Fixed Point Theory in Bicomplex Metric Spaces: A New Framework with Applications

Abstract

This paper investigates the existence of common fixed points for mappings satisfying generalized rational type contractive conditions in the framework of bicomplex valued metric spaces. Our findings extend well-established results in the existing literature. As an application of our leading result, we explore the existence and uniqueness of solutions of the Volttera integral equation of the second kind.

1. Introduction

The concept of complex numbers emerged in the 17th century, with significant contributions from Leonhard Euler, a mathematician who introduced the symbol ‘i’ to represent the imaginary unit ($\sqrt{-1}$) with the property $i^2=-1$. While Euler’s notation gained widespread adoption, earlier work by mathematicians like Gottfried Wilhelm Leibniz also played a role in the development of complex analysis.

Building upon complex numbers, bicomplex numbers were introduced by Segre [1] and offer a commutative alternative to quaternions and generalize complex numbers. Price [2] and Waugh [3] provide more detailed explorations of bicomplex numbers. Azam et al. [4] introduced the concept of complex valued metric spaces (CVMSs) as a special case of cone metric spaces, enabling the study of rational expressions not definable in the latter. This concept arises from the limitations of cone metric spaces, which rely on underlying Banach spaces that are not division rings. CVMSs allows for generalizations of fixed point theory results involving divisions. In 2017, Choi et al. [5] combined bicomplex numbers with CVMSs, introducing bicomplex valued metric spaces (bi-CVMSs) and establishing common fixed point theorems. Thereafter, Jebril et al. [6] further explored the application of bi-CVMSs by investigating the existence of common fixed points for mappings exhibiting specific contractive inequalities defined by rational expressions. In due course, Beg et al. [7] contributed by establishing fixed point results using an extrapolation technique. Gnanaprakasam et al. [8] further expanded the field by investigating contractive conditions in bi-CVMSs and their application to solving linear equations. Abdou [9] presented a novel approach to solving systems of Fredholm integral equations. Their method leverages common fixed point theorems established within the context of bi-CVMSs. Recent research in bicomplex valued metric spaces has seen a surge in activity, with works by Gu et al. [10], Tassaddiq et al. [11], Albargi et al. [12], Abdou [13], Gnanaprakasam et al. [14], Gurusamy [15], and Ramaswamy et al. [16] exploring fixed and common fixed point results, applications to integral equations, and other related structures. For a deeper understanding in this direction, we recommend consulting references [17,18,19,20,21,22].

In this research article, we establish common fixed point theorems for mappings satisfying generalized contractive conditions in the framework of bicomplex valued metric spaces. To demonstrate the originality of our main theorem, we present a novel example. Finally, we showcase the applicability of our findings by exploring the existence and uniqueness of solutions for Volterra integral equations of the second kind.

2. Preliminaries

We represent the sets of real numbers, complex numbers, and bicomplex numbers as ${\mathbb{C}}_0,$ ${\mathbb{C}}_1$, and ${\mathbb{C}}_2$, respectively. A bicomplex number, as introduced by Segre [1], can be expressed as

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

where $a_1,a_2,a_3,a_4$ belong to ${\mathbb{C}}_0,$ $i_1$ and $i_2$ are independent imaginary units satisfying $i_1^2=i_2^2=-1$ and $i_1{i}_2=i_2{i}_{1,}$ and ${\mathbb{C}}_2$ is defined as

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

that is,

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

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

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

and the product is

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

There are four idempotent members in ${\mathbb{C}}_2,$ which are $0,1,$ $e_1=\frac{1+i_1{i}_2}{2}$, and $e_2=\frac{1-i_1{i}_2}{2}.$ Notably, $e_1$ and $e_2$ are non-trivial idempotents satisfying $e_1+e_2=1$ and $e_1{e}_2=0.$ This unique property allows any bicomplex number $z_1+i_2{z}_2$ to be expressed as a specific linear combination of $e_1$ and $e_2$, i.e.,

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

This decomposition of $\varrho$ into a linear combination of idempotents ($e_1$ and $e_2$) with complex coefficients ${\varrho }_1$ $=\left(z_1-i_1{z}_2\right)$ and ${\varrho }_2=$ $\left(z_1+i_1{z}_2\right)$ is analogous to the idempotent characterization observed in other mathematical structures. We can refer to ${\varrho }_1$ and ${\varrho }_2$ as the idempotent components of $\varrho .$ A bicomplex number $\varrho$ denoted by $\varrho =z_1+i_2{z}_2$ is considered invertible within ${\mathbb{C}}_2$ if there exists another bicomplex number ħ such that their product $\varrho ħ$ equals the identity element $1.$ This element ħ is then recognized as the multiplicative inverse of $\varrho .$ Also, $\varrho$ is recognized as the multiplicative inverse of $ħ.$

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

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

Zero is the at-most member in ${\mathbb{C}}_0$ that does not possess a multiplicative inverse, and in ${\mathbb{C}}_1$, $0=0+i0$ is the at-most member that does not possess a multiplicative inverse. We represent the set of singular members of ${\mathbb{C}}_0$ and ${\mathbb{C}}_1$ by ${\aleph }_0$ and ${\aleph }_1$ in this order. There are many members in ${\mathbb{C}}_2$ that do not have a multiplicative inverse. We represent this set by ${\aleph }_2$, and evidently, ${\aleph }_0$ $={\aleph }_1\subset {\aleph }_2.$

A bicomplex number $\varrho =a_1+a_2{i}_1+a_3{i}_2+a_4{i}_1{i}_2\in {\mathbb{C}}_2$ 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. In this way ${\varrho }^{-1}$ exists, and it is degenerated. Define a norm $∥·∥:{\mathbb{C}}_2\to {\mathbb{C}}_0$ as follows:

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

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

The space ${\mathbb{C}}_2$ equipped with the norm $∥·∥$ is a Banach space. If $\varrho ,ħ\in {\mathbb{C}}_2,$ then

$$∥\varrho ħ∥\le \sqrt{2}∥\varrho ∥∥ħ∥$$

holds instead of

$$∥\varrho ħ∥\le ∥\varrho ∥∥ħ∥.$$

Therefore, ${\mathbb{C}}_2$ is not a Banach algebra. Let $\varrho =z_1+i_2{z}_2,$ $ħ={\omega }_1+i_2{\omega }_2\in {\mathbb{C}}_2;$ then we define

$$\varrho {⪯}_{i_2}ħ\iff \mathrm{Re}\left(z_1\right)⪯\mathrm{Re}\left({\omega }_1\right)\mathrm{and}\mathrm{Im}\left(z_2\right)⪯\mathrm{Im}\left({\omega }_2\right).$$

If

$$\varrho {⪯}_{i_2}ħ$$

then one of the followings conditions holds:

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

Specifically, $\varrho {⋨}_{i_2}ħ$ if $\varrho {\prec }_{i_2}ħ$ and $\varrho \ne ħ$; that is, one of (i), (ii), or (iii) holds. Also, $\varrho {\prec }_{i_2}ħ$ if only condition (iii) is satisfied. For $\varrho ,$ $ħ\in {\mathbb{C}}_2,$ we can prove the following:

(i)

$\varrho {⪯}_{i_2}ħ⟹∥\varrho ∥\le ∥ħ∥;$

(ii)

$∥\varrho +ħ∥\le ∥\varrho ∥+∥ħ∥;$

(iii)

$∥a\varrho ∥\le a∥\varrho ∥,$ where a is a non-negative real number;

(iv)

$∥\varrho ħ∥\le \sqrt{2}∥\varrho ∥∥ħ∥;$

(v)

$∥{\varrho }^{-1}∥={∥\varrho ∥}^{-1}$ if $\varrho$ is a degenerated bicomplex number with $0\prec \varrho ;$

(vi)

$∥\frac{\varrho }{ħ}∥=\frac{∥\varrho ∥}{∥ħ∥}$ if ħ is a degenerated bicomplex number.

In their work, Azam et al. [4] introduced the concept of a complex valued metric space (CVMS).

Definition 1 

([4]). Let $\mathcal{A}$ be a non-empty set and ⪯ be a partial order defined on $\mathcal{A}$. Additionally, let $d:\mathcal{A}\times \mathcal{A}\to {\mathbb{C}}_1$ be a mapping satisfying

(i) 

$0⪯d(\mathfrak{q},\kappa )$ and $d(\mathfrak{q},\kappa )=0$ if and only if $\mathfrak{q}=\kappa$;

(ii) 

$d(\mathfrak{q},\kappa )=d(\kappa ,\mathfrak{q});$

(iii) 

$d(\mathfrak{q},\kappa )⪯d(\mathfrak{q},\nu )+d(\nu ,\kappa )$ for all $\mathfrak{q},\kappa ,\nu \in \mathcal{A}.$

Then, $(\mathcal{A},⪯,d)$ is a CVMS.

Choi et al. [5] introduced the concept of a bicomplex valued metric space (bi-CVMS) with the following definition:

Definition 2 

([5]). Let $\mathcal{A}\ne \varnothing$, ${⪯}_{i_2}$ be a partial order defined on $\mathcal{A}$, and $d:\mathcal{A}\times \mathcal{A}\to {\mathbb{C}}_2$ be a mapping satisfying

(i) 

$0{⪯}_{i_2}d(\mathfrak{q},\kappa )$ and $d(\mathfrak{q},\kappa )=0$ if and only if $\mathfrak{q}=\kappa$;

(ii) 

$d(\mathfrak{q},\kappa )=d(\kappa ,\mathfrak{q});$

(iii) 

$d(\mathfrak{q},\kappa ){⪯}_{i_2}d(\mathfrak{q},\nu )+d(\nu ,\kappa )$ for all $\mathfrak{q},\kappa ,\nu \in \mathcal{A}.$

Then, $(\mathcal{A},{⪯}_{i_2},d)$ is a bi-CVMS.

Example 1 

([5]). Let $\mathcal{A}={\mathbb{C}}_2$ and $\mathfrak{q},\kappa \in \mathcal{A}.$ Define $d:\mathcal{A}\times \mathcal{A}\to {\mathbb{C}}_2$ by

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

where $\mathfrak{q}=z_1+i_2{z}_2$ and $\kappa ={\omega }_1+i_2{\omega }_2\in {\mathbb{C}}_2.$ Then $(\mathcal{A},{⪯}_{i_2},d)$ is a bi-CVMS.

Definition 3 

([5]). For a bi-CVMS $(\mathcal{A},{⪯}_{i_2},d):$

(i) 

A sequence $\left\{{\mathfrak{q}}_r\right\}$ $\subseteq \mathcal{A}$ is said to be a convergent sequence and converges to a point $\mathfrak{q}$ if for any $0{\prec }_{i_2}\ell \in {\mathbb{C}}_2$ there is a natural number $r_0\in \mathbb{N}$ such that $d({\mathfrak{q}}_r,\mathfrak{q}){\prec }_{i_2}\ell$ for all $r>r_0$, and we write ${lim}_{r\to \infty }{\mathfrak{q}}_r=\mathfrak{q}$ or ${\mathfrak{q}}_r\to \mathfrak{q}$ as $r\to \infty .$

(ii) 

A sequence $\left\{{\mathfrak{q}}_r\right\}$ $\subseteq \mathcal{A}$ is said to be a Cauchy sequence in $(\mathcal{A},d)$ if for any $0{\prec }_{i_2}\ell \in {\mathbb{C}}_2$ there is a natural number $r_0\in \mathbb{N}$ such that $d({\mathfrak{q}}_r,{\mathfrak{q}}_{r+m}){\prec }_{i_2}\ell$ for all $r,m\in \mathbb{N}$ and $r>r_0.$

(iii) 

If every Cauchy sequence in $\mathcal{A}$ is convergent in $\mathcal{A}$, then $(\mathcal{A},{⪯}_{i_2},d)$ is said to be a complete bi-CVMS.

Lemma 1 

([7]). Let $(\mathcal{A},{⪯}_{i_2},d)$ be a bi-CVMS and let $\left\{{\mathfrak{q}}_r\right\}$ $\subseteq \mathcal{A}$. Then $\left\{{\mathfrak{q}}_r\right\}$ converges to $\mathfrak{q}$ if and only if $∥d({\mathfrak{q}}_r,\mathfrak{q})∥\to 0$ as $r\to \infty .$

Lemma 2 

([7]). Let $(\mathcal{A},{⪯}_{i_2},d)$ be a bi-CVMS and let $\left\{{\mathfrak{q}}_r\right\}$ $\subseteq \mathcal{A}$. Then $\left\{{\mathfrak{q}}_r\right\}$ is a Cauchy sequence if and only if $∥d({\mathfrak{q}}_r,{\mathfrak{q}}_{r+m})∥\to 0$ as $r\to \infty ,$ where $m\in \mathbb{N}.$

3. Main Result

In the following, we state a Proposition from Tassdique et al. [11], which will be instrumental in our subsequent results.

Proposition 1 

([11]). Let $(\mathcal{A},{⪯}_{i_2},d)$ be a bi-CVMS and $\mathcal{V},\mathcal{W}:(\mathcal{A},{⪯}_{i_2},d)\to (\mathcal{A},{⪯}_{i_2},d).$ Let ${\mathfrak{q}}_0\in$ $\mathcal{A}.$ Define the sequence {${\mathfrak{q}}_r$} by

$${\mathfrak{q}}_{2r+1}=\mathcal{V}{\mathfrak{q}}_{2r}\mathit{and}{\mathfrak{q}}_{2r+2}=\mathcal{W}{\mathfrak{q}}_{2r+1}$$

(1)

for all $r=0,1,2,\dots$

Assume that there exists ${\mu }_1:\mathcal{A}\times \mathcal{A}\to [0,1)$ satisfying

$${\mu }_1\left(\mathcal{WV}\mathfrak{q},\kappa \right)\le {\mu }_1\left(\mathfrak{q},\kappa \right)\mathit{and}{\mu }_1\left(\mathfrak{q},\mathcal{VW}\kappa \right)\le {\mu }_1\left(\mathfrak{q},\kappa \right)$$

for all $\mathfrak{q},\kappa \in \mathcal{A}.$ Then

$${\mu }_1\left({\mathfrak{q}}_{2r},\kappa \right)\le {\mu }_1\left({\mathfrak{q}}_0,\kappa \right)\mathit{and}{\mu }_1\left(\mathfrak{q},{\mathfrak{q}}_{2r+1}\right)\le {\mu }_1\left(\mathfrak{q},{\mathfrak{q}}_1\right)$$

for all $\mathfrak{q},\kappa \in \mathcal{A}$ and $r=0,1,2,\dots$

Theorem 1. 

Let $(\mathcal{A},{⪯}_{i_2},d)$ be a complete bi-CVMS and $\mathcal{V},\mathcal{W}:\mathcal{A}$ $\to \mathcal{A}$. If there exist the mappings ${\mu }_i:\mathcal{A}\times \mathcal{A}\to [0,1),$ where $i=1,2,\dots 7$, such that

(a) 

${\mu }_i\left(\mathcal{WV}\mathfrak{q},\kappa \right)\le {\mu }_i\left(\mathfrak{q},\kappa \right)$ and ${\mu }_i\left(\mathfrak{q},\mathcal{VW}\kappa \right)\le {\mu }_i\left(\mathfrak{q},\kappa \right);$

(b) 

${\mu }_1\left(\mathfrak{q},\kappa \right)+\sqrt{2}\left({\mu }_2\left(\mathfrak{q},\kappa \right)+{\mu }_3\left(\mathfrak{q},\kappa \right)\right)+2\sqrt{2}\left({\mu }_4\left(\mathfrak{q},\kappa \right)+{\mu }_5\left(\mathfrak{q},\kappa \right)\right)+2\left({\mu }_6\left(\mathfrak{q},\kappa \right)+{\mu }_7\left(\mathfrak{q},\kappa \right)\right)<1;$

(c) 

$$\begin{array}{c}d\left(\mathcal{V}\mathfrak{q},\mathcal{W}\kappa \right){⪯}_{i_2}{\mu }_1\left(\mathfrak{q},\kappa \right)d\left(\mathfrak{q},\kappa \right)+{\mu }_2\left(\mathfrak{q},\kappa \right)\frac{d\left(\mathfrak{q},\mathcal{V}\mathfrak{q}\right)d\left(\kappa ,\mathcal{W}\kappa \right)}{1+d\left(\mathfrak{q},\kappa \right)}+{\mu }_3\left(\mathfrak{q},\kappa \right)\frac{d\left(\kappa ,\mathcal{V}\mathfrak{q}\right)d\left(\mathfrak{q},\mathcal{W}\kappa \right)}{1+d\left(\mathfrak{q},\kappa \right)}\\ +{\mu }_4\left(\mathfrak{q},\kappa \right)\frac{d\left(\mathfrak{q},\mathcal{V}\mathfrak{q}\right)d\left(\mathfrak{q},\mathcal{W}\kappa \right)}{1+d\left(\mathfrak{q},\kappa \right)}+{\mu }_5\left(\mathfrak{q},\kappa \right)\frac{d\left(\kappa ,\mathcal{V}\mathfrak{q}\right)d\left(\kappa ,\mathcal{W}\kappa \right)}{1+d\left(\mathfrak{q},\kappa \right)}\\ +{\mu }_6\left(\mathfrak{q},\kappa \right)\frac{d\left(\mathfrak{q},\mathcal{W}\kappa \right)\left[1+d\left(\mathfrak{q},\mathcal{V}\mathfrak{q}\right)\right]}{1+d\left(\mathfrak{q},\kappa \right)}+{\mu }_7\left(\mathfrak{q},\kappa \right)\frac{d\left(\kappa ,\mathcal{V}\mathfrak{q}\right)\left[1+d\left(\kappa ,\mathcal{W}\kappa \right)\right]}{1+d\left(\mathfrak{q},\kappa \right)}\end{array}$$

(2)

for all $\mathfrak{q},\kappa \in$ $\mathcal{A},$ then $\mathcal{V}$ and $\mathcal{W}$ possess a unique common fixed point.

Proof. 

