Pof a real Banach spaceE is called a cone if

(1)

Common Solution of Nonlinear Functional

Equations via Iterations

Muhammad Arshad, Akbar Azam and Pasquale Vetro

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-- 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.

I. INTRODUCTIONANDPRELIMINARIES 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 [11]. 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 [6] 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, 4, 5, 7, 8, 10, 13] studied the existence of fixed points, points of coincidence and common fixed points of mappings

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).

satisfying a contractive type condition in cone metric spaces. 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 space

E

is called a cone if it has the following properties:

(i)

P

is non-empty closed and

P

≠

{

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

≤

on

E

with respect to

P

by

x

≤

y

if and only if

P

x

y

−

∈

. We shall write

x

<

y

if

x

≤

y

and

y

x

≠

, while

x

<<

y

will stands for

y

−

x

∈

int

P

,

where

int

P

denotes the interior of

P

.

The cone

P

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 and

P

is a cone in

E

with

int

P

≠ ∅

and

≤

is a partial ordering with respect to

P

.

Definition 1 Let

X

be a nonempty set. Suppose that the mapping

d

:

X

×

X

→

E

satisfies:

(i)

0 ≤

d

(

x

,

y

)

for all

x

,

y

∈

X

and

d

(

x

,

y

)

=

0

if and only if

x



y

;

(ii)

d

(

x

,

y

)

=

d

(

y

,

x

)

for all

x

,

y

∈

X

;

(iii)

d

(

x

,

y

)

≤

d

(

x

,

z

)

+

d

(

z

,

y

)

for all

.

,

y

z

X

x

∈

Then

d

is called a cone metric on

X

and

(

X

,

d

)

is called a cone metric space.

(2)

each

0 <<

c

there is

n

₀

∈

N

such that for all

,

)

,

(

,

0

d

x

c

n

≥

n

<<

then



x

n



is said to be

convergent or

{ }

x

_n converges to

x

and

x

is called the

limit of



x

n



. We denote this by lim n

x

n

=

x

,

or

,

x

n

→

as

n

→

∞

.

If for each

0 <<

c

there is

n

0

∈

ℕ

such that for all

n

,

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 in

X

, then

X

is called a

complete cone metric space. Let us recall [5] that if

P

is a normal cone, then

x

n

∈

X

,

converges to

x

∈

X

if and only if

d



x

n

,

x



→

0

as

n

→



. Furthermore,

x

n

∈

X

is a Cauchy sequence if and only if

0 →

)

,

(

x

n

x

m

d

as

n

,

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. If

d

(

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

)

⊂

. Since

a

n

→

0

_{, there exists}

n

0

∈

N

be such that

a

n

∈

I



0 ,





for every

n

≥

n

₀ .

From

c

−

a

n

∈

IntP

, we deduce

c

a

x

d

(

n

,

m

)

≤

n

<<

for every

m

,

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 <

−

+

⋅

−

+

=

−

+

⋅

−

+

=

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 <

−

+

⋅

−

+

=

−

+

⋅

−

+

=

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 ON

X

ARE SAID TO BE WEAKLY COMPATIBLE IF THEY COMMUTE AT THEIR COINCIDENCE POINT (I.E.,

fTx

=

Tfx

WHENEVER

Tx

fx

=

). A POINT

y

∈

X

IS CALLED POINT OF COINCIDENCE OF

T

AND

f

IF THERE EXISTS A POINT

X

x

∈

SUCH THAT

y

=

fx

=

Tx

.

II. MAIN RESULTS

The following theorem improves/generalizes the results [1, Theorems 2.1, 2.3, 2.4], [4, Theorems 1, 2,3], [6, Theorems 1,3, 4], [7, Theorem 2.8], [10, Theorems 2.3, 2.6, 2.7, 2.8], and [12, 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 [6] 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

:

i.

α

<

1 ,

β

=

γ

=

0

( [6, Theorems 1] ), ii.

β

<

₂1

,

α

=

γ

=

0

( [6, Theorems 3] ), iii.

γ

<

₂1

,

α

=

β

=

0

( [6, Theorems 3] ).

(3)

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

(

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 point

x

₁ in

X

such that

fx

₁

=

T

m

x

₀ . This can be done since

T

(

X

)

⊆

f

(

X

)

. Similarly, choose a point

x

2 in X such that

fx

2

T

x

1

.

n

=

Continuing this process having chosen

x

_n in

X

, we obtain

x

_n₊₁ in

X

such that

fx

2k1



T

m

x

2k

fx

2k2



T

n

x

2k1

,

k



0, 1, 2,

…

.

