 Research
 Open access
 Published:
The system of mixed type additivequadratic equations and approximations
Journal of Inequalities and Applications volume 2024, Article number: 98 (2024)
Abstract
In this article, we study the structure of a multiple variable mapping. Indeed, we reduce the system of several mixed additivequadratic equations defining a multivariable mapping to obtain a single functional equation, say, the multimixed additivequadratic equation. We also show that such mappings under some conditions can be multiadditive, multiquadratic and multiadditivequadratic. Moreover, we establish the Hyers–Ulam stability of the multimixed additivequadratic equation, using the socalled 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 additivequadratic 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)
multiadditive if it satisfies the Cauchy functional equation in each variable, that is,
$$\begin{aligned} &f(v_{1},\ldots ,v_{i1},v_{i}+v'_{i},v_{i+1},\ldots ,v_{n}) \\ &=f(v_{1},\ldots ,v_{i1},v_{i},v_{i+1},\ldots ,v_{n})+f(v_{1}, \ldots ,v_{i1},v'_{i},v_{i+1},\ldots ,v_{n}) \end{aligned}$$for all \(i\in \{1,\ldots ,n\}\), where \(v_{i},v'_{i}\in V\);

(ii)
multiquadratic if it fulfills the quadratic functional equation in each of its n arguments, that is,
$$\begin{aligned} &f(v_{1},\ldots ,v_{i1},v_{i}+v'_{i},v_{i+1},\ldots ,v_{n})+f(v_{1}, \ldots ,v_{i1},v_{i}v'_{i},v_{i+1},\ldots ,v_{n}) \\ &=2f(v_{1},\ldots ,v_{i1},v_{i},v_{i+1},\ldots ,v_{n})+2f(v_{1}, \ldots ,v_{i1},v'_{i},v_{i+1},\ldots ,v_{n}) \end{aligned}$$for all \(i\in \{1,\ldots ,n\}\), where \(v_{i},v'_{i}\in V\).
It is important for us to know whether the abovementioned 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 multiadditive if and only if the equation
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 multiquadratic if and only if it fulfills the equation
for \(j\in \{1,2\}\). More information and details about the structure and stability of multiadditive, multiquadratic, and multiEuler–Lagrange quadratic mappings are available for instance in [3, 5, 9, 14, 15, 20], and [22]. Note that some results about (multi)nderivations, nhomomorphisms, 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 kadditive and \(nk\)quadratic (briefly, multiadditivequadratic) 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 multiadditivequadratic if and only if it satisfies
where \(v_{i}^{k}=(v_{1i},\ldots ,v_{ki})\in V^{k}, v_{i}^{nk}=(v_{k+1,i}, \ldots ,v_{ni})\in V^{nk}\), \(i\in \{1,2\}\).
It is known that there are much more mixed type of functional equations whose solutions are additivequadratic functions. We remember that the mixed additivequadratic functional equation
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:
The general form of (1.4) was presented in [7] as follows:
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 additivequadratic 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 additivequadratic 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 additivequadratic equation. We also show that such mappings under some conditions are multiadditive, multiquadratic, and multiadditivequadratic. In Sect. 4, we prove the generalized Hyers–Ulam stability of the multimixed additivequadratic 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 additivequadratic mapping is nonstable.
2 A characterization of multimixed additivequadratic 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 nmixed additivequadratic or multimixed additivequadratic if f satisfies (1.4) in each of its n arguments, that is,
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
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:
From now on, for the mappings introduced in Definition 2.1, we use the following conventions:
In what follows, \(\left ( \begin{array}{c} n \\ k \end{array}\right )=\frac{n!}{k!(nk)!}\) 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 \(u=(u_{1},\ldots ,u_{n})\in V^{n}\),
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
$$ f(u_{1},\ldots ,u_{j1},u_{j},u_{j+1},\ldots ,u_{n})=f(u_{1}, \ldots ,u_{j1},u_{j},u_{j+1},\ldots , u_{n}); $$ 
(iii)
f is even in the jth variable if
$$ f(u_{1},\ldots ,u_{j1},u_{j},u_{j+1},\ldots ,u_{n})=f(u_{1},\ldots ,u_{j1},u_{j},u_{j+1}, \ldots , u_{n}). $$
We shall show that every multimixed additivequadratic 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
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 \(0\leq m\leq n1\). Let \(m=0\). Putting \(v_{1}=v_{2}=\)_{0}v in (2.2), we have
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 \(f(_{m1}v)=0\). 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
The relation above implies that \((2)^{nm}f(_{m}v)=(6)^{nm}f(_{m}v)\), and so \(f(_{m}v)=0\). This completes the proof. □
Theorem 2.3
A mapping \(f: V^{n}\longrightarrow W\) is multimixed additivequadratic if and only if it satisfies equation (2.2). Furthermore,

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

(ii)
if f is even in each variable, then it is multiquadratic, 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 multiadditivequadratic, and so f fulfills (1.3).
