- Research Article
- Open Access

# Common Fixed Point Theorems in Menger Probabilistic Quasimetric Spaces

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

**2009**:546273

https://doi.org/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.

## Keywords

- Point Theorem
- Probability Distribution Function
- Maximal Element
- Cauchy Sequence
- Common Fixed Point

## 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 ;

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

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