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Probabilistic b-metric spaces and nonlinear contractions

Fixed Point Theory and Applications20172017:29

https://doi.org/10.1186/s13663-017-0624-x

  • Received: 13 June 2017
  • Accepted: 7 December 2017
  • Published:

Abstract

This work is for giving the probabilistic aspect to the known b-metric spaces (Czerwik in Atti Semin. Mat. Fis. Univ. Modena 46(2):263-276, 1998), which leads to studying the fixed point property for nonlinear contractions in this new class of spaces.

Keywords

  • probabilistic (fuzzy) b-metric spaces
  • fixed points
  • nonlinear contractions

MSC

  • 54E70
  • 54H25
  • 47S50
  • 34B15

1 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 φ-contraction 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 b-metric spaces as a generalization of metric spaces. He proved the contraction mapping principle in b-metric 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, 68]).

Let us recall the notion of a b-metric 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 b-metric iff, for all \(x,y,z \in M \), the following conditions hold:
  1. (1)

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

     
  2. (2)

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

     
  3. (3)

    \(d(x,z)\leq s[d(x,y) + d(y,z)]\).

     
In this case, the pair \((M,d)\) is called a b-metric space.

It should be noted that the class of b-metric spaces is effectively larger than the class of metric spaces since a b-metric 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) b-metric 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) φ-contraction in probabilistic (fuzzy) b-metric spaces. Subsequently, as an application of our results, in Sections 5, we provide an example and prove a fixed point theorem in b-metric spaces. Our results generalize some well-known results in the literature.

2 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, left-continuous 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 \(D^{+}\)
$$ H(t)= \textstyle\begin{cases} 0 & \mbox{if } t\leq0, \\ 1 & \mbox{if } t > 0. \end{cases} $$

Definition 2.1

Consider f and g being in \(\Delta^{+}, h\in(0, 1]\), and let \((f, g; h)\) denote the condition
$$0 \leq g(x) \leq f(x+h) + h $$
for all x in \((0, \frac{1}{h})\).
The modified Lévy distance is the function \(d_{L}\) defined on \(\Delta ^{+} \times\Delta^{+}\) by
$$d_{L}(f, g) = \inf\bigl\{ h : \mbox{both conditions } (f, g; h) \mbox{ and } (g, f;h) \mbox{ hold}\bigr\} . $$

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 \(t>0\),
$$F(t)>1-t\quad \textit{iff}\quad d_{L}(F, H)< t. $$

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:
$$\begin{aligned}& \tau(F,G) = \tau(G, F), \\& \tau(F,G) \leq \tau(K, R) \quad \mbox{whenever } F\leq K, G\leq R, \\& \tau(F,H) = F, \\& \tau\bigl(\tau(F,G), R\bigr) = \tau\bigl(F,\tau(G,R)\bigr). \end{aligned}$$
A commutative, associative and nondecreasing mapping \(T: [0,1] \times [0,1]\to[0,1]\) is called a t-norm if and only if
$$\begin{aligned} (\mathrm{i})&\quad T(a, 1) = a \quad \mbox{for all } a\in[0,1], \\ (\mathrm{ii})&\quad T(0, 0) = 0. \end{aligned}$$
As examples we mention the tree typical examples of continuous t-norms as follows: \(T_{p}(a,b) = ab\), \(T_{M}(a,b) = \operatorname{Min}(a,b)\) and \(T_{L}(a,b) = \operatorname{Max}\{ a+b-1,0\}\).
Moreover, if T is left-continuous, then the operation \(\tau_{T} : \Delta^{+}\times\Delta^{+}\to \Delta^{+}\) defined by
$$\tau_{T}(F, G) (x)= \sup\bigl\{ T\bigl(F(u), G(v)\bigr): u+v = x\bigr\} $$
is a triangle function.
We say (O Hadzić [10]) that a t-norm T is of H-type if the family \(\{T^{n}(t)\}\) is equicontinuous at \(t=1\), that is,
$$ \forall\epsilon\in(0,1)\ \exists\lambda\in(0,1) \mbox{:}\quad t>1-\lambda \Rightarrow T^{n}(t)>1-\epsilon \quad \mbox{for all } n\geq1, $$
where \(T^{1}(x) =T(x,x)\), \(T^{n}(x) = T(x,T^{n-1}(x))\) for every \(n\geq2\).

