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1.   Introduction

Throughout this paper, $\mathbb N$ and $\mathbb Q$ are the set of all positive integers and rational numbers, respectively, $\mathbb N_0: = \mathbb N\cup\{0\}, \mathbb R_+: = [0, \infty)$. Moreover, for the set $X$, we denote $\overbrace{X\times X\times\cdots\times X}^{n-\text{times}}$ by $X^n$. For any $l\in \mathbb N_0, m\in \mathbb N$, $t = (t_1, \ldots, t_m)\in \{-1, 1\}^m$ and $x = (x_1, \ldots, x_m)\in V^m$ we write $lx: = (lx_1, \ldots, lx_m)$ and $tx: = (t_1x_1, \ldots, t_mx_m)$, where $ra$ stands, as usual, for the $r$th power of an element $a$ of the commutative group $V$.

Let $V$ be a commutative group, $W$ be a linear space, and $n\geq 2$ be an integer. Recall from ^[15] that a mapping $f: V^n\longrightarrow W$ is called multi-additive if it is additive (satisfies Cauchy's functional equation $A(x+y) = A(x)+A(y)$) in each variable. Some basic facts on such mappings can be found in ^[19] and many other sources, where their application to the representation of polynomial functions is also presented. Besides, $f$ is said to be multi-quadratic if it is quadratic in each variable, i.e., it satisfies the quadratic equation

in each variable ^[14]. In ^[27], Zhao et al. proved that the mapping $f: V^n\longrightarrow W$ is multi-quadratic if and only if the following relation holds.

where $x_j = (x_{1j}, x_{2j}, \dots, x_{nj})\in V^n$ with $j\in\{1, 2\}$.

The stability of a functional equation originated from a question raised by Ulam: “when is it true that the solution of an equation differing slightly from a given one must of necessity be close to the solution of the given equation?” (see ^[26]). The first answer (in the case of Cauchy's functional equation in Banach spaces) to Ulam's question was given by Hyers in ^[18]. Following his result, a great number of papers on the the stability problems of several functional equations have been extensively published as generalizing Ulam's problem and Hyers' theorem in various directions; see for instance ^{[1,4,21,23,28]}, and the references given there.

It is worth mentioning that the fixed point theorems have been considered for various mappings, integral and fractional equations in ^[3,12,13]. Some investigations have been carried out on the stability of functional equations via fixed point theorems in ^[5,6,7,11]. Moreover, the fixed point theorem were recently applied to obtain similar stability results in ^[9,16,22,25].

In ^[14,15], Ciepliński studied the generalized Hyers-Ulam stability of multi-additive and multi-quadratic mappings in Banach spaces, respectively (see also ^[27]). Next, the stability of multi-Cauchy-Jensen mappings in non-Archimedean spaces are studied in ^[2] by applying the fixed point method, which was proved and used for the first time to investigate the Hyers-Ulam stability of functional equations in ^[11]. For more information about multi-quadratic, multi-cubic and multi-quartic mappings, we refer to ^[8,10,20,24].

In this paper, we define the generalized multi-quadratic mappings and present a characterization of such mappings. In other words, we reduce the system of $n$ equations defining the generalized multi-quadratic mappings to obtain a single functional equation. Then, we prove the generalized Hyers-Ulam stability of multi-quadratic mapping (which was recently introduced by Salimi and Bodaghi in ^[24]) in non-Archimedean normed spaces by a fixed point method.

2.   Characterization of generalized multi-quadratic mappings

From now on, let $V$ and $W$ be vector spaces over $\mathbb Q$, $n\in\mathbb N$ and $x_i^n = (x_{i1}, x_{i2}, \ldots, x_{in})\in V^n$, where $i\in\{1, 2\}$. Let $l_j\in\{1, 2\}$. Put

We shall denote $x_i^n$ and $M^n_i$ by $x_i$ and $M_i$, respectively if there is no risk of ambiguity.

A general form of (1.1), say the generalized quadratic functional equation is as follows:

where $a, b$ are the fixed non-zero numbers in $\mathbb Q$. The mapping $f:V^n\longrightarrow W$ is said to be generalized $n$-multi-quadratic or generalized multi-quadratic if $f$ is generalized quadratic in each variable.

