2.1. Main Result
In [12], 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 [6] (for one map) and of Imdad and Kumar [9] (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 selfmappings 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 selfmappings in metric spaces, under similar conditions, were considered in [14]. 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.
Definition 2.1.
Let be a cone metric space, let be a nonempty closed subset of , and let . Denote, for ,
Then is called a generalizedcontractive mapping of into if for some there exists
such that for all in
Our main result is the following.
Theorem 2.2.
Let be a complete cone metric space, let be a nonempty closed subset of such that, for each and each there exists a point such that
Suppose that is a generalized contractive mapping of into and
(i), ,
(ii),
(iii) is closed in .
Then there exists a coincidence point of in . Moreover, if the pair is coincidentally commuting, then is the unique common fixed point of and .
Proof.
We prove the theorem under the hypothesis that neither of the mappings and is necessarily a selfmapping. We proceed in several steps.
Step 1 (construction of three sequences).
The following construction is the same as the construction used in [10] in the case of hyperbolic metric spaces. It differs slightly from the constructions in [6, 9].
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 .
If , then from (i), and so there is a point such that .
If , then is a point in , such that . By (i), there is such that . Thus and .
Now we set . Since , from (ii) there is a point such that .
Note that in the case , we have and .
Continuing the foregoing procedure we construct three sequences: , and such that:
(a);
(b);
(c) if and only if ;
(d) whenever and then and .
Step 2 ( is a Cauchy sequence).
First, note that if , then , which then implies, by (b), (ii), and (a), that Also, implies that since otherwise , which then implies .
Proof of Step 2
Now we have to estimate . If for some , then it is easy to show that for all
Suppose that for all . There are three possibilities:
(1) and ;
(2), but ; and
(3) in which case and .
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 AssadKirk's induction.
Case 1.
Let and let . Then , and (observe that it is not necessarily ). Then from (2.3),
where
Clearly, there are infinitely many 's such that at least one of the following cases holds:
(I)
(II) contradicting the assumption that for each . Hence, (I) holds,
(III), that is, (I) holds.
From (I), (II), and (III) it follows that in Case 1
Case 2.
Let but . Then and . It follows that
that is, according to (2.3), , where
Again, we obtain the following three cases
(I).
(II), contradicting the assumption that for each . It follows that (I) holds.
(III), that is .
From (2.8), (I), (II), and (III), we have
in Case 2.
Case 3.
Let Then and we have and . From this and using (2.4) we get
We have to estimate and . Since , one can conclude that
in view of Case 2. Further,
where
Since
, , and , we have that
where
Substituting (2.12) and (2.16) into (2.11) we get
We have now the following four cases:

(I)

(II)

(III)

(IV)
It follows from (I), (II), (III), and (IV) that
Thus, in all Cases 1–3,
where and
It is not hard to conclude that for ,
Now, following the procedure of Assad and Kirk [8], it can be shown by induction that, for ,
where .
From (2.27) and using the triangle inequality, we have for
According to the property () from the Introduction, that is, is a Cauchy sequence.
Step 3 (Common fixed point for and ).
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.
Since and is complete, there is some point such that . Let be such that . By the construction of , there is a subsequence such that and hence .
We now prove that . We have
where
From the definition of convergence and the fact that , as , we obtain (for the given with )
In all the cases we obtain for each . According to the property (), it follows that , that is, .
Suppose now that and are coincidentally commuting. Then
Then from (2.3),
where
Hence, we obtain the following cases:
which implies that , that is, is a common fixed point of and .
Uniqueness of the common fixed point follows easily. This completes the proof of the theorem.
2.2. Examples
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).
Let , let , and , and let . The mapping is defined in the following way: , where is a fixed function, for example, . Take functions , , , so that , which map the set into . We have that is a complete cone metric space with a nonnormal cone having the nonempty interior. The topological and "metric'' notions are used in the sense of definitions from [15, 16]. For example, one easily checks the condition (2.4), that is, that for , the following holds
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).
Let , let , let , and let . The mapping is defined as , . Take the functions , , , so that , which map the set into . We have that is a complete cone metric space with a normal cone having the normal coefficient , whose interior is obviously nonempty. All the conditions of Theorem 2.2 are fulfilled. We check again the condition (2.4), that is, that for , the following holds
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
Remark 2.5.
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.
Definition 2.6.
Let be a cone metric space, let be a nonempty closed subset of , and let . Denote
Then is called a generalizedcontractive mapping from into if for some there exists
such that for all in
Our next result is the following.
Theorem 2.7.
Let be a complete cone metric space, and let be a nonempty closed subset of such that for each and there exists a point such that
Suppose that is a generalized contractive mapping of into and
(i), ,
(ii),
(iii) is closed in .
Then there exists a coincidence point of and in . Moreover, if the pair is coincidentally commuting, then is the unique common fixed point of and .
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.
Corollary 2.8.
Let be a complete cone metric space, and let be a nonempty closed subset of such that, for each and each there exists a point such that
Let be such that
for some and for all .
Suppose, further, that and satisfy the following conditions:
(i), ,
(ii),
(iii) is closed in .
Then there exists a coincidence point of and in . Moreover, if is a coincidentally commuting pair, then is the unique common fixed point of and .
Corollary 2.9.
Let be a complete cone metric space, and let be a nonempty closed subset of such that, for each and each there exists a point such that
Let be such that
for some and for all .
Suppose, further, that and satisfy the following conditions:
(i), ,
(ii),
(iii) is closed in .
Then there exists a coincidence point of and in . Moreover, if is a coincidentally commuting pair, then is the unique common fixed point of and .
Corollary 2.10.
Let be a complete cone metric space, and let be a nonempty closed subset of such that, for each and each there exists a point such that
Let be such that
for some and for all .
Suppose, further, that and satisfy the following conditions:
(i), ,
(ii),
(iii) is closed in .
Then there exists a coincidence point of and in . Moreover, if is a coincidentally commuting pair, then is the unique common fixed point of and .
Remark 2.11.
Corollaries 2.8–2.10 are the corresponding theorems of Abbas and Jungck from [2] in the case that are nonself mappings.
Remark 2.12.
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 [8]).