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Some Coupled Fixed Point Theorems in Cone Metric Spaces
Fixed Point Theory and Applications volume 2009, Article number: 125426 (2009)
Abstract
We prove some coupled fixed point theorems for mappings satisfying different contractive conditions on complete cone metric spaces.
1. Introduction
Recently, Huang and Zhang in [1] generalized the concept of metric spaces by considering vector-valued metrics (cone metrics) with values in an ordered real Banach space. They proved some fixed point theorems in cone metric spaces showing that metric spaces really doesnot provide enough space for the fixed point theory. Indeed, they gave an example of a cone metric space and proved existence of a unique fixed point for a selfmap
of
which is contractive in the category of cone metric spaces but is not contractive in the category of metric spaces. After that, cone metric spaces have been studied by many other authors (see [1–9] and the references therein).
Regarding the concept of coupled fixed point, introduced by Bhaskar and Lakshmikantham [10], we consider the corresponding definition for the mappings on complete cone metric spaces and prove some coupled fixed point theorems in the next section. First, we recall some standard notations and definitions in cone metric spaces.
A cone is a subset of a real Banach space
such that
(i) is closed, nonempty and
;
(ii)if are nonnegative real numbers and
, then
;
(iii).
For a given cone , the partial ordering
with respect to
is defined by
if and only if
. The notation
will stand for
, where
denotes the interior of
. Also, we will use
to indicate that
and
.
The cone is called normal if there exists a constant
such that for every
if
then
. The least positive number satisfying this inequality is called the normal constant of
(see [1]). The cone
is called regular if every increasing (decreasing) and bounded above (below) sequence is convergent in
. It is known that every regular cone is normal (see [1], or [7, Lemma 1.1]).
Huang and Zhang defined the concept of a cone metric space in [1] as follows.
Definition 1.1 (see [1]).
Let be a nonempty set and let
be a real Banach space equipped with the partial ordering
with respect to the cone
. Suppose that the mapping
satisfies the following conditions:
(d1) for all
and
if and only if
;
(d2) for all
;
(d3) for all
.
Then is called a cone metric on
, and
is called a cone metric space.
Definition 1.2 (see [1]).
Let be a cone metric space,
and
be a sequence in
. Then
(i) converges to
, denoted by
, if for every
with
there exists a natural number
such that
for all
;
(ii) is a Cauchy sequence if for every
with
there exists a natural number
such that
for all
.
A cone metric space is said to be complete if every Cauchy sequence in
is convergent in
. If for any sequence
in
there exists a subsequence
of
such that
is convergent in
, then the cone metric space
is called sequentially compact. Clearly, every sequentially compact cone metric space is complete. Huang and Zhang in [1] investigated the existence and uniqueness of the fixed point for a selfmap
on a cone metric space
. They considered different types of contractive conditions on
. They also assumed
to be complete when
is a normal cone, and
to be sequentially compact when
is a regular cone. Later, in [7], Rezapour and Hamlbarani improved some of the results in [1] by omitting the normality assumption of the cone
, when
is complete. See [4, 6, 7, 9] for more related results about (complete) cone metric spaces and fixed point theorems for different types of mappings on these spaces.
In the rest of this paper, we always suppose that is a real Banach space,
is a cone with
and
is partial ordering with respect to
. We also note that the relations
and
(
) always hold true.
2. Main Results
For a given partially ordered set , Bhaskar and Lakshmikantham in [10] introduced the concept of coupled fixed point of a mapping
. Later in [11] Lakshmikantham and Ćirić investigated some more coupled fixed point theorems in partially ordered sets. The following is the corresponding definition of coupled fixed point in cone metric spaces.
Definition 2.1.
Let be a cone metric space. An element
is said to be a coupled fixed point of the mapping
if
and
.
In the next theorems of this section, we investigate some coupled fixed point theorems in cone metric spaces.
Theorem 2.2.
Let be a complete cone metric space. Suppose that the mapping
satisfies the following contractive condition for all
:
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F125426/MediaObjects/13663_2009_Article_1116_Equ1_HTML.gif)
where are nonnegative constants with
. Then
has a unique coupled fixed point.
Proof.
Choose and set
,
,
. Then by (2.1) we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F125426/MediaObjects/13663_2009_Article_1116_Equ2_HTML.gif)
and similarly,
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F125426/MediaObjects/13663_2009_Article_1116_Equ3_HTML.gif)
Therefore, by letting
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F125426/MediaObjects/13663_2009_Article_1116_Equ4_HTML.gif)
we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F125426/MediaObjects/13663_2009_Article_1116_Equ5_HTML.gif)
Consequently, if we set then for each
we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F125426/MediaObjects/13663_2009_Article_1116_Equ6_HTML.gif)
If then
is a coupled fixed point of
. Now, let
. For each
we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F125426/MediaObjects/13663_2009_Article_1116_Equ7_HTML.gif)
Therefore,
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F125426/MediaObjects/13663_2009_Article_1116_Equ8_HTML.gif)
which implies that and
are Cauchy sequences in
, and there exist
such that
and
. Let
with
. For every
there exists
such that
and
for all
. Thus
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F125426/MediaObjects/13663_2009_Article_1116_Equ9_HTML.gif)
Consequently, for all
. Thus,
and hence
. Similarly, we have
meaning that
is a coupled fixed point of
.
