Solution of Fredholm Integral Equation via Common Fixed Point Theorem on Bicomplex Valued B-Metric Space

Abstract

The notion of symmetry is the main property of a metric function. The area of fixed point theory has a suitable structure for symmetry in mathematics. The goal of this paper is to find fixed point and common fixed point results in a bicomplex valued b-metric space for mixed type rational contractions with control functions. Some well-known literature findings were generalized in our main findings. We provide an example to strengthen and validate our main results. As an example, in the context of bicomplex-valued b-metric space, we develop fixed point and common fixed point results for the rational contraction mapping.

1. Introduction

Bakhtin [1] developed the notion of a “b-metric space” as a generalization of metric spaces in 1989. Rao et al. [2] presented the notion of complex-valued b-metric spaces in 2013, which was broader than the well-known complex-valued metric spaces established by Azam et al. [3] in 2011. In 1892, inspired by the work of Hamilton and Clifford, the mathematician Corrado Segre created what he termed “bicomplex numbers”, the algebra of which was identical to the algebra of tessarines. Segre [4] noted that the elements, or idempotents, play an important role in the theory of bicomplex numbers. Dragoni [5] established the first rudiments of a function theory on bicomplex numbers after Segre, which further expanded the theory of functions of bicomplex variables and was the next significant push in the study of bicomplex analyses (see [6,7]). In recent years, G.B. Price’s book [8] has become one of the most comprehensive studies of analysis in the bicomplex setting, and there has been a significant move toward the study of the properties of those functions on the ring of bicomplex numbers, the properties of which suggest a notion of bicomplex holomorphy. The resurgence of interest in this field has resulted in notable applications in mathematics, science, and technology. Several researchers have created an effective corpus of work. In 2011, Fabrizio Colombo et al. [9] presented singularities of functions of one and several bicomplex variables and proved a duality theorem for singularities of functions. In 2012, Elena Luna-Elizarraras et al. [10] showed a function theory on bicomplex numbers and a generalization of the theory of holomorphic functions for one and two complex variables. In 2012, Sitthikul and Saejung [11] proved fixed point theorems in complex valued metric spaces. In the same year, Sintunavarat and Kumam [12] proved common fixed point theorems in complex valued metric spaces. Choi et al. [13] very recently proposed the concept of a bicomplex-valued metric space, which is an extension of a complex-valued metric space and provided adequate criteria for the presence of shared fixed points in a pair of mappings meeting a contractive condition. In 2019, Jebril, Datta, Sarkar, and Biswas [14] generated a variety of fixed point outcomes using rational contractions in a bicomplex valued metric space. Beg, Kumar Datta, and Pal [15] demonstrated some fixed-point results in bicomplex valued metric spaces in 2021. Datta, Pal, Sarkar, and Saha ([16]) proved in 2020 that common fixed point theorems for contracting mappings in bicomplex with b-metric spaces exist. In 2021, Datta, Pal, Sarkar, and Manna [17] established the common fixed point theorem in bicomplex b-metric spaces. Tassaddiq, Ahmad, Eqal Al-Mazrooei, Lateef, and Lakhani [18] proved the common fixed point results in bicomplex valued metric spaces using an application. Rezapour et al. [19] proved fixed point theorems and the characteristics of the resolvent operator corresponding to the second-order integrodifferential equation. Vijayakumar et al. [20] demonstrated Gronwall’s inequality, which eliminates the necessity for compactness of the resolvent operator and the traditional fixed point theorems. Arul Joseph, Mahmoud, Gunaseelan, Cherif, and Idris [21] proved the common fixed point theorem on a bicomplex valued metric space, as follows:

Theorem 1. 

Let $(\Omega ,\varrho )$ be a complete bicomplex valued metric space and $\mathcal{S}$, Υ be self-mappings such that

$$\begin{array}{c}\hfill \varrho (\mathcal{S}\sigma ,{\rm Y}\varpi ){⪯}_{i_2}\alpha \varrho (\sigma ,\varpi )+\frac{\mu \varrho (\sigma ,\mathcal{S}\sigma )\varrho (\varpi ,{\rm Y}\varpi )+\gamma \varrho (\varpi ,\mathcal{S}\sigma )\varrho (\sigma ,{\rm Y}\varpi )}{1+\varrho (\sigma ,\varpi )},\end{array}$$

for all $\sigma ,\varpi \in \Omega$, where $\sigma ,\beta ,\gamma$ are non-negative reals with $\alpha +\sqrt{2}\beta +\sqrt{2}\gamma <1$; then, $\mathcal{S}$ and Υ have a unique common fixed point.

Motivated by the above theorem, we prove the fixed point and common fixed point theorems on a bicomplex b-metric space using control functions.

2. Preliminaries

Throughout this paper, we denote the set of real, complex, and bicomplex numbers, respectively, as ${\mathbb{C}}_0$, ${\mathbb{C}}_1$, and ${\mathbb{C}}_2$. Segre [4] defined the complex number as follows:

$$\rho ={\eta }_1+{\eta }_2{i}_1,$$

where ${\eta }_1,{\eta }_2\in {\mathbb{C}}_0$, $i_1^2=-1$. Define

$${\mathbb{C}}_1:=\{\rho :\rho ={\eta }_1+{\eta }_2{i}_1,{\eta }_1,{\eta }_2\in {\mathbb{C}}_0\}.$$

Let $\rho \in {\mathbb{C}}_1$; then, $\left|\rho \right|={({\eta }_1^2+{\eta }_2^2)}^{\frac{1}{2}}$. The norm $\left|\right|.\left|\right|:{\mathbb{C}}_1\to {\mathbb{C}}_0^{+}$ is then defined by

$$\begin{array}{cc}\hfill \left|\right|\rho \left|\right|& ={({\eta }_1^2+{\eta }_2^2)}^{\frac{1}{2}}.\hfill \end{array}$$

Segre [4] defined the bicomplex number as follows:

$$\varphi ={\eta }_1+{\eta }_2{i}_1+{\eta }_3{i}_2+{\eta }_4{i}_1{i}_2,$$

where ${\eta }_1,{\eta }_2,{\eta }_3,{\eta }_4\in {\mathbb{C}}_0$ and independent units $i_1,i_2$ satisfy $i_1^2=i_2^2=-1$ and $i_1{i}_2=i_2{i}_1$. Define

$${\mathbb{C}}_2:=\{\varphi :\varphi ={\eta }_1+{\eta }_2{i}_1+{\eta }_3{i}_2+{\eta }_4{i}_1{i}_2,{\eta }_1,{\eta }_2,{\eta }_3,{\eta }_4\in {\mathbb{C}}_0\},$$

i.e.,

$${\mathbb{C}}_2:=\{\varphi :\varphi ={\rho }_1+i_2{\rho }_2,{\rho }_1,{\rho }_2\in {\mathbb{C}}_1\},$$

where ${\rho }_1={\eta }_1+{\eta }_2{i}_1\in {\mathbb{C}}_1$ and ${\rho }_2={\eta }_3+{\eta }_4{i}_1\in {\mathbb{C}}_1$. If $\varphi ={\rho }_1+i_2{\rho }_2$ and $\theta ={\psi }_1+i_2{\psi }_2$ are any two bicomplex numbers, then the sum is $\varphi \pm \theta =({\rho }_1+i_2{\rho }_2)\pm ({\psi }_1+i_2{\psi }_2)={\rho }_1\pm {\psi }_1+i_2({\rho }_2\pm {\psi }_2)$ and the product is $\varphi .\theta =({\rho }_1+i_2{\rho }_2)({\psi }_1+i_2{\psi }_2)=({\rho }_1{\psi }_1-{\rho }_2{\psi }_2)+i_2({\rho }_1{\psi }_2+{\rho }_2{\psi }_1)$. An element $\varphi ={\rho }_1+i_2{\rho }_2\in {\mathbb{C}}_2$ is non-singular if and only if $\left|\right|{\rho }_1^2+{\rho }_2^2\left|\right|\ne 0$ and singular if and only if $\left|\right|{\rho }_1^2+{\rho }_2^2\left|\right|=0$. When it exists, the inverse of $\varphi$ is as follows:

$${\varphi }^{-1}=\theta =\frac{{\rho }_1-i_2{\rho }_2}{{\rho }_1^2+{\rho }_2^2}.$$

The norm $\left|\right|.\left|\right|:{\mathbb{C}}_2\to {\mathbb{C}}_0^{+}$ defined by

$$\begin{array}{cc}\hfill \left|\right|\varphi \left|\right|& =\left|\right|{\rho }_1+i_2{\rho }_2\left|\right|=\left\{\right||{\rho }_1{\left|\right|}^2+\left|\right|{\rho }_2{\left|\right|}^2{\}}^{\frac{1}{2}}\hfill \\ \hfill & ={\left[\frac{|{\rho }_1-i_1{\rho }_2{|}^2+{|{\rho }_1+i_1{\rho }_2|}^2}{2}\right]}^{\frac{1}{2}}\hfill \\ \hfill & ={({\eta }_1^2+{\eta }_2^2+{\eta }_3^2+{\eta }_4^2)}^{\frac{1}{2}},\hfill \end{array}$$

where $\varphi ={\eta }_1+{\eta }_2{i}_1+{\eta }_3{i}_2+{\eta }_4{i}_1{i}_2={\rho }_1+i_2{\rho }_2\in {\mathbb{C}}_2$.

The vector space ${\mathbb{C}}_2$ with respect to a defined norm is a normed linear space, and ${\mathbb{C}}_2$ is complete. Therefore, ${\mathbb{C}}_2$ is a Banach space. If $\varphi ,\theta \in {\mathbb{C}}_2$, then $\left|\right|\varphi \theta \left|\right|\le \sqrt{2}\left|\right|\varphi \left|\right|\left|\right|\theta \left|\right|$ holds instead of $\left|\right|\varphi \theta \left|\right|\le \left|\right|\varphi \left|\right|\left|\right|\theta \left|\right|$, and therefore, ${\mathbb{C}}_2$ is not a Banach algebra. For any two bicomplex numbers $\varphi ,\theta \in {\mathbb{C}}_2$, such that

• $\varphi {⪯}_{i_2}\theta \iff \left|\right|\varphi \left|\right|\le \left|\right|\theta \left|\right|$;
• $\left|\right|\varphi +\theta \left|\right|\le \left|\right|\varphi \left|\right|+\left|\right|\theta \left|\right|$;
• $\left|\right|\eta \varphi \left|\right|=\left|\eta \right|\left|\right|\varphi \left|\right|$, where $\eta \in \mathbb{R}$;
• $\left|\right|\varphi \theta \left|\right|\le \sqrt{2}\left|\right|\varphi \left|\right|\left|\right|\theta \left|\right|$, and the equality holds only when at least one of $\varphi$ and $\theta$ is degenerated;
• $\left|\right|{\varphi }^{-1}\left|\right|={\left|\right|\varphi \left|\right|}^{-1}$ if $\varphi$ is a degenerated with $0\prec \varphi \in {\mathbb{C}}_2$;
• $\left|\right|\frac{\varphi }{\theta }\left|\right|=\frac{\left|\right|\varphi \left|\right|}{\left|\right|\theta \left|\right|}$, if $\theta \in {\mathbb{C}}_2$ is a degenerated.

The partial order relation ${⪯}_{i_2}$ on ${\mathbb{C}}_2$ is defined as follows: Let $\varphi ={\rho }_1+i_2{\rho }_2$, $\theta ={\psi }_1+i_2{\psi }_2\in {\mathbb{C}}_2$. Then, $\varphi {⪯}_{i_2}\theta$ if and only if ${\rho }_1⪯{\psi }_1$, and ${\rho }_2⪯{\psi }_2$, i.e., $\varphi {⪯}_{i_2}\theta$ if one of the following postulates is fulfilled:

• ${\rho }_1={\psi }_1$, ${\rho }_2={\psi }_2$,
• ${\rho }_1\prec {\psi }_1$, ${\rho }_2={\psi }_2$,
• ${\rho }_1={\psi }_1$, ${\rho }_2\prec {\psi }_2$,
• ${\rho }_1\prec {\psi }_1$, ${\rho }_2\prec {\psi }_2$.

In particular, we can write $\varphi {⋦}_{i_2}\theta$, if $\varphi {⪯}_{i_2}\theta$ and $\varphi \ne \theta$, i.e., one of (2)–(4) is fulfilled, and we write $\varphi {\prec }_{i_2}\theta$, if only (4) is fulfilled.

Definition 1 

([16]). Let Ω be a non-empty set, and let $\mathfrak{s}\ge 1$ be a given real number. A function $\varrho :\Omega \times \Omega \to {\mathbb{C}}_2$ is called a bicomplex b-metric on Ω if for all $\sigma ,\varpi ,\beta \in \Omega$ such that

(i)

$0{⪯}_{{\mathfrak{i}}_2}\varrho (\sigma ,\varpi )$ and $\varrho (\sigma ,\varpi )=0$ if and only if $\sigma =\varpi$;

(ii)

$\varrho (\sigma ,\varpi )=\varrho (\varpi ,\sigma )$;

(iii)

$\varrho (\sigma ,\varpi ){⪯}_{{\mathfrak{i}}_2}\mathfrak{s}[\varrho (\sigma ,\beta )+\varrho (\beta ,\varpi )]$.

