Solution of Integral Equation with Neutrosophic Rectangular Triple Controlled Metric Spaces

Abstract

In this paper, we introduce the notion of neutrosophic rectangular triple-controlled metric space, relaxing the symmetry requirement of neutrosophic metric spaces, by replacing triangular inequalities with rectangular inequalities, and prove fixed point theorems. We have derived several interesting results for contraction mappings supplemented with non-trivial examples. The derived results have been applied to prove the existence of a unique analytical solution as well as a closed form of the unique solution to the integral equation.

1. Introduction

Pursuant upon the reporting of the famous Contraction Mapping Theorem (CMT) by S.Banach [1] in 1922, the study of the existence and uniqueness of fixed points and common fixed points in metric and metric-like spaces and their applications has become a subject of great interest. In 1979, Itoh [2] presented the application of fixed point results to differential equations in Banach spaces. Many authors proved the Banach contraction principle in various generalized metric spaces. In the sequel, the notion of rectangular metric space was introduced by Branciari [3] in 2000. He replaced the right-hand side of the triangular inequality of the metric space with a three-term expression and established an analogous proof of the CMT. Since then, many fixed point theorems for various contractions on rectangular metric spaces have appeared in the literature [4,5,6,7,8,9,10].

In 1965, Zadeh [11] introduced the concept of fuzzy sets, which has varied applications in logical semantics. The concept of the continuous t-norm was introduced by Schweizer et al. [12]. Kramosil and Michlek [13] were the first to introduce the notion of fuzzy metric space, using continuous t-norms as an analog to metric spaces, and analyzed the notions with the probabilistic/statistical extension of metric spaces. The concept of fuzzy sets and fuzzy metric space has varied applications in applied sciences, such as fixed point theorems, signal and image processing including medical imaging, decision making, etc. Garbiec [14] reported the fuzzy extension of the Banach contraction mapping theorem. Since then, many fixed point results have been reported by researchers using different types of contractive conditions in the setting of fuzzy metric space, dislocated fuzzy metric space, intuitionistic fuzzy metric space, etc.; see [15,16,17,18,19,20,21,22,23,24,25,26,27,28,29]. More recently, Ali et al. [30] have presented some applications of the best proximity points of non-self maps in the setting of non-Archimedian fuzzy metric spaces.

In the recent past, many researchers have reported fixed point results in the setting of fuzzy metric spaces and the like. For instance, in 2018, Mlaiki [31] presented fixed point results by defining the concept of controlled metric spaces. Konwar [32], in 2020, defined intutionistic fuzzy b-metric space and established fixed point results under various contractive conditions. Saif Ur Rehman et al. [33] proved some $\alpha -\varphi$ fuzzy cone contraction results with integral-type application.

In the sequel, the concept of neutrosophic metric spaces was introduced by Kirisci and Simsek and various fixed point results were established by them in the setting of these spaces [34,35,36]. Subsequently, Sowndrarajan et al. [37] reported several fixed point results in neutrosophic metric spaces. Sezen [38] presented the concept of controlled fuzzy metric spaces and derived various fixed point results. The concept of fuzzy double-controlled metric space was given by Saleem et al. [39] in 2021. More recently, Uddin et al. [40] established the fixed point theorem on neutrosophic double-controlled metric space and presented an application to the derived result thereon.

Inspired by the above, in the present work, we define the notion of neutrosophic rectangular triple-controlled metric space and establish fixed point theorems. Accordingly, we have organized the rest of the manuscript as follows. Some preliminaries and a monograph are presented in Section 2. In Section 3, we define the neutrosophic metric space and define the Cauchy sequence and its convergence and establish fixed point results. We support the derived results with non-trivial examples. In Section 4, we establish the existence of a unique analytical solution to the Fredholm integral equation. We have also supplemented the derived results by finding the closed form of the unique solution to the intergral equation.

2. Preliminaries

A quick review of the following definitions and monograph will be useful in the sequel.

Definition 1

([19]). A binary operation $\star :[0,1]\times [0,1]\to [0,1]$ is called a continuous triangle norm if:

(1)

$\wp \star \sigma =\sigma \star \wp ,(\forall )\wp ,\sigma \in [0,1]$;

(2)

⋆ is continuous;

(3)

$\wp \star 1=\wp ,(\forall )\wp \in [0,1]$;

(4)

$(\wp \star \sigma )\star \kappa =\wp \star (\sigma \star \kappa )$, for all $\wp ,\sigma ,\kappa \in [0,1]$;

(5)

If $\wp \le \kappa$ and $\sigma \le \nu$, with $\wp ,\sigma ,\kappa ,\nu \in [0,1]$, then $\wp \star \sigma \le \kappa \star \nu$.

Definition 2

([19]). A binary operation $\diamond :[0,1]\times [0,1]\to [0,1]$ is called a continuous triangle co-norm if:

(1)

$\wp \diamond \sigma =\sigma \diamond \wp$, for all $\wp ,\sigma \in [0,1]$;

(2)

⋄ is continuous;

(3)

$\wp \diamond 0=0$, for all $\wp \in [0,1]$;

(4)

$(\wp \diamond \sigma )\diamond \kappa =\wp \diamond (\sigma \diamond \kappa )$, for all $\wp ,\sigma ,\kappa \in [0,1]$;

(5)

If $\wp \le \kappa$ and $\sigma \le \nu$, with $\wp ,\sigma ,\kappa ,\nu \in [0,1]$, then $\wp \diamond \sigma \le \kappa \diamond \nu$.

Definition 3

([2]). Given that $\beta ,\mathrm{\Gamma }:\Im \times \Im \to [1,+\infty )$ are non-comparable functions, if $\partial :\Im \times \Im \to [0,+\infty )$ satisfies the following conditions

(a)

$\partial (\vartheta ,\mathcal{Q})=0$ iff $\vartheta =\mathcal{Q}$;

(b)

$\partial (\vartheta ,\mathcal{Q})=\partial (\mathcal{Q},\vartheta )$;

(c)

$\partial (\vartheta ,\mathcal{Q})\le \beta (\vartheta ,\psi )\partial (\vartheta ,\psi )+\mathrm{\Gamma }(\psi ,\mathcal{Q})\partial (\psi ,\mathcal{Q})$,

for all $\vartheta ,\mathcal{Q},\psi \in \Im$. Then, $(\Im ,\partial )$ is said to be a double-controlled metric space.

Definition 4

([39]). Suppose $\Im \ne \varnothing$ and $\beta ,\mathrm{\Gamma }:\Im \times \Im \to [1,+\infty )$ are two non-comparable functions, ⋆ is a continuous t-norm and $\mathcal{W}$ is a fuzzy set on $\Im \times \Im \times (0,+\infty )$. It is said to be a fuzzy double-controlled metric on ℑ, for all $\vartheta ,\mathcal{Q},\psi \in \Im$ if

(i)

$\mathcal{W}(\vartheta ,\mathcal{Q},0)=0$;

(ii)

$\mathcal{W}(\vartheta ,\mathcal{Q},\varsigma )=1$ for all $\varsigma >0$, if and only if $\vartheta =\mathcal{Q}$;

(iii)

$\mathcal{W}(\vartheta ,\mathcal{Q},\varsigma )=\mathcal{W}(\mathcal{Q},\vartheta ,\varsigma )$;

(iv)

$\mathcal{W}(\vartheta ,\psi ,\varsigma +\varsigma )\ge \mathcal{W}\left(\vartheta ,\mathcal{Q},\frac{\varsigma }{\beta (\vartheta ,\mathcal{Q})}\right)\star \mathcal{W}\left(\mathcal{Q},\psi ,\frac{\varsigma }{\mathrm{\Gamma }(\mathcal{Q},\psi )}\right)$;

(v)

$\mathcal{W}(\vartheta ,\mathcal{Q},·):(0,+\infty )\to [0,1]$ is left continuous.

Then, $(\Im ,\mathcal{W},\mathcal{E},\star )$ is said to be a fuzzy double-controlled metric space.

Definition 5

([32]). Take $\Im \ne \varnothing$. Let ⋆ be a continuous t-norm, ⋄ be a continuous t-co-norm, $\mathfrak{b}\ge 1$ and $\mathcal{W},\mathcal{E}$ be fuzzy sets on $\Im \times \Im \times (0,+\infty )$. If $(\Im ,\mathcal{W},\mathcal{E},\star ,\diamond )$ fullfils all $\vartheta ,\mathcal{Q}\in \Im$ and $\chi ,\varsigma >0$,

(I)

$\mathcal{W}(\vartheta ,\mathcal{Q},\varsigma )+\mathcal{E}(\vartheta ,\mathcal{Q},\varsigma )\le 1$;

(II)

$\mathcal{W}(\vartheta ,\mathcal{Q},\varsigma )>0$;

(III)

$\mathcal{W}(\vartheta ,\mathcal{Q},\varsigma )=1\iff \vartheta =\mathcal{Q}$;

(IV)

$\mathcal{W}(\vartheta ,\mathcal{Q},\varsigma )=\mathcal{W}(\mathcal{Q},\vartheta ,\varsigma )$;

(V)

$\mathcal{W}(\vartheta ,\psi ,\mathfrak{b}(\varsigma +\chi \left)\right)\ge \mathcal{W}(\vartheta ,\mathcal{Q},\varsigma )\star \mathcal{W}(\mathcal{Q},\psi ,\chi )$;

(VI)

$\mathcal{W}(\vartheta ,\mathcal{Q},·)$ is a non-decreasing function of ${\mathbb{R}}^{+}$ and ${lim}_{\varsigma \to +\infty }\mathcal{W}(\vartheta ,\mathcal{Q},\varsigma )=1$;

(VII)

$\mathcal{E}(\vartheta ,\mathcal{Q},\varsigma )>0$;

(VIII)

$\mathcal{E}(\vartheta ,\mathcal{Q},\varsigma )=0\iff \vartheta =\mathcal{Q}$;

(IX)

$\mathcal{E}(\vartheta ,\mathcal{Q},\varsigma )=\mathcal{E}(\mathcal{Q},\vartheta ,\varsigma )$;

(X)

$\mathcal{E}(\vartheta ,\psi ,\mathfrak{b}(\varsigma +\chi \left)\right)\le \mathcal{E}(\vartheta ,F\mathcal{Q},\varsigma )\diamond \mathcal{E}(\mathcal{Q},\psi ,\chi )$;

(XI)

$\mathcal{E}(\vartheta ,\mathcal{Q},·)$ is a non-increasing function of ${\mathbb{R}}^{+}$ and ${lim}_{\varsigma \to +\infty }\mathcal{E}(\vartheta ,\mathcal{Q},\varsigma )=0$.

Then, $(\Im ,\mathcal{W},\mathcal{E},\star ,\diamond )$ is an intuitionistic fuzzy $\mathfrak{b}$-metric space.

Definition 6

([36]). Let $\Im \ne \varnothing ,\star$ be a continuous t-norm, ⋄ be a continuous t-co-norm, and $\mathcal{W},\mathcal{E},{\rm Y}$ are neutrosophic sets on $\Im \times \Im \times (0,+\infty )$. It is said to be a neutosophic metric on ℑ, if, for all $\vartheta ,\mathcal{Q},\psi \in \Im$, the following conditions are satisfied:

(1)

$\mathcal{W}(\vartheta ,\mathcal{Q},\varsigma )+\mathcal{E}(\vartheta ,\mathcal{Q},\varsigma )+{\rm Y}(\vartheta ,\mathcal{Q},\varsigma )\le 3$;

(2)

$\mathcal{W}(\vartheta ,\mathcal{Q},\varsigma )>0$;

(3)

$\mathcal{W}(\vartheta ,\mathcal{Q},\varsigma )=1$ for all $\varsigma >0$, if and only if $\vartheta =\mathcal{Q}$;

(4)

$\mathcal{W}(\vartheta ,\mathcal{Q},\varsigma )=\mathcal{W}(\mathcal{Q},\vartheta ,\varsigma )$;

(5)

$\mathcal{W}(\vartheta ,\psi ,\varsigma +\chi )\ge \mathcal{W}(\vartheta ,\mathcal{Q},\varsigma )\star \mathcal{W}(\mathcal{Q},\psi ,\chi )$;

(6)

$\mathcal{W}(\vartheta ,\mathcal{Q},·):(0,+\infty )\to [0,1]$ is continuous and ${lim}_{\varsigma \to +\infty }\mathcal{W}(\vartheta ,\mathcal{Q},\varsigma )=1$;

(7)

$\mathcal{E}(\vartheta ,\mathcal{Q},\varsigma )<1$;

(8)

$\mathcal{E}(\vartheta ,\mathcal{Q},\varsigma )=0$ for all $\varsigma >0$, if and only if $\vartheta =\mathcal{Q}$;

(9)

$\mathcal{E}(\vartheta ,\mathcal{Q},\varsigma )=\mathcal{E}(\mathcal{Q},\vartheta ,\varsigma )$;

(10)

$\mathcal{E}(\vartheta ,\psi ,\varsigma +\chi )\le \mathcal{E}(\vartheta ,\mathcal{Q},\varsigma )\diamond \mathcal{E}(\mathcal{Q},\psi ,\chi )$;

(11)

$\mathcal{E}(\vartheta ,\mathcal{Q},·):(0,+\infty )\to [0,1]$ is continuous and ${lim}_{\varsigma \to +\infty }\mathcal{E}(\vartheta ,\mathcal{Q},\varsigma )=0$;

(12)

${\rm Y}(\vartheta ,\mathcal{Q},\varsigma )<1$;

(13)

${\rm Y}(\vartheta ,\mathcal{Q},\varsigma )=0$ for all $\varsigma >0$, if and only if $\vartheta =\mathcal{Q}$;

(14)

${\rm Y}(\vartheta ,\mathcal{Q},\varsigma )={\rm Y}(\mathcal{Q},\vartheta ,\varsigma )$;

(15)

${\rm Y}(\vartheta ,\psi ,\varsigma +\chi )\le {\rm Y}(\vartheta ,\mathcal{Q},\varsigma )\diamond {\rm Y}(\mathcal{Q},\psi ,\chi )$;

(16)

${\rm Y}(\vartheta ,\mathcal{Q},·):(0,+\infty )\to [0,1]$ is continuous and ${lim}_{\varsigma \to +\infty }{\rm Y}(\vartheta ,\mathcal{Q},\varsigma )=0$;

(17)

If $\varsigma \le 0$, then $\mathcal{W}(\vartheta ,\mathcal{Q},\varsigma )=0,\mathcal{E}(\vartheta ,\mathcal{Q},\varsigma )=0$.

Then, $(\Im ,\mathcal{W},\mathcal{E},{\rm Y},\star ,\diamond )$ is called a neutrosophic metric space.

Now, we present our main results.

3. Main Results

We commence this section by defining neutrosophic rectangular triple-controlled metric space.

Definition 7.

Let $\Im \ne \varnothing$ and $\beta ,\mathrm{\Gamma },\eta :\Im \times \Im \to [1,+\infty )$ be given non-comparable functions, ⋆ be a continuous t-norm, and ⋄ be a continuous t-co-norm. The neutrosophic set $\mathcal{W},\mathcal{E},\mathcal{G}$ on $\Im \times \Im \times (0,+\infty )$ is said to be a neutrosophic rectangular triple-controlled metric on ℑ, if, for any $\vartheta ,\psi \in \Im$ and all distinct $\mathfrak{x},\mathcal{Q}\in \Im \backslash \{\vartheta ,\psi \}$, the following conditions are satisfied:

(i)

$\mathcal{W}(\vartheta ,\mathcal{Q},\varsigma )+\mathcal{E}(\vartheta ,\mathcal{Q},\varsigma )+\mathcal{G}(\vartheta ,\mathcal{Q},\varsigma )\le 3$;

(ii)

$\mathcal{W}(\vartheta ,\mathcal{Q},\varsigma )>0$;

(iii)

$\mathcal{W}(\vartheta ,\mathcal{Q},\varsigma )=1$ for all $\varsigma >0$, if and only if $\vartheta =\mathcal{Q}$;

(iv)

$\mathcal{W}(\vartheta ,\mathcal{Q},\varsigma )=\mathcal{W}(\mathcal{Q},\vartheta ,\varsigma )$;

(v)

$\mathcal{W}(\vartheta ,\psi ,\varsigma +\chi +\check{w})\ge \mathcal{W}\left(\vartheta ,\mathcal{Q},\frac{\varsigma }{\beta (\vartheta ,\mathcal{Q})}\right)\star \mathcal{W}\left(\mathcal{Q},\mathfrak{x},\frac{\chi }{\mathrm{\Gamma }(\mathcal{Q},\mathfrak{x})}\right)\star \mathcal{W}\left(\mathfrak{x},\psi ,\frac{\check{w}}{\eta (\mathfrak{x},\psi )}\right)$;

(vi)

$\mathcal{W}(\vartheta ,\mathcal{Q},·):(0,+\infty )\to [0,1]$ is continuous and ${lim}_{\varsigma \to +\infty }\mathcal{W}(\vartheta ,\mathcal{Q},\varsigma )=1$;

(vii)

$\mathcal{E}(\vartheta ,\mathcal{Q},\varsigma )<1$;

(viii)

$\mathcal{E}(\vartheta ,\mathcal{Q},\varsigma )=0$ for all $\varsigma >0$, if and only if $\vartheta =\mathcal{Q}$;

(ix)

$\mathcal{E}(\vartheta ,\mathcal{Q},\varsigma )=\mathcal{E}(\mathcal{Q},\vartheta ,\varsigma )$;

(x)

$\mathcal{E}(\vartheta ,\psi ,\varsigma +\chi +\check{w})\le \mathcal{E}\left(\vartheta ,\mathcal{Q},\frac{\varsigma }{\beta (\vartheta ,\mathcal{Q})}\right)\diamond \mathcal{E}\left(\mathcal{Q},\mathfrak{x},\frac{\chi }{\mathrm{\Gamma }(\mathcal{Q},\mathfrak{x})}\right)\diamond \mathcal{E}\left(\mathfrak{x},\psi ,\frac{\check{w}}{\eta (\mathfrak{x},\psi )}\right)$;

(xi)

$\mathcal{E}(\vartheta ,\mathcal{Q},·):(0,+\infty )\to [0,1]$ is continuous and ${lim}_{\varsigma \to +\infty }\mathcal{E}(\vartheta ,\mathcal{Q},\varsigma )=0$;

(xii)

$\mathcal{G}(\vartheta ,\mathcal{Q},\varsigma )<1$;

(xiii)

$\mathcal{G}(\vartheta ,\mathcal{Q},\varsigma )=0$ for all $\varsigma >0$, if and only if $\vartheta =\mathcal{Q}$;

(xiv)

$\mathcal{G}(\vartheta ,\mathcal{Q},\varsigma )=\mathcal{G}(\mathcal{Q},\vartheta ,\varsigma )$;

(xv)

$\mathcal{G}(\vartheta ,\psi ,\varsigma +\chi +\check{w})\le \mathcal{G}\left(\vartheta ,\mathcal{Q},\frac{\varsigma }{\beta (\vartheta ,\mathcal{Q})}\right)\diamond \mathcal{G}\left(\mathcal{Q},\mathfrak{x},\frac{\chi }{\mathrm{\Gamma }(\mathcal{Q},\mathfrak{x})}\right)\diamond \mathcal{G}\left(\mathfrak{x},\psi ,\frac{\chi }{\eta (\mathfrak{x},\psi )}\right)$;

(xvi)

$\mathcal{G}(\vartheta ,\mathcal{Q},·):(0,+\infty )\to [0,1]$ is continuous and ${lim}_{\varsigma \to +\infty }\mathcal{G}(\vartheta ,\mathcal{Q},\varsigma )=0$;

(xvii)

If $\varsigma \le 0$, then $\mathcal{W}(\vartheta ,\mathcal{Q},\varsigma )=0,\mathcal{E}(\vartheta ,\mathcal{Q},\varsigma )=1$ and ${\rm Y}(\vartheta ,\mathcal{Q},\varsigma )=1$.

Then, $(\Im ,\mathcal{W},\mathcal{E},\mathcal{G},\star ,\diamond )$ is called a neutrosophic rectangular triple-controlled metric space.

Example 1.

Let $\Im =\{1,2,3,4\}$ and $\beta ,\mathrm{\Gamma },\eta :\Im \times \Im \to [1,+\infty )$ be three non-comparable functions given by $\beta (\vartheta ,\mathcal{Q})=\vartheta +\mathcal{Q}+1$, $\mathrm{\Gamma }(\vartheta ,\mathcal{Q})={\vartheta }^2+{\mathcal{Q}}^2+1$, and $\eta (\vartheta ,\mathcal{Q})={\vartheta }^2+{\mathcal{Q}}^2-1$. Define $\mathcal{W},\mathcal{E},\mathcal{G}:\Im \times \Im \times (0,+\infty )\to [0,1]$ as

$$\begin{array}{cc}\hfill \mathcal{W}(\vartheta ,\mathcal{Q},\varsigma )& =\left\{\begin{array}{cc}1,\hfill & \mathit{if}\vartheta =\mathcal{Q}\hfill \\ \frac{\varsigma }{\varsigma +max\{\vartheta ,\mathcal{Q}\}},\hfill & \mathit{if}\mathit{otherwise},\hfill \end{array}\right.\hfill \\ \hfill \mathcal{E}(\vartheta ,\mathcal{Q},\varsigma )& =\left\{\begin{array}{cc}0,\hfill & \mathit{if}\vartheta =\mathcal{Q}\hfill \\ \frac{max\{\vartheta ,\mathcal{Q}\}}{\varsigma +max\{\vartheta ,\mathcal{Q}\}},\hfill & \mathit{if}\mathit{otherwise},\hfill \end{array}\right.\hfill \end{array}$$

and

$$\begin{array}{c}\hfill \mathcal{G}(\vartheta ,\mathcal{Q},\varsigma )=\left\{\begin{array}{cc}0,\hfill & \mathit{if}\vartheta =\mathcal{Q}\hfill \\ \frac{max\{\vartheta ,\mathcal{Q}\}}{\varsigma },\hfill & \mathit{if}\mathit{otherwise}.\hfill \end{array}\right.\end{array}$$

Then, $(\Im ,\mathcal{W},\mathcal{E},\mathcal{G},\star ,\diamond )$ is a neutrosophic rectangular triple-controlled metric space with continuous t-norm $\wp \star \sigma =\wp \sigma$ and continuous t-co-norm, $\wp \diamond \overline{a}=max\{\wp ,\overline{a}\}$.

Proof. 

We need to prove only (v), (x) and (xv).

Let $\vartheta =1,\mathcal{Q}=2$, $\mathfrak{x}=3$ and $\psi =4$. Then

$$\begin{array}{c}\hfill \mathcal{W}(1,4,\varsigma +\chi +\check{w})=\frac{\varsigma +\chi +\check{w}}{\varsigma +\chi +\check{w}+max\{1,4\}}=\frac{\varsigma +\chi +\check{w}}{\varsigma +\chi +\check{w}+4},\end{array}$$

whereas

$$\begin{array}{c}\hfill \mathcal{W}\left(1,2,\frac{\varsigma }{\beta (1,2)}\right)=\frac{\frac{\varsigma }{\beta (1,2)}}{\frac{\varsigma }{\beta (1,2)}+max\{1,2\}}=\frac{\frac{\varsigma }{4}}{\frac{\varsigma }{4}+2}=\frac{\varsigma }{\varsigma +8},\end{array}$$

$$\begin{array}{c}\hfill \mathcal{W}\left(2,3,\frac{\chi }{\mathrm{\Gamma }(2,3)}\right)=\frac{\frac{\chi }{\mathrm{\Gamma }(2,3)}}{\frac{\chi }{\mathrm{\Gamma }(2,3)}+max\{2,3\}}=\frac{\frac{\chi }{12}}{\frac{\chi }{12}+3}=\frac{\chi }{\chi +36},\end{array}$$

and

$$\begin{array}{c}\hfill \mathcal{W}\left(3,4,\frac{\check{w}}{\eta (3,4)}\right)=\frac{\frac{\check{w}}{\eta (3,4)}}{\frac{\check{w}}{\eta (3,4)}+max\{3,4\}}=\frac{\frac{\check{w}}{24}}{\frac{\check{w}}{24}+4}=\frac{\check{w}}{\check{w}+96}.\end{array}$$

That is,

$$\begin{array}{c}\hfill \frac{\varsigma +\chi +\check{w}}{\varsigma +\chi +\check{w}+3}\ge \frac{\varsigma }{\varsigma +8}·\frac{\chi }{\chi +36}.\frac{\check{w}}{\check{w}+96}.\end{array}$$

Then, this satisfies all $\varsigma ,\chi ,\check{w}>0$. Hence,

$$\begin{array}{c}\hfill \mathcal{W}(\vartheta ,\psi ,\varsigma +\chi +\check{w})\ge \mathcal{W}\left(\vartheta ,\mathcal{Q},\frac{\varsigma }{\beta (\vartheta ,\mathcal{Q})}\right)\star \mathcal{W}\left(\mathcal{Q},\mathfrak{x},\frac{\chi }{\mathrm{\Gamma }(\mathcal{Q},\mathfrak{x})}\right)\star \mathcal{W}\left(\mathfrak{x},\psi ,\frac{\check{w}}{\eta (\mathfrak{x},\psi )}\right).\end{array}$$

Now,

$$\begin{array}{c}\hfill \mathcal{E}(1,4,\varsigma +\chi +\check{w})=\frac{max\{1,4\}}{\varsigma +\chi +\check{w}+max\{1,4\}}=\frac{4}{\varsigma +\chi +\check{w}+4},\end{array}$$

whereas

$$\begin{array}{c}\hfill \mathcal{E}\left(1,2,\frac{\varsigma }{\beta (1,2)}\right)=\frac{max\{1,2\}}{\frac{\varsigma }{\beta (1,2)}+max\{1,2\}}=\frac{2}{\frac{\varsigma }{4}+2}=\frac{8}{\varsigma +8},\end{array}$$

$$\begin{array}{c}\hfill \mathcal{E}\left(2,3,\frac{\chi }{\mathrm{\Gamma }(2,3)}\right)=\frac{max\{2,3\}}{\frac{\chi }{\mathrm{\Gamma }(2,3)}+max\{2,3\}}=\frac{3}{\frac{\chi }{12}+3}=\frac{36}{\chi +36},\end{array}$$

and

$$\begin{array}{c}\hfill \mathcal{E}\left(3,4,\frac{\check{w}}{\eta (3,4)}\right)=\frac{max\{3,4\}}{\frac{\check{w}}{\eta (3,4)}+max\{3,4\}}=\frac{4}{\frac{\check{w}}{24}+4}=\frac{96}{\check{w}+96}.\end{array}$$

That is,

$$\begin{array}{c}\hfill \frac{4}{\varsigma +\chi +\check{w}+4}\le max\left\{\frac{8}{\varsigma +8},\frac{36}{\chi +36},\frac{96}{\check{w}+96}\right\}.\end{array}$$

The above expression is true for all $\varsigma ,\chi ,\check{w}>0$.

Hence,

$$\begin{array}{c}\hfill \mathcal{E}(\vartheta ,\psi ,\varsigma +\chi +\check{w})\le \mathcal{E}\left(\vartheta ,\mathcal{Q},\frac{\varsigma }{\beta (\vartheta ,\mathcal{Q})}\right)\diamond \mathcal{E}\left(\mathfrak{x},\psi ,\frac{\chi }{\mathrm{\Gamma }(\mathfrak{x},\psi )}\right)\diamond \mathcal{E}\left(\mathfrak{x},\psi ,\frac{\check{w}}{\eta (\mathfrak{x},\psi )}\right).\end{array}$$

Now,

$$\begin{array}{c}\hfill \mathcal{G}(1,3,\varsigma +\chi +\check{w})=\frac{max\{1,3\}}{\varsigma +\chi +\check{w}}=\frac{3}{\varsigma +\chi +\check{w}},\end{array}$$

whereas

$$\begin{array}{c}\hfill \mathcal{G}\left(1,2,\frac{\varsigma }{\mathrm{\Gamma }(1,2)}\right)=\frac{max\{1,2\}}{\frac{\varsigma }{\mathrm{\Gamma }(1,2)}}=\frac{2}{\frac{\varsigma }{4}}=\frac{8}{\varsigma },\end{array}$$

$$\begin{array}{c}\hfill \mathcal{G}\left(2,3,\frac{\chi }{\mathrm{\Gamma }(2,3)}\right)=\frac{max\{2,3\}}{\frac{\chi }{\mathrm{\Gamma }(2,3)}}=\frac{3}{\frac{\chi }{12}}=\frac{36}{\chi },\end{array}$$

and

$$\begin{array}{c}\hfill \mathcal{G}\left(3,4,\frac{\check{w}}{\eta (3,4)}\right)=\frac{max\{3,4\}}{\frac{\check{w}}{\eta (3,4)}}=\frac{4}{\frac{\check{w}}{24}}=\frac{96}{\check{w}}.\end{array}$$

That is,

$$\begin{array}{c}\hfill \frac{3}{\varsigma +\chi +\check{w}}\le max\left\{\frac{8}{\varsigma },\frac{36}{\chi },\frac{96}{\check{w}}\right\}.\end{array}$$

Then, it satisfies all $\varsigma ,\chi >0$. Hence,

$$\begin{array}{c}\hfill \mathcal{G}(\vartheta ,\psi ,\varsigma +\chi +\check{w})\le \mathcal{G}\left(\vartheta ,\mathcal{Q},\frac{\varsigma }{\beta (\vartheta ,\mathcal{Q})}\right)\diamond \mathcal{G}\left(\mathcal{Q},\mathfrak{x},\frac{\chi }{\mathrm{\Gamma }(\mathcal{Q},\mathfrak{x})}\right)\diamond \mathcal{G}\left(\mathfrak{x},\psi ,\frac{\chi }{\eta (\mathfrak{x},\psi )}\right).\end{array}$$

Hence, $(\Im ,\mathcal{W},\mathcal{E},\mathcal{G},\star ,\diamond )$ is a neutrosophic rectangular triple-controlled metric space. □

Remark 1.

It can be seen that the preceding example satisfies both the continuous t-norm $\wp \star \overline{a}=min\{\wp ,\overline{a}\}$ and continuous t-co-norm $\wp \diamond \overline{a}=max\{\wp ,\overline{a}\}$.

Example 2.

Let $\Im =\mathcal{B}\cup \mathcal{U}$, where $\mathcal{B}=\{0,\frac{1}{2},\frac{1}{3},\frac{1}{4}\}$ and $\mathcal{U}=[1,2]$ and $\beta ,\mathrm{\Gamma },\eta :\Im \times \Im \to [1,+\infty )$ be given by $\beta (\vartheta ,\mathcal{Q})=1$, $\mathrm{\Gamma }(\vartheta ,\mathcal{Q})=1$ and $\eta (\vartheta ,\mathcal{Q})=1$. Define $\nu :\Im \times \Im \to [0,+\infty )$ as follows:

$$\left\{\begin{array}{c}\nu (\vartheta ,\mathcal{Q})=\nu (\mathcal{Q},\vartheta )\mathit{for}\mathit{all}\vartheta ,\mathcal{Q}\in \Im \hfill \\ \nu (\vartheta ,\mathcal{Q})=0\iff \vartheta =\mathcal{Q},\hfill \end{array}\right.$$

and

$$\left\{\begin{array}{c}\nu (0,\frac{1}{2})=\nu (\frac{1}{2},\frac{1}{3})=0.2\hfill \\ \nu (0,\frac{1}{3})=\nu (\frac{1}{3},\frac{1}{4})=0.02\hfill \\ \nu (0,\frac{1}{4})=\nu (\frac{1}{2},\frac{1}{4})=0.5\hfill \\ \nu (\vartheta ,\mathcal{Q})=|\vartheta -\mathcal{Q}|,\mathit{otherwise}.\hfill \end{array}\right.$$

Define $\mathcal{W},\mathcal{E},\mathcal{G}:\Im \times \Im \times (0,+\infty )\to [0,1]$ as

$$\begin{array}{cc}\hfill \mathcal{W}(\vartheta ,\mathcal{Q},\varsigma )& =\frac{\varsigma }{\varsigma +\nu (\vartheta ,\mathcal{Q})},\hfill \\ \hfill \mathcal{E}(\vartheta ,\mathcal{Q},\varsigma )& =\frac{\nu (\vartheta ,\mathcal{Q})}{\varsigma +\nu (\vartheta ,\mathcal{Q})},\mathcal{G}(\vartheta ,\mathcal{Q},\varsigma )=\frac{\nu (\vartheta ,\mathcal{Q})}{\varsigma }.\hfill \end{array}$$

Then, we have

$$\begin{array}{c}\hfill \mathcal{W}(\vartheta ,\psi ,\varsigma +\chi +\check{w})\ge \mathcal{W}\left(\vartheta ,\mathcal{Q},\frac{\varsigma }{\beta (\vartheta ,\mathcal{Q})}\right)\star \mathcal{W}\left(\mathcal{Q},\mathfrak{x},\frac{\chi }{\mathrm{\Gamma }(\mathcal{Q},\mathfrak{x})}\right)\star \mathcal{W}\left(\mathfrak{x},\psi ,\frac{\check{w}}{\eta (\mathfrak{x},\psi )}\right),\end{array}$$

$$\begin{array}{c}\hfill \mathcal{E}(\vartheta ,\psi ,\varsigma +\chi +\check{w})\le \mathcal{E}\left(\vartheta ,\mathcal{Q},\frac{\varsigma }{\beta (\vartheta ,\mathcal{Q})}\right)\diamond \mathcal{E}\left(\mathfrak{x},\psi ,\frac{\chi }{\mathrm{\Gamma }(\mathfrak{x},\psi )}\right)\diamond \mathcal{E}\left(\mathfrak{x},\psi ,\frac{\check{w}}{\eta (\mathfrak{x},\psi )}\right),\end{array}$$

$$\begin{array}{c}\hfill \mathcal{G}(\vartheta ,\psi ,\varsigma +\chi +\check{w})\le \mathcal{G}\left(\vartheta ,\mathcal{Q},\frac{\varsigma }{\beta (\vartheta ,\mathcal{Q})}\right)\diamond \mathcal{G}\left(\mathcal{Q},\mathfrak{x},\frac{\chi }{\mathrm{\Gamma }(\mathcal{Q},\mathfrak{x})}\right)\diamond \mathcal{G}\left(\mathfrak{x},\psi ,\frac{\chi }{\eta (\mathfrak{x},\psi )}\right).\end{array}$$

Then, $(\Im ,\mathcal{W},\mathcal{E},\mathcal{G},\star ,\diamond )$ is a neutrosophic rectangular triple-controlled metric space with continuous t-norm $\wp \star \overline{a}=\wp \overline{a}$ and continuous t-co-norm $\wp \diamond \overline{a}=max\{\wp ,\overline{a}\}$.

Definition 8.

Let $(\Im ,\mathcal{W},\mathcal{E},\mathcal{G},\star ,\diamond )$ be a neutrosophic rectangular triple-controlled metric space, an open ball with center ϑ, radius $\mathfrak{r},0<\mathfrak{r}<1$ and $\varsigma >0$ on $\mathcal{G}(\vartheta ,\mathfrak{r},\varsigma )$ is defined as below:

$$\begin{array}{c}\hfill \mathcal{G}(\vartheta ,\mathfrak{r},\varsigma )=\{\mathcal{Q}\in \Im :\mathcal{W}(\vartheta ,\mathcal{Q},\varsigma )>1-\mathfrak{r},\mathcal{E}(\vartheta ,\mathcal{Q},\varsigma )<\mathfrak{r},\mathcal{G}(\vartheta ,\mathcal{Q},\varsigma )<\mathfrak{r}\}.\end{array}$$

Definition 9.

Let $(\Im ,\mathcal{W},\mathcal{E},\mathcal{G},\star ,\diamond )$ be a neutrosophic rectangular triple-controlled metric space and $\left\{{\vartheta }_{\mathfrak{j}}\right\}$ be a sequence in ℑ. Then, $\left\{{\vartheta }_{\mathfrak{j}}\right\}$ is said to be

(a)

convergent if there exists $\vartheta \in \Im$ such that

$$\begin{array}{c}\hfill \underset{\mathfrak{j}\to +\infty }{lim}\mathcal{W}({\vartheta }_{\mathfrak{j}},\vartheta ,\varsigma )=1,\underset{\mathfrak{j}\to +\infty }{lim}\mathcal{E}({\vartheta }_{\mathfrak{j}},\vartheta ,\varsigma )=0,\underset{\mathfrak{j}\to +\infty }{lim}\mathcal{G}({\vartheta }_{\mathfrak{j}},\vartheta ,\varsigma )=0\mathit{for}\mathit{all}\varsigma >0,\end{array}$$

(b)

Cauchy, if and only if, for each $\overline{a}>0,\varsigma >0$, there exists ${\mathfrak{j}}_0\in \mathbb{N}$ such that

$$\begin{array}{c}\hfill \mathcal{W}({\vartheta }_{\mathfrak{j}},{\vartheta }_{\mathfrak{m}},\varsigma )\ge 1-\overline{a},\mathcal{E}({\vartheta }_{\mathfrak{j}},{\vartheta }_{\mathfrak{m}},\varsigma )\le \overline{a},\mathcal{E}({\vartheta }_{\mathfrak{j}},{\vartheta }_{\mathfrak{m}},\varsigma )\le \overline{a}\mathit{for}\mathit{all}\mathfrak{j},\mathfrak{m}\ge {\mathfrak{j}}_0,\end{array}$$

$(\Im ,\mathcal{W},\mathcal{E},\mathcal{G},\star ,\diamond )$ is called a complete neutrosophic rectangular triple-controlled metric space if every Cauchy seqeunce is convergent in ℑ.

Lemma 1.

Let $(\Im ,\mathcal{W},\mathcal{E},\mathcal{G},\star ,\diamond )$ be a neutrosophic rectangular triple-controlled metric space. If, for some $0<\rho <1$ and for any $\vartheta ,\mathcal{Q}\in \Im ,\varsigma >0$,

$$\begin{array}{c}\hfill \mathcal{W}(\vartheta ,\mathcal{Q},\varsigma )\ge \mathcal{W}\left(\vartheta ,\mathcal{Q},\frac{\varsigma }{\rho }\right),\mathcal{E}(\vartheta ,\mathcal{Q},\varsigma )\le \mathcal{E}\left(\vartheta ,\mathcal{Q},\frac{\varsigma }{\rho }\right),\mathcal{G}(\vartheta ,\mathcal{Q},\varsigma )\le \mathcal{G}\left(\vartheta ,\mathcal{Q},\frac{\varsigma }{\rho }\right),\end{array}$$

(1)

then, $\vartheta =\mathcal{Q}$.

Proof. 

(1) implies that

$$\begin{array}{c}\hfill \mathcal{W}(\vartheta ,\mathcal{Q},\varsigma )\ge \mathcal{W}\left(\vartheta ,\mathcal{Q},\frac{\varsigma }{{\rho }^{\mathfrak{j}}}\right),\mathcal{E}(\vartheta ,\mathcal{Q},\varsigma )\le \mathcal{E}\left(\vartheta ,\mathcal{Q},\frac{\varsigma }{{\rho }^{\mathfrak{j}}}\right),\mathcal{G}(\vartheta ,\mathcal{Q},\varsigma )\le \mathcal{G}\left(\vartheta ,\mathcal{Q},\frac{\varsigma }{{\rho }^{\mathfrak{j}}}\right),\mathfrak{j}\in \mathbb{N},\varsigma >0.\end{array}$$

Now,

$$\begin{array}{cc}\hfill \mathcal{W}(\vartheta ,\mathcal{Q},\varsigma )& \ge \underset{\mathfrak{j}\to +\infty }{lim}\mathcal{W}\left(\vartheta ,\mathcal{Q},\frac{\varsigma }{{\rho }^{\mathfrak{j}}}\right)=1,\hfill \\ \hfill \mathcal{E}(\vartheta ,\mathcal{Q},\varsigma )& \le \underset{\mathfrak{j}\to +\infty }{lim}\mathcal{E}\left(\vartheta ,\mathcal{Q},\frac{\varsigma }{{\rho }^{\mathfrak{j}}}\right)=0,\hfill \\ \hfill \mathcal{G}(\vartheta ,\mathcal{Q},\varsigma )& \le \underset{\mathfrak{j}\to +\infty }{lim}\mathcal{G}\left(\vartheta ,\mathcal{Q},\frac{\varsigma }{{\rho }^{\mathfrak{j}}}\right)=0,\varsigma >0.\hfill \end{array}$$

Moreover, from (iii), (viii), (xiii), we have $\vartheta =\mathcal{Q}$. □

Theorem 1.

