A Fixed Point Approach to Stability of k-$\text {th}$ Radical Functional Equation in Non-Archimedean $(n,\beta )$-Banach Spaces

Abstract

In this work, we prove a simple fixed point theorem in non-Archimedean $(n,\beta )$-Banach spaces, by applying this fixed point theorem, we will study the stability and the hyperstability of the kth radical-type functional equation:

$$\begin{aligned} f\left( \root k \of {x^k+y^k}\right) = f(x)+f(y), \end{aligned}$$

where f is a mapping on the set of real numbers and k is a fixed positive integer. Furthermore, we give some important consequences from our main results.

1 Introduction

In 1940, during a conference at Wisconsin University, Ulam [35] presented the following question concerning stability of group homomorphisms: Let $(G_1, \star )$, $(G_2, *)$ be two groups and $d:G_2\times G_2\rightarrow [0,\infty )$ be a metric. Given $\epsilon > 0,$ does there exist $\delta > 0$, such that if a function $g : G_1 \rightarrow G_2$ satisfies the inequality $d(g(x\star y),g(x)*g(y))\le \delta$ for all $x,y\in G_1$, then there is a homomorphism $h : G_1\rightarrow G_2$ with $d(g(x),h(x))\le \epsilon$ for all $x\in G_1?$

When the homomorphisms are stable? Therefore, we are interested in this question, that is, if a mapping is almost a homomorphism, then there exists an exact homomorphism that must be close. In following year, Hyers [24] was the first to give an affirmative answer to Ulam’s question for the case where $G_1$ and $G_2$ are Banach spaces. The famous Hyers stability result that appeared in [24] was generalized in the stability involving a sum of powers of norms by Aoki [3]. In 1978, Rassias [32] provided a generalization of Hyers’ theorem that allows the Cauchy difference to become unbounded. For the last decades, stability problems of various functional equations have been extensively investigated and generalized by many mathematicians [6, 11, 13, 15, 22, 30, 33, 34, 37]. The theory of 2-normed spaces was first developed by Gähler [20] in the mid-1960s, while that of 2-Banach spaces was studied later by Gähler [21] and White [36]. In 1897, Hensel [23] introduced a normed space which does not have the Archimedean property. It turns out that non-Archimedean spaces have many nice applications (see [4, 25, 28, 31]).

The first hyperstability result appears to be due to Bourgin [5]. However, the term hyperstability was used for the first time in [29]. Quite often, hyperstability is confused with superstability, which admits also bounded functions. Numerous papers on this subject have been published and we refer to [7, 12, 14, 16, 18, 19]. Recently, the stability problem and hyperstability results for the functional equations of the radical type in 2-Banach spaces and in some other generalized spaces have been also studied; for example, see [1, 2, 8, 10, 16,17,18, 26, 27].

The functional equation:

$$\begin{aligned} f\left( \sqrt{x^2+y^2}\right) = f(x)+f(y) \end{aligned}$$

(1.1)

is called a radical quadratic functional equation. Kim et al. [27] investigated the generalized Hyers–Ulam–Rassias stability of Eq. (1.1) in quasi-$\beta$-Banach spaces using the direct method.

In the whole paper, ${\mathbb {N}}$ and ${\mathbb {R}}$ denote the sets of all positive integers and real numbers, respectively; we put ${\mathbb {N}}_0:={\mathbb {N}}\cup \{0\},$ ${\mathbb {R}}_0:={\mathbb {R}}{\setminus } \{0\},$ and ${\mathbb {R}}_+=[0,\infty )$, and we write ${\mathcal {B}}^{\mathcal {A}}$ to mean the family of all functions mapping from a nonempty set ${\mathcal {A}}$ into a nonempty set ${\mathcal {B}}.$

This work is organized as follows: in Sect. 2, we discuss some basic definitions and lemmas used in later sections to prove the stabilities on non-Archimedean $(n,\beta )$-Banach spaces. In Sect. 3, we introduce and solve the kth radical-type functional equation:

$$\begin{aligned} f\left( \root k \of {x^k+y^k}\right) = f(x)+f(y),\quad x,y\in {\mathbb {R}}, \end{aligned}$$

