Remarks on Some Recent Fixed Point Results on Quaternion-Valued Metric Spaces

Abstract

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.

1. Introduction

Recently, Azam et al. [1] introduced the notion of complex-valued metric space, as a generalization of Banach-valued metric space which is also known as a cone metric space. The authors [1] proved several fixed point theorems in the context of complex-valued metric space. Inspired from these results, Ahmed et al. [2] defined the concept of quaternion-valued metric space, as a generalization of complex-valued metric space, and proved a common fixed point theorem in the context of such spaces.

In this paper, we announce that the quaternion-valued metric spaces, introduced by Ahmed et al. [2], are subspaces of cone metric spaces. Consequently, the fixed point results in such spaces can be concluded from the classical versions in cone metric spaces. Consequently, the fixed point results in such spaces can be concluded from the classical versions in cone metric spaces. On the other hand, several results have been reported on the equivalence of cone metric space and metric space; see, for example, [3–9]. In particular, by the help of scalarization function, Du [3] proved that several fixed point results in the context of cone metric spacecan be concluded from the existing associated results in the setting of metric space. Furthermore, if the cone is normal, then there is a metric induced by Banach-valued metric. Hence, most of the announced fixed point results in the setting cone metric space can be deduced from related existing results in the literature in the context of the metric space.

1.1. Complex-Valued Metric Spaces

First we recall the concept of complex-valued metric space which is given by Azam et al. in [1].

Let ℂ be the set of complex numbers and 𝑧₁,𝑧₂∈ℂ. Define a partial order ≾ on ℂ as follows: 𝑧₁≾𝑧₂iffRe𝑧₁≤Re𝑧₂,Im𝑧₁≤Im𝑧₂. (1) It follows that 𝑧₁≾𝑧₂ (2) if one of the following conditions is satisfied: ℎ₁Re𝑧₁=Re𝑧₂;Im𝑧₁<Im𝑧₂,ℎ₂Re𝑧₁<Re𝑧₂;Im𝑧₁=Im𝑧₂,ℎ₃Re𝑧₁<Re𝑧₂;Im𝑧₁<Im𝑧₂,ℎ₄Re𝑧₁=Re𝑧₂;Im𝑧₁=Im𝑧₂. (3) In particular, we will write 𝑧₁⋨𝑧₂ if 𝑧₁≠𝑧₂ and one of (ℎ₁), (ℎ₂), and (ℎ₃) is satisfied and we will write 𝑧₁≺𝑧₂ if only (ℎ₃) is satisfied. Note that 0≾𝑧₁⋦𝑧₂⟹𝑧₁<𝑧₂, (4) where |⋅| represents modulus or magnitude of 𝑧, and 𝑧₁≾𝑧₂,𝑧₂≺𝑧₃⟹𝑧₁≺𝑧₃. (5)

Definition 1 (see [1]). Let 𝑋 be a nonempty set. A function 𝑑∶𝑋×𝑋→ℂ is called a complex-valued metric on 𝑋, if it satisfies the following conditions: (𝑏₁):0≾𝑑(𝑥,𝑦) for all 𝑥,𝑦∈𝑋 and 𝑑(𝑥,𝑦)=0, if and only if 𝑥=𝑦,(𝑏₂):𝑑(𝑥,𝑦)=𝑑(𝑦,𝑥), for all 𝑥,𝑦∈𝑋,(𝑏₃):𝑑(𝑥,𝑦)≾𝑑(𝑥,𝑧)+𝑑(𝑦,𝑧), for all 𝑥,𝑦,𝑧∈𝑋.Here, the pair (𝑋,𝑑) is called a complex-valued metric space.

Let {𝑥_𝑛} be a sequence in 𝑋 and ∈𝑋. If for every 𝑐∈ℂ, with 0≺𝑐, there is 𝑛₀∈ℕ such that, for all 𝑛>𝑛₀, 𝑑(𝑥_𝑛,𝑥)≺𝑐, then {𝑥_𝑛} is said to be convergent, {𝑥_𝑛} converges to 𝑥, and 𝑥 is the limit point of {𝑥_𝑛}. We denote this by lim_𝑛𝑥_𝑛=𝑥, or 𝑥_𝑛→𝑥, as 𝑛→∞. If for every 𝑐∈ℂ with 0≺𝑐 there is 𝑛₀∈ℕ such that for all 𝑛>𝑛₀, 𝑑(𝑥_𝑛,𝑥_𝑛+𝑚)≺𝑐, then {𝑥_𝑛} is called a Cauchy sequence in (𝑋,𝑑). If every Cauchy sequence is convergent in (𝑋,𝑑), then (𝑋,𝑑) is called a complete complex-valued metric space.

Lemma 2 (see [1, Lemma 2, Azam et al.]). Let (𝑋,𝑑) be a complex-valued metric space and let {𝑥_𝑛} be a sequence in. Then {𝑥_𝑛} converges to 𝑥 if and only if |𝑑(𝑥_𝑛,𝑥)|→0 as 𝑛→∞.

Lemma 3 (see [1, Lemma 3, Azam et al.]). Let (𝑋,𝑑) be a complex-valued metric space and let {𝑥_𝑛} be a sequence in. Then {𝑥_𝑛} is a Cauchy sequence if and only if |𝑑(𝑥_𝑛,𝑥_𝑛+𝑚)|→0 as 𝑛→∞.

1.2. Quaternion Metric Space

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

The skew field of quaternion denoted by ℍ means to write each element 𝑞∈ℍ in the form 𝑞=𝑥₀+𝑥₁𝑖+𝑥₂𝑗+𝑥₃𝑘; 𝑥_𝑛∈ℝ, where 1, 𝑖, 𝑗, and 𝑘 are the basis elements of ℍ and 𝑛=1,2,3. For these elements we have the multiplication rules 𝑖²=𝑗²=𝑘²=−1, 𝑖𝑗=−𝑗𝑖=𝑘, 𝑘𝑗=−𝑗𝑘=−𝑖, and 𝑘𝑖=−𝑖𝑘=𝑗. The conjugate element 𝑞 is given by 𝑞=𝑥₀−𝑥₁𝑖−𝑥₂𝑗−𝑥₃𝑘.

