Solving the Fredholm Integral Equation by Common Fixed Point Results in Bicomplex Valued Metric Spaces

Abstract

The purpose of this research work is to explore the solution of the Fredholm integral equation by common fixed point results in bicomplex valued metric spaces. In this way, we develop some common fixed point theorems for generalized contractions containing point-dependent control functions in the context of bicomplex valued metric spaces. An illustrative and practical example is also given to show the novelty of the most important result.

1. Introduction

The theory of bicomplex numbers was constructed by Segre [1] in which the elements or idempotents play a significant role. These bicomplex numbers lengthen complex numbers accurately to quaternions. For a more in-depth analysis of bicomplex numbers, we point out reference [2] to the readers. In 2007, Long-Guang et al. [3] presented the notion of cone metric spaces (CMSs) as an expansion of traditional metric space (MS) and determined fixed point results for contractive mappings. Later on, Azam et al. [4] introduced the conception of a complex valued metric space (CVMS) as a particular case of a CMS. The idea to initiate a CVMS was invented to construct rational expressions which cannot be given in CMSs and consequently numerous results of this theory cannot be obtained in CMSs; thus, CVMSs form a particular class of CMS. Indeed, the concept of a CMS starts to originate the notion of Banach space that is not a division ring. However, we can investigate the extensions of numerous theorems in the theory of fixed points including divisions in CVMSs. Furthermore, this concept is also utilized to introduce the notion of complex-valued Banach spaces [5], which provide a lot of areas for supplemental exploration.

Choi et al. [6] initiated the concept of bicomplex valued metric spaces (bi-CVMSs) by combining the ideas of bicomplex numbers and CVMSs. They proved some common fixed point theorems for weakly compatible mappings. Subsequently, Jebril et al. [7] used the idea of this novel space and presented theorems for two self-mappings in the framework of bi-CVMSs. In 2021, Beg et al. [8] reinforced the conception of bi-CVMSs and proved extrapolated fixed point results. Afterwards, Gnanaprakasam et al. [9] presented results for a contractive-type condition in the framework of bi-CVMSs and explored the solution to linear equations. Recently, Asifa et al. [10] obtained common fixed point results in a bi-CVMS for contractions containing control functions of two variables. For further details in this direction, we refer the reader to [11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31].

In this research article, we develop some common fixed point results in the context of bi-CVMSs for generalized contractions containing point-dependent control functions. We explore the solution of the Fredholm integral equation as an application.

2. Preliminaries

We represent ${\mathbb{C}}_0$ as a set of real numbers, ${\mathbb{C}}_1$ as a set of complex numbers and ${\mathbb{C}}_2$ as a set of bicomplex numbers. Segre [1] defined a bicomplex number in the following way

$$\varsigma =a_1+a_2{i}_1+a_3{i}_2+a_4{i}_1{i}_2$$

where $a_1,a_2,a_3,a_4\in {\mathbb{C}}_0,$ the independent units $i_1,i_2$ are such that $i_1^2=i_2^2=-1$ and $i_1{i}_2=i_2{i}_{1,}$ and ${\mathbb{C}}_2$ is given as

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

that is

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

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 $\varsigma =z_1+i_2{z}_2$ and $\nu ={\omega }_1+i_2{\omega }_2,$ then the sum is

$$\varsigma \pm \nu =\left(z_1+i_2{z}_2\right)\pm \left({\omega }_1+i_2{\omega }_2\right)=\left(z_1\pm {\omega }_1\right)+i_2\left(z_2\pm {\omega }_2\right)$$

and the product is

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

There are four idempotent members 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 uniquely be demonstrated as the mixture of $e_1$ and $e_2$, namely

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

This description of $\varsigma$ is familiar, as the idempotent representation of $\varsigma$ and the complex coefficients ${\varsigma }_1$ $=\left(z_1-i_1{z}_2\right)$ and ${\varsigma }_2=$$\left(z_1+i_1{z}_2\right)$ are called idempotent components of $\varsigma$.

An element $\varsigma =z_1+i_2{z}_2\in {\mathbb{C}}_2$ is called invertible if ∃$\nu \in {\mathbb{C}}_2$ in such a way that $\varsigma \nu =1$, and $\nu$ is said to be the multiplicative inverse of $\varsigma .$ Hence, $\varsigma$ is said to be the multiplicative inverse of $\nu .$

An element $\varsigma =z_1+i_2{z}_2\in {\mathbb{C}}_2$ is nonsingular 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 $\varsigma$ is defined as

$${\varsigma }^{-1}=\nu =\frac{z_1-i_2{z}_2}{z_1^2+z_2^2}.$$

Zero is the only member in ${\mathbb{C}}_0$ which does not have an inverse (multiplicative) and in ${\mathbb{C}}_1$, $0=0+i0$ is the only member that does not have an inverse (multiplicative). We represent the set of singular members of ${\mathbb{C}}_0$ and ${\mathbb{C}}_1$ by ${\aleph }_0$ and ${\aleph }_1$, respectively. In ${\mathbb{C}}_2,$ there are many elements which do not possess a multiplicative inverse. Let us denote the set of singular members of ${\mathbb{C}}_2$ by ${\aleph }_2$ and thus ${\aleph }_0$$={\aleph }_1\subset {\aleph }_2.$

A bicomplex number $\varsigma =a_1+a_2{i}_1+a_3{i}_2+a_4{i}_1{i}_2\in {\mathbb{C}}_2$ is said to be degenerated if the matrix

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

is degenerated. Thus, ${\varsigma }^{-1}$ exists and $∥·∥:{\mathbb{C}}_2\to {\mathbb{C}}_0^{+}$ is given as

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

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

The space ${\mathbb{C}}_2$ with regard to the norm $∥·∥:{\mathbb{C}}_2\to {\mathbb{C}}_0^{+}$. If $\varsigma ,\nu \in {\mathbb{C}}_2,$ then

$$∥\varsigma \nu ∥\le \sqrt{2}∥\varsigma ∥∥\nu ∥$$

holds instead of

$$∥\varsigma \nu ∥\le ∥\varsigma ∥∥\nu ∥.$$

Hence, ${\mathbb{C}}_2$ is not Banach algebra. Let $\varsigma =z_1+i_2{z}_2,$$\nu ={\omega }_1+i_2{\omega }_2\in {\mathbb{C}}_2,$ then we give

$$\varsigma {⪯}_{i_2}\nu \mathrm{if}\mathrm{and}\mathrm{only}\mathrm{if}Re\left(z_1\right)⪯Re\left({\omega }_1\right)\mathrm{and}Im\left(z_2\right)⪯Im\left({\omega }_2\right).$$

This implies

$$\varsigma {⪯}_{i_2}\nu$$

if one of these assertions hold:

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

Specifically, $\varsigma {⋨}_{i_2}\nu$ if and only if $\varsigma {⪯}_{i_2}\nu$ and $\varsigma \ne \nu ,$ i.e., one of the conditions (i), (ii) and (iii) are satisfied. Furthermore, $\varsigma {\prec }_{i_2}\nu$ if only condition (iii) holds. For $\varsigma ,$$\nu \in {\mathbb{C}}_2,$ the following conditions hold,

(i) $\varsigma {⪯}_{i_2}\nu ⟹∥\varsigma ∥\le ∥\nu ∥;$

(ii) $∥\varsigma +\nu ∥\le ∥\varsigma ∥+∥\nu ∥;$

(iii) $∥a\varsigma ∥\le a∥\nu ∥,$ where a is a non negative real number;

(iv) $∥\varsigma \nu ∥\le \sqrt{2}∥\varsigma ∥∥\nu ∥;$

(v) $∥{\varsigma }^{-1}∥={∥\varsigma ∥}^{-1};$

(vi) $∥\frac{\varsigma }{\nu }∥=\frac{∥\varsigma ∥}{∥\nu ∥}$.

Azam et al. [4] defined the idea of a CVMS in this manner.

Definition 1

([4]). Let $\mathcal{Q}\ne \varnothing$, ⪯ be a partial order on $\mathbb{C}$ and $\tau :\mathcal{Q}\times \mathcal{Q}\to {\mathbb{C}}_1$ be a function such that

 (i) 

$0⪯\tau (\varsigma ,\nu )$and $\tau (\varsigma ,\nu )=0$ if and only if $\varsigma =\nu$;

 (ii) 

$\tau (\varsigma ,\nu )=\tau (\nu ,\varsigma );$

 (iii) 

$\tau (\varsigma ,\nu )⪯\tau (\varsigma ,\varphi )+\tau (\varphi ,\nu ),$

for all $\varsigma ,\nu ,\varphi \in \mathcal{Q}.$ Then, $(\mathcal{Q},\tau )$ is a CVMS.

Choi et al. [6] gave the bi-CVMS in this way.

Definition 2

([6]). Let $\mathcal{Q}\ne \varnothing ,$${⪯}_{i_2}$ be a partial order on ${\mathbb{C}}_2$ and $\tau :\mathcal{Q}\times \mathcal{Q}\to {\mathbb{C}}_2$ be a function such that

 (i) 

$0{⪯}_{i_2}\tau (\varsigma ,\nu )$and $\tau (\varsigma ,\nu )=0$ if and only if $\varsigma =\nu$;

 (ii) 

$\tau (\varsigma ,\nu )=\tau (\nu ,\varsigma );$

 (iii) 

$\tau (\varsigma ,\nu ){⪯}_{i_2}\tau (\varsigma ,\varphi )+\tau (\varphi ,\nu ),$

for all $\varsigma ,\nu ,\varphi \in \mathcal{Q}.$ Then, $(\mathcal{Q},\tau )$ is a bi-CVMS.

Example 1

([8]).

Let $\mathcal{Q}={\mathbb{C}}_2$ and $\varsigma ,\nu \in \mathcal{Q}.$ Define $\tau :\mathcal{Q}\times \mathcal{Q}\to {\mathbb{C}}_2$ by

$$\tau (\varsigma ,\nu )=\left|z_1-{\omega }_1\right|+i_2\left|z_2-{\omega }_2\right|$$

where $\varsigma =z_1+i_2{z}_2$ and $\nu ={\omega }_1+i_2{\omega }_2\in {\mathbb{C}}_2.$ Then, $(\mathcal{Q},\tau )$ is a bi-CVMS.

Lemma 1

([8]). Let $(\mathcal{Q},\tau )$be a bi-CVMS and let $\left\{{\varsigma }_{\mathfrak{o}}\right\}$$\subseteq \mathcal{Q}$. Then, $\left\{{\varsigma }_{\mathfrak{o}}\right\}$ converges to ς if and only if $∥\tau ({\varsigma }_{\mathfrak{o}},\varsigma )∥\to 0$as$\mathfrak{o}\to \infty .$

Lemma 2

([8]). Let $(\mathcal{Q},\tau )$be a bi-CVMS and let $\left\{{\varsigma }_{\mathfrak{o}}\right\}$$\subseteq \mathcal{Q}$. Then, $\left\{{\varsigma }_{\mathfrak{o}}\right\}$ is a Cauchy sequence if and only if $∥\tau ({\varsigma }_{\mathfrak{o}},{\varsigma }_{\mathfrak{o}+m})∥\to 0$as$\mathfrak{o}\to \infty ,$where $m\in \mathbb{N}.$

3. Main Result

Proposition 1.

Let $(\mathcal{Q},\tau )$be a bi-CVMS and ${\mathfrak{J}}_1,{\mathfrak{J}}_2:(\mathcal{Q},\tau )\to (\mathcal{Q},\tau ).$ Let ${\varsigma }_0\in$$\mathcal{Q}.$ Define the sequence {${\varsigma }_{\mathfrak{o}}$} by

$${\varsigma }_{2\mathfrak{o}+1}={\mathfrak{J}}_1{\varsigma }_{2\mathfrak{o}}\mathit{and}{\varsigma }_{2\mathfrak{o}+2}={\mathfrak{J}}_2{\varsigma }_{2\mathfrak{o}+1}$$

(1)

for all $\mathfrak{o}=0,1,2,\dots$

Assume that there exists $\alpha :\mathcal{Q}\times \mathcal{Q}\times \mathcal{Q}\to [0,1)$ satisfying

$$\alpha \left({\mathfrak{J}}_2{\mathfrak{J}}_1\varsigma ,\nu ,\theta \right)\le \alpha \left(\varsigma ,\nu ,\theta \right)\mathrm{and}\alpha \left(\varsigma ,{\mathfrak{J}}_1{\mathfrak{J}}_2\nu ,\theta \right)\le \alpha \left(\varsigma ,\nu ,\theta \right)$$

for all $\varsigma ,\nu \in \mathcal{Q}$ and some fixed element $\theta \in \mathcal{Q}.$ Then,

$$\alpha \left({\varsigma }_{2\mathfrak{o}},\nu ,\theta \right)\le \alpha \left({\varsigma }_0,\nu ,\theta \right)\mathrm{and}\alpha \left(\varsigma ,{\varsigma }_{2\mathfrak{o}+1},\theta \right)\le \alpha \left(\varsigma ,{\varsigma }_1,\theta \right)$$

for all $\varsigma ,\nu \in \mathcal{Q}$ and $\mathfrak{o}=0,1,2,\dots$

Proof. 

Let $\varsigma ,\nu \in \mathcal{Q}$ and $\mathfrak{o}=0,1,2,\dots$ Then, we obtain

$$\begin{array}{ccc}\hfill \alpha \left({\varsigma }_{2\mathfrak{o}},\nu ,\theta \right)& =& \alpha \left({\mathfrak{J}}_2{\mathfrak{J}}_1{\varsigma }_{2\mathfrak{o}-2},\nu ,\theta \right)\le \alpha \left({\varsigma }_{2\mathfrak{o}-2},\nu ,\theta \right)\hfill \\ & =& \alpha \left({\mathfrak{J}}_2{\mathfrak{J}}_1{\varsigma }_{2\mathfrak{o}-4},\nu ,\theta \right)\le \alpha \left({\varsigma }_{2\mathfrak{o}-4},\nu ,\theta \right)\hfill \\ & \le & ···\le \alpha \left({\varsigma }_0,\nu ,\theta \right).\hfill \end{array}$$

Similarly, we have

$$\begin{array}{ccc}\hfill \alpha \left(\varsigma ,{\varsigma }_{2\mathfrak{o}+1},\theta \right)& =& \alpha \left(\varsigma ,{\mathfrak{J}}_1{\mathfrak{J}}_2{\varsigma }_{2\mathfrak{o}-1},\theta \right)\le \alpha \left(\varsigma ,{\varsigma }_{2\mathfrak{o}-1},\theta \right)\hfill \\ & =& \alpha \left(\varsigma ,{\mathfrak{J}}_1{\mathfrak{J}}_2{\varsigma }_{2\mathfrak{o}-3},\theta \right)\le \alpha \left(\varsigma ,{\varsigma }_{2\mathfrak{o}-3},\theta \right)\hfill \\ & \le & ···\le \alpha \left(\varsigma ,{\varsigma }_1,\theta \right).\hfill \end{array}$$

□

Theorem 1.

Let $\left(\mathcal{Q},\tau \right)$ be a complete bi-CVMS and ${\mathfrak{J}}_1,{\mathfrak{J}}_2:\mathcal{Q}$$\to \mathcal{Q}$. If the functions $\alpha ,\pi ,\kappa ,\varpi ,\varkappa :{\mathcal{Q}}^3\to [0,1)$ satisfy the conditions

(a)

$\alpha \left({\mathfrak{J}}_2{\mathfrak{J}}_1\varsigma ,\nu ,\theta \right)\le \alpha \left(\varsigma ,\nu ,\theta \right)$ and $\alpha \left(\varsigma ,{\mathfrak{J}}_1{\mathfrak{J}}_2\nu ,\theta \right)\le \alpha \left(\varsigma ,\nu ,\theta \right)$

$\pi \left({\mathfrak{J}}_2{\mathfrak{J}}_1\varsigma ,\nu ,\theta \right)\le \pi \left(\varsigma ,\nu ,\theta \right)$ and $\pi \left(\varsigma ,{\mathfrak{J}}_1{\mathfrak{J}}_2\nu ,\theta \right)\le \pi \left(\varsigma ,\nu ,\theta \right)$

$\kappa \left({\mathfrak{J}}_2{\mathfrak{J}}_1\varsigma ,\nu ,\theta \right)\le \kappa \left(\varsigma ,\nu ,\theta \right)$ and $\kappa \left(\varsigma ,{\mathfrak{J}}_1{\mathfrak{J}}_2\nu ,\theta \right)\le \kappa \left(\varsigma ,\nu ,\theta \right)$

$\varpi \left({\mathfrak{J}}_2{\mathfrak{J}}_1\varsigma ,\nu ,\theta \right)\le \varpi \left(\varsigma ,\nu ,\theta \right)$ and $\varpi \left(\varsigma ,{\mathfrak{J}}_1{\mathfrak{J}}_2\nu ,\theta \right)\le \varpi \left(\varsigma ,\nu ,\theta \right)$

$\varkappa \left({\mathfrak{J}}_2{\mathfrak{J}}_1\varsigma ,\nu ,\theta \right)\le \varkappa \left(\varsigma ,\nu ,\theta \right)$ and $\varkappa \left(\varsigma ,{\mathfrak{J}}_1{\mathfrak{J}}_2\nu ,\theta \right)\le \varkappa \left(\varsigma ,\nu ,\theta \right)$

(b) $\alpha \left(\varsigma ,\nu ,\theta \right)+2\pi \left(\varsigma ,\nu ,\theta \right)+\sqrt{2}\kappa \left(\varsigma ,\nu ,\theta \right)+\sqrt{2}\varpi \left(\varsigma ,\nu ,\theta \right)+\sqrt{2}\varkappa \left(\varsigma ,\nu ,\theta \right)<1,$

(c)

$$\tau \left({\mathfrak{J}}_1\varsigma ,{\mathfrak{J}}_2\nu \right){⪯}_{i_2}\alpha \left(\varsigma ,\nu ,\theta \right)\tau \left(\varsigma ,\nu \right)+\pi \left(\varsigma ,\nu ,\theta \right)\left(\tau \left(\varsigma ,{\mathfrak{J}}_1\varsigma \right)+\tau \left(\nu ,{\mathfrak{J}}_2\nu \right)\right)$$

$$+\kappa \left(\varsigma ,\nu ,\theta \right)\frac{\tau \left(\varsigma ,{\mathfrak{J}}_1\varsigma \right)\tau \left(\nu ,{\mathfrak{J}}_2\nu \right)}{1+\tau \left(\varsigma ,\nu \right)}$$

$$+\varpi \left(\varsigma ,\nu ,\theta \right)\frac{\tau \left(\nu ,{\mathfrak{J}}_1\varsigma \right)\tau \left(\varsigma ,{\mathfrak{J}}_2\nu \right)}{1+\tau \left(\varsigma ,\nu \right)}$$

$$+\varkappa \left(\varsigma ,\nu ,\theta \right)\frac{\tau \left(\varsigma ,{\mathfrak{J}}_1\varsigma \right)\tau \left(\varsigma ,{\mathfrak{J}}_2\nu \right)+\tau \left(\nu ,{\mathfrak{J}}_2\nu \right)\tau \left(\nu ,{\mathfrak{J}}_1\varsigma \right)}{1+\tau \left(\varsigma ,{\mathfrak{J}}_2\nu \right)+\tau \left(\nu ,{\mathfrak{J}}_1\varsigma \right)},$$

(2)

for all $\varsigma ,\nu \in$$\mathcal{Q}$and for fixed element $\theta \in \mathcal{Q}$, then there exists a unique element ${\varsigma }^{*}\in \mathcal{Q}$ such that ${\mathfrak{J}}_1{\varsigma }^{*}={\mathfrak{J}}_2{\varsigma }^{*}={\varsigma }^{*}.$

Proof. 

