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Common Fixed Point Theorems for Weakly Compatible Pairs on Cone Metric Spaces
Fixed Point Theory and Applications volume 2009, Article number: 643840 (2009)
Abstract
We prove several fixed point theorems on cone metric spaces in which the cone does not need to be normal. These theorems generalize the recent results of Huang and Zhang (2007), Abbas and Jungck (2008), and Vetro (2007). Furthermore as corollaries, we obtain recent results of Rezapour and Hamlborani (2008).
1. Introduction and Preliminaries
Recently, Abbas and Jungck [1], have studied common fixed point results for noncommuting mappings without continuity in cone metric space with normal cone. In this paper, our results are related to the results of Abbas and Jungck, but our assumptions are more general, and also we generalize some results of [1–3], and [4] by omitting the assumption of normality in the results.
Let us mention that nonconvex analysis, especially ordered normed spaces, normal cones, and topical functions ([2, 4–9]) have some applications in optimization theory. In these cases, an order is introduced by using vector space cones. Huang and Zhang [2] used this approach, and they have replaced the real numbers by ordering Banach space and defining cone metric space. Consistent with Huang and Zhang [2], the following definitions and results will be needed in the sequel.
Let be a real Banach space. A subset
of
is called a cone if and only if:
(i) is closed, nonempty, and
(ii) and
imply
(iii)
Given a cone we define a partial ordering
on
with respect to
by
if and only if
We will write
to indicate that
but
while
will stand for
(interior of
). A cone
is called normal if there are a number
such that for all
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F643840/MediaObjects/13663_2008_Article_1166_Equ1_HTML.gif)
The least positive number satisfying the above inequality is called the normal constant of It is clear that
From [4] we know that there exists ordered Banach space
with cone
which is not normal but with
Definition 1.1 (see [2]).
Let be a nonempty set. Suppose that the mapping
satisfies
(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. The concept of a cone metric space is more general than of a metric space.
Definition 1.2 (see [2]).
Let be a cone metric space. We say that
is
(e)Cauchy sequence if for every in
with
there is an
such that for all
(f)convergent sequence if for every in
with
there is an
such that for all
for some fixed
in
A cone metric space is said to be complete if every Cauchy sequence in
is convergent in
The sequence
converges to
if and only if
as
It is a Cauchy if and only if
as
.
Remark 1.3.
[10] Let be an ordered Banach (normed) space. Then
is an interior point of
if and only if
is a neighborhood of
Corollary 1.4 (see, e.g., [11] without proof).
-
(1)
If
and
then
Indeed, implies
-
(2)
If
and
then
Indeed, implies
-
(3)
If
for each
, then
Remark 1.5.
If ,
and
then there exists
such that for all
we have
.
Proof.
Let be given. Choose a symmetric neighborhood
such that
Since
, there is
such that
for
This means that
for
that is,
From this it follows that: the sequence converges to
if
as
and
is a Cauchy if
as
In the situation with non-normal cone, we have only half of the lemmas 1 and 4 from [2]. Also, the fact that
if
and
is not applicable.
Remark 1.6.
Let If
and
then eventually
where
are sequence and given point in
Proof.
It follows from Remark 1.5, Corollary 1.4(1), and Definition 1.2(f).
Remark 1.7.
If and
then
for each cone
Remark 1.8.
If is a real Banach space with cone
and if
where
and
then
Proof.
The condition means that
that is,
Since
and
then also
Thus we have
and
Remark 1.9.
Let be a cone metric space. Let us remark that the family
, where
, is a subbasis for topology on
. We denote this cone topology by
, and note that
is a Hausdorff topology (see, e.g., [11] without proof).
For the proof of the last statement, we suppose that for each ,
we have
. Thus, there exists
such that
and
. Hence,
. Clearly, for each
, we have
, so
. Now,
, that is,
, and we have
.
We find it convenient to introduce the following definition.
Definition 1.10.
Let be a cone metric space and
a cone with nonempty interior. Suppose that the mappings
are such that the range of
contains the range of
, and
or
is a complete subspace of
. In this case we will say that the pair
is Abbas and Jungck's pair, or shortly AJ's pair.
Definition 1.11 (see [1]).
Let and
be self-maps of a set
(i.e.,
)
If
for some
in
then
is called a coincidence point of
and
and
is called a point of coincidence of
and
Self-maps
and
are said to be weakly compatible if they commute at their coincidence point, that is, if
for some
then
Proposition 1.12 (see [1]).
Let and
be weakly compatible self-maps of a set
If
and
have a unique point of coincidence
then
is the unique common fixed point of
and
2. Main Results
In this section we will prove some fixed point theorems of contractive mappings for cone metric space. We generalize some results of [1–4] by omitting the assumption of normality in the results.
Theorem 2.1.
Suppose that is AJ's pair, and that for some constant
and for every
there exists
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F643840/MediaObjects/13663_2008_Article_1166_Equ2_HTML.gif)
such that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F643840/MediaObjects/13663_2008_Article_1166_Equ3_HTML.gif)
Then and
have a unique coincidence point in
. Moreover if
and
are weakly compatible,
and
have a unique common fixed point.
