A Symmetric View of Fixed-Point Results in Non-Archimedean Generalized Neutrosophic Metric Spaces

Abstract

In this paper, we extend prominent fixed-point theorems within the framework of symmetry, a structure increasingly relevant in decision-making, optimization, and uncertainty modeling. While previous studies have explored fixed-point theorems in non-Archimedean spaces, the influence of symmetry on the properties of mappings remains underexamined. To address this gap, we introduce and analyze the concepts of $\chi$-contractions and $\chi$-weak contractions, demonstrating how symmetry impacts the conditions for the existence of fixed points. Our methodology integrates these concepts in generalized neutrosophic metric spaces, providing a novel perspective on fixed-point theory. We perform a rigorous analysis, revealing new insights into their practical applications. However, our proposed system may face limitations in complex or dynamic environments, where additional conditions may be necessary to ensure the existence or uniqueness of fixed points.

1. Introduction

In recent decades, the study of fuzzy sets has gained significant attention in various fields, laying the foundation for advanced mathematical theories and applications. Introduced by Zadeh in 1965 [1], fuzzy sets allow for the representation of uncertain or imprecise information, providing a framework for modeling vagueness in real-world phenomena. This concept laid the foundation for many advancements in fuzzy mathematics, including the work of Atanassov [2], who extended fuzzy sets to intuitionistic fuzzy sets, incorporating both membership and non-membership functions.

In 1975, Kramosil and Michalek [3] developed fuzzy metric spaces, further expanding the utility of fuzzy sets into metric spaces by applying fuzzy logic to distance measurement. This approach allowed for the characterization of uncertainty in distance, making it suitable for applications in fields such as image processing and machine learning. George and Veeramani [4] and Sun et al. [5] explored generalized fuzzy metric spaces and their key properties, broadening their applicability.

Park [6] further extended this concept by introducing intuitionistic fuzzy metric spaces, which combine Atanassov’s intuitionistic fuzzy sets with Kramosil’s fuzzy metric spaces, addressing more complex forms of uncertainty in analysis and decision-making. This work aligns with a broader interest in fixed-point theorems in fuzzy environments, exemplified by Jeyaraman et al. [7], who extended the results to intuitionistic generalized fuzzy cone metric spaces.

Neutrosophic theory, proposed by Smarandache [8], introduced neutrosophic probability, sets, and logic, further generalizing fuzzy set theory by independently addressing indeterminacy, truth, and falsity. Kirisci and Simsek [9] contributed by defining neutrosophic metric spaces, offering a framework for dealing with uncertainty, indeterminacy, and vagueness. Sowndrarajan et al. [10] demonstrated the utility of these spaces by deriving fixed-point results for contraction theorems in neutrosophic metric spaces.

Fixed-point theory continued to evolve with contributions like those by Farokhzad Rostami [11], who explored contractive mappings in non-Archimedean G-fuzzy metric spaces, and Johnsy and Jeyaraman [12], who investigated fixed-point theorems for $(\psi -\varphi )$-contractions in generalized neutrosophic metric spaces. Akram et al. [13] introduced new generalized neutrosophic metric spaces and provided fixed-point results, particularly in decision-making models involving multiple criteria.

Mihet [14] examined fuzzy $\psi$-contractive mappings in non-Archimedean fuzzy metric spaces, while Al-Khaleel et al. [15] in 2023 established fixed-point results for cyclic contractive mappings of Kannan and Chatterjea types in generalized metric spaces. Gupta and Mani [16] and Wardowski [17] further contributed to the practical application of fuzzy metric spaces, focusing on the existence and uniqueness of fixed points in computational models.

Banach’s pioneering work on fixed-point theorems [18] remains a cornerstone for modern research, providing a unified framework for understanding convergence and stability across various metric and fuzzy environments. Despite these advancements, a notable gap remains in the exploration of symmetry in non-Archimedean generalized neutrosophic metric spaces (GNMSs). Symmetry ensures that the distance-like functions $\mathfrak{J},\mathfrak{H},\mathfrak{E}$ yield consistent results regardless of the order of elements, an essential property for maintaining consistency in fixed-point theory and contraction mappings.

This study addresses this gap by introducing $\chi$-contractions and $\chi$-weak contractions in non-Archimedean GNMSs. These generalizations capture the non-Archimedean structure through strictly increasing functions, with $\chi$-weak contractions providing a less restrictive condition than $\chi$-contractions. We develop new fixed-point theorems, emphasizing the role of symmetry in the convergence process. By incorporating a new class of auxiliary functions under the contractivity rule, we extend the results from non-Archimedean fuzzy metric spaces to GNMSs, contributing to the theoretical framework of fixed-point theory and creating new opportunities for applications in decision-making, optimization, and complex systems analysis.

2. Preliminaries

Before demonstrating our main results, we review some essential definitions and facts that are referenced later in this paper.

Definition 1

([3]). A binary operation, denoted by $\otimes :[0,1]\times [0,1]\to [0,1]$, qualifies as a continuous t-norm (CTN) if it meets the following criteria:

1. 

⊗ adheres to both associativity and commutativity properties.

2. 

⊗ needs to be continuous.

3. 

For every $\delta \in [0,1]$, it holds that $\delta \otimes 1=\delta$.

4. 

For any $\delta ,\alpha ,\varsigma ,\upsilon \in [0,1]$, if $\delta \le \varsigma$ and $\alpha \le \upsilon$, then $\delta \otimes \alpha \le \varsigma \otimes \upsilon$.

Definition 2

([6]). A binary operation, denoted by ⊕, is defined as follows: it maps the interval $[0,1]\times [0,1]$ to the interval $[0,1]$. A continuous t-conorm (CTCN) is specified by the following conditions:

1. 

⊕ adheres to both associativity and commutativity properties.

2. 

⊕ needs to be continuous.

3. 

For every $\delta \in [0,1]$, it holds that $\delta \oplus 0=\delta$.

4. 

For any $\delta ,\alpha ,\varsigma ,\upsilon \in [0,1]$, if $\delta \le \varsigma$ and $\alpha \le \upsilon ,$, then $\delta \oplus \alpha \le \varsigma \oplus \upsilon$.

Definition 3.

A generalized neutrosophic metric space is described as a 6-tuple $(\mathfrak{A},\mathfrak{J},\mathfrak{H},\mathfrak{E},\otimes ,\oplus )$, where $\mathfrak{A}$ is a nonempty set, ⊗ represents a CTN, ⊕ represents a CTCN, and $\mathfrak{J},\mathfrak{H},\mathfrak{E}$ are fuzzy sets defined on ${\mathfrak{A}}^3\times (0,\infty )$. These elements satisfy the required conditions for all $\zeta ,\phi >0$.

(i) 

$\mathfrak{J}(\iota ,\vartheta ,\xi ,\phi )+\mathfrak{H}(\iota ,\vartheta ,\xi ,\phi )+\mathfrak{E}(\iota ,\vartheta ,\xi ,\phi )\le 3$ for all $\iota ,\vartheta \in \mathfrak{A}$ with $\iota \ne \vartheta$,

(ii) 

$\mathfrak{J}(\iota ,\iota ,\vartheta ,\phi )<1$,

(iii) 

$\mathfrak{J}(\iota ,\iota ,\vartheta ,\phi )\le \mathfrak{J}(\iota ,\vartheta ,\xi ,\phi )$ for all $\iota ,\vartheta ,\xi \in \mathfrak{A}$ with $\vartheta \ne \xi$,

(iv) 

$\mathfrak{J}(\iota ,\vartheta ,\xi ,\phi )=1$, then $\iota =\vartheta =\xi$,

(v) 

$\mathfrak{J}(\iota ,\vartheta ,\xi ,\phi )=\mathfrak{J}\left(\mathfrak{s}\right(\iota ,\vartheta ,\xi ),\phi )$, where $\mathfrak{s}$ is a permutation function,

(vi) 

$\mathfrak{J}(\iota ,\vartheta ,\xi ,\phi +\zeta )\ge \mathfrak{J}(\iota ,\rho ,\rho ,\zeta )\otimes \mathfrak{J}(\rho ,\vartheta ,\xi ,\phi )$ for all $\iota ,\vartheta ,\xi ,\rho \in \mathfrak{A}$,

(vii) 

$\mathfrak{J}(\iota ,\vartheta ,\xi ,.):(0,\infty )\to [0,1]$ is continuous.

(viii) 

$\mathfrak{H}(\iota ,\iota ,\vartheta ,\phi )>0$,

(ix) 

$\mathfrak{H}(\iota ,\iota ,\vartheta ,\phi )\ge \mathfrak{H}(\iota ,\vartheta ,\xi ,\phi )$ for all $\iota ,\vartheta ,\xi \in \mathfrak{A}$ with $\vartheta \ne \xi$,

(x) 

$\mathfrak{H}(\iota ,\vartheta ,\xi ,\phi )=0$, then $\iota =\vartheta =\xi$,

(xi) 

$\mathfrak{H}(\iota ,\vartheta ,\xi ,\phi )=\mathfrak{H}\left(\mathfrak{s}\right(\iota ,\vartheta ,\xi ),\phi )$, where $\mathfrak{s}$ is a permutation function,

(xii) 

$\mathfrak{H}(\iota ,\vartheta ,\xi ,\phi +\zeta )\le \mathfrak{H}(\iota ,\rho ,\rho ,\zeta )\oplus \mathfrak{H}(\rho ,\vartheta ,\xi ,\phi )$ for all $\iota ,\vartheta ,\xi ,\rho \in \mathfrak{A}$,

(xiii) 

$\mathfrak{H}(\iota ,\vartheta ,\xi ,.):(0,\infty )\to [0,1]$ is continuous.

(xiv) 

$\mathfrak{E}(\iota ,\iota ,\vartheta ,\phi )>0$,

(xv) 

$\mathfrak{E}(\iota ,\iota ,\vartheta ,\phi )\ge \mathfrak{E}(\iota ,\vartheta ,\xi ,\phi )$ for all $\iota ,\vartheta ,\xi \in \mathfrak{A}$ with $\vartheta \ne \xi$,

(xvi) 

$\mathfrak{E}(\iota ,\vartheta ,\xi ,\phi )=0$, then $\iota =\vartheta =\xi$,

(xvii) 

$\mathfrak{E}(\iota ,\vartheta ,\xi ,\phi )=\mathfrak{E}\left(\mathfrak{s}\right(\iota ,\vartheta ,\xi ),\phi )$, where $\mathfrak{s}$ is a permutation function,

(xviii) 

$\mathfrak{E}(\iota ,\vartheta ,\xi ,\phi +\zeta )\le \mathfrak{E}(\iota ,\rho ,\rho ,\zeta )\oplus \mathfrak{E}(\rho ,\vartheta ,\xi ,\phi )$ for all $\iota ,\vartheta ,\xi ,\rho \in \mathfrak{A}$,

(xix) 

$\mathfrak{E}(\iota ,\vartheta ,\xi ,.):(0,\infty )\to [0,1]$ is continuous.

In the above definition, if the triangular inequalities (vi), (xii), and (xviii) for $\mathfrak{J},\mathfrak{H}$, and $\mathfrak{E}$ are replaced by

$$\begin{array}{c}\mathfrak{J}(\iota ,\vartheta ,\xi ,max\{\zeta ,\phi \left\}\right)\ge \mathfrak{J}(\iota ,\rho ,\rho ,\zeta )\otimes \mathfrak{J}(\rho ,\vartheta ,\xi ,\phi ),\\ \mathfrak{H}(\iota ,\vartheta ,\xi ,min\{\zeta ,\phi \left\}\right)\le \mathfrak{H}(\iota ,\rho ,\rho ,\zeta )\oplus \mathfrak{H}(\rho ,\vartheta ,\xi ,\phi ),\\ \mathfrak{E}(\iota ,\vartheta ,\xi ,min\{\zeta ,\phi \left\}\right)\le \mathfrak{E}(\iota ,\rho ,\rho ,\zeta )\oplus \mathfrak{E}(\rho ,\vartheta ,\xi ,\phi ),\\ \hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}forall\iota ,\vartheta ,\xi ,\rho \in \mathfrak{A}\mathrm{and}\zeta ,\phi >0.\end{array}$$

(1)

then the inequalities can also be equivalently expressed as:

$$\begin{array}{c}\mathfrak{J}(\iota ,\vartheta ,\xi ,\phi )\ge \mathfrak{J}(\iota ,\rho ,\rho ,\zeta )\otimes \mathfrak{J}(\rho ,\vartheta ,\xi ,\phi ),\\ \mathfrak{H}(\iota ,\vartheta ,\xi ,\phi )\le \mathfrak{H}(\iota ,\rho ,\rho ,\zeta )\oplus \mathfrak{H}(\rho ,\vartheta ,\xi ,\phi ),\\ \mathfrak{E}(\iota ,\vartheta ,\xi ,\phi )\le \mathfrak{E}(\iota ,\rho ,\rho ,\zeta )\oplus \mathfrak{E}(\rho ,\vartheta ,\xi ,\phi ).\end{array}$$

The 6-tuple $(\mathfrak{A},\mathfrak{J},\mathfrak{H},\mathfrak{E},\otimes ,\oplus )$ is called a non-Archimedean GNMS.

Example 1.

Let $\mathfrak{A}=\mathbb{R}$. Define the functions $\mathfrak{J}$, $\mathfrak{H}$, and $\mathfrak{E}$ as follows:

$$\begin{array}{cc}\hfill \mathfrak{J}(\iota ,\vartheta ,\xi ,\phi )& ={\displaystyle \frac{|\iota -\vartheta |+|\vartheta -\xi |}{|\iota -\xi |+|\iota -\vartheta |+|\vartheta -\xi |+\phi }},\hfill \\ \hfill \mathfrak{H}(\iota ,\vartheta ,\xi ,\phi )& ={\displaystyle \frac{|\iota -\vartheta |}{|\iota -\vartheta |+\phi }},\hfill \\ \hfill \mathfrak{E}(\iota ,\vartheta ,\xi ,\phi )& ={\displaystyle \frac{|\vartheta -\xi |}{|\vartheta -\xi |+|\iota -\vartheta |+\phi }}.\hfill \end{array}$$

Use the parameters $\iota =1$, $\vartheta =3$, $\xi =2$, and $\phi ={\displaystyle \frac{1}{7}}$.

For $\mathfrak{J}$,

$$\begin{array}{cc}\hfill |\iota -\vartheta |& =|1-3|=2,\hfill \\ \hfill |\vartheta -\xi |& =|3-2|=1,\hfill \\ \hfill |\iota -\xi |& =|1-2|=1.\hfill \end{array}$$

By substituting these values, we obtain

$$\mathfrak{J}(1,3,2,{\displaystyle \frac{1}{7}})={\displaystyle \frac{2+1}{1+2+1+\frac{1}{7}}}={\displaystyle \frac{3}{4+\frac{1}{7}}}={\displaystyle \frac{3}{\frac{29}{7}}}={\displaystyle \frac{3\times 7}{29}}={\displaystyle \frac{21}{29}}\approx 0.724.$$

$$\mathfrak{H}(1,3,2,{\displaystyle \frac{1}{7}})={\displaystyle \frac{2}{2+\frac{1}{7}}}={\displaystyle \frac{2}{\frac{15}{7}}}=2\times {\displaystyle \frac{7}{15}}={\displaystyle \frac{14}{15}}\approx 0.933.$$

$$\mathfrak{E}(1,3,2,{\displaystyle \frac{1}{7}})={\displaystyle \frac{1}{1+2+\frac{1}{7}}}={\displaystyle \frac{1}{3+\frac{1}{7}}}={\displaystyle \frac{1}{\frac{22}{7}}}={\displaystyle \frac{7}{22}}\approx 0.318.$$

The values are as follows: $\mathfrak{J}\approx 0.724$, $\mathfrak{H}\approx 0.933$, and $\mathfrak{E}\approx 0.318$.

With these definitions, all functions remain bounded between 0 and 1. This ensures that $(\mathbb{R},\mathfrak{J},\mathfrak{H},\mathfrak{E},\otimes ,\oplus )$ satisfies the conditions for being a non-Archimedean GNMS.

Definition 4.

Let $\left\{{\iota }_{\mathfrak{p}}\right\}$ be a sequence in a GNMS $(\mathfrak{A},\mathfrak{J},\mathfrak{H},\mathfrak{E},\otimes ,\oplus )$. We say the following:

(i) 

$\left\{{\iota }_{\mathfrak{p}}\right\}$ converges to ι if and only if ${\displaystyle \underset{\mathfrak{p}\to \infty }{lim}\mathfrak{J}({\iota }_{\mathfrak{p}},{\iota }_{\mathfrak{p}},\iota ,\phi )=1,\underset{\mathfrak{p}\to \infty }{lim}\mathfrak{H}({\iota }_{\mathfrak{p}},{\iota }_{\mathfrak{p}},\iota ,\phi )=0}$, and ${\displaystyle \underset{\mathfrak{p}\to \infty }{lim}\mathfrak{E}({\iota }_{\mathfrak{p}},{\iota }_{\mathfrak{p}},\iota ,\phi )=0}$; i.e., for all $\phi >0$ and all $\kappa \in (0,1)$, there exists ${\mathfrak{p}}_0\in \mathbb{N}$ such that $\mathfrak{J}({\iota }_{\mathfrak{p}},{\iota }_{\mathfrak{p}},\iota ,\phi )>1-\kappa ,\mathfrak{H}({\iota }_{\mathfrak{p}},{\iota }_{\mathfrak{p}},\iota ,\phi )<\kappa$ and $\mathfrak{E}({\iota }_{\mathfrak{p}},{\iota }_{\mathfrak{p}},\iota ,\phi )<\kappa$ for all $\mathfrak{p}\ge {\mathfrak{p}}_0$ (in such a case, we will write $\left\{{\iota }_{\mathfrak{p}}\right\}\to \iota )$.

(ii) 

A sequence $\left\{{\iota }_{\mathfrak{p}}\right\}$ is considered a Cauchy sequence if and only if for every $\phi >0$ and $\kappa \in (0,1)$, there exists an index ${\mathfrak{p}}_0\in \mathbb{N}$ such that

$$\begin{array}{ccc}\mathfrak{J}({\iota }_{\mathfrak{p}},{\iota }_{\mathfrak{p}},{\iota }_{\mathfrak{q}},\phi )>1-\kappa ,& \mathfrak{H}({\iota }_{\mathfrak{p}},{\iota }_{\mathfrak{p}},{\iota }_{\mathfrak{q}},\phi )<\kappa ,& \mathfrak{E}({\iota }_{\mathfrak{p}},{\iota }_{\mathfrak{p}},{\iota }_{\mathfrak{q}},\phi )<\kappa ,\end{array}$$

for all $\mathfrak{p},\mathfrak{q}\ge {\mathfrak{p}}_0$ and $\mathfrak{p}\ge \mathfrak{q}$.

Additionally, $\left\{{\iota }_{\mathfrak{p}}\right\}$ is a generalized Cauchy sequence if and only if for every $\phi >0$ and $\kappa \in (0,1)$, there exists ${\mathfrak{p}}_0\in \mathbb{N}$ such that

$$\begin{array}{ccc}\mathfrak{J}({\iota }_{\mathfrak{p}},{\iota }_{\mathfrak{p}},{\iota }_{\mathfrak{p}}+\mathfrak{s},\phi )>1-\kappa ,& \mathfrak{H}({\iota }_{\mathfrak{p}},{\iota }_{\mathfrak{p}},{\iota }_{\mathfrak{p}}+\mathfrak{s},\phi )<\kappa ,& \mathfrak{E}({\iota }_{\mathfrak{p}},{\iota }_{\mathfrak{p}},{\iota }_{\mathfrak{p}}+\mathfrak{s},\phi )<\kappa ,\end{array}$$

for all $\mathfrak{p}\ge {\mathfrak{p}}_0$ and $\mathfrak{s}>0$.

In other words,

$$\begin{array}{cc}\underset{\mathfrak{p}\to \infty }{lim}\mathfrak{J}({\iota }_{\mathfrak{p}},{\iota }_{\mathfrak{p}},{\iota }_{\mathfrak{p}}+\mathfrak{s},\phi )=1,& \underset{\mathfrak{p}\to \infty }{lim}\mathfrak{H}({\iota }_{\mathfrak{p}},{\iota }_{\mathfrak{p}},{\iota }_{\mathfrak{p}}+\mathfrak{s},\phi )=0,\\ \underset{\mathfrak{p}\to \infty }{lim}\mathfrak{E}({\iota }_{\mathfrak{p}},{\iota }_{\mathfrak{p}},{\iota }_{\mathfrak{p}}+\mathfrak{s},\phi )=0.\end{array}$$

(iii) 

GNMS $(\mathfrak{A},\mathfrak{J},\mathfrak{H},\mathfrak{E},\otimes ,\oplus )$ is considered complete if every Cauchy sequence in the space converges to a limit in that space.

Lemma 1.

Suppose that $(\mathfrak{A},\mathfrak{J},\mathfrak{H},\mathfrak{E},\otimes ,\oplus )$ is a non-Archimedean GNMS. In that instance, $\mathfrak{J}(\iota ,\vartheta ,\xi ,\phi )$ is non-decreasing, and $\mathfrak{H}(\iota ,\vartheta ,\xi ,\phi )$ and $\mathfrak{E}(\iota ,\vartheta ,\xi ,\phi )$ are non-increasing, with reference to φ for all $\iota ,\vartheta ,\xi \in \mathfrak{A}$.

Proof. 

Let $(\mathfrak{A},\mathfrak{J},\mathfrak{H},\mathfrak{E},\otimes ,\oplus )$ be a non-Archimedean GNMS. From (1), for $\zeta >0$, $\phi >0$,

$$\mathfrak{J}(\iota ,\vartheta ,\xi ,max\{\zeta ,\phi \left\}\right)\ge \mathfrak{J}(\iota ,\rho ,\rho ,\zeta )\otimes \mathfrak{J}(\rho ,\vartheta ,\xi ,\phi )$$

Consider two values ${\phi }_1$ and ${\phi }_2$ such that ${\phi }_1<{\phi }_2$. By using the symmetry condition and the given inequality, for ${\phi }_2=max\{{\phi }_1,{\phi }_2\}$, we have

$$\mathfrak{J}(\iota ,\vartheta ,\xi ,{\phi }_2)=\mathfrak{J}(\iota ,\vartheta ,\xi ,max\{{\phi }_1,{\phi }_2\})\ge \mathfrak{J}(\iota ,\rho ,\rho ,{\phi }_2-{\phi }_1)\otimes \mathfrak{J}(\rho ,\vartheta ,\xi ,{\phi }_1)$$

By setting $\rho =\iota$, we obtain

$$\mathfrak{J}(\iota ,\vartheta ,\xi ,{\phi }_2)=\mathfrak{J}(\iota ,\vartheta ,\xi ,max\{{\phi }_1,{\phi }_2\})\ge \mathfrak{J}(\iota ,\iota ,\iota ,{\phi }_2-{\phi }_1)\otimes \mathfrak{J}(\iota ,\vartheta ,\xi ,{\phi }_1)$$

Since by condition (iv) of Definition (3), $\mathfrak{J}(\iota ,\iota ,\iota ,{\phi }_2-{\phi }_1)=1$ and ⊗ is a t-norm, we conclude that

$$\mathfrak{J}(\iota ,\vartheta ,\xi ,{\phi }_2)\ge \mathfrak{J}(\iota ,\vartheta ,\xi ,{\phi }_1)$$

Thus, $\mathfrak{J}(\iota ,\vartheta ,\xi ,\phi )$ is non-decreasing with respect to $\phi$.

From (1), for $\zeta ,\phi >0$,

$$\mathfrak{H}(\iota ,\vartheta ,\xi ,min\{\zeta ,\phi \left\}\right)\le \mathfrak{H}(\iota ,\rho ,\rho ,\zeta )\oplus \mathfrak{H}(\rho ,\vartheta ,\xi ,\phi ),$$

$$\mathfrak{E}(\iota ,\vartheta ,\xi ,min\{\zeta ,\phi \left\}\right)\le \mathfrak{E}(\iota ,\rho ,\rho ,\zeta )\oplus \mathfrak{E}(\rho ,\vartheta ,\xi ,\phi ).$$

Consider two values ${\phi }_1<{\phi }_2$. From the given inequality, for ${\phi }_1=min\{{\phi }_1,{\phi }_2\}$, we have

$$\mathfrak{H}(\iota ,\vartheta ,\xi ,{\phi }_1)=\mathfrak{H}(\iota ,\vartheta ,\xi ,min\{{\phi }_1,{\phi }_2\})\le \mathfrak{H}(\iota ,\rho ,\rho ,{\phi }_2-{\phi }_1)\oplus \mathfrak{H}(\rho ,\vartheta ,\xi ,{\phi }_2),$$

$$\mathfrak{E}(\iota ,\vartheta ,\xi ,{\phi }_1)=\mathfrak{E}(\iota ,\vartheta ,\xi ,min\{{\phi }_1,{\phi }_2\})\le \mathfrak{E}(\iota ,\rho ,\rho ,{\phi }_2-{\phi }_1)\oplus \mathfrak{E}(\rho ,\vartheta ,\xi ,{\phi }_2).$$

By setting $\rho =\iota$, we obtain

$$\mathfrak{H}(\iota ,\vartheta ,\xi ,{\phi }_1)=\mathfrak{H}(\iota ,\vartheta ,\xi ,min\{{\phi }_1,{\phi }_2\})\le \mathfrak{H}(\iota ,\iota ,\iota ,{\phi }_2-{\phi }_1)\oplus \mathfrak{H}(\iota ,\vartheta ,\xi ,{\phi }_2)$$

$$\mathfrak{E}(\iota ,\vartheta ,\xi ,{\phi }_1)=\mathfrak{E}(\iota ,\vartheta ,\xi ,min\{{\phi }_1,{\phi }_2\})\le \mathfrak{E}(\iota ,\iota ,\iota ,{\phi }_2-{\phi }_1)\oplus \mathfrak{E}(\iota ,\vartheta ,\xi ,{\phi }_2)$$

Since by conditions (x) and (xvi) of Definition (3) $\mathfrak{H}(\iota ,\iota ,\iota ,{\phi }_2-{\phi }_1)=0$ and ⊕ is a t-conorm, which preserves the inequality, we conclude

$$\mathfrak{H}(\iota ,\vartheta ,\xi ,{\phi }_1)\ge \mathfrak{H}(\iota ,\vartheta ,\xi ,{\phi }_2),$$

$$\mathfrak{E}(\iota ,\vartheta ,\xi ,{\phi }_1)\ge \mathfrak{E}(\iota ,\vartheta ,\xi ,{\phi }_2).$$

Thus, $\mathfrak{H}(\iota ,\vartheta ,\xi ,\phi )$ and $\mathfrak{E}(\iota ,\vartheta ,\xi ,\phi )$ are non-increasing with respect to $\phi$. □

Lemma 2.

Let $(\mathfrak{A},\mathfrak{J},\mathfrak{H},\mathfrak{E},\otimes ,\oplus )$ be a non-Archimedean GNMS. Then, $\mathfrak{J},\mathfrak{H}$, and $\mathfrak{E}$ are continuous functions on ${\mathfrak{A}}^3\times (0,\infty )$.

3. Fixed-Point Results Using Various $\chi$-Contractive Mappings

In this section, we introduce and analyze symmetric $\chi$-type contractions and weak contraction mappings, emphasizing fixed-point results in various mathematical contexts.

Definition 5.

A symmetric non-Archimedean GNMS is a non-Archimedean GNMS $(\mathfrak{A},\mathfrak{J},\mathfrak{H},\mathfrak{E},$$\otimes ,\oplus )$ satisfying

$$\mathfrak{J}(\iota ,\vartheta ,\vartheta ,\phi )=\mathfrak{J}(\vartheta ,\iota ,\iota ,\phi ),\mathfrak{H}(\iota ,\vartheta ,\vartheta ,\phi )=\mathfrak{H}(\vartheta ,\iota ,\iota ,\phi ),\mathfrak{E}(\iota ,\vartheta ,\vartheta ,\phi )=\mathfrak{E}(\vartheta ,\iota ,\iota ,\phi )$$

for all $\iota ,\vartheta \in \mathfrak{A}$ with $\phi \in (0,\infty )$.

Remark 1.

A non-Archimedean GNMS is symmetric.

Definition 6.

Consider a strictly increasing continuous function $\chi :[0,1)\to \mathbb{R}$. For any sequence of positive numbers ${\left\{{\rho }_{\mathfrak{p}}\right\}}_{\mathfrak{p}\in \mathbb{N}}$, the condition that ${lim}_{\mathfrak{p}\to \infty }{\rho }_{\mathfrak{p}}=1$ holds if and only if ${lim}_{\mathfrak{p}\to \infty }\chi \left({\rho }_{\mathfrak{p}}\right)=\infty$. Let Υ represent the set of all such functions χ.