Put ${\bf n}: = \{1, \ldots, n\}$, $n\in \mathbb N$. For a subset $T = \{j_1, \ldots, j_i\}$ of ${\bf n}$ with $1\leq j_1 < \ldots < j_i\leq n$ and $x = (x_1, \ldots, x_n)\in V^n$,

denotes the vector which coincides with $x$ in exactly those components, which are indexed by the elements of $T$ and whose other components are set equal zero. Note that $_{\phi}x = 0$, $_{\bf n}x = x$. We use these notations in the proof of upcoming lemma.

Let $a\in \mathbb Q$ be as in (2.2). We say the mapping $f: V^n\longrightarrow W$ satisfies the $r$-power condition in the $j$th variable if

for all $(z_1, \ldots, z_n)\in V^n$. In the sequel, $\left(\begin{array}{ccccc} n\\ k\\ \end{array}\right)$ is the binomial coefficient defined for all $n, k\in \mathbb{N}_0$ with $n\geq k$ by $n!/(k!(n-k)!)$. We shall to show that if a mapping $f: V^n\longrightarrow W$ satisfies the equation

where $a, b$ are the fixed non-zero in $\mathbb Q$ with $a+b\neq 1$, then it is generalized multi-quadratic quadratic. In order to do this, we need the next lemma.

Lemma 2.1. If the mapping $f: V^n\longrightarrow W$ satisfies the Eq. (2.3) with $2$-power condition in all variables, then $f(x) = 0$ for any $x\in V^n$ with at least one component which is equal to zero.

Proof. Putting $x_{1} = x_{2} = (0, \ldots, 0)$ in (2.3), we get

Since $a+b\neq 1$, $f(0, \ldots, 0) = 0$. Letting $x_{1k} = 0$ for all $k\in\{1, \ldots, n\}\backslash\{j\}$ and $x_{2k} = 0$ for $1\leq k\leq n$ in (2.3) and using $f(0, \ldots, 0) = 0$, we obtain

Hence, $f(0, \ldots, 0, x_{1j}, 0, \ldots, 0) = 0$. We now assume that $f(_{k-1}x_1) = 0$ for $1\leq k\leq n-1$. We are going to show that $f(_{k}x_1) = 0$. By assumptions, the above process can be repeated to obtain

where $1\leq k\leq n-1$ and so $f(_{k}x_1) = 0$. This shows that $f(x) = 0$ for any $x\in V^n$ with at least one component which is equal to zero.

Theorem 2.2. Consider the mapping $f: V^n\longrightarrow W$. Then, the following assertions are equivalent:

(ⅰ) f is generalized multi-quadratic;

(ⅱ) f satisfies Eq. (2.3) with $2$-power condition in all variables.

Proof. (ⅰ)$\Rightarrow$(ⅱ) We firstly note that it is not hard to show that $f$ satisfies 2-power condition in all variables. We now prove that $f$ satisfies Eq. (2.3) by induction on $n$. For $n = 1$, it is trivial that $f$ satisfies Eq. (2.2). Assume that (2.3) is valid for some positive integer $n > 1$. Then,

This means that (2.3) holds for $n+1$.

(ⅱ)$\Rightarrow$(ⅰ) Fix $j\in\{1, \cdots, n\}$, put $x_{2k} = 0$ for all $k\in\{1, \cdots, n\}\backslash\{j\}$. Using Lemma 2.1, we obtain

It follows from relation (2.5) that $f$ is quadratic in the $j$th variable. Since $j$ is arbitrary, we obtain the desired result.

3.   Stability Results

An special case of (2.2) is the following quadratic functional equation when $a = b = \frac{1}{2}$.

A mapping $f:V^n\longrightarrow W$ is called $n$-multi-quadratic or multi-quadratic if $f$ is quadratic in each variable (see Eq. (3.1)). It is shown in [24, Proposition 2.2] (without extra 2-power condition in each variable) that a mapping $f: V^n\longrightarrow W$ is multi-quadratic if and only if it satisfies the equation

In this section, we prove the generalized Hyers-Ulam stability of Eq. (3.2) in non-Archimedean spaces.

We recall some basic facts concerning non-Archimedean spaces and some preliminary results. By a non-Archimedean field we mean a field $\mathbb K$ equipped with a function (valuation) $|\cdot|$ from $\mathbb K$ into $[0, \infty)$ such that $|r| = 0$ if and only if $r = 0$, $|rs| = |r||s|$, and $|r+s|\leq$ max$\{|r|, |s|\}$ for all $r, s\in\mathbb K$. Clearly $|1| = |-1| = 1$ and $|n|\leq 1$ for all $n\in \mathbb N$.

