Solving System of Linear Equations Using Common Fixed Point Theorems in Bicomplex Valued Metric Spaces

Abstract

The aim of the present research article is to solve the system of linear equations using common fixed point theorems in the context of bicomplex valued metric spaces. To obtain our main objective, we introduce generalized contractive conditions in bicomplex valued metric spaces and establish common fixed point theorems for self mappings. We also give a significant example to demonstrate the legitimacy of the given theorems.

1. Introduction

Sir Carl Fredrich Gauss initiated the concept of a complex number in the 17th century, but his investigations were not on record. Augustin Louis Cauchy began his analysis of complex numbers in 1840, which is considered to be a productive initiator of complex analysis. The concept of complex numbers to solve $a{x}^2+bx+c=0$ was not beneficial for $b^2-4ac<0,$ in a set of real numbers. In this context, Euler was the first researcher who gave the sign $i,$ satisfying $i^2=-1$.

In 1892, Segre [1] began the concept of bicomplex numbers, which give a commutative alternative to the skew field of quaternions. Bicomplex numbers extend the notion of complex numbers as well as quaternions. For the excelling of far-reaching analysis regarding these numbers, we mention the researchers of reference [2]. On the other hand, Azam et al. [3] introduced the notion of complex valued metric space (CVMS) as an expansion of classical metric space and as a particular class of cone metric space. Although Choi et al. [4], in 2017, merged the ideas of CVMS and bicomplex numbers to introduce the concept of bicomplex valued metric spaces (bi-CVMS) and proved common fixed point theorems, later on, Jebril et al. [5] used this concept of bicomplex valued metric space and presented common fixed point theorems for self mappings. In due course, Beg et al. [6] reinforced the notion of bi-complex valued metric space and established some fixed point theorems. Afterwards, Gnanaprakasam et al. [7] added another term in the contractive condition of Beg et al. [6] and generalized its main result. As an application, they investigated a system of linear equations. For the specific features of CVMS and bi-CVMS, we refer the readers to references [8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26].

In the present research work, we establish common fixed points of self mappings in the framework of bicomplex valued metric spaces. We give a significant example to demonstrate the legitimacy of the given theorems. We solve the system of linear equations by applying our foremost result.

2. Preliminaries

Throughout this research article, we represent the set of real numbers, set of non-negative real numbers, set of complex numbers, and set of bicomplex numbers as ${\mathbb{C}}_0,$ ${\mathbb{C}}_0^{+},$ ${\mathbb{C}}_1$, and ${\mathbb{C}}_2$, respectively. In 1892, Segre [1] initiated the theory of a bicomplex number

$$\mathfrak{w}=a_1+a_2{i}_1+a_3{i}_2+a_4{i}_1{i}_2,$$

where $a_1,a_2,a_3,a_4\in {\mathbb{C}}_0,$ and the independent units $i_1,i_2$ are such that $i_1^2=i_2^2=-1$ and $i_1{i}_2=i_2{i}_1.$ The set of bicomplex numbers ${\mathbb{C}}_2$ is defined as

$${\mathbb{C}}_2=\left\{\mathfrak{w}:\mathfrak{w}=a_1+a_2{i}_1+a_3{i}_2+a_4{i}_1{i}_2:a_1,a_2,a_3,a_4\in {\mathbb{C}}_0\right\},$$

that is,

$${\mathbb{C}}_2=\left\{\mathfrak{w}:\mathfrak{w}=z_1+i_2{z}_2:z_1,z_2\in {\mathbb{C}}_1\right\},$$

where $z_1=a_1+a_2{i}_1\in {\mathbb{C}}_1$ and $z_2=a_3+a_4{i}_1\in {\mathbb{C}}_1.$ If $\mathfrak{w}=z_1+i_2{z}_2$ and $\mathfrak{y}={\omega }_1+i_2{\omega }_2$ are any two bicomplex numbers, then the sum is

$$\mathfrak{w}\pm \mathfrak{y}=\left(z_1+i_2{z}_2\right)\pm \left({\omega }_1+i_2{\omega }_2\right)=\left(z_1\pm {\omega }_1\right)+i_2\left(z_2\pm {\omega }_2\right),$$

and the product is

$$\mathfrak{w}·\mathfrak{y}=\left(z_1+i_2{z}_2\right)·\left({\omega }_1+i_2{\omega }_2\right)=\left(z_1{\omega }_1-z_2{\omega }_2\right)+i_2\left(z_1{\omega }_2+z_2{\omega }_1\right).$$

There are four idempotent elements in ${\mathbb{C}}_2,$ which are, $0,1,e_1=\frac{1+i_1{i}_2}{2}$ and $e_2=\frac{1-i_1{i}_2}{2}$, out of which $e_1$ and $e_2$ are nontrivial, such that $e_1+e_2=1$ and $e_1{e}_2=0.$ Every bicomplex number $z_1+i_2{z}_2$ can be uniquely given as the combination of $e_1$ and $e_2$, namely

$$\mathfrak{w}=z_1+i_2{z}_2=\left(z_1-i_1{z}_2\right)e_1+\left(z_1+i_1{z}_2\right)e_2.$$

Following this representation, $\mathfrak{w}$ is studied as the idempotent representation of ${\mathbb{C}}_2$ and the complex coefficients ${\mathfrak{w}}_1$ $=\left(z_1-i_1{z}_2\right)$ and ${\mathfrak{w}}_2=$ $\left(z_1+i_1{z}_2\right)$ are famous as idempotent components of $\mathfrak{w}$.

A member $\mathfrak{w}=z_1+i_2{z}_2\in {\mathbb{C}}_2$ is said to be invertible if there exists $\mathfrak{y}\in {\mathbb{C}}_2$, such that $\mathfrak{wy}=1$ and $\mathfrak{y}$ is said to be the multiplicative inverse of $\mathfrak{w}.$ Also, $\mathfrak{w}$ is the multiplicative inverse of $\mathfrak{y}.$ An element which has an inverse in ${\mathbb{C}}_2$ is professed to be the nonsingular element of ${\mathbb{C}}_2$ and an element which does not have an inverse in ${\mathbb{C}}_2$ is professed to be the singular element of ${\mathbb{C}}_2.$

An element $\mathfrak{w}=z_1+i_2{z}_2\in {\mathbb{C}}_2$ is non-singular if $\left|z_1^2+z_2^2\right|\ne 0$ and singular if $\left|z_1^2+z_2^2\right|=0.$ The inverse of $\mathfrak{w}$ is defined as

$${\mathfrak{w}}^{-1}=\mathfrak{y}=\frac{z_1-i_2{z}_2}{z_1^2+z_2^2}.$$

0 in ${\mathbb{C}}_0$ and $0=0+i0$ in ${\mathbb{C}}_1$ are the only elements that do not have an inverse (multiplicative). We denote by ${\mathrm{\Xi }}_0$ and ${\mathrm{\Xi }}_1$ the sets of singular members of ${\mathbb{C}}_0$ and ${\mathbb{C}}_1$ correspondingly. There are more elements in ${\mathbb{C}}_2,$ which do not have a multiplicative inverse. We denote by ${\mathrm{\Xi }}_2$ the sets of singular elements of ${\mathbb{C}}_2.$ Clearly, ${\mathrm{\Xi }}_0$ $={\mathrm{\Xi }}_1\subset {\mathrm{\Xi }}_2.$

A bicomplex number $\mathfrak{w}=a_1+a_2{i}_1+a_3{i}_2+a_4{i}_1{i}_2\in {\mathbb{C}}_2$ is called a degenerate if the matrix

$${\left(\begin{array}{cc}{a}_1& a_2\\ a_3& a_4\end{array}\right)}_{2\times 2}$$

is degenerated. In this case, ${\mathfrak{w}}^{-1}$ exists and it is also degenerated.

