Common Fixed Points for 2 Maps Azam Arshad Beg

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DOI: 10.1007/s12215-008-0032-5

Rendiconti del Circolo Matematico di Palermo 57, 433 – 441 (2008)

Akbar Azam · Muhammad Arshad ·Ismat Beg

Common fixed points of two maps in cone metric spaces

Received: July 3, 2008/ Accepted: November 7, 2008 – c Springer-Verlag 2008

Abstract. We prove the existence of points of coincidence and common fixed

points of a pair of self mappings satisfying a generalized contractive condition

in cone metric spaces. Our results generalize several well-known recent and

classical results.

Keywords Point of coincidence · Common fixed point · Contractive type

mapping · Commuting mapping · Compatible mapping · Cone metric space.

Mathematics Subject Classification (2000) 47H10 · 54H25

1 Introduction

Sessa [12] generalized the concept of commuting mappings [5] by calling self

mappings f , g on a metric space X , weakly commuting if and only if

d ( f gx , g f x ) ≤ d ( f x , gx ) for all x ∈ X .

A. Azam

Department of Mathematics, Faculty of Basic and Applied Sciences,International Islamic Uni-

versity, H-10, Islamabad, Pakistan, and Department of Mathematics, F.G. Postgraduate Col-

lege, H-8, Islamabad, Pakistan

E-mail: [email protected]

M. ArshadDepartment of Mathematics, Faculty of Basic and Applied Sciences,International Islamic Uni-

versity, H-10, Islamabad, Pakistan

E-mail: marshad−[email protected]

I. Beg (B)

Centre for Advanced Studies in Mathematics, Lahore University of Management Sciences,

54792-Lahore, Pakistan

E-mail: [email protected]



434 A. Azam et al.

Commuting mappings are weakly commuting but converse is not true in gen-

eral (see [12]). Afterwards, Jungck [6] and Pant [8] introduced some less re-strictive concepts of compatible mappings and R-weakly commuting map-

pings in order to improve the concept of weakly commuting mappings and

generalized some common fixed point theorems respectively. Later on, it has

been noticed that compatible mappings and R-weakly commuting mappings

commute at their coincidence points. Jungck and Rhoades [7], then defined a

pair of self-mappings to be weakly compatible if they commute at their coin-

cidence points.

Recently, Huang and Zhang [3] generalized the notion of metric spaces by

replacing the real numbers by ordered Banach space and define cone metric

spaces. They have proved Banach contraction mapping theorem and some

other fixed point theorems of contractive type mappings in cone metric spaces.Subsequently, Abbas and Jungck [1], Ilic and Rakocevic [4], Rezapour and

Hamlbarani [9] and Vetro [13] studied fixed point theorems for contractive

type mappings in cone metric spaces.

The aim of this paper is to obtain points of coincidence and common

fixed points of a pair of self mappings satisfying a generalized contractive

type condition in a cone metric space. Our results generalized several existing

fixed point theorems including [1–3,5,9,10].

A subset P of a real Banach space Z is called a cone if it has following

properties:

(i) P is nonempty 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 ⊆ Z , we can define a partial ordering ≤ on Z with

respect to P by x ≤ y if and only if y− x ∈ P. We shall write x < y if x ≤ y and

x = y, while x y will stands for y− x ∈ intP, where intP denotes the interior

of P. A cone P is called normal if there is a number κ > 0 such that for all

x , y ∈ Z ,

0≤ x ≤ y > x ≤ κ y . (1)

The least positive number κ satisfying (1) is called the normal constant of P.

There are no normal cones with normal constant κ < 1 [9]. Also [9, example

2.3] shows that there are non-normal cones.

In the following we always suppose that Z is a real Banach space and P is

a cone in Z with intP = φ and ≤ is a partial ordering with respect to P.

Definition 1 Let X be a nonempty set. Suppose the mapping d : X × X → Z , satisfies

1. 0 ≤ d ( x , y), for all x , y ∈ X and d ( x , y) = 0 if and only if x = y;



Common fixed points of two maps in cone metric spaces 435

2. d ( x , y) = d ( y, x ) for all x , y ∈ X ;

3. d ( x , y)≤ d ( x , z) + d ( z, y) for all x , y, z ∈ X .

