Remarks on Cone Metric Spaces and Fixed Point Theorems of Contractive Mappings
© Mohamed A. Khamsi. 2010
Received: 20 March 2010
Accepted: 4 May 2010
Published: 6 June 2010
We discuss the newly introduced concept of cone metric spaces. We also discuss the fixed point existence results of contractive mappings defined on such metric spaces. In particular, we show that most of the new results are merely copies of the classical ones.
Cone metric spaces were introduced in . A similar notion was also considered by Rzepecki in . After carefully defining convergence and completeness in cone metric spaces, the authors proved some fixed point theorems of contractive mappings. Recently, more fixed point results in cone metric spaces appeared in [3–8]. Topological questions in cone metric spaces were studied in  where it was proved that every cone metric space is first countable topological space. Hence, continuity is equivalent to sequential continuity and compactness is equivalent to sequential compactness. It is worth mentioning the pioneering work of Quilliot  who introduced the concept of generalized metric spaces. His approach was very successful and used by many (see references in ). It is our belief that cone metric spaces are a special case of generalized metric spaces. In this work, we introduce a metric type structure in cone metric spaces and show that classical proofs do carry almost identically in these metric spaces. This approach suggest that any extension of known fixed point result to cone metric spaces is redundant. Moreover the underlying Banach space and the associated cone subset are not necessary.
For more on metric fixed point theory, the reader may consult the book .
2. Basic Definitions and Results
First let us start by making some basic definitions.
The least positive number satisfying this inequality is called the normal constant of . The cone is called regular if every increasing sequence which is bounded from above is convergent. Equivalently the cone is called regular if every decreasing sequence which is bounded from below is convergent. Regular cones are normal and there exist normal cones which are not regular.
Convergence is defined as follows.
It is easy to show that the limit of a convergent sequence is unique. Cauchy sequences and completeness are defined by
Let be a cone metric space, be a sequence in . If for any with , there is such that for all , , then is called Cauchy sequence. If every Cauchy sequence is convergent in , then is called a complete cone metric space.
The basic properties of convergent and Cauchy sequences may be found at . In fact the properties and their proofs are identical to the classical metric ones. Since this work concerns the fixed point property of mappings, we will need the following property.
As we mentioned earlier cone metric spaces have a metric type structure. Indeed we have the following result.
This completes the proof of the theorem.
Note that the property (3) is discouraging since it does not give the classical triangle inequality satisfied by a distance. But there are many examples where the triangle inequality fails (see, e.g., ).
The above result suggest the following definition.
Similarly we define convergence and completeness in metric type spaces.
3. Some Fixed Point Results
Let be a complete metric type space, where satisfies instead of (3). Let be a map such that is Lipschitzian for any and . Then has a unique fixed point if and only if there exists a bounded orbit. Moreover if has a fixed point , then for any , the orbit converges to .
The connection between the above results and the main theorems of  are given in the following corollary.
In  the authors gave an example of a map which is contraction for but not for the euclidian distance. From the above corollary, we see that . Since may not be less than 1, then may not be a contraction for . This is why the above theorems were stated in terms of .
Using the ideas described above one can prove fixed point results for mappings which contracts orbits and obtain similar results as Theorem for example in .
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