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.
), 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
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
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
,
.
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
,
.
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
,
where 
are constants with
.
From the completeness of
, there exists 
such that
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.