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 Open Access
Generalized contraction mapping principle and generalized best proximity point theorems in probabilistic metric spaces
 Yongfu Su^{1},
 Wenbiao Gao^{1} and
 JenChih Yao^{2, 3}Email author
https://doi.org/10.1186/s1366301503234
© Su et al. 2015
 Received: 3 March 2015
 Accepted: 8 May 2015
 Published: 30 May 2015
Abstract
The purpose of this paper is to introduce some basic definitions about fixed point and best proximity point in two classes of probabilistic metric spaces and to prove contraction mapping principle and relevant best proximity point theorems. The first class is the socalled Sprobabilistic metric spaces. In Sprobabilistic metric spaces, the generalized contraction mapping principle and generalized best proximity point theorems have been proved by authors. These results improve and extend the recent results of Su and Zhang (Fixed Point Theory Appl. 2014:170, 2014). The second class is the socalled Menger probabilistic metric spaces. In Menger probabilistic metric spaces, the contraction mapping principle and relevant best proximity point theorems have been proved by authors. These results also improve and extend the results of many authors. In order to get the results of this paper, some new methods have been used. Meanwhile some error estimate inequalities have been established.
Keywords
 probabilistic metric spaces
 contraction
 fixed point
 best proximity point
 mathematical expectation
 bmetric spaces
1 Introduction and preliminaries
Probabilistic metric spaces were introduced in 1942 by Menger [1]. In such spaces, the notion of distance between two points x and y is replaced by a distribution function \(F_{x,y}(t)\). Thus one thinks of the distance between points as being probabilistic with \(F_{x,y}(t)\) representing the probability that the distance between x and y is less than t. Sehgal, in his PhD thesis [2], extended the notion of a contraction mapping to the setting of Menger probabilistic metric spaces. For example, a mapping T is a probabilistic contraction if T is such that for some constant \(0 < k < 1\), the probability that the distance between image points Tx and Ty is less than kt is at least as large as the probability that the distance between x and y is less than t.
In 1972, Sehgal and BharuchaReid proved the following result.
Theorem 1.1
(Sehgal and BharuchaReid [3])
The mapping T satisfying (1.1) is called a kprobabilistic contraction or a Sehgal contraction [3]. The fixed point theorem obtained by Sehgal and BharuchaReid is a generalization of the classical Banach contraction principle and is further investigated by many authors [2, 4–17]. Some results in this theory have found their applications to control theory, system theory and optimization problems.
Next we recall some wellknown definitions and results in the theory of probabilistic metric spaces which are used later in this paper. For more details, we refer the reader to [8].
Definition 1.2
 (a)
△ is associative and commutative;
 (b)
△ is continuous;
 (c)
\(\triangle(a,1)=a\) for all \(a\in[0,1]\);
 (d)
\(\triangle(a,b)\leq\triangle(c,d)\) whenever \(a \leq c\) and \(b \leq d\) for each \(a, b, c, d \in[0,1]\).

\(\triangle_{1}(a,b)=\max(a+b1,0)\);

\(\triangle_{2}(a,b)=a\cdot b\);