Let $\mathcal X$ be a vector space over a scalar field $\mathbb K$ with a non-Archimedean non-trivial valuation $|\cdot|$. A function $\|\cdot\|:\mathcal X\longrightarrow \mathbb R$ is a non-Archimedean norm (valuation) if it satisfies the following conditions:

(ⅰ) $\|x\| = 0$ if and only if $x = 0$;

(ⅱ) $\|rx\| = |r|\|x\|, \quad (x\in \mathcal X, r\in \mathbb K)$;

(ⅲ) the strong triangle inequality (ultrametric); namely,

Then $(\mathcal X, \|\cdot\|)$ is called a non-Archimedean normed space. Due to the fact that

a sequence $\{x_n\}$ is Cauchy if and only if $\{x_{n+1}-x_{n}\}$ converges to zero in a non-Archimedean normed space $\mathcal X$. By a complete non-Archimedean normed space we mean one in which every Cauchy sequence is convergent.

In ^[17], Hensel discovered the $p$-adic numbers as a number theoretical analogue of power series in complex analysis. The most interesting example of non-Archimedean normed spaces is $p$-adic numbers. A key property of $p$-adic numbers is that they do not satisfy the Archimedean axiom: for all $x, y > 0$, there exists an integer $n$ such that $x < ny$.

Let $p$ be a prime number. For any non-zero rational number $x = p^r\frac{m}{n}$ in which $m$ and $n$ are coprime to the prime number $p$. Consider the $p$-adic absolute value $|x|_p = p^{-r}$ on $\mathbb Q$. It is easy to check that $|\cdot|$ is a non-Archimedean norm on $\mathbb Q$. The completion of $\mathbb Q$ with respect to $|\cdot|$ which is denoted by $\mathbb Q_p$ is said to be the $p$-adic number field. One should remember that if $p > 2$, then $|2^n|$ = 1 in for all integers $n$.

Throughout, for two sets $A$ and $B$, the set of all mappings from $A$ to $B$ is denoted by $B^A$. The proof is based on a fixed point result that can be derived from [11, Theorem 1]. To present it, we introduce the following three hypotheses:

(H1) $E$ is a nonempty set, $Y$ is a complete non-Archimedean normed space over a non-Archimedean field of the characteristic different from 2, $j\in\mathbb N$, $g_1, \ldots, g_j:E\longrightarrow E$ and $L_1, \ldots, L_j:E\longrightarrow \mathbb R_+$,

(H2) $\mathcal T:Y^E\longrightarrow Y^E$ is an operator satisfying the inequality

(H3) $\Lambda:\mathbb R_+^E\longrightarrow \mathbb R_+^E$ is an operator defined through

Here, we highlight the following theorem which is a fundamental result in fixed point theory ^[11]. This result plays a key role in obtaining our goal in this paper.

Theorem 3.1. Let hypotheses (H1)-(H3) hold and the function $\epsilon:E\longrightarrow \mathbb R_+$ and the mapping $\varphi:E\longrightarrow Y$ fulfill the following two conditions:

Then, for every $x\in E$, the limit $\lim_{l\rightarrow\infty}\mathcal T^l\varphi(x) = :\psi(x)$ and the function $\psi\in Y^E$, defined in this way, is a fixed point of $\mathcal T$ with

Here and subsequently, given the mapping $f:V^n \longrightarrow W$, we consider the difference operator $\Gamma f:V^n\times V^n \longrightarrow W$ by

In the sequel, $S$ stands for $\{0, 1\}^n$. With these notations, we have the upcoming result.

Theorem 3.2. Let $V$ be a linear space and $W$ be a complete non-Archimedean normed space over a non-Archimedean field of the characteristic different from $2$. Suppose that $\phi:V^n\times V^n \longrightarrow \mathbb R_+$ is a mapping satisfying the equality

for all $x_1, x_2\in V^n$. Assume also $f:V^n \longrightarrow W$ is a mapping satisfying the inequality

for all $x_1, x_2\in V^n$. Then, there exists a unique multi-quadratic mapping $\mathcal Q:V^n \longrightarrow W$ such that

for all $x\in V^n$.