Let $\varsigma ,\nu \in \mathcal{Q}.$ From (2), we have

$$\tau \left({\mathfrak{J}}_1\varsigma ,{\mathfrak{J}}_2{\mathfrak{J}}_1\varsigma \right){⪯}_{i_2}\alpha \left(\varsigma ,{\mathfrak{J}}_1\varsigma ,\theta \right)\tau \left(\varsigma ,{\mathfrak{J}}_1\varsigma \right)$$

$$+\pi \left(\varsigma ,{\mathfrak{J}}_1\varsigma ,\theta \right)\left(\tau \left(\varsigma ,{\mathfrak{J}}_1\varsigma \right)+\tau \left({\mathfrak{J}}_1\varsigma ,{\mathfrak{J}}_2{\mathfrak{J}}_1\varsigma \right)\right)$$

$$+\kappa \left(\varsigma ,{\mathfrak{J}}_1\varsigma ,\theta \right)\frac{\tau \left(\varsigma ,{\mathfrak{J}}_1\varsigma \right)\tau \left({\mathfrak{J}}_1\varsigma ,{\mathfrak{J}}_2{\mathfrak{J}}_1\varsigma \right)}{1+\tau \left(\varsigma ,{\mathfrak{J}}_1\varsigma \right)}$$

$$+\varpi \left(\varsigma ,{\mathfrak{J}}_1\varsigma ,\theta \right)\frac{\tau \left({\mathfrak{J}}_1\varsigma ,{\mathfrak{J}}_1\varsigma \right)\tau \left(\varsigma ,{\mathfrak{J}}_2{\mathfrak{J}}_1\varsigma \right)}{1+\tau \left(\varsigma ,{\mathfrak{J}}_1\varsigma \right)}$$

$$+\varkappa \left(\varsigma ,{\mathfrak{J}}_1\varsigma ,\theta \right)\frac{\tau \left(\varsigma ,{\mathfrak{J}}_1\varsigma \right)\tau \left(\varsigma ,{\mathfrak{J}}_2{\mathfrak{J}}_1\varsigma \right)+\tau \left({\mathfrak{J}}_1\varsigma ,{\mathfrak{J}}_2{\mathfrak{J}}_1\varsigma \right)\tau \left({\mathfrak{J}}_1\varsigma ,{\mathfrak{J}}_1\varsigma \right)}{1+\tau \left(\varsigma ,{\mathfrak{J}}_2{\mathfrak{J}}_1\varsigma \right)+\tau \left({\mathfrak{J}}_1\varsigma ,{\mathfrak{J}}_1\varsigma \right)}$$

which implies

$$\begin{array}{ccc}\hfill ∥\tau \left({\mathfrak{J}}_1\varsigma ,{\mathfrak{J}}_2{\mathfrak{J}}_1\varsigma \right)∥& \le & \alpha \left(\varsigma ,{\mathfrak{J}}_1\varsigma ,\theta \right)∥\tau \left(\varsigma ,{\mathfrak{J}}_1\varsigma \right)∥\hfill \\ & & +\pi \left(\varsigma ,{\mathfrak{J}}_1\varsigma ,\theta \right)\left(∥\tau \left(\varsigma ,{\mathfrak{J}}_1\varsigma \right)∥+∥\tau \left({\mathfrak{J}}_1\varsigma ,{\mathfrak{J}}_2{\mathfrak{J}}_1\varsigma \right)∥\right)\hfill \\ & & +\sqrt{2}\kappa \left(\varsigma ,{\mathfrak{J}}_1\varsigma ,\theta \right)\frac{∥\tau \left(\varsigma ,{\mathfrak{J}}_1\varsigma \right)∥}{∥1+\tau \left(\varsigma ,{\mathfrak{J}}_1\varsigma \right)∥}∥\tau \left({\mathfrak{J}}_1\varsigma ,{\mathfrak{J}}_2{\mathfrak{J}}_1\varsigma \right)∥\hfill \\ & & +\sqrt{2}\varkappa \left(\varsigma ,{\mathfrak{J}}_1\varsigma ,\theta \right)∥\tau \left(\varsigma ,{\mathfrak{J}}_1\varsigma \right)∥\frac{∥\tau \left(\varsigma ,{\mathfrak{J}}_2{\mathfrak{J}}_1\varsigma \right)∥}{∥1+\tau \left(\varsigma ,{\mathfrak{J}}_2{\mathfrak{J}}_1\varsigma \right)∥},\hfill \end{array}$$

since $∥\tau \left(\varsigma ,{\mathfrak{J}}_1\varsigma \right)∥<∥1+\tau \left(\varsigma ,{\mathfrak{J}}_1\varsigma \right)∥,$ so $\frac{∥\tau \left(\varsigma ,{\mathfrak{J}}_1\varsigma \right)∥}{∥1+\tau \left(\varsigma ,{\mathfrak{J}}_1\varsigma \right)∥}<1.$ Similarly, $∥\tau \left(\varsigma ,{\mathfrak{J}}_2{\mathfrak{J}}_1\varsigma \right)∥<$$∥1+\tau \left(\varsigma ,{\mathfrak{J}}_2{\mathfrak{J}}_1\varsigma \right)∥,$ and thus $\frac{∥\tau \left(\varsigma ,{\mathfrak{J}}_2{\mathfrak{J}}_1\varsigma \right)∥}{∥1+\tau \left(\varsigma ,{\mathfrak{J}}_2{\mathfrak{J}}_1\varsigma \right)∥}<1.$ Hence, we have

$$\begin{array}{ccc}\hfill ∥\tau \left({\mathfrak{J}}_1\varsigma ,{\mathfrak{J}}_2{\mathfrak{J}}_1\varsigma \right)∥& \le & \alpha \left(\varsigma ,{\mathfrak{J}}_1\varsigma ,\theta \right)∥\tau \left(\varsigma ,{\mathfrak{J}}_1\varsigma \right)∥\hfill \\ & & +\pi \left(\varsigma ,{\mathfrak{J}}_1\varsigma ,\theta \right)\left(∥\tau \left(\varsigma ,{\mathfrak{J}}_1\varsigma \right)∥+∥\tau \left({\mathfrak{J}}_1\varsigma ,{\mathfrak{J}}_2{\mathfrak{J}}_1\varsigma \right)∥\right)\hfill \\ & & +\sqrt{2}\kappa \left(\varsigma ,{\mathfrak{J}}_1\varsigma ,\theta \right)∥\tau \left({\mathfrak{J}}_1\varsigma ,{\mathfrak{J}}_2{\mathfrak{J}}_1\varsigma \right)∥\hfill \\ & & +\sqrt{2}\varkappa \left(\varsigma ,{\mathfrak{J}}_1\varsigma ,\theta \right)∥\tau \left(\varsigma ,{\mathfrak{J}}_1\varsigma \right)∥.\hfill \end{array}$$

(3)

Similarly,

$$\tau \left({\mathfrak{J}}_1{\mathfrak{J}}_2\nu ,{\mathfrak{J}}_2\nu \right){⪯}_{i_2}\alpha \left({\mathfrak{J}}_2\nu ,\nu ,\theta \right)\tau \left({\mathfrak{J}}_2\nu ,\nu \right)+\pi \left({\mathfrak{J}}_2\nu ,\nu ,\theta \right)\left(\tau \left({\mathfrak{J}}_2\nu ,{\mathfrak{J}}_1{\mathfrak{J}}_2\nu \right)+\tau \left(\nu ,{\mathfrak{J}}_2\nu \right)\right)$$

$$+\kappa \left({\mathfrak{J}}_2\nu ,\nu ,\theta \right)\frac{\tau \left({\mathfrak{J}}_2\nu ,{\mathfrak{J}}_1{\mathfrak{J}}_2\nu \right)\tau \left(\nu ,{\mathfrak{J}}_2\nu \right)}{1+\tau \left({\mathfrak{J}}_2\nu ,\nu \right)}$$

$$+\varpi \left({\mathfrak{J}}_2\nu ,\nu ,\theta \right)\frac{\tau \left(\nu ,{\mathfrak{J}}_1{\mathfrak{J}}_2\nu \right)\tau \left({\mathfrak{J}}_2\nu ,{\mathfrak{J}}_2\nu \right)}{1+\tau \left({\mathfrak{J}}_2\nu ,\nu \right)}$$

$$+\varkappa \left({\mathfrak{J}}_2\nu ,\nu ,\theta \right)\frac{\tau \left({\mathfrak{J}}_2\nu ,{\mathfrak{J}}_1{\mathfrak{J}}_2\nu \right)\tau \left({\mathfrak{J}}_2\nu ,{\mathfrak{J}}_2\nu \right)+\tau \left(\nu ,{\mathfrak{J}}_2\nu \right)\tau \left(\nu ,{\mathfrak{J}}_1{\mathfrak{J}}_2\nu \right)}{1+\tau \left({\mathfrak{J}}_2\nu ,{\mathfrak{J}}_2\nu \right)+\tau \left(\nu ,{\mathfrak{J}}_1{\mathfrak{J}}_2\nu \right)}$$

which implies

$$\begin{array}{ccc}\hfill ∥\tau \left({\mathfrak{J}}_1{\mathfrak{J}}_2\nu ,{\mathfrak{J}}_2\nu \right)∥& \le & \alpha \left({\mathfrak{J}}_2\nu ,\nu \right)∥\tau \left({\mathfrak{J}}_2\nu ,\nu \right)∥\hfill \\ & & +\pi \left({\mathfrak{J}}_2\nu ,\nu ,\theta \right)\left(∥\tau \left({\mathfrak{J}}_2\nu ,{\mathfrak{J}}_1{\mathfrak{J}}_2\nu \right)∥+∥\tau \left(\nu ,{\mathfrak{J}}_2\nu \right)∥\right)\hfill \\ & & +\sqrt{2}\kappa \left({\mathfrak{J}}_2\nu ,\nu \right)∥\tau \left({\mathfrak{J}}_2\nu ,{\mathfrak{J}}_1{\mathfrak{J}}_2\nu \right)∥\frac{∥\tau \left(\nu ,{\mathfrak{J}}_2\nu \right)∥}{∥1+\tau \left({\mathfrak{J}}_2\nu ,\nu \right)∥}\hfill \\ & & +\sqrt{2}\varkappa \left({\mathfrak{J}}_2\nu ,\nu ,\theta \right)∥\tau \left(\nu ,{\mathfrak{J}}_2\nu \right)∥\frac{∥\tau \left(\nu ,{\mathfrak{J}}_1{\mathfrak{J}}_2\nu \right)∥}{∥1+\tau \left(\nu ,{\mathfrak{J}}_1{\mathfrak{J}}_2\nu \right)∥}.\hfill \end{array}$$

Since $∥\tau \left(\nu ,{\mathfrak{J}}_2\nu \right)∥<∥1+\tau \left({\mathfrak{J}}_2\nu ,\nu \right)∥,$ so $\frac{∥\tau \left(\nu ,{\mathfrak{J}}_2\nu \right)∥}{∥1+\tau \left({\mathfrak{J}}_2\nu ,\nu \right)∥}<1.$ Furthermore, $∥\tau \left(\nu ,{\mathfrak{J}}_1{\mathfrak{J}}_2\nu \right)∥<$$∥1+\tau \left(\nu ,{\mathfrak{J}}_1{\mathfrak{J}}_2\nu \right)∥,$ and so $\frac{∥\tau \left(\nu ,{\mathfrak{J}}_1{\mathfrak{J}}_2\nu \right)∥}{∥1+\tau \left(\nu ,{\mathfrak{J}}_1{\mathfrak{J}}_2\nu \right)∥}<1.$ Thus,

$$\begin{array}{ccc}\hfill ∥\tau \left({\mathfrak{J}}_1{\mathfrak{J}}_2\nu ,{\mathfrak{J}}_2\nu \right)∥& \le & \alpha \left({\mathfrak{J}}_2\nu ,\nu \right)∥\tau \left({\mathfrak{J}}_2\nu ,\nu \right)∥\hfill \\ & & +\pi \left({\mathfrak{J}}_2\nu ,\nu ,\theta \right)\left(∥\tau \left({\mathfrak{J}}_2\nu ,{\mathfrak{J}}_1{\mathfrak{J}}_2\nu \right)∥+∥\tau \left(\nu ,{\mathfrak{J}}_2\nu \right)∥\right)\hfill \\ & & +\sqrt{2}\kappa \left({\mathfrak{J}}_2\nu ,\nu \right)∥\tau \left({\mathfrak{J}}_2\nu ,{\mathfrak{J}}_1{\mathfrak{J}}_2\nu \right)∥\hfill \\ & & +\sqrt{2}\varkappa \left({\mathfrak{J}}_2\nu ,\nu ,\theta \right)∥\tau \left(\nu ,{\mathfrak{J}}_2\nu \right)∥.\hfill \end{array}$$

(4)

Let ${\varsigma }_0$$\in \mathcal{Q}$ and the sequence {${\varsigma }_{\mathfrak{o}}$} be defined by (1). By inequalities (3) and (4), we have

$$\begin{array}{ccc}\hfill ∥\tau \left({\varsigma }_{2\mathfrak{o}+1},{\varsigma }_{2\mathfrak{o}}\right)∥& =& ∥\tau \left({\mathfrak{J}}_1{\mathfrak{J}}_2{\varsigma }_{2\mathfrak{o}-1},{\mathfrak{J}}_2{\varsigma }_{2\mathfrak{o}-1}\right)∥\hfill \\ & \le & \alpha \left({\mathfrak{J}}_2{\varsigma }_{2\mathfrak{o}-1},{\varsigma }_{2\mathfrak{o}-1},\theta \right)∥\tau \left({\mathfrak{J}}_2{\varsigma }_{2\mathfrak{o}-1},{\varsigma }_{2\mathfrak{o}-1}\right)∥\hfill \\ & & +\pi \left({\mathfrak{J}}_2{\varsigma }_{2\mathfrak{o}-1},{\varsigma }_{2\mathfrak{o}-1},\theta \right)\left(∥\tau \left({\mathfrak{J}}_2{\varsigma }_{2\mathfrak{o}-1},{\mathfrak{J}}_1{\mathfrak{J}}_2{\varsigma }_{2\mathfrak{o}-1}\right)∥+∥\tau \left({\varsigma }_{2\mathfrak{o}-1},{\mathfrak{J}}_2{\varsigma }_{2\mathfrak{o}-1}\right)∥\right)\hfill \\ & & +\sqrt{2}\kappa \left({\mathfrak{J}}_2{\varsigma }_{2\mathfrak{o}-1},{\varsigma }_{2\mathfrak{o}-1},\theta \right)∥\tau \left({\mathfrak{J}}_2{\varsigma }_{2\mathfrak{o}-1},{\mathfrak{J}}_1{\mathfrak{J}}_2{\varsigma }_{2\mathfrak{o}-1}\right)∥\hfill \\ & & +\sqrt{2}\varkappa \left({\mathfrak{J}}_2{\varsigma }_{2\mathfrak{o}-1},{\varsigma }_{2\mathfrak{o}-1},\theta \right)∥\tau \left({\varsigma }_{2\mathfrak{o}-1},{\mathfrak{J}}_2{\varsigma }_{2\mathfrak{o}-1}\right)∥\hfill \\ & =& \alpha \left({\varsigma }_{2\mathfrak{o}},{\varsigma }_{2\mathfrak{o}-1},\theta \right)∥\tau \left({\varsigma }_{2\mathfrak{o}},{\varsigma }_{2\mathfrak{o}-1}\right)∥\hfill \\ & & +\pi \left({\varsigma }_{2\mathfrak{o}},{\varsigma }_{2\mathfrak{o}-1},\theta \right)\left(∥\tau \left({\varsigma }_{2\mathfrak{o}},{\varsigma }_{2\mathfrak{o}+1}\right)∥+∥\tau \left({\varsigma }_{2\mathfrak{o}-1},{\varsigma }_{2\mathfrak{o}}\right)∥\right)\hfill \\ & & +\sqrt{2}\kappa \left({\varsigma }_{2\mathfrak{o}},{\varsigma }_{2\mathfrak{o}-1},\theta \right)∥\tau \left({\varsigma }_{2\mathfrak{o}},{\varsigma }_{2\mathfrak{o}+1}\right)∥\hfill \\ & & +\sqrt{2}\varkappa \left({\varsigma }_{2\mathfrak{o}},{\varsigma }_{2\mathfrak{o}-1},\theta \right)∥\tau \left({\varsigma }_{2\mathfrak{o}-1},{\varsigma }_{2\mathfrak{o}}\right)∥\hfill \end{array}$$

By Proposition 1, we have

$$\begin{array}{ccc}\hfill ∥\tau \left({\varsigma }_{2\mathfrak{o}+1},{\varsigma }_{2\mathfrak{o}}\right)∥& \le & \alpha \left({\varsigma }_0,{\varsigma }_{2\mathfrak{o}-1},\theta \right)∥\tau \left({\varsigma }_{2\mathfrak{o}},{\varsigma }_{2\mathfrak{o}-1}\right)∥\hfill \\ & & +\pi \left({\varsigma }_0,{\varsigma }_{2\mathfrak{o}-1},\theta \right)\left(∥\tau \left({\varsigma }_{2\mathfrak{o}},{\varsigma }_{2\mathfrak{o}+1}\right)∥+∥\tau \left({\varsigma }_{2\mathfrak{o}-1},{\varsigma }_{2\mathfrak{o}}\right)∥\right)\hfill \\ & & +\sqrt{2}\kappa \left({\varsigma }_0,{\varsigma }_{2\mathfrak{o}-1},\theta \right)∥\tau \left({\varsigma }_{2\mathfrak{o}},{\varsigma }_{2\mathfrak{o}+1}\right)∥\hfill \\ & & +\sqrt{2}\varkappa \left({\varsigma }_0,{\varsigma }_{2\mathfrak{o}-1},\theta \right)∥\tau \left({\varsigma }_{2\mathfrak{o}-1},{\varsigma }_{2\mathfrak{o}}\right)∥\hfill \\ & \le & \alpha \left({\varsigma }_0,{\varsigma }_1,\theta \right)∥\tau \left({\varsigma }_{2\mathfrak{o}},{\varsigma }_{2\mathfrak{o}-1}\right)∥\hfill \\ & & +\pi \left({\varsigma }_0,{\varsigma }_1,\theta \right)\left(∥\tau \left({\varsigma }_{2\mathfrak{o}},{\varsigma }_{2\mathfrak{o}+1}\right)∥+∥\tau \left({\varsigma }_{2\mathfrak{o}-1},{\varsigma }_{2\mathfrak{o}}\right)∥\right)\hfill \\ & & +\sqrt{2}\kappa \left({\varsigma }_0,{\varsigma }_1,\theta \right)∥\tau \left({\varsigma }_{2\mathfrak{o}},{\varsigma }_{2\mathfrak{o}+1}\right)∥\hfill \\ & & +\sqrt{2}\varkappa \left({\varsigma }_0,{\varsigma }_1,\theta \right)∥\tau \left({\varsigma }_{2\mathfrak{o}-1},{\varsigma }_{2\mathfrak{o}}\right)∥\hfill \end{array}$$

for all $\mathfrak{o}=0,1,2,\dots$. This implies that

$$∥\tau \left({\varsigma }_{2\mathfrak{o}+1},{\varsigma }_{2\mathfrak{o}}\right)∥\le \frac{\alpha \left({\varsigma }_0,{\varsigma }_1,\theta \right)+\pi \left({\varsigma }_0,{\varsigma }_1,\theta \right)+\sqrt{2}\varkappa \left({\varsigma }_0,{\varsigma }_1,\theta \right)}{1-\pi \left({\varsigma }_0,{\varsigma }_1,\theta \right)-\sqrt{2}\kappa \left({\varsigma }_0,{\varsigma }_1,\theta \right)}∥\tau \left({\varsigma }_{2\mathfrak{o}},{\varsigma }_{2\mathfrak{o}-1}\right)∥.$$

(5)