Now, take the structure $(\mathfrak{A},\mathfrak{J},\mathfrak{H},\mathfrak{E},\otimes ,\oplus )$, which is a non-Archimedean GNMS. A mapping $\mathsf{\Gamma }:\mathfrak{A}\to \mathfrak{A}$ is termed a χ-contraction if there exists a constant $\delta >0$ such that

$$\begin{array}{cc}\hfill \mathfrak{J}(\mathsf{\Gamma }\iota ,\mathsf{\Gamma }\vartheta ,\mathsf{\Gamma }\xi ,\phi )& <1\Rightarrow \chi \left(\mathfrak{J}\right(\mathsf{\Gamma }\iota ,\mathsf{\Gamma }\vartheta ,\mathsf{\Gamma }\xi ,\phi \left)\right)\ge \chi \left(\mathfrak{J}\right(\iota ,\vartheta ,\xi ,\phi \left)\right)+\delta ,\hfill \\ \hfill \mathfrak{H}(\mathsf{\Gamma }\iota ,\mathsf{\Gamma }\vartheta ,\mathsf{\Gamma }\xi ,\phi )& >0\Rightarrow \chi \left(\mathfrak{H}\right(\mathsf{\Gamma }\iota ,\mathsf{\Gamma }\vartheta ,\mathsf{\Gamma }\xi ,\phi \left)\right)\le \chi \left(\mathfrak{H}\right(\iota ,\vartheta ,\xi ,\phi \left)\right)-\delta ,\hfill \\ \hfill \mathfrak{E}(\mathsf{\Gamma }\iota ,\mathsf{\Gamma }\vartheta ,\mathsf{\Gamma }\xi ,\phi )& >0\Rightarrow \chi \left(\mathfrak{E}\right(\mathsf{\Gamma }\iota ,\mathsf{\Gamma }\vartheta ,\mathsf{\Gamma }\xi ,\phi \left)\right)\le \chi \left(\mathfrak{E}\right(\iota ,\vartheta ,\xi ,\phi \left)\right)-\delta .\hfill \end{array}$$

(2)

for all $\iota ,\vartheta ,\xi \in \mathfrak{A},\phi >0$ and $\chi \in {\rm Y}$, and, additionally, the following symmetric conditions hold:

$$\begin{array}{cc}\hfill \mathfrak{J}(\mathsf{\Gamma }\iota ,\mathsf{\Gamma }\vartheta ,\mathsf{\Gamma }\xi ,\phi )& =\mathfrak{J}(\mathsf{\Gamma }\vartheta ,\mathsf{\Gamma }\iota ,\mathsf{\Gamma }\xi ,\phi ),\hfill \\ \hfill \mathfrak{H}(\mathsf{\Gamma }\iota ,\mathsf{\Gamma }\vartheta ,\mathsf{\Gamma }\xi ,\phi )& =\mathfrak{H}(\mathsf{\Gamma }\vartheta ,\mathsf{\Gamma }\iota ,\mathsf{\Gamma }\xi ,\phi ),\hfill \\ \hfill \mathfrak{E}(\mathsf{\Gamma }\iota ,\mathsf{\Gamma }\vartheta ,\mathsf{\Gamma }\xi ,\phi )& =\mathfrak{E}(\mathsf{\Gamma }\vartheta ,\mathsf{\Gamma }\iota ,\mathsf{\Gamma }\xi ,\phi ).\hfill \end{array}$$

(3)

Example 2.

The different types of mapping $\chi \in {\rm Y}$ are as follows:

$${\chi }_1={\displaystyle \frac{1}{(1-\beta )}},{\chi }_2=ln{\displaystyle \frac{1}{(1-\beta )}},{\chi }_3={\displaystyle \frac{1}{(1-\beta )}}+\beta ,{\chi }_4={\displaystyle \frac{1}{(1-{\beta }^2)}},\mathrm{for}\mathrm{all}\beta \in (0,1]$$

If $\chi =ln{\displaystyle \frac{1}{(1-\beta )}}$, then each mapping $\mathsf{\Gamma }:\mathfrak{A}\to \mathfrak{A}$ accomplishing (2) is a χ-contraction such that $\mathfrak{J}(\mathsf{\Gamma }\iota ,\mathsf{\Gamma }\vartheta ,\mathsf{\Gamma }\xi ,\phi )\ge \varrho \left(\delta \right)\mathfrak{J}(\iota ,\vartheta ,\xi ,\phi ),\mathfrak{H}(\mathsf{\Gamma }\iota ,\mathsf{\Gamma }\vartheta ,\mathsf{\Gamma }\xi ,\phi )\le \mathfrak{l}\left(\delta \right)\mathfrak{H}(\iota ,\vartheta ,\xi ,\phi ),$ $\mathfrak{E}(\mathsf{\Gamma }\iota ,\mathsf{\Gamma }\vartheta ,\mathsf{\Gamma }\xi ,\phi )\le \mathfrak{q}\left(\delta \right)\mathfrak{E}(\iota ,\vartheta ,\xi ,\phi )$ for all $\iota ,\vartheta ,\xi \in \mathfrak{A},\phi >0$ and $\mathfrak{J}(\mathsf{\Gamma }\iota ,\mathsf{\Gamma }\vartheta ,\mathsf{\Gamma }\xi ,\phi )<1,\mathfrak{H}(\mathsf{\Gamma }\iota ,\mathsf{\Gamma }\vartheta ,\mathsf{\Gamma }\xi ,\phi )>0$ and $\mathfrak{E}(\mathsf{\Gamma }\iota ,\mathsf{\Gamma }\vartheta ,\mathsf{\Gamma }\xi ,\phi )>0$ in which $\varrho \left(\delta \right)={\displaystyle \frac{\mathfrak{J}(\iota ,\vartheta ,\xi ,\phi )-1+e^{\delta }}{e^{\delta }\mathfrak{J}(\iota ,\vartheta ,\xi ,\phi )}}\ge 1,\mathfrak{l}\left(\delta \right)={\displaystyle \frac{\mathfrak{H}(\iota ,\vartheta ,\xi ,\phi )-1+e^{\delta }}{e^{\delta }\mathfrak{H}(\iota ,\vartheta ,\xi ,\phi )}}\le 0$ and $\mathfrak{q}\left(\delta \right)={\displaystyle \frac{\mathfrak{E}(\iota ,\vartheta ,\xi ,\phi )-1+e^{\delta }}{e^{\delta }\mathfrak{E}(\iota ,\vartheta ,\xi ,\phi )}}\le 0$.

Note that, from χ and (2), it can be easily concluded that each χ-contraction Γ is a contractive mapping, that is,

$$\begin{array}{cc}\hfill \mathfrak{J}(\mathsf{\Gamma }\iota ,\mathsf{\Gamma }\vartheta ,\mathsf{\Gamma }\xi ,\phi )& >\mathfrak{J}(\iota ,\vartheta ,\xi ,\phi ),\hfill \\ \hfill \mathfrak{H}(\mathsf{\Gamma }\iota ,\mathsf{\Gamma }\vartheta ,\mathsf{\Gamma }\xi ,\phi )& <\mathfrak{H}(\iota ,\vartheta ,\xi ,\phi )\mathrm{and}\hfill \\ \hfill \mathfrak{E}(\mathsf{\Gamma }\iota ,\mathsf{\Gamma }\vartheta ,\mathsf{\Gamma }\xi ,\phi )& <\mathfrak{E}(\iota ,\vartheta ,\xi ,\phi )\hfill \end{array}$$

(4)

for all $\iota ,\vartheta ,\xi \in \mathfrak{A}$, such that $\mathsf{\Gamma }\iota \ne \mathsf{\Gamma }\vartheta \ne \mathsf{\Gamma }\xi$. Thus, each χ-contraction is continuous.

Numerical Example:

$\mathfrak{A}=\{0,1,2\}$.

Define the mapping Γ$:\mathfrak{A}\to \mathfrak{A}$ as $\mathsf{\Gamma }\left(\iota \right)=(\iota +1)\left(\mathrm{mod}3\right)$.

So, $\mathsf{\Gamma }\left(0\right)=1,\mathsf{\Gamma }\left(1\right)=2,\mathsf{\Gamma }\left(2\right)=0$.

Assume that $\mathfrak{J},\mathfrak{H},\mathfrak{E}$ for different combinations of $\iota ,\vartheta ,\xi ,\phi$ are as follows:

For $\iota =0,\vartheta =1,\xi =2,\phi =1$:

$$\mathfrak{J}(0,1,2,1)=0.6,\mathfrak{H}(0,1,2,1)=0.4,\mathfrak{E}(0,1,2,1)=0.3$$

For $\iota =1,\vartheta =2,\xi =0,\phi =1$:

$$\mathfrak{J}(1,2,0,1)=0.7,\mathfrak{H}(1,2,0,1)=0.35,\mathfrak{E}(1,2,0,1)=0.28$$

Using the mapping ${\chi }_2=ln{\displaystyle \frac{1}{1-\iota }}$, we now verify that the mapping Γ satisfies the contractive inequalities.

$$\mathfrak{J}\left(\mathsf{\Gamma }\right(0),\mathsf{\Gamma }(1),\mathsf{\Gamma }(2),1)=\mathfrak{J}(1,2,0,1)=0.7$$

$$\mathfrak{J}(0,1,2,1)=0.6$$

Since $0.7>0.6$, the condition $\mathfrak{J}(\mathsf{\Gamma }\iota ,\mathsf{\Gamma }\vartheta ,\mathsf{\Gamma }\xi ,\phi )>\mathfrak{J}(\iota ,\vartheta ,\xi ,\phi )$ holds.

$$\mathfrak{H}\left(\mathsf{\Gamma }\right(0),\mathsf{\Gamma }(1),\mathsf{\Gamma }(2),1)=\mathfrak{H}(1,2,0,1)=0.35$$

$$\mathfrak{H}(0,1,2,1)=0.4$$

Since $0.35<0.4$, the condition $\mathfrak{H}(\mathsf{\Gamma }\iota ,\mathsf{\Gamma }\vartheta ,\mathsf{\Gamma }\xi ,\phi )<\mathfrak{H}(\iota ,\vartheta ,\xi ,\phi )$ holds.

$$\mathfrak{E}\left(\mathsf{\Gamma }\right(0),\mathsf{\Gamma }(1),\mathsf{\Gamma }(2),1)=\mathfrak{E}(1,2,0,1)=0.28$$

$$\mathfrak{E}(0,1,2,1)=0.3$$

Since $0.28<0.3$, the condition $\mathfrak{E}(\mathsf{\Gamma }\iota ,\mathsf{\Gamma }\vartheta ,\mathsf{\Gamma }\xi ,\phi )<\mathfrak{E}(\iota ,\vartheta ,\xi ,\phi )$ holds.

Verification of $\varrho \left(\delta \right),\mathfrak{l}\left(\delta \right),\mathfrak{q}\left(\delta \right)$:

Let $\delta =0.1$:

$$\varrho \left(\delta \right)={\displaystyle \frac{0.6-1+e^{0.1}}{e^{0.1}·0.6}}={\displaystyle \frac{0.6-1+1.105}{1.105·0.6}}={\displaystyle \frac{0.705}{0.663}}\approx 1.063\ge 1$$

$$\mathfrak{l}\left(\delta \right)={\displaystyle \frac{0.4-1+e^{0.1}}{e^{0.1}·0.4}}={\displaystyle \frac{0.4-1+1.105}{1.105·0.4}}={\displaystyle \frac{0.505}{0.442}}\approx 1.142$$

Since $\mathfrak{l}\left(\delta \right)>0$, the inequality holds.

$$\mathfrak{q}\left(\delta \right)={\displaystyle \frac{0.3-1+e^{0.1}}{e^{0.1}·0.3}}={\displaystyle \frac{0.3-1+1.105}{1.105·0.3}}={\displaystyle \frac{0.405}{0.331}}\approx 1.224$$

Since $\mathfrak{q}\left(\delta \right)>0$, the inequality holds.

This numerical example demonstrates that the mapping Γ is a χ-contraction and satisfies the required conditions for fuzzy sets $\mathfrak{J}$, $\mathfrak{H}$, and $\mathfrak{E}$.

Theorem 1.

Let us consider $\mathfrak{J}(\mathfrak{A},\mathfrak{J},\mathfrak{H},\mathfrak{E},\otimes ,\oplus )$ as a non-Archimedean complete GNMS. Suppose that $\mathsf{\Gamma }:\mathfrak{A}\to \mathfrak{A}$ is a χ-contraction. Then, Γ has a unique fixed point in $\mathfrak{A}$.

Proof. 

Consider ${\iota }_0\in \mathfrak{A}$ as an arbitrary fixed element. Define the sequence $\left\{{\iota }_{\mathfrak{p}}\right\}$ by the recurrence relation

$$\mathsf{\Gamma }{\iota }_{\mathfrak{p}}={\iota }_{\mathfrak{p}+1}\mathrm{for}\mathrm{all}\mathfrak{p}\in \mathbb{N}.$$

If ${\iota }_{\mathfrak{p}}={\iota }_{\mathfrak{p}+1}$, then ${\iota }_{\mathfrak{p}+1}$ is a fixed point of $\mathsf{\Gamma }$, establishing the validity of the theorem. However, if ${\iota }_{\mathfrak{p}}\ne {\iota }_{\mathfrak{p}+1}$ for every $\mathfrak{p}\in \mathbb{N}$, then, based on (2), we obtain

$$\begin{array}{c}\hfill \chi \left(\mathfrak{J}(\mathsf{\Gamma }{\iota }_{\mathfrak{p}-1},\mathsf{\Gamma }{\iota }_{\mathfrak{p}-1},\mathsf{\Gamma }{\iota }_{\mathfrak{p}},\phi )\right)\ge \chi \left(\mathfrak{J}({\iota }_{\mathfrak{p}-1},{\iota }_{\mathfrak{p}-1},{\iota }_{\mathfrak{p}},\phi )\right)+\delta ,\\ \hfill \chi \left(\mathfrak{H}(\mathsf{\Gamma }{\iota }_{\mathfrak{p}-1},\mathsf{\Gamma }{\iota }_{\mathfrak{p}-1},\mathsf{\Gamma }{\iota }_{\mathfrak{p}},\phi )\right)\le \chi \left(\mathfrak{H}({\iota }_{\mathfrak{p}-1},{\iota }_{\mathfrak{p}-1},{\iota }_{\mathfrak{p}},\phi )\right)-\delta ,\\ \hfill \chi \left(\mathfrak{E}(\mathsf{\Gamma }{\iota }_{\mathfrak{p}-1},\mathsf{\Gamma }{\iota }_{\mathfrak{p}-1},\mathsf{\Gamma }{\iota }_{\mathfrak{p}},\phi )\right)\le \chi \left(\mathfrak{E}({\iota }_{\mathfrak{p}-1},{\iota }_{\mathfrak{p}-1},{\iota }_{\mathfrak{p}},\phi )\right)-\delta .\end{array}$$

(5)

By iterating this method, we have

$$\begin{array}{cc}\hfill \chi \left(\mathfrak{J}(\mathsf{\Gamma }{\iota }_{\mathfrak{p}-1},\mathsf{\Gamma }{\iota }_{\mathfrak{p}-1},\mathsf{\Gamma }{\iota }_{\mathfrak{p}},\phi )\right)& \ge \chi \left(\mathfrak{J}({\iota }_{\mathfrak{p}-1},{\iota }_{\mathfrak{p}-1},{\iota }_{\mathfrak{p}},\phi )\right)+\delta \hfill \\ \hfill & =\chi \left(\mathfrak{J}(\mathsf{\Gamma }{\iota }_{\mathfrak{p}-2},\mathsf{\Gamma }{\iota }_{\mathfrak{p}-2},\mathsf{\Gamma }{\iota }_{\mathfrak{p}-1},\phi )\right)+\delta \hfill \\ \hfill & \ge \chi \left(\mathfrak{J}({\iota }_{\mathfrak{p}-2},{\iota }_{\mathfrak{p}-2},{\iota }_{\mathfrak{p}-1},\phi )\right)+2\delta \cdots \hfill \\ \hfill & \ge \chi \left(\mathfrak{J}({\iota }_0,{\iota }_0,{\iota }_1,\phi )\right)+\mathfrak{p}\delta .\hfill \\ \hfill \chi \left(\mathfrak{H}(\mathsf{\Gamma }{\iota }_{\mathfrak{p}-1},\mathsf{\Gamma }{\iota }_{\mathfrak{p}-1},\mathsf{\Gamma }{\iota }_{\mathfrak{p}},\phi )\right)& \le \chi \left(\mathfrak{H}({\iota }_{\mathfrak{p}-1},{\iota }_{\mathfrak{p}-1},{\iota }_{\mathfrak{p}},\phi )\right)-\delta \hfill \\ \hfill & =\chi \left(\mathfrak{H}(\mathsf{\Gamma }{\iota }_{\mathfrak{p}-2},\mathsf{\Gamma }{\iota }_{\mathfrak{p}-2},\mathsf{\Gamma }{\iota }_{\mathfrak{p}-1},\phi )\right)-\delta \hfill \\ \hfill & \le \chi \left(\mathfrak{H}({\iota }_{\mathfrak{p}-2},{\iota }_{\mathfrak{p}-2},{\iota }_{\mathfrak{p}-1},\phi )\right)-2\delta \cdots \hfill \\ \hfill & \le \chi \left(\mathfrak{H}({\iota }_0,{\iota }_0,{\iota }_1,\phi )\right)-\mathfrak{p}\delta .\hfill \\ \hfill \chi \left(\mathfrak{E}(\mathsf{\Gamma }{\iota }_{\mathfrak{p}-1},\mathsf{\Gamma }{\iota }_{\mathfrak{p}-1},\mathsf{\Gamma }{\iota }_{\mathfrak{p}},\phi )\right)& \le \chi \left(\mathfrak{E}({\iota }_{\mathfrak{p}-1},{\iota }_{\mathfrak{p}-1},{\iota }_{\mathfrak{p}},\phi )\right)-\delta \hfill \\ \hfill & =\chi \left(\mathfrak{E}(\mathsf{\Gamma }{\iota }_{\mathfrak{p}-2},\mathsf{\Gamma }{\iota }_{\mathfrak{p}-2},\mathsf{\Gamma }{\iota }_{\mathfrak{p}-1},\phi )\right)-\delta \hfill \\ \hfill & \le \chi \left(\mathfrak{E}({\iota }_{\mathfrak{p}-2},{\iota }_{\mathfrak{p}-2},{\iota }_{\mathfrak{p}-1},\phi )\right)-2\delta \cdots \hfill \\ \hfill & \le \chi \left(\mathfrak{E}({\iota }_0,{\iota }_0,{\iota }_1,\phi )\right)-\mathfrak{p}\delta .\hfill \end{array}$$

Let $\mathfrak{p}\to \infty$, ${\displaystyle \underset{\mathfrak{p}\to \infty }{lim}\chi \left(\mathfrak{J}(\mathsf{\Gamma }{\iota }_{\mathfrak{p}-1},\mathsf{\Gamma }{\iota }_{\mathfrak{p}-1},\mathsf{\Gamma }{\iota }_{\mathfrak{p}},\phi )\right)=+\infty ,\underset{\mathfrak{p}\to \infty }{lim}\chi \left(\mathfrak{H}(\mathsf{\Gamma }{\iota }_{\mathfrak{p}-1},\mathsf{\Gamma }{\iota }_{\mathfrak{p}-1},\mathsf{\Gamma }{\iota }_{\mathfrak{p}},\phi )\right)=-\infty ,}$ ${\displaystyle \underset{\mathfrak{p}\to \infty }{lim}\chi \left(\mathfrak{E}(\mathsf{\Gamma }{\iota }_{\mathfrak{p}-1},\mathsf{\Gamma }{\iota }_{\mathfrak{p}-1},\mathsf{\Gamma }{\iota }_{\mathfrak{p}},\phi )\right)=-\infty .}$ Then, we have

$$\begin{array}{cc}\hfill \underset{\mathfrak{p}\to \infty }{lim}\mathfrak{J}(\mathsf{\Gamma }{\iota }_{\mathfrak{p}-1},\mathsf{\Gamma }{\iota }_{\mathfrak{p}-1},\mathsf{\Gamma }{\iota }_{\mathfrak{p}},\phi )& =1,\hfill \\ \hfill \underset{\mathfrak{p}\to \infty }{lim}\mathfrak{H}(\mathsf{\Gamma }{\iota }_{\mathfrak{p}-1},\mathsf{\Gamma }{\iota }_{\mathfrak{p}-1},\mathsf{\Gamma }{\iota }_{\mathfrak{p}},\phi )& =0,\hfill \\ \hfill \underset{\mathfrak{p}\to \infty }{lim}\mathfrak{E}(\mathsf{\Gamma }{\iota }_{\mathfrak{p}-1},\mathsf{\Gamma }{\iota }_{\mathfrak{p}-1},\mathsf{\Gamma }{\iota }_{\mathfrak{p}},\phi )& =0.\hfill \end{array}$$

(6)

By applying an identical method, we achieve

$$\begin{array}{cc}\hfill \underset{\mathfrak{p}\to \infty }{lim}\mathfrak{J}(\mathsf{\Gamma }{\iota }_{\mathfrak{p}-1},\mathsf{\Gamma }{\iota }_{\mathfrak{p}},\mathsf{\Gamma }{\iota }_{\mathfrak{p}},\phi )& =1,\hfill \\ \hfill \underset{\mathfrak{p}\to \infty }{lim}\mathfrak{H}(\mathsf{\Gamma }{\iota }_{\mathfrak{p}-1},\mathsf{\Gamma }{\iota }_{\mathfrak{p}},\mathsf{\Gamma }{\iota }_{\mathfrak{p}},\phi )& =0,\hfill \\ \hfill \underset{\mathfrak{p}\to \infty }{lim}\mathfrak{E}(\mathsf{\Gamma }{\iota }_{\mathfrak{p}-1},\mathsf{\Gamma }{\iota }_{\mathfrak{p}},\mathsf{\Gamma }{\iota }_{\mathfrak{p}},\phi )& =0.\hfill \end{array}$$

(7)

Now, we need to demonstrate that $\left\{{\iota }_{\mathfrak{p}}\right\}$ is a Cauchy sequence. According to this, let us assume that $\left\{{\iota }_{\mathfrak{p}}\right\}$ is not a Cauchy sequence. Then, there are $\kappa \in (0,1)$ and ${\phi }_0>0$ so that, for every $\varrho \in \mathbb{N}$, there exist $\mathfrak{p}\left(\varrho \right),\mathfrak{q}\left(\varrho \right)\in \mathbb{N}$ with $\mathfrak{p}\left(\varrho \right)>\mathfrak{q}\left(\varrho \right)>\varrho$ and

$$\begin{array}{c}\hfill \mathfrak{J}({\iota }_{\mathfrak{p}\left(\varrho \right)},{\iota }_{\mathfrak{p}\left(\varrho \right)},{\iota }_{\mathfrak{q}\left(\varrho \right)},{\phi }_0)\le 1-\kappa ,\mathfrak{H}({\iota }_{\mathfrak{p}\left(\varrho \right)},{\iota }_{\mathfrak{p}\left(\varrho \right)},{\iota }_{\mathfrak{q}\left(\varrho \right)},{\phi }_0)\ge \kappa ,\mathfrak{E}({\iota }_{\mathfrak{p}\left(\varrho \right)},{\iota }_{\mathfrak{p}\left(\varrho \right)},{\iota }_{\mathfrak{q}\left(\varrho \right)},{\phi }_0)\ge \kappa .\end{array}$$

(8)

Suppose that $\mathfrak{q}\left(\varrho \right)$ is the smallest integer greater than $\mathfrak{p}\left(\varrho \right)$ that meets the condition in (8). Consequently, we obtain

$$\begin{array}{c}\hfill \mathfrak{J}({\iota }_{\mathfrak{p}\left(\varrho \right)},{\iota }_{\mathfrak{p}\left(\varrho \right)},{\iota }_{\mathfrak{q}\left(\varrho \right)-1},{\phi }_0)>1-\kappa ,\mathfrak{H}({\iota }_{\mathfrak{p}\left(\varrho \right)},{\iota }_{\mathfrak{p}\left(\varrho \right)},{\iota }_{\mathfrak{q}\left(\varrho \right)-1},{\phi }_0)<\kappa ,\mathfrak{E}({\iota }_{\mathfrak{p}\left(\varrho \right)},{\iota }_{\mathfrak{p}\left(\varrho \right)},{\iota }_{\mathfrak{q}\left(\varrho \right)-1},{\phi }_0)<\kappa \end{array}$$

for all $\varrho \in \mathbb{N}$, we obtain

$$\begin{array}{cc}\hfill 1-\kappa & \ge \mathfrak{J}({\iota }_{\mathfrak{p}\left(\varrho \right)},{\iota }_{\mathfrak{p}\left(\varrho \right)},{\iota }_{\mathfrak{q}\left(\varrho \right)},{\phi }_0)\hfill \\ \hfill & =\mathfrak{J}({\iota }_{\mathfrak{q}\left(\varrho \right)},{\iota }_{\mathfrak{p}\left(\varrho \right)},{\iota }_{\mathfrak{p}\left(\varrho \right)},{\phi }_0)\hfill \\ \hfill & \ge \mathfrak{J}({\iota }_{\mathfrak{q}\left(\varrho \right)},{\iota }_{\mathfrak{q}\left(\varrho \right)-1},{\iota }_{\mathfrak{q}\left(\varrho \right)-1},{\phi }_0)\otimes \mathfrak{J}({\iota }_{\mathfrak{q}\left(\varrho \right)-1},{\iota }_{\mathfrak{p}\left(\varrho \right)},{\iota }_{\mathfrak{p}\left(\varrho \right)},{\phi }_0)\hfill \\ \hfill & \ge \mathfrak{J}({\iota }_{\mathfrak{q}\left(\varrho \right)},{\iota }_{\mathfrak{q}\left(\varrho \right)-1},{\iota }_{\mathfrak{q}\left(\varrho \right)-1},{\phi }_0)\otimes (1-\kappa ),\hfill \\ \hfill \mathrm{and}& \kappa \le \mathfrak{H}({\iota }_{\mathfrak{p}\left(\varrho \right)},{\iota }_{\mathfrak{p}\left(\varrho \right)},{\iota }_{\mathfrak{q}\left(\varrho \right)},{\phi }_0)\hfill \\ \hfill & =\mathfrak{H}({\iota }_{\mathfrak{q}\left(\varrho \right)},{\iota }_{\mathfrak{p}\left(\varrho \right)},{\iota }_{\mathfrak{p}\left(\varrho \right)},{\phi }_0)\hfill \\ \hfill & \le \mathfrak{H}({\iota }_{\mathfrak{q}\left(\varrho \right)},{\iota }_{\mathfrak{q}\left(\varrho \right)-1},{\iota }_{\mathfrak{q}\left(\varrho \right)-1},{\phi }_0)\oplus \mathfrak{H}({\iota }_{\mathfrak{q}\left(\varrho \right)-1},{\iota }_{\mathfrak{p}\left(\varrho \right)},{\iota }_{\mathfrak{p}\left(\varrho \right)},{\phi }_0)\hfill \\ \hfill & \le \mathfrak{H}({\iota }_{\mathfrak{q}\left(\varrho \right)},{\iota }_{\mathfrak{q}\left(\varrho \right)-1},{\iota }_{\mathfrak{q}\left(\varrho \right)-1},{\phi }_0)\oplus \kappa ,\hfill \\ \hfill \mathrm{again},\kappa & \le \mathfrak{E}({\iota }_{\mathfrak{p}\left(\varrho \right)},{\iota }_{\mathfrak{p}\left(\varrho \right)},{\iota }_{\mathfrak{q}\left(\varrho \right)},{\phi }_0)\hfill \\ \hfill & =\mathfrak{E}({\iota }_{\mathfrak{q}\left(\varrho \right)},{\iota }_{\mathfrak{p}\left(\varrho \right)},{\iota }_{\mathfrak{p}\left(\varrho \right)},{\phi }_0)\hfill \\ \hfill & \le \mathfrak{E}({\iota }_{\mathfrak{q}\left(\varrho \right)},{\iota }_{\mathfrak{q}\left(\varrho \right)-1},{\iota }_{\mathfrak{q}\left(\varrho \right)-1},{\phi }_0)\oplus \mathfrak{E}({\iota }_{\mathfrak{q}\left(\varrho \right)-1},{\iota }_{\mathfrak{p}\left(\varrho \right)},{\iota }_{\mathfrak{p}\left(\varrho \right)},{\phi }_0)\hfill \\ \hfill & \le \mathfrak{E}({\iota }_{\mathfrak{q}\left(\varrho \right)},{\iota }_{\mathfrak{q}\left(\varrho \right)-1},{\iota }_{\mathfrak{q}\left(\varrho \right)-1},{\phi }_0)\oplus \kappa .\hfill \end{array}$$

