A novel stability analysis of functional equation in neutrosophic normed spaces

Abstract

The analysis of stability in functional equations (FEs) within neutrosophic normed spaces is a significant challenge due to the inherent uncertainties and complexities involved. This paper proposes a novel approach to address this challenge, offering a comprehensive framework for investigating stability properties in such contexts. Neutrosophic normed spaces are a generalization of traditional normed spaces that incorporate neutrosophic logic. By providing a systematic methodology for addressing stability concerns in neutrosophic normed spaces, our approach facilitates enhanced understanding and control of complex systems characterized by indeterminacy and uncertainty. The primary focus of this research is to propose a novel class of Euler-Lagrange additive FE and investigate its Ulam-Hyers stability in neutrosophic normed spaces. Direct and fixed point techniques are utilized to achieve the required results.

1 Introduction

Lotfi A. Zadeh, a mathematician and computer scientist, introduced the idea of fuzzy sets in his groundbreaking paper entitled “Fuzzy Sets,” published in 1965 [1]. Zadeh’s motivation was to address the limitations of classical set theory, which relies on crisp, well-defined boundaries for membership. The extension of fuzzy sets known as intuitionistic fuzzy sets (IFS) was initially proposed by Atanassov in 1983 [2, 3]. IFS provide a framework for dealing with uncertainty, vagueness, and hesitation more comprehensively than traditional fuzzy sets. A neutrosophic set is a mathematical concept introduced as an extension of classical set theory [4, 5]. Neutrosophic sets provide a way to handle indeterminacy, uncertainty, and incomplete information more flexibly.

Fuzzy normed spaces (FNS) are mathematical structures that extend the concept of normed spaces to include fuzzy numbers. Katsaras introduced the idea of FNS, a vector space equipped with a fuzzy norm, where the norm values are fuzzy numbers rather than real numbers [6]. The idea of intuitionistic fuzzy normed spaces (IFNS) was proposed in 2006 [7]. IFNS represent a fusion of concepts from fuzzy mathematics, intuitionistic fuzzy sets, and normed spaces, providing a versatile framework for handling uncertainty and imprecision in mathematical modeling and analysis. Neutrosophic normed linear spaces [8, 9] extend the ideas of neutrosophic sets to linear algebraic structures, offering a more expressive way to represent and handle uncertainty in vector spaces. Neutrosophic concepts have been applied across a wide range of mathematical fields such as groups, subgroups [10], vector spaces [11], homomorphisms of rings [12, 13], linear transformations [14], number theory [15], graph theory [16], measure theory, integral theory, probability theory [17], etc. Neutrosophic normed linear spaces find applications in various scientific and real-time applications like decision-making, control systems, optimization, image processing, pattern recognition, medical diagnosis, finance and risk management, information retrieval, and artificial intelligence [18–24].

Several mathematicians have derived the results of fixed point (F-P) theory using neutrosophic concepts. Ishtiaq et al. [25] proved F-P results in the framework of an orthogonal neutrosophic metric space. The common F-P results in the context of a neutrosophic metric space were established using contraction mapping [26]. Salama and Alblowi [27] studied the neutrosophic topological spaces, whereas Al-Omeri et al. [28] analyzed a neutrosophic cone metric space. Riaz et al. [29] discussed the F-P results for ξ-chainable neutrosophic and generalized neutrosophic cone metric spaces. Sharma et al. [30] studied the generalized summability using difference operators on neutrosophic normed spaces. One of the most prevalent stability theories was introduced by Ulam [31] and further developed by Hyers [32]. This theory is called the Ulam-Hyers stability and has applications in various branches of mathematics, including differential equations, functional analysis, and dynamic systems. Many researchers have worked on this theory and established the stability results in different normed spaces [33–40]. Recently, Agilan et al. introduced new kinds of FEs and established the Ulam-Hyers stability of the newly proposed equations in a variety of normed spaces [41–44]. Some related applications are also discussed in [45–48].

This article presents a novel category of Euler-Lagrange additive FE. The Ulam-Hyers stability of the newly introduced equation is analyzed in neutrosophic normed linear spaces using two techniques: direct and fixed point techniques. The stability analysis of the newly introduced equation holds significance due to the distinctive characteristics and potentially broad applications of neutrosophic normed spaces. It is noteworthy that, for the first time in the literature, the stability of an FE has been examined within neutrosophic normed spaces. The uniqueness of this endeavor underscores the importance of this research.

This study aims to achieve the following primary objectives:

• (a) Extend and enhance the current body of work on neutrosophic normed linear spaces.

• (b) Ascertain the uniqueness of the solution for the newly proposed Euler-Lagrange additive FE.

• (c) Derive the Ulam-Hyers stability of the newly proposed equation in neutrosophic normed linear spaces using the F-P method.

2 Definitions on neutrosophic normed spaces

Definition 2.1

The Seven-tuple $(\mathbb{A}, \mathfrak{A}_{a}, \mathfrak{B}_{a}, \mathfrak{C}_{a} \ast, \diamond,\oslash)$ is said to be a neutrosophic normed space (for short, NNS) if $\mathbb{A}$ is a vector space, ∗ is a continuous κ-norm, ⋄ and ⊘ is a continuous κ− conorm, and $\mathfrak{A}_{a}$, $\mathfrak{B}_{a}$, $\mathfrak{C}_{a}$ are fuzzy sets on $\mathbb{A}\times(0,\infty)$ satisfying the following conditions. For every $p,q \in\mathbb{A}$ and $s, \kappa> 0$,

• (A1) $\mathfrak{A}_{a}(p, \kappa)+\mathfrak{B}_{a}(p, \kappa )+\mathfrak{C}_{a}(p, \kappa) \leq3$,

• (A2) $0\leq\mathfrak{A}_{a}(p, \kappa)\leq1$, $0\leq\mathfrak {B}_{a}(p, \kappa)\leq1$, $0\leq\mathfrak{C}_{a}(p, \kappa) \leq1$,

• (A3) $\mathfrak{A}_{a}(p, \kappa)>0$,

• (A4) $\mathfrak{A}_{a}(p, \kappa)=1$, if and only if $p=0$.

• (A5) $\mathfrak{A}_{a}(\alpha p, \kappa)= \mathfrak{A}_{a} (p,\frac {\kappa}{|\alpha|} )$ for each $\alpha\ne0$,

• (A6) $\mathfrak{A}_{a}(p,\kappa) \ast\mathfrak{A}_{a}(q,s) \leq \mathfrak{A}_{a}(p+q,\kappa+s)$,

• (A7) $\mathfrak{A}_{a}(p,\cdot): (0, \infty) \to[0,1]$ is continuous,

• (A8) $\lim_{\kappa\to\infty} \mathfrak{A}_{a}(p, \kappa) = 1$ and $\lim_{\kappa\to0} \mathfrak{A}_{a}(p, \kappa) = 0$,

• (A9) $\mathfrak{B}_{a}(p, \kappa)<1$,

• (A10) $\mathfrak{B}_{a}(p, \kappa)=0$, if and only if $p=0$.

• (A11) $\mathfrak{B}_{a}(\alpha p, \kappa)= \mathfrak{B}_{a} (p,\frac {\kappa}{|\alpha|} )$ for each $\alpha\ne0$,

• (A12) $\mathfrak{B}_{a}(p,\kappa) \diamond\mathfrak{B}_{a}(q,s) \geq \mathfrak{B}_{a}(p+q,\kappa+s)$,

• (A13) $\mathfrak{B}_{a}(p,\cdot): (0, \infty) \to[0,1]$ is continuous,

• (A14) $\lim_{\kappa\to\infty} \mathfrak{B}_{a}(p, \kappa) = 0$ and $\lim_{\kappa\to0} \mathfrak{B}_{a}(p, \kappa) = 1$

• (A15) $\mathfrak{C}_{a}(p, \kappa)<1$,

• (A16) $\mathfrak{C}_{a}(p, \kappa)=0$, if and only if $p=0$.

• (A17) $\mathfrak{C}_{a}(\alpha p, \kappa)= \mathfrak{C}_{a} (p,\frac {\kappa}{|\alpha|} )$ for each $\alpha\ne0$,

• (A18) $\mathfrak{C}_{a}(p,\kappa) \oslash\mathfrak{C}_{a}(q,s) \geq \mathfrak{C}_{a}(p+q,\kappa+s)$,

• (A19) $\mathfrak{C}_{a}(p,\cdot): (0, \infty) \to[0,1]$ is continuous,

• (A20) $\lim_{\kappa\to\infty} \mathfrak{C}_{a}(p, \kappa) = 0$ and $\lim_{\kappa\to0} \mathfrak{C}_{a}(p, \kappa) = 1$.

3 Stability results: direct method

The newly proposed Euler-Lagrange additive FE is as follows:

$$\begin{aligned} &\mathfrak{T}_{1}\mathscr{A}_{1} ( \mathcal{P}_{1} \mathscr{X}_{a}+\mathcal{Q}_{1} \mathscr{V}_{a}+\mathcal{R}_{1} \mathscr{W}_{a} )+\mathcal{P}_{1} \mathscr{A}_{1} \bigl( \mathcal{Q}_{1}\mathcal{R}_{1} (\mathscr{X}_{a}- \mathscr{V}_{a} ) \bigr) \\ &\quad \quad{}+ \mathcal{Q}_{1} \mathscr{A}_{1} \bigl( \mathcal{P}_{1}\mathcal{R}_{1} (\mathscr{V}_{a}- \mathscr{W}_{a} ) \bigr)+\mathcal{R}_{1} \mathscr{A}_{1} \bigl(\mathcal{P}_{1} \mathcal{Q}_{1} (\mathscr{W}_{a}-\mathscr{X}_{a} ) \bigr) \\ &\quad= \mathfrak{T}_{1} \bigl(\mathcal{P}_{1} \mathscr{A}_{1} (\mathscr{X}_{a} ) + \mathcal{Q}_{1} \mathscr{A}_{1} (\mathscr{V}_{a} )+\mathcal{R}_{1} \mathscr{A}_{1} ( \mathscr{W}_{a} ) \bigr), \end{aligned}$$

(3.1)

where $\mathcal{P}_{1},\mathcal{Q}_{1},\mathcal{R}_{1}\in\mathbb{R}$ with $\mathcal{P}_{1}$, $\mathcal{Q}_{1}$, $\mathcal{R}_{1} \ne0$ and $\mathfrak{T}_{1}=\mathcal{P}_{1}+\mathcal{Q}_{1}+\mathcal{R}_{1}\ne0$ in neutrosophic normed space using direct and F-P methods.

Assume that $(\mathcal{M},\mathfrak{A}_{a}',\mathfrak{B}_{a}',\mathfrak {C}_{a}' )$ is a neutrosophic normed linear space and $(\mathcal {M},\mathfrak{A}_{a},\mathfrak{B}_{a},\mathfrak{C}_{a} )$ is a neutrosophic Banach space. Let $\mathcal{L}$ be a linear space. Then,

$$\begin{aligned} \mathfrak{Z}(\mathscr{X}_{a},\mathscr{V}_{a}, \mathscr{W}_{a}) &= \mathfrak{T}_{1}\mathscr{A}_{1} (\mathcal{P}_{1} \mathscr{X}_{a}+\mathcal{Q}_{1} \mathscr{V}_{a}+\mathcal{R}_{1} \mathscr{W}_{a} )+\mathcal{P}_{1} \mathscr{A}_{1} \bigl( \mathcal{Q}_{1}\mathcal{R}_{1} (\mathscr{X}_{a}- \mathscr{V}_{a} ) \bigr) \\ &\quad{}+ \mathcal{Q}_{1} \mathscr{A}_{1} \bigl( \mathcal{P}_{1}\mathcal{R}_{1} (\mathscr{V}_{a}- \mathscr{W}_{a} ) \bigr)+\mathcal{R}_{1} \mathscr{A}_{1} \bigl(\mathcal{P}_{1} \mathcal{Q}_{1} (\mathscr{W}_{a}-\mathscr{X}_{a} ) \bigr) \\ &\quad{}- \mathfrak{T}_{1} \bigl(\mathcal{P}_{1} \mathscr{A}_{1} (\mathscr{X}_{a} ) + \mathcal{Q}_{1} \mathscr{A}_{1} (\mathscr{V}_{a} )+\mathcal{R}_{1} \mathscr{A}_{1} ( \mathscr{W}_{a} ) \bigr), \end{aligned}$$

where $\mathfrak{T}_{1}=\mathcal{P}_{1}+\mathcal{Q}_{1}+\mathcal {R}_{1}$, $\mathcal{P}_{1},\mathcal{Q}_{1},\mathcal{R}_{1}\in\mathbb{R}$ and $\mathcal{P}_{1}$, $\mathcal{Q}_{1}$, $\mathcal{R}_{1} \ne0$ $\forall\mathscr{X}_{a},\mathscr{V}_{a},\mathscr{W}_{a} \in\mathcal{L}$.

Theorem 3.1

Let $N :\mathcal{L} \times\mathcal{L} \times\mathcal{L} \longrightarrow\mathcal{M}$ be a mapping with the condition $0 < (\frac{\mathscr{X}_{a}}{\mathfrak{T}_{1}} )^{\eta}< 1$, then

$$\begin{aligned} \left . \begin{aligned} &\mathfrak{A}_{a}' \bigl(N \bigl(\mathfrak{T}_{1}^{n\eta}\mathscr{X}_{a}, \mathfrak{T}_{1}^{n\eta}\mathscr{V}_{a}, \mathfrak{T}_{1}^{n\eta}\mathscr{W}_{a} \bigr), \upsilon\bigr) \geq\mathfrak{A}_{a}' \bigl({ \mathscr{X}_{a}^{n\eta}N (\mathscr{X}_{a}, \mathscr{V}_{a},\mathscr{W}_{a} ),\upsilon} \bigr), \\ &\mathfrak{B}_{a}' \bigl(N \bigl( \mathfrak{T}_{1}^{n\eta}\mathscr{X}_{a}, \mathfrak{T}_{1}^{n\eta}\mathscr{V}_{a}, \mathfrak{T}_{1}^{n\eta}\mathscr{W}_{a} \bigr), \upsilon\bigr) \leq\mathfrak{B}_{a}' \bigl({ \mathscr{X}_{a}^{n\eta}N (\mathscr{X}_{a}, \mathscr{V}_{a},\mathscr{W}_{a} ),\upsilon} \bigr), \\ &\mathfrak{C}_{a}' \bigl(N \bigl( \mathfrak{T}_{1}^{n\eta}\mathscr{X}_{a}, \mathfrak{T}_{1}^{n\eta}\mathscr{V}_{a}, \mathfrak{T}_{1}^{n\eta}\mathscr{W}_{a} \bigr), \upsilon\bigr) \leq\mathfrak{C}_{a}' \bigl({ \mathscr{X}_{a}^{n\eta}N (\mathscr{X}_{a}, \mathscr{V}_{a},\mathscr{W}_{a} ),\upsilon} \bigr) \end{aligned} \right \} \end{aligned}$$

(3.2)

and

$$\begin{aligned} \left . \begin{aligned} &\lim_{n \to\infty} \mathfrak{A}_{a}' \bigl(N \bigl(\mathfrak{T}_{1}^{{\eta}n} \mathscr{X}_{a},\mathfrak{T}_{1}^{{\eta}n} \mathscr{V}_{a},\mathfrak{T}_{1}^{{\eta}n} \mathscr{W}_{a} \bigr),a^{{\eta}n} \upsilon\bigr) = 1, \\ &\lim_{n \to\infty} \mathfrak{B}_{a}' \bigl(N \bigl(\mathfrak{T}_{1}^{{\eta}n} \mathscr{X}_{a}, \mathfrak{T}_{1}^{{\eta}n} \mathscr{V}_{a}, \mathfrak{T}_{1}^{{\eta}n} \mathscr{W}_{a} \bigr),a^{{\eta}n} \upsilon\bigr) = 0, \\ &\lim_{n \to\infty} \mathfrak{C}_{a}' \bigl(N \bigl(\mathfrak{T}_{1}^{{\eta}n} \mathscr{X}_{a}, \mathfrak{T}_{1}^{{\eta}n} \mathscr{V}_{a}, \mathfrak{T}_{1}^{{\eta}n} \mathscr{W}_{a} \bigr),a^{{\eta}n} \upsilon\bigr) = 0. \end{aligned} \right \} \end{aligned}$$

(3.3)

Assume that a mapping $\mathscr{A}_{1}:\mathcal{L} \to\mathcal{M}$ satisfies the inequality

$$\begin{aligned} \left . \begin{aligned} &\mathfrak{A}_{a} \bigl( \mathfrak{Z}(\mathscr{X}_{a},\mathscr{V}_{a}, \mathscr{W}_{a}),\upsilon\bigr) \geq\mathfrak{A}_{a}' \bigl(N (\mathscr{X}_{a},\mathscr{V}_{a}, \mathscr{W}_{a} ),\upsilon\bigr), \\ &\mathfrak{B}_{a} \bigl(\mathfrak{Z}(\mathscr{X}_{a}, \mathscr{V}_{a},\mathscr{W}_{a}),\upsilon\bigr) \leq \mathfrak{B}_{a}' \bigl(N (\mathscr{X}_{a}, \mathscr{V}_{a},\mathscr{W}_{a} ),\upsilon\bigr), \\ &\mathfrak{C}_{a} \bigl(\mathfrak{Z}(\mathscr{X}_{a}, \mathscr{V}_{a},\mathscr{W}_{a}),\upsilon\bigr) \leq \mathfrak{C}_{a}' \bigl(N (\mathscr{X}_{a}, \mathscr{V}_{a},\mathscr{W}_{a} ),\upsilon\bigr) \end{aligned} \right \} \end{aligned}$$

(3.4)

and ∃ unique additive function $\mathcal{A}_{1}:\mathcal{L} \longrightarrow\mathcal{M}$

$$\begin{aligned} \left . \begin{aligned} &\mathfrak{A}_{a} \bigl( \mathscr{A}_{1}(\mathscr{X}_{a}) - \mathcal{A}_{1}( \mathscr{X}_{a}),\upsilon\bigr) \geq\mathfrak{A}_{a}' \bigl(N (\mathscr{X}_{a},\mathscr{X}_{a}, \mathscr{X}_{a} ),\mathfrak{T}_{1} \vert \mathfrak{T}_{1}-\mathscr{X}_{a} \vert \upsilon\bigr), \\ &\mathfrak{B}_{a} \bigl(\mathscr{A}_{1}( \mathscr{X}_{a}) - \mathcal{A}_{1}(\mathscr{X}_{a}), \upsilon\bigr) \leq\mathfrak{B}_{a}' \bigl(N ( \mathscr{X}_{a},\mathscr{X}_{a},\mathscr{X}_{a} ),\mathfrak{T}_{1} \vert \mathfrak{T}_{1}- \mathscr{X}_{a} \vert \upsilon\bigr), \\ &\mathfrak{C}_{a} \bigl(\mathscr{A}_{1}( \mathscr{X}_{a}) - \mathcal{A}_{1}(\mathscr{X}_{a}), \upsilon\bigr) \leq\mathfrak{C}_{a}' \bigl(N ( \mathscr{X}_{a},\mathscr{X}_{a},\mathscr{X}_{a} ),\mathfrak{T}_{1} \vert \mathfrak{T}_{1}- \mathscr{X}_{a} \vert \upsilon\bigr) \end{aligned} \right \} \end{aligned}$$

(3.5)

with the conditions $\eta\in\{1,-1\}$, where $\mathfrak {T}_{1}=\mathcal{P}_{1}+\mathcal{Q}_{1}+\mathcal{R}_{1}$.

Proof

Let us consider $(\mathscr{X}_{a},\mathscr{V}_{a},\mathscr{W}_{a})$ by $(\mathscr{X}_{a},\mathscr{X}_{a},\mathscr{X}_{a})$ in (3.4), we reach

$$\begin{aligned} \left . \begin{aligned} &\mathfrak{A}_{a} \bigl( \mathfrak{T}_{1}\mathscr{A}_{1} (\mathfrak{T}_{1} \mathscr{X}_{a} ) - \mathfrak{T}_{1}^{2} \mathscr{A}_{1} (\mathscr{X}_{a} ),\upsilon\bigr) \geq \mathfrak{A}_{a}' \bigl(N (\mathscr{X}_{a}, \mathscr{X}_{a},\mathscr{X}_{a} ),\upsilon\bigr), \\ &\mathfrak{B}_{a} \bigl(\mathfrak{T}_{1} \mathscr{A}_{1} (\mathfrak{T}_{1}\mathscr{X}_{a} ) - \mathfrak{T}_{1}^{2} \mathscr{A}_{1} ( \mathscr{X}_{a} ),\upsilon\bigr) \leq\mathfrak{B}_{a}' \bigl(N (\mathscr{X}_{a},\mathscr{X}_{a}, \mathscr{X}_{a} ),\upsilon\bigr), \\ &\mathfrak{C}_{a} \bigl(\mathfrak{T}_{1} \mathscr{A}_{1} (\mathfrak{T}_{1}\mathscr{X}_{a} ) - \mathfrak{T}_{1}^{2} \mathscr{A}_{1} ( \mathscr{X}_{a} ),\upsilon\bigr) \leq\mathfrak{C}_{a}' \bigl(N (\mathscr{X}_{a},\mathscr{X}_{a}, \mathscr{X}_{a} ),\upsilon\bigr). \end{aligned} \right \} \end{aligned}$$

(3.6)

By applying the conditions of neutrosophic normed space, we arrive

$$\begin{aligned} \left . \begin{aligned} &\mathfrak{A}_{a} \biggl({ \frac{\mathscr{A}_{1}(\mathfrak{T}_{1}\mathscr{X}_{a})}{\mathfrak {T}_{1}}-\mathscr{A}_{1}(\mathscr{X}_{a}), \frac{\upsilon}{\mathfrak{T}_{1}^{2}}} \biggr) \geq\mathfrak{A}_{a}' \bigl(N (\mathscr{X}_{a},\mathscr{X}_{a}, \mathscr{X}_{a}),\upsilon\bigr), \\ &\mathfrak{B}_{a} \biggl({\frac{\mathscr{A}_{1}(\mathfrak{T}_{1}\mathscr{X}_{a})}{\mathfrak{T}_{1}}- \mathscr{A}_{1}(\mathscr{X}_{a}), \frac{\upsilon}{\mathfrak{T}_{1}^{2}}} \biggr) \leq\mathfrak{B}_{a}' \bigl(N ( \mathscr{X}_{a},\mathscr{X}_{a},\mathscr{X}_{a}), \upsilon\bigr), \\ &\mathfrak{C}_{a} \biggl({\frac{\mathscr{A}_{1}(\mathfrak{T}_{1}\mathscr{X}_{a})}{\mathfrak{T}_{1}}- \mathscr{A}_{1}(\mathscr{X}_{a}), \frac{\upsilon}{\mathfrak{T}_{1}^{2}}} \biggr) \leq\mathfrak{C}_{a}' \bigl(N ( \mathscr{X}_{a},\mathscr{X}_{a},\mathscr{X}_{a}), \upsilon\bigr). \end{aligned} \right \} \end{aligned}$$

(3.7)

