Definition 2.1
Consider f and g being in \(\Delta^{+}, h\in(0, 1]\), and let \((f, g; h)\) denote the condition
for all x in \((0, \frac{1}{h})\).
The modified Lévy distance is the function \(d_{L}\) defined on \(\Delta ^{+} \times\Delta^{+}\) by
Note that, for any f and g in \(\Delta^{+}\), both \((f, g; 1)\) and \((g, f; 1)\) hold, hence \(d_{L}\) is well defined and \(d_{L}(f, g) \leq1\).
Lemma 2.1
([9])
The function \(d_{L}\) is a metric on \(\Delta^{+}\).
Lemma 2.2
([9])
The metric spaces \((\Delta^{+}, d_{L})\) are compact, and hence complete.
Lemma 2.3
([9])
For any F in \(\Delta^{+}\) and \(t>0\),
Lemma 2.4
([9])
If F and G are in \(\Delta^{+}\) and \(F\leq G\), then \(d_{L}(G, H)\leq d_{L}(F, H)\).
τ on \(\Delta^{+}\) is a triangle function if it assigns a d.d.f. in \(\Delta^{+}\) to every pair of d.d.f.’s in \(\Delta^{+}\times\Delta^{+}\) and satisfies the following conditions:
A commutative, associative and nondecreasing mapping \(T: [0,1] \times [0,1]\to[0,1]\) is called a tnorm if and only if
As examples we mention the tree typical examples of continuous tnorms as follows: \(T_{p}(a,b) = ab\), \(T_{M}(a,b) = \operatorname{Min}(a,b)\) and \(T_{L}(a,b) = \operatorname{Max}\{ a+b1,0\}\).
Moreover, if T is leftcontinuous, then the operation \(\tau_{T} : \Delta^{+}\times\Delta^{+}\to \Delta^{+}\) defined by
is a triangle function.
We say (O Hadzić [10]) that a tnorm T is of Htype if the family \(\{T^{n}(t)\}\) is equicontinuous at \(t=1\), that is,
where \(T^{1}(x) =T(x,x)\), \(T^{n}(x) = T(x,T^{n1}(x))\) for every \(n\geq2\).
The tnorm \(T_{M}\) is a trivial example of tnorm of Htype (see [10]).
Finally, we also have the following.
Lemma 2.5
([9])
If T is continuous, then \(\tau_{T}\) is continuous.
Probabilistic bmetric space
Having introduced the necessary terms, we now turn to our main topic. Developing a theory of probabilistic bmetric spaces, we start with the following definition.
Definition 3.1
A probabilistic bmetric space (briefly a pbms) is a quadruple \((M,F,\tau, s)\) where M is a nonempty set, F is a function from \(M \times M\) into \(\Delta^{+}\), τ is a triangle function, \(s\geq1\) is a real number, and the following conditions are satisfied: for all \(p,r; q \in M\) and \(y> 0\),

(i)
\(F_{pp}=H\),

(ii)
\(F_{pq}=H \Rightarrow p=q\),

(iii)
\(F_{pq} = F_{qp}\),

(iv)
\(F_{pq}(sy)\geq\tau(F_{pr}, F_{rq})(y)\).
If \(\tau=\tau_{T} \) for some tnorm T, then \((M, F, \tau_{T}, s)\) is called a bMenger space.
It should be noted that if T is a continuous tnorm, then \((M, F)\) satisfies (iv) under \(\tau_{T}\) if and only if it satisfies
for all \(p,r, q \in M\) and for all \(x, y> 0 \), under T.
Recall that a probabilistic metric space is a triple \((M, F, \tau)\) satisfying (i)(iii) and the following inequality:
for all \(p,r, q \in M\).
By setting \(F_{xy}\) by \(F_{xy}(0) = 0\) and \(F_{xy}(t)=M(x; y; t)\) for \(t>0\), the fuzzy bmetric space is defined in the following manner.
Definition 3.2
The quadruple \((X; M;*, s)\) is said to be a fuzzy bmetric space if X is an arbitrary set, ∗ is a continuous tnorm, \(s\geq1\) is a real number and M is a fuzzy set on \(X^{2}\times[0;\infty)\) satisfying the following conditions:

\(M(x, y,0)=0\),

\(M(x, y, q) = 1\) for all \(q>0\) iff \(x=y\),

\(M(x, y, q)=M(y, x, q)\),

\(M(x, z, t)*M(z; y; q)\leq M(x; y; s(t+q))\),

\(M(x, y, \cdot):[0,\infty[\, \to[0,1]\) is leftcontinuous and nondecreasing
for all \(x,y, z \in X\) and \(q, t > 0\).
From [11, Lemma 2.6], \(M(x, y,\cdot)\) is a nondecreasing mapping for \(x; y\in X\). Hence, every fuzzy metric space (in the sense of Kramosil and Michalek [12]) is a fuzzy bmetric space with the constant \(s=1\).
It is clear that every probabilistic (fuzzy) metric space (PM space) is a probabilistic (fuzzy) bmetric space with \(s=1\). But the converse is not true. We confirm this by the following examples.
Example 3.1
Let \(M=\{1, 2, 3, 4\}\). Define \(F:M \times M \to\Delta^{+}\) as follows:
It is easy to check that \((M,F,\tau_{T_{M}}, 3)\) is a pbms, but \((M,F,\tau_{T_{M}})\) is not a standard probabilistic metric space because it lacks the triangle inequality:
Example 3.2
Let \(M=[0,\infty)\). Define \(F:M \times M \to\Delta^{+}\) as follows:
It is easy to check that \((M,F,\tau_{T_{M}}, 2)\) is a pbms, but \((M,F,\tau_{T_{M}})\) is not a standard probabilistic metric space because it lacks the triangle inequality:
Definition 3.3
Let \(( M, F)\) be a probabilistic semimetric space (i.e., (i), (ii) and (iii) of Definition 3.1 are satisfied). For p in M and \(t > 0 \), the strong tneighborhood of p is the set
The strong neighborhood system at p is the collection \(\wp_{p}=\{ N_{p}(t): t>0\}\), and the strong neighborhood system for M is the union \(\wp=\bigcup_{p\in M }\wp_{p}\).
An immediate consequence of Lemma 2.3 is
Definition 3.4
Let \(\{x_{n}\}\) be a sequence in a probabilistic semimetric space \((M,F)\).

(1)
A sequence \(\{x_{n}\}\) in M is said to be convergent to x in M if, for every \(\epsilon>0\) and \(\delta\in(0,1)\), there exists a positive integer \(N(\epsilon,\delta)\) such that \(F_{x_{n}x}(\epsilon )>1\delta\), whenever \(n\geq N(\epsilon,\delta)\).

(2)
A sequence \(\{x_{n}\}\) in M is called Cauchy sequence if, for every \(\epsilon>0\) and \(\delta\in(0,1)\), there exists a positive integer \(N(\epsilon,\delta)\) such that \(F_{x_{n}x_{m}}(\epsilon )>1\delta\), whenever \(n, m\geq N(\epsilon,\delta)\).

(3)
\((M,F)\) is said to be complete if every Cauchy sequence has a limit.
Every bmetric space is a probabilistic bmetric space. Moreover, we have the following.
Lemma 3.1
Let \((M,d)\) be a bmetric space (bms) with the constant s. Define \(F: M \times M \to\Delta^{+}\) by
Then

(a)
\((M,F,\tau_{T_{M}}, s)\) is a pbms.