Suppose $(\Im ,\mathcal{W},\mathcal{E},\mathcal{G},\star ,\diamond )$ is a complete neutrosophic rectangular triple-controlled metric space in the company of $\beta ,\mathrm{\Gamma },\eta :\Im \times \Im \to [1,+\infty )$ with $0<\rho <1$. Let $\mathfrak{p}:\Im \to \Im$ be a mapping satisfying

$$\begin{array}{cc}\hfill \mathcal{W}(\mathfrak{p}\vartheta ,\mathfrak{p}\mathcal{Q},\rho \varsigma )& \ge \mathcal{W}(\vartheta ,\mathcal{Q},\varsigma ),\hfill \\ \hfill \mathcal{E}(\mathfrak{p}\vartheta ,\mathfrak{p}\mathcal{Q},\rho \varsigma )\le \mathcal{E}(\vartheta ,\mathcal{Q},\varsigma )& \mathit{and}\mathcal{G}(\mathfrak{p}\vartheta ,\mathfrak{p}\mathcal{Q},\rho \varsigma )\le \mathcal{G}(\vartheta ,\mathcal{Q},\varsigma ),\hfill \end{array}$$

(2)

for all $\vartheta ,\mathcal{Q}\in \Im$ and $\varsigma >0$. Then, $\mathfrak{p}$ has a unique fixed point.

Proof. 

Let ${\vartheta }_0$∈ℑ. Define the sequence ${\vartheta }_{\mathfrak{j}}$ by ${\vartheta }_{\mathfrak{j}}={\mathfrak{p}}^{\mathfrak{j}}{\vartheta }_0=\mathfrak{p}{\vartheta }_{\mathfrak{j}-1},\mathfrak{j}\in \mathbb{N}$.

For all $\varsigma >0$, we have

$$\begin{array}{cc}\hfill \mathcal{W}({\vartheta }_{\mathfrak{j}},{\vartheta }_{\mathfrak{j}+1},\rho \varsigma )& =\mathcal{W}(\mathfrak{p}{\vartheta }_{\mathfrak{j}-1},\mathfrak{p}{\vartheta }_{\mathfrak{j}},\rho \varsigma )\ge \mathcal{W}({\vartheta }_{\mathfrak{j}-1},{\vartheta }_{\mathfrak{j}},\varsigma )\ge \mathcal{W}\left({\vartheta }_{\mathfrak{j}-2},{\vartheta }_{\mathfrak{j}-1},\frac{\varsigma }{\rho }\right)\hfill \\ \hfill & \ge \mathcal{W}\left({\vartheta }_{\mathfrak{j}-3},{\vartheta }_{\mathfrak{j}-2},\frac{\varsigma }{{\rho }^2}\right)\ge \cdots \ge \mathcal{W}\left({\vartheta }_0,{\vartheta }_1,\frac{\varsigma }{{\rho }^{\mathfrak{j}-1}}\right),\hfill \\ \hfill \mathcal{E}({\vartheta }_{\mathfrak{j}},{\vartheta }_{\mathfrak{j}+1},\rho \varsigma )& =\mathcal{E}(\mathfrak{p}{\vartheta }_{\mathfrak{j}-1},\mathfrak{p}{\vartheta }_{\mathfrak{j}},\rho \varsigma )\le \mathcal{E}({\vartheta }_{\mathfrak{j}-1},{\vartheta }_{\mathfrak{j}},\varsigma )\le \mathcal{E}\left({\vartheta }_{\mathfrak{j}-2},{\vartheta }_{\mathfrak{j}-1},\frac{\varsigma }{\rho }\right)\hfill \\ \hfill & \le \mathcal{E}\left({\vartheta }_{\mathfrak{j}-3},{\vartheta }_{\mathfrak{j}-2},\frac{\varsigma }{{\rho }^2}\right)\le \cdots \le \mathcal{E}\left({\vartheta }_0,{\vartheta }_1,\frac{\varsigma }{{\rho }^{\mathfrak{j}-1}}\right),\hfill \end{array}$$

and

$$\begin{array}{cc}\hfill \mathcal{G}({\vartheta }_{\mathfrak{j}},{\vartheta }_{\mathfrak{j}+1},\rho \varsigma )& =\mathcal{G}(\mathfrak{p}{\vartheta }_{\mathfrak{j}-1},\mathfrak{p}{\vartheta }_{\mathfrak{j}},\varsigma )\le \mathcal{G}({\vartheta }_{\mathfrak{j}-1},{\vartheta }_{\mathfrak{j}},\varsigma )\le \mathcal{G}\left({\vartheta }_{\mathfrak{j}-2},{\vartheta }_{\mathfrak{j}-1},\frac{\varsigma }{\rho }\right)\hfill \\ \hfill & \le \mathcal{G}\left({\vartheta }_{\mathfrak{j}-3},{\vartheta }_{\mathfrak{j}-2},\frac{\varsigma }{{\rho }^2}\right)\le \cdots \le \mathcal{G}\left({\vartheta }_0,{\vartheta }_1,\frac{\varsigma }{{\rho }^{\mathfrak{j}-1}}\right).\hfill \end{array}$$

We obtain

$$\begin{array}{cc}\hfill \mathcal{W}({\vartheta }_{\mathfrak{j}},{\vartheta }_{\mathfrak{j}+1},\rho \varsigma )& \ge \mathcal{W}\left({\vartheta }_0,{\vartheta }_1,\frac{\varsigma }{{\rho }^{\mathfrak{j}-1}}\right),\hfill \\ \hfill \mathcal{E}({\vartheta }_{\mathfrak{j}},{\vartheta }_{\mathfrak{j}+1},\rho \varsigma )\le \mathcal{E}\left({\vartheta }_0,{\vartheta }_1,\frac{\varsigma }{{\rho }^{\mathfrak{j}-1}}\right)& \mathrm{and}\mathcal{G}({\vartheta }_{\mathfrak{j}},{\vartheta }_{\mathfrak{j}+1},\rho \varsigma )\le \mathcal{G}\left({\vartheta }_0,{\vartheta }_1,\frac{\varsigma }{{\rho }^{\mathfrak{j}-1}}\right).\hfill \end{array}$$

(3)

Using (v), (x) and (xv), we have the following cases:

Case 1. When $\mathfrak{i}=2\mathfrak{m}+1$, i.e., $\mathfrak{i}$ is odd, then

$$\begin{array}{cc}\hfill \mathcal{W}({\vartheta }_{\mathfrak{j}},& {\vartheta }_{\mathfrak{j}+2\mathfrak{m}+1},\varsigma )\hfill \\ & \ge \mathcal{W}\left({\vartheta }_{\mathfrak{j}},{\vartheta }_{\mathfrak{j}+1},\frac{\varsigma }{3\left(\beta ({\vartheta }_{\mathfrak{j}},{\vartheta }_{\mathfrak{j}+1})\right)}\right)\star \mathcal{W}\left({\vartheta }_{\mathfrak{j}+1},{\vartheta }_{\mathfrak{j}+2},\frac{\varsigma }{3\left(\mathrm{\Gamma }({\vartheta }_{\mathfrak{j}+1},{\vartheta }_{\mathfrak{j}+2})\right)}\right)\hfill \\ \hfill & \star \mathcal{W}\left({\vartheta }_{\mathfrak{j}+2},{\vartheta }_{\mathfrak{j}+2\mathfrak{m}+1},\frac{\varsigma }{3\left(\eta ({\vartheta }_{\mathfrak{j}+2},{\vartheta }_{\mathfrak{j}+2\mathfrak{m}+1})\right)}\right)\hfill \\ \hfill & \ge \mathcal{W}\left({\vartheta }_{\mathfrak{j}},{\vartheta }_{\mathfrak{j}+1},\frac{\varsigma }{3\left(\beta ({\vartheta }_{\mathfrak{j}},{\vartheta }_{\mathfrak{j}+1})\right)}\right)\star \mathcal{W}\left({\vartheta }_{\mathfrak{j}+1},{\vartheta }_{\mathfrak{j}+2},\frac{\varsigma }{3\left(\mathrm{\Gamma }({\vartheta }_{\mathfrak{j}+1},{\vartheta }_{\mathfrak{j}+2})\right)}\right)\hfill \\ \hfill & \star \mathcal{W}\left({\vartheta }_{\mathfrak{j}+2},{\vartheta }_{\mathfrak{j}+3},\frac{\varsigma }{3^2\left(\beta ({\vartheta }_{\mathfrak{j}+2},{\vartheta }_{\mathfrak{j}+3})\eta ({\vartheta }_{\mathfrak{j}+2},{\vartheta }_{\mathfrak{j}+2\mathfrak{m}+1})\right)}\right)\hfill \\ \hfill & \star \mathcal{W}\left({\vartheta }_{\mathfrak{j}+3},{\vartheta }_{\mathfrak{j}+4},\frac{\varsigma }{3^2\left(\mathrm{\Gamma }({\vartheta }_{\mathfrak{j}+3},{\vartheta }_{\mathfrak{j}+4})\eta ({\vartheta }_{\mathfrak{j}+2},{\vartheta }_{\mathfrak{j}+2\mathfrak{m}+1})\right)}\right)\hfill \\ \hfill & \star \mathcal{W}\left({\vartheta }_{\mathfrak{j}+4},{\vartheta }_{\mathfrak{j}+2\mathfrak{m}+1},\frac{\varsigma }{3^2\left(\eta ({\vartheta }_{\mathfrak{j}+4},{\vartheta }_{\mathfrak{j}+2\mathfrak{m}+1})\eta ({\vartheta }_{\mathfrak{j}+2},{\vartheta }_{\mathfrak{j}+2\mathfrak{m}+1})\right)}\right)\hfill \\ \hfill & \ge \mathcal{W}\left({\vartheta }_{\mathfrak{j}},{\vartheta }_{\mathfrak{j}+1},\frac{\varsigma }{3\left(\beta ({\vartheta }_{\mathfrak{j}},{\vartheta }_{\mathfrak{j}+1})\right)}\right)\star \mathcal{W}\left({\vartheta }_{\mathfrak{j}+1},{\vartheta }_{\mathfrak{j}+2},\frac{\varsigma }{3\left(\mathrm{\Gamma }({\vartheta }_{\mathfrak{j}+1},{\vartheta }_{\mathfrak{j}+2})\right)}\right)\hfill \\ \hfill & \star \mathcal{W}\left({\vartheta }_{\mathfrak{j}+2},{\vartheta }_{\mathfrak{j}+3},\frac{\varsigma }{3^2\left(\beta ({\vartheta }_{\mathfrak{j}+2},{\vartheta }_{\mathfrak{j}+3})\eta ({\vartheta }_{\mathfrak{j}+2},{\vartheta }_{\mathfrak{j}+2\mathfrak{m}+1})\right)}\right)\hfill \\ \hfill & \star \mathcal{W}\left({\vartheta }_{\mathfrak{j}+3},{\vartheta }_{\mathfrak{j}+4},\frac{\varsigma }{3^2\left(\mathrm{\Gamma }({\vartheta }_{\mathfrak{j}+3},{\vartheta }_{\mathfrak{j}+4})\eta ({\vartheta }_{\mathfrak{j}+2},{\vartheta }_{\mathfrak{j}+2\mathfrak{m}+1})\right)}\right)\hfill \\ \hfill & \star \mathcal{W}\left({\vartheta }_{\mathfrak{j}+4},{\vartheta }_{\mathfrak{j}+5},\frac{\varsigma }{3^3\left(\beta ({\vartheta }_{\mathfrak{j}+4},{\vartheta }_{\mathfrak{j}+5})\eta ({\vartheta }_{\mathfrak{j}+4},{\vartheta }_{\mathfrak{j}+2\mathfrak{m}+1})\eta ({\vartheta }_{\mathfrak{j}+2},{\vartheta }_{\mathfrak{j}+2\mathfrak{m}+1})\right)}\right)\hfill \\ \hfill & \star \mathcal{W}\left({\vartheta }_{\mathfrak{j}+5},{\vartheta }_{\mathfrak{j}+6},\frac{\varsigma }{3^3\left(\mathrm{\Gamma }({\vartheta }_{\mathfrak{j}+5},{\vartheta }_{\mathfrak{j}+6})\eta ({\vartheta }_{\mathfrak{j}+4},{\vartheta }_{\mathfrak{j}+2\mathfrak{m}+1})\eta ({\vartheta }_{\mathfrak{j}+2},{\vartheta }_{\mathfrak{j}+2\mathfrak{m}+1})\right)}\right)\hfill \\ \hfill & \star \mathcal{W}\left({\vartheta }_{\mathfrak{j}+6},{\vartheta }_{\mathfrak{j}+2\mathfrak{m}+1},\frac{\varsigma }{3^3\left(\eta ({\vartheta }_{\mathfrak{j}+6},{\vartheta }_{\mathfrak{j}+2\mathfrak{m}+1})\eta ({\vartheta }_{\mathfrak{j}+4},{\vartheta }_{\mathfrak{j}+2\mathfrak{m}+1})\eta ({\vartheta }_{\mathfrak{j}+2},{\vartheta }_{\mathfrak{j}+2\mathfrak{m}+1})\right)}\right)\hfill \\ \hfill \mathcal{W}& ({\vartheta }_{\mathfrak{j}},{\vartheta }_{\mathfrak{j}+2\mathfrak{m}+1},\varsigma )\hfill \\ & \ge \mathcal{W}\left({\vartheta }_{\mathfrak{j}},{\vartheta }_{\mathfrak{j}+1},\frac{\varsigma }{3\left(\beta ({\vartheta }_{\mathfrak{j}},{\vartheta }_{\mathfrak{j}+1})\right)}\right)\star \mathcal{W}\left({\vartheta }_{\mathfrak{j}+1},{\vartheta }_{\mathfrak{j}+2},\frac{\varsigma }{3\left(\mathrm{\Gamma }({\vartheta }_{\mathfrak{j}+1},{\vartheta }_{\mathfrak{j}+2})\right)}\right)\hfill \\ \hfill & \star \mathcal{W}\left({\vartheta }_{\mathfrak{j}+2},{\vartheta }_{\mathfrak{j}+3},\frac{\varsigma }{3^2\left(\beta ({\vartheta }_{\mathfrak{j}+2},{\vartheta }_{\mathfrak{j}+3})\eta ({\vartheta }_{\mathfrak{j}+2},{\vartheta }_{\mathfrak{j}+2\mathfrak{m}+1})\right)}\right)\hfill \\ \hfill & \star \mathcal{W}\left({\vartheta }_{\mathfrak{j}+3},{\vartheta }_{\mathfrak{j}+4},\frac{\varsigma }{3^2\left(\mathrm{\Gamma }({\vartheta }_{\mathfrak{j}+3},{\vartheta }_{\mathfrak{j}+4})\eta ({\vartheta }_{\mathfrak{j}+2},{\vartheta }_{\mathfrak{j}+2\mathfrak{m}+1})\right)}\right)\hfill \\ \hfill & \star \mathcal{W}\left({\vartheta }_{\mathfrak{j}+4},{\vartheta }_{\mathfrak{j}+5},\frac{\varsigma }{3^3\left(\beta ({\vartheta }_{\mathfrak{j}+4},{\vartheta }_{\mathfrak{j}+5})\eta ({\vartheta }_{\mathfrak{j}+4},{\vartheta }_{\mathfrak{j}+2\mathfrak{m}+1})\eta ({\vartheta }_{\mathfrak{j}+2},{\vartheta }_{\mathfrak{j}+2\mathfrak{m}+1})\right)}\right)\hfill \\ \hfill & \star \mathcal{W}\left({\vartheta }_{\mathfrak{j}+5},{\vartheta }_{\mathfrak{j}+6},\frac{\varsigma }{3^3\left(\mathrm{\Gamma }({\vartheta }_{\mathfrak{j}+5},{\vartheta }_{\mathfrak{j}+6})\eta ({\vartheta }_{\mathfrak{j}+4},{\vartheta }_{\mathfrak{j}+2\mathfrak{m}+1})\eta ({\vartheta }_{\mathfrak{j}+2},{\vartheta }_{\mathfrak{j}+2\mathfrak{m}+1})\right)}\right)\hfill \\ \hfill & \star \mathcal{W}\left({\vartheta }_{\mathfrak{j}+6},{\vartheta }_{\mathfrak{j}+2\mathfrak{m}+1},\frac{\varsigma }{3^3\left(\eta ({\vartheta }_{\mathfrak{j}+6},{\vartheta }_{\mathfrak{j}+2\mathfrak{m}+1})\eta ({\vartheta }_{\mathfrak{j}+4},{\vartheta }_{\mathfrak{j}+2\mathfrak{m}+1})\eta ({\vartheta }_{\mathfrak{j}+2},{\vartheta }_{\mathfrak{j}+2\mathfrak{m}+1})\right)}\right)\hfill \\ & \star \cdots \star \hfill \\ \hfill & \mathcal{W}\left({\vartheta }_{\mathfrak{j}+2\mathfrak{m}-2},{\vartheta }_{\mathfrak{j}+2\mathfrak{m}-1},\frac{\varsigma }{3^{\mathfrak{m}}(\beta ({\vartheta }_{\mathfrak{j}+2\mathfrak{m}-2},{\vartheta }_{\mathfrak{j}+2\mathfrak{m}-1})\eta ({\vartheta }_{\mathfrak{j}+2\mathfrak{m}-2},{\vartheta }_{\mathfrak{j}+2\mathfrak{m}+1})\cdots \eta ({\vartheta }_{\mathfrak{j}+2},{\vartheta }_{\mathfrak{j}+2\mathfrak{m}+1}))}\right)\hfill \end{array}$$

$$\begin{array}{cc}\hfill & \star \mathcal{W}\left({\vartheta }_{\mathfrak{j}+2\mathfrak{m}-1},{\vartheta }_{\mathfrak{j}+2\mathfrak{m}},\frac{\varsigma }{3^{\mathfrak{m}}(\mathrm{\Gamma }({\vartheta }_{\mathfrak{j}+2\mathfrak{m}-1},{\vartheta }_{\mathfrak{j}+2\mathfrak{m}})\eta ({\vartheta }_{\mathfrak{j}+2\mathfrak{m}-2},{\vartheta }_{\mathfrak{j}+2\mathfrak{m}+1})\cdots \eta ({\vartheta }_{\mathfrak{j}+2},{\vartheta }_{\mathfrak{j}+2\mathfrak{m}+1}))}\right)\hfill \\ \hfill & \star \mathcal{W}\left({\vartheta }_{\mathfrak{j}+2\mathfrak{m}},{\vartheta }_{\mathfrak{j}+2\mathfrak{m}+1},\frac{\varsigma }{3^{\mathfrak{m}}(\eta ({\vartheta }_{\mathfrak{j}+2\mathfrak{m}},{\vartheta }_{\mathfrak{j}+2\mathfrak{m}+1})\eta ({\vartheta }_{\mathfrak{j}+2\mathfrak{m}-2},{\vartheta }_{\mathfrak{j}+2\mathfrak{m}+1})\cdots \eta ({\vartheta }_{\mathfrak{j}+2},{\vartheta }_{\mathfrak{j}+2\mathfrak{m}+1}))}\right),\hfill \end{array}$$

$$\begin{array}{cc}\hfill \mathcal{E}({\vartheta }_{\mathfrak{j}},{\vartheta }_{\mathfrak{j}+2\mathfrak{m}+1},\varsigma )& \le \mathcal{E}\left({\vartheta }_{\mathfrak{j}},{\vartheta }_{\mathfrak{j}+1},\frac{\varsigma }{3\left(\beta ({\vartheta }_{\mathfrak{j}},{\vartheta }_{\mathfrak{j}+1})\right)}\right)\diamond \mathcal{E}\left({\vartheta }_{\mathfrak{j}+1},{\vartheta }_{\mathfrak{j}+2},\frac{\varsigma }{3\left(\mathrm{\Gamma }({\vartheta }_{\mathfrak{j}+1},{\vartheta }_{\mathfrak{j}+2})\right)}\right)\hfill \\ \hfill & \diamond \mathcal{E}\left({\vartheta }_{\mathfrak{j}+2},{\vartheta }_{\mathfrak{j}+2\mathfrak{m}+1},\frac{\varsigma }{3\left(\eta ({\vartheta }_{\mathfrak{j}+2},{\vartheta }_{\mathfrak{j}+2\mathfrak{m}+1})\right)}\right)\hfill \\ \hfill & \le \mathcal{E}\left({\vartheta }_{\mathfrak{j}},{\vartheta }_{\mathfrak{j}+1},\frac{\varsigma }{3\left(\beta ({\vartheta }_{\mathfrak{j}},{\vartheta }_{\mathfrak{j}+1})\right)}\right)\diamond \mathcal{E}\left({\vartheta }_{\mathfrak{j}+1},{\vartheta }_{\mathfrak{j}+2},\frac{\varsigma }{3\left(\mathrm{\Gamma }({\vartheta }_{\mathfrak{j}+1},{\vartheta }_{\mathfrak{j}+2})\right)}\right)\hfill \\ \hfill & \diamond \mathcal{E}\left({\vartheta }_{\mathfrak{j}+2},{\vartheta }_{\mathfrak{j}+3},\frac{\varsigma }{3^2\left(\beta ({\vartheta }_{\mathfrak{j}+2},{\vartheta }_{\mathfrak{j}+3})\eta ({\vartheta }_{\mathfrak{j}+2},{\vartheta }_{\mathfrak{j}+2\mathfrak{m}+1})\right)}\right)\hfill \\ \hfill & \diamond \mathcal{E}\left({\vartheta }_{\mathfrak{j}+3},{\vartheta }_{\mathfrak{j}+4},\frac{\varsigma }{3^2\left(\mathrm{\Gamma }({\vartheta }_{\mathfrak{j}+3},{\vartheta }_{\mathfrak{j}+4})\eta ({\vartheta }_{\mathfrak{j}+2},{\vartheta }_{\mathfrak{j}+2\mathfrak{m}+1})\right)}\right)\hfill \\ \hfill & \diamond \mathcal{E}\left({\vartheta }_{\mathfrak{j}+4},{\vartheta }_{\mathfrak{j}+2\mathfrak{m}+1},\frac{\varsigma }{3^2\left(\eta ({\vartheta }_{\mathfrak{j}+4},{\vartheta }_{\mathfrak{j}+2\mathfrak{m}+1})\eta ({\vartheta }_{\mathfrak{j}+2},{\vartheta }_{\mathfrak{j}+2\mathfrak{m}+1})\right)}\right)\hfill \\ \hfill & \le \mathcal{E}\left({\vartheta }_{\mathfrak{j}},{\vartheta }_{\mathfrak{j}+1},\frac{\varsigma }{3\left(\beta ({\vartheta }_{\mathfrak{j}},{\vartheta }_{\mathfrak{j}+1})\right)}\right)\diamond \mathcal{E}\left({\vartheta }_{\mathfrak{j}+1},{\vartheta }_{\mathfrak{j}+2},\frac{\varsigma }{3\left(\mathrm{\Gamma }({\vartheta }_{\mathfrak{j}+1},{\vartheta }_{\mathfrak{j}+2})\right)}\right)\hfill \\ \hfill & \diamond \mathcal{E}\left({\vartheta }_{\mathfrak{j}+2},{\vartheta }_{\mathfrak{j}+3},\frac{\varsigma }{3^2\left(\beta ({\vartheta }_{\mathfrak{j}+2},{\vartheta }_{\mathfrak{j}+3})\eta ({\vartheta }_{\mathfrak{j}+2},{\vartheta }_{\mathfrak{j}+2\mathfrak{m}+1})\right)}\right)\hfill \\ \hfill & \diamond \mathcal{E}\left({\vartheta }_{\mathfrak{j}+3},{\vartheta }_{\mathfrak{j}+4},\frac{\varsigma }{3^2\left(\mathrm{\Gamma }({\vartheta }_{\mathfrak{j}+3},{\vartheta }_{\mathfrak{j}+4})\eta ({\vartheta }_{\mathfrak{j}+2},{\vartheta }_{\mathfrak{j}+2\mathfrak{m}+1})\right)}\right)\hfill \\ \hfill & \diamond \mathcal{E}\left({\vartheta }_{\mathfrak{j}+4},{\vartheta }_{\mathfrak{j}+5},\frac{\varsigma }{3^3\left(\beta ({\vartheta }_{\mathfrak{j}+4},{\vartheta }_{\mathfrak{j}+5})\eta ({\vartheta }_{\mathfrak{j}+4},{\vartheta }_{\mathfrak{j}+2\mathfrak{m}+1})\eta ({\vartheta }_{\mathfrak{j}+2},{\vartheta }_{\mathfrak{j}+2\mathfrak{m}+1})\right)}\right)\hfill \\ \hfill & \diamond \mathcal{E}\left({\vartheta }_{\mathfrak{j}+5},{\vartheta }_{\mathfrak{j}+6},\frac{\varsigma }{3^3\left(\mathrm{\Gamma }({\vartheta }_{\mathfrak{j}+5},{\vartheta }_{\mathfrak{j}+6})\eta ({\vartheta }_{\mathfrak{j}+4},{\vartheta }_{\mathfrak{j}+2\mathfrak{m}+1})\eta ({\vartheta }_{\mathfrak{j}+2},{\vartheta }_{\mathfrak{j}+2\mathfrak{m}+1})\right)}\right)\hfill \\ \hfill & \diamond \mathcal{E}\left({\vartheta }_{\mathfrak{j}+6},{\vartheta }_{\mathfrak{j}+2\mathfrak{m}+1},\frac{\varsigma }{3^3\left(\eta ({\vartheta }_{\mathfrak{j}+6},{\vartheta }_{\mathfrak{j}+2\mathfrak{m}+1})\eta ({\vartheta }_{\mathfrak{j}+4},{\vartheta }_{\mathfrak{j}+2\mathfrak{m}+1})\eta ({\vartheta }_{\mathfrak{j}+2},{\vartheta }_{\mathfrak{j}+2\mathfrak{m}+1})\right)}\right)\hfill \end{array}$$

$$\begin{array}{cc}\hfill \mathcal{E}& ({\vartheta }_{\mathfrak{j}},{\vartheta }_{\mathfrak{j}+2\mathfrak{m}+1},\varsigma )\hfill \\ & \le \mathcal{E}\left({\vartheta }_{\mathfrak{j}},{\vartheta }_{\mathfrak{j}+1},\frac{\varsigma }{3\left(\beta ({\vartheta }_{\mathfrak{j}},{\vartheta }_{\mathfrak{j}+1})\right)}\right)\diamond \mathcal{E}\left({\vartheta }_{\mathfrak{j}+1},{\vartheta }_{\mathfrak{j}+2},\frac{\varsigma }{3\left(\mathrm{\Gamma }({\vartheta }_{\mathfrak{j}+1},{\vartheta }_{\mathfrak{j}+2})\right)}\right)\hfill \\ \hfill & \diamond \mathcal{E}\left({\vartheta }_{\mathfrak{j}+2},{\vartheta }_{\mathfrak{j}+3},\frac{\varsigma }{3^2\left(\beta ({\vartheta }_{\mathfrak{j}+2},{\vartheta }_{\mathfrak{j}+3})\eta ({\vartheta }_{\mathfrak{j}+2},{\vartheta }_{\mathfrak{j}+2\mathfrak{m}+1})\right)}\right)\hfill \\ \hfill & \diamond \mathcal{E}\left({\vartheta }_{\mathfrak{j}+3},{\vartheta }_{\mathfrak{j}+4},\frac{\varsigma }{3^2\left(\mathrm{\Gamma }({\vartheta }_{\mathfrak{j}+3},{\vartheta }_{\mathfrak{j}+4})\eta ({\vartheta }_{\mathfrak{j}+2},{\vartheta }_{\mathfrak{j}+2\mathfrak{m}+1})\right)}\right)\hfill \\ \hfill & \diamond \mathcal{E}\left({\vartheta }_{\mathfrak{j}+4},{\vartheta }_{\mathfrak{j}+5},\frac{\varsigma }{3^3\left(\beta ({\vartheta }_{\mathfrak{j}+4},{\vartheta }_{\mathfrak{j}+5})\eta ({\vartheta }_{\mathfrak{j}+4},{\vartheta }_{\mathfrak{j}+2\mathfrak{m}+1})\eta ({\vartheta }_{\mathfrak{j}+2},{\vartheta }_{\mathfrak{j}+2\mathfrak{m}+1})\right)}\right)\hfill \\ \hfill & \diamond \mathcal{E}\left({\vartheta }_{\mathfrak{j}+5},{\vartheta }_{\mathfrak{j}+6},\frac{\varsigma }{3^3\left(\mathrm{\Gamma }({\vartheta }_{\mathfrak{j}+5},{\vartheta }_{\mathfrak{j}+6})\eta ({\vartheta }_{\mathfrak{j}+4},{\vartheta }_{\mathfrak{j}+2\mathfrak{m}+1})\eta ({\vartheta }_{\mathfrak{j}+2},{\vartheta }_{\mathfrak{j}+2\mathfrak{m}+1})\right)}\right)\hfill \\ \hfill & \diamond \mathcal{E}\left({\vartheta }_{\mathfrak{j}+6},{\vartheta }_{\mathfrak{j}+2\mathfrak{m}+1},\frac{\varsigma }{3^3\left(\eta ({\vartheta }_{\mathfrak{j}+6},{\vartheta }_{\mathfrak{j}+2\mathfrak{m}+1})\eta ({\vartheta }_{\mathfrak{j}+4},{\vartheta }_{\mathfrak{j}+2\mathfrak{m}+1})\eta ({\vartheta }_{\mathfrak{j}+2},{\vartheta }_{\mathfrak{j}+2\mathfrak{m}+1})\right)}\right)\hfill \\ & \diamond \cdots \diamond \hfill \\ \hfill & \mathcal{E}\left({\vartheta }_{\mathfrak{j}+2\mathfrak{m}-2},{\vartheta }_{\mathfrak{j}+2\mathfrak{m}-1},\frac{\varsigma }{3^{\mathfrak{m}}(\beta ({\vartheta }_{\mathfrak{j}+2\mathfrak{m}-2},{\vartheta }_{\mathfrak{j}+2\mathfrak{m}-1})\eta ({\vartheta }_{\mathfrak{j}+2\mathfrak{m}-2},{\vartheta }_{\mathfrak{j}+2\mathfrak{m}+1})\cdots \eta ({\vartheta }_{\mathfrak{j}+2},{\vartheta }_{\mathfrak{j}+2\mathfrak{m}+1}))}\right)\hfill \end{array}$$

$$\begin{array}{cc}\hfill & \diamond \mathcal{E}\left({\vartheta }_{\mathfrak{j}+2\mathfrak{m}-1},{\vartheta }_{\mathfrak{j}+2\mathfrak{m}},\frac{\varsigma }{3^{\mathfrak{m}}(\mathrm{\Gamma }({\vartheta }_{\mathfrak{j}+2\mathfrak{m}-1},{\vartheta }_{\mathfrak{j}+2\mathfrak{m}})\eta ({\vartheta }_{\mathfrak{j}+2\mathfrak{m}-2},{\vartheta }_{\mathfrak{j}+2\mathfrak{m}+1})\cdots \eta ({\vartheta }_{\mathfrak{j}+2},{\vartheta }_{\mathfrak{j}+2\mathfrak{m}+1}))}\right)\hfill \\ \hfill & \diamond \mathcal{E}\left({\vartheta }_{\mathfrak{j}+2\mathfrak{m}},{\vartheta }_{\mathfrak{j}+2\mathfrak{m}+1},\frac{\varsigma }{3^{\mathfrak{m}}(\eta ({\vartheta }_{\mathfrak{j}+2\mathfrak{m}},{\vartheta }_{\mathfrak{j}+2\mathfrak{m}+1})\eta ({\vartheta }_{\mathfrak{j}+2\mathfrak{m}-2},{\vartheta }_{\mathfrak{j}+2\mathfrak{m}+1})\cdots \eta ({\vartheta }_{\mathfrak{j}+2},{\vartheta }_{\mathfrak{j}+2\mathfrak{m}+1}))}\right),\hfill \end{array}$$

and

$$\begin{array}{cc}\hfill \mathcal{G}({\vartheta }_{\mathfrak{j}},{\vartheta }_{\mathfrak{j}+2\mathfrak{m}+1},\varsigma )& \le \mathcal{G}\left({\vartheta }_{\mathfrak{j}},{\vartheta }_{\mathfrak{j}+1},\frac{\varsigma }{3\left(\beta ({\vartheta }_{\mathfrak{j}},{\vartheta }_{\mathfrak{j}+1})\right)}\right)\diamond \mathcal{G}\left({\vartheta }_{\mathfrak{j}+1},{\vartheta }_{\mathfrak{j}+2},\frac{\varsigma }{3\left(\mathrm{\Gamma }({\vartheta }_{\mathfrak{j}+1},{\vartheta }_{\mathfrak{j}+2})\right)}\right)\hfill \\ \hfill & \diamond \mathcal{G}\left({\vartheta }_{\mathfrak{j}+2},{\vartheta }_{\mathfrak{j}+2\mathfrak{m}+1},\frac{\varsigma }{3\left(\eta ({\vartheta }_{\mathfrak{j}+2},{\vartheta }_{\mathfrak{j}+2\mathfrak{m}+1})\right)}\right)\hfill \\ \hfill & \le \mathcal{G}\left({\vartheta }_{\mathfrak{j}},{\vartheta }_{\mathfrak{j}+1},\frac{\varsigma }{3\left(\beta ({\vartheta }_{\mathfrak{j}},{\vartheta }_{\mathfrak{j}+1})\right)}\right)\diamond \mathcal{G}\left({\vartheta }_{\mathfrak{j}+1},{\vartheta }_{\mathfrak{j}+2},\frac{\varsigma }{3\left(\mathrm{\Gamma }({\vartheta }_{\mathfrak{j}+1},{\vartheta }_{\mathfrak{j}+2})\right)}\right)\hfill \\ \hfill & \diamond \mathcal{G}\left({\vartheta }_{\mathfrak{j}+2},{\vartheta }_{\mathfrak{j}+3},\frac{\varsigma }{3^2\left(\beta ({\vartheta }_{\mathfrak{j}+2},{\vartheta }_{\mathfrak{j}+3})\eta ({\vartheta }_{\mathfrak{j}+2},{\vartheta }_{\mathfrak{j}+2\mathfrak{m}+1})\right)}\right)\hfill \\ \hfill & \diamond \mathcal{G}\left({\vartheta }_{\mathfrak{j}+3},{\vartheta }_{\mathfrak{j}+4},\frac{\varsigma }{3^2\left(\mathrm{\Gamma }({\vartheta }_{\mathfrak{j}+3},{\vartheta }_{\mathfrak{j}+4})\eta ({\vartheta }_{\mathfrak{j}+2},{\vartheta }_{\mathfrak{j}+2\mathfrak{m}+1})\right)}\right)\hfill \\ \hfill & \diamond \mathcal{G}\left({\vartheta }_{\mathfrak{j}+4},{\vartheta }_{\mathfrak{j}+2\mathfrak{m}+1},\frac{\varsigma }{3^2\left(\eta ({\vartheta }_{\mathfrak{j}+4},{\vartheta }_{\mathfrak{j}+2\mathfrak{m}+1})\eta ({\vartheta }_{\mathfrak{j}+2},{\vartheta }_{\mathfrak{j}+2\mathfrak{m}+1})\right)}\right)\hfill \\ \hfill & \le \mathcal{G}\left({\vartheta }_{\mathfrak{j}},{\vartheta }_{\mathfrak{j}+1},\frac{\varsigma }{3\left(\beta ({\vartheta }_{\mathfrak{j}},{\vartheta }_{\mathfrak{j}+1})\right)}\right)\diamond \mathcal{G}\left({\vartheta }_{\mathfrak{j}+1},{\vartheta }_{\mathfrak{j}+2},\frac{\varsigma }{3\left(\mathrm{\Gamma }({\vartheta }_{\mathfrak{j}+1},{\vartheta }_{\mathfrak{j}+2})\right)}\right)\hfill \\ \hfill & \diamond \mathcal{G}\left({\vartheta }_{\mathfrak{j}+2},{\vartheta }_{\mathfrak{j}+3},\frac{\varsigma }{3^2\left(\beta ({\vartheta }_{\mathfrak{j}+2},{\vartheta }_{\mathfrak{j}+3})\eta ({\vartheta }_{\mathfrak{j}+2},{\vartheta }_{\mathfrak{j}+2\mathfrak{m}+1})\right)}\right)\hfill \\ \hfill & \diamond \mathcal{G}\left({\vartheta }_{\mathfrak{j}+3},{\vartheta }_{\mathfrak{j}+4},\frac{\varsigma }{3^2\left(\mathrm{\Gamma }({\vartheta }_{\mathfrak{j}+3},{\vartheta }_{\mathfrak{j}+4})\eta ({\vartheta }_{\mathfrak{j}+2},{\vartheta }_{\mathfrak{j}+2\mathfrak{m}+1})\right)}\right)\hfill \\ \hfill & \diamond \mathcal{G}\left({\vartheta }_{\mathfrak{j}+4},{\vartheta }_{\mathfrak{j}+5},\frac{\varsigma }{3^3\left(\beta ({\vartheta }_{\mathfrak{j}+4},{\vartheta }_{\mathfrak{j}+5})\eta ({\vartheta }_{\mathfrak{j}+4},{\vartheta }_{\mathfrak{j}+2\mathfrak{m}+1})\eta ({\vartheta }_{\mathfrak{j}+2},{\vartheta }_{\mathfrak{j}+2\mathfrak{m}+1})\right)}\right)\hfill \\ \hfill & \diamond \mathcal{G}\left({\vartheta }_{\mathfrak{j}+5},{\vartheta }_{\mathfrak{j}+6},\frac{\varsigma }{3^3\left(\mathrm{\Gamma }({\vartheta }_{\mathfrak{j}+5},{\vartheta }_{\mathfrak{j}+6})\eta ({\vartheta }_{\mathfrak{j}+4},{\vartheta }_{\mathfrak{j}+2\mathfrak{m}+1})\eta ({\vartheta }_{\mathfrak{j}+2},{\vartheta }_{\mathfrak{j}+2\mathfrak{m}+1})\right)}\right)\hfill \\ \hfill & \diamond \mathcal{G}\left({\vartheta }_{\mathfrak{j}+6},{\vartheta }_{\mathfrak{j}+2\mathfrak{m}+1},\frac{\varsigma }{3^3\left(\eta ({\vartheta }_{\mathfrak{j}+6},{\vartheta }_{\mathfrak{j}+2\mathfrak{m}+1})\eta ({\vartheta }_{\mathfrak{j}+4},{\vartheta }_{\mathfrak{j}+2\mathfrak{m}+1})\eta ({\vartheta }_{\mathfrak{j}+2},{\vartheta }_{\mathfrak{j}+2\mathfrak{m}+1})\right)}\right)\hfill \end{array}$$

$$\begin{array}{cc}\hfill \mathcal{G}& ({\vartheta }_{\mathfrak{j}},{\vartheta }_{\mathfrak{j}+2\mathfrak{m}+1},\varsigma )\hfill \\ & \le \mathcal{G}\left({\vartheta }_{\mathfrak{j}},{\vartheta }_{\mathfrak{j}+1},\frac{\varsigma }{3\left(\beta ({\vartheta }_{\mathfrak{j}},{\vartheta }_{\mathfrak{j}+1})\right)}\right)\diamond \mathcal{G}\left({\vartheta }_{\mathfrak{j}+1},{\vartheta }_{\mathfrak{j}+2},\frac{\varsigma }{3\left(\mathrm{\Gamma }({\vartheta }_{\mathfrak{j}+1},{\vartheta }_{\mathfrak{j}+2})\right)}\right)\hfill \\ \hfill & \diamond \mathcal{G}\left({\vartheta }_{\mathfrak{j}+2},{\vartheta }_{\mathfrak{j}+3},\frac{\varsigma }{3^2\left(\beta ({\vartheta }_{\mathfrak{j}+2},{\vartheta }_{\mathfrak{j}+3})\eta ({\vartheta }_{\mathfrak{j}+2},{\vartheta }_{\mathfrak{j}+2\mathfrak{m}+1})\right)}\right)\hfill \\ \hfill & \diamond \mathcal{G}\left({\vartheta }_{\mathfrak{j}+3},{\vartheta }_{\mathfrak{j}+4},\frac{\varsigma }{3^2\left(\mathrm{\Gamma }({\vartheta }_{\mathfrak{j}+3},{\vartheta }_{\mathfrak{j}+4})\eta ({\vartheta }_{\mathfrak{j}+2},{\vartheta }_{\mathfrak{j}+2\mathfrak{m}+1})\right)}\right)\hfill \\ \hfill & \diamond \mathcal{G}\left({\vartheta }_{\mathfrak{j}+4},{\vartheta }_{\mathfrak{j}+5},\frac{\varsigma }{3^3\left(\beta ({\vartheta }_{\mathfrak{j}+4},{\vartheta }_{\mathfrak{j}+5})\eta ({\vartheta }_{\mathfrak{j}+4},{\vartheta }_{\mathfrak{j}+2\mathfrak{m}+1})\eta ({\vartheta }_{\mathfrak{j}+2},{\vartheta }_{\mathfrak{j}+2\mathfrak{m}+1})\right)}\right)\hfill \\ \hfill & \diamond \mathcal{G}\left({\vartheta }_{\mathfrak{j}+5},{\vartheta }_{\mathfrak{j}+6},\frac{\varsigma }{3^3\left(\mathrm{\Gamma }({\vartheta }_{\mathfrak{j}+5},{\vartheta }_{\mathfrak{j}+6})\eta ({\vartheta }_{\mathfrak{j}+4},{\vartheta }_{\mathfrak{j}+2\mathfrak{m}+1})\eta ({\vartheta }_{\mathfrak{j}+2},{\vartheta }_{\mathfrak{j}+2\mathfrak{m}+1})\right)}\right)\hfill \\ \hfill & \diamond \mathcal{G}\left({\vartheta }_{\mathfrak{j}+6},{\vartheta }_{\mathfrak{j}+2\mathfrak{m}+1},\frac{\varsigma }{3^3\left(\eta ({\vartheta }_{\mathfrak{j}+6},{\vartheta }_{\mathfrak{j}+2\mathfrak{m}+1})\eta ({\vartheta }_{\mathfrak{j}+4},{\vartheta }_{\mathfrak{j}+2\mathfrak{m}+1})\eta ({\vartheta }_{\mathfrak{j}+2},{\vartheta }_{\mathfrak{j}+2\mathfrak{m}+1})\right)}\right)\diamond \cdots \diamond \hfill \end{array}$$

$$\begin{array}{cc}\hfill & \mathcal{G}\left({\vartheta }_{\mathfrak{j}+2\mathfrak{m}-2},{\vartheta }_{\mathfrak{j}+2\mathfrak{m}-1},\frac{\varsigma }{3^{\mathfrak{m}}(\beta ({\vartheta }_{\mathfrak{j}+2\mathfrak{m}-2},{\vartheta }_{\mathfrak{j}+2\mathfrak{m}-1})\eta ({\vartheta }_{\mathfrak{j}+2\mathfrak{m}-2},{\vartheta }_{\mathfrak{j}+2\mathfrak{m}+1})\cdots \eta ({\vartheta }_{\mathfrak{j}+2},{\vartheta }_{\mathfrak{j}+2\mathfrak{m}+1}))}\right)\hfill \\ \hfill & \diamond \mathcal{G}\left({\vartheta }_{\mathfrak{j}+2\mathfrak{m}-1},{\vartheta }_{\mathfrak{j}+2\mathfrak{m}},\frac{\varsigma }{3^{\mathfrak{m}}(\mathrm{\Gamma }({\vartheta }_{\mathfrak{j}+2\mathfrak{m}-1},{\vartheta }_{\mathfrak{j}+2\mathfrak{m}})\eta ({\vartheta }_{\mathfrak{j}+2\mathfrak{m}-2},{\vartheta }_{\mathfrak{j}+2\mathfrak{m}+1})\cdots \eta ({\vartheta }_{\mathfrak{j}+2},{\vartheta }_{\mathfrak{j}+2\mathfrak{m}+1}))}\right)\hfill \\ \hfill & \diamond \mathcal{G}\left({\vartheta }_{\mathfrak{j}+2\mathfrak{m}},{\vartheta }_{\mathfrak{j}+2\mathfrak{m}+1},\frac{\varsigma }{3^{\mathfrak{m}}(\eta ({\vartheta }_{\mathfrak{j}+2\mathfrak{m}},{\vartheta }_{\mathfrak{j}+2\mathfrak{m}+1})\eta ({\vartheta }_{\mathfrak{j}+2\mathfrak{m}-2},{\vartheta }_{\mathfrak{j}+2\mathfrak{m}+1})\cdots \eta ({\vartheta }_{\mathfrak{j}+2},{\vartheta }_{\mathfrak{j}+2\mathfrak{m}+1}))}\right).\hfill \end{array}$$