The t-norm \(T_{M}\) is a trivial example of t-norm of H-type (see [10]).

Finally, we also have the following.

Lemma 2.5

([9])

If T is continuous, then \(\tau_{T}\) is continuous.

3 Probabilistic b-metric space

Having introduced the necessary terms, we now turn to our main topic. Developing a theory of probabilistic b-metric spaces, we start with the following definition.

Definition 3.1

A probabilistic b-metric 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 \(y> 0\),
  1. (i)

    \(F_{pp}=H\),

     
  2. (ii)

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

     
  3. (iii)

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

     
  4. (iv)

    \(F_{pq}(sy)\geq\tau(F_{pr}, F_{rq})(y)\).

     
If \(\tau=\tau_{T} \) for some t-norm T, then \((M, F, \tau_{T}, s)\) is called a b-Menger space.
It should be noted that if T is a continuous t-norm, then \((M, F)\) satisfies (iv) under \(\tau_{T}\) if and only if it satisfies
$$(\mathrm{v})\quad F_{pq}\bigl(s(y+x)\bigr) \geq T\bigl(F_{pr}(y), F_{rq}(x)\bigr) $$
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:
$$(\mathrm{vi})\quad F_{pq}\geq\tau(F_{pr}, F_{rq}) \quad (\mbox{triangle inequality}) $$
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 b-metric space is defined in the following manner.

Definition 3.2

The quadruple \((X; M;*, s)\) is said to be a fuzzy b-metric space if X is an arbitrary set, is a continuous t-norm, \(s\geq1\) 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 left-continuous 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 b-metric space with the constant \(s=1\).

It is clear that every probabilistic (fuzzy) metric space (PM space) is a probabilistic (fuzzy) b-metric 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:
$$ F_{pq}(t)= \textstyle\begin{cases} H(t) & \mbox{if } p = q , \\ H(t-3) & \mbox{if } p=2 \mbox{ and } q=3 \mbox{ or } p=3 \mbox{ and } q=2 , \\ H(t-1) & \mbox{otherwise}. \end{cases} $$
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:
$$F_{32}\biggl(\frac{5}{2}\biggr)=0< 1=H\biggl(\frac{1}{4} \biggr)=\operatorname{Min}\biggl(F_{3 1}\biggl(\frac {5}{4} \biggr),F_{12}\biggl(\frac{5}{4}\biggr)\biggr). $$

Example 3.2

Let \(M=[0,\infty)\). Define \(F:M \times M \to\Delta^{+}\) as follows:
$$ F_{pq}(t)= H\bigl(t- \vert p-q \vert ^{2}\bigr). $$
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:
$$F_{32}\biggl(\frac{2}{3}\biggr)=0< 1=H\biggl(\frac{1}{12} \biggr)=\operatorname{Min}\biggl(F_{3 \frac{5}{2}}\biggl(\frac {1}{3} \biggr),F_{\frac{5}{2}2}\biggl(\frac{1}{3}\biggr)\biggr). $$

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 t-neighborhood of p is the set
$$N_{p}(t) = \bigl\{ q\in M: F_{pq}(t) > 1 - t \bigr\} . $$
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
$$N_{p}(t)= \bigl\{ q\in M: d_{L}(F_{pq}, H)< t\bigr\} . $$

Definition 3.4

Let \(\{x_{n}\}\) be a sequence in a probabilistic semimetric space \((M,F)\).
  1. (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. (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. (3)

    \((M,F)\) is said to be complete if every Cauchy sequence has a limit.

