Skip to main content

Stability of a generalized n-variable mixed-type functional equation in fuzzy modular spaces

Abstract

This research paper deals with general solution and the Hyers–Ulam stability of a new generalized n-variable mixed type of additive and quadratic functional equations in fuzzy modular spaces by using the fixed point method.

Introduction and preliminaries

A mapping \(f:U \to V\) is called additive if f satisfies the Cauchy functional equation

$$\begin{aligned} f(x+y)=f(x)+f(y) \end{aligned}$$
(1.1)

for all \(x,y \in U\). It is easy to see that the function \(f(x) = a x\) is a solution of functional equation (1.1) and every solution of functional equation (1.1) is said to be an additive mapping. A mapping \(f:U \to V\) is called quadratic if f satisfies the quadratic functional equation

$$\begin{aligned} f(x+y)+f(x-y)= 2f(x)+2f(y) \end{aligned}$$
(1.2)

for all \(x,y \in U\). It is easy to see that the quadratic function \(f(x) = a x^{2}\) is a solution of functional equation (1.2), and every solution of functional equation (1.2) is said to be a quadratic mapping. Mixed-type functional equation is the advanced development in the field of functional equations. A single functional equation, which has more than one nature, is known as mixed-type functional equation. Further, one can refer to [123] for more information on functional equations and applications.

“Let G be a group and H be a metric group with a metric \(d(\cdot,\cdot)\). Given \(\epsilon >0 \), does there exist \(\delta >0\) such that if a mapping \(f:G \rightarrow H\) satisfies \(d(f(xy),f(x)f(y)) < \delta \) for all \(x, y \in G\), then there exists a homomorphism \(a: G \rightarrow H\) with \(d(f(x),a(x))< \epsilon \) for all \(x\in G\)?” This problem for the stability of functional equations was raised by Ulam [24] and answered by Hyers [25]. Later, it was developed as Hyers–Ulam stability by Rassias [26], Rassias [27, 28], and Gavruta [29].

Definition 1.1

(Fuzzy modular space [30])

Let μ be a fuzzy set on \(X\times \mathbb{R^{+}}\), V be a complex or real vector space, γ be a zero on V, and be a continuous triangular norm. The triple \((V,\mu , \ast )\) is said to be a fuzzy modular space and μ is said to be a fuzzy modular if it satisfies the following:

  1. (i)

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

  2. (ii)

    \(\mu (x,t)=1\) if and only if \(x=\gamma \);

  3. (iii)

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

  4. (iv)

    \(\mu (a x+b y,r+t)\geq \mu (x,r)\ast \mu (y,t)\), \(a,b\geq 0\), \(a+b=1\);

  5. (v)

    the function \(\mu (x,\cdot ):(0,\infty ) \to (0, 1]\) is continuous.

Example 1.2

Let μ be a fuzzy set on \(V\times \mathbb{R^{+}}\), V be a complex or real vector space, and be a continuous triangular norm such that \(a\ast b=a\ast _{M} b=\min \{a, b\}\). Then

$$ \mu (x,t) = \textstyle\begin{cases} \frac{t}{{t + \mu (x)}} ,&t > 0 ,x \in V, \\ 0, & t \le 0 ,x \in V, \end{cases} $$

is a fuzzy modular space. This example holds even if we replace \(a\ast b\) with \(a\ast _{P} b\) and \(a\ast _{L} b\).

Definition 1.3

Let \((V,\mu , \ast )\) be a fuzzy modular space. Let \(\{z_{n}\}\) be a sequence in V.

  1. (i)

    \(\{z_{n}\}\) is said to be μ-convergent to z, denoted by \(z_{n}\xrightarrow{\mu } z\), if there exists a positive integer \(m_{0}\) such that \(\mu (z_{n}-x,t)>1-\gamma \) for all \(n\geq m_{0}\), \(t>0\), and \(\gamma \in (0,1)\).

  2. (ii)

    \(\{z_{n}\}\) is said to be a Cauchy sequence if there exists a positive integer \(m_{0}\) such that \(\mu (z_{n}-z_{m},t)>1-\gamma \) for all \(n, m\geq m_{0}\), \(t>0\), and \(\gamma \in (0,1)\).

  3. (iii)

    Every μ-convergent sequence in an FM-space is a μ-Cauchy sequence. In \((V,\mu ,\ast )\), if each μ-Cauchy sequence is μ-convergent sequence, then \((V,\mu ,\ast )\) is called a μ-complete fuzzy modular space.

Definition 1.4

([30])

If μ fulfills the property \(\mu (\gamma z, t)=\mu (z, \frac{t}{|\gamma |^{b}})\) for some fixed \(b\in (0,1]\) and a nonzero real number γ, then \((V,\mu , \ast )\) is said to be a b-homogeneous fuzzy modular space.

In 2002, J. M. Rassias [31] studied the Ulam stability of a mixed-type functional equation

$$\begin{aligned} g \Biggl(\sum_{i=1}^{3}x_{i} \Biggr)+\sum_{i=1}^{3}g(x_{i})= \sum_{1 \leq i\leq j \leq 3}g(x_{i}+x_{j}). \end{aligned}$$

Later, Nakmalachalasint [32] generalized the above functional equation and obtained an n-variable mixed-type functional equation of the form

$$\begin{aligned} g \Biggl(\sum_{i=1}^{n}x_{i} \Biggr)+(n-2)\sum_{i=1}^{n}g(x_{i})= \sum_{1\leq i\leq j \leq n}g(x_{i}+x_{j}) \end{aligned}$$

for \(n>2\) and investigated its Ulam stability.

In 2005, Jun and Kim [33] introduced a generalized additive-quadratic functional equation of the form

$$\begin{aligned} g(x+ay)+ag(x-y)=g(x-ay)+ag(x+y) \end{aligned}$$
(1.3)

for \(a\neq 0, \pm 1\).

Shen and Chen [30] introduced the concept of fuzzy modular spaces in 2013. Further, Kumam [34, 35] and Wongkum et al. [36] introduced the fixed point concept in fuzzy modular spaces and obtained some properties. Wongkum and Kumam [37] investigated the Hyers–Ulam stability of sextic functional equation in fuzzy modular spaces.

Motivated by the notion of fuzzy modular spaces and by the mixed-type functional equations, we introduce a new generalized n-variable mixed-type functional equation of the form

$$\begin{aligned} &\sum_{i=1, j=i+1}^{n-1} \bigl( f(kx_{i}+x_{j}) \bigr)+f(kx_{n}+x_{1}) \\ &\qquad {}-k \Biggl[\sum_{i=1, j=i+1}^{n-1} \bigl(f(x_{i}+x_{j}) \bigr)+f(x_{n}+x_{1}) \Biggr] \\ &\quad =\frac{(1-k)^{2}}{2}\sum_{i=1}^{n} \bigl(f(x_{i})+f(-x_{i}) \bigr)+\frac{1-k}{k^{2}-k} \sum_{i=1}^{n} \bigl(k^{2}f(x_{i})-f(kx_{i}) \bigr) \end{aligned}$$
(1.4)

for positive integers \(n, k\geq 2\) and investigate its Hyers–Ulam stability in fuzzy modular spaces.