The norm $∥·∥:{\mathbb{C}}_2\to {\mathbb{C}}_0^{+}$ is defined by

$$\begin{array}{ccc}\hfill ∥\mathfrak{w}∥& =& ∥z_1+i_2{z}_2∥={\left\{{\left|z_1\right|}^2+{\left|z_2\right|}^2\right\}}^{\frac{1}{2}}\hfill \\ & =& {\left[\frac{{\left|\left(z_1-i_1{z}_2\right)\right|}^2+{\left|\left(z_1+i_1{z}_2\right)\right|}^2}{2}\right]}^{\frac{1}{2}}\hfill \\ & =& {\left(a_1^2+a_2^2+a_3^2+a_4^2\right)}^{\frac{1}{2}},\hfill \end{array}$$

where $\mathfrak{w}=a_1+a_2{i}_1+a_3{i}_2+a_4{i}_1{i}_2=z_1+i_2{z}_2\in {\mathbb{C}}_2.$

The space ${\mathbb{C}}_2$ is the Banach space with respect to the norm given above. If $\mathfrak{w},\mathfrak{y}\in {\mathbb{C}}_2,$ then

$$∥\mathfrak{wy}∥\le \sqrt{2}∥\mathfrak{w}∥∥\mathfrak{y}∥,$$

holds instead of

$$∥\mathfrak{wy}∥\le ∥\mathfrak{w}∥∥\mathfrak{y}∥,$$

therefore, ${\mathbb{C}}_2$ is not the Banach algebra. The partial order relation ${⪯}_{i_2}$ on ${\mathbb{C}}_2$ is given as follows:

$$\mathfrak{w}{⪯}_{i_2}\mathfrak{y}\iff \mathrm{R}\mathrm{e}\left(z_1\right)⪯\mathrm{R}\mathrm{e}\left({\omega }_1\right)\mathrm{and}\mathrm{I}\mathrm{m}\left(z_2\right)⪯\mathrm{I}\mathrm{m}\left({\omega }_2\right),$$

where $\mathfrak{w}=z_1+i_2{z}_2,$ $\mathfrak{y}={\omega }_1+i_2{\omega }_2\in {\mathbb{C}}_2.$

This yields

$$\mathfrak{w}{⪯}_{i_2}\mathfrak{y},$$

if one of these postulations is satisfied:

$$\begin{array}{ccc}\hfill \left(\mathrm{i}\right)z_1& =& {\omega }_1,z_2\prec {\omega }_2,\hfill \\ \hfill \left(\mathrm{ii}\right)z_1& \prec & {\omega }_1,z_2={\omega }_2,\hfill \\ \hfill \left(\mathrm{iii}\right)z_1& \prec & {\omega }_1,z_2\prec {\omega }_2,\hfill \\ \hfill \left(\mathrm{iv}\right)z_1& =& {\omega }_1,z_2={\omega }_2.\hfill \end{array}$$

To be specific, $\mathfrak{w}{⋨}_{i_2}\mathfrak{y}$ if $\mathfrak{w}{⪯}_{i_2}\mathfrak{y}$ and $\mathfrak{w}\ne \mathfrak{y},$ i.e., one of (i), (ii), and (iii) is satisfied and $\mathfrak{w}{\prec }_{i_2}\mathfrak{y}$ if only (iii) is satisfied. For $\mathfrak{w},$ $\mathfrak{y}\in {\mathbb{C}}_2,$ we have

(i) $\mathfrak{w}{⪯}_{i_2}\mathfrak{y}⟹∥\mathfrak{w}∥\le ∥\mathfrak{y}∥,$

(ii) $∥\mathfrak{w}+\mathfrak{y}∥\le ∥\mathfrak{w}∥+∥\mathfrak{y}∥,$

(iii) $∥a\mathfrak{w}∥\le \left|a\right|∥\mathfrak{y}∥,$ where a is a non-negative real number,

(iv) $∥\mathfrak{wy}∥\le \sqrt{2}∥\mathfrak{w}∥∥\mathfrak{y}∥,$

(v) $∥{\mathfrak{w}}^{-1}∥={∥\mathfrak{w}∥}^{-1},$

(vi) $∥\frac{\mathfrak{w}}{\mathfrak{y}}∥=\frac{∥\mathfrak{w}∥}{∥\mathfrak{y}∥},$ if $\mathfrak{y}$ is a degenerated bicomplex number.

Choi et al. [4] defined the bicomplex valued metric space in this manner.

Definition 1

([4]). Let $X\ne \varnothing$ and $d:X\times X\to {\mathbb{C}}_2$ be a function satisfying

(i) 

  $0{⪯}_{i_2}d(\mathfrak{w},\mathfrak{y})$ and $d(\mathfrak{w},\mathfrak{y})=0$⟺$\mathfrak{w}=\mathfrak{y}$,

(ii) 

 $d(\mathfrak{w},\mathfrak{y})=d(\mathfrak{y},\mathfrak{w}),$

(iii) 

$d(\mathfrak{w},\mathfrak{y}){⪯}_{i_2}d(\mathfrak{w},\nu )+d(\nu ,\mathfrak{y}),$

for all $\mathfrak{w},\mathfrak{y},\nu \in X,$ then d is called a bi-complex valued metric and $(X,d)$ is a bi-complex valued metric space.

Example 1

([6]). Let $X={\mathbb{C}}_2$ and $\mathfrak{w},\mathfrak{y}\in X.$ Define $d:X\times X\to {\mathbb{C}}_2$ by

$$d(\mathfrak{w},\mathfrak{y})=\left|z_1-{\omega }_1\right|+i_2\left|z_2-{\omega }_2\right|,$$

where $\mathfrak{w}=z_1+i_2{z}_2,$ $\mathfrak{y}={\omega }_1+i_2{\omega }_2\in {\mathbb{C}}_2.$ Then $(X,d)$ is a bi-complex valued metric space.

Lemma 1

([6]). Let $(X,d)$ be a bi-complex valued metric space and let $\left\{{\mathfrak{w}}_n\right\}$ $\subseteq X$ be a sequence. Then, $\left\{{\mathfrak{w}}_n\right\}$ converges to $\mathfrak{w}$ if and only if $∥d({\mathfrak{w}}_n,\mathfrak{w})∥\to 0$ as $n\to \infty .$

Lemma 2

([6]). Let $(X,d)$ be a bi-complex valued metric space and let $\left\{{\mathfrak{w}}_n\right\}$ $\subseteq X$ be a sequence. Then, $\left\{{\mathfrak{w}}_n\right\}$ is a Cauchy sequence if and only if $∥d({\mathfrak{w}}_n,{\mathfrak{w}}_{n+m})∥\to 0$ as $n\to \infty ,$ where $m\in \mathbb{N}.$

3. Main Result

Theorem 1.

Let $(X,d)$ be a complete bi-complex valued metric space and ${\beth }_1,{\beth }_2:X\to X$. Suppose that there exist ${\mathcal{O}}_1,{\mathcal{O}}_2,{\mathcal{O}}_3,{\mathcal{O}}_4,{\mathcal{O}}_5\in [0,1)$ with ${\mathcal{O}}_1+2{\mathcal{O}}_2+2{\mathcal{O}}_3+\sqrt{2}{\mathcal{O}}_4+\sqrt{2}{\mathcal{O}}_5<1$ such that

$$d({\beth }_1\mathfrak{w},{\beth }_2\mathfrak{y}){⪯}_{i_2}{\mathcal{O}}_{1}d(\mathfrak{w},\mathfrak{y})+{\mathcal{O}}_2\left(d\left(\mathfrak{w},{\beth }_1\mathfrak{w}\right)+d\left(\mathfrak{y},{\beth }_2\mathfrak{y}\right)\right)$$

$$+{\mathcal{O}}_3\left(d\left(\mathfrak{y},{\beth }_1\mathfrak{w}\right)+d\left(\mathfrak{w},{\beth }_2\mathfrak{y}\right)\right)$$

$$+{\mathcal{O}}_4\frac{d\left(\mathfrak{w},{\beth }_1\mathfrak{w}\right)d\left(\mathfrak{y},{\beth }_2\mathfrak{y}\right)}{1+d\left(\mathfrak{w},\mathfrak{y}\right)}$$

$$+{\mathcal{O}}_5\frac{d\left(\mathfrak{y},{\beth }_1\mathfrak{w}\right)d\left(\mathfrak{w},{\beth }_2\mathfrak{y}\right)}{1+d\left(\mathfrak{w},\mathfrak{y}\right)}$$

