# Common Fixed Point Theorems on Tricomplex Valued Metric Space

Authors: Gunaseelan Mani; Arul Joseph Gnanaprakasam; Absar Ul Haq; Imran Abbas Baloch; Fahd Jarad
Journal: Mathematical Problems in Engineering
 (2022)

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## Abstract

In this paper, we introduce the notion of tricomplex valued metric space and prove some common fixed point theorems. The presented results generalize and expand some of the literature well-known results. We also explore some applications of our key results.


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## Body

## 1. Introduction

Fixed point theory plays an important role in applications of many branches of mathematics. There has been a number of generalizations of the usual notion of a metric space (see [1–8] and the references therein). Serge [9] made a pioneering attempt in the development of special algebras. He conceptualized commutative generalization of complex numbers as bicomplex numbers, tricomplex numbers, and so on as elements of an infinite set of algebra. Subsequently during the 1930s, other researchers also contributed in this area [10–12]. However, unfortunately the next fifty years failed to witness any advancement in this field. Afterward, Price [13] developed the bicomplex algebra and function theory. Recently renewed interest in this subject finds some significant applications in different fields of mathematical sciences as well as other branches of science and technology. Also, one can see the attempts in [14]. An impressive body of work has been developed by a number of researchers. Among them, an important work on elementary functions of bicomplex numbers has been done by Luna-Elizarrarás et al. [15]. Choi et al. [16] proved some common fixed point theorems in connection with two weakly compatible mappings in bicomplex valued metric spaces. Jebril et al. [17] proved some common fixed point theorems under rational contractions for a pair of mappings in bicomplex valued metric spaces. In 2021, Beg et al. [18] proved the following fixed point theorems on bicomplex valued metric spaces.Theorem 1.
LetW,φ be a complete bicomplex valued metric space with degenerated 1+φϖ,ϑ and 1+φϖ,ϑ≠0 for all ϖ,ϑ∈W and S,T:W⟶W such that(1)φSϖ,Tϑ≺¯i2λφϖ,ϑ+μφϖ,Sϖφϑ,Tϑ1+φϖ,ϑ,for all ϖ,ϑ∈W, where λandμ are nonnegative real numbers with λ+2μ<1. Then, SandT have a unique common fixed point.In this paper, inspired by Theorem1, we prove some common fixed point theorems on tricomplex metric space with applications.


