## Abstract—We obtain common fixed points and points of coincidence of a pair of mappings satisfying a generalized contractive type condition in cone metric spaces. Our results generalize some well-known recent results in the literature.

**Index Terms— coincidence point; common fixed point; **
**compatible mapping; cone metric space. **

I. INTRODUCTION

A large variety of the problems of analysis and applied mathematics reduce to finding solutions of non-linear functional equations which can be formulated in terms of finding the fixed points of a nonlinear mapping. In fact, fixed point theorems are very important tools for proving the existence and uniqueness of the solutions to various mathematical models (differential, integral and partial differential equations and variational inequalities etc.) representing phenomena arising in different fields, such as steady state temperature distribution, chemical equations, neutron transport theory, economic theories, financial analysis, epidemics, biomedical research and flow of fluids. They are also used to study the problems of optimal control related to these systems [16]. Fixed point theory concerned with ordered Banach spaces helps us in finding exact or approximate solutions of boundary value problems [2]. In 1963, S. Ghaler, generalized the idea of metric space and introduced 2-metric space which was followed by a number of papers dealing with this generalized space. A lot of materials are available in other generalized metric spaces, such as, semi metric spaces, quasi semi metric spaces and D-metric spaces. Huang and Zhang [7] introduced the concept of cone metric space and established some fixed point theorems for contractive type mappings in a cone metric space. Subsequently, some other authors [1, 3, 5, 6, 8, 10, 11, 13, 16] studied the existence of fixed points, points of coincidence and common fixed points of mappings satisfying a contractive type condition in cone metric spaces.

Manuscript received March 15, 2011, accepted March 31, 2011. This work was supported by Higher Education Commission (HEC), Pakistan.

M. Arshad is with the Department of Maths and Stats, International Islamic University, H-10, Islamabad, 44000, Pakistan (Phone: +92-3335103984, Fax: +92-51-226877,

e-mail: marshad_zia@yahoo.com).

A. Azam is with the Department of Mathematics, COMSATS Institute of Information Technology, Chak Shahzad, Islamabad, 44000, Pakistan (e-mail: akbarazam@yahoo.com)

P. Vetro is with the Università degli Studi di Palermo,Dipartimento di Matematica ed Applicazioni,Via Archirafi 34, 90123 Palermo, Italy, (e-mail: vetro@math.unipa.it).

In this paper, we obtain points of coincidence and common fixed points for a pair of mappings satisfying a more general contractive type condition. Our results improve and generalize some significant recent results.

A subset

P

of a real Banach spaceE

is called a*cone*if it has the following properties:

(i)

P

is non-empty closed andP

≠

{0

};

(ii)_{0}

≤

a

,

b

∈

R

_{ and }

x

,

y

∈

P

⇒

ax

+

by

∈

P

;

_{ }

(iii)

P

∩

(

−

P

)

=

{

**0**

}.

For a given cone

P

⊆

E

, we can define a partial ordering≤

onE

with respect toP

byx

≤

y

if and only ifP

x

y

−

∈

. We shall writex

<

y

ifx

≤

y

andy

x

≠

, whilex

<<

y

will stands fory

−

x

∈

int

P

*,*

where

int

P

denotes the interior ofP

.

The coneP

is called*normal*if there is a number

κ

≥

1

such that for all,

,

,

y

E

x

∈

.

y

x

y

x

≤

⇒

≤

κ

≤

**0**

(1)
The least number

κ

≥

1

satisfying (1) is called the*normal constant* of

P

.

In the following we always suppose that

E

is a real Banach space andP

is a cone inE

withint

P

≠ ∅

and≤

is a partial ordering with respect toP

.

**Definition 1 ** *Let *

X

*be a nonempty set. Suppose that the*

*mapping*

d

:

X

×

X

→

E

*satisfies:*

(i)

**0**

≤

d

(

x

,

y

)

for allx

,

y

∈

X

andd

(

x

,

y

)

=

**0**

if and only if x

=

y

;(ii)

d

(

x

,

y

)

=

d

(

y

,

x

)

for allx

,

y

∈

X

;(iii)

d

(

x

,

y

)

≤

d

(

x

,

z

)

+

d

(

z

,

y

)

for allx

,

y

,

z

∈

X

.

