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Fixed point and best proximity point theorems for contractions in new class of probabilistic metric spaces
Fixed Point Theory and Applications volume 2014, Article number: 170 (2014)
Abstract
The purpose of this paper is to present some definitions and basic concepts of best proximity point in a new class of probabilistic metric spaces and to prove the best proximity point theorems for the contractive mappings and weak contractive mappings. In order to get the best proximity point theorems, some new probabilistic contraction mapping principles have been proved. Meanwhile the error estimate inequalities have been established. Further, a method of the proof is also new and interesting, which is to use the mathematical expectation of the distribution function studying the related problems.
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 Ph.D. thesis [2], extended the notion of a contraction mapping to the setting of the 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], 1972)
Let (E,F,\mathrm{\u25b3}) be a complete Menger probabilistic metric space for which the triangular norm △ is continuous and satisfies \mathrm{\u25b3}(a,b)=min(a,b). If T is a mapping of E into itself such that for some 0<k<1 and all x,y\in E,
then T has a unique fixed point {x}^{\ast} in E, and for any given {x}_{0}\in X, {T}^{n}{x}_{0} converges to {x}^{\ast}.
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, 4–18]. Some results in this theory have found applications to control theory, system theory, and optimization problems.
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 [8].
Definition 1.2 A triangular norm (shortly, △norm) is a binary operation △ on [0,1] which satisfies the following conditions:

(a)
△ is associative and commutative;

(b)
△ is continuous;

(c)
\mathrm{\u25b3}(a,1)=a for all a\in [0,1];

(d)
\mathrm{\u25b3}(a,b)\le \mathrm{\u25b3}(c,d) whenever a\le c and b\le d for each a,b,c,d\in [0,1].
The following are the six basic △norms:
{\mathrm{\u25b3}}_{1}(a,b)=max(a+b1,0);
{\mathrm{\u25b3}}_{2}(a,b)=a\cdot b;
{\mathrm{\u25b3}}_{3}(a,b)=min(a,b);
{\mathrm{\u25b3}}_{4}(a,b)=max(a,b);
{\mathrm{\u25b3}}_{5}(a,b)=a+bab;
{\mathrm{\u25b3}}_{6}(a,b)=min(a+b,1).
It is easy to check that the above six △norms have the following relations:
for any a,b\in [0,1].
Definition 1.3 A function F(t):(\mathrm{\infty},+\mathrm{\infty})\to [0,1] is called a distribution function if it is nondecreasing and leftcontinuous with {lim}_{t\to \mathrm{\infty}}F(t)=0. If in addition F(0)=0 then F is called a distance distribution function.
Definition 1.4 A distance distribution function F satisfying {lim}_{t\to +\mathrm{\infty}}F(t)=1 is called a Menger distance distribution function. The set of all Menger distance distribution functions is denoted by {D}^{+}. A special Menger distance distribution function given by
Definition 1.5 A probabilistic metric space is a pair (E,F), where E is a nonempty set, 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:
(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 (\mathrm{\infty},+\mathrm{\infty});
(PM3) {F}_{x,z}(t)=1, {F}_{z,y}(s)=1 implies {F}_{x,y}(t+s)=1
for all x,y,z\in E and \mathrm{\infty}<t<+\mathrm{\infty}.
Definition 1.6 A Menger probabilistic metric space (abbreviated, Menger PM space) is a triple (E,F,\u25b3) where E is a nonempty set, △ 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 (\mathrm{\infty},+\mathrm{\infty});
(MPM3) {F}_{x,y}(t+s)\ge \mathrm{\u25b3}({F}_{x,z}(t),{F}_{z,y}(s)) for all x,y,z\in E and t>0, s>0.
Now we give a new definition of probabilistic metric space socalled Sprobabilistic metric space. This definition reflects a more probabilistic meaning and the probabilistic background. In this definition, the triangle inequality has been changed to a new form.
