The system of mixed type additive-quadratic equations and approximations

Abstract

In this article, we study the structure of a multiple variable mapping. Indeed, we reduce the system of several mixed additive-quadratic equations defining a multivariable mapping to obtain a single functional equation, say, the multimixed additive-quadratic equation. We also show that such mappings under some conditions can be multi-additive, multi-quadratic and multi-additive-quadratic. Moreover, we establish the Hyers–Ulam stability of the multimixed additive-quadratic equation, using the so-called direct (Hyers) method. Additionally, we present a concrete example (the numerical approximation) regarding the stability of some two variable mappings into real numbers. Applying some characterization results, we indicate two examples for the case that a multimixed additive-quadratic mapping (in the special cases) cannot be stable.

1 Introduction

Presenting a challenge or posing a query can be an inception point for fundamental topics in science and engineering, specially in mathematics. One of them is the existence of an exact solution, which is close to an approximate solution of functional equations, which was initiated by Ulam [25] for group homomorphisms. This is called the stability problem for functional equations. The mentioned question has been answered by Hyers [18], Aoki [2], Th. M. Rassias [24], and Găvruţa [17] for additive and linear mappings on normed spaces into Banach spaces. Here, we state some definitions of multivariable mappings.

Let V be a commutative group, W be a linear space over rational numbers, and n be an integer with $n\geq 2$. A mapping $f: V^{n}\longrightarrow W$ is called

• (i) multi-additive if it satisfies the Cauchy functional equation in each variable, that is,

  $$\begin{aligned} &f(v_{1},\ldots ,v_{i-1},v_{i}+v'_{i},v_{i+1},\ldots ,v_{n}) \\ &=f(v_{1},\ldots ,v_{i-1},v_{i},v_{i+1},\ldots ,v_{n})+f(v_{1}, \ldots ,v_{i-1},v'_{i},v_{i+1},\ldots ,v_{n}) \end{aligned}$$

  for all $i\in \{1,\ldots ,n\}$, where $v_{i},v'_{i}\in V$;

• (ii) multi-quadratic if it fulfills the quadratic functional equation in each of its n arguments, that is,

  $$\begin{aligned} &f(v_{1},\ldots ,v_{i-1},v_{i}+v'_{i},v_{i+1},\ldots ,v_{n})+f(v_{1}, \ldots ,v_{i-1},v_{i}-v'_{i},v_{i+1},\ldots ,v_{n}) \\ &=2f(v_{1},\ldots ,v_{i-1},v_{i},v_{i+1},\ldots ,v_{n})+2f(v_{1}, \ldots ,v_{i-1},v'_{i},v_{i+1},\ldots ,v_{n}) \end{aligned}$$

  for all $i\in \{1,\ldots ,n\}$, where $v_{i},v'_{i}\in V$.

It is important for us to know whether the above-mentioned mappings (defined as a system of n functional equations) can be unified and described as an equation. The answer is affirmative. In fact, Ciepliński in [13] showed that a mapping $f: V^{n}\longrightarrow W$ is multi-additive if and only if the equation

$$\begin{aligned} f(v_{1}+v_{2})=\sum _{j_{1},j_{2},\ldots ,j_{n}\in \{1,2\}}f(v_{1j_{1}},v_{2j_{2}}, \ldots ,v_{nj_{n}}) \end{aligned}$$

(1.1)

holds, where $v_{j}=(v_{1j},v_{2j},\ldots ,v_{nj})\in V^{n}$ with $j\in \{1,2\}$. Moreover, Zhao et al. [27] showed that the earlier mapping f is multi-quadratic if and only if it fulfills the equation

$$\begin{aligned} \sum _{t\in \{-1,1\}^{n}} f(v_{1}+tv_{2})=2^{n}\sum _{j_{1},j_{2}, \ldots ,j_{n}\in \{1,2\}}f(v_{1j_{1}},v_{2j_{2}},\ldots ,v_{nj_{n}}) \end{aligned}$$

(1.2)

for $j\in \{1,2\}$. More information and details about the structure and stability of multi-additive, multi-quadratic, and multi-Euler–Lagrange quadratic mappings are available for instance in [3, 5, 9, 14, 15, 20], and [22]. Note that some results about (multi-)n-derivations, n-homomorphisms, and their applications to inequalities can be found in [1, 6], and [19].

Recall from [11] that a mapping $f: V^{n}\longrightarrow W$ is said to be k-additive and $n-k$-quadratic (briefly, multi-additive-quadratic) if f is additive in each of some k variables and is quadratic in each of the other variables. Similar to the above, such mappings can be characterized by an equation. Indeed, Bahyrycz et al. [4] proved that f is multi-additive-quadratic if and only if it satisfies

$$\begin{aligned} \sum _{q\in \{-1,1\}^{n-k}} f(v_{1}^{k}+v_{2}^{k},v_{1}^{n-k}+qv_{2}^{n-k})=2^{n-k} \sum _{i_{1},i_{2},\ldots ,i_{n}\in \{1,2\}}f(v_{1i_{1}},v_{2i_{2}}, \ldots ,v_{ni_{n}}), \end{aligned}$$

(1.3)

where $v_{i}^{k}=(v_{1i},\ldots ,v_{ki})\in V^{k}, v_{i}^{n-k}=(v_{k+1,i}, \ldots ,v_{ni})\in V^{n-k}$, $i\in \{1,2\}$.

It is known that there are much more mixed type of functional equations whose solutions are additive-quadratic functions. We remember that the mixed additive-quadratic functional equation

$$\begin{aligned} h(x+2y)+h(x-2y)+8h(y)=2h(x)+4h(2y) \end{aligned}$$

(1.4)

was introduced by Zamani et al. [26] for the first time. They found the general solution of this equation and investigated its stability. Next, Najati and Moghimi [21] considered another mixed type additive and quadratic functional equation as follows:

$$\begin{aligned} h(2x+y)+h(2x-y)=h(x+y)+h(x-y)+2h(2x)-2h(x). \end{aligned}$$

(1.5)

The general form of (1.4) was presented in [7] as follows:

$$\begin{aligned} h(x+ty)+h(x-ty)=2h(x)+t^{2}h(2y)-2t^{2}h(y), \end{aligned}$$

(1.6)

where t is an integer with $t \neq 0,\pm 1$. It is easy to check that the function $\psi :\mathbb{R}\longrightarrow \mathbb{R}$ defined through $\psi (x)=ax^{2}+bx$ is a common solution of the functional equations (1.4), (1.5), and (1.6). We mention that some results on the stability of mixed type additive-quadratic functional equations are available for example in [8] and [23] and the references therein.

The rest of the paper is organized as follows. In Sect. 2, motivated by equation (1.4), we define a multimixed additive-quadratic mapping as the general system of the mixed type of additive and quadratic functional equations. Then, we characterize the structure of such mappings. Indeed, we unify this system of mixed type of additive and quadratic functional equations to a single equation and call it the multimixed additive-quadratic equation. We also show that such mappings under some conditions are multi-additive, multi-quadratic, and multi-additive-quadratic. In Sect. 4, we prove the generalized Hyers–Ulam stability of the multimixed additive-quadratic mappings, applying the direct (Hyers’) method. Furthermore, we present a concrete example (the numerical approximation) regarding the stability of some two variable mappings (as $\mathfrak {Q}(x,y)=xy^{2}$ and $g(x,y)=(x^{2}+\epsilon )(y^{2}+\epsilon )$) into real numbers. In Sect. 4, by using some characterization results, we indicate two examples for the case that a multimixed additive-quadratic mapping is nonstable.

