A System of Generalized Quadratic Functional Equations and Fuzzy Difference Results

Abstract

In this study, we present generalized multi-quadratic mappings (GM-QMs) which differ from earlier ones that were previously available in the literature. We then express these mappings (specified by a system of generalized quadratic functional equations (GQFEs)) in a single equation. The fixed-point (FP) methodology and the direct approach (Hyers) method are also used to generate a number of stability findings for a system of generalized FEs in the setting of fuzzy norm spaces (FNSs). In terms of the results obtained by the aforementioned methods, we find that in comparison to the direct method, the FP tool provides a more accurate estimate of GM-QMs while requiring fewer conditions for the proofs.

1. Introduction

A fuzzy concept, after the definition of fuzzy sets, was previously proposed by Zadeh [1]; fuzzy norms on linear spaces were first defined in [2]. In fact, Katsaras (1984), in [2], and Wu and Fang, in [3], developed and expanded the foundational structure of fuzzy normed spaces (FNSs). Their efforts included generalizations such as adapting the Kolmogorov norm and introducing fuzzy topological vector spaces. Moreover, Biswas, in [4], built a notion of fuzzy inner-product spaces within a linear-space framework. Over time, numerous scholars and researchers have conducted and explored fuzzy norms from diverse mathematical perspectives [5,6,7,8]. Cheng and Mordeson [9] contributed further by formalizing a definition of a fuzzy norm over a vector space. The formulation was structured so that the resulting fuzzy metric conformed to that of Kramosil and Michalek in [10]; Bag and Samanta, in [5], refined this definition and examined various properties associated with FNSs [11].

One of the challenging problems which has a special importance in nonlinear analysis and approximation theory is the stability of FEs; many authors are working on this problem nowadays. Stability theory was pioneered by Ulam [12], concerning the query stability of homomorphisms on groups. Hyers [13] addressed the mentioned problem for more groups, assuming that Banach spaces were the groups and that homomorphisms were the linear mappings. Let us recall that the stability of an FE is a comparison of an approximate solution with an exact solution. After that, this theory for FEs and mappings was extended on miscellaneous spaces, as can be found in many articles and books [14,15,16]. Additionally, the following studies examined the Hyers–Ulam–Rassias stability (H-U-R-Stab) of the different FEs in FNSs: Jensen FE fuzzy stability [17], Jensen FE fuzzy stability [18], fuzzy stability of the cubic mappings [19], fuzzy approximately additive functions in FNs [20], H-U-R-Stab of Jensen’s FE on fuzzy norm linear spaces [21], stability of an additive-quadratic FE with a parameter in matrix intuitionistic FNSs [22], and see [23]. Recently, the H-U-R-Stab of a Cauchy–Jensen additive FE was proved in fuzzy Banach spaces (FBSs) [24].

Let us note that Skof [25] first introduced and studied the quadratic functional equation (QFE):

$$\begin{array}{c}\hfill Q(\eta +\vartheta )+Q(\eta -\vartheta )=2Q\left(\eta \right)+2Q\left(\vartheta \right).\end{array}$$

(1)

Here, we recall that the primary tool for characterizing inner-product spaces is Equation (1). This is due to the fact that the parallelogram equality

$$P{(u+v)}^2+P{(u-v)}^2=2P{\left(u\right)}^2+2P{\left(v\right)}^2$$

is valid for a square norm on an inner-product space, where $P = \parallel ·\parallel$; for further details, see [26].

For a set $\Omega$, ${\displaystyle \stackrel{n-\mathrm{times}}{\overbrace{\Omega \times \dots \times \Omega }}}$ is denoted by ${\Omega }^n$ throughout. Let $n\in \mathbb{Z}$ with $n\ge 2$ and $(\mathfrak{G},+)$ be a commutative group and V be a linear space. A multiple-variable mapping $\mathbf{F}:{\mathfrak{G}}^n⟶V$ is said to be multi-quadratic (M-QM) if for each $i\in \{1,\dots ,n\}$,

$$\begin{array}{cc}\hfill & \mathbf{F}({\xi }_1,\dots ,{\xi }_{i-1},{\xi }_i+{\xi }_i^{\prime },{\xi }_{i+1},\dots ,{\xi }_n)+\mathbf{F}({\xi }_1,\dots ,{\xi }_{i-1},{\xi }_i-{\xi }_i^{\prime },{\xi }_{i+1},\dots ,{\xi }_n)\hfill \\ \hfill & =2\mathbf{F}({\xi }_1,\dots ,{\xi }_{i-1},{\xi }_i,{\xi }_{i+1},\dots ,{\xi }_n)+2\mathbf{F}({\xi }_1,\dots ,{\xi }_{i-1},{\xi }_i^{\prime },{\xi }_{i+1},\dots ,{\xi }_n).\hfill \end{array}$$

In the following, ${\mathbb{N}}_0:=\mathbb{N}\cup \left\{0\right\},{\mathbb{R}}_{+}:=[0,\infty )$. Regarding the stability of M-QMs, it was explored by C.-G. Park for the first time in [27], as follows.

Theorem 1.

Suppose $V_j$ represents norm spaces for $j=1,\dots ,d$ and $\mathbb{B}$ is supplied as a Banach space with $\parallel ·\parallel$. Then, the mapping $\Gamma :{\prod }_{j=1}^d{V}_j⟶\mathbb{B}$ has a function $\Lambda :{\prod }_{j=1}^d{V}_j⟶{\mathbb{R}}_{+}$ such that

$$\widehat{\Lambda }({\vartheta }_1,\dots ,{\vartheta }_d):=\sum _{j=0}^{\infty }\sum _{m=1}^d{\displaystyle \frac{1}{4^{m+jd}}}\Lambda (2^{j+1}{\sigma }_1,\dots ,2^{j+1}{\sigma }_{m-1},2^j{\sigma }_m,2^j{\sigma }_{m+1},\dots ,2^j{\sigma }_d)<\infty$$

and

$$\parallel ▵\Gamma ({\vartheta }_1,\dots ,{\vartheta }_d)\parallel \le \Lambda ({\vartheta }_1,\dots ,{\vartheta }_d),$$

for all $({\vartheta }_1,\dots ,{\vartheta }_d)\in {\prod }_{j=1}^d{V}_j^2$, where ${\vartheta }_j=({\eta }_j,{\eta }_j^{\prime }),{\sigma }_j=({\eta }_j,0)\in V^2$, whereas

$$\begin{array}{cc}\hfill ▵\Gamma ({\vartheta }_1,\dots ,{\vartheta }_d)& :=\sum _{j=1}^d\Gamma ({\eta }_1,\dots ,{\eta }_{j-1},{\eta }_j+{\eta }_j^{\prime },{\eta }_{j+1},\dots ,{\eta }_d)\hfill \\ \hfill & +\sum _{j=1}^d\Gamma ({\eta }_1,\dots ,{\eta }_{j-1},{\eta }_j+{\eta }_j^{\prime },{\eta }_{j+1},\dots ,{\eta }_d)\hfill \\ \hfill & -2\Gamma ({\eta }_1,\dots ,{\eta }_j,\dots ,{\eta }_d)\hfill \\ \hfill & -2\sum _{j=1}^d\Gamma ({\eta }_1,\dots ,{\eta }_{j-1},{\eta }_j^{\prime },{\eta }_{j+1},\dots ,{\eta }_d).\hfill \end{array}$$

Assume that $\Gamma ({\eta }_1,\dots ,{\eta }_d)=0$ if ${\eta }_j=0$ for any $j=1,\dots ,d$. Then, there exists a unique M-QM $\Phi :{\prod }_{j=1}^d{V}_j⟶\mathbb{B}$ such that

$$\parallel \Gamma ({\eta }_1,\dots ,{\eta }_d)-\Phi ({\eta }_1,\dots ,{\eta }_d)\parallel \le \widehat{\Lambda }({\theta }_1,\dots ,{\theta }_d),$$

for all $({\eta }_1,\dots ,{\eta }_d)\in {\prod }_{j=1}^d{V}_j$, where ${\theta }_j=({\eta }_j,{\eta }_j)\in V^2$ and $j\in \{1,\dots ,d\}$.

After that, Ciepliński [28] presented the stability of M-QMs as pointwise, like Park’s result, as follows.

Theorem 2.

Let $\mathbb{L}$ be a linear space and $\mathbb{B}$ be a Banach non-Archimedean space on $\mathbb{K}$ with characteristic $k\ne 2$. Also, assume that ${\Lambda }_j:{\mathbb{L}}^{n+1}⟶{\mathbb{R}}_{+}$ is a function such that for each $j=\in \{,\dots ,n\}$

$$\begin{array}{cc}\hfill \underset{l\to \infty }{lim}{\displaystyle \frac{1}{{\left|4\right|}^l}}{\Lambda }_j(2^l{\eta }_1,\dots ,{\eta }_{n+1})& =\dots =\underset{l\to \infty }{lim}{\displaystyle \frac{1}{{\left|4\right|}^l}}{\Lambda }_j({\eta }_1,\dots ,{\eta }_{j-2},2^l{\eta }_{j-1},{\eta }_j,{\eta }_{j+1}\dots ,{\eta }_{n+1})\hfill \\ \hfill & =\underset{l\to \infty }{lim}{\displaystyle \frac{1}{{\left|4\right|}^l}}{\Lambda }_j({\eta }_1,\dots ,{\eta }_{j-1},2^l{\eta }_j,2^l{\eta }_{j+1}\dots ,{\eta }_{n+1})\hfill \\ \hfill & =\underset{l\to \infty }{lim}{\displaystyle \frac{1}{{\left|4\right|}^l}}{\Lambda }_j({\eta }_1,\dots ,{\eta }_{j+1},2^l{\eta }_{j+2},{\eta }_{j+3}\dots ,{\eta }_{n+1})\hfill \\ \hfill & \dots =\underset{l\to \infty }{lim}{\displaystyle \frac{1}{{\left|4\right|}^l}}{\Lambda }_j({\eta }_1,\dots ,{\eta }_n,2^l{\eta }_{n+1})=0,\hfill \end{array}$$

