Further generalization of fixed point theorems in Menger PMspaces
 Marwan Amin Kutbi^{1},
 Dhananjay Gopal^{2},
 Calogero Vetro^{3} and
 Wutiphol Sintunavarat^{4}Email author
https://doi.org/10.1186/s1366301502794
© Kutbi et al.; licensee Springer. 2015
Received: 28 September 2014
Accepted: 4 February 2015
Published: 27 February 2015
Abstract
In this work, we establish some fixed point theorems by revisiting the notion of ψcontractive mapping in Menger PMspaces. One of our results (namely, Theorem 2.3) may be viewed as a possible answer to the problem of existence of a fixed point for generalized type contractive mappings in Mcomplete Menger PMspaces under arbitrary tnorm. Some examples are furnished to demonstrate the validity of the obtained results.
Keywords
MSC
1 Introduction and preliminaries
In 1942 Menger [1] initiated the study of probabilistic metric spaces; see also [2–4]. Successively, Sehgal and BharuchaReid [5, 6] established fixed point theorems in probabilistic metric spaces (for short, PMspaces). Indeed, by using the notion of probabilistic qcontraction, they proved a unique fixed point result, which is an extension of the celebrated Banach contraction principle [7]. For the interested reader, a comprehensive study of fixed point theory in the probabilistic metric setting can be found in the book of Hadžić and Pap [8], see also [9] for further discussion on generalizations of metric fixed point theory. Recently, Choudhury and Das [10] gave a generalized unique fixed point theorem by using an altering distance function, which was originally introduced by Khan et al. [11]. For other results in this direction, we refer to [12–19]. In particular, Dutta et al. [20] defined nonlinear generalized contractive type mappings involving altering distances (say, ψcontractive mappings) in Menger PMspaces and proved their theorem for such kind of mappings in the setting of Gcomplete Menger PMspaces.
On contributing to this study, we weaken the notion of ψcontractive mapping and establish some fixed point theorems in Gcomplete and Mcomplete Menger PMspaces, besides discussing some related results and illustrative examples. Indeed, we not only derive the result of Dutta et al. [20], Theorem 12, as a particular case of our result, but also we notice that our Theorem 2.3 may be viewed as a possible answer to the problem of existence of a fixed point for generalized type contractive mappings in Mcomplete Menger PMspaces under arbitrary tnorm.
Here, we state some allied definitions and results which are needed for the development of the present topic. We denote by ℝ the set of real numbers, by \(\mathbb{R}^{+}\) the set of nonnegative real numbers and by ℕ the set of positive integers.
Definition 1.1
A mapping \(F : \mathbb{R} \to\mathbb{R}^{+}\) is called a distribution function if it is nondecreasing and left continuous with \(\inf_{t \in \mathbb{R}}F(t) = 0\) and \(\sup_{t \in\mathbb{R}} F(t)= 1\).
Definition 1.2
([21])
 (a)
T is commutative and associative,
 (b)
T is continuous,
 (c)
\(T(a,1) = a\) for all \(a \in[0,1]\),
 (d)
\(T(a,b) \leq T(c,d)\), whenever \(a \leq c\) and \(b\leq d\), for \(a,b,c,d \in[0,1]\).
 (i)
The minimum tnorm, say \(T_{M}\), defined by \(T_{M}(a,b) = \min\{a,b\}\).
 (ii)
The product tnorm, say \(T_{p}\), defined by \(T_{p}(a,b) = a\cdot b\).
 (iii)
The Lukasiewicz tnorm, say \(T_{L}\), defined by \(T_{L}(a,b) = \max\{a+b1, 0\}\).
Definition 1.3
 (PM1)
\(F_{x,y}(t) = H(t)\) if and only if \(x =y\) for all \(t \in\mathbb{R}^{+}\),
 (PM2)
\(F_{x,y}(t) = F_{y,x}(t)\) for all \(x,y \in X\) and \(t \in\mathbb{R}^{+}\),
 (PM3)
\(F_{x,y}(t+s) \geq T (F_{x,z}(t), F_{z,y}(s) )\) for all \(x,y,z \in X\) and \(s,t \in\mathbb{R}^{+}\).
