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ψContraction and \((\phi ,\varphi )\)contraction in Menger probabilistic metric space
SpringerPlus volume 5, Article number: 233 (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.
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 triangular norm (shorter Tnorm) is a binary operation T on [0, 1] which satisfies the following conditions:

(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 \(a, b, c, d \in [0,1].\)
The following are the four basic Tnorms:
It is easy to check, the above four Tnorms have the following relations:
for any \(a,b \in [0,1]\).
Definition 2
A function \(F(t): (\infty ,+\infty )\rightarrow [0, 1]\) is called a distance distribution function if it is nondecreasing and leftcontinuous with \(\lim _{t\rightarrow \infty }F(t)=0, \quad \lim _{t\rightarrow +\infty }F(t)=1\). and \(F(0)=0\). The set of all distance distribution functions is denoted by \(D^+\). A special distance distribution function is given by
Definition 3
A Menger probabilistic metric space is a triple (E, F, T) where E is a nonempty set, T is a continuous tnorm and F is a mapping from \(E\times E\) into \(D^+\) such that, if \(F_{x,y}\) denotes the value of F at the pair (x, y), the following conditions hold:

(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
(Kutbi et al. 2015) Let (E, F, T) be a Menger probabilistic metric space.

(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
Let \(x_n=\sum _{i=1}^{n}\frac{1}{i}, \ n=1, 2, 3, \ldots\). It is easy to show, for any given m, that
as \(n\rightarrow \infty\). Hence \(\{x_n\}\) is a GCauchy sequence. But it is not a Cauchy sequence, since \(x_n\) does not converge.
Definition 5
(Kutbi et al. 2015) A function \(\phi :R^+ \rightarrow R^+\) is said to be a \(\phi\)function if it satisfies the following conditions:

(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 \(n\rightarrow \infty\).
Kutbi et al. (2015) proved the two generalized contraction mapping principles for the following socalled \(\psi\)contractive mapping T from a Menger probabilistic metric space (E, F, T) into itself:
where \(c \in (0,1)\) and \(\psi (t), \phi (t)\) are two functions with the suitable conditions. In socalled Mcomplete Menger probabilistic spaces, they have proved a generalized \(\psi\)contraction mapping principle provided that F is triangular:
for every \(x, y, z \in E\) and each \(t > 0\).
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
Let (E, F, T) be a Menger probabilistic space and \(f : E \rightarrow E\) be a mapping satisfying the following inequality
where \(c \in (0,1), \quad \phi \in \Phi , \; \varphi \in \Omega _1\). The mapping f satisfying condition (1) is called \((\phi ,\varphi )\)contractive mapping.
Definition 7
Let (E, F, T) be a Menger probabilistic space and \(f:E \rightarrow E\) be a mapping satisfying the following inequality
where \(c \in (0,1), \phi \in \Phi , \quad \psi \in \Omega _2\). The mapping f satisfying condition (2) is called \((\phi ,\psi )\)contractive mapping.
Theorem 8
Let T be a \((\phi ,\psi )\)contractive mapping, then T is also a \((\phi ,\varphi )\)contractive mapping, where
Proof
We rewrite the (2) to the following form
which can be rewritten to
That is
This completes the proof. \(\square\)
Theorem 9
Let T be a \((\phi ,\varphi )\)contractive mapping, then T is also a \((\phi ,\psi )\)contractive mapping, where
Proof
From the (3), we have
We rewrite the (1) to the following form
which can be rewritten to
That is,
This completes the proof. \(\square\)
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
For any \(x_0 \in E\), we define a sequence \(\{x_n\}\) by \(x_{n+1}=Tx_n\) for all \(n\ge 0\). From (1) and the properties of \(\phi\) and \(\varphi\) we know, for all \(t>0\), that
as \(n\rightarrow \infty\). Let \(\varepsilon >0\) be given, then by using the properties (i) and (iv) of a function \(\phi\) we can find \(t>0\) such that \(\varepsilon > \phi (t)\). It follows from (4) that
By using the triangle inequality (MPM3), we obtain
Thus, letting \(n\rightarrow \infty\) and making use of (5), for any integer p, we get
Hence \(\{x_n\}\) is a GCauchy sequence. Since (E, F, T) is Gcomplete, there exists a point \(u\in E\) such that \(x_n\rightarrow u\) as \(n\rightarrow \infty\). For any \(\varepsilon >0\), choose \(\phi (t) < \frac{\varepsilon }{2}\), we have
as \(n\rightarrow \infty\), which in turn yields that \(fu=u\). Next we show the uniqueness of the fixed point. If there exists v such that \(fv=v\), by using (3) we hvae
as \(n\rightarrow \infty\). It is easy to see \(u=v\). The proof is completed. \(\square\)
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.
In order to get the uniqueness of fixed point, authors added the following condition:
where F(f) denotes the set of all fixed points of a mapping f.
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
Let (E, F, T) be a Gcomplete Menger probabilistic space and \(f : E \rightarrow E\) be a \((\phi ,\psi )\)contractive mapping. Assume that \(\lim _{a_n\rightarrow 1}\varphi ^n (a_n)=1\). Then f has a unique fixed point, where
Proof
From Theorem 8, we know that, T is also a \((\phi ,\varphi )\)contractive mapping, where
Since \(\lim _{a_n\rightarrow 1}\varphi ^n (a_n)=1\), by using Theorem 8, we obtain the conclusion. This completes the proof. \(\square\)
Open question 14
Is the following property right?
where
If the property (6) is right, then we can obtain the following result.
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.
Conclusion 16
The property (6) is right. Therefore Theorem 15 holds.
Proof
It is not hard to show that, the property (6) is equivalent to the following proposition
where \(a_n=\frac{1}{b_n}1\) and
Next, we prove (7). Let
then we have
Now we prove
Observe
Because \(\lim _{n\rightarrow \infty }\varphi ^n (b_n)= 1\) and \(\lim _{t\rightarrow 1}A(t)=0\), we have \(\lim _{a_n\rightarrow 0}\psi ^n (a_n)=0\).
Now we prove
Observe
Because \(\lim _{n\rightarrow \infty }\psi ^n (a_n)= 0\) and \(\lim _{t\rightarrow 0}B(t)=1\), we have \(\lim _{b_n\rightarrow 1}\varphi ^n (b_n)=1\).
This completes the proof. \(\square\)
Examples
Theorem 17
Let (X, d) be a metric space, \(f: X\rightarrow X\) be a mapping satisfying the following condition:
where \(c \in (0,1)\) is a constant. Let
Then