Replacing $\mathscr{X}_{a}$ by $\mathfrak{T}_{1}^{n}\mathscr{X}_{a}$ in (3.7), we get

$$\begin{aligned} \left . \begin{aligned} &\mathfrak{A}_{a} \biggl({ \frac{\mathscr{A}_{1}(\mathfrak{T}_{1}^{n+1} \mathscr{X}_{a})}{\mathfrak {T}_{1}}-\mathscr{A}_{1}\bigl(\mathfrak{T}_{1}^{n} \mathscr{X}_{a}\bigr), \frac{\upsilon}{\mathfrak{T}_{1}^{2}}} \biggr) \geq \mathfrak{A}_{a}' \bigl(N \bigl(\mathfrak{T}_{1}^{n} \mathscr{X}_{a},\mathfrak{T}_{1}^{n} \mathscr{X}_{a},\mathfrak{T}_{1}^{n} \mathscr{X}_{a}\bigr),\upsilon\bigr), \\ &\mathfrak{B}_{a} \biggl({\frac{\mathscr{A}_{1}(\mathfrak{T}_{1}^{n+1} \mathscr{X}_{a})}{\mathfrak{T}_{1}} - \mathscr{A}_{1}\bigl(\mathfrak{T}_{1}^{n} \mathscr{X}_{a}\bigr), \frac{\upsilon}{\mathfrak{T}_{1}^{2}}} \biggr) \leq \mathfrak{B}_{a}' \bigl(N \bigl(\mathfrak{T}_{1}^{n} \mathscr{X}_{a},\mathfrak{T}_{1}^{n} \mathscr{X}_{a},\mathfrak{T}_{1}^{n} \mathscr{X}_{a}\bigr),\upsilon\bigr), \\ &\mathfrak{C}_{a} \biggl({\frac{\mathscr{A}_{1}(\mathfrak{T}_{1}^{n+1} \mathscr{X}_{a})}{\mathfrak{T}_{1}} - \mathscr{A}_{1}\bigl(\mathfrak{T}_{1}^{n} \mathscr{X}_{a}\bigr), \frac{\upsilon}{\mathfrak{T}_{1}^{2}}} \biggr) \leq \mathfrak{C}_{a}' \bigl(N \bigl(\mathfrak{T}_{1}^{n} p,\mathfrak{T}_{1}^{n} \mathscr{X}_{a}, \mathfrak{T}_{1}^{n} \mathscr{X}_{a}\bigr), \upsilon\bigr). \end{aligned} \right \} \end{aligned}$$

(3.8)

Using (3.8) and conditions of neutrosophic normed space, we arrive

$$\begin{aligned} \left . \begin{aligned} &\mathfrak{A}_{a} \biggl({ \frac{\mathscr{A}_{1}(\mathfrak{T}_{1}^{n+1} \mathscr{X}_{a})}{\mathfrak {T}_{1}^{(n+1)}}-\frac{\mathscr{A}_{1}(\mathfrak{T}_{1}^{n} \mathscr{X}_{a})}{\mathfrak{T}_{1}^{n}}, \frac{\upsilon}{\mathfrak {T}_{1}^{n+2}}} \biggr)\geq \mathfrak{A}_{a}' \biggl(N (\mathscr{X}_{a}, \mathscr{X}_{a},\mathscr{X}_{a}),\frac{\upsilon}{\mathscr{X}_{a}^{n}} \biggr), \\ &\mathfrak{B}_{a} \biggl({\frac{\mathscr{A}_{1}(\mathfrak{T}_{1}^{n+1} \mathscr{X}_{a})}{\mathfrak{T}_{1}^{(n+1)}}- \frac{\mathscr{A}_{1}(\mathfrak{T}_{1}^{n} \mathscr{X}_{a})}{\mathfrak {T}_{1}^{n}}, \frac{\upsilon}{\mathfrak{T}_{1}^{n+2}}} \biggr) \leq \mathfrak{B}_{a}' \biggl(N (\mathscr{X}_{a},\mathscr{X}_{a}, \mathscr{X}_{a}),\frac{\upsilon}{\mathscr{X}_{a}^{n}} \biggr), \\ &\mathfrak{C}_{a} \biggl({\frac{\mathscr{A}_{1}(\mathfrak{T}_{1}^{n+1} \mathscr{X}_{a})}{\mathfrak{T}_{1}^{(n+1)}}- \frac{\mathscr{A}_{1}(\mathfrak{T}_{1}^{n} \mathscr{X}_{a})}{\mathfrak {T}_{1}^{n}}, \frac{\upsilon}{\mathfrak{T}_{1}^{n+2}}} \biggr) \leq \mathfrak{C}_{a}' \biggl(N (\mathscr{X}_{a},\mathscr{X}_{a}, \mathscr{X}_{a}),\frac{\upsilon}{\mathscr{X}_{a}^{n}} \biggr). \end{aligned} \right \} \end{aligned}$$

(3.9)

Let υ by $\mathscr{X}_{a}^{n}\upsilon$ in (3.9), then

$$\begin{aligned} \left . \begin{aligned} &\mathfrak{A}_{a} \biggl({ \frac{\mathscr{A}_{1}(\mathfrak{T}_{1}^{n+1} \mathscr{X}_{a})}{\mathfrak {T}_{1}^{(n+1)}}-\frac{\mathscr{A}_{1}(\mathfrak{T}_{1}^{n} \mathscr{X}_{a})}{\mathfrak{T}_{1}^{n}}, \frac{\upsilon\cdot\mathscr{X}_{a}^{n}}{\mathfrak{T}_{1}^{n+2}}} \biggr)\geq \mathfrak{A}_{a}' \bigl(N (\mathscr{X}_{a}, \mathscr{X}_{a},\mathscr{X}_{a}),\upsilon\bigr), \\ &\mathfrak{B}_{a} \biggl({\frac{\mathscr{A}_{1}(\mathfrak{T}_{1}^{n+1} \mathscr{X}_{a})}{\mathfrak{T}_{1}^{(n+1)}}- \frac{\mathscr{A}_{1}(\mathfrak{T}_{1}^{n}\mathscr{X}_{a})}{\mathfrak {T}_{1}^{n}}, \frac{\upsilon\cdot\mathscr{X}_{a}^{n}}{\mathfrak {T}_{1}^{n+2}}} \biggr)\leq\mathfrak{B}_{a}' \bigl(N (\mathscr{X}_{a},\mathscr{X}_{a}, \mathscr{X}_{a}),\upsilon\bigr), \\ &\mathfrak{C}_{a} \biggl({\frac{\mathscr{A}_{1}(\mathfrak{T}_{1}^{n+1} \mathscr{X}_{a})}{\mathfrak{T}_{1}^{(n+1)}}- \frac{\mathscr{A}_{1}(\mathfrak{T}_{1}^{n}\mathscr{X}_{a})}{\mathfrak {T}_{1}^{n}}, \frac{\upsilon\cdot\mathscr{X}_{a}^{n}}{\mathfrak {T}_{1}^{n+2}}} \biggr)\leq\mathfrak{C}_{a}' \bigl(N (\mathscr{X}_{a},\mathscr{X}_{a}, \mathscr{X}_{a}),\upsilon\bigr). \end{aligned} \right \} \end{aligned}$$

(3.10)

It is easy to see that

$$ \frac{\mathscr{A}_{1}(\mathfrak{T}_{1}^{n}\mathscr{X}_{a})}{\mathfrak{T}_{1}^{n}} - \mathscr{A}_{1}( \mathscr{X}_{a})= \sum_{\mathcal{J}=0}^{n-1} \frac{\mathscr{A}_{1}(\mathfrak{T}_{1}^{\mathcal{J}+1}\mathscr{X}_{a})}{\mathfrak{T}_{1}^{(\mathcal{J}+1)}} - \frac{\mathscr{A}_{1}(\mathfrak{T}_{1}^{\mathcal{J}}\mathscr{X}_{a})}{\mathfrak {T}_{1}^{\mathcal{J}}}. $$

(3.11)

From (3.10) and (3.11), we reach

$$\begin{aligned} \left . \begin{aligned} & \mathfrak{A}_{a} \Biggl(\frac{\mathscr{A}_{1}(\mathfrak{T}_{1}^{n}\mathscr{X}_{a})}{\mathfrak{T}_{1}^{n}} - \mathscr{A}_{1}(\mathscr{X}_{a}), \sum_{\mathcal{J}=0}^{n-1}\frac{\mathscr{X}_{a}^{\mathcal{J}} \upsilon }{\mathfrak{T}_{1}^{\mathcal{J}+2}} \Biggr) \\ &\quad = \mathfrak{A}_{a} \Biggl(\sum_{\mathcal{J}=0}^{n-1} \frac{\mathscr{A}_{1}(\mathfrak{T}_{1}^{\mathcal{J}+1}\mathscr{X}_{a})}{\mathfrak{T}_{1}^{(\mathcal{J}+1)}} - \frac{\mathscr{A}_{1}(\mathfrak{T}_{1}^{\mathcal{J}}\mathscr{X}_{a})}{\mathfrak {T}_{1}^{\mathcal{J}}}, \sum_{\mathcal{J}=0}^{n-1} \frac{\mathscr{X}_{a}^{\mathcal{J}} \upsilon}{\mathfrak{T}_{1}^{\mathcal {J}+2}} \Biggr), \\ &\mathfrak{B}_{a} \Biggl(\frac{\mathscr{A}_{1}(\mathfrak {T}_{1}^{n}\mathscr{X}_{a})}{\mathfrak{T}_{1}^{n}} - \mathscr{A}_{1}(\mathscr{X}_{a}), \sum _{\mathcal{J}=0}^{n-1}\frac{\mathscr{X}_{a}^{\mathcal{J}} \upsilon }{\mathfrak{T}_{1}^{\mathcal{J}+2}} \Biggr) \\ &\quad = \mathfrak{B}_{a} \Biggl(\sum_{\mathcal{J}=0}^{n-1} \frac{\mathscr{A}_{1}(\mathfrak{T}_{1}^{\mathcal{J}+1}\mathscr{X}_{a})}{\mathfrak{T}_{1}^{(\mathcal{J}+1)}} - \frac{\mathscr{A}_{1}(\mathfrak{T}_{1}^{\mathcal{J}}\mathscr{X}_{a})}{\mathfrak {T}_{1}^{\mathcal{J}}}, \sum_{\mathcal{J}=0}^{n-1} \frac{\mathscr{X}_{a}^{\mathcal{J}} \upsilon}{\mathfrak{T}_{1}^{\mathcal {J}+2}} \Biggr), \\ &\mathfrak{C}_{a} \Biggl(\frac{\mathscr{A}_{1}(\mathfrak {T}_{1}^{n}\mathscr{X}_{a})}{\mathfrak{T}_{1}^{n}} - \mathscr{A}_{1}(\mathscr{X}_{a}), \sum _{\mathcal{J}=0}^{n-1}\frac{\mathscr{X}_{a}^{\mathcal{J}} \upsilon }{\mathfrak{T}_{1}^{\mathcal{J}+2}} \Biggr) \\ &\quad = \mathfrak{C}_{a} \Biggl(\sum_{\mathcal{J}=0}^{n-1} \frac{\mathscr{A}_{1}(\mathfrak{T}_{1}^{\mathcal{J}+1}\mathscr{X}_{a})}{\mathfrak{T}_{1}^{(\mathcal{J}+1)}} - \frac{\mathscr{A}_{1}(\mathfrak{T}_{1}^{\mathcal{J}}\mathscr{X}_{a})}{\mathfrak {T}_{1}^{\mathcal{J}}}, \sum_{\mathcal{J}=0}^{n-1} \frac{\mathscr{X}_{a}^{\mathcal{J}} \upsilon}{\mathfrak{T}_{1}^{\mathcal {J}+2}} \Biggr). \end{aligned} \right \} \end{aligned}$$

(3.12)

From (3.12), we arrive

$$\begin{aligned} \left . \begin{aligned} &\mathfrak{A}_{a} \Biggl( \frac{\mathscr{A}_{1}(\mathfrak{T}_{1}^{n}\mathscr{X}_{a})}{\mathfrak {T}_{1}^{n}} - \mathscr{A}_{1}(\mathscr{X}_{a}), \sum_{\mathcal{J}=0}^{n-1}\frac{\mathscr{X}_{a}^{\mathcal{J}} \upsilon }{\mathfrak{T}_{1}^{\mathcal{J}+2}} \Biggr) \\ &\quad \geq\prod_{\mathcal{J}=0}^{n-1} \mathfrak{A}_{a} \biggl(\frac{\mathscr{A}_{1}(\mathfrak{T}_{1}^{\mathcal {J}+1}\mathscr{X}_{a})}{\mathfrak{T}_{1}^{(\mathcal{J}+1)}} - \frac {\mathscr{A}_{1}(\mathfrak{T}_{1}^{\mathcal{J}}\mathscr{X}_{a})}{\mathfrak{T}_{1}^{\mathcal{J}}}, \frac{\mathscr{X}_{a}^{\mathcal{J}} \upsilon r}{ \mathfrak{T}_{1}^{\mathcal{J}+2}} \biggr), \\ &\mathfrak{B}_{a} \Biggl(\frac{\mathscr{A}_{1}(\mathfrak {T}_{1}^{n}\mathscr{X}_{a})}{\mathfrak{T}_{1}^{n}} - \mathscr{A}_{1}(\mathscr{X}_{a}), \sum _{\mathcal{J}=0}^{n-1}\frac{\mathscr{X}_{a}^{\mathcal{J}} \upsilon }{\mathfrak{T}_{1}^{\mathcal{J}+2}} \Biggr) \\ &\quad \leq \coprod_{\mathcal{J}=0}^{n-1}\mathfrak{B}_{a} \biggl(\frac{\mathscr{A}_{1}(\mathfrak{T}_{1}^{\mathcal{J}+1}\mathscr{X}_{a})}{\mathfrak{T}_{1}^{(\mathcal{J}+1)}} - \frac{\mathscr{A}_{1}(\mathfrak{T}_{1}^{\mathcal{J}}\mathscr{X}_{a})}{\mathfrak {T}_{1}^{\mathcal{J}}},\frac{\mathscr{X}_{a}^{\mathcal{J}} \upsilon}{ \mathfrak{T}_{1}^{\mathcal{J}+2}} \biggr), \\ &\mathfrak{C}_{a} \Biggl(\frac{\mathscr{A}_{1}(\mathfrak {T}_{1}^{n}\mathscr{X}_{a})}{\mathfrak{T}_{1}^{n}} - \mathscr{A}_{1}(\mathscr{X}_{a}), \sum _{\mathcal{J}=0}^{n-1}\frac{\mathscr{X}_{a}^{\mathcal{J}} \upsilon }{\mathfrak{T}_{1}^{\mathcal{J}+2}} \Biggr) \\ &\quad \leq \coprod_{\mathcal{J}=0}^{n-1}\mathfrak{C}_{a} \biggl(\frac{\mathscr{A}_{1}(\mathfrak{T}_{1}^{\mathcal{J}+1}\mathscr{X}_{a})}{\mathfrak{T}_{1}^{(\mathcal{J}+1)}} - \frac{\mathscr{A}_{1}(\mathfrak{T}_{1}^{\mathcal{J}}\mathscr{X}_{a})}{\mathfrak {T}_{1}^{\mathcal{J}}},\frac{\mathscr{X}_{a}^{\mathcal{J}} \upsilon}{ \mathfrak{T}_{1}^{\mathcal{J}+2}} \biggr), \end{aligned} \right \} \end{aligned}$$

(3.13)

where

$$\begin{aligned}& \prod_{\mathcal{J}=0}^{n-1} Q_{j} = Q_{1}\ast Q_{2}\ast\cdots\ast Q_{n} \quad \text{and} \quad\coprod_{\mathcal{J}=0}^{n-1} R_{j} = R_{1}\diamond R_{2}\diamond\cdots \diamond R_{n}\quad \text{and} \\& \coprod _{\mathcal{J}=0}^{n-1} S_{j} = S_{1}\oslash S_{2}\oslash\cdots\oslash S_{n} . \end{aligned}$$

Using the above conditions, we have

$$\begin{aligned} \left . \begin{aligned} &\mathfrak{A}_{a} \Biggl( \frac{\mathscr{A}_{1}(\mathfrak{T}_{1}^{n}\mathscr{X}_{a})}{\mathfrak {T}_{1}^{n}} - \mathscr{A}_{1}(\mathscr{X}_{a}), \sum_{\mathcal{J}=0}^{n-1}\frac{\mathscr{X}_{a}^{\mathcal{J}} \upsilon }{\mathfrak{T}_{1}^{\mathcal{J}+2}} \Biggr) \\ &\quad \geq\prod_{\mathcal{J}=0}^{n-1} \mathfrak{A}_{a}' \bigl(N (\mathscr{X}_{a}, \mathscr{X}_{a},\mathscr{X}_{a}),\upsilon\bigr) = \mathfrak{A}_{a}' \bigl(N (\mathscr{X}_{a}, \mathscr{X}_{a},\mathscr{X}_{a}), \upsilon\bigr), \\ &\mathfrak{B}_{a} \Biggl(\frac{\mathscr{A}_{1}(\mathfrak {T}_{1}^{n}\mathscr{X}_{a})}{\mathfrak{T}_{1}^{n}} - \mathscr{A}_{1}(\mathscr{X}_{a}), \sum _{\mathcal{J}=0}^{n-1}\frac{\mathscr{X}_{a}^{\mathcal{J}} \upsilon }{\mathfrak{T}_{1}^{\mathcal{J}+2}} \Biggr) \\ &\quad \leq \coprod_{\mathcal{J}=0}^{n-1}\mathfrak{B}_{a}' \bigl(N (\mathscr{X}_{a},\mathscr{X}_{a}, \mathscr{X}_{a}),\upsilon\bigr) =\mathfrak{B}_{a}' \bigl(N (\mathscr{X}_{a},\mathscr{X}_{a}, \mathscr{X}_{a}), \upsilon\bigr), \\ &\mathfrak{C}_{a} \Biggl(\frac{\mathscr{A}_{1}(\mathfrak {T}_{1}^{n}\mathscr{X}_{a})}{\mathfrak{T}_{1}^{n}} - \mathscr{A}_{1}(\mathscr{X}_{a}), \sum _{\mathcal{J}=0}^{n-1}\frac{\mathscr{X}_{a}^{\mathcal{J}} \upsilon }{\mathfrak{T}_{1}^{\mathcal{J}+2}} \Biggr) \\ &\quad \leq \coprod_{\mathcal{J}=0}^{n-1}\mathfrak{C}_{a}' \bigl(N (\mathscr{X}_{a},\mathscr{X}_{a}, \mathscr{X}_{a}),\upsilon\bigr) =\mathfrak{C}_{a}' \bigl(N (\mathscr{X}_{a},\mathscr{X}_{a}, \mathscr{X}_{a}), \upsilon\bigr). \end{aligned} \right \} \end{aligned}$$

(3.14)

Considering $\mathscr{X}_{a}$ by $\mathfrak{T}_{1}^{m}\mathscr{X}_{a}$ in (3.14) and dividing by $\mathfrak{T}_{1}^{m}$, we get

$$\begin{aligned} \left . \begin{aligned} & \mathfrak{A}_{a} \Biggl( \frac{\mathscr{A}_{1}(\mathfrak{T}_{1}^{n+m}\mathscr{X}_{a})}{\mathfrak {T}_{1}^{(n+m)}} - \frac{\mathscr{A}_{1}(\mathfrak{T}_{1}^{m}\mathscr{X}_{a})}{\mathfrak{T}_{1}^{m}}, \sum_{\mathcal{J}=0}^{n-1} \frac{\mathscr{X}_{a}^{\mathcal{J}} \upsilon}{\mathfrak {T}_{1}^{(\mathcal{J}+m+2)}} \Biggr) \\ &\quad\geq\mathfrak{A}_{a}' \bigl(N \bigl( \mathfrak{T}_{1}^{m}\mathscr{X}_{a}, \mathfrak{T}_{1}^{m}\mathscr{X}_{a}, \mathfrak{T}_{1}^{m}\mathscr{X}_{a}\bigr), \upsilon\bigr) \\ &\quad= \mathfrak{A}_{a}' \biggl(N ( \mathscr{X}_{a},\mathscr{X}_{a},\mathscr{X}_{a}), \frac{\upsilon}{\mathscr{X}_{a}^{m}} \biggr), \\ &\mathfrak{B}_{a} \Biggl(\frac{\mathscr{A}_{1}(\mathfrak {T}_{1}^{n+m}\mathscr{X}_{a})}{\mathfrak{T}_{1}^{(n+m)}} - \frac{\mathscr{A}_{1}(\mathfrak{T}_{1}^{m}\mathscr{X}_{a})}{\mathfrak {T}_{1}^{m}}, \sum_{\mathcal{J}=0}^{n-1} \frac{\mathscr{X}_{a}^{\mathcal{J}} \upsilon}{\mathfrak {T}_{1}^{(\mathcal{J}+m+2)}} \Biggr) \\ &\quad\leq\mathfrak{B}_{a}' \bigl(N \bigl( \mathfrak{T}_{1}^{m}\mathscr{X}_{a}, \mathfrak{T}_{1}^{m}\mathscr{X}_{a}, \mathfrak{T}_{1}^{m}\mathscr{X}_{a}\bigr), \upsilon\bigr) \\ &\quad= \mathfrak{B}_{a}' \biggl(N ( \mathscr{X}_{a},\mathscr{X}_{a},\mathscr{X}_{a}), \frac{\upsilon}{\mathscr{X}_{a}^{m}} \biggr), \\ &\mathfrak{C}_{a} \Biggl(\frac{\mathscr{A}_{1}(\mathfrak {T}_{1}^{n+m}\mathscr{X}_{a})}{\mathfrak{T}_{1}^{(n+m)}} - \frac{\mathscr{C}(\mathfrak{T}_{1}^{m}\mathscr{X}_{a})}{\mathfrak {T}_{1}^{m}}, \sum_{\mathcal{J}=0}^{n-1} \frac{\mathscr{X}_{a}^{\mathcal{J}} \upsilon}{\mathfrak {T}_{1}^{(\mathcal{J}+m+2)}} \Biggr) \\ &\quad\leq\mathfrak{C}_{a}' \bigl(N \bigl( \mathfrak{T}_{1}^{m}\mathscr{X}_{a}, \mathfrak{T}_{1}^{m}\mathscr{X}_{a}, \mathfrak{T}_{1}^{m}\mathscr{X}_{a}\bigr), \upsilon\bigr) \\ &\quad= \mathfrak{B}_{a}' \biggl(N ( \mathscr{X}_{a},\mathscr{X}_{a},\mathscr{X}_{a}), \frac{\upsilon}{\mathscr{X}_{a}^{m}} \biggr). \end{aligned} \right \} \end{aligned}$$