Proof. Replacing $x = x_1 = (x_{11}, \ldots, x_{1n}), x_2 = (x_{21}, \ldots, x_{2n})$ by $2x_{1}, (0, \ldots, 0)$ in (3.4), respectively, we have

for all $x\in V^n$. Inequality (3.6) implies that

for all $x\in V^n$, where $\xi(x): = \frac{1}{|2|^{2n}}\phi(2x, 0)$ and $\mathcal Tf(x): = \frac{1}{2^{2n}}\sum_{s\in S}f(2sx)$. Define $\Lambda\eta(x): =$max$_{s\in S}\frac{1}{|2|^{2n}}\eta(2sx)$ for all $\eta\in \mathbb R_+^{V^n}, x\in V^n$. It is easy to see that $\Lambda$ has the form described in (H3) with $E = V^n$, $g_i(x): = g_s(x) = 2sx$ for all $x\in V^n$ and $L_i(x) = \frac{1}{|2|^{2n}}$ for any $i$. Moreover, for each $\lambda, \mu\in W^{V^n}$ and $x\in V^n$, we get

The above inequality shows that the hypothesis (H2) holds. By induction on $l$, one can check that for any $l\in \mathbb N$ and $x\in V^n$ that

for all $x\in V^n$. Indeed, by definition of $\Lambda$, equality (3.8) is true for $l = 1$. If (3.8) holds for $l\in \mathbb N$, then

for all $x\in V^n$. Relations (3.7) and (3.8) necessitate that all assumptions of Theorem 3.1 are satisfied. Hence, there exists a unique mapping $\mathcal Q:V^n \longrightarrow W$ such that $\mathcal Q(x) = \lim_{l\rightarrow\infty}(\mathcal T^lf)(x)$ for all $x\in V^n$, and also (3.5) holds. We are going to show that

for all $x_1, x_2\in V^n$ and $l\in\mathbb N$. We argue by induction on $l$. For $l = 1$ and for all $x_1, x_2\in V^n$, we have

for all $x_1, x_2\in V^n$. Assume that (3.9) is true for an $l\in\mathbb N$. Then

for all $x_1, x_2\in V^n$. Letting $l\rightarrow\infty$ in (3.9) and applying (3.3), we arrive at $\Gamma\mathcal Q(x_1, x_2) = 0$ for all $x_1, x_2\in V^n$. This means that the mapping $\mathcal Q$ satisfies (3.2) and the proof is now completed.

The following example is an application of Theorem 3.2 concerning the stability of multi-quadratic mappings when the norm of $\Gamma f(x_1, x_2)$ is controlled by the powers sum of norms of components of vectors $x_1$ and $x_2$ in $V^n$.

Example 3.3. Let $p\in \mathbb R$ fulfills $p > 2n$. Let $V$ be a normed space and $W$ be a complete non-Archimedean normed space over a non-Archimedean field of the characteristic different from $2$ such that $|2| < 1$. Suppose that $f:V^n \longrightarrow W$ is a mapping satisfying the inequality

for all $x_1, x_2\in V^n$. Putting $\phi (x_1, x_2) = \sum_{k = 1}^{2}\sum_{j = 1}^{n}\|x_{kj}\|^p$, we have $\phi (2^lx_1, 2^lx_2) = |2|^{lp}\phi (x_1, x_2)$ and so

for all $x_1, x_2\in V^n$. On the other hand,

By Theorem 3.2, there exists a unique multi-quadratic mapping $\mathcal Q:V^n \longrightarrow W$ such that

for all $x\in V^n$.

Recall that a functional equation $\mathcal F$ is hyperstable if any mapping $f$ satisfying the equation $\mathcal F$ approximately is a true solution of $\mathcal F$. Under some conditions functional Eq. (3.2) can be hyperstable as follows.

Corollary 3.4. Suppose that $p_{kj} > 0$ for $k\in\{1, 2\}$ and $j\in\{1, \ldots, n\}$ fulfill $\sum_{k = 1}^{2}\sum_{j = 1}^{n}p_{kj} > 2n$. Let $V$ be a normed space and $W$ be a complete non-Archimedean normed space over a non-Archimedean field of the characterisitic different from $2$ such that $|2| < 1$. If $f:V^n \longrightarrow W$ is a mapping satifying the inequality

for all $x_1, x_2\in V^n$, then $f$ is multi-quadratic.

Acknowledgments

The authors sincerely thank the anonymous reviewers for their careful reading, constructive comments and suggesting some related references to improve the quality of the first draft of paper.

Conflict of interest

The authors declare no conflicts of interest.