(1.2)

in the class of functions f from ${\mathbb {R}}$ into a vector space, where $k\in {\mathbb {N}}$ is fixed. In Sect. 4, we prove the fixed point theorem [9, Theorem 1] in non-Archimedean $(n,\beta )$-Banach space. In Sect. 5, we will apply the fixed point method to study the stability and the hyperstability of (1.2) in non-Archimedean $(n,\beta )$-Banach space. In Sect. 6, we will give some consequences from our main results. Our results are improvements and generalizations of many main results referred to in [1, 2, 16,17,18, 26] on non-Archimedean $(n,\beta )$-Banach spaces.

2 Preliminaries

In this section, we will introduce some basic concepts concerning the non-Archimedean $(n,\beta )$-normed space.

Definition 2.1

By a non-Archimedean field, we mean a field ${\mathbb {K}}$ equipped with a function (valuation) $|\cdot |_{*}:{\mathbb {K}}\rightarrow [0,\infty )$ such that for all $r,s \in {\mathbb {K}}$, the following conditions hold:

• (1) $|r|_{*}=0$ if and only if $r=0$;

• (2) $|rs|_{*}=|r|_{*}|s|_{*};$

• (3) $|r+s|_{*} \le \max (|r|_{*},|s|_{*})$ for all $r,s\in {\mathbb {K}}$.

Clearly, $|1|_{*}=|-1|_{*}=1$ and $|n|_{*}\le 1$ for all $n\in {\mathbb {N}}$. The function $|\cdot |_{*}$ is called the trivial valuation if $|r|_{*}=1$, $\forall r\in {\mathbb {K}}$, $r\ne 0$, and $|0|_{*}=0$.

Definition 2.2

Let E be a vector space over a scalar field ${\mathbb {K}}$ with a non-Archimedean non-trivial valuation $|\cdot |_{*}$. A function $\Vert \cdot \Vert _{*}: E\rightarrow {\mathbb {R}}_+$ is non-Archimedean norm (valuation) if it satisfies the following conditions:

• (1) $\Vert x\Vert _{*}=0$ if and only if $x=0$;

• (2) $\Vert rx\Vert _{*}=|r|_{*}\Vert x\Vert _{*}$ for all $r\in {\mathbb {K}}$ and $x\in E$;

• (3) $\Vert x+y\Vert _{*} \le \max (\Vert x\Vert _{*},\Vert y\Vert _{*})$ for all $x,y\in E$.

Then, $(E,\Vert \cdot \Vert _{*})$ is called a non-Archimedean space or an ultrametric normed space. Due to the fact that:

$$\begin{aligned} \Vert x_{m}-x_{n}\Vert _{*}\le \max \{\Vert x_{j+1}-x_{j}\Vert _{*}:m\le j\le n-1\},\ \end{aligned}$$

in which $n>m$, the sequence $\{x_{n}\}$ is Cauchy if and only if $\{x_{n+1}-x_{n}\}$ converges to zero in a non-Archimedean normed space. In a complete non-Archimedean space, every Cauchy sequence is convergent.

Example 2.3

Fix a prime number p. For any nonzero rational number x, there exists a unique positive integer $n_x$, such that $x = \frac{a}{b}p^{n_x}$, where a and b are positive integers not divisible by p. Then, $|x|_p := p^{-{n_x}}$ defines a non-Archimedean norm on ${\mathbb {Q}}$ (the set of rational numbers). The completion of ${\mathbb {Q}}$ with respect to the metric $d(x, y) = |x - y|_p$ is denoted by ${\mathbb {Q}}_p$, which is called the p-adic number field. In fact, ${\mathbb {Q}}_p$ is the set of all formal series $x =\sum _{k\ge n_x}^{\infty } a_kp^k$, where $|a_k| \le p-1$. The addition and multiplication between any two elements of ${\mathbb {Q}}_p$ are defined naturally. The norm $\Big |\sum _{k\ge n_x}^{\infty } a_kp^k\Big |=p^{-n_{x}}$ is a non-Archimedean norm on ${\mathbb {Q}}_p$ and ${\mathbb {Q}}_p$ is a locally compact field.

