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Common Fixed Point Theorem for Weakly Contractive Mappings in Cone Metric Spaces

K. Prudhvi

Department of Mathematics, University College of Science, Saifabad, Osmania University, Hyderabad, Telangana State, India

Abstract: In this paper, we prove a unique common fixed point theorem for weakly contractive three mappings in cone metric spaces. Our results generalize and extend some recent results in the literature.

Keywords: Fixed point, common fixed point, regular cone, weakly compatible

Received July 01, 2015; Revised July 30, 2015; Accepted August 02, 2015

Citation: K. Prudhvi, Common Fixed Point Theorem for Weakly Contractive Mappings in Cone Metric Spaces. American Journal of Applied Mathematics and Statistics, 3 (4). 161-163. January 2015. doi:10.12691/ajams-3-4-5

1. Introduction

In 2007, metric space was generalized by Huang and Zhang [6] and introduced cone metric space replacing the set of real numbers by an ordered Banach space. They have proved some fixed point theorems for contractive mappings in cone metric spaces. Later on many authors have extended and generalized the fixed point theorems of Huang and Zhang [6] (see, for exampls [1,2,3,8]. In 2008, Dutta and Chudahury [4] introduced the concept of generalized weakly contractive mapping and proved some fixed point theorems. In2010 Choudhury and Meitya [5] proved some fixed point theorems for weakly contractive mapping in cone metric spaces. In 2011, Hui-ru Zhao, Byung-Soo Lee and Nan-jing Huang [7] proved some common fixed point theorems for weakly compatible mappings in cone metric space under certain contractive conditions and they have generalized the fixed point theorems of [4] and [5].In this paper, we prove a common fixed point theorem for three weakly contractive three self-mappings in cone metric spaces. Our result extends and improves the results of [7].

In this paper B stands for a real Banach space and θ is the zero element, P is a normal cone in B with where is a partial order with respect to P.

We recall some of the definitions are in [6] which are useful in the sequel.

Definition 1.1. Let P be a subset of B. Then P is called a cone if the following conditions are satisfied:

(i) P is closed and ;

(ii) , , ;

(iii) .

For a cone P, define a partial ordering with respect to P by if and only if . We shall write to indicate that but , while will stand for , where, int P is the interior of P.

Definition 1.2. The cone P in a real Banach space B is called norm if there is a number such that for all

The least positive number satisfying the above relation is called a normal constant.

The cone P is said to be regular if for every increasing sequence bounded above is convergent, that is, if is a sequence such that for some , then there exists such that as .

Similarly, if every decreasing sequence which is bounded below is convergent, then the cone P is also regular. It is well known that every regular cone is a normal cone.

Definition 1.3. Let X be a nonempty set. Let d: be a mapping satisfies the following

(i) for all and iff .

(ii) for all .

(iii) for all .

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

Definition 1.4. Let (X, d) be cone metric space. Let be a sequence in X and . If for any with , there exists a natural number N such that for all , then is said to be convergent in X. We denote this as .

Definition 1.5. Let (X, d) be cone metric space. Let {xn} be a sequence in X and . If for every with , there exists a natural number N such that for all then is called a Cauchy sequence in X.

The space (X, d) is called a complete cone metric space if every Cauchy sequence is convergent.

Definition 1.6. [7] Let I, J,T : XX be three mappings with and and be a given point. Choose such that and , then the sequence is called a T-I-J-sequence with initial point .

Definition 1.7. [9] Let X be a non-empty set and I,T:XX be two mappings. The pair of mappings I and T is said to be weakly compatible if for every whenever .

2. Main Results

In this section we obtain a common fixed point theorem for three maps in complete cone metric space.

The following theorem is extended improved the Theorem 2.1 of [7].

Theorem 2.1. Let (X, d) be a complete cone metric space, P be a regular cone, and for all with . Let I, J, T:XX be mappings with T(X) I(X) and T(X) J(X) satisfying the following condition:

[picture] | (1)

for all , where is continuous and monotone, is continuous, if and only if and if and only if . Suppose that

(i) I(X) or J(X) or T(X) is a complete subspace of X.

(ii){T, I} and {T, J} are weakly compatible.

(iii) either d(x,y) c or d(x,y) c for all and .

Then I, J and T have a unique common fixed point in X. Moreover, for any every T-I-J sequence with initial point x0 converges to the common fixed point.

Proof. First we shall prove that {Txn} is a Cauchy sequence. If Tx2n = Tx2n-1 for some n, then from (1), it follows that

By the induction, we know that Txm = Txn-1 for all with . Therefore, {Txn} is a Cauchy sequence.

Now assume for all .

From (1), we have

Since, ψ, φ are continuous,

This shows that , and so

Suppose that {Txn} is not a Cauchy sequence, then there exists with and two subsequences {T } and {T} of with such that ≥ c.

Moreover, corresponding to mk we can choose nk as the smallest integer satisfying

Thus, ,

And

Letting , we get that

By the triangle inequality

and

it is easy to see that

Now by (1)

And so , which is a contradiction to .

Therefore, {Txn} is a Cauchy sequence.

Since T(X) or I(X) or J(X) is complete and there exists such that Txn u (as ), Ixn u, Jxn u(as ).

Choose such that Iz = u such that Iz = u and Jz = u.

To prove Tz = u.

Also Tu = Iu and Tu = Ju.

Since,

We have .

.

That is, u = Tu.

Therefore, Iu = Ju = Tu = u, u is a common fixed point of I, J and T.

Uniqueness, let v be another common fixed point of I, J, and T such that Iv = Tv = Jv = v. Then

Therefore, u is a unique common fixed point of I, J and T in X.

Remark 2.2. Let be an identity mapping in the Theorem 3.1, then we get the following theorem.

Theorem 2.3. Let (X, d) be a complete cone metric space, P be a regular cone, and for all with . Let I, J, T:XX be mappings with and satisfying the following condition:

[picture] | (2)

for all , where, is continuous, and if and only if . Suppose that

(i) I(X) or J(X) or T(X) is a complete subspace of X.

(ii){T, I} and {T, J} are weakly compatible.

(iii) either d(x,y) c or d(x,y)c for all and .

Then I, J and T have a unique common fixed point in X. Moreover, for any every T-I-J sequence with initial point x0 converges to the common fixed point.

Remark 2.4. Let , I and J be identity mappings in the Theorem 3.1 , we get the following theorem.

Theorem 2.5. Let (X, d) be a complete cone metric space, P be a regular cone, and for all with . Let T:XX be mappings such that

[picture] | (3)

for all , where, is continuous, and if and only if . Suppose that

(i) I(X) or J(X) or T(X) is a complete subspace of X.

(ii){T, I} and {T, J} are weakly compatible.

(iii) either d(x,y) c or d(x,y) c for all and .

Then,T has a unique fixed point in X.

Remark 2.6. Let I, J be identity mappings in the Theorem 3.1 , we get the following theorem

Theorem 2.7. Let (X, d) be a complete cone metric space, P be a regular cone, and for all with . Let T:XX be mapping such that

[picture] | (4)

for all , where is continuous and monotone, is continuous, if and only if and if and only if . Suppose that

(i) I(X) or J(X) or T(X) is a complete subspace of X.

(ii){T, I} and {T, J} are weakly compatible.

(iii) either d(x,y) c or d(x,y) c for all and .

Then T has a unique common fixed point in X.

Remark 2.8. If we choose J = I in the above theorem then we get the Theorem 3.1 of [7].

Acknowledgement

The author thanks to referee for giving helpful comments.

References

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