     

Every b-metric space is a probabilistic b-metric space. Moreover, we have the following.

Lemma 3.1

Let \((M,d)\) be a b-metric space (bms) with the constant s. Define \(F: M \times M \to\Delta^{+}\) by
$$ F_{pq}(t)=H\bigl(t-{d(p,q)}\bigr). $$
Then
  1. (a)

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

     
  2. (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
$$ \operatorname{Min}\bigl(F_{pr}(t_{1}),F_{rq}(t_{2}) \bigr) =0, $$
then
$$ F_{pq}\bigl(s(t_{1}+t_{2})\bigr) \geq \operatorname{Min}\bigl(F_{pr}(t_{1}),F_{rq}(t_{2}) \bigr). $$
Else if
$$ \operatorname{Min}\bigl(F_{pr}(t_{1}),F_{rq}(t_{2}) \bigr)=1, $$
then \(t_{1} > d(p,r)\) and \(t_{2} > d(r,q)\). Since \((M,d)\) is a b-metric space with the constant s, we have
$$\begin{aligned} d(p,q) \leq& s\bigl(d(p,r)+d(r,q)\bigr) \\ < & s(t_{1}+t_{2}). \end{aligned}$$
Then we get
$$ F_{pq}\bigl(s(t_{1}+t_{2})\bigr)=1. $$
Thus
$$ F_{pq}\bigl(s(t_{1}+t_{2})\bigr) \geq \operatorname{Min}\bigl(F_{pr}(t_{1}),F_{rq}(t_{2}) \bigr). $$

Hence condition (v) holds. So \((M,F,\tau_{T_{M}}, s)\) is a probabilistic b-metric space.

(b) By Definition 3.2 we get, for every \(t>0\),
$$ N_{p}(t)=\bigl\{ q\in M : d(p,q)< t \bigr\} . $$
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 b-metric 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 b-metric with \(s = 2^{p-1}\).

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
$$\begin{aligned} d'(x, y)= \bigl(d(x,y)\bigr)^{p} \leq& \bigl(d(x,z)+d(z, y)\bigr)^{p} \\ \leq&2^{p-1}\bigl(d(x,z)^{p} +d(z, y)^{p}\bigr) \\ =&2^{p-1}\bigl(d'(x,z)+ d'(z,y)\bigr) \end{aligned}$$
for each \(x, y \in M\). So, condition (3) of Definition 1.1 is also satisfied and \((M,F,\tau_{T_{M}}, 2^{p-1})\) 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 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 b-metric 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:
$$ F^{a}_{pq}(t)= \textstyle\begin{cases} H(t-2a)& \mbox{if }p\mbox{ and }q\mbox{ are in } \{a,4a\}\ p \neq q, \\ H(t-|p-q|) & \mbox{otherwise}. \end{cases} $$
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:
$$F_{4a\frac{a}{2} }\biggl(\frac{7a}{2}\biggr)=0< 1=\operatorname{Min} \biggl(F_{4a a}\biggl(\frac{5a}{2}\biggr),F_{a \frac{a}{2}}(a) \biggr), $$
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 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 b-metric space \((M, F, \tau)\) the probabilistic b-metric 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:
$$ F^{a}_{pq}(t)= \textstyle\begin{cases} H(t)& \mbox{if } p = q, \\ H(t-7)& \mbox{if } p \mbox{ and } q \mbox{ are odd and } p\neq q, \\ H(t- |\frac{a}{p} - \frac{a}{q}|) &\mbox{if } p \mbox{ and } q \mbox{ are even or } pq=\infty, \\ H(t-3) &\mbox{otherwise}. \end{cases} $$
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 \(F_{2n 1}(t)=H(t-3)\neq H(t-1)=F_{\infty1}(t)\). 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
$$ d_{L}\bigl(\tau\bigl(G, G'\bigr), H\bigr)< \varrho, $$
(3.1)
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
$$ d_{L}\bigl(F'_{pq}, H\bigr)\leq d_{L}\bigl(\tau(F_{pr}, F_{rq}), H\bigr) < \varrho= d_{L}\bigl(F'_{pq}, H\bigr), $$
an impossibility. Hence \(N_{p}(t)\cap N_{q}(t)\) is empty and the proof is complete. □

