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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