## 2. Preliminaries

Throughout this paper, we denote the set of real, complex, bicomplex, and tricomplex numbers, respectively, asℂ0, ℂ1, ℂ2, and ℂ3. Price [13] defined the bicomplex number as follows:(2)σ=a1+a2i1+a3i2+a4i1i2,where a1,a2,a3,a4∈ℂ0, and independent units i1,i2 are such that i12=i22=−1 and i1i2=i2i1; we denote the set of bicomplex numbers ℂ2 as follows:(3)ℂ2=σ:σ=a1+a2i1+a3i2+a4i1i2,a1,a2,a3,a4∈ℂ0,i.e.,(4)ℂ2=σ:σ=κ1+i2κ2,κ1,κ2∈ℂ1,where κ1=a1+a2i1∈ℂ1 and κ2=a3+a4i1∈ℂ1. Price [13] defined the tricomplex number as follows:(5)ξ=a1+a2i1+a3i2+a4j1+a5i3+a6j2+a7j3+a8i4,where a1,a2,a3,a4,a5,a6,a7,a8∈ℂ0 and independent units i1,i2,i3,i4,j1,j2,andj3 are such that i12=i42=−1,i4=i1j3=i1i2i3,j2=i1i3=i3i1,j22=1,j1=i1i2=i2i1, and j12=1; we denote the set of tricomplex numbers ℂ3 as follows:(6)ℂ3=ξ:ξ=a1+a2i1+a3i2+a4j1+a5i3+a6j2+a7j3+a8i4,a1,a2,a3,a4,a5,a6,a7,a8∈ℂ0,i.e.,(7)ℂ3=ξ:ξ=σ1+i3σ2,σ1,σ2∈ℂ2,where σ1=κ1+κ2i2∈ℂ2 and σ2=κ3+κ4i2∈ℂ2. If ξ=σ1+i3σ2 and η=w1+i3w2 be any two tricomplex numbers, then the sum is ξ±η=σ1+i3σ2±w1+i3w2=σ1±w1+i3σ2±w2 and the product is ξ.η=σ1+i3σ2w1+i3w2=σ1w1−σ2w2+i3σ1w2+σ2w1.There are four idempotent elements inℂ3; they are 0,1,e1=1+j3/2,e2=1−j3/2 out of which e1 and e2 are nontrivial such that e1+e2=1 and e1e2=0. Every tricomplex number σ1+i3σ2 can be uniquely be expressed as the combination of e1 and e2, namely,(8)ξ=σ1+i3σ2=σ1−i2σ2e1+σ1+i2σ2e2.This representation ofξ is known as the idempotent representation of tricomplex number, and the complex coefficients ξ1=σ1−i2σ2 and ξ2=σ1+i2σ2 are known as idempotent components of the bicomplex number ξ.An elementξ=σ1+i3σ2∈ℂ3 is said to be invertible if there exists another element η in ℂ3 such that ξη=1 and η is said to be inverse (multiplicative) of ξ. Consequently, ξ is said to be the inverse (multiplicative) of η. An element which has an inverse in ℂ3 is said to be the nonsingular element of ℂ3 and an element which does not have an inverse in ℂ3 is said to be the singular element of ℂ3.An elementξ=σ1+i3σ2∈ℂ3 is nonsingular if and only if σ12+σ22≠0 and singular if and only if σ12+σ22=0.The inverse ofξ is defined as(9)ξ−1=η=σ1−i3σ2σ12+σ22.The norm. of ℂ3 is a positive real valued function and .:ℂ3⟶ℂ0+ is defined by(10)ξ=σ1+i3σ2=σ12+σ221/2=σ1−i2σ22+σ1+i2σ2221/2=a12+a22+a12+a32+a42+a52+a62+a72+a821/2,where ξ=a1+a2i1+a3i2+a4j1+a5i3+a6j2+a7j3+a8i4=σ1+i3σ2∈ℂ3.The linear spaceℂ3 with respect to defined norm is a norm linear space; also ℂ3 is complete; therefore, ℂ3 is the Banach space. If ξ,η∈ℂ3, then ξη≤2ξη holds instead of ξη≤ξη; therefore, ℂ3 is not the Banach algebra. The partial order relation ≺¯i3 on ℂ3 is defined as follows: let ℂ3 be the set of tricomplex numbers and ξ=σ1+i3σ2 and η=w1+i3w2∈ℂ3, then ξ≺¯i3η if and only if σ1≺¯i2w1 and σ2≺¯i2w2, i.e., ξ≺¯i3η if one of the following conditions is fulfilled:(a)
σ1=w1, σ2=w2(b)
σ1≺i2w1, σ2=w2(c)
σ1=w1, σ2≺i2w2(d)