This paper is structured as follows: In Sect. 1, we provide necessary introduction of this paper. In Sect. 2, we obtain the general solution of functional equation (1.4). In Sect. 3, we investigate the Hyers–Ulam stability of (1.4) in fuzzy modular spaces using the fixed point theory, and the conclusion is given in Sect. 4.

General solution of a mixed-type functional equation

Let U and V be real vector spaces. In this section we obtain the general solution of a generalized n-variable mixed-type functional equation (1.4).

Lemma 2.1

Let a mapping \(f:U \to V\) satisfy functional equation (1.4). If f is an even mapping, then f is quadratic.

Proof

Let a mapping \(f:U \to V\) satisfy functional equation (1.4). Substituting \((x_{1}, x_{2},\dots , x_{n})\) by \((x, 0,\dots ,0)\) in (1.4), we have

$$\begin{aligned} &f(kx)+f(x)-2kf(x) \\ &\quad =\frac{(1-k)^{2}}{2} \bigl[f(x)+f(-x) \bigr]+\frac{1-k}{k^{2}-k} \bigl[k^{2}f(x)-f(kx) \bigr] \end{aligned}$$
(2.1)

for all \(x\in U\). By the evenness of f, equation (2.1) leads to \(f(kx)=k^{2}f(x)\) for all \(x\in U\), and so f is quadratic. Hence, by the evenness of f, the mixed-type functional equation (1.4) is reduced to the following quadratic functional equation of the form:

$$\begin{aligned} &\sum_{i=1, j=i+1}^{n-1} \bigl( f(kx_{i}+x_{j}) \bigr)+f(kx_{n}+x_{1})-k \Biggl[\sum_{i=1, j=i+1}^{n-1} \bigl(f(x_{i}+x_{j}) \bigr)+f(x_{n}+x_{1}) \Biggr] \\ &\quad =(1-k)^{2}\sum_{i=1}^{n} \bigl(f(x_{i}) \bigr)+ \frac{1-k}{k^{2}-k}\sum _{i=1}^{n} \bigl(k^{2}f(x_{i})-f(kx_{i}) \bigr) \end{aligned}$$
(2.2)

for positive integers \(n, k\geq 2\). □

Lemma 2.2

Let a mapping \(f:U \to V\) satisfy functional equation (1.4). If f is an odd mapping, then f is additive.

Proof

Let a mapping \(f:U \to V\) satisfy functional equation (1.4). Substituting \((x_{1}, x_{2},\dots , x_{n})\) by \((x, 0,\dots ,0)\) in (1.4), we get (2.1). By the oddness of f, equation (2.1) leads to \(f(kx)=kf(x)\) for all \(x\in U\), and so f is additive. Hence, by the oddness of f, the mixed-type functional equation (1.4) is reduced to the following additive functional equation of the form:

$$\begin{aligned} &\sum_{i=1, j=i+1}^{n-1} \bigl( f(kx_{i}+x_{j}) \bigr)+f(kx_{n}+x_{1})-k \Biggl[\sum_{i=1, j=i+1}^{n-1} \bigl(f(x_{i}+x_{j}) \bigr)+f(x_{n}+x_{1}) \Biggr] \\ &\quad =\frac{1-k}{k^{2}-k}\sum_{i=1}^{n} \bigl(k^{2}f(x_{i})-f(kx_{i}) \bigr) \end{aligned}$$
(2.3)

for \(n \in \mathbb{N}\). □

Theorem 2.3

Let an even mapping \(f:U \to V\) satisfy functional equation (2.2), then f is quadratic.

Proof

Suppose that f is even and satisfies functional equation (2.2). Setting \(x_{1} = x_{2} = \cdots = x_{n} =0\) and replacing \((x_{1}, x_{2} , \ldots , x_{n} )\) with \((x,0,\dots ,0)\) in (2.2), we obtain \(f(0)=0\) and

$$\begin{aligned} f(k x)=k^{2} f(x), \end{aligned}$$
(2.4)

respectively, for all \(x\in U\). Replacing \((x_{1},x_{2},x_{3},\dots ,x_{n})\) with \((x_{1},x_{2},0,\dots ,0)\) in (2.2) and using (2.4), we have

$$\begin{aligned} f(kx_{1}+x_{2})-kf(x_{1}+x_{2})= \bigl(k^{2}-k\bigr)f(x_{1})+(1-k)f(x_{2}) \end{aligned}$$
(2.5)

for all \(x_{1}, x_{2} \in U\). Replacing \(x_{2}\) with \(-x_{2}\) in (2.5), using the evenness of f and again adding the resultant to (2.5), we get

$$\begin{aligned} &f(kx_{1}+x_{2})+f(kx_{1}-x_{2}) \\ & \quad =kf(x_{1}+x_{2})+kf(x_{1}-x_{2})+2 \bigl(k^{2}-k\bigr)f(x_{1})+2(1-k)f(x_{2}) \end{aligned}$$
(2.6)

for all \(x_{1}, x_{2} \in U\). Replacing \((x_{1},x_{2})\) with \((x_{1}, x_{1}+x_{2})\) in (2.6), we get

$$\begin{aligned} &f\bigl((k+1)x_{1}+x_{2}\bigr)+f \bigl((k-1)x_{1}-x_{2}\bigr) \\ &\quad = kf(2x_{1}+x_{2})+f(-x_{2})+2 \bigl(k^{2}-k\bigr)f(x_{1})+2(1-k)f(x_{1}+x_{2}) \end{aligned}$$
(2.7)

for all \(x_{1}, x_{2} \in U\). Replacing \((x_{1},x_{2})\) with \((x_{1}, -x_{2})\) in (2.7) and again adding the resultant to (2.7), we get

$$\begin{aligned} &f\bigl((k+1)x_{1}+x_{2}\bigr)+f \bigl((k+1)x_{1}-x_{2}\bigr)+f\bigl((k-1)x_{1}+x_{2} \bigr) \\ & \qquad {} +f\bigl((k-1)x_{1}-x_{2}\bigr)-k \bigl[f(2x_{1}+x_{2})+f(2x_{1}-x_{2})+2f(x_{2}) \bigr] \\ &\quad = 4\bigl(k^{2}-k\bigr)f(x_{1})+2(1-k) \bigl[f(x_{1}+x_{2})+f(x_{1}-x_{2}) \bigr] \end{aligned}$$
(2.8)

for all \(x_{1}, x_{2} \in U\). Now, by (2.6) and (2.8) and by assuming different values of k as \(k+1\), \(k-1\), and 2, we obtain (1.2). Hence the mapping f is quadratic. □

Theorem 2.4

Let an odd mapping \(f:U \to V\) satisfy functional equation (2.3). Then f is additive.

