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.
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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 . Thus one thinks of the distance between points as being probabilistic with 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 , 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 Bharucha-Reid proved the following result.
Theorem 1.1 (Sehgal and Bharucha-Reid [3], 1972)
Let be a complete Menger probabilistic metric space for which the triangular norm △ is continuous and satisfies . If T is a mapping of E into itself such that for some and all ,
then T has a unique fixed point in E, and for any given , converges to .
The mapping T satisfying (1.1) is called a k-probabilistic contraction or a Sehgal contraction [3]. The fixed point theorem obtained by Sehgal and Bharucha-Reid 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 well-known 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 which satisfies the following conditions:
-
(a)
△ is associative and commutative;
-
(b)
△ is continuous;
-
(c)
for all ;
-
(d)
whenever and for each .
The following are the six basic △-norms:
;
;
;
;
;
.
It is easy to check that the above six △-norms have the following relations:
for any .
Definition 1.3 A function is called a distribution function if it is non-decreasing and left-continuous with . If in addition then F is called a distance distribution function.
Definition 1.4 A distance distribution function F satisfying is called a Menger distance distribution function. The set of all Menger distance distribution functions is denoted by . A special Menger distance distribution function given by
Definition 1.5 A probabilistic metric space is a pair , where E is a nonempty set, F is a mapping from into such that, if denotes the value of F at the pair , the following conditions hold:
(PM-1) if and only if ;
(PM-2) for all and ;
(PM-3) , implies
for all and .
Definition 1.6 A Menger probabilistic metric space (abbreviated, Menger PM space) is a triple where E is a nonempty set, △ is a continuous t-norm and F is a mapping from into such that, if denotes the value of F at the pair , the following conditions hold:
(MPM-1) if and only if ;
(MPM-2) for all and ;
(MPM-3) for all and , .
Now we give a new definition of probabilistic metric space so-called S-probabilistic 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 S-probabilistic metric space is a pair , where E is a nonempty set, F is a mapping from into such that, if denotes the value of F at the pair , the following conditions hold:
(SPM-1) if and only if ;
(SPM-2) for all and ;
(SPM-3) ,
where is the convolution between and defined by
Example Let X be a nonempty set, S be a measurable space which consist of some metrics on the X, be a complete probabilistic measure space and be a measurable mapping. It is easy to think S is a random metric on the X, of course, is a random metric space. The following expressions of the distribution functions , , and are reasonable:
and
for all . Since
it follows from probabilistic theory that
Therefore
In addition, the conditions (SPM-1), (SPM-2) are obvious.
In this paper, both the Menger probabilistic metric spaces and the S-probabilistic metric spaces are included in the probabilistic metric spaces.
Several problems can be changed as equations of the form , where T is a given self-mapping 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 non-self-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 is minimum, where d is the distance function. In view of the fact that is at least , a best proximity point theorem guarantees the global minimization of by the requirement that an approximate solution x satisfies the condition . 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 self-mapping. 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 . The best proximity point problem is whether we can find an element such that . Since for any , in fact, the optimal solution to this problem is the one for which the value is attained.
Let A, B be two nonempty subsets of a metric space . We denote by and the following sets:
where .
It is interesting to notice that and are contained in the boundaries of A and B, respectively, provided A and B are closed subsets of a normed linear space such that [19].
In order to study the best proximity point problems, we need the following notations.
Definition 1.8 ([30])
Let be a pair of nonempty subsets of a metric space with . Then the pair is said to have the P-property if and only if for any and ,
In [31], the author proves that any pair of nonempty closed convex subsets of a real Hilbert space H satisfies P-property.
In [25, 26], P-property has been weakened to the weak P-property. An example that satisfies the P-property but not the weak P-property can be found there.
Let be a pair of nonempty subsets of a metric space with . Then the pair is said to have the weak P-property if and only if for any and ,
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 be a probabilistic metric space, be two nonempty sets. Let
which is said to be the probabilistic distance of A, B.
Example Let X be a nonempty set and , be two metrics defined on X with the probabilities , , respectively. Assume that
For any , 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 be a probabilistic metric space, be two nonempty subsets and be a mapping. We say that is a best proximity point of the mapping T if the following equality holds:
Example Let X be a nonempty set and , be two metrics defied on X with the probabilities , , respectively. Let A, B be two nonempty sets of X and be a mapping. Assume
If there exists a point , such that
then the table
is a discrete random variable with the distribution function
It is obvious that .
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 self-mapping. Let be a probabilistic metric space. Suppose that and are nonempty subsets. We define the following sets:
Definition 1.12 Let be a pair of nonempty subsets of a probabilistic metric space with . Then the pair is said to have the P-property if and only if for any and ,
Definition 1.13 Let be a pair of nonempty subsets of a probabilistic metric space with . Then the pair is said to have the weak P-property if and only if for any and ,
Definition 1.14 Let be a probabilistic metric space.
-
(1)
A sequence in E is said to converges to if for any given and , there must exist a positive integer such that whenever .
-
(2)
A sequence in E is called a Cauchy sequence if for any and , there must exists a positive integer such that , whenever .
-
(3)
is said to be complete if each Cauchy sequence in E converges to some point in E.
We denote by the converges to x. It is easy to see that if and only if for any given as .