σ1≺i2w1, σ2≺i2w2In particular, we can writeξ⋦i3η if ξ≺¯i3η and ξ≠η, i.e., one of (b), (c), and (d) is fulfilled and we will write ξ≺i3η if only (d) is fulfilled.For any two tricomplex numbersξ,η∈ℂ3, we can verify the following:(1)
ξ≺¯i3ηiffξ≤η(2)
ξ+η≤ξ+η(3)
aξ=aξ, where a is a nonnegative real number(4)
ξη≤2ξη and the equality holds only when at least one of ξ and η is nonsingular(5)
ξ−1=ξ−1 if ξ is a nonsingular(6)
ξ/η=ξ/η if η is a nonsingularNow, let us recall some basic concepts and notations, which will be used in the sequel.Definition 1.
LetW be a nonempty set and φ:W×W⟶ℂ3 such that(A1)
0≺¯i3φϖ,ϑ, for all ϖ,ϑ∈W and φϖ,ϑ=0 if and only if ϖ=ϑ(A2)
φϖ,ϑ=φϑ,ϖ for all ϖ,ϑ∈W(A3)
φϖ,ϑ≺¯i3φϖ,σ+φσ,ϑ for all ϖ,ϑ,σ∈W
Then,φ is called the tricomplex valued metric on W and W,φ is called the tricomplex valued metric space.Example 1.
LetW=0,1 and φ:W×W⟶ℂ3 be defined by φϖ,ϑ=ϖ−ϑi2i3. Then, W,φ is a tricomplex valued metric space.Definition 2.
LetW,φ be a tricomplex valued metric space. A sequence ϖk in W is said to be a convergent and converges to ϖ∈W if for every 0≺i3ϵ∈ℂ3, there exists k0∈ℕ such that φϖk,ϖ≺i3ϵ, for all k≥k0, and it is denoted by limk⟶∞ϖk=ϖ.Lemma 1.
LetW,φ be a tricomplex valued metric space. A sequence ϖk∈W converges to ϖ∈W iff limk⟶∞φϖk,ϖ=0.Proof.
Letϖk be a convergent sequence and converges to a point ϖ, and let ϵ>0 be any real number. Suppose(11)r=ε8+i1ε8+i2ε8+j1ε8+i3ε8+j2ε8+j3ε8+i4ε8.
Then,0≺i3r∈ℂ3, and for this r, there exists k0∈ℕ such that φϖk,ϖ≺i3r for all k≥k0. Therefore,(12)φϖk,ϖ<r=ε,∀k≥k0.
Hence,limk⟶∞φϖk,ϖ=0.
Conversely, letlimk⟶∞φϖk,ϖ=0. Then, for each 0≺i3r∈ℂ3, there exists a real number ϵ>0 such that for all ξ∈ℂ3,(13)ξ<ε⇒ξ≺i3r.
Then, for thisϵ>0, there exists k0∈ℕ such that(14)φϖk,ϖ<ϵ,∀k≥k0.
Therefore,(15)φϖk,ϖ≺i3r,∀k≥k0.
Hence,ϖk converges to a point ϖ.Definition 3.
LetW,φ be a tricomplex valued metric space. A sequence ϖk in W is said to be a Cauchy sequence in W,φ if for any 0≺i3ϵ∈ℂ3, there exists h∈ℕ such that φϖk,ϖk+m≺i3ε for all k,m∈ℕ and k,m≥h.Definition 4.
LetW,φ be a tricomplex valued metric space. Let ϖk be any sequence in W. Then, if every Cauchy sequence in W is convergent in W, then W,φ is said to be a complete tricomplex valued metric space.Lemma 2.
LetW,φ be a tricomplex valued metric space and ϖk be a sequence in W. Then, ϖk is a Cauchy sequence in W iff limk⟶∞φϖk,ϖk+m=0.Proof.
Letϖk is a Cauchy sequence in W. Let ϵ>0 be any real number. Suppose(16)r=ε8+i1ε8+i2ε8+j1ε8+i3ε8+j2ε8+j3ε8+i4ε8.
Then,0≺i3r∈ℂ2+, and for this r, there exists k0∈ℕ such that φϖk,ϖk+m≺i3r, for all k>k0. Therefore,(17)φϖk,ϖk+m<r=ε,for allk>k0.
And, this implies that(18)limk⟶∞φϖk,ϖk+m=0.
Conversely, letlimk⟶∞φϖk,ϖk+m=0. Then, for each 0≺i3r∈ℂ3+, there exists a real number ϵ>0 such that for all ξ∈ℂ2+,(19)ξ<ε⇒ξ≺i3r.
Then, for thisϵ>0, there exists a natural number k0∈ℕ such that(20)φϖk,ϖk+m<ε,∀k>k0.
Therefore,(21)φϖk,ϖk+m≺i3r,∀k>k0.
Hence,ϖk is a Cauchy sequence.Definition 5.
LetS and T be self mappings of nonvoid set W. A point ϖ∈W is called a common fixed point of S and T if ϖ=Sϖ=Tϖ.