2 A characterization of multimixed additive-quadratic mappings

Throughout this section, let V and W be vector spaces over the rationals $\mathbb{Q}$. Using functional equation (1.4), we commence this section with a definition as follows:

Definition 2.1

Let $n\in \mathbb{N}$ with $n\geq 2$. A multivariable mapping $f:V^{n}\longrightarrow W$ is called n-mixed additive-quadratic or multimixed additive-quadratic if f satisfies (1.4) in each of its n arguments, that is,

$$\begin{aligned} &f(v_{1},\ldots ,v_{i-1},v_{i}+2v'_{i},v_{i+1},\ldots ,v_{n})+f(v_{1}, \ldots ,v_{i-1},v_{i}-2v'_{i},v_{i+1},\ldots ,v_{n}) \\ &\qquad -4f(v_{1},\ldots ,v_{i-1},2v'_{i},v_{i+1},\ldots ,v_{n}) \\ &=2f(v_{1},\ldots ,v_{i-1},v_{i},v_{i+1},\ldots ,v_{n})-8f(v_{1}, \ldots ,v_{i-1},v'_{i},v_{i+1},\ldots ,v_{n})]. \end{aligned}$$

Let $n\in \mathbb{N}$ with $n\geq 2$ and $v_{i}^{n}=(v_{i1},v_{i2},\ldots ,v_{in})\in V^{n}$, where $i\in \{1,2\}$. We shall denote $v_{i}^{n}$ by $v_{i}$ if there is no risk of ambiguity. For $v_{1},v_{2}\in V^{n}$ and $p,q\in \mathbb{N}_{0}$ with $0\leq p,q\leq n$, put

$$\begin{aligned}& \mathcal {T}^{n}=\left \{\mathfrak {T}_{n}=(T_{1},\ldots ,T_{n})|\,\,T_{j} \in \{v_{1j}\pm 2v_{2j},2v_{2j}\}\right \},\\& \mathcal {S}^{n}=\left \{\mathfrak {S}_{n}=(S_{1},\ldots ,S_{n})|\,\,S_{j} \in \{v_{1j},v_{2j}\}\right \}, \end{aligned}$$

where $j\in \{1,\ldots ,n\}$. Consider the subsets $\mathcal {T}_{p}^{n}$, $\mathcal {S}_{q}^{n}$ of $\mathcal {T}^{n}$, $\mathcal {S}^{n}$, respectively, as follows:

$$\begin{aligned}& \mathcal {T}_{p}^{n} :=\left \{\mathfrak {T}_{n}\in \mathcal {T}^{n}|\,\, \text{Card}\{T_{j}:\, T_{j}=v_{2j}\}=p\right \},\\& \mathcal {S}_{q}^{n} :=\left \{\mathfrak {S}_{n}\in \mathcal {S}^{n}|\,\, \text{Card}\{S_{j}:\, S_{j}=v_{1j}\}=q\right \}. \end{aligned}$$

From now on, for the mappings introduced in Definition 2.1, we use the following conventions:

$$\begin{aligned} f\left (\mathcal {T}_{p}^{n}\right ):=\sum _{\mathfrak {T}_{n}\in \mathcal {T}_{p}^{n}}f(\mathfrak {T}_{n}), \qquad f\left (\mathcal {S}_{q}^{n} \right ):=\sum _{\mathfrak {S}_{n}\in \mathcal {S}_{q}^{n}}f(\mathfrak {S}_{n}), \end{aligned}$$

(2.1)

$$ f\left (\mathcal {T}_{p}^{n},v\right ):=\sum _{\mathfrak {T}_{n}\in \mathcal {T}_{p}^{n}}f(\mathfrak {T}_{n},v),\qquad f\left (\mathcal {S}_{q}^{n},v \right ):=\sum _{\mathfrak {S}_{n}\in \mathcal {S}_{q}^{n}}f(\mathfrak {S}_{n},v) \qquad (v\in V). $$

In what follows, $\left ( \begin{array}{c} n \\ k \end{array}\right )=\frac{n!}{k!(n-k)!}$ is the binomial coefficient defined for all $n, k\in \mathbb{N}_{0}$ with $n\geq k$. Set ${\mathbf{n}}:=\{1,\ldots ,n\}$. For a subset $m=\{j_{1},\ldots ,j_{i}\}$ of n with $1\leq j_{1}<\cdots < j_{i}\leq n$ and $u=(u_{1},\ldots ,u_{n})\in V^{n}$,

$$ _{m}u:=(0,\ldots ,0,u_{j_{1}},0,\ldots ,0,u_{j_{i}},0,\ldots ,0)\in V^{n} $$

denotes the vector that coincides with u in exactly those components, which are indexed by the elements of m and whose other components are set equal to zero. Note that $_{0}u=0$, $_{\mathbf{n}}u=u$.

Consider a mapping $f: V^{n}\longrightarrow W$. Then

• (i) f has zero condition if $f(u)=0$ for any $u\in V^{n}$ with at least one component that is equal to zero;

• (ii) f is odd in the jth variable if

  $$ f(u_{1},\ldots ,u_{j-1},-u_{j},u_{j+1},\ldots ,u_{n})=-f(u_{1}, \ldots ,u_{j-1},u_{j},u_{j+1},\ldots , u_{n}); $$

• (iii) f is even in the jth variable if

  $$ f(u_{1},\ldots ,u_{j-1},-u_{j},u_{j+1},\ldots ,u_{n})=f(u_{1},\ldots ,u_{j-1},u_{j},u_{j+1}, \ldots , u_{n}). $$

We shall show that every multimixed additive-quadratic mapping $f: V^{n}\longrightarrow W$ can be characterized as an equation and vice versa. For doing this, we need the next lemma.

Lemma 2.2

If a mapping $f: V^{n}\longrightarrow W$ satisfies equation

$$\begin{aligned} \sum _{p=0}^{n}(-4)^{p} f\left (\mathcal {T}_{p}^{n}\right )=\sum _{q=0}^{n}2^{q}(-8)^{n-q}f \left (\mathcal {S}_{q}^{n}\right ), \end{aligned}$$

(2.2)

then it has zero condition, where $f\left (\mathcal {T}_{p}^{n}\right )$ and $f\left (\mathcal {S}_{q}^{n}\right )$ are defined in (2.1).

