Assad-Kirk-Type Fixed Point Theorems for a Pair of Nonself Mappings on Cone Metric Spaces
© S. Jankovic et al. 2009
Received: 7 February 2009
Accepted: 27 April 2009
Published: 4 June 2009
New fixed point results for a pair of non-self mappings defined on a closed subset of a metrically convex cone metric space (which is not necessarily normal) are obtained. By adapting Assad-Kirk's method the existence of a unique common fixed point for a pair of non-self mappings is proved, using only the assumption that the cone interior is nonempty. Examples show that the obtained results are proper extensions of the existing ones.
1. Introduction and Preliminaries
Cone metric spaces were introduced by Huang and Zhang in , where they investigated the convergence in cone metric spaces, introduced the notion of their completeness, and proved some fixed point theorems for contractive mappings on these spaces. Recently, in [2–4], some common fixed point theorems have been proved for maps on cone metric spaces. However, in [1–3], the authors usually obtain their results for normal cones. In this paper we do not impose the normality condition for the cones.
We need the following definitions and results, consistent with , in the sequel.
So, in this case, the Sandwich theorem does not hold.
Example 1.1 (see ).
Definition 1.2 (see ).
Let be a sequence in , and let . If, for every in with , there is an such that for all , , then it is said that converges to , and this is denoted by , or , . If for every in with , there is an 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.
Huang and Zhang  proved that if is a normal cone then converges to if and only if , , and that is a Cauchy sequence if and only if , .
It follows from ( ) that the sequence converges to if as and is a Cauchy sequence if as . In the case when the cone is not necessarily normal, we have only one half of the statements of Lemmas??1 and??4 from . Also, in this case, the fact that if and is not applicable.
There exist a lot of fixed-point theorems for self-mappings defined on closed subsets of Banach spaces. However, for applications (numerical analysis, optimization, etc.) it is important to consider functions that are not self-mappings, and it is natural to search for sufficient conditions which would guarantee the existence of fixed points for such mappings.
Recently Imdad and Kumar  extended this result of Rhoades by considering a pair of maps in the following way.
Recall that a pair of mappings is coincidentally commuting (see, e.g., ) if they commute at their coincidence point, that is, if for some , implies .
2.1. Main Result
In , assuming only that , Theorems 1.3 and 1.4 are extended to the setting of cone metric spaces. Thus, proper generalizations of the results of Rhoades  (for one map) and of Imdad and Kumar  (for two maps) were obtained. Example 1.1 of a nonnormal cone shows that the method of proof used in [6, 8, 9] cannot be fully applied in the new setting.
The purpose of this paper is to extend the previous results to the cone metric spaces, but with new contractive conditions. This is worthwhile, since from [2, 13] we know that self-mappings that satisfy the new conditions (given below) do have a unique common fixed point. Let us note that the questions concerning common fixed points for self-mappings in metric spaces, under similar conditions, were considered in . It seems that these questions were not considered for nonself mappings. This is an additional motivation for studying these problems.
We begin with the following definition.
Our main result is the following.
Step 1 (construction of three sequences).
Let be arbitrary. We construct three sequences: and in and in in the following way. Set . Since , by (i) there exists a point such that . Since , from (ii) we conclude that . Then from (i), . Thus, there exists such that . Set and .
Proof of Step 2
Note that the estimate of in this cone version differs from those from [6, 8–11]. In the case of convex metric spaces it can be used that, for each and each , it is . In cone spaces the maximum of the set needs not to exist. Therefore, besides (2.4), we have to use here the relation " '', and to consider several cases. In cone metric spaces as well as in metric spaces the key step is Assad-Kirk's induction.
Again, we obtain the following three cases
in Case 2.
In this step we use only the definition of convergence in the terms of the relation " ''. The only assumption is that the interior of the cone is nonempty; so we use neither continuity of vector metric , nor the Sandwich theorem.
Uniqueness of the common fixed point follows easily. This completes the proof of the theorem.
We present now two examples showing that Theorem 2.2 is a proper extension of the known results. In both examples, the conditions of Theorem 2.2 are fulfilled, but in the first one (because of nonnormality of the cone) the main theorems from [6, 9] cannot be applied. This shows that Theorem 2.2 is more general, that is, the main theorems from [6, 9] can be obtained as its special cases (for ) taking , and .
Example 2.3 (The case of a nonnormal cone).
The mappings and are weakly compatible, that is, they commute in their fixed point . All the conditions of Theorem 2.2 are fulfilled, and so the nonself mappings and have a unique common fixed point .
Example 2.4 (The case of a normal cone).
The mappings and are weakly compatible, that is, they commute in their fixed point . All the conditions of Theorem 2.2 are again fulfilled. The point is the unique common fixed point for nonself mappings and .
2.3. Further Results
The following definition is a special case of Definition 2.1 when is a metric space. But when is a cone metric space, which is not a metric space, this is not true. Indeed, there may exist such that the vectors and are incomparable. For the same reason Theorems 2.2 and 2.7 (given below) are incomparable.
Our next result is the following.
The proof of this theorem is very similar to the proof of Theorem 2.2 and it is omitted.
We now list some corollaries of Theorems 2.2 and 2.7.
Corollaries 2.8–2.10 are the corresponding theorems of Abbas and Jungck from  in the case that are nonself mappings.
If is a metrically convex cone metric space, that is, if for each there is such that , we do not know whether (2.4) holds for every nonempty closed subset in (see ).
This work was supported by Grant 14021 of the Ministry of Science and Environmental Protection of Serbia.
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