Open Access

Some Common Fixed Point Results in Cone Metric Spaces

Fixed Point Theory and Applications20092009:493965

DOI: 10.1155/2009/493965

Received: 5 September 2008

Accepted: 5 February 2009

Published: 18 February 2009

Abstract

We prove a result on points of coincidence and common fixed points for three self-mappings satisfying generalized contractive type conditions in cone metric spaces. We deduce some results on common fixed points for two self-mappings satisfying contractive type conditions in cone metric spaces. These results generalize some well-known recent results.

1. Introduction

Huang and Zhang [1] recently have introduced the concept of cone metric space, where the set of real numbers is replaced by an ordered Banach space, and they have established some fixed point theorems for contractive type mappings in a normal cone metric space. Subsequently, some other authors [25] have generalized the results of Huang and Zhang [1] and have studied the existence of common fixed points of a pair of self mappings satisfying a contractive type condition in the framework of normal cone metric spaces.

Vetro [5] extends the results of Abbas and Jungck [2] and obtains common fixed point of two mappings satisfying a more general contractive type condition. Rezapour and Hamlbarani [6] prove that there aren't normal cones with normal constant and for each there are cones with normal constant . Also, omitting the assumption of normality they obtain generalizations of some results of [1]. In [7] Di Bari and Vetro obtain results on points of coincidence and common fixed points in nonnormal cone metric spaces. In this paper, we obtain points of coincidence and common fixed points for three self-mappings satisfying generalized contractive type conditions in a complete cone metric space. Our results improve and generalize the results in [1, 2, 5, 6, 8].

2. Preliminaries

We recall the definition of cone metric spaces and the notion of convergence [1]. Let be a real Banach space and be a subset of . The subset is called an order cone if it has the following properties:

(i) is nonempty, closed, and

(ii) and

(iii)

For a given cone , we can define a partial ordering on with respect to by if and only if . We will write if and , while will stands for , where denotes the interior of The cone is called normal if there is a number such that for all
(2.1)

The least number satisfying (2.1) is called the normal constant of

In the following we always suppose that is a real Banach space and is an order cone in with and is the partial ordering with respect to

Definition 2.1.

Let be a nonempty set. Suppose that the mapping satisfies

(i) for all and if and only if ;

(ii) for all ;

(iii) , for all

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

Let be a sequence in , and . If for every with there is such that for all then is said to be convergent, converges to and is the limit of We denote this by or as If for every with there is such that for all then is called a Cauchy sequence in . If every Cauchy sequence is convergent in , then is called a complete cone metric space.

3. Main Results

First, we establish the result on points of coincidence and common fixed points for three self-mappings and then show that this result generalizes some of recent results of fixed point.

A pair of self-mappings on is said to be weakly compatible if they commute at their coincidence point (i.e., whenever ). A point is called point of coincidence of a family , , of self-mappings on if there exists a point such that for all .

Lemma 3.1.

Let be a nonempty set and the mappings have a unique point of coincidence in If and are weakly compatibles, then , and have a unique common fixed point.

Proof.

Since is a point of coincidence of , and . Therefore, for some By weakly compatibility of and we have
(3.1)

It implies that (say). Then is a point of coincidence of , and . Therefore, by uniqueness. Thus is a unique common fixed point of , and

Let be a cone metric space, be self-mappings on such that and . Choose a point in such that . This can be done since . Successively, choose a point in such that Continuing this process having chosen , we choose and in such that
(3.2)

The sequence is called an - -sequence with initial point .

Proposition 3.2.

Let be a cone metric space and be an order cone. Let be such that . Assume that the following conditions hold:

(i) , for all , with , where are nonnegative real numbers with ;

(ii) , for all , whenever .

Then every - -sequence with initial point is a Cauchy sequence.

Proof.

Let be an arbitrary point in and be an - -sequence with initial point . First, we assume that for all . It implies that for all . Then,
(3.3)
It implies that
(3.4)
so
(3.5)
Similarly, we obtain
(3.6)
Now, by induction, for each we deduce
(3.7)
Let
(3.8)
Then Now, for , we have
(3.9)
In analogous way, we deduce
(3.10)
Hence, for
(3.11)

where is the integer part of .