Similarly, we have

$$\begin{array}{ccc}\hfill ∥\tau \left({\varsigma }_{2\mathfrak{o}+2},{\varsigma }_{2\mathfrak{o}+1}\right)∥& =& ∥\tau \left({\mathfrak{J}}_2{\mathfrak{J}}_1{\varsigma }_{2\mathfrak{o}},{\mathfrak{J}}_1{\varsigma }_{2\mathfrak{o}}\right)∥\hfill \\ & =& ∥\tau \left({\mathfrak{J}}_1{\varsigma }_{2\mathfrak{o}},{\mathfrak{J}}_2{\mathfrak{J}}_1{\varsigma }_{2\mathfrak{o}}\right)∥\hfill \\ & \le & \alpha \left({\varsigma }_{2\mathfrak{o}},{\mathfrak{J}}_1{\varsigma }_{2\mathfrak{o}},\theta \right)∥\tau \left({\varsigma }_{2\mathfrak{o}},{\mathfrak{J}}_1{\varsigma }_{2\mathfrak{o}}\right)∥\hfill \\ & & +\pi \left({\varsigma }_{2\mathfrak{o}},{\mathfrak{J}}_1{\varsigma }_{2\mathfrak{o}},\theta \right)\left(∥\tau \left({\varsigma }_{2\mathfrak{o}},{\mathfrak{J}}_1{\varsigma }_{2\mathfrak{o}}\right)∥+∥\tau \left({\mathfrak{J}}_1{\varsigma }_{2\mathfrak{o}},{\mathfrak{J}}_2{\mathfrak{J}}_1{\varsigma }_{2\mathfrak{o}}\right)∥\right)\hfill \\ & & +\sqrt{2}\kappa \left({\varsigma }_{2\mathfrak{o}},{\mathfrak{J}}_1{\varsigma }_{2\mathfrak{o}},\theta \right)∥\tau \left({\mathfrak{J}}_1{\varsigma }_{2\mathfrak{o}},{\mathfrak{J}}_2{\mathfrak{J}}_1{\varsigma }_{2\mathfrak{o}}\right)∥\hfill \\ & & +\sqrt{2}\varkappa \left({\varsigma }_{2\mathfrak{o}},{\mathfrak{J}}_1{\varsigma }_{2\mathfrak{o}},\theta \right)∥\tau \left({\varsigma }_{2\mathfrak{o}},{\mathfrak{J}}_1{\varsigma }_{2\mathfrak{o}}\right)∥\hfill \\ & =& \alpha \left({\varsigma }_{2\mathfrak{o}},{\varsigma }_{2\mathfrak{o}+1},\theta \right)∥\tau \left({\varsigma }_{2\mathfrak{o}},{\varsigma }_{2\mathfrak{o}+1}\right)∥\hfill \\ & & +\pi \left({\varsigma }_{2\mathfrak{o}},{\varsigma }_{2\mathfrak{o}+1},\theta \right)\left(∥\tau \left({\varsigma }_{2\mathfrak{o}},{\varsigma }_{2\mathfrak{o}+1}\right)∥+∥\tau \left({\varsigma }_{2\mathfrak{o}+1},{\varsigma }_{2\mathfrak{o}+2}\right)∥\right)\hfill \\ & & +\sqrt{2}\kappa \left({\varsigma }_{2\mathfrak{o}},{\varsigma }_{2\mathfrak{o}+1},\theta \right)∥\tau \left({\varsigma }_{2\mathfrak{o}+1},{\varsigma }_{2\mathfrak{o}+2}\right)∥\hfill \\ & & +\sqrt{2}\varkappa \left({\varsigma }_{2\mathfrak{o}},{\varsigma }_{2\mathfrak{o}+1},\theta \right)∥\tau \left({\varsigma }_{2\mathfrak{o}},{\varsigma }_{2\mathfrak{o}+1}\right)∥\hfill \\ & \le & \alpha \left({\varsigma }_0,{\varsigma }_1\right)∥\tau \left({\varsigma }_{2\mathfrak{o}},{\varsigma }_{2\mathfrak{o}+1}\right)∥+\sqrt{2}\kappa \left({\varsigma }_0,{\varsigma }_1\right)∥\tau \left({\varsigma }_{2\mathfrak{o}+1},{\varsigma }_{2\mathfrak{o}+2}\right)∥\hfill \end{array}$$

By Proposition 1, we have

$$\begin{array}{ccc}\hfill ∥\tau \left({\varsigma }_{2\mathfrak{o}+2},{\varsigma }_{2\mathfrak{o}+1}\right)∥& \le & \alpha \left({\varsigma }_0,{\varsigma }_{2\mathfrak{o}+1},\theta \right)∥\tau \left({\varsigma }_{2\mathfrak{o}},{\varsigma }_{2\mathfrak{o}+1}\right)∥\hfill \\ & & +\pi \left({\varsigma }_0,{\varsigma }_{2\mathfrak{o}+1},\theta \right)\left(∥\tau \left({\varsigma }_{2\mathfrak{o}},{\varsigma }_{2\mathfrak{o}+1}\right)∥+∥\tau \left({\varsigma }_{2\mathfrak{o}+1},{\varsigma }_{2\mathfrak{o}+2}\right)∥\right)\hfill \\ & & +\sqrt{2}\kappa \left({\varsigma }_0,{\varsigma }_{2\mathfrak{o}+1},\theta \right)∥\tau \left({\varsigma }_{2\mathfrak{o}+1},{\varsigma }_{2\mathfrak{o}+2}\right)∥\hfill \\ & & +\sqrt{2}\varkappa \left({\varsigma }_0,{\varsigma }_{2\mathfrak{o}+1},\theta \right)∥\tau \left({\varsigma }_{2\mathfrak{o}},{\varsigma }_{2\mathfrak{o}+1}\right)∥\hfill \\ & \le & \alpha \left({\varsigma }_0,{\varsigma }_1,\theta \right)∥\tau \left({\varsigma }_{2\mathfrak{o}},{\varsigma }_{2\mathfrak{o}+1}\right)∥\hfill \\ & & +\pi \left({\varsigma }_0,{\varsigma }_1,\theta \right)\left(∥\tau \left({\varsigma }_{2\mathfrak{o}},{\varsigma }_{2\mathfrak{o}+1}\right)∥+∥\tau \left({\varsigma }_{2\mathfrak{o}+1},{\varsigma }_{2\mathfrak{o}+2}\right)∥\right)\hfill \\ & & +\sqrt{2}\kappa \left({\varsigma }_0,{\varsigma }_1,\theta \right)∥\tau \left({\varsigma }_{2\mathfrak{o}+1},{\varsigma }_{2\mathfrak{o}+2}\right)∥\hfill \\ & & +\sqrt{2}\varkappa \left({\varsigma }_0,{\varsigma }_1,\theta \right)∥\tau \left({\varsigma }_{2\mathfrak{o}},{\varsigma }_{2\mathfrak{o}+1}\right)∥\hfill \end{array}$$

which implies that

$$\begin{array}{ccc}\hfill ∥\tau \left({\varsigma }_{2\mathfrak{o}+2},{\varsigma }_{2\mathfrak{o}+1}\right)∥& \le & \frac{\alpha \left({\varsigma }_0,{\varsigma }_1,\theta \right)+\pi \left({\varsigma }_0,{\varsigma }_1,\theta \right)+\sqrt{2}\varkappa \left({\varsigma }_0,{\varsigma }_1,\theta \right)}{1-\pi \left({\varsigma }_0,{\varsigma }_1,\theta \right)-\sqrt{2}\kappa \left({\varsigma }_0,{\varsigma }_1,\theta \right)}∥\tau \left({\varsigma }_{2\mathfrak{o}},{\varsigma }_{2\mathfrak{o}+1}\right)∥\hfill \\ & =& \frac{\alpha \left({\varsigma }_0,{\varsigma }_1,\theta \right)+\pi \left({\varsigma }_0,{\varsigma }_1,\theta \right)+\sqrt{2}\varkappa \left({\varsigma }_0,{\varsigma }_1,\theta \right)}{1-\pi \left({\varsigma }_0,{\varsigma }_1,\theta \right)-\sqrt{2}\kappa \left({\varsigma }_0,{\varsigma }_1,\theta \right)}∥\tau \left({\varsigma }_{2\mathfrak{o}+1},{\varsigma }_{2\mathfrak{o}}\right)∥.\hfill \end{array}$$

(6)

Let $\lambda =$$\frac{\alpha \left({\varsigma }_0,{\varsigma }_1,\theta \right)+\pi \left({\varsigma }_0,{\varsigma }_1,\theta \right)+\sqrt{2}\varkappa \left({\varsigma }_0,{\varsigma }_1,\theta \right)}{1-\pi \left({\varsigma }_0,{\varsigma }_1,\theta \right)-\sqrt{2}\kappa \left({\varsigma }_0,{\varsigma }_1,\theta \right)}<1.$ Then, from (5) and (6), we have

$$∥\tau \left({\varsigma }_{\mathfrak{o}+1},{\varsigma }_{\mathfrak{o}}\right)∥\le \lambda ∥\tau \left({\varsigma }_{\mathfrak{o}},{\varsigma }_{\mathfrak{o}-1}\right)∥$$

for all $\mathfrak{o}\in \mathbb{N}.$ Thus, deductively, we can set up a sequence $\left\{{\varsigma }_{\mathfrak{o}}\right\}$ in $\mathcal{Q}$ such that

$$\begin{array}{ccc}\hfill ∥\tau \left({\varsigma }_{\mathfrak{o}+1},{\varsigma }_{\mathfrak{o}}\right)∥& \le & \lambda ∥\tau \left({\varsigma }_{\mathfrak{o}},{\varsigma }_{\mathfrak{o}-1}\right)∥\hfill \\ & \le & {\lambda }^2∥\tau \left({\varsigma }_{\mathfrak{o}-1},{\varsigma }_{\mathfrak{o}-2}\right)∥···\le {\lambda }^{\mathfrak{o}}∥\tau \left({\varsigma }_1,{\varsigma }_0\right)∥={\lambda }^{\mathfrak{o}}∥\tau \left({\varsigma }_0,{\varsigma }_1\right)∥\hfill \end{array}$$

for all $\mathfrak{o}\in \mathbb{N}.$ Now, for $m>\mathfrak{o}$, we obtain

$$\begin{array}{ccc}\hfill ∥\tau \left({\varsigma }_{\mathfrak{o}},{\varsigma }_m\right)∥& \le & {\lambda }^{\mathfrak{o}}∥\tau \left({\varsigma }_0,{\varsigma }_1\right)∥\hfill \\ & & +{\lambda }^{\mathfrak{o}+1}∥\tau \left({\varsigma }_0,{\varsigma }_1\right)∥\hfill \\ & & +···+\hfill \\ & & {\lambda }^{m-1}∥\tau \left({\varsigma }_0,{\varsigma }_1\right)∥\hfill \\ & \le & \frac{{\lambda }^{\mathfrak{o}}}{1-\lambda }∥\tau \left({\varsigma }_0,{\varsigma }_1\right)∥.\hfill \end{array}$$

Now, by taking $\mathfrak{o},m\to \infty$, we obtain

$$∥\tau \left({\varsigma }_{\mathfrak{o}},{\varsigma }_m\right)∥\to 0.$$

Thus, the sequence $\left\{{\varsigma }_{\mathfrak{o}}\right\}$ is Cauchy by Lemma 2. Since $\mathcal{Q}$ is complete, then ∃${\varsigma }^{*}\in \mathcal{Q}$ such that ${\varsigma }_{\mathfrak{o}}\to {\varsigma }^{*}$ as $\mathfrak{o}\to \infty .$ □

Now, from (2), we have

$$\tau \left({\varsigma }^{*},{\mathfrak{J}}_1{\varsigma }^{*}\right){⪯}_{i_2}\tau \left({\varsigma }^{*},{\mathfrak{J}}_2{\varsigma }_{2\mathfrak{o}+1}\right)+\tau \left({\mathfrak{J}}_2{\varsigma }_{2\mathfrak{o}+1},{\mathfrak{J}}_1{\varsigma }^{*}\right)$$

$$=\tau \left({\varsigma }^{*},{\mathfrak{J}}_2{\varsigma }_{2\mathfrak{o}+1}\right)+\tau \left({\mathfrak{J}}_1{\varsigma }^{*},{\mathfrak{J}}_2{\varsigma }_{2\mathfrak{o}+1}\right)$$

$${⪯}_{i_2}\left(\begin{array}{c}\tau \left({\varsigma }^{*},{\varsigma }_{2\mathfrak{o}+2}\right)+\alpha \left({\varsigma }^{*},{\varsigma }_{2\mathfrak{o}+1},\theta \right)\tau \left({\varsigma }^{*},{\varsigma }_{2\mathfrak{o}+1}\right)\\ +\pi \left({\varsigma }^{*},{\varsigma }_{2\mathfrak{o}+1},\theta \right)\left(\tau \left({\varsigma }^{*},{\mathfrak{J}}_1{\varsigma }^{*}\right)+\tau \left({\varsigma }_{2\mathfrak{o}+1},{\mathfrak{J}}_2{\varsigma }_{2\mathfrak{o}+1}\right)\right)\\ +\kappa \left({\varsigma }^{*},{\varsigma }_{2\mathfrak{o}+1},\theta \right)\frac{\tau \left({\varsigma }^{*},{\mathfrak{J}}_1{\varsigma }^{*}\right)\tau \left({\varsigma }_{2\mathfrak{o}+1},{\mathfrak{J}}_2{\varsigma }_{2\mathfrak{o}+1}\right)}{1+\tau \left({\varsigma }^{*},{\varsigma }_{2\mathfrak{o}+1}\right)}\\ +\varpi \left({\varsigma }^{*},{\varsigma }_{2\mathfrak{o}+1},\theta \right)\frac{\tau \left({\varsigma }_{2\mathfrak{o}+1},{\mathfrak{J}}_1{\varsigma }^{*}\right)\tau \left({\varsigma }^{*},{\mathfrak{J}}_2{\varsigma }_{2\mathfrak{o}+1}\right)}{1+\tau \left({\varsigma }^{*},{\varsigma }_{2\mathfrak{o}+1}\right)}\\ +\varkappa \left({\varsigma }^{*},{\varsigma }_{2\mathfrak{o}+1},\theta \right)\frac{\tau \left({\varsigma }^{*},{\mathfrak{J}}_1{\varsigma }^{*}\right)\tau \left({\varsigma }^{*},{\mathfrak{J}}_2{\varsigma }_{2\mathfrak{o}+1}\right)+\tau \left({\varsigma }_{2\mathfrak{o}+1},{\mathfrak{J}}_2{\varsigma }_{2\mathfrak{o}+1}\right)\tau \left({\varsigma }_{2\mathfrak{o}+1},{\mathfrak{J}}_1{\varsigma }^{*}\right)}{1+\tau \left({\varsigma }^{*},{\mathfrak{J}}_2{\varsigma }_{2\mathfrak{o}+1}\right)+\tau \left({\varsigma }_{2\mathfrak{o}+1},{\mathfrak{J}}_1{\varsigma }^{*}\right)}\end{array}\right)$$

$${⪯}_{i_2}\left(\begin{array}{c}\tau \left({\varsigma }^{*},{\varsigma }_{2\mathfrak{o}+2}\right)+\alpha \left({\varsigma }^{*},{\varsigma }_{2\mathfrak{o}+1},\theta \right)\tau \left({\varsigma }^{*},{\varsigma }_{2\mathfrak{o}+1}\right)\\ +\pi \left({\varsigma }^{*},{\varsigma }_{2\mathfrak{o}+1},\theta \right)\left(\tau \left({\varsigma }^{*},{\mathfrak{J}}_1{\varsigma }^{*}\right)+\tau \left({\varsigma }_{2\mathfrak{o}+1},{\varsigma }_{2\mathfrak{o}+2}\right)\right)\\ +\kappa \left({\varsigma }^{*},{\varsigma }_{2\mathfrak{o}+1},\theta \right)\frac{\tau \left({\varsigma }^{*},{\mathfrak{J}}_1{\varsigma }^{*}\right)\tau \left({\varsigma }_{2\mathfrak{o}+1},{\varsigma }_{2\mathfrak{o}+2}\right)}{1+\tau \left({\varsigma }^{*},{\varsigma }_{2\mathfrak{o}+1}\right)}\\ +\varpi \left({\varsigma }^{*},{\varsigma }_{2\mathfrak{o}+1},\theta \right)\frac{\tau \left({\varsigma }_{2\mathfrak{o}+1},{\mathfrak{J}}_1{\varsigma }^{*}\right)\tau \left({\varsigma }^{*},{\varsigma }_{2\mathfrak{o}+2}\right)}{1+\tau \left({\varsigma }^{*},{\varsigma }_{2\mathfrak{o}+1}\right)}\\ +\varkappa \left({\varsigma }^{*},{\varsigma }_{2\mathfrak{o}+1},\theta \right)\frac{\tau \left({\varsigma }^{*},{\mathfrak{J}}_1{\varsigma }^{*}\right)\tau \left({\varsigma }^{*},{\varsigma }_{2\mathfrak{o}+2}\right)+\tau \left({\varsigma }_{2\mathfrak{o}+1},{\varsigma }_{2\mathfrak{o}+2}\right)\tau \left({\varsigma }_{2\mathfrak{o}+1},{\mathfrak{J}}_1{\varsigma }^{*}\right)}{1+\tau \left({\varsigma }^{*},{\varsigma }_{2\mathfrak{o}+2}\right)+\tau \left({\varsigma }_{2\mathfrak{o}+1},{\mathfrak{J}}_1{\varsigma }^{*}\right)}\end{array}\right).$$

This implies that

$$∥\tau \left({\varsigma }^{*},{\mathfrak{J}}_1{\varsigma }^{*}\right)∥\le \left(\begin{array}{c}∥\tau \left({\varsigma }^{*},{\varsigma }_{2\mathfrak{o}+2}\right)∥+\alpha \left({\varsigma }^{*},{\varsigma }_{2\mathfrak{o}+1},\theta \right)∥\tau \left({\varsigma }^{*},{\varsigma }_{2\mathfrak{o}+1}\right)∥\\ +\pi \left({\varsigma }^{*},{\varsigma }_{2\mathfrak{o}+1},\theta \right)\left(∥\tau \left({\varsigma }^{*},{\mathfrak{J}}_1{\varsigma }^{*}\right)∥+∥\tau \left({\varsigma }_{2\mathfrak{o}+1},{\varsigma }_{2\mathfrak{o}+2}\right)∥\right)\\ +\sqrt{2}\kappa \left({\varsigma }^{*},{\varsigma }_{2\mathfrak{o}+1},\theta \right)\frac{∥\tau \left({\varsigma }^{*},{\mathfrak{J}}_1{\varsigma }^{*}\right)∥∥\tau \left({\varsigma }_{2\mathfrak{o}+1},{\varsigma }_{2\mathfrak{o}+2}\right)∥}{∥1+\tau \left({\varsigma }^{*},{\varsigma }_{2\mathfrak{o}+1}\right)∥}\\ +\sqrt{2}\varpi \left({\varsigma }^{*},{\varsigma }_{2\mathfrak{o}+1},\theta \right)\frac{∥\tau \left({\varsigma }_{2\mathfrak{o}+1},{\mathfrak{J}}_1{\varsigma }^{*}\right)∥∥\tau \left({\varsigma }^{*},{\varsigma }_{2\mathfrak{o}+2}\right)∥}{∥1+\tau \left({\varsigma }^{*},{\varsigma }_{2\mathfrak{o}+1}\right)∥}\\ +\sqrt{2}\varkappa \left({\varsigma }^{*},{\varsigma }_{2\mathfrak{o}+1},\theta \right)\frac{∥\tau \left({\varsigma }^{*},{\mathfrak{J}}_1{\varsigma }^{*}\right)∥∥\tau \left({\varsigma }^{*},{\varsigma }_{2\mathfrak{o}+2}\right)∥+∥\tau \left({\varsigma }_{2\mathfrak{o}+1},{\varsigma }_{2\mathfrak{o}+2}\right)∥∥\tau \left({\varsigma }_{2\mathfrak{o}+1},{\mathfrak{J}}_1{\varsigma }^{*}\right)∥}{∥1+\tau \left({\varsigma }^{*},{\varsigma }_{2\mathfrak{o}+2}\right)+\tau \left({\varsigma }_{2\mathfrak{o}+1},{\mathfrak{J}}_1{\varsigma }^{*}\right)∥}\end{array}\right).$$

Letting $\mathfrak{o}\to \infty ,$ we have

$$\frac{1}{\pi \left({\varsigma }^{*},{\varsigma }_{2\mathfrak{o}+1},\theta \right)}∥\tau \left({\varsigma }^{*},{\mathfrak{J}}_1{\varsigma }^{*}\right)∥\le 0.$$