(1)

for all ${\mathfrak{w}}_0,\mathfrak{w},\mathfrak{y}\in \overline{B({\mathfrak{w}}_0,\varkappa )}$, $\varkappa \in {\mathbb{C}}_2$ and

$$∥d({\mathfrak{w}}_0,{\beth }_1{\mathfrak{w}}_0)∥\le (1-\lambda )∥\varkappa ∥,$$

(2)

where $\lambda =\left(\frac{{\mathcal{O}}_1+{\mathcal{O}}_2+{\mathcal{O}}_3}{1-{\mathcal{O}}_2-{\mathcal{O}}_3-\sqrt{2}{\mathcal{O}}_4}\right)<1$. Then, there exists a unique point ${\mathfrak{w}}^{*}\in \overline{B({\mathfrak{w}}_0,\varkappa )}$ such that ${\mathfrak{w}}^{*}={\beth }_1{\mathfrak{w}}^{*}={\beth }_2{\mathfrak{w}}^{*}$.

Proof. 

Let ${\mathfrak{w}}_0$ $\in X$ and define

$${\mathfrak{w}}_{2n+1}={\beth }_1{\mathfrak{w}}_{2n}\mathrm{and}{\mathfrak{w}}_{2n+2}={\beth }_2{\mathfrak{w}}_{2n+1},$$

for $n\in \mathbb{N}\cup \left\{0\right\}.$ We prove that ${\mathfrak{w}}_n\in \overline{B({\mathfrak{w}}_0,\varkappa )},$ for all $n\in \mathbb{N}$. By the fact that $\lambda =\left(\frac{{\mathcal{O}}_1+{\mathcal{O}}_2+{\mathcal{O}}_3}{1-{\mathcal{O}}_2-{\mathcal{O}}_3-\sqrt{2}{\mathcal{O}}_4}\right)<1$ and inequality (2), we have

$$∥d({\mathfrak{w}}_0,{\beth }_1{\mathfrak{w}}_0)∥\le ∥\varkappa ∥.$$

This shows that ${\mathfrak{w}}_1\in \overline{B({\mathfrak{w}}_0,\varkappa )}.$ Suppose ${\mathfrak{w}}_2,...,{\mathfrak{w}}_j\in \overline{B({\mathfrak{w}}_0,\varkappa )}$ for any $j\in \mathbb{N}$. Using (1), we obtain

$$d({\mathfrak{w}}_{2n+1},{\mathfrak{w}}_{2n+2})=d({\beth }_1{\mathfrak{w}}_{2n},{\beth }_2{\mathfrak{w}}_{2n+1})$$

$${⪯}_{i_2}{\mathcal{O}}_{1}d({\mathfrak{w}}_{2n},{\mathfrak{w}}_{2n+1})$$

$$+{\mathcal{O}}_2\left(\begin{array}{c}d\left({\mathfrak{w}}_{2n},{\beth }_1{\mathfrak{w}}_{2n}\right)\\ +d\left({\mathfrak{w}}_{2n+1},{\beth }_2{\mathfrak{w}}_{2n+1}\right)\end{array}\right)$$

$$+{\mathcal{O}}_3\left(\begin{array}{c}d\left({\mathfrak{w}}_{2n+1},{\beth }_1{\mathfrak{w}}_{2n}\right)\\ +d\left({\mathfrak{w}}_{2n},{\beth }_2{\mathfrak{w}}_{2n+1}\right)\end{array}\right)$$

$$+{\mathcal{O}}_4\frac{d\left({\mathfrak{w}}_{2n},{\beth }_1{\mathfrak{w}}_{2n}\right)d\left({\mathfrak{w}}_{2n+1},{\beth }_2{\mathfrak{w}}_{2n+1}\right)}{1+d\left({\mathfrak{w}}_{2n},{\mathfrak{w}}_{2n+1}\right)}$$

$$+{\mathcal{O}}_5\frac{d\left({\mathfrak{w}}_{2n+1},{\beth }_1{\mathfrak{w}}_{2n}\right)d\left({\mathfrak{w}}_{2n},{\beth }_2{\mathfrak{w}}_{2n+1}\right)}{1+d\left({\mathfrak{w}}_{2n},{\mathfrak{w}}_{2n+1}\right)}.$$

Now, ${\mathfrak{w}}_{2n+1}={\beth }_1{\mathfrak{w}}_{2n}$ implies that $d\left({\mathfrak{w}}_{2n+1},{\beth }_1{\mathfrak{w}}_{2n}\right)=0,$ we obtain

$$d({\mathfrak{w}}_{2n+1},{\mathfrak{w}}_{2n+2})=d({\beth }_1{\mathfrak{w}}_{2n},{\beth }_2{\mathfrak{w}}_{2n+1})$$

$${⪯}_{i_2}{\mathcal{O}}_{1}d({\mathfrak{w}}_{2n},{\mathfrak{w}}_{2n+1})$$

$$+{\mathcal{O}}_{2}d\left({\mathfrak{w}}_{2n},{\mathfrak{w}}_{2n+1}\right)+{\mathcal{O}}_{2}d\left({\mathfrak{w}}_{2n+1},{\mathfrak{w}}_{2n+2}\right)$$

$$+{\mathcal{O}}_3\left(d\left({\mathfrak{w}}_{2n},{\mathfrak{w}}_{2n+1}\right)+d\left({\mathfrak{w}}_{2n+1},{\mathfrak{w}}_{2n+2}\right)\right)$$

$$+{\mathcal{O}}_4\frac{d\left({\mathfrak{w}}_{2n},{\mathfrak{w}}_{2n+1}\right)d\left({\mathfrak{w}}_{2n+1},{\mathfrak{w}}_{2n+2}\right)}{1+d\left({\mathfrak{w}}_{2n},{\mathfrak{w}}_{2n+1}\right)},$$

which implies

$$\begin{array}{ccc}\hfill ∥d({\mathfrak{w}}_{2n+1},{\mathfrak{w}}_{2n+2})∥& \le & \left|{\mathcal{O}}_1\right|∥d({\mathfrak{w}}_{2n},{\mathfrak{w}}_{2n+1})∥\hfill \\ & & +\left|{\mathcal{O}}_2\right|∥d\left({\mathfrak{w}}_{2n},{\mathfrak{w}}_{2n+1}\right)∥+\left|{\mathcal{O}}_2\right|∥d\left({\mathfrak{w}}_{2n+1},{\mathfrak{w}}_{2n+2}\right)∥\hfill \\ & & +\left|{\mathcal{O}}_3\right|∥d\left({\mathfrak{w}}_{2n},{\mathfrak{w}}_{2n+1}\right)∥+\left|{\mathcal{O}}_3\right|∥d\left({\mathfrak{w}}_{2n+1},{\mathfrak{w}}_{2n+2}\right)∥\hfill \\ & & +\sqrt{2}\left|{\mathcal{O}}_4\right|\frac{∥d\left({\mathfrak{w}}_{2n},{\mathfrak{w}}_{2n+1}\right)∥∥d\left({\mathfrak{w}}_{2n+1},{\mathfrak{w}}_{2n+2}\right)∥}{∥1+d\left({\mathfrak{w}}_{2n},{\mathfrak{w}}_{2n+1}\right)∥}.\hfill \end{array}$$

