On fixed point theorems involving altering distances in Menger probabilistic metric spaces
 Tatjana Došenović^{1},
 Poom Kumam^{2}Email author,
 Dhananjay Gopal^{3},
 Deepesh Kumar Patel^{3} and
 Aleksandar Takači^{1}
https://doi.org/10.1186/1029242X2013576
© Došenović et al.; licensee Springer. 2013
Received: 26 August 2013
Accepted: 11 November 2013
Published: 6 December 2013
Abstract
In this paper, we show by means of an example that the results of Babačev (Appl. Anal. Discrete Math. 6:257264, 2012) do not hold for the class of tnorms $T\le {T}_{p}$. Further, we prove a fixed point theorem for quasitype contraction involving altering distance functions which is weaker than that proposed by Babačev but for any continuous tnorm in a complete Menger space.
MSC:47H10, 54H25.
Keywords
1 Introduction
Probabilistic metric spaces (in short PMspaces) 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 nonnegative real numbers. The study of PMspaces 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 PMspaces was given by Sehgal and BharuehaReid [3] wherein the notion of probabilistic Bcontraction was introduced and a generalization of the classical Banach fixed point principle to complete Menger PMspaces was given. In [3], it was proved that any Bcontraction on a complete Menger space $(S,F,{T}_{M})$, where tnorm ${T}_{M}$ is defined by ${T}_{M}(x,y)=min\{x,y\}$, has a unique fixed point. In 1982, Hadžić [4] extended the result contained in [3] for a more general class of tnorms called Htype tnorms (see also [5]).
After that several types of contractions and associated fixed point theorems have been established in PMspaces 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 PMspaces. In [14], it was proved that any probabilistic ϕcontraction on a complete Menger space $(S,F,{T}_{M})$ 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 tnorm. (However, in [16] Miheţ gave an affirmative answer to the question raised in [14] using the idea of probabilistic boundedness and Htype tnorms 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 tnorm 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 tnorms $T\le {T}_{p}$. Further, we prove a fixed point theorem for quasitype contraction involving altering distance functions in a complete Menger space for any continuous tnorm 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.
Definition 2.1 [2]
 (a)
T is commutative and associative,
 (b)
T is continuous,
 (c)
$T(a,1)=a$ for all $a\in [0,1]$,
 (d)
$T(a,b)\le T(c,d)$ whenever $a\le c$ and $b\le d$, and $a,b,c,d\in [0,1]$.
 (i)
The minimum tnorm, ${T}_{M}$, is defined by ${T}_{M}(a,b)=min\{a,b\}$.
 (ii)
The product tnorm, ${T}_{p}$, is defined by ${T}_{p}(a,b)=a\cdot b$.
 (iii)
The Lukasiewicz tnorm, ${T}_{L}$, is defined by ${T}_{L}(a,b)=max\{a+b1,0\}$.
Definition 2.2 A Menger probabilistic metric space (briefly, Menger PMspace) is a triple $(X,F,T)$, where X is a nonempty set, T is a continuous tnorm, and F is a mapping from $X\times X$ into ${\mathcal{D}}^{+}$ such that if ${F}_{xy}$ denotes the value of F at the pair $(x,y)$, the following conditions hold:
(PM1) ${F}_{xy}(t)={\u03f5}_{0}(t)$ if and only if $x=y$,
(PM2) ${F}_{xy}(t)={F}_{yx}(t)$,
(PM3) ${F}_{xy}(t+s)\ge T({F}_{xz}(t),{F}_{zy}(s))$ for all $x,y,z\in X$ and $s,t\ge 0$.
Remark 2.1 [3]
Every metric space is a PMspace. Let $(X,d)$ be a metric space and $T(a,b)=min\{a,b\}$ be a continuous tnorm. Define ${F}_{xy}(t)={\u03f5}_{0}(td(x,y))$ for all $x,y\in X$ and $t>0$. The triple $(X,F,T)$ is a PMspace induced by the metric d.
 (1)
A sequence ${\{{x}_{n}\}}_{n}$ in X is said to be convergent to x in X if for every $\u03f5>0$ and $\lambda >0$, there exists a positive integer N such that ${F}_{{x}_{n}x}(\u03f5)>1\lambda $ whenever $n\ge N$.
 (2)
A sequence ${\{{x}_{n}\}}_{n}$ in X is called a Cauchy sequence if for every $\u03f5>0$ and $\lambda >0$, there exists a positive integer N such that ${F}_{{x}_{n}{x}_{m}}(\u03f5)>1\lambda $ whenever $n,m\ge N$.
 (3)
The space X is said to be complete if every Cauchy sequence in X is convergent to a point in X.
The $(\u03f5,\lambda )$topology is a Hausdorff topology. In this topology the function f is continuous in ${x}_{0}\in X$ if and only if for every sequence ${x}_{n}\to {x}_{0}$ it holds that $f({x}_{n})\to f({x}_{0})$.
Definition 2.4 (Altering distance function [15])
 (i)
ψ is monotone increasing and continuous,
 (ii)
$\psi (t)=0$ if and only if $t=0$.
The following category of functions was introduced in [14].
Definition 2.5 [14]
 (i)
$\varphi (t)=0$ if and only if $t=0$,
 (ii)
$\varphi (t)$ is strictly monotone increasing and $\varphi (t)\to \mathrm{\infty}$ as $t\to \mathrm{\infty}$,
 (iii)
ϕ is left continuous in $(0,\mathrm{\infty})$,
 (iv)
ϕ is continuous at 0.
The class of all Φfunctions will be denoted by Φ.
Lemma 2.1 [17]
Let $(X,F,T)$ be a Menger PMspace. Let $\varphi :[0,\mathrm{\infty})\to [0,\mathrm{\infty})$ be a Φfunction. Then the following statement holds.
If for $x,y\in X$, $0<c<1$, we have ${F}_{xy}(\varphi (t))\ge {F}_{xy}(\varphi (t/c))$ for all $t>0$, then $x=y$.
Theorem 2.1 [17]
holds for every $x,y\in X$ and all $t>0$, then f has a unique fixed point in X.
3 Main results
We begin with the following example.
Then it can be easily seen that the above example satisfies Theorem 2.1 for ${T}_{p}$. Indeed,