Consider an arbitrary element ${\mathfrak{q}}_0$ belonging to the set $\mathcal{A}$. Define a sequence {${\mathfrak{q}}_r$} as follows:

$${\mathfrak{q}}_{2r+1}=\mathcal{V}{\mathfrak{q}}_{2r}\mathrm{and}{\mathfrak{q}}_{2r+2}=\mathcal{W}{\mathfrak{q}}_{2r+1}$$

for all $r=0,1,2,\dots$. By (2), we have

$$\begin{array}{cc}d\left({\mathfrak{q}}_{2r+1},{\mathfrak{q}}_{2r+2}\right)=\hfill & d\left(\mathcal{V}{\mathfrak{q}}_{2r},\mathcal{W}{\mathfrak{q}}_{2r+1}\right){⪯}_{i_2}{\mu }_1\left({\mathfrak{q}}_{2r},{\mathfrak{q}}_{2r+1}\right)d\left({\mathfrak{q}}_{2r},{\mathfrak{q}}_{2r+1}\right)\hfill \\ & +{\mu }_2\left({\mathfrak{q}}_{2r},{\mathfrak{q}}_{2r+1}\right)\frac{d\left({\mathfrak{q}}_{2r},\mathcal{V}{\mathfrak{q}}_{2r}\right)d\left({\mathfrak{q}}_{2r+1},\mathcal{W}{\mathfrak{q}}_{2r+1}\right)}{1+d\left({\mathfrak{q}}_{2r},{\mathfrak{q}}_{2r+1}\right)}\hfill \\ & +{\mu }_3\left({\mathfrak{q}}_{2r},{\mathfrak{q}}_{2r+1}\right)\frac{d\left({\mathfrak{q}}_{2r+1},\mathcal{V}{\mathfrak{q}}_{2r}\right)d\left({\mathfrak{q}}_{2r},\mathcal{W}{\mathfrak{q}}_{2r+1}\right)}{1+d\left({\mathfrak{q}}_{2r},{\mathfrak{q}}_{2r+1}\right)}\hfill \\ & +{\mu }_4\left({\mathfrak{q}}_{2r},{\mathfrak{q}}_{2r+1}\right)\frac{d\left({\mathfrak{q}}_{2r},\mathcal{V}{\mathfrak{q}}_{2r}\right)d\left({\mathfrak{q}}_{2r},\mathcal{W}{\mathfrak{q}}_{2r+1}\right)}{1+d\left({\mathfrak{q}}_{2r},{\mathfrak{q}}_{2r+1}\right)}\hfill \\ & +{\mu }_5\left({\mathfrak{q}}_{2r},{\mathfrak{q}}_{2r+1}\right)\frac{d\left({\mathfrak{q}}_{2r+1},\mathcal{V}{\mathfrak{q}}_{2r}\right)d\left({\mathfrak{q}}_{2r+1},\mathcal{W}{\mathfrak{q}}_{2r+1}\right)}{1+d\left({\mathfrak{q}}_{2r},{\mathfrak{q}}_{2r+1}\right)}\hfill \\ & +{\mu }_6\left({\mathfrak{q}}_{2r},{\mathfrak{q}}_{2r+1}\right)\frac{d\left({\mathfrak{q}}_{2r},\mathcal{W}{\mathfrak{q}}_{2r+1}\right)\left[1+d\left({\mathfrak{q}}_{2r},\mathcal{V}{\mathfrak{q}}_{2r}\right)\right]}{1+d\left({\mathfrak{q}}_{2r},{\mathfrak{q}}_{2r+1}\right)}\hfill \\ & +{\mu }_7\left({\mathfrak{q}}_{2r},{\mathfrak{q}}_{2r+1}\right)\frac{d\left({\mathfrak{q}}_{2r+1},\mathcal{V}{\mathfrak{q}}_{2r}\right)\left[1+d\left({\mathfrak{q}}_{2r+1},\mathcal{W}{\mathfrak{q}}_{2r+1}\right)\right]}{1+d\left({\mathfrak{q}}_{2r},{\mathfrak{q}}_{2r+1}\right)}\hfill \\ \hfill =& {\mu }_1\left({\mathfrak{q}}_{2r},{\mathfrak{q}}_{2r+1}\right)d\left({\mathfrak{q}}_{2r},{\mathfrak{q}}_{2r+1}\right)\hfill \\ & +{\mu }_2\left({\mathfrak{q}}_{2r},{\mathfrak{q}}_{2r+1}\right)\frac{d\left({\mathfrak{q}}_{2r},{\mathfrak{q}}_{2r+1}\right)d\left({\mathfrak{q}}_{2r+1},{\mathfrak{q}}_{2r+2}\right)}{1+d\left({\mathfrak{q}}_{2r},{\mathfrak{q}}_{2r+1}\right)}\hfill \\ & +{\mu }_3\left({\mathfrak{q}}_{2r},{\mathfrak{q}}_{2r+1}\right)\frac{d\left({\mathfrak{q}}_{2r+1},{\mathfrak{q}}_{2r+1}\right)d\left({\mathfrak{q}}_{2r},{\mathfrak{q}}_{2r+2}\right)}{1+d\left({\mathfrak{q}}_{2r},{\mathfrak{q}}_{2r+1}\right)}\hfill \\ & +{\mu }_4\left({\mathfrak{q}}_{2r},{\mathfrak{q}}_{2r+1}\right)\frac{d\left({\mathfrak{q}}_{2r},{\mathfrak{q}}_{2r+1}\right)d\left({\mathfrak{q}}_{2r},{\mathfrak{q}}_{2r+2}\right)}{1+d\left({\mathfrak{q}}_{2r},{\mathfrak{q}}_{2r+1}\right)}\hfill \\ & +{\mu }_5\left({\mathfrak{q}}_{2r},{\mathfrak{q}}_{2r+1}\right)\frac{d\left({\mathfrak{q}}_{2r+1},{\mathfrak{q}}_{2r+1}\right)d\left({\mathfrak{q}}_{2r+1},{\mathfrak{q}}_{2r+2}\right)}{1+d\left({\mathfrak{q}}_{2r},{\mathfrak{q}}_{2r+1}\right)}\hfill \\ & +{\mu }_6\left({\mathfrak{q}}_{2r},{\mathfrak{q}}_{2r+1}\right)\frac{d\left({\mathfrak{q}}_{2r},{\mathfrak{q}}_{2r+2}\right)\left[1+d\left({\mathfrak{q}}_{2r},{\mathfrak{q}}_{2r+1}\right)\right]}{1+d\left({\mathfrak{q}}_{2r},{\mathfrak{q}}_{2r+1}\right)}\hfill \\ & +{\mu }_7\left({\mathfrak{q}}_{2r},{\mathfrak{q}}_{2r+1}\right)\frac{d\left({\mathfrak{q}}_{2r+1},{\mathfrak{q}}_{2r+1}\right)\left[1+d\left({\mathfrak{q}}_{2r+1},{\mathfrak{q}}_{2r+2}\right)\right]}{1+d\left({\mathfrak{q}}_{2r},{\mathfrak{q}}_{2r+1}\right)},\hfill \end{array}$$

which implies

$$\begin{array}{c}d\left({\mathfrak{q}}_{2r+1},{\mathfrak{q}}_{2r+2}\right){⪯}_{i_2}{\mu }_1\left({\mathfrak{q}}_{2r},{\mathfrak{q}}_{2r+1}\right)d\left({\mathfrak{q}}_{2r},{\mathfrak{q}}_{2r+1}\right)\hfill \\ +{\mu }_2\left({\mathfrak{q}}_{2r},{\mathfrak{q}}_{2r+1}\right)\frac{d\left({\mathfrak{q}}_{2r},{\mathfrak{q}}_{2r+1}\right)d\left({\mathfrak{q}}_{2r+1},{\mathfrak{q}}_{2r+2}\right)}{1+d\left({\mathfrak{q}}_{2r},{\mathfrak{q}}_{2r+1}\right)}\hfill \\ +{\mu }_4\left({\mathfrak{q}}_{2r},{\mathfrak{q}}_{2r+1}\right)\frac{d\left({\mathfrak{q}}_{2r},{\mathfrak{q}}_{2r+1}\right)d\left({\mathfrak{q}}_{2r},{\mathfrak{q}}_{2r+2}\right)}{1+d\left({\mathfrak{q}}_{2r},{\mathfrak{q}}_{2r+1}\right)}\hfill \\ +{\mu }_6\left({\mathfrak{q}}_{2r},{\mathfrak{q}}_{2r+1}\right)d\left({\mathfrak{q}}_{2r},{\mathfrak{q}}_{2r+2}\right).\hfill \end{array}$$

This yields

$$\begin{array}{ccc}\hfill ∥d\left({\mathfrak{q}}_{2r+1},{\mathfrak{q}}_{2r+2}\right)∥& \le & {\mu }_1\left({\mathfrak{q}}_{2r},{\mathfrak{q}}_{2r+1}\right)∥d\left({\mathfrak{q}}_{2r},{\mathfrak{q}}_{2r+1}\right)∥\hfill \\ & & +{\mu }_2\left({\mathfrak{q}}_{2r},{\mathfrak{q}}_{2r+1}\right)∥\frac{d\left({\mathfrak{q}}_{2r},{\mathfrak{q}}_{2r+1}\right)d\left({\mathfrak{q}}_{2r+1},{\mathfrak{q}}_{2r+2}\right)}{1+d\left({\mathfrak{q}}_{2r},{\mathfrak{q}}_{2r+1}\right)}∥\hfill \\ & & +{\mu }_4\left({\mathfrak{q}}_{2r},{\mathfrak{q}}_{2r+1}\right)∥\frac{d\left({\mathfrak{q}}_{2r},{\mathfrak{q}}_{2r+1}\right)d\left({\mathfrak{q}}_{2r},{\mathfrak{q}}_{2r+2}\right)}{1+d\left({\mathfrak{q}}_{2r},{\mathfrak{q}}_{2r+1}\right)}∥\hfill \\ & & +{\mu }_6\left({\mathfrak{q}}_{2r},{\mathfrak{q}}_{2r+1}\right)∥d\left({\mathfrak{q}}_{2r},{\mathfrak{q}}_{2r+1}\right)+d\left({\mathfrak{q}}_{2r+1},{\mathfrak{q}}_{2r+2}\right)∥\hfill \end{array}$$

which implies

$$\begin{array}{ccc}\hfill ∥d\left({\mathfrak{q}}_{2r+1},{\mathfrak{q}}_{2r+2}\right)∥& \le & {\mu }_1\left({\mathfrak{q}}_{2r},{\mathfrak{q}}_{2r+1}\right)∥d\left({\mathfrak{q}}_{2r},{\mathfrak{q}}_{2r+1}\right)∥\hfill \\ & & +\sqrt{2}{\mu }_2\left({\mathfrak{q}}_{2r},{\mathfrak{q}}_{2r+1}\right)∥\frac{d\left({\mathfrak{q}}_{2r},{\mathfrak{q}}_{2r+1}\right)}{1+d\left({\mathfrak{q}}_{2r},{\mathfrak{q}}_{2r+1}\right)}∥∥d\left({\mathfrak{q}}_{2r+1},{\mathfrak{q}}_{2r+2}\right)∥\hfill \\ & & +\sqrt{2}{\mu }_4\left({\mathfrak{q}}_{2r},{\mathfrak{q}}_{2r+1}\right)∥\frac{d\left({\mathfrak{q}}_{2r},{\mathfrak{q}}_{2r+1}\right)}{1+d\left({\mathfrak{q}}_{2r},{\mathfrak{q}}_{2r+1}\right)}∥∥d\left({\mathfrak{q}}_{2r},{\mathfrak{q}}_{2r+1}\right)+d\left({\mathfrak{q}}_{2r+1},{\mathfrak{q}}_{2r+2}\right)∥\hfill \\ & & +{\mu }_6\left({\mathfrak{q}}_{2r},{\mathfrak{q}}_{2r+1}\right)∥d\left({\mathfrak{q}}_{2r},{\mathfrak{q}}_{2r+1}\right)+d\left({\mathfrak{q}}_{2r+1},{\mathfrak{q}}_{2r+2}\right)∥\hfill \\ & \le & {\mu }_1\left({\mathfrak{q}}_{2r},{\mathfrak{q}}_{2r+1}\right)∥d\left({\mathfrak{q}}_{2r},{\mathfrak{q}}_{2r+1}\right)∥\hfill \\ & & +\sqrt{2}{\mu }_2\left({\mathfrak{q}}_{2r},{\mathfrak{q}}_{2r+1}\right)∥d\left({\mathfrak{q}}_{2r+1},{\mathfrak{q}}_{2r+2}\right)∥\hfill \\ & & +\sqrt{2}{\mu }_4\left({\mathfrak{q}}_{2r},{\mathfrak{q}}_{2r+1}\right)∥d\left({\mathfrak{q}}_{2r},{\mathfrak{q}}_{2r+1}\right)+d\left({\mathfrak{q}}_{2r+1},{\mathfrak{q}}_{2r+2}\right)∥\hfill \\ & & +{\mu }_6\left({\mathfrak{q}}_{2r},{\mathfrak{q}}_{2r+1}\right)∥d\left({\mathfrak{q}}_{2r},{\mathfrak{q}}_{2r+1}\right)+d\left({\mathfrak{q}}_{2r+1},{\mathfrak{q}}_{2r+2}\right)∥.\hfill \end{array}$$

This implies

$$\begin{array}{ccc}\hfill ∥d\left({\mathfrak{q}}_{2r+1},{\mathfrak{q}}_{2r+2}\right)∥& \le & {\mu }_1\left({\mathfrak{q}}_{2r},{\mathfrak{q}}_{2r+1}\right)∥d\left({\mathfrak{q}}_{2r},{\mathfrak{q}}_{2r+1}\right)∥\hfill \\ & & +\sqrt{2}{\mu }_2\left({\mathfrak{q}}_{2r},{\mathfrak{q}}_{2r+1}\right)∥d\left({\mathfrak{q}}_{2r+1},{\mathfrak{q}}_{2r+2}\right)∥\hfill \\ & & +\sqrt{2}{\mu }_4\left({\mathfrak{q}}_{2r},{\mathfrak{q}}_{2r+1}\right)∥d\left({\mathfrak{q}}_{2r},{\mathfrak{q}}_{2r+1}\right)∥\hfill \\ & & +\sqrt{2}{\mu }_4\left({\mathfrak{q}}_{2r},{\mathfrak{q}}_{2r+1}\right)∥d\left({\mathfrak{q}}_{2r+1},{\mathfrak{q}}_{2r+2}\right)∥\hfill \\ & & +{\mu }_6\left({\mathfrak{q}}_{2r},{\mathfrak{q}}_{2r+1}\right)∥d\left({\mathfrak{q}}_{2r},{\mathfrak{q}}_{2r+1}\right)∥\hfill \\ & & +{\mu }_6\left({\mathfrak{q}}_{2r},{\mathfrak{q}}_{2r+1}\right)∥d\left({\mathfrak{q}}_{2r+1},{\mathfrak{q}}_{2r+2}\right)∥\hfill \end{array}$$

By Proposition 1, we have

$$\begin{array}{ccc}\hfill ∥d\left({\mathfrak{q}}_{2r+1},{\mathfrak{q}}_{2r+2}\right)∥& \le & {\mu }_1\left({\mathfrak{q}}_0,{\mathfrak{q}}_1\right)∥d\left({\mathfrak{q}}_{2r},{\mathfrak{q}}_{2r+1}\right)∥\hfill \\ & & +\sqrt{2}{\mu }_2\left({\mathfrak{q}}_0,{\mathfrak{q}}_1\right)∥d\left({\mathfrak{q}}_{2r+1},{\mathfrak{q}}_{2r+2}\right)∥\hfill \\ & & +\sqrt{2}{\mu }_4\left({\mathfrak{q}}_0,{\mathfrak{q}}_1\right)∥d\left({\mathfrak{q}}_{2r},{\mathfrak{q}}_{2r+1}\right)∥\hfill \\ & & +\sqrt{2}{\mu }_4\left({\mathfrak{q}}_0,{\mathfrak{q}}_1\right)∥d\left({\mathfrak{q}}_{2r+1},{\mathfrak{q}}_{2r+2}\right)∥\hfill \\ & & +{\mu }_6\left({\mathfrak{q}}_0,{\mathfrak{q}}_1\right)∥d\left({\mathfrak{q}}_{2r},{\mathfrak{q}}_{2r+1}\right)∥\hfill \\ & & +{\mu }_6\left({\mathfrak{q}}_0,{\mathfrak{q}}_1\right)∥d\left({\mathfrak{q}}_{2r+1},{\mathfrak{q}}_{2r+2}\right)∥\hfill \end{array}$$

which implies

$$∥d\left({\mathfrak{q}}_{2r+1},{\mathfrak{q}}_{2r+2}\right)∥\le \frac{{\mu }_1\left({\mathfrak{q}}_0,{\mathfrak{q}}_1\right)+\sqrt{2}{\mu }_4\left({\mathfrak{q}}_0,{\mathfrak{q}}_1\right)+{\mu }_6\left({\mathfrak{q}}_0,{\mathfrak{q}}_1\right)}{1-\sqrt{2}{\mu }_2\left({\mathfrak{q}}_0,{\mathfrak{q}}_1\right)-\sqrt{2}{\mu }_4\left({\mathfrak{q}}_0,{\mathfrak{q}}_1\right)-{\mu }_6\left({\mathfrak{q}}_0,{\mathfrak{q}}_1\right)}∥d\left({\mathfrak{q}}_{2r},{\mathfrak{q}}_{2r+1}\right)∥.$$

(3)

Similarly, by (2), we have

$$\begin{array}{cc}\hfill d\left({\mathfrak{q}}_{2r+3},{\mathfrak{q}}_{2r+2}\right)=& d\left(\mathcal{VW}{\mathfrak{q}}_{2r+1},\mathcal{W}{\mathfrak{q}}_{2r+1}\right){⪯}_{i_2}{\mu }_1\left(\mathcal{W}{\mathfrak{q}}_{2r+1},{\mathfrak{q}}_{2r+1}\right)d\left(\mathcal{W}{\mathfrak{q}}_{2r+1},{\mathfrak{q}}_{2r+1}\right)\hfill \\ & +{\mu }_2\left(\mathcal{W}{\mathfrak{q}}_{2r+1},{\mathfrak{q}}_{2r+1}\right)\frac{d\left(\mathcal{W}{\mathfrak{q}}_{2r+1},\mathcal{VW}{\mathfrak{q}}_{2r+1}\right)d\left({\mathfrak{q}}_{2r+1},\mathcal{W}{\mathfrak{q}}_{2r+1}\right)}{1+d\left(\mathcal{W}{\mathfrak{q}}_{2r+1},{\mathfrak{q}}_{2r+1}\right)}\hfill \\ & +{\mu }_3\left(\mathcal{W}{\mathfrak{q}}_{2r+1},{\mathfrak{q}}_{2r+1}\right)\frac{d\left({\mathfrak{q}}_{2r+1},\mathcal{VW}{\mathfrak{q}}_{2r+1}\right)d\left(\mathcal{W}{\mathfrak{q}}_{2r+1},\mathcal{W}{\mathfrak{q}}_{2r+1}\right)}{1+d\left(\mathcal{W}{\mathfrak{q}}_{2r+1},{\mathfrak{q}}_{2r+1}\right)}\hfill \\ & +{\mu }_4\left(\mathcal{W}{\mathfrak{q}}_{2r+1},{\mathfrak{q}}_{2r+1}\right)\frac{d\left(\mathcal{W}{\mathfrak{q}}_{2r+1},\mathcal{VW}{\mathfrak{q}}_{2r+1}\right)d\left(\mathcal{W}{\mathfrak{q}}_{2r+1},\mathcal{W}{\mathfrak{q}}_{2r+1}\right)}{1+d\left(\mathcal{W}{\mathfrak{q}}_{2r+1},{\mathfrak{q}}_{2r+1}\right)}\hfill \\ & +{\mu }_5\left(\mathcal{W}{\mathfrak{q}}_{2r+1},{\mathfrak{q}}_{2r+1}\right)\frac{d\left({\mathfrak{q}}_{2r+1},\mathcal{VW}{\mathfrak{q}}_{2r+1}\right)d\left({\mathfrak{q}}_{2r+1},\mathcal{W}{\mathfrak{q}}_{2r+1}\right)}{1+d\left(\mathcal{W}{\mathfrak{q}}_{2r+1},{\mathfrak{q}}_{2r+1}\right)}\hfill \\ & +{\mu }_6\left(\mathcal{W}{\mathfrak{q}}_{2r+1},{\mathfrak{q}}_{2r+1}\right)\frac{d\left(\mathcal{W}{\mathfrak{q}}_{2r+1},\mathcal{W}{\mathfrak{q}}_{2r+1}\right)\left[1+d\left(\mathcal{W}{\mathfrak{q}}_{2r+1},\mathcal{VW}{\mathfrak{q}}_{2r+1}\right)\right]}{1+d\left(\mathcal{W}{\mathfrak{q}}_{2r+1},{\mathfrak{q}}_{2r+1}\right)}\hfill \\ & +{\mu }_7\left(\mathcal{W}{\mathfrak{q}}_{2r+1},{\mathfrak{q}}_{2r+1}\right)\frac{d\left({\mathfrak{q}}_{2r+1},\mathcal{VW}{\mathfrak{q}}_{2r+1}\right)\left[1+d\left({\mathfrak{q}}_{2r+1},\mathcal{W}{\mathfrak{q}}_{2r+1}\right)\right]}{1+d\left(\mathcal{W}{\mathfrak{q}}_{2r+1},{\mathfrak{q}}_{2r+1}\right)}\hfill \\ \hfill =& {\mu }_1\left({\mathfrak{q}}_{2r+1},{\mathfrak{q}}_{2r+2}\right)d\left({\mathfrak{q}}_{2r+2},{\mathfrak{q}}_{2r+1}\right)\hfill \\ & +{\mu }_2\left({\mathfrak{q}}_{2r+1},{\mathfrak{q}}_{2r+2}\right)\frac{d\left({\mathfrak{q}}_{2r+2},{\mathfrak{q}}_{2r+3}\right)d\left({\mathfrak{q}}_{2r+1},{\mathfrak{q}}_{2r+2}\right)}{1+d\left({\mathfrak{q}}_{2r+2},{\mathfrak{q}}_{2r+1}\right)}\hfill \\ & +{\mu }_3\left({\mathfrak{q}}_{2r+1},{\mathfrak{q}}_{2r+2}\right)\frac{d\left({\mathfrak{q}}_{2r+1},{\mathfrak{q}}_{2r+3}\right)d\left({\mathfrak{q}}_{2r+2},{\mathfrak{q}}_{2r+2}\right)}{1+d\left({\mathfrak{q}}_{2r+2},{\mathfrak{q}}_{2r+1}\right)}\hfill \\ & +{\mu }_4\left({\mathfrak{q}}_{2r+1},{\mathfrak{q}}_{2r+2}\right)\frac{d\left({\mathfrak{q}}_{2r+2},{\mathfrak{q}}_{2r+3}\right)d\left({\mathfrak{q}}_{2r+2},{\mathfrak{q}}_{2r+2}\right)}{1+d\left({\mathfrak{q}}_{2r+2},{\mathfrak{q}}_{2r+1}\right)}\hfill \\ & +{\mu }_5\left({\mathfrak{q}}_{2r+1},{\mathfrak{q}}_{2r+2}\right)\frac{d\left({\mathfrak{q}}_{2r+1},{\mathfrak{q}}_{2r+3}\right)d\left({\mathfrak{q}}_{2r+1},{\mathfrak{q}}_{2r+2}\right)}{1+d\left({\mathfrak{q}}_{2r+2},{\mathfrak{q}}_{2r+1}\right)}\hfill \\ & +{\mu }_6\left({\mathfrak{q}}_{2r+1},{\mathfrak{q}}_{2r+2}\right)\frac{d\left({\mathfrak{q}}_{2r+2},{\mathfrak{q}}_{2r+2}\right)\left[1+d\left({\mathfrak{q}}_{2r+2},{\mathfrak{q}}_{2r+3}\right)\right]}{1+d\left({\mathfrak{q}}_{2r+2},{\mathfrak{q}}_{2r+1}\right)}\hfill \\ & +{\mu }_7\left({\mathfrak{q}}_{2r+1},{\mathfrak{q}}_{2r+2}\right)\frac{d\left({\mathfrak{q}}_{2r+1},{\mathfrak{q}}_{2r+3}\right)\left[1+d\left({\mathfrak{q}}_{2r+1},{\mathfrak{q}}_{2r+2}\right)\right]}{1+d\left({\mathfrak{q}}_{2r+2},{\mathfrak{q}}_{2r+1}\right)}\hfill \\ \hfill =& {\mu }_1\left({\mathfrak{q}}_{2r+1},{\mathfrak{q}}_{2r+2}\right)d\left({\mathfrak{q}}_{2r+2},{\mathfrak{q}}_{2r+1}\right)\hfill \\ & +{\mu }_2\left({\mathfrak{q}}_{2r+1},{\mathfrak{q}}_{2r+2}\right)\frac{d\left({\mathfrak{q}}_{2r+2},{\mathfrak{q}}_{2r+3}\right)d\left({\mathfrak{q}}_{2r+1},{\mathfrak{q}}_{2r+2}\right)}{1+d\left({\mathfrak{q}}_{2r+1},{\mathfrak{q}}_{2r+2}\right)}\hfill \\ & +{\mu }_5\left({\mathfrak{q}}_{2r+1},{\mathfrak{q}}_{2r+2}\right)\frac{d\left({\mathfrak{q}}_{2r+1},{\mathfrak{q}}_{2r+3}\right)d\left({\mathfrak{q}}_{2r+1},{\mathfrak{q}}_{2r+2}\right)}{1+d\left({\mathfrak{q}}_{2r+2},{\mathfrak{q}}_{2r+1}\right)}\hfill \\ & +{\mu }_7\left({\mathfrak{q}}_{2r+1},{\mathfrak{q}}_{2r+2}\right)d\left({\mathfrak{q}}_{2r+1},{\mathfrak{q}}_{2r+3}\right).\hfill \end{array}$$

