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: mahmed68@yahoo.com
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
yP
and nonnegative real
numbers a, b,
3)
PP
.
For a given cone P, we define a partial ordering
with respect to P by
y iff
xP .
y (resp.
y) stands for
y and
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
y leads to
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 ,
yE
,
y
implies
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
,
,ydx
2
dxy
,yx,d
iff
y,
3
,
,,y dyxdx
4
,
dxy d
,,,xz dzy
for all ,,
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
XX
X
.
Then there exists a subset Y such that
fY
X: and
YX is one-to-one.
,Definition 1.2 [4]. Let
d
,: be a cone metric space
and
gX XzX
be mappings. Then, is called
a coincidence point of f and g iff
zgz.
Definition 1.3 [1]. Let
,d
,: be a cone metric space.
The mappings
gX XzX
are
weakly compatible iff
for every coincid ence point of f and g,
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
for 1)
,
and
;
int P
int implies
xP
lim
n
; 2)
0
nx
xP
for all 3)
.
In [1], the authors established the following fixed point
theorem.
,Theorem 1.1 Let
d
,: be a cone metric space, P a
solid cone and
,
gX X. Assume that
g is a
generalized
-contraction; i.e.,
,dfx fyu
,
for all
yX
and some u where
,, ,,
,,
,, .
2
udgxgydfxgx
dgxf ydgyfx
dfygy
XgX
,
X or
Suppose that X is a
complete subspace of X, and f and g are weakly compati-
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