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Strict Contractive Conditions and Common Fixed Point Theorems in Cone Metric Spaces

Fixed Point Theory and Applications20092009:173838

https://doi.org/10.1155/2009/173838

Received: 22 June 2009

Accepted: 9 September 2009

Published: 4 October 2009

Abstract

A lot of authors have proved various common fixed-point results for pairs of self-mappings under strict contractive conditions in metric spaces. In the case of cone metric spaces, fixed point results are usually proved under assumption that the cone is normal. In the present paper we prove common fixed point results under strict contractive conditions in cone metric spaces using only the assumption that the cone interior is nonempty. We modify the definition of property (E.A), introduced recently in the work by Aamri and Moutawakil (2002), and use it instead of usual assumptions about commutativity or compatibility of the given pair. 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 [26], 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. (In fact, validity of this theorem is equivalent to the normality of the cone, see [7].)

Example 1.1 (see [7]).

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 ]).

Let be a sequence in , and . 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 , . Completeness is defined in the standard way.

It was proved in [1] that if is a normal cone, then converges to if and only if , .

Let be a cone metric space. Then the following properties are often useful (particularly when dealing with cone metric spaces in which the cone may be nonnormal):

(p1) if for each then ,

(p2) if , and , then there exists such that for all .

It follows from (p2) that the sequence converges to if as . In the case when the cone is not necessarily normal, we have only one half of the statements of Lemmas and from [1]. Also, in this case, the fact that if and is not applicable.

2. Compatible and Noncompatible Mappings in Cone Metric Spaces

In the sequel we assume only that is a Banach space and that is a cone in with . The last assumption is necessary in order to obtain reasonable results connected with convergence and continuity. In particular, with this assumption the limit of a sequence is uniquely determined. The partial ordering induced by the cone will be denoted by .

If is a pair of self-maps on the space then its well known properties, such as commutativity, weak-commutativity [8], -commutativity [9, 10], weak compatibility [11], can be introduced in the same way in metric and cone metric spaces. The only difference is that we use vectors instead of numbers. As an example, we give the following.

Definition 2.1 (see [9]).

A pair of self-mappings on a cone metric space is said to be -weakly commuting if there exists a real number such that for all , whereas the pair is said to be pointwise -weakly commuting if for each there exists such that .

Here it may be noted that at the points of coincidence, -weak commutativity is equivalent to commutativity and it remains a necessary minimal condition for the existence of a common fixed point of contractive type mappings.

Compatible mappings in the setting of metric spaces were introduced by Jungck [11, 12]. The property (E.A) was introduced in [13]. We extend these concepts to cone metric spaces and investigate their properties in this paper.

Definition 2.2.

A pair of self-mappings on a cone metric space is said to be compatible if for arbitrary sequence in such that , and for arbitrary with , there exists such that whenever . It is said to be weakly compatible if implies .

It is clear that, as in the case of metric spaces, the pair ( —the identity mapping) is both compatible and weakly compatible, for each self-map .

If , , , then these concepts reduce to the respective concepts of Jungck in metric spaces. It is known that in the case of metric spaces compatibility implies weak compatibility but that the converse is not true. We will prove that the same holds in the case of cone metric spaces.

Proposition 2.3.

If the pair of self-maps on the cone metric space is compatible, then it is also weakly compatible.

Proof.

Let for some . We have to prove that . Take the sequence with for each . It is clear that . If with , then the compatibility of the pair implies that . It follows by property (p1) that , that is, .

Example 2.4.

We show in this example that the converse in the previous proposition does not hold, neither in the case when the cone is normal nor when it is not.

Let and

(1) , (a normal cone), let , ( fixed), is a complete cone metric space,

(2) , (a nonnormal cone). Let for some fixed , for example, . is also a complete cone metric space.

Consider the pair of mappings defined as
(2.1)

and the sequence . It is , .

In both of the given cone metrics holds. Namely, in the first case, in the standard norm of the space . Also, in the same norm (since in this case the cone is normal, we can use that the cone metric is continuous).

However, . So, taking the fixed vector , we see that does not hold for each , for otherwise by (p2) this vector would reduce to . Hence, the pair is not compatible.

In the case (2) of a nonnormal cone we have in the norm of space ; in the same norm.

However, , . If we put , then is impossible since and (null function). This means that it is not , and so the pair is not compatible.

Since and , in both cases and .

Clearly, a pair of self-mappings on a cone metric space is not compatible if there exists a sequence in such that for some but is either nonzero or nonexistent.

Definition 2.5.

A pair of self-mappings on a cone metric space is said to enjoy property (E.A) if there exists a sequence in such that for some .

