 Research
 Open Access
φContraction in generalized probabilistic metric spaces
 Saud M Alsulami^{1}Email author,
 Binayak S Choudhury^{2} and
 Pradyut Das^{2}
https://doi.org/10.1186/s1366301503675
© Alsulami et al. 2015
 Received: 22 December 2014
 Accepted: 25 June 2015
 Published: 22 August 2015
Abstract
We use the gauge function introduced by Fang to gain a fixed point result in probabilistic Gmetric spaces. Our work extends some existing results. Moreover, our result is supported with an example.
Keywords
 generalized PMspace
 Hadz̆ić type tnorm
 φfunction
 Cauchy sequence
 fixed point
MSC
 47H10
 54H25
1 Introduction
Probabilistic fixed point theory originated in the work of Sehgal and BharuchaReid [1] where they introduced a contraction mapping principle in probabilistic metric spaces. After that this line of research has developed through the works of different authors over the years. A comprehensive description of this development is given in the book of Hadz̆ić and Pap [2]. Some more recent references are noted in [2–9].
Metric spaces are generalized in different ways creating spaces like fuzzy metric spaces, 2metric spaces, etc. Generalized metric space is one of such spaces in which to every triple of points a nonnegative real number is attached. Such spaces were introduced by Mustafa and Sims [10]. Fixed point theory and related topics have been developed through a good number of works in recent times, some instances of which are in [11–14].
In the same vein probabilistic generalized metric spaces have been introduced by Zhou et al. [15] wherein they also proved certain primary fixed point results in these spaces. An interesting class of problems in probabilistic fixed point theory was addressed in recent times by use of gauge functions. These are control functions which have been used to extend the Sehgal contraction in probabilistic metric spaces. Some examples of these applications are in [3, 4, 7, 9, 16, 17]. One of such gauge functions was introduced by Fang [16]. Here we use the gauge function used by Fang [16] to obtain a fixed point result in probabilistic Gmetric spaces. Our result is supported with an example.
It is perceived that the study of fixed points for contractions defined by using control functions is an important category of problems in fixed point theory. One of the causes for this interest is the particularities involved in the proofs. With this motivation we work out the results in this paper.
2 Mathematical preliminaries
In this section we discuss certain definitions and lemmas which will be necessary for establishing the results of the next section.
Definition 2.1
[18]
A mapping \(F : R \to R^{+} \) is called a distribution function if it is nondecreasing and left continuous with \(\inf_{t \in R}F(t) = 0\) and \(\sup_{t\in R}F(t) = 1\), where R is the set of real numbers and \(R^{+}\) denotes the set of all nonnegative real numbers.
Definition 2.2
 (i)
Δ is associative and commutative,
 (ii)
\(\Delta (a,1)=a\) for all \(a\in[0,1]\),
 (iii)
\(\Delta (a,b)\leq \Delta (c,d)\) whenever \(a\leq c\) and \(b\leq d\) for all \(a,b,c,d\in[0,1]\),
 (iv)
Δ is continuous.
Definition 2.3
[18]
 (i)
\(F_{x,y}(0) = 0\) for all \(x,y \in X\),
 (ii)
\(F_{x,y}(s) = 1\) for all \(s>0\) if and only if \(x=y\),
 (iii)
\(F_{x,y}(s) = F_{y,x}(s)\) for all \(s >0\), \(x,y \in X\),
 (iv)
\(F_{x,y}(u+v)\geq \Delta (F_{x,z}(u), F_{z,y}(v))\) for all \(u,v \geq 0\) and \(x,y,z \in X\).
Definition 2.4
[15]
 (i)
\(F^{*}_{x,y,z}(t)=1\) if and only if \(x=y=z\),
 (ii)
\(F^{*}_{x,x,y}(t)\geq F^{*}_{x,y,z}(t)\) with \(y \neq z\),
 (iii)
\(F^{*}_{x,y,z}(t)= F^{*}_{p(x,y,z)}(t)\), where p is a permutation function,
 (iv)
\(\Delta (F^{*}_{x,a,a}(t), F^{*}_{a,y,z}(s))\leq F^{*}_{x,y,z}(t+s)\).
Example 2.5
[15]
For more examples of probabilistic Gmetric space refer to [15].
Definition 2.6
[15]
Lemma 2.7
[15]
If \(\epsilon _{1}\leq \epsilon _{2}\) and \(\lambda _{1}\leq \lambda _{2}\), then \(N_{x_{0}}(\epsilon _{1},\lambda _{1})\subseteq N_{x_{0}}(\epsilon _{2},\lambda _{2})\).
Theorem 2.8
[15]
Let \((X,F^{*},\Delta )\) be a probabilistic Gmetric space. Then \((X,F^{*},\Delta )\) is a Hausdorff space in the topology induced by the family \(\{N_{x_{0}}(\epsilon,\lambda)\}\) of \((\epsilon,\lambda)\)neighborhoods.
Definition 2.9
[15]
 (i)A sequence \(\{x_{n}\}\subset X\) is said to converge to a point \(x \in X\) if given \(\epsilon > 0\), \(\lambda > 0\) we can find a positive integer \(N_{\epsilon,\lambda}\) such that for all \(n > N_{ \epsilon,\lambda}\),$$F_{x,x_{n},x_{n}}(\epsilon) \geq 1 \lambda. $$
 (ii)A sequence \(\{x_{n}\}\) is said to be a Cauchy sequence in X if given \(\epsilon > 0\), \(\lambda > 0 \) there exists a positive integer \(N_{\epsilon, \lambda}\) such that$$F_{x_{n},x_{m},x_{l}}(\epsilon) \geq 1\lambda\quad \mbox{for all } m, n,l > N_{\epsilon, \lambda}. $$
 (iii)
A probabilistic Gmetric space \((X, F^{*}, \Delta)\) is said to be complete if every Cauchy sequence is convergent in X.
Definition 2.10
[2]
Definition 2.11
[16]

for each \(t>0\), there exists \(r\geq t\) such that \(\lim_{n\rightarrow \infty}\varphi^{n}(r) = 0\).
Lemma 2.12
[16]
Let \(\varphi \in \Phi_{w}\), then for each \(t>0\) there exists \(r\geq t\) such that \(\varphi(r)< t\).
3 Main results
Definition 3.1
Lemma 3.2
Proof
Corollary 3.3
[16]
Proof
Define \(F_{x,y,z}^{\ast }(t)=\min \{F_{x,y}(t),F_{y,z}(t),F_{x,z}(t)\}\) for all \(x,y,z\in X\) and \(t>0\). Then \((X,F^{\ast },\Delta )\) is a probabilistic Gmetric space.
Lemma 3.4
Proof
Theorem 3.5
Let \((X,F^{\ast },\Delta )\) be a PGMspace with a continuous tnorm Δ of Htype. If the mapping f is a probabilistic φcontraction, then f has a unique fixed point in X.
Proof
Now we prove that x is a fixed point of f.
Example 3.6
Now we show that f satisfies (3.1).
Declarations
Acknowledgements
This project was funded by the Deanship of Scientific Research (DSR), King Abdulaziz University, Jeddah, under Grant No. (319/130/1434). The authors, therefore, acknowledge with thanks DSR technical and financial support. The authors thank the learned referees for suggestions, which helped to improve the paper.
Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.
Authors’ Affiliations
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