- Research Article
- Open access
- Published:
Some Fixed Point Theorems of Integral Type Contraction in Cone Metric Spaces
Fixed Point Theory and Applications volume 2010, Article number: 189684 (2010)
Abstract
We define a new concept of integral with respect to a cone. Moreover, certain fixed point theorems in those spaces are proved. Finally, an extension of Meir-Keeler fixed point in cone metric space is proved.
1. Introduction
In 2007, Huang and Zhang in [1] introduced cone metric space by substituting an ordered Banach space for the real numbers and proved some fixed point theorems in this space. Many authors study this subject and many fixed point theorems are proved; see [2–5]. In this paper, the concept of integral in this space is introduced and a fixed point theorem is proved. In order to do this, we recall some definitions, examples, and lemmas from [1, 4] as follows.
Let be a real Banach space. A subset
of
is called a cone if and only if the following hold:
(i) is closed, nonempty, and
,
(ii),
, and
imply that
(iii) and
imply that
Given a cone we define a partial ordering
with respect to
by
if and only if
We will write
to indicate that
but
, while
will stand for
int
where int
denotes the interior of
The cone
is called normal if there is a number
such that
implies
for all
The least positive number satisfying above is called the normal constant [1].
The cone is called regular if every increasing sequence which is bounded from above is convergent. That is, if
is a sequence such that
for some
, then there is
such that
. Equivalently, the cone
is regular if and only if every decreasing sequence which is bounded from below is convergent [1]. Also every regular cone is normal [4]. In addition, there are some nonnormal cones.
Example 1.1.
Suppose with the norm
and consider the cone
:
. For all
, set
and
Then
and
Since
is not normal constant of
Therefore,
is non-normal cone.
From now on, we suppose that is a real Banach space,
is a cone in
with
and
is partial ordering with respect to
. Let
be a nonempty set. As it has been defined in [1], a function
is called a cone metric on
if it satisfies the following conditions:
(i) for all
and
if and only if
(ii), for all
(iii), for all
Then is called a cone metric space.
Example 1.2.
Suppose is a metric space and
is defined by
Then
is a cone metric space and the normal constant of
is equal to
Definition 1.3.
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 to
and
is the limit of
. We denote this by
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F189684/MediaObjects/13663_2009_Article_1221_Equ1_HTML.gif)
Definition 1.4.
Let be a cone metric space and
be a sequence in
If for any
with
, there is
such that for all
then
is called a Cauchy sequence in
Definition 1.5.
Let be a cone metric space, if every Cauchy sequence is convergent in
then
is called a complete cone metric space.
Definition 1.6.
Let be a cone metric space. Let
be a self-map on
If for all sequence
in
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F189684/MediaObjects/13663_2009_Article_1221_Equ2_HTML.gif)
then is called continuous on
The following lemmas are useful for us to prove the main result.
Lemma 1.7.
Let be a cone metric space and
a normal cone with normal constant
Let
be a sequence in
Then
converges to
if and only if
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F189684/MediaObjects/13663_2009_Article_1221_Equ3_HTML.gif)
Lemma 1.8.
Let be a cone metric space and
a normal cone with normal constant
Let
be a sequence in
. Then
is a Cauchy sequence if and only if
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F189684/MediaObjects/13663_2009_Article_1221_Equ4_HTML.gif)
Lemma 1.9.
Let be a cone metric space and
a sequence in
If
is convergent, then it is a Cauchy sequence.
Lemma 1.10.
Let be a cone metric space and
be a normal cone with normal constant
. Let
and
be two sequences in
and
. Then
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F189684/MediaObjects/13663_2009_Article_1221_Equ5_HTML.gif)
The following example is a cone metric space.
Example 1.11.
Let and
Suppose that
is defined by
where
is a constant. Then
is a cone metric space.
Theorem 1.12.