Since $∥1+d\left({\mathfrak{w}}_{2n},{\mathfrak{w}}_{2n+1}\right)∥>∥d\left({\mathfrak{w}}_{2n},{\mathfrak{w}}_{2n+1}\right)∥,$ so we have

$$\begin{array}{ccc}\hfill ∥d({\mathfrak{w}}_{2n+1},{\mathfrak{w}}_{2n+2})∥|& \le & {\mathcal{O}}_1∥d({\mathfrak{w}}_{2n},{\mathfrak{w}}_{2n+1})∥\hfill \\ & & +{\mathcal{O}}_2∥d\left({\mathfrak{w}}_{2n},{\mathfrak{w}}_{2n+1}\right)∥+{\mathcal{O}}_2∥d\left({\mathfrak{w}}_{2n+1},{\mathfrak{w}}_{2n+2}\right)∥\hfill \\ & & +{\mathcal{O}}_3∥d\left({\mathfrak{w}}_{2n},{\mathfrak{w}}_{2n+1}\right)∥+{\mathcal{O}}_3∥d\left({\mathfrak{w}}_{2n+1},{\mathfrak{w}}_{2n+2}\right)∥\hfill \\ & & +\sqrt{2}{\mathcal{O}}_4∥d\left({\mathfrak{w}}_{2n+1},{\mathfrak{w}}_{2n+2}\right)∥.\hfill \end{array}$$

Thus, we have

$$∥d({\mathfrak{w}}_{2n+1},{\mathfrak{w}}_{2n+2})∥\le \left(\frac{{\mathcal{O}}_1+{\mathcal{O}}_2+{\mathcal{O}}_3}{1-{\mathcal{O}}_2-{\mathcal{O}}_3-\sqrt{2}{\mathcal{O}}_4}\right)∥d({\mathfrak{w}}_{2n},{\mathfrak{w}}_{2n+1})∥.$$

(3)

Similarly,

$$∥d({\mathfrak{w}}_{2n+2},{\mathfrak{w}}_{2n+3})∥\le \left(\frac{{\mathcal{O}}_1+{\mathcal{O}}_2+{\mathcal{O}}_3}{1-{\mathcal{O}}_2-{\mathcal{O}}_3-\sqrt{2}{\mathcal{O}}_4}\right)∥d({\mathfrak{w}}_{2n+1},{\mathfrak{w}}_{2n+2})∥.$$

(4)

Putting $\lambda =\left(\frac{{\mathcal{O}}_1+{\mathcal{O}}_2+{\mathcal{O}}_3}{1-{\mathcal{O}}_2-{\mathcal{O}}_3-\sqrt{2}{\mathcal{O}}_4}\right)<1.$ Then, from inequalities (3) and (4), we obtain

$$∥d({\mathfrak{w}}_j,{\mathfrak{w}}_{j+1})∥\le \lambda ∥d({\mathfrak{w}}_{j-1},{\mathfrak{w}}_j)∥\le ...\le {\lambda }^j∥d({\mathfrak{w}}_0,{\mathfrak{w}}_1)∥$$

(5)

for some $j\in \mathbb{N}.$ Now,

$$\begin{array}{ccc}\hfill ∥d({\mathfrak{w}}_0,{\mathfrak{w}}_{j+1})∥& \le & ∥d({\mathfrak{w}}_0,{\mathfrak{w}}_1)∥+...+∥d({\mathfrak{w}}_j,{\mathfrak{w}}_{j+1})∥\hfill \\ & \le & ∥d({\mathfrak{w}}_0,{\mathfrak{w}}_1)∥+...+{\lambda }^j∥d({\mathfrak{w}}_0,{\mathfrak{w}}_1)∥\hfill \\ & =& ∥d({\mathfrak{w}}_0,{\mathfrak{w}}_1)∥[1+...+{\lambda }^{j-1}+{\lambda }^j]\hfill \\ & \le & (1-\lambda )\left(∥\varkappa ∥\right)\frac{(1-{\lambda }^{j+1})}{1-\lambda }\hfill \\ & \le & ∥\varkappa ∥.\hfill \end{array}$$

This implies ${\mathfrak{w}}_{j+1}\in \overline{B({\mathfrak{w}}_0,\varkappa )}.$ Thus, ${\mathfrak{w}}_n\in \overline{B({\mathfrak{w}}_0,\varkappa ),}$ for all $n\in \mathbb{N}.$ Thus, using (5), we have

$$∥d({\mathfrak{w}}_n,{\mathfrak{w}}_{n+1}∥\le {\lambda }^n∥d({\mathfrak{w}}_0,{\mathfrak{w}}_1)∥,$$

for all $n.$ Using triangular inequality for $m>n$, we have

$$\begin{array}{ccc}\hfill ∥d\left({\mathfrak{w}}_n,{\mathfrak{w}}_m\right)∥& \le & {\lambda }^n∥d\left({\mathfrak{w}}_0,{\mathfrak{w}}_1\right)∥\hfill \\ & & +{\lambda }^{n+1}∥d\left({\mathfrak{w}}_0,{\mathfrak{w}}_1\right)∥\hfill \\ & & +···+\hfill \\ & & {\lambda }^{m-1}∥d\left({\mathfrak{w}}_0,{\mathfrak{w}}_1\right)∥\hfill \\ & \le & \left[{\lambda }^n+{\lambda }^{n+1}+···+{\lambda }^{m-1}\right]∥d\left({\mathfrak{w}}_0,{\mathfrak{w}}_1\right)∥.\hfill \end{array}$$

Now, by taking $n\to \infty$, we obtain

$$∥d\left({\mathfrak{w}}_n,{\mathfrak{w}}_m\right)∥\to 0.$$

Thus, the sequence $\left\{{\mathfrak{w}}_n\right\}$ is a Cauchy sequence in $\overline{B({\mathfrak{w}}_0,\varkappa )}$ using Lemma 2. As a consequence, there exists ${\mathfrak{w}}^{*}\in \overline{B({\mathfrak{w}}_0,\varkappa )}$, such that $\underset{n\to \infty }{lim}{\mathfrak{w}}_n={\mathfrak{w}}^{*}$. This implies that ${\mathfrak{w}}^{*}={\beth }_1{\mathfrak{w}}^{*},$ otherwise $d({\mathfrak{w}}^{*},{\beth }_1{\mathfrak{w}}^{*})=\upsilon \succ 0$ and we would then have

$$\upsilon {⪯}_{i_2}\left(d({\mathfrak{w}}^{*},{\mathfrak{w}}_{2n+2})+d({\mathfrak{w}}_{2n+2},{\beth }_1{\mathfrak{w}}^{*})\right)$$

$$=d({\mathfrak{w}}^{*},{\mathfrak{w}}_{2n+2})+d({\beth }_2{\mathfrak{w}}_{2n+1},{\beth }_1{\mathfrak{w}}^{*})$$

$$=d({\mathfrak{w}}^{*},{\mathfrak{w}}_{2n+2})+d({\beth }_1{\mathfrak{w}}^{*},{\beth }_2{\mathfrak{w}}_{2n+1})$$

$${⪯}_{i_2}\left(\begin{array}{c}d({\mathfrak{w}}^{*},{\mathfrak{w}}_{2n+2})+{\mathcal{O}}_{1}d({\mathfrak{w}}^{*},{\mathfrak{w}}_{2n+1})\\ +{\mathcal{O}}_2\left(d\left({\mathfrak{w}}^{*},{\beth }_1{\mathfrak{w}}^{*}\right)+d\left({\mathfrak{w}}_{2n+1},{\beth }_2{\mathfrak{w}}_{2n+1}\right)\right)\\ +{\mathcal{O}}_3\left(d\left({\mathfrak{w}}_{2n+1},{\beth }_1{\mathfrak{w}}^{*}\right)+d\left({\mathfrak{w}}^{*},{\beth }_2{\mathfrak{w}}_{2n+1}\right)\right)\\ +{\mathcal{O}}_4\frac{d\left({\mathfrak{w}}^{*},{\beth }_1{\mathfrak{w}}^{*}\right)d\left({\mathfrak{w}}_{2n+1},{\beth }_2{\mathfrak{w}}_{2n+1}\right)}{1+d\left({\mathfrak{w}}^{*},{\mathfrak{w}}_{2n+1}\right)}\\ +{\mathcal{O}}_5\frac{d\left({\mathfrak{w}}_{2n+1},{\beth }_1{\mathfrak{w}}^{*}\right)d\left({\mathfrak{w}}^{*},{\beth }_2{\mathfrak{w}}_{2n+1}\right)}{1+d\left({\mathfrak{w}}^{*},{\mathfrak{w}}_{2n+1}\right)}\end{array}\right),$$