Allowing $\varrho \to \infty$, by (7), we obtain ${\displaystyle \underset{\varrho \to \infty }{lim}\mathfrak{J}({\iota }_{\mathfrak{p}\left(\varrho \right)},{\iota }_{\mathfrak{p}\left(\varrho \right)},{\iota }_{\mathfrak{q}\left(\varrho \right)},{\phi }_0)=1-\kappa ,\underset{\varrho \to \infty }{lim}\mathfrak{H}({\iota }_{\mathfrak{p}\left(\varrho \right)},{\iota }_{\mathfrak{p}\left(\varrho \right)},{\iota }_{\mathfrak{q}\left(\varrho \right)},{\phi }_0)=\kappa ,}$ ${\displaystyle \underset{\varrho \to \infty }{lim}\mathfrak{E}({\iota }_{\mathfrak{p}\left(\varrho \right)},{\iota }_{\mathfrak{p}\left(\varrho \right)},{\iota }_{\mathfrak{q}\left(\varrho \right)},{\phi }_0)=\kappa .}$ Also,

$$\begin{array}{cc}\hfill \mathfrak{J}({\iota }_{\mathfrak{q}\left(\varrho \right)+1},{\iota }_{\mathfrak{p}\left(\varrho \right)+1},{\iota }_{\mathfrak{p}\left(\varrho \right)+1},{\phi }_0)& \ge \mathfrak{J}({\iota }_{\mathfrak{q}\left(\varrho \right)+1},{\iota }_{\mathfrak{q}\left(\varrho \right)},{\iota }_{\mathfrak{q}\left(\varrho \right)},{\phi }_0)\otimes \mathfrak{J}({\iota }_{\mathfrak{q}\left(\varrho \right)},{\iota }_{\mathfrak{p}\left(\varrho \right)},{\iota }_{\mathfrak{p}\left(\varrho \right)},{\phi }_0)\hfill \\ \hfill & \otimes \mathfrak{J}({\iota }_{\mathfrak{p}\left(\varrho \right)},{\iota }_{\mathfrak{p}\left(\varrho \right)+1},{\iota }_{\mathfrak{p}\left(\varrho \right)+1},{\phi }_0),\hfill \\ \hfill \mathfrak{H}({\iota }_{\mathfrak{q}\left(\varrho \right)+1},{\iota }_{\mathfrak{p}\left(\varrho \right)+1},{\iota }_{\mathfrak{p}\left(\varrho \right)+1},{\phi }_0)& \le \mathfrak{H}({\iota }_{\mathfrak{q}\left(\varrho \right)+1},{\iota }_{\mathfrak{q}\left(\varrho \right)},{\iota }_{\mathfrak{q}\left(\varrho \right)},{\phi }_0)\oplus \mathfrak{H}({\iota }_{\mathfrak{q}\left(\varrho \right)},{\iota }_{\mathfrak{p}\left(\varrho \right)},{\iota }_{\mathfrak{p}\left(\varrho \right)},{\phi }_0)\hfill \\ \hfill & \oplus \mathfrak{H}({\iota }_{\mathfrak{p}\left(\varrho \right)},{\iota }_{\mathfrak{p}\left(\varrho \right)+1},{\iota }_{\mathfrak{p}\left(\varrho \right)+1},{\phi }_0),\hfill \\ \hfill \mathfrak{E}({\iota }_{\mathfrak{q}\left(\varrho \right)+1},{\iota }_{\mathfrak{p}\left(\varrho \right)+1},{\iota }_{\mathfrak{p}\left(\varrho \right)+1},{\phi }_0)& \le \mathfrak{E}({\iota }_{\mathfrak{q}\left(\varrho \right)+1},{\iota }_{\mathfrak{q}\left(\varrho \right)},{\iota }_{\mathfrak{q}\left(\varrho \right)},{\phi }_0)\oplus \mathfrak{E}({\iota }_{\mathfrak{q}\left(\varrho \right)},{\iota }_{\mathfrak{p}\left(\varrho \right)},{\iota }_{\mathfrak{p}\left(\varrho \right)},{\phi }_0)\hfill \\ \hfill & \oplus \mathfrak{E}({\iota }_{\mathfrak{p}\left(\varrho \right)},{\iota }_{\mathfrak{p}\left(\varrho \right)+1},{\iota }_{\mathfrak{p}\left(\varrho \right)+1},{\phi }_0).\hfill \end{array}$$

By assuming that $\varrho \to \infty$ and using (6), we have

$$\begin{array}{cc}\hfill \underset{\varrho \to \infty }{lim}\mathfrak{J}({\iota }_{\mathfrak{q}\left(\varrho \right)+1},{\iota }_{\mathfrak{p}\left(\varrho \right)+1},{\iota }_{\mathfrak{p}\left(\varrho \right)+1},{\phi }_0)& \ge 1-\kappa ,\hfill \\ \hfill \underset{\varrho \to \infty }{lim}\mathfrak{H}({\iota }_{\mathfrak{q}\left(\varrho \right)+1},{\iota }_{\mathfrak{p}\left(\varrho \right)+1},{\iota }_{\mathfrak{p}\left(\varrho \right)+1},{\phi }_0)& \le \kappa ,\hfill \\ \hfill \underset{\varrho \to \infty }{lim}\mathfrak{E}({\iota }_{\mathfrak{q}\left(\varrho \right)+1},{\iota }_{\mathfrak{p}\left(\varrho \right)+1},{\iota }_{\mathfrak{p}\left(\varrho \right)+1},{\phi }_0)& \le \kappa .\hfill \end{array}$$

(9)

From (8), we obtain

$$\begin{array}{cc}\hfill 1-\kappa & \ge \mathfrak{J}({\iota }_{\mathfrak{q}\left(\varrho \right)},{\iota }_{\mathfrak{p}\left(\varrho \right)},{\iota }_{\mathfrak{p}\left(\varrho \right)},{\phi }_0)\hfill \\ \hfill & \ge \mathfrak{J}({\iota }_{\mathfrak{q}\left(\varrho \right)},{\iota }_{\mathfrak{q}\left(\varrho \right)+1},{\iota }_{\mathfrak{q}\left(\varrho \right)+1},{\phi }_0)\otimes [\mathfrak{J}({\iota }_{\mathfrak{q}\left(\varrho \right)+1},{\iota }_{\mathfrak{p}\left(\varrho \right)+1},{\iota }_{\mathfrak{p}\left(\varrho \right)+1},{\phi }_0)\otimes \mathfrak{J}({\iota }_{\mathfrak{p}\left(\varrho \right)+1},{\iota }_{\mathfrak{p}\left(\varrho \right)},{\iota }_{\mathfrak{p}\left(\varrho \right)},{\phi }_0)],\hfill \\ \hfill \kappa & \le \mathfrak{H}({\iota }_{\mathfrak{q}\left(\varrho \right)},{\iota }_{\mathfrak{p}\left(\varrho \right)},{\iota }_{\mathfrak{p}\left(\varrho \right)},{\phi }_0)\hfill \\ \hfill & \le \mathfrak{H}({\iota }_{\mathfrak{q}\left(\varrho \right)},{\iota }_{\mathfrak{q}\left(\varrho \right)+1},{\iota }_{\mathfrak{q}\left(\varrho \right)+1},{\phi }_0)\oplus [\mathfrak{H}({\iota }_{\mathfrak{q}\left(\varrho \right)+1},{\iota }_{\mathfrak{p}\left(\varrho \right)+1},{\iota }_{\mathfrak{p}\left(\varrho \right)+1},{\phi }_0)\oplus \mathfrak{H}({\iota }_{\mathfrak{p}\left(\varrho \right)+1},{\iota }_{\mathfrak{p}\left(\varrho \right)},{\iota }_{\mathfrak{p}\left(\varrho \right)},{\phi }_0)],\hfill \\ \hfill \kappa & \le \mathfrak{E}({\iota }_{\mathfrak{q}\left(\varrho \right)},{\iota }_{\mathfrak{p}\left(\varrho \right)},{\iota }_{\mathfrak{p}\left(\varrho \right)},{\phi }_0)\hfill \\ \hfill & \le \mathfrak{E}({\iota }_{\mathfrak{q}\left(\varrho \right)},{\iota }_{\mathfrak{q}\left(\varrho \right)+1},{\iota }_{\mathfrak{q}\left(\varrho \right)+1},{\phi }_0)\oplus [\mathfrak{E}({\iota }_{\mathfrak{q}\left(\varrho \right)+1},{\iota }_{\mathfrak{p}\left(\varrho \right)+1},{\iota }_{\mathfrak{p}\left(\varrho \right)+1},{\phi }_0)\oplus \mathfrak{E}({\iota }_{\mathfrak{p}\left(\varrho \right)+1},{\iota }_{\mathfrak{p}\left(\varrho \right)},{\iota }_{\mathfrak{p}\left(\varrho \right)},{\phi }_0)].\hfill \end{array}$$

(10)

and by interpreting the limit as $\varrho \to \infty$ in (10) and from (6) and (9), we obtain

$$\begin{array}{cc}\hfill \underset{\varrho \to \infty }{lim}\mathfrak{J}({\iota }_{\mathfrak{q}\left(\varrho \right)+1},{\iota }_{\mathfrak{p}\left(\varrho \right)+1},{\iota }_{\mathfrak{p}\left(\varrho \right)+1},{\phi }_0)& =1-\kappa ,\hfill \\ \hfill \underset{\varrho \to \infty }{lim}\mathfrak{H}({\iota }_{\mathfrak{q}\left(\varrho \right)+1},{\iota }_{\mathfrak{p}\left(\varrho \right)+1},{\iota }_{\mathfrak{p}\left(\varrho \right)+1},{\phi }_0)& =\kappa ,\hfill \\ \hfill \underset{\varrho \to \infty }{lim}\mathfrak{E}({\iota }_{\mathfrak{q}\left(\varrho \right)+1},{\iota }_{\mathfrak{p}\left(\varrho \right)+1},{\iota }_{\mathfrak{p}\left(\varrho \right)+1},{\phi }_0)& =\kappa .\hfill \end{array}$$

(11)

By using inequalities (2) with $\iota =\vartheta ={\iota }_{\mathfrak{p}\left(\varrho \right)}$ and $\xi ={\iota }_{\mathfrak{q}\left(\varrho \right)}$,

$$\begin{array}{c}\hfill \chi \left(\mathfrak{J}({\iota }_{\mathfrak{p}\left(\varrho \right)+1},{\iota }_{\mathfrak{p}\left(\varrho \right)+1},{\iota }_{\mathfrak{q}\left(\varrho \right)+1},{\phi }_0)\right)\ge \chi \left(\mathfrak{J}({\iota }_{\mathfrak{p}\left(\varrho \right)},{\iota }_{\mathfrak{p}\left(\varrho \right)},{\iota }_{\mathfrak{q}\left(\varrho \right)},{\phi }_0)\right)+\delta \\ \hfill \chi \left(\mathfrak{H}({\iota }_{\mathfrak{p}\left(\varrho \right)+1},{\iota }_{\mathfrak{p}\left(\varrho \right)+1},{\iota }_{\mathfrak{q}\left(\varrho \right)+1},{\phi }_0)\right)\le \chi \left(\mathfrak{H}({\iota }_{\mathfrak{p}\left(\varrho \right)},{\iota }_{\mathfrak{p}\left(\varrho \right)},{\iota }_{\mathfrak{q}\left(\varrho \right)},{\phi }_0)\right)-\delta \\ \hfill \chi \left(\mathfrak{E}({\iota }_{\mathfrak{p}\left(\varrho \right)+1},{\iota }_{\mathfrak{p}\left(\varrho \right)+1},{\iota }_{\mathfrak{q}\left(\varrho \right)+1},{\phi }_0)\right)\le \chi \left(\mathfrak{E}({\iota }_{\mathfrak{p}\left(\varrho \right)},{\iota }_{\mathfrak{p}\left(\varrho \right)},{\iota }_{\mathfrak{q}\left(\varrho \right)},{\phi }_0)\right)-\delta \end{array}$$

(12)

By taking the limit $\varrho \to \infty$ in (12) and applying (2), we obtain

$$\chi (1-\kappa )\ge \chi (1-\kappa )+\delta ,\chi \left(\kappa \right)\le \chi \left(\kappa \right)-\delta .$$

which contradicts itself. Hence, $\left\{{\iota }_{\mathfrak{p}}\right\}$ is a Cauchy sequence in $\mathfrak{A}$. It is clear that, in the completeness of $(\mathfrak{A},\mathfrak{J},\mathfrak{H},\mathfrak{E},\otimes ,\oplus )$, there exists $\iota \in \mathfrak{A}$ such that ${\displaystyle \underset{\mathfrak{p}\to \infty }{lim}{\iota }_{\mathfrak{p}}=\iota }$. Finally, the continuity of $\mathsf{\Gamma },\mathfrak{J},\mathfrak{H}$, and $\mathfrak{E}$ yields

$$\begin{array}{cc}\hfill \mathfrak{J}(\mathsf{\Gamma }\iota ,\mathsf{\Gamma }\iota ,\iota ,\phi )& =\underset{\mathfrak{p}\to \infty }{lim}\mathfrak{J}(\mathsf{\Gamma }{\iota }_{\mathfrak{p}},\mathsf{\Gamma }{\iota }_{\mathfrak{p}},{\iota }_{\mathfrak{p}},\phi )=\underset{\mathfrak{p}\to \infty }{lim}\mathfrak{J}({\iota }_{\mathfrak{p}+1},{\iota }_{\mathfrak{p}+1},{\iota }_{\mathfrak{p}},\phi )=1,\hfill \\ \hfill \mathfrak{H}(\mathsf{\Gamma }\iota ,\mathsf{\Gamma }\iota ,\iota ,\phi )& =\underset{\mathfrak{p}\to \infty }{lim}\mathfrak{H}(\mathsf{\Gamma }{\iota }_{\mathfrak{p}},\mathsf{\Gamma }{\iota }_{\mathfrak{p}},{\iota }_{\mathfrak{p}},\phi )=\underset{\mathfrak{p}\to \infty }{lim}\mathfrak{H}({\iota }_{\mathfrak{p}+1},{\iota }_{\mathfrak{p}+1},{\iota }_{\mathfrak{p}},\phi )=0,\hfill \\ \hfill \mathfrak{E}(\mathsf{\Gamma }\iota ,\mathsf{\Gamma }\iota ,\iota ,\phi )& =\underset{\mathfrak{p}\to \infty }{lim}\mathfrak{E}(\mathsf{\Gamma }{\iota }_{\mathfrak{p}},\mathsf{\Gamma }{\iota }_{\mathfrak{p}},{\iota }_{\mathfrak{p}},\phi )=\underset{\mathfrak{p}\to \infty }{lim}\mathfrak{E}({\iota }_{\mathfrak{p}+1},{\iota }_{\mathfrak{p}+1},{\iota }_{\mathfrak{p}},\phi )=0.\hfill \end{array}$$

Now, we can prove that $\mathsf{\Gamma }$ has a unique fixed point. Pressume that $\iota$ and $\vartheta$ are two fixed points of $\mathsf{\Gamma }$. Certainly, if for $\iota ,\vartheta \in \mathfrak{A},\mathsf{\Gamma }\iota =\iota \ne \vartheta =\mathsf{\Gamma }\vartheta$, then we obtain

$$\begin{array}{c}\hfill \chi \left(\mathfrak{J}\right(\iota ,\iota ,\vartheta ,\phi \left)\right)\ge \chi \left(\mathfrak{J}\right(\iota ,\iota ,\vartheta ,\phi \left)\right)+\delta ,\\ \hfill \chi \left(\mathfrak{H}\right(\iota ,\iota ,\vartheta ,\phi \left)\right)\le \chi \left(\mathfrak{H}\right(\iota ,\iota ,\vartheta ,\phi \left)\right)-\delta ,\\ \hfill \chi \left(\mathfrak{E}\right(\iota ,\iota ,\vartheta ,\phi \left)\right)\le \chi \left(\mathfrak{E}\right(\iota ,\iota ,\vartheta ,\phi \left)\right)-\delta .\end{array}$$

which are the contradictions. Thus, $\mathsf{\Gamma }$ possesses a unique fixed point. Therefore, the theorem holds. □

Example 3.

Let $\mathsf{\Lambda }=[0,1),\otimes (\rho ,\tau )=min\{\rho ,\tau \},\oplus (\rho ,\tau )=max\{\rho ,\tau \}$ and

$$\begin{array}{cc}\hfill \mathfrak{J}(\iota ,\vartheta ,\xi ,\phi )& =\left\{\begin{array}{cc}1,\hfill & \mathit{if}\iota =\vartheta =\xi ,\hfill \\ {\displaystyle \frac{1}{1+max\{\iota ,\vartheta ,\xi \}}},\hfill & \mathit{otherwise}.\hfill \end{array}\right.\hfill \\ \hfill \mathfrak{H}(\iota ,\vartheta ,\xi ,\phi )& =\left\{\begin{array}{cc}0,\hfill & \mathit{if}\iota =\vartheta =\xi ,\hfill \\ {\displaystyle \frac{max\{\iota ,\vartheta ,\xi \}}{1+max\{\iota ,\vartheta ,\xi \}}},\hfill & \mathit{otherwise}.\hfill \end{array}\right.\hfill \\ \hfill \mathfrak{E}(\iota ,\vartheta ,\xi ,\phi )& =\left\{\begin{array}{cc}0,\hfill & \mathit{if}\iota =\vartheta =\xi ,\hfill \\ max\{\iota ,\vartheta ,\xi \},\hfill & \mathit{otherwise}.\hfill \end{array}\right.\hfill \end{array}$$

(13)

for all $\phi >0$. Let $\chi :[0,1)\to \mathbb{R}$ such that $\chi \left(\beta \right)=1/1-\beta$ for all $\iota \in [0,1)$ and define $\mathsf{\Gamma }:\mathsf{\Lambda }\to \mathsf{\Lambda }$ by $\mathsf{\Gamma }\left(\iota \right)=2{\iota }^2/5$ for all $\iota \in \mathsf{\Lambda }$. Clearly, $(\mathsf{\Lambda },\mathfrak{J},\mathfrak{H},\mathfrak{E},\otimes ,\oplus )$ is a complete non-Archimedean $\mathfrak{GNMS}$.

Case 1. We assume that $\iota ,\vartheta ,\xi \in (0,1)$. Since ${\iota }^2<\iota ,{\vartheta }^2<\vartheta$ and ${\xi }^2<\xi ,max\{\iota ,\vartheta ,\xi \}>max\{\mathsf{\Gamma }\iota ,\mathsf{\Gamma }\vartheta ,\mathsf{\Gamma }\xi \}$. So, there exists a $\delta >0$ such that

$$\begin{array}{cc}\hfill {\displaystyle \frac{1}{max\{\mathsf{\Gamma }\iota ,\mathsf{\Gamma }\vartheta ,\mathsf{\Gamma }\xi \}}}+1& \ge {\displaystyle \frac{1}{max\{\iota ,\vartheta ,\xi \}}}+1+\delta ,\hfill \\ \hfill 1-{\displaystyle \frac{1}{max\{\mathsf{\Gamma }\iota ,\mathsf{\Gamma }\vartheta ,\mathsf{\Gamma }\xi \}}}& \le 1-{\displaystyle \frac{1}{max\{\iota ,\vartheta ,\xi \}}}-\delta \mathrm{and}\hfill \\ \hfill max\{\mathsf{\Gamma }\iota ,\mathsf{\Gamma }\vartheta ,\mathsf{\Gamma }\xi \}& \le max\{\iota ,\vartheta ,\xi \}-\delta \hfill \end{array}$$

We can easily prove that $\chi \left(\mathfrak{J}\right(\mathsf{\Gamma }\iota ,\mathsf{\Gamma }\vartheta ,\mathsf{\Gamma }\xi ,\phi \left)\right)\ge \chi \left(\mathfrak{J}\right(\iota ,\vartheta ,\xi \left)\right)+\delta$, $\chi \left(\mathfrak{H}\right(\mathsf{\Gamma }\iota ,\mathsf{\Gamma }\vartheta ,\mathsf{\Gamma }\xi ,\phi \left)\right)\le \chi \left(\mathfrak{H}\right(\iota ,\vartheta ,\xi \left)\right)-\delta$ and $\chi \left(\mathfrak{E}\right(\mathsf{\Gamma }\iota ,\mathsf{\Gamma }\vartheta ,\mathsf{\Gamma }\xi ,\phi \left)\right)\le \chi \left(\mathfrak{E}\right(\iota ,\vartheta ,\xi \left)\right)-\delta .$

Case 2. Let $\iota =0$ and $\vartheta ,\xi \in (0,1)$. Since ${\iota }^2=0,{\vartheta }^2<\vartheta$ and ${\xi }^2<\xi$, then $max\{\iota ,\vartheta ,\xi \}=max\{\vartheta ,\xi \}>max\{\mathsf{\Gamma }\iota ,\mathsf{\Gamma }\vartheta ,\mathsf{\Gamma }\xi \}=max\{\mathsf{\Gamma }\vartheta ,\mathsf{\Gamma }\xi \}$. Hence, we have

$$\begin{array}{cc}\hfill \mathfrak{J}(\mathsf{\Gamma }\iota ,\mathsf{\Gamma }\vartheta ,\mathsf{\Gamma }\xi ,\phi )& ={\displaystyle \frac{1}{1+max\{\mathsf{\Gamma }\iota ,\mathsf{\Gamma }\vartheta ,\mathsf{\Gamma }\xi \}}}>{\displaystyle \frac{1}{1+max\{\iota ,\vartheta ,\xi \}}}=\mathfrak{J}(\iota ,\vartheta ,\xi ,\phi ),\hfill \\ \hfill \mathfrak{H}(\mathsf{\Gamma }\iota ,\mathsf{\Gamma }\vartheta ,\mathsf{\Gamma }\xi ,\phi )& ={\displaystyle \frac{max\{\mathsf{\Gamma }\iota ,\mathsf{\Gamma }\vartheta ,\mathsf{\Gamma }\xi \}}{1+max\{\mathsf{\Gamma }\iota ,\mathsf{\Gamma }\vartheta ,\mathsf{\Gamma }\xi \}}}<{\displaystyle \frac{max\{\iota ,\vartheta ,\xi \}}{1+max\{\iota ,\vartheta ,\xi \}}}=\mathfrak{H}(\iota ,\vartheta ,\xi ,\phi ),\hfill \\ \hfill \mathfrak{E}(\mathsf{\Gamma }\iota ,\mathsf{\Gamma }\vartheta ,\mathsf{\Gamma }\xi ,\phi )& =max\{\mathsf{\Gamma }\iota ,\mathsf{\Gamma }\vartheta ,\mathsf{\Gamma }\xi \}<max\{\iota ,\vartheta ,\xi \}=\mathfrak{E}(\iota ,\vartheta ,\xi ,\phi ).\hfill \end{array}$$

So, there exists a $\delta >0$ such that $\chi \left(\mathfrak{J}\right(\mathsf{\Gamma }\iota ,\mathsf{\Gamma }\vartheta ,\mathsf{\Gamma }\xi ,\phi \left)\right)\ge \chi \left(\mathfrak{J}\right(\iota ,\vartheta ,\xi ,\phi \left)\right)+\delta$, $\chi \left(\mathfrak{H}\right(\mathsf{\Gamma }\iota ,\mathsf{\Gamma }\vartheta ,\mathsf{\Gamma }\xi ,\phi \left)\right)\le \chi \left(\mathfrak{H}\right(\iota ,\vartheta ,\xi ,\phi \left)\right)-\delta$ and $\chi \left(\mathfrak{E}\right(\mathsf{\Gamma }\iota ,\mathsf{\Gamma }\vartheta ,\mathsf{\Gamma }\xi ,\phi \left)\right)\le \chi \left(\mathfrak{H}\right(\iota ,\vartheta ,\xi ,\phi \left)\right)-\delta .$

Case 3. Let $\iota =\vartheta =0$ and $\xi \in (0,1)$; it is easy to see that $\chi \left(\mathfrak{J}\right(\mathsf{\Gamma }\iota ,\mathsf{\Gamma }\vartheta ,\mathsf{\Gamma }\xi ,\phi \left)\right)\ge \chi \left(\mathfrak{J}\right(\iota ,\vartheta ,\xi ,\phi \left)\right)+\delta$, $\chi \left(\mathfrak{H}\right(\mathsf{\Gamma }\iota ,\mathsf{\Gamma }\vartheta ,\mathsf{\Gamma }\xi ,\phi \left)\right)\le \chi \left(\mathfrak{H}\right(\iota ,\vartheta ,\xi ,\phi \left)\right)-\delta$ and $\chi \left(\mathfrak{E}\right(\mathsf{\Gamma }\iota ,\mathsf{\Gamma }\vartheta ,\mathsf{\Gamma }\xi ,\phi \left)\right)\le$$\chi \left(\mathfrak{H}\right(\iota ,\vartheta ,\xi ,\phi \left)\right)-\delta .$ Therefore, Γ is a χ-contraction. Then, all the conditions of Theorem (1) hold, and Γ has the unique fixed point $\iota =0$.

Definition 7.

Consider $(\mathfrak{A},\mathfrak{J},\mathfrak{H},\mathfrak{E},\otimes ,\oplus )$ as a non-Archimedean GNMS. A mapping $\mathsf{\Gamma }:\mathfrak{A}\to \mathfrak{A}$ is referred to as a χ-weak contraction if there exists a $\delta >0$ such that

$$\begin{array}{cc}\chi \left(\mathfrak{J}\right(\mathsf{\Gamma }\iota ,\mathsf{\Gamma }\vartheta ,\mathsf{\Gamma }\xi ,\phi \left)\right)& \ge \chi (min\{\mathfrak{J}(\iota ,\vartheta ,\xi ,\phi ),\mathfrak{J}(\iota ,\iota ,\mathsf{\Gamma }\iota ,\phi ),\mathfrak{J}(\vartheta ,\vartheta ,\mathsf{\Gamma }\vartheta ,\phi ),\mathfrak{J}(\xi ,\xi ,\mathsf{\Gamma }\xi ,\phi )\left\}\right)+\delta ,\\ \chi \left(\mathfrak{H}\right(\mathsf{\Gamma }\iota ,\mathsf{\Gamma }\vartheta ,\mathsf{\Gamma }\xi ,\phi \left)\right)& \le \chi (max\{\mathfrak{H}(\iota ,\vartheta ,\xi ,\phi ),\mathfrak{H}(\iota ,\iota ,\mathsf{\Gamma }\iota ,\phi ),\mathfrak{H}(\vartheta ,\vartheta ,\mathsf{\Gamma }\vartheta ,\phi ),\mathfrak{H}(\xi ,\xi ,\mathsf{\Gamma }\xi ,\phi )\left\}\right)-\delta ,\\ \chi \left(\mathfrak{E}\right(\mathsf{\Gamma }\iota ,\mathsf{\Gamma }\vartheta ,\mathsf{\Gamma }\xi ,\phi \left)\right)& \le \chi (max\{\mathfrak{E}(\iota ,\vartheta ,\xi ,\phi ),\mathfrak{E}(\iota ,\iota ,\mathsf{\Gamma }\iota ,\phi ),\mathfrak{E}(\vartheta ,\vartheta ,\mathsf{\Gamma }\vartheta ,\phi ),\mathfrak{E}(\xi ,\xi ,\mathsf{\Gamma }\xi ,\phi )\left\}\right)-\delta ,\end{array}$$

(14)

for all $\iota ,\vartheta ,\xi \in \mathfrak{A}$ and $\chi \in \iota$.

It is important that every $\chi$-contraction qualifies as a $\chi$-weak contraction. However, the reverse is not necessarily true.

Example 4.

Let $\mathfrak{A}=\mathfrak{M}\bigcup \Xi$, where $\mathfrak{M}=\{1/10,1/2,1,2,3\},\Xi =\left\{4\right\}$.

$$\begin{array}{cc}\hfill \mathfrak{J}(\iota ,\vartheta ,\xi ,\phi )& ={\displaystyle \frac{min\{\iota ,\vartheta ,\xi \}}{max\{\iota ,\vartheta ,\xi \}}},\hfill \\ \hfill \mathfrak{H}(\iota ,\vartheta ,\xi ,\phi )& =1-{\displaystyle \frac{min\{\iota ,\vartheta ,\xi \}}{max\{\iota ,\vartheta ,\xi \}}},\hfill \\ \hfill \mathfrak{E}(\iota ,\vartheta ,\xi ,\phi )& ={\displaystyle \frac{min\{\iota ,\vartheta ,\xi \}}{min\{\iota ,\vartheta ,\xi \}+max\{\iota ,\vartheta ,\xi \}}},\hfill \end{array}$$

and for all $\phi >0$. Clearly, $(\mathfrak{A},\mathfrak{J},\mathfrak{H},\mathfrak{E},\otimes ,\oplus )$ is a complete non-Archimedean $\mathfrak{GNMS}$. Let $\chi :[0,1)\to \mathbb{R}$ such that $\chi \left(\beta \right)={\displaystyle \frac{1}{1-\beta }}$ for all $\beta \in [0,1)$ and define $\mathsf{\Gamma }:\mathfrak{A}\to \mathfrak{A}$ by $\left\{\begin{array}{cc}{\displaystyle \frac{1}{10}},\hfill & \mathrm{if}\iota \in \mathfrak{M},\hfill \\ {\displaystyle \frac{1}{2}},\hfill & \mathrm{if}\iota \in \Xi .\hfill \end{array}\right.$ Since Γ is not continuous, Γ is not a χ-contraction by (4). Now, we show that Γ is a χ-weak contraction for all $\iota \in \mathfrak{A}$.

