Open Access

Common Fixed Point Theorems for Weakly Compatible Pairs on Cone Metric Spaces

Fixed Point Theory and Applications20092009:643840

DOI: 10.1155/2009/643840

Received: 17 December 2008

Accepted: 4 February 2009

Published: 18 February 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 [13], 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, 49]) 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
(1.1)

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. (1)

    If and then

     
Indeed, implies
  1. (2)

    If and then

     
Indeed, implies
  1. (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 [14] 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
(2.1)
such that
(2.2)

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
(2.3)
We have that
(2.4)
where
(2.5)

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,
(2.6)

Hence, , and we proved (2.3).

Now, we have
(2.7)
We will show that is a Cauchy sequence. For , we have
(2.8)
and we obtain
(2.9)

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
(2.10)
where
(2.11)

Let Clearly at least one of the following four cases holds for infinitely many .

(case 10)

(2.12)

(case 20)

(2.13)

(case 30)

(2.14)

(case 40)

(2.15)

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,
(2.16)
where
(2.17)

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
(2.18)
such that
(2.19)

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
(2.20)
Notice that
(2.21)
where
(2.22)

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
(2.23)
Hence, and in this case (2.20) holds. Finally, if then
(2.24)

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
(2.25)
where
(2.26)

Let Clearly at least one of the following three cases holds for infinitely many .

(case 10)

(2.27)

(case 20)

(2.28)

(case 30)

(2.29)

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
(2.30)
where
(2.31)

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

(2.32)

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

(2.33)

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

(2.34)

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
(2.35)
such that
(2.36)

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
(2.37)
such that
(2.38)

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
(2.39)

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
(2.40)
From
(2.41)
we obtain
(2.42)
Thus,
(2.43)

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
(2.44)

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,
(2.45)

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:
(2.46)
where One can see that
(2.47)

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.

Declarations

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.

Authors’ Affiliations

(1)
Department of Mathematics, Bradley University
(2)
Faculty of Mechanical Engineering, University of Belgrade
(3)
Department of Mathematics, Faculty of Science and Mathematics, University of Niš

References

  1. 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.2007.09.070MathSciNetView ArticleMATHGoogle Scholar
  2. 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.2005.03.087MathSciNetView ArticleMATHGoogle Scholar
  3. Vetro P: Common fixed points in cone metric spaces. Rendiconti del Circolo Matematico di Palermo 2007,56(3):464–468. 10.1007/BF03032097MathSciNetView ArticleMATHGoogle Scholar
  4. 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.2008.04.049MathSciNetView ArticleMATHGoogle Scholar
  5. Aliprantis CD, Tourky R: Cones and Duality, Graduate Studies in Mathematics. Volume 84. American Mathematical Society, Providence, RI, USA; 2007:xiv+279.MATHGoogle Scholar
  6. Mohebi H: Topical functions and their properties in a class of ordered Banach spaces. In Continuous Optimization, Applied Optimization. Volume 99. Springer, New York, NY, USA; 2005:343–361. 10.1007/0-387-26771-9_12View ArticleGoogle Scholar
  7. Raja P, Vaezpour SM: Some extensions of Banach's contraction principle in complete cone metric spaces. Fixed Point Theory and Applications 2008, -11.Google Scholar
  8. Ilic D, Rakocevic 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.2007.10.065MathSciNetView ArticleMATHGoogle Scholar
  9. Ilic D, Rakocevic V: Quasi-contraction on a cone metric spacestar, open. Applied Mathematics Letters 2009,22(5):728–731. 10.1016/j.aml.2008.08.011MathSciNetView ArticleMATHGoogle Scholar
  10. Wong Y-C, Ng K-F: Partially Ordered Topological Vector Spaces, Oxford Mathematical Monograph. Clarendon Press, Oxford, UK; 1973:x+217.Google Scholar
  11. Rezapour Sh: A review on topological properties of cone metric spaces. Analysis, Topology and Applications (ATA '08), May-June 2008, Vrnjacka Banja, SerbiaGoogle Scholar
  12. Rhoades BE: A comparison of various definitions of contractive mappings. Transactions of the American Mathematical Society 1977, 226: 257–290.MathSciNetView ArticleMATHGoogle Scholar

Copyright

© G. Jungck et al. 2009

This article is published under license to BioMed Central Ltd. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.