Thend

is called a*cone metric*on

X

and(

X

,

d

)

is called a*cone metric space*.

Let

x

*be a sequence in*

_{n}X

andx

∈

X

. If for | Paste Special | Picture (with “float over text” unchecked). each**0**

<<

c

there isn

_{0}

∈

N

such that for all,

)

,

(

,

0

d

x

x

c

n

n

≥

_{n}<<

then

x

*n*

is said to be*convergent or *

{ }

x

*converges to*

_{n}x

andx

is called## Common Fixed Point of Generalized

Contractive Type Mappings in Cone Metric

Spaces

Muhammad Arshad, Akbar Azam and Pasquale Vetro

IAENG International Journal of Applied Mathematics, 41:3, IJAM_41_3_12

the *limit* of

x

*n*

. We denote this by lim*n*

x

*n*

=

x

,

or,

x

x

*n*

→

asn

→

∞

.

If for each**0**

<<

c

there isn

0∈

ℕ

such that for alln

,

m

≥

n

_{0}

,

d

(

x

_{n},

x

_{m})

<<

c

,

then

x

*n*

is called a*Cauchy sequence*in

X

. If every Cauchy sequence is convergent inX

, thenX

is called a*complete cone metric space*. Let us recall [6] that if

P

is a normal cone, thenx

_{n}∈

X

,

converges tox

∈

X

if and only ifd

x

*n*

,

x

→

**0**

as n

→

∞

. Furthermore,x

*n*

∈

X

is a Cauchy sequence if and only if**0**

→

)

,

(

x

_{n}x

_{m}d

asn

,

m

→

∞

.

** Lemma** 2 *Let *

(

X

,

d

)

*be a cone metric space,*

P

*be a*

*cone. Let*

{ }

x

_{n}*be a sequence in*

X

*and*

{ }

a

_{n}*be a*

*sequence in*

P

*converging to*

**0**. Ifd

(

x

_{n},

x

_{m})

≤

a

_{n}for*every*

n

∈

N

*with*

m

>

n

*, then*

{ }

x

_{n}*is a Cauchy*

*sequence.*

**Proof**. Fix

**0**

<<

c

and choose}

:

{

)

,

(

δ

=

x

∈

E

x

<

δ

I

**0**

such that
IntP

I

c

+

(

**0**

,

δ

)

⊂

. Sincea

*n*

→

**0**

_{ , there exists }

n

0∈

N

be such thata

*n*

∈

I

**0**

,

δ

for everyn

≥

n

_{0}.

From

c

−

a

_{n}∈

IntP

, we deducec

a

x

x

d

(

_{n},

_{m})

≤

_{n}<<

for everym

,

n

≥

n

_{0}and hence

x

*n*

is a Cauchy sequence.**Remark 3 ** * Let *

A

,

B

,

C

,

D

,

E

*be non negative real*

*numbers with*

A

+

B

+

C

+

D

+

E

<

1

,

B

=

C

*or*

D

=

E

.

*If*

F

=

(

A

+

B

+

D

)(

1

−

C

−

D

)

−1*and*

1

)

1

)(

(

+

+

−

−

−=

A

C

E

B

E

G

*, then*

FG

<

1

*. In fact,*

*if*

B

=

C

*then*

,

1

1

1

1

1

<

−

−

+

+

⋅

−

−

+

+

=

−

−

+

+

⋅

−

−

+

+

=

D

C

E

B

A

E

B

D

C

A

E

B

E

C

A

D

C

D

B

A

FG

* and if *

D

=

E

*,*

.

1

1

1

1

1

<

−

−

+

+

⋅

−

−

+

+

=

−

−

+

+

⋅

−

−

+

+

=

E

B

D

C

A

D

C

E

B

A

E

B

E

C

A

D

C

D

B

A

FG

A pair

(

f

,

T

)

of self-mappings onX

are said to be weakly compatible if they commute at their coincidence point (i.e.,fTx

=

Tfx

wheneverfx

=

Tx

). A pointX

y

∈

is called point of coincidence ofT

andf

if there exists a pointx

∈

X

such thaty

=

fx

=

Tx

.

