Some Common Fixed Point Results in Cone Metric Spaces
- Muhammad Arshad^{1},
- Akbar Azam^{1, 2} and
- Pasquale Vetro^{3}Email author
https://doi.org/10.1155/2009/493965
© Muhammad Arshad et al. 2009
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 [2–5] 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:
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 ;
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.
It implies that (say). Then is a point of coincidence of , and . Therefore, by uniqueness. Thus is a unique common fixed point of , and
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 ;
Then every - -sequence with initial point is a Cauchy sequence.
Proof.
where is the integer part of .
and hence is a Cauchy sequence.
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 ;
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.
being closed, as , we deduce and so . This implies that
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.
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.
From Theorem 3.4, we deduce the followings corollaries.
Corollary 3.6.
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.
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.
Therefore, Theorem 3.4 is not applicable to obtain fixed point of or common fixed points of and .
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.
where , . Assume that are such that
Then the system of integral equations (4.1) have a unique common solution.
Proof.
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
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