\(\triangle_{3}(a,b)=\min(a,b)\).
Definition 1.3
A function \(F(t): (\infty,+\infty)\rightarrow[0, 1]\) is called a distribution function if it is nondecreasing and leftcontinuous with \(\lim_{t\rightarrow\infty}F(t)=0\). If in addition \(F(0)=0\), then F is called a distance distribution function.
Definition 1.4
Definition 1.5
 (PM1)
\(F_{x,y}(t)=H(t)\) if and only if \(x=y\);
 (PM2)
\(F_{x,y}(t)=F_{y,x}(t)\) for all \(x,y \in E\) and \(t \in (\infty,+\infty)\);
 (PM3)
\(F_{x,z}(t)=1\), \(F_{z,y}(s)=1\) implies \(F_{x,y}(t+s)=1\)
Definition 1.6
 (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)
\(F_{x,y}(t+s)\geq\triangle(F_{x,z}(t),F_{z,y}(s))\) for all \(x,y,z \in E\) and \(t>0\), \(s>0\).
In 2014, authors gave a new definition of probabilistic metric space, the socalled Sprobabilistic metric space. This definition reflects more probabilistic meaning and probabilistic background. In this definition, the triangle inequality changed to a new form.
Definition 1.7
([18])
 (SPM1)
\(F_{x,y}(t)=H(t)\) if and only if \(x=y\);
 (SPM2)
\(F_{x,y}(t)=F_{y,x}(t)\) for all \(x,y \in E\) and \(t \in (\infty,+\infty)\);
 (SPM3)\(F_{x,y}(t)\geq F_{x,z}(t)\ast F_{z,y}(t)\), \(\forall x,y,z \in E\), where \(F_{x,z}(t)\ast F_{z,y}(t)\) is the convolution between \(F_{x,z}(t)\) and \(F_{z,y}(t)\) defined by$$F_{x,z}(t)\ast F_{z,y}(t)=\int_{0}^{+\infty}F_{x,z}(tu) \, dF_{z,y}(u). $$
Example
([18])
In this paper, both the Menger probabilistic metric spaces and Sprobabilistic metric spaces are included in the probabilistic metric spaces.
On the other hand, several problems can be changed as equations of the form \(Tx = x\), where T is a given selfmapping defined on a subset of a metric space, a normed linear space, a topological vector space or some suitable space. However, if T is a nonself mapping from A to B, then the aforementioned equation does not necessarily admit a solution. In this case, it is contemplated to find an approximate solution x in A such that the error \(d(x, Tx)\) is minimum, where d is the distance function. In view of the fact that \(d(x, Tx)\) is at least \(d(A,B)\), a best proximity point theorem guarantees the global minimization of \(d(x, Tx)\) by the requirement that an approximate solution x satisfies the condition \(d(x, Tx) = d(A,B)\). Such optimal approximate solutions are called best proximity points of the mapping T. Interestingly, best proximity point theorems also serve as a natural generalization of fixed point theorems, for a best proximity point becomes a fixed point if the mapping under consideration is a selfmapping. Research on best proximity point is an important topic in the nonlinear functional analysis and applications (see [19–31]).
Let A, B be two nonempty subsets of a complete metric space and consider a mapping \(T:A\rightarrow B\). The best proximity point problem is whether we can find an element \(x_{0}\in A\) such that \(d(x_{0},Tx_{0})=\min\{d(x,Tx): x\in A\}\). Since \(d(x,Tx)\geq d(A,B)\) for any \(x\in A\), in fact, the optimal solution to this problem is the one for which the value \(d(A,B)\) is attained.
It is interesting to notice that \(A_{0}\) and \(B_{0}\) are contained in the boundaries of A and B, respectively, provided A and B are closed subsets of a normed linear space such that \(d(A, B)>0\) [19].
In order to study the best proximity point problems, we need the following notations.
Definition 1.8
([30])
In [31], the authors prove that any pair \((A,B)\) of nonempty closed convex subsets of a real Hilbert space H satisfies the Pproperty.
In [25, 26], Pproperty was weakened to weak Pproperty. And an example that satisfies Pproperty but not weak Pproperty can be found there.
Definition 1.9
Recently, many best proximity point problems with applications have been discussed and some best proximity point theorems have been proved. For more details, we refer the reader to [27].
In 2014, authors established some definitions and basic concepts of best proximity point in the framework of probabilistic metric spaces.
Definition 1.10
([18])
Example
([18])
The random variable \(\pmb{d(x,y)}\)
\(\boldsymbol{d_{1}(x,y)}\)  \(\boldsymbol{d_{2}(x,y)}\) 

0.5  0.5 
The random variable \(\pmb{d(A,B)}\)
\(\boldsymbol{d_{1}(A,B)}\)  \(\boldsymbol{d_{2}(A,B)}\) 

0.5  0.5 
Definition 1.11
([18])
Example
([18])
The random variable \(\pmb{d(x^{*},Tx^{*})}\)
\(\boldsymbol{d_{1}(x^{*},Tx^{*})}\)  \(\boldsymbol{d_{2}(x^{*},Tx^{*})}\) 

