Abstract
We discuss the newly introduced concept of cone metric spaces. We also discuss the fixed point existence results of contractive mappings defined on such metric spaces. In particular, we show that most of the new results are merely copies of the classical ones.
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1. Introduction
Cone metric spaces were introduced in [1]. A similar notion was also considered by Rzepecki in [2]. After carefully defining convergence and completeness in cone metric spaces, the authors proved some fixed point theorems of contractive mappings. Recently, more fixed point results in cone metric spaces appeared in [3–8]. Topological questions in cone metric spaces were studied in [6] where it was proved that every cone metric space is first countable topological space. Hence, continuity is equivalent to sequential continuity and compactness is equivalent to sequential compactness. It is worth mentioning the pioneering work of Quilliot [9] who introduced the concept of generalized metric spaces. His approach was very successful and used by many (see references in [10]). It is our belief that cone metric spaces are a special case of generalized metric spaces. In this work, we introduce a metric type structure in cone metric spaces and show that classical proofs do carry almost identically in these metric spaces. This approach suggest that any extension of known fixed point result to cone metric spaces is redundant. Moreover the underlying Banach space and the associated cone subset are not necessary.
For more on metric fixed point theory, the reader may consult the book [11].
2. Basic Definitions and Results
First let us start by making some basic definitions.
Definition 2.1.
Let be a real Banach space with norm
and
a subset of
. Then
is called a cone if and only if
(1) is closed, nonempty, and
, where
is the zero vector in
;
(2)if , and
, then
;
(3)if and
, then
.
Given a cone in a Banach space
, we define a partial ordering
with respect to
by
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F315398/MediaObjects/13663_2010_Article_1256_Equ1_HTML.gif)
We also write whenever
and
, while
will stand for
(where Int(
) designate the interior of
). The cone
is called normal if there is a number
, such that for all
, we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F315398/MediaObjects/13663_2010_Article_1256_Equ2_HTML.gif)
The least positive number satisfying this inequality is called the normal constant of . The cone
is called regular if every increasing sequence which is bounded from above is convergent. Equivalently the cone
is called regular if every decreasing sequence which is bounded from below is convergent. Regular cones are normal and there exist normal cones which are not regular.
Throughout the Banach space and the cone
will be omitted.
Definition 2.2.
A cone metric space is an ordered pair , where
is any set and
is a mapping satisfying
(1), that is,
, for all
, and
if and only if
;
(2) for all
;
(3), for all
.
Convergence is defined as follows.
Definition 2.3.
Let be a cone metric space, let
be a sequence in
and
. If for any
with
, there is
such that for all
,
, then
is said to be convergent. We will say
converges to
and write
.
It is easy to show that the limit of a convergent sequence is unique. Cauchy sequences and completeness are defined by
Definition 2.4.
Let be a cone metric space,
be a sequence in
. If for any
with
, there is
such that for all
,
, then
is called Cauchy sequence. If every Cauchy sequence is convergent in
, then
is called a complete cone metric space.
The basic properties of convergent and Cauchy sequences may be found at [1]. In fact the properties and their proofs are identical to the classical metric ones. Since this work concerns the fixed point property of mappings, we will need the following property.
Definition 2.5.
Let be a cone metric space. A mapping
is called Lipschitzian if there exists
such that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F315398/MediaObjects/13663_2010_Article_1256_Equ3_HTML.gif)
for all . The smallest constant
which satisfies the above inequality is called the Lipschitz constant of
, denoted
. In particular
is a contraction if
.
As we mentioned earlier cone metric spaces have a metric type structure. Indeed we have the following result.
Theorem 2.6.
Let be a metric cone over the Banach space
with the cone
which is normal with the normal constant
. The mapping
defined by
satisfies the following properties:
(1) if and only if
;
(2), for any
;
(3), for any points
,
.
Proof.
The proofs of (1) and (2) are easy and left to the reader. In order to prove (3), let be any points in
. Using the triangle inequality satisfied by
, we get
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F315398/MediaObjects/13663_2010_Article_1256_Equ4_HTML.gif)
Since is normal with constant
we get
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F315398/MediaObjects/13663_2010_Article_1256_Equ5_HTML.gif)
which implies
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F315398/MediaObjects/13663_2010_Article_1256_Equ6_HTML.gif)
This completes the proof of the theorem.
