Ulam Type Stability of ?-Quadratic Mappings in Fuzzy Modular ∗-Algebras

Abstract

In this paper, we find the solution of the following quadratic functional equation $n{\sum }_{1\le i<j\le n}Q\left(x_i-x_j\right)={\sum }_{i=1}^{n}Q\left({\sum }_{j\ne i}{x}_j-(n-1)x_i\right)$, which is derived from the gravity of the n distinct vectors $x_1,\cdots ,x_n$ in an inner product space, and prove that the stability results of the $\mathcal{A}$-quadratic mappings in $\mu$-complete convex fuzzy modular ∗-algebras without using lower semicontinuity and $\beta$-homogeneous property.

1. Introduction

A concept of stability in the case of homomorphisms between groups was formulated by S.M. Ulam [1] in 1940 in a talk at the University of Wisconsin. Let $(G_1,\ast )$ be a group and let $(G_2,\diamond ,d)$ be a metric group with the metric $d(·,·)$. Given $ϵ>0$, does there exist a $\delta \left(ϵ\right)>0$ such that if a mapping $h:G_1\to G_2$ satisfies the inequality

$$d\left(h\right(x\ast y),h(x)\diamond h(y\left)\right)<\delta$$

for all $x,y\in G_1$, then there is a homomorphism $H:G_1\to G_2$ with

$$d\left(h\right(x),H(x\left)\right)<ϵ$$

for all $x\in G_1?$

The first affirmative answer to the question of Ulam was given by Hyers [2,3] for the Cauchy functional equation in Banach spaces as follows: Let X and Y be Banach spaces. Assume that $f:X\to Y$ satisfies

$$\parallel f(x+y)-f\left(x\right)-f\left(y\right)\parallel \le \epsilon$$

for all $x,y\in X$ and for some $\epsilon \ge 0.$ Then, there exists a unique additive mapping $T:X\to Y$ such that

$$\parallel f\left(x\right)-T\left(x\right)\parallel \le \epsilon$$

for all $x\in X.$ A number of mathematicians were attracted to this result and stimulated to investigate the stability problems of various(functional, differential, difference, integral) equations in some spaces [4,5,6,7,8,9,10,11].

In 2007, Nourouzi [12] presented probabilistic modular spaces related to the theory of modular spaces. Fallahi and Nourouzi [13,14] investigated the continuity and boundedness of linear operators defined between probabilistic modular spaces in the probabilistic sense. After then, Shen and Chen [15] following the idea of probabilistic modular spaces and the definition of fuzzy metric spaces based on George and Veeramani’s sense [16], applied fuzzy concept to the classical notions of modular spaces. Using Khamsi’s fixed point theorem in modular spaces [17], Wongkum and Kumam [18] proved the stability of sextic functional equations in fuzzy modular spaces equipped necessarily with lower semicontinuity and $\beta$-homogeneous property.

In a recent paper [11], Ulam stability of the following additive functional equation

$$\sum _{\genfrac{}{}{0pt}{}{1\le i_1<\cdots <i_m\le n}{1\le k_l(\ne i_j,\forall j\in \{1,\cdots ,m\})\le n}}f\left(\frac{{\sum }_{j=1}^m{x}_{i_j}}{m}+\sum _{l=1}^{n-m}{x}_{k_l}\right)=\frac{n-m+1}{n}\left(\genfrac{}{}{0pt}{}{n}{m}\right)\sum _{i=1}^{n}f\left(x_i\right).$$

was investigated in modular algebras without using the lower semicontinuity and Fatou preperty.

In the present paper, concerning the stability problem for the following functional equation

$$\begin{array}{c}\hfill n\sum _{1\le i<j\le n}f\left(x_i-x_j\right)=\sum _{i=1}^{n}f\left(\sum _{j\ne i}{x}_j-(n-1)x_i\right)\end{array}$$

which is derived from the gravity of the n-distinct vectors in an inner product space, we investigate the stability problem for $\mathcal{A}$-quadratic mappings in $\mu$-complete convex fuzzy modular ∗-algebras of the following functional equation without using lower semicontinuity and $\beta$-homogeneous property.

2. Preliminaries

Proposition 1.

Let $X_1,X_2,\cdots ,X_n$$(n\ge 3)$ be distinct vectors in a finite n-dimensional Euclidean space $E.$ Putting $G:=\frac{{\sum }_{i=1}^n{X}_i}{n}$, the gravity of the n distinct vectors, then we get the following identity

$$\begin{array}{c}\hfill \sum _{1\le i<j\le n}\parallel \overrightarrow{X_i{X}_j}{\parallel }^2=n\sum _{i=1}^n{\parallel \overrightarrow{X_{i}G}\parallel }^2,\end{array}$$

which is equivalent to the equation

$$\begin{array}{c}\hfill n\sum _{1\le i<j\le n}{∥X_i-X_j∥}^2=\sum _{i=1}^n{∥\sum _{j\ne i}{X}_j-(n-1)X_i∥}^2\end{array}$$

(1)

for any distinct vectors $X_1,X_2,\cdots ,X_n$.

Employing the above equality (1), we introduce the new functional equation:

$$\begin{array}{c}\hfill n\sum _{1\le i<j\le n}Q\left(x_i-x_j\right)=\sum _{i=1}^{n}Q\left(\sum _{j\ne i}{x}_j-(n-1)x_i\right)\end{array}$$

(2)

for a mapping $Q:U\to V$ and for all vectors $x_1,\cdots ,x_n\in U,$ where U and V are linear spaces and $n\ge 3$ is a positive integer.

From now on, we introduce some basic definitions of fuzzy modular ∗-algebras.

Definition 1.

[18] A triangular norm (briefly, t-norm) is a function $\circ :[0,1]\times [0,1]\to [0,1]$ satisfies the following conditions:

 (1) 

$\circ$ is commutative, associative;

 (2) 

$a\circ 1=a$;

 (3) 

$a\circ b\le c\circ d$, whenever $a,b,c,d\in [0,1]$ with $a\le b,c\le d$.

Three common examples of the t-norm are (1) $a{\circ }_{M}b=min\{a,b\}$; (2) $a{\circ }_{p}b=a·b$; (3) $a{\circ }_{L}b=max\{a+b-1,0\}$. For more example, we refer to [19]. Throughout this paper, we denote that

$$\begin{array}{c}\hfill \prod _{i=1}^n{x}_i:=x_1\circ \cdots \circ x_n\end{array}$$

for all $x_1,\cdots ,x_n\in [0,1].$

Definition 2.

[18] Let X be a complex vector space and $\circ$ a t-norm, and $\mu :X\times (0,\infty )\to [0,1]$ be a function.