for all $({\eta }_1,\dots ,{\eta }_{n+1})\in {\mathbb{L}}^{n+1}$. If $\Gamma :{\mathbb{L}}^n⟶\mathbb{B}$ is a mapping satisfying

$$\Gamma ({\eta }_1,\dots ,\dots ,{\eta }_{j-1},0,\dots ,{\eta }_{j+1},\dots ,{\eta }_n)=0,$$

for all $({\eta }_1,\dots ,\dots ,{\eta }_{j-1},\dots ,{\eta }_{j+1},\dots ,{\eta }_n)\in {\mathbb{L}}^{n-1}$ and

$$\begin{array}{cc}\hfill & \parallel \Gamma ({\eta }_1,\dots ,{\eta }_{j-1},{\eta }_j+{\eta }_j^{\prime },{\eta }_{j+1},\dots ,{\eta }_n)+\Gamma ({\eta }_1,\dots ,{\eta }_{j-1},{\eta }_j-{\eta }_j^{\prime },{\eta }_{j+1},\dots ,{\eta }_n)\hfill \\ \hfill & -2\Gamma ({\eta }_1,\dots ,{\eta }_{j-1},{\eta }_j,{\eta }_{j+1},\dots ,{\eta }_n)-2\Gamma ({\eta }_1,\dots ,{\eta }_{j-1},{\eta }_j^{\prime },{\eta }_{j+1},\dots ,{\eta }_n)\parallel \hfill \\ \hfill & \le \Lambda ({\eta }_1,\dots ,{\eta }_{j-1},{\eta }_j,{\eta }_j^{\prime },{\eta }_{j+1},\dots ,{\eta }_n),\hfill \end{array}$$

also, for any $({\eta }_1,\dots ,{\eta }_{j-1},{\eta }_j,{\eta }_j^{\prime },{\eta }_{j+1},\dots ,{\eta }_n)\in {\mathbb{L}}^{n+1}$, $(j\in \{1,\dots ,n\left\}\right)$, then for $j\in \{1,\dots ,n\}$, there is a unique M-QM ${\Phi }_j:{\mathbb{L}}^n⟶\mathbb{B}$ such that

$$\begin{array}{cc}\hfill \parallel \Gamma & ({\eta }_1,\dots ,{\eta }_n)-{\Phi }_j({\eta }_1,\dots ,{\eta }_n)\parallel \hfill \\ \hfill & \le {\displaystyle \frac{1}{\left|4\right|}}\sup \left\{{\displaystyle \frac{1}{{\left|4\right|}^l}}{\Phi }_j({\eta }_1,\dots ,{\eta }_{j-1},2^l{\eta }_j,2^l{\eta }_j,{\eta }_{j+1},\dots ,{\eta }_n):l\in {\mathbb{N}}_0\right\},\hfill \end{array}$$

for all $({\eta }_1,\dots ,{\eta }_n)\in {\mathbb{L}}^n$.

Next, Zhao et al. [29] studied the structure of M-QMs and showed that $\Gamma :V^n⟶\mathbb{B}$ is an M-QM if and only if

$$\begin{array}{c}\hfill \sum _{s\in {\{-1,1\}}^n}\Gamma ({\eta }_1+s{\eta }_2)=2^n\sum _{l_1,\dots ,l_n\in \{1,2\}}\Gamma ({\eta }_{l_{1}1},\dots ,{\eta }_{l_{n}n}),\end{array}$$

where ${\eta }_j=({\eta }_{l1},\dots ,{\eta }_{ln})\in V^n$ with $l\in \{1,2\}$. We remember that they established H-U-R-Stab for M-QMs as pointwise. More information about the structures, characterizations, and the stability results for various M-QMs (which are quadratic in the upcoming list in each of variable) on miscellaneous spaces such as normed and non-Archimedean spaces are available as follows:

• M-QMs defined by the QFE $Q(\eta +\theta )+Q(\eta +\sigma (\theta \left)\right)=2Q\left(\eta \right)+2Q\left(\theta \right)$ with an involution $\sigma$ [30].
• M-QMs by using the QFE $Q(a\eta +b\theta )+Q(a\eta -b\theta )=2aQ\left(\eta \right)+2bQ\left(\theta \right)$, where $a,b\in \mathbb{Z}$ with $a\ne 0, \pm$ are fixed [31,32].
• M-QMs defined through the Euler–Lagrange QFE $Q(a\eta +b\theta )+Q(b\eta -a\theta )=(a^2+b^2)[Q\left(\eta \right)+Q\left(\theta \right)]$, where $a,b\in \mathbb{Z}$ with $a\ne 0$ are fixed [33].
• M-QMs defined via the QFE $Q(\eta +a\theta )+Q(\eta +a\theta )=2Q\left(\eta \right)+2a^2\left(\theta \right)$, where $a\in \mathbb{Z}$ with $a\ne 0,\pm$ is fixed [34].
• M-QMs by using the generalized quadratic functional equation, or briefly, GQFE:

  $$\begin{array}{cc}\hfill \mathfrak{Q}(ru+sv)+\mathfrak{Q}(ru-sv)& =\mathfrak{Q}(u+v)+\mathfrak{Q}(u-v)\hfill \\ \hfill & +2\left(r^2-1\right)\mathfrak{Q}\left(u\right)+2\left(s^2-1\right)\mathfrak{Q}\left(v\right),\hfill \end{array}$$

  (2)

  where $u,v\in V$, in which $r,s\in \mathbb{Z}$ with $r,s\ne 0,\pm 1$[35]. The function $\mathfrak{Q}\left(v\right)=v^2$ gives a solution to the equations above.

According to (2), Bodaghi et al. [35] unified the system of QFEs which defines an M-QM and indeed found an equation as the necessary and sufficient condition for a mapping being an MQM. The benefit of the presentation of such systems as a single equation allows us to prove the H-U-R-Stab of the M-QMs by means of only one functional inequality, which was not achieved in [27,28,29].

In this paper, the description of generalized multi-quadratic mappings (GM-QMs) as a single equation was inspired by the GQFE (2). Furthermore, we prove the H-U-R-Stab of the GM-QMs by applying the direct (Hyers) and FP methods in the setting of FNSs. Note that the Hyers method for proving the stability involves finding the Cauchy sequences and their convergency, obtained via recurrence, not using a fixed-point theorem. Finally, we find that the FP method provides a more exact approximation of GM-QMs in comparison to the direct method.

2. Characterization of GM-QMs

In this section, we introduce the GM-QMs and specify their structures. For any $l\in {\mathbb{N}}_0=\mathbb{N}\bigcup \left\{0\right\},n\in \mathbb{N}$, $\alpha =({\alpha }_1,\dots ,{\alpha }_n)\in {\{-1,1\}}^n$ and $\vartheta =({\vartheta }_1,\dots ,{\vartheta }_n)\in {\mathfrak{G}}^n$, we write $l\vartheta :=(l{\vartheta }_1,\dots ,l{\vartheta }_n)$ and $\alpha \vartheta :=({\alpha }_1{\vartheta }_1,\dots ,{\alpha }_n{\vartheta }_n)$, where $\mathfrak{G}$ is defined as a commutative group. Throughout the paper, $V,W$ are two linear spaces and set all mappings from $V^n$ into W, denoted by $\Xi$.

Definition 1.

A mapping $\Psi \in \Xi$ is said to be a generalized n-quadratic mapping or ageneralized multi-quadratic mapping (GM-QM) if f satisfies (2) in each of its n arguments; that is,

$$\begin{array}{cc}\hfill & \Psi ({\xi }_1,\dots ,{\xi }_{i-1},r_i{\xi }_i+s_i{\xi }_i^{\prime },{\xi }_{i+1},\dots ,{\xi }_n)+\Psi ({\xi }_1,\dots ,{\xi }_{i-1},r_i{\xi }_i-s_i{\xi }_i^{\prime },{\xi }_{i+1},\dots ,{\xi }_n)\hfill \\ \hfill & =\Psi ({\xi }_1,\dots ,{\xi }_{i-1},{\xi }_i-{\xi }_i^{\prime },{\xi }_{i+1},\dots ,{\xi }_n)+\Psi ({\xi }_1,\dots ,{\xi }_{i-1},{\xi }_i+{\xi }_i^{\prime },x_{i+1},\dots ,{\xi }_n)\hfill \\ \hfill & +R_i\Psi ({\xi }_1,\dots ,{\xi }_{i-1},{\xi }_i,{\xi }_{i+1},\dots ,{\xi }_n)+S_i\Psi ({\xi }_1,\dots ,{\xi }_{i-1},{\xi }_i^{\prime },{\xi }_{i+1},\dots ,{\xi }_n),\hfill \end{array}$$

(3)

where $r_i,s_i\in \mathbb{Z}$ are fixed with $r_i,s_i\ne 0,\pm 1$ and $R_i=2\left(r_i^2-1\right)$, $S_i=2\left(s_i^2-1\right)$, for all $i\in \{1,\dots ,n\}$.

Obviously, the function $\phi :{\mathbb{R}}^n⟶\mathbb{R}$ given via $\phi ({\alpha }_1,\dots ,{\alpha }_n)=\lambda {\prod }_{i=1}^n{\alpha }_i^2$ is a GM-QM, where $\lambda \in \mathbb{R}$ is fixed.