This implies

$$\begin{array}{ccc}\hfill ∥d\left({\mathfrak{q}}_{2r+2},{\mathfrak{q}}_{2r+3}\right)∥& \le & {\mu }_1\left({\mathfrak{q}}_{2r+1},{\mathfrak{q}}_{2r+2}\right)∥d\left({\mathfrak{q}}_{2r+2},{\mathfrak{q}}_{2r+1}\right)∥\hfill \\ & & +{\mu }_2\left({\mathfrak{q}}_{2r+1},{\mathfrak{q}}_{2r+2}\right)∥\frac{d\left({\mathfrak{q}}_{2r+2},{\mathfrak{q}}_{2r+3}\right)d\left({\mathfrak{q}}_{2r+1},{\mathfrak{q}}_{2r+2}\right)}{1+d\left({\mathfrak{q}}_{2r+1},{\mathfrak{q}}_{2r+2}\right)}∥\hfill \\ & & +{\mu }_5\left({\mathfrak{q}}_{2r+1},{\mathfrak{q}}_{2r+2}\right)∥\frac{d\left({\mathfrak{q}}_{2r+1},{\mathfrak{q}}_{2r+3}\right)d\left({\mathfrak{q}}_{2r+1},{\mathfrak{q}}_{2r+2}\right)}{1+d\left({\mathfrak{q}}_{2r+1},{\mathfrak{q}}_{2r+2}\right)}∥\hfill \\ & & +{\mu }_7\left({\mathfrak{q}}_{2r+1},{\mathfrak{q}}_{2r+2}\right)∥d\left({\mathfrak{q}}_{2r+1},{\mathfrak{q}}_{2r+3}\right)∥\hfill \end{array}$$

that is,

$$\begin{array}{ccc}\hfill ∥d\left({\mathfrak{q}}_{2r+2},{\mathfrak{q}}_{2r+3}\right)∥& \le & {\mu }_1\left({\mathfrak{q}}_{2r+1},{\mathfrak{q}}_{2r+2}\right)∥d\left({\mathfrak{q}}_{2r+2},{\mathfrak{q}}_{2r+1}\right)∥\hfill \\ & & +\sqrt{2}{\mu }_2\left({\mathfrak{q}}_{2r+1},{\mathfrak{q}}_{2r+2}\right)∥d\left({\mathfrak{q}}_{2r+2},{\mathfrak{q}}_{2r+3}\right)∥∥\frac{d\left({\mathfrak{q}}_{2r+1},{\mathfrak{q}}_{2r+2}\right)}{1+d\left({\mathfrak{q}}_{2r+1},{\mathfrak{q}}_{2r+2}\right)}∥\hfill \\ & & +\sqrt{2}{\mu }_5\left({\mathfrak{q}}_{2r+1},{\mathfrak{q}}_{2r+2}\right)∥d\left({\mathfrak{q}}_{2r+1},{\mathfrak{q}}_{2r+3}\right)∥∥\frac{d\left({\mathfrak{q}}_{2r+1},{\mathfrak{q}}_{2r+2}\right)}{1+d\left({\mathfrak{q}}_{2r+1},{\mathfrak{q}}_{2r+2}\right)}∥\hfill \\ & & +{\mu }_7\left({\mathfrak{q}}_{2r+1},{\mathfrak{q}}_{2r+2}\right)∥d\left({\mathfrak{q}}_{2r+1},{\mathfrak{q}}_{2r+3}\right)∥\hfill \end{array}$$

which further yields

$$\begin{array}{ccc}\hfill ∥d\left({\mathfrak{q}}_{2r+2},{\mathfrak{q}}_{2r+3}\right)∥& \le & {\mu }_1\left({\mathfrak{q}}_{2r+1},{\mathfrak{q}}_{2r+2}\right)∥d\left({\mathfrak{q}}_{2r+2},{\mathfrak{q}}_{2r+1}\right)∥\hfill \\ & & +\sqrt{2}{\mu }_2\left({\mathfrak{q}}_{2r+1},{\mathfrak{q}}_{2r+2}\right)∥d\left({\mathfrak{q}}_{2r+2},{\mathfrak{q}}_{2r+3}\right)∥\hfill \\ & & +\sqrt{2}{\mu }_5\left({\mathfrak{q}}_{2r+1},{\mathfrak{q}}_{2r+2}\right)∥d\left({\mathfrak{q}}_{2r+1},{\mathfrak{q}}_{2r+3}\right)∥\hfill \\ & & +{\mu }_7\left({\mathfrak{q}}_{2r+1},{\mathfrak{q}}_{2r+2}\right)∥d\left({\mathfrak{q}}_{2r+2},{\mathfrak{q}}_{2r+3}\right)∥\hfill \\ & \le & {\mu }_1\left({\mathfrak{q}}_{2r+1},{\mathfrak{q}}_{2r+2}\right)∥d\left({\mathfrak{q}}_{2r+1},{\mathfrak{q}}_{2r+2}\right)∥\hfill \\ & & +\sqrt{2}{\mu }_2\left({\mathfrak{q}}_{2r+1},{\mathfrak{q}}_{2r+2}\right)∥d\left({\mathfrak{q}}_{2r+2},{\mathfrak{q}}_{2r+3}\right)∥\hfill \\ & & +\sqrt{2}{\mu }_5\left({\mathfrak{q}}_{2r+1},{\mathfrak{q}}_{2r+2}\right)\left(∥d\left({\mathfrak{q}}_{2r+1},{\mathfrak{q}}_{2r+2}\right)∥+∥d\left({\mathfrak{q}}_{2r+2},{\mathfrak{q}}_{2r+3}\right)∥\right)\hfill \\ & & +{\mu }_7\left({\mathfrak{q}}_{2r+1},{\mathfrak{q}}_{2r+2}\right)\left(∥d\left({\mathfrak{q}}_{2r+1},{\mathfrak{q}}_{2r+2}\right)∥+∥d\left({\mathfrak{q}}_{2r+2},{\mathfrak{q}}_{2r+3}\right)∥\right).\hfill \end{array}$$

By Proposition 1, we have

$$\begin{array}{ccc}\hfill ∥d\left({\mathfrak{q}}_{2r+2},{\mathfrak{q}}_{2r+3}\right)∥& \le & {\mu }_1\left({\mathfrak{q}}_0,{\mathfrak{q}}_1\right)∥d\left({\mathfrak{q}}_{2r+1},{\mathfrak{q}}_{2r+2}\right)∥\hfill \\ & & +\sqrt{2}{\mu }_2\left({\mathfrak{q}}_0,{\mathfrak{q}}_1\right)∥d\left({\mathfrak{q}}_{2r+2},{\mathfrak{q}}_{2r+3}\right)∥\hfill \\ & & +\sqrt{2}{\mu }_5\left({\mathfrak{q}}_0,{\mathfrak{q}}_1\right)∥d\left({\mathfrak{q}}_{2r+1},{\mathfrak{q}}_{2r+2}\right)∥\hfill \\ & & +\sqrt{2}{\mu }_5\left({\mathfrak{q}}_0,{\mathfrak{q}}_1\right)∥d\left({\mathfrak{q}}_{2r+2},{\mathfrak{q}}_{2r+3}\right)∥\hfill \\ & & +{\mu }_7\left({\mathfrak{q}}_0,{\mathfrak{q}}_1\right)∥d\left({\mathfrak{q}}_{2r+1},{\mathfrak{q}}_{2r+2}\right)∥\hfill \\ & & +{\mu }_7\left({\mathfrak{q}}_0,{\mathfrak{q}}_1\right)∥d\left({\mathfrak{q}}_{2r+2},{\mathfrak{q}}_{2r+3}\right)∥\hfill \end{array}$$

which implies

$$∥d\left({\mathfrak{q}}_{2r+2},{\mathfrak{q}}_{2r+3}\right)∥\le \frac{{\mu }_1\left({\mathfrak{q}}_0,{\mathfrak{q}}_1\right)+\sqrt{2}{\mu }_5\left({\mathfrak{q}}_0,{\mathfrak{q}}_1\right)+{\mu }_7\left({\mathfrak{q}}_0,{\mathfrak{q}}_1\right)}{1-\sqrt{2}{\mu }_2\left({\mathfrak{q}}_0,{\mathfrak{q}}_1\right)-\sqrt{2}{\mu }_5\left({\mathfrak{q}}_0,{\mathfrak{q}}_1\right)-{\mu }_7\left({\mathfrak{q}}_0,{\mathfrak{q}}_1\right)}∥d\left({\mathfrak{q}}_{2r+1},{\mathfrak{q}}_{2r+2}\right)∥.$$

(4)

For all $r=0,1,2,\dots$, let

$$\lambda =max\left\{\begin{array}{c}\frac{{\mu }_1\left({\mathfrak{q}}_0,{\mathfrak{q}}_1\right)+\sqrt{2}{\mu }_4\left({\mathfrak{q}}_0,{\mathfrak{q}}_1\right)+{\mu }_6\left({\mathfrak{q}}_0,{\mathfrak{q}}_1\right)}{1-\sqrt{2}{\mu }_2\left({\mathfrak{q}}_0,{\mathfrak{q}}_1\right)-\sqrt{2}{\mu }_4\left({\mathfrak{q}}_0,{\mathfrak{q}}_1\right)-{\mu }_6\left({\mathfrak{q}}_0,{\mathfrak{q}}_1\right)},\\ \frac{{\mu }_1\left({\mathfrak{q}}_0,{\mathfrak{q}}_1\right)+\sqrt{2}{\mu }_5\left({\mathfrak{q}}_0,{\mathfrak{q}}_1\right)+{\mu }_7\left({\mathfrak{q}}_0,{\mathfrak{q}}_1\right)}{1-\sqrt{2}{\mu }_2\left({\mathfrak{q}}_0,{\mathfrak{q}}_1\right)-\sqrt{2}{\mu }_5\left({\mathfrak{q}}_0,{\mathfrak{q}}_1\right)-{\mu }_7\left({\mathfrak{q}}_0,{\mathfrak{q}}_1\right)}\end{array}\right\}<1.$$

Then from (3) and (4), we conclude that

$$∥d\left({\mathfrak{q}}_r,{\mathfrak{q}}_{r+1}\right)∥\le \lambda ∥d\left({\mathfrak{q}}_{r-1},{\mathfrak{q}}_r\right)∥$$

for all $r\in \mathbb{N}.$ Utilizing the principle of mathematical induction, we can construct a sequence $\left\{{\mathfrak{q}}_r\right\}$ in $\mathcal{A}$ satisfying the following

$$∥d\left({\mathfrak{q}}_r,{\mathfrak{q}}_{r+1}\right)∥\le \lambda ∥d\left({\mathfrak{q}}_{r-1},{\mathfrak{q}}_r\right)∥\le {\lambda }^2∥d\left({\mathfrak{q}}_{r-2},{\mathfrak{q}}_{r-1}\right)∥\le \dots \le {\lambda }^r∥d\left({\mathfrak{q}}_0,{\mathfrak{q}}_1\right)∥$$

for all $r\in \mathbb{N}.$ Now, for $m>r$, we get

$$\begin{array}{ccc}\hfill ∥d\left({\mathfrak{q}}_r,{\mathfrak{q}}_m\right)∥& \le & ∥d\left({\mathfrak{q}}_r,{\mathfrak{q}}_{r+1}\right)∥+∥d\left({\mathfrak{q}}_{r+1},{\mathfrak{q}}_{r+2}\right)∥+\dots +∥d\left({\mathfrak{q}}_{m-1},{\mathfrak{q}}_m\right)∥\hfill \\ & \le & {\lambda }^r∥d\left({\mathfrak{q}}_0,{\mathfrak{q}}_1\right)∥+{\lambda }^{r+1}∥d\left({\mathfrak{q}}_0,{\mathfrak{q}}_1\right)∥+···+{\lambda }^{m-1}∥d\left({\mathfrak{q}}_0,{\mathfrak{q}}_1\right)∥\hfill \\ & =& \left[{\lambda }^r+{\lambda }^{r+1}+\dots +{\lambda }^{m-1}\right]∥d\left({\mathfrak{q}}_0,{\mathfrak{q}}_1\right)∥\hfill \\ & =& {\lambda }^r\left[1+\lambda +{\lambda }^2\dots +{\lambda }^{m-n-1}\right]∥d\left({\mathfrak{q}}_0,{\mathfrak{q}}_1\right)∥\hfill \\ & \le & \frac{{\lambda }^r}{1-\lambda }∥d\left({\mathfrak{q}}_0,{\mathfrak{q}}_1\right)∥.\hfill \end{array}$$

Now, by taking $r,m\to \infty$, we get

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

By Lemma 2, $\left\{{\mathfrak{q}}_r\right\}$ is a Cauchy sequence in $\mathcal{A}$. As $(\mathcal{A},{⪯}_{i_2},d)$ is complete, so there exists ${\mathfrak{q}}^{*}\in \mathcal{A}$ such that ${\mathfrak{q}}_r\to {\mathfrak{q}}^{*}$ as $r\to \infty .$ Now, we show that ${\mathfrak{q}}^{*}$ is a fixed point of $\mathcal{V}.$ From (2), we have

$$\begin{array}{l}d\left({\mathfrak{q}}^{*},\mathcal{V}{\mathfrak{q}}^{*}\right){⪯}_{i_2}d\left({\mathfrak{q}}^{*},\mathcal{W}{\mathfrak{q}}_{2r+1}\right)+d\left(\mathcal{W}{\mathfrak{q}}_{2r+1},\mathcal{V}{\mathfrak{q}}^{*}\right)\\ =d\left({\mathfrak{q}}^{*},\mathcal{W}{\mathfrak{q}}_{2r+1}\right)+d\left(\mathcal{V}{\mathfrak{q}}^{*},\mathcal{W}{\mathfrak{q}}_{2r+1}\right)\\ {⪯}_{i_2}\left(\begin{array}{c}d\left({\mathfrak{q}}^{*},{\mathfrak{q}}_{2r+2}\right)+{\mu }_1\left({\mathfrak{q}}^{*},{\mathfrak{q}}_{2r+1}\right)d\left({\mathfrak{q}}^{*},{\mathfrak{q}}_{2r+1}\right)\\ +{\mu }_2\left({\mathfrak{q}}^{*},{\mathfrak{q}}_{2r+1}\right)\frac{d\left({\mathfrak{q}}^{*},\mathcal{V}{\mathfrak{q}}^{*}\right)d\left({\mathfrak{q}}_{2r+1},\mathcal{W}{\mathfrak{q}}_{2r+1}\right)}{1+d\left({\mathfrak{q}}^{*},{\mathfrak{q}}_{2r+1}\right)}\\ +{\mu }_3\left({\mathfrak{q}}^{*},{\mathfrak{q}}_{2r+1}\right)\frac{d\left({\mathfrak{q}}_{2r+1},\mathcal{V}{\mathfrak{q}}^{*}\right)d\left({\mathfrak{q}}^{*},\mathcal{W}{\mathfrak{q}}_{2r+1}\right)}{1+d\left({\mathfrak{q}}^{*},{\mathfrak{q}}_{2r+1}\right)}\\ +{\mu }_4\left({\mathfrak{q}}^{*},{\mathfrak{q}}_{2r+1}\right)\frac{d\left({\mathfrak{q}}^{*},\mathcal{V}{\mathfrak{q}}^{*}\right)d\left({\mathfrak{q}}^{*},\mathcal{W}{\mathfrak{q}}_{2r+1}\right)}{1+d\left({\mathfrak{q}}^{*},{\mathfrak{q}}_{2r+1}\right)}\\ +{\mu }_5\left({\mathfrak{q}}^{*},{\mathfrak{q}}_{2r+1}\right)\frac{d\left({\mathfrak{q}}_{2r+1},\mathcal{V}{\mathfrak{q}}^{*}\right)d\left({\mathfrak{q}}_{2r+1},\mathcal{W}{\mathfrak{q}}_{2r+1}\right)}{1+d\left({\mathfrak{q}}^{*},{\mathfrak{q}}_{2r+1}\right)}\\ +{\mu }_6\left({\mathfrak{q}}^{*},{\mathfrak{q}}_{2r+1}\right)\frac{d\left({\mathfrak{q}}^{*},\mathcal{W}{\mathfrak{q}}_{2r+1}\right)\left[1+d\left({\mathfrak{q}}^{*},\mathcal{V}{\mathfrak{q}}^{*}\right)\right]}{1+d\left({\mathfrak{q}}^{*},{\mathfrak{q}}_{2r+1}\right)}\\ +{\mu }_7\left({\mathfrak{q}}^{*},{\mathfrak{q}}_{2r+1}\right)\frac{d\left({\mathfrak{q}}_{2r+1},\mathcal{V}{\mathfrak{q}}^{*}\right)\left[1+d\left({\mathfrak{q}}_{2r+1},\mathcal{W}{\mathfrak{q}}_{2r+1}\right)\right]}{1+d\left({\mathfrak{q}}^{*},{\mathfrak{q}}_{2r+1}\right)}\end{array}\right)\\ =\left(\begin{array}{c}d\left({\mathfrak{q}}^{*},{\mathfrak{q}}_{2r+2}\right)+{\mu }_1\left({\mathfrak{q}}^{*},{\mathfrak{q}}_{2r+1}\right)d\left({\mathfrak{q}}^{*},{\mathfrak{q}}_{2r+1}\right)\\ +{\mu }_2\left({\mathfrak{q}}^{*},{\mathfrak{q}}_{2r+1}\right)\frac{d\left({\mathfrak{q}}^{*},\mathcal{V}{\mathfrak{q}}^{*}\right)d\left({\mathfrak{q}}_{2r+1},{\mathfrak{q}}_{2r+2}\right)}{1+d\left({\mathfrak{q}}^{*},{\mathfrak{q}}_{2r+1}\right)}\\ +{\mu }_3\left({\mathfrak{q}}^{*},{\mathfrak{q}}_{2r+1}\right)\frac{d\left({\mathfrak{q}}_{2r+1},\mathcal{V}{\mathfrak{q}}^{*}\right)d\left({\mathfrak{q}}^{*},{\mathfrak{q}}_{2r+2}\right)}{1+d\left({\mathfrak{q}}^{*},{\mathfrak{q}}_{2r+1}\right)}\\ +{\mu }_4\left({\mathfrak{q}}^{*},{\mathfrak{q}}_{2r+1}\right)\frac{d\left({\mathfrak{q}}^{*},\mathcal{V}{\mathfrak{q}}^{*}\right)d\left({\mathfrak{q}}^{*},{\mathfrak{q}}_{2r+2}\right)}{1+d\left({\mathfrak{q}}^{*},{\mathfrak{q}}_{2r+1}\right)}\\ +{\mu }_5\left({\mathfrak{q}}^{*},{\mathfrak{q}}_{2r+1}\right)\frac{d\left({\mathfrak{q}}_{2r+1},\mathcal{V}{\mathfrak{q}}^{*}\right)d\left({\mathfrak{q}}_{2r+1},{\mathfrak{q}}_{2r+2}\right)}{1+d\left({\mathfrak{q}}^{*},{\mathfrak{q}}_{2r+1}\right)}\\ +{\mu }_6\left({\mathfrak{q}}^{*},{\mathfrak{q}}_{2r+1}\right)\frac{d\left({\mathfrak{q}}^{*},{\mathfrak{q}}_{2r+2}\right)\left[1+d\left({\mathfrak{q}}^{*},\mathcal{V}{\mathfrak{q}}^{*}\right)\right]}{1+d\left({\mathfrak{q}}^{*},{\mathfrak{q}}_{2r+1}\right)}\\ +{\mu }_7\left({\mathfrak{q}}^{*},{\mathfrak{q}}_{2r+1}\right)\frac{d\left({\mathfrak{q}}_{2r+1},\mathcal{V}{\mathfrak{q}}^{*}\right)\left[1+d\left({\mathfrak{q}}_{2r+1},{\mathfrak{q}}_{2r+2}\right)\right]}{1+d\left({\mathfrak{q}}^{*},{\mathfrak{q}}_{2r+1}\right)}\end{array}\right).\end{array}$$