Since $\frac{1}{\pi \left({\varsigma }^{*},{\varsigma }_{2\mathfrak{o}+1},\theta \right)}\ne 0,$ then $∥\tau \left({\varsigma }^{*},{\mathfrak{J}}_1{\varsigma }^{*}\right)∥=0.$ Thus, ${\varsigma }^{*}={\mathfrak{J}}_1{\varsigma }^{*}.$ Now, we show that ${\varsigma }^{*}$ is a fixed point of ${\mathfrak{J}}_2.$ By (2), we have

$$\tau \left({\varsigma }^{*},{\mathfrak{J}}_2{\varsigma }^{*}\right){⪯}_{i_2}\left(\tau \left({\varsigma }^{*},{\mathfrak{J}}_1{\varsigma }_{2\mathfrak{o}}\right)+\tau \left({\mathfrak{J}}_1{\varsigma }_{2\mathfrak{o}},{\mathfrak{J}}_2{\varsigma }^{*}\right)\right)$$

$${⪯}_{i_2}\left(\begin{array}{c}\tau \left({\varsigma }^{*},{\mathfrak{J}}_1{\varsigma }_{2\mathfrak{o}}\right)+\alpha \left({\varsigma }_{2\mathfrak{o}},{\varsigma }^{*},\theta \right)\tau \left({\varsigma }_{2\mathfrak{o}},{\varsigma }^{*}\right)\\ +\pi \left({\varsigma }_{2\mathfrak{o}},{\varsigma }^{*},\theta \right)\left(\tau \left({\varsigma }_{2\mathfrak{o}},{\mathfrak{J}}_1{\varsigma }_{2\mathfrak{o}}\right)+\tau \left({\varsigma }^{*},{\mathfrak{J}}_2{\varsigma }^{*}\right)\right)\\ +\kappa \left({\varsigma }_{2\mathfrak{o}},{\varsigma }^{*},\theta \right)\frac{\tau \left({\varsigma }_{2\mathfrak{o}},{\mathfrak{J}}_1{\varsigma }_{2\mathfrak{o}}\right)\tau \left({\varsigma }^{*},{\mathfrak{J}}_2{\varsigma }^{*}\right)}{1+\tau \left({\varsigma }_{2\mathfrak{o}},{\varsigma }^{*}\right)}\\ +\varpi \left({\varsigma }_{2\mathfrak{o}},{\varsigma }^{*},\theta \right)\frac{\tau \left({\varsigma }^{*},{\mathfrak{J}}_1{\varsigma }_{2\mathfrak{o}}\right)\tau \left({\varsigma }_{2\mathfrak{o}},{\mathfrak{J}}_2{\varsigma }^{*}\right)}{1+\tau \left({\varsigma }_{2\mathfrak{o}},{\varsigma }^{*}\right)}\\ +\varkappa \left({\varsigma }_{2\mathfrak{o}},{\varsigma }^{*},\theta \right)\frac{\tau \left({\varsigma }_{2\mathfrak{o}},{\mathfrak{J}}_1{\varsigma }_{2\mathfrak{o}}\right)\tau \left({\varsigma }_{2\mathfrak{o}},{\mathfrak{J}}_2{\varsigma }^{*}\right)+\tau \left({\varsigma }^{*},{\mathfrak{J}}_2{\varsigma }^{*}\right)\tau \left({\varsigma }^{*},{\mathfrak{J}}_1{\varsigma }_{2\mathfrak{o}}\right)}{1+\tau \left({\varsigma }_{2\mathfrak{o}},{\mathfrak{J}}_2{\varsigma }^{*}\right)+\tau \left({\varsigma }^{*},{\mathfrak{J}}_1{\varsigma }_{2\mathfrak{o}}\right)}\end{array}\right)$$

$${⪯}_{i_2}\left(\begin{array}{c}\tau \left({\varsigma }^{*},{\varsigma }_{2\mathfrak{o}+1}\right)+\alpha \left({\varsigma }_{2\mathfrak{o}},{\varsigma }^{*},\theta \right)\tau \left({\varsigma }_{2\mathfrak{o}},{\varsigma }^{*}\right)\\ +\kappa \left({\varsigma }_{2\mathfrak{o}},{\varsigma }^{*},\theta \right)\frac{\tau \left({\varsigma }_{2\mathfrak{o}},{\varsigma }_{2\mathfrak{o}+1}\right)\tau \left({\varsigma }^{*},{\mathfrak{J}}_2{\varsigma }^{*}\right)}{1+\tau \left({\varsigma }_{2\mathfrak{o}},{\varsigma }^{*}\right)}\\ +\pi \left({\varsigma }_{2\mathfrak{o}},{\varsigma }^{*},\theta \right)\left(\tau \left({\varsigma }_{2\mathfrak{o}},{\mathfrak{J}}_1{\varsigma }_{2\mathfrak{o}}\right)+\tau \left({\varsigma }^{*},{\mathfrak{J}}_2{\varsigma }^{*}\right)\right)\\ +\varpi \left({\varsigma }_{2\mathfrak{o}},{\varsigma }^{*},\theta \right)\frac{\tau \left({\varsigma }^{*},{\varsigma }_{2\mathfrak{o}+1}\right)\tau \left({\varsigma }_{2\mathfrak{o}},{\mathfrak{J}}_2{\varsigma }^{*}\right)}{1+\tau \left({\varsigma }_{2\mathfrak{o}},{\varsigma }^{*}\right)}\\ +\varkappa \left({\varsigma }_{2\mathfrak{o}},{\varsigma }^{*},\theta \right)\frac{\tau \left({\varsigma }_{2\mathfrak{o}},{\varsigma }_{2\mathfrak{o}+1}\right)\tau \left({\varsigma }_{2\mathfrak{o}},{\mathfrak{J}}_2{\varsigma }^{*}\right)+\tau \left({\varsigma }^{*},{\mathfrak{J}}_2{\varsigma }^{*}\right)\tau \left({\varsigma }^{*},{\varsigma }_{2\mathfrak{o}+1}\right)}{1+\tau \left({\varsigma }_{2\mathfrak{o}},{\mathfrak{J}}_2{\varsigma }^{*}\right)+\tau \left({\varsigma }^{*},{\varsigma }_{2\mathfrak{o}+1}\right)}\end{array}\right).$$

This implies that

$$∥\tau \left({\varsigma }^{*},{\mathfrak{J}}_2{\varsigma }^{*}\right)∥\le \left(\begin{array}{c}∥\tau \left({\varsigma }^{*},{\varsigma }_{2\mathfrak{o}+1}\right)∥+\alpha \left({\varsigma }_{2\mathfrak{o}},{\varsigma }^{*},\theta \right)∥\tau \left({\varsigma }_{2\mathfrak{o}},{\varsigma }^{*}\right)∥\\ +\pi \left({\varsigma }_{2\mathfrak{o}},{\varsigma }^{*},\theta \right)\left(∥\tau \left({\varsigma }_{2\mathfrak{o}},{\mathfrak{J}}_1{\varsigma }_{2\mathfrak{o}}\right)∥+∥\tau \left({\varsigma }^{*},{\mathfrak{J}}_2{\varsigma }^{*}\right)∥\right)\\ +\sqrt{2}\kappa \left({\varsigma }_{2\mathfrak{o}},{\varsigma }^{*},\theta \right)\frac{∥\tau \left({\varsigma }_{2\mathfrak{o}},{\varsigma }_{2\mathfrak{o}+1}\right)∥∥\tau \left({\varsigma }^{*},{\mathfrak{J}}_2{\varsigma }^{*}\right)∥}{∥1+\tau \left({\varsigma }_{2\mathfrak{o}},{\varsigma }^{*}\right)∥}\\ +\sqrt{2}\varpi \left({\varsigma }_{2\mathfrak{o}},{\varsigma }^{*},\theta \right)\frac{∥\tau \left({\varsigma }^{*},{\varsigma }_{2\mathfrak{o}+1}\right)∥∥\tau \left({\varsigma }_{2\mathfrak{o}},{\mathfrak{J}}_2{\varsigma }^{*}\right)∥}{∥1+\tau \left({\varsigma }_{2\mathfrak{o}},{\varsigma }^{*}\right)∥}\\ +\sqrt{2}\varkappa \left({\varsigma }_{2\mathfrak{o}},{\varsigma }^{*},\theta \right)\frac{∥\tau \left({\varsigma }_{2\mathfrak{o}},{\varsigma }_{2\mathfrak{o}+1}\right)∥∥\tau \left({\varsigma }_{2\mathfrak{o}},{\mathfrak{J}}_2{\varsigma }^{*}\right)∥+∥\tau \left({\varsigma }^{*},{\mathfrak{J}}_2{\varsigma }^{*}\right)∥∥\tau \left({\varsigma }^{*},{\varsigma }_{2\mathfrak{o}+1}\right)∥}{∥1+\tau \left({\varsigma }_{2\mathfrak{o}},{\mathfrak{J}}_2{\varsigma }^{*}\right)+\tau \left({\varsigma }^{*},{\varsigma }_{2\mathfrak{o}+1}\right)∥}\end{array}\right).$$

Letting $\mathfrak{o}\to \infty ,$ we have $\frac{1}{\pi \left({\varsigma }_{2\mathfrak{o}},{\varsigma }^{*},\theta \right)}∥\tau \left({\varsigma }^{*},{\mathfrak{J}}_2{\varsigma }^{*}\right)∥\le 0$ since $\frac{1}{\pi \left({\varsigma }_{2\mathfrak{o}},{\varsigma }^{*},\theta \right)}\ne 0.$ Hence, ${\varsigma }^{*}={\mathfrak{J}}_2{\varsigma }^{*}.$ Thus, ${\varsigma }^{*}$ is a common fixed point of ${\mathfrak{J}}_1$ and ${\mathfrak{J}}_2.$ We assume that there exists another common fixed point of ${\mathfrak{J}}_1$ and ${\mathfrak{J}}_2,$ that is,

$${\varsigma }^{/}={\mathfrak{J}}_1{\varsigma }^{/}={\mathfrak{J}}_2{\varsigma }^{/}$$

but ${\varsigma }^{*}\ne {\varsigma }^{/}.$ Now, from (2), we have

$$\tau \left({\varsigma }^{*},{\varsigma }^{/}\right)=\tau \left({\mathfrak{J}}_1{\varsigma }^{*},{\mathfrak{J}}_2{\varsigma }^{/}\right){⪯}_{i_2}\alpha \left({\varsigma }^{*},{\varsigma }^{/},\theta \right)\tau \left({\varsigma }^{*},{\varsigma }^{/}\right)$$

$$+\pi \left({\varsigma }^{*},{\varsigma }^{/},\theta \right)\left(\tau \left({\varsigma }^{*},{\mathfrak{J}}_1{\varsigma }^{*}\right)+\tau \left({\varsigma }^{/},{\mathfrak{J}}_2{\varsigma }^{/}\right)\right)$$

$$+\kappa \left({\varsigma }^{*},{\varsigma }^{/},\theta \right)\frac{\tau \left({\varsigma }^{*},\mathfrak{J}{\varsigma }^{*}\right)\tau \left({\varsigma }^{/},{\mathfrak{J}}_2{\varsigma }^{/}\right)}{1+\tau \left({\varsigma }^{*},{\varsigma }^{/}\right)}$$

$$+\varpi \left({\varsigma }^{*},{\varsigma }^{/},\theta \right)\frac{\tau \left({\varsigma }^{/},{\mathfrak{J}}_1{\varsigma }^{*}\right)\tau \left({\varsigma }^{*},{\mathfrak{J}}_2{\varsigma }^{/}\right)}{1+\tau \left({\varsigma }^{*},{\varsigma }^{/}\right)}$$

$$+\varkappa \left({\varsigma }^{*},{\varsigma }^{/},\theta \right)\frac{\tau \left({\varsigma }^{*},{\mathfrak{J}}_1{\varsigma }^{*}\right)\tau \left({\varsigma }^{*},{\mathfrak{J}}_2{\varsigma }^{/}\right)+\tau \left({\varsigma }^{/},{\mathfrak{J}}_2{\varsigma }^{/}\right)\tau \left({\varsigma }^{/},{\mathfrak{J}}_1{\varsigma }^{*}\right)}{1+\tau \left({\varsigma }^{*},{\mathfrak{J}}_2{\varsigma }^{/}\right)+\tau \left({\varsigma }^{/},{\mathfrak{J}}_1{\varsigma }^{*}\right)}$$

which implies that

$$\tau \left({\varsigma }^{*},{\varsigma }^{/}\right){⪯}_{i_2}\alpha \left({\varsigma }^{*},{\varsigma }^{/},\theta \right)\tau \left({\varsigma }^{*},{\varsigma }^{/}\right)$$

$$+\varpi \left({\varsigma }^{*},{\varsigma }^{/},\theta \right)\frac{\tau \left({\varsigma }^{/},{\varsigma }^{*}\right)\tau \left({\varsigma }^{*},{\varsigma }^{/}\right)}{1+\tau \left({\varsigma }^{*},{\varsigma }^{/}\right)}.$$

This yields that

$$\begin{array}{ccc}\hfill ∥\tau \left({\varsigma }^{*},{\varsigma }^{/}\right)∥& \le & \alpha \left({\varsigma }^{*},{\varsigma }^{/},\theta \right)∥\tau \left({\varsigma }^{*},{\varsigma }^{/}\right)∥\hfill \\ & & +\sqrt{2}\varpi \left({\varsigma }^{*},{\varsigma }^{/},\theta \right)∥\tau \left({\varsigma }^{*},{\varsigma }^{/}\right)∥∥\frac{\tau \left({\varsigma }^{*},{\varsigma }^{/}\right)}{1+\tau \left({\varsigma }^{*},{\varsigma }^{/}\right)}∥\hfill \\ & \le & \alpha \left({\varsigma }^{*},{\varsigma }^{/},\theta \right)∥\tau \left({\varsigma }^{*},{\varsigma }^{/}\right)∥+\sqrt{2}\varpi \left({\varsigma }^{*},{\varsigma }^{/},\theta \right)∥\tau \left({\varsigma }^{*},{\varsigma }^{/}\right)∥\hfill \\ & =& \left(\alpha \left({\varsigma }^{*},{\varsigma }^{/},\theta \right)+\sqrt{2}\varpi \left({\varsigma }^{*},{\varsigma }^{/},\theta \right)\right)∥\tau \left({\varsigma }^{*},{\varsigma }^{/}\right)∥,\hfill \end{array}$$

that is,

$$\frac{1}{\alpha \left({\varsigma }^{*},{\varsigma }^{/},\theta \right)+\sqrt{2}\varpi \left({\varsigma }^{*},{\varsigma }^{/},\theta \right)}∥\tau \left({\varsigma }^{*},{\varsigma }^{/}\right)∥\le 0$$

As $\frac{1}{\alpha \left({\varsigma }^{*},{\varsigma }^{/},\theta \right)+\sqrt{2}\varpi \left({\varsigma }^{*},{\varsigma }^{/},\theta \right)}\ne 0,$ we have

$$∥\tau \left({\varsigma }^{*},{\varsigma }^{/}\right)∥=0.$$

Thus, ${\varsigma }^{*}={\varsigma }^{/}.$

Note:

From now onwards, we consider $\left(\mathcal{Q},\tau \right)$ as a complete bi-CVMS.

Corollary 1.

Let ${\mathfrak{J}}_1,{\mathfrak{J}}_2:\left(\mathcal{Q},\tau \right)$$\to \left(\mathcal{Q},\tau \right)$ be self-mappings. If the functions $\alpha ,\pi ,\kappa ,\varpi :{\mathcal{Q}}^3\to [0,1)$ satisfy the conditions

(a)

$\alpha \left({\mathfrak{J}}_2{\mathfrak{J}}_1\varsigma ,\nu ,\theta \right)\le \alpha \left(\varsigma ,\nu ,\theta \right)$ and $\alpha \left(\varsigma ,{\mathfrak{J}}_1{\mathfrak{J}}_2\nu ,\theta \right)\le \alpha \left(\varsigma ,\nu ,\theta \right)$

$\pi \left({\mathfrak{J}}_2{\mathfrak{J}}_1\varsigma ,\nu ,\theta \right)\le \pi \left(\varsigma ,\nu ,\theta \right)$ and $\pi \left(\varsigma ,{\mathfrak{J}}_1{\mathfrak{J}}_2\nu ,\theta \right)\le \pi \left(\varsigma ,\nu ,\theta \right)$

$\kappa \left({\mathfrak{J}}_2{\mathfrak{J}}_1\varsigma ,\nu ,\theta \right)\le \kappa \left(\varsigma ,\nu ,\theta \right)$ and $\kappa \left(\varsigma ,{\mathfrak{J}}_1{\mathfrak{J}}_2\nu ,\theta \right)\le \kappa \left(\varsigma ,\nu ,\theta \right)$

$\varpi \left({\mathfrak{J}}_2{\mathfrak{J}}_1\varsigma ,\nu ,\theta \right)\le \varpi \left(\varsigma ,\nu ,\theta \right)$ and $\varpi \left(\varsigma ,{\mathfrak{J}}_1{\mathfrak{J}}_2\nu ,\theta \right)\le \varpi \left(\varsigma ,\nu ,\theta \right);$

(b) $\alpha \left(\varsigma ,\nu ,\theta \right)+2\pi \left(\varsigma ,\nu ,\theta \right)+\sqrt{2}\kappa \left(\varsigma ,\nu ,\theta \right)+\sqrt{2}\varpi \left(\varsigma ,\nu ,\theta \right)<1;$

(c)

$$\tau \left({\mathfrak{J}}_1\varsigma ,{\mathfrak{J}}_2\nu \right){⪯}_{i_2}\alpha \left(\varsigma ,\nu ,\theta \right)\tau \left(\varsigma ,\nu \right)+\pi \left(\varsigma ,\nu ,\theta \right)\left(\tau \left(\varsigma ,{\mathfrak{J}}_1\varsigma \right)+\tau \left(\nu ,{\mathfrak{J}}_2\nu \right)\right)$$

$$+\kappa \left(\varsigma ,\nu ,\theta \right)\frac{\tau \left(\varsigma ,{\mathfrak{J}}_1\varsigma \right)\tau \left(\nu ,{\mathfrak{J}}_2\nu \right)}{1+\tau \left(\varsigma ,\nu \right)}$$

$$+\varpi \left(\varsigma ,\nu ,\theta \right)\frac{\tau \left(\nu ,{\mathfrak{J}}_1\varsigma \right)\tau \left(\varsigma ,{\mathfrak{J}}_2\nu \right)}{1+\tau \left(\varsigma ,\nu \right)}$$

for all $\varsigma ,\nu \in$$\mathcal{Q}$and for fixed element $\theta \in \mathcal{Q},$ then there exists a unique element ${\varsigma }^{*}\in \mathcal{Q}$ such that ${\mathfrak{J}}_1{\varsigma }^{*}={\mathfrak{J}}_2{\varsigma }^{*}={\varsigma }^{*}.$

Proof. 

Take $\varkappa :\mathcal{Q}\times \mathcal{Q}\times \mathcal{Q}\to [0,1)$ by $\varpi \left(\varsigma ,\nu ,\theta \right)=0$ in Theorem 1. □

Corollary 2.

Let ${\mathfrak{J}}_1,{\mathfrak{J}}_2:\left(\mathcal{Q},\tau \right)$$\to \left(\mathcal{Q},\tau \right)$ be self-mappings. If the functions $\alpha ,\pi ,\kappa ,\varkappa :{\mathcal{Q}}^3\to [0,1)$ satisfy the conditions

(a)

$\alpha \left({\mathfrak{J}}_2{\mathfrak{J}}_1\varsigma ,\nu ,\theta \right)\le \alpha \left(\varsigma ,\nu ,\theta \right)$ and $\alpha \left(\varsigma ,{\mathfrak{J}}_1{\mathfrak{J}}_2\nu ,\theta \right)\le \alpha \left(\varsigma ,\nu ,\theta \right)$

$\pi \left({\mathfrak{J}}_2{\mathfrak{J}}_1\varsigma ,\nu ,\theta \right)\le \pi \left(\varsigma ,\nu ,\theta \right)$ and $\pi \left(\varsigma ,{\mathfrak{J}}_1{\mathfrak{J}}_2\nu ,\theta \right)\le \pi \left(\varsigma ,\nu ,\theta \right)$

$\kappa \left({\mathfrak{J}}_2{\mathfrak{J}}_1\varsigma ,\nu ,\theta \right)\le \kappa \left(\varsigma ,\nu ,\theta \right)$ and $\kappa \left(\varsigma ,{\mathfrak{J}}_1{\mathfrak{J}}_2\nu ,\theta \right)\le \kappa \left(\varsigma ,\nu ,\theta \right)$

$\varkappa \left({\mathfrak{J}}_2{\mathfrak{J}}_1\varsigma ,\nu ,\theta \right)\le \varkappa \left(\varsigma ,\nu ,\theta \right)$ and $\varkappa \left(\varsigma ,{\mathfrak{J}}_1{\mathfrak{J}}_2\nu ,\theta \right)\le \varkappa \left(\varsigma ,\nu ,\theta \right);$

(b) $\alpha \left(\varsigma ,\nu ,\theta \right)+2\pi \left(\varsigma ,\nu ,\theta \right)+\sqrt{2}\kappa \left(\varsigma ,\nu ,\theta \right)+\sqrt{2}\varkappa \left(\varsigma ,\nu ,\theta \right)<1;$

(c)

$$\tau \left({\mathfrak{J}}_1\varsigma ,{\mathfrak{J}}_2\nu \right){⪯}_{i_2}\alpha \left(\varsigma ,\nu ,\theta \right)\tau \left(\varsigma ,\nu \right)+\pi \left(\varsigma ,\nu ,\theta \right)\left(\tau \left(\varsigma ,{\mathfrak{J}}_1\varsigma \right)+\tau \left(\nu ,{\mathfrak{J}}_2\nu \right)\right)$$

$$+\kappa \left(\varsigma ,\nu ,\theta \right)\frac{\tau \left(\varsigma ,{\mathfrak{J}}_1\varsigma \right)\tau \left(\nu ,{\mathfrak{J}}_2\nu \right)}{1+\tau \left(\varsigma ,\nu \right)}$$

$$+\varkappa \left(\varsigma ,\nu ,\theta \right)\frac{\tau \left(\varsigma ,{\mathfrak{J}}_1\varsigma \right)\tau \left(\varsigma ,{\mathfrak{J}}_2\nu \right)+\tau \left(\nu ,{\mathfrak{J}}_2\nu \right)\tau \left(\nu ,{\mathfrak{J}}_1\varsigma \right)}{1+\tau \left(\varsigma ,{\mathfrak{J}}_2\nu \right)+\tau \left(\nu ,{\mathfrak{J}}_1\varsigma \right)},$$

for all $\varsigma ,\nu \in$$\mathcal{Q}$and for fixed element $\theta \in \mathcal{Q},$ then there exists a unique element ${\varsigma }^{*}\in \mathcal{Q}$ such that ${\mathfrak{J}}_1{\varsigma }^{*}={\mathfrak{J}}_2{\varsigma }^{*}={\varsigma }^{*}.$

Proof. 

