The aim of this paper is twofold. First, we introduce the concept of quaternion metric spaces which generalizes both real and complex metric spaces. Further, we establish some fixed point theorems in quaternion setting. Secondly, we prove a fixed point theorem in normal cone metric spaces for four self-maps satisfying a general contraction condition.
A metric space can be thought as very basic space having a geometry, with only a few axioms. In this paper we introduce the concept of quaternion metric spaces. The paper treats material concerning quaternion metric spaces that is important for the study of fixed point theory in Clifford analysis. We introduce the basic ideas of quaternion metric spaces and Cauchy sequences and discuss the completion of a quaternion metric space.
In what follows we will work on
Quaternions can be defined in several different equivalent ways. Notice the noncommutative multiplication, their novel feature; otherwise, quaternion arithmetic has special properties. There is also more abstract possibilty of treating quaternions as simply quadruples of real numbers
Here, we give the following forms:
For more information about quaternion analysis, we refer to [
Define a partial order
In particular, we will write
The conditions from (i) to (xv) look strange but these conditions are natural generalizations to the corresponding conditions in the complex setting (see [
Azam et al. in [
Let
Then
Now, we extend the above definition to Clifford analysis.
Let
Then
Let
Now, we give the following definitions.
Point
Point
Set
Let
In this section we give some auxiliary lemmas using the concept of quaternion metric spaces; these lemmas will be used to prove some fixed point theorems of contractive mappings.
Let
Suppose that
Therefore,
Conversely, suppose that
Let
Suppose that
Conversely, suppose that
Let
Let
Let
Motivated by [
Let
It should be remarked that Definition
In this section, we prove common fixed point theorems for two pairs of weakly commuting mappings on complete quaternion metric spaces. The obtained results will be proved using generalized contractive conditions.
Now, we give the following theorem.
Let
then there exists a unique common fixed point
We construct the sequences
Let
We denote
Let us denote
or for for by induction, we get
so that
We now show that
To show that
If
Using the concept of commuting mappings (see, e.g., [
Let
then there exists a unique common fixed point
The proof is very similar to the proof of Theorem
Let
Let
then there exists a unique common fixed point
Since each element
In this section, we prove a fixed point theorem in normal cone metric spaces, including results which generalize a result due to Huang and Zhang in [
Let
Given cone
Let
Then,
Let a Cauchy sequence if there is an a convergent sequence if there is an
A cone metric space
Now, we give the following result.
Let
If
To prove uniqueness, suppose that
Now, we give the following result.
Let
Inequality (
If we put
It should be remarked that the quaternion metric space is different from cone metric space which is introduced in [
The authors declare that there is no conflict of interests regarding the publication of this paper.
The authors are grateful to Taif University, Saudi Arabia, for its financial support of this research under Project no. 1836/433/1. The first author would like to thank Professor Rashwan Ahmed Rashwan from Assiut University, Egypt, for introducing him to the subject of fixed point theory and Professor Klaus Gürlebeck from Bauhaus University, Weimar, Germany, for introducing him to the subject of Clifford analysis.