Definition 1.7 A Sprobabilistic metric space is a pair (E,F), where E is a nonempty set, 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:
(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 (\mathrm{\infty},+\mathrm{\infty});
(SPM3) {F}_{x,y}(t)\ge {F}_{x,z}(t)\ast {F}_{z,y}(t) \mathrm{\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
Example Let X be a nonempty set, S be a measurable space which consist of some metrics on the X, (\mathrm{\Omega},P) be a complete probabilistic measure space and f:\mathrm{\Omega}\to S be a measurable mapping. It is easy to think S is a random metric on the X, of course, (X,S) is a random metric space. The following expressions of the distribution functions {F}_{x,y}(t), {F}_{x,z}(t), and {F}_{z,y}(t) are reasonable:
and
for all x,y,z\in X. Since
it follows from probabilistic theory that
Therefore
In addition, the conditions (SPM1), (SPM2) are obvious.
In this paper, both the Menger probabilistic metric spaces and the Sprobabilistic metric spaces are included in the probabilistic metric spaces.
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 nonselfmapping 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 the 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\to B. The best proximity point problem is whether we can find an element {x}_{0}\in A such that d({x}_{0},T{x}_{0})=min\{d(x,Tx):x\in A\}. Since d(x,Tx)\ge 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.
Let A, B be two nonempty subsets of a metric space (X,d). We denote by {A}_{0} and {B}_{0} the following sets:
where d(A,B)=inf\{d(x,y):x\in A\text{and}y\in B\}.
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])
Let (A,B) be a pair of nonempty subsets of a metric space (X,d) with {A}_{0}\ne \mathrm{\varnothing}. Then the pair (A,B) is said to have the Pproperty if and only if for any {x}_{1},{x}_{2}\in {A}_{0} and {y}_{1},{y}_{2}\in {B}_{0},
In [31], the author proves that any pair (A,B) of nonempty closed convex subsets of a real Hilbert space H satisfies Pproperty.
In [25, 26], Pproperty has been weakened to the weak Pproperty. An example that satisfies the Pproperty but not the weak Pproperty can be found there.
Let (A,B) be a pair of nonempty subsets of a metric space (X,d) with {A}_{0}\ne \mathrm{\varnothing}. Then the pair (A,B) is said to have the weak Pproperty if and only if for any {x}_{1},{x}_{2}\in {A}_{0} and {y}_{1},{y}_{2}\in {B}_{0},
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 this paper, we establish some definitions and basic concepts of the best proximity point in the framework of probabilistic metric spaces.
Definition 1.10 Let (E,F) be a probabilistic metric space, A,B\subset E be two nonempty sets. Let
which is said to be the probabilistic distance of A, B.
Example Let X be a nonempty set and {d}_{1}, {d}_{2} be two metrics defined on X with the probabilities {p}_{1}=0.5, {p}_{2}=0.5, respectively. Assume that
For any x,y\in X, the table
is a discrete random variable with the distribution function
Let A, B be two nonempty sets of X, the table
is also a discrete random variable with the distribution function
where
It is easy to see that
Definition 1.11 Let (E,F) be a probabilistic metric space, A,B\subset E be two nonempty subsets and T:A\to B be a mapping. We say that {x}^{\ast}\in A is a best proximity point of the mapping T if the following equality holds:
Example Let X be a nonempty set and {d}_{1}, {d}_{2} be two metrics defied on X with the probabilities {p}_{1}=0.5, {p}_{2}=0.5, respectively. Let A, B be two nonempty sets of X and T:A\to B be a mapping. Assume
If there exists a point {x}^{\ast}\in A, such that
then the table
is a discrete random variable with the distribution function
It is obvious that {F}_{{x}^{\ast},T{x}^{\ast}}(t)={F}_{A,B}(t).
It is clear that the notion of a fixed point coincided with the notion of a best proximity point when the underlying mapping is a selfmapping. Let (E,F) be a probabilistic metric space. Suppose that A\subset E and B\subset E are nonempty subsets. We define the following sets:
Definition 1.12 Let (A,B) be a pair of nonempty subsets of a probabilistic metric space (E,F) with {A}_{0}\ne \mathrm{\varnothing}. Then the pair (A,B) is said to have the Pproperty if and only if for any {x}_{1},{x}_{2}\in A and {y}_{1},{y}_{2}\in B,
Definition 1.13 Let (A,B) be a pair of nonempty subsets of a probabilistic metric space (E,F) with {A}_{0}\ne \mathrm{\varnothing}. Then the pair (A,B) is said to have the weak Pproperty if and only if for any {x}_{1},{x}_{2}\in A and {y}_{1},{y}_{2}\in B,
Definition 1.14 Let (E,F) be a probabilistic metric space.

