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Common Fixed Point Theorems in Menger Probabilistic Quasimetric Spaces
Fixed Point Theory and Applications volume 2009, Article number: 546273 (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].
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
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F546273/MediaObjects/13663_2008_Article_1154_Equ1_HTML.gif)
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 :
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F546273/MediaObjects/13663_2008_Article_1154_Equ2_HTML.gif)
Each
can be extended [11] (by associativity) in a unique way to an
-ary operation taking for
the values
and
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F546273/MediaObjects/13663_2008_Article_1154_Equ3_HTML.gif)
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
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F546273/MediaObjects/13663_2008_Article_1154_Equ4_HTML.gif)
is equicontinuous at , that is,
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F546273/MediaObjects/13663_2008_Article_1154_Equ5_HTML.gif)
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]).
-
(i)
For
the following implication holds:
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F546273/MediaObjects/13663_2008_Article_1154_Equ6_HTML.gif)
-
(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.
-
(i)
The Dombi family of
is defined by
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F546273/MediaObjects/13663_2008_Article_1154_Equ7_HTML.gif)
The Aczél-Alsina family of
is defined by
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F546273/MediaObjects/13663_2008_Article_1154_Equ8_HTML.gif)
Sugeno-Weber family of
is defined by
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F546273/MediaObjects/13663_2008_Article_1154_Equ9_HTML.gif)
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:
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F546273/MediaObjects/13663_2008_Article_1154_Equ10_HTML.gif)
(b)Equivalence (1.10) holds also for the family that is,
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F546273/MediaObjects/13663_2008_Article_1154_Equ11_HTML.gif)
(c)If is the Sugeno-Weber family of
and
is a sequence of elements from
such that
then we have the following equivalence:
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F546273/MediaObjects/13663_2008_Article_1154_Equ12_HTML.gif)
Proposition 1.6.
Let be a sequence of numbers from
such that
and
-norm
is of
. Then
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F546273/MediaObjects/13663_2008_Article_1154_Equ13_HTML.gif)
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
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F546273/MediaObjects/13663_2008_Article_1154_Equ14_HTML.gif)
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
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F546273/MediaObjects/13663_2008_Article_1154_Equ15_HTML.gif)
for very , where
is a monotone increasing functions.Then the sequence
is a Cauchy sequence.
Proof.
For every and
, we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F546273/MediaObjects/13663_2008_Article_1154_Equ16_HTML.gif)
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
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F546273/MediaObjects/13663_2008_Article_1154_Equ17_HTML.gif)
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,
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F546273/MediaObjects/13663_2008_Article_1154_Equ18_HTML.gif)
Similarly, we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F546273/MediaObjects/13663_2008_Article_1154_Equ19_HTML.gif)
Hence, it follows that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F546273/MediaObjects/13663_2008_Article_1154_Equ20_HTML.gif)
for
Now by Lemma 2.1, is a Cauchy sequence. Since the space
is complete, there exists a point
such that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F546273/MediaObjects/13663_2008_Article_1154_Equ21_HTML.gif)
It follows that, there exists such that
. We prove that
. From (c), we get
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F546273/MediaObjects/13663_2008_Article_1154_Equ22_HTML.gif)
as , we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F546273/MediaObjects/13663_2008_Article_1154_Equ23_HTML.gif)
which implies that, . Moreover,
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F546273/MediaObjects/13663_2008_Article_1154_Equ24_HTML.gif)
as , we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F546273/MediaObjects/13663_2008_Article_1154_Equ25_HTML.gif)
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
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F546273/MediaObjects/13663_2008_Article_1154_Equ26_HTML.gif)
as , we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F546273/MediaObjects/13663_2008_Article_1154_Equ27_HTML.gif)
It follows that . Therefore,
. That is
is a common fixed point of
and
.
If and
are two fixed points common to
and
, then
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F546273/MediaObjects/13663_2008_Article_1154_Equ28_HTML.gif)
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
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F546273/MediaObjects/13663_2008_Article_1154_Equ29_HTML.gif)
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;
-
(d)
(2.17)
If
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F546273/MediaObjects/13663_2008_Article_1154_Equ31_HTML.gif)
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
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F546273/MediaObjects/13663_2008_Article_1154_Equ32_HTML.gif)
By (d), we get
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F546273/MediaObjects/13663_2008_Article_1154_Equ33_HTML.gif)
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
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F546273/MediaObjects/13663_2008_Article_1154_Equ34_HTML.gif)
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
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F546273/MediaObjects/13663_2008_Article_1154_Equ35_HTML.gif)
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
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F546273/MediaObjects/13663_2008_Article_1154_Equ36_HTML.gif)
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Acknowledgment
The second author is supported by MNTRRS 144012.
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Sedghi, S., Žikić-Došenović, T. & Shobe, N. Common Fixed Point Theorems in Menger Probabilistic Quasimetric Spaces. Fixed Point Theory Appl 2009, 546273 (2009). https://doi.org/10.1155/2009/546273
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DOI: https://doi.org/10.1155/2009/546273