Proof

Suppose that f is odd and satisfies functional equation(2.3). Replacing \((x_{1},x_{2},\dots ,x_{n})\) with \((0,0,\dots ,0)\) and \((x,0,\dots ,0)\) in (2.3), we obtain \(f(0)=0\) and

$$\begin{aligned} f(k x)=k f(x), \quad \forall x \in U, \end{aligned}$$
(2.9)

respectively. Replacing \((x_{1},x_{2},x_{3},x_{4},\dots ,x_{n})\) with \((x_{1},x_{2},0,0,\dots ,0)\) in (2.3), we obtain

$$\begin{aligned} f(kx_{1}+x_{2})-kf(x_{1}+x_{2})=(1-k)f(x_{2}) \end{aligned}$$
(2.10)

for all \(x_{1}, x_{2} \in U\). Replacing \(x_{2}\) with \(-x_{2}\) in (2.10), using the oddness of f and again adding the resultant to (2.10), we get

$$\begin{aligned} f(kx_{1}+x_{2})+f(kx_{1}-x_{2})=k f(x_{1}+x_{2})+kf(x_{1}-x_{2}) \end{aligned}$$
(2.11)

for all \(x_{1}, x_{2} \in U\). Replacing \((x_{1},x_{2})\) with \((x_{2}, x_{1})\) in (2.11), we get

$$\begin{aligned} f(x_{1}+kx_{2})-f(x_{1}-kx_{2})=k f(x_{1}+x_{2})-kf(x_{1}-x_{2}) \end{aligned}$$
(2.12)

for all \(x_{1}, x_{2} \in U\). Replacing \(x_{2}\) with \(kx_{2}\) in (2.11) and using (2.9), we get

$$\begin{aligned} f(x_{1} + kx_{2}) + f(x_{1} - kx_{2}) = f(x_{1}+x_{2})+f(x_{1}-x_{2}) \end{aligned}$$
(2.13)

for all \(x_{1},x_{2}\in U\). Replacing \(x_{1}\) with \(x_{1}+kx_{2}\) in (2.12), we get

$$\begin{aligned} f(x_{1} + 2kx_{2})-f(x_{1}) = kf \bigl((x_{1}+x_{2})+kx_{2} \bigr)- kf \bigl((x_{1}-x_{2}) + kx_{2} \bigr) \end{aligned}$$
(2.14)

for all \(x_{1},x_{2}\in U\). Replacing \(x_{2}\) with \(-x_{2}\) in (2.14), adding the resultant to (2.14) and using (2.12), we obtain

$$\begin{aligned} f(x_{1} + 2kx_{2}) + f(x_{1}-2kx_{2}) = k^{2} \bigl[f(x_{1}+2x_{2}) +f(x_{1}-2x_{2}) \bigr]- 2k^{2} f(x_{1})+2f(x_{1}) \end{aligned}$$
(2.15)

for all \(x_{1},x_{2}\in U\). Replacing \(x_{2}\) with \(\frac{x_{2}}{2}\) in (2.15) and using (2.13), we get

$$\begin{aligned} f(x_{1} + x_{2}) + f(x_{1}-x_{2}) =2f(x_{1}) \end{aligned}$$
(2.16)

for all \(x_{1},x_{2}\in U\). Replacing \((x_{1},x_{2})\) with \((x_{2},x_{1})\) in (2.16) and adding the resultant to (2.16), we obtain (1.1). Hence the mapping f is additive. □

Lemma 2.5

([33])

Let a mapping \(f:U \to V\) satisfy functional equation (1.3), then f is additive-quadratic.

Theorem 2.6

Let an odd mapping \(f:U \to V\) satisfy functional equation (1.4). Then f satisfies (1.3).

Proof

Suppose that an odd mapping f satisfies functional equation (1.4). Replacing \((x_{1},x_{2},\dots ,x_{n})\) with \((x_{1},x_{2},0,\dots ,0)\) in (1.4), we obtain

$$\begin{aligned} f(kx_{1}+x_{2})-kf(x_{1}+x_{2})=(1-k)f(x_{2}) \end{aligned}$$
(2.17)

for all \(x_{1}, x_{2} \in U\). Replacing \(x_{2}\) with \(-x_{2}\) in (2.17), using the oddness of f, and again adding the resultant to (2.17), we get

$$\begin{aligned} f(kx_{1}+x_{2})+f(kx_{1}-x_{2})=k f(x_{1}+x_{2})+kf(x_{1}-x_{2}) \end{aligned}$$
(2.18)

for all \(x_{1}, x_{2} \in U\). Replacing \((x_{1},x_{2})\) with \((x_{2}, x_{1})\) in (2.18), we get (1.3). □

Stability of a mixed-type functional equation

In this section, we obtain the Hyers–Ulam stability of a generalized n-variable mixed-type functional equation (1.4) in a fuzzy modular space by using the fixed point technique. For the mapping \(f: M \to (V,\mu )\), consider

$$\begin{aligned} S(x_{1},x_{2},\dots ,x_{n})={}&\sum _{i=1, j=i+1}^{n-1} \bigl( f(kx_{i}+x_{j}) \bigr)+f(kx_{n}+x_{1}) \\ &{} -k \Biggl[\sum_{i=1, j=i+1}^{n-1} \bigl(f(x_{i}+x_{j}) \bigr)+f(x_{n}+x_{1}) \Biggr] \\ &{} -\frac{(1-k)^{2}}{2}\sum_{i=1}^{n} \bigl(f(x_{i})+f(-x_{i}) \bigr)-\frac{1-k}{k^{2}-k} \sum_{i=1}^{n} \bigl(k^{2}f(x_{i})-f(kx_{i}) \bigr) \end{aligned}$$

for \(n\in \mathbb{N}\), \(k\geq 2\).

Theorem 3.1

Let M be a linear space, V be a real vector space, \((V, \mu , \ast )\) be a μ-complete b-homogeneous fuzzy modular space, and \(\alpha \in \{-1,1\}\) be fixed. Suppose that an even mapping \(f: M\to (V,\rho , \ast )\) satisfies

$$\begin{aligned} \mu \bigl(S (x_{1},x_{2},\dots ,x_{n}), t\bigr)\geq \rho (x_{1},x_{2},\dots ,x_{n},t) \end{aligned}$$
(3.1)

for all \(x_{1},x_{2},\dots ,x_{n} \in M\) and a given mapping \(\rho : M \times M \to \Delta \) such that

$$\begin{aligned} \rho \bigl(k^{a} x_{1},k^{a} x_{2},\dots , k^{a} x_{n}, k^{2b a} Nt \bigr)\geq \rho (x_{1},x_{2},\dots ,x_{n},t) \end{aligned}$$
(3.2)

for all \(x_{1},x_{2},\dots ,x_{n}\in M\) and

$$\begin{aligned} \lim_{m\to \infty }\rho \bigl(k^{am} x_{1},k^{am} x_{2}, \dots ,k^{am} x_{2}, k^{2b am} t \bigr)=1 \end{aligned}$$
(3.3)

for all \(x_{1},x_{2},\dots ,x_{n}\in M\) and a constant \(0< N<\frac{1}{ (\frac{k^{2}-2k+1}{k^{2}-k} )^{b}}\). Then there exists a unique quadratic mapping \(Q:M\to (V,\mu )\) satisfying (1.4) and

$$\begin{aligned} \rho \biggl(Q(x)-f(x), \frac{t}{k^{2b}N^{\frac{\alpha -1}{2}} (1- (\frac{k^{2}-2k+1}{k^{2}-k} )^{b} N )} \biggr)\geq \rho (x,0,\dots ,0,t) \end{aligned}$$
(3.4)

for all \(x_{1},x_{2},\dots ,x_{n} \in M\).