2 Contraction mapping principle in S-probabilistic metric spaces
Let be a S-probabilistic metric space. For any we definite
Since t is a continuous function and is a bounded variation functions, so the above integer is well definite. In fact, the above integer is just the mathematical expectation of . Throughout this paper we assume that
for all probabilistic metric spaces presented in this paper.
Next we give a new notation of convergence.
-
(1)
A sequence in E is said to converges averagely to if
-
(2)
A sequence in E is called an average Cauchy sequence if
-
(3)
is said to be average complete if each average Cauchy sequence in E converges averagely to some point in E.
We denote by the that converges averagely to x.
Theorem 2.1 Let be a S-probabilistic metric space. For any we define
Then is a metric on the E.
Proof Since () if and only if , and
we know the condition holds. The condition , for all , is obvious. Next we will prove the triangle inequality. For any , from (SPM-3) we have
By using probabilistic theory we know that
which implies that
This completes the proof. □
Theorem 2.2 Let be a complete S-probabilistic metric space. Let be a mapping satisfying the following condition:
where is a constant. Then T has a unique fixed point and for any given the iterative sequence converges to . Further, the error estimate inequality
holds for all .
Proof For any , from (2.1) we have
For any given , define for all . Observe that
Since , we have
as . Hence
as . We claim that
If not, there must exist numbers , , and subsequences , of such that , for all . In this case, we have
This is a contradiction. From (2.3) we know is a Cauchy sequence in complete S-probabilistic metric space . Hence there exists a point such that converges to in the mean of
Therefore
We claim is a fixed point of T, in fact, for any , it follows from condition (SPM-3) that
as , which implies , and hence . The is a fixed point of T. If there exists another fixed point of T, we obverse
which implies , and hence . Then the fixed point of T is unique. Meanwhile, for any given , the iterative sequence converges to . Finally, we prove the error estimate formula. Let 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 be a complete Menger probabilistic metric space. Assume
for all , . Let be a mapping satisfying the following condition:
where is a constant. Then T has a unique fixed point and for any given the iterative sequence converges to . Further, the error estimate inequality
holds for all .
Proof From (2.4) we know that is a S-probabilistic 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 P-operator , it is very useful for the proof of the best proximity point theorem. From the definitions of and , we know that for any given , there exists an element such that . Because has the weak P-property, such x is unique. We denote by the P-operator from into .
Theorem 3.1 Let be a complete S-probabilistic metric space. Let be a pair of nonempty subsets in E and be a nonempty closed subset. Suppose satisfies the weak P-property. Let be a mapping satisfying the following condition:
where is a constant. Assume that . Then T has a unique best proximity point and for any given the iterative sequence converges to . Further, the error estimate inequality
holds for all .
Proof Since the pair has the weak P-property, we have
for any . This shows that is a contraction from complete S-probabilistic metric subspace into itself. Using Theorem 2.2, we know that PT has a unique fixed point and for any given the iterative sequence converges to . Further, the error estimate inequality
holds for all . Since if and only if , so the point is a unique best proximity point of . This completes the proof. □
Theorem 3.2 Let be a complete Menger probabilistic metric space. Assume that
for all , . Let be a pair of nonempty subsets in E and be nonempty closed subset. Suppose that satisfies the weak P-property. Let be a mapping satisfying the following condition:
where is a constant. Assume . Then T has a unique best proximity point and for any given the iterative sequence converges to . Further, the error estimate inequality
holds for all .
Proof From (3.1) we know that is a S-probabilistic metric space. By using Theorem 3.1, the conclusion is proved. □
4 Best proximity point theorem for Geraghty-contractions
First, we introduce the class Γ of those functions satisfying the following condition:
Definition 4.1 Let be a probabilistic metric space. Let be a pair of nonempty subsets in E. A mapping is said to be a Geraghty-contraction if there exists such that
where
Theorem 4.2 Let be a complete S-probabilistic metric space. Let be a pair of nonempty subsets in E and be a nonempty closed subset. Suppose that satisfies the weak P-property. Let be a Geraghty-contraction. Assume . Then T has a unique best proximity point and for any given the iterative sequence converges to .
Proof From (4.1) and the weak P-property of , we get
We have proved that is a metric on the E in Theorem 2.1. For any given , define , . From (4.2) we have
Suppose that there exists such that . In this case, , which implies that is a best proximity point of T and this is the desired result. In the contrary case, suppose that , for any . By (4.3), is a decreasing sequence of nonnegative real numbers, and hence there exists such that . In the sequel, we prove that . Assume , then from (4.3) we have
for all . The last inequality implies that and since , we obtain and this contradicts with our assumption. Therefore,
In what follows, we prove that is a Cauchy sequence in metric space . In the contrary case, there exist two subsequences , such that
Without loss of generality, we still denote by , these subsequences. By using the triangular inequality,
which implies
The last inequality together with (4.4) and (4.5) give us
Therefore,
Since , we get
This is a contradiction with (4.5). Hence , the is a Cauchy sequence in metric space . By using the same method as in Theorem 2.2, we know
This shows that the is also a Cauchy sequence in S-probabilistic metric space . Since is complete, then there exists a point such that as . By using the same method as in Theorem 2.2, we know that is a unique fixed point of mapping . That is, , which is equivalent to is a unique best proximity point of T. This completes the proof. □
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This project is supported by the National Natural Science Foundation of China under grant (11071279).
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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/10.1186/1687-1812-2014-170
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DOI: https://doi.org/10.1186/1687-1812-2014-170