Proof.
Let and let
be such that
. Having defined
let
be such that
We first show that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F643840/MediaObjects/13663_2008_Article_1166_Equ4_HTML.gif)
We have that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F643840/MediaObjects/13663_2008_Article_1166_Equ5_HTML.gif)
where
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F643840/MediaObjects/13663_2008_Article_1166_Equ6_HTML.gif)
Now we have to consider the following three cases.
If then clearly (2.3) holds. If
then according to Remark 1.8
and (2.3) is immediate. Finally, suppose that
Now,
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F643840/MediaObjects/13663_2008_Article_1166_Equ7_HTML.gif)
Hence, , and we proved (2.3).
Now, we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F643840/MediaObjects/13663_2008_Article_1166_Equ8_HTML.gif)
We will show that is a Cauchy sequence. For
, we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F643840/MediaObjects/13663_2008_Article_1166_Equ9_HTML.gif)
and we obtain
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F643840/MediaObjects/13663_2008_Article_1166_Equ10_HTML.gif)
From Remark 1.5 it follows that for and large
thus, according to Corollary 1.4(1),
Hence, by Definition 1.2(e),
is a Cauchy sequence. Since
and
or
is complete, there exists a
such that
as
Consequently, we can find
such that
Let us show that For this we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F643840/MediaObjects/13663_2008_Article_1166_Equ11_HTML.gif)
where
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F643840/MediaObjects/13663_2008_Article_1166_Equ12_HTML.gif)
Let Clearly at least one of the following four cases holds for infinitely many
.
(case 10)
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F643840/MediaObjects/13663_2008_Article_1166_Equ13_HTML.gif)
(case 20)
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F643840/MediaObjects/13663_2008_Article_1166_Equ14_HTML.gif)
(case 30)
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F643840/MediaObjects/13663_2008_Article_1166_Equ15_HTML.gif)
(case 40)
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F643840/MediaObjects/13663_2008_Article_1166_Equ16_HTML.gif)
In all cases, we obtain for each
Using Corollary 1.4(3), it follows that
or
Hence, we proved that and
have a coincidence point
and a point of coincidence
such that
If
is another point of coincidence, then there is
with
Now,
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F643840/MediaObjects/13663_2008_Article_1166_Equ17_HTML.gif)
where
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F643840/MediaObjects/13663_2008_Article_1166_Equ18_HTML.gif)
Hence, that is,
Since is the unique point of coincidence of
and
and
and
are weakly compatible,
is the unique common fixed point of
and
by Proposition 1.12 [1].
In the next theorem, among other things, we generalize the well-known Zamfirescu result [12, ()].
Theorem 2.2.
Suppose that is AJ's pair, and that for some constant
and for every
there exists
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F643840/MediaObjects/13663_2008_Article_1166_Equ19_HTML.gif)
such that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F643840/MediaObjects/13663_2008_Article_1166_Equ20_HTML.gif)
Then and
have a unique coincidence point in
. Moreover, if
and
are weakly compatible,
and
have a unique common fixed point.
Proof.
Let and let
be such that
. Having defined
let
be such that
We first show that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F643840/MediaObjects/13663_2008_Article_1166_Equ21_HTML.gif)
Notice that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F643840/MediaObjects/13663_2008_Article_1166_Equ22_HTML.gif)
where
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F643840/MediaObjects/13663_2008_Article_1166_Equ23_HTML.gif)
As in Theorem 2.1, we have to consider three cases.
If , then clearly (2.20) holds. If
then from (2.19) with
and
as
we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F643840/MediaObjects/13663_2008_Article_1166_Equ24_HTML.gif)
Hence, and in this case (2.20) holds. Finally, if
then
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F643840/MediaObjects/13663_2008_Article_1166_Equ25_HTML.gif)
and (2.20) holds. Thus, we proved that in all three cases (2.20) holds.
Now, from the proof of Theorem 2.1, we know that is a Cauchy sequence. Hence, there exist
in
and
such that
,
and
Now we have to show that For this we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F643840/MediaObjects/13663_2008_Article_1166_Equ26_HTML.gif)
where
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F643840/MediaObjects/13663_2008_Article_1166_Equ27_HTML.gif)
Let Clearly at least one of the following three cases holds for infinitely many
.
(case 10)
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F643840/MediaObjects/13663_2008_Article_1166_Equ28_HTML.gif)
(case 20)
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F643840/MediaObjects/13663_2008_Article_1166_Equ29_HTML.gif)
(case 30)
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F643840/MediaObjects/13663_2008_Article_1166_Equ30_HTML.gif)
In all cases we obtain for each
Using Corollary 1.4(3), it follows that
or
Thus we showed that and
have a coincidence point
that is, point of coincidence
such that
If
is another point of coincidence then there is
with
Now from (2.19), it follows that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F643840/MediaObjects/13663_2008_Article_1166_Equ31_HTML.gif)
where
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F643840/MediaObjects/13663_2008_Article_1166_Equ32_HTML.gif)
Hence, that is,
If
and
are weakly compatible, then as in the proof of Theorem 2.1, we have that
is a unique common fixed point of
and
The assertion of the theorem follows.