4 φ-Probabilistic contraction in a probabilistic b-metric space

Over this section, the letter Ψ denotes the set of all functions \(\varphi: [0,\infty) \to[0,\infty)\) such that
$$ 0< \varphi(t)< t \quad \mbox{and}\quad \lim_{n\to\infty} \varphi^{n}(t)=0 \quad \mbox{for each } t>0. $$

Example 4.1

Let \(c \in[1, \infty)\), and let \(\varphi _{c}:[0,\infty)\to[0,\infty) \) be defined by
$$\varphi_{c}(t)= \frac{t}{c+t}. $$
Then
  1. (i)

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

     
  2. (ii)

    \(\varphi_{c}\in\Psi\).

     
(i) trivially holds. For (ii) it is very easy to check by induction that
$$\varphi^{n}_{c}(t)=\frac{t}{c^{n}+t\sum_{i=0}^{n-1}c^{i}}\quad \mbox{for all } t>0, n\in \mathbb{N}^{*}. $$
Consequently,
$$\varphi^{n}_{c}(t) \leq\frac{t}{1 +t\sum_{i=0}^{n-1}c^{i}}, $$
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 b-metric space \((M, F, \tau, s)\). We say that f is a φ-probabilistic contraction if
$$ F_{fpfq}\bigl(\varphi(t)\bigr)\geq F_{pq}(st) $$
(4.1)
for all \(p,q\in M\) and \(t>0\).

It should be noted, when \(s=1\), the above definition coincides with the concept of φ-probabilistic 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 b-metric space \((X; M;*, s)\). We say that f is a φ-fuzzy contraction if
$$ M\bigl(fp, fq, \varphi(t)\bigr)\geq M(p,q, st) $$
(4.2)
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 t-norm T of H-type and \(\operatorname{Ran}F \subset D^{+}\). Let \(\{x_{n}\}\) be a sequence in M. If there exists a function \(\varphi\in\Psi\) such that
$$ F_{x_{m+1}x_{n+1}}\bigl(\varphi(t)\bigr) \geq F_{x_{m}x_{n}}(st)\quad (n, m \geq0, t>0), $$
(4.3)
then \(\{x_{n}\}\) is a Cauchy sequence.