The pair $(\Omega ,\varrho )$ is called a bicomplex b-metric space.

Definition 2 

([16]). Let $(\Omega ,\varrho )$ be a bicomplex b-metric space. A point $\sigma \in \Omega$ is said to be an interior point of a set $\mathcal{Q}\subseteq \Omega$ whenever we can find $0\prec \omega \in \mathbb{C}$ satisfying $\mathcal{R}(\sigma ,\omega ):=\{\varpi \in \Omega :\varrho (\sigma ,\varpi ){\prec }_{{\mathfrak{i}}_2}\omega \}\subseteq \mathcal{Q}$, where $\mathcal{R}(\sigma ,\omega )$ is an open ball. Then, $\mathcal{R}(\sigma ,\omega )=\{\varpi \in \Omega :\varrho (\sigma ,\varpi ){⪯}_{{\mathfrak{i}}_2}\omega \}$ is a closed ball.

Definition 3 

([16]). Let $(\Omega ,\varrho )$ be a bicomplex b-metric space, $\left\{{\sigma }_{\lambda }\right\}$ be a sequence in Ω, and $\sigma \in \Omega$.

(i)

If for every $\mathfrak{c}\in \mathbb{C}$, with $0{\prec }_{{\mathfrak{i}}_2}\omega$, there is $\mathcal{K}\in \mathbb{N}$ satisfying for all $\lambda >\mathcal{K}$, $\varrho ({\sigma }_{\lambda },\sigma ){\prec }_{{\mathfrak{i}}_2}\mathfrak{c}$, then $\left\{{\sigma }_{\lambda }\right\}$ is said to be convergent, $\left\{{\sigma }_{\lambda }\right\}$ converges to σ, and σ is the limit point of $\left\{{\sigma }_{\lambda }\right\}$. We denote this by $li{m}_{\lambda \to \infty }{\sigma }_{\lambda }=\sigma$ or $\left\{{\sigma }_{\lambda }\right\}\to \sigma$, as $\lambda \to \infty$.

(ii)

If for every $\mathfrak{c}\in \mathbb{C}$, with $0{\prec }_{{\mathfrak{i}}_2}\omega$, there is $\mathcal{K}\in \mathbb{N}$ satisfying for all $\lambda >\mathcal{K}$ and $\varrho ({\sigma }_{\lambda },{\sigma }_{\lambda +\mathfrak{m}}){\prec }_{{\mathfrak{i}}_2}\mathfrak{c}$, where $\mathfrak{m}\in \mathbb{N}$, then $\left\{{\sigma }_{\lambda }\right\}$ is called a Cauchy sequence.

(iii)

If every Cauchy sequence in Ω is convergent, then $(\Omega ,\varrho )$ is said to be a complete bicomplex b-metric space.

Lemma 1. 

Let $(\Omega ,\varrho )$ be a bicomplex b-metric space. A sequence $\left\{{\sigma }_{\lambda }\right\}\in \Omega$ is converges to $\sigma \in \Omega$ iff $\underset{\lambda \to \infty }{lim}\varrho (\sigma ,{\sigma }_{\lambda })=0$.

Proof. 

Assume that $\left\{{\sigma }_n\right\}$ converges to $\sigma$. Let $ϵ>0$. Suppose

$$\begin{array}{c}\hfill \omega =\frac{ϵ}{2}+i_1\frac{ϵ}{2}+i_2\frac{ϵ}{2}+i_1{i}_2\frac{ϵ}{2}.\end{array}$$

Then, $0{\prec }_{i_2}\omega \in {\mathbb{C}}_2^{+}$, and we can find $\Lambda \in \mathbb{N}$ satisfying ${\sigma }_{\lambda }\in {\mathfrak{B}}_{\varrho }(\sigma ,\omega )$, for all $\lambda \ge \Lambda$, i.e., $\varrho ({\sigma }_{\lambda },\sigma ){\prec }_{i_2}\omega$. Therefore,

$$\begin{array}{c}\hfill \left|\right|\varrho ({\sigma }_{\lambda },\sigma )-0\left|\right|<ϵ,\mathrm{for}\mathrm{all}\lambda \ge \Lambda .\end{array}$$

Therefore, $\varrho ({\sigma }_{\lambda },\sigma )\to 0$, as $\lambda \to \infty$. Conversely, assume that $\varrho ({\sigma }_{\lambda },\sigma )\to 0$, as $\lambda \to \infty$. Then, for each $0{\prec }_{i_2}\omega \in {\mathbb{C}}_2^{+}$, we can find a real $ϵ>0$ satisfying for all $\vartheta \in {\mathbb{C}}_2^{+}$,

$$\begin{array}{c}\hfill \left|\right|\vartheta \left|\right|<ϵ\Rightarrow \vartheta {\prec }_{i_2}\omega .\end{array}$$

Then, for this $ϵ>0$, we can find $\Lambda \in \mathbb{N}$ satisfying

$$\begin{array}{c}\hfill \left|\right|\varrho ({\sigma }_{\lambda },\sigma )-0\left|\right|<ϵ\mathrm{for}\mathrm{all}\lambda \ge \Lambda .\end{array}$$

Therefore,

$$\begin{array}{c}\hfill \varrho ({\sigma }_{\lambda },\sigma ){\prec }_{i_2}\omega \mathrm{for}\mathrm{all}\lambda \ge \Lambda .\end{array}$$

Hence, $\left\{{\sigma }_{\lambda }\right\}$ converges to a point $\sigma$. □

Lemma 2. 

Let $(\Omega ,\varrho )$ be a bicomplex b-metric space and $\left\{{\sigma }_{\lambda }\right\}$ be a sequence in Ω. Then, $\left\{{\sigma }_{\lambda }\right\}$ is a Cauchy sequence in Ω iff $\underset{\lambda \to \infty }{lim}\varrho ({\sigma }_{\lambda },{\sigma }_{\mathfrak{m}})=0$.

Proof. 

Assume that $\left\{{\sigma }_n\right\}$ is a Cauchy sequence in $\Omega$. Let $ϵ>0$. Suppose

$$\begin{array}{c}\hfill \omega =\frac{ϵ}{2}+i_1\frac{ϵ}{2}+i_2\frac{ϵ}{2}+i_1{i}_2\frac{ϵ}{2}.\end{array}$$

Then $0{\prec }_{i_2}\omega \in {\mathbb{C}}_2^{+}$, we can find $\Lambda \in \mathbb{N}$ satisfying ${\sigma }_{\lambda }\in {\mathfrak{B}}_{\varrho }({\sigma }_{\mathfrak{m}},\omega )$, for all $\lambda ,\mathfrak{m}\ge \Lambda$ i.e., $\varrho ({\sigma }_{\lambda },{\sigma }_{\mathfrak{m}}){\prec }_{i_2}\omega$. Therefore

$$\begin{array}{c}\hfill \left|\right|\varrho ({\sigma }_{\lambda },{\sigma }_{\mathfrak{m}})-0\left|\right|<ϵ,\mathrm{for}\mathrm{all}\lambda ,\mathfrak{m}\ge \Lambda .\end{array}$$

Therefore, $\varrho ({\sigma }_{\lambda },{\sigma }_{\mathfrak{m}})\to 0$, as $\lambda ,\mathfrak{m}\to \infty$. Conversely, assume that $\varrho ({\sigma }_{\lambda },{\sigma }_{\mathfrak{m}})\to \varrho (\sigma ,\sigma )$, as $\lambda ,\mathfrak{m}\to \infty$. Then for each $0{\prec }_{i_2}\omega \in {\mathbb{C}}_2^{+}$, we can find a real $ϵ>0$ satisfying for all $\vartheta \in {\mathbb{C}}_2^{+}$,

$$\begin{array}{c}\hfill \left|\right|\vartheta \left|\right|<ϵ\Rightarrow \vartheta {\prec }_{i_2}\omega .\end{array}$$

For this $ϵ>0$, we can find $\Lambda \in \mathbb{N}$ satisfying

$$\begin{array}{c}\hfill \left|\right|\varrho ({\sigma }_{\lambda },{\sigma }_{\mathfrak{m}})-0\left|\right|<ϵ,\mathrm{for}\mathrm{all}\lambda ,\mathfrak{m}\ge \Lambda .\end{array}$$

Therefore,

$$\begin{array}{c}\hfill \varrho ({\sigma }_{\lambda },{\sigma }_{\mathfrak{m}}){\prec }_{i_2}\omega ,\mathrm{for}\mathrm{all}\lambda ,\mathfrak{m}\ge \Lambda .\end{array}$$

Hence, $\left\{{\sigma }_{\lambda }\right\}$ is a Cauchy sequence. □

3. Main Results

Now, we prove our first result.

Theorem 2. 

Let $(\Omega ,\varrho )$ be a complete bicomplex b-metric space with the coefficient $\mathfrak{s}\ge 1$ and $\Gamma :\Omega \to \Omega$. If there exists a mapping ${\lambda }_1,{\lambda }_2,{\lambda }_3,{\lambda }_4:\Omega \to [0,\frac{1}{\mathfrak{s}})$ such that for all $\sigma ,\varpi \in \Omega$:

(i)

${\lambda }_1\left(\Gamma \sigma \right)\le {\lambda }_1\left(\sigma \right),{\lambda }_2\left(\Gamma \sigma \right)\le {\lambda }_2\left(\sigma \right),{\lambda }_3\left(\Gamma \sigma \right)\le {\lambda }_3\left(\sigma \right),\mathrm{and}{\lambda }_4\left(\Gamma \sigma \right)\le {\lambda }_4\left(x\right)$;

(ii)

$({\lambda }_1+2\mathfrak{s}{\lambda }_2+{\lambda }_3+{\lambda }_4)\left(\sigma \right)<1$;

(iii)

$\varrho (\Gamma \sigma ,\Gamma \varpi ){⪯}_{{\mathfrak{i}}_2}{\lambda }_1\left(\sigma \right)\varrho (\sigma ,\varpi )+{\lambda }_2\left(\sigma \right)\frac{\varrho (\sigma ,\Gamma \varpi )(1+\varrho (\sigma ,\Gamma \sigma \left)\right)}{1+\varrho (\sigma ,\varpi )}+{\lambda }_3\left(\sigma \right)\frac{\varrho (\varpi ,\Gamma \varpi )(1+\varrho (\sigma ,\Gamma \sigma \left)\right)}{1+\varrho (\sigma ,\varpi )}+{\lambda }_4\left(\sigma \right)max\{\varrho (\sigma ,\Gamma \sigma ),\varrho (\varpi ,\Gamma \varpi )\}$,

then, Γ has a fixed point in Ω.

Proof. 

Let ${\sigma }_0\in \Omega$, define sequence $\left\{{\sigma }_{\lambda }\right\}$ in Ω satisfying ${\sigma }_{\lambda }=\Gamma {\sigma }_{\lambda -1}$.

$$\begin{array}{cc}\hfill \varrho ({\sigma }_{\lambda },{\sigma }_{\lambda +1})& =\varrho (\Gamma {\sigma }_{\lambda -1},\Gamma {\sigma }_{\lambda })\hfill \\ \hfill & {⪯}_{{\mathfrak{i}}_2}{\lambda }_1\left({\sigma }_{\lambda -1}\right)\varrho ({\sigma }_{\lambda -1},{\sigma }_{\lambda })+{\lambda }_2\left({\sigma }_{\lambda -1}\right)\frac{\varrho ({\sigma }_{\lambda -1},\Gamma {\sigma }_{\lambda })(1+\varrho ({\sigma }_{\lambda -1},\Gamma {\sigma }_{\lambda -1}))}{1+\varrho ({\sigma }_{\lambda -1},{\sigma }_{\lambda })}\hfill \\ \hfill & +{\lambda }_3\left({\sigma }_{\lambda -1}\right)\frac{\varrho ({\sigma }_{\lambda },\Gamma {\sigma }_{\lambda })(1+\varrho ({\sigma }_{\lambda -1},\Gamma {\sigma }_{\lambda -1}))}{1+\varrho ({\sigma }_{\lambda -1},{\sigma }_{\lambda })}\hfill \\ \hfill & +{\lambda }_4\left({\sigma }_{\lambda -1}\right)max\{\varrho ({\sigma }_{\lambda -1},\Gamma {\sigma }_{\lambda -1}),\varrho ({\sigma }_{\lambda },\Gamma {\sigma }_{\lambda })\}\hfill \\ \hfill & {⪯}_{{\mathfrak{i}}_2}{\lambda }_1\left(\Gamma {\sigma }_{\lambda -2}\right)\varrho ({\sigma }_{\lambda -1},{\sigma }_{\lambda })+{\lambda }_2\left(\Gamma {\sigma }_{\lambda -2}\right)\varrho ({\sigma }_{\lambda -1},{\sigma }_{\lambda +1})\hfill \\ \hfill & +{\lambda }_3\left(\Gamma {\sigma }_{\lambda -2}\right)\varrho ({\sigma }_{\lambda },{\sigma }_{\lambda +1})+{\lambda }_4\left(\Gamma {\sigma }_{\lambda -2}\right)max\{\varrho ({\sigma }_{\lambda -1},{\sigma }_{\lambda }),\varrho ({\sigma }_{\lambda },{\sigma }_{\lambda +1})\}\hfill \\ \hfill & ⋮\hfill \\ \hfill & {⪯}_{{\mathfrak{i}}_2}{\lambda }_1\left({\sigma }_0\right)\varrho ({\sigma }_{\lambda -1},{\sigma }_{\lambda })+{\lambda }_2\left({\sigma }_0\right)\varrho ({\sigma }_{\lambda -1},{\sigma }_{\lambda +1})+{\lambda }_3\left({\sigma }_0\right)\varrho ({\sigma }_{\lambda },{\sigma }_{\lambda +1})\hfill \\ \hfill & +{\lambda }_4\left({\sigma }_0\right){\mathcal{M}}_1,\hfill \end{array}$$

(1)

where ${\mathcal{M}}_1=max\{\varrho ({\sigma }_{\lambda -1},{\sigma }_{\lambda }),\varrho ({\sigma }_{\lambda },{\sigma }_{\lambda +1})\}$.

