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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
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 nonnormal 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 selfmaps 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 Selfmaps 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 selfmaps 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 1^{0})
(case 2^{0})
(case 3^{0})
(case 4^{0})
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 wellknown 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 1^{0})
(case 2^{0})
(case 3^{0})
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 wellknown 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 wellknown 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 nonnormal 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