Some Coupled Fixed Point Theorems in Cone Metric Spaces
© F. Sabetghadam et al. 2009
Received: 17 July 2009
Accepted: 28 September 2009
Published: 8 October 2009
We prove some coupled fixed point theorems for mappings satisfying different contractive conditions on complete cone metric spaces.
Recently, Huang and Zhang in  generalized the concept of metric spaces by considering vector-valued metrics (cone metrics) with values in an ordered real Banach space. They proved some fixed point theorems in cone metric spaces showing that metric spaces really doesnot provide enough space for the fixed point theory. Indeed, they gave an example of a cone metric space and proved existence of a unique fixed point for a selfmap of which is contractive in the category of cone metric spaces but is not contractive in the category of metric spaces. After that, cone metric spaces have been studied by many other authors (see [1–9] and the references therein).
Regarding the concept of coupled fixed point, introduced by Bhaskar and Lakshmikantham , we consider the corresponding definition for the mappings on complete cone metric spaces and prove some coupled fixed point theorems in the next section. First, we recall some standard notations and definitions in cone metric spaces.
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 (see ). 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 , or [7, Lemma 1.1]).
Huang and Zhang defined the concept of a cone metric space in  as follows.
Definition 1.1 (see ).
Definition 1.2 (see ).
A cone metric space is said to be complete if every Cauchy sequence in is convergent in . If for any sequence in there exists a subsequence of such that is convergent in , then the cone metric space is called sequentially compact. Clearly, every sequentially compact cone metric space is complete. Huang and Zhang in  investigated the existence and uniqueness of the fixed point for a selfmap on a cone metric space . They considered different types of contractive conditions on . They also assumed to be complete when is a normal cone, and to be sequentially compact when is a regular cone. Later, in , Rezapour and Hamlbarani improved some of the results in  by omitting the normality assumption of the cone , when is complete. See [4, 6, 7, 9] for more related results about (complete) cone metric spaces and fixed point theorems for different types of mappings on these spaces.
2. Main Results
For a given partially ordered set , Bhaskar and Lakshmikantham in  introduced the concept of coupled fixed point of a mapping . Later in  Lakshmikantham and Ćirić investigated some more coupled fixed point theorems in partially ordered sets. The following is the corresponding definition of coupled fixed point in cone metric spaces.
In the next theorems of this section, we investigate some coupled fixed point theorems in cone metric spaces.
It is worth noting that when the constants in Theorem 2.2 are equal we have the following corollary.
In this case, and are both coupled fixed points of and hence the coupled fixed point of is not unique. This shows that the condition in corollary (2.12) and hence in Theorem 2.2 are optimal conditions for the uniqueness of the coupled fixed point.
When the constants in Theorems 2.5 and 2.6 are equal, we get the following corollaries.
which is a contractive condition of the type (2.26) in Corollary 2.7 (with equal constants). Therefore, one can also reduce the proof of general case (2.15) in Theorem 2.5 to the special case of equal constants. A similar argument is valid for the contractive conditions (2.20) in Theorem 2.6 and (2.27) in Corollary 2.8.
The authors would like to thank the referees for their valuable and useful comments.
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