Fix and choose such that Since
(3.12)
there exists be such that
(3.13)
for all . The choice of assures
(3.14)
so
(3.15)
Consequently, for all , with , we have
(3.16)

and hence is a Cauchy sequence.

Now, we suppose that for some . If and , by (ii) we have
(3.17)

which implies . If we use (i) to obtain . Similarly, we deduce that and so for every . Hence is a Cauchy sequence.

Theorem 3.3.

Let be a cone metric space and be an order cone. Let be such that . Assume that the following conditions hold:

(i) , for all , with , where are nonnegative real numbers with ;

(ii) , for all , whenever .

If or is a complete subspace of , then , and have a unique point of coincidence. Moreover, if and are weakly compatibles, then , and have a unique common fixed point.

Proof.

Let be an arbitrary point in . By Proposition 3.2 every - -sequence with initial point is a Cauchy sequence. If is a complete subspace of , there exist such that (this holds also if is complete with ). From
(3.18)
we obtain
(3.19)
Fix and choose be such that
(3.20)
for all , where . Consequently and hence for every . From
(3.21)

being closed, as , we deduce and so . This implies that

Similarly, by using the inequality,
(3.22)
we can show that It implies that is a point of coincidence of , and , that is
(3.23)
Now, we show that , and have a unique point of coincidence. For this, assume that there exists another point in such that , for some in From
(3.24)
we deduce Moreover, if and are weakly compatibles, then
(3.25)

which implies (say). Then is a point of coincidence of , and therefore, by uniqueness. Thus is a unique common fixed point of , and .

From Theorem 3.3, if we choose , we deduce the following theorem.

Theorem 3.4.

Let be a cone metric space, be an order cone and be such that . Assume that the following condition holds:
(3.26)

for all where with .

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

Theorem 3.4 generalizes Theorem 1 of [5].

Remark 3.5.

In Theorem 3.4 the condition (3.26) can be replaced by
(3.27)

for all , where with .

(3.27) (3.26) is obivious. (3.26) (3.27). If in (3.26) interchanging the roles of and and adding the resultant inequality to (3.26), we obtain
(3.28)

From Theorem 3.4, we deduce the followings corollaries.

Corollary 3.6.

Let be a cone metric space, be an order cone and the mappings satisfy
(3.29)

for all where, If and is a complete subspace of , then and have a unique point of coincidence. Moreover, if the pair is weakly compatible, then and have a unique common fixed point.

Corollary 3.6 generalizes Theorem 2.1 of [2], Theorem 1 of [1], and Theorem 2.3 of [6].

Corollary 3.7.

Let be a cone metric space, be an order cone and the mappings satisfy
(3.30)

for all , where If and is a complete subspace of , then and have a unique point of coincidence. Moreover, if the pair is weakly compatible, then and have a unique common fixed point.

Corollary 3.7 generalizes Theorem 2.3 of [2], Theorem 3 of [1], and Theorem 2.6 of [6].

Example 3.8.

Let , and Define as follows:
(3.31)
Define mappings as follow:
(3.32)
Then, if
(3.33)
which implies
(3.34)

for all with .

Therefore, Theorem 3.4 is not applicable to obtain fixed point of or common fixed points of and .

Now define a constant mapping by , then for
(3.35)

It follows that all conditions of Theorem 3.3 are satisfied for and so , and have a unique point of coincidence and a unique common fixed point .

4. Applications

In this section, we prove an existence theorem for the common solutions for two Urysohn integral equations. Throughout this section let , , and for every , where is a constant. It is easily seen that is a complete cone metric space.

Theorem 4.1.

Consider the Urysohn integral equations
(4.1)

where , . Assume that are such that

(i) for each where
(4.2)
(ii)there exist such that
(4.3)

where , for every with and .

(iii)whenever
(4.4)

for every .

Then the system of integral equations (4.1) have a unique common solution.

Proof.

Define by . It is easily seen that
(4.5)
for every , with and if
(4.6)

for every . By Theorem 3.3, if is the identity map on , the Urysohn integral equations (4.1) have a unique common solution.

Authors’ Affiliations

(1)
Department of Mathematics, Faculty of Basic and Applied Sciences, International Islamic University, H-10
(2)
Department of Mathematics, F.G. Postgraduate College, H-8
(3)
Dipartimento di Matematica ed Applicazioni, Università degli Studi di Palermo

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Copyright

© Muhammad Arshad 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.