CASE(1)

Suppose that ${\mathcal{M}}_1=\varrho ({\sigma }_{\lambda -1},{\sigma }_{\lambda })$; then, we obtain

$$\begin{array}{cc}\hfill \varrho ({\sigma }_{\lambda },{\sigma }_{\lambda +1})& {⪯}_{{\mathfrak{i}}_2}\frac{{\lambda }_1\left({\sigma }_0\right)+\mathfrak{s}{\lambda }_2\left({\sigma }_0\right)+{\lambda }_4\left({\sigma }_0\right)}{1-{\lambda }_3\left({\sigma }_0\right)-\mathfrak{s}{\lambda }_2\left({\sigma }_0\right)}\varrho ({\sigma }_{\lambda -1},{\sigma }_{\lambda })\hfill \end{array}$$

$$\begin{array}{c}\hfill \left|\right|\varrho ({\sigma }_{\lambda },{\sigma }_{\lambda +1})\left|\right|\le \frac{{\lambda }_1\left({\sigma }_0\right)+\mathfrak{s}{\lambda }_2\left({\sigma }_0\right)+{\lambda }_4\left({\sigma }_0\right)}{1-{\lambda }_3\left({\sigma }_0\right)-\mathfrak{s}{\lambda }_2\left({\sigma }_0\right)}\left|\right|\varrho ({\sigma }_{\lambda -1},{\sigma }_{\lambda })\left|\right|.\end{array}$$

Since ${\lambda }_1+2\mathfrak{s}{\lambda }_2+{\lambda }_3+{\lambda }_4\left(\sigma \right)<1$, we obtain $\frac{{\lambda }_1\left({\sigma }_0\right)+\mathfrak{s}{\lambda }_2\left({\sigma }_0\right)+{\lambda }_4\left({\sigma }_0\right)}{1-{\lambda }_3\left({\sigma }_0\right)-\mathfrak{s}{\lambda }_2\left({\sigma }_0\right)}<1$.

Therefore, with $\mathfrak{k}=\frac{{\lambda }_1\left({\sigma }_0\right)+\mathfrak{s}{\lambda }_2\left({\sigma }_0\right)+{\lambda }_4\left({\sigma }_0\right)}{1-{\lambda }_3\left({\sigma }_0\right)-\mathfrak{s}{\lambda }_2\left({\sigma }_0\right)}<1$ and for all $\lambda \ge 0$, consequently, we have

$$\begin{array}{c}\hfill \left|\right|\varrho ({\sigma }_{\lambda },{\sigma }_{\lambda +1})\left|\right|\le \mathfrak{k}\left|\right|\varrho ({\sigma }_{\lambda -1},{\sigma }_{\lambda })\left|\right|\le {\mathfrak{k}}^2\left|\right|\varrho ({\sigma }_{\lambda -2},{\sigma }_{\lambda -1})\left|\right|\le \cdots \le {\mathfrak{k}}^{\lambda }\left|\right|\varrho ({\sigma }_0,{\sigma }_1)\left|\right|.\end{array}$$

That is, $\left|\right|\varrho ({\sigma }_{\lambda +1},{\sigma }_{\lambda +2})\left|\right|\le {\mathfrak{k}}^{\lambda +1}\left|\right|\varrho ({\sigma }_0,{\sigma }_1)\left|\right|.$

CASE(2)

Suppose that ${\mathcal{M}}_1=\varrho ({\sigma }_{\lambda },{\sigma }_{\lambda +1})$; then, we obtain

$$\begin{array}{cc}\hfill \varrho ({\sigma }_{\lambda },{\sigma }_{\lambda +1})& {⪯}_{{\mathfrak{i}}_2}\frac{{\lambda }_1\left({\sigma }_0\right)+\mathfrak{s}{\lambda }_2\left({\sigma }_0\right)}{1-{\lambda }_3\left({\sigma }_0\right)-{\lambda }_4\left({\sigma }_0\right)-\mathfrak{s}{\lambda }_2\left({\sigma }_0\right)}\varrho ({\sigma }_{\lambda -1},{\sigma }_{\lambda }).\hfill \end{array}$$

Therefore,

$$\begin{array}{c}\hfill \left|\right|\varrho ({\sigma }_{\lambda },{\sigma }_{\lambda +1})\left|\right|\le \frac{{\lambda }_1\left({\sigma }_0\right)+\mathfrak{s}{\lambda }_2\left({\sigma }_0\right)}{1-{\lambda }_3\left({\sigma }_0\right)-{\lambda }_4\left({\sigma }_0\right)-\mathfrak{s}{\lambda }_2\left({\sigma }_0\right)}\left|\right|\varrho ({\sigma }_{\lambda -1},{\sigma }_{\lambda })\left|\right|.\end{array}$$

Since ${\lambda }_1+2\mathfrak{s}{\lambda }_2+{\lambda }_3+{\lambda }_4\left(\sigma \right)<1$, we obtain $\frac{{\lambda }_1\left({\sigma }_0\right)+\mathfrak{s}{\lambda }_2\left({\sigma }_0\right)}{1-{\lambda }_3\left({\sigma }_0\right)-{\lambda }_4\left({\sigma }_0\right)-\mathfrak{s}{\lambda }_2\left({\sigma }_0\right)}<1$.

Therefore, with $\mathfrak{k}=\frac{{\lambda }_1\left({\sigma }_0\right)+\mathfrak{s}{\lambda }_2\left({\sigma }_0\right)}{1-{\lambda }_3\left({\sigma }_0\right)-{\lambda }_4\left({\sigma }_0\right)-\mathfrak{s}{\lambda }_2\left({\sigma }_0\right)}<1$ and for all $\lambda \ge 0$, consequently, we have

$$\begin{array}{c}\hfill \left|\right|\varrho ({\sigma }_{\lambda },{\sigma }_{\lambda +1})\left|\right|\le \mathfrak{k}\left|\right|\varrho ({\sigma }_{\lambda -1},{\sigma }_{\lambda })\left|\right|\le {\mathfrak{k}}^2\left|\right|\varrho ({\sigma }_{\lambda -2},{\sigma }_{\lambda -1})\left|\right|\le \cdots \le {\mathfrak{k}}^{\lambda }\left|\right|\varrho ({\sigma }_0,{\sigma }_1)\left|\right|.\end{array}$$

That is, $\left|\right|\varrho ({\sigma }_{\lambda +1},{\sigma }_{\lambda +2})\left|\right|\le {\mathfrak{k}}^{\lambda +1}\left|\right|\varrho ({\sigma }_0,{\sigma }_1)\left|\right|$. Let $\mathfrak{m},\lambda \in \mathbb{N},\mathfrak{m}>\lambda$,

$$\begin{array}{cc}\hfill \varrho ({\sigma }_{\lambda },{\sigma }_{\mathfrak{m}})& {⪯}_{{\mathfrak{i}}_2}\mathfrak{s}[\varrho ({\sigma }_{\lambda },{\sigma }_{\lambda +1})+\varrho ({\sigma }_{\lambda +1},{\sigma }_{\mathfrak{m}})],\hfill \\ \hfill & {⪯}_{{\mathfrak{i}}_2}\mathfrak{s}\varrho ({\sigma }_{\lambda },{\sigma }_{\lambda +1})+{\mathfrak{s}}^2[\varrho ({\sigma }_{\lambda +1},{\sigma }_{\lambda +2})+\varrho ({\sigma }_{\lambda +2},{\sigma }_{\mathfrak{m}})],\hfill \\ \hfill & {⪯}_{{\mathfrak{i}}_2}\mathfrak{s}\varrho ({\sigma }_{\lambda },{\sigma }_{\lambda +1})+{\mathfrak{s}}^2\varrho ({\sigma }_{\lambda +1},{\sigma }_{\lambda +2})+{\mathfrak{s}}^3\varrho ({\sigma }_{\lambda +2},{\sigma }_{\lambda +1})+\cdots \cdots \hfill \\ \hfill & {⪯}_{{\mathfrak{i}}_2}\mathfrak{s}{\mathfrak{k}}^{\lambda }\varrho ({\sigma }_0,{\sigma }_1)+{\mathfrak{s}}^2{\mathfrak{k}}^{\lambda +1}\varrho ({\sigma }_{\lambda +1},{\sigma }_{\lambda +2})+{\mathfrak{s}}^3{\mathfrak{k}}^{\lambda +2}\varrho ({\sigma }_0,{\sigma }_1)+\cdots \cdots \hfill \\ \hfill & {⪯}_{{\mathfrak{i}}_2}\mathfrak{s}{\mathfrak{k}}^{\lambda }\varrho ({\sigma }_0,{\sigma }_1)[1+\mathfrak{s}\mathfrak{k}+{\left(\mathfrak{s}\mathfrak{k}\right)}^2+{\left(\mathfrak{s}\mathfrak{k}\right)}^3+\cdots \cdots ]\hfill \\ \hfill & =\frac{\mathfrak{s}{\mathfrak{k}}^{\lambda }}{1-\mathfrak{s}\mathfrak{k}}\varrho ({\sigma }_0,{\sigma }_1).\hfill \end{array}$$

(2)

Therefore,

$$\begin{array}{c}\hfill \left|\right|\varrho ({\sigma }_{\lambda },{\sigma }_{\mathfrak{m}})\left|\right|\le \frac{\mathfrak{s}{\mathfrak{k}}^{\lambda }}{1-\mathfrak{s}\mathfrak{k}}\left|\right|\varrho ({\sigma }_0,{\sigma }_1)\left|\right|\to \infty ,\mathrm{as}\mathfrak{m},\lambda \to \infty .\end{array}$$

Thus, $\left\{{\sigma }_{\lambda }\right\}$ is a Cauchy sequence in Ω. Since Ω is complete, we can find $\mathfrak{u}\in \Omega$ satisfying ${\sigma }_{\lambda }\to \mathfrak{u}$ as $\lambda \to \infty$. Suppose not, then there exists $\beta \in \Omega$ such that

$$\left|\right|\varrho (\mathfrak{u},\Gamma \mathfrak{u})\left|\right|=\left|\right|\beta \left|\right|>0.$$

(3)