The quaternion modulus has the form of |𝑞|=𝑥²₀+𝑥²₁+𝑥²₂+𝑥²₃. A quaternion q may be viewed as a four-dimensional vector (𝑥₀,𝑥₁,𝑥₂,𝑥₃).

Define a partial order ≾ on ℍ as follows.

𝑞₁≾𝑞₂ if and only if Re(𝑞₁)≤Re(𝑞₂), Im_𝑠(𝑞₁)≤Im_𝑠(𝑞₂), 𝑞₁,𝑞₂∈ℍ, 𝑠=𝑖,𝑗,𝑘 where Im_𝑖=𝑥₁, Im_𝑗=𝑥₂, and Im_𝑘=𝑥₃. It follows that 𝑞₁≾𝑞₂ if one of the following conditions holds:(I)Re(𝑞₁)=Re(𝑞₂); Im_𝑠1(𝑞₁)=Im_𝑠1(𝑞₂) where 𝑠₁=𝑗,𝑘; Im_𝑖(𝑞₁)<Im_𝑖(𝑞₂),(II)Re(𝑞₁)=Re(𝑞₂); Im_𝑠2(𝑞₁)=Im_𝑠2(𝑞₂) where 𝑠₂=𝑖,𝑘; Im_𝑗(𝑞₁)<Im_𝑗(𝑞₂),(III)Re(𝑞₁)=Re(𝑞₂); Im_𝑠3(𝑞₁)=Im_𝑠3(𝑞₂) where 𝑠₃=𝑖,𝑗; Im_𝑘(𝑞₁)<Im_𝑘(𝑞₂),(IV)Re(𝑞₁)=Re(𝑞₂); Im_𝑠1(𝑞₁)<Im_𝑠1(𝑞₂); Im_𝑖(𝑞₁)=Im_𝑖(𝑞₂),(V)Re(𝑞₁)=Re(𝑞₂); Im_𝑠2(𝑞₁)<Im_𝑠2(𝑞₂); Im_𝑗(𝑞₁)=Im_𝑗(𝑞₂),(VI)Re(𝑞₁)=Re(𝑞₂); Im_𝑠3(𝑞₁)<Im_𝑠3(𝑞₂); Im_𝑘(𝑞₁)=Im_𝑘(𝑞₂),(VII)Re(𝑞₁)=Re(𝑞₂); Im_𝑠(𝑞₁)<Im_𝑠(𝑞₂),(VIII)Re(𝑞₁)<Re(𝑞₂); Im_𝑠(𝑞₁)=Im_𝑠(𝑞₂),(IX)Re(𝑞₁)<Re(𝑞₂); Im_𝑠1(𝑞₁)=Im_𝑠1(𝑞₂); Im_𝑖(𝑞₁)<Im_𝑖(𝑞₂),(X)Re(𝑞₁)<Re(𝑞₂); Im_𝑠2(𝑞₁)=Im_𝑠2(𝑞₂); Im_𝑗(𝑞₁)<Im_𝑗(𝑞₂),(XI)Re(𝑞₁)<Re(𝑞₂); Im_𝑠3(𝑞₁)=Im_𝑠3(𝑞₂); Im_𝑘(𝑞₁)<Im_𝑘(𝑞₂),(XII)Re(𝑞₁)<Re(𝑞₂); Im_𝑠1(𝑞₁)<Im_𝑠1(𝑞₂); Im_𝑖(𝑞₁)=Im_𝑖(𝑞₂),(XIII)Re(𝑞₁)<Re(𝑞₂); Im_𝑠2(𝑞₁)<Im_𝑠2(𝑞₂); Im_𝑖(𝑞₁)=Im_𝑖(𝑞₂),(XIV)Re(𝑞₁)<Re(𝑞₂); Im_𝑠3(𝑞₁)<Im_𝑠3(𝑞₂); Im_𝑖(𝑞₁)=Im_𝑖(𝑞₂),(XV)Re(𝑞₁)<Re(𝑞₂); Im_𝑠(𝑞₁)<Im_𝑠(𝑞₂),(XVI)Re(𝑞₁)=Re(𝑞₂); Im_𝑠(𝑞₁)=Im_𝑠(𝑞₂).

Remark 4. In particular, we write 𝑞₁⋦𝑞₂ if 𝑞₁≠𝑞₂ and one from (I) to (XVI) is satisfied. Also, we will write 𝑞₁<𝑞₂ if only (XV) is satisfied. It should be remarked that 𝑞₁≾𝑞₂⟹𝑞₁≤𝑞₂. (6)

Ahmed et al. [2] introduced the definition of the quaternion-valued metric space as follows.

Definition 5. Let 𝑋 be a nonempty set. A function 𝑑_ℍ∶𝑋×𝑋→ℍ is called a quaternion-valued metric on 𝑋, if it satisfies the following conditions:(𝑑₁):0≾𝑑_ℍ(𝑥,𝑦)0 for all 𝑥,𝑦∈𝑋 and 𝑑_ℍ(𝑥,𝑦)=0, if and only if, 𝑥=𝑦,(𝑑₂):𝑑_ℍ(𝑥,𝑦)=𝑑_ℍ(𝑦,𝑥), for all 𝑥,𝑦∈𝑋,(𝑑₃):𝑑_ℍ(𝑥,𝑦)≾𝑑_ℍ(𝑥,𝑧)+𝑑_ℍ(𝑦,𝑧), for all 𝑥,𝑦,𝑧∈𝑋.Then, (𝑋,𝑑_ℍ) is called a quaternion-valued metric space.

