-Contraction theorems in probabilistic metric spaces for single valued case
© Beitollahi and Azhdari; licensee Springer 2013
Received: 20 April 2012
Accepted: 9 April 2013
Published: 23 April 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 ).
for every (),
for every ,
and for every , and every .
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 .
Also, note that if the t-norm T is q-convergent, then .
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 
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 , , .
A probabilistic metric space is called sequentially complete if every Cauchy sequence is convergent.
The concept of -contraction has been introduced by Mihet .
φ is an increasing bijection;
Since every such a comparison mapping is continuous, if , then .
Definition 2.5 
In the rest of paper we suppose that ψ is increasing bijection.
Suppose that , .
Then is a probabilistic metric space .
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.
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.
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 . □
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.
If t-norm T is φ-convergent, then there exist a unique fixed point z of mapping f and .
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.
m is continuous.
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 .
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.
The authors thank the reviewers for their useful comments.
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