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Some Common Fixed Point Results in Cone Metric Spaces
Fixed Point Theory and Applications volume 2009, Article number: 493965 (2009)
Abstract
We prove a result on points of coincidence and common fixed points for three self-mappings satisfying generalized contractive type conditions in cone metric spaces. We deduce some results on common fixed points for two self-mappings satisfying contractive type conditions in cone metric spaces. These results generalize some well-known recent results.
1. Introduction
Huang and Zhang [1] recently have introduced the concept of cone metric space, where the set of real numbers is replaced by an ordered Banach space, and they have established some fixed point theorems for contractive type mappings in a normal cone metric space. Subsequently, some other authors [2–5] have generalized the results of Huang and Zhang [1] and have studied the existence of common fixed points of a pair of self mappings satisfying a contractive type condition in the framework of normal cone metric spaces.
Vetro [5] extends the results of Abbas and Jungck [2] and obtains common fixed point of two mappings satisfying a more general contractive type condition. Rezapour and Hamlbarani [6] prove that there aren't normal cones with normal constant and for each
there are cones with normal constant
. Also, omitting the assumption of normality they obtain generalizations of some results of [1]. In [7] Di Bari and Vetro obtain results on points of coincidence and common fixed points in nonnormal cone metric spaces. In this paper, we obtain points of coincidence and common fixed points for three self-mappings satisfying generalized contractive type conditions in a complete cone metric space. Our results improve and generalize the results in [1, 2, 5, 6, 8].
2. Preliminaries
We recall the definition of cone metric spaces and the notion of convergence [1]. Let be a real Banach space and
be a subset of
. The subset
is called an order cone if it has the following properties:
(i) is nonempty, closed, and
(ii) and
(iii)
For a given cone , we can define a partial ordering
on
with respect to
by
if and only if
. We will write
if
and
, while
will stands for
, where
denotes the interior of
The cone
is called normal if there is a number
such that for all
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F493965/MediaObjects/13663_2008_Article_1148_Equ1_HTML.gif)
The least number satisfying (2.1) is called the normal constant of
In the following we always suppose that is a real Banach space and
is an order cone in
with
and
is the partial ordering with respect to
Definition 2.1.
Let be a nonempty set. Suppose that the mapping
satisfies
(i) for all
and
if and only if
;
(ii) for all
;
(iii), for all
Then is called a cone metric on
, and
is called a cone metric space.
Let be a sequence in
, and
. If for every
with
there is
such that for all
then
is said to be convergent,
converges to
and
is the limit of
We denote this by
or
as
If for every
with
there is
such that for all
then
is called a Cauchy sequence in
. If every Cauchy sequence is convergent in
, then
is called a complete cone metric space.
3. Main Results
First, we establish the result on points of coincidence and common fixed points for three self-mappings and then show that this result generalizes some of recent results of fixed point.
A pair of self-mappings on
is said to be weakly compatible if they commute at their coincidence point (i.e.,
whenever
). A point
is called point of coincidence of a family
,
, of self-mappings on
if there exists a point
such that
for all
.
Lemma 3.1.
Let be a nonempty set and the mappings
have a unique point of coincidence
in
If
and
are weakly compatibles, then
, and
have a unique common fixed point.
Proof.
Since is a point of coincidence of
, and
. Therefore,
for some
By weakly compatibility of
and
we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F493965/MediaObjects/13663_2008_Article_1148_Equ2_HTML.gif)
It implies that (say). Then
is a point of coincidence of
, and
. Therefore,
by uniqueness. Thus
is a unique common fixed point of
, and
Let be a cone metric space,
be self-mappings on
such that
and
. Choose a point
in
such that
. This can be done since
. Successively, choose a point
in
such that
Continuing this process having chosen
, we choose
and
in
such that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F493965/MediaObjects/13663_2008_Article_1148_Equ3_HTML.gif)
The sequence is called an
-
-sequence with initial point
.
Proposition 3.2.
Let be a cone metric space and
be an order cone. Let
be such that
. Assume that the following conditions hold:
(i), for all
, with
, where
are nonnegative real numbers with
;
(ii), for all
, whenever
.
Then every -
-sequence with initial point
is a Cauchy sequence.
