Abstract
We extend a common fixed point theorem of Radenovic and Rhoades for four non-self-mappings in cone metric spaces.
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1. Introduction and Preliminaries
Recently, Huang and Zhang [1] generalized the concept of a metric space, replacing the set of real numbers by ordered Banach space and obtained some fixed point theorems for mappings satisfying different contractive conditions. Subsequently, the study of fixed point theorems in such spaces is followed by some other mathematicians; see [2–8]. The aim of this paper is to prove a common fixed point theorem for four non-self-mappings on cone metric spaces in which the cone need not be normal. This result generalizes the result of Radenović and Rhoades [5].
Consistent with Huang and Zhang [1], the following definitions and results will be needed in the sequel.
Let 
be a real Banach space. A subset 
of 
is called a cone if and only if
(a)
is closed, nonempty and 
;
(b)
, 
, 
implies 
;
(c)
.
Given a cone 
, we define a partial ordering 
with respect to 
by 
if and only if 
. A cone 
is called normal if there is a number 
such that for all 
,
The least positive number satisfying the above inequality is called the normal constant of 
, while 
stands for 
(interior of 
).
Definition 1.1 (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.
Definition 1.2 (see [1]).
Let 
be a cone metric space. One says that 
is
-
(e)
a Cauchy sequence if for every
with 
, there is an 
such that for all 
, 
; -
(f)
a Convergent sequence if for every
with 
, there is an 
such that for all 
, 
for some fixed 
.
with 
, there is an 
such that for all 
, 
;
with 
, there is an 
such that for all 
, 
for some fixed 
.A cone metric space 
is said to be complete if every Cauchy sequence in 
is convergent in 
. It is known that 
converges to 
if and only if 
as 
. It is a Cauchy sequence if and only if 
.
Remark 1.3 (see [9]).
Let 
be an ordered Banach (normed) space. Then 
is an interior point of 
if and only if 
is a neighborhood of 
.
Corollary 1.4 (see [10]).
-
(1)
If
and 
, then 
.Indeed, 
implies 
.
and 
, then 
.Indeed, 
implies 
.(2)If 
and 
, then 
.Indeed, 
implies 
.
-
(3)
If
for each 
, then 
.
for each 
, then 
.If 
, 
, and 
, then there exists an 
such that for all 
we have 
.
If 
is a real Banach space with cone 
and if 
where 
and 
, then 
.
We find it convenient to introduce the following definition.
Definition 1.7 (see [5]).
Let 
be a complete cone metric space and 
a nonempty closed subset of 
, and 
satisfying
where
for all 
, 
, 
, then 
is called a generalized 
-contractive mapping of 
into 
.
Definition 1.8 (see [2]).
Let 
and 
be self-maps on a set 
(i.e., 
). If 
for some 
in 
, then 
is called a coincidence point of 
and 
, and 
is called a point of coincidence of 
and 
. Self-maps 
and 
are said to be weakly compatible if they commute at their coincidence point; that is, if 
for some 
, then 
.
2. Main Result
The following theorem is Radenović and Rhoades [5] generalization of Imdad and Kumar's [12] result in cone metric spaces.
Theorem 2.1.
Let 
be a complete cone metric space and 
a nonempty closed subset of 
such that for each 
and 
there exists a point 
(the boundary of 
) such that
Suppose that 
are such that 
is a generalized 
-contractive mapping of 
into 
, and
(i)
,
(ii)
,
(iii)
is closed in 
.
Then the pair 
has a coincidence point. Moreover, if pair 
is weakly compatible, then 
and 
have a unique common fixed point.
The purpose of this paper is to extend the above theorem for four non-self-mappings in cone metric spaces. We begin with the following definition.
Definition 2.2.
Let 
be a complete cone metric space and 
a nonempty closed subset of 
, and 
satisfying
where
for all 
, 
, 
, then 
is called a generalized 
-contractive mappings pair of 
into 
.
Notice that by setting 
and 
in (2.2), one deduces the slightly generalized form of (1.3).
We state and prove our main result as follows.
Theorem 2.3.
Let 
be a complete cone metric space and 
a nonempty closed subset of 
such that for each 
and 
there exists a point 
(the boundary of 
) such that
Suppose that 
are such that 
is a generalized 
-contractive mappings pair of 
into 
, and
(I)
,
(II)
,
(III)
and 
(or 
and 
) are closed in 
.Then
(IV)
has a point of coincidence,
(V)
has a point of coincidence.
Moreover, if 
and 
are weakly compatible pairs, then 
, 
, 
, and 
have a unique common fixed point.
Proof.
Firstly, we proceed to construct two sequences 
and 
in the following way.
Let 
be arbitrary. Then (due to 
) there exists a point 
such that 
. Since 
, one concludes that 
. Thus, there exists 
such that 
. Since 
there exists a point 
such that
Suppose that 
. Then 
which implies that there exists a point 
such that 
. Otherwise, if 
, then there exists a point 
such that
Since 
there exists a point 
with 
, so that
Let 
be such that 
. Thus, repeating the foregoing arguments, one obtains two sequences 
and 
such that
(a)
, 
,
(b)
or 
,
(c)
or 
,
We denote that
Note that 
, as if 
, then 
, and one infers that 
which implies that 
. Hence 
. Similarly, one can argue that 
.
Now, we distinguish the following three cases.
Case 1.
If 
, then from (2.2)
where
Clearly, there are infinite many 
such that at least one of the following four cases holds:
-
(1)
(2.13)
-
(2)
(2.14)
-
(3)
(2.15)
-
(4)
(2.16)
which implies 
, that is,
From (1), (2), (3), and (
) it follows that
Similarly, if 
, we have
If 
, we have
Case 2.
