Open Access

# Common Fixed Point Theorems in Menger Probabilistic Quasimetric Spaces

Fixed Point Theory and Applications20092009:546273

DOI: 10.1155/2009/546273

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 [25]. It is also of fundamental importance in probabilistic functional analysis, nonlinear analysis and applications [610].

In the sequel, we will adopt usual terminology, notation, and conventions of the theory of Menger probabilistic metric spaces, as in [7, 8, 10]. Throughout this paper, the space of all probability distribution functions (in short, dfs) is denoted by is left-continuous and nondecreasing on , and and the subset is the set . Here denotes the left limit of the function at the point , . The space is partially ordered by the usual pointwise ordering of functions, that is, if and only if for all in . The maximal element for in this order is the df given by
(1.1)

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 :

(1.2)
Each    can be extended [11] (by associativity) in a unique way to an -ary operation taking for the values and
(1.3)

for and , for all .

We also mention the following families of

Definition 1.2.

It is said that the -norm is of Hadžić-type ( for short) and if the family of its iterates defined, for each in , by
(1.4)
is equicontinuous at , that is,
(1.5)

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.

Proposition 1.4 (see [13]).
1. (i)

For the following implication holds:

(1.6)
1. (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.

Example 1.5.
1. (i)

The Dombi family of is defined by

(1.7)

The Aczél-Alsina family of    is defined by

(1.8)

Sugeno-Weber family of    is defined by

(1.9)

In [13] the following results are obtained.

(a)If is the Dombi family of and is a sequence of elements from such that then we have the following equivalence:
(1.10)
(b)Equivalence (1.10) holds also for the family that is,
(1.11)
(c)If is the Sugeno-Weber family of and is a sequence of elements from such that then we have the following equivalence:
(1.12)

Proposition 1.6.

Let be a sequence of numbers from such that and -norm is of . Then
(1.13)

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

Throughout this section, a binary operation is a continuous -norm and satisfies the condition
(2.1)

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

Lemma 2.1.

Let be a Menger PQM space. If the sequence in X is such that for every
(2.2)

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

Proof.

For every and , we have
(2.3)

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.

If
(2.4)

then and have a unique common fixed point.

Proof.

Let . By (a), we can find such that and . By induction, we can define a sequence such that and . By induction again,
(2.5)

Similarly, we have

(2.6)

Hence, it follows that

(2.7)

for

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

(2.8)
It follows that, there exists such that . We prove that . From (c), we get
(2.9)
as , we have
(2.10)
which implies that, . Moreover,
(2.11)
as , we have
(2.12)
which implies that Since, the pairs and are weak compatible, we have hence it follows that Similarly, we get Now, we prove that Since, from (c) we have
(2.13)
as , we have
(2.14)

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

If and are two fixed points common to and , then

(2.15)

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.

If
(2.16)

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 ;

(c) for all and , where is monotone increasing function;
1. (d)
(2.17)

If
(2.18)

then have a unique common fixed point.

Proof.

By Corollary 2.3, if set then have a unique common fixed point in . That is, there exists , such that . We prove that , for From (c), we have
(2.19)
By (d), we get
(2.20)

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.

From equivalence (1.10) we have
(2.21)

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.

From equivalence (1.11) we have
(2.22)

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.

From equivalence (1.12) we have
(2.23)

## Declarations

### Acknowledgment

The second author is supported by MNTRRS 144012.

## Authors’ Affiliations

(1)
Department of Mathematics, Islamic Azad University-Babol Branch
(2)
Faculty of Technology, University of Novi Sad
(3)
Department of Mathematics, Islamic Azad University-Babol Branch

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