(1)
A sequence \{{x}_{n}\} in E is said to converges to x\in E if for any given \epsilon >0 and \lambda >0, there must exist a positive integer N=N(\epsilon ,\lambda ) such that {F}_{{x}_{n},x}(\epsilon )>1\lambda whenever n>N.

(2)
A sequence \{{x}_{n}\} in E is called a Cauchy sequence if for any \epsilon >0 and \lambda >0, there must exists a positive integer N=N(\epsilon ,\lambda ) such that {F}_{{x}_{n},{x}_{m}}(\epsilon )>1\lambda, whenever n,m>N.

(3)
(E,F,\mathrm{\u25b3}) is said to be complete if each Cauchy sequence in E converges to some point in E.
We denote by {x}_{n}\to x the \{{x}_{n}\} converges to x. It is easy to see that {x}_{n}\to x if and only if {F}_{{x}_{n},x}(t)\to H(t) for any given t\in (\mathrm{\infty},+\mathrm{\infty}) as n\to \mathrm{\infty}.
2 Contraction mapping principle in Sprobabilistic metric spaces
Let (E,F) be a Sprobabilistic metric space. For any x,y\in E we definite
Since t is a continuous function and {F}_{x,y} is a bounded variation functions, so the above integer is well definite. In fact, the above integer is just the mathematical expectation of {F}_{x,y}(t). Throughout this paper we assume that
for all probabilistic metric spaces (E,F) presented in this paper.
Next we give a new notation of convergence.

(1)
A sequence \{{x}_{n}\} in E is said to converges averagely to x\in E if
\underset{n\to \mathrm{\infty}}{lim}{\int}_{0}^{+\mathrm{\infty}}t\phantom{\rule{0.2em}{0ex}}d{F}_{{x}_{n},x}(t)=0. 
(2)
A sequence \{{x}_{n}\} in E is called an average Cauchy sequence if
\underset{n,m\to \mathrm{\infty}}{lim}{\int}_{0}^{+\mathrm{\infty}}t\phantom{\rule{0.2em}{0ex}}d{F}_{{x}_{n},{x}_{m}}(t)=0. 
(3)
(E,F) is said to be average complete if each average Cauchy sequence in E converges averagely to some point in E.
We denote by {x}_{n}\Rightarrow x the \{{x}_{n}\} that converges averagely to x.
Theorem 2.1 Let (E,F) be a Sprobabilistic metric space. For any x,y\in E we define
Then {d}_{F}(x,y) is a metric on the E.
Proof Since {F}_{x,y}(t)=H(t) (\mathrm{\forall}t\in R) if and only if x=y, and
we know the condition {d}_{F}(x,y)=0\iff x=y holds. The condition {d}_{F}(x,y)={d}_{F}(y,x), for all x,y\in E, is obvious. Next we will prove the triangle inequality. For any x,y,z\in E, from (SPM3) we have
By using probabilistic theory we know that
which implies that
This completes the proof. □
Theorem 2.2 Let (E,F) be a complete Sprobabilistic metric space. Let T:E\to E be a mapping satisfying the following condition:
where 0<h<1 is a constant. Then T has a unique fixed point {x}^{\ast}\in E and for any given {x}_{0}\in E the iterative sequence {x}_{n+1}=T{x}_{n} converges to {x}^{\ast}. Further, the error estimate inequality
holds for all n\ge 1.