Using (3) in the above inequalities, we have

$$\begin{array}{cc}\hfill \mathcal{W}& ({\vartheta }_{\mathfrak{j}},{\vartheta }_{\mathfrak{j}+2\mathfrak{m}+1},\varsigma )\ge \mathcal{W}\left({\vartheta }_0,{\vartheta }_1,\frac{\varsigma }{3{\rho }^{\mathfrak{j}-1}\left(\beta ({\vartheta }_{\mathfrak{j}},{\vartheta }_{\mathfrak{j}+1})\right)}\right)\star \mathcal{W}\left({\vartheta }_0,{\vartheta }_1,\frac{\varsigma }{3{\rho }^{\mathfrak{j}}\left(\mathrm{\Gamma }({\vartheta }_{\mathfrak{j}+1},{\vartheta }_{\mathfrak{j}+2})\right)}\right)\hfill \\ \hfill & \star \mathcal{W}\left({\vartheta }_0,{\vartheta }_1,\frac{\varsigma }{3^2{\rho }^{\mathfrak{j}+1}\left(\beta ({\vartheta }_{\mathfrak{j}+2},{\vartheta }_{\mathfrak{j}+3})\eta ({\vartheta }_{\mathfrak{j}+2},{\vartheta }_{\mathfrak{j}+2\mathfrak{m}+1})\right)}\right)\hfill \\ \hfill & \star \mathcal{W}\left({\vartheta }_0,{\vartheta }_1,\frac{\varsigma }{3^2{\rho }^{\mathfrak{j}+2}\left(\mathrm{\Gamma }({\vartheta }_{\mathfrak{j}+3},{\vartheta }_{\mathfrak{j}+4})\eta ({\vartheta }_{\mathfrak{j}+2},{\vartheta }_{\mathfrak{j}+2\mathfrak{m}+1})\right)}\right)\hfill \\ \hfill & \star \mathcal{W}\left({\vartheta }_0,{\vartheta }_1,\frac{\varsigma }{3^3{\rho }^{\mathfrak{j}+3}\left(\beta ({\vartheta }_{\mathfrak{j}+4},{\vartheta }_{\mathfrak{j}+5})\eta ({\vartheta }_{\mathfrak{j}+4},{\vartheta }_{\mathfrak{j}+2\mathfrak{m}+1})\eta ({\vartheta }_{\mathfrak{j}+2},{\vartheta }_{\mathfrak{j}+2\mathfrak{m}+1})\right)}\right)\hfill \\ \hfill & \star \mathcal{W}\left({\vartheta }_0,{\vartheta }_1,\frac{\varsigma }{3^3{\rho }^{\mathfrak{j}+4}\left(\mathrm{\Gamma }({\vartheta }_{\mathfrak{j}+5},{\vartheta }_{\mathfrak{j}+6})\eta ({\vartheta }_{\mathfrak{j}+4},{\vartheta }_{\mathfrak{j}+2\mathfrak{m}+1})\eta ({\vartheta }_{\mathfrak{j}+2},{\vartheta }_{\mathfrak{j}+2\mathfrak{m}+1})\right)}\right)\hfill \\ \hfill & \star \mathcal{W}\left({\vartheta }_0,{\vartheta }_1,\frac{\varsigma }{3^3{\rho }^{\mathfrak{j}+5}\left(\eta ({\vartheta }_{\mathfrak{j}+6},{\vartheta }_{\mathfrak{j}+2\mathfrak{m}+1})\eta ({\vartheta }_{\mathfrak{j}+4},{\vartheta }_{\mathfrak{j}+2\mathfrak{m}+1})\eta ({\vartheta }_{\mathfrak{j}+2},{\vartheta }_{\mathfrak{j}+2\mathfrak{m}+1})\right)}\right)\star \cdots \star \hfill \\ \hfill & \mathcal{W}\left({\vartheta }_0,{\vartheta }_1,\frac{\varsigma }{3^{\mathfrak{m}}{\rho }^{\mathfrak{j}+2\mathfrak{m}-3}(\beta ({\vartheta }_{\mathfrak{j}+2\mathfrak{m}-2},{\vartheta }_{\mathfrak{j}+2\mathfrak{m}-1})\eta ({\vartheta }_{\mathfrak{j}+2\mathfrak{m}-2},{\vartheta }_{\mathfrak{j}+2\mathfrak{m}+1})\cdots \eta ({\vartheta }_{\mathfrak{j}+2},{\vartheta }_{\mathfrak{j}+2\mathfrak{m}+1}))}\right)\hfill \\ \hfill & \star \mathcal{W}\left({\vartheta }_0,{\vartheta }_1,\frac{\varsigma }{3^{\mathfrak{m}}{\rho }^{\mathfrak{j}+2\mathfrak{m}-2}(\mathrm{\Gamma }({\vartheta }_{\mathfrak{j}+2\mathfrak{m}-1},{\vartheta }_{\mathfrak{j}+2\mathfrak{m}})\eta ({\vartheta }_{\mathfrak{j}+2\mathfrak{m}-2},{\vartheta }_{\mathfrak{j}+2\mathfrak{m}+1})\cdots \eta ({\vartheta }_{\mathfrak{j}+2},{\vartheta }_{\mathfrak{j}+2\mathfrak{m}+1}))}\right)\hfill \\ \hfill & \star \mathcal{W}\left({\vartheta }_0,{\vartheta }_1,\frac{\varsigma }{3^{\mathfrak{m}}{\rho }^{\mathfrak{j}+2\mathfrak{m}-1}(\eta ({\vartheta }_{\mathfrak{j}+2\mathfrak{m}},{\vartheta }_{\mathfrak{j}+2\mathfrak{m}+1})\eta ({\vartheta }_{\mathfrak{j}+2\mathfrak{m}-2},{\vartheta }_{\mathfrak{j}+2\mathfrak{m}+1})\cdots \eta ({\vartheta }_{\mathfrak{j}+2},{\vartheta }_{\mathfrak{j}+2\mathfrak{m}+1}))}\right),\hfill \end{array}$$

$$\begin{array}{cc}\hfill \mathcal{E}& ({\vartheta }_{\mathfrak{j}},{\vartheta }_{\mathfrak{j}+2\mathfrak{m}+1},\varsigma )\le \mathcal{E}\left({\vartheta }_0,{\vartheta }_1,\frac{\varsigma }{3{\rho }^{\mathfrak{j}-1}\left(\beta ({\vartheta }_{\mathfrak{j}},{\vartheta }_{\mathfrak{j}+1})\right)}\right)\diamond \mathcal{E}\left({\vartheta }_0,{\vartheta }_1,\frac{\varsigma }{3{\rho }^{\mathfrak{j}}\left(\mathrm{\Gamma }({\vartheta }_{\mathfrak{j}+1},{\vartheta }_{\mathfrak{j}+2})\right)}\right)\hfill \\ \hfill & \diamond \mathcal{E}\left({\vartheta }_0,{\vartheta }_1,\frac{\varsigma }{3^2{\rho }^{\mathfrak{j}+1}\left(\beta ({\vartheta }_{\mathfrak{j}+2},{\vartheta }_{\mathfrak{j}+3})\eta ({\vartheta }_{\mathfrak{j}+2},{\vartheta }_{\mathfrak{j}+2\mathfrak{m}+1})\right)}\right)\hfill \\ \hfill & \diamond \mathcal{E}\left({\vartheta }_0,{\vartheta }_1,\frac{\varsigma }{3^2{\rho }^{\mathfrak{j}+2}\left(\mathrm{\Gamma }({\vartheta }_{\mathfrak{j}+3},{\vartheta }_{\mathfrak{j}+4})\eta ({\vartheta }_{\mathfrak{j}+2},{\vartheta }_{\mathfrak{j}+2\mathfrak{m}+1})\right)}\right)\hfill \\ \hfill & \diamond \mathcal{E}\left({\vartheta }_0,{\vartheta }_1,\frac{\varsigma }{3^3{\rho }^{\mathfrak{j}+3}\left(\beta ({\vartheta }_{\mathfrak{j}+4},{\vartheta }_{\mathfrak{j}+5})\eta ({\vartheta }_{\mathfrak{j}+4},{\vartheta }_{\mathfrak{j}+2\mathfrak{m}+1})\eta ({\vartheta }_{\mathfrak{j}+2},{\vartheta }_{\mathfrak{j}+2\mathfrak{m}+1})\right)}\right)\hfill \\ \hfill & \diamond \mathcal{E}\left({\vartheta }_0,{\vartheta }_1,\frac{\varsigma }{3^3{\rho }^{\mathfrak{j}+4}\left(\mathrm{\Gamma }({\vartheta }_{\mathfrak{j}+5},{\vartheta }_{\mathfrak{j}+6})\eta ({\vartheta }_{\mathfrak{j}+4},{\vartheta }_{\mathfrak{j}+2\mathfrak{m}+1})\eta ({\vartheta }_{\mathfrak{j}+2},{\vartheta }_{\mathfrak{j}+2\mathfrak{m}+1})\right)}\right)\hfill \\ \hfill & \diamond \mathcal{E}\left({\vartheta }_0,{\vartheta }_1,\frac{\varsigma }{3^3{\rho }^{\mathfrak{j}+5}\left(\eta ({\vartheta }_{\mathfrak{j}+6},{\vartheta }_{\mathfrak{j}+2\mathfrak{m}+1})\eta ({\vartheta }_{\mathfrak{j}+4},{\vartheta }_{\mathfrak{j}+2\mathfrak{m}+1})\eta ({\vartheta }_{\mathfrak{j}+2},{\vartheta }_{\mathfrak{j}+2\mathfrak{m}+1})\right)}\right)\diamond \cdots \diamond \hfill \\ \hfill & \mathcal{E}\left({\vartheta }_0,{\vartheta }_1,\frac{\varsigma }{3^{\mathfrak{m}}{\rho }^{\mathfrak{j}+2\mathfrak{m}-3}(\beta ({\vartheta }_{\mathfrak{j}+2\mathfrak{m}-2},{\vartheta }_{\mathfrak{j}+2\mathfrak{m}-1})\eta ({\vartheta }_{\mathfrak{j}+2\mathfrak{m}-2},{\vartheta }_{\mathfrak{j}+2\mathfrak{m}+1})\cdots \eta ({\vartheta }_{\mathfrak{j}+2},{\vartheta }_{\mathfrak{j}+2\mathfrak{m}+1}))}\right)\hfill \\ \hfill & \diamond \mathcal{E}\left({\vartheta }_0,{\vartheta }_1,\frac{\varsigma }{3^{\mathfrak{m}}{\rho }^{\mathfrak{j}+2\mathfrak{m}-2}(\mathrm{\Gamma }({\vartheta }_{\mathfrak{j}+2\mathfrak{m}-1},{\vartheta }_{\mathfrak{j}+2\mathfrak{m}})\eta ({\vartheta }_{\mathfrak{j}+2\mathfrak{m}-2},{\vartheta }_{\mathfrak{j}+2\mathfrak{m}+1})\cdots \eta ({\vartheta }_{\mathfrak{j}+2},{\vartheta }_{\mathfrak{j}+2\mathfrak{m}+1}))}\right)\hfill \\ \hfill & \diamond \mathcal{E}\left({\vartheta }_0,{\vartheta }_1,\frac{\varsigma }{3^{\mathfrak{m}}{\rho }^{\mathfrak{j}+2\mathfrak{m}-1}(\eta ({\vartheta }_{\mathfrak{j}+2\mathfrak{m}},{\vartheta }_{\mathfrak{j}+2\mathfrak{m}+1})\eta ({\vartheta }_{\mathfrak{j}+2\mathfrak{m}-2},{\vartheta }_{\mathfrak{j}+2\mathfrak{m}+1})\cdots \eta ({\vartheta }_{\mathfrak{j}+2},{\vartheta }_{\mathfrak{j}+2\mathfrak{m}+1}))}\right),\hfill \end{array}$$

$$\begin{array}{cc}\hfill \mathcal{G}& ({\vartheta }_{\mathfrak{j}},{\vartheta }_{\mathfrak{j}+2\mathfrak{m}+1},\varsigma )\hfill \\ & \le \mathcal{G}\left({\vartheta }_0,{\vartheta }_1,\frac{\varsigma }{3{\rho }^{\mathfrak{j}-1}\left(\beta ({\vartheta }_{\mathfrak{j}},{\vartheta }_{\mathfrak{j}+1})\right)}\right)\diamond \mathcal{G}\left({\vartheta }_0,{\vartheta }_1,\frac{\varsigma }{3{\rho }^{\mathfrak{j}}\left(\mathrm{\Gamma }({\vartheta }_{\mathfrak{j}+1},{\vartheta }_{\mathfrak{j}+2})\right)}\right)\hfill \\ \hfill & \diamond \mathcal{G}\left({\vartheta }_0,{\vartheta }_1,\frac{\varsigma }{3^2{\rho }^{\mathfrak{j}+1}\left(\beta ({\vartheta }_{\mathfrak{j}+2},{\vartheta }_{\mathfrak{j}+3})\eta ({\vartheta }_{\mathfrak{j}+2},{\vartheta }_{\mathfrak{j}+2\mathfrak{m}+1})\right)}\right)\hfill \\ \hfill & \diamond \mathcal{G}\left({\vartheta }_0,{\vartheta }_1,\frac{\varsigma }{3^2{\rho }^{\mathfrak{j}+2}\left(\mathrm{\Gamma }({\vartheta }_{\mathfrak{j}+3},{\vartheta }_{\mathfrak{j}+4})\eta ({\vartheta }_{\mathfrak{j}+2},{\vartheta }_{\mathfrak{j}+2\mathfrak{m}+1})\right)}\right)\hfill \\ \hfill & \diamond \mathcal{G}\left({\vartheta }_0,{\vartheta }_1,\frac{\varsigma }{3^3{\rho }^{\mathfrak{j}+3}\left(\beta ({\vartheta }_{\mathfrak{j}+4},{\vartheta }_{\mathfrak{j}+5})\eta ({\vartheta }_{\mathfrak{j}+4},{\vartheta }_{\mathfrak{j}+2\mathfrak{m}+1})\eta ({\vartheta }_{\mathfrak{j}+2},{\vartheta }_{\mathfrak{j}+2\mathfrak{m}+1})\right)}\right)\hfill \\ \hfill & \diamond \mathcal{G}\left({\vartheta }_0,{\vartheta }_1,\frac{\varsigma }{3^3{\rho }^{\mathfrak{j}+4}\left(\mathrm{\Gamma }({\vartheta }_{\mathfrak{j}+5},{\vartheta }_{\mathfrak{j}+6})\eta ({\vartheta }_{\mathfrak{j}+4},{\vartheta }_{\mathfrak{j}+2\mathfrak{m}+1})\eta ({\vartheta }_{\mathfrak{j}+2},{\vartheta }_{\mathfrak{j}+2\mathfrak{m}+1})\right)}\right)\hfill \\ \hfill & \diamond \mathcal{G}\left({\vartheta }_0,{\vartheta }_1,\frac{\varsigma }{3^3{\rho }^{\mathfrak{j}+5}\left(\eta ({\vartheta }_{\mathfrak{j}+6},{\vartheta }_{\mathfrak{j}+2\mathfrak{m}+1})\eta ({\vartheta }_{\mathfrak{j}+4},{\vartheta }_{\mathfrak{j}+2\mathfrak{m}+1})\eta ({\vartheta }_{\mathfrak{j}+2},{\vartheta }_{\mathfrak{j}+2\mathfrak{m}+1})\right)}\right)\diamond \cdots \diamond \hfill \\ \hfill & \mathcal{G}\left({\vartheta }_0,{\vartheta }_1,\frac{\varsigma }{3^{\mathfrak{m}}{\rho }^{\mathfrak{j}+2\mathfrak{m}-3}(\beta ({\vartheta }_{\mathfrak{j}+2\mathfrak{m}-2},{\vartheta }_{\mathfrak{j}+2\mathfrak{m}-1})\eta ({\vartheta }_{\mathfrak{j}+2\mathfrak{m}-2},{\vartheta }_{\mathfrak{j}+2\mathfrak{m}+1})\cdots \eta ({\vartheta }_{\mathfrak{j}+2},{\vartheta }_{\mathfrak{j}+2\mathfrak{m}+1}))}\right)\hfill \\ \hfill & \diamond \mathcal{G}\left({\vartheta }_0,{\vartheta }_1,\frac{\varsigma }{3^{\mathfrak{m}}{\rho }^{\mathfrak{j}+2\mathfrak{m}-2}(\mathrm{\Gamma }({\vartheta }_{\mathfrak{j}+2\mathfrak{m}-1},{\vartheta }_{\mathfrak{j}+2\mathfrak{m}})\eta ({\vartheta }_{\mathfrak{j}+2\mathfrak{m}-2},{\vartheta }_{\mathfrak{j}+2\mathfrak{m}+1})\cdots \eta ({\vartheta }_{\mathfrak{j}+2},{\vartheta }_{\mathfrak{j}+2\mathfrak{m}+1}))}\right)\hfill \\ \hfill & \diamond \mathcal{G}\left({\vartheta }_0,{\vartheta }_1,\frac{\varsigma }{3^{\mathfrak{m}}{\rho }^{\mathfrak{j}+2\mathfrak{m}-1}(\eta ({\vartheta }_{\mathfrak{j}+2\mathfrak{m}},{\vartheta }_{\mathfrak{j}+2\mathfrak{m}+1})\eta ({\vartheta }_{\mathfrak{j}+2\mathfrak{m}-2},{\vartheta }_{\mathfrak{j}+2\mathfrak{m}+1})\cdots \eta ({\vartheta }_{\mathfrak{j}+2},{\vartheta }_{\mathfrak{j}+2\mathfrak{m}+1}))}\right).\hfill \end{array}$$

Case 2. When $\mathfrak{i}=2\mathfrak{m}$, i.e., $\mathfrak{i}$ is even, then

$$\begin{array}{cc}\hfill \mathcal{W}({\vartheta }_{\mathfrak{j}},& {\vartheta }_{\mathfrak{j}+2\mathfrak{m}},\varsigma )\hfill \\ & \ge \mathcal{W}\left({\vartheta }_{\mathfrak{j}},{\vartheta }_{\mathfrak{j}+1},\frac{\varsigma }{3\left(\beta ({\vartheta }_{\mathfrak{j}},{\vartheta }_{\mathfrak{j}+1})\right)}\right)\star \mathcal{W}\left({\vartheta }_{\mathfrak{j}+1},{\vartheta }_{\mathfrak{j}+2},\frac{\varsigma }{3\left(\mathrm{\Gamma }({\vartheta }_{\mathfrak{j}+1},{\vartheta }_{\mathfrak{j}+2})\right)}\right)\hfill \\ \hfill & \star \mathcal{W}\left({\vartheta }_{\mathfrak{j}+2},{\vartheta }_{\mathfrak{j}+2\mathfrak{m}},\frac{\varsigma }{3\left(\eta ({\vartheta }_{\mathfrak{j}+2},{\vartheta }_{\mathfrak{j}+2\mathfrak{m}})\right)}\right)\hfill \\ \hfill & \ge \mathcal{W}\left({\vartheta }_{\mathfrak{j}},{\vartheta }_{\mathfrak{j}+1},\frac{\varsigma }{3\left(\beta ({\vartheta }_{\mathfrak{j}},{\vartheta }_{\mathfrak{j}+1})\right)}\right)\star \mathcal{W}\left({\vartheta }_{\mathfrak{j}+1},{\vartheta }_{\mathfrak{j}+2},\frac{\varsigma }{3\left(\mathrm{\Gamma }({\vartheta }_{\mathfrak{j}+1},{\vartheta }_{\mathfrak{j}+2})\right)}\right)\hfill \\ \hfill & \star \mathcal{W}\left({\vartheta }_{\mathfrak{j}+2},{\vartheta }_{\mathfrak{j}+3},\frac{\varsigma }{3^2\left(\beta ({\vartheta }_{\mathfrak{j}+2},{\vartheta }_{\mathfrak{j}+3})\eta ({\vartheta }_{\mathfrak{j}+2},{\vartheta }_{\mathfrak{j}+2\mathfrak{m}})\right)}\right)\hfill \\ \hfill & \star \mathcal{W}\left({\vartheta }_{\mathfrak{j}+3},{\vartheta }_{\mathfrak{j}+4},\frac{\varsigma }{3^2\left(\mathrm{\Gamma }({\vartheta }_{\mathfrak{j}+3},{\vartheta }_{\mathfrak{j}+4})\eta ({\vartheta }_{\mathfrak{j}+2},{\vartheta }_{\mathfrak{j}+2\mathfrak{m}})\right)}\right)\hfill \\ \hfill & \star \mathcal{W}\left({\vartheta }_{\mathfrak{j}+4},{\vartheta }_{\mathfrak{j}+2\mathfrak{m}},\frac{\varsigma }{3^2\left(\eta ({\vartheta }_{\mathfrak{j}+4},{\vartheta }_{\mathfrak{j}+2\mathfrak{m}})\eta ({\vartheta }_{\mathfrak{j}+2},{\vartheta }_{\mathfrak{j}+2\mathfrak{m}})\right)}\right)\hfill \\ \hfill & \ge \mathcal{W}\left({\vartheta }_{\mathfrak{j}},{\vartheta }_{\mathfrak{j}+1},\frac{\varsigma }{3\left(\beta ({\vartheta }_{\mathfrak{j}},{\vartheta }_{\mathfrak{j}+1})\right)}\right)\star \mathcal{W}\left({\vartheta }_{\mathfrak{j}+1},{\vartheta }_{\mathfrak{j}+2},\frac{\varsigma }{3\left(\mathrm{\Gamma }({\vartheta }_{\mathfrak{j}+1},{\vartheta }_{\mathfrak{j}+2})\right)}\right)\hfill \\ \hfill & \star \mathcal{W}\left({\vartheta }_{\mathfrak{j}+2},{\vartheta }_{\mathfrak{j}+3},\frac{\varsigma }{3^2\left(\beta ({\vartheta }_{\mathfrak{j}+2},{\vartheta }_{\mathfrak{j}+3})\eta ({\vartheta }_{\mathfrak{j}+2},{\vartheta }_{\mathfrak{j}+2\mathfrak{m}})\right)}\right)\hfill \\ \hfill & \star \mathcal{W}\left({\vartheta }_{\mathfrak{j}+3},{\vartheta }_{\mathfrak{j}+4},\frac{\varsigma }{3^2\left(\mathrm{\Gamma }({\vartheta }_{\mathfrak{j}+3},{\vartheta }_{\mathfrak{j}+4})\eta ({\vartheta }_{\mathfrak{j}+2},{\vartheta }_{\mathfrak{j}+2\mathfrak{m}})\right)}\right)\hfill \\ \hfill & \star \mathcal{W}\left({\vartheta }_{\mathfrak{j}+4},{\vartheta }_{\mathfrak{j}+5},\frac{\varsigma }{3^3\left(\beta ({\vartheta }_{\mathfrak{j}+4},{\vartheta }_{\mathfrak{j}+5})\eta ({\vartheta }_{\mathfrak{j}+4},{\vartheta }_{\mathfrak{j}+2\mathfrak{m}})\eta ({\vartheta }_{\mathfrak{j}+2},{\vartheta }_{\mathfrak{j}+2\mathfrak{m}})\right)}\right)\hfill \\ \hfill & \star \mathcal{W}\left({\vartheta }_{\mathfrak{j}+5},{\vartheta }_{\mathfrak{j}+6},\frac{\varsigma }{3^3\left(\mathrm{\Gamma }({\vartheta }_{\mathfrak{j}+5},{\vartheta }_{\mathfrak{j}+6})\eta ({\vartheta }_{\mathfrak{j}+4},{\vartheta }_{\mathfrak{j}+2\mathfrak{m}})\eta ({\vartheta }_{\mathfrak{j}+2},{\vartheta }_{\mathfrak{j}+2\mathfrak{m}})\right)}\right)\hfill \\ \hfill & \star \mathcal{W}\left({\vartheta }_{\mathfrak{j}+6},{\vartheta }_{\mathfrak{j}+2\mathfrak{m}},\frac{\varsigma }{3^3\left(\eta ({\vartheta }_{\mathfrak{j}+6},{\vartheta }_{\mathfrak{j}+2\mathfrak{m}})\eta ({\vartheta }_{\mathfrak{j}+4},{\vartheta }_{\mathfrak{j}+2\mathfrak{m}})\eta ({\vartheta }_{\mathfrak{j}+2},{\vartheta }_{\mathfrak{j}+2\mathfrak{m}})\right)}\right)\hfill \end{array}$$

$$\begin{array}{cc}\hfill \mathcal{W}& ({\vartheta }_{\mathfrak{j}},{\vartheta }_{\mathfrak{j}+2\mathfrak{m}},\varsigma )\hfill \\ & \ge \mathcal{W}\left({\vartheta }_{\mathfrak{j}},{\vartheta }_{\mathfrak{j}+1},\frac{\varsigma }{3\left(\beta ({\vartheta }_{\mathfrak{j}},{\vartheta }_{\mathfrak{j}+1})\right)}\right)\star \mathcal{W}\left({\vartheta }_{\mathfrak{j}+1},{\vartheta }_{\mathfrak{j}+2},\frac{\varsigma }{3\left(\mathrm{\Gamma }({\vartheta }_{\mathfrak{j}+1},{\vartheta }_{\mathfrak{j}+2})\right)}\right)\hfill \\ \hfill & \star \mathcal{W}\left({\vartheta }_{\mathfrak{j}+2},{\vartheta }_{\mathfrak{j}+3},\frac{\varsigma }{3^2\left(\beta ({\vartheta }_{\mathfrak{j}+2},{\vartheta }_{\mathfrak{j}+3})\eta ({\vartheta }_{\mathfrak{j}+2},{\vartheta }_{\mathfrak{j}+2\mathfrak{m}})\right)}\right)\hfill \\ \hfill & \star \mathcal{W}\left({\vartheta }_{\mathfrak{j}+3},{\vartheta }_{\mathfrak{j}+4},\frac{\varsigma }{3^2\left(\mathrm{\Gamma }({\vartheta }_{\mathfrak{j}+3},{\vartheta }_{\mathfrak{j}+4})\eta ({\vartheta }_{\mathfrak{j}+2},{\vartheta }_{\mathfrak{j}+2\mathfrak{m}})\right)}\right)\hfill \\ \hfill & \star \mathcal{W}\left({\vartheta }_{\mathfrak{j}+4},{\vartheta }_{\mathfrak{j}+5},\frac{\varsigma }{3^3\left(\beta ({\vartheta }_{\mathfrak{j}+4},{\vartheta }_{\mathfrak{j}+5})\eta ({\vartheta }_{\mathfrak{j}+4},{\vartheta }_{\mathfrak{j}+2\mathfrak{m}})\eta ({\vartheta }_{\mathfrak{j}+2},{\vartheta }_{\mathfrak{j}+2\mathfrak{m}})\right)}\right)\hfill \\ \hfill & \star \mathcal{W}\left({\vartheta }_{\mathfrak{j}+5},{\vartheta }_{\mathfrak{j}+6},\frac{\varsigma }{3^3\left(\mathrm{\Gamma }({\vartheta }_{\mathfrak{j}+5},{\vartheta }_{\mathfrak{j}+6})\eta ({\vartheta }_{\mathfrak{j}+4},{\vartheta }_{\mathfrak{j}+2\mathfrak{m}})\eta ({\vartheta }_{\mathfrak{j}+2},{\vartheta }_{\mathfrak{j}+2\mathfrak{m}})\right)}\right)\hfill \\ \hfill & \star \mathcal{W}\left({\vartheta }_{\mathfrak{j}+6},{\vartheta }_{\mathfrak{j}+2\mathfrak{m}},\frac{\varsigma }{3^3\left(\eta ({\vartheta }_{\mathfrak{j}+6},{\vartheta }_{\mathfrak{j}+2\mathfrak{m}})\eta ({\vartheta }_{\mathfrak{j}+4},{\vartheta }_{\mathfrak{j}+2\mathfrak{m}})\eta ({\vartheta }_{\mathfrak{j}+2},{\vartheta }_{\mathfrak{j}+2\mathfrak{m}})\right)}\right)\star \cdots \star \hfill \\ \hfill & \mathcal{W}\left({\vartheta }_{\mathfrak{j}+2\mathfrak{m}-4},{\vartheta }_{\mathfrak{j}+2\mathfrak{m}-3},\frac{\varsigma }{3^{\mathfrak{m}-1}(\beta ({\vartheta }_{\mathfrak{j}+2\mathfrak{m}-4},{\vartheta }_{\mathfrak{j}+2\mathfrak{m}-3})\eta ({\vartheta }_{\mathfrak{j}+2\mathfrak{m}-4},{\vartheta }_{\mathfrak{j}+2\mathfrak{m}})\cdots \eta ({\vartheta }_{\mathfrak{j}+2},{\vartheta }_{\mathfrak{j}+2\mathfrak{m}}))}\right)\hfill \\ \hfill & \star \mathcal{W}\left({\vartheta }_{\mathfrak{j}+2\mathfrak{m}-3},{\vartheta }_{\mathfrak{j}+2\mathfrak{m}-2},\frac{\varsigma }{3^{\mathfrak{m}-1}(\mathrm{\Gamma }({\vartheta }_{\mathfrak{j}+2\mathfrak{m}-3},{\vartheta }_{\mathfrak{j}+2\mathfrak{m}-2})\eta ({\vartheta }_{\mathfrak{j}+2\mathfrak{m}-4},{\vartheta }_{\mathfrak{j}+2\mathfrak{m}})\cdots \eta ({\vartheta }_{\mathfrak{j}+2},{\vartheta }_{\mathfrak{j}+2\mathfrak{m}}))}\right)\hfill \\ \hfill & \star \mathcal{W}\left({\vartheta }_{\mathfrak{j}+2\mathfrak{m}-2},{\vartheta }_{\mathfrak{j}+2\mathfrak{m}},\frac{\varsigma }{3^{\mathfrak{m}-1}(\eta ({\vartheta }_{\mathfrak{j}+2\mathfrak{m}-2},{\vartheta }_{\mathfrak{j}+2\mathfrak{m}})\eta ({\vartheta }_{\mathfrak{j}+2\mathfrak{m}-4},{\vartheta }_{\mathfrak{j}+2\mathfrak{m}})\cdots \eta ({\vartheta }_{\mathfrak{j}+2},{\vartheta }_{\mathfrak{j}+2\mathfrak{m}}))}\right),\hfill \end{array}$$

$$\begin{array}{cc}\hfill \mathcal{E}({\vartheta }_{\mathfrak{j}},{\vartheta }_{\mathfrak{j}+2\mathfrak{m}},\varsigma & )\hfill \\ & \le \mathcal{E}\left({\vartheta }_{\mathfrak{j}},{\vartheta }_{\mathfrak{j}+1},\frac{\varsigma }{3\left(\beta ({\vartheta }_{\mathfrak{j}},{\vartheta }_{\mathfrak{j}+1})\right)}\right)\diamond \mathcal{E}\left({\vartheta }_{\mathfrak{j}+1},{\vartheta }_{\mathfrak{j}+2},\frac{\varsigma }{3\left(\mathrm{\Gamma }({\vartheta }_{\mathfrak{j}+1},{\vartheta }_{\mathfrak{j}+2})\right)}\right)\hfill \\ \hfill & \diamond \mathcal{E}\left({\vartheta }_{\mathfrak{j}+2},{\vartheta }_{\mathfrak{j}+2\mathfrak{m}},\frac{\varsigma }{3\left(\eta ({\vartheta }_{\mathfrak{j}+2},{\vartheta }_{\mathfrak{j}+2\mathfrak{m}})\right)}\right)\hfill \\ \hfill & \le \mathcal{E}\left({\vartheta }_{\mathfrak{j}},{\vartheta }_{\mathfrak{j}+1},\frac{\varsigma }{3\left(\beta ({\vartheta }_{\mathfrak{j}},{\vartheta }_{\mathfrak{j}+1})\right)}\right)\diamond \mathcal{E}\left({\vartheta }_{\mathfrak{j}+1},{\vartheta }_{\mathfrak{j}+2},\frac{\varsigma }{3\left(\mathrm{\Gamma }({\vartheta }_{\mathfrak{j}+1},{\vartheta }_{\mathfrak{j}+2})\right)}\right)\hfill \\ \hfill & \diamond \mathcal{E}\left({\vartheta }_{\mathfrak{j}+2},{\vartheta }_{\mathfrak{j}+3},\frac{\varsigma }{3^2\left(\beta ({\vartheta }_{\mathfrak{j}+2},{\vartheta }_{\mathfrak{j}+3})\eta ({\vartheta }_{\mathfrak{j}+2},{\vartheta }_{\mathfrak{j}+2\mathfrak{m}})\right)}\right)\hfill \\ \hfill & \diamond \mathcal{E}\left({\vartheta }_{\mathfrak{j}+3},{\vartheta }_{\mathfrak{j}+4},\frac{\varsigma }{3^2\left(\mathrm{\Gamma }({\vartheta }_{\mathfrak{j}+3},{\vartheta }_{\mathfrak{j}+4})\eta ({\vartheta }_{\mathfrak{j}+2},{\vartheta }_{\mathfrak{j}+2\mathfrak{m}})\right)}\right)\hfill \\ \hfill & \diamond \mathcal{E}\left({\vartheta }_{\mathfrak{j}+4},{\vartheta }_{\mathfrak{j}+2\mathfrak{m}},\frac{\varsigma }{3^2\left(\eta ({\vartheta }_{\mathfrak{j}+4},{\vartheta }_{\mathfrak{j}+2\mathfrak{m}})\eta ({\vartheta }_{\mathfrak{j}+2},{\vartheta }_{\mathfrak{j}+2\mathfrak{m}})\right)}\right)\hfill \\ \hfill & \le \mathcal{E}\left({\vartheta }_{\mathfrak{j}},{\vartheta }_{\mathfrak{j}+1},\frac{\varsigma }{3\left(\beta ({\vartheta }_{\mathfrak{j}},{\vartheta }_{\mathfrak{j}+1})\right)}\right)\diamond \mathcal{E}\left({\vartheta }_{\mathfrak{j}+1},{\vartheta }_{\mathfrak{j}+2},\frac{\varsigma }{3\left(\mathrm{\Gamma }({\vartheta }_{\mathfrak{j}+1},{\vartheta }_{\mathfrak{j}+2})\right)}\right)\hfill \\ \hfill & \diamond \mathcal{E}\left({\vartheta }_{\mathfrak{j}+2},{\vartheta }_{\mathfrak{j}+3},\frac{\varsigma }{3^2\left(\beta ({\vartheta }_{\mathfrak{j}+2},{\vartheta }_{\mathfrak{j}+3})\eta ({\vartheta }_{\mathfrak{j}+2},{\vartheta }_{\mathfrak{j}+2\mathfrak{m}})\right)}\right)\hfill \\ \hfill & \diamond \mathcal{E}\left({\vartheta }_{\mathfrak{j}+3},{\vartheta }_{\mathfrak{j}+4},\frac{\varsigma }{3^2\left(\mathrm{\Gamma }({\vartheta }_{\mathfrak{j}+3},{\vartheta }_{\mathfrak{j}+4})\eta ({\vartheta }_{\mathfrak{j}+2},{\vartheta }_{\mathfrak{j}+2\mathfrak{m}})\right)}\right)\hfill \\ \hfill & \diamond \mathcal{E}\left({\vartheta }_{\mathfrak{j}+4},{\vartheta }_{\mathfrak{j}+5},\frac{\varsigma }{3^3\left(\beta ({\vartheta }_{\mathfrak{j}+4},{\vartheta }_{\mathfrak{j}+5})\eta ({\vartheta }_{\mathfrak{j}+4},{\vartheta }_{\mathfrak{j}+2\mathfrak{m}})\eta ({\vartheta }_{\mathfrak{j}+2},{\vartheta }_{\mathfrak{j}+2\mathfrak{m}})\right)}\right)\hfill \\ \hfill & \diamond \mathcal{E}\left({\vartheta }_{\mathfrak{j}+5},{\vartheta }_{\mathfrak{j}+6},\frac{\varsigma }{3^3\left(\mathrm{\Gamma }({\vartheta }_{\mathfrak{j}+5},{\vartheta }_{\mathfrak{j}+6})\eta ({\vartheta }_{\mathfrak{j}+4},{\vartheta }_{\mathfrak{j}+2\mathfrak{m}})\eta ({\vartheta }_{\mathfrak{j}+2},{\vartheta }_{\mathfrak{j}+2\mathfrak{m}})\right)}\right)\hfill \\ \hfill & \diamond \mathcal{E}\left({\vartheta }_{\mathfrak{j}+6},{\vartheta }_{\mathfrak{j}+2\mathfrak{m}},\frac{\varsigma }{3^3\left(\eta ({\vartheta }_{\mathfrak{j}+6},{\vartheta }_{\mathfrak{j}+2\mathfrak{m}})\eta ({\vartheta }_{\mathfrak{j}+4},{\vartheta }_{\mathfrak{j}+2\mathfrak{m}})\eta ({\vartheta }_{\mathfrak{j}+2},{\vartheta }_{\mathfrak{j}+2\mathfrak{m}})\right)}\right)\hfill \end{array}$$