Definition 2.4

Let X be a real vector space with $dim\;X \ge n$ over a scalar field ${\mathbb {K}}$ with a non-Archimedean nontrivial valuation $|\cdot |_{*}$, where $n\in {\mathbb {N}}$ and $\beta \in (0,1]$ is a fixed number. A function $\Vert \cdot ,\ldots ,\cdot \Vert _{*,\beta }:X^n\rightarrow {\mathbb {R}}_{+}$ is called a non-Archimedean $(n,\beta )$-norm on X if and only if it satisfies:

• (N1) $\Vert x_1 ,x_2,\ldots , x_n\Vert _{*,\beta }=0$ if and only if $x_1 ,x_2,\ldots , x_n$ are linearly dependent;

• (N2) $\Vert x_1 ,x_2,\ldots , x_n\Vert _{*,\beta }$ is invariant under permutations of $x_1 ,x_2,\ldots , x_n$;

• (N3) $\Vert \lambda x_1 ,x_2,\ldots , x_n\Vert _{*\beta }=|\lambda |_{*}^{\beta }\Vert x_1 ,x_2,\ldots , x_n\Vert _{\beta }$;

• (N4) $\Vert x+y, x_2,\ldots , x_{n}\Vert _{*,\beta }\le \max \big \{\Vert x ,x_2,\ldots , x_{n}\Vert _{*,\beta },\Vert y, ,x_2,\ldots , x_{n}\Vert _{*\beta }\big \}$

for all $x,y,x_1 ,x_2,\ldots , x_n\in X$ and $\lambda \in {\mathbb {K}}$. Then, the pair $(X, \Vert \cdot ,\ldots , \cdot \Vert _{*,\beta })$ is called a non-Archimedean $(n,\beta )$-normed space.

Example 2.5

Let ${\mathbb {K}}$ be a non-Archimedean field equipped with a non-trivial valuation $|\cdot |_{*}$. For $n=2$, $\lambda \in {\mathbb {K}}$ and $x = (x_1, x_2),\; y = (y_1, y_2)\in X = {\mathbb {K}}^2$ with $x+y = (x_1+y_1, x_2+y_2)$ and $\lambda x=(\lambda x_1,\lambda x_2),$ the non-Archimedean $(2,\beta )$-norm on X is defined by:

$$\begin{aligned} \Vert x, y\Vert _{*,\beta }= |x_1y_2 - x_2y_1|_{*}^{\beta }, \end{aligned}$$

where $\beta \in (0,1]$ is a fixed number.

It follows from the preceding definition that the non-Archimedean $(n,\beta )$-normed space is a non-Archimedean n-normed space if $\beta = 1,$ and a non-Archimedean $\beta$-normed space if $n =1$, respectively.

Lemma 2.6

Let $(X,\Vert \cdot ,\ldots ,\cdot \Vert _{*,\beta })$ be a non-Archimedean $(n,\beta )$-normed space, such that $n\ge 2$ and $0<\beta \le 1$. Then:

• (1) if $x \in X$ and $\Vert x, x_2,\ldots ,x_{n}\Vert _{*,\beta } = 0$ for all $x_2,\ldots ,x_{n} \in X$, then $x = 0$;

• (2) a sequence $\{x_m\}$ in a non-Archimedean $(n,\beta )$-normed space X is a Cauchy sequence if and only if $\{x_{m+1}-x_m\}$ converges to zero in X.

