- Research Article
- Open Access
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.
- Banach Space
- Point Theorem
- Fixed Point Theorem
- Triangle Inequality
- Cauchy Sequence
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 .
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.
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 .
- Huang L-G, Zhang X: Cone metric spaces and fixed point theorems of contractive mappings. Journal of Mathematical Analysis and Applications 2007,332(2):1468–1476. 10.1016/j.jmaa.2005.03.087MathSciNetView ArticleMATHGoogle Scholar
- Rzepecki B: On fixed point theorems of Maia type. Publications de l'Institut Mathématique 1980, 28(42): 179–186.MathSciNetMATHGoogle Scholar
- Abbas M, Jungck G: Common fixed point results for noncommuting mappings without continuity in cone metric spaces. Journal of Mathematical Analysis and Applications 2008,341(1):416–420. 10.1016/j.jmaa.2007.09.070MathSciNetView ArticleMATHGoogle Scholar
- Ilić D, Rakočević V: Common fixed points for maps on cone metric space. Journal of Mathematical Analysis and Applications 2008,341(2):876–882. 10.1016/j.jmaa.2007.10.065MathSciNetView ArticleMATHGoogle Scholar
- Rezapour S, Hamlbarani R: Some notes on the paper: "Cone metric spaces and fixed point theorems of contractive mappings". Journal of Mathematical Analysis and Applications 2008,345(2):719–724. 10.1016/j.jmaa.2008.04.049MathSciNetView ArticleMATHGoogle Scholar
- Turkoglu D, Abuloha M: Cone metric spaces and fixed point theorems in diametrically contractive mappings. Acta Mathematica Sinica 2010,26(3):489–496.MathSciNetView ArticleMATHGoogle Scholar
- Turkoglu D, Abuloha M, Abdeljawad T: KKM mappings in cone metric spaces and some fixed point theorems. Nonlinear Analysis: Theory, Methods & Applications 2010,72(1):348–353. 10.1016/j.na.2009.06.058MathSciNetView ArticleMATHGoogle Scholar
- Vetro P: Common fixed points in cone metric spaces. Rendiconti del Circolo Matematico di Palermo. Serie II 2007,56(3):464–468. 10.1007/BF03032097MathSciNetView ArticleMATHGoogle Scholar
- Quilliot A: An application of the Helly property to the partially ordered sets. Journal of Combinatorial Theory. Series A 1983,35(2):185–198. 10.1016/0097-3165(83)90006-7MathSciNetView ArticleMATHGoogle Scholar
- Jawhari E, Misane D, Pouzet M: Retracts: graphs and ordered sets from the metric point of view. In Combinatorics and Ordered Sets (Arcata, Calif., 1985), Contemporary Mathematics. Volume 57. American Mathematical Society, Providence, RI, USA; 1986:175–226.View ArticleGoogle Scholar
- Khamsi MA, Kirk WA: An Introduction to Metric Spaces and Fixed Point Theory, Pure and Applied Mathematics. Wiley-Interscience, New York, NY, USA; 2001:x+302.View ArticleMATHGoogle Scholar
- Khamsi MA, Kozłowski WM, Reich S: Fixed point theory in modular function spaces. Nonlinear Analysis: Theory, Methods & Applications 1990,14(11):935–953. 10.1016/0362-546X(90)90111-SMathSciNetView ArticleMATHGoogle Scholar
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