(b)
\((M,F,\tau_{T_{M}}, s)\) is complete if and only if \((M,d)\) is complete.
Proof
(a) It is easy to check the conditions (i)(iii) of Definition 3.1. So, for condition (v), let p, r, q in M, let \(t_{1}\), \(t_{2}\) in \([0,\infty)\).
If
then
Else if
then \(t_{1} > d(p,r)\) and \(t_{2} > d(r,q)\). Since \((M,d)\) is a bmetric space with the constant s, we have
Then we get
Thus
Hence condition (v) holds. So \((M,F,\tau_{T_{M}}, s)\) is a probabilistic bmetric space.
(b) By Definition 3.2 we get, for every \(t>0\),
So \((M,F,T_{M}, s)\) is a complete pbms if and only if \((M,d)\) is a complete bms. □
By using the above lemma, we present some typical examples of a probabilistic bmetric space.
Example 3.3
Let \((M,d)\) be a metric space and \(d'(x,y) = (d(x,y))^{p}\), where \(p > 1\) is a real number. We show that \(d'\) is a bmetric with \(s = 2^{p1}\).
Obviously, conditions (1) and (2) of Definition 1.1 are satisfied. If \(1 < p < \infty\), then the convexity of the function \(f(x) = x^{p} (x > 0)\) implies
for each \(x, y \in M\). So, condition (3) of Definition 1.1 is also satisfied and \((M,F,\tau_{T_{M}}, 2^{p1})\) is a pbms with \(F_{pq}(t)= H(t d'(p, q))\).
Scheizer and Sklar [9] proved that if \((M, F, \tau) \) is a PM space with τ being continuous, then the family ℑ consisting of ∅ and all unions of elements of this strong neighborhood system for M determines a Hausdorff topology for M. Consequently, there exists a topology Λ on M such that the strong neighborhood system ℘ is a basis for Λ.
But in a probabilistic bmetric space in general the last assertion is false as shown in the following example.
Example 3.4
Let \(a> 0\), \(M_{a} = [0, a]\cup\{4a\}\). Define \(F^{a}:M_{a} \times M_{a} \to \Delta^{+}\) as follows:
It is easy to show that \((M_{a},F^{a},\tau_{T_{M}}, 2)\) is a pbms with \(\tau _{T_{M}}\) being continuous, but \((M_{a},F^{a}, \tau_{T_{M}})\) is not a standard probabilistic metric space because it lacks the triangle inequality:
in which \(N_{4a}(3a) = \{a, 4a\}\) and there does not exist \(t>0\) such that \(N_{a}(t)\subseteq N_{4a}(3a)\). Hence the strong neighborhood system ℘ is not a basis for any topology on \(M_{a}\).
It is well known that in a probabilistic metric space \((M, F, \tau)\) with τ being continuous M is endowed with the topology ℑ and \(M\times M\) with the corresponding product topology. Then the probabilistic metric F is a continuous mapping from \(M\times M \) into \(\Delta^{+}\) [9].
However, in a probabilistic bmetric space \((M, F, \tau)\) the probabilistic bmetric F is not continuous in general even though τ is continuous. The following example illustrates this fact.
Example 3.5
Let \(M = \mathbb{N}\cup\{\infty\}\), \(0< a\leq1\). Define \(F^{a}:M \times M \to \Delta^{+}\) as follows:
It is easy to show that \((M,F^{a},\tau_{T_{M}}, 4)\) is a pbms with \(\tau _{T_{M}}\) being continuous. In the sequel, we take \(a=1\). Consider the sequence \(x_{n}=2n\), \(n\in \mathbb{N}\). Then \(F_{2n\infty}(t)=H(t\frac{1}{2n})\). Therefore \(x_{n}\to\infty\), but \(F_{2n 1}(t)=H(t3)\neq H(t1)=F_{\infty1}(t)\). Hence F is not continuous at ∞.
In the following result we show that a pbms is a Hausdorff space.
Lemma 3.2
Let \((M, F, \tau, s)\) be a pbms if τ is continuous, then the strong neighborhood system ℘ satisfies: if \(p\neq q\), then there are \(t, t'> 0\) such that \(N_{p}(t)\cap N_{q}(t')=\emptyset\).
Proof
Note that since \(p\neq q\), \(F'_{pq}\neq H\) with \(F'_{pq}(t)=F_{pq}(st)\), whence \(\varrho=d_{L}(F'_{pq}, H)>0\). By the uniform continuity of τ, there exists \(t> 0\) such that
whenever \(d_{L}(G, H)< t\) and \(d_{L}(G', H)< t\). Suppose that \(N_{p}(t)\cap N_{q}(t)\neq\emptyset\). So, let \(r\in N_{p}(t)\cap N_{q}(t)\). Then \(d_{L}(F_{pr}, H)< t\) and \(d_{L}(F_{rq}, H)< t\), whence Lemma 2.4 and (3.1) yield
an impossibility. Hence \(N_{p}(t)\cap N_{q}(t)\) is empty and the proof is complete. □
φProbabilistic contraction in a probabilistic bmetric space
Over this section, the letter Ψ denotes the set of all functions \(\varphi: [0,\infty) \to[0,\infty)\) such that
Example 4.1
Let \(c \in[1, \infty)\), and let \(\varphi _{c}:[0,\infty)\to[0,\infty) \) be defined by
Then