Denote $v_i^{\left[n\right]}=(v_{i1},\dots ,v_{in})\in V^n$, where $i\in \{1,2\}$, and consider $\Psi \in \Xi$ satisfies the equation

$$\begin{array}{c}\hfill \sum _{{\alpha }_1,\dots ,{\alpha }_n\in \{-1,1\}}\Psi \left({\Phi }_1^{{\alpha }_1},\dots ,{\Phi }_n^{{\alpha }_n}\right)=\sum _{V_j\in \{v_{1j},v_{2j},v_{1j}\pm v_{2j}\}}\prod _{j=1}^n{\mathfrak{R}}_j{\mathfrak{S}}_j\Psi \left(V_1,\dots ,V_n\right),\end{array}$$

(4)

where ${\Phi }_j^{{\alpha }_j}=r_j{v}_{1j}+s_j{v}_{2j}$ and

$${\mathfrak{R}}_j=\left\{\begin{array}{c}{R}_j\mathrm{if}{V}_j=v_{1j},\hfill \\ 1\mathrm{otherwise},\hfill \end{array}\right.\mathrm{and}{\mathfrak{S}}_j=\left\{\begin{array}{c}{S}_j\mathrm{if}{V}_j=v_{2j},\hfill \\ 1\mathrm{otherwise},\hfill \end{array}\right.$$

for $j\in \{1,\dots ,n\}$.

 Definition 2. 

For $\Psi \in \Xi$, we propose the next hypotheses.

• (H1) Ψ has zero condition; that is, $\Psi \left(v^{\left[n\right]}\right)=0$ for any $v^{\left[n\right]}\in V^n$ with at least one variable equal to zero.
• (H2) Ψ satisfies the quadratic condition if for any $j\in \{1,\dots ,n\}$ we have

  $$\Psi ({\xi }_1,\dots ,{\xi }_{j-1},r_j{\xi }_j,{\xi }_{j+1},\dots ,{\xi }_n)=r_j^2\Psi ({\xi }_1,\dots ,{\xi }_{j-1},{\xi }_j,{\xi }_{j+1},\dots ,{\xi }_n),$$
  for all ${\xi }_j\in V$, where $r_j$ is the fixed integers, with $r_j\ne 0,\pm 1$.

Clearly, if a $\Psi \in \Xi$ has hypothesis (H2), then it has (H1) but not conversely. On the other hand, (H2) is a necessary condition for a mapping to be a GM-QM but not sufficient. We present the next example for correctness.

Example 1.

Fix $r_0\in \mathbb{R}$. Define the function $\Psi :{\mathbb{R}}^n⟶\mathbb{R}$ by $\Psi ({\theta }_1,\dots ,{\vartheta }_n)={\prod }_{j=1}^n{\left|{\theta }_j\right|}^2$ for all $({\theta }_1,\dots ,{\theta }_n)\in {\mathbb{R}}^n$. It is easily verified that the mapping Ψ satisfies (H2) but not a GM-QM even for $n=1$, which means that Ψ does not fulfill (2). Furthermore, consider $\Phi :{\mathbb{R}}^n⟶\mathbb{R}$ defined via $\Phi ({\theta }_1,\dots ,{\vartheta }_n)={\prod }_{j=1}^n\left|{\theta }_j\right|$ for $({\theta }_1,\dots ,{\theta }_n)\in {\mathbb{R}}^n$. Obviously, Φ fulfills (H1) but not (H2).

Theorem 3. 

For $\Psi \in \Xi$, the following axioms are equivalent:

(i) 

Ψ is a GM-QM;

(ii) 

Ψ satisfies (4) and has hypothesis (H2).

Proof. 

(i)⇒(ii) Fix $j\in \{1,\dots ,n\}$ and ${\xi }_1,\dots ,{\xi }_{j-1},{\xi }_{j+1},\dots ,{\xi }_n\in V$. By assumption, we have

$$\begin{array}{cc}\hfill & \Psi ({\xi }_1,\dots ,{\xi }_{j-1},0,{\xi }_{j+1},\dots ,{\xi }_n)+\Psi ({\xi }_1,\dots ,{\xi }_{j-1},0,{\xi }_{j+1},\dots ,{\xi }_n)\hfill \\ \hfill & =\Psi ({\xi }_1,\dots ,{\xi }_{j-1},0,{\xi }_{j+1},\dots ,{\xi }_n)+\Psi ({\xi }_1,\dots ,{\xi }_{j-1},0,{\xi }_{j+1},\dots ,{\xi }_n)\hfill \\ \hfill & +R_i\Psi ({\xi }_1,\dots ,{\xi }_{j-1},0,{\xi }_{j+1},\dots ,{\xi }_n)+S_i\Psi ({\xi }_1,\dots ,{\xi }_{j-1},0,{\xi }_{j+1},\dots ,{\xi }_n)\hfill \end{array}$$

and so

$$\begin{array}{c}\hfill \Psi ({\xi }_1,\dots ,{\xi }_{j-1},0,{\xi }_{j+1},\dots ,{\xi }_n)=0.\end{array}$$

(5)

Putting ${\xi }_j^{\prime }=0$ in (3) and using (5), we obtain

$$\begin{array}{cc}\hfill & \Psi ({\xi }_1,\dots ,{\xi }_{j-1},r_j{\xi }_j,{\xi }_{j+1},\dots ,{\xi }_n)+\Psi ({\xi }_1,\dots ,{\xi }_{j-1},r_j{\xi }_j,{\xi }_{i+1},\dots ,{\xi }_n)\hfill \\ \hfill & =\Psi ({\xi }_1,\dots ,{\xi }_{j-1},{\xi }_j,{\xi }_{j+1},\dots ,{\xi }_n)+\Psi ({\xi }_1,\dots ,{\xi }_{j-1},{\xi }_j,{\xi }_{j+1},\dots ,{\xi }_n)\hfill \\ \hfill & +R_i\Psi ({\xi }_1,\dots ,{\xi }_{j-1},{\xi }_j,{\xi }_{j+1},\dots ,{\xi }_n).\hfill \end{array}$$

The relation above necessitates that

$$\begin{array}{cc}\hfill & \Psi ({\xi }_1,\dots ,{\xi }_{j-1},r_j{\xi }_j,{\xi }_{j+1},\dots ,{\xi }_n)=r_j^2\Psi ({\xi }_1,\dots ,{\xi }_{j-1},{\xi }_j,{\xi }_{i+1},\dots ,{\xi }_n),\hfill \end{array}$$

which shows that $\Psi$ satisfies (H2). We now prove that $\Psi$ fulfills (4) by induction on m with $1\le m\le n$. For $m=1$, (2) gives us a validity of (4) for $\Psi$. Assume that (4) holds for $m-1$. Then,

$$\begin{array}{c}\hfill \sum _{{\alpha }_1,\dots ,{\alpha }_{m-1}\in \{-1,1\}}\Psi \left({\Phi }_1^{{\alpha }_1},\dots ,{\Phi }_{m-1}^{{\alpha }_{m-1}},u\right)=\sum _{V_j\in \{v_{1j},v_{2j},v_{1j}\pm v_{2j}\}}\prod _{j=1}^{m-1}{\mathfrak{R}}_j{\mathfrak{S}}_j\Psi \left(V_1,\dots ,V_{m-1},u\right),\end{array}$$

(6)

where $u\in V$. Equality (6) necessitates that

$$\begin{array}{cc}\hfill \sum _{{\alpha }_1,\dots ,{\alpha }_m\in \{-1,1\}}\Psi & \left({\Phi }_1^{{\alpha }_1},\dots ,{\Phi }_m^{{\alpha }_m}\right)\hfill \\ \hfill & =\sum _{{\alpha }_1,\dots ,{\alpha }_{m-1}\in \{-1,1\}}\Psi \left({\Phi }_1^{{\alpha }_1},\dots ,{\Phi }_{m-1}^{{\alpha }_{m-1}},v_{1m}+v_{2m}\right)\hfill \\ \hfill & +\sum _{{\alpha }_1,\dots ,{\alpha }_{m-1}\in \{-1,1\}}\Psi \left({\Phi }_1^{{\alpha }_1},\dots ,{\Phi }_{m-1}^{{\alpha }_{m-1}},v_{1m}-v_{2m}\right)\hfill \\ \hfill & +R_m\sum _{{\alpha }_1,\dots ,{\alpha }_{m-1}\in \{-1,1\}}\Psi \left({\Phi }_1^{{\alpha }_1},\dots ,{\Phi }_{m-1}^{{\alpha }_{m-1}},v_{1m}\right)\hfill \\ \hfill & +S_m\sum _{{\alpha }_1,\dots ,{\alpha }_{m-1}\in \{-1,1\}}\Psi \left({\Phi }_1^{{\alpha }_1},\dots ,{\Phi }_{m-1}^{{\alpha }_{m-1}},v_{2m}\right)\hfill \\ \hfill & =\sum _{\lambda \in \{-1,1\}}\sum _{V_j\in \{v_{1j},v_{2j},v_{1j}\pm v_{2j}\}}\prod _{j=1}^{m-1}{\mathfrak{R}}_j{\mathfrak{S}}_j\Psi \left(V_1,\dots ,V_{m-1},v_{1m}+\lambda v_{2m}\right)\hfill \\ \hfill & +R_m\sum _{V_j\in \{v_{1j},v_{2j},v_{1j}\pm v_{2j}\}}\prod _{j=1}^{m-1}{\mathfrak{R}}_j{\mathfrak{S}}_j\Psi \left(V_1,\dots ,V_{m-1},v_{1m}\right)\hfill \\ \hfill & +S_m\sum _{V_j\in \{v_{1j},v_{2j},v_{1j}\pm v_{2j}\}}\prod _{j=1}^{m-1}{\mathfrak{R}}_j{\mathfrak{S}}_j\Psi \left(V_1,\dots ,V_{m-1},v_{2m}\right)\hfill \\ \hfill & =\sum _{V_j\in \{v_{1j},v_{2j},v_{1j}\pm v_{2j}\}}\prod _{j=1}^m{\mathfrak{R}}_j{\mathfrak{S}}_j\Psi \left(V_1,\dots ,V_m\right).\hfill \end{array}$$

This implication is now finished.

