Solving Some Integral and Fractional Differential Equations via Neutrosophic Pentagonal Metric Space

Abstract

In this paper, we first introduce the notion of neutrosophic pentagonal metric space. We prove several interesting results for some classes contraction mappings and prove some fixed point theorems in neutrosophic pentagonal metric space. Finally, we prove the uniqueness and existence of the integral equation and fractional differential equation to support our main result.

1. Introduction

A fuzzy set is a category of items with a continuum of membership levels between zero and one. The concept of a fuzzy set was first introduced by Zadeh [1], who also provided a useful starting point for the development of a conceptual framework that, while similar to the framework used for sets in many ways, is more general and may have a wider range of applications, particularly in the fields of pattern classification and information processing. The study of statistical metric spaces and an examination of the continuous characteristics of the distance function were both continued by Schweizer and Sklar in [2]. Fuzzy metric spaces (shortly, Fuzzy MS) were proposed by Kramosil and Michálek in [3], who also expanded on the notion of convergence that is typically used to determine whether a generalization is appropriate. On fuzzy double-controlled MSs, Azmi [4] developed the novel idea of $(\alpha -\mathrm{\Pi })$-fuzzy contractive mappings and illustrated various fixed-point results. In the framework of extended fuzzy b-MSs, various generalized fixed point findings of Banach and Ćirić type are established by Rome et al. [5]. A Hausdorff fuzzy b-MS is described by Batul et al. [6]. A few fixed point results for multivalued mappings in G-complete fuzzy b-MSs that satisfy an appropriate contractiveness criterion are established using the novel idea. Numerous fixed point theorems in fuzzy b-MSs make up Rakić et al. [7]. They provided a necessary condition for a sequence to be Cauchy in the fuzzy b-MS, which was a significant result. By using a control function $\alpha (x,y)$ of the right-hand side of the b-triangle inequality, Mlaiki [8] created a new extension of b-MSs known as controlled metric type spaces. Controlled fuzzy MS is a brand-new development of Sezen’s [9] work on fuzzy metrics. Additionally, they demonstrated a new fixed point theorem and a Banach-type fixed point theorem for some fulfilling self-mappings. see [10,11,12,13,14,15,16]. Grabiec [17] extended two fixed point theorems of Banach and Edelstein to contractive mappings of complete and compact fuzzy MSs, respectively. In order to established some fixed point theorems, Rehman et al. [18] defined $\alpha$-admissible and $\alpha$-$\mathrm{\Pi }$-fuzzy cone contraction in fuzzy cone MS. Finally, he used theoretical results to show that a nonlinear integral equation has a solution.

Park [19] created an intuitionistic fuzzy MS delta membership and nonmembership functions. The concept of an intuitionistic fuzzy b-MS was first proposed by Konwar [20], who also demonstrated a number of fixed point theorems. Neutosophic MSs, which are initialized to handle membership, nonmembership, and naturalness, were introduced by Kirişci and Simsek in their [21] paper. Simsek and Kirişci [22] demonstrated various fixed point results in the context of neutrosophic MSs. Fixed point findings in neutrosophic MSs were demonstrated by Sowndrarajan et al. [23]. Itoh [24] showed a usage for random differential equations in Banach spaces.

In this article, we introduce the neutrosophic pentagonal MS (also known as NPMS) and demonstrate a few fixed point conclusions. The following are the primary goals of this work:

• To introduce the notion of neutrosophic pentagonal MS;
• To prove several fixed-point theorems for contraction mappings;
• Show the existence of a unique solution of an integral equation;
• Show the existence of a unique solution of a fractional differential equation.

2. Preliminaries

Here, we’ll go over some fundamental terms that will be useful for the key results.

Definition 1

([19]). Let $★:{[0,1]}^2\to [0,1]$ be a binary operation is said to be a continuous triangle norm if:

(1) 

$\widehat{\omega }★\breve{h}=\breve{h}★\widehat{\omega },\forall \widehat{\omega },\breve{h}\in [0,1]$;

(2) 

★ is continuous;

(3) 

$\widehat{\omega }★1=\widehat{\omega },\forall \widehat{\omega }\in [0,1]$;

(4) 

$(\widehat{\omega }★\breve{h})★\tilde{\mu }=\widehat{\omega }★(\breve{h}★\tilde{\mu })$, ∀ $\widehat{\omega },\breve{h},\tilde{\mu }\in [0,1]$;

(5) 

If $\widehat{\omega }\le \tilde{\mu }$ and $\breve{h}\le \varsigma$, ∀  $\widehat{\omega },\breve{h},\tilde{\mu },\varsigma \in [0,1]$, then $\widehat{\omega }★\breve{h}\le \tilde{\mu }★\varsigma$.

Definition 2

([19]). Let $•:{[0,1]}^2\to [0,1]$ be a binary operation is said to be a continuous triangle co-norm if:

(1) 

$\widehat{\omega }•\breve{h}=\breve{h}•\widehat{\omega }$, ∀ $\widehat{\omega },\breve{h}\in [0,1]$;

(2) 

• is continuous;

(3) 

$\widehat{\omega }•0=0$, ∀ $\widehat{\omega }\in [0,1]$;

(4) 

$(\widehat{\omega }•\breve{h})•\tilde{\mu }=\widehat{\omega }•(\breve{h}•\tilde{\mu })$, ∀ $\widehat{\omega },\breve{h},\tilde{\mu }\in [0,1]$;

(5) 

If $\widehat{\omega }\le \tilde{\mu }$ and  $\tilde{\mu }\le \varsigma$, with $\widehat{\omega },\breve{h},\tilde{\mu },\varsigma \in [0,1]$, then $\widehat{\omega }•\breve{h}\le \tilde{\mu }•\varsigma$.

Definition 3

([24]). Let $\chi ,\mathbf{F}:\mathit{K}\times \mathit{K}\to [1,+\infty )$ be given two non-comparable functions, if $\partial :\mathit{K}\times \mathit{K}\to [0,+\infty )$ satisfies axioms:

(a) 

$\partial (\mathit{U},\mathcal{W})=0$ iff $\mathit{U}=\mathcal{W}$;

(b) 

$\partial (\mathit{U},\mathcal{W})=\partial (\mathcal{W},\mathit{U})$;

(c) 

$\partial (\mathit{U},\mathcal{W})\le \chi (\mathit{U},\pounds )\partial (\mathit{U},\pounds )+\mathbf{F}(\pounds ,\mathcal{W})\partial (\pounds ,\mathcal{W})$;

 ∀ $\mathit{U},\mathcal{W},\pounds \in \mathit{K}$, then, $(\mathit{K},\partial )$ is known to be a double controlled MS (shortly, DCMS).

Definition 4

([25]). Let $\mathit{K}\ne \varnothing$ and $\chi ,\mathbf{F}:\mathit{K}\times \mathit{K}\to [1,+\infty )$ be known non-comparable functions. And ★ is a continuous t-norm also Λ be a fuzzy set on $\mathit{K}\times \mathit{K}\times (0,+\infty )$ is said to be fuzzy double controlled metric on $\mathit{K}$, ∀  $\mathit{U},\mathcal{W},\pounds \in \mathit{K}$ if:

(i)   

$\mathrm{\Lambda }(\mathit{U},\mathcal{W},0)=0$;

(ii)  

$\mathrm{\Lambda }(\mathit{U},\mathcal{W},\mathrm{\Gamma })=1$∀$\mathrm{\Gamma }>0$, ⇔$\mathit{U}=\mathcal{W}$;

(iii) 

$\mathrm{\Lambda }(\mathit{U},\mathcal{W},\mathrm{\Gamma })=\mathrm{\Lambda }(\mathcal{W},\mathit{U},\mathrm{\Gamma })$;

(iv)  

$\mathrm{\Lambda }(\mathit{U},\pounds ,\mathrm{\Gamma }+\mathrm{\Gamma })\ge \mathrm{\Lambda }\left(\mathit{U},\mathcal{W},\frac{\mathrm{\Gamma }}{\chi (\mathit{U},\mathcal{W})}\right)★\mathrm{\Lambda }\left(\mathcal{W},\pounds ,\frac{\mathrm{\Gamma }}{\mathbf{F}(\mathcal{W},\pounds )}\right)$;

(v)   

$\mathrm{\Lambda }(\mathit{U},\mathcal{W},·):(0,+\infty )\to [0,1]$ is continuous on left.

Then, $(\mathit{K},\mathrm{\Lambda },\mathcal{N},★)$ is known to be a fuzzy DCMS.

Definition 5

([20]). Let $\mathit{K}\ne \varnothing$. Let ★, • are the continuous t-norm, continuous t-co-norm respectively, $♭\ge 1$ and $\mathrm{\Lambda },\mathcal{N}$ be fuzzy sets on $\mathit{K}\times \mathit{K}\times (0,+\infty )$. If $(\mathit{K},\mathrm{\Lambda },\mathcal{N},★,•)$ fullfils all $\mathit{U},\mathcal{W}\in \mathit{K}$ and $\check{g},\mathrm{\Gamma }>0$:

(I)     

$\mathrm{\Lambda }(\mathit{U},\mathcal{W},\mathrm{\Gamma })+\mathrm{\Pi }(\mathit{U},\mathcal{W},\mathrm{\Gamma })\le 1$;

(II)    

$0<\mathrm{\Lambda }(\mathit{U},\mathcal{W},\mathrm{\Gamma })$;

(III)   

$\mathrm{\Lambda }(\mathit{U},\mathcal{W},\mathrm{\Gamma })=1\iff \mathit{U}=\mathcal{W}$;

(IV)    

$\mathrm{\Lambda }(\mathit{U},\mathcal{W},\mathrm{\Gamma })=\mathrm{\Lambda }(\mathcal{W},\mathit{U},\mathrm{\Gamma })$;

(V)     

$\mathrm{\Lambda }(\mathit{U},\pounds ,♭(\mathrm{\Gamma }+\check{g}))\ge \mathrm{\Lambda }(\mathit{U},\mathcal{W},\mathrm{\Gamma })★\mathrm{\Lambda }(\mathcal{W},\pounds ,\check{g})$;

(VI)    

$\mathrm{\Lambda }(\mathit{U},\mathcal{W},·)$ is a non-decreasing function of ${\mathbb{R}}^{+}$ and ${lim}_{\mathrm{\Gamma }\to +\infty }\mathrm{\Lambda }(\mathit{U},\mathcal{W},\mathrm{\Gamma })=1$;

(VII)   

$0<\mathrm{\Pi }(\mathit{U},\mathcal{W},\mathrm{\Gamma })$;

(VIII)  

$\mathrm{\Pi }(\mathit{U},\mathcal{W},\mathrm{\Gamma })=0\iff \mathit{U}=\mathcal{W}$;

(IX)    

$\mathrm{\Pi }(\mathit{U},\mathcal{W},\mathrm{\Gamma })=\mathrm{\Pi }(\mathcal{W},\mathit{U},\mathrm{\Gamma })$;

(X)     

$\mathrm{\Pi }(\mathit{U},\pounds ,♭(\mathrm{\Gamma }+\check{g}))\le \mathrm{\Pi }(\mathit{U},\mathcal{W},\mathrm{\Gamma })•\mathrm{\Pi }(\mathcal{W},\pounds ,\check{g})$;

(XI)    

$\mathrm{\Pi }(\mathit{U},\mathcal{W},·)$ is a non-increasing function of ${\mathbb{R}}^{+}$ and ${lim}_{\mathrm{\Gamma }\to +\infty }\mathrm{\Pi }(\mathit{U},\mathcal{W},\mathrm{\Gamma })=0$,

Then, $(\mathit{K},\mathrm{\Lambda },\mathrm{\Pi },★,•)$ is an intuitionistic fuzzy ♭-MS.

Definition 6

([21]). Let $\mathit{K}\ne \varnothing$, ★, • are the continuous t-norm, continuous t-co-norm respectively, and $\mathrm{\Lambda },\mathrm{\Pi },\mathcal{S}$ are neutrosophic sets (shortly, N-sets) on $\mathit{K}\times \mathit{K}\times (0,+\infty )$ is known to be a neutrosophic metric on $\mathit{K}$, if for all $\mathit{U},\mathcal{W},\pounds \in \mathit{K}$, the following axioms are fulfilled:

(1)   

$\mathrm{\Lambda }(\mathit{U},\mathcal{W},\mathrm{\Gamma })+\mathrm{\Pi }(\mathit{U},\mathcal{W},\mathrm{\Gamma })+\mathcal{S}(\mathit{U},\mathcal{W},\mathrm{\Gamma })\le 3$;

(2)   

$0<\mathrm{\Lambda }(\mathit{U},\mathcal{W},\mathrm{\Gamma })$;

(3)   

$\mathrm{\Lambda }(\mathit{U},\mathcal{W},\mathrm{\Gamma })=1$∀$\mathrm{\Gamma }>0$, ⇔$\mathit{U}=\mathcal{W}$;

(4)   

$\mathrm{\Lambda }(\mathit{U},\mathcal{W},\mathrm{\Gamma })=\mathrm{\Lambda }(\mathcal{W},\mathit{U},\mathrm{\Gamma })$;

(5)   

$\mathrm{\Lambda }(\mathit{U},\pounds ,\mathrm{\Gamma }+\check{g})\ge \mathrm{\Lambda }(\mathit{U},\mathcal{W},\mathrm{\Gamma })★\mathrm{\Lambda }(\mathcal{W},\pounds ,\check{g})$;

(6)   

$\mathrm{\Lambda }(\mathit{U},\mathcal{W},·):(0,+\infty )\to [0,1]$ is continuous and ${lim}_{\mathrm{\Gamma }\to +\infty }\mathrm{\Lambda }(\mathit{U},\mathcal{W},\mathrm{\Gamma })=1$;

(7)   

$1<\mathrm{\Pi }(\mathit{U},\mathcal{W},\mathrm{\Gamma })$;

(8)   

$\mathrm{\Pi }(\mathit{U},\mathcal{W},\mathrm{\Gamma })=0$∀$\mathrm{\Gamma }>0$, ⇔$\mathit{U}=\mathcal{W}$;

(9)   

$\mathrm{\Pi }(\mathit{U},\mathcal{W},\mathrm{\Gamma })=\mathrm{\Pi }(\mathcal{W},\mathit{U},\mathrm{\Gamma })$;

(10) 

$\mathrm{\Pi }(\mathit{U},\pounds ,\mathrm{\Gamma }+\check{g})\le \mathrm{\Pi }(\mathit{U},\mathcal{W},\mathrm{\Gamma })•\mathrm{\Pi }(\mathcal{W},\pounds ,\check{g})$;

(11) 

$\mathrm{\Pi }(\mathit{U},\mathcal{W},·):(0,+\infty )\to [0,1]$ is continuous and ${lim}_{\mathrm{\Gamma }\to +\infty }\mathrm{\Pi }(\mathit{U},\mathcal{W},\mathrm{\Gamma })=0$;

(12) 

$1<\mathcal{S}(\mathit{U},\mathcal{W},\mathrm{\Gamma })$;

(13) 

$\mathcal{S}(\mathit{U},\mathcal{W},\mathrm{\Gamma })=0$∀$\mathrm{\Gamma }>0$, ⇔$\mathit{U}=\mathcal{W}$;

(14) 

$\mathcal{S}(\mathit{U},\mathcal{W},\mathrm{\Gamma })=\mathcal{S}(\mathcal{W},\mathit{U},\mathrm{\Gamma })$;

(15) 

$\mathcal{S}(\mathit{U},\pounds ,\mathrm{\Gamma }+\check{g})\le \mathcal{S}(\mathit{U},\mathcal{W},\mathrm{\Gamma })•\mathcal{S}(\mathcal{W},\pounds ,\check{g})$;

(16) 

$\mathcal{S}(\mathit{U},\mathcal{W},·):(0,+\infty )\to [0,1]$ is continuous and ${lim}_{\mathrm{\Gamma }\to +\infty }\mathcal{S}(\mathit{U},\mathcal{W},\mathrm{\Gamma })=0$;

(17) 

If $\mathrm{\Gamma }\le 0$, then $\mathrm{\Lambda }(\mathit{U},\mathcal{W},\mathrm{\Gamma })=0,\mathrm{\Pi }(\mathit{U},\mathcal{W},\mathrm{\Gamma })=0$;

Then, $(\mathit{K},\mathrm{\Lambda },\mathrm{\Pi },\mathcal{S},★,•)$ is known to be a neutrosophic MS.

In this article, we define NPMS and demonstrate fixed point theorems.

3. Main Results

This section presents NPMS and illustrates some fixed-point theorems.

Definition 7.

Let $\mathit{K}\ne \varnothing$ and function $\chi :\mathit{K}\times \mathit{K}\to [1,+\infty )$ be non-comparable, ★, • are the continuous t-norm, continuous t-co-norm respectively, and $\mathrm{\Lambda },\mathrm{\Pi },\mathcal{M}$ be N-sets on $\mathit{K}\times \mathit{K}\times (0,+\infty )$ is known to be a neutrosophic pentagonal metric on $\mathit{K}$, if for any $\mathit{U},\pounds \in \mathit{K}$ and all distinct $\check{x},\mathcal{W},\pounds \in \mathit{K}$, the following axioms are fulfilled:

(i)    

$\mathrm{\Lambda }(\mathit{U},\mathcal{W},\mathrm{\Gamma })+\mathrm{\Pi }(\mathit{U},\mathcal{W},\mathrm{\Gamma })+\mathcal{M}(\mathit{U},\mathcal{W},\mathrm{\Gamma })\le 3$;

(ii)    

$0<\mathrm{\Lambda }(\mathit{U},\mathcal{W},\mathrm{\Gamma })$;

(iii)   

$\mathrm{\Lambda }(\mathit{U},\mathcal{W},\mathrm{\Gamma })=1$∀$\mathrm{\Gamma }>0$, ⇔$\mathit{U}=\mathcal{W}$;

(iv)    

$\mathrm{\Lambda }(\mathit{U},\mathcal{W},\mathrm{\Gamma })=\mathrm{\Lambda }(\mathcal{W},\mathit{U},\mathrm{\Gamma })$;

(v)     

$\mathrm{\Lambda }(\mathit{U},\pounds ,\mathrm{\Gamma }+\check{g}+\check{k}+\check{e})\ge \mathrm{\Lambda }\left(\mathit{U},\mathcal{W},\mathrm{\Gamma }\right)★\mathrm{\Lambda }\left(\mathcal{W},\check{x},\check{g}\right)★\mathrm{\Lambda }\left(\check{x},\check{y},\check{k}\right)★\mathrm{\Lambda }\left(\check{y},\pounds ,\check{e}\right)$;

(vi)    

$\mathrm{\Lambda }(\mathit{U},\mathcal{W},·):(0,+\infty )\to [0,1]$ is continuous and ${lim}_{\mathrm{\Gamma }\to +\infty }\mathrm{\Lambda }(\mathit{U},\mathcal{W},\mathrm{\Gamma })=1$;

(vii)   

$1<\mathrm{\Pi }(\mathit{U},\mathcal{W},\mathrm{\Gamma })$;

(viii) 

$\mathrm{\Pi }(\mathit{U},\mathcal{W},\mathrm{\Gamma })=0$∀$\mathrm{\Gamma }>0$, ⇔$\mathit{U}=\mathcal{W}$;

(ix)    

$\mathrm{\Pi }(\mathit{U},\mathcal{W},\mathrm{\Gamma })=\mathrm{\Pi }(\mathcal{W},\mathit{U},\mathrm{\Gamma })$;

(x)     

$\mathrm{\Pi }(\mathit{U},\pounds ,\mathrm{\Gamma }+\check{g}+\check{k}+\check{e})\le \mathrm{\Pi }\left(\mathit{U},\mathcal{W},\mathrm{\Gamma }\right)•\mathrm{\Pi }\left(\mathcal{W},\check{x},\check{g}\right)•\mathrm{\Pi }\left(\check{x},\check{y},\check{k}\right)•\mathrm{\Pi }\left(\check{y},\pounds ,\check{e}\right)$;

(xi)    

$\mathrm{\Pi }(\mathit{U},\mathcal{W},·):(0,+\infty )\to [0,1]$ is continuous and ${lim}_{\mathrm{\Gamma }\to +\infty }\mathrm{\Pi }(\mathit{U},\mathcal{W},\mathrm{\Gamma })=0$;

(xii)   

$1<\mathcal{M}(\mathit{U},\mathcal{W},\mathrm{\Gamma })$;

(xiii)  

$\mathcal{M}(\mathit{U},\mathcal{W},\mathrm{\Gamma })=0$∀$\mathrm{\Gamma }>0$, ⇔$\mathit{U}=\mathcal{W}$;

(xiv)   

$\mathcal{M}(\mathit{U},\mathcal{W},\mathrm{\Gamma })=\mathcal{M}(\mathcal{W},\mathit{U},\mathrm{\Gamma })$;

(xv)    

$\mathcal{M}(\mathit{U},\pounds ,\mathrm{\Gamma }+\check{g}+\check{k}+\check{e})\le \mathcal{M}\left(\mathit{U},\mathcal{W},\mathrm{\Gamma }\right)•\mathcal{M}\left(\mathcal{W},\check{x},\check{g}\right)•\mathcal{M}\left(\check{x},\check{y},\check{k}\right)•\mathcal{M}\left(\check{y},\pounds ,\check{e}\right)$;

(xvi)   

$\mathcal{M}(\mathit{U},\mathcal{W},·):(0,+\infty )\to [0,1]$ is continuous and ${lim}_{\mathrm{\Gamma }\to +\infty }\mathcal{M}(\mathit{U},\mathcal{W},\mathrm{\Gamma })=0$;

(xvii)  

If $\mathrm{\Gamma }\le 0$, then $\mathrm{\Lambda }(\mathit{U},\mathcal{W},\mathrm{\Gamma })=0,\mathrm{\Pi }(\mathit{U},\mathcal{W},\mathrm{\Gamma })=1$ and $\mathcal{S}(\mathit{U},\mathcal{W},\mathrm{\Gamma })=1$.

Then, $(\mathit{K},\mathrm{\Lambda },\mathrm{\Pi },\mathcal{M},★,•)$ is said to be a NPMS.

Example 1.

Let $\mathit{K}=\{1,2,3,4\}$. Define $\mathrm{\Lambda },\mathrm{\Pi },\mathcal{M}:\mathit{K}\times \mathit{K}\times (0,+\infty )\to [0,1]$ as

$$\begin{array}{cc}\hfill \mathrm{\Lambda }(\mathit{U},\mathcal{W},\mathrm{\Gamma })& =\left\{\begin{array}{cc}1,\hfill & \mathit{if}\mathit{U}=\mathcal{W}\hfill \\ \frac{\mathrm{\Gamma }}{\mathrm{\Gamma }+max\{\mathit{U},\mathcal{W}\}},\hfill & \mathit{otherwise},\hfill \end{array}\right.\hfill \\ \hfill \mathrm{\Pi }(\mathit{U},\mathcal{W},\mathrm{\Gamma })& =\left\{\begin{array}{cc}0,\hfill & \mathit{if}\mathit{U}=\mathcal{W}\hfill \\ \frac{max\{\mathit{U},\mathcal{W}\}}{\mathrm{\Gamma }+max\{\mathit{U},\mathcal{W}\}},\hfill & \mathit{otherwise},\hfill \end{array}\right.\hfill \end{array}$$

and

$$\begin{array}{c}\hfill \mathcal{M}(\mathit{U},\mathcal{W},\mathrm{\Gamma })=\left\{\begin{array}{cc}0,\hfill & \mathit{if}\mathit{U}=\mathcal{W}\hfill \\ \frac{max\{\mathit{U},\mathcal{W}\}}{\mathrm{\Gamma }},\hfill & \mathit{otherwise},\hfill \end{array}\right.\end{array}$$

Then, $(\mathit{K},\mathrm{\Lambda },\mathrm{\Pi },\mathcal{M},★,•)$ is a NPMS with continuous t-norm $\widehat{\omega }★\breve{h}=\widehat{\omega }\breve{h}$ and continuous t-co-norm, $\widehat{\omega }•\overline{f}=max\{\widehat{\omega },\overline{f}\}$.

Proof. 

Now, we prove the conditions (v), (x) and (xv) others are obvious.