Replacing υ by $\mathscr{X}_{a}^{m}\upsilon$ in the above inequality, we have

$$\begin{aligned}& \left . \begin{aligned} &\mathfrak{A}_{a} \Biggl( \frac{\mathscr{A}_{1}(\mathfrak{T}_{1}^{n+m}\mathscr{X}_{a})}{\mathfrak {T}_{1}^{(n+m)}} - \frac{\mathscr{A}_{1}(\mathfrak{T}_{1}^{m}\mathscr{X}_{a})}{\mathfrak{T}_{1}^{m}}, \sum_{\mathcal{J}=0}^{n-1} \frac{\mathscr{X}_{a}^{\mathcal{J}+m} \upsilon}{\mathfrak {T}_{1}^{(\mathcal{J}+m+2)}} \Biggr)\geq\mathfrak{A}_{a}' \bigl(N (\mathscr{X}_{a},\mathscr{X}_{a}, \mathscr{X}_{a}), \upsilon\bigr), \\ &\mathfrak{B}_{a} \Biggl(\frac{\mathscr{A}_{1}(\mathfrak {T}_{1}^{n+m}\mathscr{X}_{a})}{\mathfrak{T}_{1}^{(n+m)}} - \frac{\mathscr{A}_{1}(\mathfrak{T}_{1}^{m}\mathscr{X}_{a})}{\mathfrak {T}_{1}^{m}}, \sum_{\mathcal{J}=0}^{n-1} \frac{\mathscr{X}_{a}^{\mathcal{J}+m} \upsilon}{\mathfrak {T}_{1}^{(\mathcal{J}+m+2)}} \Biggr) \leq\mathfrak{B}_{a}' \bigl(N (\mathscr{X}_{a},\mathscr{X}_{a}, \mathscr{X}_{a}),\upsilon\bigr), \\ &\mathfrak{C}_{a} \Biggl(\frac{\mathscr{A}_{1}(\mathfrak {T}_{1}^{n+m}\mathscr{X}_{a})}{\mathfrak{T}_{1}^{(n+m)}} - \frac{\mathscr{A}_{1}(\mathfrak{T}_{1}^{m}\mathscr{X}_{a})}{\mathfrak {T}_{1}^{m}}, \sum_{\mathcal{J}=0}^{n-1} \frac{\mathscr{X}_{a}^{\mathcal{J}+m} \upsilon}{\mathfrak {T}_{1}^{(\mathcal{J}+m+2)}} \Biggr) \leq\mathfrak{C}_{a}' \bigl(N (\mathscr{X}_{a},\mathscr{X}_{a}, \mathscr{X}_{a}),\upsilon\bigr), \end{aligned} \right \} \end{aligned}$$

(3.15)

$$\begin{aligned}& \left . \begin{aligned} &\mathfrak{A}_{a} \biggl( \frac{\mathscr{A}_{1}(\mathfrak{T}_{1}^{n+m}\mathscr{X}_{a})}{\mathfrak {T}_{1}^{(n+m)}} - \frac{\mathscr{A}_{1}(\mathfrak{T}_{1}^{m}\mathscr{X}_{a})}{\mathfrak{T}_{1}^{m}}, \upsilon\biggr)\geq \mathfrak{A}_{a}' \biggl(N (\mathscr{X}_{a}, \mathscr{X}_{a},\mathscr{X}_{a}),\frac{\upsilon}{\sum_{\mathcal {J}=m}^{n-1}\frac{\mathscr{X}_{a}^{\mathcal{J}}}{ \mathfrak {T}_{1}^{\mathcal{J}+2}}} \biggr), \\ &\mathfrak{B}_{a} \biggl(\frac{\mathscr{A}_{1}(\mathfrak {T}_{1}^{n+m}\mathscr{X}_{a})}{\mathfrak{T}_{1}^{(n+m)}} - \frac{\mathscr{A}_{1}(\mathfrak{T}_{1}^{m}\mathscr{X}_{a})}{\mathfrak {T}_{1}^{m}}, \upsilon\biggr)\leq\mathfrak{B}_{a}' \biggl(N (\mathscr{X}_{a},\mathscr{X}_{a}, \mathscr{X}_{a}), \frac{\upsilon}{\sum_{\mathcal{J}=m}^{n-1}\frac {\mathscr{X}_{a}^{\mathcal{J}}}{ \mathfrak{T}_{1}^{\mathcal{J}+2}}} \biggr), \\ &\mathfrak{C}_{a} \biggl(\frac{\mathscr{A}_{1}(\mathfrak {T}_{1}^{n+m}\mathscr{X}_{a})}{\mathfrak{T}_{1}^{(n+m)}} - \frac{\mathscr{A}_{1}(\mathfrak{T}_{1}^{m}\mathscr{X}_{a})}{\mathfrak {T}_{1}^{m}}, \upsilon\biggr)\leq\mathfrak{C}_{a}' \biggl(N (\mathscr{X}_{a},\mathscr{X}_{a}, \mathscr{X}_{a}), \frac{\upsilon}{\sum_{\mathcal{J}=m}^{n-1}\frac {\mathscr{X}_{a}^{\mathcal{J}}}{ \mathfrak{T}_{1}^{\mathcal{J}+2}}} \biggr). \end{aligned} \right \} \end{aligned}$$

(3.16)

Since $0<\mathscr{X}_{a}<1$ and $\sum_{\mathcal{J}=0}^{n} (\frac {\mathscr{X}_{a}}{\mathfrak{T}_{1}} )^{\mathcal{J}} < \infty$. The sequence ${ \{\frac{\mathscr{A}_{1}(\mathfrak{T}_{1}^{n}\mathscr{X}_{a})}{\mathfrak{T}_{1}^{n}} \}}$ is Cauchy in $(Y,\mathfrak {A}_{a}, \mathfrak{B}_{a}, \mathfrak{C}_{a} )$. Since $({Y,\mathfrak {A}_{a},\mathfrak{B}_{a},\mathfrak{C}_{a}} )$ is a complete NSN-space, this sequence converges to some point $\mathcal {A}_{1} (\mathscr{X}_{a} ) \in Y$. Defining $\mathcal{A}_{1}:\mathcal {L} \longrightarrow\mathcal{M}$ by

$$\begin{aligned}& \lim_{n \to\infty} \mathfrak{A}_{a} \biggl( \frac{\mathscr{A}_{1}(\mathfrak{T}_{1}^{n}\mathscr{X}_{a})}{\mathfrak {T}_{1}^{n}} - \mathcal{A}_{1}(\mathscr{X}_{a}), \upsilon\biggr) =1,\\& \lim_{n \to\infty} \mathfrak{B}_{a} \biggl( \frac{\mathscr{A}_{1}(\mathfrak{T}_{1}^{n}\mathscr{X}_{a})}{\mathfrak {T}_{1}^{n}} - \mathcal{A}_{1}(\mathscr{X}_{a}), \upsilon\biggr) =0,\\& \lim_{n \to\infty} \mathfrak{C}_{a} \biggl( \frac{\mathscr{A}_{1}(\mathfrak{T}_{1}^{n}\mathscr{X}_{a})}{\mathfrak {T}_{1}^{n}} - \mathcal{A}_{1}(\mathscr{X}_{a}), \upsilon\biggr) =0. \end{aligned}$$

Finally

$$ \frac{\mathscr{A}_{1}(\mathfrak{T}_{1}^{n}\mathscr{X}_{a})}{\mathfrak {T}_{1}^{n}} \stackrel{\mathrm{NSN}}{\longrightarrow} \mathcal{A}_{1}(\mathscr{X}_{a}), \quad\text{as } n \to\infty. $$

Consider $m=0$ in (3.16), then

$$\begin{aligned} \left . \begin{aligned} &\mathfrak{A}_{a} \biggl( \frac{\mathscr{A}_{1}(\mathfrak{T}_{1}^{n}\mathscr{X}_{a})}{\mathfrak {T}_{1}^{n}} - \mathscr{A}_{1}(\mathscr{X}_{a}), \upsilon\biggr) \geq\mathfrak{A}_{a}' \biggl(N ( \mathscr{X}_{a},\mathscr{X}_{a},\mathscr{X}_{a}), \frac{\upsilon}{\sum_{\mathcal{J}=0}^{n-1}\frac{\mathscr{X}_{a}^{\mathcal{J}}}{ \mathfrak{T}_{1}^{\mathcal{J}+2}}} \biggr), \\ &\mathfrak{B}_{a} \biggl(\frac{\mathscr{A}_{1}(\mathfrak {T}_{1}^{n}\mathscr{X}_{a})}{\mathfrak{T}_{1}^{n}} - \mathscr{A}_{1}(\mathscr{X}_{a}), \upsilon\biggr) \leq \mathfrak{B}_{a}' \biggl(N (\mathscr{X}_{a}, \mathscr{X}_{a},\mathscr{X}_{a}),\frac{\upsilon}{\sum_{\mathcal {J}=0}^{n-1}\frac{\mathscr{X}_{a}^{\mathcal{J}}}{ \mathfrak {T}_{1}^{\mathcal{J}+2}}} \biggr), \\ &\mathfrak{C}_{a} \biggl(\frac{\mathscr{A}_{1}(\mathfrak {T}_{1}^{n}\mathscr{X}_{a})}{\mathfrak{T}_{1}^{n}} - \mathscr{A}_{1}(\mathscr{X}_{a}), \upsilon\biggr) \leq \mathfrak{C}_{a}' \biggl(N (\mathscr{X}_{a}, \mathscr{X}_{a},\mathscr{X}_{a}),\frac{\upsilon}{\sum_{\mathcal {J}=0}^{n-1}\frac{\mathscr{X}_{a}^{\mathcal{J}}}{ \mathfrak {T}_{1}^{\mathcal{J}+2}}} \biggr). \end{aligned} \right \} \end{aligned}$$

(3.17)

As $n\longrightarrow\infty$ in (3.17) and

$$\begin{aligned} \left . \begin{aligned} &\mathfrak{A}_{a} \bigl( \mathcal{A}_{1}(\mathscr{X}_{a}) - \mathscr{A}_{1}( \mathscr{X}_{a}), t \bigr)\geq\mathfrak{A}_{a}' \bigl(N (\mathscr{X}_{a},\mathscr{X}_{a}, \mathscr{X}_{a}), \mathfrak{T}_{1} \upsilon( \mathfrak{T}_{1}-\mathscr{X}_{a}) \bigr), \\ &\mathfrak{B}_{a} \bigl(\mathcal{A}_{1}( \mathscr{X}_{a}) - \mathscr{A}_{1}(\mathscr{X}_{a}), t \bigr)\leq\mathfrak{B}_{a}' \bigl(N ( \mathscr{X}_{a},\mathscr{X}_{a},\mathscr{X}_{a}), \mathfrak{T}_{1} \upsilon(\mathfrak{T}_{1}- \mathscr{X}_{a}) \bigr), \\ &\mathfrak{C}_{a} \bigl(\mathcal{A}_{1}( \mathscr{X}_{a}) - \mathscr{A}_{1}(\mathscr{X}_{a}), t \bigr)\leq\mathfrak{C}_{a}' \bigl(N ( \mathscr{X}_{a},\mathscr{X}_{a},\mathscr{X}_{a}), \mathfrak{T}_{1} \upsilon(\mathfrak{T}_{1}- \mathscr{X}_{a}) \bigr). \end{aligned} \right \} \end{aligned}$$

(3.18)

Finally, $\mathcal{A}_{1}$ satisfies (3.1), taking $(\mathscr{X}_{a},\mathscr{V}_{a},\mathscr{W}_{a})$ by $(\mathfrak {T}_{1}^{n}\mathscr{X}_{a},\mathfrak{T}_{1}^{n}\mathscr{V}_{a},\mathfrak {T}_{1}^{n}\mathscr{W}_{a})$ in (3.4).

$$\begin{aligned} \left . \begin{aligned} &\mathfrak{A}_{a} \biggl( \frac{1}{\mathfrak{T}_{1}^{n}}\mathfrak{Z}\bigl(\mathfrak{T}_{1}^{n} \mathscr{X}_{a},\mathfrak{T}_{1}^{n} \mathscr{V}_{a}, \mathfrak{T}_{1}^{n} \mathscr{W}_{a}\bigr),\upsilon\biggr) \geq\mathfrak{A}_{a}' \bigl(N \bigl(\mathfrak{T}_{1}^{n}\mathscr{X}_{a}, \mathfrak{T}_{1}^{n}\mathscr{V}_{a}, \mathfrak{T}_{1}^{n}\mathscr{W}_{a}\bigr), \mathfrak{T}_{1}^{n} \upsilon\bigr), \\ &\mathfrak{B}_{a} \biggl(\frac{1}{\mathfrak{T}_{1}^{n}}\mathfrak{Z}\bigl( \mathfrak{T}_{1}^{n}\mathscr{X}_{a}, \mathfrak{T}_{1}^{n}\mathscr{V}_{a}, \mathfrak{T}_{1}^{n}\mathscr{W}_{a}\bigr), \upsilon\biggr)\leq\mathfrak{B}_{a}' \bigl(N \bigl( \mathfrak{T}_{1}^{n}\mathscr{X}_{a}, \mathfrak{T}_{1}^{n}\mathscr{V}_{a}, \mathfrak{T}_{1}^{n}\mathscr{W}_{a}\bigr), \mathfrak{T}_{1}^{n} \upsilon\bigr), \\ &\mathfrak{C}_{a} \biggl(\frac{1}{\mathfrak{T}_{1}^{n}}\mathfrak{Z}\bigl( \mathfrak{T}_{1}^{n}\mathscr{X}_{a}, \mathfrak{T}_{1}^{n}\mathscr{V}_{a}, \mathfrak{T}_{1}^{n}\mathscr{W}_{a}\bigr), \upsilon\biggr)\leq\mathfrak{C}_{a}' \bigl(N \bigl( \mathfrak{T}_{1}^{n}\mathscr{X}_{a}, \mathfrak{T}_{1}^{n}\mathscr{V}_{a}, \mathfrak{T}_{1}^{n}\mathscr{W}_{a}\bigr), \mathfrak{T}_{1}^{n} \upsilon\bigr). \end{aligned} \right \} \end{aligned}$$

(3.19)

Here,

$$\begin{aligned}& \mathfrak{A}_{a} \bigl(\mathcal{P}_{1} \mathcal{A}_{1} \bigl(\mathcal{Q}_{1} \mathcal{R}_{1} (\mathscr{X}_{a}-\mathscr{V}_{a} ) \bigr) + \mathcal{Q}_{1} \mathcal{A}_{1} \bigl( \mathcal{P}_{1}\mathcal{R}_{1} (\mathscr{V}_{a}- \mathscr{W}_{a} ) \bigr) \\& \quad \quad{}+\mathcal{R}_{1} \mathcal{A}_{1} \bigl( \mathcal{P}_{1}\mathcal{Q}_{1} (\mathscr{W}_{a}- \mathscr{X}_{a} ) \bigr)+\mathfrak{T}_{1} \mathcal{A}_{1} (\mathcal{P}_{1} \mathscr{X}_{a}+ \mathcal{Q}_{1} \mathscr{V}_{a}+\mathcal{R}_{1} \mathscr{W}_{a} ) \\& \quad \quad{}-\mathfrak{T}_{1} \bigl(\mathcal{P}_{1} \mathcal{A}_{1} (\mathscr{X}_{a} ) + \mathcal{Q}_{1} \mathcal{A}_{1} (\mathscr{V}_{a} )+\mathcal{R}_{1} \mathcal{A}_{1} ( \mathscr{W}_{a} ) \bigr) \bigr) \\& \quad \geq\mathfrak{A}_{a} \biggl(\mathcal{P}_{1} \mathcal{A}_{1} \bigl(\mathcal{Q}_{1} \mathcal{R}_{1} (\mathscr{X}_{a}-\mathscr{V}_{a} ) \bigr)-\frac{\mathcal{P}_{1}}{\mathfrak{T}_{1}^{n}}\mathscr{A}_{1} \bigl( \mathcal{Q}_{1}\mathcal{R}_{1} (\mathscr{X}_{a}- \mathscr{V}_{a} ) \bigr),\frac{\upsilon}{6} \biggr) \\& \quad \quad {}\ast\mathfrak{A}_{a} \biggl(\mathcal{Q}_{1} \mathcal{A}_{1} \bigl(\mathcal{P}_{1} \mathcal{R}_{1} (\mathscr{V}_{a}-\mathscr{W}_{a} ) \bigr) -\frac{\mathcal{Q}_{1}}{\mathfrak{T}_{1}^{n}} \mathscr{A}_{1} \bigl( \mathcal{P}_{1}\mathcal{R}_{1} (\mathscr{V}_{a}- \mathscr{W}_{a} ) \bigr),\frac{\upsilon}{6} \biggr) \\& \quad \quad {}\ast\mathfrak{A}_{a} \biggl(\mathfrak{T}_{1} \mathcal{A}_{1} (\mathcal{P}_{1} \mathscr{X}_{a}+ \mathcal{Q}_{1} \mathscr{V}_{a}+\mathcal{R}_{1} \mathscr{W}_{a} ) +\frac{\mathfrak{T}_{1}}{\mathfrak{T}_{1}^{n}}\mathscr{A}_{1} (\mathcal{P}_{1} \mathscr{X}_{a}+\mathcal{Q}_{1} \mathscr{V}_{a}+\mathcal{R}_{1} \mathscr{W}_{a} ),\frac{\upsilon}{6} \biggr) \\& \quad \quad {}\ast\mathfrak{A}_{a} \biggl( -\mathfrak{T}_{1} \bigl(\mathcal{P}_{1} \mathcal{A}_{1} ( \mathscr{X}_{a} ) +\mathcal{Q}_{1} \mathcal{A}_{1} (\mathscr{V}_{a} )+\mathcal{R}_{1} \mathcal{A}_{1} (\mathscr{W}_{a} ) \bigr) \\& \quad \quad {} + \frac{\mathfrak{T}_{1}}{\mathfrak{T}_{1}^{n}}\bigl(\mathcal{P}_{1} \mathscr{A}_{1} ( \mathscr{X}_{a} ) +\mathcal{Q}_{1} \mathscr{A}_{1} (\mathscr{V}_{a} )+\mathcal{R}_{1} \mathscr{A}_{1} (\mathscr{W}_{a} ) \bigr), \frac{\upsilon}{6} \biggr) \\& \quad \quad {}\ast\mathfrak{A}_{a} \biggl( \frac{\mathcal{P}_{1}}{\mathfrak{T}_{1}^{n}}\mathscr{A}_{1} \bigl(\mathcal{Q}_{1} \mathcal{R}_{1} (\mathscr{X}_{a}-\mathscr{V}_{a} ) \bigr)+\frac{\mathcal{Q}_{1}}{\mathfrak{T}_{1}^{n}}\mathscr{A}_{1} \bigl(\mathcal{P}_{1} \mathcal{R}_{1} (\mathscr{V}_{a}-\mathscr{W}_{a} ) \bigr) \\& \quad \quad {} +\frac{\mathcal{R}_{1}}{\mathfrak{T}_{1}^{n}}\mathscr{A}_{1} \bigl( \mathcal{P}_{1}\mathcal{Q}_{1} (\mathscr{W}_{a}- \mathscr{X}_{a} ) \bigr) +\frac{\mathfrak{T}_{1}}{\mathfrak{T}_{1}^{n}}\mathscr{A}_{1} (\mathcal{P}_{1} \mathscr{X}_{a}+\mathcal{Q}_{1} \mathscr{V}_{a}+ \mathcal{R}_{1} \mathscr{W}_{a} ) \\& \quad \quad {}-\frac{\mathfrak{T}_{1}}{\mathfrak {T}_{1}^{n}} \bigl(\mathcal{P}_{1} \mathscr{A}_{1} ( \mathscr{X}_{a} ) +\mathcal{Q}_{1} \mathscr{A}_{1} (\mathscr{V}_{a} )+\mathcal{R}_{1} \mathscr{A}_{1} (\mathscr{W}_{a} ) \bigr), \frac{\upsilon}{6} \biggr) \end{aligned}$$

(3.20)

and

$$\begin{aligned}& \mathfrak{B}_{a} \bigl(\mathcal{P}_{1} \mathcal{A}_{1} \bigl(\mathcal{Q}_{1} \mathcal{R}_{1} (\mathscr{X}_{a}-\mathscr{V}_{a} ) \bigr) + \mathcal{Q}_{1} \mathcal{A}_{1} \bigl( \mathcal{P}_{1}\mathcal{R}_{1} (\mathscr{V}_{a}- \mathscr{W}_{a} ) \bigr) \\& \quad \quad{}+\mathcal{R}_{1} \mathcal{A}_{1} \bigl( \mathcal{P}_{1}\mathcal{Q}_{1} (\mathscr{W}_{a}- \mathscr{X}_{a} ) \bigr)+\mathfrak{T}_{1} \mathcal{A}_{1} (\mathcal{P}_{1} \mathscr{X}_{a}+ \mathcal{Q}_{1} \mathscr{V}_{a}+\mathcal{R}_{1} \mathscr{W}_{a} ) \\& \quad \quad{}-\mathfrak{T}_{1} \bigl(\mathcal{P}_{1} \mathcal{A}_{1} (\mathscr{X}_{a} ) + \mathcal{Q}_{1} \mathcal{A}_{1} (\mathscr{V}_{a} )+\mathcal{R}_{1} \mathcal{A}_{1} ( \mathscr{W}_{a} ) \bigr) \bigr) \\& \quad \geq\mathfrak{B}_{a} \biggl(\mathcal{P}_{1} \mathcal{A}_{1} \bigl(\mathcal{Q}_{1} \mathcal{R}_{1} (\mathscr{X}_{a}-\mathscr{V}_{a} ) \bigr)-\frac{\mathcal{P}_{1}}{\mathfrak{T}_{1}^{n}}\mathscr{A}_{1} \bigl( \mathcal{Q}_{1}\mathcal{R}_{1} (\mathscr{X}_{a}- \mathscr{V}_{a} ) \bigr),\frac{\upsilon}{6} \biggr) \\& \quad \quad{} \diamond\mathfrak{B}_{a} \biggl(\mathcal{Q}_{1} \mathcal{A}_{1} \bigl(\mathcal{P}_{1} \mathcal{R}_{1} (\mathscr{V}_{a}-\mathscr{W}_{a} ) \bigr) -\frac{\mathcal{Q}_{1}}{\mathfrak{T}_{1}^{n}} \mathscr{A}_{1} \bigl( \mathcal{P}_{1}\mathcal{R}_{1} (\mathscr{V}_{a}- \mathscr{W}_{a} ) \bigr),\frac{\upsilon}{6} \biggr) \\& \quad \quad{}\diamond\mathfrak{B}_{a} \biggl(\mathfrak{T}_{1} \mathcal{A}_{1} (\mathcal{P}_{1} \mathscr{X}_{a}+ \mathcal{Q}_{1} \mathscr{V}_{a}+\mathcal{R}_{1} \mathscr{W}_{a} ) +\frac{\mathfrak{T}_{1}}{\mathfrak{T}_{1}^{n}}\mathscr{A}_{1} (\mathcal{P}_{1} \mathscr{X}_{a}+\mathcal{Q}_{1} \mathscr{V}_{a}+\mathcal{R}_{1} \mathscr{W}_{a} ),\frac{\upsilon}{6} \biggr) \\& \quad \quad{} \diamond\mathfrak{B}_{a} \biggl( -\mathfrak{T}_{1} \bigl(\mathcal{P}_{1} \mathcal{A}_{1} ( \mathscr{X}_{a} ) +\mathcal{Q}_{1} \mathcal{A}_{1} (\mathscr{V}_{a} )+\mathcal{R}_{1} \mathcal{A}_{1} (\mathscr{W}_{a} ) \bigr) \\& \quad\quad {} + \frac{\mathfrak{T}_{1}}{\mathfrak{T}_{1}^{n}} \bigl(\mathcal{P}_{1} \mathscr{A}_{1} ( \mathscr{X}_{a} ) +\mathcal{Q}_{1} \mathscr{A}_{1} (\mathscr{V}_{a} )+\mathcal{R}_{1} \mathscr{A}_{1} (\mathscr{W}_{a} ) \bigr), \frac{\upsilon}{6} \biggr) \\& \quad \quad{}\diamond\mathfrak{B}_{a} \biggl( \frac{\mathcal{P}_{1}}{\mathfrak{T}_{1}^{n}}\mathscr{A}_{1} \bigl(\mathcal{Q}_{1} \mathcal{R}_{1} (\mathscr{X}_{a}-\mathscr{V}_{a} ) \bigr)+\frac{\mathcal{Q}_{1}}{\mathfrak{T}_{1}^{n}}\mathscr{A}_{1} \bigl(\mathcal{P}_{1} \mathcal{R}_{1} (\mathscr{V}_{a}-\mathscr{W}_{a} ) \bigr) \\& \quad\quad {} +\frac{\mathcal{R}_{1}}{\mathfrak{T}_{1}^{n}}\mathscr{A}_{1} \bigl( \mathcal{P}_{1}\mathcal{Q}_{1} (\mathscr{W}_{a}- \mathscr{X}_{a} ) \bigr) +\frac{\mathfrak{T}_{1}}{\mathfrak{T}_{1}^{n}}\mathscr{A}_{1} (\mathcal{P}_{1} \mathscr{X}_{a}+\mathcal{Q}_{1} \mathscr{V}_{a}+ \mathcal{R}_{1} \mathscr{W}_{a} ) \\& \quad \quad{}-\frac{\mathfrak{T}_{1}}{\mathfrak {T}_{1}^{n}} \bigl(\mathcal{P}_{1} \mathscr{A}_{1} ( \mathscr{X}_{a} ) +\mathcal{Q}_{1} \mathscr{A}_{1} (\mathscr{V}_{a} )+\mathcal{R}_{1} \mathscr{A}_{1} (\mathscr{W}_{a} ) \bigr), \frac{\upsilon}{6} \biggr) \end{aligned}$$