(ii)⇒(i) Set

$$\begin{array}{cc}\hfill {\Psi }^{\odot }(v_{1j},v_{2j})& :=\Psi \left(v_{11},\dots ,v_{1,j-1},v_{1j}+v_{2j},v_{1,j+1},\dots ,v_{1n}\right)\hfill \\ \hfill & +\Psi \left(v_{11},\dots ,v_{1,j-1},v_{1j}-v_{2j},v_{1,j+1},\dots ,v_{1n}\right)\hfill \end{array}$$

and

$$\begin{array}{cc}\hfill {\Psi }^{\odot }\left(v_{2j}\right)& :=\Psi \left(v_{11},\dots ,v_{1,j-1},v_{2j},v_{1,j+1},\dots ,v_{1n}\right).\hfill \end{array}$$

Putting $v_{2l}=0$ for all $l\in \{1,\dots ,n\}\setminus \left\{j\right\}$ in the left-hand side of (4) and applying our hypotheses, we obtain

$$\begin{array}{cc}\hfill & 2^{n-1}[\Psi \left(r_1{v}_{11},\dots ,r_{j-1}{v}_{1,j-1},r_j{v}_{1j}+s_j{v}_{2j},r_{j+1}{v}_{1,j+1},\dots ,r_n{v}_{1n}\right)\hfill \\ \hfill & +\Psi \left(r_1{v}_{11},\dots ,r_{j-1}{v}_{1,j-1},r_j{v}_{1j}-s_j{v}_{2j},r_{j+1}{v}_{1,j+1},\dots ,r_n{v}_{1n}\right)]\hfill \\ \hfill & =2^{n-1}\times \prod _{\stackrel{k\in \{1,\dots ,n\}}{k\ne j}}{r}_k^2[\Psi \left(v_{11},\dots ,v_{1,j-1},r_j{v}_{1j}+s_j{v}_{2j},v_{1,j+1},\dots ,v_{1n}\right)\hfill \\ \hfill & +\Psi \left(v_{11},\dots ,v_{1,j-1},r_j{v}_{1j}-s_j{v}_{2j},v_{1,j+1},\dots ,v_{1n}\right)].\hfill \end{array}$$

(7)

Once more, setting $v_{2l}=0$ for all $l\in \{1,\dots ,n\}\setminus \left\{j\right\}$ in the right-hand side of (4), we obtain

$$\begin{array}{cc}\hfill & \sum _{m=0}^{n-1}\left[\left(\begin{array}{c}n-1\\ m\end{array}\right)2^{n-m-1}\prod _{\stackrel{i\in \{1,\dots ,m\}}{R_{l_i}\in \{R_1,\dots ,R_n\}}}{R}_{l_i}\right]{\Psi }^{\odot }(v_{1j},v_{2j})\hfill \\ \hfill & +R_j\sum _{m=0}^{n-1}\left[\left(\begin{array}{c}n-1\\ m\end{array}\right)2^{n-m-1}\prod _{\stackrel{i\in \{1,\dots ,m\}}{R_{l_i}\in \{R_1,\dots ,R_n\}}}{R}_{l_i}\right]\Psi \left(v_1^{\left[n\right]}\right)\hfill \\ \hfill & +S_j\sum _{m=0}^{n-1}\left[\left(\begin{array}{c}n-1\\ m\end{array}\right)2^{n-m-1}\prod _{\stackrel{i\in \{1,\dots ,m\}}{R_{l_i}\in \{R_1,\dots ,R_n\}}}{R}_{l_i}\right]{\Psi }^{\odot }\left(v_{2j}\right)\hfill \\ \hfill & =2^{n-1}\times \prod _{\stackrel{k\in \{1,\dots ,n\}}{k\ne j}}{r}_k^2{\Psi }^{\odot }(v_{1j},v_{2j})\hfill \\ \hfill & +R_j\times 2^{n-1}\times \prod _{\stackrel{k\in \{1,\dots ,n\}}{k\ne j}}{r}_k^2\Psi \left(v_1^{\left[n\right]}\right)\hfill \\ \hfill & +S_j\times 2^{n-1}\times \prod _{\stackrel{k\in \{1,\dots ,n\}}{k\ne j}}{r}_k^2{\Psi }^{\odot }\left(v_{2j}\right),\hfill \end{array}$$

(8)

where $\left({\displaystyle \genfrac{}{}{0pt}{}{n}{r}}\right):={\displaystyle \frac{n!}{r!(n-r)!}}$. We conclude from (7) and (8) that

$$\begin{array}{cc}\hfill \Psi & \left(v_{11},\dots ,v_{1,j-1},r_j{v}_{1j}+s_j{v}_{2j},v_{1,j+1},\dots ,v_{1n}\right)\hfill \\ \hfill & +\Psi \left(v_{11},\dots ,v_{1,j-1},r_j{v}_{1j}-s_j{v}_{2j},v_{1,j+1},\dots ,v_{1n}\right)\hfill \\ \hfill & ={\Psi }^{\odot }(v_{1j},v_{2j})+R_j\Psi \left(v_1^{\left[n\right]}\right)+S_j{\Psi }^{\odot }\left(v_{2j}\right).\hfill \end{array}$$

This means that $\Psi$ is a generalized quadratic in the variable j and therefore we obtain the desired result. □

For $v_1^{\left[n\right]},v_2^{\left[n\right]}\in V^n$, we set

$${\mathbb{V}}^n=\left\{{\mathfrak{V}}_n=(V_1,\dots ,V_n)|V_j\in \{v_{1j},v_{2j},v_{1j}\pm v_{2j}\right\},$$

for all $j\in \{1,\dots ,n\}$. Additionally, for $p,q\in {\mathbb{N}}_0$ with $p,q\in \{0,\dots ,n\}$, ${\mathbb{V}}_{(p,q)}^n$ is a subset of ${\mathbb{V}}^n$ as follows:

$$\begin{array}{cc}\hfill {\mathbb{V}}_{(p,q)}^n& :=\left\{{\mathfrak{V}}_n\in {\mathbb{V}}^n|\mathrm{Card}\{V_j:V_j=v_{1j}\}=p,\mathrm{Card}\{V_j:V_j=v_{2j}\}=q\right\},\hfill \end{array}$$

where CardA is the cardinality of set A. From now on, for a GM-QM $\Psi \in \Xi$, we use the convention

$$\begin{array}{c}\hfill \Psi \left({\mathbb{V}}_{(p,q)}^n\right):=\sum _{{\mathfrak{V}}_n\in {\mathbb{V}}_{(p,q)}^n}\Psi \left({\mathfrak{V}}_n\right).\end{array}$$

(9)

In Definition 1, if $r_j=r$ and $s_j=s$ for all $j\in \{1,\dots ,n\}$, then (4) converts to

$$\begin{array}{c}\hfill \sum _{\lambda \in {\{-1,1\}}^n}\Psi \left(r{v}_1^{\left[n\right]}+\lambda s{v}_2^{\left[n\right]}\right)=\sum _{p=0}^n\sum _{q=0}^{n-p}{R}^p\times S^q\Psi \left({\mathbb{V}}_{(p,q)}^n\right),\end{array}$$

(10)

for all $v_1^{\left[n\right]},v_2^{\left[n\right]}\in V^n$ and fixed $\pm 1,0\ne r,s\in \mathbb{Z}$, where $\Psi \left({\mathbb{V}}_{(p,q)}^n\right)$ is given in (9).

3. Stability Results for Equation (10)

In the current section, we study the H-U-R-Stab of the GM-QMs in fuzzy norm spaces by the Hyers and FP methods. Note that when we say a GM-QM is stable, we mean that Equation (10) has the stability property.

 Definition 3. 

Let V be a real linear space. A function $T:V\times \mathbb{R}⟶[0,1]$ is termed a fuzzy norm on V if for all $u,v\in V$ and all $s,t\in \mathbb{R}$ the following hold:

(T1) 

$T(v,r)=0$ for $r\le 0$;

(T2) 

$v=0⟺T(v,r)=1$ for all $r>0$;

(T3) 

$T(rv,t)=T\left(v,{\displaystyle \frac{t}{\left|r\right|}}\right)$ if $r\ne 0$;

(T4) 

$T(u+v,s+t)\ge \min \left\{T\right(u,s),T(v,t\left)\right\}$;

(T5) 

$T(v,·)$ is a non-decreasing function on $\mathbb{R}$ and ${\lim }_{t\to \infty }T(v,t)=1$;

(T6) 

For $v\ne 0$, $T(v,·)$ is upper semi-continuous on $\mathbb{R}$.

It is argued that the pair $(V,T)$ is an FNS. Now, we present the following known observations for an FNS $(V,T)$. For every $t\ge 0$, a sequence ${\left\{v_j\right\}}_j$ in V converges to a $v\in V$ if ${\lim }_{j\to \infty }T(v_j-v,t)=1$. In this instance, v is denoted by $T-{lim}_{t\to \infty }{v}_j=v$ and is considered the limit of the sequence ${\left\{v_j\right\}}_j$. Furthermore, if for every $\delta >0$ and every $t>0$ there exists a $t_0\in \mathbb{N}$ such that for all $t\ge t_0$ and all $k>0$, we have $T(v_{j+k}-v_j,t)>1-\delta$, then a sequence $\left\{v_j\right\}$ in V is termed Cauchy. Like all normed spaces, the completeness of a fuzzy norm, being a fuzzy Banach space (FBS) of $(V,T)$ and continuity of $f\in \Xi$, can be defined; for more details we refer to [11].

The next example, which converts a normed linear space to an FNS, was indicated in [36]. In continuation, we apply this example to attain more results.

Example 2. 

Let $(V,\parallel ·\parallel )$ be a normed linear space. For $v\in V$, the function

$$T(v,t)=\left\{\begin{array}{c}{\displaystyle \frac{\alpha t^k}{\beta \parallel v\parallel +\alpha t^k}}t>0,\hfill \\ 0\mathit{otherwise},\hfill \end{array}\right.$$

where $\alpha ,\beta ,k\in {\mathbb{R}}^{+}$. Then, $(V,T)$ is an FNS. In particular, if $\alpha =\beta =k=1$, then $T(v,t)$ is sometimes called the standard fuzzy norm induced by norm $\parallel ·\parallel$.

In what follows, V is a linear space, $(W,T)$ is an FBS, and $(Y,T^{\prime })$ is an FNS. Moreover, we assume that each $f\in \Xi$ has the property (H1).