## 3. Main Result

In this section, we prove common fixed point theorem in a tricomplex valued metric space using rational type contraction condition.Theorem 2.
IfS and T are self mapping defined on a complete tricomplex valued metric space W,φ such that(22)φSϖ,Tϑ≺¯i3λφϖ,ϑ+μφϖ,Sϖφϑ,Tϑ+γφϑ,Sϖφϖ,Tϑ1+φϖ,ϑ,for all ϖ,ϑ∈W where λ,μ,γ are nonnegative reals with λ+2μ+2γ<1, then S and T have a unique common fixed point.Proof.
Letϖ0 be an arbitrary point in W and define ϖ2l+1=Sϖ2l, ϖ2l+2=Tϖ2l+1,k=0,1,2,⋯. Then,(23)φϖ2l+1,ϖ2l+2=φSϖ2l,Tϖ2l+1≺¯i3λφϖ2l,ϖ2l+1+μφϖ2l,Sϖ2lφϖ2l+1,Tϖ2l+1+γφϖ2l,Tϖ2l+1φϖ2l+1,Sϖ2l1+φϖ2l,ϖ2l+1.
Sinceϖ2l+1=Sϖ2l implies φϖ2l+1,Sϖ2l=0, therefore,(24)φϖ2k+1,ϖ2k+2≺¯i3λφϖ2k,ϖ2k+1+μφϖ2k,ϖ2k+1φϖ2k+1,ϖ2k+21+φϖ2k,ϖ2k+1,which implies that(25)φϖ2k+1,ϖ2k+2≤λφϖ2k,ϖ2k+1+2μφϖ2k,ϖ2k+1φϖ2k+1,ϖ2k+21+φϖ2k,ϖ2k+1.
Since1+φϖ2l,ϖ2l+1≥φϖ2l,ϖ2l+1, therefore,(26)φϖ2k+1,ϖ2k+2≤λφϖ2k,ϖ2k+1+2μφϖ2k+1,ϖ2k+2,so that(27)φϖ2k+1,ϖ2k+2≤λ1−2μφϖ2k,ϖ2k+1.
Also,(28)φϖ2k+2,ϖ2k+3=φTϖ2k+1,∫ϖ2k+2=φSϖ2k+2,Tϖ2k+1≺¯i3λφϖ2k+2,ϖ2k+1+μφϖ2k+2,Sϖ2k+2φϖ2k+1,Tϖ2k+1+γφϖ2k+1,Sϖ2k+2φϖ2k+2,Tϖ2k+11+φϖ2k+2,ϖ2k+1.
Sinceϖ2l+2=Tϖ2l+1 implies φϖ2k+2,Tϖ2k+1=0, therefore,(29)φϖ2k+2,ϖ2k+3≺¯i3λφϖ2k+2,ϖ2k+1+μφϖ2k+2,Sϖ2k+2φϖ2k+1,Tϖ2k+11+φϖ2k+2,ϖ2k+1,which implies that(30)φϖ2k+2,ϖ2k+3≤λφϖ2k+2,ϖ2k+1+2μφϖ2k+2,ϖ2k+3φϖ2k+1,ϖ2k+21+φϖ2k+2,ϖ2k+1.
As1+φϖ2k+2,ϖ2k+1≥φϖ2k+2,ϖ2k+1, therefore,(31)φϖ2k+2,ϖ2k+3≤λ1−2μφϖ2k+1,ϖ2k+2.
Puttingh=λ/1−2μ, we have (for all k)(32)φϖk,ϖk+1≤hφϖk−1,ϖk≤h2φϖk−2,ϖk−1≤…≤hkφϖ0,ϖ1.
Therefore, for anym>k, we have(33)φϖk,ϖm≤φϖk,ϖk+1+φϖk+1,ϖk+2+φϖk+2,ϖk+3⋯+φϖm−1,ϖm≤hk+hk+1+hk+2+⋯+hm−1φϖ0,ϖ1≤hk1−hφϖ0,ϖ1,which implies that(34)φϖk,ϖm≤hk1−hφϖ0,ϖ1⟶0ask⟶∞.
In view of Lemma2, the sequence ϖk is Cauchy. Since W is complete, there exists some l∈W such that ϖk⟶l as k⟶∞. On the contrary, let l≠Sk so that 0≺i3σ=φl,Sk, then(35)σ=φl,Sk≺¯i3φl,Tϖ2k+1+φTϖ2k+1,Sk≺¯i3φl,ϖ2k+2+λφl,ϖ2k+1+μφl,Skφϖ2k+1,Tϖ2k+1+γφϖ2k+1,Skφl,Tϖ2k+11+φl,ϖ2k+1≺¯i3φl,ϖ2k+2+λφl,ϖ2k+1+μσφϖ2k+1,Tϖ2k+1+γφϖ2k+1,Skφl,Tϖ2k+11+φl,ϖ2k+1.
Also, for allk, we have(36)φl,Sk≤φl,ϖ2k+2+λφl,ϖ2k+1+μσφϖ2k+1,ϖ2k+2+2γφϖ2k+1,Skφl,ϖ2k+21+φl,ϖ2k+1.
Ask⟶∞, we get(37)φl,Sk=0.
Therefore,l=Sk. Similarly, we can derive that l=Tl. Let l∗ (in W) be another common fixed point of S and T, i.e., l∗=Sl∗=Tl∗. Then,(38)φl,l∗=φSk,Tl∗≺¯i3λφl,l∗+μφl,Skφl∗,Tl∗+γφl∗,Skφl,Tl∗1+φl,l∗=λφl,l∗+γφl∗,lφl,l∗1+φl,l∗,which implies that(39)φl,l∗≤λφl,l∗+2γφl∗,lφl,l∗1+φl,l∗.
Since1+φl,l∗>φl,l∗, therefore,(40)φl,l∗≤λ+2γφl,l∗.
Therefore,l=l∗ (as λ+2γ<1).Remark 1.
(see [13]). We have(41)ℂ0⊆ℂ1⊆ℂ2⊆ℂ3.Example 2.