which implies that

$$∥\upsilon ∥⩽\left(\begin{array}{c}∥d({\mathfrak{w}}^{*},{\mathfrak{w}}_{2n+2})∥+\left|{\mathcal{O}}_1\right|∥d({\mathfrak{w}}^{*},{\mathfrak{w}}_{2n+1})∥\\ +\left|{\mathcal{O}}_2\right|∥d\left({\mathfrak{w}}^{*},{\beth }_1{\mathfrak{w}}^{*}\right)∥+\left|{\mathcal{O}}_2\right|∥d\left({\mathfrak{w}}_{2n+1},{\mathfrak{w}}_{2n+2}\right)∥\\ +\left|{\mathcal{O}}_3\right|∥d\left({\mathfrak{w}}_{2n+1},{\beth }_1{\mathfrak{w}}^{*}\right)∥+\left|{\mathcal{O}}_3\right|∥d\left({\mathfrak{w}}^{*},{\mathfrak{w}}_{2n+2}\right)∥\\ +\sqrt{2}\left|{\mathcal{O}}_4\right|\frac{∥\upsilon ∥∥d\left({\mathfrak{w}}_{2n+1},{\mathfrak{w}}_{2n+2}\right)∥}{∥1+d\left({\mathfrak{w}}^{*},{\mathfrak{w}}_{2n+1}\right)∥}\\ +\sqrt{2}\left|{\mathcal{O}}_5\right|\frac{∥d\left({\mathfrak{w}}_{2n+1},{\beth }_1{\mathfrak{w}}^{*}\right)∥∥d\left({\mathfrak{w}}^{*},{\mathfrak{w}}_{2n+2}\right)∥}{∥1+d\left({\mathfrak{w}}^{*},{\mathfrak{w}}_{2n+1}\right)∥}\end{array}\right).$$

Letting $n\to \infty ,$ we obtain $∥\upsilon ∥\le \left({\mathcal{O}}_2+{\mathcal{O}}_3\right)∥\upsilon ∥.$ If $∥\upsilon ∥\ne 0,$ then we obtain ${\mathcal{O}}_2+{\mathcal{O}}_3\ge 1$, which is a contradiction. In this way, $∥\upsilon ∥=0.$ Thus, ${\mathfrak{w}}^{*}={\beth }_1{\mathfrak{w}}^{*}.$ Likewise, one can easily prove that ${\mathfrak{w}}^{*}={\beth }_2{\mathfrak{w}}^{*}$. Now, we prove that the common fixed point of mappings ${\beth }_1$ and ${\beth }_2$ is unique. Let ${\mathfrak{w}}^{/}\in X$ be another common fixed point of the mappings ${\beth }_1$ and ${\beth }_2,$ i.e., ${\mathfrak{w}}^{/}={\beth }_1{\mathfrak{w}}^{/}={\beth }_2{\mathfrak{w}}^{/}$, such that ${\mathfrak{w}}^{*}\ne {\mathfrak{w}}^{/}.$ Using (1), we obtain

$$d({\mathfrak{w}}^{*},{\mathfrak{w}}^{/})=d({\beth }_1{\mathfrak{w}}^{*},{\beth }_2{\mathfrak{w}}^{/})$$

$${⪯}_{i_2}{\mathcal{O}}_{1}d({\mathfrak{w}}^{*},{\mathfrak{w}}^{/})$$

$$+{\mathcal{O}}_2\left(d\left({\mathfrak{w}}^{*},{\beth }_1{\mathfrak{w}}^{*}\right)+d\left({\mathfrak{w}}^{/},{\beth }_2{\mathfrak{w}}^{/}\right)\right)$$

$$+{\mathcal{O}}_3\left(d\left({\mathfrak{w}}^{/},{\beth }_1{\mathfrak{w}}^{*}\right)d\left({\mathfrak{w}}^{*},{\beth }_2{\mathfrak{w}}^{/}\right)\right)$$

$$+{\mathcal{O}}_4\frac{d\left({\mathfrak{w}}^{*},{\beth }_1{\mathfrak{w}}^{*}\right)d\left({\mathfrak{w}}^{/},{\beth }_2{\mathfrak{w}}^{/}\right)}{1+d({\mathfrak{w}}^{*},{\mathfrak{w}}^{/})}$$

$$+{\mathcal{O}}_5\frac{d\left({\mathfrak{w}}^{/},{\beth }_1{\mathfrak{w}}^{*}\right)d\left({\mathfrak{w}}^{*},{\beth }_2{\mathfrak{w}}^{/}\right)}{1+d({\mathfrak{w}}^{*},{\mathfrak{w}}^{/})}.$$

This implies that

$$d({\mathfrak{w}}^{*},{\mathfrak{w}}^{/}){⪯}_{i_2}{\mathcal{O}}_{1}d({\mathfrak{w}}^{*},{\mathfrak{w}}^{/})+{\mathcal{O}}_2\left(d\left({\mathfrak{w}}^{*},{\mathfrak{w}}^{*}\right)+d\left({\mathfrak{w}}^{/},{\mathfrak{w}}^{/}\right)\right)$$

$$+{\mathcal{O}}_3\left(d\left({\mathfrak{w}}^{/},{\mathfrak{w}}^{*}\right)+d\left({\mathfrak{w}}^{*},{\mathfrak{w}}^{/}\right)\right)$$

$$+{\mathcal{O}}_4\frac{d\left({\mathfrak{w}}^{*},{\mathfrak{w}}^{*}\right)d\left({\mathfrak{w}}^{/},{\mathfrak{w}}^{/}\right)}{1+d({\mathfrak{w}}^{*},{\mathfrak{w}}^{/})}$$

$$+{\mathcal{O}}_5\frac{d\left({\mathfrak{w}}^{/},{\mathfrak{w}}^{*}\right)d\left({\mathfrak{w}}^{*},{\mathfrak{w}}^{/}\right)}{1+d({\mathfrak{w}}^{*},{\mathfrak{w}}^{/})}.$$

This further yields,

$$d({\mathfrak{w}}^{*},{\mathfrak{w}}^{/}){⪯}_{i_2}{\mathcal{O}}_{1}d({\mathfrak{w}}^{*},{\mathfrak{w}}^{/})$$

$$+{\mathcal{O}}_3\left(d\left({\mathfrak{w}}^{/},{\mathfrak{w}}^{*}\right)+d\left({\mathfrak{w}}^{*},{\mathfrak{w}}^{/}\right)\right)$$

$$+{\mathcal{O}}_5\frac{d\left({\mathfrak{w}}^{/},{\mathfrak{w}}^{*}\right)d\left({\mathfrak{w}}^{*},{\mathfrak{w}}^{/}\right)}{1+d({\mathfrak{w}}^{*},{\mathfrak{w}}^{/})}.$$

Taking the norm on both sides, we have

$$\begin{array}{ccc}\hfill ∥d({\mathfrak{w}}^{*},{\mathfrak{w}}^{/})∥& \le & {\mathcal{O}}_1∥d({\mathfrak{w}}^{*},{\mathfrak{w}}^{/})∥\hfill \\ & & +\left|{\mathcal{O}}_3\right|∥d\left({\mathfrak{w}}^{/},{\mathfrak{w}}^{*}\right)∥+\left|{\mathcal{O}}_3\right|∥d\left({\mathfrak{w}}^{*},{\mathfrak{w}}^{/}\right)∥\hfill \\ & & +\sqrt{2}\left|{\mathcal{O}}_5\right|\frac{∥d\left({\mathfrak{w}}^{/},{\mathfrak{w}}^{*}\right)∥∥d\left({\mathfrak{w}}^{*},{\mathfrak{w}}^{/}\right)∥}{∥1+d({\mathfrak{w}}^{*},{\mathfrak{w}}^{/})∥}.\hfill \end{array}$$