By Proposition 1, we have

$$d\left({\mathfrak{q}}^{*},\mathcal{V}{\mathfrak{q}}^{*}\right){⪯}_{i_2}\left(\begin{array}{c}d\left({\mathfrak{q}}^{*},{\mathfrak{q}}_{2r+2}\right)+{\mu }_1\left({\mathfrak{q}}^{*},{\mathfrak{q}}_1\right)d\left({\mathfrak{q}}^{*},{\mathfrak{q}}_{2r+1}\right)\\ +{\mu }_2\left({\mathfrak{q}}^{*},{\mathfrak{q}}_1\right)\frac{d\left({\mathfrak{q}}^{*},\mathcal{V}{\mathfrak{q}}^{*}\right)d\left({\mathfrak{q}}_{2r+1},{\mathfrak{q}}_{2r+2}\right)}{1+d\left({\mathfrak{q}}^{*},{\mathfrak{q}}_{2r+1}\right)}\\ +{\mu }_3\left({\mathfrak{q}}^{*},{\mathfrak{q}}_1\right)\frac{d\left({\mathfrak{q}}_{2r+1},\mathcal{V}{\mathfrak{q}}^{*}\right)d\left({\mathfrak{q}}^{*},{\mathfrak{q}}_{2r+2}\right)}{1+d\left({\mathfrak{q}}^{*},{\mathfrak{q}}_{2r+1}\right)}\\ +{\mu }_4\left({\mathfrak{q}}^{*},{\mathfrak{q}}_1\right)\frac{d\left({\mathfrak{q}}^{*},\mathcal{V}{\mathfrak{q}}^{*}\right)d\left({\mathfrak{q}}^{*},{\mathfrak{q}}_{2r+2}\right)}{1+d\left({\mathfrak{q}}^{*},{\mathfrak{q}}_{2r+1}\right)}\\ +{\mu }_5\left({\mathfrak{q}}^{*},{\mathfrak{q}}_1\right)\frac{d\left({\mathfrak{q}}_{2r+1},\mathcal{V}{\mathfrak{q}}^{*}\right)d\left({\mathfrak{q}}_{2r+1},{\mathfrak{q}}_{2r+2}\right)}{1+d\left({\mathfrak{q}}^{*},{\mathfrak{q}}_{2r+1}\right)}\\ +{\mu }_6\left({\mathfrak{q}}^{*},{\mathfrak{q}}_1\right)\frac{d\left({\mathfrak{q}}^{*},{\mathfrak{q}}_{2r+2}\right)\left[1+d\left({\mathfrak{q}}^{*},\mathcal{V}{\mathfrak{q}}^{*}\right)\right]}{1+d\left({\mathfrak{q}}^{*},{\mathfrak{q}}_{2r+1}\right)}\\ +{\mu }_7\left({\mathfrak{q}}^{*},{\mathfrak{q}}_1\right)\frac{d\left({\mathfrak{q}}_{2r+1},\mathcal{V}{\mathfrak{q}}^{*}\right)\left[1+d\left({\mathfrak{q}}_{2r+1},{\mathfrak{q}}_{2r+2}\right)\right]}{1+d\left({\mathfrak{q}}^{*},{\mathfrak{q}}_{2r+1}\right)}\end{array}\right).$$

This implies that

$$\begin{array}{ccc}\hfill ∥d\left({\mathfrak{q}}^{*},\mathcal{V}{\mathfrak{q}}^{*}\right)∥& \le & \left(\begin{array}{c}∥d\left({\mathfrak{q}}^{*},{\mathfrak{q}}_{2r+2}\right)∥+{\mu }_1\left({\mathfrak{q}}^{*},{\mathfrak{q}}_1\right)∥d\left({\mathfrak{q}}^{*},{\mathfrak{q}}_{2r+1}\right)∥\\ +{\mu }_2\left({\mathfrak{q}}^{*},{\mathfrak{q}}_1\right)∥\frac{d\left({\mathfrak{q}}^{*},\mathcal{V}{\mathfrak{q}}^{*}\right)d\left({\mathfrak{q}}_{2r+1},{\mathfrak{q}}_{2r+2}\right)}{1+d\left({\mathfrak{q}}^{*},{\mathfrak{q}}_{2r+1}\right)}∥\\ +{\mu }_3\left({\mathfrak{q}}^{*},{\mathfrak{q}}_1\right)∥\frac{d\left({\mathfrak{q}}_{2r+1},\mathcal{V}{\mathfrak{q}}^{*}\right)d\left({\mathfrak{q}}^{*},{\mathfrak{q}}_{2r+2}\right)}{1+d\left({\mathfrak{q}}^{*},{\mathfrak{q}}_{2r+1}\right)}∥\\ +{\mu }_4\left({\mathfrak{q}}^{*},{\mathfrak{q}}_1\right)∥\frac{d\left({\mathfrak{q}}^{*},\mathcal{V}{\mathfrak{q}}^{*}\right)d\left({\mathfrak{q}}^{*},{\mathfrak{q}}_{2r+2}\right)}{1+d\left({\mathfrak{q}}^{*},{\mathfrak{q}}_{2r+1}\right)}∥\\ +{\mu }_5\left({\mathfrak{q}}^{*},{\mathfrak{q}}_1\right)∥\frac{d\left({\mathfrak{q}}_{2r+1},\mathcal{V}{\mathfrak{q}}^{*}\right)d\left({\mathfrak{q}}_{2r+1},{\mathfrak{q}}_{2r+2}\right)}{1+d\left({\mathfrak{q}}^{*},{\mathfrak{q}}_{2r+1}\right)}∥\\ +{\mu }_6\left({\mathfrak{q}}^{*},{\mathfrak{q}}_1\right)∥\frac{d\left({\mathfrak{q}}^{*},{\mathfrak{q}}_{2r+2}\right)\left[1+d\left({\mathfrak{q}}^{*},\mathcal{V}{\mathfrak{q}}^{*}\right)\right]}{1+d\left({\mathfrak{q}}^{*},{\mathfrak{q}}_{2r+1}\right)}∥\\ +{\mu }_7\left({\mathfrak{q}}^{*},{\mathfrak{q}}_1\right)∥\frac{d\left({\mathfrak{q}}_{2r+1},\mathcal{V}{\mathfrak{q}}^{*}\right)\left[1+d\left({\mathfrak{q}}_{2r+1},{\mathfrak{q}}_{2r+2}\right)\right]}{1+d\left({\mathfrak{q}}^{*},{\mathfrak{q}}_{2r+1}\right)}∥\end{array}\right)\hfill \\ & \le & \left(\begin{array}{c}∥d\left({\mathfrak{q}}^{*},{\mathfrak{q}}_{2r+2}\right)∥+{\mu }_1\left({\mathfrak{q}}^{*},{\mathfrak{q}}_1\right)∥d\left({\mathfrak{q}}^{*},{\mathfrak{q}}_{2r+1}\right)∥\\ +{\mu }_2\left({\mathfrak{q}}^{*},{\mathfrak{q}}_1\right)∥\frac{d\left({\mathfrak{q}}^{*},\mathcal{V}{\mathfrak{q}}^{*}\right)d\left({\mathfrak{q}}_{2r+1},{\mathfrak{q}}_{2r+2}\right)}{1+d\left({\mathfrak{q}}^{*},{\mathfrak{q}}_{2r+1}\right)}∥\\ +{\mu }_3\left({\mathfrak{q}}^{*},{\mathfrak{q}}_1\right)∥\frac{d\left({\mathfrak{q}}_{2r+1},\mathcal{V}{\mathfrak{q}}^{*}\right)d\left({\mathfrak{q}}^{*},{\mathfrak{q}}_{2r+2}\right)}{1+d\left({\mathfrak{q}}^{*},{\mathfrak{q}}_{2r+1}\right)}∥\\ +{\mu }_4\left({\mathfrak{q}}^{*},{\mathfrak{q}}_1\right)∥\frac{d\left({\mathfrak{q}}^{*},\mathcal{V}{\mathfrak{q}}^{*}\right)d\left({\mathfrak{q}}^{*},{\mathfrak{q}}_{2r+2}\right)}{1+d\left({\mathfrak{q}}^{*},{\mathfrak{q}}_{2r+1}\right)}∥\\ +{\mu }_5\left({\mathfrak{q}}^{*},{\mathfrak{q}}_1\right)∥\frac{d\left({\mathfrak{q}}_{2r+1},\mathcal{V}{\mathfrak{q}}^{*}\right)d\left({\mathfrak{q}}_{2r+1},{\mathfrak{q}}_{2r+2}\right)}{1+d\left({\mathfrak{q}}^{*},{\mathfrak{q}}_{2r+1}\right)}∥\\ +{\mu }_6\left({\mathfrak{q}}^{*},{\mathfrak{q}}_1\right)∥\frac{d\left({\mathfrak{q}}^{*},{\mathfrak{q}}_{2r+2}\right)\left[1+d\left({\mathfrak{q}}^{*},\mathcal{V}{\mathfrak{q}}^{*}\right)\right]}{1+d\left({\mathfrak{q}}^{*},{\mathfrak{q}}_{2r+1}\right)}∥\\ +\sqrt{2}{\mu }_7\left({\mathfrak{q}}^{*},{\mathfrak{q}}_1\right)∥d\left({\mathfrak{q}}_{2r+1},\mathcal{V}{\mathfrak{q}}^{*}\right)∥∥\frac{\left[1+d\left({\mathfrak{q}}_{2r+1},{\mathfrak{q}}_{2r+2}\right)\right]}{1+d\left({\mathfrak{q}}^{*},{\mathfrak{q}}_{2r+1}\right)}∥\end{array}\right).\hfill \end{array}$$

Letting $r\to \infty ,$ we have $∥d\left({\mathfrak{q}}^{*},\mathcal{V}{\mathfrak{q}}^{*}\right)∥\le \sqrt{2}{\mu }_7\left({\mathfrak{q}}^{*},{\mathfrak{q}}^{*}\right)∥d\left({\mathfrak{q}}^{*},\mathcal{V}{\mathfrak{q}}^{*}\right)∥.$ By condition $\sqrt{2}{\mu }_7\left({\mathfrak{q}}^{*},{\mathfrak{q}}^{*}\right)<1.$ Thus, $∥d\left({\mathfrak{q}}^{*},\mathcal{V}{\mathfrak{q}}^{*}\right)∥=0$ and ${\mathfrak{q}}^{*}=\mathcal{V}{\mathfrak{q}}^{*}.$ Now we prove that ${\mathfrak{q}}^{*}$ is a fixed point of $\mathcal{W}.$ By (2), we have

$$d\left({\mathfrak{q}}^{*},\mathcal{W}{\mathfrak{q}}^{*}\right){⪯}_{i_2}\left(d\left({\mathfrak{q}}^{*},\mathcal{V}{\mathfrak{q}}_{2r}\right)+d\left(\mathcal{V}{\mathfrak{q}}_{2r},\mathcal{W}{\mathfrak{q}}^{*}\right)\right)$$

$$\begin{array}{ccc}& ⪯& i_2\left(\begin{array}{c}d\left({\mathfrak{q}}^{*},\mathcal{V}{\mathfrak{q}}_{2r}\right)+{\mu }_1\left({\mathfrak{q}}_{2r},{\mathfrak{q}}^{*}\right)d\left({\mathfrak{q}}_{2r},{\mathfrak{q}}^{*}\right)\\ +{\mu }_2\left({\mathfrak{q}}_{2r},{\mathfrak{q}}^{*}\right)\frac{d\left({\mathfrak{q}}_{2r},\mathcal{V}{\mathfrak{q}}_{2r}\right)d\left({\mathfrak{q}}^{*},\mathcal{W}{\mathfrak{q}}^{*}\right)}{1+d\left({\mathfrak{q}}_{2r},{\mathfrak{q}}^{*}\right)}\\ +{\mu }_3\left({\mathfrak{q}}_{2r},{\mathfrak{q}}^{*}\right)\frac{d\left({\mathfrak{q}}^{*},\mathcal{V}{\mathfrak{q}}_{2r}\right)d\left({\mathfrak{q}}_{2r},\mathcal{W}{\mathfrak{q}}^{*}\right)}{1+d\left({\mathfrak{q}}_{2r},{\mathfrak{q}}^{*}\right)}\\ +{\mu }_4\left({\mathfrak{q}}_{2r},{\mathfrak{q}}^{*}\right)\frac{d\left({\mathfrak{q}}_{2r},\mathcal{V}{\mathfrak{q}}_{2r}\right)d\left({\mathfrak{q}}_{2r},\mathcal{W}{\mathfrak{q}}^{*}\right)}{1+d\left({\mathfrak{q}}_{2r},{\mathfrak{q}}^{*}\right)}+{\mu }_5\left({\mathfrak{q}}_{2r},{\mathfrak{q}}^{*}\right)\frac{d\left({\mathfrak{q}}^{*},\mathcal{V}{\mathfrak{q}}_{2r}\right)d\left({\mathfrak{q}}^{*},\mathcal{W}{\mathfrak{q}}^{*}\right)}{1+d\left({\mathfrak{q}}_{2r},{\mathfrak{q}}^{*}\right)}\\ +{\mu }_6\left({\mathfrak{q}}_{2r},{\mathfrak{q}}^{*}\right)\frac{d\left({\mathfrak{q}}_{2r},\mathcal{W}{\mathfrak{q}}^{*}\right)\left[1+d\left({\mathfrak{q}}_{2r},\mathcal{V}{\mathfrak{q}}_{2r}\right)\right]}{1+d\left({\mathfrak{q}}_{2r},{\mathfrak{q}}^{*}\right)}+{\mu }_7\left({\mathfrak{q}}_{2r},{\mathfrak{q}}^{*}\right)\frac{d\left({\mathfrak{q}}^{*},\mathcal{V}{\mathfrak{q}}_{2r}\right)\left[1+d\left({\mathfrak{q}}^{*},\mathcal{W}{\mathfrak{q}}^{*}\right)\right]}{1+d\left({\mathfrak{q}}_{2r},{\mathfrak{q}}^{*}\right)}\end{array}\right)\hfill \\ & =& \left(\begin{array}{c}d\left({\mathfrak{q}}^{*},{\mathfrak{q}}_{2r+1}\right)+{\mu }_1\left({\mathfrak{q}}_{2r},{\mathfrak{q}}^{*}\right)d\left({\mathfrak{q}}_{2r},{\mathfrak{q}}^{*}\right)\\ +{\mu }_2\left({\mathfrak{q}}_{2r},{\mathfrak{q}}^{*}\right)\frac{d\left({\mathfrak{q}}_{2r},{\mathfrak{q}}_{2r+1}\right)d\left({\mathfrak{q}}^{*},\mathcal{W}{\mathfrak{q}}^{*}\right)}{1+d\left({\mathfrak{q}}_{2r},{\mathfrak{q}}^{*}\right)}\\ +{\mu }_3\left({\mathfrak{q}}_{2r},{\mathfrak{q}}^{*}\right)\frac{d\left({\mathfrak{q}}^{*},{\mathfrak{q}}_{2r+1}\right)d\left({\mathfrak{q}}_{2r},\mathcal{W}{\mathfrak{q}}^{*}\right)}{1+d\left({\mathfrak{q}}_{2r},{\mathfrak{q}}^{*}\right)}\\ +{\mu }_4\left({\mathfrak{q}}_{2r},{\mathfrak{q}}^{*}\right)\frac{d\left({\mathfrak{q}}_{2r},{\mathfrak{q}}_{2r+1}\right)d\left({\mathfrak{q}}_{2r},\mathcal{W}{\mathfrak{q}}^{*}\right)}{1+d\left({\mathfrak{q}}_{2r},{\mathfrak{q}}^{*}\right)}+{\mu }_5\left({\mathfrak{q}}_{2r},{\mathfrak{q}}^{*}\right)\frac{d\left({\mathfrak{q}}^{*},{\mathfrak{q}}_{2r+1}\right)d\left({\mathfrak{q}}^{*},\mathcal{W}{\mathfrak{q}}^{*}\right)}{1+d\left({\mathfrak{q}}_{2r},{\mathfrak{q}}^{*}\right)}\\ +{\mu }_6\left({\mathfrak{q}}_{2r},{\mathfrak{q}}^{*}\right)\frac{d\left({\mathfrak{q}}_{2r},\mathcal{W}{\mathfrak{q}}^{*}\right)\left[1+d\left({\mathfrak{q}}_{2r},{\mathfrak{q}}_{2r+1}\right)\right]}{1+d\left({\mathfrak{q}}_{2r},{\mathfrak{q}}^{*}\right)}+{\mu }_7\left({\mathfrak{q}}_{2r},{\mathfrak{q}}^{*}\right)\frac{d\left({\mathfrak{q}}^{*},{\mathfrak{q}}_{2r+1}\right)\left[1+d\left({\mathfrak{q}}^{*},\mathcal{W}{\mathfrak{q}}^{*}\right)\right]}{1+d\left({\mathfrak{q}}_{2r},{\mathfrak{q}}^{*}\right)}\end{array}\right).\hfill \end{array}$$

By Proposition 1, we have

$$d\left({\mathfrak{q}}^{*},\mathcal{W}{\mathfrak{q}}^{*}\right){⪯}_{i_2}\left(\begin{array}{c}d\left({\mathfrak{q}}^{*},{\mathfrak{q}}_{2r+1}\right)+{\mu }_1\left({\mathfrak{q}}_0,{\mathfrak{q}}^{*}\right)d\left({\mathfrak{q}}_{2r},{\mathfrak{q}}^{*}\right)\\ +{\mu }_2\left({\mathfrak{q}}_0,{\mathfrak{q}}^{*}\right)\frac{d\left({\mathfrak{q}}_{2r},{\mathfrak{q}}_{2r+1}\right)d\left({\mathfrak{q}}^{*},\mathcal{W}{\mathfrak{q}}^{*}\right)}{1+d\left({\mathfrak{q}}_{2r},{\mathfrak{q}}^{*}\right)}\\ +{\mu }_3\left({\mathfrak{q}}_0,{\mathfrak{q}}^{*}\right)\frac{d\left({\mathfrak{q}}^{*},{\mathfrak{q}}_{2r+1}\right)d\left({\mathfrak{q}}_{2r},\mathcal{W}{\mathfrak{q}}^{*}\right)}{1+d\left({\mathfrak{q}}_{2r},{\mathfrak{q}}^{*}\right)}\\ +{\mu }_4\left({\mathfrak{q}}_0,{\mathfrak{q}}^{*}\right)\frac{d\left({\mathfrak{q}}_{2r},{\mathfrak{q}}_{2r+1}\right)d\left({\mathfrak{q}}_{2r},\mathcal{W}{\mathfrak{q}}^{*}\right)}{1+d\left({\mathfrak{q}}_{2r},{\mathfrak{q}}^{*}\right)}+{\mu }_5\left({\mathfrak{q}}_0,{\mathfrak{q}}^{*}\right)\frac{d\left({\mathfrak{q}}^{*},{\mathfrak{q}}_{2r+1}\right)d\left({\mathfrak{q}}^{*},\mathcal{W}{\mathfrak{q}}^{*}\right)}{1+d\left({\mathfrak{q}}_{2r},{\mathfrak{q}}^{*}\right)}\\ +{\mu }_6\left({\mathfrak{q}}_0,{\mathfrak{q}}^{*}\right)\frac{d\left({\mathfrak{q}}_{2r},\mathcal{W}{\mathfrak{q}}^{*}\right)\left[1+d\left({\mathfrak{q}}_{2r},{\mathfrak{q}}_{2r+1}\right)\right]}{1+d\left({\mathfrak{q}}_{2r},{\mathfrak{q}}^{*}\right)}+{\mu }_7\left({\mathfrak{q}}_0,{\mathfrak{q}}^{*}\right)\frac{d\left({\mathfrak{q}}^{*},{\mathfrak{q}}_{2r+1}\right)\left[1+d\left({\mathfrak{q}}^{*},\mathcal{W}{\mathfrak{q}}^{*}\right)\right]}{1+d\left({\mathfrak{q}}_{2r},{\mathfrak{q}}^{*}\right)}\end{array}\right).$$