Take $\varpi :\mathcal{Q}\times \mathcal{Q}\times \mathcal{Q}\to [0,1)$ by $\varpi \left(\varsigma ,\nu ,\theta \right)=0$ in Theorem 1. □

Corollary 3.

Let ${\mathfrak{J}}_1,{\mathfrak{J}}_2:\left(\mathcal{Q},\tau \right)$$\to \left(\mathcal{Q},\tau \right)$ be self-mappings. If the functions $\alpha ,\pi ,\varpi ,\varkappa :{\mathcal{Q}}^3\to [0,1)$ satisfy the conditions

(a)

$\alpha \left({\mathfrak{J}}_2{\mathfrak{J}}_1\varsigma ,\nu ,\theta \right)\le \alpha \left(\varsigma ,\nu ,\theta \right)$ and $\alpha \left(\varsigma ,{\mathfrak{J}}_1{\mathfrak{J}}_2\nu ,\theta \right)\le \alpha \left(\varsigma ,\nu ,\theta \right)$

$\pi \left({\mathfrak{J}}_2{\mathfrak{J}}_1\varsigma ,\nu ,\theta \right)\le \pi \left(\varsigma ,\nu ,\theta \right)$ and $\pi \left(\varsigma ,{\mathfrak{J}}_1{\mathfrak{J}}_2\nu ,\theta \right)\le \pi \left(\varsigma ,\nu ,\theta \right)$

$\varpi \left({\mathfrak{J}}_2{\mathfrak{J}}_1\varsigma ,\nu ,\theta \right)\le \varpi \left(\varsigma ,\nu ,\theta \right)$ and $\varpi \left(\varsigma ,{\mathfrak{J}}_1{\mathfrak{J}}_2\nu ,\theta \right)\le \varpi \left(\varsigma ,\nu ,\theta \right)$

$\varkappa \left({\mathfrak{J}}_2{\mathfrak{J}}_1\varsigma ,\nu ,\theta \right)\le \varkappa \left(\varsigma ,\nu ,\theta \right)$ and $\varkappa \left(\varsigma ,{\mathfrak{J}}_1{\mathfrak{J}}_2\nu ,\theta \right)\le \varkappa \left(\varsigma ,\nu ,\theta \right);$

(b) $\alpha \left(\varsigma ,\nu ,\theta \right)+2\pi \left(\varsigma ,\nu ,\theta \right)+\sqrt{2}\kappa \left(\varsigma ,\nu ,\theta \right)+\sqrt{2}\varpi \left(\varsigma ,\nu ,\theta \right)+\sqrt{2}\varkappa \left(\varsigma ,\nu ,\theta \right)<1;$

(c)

$$\tau \left({\mathfrak{J}}_1\varsigma ,{\mathfrak{J}}_2\nu \right){⪯}_{i_2}\alpha \left(\varsigma ,\nu ,\theta \right)\tau \left(\varsigma ,\nu \right)+\pi \left(\varsigma ,\nu ,\theta \right)\left(\tau \left(\varsigma ,{\mathfrak{J}}_1\varsigma \right)+\tau \left(\nu ,{\mathfrak{J}}_2\nu \right)\right)$$

$$+\varpi \left(\varsigma ,\nu ,\theta \right)\frac{\tau \left(\nu ,{\mathfrak{J}}_1\varsigma \right)\tau \left(\varsigma ,{\mathfrak{J}}_2\nu \right)}{1+\tau \left(\varsigma ,\nu \right)}$$

$$+\varkappa \left(\varsigma ,\nu ,\theta \right)\frac{\tau \left(\varsigma ,{\mathfrak{J}}_1\varsigma \right)\tau \left(\varsigma ,{\mathfrak{J}}_2\nu \right)+\tau \left(\nu ,{\mathfrak{J}}_2\nu \right)\tau \left(\nu ,{\mathfrak{J}}_1\varsigma \right)}{1+\tau \left(\varsigma ,{\mathfrak{J}}_2\nu \right)+\tau \left(\nu ,{\mathfrak{J}}_1\varsigma \right)},$$

for all $\varsigma ,\nu \in$$\mathcal{Q}$and for fixed element $\theta \in \mathcal{Q},$ then there exists a unique element ${\varsigma }^{*}\in \mathcal{Q}$ such that ${\mathfrak{J}}_1{\varsigma }^{*}={\mathfrak{J}}_2{\varsigma }^{*}={\varsigma }^{*}.$

Proof. 

Take $\kappa :\mathcal{Q}\times \mathcal{Q}\times \mathcal{Q}\to [0,1)$ by $\kappa \left(\varsigma ,\nu ,\theta \right)=0$ in Theorem 1. □

Corollary 4.

Let ${\mathfrak{J}}_1,{\mathfrak{J}}_2:\left(\mathcal{Q},\tau \right)$$\to \left(\mathcal{Q},\tau \right)$ be self-mappings. If the functions $\alpha ,\kappa ,\varpi ,\varkappa :{\mathcal{Q}}^3\to [0,1)$ satisfy the conditions

(a)

$\alpha \left({\mathfrak{J}}_2{\mathfrak{J}}_1\varsigma ,\nu ,\theta \right)\le \alpha \left(\varsigma ,\nu ,\theta \right)$ and $\alpha \left(\varsigma ,{\mathfrak{J}}_1{\mathfrak{J}}_2\nu ,\theta \right)\le \alpha \left(\varsigma ,\nu ,\theta \right)$

$\kappa \left({\mathfrak{J}}_2{\mathfrak{J}}_1\varsigma ,\nu ,\theta \right)\le \kappa \left(\varsigma ,\nu ,\theta \right)$ and $\kappa \left(\varsigma ,{\mathfrak{J}}_1{\mathfrak{J}}_2\nu ,\theta \right)\le \kappa \left(\varsigma ,\nu ,\theta \right)$

$\varpi \left({\mathfrak{J}}_2{\mathfrak{J}}_1\varsigma ,\nu ,\theta \right)\le \varpi \left(\varsigma ,\nu ,\theta \right)$ and $\varpi \left(\varsigma ,{\mathfrak{J}}_1{\mathfrak{J}}_2\nu ,\theta \right)\le \varpi \left(\varsigma ,\nu ,\theta \right)$

$\varkappa \left({\mathfrak{J}}_2{\mathfrak{J}}_1\varsigma ,\nu ,\theta \right)\le \varkappa \left(\varsigma ,\nu ,\theta \right)$ and $\varkappa \left(\varsigma ,{\mathfrak{J}}_1{\mathfrak{J}}_2\nu ,\theta \right)\le \varkappa \left(\varsigma ,\nu ,\theta \right);$

(b) $\alpha \left(\varsigma ,\nu ,\theta \right)+\sqrt{2}\kappa \left(\varsigma ,\nu ,\theta \right)+\sqrt{2}\varpi \left(\varsigma ,\nu ,\theta \right)+\sqrt{2}\varkappa \left(\varsigma ,\nu ,\theta \right)<1;$

(c)

$$\tau \left({\mathfrak{J}}_1\varsigma ,{\mathfrak{J}}_2\nu \right){⪯}_{i_2}\alpha \left(\varsigma ,\nu ,\theta \right)\tau \left(\varsigma ,\nu \right)$$

$$+\kappa \left(\varsigma ,\nu ,\theta \right)\frac{\tau \left(\varsigma ,{\mathfrak{J}}_1\varsigma \right)\tau \left(\nu ,{\mathfrak{J}}_2\nu \right)}{1+\tau \left(\varsigma ,\nu \right)}$$

$$+\varpi \left(\varsigma ,\nu ,\theta \right)\frac{\tau \left(\nu ,{\mathfrak{J}}_1\varsigma \right)\tau \left(\varsigma ,{\mathfrak{J}}_2\nu \right)}{1+\tau \left(\varsigma ,\nu \right)}$$

$$+\varkappa \left(\varsigma ,\nu ,\theta \right)\frac{\tau \left(\varsigma ,{\mathfrak{J}}_1\varsigma \right)\tau \left(\varsigma ,{\mathfrak{J}}_2\nu \right)+\tau \left(\nu ,{\mathfrak{J}}_2\nu \right)\tau \left(\nu ,{\mathfrak{J}}_1\varsigma \right)}{1+\tau \left(\varsigma ,{\mathfrak{J}}_2\nu \right)+\tau \left(\nu ,{\mathfrak{J}}_1\varsigma \right)},$$

for all $\varsigma ,\nu \in$$\mathcal{Q}$and for fixed element $\theta \in \mathcal{Q},$ then there exists a unique element ${\varsigma }^{*}\in \mathcal{Q}$ such that ${\mathfrak{J}}_1{\varsigma }^{*}={\mathfrak{J}}_2{\varsigma }^{*}={\varsigma }^{*}.$

Proof. 

Take $\pi :\mathcal{Q}\times \mathcal{Q}\times \mathcal{Q}\to [0,1)$ by $\pi \left(\varsigma ,\nu ,\theta \right)=0$ in Theorem 1. □

Corollary 5.

Let ${\mathfrak{J}}_1,{\mathfrak{J}}_2:\left(\mathcal{Q},\tau \right)$$\to \left(\mathcal{Q},\tau \right)$ be self-mappings. If the functions $\pi ,\kappa ,\varpi ,\varkappa :{\mathcal{Q}}^3\to [0,1)$ satisfy the conditions

(a) $\pi \left({\mathfrak{J}}_2{\mathfrak{J}}_1\varsigma ,\nu ,\theta \right)\le \pi \left(\varsigma ,\nu ,\theta \right)$ and $\pi \left(\varsigma ,{\mathfrak{J}}_1{\mathfrak{J}}_2\nu ,\theta \right)\le \pi \left(\varsigma ,\nu ,\theta \right)$

$\kappa \left({\mathfrak{J}}_2{\mathfrak{J}}_1\varsigma ,\nu ,\theta \right)\le \kappa \left(\varsigma ,\nu ,\theta \right)$ and $\kappa \left(\varsigma ,{\mathfrak{J}}_1{\mathfrak{J}}_2\nu ,\theta \right)\le \kappa \left(\varsigma ,\nu ,\theta \right)$

$\varpi \left({\mathfrak{J}}_2{\mathfrak{J}}_1\varsigma ,\nu ,\theta \right)\le \varpi \left(\varsigma ,\nu ,\theta \right)$ and $\varpi \left(\varsigma ,{\mathfrak{J}}_1{\mathfrak{J}}_2\nu ,\theta \right)\le \varpi \left(\varsigma ,\nu ,\theta \right)$

$\varkappa \left({\mathfrak{J}}_2{\mathfrak{J}}_1\varsigma ,\nu ,\theta \right)\le \varkappa \left(\varsigma ,\nu ,\theta \right)$ and $\varkappa \left(\varsigma ,{\mathfrak{J}}_1{\mathfrak{J}}_2\nu ,\theta \right)\le \varkappa \left(\varsigma ,\nu ,\theta \right);$

(b) $2\pi \left(\varsigma ,\nu ,\theta \right)+\sqrt{2}\kappa \left(\varsigma ,\nu ,\theta \right)+\sqrt{2}\varpi \left(\varsigma ,\nu ,\theta \right)+\sqrt{2}\varkappa \left(\varsigma ,\nu ,\theta \right)<1;$

(c)

$$\tau \left({\mathfrak{J}}_1\varsigma ,{\mathfrak{J}}_2\nu \right){⪯}_{i_2}\pi \left(\varsigma ,\nu ,\theta \right)\left(\tau \left(\varsigma ,{\mathfrak{J}}_1\varsigma \right)+\tau \left(\nu ,{\mathfrak{J}}_2\nu \right)\right)$$

$$+\kappa \left(\varsigma ,\nu ,\theta \right)\frac{\tau \left(\varsigma ,{\mathfrak{J}}_1\varsigma \right)\tau \left(\nu ,{\mathfrak{J}}_2\nu \right)}{1+\tau \left(\varsigma ,\nu \right)}$$

$$+\varpi \left(\varsigma ,\nu ,\theta \right)\frac{\tau \left(\nu ,{\mathfrak{J}}_1\varsigma \right)\tau \left(\varsigma ,{\mathfrak{J}}_2\nu \right)}{1+\tau \left(\varsigma ,\nu \right)}$$

$$+\varkappa \left(\varsigma ,\nu ,\theta \right)\frac{\tau \left(\varsigma ,{\mathfrak{J}}_1\varsigma \right)\tau \left(\varsigma ,{\mathfrak{J}}_2\nu \right)+\tau \left(\nu ,{\mathfrak{J}}_2\nu \right)\tau \left(\nu ,{\mathfrak{J}}_1\varsigma \right)}{1+\tau \left(\varsigma ,{\mathfrak{J}}_2\nu \right)+\tau \left(\nu ,{\mathfrak{J}}_1\varsigma \right)},$$

for all $\varsigma ,\nu \in$$\mathcal{Q}$ and for fixed element $\theta \in \mathcal{Q}$, then there exists a unique element ${\varsigma }^{*}\in \mathcal{Q}$ such that ${\mathfrak{J}}_1{\varsigma }^{*}={\mathfrak{J}}_2{\varsigma }^{*}={\varsigma }^{*}.$

Proof. 

Take $\alpha :\mathcal{Q}\times \mathcal{Q}\times \mathcal{Q}\to [0,1)$ by $\alpha \left(\varsigma ,\nu ,\theta \right)=0$ in Theorem 1. □

Example 2.

Let $\mathcal{Q}=[0,1]$ and $\tau :\mathcal{Q}\times \mathcal{Q}\to {\mathbb{C}}_2$ be defined by

$$\tau (\varsigma ,\nu )=(1+i_2)\left|\varsigma -\nu \right|$$

for all $\varsigma ,\nu \in$$\mathcal{Q}.$ Then, $(\mathcal{Q},\tau )$ is a complete bi-CVMS. Define ${\mathfrak{J}}_1,{\mathfrak{J}}_2:\mathcal{Q}$$\to \mathcal{Q}$ by

$${\mathfrak{J}}_1\varsigma =\frac{\varsigma }{4}\mathrm{and}{\mathfrak{J}}_2\varsigma =\frac{\varsigma }{3}.$$

Consider

$$\alpha ,\pi ,\kappa ,\varpi ,\varkappa :\mathcal{Q}\times \mathcal{Q}\times \mathcal{Q}\to [0,1)$$

by

$$\alpha (\varsigma ,\nu ,\theta )=\frac{\varsigma }{4}+\frac{\nu }{5}+\theta ,$$

$$\pi (\varsigma ,\nu ,\theta )=\frac{{\varsigma }^3{\theta }^3}{64}+\frac{{\nu }^3}{125},$$

$$\kappa (\varsigma ,\nu ,\theta )=\frac{{\varsigma }^2{\nu }^2{\theta }^2}{45},$$

$$\varpi (\varsigma ,\nu ,\theta )=\frac{{\varsigma }^2{\theta }^2}{9}+\frac{{\nu }^2{\theta }^2}{16},$$

$$\varkappa (\varsigma ,\nu ,\theta )=\frac{\varsigma }{25}+\frac{\nu }{36}+\frac{\theta }{2}$$

for all $\varsigma ,\nu \in$$\mathcal{Q}$and for fixed element $\theta \in \mathcal{Q}$. Then, evidently,

$$\alpha \left(\varsigma ,\nu ,\theta \right)+2\pi (\varsigma ,\nu ,\theta )+\sqrt{2}\kappa \left(\varsigma ,\nu ,\theta \right)+\sqrt{2}\varpi \left(\varsigma ,\nu ,\theta \right)+\sqrt{2}\varkappa (\varsigma ,\nu ,\theta )<1.$$

Now,

$$\alpha \left({\mathfrak{J}}_2{\mathfrak{J}}_1\varsigma ,\nu ,\theta \right)=\alpha \left({\mathfrak{J}}_2\left(\frac{\varsigma }{4}\right),\nu ,\theta \right)=\alpha \left(\frac{\varsigma }{12},\nu ,\theta \right)=\frac{\varsigma }{48}+\frac{\nu }{5}+\theta \le \frac{\varsigma }{4}+\frac{\nu }{5}+\theta =\alpha \left(\varsigma ,\nu ,\theta \right)$$

$$\alpha \left(\varsigma ,{\mathfrak{J}}_1{\mathfrak{J}}_2\nu ,\theta \right)=\alpha \left(\varsigma ,{\mathfrak{J}}_1\left(\frac{\nu }{3}\right),\theta \right)=\alpha \left(\varsigma ,\frac{\nu }{12},\theta \right)=\frac{\varsigma }{4}+\frac{\nu }{60}+\theta \le \frac{\varsigma }{4}+\frac{\nu }{5}+\theta =\alpha \left(\varsigma ,\nu ,\theta \right)$$

$$\pi \left({\mathfrak{J}}_2{\mathfrak{J}}_1\varsigma ,\nu ,\theta \right)=\pi \left({\mathfrak{J}}_2\left(\frac{\varsigma }{4}\right),\nu ,\theta \right)=\pi \left(\frac{\varsigma }{12},\nu ,\theta \right)=\frac{{\varsigma }^3{\theta }^3}{110592}+\frac{{\nu }^3}{125}\le \frac{{\varsigma }^3{\theta }^3}{64}+\frac{{\nu }^3}{125}=\pi \left(\varsigma ,\nu ,\theta \right)$$

$$\pi \left(\varsigma ,{\mathfrak{J}}_1{\mathfrak{J}}_2\nu ,\theta \right)=\pi \left(\varsigma ,{\mathfrak{J}}_1\left(\frac{\nu }{3}\right),\theta \right)=\pi \left(\varsigma ,\frac{\nu }{12},\theta \right)=\frac{{\varsigma }^3{\theta }^3}{64}+\frac{{\nu }^3}{216000}\le \frac{{\varsigma }^3{\theta }^3}{64}+\frac{{\nu }^3}{125}=\pi \left(\varsigma ,\nu ,\theta \right)$$

$$\kappa \left({\mathfrak{J}}_2{\mathfrak{J}}_1\varsigma ,\nu ,\theta \right)=\kappa \left({\mathfrak{J}}_2\left(\frac{\varsigma }{4}\right),\nu ,\theta \right)=\kappa \left(\frac{\varsigma }{12},\nu ,\theta \right)=\frac{{\varsigma }^2{\nu }^2{\theta }^2}{54540}\le \frac{{\varsigma }^2{\nu }^2{\theta }^2}{45}=\kappa \left(\varsigma ,\nu ,\theta \right)$$

$$\kappa \left(\varsigma ,{\mathfrak{J}}_1{\mathfrak{J}}_2\nu ,\theta \right)=\kappa \left(\varsigma ,{\mathfrak{J}}_1\left(\frac{\nu }{3}\right),\theta \right)=\kappa \left(\varsigma ,\frac{\nu }{12},\theta \right)=\frac{{\varsigma }^2{\nu }^2{\theta }^2}{54540}\le \frac{{\varsigma }^2{\nu }^2{\theta }^2}{45}=\kappa \left(\varsigma ,\nu ,\theta \right)$$

$$\varpi \left({\mathfrak{J}}_2{\mathfrak{J}}_1\varsigma ,\nu ,\theta \right)=\varpi \left({\mathfrak{J}}_2\left(\frac{\varsigma }{4}\right),\nu ,\theta \right)=\varpi \left(\frac{\varsigma }{12},\nu ,\theta \right)=\frac{{\varsigma }^2{\theta }^2}{1296}+\frac{{\nu }^2{\theta }^2}{16}\le \frac{{\varsigma }^2{\theta }^2}{9}+\frac{{\nu }^2{\theta }^2}{16}=\varpi \left(\varsigma ,\nu ,\theta \right)$$

$$\varpi \left(\varsigma ,{\mathfrak{J}}_1{\mathfrak{J}}_2\nu ,\theta \right)=\varpi \left(\varsigma ,{\mathfrak{J}}_1\left(\frac{\nu }{3}\right),\theta \right)=\varpi \left(\varsigma ,\frac{\nu }{12},\theta \right)=\frac{{\varsigma }^2{\theta }^2}{9}+\frac{{\nu }^2{\theta }^2}{2304}\le \frac{{\varsigma }^2{\theta }^2}{9}+\frac{{\nu }^2{\theta }^2}{16}=\varpi \left(\varsigma ,\nu ,\theta \right).$$