Let be a complete cone metric space and
a normal cone with normal constant
Suppose the mapping
satisfies the contractive condition
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F189684/MediaObjects/13663_2009_Article_1221_Equ6_HTML.gif)
for all where
is a constant. Then
has a unique fixed point
Also, for all
the sequence
converges to
2. Certain Integral Type Contraction Mapping in Cone Metric Space
In 2002, Branciari in [6] introduced a general contractive condition of integral type as follows.
Theorem 2.1.
Let be a complete metric space,
, and
is a mapping such that for all
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F189684/MediaObjects/13663_2009_Article_1221_Equ7_HTML.gif)
where is nonnegative and Lebesgue-integrable mapping which is summable (i.e., with finite integral) on each compact subset of
such that for each
,
, then
has a unique fixed point
, such that for each
In this section we define a new concept of integral with respect to a cone and introduce the Branciari's result in cone metric spaces.
Definition 2.2.
Suppose that is a normal cone in
. Let
and
. We define
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F189684/MediaObjects/13663_2009_Article_1221_Equ8_HTML.gif)
Definition 2.3.
The set is called a partition for
if and only if the sets
are pairwise disjoint and
Definition 2.4.
For each partition of
and each increasing function
we define cone lower summation and cone upper summation as
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F189684/MediaObjects/13663_2009_Article_1221_Equ9_HTML.gif)
respectively.
Definition 2.5.
Suppose that is a normal cone in
.
is called an integrable function on
with respect to cone
or to simplicity, Cone integrable function, if and only if for all partition
of
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F189684/MediaObjects/13663_2009_Article_1221_Equ10_HTML.gif)
where must be unique.
We show the common value by
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F189684/MediaObjects/13663_2009_Article_1221_Equ11_HTML.gif)
We denote the set of all cone integrable function by
.
Lemma 2.6.
-
(1)
If
, then
for
(2)
for
and
.
Proof.
-
(1)
Suppose that
and
are partitions for
and
respectively. That is,
(2.6)
Let
is a partition for
Therefore one can write
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F189684/MediaObjects/13663_2009_Article_1221_Equ13_HTML.gif)
So
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F189684/MediaObjects/13663_2009_Article_1221_Equ14_HTML.gif)
-
(2)
Suppose
is an partition for
, that is
(2.9)
Then
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F189684/MediaObjects/13663_2009_Article_1221_Equ16_HTML.gif)
Thus
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F189684/MediaObjects/13663_2009_Article_1221_Equ17_HTML.gif)
Definition 2.7.
The function is called subadditive cone integrable function if and only if for all
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F189684/MediaObjects/13663_2009_Article_1221_Equ18_HTML.gif)
Example 2.8.
Let ,
,
and
for all
Then for all
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F189684/MediaObjects/13663_2009_Article_1221_Equ19_HTML.gif)
Since then
. Therefore
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F189684/MediaObjects/13663_2009_Article_1221_Equ20_HTML.gif)
This shows that is an example of subadditive cone integrable function.
Theorem 2.9.
Let be a complete cone metric space and
a normal cone. Suppose that
is a nonvanishing map and a subadditive cone integrable on each
such that for each
,
. If
is a map such that, for all
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F189684/MediaObjects/13663_2009_Article_1221_Equ21_HTML.gif)
for some then
has a unique fixed point in
Proof.
Let Choose
We have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F189684/MediaObjects/13663_2009_Article_1221_Equ22_HTML.gif)
Since thus
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F189684/MediaObjects/13663_2009_Article_1221_Equ23_HTML.gif)
If then
and this is a contradiction, so
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F189684/MediaObjects/13663_2009_Article_1221_Equ24_HTML.gif)
We now show that is a Cauchy sequence. Due to this, we show that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F189684/MediaObjects/13663_2009_Article_1221_Equ25_HTML.gif)
By triangle inequality
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F189684/MediaObjects/13663_2009_Article_1221_Equ26_HTML.gif)
and by sub-additivity of we get
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F189684/MediaObjects/13663_2009_Article_1221_Equ27_HTML.gif)
Thus
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F189684/MediaObjects/13663_2009_Article_1221_Equ28_HTML.gif)
This means that is a Cauchy sequence and since
is a complete cone metric space, thus
is convergent to
Finally, since
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F189684/MediaObjects/13663_2009_Article_1221_Equ29_HTML.gif)
thus This means that
If
are two distinct fixed points of
then
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F189684/MediaObjects/13663_2009_Article_1221_Equ30_HTML.gif)
which is a contradiction. Thus has a unique fixed point
Lemma 2.10.