Since $∥1+d({\mathfrak{w}}^{*},{\mathfrak{w}}^{/})∥>∥d({\mathfrak{w}}^{*},{\mathfrak{w}}^{/})∥,$ so we have

$$∥d({\mathfrak{w}}^{*},{\mathfrak{w}}^{/})∥\le ({\mathcal{O}}_1+2{\mathcal{O}}_3+\sqrt{2}{\mathcal{O}}_5)∥d({\mathfrak{w}}^{*},{\mathfrak{w}}^{/})∥.$$

This is a contradiction to ${\mathcal{O}}_1+2{\mathcal{O}}_3+\sqrt{2}{\mathcal{O}}_5<1.$ Thus, ${\mathfrak{w}}^{/}={\mathfrak{w}}^{*}.$ Hence, ${\mathfrak{w}}^{*}$ is the unique common fixed point of mappings ${\beth }_1$ and ${\beth }_2.$ □

Corollary 1.

Let $(X,d)$ be a complete bi-complex valued metric space and $\beth :X\to X$. Assume that there exists ${\mathcal{O}}_1,{\mathcal{O}}_2,{\mathcal{O}}_3,{\mathcal{O}}_4,{\mathcal{O}}_5\in [0,1)$ with ${\mathcal{O}}_1+2{\mathcal{O}}_2+2{\mathcal{O}}_3+\sqrt{2}{\mathcal{O}}_4+\sqrt{2}{\mathcal{O}}_5<1$, such that

$$d(\beth \mathfrak{w},\beth \mathfrak{y}){⪯}_{i_2}{\mathcal{O}}_{1}d(\mathfrak{w},\mathfrak{y})+{\mathcal{O}}_2\left(d\left(\mathfrak{w},\beth \mathfrak{w}\right)+d\left(\mathfrak{y},\beth \mathfrak{y}\right)\right)$$

$$+{\mathcal{O}}_3\left(d\left(\mathfrak{y},\beth \mathfrak{w}\right)+d\left(\mathfrak{w},\beth \mathfrak{y}\right)\right)$$

$$+{\mathcal{O}}_4\frac{d\left(\mathfrak{w},\beth \mathfrak{w}\right)d\left(\mathfrak{y},\beth \mathfrak{y}\right)}{1+d\left(\mathfrak{w},\mathfrak{y}\right)}$$

$$+{\mathcal{O}}_5\frac{d\left(\mathfrak{y},\beth \mathfrak{w}\right)d\left(\mathfrak{w},\beth \mathfrak{y}\right)}{1+d\left(\mathfrak{w},\mathfrak{y}\right)},$$

for all ${\mathfrak{w}}_0,\mathfrak{w},\mathfrak{y}\in \overline{B({\mathfrak{w}}_0,\varkappa )}$, $\varkappa \in {\mathbb{C}}_2$ and

$$∥d({\mathfrak{w}}_0,\beth {\mathfrak{w}}_0)∥\le (1-\lambda )∥\varkappa ∥,$$

where $\lambda =\left(\frac{{\mathcal{O}}_1+{\mathcal{O}}_2+{\mathcal{O}}_3}{1-{\mathcal{O}}_2-{\mathcal{O}}_3-\sqrt{2}{\mathcal{O}}_4}\right)<1$. Then, there exists a unique point ${\mathfrak{w}}^{*}\in \overline{B({\mathfrak{w}}_0,\varkappa )}$, such that ${\mathfrak{w}}^{*}=\beth {\mathfrak{w}}^{*}$.

Proof. 

Taking ${\beth }_1={\beth }_2=\beth$ in Theorem 1. □

Example 2.

Let $X=[0,\infty )$, define $d:X\times X⟶{\mathbb{C}}_2$ as follows:

$$d\left({\mathfrak{w}}_{1,}{\mathfrak{w}}_2\right)=(1+i_2)\left|{\mathfrak{w}}_{1,}-{\mathfrak{w}}_2\right|.$$

Then, $(X,d)$ is a complete bi-CVMS. Take ${\mathfrak{w}}_0=\frac{1}{2}$ and $\varkappa =\frac{1}{2}+\frac{1}{2}{i}_2.$ Then, $\overline{B({\mathfrak{w}}_0,\varkappa )}=\left[0,1\right].$ Define ${\beth }_1,{\beth }_2:X\to X$ as

$${\beth }_1\mathfrak{w}={\displaystyle \frac{\mathfrak{w}}{4}}$$

and

$${\beth }_2\mathfrak{w}={\displaystyle \frac{\mathfrak{w}}{5}}.$$

Then, with ${\mathcal{O}}_1=\frac{1}{6},{\mathcal{O}}_2=\frac{1}{24},{\mathcal{O}}_3=\frac{1}{2},{\mathcal{O}}_4=\frac{1}{25}$ and ${\mathcal{O}}_5=\frac{1}{26},$ all the hypotheses of Theorem 1 are fulfilled and, hence, the mappings ${\beth }_1$ and ${\beth }_2$ have a unique common fixed point $0\in \overline{B({\mathfrak{w}}_0,\varkappa )}$.

Remark 1.

By equating ${\mathcal{O}}_1$, ${\mathcal{O}}_2$, ${\mathcal{O}}_3$, ${\mathcal{O}}_4$, and ${\mathcal{O}}_5$ to 0 in all possible combinations and taking ${\beth }_1={\beth }_2=\beth$, one can derive a host of corollaries as a generalization of the fixed point results given by Hardy-Roger [26], Reich [27], Chaterjea [28], and Kannan [29] in the framework of bi-complex valued metric spaces.

Now, we state a result which is a generalization of the leading theorem of Gnanaprakasam et al. [7].

Corollary 2.

Let $(X,d)$ be a complete bi-complex valued metric space and ${\beth }_1,{\beth }_2:X\to X$. Suppose that there exists ${\mathcal{O}}_1,{\mathcal{O}}_4,{\mathcal{O}}_5\in [0,1)$ with ${\mathcal{O}}_1+\sqrt{2}{\mathcal{O}}_4+\sqrt{2}{\mathcal{O}}_5<1$, such that

$$d({\beth }_1\mathfrak{w},{\beth }_2\mathfrak{y}){⪯}_{i_2}{\mathcal{O}}_{1}d(\mathfrak{w},\mathfrak{y})+{\mathcal{O}}_4\frac{d\left(\mathfrak{w},{\beth }_1\mathfrak{w}\right)d\left(\mathfrak{y},{\beth }_2\mathfrak{y}\right)}{1+d\left(\mathfrak{w},\mathfrak{y}\right)}+{\mathcal{O}}_5\frac{d\left(\mathfrak{y},{\beth }_1\mathfrak{w}\right)d\left(\mathfrak{w},{\beth }_2\mathfrak{y}\right)}{1+d\left(\mathfrak{w},\mathfrak{y}\right)},$$

for all ${\mathfrak{w}}_0,\mathfrak{w},\mathfrak{y}\in \overline{B({\mathfrak{w}}_0,\varkappa )}$, $\varkappa \in {\mathbb{C}}_2$ and

$$∥d({\mathfrak{w}}_0,{\beth }_1{\mathfrak{w}}_0)∥\le (1-\lambda )∥\varkappa ∥,$$

where $\lambda =\left(\frac{{\mathcal{O}}_1}{1-\sqrt{2}{\mathcal{O}}_4}\right)<1$. Then, there exists a unique point ${\mathfrak{w}}^{*}\in \overline{B({\mathfrak{w}}_0,\varkappa )}$, such that ${\mathfrak{w}}^{*}={\beth }_1{\mathfrak{w}}^{*}={\beth }_2{\mathfrak{w}}^{*}$.