## II. MAIN RESULTS

The following theorem improves/generalizes the results [1, Theorems 2.1, 2.3, 2.4], [5, Theorems 1, 2,3], [7, Theorems 1,3, 4], [8, Theorem 2.8], [13, Theorems 2.3, 2.6, 2.7, 2.8], and [15, Theorems 1, Corollary 2].

**Theorem 4 *** Let *

(

X

,

d

)

*be a complete cone metric*

*space,*

P

*be a cone and*

m

,

n

*be positive integers.*

*Assume that the mappings*

T

,

f

:

X

→

X

*satisfy:*

)]

,

(

)

,

(

[

)]

,

(

)

,

(

[

)

,

(

)

,

(

x

T

fy

d

y

T

fx

d

y

T

fy

d

x

T

fx

d

fy

fx

d

y

T

x

T

d

*m*
*n*

*n*

*m*
*n*

*m*

+

+

+

+

≤

γ

β

α

*for all *

x

,

y

∈

X

*where*

α

,

β

,

γ

*are non negative real*

*numbers with*

α

+

2

β

+

2

γ

<

1

.

*If*

T

(

X

)

⊆

f

(

X

)

*and*

f

(

X

)

*is a complete subspace of*

X

*, then*

T

*m*

,

T

*n*

*and*

f

*have a unique common point of coincidence.*

*Moreover if*

(

T

*m*

,

f

)

*and*

(

T

*n*

,

f

)

*are weakly*

*compatible, then*

T

*m*

,

T

*n and*

f

*have a unique common*

*fixed point.*

Huang and Zhang [7] proved the above result by restricting that (a)

P

is normal (b)f

=

I

*X*(c)

m

=

n

=

1

(d) one of the following is satisfied:

### 0

,

1

=

=

<

β

γ

α

( [7, Theorems 1] ),### 0

,

2

1

=

=

<

α

γ

β

( [7, Theorems 3] ),### 0

,

2

1

=

=

<

α

β

γ

( [7, Theorems 3] ).Abbas and Jungck [1] extended the results of Huang and Zhang [7] by removing restriction (b) and obtain common fixed points and points of coincidence of mappings

f

,

T

.

Meanwhile Rezapour and Hamlbarani [13] improved the results of [7] by omitting the assumption (a). Vetro [15] removed restriction (b) and replaced (d) by combining (i) and (ii). Azam, Arshad and Beg [5] and Jungck et al [9] extended these results to a generalized contractive condition by omitting the restrictions (a), (b).

The following theorem is a further generalization of Theorem 4 which removes restrictions (a), (b), (c), and replaces (d) with a more generalized contractive condition.

**Theorem** 5 *Let *

(

X

,

d

)

*be a complete cone metric space,*

P

*be a cone and*

m

,

n

*be positive integers. If the*

*mappings*

T

,

f

:

X

→

X

*satisfy:*

)

,

(

)

,

(

)

,

(

)

,

(

)

,

(

)

,

(

x

T

fy

d

E

y

T

fx

d

D

y

T

fy

Cd

x

T

fx

d

B

fy

fx

d

A

y

T

x

T

d

*m*
*n*

*n*

*m*
*n*

*m*

+

+

+

+

≤

*for all *

x

,

y

∈

X

,

*where*

A

,

B

,

C

,

D

,

E

*are non*

*negative*

*real*

*numbers*

*with*

C

B

E

D

C

B

A

+

+

+

+

<

1

,

=

*or*

D

=

E

.

*If*

)

(

)

(

X

f

X

T

⊆

*and*

f

(

X

)

*or*

T

(

X

)

*is a complete*

*subspace of*

X

*, then*

T

*m*

,

T

*n and*

f

*have a unique*

*common point of coincidence. Moreover if*

(

T

*m*

,

f

)

*and*

## IAENG International Journal of Applied Mathematics, 41:3, IJAM_41_3_12

(

T

*n*

,

f

)

*are weakly compatible, then*

T

*m*

,

T

*n and*

f

*have a unique common fixed point.*

**Proof. **Let

x

_{0}be an arbitrary point in

X

. Choose a pointx

_{1}in

X

such thatfx

_{1}

=

T

*m*

x

_{0}. This can be done since

T

(

X

)

⊆

f

(

X

)

. Similarly, choose a pointx

2 in X such thatfx

2T

x

1.

