- Research Article
- Open access
- Published:
Common Fixed Point Theorem in Partially Ordered
-Fuzzy Metric Spaces
Fixed Point Theory and Applications volume 2010, Article number: 125082 (2010)
Abstract
We introduce partially ordered -fuzzy metric spaces and prove a common fixed point theorem in these spaces.
1. Introduction
The Banach fixed point theorem for contraction mappings has been generalized and extended in many directions [1–43]. Recently Nieto and Rodríguez-López [27–29] and Ran and Reurings [33] presented some new results for contractions in partially ordered metric spaces. The main idea in [27–33] involves combining the ideas of iterative technique in the contraction mapping principle with those in the monotone technique.
Recall that if is a partially ordered set and
is such that for
implies
, then a mapping
is said to be nondecreasing. The main result of Nieto and Rodríguez-López [27–33] and Ran and Reurings [33] is the following fixed point theorem.
Theorem 1.1.
Let be a partially ordered set and suppose that there is a metric
on
such that
is a complete metric space. Suppose that
is a nondecreasing mapping with
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F125082/MediaObjects/13663_2009_Article_1201_Equ1_HTML.gif)
for all where
Also suppose the following.
(a) is continuous.
(b)If is a nondecreasing sequence with
in
then for all
hold.
If there exists an with
, then
has a fixed point.
The works of Nieto and Rodríguez-López [27, 28] and Ran and Reurings [33] have motivated Agarwal et al. [1], Bhaskar and Lakshmikantham [3], and Lakshmikantham and Ćirić [23] to undertake further investigation of fixed points in the area of ordered metric spaces. We prove the existence and approximation results for a wide class of contractive mappings in intuitionistic metric space. Our results are an extension and improvement of the results of Nieto and Rodríguez-López [27, 28] and Ran and Reurings [33] to more general class of contractive type mappings and include several recent developments.
2. Preliminaries
The notion of fuzzy sets was introduced by Zadeh [44]. Various concepts of fuzzy metric spaces were considered in [15, 16, 22, 45]. Many authors have studied fixed point theory in fuzzy metric spaces; see, for example, [7, 8, 25, 26, 39, 46–48]. In the sequel, we will adopt the usual terminology, notation, and conventions of -fuzzy metric spaces introduced by Saadati et al. [36] which are a generalization of fuzzy metric sapces [49] and intuitionistic fuzzy metric spaces [32, 37].
Definition (see [46]).
Let be a complete lattice, and
a nonempty set called a universe. An
-fuzzy set
on
is defined as a mapping
. For each
in
,
represents the degree (in
) to which
satisfies
.
Consider the set and the operation
defined by
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F125082/MediaObjects/13663_2009_Article_1201_Equ2_HTML.gif)
, and
, for every
. Then
is a complete lattice.
Classically, a triangular norm on
is defined as an increasing, commutative, associative mapping
satisfying
, for all
. These definitions can be straightforwardly extended to any lattice
. Define first
and
.
Definition.
A negation on is any strictly decreasing mapping
satisfying
and
. If
, for all
, then
is called an involutive negation.
In this paper the negation is fixed.
Definition.
A triangular norm (-norm) on
is a mapping
satisfying the following conditions:
(i)for all
(boundary condition);
(ii)for all
(commutativity);
(iii)for all
(associativity);
(iv)for all
and
(monotonicity).
A -norm
on
is said to be continuous if for any
and any sequences
and
which converge to
and
we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F125082/MediaObjects/13663_2009_Article_1201_Equ3_HTML.gif)
For example, and
are two continuous
-norms on
. A
-norm can also be defined recursively as an
-ary operation (
) by
and
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F125082/MediaObjects/13663_2009_Article_1201_Equ4_HTML.gif)
for and
.
A -norm
is said to be of Hadžić type if the family
is equicontinuous at
, that is,
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F125082/MediaObjects/13663_2009_Article_1201_Equ5_HTML.gif)
is a trivial example of a
-norm of Hadžić type, but there exist
-norms of Hadžić type weaker than
[50] where
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F125082/MediaObjects/13663_2009_Article_1201_Equ6_HTML.gif)
Definition.
The 3-tuple is said to be an
-fuzzy metric space if
is an arbitrary (nonempty) set,
is a continuous
-norm on
and
is an
-fuzzy set on
satisfying the following conditions for every
in
and
in
:
(a);
(b) for all
if and only if
;
(c);
(d);
(e) is continuous.
If the -fuzzy metric space
satisfies the condition:
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F125082/MediaObjects/13663_2009_Article_1201_Equ7_HTML.gif)
then is said to be Menger
-fuzzy metric space or for short a
-fuzzy metric space.
Let be an
-fuzzy metric space. For
, we define the open ball
with center
and radius
, as
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F125082/MediaObjects/13663_2009_Article_1201_Equ8_HTML.gif)
A subset is called open if for each
, there exist
and
such that
. Let
denote the family of all open subsets of
. Then
is called the topology induced by the
-fuzzy metric
.