Let and
Suppose that
is defined by
where
is a constant. Suppose that
is defined by
where
are two Riemann-integrable functions. Then
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F189684/MediaObjects/13663_2009_Article_1221_Equ31_HTML.gif)
Proof.
Let be a partition of set
such that
and
, then (by Definitions 2.4 and 2.5)
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F189684/MediaObjects/13663_2009_Article_1221_Equ32_HTML.gif)
Thus
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F189684/MediaObjects/13663_2009_Article_1221_Equ33_HTML.gif)
Example 2.11.
Let and
Suppose
for some constant
Firstly,
is a complete cone metric space. Secondly, if
and
are defined by
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F189684/MediaObjects/13663_2009_Article_1221_Equ34_HTML.gif)
respectively, then
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F189684/MediaObjects/13663_2009_Article_1221_Equ35_HTML.gif)
In order to obtain inequality (2.29), set and
where
Hence
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F189684/MediaObjects/13663_2009_Article_1221_Equ36_HTML.gif)
Suppose for all
and
. Thus
. By Lemma 2.10
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F189684/MediaObjects/13663_2009_Article_1221_Equ37_HTML.gif)
Since thus
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F189684/MediaObjects/13663_2009_Article_1221_Equ38_HTML.gif)
It means that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F189684/MediaObjects/13663_2009_Article_1221_Equ39_HTML.gif)
On the other side, Branciari in [6] shows that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F189684/MediaObjects/13663_2009_Article_1221_Equ40_HTML.gif)
for all . Therefore
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F189684/MediaObjects/13663_2009_Article_1221_Equ41_HTML.gif)
Thus inequalities (2.33) and (2.35) imply that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F189684/MediaObjects/13663_2009_Article_1221_Equ42_HTML.gif)
or in other words
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F189684/MediaObjects/13663_2009_Article_1221_Equ43_HTML.gif)
Thus by Theorem 2.9, has a fixed point. But, on the other hand,
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F189684/MediaObjects/13663_2009_Article_1221_Equ44_HTML.gif)
and this means that does not satisfy in Theorem 1.12.
3. Extension of Meir-Keeler Contraction in Cone Metric Space
In 2006, Suzuki in [7] proved that the integral type contraction (see [6]) is a special case of Meir-Keeler contraction (see [8]). Haghi and Rezapour in [5] extended Meir-Keeler contraction in cone metric space as follows.
Theorem 3.1 (see[5]).
Let be a complete regular cone metric space and
has the property (KMC) on
that is, for all
, there exists
such that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F189684/MediaObjects/13663_2009_Article_1221_Equ45_HTML.gif)
for all . Then
has a unique fixed point.
An extension of Theorem 3.1 is as follows.
Theorem 3.2.
Let be a complete regular cone metric space and
a mapping on
. Suppose that there exists a function
from
into itself satisfying the following:
and
for all
,
is nondecreasing and continuous function. Moreover, its inverse is continuous,
for all there exists
such that for all
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F189684/MediaObjects/13663_2009_Article_1221_Equ46_HTML.gif)
for all
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F189684/MediaObjects/13663_2009_Article_1221_Equ47_HTML.gif)
Then has a unique fixed point.
Proof.