Then, ). , ( ] [ ) , ( ] [ ) , ( ) , ( ) , ( ] [ ) , ( ) , ( ) , ( ) , ( ) , ( ) , ( ) , ( 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 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

d

D

B

A

fx

d

D

C

That is,

d



fx

2k1

,

fx

2k2





F d



fx

2k

,

fx

2k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

d

E

B

fx

d

E

C

A

x

T

x

T

d

fx

d

which implies

)

,

(

)

,

(

fx

₂_k₊₂

fx

₂_k₊₃

≤

G

d

fx

₂_k₊₁

fx

₂_k₊₂

d

with

E

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 2

fx

d

FG

F

fx

d

FG

F

fx

d

FG

fx

d

F

fx

d

k k k k k k k k k

≤

− − − + + + and

( )

(

,

).

)

,

(

)

,

(

1 0 1 2 2 1 2 3 2 2 2

fx

d

FG

fx

d

G

fx

d

k k k k k + + + + +

≤

By Remark 3

p

<

q

we have

(4)

In analogous way, we deduce

),

,

(

1 )

(

)

1 (

)

,

(

₂ ₂ ₁

d

fx

₀

fx

₁

FG

F

fx

d

p

q

p

⎥

⎦

⎤

⎢

⎣

⎡

−

+

≤

+

)

,

(

1 )

(

)

1 (

)

,

(

₂ ₂

d

fx

₀

fx

₁

FG

F

fx

d

p

q

p

⎥

⎦

⎤

⎢

⎣

⎡

−

+

≤

and

).

,

(

1 )

(

)

1 (

)

,

(

₂ ₁ ₂

d

fx

₀

fx

₁

FG

F

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

F

a

p

n

⎥

⎦

⎤

⎢

⎣

⎡

−

+

=

with

p

the integer part of

n

/

2 .

Fix

0 <<

c

and choose

}

:

{

)

,

(

δ

=

x

∈

E

x

<

δ

I 0

such that

P

I

c

+

(

0 ,

δ

)

⊂

Int

. Since

a

n

→

0

_as

n

→



_{, by}

Lemma 2, we deduce that

{

fx

n

}

is a Cauchy sequence. If

f

(

X

)

is a complete subspace of

X

, there exist

u

,

v

X

∈

such that

fx

_n

→

v

=

fu

(this holds also if

T



X



_{is complete with}

v

∈

T

(

X

)

). Fix

0 

c

and choose

n

₀

∈

_N

be such that

k

c

fx

v

d

k

c

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 max

⎭

⎬

⎫

⎩

⎨

⎧

−

+

−

+

=

E

B

C

E

B

E

A

E

B

D

k

Now,

).

,

(

)

(

)

,

(

)

,

(

)

(

)

,

(

)

1 (

)

,

(

)

,

(

)

,

(

2 1 2

1 2 2

2 2

u

T

fu

d

E

B

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

fx

d

k

fx

fu

d

k

fx

fu

d

k

u

T

fu

d

n n

m

=

+

<<

+

≤

−

3

3 )

,

(

)

,

(

)

,

(

)

,

(

2 1 2

1 2 2

Hence

p

c

u

T

fu

d

m

<<

)

,

(

for every

p

∈

_N

. From

,

)

,

(

fu

T

u

P

d

p

c

m

Int

∈

−

being

P

closed, as

p

→

∞

, we deduce

P

u

T

fu

d

m

∈

−

(

,

)

and so

d

(

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 that

v

_{is a common point of coincidence of} m n

T

,

and

f

, 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

∗ in

X

such that

∗ ∗

∗

₌

u

T

u

T

fu

v

m n for some

u

∗ in

X

.

From

),

,

(

)

(

)

,

(

)

,

(

)

,

(

)

,

(

)

,

(

)

,

(

)

,

(

∗ ∗ ∗

∗ ∗

∗ ∗ ∗

+

≤

+

≤

=

v

d

E

D

A

u

T

fu

Ed

u

T

fu

d

D

u

T

fu

Cd

u

T

fu

Bd

fu

Ad

u

T

u

T

d

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). Then

w

is a common point of coincidence of

T

m

,

T

n and

f

therefore,

v

=

w

,

by uniqueness. Thus

v

is a unique common fixed point of

T

m

,

T

n and

f

.