Proof

We argue by induction on m that $f(_{m}v)=0$ for $0\leq m\leq n-1$. Let $m=0$. Putting $v_{1}=v_{2}=$₀v in (2.2), we have

$$\begin{aligned} \sum _{p=0}^{n} \left ( \textstyle\begin{array}{c} n \\ p \end{array}\displaystyle \right )(-4)^{p}2^{n-p}f(_{0}v)=\sum _{q=0}^{n} \left ( \textstyle\begin{array}{c} n \\ q \end{array}\displaystyle \right )2^{q}(-8)^{n-q}f(_{0}v). \end{aligned}$$

A computation for the above equality shows that $(-2)^{n}f(_{0}v)=(-6)^{n}f(_{0}v)$, and so $f(_{0}v)=0$. Assume that for each $f(_{m-1}v)=0$. We prove that $f(_{m}v)=0$. Without loss of generality, we assume that the first m variables are nonzero. By our assumption, by substitution $(v_{1},v_{2})$ into $(_{m}v,0)$ in equation (2.2), we obtain

$$\begin{aligned} \sum _{p=0}^{n-m} \left ( \textstyle\begin{array}{c} n-m \\ p \end{array}\displaystyle \right )(-4)^{p}2^{n-m-p}f(_{m}v)=\sum _{q=0}^{n-m} \left ( \textstyle\begin{array}{c} n-m \\ q \end{array}\displaystyle \right )2^{q}(-8)^{n-m-q}f(_{m}v). \end{aligned}$$

The relation above implies that $(-2)^{n-m}f(_{m}v)=(-6)^{n-m}f(_{m}v)$, and so $f(_{m}v)=0$. This completes the proof. □

Theorem 2.3

A mapping $f: V^{n}\longrightarrow W$ is multimixed additive-quadratic if and only if it satisfies equation (2.2). Furthermore,

• (i) if f is odd in each variable, then it is multi-additive, and so equation (1.1) is valid for f;

• (ii) if f is even in each variable, then it is multi-quadratic, and therefore f satisfies (1.2);

• (iii) if f is odd in each of some k variables and is even in each of the other variables, then it is multi-additive-quadratic, and so f fulfills (1.3).

Proof

Suppose that f is a multimixed additive-quadratic mapping. We argue by induction on n that it fulfills (2.2). For $n=1$, it is trivial that f satisfies equation (1.4). Assume that f satisfies (2.2) for n. We have

$$\begin{aligned} &\sum _{p=0}^{n+1}(-4)^{p} f\left (\mathcal {T}_{p}^{n+1}\right ) \\ &\quad= \sum _{p=0}^{n}\sum _{t\in \{-2,2\}}(-4)^{p} f\left (\mathcal {T}_{p}^{n},v_{1n}+tv_{2n} \right )+\sum _{p=0}^{n}(-4)^{p} f\left (\mathcal {T}_{p}^{n},2v_{2n} \right ) \\ &\quad=\sum _{q=0}^{n}\sum _{t\in \{-2,2\}}2^{q}(-8)^{n-q}f\left ( \mathcal {S}_{q}^{n},v_{1n}+tv_{2n}\right )-4\sum _{q=0}^{n}2^{q}(-8)^{n-q}f \left (\mathcal {S}_{q}^{n},2v_{2n}\right ) \\ &\quad=2\sum _{q=0}^{n}2^{q}(-8)^{n-q}f\left (\mathcal {S}_{q}^{n},v_{1n} \right )-8\sum _{q=0}^{n}2^{q}(-8)^{n-q}f\left (\mathcal {S}_{q}^{n},v_{2n} \right ) \\ &\quad=\sum _{q=0}^{n+1}2^{q}(-8)^{n+1-q}f\left (\mathcal {S}_{q}^{n+1} \right ). \end{aligned}$$

Conversely, assume that f satisfies (2.2). Let $j\in \{1,\ldots ,n\}$ be arbitrary and fixed. Set

$$\begin{aligned} f^{*}_{j}(u):&=f\left (u_{1},\ldots ,u_{j-1},u,u_{j+1},\ldots ,u_{n} \right ), \end{aligned}$$

in which $u_{1},\ldots ,u_{j-1}u_{j+1},\ldots ,u_{n}$ are arbitrary and fixed vectors in V. Putting $v_{1}= (u_{1},\ldots , u_{j-1},u,u_{j+1},\ldots ,u_{n} )$, $v_{2j}=w$, and $v_{2m}=0$ for all $m\in \{1,\ldots ,n\}\backslash \{j\}$ in (2.2) and using Lemma 2.2, we get

$$\begin{aligned} &2^{n}[(f^{*}_{j}(u+2w)+f^{*}_{j}(u-2w))-4\times 2^{n-1}f^{*}_{j}(2w)=2^{n}f^{*}_{j}(u)-8 \times 2^{n-1}f^{*}_{j}(w). \end{aligned}$$

In other words, (1.4) is true for $f^{*}_{j}$. Since j is arbitrary, f is a multimixed additive-quadratic mapping.

(i) Let f be odd in the jth variable. It follows from Lemma 2.1 of [26] that $f_{j}$ is additive for all $j\in \{1,\ldots ,n\}$.

(ii) The result can be obtained from [26, Lemma 2.2] that in fact $f_{j}$ is quadratic for all $j\in \{1,\ldots ,n\}$.

(iii) This part is a direct consequence of parts (i) and (ii), which finishes the proof. □

By Theorem 2.3, it is easily checked that the function $f:\mathbb{R}^{n}\longrightarrow \mathbb{R}$ defined by $f(r_{1},\ldots , r_{n})=\prod _{j=1}^{n}(\alpha _{j}r_{j}^{2}+\beta _{j}r_{j})$ satisfies (2.2), where $\alpha _{j},\beta _{j}\in \mathbb{Q}$. Therefore, the mapping f is multimixed additive-quadratic and equation (2.2) is said to be a multimixed additive-quadratic equation.

3 Stability results for (2.2)

In this section, we prove the Găvruţa and Hyers–Ulam stability of equation (2.2) by the directed method.

For given a mapping $f:V^{n} \longrightarrow W$, to simplify, we use the notation

$$\begin{aligned} \mathfrak {D}_{aq} f(v_{1},v_{2})&:=\sum _{p=0}^{n}(-4)^{p} f\left ( \mathcal {T}_{p}^{n}\right )-\sum _{q=0}^{n}2^{q}(-8)^{n-q}f\left ( \mathcal {S}_{q}^{n}\right ) \end{aligned}$$

for all $v_{1},v_{2}\in V^{n}$, where $f\left (\mathcal {T}_{p}^{n}\right )$ and $f\left (\mathcal {S}_{q}^{n}\right )$ are defined in (2.1). In the sequel, it is assumed that any mapping $f:V^{n} \longrightarrow W$ has zero condition. With this hypothesis, we have the next Găvruţa’s stability result for equation (2.2), which is the main result in this section.

Theorem 3.1

Given $\alpha \in [0,\infty )$ and $\beta \in \{-1,1\}$ are fixed. Let V be a linear space over the rationals and W be a Banach space. Suppose that $f: V^{n}\longrightarrow W$ is an odd mapping in each of some k variables and is even in each of the other variables for which there exists a function $\phi :V^{n}\times V^{n}\longrightarrow [-\alpha ,\infty )$ such that

$$\begin{aligned} \widetilde{\phi}(v_{1},v_{2}):=\sum _{j=0}^{\infty} \frac{1}{2^{(2n-k)\beta j}}\phi \left (2^{\frac{\beta-1}{2}+\beta j}v_{1},2^{\frac{\beta-1}{2}+\beta j}v_{2} \right )< \infty \end{aligned}$$

(3.1)

and

$$\begin{aligned} \left \|\mathfrak {D}_{aq} f(v_{1},v_{2})\right \|\leq \alpha \left ( \frac{\beta +1}{2}\right )+\phi (v_{1},v_{2}) \end{aligned}$$