Let {𝑥_𝑛} be a sequence in 𝑋 and ∈𝑋. If for every 𝑐∈ℂ, with 0<𝑐, there is 𝑛₀∈ℕ such that, for all 𝑛>𝑛₀, 𝑑_ℍ(𝑥_𝑛,𝑥)<𝑐, then {𝑥_𝑛} is said to be convergent, {𝑥_𝑛} converges to 𝑥, and 𝑥 is the limit point of {𝑥_𝑛}. We denote this by lim_𝑛𝑥_𝑛=𝑥, or 𝑥_𝑛→𝑥, as 𝑛→∞. If for every 𝑐∈ℂ with 0<𝑐 there is 𝑛₀∈ℕ such that, for all 𝑛>𝑛₀, 𝑑_ℍ(𝑥_𝑛,𝑥_𝑛+𝑚)<𝑐, then {𝑥_𝑛} is called a Cauchy sequence in (𝑋,𝑑_ℍ). If every Cauchy sequence is convergent in (𝑋,𝑑_ℍ), then (𝑋,𝑑_ℍ) is called a complete quaternion-valued metric space.

Lemma 6 (see [2, Lemma 2.1, Ahmed et al.]). Let (𝑋,𝑑) be a quaternion-valued metric space and let {𝑥_𝑛} be a sequence in. Then {𝑥_𝑛} converges to 𝑥 if and only if |𝑑(𝑥_𝑛,𝑥)|→0 as 𝑛→∞.

Lemma 7 (see [2, Lemma 2.2, Ahmed et al.]). Let (𝑋,𝑑_ℍ) be a quaternion-valued metric space and let {𝑥_𝑛} be a sequence in. Then {𝑥_𝑛} is a Cauchy sequence if and only if |𝑑_ℍ(𝑥_𝑛,𝑥_𝑛+𝑚)|→0 as 𝑛→∞.

1.3. Cone Metric Space

Let 𝐸 be a real Banach space. A subset 𝑃 of 𝐸 is called a cone, if and only if the following holds:(𝑎₁):𝑃 is closed, nonempty, and 𝑃≠{0},(𝑎₂):𝑎,𝑏∈ℝ, 𝑎,𝑏≥0, and 𝑥,𝑦∈𝑃 imply that 𝑎𝑥+𝑏𝑦∈𝑃,(𝑎₃):𝑥∈𝑃 and −𝑥∈𝑃 imply that 𝑥=0.Given a cone 𝑃⊂𝐸, we define a partial ordering ≤ with respect to 𝑃 by 𝑥≤𝑦, if and only if 𝑦−𝑥∈𝑃. We write 𝑥<𝑦 to indicate that 𝑥≤𝑦 but 𝑥≠𝑦, while 𝑥≪𝑦 stands for 𝑦−𝑥∈int𝑃, where int𝑃 denotes the interior of 𝑃.

The cone 𝑃 is called normal, if there exist a number 𝐾>0 such that 0≤𝑥≤𝑦 implies ‖𝑥‖≤𝐾‖𝑦‖, for all 𝑥,𝑦∈𝐸. The least positive number satisfying this is called the normal constant [10]. It is proved that the normal constant can not be less then 1 (see [11]). For more details on cone metric space, we refer, for example, to [10–16].

In this paper, 𝐸 denotes a real Banach space, 𝑃 denotes a cone in 𝐸 with int𝑃≠0/, and ≤ denotes partial ordering with respect to 𝑃.

Definition 8 (see [10]). Let 𝑋 be a nonempty set. A function 𝑑∶𝑋×𝑋→𝐸 is called a cone metric on 𝑋, if it satisfies the following conditions: (𝑏₁):𝑑(𝑥,𝑦)≥0 for all 𝑥,𝑦∈𝑋 and 𝑑(𝑥,𝑦)=0, if and only if, 𝑥=𝑦,(𝑏₂):𝑑(𝑥,𝑦)=𝑑(𝑦,𝑥), for all 𝑥,𝑦∈𝑋,(𝑏₃):𝑑(𝑥,𝑦)≤𝑑(𝑥,𝑧)+𝑑(𝑦,𝑧), for all 𝑥,𝑦,𝑧∈𝑋.Then, (𝑋,𝑑) is called a cone metric space.

The following definitions and lemmas have been chosen from [10, 16].

Definition 9. Let (𝑋,𝑑) be a cone metric space and let {𝑥_𝑛}_𝑛∈ℕ be a sequence in 𝑋 and 𝑥∈𝑋. If, for all 𝑐∈𝐸 with 0≪𝑐, there is 𝑛₀∈ℕ such that for all 𝑛>𝑛₀, 𝑑(𝑥_𝑛,𝑥₀)≪𝑐, then {𝑥_𝑛}_𝑛∈ℕ is said to be convergent, {𝑥_𝑛}_𝑛∈ℕ converges to 𝑥, and 𝑥 is the limit of {𝑥_𝑛}_𝑛∈ℕ.

Definition 10. Let (𝑋,𝑑) be a cone metric space and let {𝑥_𝑛}_𝑛∈ℕ be a sequence in 𝑋. If for all 𝑐∈𝐸 with 0≪𝑐, there is 𝑛₀∈ℕ such that, for all 𝑚,𝑛>𝑛₀, 𝑑(𝑥_𝑛,𝑥_𝑚)≪𝑐, then {𝑥_𝑛}_𝑛∈ℕ is called a Cauchy sequence in 𝑋.

Definition 11. Let (𝑋,𝑑) be a cone metric space. If every Cauchy sequence is convergent in 𝑋, then 𝑋 is called a complete cone metric space.

