Open Access

Common Fixed Point Theorems for Four Mappings on Cone Metric Type Space

  • AleksandarS Cvetković1Email author,
  • MarijaP Stanić2,
  • Sladjana Dimitrijević2 and
  • Suzana Simić2
Fixed Point Theory and Applications20112011:589725

https://doi.org/10.1155/2011/589725

Received: 9 December 2010

Accepted: 3 February 2011

Published: 27 February 2011

Abstract

In this paper we consider the so called a cone metric type space, which is a generalization of a cone metric space. We prove some common fixed point theorems for four mappings in those spaces. Obtained results extend and generalize well-known comparable results in the literature. All results are proved in the settings of a solid cone, without the assumption of continuity of mappings.

1. Introduction

Replacing the real numbers, as the codomain of a metric, by an ordered Banach space we obtain a generalization of metric space. Such a generalized space, called a cone metric space, was introduced by Huang and Zhang in [1]. They described the convergence in cone metric space, introduced their completeness, and proved some fixed point theorems for contractive mappings on cone metric space. Cones and ordered normed spaces have some applications in optimization theory (see [2]). The initial work of Huang and Zhang [1] inspired many authors to prove fixed point theorems, as well as common fixed point theorems for two or more mappings on cone metric space, for example, [314].

In this paper we consider the so-called a cone metric type space, which is a generalization of a cone metric space and prove some common fixed point theorems for four mappings in those spaces. Obtained results are generalization of theorems proved in [13]. For some special choices of mappings we obtain theorems which generalize results from [1, 8, 15]. All results are proved in the settings of a solid cone, without the assumption of continuity of mappings.

The paper is organized as follows. In Section 2 we repeat some definitions and well-known results which will be needed in the sequel. In Section 3 we prove common fixed point theorems. Also, we presented some corollaries which show that our results are generalization of some existing results in the literature.

2. Definitions and Notation

Let be a real Banach space and a subset of . By we denote zero element of and by the interior of . The subset is called a cone if and only if

(i) is closed, nonempty and ;

(ii) , , and imply ;

(iii) .

For a given cone , a partial ordering with respect to is introduced in the following way: if and only if . One writes to indicate that , but . If , one writes .

If , the cone is called solid.

In the sequel we always suppose that is a real Banach space, is a solid cone in , and is partial ordering with respect to .

Analogously with definition of metric type space, given in [16], we consider cone metric type space.

Definition 2.1.

Let be a nonempty set and a real Banach space with cone . A vector-valued function is said to be a cone metric type function on with constant if the following conditions are satisfied:

() for all and if and only if ;

() for all ;

() for all .

The pair is called a cone metric type space (in brief CMTS).

Remark 2.2.

For in Definition 2.1 we obtain a cone metric space introduced in [1].

Definition 2.3.

Let be a CMTS and a sequence in .

() converges to if for every with there exists such that for all . We write , or , .

()If for every with there exists such that for all , then is called a Cauchy sequence in .

If every Cauchy sequence is convergent in , then is called a complete CMTS.

Example 2.4.

Let be orthonormal basis of with inner product . Let , and define
(2.1)
where represents class of element with respect to equivalence relation of functions equal almost everywhere. We choose and
(2.2)

We show that is a solid cone. Let , , with property . Since scalar product is continuous, we get , . Clearly, it must be , , and, hence, , that is, is closed. It is obvious that , and for , and all , we have , . Finally, if we have and , , and it follows that , , and, since is complete, we get . Let us choose . It is obvious that , since if not, for every there exists such that . If we choose , we conclude that it must be , hence , which is contradiction.

Finally, define by
(2.3)

Then it is obvious that is CMTS with . Let , , be functions such that , , , and , , with give , , and , which proves , but .

The following properties are well known in the case of a cone metric space, and it is easy to see that they hold also in the case of a CMTS.

Lemma 2.5.

Let be a CMTS over-ordered real Banach space with a cone . The following properties hold .

()If and , then .

()If for all , then .

()If , where and , then .

()Let in and let . Then there exists positive integer such that for each .

Definition 2.6 (see [17]).

Let be mappings of a set . If for some , then is called a coincidence point of and , and is called a point of coincidence of and .

Definition 2.7 (see [17]).

Let and be self-mappings of set and . The pair is called weakly compatible if mappings and commute at all their coincidence points, that is, if for all .

Lemma 2.8 (see [5]).

Let and be weakly compatible self-mappings of a set . If and have a unique point of coincidence , then is the unique common fixed point of and .