By the notion of a bicomplex b-metric ϱ, we have

$$\begin{array}{cc}\hfill \beta =\varrho (\mathfrak{u},\Gamma \mathfrak{u})& {⪯}_{{\mathfrak{i}}_2}\mathfrak{s}\varrho (\mathfrak{u},{\sigma }_{\mathfrak{n}+1})+\mathfrak{s}\varrho ({\sigma }_{\lambda +1},\Gamma \mathfrak{u})\hfill \\ \hfill & {⪯}_{{\mathfrak{i}}_2}\mathfrak{s}\varrho (\mathfrak{u},{\sigma }_{\lambda +1})+\mathfrak{s}\varrho (\Gamma \mathfrak{u},\Gamma {\sigma }_{\lambda })\hfill \\ \hfill & {⪯}_{{\mathfrak{i}}_2}\mathfrak{s}\varrho (\mathfrak{u},{\sigma }_{\lambda +1})+\mathfrak{s}{\lambda }_1\left(\mathfrak{u}\right)\varrho (\mathfrak{u},{\sigma }_{\lambda })+\mathfrak{s}{\lambda }_2\left(\mathfrak{u}\right)\frac{\varrho (\mathfrak{u},\Gamma {\sigma }_{\lambda })(1+\varrho (\mathfrak{u},\Gamma \mathfrak{u}))}{1+\varrho (\mathfrak{u},{\sigma }_{\lambda })}\hfill \\ \hfill & +\mathfrak{s}{\lambda }_3\left(\mathfrak{u}\right)\frac{\varrho ({\sigma }_{\lambda },\Gamma {\sigma }_{\lambda })(1+\varrho (\mathfrak{u},\Gamma \mathfrak{u}))}{1+\varrho (\mathfrak{u},{\sigma }_{\lambda })}+\mathfrak{s}{\lambda }_4\left(\mathfrak{u}\right)max\{\varrho (\mathfrak{u},\Gamma \mathfrak{u}),\varrho ({\sigma }_{\lambda },\Gamma {\sigma }_{\lambda })\}\hfill \\ \hfill & {⪯}_{{\mathfrak{i}}_2}\mathfrak{s}\varrho (\mathfrak{u},{\sigma }_{\lambda +1})+\mathfrak{s}{\lambda }_1\left(\mathfrak{u}\right)\varrho (\mathfrak{u},{\sigma }_{\lambda })+\mathfrak{s}{\lambda }_2\left(\mathfrak{u}\right)\frac{\varrho (\mathfrak{u},{\sigma }_{\lambda +1})(1+\varrho (\mathfrak{u},\Gamma \mathfrak{u}))}{1+\varrho (\mathfrak{u},{\sigma }_{\lambda })}\hfill \\ \hfill & +\mathfrak{s}{\lambda }_3\left(\mathfrak{u}\right)\frac{\varrho ({\sigma }_{\lambda },{\sigma }_{\lambda +1})(1+\varrho (\mathfrak{u},\Gamma \mathfrak{u}))}{1+\varrho (\mathfrak{u},{\sigma }_{\lambda })}+\mathfrak{s}{\lambda }_4\left(\mathfrak{u}\right)max\{\varrho (\mathfrak{u},\Gamma \mathfrak{u}),\varrho ({\sigma }_{\lambda },\Gamma {\sigma }_{\lambda })\},\hfill \end{array}$$

which implies that

$$\begin{array}{cc}\hfill \left|\right|\beta \left|\right|=\left|\right|\varrho (\mathfrak{u},\Gamma \mathfrak{u})\left|\right|& \le \mathfrak{s}\left|\right|\varrho (\mathfrak{u},{\sigma }_{\lambda +1})\left|\right|+\mathfrak{s}{\lambda }_1\left(\mathfrak{u}\right)\left|\right|\varrho (\mathfrak{u},{\sigma }_{\lambda })\left|\right|+\mathfrak{s}{\lambda }_2\left(\mathfrak{u}\right)\frac{\left|\right|\varrho (\mathfrak{u},{\sigma }_{\lambda +1})\left|\right|\left|\right|(1+\varrho (\mathfrak{u},\Gamma \mathfrak{u}))\left|\right|}{\left|\right|1+\varrho (\mathfrak{u},{\sigma }_{\lambda })\left|\right|}\hfill \\ \hfill & +\mathfrak{s}{\lambda }_3\left(\mathfrak{u}\right)\frac{\left|\right|\varrho ({\sigma }_{\lambda },{\sigma }_{\lambda +1})\left|\right|\left|\right|1+\varrho (\mathfrak{u},\Gamma \mathfrak{u})\left|\right|}{\left|\right|1+\varrho (\mathfrak{u},{\sigma }_{\lambda })\left|\right|}+\mathfrak{s}{\lambda }_4\left(\mathfrak{u}\right)max\left\{\right|\left|\varrho (\mathfrak{u},\Gamma \mathfrak{u})\right||,||\varrho ({\sigma }_{\lambda },\Gamma {\sigma }_{\lambda })\left|\right|\}.\hfill \end{array}$$

Taking $\lambda \to \infty$, we obtain

$$\begin{array}{c}\hfill \left|\right|\beta \left|\right|=\left|\right|\varrho (\mathfrak{u},\Gamma \mathfrak{u})\left|\right|\le 0,\end{array}$$

which is a contradiction to (3). Therefore, $\left|\right|\beta \left|\right|=0$. Thus, $\Gamma \mathfrak{u}=\mathfrak{u},\mathfrak{u}$ is a fixed point of Γ. □

Next, we prove our second result.

Theorem 3. 

Let $(\Omega ,\varrho )$ be a complete bicomplex b-metric space with the coefficient $\mathfrak{s}\ge 1$ and $\mathcal{S},\Gamma :\Omega \to \Omega$. If there exist mappings ${\lambda }_1,{\lambda }_2,{\lambda }_3,{\lambda }_4:\Omega \to [0,\frac{1}{\mathfrak{s}})$ such that for all $\sigma ,\varpi \in \Omega$:

(i)

${\lambda }_1\left(\Gamma \sigma \right)\le {\lambda }_1\left(\sigma \right),{\lambda }_2\left(\Gamma \sigma \right)\le {\lambda }_2\left(\sigma \right),{\lambda }_3\left(\Gamma \sigma \right)\le {\lambda }_3\left(\sigma \right)\mathrm{and}{\lambda }_4\left(\Gamma \sigma \right)\le {\lambda }_4\left(\sigma \right)$;

(ii)

${\lambda }_1\left(\mathcal{S}\sigma \right)\le {\lambda }_1\left(\sigma \right),{\lambda }_2\left(\mathcal{S}\sigma \right)\le {\lambda }_2\left(\sigma \right),{\lambda }_3\left(\mathcal{S}\sigma \right)\le {\lambda }_3\left(\sigma \right)\mathrm{and}{\lambda }_4\left(\mathcal{S}\sigma \right)\le {\lambda }_4\left(\sigma \right)$;

(iii)

$({\lambda }_1+{\lambda }_2+{\lambda }_3+{\lambda }_4)\left(\sigma \right)<1$;

(iv)

$\varrho (\mathcal{S}\sigma ,\Gamma \varpi ){⪯}_{{\mathfrak{i}}_2}{\lambda }_1\left(\sigma \right)\varrho (\sigma ,\varpi )+{\lambda }_2\left(\sigma \right)\frac{\varrho (\sigma ,\mathcal{S}\varpi )(1+\varrho (\sigma ,\Gamma \sigma \left)\right)}{1+\varrho (\sigma ,\varpi )}+{\lambda }_3\left(\sigma \right)\frac{\varrho (\varpi ,\mathcal{S}\varpi )(1+\varrho (\sigma ,\Gamma \sigma \left)\right)}{1+\varrho (\sigma ,\varpi )}+{\lambda }_4\left(\sigma \right)max\{\varrho (\sigma ,\mathcal{S}\sigma ),\varrho (\varpi ,\Gamma \varpi )\}$,

then $\mathcal{S}$ and Γ have a unique common fixed point in Ω.

Proof. 

Let ${\sigma }_0\in \Omega$. Since $\mathcal{S}\left(\Omega \right)\subseteq \Omega$ and $\Gamma \left(\Omega \right)\subseteq \Omega$, we can construct the sequence $\left\{{\sigma }_{\mathfrak{k}}\right\}\subset \Omega$ satisfying

$${\sigma }_{2\mathfrak{k}+1}=\mathcal{S}{\sigma }_{2\mathfrak{k}}\mathrm{and}{\sigma }_{2\mathfrak{k}+2}=\Gamma {\sigma }_{2\mathfrak{k}+1},$$

(4)

for all $\mathfrak{k}\ge 0$. From hypothesis and (4), we obtain

$$\begin{array}{cc}\hfill \varrho ({\sigma }_{2\mathfrak{k}+1},{\sigma }_{2\mathfrak{k}+2})& =\varrho (\mathcal{S}{\sigma }_{2\mathfrak{k}},\Gamma {\sigma }_{2\mathfrak{k}+1})\hfill \\ \hfill & {⪯}_{{\mathfrak{i}}_2}{\lambda }_1\left({\sigma }_{2\mathfrak{k}}\right)\varrho ({\sigma }_{2\mathfrak{k}},{\sigma }_{2\mathfrak{k}+1})+{\lambda }_2\left({\sigma }_{2\mathfrak{k}}\right)\frac{\varrho ({\sigma }_{2\mathfrak{k}},\mathcal{S}{\sigma }_{2\mathfrak{k}})(1+\varrho ({\sigma }_{2\mathfrak{k}},\Gamma {\sigma }_{2\mathfrak{k}}))}{1+\varrho ({\sigma }_{2\mathfrak{k}},{\sigma }_{2\mathfrak{k}+1})}\hfill \\ \hfill & +{\lambda }_3\left({\sigma }_{2\mathfrak{k}}\right)\frac{\varrho ({\sigma }_{2\mathfrak{k}+1},\mathcal{S}{\sigma }_{2\mathfrak{k}+1})(1+\varrho ({\sigma }_{2\mathfrak{k}},\Gamma {\sigma }_{2\mathfrak{k}}))}{1+\varrho ({\sigma }_{2\mathfrak{k}},{\sigma }_{2\mathfrak{k}+1})}\hfill \\ \hfill & +{\lambda }_4\left({\sigma }_{2\mathfrak{k}}\right)max\{\varrho ({\sigma }_{2\mathfrak{k}},\mathcal{S}{\sigma }_{2\mathfrak{k}}),\varrho ({\sigma }_{2\mathfrak{k}+1},\Gamma {\sigma }_{2\mathfrak{k}+1})\}\hfill \\ \hfill & {⪯}_{{\mathfrak{i}}_2}{\lambda }_1\left(\Gamma {\sigma }_{2\mathfrak{k}-1}\right)\varrho ({\sigma }_{2\mathfrak{k}},{\sigma }_{2\mathfrak{k}+1})+{\lambda }_2\left(\Gamma {\sigma }_{2\mathfrak{k}-1}\right)\varrho ({\sigma }_{2\mathfrak{k}},{\sigma }_{2\mathfrak{k}+1})\hfill \\ \hfill & +{\lambda }_3\left(\Gamma {\sigma }_{2\mathfrak{k}-1}\right)\varrho ({\sigma }_{2\mathfrak{k}+1},{\sigma }_{2\mathfrak{k}+2})+{\lambda }_4\left(\Gamma {\sigma }_{2\mathfrak{k}-1}\right)max\{\varrho ({\sigma }_{2\mathfrak{k}},{\sigma }_{2\mathfrak{k}+1}),\varrho ({\sigma }_{2\mathfrak{k}+1},{\sigma }_{2\mathfrak{k}+2})\}\hfill \\ \hfill & ⋮\hfill \\ \hfill & {⪯}_{{\mathfrak{i}}_2}{\lambda }_1\left({\sigma }_0\right)\varrho ({\sigma }_{2\mathfrak{k}},{\sigma }_{2\mathfrak{k}+1})+{\lambda }_2\left({\sigma }_0\right)\varrho ({\sigma }_{2\mathfrak{k}},{\sigma }_{2\mathfrak{k}+1})\hfill \\ \hfill & +{\lambda }_3\left({\sigma }_0\right)\varrho ({\sigma }_{2\mathfrak{k}+1},{\sigma }_{2\mathfrak{k}+2})+{\lambda }_4\left({\sigma }_0\right){\mathcal{M}}_1,\hfill \end{array}$$

(5)

where ${\mathcal{M}}_1=max\{\varrho ({\sigma }_{2\mathfrak{k}},{\sigma }_{2\mathfrak{k}+1}),\varrho ({\sigma }_{2\mathfrak{k}+1},{\sigma }_{2\mathfrak{k}+2})\}$.

CASE(1)

Suppose that ${\mathcal{M}}_1=\varrho ({\sigma }_{2\mathfrak{k}},{\sigma }_{2\mathfrak{k}+1})$; then, we obtain

$$\begin{array}{cc}\hfill \varrho ({\sigma }_{2\mathfrak{k}+1},{\sigma }_{2\mathfrak{k}+2})& {⪯}_{{\mathfrak{i}}_2}\frac{{\lambda }_1\left({\sigma }_0\right)+{\lambda }_2\left({\sigma }_0\right)+{\lambda }_4\left({\sigma }_0\right)}{1-{\lambda }_3\left({\sigma }_0\right)}\varrho ({\sigma }_{2\mathfrak{k}},{\sigma }_{2\mathfrak{k}+1}).\hfill \end{array}$$

Similarly, we obtain

$$\begin{array}{cc}\hfill \varrho ({\sigma }_{2\mathfrak{k}+2},{\sigma }_{2\mathfrak{k}+3})& {⪯}_{{\mathfrak{i}}_2}\frac{{\lambda }_1\left({\sigma }_0\right)+{\lambda }_2\left({\sigma }_0\right)+{\lambda }_4\left({\sigma }_0\right)}{1-{\lambda }_3\left({\sigma }_0\right)}\varrho ({\sigma }_{2\mathfrak{k}+1},{\sigma }_{2\mathfrak{k}+2}).\hfill \end{array}$$

Since ${\lambda }_1+{\lambda }_2+{\lambda }_3+{\lambda }_4\left(\sigma \right)<1$, we obtain $\frac{{\lambda }_1\left({\sigma }_0\right)+{\lambda }_2\left({\sigma }_0\right)+{\lambda }_4\left({\sigma }_0\right)}{1-{\lambda }_3\left({\sigma }_0\right)}<1$.