Then d is called a cone metric on X , and ( X , d ) is called a cone metric

space.

Let x ∈ X , and { x n} be a sequence in X . If for every c ∈ Z , with

0 c there is n0 ∈ N such that for all n ≥ n0, d ( x n, x ) c, then { x n} is said

to be convergent , { x n} converges to x and x is the limit of { x n}. We denote this

by limn x n = x , or x n −→ x , as n → ∞. If for every c ∈ X with 0 c there is

n0 ∈N such that for all n, m≥ n0, 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 [3] 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 →∞. Further,

x n ∈ X is a Cauchy sequence if and only if d ( x n, x m)→ 0 as n, m→∞. A point

x ∈ X is called coincidence point of two mappings T , f : X → X if f x = T x .

Definition 2 A point y ∈ X is called point of coincidence of two mappings

T , f : X → X if there exists a point x ∈ X such that y = f x = T x .

Let ( X , d ) be a complete cone metric space, P be a normal cone with nor-

mal constant κ . Suppose that the mappings T , f : X → X satisfy:

d (T x , Ty) ≤ A d ( f x , f y) + B d ( f x , T x ) +Cd ( f y, Ty)

+ D d ( f x , Ty) + E d ( f y, T x ), (2)

for all x , y ∈ X where A, B,C , D, E are non-negative real numbers.

Huang and Zhang [3] proved that T has a unique fixed point if (a) f = I , where I is the identity mapping on X (see [2,10,11])

and

(b) one of the following is satisfied:

(i) B = C = D = E = 0 with A < 1 ([3, theorem 1] ),

(ii) A = D = E = 0 with B = C < 12

([3, theorem 3]),

(iii) A = B = C = 0 with D = E < 12

([3, theorem 4]).

Abbas and Jungck [1] proved that f and T have a unique point of coinci-

dence and unique common fixed point if:

(a) the mappings f and T are weakly compatible

and

(b) one of the following is satisfied:

(i) B = C = D = E = 0 with A < 1 ([1, theorem 2.1] ),

(ii) A = D = E = 0 with B = C < 12

( [1, theorem 2.3] ),

(iii) A = B = C = 0 with D = E < 12

([1, theorem 2.4]).

Rezapour and Hamlbarani [9] generalized some results of [3] by omitting

the assumption of normality on X .



436 A. Azam et al.

2 Main results

Theorem 1 Let ( X , d ) be a cone metric space. Suppose the mappings T , f :

X → X satisfy

d (T x , Ty) ≤ A d ( f x , f y) + B [d ( f x , T x ) + d ( f y, Ty)]

+C [d ( f x , Ty) + d ( f y, T x )], (3)

for all x , y ∈ X where A, B,C are non-negative real numbers with

A + 2 B + 2C < 1. If T ( X )⊆ f ( X ) and f ( X ) or T ( X ) is a complete subspace

of X , then T and f have a unique point of coincidence.

Proof Let x 0 be an arbitrary point in X . Choose a point x 1 in X such that

f x 1 = T x 0. This can be done since T ( X ) ⊆ f ( X ). Similarly, choose a point x 2in X , such that f x 2 = T x 1. Continuing this process and having chosen x n in X ,

we obtain x n+1in X such that

f x k +1 = T x k , k = 0, 1, 2, ...

Then

d ( f x k +1, f x k +2) = d (T x k , T x k +1)

≤ A d ( f x k , f x k +1) + B [d ( f x k , T x k ) + d ( f x k +1, T x k +1)]

+C [d ( f x k , T x k +1) + d ( f x k +1, T x k )]

≤ [ A + B] d ( f x k , f x k +1) + B d ( f x k +1, f x k +2)

+C d ( f x k , f x k +2)

≤ [ A + B +C ] d ( f x k , f x k +1) + [ B +C ] d ( f x k +1, f x k +2).