Proof

For (1), suppose that $x \ne 0$. Since $\dim X \ge n,$ choose $x_2,\ldots , x_n \in X$, such that $\{x,x_2,\ldots , x_n\}$ is linearly independent and so by (N1) in Definition 2.4, we have:

$$\begin{aligned} \Vert x, x_2,\ldots ,x_{n}\Vert _{*,\beta } \ne 0. \end{aligned}$$

This is a contradiction and thus x should be a zero vector. For (2), it follows from (N4) that:

$$\begin{aligned} \Vert x_m-x_k,x_2,\ldots ,x_n\Vert _{*,\beta }\le \max \big \{\Vert x_{j+1}-x_j,x_2,\ldots ,x_n\Vert _{*,\beta }:k\le j\le m-1\big \},\;\;(m>k) \end{aligned}$$

for all $x_2,\ldots ,x_{n} \in X$. Therefore, a sequence $\{x_m\}$ is a Cauchy sequence in X if and only if $\{x_{m+1}- x_m\}$ converges to zero in X. $\square$

Definition 2.7

(a) A sequence $\{x_m\}$ in a non-Archimedean $(n, \beta )$-normed space X is called a convergent sequence if there exists an element $x \in X$, such that $\lim _{m\rightarrow \infty } \Vert x_m-x, x_2,\ldots ,x_{n}\Vert _{*,\beta } =0$ for all $x_2,\ldots ,x_{n} \in X$. In this case, we write $\lim _{m\rightarrow \infty }x_m := x,$ and we have

$$\begin{aligned} \lim _{m\rightarrow \infty } \Vert x_m, x_2,\ldots ,x_{n}\Vert _{*,\beta } = \Vert \lim _{m\rightarrow \infty }x_m, x_2,\ldots ,x_{n}\Vert _{*,\beta } \end{aligned}$$

for all $x_2,\ldots ,x_{n} \in X$.

(b) A non-Archimedean $(n, \beta )$-normed space in which every Cauchy sequence is a convergent sequence is called a non-Archimedean $(n, \beta )$-Banach space.

3 Solution of Eq. (1.2)

In this section, we give the general solution of functional equation (1.2).

Theorem 3.1

Let ${\mathcal {Y}}$ be a linear space. A function $f : {\mathbb {R}} \rightarrow {\mathcal {Y}}$ satisfies Eq. (1.2) if and only if there exists an additive function $T : {\mathbb {R}} \rightarrow {\mathcal {Y}}$, such that:

$$\begin{aligned} f(x)=T(x^k),\quad x\in {\mathbb {R}}, \end{aligned}$$

(3.1)

for each fixed $k\in {\mathbb {N}}$.

Proof

(See [8, page 127]). $\square$

Remark 3.2

• (i) The function $f(x)=cx^k$ satisfies Eq. (1.2) for all $x\in {\mathbb {R}}$, where $k\in {\mathbb {N}}$ and $c\in {\mathbb {R}}$ are fixed numbers.

• (ii) If f satisfies Eq. (1.2), then $f(r^{p/k}x)=r^pf(x)$ for all $x\in {\mathbb {R}}$ and integers p, where $r \in {\mathbb {Q}}{\setminus }\{0\}$ (${\mathbb {Q}}:=$, the set rational numbers) if k is odd and $r \in {\mathbb {Q}}^+$ (${\mathbb {Q}}^+:=$ the set of positive rational numbers) if k is even.

• (iii) If f satisfies Eq. (1.2) and continuous, then $f(x)=x^kf(1)$ for all $x\in {\mathbb {R}}$ if k is odd and $f(x)=x^kf(1)$ for all $x\in {\mathbb {R}}_+$ if k is even.

4 Fixed Point Theorem

In this section, we rewrite the fixed point theorem [9, Theorem 1] in non-Archimedean $(n,\beta )$-Banach space. For it, we need to introduce the following hypotheses.

(H1):

W is a nonempty set and X is a non-Archimedean $(n,\beta )$-Banach space.

(H2):

$f_{1},\ldots ,f_{j}:W\rightarrow W$ and $K_{1},\ldots ,K_{j}:W\times X^{n-1}\rightarrow \mathbb {R_{+}}$ are given maps.

(H3):

$\Lambda :\mathbb {R_{+}}^{W\times X^{n-1}}\rightarrow {\mathbb {R}}_{+}^{W\times X^{n-1}}$ is a non-decreasing operator defined by:

$$\begin{aligned} (\Lambda \delta )(x,x_2,\ldots ,x_n):=\max _{1\le i\le j}K_{i}(x,x_2,\ldots ,x_n)\delta (f_{i}(x),x_2,\ldots ,x_n) \end{aligned}$$

for all $\delta \in {\mathbb {R}}_{+}^{W\times X^{n-1}},$ $(x,x_2,\ldots ,x_n)\in W\times X^{n-1}$.