If 
, then 
and
which in turn yields
and hence
Now, proceeding as in Case 1, we have that (2.18) holds.
If 
, then 
. We show that
Using (2.21), we get
By noting that 
, one can conclude that
in view of Case 1.
Thus,
and we proved (2.24).
Case 3.
If 
, then 
. We show that
Since 
, then
From this, we get
By noting that 
, one can conclude that
in view of Case 1.
Thus,
and we proved (2.28).
Similarly, if 
, then 
, and
From this, we have
By noting that 
, one can conclude that
in view of Case 1.
Thus, in all Cases 1–3, there exists 
such that
and there exists 
such that
Following the procedure of Assad and Kirk [13], it can easily be shown by induction that, for 
, there exists 
such that
From (2.38) and by the triangle inequality, for 
, we have
From Remark 1.5 and Corollary 1.4(1), 
.
Thus, the sequence 
is a Cauchy sequence. Then, as noted in [14], there exists at least one subsequence 
or 
which is contained in 
or 
, respectively, and finds its limit 
Furthermore, subsequences 
and 
both converge to 
as 
is a closed subset of complete cone metric space 
. We assume that there exists a subsequence 
for each 
, then 
. Since 
as well as 
are closed in 
, and 
is Cauchy in 
, it converges to a point 
. Let 
, then 
. Similarly, 
a subsequence of Cauchy sequence 
also converges to 
as 
is closed. Using (2.2), one can write
where
Let 
. Clearly at least one of the following four cases holds for infinitely many 
:
-
(1)
(2.42)
-
(2)
(2.43)
-
(3)
(2.44)
-
(4)
(2.45)
In all cases we obtain 
for each 
. Using Corollary 1.4(3) it follows that 
or 
. Thus, 
, that is, 
is a coincidence point of 
, 
.
Further, since Cauchy sequence 
converges to 
and 
, 
, there exists 
such that 
. Again using (2.2), we get
where
Hence, we get the following cases:
Since 
, using Remark 1.6 and Corollary 1.4(3), it follows that 
; therefore, 
, that is, 
is a coincidence point of 
.
In case 
and 
are closed in 
, 
or 
. The analogous arguments establish (IV) and (V). If we assume that there exists a subsequence 
with 
as well 
being closed in 
, then noting that 
is a Cauchy sequence in 
, foregoing arguments establish (IV) and (V).
Suppose now that 
and 
are weakly compatible pairs, then
Then, from (2.2),
where
Hence, we get the following cases:
Since 
, using Remark 1.6 and Corollary 1.4(3), it follows that 
. Thus, 
.
Similarly, we can prove that 
. Therefore 
, that is, 
is a common fixed point of 
, 
, 
, and 
.
Uniqueness of the common fixed point follows easily from (2.2).
The following example shows that in general 
, 
, 
, and 
satisfying the hypotheses of Theorem 2.3 need not have a common coincidence justifying two separate conclusions (IV) and (V).
Example 2.4.
Let 
, 
, 
, 
, and 
defined by 
, where 
is a fixed function, for example, 
. Then 
is a complete cone metric space with a nonnormal cone having the nonempty interior. Define 
, 
, 
, and 
as
Since 
. Clearly, for each 
and 
there exists a point 
such that 
. Further, 
, 
, 
, and 
, 
, 
, and 
are closed in 
.
Also,
Moreover, for each 
,
that is, (2.2) is satisfied with 
.
Evidently, 
and 
. Notice that two separate coincidence points are not common fixed points as 
and 
, which shows necessity of weakly compatible property in Theorem 2.3.
Next, we furnish an illustrate example in support of our result. In doing so, we are essentially inspired by Imdad and Kumar [12].
Example 2.5.
Let 
, 
, 
, and 
defined by 
, where 
is a fixed function, for example, 
. Then 
is a complete cone metric space with a nonnormal cone having the nonempty interior. Define 
, 
, 
, and 
as
Since 
. Clearly, for each 
and 
there exists a point 
such that 
. Further, 
, 
, and 
.
Also,
Moreover, if 
and 
, then
Next, if 
, then
Finally, if 
, then
Therefore, condition (2.2) is satisfied if we choose 
. Moreover 
is a point of coincidence as 
as well as 
whereas both the pairs 
and 
are weakly compatible as 
and 
. Also, 
, 
, 
, and 
are closed in 
. Thus, all the conditions of Theorem 2.3 are satisfied and 
is the unique common fixed point of 
, 
, 
, and 
. One may note that 
is also a point of coincidence for both the pairs 
and 
.
Remark 2.6.
-
(1)
Setting
and 
in Theorem 2.3, one deduces Theorem 2.1 due to [5].
and 
in Theorem 2.3, one deduces Theorem 2.1 due to [-
(2)
Setting
and 
in Theorem 2.3, we obtain the following result.
and 
in Theorem 2.3, we obtain the following result.Corollary 2.7.
Let 
be a complete cone metric space and 
a nonempty closed subset of 
such that for each 
and 
there exists a point 
(the boundary of 
) such that
Suppose that 
satisfies the condition
where
for all 
, 
, 
, and 
has the additional property that for each 
, 
, 
has a unique fixed point.
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Acknowledgments
The authors would like to express their sincere appreciation to the referees for their very helpful suggestions and many kind comments. This project was supported by the National Natural Science Foundation of China (10461007 and 10761007) and supported partly by the Provincial Natural Science Foundation of Jiangxi, China (2008GZS0076 and 2009GZS0019).
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Huang, X., Zhu, C. & Wen, X. Common Fixed Point Theorem for Four Non-Self Mappings in Cone Metric Spaces. Fixed Point Theory Appl 2010, 983802 (2010). https://doi.org/10.1155/2010/983802
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DOI: https://doi.org/10.1155/2010/983802