Let $\mathit{U}=1,\mathcal{W}=2$, $\check{x}=3$, $\pounds =4$ and $\rho =5$. Then

$$\begin{array}{c}\hfill \mathrm{\Lambda }(1,5,\mathrm{\Gamma }+\check{g}+\check{k}+\check{q})=\frac{\mathrm{\Gamma }+\check{g}+\check{k}+\check{q}}{\mathrm{\Gamma }+\check{g}+\check{k}+\check{q}+max\{1,5\}}=\frac{\mathrm{\Gamma }+\check{g}+\check{k}+\check{q}}{\mathrm{\Gamma }+\check{g}+\check{k}+\check{q}+5}.\end{array}$$

On the other hand,

$$\begin{array}{c}\hfill \mathrm{\Lambda }\left(1,2,\mathrm{\Gamma }\right)=\frac{\mathrm{\Gamma }}{\mathrm{\Gamma }+max\{1,2\}}=\frac{\mathrm{\Gamma }}{\mathrm{\Gamma }+2},\end{array}$$

$$\begin{array}{c}\hfill \mathrm{\Lambda }\left(2,3,\check{g}\right)=\frac{\check{g}}{\check{g}+max\{2,3\}}=\frac{\check{g}}{\check{g}+3},\end{array}$$

$$\begin{array}{c}\hfill \mathrm{\Lambda }\left(3,4,\check{k}\right)=\frac{\check{k}}{\check{k}+max\{3,4\}}=\frac{\check{k}}{\check{k}+4}\end{array}$$

and

$$\begin{array}{c}\hfill \mathrm{\Lambda }\left(4,5,\check{q}\right)=\frac{\check{q}}{\check{q}+max\{4,5\}}=\frac{\check{q}}{\check{q}+5}.\end{array}$$

i.e.,

$$\begin{array}{c}\hfill \frac{\mathrm{\Gamma }+\check{g}+\check{k}+\check{q}}{\mathrm{\Gamma }+\check{g}+\check{k}+\check{q}+5}\ge \frac{\mathrm{\Gamma }}{\mathrm{\Gamma }+2}\frac{\check{g}}{\check{g}+3}\frac{\check{k}}{\check{k}+4}\frac{\check{q}}{\check{q}+5}.\end{array}$$

Then it satisfies all $\mathrm{\Gamma },\check{g},\check{k},\check{q}>0$. Hence,

$$\begin{array}{c}\hfill \mathrm{\Lambda }(\mathit{U},\rho ,\mathrm{\Gamma }+\check{g}+\check{k}+\check{q})\ge \mathrm{\Lambda }\left(\mathit{U},\mathcal{W},\mathrm{\Gamma }\right)★\mathrm{\Lambda }\left(\mathcal{W},\check{x},\check{g}\right)★\mathrm{\Lambda }\left(\check{x},\pounds ,\check{k}\right)★\mathrm{\Lambda }\left(\pounds ,\rho ,\check{q}\right).\end{array}$$

Now,

$$\begin{array}{c}\hfill \mathrm{\Pi }(1,5,\mathrm{\Gamma }+\check{g}+\check{k}+\check{q})=\frac{max\{1,5\}}{\mathrm{\Gamma }+\check{g}+\check{k}+\check{q}+max\{1,5\}}=\frac{5}{\mathrm{\Gamma }+\check{g}+\check{k}+\check{q}+5}.\end{array}$$

On the other hand,

$$\begin{array}{c}\hfill \mathrm{\Pi }\left(1,2,\mathrm{\Gamma }\right)=\frac{max\{1,2\}}{\mathrm{\Gamma }+max\{1,2\}}=\frac{2}{\mathrm{\Gamma }+2},\end{array}$$

$$\begin{array}{c}\hfill \mathrm{\Pi }\left(2,3,\check{g}\right)=\frac{max\{2,3\}}{\check{g}+max\{2,3\}}=\frac{3}{\check{g}+3},\end{array}$$

$$\begin{array}{c}\hfill \mathrm{\Pi }\left(3,4,\check{k}\right)=\frac{max\{3,4\}}{\check{k}+max\{3,4\}}=\frac{4}{\check{k}+4}\end{array}$$

and

$$\begin{array}{c}\hfill \mathrm{\Pi }\left(4,5,\check{q}\right)=\frac{max\{4,5\}}{\check{q}+max\{4,5\}}=\frac{5}{\check{q}+5}\end{array}$$

i.e.,

$$\begin{array}{c}\hfill \frac{5}{\mathrm{\Gamma }+\check{g}+\check{k}+\check{q}+5}\le max\left\{\frac{2}{\mathrm{\Gamma }+2},\frac{3}{\check{g}+3},\frac{4}{\check{k}+4},\frac{5}{\check{q}+5}\right\}.\end{array}$$

Then it satisfies all $\mathrm{\Gamma },\check{g},\check{k},\check{q}>0$. Hence,

$$\begin{array}{c}\hfill \mathrm{\Pi }(\mathit{U},\rho ,\mathrm{\Gamma }+\check{g}+\check{k}+\check{q})\le \mathrm{\Pi }\left(\mathit{U},\mathcal{W},\mathrm{\Gamma }\right)•\mathrm{\Pi }\left(\check{x},\pounds ,\widehat{s}\right)•\mathrm{\Pi }\left(\check{x},\pounds ,\widehat{w}\right)•\mathrm{\Pi }\left(\pounds ,\rho ,\widehat{y}\right).\end{array}$$

Now,

$$\begin{array}{c}\hfill \mathcal{M}(1,5,\mathrm{\Gamma }+\check{g}+\check{k}+\check{q})=\frac{max\{1,5\}}{\mathrm{\Gamma }+\check{g}+\check{k}+\check{q}}=\frac{5}{\mathrm{\Gamma }+\check{g}+\check{k}+\check{q}}.\end{array}$$

On the other hand,

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

$$\begin{array}{c}\hfill \mathcal{M}\left(2,3,\check{g}\right)=\frac{max\{2,3\}}{\check{g}}=\frac{3}{\check{g}},\end{array}$$

$$\begin{array}{c}\hfill \mathcal{M}\left(3,4,\check{k}\right)=\frac{max\{3,4\}}{\check{k}}=\frac{4}{\check{k}}\end{array}$$

and

$$\begin{array}{c}\hfill \mathcal{M}\left(4,5,\check{q}\right)=\frac{max\{4,5\}}{\check{q}}=\frac{5}{\check{q}}\end{array}$$

i.e.,

$$\begin{array}{c}\hfill \frac{5}{\mathrm{\Gamma }+\check{g}+\check{k}+\check{q}}\le max\left\{\frac{2}{\mathrm{\Gamma }},\frac{3}{\check{g}},\frac{4}{\check{k}},\frac{5}{\check{q}}\right\}.\end{array}$$

Then it satisfies all $\mathrm{\Gamma },\check{g},\check{k},\check{q}>0$. Hence,

$$\begin{array}{c}\hfill \mathcal{M}(\mathit{U},\rho ,\mathrm{\Gamma }+\check{g}+\check{k}+\check{q})\le \mathcal{M}\left(\mathit{U},\mathcal{W},\mathrm{\Gamma }\right)•\mathcal{M}\left(\mathcal{W},\check{x},\check{g}\right)•\mathcal{M}\left(\check{x},\pounds ,\check{k}\right)•\mathcal{M}\left(\pounds ,\rho ,\check{k}\right).\end{array}$$

Hence, $(\mathit{K},\mathrm{\Lambda },\mathrm{\Pi },\mathcal{M},★,•)$ is a NPMS. □

Remark 1.

The above example satisfies for continuous t-norm $\widehat{\omega }★\overline{f}=min\{\widehat{\omega },\overline{f}\}$ and continuous t-co-norm $\widehat{\omega }•\overline{f}=max\{\widehat{\omega },\overline{f}\}$.

Definition 8.

Let $(\mathit{K},\mathrm{\Lambda },\mathrm{\Pi },\mathcal{M},★,•)$ is a NPMS, an open ball is then defined $\mathcal{M}(\mathit{U},\mathtt{r},\mathrm{\Gamma })$ with center $\mathit{U}$, radius $\mathtt{r},0<\mathtt{r}<1$ and $\mathrm{\Gamma }>0$ as follows:

$$\begin{array}{c}\hfill \mathcal{M}(\mathit{U},\mathtt{r},\mathrm{\Gamma })=\{\mathcal{W}\in \mathit{K}:\mathrm{\Lambda }(\mathit{U},\mathcal{W},\mathrm{\Gamma })>1-\mathtt{r},\mathrm{\Pi }(\mathit{U},\mathcal{W},\mathrm{\Gamma })<\mathtt{r},\mathcal{M}(\mathit{U},\mathcal{W},\mathrm{\Gamma })<\mathtt{r}\}.\end{array}$$

Definition 9.

Let $(\mathit{K},\mathrm{\Lambda },\mathrm{\Pi },\mathcal{M},★,•)$ is a NPMS and $\left\{{\mathit{U}}_{{\beta }_1}\right\}$ be a sequence in $\mathit{K}$. Then $\left\{{\mathit{U}}_{{\beta }_1}\right\}$ is called:

(a) 

a convergent if exists $\mathit{U}\in \mathit{K}$ such that

$$\begin{array}{c}\hfill \underset{{\beta }_1\to +\infty }{lim}\mathrm{\Lambda }({\mathit{U}}_{{\beta }_1},\mathit{U},\mathrm{\Gamma })=1,\underset{{\beta }_1\to +\infty }{lim}\mathrm{\Pi }({\mathit{U}}_{{\beta }_1},\mathit{U},\mathrm{\Gamma })=0,\underset{{\beta }_1\to +\infty }{lim}\mathcal{M}({\mathit{U}}_{{\beta }_1},\mathit{U},\mathrm{\Gamma })=0,\forall \mathrm{\Gamma }>0,\end{array}$$

(b) 

a Cauchy sequence, if for each $\overline{f}>0,\mathrm{\Gamma }>0$, exists ${\beta }_{1_0}\in \mathbb{N}$ such that

$$\begin{array}{c}\hfill \mathrm{\Lambda }({\mathit{U}}_{{\beta }_1},{\mathit{U}}_{{\beta }_1+\mathfrak{q}},\mathrm{\Gamma })\ge 1-\overline{f},\mathrm{\Pi }({\mathit{U}}_{{\beta }_1},{\mathit{U}}_{{\beta }_1+\mathfrak{q}},\mathrm{\Gamma })\le \overline{f},\mathrm{\Pi }({\mathit{U}}_{{\beta }_1},{\mathit{U}}_{{\beta }_1+\mathfrak{q}},\mathrm{\Gamma })\le \overline{f},forall{\beta }_1,\mathfrak{m}\ge {\beta }_{1_0},\end{array}$$

If every Cauchy sequence convergent in $\mathit{K}$, then $(\mathit{K},\mathrm{\Lambda },\mathrm{\Pi },\mathcal{M},★,•)$ is said to be complete NPMS.

Lemma 1.

Let $\left\{{\mathit{U}}_{{\beta }_1}\right\}$ be a Cauchy sequence in NPMS $(\mathit{K},\mathrm{\Lambda },\mathrm{\Pi },\mathcal{M},★,•)$ such that ${\mathit{U}}_{{\beta }_1}\ne {\mathit{U}}_{\mathfrak{m}}$ whenever $\mathfrak{m},{\beta }_1\in \mathbb{N}$ with ${\beta }_1\ne \mathfrak{m}$. Then the sequence $\left\{{\mathit{U}}_{{\beta }_1}\right\}$ can converge to, at most, one limit point.

Proof. 

Contrarily, suppose that ${\mathit{U}}_{{\beta }_1}\to \mathit{U}$, ${\mathit{U}}_{{\beta }_1}\to \mathcal{W}$, and $\mathit{U}\ne \mathcal{W}$.

Then, ${lim}_{{\beta }_1\to +\infty }\mathrm{\Lambda }({\mathit{U}}_{{\beta }_1},\mathit{U},\mathrm{\Gamma })=1$, ${lim}_{{\beta }_1\to +\infty }\mathrm{\Pi }({\mathit{U}}_{{\beta }_1},\mathit{U},\mathrm{\Gamma })=0$,

${lim}_{{\beta }_1\to +\infty }\mathcal{M}({\mathit{U}}_{{\beta }_1},\mathit{U},\mathrm{\Gamma })=0$, and ${lim}_{{\beta }_1\to +\infty }\mathrm{\Lambda }({\mathit{U}}_{{\beta }_1},\mathcal{W},\mathrm{\Gamma })=1$,

${lim}_{{\beta }_1\to +\infty }\mathrm{\Pi }({\mathit{U}}_{{\beta }_1},\mathcal{W},\mathrm{\Gamma })=0$, ${lim}_{{\beta }_1\to +\infty }\mathcal{M}({\mathit{U}}_{{\beta }_1},\mathcal{W},\mathrm{\Gamma })=0$, for all $\mathrm{\Gamma }>0$. Suppose

$$\begin{array}{cc}\hfill \mathrm{\Lambda }(\mathit{U},\mathcal{W},\mathrm{\Gamma })& \ge \mathrm{\Lambda }\left(\mathit{U},{\mathit{U}}_{{\beta }_1},\frac{\mathrm{\Gamma }}{4}\right)★\mathrm{\Lambda }\left({\mathit{U}}_{{\beta }_1},{\mathit{U}}_{{\beta }_1+1},\frac{\mathrm{\Gamma }}{4}\right)★\mathrm{\Lambda }\left({\mathit{U}}_{{\beta }_1+1},{\mathit{U}}_{{\beta }_1+2},\frac{\mathrm{\Gamma }}{4}\right)★\mathrm{\Lambda }\left({\mathit{U}}_{{\beta }_1+2},\mathcal{W},\frac{\mathrm{\Gamma }}{4}\right)\hfill \\ \hfill & \to 1★1★1★1,\mathrm{as}{\beta }_1\to +\infty ,\hfill \\ \hfill \mathrm{\Pi }(\mathit{U},\mathcal{W},\mathrm{\Gamma })& \le \mathrm{\Pi }\left(\mathit{U},{\mathit{U}}_{{\beta }_1},\frac{\mathrm{\Gamma }}{4}\right)•\mathrm{\Pi }\left({\mathit{U}}_{{\beta }_1},{\mathit{U}}_{{\beta }_1+1},\frac{\mathrm{\Gamma }}{4}\right)•\mathrm{\Pi }\left({\mathit{U}}_{{\beta }_1+1},{\mathit{U}}_{{\beta }_1+2},\frac{\mathrm{\Gamma }}{4}\right)•\mathrm{\Pi }\left({\mathit{U}}_{{\beta }_1+2},\mathcal{W},\frac{\mathrm{\Gamma }}{4}\right)\hfill \\ \hfill & \to 0•0•0•0,\mathrm{as}{\beta }_1\to +\infty ,\hfill \\ \hfill \mathcal{M}(\mathit{U},\mathcal{W},\mathrm{\Gamma })& \le \mathcal{M}\left(\mathit{U},{\mathit{U}}_{{\beta }_1},\frac{\mathrm{\Gamma }}{4}\right)•\mathcal{M}\left({\mathit{U}}_{{\beta }_1},{\mathit{U}}_{{\beta }_1+1},\frac{\mathrm{\Gamma }}{4}\right)•\mathcal{M}\left({\mathit{U}}_{{\beta }_1+1},{\mathit{U}}_{{\beta }_1+2},\frac{\mathrm{\Gamma }}{4}\right)\hfill \\ \hfill & •\mathcal{M}\left({\mathit{U}}_{{\beta }_1+2},\mathcal{W},\frac{\mathrm{\Gamma }}{4}\right)\hfill \\ \hfill & \to 0•0•0•0,\mathrm{as}{\beta }_1\to +\infty .\hfill \end{array}$$

That is $\mathrm{\Lambda }(\mathit{U},\mathcal{W},\mathrm{\Gamma })\ge 1★1★1★1=1,\mathrm{\Pi }(\mathit{U},\mathcal{W},\mathrm{\Gamma })\le 0•0•0•0=0$ and $\mathcal{M}(\mathit{U},\mathcal{W},\mathrm{\Gamma })\le 0•0•0•0=0$. Hence $\mathit{U}=\mathcal{W}$, i.e., the sequence converges to at most one limit point. □

Lemma 2.

Let $(\mathit{K},\mathrm{\Lambda },\mathrm{\Pi },\mathcal{M},★,•)$ is a NPMS. If for some $0<♮<1$ and for any $\mathit{U},\mathcal{W}\in \mathit{K},\mathrm{\Gamma }>0$,

$$\begin{array}{c}\hfill \mathrm{\Lambda }(\mathit{U},\mathcal{W},\mathrm{\Gamma })\ge \mathrm{\Lambda }\left(\mathit{U},\mathcal{W},\frac{\mathrm{\Gamma }}{♮}\right),\mathrm{\Pi }(\mathit{U},\mathcal{W},\mathrm{\Gamma })\le \mathrm{\Pi }\left(\mathit{U},\mathcal{W},\frac{\mathrm{\Gamma }}{♮}\right),\mathcal{M}(\mathit{U},\mathcal{W},\mathrm{\Gamma })\le \mathcal{M}\left(\mathit{U},\mathcal{W},\frac{\mathrm{\Gamma }}{♮}\right)\end{array}$$

(1)

then $\mathit{U}=\mathcal{W}$.

Proof. 

By (1) follows that

$$\begin{array}{c}\hfill \mathrm{\Lambda }(\mathit{U},\mathcal{W},\mathrm{\Gamma })\ge \mathrm{\Lambda }\left(\mathit{U},\mathcal{W},\frac{\mathrm{\Gamma }}{{♮}^{{\beta }_1}}\right),\mathrm{\Pi }(\mathit{U},\mathcal{W},\mathrm{\Gamma })\le \mathrm{\Pi }\left(\mathit{U},\mathcal{W},\frac{\mathrm{\Gamma }}{{♮}^{{\beta }_1}}\right),\\ \hfill \mathcal{M}(\mathit{U},\mathcal{W},\mathrm{\Gamma })\le \mathcal{M}\left(\mathit{U},\mathcal{W},\frac{\mathrm{\Gamma }}{{♮}^{{\beta }_1}}\right),{\beta }_1\in \mathbb{N},\mathrm{\Gamma }>0.\end{array}$$

Now

$$\begin{array}{cc}\hfill \mathrm{\Lambda }(\mathit{U},\mathcal{W},\mathrm{\Gamma })& \ge \underset{{\beta }_1\to +\infty }{lim}\mathrm{\Lambda }\left(\mathit{U},\mathcal{W},\frac{\mathrm{\Gamma }}{{♮}^{{\beta }_1}}\right)=1,\hfill \\ \hfill \mathrm{\Pi }(\mathit{U},\mathcal{W},\mathrm{\Gamma })& \le \underset{{\beta }_1\to +\infty }{lim}\mathrm{\Pi }\left(\mathit{U},\mathcal{W},\frac{\mathrm{\Gamma }}{{♮}^{{\beta }_1}}\right)=0,\hfill \\ \hfill \mathcal{M}(\mathit{U},\mathcal{W},\mathrm{\Gamma })& \le \underset{{\beta }_1\to +\infty }{lim}\mathcal{M}\left(\mathit{U},\mathcal{W},\frac{\mathrm{\Gamma }}{{♮}^{{\beta }_1}}\right)=0,\mathrm{\Gamma }>0.\hfill \end{array}$$

Also, by definition of (iii), (viii), (xiii), i.e., $\mathit{U}=\mathcal{W}$. □

Theorem 1.

Suppose $(\mathit{K},\mathrm{\Lambda },\mathrm{\Pi },\mathcal{M},★,•)$ is a complete NPMS, $♮\in (0,1)$ and assume that

$$\begin{array}{c}\hfill \underset{\mathrm{\Gamma }\to +\infty }{lim}\mathrm{\Lambda }(\mathit{U},\mathcal{W},\mathrm{\Gamma })=1,\underset{\mathrm{\Gamma }\to +\infty }{lim}\mathrm{\Pi }(\mathit{U},\mathcal{W},\mathrm{\Gamma })=0and\underset{\mathrm{\Gamma }\to +\infty }{lim}\mathcal{M}(\mathit{U},\mathcal{W},\mathrm{\Gamma })=0,\end{array}$$

(2)

for all $\mathit{U},\mathcal{W}\in \mathit{K}$ and $\mathrm{\Gamma }>0$. Let $\wp :\mathit{K}\to \mathit{K}$ be a mapping satisfying

$$\begin{array}{cc}\hfill \mathrm{\Lambda }(\wp \mathit{U},\wp \mathcal{W},♮\mathrm{\Gamma })& \ge \mathrm{\Lambda }(\mathit{U},\mathcal{W},\mathrm{\Gamma }),\hfill \\ \hfill \mathrm{\Pi }(\wp \mathit{U},\wp \mathcal{W},♮\mathrm{\Gamma })\le \mathrm{\Pi }(\mathit{U},\mathcal{W},\mathrm{\Gamma })& and\mathcal{M}(\wp \mathit{U},\wp \mathcal{W},♮\mathrm{\Gamma })\le \mathcal{M}(\mathit{U},\mathcal{W},\mathrm{\Gamma }),\hfill \end{array}$$

(3)

for all $\mathit{U},\mathcal{W}\in \mathit{K}$ and $\mathrm{\Gamma }>0$. Then ℘ has a unique fixed point (shortly, ufp).

Proof. 

Consider a point ${\mathit{U}}_0$ of $\mathit{K}$ and define a sequence ${\mathit{U}}_{{\beta }_1}$ by ${\mathit{U}}_{{\beta }_1}={\wp }^{{\beta }_1}{\mathit{U}}_0=\wp {\mathit{U}}_{{\beta }_1-1},{\beta }_1\in \mathbb{N}$.

By utilising (3) for all $\mathrm{\Gamma }>0$, we obtain

$$\begin{array}{cc}\hfill \mathrm{\Lambda }({\mathit{U}}_{{\beta }_1},{\mathit{U}}_{{\beta }_1+1},♮\mathrm{\Gamma })& =\mathrm{\Lambda }(\wp {\mathit{U}}_{{\beta }_1-1},\wp {\mathit{U}}_{{\beta }_1},♮\mathrm{\Gamma })\ge \mathrm{\Lambda }({\mathit{U}}_{{\beta }_1-1},{\mathit{U}}_{{\beta }_1},\mathrm{\Gamma })\ge \mathrm{\Lambda }\left({\mathit{U}}_{{\beta }_1-2},{\mathit{U}}_{{\beta }_1-1},\frac{\mathrm{\Gamma }}{♮}\right)\hfill \\ \hfill & \ge \mathrm{\Lambda }\left({\mathit{U}}_{{\beta }_1-3},{\mathit{U}}_{{\beta }_1-2},\frac{\mathrm{\Gamma }}{{♮}^2}\right)\ge \cdots \ge \mathrm{\Lambda }\left({\mathit{U}}_0,{\mathit{U}}_1,\frac{\mathrm{\Gamma }}{{♮}^{{\beta }_1-1}}\right),\hfill \\ \hfill \mathrm{\Pi }({\mathit{U}}_{{\beta }_1},{\mathit{U}}_{{\beta }_1+1},♮\mathrm{\Gamma })& =\mathrm{\Pi }(\wp {\mathit{U}}_{{\beta }_1-1},\wp {\mathit{U}}_{{\beta }_1},♮\mathrm{\Gamma })\le \mathrm{\Pi }({\mathit{U}}_{{\beta }_1-1},{\mathit{U}}_{{\beta }_1},\mathrm{\Gamma })\le \mathrm{\Pi }\left({\mathit{U}}_{{\beta }_1-2},{\mathit{U}}_{{\beta }_1-1},\frac{\mathrm{\Gamma }}{♮}\right)\hfill \\ \hfill & \le \mathrm{\Pi }\left({\mathit{U}}_{{\beta }_1-3},{\mathit{U}}_{{\beta }_1-2},\frac{\mathrm{\Gamma }}{{♮}^2}\right)\le \cdots \le \mathrm{\Pi }\left({\mathit{U}}_0,{\mathit{U}}_1,\frac{\mathrm{\Gamma }}{{♮}^{{\beta }_1-1}}\right).\hfill \end{array}$$

and

$$\begin{array}{cc}\hfill \mathcal{M}({\mathit{U}}_{{\beta }_1},{\mathit{U}}_{{\beta }_1+1},♮\mathrm{\Gamma })& =\mathcal{M}(\wp {\mathit{U}}_{{\beta }_1-1},\wp {\mathit{U}}_{{\beta }_1},\mathrm{\Gamma })\le \mathcal{M}({\mathit{U}}_{{\beta }_1-1},{\mathit{U}}_{{\beta }_1},\mathrm{\Gamma })\le \mathcal{M}\left({\mathit{U}}_{{\beta }_1-2},{\mathit{U}}_{{\beta }_1-1},\frac{\mathrm{\Gamma }}{♮}\right)\hfill \\ \hfill & \le \mathcal{M}\left({\mathit{U}}_{{\beta }_1-3},{\mathit{U}}_{{\beta }_1-2},\frac{\mathrm{\Gamma }}{{♮}^2}\right)\le \cdots \le \mathcal{M}\left({\mathit{U}}_0,{\mathit{U}}_1,\frac{\mathrm{\Gamma }}{{♮}^{{\beta }_1-1}}\right).\hfill \end{array}$$

We obtain

$$\begin{array}{cc}\hfill \mathrm{\Lambda }({\mathit{U}}_{{\beta }_1},{\mathit{U}}_{{\beta }_1+1},♮\mathrm{\Gamma })& \ge \mathrm{\Lambda }\left({\mathit{U}}_0,{\mathit{U}}_1,\frac{\mathrm{\Gamma }}{{♮}^{{\beta }_1-1}}\right),\hfill \\ \hfill \mathrm{\Pi }({\mathit{U}}_{{\beta }_1},{\mathit{U}}_{{\beta }_1+1},♮\mathrm{\Gamma })\le \mathrm{\Pi }\left({\mathit{U}}_0,{\mathit{U}}_1,\frac{\mathrm{\Gamma }}{{♮}^{{\beta }_1-1}}\right)& \mathrm{and}\mathcal{M}({\mathit{U}}_{{\beta }_1},{\mathit{U}}_{{\beta }_1+1},♮\mathrm{\Gamma })\le \mathcal{M}\left({\mathit{U}}_0,{\mathit{U}}_1,\frac{\mathrm{\Gamma }}{{♮}^{{\beta }_1-1}}\right).\hfill \end{array}$$

(4)

Consequently,

$$\begin{array}{cc}\hfill \mathrm{\Lambda }({\mathit{U}}_{{\beta }_1},{\mathit{U}}_{{\beta }_1+2},♮\mathrm{\Gamma })& =\mathrm{\Lambda }(\wp {\mathit{U}}_{{\beta }_1-1},\wp {\mathit{U}}_{{\beta }_1+1},♮\mathrm{\Gamma })\ge \mathrm{\Lambda }({\mathit{U}}_{{\beta }_1-1},{\mathit{U}}_{{\beta }_1+1},\mathrm{\Gamma })\ge \mathrm{\Lambda }\left({\mathit{U}}_{{\beta }_1-2},{\mathit{U}}_{{\beta }_1},\frac{\mathrm{\Gamma }}{♮}\right)\hfill \\ \hfill & \ge \mathrm{\Lambda }\left({\mathit{U}}_{{\beta }_1-3},{\mathit{U}}_{{\beta }_1-1},\frac{\mathrm{\Gamma }}{{♮}^2}\right)\ge \cdots \ge \mathrm{\Lambda }\left({\mathit{U}}_0,{\mathit{U}}_2,\frac{\mathrm{\Gamma }}{{♮}^{{\beta }_1-1}}\right),\hfill \\ \hfill \mathrm{\Pi }({\mathit{U}}_{{\beta }_1},{\mathit{U}}_{{\beta }_1+2},♮\mathrm{\Gamma })& =\mathrm{\Pi }(\wp {\mathit{U}}_{{\beta }_1-1},\wp {\mathit{U}}_{{\beta }_1+1},♮\mathrm{\Gamma })\le \mathrm{\Pi }({\mathit{U}}_{{\beta }_1-1},{\mathit{U}}_{{\beta }_1+1},\mathrm{\Gamma })\le \mathrm{\Pi }\left({\mathit{U}}_{{\beta }_1-2},{\mathit{U}}_{{\beta }_1},\frac{\mathrm{\Gamma }}{♮}\right)\hfill \\ \hfill & \le \mathrm{\Pi }\left({\mathit{U}}_{{\beta }_1-3},{\mathit{U}}_{{\beta }_1-1},\frac{\mathrm{\Gamma }}{{♮}^2}\right)\le \cdots \le \mathrm{\Pi }\left({\mathit{U}}_0,{\mathit{U}}_2,\frac{\mathrm{\Gamma }}{{♮}^{{\beta }_1-1}}\right).\hfill \end{array}$$

and

$$\begin{array}{cc}\hfill \mathcal{M}({\mathit{U}}_{{\beta }_1},{\mathit{U}}_{{\beta }_1+2},♮\mathrm{\Gamma })& =\mathcal{M}(\wp {\mathit{U}}_{{\beta }_1-1},\wp {\mathit{U}}_{{\beta }_1+1},\mathrm{\Gamma })\le \mathcal{M}({\mathit{U}}_{{\beta }_1-1},{\mathit{U}}_{{\beta }_1+1},\mathrm{\Gamma })\le \mathcal{M}\left({\mathit{U}}_{{\beta }_1-2},{\mathit{U}}_{{\beta }_1},\frac{\mathrm{\Gamma }}{♮}\right)\hfill \\ \hfill & \le \mathcal{M}\left({\mathit{U}}_{{\beta }_1-3},{\mathit{U}}_{{\beta }_1-1},\frac{\mathrm{\Gamma }}{{♮}^2}\right)\le \cdots \le \mathcal{M}\left({\mathit{U}}_0,{\mathit{U}}_2,\frac{\mathrm{\Gamma }}{{♮}^{{\beta }_1-1}}\right).\hfill \end{array}$$