(H4):

${\mathcal {T}}:X^W\rightarrow X^W$ is an operator satisfying the inequality:

$$\begin{aligned}&\left\| {\mathcal {T}}\xi (x)-{\mathcal {T}}\mu (x),x_2,\ldots ,x_n\right\| _{*,\beta }\le \max _{1\le i\le j}K_{i}(x,x_2,\ldots ,x_n)\left\| \xi (f_{i}(x))\right. \\&\quad \left. -\mu (f_{i}(x)),x_2,\ldots ,x_n\right\| _{*,\beta } \end{aligned}$$

for all $\xi , \mu \in X^W$ and $(x,x_2,\ldots ,x_n)\in W\times X^{n-1}$.

The basic tool in this paper is the following fixed point theorem.

Theorem 4.1

Assume that hypotheses (H1)–(H4) are satisfied. Suppose that there are functions $\varepsilon :W\times X^{n-1}\rightarrow \mathbb {R_{+}}$ and $\varphi : W\rightarrow X$, such that:

$$\begin{aligned} \big \Vert {\mathcal {T}}\varphi (x)-\varphi (x),x_2,\ldots ,x_n\big \Vert _{*,\beta }\le \varepsilon (x,x_2,\ldots ,x_n),\;\; (x,x_2,\ldots ,x_n)\in W\times X^{n-1}, \end{aligned}$$

(4.1)

and

$$\begin{aligned} \lim _{m\rightarrow \infty }\Lambda ^m\varepsilon (x,x_2,\ldots ,x_n)=0, \;\; (x,x_2,\ldots ,x_n)\in W\times X^{n-1}, \end{aligned}$$

(4.2)

then, for every $x \in W$, the limit:

$$\begin{aligned} \psi (x):=\lim _{m\rightarrow \infty }({\mathcal {T}}^m\varphi )(x) \end{aligned}$$

exists and the function $\psi \in X^W,$ defined in this way, is a fixed point of ${\mathcal {T}}$ with:

$$\begin{aligned} \Vert \varphi (x)-\psi (x),x_2,\ldots ,x_n\Vert _{*,\beta }&\le \sup _{m\in {\mathbb {N}}_0}(\Lambda ^m\varepsilon )(x,x_2,\ldots ,x_n) \end{aligned}$$

(4.3)

for all $(x,x_2,\ldots ,x_n)\in W\times X^{n-1}$. Moreover, if

$$\begin{aligned} \Lambda \big (\sup _{m\in {\mathbb {N}}_0}(\Lambda ^m\varepsilon )\big )(x,x_2,\ldots ,x_n)\le \sup _{m\in {\mathbb {N}}_0}(\Lambda ^{m+1}\varepsilon )(x,x_2,\ldots ,x_n) \end{aligned}$$

for all $(x,x_2,\ldots ,x_n)\in W\times X^{n-1}$, then $\psi$ is the unique fixed point of ${\mathcal {T}}$ satisfying (4.3).

Proof

First, we show by induction that, for any $m\in {\mathbb {N}}_0$:

$$\begin{aligned} \big \Vert ({\mathcal {T}}^{m+1}\varphi )(x)-({\mathcal {T}}^{m}\varphi )(x),x_2,\ldots ,x_n\big \Vert _{*,\beta }&\le (\Lambda ^m\varepsilon )(x,x_2,\ldots ,x_n),\\ {}&\;\; (x,x_2,\ldots ,x_n)\in W\times X^{n-1}.\nonumber \end{aligned}$$

(4.4)

