On Approximate Multi-Cubic Mappings in 2-Banach Spaces

Abstract

The present article presents a system of symmetric equations defining multi-cubic mappings (M-CMs). Next, we describe how these mappings are structured and obtain an equation for describing them. Moreover, we Address the Hyers-Ulam stability (H-UStab) in the sense of Găvruţa for a symmetric multi-cubic equation through the application of the so-called Hyers (direct) method in the setting of 2-Banach spaces. For a typical case, by means of a norm, induced from a 2-norm of ${\mathbb{R}}^d$, we examine the stability and hyperstability of a mapping $f:{\mathbb{R}}^{dn}⟶{\mathbb{R}}^d$ by using a fixed point (FP) result.

1. Introduction

It is widely known that functional equations (FEQs) are a significant, necessary, and entertaining aspect of nonlinear analysis, utilizing straightforward algebraic techniques to produce fascinating solutions. Ulam stability [1], which has been proposed for group homomorphisms and answered by Hyers [2] for additive mappings on Banach algebras, is a crucial concept in studying FEQs and their solutions. Aoki later (see [3]) made a considerable generalization of Hyers’ conclusion, Th. M. Rassias [4] and Găvruţa [5] introduced some generalized version of the stability with some generic control function governs the stability.

This theory examines whether a function that roughly fulfills a specific FEQ is near a function that fulfills the equation perfectly. In other words, an FEQ $\mathcal{F}$ is called stable if any function f roughly satisfies the FEQ $\mathcal{F}$ must be colse to an exact solution. In the case that f is an exact solution of $\mathcal{F}$, we say $\mathcal{F}$ is hyperstable. In the last two decades, many researchers across various fields have explored different types of the system of FEQs (multiple variable mappings) stability, which are available, for example, in [6,7,8,9,10]. We recall the most important Theorem in Ulam’s stability for the famous additive Cauchy FEQ

$$\Gamma ({\vartheta }_1+{\vartheta }_2)=\Gamma \left({\vartheta }_1\right)+\Gamma \left({\vartheta }_2\right),{\vartheta }_1,{\vartheta }_2\in E,{\vartheta }_1+{\vartheta }_2\in X_1,$$

for $\Gamma :X_1\to X_2$, with two real normed spaces $X_1$ and $X_2$ (see, e.g., [11]).

Theorem 1.

Let $X_0:=X_1\setminus \left\{0\right\}$, $o\ge 0$, fix $1\ne \varkappa \in \mathbb{R}$, and $\Gamma :X_1\to X_2$ be such that

$$\begin{array}{c}\hfill \parallel \Gamma ({\vartheta }_1+{\vartheta }_2)-\Gamma \left({\vartheta }_1\right)-\Gamma \left({\vartheta }_2\right)\parallel \le o(\parallel {\vartheta }_1{\parallel }^{\varkappa }+\parallel {\vartheta }_2{\parallel }^{\varkappa }),{\vartheta }_1,{\vartheta }_2\in X_0.\end{array}$$

(1)

Then, the following hold:

(i) 

If $X_2$ is complete, then a unique mapping $M:X_1\to X_2$ exists:

$$\begin{array}{c}\hfill M({\vartheta }_1+{\vartheta }_2)=M\left({\vartheta }_1\right)+M\left({\vartheta }_2\right),{\vartheta }_1,{\vartheta }_2\in X_1,\end{array}$$

(2)

and

$$\begin{array}{c}\hfill \parallel \Gamma \left({\vartheta }_1\right)-M\left({\vartheta }_1\right)\parallel \le {\displaystyle \frac{o}{|1-2^{\sigma -1}|}}{\parallel {\vartheta }_1\parallel }^{\sigma },{\vartheta }_1\in X_0.\end{array}$$

(3)

(ii) 

If $\varkappa <0$, then $\Gamma \left({\vartheta }_1\right)$ is additive, i.e., it is a solution to (2).

Definition 1.

Let $V_1$ and $V_2$ be linear spaces with an integer number $i\ge 2$. A mapping $\Gamma :V_1^i⟶V_2$ is called M-CM if it is cubic (satisfies (5)) in each variable.

Here, we recall that the first cubic FEQ

$$\begin{array}{c}\hfill \Gamma ({\tau }_1+2{\tau }_2)=3\Gamma ({\tau }_1+{\tau }_2)+\Gamma ({\tau }_1-{\tau }_2)-3\Gamma \left({\tau }_1\right)+6\Gamma \left({\tau }_2\right).\end{array}$$

(4)

was introduced by J. M. Rassias [12]. After that, Jun and Kim [13,14] proposed the following cubic equations:

$$\begin{array}{c}\hfill \Gamma (2{\tau }_1+{\tau }_2)+\Gamma (2{\tau }_1-{\tau }_2)=2\Gamma ({\tau }_1+{\tau }_2)+2\Gamma ({\tau }_1-{\tau }_2)+12\Gamma \left({\tau }_1\right)\end{array}$$

(5)

and

$$\begin{array}{c}\hfill \Gamma ({\tau }_1+2{\tau }_2)+\Gamma ({\tau }_1-2{\tau }_2)=4\Gamma ({\tau }_1+{\tau }_2)+4\Gamma ({\tau }_1-{\tau }_2)-6\Gamma \left({\tau }_1\right).\end{array}$$

(6)

Moreover, they studied the stability of FEQs (5) and (6) in the setting of Banach spaces. In [14], the authors introduced a general solution of (6) and examined its Hyers–Ulam–Rassias stability. They proved the following theorem under an approximately cubic condition (with X a real vector space and Y a real Banach space).

Theorem 2.

Take a function ϕ such that ${\sum }_{i=0}^{+\infty }{\displaystyle \frac{\varphi (3^i{\tau }_1,3^i{\tau }_1)}{{27}^i}}$ converges and ${lim}_{n\to +\infty }{\displaystyle \frac{\varphi (3^n{\tau }_1,3^n{\tau }_1)}{{27}^n}}=0,\forall {\tau }_1,{\tau }_2\in X\setminus \left\{0\right\}$. Let a function $\Gamma :X\to Y$ satisfies

$$\left|\right|\Gamma ({\tau }_1+2{\tau }_2)+\Gamma ({\tau }_1-2{\tau }_2)-4\Gamma ({\tau }_1+{\tau }_2)-4\Gamma ({\tau }_1-{\tau }_2)+6\Gamma \left({\tau }_1\right)\left|\right|\le \varphi ({\tau }_1,{\tau }_2)$$

$$\left|\right|\Gamma \left(2{\tau }_1\right)+8\Gamma (-{\tau }_1)\left|\right|\le \delta \forall {\tau }_1,{\tau }_2\in X\setminus \left\{0\right\},\delta \ge 0.$$

Then, a unique cubic function $T:X\to Y$ exists, satisfies (6) and

$$\begin{array}{cc}\hfill \left|\right|\Gamma \left({\tau }_1\right)-T\left({\tau }_1\right)\left|\right|\le \sum _{i=1}^{+\infty }\left[{\displaystyle \frac{1}{2}}\right({\displaystyle \frac{1}{{27}^i}}& -{\displaystyle \frac{{(-1)}^{i-1}}{{37}^i}}\left)\varphi (3^{i-1}{\tau }_1,3^{i-1}{\tau }_1)\right)\hfill \\ \hfill +{\displaystyle \frac{1}{2}}({\displaystyle \frac{1}{{27}^i}}+& {\displaystyle \frac{{(-1)}^{i-1}}{{37}^i}}\left)\varphi (-3^{i-1}{\tau }_1,-3^{i-1}{\tau }_1)\right]+{\displaystyle \frac{2\delta +2\left|\right|f\left(0\right)\left|\right|}{13}}\hfill \end{array}$$

(7)

∀${\tau }_1\in X$. The function T is given by

$$T\left({\tau }_1\right)=\underset{n\to +\infty }{lim}{\displaystyle \frac{f\left(3^n{\tau }_1\right)}{{27}^n}},\forall {\tau }_1\in X.$$

It should be remarked that the authors in [15] proved that the following FEQ

$$\begin{array}{c}\hfill \Gamma ({\tau }_1+2{\tau }_2)+\Gamma ({\tau }_1-2{\tau }_2)=4[\Gamma ({\tau }_1+{\tau }_2)+4\Gamma ({\tau }_1-{\tau }_2)]-\Gamma 2\left({\tau }_1\right)+2\Gamma \left({\tau }_1\right)\end{array}$$

(8)

is equivalent to (5). In [16], the authors generalized the FEQ (8) according to the coefficient of the cosine function, which is somewhat different from the equations above as follows:

$$\begin{array}{cc}\hfill \Gamma ({\tau }_1+s{\tau }_2)+\Gamma ({\tau }_1-s{\tau }_2)=& 2[2cos\left({\displaystyle \frac{s\pi }{2}}\right)+s^2-1]\Gamma \left({\tau }_1\right)\hfill \\ \hfill -{\displaystyle \frac{1}{2}}[cos\left({\displaystyle \frac{s\pi }{2}}\right)+& s^2-1]\Gamma \left(2{\tau }_1\right)+s^2[\Gamma ({\tau }_1+{\tau }_2)+\Gamma ({\tau }_1-{\tau }_2)].\hfill \end{array}$$

where $s\ge 2$ is an integer. It is obvious to see that when $s=2$, we obtain (8).

We recall the Hyers–Ulams-Rasssias stability result of (5) that has been proved in [13] (X is a real vector space and Y a Banach space)

Theorem 3.