Therefore, with $\mathfrak{h}=\frac{{\lambda }_1\left({\sigma }_0\right)+{\lambda }_2\left({\sigma }_0\right)+{\lambda }_4\left({\sigma }_0\right)}{1-{\lambda }_3\left({\sigma }_0\right)}<1$ and for all $\lambda \ge 0$, consequently, we have

$$\begin{array}{c}\hfill \left|\right|\varrho ({\sigma }_{2\mathfrak{k}+1},{\sigma }_{2\mathfrak{k}+2})\left|\right|\le \mathfrak{h}\left|\right|\varrho ({\sigma }_{2\mathfrak{k}},{\sigma }_{2\mathfrak{k}+1})\left|\right|\le {\mathfrak{h}}^2\left|\right|\varrho ({\sigma }_{2\mathfrak{k}-1},{\sigma }_{2\mathfrak{k}})\left|\right|\le \cdots \le {\mathfrak{h}}^{\lambda }\left|\right|\varrho ({\sigma }_0,{\sigma }_1)\left|\right|.\end{array}$$

That is, $\left|\right|\varrho ({\sigma }_{2\mathfrak{k}+1},{\sigma }_{2\mathfrak{k}+2})\left|\right|\le {\mathfrak{h}}^{\lambda +1}\left|\right|\varrho ({\sigma }_0,{\sigma }_1)\left|\right|$.

CASE(2)

Suppose that ${\mathcal{M}}_1=\varrho ({\sigma }_{2\mathfrak{k}+1},{\sigma }_{2\mathfrak{k}+2})$; then, we obtain

$$\begin{array}{cc}\hfill \varrho ({\sigma }_{2\mathfrak{k}+1},{\sigma }_{2\mathfrak{k}+2})& {⪯}_{{\mathfrak{i}}_2}\frac{{\lambda }_1\left({\sigma }_0\right)+{\lambda }_2\left({\sigma }_0\right)}{1-{\lambda }_3\left({\sigma }_0\right)-{\lambda }_4\left({\sigma }_0\right)}\varrho ({\sigma }_{2\mathfrak{k}},{\sigma }_{2\mathfrak{k}+1}).\hfill \end{array}$$

Similarly, we obtain

$$\begin{array}{cc}\hfill \varrho ({\sigma }_{2\mathfrak{k}+2},{\sigma }_{2\mathfrak{k}+3})& {⪯}_{{\mathfrak{i}}_2}\frac{{\lambda }_1\left({\sigma }_0\right)+{\lambda }_2\left({\sigma }_0\right)}{1-{\lambda }_3\left({\sigma }_0\right)-{\lambda }_4\left({\sigma }_0\right)}\varrho ({\sigma }_{2\mathfrak{k}+1},{\sigma }_{2\mathfrak{k}+2}).\hfill \end{array}$$

Since ${\lambda }_1+{\lambda }_2+{\lambda }_3+{\lambda }_4\left(\sigma \right)<1$, we obtain $\frac{{\lambda }_1\left({\sigma }_0\right)+{\lambda }_2\left({\sigma }_0\right)}{1-{\lambda }_3\left({\sigma }_0\right)-{\lambda }_4\left({\sigma }_0\right)}<1$.

Therefore, with $\mathfrak{h}=\frac{{\lambda }_1\left({\sigma }_0\right)+{\lambda }_2\left({\sigma }_0\right)}{1-{\lambda }_3\left({\sigma }_0\right)-{\lambda }_4\left({\sigma }_0\right)}<1$ and for all $\lambda \ge 0$, consequently, we have

$$\begin{array}{c}\hfill \left|\right|\varrho ({\sigma }_{2\mathfrak{k}+1},{\sigma }_{2\mathfrak{k}+2})\left|\right|\le \mathfrak{h}\left|\right|\varrho ({\sigma }_{2\mathfrak{k}},{\sigma }_{2\mathfrak{k}+1})\left|\right|\le {\mathfrak{h}}^2\left|\right|\varrho ({\sigma }_{2\mathfrak{k}-1},{\sigma }_{2\mathfrak{k}})\left|\right|\le \cdots \le {\mathfrak{h}}^{\lambda }\left|\right|\varrho ({\sigma }_0,{\sigma }_1)\left|\right|.\end{array}$$

That is, $\left|\right|\varrho ({\sigma }_{2\mathfrak{k}+1},{\sigma }_{2\mathfrak{k}+2})\left|\right|\le {\mathfrak{h}}^{\lambda +1}\left|\right|\varrho ({\sigma }_0,{\sigma }_1)\left|\right|$. Let $\mathfrak{m},\lambda \in \mathbb{N}$, $\mathfrak{m}>\lambda$,

$$\begin{array}{cc}\hfill \varrho ({\sigma }_{\lambda },{\sigma }_{\mathfrak{m}})& {⪯}_{{\mathfrak{i}}_2}\mathfrak{s}[\varrho ({\sigma }_{\lambda },{\sigma }_{\lambda +1})+\varrho ({\sigma }_{\lambda +1},{\sigma }_{\mathfrak{m}})],\hfill \\ \hfill & {⪯}_{{\mathfrak{i}}_2}\mathfrak{s}\varrho ({\sigma }_{\lambda },{\sigma }_{\lambda +1})+{\mathfrak{s}}^2[\varrho ({\sigma }_{\lambda +1},{\sigma }_{\lambda +2})+\varrho ({\sigma }_{\lambda +2},{\sigma }_{\mathfrak{m}})],\hfill \\ \hfill & {⪯}_{{\mathfrak{i}}_2}\mathfrak{s}\varrho ({\sigma }_{\lambda },{\sigma }_{\lambda +1})+{\mathfrak{s}}^2\varrho ({\sigma }_{\lambda +1},{\sigma }_{\lambda +2})+{\mathfrak{s}}^3\varrho ({\sigma }_{\lambda +2},{\sigma }_{\lambda +1})+\cdots \cdots \hfill \\ \hfill & {⪯}_{{\mathfrak{i}}_2}\mathfrak{s}{\mathfrak{k}}^{\lambda }\varrho ({\sigma }_0,{\sigma }_1)+{\mathfrak{s}}^2{\mathfrak{k}}^{\lambda +1}\varrho ({\sigma }_{\lambda +1},{\sigma }_{\lambda +2})+{\mathfrak{s}}^3{\mathfrak{k}}^{\lambda +2}\varrho ({\sigma }_0,{\sigma }_1)+\cdots \cdots \hfill \\ \hfill & {⪯}_{{\mathfrak{i}}_2}\mathfrak{s}{\mathfrak{k}}^{\lambda }\varrho ({\sigma }_0,{\sigma }_1)[1+\mathfrak{s}\mathfrak{k}+{\left(\mathfrak{s}\mathfrak{k}\right)}^2+{\left(\mathfrak{s}\mathfrak{k}\right)}^3+\cdots \cdots ]\hfill \\ \hfill & =\frac{\mathfrak{s}{\mathfrak{k}}^{\lambda }}{1-\mathfrak{s}\mathfrak{k}}\varrho ({\sigma }_0,{\sigma }_1).\hfill \end{array}$$

(6)

Therefore,

$$\begin{array}{c}\hfill \left|\right|\varrho ({\sigma }_{\lambda },{\sigma }_{\mathfrak{m}})\left|\right|\le \frac{\mathfrak{s}{\mathfrak{k}}^{\lambda }}{1-\mathfrak{s}\mathfrak{k}}\left|\right|\varrho ({\sigma }_0,{\sigma }_1)\left|\right|\to \infty ,\mathrm{as}\mathfrak{m},\lambda \to \infty .\end{array}$$

Thus, $\left\{{\sigma }_{\lambda }\right\}$ is a Cauchy sequence in Ω. Since Ω is complete, we can find $\mathfrak{u}\in \Omega$ satisfying ${\sigma }_{\mathfrak{k}}\to \mathfrak{u}$, as $\mathfrak{k}\to \infty$.

Suppose not, then there exists $\beta \in \Omega$ such that

$$\left|\right|\varrho (\mathfrak{u},\mathcal{S}\mathfrak{u})\left|\right|=\left|\right|\beta \left|\right|>0.$$

(7)

By the notion of a bicomplex b-metric ϱ, we have

$$\begin{array}{cc}\hfill \beta =\varrho (\mathfrak{u},\mathcal{S}\mathfrak{u})& {⪯}_{{\mathfrak{i}}_2}\mathfrak{s}\varrho (\mathfrak{u},{\sigma }_{2\mathfrak{k}+2})+\mathfrak{s}\varrho ({\sigma }_{2\mathfrak{k}+2},\mathcal{S}\mathfrak{u})\hfill \\ \hfill & {⪯}_{{\mathfrak{i}}_2}\mathfrak{s}\varrho (\mathfrak{u},{\sigma }_{2\mathfrak{k}+2})+\mathfrak{s}\varrho (\mathcal{S}\mathfrak{u},\Gamma {\sigma }_{2\mathfrak{k}+1})\hfill \\ \hfill & {⪯}_{{\mathfrak{i}}_2}\mathfrak{s}\varrho (\mathfrak{u},{\sigma }_{2\mathfrak{k}+2})+\mathfrak{s}{\lambda }_1\left(\mathfrak{u}\right)\varrho (\mathfrak{u},{\sigma }_{2\mathfrak{k}+1})+\mathfrak{s}{\lambda }_2\left(\mathfrak{u}\right)\frac{\varrho (\mathfrak{u},\mathcal{S}{\sigma }_{\lambda })(1+\varrho (\mathfrak{u},\Gamma \mathfrak{u}))}{1+\varrho (\mathfrak{u},{\sigma }_{2\mathfrak{k}+1})}\hfill \\ \hfill & +\mathfrak{s}{\lambda }_3\left(\mathfrak{u}\right)\frac{\varrho ({\sigma }_{2\mathfrak{k}+1},\Gamma {\sigma }_{2\mathfrak{k}+1})(1+\varrho (\mathfrak{u},\Gamma \mathfrak{u}))}{1+\varrho (\mathfrak{u},{\sigma }_{2\mathfrak{k}+1})}+\mathfrak{s}{\lambda }_4\left(\mathfrak{u}\right)max\{\varrho (\mathfrak{u},\mathcal{S}\mathfrak{u}),\varrho ({\sigma }_{2\mathfrak{k}+1},\Gamma {\sigma }_{2\mathfrak{k}+1})\}\hfill \\ \hfill & {⪯}_{{\mathfrak{i}}_2}\mathfrak{s}\varrho (\mathfrak{u},{\sigma }_{2\mathfrak{k}+2})+\mathfrak{s}{\lambda }_1\left(\mathfrak{u}\right)\varrho (\mathfrak{u},{\sigma }_{2\mathfrak{k}+1})+\mathfrak{s}{\lambda }_2\left(\mathfrak{u}\right)\frac{\varrho (\mathfrak{u},{\sigma }_{2\mathfrak{k}+1})(1+\varrho (\mathfrak{u},\Gamma \mathfrak{u}))}{1+\varrho (\mathfrak{u},{\sigma }_{2\mathfrak{k}+1})}\hfill \\ \hfill & +\mathfrak{s}{\lambda }_3\left(\mathfrak{u}\right)\frac{\varrho ({\sigma }_{2\mathfrak{k}+1},{\sigma }_{2\mathfrak{k}+2})(1+\varrho (\mathfrak{u},\Gamma \mathfrak{u}))}{1+\varrho (\mathfrak{u},{\sigma }_{2\mathfrak{k}+1})}+\mathfrak{s}{\lambda }_4\left(\mathfrak{u}\right)max\{\varrho (\mathfrak{u},\mathcal{S}\mathfrak{u}),\varrho ({\sigma }_{2\mathfrak{k}+1},\Gamma {\sigma }_{2\mathfrak{k}+1})\},\hfill \end{array}$$

which derive that

$$\begin{array}{cc}\hfill \left|\right|\beta \left|\right|=\left|\right|\varrho (\mathfrak{u},S\mathfrak{u})\left|\right|& \le \mathfrak{s}\left|\right|\varrho (\mathfrak{u},{\sigma }_{2\mathfrak{k}+2})\left|\right|+\mathfrak{s}{\lambda }_1\left(\mathfrak{u}\right)\left|\right|\varrho (\mathfrak{u},{\sigma }_{2\mathfrak{k}+1})\left|\right|+\mathfrak{s}{\lambda }_2\left(\mathfrak{u}\right)\frac{\left|\right|\varrho (\mathfrak{u},{\sigma }_{2\mathfrak{k}+1})\left|\right|\left|\right|(1+\varrho (\mathfrak{u},\Gamma \mathfrak{u}))\left|\right|}{\left|\right|1+\varrho (\mathfrak{u},{\sigma }_{2\mathfrak{k}+1})\left|\right|}\hfill \\ \hfill & +\mathfrak{s}{\lambda }_3\left(\mathfrak{u}\right)\frac{\left|\right|\varrho ({\sigma }_{2\mathfrak{k}+1},{\sigma }_{2\mathfrak{k}+2})\left|\right|\left|\right|(1+\varrho (\mathfrak{u},\Gamma \mathfrak{u}))\left|\right|}{\left|\right|1+\varrho (\mathfrak{u},{\sigma }_{2\mathfrak{k}+1})\left|\right|}\hfill \\ \hfill & +\mathfrak{s}{\lambda }_4\left(\mathfrak{u}\right)max\left\{\right|\left|\varrho (\mathfrak{u},\mathcal{S}\mathfrak{u})\right||,||\varrho ({\sigma }_{2\mathfrak{k}+1},\Gamma {\sigma }_{2\mathfrak{k}+1})\left|\right|\}.\hfill \end{array}$$