(i)
\(\varphi_{c}\) is a strictly increasing and continuous function,

(ii)
\(\varphi_{c}\in\Psi\).
(i) trivially holds. For (ii) it is very easy to check by induction that
Consequently,
which yields \(\varphi^{n}_{c}(t)\to0\), and since \(0< \varphi_{c}(t)<t\) for each \(t>0\), we conclude that (ii) holds.
Before stating the main fixed point theorems, we introduce the following concept.
Definition 4.1
Let \(\varphi:[0,\infty)\rightarrow[0,\infty)\) be a function such that \(\varphi(t) < t\) for \(t > 0\), and f be a selfmap of a probabilistic bmetric space \((M, F, \tau, s)\). We say that f is a φprobabilistic contraction if
for all \(p,q\in M\) and \(t>0\).
It should be noted, when \(s=1\), the above definition coincides with the concept of φprobabilistic contraction according to the definition in [13] and [14].
The following definition can be considered as a fuzzy version of Definition 4.1.
Definition 4.2
Let \(\varphi:[0,\infty)\rightarrow[0,\infty)\) be a function such that \(\varphi(t) < t\) for \(t > 0\), and f be a selfmap of a fuzzy bmetric space \((X; M;*, s)\). We say that f is a φfuzzy contraction if
for all \(p,q\in X\) and \(t>0\).
In the proof of our first theorem, we use the following lemma.
Lemma 4.1
Let \((M, F, \tau_{T}, s)\) be a pbms with a tnorm T of Htype and \(\operatorname{Ran}F \subset D^{+}\). Let \(\{x_{n}\}\) be a sequence in M. If there exists a function \(\varphi\in\Psi\) such that
then \(\{x_{n}\}\) is a Cauchy sequence.
Proof
Let \(\{x_{n}\}\subset M\) be a sequence satisfying (4.3). Firstly, we prove that
Let \(t > 0\), \(n \geq1\). From (4.3) we obtain
On the other hand, let \(t>0\) and \(\delta\in(0,1)\) be given. Since \(\operatorname{Ran}F \subset D^{+}\), there exists \(t_{0}>0\) such that \(F_{x_{0}x_{1}}(t_{0})>1\delta\), and since \(\varphi^{n}(t_{0})\to0\), there exists \(n_{0}\in \mathbb{N}\) such that \(\varphi^{n}(t_{0})<\epsilon\) whenever \(n\geq n_{0}\). Using the fact that F is increasing, we get
which gives that
Thus, (4.4) is proved.
Next, let \(t>0\) and \(n\in \mathbb{N}\). We shall apply induction to show that, for any \(m\geq n\),
This is obvious for \(m=n\) since \(F_{x_{n}x_{n}}(st)=1\). Next, suppose that (4.5) is true for some positive integer \(m \geq n\). Hence, by (4.3) and the monotonicity of T, we have
So, by the induction hypothesis,
which completes the induction. Now let \(\epsilon>0\) and \(\delta\in (0,1)\) be given. Since T is a tnorm of Htype, there exists \(\lambda\in(0,1)\) such that \(T^{n}(t) > 1\delta\) for all \(n\geq1\) when \(t > 1\lambda\).
On the other hand, from (4.4) we have
Then there exists \(n_{0} \in \mathbb{N}\) such that
Therefore, from (4.5) it follows
So we conclude that \(\{x_{n}\}\) is a Cauchy sequence in M. □
Now, we can state and prove the first main fixed point theorem of this paper.
Theorem 4.1