For a $f\in \Xi$, we remark

$$\begin{array}{cc}\hfill \mathbf{D}f\left(v_1^{\left[n\right]},v_2^{\left[n\right]}\right)& :=\sum _{\lambda \in {\{-1,1\}}^n}f\left(r{v}_1^{\left[n\right]}+\lambda s{v}_2^{\left[n\right]}\right)-\sum _{p=0}^n\sum _{q=0}^{n-p}{R}^p\times S^{q}f\left({\mathbb{V}}_{(p,q)}^n\right),\hfill \end{array}$$

for all $v_1^{\left[n\right]},v_2^{\left[n\right]}\in V^n$, where $f\left({\mathbb{V}}_{(p,q)}^n\right)$ is defined in (9) and $r,s$ is a fixed integer with $r,s\ne 0,\pm 1$ and $R=2(r^2-1)$, $S=2(s^2-1)$.

3.1. Stability Results: Direct Method

In this subsection, the proof of the stability of (10) on FNSs is established by using the Hyers technique.

Theorem 4. 

Let $f\in \Xi$ and $\varphi :V^n⟶Y$ be a function with the following property.

(H3) For all $v^{\left[n\right]}\in V^n$, $\varphi \left(r{v}^{\left[n\right]}\right)=\alpha \varphi \left(v^{\left[n\right]}\right)$ for some real numbers α with $0<\left|\alpha \right|<r^{2n}$

such that

$$\begin{array}{c}\hfill T\left(\mathbf{D}f\left(v_1^{\left[n\right]},v_2^{\left[n\right]}\right),t+t^{\prime }\right)\ge min\left\{T^{\prime }\left(\varphi \left(v_1^{\left[n\right]}\right),t\right),T^{\prime }\left(\varphi \left(v_2^{\left[n\right]}\right),t^{\prime }\right)\right\},\end{array}$$

(11)

for all $v_1^{\left[n\right]},v_2^{\left[n\right]}\in V^n$ and $t,t^{\prime }>0$. Then, there exists a solution $\mathcal{Q}\in \Xi$ of (10) such that

$$\begin{array}{c}\hfill T\left(f\left(v^{\left[n\right]}\right)-\mathcal{Q}\left(v^{\left[n\right]}\right),t\right)\ge M\left(v^{\left[n\right]},{\displaystyle \frac{r^{2n}-\alpha }{2r^{2n}}}t\right),\end{array}$$

(12)

for all $v^{\left[n\right]}\in V^n$, $t>0$, where

$$\begin{array}{c}\hfill M(v^{\left[n\right]},t):=min\left\{T^{\prime }\left(\varphi \left(v^{\left[n\right]}\right),2^{n-1}\times r^{2n}t\right),T^{\prime }\left(\varphi \left(\mathbf{0}\right),2^{n-1}\times r^{2n}t\right)\right\},\end{array}$$

(13)

whereas $\mathbf{0}={\displaystyle \stackrel{n-\mathit{times}}{\overbrace{0,\dots ,0}}}$. Moreover, $\mathcal{Q}$ is a unique GM-QM provided that hypothesis (H2) is valid for it.

Proof. 

Putting $v_2^{\left[n\right]}=\mathbf{0}$ and $t^{\prime }=t$ into (11), we have

$$\begin{array}{c}\hfill T\left(2^{n}f\left(r{v}_1^{\left[n\right]}\right)-\mathfrak{R}f\left(v_1^{\left[n\right]}\right),2t\right)\ge \min \{T^{\prime }(\varphi \left(v_1^{\left[n\right]}\right),t),T^{\prime }(\varphi \left(\mathbf{0}\right),t)\},\end{array}$$

(14)

for all $v_1^{\left[n\right]}\in V^n$ and $t>0$, where

$$\begin{array}{c}\hfill \mathfrak{R}=\sum _{p=0}^n\left[\left(\begin{array}{c}n\\ p\end{array}\right)2^{n-p}\times R^p\right].\end{array}$$

(15)

For the rest, we set $v_1^{\left[n\right]}$ by $v^{\left[n\right]}$. The equivalent (14) and (15) necessitate

$$\begin{array}{c}\hfill T\left(f\left(r{v}^{\left[n\right]}\right)-r^{2n}f\left(v^{\left[n\right]}\right),t\right)\ge \min \left\{T^{\prime }\left(\varphi \left(v^{\left[n\right]}\right),2^{n-1}t\right),T^{\prime }\left(\varphi \left(\mathbf{0}\right),2^{n-1}t\right)\right\},\end{array}$$

for all $v^{\left[n\right]}\in V^n,t>0$, and thus

$$\begin{array}{c}\hfill T\left({\displaystyle \frac{f\left(r{v}^{\left[n\right]}\right)}{r^{2n}}}-f\left(v^{\left[n\right]}\right),t\right)\ge M(v^{\left[n\right]},t),\end{array}$$

(16)

for all $v^{\left[n\right]}\in V^n$ and $t>0$, where $M(v^{\left[n\right]},t)$ is given in (13). Putting $v^{\left[n\right]}$ by $r^m{v}^{\left[n\right]}$ in (16) and applying the equality

$$\begin{array}{c}\hfill M(r{v}^{\left[n\right]},t)=M\left(v^{\left[n\right]},{\displaystyle \frac{t}{\alpha }}\right),\end{array}$$

(17)

we arrive at

$$\begin{array}{cc}\hfill T\left({\displaystyle \frac{f\left(r^{m+1}{v}^{\left[n\right]}\right)}{r^{2(m+1)n}}}-{\displaystyle \frac{f\left(r^m{v}^{\left[n\right]}\right)}{r^{2mn}}},{\displaystyle \frac{{\alpha }^m}{r^{2mn}}}t\right)& =T\left({\displaystyle \frac{f\left(r^{m+1}{v}^{\left[n\right]}\right)}{r^{2n}}}-f\left(r^m{v}^{\left[n\right]}\right),{\alpha }^{m}t\right)\hfill \\ \hfill & \ge M(r^m{v}^{\left[n\right]},{\alpha }^{m}t)=M(v^{\left[n\right]},t),\hfill \end{array}$$

for all $v^{\left[n\right]}\in V^n$, $t>0$ and $m\ge 0$. Furthermore, for each $m>l>0$, we find

$$\begin{array}{cc}\hfill T& \left({\displaystyle \frac{f\left(r^m{v}^{\left[n\right]}\right)}{r^{2mn}}}-{\displaystyle \frac{f\left(r^l{v}^{\left[n\right]}\right)}{r^{2ln}}},\sum _{j=l}^{m-1}{\displaystyle \frac{{\alpha }^j}{r^{2jn}}}t\right)\hfill \\ \hfill & =T\left(\sum _{j=l}^{m-1}{\displaystyle \frac{f\left(r^{m+1}{v}^{\left[n\right]}\right)}{r^{2(m+1)n}}}-{\displaystyle \frac{f\left(r^m{v}^{\left[n\right]}\right)}{r^{2mn}}},\sum _{j=l}^{m-1}{\displaystyle \frac{{\alpha }^j}{r^{2jn}}}t\right)\hfill \\ \hfill & \ge \min \bigcup _{j=l}^{m-1}\left\{T\left({\displaystyle \frac{f\left(r^{m+1}{v}^{\left[n\right]}\right)}{r^{2(m+1)n}}}-{\displaystyle \frac{f\left(r^m{v}^{\left[n\right]}\right)}{r^{2mn}}},{\displaystyle \frac{{\alpha }^j}{r^{2jn}}}t\right)\right\}\hfill \\ \hfill & \ge M(v^{\left[n\right]},t).\hfill \end{array}$$

(18)

Take $ϵ,\delta >0$. Due to the equality ${lim}_{t\to \infty }M(v^{\left[n\right]},t)=1$, there is a $t_0>0$ such that $M(v^{\left[n\right]},t_0)>1-ϵ$. On the other hand, ${\sum }_{j=0}^{m-1}{\displaystyle \frac{{\alpha }^j}{r^{jn}}}{t}_0<\infty$, and therefore there is some $n_0\in \mathbb{N}$ such that ${\sum }_{j=l}^{m-1}{\displaystyle \frac{{\alpha }^j}{r^{2jn}}}{t}_0<\delta$ for all $m>l\ge n_0$. It can now be deduced that

$$\begin{array}{cc}\hfill T\left({\displaystyle \frac{f\left(r^m{v}^{\left[n\right]}\right)}{r^{2mn}}}-{\displaystyle \frac{f\left(r^l{v}^{\left[n\right]}\right)}{r^{2ln}}},\delta \right)& \ge T\left({\displaystyle \frac{f\left(r^m{v}^{\left[n\right]}\right)}{r^{2mn}}}-{\displaystyle \frac{f\left(r^l{v}^{\left[n\right]}\right)}{r^{2ln}}},\sum _{j=l}^{m-1}{\displaystyle \frac{{\alpha }^j}{r^{2jn}}}{t}_0\right)\hfill \\ \hfill & \ge M(v^{\left[n\right]},t_0)\ge 1-ϵ.\hfill \end{array}$$

The relation above implies that $\left\{{\displaystyle \frac{f\left(r^m{v}^{\left[n\right]}\right)}{{\lambda }^{2mn}}}\right\}$ is fuzzy Cauchy in $(W,T)$. Since $(W,T)$ is a Banach fuzzy space, the mentioned sequence converges pointwise to a mapping $\mathcal{Q}\in \Xi$; that is,

$$\mathcal{Q}\left(v^{\left[n\right]}\right)=\underset{m\to \infty }{lim}T-{\displaystyle \frac{f\left(r^m{v}^{\left[n\right]}\right)}{r^{2mn}}}.$$