Definition 12. Let (𝑋,𝑑) be a cone metric space. A self-map 𝑇 on 𝑋 is said to be continuous, if lim_𝑛→∞𝑥_𝑛=𝑥 implies lim_𝑛→∞𝑇(𝑥_𝑛)=𝑇(𝑥) for all sequence {𝑥_𝑛}_𝑛∈ℕ in 𝑋.

Lemma 13. Let (𝑋,𝑑) be a normal cone metric space and let 𝑃 be a normal cone. Let {𝑥_𝑛}_𝑛∈ℕ be a sequence in 𝑋. Then, {𝑥_𝑛}_𝑛∈ℕ converges to 𝑥, if and only if lim_𝑛→∞𝑑𝑥_𝑛,𝑥=0. (7)

Lemma 14. Let (𝑋,𝑑) be a cone metric space and let {𝑥_𝑛}_𝑛∈ℕ be a sequence in 𝑋. If {𝑥_𝑛}_𝑛∈ℕ is convergent, then it is a Cauchy sequence.

Lemma 15. Let (𝑋,𝑑) be a cone metric space and let 𝑃 be a normal cone in 𝐸. Let {𝑥_𝑛}_𝑛∈ℕ be a sequence in 𝑋. Then, {𝑥_𝑛}_𝑛∈ℕ is a Cauchy sequence, if and only if lim_{𝑚,𝑛→∞}𝑑(𝑥_𝑚,𝑥_𝑛)=0.

2. Main Result

Let (𝑋,𝑑_ℍ) be a quaternion-valued metric space where ℍ is the skew field of quaternion number 𝑞; that is, ℍ=𝑥₀+𝑥₁𝑖+𝑥₂𝑗+𝑥₃𝑘∶𝑥₀,𝑥₁,𝑥₂,𝑥₃∈ℝ⁴. (8) Define 𝒫_ℍ=𝑥₀+𝑥₁𝑖+𝑥₂𝑗+𝑥₃𝑘∶𝑥₀𝑥≥0,₁≥0,𝑥₂≥0,𝑥₃≥0. (9) It is apparent that 𝒫_ℍ⊂ℍ. Assume 𝟎_ℍ is the zero of ℍ from now on. Note that (ℍ,|⋅|) is a real Banach space.

Lemma 16. 𝒫_ℍ is a normal cone in real Banach space (ℍ,|⋅|).

Proof. Precisely, 𝒫_ℍ is nonempty, closed and 𝒫_ℍ≠(𝟎_ℍ). Also for all 𝛼,𝛽∈ℝ⁺, and 𝑝,𝑞∈𝒫_ℍ we have 𝛼𝑝+𝛽𝑞∈𝒫_ℍ and 𝒫_ℍ∩(−𝒫_ℍ)=(𝟎_ℍ). Notice that the normality of the cone 𝒫_ℍ follows from Remark 4.

Lemma 17. Any quaternion-valued metric space (𝑋,𝑑_ℍ) is a cone metric space.

Proof. For all 𝑝,𝑞∈ℍ define 𝑝⫅𝑞iff𝑞−𝑝∈𝒫_ℍ. (10)≾ defines a partial ordered on ℍ and one can easily verify that (𝑋,𝑑_ℍ) is a cone metric space with respect to ⫅.

Lemma 18. The partial ordered ≾ defined in Lemma 17 is equivalent to ⫅.

Proof. Assume 𝑝=𝑝₀+𝑝₁𝑖+𝑝₂𝑗+𝑝₃𝑘 and 𝑞=𝑞₀+𝑞₁𝑖+𝑞₂𝑗+𝑞₃𝑘. 𝑝⫅𝑞, if and only if 𝑞−𝑝∈𝒫_ℍ, if and only if 𝑞₀−𝑝₀≥0, 𝑞₁−𝑝₁≥0, 𝑞₂−𝑝₂≥0, and 𝑞₃−𝑝₃≥0. In other words, Re(𝑝)≤Re(𝑞), Im_𝑠(𝑝)≤Im_𝑠(𝑞), 𝑠=𝑖,𝑗,𝑘 where Im_𝑖(𝑝)=𝑝₁, Im_𝑗(𝑝)=𝑝₂ and Im_𝑘(𝑝)=𝑝₃ and Im_𝑖(𝑞)=𝑞₁, Im_𝑗(𝑞)=𝑞₂ and Im_𝑘(𝑞)=𝑞₃, if and only if 𝑝≾𝑞.

Lemma 19. A sequence {𝑥_𝑛} in (𝑋,𝑑_ℍ) is convergent in the context of quaternion-valued metric space if and only if {𝑥_𝑛} is convergent in the setting of of cone metric space.

Proof. Let {𝑥_𝑛} be sequence in 𝑋. {𝑥_𝑛} converges to 𝑥∈𝑋 as the concept of quaternion-valued metric space if and only if |𝑑(𝑥_𝑛,𝑥)|→0 as 𝑛→∞ (see Lemma 6) if and only if {𝑥_𝑛} converges to 𝑥 as the concept of cone metric space by considering ℍ as the Banach space endowed with the cone 𝒫_ℍ (see Lemma 13).

Let (𝑋,𝑑_ℂ) be a complex-valued metric space where ℂ is the skew field of complex number 𝑧; that is, ℂ=𝑥+𝑦𝑖∶𝑥,𝑦∈ℝ². (11) Define 𝒫_ℂ=𝑥+𝑦𝑖∶𝑥≥0,𝑦≥0. (12) It is apparent that 𝒫_ℂ⊂ℂ. Assume 𝟎_ℂ is the zero of ℂ from now on. Note that (ℂ,|⋅|) is a real Banach space.

Lemma 20. 𝒫_ℂ is a normal cone in real Banach space (ℂ,|⋅|).