We obtain

$$\begin{array}{cc}\hfill \mathrm{\Lambda }({\mathit{U}}_{{\beta }_1},{\mathit{U}}_{{\beta }_1+2},♮\mathrm{\Gamma })& \ge \mathrm{\Lambda }\left({\mathit{U}}_0,{\mathit{U}}_2,\frac{\mathrm{\Gamma }}{{♮}^{{\beta }_1-1}}\right),\hfill \\ \hfill \mathrm{\Pi }({\mathit{U}}_{{\beta }_1},{\mathit{U}}_{{\beta }_1+2},♮\mathrm{\Gamma })\le \mathrm{\Pi }\left({\mathit{U}}_0,{\mathit{U}}_2,\frac{\mathrm{\Gamma }}{{♮}^{{\beta }_1-1}}\right)& \mathrm{and}\mathcal{M}({\mathit{U}}_{{\beta }_1},{\mathit{U}}_{{\beta }_1+2},♮\mathrm{\Gamma })\le \mathcal{M}\left({\mathit{U}}_0,{\mathit{U}}_2,\frac{\mathrm{\Gamma }}{{♮}^{{\beta }_1-1}}\right).\hfill \end{array}$$

(5)

It follows that

$$\begin{array}{cc}\hfill \mathrm{\Lambda }({\mathit{U}}_{{\beta }_1},{\mathit{U}}_{{\beta }_1+3},♮\mathrm{\Gamma })& =\mathrm{\Lambda }(\wp {\mathit{U}}_{{\beta }_1-1},\wp {\mathit{U}}_{{\beta }_1+2},♮\mathrm{\Gamma })\ge \mathrm{\Lambda }({\mathit{U}}_{{\beta }_1-1},{\mathit{U}}_{{\beta }_1+2},\mathrm{\Gamma })\ge \mathrm{\Lambda }\left({\mathit{U}}_{{\beta }_1-2},{\mathit{U}}_{{\beta }_1+1},\frac{\mathrm{\Gamma }}{♮}\right)\hfill \\ \hfill & \ge \mathrm{\Lambda }\left({\mathit{U}}_{{\beta }_1-3},{\mathit{U}}_{{\beta }_1},\frac{\mathrm{\Gamma }}{{♮}^2}\right)\ge \cdots \ge \mathrm{\Lambda }\left({\mathit{U}}_0,{\mathit{U}}_3,\frac{\mathrm{\Gamma }}{{♮}^{{\beta }_1-1}}\right),\hfill \\ \hfill \mathrm{\Pi }({\mathit{U}}_{{\beta }_1},{\mathit{U}}_{{\beta }_1+3},♮\mathrm{\Gamma })& =\mathrm{\Pi }(\wp {\mathit{U}}_{{\beta }_1-1},\wp {\mathit{U}}_{{\beta }_1+2},♮\mathrm{\Gamma })\le \mathrm{\Pi }({\mathit{U}}_{{\beta }_1-1},{\mathit{U}}_{{\beta }_1+2},\mathrm{\Gamma })\le \mathrm{\Pi }\left({\mathit{U}}_{{\beta }_1-2},{\mathit{U}}_{{\beta }_1+1},\frac{\mathrm{\Gamma }}{♮}\right)\hfill \\ \hfill & \le \mathrm{\Pi }\left({\mathit{U}}_{{\beta }_1-3},{\mathit{U}}_{{\beta }_1},\frac{\mathrm{\Gamma }}{{♮}^2}\right)\le \cdots \le \mathrm{\Pi }\left({\mathit{U}}_0,{\mathit{U}}_3,\frac{\mathrm{\Gamma }}{{♮}^{{\beta }_1-1}}\right).\hfill \end{array}$$

and

$$\begin{array}{cc}\hfill \mathcal{M}({\mathit{U}}_{{\beta }_1},{\mathit{U}}_{{\beta }_1+3},♮\mathrm{\Gamma })& =\mathcal{M}(\wp {\mathit{U}}_{{\beta }_1-1},\wp {\mathit{U}}_{{\beta }_1+2},\mathrm{\Gamma })\le \mathcal{M}({\mathit{U}}_{{\beta }_1-1},{\mathit{U}}_{{\beta }_1+2},\mathrm{\Gamma })\le \mathcal{M}\left({\mathit{U}}_{{\beta }_1-2},{\mathit{U}}_{{\beta }_1+1},\frac{\mathrm{\Gamma }}{♮}\right)\hfill \\ \hfill & \le \mathcal{M}\left({\mathit{U}}_{{\beta }_1-3},{\mathit{U}}_{{\beta }_1},\frac{\mathrm{\Gamma }}{{♮}^2}\right)\le \cdots \le \mathcal{M}\left({\mathit{U}}_0,{\mathit{U}}_3,\frac{\mathrm{\Gamma }}{{♮}^{{\beta }_1-1}}\right).\hfill \end{array}$$

We obtain

$$\begin{array}{cc}\hfill \mathrm{\Lambda }({\mathit{U}}_{{\beta }_1},{\mathit{U}}_{{\beta }_1+3},♮\mathrm{\Gamma })& \ge \mathrm{\Lambda }\left({\mathit{U}}_0,{\mathit{U}}_3,\frac{\mathrm{\Gamma }}{{♮}^{{\beta }_1-1}}\right),\hfill \\ \hfill \mathrm{\Pi }({\mathit{U}}_{{\beta }_1},{\mathit{U}}_{{\beta }_1+3},♮\mathrm{\Gamma })\le \mathrm{\Pi }\left({\mathit{U}}_0,{\mathit{U}}_3,\frac{\mathrm{\Gamma }}{{♮}^{{\beta }_1-1}}\right)& \mathrm{and}\mathcal{M}({\mathit{U}}_{{\beta }_1},{\mathit{U}}_{{\beta }_1+3},♮\mathrm{\Gamma })\le \mathcal{M}\left({\mathit{U}}_0,{\mathit{U}}_3,\frac{\mathrm{\Gamma }}{{♮}^{{\beta }_1-1}}\right).\hfill \end{array}$$

Similarly, for $\mathfrak{j}=1,2,3,\dots$, we obtain

$$\begin{array}{cc}\hfill \mathrm{\Lambda }({\mathit{U}}_{{\beta }_1},{\mathit{U}}_{{\beta }_1+3\mathfrak{j}+1},♮\mathrm{\Gamma })& \ge \mathrm{\Lambda }\left({\mathit{U}}_0,{\mathit{U}}_{3\mathfrak{j}+1},\frac{\mathrm{\Gamma }}{{♮}^{{\beta }_1-1}}\right),\mathrm{\Pi }({\mathit{U}}_{{\beta }_1},{\mathit{U}}_{{\beta }_1+3\mathfrak{j}+1},♮\mathrm{\Gamma })\le \mathrm{\Pi }\left({\mathit{U}}_0,{\mathit{U}}_{3\mathfrak{j}+1},\frac{\mathrm{\Gamma }}{{♮}^{{\beta }_1-1}}\right)\hfill \\ \hfill & \mathrm{and}\mathcal{M}({\mathit{U}}_{{\beta }_1},{\mathit{U}}_{{\beta }_1+3\mathfrak{j}+1},♮\mathrm{\Gamma })\le \mathcal{M}\left({\mathit{U}}_0,{\mathit{U}}_{3\mathfrak{j}+1},\frac{\mathrm{\Gamma }}{{♮}^{{\beta }_1-1}}\right),\hfill \end{array}$$

(6)

$$\begin{array}{c}\hfill \mathrm{\Lambda }({\mathit{U}}_{{\beta }_1},{\mathit{U}}_{{\beta }_1+3\mathfrak{j}+2},♮\mathrm{\Gamma })\ge \mathrm{\Lambda }\left({\mathit{U}}_0,{\mathit{U}}_{3\mathfrak{j}+2},\frac{\mathrm{\Gamma }}{{♮}^{{\beta }_1-1}}\right),\mathrm{\Pi }({\mathit{U}}_{{\beta }_1},{\mathit{U}}_{{\beta }_1+3\mathfrak{j}+2},♮\mathrm{\Gamma })\le \mathrm{\Pi }\left({\mathit{U}}_0,{\mathit{U}}_{3\mathfrak{j}+2},\frac{\mathrm{\Gamma }}{{♮}^{{\beta }_1-1}}\right)\\ \hfill \mathrm{and}\mathcal{M}({\mathit{U}}_{{\beta }_1},{\mathit{U}}_{{\beta }_1+3\mathfrak{j}+2},♮\mathrm{\Gamma })\le \mathcal{M}\left({\mathit{U}}_0,{\mathit{U}}_{3\mathfrak{j}+2},\frac{\mathrm{\Gamma }}{{♮}^{{\beta }_1-1}}\right),\end{array}$$

$$\begin{array}{c}\hfill \mathrm{\Lambda }({\mathit{U}}_{{\beta }_1},{\mathit{U}}_{{\beta }_1+3\mathfrak{j}+3},♮\mathrm{\Gamma })\ge \mathrm{\Lambda }\left({\mathit{U}}_0,{\mathit{U}}_{3\mathfrak{j}+3},\frac{\mathrm{\Gamma }}{{♮}^{{\beta }_1-1}}\right),\mathrm{\Pi }({\mathit{U}}_{{\beta }_1},{\mathit{U}}_{{\beta }_1+3\mathfrak{j}+3},♮\mathrm{\Gamma })\le \mathrm{\Pi }\left({\mathit{U}}_0,{\mathit{U}}_{3\mathfrak{j}+3},\frac{\mathrm{\Gamma }}{{♮}^{{\beta }_1-1}}\right)\\ \hfill \mathrm{and}\mathcal{M}({\mathit{U}}_{{\beta }_1},{\mathit{U}}_{{\beta }_1+3\mathfrak{j}+3},♮\mathrm{\Gamma })\le \mathcal{M}\left({\mathit{U}}_0,{\mathit{U}}_{3\mathfrak{j}+3},\frac{\mathrm{\Gamma }}{{♮}^{{\beta }_1-1}}\right).\end{array}$$

(7)

By using (4), we obtain for each $\mathfrak{j}=1,2,3,\dots ,$

$$\begin{array}{cc}\hfill \mathrm{\Lambda }({\mathit{U}}_0,{\mathit{U}}_{3\mathfrak{j}+1},\mathrm{\Gamma })& \ge \mathrm{\Lambda }\left({\mathit{U}}_0,{\mathit{U}}_1,\frac{\mathrm{\Gamma }}{4}\right)★\mathrm{\Lambda }\left({\mathit{U}}_0,{\mathit{U}}_1,\frac{\mathrm{\Gamma }}{4}\right)★\mathrm{\Lambda }\left({\mathit{U}}_0,{\mathit{U}}_1,\frac{\mathrm{\Gamma }}{4♮}\right)★\mathrm{\Lambda }\left({\mathit{U}}_0,{\mathit{U}}_1,\frac{\mathrm{\Gamma }}{4{♮}^2}\right)\hfill \\ \hfill & ★\mathrm{\Lambda }\left({\mathit{U}}_0,{\mathit{U}}_1,\frac{\mathrm{\Gamma }}{4{♮}^3}\right)★\mathrm{\Lambda }\left({\mathit{U}}_0,{\mathit{U}}_1,\frac{\mathrm{\Gamma }}{4{♮}^4}\right)★\mathrm{\Lambda }\left({\mathit{U}}_0,{\mathit{U}}_1,\frac{\mathrm{\Gamma }}{4{♮}^5}\right)\hfill \\ \hfill & ★\cdots ★\mathrm{\Lambda }\left({\mathit{U}}_0,{\mathit{U}}_1,\frac{\mathrm{\Gamma }}{4{♮}^{3\mathfrak{j}-1}}\right),\hfill \end{array}$$

$$\begin{array}{cc}\hfill \mathrm{\Pi }({\mathit{U}}_0,{\mathit{U}}_{3\mathfrak{j}+1},\mathrm{\Gamma })& \le \mathrm{\Pi }\left({\mathit{U}}_0,{\mathit{U}}_1,\frac{\mathrm{\Gamma }}{4}\right)•\mathrm{\Pi }\left({\mathit{U}}_0,{\mathit{U}}_1,\frac{\mathrm{\Gamma }}{4}\right)•\mathrm{\Pi }\left({\mathit{U}}_0,{\mathit{U}}_1,\frac{\mathrm{\Gamma }}{4♮}\right)•\mathrm{\Pi }\left({\mathit{U}}_0,{\mathit{U}}_1,\frac{\mathrm{\Gamma }}{4{♮}^2}\right)\hfill \\ \hfill & •\mathrm{\Pi }\left({\mathit{U}}_0,{\mathit{U}}_1,\frac{\mathrm{\Gamma }}{4{♮}^3}\right)•\mathrm{\Pi }\left({\mathit{U}}_0,{\mathit{U}}_1,\frac{\mathrm{\Gamma }}{4{♮}^4}\right)•\mathrm{\Pi }\left({\mathit{U}}_0,{\mathit{U}}_1,\frac{\mathrm{\Gamma }}{4{♮}^5}\right)\hfill \\ \hfill & •\cdots •\mathrm{\Pi }\left({\mathit{U}}_0,{\mathit{U}}_1,\frac{\mathrm{\Gamma }}{4{♮}^{3\mathfrak{j}-1}}\right)\hfill \end{array}$$

and

$$\begin{array}{cc}\hfill \mathcal{M}({\mathit{U}}_0,{\mathit{U}}_{3\mathfrak{j}+1},\mathrm{\Gamma })& \le \mathcal{M}\left({\mathit{U}}_0,{\mathit{U}}_1,\frac{\mathrm{\Gamma }}{4}\right)•\mathcal{M}\left({\mathit{U}}_0,{\mathit{U}}_1,\frac{\mathrm{\Gamma }}{4}\right)•\mathcal{M}\left({\mathit{U}}_0,{\mathit{U}}_1,\frac{\mathrm{\Gamma }}{4♮}\right)•\mathcal{M}\left({\mathit{U}}_0,{\mathit{U}}_1,\frac{\mathrm{\Gamma }}{4{♮}^2}\right)\hfill \\ \hfill & •\mathcal{M}\left({\mathit{U}}_0,{\mathit{U}}_1,\frac{\mathrm{\Gamma }}{4{♮}^3}\right)•\mathcal{M}\left({\mathit{U}}_0,{\mathit{U}}_1,\frac{\mathrm{\Gamma }}{4{♮}^4}\right)•\mathcal{M}\left({\mathit{U}}_0,{\mathit{U}}_1,\frac{\mathrm{\Gamma }}{4{♮}^5}\right)\hfill \\ \hfill & •\cdots •\mathcal{M}\left({\mathit{U}}_0,{\mathit{U}}_1,\frac{\mathrm{\Gamma }}{4{♮}^{3\mathfrak{j}-1}}\right).\hfill \end{array}$$

Now, from (6), we get

$$\begin{array}{cc}\hfill \mathrm{\Lambda }({\mathit{U}}_{{\beta }_1},{\mathit{U}}_{{\beta }_1+3\mathfrak{j}+1},♮\mathrm{\Gamma })& \ge \mathrm{\Lambda }\left({\mathit{U}}_0,{\mathit{U}}_{3\mathfrak{j}+1},\frac{\mathrm{\Gamma }}{{♮}^{{\beta }_1-1}}\right)\hfill \\ \hfill & \ge \mathrm{\Lambda }\left({\mathit{U}}_0,{\mathit{U}}_1,\frac{\mathrm{\Gamma }}{4{♮}^{{\beta }_1-1}}\right)★\mathrm{\Lambda }\left({\mathit{U}}_0,{\mathit{U}}_1,\frac{\mathrm{\Gamma }}{4{♮}^{{\beta }_1-1}}\right)★\mathrm{\Lambda }\left({\mathit{U}}_0,{\mathit{U}}_1,\frac{\mathrm{\Gamma }}{4{♮}^{{\beta }_1}}\right)\hfill \\ \hfill & ★\mathrm{\Lambda }\left({\mathit{U}}_0,{\mathit{U}}_1,\frac{\mathrm{\Gamma }}{4{♮}^{{\beta }_1+1}}\right)★\mathrm{\Lambda }\left({\mathit{U}}_0,{\mathit{U}}_1,\frac{\mathrm{\Gamma }}{4{♮}^{{\beta }_1+2}}\right)★\mathrm{\Lambda }\left({\mathit{U}}_0,{\mathit{U}}_1,\frac{\mathrm{\Gamma }}{4{♮}^{{\beta }_1+3}}\right)\hfill \\ \hfill & ★\mathrm{\Lambda }\left({\mathit{U}}_0,{\mathit{U}}_1,\frac{\mathrm{\Gamma }}{4{♮}^{{\beta }_1+4}}\right)★\cdots ★\mathrm{\Lambda }\left({\mathit{U}}_0,{\mathit{U}}_1,\frac{\mathrm{\Gamma }}{4{♮}^{3\mathfrak{j}+{\beta }_1-2}}\right),\hfill \end{array}$$

(8)

$$\begin{array}{cc}\hfill \mathrm{\Pi }({\mathit{U}}_{{\beta }_1},{\mathit{U}}_{{\beta }_1+3\mathfrak{j}+1},♮\mathrm{\Gamma })& \le \mathrm{\Pi }\left({\mathit{U}}_0,{\mathit{U}}_{3\mathfrak{j}+1},\frac{\mathrm{\Gamma }}{{♮}^{{\beta }_1-1}}\right)\hfill \\ \hfill & \le \mathrm{\Pi }\left({\mathit{U}}_0,{\mathit{U}}_1,\frac{\mathrm{\Gamma }}{4{♮}^{{\beta }_1-1}}\right)•\mathrm{\Pi }\left({\mathit{U}}_0,{\mathit{U}}_1,\frac{\mathrm{\Gamma }}{4{♮}^{{\beta }_1-1}}\right)•\mathrm{\Pi }\left({\mathit{U}}_0,{\mathit{U}}_1,\frac{\mathrm{\Gamma }}{4{♮}^{{\beta }_1}}\right)\hfill \\ \hfill & •\mathrm{\Pi }\left({\mathit{U}}_0,{\mathit{U}}_1,\frac{\mathrm{\Gamma }}{4{♮}^{{\beta }_1+1}}\right)•\mathrm{\Pi }\left({\mathit{U}}_0,{\mathit{U}}_1,\frac{\mathrm{\Gamma }}{4{♮}^{{\beta }_1+2}}\right)•\mathrm{\Pi }\left({\mathit{U}}_0,{\mathit{U}}_1,\frac{\mathrm{\Gamma }}{4{♮}^{{\beta }_1+3}}\right)\hfill \\ \hfill & •\mathrm{\Pi }\left({\mathit{U}}_0,{\mathit{U}}_1,\frac{\mathrm{\Gamma }}{4{♮}^{{\beta }_1+4}}\right)•\cdots •\mathrm{\Pi }\left({\mathit{U}}_0,{\mathit{U}}_1,\frac{\mathrm{\Gamma }}{4{♮}^{3\mathfrak{j}+{\beta }_1-2}}\right)\hfill \end{array}$$

(9)

and

$$\begin{array}{cc}\hfill \mathcal{M}({\mathit{U}}_{{\beta }_1},{\mathit{U}}_{{\beta }_1+3\mathfrak{j}+1},♮\mathrm{\Gamma })& \le \mathcal{M}\left({\mathit{U}}_0,{\mathit{U}}_{3\mathfrak{j}+1},\frac{\mathrm{\Gamma }}{{♮}^{{\beta }_1-1}}\right)\hfill \\ \hfill & \le \mathcal{M}\left({\mathit{U}}_0,{\mathit{U}}_1,\frac{\mathrm{\Gamma }}{4{♮}^{{\beta }_1-1}}\right)•\mathcal{M}\left({\mathit{U}}_0,{\mathit{U}}_1,\frac{\mathrm{\Gamma }}{4{♮}^{{\beta }_1-1}}\right)•\mathcal{M}\left({\mathit{U}}_0,{\mathit{U}}_1,\frac{\mathrm{\Gamma }}{4{♮}^{{\beta }_1}}\right)\hfill \\ \hfill & •\mathcal{M}\left({\mathit{U}}_0,{\mathit{U}}_1,\frac{\mathrm{\Gamma }}{4{♮}^{{\beta }_1+1}}\right)•\mathcal{M}\left({\mathit{U}}_0,{\mathit{U}}_1,\frac{\mathrm{\Gamma }}{4{♮}^{{\beta }_1+2}}\right)•\mathcal{M}\left({\mathit{U}}_0,{\mathit{U}}_1,\frac{\mathrm{\Gamma }}{4{♮}^{{\beta }_1+3}}\right)\hfill \\ \hfill & •\mathcal{M}\left({\mathit{U}}_0,{\mathit{U}}_1,\frac{\mathrm{\Gamma }}{4{♮}^{{\beta }_1+4}}\right)•\cdots •\mathcal{M}\left({\mathit{U}}_0,{\mathit{U}}_1,\frac{\mathrm{\Gamma }}{4{♮}^{3\mathfrak{j}+{\beta }_1-2}}\right).\hfill \end{array}$$

(10)

By using (4) and (5), we obtain for each $\mathfrak{j}=1,2,3,\dots ,$

$$\begin{array}{cc}\hfill \mathrm{\Lambda }({\mathit{U}}_0,{\mathit{U}}_{3\mathfrak{j}+2},\mathrm{\Gamma })& \ge \mathrm{\Lambda }\left({\mathit{U}}_0,{\mathit{U}}_1,\frac{\mathrm{\Gamma }}{4}\right)★\mathrm{\Lambda }\left({\mathit{U}}_0,{\mathit{U}}_1,\frac{\mathrm{\Gamma }}{4}\right)★\mathrm{\Lambda }\left({\mathit{U}}_0,{\mathit{U}}_1,\frac{\mathrm{\Gamma }}{4♮}\right)★\mathrm{\Lambda }\left({\mathit{U}}_0,{\mathit{U}}_1,\frac{\mathrm{\Gamma }}{4{♮}^2}\right)\hfill \\ \hfill & ★\mathrm{\Lambda }\left({\mathit{U}}_0,{\mathit{U}}_1,\frac{\mathrm{\Gamma }}{4{♮}^3}\right)★\mathrm{\Lambda }\left({\mathit{U}}_0,{\mathit{U}}_1,\frac{\mathrm{\Gamma }}{4{♮}^4}\right)★\mathrm{\Lambda }\left({\mathit{U}}_0,{\mathit{U}}_2,\frac{\mathrm{\Gamma }}{4{♮}^5}\right)\hfill \\ \hfill & ★\cdots ★\mathrm{\Lambda }\left({\mathit{U}}_0,{\mathit{U}}_2,\frac{\mathrm{\Gamma }}{4{♮}^{3\mathfrak{j}-1}}\right),\hfill \end{array}$$

$$\begin{array}{cc}\hfill \mathrm{\Pi }({\mathit{U}}_0,{\mathit{U}}_{3\mathfrak{j}+2},\mathrm{\Gamma })& \le \mathrm{\Pi }\left({\mathit{U}}_0,{\mathit{U}}_1,\frac{\mathrm{\Gamma }}{4}\right)•\mathrm{\Pi }\left({\mathit{U}}_0,{\mathit{U}}_1,\frac{\mathrm{\Gamma }}{4}\right)•\mathrm{\Pi }\left({\mathit{U}}_0,{\mathit{U}}_1,\frac{\mathrm{\Gamma }}{4♮}\right)•\mathrm{\Pi }\left({\mathit{U}}_0,{\mathit{U}}_1,\frac{\mathrm{\Gamma }}{4{♮}^2}\right)\hfill \\ \hfill & •\mathrm{\Pi }\left({\mathit{U}}_0,{\mathit{U}}_1,\frac{\mathrm{\Gamma }}{4{♮}^3}\right)•\mathrm{\Pi }\left({\mathit{U}}_0,{\mathit{U}}_1,\frac{\mathrm{\Gamma }}{4{♮}^4}\right)•\mathrm{\Pi }\left({\mathit{U}}_0,{\mathit{U}}_2,\frac{\mathrm{\Gamma }}{4{♮}^5}\right)\hfill \\ \hfill & •\cdots •\mathrm{\Pi }\left({\mathit{U}}_0,{\mathit{U}}_2,\frac{\mathrm{\Gamma }}{4{♮}^{3\mathfrak{j}-1}}\right)\hfill \end{array}$$

and

$$\begin{array}{cc}\hfill \mathcal{M}({\mathit{U}}_0,{\mathit{U}}_{3\mathfrak{j}+2},\mathrm{\Gamma })& \le \mathcal{M}\left({\mathit{U}}_0,{\mathit{U}}_1,\frac{\mathrm{\Gamma }}{4}\right)•\mathcal{M}\left({\mathit{U}}_0,{\mathit{U}}_1,\frac{\mathrm{\Gamma }}{4}\right)•\mathcal{M}\left({\mathit{U}}_0,{\mathit{U}}_1,\frac{\mathrm{\Gamma }}{4♮}\right)•\mathcal{M}\left({\mathit{U}}_0,{\mathit{U}}_1,\frac{\mathrm{\Gamma }}{4{♮}^2}\right)\hfill \\ \hfill & •\mathcal{M}\left({\mathit{U}}_0,{\mathit{U}}_1,\frac{\mathrm{\Gamma }}{4{♮}^3}\right)•\mathcal{M}\left({\mathit{U}}_0,{\mathit{U}}_1,\frac{\mathrm{\Gamma }}{4{♮}^4}\right)•\mathcal{M}\left({\mathit{U}}_0,{\mathit{U}}_2,\frac{\mathrm{\Gamma }}{4{♮}^5}\right)\hfill \\ \hfill & •\cdots •\mathcal{M}\left({\mathit{U}}_0,{\mathit{U}}_2,\frac{\mathrm{\Gamma }}{4{♮}^{3\mathfrak{j}-1}}\right).\hfill \end{array}$$