3. Main Results

Theorem 3.1.

Let be a CMTS with constant and a solid cone. Suppose that self-mappings are such that , and that for some constant for all there exists
(3.1)
such that the following inequality
(3.2)

holds. If one of , , , or is complete subspace of , then and have a unique point of coincidence in . Moreover, if and are weakly compatible pairs, then , , , and have a unique common fixed point.

Proof.

Let us choose arbitrary. Since , there exists such that . Since , there exists such that . We continue in this manner. In general, is chosen such that , and is chosen such that .

First we prove that
(3.3)

where , which will lead us to the conclusion that is a Cauchy sequence, since (it is easy to see that ). To prove this, it is necessary to consider the cases of an odd integer and of an even .

For , , we have , and from (3.2) there exists
(3.4)

such that . Thus we have the following three cases:

(i) ;

(ii) , which, because of property , implies ;

(iii) , that is, by using ,
(3.5)

which implies .

Thus, inequality (3.3) holds in this case.

For , , we have
(3.6)
where
(3.7)

Thus we have the following three cases:

(i) ;

(ii) , which implies ;

(iii) , which implies .

So, inequality (3.3) is satisfied in this case, too.

Therefore, (3.3) is satisfied for all , and by iterating we get
(3.8)
Since , for we have
(3.9)

Now, by and , it follows that for every there exists positive integer such that for every , so is a Cauchy sequence.

Let us suppose that is complete subspace of . Completeness of implies existence of such that . Then, we have
(3.10)
that is, for any , for sufficiently large we have . Since , there exists such that . Let us prove that . From and (3.2), we have
(3.11)
where
(3.12)

Therefore we have the following four cases:

(i) , as ;

(ii) , as ;

(iii) , that is,
(3.13)
(iv) , that is, because of ,
(3.14)
which implies
(3.15)

since from and we have , and therefore .

Therefore, for each . So, by we have , that is, , is a coincidence point, and is a point of coincidence of and .

Since , there exists such that . Let us prove that . From and (3.2), we have
(3.16)
where
(3.17)

Therefore we have the following four cases:

(i) ;

(ii) ;

(iii) ;

(iv) .

By the same arguments as above, we conclude that , that is, . So, is a point of coincidence of and , too.

Now we prove that is unique point of coincidence of pairs and . Suppose that there exists which is also a point of coincidence of these four mappings, that is, . From (3.2),
(3.18)
where
(3.19)

Using we get , that is, . Therefore, is the unique point of coincidence of pairs and . If these pairs are weakly compatible, then is the unique common fixed point of , , , and , by Lemma 2.8.

Similarly, we can prove the statement in the cases when , , or is complete.

We give one simple, but illustrative, example.

Example 3.2.

Let , , and . Let us define for all . Then is a CMTS, but it is not a cone metric space since the triangle inequality is not satisfied. Starting with Minkowski inequality (see [18]) for , by using the inequality of arithmetic and geometric means, we get
(3.20)

Here, .

Let us define four mappings as follows:
(3.21)

where , , , and . Since we have trivially and . Also, is a complete space. Further, , that is, there exists such that (3.2) is satisfied.

According to Theorem 3.1, and have a unique point of coincidence in , that is, there exists unique and there exist such that . It is easy to see that , , and .

If is weakly compatible pair, we have , which implies , that is, . Similarly, if is weakly compatible pair, we have , which implies , that is, . Then , and is the unique common fixed point of these four mappings.

The following two theorems can be proved in the same way as Theorem 3.1, so we omit the proofs.

Theorem 3.3.

Let be a CMTS with constant and a solid cone. Suppose that self-mappings are such that , and that for some constant for all there exists
(3.22)
such that the following inequality
(3.23)

holds. If one of , , , or is complete subspace of , then and have a unique point of coincidence in . Moreover, if and are weakly compatible pairs, then , , , and have a unique common fixed point.

Theorem 3.4.

Let be a CMTS with constant and a solid cone. Suppose that self-mappings are such that , and that for some constant for all there exists
(3.24)
such that the following inequality
(3.25)

holds. If one of , , , or is complete subspace of , then and have a unique point of coincidence in . Moreover, if and are weakly compatible pairs, then , , , and have a unique common fixed point.

Theorems 3.1 and 3.4 are generalizations of [13, Theorem  2.2]. As a matter of fact, for , from Theorems 3.1 and 3.4, we get [13, Theorem  2.2].