Let $\Phi :X^2\to {\mathbb{R}}^{+}$ satisfying

$$\sum _{i=0}^{+\infty }{\displaystyle \frac{\Phi (2^i{\tau }_1,0)}{8^i}}\left(\sum _{i=0}^{+\infty }{8}^i\Phi ({\displaystyle \frac{{\tau }_1}{2^i}},0),\mathit{respectively}\right)$$

converges and

$$\underset{n\to +\infty }{lim}{\displaystyle \frac{\Phi (2^n{\tau }_1,2^n{\tau }_2)}{8^n}}\left(\underset{n\to +\infty }{lim}{8}^n\Phi ({\displaystyle \frac{{\tau }_1}{2^n}},{\displaystyle \frac{{\tau }_2}{2^n}})=0\right),\forall {\tau }_1,{\tau }_2\in X.$$

Suppose that a function $\Gamma :X\to Y$ satisfies

$$\left|\right|\Gamma (2{\tau }_1+{\tau }_2)+\Gamma (2{\tau }_1-{\tau }_2)=2\Gamma ({\tau }_1+{\tau }_2)+2\Gamma ({\tau }_1-{\tau }_2)-12\Gamma \left({\tau }_1\right)\left|\right|\le \Phi ({\tau }_1,{\tau }_2),\forall {\tau }_1,{\tau }_2\in X.$$

(9)

Then, a unique cubic function $T:X\to Y$ exists, satisfies (9) and the inequality

$$\left|\right|\Gamma \left({\tau }_1\right)-T\left({\tau }_1\right)\left|\right|\le {\displaystyle \frac{1}{16}}\sum _{i=0}^{+\infty }{\displaystyle \frac{\Phi (2^i{\tau }_1,0)}{8^i}}$$

$$\left(\left|\right|\Gamma \left({\tau }_1\right)-T\left({\tau }_1\right)\left|\right|\le {\displaystyle \frac{1}{16}}\sum _{i=0}^{+\infty }{8}^i\Phi ({\displaystyle \frac{{\tau }_1}{2^i}},0)\right),\forall {\tau }_1\in X.$$

The function T is provided by

$$T\left({\tau }_1\right)=\underset{n\to +\infty }{lim}{\displaystyle \frac{f\left(2^n{\tau }_1\right)}{8^n}}\left(\underset{n\to +\infty }{lim}{8}^{n}f\left({\displaystyle \frac{{\tau }_1}{2^n}}\right)\right),\forall {\tau }_1\in X.$$

In [15], Chu and Kang demonstrated that the FEQ

$$\begin{array}{c}\hfill \Gamma ({\tau }_1+2{\tau }_2)+\Gamma ({\tau }_1-2{\tau }_2)=4[\Gamma ({\tau }_1+{\tau }_2)+\Gamma ({\tau }_1-{\tau }_2)]-\Gamma \left(2{\tau }_1\right)+2\Gamma \left({\tau }_1\right)\end{array}$$

(10)

is equivalent to Equation (5). Moreover, they proved many interesting results such as the following theorem (with an integer number $n\ge 2$; a normed vector space X; and a Banach space Y).

Theorem 4.

Let $\Psi :X\to Y$ be a mapping with $\Psi \left(0\right)=0$, for which there is the function $\varphi :X^n\to [0,+\infty )$:

$$\tilde{\varphi }(a_1,a_2,\cdots ,a_n):=\sum _{j=0}^{+\infty }{8}^j\varphi (2^{-j}{a}_1,\cdots ,2^{-j}{a}_n)<+\infty$$

and (with the difference operator D)

$$\left|\right|D\Psi (a_1,a_2,\cdots ,a_n)\left|\right|\le \varphi (a_1,a_2,\cdots ,a_n),a_1,\cdots ,a_n\in X.$$

Then, $\forall m\in \{1,2,\cdots ,n-1\}$, a unique n-dimensional cubic mapping Γ exists:

$$\left|\right|\Psi \left(a\right)-\Gamma \left(a\right)\left|\right|\le {\displaystyle \frac{1}{m}}\tilde{\varphi }(\underset{m-\mathit{terms}}{\underbrace{a,a,\cdots ,a,a}},0,\cdots ,0),$$

for all $a\in X$

Such mappings were introduced in [17]. An alternative representation of M-CMs and their stability using Equation (6) has been investigated recently by Bodaghi in [18]. In [18], some mappings of multi-variables such as multicubic and some others are introduced. Additionally, a well known FPT has been applied to examine the H-UStab of multi-cubic and others in non-Archimedean normed spaces. For a certain M-CM (eq. (2.17)) in [18], the following theorem has been proved:

Theorem 5.

Fix $\alpha \in \{-1,1\}$, let $V_1$ be a linear space, and $V_2$ be a complete non-Archimidean normed space. Let $\phi :V_1^n\times V_1^n\to {\mathbb{R}}_{+}$ be a mapping:

$$\underset{t\to +\infty }{lim}{\left({\displaystyle \frac{1}{{\left|2\right|}^{(3n-k)\alpha }}}\right)}^t\phi (2^{t\alpha }{a}_1,2^{t\alpha }{a}_2)=0,\forall a_1,a_2\in V_1^n.$$

Additionally, take a mapping $f:V_1^n\to V_2$ with some special properties:

$$D_{qc}f(a_1,a_2)\le \phi (a_1,a_2),\forall a_1,a_2\in V_1^n.$$

Then, a unique multiquadratic-cubic mapping $M_{qc}:V_1^n\to V_2$:

$$\left|\right|f\left(\tau \right)-M_{qc}\left(\tau \right)\left|\right|\le \underset{t\in {\mathbb{N}}_0}{sup}{\displaystyle \frac{1}{{\left|2\right|}^{(3n-k)\frac{\alpha +1}{2}}}}{\left({\displaystyle \frac{1}{{\left|2\right|}^{(3n-k)\alpha }}}\right)}^t\phi (0,2^{t\alpha +{\displaystyle \frac{\alpha -1}{2}}}\tau )$$

for all $\tau \in V_1^n$.

In [19], the authors reduced the system of n cubic equations that defines M-CM between vector spaces to obtain a single FEQ. They proved the following stability results for an M-CM FEQ (Equation (6) in [19]).

Theorem 6.

Fix $\beta \in \{-1,1\}$, and $\alpha \in [0,+\infty )$. Take a linear space $V_1$ and a Banach space $V_2$. Let a mapping $f:V_1^n\to V_2$ for which a function $\varphi :V_1^n\times V_1^n\to [-\alpha ,+\infty )$:

$$\widehat{\varphi }(x_1,x_2):=\sum _{j=0}^{+\infty }{\displaystyle \frac{1}{2^{3n\beta j}}}\varphi \left(2^{{\displaystyle \frac{\beta -1}{2}}+\beta j}{x}_1,2^{{\displaystyle \frac{\beta -1}{2}}+\beta j}{x}_2\right)<+\infty$$

and

$$\left|\right|D_{c}f(x_1,x_2)\left|\right|\le \alpha \left({\displaystyle \frac{\beta +1}{2}}\right)+\varphi (x_1,x_2),\forall x_1,x_2\in V_1^n.$$

Then, there is a solution $H:V_1^n\to V_2$ (of (6)):

$$\left|\right|f\left(x\right)-H\left(x\right)\left|\right|\le {\displaystyle \frac{1}{8^{(\beta +1)n/2}}}\left[{\displaystyle \frac{2^{3n\beta }\alpha }{(2^{3n\beta }-1)}}\left({\displaystyle \frac{\beta +1}{2}}\right)+\widehat{\varphi }(0,x)\right],\forall x\in V_1^n.$$

Moreover, if H has the cubic condition in each component, then it is is a unique M-CM.

In [17], Bodaghi and Shojaee described the structure of M-CMs and unified the system of FEQs defining M-CMs to a single equation (calling them M-CM FEQ). They also studied H-UStab for an M-CM FEQ (Equation (6) in [17]) and proved the following theorem (with a linear space $V_1$ and a Banach space $V_2$)

Theorem 7.

Fix $\beta \in \{-1,1\}$. Let a function $\varphi :V_1^n\times V_1^n\to {\mathbb{R}}_{+}$ such that

$$\underset{l\to +\infty }{lim}{\left({\displaystyle \frac{1}{{\left|2\right|}^{3n\beta }}}\right)}^l\varphi (2^{l\beta }{a}_1,2^{l\beta }{a}_2)=0,\forall a_1,a_2\in V_1^n,$$

and

$$\Phi \left(a\right):={\displaystyle \frac{1}{2^{3n(\beta +1)/2+n}}}\sum _{l=0}^{+\infty }{\left({\displaystyle \frac{1}{2^{3n\beta }}}\right)}^l\varphi \left(2^{\beta l+{\displaystyle \frac{\beta -1}{2}}}a,0\right)<+\infty ,\forall a\in V_1^n.$$

Also assume that $\Psi :V_1^n\to V_2$ is a mapping satisfying (with a difference operator D)

$$\left|\right|D\Psi (a_1,a_2){\left|\right|}_Y\le \varphi (a_1,a_2),\forall a_1,a_2\in V_1^n.$$

Then, there is a unique M-CM $H:V_1^n\to V_2$:

$$\left|\right|\Psi \left(a\right)-H\left(a\right)\left|\right|\le \Phi \left(a\right),\forall a\in V_1^n.$$

Note that the M-CM FEQs corresponding to (4), (6), and (10) were obtained in [18,19,20], respectively.

