International Journal of Modern Nonlinear Theory and Application
Vol.2 No.2(2013), Article ID:33414,2 pages DOI:10.4236/ijmnta.2013.22019
Some Notes on the Paper “New Common Fixed Point Theorems for Maps on Cone Metric Spaces”
Department of Mathematics, Faculty of Science, Assiut University, Assiut, Egypt
Email: mahmed68@yahoo.com
Copyright © 2013 Mohamed Abd El-Rahman Ahmed. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
Received January 17, 2013; revised February 21, 2013; accepted March 19, 2013
Keywords: Cone Metric; Weakly Compatible; Fixed Point
ABSTRACT
In this paper, we show that Theorem 2.1 [1] (resp. Theorem 2.2 [1]) is a consequence of Corollary 2.1 [1] ( resp. Corollary 2.2 [1]).
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.
REFERENCES
- G. Song, X. Sun, Y. Zhao and G. Wang, “New Common Fixed Point Theorems for Maps on Cone Metric Spaces,” Applied Mathematics Letters, Vol. 23, No. 9, 2010, pp. 1033-1037. doi:10.1016/j.aml.2010.04.032
- L.-G. Huang and X. Zhang, “Cone Metric Spaces and Fixed Point Theorems of Contractive Mappings,” Journal of Mathematical Analysis and Applications, Vol. 332, No. 2, 2007, pp. 1468-1476. doi:10.1016/j.jmaa.2005.03.087
- R. H. Haghi, Sh. Rezapour and N. Shahzad, “Some Fixed Point Generalizations Are Not Real Generalizations,” Nonlinear Analysis, Theory, Methods and Applications, Vol. 74, 2011, pp. 1799-1803.
- C. Di Bari and P. Vetro, “ϕ-Pairs and Common Fixed Points in Cone Metric Spaces,” Rendiconti del Circolo Matematico di Palermo, Vol. 57, No. 2, 2008, pp. 279- 285. doi:10.1007/s12215-008-0020-9