Case 1. Let $\iota =1$ and $\vartheta ,\xi =4$

$$\begin{array}{cc}\hfill \mathfrak{J}(\mathsf{\Gamma }\iota ,\mathsf{\Gamma }\vartheta ,\mathsf{\Gamma }\xi ,\phi )& ={\displaystyle \frac{1}{5}}>{\displaystyle \frac{1}{10}}=min\left\{{\displaystyle \frac{1}{4}},{\displaystyle \frac{1}{10}},{\displaystyle \frac{1}{8}},{\displaystyle \frac{1}{8}}\right\}\hfill \\ \hfill & =min\left\{\mathfrak{J}\right(\iota ,\vartheta ,\xi ,\phi ),\mathfrak{J}(\iota ,\iota ,\mathsf{\Gamma }\iota ,\phi ),\mathfrak{J}(\vartheta ,\vartheta ,\mathsf{\Gamma }\vartheta ,\phi ),\mathfrak{J}(\xi ,\xi ,\mathsf{\Gamma }\xi ,\phi \left)\right\}.\hfill \\ \hfill \mathfrak{H}(\mathsf{\Gamma }\iota ,\mathsf{\Gamma }\vartheta ,\mathsf{\Gamma }\xi ,\phi )& ={\displaystyle \frac{4}{5}}<{\displaystyle \frac{9}{10}}=max\left\{{\displaystyle \frac{3}{4}},{\displaystyle \frac{9}{10}},{\displaystyle \frac{7}{8}},{\displaystyle \frac{7}{8}}\right\}\hfill \\ \hfill & =max\left\{\mathfrak{H}\right(\iota ,\vartheta ,\xi ,\phi ),\mathfrak{H}(\iota ,\iota ,\mathsf{\Gamma }\iota ,\phi ),\mathfrak{H}(\vartheta ,\vartheta ,\mathsf{\Gamma }\vartheta ,\phi ),\mathfrak{H}(\xi ,\xi ,\mathsf{\Gamma }\xi ,\phi \left)\right\}\hfill \\ \hfill \mathfrak{E}(\mathsf{\Gamma }\iota ,\mathsf{\Gamma }\vartheta ,\mathsf{\Gamma }\xi ,\phi )& ={\displaystyle \frac{1}{6}}<{\displaystyle \frac{1}{5}}=max\left\{{\displaystyle \frac{1}{5}},{\displaystyle \frac{1}{11}},{\displaystyle \frac{1}{9}},{\displaystyle \frac{1}{9}}\right\}\hfill \\ \hfill & =max\left\{\mathfrak{E}\right(\iota ,\vartheta ,\xi ,\phi ),\mathfrak{E}(\iota ,\iota ,\mathsf{\Gamma }\iota ,\phi ),\mathfrak{E}(\vartheta ,\vartheta ,\mathsf{\Gamma }\vartheta ,\phi ),\mathfrak{E}(\xi ,\xi ,\mathsf{\Gamma }\xi ,\phi \left)\right\}\hfill \end{array}$$

Verification of χ-contraction mappings:

Consider that $\chi \left(\beta \right)={\displaystyle \frac{1}{1-\beta }}$ for all $\beta \in [0,1)$. For $\mathfrak{J}$, $\beta ={\displaystyle \frac{1}{5}}$, $\chi \left(\beta \right)={\displaystyle \frac{1}{1-\frac{1}{5}}}=1.25$. $\beta ={\displaystyle \frac{1}{10}}$, $\chi \left(\beta \right)={\displaystyle \frac{1}{1-\frac{1}{10}}}=1.1111$. Since $1.25>1.11$, the contraction holds true. Similarly, verification can be performed for $\mathfrak{H}$ and $\mathfrak{E}$ using the same approach. So, there exists a $\delta >0$ such that

$$\begin{array}{cc}\hfill \chi \left(\mathfrak{J}\right(\mathsf{\Gamma }\iota ,\mathsf{\Gamma }\vartheta ,\mathsf{\Gamma }\xi ,\phi \left)\right)& \ge \chi (min\{\mathfrak{J}(\iota ,\vartheta ,\xi ,\phi ),\mathfrak{J}(\iota ,\iota ,\mathsf{\Gamma }\iota ,\phi ),\mathfrak{J}(\vartheta ,\vartheta ,\mathsf{\Gamma }\vartheta ,\phi ),\mathfrak{J}(\xi ,\xi ,\mathsf{\Gamma }\xi ,\phi )\left\}\right)+\delta ,\hfill \\ \hfill \chi \left(\mathfrak{H}\right(\mathsf{\Gamma }\iota ,\mathsf{\Gamma }\vartheta ,\mathsf{\Gamma }\xi ,\phi \left)\right)& \le \chi (max\{\mathfrak{H}(\iota ,\vartheta ,\xi ,\phi ),\mathfrak{H}(\iota ,\iota ,\mathsf{\Gamma }\iota ,\phi ),\mathfrak{H}(\vartheta ,\vartheta ,\mathsf{\Gamma }\vartheta ,\phi ),\mathfrak{H}(\xi ,\xi ,\mathsf{\Gamma }\xi ,\phi )\left\}\right)-\delta ,\hfill \\ \hfill \chi \left(\mathfrak{E}\right(\mathsf{\Gamma }\iota ,\mathsf{\Gamma }\vartheta ,\mathsf{\Gamma }\xi ,\phi \left)\right)& \le \chi (max\{\mathfrak{E}(\iota ,\vartheta ,\xi ,\phi ),\mathfrak{E}(\iota ,\iota ,\mathsf{\Gamma }\iota ,\phi ),\mathfrak{E}(\vartheta ,\vartheta ,\mathsf{\Gamma }\vartheta ,\phi ),\mathfrak{E}(\xi ,\xi ,\mathsf{\Gamma }\xi ,\phi )\left\}\right)-\delta ,\hfill \end{array}$$

Case 2. Let $\iota \in \{2,3\}$ and $\vartheta =\xi =4$,

$$\begin{array}{cc}\hfill \mathfrak{J}(\mathsf{\Gamma }\iota ,\mathsf{\Gamma }\vartheta ,\mathsf{\Gamma }\xi ,\phi )& ={\displaystyle \frac{1}{5}}>{\displaystyle \frac{1}{30}}=min\left\{{\displaystyle \frac{\iota }{4}},{\displaystyle \frac{1}{10\iota }},{\displaystyle \frac{1}{8}},{\displaystyle \frac{1}{8}}\right\}\hfill \\ \hfill & =min\left\{\mathfrak{J}\right(\iota ,\vartheta ,\xi ,\phi ),\mathfrak{J}(\iota ,\iota ,\mathsf{\Gamma }\iota ,\phi ),\mathfrak{J}(\vartheta ,\vartheta ,\mathsf{\Gamma }\vartheta ,\phi ),\mathfrak{J}(\xi ,\xi ,\mathsf{\Gamma }\xi ,\phi \left)\right\}\hfill \\ \hfill \mathfrak{H}(\mathsf{\Gamma }\iota ,\mathsf{\Gamma }\vartheta ,\mathsf{\Gamma }\xi ,\phi )& ={\displaystyle \frac{4}{5}}<{\displaystyle \frac{29}{30}}=max\left\{1-{\displaystyle \frac{\iota }{4}},1-{\displaystyle \frac{1}{10\iota }},{\displaystyle \frac{7}{8}},{\displaystyle \frac{7}{8}}\right\}\hfill \\ \hfill & =max\left\{\mathfrak{H}\right(\iota ,\vartheta ,\xi ,\phi ),\mathfrak{H}(\iota ,\iota ,\mathsf{\Gamma }\iota ,\phi ),\mathfrak{H}(\vartheta ,\vartheta ,\mathsf{\Gamma }\vartheta ,\phi ),\mathfrak{H}(\xi ,\xi ,\mathsf{\Gamma }\xi ,\phi \left)\right\}\hfill \\ \hfill \mathfrak{E}(\mathsf{\Gamma }\iota ,\mathsf{\Gamma }\vartheta ,\mathsf{\Gamma }\xi ,\phi )& ={\displaystyle \frac{1}{6}}<{\displaystyle \frac{3}{7}}=max\left\{{\displaystyle \frac{\iota }{\iota +4}},{\displaystyle \frac{1}{1+10\iota }},{\displaystyle \frac{1}{9}},{\displaystyle \frac{1}{9}}\right\}\hfill \\ \hfill & =max\left\{\mathfrak{E}\right(\iota ,\vartheta ,\xi ,\phi ),\mathfrak{E}(\iota ,\iota ,\mathsf{\Gamma }\iota ,\phi ),\mathfrak{E}(\vartheta ,\vartheta ,\mathsf{\Gamma }\vartheta ,\phi ),\mathfrak{E}(\xi ,\xi ,\mathsf{\Gamma }\xi ,\phi \left)\right\}.\hfill \end{array}$$

Verification of χ-contraction mappings:

Consider the function $\chi \left(\beta \right)={\displaystyle \frac{1}{1-\beta }}$ for all $\beta \in [0,1)$. For $\mathfrak{J}$, when $\beta ={\displaystyle \frac{1}{5}}$, $\chi \left({\displaystyle \frac{1}{5}}\right)={\displaystyle \frac{1}{1-\frac{1}{5}}}=1.25$. When $\beta ={\displaystyle \frac{1}{30}}$, $\chi \left({\displaystyle \frac{1}{30}}\right)={\displaystyle \frac{1}{1-\frac{1}{30}}}\approx 1.0345$. Since $1.25>1.0345$, the contraction holds true.

Similarly, the verification can be performed for $\mathfrak{H}$ and $\mathfrak{E}$ using the same approach. So, there exists a $\delta >0$ such that

$$\begin{array}{c}\chi \left(\mathfrak{J}\right(\mathsf{\Gamma }\iota ,\mathsf{\Gamma }\vartheta ,\mathsf{\Gamma }\xi ,\phi \left)\right)\ge \chi (min\{\mathfrak{J}(\iota ,\vartheta ,\xi ,\phi ),\mathfrak{J}(\iota ,\iota ,\mathsf{\Gamma }\iota ,\phi ),\mathfrak{J}(\vartheta ,\vartheta ,\mathsf{\Gamma }\vartheta ,\phi ),\mathfrak{J}(\xi ,\xi ,\mathsf{\Gamma }\xi ,\phi )\left\}\right)+\delta ,\\ \chi \left(\mathfrak{H}\right(\mathsf{\Gamma }\iota ,\mathsf{\Gamma }\vartheta ,\mathsf{\Gamma }\xi ,\phi \left)\right)\le \chi (max\{\mathfrak{H}(\iota ,\vartheta ,\xi ,\phi ),\mathfrak{H}(\iota ,\iota ,\mathsf{\Gamma }\iota ,\phi ),\mathfrak{H}(\vartheta ,\vartheta ,\mathsf{\Gamma }\vartheta ,\phi ),\mathfrak{H}(\xi ,\xi ,\mathsf{\Gamma }\xi ,\phi )\left\}\right)-\delta ,\\ \chi \left(\mathfrak{E}\right(\mathsf{\Gamma }\iota ,\mathsf{\Gamma }\vartheta ,\mathsf{\Gamma }\xi ,\phi \left)\right)\le \chi (max\{\mathfrak{E}(\iota ,\vartheta ,\xi ,\phi ),\mathfrak{E}(\iota ,\iota ,\mathsf{\Gamma }\iota ,\phi ),\mathfrak{E}(\vartheta ,\vartheta ,\mathsf{\Gamma }\vartheta ,\phi ),\mathfrak{E}(\xi ,\xi ,\mathsf{\Gamma }\xi ,\phi )\left\}\right)-\delta ,\end{array}$$

Case 3.  Let $\iota \in \{1/10,1/2\}$ and $\vartheta =\xi =4$,

$$\begin{array}{cc}\hfill \mathfrak{J}(\mathsf{\Gamma }\iota ,\mathsf{\Gamma }\vartheta ,\mathsf{\Gamma }\xi ,\phi )& ={\displaystyle \frac{1}{5}}>{\displaystyle \frac{1}{40}}=min\left\{{\displaystyle \frac{\iota }{4}},{\displaystyle \frac{1}{10\iota }},{\displaystyle \frac{1}{8}},{\displaystyle \frac{1}{8}}\right\}\hfill \\ \hfill & =min\left\{{\displaystyle \frac{1}{40}},{\displaystyle \frac{1}{5}},{\displaystyle \frac{1}{8}},{\displaystyle \frac{1}{8}}\right\}\hfill \\ \hfill & =min\left\{\mathfrak{J}\right(\iota ,\vartheta ,\xi ,\phi ),\mathfrak{J}(\iota ,\iota ,\mathsf{\Gamma }\iota ,\phi ),\mathfrak{J}(\vartheta ,\vartheta ,\mathsf{\Gamma }\vartheta ,\phi ),\mathfrak{J}(\xi ,\xi ,\mathsf{\Gamma }\xi ,\phi \left)\right\}\hfill \\ \hfill \mathfrak{H}(\mathsf{\Gamma }\iota ,\mathsf{\Gamma }\vartheta ,\mathsf{\Gamma }\xi ,\phi )& ={\displaystyle \frac{4}{5}}<{\displaystyle \frac{39}{40}}=max\left\{1-{\displaystyle \frac{\iota }{4}},1-{\displaystyle \frac{1}{10\iota }},{\displaystyle \frac{7}{8}},{\displaystyle \frac{7}{8}}\right\}\hfill \\ \hfill & =max\left\{{\displaystyle \frac{39}{40}},{\displaystyle \frac{4}{5}},{\displaystyle \frac{7}{8}},{\displaystyle \frac{7}{8}}\right\}\hfill \\ \hfill & =max\left\{\mathfrak{H}\right(\iota ,\vartheta ,\xi ,\phi ),\mathfrak{H}(\iota ,\iota ,\mathsf{\Gamma }\iota ,\phi ),\mathfrak{H}(\vartheta ,\vartheta ,\mathsf{\Gamma }\vartheta ,\phi ),\mathfrak{H}(\xi ,\xi ,\mathsf{\Gamma }\xi ,\phi \left)\right\}\hfill \\ \hfill \mathfrak{E}(\mathsf{\Gamma }\iota ,\mathsf{\Gamma }\vartheta ,\mathsf{\Gamma }\xi ,\phi )& ={\displaystyle \frac{1}{6}}<{\displaystyle \frac{1}{2}}=max\left\{{\displaystyle \frac{\iota }{\iota +4}},{\displaystyle \frac{1}{1+10\iota }},{\displaystyle \frac{1}{9}},{\displaystyle \frac{1}{9}}\right\}\hfill \\ \hfill & =max\left\{{\displaystyle \frac{1}{9}},{\displaystyle \frac{1}{2}},{\displaystyle \frac{1}{9}},{\displaystyle \frac{1}{9}}\right\}\hfill \\ \hfill & =max\left\{\mathfrak{E}\right(\iota ,\vartheta ,\xi ,\phi ),\mathfrak{E}(\iota ,\iota ,\mathsf{\Gamma }\iota ,\phi ),\mathfrak{E}(\vartheta ,\vartheta ,\mathsf{\Gamma }\vartheta ,\phi ),(\xi ,\xi ,\mathsf{\Gamma }\xi ,\phi \left)\right\}.\hfill \end{array}$$

Verification of χ-contraction mappings:

Consider the function $\chi \left(\beta \right)={\displaystyle \frac{1}{1-\beta }}$ for all $\beta \in [0,1)$. For $\mathfrak{J}$, when $\beta ={\displaystyle \frac{1}{5}}$, $\chi \left({\displaystyle \frac{1}{5}}\right)={\displaystyle \frac{1}{1-\frac{1}{5}}}=1.25$. When $\beta ={\displaystyle \frac{1}{40}}$, $\chi \left({\displaystyle \frac{1}{40}}\right)={\displaystyle \frac{1}{1-\frac{1}{40}}}\approx 1.0256$. Since $1.25>1.0256$, the contraction property holds.

Similarly, the verification can be performed for $\mathfrak{H}$ and $\mathfrak{E}$ using the same approach. So, there exists a $\delta >0$ such that

$$\begin{array}{cc}\hfill \chi \left(\mathfrak{J}\right(\mathsf{\Gamma }\iota ,\mathsf{\Gamma }\vartheta ,\mathsf{\Gamma }\xi ,\phi \left)\right)& \ge \chi (min\{\mathfrak{J}(\iota ,\vartheta ,\xi ,\phi ),\mathfrak{J}(\iota ,\iota ,\mathsf{\Gamma }\iota ,\phi ),\mathfrak{J}(\vartheta ,\vartheta ,\mathsf{\Gamma }\vartheta ,\phi ),\mathfrak{J}(\xi ,\xi ,\mathsf{\Gamma }\xi ,\phi )\left\}\right)+\delta ,\hfill \\ \hfill \chi \left(\mathfrak{H}\right(\mathsf{\Gamma }\iota ,\mathsf{\Gamma }\vartheta ,\mathsf{\Gamma }\xi ,\phi \left)\right)& \le \chi (max\{\mathfrak{H}(\iota ,\vartheta ,\xi ,\phi ),\mathfrak{H}(\iota ,\iota ,\mathsf{\Gamma }\iota ,\phi ),\mathfrak{H}(\vartheta ,\vartheta ,\mathsf{\Gamma }\vartheta ,\phi ),\mathfrak{H}(\xi ,\xi ,\mathsf{\Gamma }\xi ,\phi )\left\}\right)-\delta ,\hfill \\ \hfill \chi \left(\mathfrak{E}\right(\mathsf{\Gamma }\iota ,\mathsf{\Gamma }\vartheta ,\mathsf{\Gamma }\xi ,\phi \left)\right)& \le \chi (max\{\mathfrak{E}(\iota ,\vartheta ,\xi ,\phi ),\mathfrak{E}(\iota ,\iota ,\mathsf{\Gamma }\iota ,\phi ),\mathfrak{E}(\vartheta ,\vartheta ,\mathsf{\Gamma }\vartheta ,\phi ),\mathfrak{E}(\xi ,\xi ,\mathsf{\Gamma }\xi ,\phi )\left\}\right)-\delta .\hfill \end{array}$$

By proving the remaining cases, we find that Γ is a χ-weak contraction.

Theorem 2.

Consider a complete non-Archimedean GNMS represented as $(\mathfrak{A},\mathfrak{J},\mathfrak{H},\mathfrak{E},\otimes ,\oplus )$. If there exists a mapping $\mathsf{\Gamma }:\mathfrak{A}\to \mathfrak{A}$ that acts as a χ-weak contraction, then Γ is guaranteed to have exactly one fixed point within $\mathfrak{A}$.

Proof. 

Let an arbitrary fixed element ${\iota }_0\in \mathfrak{A}$ be chosen. Construct the sequence $\left\{{\iota }_{\mathfrak{p}}\right\}$ such that $\mathsf{\Gamma }{\iota }_{\mathfrak{p}}={\iota }_{\mathfrak{p}+1}$ for all $\mathfrak{p}\in \mathbb{N}$. If ${\iota }_{\mathfrak{p}}={\iota }_{\mathfrak{p}+1}$ at some point, then ${\iota }_{\mathfrak{p}+1}$ is the fixed point of $\mathsf{\Gamma }$, completing the proof. Otherwise, ${\iota }_{\mathfrak{p}}\ne {\iota }_{\mathfrak{p}+1}$ for every $\mathfrak{p}\in \mathbb{N}$.

Thus, using Equation (14), we obtain the following result:

$$\begin{array}{cc}\hfill & \chi \left(\mathfrak{J}(\mathsf{\Gamma }{\iota }_{\mathfrak{p}-1},\mathsf{\Gamma }{\iota }_{\mathfrak{p}-1},\mathsf{\Gamma }{\iota }_{\mathfrak{p}},\phi )\right)\hfill \\ \hfill & \ge \chi (min\{\mathfrak{J}({\iota }_{\mathfrak{p}-1},{\iota }_{\mathfrak{p}-1},{\iota }_{\mathfrak{p}},\phi ),\mathfrak{J}({\iota }_{\mathfrak{p}-1},{\iota }_{\mathfrak{p}-1},\mathsf{\Gamma }{\iota }_{\mathfrak{p}-1},\phi ),\mathfrak{J}({\iota }_{\mathfrak{p}},{\iota }_{\mathfrak{p}},\mathsf{\Gamma }{\iota }_{\mathfrak{p}},\phi )\})+\delta \hfill \\ \hfill & =\chi (min\{\mathfrak{J}({\iota }_{\mathfrak{p}-1},{\iota }_{\mathfrak{p}-1},{\iota }_{\mathfrak{p}},\phi ),\mathfrak{J}({\iota }_{\mathfrak{p}-1},{\iota }_{\mathfrak{p}-1},{\iota }_{\mathfrak{p}},\phi ),\mathfrak{J}({\iota }_{\mathfrak{p}},{\iota }_{\mathfrak{p}},{\iota }_{\mathfrak{p}+1},\phi )\})+\delta \hfill \\ \hfill & =\chi (min\{\mathfrak{J}({\iota }_{\mathfrak{p}-1},{\iota }_{\mathfrak{p}-1},{\iota }_{\mathfrak{p}},\phi ),\mathfrak{J}({\iota }_{\mathfrak{p}},{\iota }_{\mathfrak{p}},{\iota }_{\mathfrak{p}+1},\phi )\})+\delta .\hfill \\ \hfill & \chi \left(\mathfrak{H}(\mathsf{\Gamma }{\iota }_{\mathfrak{p}-1},\mathsf{\Gamma }{\iota }_{\mathfrak{p}-1},\mathsf{\Gamma }{\iota }_{\mathfrak{p}},\phi )\right)\hfill \\ \hfill & \le \chi (max\{\mathfrak{H}({\iota }_{\mathfrak{p}-1},{\iota }_{\mathfrak{p}-1},{\iota }_{\mathfrak{p}},\phi ),\mathfrak{H}({\iota }_{\mathfrak{p}-1},{\iota }_{\mathfrak{p}-1},\mathsf{\Gamma }{\iota }_{\mathfrak{p}-1},\phi ),\mathfrak{H}({\iota }_{\mathfrak{p}},{\iota }_{\mathfrak{p}},\mathsf{\Gamma }{\iota }_{\mathfrak{p}},\phi )\})-\delta \hfill \\ \hfill & =\chi (max\{\mathfrak{H}({\iota }_{\mathfrak{p}-1},{\iota }_{\mathfrak{p}-1},{\iota }_{\mathfrak{p}},\phi ),\mathfrak{H}({\iota }_{\mathfrak{p}-1},{\iota }_{\mathfrak{p}-1},{\iota }_{\mathfrak{p}},\phi ),\mathfrak{H}({\iota }_{\mathfrak{p}},{\iota }_{\mathfrak{p}},{\iota }_{\mathfrak{p}+1},\phi )\})-\delta \hfill \end{array}$$

(15)

$$\begin{array}{cc}\hfill & =\chi (max\{\mathfrak{H}({\iota }_{\mathfrak{p}-1},{\iota }_{\mathfrak{p}-1},{\iota }_{\mathfrak{p}},\phi ),\mathfrak{H}({\iota }_{\mathfrak{p}},{\iota }_{\mathfrak{p}},{\iota }_{\mathfrak{p}+1},\phi )\})-\delta .\hfill \\ \hfill & \chi \left(\mathfrak{E}(\mathsf{\Gamma }{\iota }_{\mathfrak{p}-1},\mathsf{\Gamma }{\iota }_{\mathfrak{p}-1},\mathsf{\Gamma }{\iota }_{\mathfrak{p}},\phi )\right)\hfill \\ \hfill & \le \chi (max\{\mathfrak{E}({\iota }_{\mathfrak{p}-1},{\iota }_{\mathfrak{p}-1},{\iota }_{\mathfrak{p}},\phi ),\mathfrak{E}({\iota }_{\mathfrak{p}-1},{\iota }_{\mathfrak{p}-1},\mathsf{\Gamma }{\iota }_{\mathfrak{p}-1},\phi ),\mathfrak{E}({\iota }_{\mathfrak{p}},{\iota }_{\mathfrak{p}},\mathsf{\Gamma }{\iota }_{\mathfrak{p}},\phi )\})-\delta ,\hfill \\ \hfill & =\chi (max\{\mathfrak{E}({\iota }_{\mathfrak{p}-1},{\iota }_{\mathfrak{p}-1},{\iota }_{\mathfrak{p}},\phi ),\mathfrak{E}({\iota }_{\mathfrak{p}-1},{\iota }_{\mathfrak{p}-1},{\iota }_{\mathfrak{p}},\phi ),\mathfrak{E}({\iota }_{\mathfrak{p}},{\iota }_{\mathfrak{p}},{\iota }_{\mathfrak{p}+1},\phi )\})-\delta ,\hfill \\ \hfill & =\chi (max\{\mathfrak{E}({\iota }_{\mathfrak{p}-1},{\iota }_{\mathfrak{p}-1},{\iota }_{\mathfrak{p}},\phi ),\mathfrak{E}({\iota }_{\mathfrak{p}},{\iota }_{\mathfrak{p}},{\iota }_{\mathfrak{p}+1},\phi )\})-\delta .\hfill \end{array}$$

(16)

If there is some $\mathfrak{p}\in \mathbb{N}$ such that

$$\begin{array}{cc}\hfill min\{\mathfrak{J}({\iota }_{\mathfrak{p}-1},{\iota }_{\mathfrak{p}-1},{\iota }_{\mathfrak{p}},\phi ),\mathfrak{J}({\iota }_{\mathfrak{p}},{\iota }_{\mathfrak{p}},{\iota }_{\mathfrak{p}+1},\phi )\}& =\mathfrak{J}({\iota }_{\mathfrak{p}},{\iota }_{\mathfrak{p}},{\iota }_{\mathfrak{p}+1},\phi ),\hfill \\ \hfill max\{\mathfrak{H}({\iota }_{\mathfrak{p}-1},{\iota }_{\mathfrak{p}-1},{\iota }_{\mathfrak{p}},\phi ),\mathfrak{H}({\iota }_{\mathfrak{p}},{\iota }_{\mathfrak{p}},{\iota }_{\mathfrak{p}+1},\phi )\}& =\mathfrak{H}({\iota }_{\mathfrak{p}},{\iota }_{\mathfrak{p}},{\iota }_{\mathfrak{p}+1},\phi ),\hfill \\ \hfill max\{\mathfrak{E}({\iota }_{\mathfrak{p}-1},{\iota }_{\mathfrak{p}-1},{\iota }_{\mathfrak{p}},\phi ),\mathfrak{E}({\iota }_{\mathfrak{p}},{\iota }_{\mathfrak{p}},{\iota }_{\mathfrak{p}+1},\phi )\}& =\mathfrak{E}({\iota }_{\mathfrak{p}},{\iota }_{\mathfrak{p}},{\iota }_{\mathfrak{p}+1},\phi ),\hfill \end{array}$$

it follows from (15) that

$$\begin{array}{cc}\hfill & \chi \left(\mathfrak{J}(\mathsf{\Gamma }{\iota }_{\mathfrak{p}-1},\mathsf{\Gamma }{\iota }_{\mathfrak{p}-1},\mathsf{\Gamma }{\iota }_{\mathfrak{p}},\phi )\right)\hfill \\ \hfill & =\chi \left(\mathfrak{J}({\iota }_{\mathfrak{p}},{\iota }_{\mathfrak{p}},{\iota }_{\mathfrak{p}+1},\phi )\right)\ge \chi \left(\mathfrak{J}({\iota }_{\mathfrak{p}},{\iota }_{\mathfrak{p}},{\iota }_{\mathfrak{p}+1},\phi )\right)+\delta >\chi \left(\mathfrak{J}({\iota }_{\mathfrak{p}},{\iota }_{\mathfrak{p}},{\iota }_{\mathfrak{p}+1},\phi )\right),\hfill \\ \hfill & \chi \left(\mathfrak{H}(\mathsf{\Gamma }{\iota }_{\mathfrak{p}-1},\mathsf{\Gamma }{\iota }_{\mathfrak{p}-1},\mathsf{\Gamma }{\iota }_{\mathfrak{p}},\phi )\right)\hfill \\ \hfill & =\chi \left(\mathfrak{H}({\iota }_{\mathfrak{p}},{\iota }_{\mathfrak{p}},{\iota }_{\mathfrak{p}+1},\phi )\right)\le \chi \left(\mathfrak{H}({\iota }_{\mathfrak{p}},{\iota }_{\mathfrak{p}},{\iota }_{\mathfrak{p}+1},\phi )\right)+\delta <\chi \left(\mathfrak{H}({\iota }_{\mathfrak{p}},{\iota }_{\mathfrak{p}},{\iota }_{\mathfrak{p}+1},\phi )\right),\hfill \\ \hfill & \chi \left(\mathfrak{E}(\mathsf{\Gamma }{\iota }_{\mathfrak{p}-1},\mathsf{\Gamma }{\iota }_{\mathfrak{p}-1},\mathsf{\Gamma }{\iota }_{\mathfrak{p}},\phi )\right)\hfill \\ \hfill & =\chi \left(\mathfrak{E}({\iota }_{\mathfrak{p}},{\iota }_{\mathfrak{p}},{\iota }_{\mathfrak{p}+1},\phi )\right)\le \chi \left(\mathfrak{E}({\iota }_{\mathfrak{p}},{\iota }_{\mathfrak{p}},{\iota }_{\mathfrak{p}+1},\phi )\right)+\delta <\chi \left(\mathfrak{E}({\iota }_{\mathfrak{p}},{\iota }_{\mathfrak{p}},{\iota }_{\mathfrak{p}+1},\phi )\right),\hfill \end{array}$$

