- Open Access
Fixed point and best proximity point theorems for contractions in new class of probabilistic metric spaces
© Su and Zhang; licensee Springer. 2014
- Received: 29 March 2014
- Accepted: 17 July 2014
- Published: 18 August 2014
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.
- probabilistic metric spaces
- fixed point
- best proximity point
- mathematical expectation
Probabilistic metric spaces were introduced in 1942 by Menger . 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 , 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 , 1972)
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 . 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 .
△ is associative and commutative;
△ is continuous;
for all ;
whenever and for each .
The following are the six basic △-norms:
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 1.5 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 ;
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.
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 .
In order to study the best proximity point problems, we need the following notations.
Definition 1.8 ()
In , the author proves that any pair of nonempty closed convex subsets of a real Hilbert space H satisfies P-property.
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 .
In this paper, we establish some definitions and basic concepts of the best proximity point in the framework of probabilistic metric spaces.
which is said to be the probabilistic distance of A, B.
It is obvious that .
A sequence in E is said to converges to if for any given and , there must exist a positive integer such that whenever .
A sequence in E is called a Cauchy sequence if for any and , there must exists a positive integer such that , whenever .
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 .
for all probabilistic metric spaces presented in this paper.
- (1)A sequence in E is said to converges averagely to if
- (2)A sequence in E is called an average Cauchy sequence if
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.
Then is a metric on the E.
This completes the proof. □
holds for all .
This completes the proof. □
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. □
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 .
holds for all .
holds for all . Since if and only if , so the point is a unique best proximity point of . This completes the proof. □
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. □
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 .
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. □
This project is supported by the National Natural Science Foundation of China under grant (11071279).
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