# Common Fixed Point Theorems in Menger Probabilistic Quasimetric Spaces

- Shaban Sedghi
^{1}Email author, - Tatjana Žikić-Došenović
^{2}and - Nabi Shobe
^{3}

**2009**:546273

**DOI: **10.1155/2009/546273

© Shaban Sedghi et al. 2009

**Received: **21 November 2008

**Accepted: **19 April 2009

**Published: **1 June 2009

## Abstract

We consider complete Menger probabilistic quasimetric space and prove common fixed point theorems for weakly compatible maps in this space.

## 1. Introduction and Preliminaries

K. Menger introduced the notion of a probabilistic metric space in 1942 and since then the theory of probabilistic metric spaces has developed in many directions [1]. The idea of Menger was to use distribution functions instead of nonnegative real numbers as values of the metric. The notion of a probabilistic metric space corresponds to the situations when we do not know exactly the distance between two points, we know only probabilities of possible values of this distance. Such a probabilistic generalization of metric spaces appears to be well adapted for the investigation of physiological thresholds and physical quantities particularly in connections with both string and theory; see [2–5]. It is also of fundamental importance in probabilistic functional analysis, nonlinear analysis and applications [6–10].

Definition 1.1 (see [1]).

A mapping is if is satisfying the following conditions:

(a) is commutative and associative;

(b) for all ;

(d) whenever and , and .

The following are the four basic :

for and , for all .

We also mention the following families of

Definition 1.2.

There is a nice characterization of continuous of the class [12].

(i)If there exists a strictly increasing sequence in such that and , then is of Hadžić-type.

(ii)If is continuous and , then there exists a sequence as in (i).The is an trivial example of a of but there are of Hadžić-type with (see, e.g., [13]).

Definition 1.3 (see [13]).

If is a and , then is defined recurrently by 1, if and for all . If is a sequence of numbers from then is defined as (this limit always exists) and as . In fixed point theory in probablistic metric spaces there are of particular interest the -norms and sequences such that and . Some examples of with the above property are given in the following proposition.

- (i)
For the following implication holds:

- (ii)
If , then for every sequence in I such that , one has .

Note [14, Remark 13] that if is a for which there exists such that and , then Important class of is given in the following example.

- (i)
The Dombi family of is defined by

The Aczél-Alsina family of is defined by

Sugeno-Weber family of is defined by

In [13] the following results are obtained.

Proposition 1.6.

Definition 1.7.

A *Menger Probabilistic Quasimetric space* (briefly, Menger PQM space) is a triple
, where
is a nonempty set,
is a continuous
, and
is a mapping from
into
such that, if
denotes the value of
at the pair
, then the following conditions hold, for all
in
,

(PQM1) for all if and only if ;

(PQM2) for all and .

Definition 1.8.

Let be a Menger PQM space.

(1)A sequence
in
is said to be *convergent* to
in
if, for every
and
, there exists positive integer
such that
whenever
.

(2)A sequence in is called Cauchy sequence [15] if, for every and , there exists positive integer such that whenever ( ).

(3)A Menger PQM space is said to be complete if and only if every Cauchy sequence in is convergent to a point in .

In 1998, Jungck and Rhoades [16] introduced the following concept of weak compatibility.

Definition 1.9.

Let and be mappings from a Menger PQM space into itself. Then the mappings are said to be weak compatible if they commute at their coincidence point, that is, implies that .

## 2. The Main Result

where . It is easy to see that this condition implies .

Lemma 2.1.

for very , where is a monotone increasing functions.Then the sequence is a Cauchy sequence.

Proof.

for each and . Hence sequence is Cauchy sequence.

Theorem 2.2.

Let be a complete Menger PQM space and let be maps that satisfy the following conditions:

(a) ;

(b)the pairs and are weak compatible, is closed subset of ;

(c) for all and every , where is a monotone increasing function.

then and have a unique common fixed point.

Proof.

Similarly, we have

Hence, it follows that

for

Now by Lemma 2.1, is a Cauchy sequence. Since the space is complete, there exists a point such that

It follows that . Therefore, . That is is a common fixed point of and .

If and are two fixed points common to and , then

as , which implies that and so the uniqueness of the common fixed point.

Corollary 2.3.

Let be a complete Menger PQM space and let be maps that satisfy the following conditions:

(a) ;

(b)the pair is weak compatible, is closed subset of ;

(c) for all and where is monotone increasing function.

then and have a unique common fixed point.

Proof.

It is enough, set in Theorem 2.2.

Corollary 2.4.

Let be a complete Menger PQM space and let be maps that satisfy the following conditions:

(a)

(b)the pair is weak compatible, is closed subset of ;

- (d)(2.17)

then have a unique common fixed point.

Proof.

Hence, . Thus , .

Similarly, we have .

Corollary 2.5.

Let be a complete PQM space and let satisfy conditions (a), (b), and (c) of Theorem 2.2. If is a of then there exists a unique common fixed point for the mapping and .

Proof.