Proof. Precisely, 𝒫_ℂ is nonempty, closed and 𝒫_ℂ≠(𝟎_ℂ). Also for all 𝛼,𝛽∈ℝ⁺, and 𝑝,𝑞∈𝒫_ℂ we have 𝛼𝑝+𝛽𝑞∈𝒫_ℂ and 𝒫_ℂ∩(−𝒫_ℂ)=(𝟎_ℂ).

Lemma 21. Any complex-valued metric space (𝑋,𝑑_ℂ) is a cone metric space.

Proof. For all 𝑝,𝑞∈ℂ define 𝑝⫅𝑞iff𝑞−𝑝∈𝒫_ℂ. (13)≾ defines a partial ordered on ℂ and one can easily verify that (𝑋,𝑑_ℂ) is a cone metric space with respect to ⫅.

Lemma 22. The partial ordered ≾ defined in Lemma 21 is equivalent to ⫅.

We omitted the proof of Lemma 22 since it is the mimic of the proof Lemma 18.

Lemma 23. A sequence {𝑥_𝑛} in (𝑋,𝑑_ℂ) is convergent as the concept of complex-valued metric space if and only if {𝑥_𝑛} is convergent as the concept of cone metric space.

We omit the proof of Lemma 23 above due to Lemma 19.

Definition 24. Let (𝑋,𝑑) be a complete cone metric space. For all 𝑥,𝑦∈𝑋. A cone metric space (𝑋,𝑑) is said to be metrically convex if 𝑋 has the property that, for each 𝑥,𝑦∈𝑋 with 𝑥≠𝑦, there exists 𝑧∈𝑋, 𝑥≠𝑦≠𝑧 such that 𝑑(𝑥,𝑧)+𝑑𝑧,𝑦=𝑑𝑥,𝑦. (14)

The following lemma finds immediate applications which is straightforward from [17].

Lemma 25. Let (𝑋,𝑑) be a metrically convex quaternion-valued metric space, and let 𝐾 be a nonempty closed subset of 𝑋. If 𝑥∈𝐾 and 𝑦∈𝐾, then there exists a point 𝑧∉𝜕𝐾 (where 𝜕𝐾 stands for the boundary of 𝐾) such that 𝑑𝑥,𝑦=𝑑(𝑥,𝑧)+𝑑𝑧,𝑦. (15)

Definition 26. Let 𝐾 be a nonempty subset of a cone metric space (𝑋,𝑑) and 𝐹,𝑇∶𝐾→𝑋. The pair (𝐹,𝑇) is said to be weakly commuting if, for each 𝑥,𝑦∈𝐾 such that 𝑥=𝐹𝑦 and 𝑇𝑦∈𝐾, we have 𝑑𝑇𝑥,𝐹𝑇𝑦≤𝑑𝑇𝑦,𝐹𝑦 (16) (see also [12, Hadžić and Gajić]).

Denote ℜ by the collection of all continuous and increasing mappings such that 𝜑∶[0,+∞)→[0,+∞) such that 𝜑⁻¹(0)={0}.

Lemma 27. Let 𝜑∶[0,+∞)→[0,+∞) be an increasing function. Then, 𝜑𝑡_𝑛⟶0𝑖𝑚𝑝𝑙𝑖𝑒𝑠𝑡_𝑛⟶0. (17)

Proof. Suppose that 𝜑(𝑡_𝑛)→0 and 𝑡_𝑛↛0. Then there exists 𝑛₀>0 and 𝛿>0 such that 0<𝛿<𝑡_𝑛, for all 𝑛≥𝑛₀. Since 𝜑 is increasing, we have 0<𝜑(𝛿)≤𝜑𝑡_𝑛 (18) and this is a contradiction since 𝜑(𝑡_𝑛)→0.

Definition 28. Let 𝐾 be a nonempty subset of a cone metric space (𝑋,𝑑) and let 𝐹,𝑇∶𝐾→𝑋 be two mappings. We say that 𝐹 is generalized 𝑇-contractive if 𝜑𝑑𝜑𝑑(𝑑𝜑𝑑,𝜑𝑑.𝐹𝑥,𝐹𝑦≤𝑏(‖𝑇𝑥,𝐹𝑥)‖)+𝜑𝑇𝑦,𝐹𝑦+𝑐min𝑇𝑥,𝐹𝑦𝑇𝑦,𝐹𝑥 (19) For all 𝑥,𝑦∈𝐾, with 𝑥≠𝑦, 𝑏,𝑐≥0, 2𝑏+𝑐<1, and let 𝜑∈ℜ.

Proposition 29. Let (𝑋,𝑑) be a complete Banach-valued metric space, which is metrically onvex. Let 𝐾 be a nonempty closed subset of 𝑋 and 𝜑∈ℜ, and let 𝐹,𝑇∶𝐾→𝑋 be such that 𝐹 is generalized 𝑇-contractive. Suppose also we have(i)𝜕𝐾⊂𝑇𝐾 and 𝐹𝐾⊂𝑇𝐾,(ii)𝑇𝑥∈𝜕𝐾⇒𝐹𝑥∈𝐾,(iii)𝐹 and 𝑇 are weakly commuting,(iv)𝑇 is continuous on 𝐾.Then, there exists a unique common fixed point 𝑧 in 𝐾 such that 𝑧=𝑇𝑧=𝐹𝑧.