(3.2)

for all $v_{1},v_{2}\in V^{n}$. Then there exists a multimixed additive-quadratic mapping $\mathcal {F}: V^{n}\longrightarrow W$ such that

$$\begin{aligned} \left \|f(v)-\mathcal {F}(v)\right \|\leq \frac{1}{8^{\frac{\beta +1}{2}n}\times 2^{\frac{|\beta -1|}{2}(n+k)}} \left [ \frac{2^{(2n-k)\beta}\alpha}{(2^{(2n-k)\beta}-1)}\left ( \frac{\beta +1}{2}\right )+\widetilde{\phi}(0,v)\right ] \end{aligned}$$

(3.3)

for all $v\in V^{n}$. Moreover, if $\mathcal {F}$ is odd in each of some k variables and is even in each of the other variables, then it is a unique multi-additive-quadratic mapping. In particular,

• (i) if f is an odd mapping in each variable ($k=n$), then there exists a unique multi-additive mapping $\mathcal {A}: V^{n}\longrightarrow W$ such that

  $$\begin{aligned} \left \|f(v)-\mathcal {A}(v)\right \|\leq \frac{1}{8^{\frac{\beta +1}{2}n}\times 2^{|\beta -1|n}} \left [ \frac{2^{n\beta}\alpha}{(2^{n\beta}-1)}\left ( \frac{\beta +1}{2}\right )+\widetilde{\phi}(0,v)\right ] \end{aligned}$$

  for all $v\in V^{n}$;

• (ii) if f is an even mapping in each variable ($k=0$), then there exists a unique multi-quadratic mapping $\mathcal {Q}: V^{n}\longrightarrow W$ such that

  $$\begin{aligned} \left \|f(v)-\mathcal {Q}(v)\right \|\leq \frac{1}{8^{\frac{\beta +1}{2}n}\times 2^{\frac{|\beta -1|n}{2}}}\left [ \frac{2^{2n\beta}\alpha}{(2^{2n\beta}-1)}\left (\frac{\beta +1}{2} \right )+\widetilde{\phi}(0,v)\right ] \end{aligned}$$

  for all $v\in V^{n}$.

Proof

Without loss of generality, we assume that f is odd in the k first of variables. Replacing $(v_{1},v_{2})$ by $(0,v_{1})$ in (3.2) and using the assumptions, we have

$$\begin{aligned} &\|(-4)^{k}Sf(2v_{1})-(-8)^{k}(-8)^{n-k}f(v_{1})\|\leq \alpha \left ( \frac{\beta +1}{2}\right )+\phi (0,v_{1}) \end{aligned}$$

(3.4)

for all $v_{1}\in V^{n}$ in which

$$\begin{aligned} S=\sum _{j=0}^{n-k}\left ( \textstyle\begin{array}{c} n-k \\ j \end{array}\displaystyle \right )\left (-4\right )^{j}2^{n-k-j}. \end{aligned}$$

(3.5)

For the rest, we set $v_{1}$ by v unless otherwise stated explicitly. It follows from relations (3.4) and (3.5) that

$$\begin{aligned} &\|(-1)^{n}2^{n+k}f(2v)-(-8)^{n}f(v)\|\leq \alpha \left ( \frac{\beta +1}{2}\right )+\phi (0,v), \end{aligned}$$

and so

$$\begin{aligned} \left \|\frac{f(2^{\beta }v)}{2^{(2n-k)\beta}}-f(v)\right \|\leq \frac{1}{8^{\frac{\beta +1}{2}n}\times 2^{\frac{|\beta -1|}{2}(n+k)}} \left [\alpha \left (\frac{\beta +1}{2}\right )+\phi \left (0,2^{ \frac{\beta -1}{2}} v\right )\right ] \end{aligned}$$

(3.6)

for all $v\in V^{n}$. Switching v by $2^{\beta }v$ in (3.6) and continuing this method, we obtain

$$\begin{aligned} &\left \|\frac{f(2^{\beta m}v)}{2^{(2n-k)\beta m}}-f(v)\right \| \\ &\leq \frac{1}{8^{\frac{\beta +1}{2}n}\times 2^{\frac{|\beta -1|}{2}(n+k)}} \left [\frac{\beta +1}{2}\sum _{j=0}^{m-1} \frac{\alpha}{2^{(2n-k)\beta j}}+\sum _{j=0}^{m-1} \frac{\phi \left (0,2^{\frac{\beta -1}{2}+j\beta} v\right )}{2^{(2n-k)\beta j}}\right ] \end{aligned}$$

(3.7)

for all $v\in V^{n}$. On the other hand, we can use induction to find

$$\begin{aligned} &\left \|\frac{f(2^{\beta m}v)}{2^{(2n-k)\beta m}}- \frac{f(2^{\beta l}v)}{2^{(2n-k)\beta l}}\right \| \\ &\leq \frac{1}{8^{\frac{\beta +1}{2}n}\times 2^{\frac{|\beta -1|}{2}(n+k)}} \left [\frac{\beta +1}{2}\sum _{j=l}^{m-1} \frac{\alpha}{2^{(2n-k)\beta j}}+\sum _{j=l}^{m-1} \frac{1}{2^{(2n-k)\beta j}}\phi \left (0,2^{\frac{\beta -1}{2}+\beta j}v \right )\right ] \end{aligned}$$

(3.8)

for all $v\in V^{n}$ and $m>l\geq 0$. Thus, the sequence $\left \{\frac{f(2^{\beta m}v)}{2^{(2n-k)\beta m}}\right \}$ is Cauchy by (3.1) and (3.8). Completeness of W allows us to assume that there exists a mapping $\mathcal {F}:V^{n}\longrightarrow W$ such that

$$\begin{aligned} \lim _{m\to \infty}\frac{f(2^{\beta m}v)}{2^{(2n-k)\beta m}}= \mathcal {F}(x). \end{aligned}$$

(3.9)

Taking the limit as m tends to infinity in (3.7) and using (3.9), we can see that the inequality (3.3) is valid. Now, by interchanging $v_{1},v_{2}$ into $2^{m}v_{1},2^{m}v_{2}$, respectively in (3.2), we get

$$ \frac{1}{2^{(2n-k)\beta m}}\left \|\mathfrak {D}_{aq}f\left (2^{\beta m}v_{1},2^{ \beta m}v_{2}\right )\right \|\leq \frac{\alpha}{2^{(2n-k)\beta m}} \left (\frac{\beta +1}{2}\right )+ \frac{\phi \left (2^{\beta m}v_{1},2^{\beta m}v_{2}\right )}{2^{(2n-k)\beta m}}. $$