0.5  0.5 
Definition 1.12
([18])
Definition 1.13
([18])
Definition 1.14
([3])
 (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 must exist 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 must exist a positive integer \(N = N(\varepsilon,\lambda)\) such that \(F_{x_{n},x_{m}}(\varepsilon)>1\lambda\), whenever \(n,m >N\).
 (3)
\((E,F,\triangle)\) is said to be complete if each Cauchy sequence in E converges to some point in E.
We denote by \(x_{n}\rightarrow x\) that \(\{x_{n}\}\) converges to x. It is easy to see that \(x_{n}\rightarrow x\) if and only if \(F_{x_{n},x}(t)\rightarrow H(t)\) for any given \(t \in (\infty,+\infty)\) as \(n\rightarrow\infty\).
The purpose of this paper is to introduce some basic definitions about fixed point and best proximity point in two classes of probabilistic metric spaces and to prove contraction mapping principle and relevant best proximity point theorems. The first class is the socalled Sprobabilistic metric spaces. In Sprobabilistic metric spaces, the generalized contraction mapping principle and generalized best proximity point theorems have been proved by authors. These results improve and extend the recent results of Su and Zhang [18]. The second class is the socalled Menger probabilistic metric spaces. In Menger probabilistic metric spaces, the contraction mapping principle and relevant best proximity point theorems have been proved by authors. These results also improve and extend the results of many authors. In order to get the results of this paper, some new methods have been used. Meanwhile some error estimate inequalities have been established.
2 Contraction mapping principle in Sprobabilistic metric spaces
Theorem 2.1
Proof
Now we prove the following generalized contraction mapping principle in the Sprobabilistic metric spaces which is a generalized form of the result in [18].
Theorem 2.2
 (1)
\(\psi(t)\), \(\phi(t)\) are strictly monotone increasing and continuous;
 (2)
\(\psi(t) < \phi(t)\) for all \(t>0\);
 (3)
\(\psi(0) = \phi(0)\).
Proof
Theorem 2.3
 (1)
\(\psi(t)\), \(\phi(t)\) are strictly monotone increasing and continuous;
 (2)
\(\psi(t) < \phi(t)\) for all \(t>0\);
 (3)
\(\psi(0) = \phi(0)\).
3 Best proximity point theorems in Sprobabilistic spaces
We first define the notion of Poperator \(P: B_{0}\rightarrow A_{0}\), which is very useful for the proof of the theorem. From the definitions of \(A_{0}\) and \(B_{0}\), we know that for any given \(y \in B_{0}\), there exists an element \(x \in A_{0}\) such that \(F_{x,y}(t)=F_{A,B}(t)\). Because \((A,B)\) has the weak Pproperty, so such x is unique. We denote by \(x=Py\) the Poperator from \(B_{0}\) into \(A_{0}\).
Theorem 3.1
 (1)
\(\psi(t)\), \(\phi(t)\) are strictly monotone increasing and continuous;
 (2)
\(\psi(t) < \phi(t)\) for all \(t>0\);
 (3)
\(\psi(0) = \phi(0)\).
Proof
Theorem 3.2
 (1)
\(\psi(t)\), \(\phi(t)\) are strictly monotone increasing and continuous;
 (2)
\(\psi(t) < \phi(t)\) for all \(t>0\);
 (3)
\(\psi(0) = \phi(0)\).
4 Contraction mapping principle in Menger probabilistic metric spaces
In 1973, Czerwik [32] presented a notable generalization of the classical Banach fixed point theorem in the socalled bmetric spaces.
Definition 4.1
 (BM1)
\(d(x, y) = 0\) if and only if \(x=y\);
 (BM2)
\(d(x, y) = d(y, x)\);
 (BM3)
\(d(x; y) \leq s(d(x, z) + d(z, y))\).
The notions of topology including the convergence, completeness and Cauchy sequence are similar to those of metric spaces. Now, we are in a position to present the interesting result of our paper as follows.
Theorem 4.2
Proof
Theorem 4.3
Proof
Theorem 4.4
Proof
Since \(\triangle(a,b)\geq \triangle_{1}(a,b)=\max\{a+b1,0\}\), if \((E, F, \triangle )\) is a complete Menger probabilistic metric space, so is \((E, F, \triangle _{1})\). By using Theorem 2.3, we get the conclusion of Theorem 4.4. This completes the proof. □
Corollary 4.5
(Sehgal and BharuchaReid [3], 1972)
5 Best proximity point theorems in Menger probabilistic metric spaces
We first define the notion of Poperator \(P: B_{0}\rightarrow A_{0}\), which is useful for our best proximity point theorem. From the definitions of \(A_{0}\) and \(B_{0}\), we know that for any given \(y \in B_{0}\), there exists an element \(x \in A_{0}\) such that \(F_{x,y}(t)=F_{A,B}(t)\). Because \((A,B)\) has the weak Pproperty, so such x is unique. We denote by \(x=Py\) the Poperator from \(B_{0}\) into \(A_{0}\).
Theorem 5.1
Proof
Theorem 5.2
Proof
Because \(\triangle(a,b)\geq\triangle_{1}(a,b)=\max\{a+b1,0\}\), by using Theorem 5.1, we can get the conclusion of Theorem 3.2. This completes the proof. □
Corollary 5.3
Corollary 5.4
Remark
The research for probabilistic metric spaces (probabilistic normed spaces) and relevant fixed point theory is an important topic. Many relevant results have been given by some authors. However, the profound relationship with the probabilistic theory has not been studied closely. The Sprobabilistic metric spaces and relevant probabilistic methods will play an important role in the theory and applications.
Declarations
Acknowledgements
This project is supported by the National Natural Science Foundation of China under Grant (11071279).
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
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