Abstract
In this paper, we show by means of an example that the results of Babačev (Appl. Anal. Discrete Math. 6:257-264, 2012) do not hold for the class of t-norms . Further, we prove a fixed point theorem for quasi-type contraction involving altering distance functions which is weaker than that proposed by Babačev but for any continuous t-norm in a complete Menger space.
MSC:47H10, 54H25.
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1 Introduction
Probabilistic metric spaces (in short PM-spaces) are a probabilistic generalization of metric spaces which are appropriate to carry out the study of those situations wherein distances are measured in terms of distribution functions rather than non-negative real numbers. The study of PM-spaces was initiated by Menger [1]. Schweizer and Sklar [2] further enriched this concept and provided a new impetus by proving some fundamental results on this theme.
The first result on fixed point theory in PM-spaces was given by Sehgal and Bharueha-Reid [3] wherein the notion of probabilistic B-contraction was introduced and a generalization of the classical Banach fixed point principle to complete Menger PM-spaces was given. In [3], it was proved that any B-contraction on a complete Menger space , where t-norm is defined by , has a unique fixed point. In 1982, Hadžić [4] extended the result contained in [3] for a more general class of t-norms called H-type t-norms (see also [5]).
After that several types of contractions and associated fixed point theorems have been established in PM-spaces by various authors, e.g., [6–8] (see also [9–12]). We also refer to a nice book on this topic by Hadžić and Pap [13]. In this continuation, Choudhury and Das [14] extended the classical metric fixed point result of Khan et al. [15] by introducing the idea of altering distance functions in PM-spaces. In [14], it was proved that any probabilistic ϕ-contraction on a complete Menger space has a unique fixed point.
An open problem that remains to be investigated is whether the results are valid in the cases of any arbitrary continuous t-norm. (However, in [16] Miheţ gave an affirmative answer to the question raised in [14] using the idea of probabilistic boundedness and H-type t-norms along with some additional conditions.) Very recently, Babačev in [17] extended and improved the results of Choudhury and Das [14] for a nonlinear generalized contraction wherein she used the associated t-norm as a min norm.
In this paper, we show by means of an example that Babačev’s [17] results do not hold for the class of t-norms . Further, we prove a fixed point theorem for quasi-type contraction involving altering distance functions in a complete Menger space for any continuous t-norm T.
2 Preliminaries
Consistent with Choudhury and Das [14], Choudhury et al. [18] and Babačev [17], the following definitions and results will be needed in the sequel.
In the standard notation, let be the set of all distribution functions such that F is a nondecreasing, left-continuous mapping which satisfies and . The space is partially ordered by the usual point-wise ordering of functions, i.e., if and only if for all . The maximal element for in this order is the distribution function given by
Definition 2.1 [2]
A binary operation is a continuous t-norm if it satisfies the following conditions:
-
(a)
T is commutative and associative,
-
(b)
T is continuous,
-
(c)
for all ,
-
(d)
whenever and , and .
The following are the three basic continuous t-norms:
-
(i)
The minimum t-norm, , is defined by .
-
(ii)
The product t-norm, , is defined by .
-
(iii)
The Lukasiewicz t-norm, , is defined by .
Regarding the pointwise ordering, the following inequalities hold:
Definition 2.2 A Menger probabilistic metric space (briefly, Menger PM-space) is a triple , where X is a nonempty set, T 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:
(PM1) if and only if ,
(PM2) ,
(PM3) for all and .
Remark 2.1 [3]
Every metric space is a PM-space. Let be a metric space and be a continuous t-norm. Define for all and . The triple is a PM-space induced by the metric d.
Definition 2.3 Let be a Menger PM-space.
-
(1)
A sequence in X is said to be convergent to x in X if for every and , there exists a positive integer N such that whenever .
-
(2)
A sequence in X is called a Cauchy sequence if for every and , there exists a positive integer N such that whenever .
-
(3)
The space X is said to be complete if every Cauchy sequence in X is convergent to a point in X.
The -topology [2] in a Menger space is introduced by the family of neighborhoods of a point given by
where
The -topology is a Hausdorff topology. In this topology the function f is continuous in if and only if for every sequence it holds that .
Definition 2.4 (Altering distance function [15])
The control function is called an altering distance function if it has the following properties:
-
(i)
ψ is monotone increasing and continuous,
-
(ii)
if and only if .
The following category of functions was introduced in [14].
Definition 2.5 [14]
A function is said to be a Φ-function if it satisfies the following conditions:
-
(i)
if and only if ,
-
(ii)
is strictly monotone increasing and as ,
-
(iii)
ϕ is left continuous in ,
-
(iv)
ϕ is continuous at 0.
The class of all Φ-functions will be denoted by Φ.
Lemma 2.1 [17]
Let be a Menger PM-space. Let be a Φ-function. Then the following statement holds.