(a) The triple $(X,\mu ,\circ )$ is said to be a fuzzy modular space if, for each $x,y\in X$ and $s,t>0$ and $\alpha ,\beta \in [0,\infty )$ with $\alpha +\beta =1$,

 (FM1) 

$\mu (x,t)>0$;

 (FM2) 

$\mu (x,t)=1$ for all $t>0$ if and only if $x=\theta$;

 (FM3) 

$\mu (x,t)=\mu (-x,t);$

 (FM4) 

$\mu (\alpha x+\beta y,s+t)\ge \mu (x,s)\circ \mu (y,t);$

 (FM5) 

the mapping $t\to \mu (x,t)$ is continuous at each fixed $x\in X$;

(b) alternatively, if (FM-4) is replaced by

 (FM4-1) 

$\mu (\alpha x+\beta y,s+t)\ge \mu (x,\frac{s}{\alpha })\circ \mu (y,\frac{t}{\beta }),(\mathit{where}\alpha ,\beta \ne 0);$

then we say that $(X,\mu ,\circ )$ is a convex fuzzy modular.

Now, we extend the properties (FM4) and (FM4-1) in real fields to complex scalar field acting on the space X, as follows:

(FM4)’

$\mu (\alpha x+\beta y,s+t)\ge \mu (x,s)\circ \mu (y,t);$ for $\alpha ,\beta \in \mathbb{C}$ with $\left|\alpha \right|+\left|\beta \right|=1$,

(FM4-1)’

$\mu (\alpha x+\beta y,s+t)\ge \mu (x,\frac{s}{\alpha })\circ \mu (y,\frac{t}{\beta })$ for $\alpha ,\beta \in \mathbb{C}$ with $\left|\alpha \right|+\left|\beta \right|=1$.

Next, we introduce the concept of fuzzy modular algebras based on the deifnition of fuzzy normed algebras [20,21]. If X is algebra with fuzzy modular $\mu$ subject to $\mu (xy,st)\ge \mu (x,s)\circ \mu (y,t)$ for all $x,y\in X$ and $s,t\in (0,\infty )$, then we say $(X,\mu ,\circ )$ is called a fuzzy modular algebra. In addition, a fuzzy modular algebra X is a fuzzy modular ∗-algebra if the fuzzy modular $\mu$ satisfies $\mu (z^{\ast },t)=\mu (z,t)$ for all $z\in X,t>0$.

Example 1.

Let $(X,\rho )$ be a modular ∗-algebra ([22]) and $\circ$ defined by $a\circ b:=a{\circ }_{M}b$. For every $t\in (0,\infty )$, define $\mu (x,t)=\frac{t}{t+\rho \left(x\right)}$ for all $x\in X.$ Then, $(X,\mu ,\circ )$ is a (convex) fuzzy modular ∗-algebra.

Definition 3.

(1). We say that $(X,\mu ,\circ )$ is β-homogeneous if, for every $x\in X,t>0$ and $\lambda \in \mathbb{R}\backslash \left\{0\right\}$,

$$\begin{array}{c}\hfill \mu (\lambda x,t)=\mu \left(x,{\displaystyle \frac{t}{{\left|\lambda \right|}^{\beta }}}\right),\mathit{where}\beta \in (0,1].\end{array}$$

(2). Let $n\in \mathbb{N}$. We say that $(X,\mu ,\circ )$ satisfies ${\Delta }_n$-condition if there exist ${\kappa }_n\ge n$ such that

$$\begin{array}{c}\hfill \mu (nx,t)\ge \mu \left(x,{\displaystyle \frac{t}{{\kappa }_n}}\right),\forall x\in X.\end{array}$$

Remark 1.

Let $(X,\mu ,\circ )$ be β-homogeneous for some fixed $\beta \in (0,1].$ Then, we observe that

$$\begin{array}{c}\hfill \mu (2x,t)=\mu \left(x,{\displaystyle \frac{t}{2^{\beta }}}\right)\ge \mu \left(x,{\displaystyle \frac{t}{{\kappa }_2}}\right)\end{array}$$

for all $x\in X$ and all ${\kappa }_2\ge 2\ge {\left|2\right|}^{\beta }$. Thus, β-homogeneous property implies ${\Delta }_2$-condition.

Example 2.

Let $\rho :\mathbb{R}\to \mathbb{R}, \mu :\mathbb{R}\times (0,\infty )\to (0,1]$ be defined by $\rho \left(x\right)=x^2$ and $\mu (x,t)=\frac{t}{t+\rho \left(x\right)}$. Then, we can check that $(\mu ,{\circ }_M)$ is a convex fuzzy modular on $\mathbb{R}$ but $(\mathbb{R},\mu ,{\circ }_M)$ does not satisfy β-homogeneous property. Let ${\kappa }_2\ge 4$. Then,

$$\begin{array}{c}\hfill \mu (2x,t)={\displaystyle \frac{t}{t+\rho \left(2x\right)}}=\mu \left(x,{\displaystyle \frac{t}{4}}\right)\ge \mu \left(x,{\displaystyle \frac{t}{{\kappa }_2}}\right)\end{array}$$

for all $x\in \mathbb{R}$. Thus, $(\mathbb{R},\mu ,\circ )$ satisfies ${\Delta }_2$-condition with ${\kappa }_2\ge 4$ but is not β-homogeneous.

Definition 4.

Let $(X,\mu ,\circ )$ be a fuzzy modular space and $\left\{x_n\right\}$ be a sequence in $X_{\rho }$.

 (1). 

$\left\{x_n\right\}$ is said to be μ-convergent to a point $x\in X$ if for any $t>0$,

$$\begin{array}{c}\hfill \mu (x-x_n,t)\to 1\end{array}$$

as $n\to \infty$.

 (2). 

$\left\{x_n\right\}$ is called μ-Cauchy if for each $\epsilon >0$ and each $t>0$, there exists $n_1$ such that, for all $n\ge n_1$ and all $p>0$, we have $\mu (x_{n+p}-x_n)>1-\epsilon .$

 (3). 

If each Cauchy sequence is convergent, then the fuzzy modular space is said to be complete.

3. Fuzzy Modular Stability for $\mathcal{A}$-Quadratic Mappings

First of all, we find out the general solution of (1.3) in the class of mappings between vector spaces.

Theorem 1.

Let U and V be vector spaces. A mapping $Q:U\to V$ satisfies the functional Equation (2) for each positive integer $n>2$ if and only if there exists a symmetric biadditive mapping $B:U\times U\to V$ such that $Q\left(x\right)=B(x,x)$ for all $x\in U.$

Proof. 