Gähler [21,22] pioneered the concept of a 2-normed space, and then White [23] introduced the notion of 2-Banach spaces. Next, Lewandowska defined and investigated sets in the sense of 2-normed and generalized 2-normed spaces [24,25]. Here, we recall that many H-UStab problems for various FEQs and mappings in the setting of 2-Banach spaces; see, for instance, [6,7,26,27] and other resources. In [27], the authors proved new types of stability and hyperstability results for (5) in 2-Banach spaces through the following theorem (with E representing a normed space, $(Y,||·,·|\left|\right)$ a real 2-Banach space, and $Y_0\subset Y$ containing two linearly independent vectors).

Theorem 8.

Suppose two functions $h_1,h_2:E_0\times Y_0\to {\mathbb{R}}_{+}$ exist:

$$U:=\{n\in \mathbb{N}:{\alpha }_n<1\}\ne \varphi$$

where

$${\alpha }_n:=2{\nu }_1(3n-1){\nu }_2(3n-1+2{\nu }_1(1-n){\nu }_2(1-n)+12{\nu }_1\left(n\right){\nu }_2\left(n\right)+{\nu }_1(4n-1){\nu }_2(4n-1)$$

$${\lambda }_i\left(n\right):=inf\{t\in {\mathbb{R}}_{+}:h_i(n{\tau }_1,z)\le t{h}_i({\tau }_1,z),{\tau }_1\in E_0,z\in Y_0\}\forall n\in \mathbb{N},i=1,2.$$

Let $\Gamma :X\to Y$ satisfy

$$\left|\right|\Gamma (2{\tau }_1+{\tau }_2)+\Gamma (2{\tau }_1-{\tau }_2)-2\Gamma ({\tau }_1+{\tau }_2)-2\Gamma ({\tau }_1-{\tau }_2)-12\Gamma \left({\tau }_1\right),z\left|\right|\le h_1({\tau }_1,z)h_2({\tau }_2,z)$$

for all ${\tau }_1,{\tau }_2\in E_0,z\in Y_0$ such that ${\tau }_1+{\tau }_2,{\tau }_1-{\tau }_2,2{\tau }_1+{\tau }_2,2{\tau }_1-{\tau }_2\ne 0$. Then, a unique cubic function $H:E\to Y$ exists:

$$\left|\right|\Gamma \left({\tau }_1\right)-H\left({\tau }_1\right),z\left|\right|\le {\lambda }_0{h}_1({\tau }_1,z)h_2({\tau }_1,z),\forall {\tau }_1,z\in X_0,$$

and

$${\lambda }_0:=\underset{n\in U}{inf}\left\{{\displaystyle \frac{{\nu }_1\left(n\right){\nu }_2(2n-1)}{1-{\alpha }_n}}\right\}.$$

From now on, $\mathbb{N}$, $\mathbb{Z}$, and $\mathbb{Q}$ denote the set of all positive integers, integers, and rational numbers, respectively.

Definition 2.

Let a two-dimensional (at least) real linear space Ω , over $\mathbb{R}$. Assume $\parallel ·,·\parallel :\Omega \times \Omega ⟶\mathbb{R}$ is a function fulfills the following properties ($\forall a_1,a_2,a_3\in \Omega$ and $\alpha \in \mathbb{R}$):

(i) 

$\parallel a_1,a_2\parallel =0$ if and only if $a_1$ and $a_2$ are linearly dependent;

(ii) 

$\parallel a_1,a_2\parallel =\parallel a_2,a_1\parallel ;$

(iii) 

$\parallel \alpha a_1,a_2\parallel =\left|\alpha \right|\parallel a_1,a_2\parallel$;

(iv) 

$\parallel a_1,a_2+a_3\parallel \le \parallel a_1,a_2\parallel +\parallel a_1,a_3\parallel$.

Then, $\parallel ·,·\parallel$ is called a 2-norm on Ω , and $(\Omega ,\parallel ·,·\parallel )$ is called a 2-normed space.

We recall some known basic notions concerning 2-normed spaces as follows. Consider a sequence ${\left(u_m\right)}_m\in \mathbb{N}$ of elements of a 2-normed space $(\Omega ,\parallel ·,·\parallel )$. This mentioned sequence is said to be 2-Cauchy if there are linearly independent $v,w\in \Omega$ for which

$$\underset{k,j\to +\infty }{lim}\parallel u_k-u_j,v\parallel =\underset{k,j\to +\infty }{lim}\parallel u_k-u_j,w\parallel =0.$$

Moreover, ${\left(u_m\right)}_m\in \mathbb{N}$ is called 2-convergent provided there is a $v\in \Omega$ with ${lim}_{m\to +\infty }\parallel u_m-v,w\parallel =0$. A 2-normed space $(\Omega ,\parallel ·,·\parallel )$ is called a 2-Banach space if every 2-Cauchy sequence in $\Omega$ is 2-convergent.

In the sequel, v is called the limit of the sequence ${\left(u_m\right)}_m\in \mathbb{N}$, and we denote it by ${lim}_{m\to +\infty }{u}_m.$ Of course, each convergent sequence has its own limit. The usual limit characteristics apply. Now, we recall the following results.

Lemma 1.

Let $(\Omega ,\parallel ·,·\parallel )$ be a 2-normed space. Then, the following assertions hold:

(i) 

$|\parallel {\vartheta }_1,{\vartheta }_2\parallel -|{\vartheta }_3,{\vartheta }_2\parallel |\le \parallel {\vartheta }_1-{\vartheta }_3,{\vartheta }_2\parallel$ for all ${\vartheta }_1,{\vartheta }_2,{\vartheta }_3\in \Omega$;

(ii) 

If $\parallel {\vartheta }_1,{\vartheta }_2\parallel =0$ for all ${\vartheta }_2\in \Omega$, then ${\vartheta }_1=0$;

(iii) 

For any 2-convergent sequence ${\left(u_m\right)}_m\in \mathbb{N}$ in $(\Omega ,\parallel ·,·\parallel )$, we have

$$\underset{m\to +\infty }{lim}\parallel u_m,{\vartheta }_2\parallel =\parallel \underset{m\to +\infty }{lim}{u}_m,{\vartheta }_2\parallel ,$$

for all ${\vartheta }_2\in \Omega$.

Motivated by the mentioned discussion earlier, in this study, we consider the FEQ

$$\begin{array}{cc}\hfill & \Gamma (2{\tau }_1+{\tau }_2)+\Gamma (2{\tau }_1-{\tau }_2)+\Gamma \left(2{\tau }_1\right)=2\Gamma ({\tau }_1+{\tau }_2)+2\Gamma ({\tau }_1-{\tau }_2)+20\Gamma \left({\tau }_1\right)\hfill \end{array}$$

(11)

it differs somewhat from (5). In fact, by using FEQ (11), in this paper, in contrast to the previous form, we define an alternate form of M-CM. Additionally, we describe the structure of these mappings. Thus, we reduce the system of n equations defining the M-CMs to obtain a single FEQ. Furthermore, we establish the Găvruţa and Hyers–Ulam–Rassias stability and hyperstabilty for such FEQs in 2-normed spaces by using the FP and direct methods.

2. Representation of M-CMs as an Equation

We commence this section by an elementary result. For vector spaces $V_1$ and $V_2$ over rationals, if every mapping $\Gamma :V_1⟶V_2$ fulfills either (5) or (11), then it is easy to obtain $\Gamma \left(2x\right)=8\Gamma \left(x\right)$. Therefore, we have the following lemma.

Lemma 2.

Let $V_1$ and $V_2$ be vector spaces over $\mathbb{Q}$. Then, $\Gamma :V_1⟶V_2$ satisfies (5) if and only if Equation (11) is valid for it.

Here and subsequently, $\forall l\in {\mathbb{N}}_0,k\in \mathbb{N}$, $t=(t_1,\dots ,t_k)\in {\{-1,1\}}^k$ and $v=(v_1,\dots ,v_k)\in V^k$, we write $lv:=(l{v}_1,\dots ,l{v}_k)$ and $tv:=(t_1{v}_1,\dots ,t_k{v}_k)$, where $lv$ stands, as usual, for the th power of an element v of the commutative group V.

Definition 3.

Let $V_1$ and $V_2$ be vector spaces over $\mathbb{Q}$, $n\in \mathbb{N}$. A mapping $\Gamma :V_1^n⟶V_1$ is called n-cubic or M-CM if Γ satisfies Equation (11) in each variable, that is,

$$\begin{array}{cc}\hfill & \Gamma ({\vartheta }_1,\dots ,{\vartheta }_{i-1},2{\vartheta }_i+{\vartheta }_i^{\prime },{\vartheta }_{i+1},\dots ,{\vartheta }_n)\hfill \\ \hfill & +\Gamma ({\vartheta }_1,\dots ,{\vartheta }_{i-1},2{\vartheta }_i-{\vartheta }_i^{\prime },{\vartheta }_{i+1},\dots ,{\vartheta }_n)+\Gamma ({\vartheta }_1,\dots ,{\vartheta }_{i-1},2{\vartheta }_i,{\vartheta }_{i+1},\dots ,{\vartheta }_n)\hfill \\ \hfill & =2\Gamma ({\vartheta }_1,\dots ,{\vartheta }_{i-1},{\vartheta }_i+{\vartheta }_i^{\prime },{\vartheta }_{i+1},\dots ,{\vartheta }_n)+2\Gamma ({\vartheta }_1,\dots ,{\vartheta }_{i-1},{\vartheta }_i-{\vartheta }_i^{\prime },{\vartheta }_{i+1},\dots ,{\vartheta }_n)\hfill \\ \hfill & +20\Gamma ({\vartheta }_1,\dots ,{\vartheta }_{i-1},{\vartheta }_i,{\vartheta }_{i+1},\dots ,{\vartheta }_n),\hfill \end{array}$$

for all $i\in \{1,\dots ,n\}$.