Letting the limit as $m\rightarrow \infty$ and applying (3.9), we obtain $\mathfrak {D}_{aq}\mathcal {F}(v_{1},v_{2})=0$ for all $v_{1},v_{2}\in V^{n}$. It follows from Theorem 2.3 that $\mathcal {F}$ is a multimixed additive-quadratic mapping. If now $\mathcal {F}$ is odd in each of some k variables and is even in each of the other variables, then it is a multi-additive-quadratic mapping. Let now $\mathcal {F}':V^{n}\longrightarrow W$ be another multi-additive-quadratic mapping satisfying (3.3). Then we have

$$\begin{aligned} &\left \|\mathcal {F}(v)-\mathcal {F}'(v)\right \| \\ &\quad=\frac{1}{2^{(2n-k)\beta m}}\left \|\mathcal {F}\left (2^{\beta m}v \right )-\mathcal {F}'\left (2^{\beta m}v\right )\right \| \\ &\quad\leq \frac{1}{2^{(2n-k)\beta m}}\left (\left \|\mathcal {F}\left (2^{ \beta m}v\right )-f\left (2^{\beta m}v\right )\right \|+\left \|f \left (2^{\beta m}v\right )-\mathcal {F}'\left (2^{\beta m}v\right ) \right \|\right ) \\ &\quad\leq \frac{2}{2^{(2n-k)\beta m}} \frac{1}{8^{\frac{\beta +1}{2}n}\times 2^{\frac{|\beta -1|}{2}(n+k)}} \left [\frac{2^{(2n-k)\beta}\alpha}{(2^{(2n-k)\beta}-1)}\left ( \frac{\beta +1}{2}\right )+\widetilde{\phi}\left (0,v\right )\right ] \\ &\quad=\frac{2}{2^{(2n-k)m}} \frac{1}{8^{\frac{\beta +1}{2}n}\times 2^{\frac{|\beta -1|}{2}(n+k)}} \\ &\qquad{}\times\left [ \frac{2^{(2n-k)\beta}\alpha}{(2^{(2n-k)\beta}-1)}\left ( \frac{\beta +1}{2}\right )+\sum _{j=0}^{\infty} \frac{1}{2^{(2n-k)\beta j}}\phi \left (0,2^{(j+m)\beta}v\right ) \right ] \\ &\quad=\frac{2}{2^{(2n-k)m}} \frac{1}{8^{\frac{\beta +1}{2}n}\times 2^{\frac{|\beta -1|}{2}(n+k)}} \\ &\qquad{}\times \left [ \frac{2^{(2n-k)\beta}\alpha}{(2^{(2n-k)\beta}-1)}\left ( \frac{\beta +1}{2}\right )+2^{(2n-k)\beta m}\sum _{j=m}^{\infty} \frac{1}{2^{(2n-k)\beta j}}\phi \left (0,2^{j\beta}v\right )\right ] \end{aligned}$$

for all $v\in V^{n}$. Taking $m\rightarrow \infty$ in the above inequality, we have $\mathcal {F}=\mathcal {F}'$, and hence the uniqueness of solution is shown. Letting $k=n$ and $k=0$ in (3.3), respectively, parts (i) and (ii) can be obtained by Theorem 2.3. This completes the proof. □

The upcoming corollary is a direct consequence of Theorem 3.1 concerning the Hyers–Ulam stability of (2.2).

Corollary 3.2

Given $\alpha ,\delta , r\in \mathbb{R}$ and with $r\neq 2n-k$ and $\delta ,\alpha \in [0,\infty )$. Let V be a normed space over the rationals and W be a Banach space. Suppose that $f:V^{n} \longrightarrow W$ is an odd mapping in each of some k variables and is even in each of the other variables and moreover satisfying the inequality

$$\begin{aligned} \|\mathfrak {D}_{aq} f(v_{1},v_{2})\|\leq \alpha +\delta \sum _{i=1}^{2} \sum _{j=1}^{n}\|v_{ij}\|^{r} \end{aligned}$$

for all $v_{1},v_{2}\in V^{n}$. Then there exists a multimixed additive-quadratic mapping $\mathcal {F}: V^{n}\longrightarrow W$ such that

$$\begin{aligned} \left \|f(v)-\mathcal {F}(v)\right \|\leq \textstyle\begin{cases} \frac{1}{8^{n}}\left [ \frac{2^{2n-k}\alpha}{2^{2n-k}-1}+ \frac{2^{2n-k}\delta}{2^{2n-k}-2^{r}}\sum _{j=1}^{n}\|v_{1j}\|^{r} \right ]\,\, \hspace{1.5cm} r\in \left (0,2n-k\right ) \\ \frac{\delta}{2^{n+k}(2^{r}-2^{2n-k})}\sum _{j=1}^{n} \|v_{1j}\|^{r}\,\, \hspace{2.9cm} r\in \left (2n-k,\infty \right ) \end{cases}\displaystyle \end{aligned}$$

for all $v\in V^{n}$. If also $\mathcal {F}$ is odd in each of some k variables and is even in each of the other variables, then it is a unique multi-additive-quadratic mapping. In addition,

• (i) if f is an odd mapping in each variable, then there exists a unique multi-additive mapping $\mathcal {A}: V^{n}\longrightarrow W$ such that

  $$\begin{aligned} \left \|f(v)-\mathcal {A}(v)\right \|\leq \textstyle\begin{cases} \frac{1}{8^{n}}\left [ \frac{2^{n}\alpha}{(2^{n}-1)}+ \frac{2^{n}\delta}{2^{n}-2^{r}}\sum _{j=1}^{n}\|v_{1j}\|^{r}\right ] \,\, \hspace{1.5cm} r\in \left (0,n\right ) \\ \frac{\delta}{2^{2n}(2^{r}-2^{n})}\sum _{j=1}^{n} \|v_{1j}\|^{r}\,\, \hspace{2.8cm} r\in \left (n,\infty \right ) \end{cases}\displaystyle \end{aligned}$$

  for all $v\in V^{n}$;

• (ii) if f is an even mapping in each variable, then there exists a unique multi-quadratic mapping $\mathcal {Q}: V^{n}\longrightarrow W$ such that

  $$\begin{aligned} \left \|f(v)-\mathcal {Q}(v)\right \|\leq \textstyle\begin{cases} \frac{1}{8^{n}}\left [ \frac{2^{2n}\alpha}{(2^{2n}-1)}+ \frac{2^{2n}\delta}{2^{2n}-2^{r}}\sum _{j=1}^{n}\|v_{1j}\|^{r}\right ] \,\, \hspace{1.5cm} r\in \left (0,2n\right ) \\ \frac{\delta}{2^{n}(2^{r}-2^{2n})}\sum _{j=1}^{n} \|v_{1j}\|^{r}\,\, \hspace{3cm} r\in \left (2n,\infty \right ) \end{cases}\displaystyle \end{aligned}$$

  for all $v\in V^{n}$.

Proof

Setting $\phi (v_{1},v_{2})=\delta \sum _{i=1}^{2}\sum _{j=1}^{n}\|v_{ij}\|^{r}$ in Theorem 3.1, one can obtain the desired results. □

Recall that a functional equation $\mathcal {F}$ is said to be hyperstable if any function f satisfying the equation $\mathcal {F}$ approximately is an exact solution of $\mathcal {F}$; for more details and information, we refer to [12]. Under some conditions, equation (2.2) can be hyperstable as follows.

Corollary 3.3

Suppose that $r_{ij}>0$ for $i\in \{1,2\}$ and $j\in \{1,\ldots ,n\}$ satisfy $\sum _{i=1}^{2}\sum _{j=1}^{n}r_{ij}\neq 2n-k$. Let V be a normed space and W be a Banach space. Suppose that $f:V^{n} \longrightarrow W$ is an odd mapping in each of some k variables and is even in each of the other variables and moreover satisfying the inequality

$$\begin{aligned} \|\mathfrak {D}_{aq} f(v_{1},v_{2})\|\leq \prod _{i=1}^{2}\prod _{j=1}^{n} \|v_{ij}\|^{r_{ij}} \end{aligned}$$

for all $v_{1},v_{2}\in V^{n}$, then f is a multi-additive-quadratic. Moreover, if f is an odd mapping in each component ($k=n$), it is a multi-additive mapping. Additionally, if f is an even mapping in all components ($k=0$), then it is multi-quadratic mapping.