Now, from (6), we get

$$\begin{array}{cc}\hfill \mathrm{\Lambda }({\mathit{U}}_{{\beta }_1},{\mathit{U}}_{{\beta }_1+3\mathfrak{j}+2},♮\mathrm{\Gamma })& \ge \mathrm{\Lambda }\left({\mathit{U}}_0,{\mathit{U}}_{3\mathfrak{j}+2},\frac{\mathrm{\Gamma }}{{♮}^{{\beta }_1-1}}\right)\hfill \\ \hfill & \ge \mathrm{\Lambda }\left({\mathit{U}}_0,{\mathit{U}}_1,\frac{\mathrm{\Gamma }}{4{♮}^{{\beta }_1-1}}\right)★\mathrm{\Lambda }\left({\mathit{U}}_0,{\mathit{U}}_1,\frac{\mathrm{\Gamma }}{4{♮}^{{\beta }_1-1}}\right)★\mathrm{\Lambda }\left({\mathit{U}}_0,{\mathit{U}}_1,\frac{\mathrm{\Gamma }}{4{♮}^{{\beta }_1}}\right)\hfill \\ \hfill & ★\mathrm{\Lambda }\left({\mathit{U}}_0,{\mathit{U}}_1,\frac{\mathrm{\Gamma }}{4{♮}^{{\beta }_1+1}}\right)★\mathrm{\Lambda }\left({\mathit{U}}_0,{\mathit{U}}_1,\frac{\mathrm{\Gamma }}{4{♮}^{{\beta }_1+2}}\right)★\mathrm{\Lambda }\left({\mathit{U}}_0,{\mathit{U}}_1,\frac{\mathrm{\Gamma }}{4{♮}^{{\beta }_1+3}}\right)\hfill \\ \hfill & ★\mathrm{\Lambda }\left({\mathit{U}}_0,{\mathit{U}}_2,\frac{\mathrm{\Gamma }}{4{♮}^{{\beta }_1+4}}\right)★\cdots ★\mathrm{\Lambda }\left({\mathit{U}}_0,{\mathit{U}}_2,\frac{\mathrm{\Gamma }}{4{♮}^{3\mathfrak{j}+{\beta }_1-2}}\right),\hfill \end{array}$$

(11)

$$\begin{array}{cc}\hfill \mathrm{\Pi }({\mathit{U}}_{{\beta }_1},{\mathit{U}}_{{\beta }_1+3\mathfrak{j}+2},♮\mathrm{\Gamma })& \le \mathrm{\Pi }\left({\mathit{U}}_0,{\mathit{U}}_{3\mathfrak{j}+2},\frac{\mathrm{\Gamma }}{{♮}^{{\beta }_1-1}}\right)\hfill \\ \hfill & \le \mathrm{\Pi }\left({\mathit{U}}_0,{\mathit{U}}_1,\frac{\mathrm{\Gamma }}{4{♮}^{{\beta }_1-1}}\right)•\mathrm{\Pi }\left({\mathit{U}}_0,{\mathit{U}}_1,\frac{\mathrm{\Gamma }}{4{♮}^{{\beta }_1-1}}\right)•\mathrm{\Pi }\left({\mathit{U}}_0,{\mathit{U}}_1,\frac{\mathrm{\Gamma }}{4{♮}^{{\beta }_1}}\right)\hfill \\ \hfill & •\mathrm{\Pi }\left({\mathit{U}}_0,{\mathit{U}}_1,\frac{\mathrm{\Gamma }}{4{♮}^{{\beta }_1+1}}\right)•\mathrm{\Pi }\left({\mathit{U}}_0,{\mathit{U}}_1,\frac{\mathrm{\Gamma }}{4{♮}^{{\beta }_1+2}}\right)•\mathrm{\Pi }\left({\mathit{U}}_0,{\mathit{U}}_1,\frac{\mathrm{\Gamma }}{4{♮}^{{\beta }_1+3}}\right)\hfill \\ \hfill & •\mathrm{\Pi }\left({\mathit{U}}_0,{\mathit{U}}_2,\frac{\mathrm{\Gamma }}{4{♮}^{{\beta }_1+4}}\right)•\cdots •\mathrm{\Pi }\left({\mathit{U}}_0,{\mathit{U}}_2,\frac{\mathrm{\Gamma }}{4{♮}^{3\mathfrak{j}+{\beta }_1-2}}\right)\hfill \end{array}$$

(12)

and

$$\begin{array}{cc}\hfill \mathcal{M}({\mathit{U}}_{{\beta }_1},{\mathit{U}}_{{\beta }_1+3\mathfrak{j}+2},♮\mathrm{\Gamma })& \le \mathcal{M}\left({\mathit{U}}_0,{\mathit{U}}_{3\mathfrak{j}+2},\frac{\mathrm{\Gamma }}{{♮}^{{\beta }_1-1}}\right)\hfill \\ \hfill & \le \mathcal{M}\left({\mathit{U}}_0,{\mathit{U}}_1,\frac{\mathrm{\Gamma }}{4{♮}^{{\beta }_1-1}}\right)•\mathcal{M}\left({\mathit{U}}_0,{\mathit{U}}_1,\frac{\mathrm{\Gamma }}{4{♮}^{{\beta }_1-1}}\right)•\mathcal{M}\left({\mathit{U}}_0,{\mathit{U}}_1,\frac{\mathrm{\Gamma }}{4{♮}^{{\beta }_1}}\right)\hfill \\ \hfill & •\mathcal{M}\left({\mathit{U}}_0,{\mathit{U}}_1,\frac{\mathrm{\Gamma }}{4{♮}^{{\beta }_1+1}}\right)•\mathcal{M}\left({\mathit{U}}_0,{\mathit{U}}_1,\frac{\mathrm{\Gamma }}{4{♮}^{{\beta }_1+2}}\right)•\mathcal{M}\left({\mathit{U}}_0,{\mathit{U}}_1,\frac{\mathrm{\Gamma }}{4{♮}^{{\beta }_1+3}}\right)\hfill \\ \hfill & •\mathcal{M}\left({\mathit{U}}_0,{\mathit{U}}_2,\frac{\mathrm{\Gamma }}{4{♮}^{{\beta }_1+4}}\right)•\cdots •\mathcal{M}\left({\mathit{U}}_0,{\mathit{U}}_2,\frac{\mathrm{\Gamma }}{4{♮}^{3\mathfrak{j}+{\beta }_1-2}}\right).\hfill \end{array}$$

(13)

By using (4) and (5), we obtain for each $\mathfrak{j}=1,2,3,\dots ,$

$$\begin{array}{cc}\hfill \mathrm{\Lambda }({\mathit{U}}_0,{\mathit{U}}_{3\mathfrak{j}+3},\mathrm{\Gamma })& \ge \mathrm{\Lambda }\left({\mathit{U}}_0,{\mathit{U}}_1,\frac{\mathrm{\Gamma }}{4}\right)★\mathrm{\Lambda }\left({\mathit{U}}_0,{\mathit{U}}_1,\frac{\mathrm{\Gamma }}{4}\right)★\mathrm{\Lambda }\left({\mathit{U}}_0,{\mathit{U}}_1,\frac{\mathrm{\Gamma }}{4♮}\right)★\mathrm{\Lambda }\left({\mathit{U}}_0,{\mathit{U}}_1,\frac{\mathrm{\Gamma }}{4{♮}^2}\right)\hfill \\ \hfill & ★\mathrm{\Lambda }\left({\mathit{U}}_0,{\mathit{U}}_1,\frac{\mathrm{\Gamma }}{4{♮}^3}\right)★\mathrm{\Lambda }\left({\mathit{U}}_0,{\mathit{U}}_1,\frac{\mathrm{\Gamma }}{4{♮}^4}\right)★\mathrm{\Lambda }\left({\mathit{U}}_0,{\mathit{U}}_3,\frac{\mathrm{\Gamma }}{4{♮}^5}\right)\hfill \\ \hfill & ★\cdots ★\mathrm{\Lambda }\left({\mathit{U}}_0,{\mathit{U}}_3,\frac{\mathrm{\Gamma }}{4{♮}^{3\mathfrak{j}-1}}\right),\hfill \end{array}$$

$$\begin{array}{cc}\hfill \mathrm{\Pi }({\mathit{U}}_0,{\mathit{U}}_{3\mathfrak{j}+3},\mathrm{\Gamma })& \le \mathrm{\Pi }\left({\mathit{U}}_0,{\mathit{U}}_1,\frac{\mathrm{\Gamma }}{4}\right)•\mathrm{\Pi }\left({\mathit{U}}_0,{\mathit{U}}_1,\frac{\mathrm{\Gamma }}{4}\right)•\mathrm{\Pi }\left({\mathit{U}}_0,{\mathit{U}}_1,\frac{\mathrm{\Gamma }}{4♮}\right)•\mathrm{\Pi }\left({\mathit{U}}_0,{\mathit{U}}_1,\frac{\mathrm{\Gamma }}{4{♮}^2}\right)\hfill \\ \hfill & •\mathrm{\Pi }\left({\mathit{U}}_0,{\mathit{U}}_1,\frac{\mathrm{\Gamma }}{4{♮}^3}\right)•\mathrm{\Pi }\left({\mathit{U}}_0,{\mathit{U}}_1,\frac{\mathrm{\Gamma }}{4{♮}^4}\right)•\mathrm{\Pi }\left({\mathit{U}}_0,{\mathit{U}}_3,\frac{\mathrm{\Gamma }}{4{♮}^5}\right)\hfill \\ \hfill & •\cdots •\mathrm{\Pi }\left({\mathit{U}}_0,{\mathit{U}}_3,\frac{\mathrm{\Gamma }}{4{♮}^{3\mathfrak{j}-1}}\right)\hfill \end{array}$$

and

$$\begin{array}{cc}\hfill \mathcal{M}({\mathit{U}}_0,{\mathit{U}}_{3\mathfrak{j}+3},\mathrm{\Gamma })& \le \mathcal{M}\left({\mathit{U}}_0,{\mathit{U}}_1,\frac{\mathrm{\Gamma }}{4}\right)•\mathcal{M}\left({\mathit{U}}_0,{\mathit{U}}_1,\frac{\mathrm{\Gamma }}{4}\right)•\mathcal{M}\left({\mathit{U}}_0,{\mathit{U}}_1,\frac{\mathrm{\Gamma }}{4♮}\right)•\mathcal{M}\left({\mathit{U}}_0,{\mathit{U}}_1,\frac{\mathrm{\Gamma }}{4{♮}^2}\right)\hfill \\ \hfill & •\mathcal{M}\left({\mathit{U}}_0,{\mathit{U}}_1,\frac{\mathrm{\Gamma }}{4{♮}^3}\right)•\mathcal{M}\left({\mathit{U}}_0,{\mathit{U}}_1,\frac{\mathrm{\Gamma }}{4{♮}^4}\right)•\mathcal{M}\left({\mathit{U}}_0,{\mathit{U}}_3,\frac{\mathrm{\Gamma }}{4{♮}^5}\right)\hfill \\ \hfill & •\cdots •\mathcal{M}\left({\mathit{U}}_0,{\mathit{U}}_3,\frac{\mathrm{\Gamma }}{4{♮}^{3\mathfrak{j}-1}}\right).\hfill \end{array}$$

Now, from (6), we get

$$\begin{array}{cc}\hfill \mathrm{\Lambda }({\mathit{U}}_{{\beta }_1},{\mathit{U}}_{{\beta }_1+3\mathfrak{j}+3},♮\mathrm{\Gamma })& \ge \mathrm{\Lambda }\left({\mathit{U}}_0,{\mathit{U}}_{3\mathfrak{j}+3},\frac{\mathrm{\Gamma }}{{♮}^{{\beta }_1-1}}\right)\hfill \\ \hfill & \ge \mathrm{\Lambda }\left({\mathit{U}}_0,{\mathit{U}}_1,\frac{\mathrm{\Gamma }}{4{♮}^{{\beta }_1-1}}\right)★\mathrm{\Lambda }\left({\mathit{U}}_0,{\mathit{U}}_1,\frac{\mathrm{\Gamma }}{4{♮}^{{\beta }_1-1}}\right)★\mathrm{\Lambda }\left({\mathit{U}}_0,{\mathit{U}}_1,\frac{\mathrm{\Gamma }}{4{♮}^{{\beta }_1}}\right)\hfill \\ \hfill & ★\mathrm{\Lambda }\left({\mathit{U}}_0,{\mathit{U}}_1,\frac{\mathrm{\Gamma }}{4{♮}^{{\beta }_1+1}}\right)★\mathrm{\Lambda }\left({\mathit{U}}_0,{\mathit{U}}_1,\frac{\mathrm{\Gamma }}{4{♮}^{{\beta }_1+2}}\right)★\mathrm{\Lambda }\left({\mathit{U}}_0,{\mathit{U}}_1,\frac{\mathrm{\Gamma }}{4{♮}^{{\beta }_1+3}}\right)\hfill \\ \hfill & ★\mathrm{\Lambda }\left({\mathit{U}}_0,{\mathit{U}}_3,\frac{\mathrm{\Gamma }}{4{♮}^{{\beta }_1+4}}\right)★\cdots ★\mathrm{\Lambda }\left({\mathit{U}}_0,{\mathit{U}}_3,\frac{\mathrm{\Gamma }}{4{♮}^{3\mathfrak{j}+{\beta }_1-2}}\right),\hfill \end{array}$$

(14)

$$\begin{array}{cc}\hfill \mathrm{\Pi }({\mathit{U}}_{{\beta }_1},{\mathit{U}}_{{\beta }_1+3\mathfrak{j}+3},♮\mathrm{\Gamma })& \le \mathrm{\Pi }\left({\mathit{U}}_0,{\mathit{U}}_{3\mathfrak{j}+3},\frac{\mathrm{\Gamma }}{{♮}^{{\beta }_1-1}}\right)\hfill \\ \hfill & \le \mathrm{\Pi }\left({\mathit{U}}_0,{\mathit{U}}_1,\frac{\mathrm{\Gamma }}{4{♮}^{{\beta }_1-1}}\right)•\mathrm{\Pi }\left({\mathit{U}}_0,{\mathit{U}}_1,\frac{\mathrm{\Gamma }}{4{♮}^{{\beta }_1-1}}\right)•\mathrm{\Pi }\left({\mathit{U}}_0,{\mathit{U}}_1,\frac{\mathrm{\Gamma }}{4{♮}^{{\beta }_1}}\right)\hfill \\ \hfill & •\mathrm{\Pi }\left({\mathit{U}}_0,{\mathit{U}}_1,\frac{\mathrm{\Gamma }}{4{♮}^{{\beta }_1+1}}\right)•\mathrm{\Pi }\left({\mathit{U}}_0,{\mathit{U}}_1,\frac{\mathrm{\Gamma }}{4{♮}^{{\beta }_1+1}}\right)•\mathrm{\Pi }\left({\mathit{U}}_0,{\mathit{U}}_1,\frac{\mathrm{\Gamma }}{4{♮}^{{\beta }_1+3}}\right)\hfill \\ \hfill & •\mathrm{\Pi }\left({\mathit{U}}_0,{\mathit{U}}_3,\frac{\mathrm{\Gamma }}{4{♮}^{{\beta }_1+4}}\right)•\cdots •\mathrm{\Pi }\left({\mathit{U}}_0,{\mathit{U}}_3,\frac{\mathrm{\Gamma }}{4{♮}^{3\mathfrak{j}+{\beta }_1-2}}\right)\hfill \end{array}$$

(15)

and

$$\begin{array}{cc}\hfill \mathcal{M}({\mathit{U}}_{{\beta }_1},{\mathit{U}}_{{\beta }_1+3\mathfrak{j}+3},♮\mathrm{\Gamma })& \le \mathcal{M}\left({\mathit{U}}_0,{\mathit{U}}_{3\mathfrak{j}+3},\frac{\mathrm{\Gamma }}{{♮}^{{\beta }_1-1}}\right)\hfill \\ \hfill & \le \mathcal{M}\left({\mathit{U}}_0,{\mathit{U}}_1,\frac{\mathrm{\Gamma }}{4{♮}^{{\beta }_1-1}}\right)•\mathcal{M}\left({\mathit{U}}_0,{\mathit{U}}_1,\frac{\mathrm{\Gamma }}{4{♮}^{{\beta }_1-1}}\right)•\mathcal{M}\left({\mathit{U}}_0,{\mathit{U}}_1,\frac{\mathrm{\Gamma }}{4{♮}^{{\beta }_1}}\right)\hfill \\ \hfill & •\mathcal{M}\left({\mathit{U}}_0,{\mathit{U}}_1,\frac{\mathrm{\Gamma }}{4{♮}^{{\beta }_1+1}}\right)•\mathcal{M}\left({\mathit{U}}_0,{\mathit{U}}_1,\frac{\mathrm{\Gamma }}{4{♮}^{{\beta }_1+2}}\right)•\mathcal{M}\left({\mathit{U}}_0,{\mathit{U}}_1,\frac{\mathrm{\Gamma }}{4{♮}^{{\beta }_1+3}}\right)\hfill \\ \hfill & •\mathcal{M}\left({\mathit{U}}_0,{\mathit{U}}_2,\frac{\mathrm{\Gamma }}{4{♮}^{{\beta }_1+4}}\right)•\cdots •\mathcal{M}\left({\mathit{U}}_0,{\mathit{U}}_2,\frac{\mathrm{\Gamma }}{4{♮}^{3\mathfrak{j}+{\beta }_1-2}}\right).\hfill \end{array}$$

(16)

Using (8)–(16), for each case ${\beta }_1\to +\infty$, we deduce that

$$\begin{array}{cc}\hfill \underset{{\beta }_1\to +\infty }{lim}\mathrm{\Lambda }({\mathit{U}}_{{\beta }_1},{\mathit{U}}_{{\beta }_1+\mathfrak{i}},\mathrm{\Gamma })=1★1★\cdots ★1& =1,\hfill \\ \hfill \underset{{\beta }_1\to +\infty }{lim}\mathrm{\Pi }({\mathit{U}}_{{\beta }_1},{\mathit{U}}_{{\beta }_1+\mathfrak{i}},\mathrm{\Gamma })=0•0•\cdots •0& =0\hfill \end{array}$$

and

$$\begin{array}{cc}\hfill \underset{{\beta }_1\to +\infty }{lim}\mathcal{M}({\mathit{U}}_{{\beta }_1},{\mathit{U}}_{{\beta }_1+\mathfrak{i}},\mathrm{\Gamma })=0•0•\cdots •0& =0.\hfill \end{array}$$

Which implies that $\left\{{\mathit{U}}_{{\beta }_1}\right\}$ is a Cauchy sequence. Since $(\mathit{K},\mathrm{\Lambda },\mathrm{\Pi },\mathcal{M},★,•)$ is a complete NPMS, we have

$$\begin{array}{c}\hfill \underset{{\beta }_1\to +\infty }{lim}{\mathit{U}}_{{\beta }_1}=\mathit{U}.\end{array}$$

Now, $\mathit{U}$ is a fixed point of ℘, using conditions (v), (x), (xv) and Equation (2), we obtained

$$\begin{array}{cc}\hfill \mathrm{\Lambda }(\mathit{U},\wp \mathit{U},\mathrm{\Gamma })& \ge \mathrm{\Lambda }\left(\mathit{U},{\mathit{U}}_{{\beta }_1},\frac{\mathrm{\Gamma }}{4}\right)★\mathrm{\Lambda }\left({\mathit{U}}_{{\beta }_1},{\mathit{U}}_{{\beta }_1+1},\frac{\mathrm{\Gamma }}{4}\right)★\mathrm{\Lambda }\left({\mathit{U}}_{{\beta }_1+1},{\mathit{U}}_{{\beta }_1+2},\frac{\mathrm{\Gamma }}{4}\right)★\mathrm{\Lambda }\left({\mathit{U}}_{{\beta }_1+2},\wp \mathit{U},\frac{\mathrm{\Gamma }}{4}\right)\hfill \\ \hfill & =\mathrm{\Lambda }\left(\mathit{U},{\mathit{U}}_{{\beta }_1},\frac{\mathrm{\Gamma }}{4}\right)★\mathrm{\Lambda }\left(\wp {\mathit{U}}_{{\beta }_1-1},\wp {\mathit{U}}_{{\beta }_1},\frac{\mathrm{\Gamma }}{4}\right)★\mathrm{\Lambda }\left(\wp {\mathit{U}}_{{\beta }_1},\wp {\mathit{U}}_{{\beta }_1+1},\frac{\mathrm{\Gamma }}{4}\right)\hfill \\ \hfill & ★\mathrm{\Lambda }\left(\wp {\mathit{U}}_{{\beta }_1+1},\wp \mathit{U},\frac{\mathrm{\Gamma }}{4}\right)\hfill \\ \hfill & \ge \mathrm{\Lambda }\left(\mathit{U},{\mathit{U}}_{{\beta }_1},\frac{\mathrm{\Gamma }}{4}\right)★\mathrm{\Lambda }\left({\mathit{U}}_{{\beta }_1-1},{\mathit{U}}_{{\beta }_1},\frac{\mathrm{\Gamma }}{4{♮}^{{\beta }_1-2}}\right)★\mathrm{\Lambda }\left({\mathit{U}}_{{\beta }_1},{\mathit{U}}_{{\beta }_1+1},\frac{\mathrm{\Gamma }}{4{♮}^{{\beta }_1-1}}\right)\hfill \\ \hfill & ★\mathrm{\Lambda }\left({\mathit{U}}_{{\beta }_1+1},\mathit{U},\frac{\mathrm{\Gamma }}{4♮}\right)\hfill \\ \hfill & \to 1★1★1★1=1\mathrm{as}{\beta }_1\to +\infty ,\hfill \end{array}$$

$$\begin{array}{cc}\hfill \mathrm{\Pi }(\mathit{U},\wp \mathit{U},\mathrm{\Gamma })& \le \mathrm{\Pi }\left(\mathit{U},{\mathit{U}}_{{\beta }_1},\frac{\mathrm{\Gamma }}{4}\right)•\mathrm{\Pi }\left({\mathit{U}}_{{\beta }_1},{\mathit{U}}_{{\beta }_1+1},\frac{\mathrm{\Gamma }}{4}\right)•\mathrm{\Pi }\left({\mathit{U}}_{{\beta }_1+1},{\mathit{U}}_{{\beta }_1+2},\frac{\mathrm{\Gamma }}{4}\right)•\mathrm{\Pi }\left({\mathit{U}}_{{\beta }_1+2},\wp \mathit{U},\frac{\mathrm{\Gamma }}{4}\right)\hfill \\ \hfill & =\mathrm{\Pi }\left(\mathit{U},{\mathit{U}}_{{\beta }_1},\frac{\mathrm{\Gamma }}{4}\right)•\mathrm{\Pi }\left(\wp {\mathit{U}}_{{\beta }_1-1},\wp {\mathit{U}}_{{\beta }_1},\frac{\mathrm{\Gamma }}{4}\right)•\mathrm{\Pi }\left(\wp {\mathit{U}}_{{\beta }_1},\wp {\mathit{U}}_{{\beta }_1+1},\frac{\mathrm{\Gamma }}{4}\right)\hfill \\ \hfill & •\mathrm{\Pi }\left(\wp {\mathit{U}}_{{\beta }_1+1},\wp \mathit{U},\frac{\mathrm{\Gamma }}{4}\right)\hfill \\ \hfill & \le \mathrm{\Pi }\left(\mathit{U},{\mathit{U}}_{{\beta }_1},\frac{\mathrm{\Gamma }}{4}\right)•\mathrm{\Pi }\left({\mathit{U}}_{{\beta }_1-1},{\mathit{U}}_{{\beta }_1},\frac{\mathrm{\Gamma }}{4{♮}^{{\beta }_1-2}}\right)•\mathrm{\Pi }\left({\mathit{U}}_{{\beta }_1},{\mathit{U}}_{{\beta }_1+1},\frac{\mathrm{\Gamma }}{4{♮}^{{\beta }_1-1}}\right)\hfill \\ \hfill & •\mathrm{\Pi }\left({\mathit{U}}_{{\beta }_1+1},\mathit{U},\frac{\mathrm{\Gamma }}{4♮}\right)\hfill \\ \hfill & \to 0•0•0•0=0\mathrm{as}{\beta }_1\to +\infty ,\hfill \end{array}$$

and

$$\begin{array}{cc}\hfill \mathcal{M}(\mathit{U},\wp \mathit{U},\mathrm{\Gamma })& \le \mathcal{M}\left(\mathit{U},{\mathit{U}}_{{\beta }_1},\frac{\mathrm{\Gamma }}{4}\right)•\mathcal{M}\left({\mathit{U}}_{{\beta }_1},{\mathit{U}}_{{\beta }_1+1},\frac{\mathrm{\Gamma }}{4}\right)\hfill \\ \hfill & •\mathcal{M}\left({\mathit{U}}_{{\beta }_1+1},{\mathit{U}}_{{\beta }_1+2},\frac{\mathrm{\Gamma }}{4}\right)•\mathcal{M}\left({\mathit{U}}_{{\beta }_1+2},\wp \mathit{U},\frac{\mathrm{\Gamma }}{4}\right)\hfill \\ \hfill & =\mathcal{M}\left(\mathit{U},{\mathit{U}}_{{\beta }_1},\frac{\mathrm{\Gamma }}{4}\right)•\mathcal{M}\left(\wp {\mathit{U}}_{{\beta }_1-1},\wp {\mathit{U}}_{{\beta }_1},\frac{\mathrm{\Gamma }}{4}\right)•\mathcal{M}\left(\wp {\mathit{U}}_{{\beta }_1},\wp {\mathit{U}}_{{\beta }_1+1},\frac{\mathrm{\Gamma }}{4}\right)\hfill \\ \hfill & •\mathcal{M}\left(\wp {\mathit{U}}_{{\beta }_1+1},\wp \mathit{U},\frac{\mathrm{\Gamma }}{4}\right)\hfill \\ \hfill & \le \mathcal{M}\left(\mathit{U},{\mathit{U}}_{{\beta }_1},\frac{\mathrm{\Gamma }}{4}\right)•\mathcal{M}\left({\mathit{U}}_{{\beta }_1-1},{\mathit{U}}_{{\beta }_1},\frac{\mathrm{\Gamma }}{4{♮}^{{\beta }_1-2}}\right)•\mathcal{M}\left({\mathit{U}}_{{\beta }_1},{\mathit{U}}_{{\beta }_1+1},\frac{\mathrm{\Gamma }}{4{♮}^{{\beta }_1-1}}\right)\hfill \\ \hfill & •\mathcal{M}\left({\mathit{U}}_{{\beta }_1+1},\mathit{U},\frac{\mathrm{\Gamma }}{4♮}\right)\hfill \\ \hfill & \to 0•0•0•0=0\mathrm{as}{\beta }_1\to +\infty ,\hfill \end{array}$$

Hence, $\wp \mathit{U}=\mathit{U}$. Let $\wp \tilde{\mu }=\tilde{\mu }$ for some $\tilde{\mu }\in \mathit{K}$, then

$$\begin{array}{cc}\hfill 1& \ge \mathrm{\Lambda }(\tilde{\mu },\mathit{U},\mathrm{\Gamma })=\mathrm{\Lambda }(\wp \tilde{\mu },\wp \mathit{U},\mathrm{\Gamma })\ge \mathrm{\Lambda }\left(\tilde{\mu },\mathit{U},\frac{\mathrm{\Gamma }}{♮}\right)=\mathrm{\Lambda }\left(\wp \tilde{\mu },\wp \mathit{U},\frac{\mathrm{\Gamma }}{♮}\right)\hfill \\ \hfill & \ge \mathrm{\Lambda }\left(\tilde{\mu },\mathit{U},\frac{\mathrm{\Gamma }}{{♮}^2}\right)\ge \cdots \ge \mathrm{\Lambda }\left(\tilde{\mu },\mathit{U},\frac{\mathrm{\Gamma }}{{♮}^{{\beta }_1}}\right)\to 1\mathrm{as}{\beta }_1\to +\infty ,\hfill \\ \hfill 0& \le \mathrm{\Pi }(\tilde{\mu },\mathit{U},\mathrm{\Gamma })=\mathrm{\Pi }(\wp \tilde{\mu },\wp \mathit{U},\mathrm{\Gamma })\le \mathrm{\Pi }\left(\tilde{\mu },\mathit{U},\frac{\mathrm{\Gamma }}{♮}\right)=\mathrm{\Pi }\left(\wp \tilde{\mu },\wp \mathit{U},\frac{\mathrm{\Gamma }}{♮}\right)\hfill \\ \hfill & \le \mathrm{\Pi }\left(\tilde{\mu },\mathit{U},\frac{\mathrm{\Gamma }}{{♮}^2}\right)\le \cdots \le \mathrm{\Pi }\left(\tilde{\mu },\mathit{U},\frac{\mathrm{\Gamma }}{{♮}^{{\beta }_1}}\right)\to 0\mathrm{as}{\beta }_1\to +\infty ,\hfill \end{array}$$

and

$$\begin{array}{cc}\hfill 0& \le \mathcal{M}(\tilde{\mu },\mathit{U},\mathrm{\Gamma })=\mathcal{M}(\wp \tilde{\mu },\wp \mathit{U},\mathrm{\Gamma })\le \mathcal{M}\left(\tilde{\mu },\mathit{U},\frac{\mathrm{\Gamma }}{♮}\right)=\mathcal{M}\left(\wp \tilde{\mu },\wp \mathit{U},\frac{\mathrm{\Gamma }}{♮}\right)\hfill \\ \hfill & \le \mathcal{M}\left(\tilde{\mu },\mathit{U},\frac{\mathrm{\Gamma }}{{♮}^2}\right)\le \cdots \le \mathcal{M}\left(\tilde{\mu },\mathit{U},\frac{\mathrm{\Gamma }}{{♮}^{{\beta }_1}}\right)\to 0\mathrm{as}{\beta }_1\to +\infty ,\hfill \end{array}$$

using by (iii), (viii) and (xiii), $\mathit{U}=\tilde{\mu }$. □

Definition 10.