Clearly, each noncompatible pair satisfies property (E.A). The converse is not true. Indeed, let , , , , fixed, , , . Then in the given cone metric both sequences and tend to , but

(2.2)

for each point of , that is, the pair is compatible. In other words, the set of pairs with property (E.A) contains all noncompatible pairs, and also some of the compatible ones.

3. Strict Contractive Conditions and Existence of Common Fixed Points on Cone Metric Spaces

Let be a complete cone metric space, let be a pair of self-mappings on and . Let us consider the following sets:

(3.1)

and define the following conditions:

for arbitrary there exists such that

(3.2)

for arbitrary there exists such that

(3.3)

for arbitrary there exists such that

(3.4)

These conditions are called strict contractive conditions. Since in metric spaces the following inequalities hold:

(3.5)

in this setting, condition is a special case of and is a special case of . This is not the case in the setting of cone metric spaces, since for , if and are incomparable, then also is incomparable, both with and with .

The following theorem was proved for metric spaces in [13].

Theorem 3.1.

Let the pair of weakly compatible mappings satisfy property (E.A). If condition is satisfied, , and at least one of and is complete, then and have a unique common fixed point.

Conditions and are not mentioned in [13]. We give an example of a pair of mappings satisfying and , but which have no common fixed points, neither in the setting of metric nor in the setting of cone metric spaces.

Example 3.2.

Let with the standard metric. Take and consider the functions:
(3.6)

We have to show that for each there exists such that for .

It is not hard to prove that in all possible five cases one can find a respective :

;

, ;

, ;

, ;

, .

Let now . Then and . It is clear that and all of them are complete metric spaces, so all the conditions of Theorem 3.1 except are fulfilled, but there exists no coincidence point of mappings and .

Using the previous example, it is easy to construct the respective example in the case of cone metric spaces.

Let , , , and let be defined as , for fixed . Let be the same mappings as in the previous case. Now we have the following possibilities:

;

, ;

, ;

, ;

, .

Conclusion is the same as in the metric case.

We will prove the following theorem in the setting of cone metric spaces.

Theorem 3.3.

Let and be two weakly compatible self-mappings of a cone metric space such that

(i) satisfies property (E.A);

(ii)for all there exists such that ,

(iii) .

If or is a complete subspace of , then and have a unique common fixed point.

Proof.

It follows from (i) that there exists a sequence satisfying
(3.7)

Suppose that is complete. Then for some . Also .

We will show that . Suppose that . Condition (ii) implies that there are the following three cases.

, that is, ; it follows that and so ;

; it follows that , hence , that is, and so ;

; it follows that , hence , that is, and so .

Hence, we have proved that and have a coincidence point and a point of coincidence such that If is another point of coincidence, then there is with . Now,
(3.8)
where
(3.9)

Hence, , that is, .

Since is the unique point of coincidence of and , and and are weakly compatible, is the unique common fixed point of and by [4, Proposition ].

The proof is similar when is assumed to be a complete subspace of since .

Example 3.4.

Let , , , , is a fixed function from , for example, .

Consider the mappings given by , , . Then
(3.10)

so the conditions of strict contractivity are fulfilled. Further, and it is easy to verify that the sequence satisfies the conditions , (even in the setting of cone metric spaces). All the conditions of the theorem are fulfilled. Taking , , we obtain a theorem from [13]. Note that this theorem cannot be applied directly, since the cone may not be normal in our case. So, our theorem is a proper generalization of the mentioned theorem from [13].

Example 3.5.

Let , , , , .

Take the mappings given by , . Then, since , for it is
(3.11)
that is, the conditions of strict contractivity are fulfilled. Taking we have that in the cone metric space , , , and . Indeed,
(3.12)

(in the norm of space ), which means that the pair of mappings of the cone metric space satisfies condition (E.A). The conditions of the theorem are fulfilled in the case of a normal cone .

Corollary 3.6.

If all the conditions of Theorem 3.3 are fulfilled, except that (ii) is substituted by either of the conditions
(3.13)

then and have a unique common fixed point.

Proof.

Formulas in (3.13) are clearly special cases of (ii).

Note that formulas in (3.13) are strict contractive conditions which correspond to the contractive conditions of Theorems , , and from [2].

3.1. Cone Metric Version of Das-Naik's Theorem

The following theorem was proved by Das and Naik in [14].

Theorem 3.7.

Let be a complete metric space. Let be a continuous self-map on and be any self-map on that commutes with . Further, let and there exists a constant such that for all :
(3.14)

where . Then and have a unique common fixed point.

Now we recall the definition of -quasi-contractions on cone metric spaces. Such mappings are generalizations of Das-Naik's quasi-contractions.

