Research Article Common Fixed Point Theorems on Tricomplex Valued Metric Space

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.


Introduction
Fixed point theory plays an important role in applications of many branches of mathematics.
ere has been a number of generalizations of the usual notion of a metric space (see [1][2][3][4][5][6][7][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 in nite set of algebra. Subsequently during the 1930s, other researchers also contributed in this area [10][11][12]. However, unfortunately the next fty years failed to witness any advancement in this eld. Afterward, Price [13] developed the bicomplex algebra and function theory. Recently renewed interest in this subject nds some signi cant applications in di erent elds 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 xed point theorems in connection with two weakly compatible mappings in bicomplex valued metric spaces. Jebril et al. [17] proved some common xed point theorems under rational contractions for a pair of mappings in bicomplex valued metric spaces. In 2021, Beg et al. [18] proved the following xed point theorems on bicomplex valued metric spaces. Theorem 1. Let (W, φ) be a complete bicomplex valued metric space with degenerated 1 + φ(ϖ, ϑ) and 1 + φ(ϖ, ϑ) ≠ 0 for all ϖ, ϑ ∈ W and S, T: W ⟶ W such that φ(Sϖ, Tϑ)≺ i 2 λφ(ϖ, ϑ) + μφ(ϖ, Sϖ)φ(ϑ, Tϑ) 1 + φ(ϖ, ϑ) , (1) for all ϖ, ϑ ∈ W, where λ and μ are nonnegative real numbers with λ + � 2 √ μ < 1. en, S and T have a unique common fixed point.
In this paper, inspired by eorem 1, we prove some common fixed point theorems on tricomplex metric space with applications.
An element ξ � σ 1 + i 3 σ 2 ∈ C 3 is said to be invertible if there exists another element η in C 3 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 C 3 is said to be the nonsingular element of C 3 and an element which does not have an inverse in C 3 is said to be the singular element of C 3 .
For any two tricomplex numbers ξ, η ∈ C 3 , we can verify the following: where a is a nonnegative real number (4) ‖ξη‖‖ ≤ 2‖‖ξ‖‖η‖ and the equality holds only when at least one of ξ and η is nonsingular 2 Mathematical Problems in Engineering Now, let us recall some basic concepts and notations, which will be used in the sequel.
Proof. Let ϖ k be a convergent sequence and converges to a point ϖ, and let ϵ > 0 be any real number. Suppose en, 0≺ i 3 r ∈ C 3 , and for this r, there exists en, for this ϵ > 0, there exists k 0 ∈ N such that erefore, Hence, ϖ k converges to a point ϖ.
Definition 4. Let (W, φ) be a tricomplex valued metric space. Let ϖ k be any sequence in W. en, if every Cauchy sequence in W is convergent in W, then (W, φ) is said to be a complete tricomplex valued metric space.
Proof. Let ϖ k is a Cauchy sequence in W. Let ϵ > 0 be any real number. Suppose And, this implies that Conversely, let lim k⟶∞ ‖φ(ϖ k , ϖ k+m )‖ � 0. en, for each 0≺ i 3 r ∈ C + 3 , there exists a real number ϵ > 0 such that for en, for this ϵ > 0, there exists a natural number k 0 ∈ N such that erefore, Hence, ϖ k is a Cauchy sequence.

Main Result
In this section, we prove common fixed point theorem in a tricomplex valued metric space using rational type contraction condition.
for all ϖ, ϑ ∈ W where λ, μ, c are nonnegative reals with λ + 2μ + 2c < 1, then S and T have a unique common fixed point.
en, (W, φ) is a complete tricomplex valued metric space. Every real number is a tricomplex number but every Mathematical Problems in Engineering 5 tricomplex number is not necessarily a real number. erefore, (W, φ) is not a metric space. Now, we consider a self mappings S, T: W ⟶ W defined by for all ϖ, ϑ ∈ W. en, Every real number is a tricomplex number but every tricomplex number is not necessarily a real number. erefore, we cannot find common fixed point for such mappings on metric space. us, all the hypothesis of eorem 2 are fulfilled with λ � (1/2) < 1 and μ � c � 0. Hence, S and T have a unique common fixed point.
By setting S � T in eorem 2, one deduces the following.
Proof. Consider Hence, all the hypothesis of eorem 2 are fulfilled with λ < 1 and μ � c � 0, and so the integral operators S and T defined by (85) and (86) have a unique common solution.
□ Theorem 5 Let W � C k be a complete tricomplex valued metric space with the metric of k linear equations with k unknowns has a unique solution.

Mathematical Problems in Engineering
Hence, all the conditions of Corollary 1 are satisfied with λ � 1/6, μ � 0, c � 0, and λ + 2μ + 2c < 1 and so the linear system of equation has a unique solution.

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.

Data Availability
No data were used to support this work.