Proof

Letting \((x_{1},x_{2},\dots ,x_{n})\) by \((x,0,\dots ,0)\) in (3.1), we obtain

$$\begin{aligned} \mu \biggl(\frac{k^{2}-2k+1}{k^{2}-k} \bigl(f(k x)-k^{2} f(x) \bigr),t \biggr)\geq \rho (x,0,\dots ,0,t) \end{aligned}$$
(3.5)

for all \(x \in M\), and so

$$\begin{aligned} \mu \biggl(\frac{f(k x)}{k^{2}}-f(x), t \biggr)&=\rho \biggl( \frac{k^{2}-2k+1}{k^{2}-k} \bigl(f(k x)-k^{2} f(x) \bigr), \biggl( \frac{k^{2}-2k+1}{k^{2}-k} \biggr)^{b}k^{2b} t \biggr) \\ & \geq \rho \biggl(x,0, \dots ,0, \biggl(\frac{k^{2}-2k+1}{k^{2}-k} \biggr)^{b}k^{2b}t \biggr) \end{aligned}$$
(3.6)

for all \(x \in M\). Replacing x with \(k^{-1}x\) in (3.6), we obtain

$$\begin{aligned} \mu \biggl(\frac{f(k^{-1} x)}{k^{-2}}-f(x),t \biggr)&=\mu \biggl( \frac{f( x)}{k^{2}}-f\bigl(k^{-1}x\bigr), \frac{t}{k^{2b}} \biggr) \\ & \geq \rho \biggl(k^{-1}x,0, \dots ,0, \biggl( \frac{k^{2}-2k+1}{k^{2}-k} \biggr)^{b}k^{2b}N^{-1} \frac{Nt}{k^{2b}} \biggr) \\ & \geq \rho \biggl(x,0, \dots ,0, \biggl(\frac{k^{2}-2k+1}{k^{2}-k} \biggr)^{b}k^{2b}N^{-1}t \biggr). \end{aligned}$$
(3.7)

From (3.6) and (3.7), we obtain

$$\begin{aligned} &\mu \biggl(\frac{f(k^{a} x)}{k^{2a}}-f(x), t \biggr)\geq \Psi (x,t):= \rho \biggl(x,0,\dots ,0, \biggl(\frac{k^{2}-2k+1}{k^{2}-k} \biggr)^{b}k^{2b}N^{\frac{a-1}{2}}t \biggr) \end{aligned}$$
(3.8)

for all \(x\in M\). Consider \(P:=\{h:M\to (V, \mu )|h(0)=0\}\) and define η on P as follows:

$$\begin{aligned} \eta (h)=\inf \bigl\{ l>0:\rho \bigl(h(x), lt\bigr)\geq \Psi (x,t), \forall x \in M\bigr\} . \end{aligned}$$

One can easily prove that η is modular on N and indulges the \(\Delta _{k}\)-condition with \(k^{b}=\kappa \) and the Fatou property. Additionally, N is η-complete (see [38]). Consider the mapping \(R: P_{\eta }\to P_{\eta }\) as \(RQ(x):=\frac{Q(k^{a} x)}{k^{2a}}\) for all \(Q\in P_{\eta }\).

Let \(h,j \in P_{\eta }\) and \(l>0\) be an arbitrary constant with \(\eta (h-j)\leq l\). From the definition of η, we get

$$\begin{aligned} \mu \bigl(h(x)-j(x), lt\bigr)\geq \Psi (x,t) \end{aligned}$$

for all \(x \in M\), and so

$$\begin{aligned} &\mu \bigl(Rh(x)-Rj(x), Nlt\bigr) \\ &\quad =\mu \bigl(k^{-2a}h\bigl(k^{a} x\bigr)-k^{-2a}j \bigl(k^{a} x\bigr), Nlt \bigr) \\ &\quad =\mu \bigl(h\bigl(k^{a} x\bigr)-j\bigl(k^{a} x\bigr), k^{2b a}Nlt \bigr) \\ &\quad \geq \Psi \bigl(k^{a} x, k^{2b a}Nt \bigr) \\ &\quad \geq \Psi (x, t) \end{aligned}$$

for all \(x \in M\). Hence \(\eta (Rh-Rj)\leq N \eta (h-j)\) for all \(h, j\in P_{\eta }\), which means that R is an η-strict contraction. Replacing x with \(k^{a} x\) in (3.8), we get

$$\begin{aligned} &\mu \biggl(\frac{f(k^{2a} x)}{k^{2a}}-f\bigl(k^{a} x \bigr), t \biggr)\geq \Psi \bigl(k^{a} x, t\bigr) \end{aligned}$$
(3.9)

for all \(x\in M\), and therefore

$$\begin{aligned} &\mu \bigl(k^{-2(2a)}f\bigl(k^{2a} x \bigr)-k^{-2a}f\bigl(k^{a} x\bigr), Nt \bigr) \end{aligned}$$
(3.10)
$$\begin{aligned} &\quad =\mu \bigl(k^{-2a}f\bigl(k^{2a} x\bigr)-f \bigl(k^{a}x\bigr), k^{2b a}Nt \bigr) \\ &\quad \geq \Psi \bigl(k^{a} x, k^{2b a}Nt \bigr) \\ &\quad \geq \Psi (x, t) \end{aligned}$$
(3.11)

for all \(x \in M\). Now

$$\begin{aligned} &\mu \biggl(\frac{f(k^{2a} x)}{k^{2(2a)}}-f(x), \biggl( \frac{k^{2}-2k+1}{k^{2}-k} \biggr)^{b} (Nt+t) \biggr) \\ &\quad \geq \mu \biggl(\frac{f(k^{2a} x)}{k^{2(2a)}}- \frac{f(k^{a} x)}{k^{2a}}, Nt \biggr) \wedge \mu \biggl( \frac{f(k^{a} x)}{k^{2a}}-f(x), t \biggr) \\ &\quad \geq \Psi (x,t) \end{aligned}$$
(3.12)

for all \(x \in M\). In (3.12), replacing x with \(k^{a} x\) and \((\frac{k^{2}-2k+1}{k^{2}-k} )^{b} (Nt+t)\) with \((\frac{k^{2}-2k+1}{k^{2}-k} )^{b}k^{2b a}(N^{2} t+Nt)\), we get

$$\begin{aligned} \mu \biggl(\frac{f(k^{3a} x)}{k^{2(2a)}}-f\bigl(k^{a} x \bigr), k^{2b a} \biggl( \frac{k^{2}-2k+1}{k^{2}-k} \biggr)^{b} \bigl(N^{2} t+Nt\bigr) \biggr) &\geq \Psi \bigl(k^{a} x, k^{2ba}N t \bigr)\\ &\geq \Psi (x, t) \end{aligned}$$
(3.13)