Furthermore,

$$\varkappa \left({\mathfrak{J}}_2{\mathfrak{J}}_1\varsigma ,\nu ,\theta \right)=\varkappa \left({\mathfrak{J}}_2\left(\frac{\varsigma }{4}\right),\nu ,\theta \right)=\varkappa \left(\frac{\varsigma }{12},\nu ,\theta \right)=\frac{\varsigma }{300}+\frac{\nu }{36}+\frac{\theta }{2}\le \frac{\varsigma }{25}+\frac{\nu }{36}+\frac{\theta }{2}=\varkappa \left(\varsigma ,\nu ,\theta \right)$$

$$\varkappa \left(\varsigma ,{\mathfrak{J}}_1{\mathfrak{J}}_2\nu ,\theta \right)=\varkappa \left(\varsigma ,{\mathfrak{J}}_1\left(\frac{\nu }{3}\right),\theta \right)=\varkappa \left(\varsigma ,\frac{\nu }{12},\theta \right)=\frac{\varsigma }{25}+\frac{\nu }{432}+\frac{\theta }{2}\le \frac{\varsigma }{25}+\frac{\nu }{36}+\frac{\theta }{2}=\varkappa \left(\varsigma ,\nu ,\theta \right).$$

Now, we prove the contractive condition in this way

$$\begin{array}{ccc}\hfill \tau ({\mathfrak{J}}_1\varsigma ,{\mathfrak{J}}_2\nu )& =& \tau (\frac{\varsigma }{4},\frac{\nu }{3})=(1+i_2)\left|\frac{\varsigma }{4}-\frac{\nu }{3}\right|\hfill \\ & =& (1+i_2)\left|\frac{3\varsigma -4\nu }{12}\right|\hfill \end{array}$$

$${⪯}_{i_2}(1+i_2)\left|\frac{3\varsigma -3\nu }{12}\right|$$

$$=\frac{1}{4}(1+i_2)\left|\varsigma -\nu \right|$$

$${⪯}_{i_2}\frac{13}{20}(1+i_2)\left|\varsigma -\nu \right|$$

$${⪯}_{i_2}\alpha \left(\varsigma ,\nu ,\theta \right)\tau \left(\varsigma ,\nu \right)$$

$$+\pi \left(\varsigma ,\nu ,\theta \right)\left(\tau \left(\varsigma ,{\mathfrak{J}}_1\varsigma \right)+\tau \left(\nu ,{\mathfrak{J}}_2\nu \right)\right)$$

$$+\kappa \left(\varsigma ,\nu ,\theta \right)\frac{\tau \left(\varsigma ,{\mathfrak{J}}_1\varsigma \right)\tau \left(\nu ,{\mathfrak{J}}_2\nu \right)}{1+\tau \left(\varsigma ,\nu \right)}$$

$$+\varpi \left(\varsigma ,\nu ,\theta \right)\frac{\tau \left(\nu ,{\mathfrak{J}}_1\varsigma \right)\tau \left(\varsigma ,{\mathfrak{J}}_2\nu \right)}{1+\tau \left(\varsigma ,\nu \right)}$$

$$+\varkappa \left(\varsigma ,\nu ,\theta \right)\frac{\tau \left(\varsigma ,{\mathfrak{J}}_1\varsigma \right)\tau \left(\varsigma ,{\mathfrak{J}}_2\nu \right)+\tau \left(\nu ,{\mathfrak{J}}_2\nu \right)\tau \left(\nu ,{\mathfrak{J}}_1\varsigma \right)}{1+\tau \left(\varsigma ,{\mathfrak{J}}_2\nu \right)+\tau \left(\nu ,{\mathfrak{J}}_1\varsigma \right)}.$$

Hence, all the conditions of Theorem 1 are satisfied and $0={\mathfrak{J}}_{1}0={\mathfrak{J}}_{2}0.$

Remark 1.

If we replace $\alpha ,\pi ,\kappa ,\varpi ,\varkappa :\mathcal{Q}\times \mathcal{Q}\times \mathcal{Q}\to [0,1)$ with $\alpha ,\pi ,\kappa ,\varpi ,\varkappa :\mathcal{Q}\times \mathcal{Q}\to [0,1)$ by

$$\alpha \left(\varsigma ,\nu ,\theta \right)=\alpha \left(\varsigma ,\nu \right),$$

$$\pi \left(\varsigma ,\nu ,\theta \right)=\pi \left(\varsigma ,\nu \right),$$

$$\kappa \left(\varsigma ,\nu ,\theta \right)=\kappa \left(\varsigma ,\nu \right),$$

$$\varpi \left(\varsigma ,\nu ,\theta \right)=\varpi \left(\varsigma ,\nu \right),$$

$$\varkappa \left(\varsigma ,\nu ,\theta \right)=\varkappa \left(\varsigma ,\nu \right),$$

then we have following result.

Corollary 6.

Let ${\mathfrak{J}}_1,{\mathfrak{J}}_2:\left(\mathcal{Q},\tau \right)$$\to \left(\mathcal{Q},\tau \right)$ be self-mappings. If the functions $\alpha ,\pi ,\kappa ,\varpi ,\varkappa :{\mathcal{Q}}^2\to [0,1)$ satisfy the conditions

(a) $\alpha \left({\mathfrak{J}}_2{\mathfrak{J}}_1\varsigma ,\nu \right)\le \alpha \left(\varsigma ,\nu \right)$ and $\alpha \left(\varsigma ,{\mathfrak{J}}_1{\mathfrak{J}}_2\nu \right)\le \alpha \left(\varsigma ,\nu \right)$

$\pi \left({\mathfrak{J}}_2{\mathfrak{J}}_1\varsigma ,\nu \right)\le \pi \left(\varsigma ,\nu \right)$ and $\pi \left(\varsigma ,{\mathfrak{J}}_1{\mathfrak{J}}_2\nu \right)\le \pi \left(\varsigma ,\nu \right)$

$\kappa \left({\mathfrak{J}}_2{\mathfrak{J}}_1\varsigma ,\nu \right)\le \kappa \left(\varsigma ,\nu \right)$ and $\kappa \left(\varsigma ,{\mathfrak{J}}_1{\mathfrak{J}}_2\nu \right)\le \kappa \left(\varsigma ,\nu \right)$

$\varpi \left({\mathfrak{J}}_2{\mathfrak{J}}_1\varsigma ,\nu \right)\le \varpi \left(\varsigma ,\nu \right)$ and $\varpi \left(\varsigma ,{\mathfrak{J}}_1{\mathfrak{J}}_2\nu \right)\le \varpi \left(\varsigma ,\nu \right)$

$\varkappa \left({\mathfrak{J}}_2{\mathfrak{J}}_1\varsigma ,\nu \right)\le \varkappa \left(\varsigma ,\nu \right)$ and $\varkappa \left(\varsigma ,{\mathfrak{J}}_1{\mathfrak{J}}_2\nu \right)\le \varkappa \left(\varsigma ,\nu \right);$

(b) $\alpha \left(\varsigma ,\nu \right)+2\pi \left(\varsigma ,\nu \right)+\sqrt{2}\kappa \left(\varsigma ,\nu \right)+\sqrt{2}\varpi \left(\varsigma ,\nu \right)+\sqrt{2}\varkappa \left(\varsigma ,\nu \right)<1;$

(c)

$$\tau \left({\mathfrak{J}}_1\varsigma ,{\mathfrak{J}}_2\nu \right){⪯}_{i_2}\alpha \left(\varsigma ,\nu \right)\tau \left(\varsigma ,\nu \right)+\pi \left(\varsigma ,\nu \right)\left(\tau \left(\varsigma ,{\mathfrak{J}}_1\varsigma \right)+\tau \left(\nu ,{\mathfrak{J}}_2\nu \right)\right)$$

$$+\kappa \left(\varsigma ,\nu \right)\frac{\tau \left(\varsigma ,{\mathfrak{J}}_1\varsigma \right)\tau \left(\nu ,{\mathfrak{J}}_2\nu \right)}{1+\tau \left(\varsigma ,\nu \right)}$$

$$+\varpi \left(\varsigma ,\nu \right)\frac{\tau \left(\nu ,{\mathfrak{J}}_1\varsigma \right)\tau \left(\varsigma ,{\mathfrak{J}}_2\nu \right)}{1+\tau \left(\varsigma ,\nu \right)}$$

$$+\varkappa \left(\varsigma ,\nu \right)\frac{\tau \left(\varsigma ,{\mathfrak{J}}_1\varsigma \right)\tau \left(\varsigma ,{\mathfrak{J}}_2\nu \right)+\tau \left(\nu ,{\mathfrak{J}}_2\nu \right)\tau \left(\nu ,{\mathfrak{J}}_1\varsigma \right)}{1+\tau \left(\varsigma ,{\mathfrak{J}}_2\nu \right)+\tau \left(\nu ,{\mathfrak{J}}_1\varsigma \right)},$$

for all $\varsigma ,\nu \in$$\mathcal{Q},$ then there exists a unique element ${\varsigma }^{*}\in \mathcal{Q}$ such that ${\mathfrak{J}}_1{\varsigma }^{*}={\mathfrak{J}}_2{\varsigma }^{*}={\varsigma }^{*}.$

If we define $\pi ,\varkappa :\mathcal{Q}\times \mathcal{Q}\to [0,1)$ by $\pi \left(\varsigma ,\nu \right)=\varkappa \left(\varsigma ,\nu \right)=0,$ then we achieve the key result presented by Tassaddiq et al. [10].

Corollary 7

([10]). Let ${\mathfrak{J}}_1,{\mathfrak{J}}_2:\left(\mathcal{Q},\tau \right)$$\to \left(\mathcal{Q},\tau \right)$ be self-mappings. If the functions $\alpha ,\kappa ,\varpi :{\mathcal{Q}}^2\to [0,1)$ satisfy the conditions

(a) $\alpha \left({\mathfrak{J}}_2{\mathfrak{J}}_1\varsigma ,\nu \right)\le \alpha \left(\varsigma ,\nu \right)$ and $\alpha \left(\varsigma ,{\mathfrak{J}}_1{\mathfrak{J}}_2\nu \right)\le \alpha \left(\varsigma ,\nu \right)$

$\kappa \left({\mathfrak{J}}_2{\mathfrak{J}}_1\varsigma ,\nu \right)\le \kappa \left(\varsigma ,\nu \right)$ and $\kappa \left(\varsigma ,{\mathfrak{J}}_1{\mathfrak{J}}_2\nu \right)\le \kappa \left(\varsigma ,\nu \right)$

$\varpi \left({\mathfrak{J}}_2{\mathfrak{J}}_1\varsigma ,\nu \right)\le \varpi \left(\varsigma ,\nu \right)$ and $\varpi \left(\varsigma ,{\mathfrak{J}}_1{\mathfrak{J}}_2\nu \right)\le \varpi \left(\varsigma ,\nu \right);$

(b) $\alpha \left(\varsigma ,\nu \right)+\sqrt{2}\kappa \left(\varsigma ,\nu \right)+\sqrt{2}\varpi \left(\varsigma ,\nu \right)<1;$

(c)

$$\tau \left({\mathfrak{J}}_1\varsigma ,{\mathfrak{J}}_2\nu \right){⪯}_{i_2}\alpha \left(\varsigma ,\nu \right)\tau \left(\varsigma ,\nu \right)$$

$$+\kappa \left(\varsigma ,\nu \right)\frac{\tau \left(\varsigma ,{\mathfrak{J}}_1\varsigma \right)\tau \left(\nu ,{\mathfrak{J}}_2\nu \right)}{1+\tau \left(\varsigma ,\nu \right)}$$

$$+\varpi \left(\varsigma ,\nu \right)\frac{\tau \left(\nu ,{\mathfrak{J}}_1\varsigma \right)\tau \left(\varsigma ,{\mathfrak{J}}_2\nu \right)}{1+\tau \left(\varsigma ,\nu \right)}$$

for all $\varsigma ,\nu \in$$\mathcal{Q},$ then there exists a unique ${\varsigma }^{*}\in \mathcal{Q}$ such that ${\mathfrak{J}}_1{\varsigma }^{*}={\mathfrak{J}}_2{\varsigma }^{*}={\varsigma }^{*}.$

Remark 2.

By defining $\alpha ,\pi ,\kappa ,\varpi ,\varkappa :\mathcal{Q}\times \mathcal{Q}\to [0,1)$ as 0 in all possible combinations, one can obtain all the corollaries presented by Tassaddiq et al. [10] and a host of corollaries including the Banach contraction principle and Kannan’s fixed point theorem in the setting of a complete bi-CVMS.

Corollary 8.

Let ${\mathfrak{J}}_1,{\mathfrak{J}}_2:\left(\mathcal{Q},\tau \right)$$\to \left(\mathcal{Q},\tau \right)$ be self-mappings. If the functions $\alpha ,\pi ,\kappa ,\varpi ,\varkappa :\mathcal{Q}\to [0,1)$ satisfy the conditions

(a) $\alpha \left({\mathfrak{J}}_1\varsigma \right)\le \alpha \left(\varsigma \right)$ and $\alpha \left({\mathfrak{J}}_2\varsigma \right)\le \alpha \left(\varsigma \right)$

$\pi \left({\mathfrak{J}}_1\varsigma \right)\le \pi \left(\varsigma \right)$ and $\pi \left({\mathfrak{J}}_2\varsigma \right)\le \pi \left(\varsigma \right)$

$\kappa \left({\mathfrak{J}}_1\varsigma \right)\le \kappa \left(\varsigma \right)$ and $\kappa \left({\mathfrak{J}}_2\varsigma \right)\le \kappa \left(\varsigma \right)$

$\varpi \left({\mathfrak{J}}_1\varsigma \right)\le \varpi \left(\varsigma \right)$ and $\varpi \left({\mathfrak{J}}_2\varsigma \right)\le \varpi \left(\varsigma \right)$

$\varkappa \left({\mathfrak{J}}_1\varsigma \right)\le \varkappa \left(\varsigma \right)$ and $\varkappa \left({\mathfrak{J}}_2\varsigma \right)\le \varkappa \left(\varsigma \right);$

(b) $\alpha \left(\varsigma \right)+2\pi \left(\varsigma \right)+\sqrt{2}\kappa \left(\varsigma \right)+\sqrt{2}\varpi \left(\varsigma \right)+\sqrt{2}\varkappa \left(\varsigma \right)<1;$

(c)

$$\tau \left({\mathfrak{J}}_1\varsigma ,{\mathfrak{J}}_2\nu \right){⪯}_{i_2}\alpha \left(\varsigma \right)\tau \left(\varsigma ,\nu \right)+\pi \left(\varsigma \right)\left(\tau \left(\varsigma ,{\mathfrak{J}}_1\varsigma \right)+\tau \left(\nu ,{\mathfrak{J}}_2\nu \right)\right)$$

$$+\kappa \left(\varsigma \right)\frac{\tau \left(\varsigma ,{\mathfrak{J}}_1\varsigma \right)\tau \left(\nu ,{\mathfrak{J}}_2\nu \right)}{1+\tau \left(\varsigma ,\nu \right)}$$

$$+\varpi \left(\varsigma \right)\frac{\tau \left(\nu ,{\mathfrak{J}}_1\varsigma \right)\tau \left(\varsigma ,{\mathfrak{J}}_2\nu \right)}{1+\tau \left(\varsigma ,\nu \right)}$$

$$+\varkappa \left(\varsigma \right)\frac{\tau \left(\varsigma ,{\mathfrak{J}}_1\varsigma \right)\tau \left(\varsigma ,{\mathfrak{J}}_2\nu \right)+\tau \left(\nu ,{\mathfrak{J}}_2\nu \right)\tau \left(\nu ,{\mathfrak{J}}_1\varsigma \right)}{1+\tau \left(\varsigma ,{\mathfrak{J}}_2\nu \right)+\tau \left(\nu ,{\mathfrak{J}}_1\varsigma \right)},$$

for all $\varsigma ,\nu \in$$\mathcal{Q},$ then there exists a unique element ${\varsigma }^{*}\in \mathcal{Q}$ such that ${\mathfrak{J}}_1{\varsigma }^{*}={\mathfrak{J}}_2{\varsigma }^{*}={\varsigma }^{*}.$

Proof. 