Definition 1.4
 (i)
A sequence \(\{x_{n}\}\) in X is said to be convergent to \(x \in X\) if, for every \(\epsilon> 0\) and \(\lambda> 0\), there exists a positive integer N such that \(F_{x_{n},x}(\epsilon) > 1 \lambda\) whenever \(n \geq N\).
 (ii)
A sequence \(\{x_{n}\}\) in X is called Cauchy sequence if, for every \(\epsilon> 0\) and \(\lambda> 0\), there exists a positive integer N such that \(F_{x_{n},x_{m}}(\epsilon) > 1 \lambda\) whenever \(n,m \geq N\).
 (iii)
A Menger PMspace is said to be Mcomplete if every Cauchy sequence in X is convergent to a point in X.
 (iv)
A sequence \(\{x_{n}\}\) is called GCauchy if \(\lim _{n \to\infty}F_{x_{n},x_{n+m}}(t) = 1\) for each \(m \in\mathbb {N}\) and \(t >0\).
 (v)
The space \((X, F, T)\) is called Gcomplete if every GCauchy sequence in X is convergent.
The following class of functions was introduced in [10] and will be used in proving our results in the next section.
Definition 1.5
([10])
 (i)
\(\phi(t) = 0\) if and only if \(t=0\),
 (ii)
\(\phi(t)\) is strictly increasing and \(\phi(t) \to \infty\) as \(t \to\infty\),
 (iii)
ϕ is left continuous in \((0, \infty)\),
 (iv)
ϕ is continuous at 0.
Definition 1.6
([22])
In the sequel, the class of all ϕfunctions will be denoted by Φ. Also we denote by Ψ the class of all continuous nondecreasing functions \(\psi: \mathbb{R}^{+} \to\mathbb{R}^{+}\) such that \(\psi(0) = 0\) and \(\psi^{n}(a_{n}) \to0\), whenever \(a_{n} \to0\), as \(n \to\infty\).
We conclude this section recalling the following fixed point theorem of Dutta et al., see [20], which is the main inspiration of our paper.
Theorem 1.1
A mapping \(f : X \to X\) satisfying condition (1.1) is usually called ψcontractive mapping. However, for some discussion on this notion and Theorem 1.1, the reader can refer to the recent paper of Gopal et al. [23], where analogous results are proved by using some control functions.
2 Main results
In this section, firstly we weaken the class of functions Ψ by assuming the continuity only at point \(t=0\). Precisely, we denote by \(\Psi_{0}\) the class of all nondecreasing functions \(\psi: \mathbb {R}^{+} \to\mathbb{R}^{+}\) such that ψ is continuous at 0, \(\psi(0) = 0\) and \(\psi^{n}(a_{n}) \to0\) whenever \(a_{n} \to0\) as \(n \to\infty\); then we utilize this class to prove some fixed point theorems. We start with a revised version of Theorem 1.1 useful to obtaining an affirmative answer to an existence problem of a fixed point in a Gcomplete Menger space.
Theorem 2.1
Proof
Let \(x_{0} \in X\). Define a sequence \(\{x_{n}\}\) in X so that \(x_{n+1} = fx_{n}\) for all \(n \in\mathbb{N}\cup\{0\}\). We suppose \(x_{n+1} \neq x_{n}\) for all \(n \in\mathbb{N}\), otherwise f has trivially a fixed point.
Now we show that u is a fixed point of f.
The following example illustrates our Theorem 2.1.
Example 2.1
 (∗):

\(F_{u,v}(0)=0\) if \(u,v \in \operatorname{Fix}(f)\), where \(\operatorname{Fix}(f)\) denotes the set of all fixed points of a mapping f, that is, \(\operatorname{Fix}(f):=\{x \in X : x=fx\}\).