(3.21)

and

$$\begin{aligned}& \mathfrak{C}_{a} \bigl(\mathcal{P}_{1} \mathcal{A}_{1} \bigl(\mathcal{Q}_{1} \mathcal{R}_{1} (\mathscr{X}_{a}-\mathscr{V}_{a} ) \bigr) + \mathcal{Q}_{1} \mathcal{A}_{1} \bigl( \mathcal{P}_{1}\mathcal{R}_{1} (\mathscr{V}_{a}- \mathscr{W}_{a} ) \bigr) \\& \quad \quad{}+\mathcal{R}_{1} \mathcal{A}_{1} \bigl( \mathcal{P}_{1}\mathcal{Q}_{1} (\mathscr{W}_{a}- \mathscr{X}_{a} ) \bigr)+\mathfrak{T}_{1} \mathcal{A}_{1} (\mathcal{P}_{1} \mathscr{X}_{a}+ \mathcal{Q}_{1} \mathscr{V}_{a}+\mathcal{R}_{1} \mathscr{W}_{a} ) \\& \quad \quad{}-\mathfrak{T}_{1} \bigl(\mathcal{P}_{1} \mathcal{A}_{1} (\mathscr{X}_{a} ) + \mathcal{Q}_{1} \mathcal{A}_{1} (\mathscr{V}_{a} )+\mathcal{R}_{1} \mathcal{A}_{1} ( \mathscr{W}_{a} ) \bigr) \bigr) \\& \quad \geq\mathfrak{C}_{a} \biggl(\mathcal{P}_{1} \mathcal{A}_{1} \bigl(\mathcal{Q}_{1} \mathcal{R}_{1} (\mathscr{X}_{a}-\mathscr{V}_{a} ) \bigr)-\frac{\mathcal{P}_{1}}{\mathfrak{T}_{1}^{n}}\mathscr{A}_{1} \bigl( \mathcal{Q}_{1}\mathcal{R}_{1} (\mathscr{X}_{a}- \mathscr{V}_{a} ) \bigr),\frac{\upsilon}{6} \biggr) \\& \quad\quad {}\oslash\mathfrak{C}_{a} \biggl(\mathcal{Q}_{1} \mathcal{A}_{1} \bigl(\mathcal{P}_{1} \mathcal{R}_{1} (\mathscr{V}_{a}-\mathscr{W}_{a} ) \bigr) -\frac{\mathcal{Q}_{1}}{\mathfrak{T}_{1}^{n}} \mathscr{A}_{1} \bigl( \mathcal{P}_{1}\mathcal{R}_{1} (\mathscr{V}_{a}- \mathscr{W}_{a} ) \bigr),\frac{\upsilon}{6} \biggr) \\& \quad\quad {}\oslash\mathfrak{C}_{a} \biggl(\mathfrak{T}_{1} \mathcal{A}_{1} (\mathcal{P}_{1} \mathscr{X}_{a}+ \mathcal{Q}_{1} \mathscr{V}_{a}+\mathcal{R}_{1} \mathscr{W}_{a} ) +\frac{\mathfrak{T}_{1}}{\mathfrak{T}_{1}^{n}}\mathscr{A}_{1} (\mathcal{P}_{1} \mathscr{X}_{a}+\mathcal{Q}_{1} \mathscr{V}_{a}+\mathcal{R}_{1} \mathscr{W}_{a} ),\frac{\upsilon}{6} \biggr) \\& \quad\quad {} \oslash\mathfrak{C}_{a} \biggl( -\mathfrak{T}_{1} \bigl(\mathcal{P}_{1} \mathcal{A}_{1} ( \mathscr{X}_{a} ) +\mathcal{Q}_{1} \mathcal{A}_{1} (\mathscr{V}_{a} )+\mathcal{R}_{1} \mathcal{A}_{1} (\mathscr{W}_{a} ) \bigr) \\& \quad \quad {} + \frac{\mathfrak{T}_{1}}{\mathfrak{T}_{1}^{n}}\bigl(\mathcal{P}_{1} \mathscr{A}_{1} ( \mathscr{X}_{a} ) +\mathcal{Q}_{1} \mathscr{A}_{1} (\mathscr{V}_{a} )+\mathcal{R}_{1} \mathscr{A}_{1} (\mathscr{W}_{a} ) \bigr), \frac{\upsilon}{6} \biggr) \\& \quad\quad {}\oslash\mathfrak{C}_{a} \biggl( \frac{\mathcal{P}_{1}}{\mathfrak{T}_{1}^{n}}\mathscr{A}_{1} \bigl(\mathcal{Q}_{1} \mathcal{R}_{1} (\mathscr{X}_{a}-\mathscr{V}_{a} ) \bigr)+\frac{\mathcal{Q}_{1}}{\mathfrak{T}_{1}^{n}}\mathscr{A}_{1} \bigl(\mathcal{P}_{1} \mathcal{R}_{1} (\mathscr{V}_{a}-\mathscr{W}_{a} ) \bigr) \\& \quad\quad {} +\frac{\mathcal{R}_{1}}{\mathfrak{T}_{1}^{n}}\mathscr{A}_{1} \bigl( \mathcal{P}_{1}\mathcal{Q}_{1} (\mathscr{W}_{a}- \mathscr{X}_{a} ) \bigr)+\frac{\mathfrak{T}_{1}}{\mathfrak{T}_{1}^{n}}\mathscr{A}_{1} (\mathcal{P}_{1} \mathscr{X}_{a}+\mathcal{Q}_{1} \mathscr{V}_{a}+ \mathcal{R}_{1} \mathscr{W}_{a} ) \\& \quad\quad {}-\frac{\mathfrak{T}_{1}}{\mathfrak {T}_{1}^{n}} \bigl(\mathcal{P}_{1} \mathscr{A}_{1} ( \mathscr{X}_{a} ) +\mathcal{Q}_{1} \mathscr{A}_{1} (\mathscr{V}_{a} )+\mathcal{R}_{1} \mathscr{A}_{1} (\mathscr{W}_{a} ) \bigr), \frac{\upsilon}{6} \biggr). \end{aligned}$$

(3.22)

Also,

$$\begin{aligned} \left . \begin{aligned} &\lim_{n \to\infty} \mathfrak{A}_{a} \biggl(\frac{1}{\mathfrak{T}_{1}^{n}}\mathfrak{Z}\bigl( \mathfrak{T}_{1}^{n}\mathscr{X}_{a}, \mathfrak{T}_{1}^{n}\mathscr{V}_{a}, \mathfrak{T}_{1}^{n}\mathscr{W}_{a}\bigr), \frac{\upsilon}{6} \biggr)=1, \\ &\lim_{n \to\infty}\mathfrak{B}_{a} \biggl( \frac{1}{\mathfrak{T}_{1}^{n}}\mathfrak{Z}\bigl(\mathfrak{T}_{1}^{n} \mathscr{X}_{a},\mathfrak{T}_{1}^{n} \mathscr{V}_{a},\mathfrak{T}_{1}^{n} \mathscr{W}_{a}\bigr),\frac{\upsilon}{6} \biggr)=0, \\ &\lim_{n \to\infty}\mathfrak{C}_{a} \biggl( \frac{1}{\mathfrak{T}_{1}^{n}}\mathfrak{Z}\bigl(\mathfrak{T}_{1}^{n} \mathscr{X}_{a},\mathfrak{T}_{1}^{n} \mathscr{V}_{a},\mathfrak{T}_{1}^{n} \mathscr{W}_{a}\bigr),\frac{\upsilon}{6} \biggr)=0. \end{aligned} \right \} \end{aligned}$$

(3.23)

To prove uniqueness

$$\begin{aligned} &\mathfrak{A}_{a}\bigl(\mathcal{A}_{1}( \mathscr{X}_{a})-\mathcal{A}_{1}'( \mathscr{X}_{a}), \upsilon\bigr) \\ &\quad \geq\mathfrak{A}_{a} \biggl(\mathcal{A}_{1}\bigl( \mathfrak{T}_{1}^{n}\mathscr{X}_{a}\bigr)- \mathscr{A}_{1}\bigl(\mathfrak{T}_{1}^{n} \mathscr{X}_{a}\bigr),\frac{\upsilon.\mathfrak{T}_{1}^{n}}{2} \biggr)\ast \mathfrak{A}_{a} \biggl(\mathscr{A}_{1}\bigl( \mathfrak{T}_{1}^{n}\mathscr{X}_{a}\bigr)- \mathcal{A}_{1}'\bigl(\mathfrak{T}_{1}^{n} \mathscr{X}_{a}\bigr),\frac{\upsilon.\mathfrak{T}_{1}^{n}}{2} \biggr) \\ &\quad \geq\mathfrak{A}_{a}' \biggl(N \bigl( \mathfrak{T}_{1}^{n}\mathscr{X}_{a}, \mathfrak{T}_{1}^{n}\mathscr{X}_{a}, \mathfrak{T}_{1}^{n}\mathscr{X}_{a}\bigr), \frac{\upsilon\mathfrak{T}_{1}^{n+1}}{2} \vert \mathfrak {T}_{1}-\mathscr{X}_{a} \vert \biggr) \\ &\quad \geq\mathfrak{A}_{a}' \biggl(N ( \mathscr{X}_{a},\mathscr{X}_{a},\mathscr{X}_{a}), \frac{\upsilon \mathfrak{T}_{1}^{n+1} \vert \mathfrak{T}_{1}-\mathscr{X}_{a} \vert }{2\cdot\mathscr{X}_{a}^{n}} \biggr), \\ &\mathfrak{B}_{a}\bigl(\mathcal{A}_{1}( \mathscr{X}_{a})-\mathcal{A}_{1}'( \mathscr{X}_{a}), \upsilon\bigr) \\ &\quad \leq\mathfrak{B}_{a} \biggl(\mathcal{A}_{1}\bigl( \mathfrak{T}_{1}^{n}\mathscr{X}_{a}\bigr)- \mathscr{A}_{1}\bigl(\mathfrak{T}_{1}^{n} \mathscr{X}_{a}\bigr),\frac{\upsilon.\mathfrak{T}_{1}^{n}}{2} \biggr)\diamond \mathfrak{B}_{a} \biggl(\mathscr{A}_{1}\bigl( \mathfrak{T}_{1}^{n}\mathscr{X}_{a}\bigr)- \mathcal{A}_{1}'\bigl(\mathfrak{T}_{1}^{n} \mathscr{X}_{a}\bigr),\frac{\upsilon.\mathfrak{T}_{1}^{n}}{2} \biggr) \\ &\quad \leq\mathfrak{B}_{a}' \biggl(N \bigl( \mathfrak{T}_{1}^{n}\mathscr{X}_{a}, \mathfrak{T}_{1}^{n}\mathscr{X}_{a}, \mathfrak{T}_{1}^{n}\mathscr{X}_{a}\bigr), \frac{t\mathfrak{T}_{1}^{n+1}}{2} \vert \mathfrak{T}_{1}-\mathscr{X}_{a} \vert \biggr) \\ &\quad \leq\mathfrak{B}_{a}' \biggl(N ( \mathscr{X}_{a},\mathscr{X}_{a},\mathscr{X}_{a}), \frac{\upsilon \mathfrak{T}_{1}^{n+1} \vert \mathfrak{T}_{1}-\mathscr{X}_{a} \vert }{2\cdot\mathscr{X}_{a}^{n}} \biggr), \\ &\mathfrak{C}_{a}\bigl(\mathcal{A}_{1}( \mathscr{X}_{a})-\mathcal{A}_{1}'( \mathscr{X}_{a}), \upsilon\bigr) \\ &\quad \leq\mathfrak{C}_{a} \biggl(\mathcal{A}_{1}\bigl( \mathfrak{T}_{1}^{n}\mathscr{X}_{a}\bigr)- \mathscr{A}_{1}\bigl(\mathfrak{T}_{1}^{n} \mathscr{X}_{a}\bigr),\frac{\upsilon.\mathfrak{T}_{1}^{n}}{2} \biggr)\diamond \mathfrak{C}_{a} \biggl(\mathscr{A}_{1}\bigl( \mathfrak{T}_{1}^{n}\mathscr{X}_{a}\bigr)- \mathcal{A}_{1}'\bigl(\mathfrak{T}_{1}^{n} \mathscr{X}_{a}\bigr),\frac{\upsilon.\mathfrak{T}_{1}^{n}}{2} \biggr) \\ &\quad \leq\mathfrak{C}_{a}' \biggl(N \bigl( \mathfrak{T}_{1}^{n}\mathscr{X}_{a}, \mathfrak{T}_{1}^{n}\mathscr{X}_{a}, \mathfrak{T}_{1}^{n}\mathscr{X}_{a}\bigr), \frac{\upsilon\mathfrak{T}_{1}^{n+1}}{2} \vert \mathfrak {T}_{1}-\mathscr{X}_{a} \vert \biggr) \\ &\quad \leq\mathfrak{C}_{a}' \biggl(N ( \mathscr{X}_{a},\mathscr{X}_{a},\mathscr{X}_{a}), \frac{\upsilon \mathfrak{T}_{1}^{n+1} \vert \mathfrak{T}_{1}-\mathscr{X}_{a} \vert }{2\cdot\mathscr{X}_{a}^{n}} \biggr). \end{aligned}$$

Since ${\lim_{n \rightarrow\infty}\frac{\upsilon \mathfrak {T}_{1}^{n+1}|\mathfrak{T}_{1}-\mathscr{X}_{a}|}{2 \mathscr{X}_{a}^{n}} = \infty,}$

$$\begin{aligned} \left . \begin{aligned} &\lim_{n \to\infty} \mathfrak{A}_{a}' \biggl(N (\mathscr{X}_{a},\mathscr{X}_{a}, \mathscr{X}_{a}),\frac{\upsilon \mathfrak{T}_{1}^{n+1} \vert \mathfrak{T}_{1}-\mathscr{X}_{a} \vert }{2\cdot\mathscr{X}_{a}^{n}} \biggr) =1, \\ &\lim_{n \to\infty} \mathfrak{B}_{a}' \biggl(N (\mathscr{X}_{a},\mathscr{X}_{a}, \mathscr{X}_{a}),\frac{\upsilon \mathfrak{T}_{1}^{n+1} \vert \mathfrak{T}_{1}-\mathscr{X}_{a} \vert }{2\cdot\mathscr{X}_{a}^{n}} \biggr)=0, \\ &\lim_{n \to\infty} \mathfrak{C}_{a}' \biggl(N (\mathscr{X}_{a},\mathscr{X}_{a}, \mathscr{X}_{a}),\frac{\upsilon \mathfrak{T}_{1}^{n+1} \vert \mathfrak{T}_{1}-\mathscr{X}_{a} \vert }{2\cdot\mathscr{X}_{a}^{n}} \biggr)=0. \end{aligned} \right \} \end{aligned}$$

Hence,

$$\begin{aligned} \left . \begin{aligned} &\mathfrak{A}_{a}\bigl( \mathcal{A}_{1}(\mathscr{X}_{a})-\mathcal{A}_{1}'( \mathscr{X}_{a}), \upsilon\bigr) =1, \\ &\mathfrak{B}_{a}\bigl(\mathcal{A}_{1}( \mathscr{X}_{a})-\mathcal{A}_{1}'( \mathscr{X}_{a}), \upsilon\bigr) =0, \\ &\mathfrak{B}_{a}\bigl(\mathcal{A}_{1}( \mathscr{X}_{a})-\mathcal{A}_{1}'( \mathscr{X}_{a}), \upsilon\bigr) =0. \end{aligned} \right \} \end{aligned}$$

Thus, $\mathcal{A}_{1}(\mathscr{X}_{a})=\mathcal{A}_{1}'(\mathscr{X}_{a})$. Hence, $\mathcal{A}_{1}(\mathscr{X}_{a})$ is unique.

Method 2: Assume that $\eta=-1$. Substituting p by $\frac {\mathscr{X}_{a}}{\mathfrak{T}_{1}}$ in (3.6) gives

$$\begin{aligned} \left . \begin{aligned} &\mathfrak{A}_{a} \biggl( \mathfrak{T}_{1}\mathscr{A}_{1}(\mathscr{X}_{a})- \mathfrak{T}_{1}^{2}\omega\biggl(\frac{\mathscr{X}_{a}}{\mathfrak {T}_{1}} \biggr),\upsilon\biggr) \geq\mathfrak{A}_{a}' \biggl(N \biggl(\frac{\mathscr{X}_{a}}{2},\frac{\mathscr{X}_{a}}{2},\frac{\mathscr{X}_{a}}{2} \biggr),\upsilon\biggr), \\ &\mathfrak{B}_{a} \biggl(\mathfrak{T}_{1} \mathscr{A}_{1}(\mathscr{X}_{a})- \mathfrak{T}_{1}^{2} \omega\biggl(\frac{\mathscr{X}_{a}}{\mathfrak{T}_{1}} \biggr),\upsilon \biggr) \leq \mathfrak{B}_{a}' \biggl(N \biggl( \frac{\mathscr{X}_{a}}{2},\frac{\mathscr{X}_{a}}{2},\frac{\mathscr{X}_{a}}{2} \biggr),\upsilon \biggr), \\ &\mathfrak{C}_{a} \biggl(\mathfrak{T}_{1} \mathscr{A}_{1}(\mathscr{X}_{a})- \mathfrak{T}_{1}^{2} \omega\biggl(\frac{\mathscr{X}_{a}}{\mathfrak{T}_{1}} \biggr),\upsilon \biggr) \leq \mathfrak{C}_{a}' \biggl(N \biggl( \frac{\mathscr{X}_{a}}{2},\frac{\mathscr{X}_{a}}{2},\frac{\mathscr{X}_{a}}{2} \biggr),\upsilon \biggr). \end{aligned} \right \} \end{aligned}$$

(3.24)

 □

Corollary 3.2

Let $\mathscr{A}_{1}$ be an approximately additive mapping from $(Z,\mathfrak{A}_{a}',\mathfrak{B}_{a}',\mathfrak{C}_{a}' )$ in a neutrosophic normed space and $(\mathcal{M},\mathfrak{A}_{a},\mathfrak {B}_{a},\mathfrak{C}_{a} )$ be a neutrosophic Banach space that satisfies the inequality

$$\begin{aligned} \begin{aligned} &\mathfrak{A}_{a} \bigl( \mathfrak{Z}(\mathscr{X}_{a},\mathscr{V}_{a}, \mathscr{W}_{a}),\upsilon\bigr) \\ &\quad\geq \left \{ \begin{aligned} &\mathfrak{A}_{a}' (\mathcal{S} , \upsilon), \\ &\mathfrak{A}_{a}' \bigl(\mathcal{S} \bigl( \Vert \mathscr{X}_{a} \Vert ^{\mathfrak{E}}+ \Vert \mathscr{V}_{a} \Vert ^{\mathfrak{F}}+ \Vert \mathscr{W}_{a} \Vert ^{\mathfrak{G}} \bigr), \upsilon\bigr),\quad \mathfrak{E},\mathfrak{F},\mathfrak{G} \ne1, \\ &\mathfrak{A}_{a}' \bigl(\mathcal{S} \Vert \mathscr{X}_{a} \Vert ^{\mathfrak{E}} \Vert \mathscr{V}_{a} \Vert ^{\mathfrak{F}} \Vert \mathscr{W}_{a} \Vert ^{\mathfrak{G}}, \upsilon\bigr),\quad \mathfrak{E}+\mathfrak{F}+\mathfrak{G}\ne1, \\ &\mathfrak{A}_{a}' \bigl(\mathcal{S} \bigl\{ \Vert \mathscr{X}_{a} \Vert ^{\mathfrak{E}} \Vert \mathscr{V}_{a} \Vert ^{\mathfrak{F}} \Vert \mathscr{W}_{a} \Vert ^{\mathfrak{G}} \\ &\quad {}+ \bigl( \Vert \mathscr{X}_{a} \Vert ^{\mathfrak{E}+\mathfrak{F}+\mathfrak{G}}+ \Vert \mathscr{V}_{a} \Vert ^{\mathfrak{E}+\mathfrak{F}+\mathfrak{G}}+ \Vert \mathscr{W}_{a} \Vert ^{\mathfrak{E}+\mathfrak{F}+\mathfrak{G}} \bigr) \bigr\} , \upsilon \bigr), \quad\mathfrak{E}+\mathfrak{F}+\mathfrak{G}\ne1, \end{aligned} \right . \end{aligned} \end{aligned}$$

(3.25)

$$\begin{aligned} &\mathfrak{B}_{a} \bigl(\mathfrak{Z}(\mathscr{X}_{a}, \mathscr{V}_{a},\mathscr{W}_{a}),\upsilon\bigr) \end{aligned}$$