Clearly, the mapping $\Gamma :{\mathbb{R}}^n⟶\mathbb{R}$ defined via $\Gamma (u_1,\dots ,u_n)=c{\prod }_{j=1}^n{u}_j^3$ satisfies the system of n-cubic FEQs, indicated in Definition 3, where $c\in \mathbb{R}$.

Let $n\in \mathbb{N}$ with $n\ge 2$ and $v_i^n=(v_{i1},\dots ,v_{in})\in V^n$, where $i\in \{1,2\}$. We shall denote $v_i^n$ by $v_i$ if there is no risk of ambiguity. Fix $n\in \mathbb{N}$. For $v_1,v_2\in V^n$, set

$${\mathbb{E}}^n=\left\{{\mathfrak{E}}_n=(E_1,\dots ,E_n)|E_j\in \{2v_{1j}\pm v_{2j},2v_{1j}\}\right\}$$

and

$${\mathbb{F}}^n=\left\{{\mathfrak{F}}_n=(F_1,\dots ,F_n)|F_j\in \{v_{1j}\pm v_{2j},v_{1j}\}\right\},$$

for all $j\in \{1,\dots ,n\}$. Moreover, for $p,q\in {\mathbb{N}}_0$ with $0\le p,q\le n$, consider the subset ${\mathbb{E}}_p^n$ and ${\mathbb{F}}_q^n$ of ${\mathbb{E}}^n$ and ${\mathbb{F}}^n$, respectively, as follows:

$$\begin{array}{cc}\hfill {\mathbb{E}}_p^n:& =\left\{{\mathfrak{E}}_n\in {\mathbb{E}}^n|\mathrm{Card}\{E_j:S_j=2v_{1j}\}=p\right\},\hfill \end{array}$$

$$\begin{array}{cc}\hfill {\mathbb{F}}_q^n:& =\left\{{\mathfrak{F}}_n\in {\mathbb{F}}^n|\mathrm{Card}\{F_j:F_j=v_{1j}\}=q\right\}.\hfill \end{array}$$

From now on, for M-CMs $\Gamma :V_1^n⟶V_2$, we use the following notations:

$$\begin{array}{c}\hfill \Gamma \left({\mathbb{E}}_p^n\right):=\sum _{{\mathfrak{S}}_n\in {\mathbb{E}}_p^n}\Gamma \left({\mathfrak{E}}_n\right),\Gamma \left({\mathbb{F}}_q^n\right):=\sum _{{\mathfrak{F}}_n\in {\mathbb{F}}_q^n}\Gamma \left({\mathfrak{F}}_n\right)\end{array}$$

(12)

and

$$\begin{array}{c}\hfill \Gamma \left({\mathbb{E}}_p^n,u\right):=\sum _{{\mathfrak{E}}_n\in {\mathbb{E}}_p^n}\Gamma \left({\mathfrak{S}}_n,u\right)\Gamma \left({\mathbb{F}}_p^n,u\right):=\sum _{{\mathfrak{F}}_n\in {\mathbb{F}}_q^n}\Gamma \left({\mathfrak{F}}_n,u\right)(u\in V_1).\end{array}$$

(13)

Definition 4.

Let $s\in \mathbb{Z}\setminus \left\{0\right\}$. The mapping $\Gamma :V_1^n⟶V_2$ is called to satisfy (have) the s-power condition in the jth variable if

$$\Gamma (a_1,\dots ,a_{j-1},2a_j,a_{j+1},\dots ,a_n)=2^s\Gamma (a_1,\cdots ,a_{j-1},a_j,a_{j+1},\dots ,a_n),$$

for all $a_1,\dots ,a_n\in V_1^n$. Specifically, 3-power condition is referred to as cubic condition.

Theorem 9.

Let $\Gamma :V_1^n⟶V_2$ be M-CMs. Then, it satisfies the equation

$$\begin{array}{c}\hfill \sum _{p=0}^n\Gamma \left({\mathbb{E}}_p^n\right)=\sum _{q=0}^n{2}^{n-q}\times {20}^q\Gamma \left({\mathbb{F}}_q^n\right),\end{array}$$

(14)

where $\Gamma \left({\mathbb{E}}_p^n\right)$ and $\Gamma \left({\mathbb{F}}_q^n\right)$ are defined in(12). The converse is valid provided that Γ has the cubic condition in each variable.

Proof. 

Let $\Gamma$ be M-CM. We argue this implication by induction on n that $\Gamma$ satisfies Equation (14). For $n=1$, since f fulfills (11), the first step of induction is valid. Assume that (14) is true for some positive integer $n-1$. We show that it holds for n. By (13), we have

$$\begin{array}{cc}\hfill \sum _{p=0}^n\Gamma \left({\mathbb{E}}_p^n\right)& =\sum _{p=0}^{n-1}\Gamma \left({\mathbb{E}}_p^{n-1},2v_{1,n}+v_{2,n}\right)+\sum _{p=0}^{n-1}\Gamma \left({\mathbb{E}}_p^{n-1},2v_{1,n}-v_{2,n}\right)+\sum _{p=0}^{n-1}\Gamma \left({\mathbb{E}}_p^{n-1},2v_{1,n}\right)\hfill \\ \hfill & =2\sum _{q=0}^n{2}^{n-1-q}\times {20}^q\Gamma \left({\mathbb{F}}_q^{n-1},v_{1,n}+v_{2,n}\right)+2\sum _{q=0}^n{2}^{n-1-q}\times {20}^q\Gamma \left({\mathbb{F}}_q^{n-1},v_{1,n}-v_{2,n}\right)\hfill \\ \hfill & +20\sum _{q=0}^n{2}^{n-1-q}\times {20}^q\Gamma \left({\mathbb{F}}_q^{n-1},v_{1,n}\right)\hfill \\ \hfill & =\sum _{q=0}^n{2}^{n-q}\times {20}^q\Gamma \left({\mathbb{F}}_q^n\right),\hfill \end{array}$$

which shows that Equation (14) holds for n.

Conversely, assume that $\Gamma$ satisfies (14) and has the cubic condition in each variable. We show that $\Gamma$ is cubic in the jth variable, where $j\in \{1,\dots ,n\}$ is arbitrary and fixed. Set

$$\begin{array}{cc}\hfill {\Gamma }^{*}(a_{1j},a_{2j}):& =\Gamma \left(a_{11},\cdots ,a_{1,j-1},a_{1j}+a_{2j},a_{1,j+1},\cdots ,a_{1n}\right)\hfill \\ \hfill & +\Gamma \left(a_{11},\cdots ,a_{1,j-1},a_{1j}-a_{2j},a_{1,j+1},\cdots ,a_{1n}\right),\hfill \end{array}$$

$$\begin{array}{cc}\hfill {\Gamma }^{*}(2a_{1j},a_{2j}):& =\Gamma \left(a_{11},\cdots ,a_{1,j-1},2a_{1j}+a_{2j},a_{1,j+1},\cdots ,a_{1n}\right)\hfill \\ \hfill & +\Gamma \left(a_{11},\cdots ,a_{1,j-1},2a_{1j}-a_{2j},a_{1,j+1},\cdots ,a_{1n}\right).\hfill \end{array}$$

and

$$\begin{array}{cc}\hfill {\Gamma }^{*}\left(2a_{1j}\right):& =f\left(a_{11},\cdots ,a_{1,j-1},2a_{1j},a_{1,j+1},\cdots ,a_{1n}\right)\hfill \\ \hfill & +\Gamma \left(a_{11},\cdots ,a_{1,j-1},2a_{1j}-a_{2j},a_{1,j+1},\cdots ,a_{1n}\right).\hfill \end{array}$$

Putting $a_{2k}=0$$\forall k\in \{1,\dots ,n\}\setminus \left\{j\right\}$ in (14), using the hypothesis, we have the following term for the left side of (14):

$$\begin{array}{cc}\hfill 2^{3(n-1)}\sum _{p=0}^{n-1}& \left(\begin{array}{c}n-1\\ p\end{array}\right)2^{n-1-p}{\Gamma }^{*}(2a_{1j},a_{2j})+2^{3(n-1)}\sum _{p=1}^{n-1}\left(\begin{array}{c}n-1\\ p-1\end{array}\right)2^{n-p}{\Gamma }^{*}\left(2a_{1j}\right)\hfill \\ \hfill & =2^{3(n-1)}\times 3^{n-1}\left[{\Gamma }^{*}(2a_{1j},a_{2j})+{\Gamma }^{*}\left(2a_{1j}\right)\right],\hfill \end{array}$$

(15)

where $\left(\begin{array}{c}m\\ r\end{array}\right)$ is well-known the binomial coefficient. Now, the right side of (14) becomes

$$\begin{array}{cc}\hfill \sum _{q=0}^{n-1}& \left(\begin{array}{c}n-1\\ q\end{array}\right)2^{n-1-p}\times 2^{n-q}\times {20}^q{\Gamma }^{*}(a_{1j},a_{2j})+\sum _{q=1}^n\left(\begin{array}{c}n-1\\ q-1\end{array}\right)2^{n-q}\times {20}^q\Gamma \left(a_1\right)\hfill \\ \hfill 2\sum _{q=0}^{n-1}& \left(\begin{array}{c}n-1\\ q\end{array}\right)4^{n-1-p}\times {20}^q{\Gamma }^{*}(a_{1j},a_{2j})+20\sum _{q=0}^{n-1}\left(\begin{array}{c}n-1\\ q\end{array}\right)2^{n-1-q}\times {20}^q\Gamma \left(v_1\right)\hfill \\ \hfill & =2^{3(n-1)}\times 3^{n-1}\left[2{\Gamma }^{*}(a_{1j},a_{2j})+20\Gamma \left(v_1\right)\right].\hfill \end{array}$$

(16)

Comparing relations (15) and (16), we find that

$${\Gamma }^{*}(a_{1j},a_{2j})+{\Gamma }^{*}\left(2a_{1j}\right)=2{\Gamma }^{*}(a_{1j},a_{2j})+20\Gamma \left(a_1\right),$$

which finishes the proof. □

3. Stability of M-CMs in 2-Normed Spaces

In this section, we present the Găvruţa stability of (14) from a linear space to a 2-Banach space by the direct and FP methods.