Let \((M,F,\tau_{T}, s)\) be a complete pbms under a continuous tnorm T of Htype such that \(\operatorname{Ran}F \subset D^{+}\). Let \(f : M \to M\) be a φprobabilistic contraction where \(\varphi\in\Psi\). Then f has a unique fixed point u and, for any \(x\in M\), \(\lim_{n\to\infty} f^{n}(x)= \overline{x}\).
Proof
Let \(x_{0} \in M\) be arbitrary, and we consider the sequence \(\{x_{n}\} \) defined by
By (4.1), we have
Now, by Lemma 4.1, \(\{x_{n}\}\) is a Cauchy sequence. Since M is complete, there is some \(u \in M\) such that
Now we will show that u is a fixed point of f. Let \(\epsilon>0\) and \(\delta\in(0,1)\), since \(\varphi(\epsilon) < \epsilon\), by the monotonicity of F and (4.1), we get
Since \(\{x_{n}\}\) converges to u, there exists \(n_{0}\in \mathbb{N}\) such that
So,
Then
By the inequality (v), we obtain
for all \(t>0\), \(n\geq1\).
Letting \(n\to\infty\) in (4.8) and using (4.6), (4.7), we get that
which holds unless \(F_{fuu}=H\), so u is a fixed point of f.
To prove uniqueness, suppose that there exists another fixed point v in M of f. Then, let \(t>0\), from (4.1), by using the monotonicity of F and the fact that \(\varphi(t)< t\), we get
Hence
Inductively, we obtain
Now we shall show that
Suppose, to the contrary, that there exists \(t_{0}>0\) such that \(F_{uv}(t_{0}) < 1\). Since \(F_{uv} \in D^{+}\), then \(F_{uv}(t)\to1\) as \(t \to\infty\), so there exists \(t_{1}>t_{0}\) such that
Since \(\lim_{n\to\infty}\varphi^{n}(t_{1})\to0\), there exists a positive integer \(n>1\) such that \(\varphi^{n}(t_{1})< t_{0}\). Then, by the monotonicity of \(F_{uv}\), we have
Thus
a contradiction. Therefore \(F_{uv}(t)=1\) for all \(t>0\), since \(F_{uv} \in D^{+}\). Hence \(F_{uv}=H\).
Then, in view of (ii) of Definition 3.1, we conclude that \(u = v\). This completes the proof. □
Since in the proof of Theorem 4.1 the condition \(F_{xy}(\infty)=1\) plays no role, this leads to the following.
Theorem 4.2
Let \((X; M;*; s)\) be a complete fuzzy bmetric space with the tnorm ∗ of Htype such that \(M(x, y, t)\to1\) as \(t\to\infty\) for all \(x, y\in X\). Let \(f : X \to X\) be a φfuzzy contraction where \(\varphi\in\Psi\). Then f has a unique fixed point u and, for any \(x\in M\), \(\lim_{n\to \infty} f^{n}(x)= \overline{x}\).
By taking \(s=1\) in Theorem 4.1, we obtain the following result.
Corollary 4.1
([13])
Let \((M,F,\tau_{T})\) be a complete pms under a continuous tnorm T of Htype such that \(\operatorname{Ran}F \subset D^{+}\). Let \(f : M \to M\) be a φprobabilistic contraction where \(\varphi\in\Psi\). Then f has a unique fixed point u and, for any \(x\in M\), \(\lim_{n\to\infty} f^{n}(x)= \overline{x}\).
Applications
As consequences of the above results, we can obtain the following fixed point theorems in usual bmetric spaces.
Proposition 5.1
Let \((M,d)\) be a complete bmetric space. Let f be a mapping of \((M,d)\) into itself satisfying
where the function \(\alpha:[0, \infty)\to[0, \infty)\) satisfies the following conditions:
Then f has a unique fixed point u, and \(f^{n}(x)\rightarrow u\) for all \(x\in M\).