Example (see [38]).
Let be a metric space. Denote
for all
and
in
and let
and
be fuzzy sets on
defined as follows:
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F125082/MediaObjects/13663_2009_Article_1201_Equ9_HTML.gif)
Then is an intuitionistic fuzzy metric space.
Example.
Let . Define
for all
and
in
, and let
on
be defined as follows:
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F125082/MediaObjects/13663_2009_Article_1201_Equ10_HTML.gif)
for all and
. Then
is an
-fuzzy metric space.
Lemma (see [49]).
Let be an
-fuzzy metric space. Then,
is nondecreasing with respect to
, for all
in
.
Definition.
A sequence in an
-fuzzy metric space
is called a Cauchy sequence, if for each
and
, there exists
such that for all
,
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F125082/MediaObjects/13663_2009_Article_1201_Equ11_HTML.gif)
The sequence is said to be convergent to
in the
-fuzzy metric space
(denoted by
) if
whenever
for every
. A
-fuzzy metric space is said to be complete if and only if every Cauchy sequence is convergent.
Definition.
Let be an
-fuzzy metric space.
is said to be continuous on
if
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F125082/MediaObjects/13663_2009_Article_1201_Equ12_HTML.gif)
whenever a sequence in
converges to a point
, that is,
and
.
Lemma.
Let be an
-fuzzy metric space. Then
is continuous function on
.
Proof.
The proof is the same as that for fuzzy spaces (see [35, Proposition ]).
Lemma.
If an -fuzzy metric space
satisfies the following condition:
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F125082/MediaObjects/13663_2009_Article_1201_Equ13_HTML.gif)
then one has and
.
Proof.
Let for all
. Then by
of Definition 2.5, we have
and by
of Definition 2.5, we conclude that
.
Lemma (see [50]).
Let be an
-fuzzy metric space in which
is Hadži
type. Suppose
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F125082/MediaObjects/13663_2009_Article_1201_Equ14_HTML.gif)
for some and
. Then
is a Cauchy sequence.
3. Main Results
Definition.
Suppose that is a partially ordered set and
are mappings of
into itself. We say that
is
-nondecreasing if for
,
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F125082/MediaObjects/13663_2009_Article_1201_Equ15_HTML.gif)
Now we present the main result in this paper.
Theorem.
Let be a partially ordered set and suppose that there is an
-fuzzy metric
on
such that
is a complete
-fuzzy metric space in which
is Hadži
type. Let
be two self-mappings of
such that there exist
and
such that
is a
-nondecreasing mapping and
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F125082/MediaObjects/13663_2009_Article_1201_Equ16_HTML.gif)
for all for which
and all
Also suppose that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F125082/MediaObjects/13663_2009_Article_1201_Equ17_HTML.gif)
Also suppose that is closed. If there exists an
with
, then
and
have a coincidence. Further, if
and
commute at their coincidence points, then
and
have a common fixed point.
Proof.
Let be such that
Since
we can choose
such that
Again from
we can choose
such that
Continuing this process we can choose a sequence
in
such that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F125082/MediaObjects/13663_2009_Article_1201_Equ18_HTML.gif)
Since and
we have
Then from (3.1),
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F125082/MediaObjects/13663_2009_Article_1201_Equ19_HTML.gif)
that is, by (3.4), Again from (3.1),
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F125082/MediaObjects/13663_2009_Article_1201_Equ20_HTML.gif)
that is, Continuing we obtain
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F125082/MediaObjects/13663_2009_Article_1201_Equ21_HTML.gif)
Now we will show that a sequence converges to
for each
. If
for some
and for each
, then it is easily to show that
for all
. So we suppose that
for all
We show that for each
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F125082/MediaObjects/13663_2009_Article_1201_Equ22_HTML.gif)
Since from (3.4) and (3.7) we have from (3.1) with
and
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F125082/MediaObjects/13663_2009_Article_1201_Equ23_HTML.gif)
So by (3.4),
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F125082/MediaObjects/13663_2009_Article_1201_Equ24_HTML.gif)
Since by (d) of Definition 2.5
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F125082/MediaObjects/13663_2009_Article_1201_Equ25_HTML.gif)
we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F125082/MediaObjects/13663_2009_Article_1201_Equ26_HTML.gif)
As -norm is continuous, letting
we get
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F125082/MediaObjects/13663_2009_Article_1201_Equ27_HTML.gif)
Consequently,
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F125082/MediaObjects/13663_2009_Article_1201_Equ28_HTML.gif)
By repeating the above inequality, we obtain
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F125082/MediaObjects/13663_2009_Article_1201_Equ29_HTML.gif)
Since as
it follows that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F125082/MediaObjects/13663_2009_Article_1201_Equ30_HTML.gif)
Thus we proved (3.7). By repeating the above inequality (3.7), we get
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F125082/MediaObjects/13663_2009_Article_1201_Equ31_HTML.gif)
Since as
and
, letting
in (3.17) we get
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F125082/MediaObjects/13663_2009_Article_1201_Equ32_HTML.gif)
Now we will prove that is a Cauchy sequence which means that for every
and
there exists
such that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F125082/MediaObjects/13663_2009_Article_1201_Equ33_HTML.gif)
Let and
be arbitrary. For any
we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F125082/MediaObjects/13663_2009_Article_1201_Equ34_HTML.gif)
Since is nondecreasing with respect to
, for all
in
,
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F125082/MediaObjects/13663_2009_Article_1201_Equ35_HTML.gif)
and hence, by (d) of Definition 2.5,
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F125082/MediaObjects/13663_2009_Article_1201_Equ36_HTML.gif)
From (3.17) it follows that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F125082/MediaObjects/13663_2009_Article_1201_Equ37_HTML.gif)
From (3.23) with we get
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F125082/MediaObjects/13663_2009_Article_1201_Equ38_HTML.gif)
Thus by (3.22),
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F125082/MediaObjects/13663_2009_Article_1201_Equ39_HTML.gif)
Hence we get
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F125082/MediaObjects/13663_2009_Article_1201_Equ40_HTML.gif)
From (3.26) and (3.17),
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F125082/MediaObjects/13663_2009_Article_1201_Equ41_HTML.gif)
Hence we conclude, as as
and
, that there exists
such that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F125082/MediaObjects/13663_2009_Article_1201_Equ42_HTML.gif)
Thus we proved that is a Cauchy sequence.