3.1. Stability of (14) Using the Direct Method

Here, we establish the Găvruţa and Hyers–Ulam–Rassias stability of (14) from a linear space to a 2-Banach space by the direct method.

For a given mapping $\Gamma :V_1^n⟶V_2$, for simplicity, we use the notation

$$\begin{array}{cc}\hfill {\mathfrak{D}}_c\Gamma (v_1,v_2)& :=\sum _{p=0}^n\Gamma \left({\mathbb{E}}_p^n\right)-\sum _{q=0}^n{2}^{n-q}\times {20}^q\Gamma \left({\mathbb{F}}_q^n\right),\hfill \end{array}$$

where $\Gamma \left({\mathbb{E}}_p^n\right)$ and $\Gamma \left({\mathbb{F}}_q^n\right)$ are defined in (12). Next, we investigate the Găvruţa stability result for (14).

Theorem 10.

Fix $\beta \in \{-1,1\}$. Assume a linear space $V_1$ and a 2-Banach space $V_2$. Let $\Gamma :V_1^n⟶V_2$ is a mapping for which there is a function $\varphi :V^n\times V^n\times W⟶(0,+\infty )$:

$$\begin{array}{c}\hfill \tilde{\varphi }(v_1,v_2,w):=\sum _{j=0}^{+\infty }{\displaystyle \frac{1}{2^{3n\beta j}}}\varphi \left(2^{{\displaystyle \frac{\beta -1}{2}}+\beta j}{v}_1,2^{{\displaystyle \frac{\beta -1}{2}}+\beta j}{v}_2,w\right)<+\infty ,\end{array}$$

(17)

$$\begin{array}{c}\hfill ∥{\mathfrak{D}}_c\Gamma (v_1,v_2),w∥\le \varphi (v_1,v_2,w),\forall v_1,v_2\in V_1^n,w\in V_2.\end{array}$$

(18)

Then, there is a solution $\mathrm{{\rm Y}}:V_1^n⟶V_2$ of (14) such that

$$\begin{array}{c}\hfill ∥\Gamma \left(v\right)-\mathrm{{\rm Y}}\left(v\right),w∥\le {\displaystyle \frac{1}{3^n\times 2^{3\frac{\beta +1}{2}n}}}\tilde{\varphi }(v,0,w),\end{array}$$

(19)

for all $v\in V_1^n$ and $w\in V_2$, where $0=(\stackrel{n-\mathit{times}}{\overbrace{0,\cdots ,0}})$. Moreover, if Γ has the cubic condition in each component, then it is a unique M-CMs.

Proof. 

Putting $v_2=0$ in (18), we have

$$\begin{array}{cc}\hfill & \parallel S_1\Gamma \left(2v_1\right)-S_2\Gamma \left(v_1\right),w\parallel \le \varphi (v_1,0,w),\hfill \end{array}$$

(20)

for all $v_1\in V_1^n$ and $w\in V_2$ in which

$$\begin{array}{c}\hfill S_1=\sum _{p=0}^n\left(\begin{array}{c}n\\ p\end{array}\right)2^{n-p},S_2=\sum _{q=0}^n\left(\begin{array}{c}n\\ q\end{array}\right)2^{n-q}\times 2^{n-q}\times {20}^q.\end{array}$$

It is easily verified that $S_1=3^n$ and $S_2={24}^n$. For the rest of proof, we set $v_1$ by v (unless otherwise stated). From (20) we get

$$\begin{array}{cc}\hfill & \parallel \Gamma \left(2v\right)-2^{3n}\Gamma \left(v\right),w\parallel \le {\displaystyle \frac{1}{3^n}}\varphi (v,0,w),\hfill \end{array}$$

and so the equation above can be rewritten as

$$\begin{array}{c}\hfill ∥{\displaystyle \frac{\Gamma \left(2^{\beta }v\right)}{2^{3n\beta }}}-\Gamma \left(v\right),w∥\le {\displaystyle \frac{1}{3^n\times 2^{3\frac{\beta +1}{2}n}}}\varphi \left(2^{{\displaystyle \frac{\beta -1}{2}}}v,0,w\right),\end{array}$$

(21)

for all $v\in V_1^n$ and $w\in V_2$. Substituting v into $2^{\beta }v$ in (21) and continuing this method, we obtain

$$\begin{array}{c}\hfill ∥{\displaystyle \frac{\Gamma \left(2^{\beta m}v\right)}{2^{3nm\beta }}}-\Gamma \left(v\right),w∥\le {\displaystyle \frac{1}{3^n\times 2^{3\frac{\beta +1}{2}n}}}\sum _{j=0}^{m-1}{\displaystyle \frac{\varphi \left(2^{\frac{\beta -1}{2}+j\beta }v,0,w\right)}{2^{3n\beta j}}},\end{array}$$

(22)

for all $v\in V_1^n$ and $w\in V_2$. On the other hand, by induction

$$\begin{array}{c}\hfill ∥{\displaystyle \frac{\Gamma \left(2^{\beta m}v\right)}{2^{3nm\beta }}}-{\displaystyle \frac{\Gamma \left(2^{\beta l}v\right)}{2^{3nl\beta }}},w∥\le {\displaystyle \frac{1}{3^n\times 2^{3\frac{\beta +1}{2}n}}}\sum _{j=l}^{m-1}{\displaystyle \frac{\varphi \left(2^{\frac{\beta -1}{2}+j\beta }v,0,w\right)}{2^{3n\beta j}}},\end{array}$$

(23)

for all $v\in V_1^n$, $w\in V_2$, and $m>l\ge 0$. SO, the sequence $\left\{{\displaystyle \frac{\Gamma \left(2^{\beta m}v\right)}{2^{3nm\beta }}}\right\}$ is 2-Cauchy by (17) and (23). Since $V_2$ is a 2-Banach space, one can assume that there is a mapping $\mathrm{{\rm Y}}:V_1^n⟶V_2$ such that

$$\begin{array}{c}\hfill \underset{m\to +\infty }{lim}{\displaystyle \frac{\Gamma \left(2^{\beta m}v\right)}{2^{3nm\beta }}}=\mathrm{{\rm Y}}\left(v\right).\end{array}$$

(24)

From (22) (when m tends to infinity) and using (24), clearly (19) is valid. Now, by interchanging $v_1,v_2$ into $2^m{v}_1,2^m{v}_2$, respectively, in (18) and dividing to $2^{3nm\beta }$, we obtain

$${\displaystyle \frac{1}{2^{3nm\beta }}}∥{\mathfrak{D}}_c\Gamma \left(2^{\beta m}{v}_1,2^{\beta m}{v}_2\right),w∥\le {\displaystyle \frac{\varphi \left(2^{\beta m}{v}_1,2^{\beta m}{v}_2,w\right)}{2^{3nm\beta }}}.$$

Letting the limit as $m\to +\infty$, and applying (17) and (24), we obtain ${\mathfrak{D}}_c\mathrm{{\rm Y}}(v_1,v_2)=0$ for all $v_1,v_2\in V_1^n$, and so $\mathrm{{\rm Y}}$ is a solution of (14). Assume now that $\mathrm{{\rm Y}}$ has the cubic condition in each variable, then it is M-CMs by Theorem 9. Let now ${\mathrm{{\rm Y}}}^{\prime }:V_1^n⟶V_2$ be another M-CMs satisfying (19). Then, we have

$$\begin{array}{cc}\hfill & ∥\mathrm{{\rm Y}}\left(v\right)-{\mathrm{{\rm Y}}}^{\prime }\left(v\right),w∥\hfill \\ \hfill & ={\displaystyle \frac{1}{2^{3nm\beta }}}∥\mathrm{{\rm Y}}\left(2^{\beta m}v\right)-{\mathrm{{\rm Y}}}^{\prime }\left(2^{\beta m}v\right),w∥\hfill \\ \hfill & \le {\displaystyle \frac{1}{2^{3nm\beta }}}\left(∥\mathrm{{\rm Y}}\left(2^{\beta m}v\right)-\Gamma \left(2^{\beta m}v\right),w∥+∥f\left(2^{\beta m}v\right)-{\mathrm{{\rm Y}}}^{\prime }\left(2^{\beta m}v\right),w∥\right)\hfill \\ \hfill & \le {\displaystyle \frac{2}{2^{3nm\beta }}}{\displaystyle \frac{1}{3^n\times 2^{3\frac{\beta +1}{2}n}}}\tilde{\varphi }\left(v,0,w\right)\hfill \\ \hfill & ={\displaystyle \frac{2}{2^{3nm}}}{\displaystyle \frac{1}{3^n\times 2^{3\frac{\beta +1}{2}n}}}\sum _{j=0}^{+\infty }{\displaystyle \frac{1}{2^{3n\beta j}}}\varphi \left(2^{{\displaystyle \frac{\beta -1}{2}}+j\beta }v,0,w\right)\hfill \\ \hfill & ={\displaystyle \frac{2}{3^n\times 2^{3\frac{\beta +1}{2}n}}}\sum _{j=m}^{+\infty }{\displaystyle \frac{1}{2^{3n\beta j}}}\varphi \left(2^{{\displaystyle \frac{\beta -1}{2}}+(j+m)\beta }v,0,w\right),\hfill \end{array}$$

for all $v\in V_1^n$ and $w\in V_2$. Taking $m\to +\infty$ in the above inequality, we have $\mathrm{{\rm Y}}={\mathrm{{\rm Y}}}^{\prime }$; which proves the uniqueness. □