Now as corollaries, we recover and generalize the recent results of Huang and Zhang [2], Abbas and Jungck [1], and Vetro [3]. Furthermore as corollaries, we obtain recent results of Rezapour and Hamlborani [4].
Corollary 2.3.
Suppose that is AJ's pair, and that for some constant
and for every
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F643840/MediaObjects/13663_2008_Article_1166_Equ33_HTML.gif)
Then and
have a unique coincidence point in
. Moreover if
and
are weakly compatible,
and
have a unique common fixed point.
Corollary 2.4.
Suppose that is AJ's pair, and that for some constant
and for every
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F643840/MediaObjects/13663_2008_Article_1166_Equ34_HTML.gif)
Then and
have a unique coincidence point in
. Moreover if
and
are weakly compatible,
and
have a unique common fixed point.
Corollary 2.5.
Suppose that is AJ's pair, and that for some constant
and for every
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F643840/MediaObjects/13663_2008_Article_1166_Equ35_HTML.gif)
Then and
have a unique coincidence point in
. Moreover if
and
are weakly compatible,
and
have a unique common fixed point.
In the next corollary, among other things, we generalize the well-known result [12, ()].
Corollary 2.6.
Suppose that is AJ's pair, and that for some constant
and for every
there exists
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F643840/MediaObjects/13663_2008_Article_1166_Equ36_HTML.gif)
such that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F643840/MediaObjects/13663_2008_Article_1166_Equ37_HTML.gif)
Then and
have a unique coincidence point in
. Moreover if
and
are weakly compatible,
and
have a unique common fixed point.
Now, we generalize the well-known Bianchini result [12, (5)].
Corollary 2.7.
Suppose that is AJ's pair, and that for some constant
and for every
there exists
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F643840/MediaObjects/13663_2008_Article_1166_Equ38_HTML.gif)
such that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F643840/MediaObjects/13663_2008_Article_1166_Equ39_HTML.gif)
Then and
have a unique coincidence point in
. Moreover if
and
are weakly compatible,
and
have a unique common fixed point.
When in the next theorem we set the identity map on
and
, we get the theorem of Hardy and Rogers [12, (18)].
Theorem 2.8.
Suppose that is AJ's pair, and that there exist nonnegative constants
satisfying
such that, for each
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F643840/MediaObjects/13663_2008_Article_1166_Equ40_HTML.gif)
Then and
have a unique coincidence point in
. Moreover if
and
are weakly compatible,
and
have a unique common fixed point.
Proof.
Let us define the sequences and
as in the proof of Theorem 2.1 We have to show that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F643840/MediaObjects/13663_2008_Article_1166_Equ41_HTML.gif)
From
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F643840/MediaObjects/13663_2008_Article_1166_Equ42_HTML.gif)
we obtain
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F643840/MediaObjects/13663_2008_Article_1166_Equ43_HTML.gif)
Thus,
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F643840/MediaObjects/13663_2008_Article_1166_Equ44_HTML.gif)
where and we proved (2.40).
Now, from the proof of Theorem 2.1, we know that is a Cauchy sequence. Hence, there exist
in
and
such that
,
and
We have to show that For this we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F643840/MediaObjects/13663_2008_Article_1166_Equ45_HTML.gif)
Then according to Corollary 1.4(3), , that is,
Thus we showed that and
have a coincidence point
that is, point of coincidence
such that
If
is another point of coincidence then there is
with
Now,
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F643840/MediaObjects/13663_2008_Article_1166_Equ46_HTML.gif)
According to Remark 1.8, and because we get
that is,
If
and
are weakly compatible, then as in the proof of Theorem 2.1, we have that
is a unique common fixed point of
and
The assertion of the theorem follows.
It is clear that, for the special choice of in Theorem 2.8, all the results from Corollaries 2.3, 2.4, and 2.5, could be obtained.
Finally, we add an example with Banach type contraction on non-normal cone metric space (see Corollary 2.3).
Example 2.9.
Let , and
Define
by
where
such that
It is easy to see that
is a cone metric on
Consider the mappings
in the following manner:
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F643840/MediaObjects/13663_2008_Article_1166_Equ47_HTML.gif)
where One can see that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F643840/MediaObjects/13663_2008_Article_1166_Equ48_HTML.gif)
for all where
The mappings
and
commute at
the only coincidence point. So
and
are weakly compatible. All the conditions of the Corollary 2.3 hold, then
and
have a common fixed point.
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Acknowledgments
The fourth author would like to express his gratitude to Professor Sh. Rezapour and to Professor S. M. Veazpour for the valuable comments. The second, third, and fourth authors thank the Ministry of Science and the Ministry of Environmental Protection of Serbia for their support.
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Jungck, G., Radenovic, S., Radojevic, S. et al. Common Fixed Point Theorems for Weakly Compatible Pairs on Cone Metric Spaces. Fixed Point Theory Appl 2009, 643840 (2009). https://doi.org/10.1155/2009/643840
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DOI: https://doi.org/10.1155/2009/643840