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Probabilistic bmetric spaces and nonlinear contractions
 Abderrahim Mbarki^{1}Email authorView ORCID ID profile and
 Rachid Oubrahim^{2}
https://doi.org/10.1186/s136630170624x
© The Author(s) 2017
 Received: 13 June 2017
 Accepted: 7 December 2017
 Published: 22 December 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.
Keywords
 probabilistic (fuzzy) bmetric 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 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])
 (1)
\(d(x,y) =0\) iff \(x = y\),
 (2)
\(d(x,y) = d(y,x)\),
 (3)
\(d(x,z)\leq s[d(x,y) + d(y,z)]\).
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.
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, 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 \(D^{+}\).
Definition 2.1
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 2.3
([9])
Lemma 2.4
([9])
If F and G are in \(\Delta^{+}\) and \(F\leq G\), then \(d_{L}(G, H)\leq d_{L}(F, H)\).
The tnorm \(T_{M}\) is a trivial example of tnorm of Htype (see [10]).
Finally, we also have the following.
3 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
 (i)
\(F_{pp}=H\),
 (ii)
\(F_{pq}=H \Rightarrow p=q\),
 (iii)
\(F_{pq} = F_{qp}\),
 (iv)
\(F_{pq}(sy)\geq\tau(F_{pr}, F_{rq})(y)\).
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

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

\(M(x, y, q) = 1\) 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
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
Example 3.2
Definition 3.3
Definition 3.4
 (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
 (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 \([0,\infty)\).
Hence condition (v) holds. So \((M,F,\tau_{T_{M}}, s)\) is a probabilistic bmetric space.
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 \(s = 2^{p1}\).
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
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
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
4 φProbabilistic contraction in a probabilistic bmetric space
Example 4.1
 (i)
\(\varphi_{c}\) is a strictly increasing and continuous function,
 (ii)
\(\varphi_{c}\in\Psi\).
Before stating the main fixed point theorems, we introduce the following concept.
Definition 4.1
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
In the proof of our first theorem, we use the following lemma.
Lemma 4.1
Proof
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
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 \(x\in M\), \(\lim_{n\to\infty} f^{n}(x)= \overline{x}\).
5 Applications
As consequences of the above results, we can obtain the following fixed point theorems in usual bmetric spaces.
Proposition 5.1
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.
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])
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.
Thus, f satisfies the \(\varphi_{s^{n1}}\)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 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.
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
References
 Czerwik, S: Nonlinear setvalued contraction mappings in bmetric spaces. Atti Semin. Mat. Fis. Univ. Modena 46(2), 263276 (1998) MathSciNetMATHGoogle Scholar
 Deimling, K: Nonlinear Functional Analysis. Springer, Berlin (1985) View ArticleMATHGoogle Scholar
 Banach, S: Sur les operations dans les ensembles abstrait et leur application aux equations, integrals. Fundam. Math. 3, 133181 (1922) View ArticleMATHGoogle Scholar
 Browder, F: On the convergence of successive approximations for nonlinear functional equations. Indag. Math. 30, 2735 (1968) MathSciNetView ArticleMATHGoogle Scholar
 Bakhtin, IA: The contraction principle in quasimetric spaces. In: Functional Analysis, vol. 30, pp. 2637. Ul’yanovsk. Gos. Ped. Inst., Ul’yanovsk (1989) (in Russian) Google Scholar
 Aydi, H, Bota, MF, Karapınar, E, Mitrović, S: A fixed point theorem for set valued quasicontractions in bmetric spaces. Fixed Point Theory Appl. 2012, 88 (2012) View ArticleMATHGoogle Scholar
 Hussain, N, Dorić, D, Kadelburg, Z, Radenović, S: Suzukitype fixed point results in metric type spaces. Fixed Point Theory Appl. 2012, 126 (2012) MathSciNetView ArticleMATHGoogle Scholar
 Kir, M, Kiziltunc, H: On some well known fixed point theorems in bmetric spaces. Turk. J. Anal. Number Theory 1(1), 1316 (2013) View ArticleGoogle Scholar
 Schweizer, B, Sklar, A: Probabilistic Metric Spaces. NorthHolland Series in Probability and Applied Mathimatics, vol. 5 (1983) MATHGoogle Scholar
 Hadzić, O: A fixed point theorem in Menger spaces. Publ. Inst. Math. (Belgr.) 20, 107112 (1979) MathSciNetMATHGoogle Scholar
 Gregori, V, Sapena, A: On fixedpoint theorems in fuzzy metric spaces. Fuzzy Sets Syst. 125, 245252 (2002) View ArticleMATHGoogle Scholar
 Kramosil, O, Michalek, J: Fuzzy metric and statistical metric spaces. Kybernetika 11, 326334 (1975) MathSciNetMATHGoogle Scholar
 Jachymski, J: On probabilistic φcontractions on Menger spaces. Nonlinear Anal. 73, 21992203 (2010) MathSciNetView ArticleMATHGoogle Scholar
 Mbarki, A, Naciri, R: Probabilistic generalized metric spaces and nonlinear contractions. Demonstr. Math. 49(4), 437452 (2016) MathSciNetMATHGoogle Scholar
 Jachymski, JR: Equivalence of some contractivity properties over metrical structures. Proc. Am. Math. Soc. 125(8), 23272335 (1997) MathSciNetView ArticleMATHGoogle Scholar