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Advances in Pure Mathematics, 2012, 2, 6-7
http://dx.doi.org/10.4236/apm.2012.21002 Published Online January 2012 (http://www.SciRP.org/journal/apm)
A Note on the Paper “Generalized
-Contraction for a
Pair of Mappings on Cone Metric Spaces”
Mohamed Abd El-Rahman Ahmed
Department of Mathematics, Faculty of Science, Assiut University, Assiut, Egypt
Email: [email protected]
Received August 10, 2011; revised September 20, 2011; accepted September 28, 2011
ABSTRACT
We note that Theorem 2.3 [1] is a consequence of the same theorem for one map.
Keywords: Generalized
-Contraction; Weakly Compatible; Solid Cone
1. Introduction
Huang and Zhang [2 ] initiated fixed po int theory in cone
metric spaces. On the other hand, the authors [3] gave a
lemma and show ed that some fixed point generalizations
are not real generalizations. In this note, we show that
Theorem 2.3 [1 ] is so.
Following [2], let E be a real Banach space and
be
the zero vector in E, and . P is called cone iff PE

P1) P is closed, nonempty and
by P ,,
2) for all
ax
x
yP
and nonnegative real
numbers a, b,
3)

PP
 .
For a given cone P, we define a partial ordering
with respect to P by
x
y iff
xP .
x
y (resp.
x
y) stands for
x
y and
x
y (resp. yx
in
), where denotes the interior of P. In the
paper we always assume that P is solid, i.e.,
tP
 
int P
int P
.
It is clear that
x
y leads to
x
y
>0K
but the reverse
need not to be true.
The cone P is called normal if there exists a number
such that for all ,
x
yE
,
x
y
 implies
x
Ky
E
.
The least positive number satisfying above is called
the normal constant of P.
Definition 1.1 [2]. Let X be a nonempty set. A func-
tion is called cone metric iff
:dX X

1
M
,

,ydx

2
M

dxy

,yx,d
 iff
x
y,

3
M
,
 
,,y dyxdx

4
M
,

dxy d

,,,xz dzy
for all ,,
x
yz X
,
d
:
is said to be a cone metric
space.
In [3], the authors gave the following important lem-
ma.
Lemma 1.1 Let X be a nonempty and
f
XX
X
.
Then there exists a subset Y such that
fY
f
X: and
f
YX is one-to-one.
,Definition 1.2 [4]. Let
X
d
,: be a cone metric space
and
f
gX XzX
 
be mappings. Then, is called
a coincidence point of f and g iff
f
zgz.
Definition 1.3 [1]. Let
X
,d
,: be a cone metric space.
The mappings
f
gX XzX
are
weakly compatible iff
for every coincid ence point of f and g,
f
gx gfx
:PP
.
Definition 1.4 (see [1]). Let P be a solid cone in a real
Banach space E. A nondecreasing function
is called a compari son function iff
xP
x
x

 for 1)
,
X

and
 ;
int P
 
int implies
x
x
xP

lim
n
; 2)
0
nx
xP
for all 3)
 .
In [1], the authors established the following fixed point
theorem.
,Theorem 1.1 Let
X
d
,: be a cone metric space, P a
solid cone and
,
f
gX X. Assume that
f
g is a
generalized
 

-contraction; i.e.,
,dfx fyu
,
for all
x
yX
and some u where
 

 

 

 

 

,, ,,
,,
,, .
2
udgxgydfxgx
dgxf ydgyfx
dfygy
XgX

,
f
f
X or
g
Suppose that X is a
complete subspace of X, and f and g are weakly compati-
C
opyright © 2012 SciRes. APM
M. ABD EL-R. AHMED 7

ble. Then the mappings f and g have a unique common
fixed poi n t in X.
f
X or
g
2. Main Result
In Theorem 1.1, if we choose X
g
I (X
I
:= the identity
map on X), then we have the following theorem.
Theorem 2.1 Let

,
X
d
: be a cone metric space, P a
solid cone and
f
XX. Assume that f is a general-
ized
-contraction; i.e.,


fy u
,dfx for all
,
x
yX and some u where
 







,, ,,
,,
2
udxydxfx
dxfx dyf
, ,
.
dyfy
y
Suppose that
f
X
YX

or X is a complete subspace of X.
Then the mapping f has a unique fixed point in X.
Now, we state and prove our main result in the fol-
lowing way.
Theorem 2.2 Theorem 1.1 is a consequence of Theo-
rem 2.1.
Proof. By Lemma 1.1, there exists such that
 
g
YgX and :
g
YX is one-to-one. Define a
map
:hgYgY by


xf xhg for each

gY

. Since g is one-to-one on Y, then h is well-
defined.

u
,dfx fy for all ,
x
yX
and
some u where
 

 

 

 

Since YgX is complete, by us-
ing Theorem 2.1, there exists 0
x
X such that
hgxgxf x
000
. Hence, f and g have a point
of coincidence which is also unique. Since f and g are
weakly compatible, then f and g have a unique common
fixed poi nt.
Remark 2.1 Since Theorem 1 [4] is a special case of
Theorem 1.1, then it is a consequence of Theorem 2.1,
too.
REFERENCES
[1] A. Razani, V. Rakocevic and Z. Goodarzi, “Generalized
-Contraction for a Pair of Mappings on Cone Metric
Spaces,” Applied Mathematics and Computation, Vol. 217,
No. 22, 2011, pp. 8899-8906.
[2] 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
[3] R. H. Haghi, Sh. Rezapour and N. Shahzad, “Some Fixed
Point Generalizations Are Not Real Generalizations,”
Nonlinear Analysi s, Vol. 74, 2011, pp. 1799-1803.
[4] C. D. Bari and P. Vetro, “
-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
 

,, ,,
,, 2
udgxgydfxgx
dgxf ydg
dfygy
,,
.
yfx
Copyright © 2012 SciRes. APM