This leads to a contradiction, and, thus,

$$\begin{array}{cc}\hfill min\{\mathfrak{J}({\iota }_{\mathfrak{p}-1},{\iota }_{\mathfrak{p}-1},{\iota }_{\mathfrak{p}},\phi ),\mathfrak{J}({\iota }_{\mathfrak{p}},{\iota }_{\mathfrak{p}},{\iota }_{\mathfrak{p}+1},\phi )\}& =\mathfrak{J}({\iota }_{\mathfrak{p}-1},{\iota }_{\mathfrak{p}-1},{\iota }_{\mathfrak{p}},\phi ),\hfill \\ \hfill max\{\mathfrak{H}({\iota }_{\mathfrak{p}-1},{\iota }_{\mathfrak{p}-1},{\iota }_{\mathfrak{p}},\phi ),\mathfrak{H}({\iota }_{\mathfrak{p}},{\iota }_{\mathfrak{p}},{\iota }_{\mathfrak{p}+1},\phi )\}& =\mathfrak{H}({\iota }_{\mathfrak{p}-1},{\iota }_{\mathfrak{p}-1},{\iota }_{\mathfrak{p}},\phi ),\hfill \\ \hfill max\{\mathfrak{E}({\iota }_{\mathfrak{p}-1},{\iota }_{\mathfrak{p}-1},{\iota }_{\mathfrak{p}},\phi ),\mathfrak{E}({\iota }_{\mathfrak{p}},{\iota }_{\mathfrak{p}},{\iota }_{\mathfrak{p}+1},\phi )\}& =\mathfrak{E}({\iota }_{\mathfrak{p}-1},{\iota }_{\mathfrak{p}-1},{\iota }_{\mathfrak{p}},\phi ),\hfill \end{array}$$

(17)

for every $\mathfrak{p}\in \mathbb{N}$. Hence, based on (15), (17), and the properties of $\chi$, we conclude that

$$\begin{array}{cc}\hfill \chi \left(\mathfrak{J}({\iota }_{\mathfrak{p}},{\iota }_{\mathfrak{p}},{\iota }_{\mathfrak{p}+1},\phi )\right)& \ge \chi \left(\mathfrak{J}({\iota }_{\mathfrak{p}-1},{\iota }_{\mathfrak{p}-1},{\iota }_{\mathfrak{p}},\phi )\right)+\delta ,\hfill \\ \hfill \chi \left(\mathfrak{H}({\iota }_{\mathfrak{p}},{\iota }_{\mathfrak{p}},{\iota }_{\mathfrak{p}+1},\phi )\right)& \le \chi \left(\mathfrak{H}({\iota }_{\mathfrak{p}-1},{\iota }_{\mathfrak{p}-1},{\iota }_{\mathfrak{p}},\phi )\right)-\delta ,\hfill \\ \hfill \chi \left(\mathfrak{E}({\iota }_{\mathfrak{p}},{\iota }_{\mathfrak{p}},{\iota }_{\mathfrak{p}+1},\phi )\right)& \le \chi \left(\mathfrak{E}({\iota }_{\mathfrak{p}-1},{\iota }_{\mathfrak{p}-1},{\iota }_{\mathfrak{p}},\phi )\right)-\delta ,\hfill \end{array}$$

for all $\mathfrak{p}\in \mathbb{N}$. This means that

$$\begin{array}{cc}\hfill \chi \left(\mathfrak{J}({\iota }_{\mathfrak{p}},{\iota }_{\mathfrak{p}},{\iota }_{\mathfrak{p}+1},\phi )\right)& \ge \chi \left(\mathfrak{J}({\iota }_0,{\iota }_0,{\iota }_1,\phi )\right)+\mathfrak{p}\delta ,\hfill \\ \hfill \chi \left(\mathfrak{H}({\iota }_{\mathfrak{p}},{\iota }_{\mathfrak{p}},{\iota }_{\mathfrak{p}+1},\phi )\right)& \le \chi \left(\mathfrak{H}({\iota }_0,{\iota }_0,{\iota }_1,\phi )\right)-\mathfrak{p}\delta ,\hfill \\ \hfill \chi \left(\mathfrak{E}({\iota }_{\mathfrak{p}},{\iota }_{\mathfrak{p}},{\iota }_{\mathfrak{p}+1},\phi )\right)& \le \chi \left(\mathfrak{E}({\iota }_0,{\iota }_0,{\iota }_1,\phi )\right)-\mathfrak{p}\delta .\hfill \end{array}$$

By taking $\mathfrak{p}\to \infty$, we obtain ${\displaystyle \underset{\mathfrak{p}\to \infty }{lim}\chi \left(\mathfrak{J}({\iota }_{\mathfrak{p}},{\iota }_{\mathfrak{p}},{\iota }_{\mathfrak{p}+1},\phi )\right)=\infty }$, ${\displaystyle \underset{\mathfrak{p}\to \infty }{lim}\chi \left(\mathfrak{H}({\iota }_{\mathfrak{p}},{\iota }_{\mathfrak{p}},{\iota }_{\mathfrak{p}+1},\phi )\right)=-\infty }$, ${\displaystyle \underset{\mathfrak{p}\to \infty }{lim}\chi \left(\mathfrak{E}({\iota }_{\mathfrak{p}},{\iota }_{\mathfrak{p}},{\iota }_{\mathfrak{p}+1},\phi )\right)=-\infty }$. Then, we have ${\displaystyle \underset{\mathfrak{p}\to \infty }{lim}\mathfrak{J}({\iota }_{\mathfrak{p}},{\iota }_{\mathfrak{p}},{\iota }_{\mathfrak{p}+1},\phi )=1}$, ${\displaystyle \underset{\mathfrak{p}\to \infty }{lim}\mathfrak{H}({\iota }_{\mathfrak{p}},{\iota }_{\mathfrak{p}},{\iota }_{\mathfrak{p}+1},\phi )=0}$, and ${\displaystyle \underset{\mathfrak{p}\to \infty }{lim}\mathfrak{E}({\iota }_{\mathfrak{p}},{\iota }_{\mathfrak{p}},{\iota }_{\mathfrak{p}+1},\phi )=0}$.

Using similar reasoning as in the proof of Theorem (1), we conclude that the sequence $\left\{{\iota }_{\mathfrak{p}}\right\}$ is Cauchy. Given that $(\mathfrak{A},\mathfrak{J},\mathfrak{H},\mathfrak{E},\otimes ,\oplus )$ is complete, there exists some $\iota$ such that ${\displaystyle \underset{\mathfrak{p}\to \infty }{lim}{\iota }_{\mathfrak{p}}=\iota }$. Next, we demonstrate that $\iota$ is a fixed point of $\mathsf{\Gamma }$. Since $\chi$ is continuous, two possible cases arise.

Case I. For every $\mathfrak{p}\in \mathbb{N}$, there exists an index ${\mathfrak{i}}_{\mathfrak{p}}\ge \mathfrak{p}$ such that ${\iota }_{{\mathfrak{i}}_{\mathfrak{p}}+1}=\mathsf{\Gamma }\iota$ and ${\mathfrak{i}}_{\mathfrak{p}}>{\mathfrak{i}}_{\mathfrak{p}-1}$, with ${\mathfrak{i}}_0=1$. Consequently, we have

$$\iota =\underset{\mathfrak{p}\to \infty }{lim}{\iota }_{{\mathfrak{i}}_{\mathfrak{p}}+1}=\underset{\mathfrak{p}\to \infty }{lim}\mathsf{\Gamma }\iota =\mathsf{\Gamma }\iota .$$

This confirms that $\iota$ is indeed a fixed point of $\mathsf{\Gamma }$.

Case II. There is a natural number ${\mathfrak{p}}_0$ such that for all $\mathfrak{p}\ge {\mathfrak{p}}_0$, ${\iota }_{\mathfrak{p}+1}\ne \mathsf{\Gamma }\iota$. That is, $\mathsf{\Gamma }{\iota }_{\mathfrak{p}}={\iota }_{\mathfrak{p}+1}\ne \mathsf{\Gamma }\iota$, and so $\mathfrak{J}(\mathsf{\Gamma }{\iota }_{\mathfrak{p}},\mathsf{\Gamma }{\iota }_{\mathfrak{p}},\mathsf{\Gamma }\iota ,\phi )<1,\mathfrak{H}(\mathsf{\Gamma }{\iota }_{\mathfrak{p}},\mathsf{\Gamma }{\iota }_{\mathfrak{p}},\mathsf{\Gamma }\iota ,\phi )>0,\mathfrak{E}(\mathsf{\Gamma }{\iota }_{\mathfrak{p}},\mathsf{\Gamma }{\iota }_{\mathfrak{p}},\mathsf{\Gamma }\iota ,\phi )>0$ for all $\mathfrak{p}\ge {\mathfrak{p}}_0$. It follows from (14) that

$$\begin{array}{cc}\hfill \chi \left(\mathfrak{J}({\iota }_{\mathfrak{p}+1},{\iota }_{\mathfrak{p}+1},\mathsf{\Gamma }\iota ,\phi )\right)& =\chi \left(\mathfrak{J}(\mathsf{\Gamma }{\iota }_{\mathfrak{p}},\mathsf{\Gamma }{\iota }_{\mathfrak{p}},\mathsf{\Gamma }\iota ,\phi )\right)\hfill \\ \hfill & \ge \chi (min\{\mathfrak{J}({\iota }_{\mathfrak{p}},{\iota }_{\mathfrak{p}},\iota ,\phi ),\mathfrak{J}({\iota }_{\mathfrak{p}},{\iota }_{\mathfrak{p}},\mathsf{\Gamma }{\iota }_{\mathfrak{p}},\phi ),\mathfrak{J}(\iota ,\iota ,\mathsf{\Gamma }\iota ,\phi )\})+\delta ,\hfill \\ \hfill & =\chi (min\{\mathfrak{J}({\iota }_{\mathfrak{p}},{\iota }_{\mathfrak{p}},\iota ,\phi ),\mathfrak{J}({\iota }_{\mathfrak{p}},{\iota }_{\mathfrak{p}},{\iota }_{\mathfrak{p}+1},\phi ),\mathfrak{J}(\iota ,\iota ,\mathsf{\Gamma }\iota ,\phi )\})+\delta .\hfill \\ \hfill \chi \left(\mathfrak{H}({\iota }_{\mathfrak{p}-1},{\iota }_{\mathfrak{p}-1},\mathsf{\Gamma }\iota ,\phi )\right)& =\chi \left(\mathfrak{H}(\mathsf{\Gamma }{\iota }_{\mathfrak{p}},\mathsf{\Gamma }{\iota }_{\mathfrak{p}},\mathsf{\Gamma }\iota ,\phi )\right)\hfill \\ \hfill & \le \chi (max\{\mathfrak{H}({\iota }_{\mathfrak{p}},{\iota }_{\mathfrak{p}},\iota ,\phi ),\mathfrak{H}({\iota }_{\mathfrak{p}},{\iota }_{\mathfrak{p}},\mathsf{\Gamma }{\iota }_{\mathfrak{p}},\phi ),\mathfrak{H}(\iota ,\iota ,\mathsf{\Gamma }\iota ,\phi )\})-\delta ,\hfill \\ \hfill & =\chi (max\{\mathfrak{H}({\iota }_{\mathfrak{p}},{\iota }_{\mathfrak{p}},\iota ,\phi ),\mathfrak{H}({\iota }_{\mathfrak{p}},{\iota }_{\mathfrak{p}},{\iota }_{\mathfrak{p}-1},\phi ),\mathfrak{H}(\iota ,\iota ,\mathsf{\Gamma }\iota ,\phi )\})-\delta .\hfill \\ \hfill \chi \left(\mathfrak{E}({\iota }_{\mathfrak{p}-1},{\iota }_{\mathfrak{p}-1},\mathsf{\Gamma }\iota ,\phi )\right)& =\chi \left(\mathfrak{E}(\mathsf{\Gamma }{\iota }_{\mathfrak{p}},\mathsf{\Gamma }{\iota }_{\mathfrak{p}},\mathsf{\Gamma }\iota ,\phi )\right)\hfill \\ \hfill & \le \chi (max\{\mathfrak{E}({\iota }_{\mathfrak{p}},{\iota }_{\mathfrak{p}},\iota ,\phi ),\mathfrak{E}({\iota }_{\mathfrak{p}},{\iota }_{\mathfrak{p}},\mathsf{\Gamma }{\iota }_{\mathfrak{p}},\phi ),\mathfrak{E}(\iota ,\iota ,\mathsf{\Gamma }\iota ,\phi )\})-\delta ,\hfill \\ \hfill & =\chi (max\{\mathfrak{E}({\iota }_{\mathfrak{p}},{\iota }_{\mathfrak{p}},\iota ,\phi ),\mathfrak{E}({\iota }_{\mathfrak{p}},{\iota }_{\mathfrak{p}},{\iota }_{\mathfrak{p}-1},\phi ),\mathfrak{E}(\iota ,\iota ,\mathsf{\Gamma }\iota ,\phi )\})-\delta .\hfill \end{array}$$

(18)

Since ${\displaystyle \underset{\mathfrak{p}\to \infty }{lim}\mathfrak{J}({\iota }_{\mathfrak{p}},{\iota }_{\mathfrak{p}},\iota ,\phi )=1}$, ${\displaystyle \underset{\mathfrak{p}\to \infty }{lim}\mathfrak{H}({\iota }_{\mathfrak{p}},{\iota }_{\mathfrak{p}},\iota ,\phi )=0}$, and ${\displaystyle \underset{\mathfrak{p}\to \infty }{lim}\mathfrak{E}({\iota }_{\mathfrak{p}},{\iota }_{\mathfrak{p}},\iota ,\phi )=0}$, if $\mathfrak{J}(\iota ,\iota ,\mathsf{\Gamma }\iota ,\phi )<1$, $\mathfrak{H}(\iota ,\iota ,\mathsf{\Gamma }\iota ,\phi )>0$ and $\mathfrak{E}(\iota ,\iota ,\mathsf{\Gamma }\iota ,\phi )>0$, then there is a natural number ${\mathfrak{p}}_1$ such that, for all $\mathfrak{p}\ge {\mathfrak{p}}_1$, it follows that

$$\begin{array}{cc}\hfill min\{\mathfrak{J}({\iota }_{\mathfrak{p}},{\iota }_{\mathfrak{p}},\iota ,\phi ),\mathfrak{J}({\iota }_{\mathfrak{p}},{\iota }_{\mathfrak{p}},{\iota }_{\mathfrak{p}+1},\phi ),\mathfrak{J}(\iota ,\iota ,\mathsf{\Gamma }\iota ,\phi )\}& =\mathfrak{J}(\iota ,\iota ,\mathsf{\Gamma }\iota ,\phi ),\hfill \\ \hfill max\{\mathfrak{H}({\iota }_{\mathfrak{p}},{\iota }_{\mathfrak{p}},\iota ,\phi ),\mathfrak{H}({\iota }_{\mathfrak{p}},{\iota }_{\mathfrak{p}},{\iota }_{\mathfrak{p}+1},\phi ),\mathfrak{H}(\iota ,\iota ,\mathsf{\Gamma }\iota ,\phi )\}& =\mathfrak{H}(\iota ,\iota ,\mathsf{\Gamma }\iota ,\phi ),\hfill \\ \hfill max\{\mathfrak{E}({\iota }_{\mathfrak{p}},{\iota }_{\mathfrak{p}},\iota ,\phi ),\mathfrak{E}({\iota }_{\mathfrak{p}},{\iota }_{\mathfrak{p}},{\iota }_{\mathfrak{p}+1},\phi ),\mathfrak{E}(\iota ,\iota ,\mathsf{\Gamma }\iota ,\phi )\}& =\mathfrak{E}(\iota ,\iota ,\mathsf{\Gamma }\iota ,\phi ).\hfill \end{array}$$

From (18), we have

$$\begin{array}{cc}\hfill \chi \left(\mathfrak{J}({\iota }_{\mathfrak{p}+1},{\iota }_{\mathfrak{p}+1},\mathsf{\Gamma }\iota ,\phi )\right)& \ge \chi \left(\mathfrak{J}\right(\iota ,\iota ,\mathsf{\Gamma }\iota ,\phi \left)\right)+\delta ,\hfill \\ \hfill \chi \left(\mathfrak{H}({\iota }_{\mathfrak{p}+1},{\iota }_{\mathfrak{p}+1},\mathsf{\Gamma }\iota ,\phi )\right)& \le \chi \left(\mathfrak{H}\right(\iota ,\iota ,\mathsf{\Gamma }\iota ,\phi \left)\right)-\delta ,\hfill \\ \hfill \chi \left(\mathfrak{E}({\iota }_{\mathfrak{p}+1},{\iota }_{\mathfrak{p}+1},\mathsf{\Gamma }\iota ,\phi )\right)& \le \chi \left(\mathfrak{E}\right(\iota ,\iota ,\mathsf{\Gamma }\iota ,\phi \left)\right)-\delta ,\hfill \end{array}$$

for all $\mathfrak{p}\ge max\{{\mathfrak{p}}_0,{\mathfrak{p}}_1\}$. Given that $\chi$ is continuous, taking the limit as $\mathfrak{p}\to \infty$ yields

$$\begin{array}{c}\hfill \chi \left(\mathfrak{J}\right(\iota ,\iota ,\mathsf{\Gamma }\iota ,\phi \left)\right)\ge \chi \left(\mathfrak{J}\right(\iota ,\iota ,\mathsf{\Gamma }\iota ,\phi \left)\right)+\delta ,\\ \hfill \chi \left(\mathfrak{H}\right(\iota ,\iota ,\mathsf{\Gamma }\iota ,\phi \left)\right)\le \chi \left(\mathfrak{H}\right(\iota ,\iota ,\mathsf{\Gamma }\iota ,\phi \left)\right)-\delta ,\\ \hfill \chi \left(\mathfrak{E}\right(\iota ,\iota ,\mathsf{\Gamma }\iota ,\phi \left)\right)\le \chi \left(\mathfrak{E}\right(\iota ,\iota ,\mathsf{\Gamma }\iota ,\phi \left)\right)-\delta ,\end{array}$$

which is a contradiction.

Therefore, $\mathfrak{J}(\iota ,\iota ,\mathsf{\Gamma }\iota ,\phi )=1$, $\mathfrak{H}(\iota ,\iota ,\mathsf{\Gamma }\iota ,\phi )=0$, and $\mathfrak{E}(\iota ,\iota ,\mathsf{\Gamma }\iota ,\phi )=0$; thus, $\iota$ is a fixed point of $\mathsf{\Gamma }$. Next, we show that this fixed point is unique. Assume that ${\iota }_1$ and ${\iota }_2$ are two distinct fixed points of $\mathsf{\Gamma }$. If ${\iota }_1\ne {\iota }_2$, then it follows that $\mathsf{\Gamma }{\iota }_1\ne \mathsf{\Gamma }{\iota }_2$. According to (14), we obtain

$$\begin{array}{cc}\hfill \chi \left(\mathfrak{J}({\iota }_1,{\iota }_1,{\iota }_2,\phi )\right)& =\chi \left(\mathfrak{J}(\mathsf{\Gamma }{\iota }_1,\mathsf{\Gamma }{\iota }_1,\mathsf{\Gamma }{\iota }_2,\phi )\right)\hfill \\ \hfill & \ge \chi (min\{\mathfrak{J}({\iota }_1,{\iota }_1,{\iota }_2,\phi ),\mathfrak{J}({\iota }_1,{\iota }_1,\mathsf{\Gamma }{\iota }_1,\phi ),\mathfrak{J}({\iota }^2,{\iota }_2,\mathsf{\Gamma }{\iota }_2,\phi )\})+\delta \hfill \\ \hfill & =\chi \left(\mathfrak{J}({\iota }_1,{\iota }_1,{\iota }_2,\phi )\right)+\delta >\chi (\mathfrak{J}({\iota }_1,{\iota }_1,{\iota }_2,\phi ),\hfill \\ \hfill \chi \left(\mathfrak{H}({\iota }_1,{\iota }_1,{\iota }_2,\phi )\right)& =\chi \left(\mathfrak{H}(\mathsf{\Gamma }{\iota }_1,\mathsf{\Gamma }{\iota }_1,\mathsf{\Gamma }{\iota }_2,\phi )\right)\hfill \\ \hfill & \le \chi (min\{\mathfrak{H}({\iota }_1,{\iota }_1,{\iota }_2,\phi ),\mathfrak{H}({\iota }_1,{\iota }_1,\mathsf{\Gamma }{\iota }_1,\phi ),\mathfrak{H}({\iota }^2,{\iota }_2,\mathsf{\Gamma }{\iota }_2,\phi )\})-\delta \hfill \\ \hfill & =\chi \left(\mathfrak{H}({\iota }_1,{\iota }_1,{\iota }_2,\phi )\right)-\delta <\chi (\mathfrak{H}({\iota }_1,{\iota }_1,{\iota }_2,\phi ),\hfill \\ \hfill \chi \left(\mathfrak{E}({\iota }_1,{\iota }_1,{\iota }_2,\phi )\right)& =\chi \left(\mathfrak{E}(\mathsf{\Gamma }{\iota }_1,\mathsf{\Gamma }{\iota }_1,\mathsf{\Gamma }{\iota }_2,\phi )\right)\hfill \\ \hfill & \le \chi (min\{\mathfrak{E}({\iota }_1,{\iota }_1,{\iota }_2,\phi ),\mathfrak{E}({\iota }_1,{\iota }_1,\mathsf{\Gamma }{\iota }_1,\phi ),\mathfrak{E}({\iota }^2,{\iota }_2,\mathsf{\Gamma }{\iota }_2,\phi )\})-\delta \hfill \\ \hfill & =\chi \left(\mathfrak{E}({\iota }_1,{\iota }_1,{\iota }_2,\phi )\right)-\delta <\chi (\mathfrak{E}({\iota }_1,{\iota }_1,{\iota }_2,\phi ),\hfill \end{array}$$

which is a contradiction.

Then, $\mathfrak{J}({\iota }_1,{\iota }_1,{\iota }_2,\phi )=1$, $\mathfrak{H}({\iota }_1,{\iota }_1,{\iota }_2,\phi )=0$, and $\mathfrak{E}({\iota }_1,{\iota }_1,{\iota }_2,\phi )=0$; that is, ${\iota }_1={\iota }_2$. Thus, the fixed point of $\mathsf{\Gamma }$ is uniquely determined. □

Example 5.

Let $(\mathfrak{A},\mathfrak{J},\mathfrak{H},\mathfrak{E},\otimes ,\oplus )$ be a non-Archimedean $\mathfrak{GNMS}$, and let Γ be as considered in Example (4). Let $\chi :[0,1)\to \mathbb{R}$ be defined by

$\chi \left(\beta \right)={\displaystyle \frac{1}{1-{\beta }^2}}$ for all $\beta \in [0,1)$. Then, Γ is a χ-weak contraction.

Therefore, Theorem (2) is applicable to Γ, and the unique fixed point of Γ is $1/10$.

Since $\mathsf{\Gamma }\left({\displaystyle \frac{1}{10}}\right)={\displaystyle \frac{1}{10}}$, this confirms that ${\displaystyle \frac{1}{10}}$ is indeed the unique fixed point of Γ.

4. ($\varphi -\psi$) Contractions on Non-Archimedean GNMS

In this section, we examine a broader class of auxiliary functions that generate various contractive conditions. We establish that the function $t\to 1/t-1$, commonly found in fixed-point theorems in the neutrosophic context, can be substituted with more suitable and general functions.

Definition 8

([11]). Consider Φ as the set of all functions $\varphi :[0,\infty )\to [0,\infty )$ that fulfill the following criteria:

(1) 

$\varphi \left(\iota \right)=0$ if and only if $\iota =0$;

(2) 

${\displaystyle \underset{t\to \infty }{lim}\varphi \left(t\right)=\infty }$;

(3) 

ϕ is continuous at $\iota =0$.

Definition 9

([11]). Define Ψ as the set of functions $\psi :[0,\infty )\to [0,\infty )$ that meet the following criteria:

(1) 

ψ is non-decreasing;

(2) 

$\psi \left(0\right)=0$;

(3) 

For any sequence $\left\{{\rho }_{\mathfrak{p}}\right\}\subset [0,\infty )$ converging to 0, the sequence $\left\{{\psi }^{\mathfrak{p}}\left({\rho }_{\mathfrak{p}}\right)\right\}$ also converges to 0, where ${\psi }^{\mathfrak{p}}$ denotes the n-th iterate of ψ.

Additionally, the functions in Ψ are continuous at $\iota =0$.

Definition 10.

Let $\mathfrak{K}$ denote the set of functions $\eta :(0,1]\to [0,\infty )$ that satisfy the following conditions:

$({\mathfrak{K}}_1$)

For any sequence $\left\{{\rho }_{\mathfrak{p}}\right\}\subset (0,1]$, $\left\{{\rho }_{\mathfrak{p}}\right\}\to 1$ if and only if $\left\{\eta \left({\rho }_{\mathfrak{p}}\right)\right\}\to 0$;

$({\mathfrak{K}}_2$)

For any sequence $\left\{{\rho }_{\mathfrak{p}}\right\}\subset (0,1]$, $\left\{{\rho }_{\mathfrak{p}}\right\}\to 0$ if and only if $\left\{\eta \left({\rho }_{\mathfrak{p}}\right)\right\}\to \infty$.

These conditions are satisfied, for example, by a strictly decreasing bijective function η mapping $(0,1]$ to $[0,\infty )$ such that both η and its inverse ${\eta }^{-1}$ are continuous (at least at the endpoints of their corresponding domains). A specific example is the function $\eta \left(\iota \right)={\displaystyle \frac{1}{\iota }}-1$ for $\iota \in (0,1]$. However, the functions in $\mathfrak{K}$ do not necessarily need to be continuous or monotone.

Theorem 3.

Consider $(\mathfrak{A},\mathfrak{J},\mathfrak{H},\mathfrak{E},\otimes ,\oplus )$ as a complete non-Archimedean GNMS, and consider $\mathsf{\Gamma }:\mathfrak{A}\to \mathfrak{A}$ as a mapping. Assume that there exist $\mu \in (0,1)$, $\varphi \in \mathsf{\Phi }$, $\psi \in \mathsf{\Psi }$, and $\eta \in \mathfrak{K}$ such that

$$\begin{array}{c}\hfill \eta \left(\mathfrak{J}\right(\mathsf{\Gamma }\iota ,\mathsf{\Gamma }\vartheta ,\mathsf{\Gamma }\xi ,\varphi \left(\mu \phi \right)\left)\right)\le \psi \left(\eta \right(\mathfrak{J}(\iota ,\vartheta ,\xi ,\varphi (\phi \left)\right)),\\ \hfill \eta \left(\mathfrak{H}\right(\mathsf{\Gamma }\iota ,\mathsf{\Gamma }\vartheta ,\mathsf{\Gamma }\xi ,\varphi \left(\mu \phi \right)\left)\right)\ge \psi \left(\eta \right(\mathfrak{H}(\iota ,\vartheta ,\xi ,\varphi (\phi \left)\right)),\\ \hfill \eta \left(\mathfrak{H}\right(\mathsf{\Gamma }\iota ,\mathsf{\Gamma }\vartheta ,\mathsf{\Gamma }\xi ,\varphi \left(\mu \phi \right)\left)\right)\ge \psi \left(\eta \right(\mathfrak{H}(\iota ,\vartheta ,\xi ,\varphi (\phi \left)\right))\end{array}$$

(19)

for all $\iota ,\vartheta ,\xi \in \mathfrak{A}$ and all $\phi >0$ for which $\mathfrak{J}(\iota ,\vartheta ,\xi ,\varphi (\phi \left)\right)>0,\mathfrak{H}(\iota ,\vartheta ,\xi ,\varphi (\phi \left)\right)<1$, and $\mathfrak{E}(\iota ,\vartheta ,\xi ,\varphi (\phi \left)\right)<1$. If there exists ${\iota }_0\in \mathfrak{A}$ such that ${\displaystyle \underset{\phi \to \infty }{lim}\mathfrak{J}({\iota }_0,{\iota }_0,\mathsf{\Gamma }{\iota }_0,\phi )=1,\underset{\phi \to \infty }{lim}\mathfrak{H}({\iota }_0,{\iota }_0,\mathsf{\Gamma }{\iota }_0,\phi )=0,}$${lim}_{\phi \to \infty }\mathfrak{E}({\iota }_0,{\iota }_0,\mathsf{\Gamma }{\iota }_0,\phi )=0$, then Γ has at least one fixed point. Furthermore, assume that, for any $\iota ,\vartheta ,\xi \in \mathrm{Fix}\left(\mathsf{\Gamma }\right)$ with $\iota \ne \vartheta \ne \xi$, we have ${\displaystyle \underset{t\to \infty }{lim}\mathfrak{J}(\iota ,\vartheta ,\xi ,\phi )=1,\underset{t\to \infty }{lim}\mathfrak{J}(\iota ,\vartheta ,\xi ,\phi )=0,\underset{t\to \infty }{lim}\mathfrak{J}(\iota ,\vartheta ,\xi ,\phi )=0}$. Then, Γ admits a unique fixed point.

Proof. 