Proof
Suppose that f is a multimixed additivequadratic 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
Conversely, assume that f satisfies (2.2). Let \(j\in \{1,\ldots ,n\}\) be arbitrary and fixed. Set
in which \(u_{1},\ldots ,u_{j1}u_{j+1},\ldots ,u_{n}\) are arbitrary and fixed vectors in V. Putting \(v_{1}= (u_{1},\ldots , u_{j1},u,u_{j+1},\ldots ,u_{n} )\), \(v_{2j}=w\), and \(v_{2m}=0\) for all \(m\in \{1,\ldots ,n\}\backslash \{j\}\) in (2.2) and using Lemma 2.2, we get
In other words, (1.4) is true for \(f^{*}_{j}\). Since j is arbitrary, f is a multimixed additivequadratic 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 additivequadratic and equation (2.2) is said to be a multimixed additivequadratic 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
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
and
for all \(v_{1},v_{2}\in V^{n}\). Then there exists a multimixed additivequadratic mapping \(\mathcal {F}: V^{n}\longrightarrow W\) such that
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 multiadditivequadratic mapping. In particular,

(i)
if f is an odd mapping in each variable (\(k=n\)), then there exists a unique multiadditive mapping \(\mathcal {A}: V^{n}\longrightarrow W\) such that
$$\begin{aligned} \left \f(v)\mathcal {A}(v)\right \\leq \frac{1}{8^{\frac{\beta +1}{2}n}\times 2^{\beta 1n}} \left [ \frac{2^{n\beta}\alpha}{(2^{n\beta}1)}\left ( \frac{\beta +1}{2}\right )+\widetilde{\phi}(0,v)\right ] \end{aligned}$$for all \(v\in V^{n}\);

(ii)
if f is an even mapping in each variable (\(k=0\)), then there exists a unique multiquadratic mapping \(\mathcal {Q}: V^{n}\longrightarrow W\) such that
$$\begin{aligned} \left \f(v)\mathcal {Q}(v)\right \\leq \frac{1}{8^{\frac{\beta +1}{2}n}\times 2^{\frac{\beta 1n}{2}}}\left [ \frac{2^{2n\beta}\alpha}{(2^{2n\beta}1)}\left (\frac{\beta +1}{2} \right )+\widetilde{\phi}(0,v)\right ] \end{aligned}$$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
for all \(v_{1}\in V^{n}\) in which
For the rest, we set \(v_{1}\) by v unless otherwise stated explicitly. It follows from relations (3.4) and (3.5) that
and so
for all \(v\in V^{n}\). Switching v by \(2^{\beta }v\) in (3.6) and continuing this method, we obtain
for all \(v\in V^{n}\). On the other hand, we can use induction to find
for all \(v\in V^{n}\) and \(m>l\geq 0\). Thus, the sequence \(\left \{\frac{f(2^{\beta m}v)}{2^{(2nk)\beta m}}\right \}\) 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
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
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 additivequadratic 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 multiadditivequadratic mapping. Let now \(\mathcal {F}':V^{n}\longrightarrow W\) be another multiadditivequadratic mapping satisfying (3.3). Then we have
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 \(r\neq 2nk\) 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
for all \(v_{1},v_{2}\in V^{n}\). Then there exists a multimixed additivequadratic mapping \(\mathcal {F}: V^{n}\longrightarrow W\) such that
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 multiadditivequadratic mapping. In addition,

(i)
if f is an odd mapping in each variable, then there exists a unique multiadditive 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 multiquadratic 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 \(\sum _{i=1}^{2}\sum _{j=1}^{n}r_{ij}\neq 2nk\). 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
for all \(v_{1},v_{2}\in V^{n}\), then f is a multiadditivequadratic. Moreover, if f is an odd mapping in each component (\(k=n\)), it is a multiadditive mapping. Additionally, if f is an even mapping in all components (\(k=0\)), then it is multiquadratic 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 multiadditivequadratic mapping \(\mathcal {F}: V^{n}\longrightarrow W\) such that
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 multiadditive 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 multiquadratic 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
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
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 multiquadratic mapping \(\mathcal {Q}: V^{n}\longrightarrow W\) such that
for all \(r^{n}\in \mathbb{R}^{n}\). For instance, set \(\varepsilon =0.01\), and for the case \(n=2\), we have
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 multiquadratic 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 \([0.06,0.06]\times [0.06,0.06]\).