Putting $l=0$ into (18), we obtain

$$T\left({\displaystyle \frac{f\left(r^m{v}^{\left[n\right]}\right)}{r^{2mn}}}-f\left(v^{\left[n\right]}\right),\sum _{j=0}^{m-1}{\displaystyle \frac{{\alpha }^j}{r^{jn}}}t\right)\ge M(v^{\left[n\right]},t),$$

for all $v^{\left[n\right]}\in V^n$ and $t>0$. The last inequality necessitates that

$$\begin{array}{c}\hfill T\left({\displaystyle \frac{f\left(r^m{v}^{\left[n\right]}\right)}{r^{2mn}}}-f\left(v^{\left[n\right]}\right),t\right)\ge M\left(v^{\left[n\right]},{\displaystyle \frac{t}{{\sum }_{j=0}^{m-1}\frac{{\alpha }^j}{r^{2jn}}}}\right),\end{array}$$

(19)

for all $v^{\left[n\right]}\in V^n$ and $t>0$. Consider the function $\mathbf{1}:{\mathbb{R}}^n⟶\mathbb{R}$, whose range is the set$\left\{1\right\}$. Let $\Omega =2^n+\left|{\sum }_{p=0}^n{\sum }_{q=0}^{n-p}{R}^p\times S^q\mathbf{1}\left({\mathbb{V}}_{(p,q)}^n\right)\right|$. We have

$$\begin{array}{cc}\hfill & T\left(\mathbf{D}\mathcal{Q}(v_1^{\left[n\right]},v_2^{\left[n\right]}),t\right)\hfill \\ \hfill & \ge \min \{\bigcup _{\lambda \in {\{-1,1\}}^n}T\left(\mathcal{Q}\left(r{v}_1^{\left[n\right]}+\lambda s{v}_2^{\left[n\right]}\right)-{\displaystyle \frac{f\left(r^{m+1}{v}_1^{\left[n\right]}+\lambda s{r}^m{v}_2^{\left[n\right]}\right)}{r^{2mn}}},{\displaystyle \frac{t}{\Omega }}\right),\hfill \\ \hfill & \bigcup _{p=0}^n\bigcup _{q=0}^{n-p}T\left(R^p\times S^q\mathcal{Q}\left({\mathbb{V}}_{(p,q)}^n\right)-R^p\times S^q{\displaystyle \frac{f\left(r^m{\mathbb{V}}_{(p,q)}^n\right)}{r^{2mn}}},{\displaystyle \frac{t}{\Omega }}\right),\hfill \\ \hfill & \bigcup _{\lambda \in {\{-1,1\}}^n}\bigcup _{p=0}^n\bigcup _{q=0}^{n-p}T\left({\displaystyle \frac{f\left(r^{m+1}{v}_1^{\left[n\right]}+\lambda s{r}^m{v}_2^{\left[n\right]}\right)}{r^{2mn}}}-R^p\times S^q{\displaystyle \frac{f\left(r^m{\mathbb{V}}_{(p,q)}^n\right)}{r^{2mn}}},{\displaystyle \frac{t}{\Omega }}\right)\},\hfill \end{array}$$

$\forall v_1^{\left[n\right]},v_2^{\left[n\right]}\in V^n$ and $t>0$, where $r^m{\mathbb{V}}_{(p,q)}^n=\left\{r^m{\mathfrak{V}}_n|{\mathfrak{V}}_n\in {\mathbb{V}}_{(p,q)}^n\right\}.$ In addition, $\mathcal{Q}$, defined as the member of the first and second unions on the right-hand side of the above inequality, goes to 1, and when $m\to \infty$ is a member of the third union, (11) is greater than or equal to

$$\begin{array}{cc}\hfill & \min \left\{T^{\prime }\left(\varphi \left(r^m{v}_1^{\left[n\right]}\right),{\displaystyle \frac{r^{2mn}}{2\Omega }}t\right),T^{\prime }\left(\varphi \left(r^m{v}_2^{\left[n\right]}\right),{\displaystyle \frac{r^{2mn}}{2\Omega }}t\right)\right\}\hfill \\ \hfill & =\min \left\{T^{\prime }\left(\varphi \left(v_1^{\left[n\right]}\right),{\displaystyle \frac{r^{2mn}}{2{\alpha }^m\Omega }}t\right),T^{\prime }\left(\varphi \left(v_2^{\left[n\right]}\right),{\displaystyle \frac{r^{2mn}}{2{\alpha }^m\Omega }}t\right)\right\},\hfill \end{array}$$

which goes to 1 when $m\to \infty$. So,

$$T\left(\mathbf{D}\mathcal{Q}\left(v_1^{\left[n\right]},v_2^{\left[n\right]}\right),t\right)=1,$$

for all $v_1^{\left[n\right]},v_2^{\left[n\right]}\in V^n$ and $t>0$. Hence, $\mathcal{Q}$ gives (10). If now $\mathcal{Q}$ has (H2), then it is a GM-QM by Theorem 3. For the fuzzy difference between f and $\mathcal{Q}$, by inequality (19), for $v^{\left[n\right]}\in V^n$ and $t>0$, this can be deduced as

$$\begin{array}{cc}\hfill T& \left(\mathcal{Q}\left(v^{\left[n\right]}\right)-f\left(v^{\left[n\right]}\right),t\right)\hfill \\ \hfill & \ge \min \left\{T\left(\mathcal{Q}\left(v^{\left[n\right]}\right)-{\displaystyle \frac{f\left(r^m{v}^{\left[n\right]}\right)}{r^{2mn}}},{\displaystyle \frac{t}{2}}\right),T\left({\displaystyle \frac{f\left(r^m{v}^{\left[n\right]}\right)}{r^{2mn}}}-f\left(v^{\left[n\right]}\right),{\displaystyle \frac{t}{2}}\right)\right\}\hfill \\ \hfill & \ge M\left(v^{\left[n\right]},{\displaystyle \frac{t}{2{\sum }_{j=0}^{\infty }\frac{{\alpha }^j}{r^{2jn}}}}\right)\hfill \\ \hfill & =M\left(v^{\left[n\right]},{\displaystyle \frac{r^{2n}-\alpha }{2r^{2n}}}t\right).\hfill \end{array}$$

Let $\mathfrak{Q}$ be another GM-QM in $\Xi$ such that inequality (24) is true for it. For each $v^{\left[n\right]}\in V^n$ and $t>0$, we reach

$$\begin{array}{cc}\hfill T\left(\mathcal{Q}\left(v^{\left[n\right]}\right)-\mathfrak{Q}\left(v^{\left[n\right]}\right),t\right)& \ge \min \left\{T\left(\mathcal{Q}\left(v^{\left[n\right]}\right)-f\left(v^{\left[n\right]}\right),{\displaystyle \frac{t}{2}}\right),T\left(\mathfrak{Q}\left(v^{\left[n\right]}\right)-f\left(v^{\left[n\right]}\right),{\displaystyle \frac{t}{2}}\right)\right\}\hfill \\ \hfill & \ge M\left(v^{\left[n\right]},{\displaystyle \frac{r^{2n}-\alpha }{2r^{2n}}}t\right).\hfill \end{array}$$

However, $\mathcal{Q}$ and $\mathfrak{Q}$ are GM-QMs and so

$$\begin{array}{cc}\hfill T\left(\mathcal{Q}\left(v^{\left[n\right]}\right)-\mathfrak{Q}\left(v^{\left[n\right]}\right),t\right)& =T\left(\mathcal{Q}\left(r^m{v}^{\left[n\right]}\right),r^{2mn}t\right)\hfill \\ & \ge M\left(v^{\left[n\right]},{\displaystyle \frac{r^{2n}-\alpha }{2r^{2n}}}{\left({\displaystyle \frac{r^{2n}}{\alpha }}\right)}^{m}t\right),\hfill \end{array}$$

for all $v^{\left[n\right]}\in V^n$, $t>0$, and $m\in \mathbb{N}$. Since ${lim}_{m\to \infty }{\left({\displaystyle \frac{r^{2n}}{\alpha }}\right)}^m=+\infty$, we have

$$\underset{m\to \infty }{lim}M\left(v^{\left[n\right]},{\displaystyle \frac{r^{2n}-\alpha }{2r^{2n}}}t\right)=1.$$

The proof is now finished because it demonstrates that $\mathcal{Q}\left(v^{\left[n\right]}\right)=\mathfrak{Q}\left(v^{\left[n\right]}\right)$ for all $v^{\left[n\right]}\in V^n$. □

The following theorem is analogous to the prior one. The assumptions are the same but the results are different. We provide a sketch of the proof because the methods are exactly the same as those of Theorem 4.

Theorem 5.

Let $f\in \Xi$ and $\varphi :V^n⟶Y$ be a function with hypothesis (H3) such that

$$\begin{array}{c}\hfill T\left(\mathbf{D}f(v_1^{\left[n\right]},v_2^{\left[n\right]}),t+s\right)\ge min\{T^{\prime }(\varphi \left(v_1^{\left[n\right]}\right),t),T^{\prime }(\varphi \left(v_2^{\left[n\right]}\right),s)\},\end{array}$$

for all $v_1^{\left[n\right]},v_2^{\left[n\right]}\in V^n$ and $t,s>0$. Then, there exists a solution $\mathcal{Q}\in \Xi$ of (10) such that

$$\begin{array}{c}\hfill T\left(f\left(v^{\left[n\right]}\right)-\mathcal{Q}\left(v^{\left[n\right]}\right),t\right)\ge N\left(v^{\left[n\right]},{\displaystyle \frac{\alpha -r^{2n}}{2\alpha }}t\right),\end{array}$$

for all $v^{\left[n\right]}\in V^n$ and $t>0$, where

$$\begin{array}{c}\hfill N(v^{\left[n\right]},t):=min\left\{T^{\prime }\left(\varphi \left(v^{\left[n\right]}\right),2^{n-1}\alpha t\right),T^{\prime }\left(\varphi \left(\mathbf{0}\right),2^{n-1}\alpha t\right)\right\}.\end{array}$$

(20)

In addition, $\mathcal{Q}$ is a unique GM-QM if it satisfies hypothesis (H2).

Proof. 