(3.26)

$$\begin{aligned} &\quad\leq \left \{ \begin{aligned} &\mathfrak{B}_{a}' (\mathcal{S} , \upsilon), \\ &\mathfrak{B}_{a}' \bigl(\mathcal{S} \bigl( \Vert \mathscr{X}_{a} \Vert ^{\mathfrak{E}}+ \Vert \mathscr{V}_{a} \Vert ^{\mathfrak{F}}+ \Vert \mathscr{W}_{a} \Vert ^{\mathfrak{G}} \bigr), \upsilon\bigr),\quad \mathfrak{E},\mathfrak{F},\mathfrak{G} \ne1, \\ &\mathfrak{B}_{a}' \bigl(\mathcal{S} \Vert \mathscr{X}_{a} \Vert ^{\mathfrak{E}} \Vert \mathscr{X}_{a} \Vert ^{\mathfrak{F}} \Vert \mathscr{V}_{a} \Vert ^{\mathfrak{G}}, \upsilon\bigr),\quad \mathfrak{E}+\mathfrak{F}+\mathfrak{G}\ne1, \\ &\mathfrak{B}_{a}' \bigl(\mathcal{S} \bigl\{ \Vert \mathscr{X}_{a} \Vert ^{\mathfrak{E}} \Vert \mathscr{V}_{a} \Vert ^{\mathfrak{F}} \Vert \mathscr{W}_{a} \Vert ^{\mathfrak{G}} \\ &\quad {}+ \bigl( \Vert \mathscr{X}_{a} \Vert ^{\mathfrak{E}+\mathfrak{F}+\mathfrak{G}}+ \Vert \mathscr{V}_{a} \Vert ^{\mathfrak{E}+\mathfrak{F}+\mathfrak{G}}+ \Vert \mathscr{W}_{a} \Vert ^{\mathfrak{E}+\mathfrak{F}+\mathfrak{G}} \bigr) \bigr\} , \upsilon \bigr), \quad\mathfrak{E}+\mathfrak{F}+\mathfrak{G}\ne1, \end{aligned} \right . \end{aligned}$$

(3.27)

$$\begin{aligned} &\mathfrak{C}_{a} \bigl(\mathfrak{Z}(\mathscr{X}_{a}, \mathscr{V}_{a},\mathscr{W}_{a}),\upsilon\bigr) \end{aligned}$$

(3.28)

$$\begin{aligned} &\quad\leq \left \{ \begin{aligned} &\mathfrak{C}_{a}' (\mathcal{S} , \upsilon), \\ &\mathfrak{C}_{a}' \bigl(\mathcal{S} \bigl( \Vert \mathscr{X}_{a} \Vert ^{\mathfrak{E}}+ \Vert \mathscr{V}_{a} \Vert ^{\mathfrak{F}}+ \Vert \mathscr{W}_{a} \Vert ^{\mathfrak{G}} \bigr), \upsilon\bigr),\quad \mathfrak{E},\mathfrak{F},\mathfrak{G} \ne1, \\ &\mathfrak{C}_{a}' \bigl(\mathcal{S} \Vert \mathscr{X}_{a} \Vert ^{\mathfrak{E}} \Vert \mathscr{X}_{a} \Vert ^{\mathfrak{F}} \Vert \mathscr{V}_{a} \Vert ^{\mathfrak{G}}, \upsilon\bigr),\quad \mathfrak{E}+\mathfrak{F}+\mathfrak{G}\ne1, \\ &\mathfrak{C}_{a}' \bigl(\mathcal{S} \bigl\{ \Vert \mathscr{X}_{a} \Vert ^{\mathfrak{E}} \Vert \mathscr{V}_{a} \Vert ^{\mathfrak{F}} \Vert \mathscr{W}_{a} \Vert ^{\mathfrak{G}} \\ &\quad {}+ \bigl( \Vert \mathscr{X}_{a} \Vert ^{\mathfrak{E}+\mathfrak{F}+\mathfrak{G}}+ \Vert \mathscr{V}_{a} \Vert ^{\mathfrak{E}+\mathfrak{F}+\mathfrak{G}}+ \Vert \mathscr{W}_{a} \Vert ^{\mathfrak{E}+\mathfrak{F}+\mathfrak{G}} \bigr) \bigr\} , \upsilon \bigr), \quad\mathfrak{E}+\mathfrak{F}+\mathfrak{G}\ne1 \end{aligned} \right . \end{aligned}$$

(3.29)

such that

$$\begin{aligned} &\mathfrak{A}_{a}\bigl( \mathscr{A}_{1}(\mathscr{X}_{a}) - \mathcal{A}_{1}( \mathscr{X}_{a}),\upsilon\bigr) \\ &\quad\geq \left \{ \begin{aligned} &\mathfrak{A}_{a}' \bigl(\mathcal{S}, \mathfrak{T}_{1} t\upsilon \vert \mathfrak{T}_{1}-1 \vert \bigr), \\ &\mathfrak{A}_{a}'\bigl(\bigl[\mathcal{S} \Vert \mathscr{X}_{a} \Vert ^{\mathfrak{E}} \vert \mathfrak{T}_{1} \vert ^{\mathfrak{E}} + \mathcal{S} \Vert \mathscr{X}_{a} \Vert ^{\mathfrak{F}} \vert \mathfrak{T}_{1} \vert ^{\mathfrak{F}}+\mathcal{S} \Vert \mathscr{X}_{a} \Vert ^{\mathfrak{G}} \vert \mathfrak{T}_{1} \vert ^{\mathfrak{G}}\bigr], \\ &\quad\mathfrak{T}_{1} \upsilon\bigl[ \bigl\vert \mathfrak{T}_{1}-\mathfrak{T}_{1}^{\mathfrak{E}} \bigr\vert + \bigl\vert \mathfrak{T}_{1}-\mathfrak{T}_{1}^{\mathfrak{F}} \bigr\vert + \bigl\vert \mathfrak{T}_{1}-\mathfrak{T}_{1}^{\mathfrak{G}} \bigr\vert \bigr]\bigr), \\ &\mathfrak{A}_{a}'\bigl(\mathcal{S} \Vert \mathscr{X}_{a} \Vert ^{\mathfrak{E}+\mathfrak{F}+\mathfrak{G}} \vert \mathfrak{T}_{1} \vert ^{\mathfrak{E}+\mathfrak{F}+\mathfrak{G}}, \mathfrak{T}_{1} \upsilon\bigl\vert \mathfrak{T}_{1}- \mathfrak{T}_{1}^{\mathfrak{E}+\mathfrak{F}+\mathfrak{G}} \bigr\vert \bigr), \\ &\mathfrak{A}_{a}'\bigl(\mathcal{S} \Vert \mathscr{X}_{a} \Vert ^{\mathfrak{E}+\mathfrak{F}+\mathfrak{G}} \vert \mathfrak{T}_{1} \vert ^{\mathfrak{E}+\mathfrak{F}+\mathfrak{G}} \\ &\quad{}+\bigl[\mathcal{S} \Vert \mathscr{X}_{a} \Vert ^{\mathfrak{E}} \vert \mathfrak{T}_{1} \vert ^{\mathfrak{E}} + \mathcal{S} \Vert \mathscr{X}_{a} \Vert ^{\mathfrak{F}} \vert \mathfrak{T}_{1} \vert ^{\mathfrak{F}}+ \mathcal{S} \Vert \mathscr{X}_{a} \Vert ^{\mathfrak{G}} \vert \mathfrak{T}_{1} \vert ^{\mathfrak{G}}\bigr], \\ &\quad\mathfrak{T}_{1} \upsilon\bigl\vert \mathfrak{T}_{1}- \mathfrak{T}_{1}^{\mathfrak{E}+\mathfrak{F}+\mathfrak{G}} \bigr\vert \bigr), \end{aligned} \right . \\ &\mathfrak{B}_{a} \bigl(\mathscr{A}_{1}( \mathscr{X}_{a}) - \mathcal{A}_{1}(\mathscr{X}_{a}), \upsilon\bigr) \\ &\quad\leq \left \{ \begin{aligned} & \mathfrak{B}_{a}' \bigl(\mathcal{S}, \mathfrak{T}_{1} \upsilon \vert \mathfrak{T}_{1}-1 \vert \bigr), \\ &\mathfrak{B}_{a}' \bigl( \bigl[\mathcal{S} \Vert \mathscr{X}_{a} \Vert ^{\mathfrak{E}} \vert \mathfrak{T}_{1} \vert ^{\mathfrak{E}} + \mathcal{S} \Vert \mathscr{X}_{a} \Vert ^{\mathfrak{F}} \vert \mathfrak{T}_{1} \vert ^{\mathfrak{F}}+\mathcal{S} \Vert \mathscr{X}_{a} \Vert ^{\mathfrak{G}} \vert \mathfrak{T}_{1} \vert ^{\mathfrak{G}} \bigr], \\ &\quad\mathfrak{T}_{1} \upsilon\bigl[ \bigl\vert \mathfrak{T}_{1}-\mathfrak{T}_{1}^{\mathfrak{E}} \bigr\vert + \bigl\vert \mathfrak{T}_{1}-\mathfrak{T}_{1}^{\mathfrak{F}} \bigr\vert + \bigl\vert \mathfrak{T}_{1}-\mathfrak{T}_{1}^{\mathfrak{G}} \bigr\vert \bigr] \bigr), \\ &\mathfrak{B}_{a}' \bigl(\mathcal{S} \Vert \mathscr{X}_{a} \Vert ^{\mathfrak{E}+\mathfrak{F}+\mathfrak{G}} \vert \mathfrak{T}_{1} \vert ^{\mathfrak{E}+\mathfrak{F}+\mathfrak{G}}, \mathfrak{T}_{1} \upsilon\bigl\vert \mathfrak{T}_{1}- \mathfrak{T}_{1}^{\mathfrak{E}+\mathfrak{F}+\mathfrak{G}} \bigr\vert \bigr), \\ &\mathfrak{B}_{a}' \bigl(\mathcal{S} \Vert \mathscr{X}_{a} \Vert ^{\mathfrak{E}+\mathfrak{F}+\mathfrak{G}} \vert \mathfrak{T}_{1} \vert ^{\mathfrak{E}+\mathfrak{F}+\mathfrak{G}} \\ &\quad{}+ \bigl[\mathcal{S} \Vert \mathscr{X}_{a} \Vert ^{\mathfrak{E}} \vert \mathfrak{T}_{1} \vert ^{\mathfrak{E}} + \mathcal{S} \Vert \mathscr{X}_{a} \Vert ^{\mathfrak{F}} \vert \mathfrak{T}_{1} \vert ^{\mathfrak{F}} + \mathcal{S} \Vert \mathscr{X}_{a} \Vert ^{\mathfrak{G}} \vert \mathfrak{T}_{1} \vert ^{\mathfrak{G}} \bigr], \\ &\quad\mathfrak{T}_{1} \upsilon\bigl\vert \mathfrak{T}_{1}- \mathfrak{T}_{1}^{\mathfrak{E}+\mathfrak{F}+\mathfrak{G}} \bigr\vert \bigr), \end{aligned} \right . \end{aligned}$$

(3.30)

$$\begin{aligned} &\mathfrak{C}_{a} \bigl(\mathscr{A}_{1}( \mathscr{X}_{a}) - \mathcal{A}_{1}(\mathscr{X}_{a}), \upsilon\bigr) \\ &\quad\leq \left \{ \begin{aligned} & \mathfrak{C}_{a}' \bigl(\mathcal{S}, \mathfrak{T}_{1} \upsilon \vert \mathfrak{T}_{1}-1 \vert \bigr), \\ &\mathfrak{C}_{a}'\bigl(\bigl[\mathcal{S} \Vert \mathscr{X}_{a} \Vert ^{\mathfrak{E}} \vert \mathfrak{T}_{1} \vert ^{\mathfrak{E}} + \mathcal{S} \Vert \mathscr{X}_{a} \Vert ^{\mathfrak{F}} \vert \mathfrak{T}_{1} \vert ^{\mathfrak{F}}+\mathcal{S} \Vert \mathscr{X}_{a} \Vert ^{\mathfrak{G}} \vert \mathfrak{T}_{1} \vert ^{\mathfrak{G}}\bigr], \\ &\quad\mathfrak{T}_{1} \upsilon\bigl[ \bigl\vert \mathfrak{T}_{1}-\mathfrak{T}_{1}^{\mathfrak{E}} \bigr\vert + \bigl\vert \mathfrak{T}_{1}-\mathfrak{T}_{1}^{\mathfrak{F}} \bigr\vert + \bigl\vert \mathfrak{T}_{1}-\mathfrak{T}_{1}^{\mathfrak{G}} \bigr\vert \bigr]\bigr), \\ &\mathfrak{C}_{a}'\bigl(\mathcal{S} \Vert \mathscr{X}_{a} \Vert ^{\mathfrak{E}+\mathfrak{F}+\mathfrak{G}} \vert \mathfrak{T}_{1} \vert ^{\mathfrak{E}+\mathfrak{F}+\mathfrak{G}}, \mathfrak{T}_{1} \upsilon\bigl\vert \mathfrak{T}_{1}-\mathfrak{T}_{1}^{\mathfrak{E}+\mathfrak{F}+\mathfrak {G}} \bigr\vert \bigr), \\ &\mathfrak{C}_{a}'\bigl(\mathcal{S} \Vert \mathscr{X}_{a} \Vert ^{\mathfrak{E}+\mathfrak{F}+\mathfrak{G}} \vert \mathfrak{T}_{1} \vert ^{\mathfrak{E}+\mathfrak{F}+\mathfrak{G}} \\ &\quad{}+\bigl[\mathcal{S} \Vert \mathscr{X}_{a} \Vert ^{\mathfrak{E}} \vert \mathfrak{T}_{1} \vert ^{\mathfrak{E}} + \mathcal{S} \Vert \mathscr{X}_{a} \Vert ^{\mathfrak{F}} \vert \mathfrak{T}_{1} \vert ^{\mathfrak{F}}+ \mathcal{S} \Vert \mathscr{X}_{a} \Vert ^{\mathfrak{G}} \vert \mathfrak{T}_{1} \vert ^{\mathfrak{G}}\bigr], \\ &\quad\mathfrak{T}_{1} \upsilon\bigl\vert \mathfrak{T}_{1}- \mathfrak{T}_{1}^{\mathfrak{E}+\mathfrak{F}+\mathfrak{G}} \bigr\vert \bigr). \end{aligned} \right . \end{aligned}$$

Proof

Let

$$ N (\mathscr{X}_{a},\mathscr{V}_{a}, \mathscr{W}_{a} ) = \left \{ \begin{aligned} &\mathcal{S}, \\ &\mathcal{S} \bigl( \Vert \mathscr{X}_{a} \Vert ^{\mathfrak{E}}+ \Vert \mathscr{V}_{a} \Vert ^{\mathfrak{F}}+ \Vert \mathscr{W}_{a} \Vert ^{\mathfrak{G}} \bigr), \\ &\mathcal{S} \Vert \mathscr{X}_{a} \Vert ^{\mathfrak{E}} \Vert y \Vert ^{\mathfrak{F}} \Vert z \Vert ^{\mathfrak{G}}, \\ &\mathcal{S} \bigl\{ \Vert \mathscr{X}_{a} \Vert ^{\mathfrak{E}} \Vert \mathscr{V}_{a} \Vert ^{\mathfrak{F}} \Vert \mathscr{W}_{a} \Vert ^{\mathfrak{G}} \\ &\quad {}+ \bigl( \Vert \mathscr{X}_{a} \Vert ^{\mathfrak{E}+\mathfrak{F}+\mathfrak{G}}+ \Vert \mathscr{V}_{a} \Vert ^{\mathfrak{E}+\mathfrak{F}+\mathfrak{G}}+ \Vert \mathscr{W}_{a} \Vert ^{\mathfrak{E}+\mathfrak{F}+\mathfrak{G}} \bigr) \bigr\} , \end{aligned} \right . $$

and

$$ \mathscr{X}_{a} = \left \{ \begin{aligned} &\mathfrak{T}_{1}^{0}, \\ &\mathfrak{T}_{1}^{A}+\mathfrak{T}_{1}^{B}+ \mathfrak{T}_{1}^{C}, \\ &\mathfrak{T}_{1}^{\mathfrak{E}+\mathfrak{F}+\mathfrak{G}}, \\ &\mathfrak{T}_{1}^{\mathfrak{E}+\mathfrak{F}+\mathfrak{G}}. \end{aligned} \right . $$

 □

4 Stability results: fixed point method [49]

Theorem 4.1

Let $N :\mathcal{L} \times\mathcal{L} \times\mathcal{L} \longrightarrow\mathcal{M}$ such that

$$\begin{aligned} \left . \begin{aligned} &\lim_{n \to\infty} \mathfrak{A}_{a}' \bigl(K \bigl(\mathfrak{D}_{i}^{n} \mathscr{X}_{a},\mathfrak{D}_{i}^{n} \mathscr{V}_{a},\mathfrak{D}_{i}^{n} \mathscr{W}_{a} \bigr),\mathfrak{D}^{n} \upsilon\bigr) = 1, \\ &\lim_{n \to\infty} \mathfrak{B}_{a}' \bigl(K \bigl(\mathfrak{D}_{i}^{n}\mathscr{X}_{a}, \mathfrak{D}_{i}^{n}\mathscr{V}_{a}, \mathfrak{D}_{i}^{n}\mathscr{W}_{a} \bigr), \mathfrak{D}^{n} \upsilon\bigr) = 0, \\ &\lim_{n \to\infty} \mathfrak{C}_{a}' \bigl(K \bigl(\mathfrak{D}_{i}^{n}\mathscr{X}_{a}, \mathfrak{D}_{i}^{n}\mathscr{V}_{a}, \mathfrak{D}_{i}^{n}\mathscr{W}_{a} \bigr), \mathfrak{D}^{n} \upsilon\bigr) = 0, \end{aligned} \right \} \end{aligned}$$

(4.1)

where

$$\begin{aligned} \mathfrak{D}_{i} = \textstyle\begin{cases} \mathfrak{T}_{1} &\textit{if }i=0, \\ \frac{1}{\mathfrak{T}_{1}}&\textit{if }i=1, \end{cases}\displaystyle \end{aligned}$$

(4.2)

then

$$\begin{aligned} \left . \begin{aligned} &\mathfrak{A}_{a} \bigl( \mathfrak{Z}(\mathscr{X}_{a},\mathscr{V}_{a}, \mathscr{W}_{a}),\upsilon\bigr) \geq\mathfrak{A}_{a}' \bigl(K (\mathscr{X}_{a},\mathscr{V}_{a}, \mathscr{W}_{a} ),\upsilon\bigr), \\ &\mathfrak{B}_{a} \bigl(\mathfrak{Z}(\mathscr{X}_{a}, \mathscr{V}_{a},\mathscr{W}_{a}),\upsilon\bigr) \leq \mathfrak{B}_{a}' \bigl(K (\mathscr{X}_{a}, \mathscr{V}_{a},\mathscr{W}_{a} ),\upsilon\bigr), \\ &\mathfrak{C}_{a} \bigl(\mathfrak{Z}(\mathscr{X}_{a}, \mathscr{V}_{a},\mathscr{W}_{a}),\upsilon\bigr) \leq \mathfrak{C}_{a}' \bigl(K (\mathscr{X}_{a}, \mathscr{V}_{a},\mathscr{W}_{a} ),\upsilon\bigr). \end{aligned} \right \} \end{aligned}$$

(4.3)

If $L=L(i)$ and

$$ \mathscr{A}_{1}(\mathscr{X}_{a})= \frac{1}{\mathfrak{T}_{1}} K \biggl(\frac{\mathscr{X}_{a}}{\mathfrak {T}_{1}},\frac{\mathscr{X}_{a}}{\mathfrak{T}_{1}}, \frac{\mathscr{X}_{a}}{\mathfrak{T}_{1}} \biggr), $$

(4.4)

and

$$\begin{aligned} \left . \begin{aligned} &\mathfrak{A}_{a}' \biggl(L\frac{\mathscr{A}_{1}(\mathfrak{D}_{i}\mathscr{X}_{a})}{\mathfrak{D}_{i}}, \upsilon\biggr) = \mathfrak{A}_{a}' \bigl(\mathscr{A}_{1}(\mathscr{X}_{a}),\upsilon\bigr), \\ &\mathfrak{B}_{a}' \biggl(L\frac{\mathscr{A}_{1}(\mathfrak{D}_{i}\mathscr{X}_{a})}{\mathfrak{D}_{i}}, \upsilon\biggr) = \mathfrak{B}_{a}' \bigl( \mathscr{A}_{1}(\mathscr{X}_{a}),\upsilon\bigr), \\ &\mathfrak{C}_{a}' \biggl(L\frac{\mathscr{A}_{1}(\mathfrak{D}_{i}\mathscr{X}_{a})}{\mathfrak{D}_{i}}, \upsilon\biggr) = \mathfrak{C}_{a}' \bigl( \mathscr{A}_{1}(\mathscr{X}_{a}),\upsilon\bigr) \end{aligned} \right \} \end{aligned}$$

(4.5)

also

$$\begin{aligned} \left . \begin{aligned} &\mathfrak{A}_{a} \bigl( \mathscr{A}_{1}(\mathscr{X}_{a})-\mathcal{A}_{1}( \mathscr{X}_{a}), \upsilon\bigr) \geq\mathfrak{A}_{a}' \biggl(\mathscr{A}_{1}(\mathscr{X}_{a}), \frac{L^{1-i}}{1-L}\upsilon\biggr), \\ &\mathfrak{B}_{a} \bigl(\mathscr{A}_{1}( \mathscr{X}_{a})-\mathcal{A}_{1}(\mathscr{X}_{a}), \upsilon\bigr) \leq\mathfrak{B}_{a}' \biggl( \mathscr{A}_{1}(\mathscr{X}_{a}),\frac{L^{1-i}}{1-L} \upsilon\biggr), \\ &\mathfrak{C}_{a} \bigl(\mathscr{A}_{1}( \mathscr{X}_{a})-\mathcal{A}_{1}(\mathscr{X}_{a}), \upsilon\bigr) \leq\mathfrak{C}_{a}' \biggl( \mathscr{A}_{1}(\mathscr{X}_{a}),\frac{L^{1-i}}{1-L} \upsilon\biggr). \end{aligned} \right \} \end{aligned}$$

(4.6)

Proof

Let

$$\begin{aligned}& \mathcal{S}=\bigl\{ h_{1}\mid h_{1}:{ \mathscr{X}_{a}}\longrightarrow{Y}, \mathscr{A}_{1}(0)=0 \bigr\} , \\& d(h_{1},f_{1})=\inf\left\{L \in(0,\infty):\left \{ \begin{aligned} &\mathfrak{A}_{a}(\mathscr{A}_{1}(\mathscr{X}_{a})-\mathscr{A}_{1}(\mathscr{X}_{a}),\upsilon)\geq\mathfrak{A}_{a}'(\mathscr{A}_{1}(\mathscr{X}_{a}), L\upsilon), \upsilon> 0, \\ &\mathfrak{B}_{a}(\mathscr{A}_{1}(\mathscr{X}_{a})-\mathscr{A}_{1}(\mathscr{X}_{a}), \upsilon)\leq\mathfrak{B}_{a}'(\mathscr{A}_{1}(\mathscr{X}_{a}), L\upsilon), \upsilon> 0, \\ &\mathfrak{C}_{a}(\mathscr{A}_{1}(\mathscr{X}_{a})-\mathscr{A}_{1}(\mathscr{X}_{a}), \upsilon)\leq\mathfrak{B}_{a}'(\mathscr{A}_{1}(\mathscr{X}_{a}), L\upsilon), \upsilon> 0 \end{aligned} \right \} \right\} . \end{aligned}$$

(4.7)