$$\begin{array}{cc}\hfill \mathcal{E}& ({\vartheta }_{\mathfrak{j}},{\vartheta }_{\mathfrak{j}+2\mathfrak{m}},\varsigma )\hfill \\ & \le \mathcal{E}\left({\vartheta }_{\mathfrak{j}},{\vartheta }_{\mathfrak{j}+1},\frac{\varsigma }{3\left(\beta ({\vartheta }_{\mathfrak{j}},{\vartheta }_{\mathfrak{j}+1})\right)}\right)\diamond \mathcal{E}\left({\vartheta }_{\mathfrak{j}+1},{\vartheta }_{\mathfrak{j}+2},\frac{\varsigma }{3\left(\mathrm{\Gamma }({\vartheta }_{\mathfrak{j}+1},{\vartheta }_{\mathfrak{j}+2})\right)}\right)\hfill \\ \hfill & \diamond \mathcal{E}\left({\vartheta }_{\mathfrak{j}+2},{\vartheta }_{\mathfrak{j}+3},\frac{\varsigma }{3^2\left(\beta ({\vartheta }_{\mathfrak{j}+2},{\vartheta }_{\mathfrak{j}+3})\eta ({\vartheta }_{\mathfrak{j}+2},{\vartheta }_{\mathfrak{j}+2\mathfrak{m}})\right)}\right)\hfill \\ \hfill & \diamond \mathcal{E}\left({\vartheta }_{\mathfrak{j}+3},{\vartheta }_{\mathfrak{j}+4},\frac{\varsigma }{3^2\left(\mathrm{\Gamma }({\vartheta }_{\mathfrak{j}+3},{\vartheta }_{\mathfrak{j}+4})\eta ({\vartheta }_{\mathfrak{j}+2},{\vartheta }_{\mathfrak{j}+2\mathfrak{m}})\right)}\right)\hfill \\ \hfill & \diamond \mathcal{E}\left({\vartheta }_{\mathfrak{j}+4},{\vartheta }_{\mathfrak{j}+5},\frac{\varsigma }{3^3\left(\beta ({\vartheta }_{\mathfrak{j}+4},{\vartheta }_{\mathfrak{j}+5})\eta ({\vartheta }_{\mathfrak{j}+4},{\vartheta }_{\mathfrak{j}+2\mathfrak{m}})\eta ({\vartheta }_{\mathfrak{j}+2},{\vartheta }_{\mathfrak{j}+2\mathfrak{m}})\right)}\right)\hfill \\ \hfill & \diamond \mathcal{E}\left({\vartheta }_{\mathfrak{j}+5},{\vartheta }_{\mathfrak{j}+6},\frac{\varsigma }{3^3\left(\mathrm{\Gamma }({\vartheta }_{\mathfrak{j}+5},{\vartheta }_{\mathfrak{j}+6})\eta ({\vartheta }_{\mathfrak{j}+4},{\vartheta }_{\mathfrak{j}+2\mathfrak{m}})\eta ({\vartheta }_{\mathfrak{j}+2},{\vartheta }_{\mathfrak{j}+2\mathfrak{m}})\right)}\right)\hfill \\ \hfill & \diamond \mathcal{E}\left({\vartheta }_{\mathfrak{j}+6},{\vartheta }_{\mathfrak{j}+2\mathfrak{m}},\frac{\varsigma }{3^3\left(\eta ({\vartheta }_{\mathfrak{j}+6},{\vartheta }_{\mathfrak{j}+2\mathfrak{m}})\eta ({\vartheta }_{\mathfrak{j}+4},{\vartheta }_{\mathfrak{j}+2\mathfrak{m}})\eta ({\vartheta }_{\mathfrak{j}+2},{\vartheta }_{\mathfrak{j}+2\mathfrak{m}})\right)}\right)\diamond \cdots \diamond \hfill \\ \hfill & \mathcal{E}\left({\vartheta }_{\mathfrak{j}+2\mathfrak{m}-4},{\vartheta }_{\mathfrak{j}+2\mathfrak{m}-3},\frac{\varsigma }{3^{\mathfrak{m}-1}(\beta ({\vartheta }_{\mathfrak{j}+2\mathfrak{m}-4},{\vartheta }_{\mathfrak{j}+2\mathfrak{m}-3})\eta ({\vartheta }_{\mathfrak{j}+2\mathfrak{m}-4},{\vartheta }_{\mathfrak{j}+2\mathfrak{m}})\cdots \eta ({\vartheta }_{\mathfrak{j}+2},{\vartheta }_{\mathfrak{j}+2\mathfrak{m}}))}\right)\hfill \\ \hfill & \diamond \mathcal{E}\left({\vartheta }_{\mathfrak{j}+2\mathfrak{m}-3},{\vartheta }_{\mathfrak{j}+2\mathfrak{m}-2},\frac{\varsigma }{3^{\mathfrak{m}-1}(\mathrm{\Gamma }({\vartheta }_{\mathfrak{j}+2\mathfrak{m}-3},{\vartheta }_{\mathfrak{j}+2\mathfrak{m}-2})\eta ({\vartheta }_{\mathfrak{j}+2\mathfrak{m}-4},{\vartheta }_{\mathfrak{j}+2\mathfrak{m}})\cdots \eta ({\vartheta }_{\mathfrak{j}+2},{\vartheta }_{\mathfrak{j}+2\mathfrak{m}}))}\right)\hfill \\ \hfill & \diamond \mathcal{E}\left({\vartheta }_{\mathfrak{j}+2\mathfrak{m}-2},{\vartheta }_{\mathfrak{j}+2\mathfrak{m}},\frac{\varsigma }{3^{\mathfrak{m}-1}(\eta ({\vartheta }_{\mathfrak{j}+2\mathfrak{m}-2},{\vartheta }_{\mathfrak{j}+2\mathfrak{m}})\eta ({\vartheta }_{\mathfrak{j}+2\mathfrak{m}-4},{\vartheta }_{\mathfrak{j}+2\mathfrak{m}})\cdots \eta ({\vartheta }_{\mathfrak{j}+2},{\vartheta }_{\mathfrak{j}+2\mathfrak{m}}))}\right),\hfill \end{array}$$

and

$$\begin{array}{cc}\hfill \mathcal{G}({\vartheta }_{\mathfrak{j}},{\vartheta }_{\mathfrak{j}+2\mathfrak{m}},\varsigma & )\hfill \\ & \le \mathcal{G}\left({\vartheta }_{\mathfrak{j}},{\vartheta }_{\mathfrak{j}+1},\frac{\varsigma }{3\left(\beta ({\vartheta }_{\mathfrak{j}},{\vartheta }_{\mathfrak{j}+1})\right)}\right)\diamond \mathcal{G}\left({\vartheta }_{\mathfrak{j}+1},{\vartheta }_{\mathfrak{j}+2},\frac{\varsigma }{3\left(\mathrm{\Gamma }({\vartheta }_{\mathfrak{j}+1},{\vartheta }_{\mathfrak{j}+2})\right)}\right)\hfill \\ \hfill & \diamond \mathcal{G}\left({\vartheta }_{\mathfrak{j}+2},{\vartheta }_{\mathfrak{j}+2\mathfrak{m}},\frac{\varsigma }{3\left(\eta ({\vartheta }_{\mathfrak{j}+2},{\vartheta }_{\mathfrak{j}+2\mathfrak{m}})\right)}\right)\hfill \\ \hfill & \le \mathcal{G}\left({\vartheta }_{\mathfrak{j}},{\vartheta }_{\mathfrak{j}+1},\frac{\varsigma }{3\left(\beta ({\vartheta }_{\mathfrak{j}},{\vartheta }_{\mathfrak{j}+1})\right)}\right)\diamond \mathcal{G}\left({\vartheta }_{\mathfrak{j}+1},{\vartheta }_{\mathfrak{j}+2},\frac{\varsigma }{3\left(\mathrm{\Gamma }({\vartheta }_{\mathfrak{j}+1},{\vartheta }_{\mathfrak{j}+2})\right)}\right)\hfill \\ \hfill & \diamond \mathcal{G}\left({\vartheta }_{\mathfrak{j}+2},{\vartheta }_{\mathfrak{j}+3},\frac{\varsigma }{3^2\left(\beta ({\vartheta }_{\mathfrak{j}+2},{\vartheta }_{\mathfrak{j}+3})\eta ({\vartheta }_{\mathfrak{j}+2},{\vartheta }_{\mathfrak{j}+2\mathfrak{m}})\right)}\right)\hfill \\ \hfill & \diamond \mathcal{G}\left({\vartheta }_{\mathfrak{j}+3},{\vartheta }_{\mathfrak{j}+4},\frac{\varsigma }{3^2\left(\mathrm{\Gamma }({\vartheta }_{\mathfrak{j}+3},{\vartheta }_{\mathfrak{j}+4})\eta ({\vartheta }_{\mathfrak{j}+2},{\vartheta }_{\mathfrak{j}+2\mathfrak{m}})\right)}\right)\hfill \\ \hfill & \diamond \mathcal{G}\left({\vartheta }_{\mathfrak{j}+4},{\vartheta }_{\mathfrak{j}+2\mathfrak{m}},\frac{\varsigma }{3^2\left(\eta ({\vartheta }_{\mathfrak{j}+4},{\vartheta }_{\mathfrak{j}+2\mathfrak{m}})\eta ({\vartheta }_{\mathfrak{j}+2},{\vartheta }_{\mathfrak{j}+2\mathfrak{m}})\right)}\right)\hfill \\ \hfill & \le \mathcal{G}\left({\vartheta }_{\mathfrak{j}},{\vartheta }_{\mathfrak{j}+1},\frac{\varsigma }{3\left(\beta ({\vartheta }_{\mathfrak{j}},{\vartheta }_{\mathfrak{j}+1})\right)}\right)\diamond \mathcal{G}\left({\vartheta }_{\mathfrak{j}+1},{\vartheta }_{\mathfrak{j}+2},\frac{\varsigma }{3\left(\mathrm{\Gamma }({\vartheta }_{\mathfrak{j}+1},{\vartheta }_{\mathfrak{j}+2})\right)}\right)\hfill \\ \hfill & \diamond \mathcal{G}\left({\vartheta }_{\mathfrak{j}+2},{\vartheta }_{\mathfrak{j}+3},\frac{\varsigma }{3^2\left(\beta ({\vartheta }_{\mathfrak{j}+2},{\vartheta }_{\mathfrak{j}+3})\eta ({\vartheta }_{\mathfrak{j}+2},{\vartheta }_{\mathfrak{j}+2\mathfrak{m}})\right)}\right)\hfill \\ \hfill & \diamond \mathcal{G}\left({\vartheta }_{\mathfrak{j}+3},{\vartheta }_{\mathfrak{j}+4},\frac{\varsigma }{3^2\left(\mathrm{\Gamma }({\vartheta }_{\mathfrak{j}+3},{\vartheta }_{\mathfrak{j}+4})\eta ({\vartheta }_{\mathfrak{j}+2},{\vartheta }_{\mathfrak{j}+2\mathfrak{m}})\right)}\right)\hfill \\ \hfill & \diamond \mathcal{G}\left({\vartheta }_{\mathfrak{j}+4},{\vartheta }_{\mathfrak{j}+5},\frac{\varsigma }{3^3\left(\beta ({\vartheta }_{\mathfrak{j}+4},{\vartheta }_{\mathfrak{j}+5})\eta ({\vartheta }_{\mathfrak{j}+4},{\vartheta }_{\mathfrak{j}+2\mathfrak{m}})\eta ({\vartheta }_{\mathfrak{j}+2},{\vartheta }_{\mathfrak{j}+2\mathfrak{m}})\right)}\right)\hfill \\ \hfill & \diamond \mathcal{G}\left({\vartheta }_{\mathfrak{j}+5},{\vartheta }_{\mathfrak{j}+6},\frac{\varsigma }{3^3\left(\mathrm{\Gamma }({\vartheta }_{\mathfrak{j}+5},{\vartheta }_{\mathfrak{j}+6})\eta ({\vartheta }_{\mathfrak{j}+4},{\vartheta }_{\mathfrak{j}+2\mathfrak{m}})\eta ({\vartheta }_{\mathfrak{j}+2},{\vartheta }_{\mathfrak{j}+2\mathfrak{m}})\right)}\right)\hfill \\ \hfill & \diamond \mathcal{G}\left({\vartheta }_{\mathfrak{j}+6},{\vartheta }_{\mathfrak{j}+2\mathfrak{m}},\frac{\varsigma }{3^3\left(\eta ({\vartheta }_{\mathfrak{j}+6},{\vartheta }_{\mathfrak{j}+2\mathfrak{m}})\eta ({\vartheta }_{\mathfrak{j}+4},{\vartheta }_{\mathfrak{j}+2\mathfrak{m}})\eta ({\vartheta }_{\mathfrak{j}+2},{\vartheta }_{\mathfrak{j}+2\mathfrak{m}})\right)}\right)\hfill \end{array}$$

$$\begin{array}{cc}\hfill \mathcal{G}& ({\vartheta }_{\mathfrak{j}},{\vartheta }_{\mathfrak{j}+2\mathfrak{m}},\varsigma )\hfill \\ & \le \mathcal{G}\left({\vartheta }_{\mathfrak{j}},{\vartheta }_{\mathfrak{j}+1},\frac{\varsigma }{3\left(\beta ({\vartheta }_{\mathfrak{j}},{\vartheta }_{\mathfrak{j}+1})\right)}\right)\diamond \mathcal{G}\left({\vartheta }_{\mathfrak{j}+1},{\vartheta }_{\mathfrak{j}+2},\frac{\varsigma }{3\left(\mathrm{\Gamma }({\vartheta }_{\mathfrak{j}+1},{\vartheta }_{\mathfrak{j}+2})\right)}\right)\hfill \\ \hfill & \diamond \mathcal{G}\left({\vartheta }_{\mathfrak{j}+2},{\vartheta }_{\mathfrak{j}+3},\frac{\varsigma }{3^2\left(\beta ({\vartheta }_{\mathfrak{j}+2},{\vartheta }_{\mathfrak{j}+3})\eta ({\vartheta }_{\mathfrak{j}+2},{\vartheta }_{\mathfrak{j}+2\mathfrak{m}})\right)}\right)\hfill \\ \hfill & \diamond \mathcal{G}\left({\vartheta }_{\mathfrak{j}+3},{\vartheta }_{\mathfrak{j}+4},\frac{\varsigma }{3^2\left(\mathrm{\Gamma }({\vartheta }_{\mathfrak{j}+3},{\vartheta }_{\mathfrak{j}+4})\eta ({\vartheta }_{\mathfrak{j}+2},{\vartheta }_{\mathfrak{j}+2\mathfrak{m}})\right)}\right)\hfill \\ \hfill & \diamond \mathcal{G}\left({\vartheta }_{\mathfrak{j}+4},{\vartheta }_{\mathfrak{j}+5},\frac{\varsigma }{3^3\left(\beta ({\vartheta }_{\mathfrak{j}+4},{\vartheta }_{\mathfrak{j}+5})\eta ({\vartheta }_{\mathfrak{j}+4},{\vartheta }_{\mathfrak{j}+2\mathfrak{m}})\eta ({\vartheta }_{\mathfrak{j}+2},{\vartheta }_{\mathfrak{j}+2\mathfrak{m}})\right)}\right)\hfill \\ \hfill & \diamond \mathcal{G}\left({\vartheta }_{\mathfrak{j}+5},{\vartheta }_{\mathfrak{j}+6},\frac{\varsigma }{3^3\left(\mathrm{\Gamma }({\vartheta }_{\mathfrak{j}+5},{\vartheta }_{\mathfrak{j}+6})\eta ({\vartheta }_{\mathfrak{j}+4},{\vartheta }_{\mathfrak{j}+2\mathfrak{m}})\eta ({\vartheta }_{\mathfrak{j}+2},{\vartheta }_{\mathfrak{j}+2\mathfrak{m}})\right)}\right)\hfill \\ \hfill & \diamond \mathcal{G}\left({\vartheta }_{\mathfrak{j}+6},{\vartheta }_{\mathfrak{j}+2\mathfrak{m}},\frac{\varsigma }{3^3\left(\eta ({\vartheta }_{\mathfrak{j}+6},{\vartheta }_{\mathfrak{j}+2\mathfrak{m}})\eta ({\vartheta }_{\mathfrak{j}+4},{\vartheta }_{\mathfrak{j}+2\mathfrak{m}})\eta ({\vartheta }_{\mathfrak{j}+2},{\vartheta }_{\mathfrak{j}+2\mathfrak{m}})\right)}\right)\diamond \cdots \diamond \hfill \\ \hfill & \mathcal{G}\left({\vartheta }_{\mathfrak{j}+2\mathfrak{m}-4},{\vartheta }_{\mathfrak{j}+2\mathfrak{m}-3},\frac{\varsigma }{3^{\mathfrak{m}-1}(\beta ({\vartheta }_{\mathfrak{j}+2\mathfrak{m}-4},{\vartheta }_{\mathfrak{j}+2\mathfrak{m}-3})\eta ({\vartheta }_{\mathfrak{j}+2\mathfrak{m}-4},{\vartheta }_{\mathfrak{j}+2\mathfrak{m}})\cdots \eta ({\vartheta }_{\mathfrak{j}+2},{\vartheta }_{\mathfrak{j}+2\mathfrak{m}}))}\right)\hfill \\ \hfill & \diamond \mathcal{G}\left({\vartheta }_{\mathfrak{j}+2\mathfrak{m}-3},{\vartheta }_{\mathfrak{j}+2\mathfrak{m}-2},\frac{\varsigma }{3^{\mathfrak{m}-1}(\mathrm{\Gamma }({\vartheta }_{\mathfrak{j}+2\mathfrak{m}-3},{\vartheta }_{\mathfrak{j}+2\mathfrak{m}-2})\eta ({\vartheta }_{\mathfrak{j}+2\mathfrak{m}-4},{\vartheta }_{\mathfrak{j}+2\mathfrak{m}})\cdots \eta ({\vartheta }_{\mathfrak{j}+2},{\vartheta }_{\mathfrak{j}+2\mathfrak{m}}))}\right)\hfill \\ \hfill & \diamond \mathcal{G}\left({\vartheta }_{\mathfrak{j}+2\mathfrak{m}-2},{\vartheta }_{\mathfrak{j}+2\mathfrak{m}},\frac{\varsigma }{3^{\mathfrak{m}-1}(\eta ({\vartheta }_{\mathfrak{j}+2\mathfrak{m}-2},{\vartheta }_{\mathfrak{j}+2\mathfrak{m}})\eta ({\vartheta }_{\mathfrak{j}+2\mathfrak{m}-4},{\vartheta }_{\mathfrak{j}+2\mathfrak{m}})\cdots \eta ({\vartheta }_{\mathfrak{j}+2},{\vartheta }_{\mathfrak{j}+2\mathfrak{m}}))}\right).\hfill \end{array}$$

Using (3) in the above inequalities, we have

$$\begin{array}{cc}\hfill \mathcal{W}& ({\vartheta }_{\mathfrak{j}},{\vartheta }_{\mathfrak{j}+2\mathfrak{m}},\varsigma )\hfill \\ & \ge \mathcal{W}\left({\vartheta }_0,{\vartheta }_1,\frac{\varsigma }{3{\rho }^{\mathfrak{j}-1}\beta ({\vartheta }_{\mathfrak{j}},{\vartheta }_{\mathfrak{j}+1})}\right)\star \mathcal{W}\left({\vartheta }_0,{\vartheta }_1,\frac{\varsigma }{3{\rho }^{\mathfrak{j}}\left(\mathrm{\Gamma }({\vartheta }_{\mathfrak{j}+1},{\vartheta }_{\mathfrak{j}+2})\right)}\right)\hfill \\ \hfill & \star \mathcal{W}\left({\vartheta }_0,{\vartheta }_1,\frac{\varsigma }{3^2{\rho }^{\mathfrak{j}+1}\left(\beta ({\vartheta }_{\mathfrak{j}+2},{\vartheta }_{\mathfrak{j}+3})\eta ({\vartheta }_{\mathfrak{j}+2},{\vartheta }_{\mathfrak{j}+2\mathfrak{m}})\right)}\right)\hfill \\ \hfill & \star \mathcal{W}\left({\vartheta }_0,{\vartheta }_1,\frac{\varsigma }{3^2{\rho }^{\mathfrak{j}+2}\left(\mathrm{\Gamma }({\vartheta }_{\mathfrak{j}+3},{\vartheta }_{\mathfrak{j}+4})\eta ({\vartheta }_{\mathfrak{j}+2},{\vartheta }_{\mathfrak{j}+2\mathfrak{m}})\right)}\right)\hfill \\ \hfill & \star \mathcal{W}\left({\vartheta }_0,{\vartheta }_1,\frac{\varsigma }{3^3{\rho }^{\mathfrak{j}+3}\left(\beta ({\vartheta }_{\mathfrak{j}+4},{\vartheta }_{\mathfrak{j}+5})\eta ({\vartheta }_{\mathfrak{j}+4},{\vartheta }_{\mathfrak{j}+2\mathfrak{m}})\eta ({\vartheta }_{\mathfrak{j}+2},{\vartheta }_{\mathfrak{j}+2\mathfrak{m}})\right)}\right)\hfill \\ \hfill & \star \mathcal{W}\left({\vartheta }_0,{\vartheta }_1,\frac{\varsigma }{3^3{\rho }^{\mathfrak{j}+4}\left(\mathrm{\Gamma }({\vartheta }_{\mathfrak{j}+5},{\vartheta }_{\mathfrak{j}+6})\eta ({\vartheta }_{\mathfrak{j}+4},{\vartheta }_{\mathfrak{j}+2\mathfrak{m}})\eta ({\vartheta }_{\mathfrak{j}+2},{\vartheta }_{\mathfrak{j}+2\mathfrak{m}})\right)}\right)\hfill \\ \hfill & \star \mathcal{W}\left({\vartheta }_0,{\vartheta }_1,\frac{\varsigma }{3^3{\rho }^{\mathfrak{j}+5}\left(\eta ({\vartheta }_{\mathfrak{j}+6},{\vartheta }_{\mathfrak{j}+2\mathfrak{m}})\eta ({\vartheta }_{\mathfrak{j}+4},{\vartheta }_{\mathfrak{j}+2\mathfrak{m}})\eta ({\vartheta }_{\mathfrak{j}+2},{\vartheta }_{\mathfrak{j}+2\mathfrak{m}})\right)}\right)\star \cdots \star \hfill \\ \hfill & \mathcal{W}\left({\vartheta }_0,{\vartheta }_1,\frac{\varsigma }{3^{\mathfrak{m}-1}{\rho }^{\mathfrak{j}+2\mathfrak{m}-5}(\beta ({\vartheta }_{\mathfrak{j}+2\mathfrak{m}-4},{\vartheta }_{\mathfrak{j}+2\mathfrak{m}-3})\eta ({\vartheta }_{\mathfrak{j}+2\mathfrak{m}-4},{\vartheta }_{\mathfrak{j}+2\mathfrak{m}})\cdots \eta ({\vartheta }_{\mathfrak{j}+2},{\vartheta }_{\mathfrak{j}+2\mathfrak{m}}))}\right)\hfill \\ \hfill & \star \mathcal{W}\left({\vartheta }_0,{\vartheta }_1,\frac{\varsigma }{3^{\mathfrak{m}-1}{\rho }^{\mathfrak{j}+2\mathfrak{m}-4}(\mathrm{\Gamma }({\vartheta }_{\mathfrak{j}+2\mathfrak{m}-3},{\vartheta }_{\mathfrak{j}+2\mathfrak{m}-2})\eta ({\vartheta }_{\mathfrak{j}+2\mathfrak{m}-4},{\vartheta }_{\mathfrak{j}+2\mathfrak{m}})\cdots \eta ({\vartheta }_{\mathfrak{j}+2},{\vartheta }_{\mathfrak{j}+2\mathfrak{m}}))}\right)\hfill \\ \hfill & \star \mathcal{W}\left({\vartheta }_0,{\vartheta }_1,\frac{\varsigma }{3^{\mathfrak{m}-1}{\rho }^{\mathfrak{j}+2\mathfrak{m}-3}(\eta ({\vartheta }_{\mathfrak{j}+2\mathfrak{m}-2},{\vartheta }_{\mathfrak{j}+2\mathfrak{m}})\eta ({\vartheta }_{\mathfrak{j}+2\mathfrak{m}-4},{\vartheta }_{\mathfrak{j}+2\mathfrak{m}})\cdots \eta ({\vartheta }_{\mathfrak{j}+2},{\vartheta }_{\mathfrak{j}+2\mathfrak{m}}))}\right),\hfill \end{array}$$

$$\begin{array}{cc}\hfill \mathcal{E}({\vartheta }_{\mathfrak{j}},{\vartheta }_{\mathfrak{j}+2\mathfrak{m}},\varsigma )& \le \mathcal{E}\left({\vartheta }_0,{\vartheta }_1,\frac{\varsigma }{3{\rho }^{\mathfrak{j}-1}\left(\beta ({\vartheta }_{\mathfrak{j}},{\vartheta }_{\mathfrak{j}+1})\right)}\right)\diamond \mathcal{E}\left({\vartheta }_{\mathfrak{j}+1},{\vartheta }_{\mathfrak{j}+2},\frac{\varsigma }{3{\rho }^{\mathfrak{j}}\left(\mathrm{\Gamma }({\vartheta }_{\mathfrak{j}+1},{\vartheta }_{\mathfrak{j}+2})\right)}\right)\hfill \\ \hfill & \diamond \mathcal{E}\left({\vartheta }_0,{\vartheta }_1,\frac{\varsigma }{3^2{\rho }^{\mathfrak{j}+1}\left(\beta ({\vartheta }_{\mathfrak{j}+2},{\vartheta }_{\mathfrak{j}+3})\eta ({\vartheta }_{\mathfrak{j}+2},{\vartheta }_{\mathfrak{j}+2\mathfrak{m}})\right)}\right)\hfill \\ \hfill & \diamond \mathcal{E}\left({\vartheta }_0,{\vartheta }_1,\frac{\varsigma }{3^2{\rho }^{\mathfrak{j}+2}\left(\mathrm{\Gamma }({\vartheta }_{\mathfrak{j}+3},{\vartheta }_{\mathfrak{j}+4})\eta ({\vartheta }_{\mathfrak{j}+2},{\vartheta }_{\mathfrak{j}+2\mathfrak{m}})\right)}\right)\hfill \end{array}$$

$$\begin{array}{cc}\hfill & \diamond \mathcal{E}\left({\vartheta }_0,{\vartheta }_1,\frac{\varsigma }{3^3{\rho }^{\mathfrak{j}+3}\left(\beta ({\vartheta }_{\mathfrak{j}+4},{\vartheta }_{\mathfrak{j}+5})\eta ({\vartheta }_{\mathfrak{j}+4},{\vartheta }_{\mathfrak{j}+2\mathfrak{m}})\eta ({\vartheta }_{\mathfrak{j}+2},{\vartheta }_{\mathfrak{j}+2\mathfrak{m}})\right)}\right)\hfill \\ \hfill & \diamond \mathcal{E}\left({\vartheta }_0,{\vartheta }_1,\frac{\varsigma }{3^3{\rho }^{\mathfrak{j}+4}\left(\mathrm{\Gamma }({\vartheta }_{\mathfrak{j}+5},{\vartheta }_{\mathfrak{j}+6})\eta ({\vartheta }_{\mathfrak{j}+4},{\vartheta }_{\mathfrak{j}+2\mathfrak{m}})\eta ({\vartheta }_{\mathfrak{j}+2},{\vartheta }_{\mathfrak{j}+2\mathfrak{m}})\right)}\right)\hfill \\ \hfill & \diamond \mathcal{E}\left({\vartheta }_0,{\vartheta }_1,\frac{\varsigma }{3^3{\rho }^{\mathfrak{j}+5}\left(\eta ({\vartheta }_{\mathfrak{j}+6},{\vartheta }_{\mathfrak{j}+2\mathfrak{m}})\eta ({\vartheta }_{\mathfrak{j}+4},{\vartheta }_{\mathfrak{j}+2\mathfrak{m}})\eta ({\vartheta }_{\mathfrak{j}+2},{\vartheta }_{\mathfrak{j}+2\mathfrak{m}})\right)}\right)\diamond \cdots \diamond \hfill \\ \hfill & \mathcal{E}\left({\vartheta }_0,{\vartheta }_1,\frac{\varsigma }{3^{\mathfrak{m}-1}{\rho }^{\mathfrak{j}+2\mathfrak{m}-5}(\beta ({\vartheta }_{\mathfrak{j}+2\mathfrak{m}-4},{\vartheta }_{\mathfrak{j}+2\mathfrak{m}-3})\eta ({\vartheta }_{\mathfrak{j}+2\mathfrak{m}-4},{\vartheta }_{\mathfrak{j}+2\mathfrak{m}})\cdots \eta ({\vartheta }_{\mathfrak{j}+2},{\vartheta }_{\mathfrak{j}+2\mathfrak{m}}))}\right)\hfill \\ \hfill & \diamond \mathcal{E}\left({\vartheta }_0,{\vartheta }_1,\frac{\varsigma }{3^{\mathfrak{m}-1}{\rho }^{\mathfrak{j}+2\mathfrak{m}-4}(\mathrm{\Gamma }({\vartheta }_{\mathfrak{j}+2\mathfrak{m}-3},{\vartheta }_{\mathfrak{j}+2\mathfrak{m}-2})\eta ({\vartheta }_{\mathfrak{j}+2\mathfrak{m}-4},{\vartheta }_{\mathfrak{j}+2\mathfrak{m}})\cdots \eta ({\vartheta }_{\mathfrak{j}+2},{\vartheta }_{\mathfrak{j}+2\mathfrak{m}}))}\right)\hfill \\ \hfill & \diamond \mathcal{E}\left({\vartheta }_0,{\vartheta }_1,\frac{\varsigma }{3^{\mathfrak{m}-1}{\rho }^{\mathfrak{j}+2\mathfrak{m}-3}(\eta ({\vartheta }_{\mathfrak{j}+2\mathfrak{m}-2},{\vartheta }_{\mathfrak{j}+2\mathfrak{m}})\eta ({\vartheta }_{\mathfrak{j}+2\mathfrak{m}-4},{\vartheta }_{\mathfrak{j}+2\mathfrak{m}})\cdots \eta ({\vartheta }_{\mathfrak{j}+2},{\vartheta }_{\mathfrak{j}+2\mathfrak{m}}))}\right),\hfill \end{array}$$

and,

$$\begin{array}{cc}\hfill \mathcal{G}& ({\vartheta }_{\mathfrak{j}},{\vartheta }_{\mathfrak{j}+2\mathfrak{m}},\varsigma )\le \mathcal{G}\left({\vartheta }_0,{\vartheta }_1,\frac{\varsigma }{3{\rho }^{\mathfrak{j}-1}\left(\beta ({\vartheta }_{\mathfrak{j}},{\vartheta }_{\mathfrak{j}+1})\right)}\right)\diamond \mathcal{G}\left({\vartheta }_{\mathfrak{j}+1},{\vartheta }_{\mathfrak{j}+2},\frac{\varsigma }{3{\rho }^{\mathfrak{j}}\left(\mathrm{\Gamma }({\vartheta }_{\mathfrak{j}+1},{\vartheta }_{\mathfrak{j}+2})\right)}\right)\hfill \\ \hfill & \diamond \mathcal{G}\left({\vartheta }_0,{\vartheta }_1,\frac{\varsigma }{3^2{\rho }^{\mathfrak{j}+1}\left(\beta ({\vartheta }_{\mathfrak{j}+2},{\vartheta }_{\mathfrak{j}+3})\eta ({\vartheta }_{\mathfrak{j}+2},{\vartheta }_{\mathfrak{j}+2\mathfrak{m}})\right)}\right)\hfill \\ \hfill & \diamond \mathcal{G}\left({\vartheta }_0,{\vartheta }_1,\frac{\varsigma }{3^2{\rho }^{\mathfrak{j}+2}\left(\mathrm{\Gamma }({\vartheta }_{\mathfrak{j}+3},{\vartheta }_{\mathfrak{j}+4})\eta ({\vartheta }_{\mathfrak{j}+2},{\vartheta }_{\mathfrak{j}+2\mathfrak{m}})\right)}\right)\hfill \\ \hfill & \diamond \mathcal{G}\left({\vartheta }_0,{\vartheta }_1,\frac{\varsigma }{3^3{\rho }^{\mathfrak{j}+3}\left(\beta ({\vartheta }_{\mathfrak{j}+4},{\vartheta }_{\mathfrak{j}+5})\eta ({\vartheta }_{\mathfrak{j}+4},{\vartheta }_{\mathfrak{j}+2\mathfrak{m}})\eta ({\vartheta }_{\mathfrak{j}+2},{\vartheta }_{\mathfrak{j}+2\mathfrak{m}})\right)}\right)\hfill \\ \hfill & \diamond \mathcal{G}\left({\vartheta }_0,{\vartheta }_1,\frac{\varsigma }{3^3{\rho }^{\mathfrak{j}+4}\left(\mathrm{\Gamma }({\vartheta }_{\mathfrak{j}+5},{\vartheta }_{\mathfrak{j}+6})\eta ({\vartheta }_{\mathfrak{j}+4},{\vartheta }_{\mathfrak{j}+2\mathfrak{m}})\eta ({\vartheta }_{\mathfrak{j}+2},{\vartheta }_{\mathfrak{j}+2\mathfrak{m}})\right)}\right)\hfill \\ \hfill & \diamond \mathcal{G}\left({\vartheta }_0,{\vartheta }_1,\frac{\varsigma }{3^3{\rho }^{\mathfrak{j}+5}\left(\eta ({\vartheta }_{\mathfrak{j}+6},{\vartheta }_{\mathfrak{j}+2\mathfrak{m}})\eta ({\vartheta }_{\mathfrak{j}+4},{\vartheta }_{\mathfrak{j}+2\mathfrak{m}})\eta ({\vartheta }_{\mathfrak{j}+2},{\vartheta }_{\mathfrak{j}+2\mathfrak{m}})\right)}\right)\diamond \cdots \diamond \hfill \\ \hfill & \mathcal{G}\left({\vartheta }_0,{\vartheta }_1,\frac{\varsigma }{3^{\mathfrak{m}-1}{\rho }^{\mathfrak{j}+2\mathfrak{m}-5}(\beta ({\vartheta }_{\mathfrak{j}+2\mathfrak{m}-4},{\vartheta }_{\mathfrak{j}+2\mathfrak{m}-3})\eta ({\vartheta }_{\mathfrak{j}+2\mathfrak{m}-4},{\vartheta }_{\mathfrak{j}+2\mathfrak{m}})\cdots \eta ({\vartheta }_{\mathfrak{j}+2},{\vartheta }_{\mathfrak{j}+2\mathfrak{m}}))}\right)\hfill \\ \hfill & \diamond \mathcal{G}\left({\vartheta }_0,{\vartheta }_1,\frac{\varsigma }{3^{\mathfrak{m}-1}{\rho }^{\mathfrak{j}+2\mathfrak{m}-4}(\mathrm{\Gamma }({\vartheta }_{\mathfrak{j}+2\mathfrak{m}-3},{\vartheta }_{\mathfrak{j}+2\mathfrak{m}-2})\eta ({\vartheta }_{\mathfrak{j}+2\mathfrak{m}-4},{\vartheta }_{\mathfrak{j}+2\mathfrak{m}})\cdots \eta ({\vartheta }_{\mathfrak{j}+2},{\vartheta }_{\mathfrak{j}+2\mathfrak{m}}))}\right)\hfill \\ \hfill & \diamond \mathcal{G}\left({\vartheta }_0,{\vartheta }_1,\frac{\varsigma }{3^{\mathfrak{m}-1}{\rho }^{\mathfrak{j}+2\mathfrak{m}-3}(\eta ({\vartheta }_{\mathfrak{j}+2\mathfrak{m}-2},{\vartheta }_{\mathfrak{j}+2\mathfrak{m}})\eta ({\vartheta }_{\mathfrak{j}+2\mathfrak{m}-4},{\vartheta }_{\mathfrak{j}+2\mathfrak{m}})\cdots \eta ({\vartheta }_{\mathfrak{j}+2},{\vartheta }_{\mathfrak{j}+2\mathfrak{m}}))}\right).\hfill \end{array}$$

As $\mathfrak{j}\to +\infty$, we have

$$\begin{array}{cc}\hfill \underset{\mathfrak{j}\to +\infty }{lim}\mathcal{W}({\vartheta }_{\mathfrak{j}},{\vartheta }_{\mathfrak{j}+\mathfrak{i}},\varsigma )=1\star 1\star \cdots \star 1& =1,\hfill \\ \hfill \underset{\mathfrak{j}\to +\infty }{lim}\mathcal{E}({\vartheta }_{\mathfrak{j}},{\vartheta }_{\mathfrak{j}+\mathfrak{i}},\varsigma )=0\diamond 0\diamond \cdots \diamond 0& =0,\hfill \end{array}$$

and

$$\begin{array}{cc}\hfill \underset{\mathfrak{j}\to +\infty }{lim}\mathcal{G}({\vartheta }_{\mathfrak{j}},{\vartheta }_{\mathfrak{j}+\mathfrak{i}},\varsigma )=0\diamond 0\diamond \cdots \diamond 0& =0.\hfill \end{array}$$

Therefore, $\left\{{\vartheta }_{\mathfrak{j}}\right\}$ is a Cauchy sequence. Since $(\Im ,\mathcal{W},\mathcal{E},\mathcal{G},\star ,\diamond )$ is a complete neutrosophic rectangular triple-controlled metric space, we have

$$\begin{array}{c}\hfill \underset{\mathfrak{j}\to +\infty }{lim}{\vartheta }_{\mathfrak{j}}=\vartheta .\end{array}$$

Now, due to the fact that ϑ is a fixed point of $\mathfrak{p}$, utilizing e, j, and o, we obtain

$$\begin{array}{cc}\hfill \mathcal{W}(\vartheta ,\mathfrak{p}\vartheta ,\varsigma )& \ge \mathcal{W}\left(\vartheta ,{\vartheta }_{\mathfrak{j}},\frac{\varsigma }{3\left(\beta (\vartheta ,{\vartheta }_{\mathfrak{j}})\right)}\right)\star \mathcal{W}\left({\vartheta }_{\mathfrak{j}},{\vartheta }_{\mathfrak{j}+1},\frac{\varsigma }{3\left(\mathrm{\Gamma }({\vartheta }_{\mathfrak{j}},{\vartheta }_{\mathfrak{j}+1})\right)}\right)\hfill \\ & \star \mathcal{W}\left({\vartheta }_{\mathfrak{j}+1},\mathfrak{p}\vartheta ,\frac{\varsigma }{3\left(\eta ({\vartheta }_{\mathfrak{j}+1},\mathfrak{p}\vartheta )\right)}\right)\hfill \\ \hfill & =\mathcal{W}\left(\vartheta ,{\vartheta }_{\mathfrak{j}},\frac{\varsigma }{3\left(\beta (\vartheta ,{\vartheta }_{\mathfrak{j}})\right)}\right)\star \mathcal{W}\left(\mathfrak{p}{\vartheta }_{\mathfrak{j}-1},\mathfrak{p}{\vartheta }_{\mathfrak{j}},\frac{\varsigma }{3\left(\mathrm{\Gamma }({\vartheta }_{\mathfrak{j}},{\vartheta }_{\mathfrak{j}+1})\right)}\right)\hfill \\ & \star \mathcal{W}\left(\mathfrak{p}{\vartheta }_{\mathfrak{j}},\mathfrak{p}\vartheta ,\frac{\varsigma }{3\left(\eta ({\vartheta }_{\mathfrak{j}+1},\mathfrak{p}\vartheta )\right)}\right)\hfill \\ \hfill & \ge \mathcal{W}\left(\vartheta ,{\vartheta }_{\mathfrak{j}},\frac{\varsigma }{3\left(\beta (\vartheta ,{\vartheta }_{\mathfrak{j}})\right)}\right)\star \mathcal{W}\left({\vartheta }_{\mathfrak{j}-1},{\vartheta }_{\mathfrak{j}},\frac{\varsigma }{3\left(\mathrm{\Gamma }({\vartheta }_{\mathfrak{j}},{\vartheta }_{\mathfrak{j}+1})\right)}\right)\hfill \\ & \star \mathcal{W}\left({\vartheta }_{\mathfrak{j}},\vartheta ,\frac{\varsigma }{3\left(\eta ({\vartheta }_{\mathfrak{j}+1},\mathfrak{p}\vartheta )\right)}\right)\hfill \\ \hfill & \to 1\star 1\star 1=1\mathrm{as}\mathfrak{j}\to +\infty ,\hfill \end{array}$$

$$\begin{array}{cc}\hfill \mathcal{E}(\vartheta ,\mathfrak{p}\vartheta ,& \varsigma )\le \mathcal{E}\left(\vartheta ,{\vartheta }_{\mathfrak{j}},\frac{\varsigma }{3\left(\beta (\vartheta ,{\vartheta }_{\mathfrak{j}})\right)}\right)\diamond \mathcal{E}\left({\vartheta }_{\mathfrak{j}},{\vartheta }_{\mathfrak{j}+1},\frac{\varsigma }{3\left(\mathrm{\Gamma }({\vartheta }_{\mathfrak{j}},{\vartheta }_{\mathfrak{j}+1})\right)}\right)\diamond \mathcal{E}\left({\vartheta }_{\mathfrak{j}+1},\mathfrak{p}\vartheta ,\frac{\varsigma }{3\left(\eta ({\vartheta }_{\mathfrak{j}+1},\mathfrak{p}\vartheta )\right)}\right)\hfill \\ \hfill & =\mathcal{E}\left(\vartheta ,{\vartheta }_{\mathfrak{j}},\frac{\varsigma }{3\left(\beta (\vartheta ,{\vartheta }_{\mathfrak{j}})\right)}\right)\diamond \mathcal{E}\left(\mathfrak{p}{\vartheta }_{\mathfrak{j}-1},\mathfrak{p}{\vartheta }_{\mathfrak{j}},\frac{\varsigma }{3\left(\mathrm{\Gamma }({\vartheta }_{\mathfrak{j}},{\vartheta }_{\mathfrak{j}+1})\right)}\right)\diamond \mathcal{E}\left(\mathfrak{p}{\vartheta }_{\mathfrak{j}},\mathfrak{p}\vartheta ,\frac{\varsigma }{3\left(\eta ({\vartheta }_{\mathfrak{j}+1},\mathfrak{p}\vartheta )\right)}\right)\hfill \\ \hfill & \le \mathcal{E}\left(\vartheta ,{\vartheta }_{\mathfrak{j}},\frac{\varsigma }{3\left(\beta (\vartheta ,{\vartheta }_{\mathfrak{j}})\right)}\right)\diamond \mathcal{E}\left({\vartheta }_{\mathfrak{j}-1},{\vartheta }_{\mathfrak{j}},\frac{\varsigma }{3\left(\mathrm{\Gamma }({\vartheta }_{\mathfrak{j}},{\vartheta }_{\mathfrak{j}+1})\right)}\right)\diamond \mathcal{E}\left({\vartheta }_{\mathfrak{j}},\vartheta ,\frac{\varsigma }{3\left(\eta ({\vartheta }_{\mathfrak{j}+1},\mathfrak{p}\vartheta )\right)}\right)\hfill \\ \hfill & \to 0\diamond 0\diamond 0=0\mathrm{as}\mathfrak{j}\to +\infty ,\hfill \end{array}$$

and

$$\begin{array}{cc}\hfill \mathcal{G}(\vartheta ,\mathfrak{p}\vartheta ,& \varsigma )\le \mathcal{G}\left(\vartheta ,{\vartheta }_{\mathfrak{j}},\frac{\varsigma }{3\left(\beta (\vartheta ,{\vartheta }_{\mathfrak{j}})\right)}\right)\diamond \mathcal{G}\left({\vartheta }_{\mathfrak{j}},{\vartheta }_{\mathfrak{j}+1},\frac{\varsigma }{3\left(\mathrm{\Gamma }({\vartheta }_{\mathfrak{j}},{\vartheta }_{\mathfrak{j}+1})\right)}\right)\diamond \mathcal{G}\left({\vartheta }_{\mathfrak{j}+1},\mathfrak{p}\vartheta ,\frac{\varsigma }{3\left(\eta ({\vartheta }_{\mathfrak{j}+1},\mathfrak{p}\vartheta )\right)}\right)\hfill \\ \hfill & =\mathcal{G}\left(\vartheta ,{\vartheta }_{\mathfrak{j}},\frac{\varsigma }{3\left(\beta (\vartheta ,{\vartheta }_{\mathfrak{j}})\right)}\right)\diamond \mathcal{G}\left(\mathfrak{p}{\vartheta }_{\mathfrak{j}-1},\mathfrak{p}{\vartheta }_{\mathfrak{j}},\frac{\varsigma }{3\left(\mathrm{\Gamma }({\vartheta }_{\mathfrak{j}},{\vartheta }_{\mathfrak{j}+1})\right)}\right)\diamond \mathcal{G}\left(\mathfrak{p}{\vartheta }_{\mathfrak{j}},\mathfrak{p}\vartheta ,\frac{\varsigma }{3\left(\eta ({\vartheta }_{\mathfrak{j}+1},\mathfrak{p}\vartheta )\right)}\right)\hfill \\ \hfill & \le \mathcal{G}\left(\vartheta ,{\vartheta }_{\mathfrak{j}},\frac{\varsigma }{3\left(\beta (\vartheta ,{\vartheta }_{\mathfrak{j}})\right)}\right)\diamond \mathcal{G}\left({\vartheta }_{\mathfrak{j}-1},{\vartheta }_{\mathfrak{j}},\frac{\varsigma }{3\left(\mathrm{\Gamma }({\vartheta }_{\mathfrak{j}},{\vartheta }_{\mathfrak{j}+1})\right)}\right)\diamond \mathcal{G}\left({\vartheta }_{\mathfrak{j}},\vartheta ,\frac{\varsigma }{3\left(\eta ({\vartheta }_{\mathfrak{j}+1},\mathfrak{p}\vartheta )\right)}\right)\hfill \\ \hfill & \to 0\diamond 0\diamond 0=0\mathrm{as}\mathfrak{j}\to +\infty .\hfill \end{array}$$

Hence, $\mathfrak{p}\vartheta =\vartheta$.