for all \(x \in E\). Therefore,

$$\begin{aligned} \mu \biggl(\frac{f(k^{3a} x)}{k^{3(2a)}}-\frac{f(k^{a} x)}{k^{2 a}}, \biggl( \frac{k^{2}-2k+1}{k^{2}-k} \biggr)^{b} \bigl(N^{2} t+Nt\bigr) \biggr) \geq \Psi (x, t) \end{aligned}$$
(3.14)

for all \(x \in M\), and so

$$\begin{aligned} &\mu \biggl(\frac{f(k^{3a} x)}{k^{3(2a)}}-f(x), \biggl( \frac{k^{2}-2k+1}{k^{2}-k} \biggr)^{b} \biggl( \biggl( \frac{k^{2}-2k+1}{k^{2}-k} \biggr)^{b} \bigl(N^{2} t+Nt\bigr)+t \biggr) \biggr) \\ &\quad \geq \mu \biggl(\frac{f(k^{3a} x)}{k^{3(2a)}}- \frac{f(k^{a} x)}{k^{2a}}, \biggl( \frac{k^{2}-2k+1}{k^{2}-k} \biggr)^{b} \bigl(N^{2} t+Nt \bigr) \biggr) \wedge \mu \biggl( \frac{f(k^{a} x)}{k^{2a}}-f(x), t \biggr) \\ &\quad \geq \Psi (x,t) \end{aligned}$$
(3.15)

for all \(x\in M\). Generalizing the above inequality, we obtain

$$\begin{aligned} &\mu \Biggl(\frac{f(k^{am} x)}{k^{2(am)}}-f(x), \\ &\qquad {} \Biggl( \biggl( \biggl(\frac{k^{2}-2k+1}{k^{2}-k} \biggr)^{b} N \biggr)^{m-1}+ \biggl(\frac{k^{2}-2k+1}{k^{2}-k} \biggr)^{b}\sum _{i=1}^{m-1} \biggl( \biggl( \frac{k^{2}-2k+1}{k^{2}-k} \biggr)^{b} N \biggr)^{i-1} \Biggr)t \Biggr) \\ & \quad \geq \Psi (x,t) \end{aligned}$$
(3.16)

for all \(x\in M\) and a positive integer m. Hence we have

$$\begin{aligned} &\eta \bigl(R^{m} f-f\bigr) \\ &\quad \leq \biggl( \biggl(\frac{k^{2}-2k+1}{k^{2}-k} \biggr)^{b} N \biggr)^{m-1}+ \biggl(\frac{k^{2}-2k+1}{k^{2}-k} \biggr)^{b}\sum _{i=1}^{m-1} \biggl( \biggl( \frac{k^{2}-2k+1}{k^{2}-k} \biggr)^{b} N \biggr)^{i-1} \\ &\quad \leq \biggl(\frac{k^{2}-2k+1}{k^{2}-k} \biggr)^{b} \sum _{i=1}^{m} \biggl( \biggl(\frac{k^{2}-2k+1}{k^{2}-k} \biggr)^{b} N \biggr)^{i-1}\leq \frac{ (\frac{k^{2}-2k+1}{k^{2}-k} )^{b}}{1- (\frac{k^{2}-2k+1}{k^{2}-k} )^{b} N}. \end{aligned}$$
(3.17)

Now, one can easily prove that \(\{R^{m}(f)\}\) is η-convergent to \(Q\in P_{\eta }\) (see [37]). Therefore, (3.17) becomes

$$\begin{aligned} &\eta (Q-f)\leq \frac{ (\frac{k^{2}-2k+1}{k^{2}-k} )^{b}}{1- (\frac{k^{2}-2k+1}{k^{2}-k} )^{b} N}, \end{aligned}$$
(3.18)

which implies

$$\begin{aligned} &\mu \biggl(Q(x)-f(x), \frac{ (\frac{k^{2}-2k+1}{k^{2}-k} )^{b}}{1- (\frac{k^{2}-2k+1}{k^{2}-k} )^{b} N}t \biggr) \\ &\quad \geq \Psi (x,t)=\rho \biggl(x,0,\dots ,0, \biggl( \frac{k^{2}-2k+1}{k^{2}-k} \biggr)^{b}k^{2b}N^{\frac{a-1}{2}}t \biggr) \end{aligned}$$
(3.19)

for all \(x\in M\), and hence we have

$$\begin{aligned} &\mu \biggl(Q(x)-f(x), \frac{t}{k^{2b}N^{\frac{a-1}{2}} (1- (\frac{k^{2}-2k+1}{k^{2}-k} )^{b} N )} \biggr)\geq \rho (x,0,\dots ,0,t) \end{aligned}$$
(3.20)

for all \(x\in M\), and so inequality (3.4) holds. One can easily prove the uniqueness of Q (see [37]). □

Theorem 3.2

Let M be a linear space, V be a real vector space, \((V, \mu , \ast )\) be a μ-complete b-homogeneous fuzzy modular space, and \(\alpha \in \{-1,1\}\) be fixed. Suppose that an odd mapping \(f: M\to (V,\rho , \ast )\) satisfies

$$\begin{aligned} \mu \bigl(S(x_{1},x_{2},\dots ,x_{n}), t\bigr)\geq \rho (x_{1},x_{2},\dots ,x_{n},t) \end{aligned}$$
(3.21)

for all \(x_{1},x_{2},\dots ,x_{n} \in M\) and a given mapping \(\rho : M \times M \to \Delta \) such that

$$\begin{aligned} \rho \bigl(k^{a} x_{1},k^{a} x_{2},\dots , k^{a} x_{n}, k^{b a} Nt \bigr)\geq \rho (x_{1},x_{2},\dots ,x_{n},t) \end{aligned}$$
(3.22)

for all \(x_{1},x_{2},\dots ,x_{n}\in M\) and

$$\begin{aligned} \lim_{m\to \infty }\rho \bigl(k^{am} x_{1},k^{am} x_{2}, \dots ,k^{am} x_{2}, k^{b am} t \bigr)=1 \end{aligned}$$
(3.23)

for all \(x_{1},x_{2},\dots ,x_{n}\in M\) and a constant \(0< N<\frac{1}{ (\frac{k^{2}-2k+1}{k^{2}-k} )^{b}}\). Then there exists a unique additive mapping \(A:M\to (V,\mu )\) satisfying (1.4) and

$$\begin{aligned} \rho \biggl(A(x)-f(x), \frac{t}{k^{b}N^{\frac{\alpha -1}{2}} (1- (\frac{k^{2}-2k+1}{k^{2}-k} )^{b} N )} \biggr)\geq \rho (x,0,\dots ,0,t) \end{aligned}$$
(3.24)

for all \(x_{1},x_{2},\dots ,x_{n} \in M\).