Proof.
Let be an arbitrary point in
and
be an
-
-sequence with initial point
. First, we assume that
for all
. It implies that
for all
. Then,
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F493965/MediaObjects/13663_2008_Article_1148_Equ4_HTML.gif)
It implies that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F493965/MediaObjects/13663_2008_Article_1148_Equ5_HTML.gif)
so
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F493965/MediaObjects/13663_2008_Article_1148_Equ6_HTML.gif)
Similarly, we obtain
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F493965/MediaObjects/13663_2008_Article_1148_Equ7_HTML.gif)
Now, by induction, for each we deduce
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F493965/MediaObjects/13663_2008_Article_1148_Equ8_HTML.gif)
Let
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F493965/MediaObjects/13663_2008_Article_1148_Equ9_HTML.gif)
Then Now, for
, we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F493965/MediaObjects/13663_2008_Article_1148_Equ10_HTML.gif)
In analogous way, we deduce
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F493965/MediaObjects/13663_2008_Article_1148_Equ11_HTML.gif)
Hence, for
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F493965/MediaObjects/13663_2008_Article_1148_Equ12_HTML.gif)
where is the integer part of
.
Fix and choose
such that
Since
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F493965/MediaObjects/13663_2008_Article_1148_Equ13_HTML.gif)
there exists be such that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F493965/MediaObjects/13663_2008_Article_1148_Equ14_HTML.gif)
for all . The choice of
assures
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F493965/MediaObjects/13663_2008_Article_1148_Equ15_HTML.gif)
so
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F493965/MediaObjects/13663_2008_Article_1148_Equ16_HTML.gif)
Consequently, for all , with
, we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F493965/MediaObjects/13663_2008_Article_1148_Equ17_HTML.gif)
and hence is a Cauchy sequence.
Now, we suppose that for some
. If
and
, by (ii) we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F493965/MediaObjects/13663_2008_Article_1148_Equ18_HTML.gif)
which implies . If
we use (i) to obtain
. Similarly, we deduce that
and so
for every
. Hence
is a Cauchy sequence.
Theorem 3.3.
Let be a cone metric space and
be an order cone. Let
be such that
. Assume that the following conditions hold:
(i), for all
, with
, where
are nonnegative real numbers with
;
(ii), for all
, whenever
.
If or
is a complete subspace of
, then
, and
have a unique point of coincidence. Moreover, if
and
are weakly compatibles, then
, and
have a unique common fixed point.
Proof.
Let be an arbitrary point in
. By Proposition 3.2 every
-
-sequence
with initial point
is a Cauchy sequence. If
is a complete subspace of
, there exist
such that
(this holds also if
is complete with
). From
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F493965/MediaObjects/13663_2008_Article_1148_Equ19_HTML.gif)
we obtain
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F493965/MediaObjects/13663_2008_Article_1148_Equ20_HTML.gif)
Fix and choose
be such that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F493965/MediaObjects/13663_2008_Article_1148_Equ21_HTML.gif)
for all , where
. Consequently
and hence
for every
. From
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F493965/MediaObjects/13663_2008_Article_1148_Equ22_HTML.gif)
being closed, as
, we deduce
and so
. This implies that
Similarly, by using the inequality,
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F493965/MediaObjects/13663_2008_Article_1148_Equ23_HTML.gif)
we can show that It implies that
is a point of coincidence of
, and
, that is
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F493965/MediaObjects/13663_2008_Article_1148_Equ24_HTML.gif)
Now, we show that , and
have a unique point of coincidence. For this, assume that there exists another point
in
such that
, for some
in
From
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F493965/MediaObjects/13663_2008_Article_1148_Equ25_HTML.gif)
we deduce Moreover, if
and
are weakly compatibles, then
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F493965/MediaObjects/13663_2008_Article_1148_Equ26_HTML.gif)
which implies (say). Then
is a point of coincidence of
, and
therefore,
by uniqueness. Thus
is a unique common fixed point of
, and
.
From Theorem 3.3, if we choose , we deduce the following theorem.
Theorem 3.4.
