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-Contraction theorems in probabilistic metric spaces for single valued case
Fixed Point Theory and Applications volume 2013, Article number: 109 (2013)
In this article, we prove some fixed-point theorems for -contraction in probabilistic metric spaces for single valued case. We will generalize the definition of -contraction and present fixed-point theorem in the generalized -contraction.
The probabilistic metric space was introduced by Menger . Mihet presented the class of -contraction for a single valued case in fuzzy metric spaces [2, 3]. This class is a generalization of the -contraction which was introduced in . We defined the class of -contraction for the multi-valued case in a probabilistic metric space before . Now, we obtain two fixed-point theorems of -contraction for single valued case. Also, we extend the concept of -contraction to the generalized -contraction.
The structure of this paper is as follows: Section 2 is a review of some concepts in probabilistic metric spaces and probabilistic contractions. In Section 3, we will show two theorems for -contraction in the single-valued case and explain the generalized -contraction.
2 Preliminary notes
Let be the set of all distribution of functions F such that (F is a non-decreasing, left continuous mapping from ℝ into such that ).
The ordered pair is said to be a probabilistic metric space if S is a nonempty set and ( written by for every ) satisfies the following conditions:
for every (),
for every ,
and for every , and every .
A Menger space is a triple where is a probabilistic metric space, T is a triangular norm (abbreviated t-norm) and the following inequality holds:
Recall the mapping is called a triangular norm (a t-norm) if the following conditions are satisfied: for every ; for every ; , ; , . Basic examples of t-norms are (Lukasiewicz t-norm), and , defined by , and . If T is a t-norm and (), one can define recurrently for all . One can also extend T to a countable infinitary operation by defining for any sequence as .
If is given, we say that the t-norm T is q-convergent if . We remark that if T is q-convergent, then
Also, note that if the t-norm T is q-convergent, then .
Proposition 2.1 Let be a Menger space. If , then the family , where
is a base for a metrizable uniformity on S, called the F-uniformity [6–8]. The F-uniformity naturally determines a metrizable topology on S, called the strong topology or F-topology , a subset O of S is F-open if for every there exists such that .
Definition 2.1 
A sequence is called an F-convergent sequence to if for every and there exists such that , .
Definition 2.2 
Let be a mapping, we say that the t-norm T is φ-convergent if
Definition 2.3 
A sequence is called a convergent sequence to if for every and there exists such that , .
Definition 2.4 
A sequence is called a Cauchy sequence if for every and there exists such that , , .
We also have
A probabilistic metric space is called sequentially complete if every Cauchy sequence is convergent.
The concept of -contraction has been introduced by Mihet .
We will consider comparison functions from the class ϕ of all mapping with the properties:
φ is an increasing bijection;
Since every such a comparison mapping is continuous, if , then .
Definition 2.5 
Let be a probabilistic space, and ψ be a map from to . A mapping is called a -contraction on S if it satisfies in the following condition:
In the rest of paper we suppose that ψ is increasing bijection.
Example 2.1 Let and (for )
Suppose that , .
Then is a probabilistic metric space .
Let x, y, ϵ, λ be such that .
If , then .
This implies , that is,
If then , hence again . Thus, the mapping f is a -contraction on S with and .
3 Main results
In this section, we will show -contraction is continuous. By using this assumption, we will also prove two theorems.
Definition 3.1 Let F be a probabilistic distance on S. A mapping is called continuous if for every there exists such that
Before we start to present the theorems, we will explain the following lemma.
Lemma 3.1 Every -contraction is continuous.
Proof Suppose that be given and be such that and since ψ is increasing bijection then . If then, by -contraction we have , from where we obtain that . So f is continuous. □
Theorem 3.1 Let be a complete Menger space and T a t-norm satisfies in . Also, a -contraction where for every . If for some and all , then there exists a unique fixed point x of the mapping f and .
Proof Let , . We shall prove that is a Cauchy sequence.
Let , , . Since , it follows that for every there exists such that and by induction for all . By choosing n such that and , we obtain
Hence, is a Cauchy sequence and since S is complete, it follows the existence of such that . By continuity of f and for every , when , we obtain that . □
Example 3.1 Let be a complete Menger space where , and is defined as
is given by and . If we take , , then f is a -contraction where for every and if we set , then for all , we have and , so is the unique fixed point for f.
Theorem 3.2 Let be a complete Menger space, T be a t-norm such that and a -contraction where the series is convergent for all and suppose that for some and
If t-norm T is φ-convergent, then there exist a unique fixed point z of mapping f and .
Proof Choose and . Let , . We shall prove that is a Cauchy sequence. It means we prove that there exists such that
Suppose that , are such that
Let be such that
From (1), it follows that
Since f is -contraction, we derived by induction . Since the series is convergent, there exists such that . We know for every .
Since T is φ-convergent, we conclude that is a Cauchy sequence. On the other hand, S is complete, therefore, there is a such that . By the continuity of the mapping f and when , it follows that . □
Example 3.2 Let and the mappings f, ψ and φ be the same as in Example 3.1. Since for all and if we set , then for every or , so is the unique fixed point for f.
Mihet in  showed, if is a -contraction and is a complete fuzzy metric space, then f has an unique fixed point. Now we present a generalization of the -contraction. First, we define the class of functions ℵ as follows.
Let ℵ be the family of all the mappings such that the following conditions are satisfied:
m is continuous.
Definition 3.2 Let be a probabilistic metric space and . The mapping f is a generalized -contraction if there exist a continuous, decreasing function such that , , and such that the following implication holds for every and for every :
If , and for every , we obtain the Mihet definition.
Theorem 3.3 Let be a complete Menger space with t-norm T such that and be a generalized -contraction such that ψ is continuous on and for every . Suppose that there exists such that and φ, ψ satisfy . Then is the unique fixed point of the mapping f for an arbitrary .
Proof First we shall prove that f is uniformly continuous. Let and . We have to prove that there exists such that
Let ϵ be such that and such that
Since and are continuous at zero, and such numbers ϵ and λ exist. We prove that , . If , we have
Since h is decreasing, it follows that . Hence,
Using (2), we conclude that
and since h is decreasing we have
Therefore, if . We prove that for every and there exists such that for every
By assumption, there is a such that . From , it follows that
which implies that , and continuing in this way we obtain that for every
Let be a natural number such that and , for every . Then implies that
If , from (3) we obtain that
Relation (4) means that is a Cauchy sequence, and since S is complete there exists , which is obviously a fixed point of f since f is continuous.
For every and such that and we have for every that , and, therefore, from (3) we have for every and . This implies that for every and, therefore, . □
Example 3.3 Let and the mappings f, ψ and φ be the same as in Example 3.1. Set for every and . The mapping f is generalized -contraction and for every . On the other hand, there exists such that and . So is the unique fixed point for f.
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The authors thank the reviewers for their useful comments.
The authors declare that they have no competing interests.
PA defined the definitions and wrote the Introduction, preliminaries and abstract. AB proved the theorems. AB has approved the final manuscript. Also, PA has verified the final manuscript.
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Beitollahi, A., Azhdari, P. -Contraction theorems in probabilistic metric spaces for single valued case. Fixed Point Theory Appl 2013, 109 (2013). https://doi.org/10.1186/1687-1812-2013-109
- probabilistic metric spaces