Taking $\mathfrak{k}\to \infty$, we obtain

$$\begin{array}{c}\hfill \left|\right|\beta \left|\right|=\left|\right|\varrho (\mathfrak{u},\mathcal{S}\mathfrak{u})\left|\right|\le 0,\end{array}$$

which is an absurdity to (7). So, $\left|\beta \right|=0$. Hence, $\mathcal{S}\mathfrak{u}=\mathfrak{u}$. Similarly, we can derive that $\Gamma \mathfrak{u}=\mathfrak{u}$. Therefore, $\mathfrak{u}$ is a common fixed point of $\mathcal{S}$ and Γ. Assume that there exists another common fixed point ${\mathfrak{u}}_1$, that is, ${\mathfrak{u}}_1=\mathcal{S}{\mathfrak{u}}_1=\Gamma {\mathfrak{u}}_1$. Then,

$$\begin{array}{cc}\hfill \varrho (\mathfrak{u},{\mathfrak{u}}_1)& =\varrho (\mathcal{S}\mathfrak{u},\Gamma {\mathfrak{u}}_1)\hfill \\ \hfill & {⪯}_{{\mathfrak{i}}_2}{\lambda }_1\left(\mathfrak{u}\right)\varrho (\mathfrak{u},{\mathfrak{u}}_1)+{\lambda }_2\left(\mathfrak{u}\right)\frac{\varrho (\mathfrak{u},\mathcal{S}\mathfrak{u})(1+\varrho (\mathfrak{u},\Gamma \mathfrak{u}\left)\right)}{1+\varrho (\mathfrak{u},{\mathfrak{u}}_1)}\hfill \\ \hfill & +{\lambda }_3\left(\mathfrak{u}\right)\frac{\varrho ({\mathfrak{u}}_1,\mathcal{S}{\mathfrak{u}}_1)(1+\varrho (\mathfrak{u},\Gamma \mathfrak{u}))}{1+\varrho (\mathfrak{u},{\mathfrak{u}}_1)}+{\lambda }_4\left(\mathfrak{u}\right)max\{\varrho (\mathfrak{u},\mathcal{S}\mathfrak{u}),\varrho ({\mathfrak{u}}_1,\Gamma {\mathfrak{u}}_1)\}\hfill \\ \hfill & {⪯}_{{\mathfrak{i}}_2}{\lambda }_1\left(\mathfrak{u}\right)\varrho (\mathfrak{u},{\mathfrak{u}}_1)+{\lambda }_2\left(\mathfrak{u}\right)\frac{\varrho (\mathfrak{u},\mathfrak{u})(1+\varrho (\mathfrak{u},\mathfrak{u}\left)\right)}{1+\varrho (\mathfrak{u},{\mathfrak{u}}_1)}\hfill \\ \hfill & +{\lambda }_3\left(\mathfrak{u}\right)\frac{\varrho ({\mathfrak{u}}_1,{\mathfrak{u}}_1)(1+\varrho (\mathfrak{u},\mathfrak{u}))}{1+\varrho (\mathfrak{u},{\mathfrak{u}}_1)}+{\lambda }_4\left(\mathfrak{u}\right)max\{\varrho (\mathfrak{u},\mathfrak{u}),\varrho ({\mathfrak{u}}_1,{\mathfrak{u}}_1)\}\hfill \\ \hfill & ={\lambda }_1\left(\mathfrak{u}\right)\varrho (\mathfrak{u},{\mathfrak{u}}_1),\hfill \end{array}$$

which implies that $\left|\right|\varrho (\mathfrak{u},{\mathfrak{u}}_1)\left|\right|\le {\lambda }_1\left(\mathfrak{u}\right)\left|\right|\varrho (\mathfrak{u},{\mathfrak{u}}_1)\left|\right|$. Since, ${\lambda }_1\left(\mathfrak{u}\right)\in [0,\frac{1}{\mathfrak{s}})$, we have $\left|\right|\varrho (\mathfrak{u},{\mathfrak{u}}_1)\left|\right|=0$. Therefore, $\mathfrak{u}={\mathfrak{u}}_1$. Hence, $\mathfrak{u}$ is a unique common fixed point of $\mathcal{S}$ and Γ. □

Example 1. 

Let $\Omega =[0,1]$ and $\varrho :\Omega \times \Omega \to {\mathbb{C}}_2$ given by $\varrho (\sigma ,\varpi )={|\sigma -\varpi |}^2+{\mathfrak{i}}_2{|\sigma -\varpi |}^2$. Then, $(\Omega ,\varrho )$ is a complete bicomplex b-metric space with $\mathfrak{s}\ge 1$. Define $\Gamma :\Omega \to \Omega$ by

$$\Gamma \left(\sigma \right)=\left\{\begin{array}{cc}\frac{1}{5},\hfill & \mathrm{if}\sigma \in [0,\frac{1}{2}],\hfill \\ \frac{2}{3},\hfill & \mathrm{if}\sigma \in (\frac{1}{2},1].\hfill \end{array}\right.$$

(8)

We define the functions ${\lambda }_1,{\lambda }_2,{\lambda }_3,{\lambda }_4:\Omega \to [0,\frac{1}{\mathfrak{s}}]$ by ${\lambda }_1\left(\sigma \right)=\frac{\sigma }{4},{\lambda }_2\left(\sigma \right)=\frac{\sigma }{6},{\lambda }_3\left(\sigma \right)=\frac{\sigma }{3},{\lambda }_4\left(\sigma \right)=\frac{\sigma }{7}$.Clearly, ${\lambda }_1+2\mathfrak{s}{\lambda }_2+{\lambda }_3+{\lambda }_4\left(\sigma \right)<1$, for all $\sigma ,\varpi \in \Omega$.

Now, consider

${\lambda }_1\left(\Gamma \sigma \right)={\lambda }_1\left(\frac{1}{5}\right)=\frac{1}{20}\le \frac{\sigma }{4}={\lambda }_1\left(\sigma \right)$.

That is, ${\lambda }_1\left(\Gamma \sigma \right)\le {\lambda }_1\left(\sigma \right)$, for all $\sigma \in [0,\frac{1}{2}]$.

Additionally,

${\lambda }_1\left(\Gamma \sigma \right)={\lambda }_1\left(\frac{2}{3}\right)=\frac{1}{6}\le \frac{\sigma }{4}={\lambda }_1\left(\sigma \right)$.

That is, ${\lambda }_1\left(\Gamma \sigma \right)\le {\lambda }_1\left(\sigma \right)$, for all $\sigma \in (\frac{1}{2},1]$.

Similarly, we can show that ${\lambda }_2\left(\Gamma \sigma \right)\le {\lambda }_2\left(\sigma \right),{\lambda }_3\left(\Gamma \sigma \right)\le {\lambda }_3\left(\sigma \right),\mathrm{and}{\lambda }_4\left(\Gamma \sigma \right)\le {\lambda }_4\left(x\right)$. Before discussing different cases, one needs to notice that for all $\sigma ,\varpi \in \Omega$,

$0{⪯}_{{\mathfrak{i}}_2}\{\varrho (\sigma ,\varpi ),\frac{\varrho (\sigma ,\Gamma \varpi )(1+\varrho (\sigma ,\Gamma \sigma \left)\right)}{1+\varrho (\sigma ,\varpi )},\frac{\varrho (\varpi ,\Gamma \varpi )(1+\varrho (\sigma ,\Gamma \sigma \left)\right)}{1+\varrho (\sigma ,\varpi )},max\{\varrho (\sigma ,\Gamma \sigma ),\varrho (\varpi ,\Gamma \varpi )\}\}$ in all aspects.

It is sufficient to show that

$$\begin{array}{c}\hfill \varrho (\Gamma \sigma ,\Gamma \varpi ){⪯}_{{\mathfrak{i}}_2}{\lambda }_1\left(\sigma \right)\varrho (\sigma ,\varpi ).\end{array}$$

CASE (1) 

For all $\sigma ,\varpi \in [0,\frac{1}{2}]$.

$$\begin{array}{cc}\hfill \varrho (\Gamma \sigma ,\Gamma \varpi )& =|\frac{1}{5}-\frac{1}{5}{|}^2+{\mathfrak{i}}_2{|\frac{1}{5}-\frac{1}{5}|}^2\hfill \\ \hfill & {⪯}_{{\mathfrak{i}}_2}\frac{\sigma }{4}{\left(\right|\sigma -\varpi |}^2{+\mathfrak{i}|\sigma -\varpi |}^2)\hfill \\ \hfill & ={\lambda }_1\left(\sigma \right)\varrho (\sigma ,\varpi ).\hfill \end{array}$$

That is, $\varrho (\Gamma \sigma ,\Gamma \varpi ){⪯}_{{\mathfrak{i}}_2}{\lambda }_1\left(\sigma \right)\varrho (\sigma ,\varpi )$, for all $\sigma ,\varpi \in [0,\frac{1}{2}]$.

CASE (2) 

For all $\sigma ,\varpi \in (\frac{1}{2},1]$.

$$\begin{array}{cc}\hfill \varrho (\Gamma \sigma ,\Gamma \varpi )& =|\frac{2}{3}-\frac{2}{3}{|}^2+{\mathfrak{i}}_2{|\frac{2}{3}-\frac{2}{3}|}^2\hfill \\ \hfill & =0\hfill \\ \hfill & {⪯}_{{\mathfrak{i}}_2}\frac{\sigma }{4}{\left(\right|\sigma -\varpi |}^2+{\mathfrak{i}}_2{|\sigma -\varpi |}^2)\hfill \\ \hfill & ={\lambda }_1\left(\sigma \right)\varrho (\sigma ,\varpi ).\hfill \end{array}$$

That is, $\varrho (\Gamma \sigma ,\Gamma \varpi ){⪯}_{{\mathfrak{i}}_2}{\lambda }_1\left(\sigma \right)\varrho (\sigma ,\varpi )$, for all $\sigma ,\varpi \in (\frac{1}{2},1]$.

CASE (3) 

For all $\sigma \in [0,\frac{1}{2}]$, $\varpi \in (\frac{1}{2},1]$.

$$\begin{array}{cc}\hfill \varrho (\Gamma \sigma ,\Gamma \varpi )& =|\frac{1}{5}-\frac{2}{3}{|}^2+{\mathfrak{i}}_2{|\frac{1}{5}-\frac{2}{3}|}^2\hfill \\ \hfill & =\frac{49}{225}(1+{\mathfrak{i}}_2)\hfill \\ \hfill & {⪯}_{{\mathfrak{i}}_2}\frac{\sigma }{4}{\left(\right|\sigma -\varpi |}^2+{\mathfrak{i}}_2{|\sigma -\varpi |}^2)\hfill \\ \hfill & ={\lambda }_1\left(\sigma \right)\varrho (\sigma ,\varpi ).\hfill \end{array}$$

That is, $\varrho (\Gamma \sigma ,\Gamma \varpi ){⪯}_{{\mathfrak{i}}_2}{\lambda }_1\left(\sigma \right)\varrho (\sigma ,\varpi )$, for all $\sigma \in [0,\frac{1}{2}]$, $\varpi \in (\frac{1}{2},1]$.

Therefore, all the hypothesis of Theorem 2 are fulfilled. Moreover, $\frac{1}{5}$ and $\frac{2}{3}$ are the two fixed points of Γ.

Example 2. 

Let $\Omega =[0,1]$ and $\varrho :\Omega \times \Omega \to {\mathbb{C}}_2$ given by $\varrho (\sigma ,\varpi )={|\sigma -\varpi |}^2{\mathfrak{e}}^{i_2\frac{\pi }{6}}$. Then, $(\Omega ,\varrho )$ is a complete bicomplex b-metric space with $\mathfrak{s}\ge 1$. Define $\Gamma :\Omega \to \Omega$ by

$$\Gamma \left(\sigma \right)=\left\{\begin{array}{cc}\frac{1}{6},\hfill & \mathrm{if}\sigma \in [0,\frac{1}{2}],\hfill \\ \frac{4}{5},\hfill & \mathrm{if}\sigma \in (\frac{1}{2},1].\hfill \end{array}\right.$$

(9)

We define the functions ${\lambda }_1,{\lambda }_2,{\lambda }_3,{\lambda }_4:\Omega \to [0,\frac{1}{\mathfrak{s}}]$ by ${\lambda }_1\left(\sigma \right)=\frac{\sigma }{8},{\lambda }_2\left(\sigma \right)=\frac{\sigma }{6},{\lambda }_3\left(\sigma \right)=\frac{\sigma }{9},{\lambda }_4\left(\sigma \right)=\frac{\sigma }{7}$.