Let be a cone metric space,
be an order cone and
be such that
. Assume that the following condition holds:
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F493965/MediaObjects/13663_2008_Article_1148_Equ27_HTML.gif)
for all where
with
.
If or
is a complete subspace of
, then
and
have a unique point of coincidence. Moreover, if the pair
is weakly compatible, then
and
have a unique common fixed point.
Theorem 3.4 generalizes Theorem 1 of [5].
Remark 3.5.
In Theorem 3.4 the condition (3.26) can be replaced by
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F493965/MediaObjects/13663_2008_Article_1148_Equ28_HTML.gif)
for all , where
with
.
(3.27)(3.26) is obivious. (3.26)
(3.27). If in (3.26) interchanging the roles of
and
and adding the resultant inequality to (3.26), we obtain
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F493965/MediaObjects/13663_2008_Article_1148_Equ29_HTML.gif)
From Theorem 3.4, we deduce the followings corollaries.
Corollary 3.6.
Let be a cone metric space,
be an order cone and the mappings
satisfy
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F493965/MediaObjects/13663_2008_Article_1148_Equ30_HTML.gif)
for all where,
If
and
is a complete subspace of
, then
and
have a unique point of coincidence. Moreover, if the pair
is weakly compatible, then
and
have a unique common fixed point.
Corollary 3.6 generalizes Theorem 2.1 of [2], Theorem 1 of [1], and Theorem 2.3 of [6].
Corollary 3.7.
Let be a cone metric space,
be an order cone and the mappings
satisfy
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F493965/MediaObjects/13663_2008_Article_1148_Equ31_HTML.gif)
for all , where
If
and
is a complete subspace of
, then
and
have a unique point of coincidence. Moreover, if the pair
is weakly compatible, then
and
have a unique common fixed point.
Corollary 3.7 generalizes Theorem 2.3 of [2], Theorem 3 of [1], and Theorem 2.6 of [6].
Example 3.8.
Let ,
and
Define
as follows:
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F493965/MediaObjects/13663_2008_Article_1148_Equ32_HTML.gif)
Define mappings as follow:
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F493965/MediaObjects/13663_2008_Article_1148_Equ33_HTML.gif)
Then, if
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F493965/MediaObjects/13663_2008_Article_1148_Equ34_HTML.gif)
which implies
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F493965/MediaObjects/13663_2008_Article_1148_Equ35_HTML.gif)
for all with
.
Therefore, Theorem 3.4 is not applicable to obtain fixed point of or common fixed points of
and
.
Now define a constant mapping by
, then for
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F493965/MediaObjects/13663_2008_Article_1148_Equ36_HTML.gif)
It follows that all conditions of Theorem 3.3 are satisfied for and so
, and
have a unique point of coincidence and a unique common fixed point
.
4. Applications
In this section, we prove an existence theorem for the common solutions for two Urysohn integral equations. Throughout this section let ,
, and
for every
, where
is a constant. It is easily seen that
is a complete cone metric space.
Theorem 4.1.
Consider the Urysohn integral equations
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F493965/MediaObjects/13663_2008_Article_1148_Equ37_HTML.gif)
where ,
. Assume that
are such that
(i) for each
where
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F493965/MediaObjects/13663_2008_Article_1148_Equ38_HTML.gif)
(ii)there exist such that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F493965/MediaObjects/13663_2008_Article_1148_Equ39_HTML.gif)
where , for every
with
and
.
(iii)whenever
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F493965/MediaObjects/13663_2008_Article_1148_Equ40_HTML.gif)
for every .
Then the system of integral equations (4.1) have a unique common solution.
Proof.
Define by
. It is easily seen that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F493965/MediaObjects/13663_2008_Article_1148_Equ41_HTML.gif)
for every , with
and if
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F493965/MediaObjects/13663_2008_Article_1148_Equ42_HTML.gif)
for every . By Theorem 3.3, if
is the identity map on
, the Urysohn integral equations (4.1) have a unique common solution.
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Arshad, M., Azam, A. & Vetro, P. Some Common Fixed Point Results in Cone Metric Spaces. Fixed Point Theory Appl 2009, 493965 (2009). https://doi.org/10.1155/2009/493965
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DOI: https://doi.org/10.1155/2009/493965