Note that hypothesis (H3) for $\varphi$ implies that $\varphi \left({\displaystyle \frac{v^{\left[n\right]}}{r}}\right)={\displaystyle \frac{1}{\alpha }}\varphi \left(v^{\left[n\right]}\right)$, in which $\left|\alpha \right|>r^{2n}$. The proof of Theorem 4 gives

$$\begin{array}{c}\hfill T\left(f\left(r{v}^{\left[n\right]}\right)-r^{2n}f\left(v^{\left[n\right]}\right),t\right)\ge \min \left\{T^{\prime }\left(\varphi \left(v^{\left[n\right]}\right),2^{n-1}t\right),T^{\prime }\left(\varphi \left(\mathbf{0}\right),2^{n-1}t\right)\right\},\end{array}$$

for all $v^{\left[n\right]}\in V^n$ and $t>0$. A direct result from the inequality is as follows:

$$\begin{array}{c}\hfill T\left(f\left(v^{\left[n\right]}\right)-r^{2n}f\left({\displaystyle \frac{v^{\left[n\right]}}{r}}\right),t\right)\ge N(v,t),\end{array}$$

(21)

for all $v^{\left[n\right]}\in V^n$ and $t>0$, where $N(v,t)$ is defined in (20). Interchanging $v^{\left[n\right]}$ into $r^m{v}^{\left[n\right]}$ in (21) and applying the property $N\left({\displaystyle \frac{v^{\left[n\right]}}{r}},t\right)=N\left(v^{\left[n\right]},\alpha t\right)$, we obtain

$$\begin{array}{cc}\hfill T\left(r^{2mn}f\left({\displaystyle \frac{v^{\left[n\right]}}{r^m}}\right)-r^{(m+1)n}f\left({\displaystyle \frac{v^{\left[n\right]}}{r^{m+1}}}\right),{\displaystyle \frac{r^{2mn}}{{\alpha }^m}}t\right)& \ge N(v,t),\hfill \end{array}$$

for all $v^{\left[n\right]}\in V^n$, $t>0$, and $m\ge 0$. Moreover, for $m>l>0$, we obtain

$$\begin{array}{c}\hfill T\left(r^{2ln}f\left({\displaystyle \frac{v^{\left[n\right]}}{r^l}}\right)-r^{2mn}f\left({\displaystyle \frac{v^{\left[n\right]}}{r^m}}\right),\sum _{j=l}^{m-1}{\displaystyle \frac{r^{2jn}}{{\alpha }^j}}t\right)\ge N(v^{\left[n\right]},t).\end{array}$$

(22)

Let $ϵ,\delta >0$. Since ${lim}_{t\to \infty }N(v^{\left[n\right]},t)=1$, there is a $t_0>0$ so that $N(v^{\left[n\right]},t_0)>1-ϵ$. Further, ${\sum }_{j=0}^{m-1}{\displaystyle \frac{r^{2jn}}{{\alpha }^j}}{t}_0<\infty$, and so for some $n_0\in \mathbb{N}$, we find ${\sum }_{j=l}^{m-1}{\displaystyle \frac{r^{2jn}}{{\alpha }^j}}{t}_0<\delta$ for all $m>l\ge n_0$. These arguments imply that

$$\begin{array}{c}\hfill T\left(r^{2ln}f\left({\displaystyle \frac{v^{\left[n\right]}}{r^l}}\right)-r^{2mn}f\left({\displaystyle \frac{v^{\left[n\right]}}{r^m}}\right),\delta \right)\ge N(v^{\left[n\right]},t_0)\ge 1-ϵ.\end{array}$$

Thus, the sequence $\left\{r^{2mn}f\left({\displaystyle \frac{v^{\left[n\right]}}{r^m}}\right)\right\}$ is fuzzy Cauchy in $(W,T)$. Due to the completeness of $(W,T^{\prime })$, this sequence converges pointwise to a $\mathcal{Q}\in \Xi$ such that

$$\mathcal{Q}\left(v\right)=\underset{m\to \infty }{lim}T-r^{2mn}f\left({\displaystyle \frac{v^{\left[n\right]}}{r^m}}\right).$$

Putting $l=0$ into (22), we obtain

$$T\left(f\left(v^{\left[n\right]}\right)-r^{2mn}f\left({\displaystyle \frac{v^{\left[n\right]}}{r^m}}\right),\sum _{j=0}^{m-1}{\displaystyle \frac{r^{2jn}}{{\alpha }^j}}t\right)\ge N(v^{\left[n\right]},t),$$

for all $v^{\left[n\right]}\in V^n$ and $t>0$. The last inequality shows that

$$\begin{array}{c}\hfill T\left(f\left(v^{\left[n\right]}\right)-r^{2mn}f\left({\displaystyle \frac{v^{\left[n\right]}}{r^m}}\right),t\right)\ge N\left(v^{\left[n\right]},{\displaystyle \frac{t}{{\sum }_{j=0}^{m-1}\frac{r^{2jn}}{{\alpha }^j}}}\right),\end{array}$$

for all $v^{\left[n\right]}\in V^n$ and $t>0$. The proof of being a solution of (10) and the uniqueness of $\mathcal{Q}$ can be a achieved in a standard fashion taken from the proof of Theorem 4. □

The upcoming examples can be deduced from Theorems 4 and 5.

Example 3.

Suppose V is a normed space, $(W,T)$ is an FBS, and $(Y,T^{\prime })$ is an FNS, where T and $T^{\prime }$ are both the standard fuzzy norm as considered in Example 2. Moreover, $f\in \Xi$ satisfies

$$\begin{array}{cc}& T\left(\mathbf{D}f(v_1^{\left[n\right]},v_2^{\left[n\right]}),t+t^{\prime }\right)\ge \min \left\{T^{\prime }\left({\displaystyle {\sum }_{j=1}^n}{\parallel v_{1j}\parallel }^{\rho }{y}_0,t\right),T^{\prime }\left({\displaystyle {\sum }_{j=1}^n}{\parallel v_{2j}\parallel }^{\rho }{y}_0,t^{\prime }\right)\right\},\end{array}$$

for all $v_1^{\left[n\right]},v_2^{\left[n\right]}\in V^n$, and $t,t^{\prime }>0$, where $y_0$ is a fixed vector in Y. If $0<r\ne 2n$, then by Theorems 4 and 5 there exists a solution $\mathcal{Q}\in \Xi$ of (10) such that

$$\begin{array}{c}\hfill T\left(f\left(v^{\left[n\right]}\right)-\mathcal{Q}\left(v^{\left[n\right]}\right),t\right)\ge {\displaystyle \frac{2^{n-2}|r^{2n}-r^{\rho }|t}{2^{n-2}|r^{2n}-r^{\rho }|t + \parallel y_0\parallel {\sum }_{j=1}^n{\parallel v_{1j}\parallel }^{\rho }}},\end{array}$$

(23)

for all $v^{\left[n\right]}:=v_1^{\left[n\right]}\in V^n$ and $t>0$. Additionally, $\mathcal{Q}$ is a unique GM-QM provided that it has property (H2).

From the definition of a GM-QM, consider a large number for n, then the right-hand side of inequality (23) goes to 1 and the fuzzy difference goes to zero. In the following example, the study of the Hyers stability of the GM-QMs gives some examples like the following example.

Example 4. 

Given $\delta >0$. Suppose that $f\in \Xi$ is a mapping satisfying

$$\begin{array}{cc}& T\left(\mathbf{D}f(v_1^{\left[n\right]},v_2^{\left[n\right]}),t+s\right)\ge \min \{T^{\prime }\left(\delta y_0,t\right),T^{\prime }\left(\delta y_0,s\right)\},\end{array}$$

for all $v_1^{\left[n\right]},v_2^{\left[n\right]}\in V^n$ and $t,s>0$, where $y_0$ is a fixed vector in Y. By Theorem 4, there exists a solution $\mathcal{Q}:V^n⟶W$ of (10) such that

$$\begin{array}{c}\hfill T\left(f\left(v^{\left[n\right]}\right)-\mathcal{Q}\left(v^{\left[n\right]}\right),t\right)\ge T^{\prime }\left(y_0,{\displaystyle \frac{2^{n-2}(r^{2n}-1)}{\delta }}t\right),\end{array}$$

for all $v^{\left[n\right]}\in V^n$ and $t>0$.

3.2. Stability Results: The FP Approach

We present the H-U-R-Stab of GM-QMs using an FP technique in this subsection. Prior to outlining the primary findings, it is important to remember that the range of the generalized metric (GM) d on a set X encompasses infinity, which is the only significant distinction between it and the metric $d^{\prime }$ on X. We present a fundamental result in FP theory in the following theorem, which serves our objectives. We note that [37] presented an extension of the result.

Theorem 6 

([38]). Let $(\Delta ,d)$ be a complete GMS and $\mathcal{T}:\Delta ⟶\Delta$ be a mapping that is Lipschitz constant (the definition of which is available in the literature), $0<L<1$. Then, for each element $\eta \in \Delta$, we have the following assertions:

• $d({\mathcal{T}}^n\eta ,{\mathcal{T}}^{n+1}\eta )=\infty$ for all $n\ge 0$;

or

• there exists a $n_0\in \mathbb{N}$ such that the following hold:

(1) 

$d({\mathcal{T}}^n\eta ,{\mathcal{T}}^{n+1}\eta )<\infty$ for all $n\ge n_0$;

(2) 

the sequence $\left\{{\mathcal{T}}^n\eta \right\}$ converges to an element ${\eta }^{*}$ of $\mathcal{T}$;

(3) 

${\eta }^{*}$ is the unique fixed point of $\mathcal{J}\in {\Delta }_0=\{\eta \in \Delta :d({\mathcal{T}}^{n_0}y,y)<\infty \}$;

(4) 

$d(\eta ,{\eta }^{*})\le {\displaystyle \frac{1}{1-L}}d(\eta ,\mathcal{T}\eta )$ for all $\eta \in {\Delta }_0$.

Theorem 7.