Proof

Let \(\{x_{n}\}\subset M\) be a sequence satisfying (4.3). Firstly, we prove that
$$ F_{x_{n}x_{n+1}}(t)\to1 \quad \mbox{as } n\to\infty\mbox{ for all } t> 0. $$
(4.4)
Let \(t > 0\), \(n \geq1\). From (4.3) we obtain
$$\begin{aligned} F_{x_{n}x_{n+1}}\bigl(\varphi^{n}(t)\bigr) \geq& F_{x_{n-1}x_{n}} \bigl(s\varphi ^{n-1}(t)\bigr) \\ \geq& F_{x_{n-1}x_{n}}\bigl(\varphi^{n-1}(t)\bigr) \\ \geq& F_{x_{n-2}x_{n-1}}\bigl(s\varphi^{n-2}(t)\bigr) \\ \geq& F_{x_{n-2}x_{n-1}}\bigl(\varphi^{n-2}(t)\bigr) \\ \vdots& \\ \geq& F_{x_{0}x_{1}}(st) \\ \geq& F_{x_{0}x_{1}}(t). \end{aligned}$$
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
$$ F_{x_{n}x_{n+1}}(t) \geq F_{x_{n}x_{n+1}}\bigl(\varphi^{n}(t_{0}) \bigr)\quad \mbox{whenever } n\geq n_{0}, $$
which gives that
$$ F_{x_{n}x_{n+1}}(t) \geq F_{x_{0}x_{1}}(t_{0}) > 1-\delta \quad \mbox{whenever } n\geq n_{0}. $$
Thus, (4.4) is proved.
Next, let \(t>0\) and \(n\in \mathbb{N}\). We shall apply induction to show that, for any \(m\geq n\),
$$ F_{x_{n}x_{m}}(st)\geq T^{m-n}\bigl(F_{x_{n}x_{n+1}}\bigl(t- \varphi(t)\bigr)\bigr). $$
(4.5)
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
$$\begin{aligned} F_{x_{n}x_{m+1}}(st) =& F_{x_{n}x_{m+1}}\bigl(s\bigl(t-\varphi(t)\bigr)+s \varphi (t)\bigr) \\ \geq& T\bigl( F_{x_{n}x_{n+1}}\bigl(t-\varphi(t)\bigr),F_{x_{n+1}x_{m+1}}\bigl( \varphi (t)\bigr)\bigr) \\ \geq& T\bigl( F_{x_{n}x_{n+1}}\bigl(t-\varphi(t)\bigr),F_{x_{n}x_{m}}(st) \bigr). \end{aligned}$$
So, by the induction hypothesis,
$$\begin{aligned} F_{x_{n}x_{m+1}}(st) \geq& T\bigl( F_{x_{n}x_{n+1}}\bigl(t-\varphi (t) \bigr),F_{x_{n}x_{m}}(st)\bigr) \\ \geq& T\bigl( F_{x_{n}x_{n+1}}\bigl(t-\varphi (t)\bigr),T^{m-n} \bigl(F_{x_{n}x_{n+1}}\bigl(t-\varphi(t)\bigr)\bigr)\bigr) \\ \geq& T^{m+1-n}\bigl(F_{x_{n}x_{n+1}}\bigl(t-\varphi(t)\bigr)\bigr), \end{aligned}$$
which completes the induction. Now let \(\epsilon>0\) and \(\delta\in (0,1)\) be given. Since T is a t-norm of H-type, 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
$$ F_{x_{n}x_{n+1}}\biggl(\frac{\epsilon}{s}-\varphi\biggl(\frac{\epsilon}{s} \biggr)\biggr)\to 1\quad \mbox{as } n\to\infty. $$
Then there exists \(n_{0} \in \mathbb{N}\) such that
$$ F_{x_{n}x_{n+1}}\biggl(\frac{\epsilon}{s}-\varphi\biggl(\frac{\epsilon}{s} \biggr)\biggr)> 1-\lambda \quad \mbox{for all } n \geq n_{0}. $$
Therefore, from (4.5) it follows
$$ F_{x_{n}x_{m}}(\epsilon) > 1-\delta \quad \mbox{whenever } m > n \geq n_{0}. $$
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 t-norm T of H-type such that \(\operatorname{Ran}F \subset D^{+}\). Let \(f : M \to M\) be a φ-probabilistic 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
$$ x_{n}=f(x_{n-1})=f^{n}(x_{0}) \quad \mbox{for each } n\geq1. $$
By (4.1), we have
$$ F_{x_{m+1}x_{n+1}}\bigl(\varphi(t)\bigr) \geq F_{x_{m}x_{n}}(st). $$
Now, by Lemma 4.1, \(\{x_{n}\}\) is a Cauchy sequence. Since M is complete, there is some \(u \in M\) such that
$$ x_{n}\to u\quad \mbox{as } n\to\infty. $$
(4.6)
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
$$\begin{aligned} \begin{aligned} F_{x_{n+1}fu}(\epsilon) & \geq F_{x_{n+1}fu}\bigl(\varphi(\epsilon)\bigr) \\ & = F_{fx_{n}fu}\bigl(\varphi(\epsilon)\bigr) \\ & \geq F_{x_{n}u}(s\epsilon). \end{aligned} \end{aligned}$$
Since \(\{x_{n}\}\) converges to u, there exists \(n_{0}\in \mathbb{N}\) such that
$$ F_{x_{n}u}(s\epsilon) > 1-\delta \quad \mbox{for each } n \geq n_{0}. $$
So,
$$ F_{x_{n+1}fu}(\epsilon) > 1-\delta \quad \mbox{for each } n \geq n_{0}. $$
Then
$$ \lim_{n\to\infty}x_{n+1}=fu. $$
(4.7)
By the inequality (v), we obtain
$$ F_{fuu}(t)\geq T\biggl(F_{fux_{n+1}}\biggl(\frac{t}{2s} \biggr), F_{x_{n+1}u}\biggl(\frac {t}{2s}\biggr)\biggr) $$
(4.8)
for all \(t>0\), \(n\geq1\).
Letting \(n\to\infty\) in (4.8) and using (4.6), (4.7), we get that
$$F_{fuu}(t)\geq1\quad \mbox{for all } t>0, $$
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
$$\begin{aligned} F_{uv}\bigl(\varphi(t)\bigr) = & F_{fufv}\bigl(\varphi(t) \bigr) \\ \geq& F_{uv}(st) \\ \geq& F_{uv}(t) \\ \geq& F_{uv}\bigl(\varphi(t)\bigr). \end{aligned}$$
Hence
$$ F_{uv}\bigl(\varphi(t)\bigr)= F_{uv}(t). $$
Inductively, we obtain
$$ F_{uv}\bigl(\varphi^{n}(t)\bigr)= F_{uv}(t). $$
Now we shall show that
$$ F_{uv}(t)=1 \quad \mbox{for all } t>0. $$
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
$$ F_{uv}(t_{1})>F_{uv}(t_{0}). $$
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
$$ F_{uv}\bigl(\varphi^{n}(t_{1})\bigr)\leq F_{uv}(t_{0}). $$
Thus
$$ F_{uv}(t_{1})=F_{uv}\bigl(\varphi^{n}(t_{1}) \bigr)\leq F_{uv}(t_{0}), $$
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 b-metric space with the t-norm of H-type such that \(M(x, y, t)\to1\) as \(t\to\infty\) for all \(x, y\in X\). Let \(f : X \to X\) be a φ-fuzzy 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 t-norm T of H-type such that \(\operatorname{Ran}F \subset D^{+}\). Let \(f : M \to M\) be a φ-probabilistic contraction where \(\varphi\in\Psi\). Then f has a unique fixed point u and, for any \(x\in M\), \(\lim_{n\to\infty} f^{n}(x)= \overline{x}\).

