Common Fixed Point Theorem for Four Non-Self Mappings in Cone Metric Spaces
© The Author(s). 2010
Received: 13 June 2009
Accepted: 18 April 2010
Published: 23 May 2010
We extend a common fixed point theorem of Radenovic and Rhoades for four non-self-mappings in cone metric spaces.
1. Introduction and Preliminaries
Recently, Huang and Zhang  generalized the concept of a metric space, replacing the set of real numbers by ordered Banach space and obtained some fixed point theorems for mappings satisfying different contractive conditions. Subsequently, the study of fixed point theorems in such spaces is followed by some other mathematicians; see [2–8]. The aim of this paper is to prove a common fixed point theorem for four non-self-mappings on cone metric spaces in which the cone need not be normal. This result generalizes the result of Radenović and Rhoades .
Consistent with Huang and Zhang , the following definitions and results will be needed in the sequel.
Definition 1.1 (see ).
The concept of a cone metric space is more general than that of a metric space.
Definition 1.2 (see ).
Remark 1.3 (see ).
We find it convenient to introduce the following definition.
Definition 1.7 (see ).
Definition 1.8 (see ).
Let and be self-maps on 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 .
2. Main Result
The purpose of this paper is to extend the above theorem for four non-self-mappings in cone metric spaces. We begin with the following definition.
We state and prove our main result as follows.
Now, we distinguish the following three cases.
Now, proceeding as in Case 1, we have that (2.18) holds.
in view of Case 1.
and we proved (2.24).
in view of Case 1.
and we proved (2.28).
in view of Case 1.
In case and are closed in , or . The analogous arguments establish (IV) and (V). If we assume that there exists a subsequence with as well being closed in , then noting that is a Cauchy sequence in , foregoing arguments establish (IV) and (V).
Uniqueness of the common fixed point follows easily from (2.2).
Next, we furnish an illustrate example in support of our result. In doing so, we are essentially inspired by Imdad and Kumar .
Therefore, condition (2.2) is satisfied if we choose . Moreover is a point of coincidence as as well as whereas both the pairs and are weakly compatible as and . Also, , , , and are closed in . Thus, all the conditions of Theorem 2.3 are satisfied and is the unique common fixed point of , , , and . One may note that is also a point of coincidence for both the pairs and .
Setting and in Theorem 2.3, one deduces Theorem 2.1 due to .
The authors would like to express their sincere appreciation to the referees for their very helpful suggestions and many kind comments. This project was supported by the National Natural Science Foundation of China (10461007 and 10761007) and supported partly by the Provincial Natural Science Foundation of Jiangxi, China (2008GZS0076 and 2009GZS0019).
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