(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
(1) We prove \((X,F,T_4)\) is a Menger probabilistic metric space. The conditions (MPM1) and (MPM2) obviously hold. We prove the condition (MPM3). For any \(x,y,z \in X\) and \(t>0, s>0\), we claim that
If not, we have
which is equivalent to
Adding the above two inequalities, we get
which implies
This is a contradiction which implies the condition (MPM3) holds.
(2) From (8) we have
and hence
We rewrite inequality (9) to the following form
That is,
where \(\phi (t)=t^2, \varphi (t)=t^2.\)
(3) By using Theorem 9, we know that, T is also a \((\phi ,\psi )\)contractive mapping with
That is \(\psi (t)=\frac{1}{ (\frac{1}{t+1})^2}1=t^2+t\). This completes the proof. \(\square\)
Theorem 18
Let (X, d) be a metric space, \(f: X\rightarrow X\) be a nonexpansive mapping. Let
Then

(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
(1) It is a conclusion of Theorem 17. (2) Since T is nonexpansive, let \(c \in (0,1)\) be a constant such that \(3c^2\ge 2\), we have
and hence
We rewrite inequality (10) to the following form
That is,
where \(\phi (t)=t^2, \quad \varphi (t)=\frac{(1+t)t}{2}.\) (3) By using Theorem 9, we know that, T is also a \((\phi ,\psi )\)contractive mapping with
That is,
This completes the proof. \(\square\)
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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.
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The authors declare that they have no competing interests.
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Ma, P., Guan, J., Tang, Y. et al. ψContraction and \((\phi ,\varphi )\)contraction in Menger probabilistic metric space. SpringerPlus 5, 233 (2016). https://doi.org/10.1186/s4006401617953
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Keywords
 Probabilistic metric spaces
 \((\phi , \varphi )\)Contraction
 \(\psi\)Contraction
 Contraction mapping principle