*n*

=

Continuing this process having chosenx

*in*

_{n}X

, we obtainx

_{n}_{+}

_{1}in

X

such thatfx

2*k*+1

=

T

*m*

x

2*k*

fx

_{2}

*+*

_{k}_{2}

=

T

*n*

x

_{2}

*+*

_{k}_{1}

,

k

=

0, 1, 2,

…

.

Then,).

,

(

]

[

)

,

(

]

[

)

,

(

)

,

(

)

,

(

]

[

)

,

(

)

,

(

)

,

(

)

,

(

)

,

(

)

,

(

)

,

(

2 2 1 2 1 2 2 2 2 2 2 2 1 2 1 2 2 2 1 2 1 2 2 1 2 1 2 2 2 1 2 2 1 2 2 2 2 1 2 + + + + + + + + + + + + + + ++

+

+

+

≤

+

+

+

≤

+

+

+

+

≤

=

*k*

*k*

*k*

*k*

*k*

*k*

*k*

*k*

*k*

*k*

*k*

*m*

*k*

*k*

*n*

*k*

*k*

*n*

*k*

*k*

*m*

*k*

*k*

*k*

*k*

*n*

*k*

*m*

*k*

*k*

fx

fx

d

D

C

fx

fx

d

D

B

A

fx

fx

d

D

fx

fx

Cd

fx

fx

d

B

A

x

T

fx

Ed

x

T

fx

Dd

x

T

fx

Cd

x

T

fx

Bd

fx

fx

Ad

x

T

x

T

d

fx

fx

d

It implies that

).

,

(

]

[

)

,

(

]

1

[

1 2 2 2 2 1 2 + + ++

+

≤

−

−

*k*

*k*

*k*

*k*

fx

fx

d

D

B

A

fx

fx

d

D

C

That is,d

fx

2*k*+1

,

fx

2*k*+2

F d

fx

2*k*

,

fx

2*k*+1

,

where

D

C

D

B

A

F

−

−

+

+

=

1

.Similarly we obtain,

),

,

(

]

[

)

,

(

]

[

)

,

(

)

,

(

3 2 2 2 2 2 1 2 1 2 2 2 3 2 2 2 + + + + + + + ++

+

+

+

≤

=

*k*

*k*

*k*

*k*

*k*

*n*

*k*

*m*

*k*

*k*

fx

fx

d

E

B

fx

fx

d

E

C

A

x

T

x

T

d

fx

fx

d

which implies)

,

(

)

,

(

fx

2*k*+2

fx

2*k*+3

≤

G

d

fx

2*k*+1

fx

2*k*+2

d

withE

B

E

C

A

G

−

−

+

+

=

1

.Now by induction, we obtain for each

k

=

### 0

,

1

,

2

,

…

)

,

(

)

(

)

,

(

)

(

)

,

(

)

(

)

,

(

)

,

(

1 0 1 2 2 2 2 1 2 1 2 2 2 2 1 2fx

fx

d

FG

F

fx

fx

d

FG

F

fx

fx

d

FG

fx

fx

d

F

fx

fx

d

*k*

*k*

*k*

*k*

*k*

*k*

*k*

*k*

*k*

≤

≤

≤

≤

≤

− − − + + + and(

)

(

,

).

)

,

(

)

,

(

1 0 1 2 2 1 2 3 2 2 2fx

fx

d

FG

fx

fx

d

G

fx

fx

d

*k*

*k*

*k*

*k*

*k*+ + + + +

≤

≤

≤

By Remark 3

p

<

q

we have(

)

(

)

).

