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Assad-Kirk-Type Fixed Point Theorems for a Pair of Nonself Mappings on Cone Metric Spaces

Fixed Point Theory and Applications20092009:761086

DOI: 10.1155/2009/761086

Received: 7 February 2009

Accepted: 27 April 2009

Published: 4 June 2009

Abstract

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 [1], 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 [24], some common fixed point theorems have been proved for maps on cone metric spaces. However, in [13], 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 [1], in the sequel.

Let be a real Banach space. A subset of is a cone if

(i) is closed, nonempty and

(ii) , and imply

(iii) .

Given a cone , we define the partial ordering with respect to by if and only if . We write to indicate that but , while stands for (the interior of ).

There exist two kinds of cones: normal and nonnormal cones. A cone is a normal cone if
(1.1)
or, equivalently, if there is a number such that for all ,
(1.2)

The least positive number satisfying (1.2) is called the normal constant of . It is clear that .

It follows from (1.1) that is nonnormal if and only if there exist sequences such that
(1.3)

So, in this case, the Sandwich theorem does not hold.

Example 1.1 (see [5]).

Let with and . This cone is not normal. Consider, for example,
(1.4)

Then and .

Definition 1.2 (see [1]).

Let be a nonempty set. Suppose that the mapping satisfies

(d1) for all and if and only if ;

(d2) for all ;

(d3) for all .

Then is called a cone metric on and is called a cone metric space.

The concept of a cone metric space is more general than that of a metric space, because each metric space is a cone metric space with and (see [1, Example??1] and [4, Examples??1.2 and??2.2]).

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 [1] proved that if is a normal cone then converges to if and only if , , and that is a Cauchy sequence if and only if , .

Let be a cone metric space. Then the following properties are often useful (particulary when dealing with cone metric spaces in which the cone needs not to be normal):

() if and , then

() if for each then

() if for each then

() if , and , then

() if for each and , , then

() if and , then where and are, respectively, a sequence and a given point in

() if is a real Banach space with a cone and if where and , then

() if , and , then there exists such that for all we have .

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 [1]. 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.

In what follows we suppose only that is a Banach space, that is a cone in with and that is the partial ordering with respect to .

Rhoades [6] proved the following result, generalizing theorems of Assad [7] and Assad and Kirk [8].

Theorem 1.3.

Let be a Banach space, a nonempty closed subset of and let be a mapping from into satisfying the condition
(1.5)

for some , , and for all in . Let have the additional property that for each , the boundary of , . Then has the unique fixed point.

Recently Imdad and Kumar [9] extended this result of Rhoades by considering a pair of maps in the following way.

Theorem 1.4.

Let be a Banach space, let be a nonempty closed subset of and let be two mappings satisfying the condition
(1.6)

for some , , and for all and suppose

(i) , ,

(ii) ,

(iii) is closed in .

Then there exists a coincidence point of in . Moreover, if and are coincidentally commuting, then is the unique common fixed point of and .

Recall that a pair of mappings is coincidentally commuting (see, e.g., [2]) if they commute at their coincidence point, that is, if for some , implies .

In [10, 11] these results were extended using complete metric spaces of hyperbolic type, instead of Banach spaces.

2. Results

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 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 [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 ,
(2.1)
Then is called a generalized -contractive mapping of into if for some there exists
(2.2)
such that for all in
(2.3)

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
(2.4)

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 self-mapping. 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, 811]. 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.

Case 1.

Let and let . Then , and (observe that it is not necessarily ). Then from (2.3),
(2.5)
where
(2.6)

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
(2.7)

Case 2.

Let but . Then and . It follows that
(2.8)
that is, according to (2.3), , where
(2.9)

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
(2.10)

in Case 2.

Case 3.

Let Then and we have and . From this and using (2.4) we get
(2.11)
We have to estimate and . Since , one can conclude that
(2.12)
in view of Case 2. Further,
(2.13)
where
(2.14)
Since
(2.15)
, , and , we have that
(2.16)
where
(2.17)
Substituting (2.12) and (2.16) into (2.11) we get
(2.18)
We have now the following four cases:
  1. (I)
    (2.19)
     
  1. (II)
    (2.20)
     
  1. (III)
    (2.21)
     
  1. (IV)
    (2.22)
     
It follows from (I), (II), (III), and (IV) that
(2.23)
Thus, in all Cases 1–3,
(2.24)
where and
(2.25)
It is not hard to conclude that for ,
(2.26)
Now, following the procedure of Assad and Kirk [8], it can be shown by induction that, for ,
(2.27)

where .

From (2.27) and using the triangle inequality, we have for
(2.28)

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
(2.29)
where
(2.30)
From the definition of convergence and the fact that , as , we obtain (for the given with )
(2.31)

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
(2.32)
Then from (2.3),
(2.33)
where
(2.34)
Hence, we obtain the following cases:
(2.35)

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
(2.36)

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
(2.37)

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
(2.38)
Then is called a generalized -contractive mapping from into if for some there exists
(2.39)
such that for all in
(2.40)

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
(2.41)

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
(2.42)
Let be such that
(2.43)

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
(2.44)
Let be such that
(2.45)

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
(2.46)
Let be such that
(2.47)

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]).

Declarations

Acknowledgment

This work was supported by Grant 14021 of the Ministry of Science and Environmental Protection of Serbia.

Authors’ Affiliations

(1)
Mathematical Institute SANU
(2)
Faculty of Mathematics, University of Belgrade
(3)
Faculty of Mechanical Engineering, University of Belgrade
(4)
Department of Mathematics, Indiana University

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Copyright

© S. Jankovic et al. 2009

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