Some Notes on the Paper “New Common Fixed Point Theorems for Maps on Cone Metric Spaces” ()
1. Introduction
In 2007, Huang and Zhang [2] initiated fixed point theory in cone metric spaces. On the other hand, in 2011, Haghi, Rezapour and Shahzad [3] gave a lemma and showed that some fixed point generalizations are not real generalizations. In this note, we show that Theorem 2.1 [1] and Theorem 2.2 [1] are so.
Following [2], let
be a real Banach space and
be the zero vector in
, and
.
is called cone iff 1)
is closed, nonempty and
2)
for all
and nonnegative real numbers
3)
.
For a given cone
, we define a partial ordering
with respect to
by
iff
.
(resp.
) stands for
and
(resp.![](https://www.scirp.org/html/6-2340032\454c5cb5-9d00-405b-b870-967b6625c212.jpg)
), where
denotes the interior of
. In the paper we always assume that
is solid, i.e.,
. It is clear that
leads to
but the reverse need not to be true.
The cone
is called normal if there exists a number
such that for all
,
implies
.
The least positive number satisfying above is called the normal constant of
.
Definition 1.1 [2]. Let
be a nonempty set. A function
is called cone metric iff
(M1)
(M2)
iff
,
(M3)
,
(M4)
for all
.
is said to be a cone metric space.
Lemma 1.1 [3]. Let
be a nonempty and
. Then there exists a subset
such that
and
is one-to-one.
Definition 1.2 [4]. Let
be a cone metric space and
be mappings. Then,
is called a coincidence point of
and
iff
.
Definition 1.3 [4]. Let
be a cone metric space. The mappings
are weakly compatible iff for every coincidence point
of
and
,
.
Theorem 1.1 (Theorem 2.1 [1]). Let
be a cone metric space and let
be constants with
. Suppose that the mappings
satisfy the condition
![](https://www.scirp.org/html/6-2340032\8af9ad69-fb1c-417f-b7ee-a8ba568ae8d4.jpg)
for all
.
If the range of
contains the range of
and
is a complete subspace, then
and
have a unique point of coincidence in
. Moreover, if
and
are weakly compatible, then
and
have a unique fixed point.
Theorem 1.2 (Corollary 2.1 [1]). Let
be a complete cone metric space and let
i = (1,2,3,4,5) be constants with
. Suppose that the mapping
satisfies the condition
![](https://www.scirp.org/html/6-2340032\de82ff6d-5fcd-40e9-bf86-8bb734747f2b.jpg)
for all
.
Then
has a unique fixed point
in
.
Theorem 1.3 (Theorem 2.2 [1]). Let
be a cone metric space and let the mappings
satisfy the condition
, for all
where
![](https://www.scirp.org/html/6-2340032\10052b3b-7fd1-4ece-91bc-c2c2f70deec8.jpg)
,
.
If the range of
contains the range of
and
is a complete subspace, then
and
have a unique point of coincidence in
. Moreover, if
and
are weakly compatible, then
and
have a unique fixed point.
Theorem 1.4 (Corollary 2.2 [1]). Let
be a complete cone metric space and let the mapping
satisfies the condition
, for all
where
![](https://www.scirp.org/html/6-2340032\1a7e9724-899d-43a9-a4f7-65444806c25e.jpg)
,
.
Then
has a unique fixed point
in
.
2. Main Result
In this section, we show that that Theorem 1.1 (resp. Theorem 1.3) is a consequence of Theorem 1.2 (resp. Theorem 1.4).
Theorem 2.1. Theorem 1.1 is a consequence of Theorem 1.2.
Proof. By Lemma 1.1, there exists
such that
and
is one-to-one. Define a map
by
for each
. Since
is one-to-one on
, then
is well-defined. Also, for arbitrary
,
![](https://www.scirp.org/html/6-2340032\f16a10c2-ef08-4c27-8b78-5600775e6455.jpg)
where
are constants with
.
From the completeness of
, there exists
such that
![](https://www.scirp.org/html/6-2340032\9fc85030-955f-4589-b13e-ca32fe55e513.jpg)
by Theorem 1.2. Hence,
and
have a point of coincidence which is also unique. Since
and
are weakly compatible, then
and
have a unique common fixed point.
Theorem 2.2. Theorem 1.3 is a consequence of Theorem 1.4.