Suppose that $\Gamma \in \Xi$ and $\Lambda :V^n⟶Y$ is a function satisfying (11). If property (H3) in Theorem 4 is true for Λ, then there exists a solution $\mathcal{Q}\in \Xi$ of (10) such that

$$\begin{array}{c}\hfill T\left(\Gamma \left(v^{\left[n\right]}\right)-\mathcal{Q}\left(v^{\left[n\right]}\right),t\right)\ge M\left(v^{\left[n\right]},{\displaystyle \frac{r^{2n}-\alpha }{r^{2n}}}t\right),\end{array}$$

(24)

for all $v^{\left[n\right]}\in V^n$ and $t>0$, where $M(v^{\left[n\right]},t)$ is defined in (13). In addition, $\mathcal{Q}$ is a unique GM-QM if it has property (H2).

 Proof. 

Consider the set

$$\Delta :=\{\Gamma \in \Xi |\Gamma \mathrm{has}\mathrm{the}\mathrm{property}\left(\mathrm{H}1\right)\}.$$

Let us define a mapping $d_F:V^n\times V^n⟶[0,\infty ]$ via

$$d_F(\varphi ,\psi ):=\inf \left\{\mu \in [0,\infty ):T(\varphi \left(v^{\left[n\right]}\right)-\psi \left(v^{\left[n\right]}\right),\mu t)\ge M(v,t),\forall v^{\left[n\right]}\in V^n,t>0\right\},$$

where, as usual, inf$\varnothing =+\infty$, for which $\varphi ,\psi \in \Delta$, where $M(v^{\left[n\right]},t)$ is defined in (13). It is verified that $d_F$ is a GM on $\Delta$. Moreover, $(\Delta ,d_F)$ is a complete GMS; see ([39] Theorem 2.6) and ([18] Theorem 2.1). Define the mapping $\mathcal{T}:\Delta ⟶\Delta$ through

$$\mathcal{T}\Gamma \left(v^{\left[n\right]}\right):={\displaystyle \frac{1}{r^{2n}}}\Gamma \left(r{v}^{\left[n\right]}\right),$$

for all $v^{\left[n\right]}\in V^n$. Take $\varphi ,\psi \in \Delta$, $v^{\left[n\right]}\in V^n$ and $\mu \in [0,\infty ]$, with $d_F(\varphi ,\psi )\le \mu$. Then, $T(\varphi \left(v^{\left[n\right]}\right)-\psi \left(v^{\left[n\right]}\right),\mu t)\ge M(v^{\left[n\right]},t)$, and so by the property (17), we have

$$\begin{array}{cc}\hfill T\left(\mathcal{T}\varphi \left(v^{\left[n\right]}\right)-\mathcal{T}\psi \left(v^{\left[n\right]}\right),{\displaystyle \frac{\alpha }{r^{2n}}}\mu t\right)& =T\left({\displaystyle \frac{1}{r^{2n}}}\varphi \left(r{v}^{\left[n\right]}\right)-{\displaystyle \frac{1}{r^{2n}}}\psi \left(r{v}^{\left[n\right]}\right),{\displaystyle \frac{\alpha }{2^n}}\mu t\right)\hfill \\ \hfill & =T\left(\varphi \left(r{v}^{\left[n\right]}\right)-\psi \left(r{v}^{\left[n\right]}\right),\alpha \mu t\right)\hfill \\ \hfill & \ge M\left(r{v}^{\left[n\right]},\alpha t\right)=M(v^{\left[n\right]},t),\hfill \end{array}$$

for all $v^{\left[n\right]}\in V^n$ and $t>0$. Therefore, $d_F(\mathcal{T}\varphi ,\mathcal{T}\psi )\le {\displaystyle \frac{\alpha }{r^{2n}}}\mu$. This shows that $d(\mathcal{T}\varphi ,\mathcal{T}\psi )\le {\displaystyle \frac{\alpha }{r^{2n}}}d(\varphi ,\psi )$. By the proof of Theorem 4, we obtain

$$\begin{array}{c}\hfill T\left({\displaystyle \frac{\Gamma \left(r{v}^{\left[n\right]}\right)}{r^{2n}}}-\Gamma \left(v^{\left[n\right]}\right),t\right)\ge M(v^{\left[n\right]},t),\end{array}$$

for all $v^{\left[n\right]}\in V^n$ and $t>0$, where $M(v^{\left[n\right]},t)$ is defined in (13). Therefore, $d_F(\mathcal{T}\Gamma ,\Gamma )\le 1.$ By Theorem 6, we find that the sequence ${({\mathcal{T}}^m\Gamma )}_{m\in \mathbb{N}}$ is convergent in $(\Delta ,d_F)$ and its pointwise limit, namely, $\mathcal{Q}$, is an FP of $\mathcal{T}$, and additionally $\mathcal{Q}\left(v^{\left[n\right]}\right)={\displaystyle \frac{1}{r^{2n}}}\mathcal{Q}\left(r{v}^{\left[n\right]}\right)$. Moreover, we have $d_F\left({\mathcal{T}}^{m}f,\mathcal{Q}\right)\to 0$, which implies that

$$\begin{array}{c}\hfill \mathcal{Q}\left(v^{\left[n\right]}\right)=T-\underset{l\to \infty }{lim}{\displaystyle \frac{\Gamma \left(r^m{v}^{\left[n\right]}\right)}{r^{2mn}}},\end{array}$$

for all $v^{\left[n\right]}\in V^n$. Furthermore, $d_F(\Gamma ,\mathcal{Q})\le {\displaystyle \frac{1}{1-L}}{d}_F(\Gamma ,\mathcal{T}\Gamma )$, which necessitates that

$$d_F(\Gamma ,\mathcal{Q})\le {\displaystyle \frac{1}{1-\frac{\alpha }{r^{2n}}}}={\displaystyle \frac{r^{2n}}{r^{2n}-\alpha }}.$$

If $\left\{{\delta }_l\right\}$ is a decreasing sequence converging to ${\displaystyle \frac{r^{2n}}{r^{2n}-\alpha }}$, then

$$T\left(\Gamma \left(v^{\left[n\right]}\right)-\mathcal{Q}\left(v^{\left[n\right]}\right),{\delta }_{l}t\right)\ge M(v,t),$$

for all $v^{\left[n\right]}\in V^n$, $t>0$, and $l\in \mathbb{N}$. From this we conclude that

$$T\left(\Gamma \left(v^{\left[n\right]}\right)-\mathcal{Q}\left(v^{\left[n\right]}\right),t\right)\ge M\left(v^{\left[n\right]},{\displaystyle \frac{t}{{\delta }_l}}\right),$$

for all $v^{\left[n\right]}\in V^n$, $t>0$, and $l\in \mathbb{N}$. Since M is left continuous by (T6), we deduce that

$$T\left(\Gamma \left(v^{\left[n\right]}\right)-\mathcal{Q}\left(v^{\left[n\right]}\right),t\right)\ge M\left(v^{\left[n\right]},{\displaystyle \frac{r^{2n}}{r^{2n}-\alpha }}\right),$$

for all $v^{\left[n\right]}\in V^n$, $t>0$, and $l\in \mathbb{N}$. Here, it can be proven like in Theorem 4 that $\mathcal{Q}$ is a solution of (10). In addition, the uniqueness of $\mathcal{Q}$ follows from the fact that $\mathcal{Q}$ is the unique FP of $\mathcal{T}$ with the property that there exists $\zeta \in (0,\infty )$ such that

$$T\left(\Gamma \left(v^{\left[n\right]}\right)-\mathcal{Q}\left(v^{\left[n\right]}\right),\zeta t\right)\ge M(v^{\left[n\right]},t),$$

for all $v^{\left[n\right]}\in V^n$ and $t>0$. This completes the proof. □

The proof of Theorem 7 shows that for proving the uniqueness of the solution, we do not need the property (H2) for a $f\in \Xi$, and this property is used only for being a GM-QM, and so the property (H2) is a redundant condition in Theorem 7.

The upcoming theorem is analogous to Theorem 7 in which we find a different approximation for GM-QEs in FBSs. Since the proof is similar, it is omitted.

Theorem 8. 

Suppose that $\Gamma \in \Xi$ and $\Lambda :V^n⟶Y$ is a function satisfying (11). If hypothesis (H3) holds for Λ, then there exists a solution $\mathcal{Q}\in \Xi$ of (10) such that

$$\begin{array}{c}\hfill T\left(\Gamma \left(v^{\left[n\right]}\right)-\mathcal{Q}\left(v^{\left[n\right]}\right),t\right)\ge N\left(v^{\left[n\right]},{\displaystyle \frac{\alpha -r^{2n}}{\alpha }}t\right),\end{array}$$

for all $v^{\left[n\right]}\in V^n$ and $t>0$, where $N(v^{\left[n\right]},t)$ is defined in (20).

Remark 1. 

In view of the results in this section, we deduce that the FP method give us a more exact approximation in comparison to the direct method when used in Theorems 4 and 5. For the accuracy, we return to property (T5) of Definition 3, which states that $T(v,·)$ is a non-decreasing function on $\mathbb{R}$ and ${\lim }_{t\to \infty }T(v,t)=1$. In light of the obtained results in Theorems 4 and 7, we observe that ${\displaystyle \frac{r^{2n}-\alpha }{r^{2n}}}t$ is greater than ${\displaystyle \frac{r^{2n}-\alpha }{2r^{2n}}}t$ and so the fuzzy difference in Theorem 7 is smaller.

4. Conclusions and Future Works

We have introduced generalized multi-quadratic mappings as a system of symmetric equations, which differs from earlier ones that were previously available in the literature and are quadratic in each component. We have expressed such mappings in a single equation. Note that the benefit of the presentation of such systems as a single equation is that it allows us to prove the H-U-R-Stab of the M-QMs by means of only one single functional inequality. We also used the fixed-point methodology and the direct approach (Hyers) method to generate a number of stability findings for a system of generalized quadratic functional equations in the setting of fuzzy norm spaces. Moreover, we compared the aforementioned methods and found two different results. Here, we demonstrate to the reader how to define the new multiple variable mappings by means of various FEs, to represent them as one single equation, and present an investigation of their varied stability in the setting of fuzzy norm spaces.