Now, we examine the uniqueness. Let $\mathfrak{p}\kappa =\kappa$ for some $\kappa \in \Im$, then

$$\begin{array}{cc}\hfill 1& \ge \mathcal{W}(\kappa ,\vartheta ,\varsigma )=\mathcal{W}(\mathfrak{p}\kappa ,\mathfrak{p}\vartheta ,\varsigma )\ge \mathcal{W}\left(\kappa ,\vartheta ,\frac{\varsigma }{\rho }\right)=\mathcal{W}\left(\mathfrak{p}\kappa ,\mathfrak{p}\vartheta ,\frac{\varsigma }{\rho }\right)\hfill \\ \hfill & \ge \mathcal{W}\left(\kappa ,\vartheta ,\frac{\varsigma }{{\rho }^2}\right)\ge \cdots \ge \mathcal{W}\left(\kappa ,\vartheta ,\frac{\varsigma }{{\rho }^{\mathfrak{j}}}\right)\to 1\mathrm{as}\mathfrak{j}\to +\infty ,\hfill \\ \hfill 0& \le \mathcal{E}(\kappa ,\vartheta ,\varsigma )=\mathcal{E}(\mathfrak{p}\kappa ,\mathfrak{p}\vartheta ,\varsigma )\le \mathcal{E}\left(\kappa ,\vartheta ,\frac{\varsigma }{\rho }\right)=\mathcal{E}\left(\mathfrak{p}\kappa ,\mathfrak{p}\vartheta ,\frac{\varsigma }{\rho }\right)\hfill \\ \hfill & \le \mathcal{E}\left(\kappa ,\vartheta ,\frac{\varsigma }{{\rho }^2}\right)\le \cdots \le \mathcal{E}\left(\kappa ,\vartheta ,\frac{\varsigma }{{\rho }^{\mathfrak{j}}}\right)\to 0\mathrm{as}\mathfrak{j}\to +\infty ,\hfill \end{array}$$

and

$$\begin{array}{cc}\hfill 0& \le \mathcal{G}(\kappa ,\vartheta ,\varsigma )=\mathcal{G}(\mathfrak{p}\kappa ,\mathfrak{p}\vartheta ,\varsigma )\le \mathcal{G}\left(\kappa ,\vartheta ,\frac{\varsigma }{\rho }\right)=\mathcal{G}\left(\mathfrak{p}\kappa ,\mathfrak{p}\vartheta ,\frac{\varsigma }{\rho }\right)\hfill \\ \hfill & \le \mathcal{G}\left(\kappa ,\vartheta ,\frac{\varsigma }{{\rho }^2}\right)\le \cdots \le \mathcal{G}\left(\kappa ,\vartheta ,\frac{\varsigma }{{\rho }^{\mathfrak{j}}}\right)\to 0\mathrm{as}\mathfrak{j}\to +\infty ,\hfill \end{array}$$

by using (iii), (viii) and (xiii), $\vartheta =\kappa$. □

Definition 10.

Let $(\Im ,\mathcal{W},\mathcal{E},\mathcal{G},\star ,\diamond )$ be a neutrosophic rectangular triple-controlled metric space. A map $\mathfrak{p}:\Im \to \Im$ is an NRT(neutrosophic rectangular triple)-controlled contraction if there exists $0<\rho <1$, such that

$$\begin{array}{cc}\hfill \frac{1}{\mathcal{W}(\mathfrak{p}\vartheta ,\mathfrak{p}\mathcal{Q},\varsigma )}-1& \le \rho \left[\frac{1}{\mathcal{W}(\vartheta ,\mathcal{Q},\varsigma )}-1\right]\hfill \end{array}$$

(4)

$$\begin{array}{cc}\hfill \mathcal{E}(\mathfrak{p}\vartheta ,\mathfrak{p}\mathcal{Q},\varsigma )& \le \rho \mathcal{E}(\vartheta ,\mathcal{Q},\varsigma ),\hfill \end{array}$$

(5)

and

$$\begin{array}{c}\hfill \mathcal{G}(\mathfrak{p}\vartheta ,\mathfrak{p}\mathcal{Q},\varsigma )\le \rho \mathcal{G}(\vartheta ,\mathcal{Q},\varsigma ),\end{array}$$

(6)

for all $\vartheta ,\mathcal{Q}\in \Im$ and $\varsigma >0$.

We now present the fixed point result for an NRT (neutrosophic rectangular triple)-controlled contraction.

Theorem 2.

Let $(\Im ,\mathcal{W},\mathcal{E},\mathcal{G},\star ,\diamond )$ be a complete neutrosophic rectangular triple-controlled metric space with $\beta ,\mathrm{\Gamma },\eta :\Im \times \Im \to [1,+\infty )$. Let $\mathfrak{p}:\Im \to \Im$ be an ND-controlled contraction. Further, suppose that for an arbitrary ${\vartheta }_0\in \Im$, and $\mathfrak{j},\mathfrak{q}\in \mathbb{N}$, where ${\vartheta }_{\mathfrak{j}}={\mathfrak{p}}^{\mathfrak{j}}{\vartheta }_0=\mathfrak{p}{\vartheta }_{\mathfrak{j}-1}$. Then, $\mathfrak{p}$ has a unique fixed point.

Proof. 

Let ${\vartheta }_0$∈ℑ. Define ${\vartheta }_{\mathfrak{j}}$ by ${\vartheta }_{\mathfrak{j}}={\mathfrak{p}}^{\mathfrak{j}}{\vartheta }_0=\mathfrak{p}{\vartheta }_{\mathfrak{j}-1},\mathfrak{j}\in \mathbb{N}$. From (4)–(6), for all $\varsigma >0,\mathfrak{j}>\mathfrak{q}$, we deduce

$$\begin{array}{cc}\hfill \frac{1}{\mathcal{W}({\vartheta }_{\mathfrak{j}},{\vartheta }_{\mathfrak{j}+1},\varsigma )}-1& =\frac{1}{\mathcal{W}(\mathfrak{p}{\vartheta }_{\mathfrak{j}-1},\mathfrak{p}{\vartheta }_{\mathfrak{j}},\varsigma )}-1\le \rho \left[\frac{1}{\mathcal{W}({\vartheta }_{\mathfrak{j}-1},{\vartheta }_{\mathfrak{j}},\varsigma )}\right]=\frac{\rho }{\mathcal{W}({\vartheta }_{\mathfrak{j}-1},{\vartheta }_{\mathfrak{j}},\varsigma )}-\rho \hfill \\ \hfill \Rightarrow \frac{1}{\mathcal{W}({\vartheta }_{\mathfrak{j}},{\vartheta }_{\mathfrak{j}+1},\varsigma )}& \le \frac{\rho }{\mathcal{W}({\vartheta }_{\mathfrak{j}-1},{\vartheta }_{\mathfrak{j}},\varsigma )}+(1-\rho )\le \frac{{\rho }^2}{\mathcal{W}({\vartheta }_{\mathfrak{j}-2},{\vartheta }_{\mathfrak{j}-1},\varsigma )}+\rho (1-\rho )+(1-\rho ).\hfill \end{array}$$

Proceeding in this way, we have

$$\begin{array}{cc}\hfill \frac{1}{\mathcal{W}({\vartheta }_{\mathfrak{j}},{\vartheta }_{\mathfrak{j}+1},\varsigma )}& \le \frac{{\rho }^{\mathfrak{j}}}{\mathcal{W}({\vartheta }_0,{\vartheta }_1,\varsigma )}+{\rho }^{\mathfrak{j}-1}(1-\rho )+{\rho }^{\mathfrak{j}-2}(1-\rho )+\cdots +\rho (1-\rho )+(1-\rho )\hfill \\ \hfill & \le \frac{{\rho }^{\mathfrak{j}}}{\mathcal{W}({\vartheta }_0,{\vartheta }_1,\varsigma )}+({\rho }^{\mathfrak{j}-1}+{\rho }^{\mathfrak{j}-2}+\cdots +1)(1-\rho )\le \frac{{\rho }^{\mathfrak{j}}}{\mathcal{W}({\vartheta }_0,{\vartheta }_1,\varsigma )}+(1-{\rho }^{\mathfrak{j}}).\hfill \end{array}$$

We obtain

$$\begin{array}{c}\hfill \frac{1}{\frac{{\rho }^{\mathfrak{j}}}{\mathcal{W}({\vartheta }_0,{\vartheta }_1,\varsigma )}+(1-{\rho }^{\mathfrak{j}})}\le \mathcal{W}({\vartheta }_{\mathfrak{j}},{\vartheta }_{\mathfrak{j}+1},\varsigma )\end{array}$$

(7)

$$\begin{array}{cc}\hfill \mathcal{E}({\vartheta }_{\mathfrak{j}},{\vartheta }_{\mathfrak{j}+1},\varsigma )& =\mathcal{E}(\mathfrak{p}{\vartheta }_{\mathfrak{j}-1},\mathfrak{p}{\vartheta }_{\mathfrak{j}},\varsigma )\le \rho \mathcal{E}({\vartheta }_{\mathfrak{j}-1},{\vartheta }_{\mathfrak{j}},\varsigma )=\mathcal{E}(\mathfrak{p}{\vartheta }_{\mathfrak{j}-2},\mathfrak{p}{\vartheta }_{\mathfrak{j}-1},\varsigma )\hfill \\ \hfill & \le {\rho }^2\mathcal{E}({\vartheta }_{\mathfrak{j}-2},{\vartheta }_{\mathfrak{j}-1},\varsigma )\le \cdots \le {\rho }^{\mathfrak{j}}\mathcal{E}({\vartheta }_0,{\vartheta }_1,\varsigma ),\hfill \end{array}$$

(8)

and

$$\begin{array}{cc}\hfill \mathcal{G}({\vartheta }_{\mathfrak{j}},{\vartheta }_{\mathfrak{j}+1},\varsigma )& =\mathcal{G}(\mathfrak{p}{\vartheta }_{\mathfrak{j}-1},\mathfrak{p}{\vartheta }_{\mathfrak{j}},\varsigma )\le \rho \mathcal{G}({\vartheta }_{\mathfrak{j}-1},{\vartheta }_{\mathfrak{j}},\varsigma )=\mathcal{G}(\mathfrak{p}{\vartheta }_{\mathfrak{j}-2},\mathfrak{p}{\vartheta }_{\mathfrak{j}-1},\varsigma )\hfill \\ \hfill & \le {\rho }^2\mathcal{G}({\vartheta }_{\mathfrak{j}-2},{\vartheta }_{\mathfrak{j}-1},\varsigma )\le \cdots \le {\rho }^{\mathfrak{j}}\mathcal{G}({\vartheta }_0,{\vartheta }_1,\varsigma ).\hfill \end{array}$$

(9)

Using (v), (x) and (xv), we have the following cases:

Case 1. When $\mathfrak{i}=2\mathfrak{m}+1$, i.e., $\mathfrak{i}$ is odd, then

$$\begin{array}{cc}\hfill \mathcal{W}({\vartheta }_{\mathfrak{j}},{\vartheta }_{\mathfrak{j}+2\mathfrak{m}+1},\varsigma )& \ge \mathcal{W}\left({\vartheta }_{\mathfrak{j}},{\vartheta }_{\mathfrak{j}+1},\frac{\varsigma }{3\left(\beta ({\vartheta }_{\mathfrak{j}},{\vartheta }_{\mathfrak{j}+1})\right)}\right)\star \mathcal{W}\left({\vartheta }_{\mathfrak{j}+1},{\vartheta }_{\mathfrak{j}+2},\frac{\varsigma }{3\left(\mathrm{\Gamma }({\vartheta }_{\mathfrak{j}+1},{\vartheta }_{\mathfrak{j}+2})\right)}\right)\hfill \\ \hfill & \star \mathcal{W}\left({\vartheta }_{\mathfrak{j}+2},{\vartheta }_{\mathfrak{j}+2\mathfrak{m}+1},\frac{\varsigma }{3\left(\eta ({\vartheta }_{\mathfrak{j}+2},{\vartheta }_{\mathfrak{j}+2\mathfrak{m}+1})\right)}\right)\hfill \\ \hfill & \ge \mathcal{W}\left({\vartheta }_{\mathfrak{j}},{\vartheta }_{\mathfrak{j}+1},\frac{\varsigma }{3\left(\beta ({\vartheta }_{\mathfrak{j}},{\vartheta }_{\mathfrak{j}+1})\right)}\right)\star \mathcal{W}\left({\vartheta }_{\mathfrak{j}+1},{\vartheta }_{\mathfrak{j}+2},\frac{\varsigma }{3\left(\mathrm{\Gamma }({\vartheta }_{\mathfrak{j}+1},{\vartheta }_{\mathfrak{j}+2})\right)}\right)\hfill \\ \hfill & \star \mathcal{W}\left({\vartheta }_{\mathfrak{j}+2},{\vartheta }_{\mathfrak{j}+3},\frac{\varsigma }{3^2\left(\beta ({\vartheta }_{\mathfrak{j}+2},{\vartheta }_{\mathfrak{j}+3})\eta ({\vartheta }_{\mathfrak{j}+2},{\vartheta }_{\mathfrak{j}+2\mathfrak{m}+1})\right)}\right)\hfill \\ \hfill & \star \mathcal{W}\left({\vartheta }_{\mathfrak{j}+3},{\vartheta }_{\mathfrak{j}+4},\frac{\varsigma }{3^2\left(\mathrm{\Gamma }({\vartheta }_{\mathfrak{j}+3},{\vartheta }_{\mathfrak{j}+4})\eta ({\vartheta }_{\mathfrak{j}+2},{\vartheta }_{\mathfrak{j}+2\mathfrak{m}+1})\right)}\right)\hfill \\ \hfill & \star \mathcal{W}\left({\vartheta }_{\mathfrak{j}+4},{\vartheta }_{\mathfrak{j}+2\mathfrak{m}+1},\frac{\varsigma }{3^2\left(\eta ({\vartheta }_{\mathfrak{j}+4},{\vartheta }_{\mathfrak{j}+2\mathfrak{m}+1})\eta ({\vartheta }_{\mathfrak{j}+2},{\vartheta }_{\mathfrak{j}+2\mathfrak{m}+1})\right)}\right)\hfill \\ \hfill & \ge \mathcal{W}\left({\vartheta }_{\mathfrak{j}},{\vartheta }_{\mathfrak{j}+1},\frac{\varsigma }{3\left(\beta ({\vartheta }_{\mathfrak{j}},{\vartheta }_{\mathfrak{j}+1})\right)}\right)\star \mathcal{W}\left({\vartheta }_{\mathfrak{j}+1},{\vartheta }_{\mathfrak{j}+2},\frac{\varsigma }{3\left(\mathrm{\Gamma }({\vartheta }_{\mathfrak{j}+1},{\vartheta }_{\mathfrak{j}+2})\right)}\right)\hfill \\ \hfill & \star \mathcal{W}\left({\vartheta }_{\mathfrak{j}+2},{\vartheta }_{\mathfrak{j}+3},\frac{\varsigma }{3^2\left(\beta ({\vartheta }_{\mathfrak{j}+2},{\vartheta }_{\mathfrak{j}+3})\eta ({\vartheta }_{\mathfrak{j}+2},{\vartheta }_{\mathfrak{j}+2\mathfrak{m}+1})\right)}\right)\hfill \\ \hfill & \star \mathcal{W}\left({\vartheta }_{\mathfrak{j}+3},{\vartheta }_{\mathfrak{j}+4},\frac{\varsigma }{3^2\left(\mathrm{\Gamma }({\vartheta }_{\mathfrak{j}+3},{\vartheta }_{\mathfrak{j}+4})\eta ({\vartheta }_{\mathfrak{j}+2},{\vartheta }_{\mathfrak{j}+2\mathfrak{m}+1})\right)}\right)\hfill \end{array}$$

$$\begin{array}{cc}\hfill & \star \mathcal{W}\left({\vartheta }_{\mathfrak{j}+4},{\vartheta }_{\mathfrak{j}+5},\frac{\varsigma }{3^3\left(\beta ({\vartheta }_{\mathfrak{j}+4},{\vartheta }_{\mathfrak{j}+5})\eta ({\vartheta }_{\mathfrak{j}+4},{\vartheta }_{\mathfrak{j}+2\mathfrak{m}+1})\eta ({\vartheta }_{\mathfrak{j}+2},{\vartheta }_{\mathfrak{j}+2\mathfrak{m}+1})\right)}\right)\hfill \\ \hfill & \star \mathcal{W}\left({\vartheta }_{\mathfrak{j}+5},{\vartheta }_{\mathfrak{j}+6},\frac{\varsigma }{3^3\left(\mathrm{\Gamma }({\vartheta }_{\mathfrak{j}+5},{\vartheta }_{\mathfrak{j}+6})\eta ({\vartheta }_{\mathfrak{j}+4},{\vartheta }_{\mathfrak{j}+2\mathfrak{m}+1})\eta ({\vartheta }_{\mathfrak{j}+2},{\vartheta }_{\mathfrak{j}+2\mathfrak{m}+1})\right)}\right)\hfill \\ \hfill & \star \mathcal{W}\left({\vartheta }_{\mathfrak{j}+6},{\vartheta }_{\mathfrak{j}+2\mathfrak{m}+1},\frac{\varsigma }{3^3\left(\eta ({\vartheta }_{\mathfrak{j}+6},{\vartheta }_{\mathfrak{j}+2\mathfrak{m}+1})\eta ({\vartheta }_{\mathfrak{j}+4},{\vartheta }_{\mathfrak{j}+2\mathfrak{m}+1})\eta ({\vartheta }_{\mathfrak{j}+2},{\vartheta }_{\mathfrak{j}+2\mathfrak{m}+1})\right)}\right)\hfill \end{array}$$

$$\begin{array}{cc}\hfill \mathcal{W}& ({\vartheta }_{\mathfrak{j}},{\vartheta }_{\mathfrak{j}+2\mathfrak{m}+1},\varsigma )\hfill \\ & \ge \mathcal{W}\left({\vartheta }_{\mathfrak{j}},{\vartheta }_{\mathfrak{j}+1},\frac{\varsigma }{3\left(\beta ({\vartheta }_{\mathfrak{j}},{\vartheta }_{\mathfrak{j}+1})\right)}\right)\star \mathcal{W}\left({\vartheta }_{\mathfrak{j}+1},{\vartheta }_{\mathfrak{j}+2},\frac{\varsigma }{3\left(\mathrm{\Gamma }({\vartheta }_{\mathfrak{j}+1},{\vartheta }_{\mathfrak{j}+2})\right)}\right)\hfill \\ \hfill & \star \mathcal{W}\left({\vartheta }_{\mathfrak{j}+2},{\vartheta }_{\mathfrak{j}+3},\frac{\varsigma }{3^2\left(\beta ({\vartheta }_{\mathfrak{j}+2},{\vartheta }_{\mathfrak{j}+3})\eta ({\vartheta }_{\mathfrak{j}+2},{\vartheta }_{\mathfrak{j}+2\mathfrak{m}+1})\right)}\right)\hfill \\ \hfill & \star \mathcal{W}\left({\vartheta }_{\mathfrak{j}+3},{\vartheta }_{\mathfrak{j}+4},\frac{\varsigma }{3^2\left(\mathrm{\Gamma }({\vartheta }_{\mathfrak{j}+3},{\vartheta }_{\mathfrak{j}+4})\eta ({\vartheta }_{\mathfrak{j}+2},{\vartheta }_{\mathfrak{j}+2\mathfrak{m}+1})\right)}\right)\hfill \\ \hfill & \star \mathcal{W}\left({\vartheta }_{\mathfrak{j}+4},{\vartheta }_{\mathfrak{j}+5},\frac{\varsigma }{3^3\left(\beta ({\vartheta }_{\mathfrak{j}+4},{\vartheta }_{\mathfrak{j}+5})\eta ({\vartheta }_{\mathfrak{j}+4},{\vartheta }_{\mathfrak{j}+2\mathfrak{m}+1})\eta ({\vartheta }_{\mathfrak{j}+2},{\vartheta }_{\mathfrak{j}+2\mathfrak{m}+1})\right)}\right)\hfill \\ \hfill & \star \mathcal{W}\left({\vartheta }_{\mathfrak{j}+5},{\vartheta }_{\mathfrak{j}+6},\frac{\varsigma }{3^3\left(\mathrm{\Gamma }({\vartheta }_{\mathfrak{j}+5},{\vartheta }_{\mathfrak{j}+6})\eta ({\vartheta }_{\mathfrak{j}+4},{\vartheta }_{\mathfrak{j}+2\mathfrak{m}+1})\eta ({\vartheta }_{\mathfrak{j}+2},{\vartheta }_{\mathfrak{j}+2\mathfrak{m}+1})\right)}\right)\hfill \\ \hfill & \star \mathcal{W}\left({\vartheta }_{\mathfrak{j}+6},{\vartheta }_{\mathfrak{j}+2\mathfrak{m}+1},\frac{\varsigma }{3^3\left(\eta ({\vartheta }_{\mathfrak{j}+6},{\vartheta }_{\mathfrak{j}+2\mathfrak{m}+1})\eta ({\vartheta }_{\mathfrak{j}+4},{\vartheta }_{\mathfrak{j}+2\mathfrak{m}+1})\eta ({\vartheta }_{\mathfrak{j}+2},{\vartheta }_{\mathfrak{j}+2\mathfrak{m}+1})\right)}\right)\star \cdots \star \hfill \\ \hfill & \mathcal{W}\left({\vartheta }_{\mathfrak{j}+2\mathfrak{m}-2},{\vartheta }_{\mathfrak{j}+2\mathfrak{m}-1},\frac{\varsigma }{3^{\mathfrak{m}}(\beta ({\vartheta }_{\mathfrak{j}+2\mathfrak{m}-2},{\vartheta }_{\mathfrak{j}+2\mathfrak{m}-1})\eta ({\vartheta }_{\mathfrak{j}+2\mathfrak{m}-2},{\vartheta }_{\mathfrak{j}+2\mathfrak{m}+1})\cdots \eta ({\vartheta }_{\mathfrak{j}+2},{\vartheta }_{\mathfrak{j}+2\mathfrak{m}+1}))}\right)\hfill \\ \hfill & \star \mathcal{W}\left({\vartheta }_{\mathfrak{j}+2\mathfrak{m}-1},{\vartheta }_{\mathfrak{j}+2\mathfrak{m}},\frac{\varsigma }{3^{\mathfrak{m}}(\mathrm{\Gamma }({\vartheta }_{\mathfrak{j}+2\mathfrak{m}-1},{\vartheta }_{\mathfrak{j}+2\mathfrak{m}})\eta ({\vartheta }_{\mathfrak{j}+2\mathfrak{m}-2},{\vartheta }_{\mathfrak{j}+2\mathfrak{m}+1})\cdots \eta ({\vartheta }_{\mathfrak{j}+2},{\vartheta }_{\mathfrak{j}+2\mathfrak{m}+1}))}\right)\hfill \\ \hfill & \star \mathcal{W}\left({\vartheta }_{\mathfrak{j}+2\mathfrak{m}},{\vartheta }_{\mathfrak{j}+2\mathfrak{m}+1},\frac{\varsigma }{3^{\mathfrak{m}}(\eta ({\vartheta }_{\mathfrak{j}+2\mathfrak{m}},{\vartheta }_{\mathfrak{j}+2\mathfrak{m}+1})\eta ({\vartheta }_{\mathfrak{j}+2\mathfrak{m}-2},{\vartheta }_{\mathfrak{j}+2\mathfrak{m}+1})\cdots \eta ({\vartheta }_{\mathfrak{j}+2},{\vartheta }_{\mathfrak{j}+2\mathfrak{m}+1}))}\right),\hfill \end{array}$$

$$\begin{array}{cc}\hfill \mathcal{E}({\vartheta }_{\mathfrak{j}},{\vartheta }_{\mathfrak{j}+2\mathfrak{m}+1},\varsigma )& \le \mathcal{E}\left({\vartheta }_{\mathfrak{j}},{\vartheta }_{\mathfrak{j}+1},\frac{\varsigma }{3\left(\beta ({\vartheta }_{\mathfrak{j}},{\vartheta }_{\mathfrak{j}+1})\right)}\right)\diamond \mathcal{E}\left({\vartheta }_{\mathfrak{j}+1},{\vartheta }_{\mathfrak{j}+2},\frac{\varsigma }{3\left(\mathrm{\Gamma }({\vartheta }_{\mathfrak{j}+1},{\vartheta }_{\mathfrak{j}+2})\right)}\right)\hfill \\ \hfill & \diamond \mathcal{E}\left({\vartheta }_{\mathfrak{j}+2},{\vartheta }_{\mathfrak{j}+2\mathfrak{m}+1},\frac{\varsigma }{3\left(\eta ({\vartheta }_{\mathfrak{j}+2},{\vartheta }_{\mathfrak{j}+2\mathfrak{m}+1})\right)}\right)\hfill \\ \hfill & \le \mathcal{E}\left({\vartheta }_{\mathfrak{j}},{\vartheta }_{\mathfrak{j}+1},\frac{\varsigma }{3\left(\beta ({\vartheta }_{\mathfrak{j}},{\vartheta }_{\mathfrak{j}+1})\right)}\right)\diamond \mathcal{E}\left({\vartheta }_{\mathfrak{j}+1},{\vartheta }_{\mathfrak{j}+2},\frac{\varsigma }{3\left(\mathrm{\Gamma }({\vartheta }_{\mathfrak{j}+1},{\vartheta }_{\mathfrak{j}+2})\right)}\right)\hfill \\ \hfill & \diamond \mathcal{E}\left({\vartheta }_{\mathfrak{j}+2},{\vartheta }_{\mathfrak{j}+3},\frac{\varsigma }{3^2\left(\beta ({\vartheta }_{\mathfrak{j}+2},{\vartheta }_{\mathfrak{j}+3})\eta ({\vartheta }_{\mathfrak{j}+2},{\vartheta }_{\mathfrak{j}+2\mathfrak{m}+1})\right)}\right)\hfill \\ \hfill & \diamond \mathcal{E}\left({\vartheta }_{\mathfrak{j}+3},{\vartheta }_{\mathfrak{j}+4},\frac{\varsigma }{3^2\left(\mathrm{\Gamma }({\vartheta }_{\mathfrak{j}+3},{\vartheta }_{\mathfrak{j}+4})\eta ({\vartheta }_{\mathfrak{j}+2},{\vartheta }_{\mathfrak{j}+2\mathfrak{m}+1})\right)}\right)\hfill \\ \hfill & \diamond \mathcal{E}\left({\vartheta }_{\mathfrak{j}+4},{\vartheta }_{\mathfrak{j}+2\mathfrak{m}+1},\frac{\varsigma }{3^2\left(\eta ({\vartheta }_{\mathfrak{j}+4},{\vartheta }_{\mathfrak{j}+2\mathfrak{m}+1})\eta ({\vartheta }_{\mathfrak{j}+2},{\vartheta }_{\mathfrak{j}+2\mathfrak{m}+1})\right)}\right)\hfill \\ \hfill & \le \mathcal{E}\left({\vartheta }_{\mathfrak{j}},{\vartheta }_{\mathfrak{j}+1},\frac{\varsigma }{3\left(\beta ({\vartheta }_{\mathfrak{j}},{\vartheta }_{\mathfrak{j}+1})\right)}\right)\diamond \mathcal{E}\left({\vartheta }_{\mathfrak{j}+1},{\vartheta }_{\mathfrak{j}+2},\frac{\varsigma }{3\left(\mathrm{\Gamma }({\vartheta }_{\mathfrak{j}+1},{\vartheta }_{\mathfrak{j}+2})\right)}\right)\hfill \\ \hfill & \diamond \mathcal{E}\left({\vartheta }_{\mathfrak{j}+2},{\vartheta }_{\mathfrak{j}+3},\frac{\varsigma }{3^2\left(\beta ({\vartheta }_{\mathfrak{j}+2},{\vartheta }_{\mathfrak{j}+3})\eta ({\vartheta }_{\mathfrak{j}+2},{\vartheta }_{\mathfrak{j}+2\mathfrak{m}+1})\right)}\right)\hfill \\ \hfill & \diamond \mathcal{E}\left({\vartheta }_{\mathfrak{j}+3},{\vartheta }_{\mathfrak{j}+4},\frac{\varsigma }{3^2\left(\mathrm{\Gamma }({\vartheta }_{\mathfrak{j}+3},{\vartheta }_{\mathfrak{j}+4})\eta ({\vartheta }_{\mathfrak{j}+2},{\vartheta }_{\mathfrak{j}+2\mathfrak{m}+1})\right)}\right)\hfill \end{array}$$

$$\begin{array}{cc}\hfill & \diamond \mathcal{E}\left({\vartheta }_{\mathfrak{j}+4},{\vartheta }_{\mathfrak{j}+5},\frac{\varsigma }{3^3\left(\beta ({\vartheta }_{\mathfrak{j}+4},{\vartheta }_{\mathfrak{j}+5})\eta ({\vartheta }_{\mathfrak{j}+4},{\vartheta }_{\mathfrak{j}+2\mathfrak{m}+1})\eta ({\vartheta }_{\mathfrak{j}+2},{\vartheta }_{\mathfrak{j}+2\mathfrak{m}+1})\right)}\right)\hfill \\ \hfill & \diamond \mathcal{E}\left({\vartheta }_{\mathfrak{j}+5},{\vartheta }_{\mathfrak{j}+6},\frac{\varsigma }{3^3\left(\mathrm{\Gamma }({\vartheta }_{\mathfrak{j}+5},{\vartheta }_{\mathfrak{j}+6})\eta ({\vartheta }_{\mathfrak{j}+4},{\vartheta }_{\mathfrak{j}+2\mathfrak{m}+1})\eta ({\vartheta }_{\mathfrak{j}+2},{\vartheta }_{\mathfrak{j}+2\mathfrak{m}+1})\right)}\right)\hfill \\ \hfill & \diamond \mathcal{E}\left({\vartheta }_{\mathfrak{j}+6},{\vartheta }_{\mathfrak{j}+2\mathfrak{m}+1},\frac{\varsigma }{3^3\left(\eta ({\vartheta }_{\mathfrak{j}+6},{\vartheta }_{\mathfrak{j}+2\mathfrak{m}+1})\eta ({\vartheta }_{\mathfrak{j}+4},{\vartheta }_{\mathfrak{j}+2\mathfrak{m}+1})\eta ({\vartheta }_{\mathfrak{j}+2},{\vartheta }_{\mathfrak{j}+2\mathfrak{m}+1})\right)}\right)\hfill \end{array}$$

$$\begin{array}{cc}\hfill \mathcal{E}& ({\vartheta }_{\mathfrak{j}},{\vartheta }_{\mathfrak{j}+2\mathfrak{m}+1},\varsigma )\hfill \\ & \le \mathcal{E}\left({\vartheta }_{\mathfrak{j}},{\vartheta }_{\mathfrak{j}+1},\frac{\varsigma }{3\left(\beta ({\vartheta }_{\mathfrak{j}},{\vartheta }_{\mathfrak{j}+1})\right)}\right)\diamond \mathcal{E}\left({\vartheta }_{\mathfrak{j}+1},{\vartheta }_{\mathfrak{j}+2},\frac{\varsigma }{3\left(\mathrm{\Gamma }({\vartheta }_{\mathfrak{j}+1},{\vartheta }_{\mathfrak{j}+2})\right)}\right)\hfill \\ \hfill & \diamond \mathcal{E}\left({\vartheta }_{\mathfrak{j}+2},{\vartheta }_{\mathfrak{j}+3},\frac{\varsigma }{3^2\left(\beta ({\vartheta }_{\mathfrak{j}+2},{\vartheta }_{\mathfrak{j}+3})\eta ({\vartheta }_{\mathfrak{j}+2},{\vartheta }_{\mathfrak{j}+2\mathfrak{m}+1})\right)}\right)\hfill \\ \hfill & \diamond \mathcal{E}\left({\vartheta }_{\mathfrak{j}+3},{\vartheta }_{\mathfrak{j}+4},\frac{\varsigma }{3^2\left(\mathrm{\Gamma }({\vartheta }_{\mathfrak{j}+3},{\vartheta }_{\mathfrak{j}+4})\eta ({\vartheta }_{\mathfrak{j}+2},{\vartheta }_{\mathfrak{j}+2\mathfrak{m}+1})\right)}\right)\hfill \\ \hfill & \diamond \mathcal{E}\left({\vartheta }_{\mathfrak{j}+4},{\vartheta }_{\mathfrak{j}+5},\frac{\varsigma }{3^3\left(\beta ({\vartheta }_{\mathfrak{j}+4},{\vartheta }_{\mathfrak{j}+5})\eta ({\vartheta }_{\mathfrak{j}+4},{\vartheta }_{\mathfrak{j}+2\mathfrak{m}+1})\eta ({\vartheta }_{\mathfrak{j}+2},{\vartheta }_{\mathfrak{j}+2\mathfrak{m}+1})\right)}\right)\hfill \\ \hfill & \diamond \mathcal{E}\left({\vartheta }_{\mathfrak{j}+5},{\vartheta }_{\mathfrak{j}+6},\frac{\varsigma }{3^3\left(\mathrm{\Gamma }({\vartheta }_{\mathfrak{j}+5},{\vartheta }_{\mathfrak{j}+6})\eta ({\vartheta }_{\mathfrak{j}+4},{\vartheta }_{\mathfrak{j}+2\mathfrak{m}+1})\eta ({\vartheta }_{\mathfrak{j}+2},{\vartheta }_{\mathfrak{j}+2\mathfrak{m}+1})\right)}\right)\hfill \\ \hfill & \diamond \mathcal{E}\left({\vartheta }_{\mathfrak{j}+6},{\vartheta }_{\mathfrak{j}+2\mathfrak{m}+1},\frac{\varsigma }{3^3\left(\eta ({\vartheta }_{\mathfrak{j}+6},{\vartheta }_{\mathfrak{j}+2\mathfrak{m}+1})\eta ({\vartheta }_{\mathfrak{j}+4},{\vartheta }_{\mathfrak{j}+2\mathfrak{m}+1})\eta ({\vartheta }_{\mathfrak{j}+2},{\vartheta }_{\mathfrak{j}+2\mathfrak{m}+1})\right)}\right)\diamond \cdots \diamond \hfill \\ \hfill & \mathcal{E}\left({\vartheta }_{\mathfrak{j}+2\mathfrak{m}-2},{\vartheta }_{\mathfrak{j}+2\mathfrak{m}-1},\frac{\varsigma }{3^{\mathfrak{m}}(\beta ({\vartheta }_{\mathfrak{j}+2\mathfrak{m}-2},{\vartheta }_{\mathfrak{j}+2\mathfrak{m}-1})\eta ({\vartheta }_{\mathfrak{j}+2\mathfrak{m}-2},{\vartheta }_{\mathfrak{j}+2\mathfrak{m}+1})\cdots \eta ({\vartheta }_{\mathfrak{j}+2},{\vartheta }_{\mathfrak{j}+2\mathfrak{m}+1}))}\right)\hfill \\ \hfill & \diamond \mathcal{E}\left({\vartheta }_{\mathfrak{j}+2\mathfrak{m}-1},{\vartheta }_{\mathfrak{j}+2\mathfrak{m}},\frac{\varsigma }{3^{\mathfrak{m}}(\mathrm{\Gamma }({\vartheta }_{\mathfrak{j}+2\mathfrak{m}-1},{\vartheta }_{\mathfrak{j}+2\mathfrak{m}})\eta ({\vartheta }_{\mathfrak{j}+2\mathfrak{m}-2},{\vartheta }_{\mathfrak{j}+2\mathfrak{m}+1})\cdots \eta ({\vartheta }_{\mathfrak{j}+2},{\vartheta }_{\mathfrak{j}+2\mathfrak{m}+1}))}\right)\hfill \\ \hfill & \diamond \mathcal{E}\left({\vartheta }_{\mathfrak{j}+2\mathfrak{m}},{\vartheta }_{\mathfrak{j}+2\mathfrak{m}+1},\frac{\varsigma }{3^{\mathfrak{m}}(\eta ({\vartheta }_{\mathfrak{j}+2\mathfrak{m}},{\vartheta }_{\mathfrak{j}+2\mathfrak{m}+1})\eta ({\vartheta }_{\mathfrak{j}+2\mathfrak{m}-2},{\vartheta }_{\mathfrak{j}+2\mathfrak{m}+1})\cdots \eta ({\vartheta }_{\mathfrak{j}+2},{\vartheta }_{\mathfrak{j}+2\mathfrak{m}+1}))}\right),\hfill \end{array}$$

