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ψContraction and \((\phi ,\varphi )\)contraction in Menger probabilistic metric space
 Pengcheng Ma^{1},
 Jinyu Guan^{1},
 Yanxia Tang^{1},
 Yongchun Xu^{1} and
 Yongfu Su^{1, 2}Email authorView ORCID ID profile
 Received: 20 November 2015
 Accepted: 12 February 2016
 Published: 29 February 2016
Abstract
The purpose of this paper is to present the definition of \((\phi ,\varphi )\)contractive mapping and to discuss the relation of \(\psi\)contractive mappings and \((\phi ,\varphi )\)contractive mappings. Furthermore, the generalized \((\phi ,\varphi )\)contraction mapping principle has been proved without the uniqueness condition. Meanwhile, the generalized \(\psi\)contraction mapping principle has been obtained by using an ingenious method.
Keywords
 Probabilistic metric spaces
 \((\phi , \varphi )\)Contraction
 \(\psi\)Contraction
 Contraction mapping principle
Introduction and preliminaries
Sometimes, it is found appropriate to assign the average of several measurements as a measure to ascertain the distance between two points. Inspired from this line of thinking, Menger (1942, 1951) introduced the notion of probabilistic metric spaces as a generalization of metric spaces. In fact, he replaced the distance function d(x, y) with a distribution function \(F_{x,y}:X\times X\rightarrow R\) wherein for any number t, the value \(F_{x,y}(t)\) describes the probability that the distance between x and y is less than t. In fact the study of such spaces received an impetus with the pioneering work of Schweizer and Sklar (1983). The theory of probabilistic metric spaces is of paramount importance in random functional analysis especially due to its extensive applications in random differential as well as random integral equations (Chang et al. 1994). Sehgal and BharuchaReid (1972; Sehgal 1966) established fixed point theorems in probabilistic metric spaces (for short, PMspaces). Indeed, by using the notion of probabilistic qcontraction, they proved a unique fixed point result, which is an extension of the celebrated Banach contraction principle (Banach 1922). For the interested reader, a comprehensive study of fixed point theory in the probabilistic metric setting can be found in the book of Hadǐić and Pap (2001), see also Van An et al. (2014) for further discussion on generalizations of metric fixed point theory. Recently, Choudhury and Das (2008) gave a generalized unique fixed point theorem by using an altering distance function, which was originally introduced by Khan et al. (1984). For other results in this direction, we refer to Chauhan et al. (2013, 2014a, b, c, d), Choudhury et al. (2008), Choudhury and Das (2009), Ćirić (1975), Gajić and Rakoćević (2007), Mihet (2009), Dutta et al. (2009), Hadzi and Pap (2001), Kutbi et al. (2015). In particular, Dutta et al. (2009) defined nonlinear generalized contractive type mappings involving altering distances (say, \(\psi\)contractive mappings) in Menger PMspaces and proved their theorem for such kind of mappings in the setting of Gcomplete Menger PMspaces. On contributing to this study, In 2015, Marwan Amin Kutbi et al. weakened the notion of \(\psi\)contractive mapping and establish some fixed point theorems in Gcomplete and Mcomplete Menger PMspaces, besides discussing some related results and illustrative examples.
Next we shall recall some wellknown definitions and results in the theory of probabilistic metric spaces which are used later on in this paper. For more details, we refer the reader to Chauhan et al. (2014a, b), Kutbi et al. (2015), Xu et al. (2015a, b), Chauhan and Pant (2014), Su and Zhang (2014), Su et al. (2015).
Definition 1
 (a)
T is associative and commutative;
 (b)
T is continuous;
 (c)
\(T(a,1)=a\) for all \(a\in [0,1];\)
 (d)
\(T(a,b)\le T(c,d)\) whenever \(a \le c\) and \(b \le d\) for each \(a, b, c, d \in [0,1].\)
Definition 2
Definition 3

(MPM1) \(F_{x,y}(t)=H(t)\) if and only if \(x=y\);

(MPM2) \(F_{x,y}(t)=F_{y,x}(t)\) for all \(x,y \in E\) and \(t \in (\infty ,+\infty);\)

