#### Abstract

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.

#### 1. Introduction and Preliminaries

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 , the skew field of quaternions. This means we can write each element in the form , , where , , , are the basis elements of and . For these elements we have the multiplication rules , , , and . The conjugate element is given by . The quaternion modulus has the form of .

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 , with operation of addition and multiplication suitably defined. The components naturally group into the imaginary part , for which we take this part as a vector and the purely real part, , which called a scalar. Sometimes, we write a quaternion as with .

Here, we give the following forms: Thus a quaternion may be viewed as a four-dimensional vector .

For more information about quaternion analysis, we refer to [1–4] and others.

Define a partial order on as follows.

if and only if , , ; , where ; ; . It follows that , if one of the following conditions is satisfied.(i); , where ; .(ii); , where ; .(iii); , where ; .(iv); ; .(v); ; .(vi); ; .(vii); .(viii); .(ix); ; .(x); ; .(xi); ; .(xii); ; .(xiii); ; .(xiv); ; .(xv); .(xvi); .

*Remark 1. *In particular, we will write if and one from (i) to (xvi) is satisfied. Also, we will write if only (xv) is satisfied. It should be remarked that

*Remark 2. *The conditions from (i) to (xv) look strange but these conditions are natural generalizations to the corresponding conditions in the complex setting (see [5]). So, the number of these conditions is related to the number of units in the working space. For our quaternion setting we have four units (one real and three imaginary); then we have conditions. But in the complex setting there were conditions.

Azam et al. in [5] introduced the definition of the complex metric space as follows.

*Definition 3 ([5]). *Let be a nonempty set and suppose that the mapping satisfies the following. (d_{1}) , for all and if and only if .(d_{2}) for all .(d_{3}) for all .Then is called a complex metric space.

Now, we extend the above definition to Clifford analysis.

*Definition 4. *Let be a nonempty set. Suppose that the mapping satisfies (1) for all and if and only if ,(2) for all ,(3) for all .

Then is called a quaternion valued metric on , and is called a quaternion valued metric space.

*Example 5. *Let be a quaternion valued function defined as , where with
Then is a quaternion metric space.

Now, we give the following definitions.

*Definition 6. *Point is said to be an interior point of set whenever there exists such that

*Definition 7. *Point is said to be a limit point of whenever for every

*Definition 8. *Set is called an open set whenever each element of is an interior point of . Subset is called a closed set whenever each limit point of belongs to . The family
is a subbase for Hausdroff topology on .

*Definition 9. *Let be a sequence in and . If for every with there is such that for all , , then is said to be convergent if converges to the limit point ; that is, as or . If for every with there is such that for all , , then is called Cauchy sequence in . If every Cauchy sequence is convergent in , then is called a complete quaternion valued metric space.

#### 2. Convergence in Quaternion Metric Spaces

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.

Lemma 10. *Let be a quaternion valued metric space and let be a sequence in . Then converges to if and only if as .*

*Proof. *Suppose that converges to . For a given real number , let
Then and there is a natural number such that for all .

Therefore, for all . Hence as .

Conversely, suppose that as . Then, given with , there exists a real number , such that, for ,
For this , there is a natural number such that for all . Implying that for all , hence converges to .

Lemma 11. *Let be a quaternion valued metric space and let be a sequence in . Then is a Cauchy sequence if and only if as .*

*Proof. *Suppose that is a Cauchy sequence. For a given real number , let
Then, and there is a natural number such that for all . Therefore for all . Hence as .

Conversely, suppose that as . Then, given with , there exists a real number , such that for , we have that
For this , there is a natural number such that for all , which implies that for all . Hence is a Cauchy sequence. This completes the proof of Lemma 11.

*Definition 12. *Let be a complete quaternion valued metric space. For all , represents . A quaternion valued metric space is said to be metrically convex if has the property that, for each with , there exists , such that
The following lemma finds immediate applications which is straightforward from [6].

Lemma 13. *Let be a metrically convex quaternion valued metric space and a nonempty closed subset of . If and , then there exists a point (where stands for the boundary of ) such that
*

*Definition 14. *Let be a nonempty subset of a quaternion valued metric space and let satisfy the condition
For all , with , , and let be an increasing continuous function for which the following property holds:
We call function satisfying condition (13) generalized -contractive.

Motivated by [7, 8], we construct the following definition.

*Definition 15. *Let be a nonempty subset of a quaternion valued metric space and . The pair is said to be weakly commuting if, for each such that and , we have
It follows that

*Remark 16. *It should be remarked that Definition 15 extends and generalizes the definition of weakly commuting mappings which are introduced in [7].

#### 3. Common Fixed Point Theorems in Quaternion Analysis

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.

Theorem 17. *Let be a complete quaternion valued metric space, a nonempty closed subset of , and an increasing continuous function satisfying (13) and (14). Let be such that is generalized -contractive satisfying the conditions: *(i)*, ,*(ii)*,
*(iii)* and are weakly commuting mappings,*(iv)* is continuous at ,**then there exists a unique common fixed point in , such that .*

*Proof. *We construct the sequences and in the following way.

Let . Then there exists a point such that as . From and the implication , we conclude that . Now, let be such that
Let and assume that ; then
which implies that there exists a point such that . Suppose ; then there exists a point (using Lemma 13) such that
Since , there exists a point such that and so
Let . Thus, repeating the forgoing arguments, we obtain two sequences and such that(i),(ii), or(iii),
We denote
Obviously, the two consecutive terms of cannot lie in .