This implies that

$$\begin{array}{ccc}\hfill ∥d\left({\mathfrak{q}}^{*},\mathcal{W}{\mathfrak{q}}^{*}\right)∥& \le & \left(\begin{array}{c}∥d\left({\mathfrak{q}}^{*},{\mathfrak{q}}_{2r+1}\right)∥+{\mu }_1\left({\mathfrak{q}}_0,{\mathfrak{q}}^{*}\right)∥d\left({\mathfrak{q}}_{2r},{\mathfrak{q}}^{*}\right)∥\\ +{\mu }_2\left({\mathfrak{q}}_0,{\mathfrak{q}}^{*}\right)∥\frac{d\left({\mathfrak{q}}_{2r},{\mathfrak{q}}_{2r+1}\right)d\left({\mathfrak{q}}^{*},\mathcal{W}{\mathfrak{q}}^{*}\right)}{1+d\left({\mathfrak{q}}_{2r},{\mathfrak{q}}^{*}\right)}∥\\ +{\mu }_3\left({\mathfrak{q}}_0,{\mathfrak{q}}^{*}\right)∥\frac{d\left({\mathfrak{q}}^{*},{\mathfrak{q}}_{2r+1}\right)d\left({\mathfrak{q}}_{2r},\mathcal{W}{\mathfrak{q}}^{*}\right)}{1+d\left({\mathfrak{q}}_{2r},{\mathfrak{q}}^{*}\right)}∥\\ +{\mu }_4\left({\mathfrak{q}}_0,{\mathfrak{q}}^{*}\right)∥\frac{d\left({\mathfrak{q}}_{2r},{\mathfrak{q}}_{2r+1}\right)d\left({\mathfrak{q}}_{2r},\mathcal{W}{\mathfrak{q}}^{*}\right)}{1+d\left({\mathfrak{q}}_{2r},{\mathfrak{q}}^{*}\right)}∥\\ +{\mu }_5\left({\mathfrak{q}}_0,{\mathfrak{q}}^{*}\right)∥\frac{d\left({\mathfrak{q}}^{*},{\mathfrak{q}}_{2r+1}\right)d\left({\mathfrak{q}}^{*},\mathcal{W}{\mathfrak{q}}^{*}\right)}{1+d\left({\mathfrak{q}}_{2r},{\mathfrak{q}}^{*}\right)}∥\\ +{\mu }_6\left({\mathfrak{q}}_0,{\mathfrak{q}}^{*}\right)∥\frac{d\left({\mathfrak{q}}_{2r},\mathcal{W}{\mathfrak{q}}^{*}\right)\left[1+d\left({\mathfrak{q}}_{2r},{\mathfrak{q}}_{2r+1}\right)\right]}{1+d\left({\mathfrak{q}}_{2r},{\mathfrak{q}}^{*}\right)}∥\\ +{\mu }_7\left({\mathfrak{q}}_0,{\mathfrak{q}}^{*}\right)∥\frac{d\left({\mathfrak{q}}^{*},{\mathfrak{q}}_{2r+1}\right)\left[1+d\left({\mathfrak{q}}^{*},\mathcal{W}{\mathfrak{q}}^{*}\right)\right]}{1+d\left({\mathfrak{q}}_{2r},{\mathfrak{q}}^{*}\right)}∥\end{array}\right)\hfill \\ & \le & \left(\begin{array}{c}∥d\left({\mathfrak{q}}^{*},{\mathfrak{q}}_{2r+1}\right)∥+{\mu }_1\left({\mathfrak{q}}_0,{\mathfrak{q}}^{*}\right)∥d\left({\mathfrak{q}}_{2r},{\mathfrak{q}}^{*}\right)∥\\ +{\mu }_2\left({\mathfrak{q}}_0,{\mathfrak{q}}^{*}\right)∥\frac{d\left({\mathfrak{q}}_{2r},{\mathfrak{q}}_{2r+1}\right)d\left({\mathfrak{q}}^{*},\mathcal{W}{\mathfrak{q}}^{*}\right)}{1+d\left({\mathfrak{q}}_{2r},{\mathfrak{q}}^{*}\right)}∥\\ +{\mu }_3\left({\mathfrak{q}}_0,{\mathfrak{q}}^{*}\right)∥\frac{d\left({\mathfrak{q}}^{*},{\mathfrak{q}}_{2r+1}\right)d\left({\mathfrak{q}}_{2r},\mathcal{W}{\mathfrak{q}}^{*}\right)}{1+d\left({\mathfrak{q}}_{2r},{\mathfrak{q}}^{*}\right)}∥\\ +{\mu }_4\left({\mathfrak{q}}_0,{\mathfrak{q}}^{*}\right)∥\frac{d\left({\mathfrak{q}}_{2r},{\mathfrak{q}}_{2r+1}\right)d\left({\mathfrak{q}}_{2r},\mathcal{W}{\mathfrak{q}}^{*}\right)}{1+d\left({\mathfrak{q}}_{2r},{\mathfrak{q}}^{*}\right)}∥\\ +{\mu }_5\left({\mathfrak{q}}_0,{\mathfrak{q}}^{*}\right)∥\frac{d\left({\mathfrak{q}}^{*},{\mathfrak{q}}_{2r+1}\right)d\left({\mathfrak{q}}^{*},\mathcal{W}{\mathfrak{q}}^{*}\right)}{1+d\left({\mathfrak{q}}_{2r},{\mathfrak{q}}^{*}\right)}∥\\ +\sqrt{2}{\mu }_6\left({\mathfrak{q}}_0,{\mathfrak{q}}^{*}\right)∥d\left({\mathfrak{q}}_{2r},\mathcal{W}{\mathfrak{q}}^{*}\right)∥∥\frac{\left[1+d\left({\mathfrak{q}}_{2r},{\mathfrak{q}}_{2r+1}\right)\right]}{1+d\left({\mathfrak{q}}_{2r},{\mathfrak{q}}^{*}\right)}∥\\ +{\mu }_7\left({\mathfrak{q}}_0,{\mathfrak{q}}^{*}\right)∥\frac{d\left({\mathfrak{q}}^{*},{\mathfrak{q}}_{2r+1}\right)\left[1+d\left({\mathfrak{q}}^{*},\mathcal{W}{\mathfrak{q}}^{*}\right)\right]}{1+d\left({\mathfrak{q}}_{2r},{\mathfrak{q}}^{*}\right)}∥\end{array}\right)\hfill \end{array}$$

Letting $r\to \infty ,$ we have $\frac{1}{1-\sqrt{2}{\mu }_6\left({\mathfrak{q}}_0,{\mathfrak{q}}^{*}\right)}∥d\left({\mathfrak{q}}^{*},\mathcal{W}{\mathfrak{q}}^{*}\right)∥=0.$ Thus, ${\mathfrak{q}}^{*}=\mathcal{W}{\mathfrak{q}}^{*}.$ Thus, ${\mathfrak{q}}^{*}$ is a common fixed point of $\mathcal{V}$ and $\mathcal{W}.$ Now we prove that ${\mathfrak{q}}^{*}$ is unique. We suppose that

$${\mathfrak{q}}^{/}=\mathcal{V}{\mathfrak{q}}^{/}=\mathcal{W}{\mathfrak{q}}^{/}$$

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

$$d\left({\mathfrak{q}}^{*},{\mathfrak{q}}^{/}\right)=d\left(\mathcal{V}{\mathfrak{q}}^{*},\mathcal{W}{\mathfrak{q}}^{/}\right)$$

$${⪯}_{i_2}{\mu }_1\left({\mathfrak{q}}^{*},{\mathfrak{q}}^{/}\right)d\left({\mathfrak{q}}^{*},{\mathfrak{q}}^{/}\right)+{\mu }_2\left({\mathfrak{q}}^{*},{\mathfrak{q}}^{/}\right)\frac{d\left({\mathfrak{q}}^{*},\mathcal{V}{\mathfrak{q}}^{*}\right)d\left({\mathfrak{q}}^{/},\mathcal{W}{\mathfrak{q}}^{/}\right)}{1+d\left({\mathfrak{q}}^{*},{\mathfrak{q}}^{/}\right)}$$

$$\begin{array}{ccc}& & +{\mu }_3\left({\mathfrak{q}}^{*},{\mathfrak{q}}^{/}\right)\frac{d\left({\mathfrak{q}}^{/},\mathcal{V}{\mathfrak{q}}^{*}\right)d\left({\mathfrak{q}}^{*},\mathcal{W}{\mathfrak{q}}^{/}\right)}{1+d\left({\mathfrak{q}}^{*},{\mathfrak{q}}^{/}\right)}\hfill \\ & & +{\mu }_4\left({\mathfrak{q}}^{*},{\mathfrak{q}}^{/}\right)\frac{d\left({\mathfrak{q}}^{*},\mathcal{V}{\mathfrak{q}}^{*}\right)d\left({\mathfrak{q}}^{*},\mathcal{W}{\mathfrak{q}}^{/}\right)}{1+d\left({\mathfrak{q}}^{*},{\mathfrak{q}}^{/}\right)}+{\mu }_5\left({\mathfrak{q}}^{*},{\mathfrak{q}}^{/}\right)\frac{d\left({\mathfrak{q}}^{/},\mathcal{V}{\mathfrak{q}}^{*}\right)d\left({\mathfrak{q}}^{/},\mathcal{W}{\mathfrak{q}}^{/}\right)}{1+d\left({\mathfrak{q}}^{*},{\mathfrak{q}}^{/}\right)}\hfill \\ & & +{\mu }_6\left({\mathfrak{q}}^{*},{\mathfrak{q}}^{/}\right)\frac{d\left({\mathfrak{q}}^{*},\mathcal{W}{\mathfrak{q}}^{/}\right)\left[1+d\left({\mathfrak{q}}^{*},\mathcal{V}{\mathfrak{q}}^{*}\right)\right]}{1+d\left({\mathfrak{q}}^{*},{\mathfrak{q}}^{/}\right)}+{\mu }_7\left({\mathfrak{q}}^{*},{\mathfrak{q}}^{/}\right)\frac{d\left({\mathfrak{q}}^{/},\mathcal{V}{\mathfrak{q}}^{*}\right)\left[1+d\left({\mathfrak{q}}^{/},\mathcal{W}{\mathfrak{q}}^{/}\right)\right]}{1+d\left({\mathfrak{q}}^{*},{\mathfrak{q}}^{/}\right)}\hfill \end{array}$$

$$\begin{array}{ccc}& =& {\mu }_1\left({\mathfrak{q}}^{*},{\mathfrak{q}}^{/}\right)d\left({\mathfrak{q}}^{*},{\mathfrak{q}}^{/}\right)+{\mu }_2\left({\mathfrak{q}}^{*},{\mathfrak{q}}^{/}\right)\frac{d\left({\mathfrak{q}}^{*},{\mathfrak{q}}^{*}\right)d\left({\mathfrak{q}}^{/},{\mathfrak{q}}^{/}\right)}{1+d\left({\mathfrak{q}}^{*},{\mathfrak{q}}^{/}\right)}+{\mu }_3\left({\mathfrak{q}}^{*},{\mathfrak{q}}^{/}\right)\frac{d\left({\mathfrak{q}}^{/},{\mathfrak{q}}^{*}\right)d\left({\mathfrak{q}}^{*},{\mathfrak{q}}^{/}\right)}{1+d\left({\mathfrak{q}}^{*},{\mathfrak{q}}^{/}\right)}\hfill \\ & & +{\mu }_4\left({\mathfrak{q}}^{*},{\mathfrak{q}}^{/}\right)\frac{d\left({\mathfrak{q}}^{*},{\mathfrak{q}}^{*}\right)d\left({\mathfrak{q}}^{*},{\mathfrak{q}}^{/}\right)}{1+d\left({\mathfrak{q}}^{*},{\mathfrak{q}}^{/}\right)}+{\mu }_5\left({\mathfrak{q}}^{*},{\mathfrak{q}}^{/}\right)\frac{d\left({\mathfrak{q}}^{/},{\mathfrak{q}}^{*}\right)d\left({\mathfrak{q}}^{/},{\mathfrak{q}}^{/}\right)}{1+d\left({\mathfrak{q}}^{*},{\mathfrak{q}}^{/}\right)}\hfill \\ & & +{\mu }_6\left({\mathfrak{q}}^{*},{\mathfrak{q}}^{/}\right)\frac{d\left({\mathfrak{q}}^{*},{\mathfrak{q}}^{/}\right)\left[1+d\left({\mathfrak{q}}^{*},{\mathfrak{q}}^{*}\right)\right]}{1+d\left({\mathfrak{q}}^{*},{\mathfrak{q}}^{/}\right)}+{\mu }_7\left({\mathfrak{q}}^{*},{\mathfrak{q}}^{/}\right)\frac{d\left({\mathfrak{q}}^{/},{\mathfrak{q}}^{*}\right)\left[1+d\left({\mathfrak{q}}^{/},{\mathfrak{q}}^{/}\right)\right]}{1+d\left({\mathfrak{q}}^{*},{\mathfrak{q}}^{/}\right)}\hfill \\ & =& {\mu }_1\left({\mathfrak{q}}^{*},{\mathfrak{q}}^{/}\right)d\left({\mathfrak{q}}^{*},{\mathfrak{q}}^{/}\right)+{\mu }_3\left({\mathfrak{q}}^{*},{\mathfrak{q}}^{/}\right)\frac{d\left({\mathfrak{q}}^{/},{\mathfrak{q}}^{*}\right)d\left({\mathfrak{q}}^{*},{\mathfrak{q}}^{/}\right)}{1+d\left({\mathfrak{q}}^{*},{\mathfrak{q}}^{/}\right)}+{\mu }_6\left({\mathfrak{q}}^{*},{\mathfrak{q}}^{/}\right)\frac{d\left({\mathfrak{q}}^{*},{\mathfrak{q}}^{/}\right)}{1+d\left({\mathfrak{q}}^{*},{\mathfrak{q}}^{/}\right)}\hfill \\ & & +{\mu }_7\left({\mathfrak{q}}^{*},{\mathfrak{q}}^{/}\right)\frac{d\left({\mathfrak{q}}^{/},{\mathfrak{q}}^{*}\right)}{1+d\left({\mathfrak{q}}^{*},{\mathfrak{q}}^{/}\right)}.\hfill \end{array}$$

This implies that

$$\begin{array}{ccc}\hfill ∥d\left({\mathfrak{q}}^{*},{\mathfrak{q}}^{/}\right)∥& \le & {\mu }_1\left({\mathfrak{q}}^{*},{\mathfrak{q}}^{/}\right)∥d\left({\mathfrak{q}}^{*},{\mathfrak{q}}^{/}\right)∥\hfill \\ & & +\sqrt{2}{\mu }_3\left({\mathfrak{q}}^{*},{\mathfrak{q}}^{/}\right)∥d\left({\mathfrak{q}}^{*},{\mathfrak{q}}^{/}\right)∥\frac{∥d\left({\mathfrak{q}}^{*},{\mathfrak{q}}^{/}\right)∥}{∥1+d\left({\mathfrak{q}}^{*},{\mathfrak{q}}^{/}\right)∥}\hfill \\ & & +{\mu }_6\left({\mathfrak{q}}^{*},{\mathfrak{q}}^{/}\right)\frac{∥d\left({\mathfrak{q}}^{*},{\mathfrak{q}}^{/}\right)∥}{∥1+d\left({\mathfrak{q}}^{*},{\mathfrak{q}}^{/}\right)∥}+{\mu }_7\left({\mathfrak{q}}^{*},{\mathfrak{q}}^{/}\right)\frac{∥d\left({\mathfrak{q}}^{/},{\mathfrak{q}}^{*}\right)∥}{∥1+d\left({\mathfrak{q}}^{*},{\mathfrak{q}}^{/}\right)∥}.\hfill \end{array}$$

Since $∥d\left({\mathfrak{q}}^{*},{\mathfrak{q}}^{/}\right)∥\le ∥1+d\left({\mathfrak{q}}^{*},{\mathfrak{q}}^{/}\right)∥,$ so $\frac{∥d\left({\mathfrak{q}}^{*},{\mathfrak{q}}^{/}\right)∥}{∥1+d\left({\mathfrak{q}}^{*},{\mathfrak{q}}^{/}\right)∥}\le 1$ and $\frac{1}{∥1+d\left({\mathfrak{q}}^{*},{\mathfrak{q}}^{/}\right)∥}\le 1.$ Thus,

$$\begin{array}{ccc}\hfill ∥d\left({\mathfrak{q}}^{*},{\mathfrak{q}}^{/}\right)∥& \le & {\mu }_1\left({\mathfrak{q}}^{*},{\mathfrak{q}}^{/}\right)∥d\left({\mathfrak{q}}^{*},{\mathfrak{q}}^{/}\right)∥+\sqrt{2}{\mu }_3\left({\mathfrak{q}}^{*},{\mathfrak{q}}^{/}\right)∥d\left({\mathfrak{q}}^{*},{\mathfrak{q}}^{/}\right)∥\hfill \\ & & +{\mu }_6\left({\mathfrak{q}}^{*},{\mathfrak{q}}^{/}\right)∥d\left({\mathfrak{q}}^{*},{\mathfrak{q}}^{/}\right)∥+{\mu }_7\left({\mathfrak{q}}^{*},{\mathfrak{q}}^{/}\right)∥d\left({\mathfrak{q}}^{*},{\mathfrak{q}}^{/}\right)∥\hfill \\ & =& \left({\mu }_1\left({\mathfrak{q}}^{*},{\mathfrak{q}}^{/}\right)+\sqrt{2}{\mu }_3\left({\mathfrak{q}}^{*},{\mathfrak{q}}^{/}\right)+{\mu }_6\left({\mathfrak{q}}^{*},{\mathfrak{q}}^{/}\right)+{\mu }_7\left({\mathfrak{q}}^{*},{\mathfrak{q}}^{/}\right)\right)∥d\left({\mathfrak{q}}^{*},{\mathfrak{q}}^{/}\right)∥.\hfill \end{array}$$

As $\left({\mu }_1\left({\mathfrak{q}}^{*},{\mathfrak{q}}^{/}\right)+\sqrt{2}{\mu }_3\left({\mathfrak{q}}^{*},{\mathfrak{q}}^{/}\right)+{\mu }_6\left({\mathfrak{q}}^{*},{\mathfrak{q}}^{/}\right)+{\mu }_7\left({\mathfrak{q}}^{*},{\mathfrak{q}}^{/}\right)\right)<1,$ we have

$$∥d\left({\mathfrak{q}}^{*},{\mathfrak{q}}^{/}\right)∥=0.$$

Thus, ${\mathfrak{q}}^{*}={\mathfrak{q}}^{/}.$ □

Corollary 1. 

