- Research Article
- Open Access
Common Fixed Point Theorems for Weakly Compatible Pairs on Cone Metric Spaces
© G. Jungck et al. 2009
Received: 17 December 2008
Accepted: 4 February 2009
Published: 18 February 2009
We prove several fixed point theorems on cone metric spaces in which the cone does not need to be normal. These theorems generalize the recent results of Huang and Zhang (2007), Abbas and Jungck (2008), and Vetro (2007). Furthermore as corollaries, we obtain recent results of Rezapour and Hamlborani (2008).
1. Introduction and Preliminaries
Recently, Abbas and Jungck , have studied common fixed point results for noncommuting mappings without continuity in cone metric space with normal cone. In this paper, our results are related to the results of Abbas and Jungck, but our assumptions are more general, and also we generalize some results of [1–3], and  by omitting the assumption of normality in the results.
Let us mention that nonconvex analysis, especially ordered normed spaces, normal cones, and topical functions ([2, 4–9]) have some applications in optimization theory. In these cases, an order is introduced by using vector space cones. Huang and Zhang  used this approach, and they have replaced the real numbers by ordering Banach space and defining cone metric space. Consistent with Huang and Zhang , the following definitions and results will be needed in the sequel.
The least positive number satisfying the above inequality is called the normal constant of It is clear that From  we know that there exists ordered Banach space with cone which is not normal but with
Definition 1.1 (see ).
Definition 1.2 (see ).
 Let be an ordered Banach (normed) space. Then is an interior point of if and only if is a neighborhood of
From this it follows that: the sequence converges to if as and is a Cauchy if as In the situation with non-normal cone, we have only half of the lemmas 1 and 4 from . Also, the fact that if and is not applicable.
It follows from Remark 1.5, Corollary 1.4(1), and Definition 1.2(f).
Let be a cone metric space. Let us remark that the family , where , is a subbasis for topology on . We denote this cone topology by , and note that is a Hausdorff topology (see, e.g.,  without proof).
We find it convenient to introduce the following definition.
Let be a cone metric space and a cone with nonempty interior. Suppose that the mappings are such that the range of contains the range of , and or is a complete subspace of . In this case we will say that the pair is Abbas and Jungck's pair, or shortly AJ's pair.
Definition 1.11 (see ).
Let and be self-maps of a set (i.e., ) If for some in then is called a coincidence point of and and is called a point of coincidence of and Self-maps and are said to be weakly compatible if they commute at their coincidence point, that is, if for some then
Proposition 1.12 (see ).
2. Main Results
Now we have to consider the following three cases.
From Remark 1.5 it follows that for and large thus, according to Corollary 1.4(1), Hence, by Definition 1.2(e), is a Cauchy sequence. Since and or is complete, there exists a such that as Consequently, we can find such that
Since is the unique point of coincidence of and and and are weakly compatible, is the unique common fixed point of and by Proposition 1.12 .
In the next theorem, among other things, we generalize the well-known Zamfirescu result [12, ( )].
As in Theorem 2.1, we have to consider three cases.
and (2.20) holds. Thus, we proved that in all three cases (2.20) holds.
Now as corollaries, we recover and generalize the recent results of Huang and Zhang , Abbas and Jungck , and Vetro . Furthermore as corollaries, we obtain recent results of Rezapour and Hamlborani .
In the next corollary, among other things, we generalize the well-known result [12, ( )].
Now, we generalize the well-known Bianchini result [12, (5)].
When in the next theorem we set the identity map on and , we get the theorem of Hardy and Rogers [12, (18)].
According to Remark 1.8, and because we get that is, If and are weakly compatible, then as in the proof of Theorem 2.1, we have that is a unique common fixed point of and The assertion of the theorem follows.
Finally, we add an example with Banach type contraction on non-normal cone metric space (see Corollary 2.3).
The fourth author would like to express his gratitude to Professor Sh. Rezapour and to Professor S. M. Veazpour for the valuable comments. The second, third, and fourth authors thank the Ministry of Science and the Ministry of Environmental Protection of Serbia for their support.
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