,

(

1

)

(

)

1

(

)

,

(

1

)

(

1

)

(

)

,

(

)

,

(

)

,

(

)

,

(

)

,

(

)

,

(

1 0 1 0 1 1 0 1 1 1 2 2 4 2 3 2 3 2 2 2 2 2 1 2 1 2 1 2fx

fx

d

FG

FG

F

fx

fx

d

FG

FG

FG

FG

F

fx

fx

d

FG

FG

F

fx

fx

d

fx

fx

d

fx

fx

d

fx

fx

d

fx

fx

d

*p*

*p*

*p*

*i*

*q*

*p*

*i*

*i*

*q*

*p*

*i*

*q*

*q*

*p*

*p*

*p*

*p*

*p*

*p*

*q*

*p*

−

+

≤

×

−

+

−

≤

×

+

≤

+

+

+

+

≤

+ + = − = + + + + + + + + +∑

∑

In analogous way, we deduce

),

,

(

1

)

(

)

1

(

)

,

(

_{2}

_{2}

_{1}

d

fx

_{0}

fx

_{1}

FG

FG

F

fx

fx

d

*p*

*q*

*p*

−

+

≤

+)

,

(

1

)

(

)

1

(

)

,

(

_{2}

_{2}

d

fx

_{0}

fx

_{1}

FG

FG

F

fx

fx

d

*p*

*q*

*p*

−

+

≤

and).

,

(

1

)

(

)

1

(

)

,

(

_{2}

_{1}

_{2}

d

fx

_{0}

fx

_{1}

FG

FG

F

fx

fx

d

*p*

*q*

*p*

−

+

≤

+Hence, for

### 0

<

n

<

m

,

)

,

(

fx

_{n}fx

_{m}a

_{n}d

≤

where

(

,

)

1

)

(

)

1

(

d

fx

_{0}

fx

_{1}

FG

FG

F

a

*p*

*n*

−

+

=

withp

theinteger part of

n

/

2

.

Fix

**0**

<<

c

and chooseI

(

**0**

,

δ

)

=

{

x

∈

E

:

x

x

<

δ

}

such thatc

+

I

(

_{0}

_{0}

,

δ

)

⊂

Int

P

_{ . Since }

a

*n*

→

**0**

_{ as }

n

→

∞

_{ , by Lemma 2, we deduce that }

{

}

*n*

fx

is a Cauchy sequence.If

f

(

X

)

is a complete subspace ofX

, there existu

,

v

X

∈

such thatfx

*n*

→

v

=

fu

(this holds also ifT

X

_{ is complete with }

v

∈

T

(

X

)

_{ ). Fix }

_{0}

_{0}

_{ }

≪

c

_{ and }

choose

∈

N

0

n

be such thatIAENG International Journal of Applied Mathematics, 41:3, IJAM_41_3_12

k

c

fx

v

d

k

c

fx

fx

d

k

c

fx

v

d

*n*

*n*

*n*

*n*

3

)

,

(

,

3

)

,

(

,

3

)

,

(

1 2 2 1 2 2<<

<<

<<

− −for all

n

≥

n

_{0}, where

.

1

,

1

,

1

1

max

−

−

−

−

+

−

−

+

=

E

B

C

E

B

E

A

E

B

D

k

Now,).

,

(

)

(

)

,

(

)

,

(

)

(

)

,

(

)

1

(

)

,

(

)

,

(

)

,

(

2 1 2 1 2 2 2 2u

T

fu

d

E

B

fx

fx

Cd

fx

fu

d

E

A

fx

fu

d

D

u

T

fx

d

fx

fu

d

u

T

fu

d

*m*

*n*

*n*

*n*

*n*

*m*

*n*

*n*

*m*

+

+

+

+

+

+

≤

+

≤

− − So,c

c

c

c

fx

fx

d

k

fx

fu

d

k

fx

fu

d

k

u

T

fu

d

*n*

*n*

*n*

*n*

*m*

=

+

+

<<

+

+

≤

− −3

3

3

)

,

(

)

,

(

)

,

(

)

,

(

2 1 2 1 2 2 Hencep

c

u

T

fu

d

(

,

*m*

)

<<

for every

p

∈

N

_{ . From }

,

)

,

(

fu

T

u

P

d

p

c

*m*

Int

∈

−

being

P

closed, asp

→

∞

, we deduceP

u

T

fu

d

*m*

∈

−

(

,

)

and sod

(

fu

,

T

*m*

u

)

=

**0**

. This
implies that fu

=

T

*m*

u

.