Let $(\mathcal{A},{⪯}_{i_2},d)$ be a complete bi-CVMS and $\mathcal{V}:\mathcal{A}$ $\to \mathcal{A}$. If there exist the mappings ${\mu }_i:\mathcal{A}\times \mathcal{A}\to [0,1),$ where $i=1,2,\dots 7$ such that

(a) 

${\mu }_i\left(\mathcal{V}\mathfrak{q},\kappa \right)\le {\mu }_i\left(\mathfrak{q},\kappa \right)$ and ${\mu }_i\left(\mathfrak{q},\mathcal{V}\kappa \right)\le {\mu }_i\left(\mathfrak{q},\kappa \right)$;

(b) 

${\mu }_1\left(\mathfrak{q},\kappa \right)+\sqrt{2}\left({\mu }_2\left(\mathfrak{q},\kappa \right)+{\mu }_3\left(\mathfrak{q},\kappa \right)\right)+2\sqrt{2}\left({\mu }_4\left(\mathfrak{q},\kappa \right)+{\mu }_5\left(\mathfrak{q},\kappa \right)\right)+2\left({\mu }_6\left(\mathfrak{q},\kappa \right)+{\mu }_7\left(\mathfrak{q},\kappa \right)\right)<1;$

(c) 

$$\begin{array}{c}d\left(\mathcal{V}\mathfrak{q},\mathcal{V}\kappa \right){⪯}_{i_2}{\mu }_1\left(\mathfrak{q},\kappa \right)d\left(\mathfrak{q},\kappa \right)+{\mu }_2\left(\mathfrak{q},\kappa \right)\frac{d\left(\mathfrak{q},\mathcal{V}\mathfrak{q}\right)d\left(\kappa ,\mathcal{V}\kappa \right)}{1+d\left(\mathfrak{q},\kappa \right)}+{\mu }_3\left(\mathfrak{q},\kappa \right)\frac{d\left(\kappa ,\mathcal{V}\mathfrak{q}\right)d\left(\mathfrak{q},\mathcal{V}\kappa \right)}{1+d\left(\mathfrak{q},\kappa \right)}\\ +{\mu }_4\left(\mathfrak{q},\kappa \right)\frac{d\left(\mathfrak{q},\mathcal{V}\mathfrak{q}\right)d\left(\mathfrak{q},\mathcal{V}\kappa \right)}{1+d\left(\mathfrak{q},\kappa \right)}+{\mu }_5\left(\mathfrak{q},\kappa \right)\frac{d\left(\kappa ,\mathcal{V}\mathfrak{q}\right)d\left(\kappa ,\mathcal{V}\kappa \right)}{1+d\left(\mathfrak{q},\kappa \right)}\\ +{\mu }_6\left(\mathfrak{q},\kappa \right)\frac{d\left(\mathfrak{q},\mathcal{V}\kappa \right)\left[1+d\left(\mathfrak{q},\mathcal{V}\mathfrak{q}\right)\right]}{1+d\left(\mathfrak{q},\kappa \right)}+{\mu }_7\left(\mathfrak{q},\kappa \right)\frac{d\left(\kappa ,\mathcal{V}\mathfrak{q}\right)\left[1+d\left(\kappa ,\mathcal{V}\kappa \right)\right]}{1+d\left(\mathfrak{q},\kappa \right)}\end{array}$$

for all $\mathfrak{q},\kappa \in$ $\mathcal{A},$ then $\mathcal{V}$ has a unique fixed point.

Proof. 

Take $\mathcal{W}=\mathcal{V}$ in Theorem 1. □

Corollary 2. 

Let $(\mathcal{A},{⪯}_{i_2},d)$ be a complete bi-CVMS and $\mathcal{V},\mathcal{W}:\mathcal{A}$ $\to \mathcal{A}$. If there exist the mappings ${\mu }_i:\mathcal{A}\times \mathcal{A}\to [0,1),$ where $i=1,2,\dots 6$, such that

(a) 

${\mu }_i\left(\mathcal{WV}\mathfrak{q},\kappa \right)\le {\mu }_i\left(\mathfrak{q},\kappa \right)$ and ${\mu }_i\left(\mathfrak{q},\mathcal{VW}\kappa \right)\le {\mu }_i\left(\mathfrak{q},\kappa \right)$;

(b) 

${\mu }_1\left(\mathfrak{q},\kappa \right)+\sqrt{2}\left({\mu }_2\left(\mathfrak{q},\kappa \right)+{\mu }_3\left(\mathfrak{q},\kappa \right)\right)+2\sqrt{2}\left({\mu }_4\left(\mathfrak{q},\kappa \right)+{\mu }_5\left(\mathfrak{q},\kappa \right)\right){+2\mu }_6\left(\mathfrak{q},\kappa \right)<1;$

(c) 

$$\begin{array}{c}d\left(\mathcal{V}\mathfrak{q},\mathcal{W}\kappa \right){⪯}_{i_2}{\mu }_1\left(\mathfrak{q},\kappa \right)d\left(\mathfrak{q},\kappa \right)+{\mu }_2\left(\mathfrak{q},\kappa \right)\frac{d\left(\mathfrak{q},\mathcal{V}\mathfrak{q}\right)d\left(\kappa ,\mathcal{W}\kappa \right)}{1+d\left(\mathfrak{q},\kappa \right)}+{\mu }_3\left(\mathfrak{q},\kappa \right)\frac{d\left(\kappa ,\mathcal{V}\mathfrak{q}\right)d\left(\mathfrak{q},\mathcal{W}\kappa \right)}{1+d\left(\mathfrak{q},\kappa \right)}\\ +{\mu }_4\left(\mathfrak{q},\kappa \right)\frac{d\left(\mathfrak{q},\mathcal{V}\mathfrak{q}\right)d\left(\mathfrak{q},\mathcal{W}\kappa \right)}{1+d\left(\mathfrak{q},\kappa \right)}+{\mu }_5\left(\mathfrak{q},\kappa \right)\frac{d\left(\kappa ,\mathcal{V}\mathfrak{q}\right)d\left(\kappa ,\mathcal{W}\kappa \right)}{1+d\left(\mathfrak{q},\kappa \right)}+{\mu }_6\left(\mathfrak{q},\kappa \right)\frac{d\left(\mathfrak{q},\mathcal{W}\kappa \right)\left[1+d\left(\mathfrak{q},\mathcal{V}\mathfrak{q}\right)\right]}{1+d\left(\mathfrak{q},\kappa \right)}\end{array}$$

for all $\mathfrak{q},\kappa \in$ $\mathcal{A},$ then $\mathcal{V}$ and $\mathcal{W}$ have a unique common fixed point.

Proof. 

Setting ${\mu }_7:\mathcal{A}\times \mathcal{A}\to [0,1)$ by ${\mu }_7\left(\mathfrak{q},\kappa \right)=0$ in Theorem 1. □

Corollary 3. 

Let $(\mathcal{A},{⪯}_{i_2},d)$ be a complete bi-CVMS and $\mathcal{V},\mathcal{W}:\mathcal{A}$ $\to \mathcal{A}$. If there exist the mappings ${\mu }_i:\mathcal{A}\times \mathcal{A}\to [0,1),$ where $i=1,2,\dots 5$, such that

(a) 

${\mu }_i\left(\mathcal{WV}\mathfrak{q},\kappa \right)\le {\mu }_i\left(\mathfrak{q},\kappa \right)$ and ${\mu }_i\left(\mathfrak{q},\mathcal{VW}\kappa \right)\le {\mu }_i\left(\mathfrak{q},\kappa \right)$;

(b) 

${\mu }_1\left(\mathfrak{q},\kappa \right)+\sqrt{2}\left({\mu }_2\left(\mathfrak{q},\kappa \right)+{\mu }_3\left(\mathfrak{q},\kappa \right)\right)+2\sqrt{2}\left({\mu }_4\left(\mathfrak{q},\kappa \right)+{\mu }_5\left(\mathfrak{q},\kappa \right)\right)<1$;

(c) 

$$\begin{array}{c}d\left(\mathcal{V}\mathfrak{q},\mathcal{W}\kappa \right){⪯}_{i_2}{\mu }_1\left(\mathfrak{q},\kappa \right)d\left(\mathfrak{q},\kappa \right)+{\mu }_2\left(\mathfrak{q},\kappa \right)\frac{d\left(\mathfrak{q},\mathcal{V}\mathfrak{q}\right)d\left(\kappa ,\mathcal{W}\kappa \right)}{1+d\left(\mathfrak{q},\kappa \right)}+{\mu }_3\left(\mathfrak{q},\kappa \right)\frac{d\left(\kappa ,\mathcal{V}\mathfrak{q}\right)d\left(\mathfrak{q},\mathcal{W}\kappa \right)}{1+d\left(\mathfrak{q},\kappa \right)}\\ +{\mu }_4\left(\mathfrak{q},\kappa \right)\frac{d\left(\mathfrak{q},\mathcal{V}\mathfrak{q}\right)d\left(\mathfrak{q},\mathcal{W}\kappa \right)}{1+d\left(\mathfrak{q},\kappa \right)}+{\mu }_5\left(\mathfrak{q},\kappa \right)\frac{d\left(\kappa ,\mathcal{V}\mathfrak{q}\right)d\left(\kappa ,\mathcal{W}\kappa \right)}{1+d\left(\mathfrak{q},\kappa \right)}\end{array}$$

for all $\mathfrak{q},\kappa \in$ $\mathcal{A},$ then $\mathcal{V}$ and $\mathcal{W}$ have a unique common fixed point.

Proof. 

Setting ${\mu }_6,{\mu }_7:\mathcal{A}\times \mathcal{A}\to [0,1)$ by ${\mu }_6\left(\mathfrak{q},\kappa \right)={\mu }_7\left(\mathfrak{q},\kappa \right)=0$ in Theorem 1. □

Corollary 4. 

Let $(\mathcal{A},{⪯}_{i_2},d)$ be a complete bi-CVMS and $\mathcal{V},\mathcal{W}:\mathcal{A}\to \mathcal{A}$. If there exist the mappings ${\mu }_i:\mathcal{A}\times \mathcal{A}\to [0,1),$ where $i=1,2,\dots 4$, such that

(a) 

${\mu }_i\left(\mathcal{WV}\mathfrak{q},\kappa \right)\le {\mu }_i\left(\mathfrak{q},\kappa \right)$ and ${\mu }_i\left(\mathfrak{q},\mathcal{VW}\kappa \right)\le {\mu }_i\left(\mathfrak{q},\kappa \right)$;

(b) 

${\mu }_1\left(\mathfrak{q},\kappa \right)+\sqrt{2}\left({\mu }_2\left(\mathfrak{q},\kappa \right)+{\mu }_3\left(\mathfrak{q},\kappa \right)\right)+2\sqrt{2}{\mu }_4\left(\mathfrak{q},\kappa \right)<1$;

(c) 

$$\begin{array}{c}d\left(\mathcal{V}\mathfrak{q},\mathcal{W}\kappa \right){⪯}_{i_2}{\mu }_1\left(\mathfrak{q},\kappa \right)d\left(\mathfrak{q},\kappa \right)+{\mu }_2\left(\mathfrak{q},\kappa \right)\frac{d\left(\mathfrak{q},\mathcal{V}\mathfrak{q}\right)d\left(\kappa ,\mathcal{W}\kappa \right)}{1+d\left(\mathfrak{q},\kappa \right)}\\ +{\mu }_3\left(\mathfrak{q},\kappa \right)\frac{d\left(\kappa ,\mathcal{V}\mathfrak{q}\right)d\left(\mathfrak{q},\mathcal{W}\kappa \right)}{1+d\left(\mathfrak{q},\kappa \right)}+{\mu }_4\left(\mathfrak{q},\kappa \right)\frac{d\left(\mathfrak{q},\mathcal{V}\mathfrak{q}\right)d\left(\mathfrak{q},\mathcal{W}\kappa \right)}{1+d\left(\mathfrak{q},\kappa \right)}\end{array}$$

for all $\mathfrak{q},\kappa \in$ $\mathcal{A},$ then $\mathcal{V}$ and $\mathcal{W}$ have a unique common fixed point.

Proof. 

Setting ${\mu }_5,{\mu }_6,{\mu }_7:\mathcal{A}\times \mathcal{A}\to [0,1)$ by ${\mu }_6\left(\mathfrak{q},\kappa \right)={\mu }_7\left(\mathfrak{q},\kappa \right)=0$ in Theorem 1. □

Our main result generalizes a key finding by Tassaddiq et al. [11] in this way.

Corollary 5 

([11]). Let $(\mathcal{A},{⪯}_{i_2},d)$ be a complete bi-CVMS and $\mathcal{V},\mathcal{W}:\mathcal{A}\to \mathcal{A}$. If there exist the mappings ${\mu }_i:\mathcal{A}\times \mathcal{A}\to [0,1),$ where $i=1,2,3$, such that:

(a) 

${\mu }_i\left(\mathcal{WV}\mathfrak{q},\kappa \right)\le {\mu }_i\left(\mathfrak{q},\kappa \right)$ and ${\mu }_i\left(\mathfrak{q},\mathcal{VW}\kappa \right)\le {\mu }_i\left(\mathfrak{q},\kappa \right)$;

(b) 

${\mu }_1\left(\mathfrak{q},\kappa \right)+\sqrt{2}\left({\mu }_2\left(\mathfrak{q},\kappa \right)+{\mu }_3\left(\mathfrak{q},\kappa \right)\right)<1$;

(c) 

$$d\left(\mathcal{V}\mathfrak{q},\mathcal{W}\kappa \right){⪯}_{i_2}{\mu }_1\left(\mathfrak{q},\kappa \right)d\left(\mathfrak{q},\kappa \right)+{\mu }_2\left(\mathfrak{q},\kappa \right)\frac{d\left(\mathfrak{q},\mathcal{V}\mathfrak{q}\right)d\left(\kappa ,\mathcal{W}\kappa \right)}{1+d\left(\mathfrak{q},\kappa \right)}+{\mu }_3\left(\mathfrak{q},\kappa \right)\frac{d\left(\kappa ,\mathcal{V}\mathfrak{q}\right)d\left(\mathfrak{q},\mathcal{W}\kappa \right)}{1+d\left(\mathfrak{q},\kappa \right)}$$

for all $\mathfrak{q},\kappa \in$ $\mathcal{A},$ then $\mathcal{V}$ and $\mathcal{W}$ have a unique common fixed point.

Proof. 

Setting ${\mu }_4,{\mu }_5,{\mu }_6,{\mu }_7:\mathcal{A}\times \mathcal{A}\to [0,1)$ by ${\mu }_4\left(\mathfrak{q},\kappa \right)={\mu }_5\left(\mathfrak{q},\kappa \right)={\mu }_6\left(\mathfrak{q},\kappa \right)={\mu }_7\left(\mathfrak{q},\kappa \right)=0$ in Theorem 1. □

Corollary 6. 

Let $(\mathcal{A},{⪯}_{i_2},d)$ be a complete bi-CVMS and $\mathcal{V},\mathcal{W}:\mathcal{A}\to \mathcal{A}$. If there exist the mappings ${\mu }_i:\mathcal{A}\times \mathcal{A}\to [0,1),$ where $i=1,2$, such that

(a) 

${\mu }_i\left(\mathcal{WV}\mathfrak{q},\kappa \right)\le {\mu }_i\left(\mathfrak{q},\kappa \right)$ and ${\mu }_i\left(\mathfrak{q},\mathcal{VW}\kappa \right)\le {\mu }_i\left(\mathfrak{q},\kappa \right)$;

(b) 

${\mu }_1\left(\mathfrak{q},\kappa \right)+\sqrt{2}{\mu }_2\left(\mathfrak{q},\kappa \right)<1$;

(c) 

$$d\left(\mathcal{V}\mathfrak{q},\mathcal{W}\kappa \right){⪯}_{i_2}{\mu }_1\left(\mathfrak{q},\kappa \right)d\left(\mathfrak{q},\kappa \right)+{\mu }_2\left(\mathfrak{q},\kappa \right)\frac{d\left(\mathfrak{q},\mathcal{V}\mathfrak{q}\right)d\left(\kappa ,\mathcal{W}\kappa \right)}{1+d\left(\mathfrak{q},\kappa \right)}$$

for all $\mathfrak{q},\kappa \in$ $\mathcal{A},$ then $\mathcal{V}$ and $\mathcal{W}$ have a unique common fixed point.

Proof. 

Setting ${\mu }_3,{\mu }_4,{\mu }_5,{\mu }_6,{\mu }_7:\mathcal{A}\times \mathcal{A}\to [0,1)$ by ${\mu }_3\left(\mathfrak{q},\kappa \right)={\mu }_4\left(\mathfrak{q},\kappa \right)={\mu }_5\left(\mathfrak{q},\kappa \right)={\mu }_6\left(\mathfrak{q},\kappa \right)={\mu }_7\left(\mathfrak{q},\kappa \right)=0$ in Theorem 1. □

Example 2. 

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

$$d(\mathfrak{q},\kappa )=\left|\mathfrak{q}-\kappa \right|+i_2\left|\mathfrak{q}-\kappa \right|$$

for all $\mathfrak{q},\kappa \in$ $\mathcal{A}.$

Then $(\mathcal{A},{⪯}_{i_2},d)$ is a complete bi-CVMS. Define $\mathcal{V},\mathcal{W}:\mathcal{A}$ $\to \mathcal{A}$ by

$$\mathcal{V}\mathfrak{q}=\frac{\mathfrak{q}}{3}\mathit{and}\mathcal{W}\mathfrak{q}=\frac{\mathfrak{q}}{2}$$

Consider

$${\mu }_i:\mathcal{A}\times \mathcal{A}\to [0,1)$$

for $i=1,2,\dots ,5$ by

$${\mu }_1(\mathfrak{q},\kappa )=\frac{\mathfrak{q}}{5}+\frac{\kappa }{6}$$

$${\mu }_2(\mathfrak{q},\kappa )=\frac{\mathfrak{q}}{8}+\frac{\kappa }{9}$$

$${\mu }_3(\mathfrak{q},\kappa )=\frac{{\mathfrak{q}}^2{\kappa }^2}{32}$$

$${\mu }_4(\mathfrak{q},\kappa )=\frac{{\mathfrak{q}}^2}{16}+\frac{{\kappa }^2}{25}.$$

Then, evidently,

$${\mu }_1\left(\mathfrak{q},\kappa \right)+\sqrt{2}\left({\mu }_2\left(\mathfrak{q},\kappa \right)+{\mu }_3\left(\mathfrak{q},\kappa \right)\right)+2\sqrt{2}{\mu }_4\left(\mathfrak{q},\kappa \right)<1.$$

Now

$${\mu }_1\left(\mathcal{WV}\mathfrak{q},\kappa \right)={\mu }_1\left(\mathcal{W}\left(\frac{\mathfrak{q}}{3}\right),\kappa \right)={\mu }_1\left(\frac{\mathfrak{q}}{6},\kappa \right)=\frac{\mathfrak{q}}{30}+\frac{\kappa }{6}\le \frac{\mathfrak{q}}{5}+\frac{\kappa }{6}={\mu }_1\left(\mathfrak{q},\kappa \right)$$

and

$${\mu }_1\left(\mathfrak{q},\mathcal{VW}\kappa \right)={\mu }_1\left(\mathfrak{q},\mathcal{V}\left(\frac{\kappa }{2}\right)\right)={\mu }_1\left(\mathfrak{q},\frac{\kappa }{6}\right)=\frac{\mathfrak{q}}{5}+\frac{\kappa }{36}\le \frac{\mathfrak{q}}{5}+\frac{\kappa }{6}={\mu }_1\left(\mathfrak{q},\kappa \right).$$

Also,

$${\mu }_2\left(\mathcal{WV}\mathfrak{q},\kappa \right)={\mu }_2\left(\mathcal{W}\left(\frac{\mathfrak{q}}{3}\right),\kappa \right)={\mu }_2\left(\frac{\mathfrak{q}}{6},\kappa \right)=\frac{\mathfrak{q}}{48}+\frac{\kappa }{9}\le \frac{\mathfrak{q}}{8}+\frac{\kappa }{9}={\mu }_2\left(\mathfrak{q},\kappa \right)$$

and

$${\mu }_2\left(\mathfrak{q},\mathcal{VW}\kappa \right)={\mu }_2\left(\mathfrak{q},\mathcal{V}\left(\frac{\kappa }{2}\right)\right)={\mu }_2\left(\mathfrak{q},\frac{\kappa }{6}\right)=\frac{\mathfrak{q}}{8}+\frac{\kappa }{54}\le \frac{\mathfrak{q}}{8}+\frac{\kappa }{9}={\mu }_2\left(\mathfrak{q},\kappa \right)$$

and

$${\mu }_3\left(\mathcal{WV}\mathfrak{q},\kappa \right)={\mu }_3\left(\mathcal{W}\left(\frac{\mathfrak{q}}{3}\right),\kappa \right)={\mu }_3\left(\frac{\mathfrak{q}}{6},\kappa \right)=\frac{{\mathfrak{q}}^2{\kappa }^2}{192}\le \frac{{\mathfrak{q}}^2{\kappa }^2}{32}={\mu }_3\left(\mathfrak{q},\kappa \right)$$

$${\mu }_3\left(\mathfrak{q},\mathcal{VW}\kappa \right)={\mu }_3\left(\mathfrak{q},\mathcal{V}\left(\frac{\kappa }{2}\right)\right)={\mu }_3\left(\mathfrak{q},\frac{\kappa }{6}\right)=\frac{{\mathfrak{q}}^2{\kappa }^2}{192}\le \frac{{\mathfrak{q}}^2{\kappa }^2}{32}={\mu }_3\left(\mathfrak{q},\kappa \right)$$

and

$${\mu }_4\left(\mathcal{WV}\mathfrak{q},\kappa \right)={\mu }_4\left(\mathcal{W}\left(\frac{\mathfrak{q}}{3}\right),\kappa \right)={\mu }_4\left(\frac{\mathfrak{q}}{6},\kappa \right)=\frac{{\mathfrak{q}}^2}{96}+\frac{{\kappa }^2}{25}\le \frac{{\mathfrak{q}}^2}{16}+\frac{{\kappa }^2}{25}={\mu }_4\left(\mathfrak{q},\kappa \right)$$

$${\mu }_4\left(\mathfrak{q},\mathcal{VW}\kappa \right)={\mu }_4\left(\mathfrak{q},\mathcal{V}\left(\frac{\kappa }{2}\right)\right)={\mu }_4\left(\mathfrak{q},\frac{\kappa }{6}\right)=\frac{{\mathfrak{q}}^2}{96}+\frac{{\kappa }^2}{150}\le \frac{{\mathfrak{q}}^2}{16}+\frac{{\kappa }^2}{25}={\mu }_4\left(\mathfrak{q},\kappa \right).$$