Let $(\mathit{K},\mathrm{\Lambda },\mathrm{\Pi },\mathcal{M},★,•)$ be a NPMS. A function $\wp :\mathit{K}\to \mathit{K}$ is an NPC (neutrosophic pentagonal contraction) if $\exists$ $0<♮<1$, such that

$$\begin{array}{cc}\hfill \frac{1}{\mathrm{\Lambda }(\mathcal{P}\mathit{U},\mathcal{P}\mathcal{W},\mathrm{\Gamma })}-1& \le ♮\left[\frac{1}{\mathrm{\Lambda }(\mathit{U},\mathcal{W},\mathrm{\Gamma })}-1\right]\hfill \end{array}$$

(17)

$$\begin{array}{cc}\hfill \mathrm{\Pi }(\mathcal{P}\mathit{U},\mathcal{P}\mathcal{W},\mathrm{\Gamma })& \le ♮\mathrm{\Pi }(\mathit{U},\mathcal{W},\mathrm{\Gamma }),\hfill \end{array}$$

(18)

and

$$\begin{array}{c}\hfill \mathcal{M}(\mathcal{P}\mathit{U},\mathcal{P}\mathcal{W},\mathrm{\Gamma })\le ♮\mathcal{M}(\mathit{U},\mathcal{W},\mathrm{\Gamma }),\end{array}$$

(19)

for all $\mathit{U},\mathcal{W}\in \mathit{K}$ and $\mathrm{\Gamma }>0$.

Theorem 2.

Let $(\mathit{K},\mathrm{\Lambda },\mathrm{\Pi },\mathcal{M},★,•)$ be a complete NPMS with $\chi :\mathit{K}\times \mathit{K}\to [1,+\infty )$ and assume that

$$\begin{array}{c}\hfill \underset{\mathrm{\Gamma }\to +\infty }{lim}\mathcal{M}(\mathit{U},\mathcal{W},\mathrm{\Gamma })=0,\underset{\mathrm{\Gamma }\to +\infty }{lim}\mathrm{\Pi }(\mathit{U},\mathcal{W},\mathrm{\Gamma })=0and\underset{\mathrm{\Gamma }\to +\infty }{lim}\mathrm{\Lambda }(\mathit{U},\mathcal{W},\mathrm{\Gamma })=1,\end{array}$$

(20)

for all $\mathit{U},\mathcal{W}\in \mathit{K}$ and $\mathrm{\Gamma }>0$. Let $\wp :\mathit{K}\to \mathit{K}$ be a ND-controlled contraction. Furthermore, assume that for an arbitrary ${\mathit{U}}_0\in \mathit{K}$, and ${\beta }_1,\mathfrak{q}\in \mathbb{N}$, where ${\mathit{U}}_{{\beta }_1}={\wp }^{{\beta }_1}{\mathit{U}}_0=\wp {\mathit{U}}_{{\beta }_1-1}$. Then, ℘ has a ufp.

Proof. 

Suppose ${\mathit{U}}_0$ be a point of $\mathit{K}$ and define a sequence $\left\{{\mathit{U}}_{{\beta }_1}\right\}$ by ${\mathit{U}}_{{\beta }_1}={\wp }^{{\beta }_1}{\mathit{U}}_0=\wp {\mathit{U}}_{{\beta }_1-1},{\beta }_1\in \mathbb{N}$. Using by (17)–(19) for all $\mathrm{\Gamma }>0,{\beta }_1>\mathfrak{q}$, we deduce

$$\begin{array}{cc}\hfill \frac{1}{\mathrm{\Lambda }({\mathit{U}}_{{\beta }_1},{\mathit{U}}_{{\beta }_1+1},\mathrm{\Gamma })}-1& =\frac{1}{\mathrm{\Lambda }(\wp {\mathit{U}}_{{\beta }_1-1},\wp {\mathit{U}}_{{\beta }_1},\mathrm{\Gamma })}-1\hfill \\ \hfill & \le ♮\left[\frac{1}{\mathrm{\Lambda }({\mathit{U}}_{{\beta }_1-1},{\mathit{U}}_{{\beta }_1},\mathrm{\Gamma })}\right]=\frac{♮}{\mathrm{\Lambda }({\mathit{U}}_{{\beta }_1-1},{\mathit{U}}_{{\beta }_1},\mathrm{\Gamma })}-♮.\hfill \end{array}$$

This implies

$$\begin{array}{cc}\hfill \frac{1}{\mathrm{\Lambda }({\mathit{U}}_{{\beta }_1},{\mathit{U}}_{{\beta }_1+1},\mathrm{\Gamma })}& \le \frac{♮}{\mathrm{\Lambda }({\mathit{U}}_{{\beta }_1-1},{\mathit{U}}_{{\beta }_1},\mathrm{\Gamma })}+(1-♮)\hfill \\ \hfill & \le \frac{{♮}^2}{\mathrm{\Lambda }({\mathit{U}}_{{\beta }_1-2},{\mathit{U}}_{{\beta }_1-1},\mathrm{\Gamma })}+♮(1-♮)+(1-♮).\hfill \end{array}$$

In this manner, we conclude that

$$\begin{array}{cc}\hfill \frac{1}{\mathrm{\Lambda }({\mathit{U}}_{{\beta }_1},{\mathit{U}}_{{\beta }_1+1},\mathrm{\Gamma })}& \le \frac{{♮}^{{\beta }_1}}{\mathrm{\Lambda }({\mathit{U}}_0,{\mathit{U}}_1,\mathrm{\Gamma })}+{♮}^{{\beta }_1-1}(1-♮)+{♮}^{{\beta }_1-2}(1-♮)\hfill \\ \hfill & +\cdots +♮(1-♮)+(1-♮)\hfill \\ \hfill & \le \frac{{♮}^{{\beta }_1}}{\mathrm{\Lambda }({\mathit{U}}_0,{\mathit{U}}_1,\mathrm{\Gamma })}+({♮}^{{\beta }_1-1}+{♮}^{{\beta }_1-2}+\cdots +1)(1-♮)\hfill \\ \hfill & \le \frac{{♮}^{{\beta }_1}}{\mathrm{\Lambda }({\mathit{U}}_0,{\mathit{U}}_1,\mathrm{\Gamma })}+(1-{♮}^{{\beta }_1}).\hfill \end{array}$$

We obtain

$$\begin{array}{c}\hfill \frac{1}{\frac{{♮}^{{\beta }_1}}{\mathrm{\Lambda }({\mathit{U}}_0,{\mathit{U}}_1,\mathrm{\Gamma })}+(1-{♮}^{{\beta }_1})}\le \mathrm{\Lambda }({\mathit{U}}_{{\beta }_1},{\mathit{U}}_{{\beta }_1+1},\mathrm{\Gamma })\end{array}$$

$$\begin{array}{cc}\hfill \mathrm{\Pi }({\mathit{U}}_{{\beta }_1},{\mathit{U}}_{{\beta }_1+1},\mathrm{\Gamma })& =\mathrm{\Pi }(\wp {\mathit{U}}_{{\beta }_1-1},\wp {\mathit{U}}_{{\beta }_1},\mathrm{\Gamma })\le ♮\mathrm{\Pi }({\mathit{U}}_{{\beta }_1-1},{\mathit{U}}_{{\beta }_1},\mathrm{\Gamma })=\mathrm{\Pi }(\wp {\mathit{U}}_{{\beta }_1-2},\wp {\mathit{U}}_{{\beta }_1-1},\mathrm{\Gamma })\hfill \\ \hfill & \le {♮}^2\mathrm{\Pi }({\mathit{U}}_{{\beta }_1-2},{\mathit{U}}_{{\beta }_1-1},\mathrm{\Gamma })\le \cdots \le {♮}^{{\beta }_1}\mathrm{\Pi }({\mathit{U}}_0,{\mathit{U}}_1,\mathrm{\Gamma })\hfill \end{array}$$

and

$$\begin{array}{cc}\hfill \mathcal{M}({\mathit{U}}_{{\beta }_1},{\mathit{U}}_{{\beta }_1+1},\mathrm{\Gamma })& =\mathcal{M}(\wp {\mathit{U}}_{{\beta }_1-1},\wp {\mathit{U}}_{{\beta }_1},\mathrm{\Gamma })\le ♮\mathcal{M}({\mathit{U}}_{{\beta }_1-1},{\mathit{U}}_{{\beta }_1},\mathrm{\Gamma })=\mathcal{M}(\wp {\mathit{U}}_{{\beta }_1-2},\wp {\mathit{U}}_{{\beta }_1-1},\mathrm{\Gamma })\hfill \\ \hfill & \le {♮}^2\mathcal{M}({\mathit{U}}_{{\beta }_1-2},{\mathit{U}}_{{\beta }_1-1},\mathrm{\Gamma })\le \cdots \le {♮}^{{\beta }_1}\mathcal{M}({\mathit{U}}_0,{\mathit{U}}_1,\mathrm{\Gamma }).\hfill \end{array}$$

(21)

It again follows that

$$\begin{array}{cc}\hfill \frac{1}{\mathrm{\Lambda }({\mathit{U}}_{{\beta }_1},{\mathit{U}}_{{\beta }_1+2},\mathrm{\Gamma })}-1& =\frac{1}{\mathrm{\Lambda }(\wp {\mathit{U}}_{{\beta }_1-1},\wp {\mathit{U}}_{{\beta }_1+1},\mathrm{\Gamma })}-1\hfill \\ \hfill & \le ♮\left[\frac{1}{\mathrm{\Lambda }({\mathit{U}}_{{\beta }_1-1},{\mathit{U}}_{{\beta }_1+1},\mathrm{\Gamma })}\right]\hfill \\ \hfill & =\frac{♮}{\mathrm{\Lambda }({\mathit{U}}_{{\beta }_1-1},{\mathit{U}}_{{\beta }_1+1},\mathrm{\Gamma })}-♮\hfill \\ \hfill \Rightarrow \frac{1}{\mathrm{\Lambda }({\mathit{U}}_{{\beta }_1},{\mathit{U}}_{{\beta }_1+2},\mathrm{\Gamma })}& \le \frac{♮}{\mathrm{\Lambda }({\mathit{U}}_{{\beta }_1-1},{\mathit{U}}_{{\beta }_1+1},\mathrm{\Gamma })}+(1-♮)\hfill \\ \hfill & \le \frac{{♮}^2}{\mathrm{\Lambda }({\mathit{U}}_{{\beta }_1-2},{\mathit{U}}_{{\beta }_1},\mathrm{\Gamma })}+♮(1-♮)+(1-♮).\hfill \end{array}$$

In this manner, we conclude that

$$\begin{array}{cc}\hfill \frac{1}{\mathrm{\Lambda }({\mathit{U}}_{{\beta }_1},{\mathit{U}}_{{\beta }_1+2},\mathrm{\Gamma })}& \le \frac{{♮}^{{\beta }_1}}{\mathrm{\Lambda }({\mathit{U}}_0,{\mathit{U}}_2,\mathrm{\Gamma })}+{♮}^{{\beta }_1-1}(1-♮)+{♮}^{{\beta }_1-2}(1-♮)\hfill \\ \hfill & +\cdots +♮(1-♮)+(1-♮)\hfill \\ \hfill & \le \frac{{♮}^{{\beta }_1}}{\mathrm{\Lambda }({\mathit{U}}_0,{\mathit{U}}_2,\mathrm{\Gamma })}+({♮}^{{\beta }_1-1}+{♮}^{{\beta }_1-2}+\cdots +1)(1-♮)\hfill \\ \hfill & \le \frac{{♮}^{{\beta }_1}}{\mathrm{\Lambda }({\mathit{U}}_0,{\mathit{U}}_2,\mathrm{\Gamma })}+(1-{♮}^{{\beta }_1}).\hfill \end{array}$$

We obtain

$$\begin{array}{c}\hfill \frac{1}{\frac{{♮}^{{\beta }_1}}{\mathrm{\Lambda }({\mathit{U}}_0,{\mathit{U}}_2,\mathrm{\Gamma })}+(1-{♮}^{{\beta }_1})}\le \mathrm{\Lambda }({\mathit{U}}_{{\beta }_1},{\mathit{U}}_{{\beta }_1+2},\mathrm{\Gamma })\end{array}$$

$$\begin{array}{cc}\hfill \mathrm{\Pi }({\mathit{U}}_{{\beta }_1},{\mathit{U}}_{{\beta }_1+2},\mathrm{\Gamma })& =\mathrm{\Pi }(\wp {\mathit{U}}_{{\beta }_1-1},\wp {\mathit{U}}_{{\beta }_1+1},\mathrm{\Gamma })\le ♮\mathrm{\Pi }({\mathit{U}}_{{\beta }_1-1},{\mathit{U}}_{{\beta }_1+1},\mathrm{\Gamma })=\mathrm{\Pi }(\wp {\mathit{U}}_{{\beta }_1-2},\wp {\mathit{U}}_{{\beta }_1},\mathrm{\Gamma })\hfill \\ \hfill & \le {♮}^2\mathrm{\Pi }({\mathit{U}}_{{\beta }_1-2},{\mathit{U}}_{{\beta }_1},\mathrm{\Gamma })\le \cdots \le {♮}^{{\beta }_1}\mathrm{\Pi }({\mathit{U}}_0,{\mathit{U}}_2,\mathrm{\Gamma })\hfill \end{array}$$

and

$$\begin{array}{cc}\hfill \mathcal{M}({\mathit{U}}_{{\beta }_1},{\mathit{U}}_{{\beta }_1+2},\mathrm{\Gamma })& =\mathcal{M}(\wp {\mathit{U}}_{{\beta }_1-1},\wp {\mathit{U}}_{{\beta }_1+1},\mathrm{\Gamma })\le ♮\mathcal{M}({\mathit{U}}_{{\beta }_1-1},{\mathit{U}}_{{\beta }_1+1},\mathrm{\Gamma })=\mathcal{M}(\wp {\mathit{U}}_{{\beta }_1-2},\wp {\mathit{U}}_{{\beta }_1},\mathrm{\Gamma })\hfill \\ \hfill & \le {♮}^2\mathcal{M}({\mathit{U}}_{{\beta }_1-2},{\mathit{U}}_{{\beta }_1},\mathrm{\Gamma })\le \cdots \le {♮}^{{\beta }_1}\mathcal{M}({\mathit{U}}_0,{\mathit{U}}_2,\mathrm{\Gamma }).\hfill \end{array}$$

Consequently,

$$\begin{array}{cc}\hfill \frac{1}{\mathrm{\Lambda }({\mathit{U}}_{{\beta }_1},{\mathit{U}}_{{\beta }_1+3},\mathrm{\Gamma })}-1& =\frac{1}{\mathrm{\Lambda }(\wp {\mathit{U}}_{{\beta }_1-1},\wp {\mathit{U}}_{{\beta }_1+2},\mathrm{\Gamma })}-1\hfill \\ \hfill & \le ♮\left[\frac{1}{\mathrm{\Lambda }({\mathit{U}}_{{\beta }_1-1},{\mathit{U}}_{{\beta }_1+2},\mathrm{\Gamma })}\right]\hfill \\ \hfill & =\frac{♮}{\mathrm{\Lambda }({\mathit{U}}_{{\beta }_1-1},{\mathit{U}}_{{\beta }_1+2},\mathrm{\Gamma })}-♮.\hfill \end{array}$$

This implies

$$\begin{array}{cc}\hfill \frac{1}{\mathrm{\Lambda }({\mathit{U}}_{{\beta }_1},{\mathit{U}}_{{\beta }_1+3},\mathrm{\Gamma })}& \le \frac{♮}{\mathrm{\Lambda }({\mathit{U}}_{{\beta }_1-1},{\mathit{U}}_{{\beta }_1+2},\mathrm{\Gamma })}+(1-♮)\hfill \\ \hfill & \le \frac{{♮}^2}{\mathrm{\Lambda }({\mathit{U}}_{{\beta }_1-2},{\mathit{U}}_{{\beta }_1+1},\mathrm{\Gamma })}+♮(1-♮)+(1-♮).\hfill \end{array}$$

In this manner, we conclude that

$$\begin{array}{cc}\hfill \frac{1}{\mathrm{\Lambda }({\mathit{U}}_{{\beta }_1},{\mathit{U}}_{{\beta }_1+3},\mathrm{\Gamma })}& \le \frac{{♮}^{{\beta }_1}}{\mathrm{\Lambda }({\mathit{U}}_0,{\mathit{U}}_3,\mathrm{\Gamma })}+{♮}^{{\beta }_1-1}(1-♮)+{♮}^{{\beta }_1-2}(1-♮)\hfill \\ \hfill & +\cdots +♮(1-♮)+(1-♮)\hfill \\ \hfill & \le \frac{{♮}^{{\beta }_1}}{\mathrm{\Lambda }({\mathit{U}}_0,{\mathit{U}}_3,\mathrm{\Gamma })}+({♮}^{{\beta }_1-1}+{♮}^{{\beta }_1-2}+\cdots +1)(1-♮)\hfill \\ \hfill & \le \frac{{♮}^{{\beta }_1}}{\mathrm{\Lambda }({\mathit{U}}_0,{\mathit{U}}_3,\mathrm{\Gamma })}+(1-{♮}^{{\beta }_1}).\hfill \end{array}$$

We obtain

$$\begin{array}{c}\hfill \frac{1}{\frac{{♮}^{{\beta }_1}}{\mathrm{\Lambda }({\mathit{U}}_0,{\mathit{U}}_3,\mathrm{\Gamma })}+(1-{♮}^{{\beta }_1})}\le \mathrm{\Lambda }({\mathit{U}}_{{\beta }_1},{\mathit{U}}_{{\beta }_1+3},\mathrm{\Gamma }),\end{array}$$

$$\begin{array}{cc}\hfill \mathrm{\Pi }({\mathit{U}}_{{\beta }_1},{\mathit{U}}_{{\beta }_1+3},\mathrm{\Gamma })& =\mathrm{\Pi }(\wp {\mathit{U}}_{{\beta }_1-1},\wp {\mathit{U}}_{{\beta }_1+2},\mathrm{\Gamma })\le ♮\mathrm{\Pi }({\mathit{U}}_{{\beta }_1-1},{\mathit{U}}_{{\beta }_1+2},\mathrm{\Gamma })=\mathrm{\Pi }(\wp {\mathit{U}}_{{\beta }_1-2},\wp {\mathit{U}}_{{\beta }_1+1},\mathrm{\Gamma })\hfill \\ \hfill & \le {♮}^2\mathrm{\Pi }({\mathit{U}}_{{\beta }_1-2},{\mathit{U}}_{{\beta }_1+1},\mathrm{\Gamma })\le \cdots \le {♮}^{{\beta }_1}\mathrm{\Pi }({\mathit{U}}_0,{\mathit{U}}_3,\mathrm{\Gamma })\hfill \end{array}$$

and

$$\begin{array}{cc}\hfill \mathcal{M}({\mathit{U}}_{{\beta }_1},{\mathit{U}}_{{\beta }_1+3},\mathrm{\Gamma })& =\mathcal{M}(\wp {\mathit{U}}_{{\beta }_1-1},\wp {\mathit{U}}_{{\beta }_1+2},\mathrm{\Gamma })\le ♮\mathcal{M}({\mathit{U}}_{{\beta }_1-1},{\mathit{U}}_{{\beta }_1+2},\mathrm{\Gamma })=\mathcal{M}(\wp {\mathit{U}}_{{\beta }_1-2},\wp {\mathit{U}}_{{\beta }_1+1},\mathrm{\Gamma })\hfill \\ \hfill & \le {♮}^2\mathcal{M}({\mathit{U}}_{{\beta }_1-2},{\mathit{U}}_{{\beta }_1+1},\mathrm{\Gamma })\le \cdots \le {♮}^{{\beta }_1}\mathcal{M}({\mathit{U}}_0,{\mathit{U}}_3,\mathrm{\Gamma }).\hfill \end{array}$$

Similarly, for $\mathfrak{j}=1,2,3,\dots$, we obtain

$$\begin{array}{cc}\hfill \frac{1}{\frac{{♮}^{{\beta }_1}}{\mathrm{\Lambda }({\mathit{U}}_0,{\mathit{U}}_{3\mathfrak{j}+1},\mathrm{\Gamma })}+(1-{♮}^{{\beta }_1})}& \le \mathrm{\Lambda }({\mathit{U}}_{{\beta }_1},{\mathit{U}}_{{\beta }_1+3\mathfrak{j}+1},\mathrm{\Gamma })\hfill \\ \hfill \mathrm{\Pi }({\mathit{U}}_{{\beta }_1},{\mathit{U}}_{{\beta }_1+3\mathfrak{j}+1},♮\mathrm{\Gamma })\le {♮}^{{\beta }_1}\mathrm{\Pi }({\mathit{U}}_0,{\mathit{U}}_{3\mathfrak{j}+1},\mathrm{\Gamma })& \mathrm{and}\mathcal{M}({\mathit{U}}_{{\beta }_1},{\mathit{U}}_{{\beta }_1+3\mathfrak{j}+1},♮\mathrm{\Gamma })\le {♮}^{{\beta }_1}\mathcal{M}({\mathit{U}}_0,{\mathit{U}}_{3\mathfrak{j}+1},\mathrm{\Gamma }),\hfill \\ \hfill \frac{1}{\frac{{♮}^{{\beta }_1}}{\mathrm{\Lambda }({\mathit{U}}_0,{\mathit{U}}_{3\mathfrak{j}+2},\mathrm{\Gamma })}+(1-{♮}^{{\beta }_1})}& \le \mathrm{\Lambda }({\mathit{U}}_{{\beta }_1},{\mathit{U}}_{{\beta }_1+3\mathfrak{j}+2},\mathrm{\Gamma })\hfill \\ \hfill \mathrm{\Pi }({\mathit{U}}_{{\beta }_1},{\mathit{U}}_{{\beta }_1+3\mathfrak{j}+2},♮\mathrm{\Gamma })\le {♮}^{{\beta }_1}\mathrm{\Pi }({\mathit{U}}_0,{\mathit{U}}_{3\mathfrak{j}+2},\mathrm{\Gamma })& \mathrm{and}\mathcal{M}({\mathit{U}}_{{\beta }_1},{\mathit{U}}_{{\beta }_1+3\mathfrak{j}+2},♮\mathrm{\Gamma })\le {♮}^{{\beta }_1}\mathcal{M}({\mathit{U}}_0,{\mathit{U}}_{3\mathfrak{j}+2},\mathrm{\Gamma }),\hfill \end{array}$$

$$\begin{array}{cc}\hfill \frac{1}{\frac{{♮}^{{\beta }_1}}{\mathrm{\Lambda }({\mathit{U}}_0,{\mathit{U}}_{3\mathfrak{j}+3},\mathrm{\Gamma })}+(1-{♮}^{{\beta }_1})}& \le \mathrm{\Lambda }({\mathit{U}}_{{\beta }_1},{\mathit{U}}_{{\beta }_1+3\mathfrak{j}+3},\mathrm{\Gamma })\hfill \\ \hfill \mathrm{\Pi }({\mathit{U}}_{{\beta }_1},{\mathit{U}}_{{\beta }_1+3\mathfrak{j}+3},♮\mathrm{\Gamma })\le {♮}^{{\beta }_1}\mathrm{\Pi }({\mathit{U}}_0,{\mathit{U}}_{3\mathfrak{j}+3},\mathrm{\Gamma })& \mathrm{and}\mathcal{M}({\mathit{U}}_{{\beta }_1},{\mathit{U}}_{{\beta }_1+3\mathfrak{j}+3},♮\mathrm{\Gamma })\le {♮}^{{\beta }_1}\mathcal{M}({\mathit{U}}_0,{\mathit{U}}_{3\mathfrak{j}+3},\mathrm{\Gamma }).\hfill \end{array}$$

