Strict Contractive Conditions and Common Fixed Point Theorems in Cone Metric Spaces
© Z. Kadelburg et al. 2009
Received: 22 June 2009
Accepted: 9 September 2009
Published: 4 October 2009
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 , 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 [2–6], 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. (In fact, validity of this theorem is equivalent to the normality of the cone, see .)
Example 1.1 (see ).
Definition 1.2 (see ).
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 ]).
It was proved in  that if is a normal cone, then converges to if and only if , .
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 . 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 , -commutativity [9, 10], weak compatibility , 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 ).
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 . We extend these concepts to cone metric spaces and investigate their properties in this paper.
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 .
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.
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).
3. Strict Contractive Conditions and Existence of Common Fixed Points on Cone Metric Spaces
and define the following conditions:
These conditions are called strict contractive conditions. Since in metric spaces the following inequalities hold:
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 .
Conditions and are not mentioned in . 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.
Using the previous example, it is easy to construct the respective example in the case of cone metric spaces.
Conclusion is the same as in the metric case.
We will prove the following theorem in the setting of cone metric spaces.
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 ].
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 . 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 .
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 .
3.1. Cone Metric Version of Das-Naik's Theorem
The following theorem was proved by Das and Naik in .
Definition 3.8 (see ).
The following theorem was proved in .
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.
The uniqueness follows easily. The theorem is proved.
Taking into account [15, Theorem ] and results from , 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 [16–21]. As an example we state the following result.
Theorem 3.12 (see ).
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 .
4. Strict Contractivity and the Hardy-Rogers Theorem
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.
The version of Hardy-Rogers' theorem for metric spaces from  is obtained taking , , , .
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.
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