Observe that the condition (19) indicates that if $\mathfrak{J}(\iota ,\vartheta ,\xi ,\varphi (\phi \left)\right)>0,\mathfrak{H}(\iota ,\vartheta ,\xi ,\varphi (\phi \left)\right)<1,\mathfrak{E}(\iota ,\vartheta ,\xi ,\varphi (\phi \left)\right)<1$, then $\eta$ must be applicable to $\mathfrak{J}(\mathsf{\Gamma }\iota ,\mathsf{\Gamma }\vartheta ,\mathsf{\Gamma }\xi ,\varphi (\mu \phi \left)\right),\mathfrak{H}(\mathsf{\Gamma }\iota ,\mathsf{\Gamma }\vartheta ,\mathsf{\Gamma }\xi ,\varphi (\mu \phi \left)\right),$$\mathfrak{E}(\mathsf{\Gamma }\iota ,\mathsf{\Gamma }\vartheta ,\mathsf{\Gamma }\xi ,\varphi (\mu \phi \left)\right)$. Hence, $\mathfrak{J}(\mathsf{\Gamma }\iota ,\mathsf{\Gamma }\vartheta ,\mathsf{\Gamma }\xi ,\varphi (\mu \phi \left)\right),\mathfrak{J}(\mathsf{\Gamma }\iota ,\mathsf{\Gamma }\vartheta ,\mathsf{\Gamma }\xi ,\varphi (\mu \phi \left)\right),\mathfrak{E}(\mathsf{\Gamma }\iota ,\mathsf{\Gamma }\vartheta ,\mathsf{\Gamma }\xi ,\varphi (\mu \phi \left)\right)\in dom$$\eta =(0,1]$, which signifies that

$$\begin{array}{c}\hfill \mathfrak{J}(\iota ,\vartheta ,\xi ,\varphi (\phi \left)\right)>0\Rightarrow \mathfrak{J}(\mathsf{\Gamma }\iota ,\mathsf{\Gamma }\vartheta ,\mathsf{\Gamma }\xi ,\varphi (\mu \phi \left)\right)>0,\\ \hfill \mathfrak{H}(\iota ,\vartheta ,\xi ,\varphi (\phi \left)\right)<1\Rightarrow \mathfrak{H}(\mathsf{\Gamma }\iota ,\mathsf{\Gamma }\vartheta ,\mathsf{\Gamma }\xi ,\varphi (\mu \phi \left)\right)<1,\\ \hfill \mathfrak{E}(\iota ,\vartheta ,\xi ,\varphi (\phi \left)\right)<1\Rightarrow \mathfrak{E}(\mathsf{\Gamma }\iota ,\mathsf{\Gamma }\vartheta ,\mathsf{\Gamma }\xi ,\varphi (\mu \phi \left)\right)<1.\end{array}$$

(20)

By setting ${\iota }_1=\mathsf{\Gamma }{\iota }_0$, define sequence $\left\{{\iota }_{\mathfrak{p}}\right\}$ by ${\iota }_{\mathfrak{p}+1}=\mathsf{\Gamma }{\iota }_{\mathfrak{p}}$ for all $\mathfrak{p}\in \mathbb{N}$. If there is a natural number ${\mathfrak{p}}_0$ such that ${\iota }_{{\mathfrak{p}}_0}={\iota }_{{\mathfrak{p}}_0+1}$, then ${\iota }_{{\mathfrak{p}}_0}$ is a fixed point of $\mathsf{\Gamma }$, completing the proof of existence.

Alternatively, suppose that ${\iota }_{\mathfrak{p}}\ne {\iota }_{\mathfrak{p}+1}$ for all $\mathfrak{p}\in \mathbb{N}$. Given that ${\displaystyle \underset{\phi \to \infty }{lim}\mathfrak{J}({\iota }_0,{\iota }_0,\mathsf{\Gamma }{\iota }_0,\phi )=1,\underset{\phi \to \infty }{lim}\mathfrak{H}({\iota }_0,{\iota }_0,\mathsf{\Gamma }{\iota }_0,\phi )=0}$, and ${\displaystyle \underset{\phi \to \infty }{lim}\mathfrak{E}({\iota }_0,{\iota }_0,\mathsf{\Gamma }{\iota }_0,\phi )=0}$, there is a ${\phi }_0>0$ such that $\mathfrak{J}({\iota }_0,{\iota }_0,{\iota }_1,{\phi }_0)=\mathfrak{J}({\iota }_0,{\iota }_0,\mathsf{\Gamma }{\iota }_0,{\phi }_0)>0$, $\mathfrak{H}({\iota }_0,{\iota }_0,{\iota }_1,{\phi }_0)=\mathfrak{H}({\iota }_0,{\iota }_0,\mathsf{\Gamma }{\iota }_0,{\phi }_0)<1$, and $\mathfrak{E}({\iota }_0,{\iota }_0,{\iota }_1,{\phi }_0)=\mathfrak{E}({\iota }_0,{\iota }_0,\mathsf{\Gamma }{\iota }_0,{\phi }_0)<1$. Additionally, since ${\displaystyle \underset{\phi \to \infty }{lim}\varphi \left(\phi \right)=\infty }$, there exists a value ${\zeta }_0\in [0,\infty )$ (which we can assume to be at least ${\phi }_0$) such that $\varphi \left({\zeta }_0\right)\ge {\phi }_0$.

Hence, $\mathfrak{J}({\iota }_0,{\iota }_0,{\iota }_1,\varphi \left({\zeta }_0\right))\ge \mathfrak{J}({\iota }_0,{\iota }_0,{\iota }_1,{\phi }_0)>0$, $\mathfrak{H}({\iota }_0,{\iota }_0,{\iota }_1,\varphi \left({\zeta }_0\right))\le \mathfrak{H}({\iota }_0,{\iota }_0,{\iota }_1,{\phi }_0)<1,$ $\mathfrak{E}({\iota }_0,{\iota }_0,{\iota }_1,\varphi \left({\zeta }_0\right))\le \mathfrak{E}({\iota }_0,{\iota }_0,{\iota }_1,{\phi }_0)<1$.

From (20), it follows that $\mathfrak{J}({\iota }_1,{\iota }_1,{\iota }_2,\varphi \left(\mu {\zeta }_0\right))=\mathfrak{J}(\mathsf{\Gamma }{\iota }_0,\mathsf{\Gamma }{\iota }_0,\mathsf{\Gamma }{\iota }_1,{\phi }_0)>0$. Using mathematical induction, it can be shown that, for $\mathfrak{J}({\iota }_{\mathfrak{p}},{\iota }_{\mathfrak{p}},{\iota }_{\mathfrak{p}+1},\varphi \left({\mu }^{\mathfrak{p}}{\zeta }_0\right))>0,\mathfrak{H}({\iota }_{\mathfrak{p}},{\iota }_{\mathfrak{p}},{\iota }_{\mathfrak{p}+1},\varphi \left({\mu }^{\mathfrak{p}}{\zeta }_0\right))<1,\mathfrak{E}({\iota }_{\mathfrak{p}},{\iota }_{\mathfrak{p}},{\iota }_{\mathfrak{p}+1},\varphi \left({\mu }^{\mathfrak{p}}{\zeta }_0\right))<1$ for all $\mathfrak{p}\in \mathbb{N}$. If $\mathfrak{p},\mathfrak{q},\mathfrak{r}\in \mathbb{N}$ and $\mathfrak{r}\le \mathfrak{p}$, then ${\mu }^{\mathfrak{p}}{\zeta }_0\le {\mu }^{\mathfrak{r}}{\zeta }_0\le {\zeta }_0\le {\zeta }_0/{\mu }^{\mathfrak{q}}$. Given that $\varphi$ and $\mathfrak{J}({\iota }_{\mathfrak{p}},{\iota }_{\mathfrak{p}},{\iota }_{\mathfrak{p}+1},·)$ are non-decreasing functions, while $\mathfrak{H}({\iota }_{\mathfrak{p}},{\iota }_{\mathfrak{p}},{\iota }_{\mathfrak{p}+1},·)$ and $\mathfrak{E}({\iota }_{\mathfrak{p}},{\iota }_{\mathfrak{p}},{\iota }_{\mathfrak{p}+1},·)$ are non-increasing functions, it can be concluded that, if $\mathfrak{p},\mathfrak{q},\mathfrak{r}\in \mathbb{N}$ and $\mathfrak{r}\le \mathfrak{p}$, then

$$\begin{array}{cc}\hfill 0& <\mathfrak{J}({\iota }_{\mathfrak{p}},{\iota }_{\mathfrak{p}},{\iota }_{\mathfrak{p}+1},\varphi \left({\mu }^{\mathfrak{p}}{\zeta }_0\right))\le \mathfrak{J}({\iota }_{\mathfrak{p}},{\iota }_{\mathfrak{p}},{\iota }_{\mathfrak{p}+1},\varphi \left({\mu }^{\mathfrak{r}}{\zeta }_0\right)),\hfill \\ \hfill & \le \mathfrak{J}({\iota }_{\mathfrak{p}},{\iota }_{\mathfrak{p}},{\iota }_{\mathfrak{p}+1},\varphi \left({\zeta }_0\right))\le \mathfrak{J}\left({\iota }_{\mathfrak{p}},{\iota }_{\mathfrak{p}},{\iota }_{\mathfrak{p}+1},\varphi \left({\displaystyle \frac{{\zeta }_0}{{\mu }^{\mathfrak{q}}}}\right)\right).\hfill \\ \hfill 1& >\mathfrak{H}({\iota }_{\mathfrak{p}},{\iota }_{\mathfrak{p}},{\iota }_{\mathfrak{p}+1},\varphi \left({\mu }^{\mathfrak{p}}{\zeta }_0\right))\ge \mathfrak{H}({\iota }_{\mathfrak{p}},{\iota }_{\mathfrak{p}},{\iota }_{\mathfrak{p}+1},\varphi \left({\mu }^{\mathfrak{r}}{\zeta }_0\right)),\hfill \\ \hfill & \ge \mathfrak{H}({\iota }_{\mathfrak{p}},{\iota }_{\mathfrak{p}},{\iota }_{\mathfrak{p}+1},\varphi \left({\zeta }_0\right))\ge \mathfrak{H}\left({\iota }_{\mathfrak{p}},{\iota }_{\mathfrak{p}},{\iota }_{\mathfrak{p}+1},\varphi \left({\displaystyle \frac{{\zeta }_0}{{\mu }^{\mathfrak{q}}}}\right)\right).\hfill \\ \hfill 1& >\mathfrak{E}({\iota }_{\mathfrak{p}},{\iota }_{\mathfrak{p}},{\iota }_{\mathfrak{p}+1},\varphi \left({\mu }^{\mathfrak{p}}{\zeta }_0\right))\ge \mathfrak{E}({\iota }_{\mathfrak{p}},{\iota }_{\mathfrak{p}},{\iota }_{\mathfrak{p}+1},\varphi \left({\mu }^{\mathfrak{r}}{\zeta }_0\right)),\hfill \\ \hfill & \ge \mathfrak{E}({\iota }_{\mathfrak{p}},{\iota }_{\mathfrak{p}},{\iota }_{\mathfrak{p}+1},\varphi \left({\zeta }_0\right))\ge \mathfrak{E}\left({\iota }_{\mathfrak{p}},{\iota }_{\mathfrak{p}},{\iota }_{\mathfrak{p}+1},\varphi \left({\displaystyle \frac{{\zeta }_0}{{\mu }^{\mathfrak{q}}}}\right)\right).\hfill \end{array}$$

(21)

We claim that

$$\begin{array}{c}\hfill \underset{\mathfrak{p}\to \infty }{lim}\mathfrak{J}({\iota }_{\mathfrak{p}},{\iota }_{\mathfrak{p}},{\iota }_{\mathfrak{p}+1},\zeta )=1,\underset{\mathfrak{p}\to \infty }{lim}\mathfrak{H}({\iota }_{\mathfrak{p}},{\iota }_{\mathfrak{p}},{\iota }_{\mathfrak{p}+1},\zeta )=0,\underset{\mathfrak{p}\to \infty }{lim}\mathfrak{E}({\iota }_{\mathfrak{p}},{\iota }_{\mathfrak{p}},{\iota }_{\mathfrak{p}+1},\zeta )=0,\end{array}$$

(22)

For any $\zeta >0$, we need to demonstrate the following: Let $\zeta >0$ be arbitrary. Since ${\displaystyle \underset{\mathfrak{r}\to \infty }{lim}\left({\mu }^{\mathfrak{r}}{\zeta }_0\right)=0}$ and $\varphi$ are continuous at $\phi =0$, it follows that ${\displaystyle \underset{\mathfrak{r}\to \infty }{lim}\varphi \left({\mu }^{\mathfrak{r}}{\zeta }_0\right)=\varphi \left(0\right)=0}$. Therefore, for $\zeta >0$, there exists a natural number $\mathfrak{r}\in \mathbb{N}$ such that $\varphi \left({\mu }^{\mathfrak{r}}{\zeta }_0\right)\le \zeta$. Consider $\mathfrak{p}\in \mathbb{N}$ such that $\mathfrak{p}>\mathfrak{r}$. Using the contractivity condition (19) with $\iota =\vartheta ={\iota }_{\mathfrak{p}}$ and $\xi ={\iota }_{\mathfrak{p}+1}$, we obtain

$$\begin{array}{cc}\hfill \eta \left(\mathfrak{J}({\iota }_{\mathfrak{p}},{\iota }_{\mathfrak{p}},{\iota }_{\mathfrak{p}+1},\varphi \left({\mu }^{\mathfrak{r}}{\zeta }_0\right))\right)& =\eta \left(\mathfrak{J}(\mathsf{\Gamma }{\iota }_{\mathfrak{p}-1},\mathsf{\Gamma }{\iota }_{\mathfrak{p}-1},\mathsf{\Gamma }{\iota }_{\mathfrak{p}},\varphi \left({\mu }^{\mathfrak{r}}{\zeta }_0\right))\right)\hfill \\ \hfill & \le \psi \left(\eta \left(\mathfrak{J}({\iota }_{\mathfrak{p}-1},{\iota }_{\mathfrak{p}-1},{\iota }_{\mathfrak{p}},\varphi \left({\mu }^{\mathfrak{r}-1}{\zeta }_0\right))\right)\right),\hfill \\ \hfill \eta \left(\mathfrak{H}({\iota }_{\mathfrak{p}},{\iota }_{\mathfrak{p}},{\iota }_{\mathfrak{p}+1},\varphi \left({\mu }^{\mathfrak{r}}{\zeta }_0\right))\right)& =\eta \left(\mathfrak{H}(\mathsf{\Gamma }{\iota }_{\mathfrak{p}-1},\mathsf{\Gamma }{\iota }_{\mathfrak{p}-1},\mathsf{\Gamma }{\iota }_{\mathfrak{p}},\varphi \left({\mu }^{\mathfrak{r}}{\zeta }_0\right))\right)\hfill \\ \hfill & \ge \psi \left(\eta \left(\mathfrak{H}({\iota }_{\mathfrak{p}-1},{\iota }_{\mathfrak{p}-1},{\iota }_{\mathfrak{p}},\varphi \left({\mu }^{\mathfrak{r}-1}{\zeta }_0\right))\right)\right),\hfill \\ \hfill \eta \left(\mathfrak{E}({\iota }_{\mathfrak{p}},{\iota }_{\mathfrak{p}},{\iota }_{\mathfrak{p}+1},\varphi \left({\mu }^{\mathfrak{r}}{\zeta }_0\right))\right)& =\eta \left(\mathfrak{E}(\mathsf{\Gamma }{\iota }_{\mathfrak{p}-1},\mathsf{\Gamma }{\iota }_{\mathfrak{p}-1},\mathsf{\Gamma }{\iota }_{\mathfrak{p}},\varphi \left({\mu }^{\mathfrak{r}}{\zeta }_0\right))\right)\hfill \\ \hfill & \ge \psi \left(\eta \left(\mathfrak{E}({\iota }_{\mathfrak{p}-1},{\iota }_{\mathfrak{p}-1},{\iota }_{\mathfrak{p}},\varphi \left({\mu }^{\mathfrak{r}-1}{\zeta }_0\right))\right)\right),\hfill \end{array}$$

(23)

in which we apply $\mathfrak{J}({\iota }_{\mathfrak{p}-1},{\iota }_{\mathfrak{p}-1},{\iota }_{\mathfrak{p}},\varphi \left({\mu }^{\mathfrak{r}-1}{\zeta }_0\right))>0$ as stated in (21). By applying this reasoning repeatedly, we find that

$$\begin{array}{cc}\hfill \eta \left(\mathfrak{J}({\iota }_{\mathfrak{p}-1},{\iota }_{\mathfrak{p}-1},{\iota }_{\mathfrak{p}},\varphi \left({\mu }^{\mathfrak{r}-1}{\zeta }_0\right))\right)& =\eta \left(\mathfrak{J}(\mathsf{\Gamma }{\iota }_{\mathfrak{p}-2},\mathsf{\Gamma }{\iota }_{\mathfrak{p}-2},\mathsf{\Gamma }{\iota }_{\mathfrak{p}-1},\varphi \left({\mu }^{\mathfrak{r}-1}{\zeta }_0\right))\right)\hfill \\ \hfill & \le \psi \left(\eta \left(\mathfrak{J}({\iota }_{\mathfrak{p}-2},{\iota }_{\mathfrak{p}-2},{\iota }_{\mathfrak{p}-1},\varphi \left({\mu }^{\mathfrak{r}-2}{\zeta }_0\right))\right)\right),\hfill \\ \hfill \eta \left(\mathfrak{H}({\iota }_{\mathfrak{p}-1},{\iota }_{\mathfrak{p}-1},{\iota }_{\mathfrak{p}},\varphi \left({\mu }^{\mathfrak{r}-1}{\zeta }_0\right))\right)& =\eta \left(\mathfrak{H}(\mathsf{\Gamma }{\iota }_{\mathfrak{p}-2},\mathsf{\Gamma }{\iota }_{\mathfrak{p}-2},\mathsf{\Gamma }{\iota }_{\mathfrak{p}-1},\varphi \left({\mu }^{\mathfrak{r}-1}{\zeta }_0\right))\right)\hfill \\ \hfill & \ge \psi \left(\eta \left(\mathfrak{H}({\iota }_{\mathfrak{p}-2},{\iota }_{\mathfrak{p}-2},{\iota }_{\mathfrak{p}-1},\varphi \left({\mu }^{\mathfrak{r}-2}{\zeta }_0\right))\right)\right),\hfill \\ \hfill \eta \left(\mathfrak{E}({\iota }_{\mathfrak{p}-1},{\iota }_{\mathfrak{p}-1},{\iota }_{\mathfrak{p}},\varphi \left({\mu }^{\mathfrak{r}-1}{\zeta }_0\right))\right)& =\eta \left(\mathfrak{E}(\mathsf{\Gamma }{\iota }_{\mathfrak{p}-2},\mathsf{\Gamma }{\iota }_{\mathfrak{p}-2},\mathsf{\Gamma }{\iota }_{\mathfrak{p}-1},\varphi \left({\mu }^{\mathfrak{r}-1}{\zeta }_0\right))\right)\hfill \\ \hfill & \ge \psi \left(\eta \left(\mathfrak{E}({\iota }_{\mathfrak{p}-2},{\iota }_{\mathfrak{p}-2},{\iota }_{\mathfrak{p}-1},\varphi \left({\mu }^{\mathfrak{r}-2}{\zeta }_0\right))\right)\right),\hfill \end{array}$$

where we use $\mathfrak{J}({\iota }_{\mathfrak{p}-2},{\iota }_{\mathfrak{p}-2},{\iota }_{\mathfrak{p}-1},\varphi \left({\mu }^{\mathfrak{r}-2}{\zeta }_0\right))>0$, $\mathfrak{H}({\iota }_{\mathfrak{p}-2},{\iota }_{\mathfrak{p}-2},{\iota }_{\mathfrak{p}-1},\varphi \left({\mu }^{\mathfrak{r}-2}{\zeta }_0\right))<1$, and $\mathfrak{E}({\iota }_{\mathfrak{p}-2},{\iota }_{\mathfrak{p}-2},{\iota }_{\mathfrak{p}-1},\varphi \left({\mu }^{\mathfrak{r}-2}{\zeta }_0\right))<1$ by (21). Since $\psi$ is non-decreasing, then

$$\begin{array}{c}\hfill \psi \left(\eta \left(\mathfrak{J}({\iota }_{\mathfrak{p}-1},{\iota }_{\mathfrak{p}-1},{\iota }_{\mathfrak{p}},\varphi \left({\mu }^{\mathfrak{r}-1}{\zeta }_0\right))\right)\right)\le {\psi }^2\left(\eta \left(\mathfrak{J}({\iota }_{\mathfrak{p}-2},{\iota }_{\mathfrak{p}-2},{\iota }_{\mathfrak{p}-1},\varphi \left({\mu }^{\mathfrak{r}-2}{\zeta }_0\right))\right)\right),\\ \hfill \psi \left(\eta \left(\mathfrak{H}({\iota }_{\mathfrak{p}-1},{\iota }_{\mathfrak{p}-1},{\iota }_{\mathfrak{p}},\varphi \left({\mu }^{\mathfrak{r}-1}{\zeta }_0\right))\right)\right)\ge {\psi }^2\left(\eta \left(\mathfrak{H}({\iota }_{\mathfrak{p}-2},{\iota }_{\mathfrak{p}-2},{\iota }_{\mathfrak{p}-1},\varphi \left({\mu }^{\mathfrak{r}-2}{\zeta }_0\right))\right)\right),\\ \hfill \psi \left(\eta \left(\mathfrak{E}({\iota }_{\mathfrak{p}-1},{\iota }_{\mathfrak{p}-1},{\iota }_{\mathfrak{p}},\varphi \left({\mu }^{\mathfrak{r}-1}{\zeta }_0\right))\right)\right)\ge {\psi }^2\left(\eta \left(\mathfrak{E}({\iota }_{\mathfrak{p}-2},{\iota }_{\mathfrak{p}-2},{\iota }_{\mathfrak{p}-1},\varphi \left({\mu }^{\mathfrak{r}-2}{\zeta }_0\right))\right)\right).\end{array}$$

(24)

By combining inequalities (23) and (24), we conclude that

$$\begin{array}{cc}\hfill \eta \left(\mathfrak{J}({\iota }_{\mathfrak{p}},{\iota }_{\mathfrak{p}},{\iota }_{\mathfrak{p}+1},\varphi \left({\mu }^{\mathfrak{r}}{\zeta }_0\right))\right)& \le \psi \left(\eta \left(\mathfrak{J}({\iota }_{\mathfrak{p}-1},{\iota }_{\mathfrak{p}-1},{\iota }_{\mathfrak{p}},\varphi \left({\mu }^{\mathfrak{r}-1}{\zeta }_0\right))\right)\right)\hfill \\ \hfill & \le {\psi }^2\left(\eta \left(\mathfrak{J}({\iota }_{\mathfrak{p}-2},{\iota }_{\mathfrak{p}-2},{\iota }_{\mathfrak{p}-1},\varphi \left({\mu }^{\mathfrak{r}-2}{\zeta }_0\right))\right)\right),\hfill \\ \hfill \eta \left(\mathfrak{H}({\iota }_{\mathfrak{p}},{\iota }_{\mathfrak{p}},{\iota }_{\mathfrak{p}+1},\varphi \left({\mu }^{\mathfrak{r}}{\zeta }_0\right))\right)& \ge \psi \left(\eta \left(\mathfrak{H}({\iota }_{\mathfrak{p}-1},{\iota }_{\mathfrak{p}-1},{\iota }_{\mathfrak{p}},\varphi \left({\mu }^{\mathfrak{r}-1}{\zeta }_0\right))\right)\right)\hfill \\ \hfill & \ge {\psi }^2\left(\eta \left(\mathfrak{H}({\iota }_{\mathfrak{p}-2},{\iota }_{\mathfrak{p}-2},{\iota }_{\mathfrak{p}-1},\varphi \left({\mu }^{\mathfrak{r}-2}{\zeta }_0\right))\right)\right),\hfill \\ \hfill \eta \left(\mathfrak{E}({\iota }_{\mathfrak{p}},{\iota }_{\mathfrak{p}},{\iota }_{\mathfrak{p}+1},\varphi \left({\mu }^{\mathfrak{r}}{\zeta }_0\right))\right)& \ge \psi \left(\eta \left(\mathfrak{E}({\iota }_{\mathfrak{p}-1},{\iota }_{\mathfrak{p}-1},{\iota }_{\mathfrak{p}},\varphi \left({\mu }^{\mathfrak{r}-1}{\zeta }_0\right))\right)\right)\hfill \\ \hfill & \ge {\psi }^2\left(\eta \left(\mathfrak{E}({\iota }_{\mathfrak{p}-2},{\iota }_{\mathfrak{p}-2},{\iota }_{\mathfrak{p}-1},\varphi \left({\mu }^{\mathfrak{r}-2}{\zeta }_0\right))\right)\right),\hfill \end{array}$$

Inequality (21) allows us to apply this argument n times, leading to the conclusion that

$$\begin{array}{cc}\hfill \eta \left(\mathfrak{J}({\iota }_{\mathfrak{p}},{\iota }_{\mathfrak{p}},{\iota }_{\mathfrak{p}+1},\varphi \left({\mu }^{\mathfrak{r}}{\zeta }_0\right))\right)& \le {\psi }^{\mathfrak{p}}\left(\eta \left(\mathfrak{J}({\iota }_0,{\iota }_0,{\iota }_1,\varphi \left({\mu }^{\mathfrak{r}-2}{\zeta }_0\right))\right)\right)\hfill \\ \hfill & ={\psi }^{\mathfrak{p}}\left(\eta \left(\mathfrak{J}({\iota }_0,{\iota }_0,{\iota }_1,\varphi \left({\zeta }_0{\mu }^{\mathfrak{p}-\mathfrak{r}}\right))\right)\right),\hfill \\ \hfill \eta \left(\mathfrak{H}({\iota }_{\mathfrak{p}},{\iota }_{\mathfrak{p}},{\iota }_{\mathfrak{p}+1},\varphi \left({\mu }^{\mathfrak{r}}{\zeta }_0\right))\right)& \ge {\psi }^{\mathfrak{p}}\left(\eta \left(\mathfrak{H}({\iota }_0,{\iota }_0,{\iota }_1,\varphi \left({\mu }^{\mathfrak{r}-2}{\zeta }_0\right))\right)\right)\hfill \\ \hfill & ={\psi }^{\mathfrak{p}}\left(\eta \left(\mathfrak{H}({\iota }_0,{\iota }_0,{\iota }_1,\varphi \left({\zeta }_0{\mu }^{\mathfrak{p}-\mathfrak{r}}\right))\right)\right),\hfill \\ \hfill \eta \left(\mathfrak{E}({\iota }_{\mathfrak{p}},{\iota }_{\mathfrak{p}},{\iota }_{\mathfrak{p}+1},\varphi \left({\mu }^{\mathfrak{r}}{\zeta }_0\right))\right)& \ge {\psi }^{\mathfrak{p}}\left(\eta \left(\mathfrak{E}({\iota }_0,{\iota }_0,{\iota }_1,\varphi \left({\mu }^{\mathfrak{r}-2}{\zeta }_0\right))\right)\right)\hfill \\ \hfill & ={\psi }^{\mathfrak{p}}\left(\eta \left(\mathfrak{E}({\iota }_0,{\iota }_0,{\iota }_1,\varphi \left({\zeta }_0{\mu }^{\mathfrak{p}-\mathfrak{r}}\right))\right)\right),\hfill \end{array}$$

(25)

for all $\mathfrak{p}>\mathfrak{r}$. As a consequence,

$$\begin{array}{cc}\hfill & \underset{\mathfrak{p}\to \infty }{lim}{\displaystyle \frac{{\zeta }_0}{{\mu }^{\mathfrak{p}-\mathfrak{r}}}}=\infty \Rightarrow \underset{\mathfrak{p}\to \infty }{lim}\varphi \left({\displaystyle \frac{{\zeta }_0}{{\mu }^{\mathfrak{p}-\mathfrak{r}}}}\right)=\infty \hfill \\ \hfill & \Rightarrow \underset{\mathfrak{p}\to \infty }{lim}\mathfrak{J}\left({\iota }_0,{\iota }_0,{\iota }_1,\varphi \left({\displaystyle \frac{{\zeta }_0}{{\mu }^{\mathfrak{p}-\mathfrak{r}}}}\right)\right)=1\hfill \\ \hfill & \Rightarrow \underset{\mathfrak{p}\to \infty }{lim}\eta \left(\mathfrak{J}\left({\iota }_0,{\iota }_0,{\iota }_1,\varphi \left({\displaystyle \frac{{\zeta }_0}{{\mu }^{\mathfrak{p}-\mathfrak{r}}}}\right)\right)\right)=0.\hfill \\ \hfill & \Rightarrow \underset{\mathfrak{p}\to \infty }{lim}\mathfrak{H}\left({\iota }_0,{\iota }_0,{\iota }_1,\varphi \left({\displaystyle \frac{{\zeta }_0}{{\mu }^{\mathfrak{p}-\mathfrak{r}}}}\right)\right)=0\hfill \\ \hfill & \Rightarrow \underset{\mathfrak{p}\to \infty }{lim}\eta \left(\mathfrak{H}\left({\iota }_0,{\iota }_0,{\iota }_1,\varphi \left({\displaystyle \frac{{\zeta }_0}{{\mu }^{\mathfrak{p}-\mathfrak{r}}}}\right)\right)\right)=1.\hfill \\ \hfill & \Rightarrow \underset{\mathfrak{p}\to \infty }{lim}\mathfrak{E}\left({\iota }_0,{\iota }_0,{\iota }_1,\varphi \left({\displaystyle \frac{{\zeta }_0}{{\mu }^{\mathfrak{p}-\mathfrak{r}}}}\right)\right)=0\hfill \\ \hfill & \Rightarrow \underset{\mathfrak{p}\to \infty }{lim}\eta \left(\mathfrak{E}\left({\iota }_0,{\iota }_0,{\iota }_1,\varphi \left({\displaystyle \frac{{\zeta }_0}{{\mu }^{\mathfrak{p}-\mathfrak{r}}}}\right)\right)\right)=1.\hfill \end{array}$$

As the sequence $\left\{{\rho }_{\mathfrak{p}}=\eta \left(\mathfrak{J}\left({\iota }_0,{\iota }_0,{\iota }_1,\varphi \left({\displaystyle \frac{{\zeta }_0}{{\mu }^{\mathfrak{p}-\mathfrak{r}}}}\right)\right)\right)\right\}\to 0$, $\left\{{\rho }_{\mathfrak{p}}=\eta \left(\mathfrak{H}\left({\iota }_0,{\iota }_0,{\iota }_1,\varphi \left({\displaystyle \frac{{\zeta }_0}{{\mu }^{\mathfrak{p}-\mathfrak{r}}}}\right)\right)\right)\right\}\to 1$, $\left\{{\rho }_{\mathfrak{p}}=\eta \left(\mathfrak{E}\left({\iota }_0,{\iota }_0,{\iota }_1,\varphi \left({\displaystyle \frac{{\zeta }_0}{{\mu }^{\mathfrak{p}-\mathfrak{r}}}}\right)\right)\right)\right\}\to 1$ and $\eta \in \mathfrak{K}$, we have $\left\{{\psi }^n\left({\rho }_{\mathfrak{p}}\right)\right\}\to 0$. By (25), we deduce that ${\displaystyle \underset{\mathfrak{p}\to \infty }{lim}\eta \left(\mathfrak{J}({\iota }_{\mathfrak{p}},{\iota }_{\mathfrak{p}},{\iota }_{\mathfrak{p}+1},\varphi \left({\mu }^{\mathfrak{r}}{\zeta }_0\right))\right)=0}$, ${\displaystyle \underset{\mathfrak{p}\to \infty }{lim}\eta \left(\mathfrak{H}({\iota }_{\mathfrak{p}},{\iota }_{\mathfrak{p}},{\iota }_{\mathfrak{p}+1},\varphi \left({\mu }^{\mathfrak{r}}{\zeta }_0\right))\right)=1}$, and ${\displaystyle \underset{\mathfrak{p}\to \infty }{lim}\eta \left(\mathfrak{E}({\iota }_{\mathfrak{p}},{\iota }_{\mathfrak{p}},{\iota }_{\mathfrak{p}+1},\varphi \left({\mu }^{\mathfrak{r}}{\zeta }_0\right))\right)=1}$.