Let us denote . We have the following three cases.*Case 1.* If , then
Thus,
*Case 2.* If , , note that
or
which implies that
Hence
Therefore,
Hence,
*Case 3.* If , , so . Since is a convex linear combination of and , it follows that
If , then implying that
Hence,
It follows that
Since
then
Therefore,
Now, proceeding as in Case 2 (because , ), we obtain that
Also from (31), if , then
which implies that ; hence
Therefore, noting that, by Case 2, , we conclude that
Thus, in all cases we get either or for , we have ; for , we have ; by induction, we get . Letting , we have and by (14), we have
so that is a Cauchy sequence and hence it converges to point in . Now there exists a subsequence of such that it is contained in . Without loss of generality, we may denote . Since is continuous, converges to .

We now show that and have common fixed point (). Using the weak commutativity of and , we obtain that
then
This implies that
On letting , we obtain
which means that
Now, consider
Letting yields
a contradiction, thus giving which implies , so that and hence .

To show that , consider
Letting, we obtain
a contradiction, thereby giving which implies , so that and hence . Thus, we have shown that , so is a common fixed point of and . To show that is unique, let be another fixed point of and ; then
a contradiction, therefore giving which implies that , so that ; thus . This completes the proof.

*Remark 18. *If , then and therefore, we obtain the same results in [9]. So, our theorem is more general than Theorem 3.1 in [9] for a pair of weakly commuting mappings.

Using the concept of commuting mappings (see, e.g., [10]), we can give the following result.

Theorem 19. *Let be a complete quaternion valued metric space, a nonempty closed subset of , and an increasing continuous function satisfying (13) and (14). Let be such that is generalized -contractive satisfying the conditions:*(i)*, ,*(ii)*,*(iii)* and are commuting mappings,*(iv)* is continuous at ;**then there exists a unique common fixed point in such that .*

*Proof. *The proof is very similar to the proof of Theorem 17 with some simple modifications; so it will be omitted.

Corollary 20. *Let be a complete complex valued metric space, a nonempty closed subset of , and an increasing continuous function satisfying (14) and
**
for all , with ; , .**Let be such that is generalized -contractive satisfying the conditions:*(i)*, ,*(ii)*,*(iii)* and are weakly mappings,*(iv)* is continuous at ;**then there exists a unique common fixed point in such that .*

*Proof. *Since each element can be written in the form , , where , , , are the basis elements of and . Putting , we obtain an element in . So, the proof can be obtained from Theorem 17 directly.

#### 4. Fixed Points in Normal Cone Metric Spaces

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 [11] as well as a result due to Abbas and Rhoades [12]. The obtained result gives a fixed point theorem for four mappings without appealing to commutativity conditions, defined on a cone metric space.

Let be a real Banach space. A subset of is called a cone if and only if(a) is closed and nonempty and ;(b), , implies that ;(c).Given cone , we define a partial ordering with respect to by if and only if . Cone is called normal if there is a number such that, for all , The least positive number satisfying the above inequality is called the normal constant of , while stands for (interior of ). We will write to indicate that but .

*Definition 21 (see [11]). *Let be a nonempty set. Suppose that the mapping satisfies (d_{1}) for all and if and only if ;(d_{2}) for all ;(d_{3}) for all .Then, is called a cone metric on and is called a cone metric space.

*Definition 22 (see [11]). *Let be a cone metric space, a sequence in , and . For every with , we say that is (1)a Cauchy sequence if there is an such that, for all , ;(2)a convergent sequence if there is an such that, for all , ; for some , for all .A cone metric space is said to be complete if every Cauchy sequence in is convergent in .

Now, we give the following result.

Theorem 23. *Let be a complete cone metric space and a normal cone with normal constant . Suppose that the mappings , , , are four self-maps of satisfying
**
for all , where and with . Then, , , , and have a unique common fixed point in .*

*Proof. *If or , the proof is already known from [12]. So, we consider the case when . Suppose is an arbitrary point of , and define by and ; . Then, we have
this yields that
Therefore, for all , we deduce that
Now, for , we obtain that
From (54), we have
which implies that as . Hence, is a Cauchy sequence. Since is complete there exists a point such that as . Now using (55), we obtain that
Now, using (54) and (61), we obtain that
Therefore
Since the right hand side of the above inequality approaches zero as , hence , and then . Also, we have
which implies that
Letting , we deduce that , and then . Similarly, by replacing by and by in (61), we deduce that
where , as . Hence , and then . Also,
Letting , we deduce that , and then .

To prove uniqueness, suppose that is another fixed point of , , , and , and then
which gives , and hence . This completes the proof.

Now, we give the following result.

Corollary 24. *Let be a complete cone metric space and a normal cone with normal constant . Suppose that the mappings , , , are four self-maps of satisfying
**
for all , where and with . Then, , , , and have a unique common fixed point in ; .*

*Proof. *Inequality (69) is obtained from (55) by setting , , and , . Then the result follows from Theorem 23.

*Remark 25. *If we put or in Theorem 23, we obtain Theorem 2.1 in [12].

*Remark 26. *It should be remarked that the quaternion metric space is different from cone metric space which is introduced in [11] (see also [12–19]). The elements in or form an algebra while the same is not true in .

*
Open Problem.* It is still an open problem to extend our results in this paper for some kinds of mappings like biased, weakly biased, weakly compatible mappings, compatible mappings of type , and -weak** commuting mappings. See [20–30] for more studies on these mentioned mappings.

#### Conflict of Interests

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

#### Acknowledgments

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.