Abstract

In this paper, we prove a common fixed point theorem for coincidentally commuting non-self mappings for a generalized contraction condition in cone b-metric space.

1. Introduction

Fixed point theory has various equal on fixed point theorems for self-mappings in metric and Banach spaces. Huang and Zhang ¹ originated the conception of cone metric space by reconstituting the collection of real numbers by an ordered Banach space and attained few fixed point theorems for mappings gratifying disparate contractive conditions. Various originators like Abbas and Jungck ², Rhoades ³, Raja and Vaezpour ⁴ have generalized the result of Huang and Zhang ¹ and analyzed the origination of common fixed point in cone metric spaces. In ⁵, Bhakhtin acquainted b-metric spaces as a generalization of metric spaces and verified the contraction mappings equal in b-metric space that generalizes the familiar Banach contraction results in metric spaces. The analysis of fixed point results for non-self mappings in metrically convex metric space was initiated by Assad and Kirk ⁶. B.E. Rhoades and S. Radenovic ⁷ have manifested fixed point theorems for non-self mappings satisfying generalized contraction condition in cone metric spaces. In this paper, we prove a common fixed point theorem for non-self mappings convincing contraction condition in cone b-metric space.

2. Preliminaries

Definition 2.1

Let B be a real Banach space. A subset C of B is called a cone if and only if

a) C is closed, nonempty and

b) p, qR, p, shows that

c)

In a cone we imply a partial ordering with respect to C by which implies A cone C is called normal if there is a number such that for all shows

The smallest positive number convincing the above inequality is called normal constant of C, while stands for int C (interior of C).

Definition 2.2

If is a non-void set then the mapping d: ×→ Econvincing these conditions

a) 0≤d (a,b) for all and if and only if

b) for all

c) for all

Called a cone metric in and (,d) is called a cone metric in and (,d) is called a cone metric space. The idea of a cone metric space is more familiar than that of a metric space.

Definition 2.3 ⁸

If is a non-void set and be a given real number then the mapping d: ×→E is said to be cone b-metric if and only if for all the following conditions are satisfied:

a) for all and if and only if

b) for all

c) for all

The pair(,d)is said to be a cone b-metric space.

Definition 2.4

If (,d) is a cone b-metric space then we say that is

a) A cauchy sequence, if for every k in E with there is N such that for all

b) A convergent sequence, if for every k in E with there is N such that for all , for some fixed pointain .

We say that a cone b-metric space is said to be complete only if every Cauchy sequence in is convergent in . Also is convergent to in if and only if

Remarks 2.5 ⁴:

1. If

2. If

3. If

Remarks 2.6 ⁴

If and then there existan such that for all

3. Main Result

Theorem 3.1

If (,d) is a complete cone b-metric space and Ṁ a non-empty closed subset of X such that for each and there exist a point such that

(1)

Suppose that f,T: → are two non-self mappings satisfying for all

(2)

For every a, b in Ṁ and are positive real numbers such that and

where and Also assume that

(1)

(2) implies

(3) is closed in .

Then there exist a coincidence point of Moreover if are weakly compatible, then have a unique common fixed point in

Proof:

We construct the sequence and in M and a sequence in Let Set up a point Also and from condition (1) we have for some Now, since from condition (2) we conclude that Also clearly Therefore and from (1) Therefore for some we have

Set up If then condition (1) implies Therefore for some point we have Suppose if then we denote a point in such that

Next we set Therefore for some point we have Therefore, if then we have and If we continue the process, we obtain three sequences which is in X, such a way that

a)

b)

c) if and only if

d) If whenever and then from equation (1), and

If then and We discuss the case about ( If then it is clear that for all Now, if for all n, then three cases are distinguished.

Case (i)

If and then and from (a)

And from (b). Using contraction (2), we have,

Now, three subcases arises,

i)

(ii)

(iii)

d(

from subcases (i), (ii), (iii) we get

where h=max

Case ii

Let and Then and

Therefore,

Using contraction (2) we have,

Now, three subcases arises,

i)

d(

(ii)

d(

(iii)

d(

d(

from subcases (i),(ii),(iii) we get

where h=max

Case (iii)

Let Then and

(3)

and we have and

Now using triangle inequality, we get

(4)

We need to find and Now using contraction (2) we find

As and we have,

(5)

Next we have to find

Now, we find separately what is from above equation

As and

We get,

(6)

Substituting (5) and (6) in (4), we get

Again three subcases follows,

(i)

Therefore,

We get,

(ii)

Therefore,

(iii)

Therefore,

From all the above three subcases we have,

Where

In all cases (i), (ii), (iii) we get

Where

and

Now, following the induction procedure of Assad & Kirk [3] it can be showed that by induction for n>1,

(7)

Where, from (6) and by triangle inequality for we have

From remark 2.5 and 2.6, where is constant, (i.e), is a Cauchy sequence. Since and is complete, there is some point such that Let be such that fw = c. By construction of there is subsequence such that and so

And on using definition and the fact that

we obtain again three subcases.

Sub case (i)

Sub case (ii)

Sub case (iii)

In all subcases (i), (ii), (iii), we obtain for each int C and using result, it follows that d(Tw,c)=0 or Tw=c. If T and f are coincidentally commuting then c=Tw=fw which implies Tc=Tfw=fTw=fc. From contraction (2) we have,

Since as mentioned in the conditions of theorem, it follows that Tc=c, that is c is a common fixed point of f and T. Hence, Uniqueness of common fixed point easily follows from contraction (2).

Corollary 3.2

If (,d) is a complete cone b-metric space and Ṁ be a nonempty closed subset of such that for each and there exist a point such that

(1)

Suppose that T: Ṁ→ satisfying the condition

(2)

For all a,b in and are positive real numbers such that If T has additional property that for each then T has a unique fixed point.

References