Proof

Replacing \((x_{1},x_{2},\dots ,x_{n})\) with \((x,0,\dots ,0)\) in (3.21), we obtain

$$\begin{aligned} \mu \biggl(\frac{k^{2}-2k+1}{k^{2}-k}f(k x)-k f(x),t \biggr)\geq \rho (x,0,\dots ,0,t) \end{aligned}$$
(3.25)

for all \(x \in M\), and so

$$\begin{aligned} \mu \biggl(\frac{f(k x)}{k}-f(x), t \biggr)&=\rho \biggl( \frac{k^{2}-2k+1}{k^{2}-k}f(k x)-k f(x), \biggl( \frac{k(k^{2}-2k+1)}{k^{2}-k} \biggr)^{b}t \biggr) \\ & \geq \rho \biggl(x,0, \dots ,0, \biggl(\frac{k(k^{2}-2k+1)}{k^{2}-k} \biggr)^{b}t \biggr) \end{aligned}$$
(3.26)

for all \(x \in M\). Replacing x with \(k^{-1}x\) in (3.26), we obtain

$$\begin{aligned} \mu \biggl(\frac{f(k^{-1} x)}{k^{-1}}-f(x),t \biggr)&=\mu \biggl( \frac{f( x)}{k}-f\bigl(k^{-1}x\bigr), \frac{t}{ (\frac{k(k^{2}-2k+1)}{k^{2}-k} )^{b}} \biggr) \\ & \geq \rho \biggl(k^{-1}x,0, \dots ,0, \biggl( \frac{k(k^{2}-2k+1)}{k^{2}-k} \biggr)^{b}N^{-1} \frac{Nt}{ (\frac{k(k^{2}-2k+1)}{k^{2}-k} )^{b}} \biggr) \\ & \geq \rho \biggl(x,0, \dots ,0, \biggl(\frac{k(k^{2}-2k+1)}{k^{2}-k} \biggr)^{b}N^{-1}t \biggr). \end{aligned}$$
(3.27)

From (3.26) and (3.27), we obtain

$$\begin{aligned} &\mu \biggl(\frac{f(k^{a} x)}{k^{a}}-f(x), t \biggr)\geq \Psi (x,t):= \rho \biggl(x,0,\dots ,0, \biggl(\frac{k(k^{2}-2k+1)}{k^{2}-k} \biggr)^{b}N^{\frac{a-1}{2}}t \biggr) \end{aligned}$$
(3.28)

for all \(x\in M\). Consider \(P:=\{h:M\to (V, \mu )|h(0)=0\}\) and define η on P as follows:

$$\begin{aligned} \eta (h)=\inf \bigl\{ l>0:\rho \bigl(h(x), lt\bigr)\geq \Psi (x,t), \forall x \in M\bigr\} . \end{aligned}$$

One can easily prove that η is modular on N and indulges the \(\Delta _{k}\)-condition with \(k^{b}=\kappa \) and the Fatou property. Additionally, N is η-complete (see [38]). Consider the mapping \(R: P_{\eta }\to P_{\eta }\) as \(RA(x):=\frac{A(k^{a} x)}{k^{a}}\) for all \(A\in P_{\eta }\).

Let \(h,j \in P_{\eta }\) and \(l>0\) be an arbitrary constant with \(\eta (h-j)\leq l\). From the definition of η, we get

$$\begin{aligned} \mu \bigl(h(x)-j(x), lt\bigr)\geq \Psi (x,t) \end{aligned}$$

for all \(x \in M\), and so

$$\begin{aligned} &\mu \bigl(Rh(x)-Rj(x), Nlt\bigr) \\ &\quad =\mu \bigl(k^{-a}h\bigl(k^{a} x\bigr)-k^{-a}j \bigl(k^{a} x\bigr), Nlt \bigr) \\ &\quad =\mu \bigl(h\bigl(k^{a} x\bigr)-j\bigl(k^{a} x\bigr), k^{b a}Nlt \bigr) \\ &\quad \geq \Psi \bigl(k^{a} x, k^{b a}Nt \bigr) \\ &\quad \geq \Psi (x, t) \end{aligned}$$

for all \(x \in M\). Hence \(\eta (Rh-Rj)\leq N \eta (h-j)\) for all \(h, j\in P_{\eta }\), which means that R is an η-strict contraction. Replacing x with \(k^{a} x\) in (3.28), we have

$$\begin{aligned} &\mu \biggl(\frac{f(k^{2a} x)}{k^{a}}-f\bigl(k^{a} x \bigr), t \biggr)\geq \Psi \bigl(k^{a} x, t\bigr) \end{aligned}$$
(3.29)

for all \(x\in M\), and therefore

$$\begin{aligned} &\mu \bigl(k^{-2a}f\bigl(k^{2a} x \bigr)-k^{-a}f\bigl(k^{a} x\bigr), Nt \bigr) \\ &\quad =\mu \bigl(k^{-a}f\bigl(k^{2a} x\bigr)-f \bigl(k^{a}x\bigr), k^{b a}Nt \bigr) \\ &\quad \geq \Psi \bigl(k^{a} x, k^{b a}Nt \bigr)\geq \Psi (x, t) \end{aligned}$$
(3.30)

for all \(x \in M\). Now

$$\begin{aligned} &\mu \biggl(\frac{f(k^{2a} x)}{k^{2a}}-f(x), \biggl( \frac{k^{2}-2k+1}{k^{2}-k} \biggr)^{b} (Nt+t) \biggr) \\ &\quad \geq \mu \biggl(\frac{f(k^{2a} x)}{k^{2a}}- \frac{f(k^{a} x)}{k^{a}}, Nt \biggr) \wedge \mu \biggl( \frac{f(k^{a} x)}{k^{a}}-f(x), t \biggr) \\ &\quad \geq \Psi (x,t) \end{aligned}$$
(3.31)

for all \(x \in M\). In (3.31), replacing x with \(k^{a} x\) and \((\frac{k^{2}-2k+1}{k^{2}-k} )^{b} (Nt+t)\) with \(k^{b a} (\frac{k^{2}-2k+1}{k^{2}-k} )^{b}(N^{2} t+Nt)\), we obtain

$$\begin{aligned} &\mu \biggl(\frac{f(k^{3a} x)}{k^{2a}}-f\bigl(k^{a} x \bigr), k^{b a} \biggl( \frac{k^{2}-2k+1}{k^{2}-k} \biggr)^{b} \bigl(N^{2} t+Nt\bigr) \biggr) \\ &\quad \geq \Psi \bigl(k^{a} x, k^{b a} N t \bigr)\geq \Psi (x, t) \end{aligned}$$
(3.32)

for all \(x \in E\). Therefore,

$$\begin{aligned} \mu \biggl(\frac{f(k^{3a} x)}{k^{3a}}-\frac{f(k^{a} x)}{k^{ a}}, \biggl( \frac{k^{2}-2k+1}{k^{2}-k} \biggr)^{b}\bigl(N^{2} t+Nt\bigr) \biggr) \geq \Psi (x, t) \end{aligned}$$
(3.33)

for all \(x \in M\), and so

$$\begin{aligned} &\mu \biggl(\frac{f(k^{3a} x)}{k^{3a}}-f(x), \biggl( \frac{k^{2}-2k+1}{k^{2}-k} \biggr)^{b} \biggl( \biggl( \frac{k^{2}-2k+1}{k^{2}-k} \biggr)^{b} \bigl(N^{2} t+Nt\bigr)+t \biggr) \biggr) \\ &\quad \geq \mu \biggl(\frac{f(k^{3a} x)}{k^{3a}}- \frac{f(k^{a} x)}{k^{a}}, \biggl( \frac{k^{2}-2k+1}{k^{2}-k} \biggr)^{b} \bigl(N^{2} t+Nt \bigr) \biggr) \wedge \mu \biggl( \frac{f(k^{a} x)}{k^{a}}-f(x), t \biggr) \\ &\quad \geq \Psi (x,t) \end{aligned}$$
(3.34)