and

$$\begin{array}{cc}\hfill \mathcal{G}({\vartheta }_{\mathfrak{j}},{\vartheta }_{\mathfrak{j}+2\mathfrak{m}+1},\varsigma )& \le \mathcal{G}\left({\vartheta }_{\mathfrak{j}},{\vartheta }_{\mathfrak{j}+1},\frac{\varsigma }{3\left(\beta ({\vartheta }_{\mathfrak{j}},{\vartheta }_{\mathfrak{j}+1})\right)}\right)\diamond \mathcal{G}\left({\vartheta }_{\mathfrak{j}+1},{\vartheta }_{\mathfrak{j}+2},\frac{\varsigma }{3\left(\mathrm{\Gamma }({\vartheta }_{\mathfrak{j}+1},{\vartheta }_{\mathfrak{j}+2})\right)}\right)\hfill \\ \hfill & \diamond \mathcal{G}\left({\vartheta }_{\mathfrak{j}+2},{\vartheta }_{\mathfrak{j}+2\mathfrak{m}+1},\frac{\varsigma }{3\left(\eta ({\vartheta }_{\mathfrak{j}+2},{\vartheta }_{\mathfrak{j}+2\mathfrak{m}+1})\right)}\right)\hfill \\ \hfill & \le \mathcal{G}\left({\vartheta }_{\mathfrak{j}},{\vartheta }_{\mathfrak{j}+1},\frac{\varsigma }{3\left(\beta ({\vartheta }_{\mathfrak{j}},{\vartheta }_{\mathfrak{j}+1})\right)}\right)\diamond \mathcal{G}\left({\vartheta }_{\mathfrak{j}+1},{\vartheta }_{\mathfrak{j}+2},\frac{\varsigma }{3\left(\mathrm{\Gamma }({\vartheta }_{\mathfrak{j}+1},{\vartheta }_{\mathfrak{j}+2})\right)}\right)\hfill \\ \hfill & \diamond \mathcal{G}\left({\vartheta }_{\mathfrak{j}+2},{\vartheta }_{\mathfrak{j}+3},\frac{\varsigma }{3^2\left(\beta ({\vartheta }_{\mathfrak{j}+2},{\vartheta }_{\mathfrak{j}+3})\eta ({\vartheta }_{\mathfrak{j}+2},{\vartheta }_{\mathfrak{j}+2\mathfrak{m}+1})\right)}\right)\hfill \\ \hfill & \diamond \mathcal{G}\left({\vartheta }_{\mathfrak{j}+3},{\vartheta }_{\mathfrak{j}+4},\frac{\varsigma }{3^2\left(\mathrm{\Gamma }({\vartheta }_{\mathfrak{j}+3},{\vartheta }_{\mathfrak{j}+4})\eta ({\vartheta }_{\mathfrak{j}+2},{\vartheta }_{\mathfrak{j}+2\mathfrak{m}+1})\right)}\right)\hfill \\ \hfill & \diamond \mathcal{G}\left({\vartheta }_{\mathfrak{j}+4},{\vartheta }_{\mathfrak{j}+2\mathfrak{m}+1},\frac{\varsigma }{3^2\left(\eta ({\vartheta }_{\mathfrak{j}+4},{\vartheta }_{\mathfrak{j}+2\mathfrak{m}+1})\eta ({\vartheta }_{\mathfrak{j}+2},{\vartheta }_{\mathfrak{j}+2\mathfrak{m}+1})\right)}\right)\hfill \\ \hfill & \le \mathcal{G}\left({\vartheta }_{\mathfrak{j}},{\vartheta }_{\mathfrak{j}+1},\frac{\varsigma }{3\left(\beta ({\vartheta }_{\mathfrak{j}},{\vartheta }_{\mathfrak{j}+1})\right)}\right)\diamond \mathcal{G}\left({\vartheta }_{\mathfrak{j}+1},{\vartheta }_{\mathfrak{j}+2},\frac{\varsigma }{3\left(\mathrm{\Gamma }({\vartheta }_{\mathfrak{j}+1},{\vartheta }_{\mathfrak{j}+2})\right)}\right)\hfill \\ \hfill & \diamond \mathcal{G}\left({\vartheta }_{\mathfrak{j}+2},{\vartheta }_{\mathfrak{j}+3},\frac{\varsigma }{3^2\left(\beta ({\vartheta }_{\mathfrak{j}+2},{\vartheta }_{\mathfrak{j}+3})\eta ({\vartheta }_{\mathfrak{j}+2},{\vartheta }_{\mathfrak{j}+2\mathfrak{m}+1})\right)}\right)\hfill \\ \hfill & \diamond \mathcal{G}\left({\vartheta }_{\mathfrak{j}+3},{\vartheta }_{\mathfrak{j}+4},\frac{\varsigma }{3^2\left(\mathrm{\Gamma }({\vartheta }_{\mathfrak{j}+3},{\vartheta }_{\mathfrak{j}+4})\eta ({\vartheta }_{\mathfrak{j}+2},{\vartheta }_{\mathfrak{j}+2\mathfrak{m}+1})\right)}\right)\hfill \end{array}$$

$$\begin{array}{cc}\hfill & \diamond \mathcal{G}\left({\vartheta }_{\mathfrak{j}+4},{\vartheta }_{\mathfrak{j}+5},\frac{\varsigma }{3^3\left(\beta ({\vartheta }_{\mathfrak{j}+4},{\vartheta }_{\mathfrak{j}+5})\eta ({\vartheta }_{\mathfrak{j}+4},{\vartheta }_{\mathfrak{j}+2\mathfrak{m}+1})\eta ({\vartheta }_{\mathfrak{j}+2},{\vartheta }_{\mathfrak{j}+2\mathfrak{m}+1})\right)}\right)\hfill \\ \hfill & \diamond \mathcal{G}\left({\vartheta }_{\mathfrak{j}+5},{\vartheta }_{\mathfrak{j}+6},\frac{\varsigma }{3^3\left(\mathrm{\Gamma }({\vartheta }_{\mathfrak{j}+5},{\vartheta }_{\mathfrak{j}+6})\eta ({\vartheta }_{\mathfrak{j}+4},{\vartheta }_{\mathfrak{j}+2\mathfrak{m}+1})\eta ({\vartheta }_{\mathfrak{j}+2},{\vartheta }_{\mathfrak{j}+2\mathfrak{m}+1})\right)}\right)\hfill \\ \hfill & \diamond \mathcal{G}\left({\vartheta }_{\mathfrak{j}+6},{\vartheta }_{\mathfrak{j}+2\mathfrak{m}+1},\frac{\varsigma }{3^3\left(\eta ({\vartheta }_{\mathfrak{j}+6},{\vartheta }_{\mathfrak{j}+2\mathfrak{m}+1})\eta ({\vartheta }_{\mathfrak{j}+4},{\vartheta }_{\mathfrak{j}+2\mathfrak{m}+1})\eta ({\vartheta }_{\mathfrak{j}+2},{\vartheta }_{\mathfrak{j}+2\mathfrak{m}+1})\right)}\right)\hfill \end{array}$$

$$\begin{array}{cc}\hfill \mathcal{G}& ({\vartheta }_{\mathfrak{j}},{\vartheta }_{\mathfrak{j}+2\mathfrak{m}+1},\varsigma )\le \mathcal{G}\left({\vartheta }_{\mathfrak{j}},{\vartheta }_{\mathfrak{j}+1},\frac{\varsigma }{3\left(\beta ({\vartheta }_{\mathfrak{j}},{\vartheta }_{\mathfrak{j}+1})\right)}\right)\diamond \mathcal{G}\left({\vartheta }_{\mathfrak{j}+1},{\vartheta }_{\mathfrak{j}+2},\frac{\varsigma }{3\left(\mathrm{\Gamma }({\vartheta }_{\mathfrak{j}+1},{\vartheta }_{\mathfrak{j}+2})\right)}\right)\hfill \\ \hfill & \diamond \mathcal{G}\left({\vartheta }_{\mathfrak{j}+2},{\vartheta }_{\mathfrak{j}+3},\frac{\varsigma }{3^2\left(\beta ({\vartheta }_{\mathfrak{j}+2},{\vartheta }_{\mathfrak{j}+3})\eta ({\vartheta }_{\mathfrak{j}+2},{\vartheta }_{\mathfrak{j}+2\mathfrak{m}+1})\right)}\right)\hfill \\ \hfill & \diamond \mathcal{G}\left({\vartheta }_{\mathfrak{j}+3},{\vartheta }_{\mathfrak{j}+4},\frac{\varsigma }{3^2\left(\mathrm{\Gamma }({\vartheta }_{\mathfrak{j}+3},{\vartheta }_{\mathfrak{j}+4})\eta ({\vartheta }_{\mathfrak{j}+2},{\vartheta }_{\mathfrak{j}+2\mathfrak{m}+1})\right)}\right)\hfill \\ \hfill & \diamond \mathcal{G}\left({\vartheta }_{\mathfrak{j}+4},{\vartheta }_{\mathfrak{j}+5},\frac{\varsigma }{3^3\left(\beta ({\vartheta }_{\mathfrak{j}+4},{\vartheta }_{\mathfrak{j}+5})\eta ({\vartheta }_{\mathfrak{j}+4},{\vartheta }_{\mathfrak{j}+2\mathfrak{m}+1})\eta ({\vartheta }_{\mathfrak{j}+2},{\vartheta }_{\mathfrak{j}+2\mathfrak{m}+1})\right)}\right)\hfill \\ \hfill & \diamond \mathcal{G}\left({\vartheta }_{\mathfrak{j}+5},{\vartheta }_{\mathfrak{j}+6},\frac{\varsigma }{3^3\left(\mathrm{\Gamma }({\vartheta }_{\mathfrak{j}+5},{\vartheta }_{\mathfrak{j}+6})\eta ({\vartheta }_{\mathfrak{j}+4},{\vartheta }_{\mathfrak{j}+2\mathfrak{m}+1})\eta ({\vartheta }_{\mathfrak{j}+2},{\vartheta }_{\mathfrak{j}+2\mathfrak{m}+1})\right)}\right)\hfill \\ \hfill & \diamond \mathcal{G}\left({\vartheta }_{\mathfrak{j}+6},{\vartheta }_{\mathfrak{j}+2\mathfrak{m}+1},\frac{\varsigma }{3^3\left(\eta ({\vartheta }_{\mathfrak{j}+6},{\vartheta }_{\mathfrak{j}+2\mathfrak{m}+1})\eta ({\vartheta }_{\mathfrak{j}+4},{\vartheta }_{\mathfrak{j}+2\mathfrak{m}+1})\eta ({\vartheta }_{\mathfrak{j}+2},{\vartheta }_{\mathfrak{j}+2\mathfrak{m}+1})\right)}\right)\diamond \cdots \diamond \hfill \\ \hfill & \mathcal{G}\left({\vartheta }_{\mathfrak{j}+2\mathfrak{m}-2},{\vartheta }_{\mathfrak{j}+2\mathfrak{m}-1},\frac{\varsigma }{3^{\mathfrak{m}}(\beta ({\vartheta }_{\mathfrak{j}+2\mathfrak{m}-2},{\vartheta }_{\mathfrak{j}+2\mathfrak{m}-1})\eta ({\vartheta }_{\mathfrak{j}+2\mathfrak{m}-2},{\vartheta }_{\mathfrak{j}+2\mathfrak{m}+1})\cdots \eta ({\vartheta }_{\mathfrak{j}+2},{\vartheta }_{\mathfrak{j}+2\mathfrak{m}+1}))}\right)\hfill \\ \hfill & \diamond \mathcal{G}\left({\vartheta }_{\mathfrak{j}+2\mathfrak{m}-1},{\vartheta }_{\mathfrak{j}+2\mathfrak{m}},\frac{\varsigma }{3^{\mathfrak{m}}(\mathrm{\Gamma }({\vartheta }_{\mathfrak{j}+2\mathfrak{m}-1},{\vartheta }_{\mathfrak{j}+2\mathfrak{m}})\eta ({\vartheta }_{\mathfrak{j}+2\mathfrak{m}-2},{\vartheta }_{\mathfrak{j}+2\mathfrak{m}+1})\cdots \eta ({\vartheta }_{\mathfrak{j}+2},{\vartheta }_{\mathfrak{j}+2\mathfrak{m}+1}))}\right)\hfill \\ \hfill & \diamond \mathcal{G}\left({\vartheta }_{\mathfrak{j}+2\mathfrak{m}},{\vartheta }_{\mathfrak{j}+2\mathfrak{m}+1},\frac{\varsigma }{3^{\mathfrak{m}}(\eta ({\vartheta }_{\mathfrak{j}+2\mathfrak{m}},{\vartheta }_{\mathfrak{j}+2\mathfrak{m}+1})\eta ({\vartheta }_{\mathfrak{j}+2\mathfrak{m}-2},{\vartheta }_{\mathfrak{j}+2\mathfrak{m}+1})\cdots \eta ({\vartheta }_{\mathfrak{j}+2},{\vartheta }_{\mathfrak{j}+2\mathfrak{m}+1}))}\right).\hfill \end{array}$$

Using (3) in the above inequalities, we deduce

$$\begin{array}{cc}\hfill \mathcal{W}({\vartheta }_{\mathfrak{j}},{\vartheta }_{\mathfrak{j}+2\mathfrak{m}+1},& \varsigma )\ge \frac{1}{\frac{{\rho }^{\mathfrak{j}}}{\mathcal{W}\left({\vartheta }_0,{\vartheta }_1,\frac{\varsigma }{3\left(\beta ({\vartheta }_{\mathfrak{j}},{\vartheta }_{\mathfrak{j}+1})\right)}\right)}+(1-{\rho }^{\mathfrak{j}})}\star \frac{1}{\frac{{\rho }^{\mathfrak{j}+1}}{\mathcal{W}\left({\vartheta }_0,{\vartheta }_1,\frac{\varsigma }{3\left(\mathrm{\Gamma }({\vartheta }_{\mathfrak{j}+1},{\vartheta }_{\mathfrak{j}+2})\right)}\right)}+(1-{\rho }^{\mathfrak{j}+1})}\hfill \\ \hfill & \star \frac{1}{\frac{{\rho }^{\mathfrak{j}+2}}{\mathcal{W}\left({\vartheta }_0,{\vartheta }_1,\frac{\varsigma }{3^2\left(\beta ({\vartheta }_{\mathfrak{j}+2},{\vartheta }_{\mathfrak{j}+3})\eta ({\vartheta }_{\mathfrak{j}+2},{\vartheta }_{\mathfrak{j}+2\mathfrak{m}+1})\right)}\right)}+(1-{\rho }^{\mathfrak{j}+2})}\hfill \\ \hfill & \star \frac{1}{\frac{{\rho }^{\mathfrak{j}+3}}{\mathcal{W}\left({\vartheta }_0,{\vartheta }_1,\frac{\varsigma }{3^2\left(\mathrm{\Gamma }({\vartheta }_{\mathfrak{j}+3},{\vartheta }_{\mathfrak{j}+4})\eta ({\vartheta }_{\mathfrak{j}+2},{\vartheta }_{\mathfrak{j}+2\mathfrak{m}+1})\right)}\right)}+(1-{\rho }^{\mathfrak{j}+3})}\hfill \\ \hfill & \star \frac{1}{\frac{{\rho }^{\mathfrak{j}+4}}{\mathcal{W}\left({\vartheta }_0,{\vartheta }_1,\frac{\varsigma }{3^3\left(\beta ({\vartheta }_{\mathfrak{j}+4},{\vartheta }_{\mathfrak{j}+5})\eta ({\vartheta }_{\mathfrak{j}+4},{\vartheta }_{\mathfrak{j}+2\mathfrak{m}+1})\eta ({\vartheta }_{\mathfrak{j}+2},{\vartheta }_{\mathfrak{j}+2\mathfrak{m}+1})\right)}\right)}+(1-{\rho }^{\mathfrak{j}+4})}\hfill \\ \hfill & \star \frac{1}{\frac{{\rho }^{\mathfrak{j}+5}}{\mathcal{W}\left({\vartheta }_0,{\vartheta }_1,\frac{\varsigma }{3^3\left(\mathrm{\Gamma }({\vartheta }_{\mathfrak{j}+5},{\vartheta }_{\mathfrak{j}+6})\eta ({\vartheta }_{\mathfrak{j}+4},{\vartheta }_{\mathfrak{j}+2\mathfrak{m}+1})\eta ({\vartheta }_{\mathfrak{j}+2},{\vartheta }_{\mathfrak{j}+2\mathfrak{m}+1})\right)}\right)}+(1-{\rho }^{\mathfrak{j}+5})}\hfill \end{array}$$

$$\begin{array}{cc}\hfill & \star \frac{1}{\frac{{\rho }^{\mathfrak{j}+6}}{\mathcal{W}\left({\vartheta }_0,{\vartheta }_1,\frac{\varsigma }{3^3\left(\eta ({\vartheta }_{\mathfrak{j}+6},{\vartheta }_{\mathfrak{j}+2\mathfrak{m}+1})\eta ({\vartheta }_{\mathfrak{j}+4},{\vartheta }_{\mathfrak{j}+2\mathfrak{m}+1})\eta ({\vartheta }_{\mathfrak{j}+2},{\vartheta }_{\mathfrak{j}+2\mathfrak{m}+1})\right)}\right)}+(1-{\rho }^{\mathfrak{j}+6})}\star \cdots \star \hfill \\ \hfill & \star \frac{1}{\frac{{\rho }^{\mathfrak{j}+2\mathfrak{m}-2}}{\mathcal{W}\left({\vartheta }_0,{\vartheta }_1,\frac{\varsigma }{3^{\mathfrak{m}}(\beta ({\vartheta }_{\mathfrak{j}+2\mathfrak{m}-2},{\vartheta }_{\mathfrak{j}+2\mathfrak{m}-1})\eta ({\vartheta }_{\mathfrak{j}+2\mathfrak{m}-2},{\vartheta }_{\mathfrak{j}+2\mathfrak{m}+1})\cdots \eta ({\vartheta }_{\mathfrak{j}+2},{\vartheta }_{\mathfrak{j}+2\mathfrak{m}+1}))}\right)}+(1-{\rho }^{\mathfrak{j}+2\mathfrak{m}-2})}\hfill \\ \hfill & \star \frac{1}{\frac{{\rho }^{\mathfrak{j}+2\mathfrak{m}-1}}{\mathcal{W}\left({\vartheta }_0,{\vartheta }_1,\frac{\varsigma }{3^{\mathfrak{m}}(\mathrm{\Gamma }({\vartheta }_{\mathfrak{j}+2\mathfrak{m}-1},{\vartheta }_{\mathfrak{j}+2\mathfrak{m}})\eta ({\vartheta }_{\mathfrak{j}+2\mathfrak{m}-2},{\vartheta }_{\mathfrak{j}+2\mathfrak{m}+1})\cdots \eta ({\vartheta }_{\mathfrak{j}+2},{\vartheta }_{\mathfrak{j}+2\mathfrak{m}+1}))}\right)}+(1-{\rho }^{\mathfrak{j}+2\mathfrak{m}-1})}\hfill \\ \hfill & \star \frac{1}{\frac{{\rho }^{\mathfrak{j}+2\mathfrak{m}}}{\mathcal{W}\left({\vartheta }_0,{\vartheta }_1,\frac{\varsigma }{3^{\mathfrak{m}}(\eta ({\vartheta }_{\mathfrak{j}+2\mathfrak{m}},{\vartheta }_{\mathfrak{j}+2\mathfrak{m}+1})\eta ({\vartheta }_{\mathfrak{j}+2\mathfrak{m}-2},{\vartheta }_{\mathfrak{j}+2\mathfrak{m}+1})\cdots \eta ({\vartheta }_{\mathfrak{j}+2},{\vartheta }_{\mathfrak{j}+2\mathfrak{m}+1}))}\right)}+(1-{\rho }^{\mathfrak{j}+2\mathfrak{m}})},\hfill \end{array}$$

$$\begin{array}{cc}\hfill \mathcal{E}& ({\vartheta }_{\mathfrak{j}},{\vartheta }_{\mathfrak{j}+2\mathfrak{m}+1},\varsigma )\hfill \\ & \le {\rho }^{\mathfrak{j}}\mathcal{E}\left({\vartheta }_0,{\vartheta }_1,\frac{\varsigma }{3\left(\beta ({\vartheta }_{\mathfrak{j}},{\vartheta }_{\mathfrak{j}+1})\right)}\right)\diamond {\rho }^{\mathfrak{j}+1}\mathcal{E}\left({\vartheta }_0,{\vartheta }_1,\frac{\varsigma }{3\left(\mathrm{\Gamma }({\vartheta }_{\mathfrak{j}+1},{\vartheta }_{\mathfrak{j}+2})\right)}\right)\hfill \\ \hfill & \diamond {\rho }^{\mathfrak{j}+2}\mathcal{E}\left({\vartheta }_0,{\vartheta }_1,\frac{\varsigma }{3^2\left(\beta ({\vartheta }_{\mathfrak{j}+2},{\vartheta }_{\mathfrak{j}+3})\eta ({\vartheta }_{\mathfrak{j}+2},{\vartheta }_{\mathfrak{j}+2\mathfrak{m}+1})\right)}\right)\hfill \\ \hfill & \diamond {\rho }^{\mathfrak{j}+3}\mathcal{E}\left({\vartheta }_0,{\vartheta }_1,\frac{\varsigma }{3^2\left(\mathrm{\Gamma }({\vartheta }_{\mathfrak{j}+3},{\vartheta }_{\mathfrak{j}+4})\eta ({\vartheta }_{\mathfrak{j}+2},{\vartheta }_{\mathfrak{j}+2\mathfrak{m}+1})\right)}\right)\hfill \\ \hfill & \diamond {\rho }^{\mathfrak{j}+4}\mathcal{E}\left({\vartheta }_0,{\vartheta }_1,\frac{\varsigma }{3^3\left(\beta ({\vartheta }_{\mathfrak{j}+4},{\vartheta }_{\mathfrak{j}+5})\eta ({\vartheta }_{\mathfrak{j}+4},{\vartheta }_{\mathfrak{j}+2\mathfrak{m}+1})\eta ({\vartheta }_{\mathfrak{j}+2},{\vartheta }_{\mathfrak{j}+2\mathfrak{m}+1})\right)}\right)\hfill \\ \hfill & \diamond {\rho }^{\mathfrak{j}+5}\mathcal{E}\left({\vartheta }_0,{\vartheta }_1,\frac{\varsigma }{3^3\left(\mathrm{\Gamma }({\vartheta }_{\mathfrak{j}+5},{\vartheta }_{\mathfrak{j}+6})\eta ({\vartheta }_{\mathfrak{j}+4},{\vartheta }_{\mathfrak{j}+2\mathfrak{m}+1})\eta ({\vartheta }_{\mathfrak{j}+2},{\vartheta }_{\mathfrak{j}+2\mathfrak{m}+1})\right)}\right)\hfill \\ \hfill & \diamond {\rho }^{\mathfrak{j}+6}\mathcal{E}\left({\vartheta }_0,{\vartheta }_1,\frac{\varsigma }{3^3\left(\eta ({\vartheta }_{\mathfrak{j}+6},{\vartheta }_{\mathfrak{j}+2\mathfrak{m}+1})\eta ({\vartheta }_{\mathfrak{j}+4},{\vartheta }_{\mathfrak{j}+2\mathfrak{m}+1})\eta ({\vartheta }_{\mathfrak{j}+2},{\vartheta }_{\mathfrak{j}+2\mathfrak{m}+1})\right)}\right)\diamond \cdots \diamond \hfill \\ \hfill & {\rho }^{\mathfrak{j}+2\mathfrak{m}-2}\mathcal{E}\left({\vartheta }_0,{\vartheta }_1,\frac{\varsigma }{3^{\mathfrak{m}}(\beta ({\vartheta }_{\mathfrak{j}+2\mathfrak{m}-2},{\vartheta }_{\mathfrak{j}+2\mathfrak{m}-1})\eta ({\vartheta }_{\mathfrak{j}+2\mathfrak{m}-2},{\vartheta }_{\mathfrak{j}+2\mathfrak{m}+1})\cdots \eta ({\vartheta }_{\mathfrak{j}+2},{\vartheta }_{\mathfrak{j}+2\mathfrak{m}+1}))}\right)\hfill \\ \hfill & \diamond {\rho }^{\mathfrak{j}+2\mathfrak{m}-1}\mathcal{E}\left({\vartheta }_0,{\vartheta }_1,\frac{\varsigma }{3^{\mathfrak{m}}(\mathrm{\Gamma }({\vartheta }_{\mathfrak{j}+2\mathfrak{m}-1},{\vartheta }_{\mathfrak{j}+2\mathfrak{m}})\eta ({\vartheta }_{\mathfrak{j}+2\mathfrak{m}-2},{\vartheta }_{\mathfrak{j}+2\mathfrak{m}+1})\cdots \eta ({\vartheta }_{\mathfrak{j}+2},{\vartheta }_{\mathfrak{j}+2\mathfrak{m}+1}))}\right)\hfill \\ \hfill & \diamond {\rho }^{\mathfrak{j}+2\mathfrak{m}}\mathcal{E}\left({\vartheta }_0,{\vartheta }_1,\frac{\varsigma }{3^{\mathfrak{m}}(\eta ({\vartheta }_{\mathfrak{j}+2\mathfrak{m}},{\vartheta }_{\mathfrak{j}+2\mathfrak{m}+1})\eta ({\vartheta }_{\mathfrak{j}+2\mathfrak{m}-2},{\vartheta }_{\mathfrak{j}+2\mathfrak{m}+1})\cdots \eta ({\vartheta }_{\mathfrak{j}+2},{\vartheta }_{\mathfrak{j}+2\mathfrak{m}+1}))}\right),\hfill \end{array}$$

and

$$\begin{array}{cc}\hfill \mathcal{G}({\vartheta }_{\mathfrak{j}},{\vartheta }_{\mathfrak{j}+2\mathfrak{m}+1},\varsigma )& \le {\rho }^{\mathfrak{j}}\mathcal{G}\left({\vartheta }_0,{\vartheta }_1,\frac{\varsigma }{3\left(\beta ({\vartheta }_{\mathfrak{j}},{\vartheta }_{\mathfrak{j}+1})\right)}\right)\diamond {\rho }^{\mathfrak{j}+1}\mathcal{G}\left({\vartheta }_0,{\vartheta }_1,\frac{\varsigma }{3\left(\mathrm{\Gamma }({\vartheta }_{\mathfrak{j}+1},{\vartheta }_{\mathfrak{j}+2})\right)}\right)\hfill \\ \hfill & \diamond {\rho }^{\mathfrak{j}+2}\mathcal{G}\left({\vartheta }_0,{\vartheta }_1,\frac{\varsigma }{3^2\left(\beta ({\vartheta }_{\mathfrak{j}+2},{\vartheta }_{\mathfrak{j}+3})\eta ({\vartheta }_{\mathfrak{j}+2},{\vartheta }_{\mathfrak{j}+2\mathfrak{m}+1})\right)}\right)\hfill \\ \hfill & \diamond {\rho }^{\mathfrak{j}+3}\mathcal{G}\left({\vartheta }_0,{\vartheta }_1,\frac{\varsigma }{3^2\left(\mathrm{\Gamma }({\vartheta }_{\mathfrak{j}+3},{\vartheta }_{\mathfrak{j}+4})\eta ({\vartheta }_{\mathfrak{j}+2},{\vartheta }_{\mathfrak{j}+2\mathfrak{m}+1})\right)}\right)\hfill \\ \hfill & \diamond {\rho }^{\mathfrak{j}+4}\mathcal{G}\left({\vartheta }_0,{\vartheta }_1,\frac{\varsigma }{3^3\left(\beta ({\vartheta }_{\mathfrak{j}+4},{\vartheta }_{\mathfrak{j}+5})\eta ({\vartheta }_{\mathfrak{j}+4},{\vartheta }_{\mathfrak{j}+2\mathfrak{m}+1})\eta ({\vartheta }_{\mathfrak{j}+2},{\vartheta }_{\mathfrak{j}+2\mathfrak{m}+1})\right)}\right)\hfill \\ \hfill & \diamond {\rho }^{\mathfrak{j}+5}\mathcal{G}\left({\vartheta }_0,{\vartheta }_1,\frac{\varsigma }{3^3\left(\mathrm{\Gamma }({\vartheta }_{\mathfrak{j}+5},{\vartheta }_{\mathfrak{j}+6})\eta ({\vartheta }_{\mathfrak{j}+4},{\vartheta }_{\mathfrak{j}+2\mathfrak{m}+1})\eta ({\vartheta }_{\mathfrak{j}+2},{\vartheta }_{\mathfrak{j}+2\mathfrak{m}+1})\right)}\right)\hfill \end{array}$$

$$\begin{array}{cc}\hfill & \diamond {\rho }^{\mathfrak{j}+6}\mathcal{G}\left({\vartheta }_0,{\vartheta }_1,\frac{\varsigma }{3^3\left(\eta ({\vartheta }_{\mathfrak{j}+6},{\vartheta }_{\mathfrak{j}+2\mathfrak{m}+1})\eta ({\vartheta }_{\mathfrak{j}+4},{\vartheta }_{\mathfrak{j}+2\mathfrak{m}+1})\eta ({\vartheta }_{\mathfrak{j}+2},{\vartheta }_{\mathfrak{j}+2\mathfrak{m}+1})\right)}\right)\diamond \cdots \diamond \hfill \\ \hfill & {\rho }^{\mathfrak{j}+2\mathfrak{m}-2}\mathcal{G}\left({\vartheta }_0,{\vartheta }_1,\frac{\varsigma }{3^{\mathfrak{m}}(\beta ({\vartheta }_{\mathfrak{j}+2\mathfrak{m}-2},{\vartheta }_{\mathfrak{j}+2\mathfrak{m}-1})\eta ({\vartheta }_{\mathfrak{j}+2\mathfrak{m}-2},{\vartheta }_{\mathfrak{j}+2\mathfrak{m}+1})\cdots \eta ({\vartheta }_{\mathfrak{j}+2},{\vartheta }_{\mathfrak{j}+2\mathfrak{m}+1}))}\right)\hfill \\ \hfill & \diamond {\rho }^{\mathfrak{j}+2\mathfrak{m}-1}\mathcal{G}\left({\vartheta }_0,{\vartheta }_1,\frac{\varsigma }{3^{\mathfrak{m}}(\mathrm{\Gamma }({\vartheta }_{\mathfrak{j}+2\mathfrak{m}-1},{\vartheta }_{\mathfrak{j}+2\mathfrak{m}})\eta ({\vartheta }_{\mathfrak{j}+2\mathfrak{m}-2},{\vartheta }_{\mathfrak{j}+2\mathfrak{m}+1})\cdots \eta ({\vartheta }_{\mathfrak{j}+2},{\vartheta }_{\mathfrak{j}+2\mathfrak{m}+1}))}\right)\hfill \\ \hfill & \diamond {\rho }^{\mathfrak{j}+2\mathfrak{m}}\mathcal{G}\left({\vartheta }_0,{\vartheta }_1,\frac{\varsigma }{3^{\mathfrak{m}}(\eta ({\vartheta }_{\mathfrak{j}+2\mathfrak{m}},{\vartheta }_{\mathfrak{j}+2\mathfrak{m}+1})\eta ({\vartheta }_{\mathfrak{j}+2\mathfrak{m}-2},{\vartheta }_{\mathfrak{j}+2\mathfrak{m}+1})\cdots \eta ({\vartheta }_{\mathfrak{j}+2},{\vartheta }_{\mathfrak{j}+2\mathfrak{m}+1}))}\right).\hfill \end{array}$$

Case 2. When $\mathfrak{i}=2\mathfrak{m}$, i.e., $\mathfrak{i}$ is even, then

$$\begin{array}{cc}\hfill \mathcal{W}({\vartheta }_{\mathfrak{j}},{\vartheta }_{\mathfrak{j}+2\mathfrak{m}},\varsigma )& \ge \mathcal{W}\left({\vartheta }_{\mathfrak{j}},{\vartheta }_{\mathfrak{j}+1},\frac{\varsigma }{3\left(\beta ({\vartheta }_{\mathfrak{j}},{\vartheta }_{\mathfrak{j}+1})\right)}\right)\star \mathcal{W}\left({\vartheta }_{\mathfrak{j}+1},{\vartheta }_{\mathfrak{j}+2},\frac{\varsigma }{3\left(\mathrm{\Gamma }({\vartheta }_{\mathfrak{j}+1},{\vartheta }_{\mathfrak{j}+2})\right)}\right)\hfill \\ \hfill & \star \mathcal{W}\left({\vartheta }_{\mathfrak{j}+2},{\vartheta }_{\mathfrak{j}+2\mathfrak{m}},\frac{\varsigma }{3\left(\eta ({\vartheta }_{\mathfrak{j}+2},{\vartheta }_{\mathfrak{j}+2\mathfrak{m}})\right)}\right)\hfill \\ \hfill & \ge \mathcal{W}\left({\vartheta }_{\mathfrak{j}},{\vartheta }_{\mathfrak{j}+1},\frac{\varsigma }{3\left(\beta ({\vartheta }_{\mathfrak{j}},{\vartheta }_{\mathfrak{j}+1})\right)}\right)\star \mathcal{W}\left({\vartheta }_{\mathfrak{j}+1},{\vartheta }_{\mathfrak{j}+2},\frac{\varsigma }{3\left(\mathrm{\Gamma }({\vartheta }_{\mathfrak{j}+1},{\vartheta }_{\mathfrak{j}+2})\right)}\right)\hfill \\ \hfill & \star \mathcal{W}\left({\vartheta }_{\mathfrak{j}+2},{\vartheta }_{\mathfrak{j}+3},\frac{\varsigma }{3^2\left(\beta ({\vartheta }_{\mathfrak{j}+2},{\vartheta }_{\mathfrak{j}+3})\eta ({\vartheta }_{\mathfrak{j}+2},{\vartheta }_{\mathfrak{j}+2\mathfrak{m}})\right)}\right)\hfill \\ \hfill & \star \mathcal{W}\left({\vartheta }_{\mathfrak{j}+3},{\vartheta }_{\mathfrak{j}+4},\frac{\varsigma }{3^2\left(\mathrm{\Gamma }({\vartheta }_{\mathfrak{j}+3},{\vartheta }_{\mathfrak{j}+4})\eta ({\vartheta }_{\mathfrak{j}+2},{\vartheta }_{\mathfrak{j}+2\mathfrak{m}})\right)}\right)\hfill \\ \hfill & \star \mathcal{W}\left({\vartheta }_{\mathfrak{j}+4},{\vartheta }_{\mathfrak{j}+2\mathfrak{m}},\frac{\varsigma }{3^2\left(\eta ({\vartheta }_{\mathfrak{j}+4},{\vartheta }_{\mathfrak{j}+2\mathfrak{m}})\eta ({\vartheta }_{\mathfrak{j}+2},{\vartheta }_{\mathfrak{j}+2\mathfrak{m}})\right)}\right)\hfill \\ \hfill & \ge \mathcal{W}\left({\vartheta }_{\mathfrak{j}},{\vartheta }_{\mathfrak{j}+1},\frac{\varsigma }{3\left(\beta ({\vartheta }_{\mathfrak{j}},{\vartheta }_{\mathfrak{j}+1})\right)}\right)\star \mathcal{W}\left({\vartheta }_{\mathfrak{j}+1},{\vartheta }_{\mathfrak{j}+2},\frac{\varsigma }{3\left(\mathrm{\Gamma }({\vartheta }_{\mathfrak{j}+1},{\vartheta }_{\mathfrak{j}+2})\right)}\right)\hfill \\ \hfill & \star \mathcal{W}\left({\vartheta }_{\mathfrak{j}+2},{\vartheta }_{\mathfrak{j}+3},\frac{\varsigma }{3^2\left(\beta ({\vartheta }_{\mathfrak{j}+2},{\vartheta }_{\mathfrak{j}+3})\eta ({\vartheta }_{\mathfrak{j}+2},{\vartheta }_{\mathfrak{j}+2\mathfrak{m}})\right)}\right)\hfill \\ \hfill & \star \mathcal{W}\left({\vartheta }_{\mathfrak{j}+3},{\vartheta }_{\mathfrak{j}+4},\frac{\varsigma }{3^2\left(\mathrm{\Gamma }({\vartheta }_{\mathfrak{j}+3},{\vartheta }_{\mathfrak{j}+4})\eta ({\vartheta }_{\mathfrak{j}+2},{\vartheta }_{\mathfrak{j}+2\mathfrak{m}})\right)}\right)\hfill \\ \hfill & \star \mathcal{W}\left({\vartheta }_{\mathfrak{j}+4},{\vartheta }_{\mathfrak{j}+5},\frac{\varsigma }{3^3\left(\beta ({\vartheta }_{\mathfrak{j}+4},{\vartheta }_{\mathfrak{j}+5})\eta ({\vartheta }_{\mathfrak{j}+4},{\vartheta }_{\mathfrak{j}+2\mathfrak{m}})\eta ({\vartheta }_{\mathfrak{j}+2},{\vartheta }_{\mathfrak{j}+2\mathfrak{m}})\right)}\right)\hfill \\ \hfill & \star \mathcal{W}\left({\vartheta }_{\mathfrak{j}+5},{\vartheta }_{\mathfrak{j}+6},\frac{\varsigma }{3^3\left(\mathrm{\Gamma }({\vartheta }_{\mathfrak{j}+5},{\vartheta }_{\mathfrak{j}+6})\eta ({\vartheta }_{\mathfrak{j}+4},{\vartheta }_{\mathfrak{j}+2\mathfrak{m}})\eta ({\vartheta }_{\mathfrak{j}+2},{\vartheta }_{\mathfrak{j}+2\mathfrak{m}})\right)}\right)\hfill \\ \hfill & \star \mathcal{W}\left({\vartheta }_{\mathfrak{j}+6},{\vartheta }_{\mathfrak{j}+2\mathfrak{m}},\frac{\varsigma }{3^3\left(\eta ({\vartheta }_{\mathfrak{j}+6},{\vartheta }_{\mathfrak{j}+2\mathfrak{m}})\eta ({\vartheta }_{\mathfrak{j}+4},{\vartheta }_{\mathfrak{j}+2\mathfrak{m}})\eta ({\vartheta }_{\mathfrak{j}+2},{\vartheta }_{\mathfrak{j}+2\mathfrak{m}})\right)}\right)\hfill \end{array}$$

$$\begin{array}{cc}\hfill \mathcal{W}& ({\vartheta }_{\mathfrak{j}},{\vartheta }_{\mathfrak{j}+2\mathfrak{m}},\varsigma )\ge \mathcal{W}\left({\vartheta }_{\mathfrak{j}},{\vartheta }_{\mathfrak{j}+1},\frac{\varsigma }{3\left(\beta ({\vartheta }_{\mathfrak{j}},{\vartheta }_{\mathfrak{j}+1})\right)}\right)\star \mathcal{W}\left({\vartheta }_{\mathfrak{j}+1},{\vartheta }_{\mathfrak{j}+2},\frac{\varsigma }{3\left(\mathrm{\Gamma }({\vartheta }_{\mathfrak{j}+1},{\vartheta }_{\mathfrak{j}+2})\right)}\right)\hfill \\ \hfill & \star \mathcal{W}\left({\vartheta }_{\mathfrak{j}+2},{\vartheta }_{\mathfrak{j}+3},\frac{\varsigma }{3^2\left(\beta ({\vartheta }_{\mathfrak{j}+2},{\vartheta }_{\mathfrak{j}+3})\eta ({\vartheta }_{\mathfrak{j}+2},{\vartheta }_{\mathfrak{j}+2\mathfrak{m}})\right)}\right)\hfill \\ \hfill & \star \mathcal{W}\left({\vartheta }_{\mathfrak{j}+3},{\vartheta }_{\mathfrak{j}+4},\frac{\varsigma }{3^2\left(\mathrm{\Gamma }({\vartheta }_{\mathfrak{j}+3},{\vartheta }_{\mathfrak{j}+4})\eta ({\vartheta }_{\mathfrak{j}+2},{\vartheta }_{\mathfrak{j}+2\mathfrak{m}})\right)}\right)\hfill \\ \hfill & \star \mathcal{W}\left({\vartheta }_{\mathfrak{j}+4},{\vartheta }_{\mathfrak{j}+5},\frac{\varsigma }{3^3\left(\beta ({\vartheta }_{\mathfrak{j}+4},{\vartheta }_{\mathfrak{j}+5})\eta ({\vartheta }_{\mathfrak{j}+4},{\vartheta }_{\mathfrak{j}+2\mathfrak{m}})\eta ({\vartheta }_{\mathfrak{j}+2},{\vartheta }_{\mathfrak{j}+2\mathfrak{m}})\right)}\right)\hfill \\ \hfill & \star \mathcal{W}\left({\vartheta }_{\mathfrak{j}+5},{\vartheta }_{\mathfrak{j}+6},\frac{\varsigma }{3^3\left(\mathrm{\Gamma }({\vartheta }_{\mathfrak{j}+5},{\vartheta }_{\mathfrak{j}+6})\eta ({\vartheta }_{\mathfrak{j}+4},{\vartheta }_{\mathfrak{j}+2\mathfrak{m}})\eta ({\vartheta }_{\mathfrak{j}+2},{\vartheta }_{\mathfrak{j}+2\mathfrak{m}})\right)}\right)\hfill \\ \hfill & \star \mathcal{W}\left({\vartheta }_{\mathfrak{j}+6},{\vartheta }_{\mathfrak{j}+2\mathfrak{m}},\frac{\varsigma }{3^3\left(\eta ({\vartheta }_{\mathfrak{j}+6},{\vartheta }_{\mathfrak{j}+2\mathfrak{m}})\eta ({\vartheta }_{\mathfrak{j}+4},{\vartheta }_{\mathfrak{j}+2\mathfrak{m}})\eta ({\vartheta }_{\mathfrak{j}+2},{\vartheta }_{\mathfrak{j}+2\mathfrak{m}})\right)}\right)\star \cdots \star \hfill \end{array}$$