By Proposition 1.6 all the conditions of the Theorem 2.2 are satisfied.

Corollary 2.6.

Let for some be a complete PQM space and let satisfy conditions (a), (b), and (c) of Theorem 2.2. If then there exists a unique common fixed point for the mapping and .

Proof.

Corollary 2.7.

Let for some be a complete PQM space and let satisfy conditions (a), (b), and (c) of Theorem 2.2. If then there exists a unique common fixed point for the mapping and .

Proof.

Corollary 2.8.

Let for some be a complete PQM space and let satisfy conditions (a), (b), and (c) of Theorem 2.2. If then there exists a unique common fixed point for the mapping and .

Proof.

## Declarations

### Acknowledgment

The second author is supported by MNTRRS 144012.

## Authors’ Affiliations

## References

- Schweizer B, Sklar A:
*Probabilistic Metric Spaces, North-Holland Series in Probability and Applied Mathematics*. North-Holland, New York, NY, USA; 1983:xvi+275.Google Scholar - El Naschie MS:
**On the uncertainty of Cantorian geometry and the two-slit experiment.***Chaos, Solitons & Fractals*1998,**9**(3):517–529. 10.1016/S0960-0779(97)00150-1MathSciNetView ArticleMATHGoogle Scholar - El Naschie MS:
**A review of infinity theory and the mass spectrum of high energy particle physics.***Chaos, Solitons & Fractals*2004,**19**(1):209–236. 10.1016/S0960-0779(03)00278-9View ArticleMATHGoogle Scholar - El Naschie MS:
**On a fuzzy Kähler-like manifold which is consistent with the two slit experiment.***International Journal of Nonlinear Sciences and Numerical Simulation*2005,**6**(2):95–98. 10.1515/IJNSNS.2005.6.2.95View ArticleGoogle Scholar - El Naschie MS:
**The idealized quantum two-slit gedanken experiment revisited-Criticism and reinterpretation.***Chaos, Solitons & Fractals*2006,**27**(4):843–849. 10.1016/j.chaos.2005.06.002MathSciNetView ArticleMATHGoogle Scholar - Chang SS, Lee BS, Cho YJ, Chen YQ, Kang SM, Jung JS:
**Generalized contraction mapping principle and differential equations in probabilistic metric spaces.***Proceedings of the American Mathematical Society*1996,**124**(8):2367–2376. 10.1090/S0002-9939-96-03289-3MathSciNetView ArticleMATHGoogle Scholar - Chang S-S, Cho YJ, Kang SM:
*Nonlinear Operator Theory in Probabilistic Metric Spaces*. Nova Science Publishers, Huntington, NY, USA; 2001:x+338.MATHGoogle Scholar - Khamsi MA, Kreinovich VY:
**Fixed point theorems for dissipative mappings in complete probabilistic metric spaces.***Mathematica Japonica*1996,**44**(3):513–520.MathSciNetMATHGoogle Scholar - Razani A:
**A contraction theorem in fuzzy metric spaces.***Fixed Point Theory and Applications*2005,**2005**(3):257–265. 10.1155/FPTA.2005.257MathSciNetView ArticleMATHGoogle Scholar - Schweizer B, Sherwood H, Tardiff RM:
**Contractions on probabilistic metric spaces: examples and counterexamples.***Stochastica*1988,**12**(1):5–17.MathSciNetMATHGoogle Scholar - Klement EP, Mesiar R, Pap E:
*Triangular Norms, Trends in Logic—Studia Logica Library*.*Volume 8*. Kluwer Academic Publishers, Dordrecht, The Netherlands; 2000:xx+385.Google Scholar - Radu V:
*Lectures on Probabilistic Analysis, Surveys, Lecture Notes and Monographs.Series on Probability, Statistics and Applied Mathematics*.*Volume 2*. Universitatea din Timisoara, Timisoara, Romania; 1994.Google Scholar - Hadžić O, Pap E:
*Fixed Point Theory in Probabilistic Metric Spaces, Mathematics and Its Applications*.*Volume 536*. Kluwer Academic Publishers, Dordrecht, The Netherlands; 2001:x+273.Google Scholar - Hadžić O, Pap E:
**New classes of probabilistic contractions and applications to random operators.**In*Fixed Point Theory and Applications (Chinju/Masan, 2001)*.*Volume 4*. Nova Science Publishers, Hauppauge, NY, USA; 2003:97–119.Google Scholar - Reilly IL, Subrahmanyam PV, Vamanamurthy MK:
**Cauchy sequences in quasipseudometric spaces.***Monatshefte für Mathematik*1982,**93**(2):127–140. 10.1007/BF01301400MathSciNetView ArticleMATHGoogle Scholar - Jungck G, Rhoades BE:
**Fixed points for set valued functions without continuity.***Indian Journal of Pure and Applied Mathematics*1998,**29**(3):227–238.MathSciNetMATHGoogle Scholar

## Copyright

This article is published under license to BioMed Central Ltd. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.