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Common Fixed Point Theorems for Two Self-maps in Cone Metric Spaces under Different Contractive Conditions

P. Rama Bhadra Murthy*, M. Rangamma

Department of Mathematics, University College of Science, Osmania University, Hyderabad, India

Abstract: The existence of unique common fixed point theorems for two weakly compatible self-maps satisfying different contractive conditions in cone metric spaces without using normality.

Keywords: weakly compatible maps, common fixed points, cone metric spaces, coincidence points

Received January 12, 2015; Revised January 19, 2015; Accepted January 27, 2015

Citation: P. Rama Bhadra Murthy, M. Rangamma, Common Fixed Point Theorems for Two Self-maps in Cone Metric Spaces under Different Contractive Conditions. American Journal of Mathematical Analysis, 3 (1). 5-9. January 2015. doi:10.12691/ajma-3-1-2

1. Introduction

The fundamental work in metric fixed point theory, by Stefan Banach in 1922 is famous as Banach Contraction Principle and found many applications viz., in proving the existence and uniqueness of solutions of Differential, Integral, Integro-Differential, Impulsive differential equations, etc. A number of authors introduced different contractive type mappings and proved many fixed point theorems extending the theory. B.E.Rhoades [1], Paula Collaco and Jaime Carvalho E Silva [2] compared various definitions of contractive mappings. In 2007, Huang and Zhang [3] introduced the concept of cone metric spaces by replacing the codomain with Banach spaces in a metric function whose range satisfy the properties of a cone. Subsequently, Abbas and Jungck [4], Abbas and Rhoades [5] have studied common fixed point theorems in cone metric spaces for normal cones with the assumption of normality. Sh.Rezapour and R.Hamlbarani [6] proved some fixed point theorems for any cone, omitting the assumption of normality. Recently, several authors have proved and proving many common fixed point theorems. See [7-12]. The purpose of this paper is to prove the common fixed point theorems for all cones which were proved by Abbas and Jungck in [4] only for normal cones.

The following definitions are taken from [3,4,5]. In the entire paper, E and R represents Banach space and Real Numbers respectively.

Definition 1.1:([3]) Let be a Banach space and is called a cone if it satisfies the following properties

1.1(i) is closed, non-empty and

1.1(ii)

1.1(iii)

Examples 1.2:

1.2(i) is a cone.

1.2(ii) Every non-zero Banach Space acts as a cone to itself.

Definition 1.3: ([3]) For a given cone , define a partial ordering w.r.t. by

1.3(i)

1.3(ii)

1.3(iii) denotes interior of .

Definition 1.4:([3]) Let be a non-empty set and E be a Banach Space. Suppose the mapping satisfy

1.4(i)

1.4(ii)

1.4(iii)

1.4(iv)

Then is called a cone metric on and is called a cone metric space.

Example 1.5: , and

Then is a cone metric space.

Definition 1.6: ([3]) Let be a cone metric space, and be a sequence in . Then

1.6(i) whenever for every with a natural number N such that

1.6(ii) is a Cauchy Sequence if for every with there is a natural number N such that .

1.6(iii) is a complete cone metric space if every Cauchy sequence in is convergent in

The following lemma’s are taken from [3].

Lemma 1.7: ([3]) Let be a cone metric space. Let be a normal cone with normal constant K. Let be a sequence in Then

1.7(i)

1.7(ii) is a Cauchy sequence iff as

Lemma 1.8: ([3]) Let be a cone metric space. Let be a sequence in If converges to and converges to , then i.e., the limit of the sequence is unique.

For the definition and examples of normal and non-normal cones, see [3,4,5].

Definition 1.9: ([4]) Let be a non-empty set and two self maps on Then

1.9(i) If for some , then is called a coincidence point of and . Also is called a point of coincidence of and .

1.9(ii) If for some , then is called a common fixed point of and .

Definition 1.10: ([4]) Let be a non-empty set and two self maps on The pair is said to be weakly compatible if whenever for some

Lemma 1.11: ([4]) Let and be weakly compatible self maps of a set If and have a unique point of coincidence then is the unique common fixed point of and

2. Main Results

Theorem 2.1: Let be a cone metric space. Suppose the self-maps satisfy the contractive condition

[picture] | 2.1(1)

where is a constant. If the range of is contained in range of and range of is a complete subspace of More over if and are weakly compatible, and have a unique common fixed point.

