- Research Article
- Open Access
© F. Sabetghadam and H. P. Masiha. 2010
- Received: 19 October 2009
- Accepted: 10 January 2010
- Published: 24 February 2010
- Natural Number
- Differential Geometry
- Arbitrary Point
- Fixed Point Theorem
- Contraction Condition
Huang and Zhang  recently introduced the concept of cone metric spaces and established some fixed point theorems for contractive mappings in these spaces. Afterwards, Rezapour and Hamlbarani  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  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 , Di Bari and Vetro , and Abbas and Jungck . First, we recall some standard notations and definitions in cone metric spaces.
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 ).
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  (see also [2, Lemma ]).
Definition 1.2 (see ).
Definition 1.3 (see ).
The concept of weakly compatible mappings is introduced as follows.
Definition 1.4 (see ).
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  if
Now, we are in the position to state the following theorem.
Corollary 2.7 generalizes Theorem in . Also, if we choose the -mapping defined by , where is a constant, then Theorem 2.3 generalizes Theorem in . Furthermore, if we let be the identity map of , then we obtain Theorem in , that is, the extension of the Banach fixed point theorem for cone metric spaces.
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 .
The authors would like to thank the referees for their valuable and useful comments.
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