Case I. If $x=y$, then inequality (2.1) is obviously true.

Case II. If $x\ne y$ and $x<y$, then we have $x+1<y+1$, $x<x+1$, $y<y+1$, $x<y+1$.
Clearly, the inequality holds with the minimum value $x/(y+1)$.
Then the inequality holds with the minimum value $x/(y+1)$.
The inequality holds with the minimum value $x/(y+1)$.

Case III. If $x\ne y$ and $x>y$, then we have $x+1>y+1$, $x+1>y$.
The inequality holds with the minimum value $y/(x+1)$.
The inequality holds with the minimum value $y/(x+1)$.
Again inequality holds with the minimum value $y/(x+1)$.
Thus the above example satisfies all the conditions of Theorem 2.1 with the tnorm ${T}_{p}$, but here the mapping f has no fixed point. Therefore, Theorem 2.1 cannot be generalized for $T\le {T}_{p}$.
Now, we are motivated to introduce our result.
holds for every $x,y\in X$ and all $t>0$, then f has a unique fixed point in X.
for all $t>0$.
for all $t>0$.
So, we get a contradiction.
Accordingly, $min\{{F}_{{x}_{n1}{x}_{n}}(\varphi (p/{c}^{2})),{F}_{{x}_{n}{x}_{n+1}}(\varphi (p/{c}^{2}))\}={F}_{{x}_{n}{x}_{n+1}}(\varphi (p/{c}^{2}))$.
as $n\to \mathrm{\infty}$.
for all $s>0$.
for all $p\in X$, $\lambda >0$, and $0<{\u03f5}^{\u2033}<\u03f5$. It follows that whenever the above construction is possible for $\u03f5>0$, $\lambda >0$, it is also possible to construct $\{{x}_{m(k)}\}$ and $\{{x}_{n(k)}\}$ satisfying (3.6) and (3.7) corresponding to ${\u03f5}^{\u2033}>0$, $\lambda >0$ whenever ${\u03f5}^{\u2033}<\u03f5$.
which is a contradiction. Therefore $\{{x}_{n}\}$ is a Cauchy sequence in a complete Menger PMspace X, thus there exists $z\in X$ such that $z={lim}_{n\to \mathrm{\infty}}{x}_{n}$.
and applying Lemma 2.1, we get $z=fz$.
From Lemma 2.1, it follows that $z=w$, i.e., z is the unique fixed point of f. □
4 Connection with metric spaces
is a Φfunction (see [18]).
$c\in (0,1)$ and $x,y\in X$ in a metric space.
and we get a contradiction.
Declarations
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 MNTRRS174009. The third author gratefully acknowledges the support from the CSIR, Govt. of India, Grant No.25(0215)/13/EMRII. The fourth author is thankful to S. V. National Institute of Technology, Surat, India for awarding Senior Research Fellow.
Authors’ Affiliations
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