Note that the property (3) is discouraging since it does not give the classical triangle inequality satisfied by a distance. But there are many examples where the triangle inequality fails (see, e.g., [12]).
The above result suggest the following definition.
Definition 2.7.
Let be a set. Let
be a function which satisfies
(1) if and only if
;
(2), for any
;
(3), for any points
,
, for some constant
.
The pair is called a metric type space.
Similarly we define convergence and completeness in metric type spaces.
Definition 2.8.
Let be a metric type space.
(1)The sequence converges to
if and only if
.
(2)The sequence is Cauchy if and only if
.
is complete if and only if any Cauchy sequence in
is convergent.
3. Some Fixed Point Results
Let be a map.
is called Lipschitzian if there exists a constant
such that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F315398/MediaObjects/13663_2010_Article_1256_Equ7_HTML.gif)
for any . The smallest constant
will be denoted
.
Theorem 3.1.
Let be a complete metric type space. Let
be a map such
is Lipschitzian for all
and that
. Then
has a unique fixed point
. Moreover for any
, the orbit
converges to
.
Proof.
Let . For any
, we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F315398/MediaObjects/13663_2010_Article_1256_Equ8_HTML.gif)
Hence
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F315398/MediaObjects/13663_2010_Article_1256_Equ9_HTML.gif)
Since is convergent, then
. This forces
to be a Cauchy sequence. Since
is complete, then
converges to some point
. First let us show that
is a fixed point of
. Since
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F315398/MediaObjects/13663_2010_Article_1256_Equ10_HTML.gif)
we get
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F315398/MediaObjects/13663_2010_Article_1256_Equ11_HTML.gif)
If we let , we get
, or
. Next we show that
has at most one fixed point. Indeed let
and
be two fixed points of
. Then we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F315398/MediaObjects/13663_2010_Article_1256_Equ12_HTML.gif)
for any . Since
, we get
, or
. Therefore we have
for any
, which completes the proof of the theorem.
The condition is needed because of the condition (3) satisfied by
. In fact a more natural condition should be
(), for any points
, for some constant
.
An example of such satisfying
is given below.
Example 3.2.
Let be the set of Lebesgue measurable functions on
such that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F315398/MediaObjects/13663_2010_Article_1256_Equ13_HTML.gif)
Define by
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F315398/MediaObjects/13663_2010_Article_1256_Equ14_HTML.gif)
Then satisfies the following properties:
if and only if
;
, for any
;
, for any points
.
In the next result we consider the case of metric type spaces when
satisfies
. Recall that a subset
of
is said to be bounded whenever
.
Theorem 3.3.
Let be a complete metric type space, where
satisfies
instead of (3). Let
be a map such that
is Lipschitzian for any
and
. Then
has a unique fixed point if and only if there exists a bounded orbit. Moreover if
has a fixed point
, then for any
, the orbit
converges to
.
Proof.
Clearly if has a fixed point, then its orbit is bounded. Conversely let
such that
is bounded, that is, there exists
such that
, for any
. Let
, we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F315398/MediaObjects/13663_2010_Article_1256_Equ15_HTML.gif)
Since , then
is a Cauchy sequence. Hence
converges to some point
since
is complete. The remaining part of the proof follows the same as in the previous theorem.
The connection between the above results and the main theorems of [1] are given in the following corollary.
Corollary 3.4.
Let be a metric cone over the Banach space
with the cone
which is normal with the normal constant
. Consider
defined by
. Let
be a contraction with constant
. Then
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F315398/MediaObjects/13663_2010_Article_1256_Equ16_HTML.gif)
for any and
. Hence
, for any
. Therefore
is convergent, which implies
has a unique fixed point
, and any orbit converges to
.
From the definition of in the above Corollary, we easily see that
-convergence and
-convergence are identical.
Remark 3.5.
In [1] the authors gave an example of a map which is contraction for
but not for the euclidian distance. From the above corollary, we see that
. Since
may not be less than 1, then
may not be a contraction for
. This is why the above theorems were stated in terms of
.
Using the ideas described above one can prove fixed point results for mappings which contracts orbits and obtain similar results as Theorem for example in [1].
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Khamsi, M. Remarks on Cone Metric Spaces and Fixed Point Theorems of Contractive Mappings. Fixed Point Theory Appl 2010, 315398 (2010). https://doi.org/10.1155/2010/315398
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DOI: https://doi.org/10.1155/2010/315398