Solving system of linear equations via bicomplex valued metric space

: In this paper, we prove some common ﬁ xed point theorems on bicomplex metric space. Our results generalize and expand some of the literature ’ s well - known results. We also explore some of the applications of our key results.


Introduction
Serge [1] made a pioneering attempt in the development of special algebras. He conceptualized commutative generalization of complex numbers as bicomplex numbers, bicomplex numbers, etc. as elements of an infinite set of algebras. Subsequently during the 1930s, other researchers also contributed in this area [2][3][4]. Priestley [5] proved Cauchy's integral formula as follows. But unfortunately the next 50 years failed to witness any advancement in this field. Afterward, Price [6] 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 [7]. 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 carried out by Luna-Elizaarrarás et al. [8]. Choi et al. [9] proved some common fixed point theorems in connection with two weakly compatible mappings in bicomplex valued metric spaces. Jebril [10] proved some common fixed point theorems under rational contractions for a pair of mappings in bicomplex valued metric spaces.
In this paper, inspired by Theorem 1.2, we prove some common fixed point theorems on bicomplex metric space with an application.

Preliminaries
Throughout this paper, we denote the set of real, complex, and bicomplex numbers, respectively, as 0 , 1 , and 2 . Segre [1] defined the bicomplex number as: = , we denote the set of bicomplex numbers 2 is defined as: = + w w be any two bicomplex numbers, then the sum is σ ϑ There are four idempotent elements in 2 , they are 0, 1, , This representation of σ is known as the idempotent representation of bicomplex number and the complex coefficients σ φ iφ are known as idempotent components of the bicomplex number σ.
An element σ φ i φ is said to be invertible if there exists another element ϑ in 2 such that σϑ 1 = and ϑ is said to be inverse (multiplicative) of σ. Consequently, σ is said to be the inverse (multiplicative) of ϑ. An element which has an inverse in 2 is said to be the nonsingular element of 2 and an element which does not have an inverse in 2 is said to be the singular element of 2 .
The linear space 2 with respect to defined norm is a norm linear space, also 2 is complete, therefore 2 is the Banach space. If σ ϑ , 2 ∈ , then σϑ σ ϑ 2 ‖ ‖ ≤ ‖ ‖‖ ‖ holds instead of σϑ σ ϑ ‖ ‖ ≤ ‖ ‖‖ ‖, therefore, 2 is not the Banach algebra. The partial order relation i2 ≼ on 2 is defined as: Let 2 be the set of bicomplex numbers and σ φ i φ if one of the following conditions is satisfied: In particular, we can write σ ϑ i2 ⋦ if σ ϑ i2 ≼ and σ ϑ ≠ , i.e., one of (B), (C), and (D) is satisfied and we will write σ ϑ i2 ≺ if only (D) is satisfied. For any two bicomplex numbers σ ϑ , 2 ∈ we can verify the followings: where a is a nonnegative real number; (S4) σϑ σ ϑ 2 ‖ ‖ ≤ ‖ ‖‖ ‖ and the equality holds only when at least one of σ and ϑ is degenerated; Now, let us recall some basic concepts and notations, which will be used in the sequel.
Definition 2.1. [15] Let be a nonvoid set, whereas 2 be the set of bicomplex numbers. Suppose that the mapping ϱ : 2 × → satisfies the following conditions: ∈ . Then ϱ is called the bicomplex valued metric on , and , ϱ ( ) b is called the bicomplex valued metric space.
⊆ is called open whenever each element of is an interior point of . A subset ⊆ is called closed whenever each limit point of belongs to . The family , : , is a sub-basis for a topology on . We denote this bicomplex topology by τ trc . Indeed, the topology τ trc is Hausdorff.
Let , ϱ ( ) be a bicomplex valued metric space. A sequence { } ℵ k in is said to be a convergent and converges to ℵ ∈ if for every ε 0 i and , ≥ k m h.
Definition 2.5. [15] Let , ϱ ( ) be a bicomplex valued metric space. Let { } ℵ k be any sequence in . Then, if every Cauchy sequence in is convergent in , then , ϱ ( ) is said to be a complete bicomplex valued metric space.
Definition 2.6. Let and be self-mappings of nonvoid set . A point ℵ ∈ is called a common fixed point of and if ℵ = ℵ = ℵ .

Main result
In this section, we prove common fixed point theorem in a bicomplex valued metric space using rationaltype contraction condition.
Proof. Let 0 ℵ be an arbitrary point in and define 2 1 Therefore, for any > m k, we have which is a contradiction so that = l l . Similarly, one can also show that = l l . To prove the uniqueness of common fixed point, let * l (in ) be another common fixed point of and , i.e., = = * * * l l l . Then l l l l l l l l l l l l l l l l l l l l l l l l which implies that which is a contradiction so that = * l l (as λ γ 2 1 + < ). This completes the proof of the theorem. □ ≼ is a partial order in . Define the functions , : → by Clearly, , ϱ ( ) is a complete bicomplex valued metric space. Now, we consider four cases: Case I: It is easy to verify that, all the above cases, the following conditions hold.
Hence, the conditions of Theorem 3.1 are satisfied. Therefore, 0 is the unique common fixed point of and .
So, for any > m k, we have ϱ , ϱ , ϱ , ϱ , ϱ , 1 ϱ , , , then also proof can be completed in the preceding lines. This completes the proof of the theorem. □ By setting = , we get the following.
) be a complete bicomplex valued metric space and let the mapping : → satisfy:  + + = + + < . Then, clearly Hence, the conditions of Corollary 3.3 are satisfied. Therefore, 0 is the unique fixed point of .
≼ is a partial order in . Define ϱ : a complete bicomplex valued metric space, where is a closed path in containing a zero. We prove that ϱ is a bicomplex metric space. For this,

Application
In this section, we give an application using Corollary 3.3.

Conclusion and future work
In this paper, we proved some common fixed point theorems on bicomplex valued metric space. An illustrative example and application on bicomplex valued metric space is given. Furthermore, one can prove common fixed theorems on bicomplex b-metric space, bicomplex metric-like space, bicomplex b-metriclike space, bicomplex bipolar metric space, bicomplex partial metric space, bicomplex partial b-metric space under some rational-type contraction mappings.
Author contributions: All authors contributed equally to the writing of this paper. All authors read and approved the final manuscript.

Conflict of interest:
The authors declare that there is no competing interest regarding the publication of this manuscript.