Example 6 Let

X

=

{

1 ,

2 ,

3 },

E

=

_R

2 and

( )

,

:

,

0 }.

{

∈

≥

=

x

y

E

x

y

(5)

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 5

X

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

,

f

=

⎩

⎨

⎧

=

≠

=

.

2 if

3

2 if

1 )

(

x

T

Note that

T

2

(

x

)

=

1

for each

x

∈

X

,

.

3

10 ,

7

5 ))

2 (

),

3 (

(

2

⎟

⎠

⎞

⎜

⎝

⎛

=

T

d

Then, if

α

+

2 β

+

2 γ

<

1

we have

(

)

₍

₎

).

2 (

),

3 (

(

3

10 ,

7

5

2

3

10 ,

7

2

5

3

20

10 ,

7

10

5

3

14

18

8 ,

7

9

4

2

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 2

T

d

T

f

d

T

f

d

T

f

d

T

f

d

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, 2.3, 2.4], [4, Theorems 1, 2, 3], [6, Theorem 2.8], [10, Theorems 2.3, 2.6, 2.7, 2.8] and [12, Theorem 1, Corollary 2]) are not applicable. From

(

)

⎩

⎨

⎧

=

≠

=

2 if

,

2 if

)

,

(

3 10 7 5 2

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

and

A

=

B

=

C

=

D

=

0 ,

E

=

₇5 , it follows that all conditions of Theorem 5are satisfied for

7 5

,

0 =

=

B

C

D

E

A

and so

T

and

f

have a

unique 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

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

Cd

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

and

[ ]

( ) ( )

[ ]

( ) ( )

⎟

_⎠

⎞

⎜

⎝

⎛

₋

=

∈

x

t

y

t

a

x

t

y

t

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

T

:

→

by

( )

4 (

( )

2

)

1

.

1

du

e

u

x

t

x

T

u t −

+

=

∫

(6)

[ ]

( )

[ ]

( )

[ ]

(

( ) ( )

)

[ ]

(

( ) ( )

)

( )

,

.

2 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 )

,

(

2

d

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

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

.

1

du

e

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

du

x

t

C

a

b

+

∫

=

∈

_R

REFERENCES

[1] M. Abbas and G. Jungck, Common fixed point results for non commuting mappings without continuity in one

metric spaces, J. Math. Anal. Appl. 341 (2008) 416--420.

[2] H. Amann, Fixed point Equations and non-linear eigen-value problems in ordered Banach spaces, SIAM Review, 18 (1976) 620--709.

[3] M. Arshad, A. Azam and P. Vetro, Some common fixed point results in cone metric spaces, Fixed Point Theory Appl. 2009 (2009), 11 pp., Article ID 493965. [4] A. Azam, M. Arshad and I. Beg, Common fixed points

of two maps in cone metric spaces, Rend. Circ. Mat. Palermo 57 (2008) 433--441.

[5] C. Di Bari and P. Vetro, Weakly -pairs and common fixed points in cone metric spaces, Rend. Circ. Mat. Palermo 58 (2009) 125--132.

[6] L.-G. Huang and X. Zhang, Cone metric spaces and fixed point theorems of contractive mappings, J. Math. Anal. Appl. 332 (2007) 1468--1476.

[7] 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. 2009 (2009), 13 pp., Article ID 643840. [8] D. Klim,. Wardowski, Dynamic processes and fixed

points of set-valued nonlinear contractions in cone metric spaces, Nonlinear Analysis (2009) doi:10.1016/j.na.2009.04.001.

[9] V.Popa, A general fixed point theorem for weakly compatible mappings in compact metric spaces,Turk J. Math.25 (2001) 465-474.

[10] S. Rezapour and R. Hamlbarani, Some notes on paper "Cone metric spaces and fixed point theorems of contractive mappings.", J. Math. Anal. Appl., 345 (2008) 719--724.

[11] N. Shahzad, Fixed points for generalized contractions and applications to control theory, Nonlinear Analysis 68 (8) (2008) 867--876.

[12] P. Vetro, Common fixed points in cone metric spaces, Rend. Circ. Mat. Palermo, 56 (2007) 464--468.

[13] D. Wardowski, Endpoints and fixed points of set-valued contractions in cone metric spaces, Nonlinear Analysis 71 (2009) 512--516.