Proof. We construct the sequences {𝑥_𝑛} and {𝑦_𝑛} in the following way.
Let 𝑥∈𝐾. Then there exists a point 𝑥₀∈𝐾 such that 𝑥=𝑇𝑥₀ as 𝜕𝐾⊆𝑇𝐾. From 𝑇𝑥₀∈𝜕𝐾 and the implication 𝑇𝑥∈𝜕𝐾⇒𝐹𝑥∈𝐾, we conclude that 𝐹𝑥₀∈𝐾∩𝐹𝐾⊆𝑇𝐾. Now, let 𝑥₁∈𝐾 be such that 𝑦₁=𝑇𝑥₁=𝐹𝑥₀∈𝐾. (20) Let 𝑦₂=𝐹𝑥₁ and assume that 𝑦₂∈𝐾, and then 𝑦₂∈𝐾∩𝐹𝐾⊂𝑇𝐾 (21) which implies that there exists a point 𝑥₂∈𝐾 such that 𝑦₂=𝑇𝑥₂. Suppose 𝑦₂∉𝐾, and then there exists a point 𝑝∈𝜕𝐾 (using Lemma 25), such that 𝑑𝑇𝑥₁,𝑝+𝑑𝑝,𝑦₂=𝑑𝑇𝑥₁,𝑦₂. (22) Since 𝑝∈𝜕𝐾⊆𝑇𝐾, there exists a point 𝑥₂∈𝐾 such that 𝑝=𝑇𝑥₂ and so 𝑑𝑇𝑥₁,𝑇𝑥₂+𝑑𝑇𝑥₂,𝑦₂=𝑑𝑇𝑥₁,𝑦₂. (23) Let 𝑦₃=𝐹𝑥₂. Thus, repeating the forgoing arguments, we obtain two sequences {𝑥_𝑛} and {𝑦_𝑛} such that (i)𝑦_𝑛+1=𝐹𝑥_𝑛,(ii)𝑦_𝑛∈𝐾⇒𝑦_𝑛=𝑇𝑥_𝑛, (iii)𝑦_𝑛∉𝐾⇒𝑇𝑥_𝑛∈𝜕𝐾, and 𝑑𝑇𝑥_𝑛−1,𝑇𝑥_𝑛+𝑑𝑇𝑥_𝑛,𝑦_𝑛=𝑑𝑇𝑥_𝑛−1,𝑦_𝑛. (24)
Denote 𝑃=𝑇𝑥_𝑖∈𝑇𝑥_𝑛∶𝑇𝑥_𝑖=𝑦_𝑖,𝑄=𝑇𝑥_𝑖∈𝑇𝑥_𝑛∶𝑇𝑥_𝑖≠𝑦_𝑖. (25) Obviously, the two consecutive terms of {𝑇𝑥_𝑛} cannot lie in 𝑄. Let us denote 𝑟_𝑛=𝑑(𝑇𝑥_𝑛;𝑇𝑥_𝑛+1). We have the following three cases.
Case 1. If 𝑇𝑥_𝑛,𝑇𝑥_𝑛+1∈𝑃.
Case 2. If 𝑇𝑥_𝑛∈𝑃 and 𝑇𝑥_𝑛+1∈𝑄.
Case 3. If 𝑇𝑥_𝑛∈𝑄  𝑇𝑥_𝑛+1∈𝑃 and so 𝑇𝑥_𝑛−1∈𝑃.
Proving the above cases are similar to [2, Theorem 3.1]. Also we see that for all 𝑛∈ℕ we get 𝜑𝑡_𝑛≤𝑏+𝑐1−𝑏^𝑛𝜑𝑡₀. (26) Letting 𝑛→∞, we have 𝜑(‖𝑡_𝑛‖)→0. Since 𝜑∈ℜ, we have ‖𝑡_𝑛‖→0. So that {𝑇𝑥_𝑛} is a Cauchy sequence and hence it converges to a point 𝑧∈𝐾. Now there exists a subsequence {𝑇𝑥_𝑛𝑘} of {𝑇𝑥_𝑛} which is contained in 𝑃. Without loss of generality, we may denote {𝑇𝑥_𝑛𝑘}={𝑇𝑥_𝑛}. Since 𝑇 is continuous, {𝑇𝑇𝑥_𝑛} converges to 𝑇𝑧. We are going now to show that 𝑇 and 𝐹 have common fixed point (𝑇𝑧=𝐹𝑧). Using the weak commutativity of 𝑇 and 𝐹, we obtain that 𝑇𝑥_𝑛=𝐹𝑥_𝑛−1,𝑇𝑥_𝑛−1∈𝐾, (27) and then 𝑑𝑇𝑇𝑥_𝑛,𝐹𝑇𝑥_𝑛−1≤𝑑𝑇𝑥_𝑛−1,𝐹𝑥_𝑛−1=𝑑𝑇𝑥_𝑛,𝑇𝑥_𝑛−1. (28) This implies that 𝑑𝑇𝑇𝑥_𝑛,𝐹𝑇𝑥_𝑛−1≤𝑑𝑇𝑥_𝑛,𝑇𝑥_𝑛−1. (29) On letting 𝑛→∞, we obtain 𝑑𝑇𝑧,𝐹𝑇𝑥_𝑛−1⟶0. (30) It means that 𝐹𝑇𝑥_𝑛−1→𝑇𝑧.
Now, consider 𝜑𝑑𝐹𝑇𝑥_𝑛−1,𝐹𝑧≤𝑏𝜑𝑑𝑇𝑇𝑥_𝑛−1,𝐹𝑇𝑥_𝑛−1𝑑(𝜑𝑑+𝜑(‖𝑇𝑧,𝐹𝑧)‖)+𝑐min𝑇𝑇𝑥_𝑛−1,𝜑𝑑,𝐹𝑧𝑇𝑧,𝐹𝑇𝑥_𝑛−1. (31) Taking limit on both sides of (31) yields 𝜑(‖𝑑(𝑇𝑧,𝐹𝑧)‖)≤𝑏𝜑(‖𝑑(𝑇𝑧,𝐹𝑧)‖), (32) which is a contradiction, thus giving 𝜑(||𝑑(𝑇𝑧,𝐹𝑧)||)=0 which implies ||𝑑(𝑇𝑧,𝐹𝑧)||=0, so that 𝑑(𝑇𝑧,𝐹𝑧)=0 and hence 𝑇𝑧=𝐹𝑧.
To show that 𝑇𝑧=𝑧, consider 𝜑𝑑𝑇𝑥_𝑛,𝑇𝑧=𝜑𝑑𝐹𝑥_𝑛−1𝜑𝑑,𝐹𝑧≤𝑏𝑇𝑥_𝑛−1,𝐹𝑥_𝑛−1𝑑(𝜑𝑑+𝜑(‖𝑇𝑧,𝐹𝑧)‖)+𝑐min𝑇𝑥_𝑛−1,𝜑𝑑,𝐹𝑧𝑇𝑧,𝐹𝑥_𝑛−1. (33) Taking limit on both sides of (33) yields 𝜑(‖𝑑(𝑇𝑧,𝑧)‖≤𝑐𝜑(‖𝑑(𝑇𝑧,𝑧)‖), (34) which is a contradiction, thereby giving 𝜑(‖𝑑(𝑧,𝑇𝑧)‖)=0 which implies ‖𝑑(𝑧,𝑇𝑧)‖=0, so that 𝑑(𝑧,𝑇𝑧)=0 and hence 𝑧=𝑇𝑧.
Thus, we have shown that 𝑧=𝑇𝑧=𝐹𝑧, so 𝑧 is a common fixed point of 𝐹 and 𝑇. To show that 𝑧 is unique, let 𝑤 be another fixed point of 𝐹 and 𝑇, and then 𝜑𝑑(𝑑(𝑑(𝜑𝑑(𝑑(𝜑𝑑(,𝜑𝑑(𝑑(,(‖𝑤,𝑧)‖)=𝜑(‖𝑇𝑤,𝑇𝑧)‖)=𝜑(‖𝐹𝑤,𝐹𝑧)‖)≤𝑏(‖𝑇𝑧,𝐹𝑧)‖)+𝜑(‖𝑇𝑤,𝐹𝑤)‖)+𝑐min(‖𝑇𝑧,𝐹𝑤)‖)(‖𝑇𝑤,𝐹𝑧)‖)=𝑐𝜑(‖𝑤,𝑧)‖) (35) which is a contradiction, therefore giving 𝜑(‖𝑑(𝑤,𝑧)‖)=0 which implies that ‖𝑑(𝑤,𝑧)‖=0, so that 𝑑(𝑤,𝑧)=0; thus, 𝑤=𝑧.