Hence, by (4.7)

$$\begin{aligned} \inf\left\{ L \in(0,\infty): \begin{aligned} &\left \{ \begin{aligned} &\mathfrak{A}_{a}\bigl(\mathscr{A}_{1}( \mathscr{X}_{a})-\mathscr{A}_{1}(\mathscr{X}_{a}), \upsilon\bigr)\geq\mathfrak{A}_{a}'\bigl( \mathscr{A}_{1}(\mathscr{X}_{a}), \upsilon\bigr)\bigr\} , \\ &\mathfrak{A}_{a}\biggl(\frac{1}{\mathfrak{D}_{i} }\mathscr{A}_{1}( \mathfrak{D}_{i} \mathscr{X}_{a})-\frac{1}{\mathfrak{D}_{i} } \mathscr{A}_{1}(\mathfrak{D}_{i} \mathscr{X}_{a}), \upsilon\biggr)\geq\mathfrak{A}_{a}'\bigl( \mathscr{A}_{1}(\mathfrak{D}_{i} \mathscr{X}_{a}), \mathfrak{D}_{i} \upsilon\bigr)\bigr\} , \\ & \mathfrak{A}_{a}\biggl(\frac{1}{\mathfrak{D}_{i} }\mathscr{A}_{1}( \mathfrak{D}_{i} \mathscr{X}_{a})-\frac{1}{\mathfrak{D}_{i} } \mathscr{A}_{1}(\mathfrak{D}_{i} \mathscr{X}_{a}), \upsilon\biggr)\geq\mathfrak{A}_{a}'\bigl( \mathscr{A}_{1}(\mathscr{X}_{a}), L \upsilon\bigr)\}, \\ & \mathfrak{A}_{a}\bigl(J\mathscr{A}_{1}( \mathscr{X}_{a})-J\mathscr{A}_{1}(\mathscr{X}_{a}), \upsilon\bigr)\geq\mathfrak{A}_{a}'\bigl( \mathscr{A}_{1}(\mathscr{X}_{a}), L \upsilon\bigr)\bigr\} , \end{aligned} \right . \\ &\left \{ \begin{aligned} & \mathfrak{B}_{a}\bigl( \mathscr{A}_{1}(\mathscr{X}_{a})-\mathscr{A}_{1}( \mathscr{X}_{a}), \upsilon\bigr)\leq\mathfrak{B}_{a}' \bigl(\mathscr{A}_{1}(\mathscr{X}_{a}), \upsilon\bigr) \bigr\} , \\ & \mathfrak{B}_{a}\biggl(\frac{1}{\mathfrak{D}_{i} }\mathscr{A}_{1}( \mathfrak{D}_{i} \mathscr{X}_{a})-\frac{1}{\mathfrak{D}_{i} } \mathscr{A}_{1}(\mathfrak{D}_{i} \mathscr{X}_{a}), \upsilon\biggr)\leq\mathfrak{B}_{a}'\bigl( \mathscr{A}_{1}(\mathfrak{D}_{i} \mathscr{X}_{a}), \mathfrak{D}_{i} \upsilon\bigr)\bigr\} , \\ & \mathfrak{B}_{a}\biggl(\frac{1}{\mathfrak{D}_{i} }\mathscr{A}_{1}( \mathfrak{D}_{i} \mathscr{X}_{a})-\frac{1}{\mathfrak{D}_{i} } \mathscr{A}_{1}(\mathfrak{D}_{i} \mathscr{X}_{a}), \upsilon\biggr)\leq\mathfrak{B}_{a}'\bigl( \mathscr{A}_{1}(\mathscr{X}_{a}), L\upsilon\bigr)\bigr\} , \\ &\mathfrak{B}_{a}\bigl(J\mathscr{A}_{1}( \mathscr{X}_{a})-J\mathscr{A}_{1}(\mathscr{X}_{a}), \upsilon\bigr)\leq\mathfrak{B}_{a}'\bigl( \mathscr{A}_{1}(\mathscr{X}_{a}), L \upsilon\bigr)\bigr\} , \end{aligned} \right . \\ &\left \{ \begin{aligned} & \mathfrak{C}_{a}\bigl( \mathscr{A}_{1}(\mathscr{X}_{a})-\mathscr{A}_{1}( \mathscr{X}_{a}), \upsilon\bigr)\leq\mathfrak{C}_{a}' \bigl(\mathscr{A}_{1}(\mathscr{X}_{a}), \upsilon\bigr) \bigr\} , \\ & \mathfrak{C}_{a}\biggl(\frac{1}{\mathfrak{D}_{i} }\mathscr{A}_{1}( \mathfrak{D}_{i} \mathscr{X}_{a})-\frac{1}{\mathfrak{D}_{i} } \mathscr{A}_{1}(\mathfrak{D}_{i} \mathscr{X}_{a}), \upsilon\biggr)\leq\mathfrak{C}_{a}'\bigl( \mathscr{A}_{1}(\mathfrak{D}_{i} \mathscr{X}_{a}), \mathfrak{D}_{i} \upsilon\bigr)\bigr\} , \\ & \mathfrak{C}_{a}\biggl(\frac{1}{\mathfrak{D}_{i} }\mathscr{A}_{1}( \mathfrak{D}_{i} \mathscr{X}_{a})-\frac{1}{\mathfrak{D}_{i} } \mathscr{A}_{1}(\mathfrak{D}_{i} \mathscr{X}_{a}), \upsilon\biggr)\leq\mathfrak{C}_{a}'\bigl( \mathscr{A}_{1}(\mathscr{X}_{a}), L\upsilon\bigr)\bigr\} , \\ &\mathfrak{C}_{a}\bigl(J\mathscr{A}_{1}( \mathscr{X}_{a})-J\mathscr{A}_{1}(\mathscr{X}_{a}), \upsilon\bigr)\leq\mathfrak{C}_{a}'\bigl( \mathscr{A}_{1}(\mathscr{X}_{a}), L \upsilon\bigr)\bigr\} \end{aligned} \right . \end{aligned} \right\} \end{aligned}$$

after that

$$ \inf\left\{ 1 \in(0,\infty): \left\{ \begin{aligned} &\mathfrak{A}_{a} \bigl( \mathscr{A}_{1}(\mathfrak{T}_{1}\mathscr{X}_{a})- \mathfrak{T}_{1}\mathscr{A}_{1}(\mathscr{X}_{a}), \upsilon\bigr)\geq\mathfrak{A}_{a}' \bigl(K( \mathscr{X}_{a},\mathscr{X}_{a},\mathscr{X}_{a}), \mathfrak{T}_{1}\upsilon\bigr), \\ &\mathfrak{B}_{a} \bigl(\mathscr{A}_{1}( \mathfrak{T}_{1}\mathscr{X}_{a})-\mathfrak{T}_{1} \mathscr{A}_{1}(\mathscr{X}_{a}), \upsilon\bigr)\leq \mathfrak{B}_{a}' \bigl(K(\mathscr{X}_{a}, \mathscr{X}_{a},\mathscr{X}_{a}),\mathfrak{T}_{1} \upsilon\bigr), \\ &\mathfrak{C}_{a} \bigl(\mathscr{A}_{1}( \mathfrak{T}_{1}\mathscr{X}_{a})-\mathfrak{T}_{1} \mathscr{A}_{1}(\mathscr{X}_{a}), \upsilon\bigr)\leq \mathfrak{C}_{a}' \bigl(K(\mathscr{X}_{a}, \mathscr{X}_{a},\mathscr{X}_{a}),\mathfrak{T}_{1} \upsilon\bigr) \end{aligned} \right\} \right\}. $$

(4.8)

Assuming $i=0$,

$$\begin{aligned}& \inf\left\{L^{1-0} \in(0,\infty): \left \{ \begin{aligned} &\mathfrak{A}_{a} \bigl( \mathscr{A}_{1}(\mathfrak{T}_{1}\mathscr{X}_{a})- \mathfrak{T}_{1}\mathscr{A}_{1}(\mathscr{X}_{a}), \upsilon\bigr) \geq\mathfrak{A}_{a}' \bigl(K( \mathscr{X}_{a},\mathscr{X}_{a},\mathscr{X}_{a}), \mathfrak{T}_{1}\upsilon\bigr), \\ &\mathfrak{A}_{a} \biggl(\frac{\mathscr{A}_{1}(\mathfrak{T}_{1}\mathscr{X}_{a})}{\mathfrak{T}_{1}}-\mathscr{A}_{1}( \mathscr{X}_{a}),\upsilon\biggr) \geq\mathfrak{A}_{a}' \bigl(K(\mathscr{X}_{a},\mathscr{X}_{a}, \mathscr{X}_{a}),\mathfrak{T}_{1}^{2} \upsilon\bigr), \\ &\mathfrak{A}_{a} \bigl(J\mathscr{A}_{1}( \mathscr{X}_{a})-\mathscr{A}_{1}(\mathscr{X}_{a}), \upsilon\bigr)\geq\mathfrak{A}_{a}' \bigl( \mathscr{A}_{1}(\mathscr{X}_{a}),L\upsilon\bigr), \\ &\mathfrak{A}_{a} \bigl(J\mathscr{A}_{1}( \mathscr{X}_{a})-\mathscr{A}_{1}(\mathscr{X}_{a}), \upsilon\bigr)\geq\mathfrak{A}_{a}' \bigl( \mathscr{A}_{1}(\mathscr{X}_{a}),L\upsilon\bigr), \\ &\mathfrak{A}_{a} \bigl(J\mathscr{A}_{1}( \mathscr{X}_{a})-\mathscr{A}_{1}(\mathscr{X}_{a}), \upsilon\bigr)\geq\mathfrak{A}_{a}' \bigl( \mathscr{A}_{1}(\mathscr{X}_{a}),L\upsilon\bigr) \end{aligned} \right \} \right\} , \end{aligned}$$

(4.9)

$$\begin{aligned}& \inf\left\{L^{1-0} \in(0,\infty): \left \{ \begin{aligned} &\mathfrak{B}_{a} \bigl(\mathscr{A}_{1}(\mathfrak{T}_{1} \mathscr{X}_{a})-\mathfrak{T}_{1}\mathscr{A}_{1}( \mathscr{X}_{a}), t \bigr) \leq\mathfrak{B}_{a}' \bigl(K(\mathscr{X}_{a},\mathscr{X}_{a}, \mathscr{X}_{a}),\mathfrak{T}_{1}\upsilon\bigr), \\ &\mathfrak{B}_{a} \biggl(\frac{\mathscr{A}_{1}(\mathfrak{T}_{1}\mathscr{X}_{a})}{\mathfrak{T}_{1}}-\mathscr{A}_{1}( \mathscr{X}_{a}),\upsilon\biggr)\leq\mathfrak{B}_{a}' \bigl(K(\mathscr{X}_{a},\mathscr{X}_{a}, \mathscr{X}_{a}),\mathfrak{T}_{1}^{2} \upsilon\bigr), \\ &\mathfrak{B}_{a} \bigl(J\mathscr{A}_{1}( \mathscr{X}_{a})-\mathscr{A}_{1}(\mathscr{X}_{a}), \upsilon\bigr)\leq\mathfrak{B}_{a}' \bigl( \mathscr{A}_{1}(\mathscr{X}_{a}),L\upsilon\bigr), \\ &\mathfrak{B}_{a} \bigl(J\mathscr{A}_{1}( \mathscr{X}_{a})-\mathscr{A}_{1}(\mathscr{X}_{a}), \upsilon\bigr)\leq\mathfrak{B}_{a}' \bigl( \mathscr{A}_{1}(\mathscr{X}_{a}),L\upsilon\bigr), \\ &\mathfrak{B}_{a} \bigl(J\mathscr{A}_{1}( \mathscr{X}_{a})-\mathscr{A}_{1}(\mathscr{X}_{a}), \upsilon\bigr)\leq\mathfrak{B}_{a}' \bigl( \mathscr{A}_{1}(\mathscr{X}_{a}),L\upsilon\bigr) \end{aligned} \right \} \right\} , \end{aligned}$$

(4.10)

$$\begin{aligned}& \inf\left\{L^{1-0} \in(0,\infty): \left \{ \begin{aligned} &\mathfrak{C}_{a} \bigl(\mathscr{A}_{1}(\mathfrak{T}_{1} \mathscr{X}_{a})-\mathfrak{T}_{1}\mathscr{A}_{1}( \mathscr{X}_{a}), t \bigr) \leq\mathfrak{B}_{a}' \bigl(K(\mathscr{X}_{a},\mathscr{X}_{a}, \mathscr{X}_{a}),\mathfrak{T}_{1}\upsilon\bigr), \\ &\mathfrak{C}_{a} \biggl(\frac{\mathscr{A}_{1}(\mathfrak{T}_{1}\mathscr{X}_{a})}{\mathfrak{T}_{1}}-\mathscr{A}_{1}( \mathscr{X}_{a}),\upsilon\biggr)\leq\mathfrak{C}_{a}' \bigl(K(\mathscr{X}_{a},\mathscr{X}_{a}, \mathscr{X}_{a}),\mathfrak{T}_{1}^{2} \upsilon\bigr), \\ &\mathfrak{C}_{a} \bigl(J\mathscr{A}_{1}( \mathscr{X}_{a})-\mathscr{A}_{1}(\mathscr{X}_{a}), \upsilon\bigr)\leq\mathfrak{C}_{a}' \bigl( \mathscr{A}_{1}(\mathscr{X}_{a}),L\upsilon\bigr), \\ &\mathfrak{C}_{a} \bigl(J\mathscr{A}_{1}( \mathscr{X}_{a})-\mathscr{A}_{1}(\mathscr{X}_{a}), \upsilon\bigr)\leq\mathfrak{C}_{a}' \bigl( \mathscr{A}_{1}(\mathscr{X}_{a}),L\upsilon\bigr), \\ &\mathfrak{C}_{a} \bigl(J\mathscr{A}_{1}( \mathscr{X}_{a})-\mathscr{A}_{1}(\mathscr{X}_{a}), \upsilon\bigr)\leq\mathfrak{C}_{a}' \bigl( \mathscr{A}_{1}(\mathscr{X}_{a}),L\upsilon\bigr) \end{aligned} \right \} \right\} . \end{aligned}$$

(4.11)

If $i=1$, and

$$\begin{aligned}& \inf\bigl\{ L^{1-1} \in(0,\infty): \\& \left. \begin{aligned} &\quad \left\{ \begin{aligned} &\mathfrak{A}_{a} \biggl(\mathscr{A}_{1}(\mathscr{X}_{a})- \mathfrak{T}_{1}\omega\biggl(\frac{\mathscr{X}_{a}}{\mathfrak{T}_{1}} \biggr),\upsilon \biggr) \geq\mathfrak{A}_{a}' \biggl(K \biggl( \frac{\mathscr{X}_{a}}{\mathfrak{T}_{1}},\frac{\mathscr{X}_{a}}{\mathfrak{T}_{1}} \biggr),\mathfrak{T}_{1} \upsilon\biggr), \\ &\mathfrak{A}_{a} \bigl(\mathscr{A}_{1}( \mathscr{X}_{a})-J\mathscr{A}_{1}(\mathscr{X}_{a}), \upsilon\bigr)\geq\mathfrak{A}_{a}' \bigl( \mathscr{A}_{1}(\mathscr{X}_{a}),\upsilon\bigr), \\ &\mathfrak{A}_{a} \bigl(\mathscr{A}_{1}( \mathscr{X}_{a})-J\mathscr{A}_{1}(\mathscr{X}_{a}), \upsilon\bigr)\geq\mathfrak{A}_{a}' \bigl( \mathscr{A}_{1}(\mathscr{X}_{a}),\upsilon\bigr), \\ &\mathfrak{A}_{a} \bigl(\mathscr{A}_{1}( \mathscr{X}_{a})-J\mathscr{A}_{1}(\mathscr{X}_{a}), \upsilon\bigr)\geq\mathfrak{A}_{a}' \bigl( \mathscr{A}_{1}(\mathscr{X}_{a}),\upsilon\bigr), \end{aligned} \right. \\ &\quad \left\{ \begin{aligned} &\mathfrak{B}_{a} \biggl( \mathscr{A}_{1}(\mathscr{X}_{a})-\mathfrak{T}_{1} \omega\biggl(\frac{\mathscr{X}_{a}}{\mathfrak{T}_{1}},\frac{\mathscr{X}_{a}}{\mathfrak{T}_{1}} \biggr), t \biggr) \leq \mathfrak{B}_{a}' \biggl(K \biggl( \frac{\mathscr{X}_{a}}{\mathfrak{T}_{1}} \biggr),\mathfrak {T}_{1}\upsilon\biggr), \\ &\mathfrak{B}_{a} \bigl(\mathscr{A}_{1}( \mathscr{X}_{a})-J\mathscr{A}_{1}(\mathscr{X}_{a}), \upsilon\bigr)\leq\mathfrak{B}_{a}' \bigl( \mathscr{A}_{1}(\mathscr{X}_{a}),\upsilon\bigr), \\ &\mathfrak{B}_{a} \bigl(\mathscr{A}_{1}( \mathscr{X}_{a})-J\mathscr{A}_{1}(\mathscr{X}_{a}), \upsilon\bigr)\leq\mathfrak{B}_{a}' \bigl( \mathscr{A}_{1}(\mathscr{X}_{a}),\upsilon\bigr), \\ &\mathfrak{B}_{a} \bigl(\mathscr{A}_{1}( \mathscr{X}_{a})-J\mathscr{A}_{1}(\mathscr{X}_{a}), \upsilon\bigr)\leq\mathfrak{B}_{a}' \bigl( \mathscr{A}_{1}(\mathscr{X}_{a}),\upsilon\bigr), \end{aligned} \right. \\ &\quad \left\{ \begin{aligned} &\mathfrak{C}_{a} \biggl( \mathscr{A}_{1}(\mathscr{X}_{a})-\mathfrak{T}_{1} \omega\biggl(\frac{\mathscr{X}_{a}}{\mathfrak{T}_{1}},\frac{\mathscr{X}_{a}}{\mathfrak{T}_{1}} \biggr), t \biggr) \leq \mathfrak{C}_{a}' \biggl(K \biggl( \frac{\mathscr{X}_{a}}{\mathfrak{T}_{1}} \biggr),\mathfrak {T}_{1}\upsilon\biggr), \\ &\mathfrak{C}_{a} \bigl(\mathscr{A}_{1}( \mathscr{X}_{a})-J\mathscr{A}_{1}(\mathscr{X}_{a}), \upsilon\bigr)\leq\mathfrak{C}_{a}' \bigl( \mathscr{A}_{1}(\mathscr{X}_{a}),\upsilon\bigr), \\ &\mathfrak{C}_{a} \bigl(\mathscr{A}_{1}( \mathscr{X}_{a})-J\mathscr{A}_{1}(\mathscr{X}_{a}), \upsilon\bigr)\leq\mathfrak{C}_{a}' \bigl( \mathscr{A}_{1}(\mathscr{X}_{a}),\upsilon\bigr), \\ &\mathfrak{C}_{a} \bigl(\mathscr{A}_{1}( \mathscr{X}_{a})-J\mathscr{A}_{1}(\mathscr{X}_{a}), \upsilon\bigr)\leq\mathfrak{C}_{a}' \bigl( \mathscr{A}_{1}(\mathscr{X}_{a}),\upsilon\bigr) \end{aligned} \right. \end{aligned} \right\} \end{aligned}$$

(4.12)

and

$$\begin{aligned} &\inf\left\{L^{1-i} \in(0,\infty):\left\{ \begin{aligned} &\mathfrak{A}_{a}\bigl(\mathscr{A}_{1}( \mathscr{X}_{a})-J\mathscr{A}_{1}(\mathscr{X}_{a}), \upsilon\bigr)\geq\mathfrak{A}_{a}'\bigl( \mathscr{A}_{1}(\mathscr{X}_{a}), L^{1-i} \upsilon\bigr), \\ &\mathfrak{B}_{a}\bigl(\mathscr{A}_{1}( \mathscr{X}_{a})-J\mathscr{A}_{1}(\mathscr{X}_{a}), \upsilon\bigr)\leq\mathfrak{B}_{a}'\bigl( \mathscr{A}_{1}(\mathscr{X}_{a}), L^{1-i} \upsilon\bigr), \\ &\mathfrak{C}_{a}\bigl(\mathscr{A}_{1}( \mathscr{X}_{a})-J\mathscr{A}_{1}(\mathscr{X}_{a}), \upsilon\bigr)\leq\mathfrak{C}_{a}'\bigl( \mathscr{A}_{1}(\mathscr{X}_{a}), L^{1-i} \upsilon\bigr) \end{aligned} \right\}\right\}. \end{aligned}$$

(4.13)

Hence property [49] Condition :1 holds.

According to the Condition :2, [49] there is a alternative fixed point $\mathcal{A}_{1}$ of J in $\mathcal {S}$ then

$$\begin{aligned}& \lim_{n \to\infty} \mathfrak{A}_{a} \biggl( \frac{\mathscr{A}_{1}(\mathfrak{D}_{i}^{n}\mathscr{X}_{a})}{\mathfrak {D}_{i}^{n}} - \mathcal{A}_{1}(\mathscr{X}_{a}), \upsilon\biggr) =1, \\& \lim_{n \to\infty} \mathfrak{B}_{a} \biggl( \frac{\mathscr{A}_{1}(\mathfrak{D}_{i}^{n}\mathscr{X}_{a})}{\mathfrak {D}_{i}^{n}} - \mathcal{A}_{1}(\mathscr{X}_{a}), \upsilon\biggr) =0, \\& \lim_{n \to\infty} \mathfrak{C}_{a} \biggl( \frac{\mathscr{A}_{1}(\mathfrak{D}_{i}^{n}\mathscr{X}_{a})}{\mathfrak {D}_{i}^{n}} - \mathcal{A}_{1}(\mathscr{X}_{a}), \upsilon\biggr) =0. \end{aligned}$$

For $\mathscr{A}_{1}$ is additive, taking $(\mathscr{X}_{a},\mathscr{V}_{a},\mathscr{W}_{a} )$ by $(\mathfrak{D}_{i}^{n} \mathscr{X}_{a}, \mathfrak{D}_{i}^{n} \mathscr{V}_{a},\mathfrak{D}_{i}^{n} \mathscr{W}_{a} )$.