$$\begin{array}{cc}\hfill & \mathcal{W}\left({\vartheta }_{\mathfrak{j}+2\mathfrak{m}-4},{\vartheta }_{\mathfrak{j}+2\mathfrak{m}-3},\frac{\varsigma }{3^{\mathfrak{m}-1}(\beta ({\vartheta }_{\mathfrak{j}+2\mathfrak{m}-4},{\vartheta }_{\mathfrak{j}+2\mathfrak{m}-3})\eta ({\vartheta }_{\mathfrak{j}+2\mathfrak{m}-4},{\vartheta }_{\mathfrak{j}+2\mathfrak{m}})\cdots \eta ({\vartheta }_{\mathfrak{j}+2},{\vartheta }_{\mathfrak{j}+2\mathfrak{m}}))}\right)\hfill \\ \hfill & \star \mathcal{W}\left({\vartheta }_{\mathfrak{j}+2\mathfrak{m}-3},{\vartheta }_{\mathfrak{j}+2\mathfrak{m}-2},\frac{\varsigma }{3^{\mathfrak{m}-1}(\mathrm{\Gamma }({\vartheta }_{\mathfrak{j}+2\mathfrak{m}-3},{\vartheta }_{\mathfrak{j}+2\mathfrak{m}-2})\eta ({\vartheta }_{\mathfrak{j}+2\mathfrak{m}-4},{\vartheta }_{\mathfrak{j}+2\mathfrak{m}})\cdots \eta ({\vartheta }_{\mathfrak{j}+2},{\vartheta }_{\mathfrak{j}+2\mathfrak{m}}))}\right)\hfill \\ \hfill & \star \mathcal{W}\left({\vartheta }_{\mathfrak{j}+2\mathfrak{m}-2},{\vartheta }_{\mathfrak{j}+2\mathfrak{m}},\frac{\varsigma }{3^{\mathfrak{m}-1}(\eta ({\vartheta }_{\mathfrak{j}+2\mathfrak{m}-2},{\vartheta }_{\mathfrak{j}+2\mathfrak{m}})\eta ({\vartheta }_{\mathfrak{j}+2\mathfrak{m}-4},{\vartheta }_{\mathfrak{j}+2\mathfrak{m}+1})\cdots \eta ({\vartheta }_{\mathfrak{j}+2},{\vartheta }_{\mathfrak{j}+2\mathfrak{m}}))}\right),\hfill \end{array}$$

$$\begin{array}{cc}\hfill \mathcal{E}({\vartheta }_{\mathfrak{j}},{\vartheta }_{\mathfrak{j}+2\mathfrak{m}},\varsigma )& \le \mathcal{E}\left({\vartheta }_{\mathfrak{j}},{\vartheta }_{\mathfrak{j}+1},\frac{\varsigma }{3\left(\beta ({\vartheta }_{\mathfrak{j}},{\vartheta }_{\mathfrak{j}+1})\right)}\right)\diamond \mathcal{E}\left({\vartheta }_{\mathfrak{j}+1},{\vartheta }_{\mathfrak{j}+2},\frac{\varsigma }{3\left(\mathrm{\Gamma }({\vartheta }_{\mathfrak{j}+1},{\vartheta }_{\mathfrak{j}+2})\right)}\right)\hfill \\ \hfill & \diamond \mathcal{E}\left({\vartheta }_{\mathfrak{j}+2},{\vartheta }_{\mathfrak{j}+2\mathfrak{m}},\frac{\varsigma }{3\left(\eta ({\vartheta }_{\mathfrak{j}+2},{\vartheta }_{\mathfrak{j}+2\mathfrak{m}})\right)}\right)\hfill \\ \hfill & \le \mathcal{E}\left({\vartheta }_{\mathfrak{j}},{\vartheta }_{\mathfrak{j}+1},\frac{\varsigma }{3\left(\beta ({\vartheta }_{\mathfrak{j}},{\vartheta }_{\mathfrak{j}+1})\right)}\right)\diamond \mathcal{E}\left({\vartheta }_{\mathfrak{j}+1},{\vartheta }_{\mathfrak{j}+2},\frac{\varsigma }{3\left(\mathrm{\Gamma }({\vartheta }_{\mathfrak{j}+1},{\vartheta }_{\mathfrak{j}+2})\right)}\right)\hfill \\ \hfill & \diamond \mathcal{E}\left({\vartheta }_{\mathfrak{j}+2},{\vartheta }_{\mathfrak{j}+3},\frac{\varsigma }{3^2\left(\beta ({\vartheta }_{\mathfrak{j}+2},{\vartheta }_{\mathfrak{j}+3})\eta ({\vartheta }_{\mathfrak{j}+2},{\vartheta }_{\mathfrak{j}+2\mathfrak{m}})\right)}\right)\hfill \\ \hfill & \diamond \mathcal{E}\left({\vartheta }_{\mathfrak{j}+3},{\vartheta }_{\mathfrak{j}+4},\frac{\varsigma }{3^2\left(\mathrm{\Gamma }({\vartheta }_{\mathfrak{j}+3},{\vartheta }_{\mathfrak{j}+4})\eta ({\vartheta }_{\mathfrak{j}+2},{\vartheta }_{\mathfrak{j}+2\mathfrak{m}})\right)}\right)\hfill \\ \hfill & \diamond \mathcal{E}\left({\vartheta }_{\mathfrak{j}+4},{\vartheta }_{\mathfrak{j}+2\mathfrak{m}},\frac{\varsigma }{3^2\left(\eta ({\vartheta }_{\mathfrak{j}+4},{\vartheta }_{\mathfrak{j}+2\mathfrak{m}})\eta ({\vartheta }_{\mathfrak{j}+2},{\vartheta }_{\mathfrak{j}+2\mathfrak{m}})\right)}\right)\hfill \\ \hfill & \le \mathcal{E}\left({\vartheta }_{\mathfrak{j}},{\vartheta }_{\mathfrak{j}+1},\frac{\varsigma }{3\left(\beta ({\vartheta }_{\mathfrak{j}},{\vartheta }_{\mathfrak{j}+1})\right)}\right)\diamond \mathcal{E}\left({\vartheta }_{\mathfrak{j}+1},{\vartheta }_{\mathfrak{j}+2},\frac{\varsigma }{3\left(\mathrm{\Gamma }({\vartheta }_{\mathfrak{j}+1},{\vartheta }_{\mathfrak{j}+2})\right)}\right)\hfill \\ \hfill & \diamond \mathcal{E}\left({\vartheta }_{\mathfrak{j}+2},{\vartheta }_{\mathfrak{j}+3},\frac{\varsigma }{3^2\left(\beta ({\vartheta }_{\mathfrak{j}+2},{\vartheta }_{\mathfrak{j}+3})\eta ({\vartheta }_{\mathfrak{j}+2},{\vartheta }_{\mathfrak{j}+2\mathfrak{m}})\right)}\right)\hfill \\ \hfill & \diamond \mathcal{E}\left({\vartheta }_{\mathfrak{j}+3},{\vartheta }_{\mathfrak{j}+4},\frac{\varsigma }{3^2\left(\mathrm{\Gamma }({\vartheta }_{\mathfrak{j}+3},{\vartheta }_{\mathfrak{j}+4})\eta ({\vartheta }_{\mathfrak{j}+2},{\vartheta }_{\mathfrak{j}+2\mathfrak{m}})\right)}\right)\hfill \\ \hfill & \diamond \mathcal{E}\left({\vartheta }_{\mathfrak{j}+4},{\vartheta }_{\mathfrak{j}+5},\frac{\varsigma }{3^3\left(\beta ({\vartheta }_{\mathfrak{j}+4},{\vartheta }_{\mathfrak{j}+5})\eta ({\vartheta }_{\mathfrak{j}+4},{\vartheta }_{\mathfrak{j}+2\mathfrak{m}})\eta ({\vartheta }_{\mathfrak{j}+2},{\vartheta }_{\mathfrak{j}+2\mathfrak{m}})\right)}\right)\hfill \\ \hfill & \diamond \mathcal{E}\left({\vartheta }_{\mathfrak{j}+5},{\vartheta }_{\mathfrak{j}+6},\frac{\varsigma }{3^3\left(\mathrm{\Gamma }({\vartheta }_{\mathfrak{j}+5},{\vartheta }_{\mathfrak{j}+6})\eta ({\vartheta }_{\mathfrak{j}+4},{\vartheta }_{\mathfrak{j}+2\mathfrak{m}})\eta ({\vartheta }_{\mathfrak{j}+2},{\vartheta }_{\mathfrak{j}+2\mathfrak{m}})\right)}\right)\hfill \\ \hfill & \diamond \mathcal{E}\left({\vartheta }_{\mathfrak{j}+6},{\vartheta }_{\mathfrak{j}+2\mathfrak{m}},\frac{\varsigma }{3^3\left(\eta ({\vartheta }_{\mathfrak{j}+6},{\vartheta }_{\mathfrak{j}+2\mathfrak{m}})\eta ({\vartheta }_{\mathfrak{j}+4},{\vartheta }_{\mathfrak{j}+2\mathfrak{m}})\eta ({\vartheta }_{\mathfrak{j}+2},{\vartheta }_{\mathfrak{j}+2\mathfrak{m}})\right)}\right)\hfill \end{array}$$

$$\begin{array}{cc}\hfill \mathcal{E}& ({\vartheta }_{\mathfrak{j}},{\vartheta }_{\mathfrak{j}+2\mathfrak{m}},\varsigma )\le \mathcal{E}\left({\vartheta }_{\mathfrak{j}},{\vartheta }_{\mathfrak{j}+1},\frac{\varsigma }{3\left(\beta ({\vartheta }_{\mathfrak{j}},{\vartheta }_{\mathfrak{j}+1})\right)}\right)\diamond \mathcal{E}\left({\vartheta }_{\mathfrak{j}+1},{\vartheta }_{\mathfrak{j}+2},\frac{\varsigma }{3\left(\mathrm{\Gamma }({\vartheta }_{\mathfrak{j}+1},{\vartheta }_{\mathfrak{j}+2})\right)}\right)\hfill \\ \hfill & \diamond \mathcal{E}\left({\vartheta }_{\mathfrak{j}+2},{\vartheta }_{\mathfrak{j}+3},\frac{\varsigma }{3^2\left(\beta ({\vartheta }_{\mathfrak{j}+2},{\vartheta }_{\mathfrak{j}+3})\eta ({\vartheta }_{\mathfrak{j}+2},{\vartheta }_{\mathfrak{j}+2\mathfrak{m}})\right)}\right)\hfill \\ \hfill & \diamond \mathcal{E}\left({\vartheta }_{\mathfrak{j}+3},{\vartheta }_{\mathfrak{j}+4},\frac{\varsigma }{3^2\left(\mathrm{\Gamma }({\vartheta }_{\mathfrak{j}+3},{\vartheta }_{\mathfrak{j}+4})\eta ({\vartheta }_{\mathfrak{j}+2},{\vartheta }_{\mathfrak{j}+2\mathfrak{m}})\right)}\right)\hfill \\ \hfill & \diamond \mathcal{E}\left({\vartheta }_{\mathfrak{j}+4},{\vartheta }_{\mathfrak{j}+5},\frac{\varsigma }{3^3\left(\beta ({\vartheta }_{\mathfrak{j}+4},{\vartheta }_{\mathfrak{j}+5})\eta ({\vartheta }_{\mathfrak{j}+4},{\vartheta }_{\mathfrak{j}+2\mathfrak{m}})\eta ({\vartheta }_{\mathfrak{j}+2},{\vartheta }_{\mathfrak{j}+2\mathfrak{m}})\right)}\right)\hfill \\ \hfill & \diamond \mathcal{E}\left({\vartheta }_{\mathfrak{j}+5},{\vartheta }_{\mathfrak{j}+6},\frac{\varsigma }{3^3\left(\mathrm{\Gamma }({\vartheta }_{\mathfrak{j}+5},{\vartheta }_{\mathfrak{j}+6})\eta ({\vartheta }_{\mathfrak{j}+4},{\vartheta }_{\mathfrak{j}+2\mathfrak{m}})\eta ({\vartheta }_{\mathfrak{j}+2},{\vartheta }_{\mathfrak{j}+2\mathfrak{m}})\right)}\right)\hfill \\ \hfill & \diamond \mathcal{E}\left({\vartheta }_{\mathfrak{j}+6},{\vartheta }_{\mathfrak{j}+2\mathfrak{m}},\frac{\varsigma }{3^3\left(\eta ({\vartheta }_{\mathfrak{j}+6},{\vartheta }_{\mathfrak{j}+2\mathfrak{m}})\eta ({\vartheta }_{\mathfrak{j}+4},{\vartheta }_{\mathfrak{j}+2\mathfrak{m}})\eta ({\vartheta }_{\mathfrak{j}+2},{\vartheta }_{\mathfrak{j}+2\mathfrak{m}})\right)}\right)\diamond \cdots \diamond \hfill \\ \hfill & \mathcal{E}\left({\vartheta }_{\mathfrak{j}+2\mathfrak{m}-4},{\vartheta }_{\mathfrak{j}+2\mathfrak{m}-3},\frac{\varsigma }{3^{\mathfrak{m}-1}(\beta ({\vartheta }_{\mathfrak{j}+2\mathfrak{m}-4},{\vartheta }_{\mathfrak{j}+2\mathfrak{m}-3})\eta ({\vartheta }_{\mathfrak{j}+2\mathfrak{m}-4},{\vartheta }_{\mathfrak{j}+2\mathfrak{m}+1})\cdots \eta ({\vartheta }_{\mathfrak{j}+2},{\vartheta }_{\mathfrak{j}+2\mathfrak{m}}))}\right)\hfill \end{array}$$

$$\begin{array}{cc}\hfill & \diamond \mathcal{E}\left({\vartheta }_{\mathfrak{j}+2\mathfrak{m}-3},{\vartheta }_{\mathfrak{j}+2\mathfrak{m}-2},\frac{\varsigma }{3^{\mathfrak{m}-1}(\mathrm{\Gamma }({\vartheta }_{\mathfrak{j}+2\mathfrak{m}-3},{\vartheta }_{\mathfrak{j}+2\mathfrak{m}-2})\eta ({\vartheta }_{\mathfrak{j}+2\mathfrak{m}-4},{\vartheta }_{\mathfrak{j}+2\mathfrak{m}})\cdots \eta ({\vartheta }_{\mathfrak{j}+2},{\vartheta }_{\mathfrak{j}+2\mathfrak{m}}))}\right)\hfill \\ \hfill & \diamond \mathcal{E}\left({\vartheta }_{\mathfrak{j}+2\mathfrak{m}-2},{\vartheta }_{\mathfrak{j}+2\mathfrak{m}},\frac{\varsigma }{3^{\mathfrak{m}-1}(\eta ({\vartheta }_{\mathfrak{j}+2\mathfrak{m}-2},{\vartheta }_{\mathfrak{j}+2\mathfrak{m}})\eta ({\vartheta }_{\mathfrak{j}+2\mathfrak{m}-4},{\vartheta }_{\mathfrak{j}+2\mathfrak{m}+1})\cdots \eta ({\vartheta }_{\mathfrak{j}+2},{\vartheta }_{\mathfrak{j}+2\mathfrak{m}}))}\right),\hfill \end{array}$$

and

$$\begin{array}{cc}\hfill \mathcal{G}({\vartheta }_{\mathfrak{j}},{\vartheta }_{\mathfrak{j}+2\mathfrak{m}},\varsigma )& \le \mathcal{G}\left({\vartheta }_{\mathfrak{j}},{\vartheta }_{\mathfrak{j}+1},\frac{\varsigma }{3\left(\beta ({\vartheta }_{\mathfrak{j}},{\vartheta }_{\mathfrak{j}+1})\right)}\right)\diamond \mathcal{G}\left({\vartheta }_{\mathfrak{j}+1},{\vartheta }_{\mathfrak{j}+2},\frac{\varsigma }{3\left(\mathrm{\Gamma }({\vartheta }_{\mathfrak{j}+1},{\vartheta }_{\mathfrak{j}+2})\right)}\right)\hfill \\ \hfill & \diamond \mathcal{G}\left({\vartheta }_{\mathfrak{j}+2},{\vartheta }_{\mathfrak{j}+2\mathfrak{m}},\frac{\varsigma }{3\left(\eta ({\vartheta }_{\mathfrak{j}+2},{\vartheta }_{\mathfrak{j}+2\mathfrak{m}})\right)}\right)\hfill \\ \hfill & \le \mathcal{G}\left({\vartheta }_{\mathfrak{j}},{\vartheta }_{\mathfrak{j}+1},\frac{\varsigma }{3\left(\beta ({\vartheta }_{\mathfrak{j}},{\vartheta }_{\mathfrak{j}+1})\right)}\right)\diamond \mathcal{G}\left({\vartheta }_{\mathfrak{j}+1},{\vartheta }_{\mathfrak{j}+2},\frac{\varsigma }{3\left(\mathrm{\Gamma }({\vartheta }_{\mathfrak{j}+1},{\vartheta }_{\mathfrak{j}+2})\right)}\right)\hfill \\ \hfill & \diamond \mathcal{G}\left({\vartheta }_{\mathfrak{j}+2},{\vartheta }_{\mathfrak{j}+3},\frac{\varsigma }{3^2\left(\beta ({\vartheta }_{\mathfrak{j}+2},{\vartheta }_{\mathfrak{j}+3})\eta ({\vartheta }_{\mathfrak{j}+2},{\vartheta }_{\mathfrak{j}+2\mathfrak{m}})\right)}\right)\hfill \\ \hfill & \diamond \mathcal{G}\left({\vartheta }_{\mathfrak{j}+3},{\vartheta }_{\mathfrak{j}+4},\frac{\varsigma }{3^2\left(\mathrm{\Gamma }({\vartheta }_{\mathfrak{j}+3},{\vartheta }_{\mathfrak{j}+4})\eta ({\vartheta }_{\mathfrak{j}+2},{\vartheta }_{\mathfrak{j}+2\mathfrak{m}})\right)}\right)\hfill \\ \hfill & \diamond \mathcal{G}\left({\vartheta }_{\mathfrak{j}+4},{\vartheta }_{\mathfrak{j}+2\mathfrak{m}},\frac{\varsigma }{3^2\left(\eta ({\vartheta }_{\mathfrak{j}+4},{\vartheta }_{\mathfrak{j}+2\mathfrak{m}})\eta ({\vartheta }_{\mathfrak{j}+2},{\vartheta }_{\mathfrak{j}+2\mathfrak{m}})\right)}\right)\hfill \\ \hfill & \le \mathcal{G}\left({\vartheta }_{\mathfrak{j}},{\vartheta }_{\mathfrak{j}+1},\frac{\varsigma }{3\left(\beta ({\vartheta }_{\mathfrak{j}},{\vartheta }_{\mathfrak{j}+1})\right)}\right)\diamond \mathcal{G}\left({\vartheta }_{\mathfrak{j}+1},{\vartheta }_{\mathfrak{j}+2},\frac{\varsigma }{3\left(\mathrm{\Gamma }({\vartheta }_{\mathfrak{j}+1},{\vartheta }_{\mathfrak{j}+2})\right)}\right)\hfill \\ \hfill & \diamond \mathcal{G}\left({\vartheta }_{\mathfrak{j}+2},{\vartheta }_{\mathfrak{j}+3},\frac{\varsigma }{3^2\left(\beta ({\vartheta }_{\mathfrak{j}+2},{\vartheta }_{\mathfrak{j}+3})\eta ({\vartheta }_{\mathfrak{j}+2},{\vartheta }_{\mathfrak{j}+2\mathfrak{m}})\right)}\right)\hfill \\ \hfill & \diamond \mathcal{G}\left({\vartheta }_{\mathfrak{j}+3},{\vartheta }_{\mathfrak{j}+4},\frac{\varsigma }{3^2\left(\mathrm{\Gamma }({\vartheta }_{\mathfrak{j}+3},{\vartheta }_{\mathfrak{j}+4})\eta ({\vartheta }_{\mathfrak{j}+2},{\vartheta }_{\mathfrak{j}+2\mathfrak{m}})\right)}\right)\hfill \\ \hfill & \diamond \mathcal{G}\left({\vartheta }_{\mathfrak{j}+4},{\vartheta }_{\mathfrak{j}+5},\frac{\varsigma }{3^3\left(\beta ({\vartheta }_{\mathfrak{j}+4},{\vartheta }_{\mathfrak{j}+5})\eta ({\vartheta }_{\mathfrak{j}+4},{\vartheta }_{\mathfrak{j}+2\mathfrak{m}})\eta ({\vartheta }_{\mathfrak{j}+2},{\vartheta }_{\mathfrak{j}+2\mathfrak{m}})\right)}\right)\hfill \\ \hfill & \diamond \mathcal{G}\left({\vartheta }_{\mathfrak{j}+5},{\vartheta }_{\mathfrak{j}+6},\frac{\varsigma }{3^3\left(\mathrm{\Gamma }({\vartheta }_{\mathfrak{j}+5},{\vartheta }_{\mathfrak{j}+6})\eta ({\vartheta }_{\mathfrak{j}+4},{\vartheta }_{\mathfrak{j}+2\mathfrak{m}})\eta ({\vartheta }_{\mathfrak{j}+2},{\vartheta }_{\mathfrak{j}+2\mathfrak{m}})\right)}\right)\hfill \\ \hfill & \diamond \mathcal{G}\left({\vartheta }_{\mathfrak{j}+6},{\vartheta }_{\mathfrak{j}+2\mathfrak{m}},\frac{\varsigma }{3^3\left(\eta ({\vartheta }_{\mathfrak{j}+6},{\vartheta }_{\mathfrak{j}+2\mathfrak{m}})\eta ({\vartheta }_{\mathfrak{j}+4},{\vartheta }_{\mathfrak{j}+2\mathfrak{m}})\eta ({\vartheta }_{\mathfrak{j}+2},{\vartheta }_{\mathfrak{j}+2\mathfrak{m}})\right)}\right)\hfill \end{array}$$

$$\begin{array}{cc}\hfill \mathcal{G}& ({\vartheta }_{\mathfrak{j}},{\vartheta }_{\mathfrak{j}+2\mathfrak{m}},\varsigma )\hfill \\ & \le \mathcal{G}\left({\vartheta }_{\mathfrak{j}},{\vartheta }_{\mathfrak{j}+1},\frac{\varsigma }{3\left(\beta ({\vartheta }_{\mathfrak{j}},{\vartheta }_{\mathfrak{j}+1})\right)}\right)\diamond \mathcal{G}\left({\vartheta }_{\mathfrak{j}+1},{\vartheta }_{\mathfrak{j}+2},\frac{\varsigma }{3\left(\mathrm{\Gamma }({\vartheta }_{\mathfrak{j}+1},{\vartheta }_{\mathfrak{j}+2})\right)}\right)\hfill \\ \hfill & \diamond \mathcal{G}\left({\vartheta }_{\mathfrak{j}+2},{\vartheta }_{\mathfrak{j}+3},\frac{\varsigma }{3^2\left(\beta ({\vartheta }_{\mathfrak{j}+2},{\vartheta }_{\mathfrak{j}+3})\eta ({\vartheta }_{\mathfrak{j}+2},{\vartheta }_{\mathfrak{j}+2\mathfrak{m}})\right)}\right)\hfill \\ \hfill & \diamond \mathcal{G}\left({\vartheta }_{\mathfrak{j}+3},{\vartheta }_{\mathfrak{j}+4},\frac{\varsigma }{3^2\left(\mathrm{\Gamma }({\vartheta }_{\mathfrak{j}+3},{\vartheta }_{\mathfrak{j}+4})\eta ({\vartheta }_{\mathfrak{j}+2},{\vartheta }_{\mathfrak{j}+2\mathfrak{m}})\right)}\right)\hfill \\ \hfill & \diamond \mathcal{G}\left({\vartheta }_{\mathfrak{j}+4},{\vartheta }_{\mathfrak{j}+5},\frac{\varsigma }{3^3\left(\beta ({\vartheta }_{\mathfrak{j}+4},{\vartheta }_{\mathfrak{j}+5})\eta ({\vartheta }_{\mathfrak{j}+4},{\vartheta }_{\mathfrak{j}+2\mathfrak{m}})\eta ({\vartheta }_{\mathfrak{j}+2},{\vartheta }_{\mathfrak{j}+2\mathfrak{m}})\right)}\right)\hfill \\ \hfill & \diamond \mathcal{G}\left({\vartheta }_{\mathfrak{j}+5},{\vartheta }_{\mathfrak{j}+6},\frac{\varsigma }{3^3\left(\mathrm{\Gamma }({\vartheta }_{\mathfrak{j}+5},{\vartheta }_{\mathfrak{j}+6})\eta ({\vartheta }_{\mathfrak{j}+4},{\vartheta }_{\mathfrak{j}+2\mathfrak{m}})\eta ({\vartheta }_{\mathfrak{j}+2},{\vartheta }_{\mathfrak{j}+2\mathfrak{m}})\right)}\right)\hfill \\ \hfill & \diamond \mathcal{G}\left({\vartheta }_{\mathfrak{j}+6},{\vartheta }_{\mathfrak{j}+2\mathfrak{m}},\frac{\varsigma }{3^3\left(\eta ({\vartheta }_{\mathfrak{j}+6},{\vartheta }_{\mathfrak{j}+2\mathfrak{m}})\eta ({\vartheta }_{\mathfrak{j}+4},{\vartheta }_{\mathfrak{j}+2\mathfrak{m}})\eta ({\vartheta }_{\mathfrak{j}+2},{\vartheta }_{\mathfrak{j}+2\mathfrak{m}})\right)}\right)\diamond \cdots \diamond \hfill \\ \hfill & \mathcal{G}\left({\vartheta }_{\mathfrak{j}+2\mathfrak{m}-4},{\vartheta }_{\mathfrak{j}+2\mathfrak{m}-3},\frac{\varsigma }{3^{\mathfrak{m}-1}(\beta ({\vartheta }_{\mathfrak{j}+2\mathfrak{m}-4},{\vartheta }_{\mathfrak{j}+2\mathfrak{m}-3})\eta ({\vartheta }_{\mathfrak{j}+2\mathfrak{m}-4},{\vartheta }_{\mathfrak{j}+2\mathfrak{m}})\cdots \eta ({\vartheta }_{\mathfrak{j}+2},{\vartheta }_{\mathfrak{j}+2\mathfrak{m}}))}\right)\hfill \end{array}$$

$$\begin{array}{cc}\hfill & \diamond \mathcal{G}\left({\vartheta }_{\mathfrak{j}+2\mathfrak{m}-3},{\vartheta }_{\mathfrak{j}+2\mathfrak{m}-2},\frac{\varsigma }{3^{\mathfrak{m}-1}(\mathrm{\Gamma }({\vartheta }_{\mathfrak{j}+2\mathfrak{m}-3},{\vartheta }_{\mathfrak{j}+2\mathfrak{m}-2})\eta ({\vartheta }_{\mathfrak{j}+2\mathfrak{m}-4},{\vartheta }_{\mathfrak{j}+2\mathfrak{m}})\cdots \eta ({\vartheta }_{\mathfrak{j}+2},{\vartheta }_{\mathfrak{j}+2\mathfrak{m}}))}\right)\hfill \\ \hfill & \diamond \mathcal{G}\left({\vartheta }_{\mathfrak{j}+2\mathfrak{m}-2},{\vartheta }_{\mathfrak{j}+2\mathfrak{m}},\frac{\varsigma }{3^{\mathfrak{m}-1}(\eta ({\vartheta }_{\mathfrak{j}+2\mathfrak{m}-2},{\vartheta }_{\mathfrak{j}+2\mathfrak{m}})\eta ({\vartheta }_{\mathfrak{j}+2\mathfrak{m}-4},{\vartheta }_{\mathfrak{j}+2\mathfrak{m}+1})\cdots \eta ({\vartheta }_{\mathfrak{j}+2},{\vartheta }_{\mathfrak{j}+2\mathfrak{m}}))}\right).\hfill \end{array}$$

Using (3) in the above inequalities, we deduce

$$\begin{array}{cc}\hfill \mathcal{W}({\vartheta }_{\mathfrak{j}},{\vartheta }_{\mathfrak{j}+2\mathfrak{m}},\varsigma )& \ge \frac{1}{\frac{{\rho }^{\mathfrak{j}}}{\mathcal{W}\left({\vartheta }_0,{\vartheta }_1,\frac{\varsigma }{3\left(\beta ({\vartheta }_{\mathfrak{j}},{\vartheta }_{\mathfrak{j}+1})\right)}\right)}+(1-{\rho }^{\mathfrak{j}})}\star \frac{1}{\frac{{\rho }^{\mathfrak{j}+1}}{\mathcal{W}\left({\vartheta }_0,{\vartheta }_1,\frac{\varsigma }{3\left(\mathrm{\Gamma }({\vartheta }_{\mathfrak{j}+1},{\vartheta }_{\mathfrak{j}+2})\right)}\right)}+(1-{\rho }^{\mathfrak{j}+1})}\hfill \\ \hfill & \star \frac{1}{\frac{{\rho }^{\mathfrak{j}+2}}{\mathcal{W}\left({\vartheta }_0,{\vartheta }_1,\frac{\varsigma }{3^2\left(\beta ({\vartheta }_{\mathfrak{j}+2},{\vartheta }_{\mathfrak{j}+3})\eta ({\vartheta }_{\mathfrak{j}+2},{\vartheta }_{\mathfrak{j}+2\mathfrak{m}})\right)}\right)}+(1-{\rho }^{\mathfrak{j}+2})}\hfill \\ \hfill & \star \frac{1}{\frac{{\rho }^{\mathfrak{j}+3}}{\mathcal{W}\left({\vartheta }_0,{\vartheta }_1,\frac{\varsigma }{3^2\left(\mathrm{\Gamma }({\vartheta }_{\mathfrak{j}+3},{\vartheta }_{\mathfrak{j}+4})\eta ({\vartheta }_{\mathfrak{j}+2},{\vartheta }_{\mathfrak{j}+2\mathfrak{m}})\right)}\right)}+(1-{\rho }^{\mathfrak{j}+3})}\hfill \\ \hfill & \star \frac{1}{\frac{{\rho }^{\mathfrak{j}+4}}{\mathcal{W}\left({\vartheta }_0,{\vartheta }_1,\frac{\varsigma }{3^3\left(\beta ({\vartheta }_{\mathfrak{j}+4},{\vartheta }_{\mathfrak{j}+5})\eta ({\vartheta }_{\mathfrak{j}+4},{\vartheta }_{\mathfrak{j}+2\mathfrak{m}})\eta ({\vartheta }_{\mathfrak{j}+2},{\vartheta }_{\mathfrak{j}+2\mathfrak{m}})\right)}\right)}+(1-{\rho }^{\mathfrak{j}+4})}\hfill \\ \hfill & \star \frac{1}{\frac{{\rho }^{\mathfrak{j}+5}}{\mathcal{W}\left({\vartheta }_0,{\vartheta }_1,\frac{\varsigma }{3^3\left(\mathrm{\Gamma }({\vartheta }_{\mathfrak{j}+5},{\vartheta }_{\mathfrak{j}+6})\eta ({\vartheta }_{\mathfrak{j}+4},{\vartheta }_{\mathfrak{j}+2\mathfrak{m}})\eta ({\vartheta }_{\mathfrak{j}+2},{\vartheta }_{\mathfrak{j}+2\mathfrak{m}})\right)}\right)}+(1-{\rho }^{\mathfrak{j}+5})}\hfill \\ \hfill & \star \frac{1}{\frac{{\rho }^{\mathfrak{j}+6}}{\mathcal{W}\left({\vartheta }_0,{\vartheta }_1,\frac{\varsigma }{3^3\left(\eta ({\vartheta }_{\mathfrak{j}+6},{\vartheta }_{\mathfrak{j}+2\mathfrak{m}})\eta ({\vartheta }_{\mathfrak{j}+4},{\vartheta }_{\mathfrak{j}+2\mathfrak{m}})\eta ({\vartheta }_{\mathfrak{j}+2},{\vartheta }_{\mathfrak{j}+2\mathfrak{m}})\right)}\right)}+(1-{\rho }^{\mathfrak{j}+6})}\star \cdots \star \hfill \\ \hfill & \star \frac{1}{\frac{{\rho }^{\mathfrak{j}+2\mathfrak{m}-4}}{\mathcal{W}\left({\vartheta }_0,{\vartheta }_1,\frac{\varsigma }{3^{\mathfrak{m}-1}(\beta ({\vartheta }_{\mathfrak{j}+2\mathfrak{m}-4},{\vartheta }_{\mathfrak{j}+2\mathfrak{m}-3})\eta ({\vartheta }_{\mathfrak{j}+2\mathfrak{m}-4},{\vartheta }_{\mathfrak{j}+2\mathfrak{m}})\cdots \eta ({\vartheta }_{\mathfrak{j}+2},{\vartheta }_{\mathfrak{j}+2\mathfrak{m}}))}\right)}+(1-{\rho }^{\mathfrak{j}+2\mathfrak{m}-4})}\hfill \\ \hfill & \star \frac{1}{\frac{{\rho }^{\mathfrak{j}+2\mathfrak{m}-3}}{\mathcal{W}\left({\vartheta }_0,{\vartheta }_1,\frac{\varsigma }{3^{\mathfrak{m}-1}(\mathrm{\Gamma }({\vartheta }_{\mathfrak{j}+2\mathfrak{m}-3},{\vartheta }_{\mathfrak{j}+2\mathfrak{m}-2})\eta ({\vartheta }_{\mathfrak{j}+2\mathfrak{m}-4},{\vartheta }_{\mathfrak{j}+2\mathfrak{m}})\cdots \eta ({\vartheta }_{\mathfrak{j}+2},{\vartheta }_{\mathfrak{j}+2\mathfrak{m}}))}\right)}+(1-{\rho }^{\mathfrak{j}+2\mathfrak{m}-3})}\hfill \\ \hfill & \star \frac{1}{\frac{{\rho }^{\mathfrak{j}+2\mathfrak{m}-2}}{\mathcal{W}\left({\vartheta }_0,{\vartheta }_1,\frac{\varsigma }{3^{\mathfrak{m}-1}(\eta ({\vartheta }_{\mathfrak{j}+2\mathfrak{m}-2},{\vartheta }_{\mathfrak{j}+2\mathfrak{m}})\eta ({\vartheta }_{\mathfrak{j}+2\mathfrak{m}-4},{\vartheta }_{\mathfrak{j}+2\mathfrak{m}+1})\cdots \eta ({\vartheta }_{\mathfrak{j}+2},{\vartheta }_{\mathfrak{j}+2\mathfrak{m}}))}\right)}+(1-{\rho }^{\mathfrak{j}+2\mathfrak{m}-2})}\hfill \end{array}$$

$$\begin{array}{cc}\hfill \mathcal{G}& ({\vartheta }_{\mathfrak{j}},{\vartheta }_{\mathfrak{j}+2\mathfrak{m}},\varsigma )\hfill \\ & \le {\rho }^{\mathfrak{j}}\mathcal{G}\left({\vartheta }_0,{\vartheta }_1,\frac{\varsigma }{3\left(\beta ({\vartheta }_{\mathfrak{j}},{\vartheta }_{\mathfrak{j}+1})\right)}\right)\diamond {\rho }^{\mathfrak{j}+1}\mathcal{G}\left({\vartheta }_{\mathfrak{j}+1},{\vartheta }_{\mathfrak{j}+2},\frac{\varsigma }{3\left(\mathrm{\Gamma }({\vartheta }_{\mathfrak{j}+1},{\vartheta }_{\mathfrak{j}+2})\right)}\right)\hfill \\ \hfill & \diamond {\rho }^{\mathfrak{j}+2}\mathcal{G}\left({\vartheta }_0,{\vartheta }_1,\frac{\varsigma }{3^2\left(\beta ({\vartheta }_{\mathfrak{j}+2},{\vartheta }_{\mathfrak{j}+3})\eta ({\vartheta }_{\mathfrak{j}+2},{\vartheta }_{\mathfrak{j}+2\mathfrak{m}})\right)}\right)\hfill \\ \hfill & \diamond {\rho }^{\mathfrak{j}+3}\mathcal{G}\left({\vartheta }_0,{\vartheta }_1,\frac{\varsigma }{3^2\left(\mathrm{\Gamma }({\vartheta }_{\mathfrak{j}+3},{\vartheta }_{\mathfrak{j}+4})\eta ({\vartheta }_{\mathfrak{j}+2},{\vartheta }_{\mathfrak{j}+2\mathfrak{m}})\right)}\right)\hfill \end{array}$$

$$\begin{array}{cc}\hfill & \diamond {\rho }^{\mathfrak{j}+4}\mathcal{G}\left({\vartheta }_0,{\vartheta }_1,\frac{\varsigma }{3^3\left(\beta ({\vartheta }_{\mathfrak{j}+4},{\vartheta }_{\mathfrak{j}+5})\eta ({\vartheta }_{\mathfrak{j}+4},{\vartheta }_{\mathfrak{j}+2\mathfrak{m}})\eta ({\vartheta }_{\mathfrak{j}+2},{\vartheta }_{\mathfrak{j}+2\mathfrak{m}})\right)}\right)\hfill \\ \hfill & \diamond {\rho }^{\mathfrak{j}+5}\mathcal{G}\left({\vartheta }_0,{\vartheta }_1,\frac{\varsigma }{3^3\left(\mathrm{\Gamma }({\vartheta }_{\mathfrak{j}+5},{\vartheta }_{\mathfrak{j}+6})\eta ({\vartheta }_{\mathfrak{j}+4},{\vartheta }_{\mathfrak{j}+2\mathfrak{m}})\eta ({\vartheta }_{\mathfrak{j}+2},{\vartheta }_{\mathfrak{j}+2\mathfrak{m}})\right)}\right)\hfill \\ \hfill & \diamond {\rho }^{\mathfrak{j}+6}\mathcal{G}\left({\vartheta }_0,{\vartheta }_1,\frac{\varsigma }{3^3\left(\eta ({\vartheta }_{\mathfrak{j}+6},{\vartheta }_{\mathfrak{j}+2\mathfrak{m}})\eta ({\vartheta }_{\mathfrak{j}+4},{\vartheta }_{\mathfrak{j}+2\mathfrak{m}})\eta ({\vartheta }_{\mathfrak{j}+2},{\vartheta }_{\mathfrak{j}+2\mathfrak{m}})\right)}\right)\diamond \cdots \diamond \hfill \\ \hfill & {\rho }^{\mathfrak{j}+2\mathfrak{m}-4}\mathcal{G}\left({\vartheta }_0,{\vartheta }_1,\frac{\varsigma }{3^{\mathfrak{m}-1}(\beta ({\vartheta }_{\mathfrak{j}+2\mathfrak{m}-4},{\vartheta }_{\mathfrak{j}+2\mathfrak{m}-3})\eta ({\vartheta }_{\mathfrak{j}+2\mathfrak{m}-4},{\vartheta }_{\mathfrak{j}+2\mathfrak{m}})\cdots \eta ({\vartheta }_{\mathfrak{j}+2},{\vartheta }_{\mathfrak{j}+2\mathfrak{m}}))}\right)\hfill \\ \hfill & \diamond {\rho }^{\mathfrak{j}+2\mathfrak{m}-3}\mathcal{G}\left({\vartheta }_0,{\vartheta }_1,\frac{\varsigma }{3^{\mathfrak{m}-1}(\mathrm{\Gamma }({\vartheta }_{\mathfrak{j}+2\mathfrak{m}-3},{\vartheta }_{\mathfrak{j}+2\mathfrak{m}-2})\eta ({\vartheta }_{\mathfrak{j}+2\mathfrak{m}-4},{\vartheta }_{\mathfrak{j}+2\mathfrak{m}})\cdots \eta ({\vartheta }_{\mathfrak{j}+2},{\vartheta }_{\mathfrak{j}+2\mathfrak{m}}))}\right)\hfill \\ \hfill & \diamond {\rho }^{\mathfrak{j}+2\mathfrak{m}-2}\mathcal{G}\left({\vartheta }_0,{\vartheta }_1,\frac{\varsigma }{3^{\mathfrak{m}-1}(\eta ({\vartheta }_{\mathfrak{j}+2\mathfrak{m}-2},{\vartheta }_{\mathfrak{j}+2\mathfrak{m}})\eta ({\vartheta }_{\mathfrak{j}+2\mathfrak{m}-4},{\vartheta }_{\mathfrak{j}+2\mathfrak{m}})\cdots \eta ({\vartheta }_{\mathfrak{j}+2},{\vartheta }_{\mathfrak{j}+2\mathfrak{m}}))}\right).\hfill \end{array}$$

$$\begin{array}{cc}\hfill \mathcal{E}& ({\vartheta }_{\mathfrak{j}},{\vartheta }_{\mathfrak{j}+2\mathfrak{m}},\varsigma )\hfill \\ & \le {\rho }^{\mathfrak{j}}\mathcal{E}\left({\vartheta }_0,{\vartheta }_1,\frac{\varsigma }{3\left(\beta ({\vartheta }_{\mathfrak{j}},{\vartheta }_{\mathfrak{j}+1})\right)}\right)\diamond {\rho }^{\mathfrak{j}+1}\mathcal{E}\left({\vartheta }_{\mathfrak{j}+1},{\vartheta }_{\mathfrak{j}+2},\frac{\varsigma }{3\left(\mathrm{\Gamma }({\vartheta }_{\mathfrak{j}+1},{\vartheta }_{\mathfrak{j}+2})\right)}\right)\hfill \\ \hfill & \diamond {\rho }^{\mathfrak{j}+2}\mathcal{E}\left({\vartheta }_0,{\vartheta }_1,\frac{\varsigma }{3^2\left(\beta ({\vartheta }_{\mathfrak{j}+2},{\vartheta }_{\mathfrak{j}+3})\eta ({\vartheta }_{\mathfrak{j}+2},{\vartheta }_{\mathfrak{j}+2\mathfrak{m}})\right)}\right)\hfill \\ \hfill & \diamond {\rho }^{\mathfrak{j}+3}\mathcal{E}\left({\vartheta }_0,{\vartheta }_1,\frac{\varsigma }{3^2\left(\mathrm{\Gamma }({\vartheta }_{\mathfrak{j}+3},{\vartheta }_{\mathfrak{j}+4})\eta ({\vartheta }_{\mathfrak{j}+2},{\vartheta }_{\mathfrak{j}+2\mathfrak{m}})\right)}\right)\hfill \\ \hfill & \diamond {\rho }^{\mathfrak{j}+4}\mathcal{E}\left({\vartheta }_0,{\vartheta }_1,\frac{\varsigma }{3^3\left(\beta ({\vartheta }_{\mathfrak{j}+4},{\vartheta }_{\mathfrak{j}+5})\eta ({\vartheta }_{\mathfrak{j}+4},{\vartheta }_{\mathfrak{j}+2\mathfrak{m}})\eta ({\vartheta }_{\mathfrak{j}+2},{\vartheta }_{\mathfrak{j}+2\mathfrak{m}})\right)}\right)\hfill \\ \hfill & \diamond {\rho }^{\mathfrak{j}+5}\mathcal{E}\left({\vartheta }_0,{\vartheta }_1,\frac{\varsigma }{3^3\left(\mathrm{\Gamma }({\vartheta }_{\mathfrak{j}+5},{\vartheta }_{\mathfrak{j}+6})\eta ({\vartheta }_{\mathfrak{j}+4},{\vartheta }_{\mathfrak{j}+2\mathfrak{m}})\eta ({\vartheta }_{\mathfrak{j}+2},{\vartheta }_{\mathfrak{j}+2\mathfrak{m}})\right)}\right)\hfill \\ \hfill & \diamond {\rho }^{\mathfrak{j}+6}\mathcal{E}\left({\vartheta }_0,{\vartheta }_1,\frac{\varsigma }{3^3\left(\eta ({\vartheta }_{\mathfrak{j}+6},{\vartheta }_{\mathfrak{j}+2\mathfrak{m}})\eta ({\vartheta }_{\mathfrak{j}+4},{\vartheta }_{\mathfrak{j}+2\mathfrak{m}})\eta ({\vartheta }_{\mathfrak{j}+2},{\vartheta }_{\mathfrak{j}+2\mathfrak{m}})\right)}\right)\diamond \cdots \diamond \hfill \\ \hfill & {\rho }^{\mathfrak{j}+2\mathfrak{m}-4}\mathcal{E}\left({\vartheta }_0,{\vartheta }_1,\frac{\varsigma }{3^{\mathfrak{m}-1}(\beta ({\vartheta }_{\mathfrak{j}+2\mathfrak{m}-4},{\vartheta }_{\mathfrak{j}+2\mathfrak{m}-3})\eta ({\vartheta }_{\mathfrak{j}+2\mathfrak{m}-4},{\vartheta }_{\mathfrak{j}+2\mathfrak{m}})\cdots \eta ({\vartheta }_{\mathfrak{j}+2},{\vartheta }_{\mathfrak{j}+2\mathfrak{m}}))}\right)\hfill \\ \hfill & \diamond {\rho }^{\mathfrak{j}+2\mathfrak{m}-3}\mathcal{E}\left({\vartheta }_0,{\vartheta }_1,\frac{\varsigma }{3^{\mathfrak{m}-1}(\mathrm{\Gamma }({\vartheta }_{\mathfrak{j}+2\mathfrak{m}-3},{\vartheta }_{\mathfrak{j}+2\mathfrak{m}-2})\eta ({\vartheta }_{\mathfrak{j}+2\mathfrak{m}-4},{\vartheta }_{\mathfrak{j}+2\mathfrak{m}})\cdots \eta ({\vartheta }_{\mathfrak{j}+2},{\vartheta }_{\mathfrak{j}+2\mathfrak{m}}))}\right)\hfill \\ \hfill & \diamond {\rho }^{\mathfrak{j}+2\mathfrak{m}-2}\mathcal{E}\left({\vartheta }_0,{\vartheta }_1,\frac{\varsigma }{3^{\mathfrak{m}-1}(\eta ({\vartheta }_{\mathfrak{j}+2\mathfrak{m}-2},{\vartheta }_{\mathfrak{j}+2\mathfrak{m}})\eta ({\vartheta }_{\mathfrak{j}+2\mathfrak{m}-4},{\vartheta }_{\mathfrak{j}+2\mathfrak{m}})\cdots \eta ({\vartheta }_{\mathfrak{j}+2},{\vartheta }_{\mathfrak{j}+2\mathfrak{m}}))}\right).\hfill \end{array}$$