Proof
From Lemma 3.1, \((M,F,\tau_{T_{M}}, s)\) is a complete probabilistic bmetric space, where \(F_{xy}(t)=H(td(x,y))\) for all \(x, y \in M\). Let f be a mapping such that there exists α satisfying the conditions of Proposition 5.1.
By [15, Lemma 1], there exists a strictly increasing and continuous function \(\varphi:[0, \infty)\to[0, \infty)\) such that
for all \(t>0\).
It is clear that \(0< \varphi(t)<t\) and \(\lim_{n\to\infty}\varphi ^{n}(t)=0\) for each \(t>0\). Then
for all \(x, y \in M\). Now, we prove that f is a φprobabilistic contraction in \((M,F,\tau_{T_{M}}, s)\). Indeed, let \(t>0\) and \(x, y\in M\), from (5.1) and the monotonicity of H, we have
Hence f is a φprobabilistic contraction in \((M,F,\tau_{T_{M}}, s)\). The existence and uniqueness of the fixed point follow immediately by Theorem 4.1. □
If in Proposition 5.1 we take the function \(\alpha(t)=skt\), then we have the following corollary.
Corollary 5.1
([8])
Let \((M,d)\) be a complete bmetric space. Let f be a mapping of \((M,d)\) into itself satisfying
with the restrictions \(k \in[0,1)\) and \(ks < 1\). Then f has a unique fixed point z, and \(f^{n}(x)\rightarrow z\) for all \(x\in M\).
Example 5.1
Let \(M = [0,1]\), \(n\in \mathbb{N}^{*}\{1\}\) and F be defined by \(F_{xy}(t)= H(t xy^{n})\). Then \((M,F,\tau_{T_{M}})\) is a complete probabilistic bmetric space with \(s=2^{n1}\). But in general it is not a probabilistic metric space.
Now we define the mapping \(f : [0, 1]\to[0, 1]\) by
For all \(x; y \in M\) with \(x \geq y\), we have
Similarly, for \(x \leq y\), we also conclude that
Now, suppose that \(F_{xy}(st)=H(stxy^{n})>0\), this implies that \(st> xy^{n} \). Then from Example 4.1 we have
It follows from (5.2) and (5.3) that
From the previous inequality, we get
Hence
Thus, f satisfies the \(\varphi_{s^{n1}}\)probabilistic contraction of Theorem 4.1 and 0 is the unique fixed point of f.
Conclusion
The paper deals with the achievement of introducing the notion of probabilistic bmetric space as a generalization of probabilistic metric space and bmetric space and studying some of its topological properties. Also, here we define φcontraction maps for such spaces. Moreover, we investigate some fixed points for mappings satisfying such conditions in the new framework. Our main theorems extend and unify the existing results in the recent literature. An example is constructed to support our result.
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Acknowledgements
The authors are thankful to the referees and the editor for their constructive comments and suggestions which have been useful for the improvement of the paper.
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Correspondence to Abderrahim Mbarki.
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MSC
 54E70
 54H25
 47S50
 34B15
Keywords
 probabilistic (fuzzy) bmetric spaces
 fixed points
 nonlinear contractions