Clearly, by (4.1), the case $m=0$ is trivial. Now, fix $m\in {\mathbb {N}}_0$ and suppose that (4.4) is valid. Then, using (H3) and (H4), for any $(x,x_2,\ldots ,x_n)\in W\times X^{n-1},$ we obtain:

$$\begin{aligned}&\big \Vert ({\mathcal {T}}^{m+2}\varphi )(x)-({\mathcal {T}}^{m+1}\varphi )(x),x_2,\ldots ,x_n\big \Vert _{*,\beta }= \big \Vert {\mathcal {T}}({\mathcal {T}}^{m+1}\varphi )(x)-{\mathcal {T}}({\mathcal {T}}^{m}\varphi )(x),x_2,\ldots ,x_n\big \Vert _{*,\beta } \\&\quad \le \max _{1\le i\le j}K_i(x,x_2,\ldots ,x_n)\big \Vert {\mathcal {T}}^{m+1}\varphi (f_i(x))-{\mathcal {T}}^{m}\varphi (f_i(x)),x_2,\ldots ,x_n\big \Vert _{*,\beta } \\&\quad \le \max _{1\le i\le j}K_i(x,x_2,\ldots ,x_n)(\Lambda ^m\varepsilon )(f_i(x),x_2,\ldots ,x_n)\\&\quad =(\Lambda ^{m+1}\varepsilon )(x,x_2,\ldots ,x_n), \end{aligned}$$

and therefore, (4.4) holds for every $m\in {\mathbb {N}}_0$.

By (4.2), (4.4) and Lemma 2.6, we get $\big \{({\mathcal {T}}^{m}\varphi )(x)\big \}_{m\in {\mathbb {N}}}$ is a Cauchy sequence in X. Thus, the fact that X is a non-Archimedean $(n,\beta )$-Banach space implies that the limit $\psi (x)$ exists for every $x\in W$, i.e., $\psi (x):=\lim _{m\rightarrow \infty }({\mathcal {T}}^m\varphi )(x)$ for any $x\in W$. Moreover, (4.4) shows that, for any $k\in {\mathbb {N}}$, $m\in {\mathbb {N}}_0$ and $(x,x_2,\ldots ,x_n)\in W\times X^{n-1}:$

$$\begin{aligned}&\big \Vert ({\mathcal {T}}^{m}\varphi )(x)-({\mathcal {T}}^{m+k}\varphi )(x),x_2,\ldots ,x_n\big \Vert _{*,\beta }\\&\quad \le \max _{\ell \in \{0,\cdots k-1\}}\big \Vert ({\mathcal {T}}^{m+\ell }\varphi )(x)-({\mathcal {T}}^{m+\ell +1}\varphi )(x),x_2,\ldots ,x_n\big \Vert _{*,\beta }\\&\quad \le \max _{\ell \in \{0,\cdots k-1\}}(\Lambda ^{m+\ell }\varepsilon )(x,x_2,\ldots ,x_n) \\&\quad \le \sup _{\ell \ge m}(\Lambda ^{\ell }\varepsilon )(x,x_2,\ldots ,x_n). \end{aligned}$$

Letting now $k\rightarrow \infty$, we see that for any $m\in {\mathbb {N}}_0$ and $(x,x_2,\ldots ,x_n)\in W\times X^{n-1}$, we have:

$$\begin{aligned} \big \Vert ({\mathcal {T}}^{m}\varphi )(x)-\psi (x),x_2,\ldots ,x_n\big \Vert _{*,\beta }\le \sup _{\ell \ge m}(\Lambda ^{\ell }\varepsilon )(x,x_2,\ldots ,x_n). \end{aligned}$$

(4.5)

Putting $m = 0$ in (4.5), we see that inequality (4.3) holds. Moreover, using (H3)–(H4) and (4.5), we obtain:

$$\begin{aligned} \big \Vert ({\mathcal {T}}\psi )(x)-({\mathcal {T}}^{m+1}\varphi )(x)&,x_2,\ldots ,x_n\big \Vert _{*,\beta }\nonumber \\ {}&\le \max _{1\le i\le j}K_i(x,x_2,\ldots ,x_n)\big \Vert \psi (f_i(x))-{\mathcal {T}}^{m}\varphi (f_i(x)),x_2,\ldots ,x_n\big \Vert _{*,\beta } \nonumber \\&\le \max _{1\le i\le j}K_i(x,x_2,\ldots ,x_n)\sup _{\ell \ge m}(\Lambda ^{\ell }\varepsilon )(f_i(x),x_2,\ldots ,x_n) \nonumber \\ {}&= \Lambda \big (\sup _{\ell \ge m}(\Lambda ^{\ell }\varepsilon )\big )(x,x_2,\ldots ,x_n) \end{aligned}$$