Theorem 2.2
Adding condition (∗) to the hypotheses of Theorem 2.1, we obtain uniqueness of the fixed point.
Proof
It follows that \(F_{u,v}(t) =H(t)\) for all \(t>0\). In fact, if t is not in range of ϕ, since ϕ is continuous at 0, then there exists \(s>0\) such that \(\phi(s)< t\). This implies \(F_{u,v}(t) \geq F_{u,v}(\phi(s))=1\), yielding thereby \(u =v\). □
Our next step is to furnish a fixed point theorem in an Mcomplete Menger PMspace.
Theorem 2.3
Let \((X,F,T)\) be an Mcomplete Menger PMspace and \(f : X\to X\) be a ψcontractive mapping, where the function \(\psi: \mathbb{R}^{+} \to\mathbb{R}^{+}\) is nondecreasing, continuous at 0, \(\psi(0) = 0\) and \(\sum_{n=1}^{\infty}\psi^{n}(a_{n}) < \infty\), whenever \(a_{n} \to0\) as \(n \to\infty\). Then f has a fixed point provided that F is triangular.
Proof
Clearly, on the same lines of Theorem 2.2 one can solve the uniqueness problem of a fixed point in an Mcomplete Menger space. To avoid repetition, we give the statement of this theorem without the proof.
Theorem 2.4
Adding condition (∗) to the hypotheses of Theorem 2.3, we obtain uniqueness of the fixed point.
Remark 2.1
Our Theorem 2.3 is proved in an Mcomplete Menger PMspace under arbitrary tnorm, therefore Theorem 2.3 can be realized as a possible answer to the problem of existence of a fixed point for generalized type contractive mappings in Menger PMspaces.
As an application of Theorems 2.1 and 2.2, we prove the following common fixed point theorem for a finite family of mappings which runs as follows.
Theorem 2.5
Let \((X,F,T)\) be a Gcomplete Menger PMspace, \(\{f_{i}\}_{1}^{m} \) be a finite family of selfmappings defined on X and denote \(f=f_{1}f_{2}f_{3}\cdots f_{m}\). If \(f:X\rightarrow X\) satisfies all the hypotheses of Theorem 2.2, then the family \(\{f_{i}\}_{1}^{m} \) has a unique common fixed point provided that \(f_{i}f_{j}=f_{j}f_{i}\) whenever \(i\neq j\), with \(i,j\in\{ 1,2,\ldots,m\}\).
Proof
By setting \(f_{1}=f_{2}= \cdots=f_{m}=g\) in Theorem 2.5, we deduce the following fixed point theorem for mth iterates of a mapping g.
Corollary 2.1
Let \((X,F,T)\) be a Gcomplete Menger PMspace and \(g:X\rightarrow X\) be a mapping such that \(g^{m}\) satisfies all the hypotheses of Theorem 2.2. Then g has a unique fixed point.
Remark 2.2
Results similar to Theorem 2.5 and Corollary 2.1 can be outlined in respect of Theorems 2.3 and 2.4.
Finally, by using the following example, we show that Corollary 2.1 can be situationally more useful than Theorems 2.1 and 2.2.
Example 2.2
Thus we conclude that f does not meet the requirements of Theorem 2.1, whereas the power mapping \(f^{2}\) satisfies all the conditions of Corollary 2.1 substantiating the utility of Corollary 2.1 (and hence Theorem 2.5) over Theorem 2.1.
Declarations
Acknowledgements
We would like to acknowledge the anonymous reviewers for their helpful comments and detailed suggestions. The first author gratefully acknowledges the support from the Deanship of Scientific Research (DSR) at King Abdulaziz University (KAU) during this research. The fourth author would like to thank the Thailand Research Fund and Thammasat University under Grant No. TRG5780013 for financial support during the preparation of this manuscript.
Open Access This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly credited.
Authors’ Affiliations
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