Since is closed and as
, there is some
such that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F125082/MediaObjects/13663_2009_Article_1201_Equ43_HTML.gif)
Now we show that is a coincidence of
and
Since from (3.3) and (3.29) we have
for all
then from (3.2) and by (d) of Definition 2.5 we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F125082/MediaObjects/13663_2009_Article_1201_Equ44_HTML.gif)
Letting we get
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F125082/MediaObjects/13663_2009_Article_1201_Equ45_HTML.gif)
for all Therefore,
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F125082/MediaObjects/13663_2009_Article_1201_Equ46_HTML.gif)
Hence we get
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F125082/MediaObjects/13663_2009_Article_1201_Equ47_HTML.gif)
Hence we conclude that for all
Then by (b) of Definition 2.5 we have
Thus we proved that
and
have a coincidence.
Suppose now that and
commute at
. Set
Then
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F125082/MediaObjects/13663_2009_Article_1201_Equ48_HTML.gif)
Since from (3.3) we have and as
and
from (3.2) we get
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F125082/MediaObjects/13663_2009_Article_1201_Equ49_HTML.gif)
Letting we get
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F125082/MediaObjects/13663_2009_Article_1201_Equ50_HTML.gif)
Hence, similarly as above, we get
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F125082/MediaObjects/13663_2009_Article_1201_Equ51_HTML.gif)
Hence we conclude that Since
we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F125082/MediaObjects/13663_2009_Article_1201_Equ52_HTML.gif)
Thus we proved that and
have a common fixed point.
Remark.
Note that is
-nondecreasing and can be replaced by
which is
-non-increasing in Theorem 3.2 provided that
is replaced by
in Theorem 3.2.
Corollary 3.4.
Let be a partially ordered set and suppose that there is an
-fuzzy metric
on
such that
is a complete
-fuzzy metric space in which
is Hadži
type. Let
be a nondecreasing self-mappings of
such that there exist
and
such that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F125082/MediaObjects/13663_2009_Article_1201_Equ53_HTML.gif)
for all for which
and all
Also suppose the following.
(i)If is a nondecreasing sequence with
in
, then
for all
hold.
(ii) is continuous.
If there exists an with
, then
has a fixed point.
Proof.
Taking (
= the identity mapping) in Theorem 3.2, then (3.3) reduces to the hypothesis
Suppose now that is continuous. Since from (3.4) we have
for all
and as from (3.29),
then
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F125082/MediaObjects/13663_2009_Article_1201_Equ54_HTML.gif)
Corollary 3.5.
Let be a partially ordered set and suppose that there is an
-fuzzy metric
on
such that
is a complete
-fuzzy metric space in which
is Hadži
type. Let
be a nondecreasing self-mappings of
such that there exist
and
such that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F125082/MediaObjects/13663_2009_Article_1201_Equ55_HTML.gif)
for all for which
and all
Also suppose the following.
(i)If is a nondecreasing sequence with
in
, then
for all
hold.
(ii) is continuous.
If there exists an with
, then
has a fixed point.
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Acknowledgments
This research is supported by Young research Club, Islamic Azad University-Ayatollah Amoli Branch, Amol, Iran. The authors would like to thank Professor J. J. Nieto for giving useful suggestions for the improvement of this paper.
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Shakeri, S., Ćirić, L. & Saadati, R. Common Fixed Point Theorem in Partially Ordered -Fuzzy Metric Spaces.
Fixed Point Theory Appl 2010, 125082 (2010). https://doi.org/10.1155/2010/125082
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DOI: https://doi.org/10.1155/2010/125082