Let Q satisfy Equation (2). One finds that $Q\left(0\right)=0$ and $Q\left(ax\right)=a^{2}Q\left(x\right)$ by changing $(x,y)$ to $(0,0)$ and $(x,0)$ in (3), respectively, where $a:=n-1$ is a positive integer with $a\ge 2.$ Putting $x_1:=x$, $x_2:=y$ and $x_i:=0$ for all $i=3,\cdots ,n$ in (2), we get

$$\begin{array}{ccc}\hfill & & Q(x-ay)+Q(ax-y)+(a-1)Q(x+y)\hfill \\ \hfill & & =(a+1)Q(x-y)+(a^2-1)[Q\left(x\right)+Q\left(y\right)]\hfill \end{array}$$

(3)

for all $x,y\in U$. Using [23] [Theorem 1], we obtain that Q is a generalized polynomial map of degree at most 4. Therefore,

$$\begin{array}{c}\hfill Q\left(x\right)=A_0+A_1\left(x\right)+A_2(x,x)+A_3(x,x,)+A_4(x,x,x,x)\end{array}$$

for all $x\in U$, where $A_k:U^k\to V$ is a k-additive symmetric map ($k=1,\cdots ,4$) and $A_0\in V$. Since a is an integer, we get

$$\begin{array}{c}\hfill (a^2-1)A_0+(a^2-a)A_1\left(x\right)+(a^2-a^3)A_3(x,x,x)+(a^2-a^4)A_4(x,x,x,x)=0\end{array}$$

for all $x\in U$ by $Q\left(ax\right)=a^{2}Q\left(x\right)$. This yields that $Q\left(x\right)=A_2(x,x)$ for all $x\in U$. □

Let $\mathcal{A}$ be a complex ∗-algebra with unit and let M be a left $\mathcal{A}$-module. We call a mapping $Q:M\to \mathcal{A}$ an $\mathcal{A}$-quadratic mapping if both relations $Q\left(ax\right)=aQ\left(x\right)a^{\ast }$ and $Q(x+y)+Q(x-y)=2Q\left(x\right)+2Q\left(y\right)$ are fulfilled for all $a\in \mathcal{A},x,y\in M$ [24]. For the sake of convenience, we define the following:

$$\begin{array}{ccc}\hfill {\mathcal{D}}_{u}f(x_1,\cdots ,x_n)& :=& n\sum _{1\le i<j\le n}f(u{x}_i-u{x}_j)-\sum _{i=1}^{n}uf\left(\sum _{j\ne i}{x}_j-(n-1)x_i\right)u^{\ast },\hfill \\ \hfill {\epsilon }_i\left(x\right)& :=& \epsilon (0,\cdots ,0,\underset{i-th}{\underbrace{x}},0,\cdots ,0),\hfill \\ \hfill \mathcal{J}& :=& \left\{\begin{array}{cc}\{1,\cdots ,n-1\}\times \{1,\cdots ,n\},\hfill & \mathrm{if}n>3,\hfill \\ \left\{2\right\}\times \{1,2,3\}\hfill & \mathrm{if}n=3.\hfill \end{array}\right.\hfill \end{array}$$

In addition, let $\circ$ be defined by minimum t-norm and ${\mathcal{A}}^M$ be the set of all mapping from M to $\mathcal{A}$, $Q_{\mathcal{A}}(M,\mathcal{A})$ be the set of all $\mathcal{A}$-quadratic mappings from M to $\mathcal{A}$.

Now, we present a stability of the $\mathcal{A}$-quadratic mapping concerning Equation (2) in $\mu$-complete convex fuzzy modular ∗-algebras without using $\beta$-homogeneous properties.

Theorem 2.

Let $(\mathcal{A},\mu ,\circ )$ be μ-complete convex fuzzy modular ∗-algebra with norm $\parallel ·\parallel$ and M be a left $\mathcal{A}$-module, $(X,{\mu }^{\prime },\circ )$ fuzzy modular space, $\mathcal{U}\left(\mathcal{A}\right)$ the unitary group of $\mathcal{A}$. Assume that there exist two mappings $f\in {\mathcal{A}}^M$ and $\epsilon \in X^{M^n}$ such that

$$\begin{array}{ccc}\hfill \mu ({\mathcal{D}}_{u}f(x_1,\cdots ,x_n),t)& \ge & {\mu }^{\prime }(\epsilon (x_1,\cdots ,x_n),t),\hfill \\ \hfill {\mu }^{\prime }\left(\epsilon ((n-1)x_1,\cdots ,(n-1)x_n),t\right)& \ge & {\mu }^{\prime }\left(\epsilon (x_1,\cdots ,x_n),{\displaystyle \frac{t}{\beta }}\right)\hfill \end{array}$$

(4)

for all $(x_1,\cdots ,x_n)\in X^n,u\in \mathcal{U}\left(\mathcal{A}\right)$, where $2\le 2\beta <{(n-1)}^2$, and either f is measurable or $f\left(tx\right)$ is continuous in $t\in R$ for each fixed $x\in M$. Then, there exists a unique mapping $Q\in Q_{\mathcal{A}}(M,\mathcal{A})$ that satisfies Equation (2) and the inequality

$$\begin{array}{c}\hfill \mu \left(f\left(x\right)+{\displaystyle \frac{(n-1)f\left(0\right)}{2}}-Q\left(x\right),t\right)\ge \mathsf{\Phi }\left(x,{\displaystyle \frac{{(n-1)}^{2}t}{2\beta }}\right)\end{array}$$

(5)

for all $x\in M$ and $t>0$, where

$$\begin{array}{ccc}\hfill \mathsf{\Phi }(x,t)& :=& \underset{(i,j)\in \mathcal{J}}{max}\{{\mu }^{\prime }\left({\epsilon }_j(-x),{\displaystyle \frac{{(n-1)}^{2}t}{6}}\right)\circ {\mu }^{\prime }\left({\epsilon }_i\left(x\right),{\displaystyle \frac{{(n-1)}^{2}nt}{6(n^2-(i+1)n+1)}}\right)\hfill \\ & & \circ {\mu }^{\prime }\left({\epsilon }_{i+1}\left(x\right),{\displaystyle \frac{{(n-1)}^{2}nt}{6(n^2-(i+1)n+1)}}\right)\}.\hfill \end{array}$$

Proof. 