Fixed point Condition :3 of [49], $\mathcal{A}_{1}$ is the alternative fixed point of J in the set $\Delta=\{\mathcal{A}_{1} \in\mathcal{S} : d(\mathscr{A}_{1},A)<\infty\}$, and $\mathcal{A}_{1}$ is a unique mapping then

$$ \left . \begin{aligned} &\mathfrak{A}_{a}\bigl( \mathscr{A}_{1}(\mathscr{X}_{a})-\mathcal{A}_{1}( \mathscr{X}_{a}), \upsilon\bigr)\geq\mathfrak{A}_{a}' \bigl(\mathscr{A}_{1}(\mathscr{X}_{a}), L^{1-i}\upsilon\bigr),\quad \mathscr{X}_{a}\in{\mathcal{L}}, \\ &\mathfrak{B}_{a}\bigl(\mathscr{A}_{1}( \mathscr{X}_{a})-\mathcal{A}_{1}(\mathscr{X}_{a}), \upsilon\bigr)\leq\mathfrak{B}_{a}'\bigl( \mathscr{A}_{1}(\mathscr{X}_{a}), L^{1-i} \upsilon\bigr),\quad \mathscr{X}_{a}\in{\mathcal{L}}, \\ &\mathfrak{C}_{a}\bigl(\mathscr{A}_{1}( \mathscr{X}_{a})-\mathcal{A}_{1}(\mathscr{X}_{a}), \upsilon\bigr)\leq\mathfrak{C}_{a}'\bigl( \mathscr{A}_{1}(\mathscr{X}_{a}), L^{1-i} \upsilon\bigr),\quad \mathscr{X}_{a}\in{\mathcal{L}}. \end{aligned} \right \} $$

By applying [49] Condition :4

$$ \left . \begin{aligned} &\mathfrak{A}_{a} \bigl( \mathscr{A}_{1}(\mathscr{X}_{a})-\mathcal{A}_{1}( \mathscr{X}_{a}), \upsilon\bigr) \geq\mathfrak{A}_{a}' \biggl(\mathscr{A}_{1}(\mathscr{X}_{a}), \frac{L^{1-i}}{1-L}\upsilon\biggr), \\ &\mathfrak{B}_{a} \bigl(\mathscr{A}_{1}( \mathscr{X}_{a})-\mathcal{A}_{1}(\mathscr{X}_{a}), \upsilon\bigr) \leq\mathfrak{B}_{a}' \biggl( \mathscr{A}_{1}(\mathscr{X}_{a}),\frac{L^{1-i}}{1-L} \upsilon\biggr), \\ &\mathfrak{C}_{a} \bigl(\mathscr{A}_{1}( \mathscr{X}_{a})-\mathcal{A}_{1}(\mathscr{X}_{a}), \upsilon\bigr) \leq\mathfrak{C}_{a}' \biggl( \mathscr{A}_{1}(\mathscr{X}_{a}),\frac{L^{1-i}}{1-L} \upsilon\biggr). \end{aligned} \right \} $$

Hence proved. □

Corollary 4.2

Let $\mathscr{A}_{1}$ be an approximately additive mapping from $(Z,\mathfrak{A}_{a}',\mathfrak{B}_{a}',\mathfrak{C}_{a}' )$ in a neutrosophic normed space and $(\mathcal{M},\mathfrak{A}_{a},\mathfrak {B}_{a},\mathfrak{C}_{a} )$ be a neutrosophic Banach space satisfying the inequality

$$\begin{aligned}& \mathfrak{A}_{a} \bigl(\mathfrak{Z}( \mathscr{X}_{a},\mathscr{V}_{a},\mathscr{W}_{a}), \upsilon\bigr) \\& \quad\geq \left \{ \begin{aligned} &\mathfrak{A}_{a}' (\mathcal{S} , \upsilon), \\ &\mathfrak{A}_{a}' \bigl(\mathcal{S} \bigl( \Vert \mathscr{X}_{a} \Vert ^{\mathfrak{E}}+ \Vert \mathscr{V}_{a} \Vert ^{\mathfrak{E}}+ \Vert \mathscr{W}_{a} \Vert ^{\mathfrak{E}} \bigr), \upsilon\bigr),\quad \mathfrak{E}\ne1, \\ &\mathfrak{A}_{a}' \bigl(\mathcal{S} \Vert \mathscr{X}_{a} \Vert ^{\mathfrak{E}} \Vert \mathscr{V}_{a} \Vert ^{\mathfrak{E}} \Vert \mathscr{W}_{a} \Vert ^{\mathfrak{E}}, \upsilon\bigr), \quad3 \mathfrak{E}\ne1, \\ &\mathfrak{A}_{a}' \bigl(\mathcal{S} \bigl\{ \Vert \mathscr{X}_{a} \Vert ^{\mathfrak{E}} \Vert \mathscr{V}_{a} \Vert ^{\mathfrak{E}} \Vert \mathscr{W}_{a} \Vert ^{\mathfrak{E}}+ \bigl( \Vert \mathscr{X}_{a} \Vert ^{3\mathfrak{E}}+ \Vert \mathscr{V}_{a} \Vert ^{3\mathfrak{E}}+ \Vert \mathscr{W}_{a} \Vert ^{3\mathfrak{E}} \bigr) \bigr\} , \upsilon \bigr),\quad3\mathfrak{E}\ne1, \end{aligned} \right . \\& \mathfrak{B}_{a} \bigl(\mathfrak{Z}(\mathscr{X}_{a}, \mathscr{V}_{a},\mathscr{W}_{a}),\upsilon\bigr) \\& \quad\leq \left \{ \begin{aligned} &\mathfrak{B}_{a}' (\mathcal{S} , \upsilon), \\ &\mathfrak{B}_{a}' \bigl(\mathcal{S} \bigl( \Vert \mathscr{X}_{a} \Vert ^{\mathfrak{E}}+ \Vert \mathscr{V}_{a} \Vert ^{\mathfrak{E}}+ \Vert \mathscr{W}_{a} \Vert ^{\mathfrak{E}} \bigr), \upsilon\bigr),\quad \mathfrak{E}\ne1, \\ &\mathfrak{B}_{a}' \bigl(\mathcal{S} \Vert \mathscr{X}_{a} \Vert ^{\mathfrak{E}} \Vert \mathscr{V}_{a} \Vert ^{\mathfrak{E}} \Vert \mathscr{W}_{a} \Vert ^{\mathfrak{E}}, \upsilon\bigr), \quad3 \mathfrak{E}\ne1, \\ &\mathfrak{B}_{a}' \bigl(\mathcal{S} \bigl\{ \Vert \mathscr{X}_{a} \Vert ^{\mathfrak{E}} \Vert \mathscr{V}_{a} \Vert ^{\mathfrak{E}} \Vert \mathscr{W}_{a} \Vert ^{\mathfrak{E}}+ \bigl( \Vert \mathscr{X}_{a} \Vert ^{3\mathfrak{E}}+ \Vert \mathscr{V}_{a} \Vert ^{3\mathfrak{E}}+ \Vert \mathscr{W}_{a} \Vert ^{3\mathfrak{E}} \bigr) \bigr\} , \upsilon \bigr),\quad3\mathfrak{E}\ne1, \end{aligned} \right . \\& \mathfrak{C}_{a} \bigl(\mathfrak{Z}(\mathscr{X}_{a}, \mathscr{V}_{a},\mathscr{W}_{a}),\upsilon\bigr) \\& \quad\leq \left \{ \begin{aligned} &\mathfrak{C}_{a}' (\mathcal{S} , \upsilon), \\ &\mathfrak{C}_{a}' \bigl(\mathcal{S} \bigl( \Vert \mathscr{X}_{a} \Vert ^{\mathfrak{E}}+ \Vert \mathscr{V}_{a} \Vert ^{\mathfrak{E}}+ \Vert \mathscr{W}_{a} \Vert ^{\mathfrak{E}} \bigr), \upsilon\bigr),\quad \mathfrak{E}\ne1, \\ &\mathfrak{C}_{a}' \bigl(\mathcal{S} \Vert \mathscr{X}_{a} \Vert ^{\mathfrak{E}} \Vert \mathscr{V}_{a} \Vert ^{\mathfrak{E}} \Vert \mathscr{W}_{a} \Vert ^{\mathfrak{E}}, \upsilon\bigr), \quad3 \mathfrak{E}\ne1, \\ &\mathfrak{C}_{a}' \bigl(\mathcal{S} \bigl\{ \Vert \mathscr{X}_{a} \Vert ^{\mathfrak{E}} \Vert \mathscr{V}_{a} \Vert ^{\mathfrak{E}} \Vert \mathscr{W}_{a} \Vert ^{\mathfrak{E}}+ \bigl( \Vert \mathscr{X}_{a} \Vert ^{3\mathfrak{E}}+ \Vert \mathscr{V}_{a} \Vert ^{3\mathfrak{E}}+ \Vert \mathscr{W}_{a} \Vert ^{3\mathfrak{E}} \bigr) \bigr\} , \upsilon \bigr),\quad3\mathfrak{E}\ne1 \end{aligned} \right . \end{aligned}$$

such that

$$\begin{aligned} \begin{aligned} &\mathfrak{A}_{a} \bigl( \mathscr{A}_{1}(\mathscr{X}_{a}) - \mathcal{A}_{1}( \mathscr{X}_{a}),\upsilon\bigr) \geq \left \{ \begin{aligned} &\mathfrak{A}_{a}' \biggl(\frac{\mathcal{S}}{\mathfrak{T}_{1}}, \frac{\mathfrak{T}_{1}}{1-\mathfrak{T}_{1}}\upsilon\biggr), \\ &\mathfrak{A}_{a}' \biggl(\frac{\mathcal{S} \Vert \mathscr{X}_{a} \Vert ^{\mathfrak{E}}}{\mathfrak{T}_{1}} \frac{3}{ \vert \mathfrak{T}_{1} \vert ^{\mathfrak{E}}},\frac {\mathfrak{T}_{1}}{ \mathfrak{T}_{1}^{\mathfrak{E}}-\mathfrak{T}_{1}}\upsilon\biggr), \\ &\mathfrak{A}_{a}' \biggl(\frac{\mathcal{S} \Vert \mathscr{X}_{a} \Vert ^{3\mathfrak{E}}}{\mathfrak{T}_{1}} \frac{1}{ \vert \mathfrak{T}_{1} \vert ^{3\mathfrak{E}}},\frac {\mathfrak{T}_{1}}{ \mathfrak{T}_{1}^{3\mathfrak{E}}-\mathfrak{T}_{1}}\upsilon\biggr), \\ &\mathfrak{A}_{a}' \biggl(\frac{\mathcal{S} \Vert \mathscr{X}_{a} \Vert ^{3\mathfrak{E}}}{\mathfrak{T}_{1}} \biggl(\frac{3}{ \vert \mathfrak{T}_{1} \vert ^{3\mathfrak {E}}}+\frac{1}{ \vert \mathfrak{T}_{1} \vert ^{3\mathfrak{E}}} ,\frac{\mathfrak{T}_{1}}{\mathfrak{T}_{1}^{3\mathfrak{E}}-\mathfrak {T}_{1}}\upsilon \biggr) \biggr), \end{aligned} \right . \\ &\mathfrak{B}_{a} \bigl(\mathscr{A}_{1}( \mathscr{X}_{a}) - \mathcal{A}_{1}(\mathscr{X}_{a}), \upsilon\bigr) \leq \left \{ \begin{aligned} &\mathfrak{B}_{a}' \biggl(\frac{\mathcal{S}}{\mathfrak{T}_{1}},\frac{\mathfrak {T}_{1}}{1-\mathfrak{T}_{1}}\upsilon\biggr), \\ &\mathfrak{B}_{a}' \biggl(\frac{\mathcal{S} \Vert \mathscr{X}_{a} \Vert ^{\mathfrak{E}}}{\mathfrak{T}_{1}} \frac{3}{ \vert \mathfrak{T}_{1} \vert ^{\mathfrak{E}}},\frac {\mathfrak{T}_{1}}{\mathfrak{T}_{1}^{\mathfrak{E}}-\mathfrak {T}_{1}}\upsilon\biggr), \\ &\mathfrak{B}_{a}' \biggl(\frac{\mathcal{S} \Vert \mathscr{X}_{a} \Vert ^{3\mathfrak{E}}}{\mathfrak{T}_{1}} \frac{1}{ \vert \mathfrak{T}_{1} \vert ^{3\mathfrak{E}}},\frac {\mathfrak{T}_{1}}{\mathfrak{T}_{1}^{3\mathfrak{E}}-\mathfrak {T}_{1}}\upsilon\biggr), \\ &\mathfrak{B}_{a}' \biggl(\frac{\mathcal{S} \Vert \mathscr{X}_{a} \Vert ^{3\mathfrak{E}}}{\mathfrak{T}_{1}} \biggl(\frac{3}{ \vert \mathfrak{T}_{1} \vert ^{3\mathfrak {E}}}+\frac{1}{ \vert \mathfrak{T}_{1} \vert ^{3\mathfrak{E}}} ,\frac{\mathfrak{T}_{1}}{\mathfrak{T}_{1}^{3\mathfrak{E}}-\mathfrak {T}_{1}}\upsilon \biggr) \biggr), \end{aligned} \right . \\ &\mathfrak{C}_{a} \bigl(\mathscr{A}_{1}( \mathscr{X}_{a}) - \mathcal{A}_{1}(\mathscr{X}_{a}), \upsilon\bigr) \leq \left \{ \begin{aligned} &\mathfrak{C}_{a}' \biggl(\frac{\mathcal{S}}{\mathfrak{T}_{1}},\frac{\mathfrak {T}_{1}}{1-\mathfrak{T}_{1}}\upsilon\biggr), \\ &\mathfrak{C}_{a}' \biggl(\frac{\mathcal{S} \Vert \mathscr{X}_{a} \Vert ^{\mathfrak{E}}}{\mathfrak{T}_{1}} \frac{3}{ \vert \mathfrak{T}_{1} \vert ^{\mathfrak{E}}},\frac {\mathfrak{T}_{1}}{\mathfrak{T}_{1}^{\mathfrak{E}}-\mathfrak {T}_{1}}\upsilon\biggr), \\ &\mathfrak{C}_{a}' \biggl(\frac{\mathcal{S} \Vert \mathscr{X}_{a} \Vert ^{3\mathfrak{E}}}{\mathfrak{T}_{1}} \frac{1}{ \vert \mathfrak{T}_{1} \vert ^{3\mathfrak{E}}},\frac {\mathfrak{T}_{1}}{\mathfrak{T}_{1}^{3\mathfrak{E}}-\mathfrak {T}_{1}}\upsilon\biggr), \\ &\mathfrak{C}_{a}' \biggl(\frac{\mathcal{S} \Vert \mathscr{X}_{a} \Vert ^{3\mathfrak{E}}}{\mathfrak{T}_{1}} \biggl(\frac{3}{ \vert \mathfrak{T}_{1} \vert ^{3\mathfrak {E}}}+\frac{1}{ \vert \mathfrak{T}_{1} \vert ^{3\mathfrak{E}}} ,\frac{\mathfrak{T}_{1}}{\mathfrak{T}_{1}^{3\mathfrak{E}}-\mathfrak {T}_{1}}\upsilon \biggr) \biggr). \end{aligned} \right . \end{aligned} \end{aligned}$$

(4.14)

Proof

Let

$$\begin{aligned} &\mathfrak{A}_{a}' \bigl(K\bigl( \mathfrak{D}_{i}^{n} \mathscr{X}_{a}, \mathfrak{D}_{i}^{n} \mathscr{V}_{a}, \mathfrak{D}_{i}^{n} \mathscr{W}_{a}\bigr), \mathfrak{D}_{i}^{\mathscr{L}_{1}} \upsilon\bigr) \\ &\quad = \left \{ \begin{aligned} &\mathfrak{A}_{a}' \bigl(\mathcal{S} , \mathfrak{D}_{i}^{k} \upsilon \bigr), \\ &\mathfrak{A}_{a}' \bigl(\mathcal{S} \bigl( \Vert \mathscr{X}_{a} \Vert ^{\mathfrak{E}}+ \Vert \mathscr{V}_{a} \Vert ^{\mathfrak{E}}+ \Vert \mathscr{W}_{a} \Vert ^{\mathfrak{E}} \bigr), \mathfrak{D}_{i}^{\mathscr{L}_{1}-\mathfrak{E}} \upsilon\bigr), \\ &\mathfrak{A}_{a}' \bigl(\mathcal{S} \Vert \mathscr{X}_{a} \Vert ^{\mathfrak{E}} \Vert \mathscr{V}_{a} \Vert ^{\mathfrak{E}} \Vert \mathscr{W}_{a} \Vert ^{\mathfrak{E}}, \mathfrak{D}_{i}^{\mathscr{L}_{1}-3\mathfrak{E}} \upsilon\bigr), \\ &\mathfrak{A}_{a}' \bigl(\mathcal{S} \bigl\{ \Vert \mathscr{X}_{a} \Vert ^{\mathfrak{E}} \Vert \mathscr{V}_{a} \Vert ^{\mathfrak{E}} \Vert \mathscr{W}_{a} \Vert ^{\mathfrak{E}}+ \bigl( \Vert \mathscr{X}_{a} \Vert ^{3\mathfrak{E}}+ \Vert \mathscr{V}_{a} \Vert ^{3\mathfrak{E}}+ \Vert \mathscr{W}_{a} \Vert ^{3\mathfrak{E}} \bigr) \bigr\} , \mathfrak{D}_{i}^{\mathscr{L}_{1}-3\mathfrak{E}} \upsilon\bigr) \end{aligned} \right . \\ &\quad =\left \{ \begin{aligned} &\rightarrow1 \quad\text{as } \mathscr{L}_{1} \rightarrow\infty, \\ &\rightarrow1 \quad\text{as }\mathscr{L}_{1} \rightarrow \infty, \\ &\rightarrow1 \quad\text{as }\mathscr{L}_{1} \rightarrow \infty, \\ &\rightarrow1 \quad\text{as }\mathscr{L}_{1} \rightarrow \infty \end{aligned} \right . \\ &\mathfrak{B}_{a}' \bigl(K\bigl( \mathfrak{D}_{i}^{n} \mathscr{X}_{a}, \mathfrak{D}_{i}^{n} \mathscr{V}_{a}, \mathfrak{D}_{i}^{n} \mathscr{W}_{a}\bigr), \mathfrak{D}_{i}^{\mathscr{L}_{1}} \upsilon\bigr) \\ &\quad = \left \{ \begin{aligned} &\mathfrak{B}_{a}' \bigl(\mathcal{S} , \mathfrak{D}_{i}^{\mathscr{L}_{1}} \upsilon \bigr), \\ &\mathfrak{B}_{a}' \bigl(\mathcal{S} \bigl( \Vert \mathscr{X}_{a} \Vert ^{\mathfrak{E}}+ \Vert \mathscr{V}_{a} \Vert ^{\mathfrak{E}}+ \Vert \mathscr{W}_{a} \Vert ^{\mathfrak{E}} \bigr), \mathfrak{D}_{i}^{\mathscr{L}_{1}-\mathfrak{E}} \upsilon\bigr), \\ &\mathfrak{B}_{a}' \bigl(\mathcal{S} \Vert \mathscr{X}_{a} \Vert ^{\mathfrak{E}} \Vert \mathscr{V}_{a} \Vert ^{\mathfrak{E}} \Vert \mathscr{W}_{a} \Vert ^{\mathfrak{E}}, \mathfrak{D}_{i}^{\mathscr{L}_{1}-3\mathfrak{E}} \upsilon\bigr), \\ &\mathfrak{B}_{a}' \bigl(\mathcal{S} \bigl\{ \Vert \mathscr{X}_{a} \Vert ^{\mathfrak{E}} \Vert \mathscr{V}_{a} \Vert ^{\mathfrak{E}} \Vert \mathscr{W}_{a} \Vert ^{\mathfrak{E}}+ \bigl( \Vert \mathscr{X}_{a} \Vert ^{3\mathfrak{E}}+ \Vert \mathscr{V}_{a} \Vert ^{3\mathfrak{E}}+ \Vert \mathscr{W}_{a} \Vert ^{3\mathfrak{E}} \bigr) \bigr\} , \mathfrak{D}_{i}^{\mathscr{L}_{1}-3\mathfrak{E}} \upsilon\bigr) \end{aligned} \right . \\ &\quad =\left \{ \begin{aligned} &\rightarrow0 \quad\text{as } \mathscr{L}_{1} \rightarrow\infty, \\ &\rightarrow0 \quad\text{as }\mathscr{L}_{1} \rightarrow \infty, \\ &\rightarrow0 \quad\text{as }\mathscr{L}_{1} \rightarrow \infty, \\ &\rightarrow0 \quad\text{as }\mathscr{L}_{1} \rightarrow \infty \end{aligned} \right . \\ &\mathfrak{C}_{a}' \bigl(K\bigl( \mathfrak{D}_{i}^{n} \mathscr{X}_{a}, \mathfrak{D}_{i}^{n} \mathscr{V}_{a}, \mathfrak{D}_{i}^{n} \mathscr{W}_{a}\bigr), \mathfrak{D}_{i}^{\mathscr{L}_{1}} \upsilon\bigr) \\ &\quad = \left \{ \begin{aligned} &\mathfrak{C}_{a}' \bigl(\mathcal{S} , \mathfrak{D}_{i}^{\mathscr{L}_{1}} \upsilon \bigr), \\ &\mathfrak{C}_{a}' \bigl(\mathcal{S} \bigl( \Vert \mathscr{X}_{a} \Vert ^{\mathfrak{E}}+ \Vert \mathscr{V}_{a} \Vert ^{\mathfrak{E}}+ \Vert \mathscr{W}_{a} \Vert ^{\mathfrak{E}} \bigr), \mathfrak{D}_{i}^{\mathscr{L}_{1}-\mathfrak{E}} \upsilon\bigr), \\ &\mathfrak{C}_{a}' \bigl(\mathcal{S} \Vert \mathscr{X}_{a} \Vert ^{\mathfrak{E}} \Vert \mathscr{V}_{a} \Vert ^{\mathfrak{E}} \Vert \mathscr{W}_{a} \Vert ^{\mathfrak{E}}, \mathfrak{D}_{i}^{\mathscr{L}_{1}-3\mathfrak{E}} \upsilon\bigr), \\ &\mathfrak{C}_{a}' \bigl(\mathcal{S} \bigl\{ \Vert \mathscr{X}_{a} \Vert ^{\mathfrak{E}} \Vert \mathscr{V}_{a} \Vert ^{\mathfrak{E}} \Vert \mathscr{W}_{a} \Vert ^{\mathfrak{E}}+ \bigl( \Vert \mathscr{X}_{a} \Vert ^{3\mathfrak{E}}+ \Vert y \Vert ^{3\mathfrak{E}}+ \Vert z \Vert ^{3\mathfrak{E}} \bigr) \bigr\} , \mathfrak{D}_{i}^{\mathscr{L}_{1}-3\mathfrak{E}} \upsilon\bigr) \end{aligned} \right . \\ &\quad =\left \{ \begin{aligned} &\rightarrow0 \quad\text{as } \mathscr{L}_{1} \rightarrow\infty, \\ &\rightarrow0 \quad\text{as }\mathscr{L}_{1} \rightarrow \infty, \\ &\rightarrow0 \quad\text{as }\mathscr{L}_{1} \rightarrow \infty, \\ &\rightarrow0 \quad\text{as }\mathscr{L}_{1} \rightarrow \infty. \end{aligned} \right . \end{aligned}$$