Let $(W,\parallel ·,·\parallel )$ be a 2-normed space and $\{w_1,w_2\}$ be a linearly independent set in W. It was proved in ([28], Theorem 8) that the function $\parallel ·\parallel :W⟶[0,+\infty )$ defined by $\parallel w\parallel =\parallel w,w_1\parallel +\parallel w,w_2\parallel$ is a norm on W.

Next corollary is The following corollary is a direct result of Theorem 10 concerning the Ulam–Rassias stability of (14).

Corollary 1.

Given $r,\alpha \in (0,+\infty )$ with $r\ne 3n$, let $V_1$ be a normed space, $V_2$ be a 2-Banach space, and $\{w_1,w_2\}$ be a linearly independent set in $V_2$. Suppose that a mapping $\Gamma :V_1^n⟶V_2$ satisfies the inequality

$$\begin{array}{c}\hfill \parallel {\mathfrak{D}}_c\Gamma (v_1,v_2){,w\parallel \le \parallel w\parallel }^{\alpha }\sum _{i=1}^2\sum _{j=1}^n{\parallel v_{ij}\parallel }^r,\forall v_1,v_2\in V_1^n,w\in V_2.\end{array}$$

Then, there is a solution $\mathrm{{\rm Y}}:V_1^n⟶V_2$ of (14) such that

$$\begin{array}{c}\hfill ∥\Gamma \left(v\right)-\mathrm{{\rm Y}}\left(v\right),w∥\le {\displaystyle \frac{{\parallel w\parallel }^{\alpha }}{3^n|2^{3n}-2^r|}}\sum _{j=1}^n{\parallel v_{1j}\parallel }^r,\end{array}$$

for all $v:=v_1\in V_1^n$ and $w\in V_2$. Furthermore, if Υ has the cubic condition in all variables, then it is a unique M-CM.

The proof is quite simple as follows. By setting $\varphi (v_1,v_2,w)={\parallel w\parallel }^{\alpha }{\sum }_{i=1}^2{\sum }_{j=1}^n{\parallel v_{ij}\parallel }^r$ in Theorem 10, one can obtain the results.

Suppose that $(V,\parallel ·,·\parallel )$ is a 2-normed space. Moreover, it is assumed that V has dimension d, where $2\le d<+\infty$. Fix $B=\{v_1,\dots ,v_d\}$ to be a basis for V. With respect to the basis B, one can define a norm on V, denoted by ${\parallel ·\parallel }_{+\infty }$, as follows:

$${\parallel v\parallel }_{+\infty }:=\max \{\parallel v,v_j\parallel :1\le j\le d\}.$$

For the case that $1\le p<+\infty$, the function ${\parallel ·\parallel }_p:V⟶[0,+\infty )$ defined through

$${\parallel v\parallel }_p:={\left(\sum _{j=1}^d{\parallel v,v_j\parallel }^p\right)}^{{\displaystyle \frac{1}{p}}},$$

is also a norm on V. Such norms were defined and studied in [29]. Note that all norms introduced above are equivalent, and moreover the choice of the basis is not essential. In other words, by choosing another basis, the resultant norm will be equivalent to the one considered with respect to B.

Corollary 2.

Fix $\alpha >0$ and $r\in (0,+\infty )$ with $r\ne 3n$, let $V_1$ be a normed space, $V_2$ be a 2-Banach space, and ${\parallel ·\parallel }_p$ be the norm on $V_2$ for $1\le p\le +\infty$. Suppose that $\Gamma :V_1^n⟶V_2$ satisfies the inequality

$$\begin{array}{c}\hfill \parallel {\mathfrak{D}}_c\Gamma (v_1,v_2){,w\parallel \le \parallel w\parallel }_p^{\alpha }\sum _{i=1}^2\sum _{j=1}^n{\parallel v_{ij}\parallel }^r,\forall v_1,v_2\in V_1^n,\end{array}$$

and $w\in V_2$. Then, there is a solution $\mathrm{{\rm Y}}:V_1^n⟶V_2$ of (14) such that

$$\begin{array}{c}\hfill ∥\Gamma \left(v\right)-\mathrm{{\rm Y}}\left(v\right),w∥\le {\displaystyle \frac{{\parallel w\parallel }_p^{\alpha }}{3^n|2^{3n}-2^r|}}\sum _{j=1}^n{\parallel v_{1j}\parallel }^r,\end{array}$$

for all $v:=v_1\in V_1^n$ and $w\in V_2$. Furthermore, if Υ has the cubic condition in all variables, then it is a unique M-CMs.

The proof is simple by setting $\varphi (v_1,v_2,w)={\parallel w\parallel }^{\alpha }{\sum }_{i=1}^2{\sum }_{j=1}^n{\parallel v_{ij}\parallel }^r$ in Theorem 10.

In the next result, the Hyers stability of (14), which is a direct consequence of Theorem 10, is indicated.

Corollary 3.

Take $\delta \in [0,+\infty )$. Let $V_1$ be a normed space and $V_2$ be a 2-Banach space. Let the mapping $f:V_1^n⟶V_2$ satisfies

$$\begin{array}{c}\hfill \parallel {\mathfrak{D}}_c\Gamma (v_1,v_2),w\parallel \le \delta ,\forall v_1,v_2\in V_1^n,\end{array}$$

and $w\in V_2$. Then, there is a solution $\Gamma :V_1^n⟶V_2$ of (14) such that

$$\begin{array}{c}\hfill ∥\Gamma \left(v\right)-\Gamma \left(v\right),w∥\le {\displaystyle \frac{\delta }{3^n(2^{3n}-1)}},\end{array}$$

for all $v\in V_1^n$ and $w\in V_2$. Furthermore, if Γ has the cubic condition in all variables, then it is a unique M-CM.

Setting $\varphi (v_1,v_2)=\delta$ in Theorem 10, with $\beta =1$, we can easily prove the result.

Based on some conditions, (14) can be hyperstable as follows.

Corollary 4.

Fix $\delta \in [0,+\infty )$ and $\alpha >0$. Given $r_{ij}>0$ and $s_{ij}>0$ for $i\in \{1,2\}$ and $j\in \{1,\dots ,n\}$ with ${\sum }_{i=1}^2{\sum }_{j=1}^n{r}_{ij}\ne 3n$ and ${\sum }_{i=0}^d{s}_{ij}\ne 3n$ $(1\le j\le n)$, let $V_1$ be a normed space and $V_2$ be a 2-Banach space with ${\parallel ·\parallel }_p$ as the norm on $V_2$ for $1\le p\le +\infty$. Suppose that a mapping $f:V_1^n⟶V_2$ satisfies the inequality

$$\begin{array}{c}\hfill \parallel {\mathfrak{D}}_c\Gamma (v_1,v_2),w\parallel \le \left\{\begin{array}{c}{\delta \parallel w\parallel }_p^{\alpha }\prod _{i=1}^2\prod _{j=1}^n{\parallel v_{ij}\parallel }^{r_{ij}},\hfill \\ or\hfill \\ {\delta \parallel w\parallel }_p^{\alpha }\sum _{j=1}^n\prod _{i=0}^2{\parallel v_{ij}\parallel }^{s_{ij}}.\hfill \end{array}\right.\end{array}$$

for all $v_1,v_2\in V_1^n$ and $w\in V_2$. Then, Γ is a solution of (14). Furthermore, if Γ has the cubic condition in all variables, then it is a unique M-CMs.

The result can be obtained by Theorem 10 when $\varphi (v_1,v_2)={\delta \parallel w\parallel }_p^{\alpha }{\prod }_{i=1}^2{\prod }_{j=1}^n{\parallel v_{ij}\parallel }^{r_{ij}}$ or $\varphi (v_1,v_2)={\delta \parallel w\parallel }_p^{\alpha }{\sum }_{j=1}^n{\prod }_{i=0}^2{\parallel v_{ij}\parallel }^{s_{ij}}$.

3.2. Stability of (14): The FP Method

Here, we present the Găvruţa and Hyers–Ulam–Rassias stability of (14) for a mapping $f:{\mathbb{R}}^{dn}⟶{\mathbb{R}}^d$ by an FP method.

Definition 5.