Define a mapping $g:M\to \mathcal{A}$ by $g\left(x\right):=f\left(x\right)+\frac{(n-1)f\left(0\right)}{2}$ for all $x\in M$. Then, for each $x\in M,$ the following equation is obtained:

$$\begin{array}{c}\hfill g\left((n-1)x\right)-{(n-1)}^{2}g\left(x\right)={\mathcal{D}}_u{f}_j(-x)+\left({\displaystyle \frac{n^2-(i+1)n+1}{n}}\right)[{\mathcal{D}}_1{f}_i\left(x\right)-{\mathcal{D}}_1{f}_{i+1}\left(x\right)]\end{array}$$

(6)

for all $i=1,\cdots ,n-1$ and for all $j=1,\cdots ,n$, where

$$\begin{array}{c}\hfill {\mathcal{D}}_1{f}_i\left(x\right)={\mathcal{D}}_{1}f(0,\cdots ,0,\underset{i-th}{\underbrace{x}},0,\cdots ,0).\end{array}$$

For each fixed $(i,j)\in \mathcal{J}$, one obtains from ${\sum }_{k=1}^m\frac{1+2\left(\frac{n^2-(i+1)n+1}{n}\right)}{{(n-1)}^{2k}}\le 1$ that

$$\begin{array}{ccc}\hfill \mu \left(g\left(x\right)-{\displaystyle \frac{g\left({(n-1)}^{m}x\right)}{{(n-1)}^{2m}}},t\right)& \ge & \mu \left(\sum _{k=1}^m{\displaystyle \frac{{(n-1)}^{2}g\left({(n-1)}^{k-1}x\right)-g\left({(n-1)}^{k}x\right)}{{(n-1)}^{2k}}},\sum _{k=1}^m{\displaystyle \frac{t}{2^k}}\right)\hfill \\ \hfill & \ge & \prod _{k=1}^m({\mu }^{\prime }({\epsilon }_j(-x),{\displaystyle \frac{{(n-1)}^{2k}t}{3·2^k{\beta }^{k-1}}})\hfill \\ & & \circ {\mu }^{\prime }\left({\epsilon }_i\left(x\right),{\displaystyle \frac{{(n-1)}^{2k}nt}{3·2^k{\beta }^{k-1}(n^2-(i+1)n+1)}}\right)\hfill \\ \hfill & & \circ {\mu }^{\prime }\left({\epsilon }_{i+1}\left(x\right),{\displaystyle \frac{{(n-1)}^{2k}nt}{3·2^k{\beta }^{k-1}(n^2-(i+1)n+1)}}\right))\hfill \\ \hfill & =& {\mu }^{\prime }\left({\epsilon }_j(-x),{\displaystyle \frac{{(n-1)}^{2}t}{6}}\right)\circ {\mu }^{\prime }\left({\epsilon }_i\left(x\right),{\displaystyle \frac{{(n-1)}^{2}nt}{6(n^2-(i+1)n+1)}}\right)\hfill \\ \hfill & & \circ {\mu }^{\prime }\left({\epsilon }_{i+1}\left(x\right),{\displaystyle \frac{{(n-1)}^{2}nt}{6(n^2-(i+1)n+1)}}\right)\hfill \end{array}$$

for all $t>0$ and $x\in M$, $m\in \mathbb{N}$. Then, it follows from the above inequality that

$$\begin{array}{c}\hfill \mu \left(g\left(x\right)-{\displaystyle \frac{g\left({(n-1)}^{m}x\right)}{{(n-1)}^{2m}}},t\right)\ge \mathsf{\Phi }(x,t)\end{array}$$

for all $x\in M$ and $t>0.$ Therefore, we prove from this relation that, for any integers $m,p$,

$$\begin{array}{ccc}\hfill \mu \left({\displaystyle \frac{g\left({(n-1)}^{m}x\right)}{{(n-1)}^{2m}}}-{\displaystyle \frac{g\left({(n-1)}^{m+p}x\right)}{{(n-1)}^{2(m+p)}}},t\right)& \ge & \mu \left(g\left({(n-1)}^{m}x\right)-{\displaystyle \frac{g({(n-1)}^p·{(n-1)}^{m}x)}{{(n-1)}^p}},{(n-1)}^{2m}t\right)\hfill \\ & \ge & \mathsf{\Phi }({(n-1)}^{m}x,{(n-1)}^{2m}t)\ge \mathsf{\Phi }\left(x,{\left({\displaystyle \frac{{(n-1)}^2}{\beta }}\right)}^{m}t\right)\hfill \end{array}$$

for all $t>0,x\in M.$ Since the right-hand side of the above inequality tends to 1 as $m\to \infty$, the sequence $\left\{\frac{g\left({(n-1)}^{m}x\right)}{{(n-1)}^{2m}}\right\}$ is $\mu$-Cauchy and thus converges in $\mathcal{A}$. Hence, we may define a mapping $Q:M\to \mathcal{A}$ as

$$\begin{array}{c}\hfill Q\left(x\right):=\mu -\underset{m\to \infty }{lim}{\displaystyle \frac{g\left({(n-1)}^{m}x\right)}{{(n-1)}^{2m}}}\left(\iff \underset{m\to \infty }{lim}\mu \left(Q\left(x\right)-{\displaystyle \frac{g\left({(n-1)}^{m}x\right)}{{(n-1)}^{2m}}},t\right)=1\right)\end{array}$$

for all $x\in M$ and $t>0.$ In addition, we claim that the mapping Q satisfies (2). For this purpose, we calculate the following inequality:

$$\begin{array}{ccc}\hfill \mu \left({\displaystyle \frac{{\mathcal{D}}_{u}Q(x_1,\cdots ,x)}{L}},t\right)& \ge & \prod _{1\le i<j\le n}(\mu \left(Q(u{x}_i-u{x}_j)-{\displaystyle \frac{g\left({(n-1)}^m(u{x}_i-u{x}_j)\right)}{{(n-1)}^{2m}}},{\displaystyle \frac{Lt}{2^{i+j}n}}\right)\hfill \\ & & \circ \mu \left(uQ(\sum _{j=1}^n{x}_j-n{x}_i)u^{\ast }-{\displaystyle \frac{ug\left({(n-1)}^m({\sum }_{j=1}^n{x}_j-n{x}_i)\right)u^{\ast }}{{(n-1)}^{2m}}},{\displaystyle \frac{Lt}{2^{i+j}}}\right))\hfill \\ & & \circ {\mu }^{\prime }\left(\epsilon (x_1,\cdots ,x_n),{\left({\displaystyle \frac{{(n-1)}^2}{\beta }}\right)}^m·{\displaystyle \frac{Lt}{2^{i+j}}}\right)\hfill \end{array}$$

for all $x\in M,u\in \mathcal{U}\left(\mathcal{A}\right),m\in \mathbb{N},t>0$, where $L:={\displaystyle \frac{n^3-n^2+2n+2}{2}}$. This means that ${\mathcal{D}}_{u}Q(x_1,\cdots ,x_n)=0$ for all $x_1,\cdots x_n\in M,u\in \mathcal{U}\left(\mathcal{A}\right)$. Hence, the mapping Q satisfies (2) and so $Q\left((n-1)x\right)={(n-1)}^{2}Q\left(x\right)$ for all $x\in M$. It follows that