Similarly, by using the inequality,

),

,

(

)

,

(

)

,

(

fu

T

u

d

fu

x

_{2}

_{1}

d

fx

_{2}

_{1}

T

u

d

*n*

*n*

*n*

*n*

+ +

+

≤

we can show that

fu

=

T

*n*

u

,

which in turn implies thatv

_{ is a common point of coincidence of }

*m*

*n*

T

T

,

andf

, that is.

u

T

u

T

fu

v

=

=

*m*

=

*n*

Now we show that

f

,

T

*m*and

T

*n*have a unique common point of coincidence. For this, assume that there exists another point

v

∗ inX

such that∗ ∗ ∗ ∗

=

=

=

u

T

u

T

fu

v

*m*

*n*

for some

u

∗ inX

.

From),

,

(

)

(

)

,

(

)

,

(

)

,

(

)

,

(

)

,

(

)

,

(

)

,

(

∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗+

+

≤

+

+

+

+

≤

=

v

v

d

E

D

A

u

T

fu

Ed

u

T

fu

d

D

u

T

fu

Cd

u

T

fu

Bd

fu

fu

Ad

u

T

u

T

d

v

v

d

*m*

*n*

*n*

*m*

*n*

*m*

we obtain that

v

∗=

v

.

Moreover,(

T

*m*

,

f

)

and(

T

*n*

,

f

)

are weakly compatible, then,

and

fv

u

fT

fu

T

v

T

fv

u

fT

fu

T

v

T

*n*

*n*

*n*

*m*

*m*

*m*

=

=

=

=

=

=

which implies

T

*m*

v

=

T

*n*

v

=

fv

=

w

(say). Thenw

is a common point of coincidence ofT

*m*

,

T

*n*and

f

therefore,v

=

w

,

by uniqueness. Thusv

is a unique common fixed point ofT

*m*

,

T

*n*and

f

.**Example 6 ***Let *

X

=

{

1

,

2

,

3

},

E

=

R

2*and*

(

,

)

:

,

### 0

}.

{

∈

≥

=

x

y

E

x

y

P

*Define *

d

:

X

×

X

→

R

2*as follows:*

(

)

(

)

(

)

−

∈

≠

−

∈

≠

−

∈

≠

=

=

}.

1

{

,

and

if

,

}

3

{

,

and

if

,

1

}

2

{

,

and

if

,

if

)

,

(

3 8 7 4 3 14 3 10 7 5X

y

x

y

x

X

y

x

y

x

X

y

x

y

x

y

x

y

x

d

**0**

*Define the mappings *

T

,

f

:

X

→

X

*as follows:*

( )

x

x

,

f

=

=

≠

=

.

2

if

3

2

if

1

)

(

x

x

x

T

*Note that *

T

2(

x

)

=

1

*for each*

x

∈

X

,

.

3

10

,

7

5

))

2

(

),

3

(

(

2

=

T

T

d

*Then, if *

α

+

2

β

+

2

γ

<

1

*we have*

(

)

(

)

).

2

(

),

3

(

(

3

10

,

7

5

2

2

3

10

,

7

2

2

5

3

20

20

10

,

7

10

10

5

3

14

18

8

,

7

7

9

4

2T

T

d

=

<

+

+

+

+

≤

+

+

+

+

<

+

+

+

+

γ

β

α

γ

β

α

γ

β

α

γ

β

α

γ

β

α

γ

β

α

*This implies*

( ) ( )

( )

( )

( )

( )

[

]

[

]

[

]

)

2

(

),

3

(

(

3

10

,

7

5

)

1

,

2

(

)

3

,

3

(

)

3

,

2

(

)

1

,

3

(

)

2

,

3

(

))

3

(

,

2

(

))

2

(

,

3

(

))

2

(

,

2

(

))

3

(

,

3

(

)

2

,

3

(

2 2 2T

T

d

d

d

d

d

d

T

f

d

T

f

d

T

f

d

T

f

d

f

f

d

=

<

+

+

+

+

=

+

+

+

+

γ

β

α

γ

β

α

*for all *

α

,

β

,

γ

∈

[

### 0

,

1

)