Let Ω be a set. A function $\mathsf{ϝ}:\Omega \times \Omega ⟶[0,+\infty ]$ is called a generalized metric on Ω if and only if ϝ satisfies the statements

(i) 

$\mathsf{ϝ}({\tau }_1,{\tau }_2)=0$ if and only if ${\tau }_1={\tau }_2$ for all ${\tau }_1,{\tau }_2\in \Omega$;

(ii) 

$\mathsf{ϝ}({\tau }_1,{\tau }_2)=\mathsf{ϝ}({\tau }_2,{\tau }_1)$ for all ${\tau }_1,{\tau }_2\in \Omega$;

(iii) 

$\mathsf{ϝ}({\tau }_1,{\tau }_3)\le \mathsf{ϝ}({\tau }_1,{\tau }_2)+\mathsf{ϝ}({\tau }_2,{\tau }_3)$ for all ${\tau }_1,{\tau }_2,{\tau }_3\in \Omega$.

Obviously, the generalized metric space’s range covers the infinity, which is the sole significant difference between it and the usual metric space.

The theorem brings a fundamental result in FP theory, which is useful to our purpose in this subsection (an extension of the result was stated in [30]). The next theorem is the FP alternative [31], which will play a vital role in our study.

Theorem 11.

Assume a complete generalized metric space $(\Omega ,\mu )$ and a lipschitz continuous mapping $\Xi :\Omega \to \Omega$ (with Lipschitz constant $\varrho <1$). Then, $\forall y\in \Omega$, either $\mu ({\Xi }^{n}y,{\Xi }^{n+1}y)=+\infty$$\forall n\ge 0,$ or a natural number $n_0$ exists:

(i) 

$\mu ({\Xi }^{n}y,{\Xi }^{n+1}y)<+\infty$∀$n\ge n_0$;

(ii) 

The sequence $\left\{{\Xi }^{n}y\right\}$ is convergent to an FP $y^{*}$ of Ξ;

(iii) 

$y^{*}$ is the unique FP of Ξ in the set

$$\Lambda =\{y\in \Omega :\mu ({\Xi }^{n_0}y,y)<+\infty \};$$

(iv) 

$\mu (y,y^{*})\le {\displaystyle \frac{1}{1-\varrho }}\mu (y,\Xi y)$ for all $y\in \Lambda$.

For ${\mathbb{R}}^d$, define $\parallel ·,·\parallel :{\mathbb{R}}^d\times {\mathbb{R}}^d⟶[0,+\infty )$ via the formula

$$\begin{array}{c}\hfill \parallel x,y\parallel ={\left(\left(\sum _{j=1}^d{x}_j^2\right)\left(\sum _{j=1}^d{y}_j^2\right)-{\left(\sum _{j=1}^d{x}_j{y}_j\right)}^2\right)}^{{\displaystyle \frac{1}{2}}},\end{array}$$

(25)

where $x=(x_1,\dots ,x_d),y=(y_1,\dots ,y_d)\in {\mathbb{R}}^d$. Then, $\parallel {\tau }_1,{\tau }_2\parallel$ is the area of the parallelogram spanned by ${\tau }_1$ and ${\tau }_2$ is 2-norm on ${\mathbb{R}}^d$ [29]. If $\{e_1,\dots ,e_d\}$ is the standard basis for ${\mathbb{R}}^d$, then for $j=1,\dots ,d$, we have $\parallel x,e_j\parallel ={\left(\left({\sum }_{j=1}^d{x}_j^2\right)-x_j^2\right)}^{{\displaystyle \frac{1}{2}}},$ and therefore

$$\begin{array}{c}\hfill {\parallel x\parallel }_{+\infty }=\max \left\{{\left(\left(\sum _{j=1}^d{x}_j^2\right)-x_j^2\right)}^{{\displaystyle \frac{1}{2}}}:j=1,\dots ,d\right\}.\end{array}$$

(26)

Next, a stability result for FEQ (14) (for a mapping $f:{\mathbb{R}}^{dn}⟶{\mathbb{R}}^d$) when $\parallel {\mathfrak{D}}_{c}f(x_1,x_2),y\parallel$ is controlled by an arbitrary function $\forall x_1,x_2\in {\mathbb{R}}^{dn}$ and $y\in {\mathbb{R}}^d$. In other words, ${\mathfrak{D}}_{c}f$ is assumed to be pointwise-bounded.

Theorem 12.

Let $\sigma \in \{-1,1\}$. Suppose also that ${\mathbb{R}}^d$ is considered as a 2-Banach space equipped with 2-norm (25). Let $\varphi :{\mathbb{R}}^{dn}\times {\mathbb{R}}^{dn}\times {\mathbb{R}}^d⟶[0,+\infty )$ be a mapping fulfilling the inequality

$$\begin{array}{c}\hfill {\displaystyle \frac{1}{2^{3n\sigma }}}\varphi \left(2^{\sigma }{x}_1,2^{\sigma }{x}_2,y\right)\le L\varphi (x_1,x_2,y)\end{array}$$

(27)

and

$$\begin{array}{c}\hfill \underset{l\to +\infty }{lim}{\displaystyle \frac{1}{2^{3nl}}}\varphi \left(2^l{x}_1,2^l{x}_2,y\right)=0,\end{array}$$

(28)

for all $x_1,x_2\in {\mathbb{R}}^{dn}$, $y\in {\mathbb{R}}^d$ and for some $0<L<1$, where $x_i=(x_{i1},\dots ,x_{in})$, whereas $x_{ij}\in {\mathbb{R}}^d$ with $i\in \{1,2\}$ and $j\in \{1,\dots ,n\}$. Assume a mapping $\Gamma :{\mathbb{R}}^{dn}⟶{\mathbb{R}}^d$ satisfying

$$\begin{array}{c}\hfill \parallel {\mathfrak{D}}_c\Gamma (x_1,x_2),y\parallel \le \varphi (x_1,x_2,y),\forall x_1,x_2\in {\mathbb{R}}^{dn},y\in {\mathbb{R}}^d.\end{array}$$

(29)

Then, there is a solution $\mathrm{{\rm Y}}:{\mathbb{R}}^{dn}⟶{\mathbb{R}}^d$ of (14) such that

$$\begin{array}{c}\hfill ∥\Gamma \left({\tau }_1\right)-\mathrm{{\rm Y}}\left({\tau }_1\right),y∥\le {\displaystyle \frac{L^{\frac{|\sigma -1|}{2}}}{{24}^n(1-L)}}\varphi ({\tau }_1,0,y),\end{array}$$

(30)

for all ${\tau }_1\in {\mathbb{R}}^{dn}$, $y\in {\mathbb{R}}^d$, where $0=(\stackrel{n-\mathit{times}}{\overbrace{0,\cdots ,0}})$. Additionally, if Υ has the cubic condition in each component, then it is a unique M-CM.

Proof. 

We establish the result for the case $\sigma =1$. Other cases can be obtained similarly. Let us define a generalized metric space $(\Omega ,\mu )$, where $\Omega :={\left({\mathbb{R}}^d\right)}^{{\mathbb{R}}^{dn}}$ (the set of all mappings from ${\mathbb{R}}^{dn}$ to ${\mathbb{R}}^d$) and

$$\mu (g,h):=\inf \left\{C\in [0,+\infty ):\parallel g\left({\tau }_1\right)-h\left({\tau }_1\right),y\parallel \le C\varphi ({\tau }_1,0,y),\forall {\tau }_1\in {\mathbb{R}}^{dn},y\in {\mathbb{R}}^d\right\},$$

where, as usual, inf$\varnothing =+\infty$, for which $g,h\in \Omega$. Similar to the proof of ([32], Theorem 3.1) (see also [33]), one can show that $(\Omega ,\mu )$ is a complete generalized metric space. Consider a mapping $\mathcal{J}:\Omega ⟶\Omega$ defined through

$$\mathcal{J}f\left(2{\tau }_1\right):={\displaystyle \frac{1}{2^{3n}}}\Gamma \left({\tau }_1\right).$$

for all ${\tau }_1\in {\mathbb{R}}^{dn}$. We claim that $\mathcal{J}$ is a strictly contractive operator with the Lipschitz constant L. Take $g,h\in \Omega$, ${\tau }_1\in {\mathbb{R}}^{dn}$ and $C\in [0,+\infty ]$ with $\mu (g,h)\le C$. Then, $\parallel g\left({\tau }_1\right)-h\left({\tau }_1\right),y\parallel \le C\varphi ({\tau }_1,0,y)$ for all ${\tau }_1\in {\mathbb{R}}^{dn}$ and $y\in {\mathbb{R}}^d$. We have

$$\begin{array}{c}\hfill \parallel \mathcal{J}g\left({\tau }_1\right)-\mathcal{J}h\left({\tau }_1\right),y\parallel \le {\displaystyle \frac{1}{2^{3n}}}\parallel g\left(2{\tau }_1\right)-h\left(2{\tau }_1\right),y\parallel \le {\displaystyle \frac{1}{2^{3n}}}C\varphi (2{\tau }_1,0,y)\le CL\varphi ({\tau }_1,0,y),\end{array}$$

for all ${\tau }_1\in {\mathbb{R}}^{dn}$ and $y\in {\mathbb{R}}^d$. Therefore, $\mu (\mathcal{J}g,\mathcal{J}h)\le CL$. This shows that $d(\mathcal{J}g,\mathcal{J}h)\le Ld(g,h)$, as claimed. Putting $x_2=0$ in (29), similar to the proof of Theorem 10 (relation (20)), we have