4 Nonstability examples
In this section, we give two nonstability examples for multiadditive and multiquadratic 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 padditive function. Then there exist constants \(c_{j_{1}\ldots j_{d}}\in \mathbb{R}\), \(j_{1},\ldots ,j_{d}=1,\ldots ,N\), such that
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 multiadditive and multiquadratic 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
and \(\mu =\frac{2^{n}1}{2^{2n}M}\delta \). Define the function \(\psi :\mathbb{Q}^{n}\longrightarrow \mathbb{Q}\) through
Moreover, define the function \(f:\mathbb{Q}^{n}\longrightarrow \mathbb{Q}\) via
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
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
Thus, there exists a positive integer N such that
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, \(2^{N1}x_{ij}< 1\). If \(y_{1},y_{2}\in \{x_{ij}\, i\in \{1,2\},\,\, j\in \{1,\ldots ,n\}\}\), then
Since ψ is a multiadditive function on \((1,1)^{n}\), \(\mathfrak {D}_{aq}\psi \left (2^{l}x_{1},2^{l}x_{2}\right )=0\) for all \(l\in \{0,1,2,\ldots , N1\}\). We conclude from the last equality and (4.4) that
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
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 multiadditive function \(\mathcal {A}:\mathbb{Q}^{n}\longrightarrow \mathbb{Q}\) such that
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
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
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
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
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 multiquadratic 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
Using the function Φ, we consider the function \(f:\mathbb{R}^{n}\longrightarrow \mathbb{R}\) defined through
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
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
It follows from relation (4.8) that there is a positive integer N such that
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 \(2^{N1}x_{ij}< 1\). Similar to Example 4.3, the inequalities in (4.5) are true. By the definition of Φ, it is a multiquadratic function on \((1,1)^{n}\), and so for each \(l\in \{0,1,2,\ldots , N1\}\) we get \(\mathfrak {D}_{aq}\Phi \left (2^{l}x_{1},2^{l}x_{2}\right )=0\). This equality and (4.9) necessitate that
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
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 multiquadratic 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
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 \(r_{j}\in \left (0, \frac{1}{2^{N1}}\right )\) for all \(j\in \{1,\ldots ,n\}\). This means that \(2^{l}r_{j}\in (0,1)\) for all \(l=0,1,\ldots ,N1\), and therefore
The above relation contradicts (4.10).
Data availability
Not applicable. In fact, all results are obtained without any software and found by manual computations. In other words, the manuscript is in the pure mathematics (mathematical analysis) category.
References
Alinejad, A., Khodaei, H., Rostami, M.: nDerivations and functional inequalities with applications. Math. Inequal. Appl. 32(4), 1343–1360 (2020)
Aoki, T.: On the stability of the linear transformation in Banach spaces. J. Math. Soc. Jpn. 2, 64–66 (1950)
Bahyrycz, A.: On stability and hyperstability of an equation characterizing multiadditive mappings. Fixed Point Theory 18(2), 445–456 (2017)
Bahyrycz, A., Ciepliński, K., Olko, J.: On an equation characterizing multiadditivequadratic mappings and its HyersUlam stability. Appl. Math. Comput. 265, 448–455 (2015)
Bodaghi, A.: Functional inequalities for generalized multiquadratic mappings. J. Inequal. Appl. 2021, Article ID 145 (2021). https://doi.org/10.1186/s1366002102682z
Bodaghi, A., Feizabadi, H.: Multiderivations and approximations. Commun. Korean Math. Soc. 37(3), 801–812 (2022)
Bodaghi, A., Kim, S.O.: Stability of a functional equation deriving from quadratic and additive functions in nonArchimedean normed spaces. Abstr. Appl. Anal. 2013, Article ID 198018 (2013)
Bodaghi, A., Kim, S.O.: Ulam’s type stability of a functional equation deriving from quadratic and additive functions. J. Math. Inequal. 9(1), 73–84 (2015)
Bodaghi, A., Moshtagh, H., Dutta, H.: Characterization and stability analysis of advanced multiquadratic functional equations. Adv. Differ. Equ. 2021, 380 (2021). https://doi.org/10.1186/s13662021035413