*with*

α

+

2

β

+

2

γ

<

1

*.*

Therefore, Theorem 4 and its corollaries (, Theorems 2.1,

## IAENG International Journal of Applied Mathematics, 41:3, IJAM_41_3_12

2.3, 2.4], [5, Theorems 1, 2, 3], [7, Theorem 2.8], [13, Theorems 2.3, 2.6, 2.7, 2.8] and [15, Theorem 1, Corollary 2]) are not applicable. From

(

)

=

≠

=

2

if

,

2

if

)

,

(

3 10 7 5 2

y

y

Ty

x

T

d

**0**

and

=

+

+

+

+

3

10

,

7

5

)

,

(

)

,

(

)

,

(

)

,

(

)

,

(

2 2

x

T

fy

d

E

Ty

fx

d

D

Ty

fy

Cd

x

T

fx

d

B

fy

fx

d

A

for

y

=

2

andA

=

B

=

C

=

D

=

### 0

,

E

=

5_{7}, it follows that all conditions of Theorem 5are satisfied for

7 5

,

### 0

=

=

=

=

=

B

C

D

E

A

and soT

andf

have aunique common point of coincidence and a unique common fixed point.

**Corollary 7 ** *Let *

(

X

,

d

)

*be a complete cone metric*

*space,*

P

*be a cone and*

m

,

n

*be positive integers. If a*

*mapping*

T

:

X

→

X

*satisfies:*

)

,

(

)

,

(

)

,

(

)

,

(

)

,

(

)

,

(

x

T

y

d

E

y

T

x

d

D

y

T

y

Cd

x

T

x

d

B

y

x

d

A

y

T

x

T

d

*m*
*n*

*n*

*m*
*n*

*m*

+

+

+

+

≤

*for all *

x

,

y

∈

X

*, where*

A

,

B

,

C

,

D

,

E

*are non negative*

*real numbers with*

A

+

B

+

C

+

D

+

E

<

1

,

B

=

C

*or*

D

=

E

.

*Then*

T

*has a unique fixed point.*

**Proof. ** By Theorem 5, we get

x

∈

X

such that.

x

x

T

x

T

*m*

=

*n*

=

The result then follows from the fact that),

,

(

)

(

)

,

(

)

,

(

)

,

(

)

,

(

)

,

(

)

,

(

)

,

(

)

,

(

)

,

(

)

,

(

)

,

(

)

,

(

)

,

(

x

Tx

d

E

D

A

Tx

x

Ed

x

Tx

Dd

x

x

Cd

Tx

Tx

Bd

x

Tx

d

A

Tx

T

x

Ed

x

T

Tx

Dd

x

T

x

Cd

Tx

T

Tx

Bd

x

Tx

Ad

x

T

Tx

T

d

x

T

x

TT

d

x

Tx

d

*m*
*n*

*n*
*m*

*n*
*m*
*n*

*m*

+

+

=

+

+

+

+

≤

+

+

+

+

≤

=

=

which implies

Tx

=

x

.

**Example ** **(Applications) ** **8 ** *Let *

### 0

,

),

],

3

,

1

([

=

2>

=

C

E

a

X

R

R

*and*

[ ]

( )

( )

[ ]( )

( )

−

−

=

∈ ∈

t

y

t

x

a

t

y

t

x

y

x

d

*t*

*t* 1,3 1,3

sup

,

sup

)

,

(

*for * *every *

x

,

y

∈

X

,

*and*

(

,

)

:

,

### 0

}.

{

∈

2≥

=

u

v

u

v

P

R

*It is easily seen that*

)

,

(

X

d

*is a complete cone metric space. Define*

X

X

T

:

→

*by*

( )

(

)

4

(

( )

2)

1.

1du

e

u

u

x

t

x

T

*u*

*t*

−

+

+

=

∫

*For *

x

,

y

∈

X

[ ]

( )

( )

[ ]( )

( )

[ ]

(

( )

( )

)

[ ]

(

( )

( )

)

(

,

)

.

2

sup

,

sup

sup

,

sup

)

,

(

2

2

3 , 1 3

1

2

3 , 1 3

1 3 , 1 3 , 1

y

x

d

e

du

e

u

y

u

x

a

du

e

u

y

u

x

t

Ty

t

Tx

a

t

Ty

t

Tx

Ty

Tx

d

*t*
*t*

*t*
*t*

=

−

−

≤

−

−

=

∈ ∈

∈ ∈

∫

∫

*Similarly, *

(

,

)

.