$$\begin{array}{ccc}\hfill \mu \left(Q\left(x\right)-g\left(x\right),t\right)& \ge & \mu ({\displaystyle \frac{Q\left(\right(n-1\left)x\right)}{{(n-1)}^2}}-{\displaystyle \frac{g\left({(n-1)}^{m+1}x\right)}{{(n-1)}^{2m+2}}}\hfill \\ & & +\sum _{k=1}^m{\displaystyle \frac{{(n-1)}^{2}g\left({(n-1)}^{k-1}x\right)-g\left({(n-1)}^{k}x\right)}{{(n-1)}^{2k}}},t)\hfill \\ & \ge & \mathsf{\Phi }\left((n-1)x,{\displaystyle \frac{{(n-1)}^{2}t}{2}}\right)\circ \prod _{k=1}^m\mathsf{\Phi }\left(x,{\displaystyle \frac{{(n-1)}^{2k}t}{2{\beta }^{k-1}}}\right)\hfill \\ & \ge & \mathsf{\Phi }\left(x,{\displaystyle \frac{{(n-1)}^2}{2\beta }}t\right)\hfill \end{array}$$

for all $x\in M,t>0$.

To prove the uniqueness, let $Q^{\prime }$ be another mapping satisfying (2) and

$$\begin{array}{c}\hfill \mu \left(g\left(x\right)-Q^{\prime }\left(x\right),t\right)\ge \mathsf{\Phi }\left(x,{\displaystyle \frac{{(n-1)}^2}{2\beta }}t\right)\end{array}$$

for all $x\in M.$ Thus, we have

$$\begin{array}{ccc}\hfill \mu \left({\displaystyle \frac{1}{2}}\left(Q\left(x\right)-Q^{\prime }\left(x\right)\right),t\right)& \ge & \mu \left({\displaystyle \frac{Q\left({(n-1)}^{m}x\right)-g\left({(n-1)}^{m}x\right)}{{(n-1)}^{2m}}},t\right)\hfill \\ & & \circ \mu \left({\displaystyle \frac{g\left({(n-1)}^{m}x\right)-Q^{\prime }\left({(n-1)}^{m}x\right)}{{(n-1)}^{2m}}},t\right)\hfill \\ & \ge & \mathsf{\Phi }\left(x,{\displaystyle \frac{{(n-1)}^{2m}}{2\beta }}t\right)\hfill \end{array}$$

for all $x\in M,t>0.$ Taking the limit as $m\to \infty$, then we conclude that $Q\left(x\right)=Q^{\prime }\left(x\right)$ for all $x\in M.$

Under the assumption that either f is measurable or $f\left(tx\right)$ is continuous in $t\in \mathbb{R}$ for each fixed $x\in M$, the quadratic mapping Q satisfies $Q\left(tx\right)=t^{2}Q\left(x\right)$ for all $x\in M$ and for all $t\in \mathbb{R}$ by the same reasoning as the proof of [25]. That is, Q is $\mathbb{R}$-quadratic. Let $P:=\frac{n^4-2n^3+3n^2-3n+14}{4}$. Putting $x_1:=-{(n-1)}^{k}x$ and $x_i:=0$ for all $i=2,\cdots ,n$ in (4) and dividing the resulting inequality by ${(n-1)}^{2k},$ we have

$$\begin{array}{ccc}& & \mu \left({\displaystyle \frac{1}{P}}\left(n(n-1)Q(-ux)-uQ\left((n-1)x\right)u^{\ast }-(n-1)uQ(-x)u^{\ast }\right),4t\right)\hfill \\ & & \ge \mu \left(Q(-ux)-{\displaystyle \frac{g(-u{(n-1)}^{k}x)}{{(n-1)}^{2k}}},{\displaystyle \frac{Pt}{n(n-1)}}\right)\hfill \\ & & \circ \mu \left(uQ(-(n-1)x)u^{\ast }-u{\displaystyle \frac{g(-\lambda ·{(n-1)}^{k}x)}{{(n-1)}^{2k}}}{u}^{\ast },Pt\right)\hfill \\ & & \circ \mu \left(uQ\left(x\right)u^{\ast }-u{\displaystyle \frac{g\left({(n-1)}^{k}x\right)}{{(n-1)}^{2k}}}{u}^{\ast },{\displaystyle \frac{Pt}{n(n-1)}}\right)\hfill \\ & & \circ \mu \left({\mathcal{D}}_{u}f(-{(n-1)}^{k}x,0,\cdots ,0),{(n-1)}^{2k}Pt\right)\hfill \\ & & \circ \mu \left(f\left(0\right),{\displaystyle \frac{4{(n-1)}^{2k}Pt}{(n-2)(n-1)n(n+1)}}\right)\hfill \end{array}$$

for all $x\in M,u\in \mathcal{U}\left(\mathcal{A}\right),t>0.$ Taking $k\to \infty$ and using the evenness of Q, we obtain that $Q\left(ux\right)=uQ\left(x\right)u^{\ast }$ for all $x\in M$ and for each $u\in \mathcal{U}\left(\mathcal{A}\right).$ The last relation is also true for $u=0.$

Now, let a be a nonzero element in $\mathcal{A}$ and K a positive integer greater than $4\parallel a\parallel .$ Then, we have $\frac{\parallel a\parallel }{K}<\frac{1}{4}<1-\frac{2}{3}.$ By [26] [Theorem 1], there exist three elements $u_1,u_2,u_3\in \mathcal{U}\left(\mathcal{A}\right)$ such that $3\frac{a}{K}=u_1+u_2+u_3.$ Thus, we calculate in conjunction with [27] [Lemma 2.1] that

$$\begin{array}{ccc}\hfill Q\left(ax\right)& =& Q\left(\frac{K}{3}3\frac{a}{K}x\right)={\left(\frac{K}{3}\right)}^{2}Q(u_{1}x+u_{2}x+u_{3}x)\hfill \\ & =& {\left(\frac{K}{3}\right)}^{2}B(u_{1}x+u_{2}x+u_{3}x,u_{1}x+u_{2}x+u_{3}x)\hfill \\ & =& {\left(\frac{K}{3}\right)}^2(u_1+u_2+u_3)B(x,x)(u_1^{\ast }+u_2^{\ast }+u_3^{\ast })\hfill \\ & =& {\left(\frac{K}{3}\right)}^{2}3\frac{a}{K}Q\left(x\right)3\frac{a^{\ast }}{K}=aQ\left(x\right)a^{\ast }\hfill \end{array}$$

for all $a\in \mathcal{A}(a\ne 0)$ and for all $x\in M.$ Thus, the unique $\mathbb{R}$-quadratic mapping Q is also $\mathcal{A}$-quadratic, as desired. This completes the proof. □

Corollary 1.