Bodaghi, A., Moshtagh, H., Mousivand, A.: Characterization and stability of multiEulerLagrange quadratic functional equations. J. Funct. Spaces 2022, Article ID 3021457 (2022). https://doi.org/10.1155/2022/3021457
Brzdȩk, J., Ciepliński, K.: Remarks on the HyersUlam stability of some systems of functional equations. Appl. Math. Comput. 219, 4096–4105 (2012)
Brzdȩk, J., Ciepliński, K.: Hyperstability and superstability. Abstr. Appl. Anal. 2013, Article ID 401756 (2013)
Ciepliński, K.: Generalized stability of multiadditive mappings. Appl. Math. Lett. 23, 1291–1294 (2010)
Ciepliński, K.: On the generalized HyersUlam stability of multiquadratic mappings. Comput. Math. Appl. 62, 3418–3426 (2011)
Ciepliński, K.: Ulam stability of functional equations in 2Banach spaces via the fixed point method. J. Fixed Point Theory Appl. 23, Article ID 33 (2021)
Gajda, Z.: On stability of additive mappings. Int. J. Math. Math. Sci. 14(3), 431–434 (1991)
Găvruţa, P.: A Generalization of the HyersUlamRassias stability of approximately additive mappings. J. Math. Anal. Appl. 184(3), 431–436 (1994)
Hyers, D.H.: On the stability of the linear functional equation. Proc. Natl. Acad. Sci. 27, 222–224 (1941)
Khodaei, H.: Asymptotic behavior of nJordan homomorphisms. Mediterr. J. Math. 17, Article ID 143 (2020)
Kuczma, M.: An Introduction to the Theory of Functional Equations and Inequalities. Cauchy’s Equation and Jensen’s Inequality. Birkhäuser, Basel (2009)
Najati, A., Moghimi, M.B.: Stability of a functional equation deriving from quadratic and additive functions in quasiBanach spaces. J. Math. Anal. Appl. 337, 399–415 (2008)
Park, C.G.: Multiquadratic mappings in Banach spaces. Proc. Am. Math. Soc. 131, 2501–2504 (2002)
Park, W.G., Bae, J.H., Chung, B.H.: On an additivequadratic functional equation and its stability. J. Appl. Math. Comput. 18, 563–572 (2005)
Rassias, T.M.: On the stability of the linear mapping in Banach spaces. Proc. Am. Math. Soc. 72(2), 297–300 (1978)
Ulam, S.M.: Problems in Modern Mathematic, Science Editions. Wiley, New York (1964)
Zamani Eskandani, G., Vaezi, H., Dehghan, Y.N.: Stability of a mixed additive and quadratic functional equation in nonArchimedean Banach modules. Taiwan. J. Math. 14(4), 1309–1324 (2010)
Zhao, X., Yang, X., Pang, C.T.: Solution and stability of the multiquadratic functional equation. Abstr. Appl. Anal. 2013, Article ID 415053 (2013)
Acknowledgements
The authors sincerely thank the anonymous reviewers for their careful reading, constructive comments, and suggesting some related references that improved the manuscript substantially.
Funding
Not applicable.
Author information
Authors and Affiliations
Contributions
All authors conceived of the study, participated in its design and coordination, drafted the manuscript, participated in the sequence alignment, and read and approved the final manuscript.
Corresponding author
Ethics declarations
Ethics approval and consent to participate
The authors declare that this manuscript is original, has not been published before, and is not under consideration for publication elsewhere. Moreover, we consent to publish when the paper is accepted.
Competing interests
The authors declare no competing interests.
Additional information
Publisher’s Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
Rights and permissions
Open Access This article is licensed under a Creative Commons Attribution 4.0 International License, which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons licence, and indicate if changes were made. The images or other third party material in this article are included in the article’s Creative Commons licence, unless indicated otherwise in a credit line to the material. If material is not included in the article’s Creative Commons licence and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder. To view a copy of this licence, visit http://creativecommons.org/licenses/by/4.0/.
About this article
Cite this article
Bodaghi, A., Mahzoon, H. & Mikaeilvand, N. The system of mixed type additivequadratic equations and approximations. J Inequal Appl 2024, 98 (2024). https://doi.org/10.1186/s13660024031808
Received:
Accepted:
Published:
DOI: https://doi.org/10.1186/s13660024031808