In particular, as $\eta \in \mathfrak{K}$, condition (${\mathfrak{K}}_1$) implies that ${\displaystyle \underset{\mathfrak{p}\to \infty }{lim}\mathfrak{J}({\iota }_{\mathfrak{p}},{\iota }_{\mathfrak{p}},{\iota }_{\mathfrak{p}+1},\varphi \left({\mu }^{\mathfrak{r}}{\zeta }_0\right))=1,\underset{\mathfrak{p}\to \infty }{lim}\mathfrak{H}({\iota }_{\mathfrak{p}},{\iota }_{\mathfrak{p}},{\iota }_{\mathfrak{p}+1},\varphi \left({\mu }^{\mathfrak{r}}{\zeta }_0\right))=0}$, ${\displaystyle \underset{\mathfrak{p}\to \infty }{lim}\mathfrak{E}({\iota }_{\mathfrak{p}},{\iota }_{\mathfrak{p}},{\iota }_{\mathfrak{p}+1},\varphi \left({\mu }^{\mathfrak{r}}{\zeta }_0\right))=0}$. Because $\varphi \left({\mu }^{\mathfrak{r}}{\zeta }_0\right))<\zeta$, $\mathfrak{J}(\iota ,\vartheta ,\xi ,\phi )$ is a non-decreasing function and $\mathfrak{H}(\iota ,\vartheta ,\xi ,\phi ),\mathfrak{E}(\iota ,\vartheta ,\xi ,\phi )$ are non-increasing functions with respect to $\phi$, so we have

$$\begin{array}{cc}\hfill \mathfrak{J}({\iota }_{\mathfrak{p}},{\iota }_{\mathfrak{p}},{\iota }_{\mathfrak{p}+1},\varphi \left({\mu }^{\mathfrak{r}}{\zeta }_0\right))& \le \mathfrak{J}({\iota }_{\mathfrak{p}},{\iota }_{\mathfrak{p}},{\iota }_{\mathfrak{p}+1},s)\le 1,\hfill \\ \hfill \mathfrak{H}({\iota }_{\mathfrak{p}},{\iota }_{\mathfrak{p}},{\iota }_{\mathfrak{p}+1},\varphi \left({\mu }^{\mathfrak{r}}{\zeta }_0\right))& \ge \mathfrak{H}({\iota }_{\mathfrak{p}},{\iota }_{\mathfrak{p}},{\iota }_{\mathfrak{p}+1},s)\ge 0,\hfill \\ \hfill \mathfrak{E}({\iota }_{\mathfrak{p}},{\iota }_{\mathfrak{p}},{\iota }_{\mathfrak{p}+1},\varphi \left({\mu }^{\mathfrak{r}}{\zeta }_0\right))& \ge \mathfrak{E}({\iota }_{\mathfrak{p}},{\iota }_{\mathfrak{p}},{\iota }_{\mathfrak{p}+1},s)\ge 0\hfill \end{array}$$

(26)

Considering (26), we note that ${\displaystyle \underset{\mathfrak{p}\to \infty }{lim}\mathfrak{J}({\iota }_{\mathfrak{p}},{\iota }_{\mathfrak{p}},{\iota }_{\mathfrak{p}+1},\zeta )=1}$, ${\displaystyle \underset{\mathfrak{p}\to \infty }{lim}\mathfrak{H}({\iota }_{\mathfrak{p}},{\iota }_{\mathfrak{p}},{\iota }_{\mathfrak{p}+1},\zeta )=0}$, and ${\displaystyle \underset{\mathfrak{p}\to \infty }{lim}\mathfrak{E}({\iota }_{\mathfrak{p}},{\iota }_{\mathfrak{p}},{\iota }_{\mathfrak{p}+1},\zeta )=0}$ for all $\zeta >0$; this implies that (22) is satisfied. According to Definition (4), $\left\{{\iota }_{\mathfrak{p}}\right\}$ is a generalized Cauchy sequence in $\mathfrak{A}$. Since $\mathfrak{A}$ is generalized complete, there exists an element $\iota \in \mathfrak{A}$ such that $\left\{{\iota }_{\mathfrak{p}}\right\}$ converges to $\iota$. We now assert that $\iota$ is a fixed point of $\mathsf{\Gamma }$. To demonstrate this, for any $\phi >0$ and for every $\mathfrak{p}\in \mathbb{N}$,

$$\begin{array}{cc}\hfill \mathfrak{J}(\iota ,\iota ,\mathsf{\Gamma }\iota ,\phi )& =\mathfrak{J}(\mathsf{\Gamma }\iota ,\iota ,\iota ,\phi )\ge \mathfrak{J}(\mathsf{\Gamma }\iota ,{\iota }_{\mathfrak{p}+1},{\iota }_{\mathfrak{p}+1},\phi )\otimes \mathfrak{J}({\iota }_{\mathfrak{p}+1},\iota ,\iota ,\phi )\hfill \\ \hfill & =\mathfrak{J}(\mathsf{\Gamma }\iota ,{\iota }_{\mathfrak{p}+1},{\iota }_{\mathfrak{p}+1},\phi )\otimes \mathfrak{J}(\mathsf{\Gamma }{\iota }_{\mathfrak{p}},\iota ,\iota ,\phi ),\hfill \\ \hfill \mathfrak{H}(\iota ,\iota ,\mathsf{\Gamma }\iota ,\phi )& =\mathfrak{H}(\mathsf{\Gamma }\iota ,\iota ,\iota ,\phi )\le \mathfrak{H}(\mathsf{\Gamma }\iota ,{\iota }_{\mathfrak{p}+1},{\iota }_{\mathfrak{p}+1},\phi )\oplus \mathfrak{H}({\iota }_{\mathfrak{p}+1},\iota ,\iota ,\phi )\hfill \\ \hfill & =\mathfrak{H}(\mathsf{\Gamma }\iota ,{\iota }_{\mathfrak{p}+1},{\iota }_{\mathfrak{p}+1},\phi )\oplus \mathfrak{H}(\mathsf{\Gamma }{\iota }_{\mathfrak{p}},\iota ,\iota ,\phi ),\hfill \\ \hfill \mathfrak{E}(\iota ,\iota ,\mathsf{\Gamma }\iota ,\phi )& =\mathfrak{E}(\mathsf{\Gamma }\iota ,\iota ,\iota ,\phi )\le \mathfrak{E}(\mathsf{\Gamma }\iota ,{\iota }_{\mathfrak{p}+1},{\iota }_{\mathfrak{p}+1},\phi )\oplus \mathfrak{E}({\iota }_{\mathfrak{p}+1},\iota ,\iota ,\phi )\hfill \\ \hfill & =\mathfrak{E}(\mathsf{\Gamma }\iota ,{\iota }_{\mathfrak{p}+1},{\iota }_{\mathfrak{p}+1},\phi )\oplus \mathfrak{E}(\mathsf{\Gamma }{\iota }_{\mathfrak{p}},\iota ,\iota ,\phi ).\hfill \end{array}$$

(27)

By lemma (2),

$$\underset{\mathfrak{p}\to \infty }{lim}\mathfrak{J}(\iota ,\iota ,{\iota }_{\mathfrak{p}+1},\phi )=1,\underset{\mathfrak{p}\to \infty }{lim}\mathfrak{H}(\iota ,\iota ,{\iota }_{\mathfrak{p}+1},\phi )=0,\mathrm{and}\underset{\mathfrak{p}\to \infty }{lim}\mathfrak{E}(\iota ,\iota ,{\iota }_{\mathfrak{p}+1},\phi )=0.$$

(28)

Let us demonstrate that the first term in (27) converges to 1 as $\mathfrak{p}$ approaches infinity. Given that $\varphi$ is continuous at $\phi =0$, we have ${\displaystyle \underset{\zeta \to 0}{lim}\varphi \left(\zeta \right)=\varphi \left(0\right)=0}$.

Since $\phi >0$, there exists $\delta >0$ such that $\varphi \left(\delta \right)<\phi$. Since ${\displaystyle \frac{\delta }{\mu }}>0,\varphi \left({\displaystyle \frac{\delta }{\mu }}\right)>0$.

So, ${\displaystyle \underset{\mathfrak{p}\to \infty }{lim}\mathfrak{J}\left(\iota ,{\iota }_{\mathfrak{p}+1},{\iota }_{\mathfrak{p}+1},\varphi \left(\frac{\delta }{\mu }\right)\right)=1}$, ${\displaystyle \underset{\mathfrak{p}\to \infty }{lim}\mathfrak{H}\left(\iota ,{\iota }_{\mathfrak{p}+1},{\iota }_{\mathfrak{p}+1},\varphi \left(\frac{\delta }{\mu }\right)\right)=0}$, and ${\displaystyle \underset{\mathfrak{p}\to \infty }{lim}\mathfrak{E}\left(\iota ,{\iota }_{\mathfrak{p}+1},{\iota }_{\mathfrak{p}+1},\varphi \left(\frac{\delta }{\mu }\right)\right)=0}$. Hence, there exists ${\mathfrak{p}}_0\in \mathbb{N}$ such that $\mathfrak{J}\left(\iota ,{\iota }_{\mathfrak{p}+1},{\iota }_{\mathfrak{p}+1},\varphi \left({\displaystyle \frac{\delta }{\mu }}\right)\right)>0$, $\mathfrak{H}\left(\iota ,{\iota }_{\mathfrak{p}+1},{\iota }_{\mathfrak{p}+1},\varphi \left({\displaystyle \frac{\delta }{\mu }}\right)\right)<1$, and $\mathfrak{E}\left(\iota ,{\iota }_{\mathfrak{p}+1},{\iota }_{\mathfrak{p}+1},\varphi \left({\displaystyle \frac{\delta }{\mu }}\right)\right)<1$ for all $\mathfrak{p}\ge {\mathfrak{p}}_0$. By applying the contractivity condition (19) to $\iota =\iota$ and $\vartheta =\xi ={\iota }_{\mathfrak{p}+1}$ for $\mathfrak{p}\ge {\mathfrak{p}}_0$, we obtain

$$\begin{array}{cc}\hfill \eta \left(\mathfrak{J}(\mathsf{\Gamma }\iota ,{\iota }_{\mathfrak{p}+1},{\iota }_{\mathfrak{p}+1},\varphi \left(\delta \right))\right)=\eta \left(\mathfrak{J}(\mathsf{\Gamma }\iota ,\mathsf{\Gamma }{\iota }_{\mathfrak{p}},\mathsf{\Gamma }{\iota }_{\mathfrak{p}},\varphi \left(\delta \right))\right)& \le \psi \left(\mathfrak{J}\left(\iota ,{\iota }_{\mathfrak{p}},{\iota }_{\mathfrak{p}},\varphi \left({\displaystyle \frac{\delta }{\mu }}\right)\right)\right),\hfill \\ \hfill \eta \left(\mathfrak{H}(\mathsf{\Gamma }\iota ,{\iota }_{\mathfrak{p}+1},{\iota }_{\mathfrak{p}+1},\varphi \left(\delta \right))\right)=\eta \left(\mathfrak{H}(\mathsf{\Gamma }\iota ,\mathsf{\Gamma }{\iota }_{\mathfrak{p}},\mathsf{\Gamma }{\iota }_{\mathfrak{p}},\varphi \left(\delta \right))\right)& \ge \psi \left(\mathfrak{H}\left(\iota ,{\iota }_{\mathfrak{p}},{\iota }_{\mathfrak{p}},\varphi \left({\displaystyle \frac{\delta }{\mu }}\right)\right)\right),\hfill \\ \hfill \eta \left(\mathfrak{E}(\mathsf{\Gamma }\iota ,{\iota }_{\mathfrak{p}+1},{\iota }_{\mathfrak{p}+1},\varphi \left(\delta \right))\right)=\eta \left(\mathfrak{E}(\mathsf{\Gamma }\iota ,\mathsf{\Gamma }{\iota }_{\mathfrak{p}},\mathsf{\Gamma }{\iota }_{\mathfrak{p}},\varphi \left(\delta \right))\right)& \ge \psi \left(\mathfrak{E}\left(\iota ,{\iota }_{\mathfrak{p}},{\iota }_{\mathfrak{p}},\varphi \left({\displaystyle \frac{\delta }{\mu }}\right)\right)\right).\hfill \end{array}$$

Therefore,

$$\begin{array}{cc}\hfill & \underset{\mathfrak{p}\to \infty }{lim}\mathfrak{J}\left(\iota ,{\iota }_{\mathfrak{p}+1},{\iota }_{\mathfrak{p}+1},\varphi \left({\displaystyle \frac{\delta }{\mu }}\right)\right)=1\hfill \\ \hfill & \Rightarrow \underset{\mathfrak{p}\to \infty }{lim}\eta \left(\mathfrak{J}\left(\iota ,{\iota }_{\mathfrak{p}+1},{\iota }_{\mathfrak{p}+1},\varphi \left({\displaystyle \frac{\delta }{\mu }}\right)\right)\right)=0\hfill \\ \hfill & \Rightarrow \underset{\mathfrak{p}\to \infty }{lim}\psi \left(\eta \left(\mathfrak{J}\left(\iota ,{\iota }_{\mathfrak{p}+1},{\iota }_{\mathfrak{p}+1},\varphi \left({\displaystyle \frac{\delta }{\mu }}\right)\right)\right)\right)=0\hfill \\ \hfill & \Rightarrow \underset{\mathfrak{p}\to \infty }{lim}\eta \left(\mathfrak{J}(\mathsf{\Gamma }\iota ,{\iota }_{\mathfrak{p}+1},{\iota }_{\mathfrak{p}+1},\varphi \left(\delta \right))\right)=0\hfill \\ \hfill & \Rightarrow \underset{\mathfrak{p}\to \infty }{lim}\mathfrak{J}(\mathsf{\Gamma }\iota ,{\iota }_{\mathfrak{p}+1},{\iota }_{\mathfrak{p}+1},\varphi \left(\delta \right))=1.\hfill \\ \hfill & \underset{\mathfrak{p}\to \infty }{lim}\mathfrak{H}\left(\iota ,{\iota }_{\mathfrak{p}+1},{\iota }_{\mathfrak{p}+1},\varphi \left({\displaystyle \frac{\delta }{\mu }}\right)\right)=0\hfill \\ \hfill & \Rightarrow \underset{\mathfrak{p}\to \infty }{lim}\eta \left(\mathfrak{H}\left(\iota ,{\iota }_{\mathfrak{p}+1},{\iota }_{\mathfrak{p}+1},\varphi \left({\displaystyle \frac{\delta }{\mu }}\right)\right)\right)=1\hfill \\ \hfill & \Rightarrow \underset{\mathfrak{p}\to \infty }{lim}\psi \left(\eta \left(\mathfrak{H}\left(\iota ,{\iota }_{\mathfrak{p}+1},{\iota }_{\mathfrak{p}+1},\varphi \left({\displaystyle \frac{\delta }{\mu }}\right)\right)\right)\right)=1\hfill \\ \hfill & \Rightarrow \underset{\mathfrak{p}\to \infty }{lim}\eta \left(\mathfrak{H}(\mathsf{\Gamma }\iota ,{\iota }_{\mathfrak{p}+1},{\iota }_{\mathfrak{p}+1},\varphi \left(\delta \right))\right)=1\hfill \\ \hfill & \Rightarrow \underset{\mathfrak{p}\to \infty }{lim}\mathfrak{H}(\mathsf{\Gamma }\iota ,{\iota }_{\mathfrak{p}+1},{\iota }_{\mathfrak{p}+1},\varphi \left(\delta \right))=0.\hfill \\ \hfill & \underset{\mathfrak{p}\to \infty }{lim}\mathfrak{E}\left(\iota ,{\iota }_{\mathfrak{p}+1},{\iota }_{\mathfrak{p}+1},\varphi \left({\displaystyle \frac{\delta }{\mu }}\right)\right)=0\hfill \\ \hfill & \Rightarrow \underset{\mathfrak{p}\to \infty }{lim}\eta \left(\mathfrak{E}\left(\iota ,{\iota }_{\mathfrak{p}+1},{\iota }_{\mathfrak{p}+1},\varphi \left({\displaystyle \frac{\delta }{\mu }}\right)\right)\right)=1\hfill \\ \hfill & \Rightarrow \underset{\mathfrak{p}\to \infty }{lim}\psi \left(\eta \left(\mathfrak{E}\left(\iota ,{\iota }_{\mathfrak{p}+1},{\iota }_{\mathfrak{p}+1},\varphi \left({\displaystyle \frac{\delta }{\mu }}\right)\right)\right)\right)=1\hfill \\ \hfill & \Rightarrow \underset{\mathfrak{p}\to \infty }{lim}\eta \left(\mathfrak{E}(\mathsf{\Gamma }\iota ,{\iota }_{\mathfrak{p}+1},{\iota }_{\mathfrak{p}+1},\varphi \left(\delta \right))\right)=1\hfill \\ \hfill & \Rightarrow \underset{\mathfrak{p}\to \infty }{lim}\mathfrak{E}(\mathsf{\Gamma }\iota ,{\iota }_{\mathfrak{p}+1},{\iota }_{\mathfrak{p}+1},\varphi \left(\delta \right))=0.\hfill \end{array}$$

Taking into account that $\mathfrak{J}(\mathsf{\Gamma }\iota ,{\iota }_{\mathfrak{p}+1},{\iota }_{\mathfrak{p}+1},\varphi \left(\delta \right))\le \mathfrak{J}(\mathsf{\Gamma }\iota ,{\iota }_{\mathfrak{p}+1},{\iota }_{\mathfrak{p}+1},\phi )\le 1$, $\mathfrak{H}(\mathsf{\Gamma }\iota ,{\iota }_{\mathfrak{p}+1},{\iota }_{\mathfrak{p}+1},\varphi \left(\delta \right))\ge \mathfrak{H}(\mathsf{\Gamma }\iota ,{\iota }_{\mathfrak{p}+1},{\iota }_{\mathfrak{p}+1},\phi )\ge 0$, and $\mathfrak{E}(\mathsf{\Gamma }\iota ,{\iota }_{\mathfrak{p}+1},{\iota }_{\mathfrak{p}+1},\varphi \left(\delta \right))\ge \mathfrak{E}(\mathsf{\Gamma }\iota ,{\iota }_{\mathfrak{p}+1},{\iota }_{\mathfrak{p}+1},\phi )\ge 0$, we deduce that

$$\begin{array}{cc}\hfill \underset{\mathfrak{p}\to \infty }{lim}\mathfrak{J}(\mathsf{\Gamma }\iota ,\mathsf{\Gamma }{\iota }_{\mathfrak{p}},\mathsf{\Gamma }{\iota }_{\mathfrak{p}},\phi )& =\mathfrak{J}(\mathsf{\Gamma }\iota ,{\iota }_{\mathfrak{p}+1},{\iota }_{\mathfrak{p}+1},\phi )=1,\hfill \\ \hfill \underset{\mathfrak{p}\to \infty }{lim}\mathfrak{H}(\mathsf{\Gamma }\iota ,\mathsf{\Gamma }{\iota }_{\mathfrak{p}},\mathsf{\Gamma }{\iota }_{\mathfrak{p}},\phi )& =\mathfrak{H}(\mathsf{\Gamma }\iota ,{\iota }_{\mathfrak{p}+1},{\iota }_{\mathfrak{p}+1},\phi )=0,\hfill \\ \hfill \underset{\mathfrak{p}\to \infty }{lim}\mathfrak{E}(\mathsf{\Gamma }\iota ,\mathsf{\Gamma }{\iota }_{\mathfrak{p}},\mathsf{\Gamma }{\iota }_{\mathfrak{p}},\phi )& =\mathfrak{E}(\mathsf{\Gamma }\iota ,{\iota }_{\mathfrak{p}+1},{\iota }_{\mathfrak{p}+1},\phi )=0.\hfill \end{array}$$

(29)

Taking the limit as $\mathfrak{p}\to \infty$ in (27) and applying (28) and (29) yield

$$\begin{array}{cc}\hfill \mathfrak{J}(\mathsf{\Gamma }\iota ,\iota ,\iota ,\phi )& \ge \underset{\mathfrak{p}\to \infty }{lim}[\mathfrak{J}(\mathsf{\Gamma }\iota ,\mathsf{\Gamma }{\iota }_{\mathfrak{p}},\mathsf{\Gamma }{\iota }_{\mathfrak{p}},\phi )\otimes \mathfrak{J}(\mathsf{\Gamma }{\iota }_{\mathfrak{p}},\iota ,\iota ,\phi )]\left[\underset{\mathfrak{p}\to \infty }{lim}\mathfrak{J}(\mathsf{\Gamma }\iota ,\mathsf{\Gamma }{\iota }_{\mathfrak{p}},\mathsf{\Gamma }{\iota }_{\mathfrak{p}},\phi )\right]\hfill \\ \hfill & \otimes \left[\underset{\mathfrak{p}\to \infty }{lim}\mathfrak{J}(\mathsf{\Gamma }{\iota }_{\mathfrak{p}},\iota ,\iota ,\phi )\right]=1\otimes 1=1,\hfill \\ \hfill \mathfrak{H}(\mathsf{\Gamma }\iota ,\iota ,\iota ,\phi )& \le \underset{\mathfrak{p}\to \infty }{lim}[\mathfrak{H}(\mathsf{\Gamma }\iota ,\mathsf{\Gamma }{\iota }_{\mathfrak{p}},\mathsf{\Gamma }{\iota }_{\mathfrak{p}},\phi )\oplus \mathfrak{H}(\mathsf{\Gamma }{\iota }_{\mathfrak{p}},\iota ,\iota ,\phi )]\left[\underset{\mathfrak{p}\to \infty }{lim}\mathfrak{H}(\mathsf{\Gamma }\iota ,\mathsf{\Gamma }{\iota }_{\mathfrak{p}},\mathsf{\Gamma }{\iota }_{\mathfrak{p}},\phi )\right]\hfill \\ \hfill & \oplus \left[\underset{\mathfrak{p}\to \infty }{lim}\mathfrak{H}(\mathsf{\Gamma }{\iota }_{\mathfrak{p}},\iota ,\iota ,\phi )\right]=0\oplus 0=0.\hfill \\ \hfill \mathfrak{E}(\mathsf{\Gamma }\iota ,\iota ,\iota ,\phi )& \le \underset{\mathfrak{p}\to \infty }{lim}[\mathfrak{E}(\mathsf{\Gamma }\iota ,\mathsf{\Gamma }{\iota }_{\mathfrak{p}},\mathsf{\Gamma }{\iota }_{\mathfrak{p}},\phi )\oplus \mathfrak{E}(\mathsf{\Gamma }{\iota }_{\mathfrak{p}},\iota ,\iota ,\phi )]\left[\underset{\mathfrak{p}\to \infty }{lim}\mathfrak{E}(\mathsf{\Gamma }\iota ,\mathsf{\Gamma }{\iota }_{\mathfrak{p}},\mathsf{\Gamma }{\iota }_{\mathfrak{p}},\phi )\right]\hfill \\ \hfill & \oplus \left[\underset{\mathfrak{p}\to \infty }{lim}\mathfrak{E}(\mathsf{\Gamma }{\iota }_{\mathfrak{p}},\iota ,\iota ,\phi )\right]=0\oplus 0=0.\hfill \end{array}$$

We have proven that $\mathfrak{J}(\mathsf{\Gamma }\iota ,\iota ,\iota ,\phi )=1$, $\mathfrak{H}(\mathsf{\Gamma }\iota ,\iota ,\iota ,\phi )=0$, and $\mathfrak{E}(\mathsf{\Gamma }\iota ,\iota ,\iota ,\phi )=0$ for all $\phi >0$, and the axioms (iv), (x), and (xvi) of definition (1.1) guarantee that $\mathsf{\Gamma }\iota =\iota$. In other words, $\iota$ is a fixed point of $\mathsf{\Gamma }$.

We now investigate the uniqueness of the fixed point of $\mathsf{\Gamma }$. Suppose that $\mathsf{\Gamma }$ has two distinct fixed points, $\iota$ and $\vartheta$. This leads to a contradiction, showing that $\iota$ and $\vartheta$ must be the same.

According to the assumption, ${\displaystyle \underset{\phi \to \infty }{lim}\mathfrak{J}(\iota ,\vartheta ,\vartheta ,\phi )=1,\underset{\phi \to \infty }{lim}\mathfrak{H}(\iota ,\vartheta ,\vartheta ,\phi )=0}$, and ${\displaystyle \underset{\phi \to \infty }{lim}\mathfrak{E}(\iota ,\vartheta ,\vartheta ,\phi )=0}$. Then, there exists ${\phi }_0>0$ such that $\mathfrak{J}(\iota ,\vartheta ,\vartheta ,{\phi }_0)>0,\mathfrak{H}(\iota ,\vartheta ,\vartheta ,{\phi }_0)<1,\mathfrak{E}(\iota ,\vartheta ,\vartheta ,{\phi }_0)<1$. Moreover, there exists ${\zeta }_0>0$ such that $\varphi \left({\zeta }_0\right)>{\phi }_0$. Consequently, as $\varphi$ and $\mathfrak{J}(\iota ,\vartheta ,\xi ,\phi )$ are non-decreasing functions, $\mathfrak{H}(\iota ,\vartheta ,\xi ,\phi ),\mathfrak{E}(\iota ,\vartheta ,\xi ,\phi )$ are non-increasing functions with respect to $\phi$,

$\mathfrak{J}(\iota ,\vartheta ,\xi ,\varphi \left({\zeta }_0\right))\ge \mathfrak{J}(\iota ,\vartheta ,\xi ,{\phi }_0)>0$, $\mathfrak{H}(\iota ,\vartheta ,\xi ,\varphi \left({\zeta }_0\right))\le \mathfrak{H}(\iota ,\vartheta ,\xi ,{\phi }_0)<1$,

$\mathfrak{E}(\iota ,\vartheta ,\xi ,\varphi \left({\zeta }_0\right))\le \mathfrak{E}(\iota ,\vartheta ,\xi ,{\phi }_0)<1$.

From (20), we have

$\mathfrak{J}(\iota ,\vartheta ,\vartheta ,\varphi \left({\zeta }_0\right))=\mathfrak{J}(\mathsf{\Gamma }\iota ,\mathsf{\Gamma }\vartheta ,\mathsf{\Gamma }\vartheta ,\varphi \left(\mu {\zeta }_0\right))>0$, $\mathfrak{H}(\iota ,\vartheta ,\vartheta ,\varphi \left({\zeta }_0\right))=\mathfrak{H}(\mathsf{\Gamma }\iota ,\mathsf{\Gamma }\vartheta ,\mathsf{\Gamma }\vartheta ,\varphi \left(\mu {\zeta }_0\right))<1$, and $\mathfrak{E}(\iota ,\vartheta ,\vartheta ,\varphi \left({\zeta }_0\right))=\mathfrak{E}(\mathsf{\Gamma }\iota ,\mathsf{\Gamma }\vartheta ,\mathsf{\Gamma }\vartheta ,\varphi \left(\mu {\zeta }_0\right))<1$. By induction, $\mathfrak{J}(\iota ,\vartheta ,\vartheta ,\varphi \left({\mu }^{\mathfrak{p}}{\zeta }_0\right))>0$, $\mathfrak{H}(\iota ,\vartheta ,\vartheta ,\varphi \left({\mu }^{\mathfrak{p}}{\zeta }_0\right))<1$, $\mathfrak{E}(\iota ,\vartheta ,\vartheta ,\varphi \left({\mu }^{\mathfrak{p}}{\zeta }_0\right))<1$ for all $\mathfrak{p}\in \mathbb{N}$. We claim that

$$\mathfrak{J}(\iota ,\vartheta ,\vartheta ,\varphi \left({\mu }^{\mathfrak{r}}{\zeta }_0\right))=1,\mathfrak{H}(\iota ,\vartheta ,\vartheta ,\varphi \left({\mu }^{\mathfrak{r}}{\zeta }_0\right))=0,\mathfrak{E}(\iota ,\vartheta ,\vartheta ,\varphi \left({\mu }^{\mathfrak{r}}{\zeta }_0\right))=0,\mathrm{for}\mathrm{all}\mathfrak{r}\in \mathbb{N}.$$

(30)

To demonstrate this, consider an arbitrary $\mathfrak{r}\in \mathbb{N}$ and assume that $\mathfrak{p},\mathfrak{q}\in \mathbb{N}$ such that $\mathfrak{p}>\mathfrak{r}$.

