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Strict Contractive Conditions and Common Fixed Point Theorems in Cone Metric Spaces
Fixed Point Theory and Applications volume 2009, Article number: 173838 (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 [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 [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
or, equivalently, if there is a number 
such that for all 
,
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
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,
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
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
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:
and define the following conditions:
for arbitrary 
there exists 
such that
for arbitrary 
there exists 
such that
for arbitrary 
there exists 
such that
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 [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:
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
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,
where
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
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
that is, the conditions of strict contractivity are fulfilled. Taking 
we have that in the cone metric space 
, 
, 
, and 
. Indeed,
(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
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 
:
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
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
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
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 [16–21]. 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 
, 
,
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
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 
,
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
Hence,
that is, denoting 
, 
,
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
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 
, 
, 
, 
.
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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.
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Kadelburg, Z., Radenović, S. & Rosić, B. Strict Contractive Conditions and Common Fixed Point Theorems in Cone Metric Spaces. Fixed Point Theory Appl 2009, 173838 (2009). https://doi.org/10.1155/2009/173838
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DOI: https://doi.org/10.1155/2009/173838