Now,

$$\begin{array}{cc}\hfill d(\mathcal{V}\mathfrak{q},\mathcal{W}\kappa )& =d(\frac{\mathfrak{q}}{3},\frac{\kappa }{2})=\left|\frac{\mathfrak{q}}{3}-\frac{\kappa }{2}\right|+i_2\left|\frac{\mathfrak{q}}{3}-\frac{\kappa }{2}\right|\hfill \\ & =\left|\frac{2\mathfrak{q}-3\kappa }{6}\right|+i_2\left|\frac{2\mathfrak{q}-3\kappa }{6}\right|\hfill \\ & {⪯}_{i_2}\left|\frac{2\mathfrak{q}-2\kappa }{6}\right|+i_2\left|\frac{2\mathfrak{q}-2\kappa }{3}\right|\hfill \\ & =\frac{1}{3}\left(\left|\mathfrak{q}-\kappa \right|+i_2\left|\mathfrak{q}-\kappa \right|\right)\hfill \\ & {⪯}_{i_2}\frac{11}{30}\left(\left|\mathfrak{q}-\kappa \right|+i_2\left|\mathfrak{q}-\kappa \right|\right)\hfill \\ & {⪯}_{i_2}{\mu }_1\left(\mathfrak{q},\kappa \right)d\left(\mathfrak{q},\kappa \right)+{\mu }_2\left(\mathfrak{q},\kappa \right)\frac{d\left(\mathfrak{q},\mathcal{V}\mathfrak{q}\right)d\left(\kappa ,\mathcal{W}\kappa \right)}{1+d\left(\mathfrak{q},\kappa \right)}\hfill \\ & +{\mu }_3\left(\mathfrak{q},\kappa \right)\frac{d\left(\kappa ,\mathcal{V}\mathfrak{q}\right)d\left(\mathfrak{q},\mathcal{W}\kappa \right)}{1+d\left(\mathfrak{q},\kappa \right)}+{\mu }_4\left(\mathfrak{q},\kappa \right)\frac{d\left(\mathfrak{q},\mathcal{V}\mathfrak{q}\right)d\left(\mathfrak{q},\mathcal{W}\kappa \right)}{1+d\left(\mathfrak{q},\kappa \right)}.\hfill \end{array}$$

Then it is very simple to prove that all the conditions of Corollary 4 are satisfied and 0 is a common fixed point of mappings $\mathcal{V}$ and $\mathcal{W}.$

Corollary 7. 

Let $(\mathcal{A},{⪯}_{i_2},d)$ be a complete bi-CVMS and $\mathcal{V}:\mathcal{A}\to \mathcal{A}$. If there exist the mappings ${\mu }_i:\mathcal{A}\times \mathcal{A}\to [0,1),$ where $i=1,2,\dots 7$, such that

(a) 

${\mu }_i\left(\mathcal{V}\mathfrak{q},\kappa \right)\le {\mu }_i\left(\mathfrak{q},\kappa \right)$ and ${\mu }_i\left(\mathfrak{q},\mathcal{V}\kappa \right)\le {\mu }_i\left(\mathfrak{q},\kappa \right)$;

(b) 

${\mu }_1\left(\mathfrak{q},\kappa \right)+\sqrt{2}\left({\mu }_2\left(\mathfrak{q},\kappa \right)+{\mu }_3\left(\mathfrak{q},\kappa \right)\right)+2\sqrt{2}\left({\mu }_4\left(\mathfrak{q},\kappa \right)+{\mu }_5\left(\mathfrak{q},\kappa \right)\right)+2\left({\mu }_6\left(\mathfrak{q},\kappa \right)+{\mu }_7\left(\mathfrak{q},\kappa \right)\right)<1,$;

(c) 

$$\begin{array}{c}d\left({\mathcal{V}}^r\mathfrak{q},{\mathcal{V}}^r\kappa \right){⪯}_{i_2}{\mu }_1\left(\mathfrak{q},\kappa \right)d\left(\mathfrak{q},\kappa \right)+{\mu }_2\left(\mathfrak{q},\kappa \right)\frac{d\left(\mathfrak{q},{\mathcal{V}}^r\mathfrak{q}\right)d\left(\kappa ,{\mathcal{V}}^r\kappa \right)}{1+d\left(\mathfrak{q},\kappa \right)}+{\mu }_3\left(\mathfrak{q},\kappa \right)\frac{d\left(\kappa ,{\mathcal{V}}^r\mathfrak{q}\right)d\left(\mathfrak{q},{\mathcal{V}}^r\kappa \right)}{1+d\left(\mathfrak{q},\kappa \right)}\\ +{\mu }_4\left(\mathfrak{q},\kappa \right)\frac{d\left(\mathfrak{q},{\mathcal{V}}^r\mathfrak{q}\right)d\left(\mathfrak{q},{\mathcal{V}}^r\kappa \right)}{1+d\left(\mathfrak{q},\kappa \right)}+{\mu }_5\left(\mathfrak{q},\kappa \right)\frac{d\left(\kappa ,{\mathcal{V}}^r\mathfrak{q}\right)d\left(\kappa ,{\mathcal{V}}^r\kappa \right)}{1+d\left(\mathfrak{q},\kappa \right)}\\ +{\mu }_6\left(\mathfrak{q},\kappa \right)\frac{d\left(\mathfrak{q},{\mathcal{V}}^r\kappa \right)\left[1+d\left(\mathfrak{q},{\mathcal{V}}^r\mathfrak{q}\right)\right]}{1+d\left(\mathfrak{q},\kappa \right)}+{\mu }_7\left(\mathfrak{q},\kappa \right)\frac{d\left(\kappa ,{\mathcal{V}}^r\mathfrak{q}\right)\left[1+d\left(\kappa ,{\mathcal{V}}^r\kappa \right)\right]}{1+d\left(\mathfrak{q},\kappa \right)}\end{array}$$

for all $\mathfrak{q},\kappa \in$ $\mathcal{A}$, then $\mathcal{V}$ has a unique fixed point.

Proof. 

From Corollary 1, we have $\mathfrak{q}\in \mathcal{A}$ such that ${\mathcal{V}}^r\mathfrak{q}=\mathfrak{q}.$ Now, from

$$\begin{array}{ccc}\hfill d\left(\mathcal{V}\mathfrak{q},\mathfrak{q}\right)& =& d\left({\mathcal{VV}}^r\mathfrak{q},{\mathcal{V}}^r\mathfrak{q}\right)\hfill \\ & =& d\left({\mathcal{V}}^r\mathcal{V}\mathfrak{q},{\mathcal{V}}^r\mathfrak{q}\right){⪯}_{i_2}{\mu }_1\left(\mathcal{V}\mathfrak{q},\mathfrak{q}\right)d\left(\mathcal{V}\mathfrak{q},\mathfrak{q}\right)+{\mu }_2\left(\mathcal{V}\mathfrak{q},\mathfrak{q}\right)\frac{d\left(\mathcal{V}\mathfrak{q},{\mathcal{V}}^r\mathcal{V}\mathfrak{q}\right)d\left(\mathfrak{q},\mathfrak{q}\right)}{1+d\left(\mathcal{V}\mathfrak{q},\mathfrak{q}\right)}\hfill \\ & & +{\mu }_3\left(\mathcal{V}\mathfrak{q},\mathfrak{q}\right)\frac{d\left(\mathfrak{q},{\mathcal{V}}^r\mathcal{V}\mathfrak{q}\right)d\left(\mathcal{V}\mathfrak{q},{\mathcal{V}}^r\mathfrak{q}\right)}{1+d\left(\mathcal{V}\mathfrak{q},\mathfrak{q}\right)}\hfill \\ & & +{\mu }_4\left(\mathcal{V}\mathfrak{q},\mathfrak{q}\right)\frac{d\left(\mathcal{V}\mathfrak{q},{\mathcal{V}}^r\mathcal{V}\mathfrak{q}\right)d\left(\mathcal{V}\mathfrak{q},{\mathcal{V}}^r\mathfrak{q}\right)}{1+d\left(\mathcal{V}\mathfrak{q},\mathfrak{q}\right)}+{\mu }_5\left(\mathcal{V}\mathfrak{q},\mathfrak{q}\right)\frac{d\left(\mathfrak{q},{\mathcal{V}}^r\mathcal{V}\mathfrak{q}\right)d\left(\mathfrak{q},{\mathcal{V}}^r\mathfrak{q}\right)}{1+d\left(\mathcal{V}\mathfrak{q},\mathfrak{q}\right)}\hfill \\ & & +{\mu }_6\left(\mathcal{V}\mathfrak{q},\mathfrak{q}\right)\frac{d\left(\mathcal{V}\mathfrak{q},{\mathcal{V}}^r\mathfrak{q}\right)\left[1+d\left(\mathcal{V}\mathfrak{q},{\mathcal{V}}^r\mathcal{V}\mathfrak{q}\right)\right]}{1+d\left(\mathcal{V}\mathfrak{q},\mathfrak{q}\right)}+{\mu }_7\left(\mathcal{V}\mathfrak{q},\mathfrak{q}\right)\frac{d\left(\mathfrak{q},{\mathcal{V}}^r\mathcal{V}\mathfrak{q}\right)\left[1+d\left(\mathfrak{q},{\mathcal{V}}^r\mathfrak{q}\right)\right]}{1+d\left(\mathcal{V}\mathfrak{q},\mathfrak{q}\right)}\hfill \\ & =& {\mu }_1\left(\mathcal{V}\mathfrak{q},\mathfrak{q}\right)d\left(\mathcal{V}\mathfrak{q},\mathfrak{q}\right)+{\mu }_2\left(\mathcal{V}\mathfrak{q},\mathfrak{q}\right)\frac{d\left(\mathcal{V}\mathfrak{q},\mathcal{V}\mathfrak{q}\right)d\left(\mathfrak{q},\mathfrak{q}\right)}{1+d\left(\mathcal{V}\mathfrak{q},\mathfrak{q}\right)}\hfill \\ & & +{\mu }_3\left(\mathcal{V}\mathfrak{q},\mathfrak{q}\right)\frac{d\left(\mathfrak{q},\mathcal{V}\mathfrak{q}\right)d\left(\mathcal{V}\mathfrak{q},\mathfrak{q}\right)}{1+d\left(\mathcal{V}\mathfrak{q},\mathfrak{q}\right)}\hfill \\ & & +{\mu }_4\left(\mathcal{V}\mathfrak{q},\mathfrak{q}\right)\frac{d\left(\mathcal{V}\mathfrak{q},\mathcal{V}\mathfrak{q}\right)d\left(\mathcal{V}\mathfrak{q},\mathfrak{q}\right)}{1+d\left(\mathcal{V}\mathfrak{q},\mathfrak{q}\right)}+{\mu }_5\left(\mathcal{V}\mathfrak{q},\mathfrak{q}\right)\frac{d\left(\mathfrak{q},\mathcal{V}\mathfrak{q}\right)d\left(\mathfrak{q},\mathfrak{q}\right)}{1+d\left(\mathcal{V}\mathfrak{q},\mathfrak{q}\right)}\hfill \\ & & +{\mu }_6\left(\mathcal{V}\mathfrak{q},\mathfrak{q}\right)\frac{d\left(\mathcal{V}\mathfrak{q},\mathfrak{q}\right)\left[1+d\left(\mathcal{V}\mathfrak{q},\mathcal{V}\mathfrak{q}\right)\right]}{1+d\left(\mathcal{V}\mathfrak{q},\mathfrak{q}\right)}+{\mu }_7\left(\mathcal{V}\mathfrak{q},\mathfrak{q}\right)\frac{d\left(\mathfrak{q},\mathcal{V}\mathfrak{q}\right)\left[1+d\left(\mathfrak{q},\mathfrak{q}\right)\right]}{1+d\left(\mathcal{V}\mathfrak{q},\mathfrak{q}\right)}\hfill \\ & =& {\mu }_1\left(\mathcal{V}\mathfrak{q},\mathfrak{q}\right)d\left(\mathcal{V}\mathfrak{q},\mathfrak{q}\right)+{\mu }_3\left(\mathcal{V}\mathfrak{q},\mathfrak{q}\right)\frac{d\left(\mathfrak{q},\mathcal{V}\mathfrak{q}\right)d\left(\mathcal{V}\mathfrak{q},\mathfrak{q}\right)}{1+d\left(\mathcal{V}\mathfrak{q},\mathfrak{q}\right)}+{\mu }_6\left(\mathcal{V}\mathfrak{q},\mathfrak{q}\right)\frac{d\left(\mathcal{V}\mathfrak{q},\mathfrak{q}\right)}{1+d\left(\mathcal{V}\mathfrak{q},\mathfrak{q}\right)}\hfill \\ & & +{\mu }_7\left(\mathcal{V}\mathfrak{q},\mathfrak{q}\right)\frac{d\left(\mathfrak{q},\mathcal{V}\mathfrak{q}\right)}{1+d\left(\mathcal{V}\mathfrak{q},\mathfrak{q}\right)}\hfill \end{array}$$

which implies that

$$\begin{array}{ccc}\hfill ∥d\left(\mathcal{V}\mathfrak{q},\mathfrak{q}\right)∥& \le & {\mu }_1\left(\mathcal{V}\mathfrak{q},\mathfrak{q}\right)∥d\left(\mathcal{V}\mathfrak{q},\mathfrak{q}\right)∥+\sqrt{2}{\mu }_3\left(\mathcal{V}\mathfrak{q},\mathfrak{q}\right)∥d\left(\mathfrak{q},\mathcal{V}\mathfrak{q}\right)∥\frac{∥d\left(\mathcal{V}\mathfrak{q},\mathfrak{q}\right)∥}{∥1+d\left(\mathcal{V}\mathfrak{q},\mathfrak{q}\right)∥}\hfill \\ & & +{\mu }_6\left(\mathcal{V}\mathfrak{q},\mathfrak{q}\right)\frac{∥d\left(\mathcal{V}\mathfrak{q},\mathfrak{q}\right)∥}{∥1+d\left(\mathcal{V}\mathfrak{q},\mathfrak{q}\right)∥}+{\mu }_7\left(\mathcal{V}\mathfrak{q},\mathfrak{q}\right)\frac{∥d\left(\mathfrak{q},\mathcal{V}\mathfrak{q}\right)∥}{∥1+d\left(\mathcal{V}\mathfrak{q},\mathfrak{q}\right)∥}.\hfill \end{array}$$

Since $∥d\left(\mathcal{V}\mathfrak{q},\mathfrak{q}\right)∥\le ∥1+d\left(\mathcal{V}\mathfrak{q},\mathfrak{q}\right)∥$ and $1\le ∥1+d\left(\mathcal{V}\mathfrak{q},\mathfrak{q}\right)∥$, so $\frac{∥d\left(\mathcal{V}\mathfrak{q},\mathfrak{q}\right)∥}{∥1+d\left(\mathcal{V}\mathfrak{q},\mathfrak{q}\right)∥}\le 1$ and $\frac{1}{∥1+d\left(\mathcal{V}\mathfrak{q},\mathfrak{q}\right)∥}\le 1.$ Thus, we have

$$\begin{array}{ccc}\hfill ∥d\left(\mathcal{V}\mathfrak{q},\mathfrak{q}\right)∥& \le & {\mu }_1\left(\mathcal{V}\mathfrak{q},\mathfrak{q}\right)∥d\left(\mathcal{V}\mathfrak{q},\mathfrak{q}\right)∥+\sqrt{2}{\mu }_3\left(\mathcal{V}\mathfrak{q},\mathfrak{q}\right)∥d\left(\mathfrak{q},\mathcal{V}\mathfrak{q}\right)∥\hfill \\ & & +{\mu }_6\left(\mathcal{V}\mathfrak{q},\mathfrak{q}\right)∥d\left(\mathcal{V}\mathfrak{q},\mathfrak{q}\right)∥+{\mu }_7\left(\mathcal{V}\mathfrak{q},\mathfrak{q}\right)∥d\left(\mathfrak{q},\mathcal{V}\mathfrak{q}\right)∥\hfill \\ & =& \left({\mu }_1\left(\mathcal{V}\mathfrak{q},\mathfrak{q}\right)+\sqrt{2}{\mu }_3\left(\mathcal{V}\mathfrak{q},\mathfrak{q}\right)+{\mu }_6\left(\mathcal{V}\mathfrak{q},\mathfrak{q}\right)+{\mu }_7\left(\mathcal{V}\mathfrak{q},\mathfrak{q}\right)\right)∥d\left(\mathfrak{q},\mathcal{V}\mathfrak{q}\right)∥.\hfill \end{array}$$

As $\left({\mu }_1\left(\mathcal{V}\mathfrak{q},\mathfrak{q}\right)+\sqrt{2}{\mu }_3\left(\mathcal{V}\mathfrak{q},\mathfrak{q}\right)+{\mu }_6\left(\mathcal{V}\mathfrak{q},\mathfrak{q}\right)+{\mu }_7\left(\mathcal{V}\mathfrak{q},\mathfrak{q}\right)\right)<1,$ so it is possible only if $∥d\left(\mathcal{V}\mathfrak{q},\mathfrak{q}\right)∥=0.$ Thus, $\mathcal{V}\mathfrak{q}=\mathfrak{q}.$ □

4. Inferred Findings

Corollary 8. 

Let $(\mathcal{A},{⪯}_{i_2},d)$ be a complete bi-CVMS and $\mathcal{V},\mathcal{W}:\mathcal{A}$ $\to \mathcal{A}$. If there exist the mappings ${\mu }_i:\mathcal{A}\to [0,1),$ where $i=1,2,\dots 7$, such that

(a) 

${\mu }_i\left(\mathcal{WV}\mathfrak{q}\right)\le {\mu }_i\left(\mathfrak{q}\right)$ and ${\mu }_i\left(\mathcal{VW}\kappa \right)\le {\mu }_i\left(\kappa \right)$;

(b) 

${\mu }_1\left(\mathfrak{q}\right)+\sqrt{2}\left({\mu }_2\left(\mathfrak{q}\right)+{\mu }_3\left(\mathfrak{q}\right)\right)+2\sqrt{2}\left({\mu }_4\left(\mathfrak{q}\right)+{\mu }_5\left(\mathfrak{q}\right)\right)+2\left({\mu }_6\left(\mathfrak{q}\right)+{\mu }_7\left(\mathfrak{q}\right)\right)<1$;

(c) 

$$\begin{array}{c}d\left(\mathcal{V}\mathfrak{q},\mathcal{W}\kappa \right){⪯}_{i_2}{\mu }_1\left(\mathfrak{q}\right)d\left(\mathfrak{q},\kappa \right)+{\mu }_2\left(\mathfrak{q}\right)\frac{d\left(\mathfrak{q},\mathcal{V}\mathfrak{q}\right)d\left(\kappa ,\mathcal{W}\kappa \right)}{1+d\left(\mathfrak{q},\kappa \right)}+{\mu }_3\left(\mathfrak{q}\right)\frac{d\left(\kappa ,\mathcal{V}\mathfrak{q}\right)d\left(\mathfrak{q},\mathcal{W}\kappa \right)}{1+d\left(\mathfrak{q},\kappa \right)}\\ +{\mu }_4\left(\mathfrak{q}\right)\frac{d\left(\mathfrak{q},\mathcal{V}\mathfrak{q}\right)d\left(\mathfrak{q},\mathcal{W}\kappa \right)}{1+d\left(\mathfrak{q},\kappa \right)}+{\mu }_5\left(\mathfrak{q}\right)\frac{d\left(\kappa ,\mathcal{V}\mathfrak{q}\right)d\left(\kappa ,\mathcal{W}\kappa \right)}{1+d\left(\mathfrak{q},\kappa \right)}\\ +{\mu }_6\left(\mathfrak{q}\right)\frac{d\left(\mathfrak{q},\mathcal{W}\kappa \right)\left[1+d\left(\mathfrak{q},\mathcal{V}\mathfrak{q}\right)\right]}{1+d\left(\mathfrak{q},\kappa \right)}+{\mu }_7\left(\mathfrak{q}\right)\frac{d\left(\kappa ,\mathcal{V}\mathfrak{q}\right)\left[1+d\left(\kappa ,\mathcal{W}\kappa \right)\right]}{1+d\left(\mathfrak{q},\kappa \right)},\end{array}$$

for all $\mathfrak{q},\kappa \in$ $\mathcal{A},$ then $\mathcal{V}$ and $\mathcal{W}$ have a unique common fixed point.