If we choose and , from Theorems 3.1, 3.3, and 3.4 we get the following results for two mappings on CMTS.

Corollary 3.5.

Let be a CMTS with constant and a solid cone. Suppose that self-mappings are such that and that for some constant for all there exists
(3.26)
such that the following inequality
(3.27)

holds. If or is complete subspace of , then and have a unique point of coincidence in . Moreover, if is a weakly compatible pair, then and have a unique common fixed point.

Corollary 3.6.

Let be a CMTS with constant and a solid cone. Suppose that self-mappings are such that and that for some constant for all there exists
(3.28)
such that the following inequality
(3.29)

holds. If or is complete subspace of , then and have a unique point of coincidence in . Moreover, if is a weakly compatible pair, then and have a unique common fixed point.

Corollary 3.7.

Let be a CMTS with constant and a solid cone. Suppose that self-mappings are such that and that for some constant for all there exists
(3.30)
such that the following inequality
(3.31)

holds. If or is complete subspace of , then and have a unique point of coincidence in . Moreover, if is a weakly compatible pair, then and have a unique common fixed point.

Theorem 3.8.

Let be a CMTS with constant and a solid cone. Suppose that self-mappings are such that , and that there exist nonnegative constants , , satisfying
(3.32)
such that for all inequality
(3.33)

holds. If one of , , , or is complete subspace of , then and have a unique point of coincidence in . Moreover, if and are weakly compatible pairs, then , , , and have a unique common fixed point.

Proof.

We define sequences and as in the proof of Theorem 3.1. First we prove that
(3.34)
where
(3.35)

which implies that is a Cauchy sequence, since, because of (3.32), it is easy to check that . To prove this, it is necessary to consider the cases of an odd and of an even integer .

For , , we have , and from (3.33) we have
(3.36)
that is,
(3.37)
Therefore,
(3.38)

that is, inequality (3.34) holds in this case.

Similarly, for , , we have , and from (3.33) we have
(3.39)
that is,
(3.40)
Thus,
(3.41)

and inequality (3.34) holds in this case, too.

By the same arguments as in Theorem 3.1 we conclude that is a Cauchy sequence.

Let us suppose that is complete subspace of . Completeness of implies existence of such that . Then, we have
(3.42)
that is, for any , for sufficiently large we have . Since , there exists such that . Let us prove that . From and (3.33), we have
(3.43)
The sequence converges to , so for each there exists such that for every
(3.44)

because of (3.32). Now, by it follows that , that is, . So, we have , that is, is a coincidence point, and is a point of coincidence of mappings and .

Since , there exists such that . Let us prove that , too. From and (3.33), we have
(3.45)

and by the same arguments as above, we conclude that , that is, . Thus, is a point of coincidence of mappings and , too.

Suppose that there exists which is also a point of coincidence of these four mappings, that is, . From (3.33) we have
(3.46)

and (because of ) it follows that . Therefore, is the unique point of coincidence of pairs and , and we have . If and are weakly compatible pairs, then is the unique common fixed point of , , , and , by Lemma 2.8.

The proofs for the cases in which , , or is complete are similar.

Theorem 3.8 is a generalization of [13, Theorem  2.8]. Choosing from Theorem 3.8 we get the following corollary.

Corollary 3.9.

Let be cone metric space and a solid cone. Suppose that self-mappings are such that , and that there exist nonnegative constants , , satisfying , such that for all inequality
(3.47)

holds. If one of , , , or is complete subspace of , then and have a unique point of coincidence in . Moreover, if and are weakly compatible pairs, then , , , and have a unique common fixed point.

If we choose and , from Theorem 3.8, we get the following result for two mappings on CMTS.

Corollary 3.10.

Let be a CMTS with constant and a solid cone. Suppose that self-mappings are such that and that there exist nonnegative constants , , satisfying
(3.48)
such that for all inequality
(3.49)

holds. If one of or is complete subspace of , then and have a unique point of coincidence in . Moreover, if is a weakly compatible pair, then and have a unique common fixed point.

Declarations

Acknowledgments

The authors are indebted to the referees for their valuable suggestions, which have contributed to improve the presentation of the paper. The first two authors were supported in part by the Serbian Ministry of Science and Technological Developments (Grant no. 174015).

Authors’ Affiliations

(1)
Department of Mathematics, Faculty of Mechanical Engineering, University of Belgrade
(2)
Department of Mathematics and Informatics, Faculty of Science, University of Kragujevac

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© Aleksandar S. Cvetković et al. 2011

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.