Let $(\mathcal{A},\rho )$ be a ρ-complete convex modular ∗-algebra with norm $\parallel ·\parallel$ and M be a left $\mathcal{A}$-module, $\mathcal{U}\left(\mathcal{A}\right)$ the unitary group of $\mathcal{A}$. Assume that there exist two mappings $f\in {\mathcal{A}}^M$ and $\epsilon \in {\mathbb{R}}^{M^n}$ such that

$$\begin{array}{ccc}\hfill \rho \left({\mathcal{D}}_{u}f(x_1,\cdots ,x_n)\right)& \le & \epsilon (x_1,\cdots ,x_n),\hfill \\ \hfill \epsilon ((n-1)x_1,\cdots ,(n-1)x_n)& \le & \beta \epsilon (x_1,\cdots ,x_n)\hfill \end{array}$$

(7)

for all $(x_1,\cdots ,x_n)\in X^n,u\in \mathcal{U}\left(\mathcal{A}\right)$, where $2\le 2\beta <{(n-1)}^2$, and either f is measurable or $f\left(tx\right)$ is continuous in $t\in \mathbb{R}$ for each fixed $x\in M$. Then, there exists a unique mapping $Q\in Q_{\mathcal{A}}(M,\mathcal{A})$ which satisfies Equation (2) and the inequality

$$\begin{array}{ccc}\hfill \rho \left(f\left(x\right)+{\displaystyle \frac{(n-1)f\left(0\right)}{2}}-Q\left(x\right)\right)& \le & {\displaystyle \frac{12\beta }{{(n-1)}^4}}\underset{(i,j)\in \mathcal{J}}{min}\{max\{{\epsilon }_j(-x),{\displaystyle \frac{(n^2-(i+1)n+1)}{n}}{\epsilon }_i\left(x\right)\hfill \\ & & ,{\displaystyle \frac{(n^2-(i+1)n+1)}{n}}{\epsilon }_{i+1}\left(x\right)\left\}\right\}\hfill \end{array}$$

(8)

for all $x\in M$.

Proof. 

Let $X=\mathbb{R}$ with the fuzzy modular ${\mu }^{\prime }:X\times (0,\infty )\to \mathbb{R}$ as

$$\begin{array}{c}\hfill {\mu }^{\prime }(z,t)={\displaystyle \frac{t}{t+\left|z\right|}}\end{array}$$

for all $z\in \mathbb{R},t>0.$ In addition, define the following convex fuzzy modular $\mu$ as

$$\begin{array}{c}\hfill \mu (y,t)={\displaystyle \frac{t}{t+\rho \left(y\right)}},\end{array}$$

for all $y\in M,t>0.$ As noted in Example 1, $(\mathcal{A},\mu ,{\circ }_M)$ is a $\mu$-complete convex fuzzy modular ∗-algebra and $(\mathbb{R},{\mu }^{\prime }.{\circ }_M)$ is a fuzzy modular space. The result follows from the fact that (4) and (5) are equivalent to (7) and (8), respectively. □

Corollary 2.

Let $(\mathcal{A},\parallel ·\parallel )$ be a Banach ∗-algebra and M be a left $\mathcal{A}$-module and $\theta >0,p\in (0,2-{log}_{\lambda }2)$. Assume that there exists a mapping $f\in {\mathcal{A}}^M$ such that

$$\begin{array}{ccc}& & \parallel {\mathcal{D}}_{u}f(x_1,\cdots ,x_n)\parallel \le \theta (\parallel x_1{\parallel }^p+\cdots +\parallel x_n{\parallel }^p)\hfill \end{array}$$

for all $(x_1,\cdots ,x_n)\in X^n,u\in \mathcal{U}\left(\mathcal{A}\right)$, and either f is measurable or $f\left(tx\right)$ is continuous in $t\in \mathbb{R}$ for each fixed $x\in M$. Then, there exists a unique quadratic mapping $Q\in Q_{\mathcal{A}}(M,\mathcal{A})$ which satisfies Equation (2) and the inequality

$$\begin{array}{ccc}\hfill \parallel f\left(x\right)+{\displaystyle \frac{(n-1)f\left(0\right)}{2}}-Q\left(x\right)\parallel & \le & {\displaystyle \frac{12}{{(n-1)}^{4-p}}}\epsilon \theta {\parallel x\parallel }^p\hfill \end{array}$$

for all $x\in M$, where ε is a real number defined by

$$\begin{array}{ccc}\hfill \epsilon & :=& \left\{\begin{array}{cc}min\left\{{\displaystyle \frac{n^2-(i+1)n+1}{n}}\ge 1|i=1,\cdots ,n-1\right\},\hfill & ifn>3,\hfill \\ 1,\hfill & ifn=3.\hfill \end{array}\right.\hfill \end{array}$$

Proof. 

Letting $\epsilon (x_1,\cdots ,x_n):=\theta (\parallel x_1{\parallel }^p+\cdots +\parallel x_n{\parallel }^p)$, $\beta :={(n-1)}^p$ and applying Corollary 1, we obtain the desired result, as claimed. □

Next, we provide an alternative stability theorem of Theorem 2 equipped with ${\Delta }_{n-1}$-condition in $\mu$-complete convex fuzzy modular ∗-algebras.

Theorem 3.

Let $(\mathcal{A},\mu ,\circ )$ be a μ-complete convex fuzzy modular ∗-algebra with ${\Delta }_{n-1}$-condition and norm $\parallel ·\parallel$ and M be a $\mathcal{A}$-left module, $(X,{\mu }^{\prime },\circ )$ fuzzy modular space. Assume that there exist two mappings $f\in {\mathcal{A}}^M$ and $\epsilon \in X^{M^n}$ such that

$$\begin{array}{ccc}\hfill \mu ({\mathcal{D}}_{u}f(x_1,\cdots ,x_n),t)& \ge & {\mu }^{\prime }(\epsilon (x_1,\cdots ,x_n),t),\hfill \\ \hfill {\mu }^{\prime }\left(\epsilon \left({\displaystyle \frac{x_1}{n-1}},\cdots ,{\displaystyle \frac{x_n}{n-1}}\right),t\right)& \ge & {\mu }^{\prime }(\epsilon (x_1,\cdots ,x_n),\gamma t)\hfill \end{array}$$