Theorem 30 (see [2]). Let (𝑋,𝑑_ℍ) be a complete quaternion-valued metric space which is metrically convex and 𝐾 a nonempty closed subset of 𝑋 and 𝜑∈ℜ. Let 𝐹,𝑇∶𝐾→𝑋 be such that 𝐹 is generalized 𝑇-contractive satisfying the following conditions: (i)𝜕𝐾⊂𝑇𝐾 and 𝐹𝐾⊂𝑇𝐾,(ii)𝑇𝑥∈𝜕𝐾⇒𝐹𝑥∈𝐾,(iii)𝐹 and 𝑇 are weakly commuting,(iv)𝑇 is continuous on 𝐾, and then there exists a unique common fixed point 𝑧 in 𝐾 such that 𝑧=𝑇𝑧=𝐹𝑧.

Theorem 31. Theorem 30 is a consequence of Proposition 29.

Proof. Let (𝑋,𝑑_ℍ) be a complete quaternion-valued metric space and 𝐾 a nonempty closed subset of. Then (𝑋,𝑑_ℍ) is a complete cone-valued metric space with cone 𝒫_ℍ={𝑥₀+𝑥₁𝑖+𝑥₂𝑗+𝑥₃𝑘∶𝑥₀≥0,𝑥₁≥0,𝑥₂≥0,𝑥₃≥0}. Further, we have that 𝐹 is generalized 𝑇-contractive in cone metric space if and only if 𝐹 is generalized 𝑇-contractive in quaternion metric space. The rest follows from Proposition 29.

3. Further Comment on Cone Metric Spaces

3.1. By Using a Scalarization Function

In 2010, Du [3] introduced the notion of TVS-valued metric space, also known as TVS-cone metric space (TVS-CMS), as a real generalization of Banach-valued metric space. Let 𝑌 be a locally convex Hausdorff t.v.s. with its zero vector 𝜃, 𝐾 a proper, closed, convex pointed cone in 𝑌 with 𝐾≠0/, 𝑒∈int(𝐾), and ≾_𝐾 a partial ordering with respect to 𝐾.

Definition 32 (see [3]). Let 𝑋 be a nonempty set. Suppose that a vector-valued function 𝑝∶𝑋×𝑋→𝑌 satisfies the following: (𝑐₁):𝜃≾_𝐾𝑝(𝑥,𝑦) for all 𝑥,𝑦∈𝑋 and 𝑝(𝑥,𝑦)=𝜃 if and only if 𝑥=𝑦,(𝑐₂):𝑝(𝑥,𝑦)=𝑝(𝑦,𝑥) for all 𝑥,𝑦∈𝑋,(𝑐₃):𝑝(𝑥,𝑦)≾_𝐾𝑝(𝑥,𝑧)+𝑝(𝑧,𝑦), forall 𝑥,𝑦,𝑧∈𝑋.Then, the function 𝑝 is called TVS-cone metric on 𝑋. Furthermore, the pair (𝑋,𝑝) is called a TVS-cone metric space (in short, TVS-CMS).

On his paper, Du [3] concluded that, for a TVS-valued metric space (𝑋,𝑝), one can define a function 𝑑_𝑝∶𝑋×𝑋→[0,∞) by 𝑑_𝑝=𝜉_𝑒∘𝑝 forming a metric, where 𝜉_𝑒∶𝑌→ℝ, defined by 𝜉_𝑒𝑦=inf𝑟∈ℝ∶𝑦∈𝑟𝑒−𝐾,∀𝑦∈𝑌, (36) is a nonlinear scalarization function (see e.g., [18–20]). In this part, whenever we write ≾, we mean ≾_𝐾.