Proof. 

Taking ${\mathcal{O}}_2={\mathcal{O}}_3=0$ in Theorem 1. □

Now, we state a corollary which is a generalization of the main theorem of Beg et al. [6].

Corollary 3.

Let $(X,d)$ be a complete bi-complex valued metric space and ${\beth }_1,{\beth }_2:X\to X$. Suppose that there exists ${\mathcal{O}}_1,{\mathcal{O}}_4\in [0,1)$ with ${\mathcal{O}}_1+\sqrt{2}{\mathcal{O}}_4<1$, such that

$$d({\beth }_1\mathfrak{w},{\beth }_2\mathfrak{y}){⪯}_{i_2}{\mathcal{O}}_{1}d(\mathfrak{w},\mathfrak{y})+{\mathcal{O}}_4\frac{d\left(\mathfrak{w},{\beth }_1\mathfrak{w}\right)d\left(\mathfrak{y},{\beth }_2\mathfrak{y}\right)}{1+d\left(\mathfrak{w},\mathfrak{y}\right)},$$

for all ${\mathfrak{w}}_0,\mathfrak{w},\mathfrak{y}\in \overline{B({\mathfrak{w}}_0,\varkappa )}$, $\varkappa \in {\mathbb{C}}_2$ and

$$∥d({\mathfrak{w}}_0,{\beth }_1{\mathfrak{w}}_0)∥\le (1-\lambda )∥\varkappa ∥,$$

where $\lambda =\left(\frac{{\mathcal{O}}_1}{1-\sqrt{2}{\mathcal{O}}_4}\right)<1$. Then, there exists a unique point ${\mathfrak{w}}^{*}\in \overline{B({\mathfrak{w}}_0,\varkappa )}$, such that ${\mathfrak{w}}^{*}={\beth }_1{\mathfrak{w}}^{*}={\beth }_2{\mathfrak{w}}^{*}$.

Proof. 

Taking ${\mathcal{O}}_2={\mathcal{O}}_3={\mathcal{O}}_5=0$ in Theorem 1. □

For two finite families of mappings, we prove the following result by applying Theorem 1.

Theorem 2.

If ${\left\{{\Im }_i\right\}}_1^m$ and ${\left\{{\Re }_i\right\}}_1^n$ are two finite pairwise commuting finite families of self-mapping defined on a bi-complex valued metric space, such that the mappings ℜ and ℑ (with $\Im ={\Im }_1{\Im }_2···{\Im }_m$ and $\Re ={\Re }_1{\Re }_2···{\Re }_n)$ satisfy (1) and (2), then the component mappings of these ${\left\{{\Im }_i\right\}}_1^m$ and ${\left\{{\Re }_i\right\}}_1^n$ have a unique common fixed point.

Proof. 

By applying Theorem 1, we have $\Im {\mathfrak{w}}^{*}=\Re {\mathfrak{w}}^{*}={\mathfrak{w}}^{*},$ which is unique. Since ${\left\{{\Im }_i\right\}}_1^m$ and ${\left\{{\Re }_i\right\}}_1^n$ (for every $1\le k\le m)$ are pairwise commutative, ${\Im }_k{\mathfrak{w}}^{*}={\Im }_k\Im {\mathfrak{w}}^{*}=\Im {\Im }_k{\mathfrak{w}}^{*}$ and ${\Im }_k{\mathfrak{w}}^{*}={\Im }_k\Re {\mathfrak{w}}^{*}=\Re {\Im }_k{\mathfrak{w}}^{*}$, which indicates that ${\Im }_k{\mathfrak{w}}^{*}$, for all k is also a common fixed point of mappings ℑ and ℜ. Now, using the uniqueness, so ${\Im }_k{\mathfrak{w}}^{*}={\mathfrak{w}}^{*}$ (for each k) that proves ${\mathfrak{w}}^{*}$ is the common fixed point of ${\left\{{\Im }_i\right\}}_1^m.$ Likewise, we can show that ${\Re }_k{\mathfrak{w}}^{*}={\mathfrak{w}}^{*}$ ($1\le k\le n)$. Thus, ${\left\{{\Im }_i\right\}}_1^m$ and ${\left\{{\Re }_i\right\}}_1^n$ possess a unique common fixed point. □

Corollary 4.

Let $(X,d)$ be a complete bi-complex valued metric space and ${\mathfrak{F}}_1,{\mathfrak{F}}_2:X\to X$. Suppose that there exists ${\mathcal{O}}_1,{\mathcal{O}}_2,{\mathcal{O}}_3,{\mathcal{O}}_4$, ${\mathcal{O}}_5\in [0,1)$ with ${\mathcal{O}}_1+2{\mathcal{O}}_2+2{\mathcal{O}}_3+\sqrt{2}{\mathcal{O}}_4+\sqrt{2}{\mathcal{O}}_5<1$ and ${\mathfrak{F}}_1$, ${\mathfrak{F}}_2$ satisfy

$$d({\mathfrak{F}}_1^m\mathfrak{w},{\mathfrak{F}}_2^n\mathfrak{y}){⪯}_{i_2}{\mathcal{O}}_{1}d(\mathfrak{w},\mathfrak{y})+{\mathcal{O}}_2\left(d\left(\mathfrak{w},{\mathfrak{F}}_1^m\mathfrak{w}\right)+d\left(\mathfrak{y},{\mathfrak{F}}_2^n\mathfrak{y}\right)\right)$$

$$+{\mathcal{O}}_3\left(d\left(\mathfrak{y},{\mathfrak{F}}_1^m\mathfrak{w}\right)+d\left(\mathfrak{w},{\mathfrak{F}}_2^n\mathfrak{y}\right)\right)$$

$$+{\mathcal{O}}_4\frac{d\left(\mathfrak{w},{\mathfrak{F}}_1^m\mathfrak{w}\right)d\left(\mathfrak{y},{\mathfrak{F}}_2^n\mathfrak{y}\right)}{1+d\left(\mathfrak{w},\mathfrak{y}\right)}$$

$$+{\mathcal{O}}_5\frac{d\left(\mathfrak{y},{\mathfrak{F}}_1^m\mathfrak{w}\right)d\left(\mathfrak{w},{\mathfrak{F}}_2^n\mathfrak{y}\right)}{1+d\left(\mathfrak{w},\mathfrak{y}\right)},$$

for all ${\mathfrak{w}}_0,\mathfrak{w},\mathfrak{y}\in \overline{B({\mathfrak{w}}_0,\varkappa )}$, $0\prec \varkappa \in {\mathbb{C}}_2$ and

$$∥d({\mathfrak{w}}_0,{\mathfrak{F}}_2^n{\mathfrak{w}}_0)∥\le (1-\lambda )∥\varkappa ∥,$$

where $\lambda =\left(\frac{{\mathcal{O}}_1+{\mathcal{O}}_2+{\mathcal{O}}_3}{1-{\mathcal{O}}_2-{\mathcal{O}}_3-\sqrt{2}{\mathcal{O}}_4}\right)<1$. Then, there exists a unique point ${\mathfrak{w}}^{*}\in \overline{B({\mathfrak{w}}_0,\varkappa )}$, such that ${\mathfrak{w}}^{*}={\mathfrak{F}}_1{\mathfrak{w}}^{*}={\mathfrak{F}}_2{\mathfrak{w}}^{*}$.

Proof. 

Taking ${\Im }_1={\Im }_2=···={\Im }_m={\mathfrak{F}}_1$ and ${\Re }_1={\Re }_2=···={\Re }_n={\mathfrak{F}}_2,$ in Theorem 2. □

Corollary 5.