(9)

for all $(x_1,\cdots ,x_n)\in X^n,u\in \mathcal{U}\left(\mathcal{A}\right)$, where ${(n-1)}^2\gamma >2{\kappa }_{n-1}^4$, and either f is measurable or $f\left(tx\right)$ is continuous in $t\in \mathbb{R}$ for each fixed $x\in M$. Then, there exists a unique mapping $Q\in Q_{\mathcal{A}}(M,\mathcal{A})$ which satisfies Equation (2) and the inequality

$$\begin{array}{c}\hfill \mu (f\left(x\right)-Q\left(x\right),t)\ge \mathsf{\Psi }\left(x,{\displaystyle \frac{(n-1)t}{2{\kappa }_{n-1}}}\right)\end{array}$$

(10)

for all $x\in M,t>0$, where

$$\begin{array}{ccc}\hfill \mathsf{\Psi }(x,t)& =& \underset{(i,j)\in \mathcal{J}}{max}\{{\mu }^{\prime }\left({\epsilon }_j\left(-x\right),{\displaystyle \frac{\gamma {(n-1)}^{2}t}{6{\kappa }_{n-1}^2}}\right)\circ {\mu }^{\prime }\left({\epsilon }_i\left(x\right),{\displaystyle \frac{\gamma {(n-1)}^{2}nt}{6{\kappa }_{n-1}^2(n^2-(i+1)n+1)}}\right)\hfill \\ & & \circ {\mu }^{\prime }\left({\epsilon }_{i+1}\left(x\right),{\displaystyle \frac{\gamma {(n-1)}^{2}nt}{6{\kappa }_{n-1}^2(n^2-(i+1)n+1)}}\right)\}.\hfill \end{array}$$

Proof. 

Letting $(x_1,\cdots ,x_n):=(0,\cdots ,0)$ in (9) and using it, we get

$$\begin{array}{c}\hfill {\mu }^{\prime }(\epsilon (0,\cdots ,0),t)\ge {\mu }^{\prime }(\epsilon (0,\cdots ,0),{\gamma }^{m}t)\end{array}$$

for all $t>0,m\in \mathbb{N}.$ Thus, $\epsilon (0,\cdots ,0)=0$ and

$$\begin{array}{c}\hfill \mu \left({\displaystyle \frac{n{(n-1)}^2}{2}}f\left(0\right),t\right)=\mu ({\mathcal{D}}_u\delta (0,\cdots ,0),t)\ge {\mu }^{\prime }(\epsilon (0,\cdots ,0),t)=1\end{array}$$

for all $t>0$, which implies $f\left(0\right)=0$. From Equation (6), we get the following equality

$$\begin{array}{ccc}\hfill & & f\left(x\right)-{(n-1)}^{2}f\left({\displaystyle \frac{x}{n-1}}\right)\hfill \\ \hfill & & ={\mathcal{D}}_1{f}_j\left(-{\displaystyle \frac{x}{n-1}}\right)+\left({\displaystyle \frac{n^2-(i+1)n+1}{n}}\right)\left[{\mathcal{D}}_1{f}_i\left({\displaystyle \frac{x}{n-1}}\right)-{\mathcal{D}}_1{f}_{i+1}\left({\displaystyle \frac{x}{n-1}}\right)\right]\hfill \end{array}$$

(11)

for all $(i,j)\in \mathcal{J}.$ Using (11) and ${\Delta }_{n-1}$-condition of $\mu$, one gets

$$\begin{array}{ccc}& & \mu \left(f\left(x\right)-{(n-1)}^{2m}f\left({\displaystyle \frac{x}{{(n-1)}^m}}\right),t\right)\hfill \\ & & \ge \mu \left(\sum _{k=1}^m{\displaystyle \frac{{(n-1)}^{4k-2}}{{(n-1)}^{2k}}}\left(f\left({\displaystyle \frac{x}{{(n-1)}^k}}\right)-{(n-1)}^{2}f\left({\displaystyle \frac{x}{{(n-1)}^k}}\right)\right),\sum _{k=1}^m{\displaystyle \frac{t}{2^k}}\right)\hfill \\ \\ & & \ge \prod _{k=1}^m({\mu }^{\prime }\left({\epsilon }_j\left(-x\right),{\left({\displaystyle \frac{\gamma {(n-1)}^2}{2{\kappa }_{n-1}^4}}\right)}^k·{\displaystyle \frac{{\kappa }_{n-1}^{2}t}{3}}\right)\hfill \\ & & \circ {\mu }^{\prime }\left({\epsilon }_i\left(x\right),{\left({\displaystyle \frac{\gamma {(n-1)}^2}{2{\kappa }_{n-1}^4}}\right)}^k·{\displaystyle \frac{{\kappa }_{n-1}^{2}nt}{3(n^2-(i+1)n+1)}}\right)\hfill \\ & & \circ \mu \left({\epsilon }_{i+1}\left(x\right),{\left({\displaystyle \frac{\gamma {(n-1)}^2}{2{\kappa }_{n-1}^4}}\right)}^k{\displaystyle \frac{{\kappa }_{n-1}^{2}nt}{3(n^2-(i+1)n+1)}}\right))\hfill \\ & & ={\mu }^{\prime }\left({\epsilon }_j\left(-x\right),{\displaystyle \frac{\gamma {(n-1)}^{2}t}{6{\kappa }_{n-1}^2}}\right)\circ {\mu }^{\prime }\left({\epsilon }_i\left(x\right),{\displaystyle \frac{\gamma {(n-1)}^{2}nt}{6{\kappa }_{n-1}^2(n^2-(i+1)n+1)}}\right)\hfill \\ & & \circ {\mu }^{\prime }\left({\epsilon }_{i+1}\left(x\right),{\displaystyle \frac{\gamma {(n-1)}^{2}nt}{6{\kappa }_{n-1}^2(n^2-(i+1)n+1)}}\right)\hfill \end{array}$$

for all $x\in M,t>0,(i,j)\in \mathcal{J}.$ This relation leads to

$$\begin{array}{c}\hfill \mu \left(f\left(x\right)-{(n-1)}^{2m}f\left({\displaystyle \frac{x}{{(n-1)}^m}}\right),t\right)\ge \mathsf{\Psi }(x,t)\end{array}$$