ConsideringW=ℝ, define a mapping φ:W×W⟶ℂ3 by φϖ,ϑ=i2i3ϖ−ϑ, for all ϖ,ϑ∈W, where . is the usual real modulus. Then, W,φ is a complete tricomplex valued metric space. Every real number is a tricomplex number but every tricomplex number is not necessarily a real number. Therefore, W,φ is not a metric space. Now, we consider a self mapping S,T:W⟶W defined by(42)Sϖ=ϖ4,Tϑ=ϑ2,for all ϖ,ϑ∈W. Then,(43)φSϖ,Tϑ=φϖ4,ϑ2=i2i3ϖ4−ϑ2=i2i32ϖ2−ϑ≺¯i3i2i32ϖ−ϑ=12φϖ,ϑ.
Thus, all the hypothesis of Theorem2 are fulfilled with λ=1/2<1 and μ=γ=0. Hence, S and T have a unique common fixed point.Example 3.
ConsideringW=ℂ3, define a mapping φ:W×W⟶ℂ3 by φϖ,ϑ=1+i1i3ϖ−ϑ, for all ϖ,ϑ∈W. Then, W,φ is a complete tricomplex valued metric space. Every real number is a tricomplex number but every tricomplex number is not necessarily a real number. Therefore, W,φ is not a metric space. Now, we consider a self mappings S,T:W⟶W defined by(44)Sϖ=i1i3ϖ4,Tϑ=i1i3ϑ2,for all ϖ,ϑ∈W. Then,(45)φSϖ,Tϑ=φi1i3ϖ4,i1i3ϑ2=1+i1i3i1i3ϖ4−i1i3ϑ2=1+i1i32ϖ2−ϑ≺¯i31+i1i32ϖ−ϑ=12φϖ,ϑ.
Every real number is a tricomplex number but every tricomplex number is not necessarily a real number. Therefore, we cannot find common fixed point for such mappings on metric space. Thus, all the hypothesis of Theorem2 are fulfilled with λ=1/2<1 and μ=γ=0. Hence, S and T have a unique common fixed point.
By settingS=T in Theorem 2, one deduces the following.Corollary 1.
IfT:W⟶W is a self mapping defined on a complete tricomplex valued metric space W,φ such that(46)φTϖ,Tϑ≺¯i3λφϖ,ϑ+μφϖ,Tϖφϑ,Tϑ+γφϑ,Tϖφϖ,Tϑ1+φϖ,ϑ,for all ϖ,ϑ∈W, where λ,μ,γ are nonnegative reals with λ+2μ+2γ<1, then T has a unique fixed point.Theorem 3.
LetW,φ be a complete tricomplex valued metric space and the mappings S,T:W⟶W such that(47)φSϖ,Tϑ≺¯i3λφϖ,ϑ+μφϖ,Sϖφϑ,Tϑ+φϑ,Sϖφϖ,TϑφSϖ,ϖ+φTϑ,ϑ,+γφϖ,Sϖφϖ,Tϑ+φϑ,Sϖφϑ,TϑφSϖ,ϑ+φTϑ,ϖ,ifD≠0,D1≠0,0,ifD=0orD1=0,.for all ϖ,ϑ∈W, where D=φSϖ,ϖ+φTϑ,ϑ and D1=φSϖ,ϑ+φTϑ,ϖ and λ,μ,γ are nonnegative reals with λ+2μ+γ<1. Then, S, T have a unique common fixed point.Proof.
Letϖ0 be an arbitrary point in W. Define ϖ2l+1=Sϖ2l and ϖ2l+2=Tϖ2l+1, k=0,1,2,… Now, we distinguish two cases. First, if (for k=0,1,2,…) φSϖ2k,ϖ2k+φTϖ2k+1,ϖ2k+1≠0 and φSϖ2k,ϖ2k+1+φTϖ2k+1,ϖ2k≠0, then(48)φϖ2k+1,ϖ2k+2=φSϖ2k,Tϖ2k+1≺¯i3λφϖ2k,ϖ2k+1+μφϖ2k,Sϖ2kφϖ2k+1,Tϖ2k+1+φϖ2k+1,Sϖ2kφϖ2k,Tϖ2k+1φSϖ2k,ϖ2k+φTϖ2k+1,ϖ2k+1+γφϖ2k,Sϖ2kφϖ2k,Tϖ2k+1+φϖ2k+1,Sϖ2kφϖ2k+1,Tϖ2k+1φSϖ2k,ϖ2k+1+φTϖ2k+1,ϖ2k.
Sinceϖ2l+1=Sϖ2l and ϖ2l+2=Tϖ2l+1, therefore,(49)φϖ2k+1,ϖ2k+2≺¯i3λφϖ2k,ϖ2k+1+μφϖ2k,ϖ2k+1φϖ2k+1,ϖ2k+2+φϖ2k+1,ϖ2k+1φϖ2k,ϖ2k+2φϖ2k+1,ϖ2k+φϖ2k+2,ϖ2k+1+γφϖ2k,ϖ2k+1φϖ2k,ϖ2k+2+φϖ2k+1,ϖ2k+1φϖ2k+1,ϖ2k+2φϖ2k+1,ϖ2k+1+φϖ2k+2,ϖ2k,(50)orφϖ2k+1,ϖ2k+2≺¯i3λφϖ2k,ϖ2k+1+μφϖ2k,ϖ2k+1φϖ2k+1,ϖ2k+2φϖ2k+1,ϖ2k+φϖ2k+2,ϖ2k+1+γφϖ2k,ϖ2k+1φϖ2k,ϖ2k+2φϖ2k+2,ϖ2k,which implies that(51)φϖ2k+1,ϖ2k+2≤λφϖ2k,ϖ2k+1+2μφϖ2k,ϖ2k+1φϖ2k+1,ϖ2k+2φϖ2k+1,ϖ2k+φϖ2k+2,ϖ2k+1+γφϖ2k,ϖ2k+1.