for all \(x\in M\). Generalizing the above inequality, we get

$$\begin{aligned} &\mu \Biggl(\frac{f(k^{am} x)}{k^{am}}-f(x), \\ &\qquad {} \Biggl( \biggl( \biggl(\frac{k^{2}-2k+1}{k^{2}-k} \biggr)^{b} N \biggr)^{m-1}+ \biggl(\frac{k^{2}-2k+1}{k^{2}-k} \biggr)^{b}\sum _{i=1}^{m-1} \biggl( \biggl( \frac{k^{2}-2k+1}{k^{2}-k} \biggr)^{b} N \biggr)^{i-1} \Biggr)t \Biggr) \\ &\quad \geq \Psi (x,t) \end{aligned}$$
(3.35)

for all \(x\in M\) and a positive integer m. Hence we have

$$\begin{aligned} &\eta \bigl(R^{m} f-f\bigr) \\ &\quad \leq \biggl( \biggl(\frac{k^{2}-2k+1}{k^{2}-k} \biggr)^{b} N \biggr)^{m-1}+ \biggl(\frac{k^{2}-2k+1}{k^{2}-k} \biggr)^{b}\sum _{i=1}^{m-1} \biggl( \biggl( \frac{k^{2}-2k+1}{k^{2}-k} \biggr)^{b} N \biggr)^{i-1} \\ &\quad \leq \biggl(\frac{k^{2}-2k+1}{k^{2}-k} \biggr)^{b}\sum _{i=1}^{m} \biggl( \biggl(\frac{k^{2}-2k+1}{k^{2}-k} \biggr)^{b} N \biggr)^{i-1} \leq \frac{ (\frac{k^{2}-2k+1}{k^{2}-k} )^{b}}{1- (\frac{k^{2}-2k+1}{k^{2}-k} )^{b} N}. \end{aligned}$$
(3.36)

Now, one can easily prove that \(\{R^{m}(f)\}\) η-converges to \(A\in P_{\eta }\) (see [37]). Therefore, (3.36) becomes

$$\begin{aligned} &\eta (A-f)\leq \frac{ (\frac{k^{2}-2k+1}{k^{2}-k} )^{b}}{1- (\frac{k^{2}-2k+1}{k^{2}-k} )^{b} N}, \end{aligned}$$
(3.37)

which implies

$$\begin{aligned} \mu \biggl(A(x)-f(x), \frac{ (\frac{k^{2}-2k+1}{k^{2}-k} )^{b}}{1- (\frac{k^{2}-2k+1}{k^{2}-k} )^{b} N}t \biggr)&\geq \Psi (x,t) \\ & =\rho \biggl(x,0,\dots ,0, \biggl(\frac{k^{2}-2k+1}{k^{2}-k} \biggr)^{b}k^{b}N^{\frac{a-1}{2}}t \biggr) \end{aligned}$$
(3.38)

for all \(x\in M\), and hence we have

$$ \mu \biggl(A(x)-f(x), \frac{t}{k^{b}N^{\frac{a-1}{2}} (1- (\frac{k^{2}-2k+1}{k^{2}-k} )^{b} N )} \biggr)\geq \rho (x,0,\dots ,0,t) $$

for all \(x\in M\), and hence inequality (3.24) holds. One can easily prove the uniqueness of A (see [37]). □

Conclusion

In this paper, we introduced a new n-variable mixed-type functional equation which satisfies \(f(x)=x+x^{2}\). Mainly, we obtained its general solution and investigated its Hyers–Ulam stability in fuzzy modular spaces by using the fixed point method, and we hope that this research work is a further improvement in the field of functional equations.

Availability of data and materials

Not applicable.

References

  1. 1.

    Govindan, V., Park, C., Pinelas, S., Baskaran, S.: Solution of a 3-D cubic functional equation and its stability. AIMS Math. 5, 1693–1705 (2020)

    MathSciNet  Article  Google Scholar 

  2. 2.

    Jung, S., Popa, D., Rassias, M.T.: On the stability of the linear functional equation in a single variable on complete metric spaces. J. Glob. Optim. 59, 13–16 (2014)

    Article  Google Scholar 

  3. 3.

    Lee, J., Kim, J., Park, C.: A fixed point approach to the stability of an additive-quadratic-cubic-quartic functional equation. Fixed Point Theory Appl. 2010, Article ID 185780 (2010)

    MathSciNet  Article  Google Scholar 

  4. 4.

    Lee, Y., Jung, S., Rassias, M.T.: Uniqueness theorems on functional inequalities concerning cubic-quadratic-additive equation. J. Math. Inequal. 12, 43–61 (2018)

    MathSciNet  Article  Google Scholar 

  5. 5.

    Park, C., Jang, S., Lee, J., Shin, D.: On the stability of an \(AQCQ\)-functional equation in random normed spaces. J. Inequal. Appl. 2011, 34 (2011)

    MathSciNet  Article  Google Scholar 

  6. 6.

    Pinelas, P., Govindan, V., Tamilvanan, K.: Stability of a quartic functional equation. J. Fixed Point Theory Appl. 20, 148 (2018)

    MathSciNet  Article  Google Scholar 

  7. 7.

    Senthil Kumar, B.V., Al-Shaqsi, K., Dutta, H.: Classical stabilities of multiplicative inverse difference and adjoint functional equations. Adv. Differ. Equ. 2020, 215 (2020)

    MathSciNet  Article  Google Scholar 

  8. 8.

    Sayar, K.Y.N., Bergam, A.: Approximate solution of a quadratic functional equation in 2-Banach spaces using fixed point theorem. J. Fixed Point Theory Appl. 22, 3 (2020)

    MathSciNet  Article  Google Scholar 

  9. 9.

    Xu, T.Z., Rassias, J.M., Xu, W.X.: Generalized Ulam-Hyers stability of a general mixed \(AQCQ\)-functional equation in multi-Banach spaces: a fixed point approach. Eur. J. Pure Appl. Math. 3(6), 1032–1047 (2010)

    MathSciNet  MATH  Google Scholar 

  10. 10.

    Park, C., Bodaghi, A.: Two multi-cubic functional equations and some results on the stability in modular spaces. J. Inequal. Appl. 2020, 6 (2020)

    MathSciNet  Article  Google Scholar 

  11. 11.

    Aczel, J., Dhombres, J.: Functional Equations in Several Variables. Cambridge University Press, Cambridge (1989)

    Google Scholar 

  12. 12.

    Bourgin, D.G.: Classes of transformations and bordering transforms. Bull. Am. Math. Soc. 57, 223–237 (1951)

    Article  Google Scholar 

  13. 13.