(12)

for all $x\in M$ and $t>0.$ Now, replacing x by $\frac{x}{{(n-1)}^m}$ in (12), we have

$$\begin{array}{ccc}& & \mu \left({(n-1)}^{2m}f\left({\displaystyle \frac{x}{{(n-1)}^m}}\right)-{(n-1)}^{2m+2p}f\left({\displaystyle \frac{x}{{(n-1)}^{m+p}}}\right),t\right)\hfill \\ & & \ge \mu \left(f\left({\displaystyle \frac{x}{{(n-1)}^m}}\right)-{(n-1)}^{2p}f\left({\displaystyle \frac{x}{{(n-1)}^{m+p}}}\right),{\displaystyle \frac{t}{{\kappa }_{n-1}^{2m}}}\right)\hfill \\ & & \ge \mathsf{\Psi }\left({\displaystyle \frac{x}{{(n-1)}^m}},{\displaystyle \frac{t}{{\kappa }_{n-1}^{2m}}}\right)\ge \mathsf{\Psi }\left(x,{\left({\displaystyle \frac{\gamma }{{\kappa }_{n-1}^2}}\right)}^{m}t\right)\hfill \end{array}$$

which converges to zero as $m\to \infty$. Thus, $\left\{{(n-1)}^{2m}f(x/{(n-1)}^m)\right\}$ is $\mu$-Cauchy for all $x\in M$, and so it is $\mu$-convergent in $\mathcal{A}$ since the space $\mathcal{A}$ is $\mu$-complete. Thus, we may define a mapping $Q:M\to \mathcal{A}$ as

$$\begin{array}{ccc}& & Q\left(x\right):=\mu -\underset{m\to \infty }{lim}{(n-1)}^{2m}f\left({\displaystyle \frac{x}{{(n-1)}^m}}\right)\hfill \\ & & \left(⟺\underset{m\to \infty }{lim}{(n-1)}^{2m}\mu \left(Q\left(x\right)-f\left({\displaystyle \frac{x}{{(n-1)}^m}}\right),t\right)=1\right)\hfill \end{array}$$

for all $x\in M$ and all $t>0.$ Using ${\Delta }_{n-1}$-condition and convexity of $\mu$, we find the following inequality

$$\begin{array}{ccc}\hfill \mu \left(f\left(x\right)-Q\left(x\right),t\right)& \ge & \mu \left(f\left(x\right)-{(n-1)}^{2m}f\left({\displaystyle \frac{x}{{(n-1)}^{2m}}}\right),{\displaystyle \frac{(n-1)t}{2{\kappa }_{n-1}}}\right)\hfill \\ & & \circ \mu \left({(n-1)}^{2m}f\left({\displaystyle \frac{x}{{(n-1)}^{2m}}}\right)-Q\left(x\right)\right),{\displaystyle \frac{(n-1)t}{2{\kappa }_{n-1}}})\hfill \\ & \ge & \mathsf{\Psi }\left(x,{\displaystyle \frac{(n-1)t}{2{\kappa }_{n-1}}}\right)\hfill \end{array}$$

for all $x\in M,t>0$ and for enough large $m\in \mathbb{N}.$ By the similar way of the proof of Theorem 2, we get Q is $\mathcal{A}$-quadratic functional equation.

To prove the uniqueness, let T be another $\mathcal{A}$-quadratic mapping satisfying (10). Then, we get $T\left({(n-1)}^{m}x\right)={(n-1)}^{2m}T\left(x\right)$ for all $x\in M$ and all $m\in \mathbb{N}$. Thus, we have

$$\begin{array}{ccc}\hfill \mu \left({\displaystyle \frac{T\left(x\right)-Q\left(x\right)}{2}},t\right)& \ge & \mu \left(T\left({\displaystyle \frac{x}{{(n-1)}^m}}\right)-f\left({\displaystyle \frac{x}{{(n-1)}^m}}\right),{\displaystyle \frac{t}{{\kappa }_{n-1}^{2m}}}\right)\hfill \\ & & \circ \mu \left(f\left({\displaystyle \frac{x}{{(n-1)}^m}}\right)-Q\left({\displaystyle \frac{x}{{(n-1)}^m}}\right),{\displaystyle \frac{t}{{\kappa }_{n-1}^{2m}}}\right)\hfill \\ & \ge & \mathsf{\Psi }\left({\displaystyle \frac{x}{{(n-1)}^m}},{\displaystyle \frac{(n-1)t}{{\kappa }_{n-1}^{2m+1}}}\right)\ge \mathsf{\Psi }\left(x,{\displaystyle \frac{(n-1){\gamma }^{m}t}{{\kappa }_{n-1}^{2m+1}}}\right)\hfill \end{array}$$

Taking the limit as $m\to \infty$, then we conclude that $T\left(x\right)=Q\left(x\right)$ for all $x\in M$. This completes the proof. □

Corollary 3.

Let ($\mathcal{A},\rho$) be a ρ-complete convex modular ∗-algebra with ${\Delta }_{n-1}$-condition and norm $\parallel ·\parallel$. Assume that there exist two mappings $f\in {\mathcal{A}}^M$ and $\epsilon \in {\mathbb{R}}^{M^n}$ such that

$$\begin{array}{ccc}\hfill \rho \left({\mathcal{D}}_{u}f(x_1,\cdots ,x_n)\right)& \le & \epsilon (x_1,\cdots ,x_n),\hfill \\ \hfill \epsilon \left({\displaystyle \frac{x_1}{n-1}},\cdots {\displaystyle \frac{x_n}{n-1}}\right)& \le & {\displaystyle \frac{1}{\gamma }}\epsilon (x_1,\cdots ,x_n)\hfill \end{array}$$

for all $(x_1,\cdots ,x_n)\in X^n,u\in \mathcal{U}\left(\mathcal{A}\right)$, where $\gamma {(n-1)}^2>2{\kappa }_{n-1}^4$ and either f is measurable or $f\left(tx\right)$ is continuous in $t\in \mathbb{R}$ for each fixed $x\in M$. Then, there exists a unique mapping $Q\in Q_{\mathcal{A}}(M,\mathcal{A})$ which satisfies Equation (2) and the inequality

$$\begin{array}{ccc}\hfill \rho \left(f\right(x)-Q(x\left)\right)& \le & {\displaystyle \frac{12{\kappa }_{n-1}^3}{\gamma {(n-1)}^3}}\underset{(i,j)\in \mathcal{J}}{min}\{max\{{\epsilon }_j\left(-x\right),{\displaystyle \frac{(n^2-(i+1)n+1)}{n}}{\epsilon }_i\left(x\right)\hfill \\ & & ,{\displaystyle \frac{(n^2-(i+1)n+1)}{n}}{\epsilon }_{i+1}\left(x\right)\left\}\right\}\hfill \end{array}$$

for all $x\in M$.

4. Conclusions

We have studied a quadratic functional equation from the gravity of the n-distinct vectors and obtained the solution of the quadratic functional equation and investigated the stability results of a $\mathcal{A}$-quadratic mapping on $\mu$-complete convex fuzzy modular ∗-algebras without using $\beta$-homogeneous property and lower semicontinuity. Furthermore, as corollaries, we have presented the stability results of the $\mathcal{A}$-quadratic mapping in $\rho$-complete convex modular ∗-algebras and Banach ∗-algebras, respectively.