!

2

)

,

(

2d

x

y

n

e

y

T

x

T

d

*n*
*n*
*n*

*n*

≤

*Note that*

=

=

=

=

=

.

38

if

53

.

### 0

37

if

31

.

1

4

if

1987

2

if

109

!

2

2n

n

n

n

n

e

*n*
*n*

*Thus for *

,

38

,

### 0

,

53

.

### 0

=

=

=

=

=

=

=

B

C

D

E

m

n

A

*all*

*conditions of Corollary 7 are satisfied and so *

T

*has a*

*unique fixed point, which is the unique solution of the*

*integral equation:*

( )

4

(

( )

2)

1.

1du

e

u

u

x

t

x

*u*

*t*

−

+

+

=

∫

*or the differential equation: *

( )

=

(

+

2)

1,

∈

[

1

,

3

],

( )

1

=

4

.

′

−x

t

e

t

x

t

x

*t*

*Hence, the use of Corollary 7 is a delightful way of showing *
*the existence and uniqueness of solutions for the following *
*class of integral equations: *

).

],

,

([

)

(

)

),

(

(

*n*

*t*

*a*

K

x

u

u

du

x

t

C

a

b

b

+

∫

=

∈

R

## REFERENCES

[1] M. Abbas and G. Jungck, “Common fixed point results for non commuting mappings without continuity in cone metric spaces,” J. Math. Anal. Appl., vol 341, pp. 416—420, 2008.

[2] H. Amann, “Fixed point Equations and non-linear eigen-value problems in ordered Banach spaces,” SIAM Review, vol. 18,pp. 620—709, 1976.

[3] M. Arshad, A. Azam and P. Vetro, “Some common fixed point results in cone metric spaces,” Fixed Point Theory Appl., vol. 11, Article ID 493965, 2009.

[4] M. Arshad, A. Azam and P. Vetro, “Common Solution of Nonlinear Functional Equations via Iterations,” Lecture Notes in Engineering and Computer Science: Proceedings of The World Congress on Engineering 2011, WCE 2011, 6-8 July, 2011, London, U.K.,

pp. 76—81.

[5] A. Azam, M. Arshad and I. Beg, “Common fixed points of two maps in cone metric spaces,” Rend. Circ. Mat. Palermo, vol. 57, pp. 433— 441, 2008.

[6] C. Di Bari and P. Vetro, “Weakly ϕ-pairs and common fixed points in cone metric spaces,” Rend. Circ. Mat. Palermo, vol. 58, pp. 125— 132, 2009.

[7] L.-G. Huang and X. Zhang, “Cone metric spaces and fixed point theorems of contractive mappings,” J. Math. Anal. Appl., vol. 332, pp. 1468—1476, 2007.

## IAENG International Journal of Applied Mathematics, 41:3, IJAM_41_3_12

[8] S. Jankovi´ca, Z. Kadelburgb and S. Radenovi´cc, “On cone metric spaces: A survey,” Nonlinear Analysis, vol. 74, pp. 2591—2601, 2011.

[9] G. Jungck, S. Radenovic, S. Radojevic and V. Rakocevic, “Common fixed point theorems for weakly compatible pairs on cone metric spaces,” Fixed Point Theory Appl., vol. 13, Article ID 643840, 2009. [10] M. Khani, M. Pourmahdian, “On the metrizability of cone metric

spaces,” Topology Appl., vol. 158 (2), pp. 190—193, 2011. [11] D. Klim,. Wardowski, “Dynamic processes and fixed points of

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[13] S. Rezapour and R. Hamlbarani, “Some notes on paper "Cone metric spaces and fixed point theorems of contractive mappings,” J. Math. Anal. Appl., vol. 345, pp. 719—724, 2008.

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[15] P. Vetro, “Common fixed points in cone metric spaces,” Rend. Circ. Mat. Palermo, vol. 56, pp. 464—468, 2007.

[16] D. Wardowski, “Endpoints and fixed points of set-valued contractions in cone metric spaces,” Nonlinear Analysis, vol. 71, pp. 512—516, 2009 .