Definition 3.8 (see [3]).

Let be a cone metric space, and let . Then is called a -quasicontraction, if for some constant and for every , there exists such that
(3.15)

The following theorem was proved in [3].

Theorem 3.9.

Let be a complete cone metric space with a normal cone. Let , is a -quasicontraction that commutes with , one of the mappings and is continuous, and they satisfy . Then and have a unique common fixed point in .

Using property (E.A) of the pair instead of commutativity and continuity, we can prove the existence of a common fixed point without normality condition. Then, Theorem 3.7 for metric spaces follows as a consequence.

Theorem 3.10.

Let and be two weakly compatible self-mappings of a cone metric space such that

(i) satisfies property (E.A);

(ii) is a -quasicontraction;

(iii) .

If or is a complete subspace of , then and have a unique common fixed point.

Proof.

Let be such that , . It follows from (iii) and the completeness of one of , that there exists such that . Hence, . We will prove first that . Putting and in (3.15) we obtain that
(3.16)

for some . We have to consider the following cases:

;

which implies ;

which implies ;

) , since ;

which implies .

Thus, in all possible cases, for each and so . The uniqueness of limits (which is a consequence of the condition without using normality of the cone) implies that .

Since and are weakly compatible it follows that . Let us prove that is a common fixed point of the pair . Suppose . Putting in (3.15) , , we obtain that
(3.17)

where . So, we have only two possible cases:

implying and ;

implying and .

The uniqueness follows easily. The theorem is proved.

Note that in Theorems 3.3 and 3.10 condition that one of the subspaces , is complete can be replaced by the condition that one of them is closed in the cone metric space .

Corollary 3.11.

The conclusion in Theorem 3.7 remains valid if the conditions of commutativity and continuity of one of the mappings are replaced by the condition (E.A) for the pair .

Proof.

This follows easily by taking , , .

Taking into account [15, Theorem ] and results from [5], it can be seen that the question of existence of fixed points for quasicontractions on complete cone metric spaces without normality condition is still open in the case when . Theorem 3.10 answers this question when property (E.A) is fulfilled.

Note that the common fixed point problem for a weak compatible pair with property (E.A) under strict conditions in symmetric spaces was investigated in [1621]. As an example we state the following result.

Theorem 3.12 (see [19]).

Let be a symmetric (semimetric) space that enjoys property ( ) (the Hausdorffness of the topology ). Let and be two self-mappings on such that

(i) satisfies property (E.A),

(ii)for all , ,

(3.18)

for some , . If is a -closed ( -closed) subset of , then and have a point of coincidence.

This result can be proved in the setting of cone metric spaces putting " " instead of " ," and also for the symmetric space associated with a complete cone metric space with a normal cone, introduced in [22].

4. Strict Contractivity and the Hardy-Rogers Theorem

It was proved in [23] (see also [24]) that a self-map of a complete metric space has a unique fixed point if for some nonnegative scalars , with and for all , the inequality

(4.1)

holds. In [4, Theorem ], this result was proved in the setting of cone metric spaces, but in a generalized version–-for a pair of self-mappings satisfying certain conditions.

Assuming property (E.A), we can prove the following theorem.

Theorem 4.1.

Let be a cone metric space and let be a weakly compatible pair of self-mappings on satisfying condition (E.A). Suppose that there exist nonnegative scalars , such that and that for each ,
(4.2)

If and if at least one of and is a complete subspace of , then and have a unique common fixed point.

Proof.

There exists a sequence such that , in the cone metric , for some . Let, for example, be complete. Then there exists such that and converge to . Let us prove that . Putting in (4.2) and instead of and , respectively, we obtain
(4.3)
Hence,
(4.4)
that is, denoting , ,
(4.5)

Thus, , that is, . The uniqueness of limit in cone metric spaces (when the cone has nonempty interior) implies that .

Since the mappings , are weakly compatible, this implies that . Hence, we obtain that is the unique common fixed point of the mappings and . Namely, suppose that . Putting in (4.2) and instead of , respectively, we obtain
(4.6)

a contradiction.

Since , the proof is the same if we assume that is complete.

The version of Hardy-Rogers' theorem for metric spaces from [24] is obtained taking , , , .

Declarations

Acknowledgments

The authors are grateful to the referees for valuable comments which improved the exposition of the paper. This work is supported by Grant no. 14021 of the Ministry of Science and Environmental Protection of Serbia.

Authors’ Affiliations

(1)
Faculty of Mathematics, University of Belgrade
(2)
Faculty of Mechanical Engineering, University of Belgrade

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Copyright

© Z. Kadelburg et al. 2009

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