Proposition 33 (see [3, 4]). Let (𝑋,𝑝) be a TVS-CMS, 𝑥∈𝑋, and {𝑥_𝑛}_𝑛∈ℕ a sequence in 𝑋. Set 𝑑_𝑝=𝜉_𝑒∘𝑝. Then the following statements hold:(i){𝑥_𝑛}_𝑛∈ℕ converges to 𝑥 in TVS-CMS (𝑋,𝑝) if and only if 𝑑_𝑝(𝑥_𝑛,𝑥)→0 as 𝑛→∞,(ii){𝑥_𝑛}_𝑛∈ℕ is a Cauchy sequence in TVS-CMS (𝑋,𝑝) if and only if {𝑥_𝑛}_𝑛∈ℕ is a Cauchy sequence in (𝑋,𝑑_𝑝),(iii)(𝑋,𝑝) is a complete TVS-CMS if and only if (𝑋,𝑑_𝑝) is a complete metric space.

From Proposition 33, the following result was derived easily.

Proposition 34 (see [3]). Let (𝑋,𝑝) is complete TVS-CMS and 𝑇∶𝑋→𝑋 satisfies the contractive condition: 𝑝𝑇𝑥,𝑇𝑦≾𝑘𝑝𝑥,𝑦 (37) for all 𝑥,𝑦∈𝑋 and 0≤𝑘<1. Then, 𝑇 has a unique fixed point in 𝑋. Moreover, for each 𝑥∈𝑋, the iterative sequence {𝑇^𝑛𝑥}^∞_𝑛=1 converges to fixed point.

Proposition 35 (see [3]). The Banach contraction principle and Proposition 34 are equivalent.

Definition 36. Let 𝐾 be a nonempty subset of a metric space (𝑋,𝑑) and let 𝐹,𝑇∶𝐾→𝑋 be two mappings. We say that 𝐹 is generalized 𝑇-contractive of type A if 𝜑𝑑𝜑(𝑑(𝑑𝜑𝑑𝑑.𝐹𝑥,𝐹𝑦≤𝑏𝑇𝑥,𝐹𝑥))+𝜑𝑇𝑦,𝐹𝑦+𝑐min𝑇𝑥,𝐹𝑦,𝜑𝑇𝑦,𝐹𝑥 (38) For all 𝑥,𝑦∈𝐾, with 𝑥≠𝑦, 𝑏,𝑐≥0, 2𝑏+𝑐<1, and let 𝜑∈ℜ.

Proposition 37. Let (𝑋,𝑑) be a complete metric space, let 𝐾 be a nonempty closed subset of 𝑋 and 𝜑∈ℜ, and let 𝐹,𝑇∶𝐾→𝑋 be such that 𝐹 is generalized 𝑇-contractive of type A. If the following are satisfied:(i)𝜕𝐾⊂𝑇𝐾 and 𝐹𝐾⊂𝑇𝐾,(ii)𝑇𝑥∈𝜕𝐾⇒𝐹𝑥∈𝐾,(iii)𝐹 and 𝑇 are weakly commuting,(iv)𝑇 is continuous on 𝐾,then there exists a unique common fixed point 𝑧 in 𝐾 such that 𝑧=𝑇𝑧=𝐹𝑧.

We skip the proof of Proposition 37 since it can be derived by the mimic of Proposition 29. On the other hand, regarding Proposition 35, we can conclude that Proposition 37 implies Proposition 29.

3.2. By Using a Metric-Type Space

Definition 38 (see, e.g., [5]). Let 𝑋 be a set. Let 𝐷∶𝑋×𝑋→[0,∞) be a function which satisfies(1)𝐷(𝑥,𝑦)=0 if and only if 𝑥=𝑦,(2)𝐷(𝑥,𝑦)=𝐷(𝑦,𝑥), for any 𝑥,𝑦∈𝑋,(3)𝐷(𝑥,𝑦)≤𝐾(𝐷(𝑥,𝑧₁)+𝐷(𝑧₂,𝑧₃)+⋯+𝐷(𝑧_𝑛−1,𝑧_𝑛)+𝐷(𝑧_𝑛,𝑦)) for any 𝑥,𝑦,𝑧₁,…,𝑧_𝑛∈𝑋,for some constant 𝐾>0. The pair (𝑋,𝐷) is called a metric-type space.

Proposition 39. Let (𝑋,𝑑) be a metric cone over the Banach space 𝐸 with the cone 𝑃 which is normal with the normal constant 𝐾. The mapping 𝐷∶𝑋×𝑋→[0,∞) defined by 𝐷(𝑥,𝑦)=‖𝑑(𝑥,𝑦)‖ is a function which satisfies(1)𝐷(𝑥,𝑦)=0 if and only if 𝑥=𝑦,(2)𝐷(𝑥,𝑦)=𝐷(𝑦,𝑥), for any 𝑥,𝑦∈𝑋,(3)𝐷(𝑥,𝑦)≤𝐾(𝐷(𝑥,𝑧₁)+𝐷(𝑧₂,𝑧₃)+⋯+𝐷(𝑧_𝑛−1,𝑧_𝑛)+𝐷(𝑧_𝑛,𝑦)) for any 𝑥,𝑦,𝑧₁,…,𝑧_𝑛∈𝑋.

Remark 40. In Definition 38, (3), the term 𝑧_𝑖 needs not to be distinct. Hence, metric type space turns into 𝑏-metric space when we deal with cone metric space (see, e.g., [21, 22]).

Remark 41. Furthermore, by Lemma 20, 𝒫_ℍ is a normal cone. Hence, some resuls of [2] and Theorem 30, are equivalent to the corresponding results in the context of metric-type space (see also [5]).

Conflict of Interests

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

Authors’ Contribution

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

Acknowledgments

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.