(MPM3) \(F_{x,y}(t+s)\ge T (F_{x,z}(t),F_{z,y}(s))\) for all \(x,y,z \in E\) and \(t>0, s>0.\)
Definition 4
 (1)
A sequence \(\{x_n\}\) in E is said to converge to \(x\in E\) if for any given \(\varepsilon > 0\) and \(\lambda >0\), there exists a positive integer \(N = N(\varepsilon , \lambda )\) such that \(F_{x_n,x}(\varepsilon )>1\lambda\) whenever \(n>N\).
 (2)
A sequence \(\{x_n\}\) in E is called a Cauchy sequence if for any \(\varepsilon >0\) and \(\lambda >0\), there exists a positive integer \(N = N(\varepsilon ,\lambda )\) such that \(F_{x_n,x_m}(\varepsilon )>1\lambda\), whenever \(n,m >N\).
 (3)
(E, F, T) is said to be Mcomplete if each Cauchy sequence in E converges to some point in E.
 (4)
A sequence \(\{x_n\}\) in E is called a GCauchy sequence if \(\lim _{n\rightarrow \infty }F_{x_n,x_{n+m}}(t)=0\) for any given positive integer m and \(t>0\).
 (5)
(E, F, T) is said to be Gcomplete if each GCauchy sequence is convergent in E.
Example
Definition 5
 (i)
\(\phi (t) =0\) if and only if \(t = 0\);
 (ii)
\(\phi (t)\) is strictly increasing and \(\phi (t)\rightarrow \infty\) as \(t\rightarrow \infty\);
 (iii)
\(\phi (t)\) is left continuous in \((0,+\infty )\);
 (iv)
\(\phi (t)\) is continuous at 0.
In the sequel, the class of all \(\phi\)functions will be denoted by \(\Phi\). We denote by \(\Psi\) the class of all continuous nondecreasing functions \(\psi : R^+ \rightarrow R^+\) such that \(\psi (0) = 0\) and \(\psi ^n(a_n)\rightarrow 0\), whenever \(a_n\rightarrow 0\), as \(n\rightarrow \infty\).
The purpose of this paper is to present the definition of \((\phi ,\varphi )\)contractive mapping and to discuss the relation of \(\psi\)contractive mappings and \((\phi ,\varphi )\)contractive mappings. Furthermore, the generalized \((\phi ,\varphi )\)contraction mapping principle has been proved without the uniqueness condition. Meanwhile, the generalized \(\psi\)contraction mapping principle has been obtained by using an ingenious method.
The equivalence of \((\phi ,\varphi )\)contractive and \((\phi ,\psi )\)contractive
We denote by \(\Omega _1\) the class of all continuous nondecreasing functions \(\varphi : (0,1]\rightarrow (0,1]\) such that \(\lim _{t\rightarrow 0}\varphi (t)=0\) and \(\varphi (1)=1\). We denote by \(\Omega _2\) the class of all continuous nondecreasing functions \(\psi : [0,+\infty )\rightarrow [0,+\infty )\) such that \(\psi (0)=0\) and \(\lim _{t\rightarrow +\infty }\varphi (t)=+\infty\). Further we give the following definition.
Definition 6
Definition 7
Theorem 8
Proof
Theorem 9
Proof
In this paper, we prove the following contraction mapping principle for the \((\phi ,\varphi )\)contractive mappings in a Gcomplete Menger probabilistic space. Meanwhile, we do not need to add the uniqueness condition of fixed point (see Kutbi et al. 2015).
Theorem 10
Let (E, F, T) be a Gcomplete Menger probabilistic space and \(f : E \rightarrow E\) be a \((\phi ,\varphi )\)contractive mapping. Assume that \(\lim _{a_n\rightarrow 1}\varphi ^n (a_n)=1\). Then f has a unique fixed point.
Proof
Kutbi et al. (2015) proved the following fixed point theorem for the \((\phi ,\psi )\)contractive mappings in a Gcomplete Menger probabilistic space. Meanwhile, they need to add the uniqueness condition of fixed point (see Xu et al. 2015). In order to clearly show the content of theorem, we use a clear form to write this theorem.
Theorem 11
(Kutbi et al. 2015) Let (E, F, T) be a Gcomplete Menger probabilistic space and \(f : E \rightarrow E\) be a \((\phi ,\psi )\)contractive mapping. Assume that \(\lim _{a_n\rightarrow 0}\psi ^n (a_n)=0\). Then f has a fixed point.
Theorem 12
(Kutbi et al. 2015) Adding condition \((*)\) to the hypotheses of Theorem 11, we obtain uniqueness of the fixed point.
By using Theorem 10, we can get the following contraction mapping principle for the \((\phi ,\psi )\)contractive mappings in a Gcomplete Menger probabilistic space.
Theorem 13
Proof
Open question 14
Theorem 15
Let (E, F, T) be a Gcomplete Menger probabilistic space and \(f : E \rightarrow E\) be a \((\phi ,\psi )\)contractive mapping. Assume that \(\lim _{a_n\rightarrow 0}\psi ^n (a_n)=0\). Then f has a unique fixed point.
Proof
This completes the proof. \(\square\)
Examples
Theorem 17
 (1)
\((X,F,T_4)\) is a Menger probabilistic metric space;
 (2)
T is a \((\phi ,\varphi )\)contractive mapping, where \(\phi (t)=\varphi (t)=t^2\);
 (3)
T is also a \((\phi ,\psi )\)contractive mapping, where \(\phi (t)=t^2, \psi (t)=t^2+2t\).
Proof
Theorem 18
 (1)
\((X,F,T_4)\) is a Menger probabilistic metric space;
 (2)
T is a \((\phi ,\varphi )\)contractive mapping, where \(\phi (t)=t^2, \varphi (t)=\frac{(1+t)t}{2}\);
 (3)
T is also a \((\phi ,\psi )\)contractive mapping, where \(\phi (t)=t^2,\quad \psi (t)=\frac{2t^2+3t}{2+t}\).
Proof
Declarations
Authors’ contributions
PM, JG, YT, YX and YS authors contributed equally and significantly in writing this article. All authors read and approved the final manuscript.
Acknowledgements
This project is supported by the major project of Hebei North University under Grant No. ZD201304.
Competing interests
The authors declare that they have no competing interests.
Open AccessThis 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
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