By using (21), we obtain for each $\mathfrak{j}=1,2,3,....,$

$$\begin{array}{cc}\hfill \mathrm{\Lambda }({\mathit{U}}_0,{\mathit{U}}_{3\mathfrak{j}+1},\mathrm{\Gamma })& \ge \frac{1}{\frac{1}{\mathrm{\Lambda }({\mathit{U}}_0,{\mathit{U}}_1,\frac{\mathrm{\Gamma }}{4})}}★\frac{1}{\frac{♮}{\mathrm{\Lambda }({\mathit{U}}_0,{\mathit{U}}_1,\frac{\mathrm{\Gamma }}{4})}+(1-♮)}★\frac{1}{\frac{{♮}^2}{\mathrm{\Lambda }({\mathit{U}}_0,{\mathit{U}}_1,\frac{\mathrm{\Gamma }}{4})}+(1-{♮}^2)}\hfill \\ \hfill & ★\frac{1}{\frac{{♮}^3}{\mathrm{\Lambda }({\mathit{U}}_0,{\mathit{U}}_1,\frac{\mathrm{\Gamma }}{4})}+(1-{♮}^3)}★\frac{1}{\frac{{♮}^4}{\mathrm{\Lambda }({\mathit{U}}_0,{\mathit{U}}_1,\frac{\mathrm{\Gamma }}{4})}+(1-{♮}^4)}\hfill \\ \hfill & ★\frac{1}{\frac{{♮}^5}{\mathrm{\Lambda }({\mathit{U}}_0,{\mathit{U}}_1,\frac{\mathrm{\Gamma }}{4})}+(1-{♮}^5)}★\frac{1}{\frac{{♮}^6}{\mathrm{\Lambda }({\mathit{U}}_0,{\mathit{U}}_1,\frac{\mathrm{\Gamma }}{4})}+(1-{♮}^6)}\hfill \\ \hfill & ★\cdots ★\frac{1}{\frac{{♮}^{3\mathfrak{j}}}{\mathrm{\Lambda }({\mathit{U}}_0,{\mathit{U}}_1,\frac{\mathrm{\Gamma }}{4})}+(1-{♮}^{3\mathfrak{j}})}\hfill \end{array}$$

$$\begin{array}{cc}\hfill \mathrm{\Pi }({\mathit{U}}_0,{\mathit{U}}_{3\mathfrak{j}+1},\mathrm{\Gamma })& \le \mathrm{\Pi }\left({\mathit{U}}_0,{\mathit{U}}_1,\frac{\mathrm{\Gamma }}{4}\right)•♮\mathrm{\Pi }\left({\mathit{U}}_0,{\mathit{U}}_1,\frac{\mathrm{\Gamma }}{4}\right)•{♮}^2\mathrm{\Pi }\left({\mathit{U}}_0,{\mathit{U}}_1,\frac{\mathrm{\Gamma }}{4}\right)\hfill \\ \hfill & •{♮}^3\mathrm{\Pi }\left({\mathit{U}}_0,{\mathit{U}}_1,\frac{\mathrm{\Gamma }}{4}\right)•{♮}^4\mathrm{\Pi }\left({\mathit{U}}_0,{\mathit{U}}_1,\frac{\mathrm{\Gamma }}{4}\right)•{♮}^5\mathrm{\Pi }\left({\mathit{U}}_0,{\mathit{U}}_1,\frac{\mathrm{\Gamma }}{4}\right)\hfill \\ \hfill & •{♮}^6\mathrm{\Pi }\left({\mathit{U}}_0,{\mathit{U}}_1,\frac{\mathrm{\Gamma }}{4}\right)•\cdots •{♮}^{3\mathfrak{j}}\mathrm{\Pi }\left({\mathit{U}}_0,{\mathit{U}}_1,\frac{\mathrm{\Gamma }}{4}\right)\hfill \end{array}$$

and

$$\begin{array}{cc}\hfill \mathcal{M}({\mathit{U}}_0,{\mathit{U}}_{3\mathfrak{j}+1},\mathrm{\Gamma })& \le \mathcal{M}\left({\mathit{U}}_0,{\mathit{U}}_1,\frac{\mathrm{\Gamma }}{4}\right)•♮\mathcal{M}\left({\mathit{U}}_0,{\mathit{U}}_1,\frac{\mathrm{\Gamma }}{4}\right)•{♮}^2\mathcal{M}\left({\mathit{U}}_0,{\mathit{U}}_1,\frac{\mathrm{\Gamma }}{4}\right)\hfill \\ \hfill & •{♮}^3\mathcal{M}\left({\mathit{U}}_0,{\mathit{U}}_1,\frac{\mathrm{\Gamma }}{4}\right)•{♮}^4\mathcal{M}\left({\mathit{U}}_0,{\mathit{U}}_1,\frac{\mathrm{\Gamma }}{4}\right)•{♮}^5\mathcal{M}\left({\mathit{U}}_0,{\mathit{U}}_1,\frac{\mathrm{\Gamma }}{4}\right)\hfill \\ \hfill & •{♮}^6\mathcal{M}\left({\mathit{U}}_0,{\mathit{U}}_1,\frac{\mathrm{\Gamma }}{4}\right)•\cdots •{♮}^{3\mathfrak{j}}\mathcal{M}\left({\mathit{U}}_0,{\mathit{U}}_1,\frac{\mathrm{\Gamma }}{4}\right).\hfill \end{array}$$

Now, from (21), we get

$$\begin{array}{cc}\hfill \mathrm{\Lambda }({\mathit{U}}_{{\beta }_1},{\mathit{U}}_{{\beta }_1+3\mathfrak{j}+1},♮\mathrm{\Gamma })& \ge \mathrm{\Lambda }\left({\mathit{U}}_0,{\mathit{U}}_{3\mathfrak{j}+1},\frac{\mathrm{\Gamma }}{{♮}^{{\beta }_1-1}}\right)\hfill \\ \hfill & \ge \frac{1}{\frac{1}{\mathrm{\Lambda }({\mathit{U}}_0,{\mathit{U}}_1,\frac{\mathrm{\Gamma }}{4{♮}^{{\beta }_1-1}})}}★\frac{1}{\frac{♮}{\mathrm{\Lambda }({\mathit{U}}_0,{\mathit{U}}_1,\frac{\mathrm{\Gamma }}{4{♮}^{{\beta }_1-1}})}+(1-♮)}\hfill \\ \hfill & ★\frac{1}{\frac{{♮}^2}{\mathrm{\Lambda }({\mathit{U}}_0,{\mathit{U}}_1,\frac{\mathrm{\Gamma }}{4{♮}^{{\beta }_1-1}})}+(1-{♮}^2)}\hfill \\ \hfill & ★\frac{1}{\frac{{♮}^3}{\mathrm{\Lambda }({\mathit{U}}_0,{\mathit{U}}_1,\frac{\mathrm{\Gamma }}{4{♮}^{{\beta }_1-1}})}+(1-{♮}^3)}★\frac{1}{\frac{{♮}^4}{\mathrm{\Lambda }({\mathit{U}}_0,{\mathit{U}}_1,\frac{\mathrm{\Gamma }}{4{♮}^{{\beta }_1-1}})}+(1-{♮}^4)}\hfill \\ \hfill & ★\frac{1}{\frac{{♮}^5}{\mathrm{\Lambda }({\mathit{U}}_0,{\mathit{U}}_1,\frac{\mathrm{\Gamma }}{4{♮}^{{\beta }_1-1}})}+(1-{♮}^5)}★\frac{1}{\frac{{♮}^6}{\mathrm{\Lambda }({\mathit{U}}_0,{\mathit{U}}_1,\frac{\mathrm{\Gamma }}{4{♮}^{{\beta }_1-1}})}+(1-{♮}^6)}\hfill \\ \hfill & ★\cdots ★\frac{1}{\frac{{♮}^{3\mathfrak{j}}}{\mathrm{\Lambda }({\mathit{U}}_0,{\mathit{U}}_1,\frac{\mathrm{\Gamma }}{4{♮}^{{\beta }_1-1}})}+(1-{♮}^{3\mathfrak{j}})},\hfill \end{array}$$

(22)

$$\begin{array}{cc}\hfill \mathrm{\Pi }({\mathit{U}}_{{\beta }_1},{\mathit{U}}_{{\beta }_1+3\mathfrak{j}+1},♮\mathrm{\Gamma })& \le \mathrm{\Pi }\left({\mathit{U}}_0,{\mathit{U}}_{3\mathfrak{j}+1},\frac{\mathrm{\Gamma }}{{♮}^{{\beta }_1-1}}\right)\hfill \\ \hfill & \le \mathrm{\Pi }\left({\mathit{U}}_0,{\mathit{U}}_1,\frac{\mathrm{\Gamma }}{4{♮}^{{\beta }_1-1}}\right)•♮\mathrm{\Pi }\left({\mathit{U}}_0,{\mathit{U}}_1,\frac{\mathrm{\Gamma }}{4{♮}^{{\beta }_1-1}}\right)•{♮}^2\mathrm{\Pi }\left({\mathit{U}}_0,{\mathit{U}}_1,\frac{\mathrm{\Gamma }}{4{♮}^{{\beta }_1-1}}\right)\hfill \\ \hfill & •{♮}^3\mathrm{\Pi }\left({\mathit{U}}_0,{\mathit{U}}_1,\frac{\mathrm{\Gamma }}{4{♮}^{{\beta }_1-1}}\right)•{♮}^4\mathrm{\Pi }\left({\mathit{U}}_0,{\mathit{U}}_1,\frac{\mathrm{\Gamma }}{4{♮}^{{\beta }_1-1}}\right)\hfill \\ \hfill & •{♮}^5\mathrm{\Pi }\left({\mathit{U}}_0,{\mathit{U}}_1,\frac{\mathrm{\Gamma }}{4{♮}^{{\beta }_1-1}}\right)•{♮}^6\mathrm{\Pi }\left({\mathit{U}}_0,{\mathit{U}}_1,\frac{\mathrm{\Gamma }}{4{♮}^{{\beta }_1-1}}\right)\hfill \\ \hfill & •\cdots •{♮}^{3\mathfrak{j}}\mathrm{\Pi }\left({\mathit{U}}_0,{\mathit{U}}_1,\frac{\mathrm{\Gamma }}{4{♮}^{{\beta }_1-1}}\right)\hfill \end{array}$$

(23)

and

$$\begin{array}{cc}\hfill \mathcal{M}({\mathit{U}}_{{\beta }_1},{\mathit{U}}_{{\beta }_1+3\mathfrak{j}+1},♮\mathrm{\Gamma })& \le \mathcal{M}\left({\mathit{U}}_0,{\mathit{U}}_{3\mathfrak{j}+1},\frac{\mathrm{\Gamma }}{{♮}^{{\beta }_1-1}}\right)\hfill \\ \hfill & \le \mathcal{M}\left({\mathit{U}}_0,{\mathit{U}}_1,\frac{\mathrm{\Gamma }}{4{♮}^{{\beta }_1-1}}\right)•♮\mathcal{M}\left({\mathit{U}}_0,{\mathit{U}}_1,\frac{\mathrm{\Gamma }}{4{♮}^{{\beta }_1-1}}\right)\hfill \\ \hfill & •{♮}^2\mathcal{M}\left({\mathit{U}}_0,{\mathit{U}}_1,\frac{\mathrm{\Gamma }}{4{♮}^{{\beta }_1-1}}\right)•{♮}^3\mathcal{M}\left({\mathit{U}}_0,{\mathit{U}}_1,\frac{\mathrm{\Gamma }}{4{♮}^{{\beta }_1-1}}\right)\hfill \\ \hfill & •{♮}^4\mathcal{M}\left({\mathit{U}}_0,{\mathit{U}}_1,\frac{\mathrm{\Gamma }}{4{♮}^{{\beta }_1-1}}\right)•{♮}^5\mathcal{M}\left({\mathit{U}}_0,{\mathit{U}}_1,\frac{\mathrm{\Gamma }}{4{♮}^{{\beta }_1-1}}\right)\hfill \\ \hfill & •{♮}^6\mathcal{M}\left({\mathit{U}}_0,{\mathit{U}}_1,\frac{\mathrm{\Gamma }}{4{♮}^{{\beta }_1-1}}\right)•\cdots •{♮}^{3\mathfrak{j}}\mathcal{M}\left({\mathit{U}}_0,{\mathit{U}}_1,\frac{\mathrm{\Gamma }}{4{♮}^{{\beta }_1-1}}\right).\hfill \end{array}$$

By using (21), we obtain for each $\mathfrak{j}=1,2,3,\dots ,$

$$\begin{array}{cc}\hfill \mathrm{\Lambda }({\mathit{U}}_0,{\mathit{U}}_{3\mathfrak{j}+2},\mathrm{\Gamma })& \ge \frac{1}{\frac{1}{\mathrm{\Lambda }({\mathit{U}}_0,{\mathit{U}}_1,\frac{\mathrm{\Gamma }}{4})}}★\frac{1}{\frac{♮}{\mathrm{\Lambda }({\mathit{U}}_0,{\mathit{U}}_1,\frac{\mathrm{\Gamma }}{4})}+(1-♮)}★\frac{1}{\frac{{♮}^2}{\mathrm{\Lambda }({\mathit{U}}_0,{\mathit{U}}_1,\frac{\mathrm{\Gamma }}{4})}+(1-{♮}^2)}\hfill \\ \hfill & ★\frac{1}{\frac{{♮}^3}{\mathrm{\Lambda }({\mathit{U}}_0,{\mathit{U}}_1,\frac{\mathrm{\Gamma }}{4})}+(1-{♮}^3)}★\frac{1}{\frac{{♮}^4}{\mathrm{\Lambda }({\mathit{U}}_0,{\mathit{U}}_1,\frac{\mathrm{\Gamma }}{4})}+(1-{♮}^4)}\hfill \\ \hfill & ★\frac{1}{\frac{{♮}^5}{\mathrm{\Lambda }({\mathit{U}}_0,{\mathit{U}}_1,\frac{\mathrm{\Gamma }}{4})}+(1-{♮}^5)}★\frac{1}{\frac{{♮}^6}{\mathrm{\Lambda }({\mathit{U}}_0,{\mathit{U}}_2,\frac{\mathrm{\Gamma }}{4})}+(1-{♮}^6)}\hfill \\ \hfill & ★\cdots ★\frac{1}{\frac{{♮}^{3\mathfrak{j}}}{\mathrm{\Lambda }({\mathit{U}}_0,{\mathit{U}}_1,\frac{\mathrm{\Gamma }}{4})}+(1-{♮}^{3\mathfrak{j}})}\hfill \end{array}$$

$$\begin{array}{cc}\hfill \mathrm{\Pi }({\mathit{U}}_0,{\mathit{U}}_{3\mathfrak{j}+2},\mathrm{\Gamma })& \le \mathrm{\Pi }\left({\mathit{U}}_0,{\mathit{U}}_1,\frac{\mathrm{\Gamma }}{4}\right)•♮\mathrm{\Pi }\left({\mathit{U}}_0,{\mathit{U}}_1,\frac{\mathrm{\Gamma }}{4}\right)•{♮}^2\mathrm{\Pi }\left({\mathit{U}}_0,{\mathit{U}}_1,\frac{\mathrm{\Gamma }}{4}\right)\hfill \\ \hfill & •{♮}^3\mathrm{\Pi }\left({\mathit{U}}_0,{\mathit{U}}_1,\frac{\mathrm{\Gamma }}{4}\right)•{♮}^4\mathrm{\Pi }\left({\mathit{U}}_0,{\mathit{U}}_1,\frac{\mathrm{\Gamma }}{4}\right)•{♮}^5\mathrm{\Pi }\left({\mathit{U}}_0,{\mathit{U}}_1,\frac{\mathrm{\Gamma }}{4}\right)\hfill \\ \hfill & •{♮}^6\mathrm{\Pi }\left({\mathit{U}}_0,{\mathit{U}}_2,\frac{\mathrm{\Gamma }}{4}\right)•\cdots •{♮}^{3\mathfrak{j}}\mathrm{\Pi }\left({\mathit{U}}_0,{\mathit{U}}_2,\frac{\mathrm{\Gamma }}{4}\right)\hfill \end{array}$$

and

$$\begin{array}{cc}\hfill \mathcal{M}({\mathit{U}}_0,{\mathit{U}}_{3\mathfrak{j}+2},\mathrm{\Gamma })& \le \mathcal{M}\left({\mathit{U}}_0,{\mathit{U}}_1,\frac{\mathrm{\Gamma }}{4}\right)•♮\mathcal{M}\left({\mathit{U}}_0,{\mathit{U}}_1,\frac{\mathrm{\Gamma }}{4}\right)•{♮}^2\mathcal{M}\left({\mathit{U}}_0,{\mathit{U}}_1,\frac{\mathrm{\Gamma }}{4}\right)\hfill \\ \hfill & •{♮}^3\mathcal{M}\left({\mathit{U}}_0,{\mathit{U}}_1,\frac{\mathrm{\Gamma }}{4}\right)•{♮}^4\mathcal{M}\left({\mathit{U}}_0,{\mathit{U}}_1,\frac{\mathrm{\Gamma }}{4}\right)•{♮}^5\mathcal{M}\left({\mathit{U}}_0,{\mathit{U}}_1,\frac{\mathrm{\Gamma }}{4}\right)\hfill \\ \hfill & •{♮}^6\mathcal{M}\left({\mathit{U}}_0,{\mathit{U}}_2,\frac{\mathrm{\Gamma }}{4}\right)•\cdots •{♮}^{3\mathfrak{j}}\mathcal{M}\left({\mathit{U}}_0,{\mathit{U}}_2,\frac{\mathrm{\Gamma }}{4}\right).\hfill \end{array}$$

Now, from (21), we get

$$\begin{array}{cc}\hfill \mathrm{\Lambda }({\mathit{U}}_{{\beta }_1},{\mathit{U}}_{{\beta }_1+3\mathfrak{j}+2},♮\mathrm{\Gamma })& \ge \mathrm{\Lambda }\left({\mathit{U}}_0,{\mathit{U}}_{3\mathfrak{j}+2},\frac{\mathrm{\Gamma }}{{♮}^{{\beta }_1-1}}\right)\hfill \\ \hfill & \ge \frac{1}{\frac{1}{\mathrm{\Lambda }({\mathit{U}}_0,{\mathit{U}}_1,\frac{\mathrm{\Gamma }}{4{♮}^{{\beta }_1-1}})}}★\frac{1}{\frac{♮}{\mathrm{\Lambda }({\mathit{U}}_0,{\mathit{U}}_1,\frac{\mathrm{\Gamma }}{4{♮}^{{\beta }_1-1}})}+(1-♮)}\hfill \\ \hfill & ★\frac{1}{\frac{{♮}^2}{\mathrm{\Lambda }({\mathit{U}}_0,{\mathit{U}}_1,\frac{\mathrm{\Gamma }}{4{♮}^{{\beta }_1-1}})}+(1-{♮}^2)}★\frac{1}{\frac{{♮}^3}{\mathrm{\Lambda }({\mathit{U}}_0,{\mathit{U}}_1,\frac{\mathrm{\Gamma }}{4{♮}^{{\beta }_1-1}})}+(1-{♮}^3)}\hfill \\ \hfill & ★\frac{1}{\frac{{♮}^4}{\mathrm{\Lambda }({\mathit{U}}_0,{\mathit{U}}_1,\frac{\mathrm{\Gamma }}{4{♮}^{{\beta }_1-1}})}+(1-{♮}^4)}★\frac{1}{\frac{{♮}^5}{\mathrm{\Lambda }({\mathit{U}}_0,{\mathit{U}}_1,\frac{\mathrm{\Gamma }}{4{♮}^{{\beta }_1-1}})}+(1-{♮}^5)}\hfill \\ \hfill & ★\frac{1}{\frac{{♮}^6}{\mathrm{\Lambda }({\mathit{U}}_0,{\mathit{U}}_2,\frac{\mathrm{\Gamma }}{4{♮}^{{\beta }_1-1}})}+(1-{♮}^6)}★\cdots ★\frac{1}{\frac{{♮}^{3\mathfrak{j}}}{\mathrm{\Lambda }({\mathit{U}}_0,{\mathit{U}}_1,\frac{\mathrm{\Gamma }}{4{♮}^{{\beta }_1-1}})}+(1-{♮}^{3\mathfrak{j}})}\hfill \end{array}$$

(24)

$$\begin{array}{cc}\hfill \mathrm{\Pi }({\mathit{U}}_{{\beta }_1},{\mathit{U}}_{{\beta }_1+3\mathfrak{j}+2},♮\mathrm{\Gamma })& \le \mathrm{\Pi }\left({\mathit{U}}_0,{\mathit{U}}_{3\mathfrak{j}+2},\frac{\mathrm{\Gamma }}{{♮}^{{\beta }_1-1}}\right)\hfill \\ \hfill & \le \mathrm{\Pi }\left({\mathit{U}}_0,{\mathit{U}}_1,\frac{\mathrm{\Gamma }}{4{♮}^{{\beta }_1-1}}\right)•♮\mathrm{\Pi }\left({\mathit{U}}_0,{\mathit{U}}_1,\frac{\mathrm{\Gamma }}{4{♮}^{{\beta }_1-1}}\right)\hfill \\ \hfill & •{♮}^2\mathrm{\Pi }\left({\mathit{U}}_0,{\mathit{U}}_1,\frac{\mathrm{\Gamma }}{4{♮}^{{\beta }_1-1}}\right)•{♮}^3\mathrm{\Pi }\left({\mathit{U}}_0,{\mathit{U}}_1,\frac{\mathrm{\Gamma }}{4{♮}^{{\beta }_1-1}}\right)\hfill \\ \hfill & •{♮}^4\mathrm{\Pi }\left({\mathit{U}}_0,{\mathit{U}}_1,\frac{\mathrm{\Gamma }}{4{♮}^{{\beta }_1-1}}\right)•{♮}^5\mathrm{\Pi }\left({\mathit{U}}_0,{\mathit{U}}_1,\frac{\mathrm{\Gamma }}{4{♮}^{{\beta }_1-1}}\right)\hfill \\ \hfill & •{♮}^6\mathrm{\Pi }\left({\mathit{U}}_0,{\mathit{U}}_2,\frac{\mathrm{\Gamma }}{4{♮}^{{\beta }_1-1}}\right)•\cdots •{♮}^{3\mathfrak{j}}\mathrm{\Pi }\left({\mathit{U}}_0,{\mathit{U}}_2,\frac{\mathrm{\Gamma }}{4{♮}^{{\beta }_1-1}}\right)\hfill \end{array}$$

(25)

and

$$\begin{array}{cc}\hfill \mathcal{M}({\mathit{U}}_{{\beta }_1},{\mathit{U}}_{{\beta }_1+3\mathfrak{j}+2},♮\mathrm{\Gamma })& \le \mathcal{M}\left({\mathit{U}}_0,{\mathit{U}}_{3\mathfrak{j}+2},\frac{\mathrm{\Gamma }}{{♮}^{{\beta }_1-1}}\right)\hfill \\ \hfill & \le \mathcal{M}\left({\mathit{U}}_0,{\mathit{U}}_1,\frac{\mathrm{\Gamma }}{4{♮}^{{\beta }_1-1}}\right)•♮\mathcal{M}\left({\mathit{U}}_0,{\mathit{U}}_1,\frac{\mathrm{\Gamma }}{4{♮}^{{\beta }_1-1}}\right)\hfill \\ \hfill & •{♮}^2\mathcal{M}\left({\mathit{U}}_0,{\mathit{U}}_1,\frac{\mathrm{\Gamma }}{4{♮}^{{\beta }_1-1}}\right)•{♮}^3\mathcal{M}\left({\mathit{U}}_0,{\mathit{U}}_1,\frac{\mathrm{\Gamma }}{4{♮}^{{\beta }_1-1}}\right)\hfill \\ \hfill & •{♮}^4\mathcal{M}\left({\mathit{U}}_0,{\mathit{U}}_1,\frac{\mathrm{\Gamma }}{4{♮}^{{\beta }_1-1}}\right)•{♮}^5\mathcal{M}\left({\mathit{U}}_0,{\mathit{U}}_1,\frac{\mathrm{\Gamma }}{4{♮}^{{\beta }_1-1}}\right)\hfill \\ \hfill & •{♮}^6\mathcal{M}\left({\mathit{U}}_0,{\mathit{U}}_2,\frac{\mathrm{\Gamma }}{4{♮}^{{\beta }_1-1}}\right)•\cdots •{♮}^{3\mathfrak{j}}\mathcal{M}\left({\mathit{U}}_0,{\mathit{U}}_2,\frac{\mathrm{\Gamma }}{4{♮}^{{\beta }_1-1}}\right).\hfill \end{array}$$

(26)

By using (21), we obtain for each $\mathfrak{j}=1,2,3,\dots ,$

$$\begin{array}{cc}\hfill \mathrm{\Lambda }({\mathit{U}}_0,{\mathit{U}}_{3\mathfrak{j}+3},\mathrm{\Gamma })& \ge \frac{1}{\frac{1}{\mathrm{\Lambda }({\mathit{U}}_0,{\mathit{U}}_1,\frac{\mathrm{\Gamma }}{4})}}★\frac{1}{\frac{♮}{\mathrm{\Lambda }({\mathit{U}}_0,{\mathit{U}}_1,\frac{\mathrm{\Gamma }}{4})}+(1-♮)}★\frac{1}{\frac{{♮}^2}{\mathrm{\Lambda }({\mathit{U}}_0,{\mathit{U}}_1,\frac{\mathrm{\Gamma }}{4})}+(1-{♮}^2)}\hfill \\ \hfill & ★\frac{1}{\frac{{♮}^3}{\mathrm{\Lambda }({\mathit{U}}_0,{\mathit{U}}_1,\frac{\mathrm{\Gamma }}{4})}+(1-{♮}^3)}★\frac{1}{\frac{{♮}^4}{\mathrm{\Lambda }({\mathit{U}}_0,{\mathit{U}}_1,\frac{\mathrm{\Gamma }}{4})}+(1-{♮}^4)}\hfill \\ \hfill & ★\frac{1}{\frac{{♮}^5}{\mathrm{\Lambda }({\mathit{U}}_0,{\mathit{U}}_1,\frac{\mathrm{\Gamma }}{4})}+(1-{♮}^5)}★\frac{1}{\frac{{♮}^6}{\mathrm{\Lambda }({\mathit{U}}_0,{\mathit{U}}_3,\frac{\mathrm{\Gamma }}{4})}+(1-{♮}^6)}\hfill \\ \hfill & ★\cdots ★\frac{1}{\frac{{♮}^{3\mathfrak{j}}}{\mathrm{\Lambda }({\mathit{U}}_0,{\mathit{U}}_3,\frac{\mathrm{\Gamma }}{4})}+(1-{♮}^{3\mathfrak{j}})}\hfill \end{array}$$

$$\begin{array}{cc}\hfill \mathrm{\Pi }({\mathit{U}}_0,{\mathit{U}}_{3\mathfrak{j}+3},\mathrm{\Gamma })& \le \mathrm{\Pi }\left({\mathit{U}}_0,{\mathit{U}}_1,\frac{\mathrm{\Gamma }}{4}\right)•♮\mathrm{\Pi }\left({\mathit{U}}_0,{\mathit{U}}_1,\frac{\mathrm{\Gamma }}{4}\right)•{♮}^2\mathrm{\Pi }\left({\mathit{U}}_0,{\mathit{U}}_1,\frac{\mathrm{\Gamma }}{4}\right)\hfill \\ \hfill & •{♮}^3\mathrm{\Pi }\left({\mathit{U}}_0,{\mathit{U}}_1,\frac{\mathrm{\Gamma }}{4}\right)•{♮}^4\mathrm{\Pi }\left({\mathit{U}}_0,{\mathit{U}}_1,\frac{\mathrm{\Gamma }}{4}\right)•{♮}^5\mathrm{\Pi }\left({\mathit{U}}_0,{\mathit{U}}_1,\frac{\mathrm{\Gamma }}{4}\right)\hfill \\ \hfill & •{♮}^6\mathrm{\Pi }\left({\mathit{U}}_0,{\mathit{U}}_3,\frac{\mathrm{\Gamma }}{4}\right)•\cdots •{♮}^{3\mathfrak{j}}\mathrm{\Pi }\left({\mathit{U}}_0,{\mathit{U}}_3,\frac{\mathrm{\Gamma }}{4}\right)\hfill \end{array}$$

and

$$\begin{array}{cc}\hfill \mathcal{M}({\mathit{U}}_0,{\mathit{U}}_{3\mathfrak{j}+3},\mathrm{\Gamma })& \le \mathcal{M}\left({\mathit{U}}_0,{\mathit{U}}_1,\frac{\mathrm{\Gamma }}{4}\right)•♮\mathcal{M}\left({\mathit{U}}_0,{\mathit{U}}_1,\frac{\mathrm{\Gamma }}{4}\right)•{♮}^2\mathcal{M}\left({\mathit{U}}_0,{\mathit{U}}_1,\frac{\mathrm{\Gamma }}{4}\right)\hfill \\ \hfill & •{♮}^3\mathcal{M}\left({\mathit{U}}_0,{\mathit{U}}_1,\frac{\mathrm{\Gamma }}{4}\right)•{♮}^4\mathcal{M}\left({\mathit{U}}_0,{\mathit{U}}_1,\frac{\mathrm{\Gamma }}{4}\right)•{♮}^5\mathcal{M}\left({\mathit{U}}_0,{\mathit{U}}_1,\frac{\mathrm{\Gamma }}{4}\right)\hfill \\ \hfill & •{♮}^6\mathcal{M}\left({\mathit{U}}_0,{\mathit{U}}_3,\frac{\mathrm{\Gamma }}{4}\right)•\cdots •{♮}^{3\mathfrak{j}}\mathcal{M}\left({\mathit{U}}_0,{\mathit{U}}_3,\frac{\mathrm{\Gamma }}{4}\right).\hfill \end{array}$$

