^{1, 2}

^{3}

^{3, 4}

^{5}

^{1}

^{2}

^{3}

^{4}

^{5}

Very recently, Ahmed et al. introduced the notion of quaternion-valued metric as a generalization of metric and proved a common fixed point theorem in the context of quaternion-valued metric space. In this paper, we will show that the quaternion-valued metric spaces are subspaces of cone metric spaces. Consequently, the fixed point results in such spaces can be derived as a consequence of the corresponding existing fixed point result in the setting cone metric spaces.

Recently, Azam et al. [

In this paper, we announce that the quaternion-valued metric spaces, introduced by Ahmed et al. [

First we recall the concept of complex-valued metric space which is given by Azam et al. in [

Let

Let

Let

Let

Let

Now, we recollect the basic definitions and concept on quaternion-valued metric spaces.

The skew field of quaternion denoted by

The quaternion modulus has the form of

Define a partial order

In particular, we write

Ahmed et al. [

Let

Let

Let

Let

Let

The cone

In this paper,

Let

The following definitions and lemmas have been chosen from [

Let

Let

Let

Let

Let

Let

Let

Let

Precisely,

Any quaternion-valued metric space

For all

The partial ordered

Assume

A sequence

Let

Let

Precisely,

Any complex-valued metric space

For all

The partial ordered

We omitted the proof of Lemma

A sequence

We omit the proof of Lemma

Let

The following lemma finds immediate applications which is straightforward from [

Let

Let

Denote

Let

Suppose that

Let

Let

We construct the sequences

Let

Denote

Proving the above cases are similar to [

Now, consider

To show that

Thus, we have shown that

Let

Theorem

Let

In 2010, Du [

Let

On his paper, Du [

Let

From Proposition

Let

The Banach contraction principle and Proposition

Let

Let

We skip the proof of Proposition

Let

Let

In Definition

Furthermore, by Lemma

The authors declare that there is no conflict of interests regarding the publication of this paper.

All authors contributed equally and significantly in writing this paper. All authors read and approved the final paper.

This research was supported by the Deanship of Scientific Research (DSR), King Abdulaziz University, Jeddah, Saudi Arabia. The authors thank the anonymous referees for their remarkable comments, suggestions, and ideas that helped to improve this paper.