Let $(X,d)$ be a complete bi-complex valued metric space and $\beth :X\to X.$ Suppose that there exists ${\mathcal{O}}_1,{\mathcal{O}}_2,{\mathcal{O}}_3,{\mathcal{O}}_4$, ${\mathcal{O}}_5\in [0,1)$ with ${\mathcal{O}}_1+2{\mathcal{O}}_2+2{\mathcal{O}}_3+\sqrt{2}{\mathcal{O}}_4+\sqrt{2}{\mathcal{O}}_5<1$ and ℶ satisfies

$$d({\beth }^n\mathfrak{w},{\beth }^n\mathfrak{y}){⪯}_{i_2}{\mathcal{O}}_{1}d(\mathfrak{w},\mathfrak{y})+{\mathcal{O}}_2\left(d\left(\mathfrak{w},{\beth }^n\mathfrak{w}\right)+d\left(\mathfrak{y},{\beth }^n\mathfrak{y}\right)\right)$$

$$+{\mathcal{O}}_3\left(d\left(\mathfrak{y},{\beth }^n\mathfrak{w}\right)+d\left(\mathfrak{w},{\beth }^n\mathfrak{y}\right)\right)$$

$$+{\mathcal{O}}_4\frac{d\left(\mathfrak{w},{\beth }^n\mathfrak{w}\right)d\left(\mathfrak{y},{\beth }^n\mathfrak{y}\right)}{1+d\left(\mathfrak{w},\mathfrak{y}\right)}$$

$$+{\mathcal{O}}_5\frac{d\left(\mathfrak{y},{\beth }^n\mathfrak{w}\right)d\left(\mathfrak{w},{\beth }^n\mathfrak{y}\right)}{1+d\left(\mathfrak{w},\mathfrak{y}\right)},$$

for all ${\mathfrak{w}}_0,\mathfrak{w},\mathfrak{y}\in \overline{B({\mathfrak{w}}_0,\varkappa )}$, $0\prec \varkappa \in {\mathbb{C}}_2$ and

$$∥d({\mathfrak{w}}_0,{\beth }^n{\mathfrak{w}}_0)∥\le (1-\lambda )∥\varkappa ∥,$$

where $\lambda =\left(\frac{{\mathcal{O}}_1+{\mathcal{O}}_2+{\mathcal{O}}_3}{1-{\mathcal{O}}_2-{\mathcal{O}}_3-\sqrt{2}{\mathcal{O}}_4}\right)<1$. Then, there exists a unique point ${\mathfrak{w}}^{*}\in \overline{B({\mathfrak{w}}_0,\varkappa )}$, such that ${\mathfrak{w}}^{*}=\beth {\mathfrak{w}}^{*}$.

Proof. 

Taking $m=n$ and ${\mathfrak{F}}_1={\mathfrak{F}}_2=\beth$ in Corollary 4. □

4. Applications

Theorem 3.

Let $X=C^n$ be a bi-complex valued metric space with the metric

$$d(\mathfrak{w},\mathfrak{y})=\sum _{i=1}^n\left(\left|\mathfrak{w}\left(t\right)-\mathfrak{y}\left(t\right)\right|+i_3\left|\mathfrak{w}\left(t\right)-\mathfrak{y}\left(t\right)\right|\right)$$

for all $\mathfrak{w},\mathfrak{y}\in$ X. If

$$\sum _{j=1}^n\left|{\mathcal{O}}_{ij}\right|{⪯}_{i_2}\mathcal{O}<1,$$

for all $i=1,2,...,n,$ then the linear system

$$\left\{\begin{array}{c}{b}_1=a_{11}{\mathfrak{w}}_1+a_{12}{\mathfrak{w}}_2+...+a_{1n}{\mathfrak{w}}_n\\ b_2=a_{21}{\mathfrak{w}}_1+a_{22}{\mathfrak{w}}_2+...+a_{2n}{\mathfrak{w}}_n\\ ·\\ ·\\ ·\\ b_n=a_{n1}{\mathfrak{w}}_1+a_{n2}{\mathfrak{w}}_2+...+a_{nn}{\mathfrak{w}}_n\end{array}\right.$$

of n linear equations in n unknowns has a unique solution.

Proof. 

Define $\beth :X\to X$ by

$$\beth \left(\mathfrak{w}\right)=A\mathfrak{w}+b$$

where $\mathfrak{w}=\left({\mathfrak{w}}_1,{\mathfrak{w}}_2,...,{\mathfrak{w}}_n\right)\in {\mathbb{C}}^n,$ $b=\left(b_1,b_2,...,b_n\right)\in {\mathbb{C}}^n$ and

$$A=\left(\begin{array}{cccc}{a}_{11}& a_{12}& ...& a_{1n}\\ a_{21}& a_{22}& ...& a_{2n}\\ .& .& .& .\\ .& .& .& .\\ .& .& .& .\\ a_{n1}& a_{n2}& .& a_{nn}\end{array}\right).$$

Now,

$$d(\beth \left(\mathfrak{w}\right),\beth \left(\mathfrak{y}\right))=\sum _{j=1}^n\left(\left|{\mathcal{O}}_{ij}\left({\mathfrak{w}}_j-{\mathfrak{y}}_j\right)\right|+i_3\left|{\mathcal{O}}_{ij}\left({\mathfrak{w}}_j-{\mathfrak{y}}_j\right)\right|\right)$$

$${⪯}_{i_2}\sum _{j=1}^n\left|{\mathcal{O}}_{ij}\right|\left(\sum _{j=1}^n\left(\left|\left({\mathfrak{w}}_j-{\mathfrak{y}}_j\right)\right|+i_3\left|\left({\mathfrak{w}}_j-{\mathfrak{y}}_j\right)\right|\right)\right)$$

$${⪯}_{i_2}\mathcal{O}\left(\sum _{j=1}^n\left(\left|\left({\mathfrak{w}}_j-{\mathfrak{y}}_j\right)\right|+i_3\left|\left({\mathfrak{w}}_j-{\mathfrak{y}}_j\right)\right|\right)\right)$$

$$\begin{array}{ccc}\hfill & =& {\mathcal{O}}_{1}d(\mathfrak{w},\mathfrak{y})+{\mathcal{O}}_2\left(d\left(\mathfrak{w},\beth \mathfrak{w}\right)+d\left(\mathfrak{y},\beth \mathfrak{y}\right)\right)\hfill \\ & & +{\mathcal{O}}_3\left(d\left(\mathfrak{y},\beth \mathfrak{w}\right)+d\left(\mathfrak{w},\beth \mathfrak{y}\right)\right)\hfill \\ & & +{\mathcal{O}}_4\frac{d\left(\mathfrak{w},\beth \mathfrak{w}\right)d\left(\mathfrak{y},\beth \mathfrak{y}\right)}{1+d\left(\mathfrak{w},\mathfrak{y}\right)}\hfill \\ & & +{\mathcal{O}}_5\frac{d\left(\mathfrak{y},\beth \mathfrak{w}\right)d\left(\mathfrak{w},\beth \mathfrak{y}\right)}{1+d\left(\mathfrak{w},\mathfrak{y}\right)}.\hfill \end{array}$$

Hence, all the assertions of Corollary 1 are satisfied with ${\mathcal{O}}_1=\mathcal{O}=\frac{4}{5},$ ${\mathcal{O}}_2={\mathcal{O}}_3={\mathcal{O}}_4={\mathcal{O}}_5=0,$ and the linear system of the equation has a unique solution. □

5. Conclusions

In the present research work, we have established common fixed point results for locally contractive mappings in the background of a bicomplex valued metric space. In such a manner, we have generalized the leading theorems of Beg et al. [6], Gnanaprakasam et al. [7], Hardy-Roger [26], Reich [27], Chatterjea [28], and Kannan [29] in the framework of bicomplex valued metric spaces. We hope that the established results in this research article will form new connections for researchers who are working in the framework of bi-CVMS.

Common fixed points of self and set valued mappings in the background of bi-CVMS can be obtained for future work. Integral equations and inclusions can be examined as utilizations of the established results.