Proof: Let be arbitrary. Choose such that This possible since Continuing this process, choose such that

[picture] | 2.1(2)

Now,

using 2.1(1)

By Repeated application of 2.1(1), we get

[picture] | 2.1(3)

For

[picture] | 2.1(4)

Let be given. Choose such that

[picture] | 2.1(5)

From 2.1(4) and 2.1(5), we get

is a Cauchy sequence in , be def.of 1.6(ii)

is a convergent sequence in since is complete in

Let is convergent to Consquently, there is

[picture] | 2.1(6)

For the same given choose such that

[picture] | 2.1(7)

Hence, using 2.1(7)

Hence converges to both q and fp. By uniqueness property of limit

[picture] | 2.1(8)

is the point of coincidence of and

Let be any other coincidence point of and

be the point of coincidence of and 2.1(9)

Now,

Multiplying with positive real number , we get But, we have

From the definition of cone and cone metric, we get

[picture] | 2.1(10)

From 2.1(8,9,10), and have unique point of coincidence.

Finally, let and are weakly compatible self-maps having unique point of coincidence. Using the lemma 1.11, and have a unique common fixed point.

Example 2.1.1: Let

Define

where

Clearly

where is a cone metric space.

Moreover, is coincidence point for and But and are not weakly compatible at Hence and do not have common fixed point.

Theorem 2.2 : Let be a cone metric space. Suppose the maps satisfy the contractive condition

[picture] | 2.2(1)

where is a constant. If the range of is contained in range of and the range of is a complete subspace of , then and have a unique coincidence point in Moreover, if and are weakly compatible, and have a unique common fixed point.

Proof: Let be arbitrary. Choose such that whitch is possible since . Continuing this process, choose such that

[picture] | 2.2(2)

Now,

By repeated application of 2.2(1), we get

[picture] | 2.2(3)

For

[picture] | 2.2(4)

Let be given. Choose such that

[picture] | 2.2(5)

From 2.2(4), 2.2(5), we get

is a Cauchy sequence in

is a convergent sequence in since is complete in X.

Let is convergent to Consequently there is such that

[picture] | 2.2(6)

For the same given choose such that

and

[picture] | 2.2(7)

Hence,

Applying 2.2(7), we get

[picture] | 2.2(8)

converges to both and .

By uniqueness property of limit,

[picture] | 2.2(9)

is point of coincidence of and

Let be any other coincidence point of and

is the point of coincidence of and

Now

We have

From the definition of cone and cone metric, we get

[picture] | 2.2(10)

From 2.2(8,9,10), and have unique point of coincidence.

Finally, let and are weakly compatible self-maps having unique point of coincidence. Using the lemma 1.11, and have a unique common fixed point.

Example 2.2.1: Since , example 2.1.1 also satisfies the contractive condition 2.2(1) with the same

Theorem 2.3: Let be a cone metric space. Suppose the mappings satisfy the contra

[picture] | 2.3(1)

where s constant. If the range of is contained in range of and is a complete subspace of , then and have a unique coincidence point in Moreover, if and are weakly compatible, and have a unique common fixed point.

Proof: Let be arbitrary. Choose such that which is possible since

Continuing this process, choose such that

[picture] | 2.3(2)

Now,

By repeated application of 2.2(1), we get

[picture] | 2.3(3)

For

[picture] | 2.3(4)

Let be given. choose such that

[picture] | 2.3(5)

From 2.3(4), 2.3(5), we get

is a Cauchy sequence in

is a convergent sequence in since is complete in X.

Let is convergent to . Consequently there is such that

[picture] | 2.3(6)

For the same given choose such that

and

[picture] | 2.3(7)

Hence,

[picture] | 2.3(8)

converges to both and .

By uniqueness property of limit,

[picture] | 2.3(9)

is point of coincidence of and .

Let be any other coincidence point of and

is the point of coincidence of and .

Now

Multiplying with positive real number we get But we have

From the definition of cone and cone metric, we get

[picture] | 2.3(10)

From 2.2(8,9,10), and have unique point of coincidence.

Finally, let and are weakly compatible self-maps having unique point of coincidence. Using the lemma 1.11, and have a unique common fixed point.

Example 2.3.1: Let

Define and

Clearly and satisfies the contractive condition 2.3(1) with Moreover, is coincidence point for and . But and are not weakly compatible at Hence and do not have common fixed point.

Acknowledgement

The author thanks the assistance provided by the CSIR in all the respects.

References

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