If for , , we have for all , then .
Theorem 2.1 [17]
Let be a complete Menger PM-space with a continuous t-norm T which satisfies for every . Let be fixed. If for a Φ-function ϕ and a self-mapping f on X,
holds for every and all , then f has a unique fixed point in X.
3 Main results
We begin with the following example.
Example 3.1 Let and . For each , define
It is clear that is a complete Menger PM-space (see [19]) (but here is not a PM-space). Let us consider the function
Then it can be easily seen that the above example satisfies Theorem 2.1 for . Indeed,
-
Case I. If , then inequality (2.1) is obviously true.
-
Case II. If and , then we have , , , .
For any function and , inequality (2.1) becomes
If , then we have
Clearly, the inequality holds with the minimum value .
If , then we have
Then the inequality holds with the minimum value .
And if , then we have
The inequality holds with the minimum value .
-
Case III. If and , then we have , .
By inequality (2.1), we have
If , then we have
The inequality holds with the minimum value .
If , then we have
The inequality holds with the minimum value .
Finally, if , then we have
Again inequality holds with the minimum value .
Thus the above example satisfies all the conditions of Theorem 2.1 with the t-norm , but here the mapping f has no fixed point. Therefore, Theorem 2.1 cannot be generalized for .
Now, we are motivated to introduce our result.
Theorem 3.1 Let be a complete Menger PM-space with a continuous t-norm, and let be fixed. If for a Φ-function ϕ and a self-mapping f on X,
holds for every and all , then f has a unique fixed point in X.
Proof Let . Now, construct a sequence in X as follows:
Applying (3.1) for and , we have
for all .
We should prove that
for all .
If it is not, there exists such that
Then, using (3.1), we have
If
then by (3.3) and (3.4)
So, we get a contradiction.
Accordingly, .
Now, we have
Repeating the same procedure, we conclude that
Since , , we get a contradiction. Accordingly, (3.2) is true. Therefore,
as .
By the property of ϕ, given , there exists such that . Thus,
for all .
Now, we claim that is a Cauchy sequence. If not, then and , and subsequences and such that and
Since F is non-decreasing, we have
for all , , and . It follows that whenever the above construction is possible for , , it is also possible to construct and satisfying (3.6) and (3.7) corresponding to , whenever .
Since ϕ is continuous at 0 and strictly monotonic increasing with , it is possible to obtain such that . Then, by the above argument, it is possible to obtain increasing sequences of integers and with such that
Since and , we can choose such that . Since ϕ is strictly increasing, therefore
From (3.9), we get
By (3.5), for , it is possible to find a positive integer such that for all ,
By (PM3), we have
Let be arbitrary. Then by (3.5) there exists a positive integer such that for all ,
Now, using (3.10), (3.12), and (3.13), we have, for all ,
As is arbitrary and T is continuous, we have
Now, using (3.1), (3.9), (3.11), and (3.14), we have
which is a contradiction. Therefore is a Cauchy sequence in a complete Menger PM-space X, thus there exists such that .
Now, we will show that z is a fixed point of f. Since , we have that for every and all , there exists such that and such that for all ,
Since , thus . Also, since , for arbitrary , we have
Hence, from (3.15) and (3.16), we get
Since is arbitrary and the t-norm T is continuous, we get
Letting in the above inequality and using the fact that the t-norm T is continuous, we obtain
and applying Lemma 2.1, we get .
Next, we prove the uniqueness of a fixed point. Let be another fixed point of f, i.e., . Since , for all , there exists such that . Then we have
From Lemma 2.1, it follows that , i.e., z is the unique fixed point of f. □
4 Connection with metric spaces
It is well known that every metric space is also a Menger space if F is defined in the following way:
If ψ is an altering distance function defined in Definition 2.4 with additional property as , then the function
is a Φ-function (see [18]).
We will present that (3.1) in this case implies
and in a metric space.
Suppose the contrary, i.e., there exists such that and all of
So, implies that , and since ψ is continuous, we have
Similarly, since , we have that , which implies
Also, we have the following:
Thus, we have
and we get a contradiction.
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Acknowledgements
The authors would like to express their thanks to the referees for their helpful comments and suggestions. The first and fifth author are supported by MNTRRS-174009. The third author gratefully acknowledges the support from the CSIR, Govt. of India, Grant No.-25(0215)/13/EMR-II. The fourth author is thankful to S. V. National Institute of Technology, Surat, India for awarding Senior Research Fellow.
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Došenović, T., Kumam, P., Gopal, D. et al. On fixed point theorems involving altering distances in Menger probabilistic metric spaces. J Inequal Appl 2013, 576 (2013). https://doi.org/10.1186/1029-242X-2013-576
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DOI: https://doi.org/10.1186/1029-242X-2013-576