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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)
Abstract
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.
1 Introduction
The probabilistic metric space was introduced by Menger [1]. 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 [4]. We defined the class of -contraction for the multi-valued case in a probabilistic metric space before [5]. 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
We recall some concepts from probabilistic metric space, convergence and contraction. For more details, we refer the reader to [6–8].
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:
-
(1)
for every (),
-
(2)
for every ,
-
(3)
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 [9], a subset O of S is F-open if for every there exists such that .
Definition 2.1 [6]
A sequence is called an F-convergent sequence to if for every and there exists such that , .
Definition 2.2 [6]
Let be a mapping, we say that the t-norm T is φ-convergent if
Definition 2.3 [6]
A sequence is called a convergent sequence to if for every and there exists such that , .
Definition 2.4 [6]
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 [3].
We will consider comparison functions from the class ϕ of all mapping with the properties:
-
(1)
φ is an increasing bijection;
-
(2)
.
Since every such a comparison mapping is continuous, if , then .
Definition 2.5 [3]
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 [10].
Let x, y, ϵ, λ be such that .
-
(i)
If , then .
This implies , that is,
-
(ii)
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
and
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 .
Now
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 [3] 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:
-
(1)
;
-
(2)
;
-
(3)
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.
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Authors’ contributions
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
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DOI: https://doi.org/10.1186/1687-1812-2013-109