$$\begin{array}{cc}\hfill & \parallel 3^n\Gamma \left(2x_1\right)-{24}^n\Gamma \left(x_1\right),y\parallel \le \varphi (x_1,0,y),\hfill \end{array}$$

for all $x_1\in {\mathbb{R}}^{dn}$ and $y\in {\mathbb{R}}^d$. For the remaining of the proof, we set $x_1$ by ${\tau }_1$ Unless specifically indicated otherwise. The last inequality implies that

$$\begin{array}{cc}\hfill & ∥{\displaystyle \frac{1}{2^{3n}}}\Gamma \left(2{\tau }_1\right)-\Gamma \left({\tau }_1\right),y∥\le {\displaystyle \frac{1}{{24}^n}}\varphi ({\tau }_1,0,y),\hfill \end{array}$$

for all ${\tau }_1\in {\mathbb{R}}^{dn}$ and $y\in {\mathbb{R}}^d$. In other words,

$$\begin{array}{cc}\hfill & \parallel \mathcal{J}\Gamma \left({\tau }_1\right)-\Gamma \left({\tau }_1\right),y\parallel \le {\displaystyle \frac{1}{{24}^n}}\varphi ({\tau }_1,0,y),\hfill \end{array}$$

for all ${\tau }_1\in {\mathbb{R}}^{dn}$ and $y\in {\mathbb{R}}^d$. This means that

$$\begin{array}{c}\hfill d(\mathcal{J}\Gamma ,\Gamma )\le {\displaystyle \frac{1}{{24}^n}}.\end{array}$$

(31)

Now, we apply Theorem 11 for the space $(\Omega ,d)$, the operator $\mathcal{J}$, $n_0=0$, and $y=f$ to deduce that the sequence ${({\mathcal{J}}^l\Gamma )}_{l\in \mathbb{N}}$ is convergent in $(\Omega ,d)$ and its limit, $\mathrm{{\rm Y}}$, is an FP of $\mathcal{J}$. In fact,

$$\begin{array}{c}\hfill \mathrm{{\rm Y}}\left({\tau }_1\right)={\displaystyle \frac{1}{2^{3n}}}\mathrm{{\rm Y}}\left(2{\tau }_1\right)=\underset{l\to +\infty }{lim}{\mathcal{J}}^l\Gamma \left({\tau }_1\right),\end{array}$$

(32)

for all ${\tau }_1\in {\mathbb{R}}^{dn}$. By induction on l, one can easily show for all $v\in {\mathbb{R}}^{dn}$ that

$$\begin{array}{c}\hfill {\mathcal{J}}^l\Gamma \left({\tau }_1\right):={\left({\displaystyle \frac{1}{2^{3n}}}\right)}^l\Gamma \left(2^l{\tau }_1\right),\end{array}$$

(33)

for all ${\tau }_1\in {\mathbb{R}}^{dn}$. Since $\Gamma \in \Lambda$, part (iv) of Theorem 11 and (31) necessitate that

$$\mu (\Gamma ,\mathrm{{\rm Y}})\le {\displaystyle \frac{d(\mathcal{J}\Gamma ,\Gamma )}{1-L}}\le {\displaystyle \frac{1}{{24}^n(1-L)}},$$

which proves (30). It follows from (28), (29) and (33) that

$$\begin{array}{cc}\hfill \parallel {\mathfrak{D}}_c\mathrm{{\rm Y}}(x_1,x_2),y\parallel & =\underset{l\to +\infty }{lim}{\displaystyle \frac{1}{2^{3nl}}}∥{\mathfrak{D}}_c\Gamma (2^l{x}_1,2^l{x}_2),y∥\hfill \\ \hfill \le \underset{l\to +\infty }{lim}& {\displaystyle \frac{1}{2^{3nl}}}\varphi \left(2^l{x}_1,2^l{x}_2,y\right)=0,\forall x_1,x_2\in {\mathbb{R}}^{dn}.\hfill \end{array}$$

The relation above and part (ii) of Lemma 1 implies that ${\mathfrak{D}}_c\mathrm{{\rm Y}}(x_1,x_2)=0$$\forall x_1,x_2\in {\mathbb{R}}^{dn}$. Thus $\Gamma$ fulfills Equation (14). If now $\Gamma$ has the cubic condition in each variable, then it is M-CMs by Theorem 9. Let us lastly assume that $\mathfrak{C}:{\mathbb{R}}^{dn}⟶{\mathbb{R}}^d$ is a solution of (14): inequality (30) is valid. Then, $\mathrm{{\rm Y}}$ fulfills (32), and thus it is an FP of the operator $\mathcal{J}$. Furthermore, by (30), we find

$$d(\Gamma ,\mathrm{{\rm Y}})\le {\displaystyle \frac{1}{{24}^n(1-L)}}<+\infty ,$$

and consequently $\mathrm{{\rm Y}}\in \Lambda$. Now, part (ii) of Theorem 11 implies that $\mathrm{{\rm Y}}=\Gamma$. □

The upcoming stability result is deduced from Theorem 12. We bring only the sketch of the proof.

Corollary 5.

Fix $\delta ,\theta \in [0,+\infty )$ and $\alpha ,\beta >0$. Given $r\in (0,+\infty )$ with $r\ne 3n$. Given $r_{ij}>0$ and $s_{ij}>0$ for $i\in \{1,2\}$ and $j\in \{1,\dots ,n\}$ with ${\sum }_{i=1}^2{\sum }_{j=1}^n{r}_{ij}\ne 3n$ and ${\sum }_{i=0}^d{s}_{ij}\ne 3n$ $(1\le j\le n)$. Also, let ${\mathbb{R}}^d$ be a 2-Banach space equipped with 2-norm (25). Also, assume that $\Gamma :{\mathbb{R}}^{dn}⟶{\mathbb{R}}^d$ is a mapping satisfying

$$\begin{array}{c}\hfill \parallel {\mathfrak{D}}_c\Gamma (x_1,x_2),y\parallel \le \left\{\begin{array}{c}{\delta \parallel y\parallel }_{+\infty }^{\alpha }\sum _{i=1}^2\sum _{j=1}^n\parallel x_{ij}{\parallel }^r+{\theta \parallel y\parallel }_{+\infty }^{\beta }\prod _{i=1}^2\prod _{j=1}^n{\parallel x_{ij}\parallel }^{r_{ij}},\hfill \\ or\hfill \\ {\delta \parallel y\parallel }_{+\infty }^{\alpha }\sum _{i=1}^2\sum _{j=1}^n\parallel x_{ij}{\parallel }^r+{\theta \parallel y\parallel }_{+\infty }^{\beta }\sum _{j=1}^n\prod _{i=0}^2{\parallel x_{ij}\parallel }^{s_{ij}}.\hfill \end{array}\right.\end{array}$$

for all $x_1,x_2\in {\mathbb{R}}^{dn}$, $y\in {\mathbb{R}}^d$, where $\parallel {\tau }_1{\parallel }_{+\infty }$ is defined in (26). Then, a solution $\mathrm{{\rm Y}}:{\mathbb{R}}^{dn}⟶{\mathbb{R}}^d$ of (14) exists:

$$\begin{array}{c}\hfill ∥\Gamma \left({\tau }_1\right)-\mathrm{{\rm Y}}\left({\tau }_1\right),y∥\le {\displaystyle \frac{{\parallel w\parallel }_p^{\alpha }}{3^n|2^{3n}-2^r|}}\sum _{j=1}^n{\parallel x_{1j}\parallel }^r,\end{array}$$

for all ${\tau }_1:=x_1\in {\mathbb{R}}^{dn}$, $y\in {\mathbb{R}}^d$. Furthermore, if Υ has the cubic condition in all variables, then it is a unique M-CM.

The proof is quite simple. Set $\varphi (x_1,x_2,y)={\delta \parallel y\parallel }_{+\infty }^{\alpha }{\sum }_{i=1}^2{\sum }_{j=1}^n\parallel x_{ij}{\parallel }^r+\theta {\parallel y\parallel }_{+\infty }^{\beta }$${\prod }_{i=1}^2{\prod }_{j=1}^n{\parallel x_{ij}\parallel }^{r_{ij}}$ and $\varphi (x_1,x_2,y)={\delta \parallel y\parallel }_{+\infty }^{\alpha }{\sum }_{i=1}^2{\sum }_{j=1}^n\parallel x_{ij}{\parallel }^r+{\theta \parallel y\parallel }_{+\infty }^{\beta }{\sum }_{j=1}^n{\prod }_{i=0}^2{\parallel x_{ij}\parallel }^{s_{ij}}$ in Theorem 12. Then, in the case $\beta =1$ (resp. $\beta =-1$), we have $L={\displaystyle \frac{2^r}{2^{3n}}}$ (resp. $L={\displaystyle \frac{2^{3n}}{2^r}}$).

Remark 1.

Under the hypotheses in Corollary 5, if $\theta =0$, in view of the proof of Theorem 12, one can obtain the same result. On the other hand, if we put $\delta =0$ in Corollary 5, then Γ is a unique M-CM.

4. Conclusions

We introduced a system of symmetric equations defining M-CMs. Then, we characterized the structure of such mappings and obtained an equation for describing them.

In addition, We examined the H-UStab in the spirit of Gavruta for a symmetric M-CM equation by applying the so-called direct (Hyers) method in the setting of 2-Banach spaces. Moreover, we employed some FPTs to examine the stability and hyperstability of a certain mapping. Examining the stability in Ulam sense for a few additional complicated FEQs could be some possible future research.