- F Sabetghadam
^{1}and - HP Masiha
^{1}Email author

**2010**:718340

https://doi.org/10.1155/2010/718340

© F. Sabetghadam and H. P. Masiha. 2010

**Received: **19 October 2009

**Accepted: **10 January 2010

**Published: **24 February 2010

## Abstract

## 1. Introduction

Huang and Zhang [1] recently introduced the concept of cone metric spaces and established some fixed point theorems for contractive mappings in these spaces. Afterwards, Rezapour and Hamlbarani [2] studied fixed point theorems of contractive type mappings by omitting the assumption of normality in cone metric spaces. Also, other authors proved the existence of points of coincidence, common fixed point, and coupled fixed point for mappings satisfying different contraction conditions in cone metric spaces (see [1–12]). In [6] Di Bari and Vetro introduced the concept of -map and proved a main theorem generalizing some known results. We define the concept of generalized -mappings and prove some results about common fixed points for such mappings. Our results generalize some results of Huang and Zhang [1], Di Bari and Vetro [6], and Abbas and Jungck [3]. First, we recall some standard notations and definitions in cone metric spaces.

Let be a real Banach space and let denote the zero element in . A cone is a subset of such that

(i) is closed, nonempty, and ,

(ii)if are nonnegative real numbers and , then ,

For a given cone , the partial ordering with respect to is defined by if and only if . The notation will stand for but . Also, we will use to indicate that where denotes the interior of . Using these notations, we have the following definition of a cone metric space.

Definition 1.1 (see [1]).

Let be a nonempty set and let be a real Banach space equipped with the partial ordering with respect to the cone . Suppose that the mapping satisfies the following conditions:

Then is called a cone metric on , and is called a cone metric space.

The cone is called normal if there exists a constant such that for every if then . The least positive number satisfying this inequality is called the normal constant of . The cone is called regular if every increasing (decreasing) and bounded above (below) sequence is convergent in . It is known that every regular cone is normal [1] (see also [2, Lemma ]).

Definition 1.2 (see [1]).

Let be a cone metric space, let be a sequence in and let .

(i) is said to be Cauchy sequence if for every with there exists such that for all , .

(ii) is said to be convergent to , denoted by or as if for every with there exists such that for all , .

(iii) is said to be complete if every Cauchy sequence in is convergent in .

(iv) is said to be sequentially compact if for every sequence in there exists a subsequence of such that is convergent in .

Clearly, every sequentially compact cone metric space is complete (see [1–12]) for more related results about complete cone metric spaces). We also note that the relations and always hold true.

Definition 1.3 (see [13]).

Let and be self-mappings of a cone metric space . One says that and are compatible if , whenever is a sequence in such that for some .

The concept of weakly compatible mappings is introduced as follows.

Definition 1.4 (see [13]).

The self-mappings and of a cone metric space are said to be weakly compatible if they commute at their coincidence points, that is, if for some , then .

## 2. Main Results

In this section, we introduce the notation of generalized -mapping and a contractive condition called generalized -pair. We prove some results on common fixed points of these mappings on cone metric spaces.

Let be a cone. A nondecreasing mapping is called a -mapping [6] if

Definition 2.1.

Let be a cone and let be a sequence in . One says that if for every with there exists such that for all .

For a nondecreasing mapping we define the following conditions which will be used in the sequal:

Definition 2.2.

Now, we are in the position to state the following theorem.

Theorem 2.3.

Let be a cone metric space and let be a generalized -pair. Suppose that and are weakly compatible with such that or is complete. Then the self-mappings and have a unique common fixed point in .

Proof.

which implies that . So is a common fixed point of and . The uniqueness of the common fixed point is clear.

Example 2.4.

Let and let be a normal cone. Let with usual metric . Define by and , for all . Also, define by and , for all . Then

(1) and are weakly compatible,

Example 2.5.

Let and let be a normal cone. Let with metric . Define by and , for all . Also, define by and , for all . Then

(1) and are weakly compatible,

Example 2.6.

Let and let be a normal cone. Let with metric . Define by and , for all . Also, define by and , for all . Then

(1) and are weakly compatible,

If we let the mapping be the identity mapping in Theorem 2.3, then we obtain the following corollary.

Corollary 2.7.

for all . If , and are weakly compatible, and or is complete, then and have a unique common fixed point in .

Remark 2.8.

Corollary 2.7 generalizes Theorem in [6]. Also, if we choose the -mapping defined by , where is a constant, then Theorem 2.3 generalizes Theorem in [3]. Furthermore, if we let be the identity map of , then we obtain Theorem in [1], that is, the extension of the Banach fixed point theorem for cone metric spaces.

If we replace the condition with the following condition:

there exists such that for and , then we have the following theorems.

Theorem 2.9.

for all where is a nondecreasing mapping from into satisfying the conditions , and and is a nondecreasing mapping satisfying the conditions . Suppose that and are weakly compatible, and or is complete. Then the mappings and have a unique common fixed point in .

Proof.

which gives and hence, . So is a common fixed point for and . The uniqueness of common fixed point is clear.

If in Theorem 2.9 we let be and let the -mapping be , where is a constant, then we obtain the following corollary.

Corollary 2.10.

for all , where is a constant. Suppose that and are weakly compatible, the range of contains the range of and or is complete. Then the mappings and have a unique common fixed point in .

Remark 2.11.

Corollary 2.10 generalizes Theorem of [3]. If in Corollary 2.10 we let be the identity map on , then we obtain Theorem of [1].

Theorem 2.12.

for all . Suppose that and are weakly compatible, the range of contains the range of and or is complete. Then the mappings and have a unique common fixed point in .

Proof.

Following an argument similar to that one given in Theorem 2.9, we obtain a unique common fixed point of and .

If in Theorem 2.12 we let be the identity map on and let the -map be , where is a constant, then we obtain the following corollary.

Corollary 2.13.

for all , where is a constant. Suppose that and are weakly compatible, the range of contains the range of and or is complete. Then the mappings and have a unique common fixed point in .

Remark 2.14.

Corollary 2.13 generalizes Theorem of [3] and if in Corollary 2.13 we let be the identity map on , then we obtain Theorem of [1].

## Declarations

### Acknowledgment

The authors would like to thank the referees for their valuable and useful comments.

## Authors’ Affiliations

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