As $\mathfrak{j}\to +\infty$, we deduce

$$\begin{array}{cc}\hfill \underset{\mathfrak{j}\to +\infty }{lim}\mathcal{W}({\vartheta }_{\mathfrak{j}},{\vartheta }_{\mathfrak{j}+\mathfrak{q}},\varsigma )& =1\star 1\star \cdots \star =1,\hfill \\ \hfill \underset{\mathfrak{j}\to +\infty }{lim}\mathcal{E}({\vartheta }_{\mathfrak{j}},{\vartheta }_{\mathfrak{j}+\mathfrak{q}},\varsigma )& =0\diamond 0\diamond \cdots \diamond 0=0,\hfill \end{array}$$

and

$$\begin{array}{cc}\hfill \underset{\mathfrak{j}\to +\infty }{lim}\mathcal{G}({\vartheta }_{\mathfrak{j}},{\vartheta }_{\mathfrak{j}+\mathfrak{q}},\varsigma )& =0\diamond 0\diamond \cdots \diamond 0=0.\hfill \end{array}$$

Therefore, $\left\{{\vartheta }_{\mathfrak{j}}\right\}$ is a Cauchy sequence. Since $(\Im ,\mathcal{W},\mathcal{E},\mathcal{G},\star ,\diamond )$ is a complete neutrosophic rectangular triple-controlled metric space, we have

$$\begin{array}{c}\hfill \underset{\mathfrak{j}\to +\infty }{lim}{\vartheta }_{\mathfrak{j}}=\vartheta .\end{array}$$

Now, we examine that ϑ is a fixed point of $\mathfrak{p}$. Utilizing (v), (x) and (xv), we obtain

$$\begin{array}{cc}\hfill \frac{1}{\mathcal{W}(\mathfrak{p}{\vartheta }_{\mathfrak{j}},\mathfrak{p}\vartheta ,\varsigma )}-1& \le \rho \left[\frac{1}{\mathcal{W}({\vartheta }_{\mathfrak{j}},\vartheta ,\varsigma )}-1\right]=\frac{\rho }{\mathcal{W}({\vartheta }_{\mathfrak{j}},\vartheta ,\varsigma )}-\rho \hfill \\ \hfill & \Rightarrow \frac{1}{\frac{\rho }{\mathcal{W}({\vartheta }_{\mathfrak{j}},\vartheta ,\varsigma )}+(1-\rho )}\le \mathcal{W}(\mathfrak{p}{\vartheta }_{\mathfrak{j}},\mathfrak{p}\vartheta ,\varsigma ).\hfill \end{array}$$

Using the above inequality, we obtain

$$\begin{array}{cc}\hfill \mathcal{W}(\vartheta ,\mathfrak{p}\vartheta ,\varsigma )& \ge \mathcal{W}\left(\vartheta ,{\vartheta }_{\mathfrak{j}},\frac{\varsigma }{3\beta (\vartheta ,{\vartheta }_{\mathfrak{j}})}\right)\star \mathcal{W}\left({\vartheta }_{\mathfrak{j}},{\vartheta }_{\mathfrak{j}+1},\frac{\varsigma }{3\mathrm{\Gamma }({\vartheta }_{\mathfrak{j}},{\vartheta }_{\mathfrak{j}+1})}\right)\star \mathcal{W}\left({\vartheta }_{\mathfrak{j}+1},\mathfrak{p}\vartheta ,\frac{\varsigma }{3\eta ({\vartheta }_{\mathfrak{j}+1},\mathfrak{p}\vartheta )}\right)\hfill \\ \hfill & \ge \mathcal{W}\left(\vartheta ,{\vartheta }_{\mathfrak{j}},\frac{\varsigma }{3\beta (\vartheta ,{\vartheta }_{\mathfrak{j}})}\right)\star \mathcal{W}\left(\mathfrak{p}{\vartheta }_{\mathfrak{j}-1},\mathfrak{p}{\vartheta }_{\mathfrak{j}},\frac{\varsigma }{3\mathrm{\Gamma }({\vartheta }_{\mathfrak{j}},{\vartheta }_{\mathfrak{j}+1})}\right)\star \mathcal{W}\left(\mathfrak{p}{\vartheta }_{\mathfrak{j}},\mathfrak{p}\vartheta ,\frac{\varsigma }{3\eta ({\vartheta }_{\mathfrak{j}+1},\mathfrak{p}\vartheta )}\right)\hfill \\ \hfill & \ge \mathcal{W}\left(\vartheta ,{\vartheta }_{\mathfrak{j}},\frac{\varsigma }{\left(3\beta (\vartheta ,{\vartheta }_{\mathfrak{j}})\right)}\right)\star \frac{1}{\frac{{\rho }^{\mathfrak{j}}}{\mathcal{W}({\vartheta }_0,{\vartheta }_1,\frac{\varsigma }{3\mathrm{\Gamma }({\vartheta }_{\mathfrak{j}},{\vartheta }_{\mathfrak{j}+1})})+(1-{\rho }^{\mathfrak{j}})}}\star \frac{1}{\frac{\rho }{\mathcal{W}\left({\vartheta }_{\mathfrak{j}},\vartheta ,\frac{\varsigma }{3\eta ({\vartheta }_{\mathfrak{j}+1},\mathfrak{p}\vartheta )}\right)+(1-\rho )}}\hfill \\ \hfill & \to 1\star 1\star 1=1\mathrm{as}\mathfrak{j}\to +\infty ,\hfill \end{array}$$

$$\begin{array}{cc}\hfill \mathcal{E}(\vartheta ,\mathfrak{p}\vartheta ,\varsigma )& \le \mathcal{E}\left(\vartheta ,{\vartheta }_{\mathfrak{j}},\frac{\varsigma }{3\beta (\vartheta ,{\vartheta }_{\mathfrak{j}})}\right)\diamond \mathcal{E}\left({\vartheta }_{\mathfrak{j}},{\vartheta }_{\mathfrak{j}+1},\frac{\varsigma }{3\mathrm{\Gamma }({\vartheta }_{\mathfrak{j}},{\vartheta }_{\mathfrak{j}+1})}\right)\diamond \mathcal{E}\left({\vartheta }_{\mathfrak{j}+1},\mathfrak{p}\vartheta ,\frac{\varsigma }{3\eta ({\vartheta }_{\mathfrak{j}+1},\mathfrak{p}\vartheta )}\right)\hfill \\ \hfill & \le \mathcal{E}\left(\vartheta ,{\vartheta }_{\mathfrak{j}},\frac{\varsigma }{3\beta (\vartheta ,{\vartheta }_{\mathfrak{j}})}\right)\diamond \mathcal{E}\left(\mathfrak{p}{\vartheta }_{\mathfrak{j}-1},\mathfrak{p}{\vartheta }_{\mathfrak{j}},\frac{\varsigma }{3\mathrm{\Gamma }({\vartheta }_{\mathfrak{j}},{\vartheta }_{\mathfrak{j}+1})}\right)\diamond \mathcal{E}\left(\mathfrak{p}{\vartheta }_{\mathfrak{j}},\mathfrak{p}\vartheta ,\frac{\varsigma }{3\eta ({\vartheta }_{\mathfrak{j}+1},\mathfrak{p}\vartheta )}\right)\hfill \\ \hfill & \le \mathcal{E}\left(\vartheta ,{\vartheta }_{\mathfrak{j}},\frac{\varsigma }{3\beta (\vartheta ,{\vartheta }_{\mathfrak{j}})}\right)\diamond {\rho }^{\mathfrak{j}-1}\mathcal{E}\left({\vartheta }_{\mathfrak{j}-1},{\vartheta }_{\mathfrak{j}},\frac{\varsigma }{3\mathrm{\Gamma }({\vartheta }_{\mathfrak{j}},{\vartheta }_{\mathfrak{j}+1})}\right)\diamond \rho \mathcal{E}\left({\vartheta }_{\mathfrak{j}},\vartheta ,\frac{\varsigma }{3\eta ({\vartheta }_{\mathfrak{j}+1},\vartheta )}\right)\hfill \\ \hfill & \to 0\diamond 0\diamond 0=0\mathrm{as}\mathfrak{j}\to +\infty ,\hfill \end{array}$$

and

$$\begin{array}{cc}\hfill \mathcal{G}(\vartheta ,\mathfrak{p}\vartheta ,\varsigma )& \le \mathcal{G}\left(\vartheta ,{\vartheta }_{\mathfrak{j}},\frac{\varsigma }{3\beta (\vartheta ,{\vartheta }_{\mathfrak{j}})}\right)\diamond \mathcal{G}\left({\vartheta }_{\mathfrak{j}},{\vartheta }_{\mathfrak{j}+1},\frac{\varsigma }{3\mathrm{\Gamma }({\vartheta }_{\mathfrak{j}},{\vartheta }_{\mathfrak{j}+1})}\right)\diamond \mathcal{G}\left({\vartheta }_{\mathfrak{j}+1},\mathfrak{p}\vartheta ,\frac{\varsigma }{3\eta ({\vartheta }_{\mathfrak{j}+1},\mathfrak{p}\vartheta )}\right)\hfill \\ \hfill & \le \mathcal{G}\left(\vartheta ,{\vartheta }_{\mathfrak{j}},\frac{\varsigma }{3\beta (\vartheta ,{\vartheta }_{\mathfrak{j}})}\right)\diamond \mathcal{G}\left(\mathfrak{p}{\vartheta }_{\mathfrak{j}-1},\mathfrak{p}{\vartheta }_{\mathfrak{j}},\frac{\varsigma }{3\mathrm{\Gamma }({\vartheta }_{\mathfrak{j}},{\vartheta }_{\mathfrak{j}+1})}\right)\diamond \mathcal{G}\left(\mathfrak{p}{\vartheta }_{\mathfrak{j}},\mathfrak{p}\vartheta ,\frac{\varsigma }{3\eta ({\vartheta }_{\mathfrak{j}+1},\mathfrak{p}\vartheta )}\right)\hfill \\ \hfill & \le \mathcal{G}\left(\vartheta ,{\vartheta }_{\mathfrak{j}},\frac{\varsigma }{3\beta (\vartheta ,{\vartheta }_{\mathfrak{j}})}\right)\diamond {\rho }^{\mathfrak{j}-1}\mathcal{G}\left({\vartheta }_{\mathfrak{j}-1},{\vartheta }_{\mathfrak{j}},\frac{\varsigma }{3\mathrm{\Gamma }({\vartheta }_{\mathfrak{j}},{\vartheta }_{\mathfrak{j}+1})}\right)\diamond \rho \mathcal{G}\left({\vartheta }_{\mathfrak{j}},\vartheta ,\frac{\varsigma }{3\eta ({\vartheta }_{\mathfrak{j}+1},\vartheta )}\right)\hfill \\ \hfill & \to 0\diamond 0\diamond 0=0\mathrm{as}\mathfrak{j}\to +\infty .\hfill \end{array}$$

Hence, $\mathfrak{p}\vartheta =\vartheta$. For uniqueness, consider that $\mathfrak{p}\kappa =\kappa$ for some $\kappa \in \Im$. Then,

$$\begin{array}{cc}\hfill \frac{1}{\mathcal{W}(\vartheta ,\kappa ,\varsigma )}-1& =\frac{1}{\mathcal{W}(\mathfrak{p}\vartheta ,\mathfrak{p}\kappa ,\varsigma )}-1\hfill \\ \hfill & \le \rho \left[\frac{1}{\mathcal{W}(\vartheta ,\kappa ,\varsigma )}-1\right]<\frac{1}{\mathcal{W}(\vartheta ,\kappa ,\varsigma )}-1,\hfill \end{array}$$

$$\begin{array}{c}\hfill \mathcal{E}(\vartheta ,\kappa ,\varsigma )=\mathcal{E}(\mathfrak{p}\vartheta ,\mathfrak{p}\kappa ,\varsigma )\le \rho \mathcal{E}(\vartheta ,\kappa ,\varsigma )<\mathcal{E}(\vartheta ,\kappa ,\varsigma ),\end{array}$$

and

$$\begin{array}{c}\hfill \mathcal{G}(\vartheta ,\kappa ,\varsigma )=\mathcal{G}(\mathfrak{p}\vartheta ,\mathfrak{p}\kappa ,\varsigma )\le \rho \mathcal{G}(\vartheta ,\kappa ,\varsigma )<\mathcal{G}(\vartheta ,\kappa ,\varsigma ),\end{array}$$

which are contradictions.

Thus, we have $\mathcal{W}(\vartheta ,\kappa ,\varsigma )=1,\mathcal{E}(\vartheta ,\kappa ,\varsigma )=0$ and $\mathcal{G}(\vartheta ,\kappa ,\varsigma )=0$, and accordingly, $\vartheta =\kappa$. □

Example 3.

Let $\Im =[0,1]$ and $\beta ,\mathrm{\Gamma },\eta :\Im \times \Im \to [1,+\infty )$ be three non-comparable functions given by

$$\begin{array}{c}\hfill \beta (\vartheta ,\mathcal{Q})=\left\{\begin{array}{cc}1\hfill & \mathit{if}\vartheta =\mathcal{Q},\hfill \\ \frac{1+max\{\vartheta ,\mathcal{Q}\}}{1+min\{\vartheta ,\mathcal{Q}\}}\hfill & \mathit{if}\vartheta \ne \mathcal{Q},\hfill \end{array}\right.\end{array}$$

$$\begin{array}{c}\hfill \mathrm{\Gamma }(\vartheta ,\mathcal{Q})=\left\{\begin{array}{cc}1\hfill & \mathit{if}\vartheta =\mathcal{Q},\hfill \\ \frac{1+max\{{\vartheta }^2,{\mathcal{Q}}^2\}}{1+min\{{\vartheta }^2,{\mathcal{Q}}^2\}}\hfill & \mathit{if}\vartheta \ne \mathcal{Q},\hfill \end{array}\right.\end{array}$$

and

$$\begin{array}{c}\hfill \eta (\vartheta ,\mathcal{Q})=\left\{\begin{array}{cc}1\hfill & \mathit{if}\vartheta =\mathcal{Q},\hfill \\ \frac{1+max\{{\vartheta }^2,\mathcal{Q}\}}{1+min\{{\vartheta }^2,\mathcal{Q}\}}\hfill & \mathit{if}\vartheta \ne \mathcal{Q}.\hfill \end{array}\right.\end{array}$$

Define $\mathcal{W},\mathcal{E},\mathcal{G}:\Im \times \Im \times (0,+\infty )\to [0,1]$ as

$$\begin{array}{cc}\hfill \mathcal{W}(\vartheta ,\mathcal{Q},\varsigma )& =\frac{\varsigma }{\varsigma +|\vartheta -\mathcal{Q}|},\hfill \\ \hfill \mathcal{E}(\vartheta ,\mathcal{Q},\varsigma )& =\frac{|\vartheta -\mathcal{Q}|}{\varsigma +|\vartheta -\mathcal{Q}|},\hfill \\ \hfill \mathcal{G}(\vartheta ,\mathcal{Q},\varsigma )& =\frac{|\vartheta -\mathcal{Q}|}{\varsigma }.\hfill \end{array}$$

Then, $(\Im ,\mathcal{W},\mathcal{E},\mathcal{G},\star ,\diamond )$ is a complete neutrosophic rectangular triple-controlled metric space with continuous t-norm $\wp \star \sigma =\wp \sigma$ and continuous t-co-norm $\wp \diamond \sigma =max\{\wp ,\sigma \}$.

Define $\mathfrak{p}:\Im \to \Im$ by $\mathfrak{p}\left(\vartheta \right)=\frac{1-3^{-\vartheta }}{5}$ and take $\rho \in [\frac{1}{2},1)$, then

$$\begin{array}{cc}\hfill \mathcal{W}(\mathfrak{p}\vartheta ,\mathfrak{p}\mathcal{Q},\rho \varsigma )& =\mathcal{W}\left(\frac{1-3^{-\vartheta }}{5},\frac{1-3^{-\mathcal{Q}}}{5},\rho \varsigma \right)\hfill \\ \hfill & =\frac{\rho \varsigma }{\rho \varsigma +|\frac{1-3^{-\vartheta }}{5}-\frac{1-3^{-\mathcal{Q}}}{5}|}=\frac{\rho \varsigma }{\rho \varsigma +\frac{|3^{-\vartheta }-3^{-\mathcal{Q}}|}{5}}\hfill \\ \hfill & \ge \frac{\rho \varsigma }{\rho \varsigma +\frac{|\vartheta -\mathcal{Q}|}{5}}=\frac{5\rho \varsigma }{5\rho \varsigma +|\vartheta -\mathcal{Q}|}\ge \frac{\varsigma }{\varsigma +|\vartheta -\mathcal{Q}|}=\mathcal{W}(\vartheta ,\mathcal{Q},\varsigma ),\hfill \end{array}$$

$$\begin{array}{cc}\hfill \mathcal{E}(\mathfrak{p}\vartheta ,\mathfrak{p}\mathcal{Q},\rho \varsigma )& =\mathcal{E}\left(\frac{1-3^{-\vartheta }}{5},\frac{1-3^{-\mathcal{Q}}}{5},\rho \varsigma \right)\hfill \\ \hfill & =\frac{|\frac{1-3^{-\vartheta }}{5}-\frac{1-3^{-\mathcal{Q}}}{5}|}{\rho \varsigma +|\frac{1-3^{-\vartheta }}{5}-\frac{1-3^{-\mathcal{Q}}}{5}|}=\frac{\frac{|3^{-\vartheta }-3^{-\mathcal{Q}}|}{5}}{\rho \varsigma +\frac{|3^{-\vartheta }-3^{-\mathcal{Q}}|}{5}}\hfill \\ \hfill & =\frac{|3^{-\vartheta }-3^{-\mathcal{Q}}|}{5\rho \varsigma +|3^{-\vartheta }-3^{-\mathcal{Q}}|}\le \frac{|\vartheta -\mathcal{Q}|}{5\rho \varsigma +|\vartheta -\mathcal{Q}|}\le \frac{|\vartheta -\mathcal{Q}|}{\varsigma +|\vartheta -\mathcal{Q}|}=\mathcal{E}(\vartheta ,\mathcal{Q},\varsigma ),\hfill \end{array}$$

and

$$\begin{array}{cc}\hfill \mathcal{G}(\mathfrak{p}\vartheta ,\mathfrak{p}\mathcal{Q},\rho \varsigma )& =\mathcal{G}\left(\frac{1-3^{-\vartheta }}{5},\frac{1-3^{-\mathcal{Q}}}{5},\rho \varsigma \right)\hfill \\ \hfill & =\frac{|\frac{1-3^{-\vartheta }}{5}-\frac{1-3^{-\mathcal{Q}}}{5}|}{\rho \varsigma }=\frac{\frac{|3^{-\vartheta }-3^{-\mathcal{Q}}|}{5}}{\rho \varsigma }\hfill \\ \hfill & =\frac{|3^{-\vartheta }-3^{-\mathcal{Q}}|}{5\rho \varsigma }\le \frac{|\vartheta -\mathcal{Q}|}{5\rho \varsigma }\le \frac{|\vartheta -\mathcal{Q}|}{\varsigma }=\mathcal{G}(\vartheta ,\mathcal{Q},\varsigma ).\hfill \end{array}$$

Thus, all conditions of Theorem 1 are satisfied with 0 as the unique fixed point for $\mathfrak{p}$.

Application

Let $\Im =\mathcal{C}\left(\right[\mathfrak{c},\mathfrak{a}],\mathbb{R})$, the set of real-valued continuous functions defined on $[\mathfrak{c},\mathfrak{a}]$.

Consider the integral equation

$$\begin{array}{c}\hfill \vartheta \left(\tau \right)=\wedge \left(\tau \right)+\delta {\int }_{\mathfrak{c}}^{\mathfrak{a}}\mho (\tau ,\mathfrak{v})\vartheta \left(\tau \right)\nu \mathfrak{v}\mathrm{for}\tau ,\mathfrak{v}\in [\mathfrak{c},\mathfrak{a}]\end{array}$$

(10)

where $\delta >0,\wedge \left(\mathfrak{v}\right)$ is a fuzzy function of $\mathfrak{v}:\mathfrak{v}\in [\mathfrak{c},\mathfrak{a}]$ and $\mho :\mathcal{C}([\mathfrak{c},\mathfrak{a}]\times \mathbb{R})\to {\mathbb{R}}^{+}$. Define $\mathcal{W}$ and $\mathcal{E}$ by means of

$$\begin{array}{cc}\hfill \mathcal{W}\left(\vartheta \right(\tau ),\mathcal{Q}(\tau ),\varsigma )& =\underset{\tau \in [\mathfrak{c},\mathfrak{a}]}{sup}\frac{\varsigma }{\varsigma +\left|\vartheta \right(\tau )-\mathcal{Q}(\tau \left)\right|}\mathrm{for}\mathrm{all}\vartheta ,\mathcal{Q}\in \Im \mathrm{and}\varsigma >0,\hfill \\ \hfill \mathcal{E}\left(\vartheta \right(\tau ),\mathcal{Q}(\tau ),\varsigma )& =1-\underset{\tau \in [\mathfrak{c},\mathfrak{a}]}{sup}\frac{\varsigma }{\varsigma +\left|\vartheta \right(\tau )-\mathcal{Q}(\tau \left)\right|}\mathrm{for}\mathrm{all}\vartheta ,\mathcal{Q}\in \Im \mathrm{and}\varsigma >0,\hfill \end{array}$$

and

$$\begin{array}{cc}\hfill \mathcal{G}\left(\vartheta \right(\tau ),\mathcal{Q}(\tau ),\varsigma )& =\underset{\tau \in [\mathfrak{c},\mathfrak{a}]}{sup}\frac{\left|\vartheta \right(\tau )-\mathcal{Q}(\tau \left)\right|}{\varsigma }\mathrm{for}\mathrm{all}\vartheta ,\mathcal{Q}\in \Im \mathrm{and}\varsigma >0,\hfill \end{array}$$

with the continuous t-norm and continuous t-co-norm defined by $\wp \star \sigma =\wp ·\sigma$ and $\wp \diamond \sigma =max\{\wp ,\sigma \}$, respectively. Define $\beta ,\mathrm{\Gamma },\eta :\Im \times \Im \to [1,+\infty )$ as

$$\begin{array}{c}\hfill \beta (\vartheta ,\mathcal{Q})=\left\{\begin{array}{cc}1\hfill & \mathrm{if}\vartheta =\mathcal{Q},\hfill \\ \frac{1+max\{\vartheta ,\mathcal{Q}\}}{1+min\{\vartheta ,\mathcal{Q}\}}\hfill & \mathrm{if}\vartheta \ne \mathcal{Q},\hfill \end{array}\right.\end{array}$$

$$\begin{array}{c}\hfill \mathrm{\Gamma }(\vartheta ,\mathcal{Q})=\left\{\begin{array}{cc}1\hfill & \mathrm{if}\vartheta =\mathcal{Q},\hfill \\ \frac{1+max\{{\vartheta }^2,{\mathcal{Q}}^2\}}{1+min\{{\vartheta }^2,{\mathcal{Q}}^2\}}\hfill & \mathrm{if}\vartheta \ne \mathcal{Q},\hfill \end{array}\right.\end{array}$$

and

$$\begin{array}{c}\hfill \eta (\vartheta ,\mathcal{Q})=\left\{\begin{array}{cc}1\hfill & \mathrm{if}\vartheta =\mathcal{Q},\hfill \\ \frac{1+max\{{\vartheta }^2,\mathcal{Q}\}}{1+min\{{\vartheta }^2,\mathcal{Q}\}}\hfill & \mathrm{if}\vartheta \ne \mathcal{Q}.\hfill \end{array}\right.\end{array}$$

Then, $(\Im ,\mathcal{W},\mathcal{E},\mathcal{G},\star ,\diamond )$ is a complete neutrosophic rectangular triple-controlled metric space. Let $|\mho (\tau ,\mathfrak{v}\left)\vartheta \right(\tau )-\mho (\tau ,\mathfrak{v}\left)\mathcal{Q}\right(\tau \left)\right|\le \left|\vartheta \right(\tau )-\mathcal{Q}(\tau \left)\right|$ for $\vartheta ,\mathcal{Q}\in \Im ,\rho \in (0,1)$ and for all$\tau ,\mathfrak{v}\in [\mathfrak{c},\mathfrak{a}]$. Moreover, let $\mho (\tau ,\mathfrak{v})\left(\delta {\int }_{\mathfrak{c}}^{\mathfrak{a}}\nu \mathfrak{v}\right)\le \rho <1$. Then, the integral Equation (10) has a unique solution.

Proof. 

Define $\mathfrak{p}:\Im \to \Im$ by

$$\begin{array}{c}\hfill \mathfrak{p}\vartheta \left(\tau \right)=\wedge \left(\tau \right)+\delta {\int }_{\mathfrak{c}}^{\mathfrak{a}}\mho (\tau ,\mathfrak{v})\vartheta \left(\tau \right)\nu \mathfrak{v}\mathrm{for}\mathrm{all}\tau ,\mathfrak{v}\in [\mathfrak{c},\mathfrak{a}].\end{array}$$

Now, for all $\vartheta ,\mathcal{Q}\in \Im$, we deduce

$$\begin{array}{cc}\hfill \mathcal{W}\left(\mathfrak{p}\vartheta \right(\tau ),& \mathfrak{p}\mathcal{Q}\left(\tau \right),\rho \varsigma )=\underset{\tau \in [\mathfrak{c},\mathfrak{a}]}{sup}\frac{\rho \varsigma }{\rho \varsigma +\left|\mathfrak{p}\vartheta \right(\tau )-\mathfrak{p}\mathcal{Q}(\tau \left)\right|}\hfill \\ \hfill & =\underset{\tau \in [\mathfrak{c},\mathfrak{a}]}{sup}\frac{\rho \varsigma }{\rho \varsigma +|\wedge \left(\tau \right)+\delta {\int }_{\mathfrak{c}}^{\mathfrak{a}}\mho (\tau ,\mathfrak{v})\vartheta \left(\tau \right)\nu \mathfrak{v}-\wedge \left(\tau \right)-\delta {\int }_{\mathfrak{c}}^{\mathfrak{a}}\mho (\tau ,\mathfrak{v})\vartheta \left(\tau \right)\nu \mathfrak{v}|}\hfill \\ \hfill & =\underset{\tau \in [\mathfrak{c},\mathfrak{a}]}{sup}\frac{\rho \varsigma }{\rho \varsigma +|\delta {\int }_{\mathfrak{c}}^{\mathfrak{a}}\mho (\tau ,\mathfrak{v})\vartheta \left(\tau \right)\nu \mathfrak{v}-\delta {\int }_{\mathfrak{c}}^{\mathfrak{a}}\mho (\tau ,\mathfrak{v})\vartheta \left(\tau \right)\nu \mathfrak{v}|}\hfill \\ \hfill & =\underset{\tau \in [\mathfrak{c},\mathfrak{a}]}{sup}\frac{\rho \varsigma }{\rho \varsigma +|\mho (\tau ,\mathfrak{v})\vartheta \left(\tau \right)-\mho (\tau ,\mathfrak{v})\mathcal{Q}\left(\tau \right)|\left(\delta {\int }_{\mathfrak{c}}^{\mathfrak{a}}\nu \mathfrak{v}\right)}\hfill \\ \hfill & \ge \underset{\tau \in [\mathfrak{c},\mathfrak{a}]}{sup}\frac{\varsigma }{\varsigma +\left|\vartheta \right(\tau )-\mathcal{Q}(\tau \left)\right|}\hfill \\ \hfill & \ge \mathcal{W}\left(\vartheta \right(\tau ),\mathcal{Q}(\tau ),\varsigma ),\hfill \end{array}$$

$$\begin{array}{cc}\hfill \mathcal{E}\left(\mathfrak{p}\vartheta \right(\tau ),& \mathfrak{p}\mathcal{Q}\left(\tau \right),\rho \varsigma )=1-\underset{\tau \in [\mathfrak{c},\mathfrak{a}]}{sup}\frac{\rho \varsigma }{\rho \varsigma +\left|\mathfrak{p}\vartheta \right(\tau )-\mathfrak{p}\mathcal{Q}(\tau \left)\right|}\hfill \\ \hfill & =1-\underset{\tau \in [\mathfrak{c},\mathfrak{a}]}{sup}\frac{\rho \varsigma }{\rho \varsigma +|\wedge \left(\tau \right)+\delta {\int }_{\mathfrak{c}}^{\mathfrak{a}}\mho (\tau ,\mathfrak{v})\vartheta \left(\tau \right)\nu \mathfrak{v}-\wedge \left(\tau \right)-\delta {\int }_{\mathfrak{c}}^{\mathfrak{a}}\mho (\tau ,\mathfrak{v})\vartheta \left(\tau \right)\nu \mathfrak{v}|}\hfill \\ \hfill & =1-\underset{\tau \in [\mathfrak{c},\mathfrak{a}]}{sup}\frac{\rho \varsigma }{\rho \varsigma +|\delta {\int }_{\mathfrak{c}}^{\mathfrak{a}}\mho (\tau ,\mathfrak{v})\vartheta \left(\tau \right)\nu \mathfrak{v}-\delta {\int }_{\mathfrak{c}}^{\mathfrak{a}}\mho (\tau ,\mathfrak{v})\vartheta \left(\tau \right)\nu \mathfrak{v}|}\hfill \\ \hfill & =1-\underset{\tau \in [\mathfrak{c},\mathfrak{a}]}{sup}\frac{\rho \varsigma }{\rho \varsigma +|\mho (\tau ,\mathfrak{v})\vartheta \left(\tau \right)-\mho (\tau ,\mathfrak{v})\mathcal{Q}\left(\tau \right)|\left(\delta {\int }_{\mathfrak{c}}^{\mathfrak{a}}\nu \mathfrak{v}\right)}\hfill \\ \hfill & \le 1-\underset{\tau \in [\mathfrak{c},\mathfrak{a}]}{sup}\frac{\varsigma }{\varsigma +\left|\vartheta \right(\tau )-\mathcal{Q}(\tau \left)\right|}\hfill \\ \hfill & \le \mathcal{E}\left(\vartheta \right(\tau ),\mathcal{Q}(\tau ),\varsigma ),\hfill \end{array}$$

and

$$\begin{array}{cc}\hfill \mathcal{G}\left(\mathfrak{p}\vartheta \right(\tau ),& \mathfrak{p}\mathcal{Q}\left(\tau \right),\rho \varsigma )=\underset{\tau \in [\mathfrak{c},\mathfrak{a}]}{sup}\frac{\rho \varsigma }{\rho \varsigma +\left|\mathfrak{p}\vartheta \right(\tau )-\mathfrak{p}\mathcal{Q}(\tau \left)\right|}\hfill \\ \hfill & =\underset{\tau \in [\mathfrak{c},\mathfrak{a}]}{sup}\frac{\rho \varsigma }{\rho \varsigma +|\wedge \left(\tau \right)+\delta {\int }_{\mathfrak{c}}^{\mathfrak{a}}\mho (\tau ,\mathfrak{v})\vartheta \left(\tau \right)\nu \mathfrak{v}-\wedge \left(\tau \right)-\delta {\int }_{\mathfrak{c}}^{\mathfrak{a}}\mho (\tau ,\mathfrak{v})\vartheta \left(\tau \right)\nu \mathfrak{v}|}\hfill \\ \hfill & =\underset{\tau \in [\mathfrak{c},\mathfrak{a}]}{sup}\frac{\rho \varsigma }{\rho \varsigma +|\delta {\int }_{\mathfrak{c}}^{\mathfrak{a}}\mho (\tau ,\mathfrak{v})\vartheta \left(\tau \right)\nu \mathfrak{v}-\delta {\int }_{\mathfrak{c}}^{\mathfrak{a}}\mho (\tau ,\mathfrak{v})\vartheta \left(\tau \right)\nu \mathfrak{v}|}\hfill \\ \hfill & =\underset{\tau \in [\mathfrak{c},\mathfrak{a}]}{sup}\frac{\rho \varsigma }{\rho \varsigma +|\mho (\tau ,\mathfrak{v})\vartheta \left(\tau \right)-\mho (\tau ,\mathfrak{v})\mathcal{Q}\left(\tau \right)|\left(\delta {\int }_{\mathfrak{c}}^{\mathfrak{a}}\nu \mathfrak{v}\right)}\hfill \\ \hfill & \le \underset{\tau \in [\mathfrak{c},\mathfrak{a}]}{sup}\frac{\varsigma }{\varsigma +\left|\vartheta \right(\tau )-\mathcal{Q}(\tau \left)\right|}\hfill \\ \hfill & \le \mathcal{W}\left(\vartheta \right(\tau ),\mathcal{Q}(\tau ),\varsigma ),\hfill \end{array}$$

Thus, all the conditions of Theorem (1) are satisfied and operator $\mathfrak{p}$ has a unique fixed point. This proves that (10) has a unique solution. □

Example 4.

Consider the integral equation

$$\begin{array}{c}\hfill \vartheta \left(\tau \right)=|cos\tau |+\frac{1}{7}{\int }_0^1\mathfrak{v}\vartheta \left(\mathfrak{v}\right)\nu \mathfrak{v},\mathit{for}\mathit{all}\mathfrak{v}\in [0,1].\end{array}$$

Then, it has a solution in ℑ.

Proof. 

Let $\mathfrak{p}:\Im \to \Im$ be defined by

$$\begin{array}{c}\hfill \mathfrak{p}\vartheta \left(\tau \right)=|cos\tau |+\frac{1}{7}{\int }_0^1\mathfrak{v}\vartheta \left(\mathfrak{v}\right)\nu \mathfrak{v},\end{array}$$

and set $\mho (\tau ,\mathfrak{v})\vartheta \left(\tau \right)=\frac{1}{7}\mathfrak{v}\vartheta \left(\mathfrak{v}\right)$ and $\mho (\tau ,\mathfrak{v})\mathcal{Q}\left(\tau \right)=\frac{1}{7}\mathfrak{v}\mathcal{Q}\left(\mathfrak{v}\right)$, where $\vartheta ,\mathcal{Q}\in \Im$, and for all $\tau ,\mathfrak{v}\in [0,1]$. Then, we have

$$\begin{array}{cc}\hfill |\mho (\tau ,\mathfrak{v}\left)\vartheta \right(\tau )-\mho (\tau ,\mathfrak{v}\left)\mathcal{Q}\right(\tau \left)\right|& =|\frac{1}{7}\mathfrak{v}\vartheta \left(\mathfrak{v}\right)-\frac{1}{7}\mathfrak{v}\mathcal{Q}\left(\mathfrak{v}\right)|\hfill \\ \hfill & =\frac{\mathfrak{v}}{7}|\vartheta \left(\mathfrak{v}\right)-\mathcal{Q}\left(\mathfrak{v}\right)|\le |\vartheta \left(\mathfrak{v}\right)-\mathcal{Q}\left(\mathfrak{v}\right)|.\hfill \end{array}$$

Furthermore, see that $\frac{1}{7}{\int }_0^1\mathfrak{v}\nu \mathfrak{v}=\frac{1}{7}\left(\frac{{\left(1\right)}^2}{2}-\frac{{\left(0\right)}^2}{2}\right)=\rho <1$, where $\delta =\frac{1}{7}$. It is easy to verify all other conditions of the preceding application and hence a solution exists in ℑ. □

Indeed, the closed form of the unique solution for the integral equation of Example 4 using the software is found to be

$$\mathfrak{a}\left(\tau \right)=|cos\tau |+0.0574712,$$

and the graph of the solution is shown in Figure 1.

4. Conclusions

In the above work, we have defined neutrosophic rectangular triple-controlled metric spaces and defined some topological properties of such spaces. We have proven the existence of a unique fixed point under various contractive conditions in these spaces, supported with non-trivial examples. To substantiate the derived results, we have presented the existence of a unique analytical solution to the Fredholm integral equations, outperforming those present in the literature. We have also presented the closed form of the unique solution to Example 4 to supplement the derived results. It is an open problem to explore the possibility of extending our results thorugh various contractive conditions, such as Meir-Keeler contractions, Ciric-type contractions or by using more generalized verisons of the defined spaces.