Proof For any x,y\in E, from (2.1) we have
For any given {x}_{0}\in E, define {x}_{n+1}=T{x}_{n} for all n=0,1,2,\dots . Observe that
Since 0<h<1, we have
as n\to \mathrm{\infty}. Hence
as n\to \mathrm{\infty}. We claim that
If not, there must exist numbers {t}_{0}>0, 0<{\lambda}_{0}<1, and subsequences \{{n}_{k}\}, \{{m}_{k}\} of \{n\} such that {F}_{{x}_{{n}_{k}},{x}_{{n}_{k}+{m}_{k}}}({t}_{0})\le {\lambda}_{0}, for all k\ge 1. In this case, we have
This is a contradiction. From (2.3) we know \{{x}_{n}\} is a Cauchy sequence in complete Sprobabilistic metric space (E,F). Hence there exists a point {x}^{\ast}\in E such that \{{x}_{n}\} converges to {x}^{\ast} in the mean of
Therefore
We claim {x}^{\ast} is a fixed point of T, in fact, for any t>0, it follows from condition (SPM3) that
as n\to \mathrm{\infty}, which implies {F}_{{x}^{\ast},T{x}^{\ast}}(t)=H(t), and hence {x}^{\ast}=T{x}^{\ast}. The {x}^{\ast} is a fixed point of T. If there exists another fixed point {x}^{\ast \ast} of T, we obverse
which implies {F}_{{x}^{\ast},{x}^{\ast \ast}}(t)=H(t) \mathrm{\forall}t\in R, and hence {x}^{\ast}={x}^{\ast \ast}. Then the fixed point of T is unique. Meanwhile, for any given {x}_{0}, the iterative sequence {x}_{n}={T}^{n}{x}_{0} converges to {x}^{\ast}. Finally, we prove the error estimate formula. Let m\to \mathrm{\infty} in the inequality (2.2); we get
which can be rewritten as the following error estimate formula:
This completes the proof. □
Theorem 2.3 Let (E,F,\u25b3) be a complete Menger probabilistic metric space. Assume
for all x,y,z\in E, t>0. Let T:E\to E be a mapping satisfying the following condition:
where 0<h<1 is a constant. Then T has a unique fixed point {x}^{\ast}\in E and for any given {x}_{0}\in E the iterative sequence {x}_{n+1}=T{x}_{n} converges to {x}^{\ast}. Further, the error estimate inequality
holds for all n\ge 1.
Proof From (2.4) we know that (E,F,\mathrm{\u25b3}) is a Sprobabilistic metric space. This together with (2.5), by using Theorem 2.2, shows that the conclusion is proved. □
3 Best proximity point theorems for contractions
We first define the notion of Poperator P:{B}_{0}\to {A}_{0}, it is very useful for the proof of the 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, such x is unique. We denote by x=Py the Poperator from {B}_{0} into {A}_{0}.
Theorem 3.1 Let (E,F) be a complete Sprobabilistic metric space. Let (A,B) be a pair of nonempty subsets in E and {A}_{0} be a nonempty closed subset. Suppose (A,B) satisfies the weak Pproperty. Let T:A\to B be a mapping satisfying the following condition:
where 0<h<1 is a constant. Assume that T({A}_{0})\subset {B}_{0}. Then T has a unique best proximity point {x}^{\ast}\in A and for any given {x}_{0}\in E the iterative sequence {x}_{n+1}=PT{x}_{n} converges to {x}^{\ast}. Further, the error estimate inequality
holds for all n\ge 1.
Proof Since the pair (A,B) has the weak Pproperty, we have
for any {x}_{1},{x}_{2}\in {A}_{0}. This shows that PT:{A}_{0}\to {A}_{0} is a contraction from complete Sprobabilistic metric subspace {A}_{0} into itself. Using Theorem 2.2, we know that PT has a unique fixed point {x}^{\ast} and for any given {x}_{0}\in E the iterative sequence {x}_{n+1}=PT{x}_{n} converges to {x}^{\ast}. Further, the error estimate inequality
holds for all n\ge 1. Since PT{x}^{\ast}={x}^{\ast} if and only if {F}_{{x}^{\ast},T{x}^{\ast}}(t)={F}_{A,B}(t), so the point {x}^{\ast} is a unique best proximity point of T:A\to B. This completes the proof. □
Theorem 3.2 Let (E,F,\mathrm{\u25b3}) be a complete Menger probabilistic metric space. Assume that
for all x,y,z\in E, t>0. Let (A,B) be a pair of nonempty subsets in E and {A}_{0} be nonempty closed subset. Suppose that (A,B) satisfies the weak Pproperty. Let T:A\to B be a mapping satisfying the following condition:
where 0<h<1 is a constant. Assume T({A}_{0})\subset {B}_{0}. Then T has a unique best proximity point {x}^{\ast}\in A and for any given {x}_{0}\in E the iterative sequence {x}_{n+1}=PT{x}_{n} converges to {x}^{\ast}. Further, the error estimate inequality
holds for all n\ge 1.