Sinceφϖ2k+1,ϖ2k+φϖ2k+2,ϖ2k+1>φϖ2k+1,ϖ2k, therefore,(52)φϖ2k+1,ϖ2k+2≤λφϖ2k,ϖ2k+1+2μφϖ2k+1,ϖ2k+2+γφϖ2k,ϖ2k+1,so that(53)φϖ2k+1,ϖ2k+2≤λ+γ1−2μφϖ2k,ϖ2k+1.
Also,(54)φϖ2k+2,ϖ2k+3=φSϖ2k+2,Tϖ2k+1≺¯i3λφϖ2k+2,ϖ2k+1+μφϖ2k+2,Sϖ2k+2φϖ2k+1,Tϖ2k+1+φϖ2k+1,Sϖ2k+2φϖ2k+2,Tϖ2k+1φSϖ2k+2,ϖ2k+2+φTϖ2k+1,ϖ2k+1+γφϖ2k+2,Sϖ2k+2φϖ2k+2,Tϖ2k+1+φϖ2k+1,Sϖ2k+2φϖ2k+1,Tϖ2k+1φSϖ2k+2,ϖ2k+1+φTϖ2k+1,ϖ2k+2.
Sinceϖ2k+3=Sϖ2k+2 and ϖ2k+2=Tϖ2k+1, we get(55)φϖ2k+2,ϖ2k+3≺¯i3λφϖ2k+2,ϖ2k+1+μφϖ2k+2,ϖ2k+3φϖ2k+1,ϖ2k+2+φϖ2k+1,ϖ2k+3φϖ2k+2,ϖ2k+2φϖ2k+3,ϖ2k+2+φϖ2k+2,ϖ2k+1+γφϖ2k+2,ϖ2k+3φϖ2k+2,ϖ2k+2+φϖ2k+1,ϖ2k+3φϖ2k+1,ϖ2k+2φϖ2k+3,ϖ2k+1+φϖ2k+2,ϖ2k+2,(56)orφϖ2k+2,ϖ2k+3≺¯i3λφϖ2k+2,ϖ2k+1+μφϖ2k+2,ϖ2k+3φϖ2k+1,ϖ2k+2φϖ2k+3,ϖ2k+2+φϖ2k+2,ϖ2k+1+γφϖ2k+1,ϖ2k+3φϖ2k+1,ϖ2k+2φϖ2k+1,ϖ2k+3,which implies that(57)φϖ2k+2,ϖ2k+3≤λφϖ2k+2,ϖ2k+1+2μφϖ2k+2,ϖ2k+3φϖ2k+1,ϖ2k+2φϖ2k+3,ϖ2k+2+φϖ2k+2,ϖ2k+1+γφϖ2k+1,ϖ2k+2.
Sinceφϖ2k+3,ϖ2k+2+φϖ2k+2,ϖ2k+1>φϖ2k+1,ϖ2k+2, therefore,(58)φϖ2k+2,ϖ2k+3≤λφϖ2k+2,ϖ2k+1+2μφϖ2k+2,ϖ2k+3φϖ2k+1,ϖ2k+2φϖ2k+2,ϖ2k+1+γφϖ2k+1,ϖ2k+2,(59)orφϖ2k+2,ϖ2k+3≤λ+γ1−2μφϖ2k+1,ϖ2k+2.
Now, withh=λ+γ/1−2μ, we have (for all k)(60)φϖk,ϖk+1≤hφϖk−1,ϖk≤⋯≤hkφϖ0,ϖ1.
So, for anym>k, we have(61)φϖk,ϖm≤φϖk,ϖk+1+φϖk+1,ϖk+2+⋯+φϖm−1,ϖm≤hk+hk+1+⋯+hm−1φϖ0,ϖ1≤hk1−hφϖ0,ϖ1,and henceforth(62)φϖk,ϖm≤hk1−hφϖ0,ϖ1⟶0ask⟶∞.
On using Lemma2, we conclude that ϖk is a Cauchy sequence. Since W is a complete, then there exists l∈W such that ϖk⟶l as k⟶∞. Now, we assert that l=Sk; otherwise, 0≺i3σ=φl,Sk and we have(63)σ=φl,Sk≺¯i3φl,Tϖ2k+1+φTϖ2k+1,Sk≺¯i3φl,ϖ2k+2+λφl,ϖ2k+1+μφl,Skφϖ2k+1,Tϖ2k+1+φϖ2k+1,Skφl,Tϖ2k+1φSk,l+φTϖ2k+1,ϖ2k+1+γφl,Skφl,Tϖ2k+1+φϖ2k+1,Skφϖ2k+1,Tϖ2k+1φSk,ϖ2k+1+φϖ2k+2,l,which implies that(64)σ=φl,Sk≤φl,ϖ2k+2+λφl,ϖ2k+1+μσφϖ2k+1,ϖ2k+2+2φϖ2k+1,Skφl,ϖ2k+2φSk,l+φϖ2k+2,ϖ2k+1+γσφl,ϖ2k+2+2φϖ2k+1,Skφϖ2k+1,ϖ2k+2φSk,ϖ2k+1+φϖ2k+2,l.
Therefore,σ=φl,Sk=0, i.e., l=Sk. Similarly, we can derive that l=Tl. Assume that l∗ in W is an another common fixed point of S and T. Then,(65)Sl∗=Tl∗=l∗.
SinceD=φSk,l+φTl∗,l∗=0, then(66)φl,l∗=φSk,Tl∗=0.
Hence,l=l∗.
Second, we consider that the caseφSϖ2k,ϖ2k+φTϖ2k+1,ϖ2k+1×φSϖ2k,ϖ2k+1+φTϖ2k+1,ϖ2k=0 (for any k) implies φSϖ2k,Tϖ2k+1=0. Now, if φSϖ2k,ϖ2k+φTϖ2k+1,ϖ2k+1=0, then ϖ2k=Sϖ2k=ϖ2k+1=Tϖ2k=ϖ2k+2. Thus, we have ϖ2k+1=Sϖ2k=ϖ2k, so there exist k1 and m1 such that k1=Sm1=m1. Similarly, we can derive that there exist k2 and m2 such that k2=Tm2=m2. As φSm1,m1+φTm2,m2=0 (due to definition), it implies φSm1,Tm2=0, so that k1=Sm1=Tm2=k2 which in turn yields that k1=Sm1=Sk1. Similarly, one can also have k2=Tk2. As k1=k2, it implies Sk1=Tk1=k1; therefore, k1=k2, is a common fixed point of S and T. Assume that k1∗ in W is an another common fixed point of S and T. Then,(67)Sk1∗=Tk1∗=k1∗.