Now, from (21), we get

$$\begin{array}{cc}\hfill \mathrm{\Lambda }({\mathit{U}}_{{\beta }_1},{\mathit{U}}_{{\beta }_1+3\mathfrak{j}+3},♮\mathrm{\Gamma })& \ge \mathrm{\Lambda }\left({\mathit{U}}_0,{\mathit{U}}_{3\mathfrak{j}+3},\frac{\mathrm{\Gamma }}{{♮}^{{\beta }_1-1}}\right)\hfill \\ \hfill & \ge \frac{1}{\frac{1}{\mathrm{\Lambda }({\mathit{U}}_0,{\mathit{U}}_1,\frac{\mathrm{\Gamma }}{4{♮}^{{\beta }_1-1}})}}★\frac{1}{\frac{♮}{\mathrm{\Lambda }({\mathit{U}}_0,{\mathit{U}}_1,\frac{\mathrm{\Gamma }}{4{♮}^{{\beta }_1-1}})}+(1-♮)}\hfill \\ \hfill & ★\frac{1}{\frac{{♮}^2}{\mathrm{\Lambda }({\mathit{U}}_0,{\mathit{U}}_1,\frac{\mathrm{\Gamma }}{4{♮}^{{\beta }_1-1}})}+(1-{♮}^2)}★\frac{1}{\frac{{♮}^3}{\mathrm{\Lambda }({\mathit{U}}_0,{\mathit{U}}_1,\frac{\mathrm{\Gamma }}{4{♮}^{{\beta }_1-1}})}+(1-{♮}^3)}\hfill \\ \hfill & ★\frac{1}{\frac{{♮}^4}{\mathrm{\Lambda }({\mathit{U}}_0,{\mathit{U}}_1,\frac{\mathrm{\Gamma }}{4{♮}^{{\beta }_1-1}})}+(1-{♮}^4)}★\frac{1}{\frac{{♮}^5}{\mathrm{\Lambda }({\mathit{U}}_0,{\mathit{U}}_1,\frac{\mathrm{\Gamma }}{4{♮}^{{\beta }_1-1}})}+(1-{♮}^5)}\hfill \\ \hfill & ★\frac{1}{\frac{{♮}^6}{\mathrm{\Lambda }({\mathit{U}}_0,{\mathit{U}}_3,\frac{\mathrm{\Gamma }}{4{♮}^{{\beta }_1-1}})}+(1-{♮}^6)}\hfill \\ \hfill & ★\cdots ★\frac{1}{\frac{{♮}^{3\mathfrak{j}}}{\mathrm{\Lambda }({\mathit{U}}_0,{\mathit{U}}_3,\frac{\mathrm{\Gamma }}{4{♮}^{{\beta }_1-1}})}+(1-{♮}^{3\mathfrak{j}})},\hfill \end{array}$$

(27)

$$\begin{array}{cc}\hfill \mathrm{\Pi }({\mathit{U}}_{{\beta }_1},{\mathit{U}}_{{\beta }_1+3\mathfrak{j}+3},♮\mathrm{\Gamma })& \le \mathrm{\Pi }\left({\mathit{U}}_0,{\mathit{U}}_{3\mathfrak{j}+3},\frac{\mathrm{\Gamma }}{{♮}^{{\beta }_1-1}}\right)\hfill \\ \hfill & \le \mathrm{\Pi }\left({\mathit{U}}_0,{\mathit{U}}_1,\frac{\mathrm{\Gamma }}{4{♮}^{{\beta }_1-1}}\right)•♮\mathrm{\Pi }\left({\mathit{U}}_0,{\mathit{U}}_1,\frac{\mathrm{\Gamma }}{4{♮}^{{\beta }_1-1}}\right)\hfill \\ \hfill & •{♮}^2\mathrm{\Pi }\left({\mathit{U}}_0,{\mathit{U}}_1,\frac{\mathrm{\Gamma }}{4{♮}^{{\beta }_1-1}}\right)•{♮}^3\mathrm{\Pi }\left({\mathit{U}}_0,{\mathit{U}}_1,\frac{\mathrm{\Gamma }}{4{♮}^{{\beta }_1-1}}\right)\hfill \\ \hfill & •{♮}^4\mathrm{\Pi }\left({\mathit{U}}_0,{\mathit{U}}_1,\frac{\mathrm{\Gamma }}{4{♮}^{{\beta }_1-1}}\right)•{♮}^5\mathrm{\Pi }\left({\mathit{U}}_0,{\mathit{U}}_1,\frac{\mathrm{\Gamma }}{4{♮}^{{\beta }_1-1}}\right)\hfill \\ \hfill & •{♮}^6\mathrm{\Pi }\left({\mathit{U}}_0,{\mathit{U}}_3,\frac{\mathrm{\Gamma }}{4{♮}^{{\beta }_1-1}}\right)•\cdots •{♮}^{3\mathfrak{j}}\mathrm{\Pi }\left({\mathit{U}}_0,{\mathit{U}}_3,\frac{\mathrm{\Gamma }}{4{♮}^{{\beta }_1-1}}\right)\hfill \end{array}$$

(28)

and

$$\begin{array}{cc}\hfill \mathcal{M}({\mathit{U}}_{{\beta }_1},{\mathit{U}}_{{\beta }_1+3\mathfrak{j}+3},♮\mathrm{\Gamma })& \le \mathcal{M}\left({\mathit{U}}_0,{\mathit{U}}_{3\mathfrak{j}+3},\frac{\mathrm{\Gamma }}{{♮}^{{\beta }_1-1}}\right)\hfill \\ \hfill & \le \mathcal{M}\left({\mathit{U}}_0,{\mathit{U}}_1,\frac{\mathrm{\Gamma }}{4{♮}^{{\beta }_1-1}}\right)•♮\mathcal{M}\left({\mathit{U}}_0,{\mathit{U}}_1,\frac{\mathrm{\Gamma }}{4{♮}^{{\beta }_1-1}}\right)\hfill \\ \hfill & •{♮}^2\mathcal{M}\left({\mathit{U}}_0,{\mathit{U}}_1,\frac{\mathrm{\Gamma }}{4{♮}^{{\beta }_1-1}}\right)•{♮}^3\mathcal{M}\left({\mathit{U}}_0,{\mathit{U}}_1,\frac{\mathrm{\Gamma }}{4{♮}^{{\beta }_1-1}}\right)\hfill \\ \hfill & •{♮}^4\mathcal{M}\left({\mathit{U}}_0,{\mathit{U}}_1,\frac{\mathrm{\Gamma }}{4{♮}^{{\beta }_1-1}}\right)•{♮}^5\mathcal{M}\left({\mathit{U}}_0,{\mathit{U}}_1,\frac{\mathrm{\Gamma }}{4{♮}^{{\beta }_1-1}}\right)\hfill \\ \hfill & •{♮}^6\mathcal{M}\left({\mathit{U}}_0,{\mathit{U}}_3,\frac{\mathrm{\Gamma }}{4{♮}^{{\beta }_1-1}}\right)•\cdots •{♮}^{3\mathfrak{j}}\mathcal{M}\left({\mathit{U}}_0,{\mathit{U}}_3,\frac{\mathrm{\Gamma }}{4{♮}^{{\beta }_1-1}}\right).\hfill \end{array}$$

(29)

Using (22)–(29), for each case ${\beta }_1\to +\infty$, we deduce

$$\begin{array}{cc}\hfill \underset{{\beta }_1\to +\infty }{lim}\mathrm{\Lambda }({\mathit{U}}_{{\beta }_1},{\mathit{U}}_{{\beta }_1+\mathfrak{q}},\mathrm{\Gamma })& =1★1★\cdots ★1=1,\hfill \\ \hfill \underset{{\beta }_1\to +\infty }{lim}\mathrm{\Pi }({\mathit{U}}_{{\beta }_1},{\mathit{U}}_{{\beta }_1+\mathfrak{q}},\mathrm{\Gamma })& =0•0•\cdots •0=0,\hfill \end{array}$$

and

$$\begin{array}{cc}\hfill \underset{{\beta }_1\to +\infty }{lim}\mathcal{M}({\mathit{U}}_{{\beta }_1},{\mathit{U}}_{{\beta }_1+\mathfrak{q}},\mathrm{\Gamma })& =0•0•\cdots •0=0.\hfill \end{array}$$

It follows that $\left\{{\mathit{U}}_{{\beta }_1}\right\}$ is a Cauchy sequence. Since $(\mathit{K},\mathrm{\Lambda },\mathrm{\Pi },\mathcal{M},★,•)$ be a complete NPMS, there exists

$$\begin{array}{c}\hfill \underset{{\beta }_1\to +\infty }{lim}{\mathit{U}}_{{\beta }_1}=\mathit{U}.\end{array}$$

Since (v), (x) and (xv), we get

$$\begin{array}{cc}\hfill \frac{1}{\mathrm{\Lambda }(\wp {\mathit{U}}_{{\beta }_1},\wp \mathit{U},\mathrm{\Gamma })}-1& \le ♮\left[\frac{1}{\mathrm{\Lambda }({\mathit{U}}_{{\beta }_1},\mathit{U},\mathrm{\Gamma })}-1\right]=\frac{♮}{\mathrm{\Lambda }({\mathit{U}}_{{\beta }_1},\mathit{U},\mathrm{\Gamma })}-♮\hfill \\ \hfill & \Rightarrow \frac{1}{\frac{♮}{\mathrm{\Lambda }({\mathit{U}}_{{\beta }_1},\mathit{U},\mathrm{\Gamma })}+(1-♮)}\le \mathrm{\Lambda }(\wp {\mathit{U}}_{{\beta }_1},\wp \mathit{U},\mathrm{\Gamma }).\hfill \end{array}$$

By the above inequality, we have

$$\begin{array}{cc}\hfill \mathrm{\Lambda }(\mathit{U},\wp \mathit{U},\mathrm{\Gamma })& \ge \mathrm{\Lambda }\left(\mathit{U},{\mathit{U}}_{{\beta }_1},\frac{\mathrm{\Gamma }}{4}\right)★\mathrm{\Lambda }\left({\mathit{U}}_{{\beta }_1},{\mathit{U}}_{{\beta }_1+1},\frac{\mathrm{\Gamma }}{4}\right)★\mathrm{\Lambda }\left({\mathit{U}}_{{\beta }_1+1},\wp {\mathit{U}}_{{\beta }_1+2},\frac{\mathrm{\Gamma }}{4}\right)\hfill \\ \hfill & ★\mathrm{\Lambda }\left({\mathit{U}}_{{\beta }_1+2},\wp \mathit{U},\frac{\mathrm{\Gamma }}{4}\right)\hfill \\ \hfill & \ge \mathrm{\Lambda }\left(\mathit{U},{\mathit{U}}_{{\beta }_1},\frac{\mathrm{\Gamma }}{4}\right)★\mathrm{\Lambda }\left(\wp {\mathit{U}}_{{\beta }_1-1},\wp {\mathit{U}}_{{\beta }_1},\frac{\mathrm{\Gamma }}{4}\right)★\mathrm{\Lambda }\left(\wp {\mathit{U}}_{{\beta }_1},\wp {\mathit{U}}_{{\beta }_1+1},\frac{\mathrm{\Gamma }}{4}\right)\hfill \\ \hfill & ★\mathrm{\Lambda }\left(\wp {\mathit{U}}_{{\beta }_1+1},\wp \mathit{U},\frac{\mathrm{\Gamma }}{4}\right)\hfill \\ \hfill & \ge \mathrm{\Lambda }\left(\mathit{U},{\mathit{U}}_{{\beta }_1},\frac{\mathrm{\Gamma }}{4}\right)★\frac{1}{\frac{{♮}^{{\beta }_1-1}}{\mathrm{\Lambda }({\mathit{U}}_{{\beta }_1-1},{\mathit{U}}_{{\beta }_1},\frac{\mathrm{\Gamma }}{4})}+(1-{♮}^{{\beta }_1-1})}★\frac{1}{\frac{{♮}^{{\beta }_1}}{\mathrm{\Lambda }({\mathit{U}}_{{\beta }_1},{\mathit{U}}_{{\beta }_1+1},\frac{\mathrm{\Gamma }}{4})}+(1-{♮}^{{\beta }_1})}\hfill \\ \hfill & ★\frac{1}{\frac{♮}{\mathrm{\Lambda }({\mathit{U}}_{{\beta }_1+1},\mathit{U},\frac{\mathrm{\Gamma }}{4})}+(1-♮)}\hfill \\ \hfill & \to 1★1★1★1=1\mathrm{as}{\beta }_1\to +\infty ,\hfill \end{array}$$

$$\begin{array}{cc}\hfill \mathrm{\Pi }(\mathit{U},\wp \mathit{U},\mathrm{\Gamma })& \le \mathrm{\Pi }\left(\mathit{U},{\mathit{U}}_{{\beta }_1},\frac{\mathrm{\Gamma }}{4}\right)•\mathrm{\Pi }\left({\mathit{U}}_{{\beta }_1},{\mathit{U}}_{{\beta }_1+1},\frac{\mathrm{\Gamma }}{4}\right)•\mathrm{\Pi }\left({\mathit{U}}_{{\beta }_1+1},\wp {\mathit{U}}_{{\beta }_1+2},\frac{\mathrm{\Gamma }}{4}\right)\hfill \\ \hfill & •\mathrm{\Pi }\left({\mathit{U}}_{{\beta }_1+2},\wp \mathit{U},\frac{\mathrm{\Gamma }}{4}\right)\hfill \\ \hfill & \le \mathrm{\Pi }\left(\mathit{U},{\mathit{U}}_{{\beta }_1},\frac{\mathrm{\Gamma }}{4}\right)•\mathrm{\Pi }\left(\wp {\mathit{U}}_{{\beta }_1-1},\wp {\mathit{U}}_{{\beta }_1},\frac{\mathrm{\Gamma }}{4}\right)•\mathrm{\Pi }\left(\wp {\mathit{U}}_{{\beta }_1},\wp {\mathit{U}}_{{\beta }_1+1},\frac{\mathrm{\Gamma }}{4}\right)\hfill \\ \hfill & •\mathrm{\Pi }\left(\wp {\mathit{U}}_{{\beta }_1+1},\wp \mathit{U},\frac{\mathrm{\Gamma }}{4}\right)\hfill \\ \hfill & \le \mathrm{\Pi }\left(\mathit{U},{\mathit{U}}_{{\beta }_1},\frac{\mathrm{\Gamma }}{4}\right)•{♮}^{{\beta }_1-1}\mathrm{\Pi }\left({\mathit{U}}_{{\beta }_1-1},{\mathit{U}}_{{\beta }_1},\frac{\mathrm{\Gamma }}{4}\right)•{♮}^{{\beta }_1}\mathrm{\Pi }\left({\mathit{U}}_{{\beta }_1},{\mathit{U}}_{{\beta }_1+1},\frac{\mathrm{\Gamma }}{4}\right)\hfill \\ \hfill & •♮\mathrm{\Pi }\left({\mathit{U}}_{{\beta }_1+1},\mathit{U},\frac{\mathrm{\Gamma }}{4}\right)\hfill \\ \hfill & \to 0•0•0•0=0\mathrm{as}{\beta }_1\to +\infty \hfill \end{array}$$

and

$$\begin{array}{cc}\hfill \mathcal{M}(\mathit{U},\wp \mathit{U},\mathrm{\Gamma })& \le \mathcal{M}\left(\mathit{U},{\mathit{U}}_{{\beta }_1},\frac{\mathrm{\Gamma }}{4}\right)•\mathcal{M}\left({\mathit{U}}_{{\beta }_1},{\mathit{U}}_{{\beta }_1+1},\frac{\mathrm{\Gamma }}{4}\right)•\mathcal{M}\left({\mathit{U}}_{{\beta }_1+1},{\mathit{U}}_{{\beta }_1+2},\frac{\mathrm{\Gamma }}{4}\right)\hfill \\ \hfill & •\mathcal{M}\left({\mathit{U}}_{{\beta }_1+2},\wp \mathit{U},\frac{\mathrm{\Gamma }}{4}\right)\hfill \\ \hfill & \le \mathcal{M}\left(\mathit{U},{\mathit{U}}_{{\beta }_1},\frac{\mathrm{\Gamma }}{4}\right)•\mathcal{M}\left(\wp {\mathit{U}}_{{\beta }_1-1},\wp {\mathit{U}}_{{\beta }_1},\frac{\mathrm{\Gamma }}{4}\right)•\mathcal{M}\left(\wp {\mathit{U}}_{{\beta }_1},\wp {\mathit{U}}_{{\beta }_1+1},\frac{\mathrm{\Gamma }}{4}\right)\hfill \\ \hfill & •\mathcal{M}\left(\wp {\mathit{U}}_{{\beta }_1},\wp \mathit{U},\frac{\mathrm{\Gamma }}{4}\right)\hfill \\ \hfill & \le \mathcal{M}\left(\mathit{U},{\mathit{U}}_{{\beta }_1},\frac{\mathrm{\Gamma }}{4}\right)•{♮}^{{\beta }_1-1}\mathcal{M}\left({\mathit{U}}_{{\beta }_1-1},{\mathit{U}}_{{\beta }_1},\frac{\mathrm{\Gamma }}{4}\right)•{♮}^{{\beta }_1}\mathcal{M}\left({\mathit{U}}_{{\beta }_1},{\mathit{U}}_{{\beta }_1+1},\frac{\mathrm{\Gamma }}{4}\right)\hfill \\ \hfill & •♮\mathcal{M}\left({\mathit{U}}_{{\beta }_1},\mathit{U},\frac{\mathrm{\Gamma }}{4}\right)\hfill \\ \hfill & \to 0•0•0•0=0\mathrm{as}{\beta }_1\to +\infty .\hfill \end{array}$$

Hence, $\wp \mathit{U}=\mathit{U}$. Let $\wp \tilde{\mu }=\tilde{\mu }$ for some $\tilde{\mu }\in \mathit{K}$, then

$$\begin{array}{cc}\hfill \frac{1}{\mathrm{\Lambda }(\mathit{U},\tilde{\mu },\mathrm{\Gamma })}-1& =\frac{1}{\mathrm{\Lambda }(\wp \mathit{U},\wp \tilde{\mu },\mathrm{\Gamma })}-1\hfill \\ \hfill & \le ♮\left[\frac{1}{\mathrm{\Lambda }(\mathit{U},\tilde{\mu },\mathrm{\Gamma })}-1\right]<\frac{1}{\mathrm{\Lambda }(\mathit{U},\tilde{\mu },\mathrm{\Gamma })}-1,\hfill \end{array}$$

which is a contradiction.

$$\begin{array}{c}\hfill \mathrm{\Pi }(\mathit{U},\tilde{\mu },\mathrm{\Gamma })=\mathrm{\Pi }(\wp \mathit{U},\wp \tilde{\mu },\mathrm{\Gamma })\le ♮\mathrm{\Pi }(\mathit{U},\tilde{\mu },\mathrm{\Gamma })<\mathrm{\Pi }(\mathit{U},\tilde{\mu },\mathrm{\Gamma }),\end{array}$$

which is a contradiction and

$$\begin{array}{c}\hfill \mathcal{M}(\mathit{U},\tilde{\mu },\mathrm{\Gamma })=\mathcal{M}(\wp \mathit{U},\wp \tilde{\mu },\mathrm{\Gamma })\le ♮\mathcal{M}(\mathit{U},\tilde{\mu },\mathrm{\Gamma })<\mathcal{M}(\mathit{U},\tilde{\mu },\mathrm{\Gamma }),\end{array}$$

which is a contradiction. Therefore, we obtain $\mathrm{\Lambda }(\mathit{U},\tilde{\mu },\mathrm{\Gamma })=1,\mathrm{\Pi }(\mathit{U},\tilde{\mu },\mathrm{\Gamma })=0$ and $\mathcal{M}(\mathit{U},\tilde{\mu },\mathrm{\Gamma })=0$, hence, $\mathit{U}=\tilde{\mu }$. □

Example 2.

Let $\mathit{K}=[0,1]$. Define $\mathrm{\Lambda },\mathrm{\Pi },\mathcal{M}:\mathit{K}\times \mathit{K}\times (0,+\infty )\to [0,1]$ as

$$\begin{array}{cc}\hfill \mathrm{\Lambda }(\mathit{U},\mathcal{W},\mathrm{\Gamma })& =\frac{\mathrm{\Gamma }}{\mathrm{\Gamma }+|\mathit{U}-\mathcal{W}|},\hfill \\ \hfill \mathrm{\Pi }(\mathit{U},\mathcal{W},\mathrm{\Gamma })& =\frac{|\mathit{U}-\mathcal{W}|}{\mathrm{\Gamma }+|\mathit{U}-\mathcal{W}|},\hfill \\ \hfill \mathcal{M}(\mathit{U},\mathcal{W},\mathrm{\Gamma })& =\frac{|\mathit{U}-\mathcal{W}|}{\mathrm{\Gamma }}.\hfill \end{array}$$

Then, $(\mathit{K},\mathrm{\Lambda },\mathrm{\Pi },\mathcal{M},★,•)$ is a complete NPMS with continuous t-norm and t-co-norm, i.e., $\widehat{\omega }★\breve{h}=\widehat{\omega }\breve{h}$ and $\widehat{\omega }•\breve{h}=max\{\widehat{\omega },\breve{h}\}$.

Define $\wp :\mathit{K}\to \mathit{K}$ by $\wp \left(\mathit{U}\right)=\frac{1-3^{-\mathit{U}}}{11}$ and set $♮\in [\frac{1}{2},1)$, then

$$\begin{array}{cc}\hfill \mathrm{\Lambda }(\wp \mathit{U},\wp \mathcal{W},♮\mathrm{\Gamma })& =\mathrm{\Lambda }\left(\frac{1-3^{-\mathit{U}}}{11},\frac{1-3^{-\mathcal{W}}}{11},♮\mathrm{\Gamma }\right)\hfill \\ \hfill & =\frac{♮\mathrm{\Gamma }}{♮\mathrm{\Gamma }+|\frac{1-3^{-\mathit{U}}}{11}-\frac{1-3^{-\mathcal{W}}}{11}|}=\frac{♮\mathrm{\Gamma }}{♮\mathrm{\Gamma }+\frac{|3^{-\mathit{U}}-3^{-\mathcal{W}}|}{11}}\hfill \\ \hfill & \ge \frac{♮\mathrm{\Gamma }}{♮\mathrm{\Gamma }+\frac{|\mathit{U}-\mathcal{W}|}{11}}=\frac{11♮\mathrm{\Gamma }}{11♮\mathrm{\Gamma }+|\mathit{U}-\mathcal{W}|}\ge \frac{\mathrm{\Gamma }}{\mathrm{\Gamma }+|\mathit{U}-\mathcal{W}|}=\mathrm{\Lambda }(\mathit{U},\mathcal{W},\mathrm{\Gamma }),\hfill \end{array}$$

$$\begin{array}{cc}\hfill \mathrm{\Pi }(\wp \mathit{U},\wp \mathcal{W},♮\mathrm{\Gamma })& =\mathrm{\Pi }\left(\frac{1-3^{-\mathit{U}}}{11},\frac{1-3^{-\mathcal{W}}}{11},♮\mathrm{\Gamma }\right)\hfill \\ \hfill & =\frac{|\frac{1-3^{-\mathit{U}}}{11}-\frac{1-3^{-\mathcal{W}}}{11}|}{♮\mathrm{\Gamma }+|\frac{1-3^{-\mathit{U}}}{11}-\frac{1-3^{-\mathcal{W}}}{11}|}=\frac{\frac{|3^{-\mathit{U}}-3^{-\mathcal{W}}|}{11}}{♮\mathrm{\Gamma }+\frac{|3^{-\mathit{U}}-3^{-\mathcal{W}}|}{11}}\hfill \\ \hfill & =\frac{|3^{-\mathit{U}}-3^{-\mathcal{W}}|}{11♮\mathrm{\Gamma }+|3^{-\mathit{U}}-3^{-\mathcal{W}}|}\le \frac{|\mathit{U}-\mathcal{W}|}{11♮\mathrm{\Gamma }+|\mathit{U}-\mathcal{W}|}\le \frac{|\mathit{U}-\mathcal{W}|}{\mathrm{\Gamma }+|\mathit{U}-\mathcal{W}|}=\mathrm{\Pi }(\mathit{U},\mathcal{W},\mathrm{\Gamma })\hfill \end{array}$$

and

$$\begin{array}{cc}\hfill \mathcal{M}(\wp \mathit{U},\wp \mathcal{W},♮\mathrm{\Gamma })& =\mathcal{M}\left(\frac{1-3^{-\mathit{U}}}{11},\frac{1-3^{-\mathcal{W}}}{11},♮\mathrm{\Gamma }\right)\hfill \\ \hfill & =\frac{|\frac{1-3^{-\mathit{U}}}{11}-\frac{1-3^{-\mathcal{W}}}{11}|}{♮\mathrm{\Gamma }}=\frac{\frac{|3^{-\mathit{U}}-3^{-\mathcal{W}}|}{11}}{♮\mathrm{\Gamma }}\hfill \\ \hfill & =\frac{|3^{-\mathit{U}}-3^{-\mathcal{W}}|}{11♮\mathrm{\Gamma }}\le \frac{|\mathit{U}-\mathcal{W}|}{11♮\mathrm{\Gamma }}\le \frac{|\mathit{U}-\mathcal{W}|}{\mathrm{\Gamma }}=\mathcal{M}(\mathit{U},\mathcal{W},\mathrm{\Gamma }).\hfill \end{array}$$

As a result, all of Theorem 1 criteria are satisfied, and 0 is the only fixed point for ℘.