It implies that

[1− B−C ]d ( f x k +1, f x k +2)≤ [ A + B +C ] d ( f x k , f x k +1).

That is

d ( f x k +1, f x k +2)≤

A + B +C

1− B−C

d ( f x k , f x k +1).

Moreover,

d ( f x k +1, f x k +2) ≤ A + B +C

1− B−C 2

d ( f x k −1, f x k )

≤ · · · ≤

A + B +C

1− B−C

k +1

d ( f x 0, f x 1).

Putting,

yn = f x n and λ =

A + B +C

1− B−C

.




We have,

d ( yn, yn+1)≤ λ nd ( y0, y1).

For n > m

d ( yn, ym) ≤ d ( yn, yn−1) + d ( yn−1, yn−2) + · · ·+ d ( ym+1, ym)

≤λ n−1 +λ n−2 + · · ·+λ m

d ( y0, y1)

≤ λ m

1−λ d ( y0, y1).

Let 0 c be given. Choose δ > 0 such that

c +{ x ∈ Z : x < δ } ⊆ P.

Also choose a natural number N 1 such that

λ m

1−λ d ( y0, y1) ∈ { x ∈ Z : x < δ } , for all m ≥ N 1.

Thenλ m

1−λ d ( y0, y1) c, for all m ≥ N 1.

Thus,

n > m>d ( yn, ym) ≤ λ m

1−λ d ( y0, y1) c,

which implies that { yn} is a Cauchy sequence. We assume that f ( X ) is com-

plete, then there exists u, v ∈ X such that yn → v = f u.

Choose a natural number N 2 such that for all n ≥ N 2

d ( yn−1, yn)

c(1− B−C )

3 B

, d ( yn−1, v)

c(1− B−C )

3( A +C )

and

d ( yn, v)

c(1− B−C )

3(1 +C )

.

Now, inequality (3) implies that

d ( f u, Tu) ≤ d ( f u, yn) + d ( yn, Tu)

≤ d (v, yn) + d (T x n−1, Tu)

≤ d (v, yn) + Ad ( f u, f x n−1) + B[d ( f u, Tu) + d ( f x n−1, T x n−1)]

+C [d ( f u, T x n−1) + d ( f x n−1, Tu)]

≤ d (v, yn) + Ad (v, yn−1) + B[d ( f u, Tu) + d ( yn−1, yn)]

+C [d (v, yn) + d ( yn−1, v) + d ( f u, Tu)]

≤ (1 +C )d (v, yn) + ( A +C )d (v, yn−1) + Bd ( yn−1, yn)

+( B +C ) d ( f u, Tu).



438 A. Azam et al.

Consequently,

d ( f u, Tu) ≤

1 +C

1− B−C

d (v, yn) +

A +C

1− B−C

d (v, yn−1)

+

B

1− B−C

d ( yn−1, yn).

It further implies that

d ( f u, Tu) c

3 +

c

3 +

c

3 = c.

Thus,

d ( f u, Tu) c

m, for all m ≥ 1.

So, cm − d ( f u, Tu) ∈ P, for all m ≥ 1. Since c

m → 0 (as m → ∞) and P isclosed,−d ( f u, Tu)∈P. But P∩(−P) ={0}. Therefore, d ( f u, Tu) = 0. Hence

v = f u = T u.

Next we show that f and T have a unique point of coincidence. For this,

assume that there exists another point v∗ in X such that v∗ = f u∗ = T u∗ for

some u∗ in X . Now

d (v, v∗) = d (Tu, Tu∗)

≤ Ad ( f u, f u∗) + B[d ( f u, Tu) + d ( f u∗, Tu∗)]

+C [d ( f u, Tu∗) + d ( f u∗, Tu)]

≤ Ad (v, v∗) +C [d (v, v∗) + d (v∗, v)]

≤ ( A + 2C ) d (v, v∗),

hence v = v∗.

On the other hand, if we assume that T ( X ) is complete, then the Cauchy

sequence

yn = f x n = T x n−1

converges to v ∈ T X . But T X ⊆ f X which allows us to obtain u ∈ f X such

that v = f u. The rest of the proof is similar to the previous case.