First, note that for all
with
Since
exists, thus
for all
with
Now Let
Set
for all
If, there is a natural
such that
then
and so
has a fixed point. If
for all
then
Hence, according to regularity of
there exists
such that
We claim
If
then according to
there is
such that
for all
with
Choose
such that
and take the natural number
such that
for all
We obtain
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F189684/MediaObjects/13663_2009_Article_1221_Equ48_HTML.gif)
Thus
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F189684/MediaObjects/13663_2009_Article_1221_Equ49_HTML.gif)
So, Since
has the property
for all
. This is a contradiction because
for all
Thus
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F189684/MediaObjects/13663_2009_Article_1221_Equ50_HTML.gif)
Now, we show that is a Cauchy sequence. If this is not, then there is a
such that for all natural number
there are
so that the relation
does not hold. Since
has continuous inverse thus there exists
such that for all natural number
there are
so that the relation
does not hold. For each
there exists
such that
for all
with
Choose a natural number
such that
for all
Also, take
so that the relation
does not hold. Then
yields
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F189684/MediaObjects/13663_2009_Article_1221_Equ51_HTML.gif)
Hence, Similarly,
Thus
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F189684/MediaObjects/13663_2009_Article_1221_Equ52_HTML.gif)
which is a contradiction. Therefore is a Cauchy sequence. Since
is a complete cone metric space, there is
such that
. Since
, for all
with
, thus for each
, there is a natural number
such that for all
,
Since
thus
for all
. It means that
In the other side,
and the limit point is unique in cone metric spaces. Thus
has at least one fixed point. Now, if
are two distinct fixed points for
then
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F189684/MediaObjects/13663_2009_Article_1221_Equ53_HTML.gif)
which is a contradiction. Therefore has a unique fixed point.
Remark 3.3.
Set
, then Theorem 3.1 is a direct result of Theorem 3.2.
Let
be a nonvanishing map and a subadditive cone integrable on each
such that for each
,
If
then
satisfies all conditions of Theorem 3.2. In other words, Theorem 2.9 is a direct result of Theorem 3.2.
References
Huang L-G, Zhang X: Cone metric spaces and fixed point theorems of contractive mappings. Journal of Mathematical Analysis and Applications 2007,332(2):1468–1476. 10.1016/j.jmaa.2005.03.087
Abbas M, Jungck G: Common fixed point results for noncommuting mappings without continuity in cone metric spaces. Journal of Mathematical Analysis and Applications 2008,341(1):416–420. 10.1016/j.jmaa.2007.09.070
Ilić D, Rakočević V: Common fixed points for maps on cone metric space. Journal of Mathematical Analysis and Applications 2008,341(2):876–882. 10.1016/j.jmaa.2007.10.065
Rezapour Sh, Hamlbarani R: Some notes on the paper: "Cone metric spaces and fixed point theorems of contractive mappings". Journal of Mathematical Analysis and Applications 2008,345(2):719–724. 10.1016/j.jmaa.2008.04.049
Haghi RH, Rezapour Sh: Fixed points of multifunctions on regular cone metric spaces. Expositiones Mathematicae 2010,28(1):71–77. 10.1016/j.exmath.2009.04.001
Branciari A: A fixed point theorem for mappings satisfying a general contractive condition of integral type. International Journal of Mathematics and Mathematical Sciences 2002,29(9):531–536. 10.1155/S0161171202007524
Suzuki T: Meir-Keeler contractions of integral type are still Meir-Keeler contractions. International Journal of Mathematics and Mathematical Sciences 2007, 2007:-6.
Meir A, Keeler E: A theorem on contraction mappings. Journal of Mathematical Analysis and Applications 1969, 28: 326–329. 10.1016/0022-247X(69)90031-6
Acknowledgment
The third author would like to thank the School of Mathematics of the Institute for Research in Fundamental Sciences, Teheran, Iran, for supporting this research (Grant no. 88470119).
Author information
Authors and Affiliations
Corresponding author
Additional information
An erratum to this article is available at http://dx.doi.org/10.1155/2011/346059.
Rights and permissions
Open Access This article is distributed under the terms of the Creative Commons Attribution 2.0 International License (https://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
About this article
Cite this article
Khojasteh, F., Goodarzi, Z. & Razani, A. Some Fixed Point Theorems of Integral Type Contraction in Cone Metric Spaces. Fixed Point Theory Appl 2010, 189684 (2010). https://doi.org/10.1155/2010/189684
Received:
Accepted:
Published:
DOI: https://doi.org/10.1155/2010/189684