4. Applications

4.1. Application to Fredholm Integral Equation

Let $\mathit{K}=\mathcal{C}\left(\right[𝚥,\delta ],\mathbb{R})$ be the set of real value continuous functions on $[𝚥,\delta ]$.

Consider the integral equation:

$$\begin{array}{c}\hfill \mathit{U}(\aleph )=\wedge (\aleph )+\mathsf{\ell }{\int }_{𝚥}^{\delta }\mho (\aleph ,\kappa )\mathit{U}(\aleph )\varsigma \kappa \mathrm{for}\aleph ,\kappa \in [𝚥,\delta ],\end{array}$$

(30)

where $\wedge \left(\kappa \right)$ is a fuzzy function of $\kappa \in [𝚥,\delta ]$, $\mathsf{\ell }>0$ and $\mho :\mathcal{C}([𝚥,\delta ]\times \mathbb{R})\to {\mathbb{R}}^{+}$. Define $\mathrm{\Lambda }$, $\mathrm{\Pi }$ and $\mathcal{M}$ by means of

$$\begin{array}{cc}\hfill \mathrm{\Lambda }\left(\mathit{U}\right(\aleph ),\mathcal{W}(\aleph ),\mathrm{\Gamma })& =\underset{\aleph \in [𝚥,\delta ]}{sup}\frac{\mathrm{\Gamma }}{\mathrm{\Gamma }+\left|\mathit{U}\right(\aleph )-\mathcal{W}(\aleph \left)\right|}\forall \mathit{U},\mathcal{W}\in \mathit{K}\mathrm{and}\mathrm{\Gamma }>0,\hfill \\ \hfill \mathrm{\Pi }\left(\mathit{U}\right(\aleph ),\mathcal{W}(\aleph ),\mathrm{\Gamma })& =1-\underset{\aleph \in [𝚥,\delta ]}{sup}\frac{\mathrm{\Gamma }}{\mathrm{\Gamma }+\left|\mathit{U}\right(\aleph )-\mathcal{W}(\aleph \left)\right|}\forall \mathit{U},\mathcal{W}\in \mathit{K}\mathrm{and}\mathrm{\Gamma }>0,\hfill \end{array}$$

and

$$\begin{array}{cc}\hfill \mathcal{M}\left(\mathit{U}\right(\aleph ),\mathcal{W}(\aleph ),\mathrm{\Gamma })& =\underset{\aleph \in [𝚥,\delta ]}{sup}\frac{\left|\mathit{U}\right(\aleph )-\mathcal{W}(\aleph \left)\right|}{\mathrm{\Gamma }}\forall \mathit{U},\mathcal{W}\in \mathit{K}\mathrm{and}\mathrm{\Gamma }>0,\hfill \end{array}$$

by continuous t-norm and continuous t-co-norm define by $\widehat{\omega }★\breve{h}=\widehat{\omega }·\breve{h}$ and $\widehat{\omega }•\breve{h}=max\{\widehat{\omega },\breve{h}\}$. Then $(\mathit{K},\mathrm{\Lambda },\mathrm{\Pi },\mathcal{M},★,•)$ is a complete NPMS. Suppose that $|\mho (\aleph ,\kappa \left)\mathit{U}\right(\aleph )-\mho (\aleph ,\kappa \left)\mathcal{W}\right(\aleph \left)\right|\le \left|\mathit{U}\right(\aleph )-\mathcal{W}(\aleph \left)\right|$ for $\mathit{U},\mathcal{W}\in \mathit{K},0<♮<1$ and $\forall \aleph ,\kappa \in [𝚥,\delta ]$. Let $\mho (\aleph ,\kappa )(\mathsf{\ell }{\int }_{𝚥}^{\delta }d\kappa )\le ♮<1$. Then Equation (30) has a unique solution.

Proof. 

Define $\wp :\mathit{K}\to \mathit{K}$ by

$$\begin{array}{c}\hfill \wp \mathit{U}(\aleph )=\wedge (\aleph )+\mathsf{\ell }{\int }_{𝚥}^{\delta }\mho (\aleph ,\kappa )\mathit{U}(\aleph )\varsigma \kappa ,\mathrm{for}\mathrm{all}\aleph ,\kappa \in [𝚥,\delta ].\end{array}$$

Now, for all $\mathit{U},\mathcal{W}\in \mathit{K}$, we deduce

$$\begin{array}{cc}\hfill & \mathrm{\Lambda }(\wp \mathit{U}(\aleph ),\wp \mathcal{W}(\aleph ),♮\mathrm{\Gamma })=\underset{\aleph \in [𝚥,\delta ]}{sup}\frac{♮\mathrm{\Gamma }}{♮\mathrm{\Gamma }+|\wp \mathit{U}(\aleph )-\wp \mathcal{W}(\aleph \left)\right|}\hfill \\ \hfill & =\underset{\aleph \in [𝚥,\delta ]}{sup}\frac{♮\mathrm{\Gamma }}{♮\mathrm{\Gamma }+|\wedge (\aleph )+\mathsf{\ell }{\int }_{𝚥}^{\delta }\mho (\aleph ,\kappa )\mathit{U}(\aleph )\varsigma \kappa -\wedge (\aleph )-\mathsf{\ell }{\int }_{𝚥}^{\delta }\mho (\aleph ,\kappa )\mathit{U}(\aleph )\varsigma \kappa |}\hfill \\ \hfill & =\underset{\aleph \in [𝚥,\delta ]}{sup}\frac{♮\mathrm{\Gamma }}{♮\mathrm{\Gamma }+|\mathsf{\ell }{\int }_{𝚥}^{\delta }\mho (\aleph ,\kappa )\mathit{U}(\aleph )\varsigma \kappa -\mathsf{\ell }{\int }_{𝚥}^{\delta }\mho (\aleph ,\kappa )\mathit{U}(\aleph )\varsigma \kappa |}\hfill \\ \hfill & =\underset{\aleph \in [𝚥,\delta ]}{sup}\frac{♮\mathrm{\Gamma }}{♮\mathrm{\Gamma }+|\mho (\aleph ,\kappa )\mathit{U}(\aleph )-\mho (\aleph ,\kappa )\mathcal{W}(\aleph )|(\mathsf{\ell }{\int }_{𝚥}^{\delta }\varsigma \kappa )}\hfill \\ \hfill & \ge \underset{\aleph \in [𝚥,\delta ]}{sup}\frac{\mathrm{\Gamma }}{\mathrm{\Gamma }+\left|\mathit{U}\right(\aleph )-\mathcal{W}(\aleph \left)\right|}\hfill \\ \hfill & \ge \mathrm{\Lambda }\left(\mathit{U}\right(\aleph ),\mathcal{W}(\aleph ),\mathrm{\Gamma }),\hfill \end{array}$$

$$\begin{array}{cc}\hfill & \mathrm{\Pi }(\wp \mathit{U}(\aleph ),\wp \mathcal{W}(\aleph ),♮\mathrm{\Gamma })=1-\underset{\aleph \in [𝚥,\delta ]}{sup}\frac{♮\mathrm{\Gamma }}{♮\mathrm{\Gamma }+|\wp \mathit{U}(\aleph )-\wp \mathcal{W}(\aleph \left)\right|}\hfill \\ \hfill & =1-\underset{\aleph \in [𝚥,\delta ]}{sup}\frac{♮\mathrm{\Gamma }}{♮\mathrm{\Gamma }+|\wedge (\aleph )+\mathsf{\ell }{\int }_{𝚥}^{\delta }\mho (\aleph ,\kappa )\mathit{U}(\aleph )\varsigma \kappa -\wedge (\aleph )-\mathsf{\ell }{\int }_{𝚥}^{\delta }\mho (\aleph ,\kappa )\mathit{U}(\aleph )\varsigma \kappa |}\hfill \\ \hfill & =1-\underset{\aleph \in [𝚥,\delta ]}{sup}\frac{♮\mathrm{\Gamma }}{♮\mathrm{\Gamma }+|\mathsf{\ell }{\int }_{𝚥}^{\delta }\mho (\aleph ,\kappa )\mathit{U}(\aleph )\varsigma \kappa -\mathsf{\ell }{\int }_{𝚥}^{\delta }\mho (\aleph ,\kappa )\mathit{U}(\aleph )\varsigma \kappa |}\hfill \\ \hfill & =1-\underset{\aleph \in [𝚥,\delta ]}{sup}\frac{♮\mathrm{\Gamma }}{♮\mathrm{\Gamma }+|\mho (\aleph ,\kappa )\mathit{U}(\aleph )-\mho (\aleph ,\kappa )\mathcal{W}(\aleph )|(\mathsf{\ell }{\int }_{𝚥}^{\delta }\varsigma \kappa )}\hfill \\ \hfill & \le 1-\underset{\aleph \in [𝚥,\delta ]}{sup}\frac{\mathrm{\Gamma }}{\mathrm{\Gamma }+\left|\mathit{U}\right(\aleph )-\mathcal{W}(\aleph \left)\right|}\hfill \\ \hfill & \le \mathrm{\Pi }\left(\mathit{U}\right(\aleph ),\mathcal{W}(\aleph ),\mathrm{\Gamma }),\hfill \end{array}$$

and

$$\begin{array}{cc}\hfill & \mathcal{M}(\wp \mathit{U}(\aleph ),\wp \mathcal{W}(\aleph ),♮\mathrm{\Gamma })=\underset{\aleph \in [𝚥,\delta ]}{sup}\frac{|\wp \mathit{U}(\aleph )-\wp \mathcal{W}(\aleph \left)\right|}{♮\mathrm{\Gamma }}\hfill \\ \hfill & =\underset{\aleph \in [𝚥,\delta ]}{sup}\frac{|\wedge (\aleph )+\mathsf{\ell }{\int }_{𝚥}^{\delta }\mho (\aleph ,\kappa )\mathit{U}(\aleph )\varsigma \kappa -\wedge (\aleph )-\mathsf{\ell }{\int }_{𝚥}^{\delta }\mho (\aleph ,\kappa )\mathit{U}(\aleph )\varsigma \kappa |}{♮\mathrm{\Gamma }}\hfill \\ \hfill & =\underset{\aleph \in [𝚥,\delta ]}{sup}\frac{|\mathsf{\ell }{\int }_{𝚥}^{\delta }\mho (\aleph ,\kappa )\mathit{U}(\aleph )\varsigma \kappa -\mathsf{\ell }{\int }_{𝚥}^{\delta }\mho (\aleph ,\kappa )\mathit{U}(\aleph )\varsigma \kappa |}{♮\mathrm{\Gamma }}\hfill \\ \hfill & =\underset{\aleph \in [𝚥,\delta ]}{sup}\frac{|\mho (\aleph ,\kappa )\mathit{U}(\aleph )-\mho (\aleph ,\kappa )\mathcal{W}(\aleph )|(\mathsf{\ell }{\int }_{𝚥}^{\delta }\varsigma \kappa )}{♮\mathrm{\Gamma }}\hfill \\ \hfill & \le \underset{\aleph \in [𝚥,\delta ]}{sup}\frac{\left|\mathit{U}\right(\aleph )-\mathcal{W}(\aleph \left)\right|}{\mathrm{\Gamma }}\hfill \\ \hfill & \le \mathcal{M}\left(\mathit{U}\right(\aleph ),\mathcal{W}(\aleph ),\mathrm{\Gamma }),\hfill \end{array}$$

As a result, ℘ has a ufp and all of the requirements of Theorem 1 are fulfilled. It is obvious that the Equation (30) has only one solution. □

Example 3.

Consider the non-linear integral equation.

$$\begin{array}{c}\hfill \mathit{U}(\aleph )=|cos\aleph |+\frac{1}{11}{\int }_0^1\kappa \mathit{U}\left(\kappa \right)\varsigma \kappa ,forall\kappa \in [0,1].\end{array}$$

Then it has a solution in $\mathit{K}$.

Proof. 

Let $\wp :\mathit{K}\to \mathit{K}$ be defined by

$$\begin{array}{c}\hfill \wp \mathit{U}(\aleph )=|cos\aleph |+\frac{1}{11}{\int }_0^1\kappa \mathit{U}\left(\kappa \right)\varsigma \kappa ,\end{array}$$

and set $\mho (\aleph ,\kappa )\mathit{U}(\aleph )=\frac{1}{11}\kappa \mathit{U}\left(\kappa \right)$ and $\mho (\aleph ,\kappa )\mathcal{W}(\aleph )=\frac{1}{11}\kappa \mathcal{W}\left(\kappa \right)$, where $\mathit{U},\mathcal{W}\in \mathit{K}$, and $\forall \aleph ,\kappa \in [0,1]$. Then we obtain

$$\begin{array}{cc}\hfill |\mho (\aleph ,\kappa \left)\mathit{U}\right(\aleph )-\mho (\aleph ,\kappa \left)\mathcal{W}\right(\aleph \left)\right|& =|\frac{1}{11}\kappa \mathit{U}\left(\kappa \right)-\frac{1}{11}\kappa \mathcal{W}\left(\kappa \right)|\hfill \\ \hfill & =\frac{\kappa }{11}|\mathit{U}\left(\kappa \right)-\mathcal{W}\left(\kappa \right)|\le |\mathit{U}\left(\kappa \right)-\mathcal{W}\left(\kappa \right)|.\hfill \end{array}$$

Furthermore, see that $\frac{1}{11}{\int }_0^1\kappa \varsigma \kappa =\frac{1}{11}\left(\frac{{\left(1\right)}^2}{2}-\frac{{\left(0\right)}^2}{2}\right)=\frac{1}{11}=♮\le 1$, where $\mathsf{\ell }=\frac{1}{11}$. Then, it follows that all criteria of the above application are easily verified and the above problem has a solution in $\mathit{K}$. □

4.2. Application to Fractional Differential Equations

In order to start, we need to review some basic definitions from the theory of fractional calculus.

For a function $\mathit{U}\in \mathcal{C}[0,1]$, the Reiman-Liouville fractional derivative of order $\mathsf{\ell }>0$ is given by

$$\begin{array}{c}\hfill \frac{1}{\mathbf{F}({\beta }_1-\mathsf{\ell })}\frac{d^{{\beta }_1}}{d{\aleph }^{{\beta }_1}}{\int }_0^{\aleph }\frac{\mathit{U}\left(𝚥\right)d𝚥}{{(\aleph -𝚥)}^{\mathsf{\ell }-{\beta }_1+1}}={\mathcal{D}}^{\mathsf{\ell }}\mathit{U}(\aleph ),\end{array}$$

For as long as the right hand side is pointwise defined on $[0,1]$, $[\mathsf{\ell }]$ is the integer portion of the number $\mathsf{\ell },\mathbf{F}$ is the Euler gamma function.

Consider the following fractional differential equation

$$\begin{array}{cc}\hfill & {}^{𝚥}{\mathcal{D}}^{\chi }\mathit{U}(\aleph )+\mathfrak{f}(\aleph ,\mathit{U}(\aleph ))=0,1\le \aleph \le 0,2\le \chi >1;\hfill \\ \hfill & \mathit{U}\left(0\right)=\mathit{U}\left(1\right)=0,\hfill \end{array}$$

(31)

where $\mathfrak{f}$ is a continuous function from $[0,1]\times \mathbb{R}$ to $\mathbb{R}$ and ${}^{𝚥}{\mathcal{D}}^{\chi }$ represents the Caputo fractional derivative of order $\chi$ and it is defined by

$$\begin{array}{c}\hfill {}^{𝚥}{\mathcal{D}}^{\chi }=\frac{1}{\mathbf{F}({\beta }_1-\chi )}{\int }_0^{\aleph }\frac{{\mathit{U}}^{{\beta }_1}\left(𝚥\right)d𝚥}{{(\aleph -𝚥)}^{\chi -{\beta }_1+1}}.\end{array}$$

The given fractional differential Equation (31) is equivalent

$$\begin{array}{c}\hfill \mathit{U}(\aleph )={\int }_0^1{\rm Y}(\aleph ,𝚥)\mathfrak{f}(\aleph ,\mathit{U}\left(𝚥\right))d𝚥,\end{array}$$

for all $\mathit{U}\in \mathcal{Y}$ and $\aleph \in [0,1]$, where

$$\begin{array}{c}\hfill {\rm Y}(\aleph ,𝚥)=\left\{\begin{array}{cc}\frac{{[\aleph (1-𝚥)]}^{\chi -1}-{(\aleph -𝚥)}^{\chi -1}}{\mathbf{F}\left(\chi \right)},\hfill & 0\le 𝚥\le \aleph \le 1,\hfill \\ \frac{{[\aleph (1-𝚥)]}^{\chi -1}}{\mathbf{F}\left(\chi \right)},\hfill & 0\le \aleph \le 𝚥\le 1.\hfill \end{array}\right.\end{array}$$

Let $\mathcal{C}\left(\right[0,1],\mathbb{R})=\mathit{K}$ be the space of all continuous functions defined on $[0,1]$. Define $\mathrm{\Lambda }$, $\mathrm{\Pi }$ and $\mathcal{M}$ by means of

$$\begin{array}{cc}\hfill \mathrm{\Lambda }\left(\mathit{U}\right(\aleph ),\mathcal{W}(\aleph ),\mathrm{\Gamma })& =\underset{\aleph \in [0,1]}{sup}\frac{\mathrm{\Gamma }}{\mathrm{\Gamma }+\left|\mathit{U}\right(\aleph )-\mathcal{W}(\aleph \left)\right|},\mathrm{for}\mathrm{all}\mathit{U},\mathcal{W}\in \mathit{K}\mathrm{and}\mathrm{\Gamma }>0,\hfill \\ \hfill \mathrm{\Pi }\left(\mathit{U}\right(\aleph ),\mathcal{W}(\aleph ),\mathrm{\Gamma })& =1-\underset{\aleph \in [0,1]}{sup}\frac{\mathrm{\Gamma }}{\mathrm{\Gamma }+\left|\mathit{U}\right(\aleph )-\mathcal{W}(\aleph \left)\right|},\mathrm{for}\mathrm{all}\mathit{U},\mathcal{W}\in \mathit{K}\mathrm{and}\mathrm{\Gamma }>0\hfill \end{array}$$

and

$$\begin{array}{cc}\hfill \mathcal{M}\left(\mathit{U}\right(\aleph ),\mathcal{W}(\aleph ),\mathrm{\Gamma })& =\underset{\aleph \in [0,1]}{sup}\frac{\left|\mathit{U}\right(\aleph )-\mathcal{W}(\aleph \left)\right|}{\mathrm{\Gamma }},\mathrm{for}\mathrm{all}\mathit{U},\mathcal{W}\in \mathit{K}\mathrm{and}\mathrm{\Gamma }>0,\hfill \end{array}$$

with continuous t-norm and continuous t-co-norm define by $\widehat{\omega }★\breve{h}=\widehat{\omega }·\breve{h}$ and $\widehat{\omega }•\breve{h}=max\{\widehat{\omega },\breve{h}\}$. Then $(\mathit{K},\mathrm{\Lambda },\mathrm{\Pi },\mathcal{M},★,•)$ is a complete NPMS.

Theorem 3.

Consider the nonlinear fractional differential Equation (31). Let us assume that the following claims are true:

(i)

there exist $\aleph \in [0,1]$ and $\mathit{U},\mathcal{W}\in \mathcal{C}\left(\right[0,1],\mathbb{R})$ such that

$$\begin{array}{c}\hfill \left|\mathfrak{f}\right(\aleph ,\mathit{U})-\mathfrak{f}(\aleph ,\mathcal{W}\left)\right|\le \left|\mathit{U}\right(\aleph )-\mathcal{W}(\aleph \left)\right|;\end{array}$$

(ii)

${sup}_{\aleph \in [0,1]}{\int }_0^1{\rm Y}(\aleph ,𝚥)\varsigma \aleph \le ♮<1.$

Then the fractional differential Equation (31) has a unique solution in $\mathit{K}$.

Proof. 

Let us consider the function $\wp :\mathit{K}\to \mathit{K}$ defined by

$$\begin{array}{c}\hfill \mathit{K}\mathit{U}(\aleph )={\int }_0^1{\rm Y}(\aleph ,𝚥)\mathfrak{f}(\aleph ,\mathit{U}\left(𝚥\right))d𝚥.\end{array}$$

It is obvious that ${\mathit{U}}^{*}$ is a solution to the problem (31) if ${\mathit{U}}^{*}$ is a fixed point of ℘.

Now, for all $\mathit{U},\mathcal{W}\in \mathit{K}$, we deduce

$$\begin{array}{cc}\hfill \mathrm{\Lambda }(\wp \mathit{U}(\aleph ),\wp \mathcal{W}(\aleph ),♮\mathrm{\Gamma })& =\underset{\aleph \in [0,1]}{sup}\frac{♮\mathrm{\Gamma }}{♮\mathrm{\Gamma }+|\wp \mathit{U}(\aleph )-\wp \mathcal{W}(\aleph \left)\right|}\hfill \\ \hfill & =\underset{\aleph \in [0,1]}{sup}\frac{♮\mathrm{\Gamma }}{♮\mathrm{\Gamma }+|{\int }_0^1{\rm Y}(\aleph ,𝚥)\mathfrak{f}(\aleph ,\mathit{U}\left(𝚥\right))d𝚥-{\int }_0^1{\rm Y}(\aleph ,𝚥)\mathfrak{f}(\aleph ,\mathcal{W}\left(𝚥\right))d𝚥|}\hfill \\ \hfill & =\underset{\aleph \in [0,1]}{sup}\frac{♮\mathrm{\Gamma }}{♮\mathrm{\Gamma }+{\int }_0^1{\rm Y}(\aleph ,𝚥)|\mathfrak{f}(\aleph ,\mathit{U}\left(𝚥\right))-\mathfrak{f}(\aleph ,\mathcal{W}\left(𝚥\right))|d𝚥}\hfill \\ \hfill & \ge \underset{\aleph \in [0,1]}{sup}\frac{\mathrm{\Gamma }}{\mathrm{\Gamma }+\left|\mathit{U}\right(\aleph )-\mathcal{W}(\aleph \left)\right|}\hfill \\ \hfill & \ge \mathrm{\Lambda }\left(\mathit{U}\right(\aleph ),\mathcal{W}(\aleph ),\mathrm{\Gamma }),\hfill \end{array}$$

$$\begin{array}{cc}\hfill \mathrm{\Pi }(\wp \mathit{U}(\aleph ),\wp \mathcal{W}(\aleph ),♮\mathrm{\Gamma })& =1-\underset{\aleph \in [0,1]}{sup}\frac{♮\mathrm{\Gamma }}{♮\mathrm{\Gamma }+|\wp \mathit{U}(\aleph )-\wp \mathcal{W}(\aleph \left)\right|}\hfill \\ \hfill & =1-\underset{\aleph \in [0,1]}{sup}\frac{♮\mathrm{\Gamma }}{♮\mathrm{\Gamma }+|{\int }_0^1{\rm Y}(\aleph ,𝚥)\mathfrak{f}(\aleph ,\mathit{U}\left(𝚥\right))d𝚥-{\int }_0^1{\rm Y}(\aleph ,𝚥)\mathfrak{f}(\aleph ,\mathcal{W}\left(𝚥\right))d𝚥|}\hfill \\ \hfill & =1-\underset{\aleph \in [0,1]}{sup}\frac{♮\mathrm{\Gamma }}{♮\mathrm{\Gamma }+{\int }_0^1{\rm Y}(\aleph ,𝚥)|\mathfrak{f}(\aleph ,\mathit{U}\left(𝚥\right))-\mathfrak{f}(\aleph ,\mathcal{W}\left(𝚥\right))|d𝚥}\hfill \\ \hfill & \le 1-\underset{\aleph \in [0,1]}{sup}\frac{\mathrm{\Gamma }}{\mathrm{\Gamma }+\left|\mathit{U}\right(\aleph )-\mathcal{W}(\aleph \left)\right|}\hfill \\ \hfill & \le \mathrm{\Pi }\left(\mathit{U}\right(\aleph ),\mathcal{W}(\aleph ),\mathrm{\Gamma })\hfill \end{array}$$

and

$$\begin{array}{cc}\hfill \mathcal{M}(\wp \mathit{U}(\aleph ),\wp \mathcal{W}(\aleph ),♮\mathrm{\Gamma })& =\underset{\aleph \in [0,1]}{sup}\frac{|\wp \mathit{U}(\aleph )-\wp \mathcal{W}(\aleph \left)\right|}{♮\mathrm{\Gamma }}\hfill \\ \hfill & =\underset{\aleph \in [0,1]}{sup}\frac{|{\int }_0^1{\rm Y}(\aleph ,𝚥)\mathfrak{f}(\aleph ,\mathit{U}\left(𝚥\right))d𝚥-{\int }_0^1{\rm Y}(\aleph ,𝚥)\mathfrak{f}(\aleph ,\mathcal{W}\left(𝚥\right))d𝚥|}{♮\mathrm{\Gamma }}\hfill \\ \hfill & =\underset{\aleph \in [0,1]}{sup}\frac{{\int }_0^1{\rm Y}(\aleph ,𝚥)|\mathfrak{f}(\aleph ,\mathit{U}\left(𝚥\right))-\mathfrak{f}(\aleph ,\mathcal{W}\left(𝚥\right))|d𝚥}{♮\mathrm{\Gamma }}\hfill \\ \hfill & \le \underset{\aleph \in [0,1]}{sup}\frac{\left|\mathit{U}\right(\aleph )-\mathcal{W}(\aleph \left)\right|}{\mathrm{\Gamma }}\hfill \\ \hfill & \le \mathcal{M}\left(\mathit{U}\right(\aleph ),\mathcal{W}(\aleph ),\mathrm{\Gamma }).\hfill \end{array}$$

As a result, ℘ has a ufp and all of Theorem 1 requirements are satisfied. It implies that there is only one solution to the Equation (30). □

5. Conclusions

We proposed the idea of neutrosophic pentagonal MS in this study and proved new varieties of fixed-point theorems. By applying a new methodology to an application based on the literature, we have shown that it outperforms our results.