Some Common Fixed Point Results in Cone Metric Spaces
© Muhammad Arshad et al. 2009
Received: 5 September 2008
Accepted: 5 February 2009
Published: 18 February 2009
We prove a result on points of coincidence and common fixed points for three self-mappings satisfying generalized contractive type conditions in cone metric spaces. We deduce some results on common fixed points for two self-mappings satisfying contractive type conditions in cone metric spaces. These results generalize some well-known recent results.
Huang and Zhang  recently have introduced the concept of cone metric space, where the set of real numbers is replaced by an ordered Banach space, and they have established some fixed point theorems for contractive type mappings in a normal cone metric space. Subsequently, some other authors [2–5] have generalized the results of Huang and Zhang  and have studied the existence of common fixed points of a pair of self mappings satisfying a contractive type condition in the framework of normal cone metric spaces.
Vetro  extends the results of Abbas and Jungck  and obtains common fixed point of two mappings satisfying a more general contractive type condition. Rezapour and Hamlbarani  prove that there aren't normal cones with normal constant and for each there are cones with normal constant . Also, omitting the assumption of normality they obtain generalizations of some results of . In  Di Bari and Vetro obtain results on points of coincidence and common fixed points in nonnormal cone metric spaces. In this paper, we obtain points of coincidence and common fixed points for three self-mappings satisfying generalized contractive type conditions in a complete cone metric space. Our results improve and generalize the results in [1, 2, 5, 6, 8].
We recall the definition of cone metric spaces and the notion of convergence . Let be a real Banach space and be a subset of . The subset is called an order cone if it has the following properties:
Let be a sequence in , and . If for every with there is such that for all then is said to be convergent, converges to and is the limit of We denote this by or as If for every with there is such that for all then is called a Cauchy sequence in . If every Cauchy sequence is convergent in , then is called a complete cone metric space.
3. Main Results
First, we establish the result on points of coincidence and common fixed points for three self-mappings and then show that this result generalizes some of recent results of fixed point.
A pair of self-mappings on is said to be weakly compatible if they commute at their coincidence point (i.e., whenever ). A point is called point of coincidence of a family , , of self-mappings on if there exists a point such that for all .
Theorem 3.4 generalizes Theorem 1 of .
From Theorem 3.4, we deduce the followings corollaries.
In this section, we prove an existence theorem for the common solutions for two Urysohn integral equations. Throughout this section let , , and for every , where is a constant. It is easily seen that is a complete cone metric space.
Then the system of integral equations (4.1) have a unique common solution.
for every . By Theorem 3.3, if is the identity map on , the Urysohn integral equations (4.1) have a unique common solution.
- 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
- 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
- Abbas M, Rhoades BE: Fixed and periodic point results in cone metric spaces. Applied Mathematics Letters 2009,22(4):511–515. 10.1016/j.aml.2008.07.001MathSciNetView 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
- 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
- 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
- Di Bari C, Vetro P: -pairs and common fixed points in cone metric spaces. Rendiconti del Circolo Matematico di Palermo 2008,57(2):279–285. 10.1007/s12215-008-0020-9MathSciNetView ArticleMATHGoogle Scholar
- Raja P, Vaezpour SM: Some extensions of Banach's contraction principle in complete cone metric spaces. Fixed Point Theory and Applications 2008, 2008:-11.Google Scholar
This article is published under license to BioMed Central Ltd. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.