Now, if is another coupled fixed point of
then
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F125426/MediaObjects/13663_2009_Article_1116_Equ10_HTML.gif)
and therefore,
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F125426/MediaObjects/13663_2009_Article_1116_Equ11_HTML.gif)
Since , (2.11) implies that
. Hence, we have
and the proof of the theorem is complete.
It is worth noting that when the constants in Theorem 2.2 are equal we have the following corollary.
Corollary 2.3.
Let be a complete cone metric space. Suppose that the mapping
satisfies the following contractive condition for all
:
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F125426/MediaObjects/13663_2009_Article_1116_Equ12_HTML.gif)
where is a constant. Then
has a unique coupled fixed point.
Example 2.4.
Let ,
and
. Define
with
. Then
is a complete cone metric space. Consider the mapping
with
. Then
satisfies the contractive condition (2.12) for
, that is,
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F125426/MediaObjects/13663_2009_Article_1116_Equ13_HTML.gif)
Therefore, by Corollary 2.3, has a unique coupled fixed point, which in this case is
. Note that if the mapping
is given by
, then
satisfies the contractive condition (2.12) for
, that is,
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F125426/MediaObjects/13663_2009_Article_1116_Equ14_HTML.gif)
In this case, and
are both coupled fixed points of
and hence the coupled fixed point of
is not unique. This shows that the condition
in corollary (2.12) and hence
in Theorem 2.2 are optimal conditions for the uniqueness of the coupled fixed point.
Theorem 2.5.
Let be a complete cone metric space. Suppose that the mapping
satisfies the following contractive condition for all
:
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F125426/MediaObjects/13663_2009_Article_1116_Equ15_HTML.gif)
where are nonnegative constants with
. Then
has a unique coupled fixed point.
Proof.
Choose and set
,
,
. Then by applying (2.15) we get
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F125426/MediaObjects/13663_2009_Article_1116_Equ16_HTML.gif)
where . This implies that
and
are Cauchy sequences in
and therefore by the completeness of
, there exist
,
such that
and
. Let
and choose a natural number
such that
for all
. Thus,
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F125426/MediaObjects/13663_2009_Article_1116_Equ17_HTML.gif)
which implies that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F125426/MediaObjects/13663_2009_Article_1116_Equ18_HTML.gif)
Since was arbitrary,
or equivalently
. Similarly, one can get
showing that
is a coupled fixed point of
.
Now, if is another coupled fixed point of
then by applying (2.15) we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F125426/MediaObjects/13663_2009_Article_1116_Equ19_HTML.gif)
and therefore . Similarly, we can get
and hence
.
Theorem 2.6.
Let be a complete cone metric space. Suppose that the mapping
satisfies the following contractive condition for all
,
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F125426/MediaObjects/13663_2009_Article_1116_Equ20_HTML.gif)
where are nonnegative constants with
. Then
has a unique coupled fixed point.
Proof.
First, note that the uniqueness of the coupled fixed point is an obvious result of in (2.20). To prove the existence of the fixed point, let
and choose the sequence
and
like in the proof of Theorem 2.5, that is
,
,
. Then by applying (2.20) we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F125426/MediaObjects/13663_2009_Article_1116_Equ21_HTML.gif)
which implies
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F125426/MediaObjects/13663_2009_Article_1116_Equ22_HTML.gif)
Similarly, one can get
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F125426/MediaObjects/13663_2009_Article_1116_Equ23_HTML.gif)
Therefore, and
are Cauchy sequences in
and hence by the completeness of
, there exist
such that
and
. Let
with
and for each
choose a natural number
such that
for all
. Thus,
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F125426/MediaObjects/13663_2009_Article_1116_Equ24_HTML.gif)
which implies
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F125426/MediaObjects/13663_2009_Article_1116_Equ25_HTML.gif)
Since was arbitrary,
or equivalently
. Similarly, one can get
and hence
is a coupled fixed point of
.
When the constants in Theorems 2.5 and 2.6 are equal, we get the following corollaries.
Corollary 2.7.
Let be a complete cone metric space. Suppose that the mapping
satisfies the following contractive condition for all
:
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F125426/MediaObjects/13663_2009_Article_1116_Equ26_HTML.gif)
where is a constant. Then
has a unique coupled fixed point.
Corollary 2.8.
Let be a complete cone metric space. Suppose that the mapping
satisfies the following contractive condition for all
:
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F125426/MediaObjects/13663_2009_Article_1116_Equ27_HTML.gif)
where is a constant. Then
has a unique coupled fixed point.
Remark 2.9.
Note that in Theorem 2.5, if the mapping satisfies the contractive condition (2.15) for all
, then
also satisfies the following contractive condition:
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F125426/MediaObjects/13663_2009_Article_1116_Equ28_HTML.gif)
Consequently, by adding (2.15) and (2.28), also satisfies the following:
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F125426/MediaObjects/13663_2009_Article_1116_Equ29_HTML.gif)
which is a contractive condition of the type (2.26) in Corollary 2.7 (with equal constants). Therefore, one can also reduce the proof of general case (2.15) in Theorem 2.5 to the special case of equal constants. A similar argument is valid for the contractive conditions (2.20) in Theorem 2.6 and (2.27) in Corollary 2.8.
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The authors would like to thank the referees for their valuable and useful comments.
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Sabetghadam, F., Masiha, H.P. & Sanatpour, A.H. Some Coupled Fixed Point Theorems in Cone Metric Spaces. Fixed Point Theory Appl 2009, 125426 (2009). https://doi.org/10.1155/2009/125426
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DOI: https://doi.org/10.1155/2009/125426