(4.6)

for all $(x,x_2,\ldots ,x_n)\in W\times X^{n-1}$. From (4.2) and (4.6), we get:

$$\begin{aligned} {\mathcal {T}}(\psi )(x)=\lim _{m\rightarrow \infty }({\mathcal {T}}^{m+1}\varphi )(x)=\psi (x),\;\;x\in W. \end{aligned}$$

To prove the statement on the uniqueness of $\psi$, suppose that $\psi _1,\;\psi _2\in X^W$ are two fixed points of ${\mathcal {T}}$ satisfies (4.3). Then, for each $(x,x_2,\ldots ,x_n)\in W\times X^{n-1}$, we have:

$$\begin{aligned} \Vert \psi _1(x)-\psi _2(x),x_2,\ldots ,x_n\Vert _{*,\beta }\le \sup _{m\in {\mathbb {N}}_0}(\Lambda ^m\varepsilon )(x,x_2,\ldots ,x_n), \end{aligned}$$

and as in the proof of (4.4), for any $(x,x_2,\ldots ,x_n)\in W\times X^{n-1}$ and $k \in {\mathbb {N}}_0,$ we get:

$$\begin{aligned} \Vert \psi _1(x)-\psi _2(x),x_2,\ldots ,x_n\Vert _{*,\beta }&= \Vert ({\mathcal {T}}^{k}\psi _1)(x)-({\mathcal {T}}^{k}\psi _2)(x),x_2,\ldots ,x_n\Vert _{*,\beta } \nonumber \\&\le \sup _{m\in {\mathbb {N}}_0}(\Lambda ^{m+k}\varepsilon )(x,x_2,\ldots ,x_n). \end{aligned}$$

(4.7)

Letting $m\rightarrow \infty$ in (4.7) and from (4.2), we finally get $\psi _1=\psi _2$. $\square$

5 A New Stability Result for Eq. (1.2)

The following theorem is the main result of this paper. It has been motivated by the issue of Ulam stability, which concerns approximate solutions of a functional equation (1.2) in non-Archimedean $(n,\beta )$-Banach spaces by applying the fixed point theorem 4.1.

Theorem 5.1

Let X be a non-Archimedean $(n,\beta )$-Banach space. Let $f : {\mathbb {R}} \rightarrow X$, $c:{\mathbb {N}}\rightarrow {\mathbb {R}}_+$ and $L:{\mathbb {R}}_{0}\times {\mathbb {R}}_{0} \times X^{n-1}\rightarrow {\mathbb {R}}_+$ be functions satisfying the following three conditions:

$$\begin{aligned}&{\mathcal {M}}:=\{m\in {\mathbb {N}}\mid a_m:= \max \big \{c(m^k) ,c(1+m^k)\big \}<1\}\ne \emptyset , \end{aligned}$$

(5.1)

$$\begin{aligned}&L\big (tx^k,ty^k,x_2,\ldots ,x_n\big )\le c(t)L\big (x^k,y^k,x_2,\ldots ,x_n\big ), \nonumber \\&t\in \big \{m^k,1+m^k\big \},\;\;m\in {\mathcal {M}}, \end{aligned}$$

(5.2)

$$\begin{aligned}&\left\| f\left( \root k \of {x^k+y^k}\right) -f(x)-f(y),x_2,\ldots ,x_n \right\| _{*,\beta }\le L\big (x^k,y^k,x_2,\ldots ,x_n\big )\nonumber \\ \end{aligned}$$