5 Applications

As consequences of the above results, we can obtain the following fixed point theorems in usual b-metric spaces.

Proposition 5.1

Let \((M,d)\) be a complete b-metric space. Let f be a mapping of \((M,d)\) into itself satisfying
$$ d(fx,fy) \leq\alpha\biggl(\frac{d(x,y)}{s}\biggr),\quad x, y \in M, $$
where the function \(\alpha:[0, \infty)\to[0, \infty)\) satisfies the following conditions:
$$\alpha(0)=0,\qquad \alpha(t)< t,\quad \textit{and}\quad \limsup _{r\to t}\alpha(r)< t\quad \textit{for each } t>0. $$
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 b-metric space, where \(F_{xy}(t)=H(t-d(x,y))\) 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
$$\alpha(t)< \varphi(t)< t $$
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
$$ d(fx,fy) \leq\alpha\biggl(\frac{d(x,y)}{s}\biggr)\leq\varphi \biggl(\frac{d(x,y)}{s}\biggr) $$
(5.1)
for all \(x, y \in M\). Now, we prove that f is a φ-probabilistic 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
$$\begin{aligned} F_{fxfy}\bigl(\varphi(t)\bigr) =& H\bigl(\varphi(t)- d(fx,fy)\bigr) \\ \geq& H\bigl(t- \varphi^{-1}\bigl(d(fx,fy)\bigr)\bigr) \\ \geq& H\biggl(t- \frac{d(x,y)}{s}\biggr) \\ \geq& H\bigl(st- d(x,y)\bigr) \\ \geq& F_{xy}(st). \end{aligned}$$