Proof

The result follows from part (iii) of Theorem 2.3 and Theorem 3.1 by letting $\alpha =0$, $\delta =1$, and $\phi (v_{1},v_{2})=\delta \prod _{i=1}^{2}\prod _{j=1}^{n}\|v_{ij}\|^{r}$. □

Remark 3.4

Putting ϕ as the zero function and considering the case $\beta =1$ in Theorem 3.1, we find that there exists a unique multi-additive-quadratic mapping $\mathcal {F}: V^{n}\longrightarrow W$ such that

$$\begin{aligned} \left \|f(v)-\mathcal {F}(v)\right \|\leq \frac{\alpha}{2^{n+k}(2^{2n-k}-1)} \end{aligned}$$

for all $v\in V^{n}$. In particular,

• (i) if f is an odd mapping in each component ($k=n$), then there exists a unique multi-additive mapping $\mathcal {A}: V^{n}\longrightarrow W$ such that

  $$\begin{aligned} \left \|f(v)-\mathcal {A}(v)\right \|\leq \frac{\alpha}{2^{2n}(2^{n}-1)} \end{aligned}$$

  for all $v\in V^{n}$;

• (ii) if f is an even mapping in all components ($k=0$), then there exists a unique multi-quadratic mapping $\mathcal {Q}: V^{n}\longrightarrow W$ such that

  $$\begin{aligned} \left \|f(v)-\mathcal {Q}(v)\right \|\leq \frac{\alpha}{2^{n}(2^{2n}-1)} \end{aligned}$$

  for all $v\in V^{n}$.

We bring a concrete example regarding the above results such that all hypotheses of Theorem 3.1 can happen.

Example 3.5

Define the mapping $f:\mathbb{R}^{n}\longrightarrow \mathbb{R}$ through

$$ f(r_{1},\ldots ,r_{n})= \textstyle\begin{cases} \prod _{j=1}^{k}\prod _{i=k+1}^{n}r_{j}(r_{i}^{2}+\epsilon )\,\, \hspace{1.3cm} \forall r_{i},r_{j}\neq 0, \\ 0\,\, \hspace{3cm} \text{otherwise}. \end{cases} $$

It is easy to see that an odd mapping in each of some k variables is even in each of the other variables. On the other hand, in Remark 3.4, define the mapping $g:\mathbb{R}^{n}\longrightarrow \mathbb{R}$ through

$$ g(r_{1},\ldots ,r_{n})=\prod _{j=1}^{n}(r_{i}^{2}+\epsilon ). $$

By a simple computation, it concludes that $|\mathcal {D}g(r_{1}^{n},r_{2}^{n})|\leq (5\epsilon )^{n}=\alpha$ for all $r_{1}^{n},r_{2}^{n}\in \mathbb{R}^{n}$. Therefore, it follows from Theorem 3.1 that there exists a unique multi-quadratic mapping $\mathcal {Q}: V^{n}\longrightarrow W$ such that

$$\begin{aligned} \left \|g(r^{n})-\mathcal {Q}(r^{n})\right \|\leq \frac{(5\epsilon )^{n}}{2^{n}(2^{2n}-1)} \end{aligned}$$

for all $r^{n}\in \mathbb{R}^{n}$. For instance, set $\varepsilon =0.01$, and for the case $n=2$, we have

$$\begin{aligned}& g(x,y)=(x^{2}+0.01)(y^{2}+0.01)\\& \left \|g(x,y)-\mathfrak {Q}(x,y)\right \|\leq 0.000042 \end{aligned}$$

for all $x,y\in \mathbb{R}$, in which $\mathfrak {Q}(x,y)=x^{2}y^{2}$. Note that Theorem 3.1 shows the existence of a multi-quadratic function, while the function $\mathfrak {Q}$ is not necessarily the same as in this theorem. We have Figs. 1 and 2 for $f:=g$ $[g(x,y)=(x^{2}+0.01)(y^{2}+0.01)]$ and $\mathfrak {Q}$ on interval $[-0.06,0.06]\times [-0.06,0.06]$.

Figure 1

(Main figure)

Figure 2

(Image on yz-plan)

4 Nonstability examples

In this section, we give two nonstability examples for multi-additive and multi-quadratic mappings on $\mathbb{R}^{n}$. For the first case, we bring the next result, presented in [20, Theorem 13.4.3].

Theorem 4.1

Let $h:\mathbb{R}^{d^{N}}\longrightarrow \mathbb{R}$ be a continuous p-additive function. Then there exist constants $c_{j_{1}\ldots j_{d}}\in \mathbb{R}$, $j_{1},\ldots ,j_{d}=1,\ldots ,N$, such that

$$\begin{aligned} h(x_{1},\ldots ,x_{d})=\sum _{j_{1}=1}^{N}\ldots \sum _{j_{d}=1}^{N}c_{j_{1} \ldots j_{d}}x_{1j_{1}}\ldots x_{dj_{d}} \end{aligned}$$

for all $x_{i}=(x_{i1},\ldots ,x_{iN})$ and $i=1,\ldots ,d$.

Remark 4.2

In the proof of Theorem 4.1 only the continuity of h with respect to each variable separately was used. Therefore, the result is true if and only if h is supposed to be separately continuous with respect to each variable. On the other hand, in virtue of the proof of Theorem 4.1, if the continuity condition of g is removed, then the theorem remains valid for a function $h:\mathbb{Q}^{d}\longrightarrow \mathbb{Q}$ in the case $N=1$. We use this fact to make a nonstable example.

In continuation, we present two counterexamples such that the hypotheses $r\neq n$ and $r\neq 2n$ are necessary and cannot be removed in Corollary 3.2 (parts (i) and (ii)) for multi-additive and multi-quadratic mappings. Note that the idea of both are taken from [16].

Example 4.3

Let $\delta >0$ and $n\in \mathbb{N}$. Consider the constant function ${\mathbf{1}}:\mathbb{Q}^{n}\longrightarrow \mathbb{Q}$ whose range is 1. Set

$$\begin{aligned} |\mathfrak {D}_{aq}{\mathbf{1}}|=M \end{aligned}$$

(4.1)

and $\mu =\frac{2^{n}-1}{2^{2n}M}\delta$. Define the function $\psi :\mathbb{Q}^{n}\longrightarrow \mathbb{Q}$ through

$$ \psi (r_{1},\ldots ,r_{n})= \textstyle\begin{cases} \mu \prod _{j=1}^{n}r_{j}\,\, \hspace{.8cm} \text{for all $r_{j}$ with}\,\, |r_{j}|< 1, \\ \mu \,\, \hspace{3cm} \text{otherwise}. \end{cases} $$

Moreover, define the function $f:\mathbb{Q}^{n}\longrightarrow \mathbb{Q}$ via

$$ f(r_{1},\ldots ,r_{n})=\sum _{l=0}^{\infty} \frac{\psi (2^{l}r_{1},\ldots ,2^{l}r_{n})}{2^{nl}},\qquad (r_{j}\in \mathbb{Q}). $$