Proof. 

Define ${\mu }_i:\mathcal{A}\times \mathcal{A}\to [0,1)$ by

$${\mu }_i\left(\mathfrak{q},\kappa \right)={\mu }_i\left(\mathfrak{q}\right)$$

for all $\mathfrak{q},\kappa \in \mathcal{A}.$ Then for all $\mathfrak{q},\kappa \in \mathcal{A},$ we have

(a)

${\mu }_i\left(\mathcal{WV}\mathfrak{q},\kappa \right)={\mu }_i\left(\mathcal{WV}\mathfrak{q}\right)\le {\mu }_i\left(\mathfrak{q}\right)={\mu }_i\left(\mathfrak{q},\kappa \right)$ and ${\mu }_i\left(\mathfrak{q},\mathcal{VW}\kappa \right)={\mu }_i\left(\mathfrak{q}\right)={\mu }_i\left(\mathfrak{q},\kappa \right);$

(b)

$$\begin{array}{c}{\mu }_1\left(\mathfrak{q},\kappa \right)+\sqrt{2}\left({\mu }_2\left(\mathfrak{q},\kappa \right)+{\mu }_3\left(\mathfrak{q},\kappa \right)\right)+2\sqrt{2}\left({\mu }_4\left(\mathfrak{q},\kappa \right)+{\mu }_5\left(\mathfrak{q},\kappa \right)\right)+2\left({\mu }_6\left(\mathfrak{q},\kappa \right)+{\mu }_7\left(\mathfrak{q},\kappa \right)\right)\\ ={\mu }_1\left(\mathfrak{q}\right)+\sqrt{2}\left({\mu }_2\left(\mathfrak{q}\right)+{\mu }_3\left(\mathfrak{q}\right)\right)+2\sqrt{2}\left({\mu }_4\left(\mathfrak{q}\right)+{\mu }_5\left(\mathfrak{q}\right)\right)+2\left({\mu }_6\left(\mathfrak{q}\right)+{\mu }_7\left(\mathfrak{q}\right)\right)<1;\end{array}$$

(c)

$$\begin{array}{ccc}\hfill d\left(\mathcal{V}\mathfrak{q},\mathcal{W}\kappa \right)& ⪯& i_2{\mu }_1\left(\mathfrak{q}\right)d\left(\mathfrak{q},\kappa \right)+{\mu }_2\left(\mathfrak{q}\right)\frac{d\left(\mathfrak{q},\mathcal{V}\mathfrak{q}\right)d\left(\kappa ,\mathcal{W}\kappa \right)}{1+d\left(\mathfrak{q},\kappa \right)}+{\mu }_3\left(\mathfrak{q}\right)\frac{d\left(\kappa ,\mathcal{V}\mathfrak{q}\right)d\left(\mathfrak{q},\mathcal{W}\kappa \right)}{1+d\left(\mathfrak{q},\kappa \right)}\hfill \\ & & +{\mu }_4\left(\mathfrak{q}\right)\frac{d\left(\mathfrak{q},\mathcal{V}\mathfrak{q}\right)d\left(\mathfrak{q},\mathcal{W}\kappa \right)}{1+d\left(\mathfrak{q},\kappa \right)}+{\mu }_5\left(\mathfrak{q}\right)\frac{d\left(\kappa ,\mathcal{V}\mathfrak{q}\right)d\left(\kappa ,\mathcal{W}\kappa \right)}{1+d\left(\mathfrak{q},\kappa \right)}\hfill \\ & & +{\mu }_6\left(\mathfrak{q}\right)\frac{d\left(\mathfrak{q},\mathcal{W}\kappa \right)\left[1+d\left(\mathfrak{q},\mathcal{V}\mathfrak{q}\right)\right]}{1+d\left(\mathfrak{q},\kappa \right)}+{\mu }_7\left(\mathfrak{q}\right)\frac{d\left(\kappa ,\mathcal{V}\mathfrak{q}\right)\left[1+d\left(\kappa ,\mathcal{W}\kappa \right)\right]}{1+d\left(\mathfrak{q},\kappa \right)}\hfill \\ & =& {\mu }_1\left(\mathfrak{q},\kappa \right)d\left(\mathfrak{q},\kappa \right)+{\mu }_2\left(\mathfrak{q},\kappa \right)\frac{d\left(\mathfrak{q},\mathcal{V}\mathfrak{q}\right)d\left(\kappa ,\mathcal{W}\kappa \right)}{1+d\left(\mathfrak{q},\kappa \right)}+{\mu }_3\left(\mathfrak{q},\kappa \right)\frac{d\left(\kappa ,\mathcal{V}\mathfrak{q}\right)d\left(\mathfrak{q},\mathcal{W}\kappa \right)}{1+d\left(\mathfrak{q},\kappa \right)}\hfill \\ & & +{\mu }_4\left(\mathfrak{q},\kappa \right)\frac{d\left(\mathfrak{q},\mathcal{V}\mathfrak{q}\right)d\left(\mathfrak{q},\mathcal{W}\kappa \right)}{1+d\left(\mathfrak{q},\kappa \right)}+{\mu }_5\left(\mathfrak{q},\kappa \right)\frac{d\left(\kappa ,\mathcal{V}\mathfrak{q}\right)d\left(\kappa ,\mathcal{W}\kappa \right)}{1+d\left(\mathfrak{q},\kappa \right)}\hfill \\ & & +{\mu }_6\left(\mathfrak{q},\kappa \right)\frac{d\left(\mathfrak{q},\mathcal{W}\kappa \right)\left[1+d\left(\mathfrak{q},\mathcal{V}\mathfrak{q}\right)\right]}{1+d\left(\mathfrak{q},\kappa \right)}+{\mu }_7\left(\mathfrak{q},\kappa \right)\frac{d\left(\kappa ,\mathcal{V}\mathfrak{q}\right)\left[1+d\left(\kappa ,\mathcal{W}\kappa \right)\right]}{1+d\left(\mathfrak{q},\kappa \right)}.\hfill \end{array}$$

By Theorem 1, $\mathcal{V}$ and $\mathcal{W}$ have a unique common fixed point. □

Corollary 9. 

Let $(\mathcal{A},{⪯}_{i_2},d)$ be a complete bi-CVMS, and let $\mathcal{V},\mathcal{W}:\mathcal{A}$ $\to \mathcal{A}$. If there exist the constants ${\mu }_i\in [0,1)$ for $i=1,2,\dots ,7$ with ${\mu }_1+\sqrt{2}\left({\mu }_2+{\mu }_3\right)+2\sqrt{2}\left({\mu }_4+{\mu }_5\right)+2\left({\mu }_6+{\mu }_7\right)<1$, such that

$$\begin{array}{c}d\left(\mathcal{V}\mathfrak{q},\mathcal{W}\kappa \right){⪯}_{i_2}{\mu }_{1}d\left(\mathfrak{q},\kappa \right)+{\mu }_2\frac{d\left(\mathfrak{q},\mathcal{V}\mathfrak{q}\right)d\left(\kappa ,\mathcal{W}\kappa \right)}{1+d\left(\mathfrak{q},\kappa \right)}+{\mu }_3\frac{d\left(\kappa ,\mathcal{V}\mathfrak{q}\right)d\left(\mathfrak{q},\mathcal{W}\kappa \right)}{1+d\left(\mathfrak{q},\kappa \right)}\\ +{\mu }_4\frac{d\left(\mathfrak{q},\mathcal{V}\mathfrak{q}\right)d\left(\mathfrak{q},\mathcal{W}\kappa \right)}{1+d\left(\mathfrak{q},\kappa \right)}+{\mu }_5\frac{d\left(\kappa ,\mathcal{V}\mathfrak{q}\right)d\left(\kappa ,\mathcal{W}\kappa \right)}{1+d\left(\mathfrak{q},\kappa \right)}\\ +{\mu }_6\frac{d\left(\mathfrak{q},\mathcal{W}\kappa \right)\left[1+d\left(\mathfrak{q},\mathcal{V}\mathfrak{q}\right)\right]}{1+d\left(\mathfrak{q},\kappa \right)}+{\mu }_7\frac{d\left(\kappa ,\mathcal{V}\mathfrak{q}\right)\left[1+d\left(\kappa ,\mathcal{W}\kappa \right)\right]}{1+d\left(\mathfrak{q},\kappa \right)},\end{array}$$

for all $\mathfrak{q},\kappa \in$ $\mathcal{A}$, then $\mathcal{V}$ and $\mathcal{W}$ have a unique common fixed point.

Proof. 

Take ${\mu }_i\left(·\right)={\mu }_i,$ where $i=1,2,\dots ,7$, in Corollary 8. □

Now we present more general results, which, subsequently, recover some results from Gnanaprakasam et al. [8] as a special case.

Corollary 10 

([8]). Let $(\mathcal{A},{⪯}_{i_2},d)$ be a complete bi-CVMS, and let $\mathcal{V},\mathcal{W}:\mathcal{A}$ $\to \mathcal{A}$. If there exist the constants ${\mu }_1,{\mu }_2,{\mu }_3\in [0,1)$ with ${\mu }_1+\sqrt{2}{\mu }_2+\sqrt{2}{\mu }_3<1$ such that

$$d\left(\mathcal{V}\mathfrak{q},\mathcal{W}\kappa \right){⪯}_{i_2}{\mu }_{1}d\left(\mathfrak{q},\kappa \right)+{\mu }_2\frac{d\left(\mathfrak{q},\mathcal{V}\mathfrak{q}\right)d\left(\kappa ,\mathcal{W}\kappa \right)}{1+d\left(\mathfrak{q},\kappa \right)}+{\mu }_3\frac{d\left(\kappa ,\mathcal{V}\mathfrak{q}\right)d\left(\mathfrak{q},\mathcal{W}\kappa \right)}{1+d\left(\mathfrak{q},\kappa \right)}$$

for all $\mathfrak{q},\kappa \in$ $\mathcal{A}$, then $\mathcal{V}$ and $\mathcal{W}$ have a unique common fixed point.

Proof. 

Take ${\mu }_i=0$, where $i=4,5,6,7$, in Corollary 9. □

Corollary 11 

([8]). Let $(\mathcal{A},{⪯}_{i_2},d)$ be a complete bi-CVMS, and let $\mathcal{V}:\mathcal{A}$ $\to \mathcal{A}$. If there exist the constants ${\mu }_1,{\mu }_2,{\mu }_3\in [0,1)$, with ${\mu }_1+\sqrt{2}{\mu }_2+\sqrt{2}{\mu }_3<1$, such that

$$d\left(\mathcal{V}\mathfrak{q},\mathcal{V}\kappa \right){⪯}_{i_2}{\mu }_{1}d\left(\mathfrak{q},\kappa \right)+{\mu }_2\frac{d\left(\mathfrak{q},\mathcal{V}\mathfrak{q}\right)d\left(\kappa ,\mathcal{V}\kappa \right)}{1+d\left(\mathfrak{q},\kappa \right)}+{\mu }_3\frac{d\left(\kappa ,\mathcal{V}\mathfrak{q}\right)d\left(\mathfrak{q},\mathcal{V}\kappa \right)}{1+d\left(\mathfrak{q},\kappa \right)}$$

for all $\mathfrak{q},\kappa \in$ $\mathcal{A}$, then $\mathcal{V}$ has a unique fixed point.

Proof. 

Set $\mathcal{W}=\mathcal{V}$ in Corollary 10. □

In the following way, we derive the principal result established by Beg et al. [7].

Corollary 12 

([7]). Let $(\mathcal{A},{⪯}_{i_2},d)$ be a complete bi-CVMS, and let $\mathcal{V},\mathcal{W}:\mathcal{A}$ $\to \mathcal{A}$. If there exist ${\mu }_1,{\mu }_2\in [0,1)$, with ${\mu }_1+\sqrt{2}{\mu }_2<1$, such that

$$d\left(\mathcal{V}\mathfrak{q},\mathcal{W}\kappa \right){⪯}_{i_2}{\mu }_{1}d\left(\mathfrak{q},\kappa \right)+{\mu }_2\frac{d\left(\mathfrak{q},\mathcal{V}\mathfrak{q}\right)d\left(\kappa ,\mathcal{W}\kappa \right)}{1+d\left(\mathfrak{q},\kappa \right)}$$

for all $\mathfrak{q},\kappa \in$ $\mathcal{A}$, then $\mathcal{V}$ and $\mathcal{W}$ have a unique common fixed point.

Proof. 

Take ${\mu }_i=0$, where $i=3,4,5,6,7$, in Corollary 9. □

5. Applications

The powerful tools of fixed point theory have proven immensely valuable for tackling various problems across diverse mathematical disciplines. In this section, we demonstrate the applicability of the established fixed point results to the realm of integral equations. These integral equations play a crucial role in numerous scientific and engineering fields, including population dynamics, heat transfer, finance, and many more. These equations often lack closed-form solutions, necessitating the use of alternative methods to establish their existence and uniqueness. Here, we will use the framework developed in the previous section (Corollary 4) to address a specific type of integral equation named the Volterra integral equation of the second kind. By translating the integral equation into a fixed point problem, we will demonstrate how our fixed point theorems can be utilized to guarantee the existence and, potentially, the uniqueness of solutions for this equation.

We consider the space $\mathcal{A}$ consisting of all real-valued continuous functions defined on the closed interval $[0,1],$ denoted by $\left(C[0,1],\mathbb{R}\right).$ Furthermore, define a distance function $d:\mathcal{A}\times \mathcal{A}\to {\mathbb{C}}_2$ as follows:

$$d(\mathfrak{q},\kappa )=\underset{t\in [0,1]}{max}\left(1+i\right)\left(\left|\mathfrak{q}\left(t\right)-\kappa \left(t\right)\right|\right)$$

for all $\mathfrak{q},\kappa \in$ $\mathcal{A}$ and $t\in [0,1].$ Then $(\mathcal{A},{⪯}_{i_2},d)$ qualifies as a complete bi-CVMS. Consider

$$\mathfrak{q}\left(t\right)={\int }_0^1{K}_1(t,s,\mathfrak{q}\left(s\right))ds+g\left(t\right),$$

(5)

$$\mathfrak{q}\left(t\right)={\int }_0^1{K}_2(t,s,\mathfrak{q}\left(s\right))ds+g\left(t\right),$$

(6)

where $g:$ $[0,1]\to \mathbb{R}$ and $K_1,K_2:[0,1]\times [0,1]\times \mathbb{R}\to \mathbb{R}$ are continuous for $t\in [0,1]$. Within the set of bicomplex numbers ${\mathbb{C}}_2,$ we introduce a partial order denoted by ${⪯}_{i_2}.$ This relation holds between two elements $\mathfrak{q}\left(t\right)$ and $\kappa \left(t\right)$ if and only if $\mathfrak{q}\le \kappa .$

Theorem 2. 

Suppose the following condition

$$\left|K_1(t,s,\mathfrak{q}\left(s\right))-K_2(t,s,\kappa \left(s\right))\right|\le {\mu }_1\left(\mathfrak{q},\kappa \right)\left|\mathfrak{q}\left(s\right)-\kappa \left(s\right)\right|$$

holds for all $\mathfrak{q},\kappa \in \mathcal{A}$, with $\mathfrak{q}\ne \kappa$ and for some control function ${\mu }_1:\mathcal{A}\times \mathcal{A}\to [0,1)$; then the integral operators defined by (5) and (6) possess a unique solution that is common to both operators.

Proof. 

Define the continuous mappings $\mathcal{V},\mathcal{W}:\mathcal{A}$ $\to \mathcal{A}$ by

$$\mathcal{V}\mathfrak{q}\left(t\right)={\int }_0^1{K}_1(t,s,\mathfrak{q}\left(s\right))ds+g\left(t\right),$$

$$\mathcal{W}\mathfrak{q}\left(t\right)={\int }_0^1{K}_2(t,s,\mathfrak{q}\left(s\right))ds+g\left(t\right),$$

for all $t\in [0,1].$ Consider

$$\begin{array}{cc}\hfill d\left(\mathcal{V}\mathfrak{q},\mathcal{W}\kappa \right)& =\underset{t\in [0,1]}{max}\left(1+i_2\right)\left|\mathcal{V}\mathfrak{q}\left(t\right)-T\kappa \left(t\right)\right|\hfill \\ & =\underset{t\in [0,1]}{max}\left(1+i_2\right)\left(\left|{\int }_0^1{K}_1(t,s,\mathfrak{q}\left(s\right))ds-{\int }_0^1{K}_2(t,s,\kappa \left(s\right))ds\right|\right)\hfill \\ & {⪯}_{i_2}\underset{t\in [0,1]}{max}\left(1+i_2\right)\left({\int }_0^1\left|K_1(t,s,\mathfrak{q}\left(s\right))-K_2(t,s,\kappa \left(s\right))\right|ds\right)\hfill \\ & {⪯}_{i_2}\underset{t\in [0,1]}{max}\left(1+i_2\right)\left({\mu }_1\left(\mathfrak{q},\kappa \right){\int }_0^1\left|\mathfrak{q}\left(s\right)-\kappa \left(s\right)\right|ds\right)\hfill \\ & {⪯}_{i_2}{\mu }_1\left(\mathfrak{q},\kappa \right)d\left(\mathfrak{q},\kappa \right)+{\mu }_2\left(\mathfrak{q},\kappa \right)\frac{d\left(\mathfrak{q},\mathcal{V}\mathfrak{q}\right)d\left(\kappa ,\mathcal{W}\kappa \right)}{1+d\left(\mathfrak{q},\kappa \right)}\hfill \\ & +{\mu }_3\left(\mathfrak{q},\kappa \right)\frac{d\left(\kappa ,\mathcal{V}\mathfrak{q}\right)d\left(\mathfrak{q},\mathcal{W}\kappa \right)}{1+d\left(\mathfrak{q},\kappa \right)}+{\mu }_4\left(\mathfrak{q},\kappa \right)\frac{d\left(\mathfrak{q},\mathcal{V}\mathfrak{q}\right)d\left(\mathfrak{q},\mathcal{W}\kappa \right)}{1+d\left(\mathfrak{q},\kappa \right)}\hfill \\ & +{\mu }_5\left(\mathfrak{q},\kappa \right)\frac{d\left(\kappa ,\mathcal{V}\mathfrak{q}\right)d\left(\kappa ,\mathcal{W}\kappa \right)}{1+d\left(\mathfrak{q},\kappa \right)}+{\mu }_6\left(\mathfrak{q},\kappa \right)\frac{d\left(\mathfrak{q},\mathcal{W}\kappa \right)\left[1+d\left(\mathfrak{q},\mathcal{V}\mathfrak{q}\right)\right]}{1+d\left(\mathfrak{q},\kappa \right)}\hfill \\ & +{\mu }_7\left(\mathfrak{q},\kappa \right)\frac{d\left(\kappa ,\mathcal{V}\mathfrak{q}\right)\left[1+d\left(\kappa ,\mathcal{W}\kappa \right)\right]}{1+d\left(\mathfrak{q},\kappa \right)}\hfill \end{array}$$

for any control functions ${\mu }_2,{\mu }_3,{\mu }_4,{\mu }_5,{\mu }_6,{\mu }_7:\mathcal{A}\times \mathcal{A}\to [0,1).$ Hence, all the conditions stipulated in Theorem 1 are met; the integral equations defined by (5) and (6) possess a unique common solution. □

6. Conclusions

In the present research article, we have obtained common fixed point results for rational contractions involving control functions of two variables in the framework of bi-CVMS. In this way, we have derived the leading results of Beg et al. [7], Gnanaprakasam et al. [8], and Tassaddiq et al. [11]. Furthermore, to substantiate the validity of our findings, we have presented a non-trivial example that demonstrates the effectiveness of the obtained results. To illustrate the applicability of our main result, we further investigated the existence and uniqueness of solutions for Volterra integral equations of the second kind.

Building upon the recent work of Özgür et al. [23,24] on the fixed-circle problem in metric spaces, a compelling avenue for future research would be to investigate this problem within the context of complete bi-CVMS. Even more generally, exploring suitable contractive conditions to establish existence and uniqueness theorems for fixed circles of self-mappings in these spaces holds significant promise for uncovering geometric interpretations and expanding the theory.