    Czerwik, S.: Functional Equations and Inequalities in Several Variables. World Scientific, River Edge (2002)

    Google Scholar 

  14. 14.

    Czerwik, S.: On the stability of the quadratic mapping in normed spaces. Abh. Math. Semin. Univ. Hamb. 62, 59–64 (1992)

    MathSciNet  Article  Google Scholar 

  15. 15.

    Gajda, Z.: On stability of additive mappings. Int. J. Math. Math. Sci. 14, 431–434 (1991)

    MathSciNet  Article  Google Scholar 

  16. 16.

    Hyers, D.H., Isac, G., Rassias, T.M.: Stability of Functional Equations in Several Variables. Birkhäuser, Boston (1998)

    Google Scholar 

  17. 17.

    Hyers, D.H., Rassias, T.M.: Approximate homomorphisms. Aequ. Math. 44, 125–153 (1992)

    MathSciNet  Article  Google Scholar 

  18. 18.

    Jung, S.: Hyers–Ulam–Rassias Stability of Functional Equations in Nonlinear Analysis. Springer, New York (2011)

    Google Scholar 

  19. 19.

    Kannappan, P.: Functional Equations and Inequalities with Applications. Springer, New York (2009)

    Google Scholar 

  20. 20.

    Lee, Y., Jung, S., Rassias, M.T.: On an n-dimensional mixed type additive and quadratic functional equation. Appl. Math. Comput. 228, 13–16 (2014)

    MathSciNet  MATH  Google Scholar 

  21. 21.

    Park, C., Rassias, M.T.: Additive functional equations and partial multipliers in \(C^{*}\)-algebras. Rev. R. Acad. Cienc. Exactas Fís. Nat., Ser. A Mat. 113, 2261–2275 (2019)

    MathSciNet  Article  Google Scholar 

  22. 22.

    Rassias, T.M.: Functional Equations and Inequalities. Kluwer Academic, Dordrecht (2000)

    Google Scholar 

  23. 23.

    Sahoo, P.K., Kannappan, P.: Introduction to Functional Equations. CRC Press, Boca Raton (2011)

    Google Scholar 

  24. 24.

    Ulam, S.M.: Problems in Modern Mathematics. Wiley, New York (1940)

    Google Scholar 

  25. 25.

    Hyers, D.H.: On the stability of the linear functional equation. Proc. Natl. Acad. Sci. USA 27, 222–224 (1941)

    MathSciNet  Article  Google Scholar 

  26. 26.

    Rassias, T.M.: On the stability of the linear mapping in Banach spaces. Proc. Am. Math. Soc. 72, 297–300 (1978)

    MathSciNet  Article  Google Scholar 

  27. 27.

    Rassias, J.M.: On approximation of approximately linear mappings by linear mappings. J. Funct. Anal. 46, 126–130 (1982)

    MathSciNet  Article  Google Scholar 

  28. 28.

    Ravi, K., Arunkumar, M., Rassias, J.M.: On the Ulam stability for the orthogonally general Euler–Lagrange type functional equation. Int. J. Math. Sci. 3(8), 36–47 (2008)

    MathSciNet  MATH  Google Scholar 

  29. 29.

    Gavruta, P.: A generalization of the Hyers–Ulam–Rassias stability of approximately additive mappings. J. Math. Anal. Appl. 184, 431–436 (1994)

    MathSciNet  Article  Google Scholar 

  30. 30.

    Shen, Y., Chen, W.: On fuzzy modular spaces. J. Appl. Math. 2013, Article ID 576237 (2013)

    MathSciNet  MATH  Google Scholar 

  31. 31.

    Rassias, J.M.: On the Ulam stability of the mixed type mappings on restricted domains. J. Math. Anal. Appl. 276, 747–762 (2002)

    MathSciNet  Article  Google Scholar 

  32. 32.

    Nakmahachalasint, P.: On the generalized Ulam–Gavruta–Rassias stability of mixed-type linear and Euler–Lagrange–Rassias functional equation. Int. J. Math. Math. Sci. 2007, Article ID 63239 (2007)

    MathSciNet  MATH  Google Scholar 

  33. 33.

    Jun, K., Kim, H.: On the Hyers–Ulam–Rassias stability of a generalized quadratic and additive functional equation. Bull. Korean Math. Soc. 42, 133–148 (2005)

    MathSciNet  Article  Google Scholar 

  34. 34.

    Kumam, P.: Some geometric properties and fixed point theorem in modular spaces. In: Garcia Falset, J., Fuster, L., Sims, B. (eds.) Fixed Point Theorem and Its Applications, pp. 173–188. Yokohama Pub., Yokohama (2004)

    Google Scholar 

  35. 35.

    Kumam, P.: Fixed point theorems for nonexpansive mappings in modular spaces. Arch. Math. 40, 345–353 (2004)

    MathSciNet  MATH  Google Scholar 

  36. 36.

    Wongkum, K., Chaipunya, P., Kumam, P.: Some analogies of the Banach contraction principle in fuzzy modular spaces. Sci. World J. 2013, Article ID 205275 (2013)

    Article  Google Scholar 

  37. 37.

    Wongkum, K., Kumam, P.: The stability of sextic functional equation in fuzzy modular spaces. J. Nonlinear Sci. Appl. 9, 3555–3569 (2016)

    MathSciNet  Article  Google Scholar 

  38. 38.

    Zolfaghari, S., Ebadian, A., Ostadbashi, S., de la Sen, M., Eshaghi Gordji, M.: A fixed point approach to the Hyers–Ulam stability of an AQ functional equation in probabilistic modular spaces. Int. J. Nonlinear Anal. Appl. 4(2), 89–101 (2013)

    MATH  Google Scholar 

Download references

Acknowledgements

We would like to express our sincere gratitude to the anonymous referee for his/her helpful comments that helped to improve the quality of the manuscript.

Funding

Not applicable.

Author information

Affiliations

Authors

Contributions

The authors equally conceived of the study, participated in its design and coordination, drafted the manuscript, participated in the sequence alignment, and read and approved the final manuscript.

Corresponding authors

Correspondence to Choonkil Park or Jung Rye Lee.

Ethics declarations

Competing interests

The authors declare that they have no competing interests.

Rights and permissions

Open Access This article is licensed under a Creative Commons Attribution 4.0 International License, which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons licence, and indicate if changes were made. The images or other third party material in this article are included in the article’s Creative Commons licence, unless indicated otherwise in a credit line to the material. If material is not included in the article’s Creative Commons licence and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder. To view a copy of this licence, visit http://creativecommons.org/licenses/by/4.0/.

Reprints and Permissions

About this article

Verify currency and authenticity via CrossMark

Cite this article

Ramdoss, M., Pachaiyappan, D., Park, C. et al. Stability of a generalized n-variable mixed-type functional equation in fuzzy modular spaces. J Inequal Appl 2021, 61 (2021). https://doi.org/10.1186/s13660-021-02594-y

Download citation

MSC

  • 39B72
  • 68U10
  • 94A08
  • 47H10
  • 39B52

Keywords

  • Hyers–Ulam stability
  • Additive and quadratic functional equation
  • Fuzzy modular space
  • Fixed point method
\