Clearly, ${\lambda }_1+2\mathfrak{s}{\lambda }_2+{\lambda }_3+{\lambda }_4\left(\sigma \right)<1$, for all $\sigma ,\varpi \in \Omega$.

Now, consider

${\lambda }_1\left(\Gamma \sigma \right)={\lambda }_1\left(\frac{1}{6}\right)=\frac{1}{48}\le \frac{\sigma }{8}={\lambda }_1\left(\sigma \right)$.

That is, ${\lambda }_1\left(\Gamma \sigma \right)\le {\lambda }_1\left(\sigma \right)$, for all $\sigma \in [0,\frac{1}{2}]$.

Additionally,

${\lambda }_1\left(\Gamma \sigma \right)={\lambda }_1\left(\frac{4}{5}\right)=\frac{1}{10}\le \frac{\sigma }{8}={\lambda }_1\left(\sigma \right)$.

That is, ${\lambda }_1\left(\Gamma \sigma \right)\le {\lambda }_1\left(\sigma \right)$, for all $\sigma \in (\frac{1}{2},1]$.

Similarly, we can show that ${\lambda }_2\left(\Gamma \sigma \right)\le {\lambda }_2\left(\sigma \right),{\lambda }_3\left(\Gamma \sigma \right)\le {\lambda }_3\left(\sigma \right),\mathrm{and}{\lambda }_4\left(\Gamma \sigma \right)\le {\lambda }_4\left(x\right)$. Before discussing different cases, one needs to notice that for all $\sigma ,\varpi \in \Omega$,

$0{⪯}_{{\mathfrak{i}}_2}\{\varrho (\sigma ,\varpi ),\frac{\varrho (\sigma ,\Gamma \varpi )(1+\varrho (\sigma ,\Gamma \sigma \left)\right)}{1+\varrho (\sigma ,\varpi )},\frac{\varrho (\varpi ,\Gamma \varpi )(1+\varrho (\sigma ,\Gamma \sigma \left)\right)}{1+\varrho (\sigma ,\varpi )},max\{\varrho (\sigma ,\Gamma \sigma ),\varrho (\varpi ,\Gamma \varpi )\}\}$ in all aspects.

It is sufficient to show that

$$\begin{array}{c}\hfill \varrho (\Gamma \sigma ,\Gamma \varpi ){⪯}_{{\mathfrak{i}}_2}{\lambda }_1\left(\sigma \right)\varrho (\sigma ,\varpi ).\end{array}$$

CASE (1) 

For all $\sigma ,\varpi \in [0,\frac{1}{2}]$.

$$\begin{array}{cc}\hfill \varrho (\Gamma \sigma ,\Gamma \varpi )& =|\frac{1}{6}-\frac{1}{6}{|}^2{\mathfrak{e}}^{i_2\frac{\pi }{6}}\hfill \\ \hfill & {⪯}_{{\mathfrak{i}}_2}\frac{\sigma }{8}{\left(\right|\sigma -\varpi |}^2{\mathfrak{e}}^{i_2\frac{\pi }{6}}\hfill \\ \hfill & ={\lambda }_1\left(\sigma \right)\varrho (\sigma ,\varpi ).\hfill \end{array}$$

That is, $\varrho (\Gamma \sigma ,\Gamma \varpi ){⪯}_{{\mathfrak{i}}_2}{\lambda }_1\left(\sigma \right)\varrho (\sigma ,\varpi )$, for all $\sigma ,\varpi \in [0,\frac{1}{2}]$.

CASE (2) 

For all $\sigma ,\varpi \in (\frac{1}{2},1]$.

$$\begin{array}{cc}\hfill \varrho (\Gamma \sigma ,\Gamma \varpi )& =|\frac{4}{5}-\frac{4}{5}{|}^2{\mathfrak{e}}^{i_2\frac{\pi }{6}}\hfill \\ \hfill & =0\hfill \\ \hfill & {⪯}_{{\mathfrak{i}}_2}\frac{\sigma }{8}{\left(\right|\sigma -\varpi |}^2{\mathfrak{e}}^{i_2\frac{\pi }{6}})\hfill \\ \hfill & ={\lambda }_1\left(\sigma \right)\varrho (\sigma ,\varpi ).\hfill \end{array}$$

That is, $\varrho (\Gamma \sigma ,\Gamma \varpi ){⪯}_{{\mathfrak{i}}_2}{\lambda }_1\left(\sigma \right)\varrho (\sigma ,\varpi )$, for all $\sigma ,\varpi \in (\frac{1}{2},1]$.

CASE (3) 

For all $\sigma \in [0,\frac{1}{2}]$, $\varpi \in (\frac{1}{2},1]$.

$$\begin{array}{cc}\hfill \varrho (\Gamma \sigma ,\Gamma \varpi )& =|\frac{1}{6}-\frac{4}{5}{|}^2{\mathfrak{e}}^{i_2\frac{\pi }{6}}\hfill \\ \hfill & {⪯}_{{\mathfrak{i}}_2}\frac{\sigma }{8}{\left(\right|\sigma -\varpi |}^2{\mathfrak{e}}^{i_2\frac{\pi }{6}})\hfill \\ \hfill & ={\lambda }_1\left(\sigma \right)\varrho (\sigma ,\varpi ).\hfill \end{array}$$

That is, $\varrho (\Gamma \sigma ,\Gamma \varpi ){⪯}_{{\mathfrak{i}}_2}{\lambda }_1\left(\sigma \right)\varrho (\sigma ,\varpi )$, for all $\sigma \in [0,\frac{1}{2}]$, $\varpi \in (\frac{1}{2},1]$.

Therefore, all the hypothesis of Theorem 2 are fulfilled. Moreover, $\frac{1}{6}$ and $\frac{4}{5}$ are the two fixed points of Γ.

Example 3. 

Let $\Omega =[0,1]$ and $\varrho :\Omega \times \Omega \to {\mathbb{C}}_2$ be defined by $\varrho (\sigma ,\varpi )={|\sigma -\varpi |}^2+{\mathfrak{i}}_2{|\sigma -\varpi |}^2$. Then, $(\Omega ,\varrho )$ is a complete bicomplex b-metric space with $\mathfrak{s}\ge 1$. Now, we define self mappings $\mathcal{S},\Gamma :\Omega \to \Omega$ by

$$\begin{array}{c}\hfill \mathcal{S}\sigma =\frac{\sigma }{2},\Gamma \varpi =\frac{\varpi }{8},\mathrm{for}\mathrm{all}\sigma ,\varpi \in \Omega .\end{array}$$

We define the functions ${\lambda }_1,{\lambda }_2,{\lambda }_3,{\lambda }_4:\Omega \to [0,\frac{1}{\mathfrak{s}}]$ by

$$\begin{array}{c}\hfill {\lambda }_1\left(\sigma \right)=\frac{\sigma }{4},{\lambda }_2\left(\sigma \right)=\frac{\sigma }{6},{\lambda }_3\left(\sigma \right)=\frac{\sigma }{3},{\lambda }_4\left(\sigma \right)=\frac{\sigma }{7},\mathrm{for}\mathrm{all}\sigma \in \Omega .\end{array}$$

Clearly, $({\lambda }_1+2\mathfrak{s}{\lambda }_2+{\lambda }_3+{\lambda }_4)\left(\sigma \right)<1$, for all $\sigma ,\varpi \in \Omega$.

Now, consider ${\lambda }_1\left(\mathcal{S}\sigma \right)={\lambda }_1\left(\frac{\sigma }{2}\right)=\frac{\sigma }{8}\le \frac{\sigma }{4}={\lambda }_1\left(\sigma \right)$.

That is, ${\lambda }_1\left(\mathcal{S}\sigma \right)\le {\lambda }_1\left(\sigma \right)$, for all $\sigma \in \Omega$.

Additionally,

${\lambda }_1\left(\Gamma \varpi \right)={\lambda }_1\left(\frac{\varpi }{8}\right)=\frac{\varpi }{32}\le \frac{\varpi }{4}={\lambda }_1\left(\varpi \right)$.

That is, ${\lambda }_1\left(\Gamma \varpi \right)\le {\lambda }_1\left(\varpi \right)$, for all $\varpi \in \Omega$.

Similarly, we can show that ${\lambda }_2\left(\Gamma \sigma \right)\le {\lambda }_2\left(\sigma \right),{\lambda }_3\left(\Gamma \sigma \right)\le {\lambda }_3\left(\sigma \right)\mathrm{and}{\lambda }_4\left(\Gamma \sigma \right)\le {\lambda }_4\left(\sigma \right)$, ${\lambda }_2\left(\mathcal{S}\sigma \right)\le {\lambda }_2\left(\sigma \right),{\lambda }_3\left(\mathcal{S}\sigma \right)\le {\lambda }_3\left(\sigma \right),\mathrm{and}{\lambda }_4\left(\mathcal{S}\sigma \right)\le {\lambda }_4\left(\sigma \right)$. Before discussing different cases, one needs to notice that for all $\sigma ,\varpi \in \Omega$,

$0{⪯}_{{\mathfrak{i}}_2}\{\varrho (\sigma ,\varpi ),\frac{\varrho (\sigma ,\mathcal{S}\varpi )(1+\varrho (\sigma ,\Gamma \sigma \left)\right)}{1+\varrho (\sigma ,\varpi )},\frac{\varrho (\varpi ,\mathcal{S}\varpi )(1+\varrho (\sigma ,\Gamma \sigma \left)\right)}{1+\varrho (\sigma ,\varpi )},max\{\varrho (\sigma ,\mathcal{S}\sigma ),\varrho (\varpi ,\Gamma \varpi )\}\}$ in all aspects.

It is sufficient to show that

$$\begin{array}{c}\hfill \varrho (\mathcal{S}\sigma ,\Gamma \varpi ){⪯}_{{\mathfrak{i}}_2}{\lambda }_1\left(\sigma \right)\varrho (\sigma ,\varpi ).\end{array}$$

Consider

$$\begin{array}{cc}\hfill \varrho (\mathcal{S}\sigma ,\Gamma \varpi )& =\varrho (\frac{\sigma }{2},\frac{\varpi }{8})\hfill \\ \hfill & =|\frac{\sigma }{2}-\frac{\varpi }{8}{|}^2+{\mathfrak{i}}_2{|\frac{\sigma }{2}-\frac{\varpi }{8}|}^2\hfill \\ \hfill & {⪯}_{{\mathfrak{i}}_2}\frac{\sigma }{4}{\left(\right|\sigma -\varpi |}^2+{\mathfrak{i}}_2{|\sigma -\varpi |}^2)\hfill \\ \hfill & ={\lambda }_1\left(\sigma \right)\varrho (\sigma ,\varpi ).\hfill \end{array}$$

That is, $\varrho (\mathcal{S}\sigma ,\Gamma \varpi ){⪯}_{{\mathfrak{i}}_2}{\lambda }_1\left(\sigma \right)\varrho (\sigma ,\varpi )$, for all $\sigma ,\varpi \in \Omega$. Therefore, all the conditions of Theorem 3 are satisfied, and $\sigma =0$ remains fixed under $\mathcal{S}$ and Γ and is indeed unique.

Example 4. 

Let $\Omega =[0,1]$ and $\varrho :\Omega \times \Omega \to {\mathbb{C}}_2$ be defined by $\varrho (\sigma ,\varpi )={|\sigma -\varpi |}^2{\mathfrak{e}}^{i_2\frac{\pi }{6}}$ and be a complete bicomplex b-metric space with $\mathfrak{s}\ge 1$. Now, we define self mappings $\mathcal{S},\Gamma :\Omega \to \Omega$ by

$$\begin{array}{c}\hfill \mathcal{S}\sigma =\frac{\sigma }{3},\Gamma \varpi =\frac{\varpi }{6},\mathrm{for}\mathrm{all}\sigma ,\varpi \in \Omega .\end{array}$$

We define the functions ${\lambda }_1,{\lambda }_2,{\lambda }_3,{\lambda }_4:\Omega \to [0,\frac{1}{\mathfrak{s}}]$ by

$$\begin{array}{c}\hfill {\lambda }_1\left(\sigma \right)=\frac{\sigma }{7},{\lambda }_2\left(\sigma \right)=\frac{\sigma }{8},{\lambda }_3\left(\sigma \right)=\frac{\sigma }{9},{\lambda }_4\left(\sigma \right)=\frac{\sigma }{5},\mathrm{for}\mathrm{all}\sigma \in \Omega .\end{array}$$

Clearly, $({\lambda }_1+2\mathfrak{s}{\lambda }_2+{\lambda }_3+{\lambda }_4)\left(\sigma \right)<1$, for all $\sigma ,\varpi \in \Omega$.

Now, consider ${\lambda }_1\left(\mathcal{S}\sigma \right)={\lambda }_1\left(\frac{\sigma }{3}\right)=\frac{\sigma }{21}\le \frac{\sigma }{7}={\lambda }_1\left(\sigma \right)$.

That is, ${\lambda }_1\left(\mathcal{S}\sigma \right)\le {\lambda }_1\left(\sigma \right)$, for all $\sigma \in \Omega$.

