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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 
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 
and 
then 
Indeed, 
implies 
-
(2)
If
and 
then 
and 
then 
Indeed, 
implies 
-
(3)
If
for each 
, then 
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
such that
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
We have that
where
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,
Hence, 
, and we proved (2.3).
Now, we have
We will show that 
is a Cauchy sequence. For 
, we have
and we obtain
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
where
Let 
Clearly at least one of the following four cases holds for infinitely many 
.
(case 10)
(case 20)
(case 30)
(case 40)
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,
where
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
such that
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
Notice that
where
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
Hence, 
and in this case (2.20) holds. Finally, if 
then
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
where
Let 
Clearly at least one of the following three cases holds for infinitely many 
.
(case 10)
(case 20)
(case 30)
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
where
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 
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 
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 
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
such that
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
such that
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 
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
From
we obtain
Thus,
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
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,
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:
where 
One can see that
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