Define $\alpha ,\pi ,\kappa ,\varpi ,\varkappa :\mathcal{Q}\times \mathcal{Q}\times \mathcal{Q}\to [0,1)$ by

$$\alpha \left(\varsigma ,\nu ,\theta \right)=\alpha \left(\varsigma \right),$$

$$\pi \left(\varsigma ,\nu ,\theta \right)=\pi \left(\varsigma \right),$$

$$\kappa \left(\varsigma ,\nu ,\theta \right)=\kappa \left(\varsigma \right),$$

$$\varpi \left(\varsigma ,\nu ,\theta \right)=\varpi \left(\varsigma \right),$$

$$\varkappa \left(\varsigma ,\nu ,\theta \right)=\varkappa \left(\varsigma \right).$$

Then, for all $\varsigma ,\nu \in$$\mathcal{Q}$ and for a fixed element $\theta \in$$\mathcal{Q},$ we have

(a)

$\alpha \left({\mathfrak{J}}_2{\mathfrak{J}}_1\varsigma ,\nu ,\theta \right)=\alpha \left({\mathfrak{J}}_2{\mathfrak{J}}_1\varsigma \right)\le \alpha \left({\mathfrak{J}}_1\varsigma \right)\le \alpha \left(\varsigma \right)=\alpha \left(\varsigma ,\nu ,\theta \right)$ and $\alpha \left(\varsigma ,{\mathfrak{J}}_1{\mathfrak{J}}_2\nu ,\theta \right)=\alpha \left(\varsigma \right)=\alpha \left(\varsigma ,\nu ,\theta \right)$

$\pi \left({\mathfrak{J}}_2{\mathfrak{J}}_1\varsigma ,\nu ,\theta \right)=\pi \left({\mathfrak{J}}_2{\mathfrak{J}}_1\varsigma \right)\le \pi \left({\mathfrak{J}}_1\varsigma \right)\le \pi \left(\varsigma \right)=\pi \left(\varsigma ,\nu ,\theta \right)$ and $\pi \left(\varsigma ,{\mathfrak{J}}_1{\mathfrak{J}}_2\nu ,\theta \right)=\pi \left(\varsigma \right)=\pi \left(\varsigma ,\nu ,\theta \right)$

$\kappa \left({\mathfrak{J}}_2{\mathfrak{J}}_1\varsigma ,\nu ,\theta \right)=\kappa \left({\mathfrak{J}}_2{\mathfrak{J}}_1\varsigma \right)\le \kappa \left({\mathfrak{J}}_1\varsigma \right)\le \kappa \left(\varsigma \right)=\kappa \left(\varsigma ,\nu ,\theta \right)$ and $\kappa \left(\varsigma ,{\mathfrak{J}}_1{\mathfrak{J}}_2\nu ,\theta \right)=\kappa \left(\varsigma \right)=\kappa \left(\varsigma ,\nu ,\theta \right)$

$\varpi \left({\mathfrak{J}}_2{\mathfrak{J}}_1\varsigma ,\nu ,\theta \right)=\varpi \left({\mathfrak{J}}_2{\mathfrak{J}}_1\varsigma \right)\le \varpi \left({\mathfrak{J}}_1\varsigma \right)\le \varpi \left(\varsigma \right)=\varpi \left(\varsigma ,\nu ,\theta \right)$ and $\varpi \left(\varsigma ,{\mathfrak{J}}_1{\mathfrak{J}}_2\nu ,\theta \right)=\varpi \left(\varsigma \right)=\varpi \left(\varsigma ,\nu ,\theta \right)$

$\varkappa \left({\mathfrak{J}}_2{\mathfrak{J}}_1\varsigma ,\nu ,\theta \right)=\varkappa \left({\mathfrak{J}}_2{\mathfrak{J}}_1\varsigma \right)\le \varkappa \left({\mathfrak{J}}_1\varsigma \right)\le \varkappa \left(\varsigma \right)=\varkappa \left(\varsigma ,\nu ,\theta \right)$ and $\varkappa \left(\varsigma ,{\mathfrak{J}}_1{\mathfrak{J}}_2\nu ,\theta \right)=\varkappa \left(\varsigma \right)=\varkappa \left(\varsigma ,\nu ,\theta \right);$

(b)

$$\begin{array}{ccc}& & \alpha \left(\varsigma ,\nu ,\theta \right)+2\pi \left(\varsigma ,\nu ,\theta \right)+\sqrt{2}\kappa \left(\varsigma ,\nu ,\theta \right)+\sqrt{2}\varpi \left(\varsigma ,\nu ,\theta \right)+\sqrt{2}\varkappa \left(\varsigma ,\nu ,\theta \right)\hfill \\ & =& \alpha \left(\varsigma \right)+2\pi \left(\varsigma \right)+\sqrt{2}\kappa \left(\varsigma \right)+\sqrt{2}\varpi \left(\varsigma \right)+\sqrt{2}\varkappa \left(\varsigma \right)<1;\hfill \end{array}$$

(c)

$$\tau \left({\mathfrak{J}}_1\varsigma ,{\mathfrak{J}}_2\nu \right){⪯}_{i_2}\alpha \left(\varsigma \right)\tau \left(\varsigma ,\nu \right)+\pi \left(\varsigma \right)\left(\tau \left(\varsigma ,{\mathfrak{J}}_1\varsigma \right)+\tau \left(\nu ,{\mathfrak{J}}_2\nu \right)\right)$$

$$+\kappa \left(\varsigma \right)\frac{\tau \left(\varsigma ,{\mathfrak{J}}_1\varsigma \right)\tau \left(\nu ,{\mathfrak{J}}_2\nu \right)}{1+\tau \left(\varsigma ,\nu \right)}$$

$$+\varpi \left(\varsigma \right)\frac{\tau \left(\nu ,{\mathfrak{J}}_1\varsigma \right)\tau \left(\varsigma ,{\mathfrak{J}}_2\nu \right)}{1+\tau \left(\varsigma ,\nu \right)}$$

$$+\varkappa \left(\varsigma \right)\frac{\tau \left(\varsigma ,{\mathfrak{J}}_1\varsigma \right)\tau \left(\varsigma ,{\mathfrak{J}}_2\nu \right)+\tau \left(\nu ,{\mathfrak{J}}_2\nu \right)\tau \left(\nu ,{\mathfrak{J}}_1\varsigma \right)}{1+\tau \left(\varsigma ,{\mathfrak{J}}_2\nu \right)+\tau \left(\nu ,{\mathfrak{J}}_1\varsigma \right)}.$$

$$\begin{array}{ccc}\hfill & =& \alpha \left(\varsigma ,\nu ,\theta \right)\tau \left(\varsigma ,\nu \right)\hfill \\ & & +\pi \left(\varsigma ,\nu ,\theta \right)\left(\tau \left(\varsigma ,{\mathfrak{J}}_1\varsigma \right)+\tau \left(\nu ,{\mathfrak{J}}_2\nu \right)\right)\hfill \\ & & +\kappa \left(\varsigma ,\nu ,\theta \right)\frac{\tau \left(\varsigma ,{\mathfrak{J}}_1\varsigma \right)\tau \left(\nu ,{\mathfrak{J}}_2\nu \right)}{1+\tau \left(\varsigma ,\nu \right)}\hfill \\ & & +\varpi \left(\varsigma ,\nu ,\theta \right)\frac{\tau \left(\nu ,{\mathfrak{J}}_1\varsigma \right)\tau \left(\varsigma ,{\mathfrak{J}}_2\nu \right)}{1+\tau \left(\varsigma ,\nu \right)}\hfill \\ & & +\varkappa \left(\varsigma ,\nu ,\theta \right)\frac{\tau \left(\varsigma ,{\mathfrak{J}}_1\varsigma \right)\tau \left(\varsigma ,{\mathfrak{J}}_2\nu \right)+\tau \left(\nu ,{\mathfrak{J}}_2\nu \right)\tau \left(\nu ,{\mathfrak{J}}_1\varsigma \right)}{1+\tau \left(\varsigma ,{\mathfrak{J}}_2\nu \right)+\tau \left(\nu ,{\mathfrak{J}}_1\varsigma \right)}.\hfill \end{array}$$

Then, by Theorem 1, there exists ${\varsigma }^{*}\in \mathcal{Q}$ such that ${\mathfrak{J}}_1{\varsigma }^{*}={\mathfrak{J}}_2{\varsigma }^{*}={\varsigma }^{*}.$ □

Corollary 9.

Let ${\mathfrak{J}}_1,{\mathfrak{J}}_2:\left(\mathcal{Q},\tau \right)$$\to \left(\mathcal{Q},\tau \right)$ be self-mappings. If there exist constants $\alpha ,\pi ,\kappa ,\varpi ,\varkappa \in [0,1)$ such that

$\alpha +2\pi +\sqrt{2}\kappa +\sqrt{2}\varpi +\sqrt{2}\varkappa <1$ and

$$\tau \left({\mathfrak{J}}_1\varsigma ,{\mathfrak{J}}_2\nu \right){⪯}_{i_2}\alpha \tau \left(\varsigma ,\nu \right)+\pi \left(\tau \left(\varsigma ,{\mathfrak{J}}_1\varsigma \right)+\tau \left(\nu ,{\mathfrak{J}}_2\nu \right)\right)$$

$$+\kappa \frac{\tau \left(\varsigma ,{\mathfrak{J}}_1\varsigma \right)\tau \left(\nu ,{\mathfrak{J}}_2\nu \right)}{1+\tau \left(\varsigma ,\nu \right)}$$

$$+\varpi \frac{\tau \left(\nu ,{\mathfrak{J}}_1\varsigma \right)\tau \left(\varsigma ,{\mathfrak{J}}_2\nu \right)}{1+\tau \left(\varsigma ,\nu \right)}$$

$$+\varkappa \frac{\tau \left(\varsigma ,{\mathfrak{J}}_1\varsigma \right)\tau \left(\varsigma ,{\mathfrak{J}}_2\nu \right)+\tau \left(\nu ,{\mathfrak{J}}_2\nu \right)\tau \left(\nu ,{\mathfrak{J}}_1\varsigma \right)}{1+\tau \left(\varsigma ,{\mathfrak{J}}_2\nu \right)+\tau \left(\nu ,{\mathfrak{J}}_1\varsigma \right)},$$

for all $\varsigma ,\nu \in$$\mathcal{Q},$ then there exists a unique ${\varsigma }^{*}\in \mathcal{Q}$ such that ${\mathfrak{J}}_1{\varsigma }^{*}={\mathfrak{J}}_2{\varsigma }^{*}={\varsigma }^{*}.$

Proof. 

Define $\alpha ,\pi ,\kappa ,\varpi ,\varkappa :\mathcal{Q}\to [0,1)$ by $\alpha \left(·\right)=\alpha ,\pi \left(·\right)=\pi ,\kappa \left(·\right)=\kappa ,\varpi \left(·\right)$ and $\varkappa \left(·\right)=\varkappa$ in Corollary 8. □

If we consider $\pi =\varkappa =0$ in Corollary 9, then we obtain the key result of Gnanaprakasam et al. [9] in this manner.

Corollary 10

([9]). Let ${\mathfrak{J}}_1,{\mathfrak{J}}_2:\left(\mathcal{Q},\tau \right)$$\to \left(\mathcal{Q},\tau \right)$ be self-mappings. If there exist $\alpha ,\kappa ,\varpi \in [0,1)$ such that $\alpha +\sqrt{2}\kappa +\sqrt{2}\varpi <1$ and

$$\tau \left({\mathfrak{J}}_1\varsigma ,{\mathfrak{J}}_2\nu \right){⪯}_{i_2}\alpha \tau \left(\varsigma ,\nu \right)+\kappa \frac{\tau \left(\varsigma ,{\mathfrak{J}}_1\varsigma \right)\tau \left(\nu ,{\mathfrak{J}}_2\nu \right)}{1+\tau \left(\varsigma ,\nu \right)}+\varpi \frac{\tau \left(\nu ,{\mathfrak{J}}_1\varsigma \right)\tau \left(\varsigma ,{\mathfrak{J}}_2\nu \right)}{1+\tau \left(\varsigma ,\nu \right)},$$

for all $\varsigma ,\nu \in$$\mathcal{Q},$ then there exists a unique element ${\varsigma }^{*}\in \mathcal{Q}$ such that ${\mathfrak{J}}_1{\varsigma }^{*}={\mathfrak{J}}_2{\varsigma }^{*}={\varsigma }^{*}.$

Corollary 11.

Let $\mathfrak{J}:\left(\mathcal{Q},\tau \right)$$\to \left(\mathcal{Q},\tau \right)$ be a self-mapping. If the functions $\alpha ,\pi ,\kappa ,\varpi ,\varkappa :\mathcal{Q}\to [0,1)$ satisfy the conditions

(a) $\alpha \left(\mathfrak{J}\varsigma \right)\le \alpha \left(\varsigma \right)$,

$\pi \left(\mathfrak{J}\varsigma \right)\le \pi \left(\varsigma \right)$,

$\kappa \left(\mathfrak{J}\varsigma \right)\le \kappa \left(\varsigma \right)$,

$\varpi \left(\mathfrak{J}\varsigma \right)\le \varpi \left(\varsigma \right)$,

$\varkappa \left(\mathfrak{J}\varsigma \right)\le \varkappa \left(\varsigma \right)$;

(b) $\alpha \left(\varsigma \right)+2\pi \left(\varsigma \right)+\sqrt{2}\kappa \left(\varsigma \right)+\sqrt{2}\varpi \left(\varsigma \right)+\sqrt{2}\varkappa \left(\varsigma \right)<1;$

(c)

$$\tau \left(\mathfrak{J}\varsigma ,\mathfrak{J}\nu \right){⪯}_{i_2}\alpha \left(\varsigma \right)\tau \left(\varsigma ,\nu \right)+\pi \left(\varsigma \right)\left(\tau \left(\varsigma ,\mathfrak{J}\varsigma \right)+\tau \left(\nu ,\mathfrak{J}\nu \right)\right)$$

$$+\kappa \left(\varsigma \right)\frac{\tau \left(\varsigma ,\mathfrak{J}\varsigma \right)\tau \left(\nu ,\mathfrak{J}\nu \right)}{1+\tau \left(\varsigma ,\nu \right)}$$

$$+\varpi \left(\varsigma \right)\frac{\tau \left(\nu ,\mathfrak{J}\varsigma \right)\tau \left(\varsigma ,\mathfrak{J}\nu \right)}{1+\tau \left(\varsigma ,\nu \right)}$$

$$+\varkappa \left(\varsigma \right)\frac{\tau \left(\varsigma ,\mathfrak{J}\varsigma \right)\tau \left(\varsigma ,\mathfrak{J}\nu \right)+\tau \left(\nu ,\mathfrak{J}\nu \right)\tau \left(\nu ,\mathfrak{J}\varsigma \right)}{1+\tau \left(\varsigma ,\mathfrak{J}\nu \right)+\tau \left(\nu ,\mathfrak{J}\varsigma \right)},$$

for all $\varsigma ,\nu \in$$\mathcal{Q},$ then there exists a unique element ${\varsigma }^{*}\in \mathcal{Q}$ such that $\mathfrak{J}{\varsigma }^{*}={\varsigma }^{*}$.

Proof. 

Take ${\mathfrak{J}}_1={\mathfrak{J}}_2$$=\mathfrak{J}$ in Corollary 8. □

Corollary 12.

Let $\mathfrak{J}:\left(\mathcal{Q},\tau \right)$$\to \left(\mathcal{Q},\tau \right)$ be a self-mapping. If there exist $\alpha ,\pi ,\kappa ,\varpi ,\varkappa \in [0,1)$ such that $\alpha +2\pi +\sqrt{2}\kappa +\sqrt{2}\varpi +\sqrt{2}\varkappa <1$ and

$$\tau \left(\mathfrak{J}\varsigma ,\mathfrak{J}\nu \right){⪯}_{i_2}\alpha \tau \left(\varsigma ,\nu \right)+\pi \left(\tau \left(\varsigma ,\mathfrak{J}\varsigma \right)+\tau \left(\nu ,\mathfrak{J}\nu \right)\right)$$

$$+\kappa \frac{\tau \left(\varsigma ,\mathfrak{J}\varsigma \right)\tau \left(\nu ,\mathfrak{J}\nu \right)}{1+\tau \left(\varsigma ,\nu \right)}$$

$$+\varpi \frac{\tau \left(\nu ,\mathfrak{J}\varsigma \right)\tau \left(\varsigma ,\mathfrak{J}\nu \right)}{1+\tau \left(\varsigma ,\nu \right)}$$

$$+\varkappa \frac{\tau \left(\varsigma ,\mathfrak{J}\varsigma \right)\tau \left(\varsigma ,\mathfrak{J}\nu \right)+\tau \left(\nu ,\mathfrak{J}\nu \right)\tau \left(\nu ,\mathfrak{J}\varsigma \right)}{1+\tau \left(\varsigma ,\mathfrak{J}\nu \right)+\tau \left(\nu ,\mathfrak{J}\varsigma \right)},$$

for all $\varsigma ,\nu \in$$\mathcal{Q},$ then there exists a unique element ${\varsigma }^{*}\in \mathcal{Q}$ such that $\mathfrak{J}{\varsigma }^{*}={\varsigma }^{*}$.

Proof. 

Define $\alpha ,\pi ,\kappa ,\varpi ,\varkappa :\mathcal{Q}\to [0,1)$ by $\alpha \left(·\right)=\alpha ,\pi \left(·\right)=\pi ,\kappa \left(·\right)=\kappa ,\varpi \left(·\right)$ and $\varkappa \left(·\right)=\varkappa$ in Corollary 11. □

Corollary 13.

Let $\mathfrak{J}:\left(\mathcal{Q},\tau \right)$$\to \left(\mathcal{Q},\tau \right)$ be a self-mapping. If there exist $\alpha ,\pi ,\kappa ,\varpi ,\varkappa \in [0,1)$ such that $\alpha +2\pi +\sqrt{2}\kappa +\sqrt{2}\varpi +\sqrt{2}\varkappa <1$ and

$$\tau \left({\mathfrak{J}}^n\varsigma ,{\mathfrak{J}}^n\nu \right){⪯}_{i_2}\alpha \tau \left(\varsigma ,\nu \right)+\pi \left(\tau \left(\varsigma ,{\mathfrak{J}}^n\varsigma \right)+\tau \left(\nu ,{\mathfrak{J}}^n\nu \right)\right)$$

$$+\kappa \frac{\tau \left(\varsigma ,{\mathfrak{J}}^n\varsigma \right)\tau \left(\nu ,{\mathfrak{J}}^n\nu \right)}{1+\tau \left(\varsigma ,\nu \right)}$$

$$+\varpi \frac{\tau \left(\nu ,{\mathfrak{J}}^n\varsigma \right)\tau \left(\varsigma ,{\mathfrak{J}}^n\nu \right)}{1+\tau \left(\varsigma ,\nu \right)}$$

$$+\varkappa \frac{\tau \left(\varsigma ,{\mathfrak{J}}^n\varsigma \right)\tau \left(\varsigma ,{\mathfrak{J}}^n\nu \right)+\tau \left(\nu ,{\mathfrak{J}}^n\nu \right)\tau \left(\nu ,{\mathfrak{J}}^n\varsigma \right)}{1+\tau \left(\varsigma ,{\mathfrak{J}}^n\nu \right)+\tau \left(\nu ,{\mathfrak{J}}^n\varsigma \right)},$$

for all $\varsigma ,\nu \in$$\mathcal{Q},$ then there exists a unique element ${\varsigma }^{*}\in \mathcal{Q}$ such that $\mathfrak{J}{\varsigma }^{*}={\varsigma }^{*}$.

Proof. 

By Corollary 12, we can obtain $\varsigma \in \mathcal{Q}$ such that ${\mathfrak{J}}^n\varsigma =\varsigma .$ Now,

$$\begin{array}{ccc}\hfill \tau \left(\mathfrak{J}\varsigma ,\varsigma \right)& =& \tau \left({\mathfrak{JJ}}^n\varsigma ,{\mathfrak{J}}^n\varsigma \right)\hfill \\ & =& \tau \left({\mathfrak{J}}^n\mathfrak{J}\varsigma ,{\mathfrak{J}}^n\varsigma \right)\hfill \end{array}$$

$${⪯}_{i_2}\alpha \tau \left(\mathfrak{J}\varsigma ,\varsigma \right)+\pi \left(\tau \left(\mathfrak{J}\varsigma ,{\mathfrak{J}}^n\mathfrak{J}\varsigma \right)+\tau \left(\varsigma ,{\mathfrak{J}}^n\varsigma \right)\right)$$

$$+\kappa \frac{\tau \left(\mathfrak{J}\varsigma ,{\mathfrak{J}}^n\mathfrak{J}\varsigma \right)\tau \left(\varsigma ,{\mathfrak{J}}^n\varsigma \right)}{1+\tau \left(\mathfrak{J}\varsigma ,\varsigma \right)}$$

$$+\varpi \frac{\tau \left(\varsigma ,{\mathfrak{J}}^n\mathfrak{J}\varsigma \right)\tau \left(\mathfrak{J}\varsigma ,{\mathfrak{J}}^n\varsigma \right)}{1+\tau \left(\mathfrak{J}\varsigma ,\varsigma \right)}$$

$$+\varkappa \frac{\tau \left(\mathfrak{J}\varsigma ,{\mathfrak{J}}^n\mathfrak{J}\varsigma \right)\tau \left(\mathfrak{J}\varsigma ,{\mathfrak{J}}^n\varsigma \right)+\tau \left(\varsigma ,{\mathfrak{J}}^n\varsigma \right)\tau \left(\varsigma ,{\mathfrak{J}}^n\mathfrak{J}\varsigma \right)}{1+\tau \left(\mathfrak{J}\varsigma ,{\mathfrak{J}}^n\varsigma \right)+\tau \left(\varsigma ,{\mathfrak{J}}^n\mathfrak{J}\varsigma \right)}$$

$${⪯}_{i_2}\alpha \tau \left(\mathfrak{J}\varsigma ,\varsigma \right)+\pi \left(\tau \left(\mathfrak{J}\varsigma ,\mathfrak{J}\varsigma \right)+\tau \left(\varsigma ,\varsigma \right)\right)$$

$$+\kappa \frac{\tau \left(\mathfrak{J}\varsigma ,\mathfrak{J}\varsigma \right)\tau \left(\varsigma ,\varsigma \right)}{1+\tau \left(\mathfrak{J}\varsigma ,\varsigma \right)}$$

$$+\varpi \frac{\tau \left(\varsigma ,\mathfrak{J}\varsigma \right)\tau \left(\mathfrak{J}\varsigma ,\varsigma \right)}{1+\tau \left(\mathfrak{J}\varsigma ,\varsigma \right)}$$

$$+\varkappa \frac{\tau \left(\mathfrak{J}\varsigma ,\mathfrak{J}\varsigma \right)\tau \left(\mathfrak{J}\varsigma ,\varsigma \right)+\tau \left(\varsigma ,\varsigma \right)\tau \left(\varsigma ,\mathfrak{J}\varsigma \right)}{1+\tau \left(\mathfrak{J}\varsigma ,\varsigma \right)+\tau \left(\varsigma ,\mathfrak{J}\varsigma \right)}$$

$$=\alpha \tau \left(\mathfrak{J}\varsigma ,\varsigma \right)+\varpi \frac{\tau \left(\varsigma ,\mathfrak{J}\varsigma \right)\tau \left(\mathfrak{J}\varsigma ,\varsigma \right)}{1+\tau \left(\mathfrak{J}\varsigma ,\varsigma \right)}$$

which implies that

$$\begin{array}{ccc}\hfill ∥\tau \left(\mathfrak{J}\varsigma ,\varsigma \right)∥& \le & \alpha ∥\tau \left(\mathfrak{J}\varsigma ,\varsigma \right)∥+\sqrt{2}\varpi ∥\tau \left(\varsigma ,\mathfrak{J}\varsigma \right)∥∥\frac{\tau \left(\mathfrak{J}\varsigma ,\varsigma \right)}{1+\tau \left(\mathfrak{J}\varsigma ,\varsigma \right)}∥\hfill \\ & \le & \alpha ∥\tau \left(\mathfrak{J}\varsigma ,\varsigma \right)∥+\sqrt{2}\varpi ∥\tau \left(\varsigma ,\mathfrak{J}\varsigma \right)∥\hfill \\ & =& \left(\alpha +\sqrt{2}\varpi \right)∥\tau \left(\varsigma ,\mathfrak{J}\varsigma \right)∥\hfill \end{array}$$

which is possible only whenever $∥\tau \left(\mathfrak{J}\varsigma ,\varsigma \right)∥=0.$ Thus, $\mathfrak{J}\varsigma =\varsigma .$ □

Corollary 14

([8]). Let $\mathfrak{J}:\left(\mathcal{Q},\tau \right)$$\to \left(\mathcal{Q},\tau \right)$ be a self-mapping. If there exist $\alpha ,\kappa \in [0,1)$ such that $\alpha +\sqrt{2}\kappa <1$ and for all $\varsigma ,\nu \in$$\mathcal{Q},$

$$\tau \left(\mathfrak{J}\varsigma ,\mathfrak{J}\nu \right){⪯}_{i_2}\alpha \tau \left(\varsigma ,\nu \right)+\kappa \frac{\tau \left(\varsigma ,\mathfrak{J}\varsigma \right)\tau \left(\nu ,\mathfrak{J}\nu \right)}{1+\tau \left(\varsigma ,\nu \right)}$$

then there exists a unique element ${\varsigma }^{*}\in \mathcal{Q}$ such that $\mathfrak{J}{\varsigma }^{*}={\varsigma }^{*}$.

