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