Hence f is a φ-probabilistic 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 b-metric space. Let f be a mapping of \((M,d)\) into itself satisfying
$$ d(fx,fy) \leq kd(x,y),\quad x , y \in M , $$
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 \(F_{xy}(t)= H(t- |x-y|^{n})\). Then \((M,F,\tau_{T_{M}})\) is a complete probabilistic b-metric space with \(s=2^{n-1}\). But in general it is not a probabilistic metric space.

Now we define the mapping \(f : [0, 1]\to[0, 1]\) by
$$f(x)= \frac{x}{s+x}. $$
For all \(x; y \in M\) with \(x \geq y\), we have
$$\begin{aligned} \begin{aligned} \vert fx - fy \vert ^{n} & = \biggl\vert \frac{x}{s+x} - \frac{y}{s+y} \biggr\vert ^{n} \\ & \leq \frac{s^{n}(x-y)^{n}}{[(s+x)(s+y)]^{n}} \\ & \leq \frac{s^{n}(x-y)^{n}}{[s^{2} + s(x+y)+xy]^{n}} \\ & \leq \frac{s^{n}(x-y)^{n}}{[s^{2} + s(x-y)]^{n}} \\ & \leq \frac{s^{n}(x-y)^{n}}{s^{2n} + s^{n}(x-y)^{n}}. \end{aligned} \end{aligned}$$
Similarly, for \(x \leq y\), we also conclude that
$$ \vert fx - fy \vert ^{n} \leq \frac{s^{n} \vert x-y \vert ^{n}}{s^{2n} + s^{n} \vert x-y \vert ^{n}} = \varphi _{s^{n-1}}\biggl(\frac{ \vert x-y \vert ^{n}}{s}\biggr). $$
(5.2)
Now, suppose that \(F_{xy}(st)=H(st-|x-y|^{n})>0\), this implies that \(st> |x-y|^{n} \). Then from Example 4.1 we have
$$ \varphi_{s^{n-1}}(t)> \varphi_{s^{n-1}}\biggl(\frac{ \vert x-y \vert ^{n}}{s} \biggr). $$
(5.3)
It follows from (5.2) and (5.3) that
$$\varphi_{s^{n-1}}(t) > |fx - fy|^{n}. $$
From the previous inequality, we get
$$F_{xy}\bigl(\varphi_{s^{n-1}}(t)\bigr)=H\bigl( \varphi_{s^{n-1}}(t)- \vert fx-fy \vert ^{n}\bigr)>0. $$
Hence
$$F_{fxfy}\bigl(\varphi_{s^{n-1}}(t)\bigr)\geq F_{xy}(st). $$

Thus, f satisfies the \(\varphi_{s^{n-1}}\)-probabilistic contraction of Theorem 4.1 and 0 is the unique fixed point of f.

6 Conclusion

The paper deals with the achievement of introducing the notion of probabilistic b-metric space as a generalization of probabilistic metric space and b-metric space and studying some of its topological properties. Also, here we define φ-contraction 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.

Declarations

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.

Funding

No sources of funding may be declared.

Authors’ contributions

Each author equally contributed to this paper, read and approved the final manuscript.

Competing interests

The authors declare that they do not have any competing interests.

Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.

Authors’ Affiliations

(1)
ANO Laboratory, National School of Applied Sciences, Oujda University, P.O. Box 669, Oujda, Morocco
(2)
ANO Laboratory, Faculty of Sciences, Oujda University, Oujda, Morocco

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