Since ${\mu }^{\mathfrak{p}}{\zeta }_0\le {\mu }^{\mathfrak{r}}{\zeta }_0\le {\zeta }_0\le {\displaystyle \frac{{\zeta }_0}{{\mu }^{\mathfrak{q}}}}$, and given that $\varphi$ and $\mathfrak{J}(\iota ,\vartheta ,\xi ,\phi )$ are non-decreasing functions while $\mathfrak{H}(\iota ,\vartheta ,\xi ,\phi )$ and $\mathfrak{E}(\iota ,\vartheta ,\xi ,\phi )$ are non-increasing functions with respect to $\phi$, it follows that, if $\mathfrak{p},\mathfrak{q}\in \mathbb{N}$ and $\mathfrak{r}\le \mathfrak{p}$, then

$$\begin{array}{c}\hfill 0<\mathfrak{J}(\iota ,\vartheta ,\vartheta ,\varphi \left({\mu }^{\mathfrak{p}}{\zeta }_0\right))\le \mathfrak{J}(\iota ,\vartheta ,\vartheta ,\varphi \left({\mu }^{\mathfrak{r}}{\zeta }_0\right))\le \mathfrak{J}(\iota ,\vartheta ,\vartheta ,\varphi \left({\zeta }_0\right))\le \mathfrak{J}\left(\iota ,\vartheta ,\vartheta ,\varphi \left({\displaystyle \frac{{\zeta }_0}{{\mu }^{\mathfrak{q}}}}\right)\right),\\ \hfill 1>\mathfrak{H}(\iota ,\vartheta ,\vartheta ,\varphi \left({\mu }^{\mathfrak{p}}{\zeta }_0\right))\ge \mathfrak{H}(\iota ,\vartheta ,\vartheta ,\varphi \left({\mu }^{\mathfrak{r}}{\zeta }_0\right))\ge \mathfrak{H}(\iota ,\vartheta ,\vartheta ,\varphi \left({\zeta }_0\right))\ge \mathfrak{H}\left(\iota ,\vartheta ,\vartheta ,\varphi \left({\displaystyle \frac{{\zeta }_0}{{\mu }^{\mathfrak{q}}}}\right)\right),\\ \hfill 1>\mathfrak{E}(\iota ,\vartheta ,\vartheta ,\varphi \left({\mu }^{\mathfrak{p}}{\zeta }_0\right))\ge \mathfrak{E}(\iota ,\vartheta ,\vartheta ,\varphi \left({\mu }^{\mathfrak{r}}{\zeta }_0\right))\ge \mathfrak{E}(\iota ,\vartheta ,\vartheta ,\varphi \left({\zeta }_0\right))\ge \mathfrak{E}\left(\iota ,\vartheta ,\vartheta ,\varphi \left({\displaystyle \frac{{\zeta }_0}{{\mu }^{\mathfrak{q}}}}\right)\right).\end{array}$$

(31)

Using the contractivity condition (19) for $\iota$ and $\vartheta$ leads to the conclusion that

$$\begin{array}{cc}\hfill \eta \left(\mathfrak{J}(\iota ,\vartheta ,\vartheta ,\varphi \left({\mu }^{\mathfrak{r}}{\zeta }_0\right))\right)& =\eta \left(\mathfrak{J}(\mathsf{\Gamma }\iota ,\mathsf{\Gamma }\vartheta ,\mathsf{\Gamma }\vartheta ,\varphi \left({\mu }^{\mathfrak{r}}{\zeta }_0\right))\right)\le \psi \left(\eta \left(\mathfrak{J}(\iota ,\vartheta ,\vartheta ,\varphi \left({\mu }^{\mathfrak{r}-1}{\zeta }_0\right))\right)\right),\hfill \\ \hfill \eta \left(\mathfrak{H}(\iota ,\vartheta ,\vartheta ,\varphi \left({\mu }^{\mathfrak{r}}{\zeta }_0\right))\right)& =\eta \left(\mathfrak{H}(\mathsf{\Gamma }\iota ,\mathsf{\Gamma }\vartheta ,\mathsf{\Gamma }\vartheta ,\varphi \left({\mu }^{\mathfrak{r}}{\zeta }_0\right))\right)\ge \psi \left(\eta \left(\mathfrak{H}(\iota ,\vartheta ,\vartheta ,\varphi \left({\mu }^{\mathfrak{r}-1}{\zeta }_0\right))\right)\right),\hfill \\ \hfill \eta \left(\mathfrak{E}(\iota ,\vartheta ,\vartheta ,\varphi \left({\mu }^{\mathfrak{r}}{\zeta }_0\right))\right)& =\eta \left(\mathfrak{E}(\mathsf{\Gamma }\iota ,\mathsf{\Gamma }\vartheta ,\mathsf{\Gamma }\vartheta ,\varphi \left({\mu }^{\mathfrak{r}}{\zeta }_0\right))\right)\ge \psi \left(\eta \left(\mathfrak{E}(\iota ,\vartheta ,\vartheta ,\varphi \left({\mu }^{\mathfrak{r}-1}{\zeta }_0\right))\right)\right).\hfill \end{array}$$

(32)

where we use $\mathfrak{J}(\iota ,\vartheta ,\vartheta ,\varphi \left({\mu }^{\mathfrak{r}-1}{\zeta }_0\right))>0,\mathfrak{H}(\iota ,\vartheta ,\vartheta ,\varphi \left({\mu }^{\mathfrak{r}-1}{\zeta }_0\right))<1,\mathfrak{E}(\iota ,\vartheta ,\vartheta ,\varphi \left({\mu }^{\mathfrak{r}-1}{\zeta }_0\right))<1$ by (31). By applying this reasoning repeatedly, we find that

$$\begin{array}{c}\hfill \eta \left(\mathfrak{J}(\iota ,\vartheta ,\vartheta ,\varphi \left({\mu }^{\mathfrak{r}-1}{\zeta }_0\right))\right)=\eta \left(\mathfrak{J}(\mathsf{\Gamma }\iota ,\mathsf{\Gamma }\vartheta ,\mathsf{\Gamma }\vartheta ,\varphi \left({\mu }^{\mathfrak{r}-1}{\zeta }_0\right))\right)\le \psi \left(\eta \left(\mathfrak{J}(\iota ,\vartheta ,\vartheta ,\varphi \left({\mu }^{\mathfrak{r}-2}{\zeta }_0\right))\right)\right),\\ \hfill \eta \left(\mathfrak{H}(\iota ,\vartheta ,\vartheta ,\varphi \left({\mu }^{\mathfrak{r}-1}{\zeta }_0\right))\right)=\eta \left(\mathfrak{H}(\mathsf{\Gamma }\iota ,\mathsf{\Gamma }\vartheta ,\mathsf{\Gamma }\vartheta ,\varphi \left({\mu }^{\mathfrak{r}-1}{\zeta }_0\right))\right)\ge \psi \left(\eta \left(\mathfrak{H}(\iota ,\vartheta ,\vartheta ,\varphi \left({\mu }^{\mathfrak{r}-2}{\zeta }_0\right))\right)\right),\\ \hfill \eta \left(\mathfrak{E}(\iota ,\vartheta ,\vartheta ,\varphi \left({\mu }^{\mathfrak{r}-1}{\zeta }_0\right))\right)=\eta \left(\mathfrak{E}(\mathsf{\Gamma }\iota ,\mathsf{\Gamma }\vartheta ,\mathsf{\Gamma }\vartheta ,\varphi \left({\mu }^{\mathfrak{r}-1}{\zeta }_0\right))\right)\ge \psi \left(\eta \left(\mathfrak{E}(\iota ,\vartheta ,\vartheta ,\varphi \left({\mu }^{\mathfrak{r}-2}{\zeta }_0\right))\right)\right),\end{array}$$

where we use $\mathfrak{J}(\iota ,\vartheta ,\vartheta ,\varphi \left({\mu }^{\mathfrak{r}-2}{\zeta }_0\right))>0,\mathfrak{H}(\iota ,\vartheta ,\vartheta ,\varphi \left({\mu }^{\mathfrak{r}-2}{\zeta }_0\right))<1,\mathfrak{E}(\iota ,\vartheta ,\vartheta ,\varphi \left({\mu }^{\mathfrak{r}-2}{\zeta }_0\right))<1$ by (31).

Given that $\psi$ is non-decreasing, it follows that

$$\begin{array}{cc}\hfill \psi \left(\eta \left(\mathfrak{J}(\mathsf{\Gamma }\iota ,\mathsf{\Gamma }\vartheta ,\mathsf{\Gamma }\vartheta ,\varphi \left({\mu }^{\mathfrak{r}-1}{\zeta }_0\right))\right)\right)& \le {\psi }^2\left(\eta \left(\mathfrak{J}(\iota ,\vartheta ,\vartheta ,\varphi \left({\mu }^{\mathfrak{r}-2}{\zeta }_0\right))\right)\right),\hfill \\ \hfill \psi \left(\eta \left(\mathfrak{H}(\mathsf{\Gamma }\iota ,\mathsf{\Gamma }\vartheta ,\mathsf{\Gamma }\vartheta ,\varphi \left({\mu }^{\mathfrak{r}-1}{\zeta }_0\right))\right)\right)& \ge {\psi }^2\left(\eta \left(\mathfrak{H}(\iota ,\vartheta ,\vartheta ,\varphi \left({\mu }^{\mathfrak{r}-2}{\zeta }_0\right))\right)\right),\hfill \\ \hfill \psi \left(\eta \left(\mathfrak{E}(\mathsf{\Gamma }\iota ,\mathsf{\Gamma }\vartheta ,\mathsf{\Gamma }\vartheta ,\varphi \left({\mu }^{\mathfrak{r}-1}{\zeta }_0\right))\right)\right)& \ge {\psi }^2\left(\eta \left(\mathfrak{E}(\iota ,\vartheta ,\vartheta ,\varphi \left({\mu }^{\mathfrak{r}-2}{\zeta }_0\right))\right)\right)\hfill \end{array}$$

(33)

By merging inequalities (32) and (33), we derive

$$\begin{array}{cc}\hfill \eta \left(\mathfrak{J}(\iota ,\vartheta ,\vartheta ,\varphi \left({\mu }^{\mathfrak{r}}{\zeta }_0\right))\right)& \le \psi \left(\eta \left(\mathfrak{J}(\mathsf{\Gamma }\iota ,\mathsf{\Gamma }\vartheta ,\mathsf{\Gamma }\vartheta ,\varphi \left({\mu }^{\mathfrak{r}-1}{\zeta }_0\right))\right)\right)\le {\psi }^2\left(\eta \left(\mathfrak{J}(\iota ,\vartheta ,\vartheta ,\varphi \left({\mu }^{\mathfrak{r}-2}{\zeta }_0\right))\right)\right),\hfill \\ \hfill \eta \left(\mathfrak{H}(\iota ,\vartheta ,\vartheta ,\varphi \left({\mu }^{\mathfrak{r}}{\zeta }_0\right))\right)& \ge \psi \left(\eta \left(\mathfrak{H}(\mathsf{\Gamma }\iota ,\mathsf{\Gamma }\vartheta ,\mathsf{\Gamma }\vartheta ,\varphi \left({\mu }^{\mathfrak{r}-1}{\zeta }_0\right))\right)\right)\ge {\psi }^2\left(\eta \left(\mathfrak{H}(\iota ,\vartheta ,\vartheta ,\varphi \left({\mu }^{\mathfrak{r}-2}{\zeta }_0\right))\right)\right),\hfill \\ \hfill \eta \left(\mathfrak{E}(\iota ,\vartheta ,\vartheta ,\varphi \left({\mu }^{\mathfrak{r}}{\zeta }_0\right))\right)& \ge \psi \left(\eta \left(\mathfrak{E}(\mathsf{\Gamma }\iota ,\mathsf{\Gamma }\vartheta ,\mathsf{\Gamma }\vartheta ,\varphi \left({\mu }^{\mathfrak{r}-1}{\zeta }_0\right))\right)\right)\ge {\psi }^2\left(\eta \left(\mathfrak{E}(\iota ,\vartheta ,\vartheta ,\varphi \left({\mu }^{\mathfrak{r}-2}{\zeta }_0\right))\right)\right).\hfill \end{array}$$

Inequality (31) allows us to apply this argument n times, leading to the result that

$$\begin{array}{c}\hfill \eta \left(\mathfrak{J}(\iota ,\vartheta ,\vartheta ,\varphi \left({\mu }^{\mathfrak{r}}{\zeta }_0\right))\right)\le {\psi }^{\mathfrak{p}}\left(\eta \left(\mathfrak{J}(\iota ,\vartheta ,\vartheta ,\varphi ({\mu }^{\mathfrak{r}}-\mathfrak{p}{\zeta }_0))\right)\right)={\psi }^{\mathfrak{p}}\left(\eta \left(\mathfrak{J}\left(\iota ,\vartheta ,\vartheta ,\varphi \left({\displaystyle \frac{{\zeta }_0}{{\mu }^{\mathfrak{p}-1}}}\right)\right)\right)\right),\\ \hfill \eta \left(\mathfrak{H}(\iota ,\vartheta ,\vartheta ,\varphi \left({\mu }^{\mathfrak{r}}{\zeta }_0\right))\right)\ge {\psi }^{\mathfrak{p}}\left(\eta \left(\mathfrak{H}(\iota ,\vartheta ,\vartheta ,\varphi ({\mu }^{\mathfrak{r}}-\mathfrak{p}{\zeta }_0))\right)\right)={\psi }^{\mathfrak{p}}\left(\eta \left(\mathfrak{H}\left(\iota ,\vartheta ,\vartheta ,\varphi \left({\displaystyle \frac{{\zeta }_0}{{\mu }^{\mathfrak{p}-1}}}\right)\right)\right)\right),\\ \hfill \eta \left(\mathfrak{E}(\iota ,\vartheta ,\vartheta ,\varphi \left({\mu }^{\mathfrak{r}}{\zeta }_0\right))\right)\ge {\psi }^{\mathfrak{p}}\left(\eta \left(\mathfrak{E}(\iota ,\vartheta ,\vartheta ,\varphi ({\mu }^{\mathfrak{r}}-\mathfrak{p}{\zeta }_0))\right)\right)={\psi }^{\mathfrak{p}}\left(\eta \left(\mathfrak{E}\left(\iota ,\vartheta ,\vartheta ,\varphi \left({\displaystyle \frac{{\zeta }_0}{{\mu }^{\mathfrak{p}-1}}}\right)\right)\right)\right),\end{array}$$

(34)

for all $\mathfrak{p}>\mathfrak{r}$. As a consequence,

$$\begin{array}{cc}\hfill & \underset{\mathfrak{p}\to \infty }{lim}{\displaystyle \frac{{\zeta }_0}{{\mu }^{\mathfrak{p}-\mathfrak{r}}}}\Rightarrow \underset{\mathfrak{p}\to \infty }{lim}\varphi \left({\displaystyle \frac{{\zeta }_0}{{\mu }^{\mathfrak{p}-\mathfrak{r}}}}\right)=\infty \hfill \\ \hfill & \Rightarrow \underset{\mathfrak{p}\to \infty }{lim}\mathfrak{J}\left(\iota ,\vartheta ,\vartheta ,\varphi \left({\displaystyle \frac{{\zeta }_0}{{\mu }^{\mathfrak{p}-\mathfrak{r}}}}\right)\right)=1\hfill \\ \hfill & \Rightarrow \underset{\mathfrak{p}\to \infty }{lim}\eta \left(\mathfrak{J}\left(\iota ,\vartheta ,\phi ,\varphi \left({\displaystyle \frac{{\zeta }_0}{{\mu }^{\mathfrak{p}-\mathfrak{r}}}}\right)\right)\right)=0.\hfill \\ \hfill & \underset{\mathfrak{p}\to \infty }{lim}{\displaystyle \frac{{\zeta }_0}{{\mu }^{\mathfrak{p}-\mathfrak{r}}}}\Rightarrow \underset{\mathfrak{p}\to \infty }{lim}\varphi \left({\displaystyle \frac{{\zeta }_0}{{\mu }^{\mathfrak{p}-\mathfrak{r}}}}\right)=\infty \hfill \\ \hfill & \Rightarrow \underset{\mathfrak{p}\to \infty }{lim}\mathfrak{H}\left(\iota ,\vartheta ,\vartheta ,\varphi \left({\displaystyle \frac{{\zeta }_0}{{\mu }^{\mathfrak{p}-\mathfrak{r}}}}\right)\right)=0\hfill \\ \hfill & \Rightarrow \underset{\mathfrak{p}\to \infty }{lim}\eta \left(\mathfrak{H}\left(\iota ,\vartheta ,\phi ,\varphi \left({\displaystyle \frac{{\zeta }_0}{{\mu }^{\mathfrak{p}-\mathfrak{r}}}}\right)\right)\right)=1.\hfill \\ \hfill & \underset{\mathfrak{p}\to \infty }{lim}{\displaystyle \frac{{\zeta }_0}{{\mu }^{\mathfrak{p}-\mathfrak{r}}}}\Rightarrow \underset{\mathfrak{p}\to \infty }{lim}\varphi \left({\displaystyle \frac{{\zeta }_0}{{\mu }^{\mathfrak{p}-\mathfrak{r}}}}\right)=\infty \hfill \\ \hfill & \Rightarrow \underset{\mathfrak{p}\to \infty }{lim}\mathfrak{E}\left(\iota ,\vartheta ,\vartheta ,\varphi \left({\displaystyle \frac{{\zeta }_0}{{\mu }^{\mathfrak{p}-\mathfrak{r}}}}\right)\right)=0\hfill \\ \hfill & \Rightarrow \underset{\mathfrak{p}\to \infty }{lim}\eta \left(\mathfrak{E}\left(\iota ,\vartheta ,\phi ,\varphi \left({\displaystyle \frac{{\zeta }_0}{{\mu }^{\mathfrak{p}-\mathfrak{r}}}}\right)\right)\right)=1.\hfill \end{array}$$

As the sequence $\left\{{\rho }_{\mathfrak{p}}=\eta \left(\mathfrak{J}\left(\iota ,\vartheta ,\vartheta ,\varphi \left({\displaystyle \frac{{\zeta }_0}{{\mu }^{\mathfrak{p}-\mathfrak{r}}}}\right)\right)\right)\right\}\to 0,\left\{{\rho }_{\mathfrak{p}}=\eta \left(\mathfrak{H}\left(\iota ,\vartheta ,\vartheta ,\varphi \left({\displaystyle \frac{{\zeta }_0}{{\mu }^{\mathfrak{p}-\mathfrak{r}}}}\right)\right)\right)\right\}\to 1$, $\left\{{\rho }_{\mathfrak{p}}=\eta \left(\mathfrak{E}\left(\iota ,\vartheta ,\vartheta ,\varphi \left({\displaystyle \frac{{\zeta }_0}{{\mu }^{\mathfrak{p}-\mathfrak{r}}}}\right)\right)\right)\right\}\to 1$ and $\eta \in \mathfrak{K}$, we have $\left\{{\psi }^{\mathfrak{p}}({\rho }_{\mathfrak{p})}\right\}\to 0$.

By (34), we deduce that $\eta \left(\mathfrak{J}(\iota ,\vartheta ,\vartheta ,\varphi \left({\mu }^{\mathfrak{r}}{\zeta }_0\right))\right)=0,\eta \left(\mathfrak{H}(\iota ,\vartheta ,\vartheta ,\varphi \left({\mu }^{\mathfrak{r}}{\zeta }_0\right))\right)=1,$$\eta \left(\mathfrak{E}\right(\iota ,\vartheta ,\vartheta ,$$\varphi \left({\mu }^{\mathfrak{r}}{\zeta }_0\right)\left)\right)=1$. In particular, as $\eta \in \mathfrak{K}$, Definition (10) implies that $\mathfrak{J}(\iota ,\vartheta ,\vartheta ,\varphi \left({\mu }^{\mathfrak{r}}{\zeta }_0\right))=1,\mathfrak{J}(\iota ,\vartheta ,\vartheta ,\varphi \left({\mu }^{\mathfrak{r}}{\zeta }_0\right))=0,\mathfrak{E}(\iota ,\vartheta ,\vartheta ,\varphi \left({\mu }^{\mathfrak{r}}{\zeta }_0\right))=0$. This indicates that (30) is satisfied.

To proceed, we need to demonstrate that $\mathfrak{J}(\iota ,\vartheta ,\vartheta ,\phi )=1$ for every $\phi >0$. Consider an arbitrary $\phi >0$. Since ${\displaystyle \underset{\mathfrak{p}\to \infty }{lim}\left({\mu }^{\mathfrak{p}}{\zeta }_0\right)=0}$ and ${\displaystyle \underset{\mathfrak{p}\to \infty }{lim}\varphi \left({\mu }^{\mathfrak{p}}{\zeta }_0\right)=\varphi \left(0\right)=0}$, there exists a natural number $\mathfrak{r}$ such that $\varphi \left({\mu }^{\mathfrak{r}}{\zeta }_0\right)<\phi$. Hence, $1=\mathfrak{J}(\iota ,\vartheta ,\vartheta ,\varphi \left({\mu }^{\mathfrak{r}}{\zeta }_0\right))\le \mathfrak{J}(\iota ,\vartheta ,\vartheta ,\phi )=1$, $0=\mathfrak{H}(\iota ,\vartheta ,\vartheta ,\varphi \left({\mu }^{\mathfrak{r}}{\zeta }_0\right))\le \mathfrak{H}(\iota ,\vartheta ,\vartheta ,\phi )=0$, and $0=\mathfrak{E}(\iota ,\vartheta ,\vartheta ,\varphi \left({\mu }^{\mathfrak{r}}{\zeta }_0\right))\le \mathfrak{E}(\iota ,\vartheta ,\vartheta ,\phi )=0$, so $\mathfrak{J}(\iota ,\vartheta ,\vartheta ,\phi )=1$, $\mathfrak{H}(\iota ,\vartheta ,\vartheta ,\phi )=0$ and $\mathfrak{E}(\iota ,\vartheta ,\vartheta ,\phi )=0$. By varying $\phi >0$, we can infer that $\iota =\vartheta$ based on the definiton of a GNMS, which contradicts the assumption that $\iota \ne \vartheta$. Therefore, $\mathsf{\Gamma }$ must have exactly one fixed point. □

Example 6.

Let $(\mathfrak{A},\mathfrak{J},\mathfrak{H},\mathfrak{E},\otimes ,\oplus )$ be a complete non-Archimedean GNMS, where

$\mathfrak{A}=[0,1]$, the set of real numbers between 0 and 1. The neutrosophic metric functions $\mathfrak{J}$, $\mathfrak{H}$, and $\mathfrak{E}$ are defined as

$$\mathfrak{J}(\alpha ,\omega ,\gamma ,u)={\displaystyle \frac{|\alpha -\omega |+|\omega -\gamma |}{u}},\mathfrak{H}(\alpha ,\omega ,\gamma ,u)={\displaystyle \frac{|\alpha -\gamma |}{u}},\mathfrak{E}(\alpha ,\omega ,\gamma ,u)={\displaystyle \frac{|\omega -\gamma |}{u}},$$

for all $\alpha ,\omega ,\gamma \in \mathfrak{A}$ and $u>0$.

Let the mapping $\mathsf{\Gamma }:\mathfrak{A}\to \mathfrak{A}$ be defined as $\mathsf{\Gamma }\left(\alpha \right)=1-\alpha$ for all $\alpha \in \mathfrak{A}$.

Now, choose $\mu =0.75$, $\varphi \left(x\right)=x^3$, and $\psi \left(x\right)=0.75x$, with $\eta =\mathfrak{H}$.

Verifying the Conditions:

1. Condition for $\mathfrak{J}$:

$$\eta \left(\mathfrak{J}\right(\mathsf{\Gamma }\alpha ,\mathsf{\Gamma }\omega ,\mathsf{\Gamma }\gamma ,\varphi \left(\mu \phi \right)\left)\right)\le \psi \left(\eta \right(\mathfrak{J}(\alpha ,\omega ,\gamma ,\varphi (\phi \left)\right)\left)\right).$$

Substituting the values:

$$\mathfrak{H}\left({\displaystyle \frac{\left|\right(1-\alpha )-(1-\omega \left)\right|+\left|\right(1-\omega )-(1-\gamma \left)\right|}{{\left(0.75\phi \right)}^3}}\right)\le 0.75·\mathfrak{H}\left({\displaystyle \frac{|\alpha -\omega |+|\omega -\gamma |}{{\phi }^3}}\right).$$

Since $\mathsf{\Gamma }\left(\alpha \right)=1-\alpha$ maintains a bounded distance, the inequality holds.

2. Condition for $\mathfrak{H}$:

$$\eta \left(\mathfrak{H}\right(\mathsf{\Gamma }\alpha ,\mathsf{\Gamma }\omega ,\mathsf{\Gamma }\gamma ,\varphi \left(\mu \phi \right)\left)\right)\ge \psi \left(\eta \right(\mathfrak{H}(\alpha ,\omega ,\gamma ,\varphi (\phi \left)\right)\left)\right).$$

Substituting the values:

$$\mathfrak{H}\left({\displaystyle \frac{\left|\right(1-\alpha )-(1-\gamma \left)\right|}{{\left(0.75\phi \right)}^3}}\right)\ge 0.75·\mathfrak{H}\left({\displaystyle \frac{|\alpha -\gamma |}{{\phi }^3}}\right).$$

This inequality holds because the difference $|1-\alpha |$ flips the values of α but does not increase the overall distance.

3. Condition for $\mathfrak{E}$:

$$\eta \left(\mathfrak{E}\right(\mathsf{\Gamma }\alpha ,\mathsf{\Gamma }\omega ,\mathsf{\Gamma }\gamma ,\varphi \left(\mu \phi \right)\left)\right)\ge \psi \left(\eta \right(\mathfrak{E}(\alpha ,\omega ,\gamma ,\varphi (\phi \left)\right)\left)\right).$$

Substituting the values:

$$\mathfrak{H}\left({\displaystyle \frac{\left|\right(1-\omega )-(1-\gamma \left)\right|}{{\left(0.75\phi \right)}^3}}\right)\ge 0.75·\mathfrak{H}\left({\displaystyle \frac{|\omega -\gamma |}{{\phi }^3}}\right).$$

This is satisfied for the same reason as above, where the flipping does not affect the distance significantly.

Existence of a Fixed Point:

For ${\alpha }_0=0.5$, we have

$$\underset{\phi \to \infty }{lim}\mathfrak{J}({\alpha }_0,{\alpha }_0,\mathsf{\Gamma }{\alpha }_0,\phi )=1,\underset{\phi \to \infty }{lim}\mathfrak{H}({\alpha }_0,{\alpha }_0,\mathsf{\Gamma }{\alpha }_0,\phi )=0,\underset{\phi \to \infty }{lim}\mathfrak{E}({\alpha }_0,{\alpha }_0,\mathsf{\Gamma }{\alpha }_0,\phi )=0.$$

Thus, $\mathsf{\Gamma }\left(\alpha \right)$ has a fixed point at $\alpha =0.5$.

Since $\mathsf{\Gamma }\left(\alpha \right)$ satisfies all the conditions of the theorem, the fixed point at $\alpha =0.5$ is unique.

5. Conclusions

In this study, we established several significant fixed-point theorems for self-mappings in non-Archimedean GNMSs, focusing on $\chi$-contractions and $\chi$-weak contractions. Our results demonstrate that the proposed class of contractive conditions for auxiliary functions effectively extends the existing framework. This advancement not only broadens the scope of our findings but also enhances their applicability in contexts where symmetry plays a crucial role in the behaviour of mappings. The results underscore the importance of symmetry in ensuring the existence and uniqueness of fixed points, providing a robust foundation for future applications in metric space theory. Furthermore, our findings show that the developed conditions can yield valuable insights across various mathematical domains, reinforcing their versatility. However, the applicability of these results is limited to symmetric settings, and both $\chi$-contractions and $\chi$-weak contractions must be strictly increasing functions; otherwise, the conditions may not hold. Looking ahead, future research could explore the applicability of these generalized results in more dynamic and complex systems, which would further enhance both their practical utility and theoretical significance. Additionally, investigating the role of symmetry in influencing the dynamics of fixed-point behaviour could offer deeper insights and contribute to advancements in mathematical theory and its applications.