Additionally,

${\lambda }_1\left(\Gamma \varpi \right)={\lambda }_1\left(\frac{\varpi }{6}\right)=\frac{\varpi }{42}\le \frac{\varpi }{7}={\lambda }_1\left(\varpi \right)$.

That is, ${\lambda }_1\left(\Gamma \varpi \right)\le {\lambda }_1\left(\varpi \right)$, for all $\varpi \in \Omega$.

Similarly, we can show that ${\lambda }_2\left(\Gamma \sigma \right)\le {\lambda }_2\left(\sigma \right),{\lambda }_3\left(\Gamma \sigma \right)\le {\lambda }_3\left(\sigma \right)\mathrm{and}{\lambda }_4\left(\Gamma \sigma \right)\le {\lambda }_4\left(\sigma \right)$, ${\lambda }_2\left(\mathcal{S}\sigma \right)\le {\lambda }_2\left(\sigma \right),{\lambda }_3\left(\mathcal{S}\sigma \right)\le {\lambda }_3\left(\sigma \right),\mathrm{and}{\lambda }_4\left(\mathcal{S}\sigma \right)\le {\lambda }_4\left(\sigma \right)$. Before discussing different cases, one needs to notice that for all $\sigma ,\varpi \in \Omega$,

$0{⪯}_{{\mathfrak{i}}_2}\{\varrho (\sigma ,\varpi ),\frac{\varrho (\sigma ,\mathcal{S}\varpi )(1+\varrho (\sigma ,\Gamma \sigma \left)\right)}{1+\varrho (\sigma ,\varpi )},\frac{\varrho (\varpi ,\mathcal{S}\varpi )(1+\varrho (\sigma ,\Gamma \sigma \left)\right)}{1+\varrho (\sigma ,\varpi )},max\{\varrho (\sigma ,\mathcal{S}\sigma ),\varrho (\varpi ,\Gamma \varpi )\}\}$ in all aspects.

It is sufficient to show that

$$\begin{array}{c}\hfill \varrho (\mathcal{S}\sigma ,\Gamma \varpi ){⪯}_{{\mathfrak{i}}_2}{\lambda }_1\left(\sigma \right)\varrho (\sigma ,\varpi ).\end{array}$$

Consider

$$\begin{array}{cc}\hfill \varrho (\mathcal{S}\sigma ,\Gamma \varpi )& =\varrho (\frac{\sigma }{3},\frac{\varpi }{6})\hfill \\ \hfill & =|\frac{\sigma }{3}-\frac{\varpi }{6}{|}^2{\mathfrak{e}}^{i_2\frac{\pi }{6}}\hfill \\ \hfill & {⪯}_{{\mathfrak{i}}_2}\frac{\sigma }{7}{|\sigma -\varpi |}^2{\mathfrak{e}}^{i_2\frac{\pi }{6}}\hfill \\ \hfill & ={\lambda }_1\left(\sigma \right)\varrho (\sigma ,\varpi ).\hfill \end{array}$$

That is, $\varrho (\mathcal{S}\sigma ,\Gamma \varpi ){⪯}_{{\mathfrak{i}}_2}{\lambda }_1\left(\sigma \right)\varrho (\sigma ,\varpi )$, for all $\sigma ,\varpi \in \Omega$. Therefore, all the conditions of Theorem 3 are satisfied, and $\sigma =0$ remains fixed under $\mathcal{S}$ and Γ and is indeed unique.

4. Application

In this section, we give an application using Theorem 3.

Let $\Omega =C[{\aleph }_1,{\aleph }_2]$ be a set of all real continuous functions on $[{\aleph }_1,{\aleph }_2]$ equipped with the metric $\varrho (\sigma ,\varpi )=(1+i_2)\left(\right|\sigma \left({\rm Y}\right)-\varpi \left({\rm Y}\right){|}^2),$ for all $\sigma ,\varpi \in C[{\aleph }_1,{\aleph }_2]$ and ${\rm Y}\in [{\aleph }_1,{\aleph }_2]$, where $|.|$ is the usual real modulus. Define the functions ${\lambda }_1,{\lambda }_2,{\lambda }_3,{\lambda }_4:\Omega \to [0,\frac{1}{\mathfrak{s}}]$ by

$$\begin{array}{c}\hfill {\lambda }_1\left(\sigma \right)=\frac{\sigma }{4},{\lambda }_2\left(\sigma \right)=\frac{\sigma }{6},{\lambda }_3\left(\sigma \right)=\frac{\sigma }{3},{\lambda }_4\left(\sigma \right)=\frac{\sigma }{7},\mathrm{for}\mathrm{all}\sigma \in \Omega .\end{array}$$

One can easily verify ${\lambda }_1\left(\Gamma \sigma \right)\le {\lambda }_1\left(\sigma \right),{\lambda }_2\left(\Gamma \sigma \right)\le {\lambda }_2\left(\sigma \right),{\lambda }_3\left(\Gamma \sigma \right)\le {\lambda }_3\left(\sigma \right),{\lambda }_4\left(\Gamma \sigma \right)\le {\lambda }_4\left(\sigma \right)$, ${\lambda }_1\left(\mathcal{S}\sigma \right)\le {\lambda }_1\left(\sigma \right),{\lambda }_2\left(\mathcal{S}\sigma \right)\le {\lambda }_2\left(\sigma \right),{\lambda }_3\left(\mathcal{S}\sigma \right)\le {\lambda }_3\left(\sigma \right),{\lambda }_4\left(\mathcal{S}\sigma \right)\le {\lambda }_4\left(\sigma \right)$, and $({\lambda }_1+{\lambda }_2+{\lambda }_3+{\lambda }_4)\left(\sigma \right)<1$. Then, $(\Omega ,\varrho )$ is a complete bicomplex valued b-metric space. Consider

$$\begin{array}{c}\hfill \sigma \left({\rm Y}\right)=\mathfrak{v}\left({\rm Y}\right)+\frac{1}{{\aleph }_2-{\aleph }_1}{\int }_{{\aleph }_1}^{{\aleph }_2}{\mathfrak{K}}_1({\rm Y},\mathfrak{s},\sigma \left(\mathfrak{s}\right))d\mathfrak{s},\end{array}$$

and

$$\begin{array}{c}\hfill \sigma \left({\rm Y}\right)=\mathfrak{v}\left({\rm Y}\right)+\frac{1}{{\aleph }_2-{\aleph }_1}{\int }_{{\aleph }_1}^{{\aleph }_2}{\mathfrak{K}}_2({\rm Y},\mathfrak{s},\sigma \left(\mathfrak{s}\right))d\mathfrak{s},\end{array}$$

where ${\rm Y},\mathfrak{s}\in [{\aleph }_1,{\aleph }_2]$. Assume that ${\mathfrak{K}}_1,{\mathfrak{K}}_2:[{\aleph }_1,{\aleph }_2]\times [{\aleph }_1,{\aleph }_2]\times \Omega \to \mathbb{R}$ and $\mathfrak{v}:[{\aleph }_1,{\aleph }_2]\to \mathbb{R}$ are continuous, where $\mathfrak{v}\left({\rm Y}\right)$ is a given function in $\Omega$. We define a partial order ${⪯}_{i_2}$ in ${\mathbb{C}}_2$ as $\sigma {⪯}_{i_2}\varpi$ iff $\sigma \le \varpi .$

Theorem 4. 

Suppose that $(\Omega ,\varrho )$ is a complete bicomplex valued b-metric space equipped with the metric $\varrho (\sigma ,\varpi )=(1+i_2)\left(\right|\sigma \left({\rm Y}\right)-\varpi \left({\rm Y}\right){|}^2)$ for all $\sigma ,\varpi \in \Omega$, ${\rm Y}\in [{\aleph }_1,{\aleph }_2]$, and that $\mathcal{S},{\rm Y}:\Omega \to \Omega$ is a continuous operator on Ω defined by

$$\begin{array}{c}\hfill \mathcal{S}\sigma \left({\rm Y}\right)=\mathfrak{v}\left({\rm Y}\right)+\frac{1}{{\aleph }_2-{\aleph }_1}{\int }_{{\aleph }_1}^{{\aleph }_2}{\mathfrak{K}}_1({\rm Y},\mathfrak{s},\sigma \left(\mathfrak{s}\right))d\mathfrak{s},\end{array}$$

(10)

and

$$\begin{array}{c}\hfill \Gamma \sigma \left({\rm Y}\right)=\mathfrak{v}\left({\rm Y}\right)+\frac{1}{{\aleph }_2-{\aleph }_1}{\int }_{{\aleph }_1}^{{\aleph }_2}{\mathfrak{K}}_2({\rm Y},\mathfrak{s},\sigma \left(\mathfrak{s}\right))d\mathfrak{s}.\end{array}$$

(11)

If there exists ${\lambda }_1<1$ satisfying for all $\sigma ,\varpi \in \Omega$, with $\sigma \ne \varpi$ and $\mathfrak{s},{\rm Y}\in [{\aleph }_1,{\aleph }_2]$ satisfying the following inequality

$$\begin{array}{cc}\hfill |{\mathfrak{K}}_1({\rm Y},\mathfrak{s},\sigma \left(s\right))-{\mathfrak{K}}_2({\rm Y},\mathfrak{s},\varpi \left(\mathfrak{s}\right))|\le & \sqrt{{\lambda }_1\left(\sigma \right)}|\sigma \left({\rm Y}\right)-\varpi \left({\rm Y}\right)|,\hfill \end{array}$$

(12)

then Equations (10) and (11) have a unique common solution.

Proof. 

Now,

$$\begin{array}{cc}\hfill (1+i_2)\left(\right|\mathcal{S}\sigma \left({\rm Y}\right)-\Gamma \varpi \left({\rm Y}\right){|}^2)& =\frac{(1+i_2)}{|{\aleph }_2-{\aleph }_1|}\left(\right|{\int }_{{\aleph }_1}^{{\aleph }_2}{\mathfrak{K}}_1({\rm Y},\mathfrak{s},\sigma \left(\mathfrak{s}\right))d\mathfrak{s}\hfill \\ \hfill & -{\int }_{{\aleph }_1}^{{\aleph }_2}{\mathfrak{K}}_2({\rm Y},\mathfrak{s},\varpi \left(\mathfrak{s}\right))d\mathfrak{s}{|}^2)\hfill \\ \hfill & \le \frac{(1+i_2)}{|{\aleph }_2-{\aleph }_1|}({\int }_{{\aleph }_1}^{{\aleph }_2}|{\mathfrak{K}}_1({\rm Y},\mathfrak{s},\sigma \left(\mathfrak{s}\right))\hfill \\ \hfill & -{\mathfrak{K}}_2{({\rm Y},\mathfrak{s},\varpi \left(\mathfrak{s}\right))|}^{2}d\mathfrak{s})\hfill \\ \hfill & \le \frac{(1+i_2){\lambda }_1\left(\sigma \right)}{|{\aleph }_2-{\aleph }_1|}\left({\int }_{{\aleph }_1}^{{\aleph }_2}{|\sigma \left({\rm Y}\right)-\varpi \left({\rm Y}\right)|}^{2}d\mathfrak{s}\right)\hfill \\ \hfill & \le \frac{{\lambda }_1\left(\sigma \right)}{|{\aleph }_2-{\aleph }_1|}{\int }_{{\aleph }_1}^{{\aleph }_2}(1+i_2){|\sigma \left({\rm Y}\right)-\varpi \left({\rm Y}\right)|}^{2}d\mathfrak{s}\hfill \\ \hfill & \le \frac{{\lambda }_1\left(\sigma \right)(1+i_2){|\sigma \left({\rm Y}\right)-\varpi \left({\rm Y}\right)|}^2}{|{\aleph }_2-{\aleph }_1|}{\int }_{{\aleph }_1}^{{\aleph }_2}d\mathfrak{s}.\hfill \end{array}$$

Therefore,

$$\begin{array}{cc}\hfill \varrho (\mathcal{S}\sigma ,\Gamma \varpi ))& \le {\lambda }_1\left(\sigma \right)\varrho (\sigma ,\varpi ).\hfill \end{array}$$

Therefore, all the hypothesis of Theorem 3 are fulfilled with ${\lambda }_2\left(\sigma \right)={\lambda }_3\left(\sigma \right)={\lambda }_4\left(\sigma \right)=0$. Hence, $\mathcal{S}$ and $\Gamma$ have a unique common solution. □

5. Conclusions

In this paper, we used the concept of a bicomplex valued b-metric space to obtain common fixed point results for mixed-type rational contractions involving control functions. We derived common fixed points and fixed points for contractions involving control functions of variables and constants. In 2022, Guan, Li, and Hao [22] proved common fixed point theorems for weak contractions in rectangular b-metric spaces. It is an interesting open problem to prove common fixed point theorems for weak contractions in bicomplex rectangular b-metric spaces. In 2022, Haque, Azmi, and Mlaiki [23] proved the Fredholm-type integral equation in controlled rectangular metric-like spaces. It is also an interesting open problem to prove fixed point theorems on controlled bicomplex rectangular metric-like spaces.