Here (4.1) exists, then

$$\begin{aligned} \left . \begin{aligned} &\mathfrak{A}_{a}' \biggl( \frac{1}{\mathfrak{T}_{1}}K \biggl(\frac{\mathscr{X}_{a}}{\mathfrak {T}_{1}},\frac{\mathscr{X}_{a}}{\mathfrak{T}_{1}}, \frac{\mathscr{X}_{a}}{\mathfrak{T}_{1}} \biggr),\upsilon\biggr) = \left \{ \begin{aligned} &\mathfrak{A}_{a}' \biggl(\frac{\mathcal{S}}{\mathfrak{T}_{1}}, \upsilon\biggr), \\ &\mathfrak{A}_{a}' \biggl(\frac{\mathcal{S} \Vert \mathscr{X}_{a} \Vert ^{\mathfrak{E}}}{\mathfrak{T}_{1}} \frac{3}{ \vert \mathfrak{T}_{1} \vert ^{\mathfrak{E}}},\upsilon \biggr), \\ &\mathfrak{A}_{a}' \biggl(\frac{\mathcal{S} \Vert \mathscr{X}_{a} \Vert ^{3\mathfrak{E}}}{\mathfrak{T}_{1}} \frac{1}{ \vert \mathfrak{T}_{1} \vert ^{3\mathfrak{E}}},\upsilon \biggr), \\ &\mathfrak{A}_{a}' \biggl(\frac{\mathcal{S} \Vert \mathscr{X}_{a} \Vert ^{3\mathfrak{E}}}{\mathfrak{T}_{1}} \biggl(\frac{3}{ \vert \mathfrak{T}_{1} \vert ^{3\mathfrak {E}}}+\frac{1}{ \vert \mathfrak{T}_{1} \vert ^{3\mathfrak{E}}} \biggr) ,\upsilon\biggr), \end{aligned} \right . \\ &\mathfrak{B}_{a}' \biggl(\frac{1}{\mathfrak{T}_{1}}K \biggl(\frac{\mathscr{X}_{a}}{\mathfrak{T}_{1}},v,\frac{\mathscr{X}_{a}}{\mathfrak{T}_{1}} \biggr),\upsilon\biggr) = \left \{ \begin{aligned} &\mathfrak{B}_{a}' \biggl( \frac{\mathcal{S}}{\mathfrak{T}_{1}},\upsilon\biggr), \\ &\mathfrak{B}_{a}' \biggl(\frac{\mathcal{S} \Vert \mathscr{X}_{a} \Vert ^{\mathfrak{E}}}{\mathfrak{T}_{1}} \frac{3}{ \vert \mathfrak{T}_{1} \vert ^{\mathfrak{E}}},\upsilon \biggr), \\ &\mathfrak{B}_{a}' \biggl(\frac{\mathcal{S} \Vert \mathscr{X}_{a} \Vert ^{3\mathfrak{E}}}{\mathfrak{T}_{1}} \frac{1}{ \vert \mathfrak{T}_{1} \vert ^{3\mathfrak{E}}},\upsilon \biggr), \\ &\mathfrak{B}_{a}' \biggl(\frac{\mathcal{S} \Vert \mathscr{X}_{a} \Vert ^{3\mathfrak{E}}}{\mathfrak{T}_{1}} \biggl(\frac{3}{ \vert \mathfrak{T}_{1} \vert ^{3\mathfrak {E}}}+\frac{1}{ \vert \mathfrak{T}_{1} \vert ^{3\mathfrak{E}}} \biggr) ,\upsilon\biggr), \end{aligned} \right . \\ &\mathfrak{C}_{a}' \biggl(\frac{1}{\mathfrak{T}_{1}}K \biggl(\frac{\mathscr{X}_{a}}{\mathfrak{T}_{1}},\frac{\mathscr{X}_{a}}{\mathfrak{T}_{1}},\frac{\mathscr{X}_{a}}{\mathfrak{T}_{1}} \biggr), \upsilon\biggr) = \left \{ \begin{aligned} &\mathfrak{C}_{a}' \biggl(\frac{\mathcal{S}}{\mathfrak{T}_{1}},\upsilon\biggr), \\ &\mathfrak{C}_{a}' \biggl(\frac{\mathcal{S} \Vert \mathscr{X}_{a} \Vert ^{\mathfrak{E}}}{\mathfrak{T}_{1}} \frac{3}{ \vert \mathfrak{T}_{1} \vert ^{\mathfrak{E}}},\upsilon \biggr), \\ &\mathfrak{C}_{a}' \biggl(\frac{\mathcal{S} \Vert \mathscr{X}_{a} \Vert ^{3\mathfrak{E}}}{\mathfrak{T}_{1}} \frac{1}{ \vert \mathfrak{T}_{1} \vert ^{3\mathfrak{E}}},\upsilon \biggr), \\ &\mathfrak{C}_{a}' \biggl(\frac{\mathcal{S} \Vert \mathscr{X}_{a} \Vert ^{3\mathfrak{E}}}{\mathfrak{T}_{1}} \biggl(\frac{3}{ \vert \mathfrak{T}_{1} \vert ^{3\mathfrak {E}}}+\frac{1}{ \vert \mathfrak{T}_{1} \vert ^{3\mathfrak{E}}} \biggr) ,\upsilon\biggr). \end{aligned} \right . \end{aligned} \right \} \end{aligned}$$

From (4.5),

$$\begin{aligned} \left . \begin{aligned} &\mathfrak{A}_{a}' \biggl( \frac{\mathscr{A}_{1}(\mathfrak{D}_{i}\mathscr{X}_{a})}{\mathfrak {D}_{i}}, \upsilon\biggr) = \left \{ \begin{aligned} &\mathfrak{A}_{a}' \bigl(\mathcal{S},{ \mathfrak{D}_{i}}\upsilon\bigr), \\ &\mathfrak{A}_{a}' \biggl(\frac{\mathcal{S} \Vert \mathscr{X}_{a} \Vert ^{\mathfrak{E}}}{\mathfrak{T}_{1}} \frac{3}{ \vert \mathfrak{T}_{1} \vert ^{\mathfrak{E}}},{\mathfrak {D}_{i}}^{1-\mathfrak{E}}\upsilon \biggr), \\ &\mathfrak{A}_{a}' \biggl(\frac{\mathcal{S} \Vert \mathscr{X}_{a} \Vert ^{3\mathfrak{E}}}{\mathfrak{T}_{1}} \frac{1}{ \vert \mathfrak{T}_{1} \vert ^{3\mathfrak{E}}},{\mathfrak {D}_{i}}^{1-3\mathfrak{E}}\upsilon \biggr), \\ &\mathfrak{A}_{a}' \biggl(\frac{\mathcal{S} \Vert \mathscr{X}_{a} \Vert ^{3\mathfrak{E}}}{\mathfrak{T}_{1}} \biggl(\frac{3}{ \vert \mathfrak{T}_{1} \vert ^{3\mathfrak {E}}}+\frac{1}{ \vert \mathfrak{T}_{1} \vert ^{3\mathfrak{E}}} \biggr),{\mathfrak{D}_{i}}^{1-3\mathfrak{E}} \upsilon\biggr), \end{aligned} \right . \\ &\mathfrak{B}_{a}' \biggl(\frac{\mathscr{A}_{1}(\mathfrak{D}_{i}\mathscr{X}_{a})}{\mathfrak{D}_{i}}, \upsilon\biggr) = \left \{ \begin{aligned} &\mathfrak{B}_{a}' \bigl(\mathcal{S},{\mathfrak{D}_{i}}\upsilon\bigr), \\ &\mathfrak{B}_{a}' \biggl(\frac{\mathcal{S} \Vert \mathscr{X}_{a} \Vert ^{\mathfrak{E}}}{\mathfrak{T}_{1}} \frac{3}{ \vert \mathfrak{T}_{1} \vert ^{\mathfrak{E}}},{\mathfrak {D}_{i}}^{1-\mathfrak{E}}\upsilon \biggr), \\ &\mathfrak{B}_{a}' \biggl(\frac{\mathcal{S} \Vert \mathscr{X}_{a} \Vert ^{3\mathfrak{E}}}{\mathfrak{T}_{1}} \frac{1}{ \vert \mathfrak{T}_{1} \vert ^{3\mathfrak{E}}},{\mathfrak {D}_{i}}^{1-3\mathfrak{E}}\upsilon \biggr), \\ &\mathfrak{B}_{a}' \biggl(\frac{\mathcal{S} \Vert \mathscr{X}_{a} \Vert ^{3\mathfrak{E}}}{\mathfrak{T}_{1}} \biggl(\frac{3}{ \vert \mathfrak{T}_{1} \vert ^{3\mathfrak {E}}}+\frac{1}{ \vert \mathfrak{T}_{1} \vert ^{3\mathfrak{E}}} \biggr),{\mathfrak{D}_{i}}^{1-3\mathfrak{E}} \upsilon\biggr), \end{aligned} \right . \\ &\mathfrak{C}_{a}' \biggl(\frac{\mathscr{A}_{1}(\mathfrak{D}_{i}\mathscr{X}_{a})}{\mathfrak{D}_{i}}, \upsilon\biggr) = \left \{ \begin{aligned} &\mathfrak{B}_{a}' \bigl(\mathcal{S},{\mathfrak{D}_{i}}\upsilon\bigr), \\ &\mathfrak{C}_{a}' \biggl(\frac{\mathcal{S} \Vert \mathscr{X}_{a} \Vert ^{\mathfrak{E}}}{\mathfrak{T}_{1}} \frac{3}{ \vert \mathfrak{T}_{1} \vert ^{\mathfrak{E}}},{\mathfrak {D}_{i}}^{1-\mathfrak{E}}\upsilon \biggr), \\ &\mathfrak{C}_{a}' \biggl(\frac{\mathcal{S} \Vert \mathscr{X}_{a} \Vert ^{3\mathfrak{E}}}{\mathfrak{T}_{1}} \frac{1}{ \vert \mathfrak{T}_{1} \vert ^{3\mathfrak{E}}},{\mathfrak {D}_{i}}^{1-3\mathfrak{E}}\upsilon \biggr), \\ &\mathfrak{C}_{a}' \biggl(\frac{\mathcal{S} \Vert \mathscr{X}_{a} \Vert ^{3\mathfrak{E}}}{\mathfrak{T}_{1}} \biggl(\frac{3}{ \vert \mathfrak{T}_{1} \vert ^{3\mathfrak {E}}}+\frac{1}{ \vert \mathfrak{T}_{1} \vert ^{3\mathfrak{E}}} \biggr),{\mathfrak{D}_{i}}^{1-3\mathfrak{E}} \upsilon\biggr). \end{aligned} \right . \end{aligned} \right \} \end{aligned}$$

Hence (4.6), we have

$$\begin{array}{rlrlrl}& L& \mathfrak{E},i=0& & L& \mathfrak{E},i=1\\ 1.& {\mathfrak{T}}_1& 0& & {\mathfrak{T}}_1^{-1}& 0\\ 2.& {\mathfrak{T}}_1^{1-\mathfrak{E}}& \mathfrak{E}<1& & {\mathfrak{T}}_1^{\mathfrak{E}-1}& \mathfrak{E}>1\\ 3.& {\mathfrak{T}}_1^{1-3\mathfrak{E}}& 3\mathfrak{E}<1& & {\mathfrak{T}}_1^{3\mathfrak{E}-1}& 3\mathfrak{E}>1\\ 4.& {\mathfrak{T}}_1^{3\mathfrak{E}-1}& 3\mathfrak{E}<1& & {\mathfrak{T}}_1^{3\mathfrak{E}-1}& 3\mathfrak{E}>1.\end{array}$$

Method: 1 For $i=0$,

$$\begin{aligned} \left . \begin{aligned} &\mathfrak{A}_{a} \bigl( \mathscr{A}_{1}(\mathscr{X}_{a})-\mathcal{A}_{1}( \mathscr{X}_{a}), \upsilon\bigr) \geq\mathfrak{A}_{a}' \biggl(\mathscr{A}_{1}(\mathscr{X}_{a}), \frac{\mathfrak{T}_{1}^{1-0}}{1-\mathfrak{T}_{1}}\upsilon\biggr)= \mathfrak{A}_{a}' \biggl(\frac{\mathcal{S}}{\mathfrak{T}_{1}},\frac{\mathfrak {T}_{1}}{1-\mathfrak{T}_{1}}\upsilon\biggr), \\ &\mathfrak{B}_{a} \bigl(\mathscr{A}_{1}( \mathscr{X}_{a})-\mathcal{A}_{1}(\mathscr{X}_{a}), \upsilon\bigr) \leq\mathfrak{B}_{a}' \biggl( \mathscr{A}_{1}(\mathscr{X}_{a}),\frac{\mathfrak {T}_{1}^{1-0}}{1-\mathfrak{T}_{1}} \upsilon\biggr)=\mathfrak{B}_{a}' \biggl( \frac{\mathcal{S}}{\mathfrak{T}_{1}},\frac{\mathfrak{T}_{1}}{1-\mathfrak {T}_{1}}\upsilon\biggr), \\ &\mathfrak{C}_{a} \bigl(\mathscr{A}_{1}( \mathscr{X}_{a})-\mathcal{A}_{1}(\mathscr{X}_{a}), \upsilon\bigr) \leq\mathfrak{C}_{a}' \biggl( \mathscr{A}_{1}(\mathscr{X}_{a}),\frac{\mathfrak {T}_{1}^{1-0}}{1-\mathfrak{T}_{1}} \upsilon\biggr)=\mathfrak{C}_{a}' \biggl( \frac{\mathcal{S}}{\mathfrak{T}_{1}},\frac{\mathfrak{T}_{1}}{1-\mathfrak {T}_{1}}\upsilon\biggr). \end{aligned} \right \} \end{aligned}$$

Method: 1 For $i=1$,

$$\begin{aligned} \left . \begin{aligned}& \mathfrak{A}_{a} \bigl( \mathscr{A}_{1}(\mathscr{X}_{a})-\mathcal{A}_{1}( \mathscr{X}_{a}), \upsilon\bigr) \geq\mathfrak{A}_{a}' \biggl(\mathscr{A}_{1}(\mathscr{X}_{a}), \frac{((\mathfrak{T}_{1})^{-1})^{1-1}}{1-(\mathfrak {T}_{1})^{-1}}\upsilon\biggr) = \mathfrak{A}_{a}' \biggl(\frac{\mathcal{S}}{\mathfrak{T}_{1}},\frac{\mathfrak {T}_{1}}{\mathfrak{T}_{1}-1} \upsilon\biggr), \\ &\mathfrak{B}_{a} \bigl(\mathscr{A}_{1}( \mathscr{X}_{a})-\mathcal{A}_{1}(\mathscr{X}_{a}), \upsilon\bigr) \leq\mathfrak{B}_{a}' \biggl( \mathscr{A}_{1}(\mathscr{X}_{a}),\frac{((\mathfrak {T}_{1})^{-1})^{1-1}}{1-(\mathfrak{T}_{1})^{-1}} \upsilon\biggr)= \mathfrak{B}_{a}' \biggl( \frac{\mathcal{S}}{\mathfrak{T}_{1}},\frac{\mathfrak{T}_{1}}{\mathfrak {T}_{1}-1}\upsilon\biggr), \\ &\mathfrak{C}_{a} \bigl(\mathscr{A}_{1}( \mathscr{X}_{a})-\mathcal{A}_{1}(\mathscr{X}_{a}), \upsilon\bigr) \leq\mathfrak{C}_{a}' \biggl( \mathscr{A}_{1}(\mathscr{X}_{a}),\frac{((\mathfrak {T}_{1})^{-1})^{1-1}}{1-(\mathfrak{T}_{1})^{-1}} \upsilon\biggr)= \mathfrak{C}_{a}' \biggl( \frac{\mathcal{S}}{\mathfrak{T}_{1}},\frac{\mathfrak{T}_{1}}{\mathfrak {T}_{1}-1}\upsilon\biggr). \end{aligned} \right \} \end{aligned}$$

Method: 2 For $i=0$,

$$\begin{aligned} \left . \begin{aligned} &\mathfrak{A}_{a} \bigl( \mathscr{A}_{1}(\mathscr{X}_{a})-\mathcal{A}_{1}( \mathscr{X}_{a}), \upsilon\bigr) \geq\mathfrak{A}_{a}' \biggl(\mathscr{A}_{1}(\mathscr{X}_{a}), \frac{(\mathfrak{T}_{1}^{1-\mathfrak{E}})^{1-0}}{1-(\mathfrak {T}_{1}^{1-\mathfrak{E}})}\upsilon\biggr) \\ &\hphantom{\mathfrak{A}_{a} \bigl( \mathscr{A}_{1}(\mathscr{X}_{a})-\mathcal{A}_{1}( \mathscr{X}_{a}), \upsilon\bigr)}=\mathfrak{A}_{a}' \biggl(\frac{\mathcal{S} \Vert \mathscr{X}_{a} \Vert ^{\mathfrak {E}}}{\mathfrak{T}_{1}}\frac{3}{ \vert \mathfrak{T}_{1} \vert ^{\mathfrak{E}}},\frac{\mathfrak{T}_{1}}{\mathfrak{T}_{1}^{\mathfrak {E}}-\mathfrak{T}_{1}^{k}}\upsilon \biggr), \\ &\mathfrak{B}_{a} \bigl(\mathscr{A}_{1}( \mathscr{X}_{a})-\mathcal{A}_{1}(\mathscr{X}_{a}), \upsilon\bigr) \leq\mathfrak{B}_{a}' \biggl( \mathscr{A}_{1}(\mathscr{X}_{a}),\frac{(\mathfrak{T}_{1}^{1-\mathfrak {E}})^{1-0}}{1-(\mathfrak{T}_{1}^{1-\mathfrak{E}})} \upsilon\biggr) \\ &\hphantom{\mathfrak{B}_{a} \bigl(\mathscr{A}_{1}( \mathscr{X}_{a})-\mathcal{A}_{1}(\mathscr{X}_{a}), \upsilon\bigr)} =\mathfrak{B}_{a}' \biggl( \frac{\mathcal{S} \Vert \mathscr{X}_{a} \Vert ^{\mathfrak {E}}}{\mathfrak{T}_{1}}\frac{3}{ \vert \mathfrak{T}_{1} \vert ^{\mathfrak{E}}},\frac{\mathfrak{T}_{1}}{\mathfrak{T}_{1}^{\mathfrak {E}}-\mathfrak{T}_{1}}\upsilon\biggr), \\ &\mathfrak{C}_{a} \bigl(\mathscr{A}_{1}( \mathscr{X}_{a})-\mathcal{A}_{1}(\mathscr{X}_{a}), \upsilon\bigr) \leq\mathfrak{C}_{a}' \biggl( \mathscr{A}_{1}(\mathscr{X}_{a}),\frac{(\mathfrak{T}_{1}^{1-\mathfrak {E}})^{1-0}}{1-(\mathfrak{T}_{1}^{1-\mathfrak{E}})} \upsilon\biggr) \\ &\hphantom{\mathfrak{C}_{a} \bigl(\mathscr{A}_{1}( \mathscr{X}_{a})-\mathcal{A}_{1}(\mathscr{X}_{a}), \upsilon\bigr)}=\mathfrak{C}_{a}' \biggl( \frac{\mathcal{S} \Vert \mathscr{X}_{a} \Vert ^{\mathfrak {E}}}{\mathfrak{T}_{1}}\frac{3}{ \vert \mathfrak{T}_{1} \vert ^{\mathfrak{E}}},\frac{\mathfrak{T}_{1}}{\mathfrak{T}_{1}^{\mathfrak {E}}-\mathfrak{T}_{1}}\upsilon\biggr). \end{aligned} \right \} \end{aligned}$$

Method: 2 For $i=1$,

$$\begin{aligned} \left . \begin{aligned} &\mathfrak{A}_{a} \bigl( \mathscr{A}_{1}(\mathscr{X}_{a})-\mathcal{A}_{1}( \mathscr{X}_{a}), \upsilon\bigr) \geq\mathfrak{A}_{a}' \biggl(\mathscr{A}_{1}(\mathscr{X}_{a}), \frac{(\mathfrak{T}_{1}^{A-1})^{1-1}}{1-(\mathfrak {T}_{1}^{A-1})}\upsilon\biggr) \\ &\hphantom{\mathfrak{A}_{a} \bigl( \mathscr{A}_{1}(\mathscr{X}_{a})-\mathcal{A}_{1}( \mathscr{X}_{a}), \upsilon\bigr)}=\mathfrak{A}_{a}' \biggl(\frac{\mathcal{S} \Vert \mathscr{X}_{a} \Vert ^{\mathfrak {E}}}{\mathfrak{T}_{1}}\frac{3}{ \vert \mathfrak{T}_{1} \vert ^{\mathfrak{E}}},\frac{\mathfrak{T}_{1}}{\mathfrak{T}_{1}^{k}-\mathfrak {T}_{1}^{a}}\upsilon \biggr), \\ &\mathfrak{B}_{a} \bigl(\mathscr{A}_{1}( \mathscr{X}_{a})-\mathcal{A}_{1}(\mathscr{X}_{a}), \upsilon\bigr) \leq\mathfrak{B}_{a}' \biggl( \mathscr{A}_{1}(\mathscr{X}_{a}),\frac{(\mathfrak {T}_{1}^{A-1})^{1-1}}{1-(\mathfrak{T}_{1}^{A-1})} \upsilon\biggr) \\ &\hphantom{\mathfrak{B}_{a} \bigl(\mathscr{A}_{1}( \mathscr{X}_{a})-\mathcal{A}_{1}(\mathscr{X}_{a}), \upsilon\bigr)}=\mathfrak{B}_{a}' \biggl( \frac{\mathcal{S} \Vert \mathscr{X}_{a} \Vert ^{\mathfrak {E}}}{\mathfrak{T}_{1}}\frac{3}{ \vert \mathfrak{T}_{1} \vert ^{\mathfrak{E}}},\frac{\mathfrak{T}_{1}}{\mathfrak{T}_{1}-\mathfrak {T}_{1}^{a}}\upsilon\biggr), \\ &\mathfrak{C}_{a} \bigl(\mathscr{A}_{1}( \mathscr{X}_{a})-\mathcal{A}_{1}(\mathscr{X}_{a}), \upsilon\bigr) \leq\mathfrak{C}_{a}' \biggl( \mathscr{A}_{1}(\mathscr{X}_{a}),\frac{(\mathfrak {T}_{1}^{A-1})^{1-1}}{1-(\mathfrak{T}_{1}^{A-1})} \upsilon\biggr) \\ &\hphantom{\mathfrak{C}_{a} \bigl(\mathscr{A}_{1}( \mathscr{X}_{a})-\mathcal{A}_{1}(\mathscr{X}_{a}), \upsilon\bigr)}=\mathfrak{C}_{a}' \biggl( \frac{\mathcal{S} \Vert \mathscr{X}_{a} \Vert ^{\mathfrak {E}}}{\mathfrak{T}_{1}}\frac{3}{ \vert \mathfrak{T}_{1} \vert ^{\mathfrak{E}}},\frac{\mathfrak{T}_{1}}{\mathfrak{T}_{1}-\mathfrak {T}_{1}^{a}}\upsilon\biggr). \end{aligned} \right \} \end{aligned}$$

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5 Conclusion

This paper has presented a novel approach for analyzing the stability of the Euler-Lagrange additive FE within neutrosophic normed spaces, addressing a critical need in the realm of uncertainty modeling and functional analysis. The investigation delves into the existence of solutions for this equation and explores its Ulam-Hyers stability within neutrosophic normed spaces. The results are obtained by applying two distinct approaches: direct and fixed point techniques. The findings presented in this article establish the relationship between four distinct areas of research: FEs, neutrosophic normed spaces, Ulam-Hyers stability, and fixed point theory. The stability analysis of this equation in neutrosophic normed spaces is unique, given the absence of prior research on the stability of equations employing neutrosophic concepts. This paper contributes to the broader advancement of neutrosophic mathematics by offering new perspectives and methodologies for analyzing functional equations in nonclassical settings. Our work opens up avenues for further research and exploration, inviting interdisciplinary collaboration and innovation in the fields of mathematics, uncertainty modeling, and system analysis.