SinceD=φSk1,k1+φTk1∗,k1∗=0, then(68)φk1,k1∗=φSk1,Tk1∗=0.
This implies thatk1∗=k1.
IfφSϖ2k,ϖ2k+1+φTϖ2k+1,ϖ2k=0 implies that φSϖ2k,Tϖ2k+1=0, then also proof can be completed on the preceding lines.Example 4.
ConsideringW=0,∞, define a mapping φ:W×W⟶ℂ3 by φϖ,ϑ=1+i3ϖ−ϑ, for all ϖ,ϑ∈W, where . is the usual real modulus. Then, W,φ is a complete tricomplex valued metric space. Now, we consider a self mapping T:W⟶W defined by(69)Tϖ=ϖ6,∀ϖ∈W.
Letλ=1/6, μ=0 and λ=0, then λ+2μ+2γ=1/6<1. Let ϖ,ϑ∈W, then(70)φTϖ,Tϑ=1+i36ϖ−ϑ≺¯i31+i32ϖ−ϑ.
Therefore,(71)φTϖ,Tϑ≺¯i3λφϖ,ϑ+μφϖ,Tϖφϑ,Tϑ+γφϑ,Tϖφϖ,Tϑ1+φϖ,ϑ.
Hence, the conditions of Corollary1 are fulfilled. Therefore, 0 is the unique fixed point of T.Example 5.
LetW=ℬ0,q, q>1, for all ϖ,ϑ∈W. Define φ:W×W⟶ℂ3 by(72)φϖt,ϑt=i32π∫Εϖtt−∫Εϑtt,
whereE is a closed path in W containing a zero. Then, then we prove that (W, φ) is a complete tricomplex valued metric space. We have(73)φϖt,ϑt=i32π∫Εϖtt−∫Εϑtt,=i32π∫Εϖtt−∫Εσtt+∫Εσtt−∫Εϑtt≺¯i3i32π∫Εϖtt−∫Εσtt+i32π∫Εσtt−∫Εϑtt=φϖt,σt+φσt,ϑt.
Now, we define the mappingS,T:W⟶W by(74)Sϖt=t,Tϑt=et−1.
Using the Cauchy integral formula when the mappingsS and T are analytic, we get(75)φSϖt,Tϑt=i32π∫Ett−∫Eet−1t=0,φϖt,ϑt=i32π∫Eϖtt−∫Eϑtt,φϖt,Sϖt=i32π∫Eϖtt−∫Ett=i32π∫Eϖtt,φϑt,Tϑt=i32π∫Eϑtt−∫Eet−1tv=i32π∫Eϑtt,φϖt,Sϑt=i32π∫Eϖtt−∫Ett=i32π∫Eϖtt,φϖt,Tϑt=i32π∫Eϖtt−∫Eet−1t=i32π∫Eϖtt.
Clearly,(76)φSϖ,Tϑ≺¯i3λφϖ,ϑ+μφϖ,Sϖφϑ,Tϑ+γφϑ,Sϖφϖ,Tϑ1+φϖ,ϑ,for all ϖ,ϑ∈W, where λ,μ,γ are nonnegative reals with λ+2μ+2γ<1. Therefore, all the conditions of Theorem 2 are fulfilled, then the mappings S and T have a unique common fixed point in W.Definition 6.
(see [19]). Two self mappings T and S of a metric space W,φ are said to be commuting iff TSϖ=STϖ for all ϖ in W.Example 6.
ConsideringW=ℝ, define a mapping φ:W×W⟶ℂ3 by φϖ,ϑ=1+i3ϖ−ϑ, for all ϖ,ϑ∈W, where . is the usual real modulus. Then, W,φ is a complete tricomplex valued metric space. Now, we consider a self mappings T,S:W⟶W defined by(77)Tϖ=7ϖ,Sϑ=11ϑ.
Then,(78)TSϖ=T11ϖ=117ϖ=S7ϖ=STϖ.
So,T and S are commute. Now,(79)φTϖ,Tϑ=φ7ϖ,7ϑ=71+i3ϖ−ϑ,(80)φSϖ,Sϑ=φ11ϖ,11ϑ=111+i3ϖ−ϑ.
Therefore,(81)φTϖ,Tϑ≺¯i323φSϖ,Sϑ=λ=23φSϖ,Sϑ.
However,(82)φTϖ,Tϑ=71+i3ϖ−ϑ⪵i3λϖ−ϑ.
Thus,φ is not a contraction. Hence, T and S have a unique common fixed point.
We can prove that common fixed point theorems on tricomplex valued metric space under commuting mappings.Definition 7.
(see [19]). Two self mappings T and S of a metric space W,φ are said to be weakly compatible if the mappings commute at their coincidence points, i.e., Tϖ=Sϖ for some ϖ∈W implies TSϖ=STϖ.
We can prove common fixed point theorems on tricomplex valued metric space under weakly compatible mappings.