Obviously, ψ is bounded by μ. Indeed, for each $(r_{1},\ldots ,r_{n})\in \mathbb{Q}^{n}$, we have $|f(r_{1},\ldots ,r_{n})|\leq \frac{2^{n}}{2^{n}-1}\mu$. Put $x_{i}=(x_{i1},\ldots ,x_{in})$, where $i\in \{1,2\}$. We claim that

$$\begin{aligned} \left |\mathfrak {D}_{aq} f\left (x_{1},x_{2}\right )\right |\leq \delta \sum _{i=1}^{2}\sum _{j=1}^{n}|x_{ij}|^{n} \end{aligned}$$

(4.2)

for all $x_{1},x_{2}\in \mathbb{Q}^{n}$, where $x_{j}=(x_{j1},\ldots ,x_{jn})\in \mathbb{Q}^{n}$ with $j\in \{1,2\}$. It is clear that (4.2) holds for $x_{1}=x_{2}=0$. Let $x_{1},x_{2}\in \mathbb{Q}^{n}$ with

$$\begin{aligned} \sum _{i=1}^{2}\sum _{j=1}^{n}|x_{ij}|^{n}< \frac{1}{2^{n}}. \end{aligned}$$

(4.3)

Thus, there exists a positive integer N such that

$$\begin{aligned} \frac{1}{2^{n(N+1)}}< \sum _{i=1}^{2}\sum _{j=1}^{n}|x_{ij}|^{n}< \frac{1}{2^{nN}}, \end{aligned}$$

(4.4)

and hence $|x_{ij}|^{n}<\sum _{i=1}^{2}\sum _{j=1}^{n}|x_{ij}|^{n}< \frac{1}{2^{nN}}$. The last relation implies that $2^{N}|x_{ij}|< 1$ for all $i\in \{1,2\}$ and $j\in \{1,\ldots ,n\}$. Therefore, $2^{N-1}|x_{ij}|< 1$. If $y_{1},y_{2}\in \{x_{ij}|\, i\in \{1,2\},\,\, j\in \{1,\ldots ,n\}\}$, then

$$\begin{aligned} 2^{N-1}|\, y_{1}\pm y_{2}|< 1\,\,\text{ and}\,\ 2^{N-1}|\, y_{1}\pm 2y_{2}|< 1. \end{aligned}$$

(4.5)

Since ψ is a multi-additive function on $(-1,1)^{n}$, $\mathfrak {D}_{aq}\psi \left (2^{l}x_{1},2^{l}x_{2}\right )=0$ for all $l\in \{0,1,2,\ldots , N-1\}$. We conclude from the last equality and (4.4) that

$$\begin{aligned} \frac{\left |\mathfrak {D}_{aq} f\left (2^{l}x_{1},2^{l}x_{2}\right )\right |}{\sum _{i=1}^{2}\sum _{j=1}^{n}|x_{ij}|^{n}}& \leq \sum _{l=N}^{\infty} \frac{\left |\mathfrak {D}_{aq}\psi \left (2^{l}x_{1},2^{l}x_{2}\right )\right |}{2^{nl}\sum _{i=1}^{2}\sum _{j=1}^{n}|x_{ij}|^{n}} \\ &\leq \sum _{l=0}^{\infty} \frac{\mu M}{2^{n(l+N)}\sum _{i=1}^{2}\sum _{j=1}^{n}|x_{ij}|^{n}} \\ &\leq \mu 2^{n}M\sum _{l=0}^{\infty}\frac{1}{2^{nl}} \\ &=\mu M\frac{2^{2n}}{2^{n}-1}=\delta \end{aligned}$$

for all $x_{1},x_{2}\in \mathbb{Q}^{n}$, and thus (4.2) is true when (4.3) happens. If $\sum _{i=1}^{2}\sum _{j=1}^{n}|x_{ij}|^{n}\geq \frac{1}{2^{n}}$, then

$$ \frac{\left |\mathfrak {D}_{aq} f\left (2^{l}x_{1},2^{l}x_{2}\right )\right |}{\sum _{i=1}^{2}\sum _{j=1}^{n}|x_{ij}|^{n}} \leq 2^{n}\frac{2^{n}}{2^{n}-1}\mu M=\delta . $$

Therefore, f satisfies in (4.2) for all $x_{1},x_{2}\in \mathbb{Q}^{n}$. Now, suppose the assertion is false, that there exist a number $\lambda \in [0, \infty )$ and a multi-additive function $\mathcal {A}:\mathbb{Q}^{n}\longrightarrow \mathbb{Q}$ such that

$$ |f(r_{1},\ldots ,r_{n})-\mathcal {A}(r_{1},\ldots ,r_{n})| \leq \lambda \sum _{j=1}^{n}|r_{j}|^{n} $$

for all $(r_{1},\ldots ,r_{n})\in \mathbb{Q}^{n}$. Since n is a fixed positive integer, without loss of generality, one can take a number $b\in [0, \infty )$ so that

$$ \lambda \sum _{j=1}^{n}|r_{j}|^{n}\leq b\prod _{j=1}^{n}|r_{j}|. $$

Hence, $|f(r_{1},\ldots ,r_{n})-\mathcal {A}(r_{1},\ldots ,r_{n})|< b\prod _{j=1}^{n}|r_{j}|$ for all $(r_{1},\ldots ,r_{n})\in \mathbb{Q}^{n}$. It follows now from Remark 4.2 that there is a constant $c\in \mathbb{R}$ such that $\mathcal {A}(r_{1},\ldots ,r_{n})=c\prod _{j=1}^{n}r_{j}$ for all $(r_{1},\ldots ,r_{n})\in \mathbb{Q}^{n}$, and therefore

$$ |f(r_{1},\ldots ,r_{n})|\leq (|c|+b)\prod _{j=1}^{n}|r_{j}| $$

(4.6)

for all $(r_{1},\ldots ,r_{n})\in \mathbb{Q}^{n}$. On the other hand, one can choose $N\in \mathbb{N}$ such that $(N+1)\mu >|c|+b$. If $r=(r_{1},\ldots ,r_{n})\in \mathbb{Q}^{n}$ such that $r_{j}\in \left (0, \frac{1}{2^{N}}\right )$ for all $j\in \{1,\ldots ,n\}$, then $2^{l}r_{j}\in (0,1)$ for all $l=0,1,\ldots ,N$. Hence

$$\begin{aligned} |f(r_{1},\ldots ,r_{n})|&=\left |\sum _{l=0}^{\infty} \frac{\psi \left (2^{l}r_{1},\ldots ,2^{l}r_{2}\right )}{2^{nl}} \right | \\ &=\left |\sum _{l=0}^{N} \frac{\mu 2^{nl}\prod _{j=1}^{n}r_{j}}{2^{nl}}\right | \\ &= (N+1)\mu \prod _{j=1}^{n}|r_{j}| \\ &>(|c|+b)\prod _{j=1}^{n}|r_{j}|, \end{aligned}$$

that leads us to a contradiction with (4.6).

We bring the following result, which was proved in [10, Proposition 14].

Proposition 4.4

Let $f:\mathbb{R}^{n}\longrightarrow \mathbb{R}$ be a continuous function satisfying (1.2). Then f has the form

$$\begin{aligned} f(r_{1},\ldots ,r_{n})=c r_{1}^{2}\ldots r_{n}^{2}, \qquad (r_{1}, \ldots ,r_{n}\in \mathbb{R}), \end{aligned}$$

where c is a constant in $\mathbb{R}$.