Theorem 2 If in addition to the hypotheses of Theorem 1 the mappings T , f :

X → X are weakly compatible, then T and f have a unique common fixed

point.

Proof As in the proof of Theorem 1, there is a unique point of coincidence v

of f and T . Now T , f are weakly compatible, therefore

T v = T f u = f Tu = f v.

It implies that T v = f v = w (say). Then w is a point of coincidence of T and

f , therefore v = w by uniqueness. Thus v is a unique common fixed point of

T and f .




Theorem 3 Let ( X , d ) be a cone metric space. Suppose that the mappings

T , f : X → X satisfy (2), for all x , y ∈ X where A, B,C,D and E are non-negative real numbers with A + B + C + D + E < 1. If T ( X ) ⊆ f ( X ) and

f ( X ) or T ( X ) is a complete subspace of X, then T and f have a unique point

of coincidence. Moreover, if T , f are weakly compatible, then T and f have

a unique common fixed point.

Proof By hypothesis for all x , y ∈ X , we get,

d (Ty, T x ) ≤ A d ( f y, f x ) + B d ( f y, Ty) +Cd ( f x , T x )

+ D d ( f y, T x ) + Ed ( f x , Ty).

It follows that,

d (T x , Ty) ≤ A d ( f x , f y) +

B +C

2

[d ( f x , T x ) + d ( f y, Ty)]

+

D + E

2

[d ( f x , Ty) + Ed ( f y, T x )] .

The required result follows from Theorems 1 and 2.

Example 1 Let X = R, Z = R2, d ( x , y) = (| x − y| ,β | x − y|) ,β > 0,

P = {( x , y) : x , y ≥ 0}, T ( x ) = 2 x 2 + 4 x + 3 and f ( x ) = 3 x 2 + 6 x + 4. Then

T X = f X = [1,∞)

and all the conditions of Theorem 1 are satisfied for

A ∈

2

3, 1

, B = C = 0

as we obtain 1 ∈ X as a unique point of coincidence

1 = f (−1) = T (−1).

Remark 1

(i) Note that in Example 1

T f (−1) = T (1) = 9 and f T (−1) = f (1) = 13.

Thus T and f are not weakly compatible. It follows that except the weak

compatibility of T and f all other hypotheses of Theorem 2 are satisfied

but

1 = f (1) = T (1).

It shows that the weak compatibility for T and f in Theorem 2 is an

essential condition.



440 A. Azam et al.

(ii) In Example 1 if we assume T ( x ) = 2 x 2 + 4 x + 1 and f ( x ) = 3 x 2 + 6 x + 2

then T and f become weakly compatible and all conditions of Theorems1, 2 and 3 are satisfied to obtain a unique point of coincidence and a

unique common fixed point −1 = f (−1) = T (−1).

Our next example demonstrates the crucial role of the condition

T ( X ) ⊆ f ( X ) in our results.

Example 2 Let X = R+ (the set of all non-negative real numbers), Z = R2,

d ( x , y) = (| x − y| , e | x − y|) , P = {( x , y) : x , y ≥ 0}, T ( x ) = e x and f x = e x +1.

Then

T X = (0,∞) [e,∞) = f X ,

d (T x , Ty) = (|e x −e y| ,e x +1−e y+1

)

= 1

e(e x +1−e y+1

,e x +2−e y+2

).

= 1

ed ( f x , f y).

It follows that all the assumptions of Theorem 1 except T X ⊆ f X are satisfied

for A = 1e

, B = C = 0. But T and f do not have a point of coincidence in X .

Remark 2 Our results generalized several known results included among them

are [1, theorems 2.1, 2.3, 2.4], [3, theorems 1, 3, 4] and [9, theorems 2.3,

2.6, 2.7, 2.8].

Acknowledgements The present version of the paper owes much to the precise and kind

remarks of the learned referee.

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Common Fixed Points for 2 Maps Azam Arshad Beg

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