Proof. 

Take $\pi =\varpi =\varkappa =0$ in Corollary 12. □

Remark 3.

It is notable that (a) and (b) of Theorem 1 above can be weakened by the condition

$$\alpha \left({\mathfrak{J}}_2{\mathfrak{J}}_1\varsigma \right)\le \alpha \left(\varsigma \right)$$

$$\pi \left({\mathfrak{J}}_2{\mathfrak{J}}_1\varsigma \right)\le \pi \left(\varsigma \right)$$

$$\kappa \left({\mathfrak{J}}_2{\mathfrak{J}}_1\varsigma \right)\le \kappa \left(\varsigma \right)$$

$$\varpi \left({\mathfrak{J}}_2{\mathfrak{J}}_1\varsigma \right)\le \varpi \left(\varsigma \right)$$

$$\varkappa \left({\mathfrak{J}}_2{\mathfrak{J}}_1\varsigma \right)\le \varkappa \left(\varsigma \right)$$

for all $\varsigma \in \mathcal{Q}.$

Corollary 15.

Let ${\mathfrak{J}}_1,{\mathfrak{J}}_2:\left(\mathcal{Q},\tau \right)$$\to \left(\mathcal{Q},\tau \right)$ be self-mappings. If there exist $\alpha ,\pi ,\kappa ,\varpi ,\varkappa :\mathcal{Q}\to [0,1)$ such that for all $\varsigma ,\nu \in$$\mathcal{Q},$

(a) $\alpha \left({\mathfrak{J}}_2{\mathfrak{J}}_1\varsigma \right)\le \alpha \left(\varsigma \right)$,

$\pi \left({\mathfrak{J}}_2{\mathfrak{J}}_1\varsigma \right)\le \pi \left(\varsigma \right)$,

$\kappa \left({\mathfrak{J}}_2{\mathfrak{J}}_1\varsigma \right)\le \kappa \left(\varsigma \right)$,

$\varpi \left({\mathfrak{J}}_2{\mathfrak{J}}_1\varsigma \right)\le \varpi \left(\varsigma \right)$,

$\varkappa \left({\mathfrak{J}}_2{\mathfrak{J}}_1\varsigma \right)\le \varkappa \left(\varsigma \right)$;

(b) $\alpha \left(\varsigma \right)+2\pi \left(\varsigma \right)+\sqrt{2}\kappa \left(\varsigma \right)+\sqrt{2}\varpi \left(\varsigma \right)+\sqrt{2}\varkappa \left(\varsigma \right)<1;$

(c)

$$\tau \left({\mathfrak{J}}_1\varsigma ,{\mathfrak{J}}_2\nu \right){⪯}_{i_2}\alpha \left(\varsigma \right)\tau \left(\varsigma ,\nu \right)+\pi \left(\varsigma \right)\left(\tau \left(\varsigma ,{\mathfrak{J}}_1\varsigma \right)+\tau \left(\nu ,{\mathfrak{J}}_2\nu \right)\right)$$

$$+\kappa \left(\varsigma \right)\frac{\tau \left(\varsigma ,{\mathfrak{J}}_1\varsigma \right)\tau \left(\nu ,{\mathfrak{J}}_2\nu \right)}{1+\tau \left(\varsigma ,\nu \right)}$$

$$+\varpi \left(\varsigma \right)\frac{\tau \left(\nu ,{\mathfrak{J}}_1\varsigma \right)\tau \left(\varsigma ,{\mathfrak{J}}_2\nu \right)}{1+\tau \left(\varsigma ,\nu \right)}$$

$$+\varkappa \left(\varsigma \right)\frac{\tau \left(\varsigma ,{\mathfrak{J}}_1\varsigma \right)\tau \left(\varsigma ,{\mathfrak{J}}_2\nu \right)+\tau \left(\nu ,{\mathfrak{J}}_2\nu \right)\tau \left(\nu ,{\mathfrak{J}}_1\varsigma \right)}{1+\tau \left(\varsigma ,{\mathfrak{J}}_2\nu \right)+\tau \left(\nu ,{\mathfrak{J}}_1\varsigma \right)},$$

then there exists a unique element ${\varsigma }^{*}\in \mathcal{Q}$ such that ${\mathfrak{J}}_1{\varsigma }^{*}={\mathfrak{J}}_2{\varsigma }^{*}={\varsigma }^{*}.$

Proof. 

Define $\alpha ,\pi ,\kappa ,\varpi ,\varkappa :\mathcal{Q}\times \mathcal{Q}\times \mathcal{Q}\to [0,1)$ by

$$\alpha \left(\varsigma ,\nu ,\theta \right)=\alpha \left(\varsigma \right),$$

$$\pi \left(\varsigma ,\nu ,\theta \right)=\pi \left(\varsigma \right),$$

$$\kappa \left(\varsigma ,\nu ,\theta \right)=\kappa \left(\varsigma \right),$$

$$\varpi \left(\varsigma ,\nu ,\theta \right)=\varpi \left(\varsigma \right),$$

$$\varkappa \left(\varsigma ,\nu ,\theta \right)=\varkappa \left(\varsigma \right).$$

Then, for all $\varsigma ,\nu \in$$\mathcal{Q},$ we have

(a)

$\alpha \left({\mathfrak{J}}_2{\mathfrak{J}}_1\varsigma ,\nu ,\theta \right)=\alpha \left({\mathfrak{J}}_2{\mathfrak{J}}_1\varsigma \right)\le \alpha \left({\mathfrak{J}}_1\varsigma \right)\le \alpha \left(\varsigma \right)=\alpha \left(\varsigma ,\nu ,\theta \right)$ and $\alpha \left(\varsigma ,{\mathfrak{J}}_1{\mathfrak{J}}_2\nu ,\theta \right)=\alpha \left(\varsigma \right)=\alpha \left(\varsigma ,\nu ,\theta \right)$

$\pi \left({\mathfrak{J}}_2{\mathfrak{J}}_1\varsigma ,\nu ,\theta \right)=\pi \left({\mathfrak{J}}_2{\mathfrak{J}}_1\varsigma \right)\le \pi \left({\mathfrak{J}}_1\varsigma \right)\le \pi \left(\varsigma \right)=\pi \left(\varsigma ,\nu ,\theta \right)$ and $\pi \left(\varsigma ,{\mathfrak{J}}_1{\mathfrak{J}}_2\nu ,\theta \right)=\pi \left(\varsigma \right)=\pi \left(\varsigma ,\nu ,\theta \right)$

$\kappa \left({\mathfrak{J}}_2{\mathfrak{J}}_1\varsigma ,\nu ,\theta \right)=\kappa \left({\mathfrak{J}}_2{\mathfrak{J}}_1\varsigma \right)\le \kappa \left({\mathfrak{J}}_1\varsigma \right)\le \kappa \left(\varsigma \right)=\kappa \left(\varsigma ,\nu ,\theta \right)$ and $\kappa \left(\varsigma ,{\mathfrak{J}}_1{\mathfrak{J}}_2\nu ,\theta \right)=\kappa \left(\varsigma \right)=\kappa \left(\varsigma ,\nu ,\theta \right)$

$\varpi \left({\mathfrak{J}}_2{\mathfrak{J}}_1\varsigma ,\nu ,\theta \right)=\varpi \left({\mathfrak{J}}_2{\mathfrak{J}}_1\varsigma \right)\le \varpi \left({\mathfrak{J}}_1\varsigma \right)\le \varpi \left(\varsigma \right)=\varpi \left(\varsigma ,\nu ,\theta \right)$ and $\varpi \left(\varsigma ,{\mathfrak{J}}_1{\mathfrak{J}}_2\nu ,\theta \right)=\varpi \left(\varsigma \right)=\varpi \left(\varsigma ,\nu ,\theta \right)$

$\varkappa \left({\mathfrak{J}}_2{\mathfrak{J}}_1\varsigma ,\nu ,\theta \right)=\varkappa \left({\mathfrak{J}}_2{\mathfrak{J}}_1\varsigma \right)\le \varkappa \left({\mathfrak{J}}_1\varsigma \right)\le \varkappa \left(\varsigma \right)=\varkappa \left(\varsigma ,\nu ,\theta \right)$ and $\varkappa \left(\varsigma ,{\mathfrak{J}}_1{\mathfrak{J}}_2\nu ,\theta \right)=\varkappa \left(\varsigma \right)=\varkappa \left(\varsigma ,\nu ,\theta \right);$

(b)

$$\begin{array}{ccc}& & \alpha \left(\varsigma ,\nu ,\theta \right)+2\pi \left(\varsigma ,\nu ,\theta \right)+\sqrt{2}\kappa \left(\varsigma ,\nu ,\theta \right)+\sqrt{2}\varpi \left(\varsigma ,\nu ,\theta \right)+\sqrt{2}\varkappa \left(\varsigma ,\nu ,\theta \right)\hfill \\ & =& \alpha \left(\varsigma \right)+2\pi \left(\varsigma \right)+\sqrt{2}\kappa \left(\varsigma \right)+\sqrt{2}\varpi \left(\varsigma \right)+\sqrt{2}\varkappa \left(\varsigma \right)<1;\hfill \end{array}$$

(c)

$$\tau \left({\mathfrak{J}}_1\varsigma ,{\mathfrak{J}}_2\nu \right){⪯}_{i_2}\alpha \left(\varsigma \right)\tau \left(\varsigma ,\nu \right)+\pi \left(\varsigma \right)\left(\tau \left(\varsigma ,{\mathfrak{J}}_1\varsigma \right)+\tau \left(\nu ,{\mathfrak{J}}_2\nu \right)\right)$$

$$+\kappa \left(\varsigma \right)\frac{\tau \left(\varsigma ,{\mathfrak{J}}_1\varsigma \right)\tau \left(\nu ,{\mathfrak{J}}_2\nu \right)}{1+\tau \left(\varsigma ,\nu \right)}$$

$$+\varpi \left(\varsigma \right)\frac{\tau \left(\nu ,{\mathfrak{J}}_1\varsigma \right)\tau \left(\varsigma ,{\mathfrak{J}}_2\nu \right)}{1+\tau \left(\varsigma ,\nu \right)}$$

$$+\varkappa \left(\varsigma \right)\frac{\tau \left(\varsigma ,{\mathfrak{J}}_1\varsigma \right)\tau \left(\varsigma ,{\mathfrak{J}}_2\nu \right)+\tau \left(\nu ,{\mathfrak{J}}_2\nu \right)\tau \left(\nu ,{\mathfrak{J}}_1\varsigma \right)}{1+\tau \left(\varsigma ,{\mathfrak{J}}_2\nu \right)+\tau \left(\nu ,{\mathfrak{J}}_1\varsigma \right)}$$

$$\begin{array}{ccc}\hfill & =& \alpha \left(\varsigma ,\nu ,\theta \right)\tau \left(\varsigma ,\nu \right)\hfill \\ & & +\pi \left(\varsigma ,\nu ,\theta \right)\left(\tau \left(\varsigma ,{\mathfrak{J}}_1\varsigma \right)+\tau \left(\nu ,{\mathfrak{J}}_2\nu \right)\right)\hfill \\ & & +\kappa \left(\varsigma ,\nu ,\theta \right)\frac{\tau \left(\varsigma ,{\mathfrak{J}}_1\varsigma \right)\tau \left(\nu ,{\mathfrak{J}}_2\nu \right)}{1+\tau \left(\varsigma ,\nu \right)}\hfill \\ & & +\varpi \left(\varsigma ,\nu ,\theta \right)\frac{\tau \left(\nu ,{\mathfrak{J}}_1\varsigma \right)\tau \left(\varsigma ,{\mathfrak{J}}_2\nu \right)}{1+\tau \left(\varsigma ,\nu \right)}\hfill \\ & & +\varkappa \left(\varsigma ,\nu ,\theta \right)\frac{\tau \left(\varsigma ,{\mathfrak{J}}_1\varsigma \right)\tau \left(\varsigma ,{\mathfrak{J}}_2\nu \right)+\tau \left(\nu ,{\mathfrak{J}}_2\nu \right)\tau \left(\nu ,{\mathfrak{J}}_1\varsigma \right)}{1+\tau \left(\varsigma ,{\mathfrak{J}}_2\nu \right)+\tau \left(\nu ,{\mathfrak{J}}_1\varsigma \right)}.\hfill \end{array}$$

Then, by Theorem 1, there exists a unique ${\varsigma }^{*}\in \mathcal{Q}$ such that ${\mathfrak{J}}_1{\varsigma }^{*}={\mathfrak{J}}_2{\varsigma }^{*}={\varsigma }^{*}.$ □

4. Applications

Let $C[a,b]$ represent the class of all real continuous functions defined on $[a,b]$ and$\tau :C\left([a,b]\right)\times C\left([a,b]\right)\to {\mathbb{C}}_2$ be defined as follows

$$\tau (\varsigma ,\nu )=\underset{t\in [a,b]}{max}\left(1+i\right)\left(\left|\varsigma \left(t\right)-\nu \left(t\right)\right|\right)$$

for all $\varsigma ,\nu \in$$C\left(\right[a,b\left]\right)$ and $t\in [a,b].$ Then, ($C\left(\right[a,b],\mathbb{R}),\tau$) is a complete bi-CVMS. Take the integral equations

$$\varsigma \left(t\right)={\int }_a^b{K}_1(t,s,\varsigma \left(s\right))\tau s+g\left(t\right),$$

(7)

$$\varsigma \left(t\right)={\int }_a^b{K}_2(t,s,\varsigma \left(s\right))\tau s+g\left(t\right),$$

(8)

where $g:$$[a,b]\to \mathbb{R}$ and $K_1,K_2:[a,b]\times [a,b]\times \mathbb{R}\to \mathbb{R}$ are continuous for $t\in [a,b]$. In ${\mathbb{C}}_2,$ we define ${⪯}_{i_2}$ in this way

$$\varsigma \left(t\right){⪯}_{i_2}\nu \left(t\right)⟺\varsigma \le \nu .$$

Theorem 2.

Suppose there exists some fixed element $\theta \in \mathcal{Q}$ such that the following condition

$$\left|K_1(t,s,\varsigma \left(s\right))-K_2(t,s,\nu \left(s\right))\right|\le \alpha \left(\varsigma ,\nu ,\theta \right)\left|\varsigma \left(s\right)-\nu \left(s\right)\right|$$

holds for all $\varsigma ,\nu \in \mathcal{Q}$ with $\varsigma \ne \nu$ and $\alpha :\mathcal{Q}\times \mathcal{Q}\times \mathcal{Q}\to [0,1)$. Then, (7) and (8) have a unique common solution.

Proof. 

Define ${\mathfrak{J}}_1,{\mathfrak{J}}_2:\mathcal{Q}$$\to \mathcal{Q}$ by

$${\mathfrak{J}}_1\varsigma \left(t\right)=\frac{1}{b-a}{\int }_a^b{K}_1(t,s,\varsigma \left(s\right))\tau s+g\left(t\right),$$

$${\mathfrak{J}}_2\varsigma \left(t\right)=\frac{1}{b-a}{\int }_a^b{K}_2(t,s,\varsigma \left(s\right))\tau s+g\left(t\right),$$

for all$t\in [a,b].$ Consider

$$\begin{array}{ccc}\hfill \tau \left({\mathfrak{J}}_1\varsigma ,{\mathfrak{J}}_2\nu \right)& =& \underset{t\in [a,b]}{max}\left(1+i_2\right)\left|{\mathfrak{J}}_1\varsigma \left(t\right)-{\mathfrak{J}}_{2}h\left(t\right)\right|\hfill \\ & =& \underset{t\in [a,b]}{max}\left(1+i_2\right)\left(\frac{1}{b-a}\left|{\int }_a^b{K}_1(t,s,\varsigma \left(s\right))\tau s-{\int }_a^b{K}_2(t,s,h\left(s\right))\tau s\right|\right)\hfill \end{array}$$

$${⪯}_{i_2}\underset{t\in [a,b]}{max}\left(1+i_2\right)\left(\frac{1}{b-a}{\int }_a^b\left|K_1(t,s,\varsigma \left(s\right))-K_2(t,s,h\left(s\right))\right|\tau s\right)$$

$${⪯}_{i_2}\underset{t\in [a,b]}{max}\left(1+i_2\right)\left(\frac{\alpha \left(\varsigma ,\nu ,\theta \right)}{b-a}{\int }_a^b\left|\varsigma \left(s\right)-\nu \left(s\right)\right|\tau s\right).$$

Thus,

$$\tau \left({\mathfrak{J}}_1\varsigma ,{\mathfrak{J}}_2\nu \right){⪯}_{i_2}\alpha \left(\varsigma ,\nu ,\theta \right)\tau (\varsigma ,\nu ).$$

Now, with $\pi ,\kappa ,\varpi ,\varkappa :\mathcal{Q}\times \mathcal{Q}\times \mathcal{Q}\to [0,1)$ defined by

$$\pi \left(\varsigma ,\nu ,\theta \right)=\kappa \left(\varsigma ,\nu ,\theta \right)=\varpi \left(\varsigma ,\nu ,\theta \right)=\varkappa \left(\varsigma ,\nu ,\theta \right)=0$$

for every $\varsigma ,\nu \in \mathcal{Q}$, all the hypotheses of Theorem 1 are fulfilled and the integral Equations (7) and (8) have a unique common solution. □

5. Conclusions

Complex-valued metric spaces and their several generalizations allow us to consider the distances between points in a set, either classically or non-classically. In this draft, we have obtained common fixed-point results for rational contractions involving point-dependent control functions in bi-CVMSs. In this way, we have derived the key results of Beg et al. [8], Gnanaprakasam et al. [9] and Tassaddiq et al. [10] from our results. We apply our result to solve the Fredholm integral equation as an application.

For future work, one can expand the notion of bi-CVMSs to hypercomplex-valued metric spaces. Moreover, the results established in this paper can be lengthened to set-valued mappings. Some integral and differential inclusions can be explored to apply fixed-point theorems for set-valued mappings in the framework of bi-CVMSs.