Proof From (3.1) we know that (E,F,\mathrm{\u25b3}) is a Sprobabilistic metric space. By using Theorem 3.1, the conclusion is proved. □
4 Best proximity point theorem for Geraghtycontractions
First, we introduce the class Γ of those functions \beta :[0,+\mathrm{\infty})\to [0,1) satisfying the following condition:
Definition 4.1 Let (E,F) be a probabilistic metric space. Let (A,B) be a pair of nonempty subsets in E. A mapping T:A\to B is said to be a Geraghtycontraction if there exists \beta \in \Gamma such that
where
Theorem 4.2 Let (E,F) be a complete Sprobabilistic metric space. Let (A,B) be a pair of nonempty subsets in E and {A}_{0} be a nonempty closed subset. Suppose that (A,B) satisfies the weak Pproperty. Let T:A\to B be a Geraghtycontraction. Assume T({A}_{0})\subset {B}_{0}. Then T has a unique best proximity point {x}^{\ast}\in A and for any given {x}_{0}\in E the iterative sequence {x}_{n+1}=PT{x}_{n} converges to {x}^{\ast}.
Proof From (4.1) and the weak Pproperty of (A,B), we get
We have proved that {d}_{F}(\cdot ,\cdot ) is a metric on the E in Theorem 2.1. For any given {x}_{0}\in E, define {x}_{n+1}=PT{x}_{n}, n=0,1,2,\dots . From (4.2) we have
Suppose that there exists {n}_{0} such that {d}_{F}({x}_{{n}_{0}},{x}_{{n}_{0}+1})=0. In this case, PT{x}_{{n}_{0}}={x}_{{n}_{0}}, which implies that {x}_{{n}_{0}} is a best proximity point of T and this is the desired result. In the contrary case, suppose that {d}_{F}({x}_{n},{x}_{n+1})>0, for any n\ge 0. By (4.3), {d}_{F}({x}_{n},{x}_{n+1}) is a decreasing sequence of nonnegative real numbers, and hence there exists r\ge 0 such that {lim}_{n\to \mathrm{\infty}}{d}_{F}({x}_{n},{x}_{n+1})=r. In the sequel, we prove that r=0. Assume r>0, then from (4.3) we have
for all n\ge 0. The last inequality implies that {lim}_{n\to \mathrm{\infty}}\beta ({d}_{F}({x}_{n1},{x}_{n}))=1 and since \beta \in \Gamma, we obtain r=0 and this contradicts with our assumption. Therefore,
In what follows, we prove that \{{x}_{n}\} is a Cauchy sequence in metric space (E,{d}_{F}(\cdot ,\cdot )). In the contrary case, there exist two subsequences \{{x}_{{n}_{k}}\}, \{{x}_{{m}_{k}}\} such that
Without loss of generality, we still denote by \{{x}_{n}\}, \{{x}_{m}\} these subsequences. By using the triangular inequality,
which implies
The last inequality together with (4.4) and (4.5) give us
Therefore,
Since \beta \in \Gamma, we get
This is a contradiction with (4.5). Hence {lim}_{n,m\to \mathrm{\infty}}{d}_{F}({x}_{n},{x}_{m})=0, the \{{x}_{n}\} is a Cauchy sequence in metric space (E,{d}_{F}(\cdot ,\cdot )). By using the same method as in Theorem 2.2, we know
This shows that the \{{x}_{n}\} is also a Cauchy sequence in Sprobabilistic metric space (E,F). Since (E,F) is complete, then there exists a point {x}^{\ast}\in E such that {x}_{n}\to {x}^{\ast} as n\to \mathrm{\infty}. By using the same method as in Theorem 2.2, we know that {x}^{\ast} is a unique fixed point of mapping PT:{A}_{0}\to {A}_{0}. That is, PT{x}^{\ast}={x}^{\ast}, which is equivalent to {x}^{\ast} is a unique best proximity point of T. This completes the proof. □
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Su, Y., Zhang, J. Fixed point and best proximity point theorems for contractions in new class of probabilistic metric spaces. Fixed Point Theory Appl 2014, 170 (2014). https://doi.org/10.1186/168718122014170
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DOI: https://doi.org/10.1186/168718122014170
Keywords
 probabilistic metric spaces
 contraction
 fixed point
 best proximity point
 mathematical expectation