In analogy with Example 4.3 and similar to Example 1 from [10], we indicate the upcoming example to show the nonstability of multi-quadratic mappings on $\mathbb{R}^{n}$. Indeed, we show that the hypothesis $r\neq 2n$ cannot be eliminated in Corollary 3.2. The argument is similar to the mentioned example, but we include it completely for the sake of completeness.

Example 4.5

Fix $n\in \mathbb{N}$ and $\delta >0$. Put $\lambda :=\frac{2^{2n}-1}{2^{4n}M}\delta$, where is defined in (4.1). Define the function $\Phi :\mathbb{R}^{n}\longrightarrow \mathbb{R}$ is defined by

$$ \Phi (r_{1},\ldots ,r_{n})= \textstyle\begin{cases} \lambda \prod _{j=1}^{n}r_{j}^{2}\,\, \hspace{.8cm} \text{for all $r_{j}$ with}\,\, |r_{j}|< 1, \\ \lambda \,\, \hspace{3cm} \text{otherwise}. \end{cases} $$

Using the function Φ, we consider the function $f:\mathbb{R}^{n}\longrightarrow \mathbb{R}$ defined through

$$ f(r_{1},\ldots ,r_{n})=\sum _{l=0}^{\infty} \frac{\Phi (2^{l}r_{1},\ldots ,2^{l}r_{n})}{2^{2nl}}. $$

It is clear that the function f is even in each component. Moreover, Φ is bounded by λ and continuous as well. It is also shown that f is a uniformly convergent series of continuous functions, and therefore it is continuous and bounded by $\frac{2^{2n}}{2^{2n}-1}\mu$ for all $(r_{1},\ldots ,r_{n})\in \mathbb{R}^{n}$. Take $x_{i}=(x_{i1},\ldots ,x_{in})$ for $i\in \{1,2\}$. We claim that

$$\begin{aligned} \left |\mathfrak {D}_{aq} f\left (x_{1},x_{2}\right )\right |\leq \delta \sum _{i=1}^{2}\sum _{j=1}^{n}x_{ij}^{2n} \end{aligned}$$

(4.7)

for all $x_{1},x_{2}\in \mathbb{R}^{n}$. Obviously, (4.7) is valid for $x_{1}=x_{2}=0$. Given $x_{1},x_{2}\in \mathbb{R}^{n}$ with

$$\begin{aligned} \sum _{i=1}^{2}\sum _{j=1}^{n}x_{ij}^{2n}< \frac{1}{2^{2n}}. \end{aligned}$$

(4.8)

It follows from relation (4.8) that there is a positive integer N such that

$$\begin{aligned} \frac{1}{2^{2n(N+1)}}< \sum _{i=1}^{2}\sum _{j=1}^{n}x_{ij}^{2n}< \frac{1}{2^{2nN}}, \end{aligned}$$

(4.9)

and hence $x_{ij}^{2n}<\sum _{i=1}^{2}\sum _{j=1}^{n}x_{ij}^{2n}< \frac{1}{2^{2nN}}$. The last relation implies that $2^{N}|x_{ij}|< 1$ for all $i\in \{1,2\}$ and $j\in \{1,\ldots ,n\}$, and so $2^{N-1}|x_{ij}|< 1$. Similar to Example 4.3, the inequalities in (4.5) are true. By the definition of Φ, it is a multi-quadratic function on $(-1,1)^{n}$, and so for each $l\in \{0,1,2,\ldots , N-1\}$ we get $\mathfrak {D}_{aq}\Phi \left (2^{l}x_{1},2^{l}x_{2}\right )=0$. This equality and (4.9) necessitate that

$$\begin{aligned} \frac{\left |\mathfrak {D}_{aq} f\left (2^{l}x_{1},2^{l}x_{2}\right )\right |}{\sum _{i=1}^{2}\sum _{j=1}^{n}x_{ij}^{2n}}& \leq \sum _{l=N}^{\infty} \frac{\left |\mathfrak {D}_{aq}\psi \left (2^{l}x_{1},2^{l}x_{2}\right )\right |}{2^{2nl}\sum _{i=1}^{2}\sum _{j=1}^{n}x_{ij}^{2n}} \\ &\leq \sum _{l=0}^{\infty} \frac{\mu M}{2^{2n(l+N)}\sum _{i=1}^{2}\sum _{j=1}^{n}x_{ij}^{2n}} \\ &\leq \mu M\sum _{l=0}^{\infty}\frac{1}{2^{2nl}} \\ &\leq \mu M2^{2n}\frac{2^{2n}}{2^{2n}-1} \\ &=\mu M\frac{2^{4n}}{2^{2n}-1}=\delta \end{aligned}$$

for all $x_{1},x_{2}\in \mathbb{R}^{n}$. Hence, the validity of (4.7) is proved for case (4.8). In the case of $\sum _{i=1}^{2}\sum _{j=1}^{n}x_{ij}^{2n}\geq \frac{1}{2^{2n}}$, we have

$$ \frac{\left |\mathfrak {D}_{aq} f\left (2^{l}x_{1},2^{l}x_{2}\right )\right |}{\sum _{i=1}^{2}\sum _{j=1}^{n}x_{ij}^{2n}} \leq 2^{2n}\frac{2^{2n}}{2^{2n}-1}\mu M=\delta . $$

Therefore, (4.7) holds for all $x_{1},x_{2}\in \mathbb{R}^{n}$. Similar to the argument in Example 4.3, suppose contrary to our claim that there exist a number $b\in [0, \infty )$ and a multi-quadratic function $\mathcal {Q}:\mathbb{R}^{n}\longrightarrow \mathbb{R}$ so that $|f(r_{1},\ldots ,r_{n})-\mathcal {Q}(r_{1},\ldots ,r_{n})|< b\prod _{j=1}^{n}r_{j}^{2}$ holds for all $(r_{1},\ldots ,r_{n})\in \mathbb{R}^{n}$. It follows from Proposition 4.4 that there is a constant $c\in \mathbb{R}$ such that $\mathcal {Q}(r_{1},\ldots ,r_{n})=c\prod _{j=1}^{n}r_{j}^{2}$, and so

$$ f(r_{1},\ldots ,r_{n})\leq (|c|+b)\prod _{j=1}^{n}r_{j}^{2} $$

(4.10)

for all $r_{1},\ldots ,r_{n}\in \mathbb{R}^{n}$. In addition, take $N\in \mathbb{N}$ such that $N\mu >|c|+b$. Consider $r=(r_{1},\ldots ,r_{n})\in \mathbb{R}^{n}$ such that $r_{j}\in \left (0, \frac{1}{2^{N-1}}\right )$ for all $j\in \{1,\ldots ,n\}$. This means that $2^{l}r_{j}\in (0,1)$ for all $l=0,1,\ldots ,N-1$, and therefore

$$ f(r_{1},\ldots ,r_{n})=\sum _{l=0}^{\infty} \frac{\psi \left (2^{l}r_{1},\ldots ,2^{l}r_{2}\right )}{2^{2nl}} \geq \sum _{l=0}^{N-1} \frac{\mu 2^{2nl}\prod _{j=1}^{n}r_{j}^{2}}{2^{2nl}}=N\mu \prod _{j=1}^{n}r_{j}^{2}>(|c|+b) \prod _{j=1}^{n}r_{j}^{2}. $$

The above relation contradicts (4.10).