## 4. Applications

In this section, we give applications using Theorem2 and Corollary 1.LetW=C⋋1,⋋2 be a set of all real continuous functions on ⋋1,⋋2 equipped with metric φϖ,ϑ=1+i3ϖ⊔−ϑ⊔ for all ϖ,ϑ∈C⋋1,⋋2 and ⊔∈⋋1,⋋2, where . is the usual real modulus. Then, W,φ is a complete tricomplex valued metric space. Now, we consider the system of nonlinear Fredholm integral equation as follows:(83)ϖ⊔=v⊔+1⋋2−⋋1∫⋋1⋋2K1⊔,s,ϖsds,(84)ϖ⊔=v⊔+1⋋2−⋋1∫⋋1⋋2K2⊔,s,ϖsds,where ⊔,s∈⋋1,⋋2. Assume that K1,K2:⋋1,⋋2×⋋1,⋋2×W⟶ℝ and v:⋋1,⋋2⟶ℝ are continuous, where v⊔ is a given function in W. We define a partial order ⊔i3 in ℂ3 as ϖ⊔i3ϑ iff ϖ≤ϑ.Theorem 4.
Suppose thatW,φ is a complete tricomplex valued metric space equipped with metric φϖ,ϑ=1+i3ϖ⊔−ϑ⊔ for all ϖ,ϑ∈W, ⊔∈⋋1,⋋2 and S,T:W⟶W be continuous operator on W defined by(85)Sϖ⊔=v⊔+1⋋2−⋋1∫⋋1⋋2K1⊔,s,ϖsds,(86)Tϖ⊔=v⊔+1⋋2−⋋1∫⋋1⋋2K2⊔,s,ϖsds.If there existsλ<1 such that for all ϖ,ϑ∈W with ϖ≠ϑ and s,⊔∈⋋1,⋋2 satisfying the following inequality:(87)K1⊔,s,ϖs−K2⊔,s,ϑs≤λϖ⊔−ϑ⊔,then the integral operators defined by (85) and (86) have a unique common solution.Proof.
Consider(88)1+i3Sϖ⊔−Tϑ⊔=1+i3⋋2−⋋1∫⋋1⋋2K1⊔,s,ϖsds∫⋋1⋋2K2⊔,s,ϑsds≤1+i3⋋2−⋋1∫⋋1⋋2|K1⊔,s,ϖsK2⊔,s,ϑsds≤1+i3λ⋋2−⋋1∫⋋1⋋2ϖ⊔−ϑ⊔ds≤λ⋋2−⋋1∫⋋1⋋21+i3ϖ⊔−ϑ⊔ds≤λ1+i3ϖ⊔−ϑ⊔⋋2−⋋1∫⋋1⋋2ds.
Therefore,(89)φSϖ,Tϑ≤λφϖ,ϑ.
Hence, all the hypothesis of Theorem2 are fulfilled with λ<1 and μ=γ=0, and so the integral operators S and T defined by (85) and (86) have a unique common solution.Theorem 5.
LetW=ℂk be a complete tricomplex valued metric space with the metric(90)φϖ,ϑ=∑i=1kϖi−ϑi|+i3|ϖi−ϑi,where ϖ,ϑ∈W. If(91)∑j=1kλij≺¯i3λ<1,for alli=1,2,…k,then the linear system(92)b1=a11ϖ1+a12ϖ2+…+a1kϖk,b2=a21ϖ1+a22ϖ2+…+a2kϖk,⋮bk=ak1ϖ1+ak2ϖ2+…+akkϖk,of k linear equations with k unknowns has a unique solution.Proof.
DefineT:W⟶W by(93)Tϖ=Aϖ+b,where ϖ=ϖ1,ϖ2,ϖ3,…,ϖk∈ℂk, b=b1,b2,…,bk∈ℂk, and(94)A=a11a11…a1ka21a22…a2k⋮⋮⋮⋮ak1ak2…akk.
Now,(95)φTϖ,Tϑ=∑j=1kλijϖj−ϑj+i3λijϖj−ϑj≺¯i3∑j=1kλij∑j=1kϖj−ϑj+i3ϖj−ϑj=λφϖ,ϑ=λφϖ,ϑ+μφϖ,Tϖφϑ,Tϑ+γφϑ,Tϖφϖ,Tϑ1+φϖ,ϑ.
Hence, all the conditions of Corollary1 are satisfied with λ=1/6,μ=0,γ=0, and λ+2μ+2γ<1 and so the linear system of equation has a unique solution.


## 5. Conclusion and Future Work

In this paper, we introduce the notion of tricomplex valued metric space and proved some common fixed point theorems on tricomplex valued metric space. An illustrative example and applications on tricomplex valued metric space is given. In 2013, Patel et al. [20] proved common fixed points for a pair of maps on metric space. It is an interesting open problem to study the tricomplex valued metric space instead of metric space and obtain common fixed points for a pair of maps on tricomplex valued metric space. In 2013, Rouzkard et al. [21] proved existence and uniqueness theorems on ordered metric spaces via generalized distances. It is an interesting open problem to study the ordered tricomplex valued metric space instead of ordered metric space and obtain fixed point theorems on ordered tricomplex valued metric space. Recently, Altun et al. [22, 23] proved the best proximity point theorems on complete metric space. It is an interesting open problem to study the best proximity theorems on tricomplex valued metric space instead of best proximity point theorems on complete metric space.



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