(5.3)

for all $x,y\in {\mathbb {R}}_0$ and $x_2,\ldots ,x_n\in X,$ with $k\in {\mathbb {N}}$ is fixed. Then, there exists a unique additive function $T : {\mathbb {R}} \rightarrow X$ (i.e., $T(x+y)=T(x)+T(y)$ for all $x,y\in {\mathbb {R}})$, such that:

$$\begin{aligned} \left\| f(x)-T(x^k),x_2,\ldots ,x_n\right\| _{*,\beta }\le \phi _L(x,x_2,\ldots ,x_n), \quad x\in {\mathbb {R}}_0,\;\; x_2,\ldots ,x_n\in X, \end{aligned}$$

(5.4)

where:

$$\begin{aligned} \phi _L(x,x_2,\ldots ,x_n):=\inf _{m\in {\mathcal {M}}}L\big (x^k,m^kx^k,x_2,\ldots ,x_n\big ). \end{aligned}$$

(5.5)

Proof

Taking $y=m x$ in (5.3), we get:

$$\begin{aligned} \left\| f\left( \root k \of {(1+m^k)x^k}\right) -f(mx)-f(x),x_2,\ldots ,x_n\right\| _{*,\beta }&\le L\big (x^k,m^kx^k,x_2,\ldots ,x_n\big )\nonumber \\&=:\varepsilon _ m(x,x_2,\ldots ,x_n) \end{aligned}$$

(5.6)

for all $x\in {\mathbb {R}}_0$ and $x_2,\ldots ,x_n\in X,$ with $m\in {\mathbb {N}}$. For each $m\in {\mathbb {N}},$ we will define an operator ${\mathcal {T}}_m: X^{{\mathbb {R}}}\rightarrow X^{{\mathbb {R}}}$ by:

$$\begin{aligned} {\mathcal {T}}_m\xi (x):= \xi \left( \root k \of {(1+m^k)x^k}\right) -\xi (mx),\quad \xi \in X^{{\mathbb {R}}},\quad x\in {\mathbb {R}}. \end{aligned}$$

Then:

$$\begin{aligned} {\mathcal {T}}_m^{\ell }f(0)=0,\;\;\ell ,m\in {\mathbb {N}}, \end{aligned}$$

(5.7)

and inequality (5.6) takes the form:

$$\begin{aligned} \Vert {\mathcal {T}}_mf(x)-f(x),x_2,\ldots ,x_n\Vert _{*,\beta }\le \varepsilon _m(x,x_2,\ldots ,x_n) \end{aligned}$$

for all $x\in {\mathbb {R}}_0$, $x_2,\ldots ,x_n\in X$, and $m\in {\mathbb {N}}$.

Let $\Lambda _m:{\mathbb {R}}_+^{{\mathbb {R}}_0\times X^{n-1}}\rightarrow {\mathbb {R}}_+^{{\mathbb {R}}_0\times X^{n-1}}$ be an operator which is defined by:

$$\begin{aligned} \Lambda _m\delta (x,x_2,\ldots ,x_n)=\max \left\{ \delta \left( \root k \of {(1+m^k)x^k},x_2,\ldots ,x_n\right) ,\delta (mx,x_2,\ldots ,x_n)\right\} \end{aligned}$$

for all $\delta \in {\mathbb {R}}_+^{{\mathbb {R}}_0},$ $x\in {\mathbb {R}}_0$ and $x_2,\ldots ,x_n\in X$. Then, it is easily seen that, for each $m\in {\mathbb {N}}$, the operator $\Lambda :=\Lambda _m$ has the form described in (H3), with $j=2$, $W={\mathbb {R}}_0$ and:

$$\begin{aligned} f_{1}(x)=\root k \of {(1+m^k)x^k},\quad f_{2}(x)=mx, \quad K_{1}(x,x_2,\ldots ,x_n)=K_{2}(x,x_2,\ldots ,x_n)=1 \end{aligned}$$

for all $x\in {\mathbb {R}}_0$ and $x_2,\ldots ,x_n\in X$. Moreover, for every $\xi , \mu \in X^{{\mathbb {R}}_0}$, $m\in {\mathbb {N}},$ $x\in {\mathbb {R}}_0$, and $x_2,\ldots ,x_n\in X$, we obtain:

$$\begin{aligned} \big \Vert&{\mathcal {T}}_m\xi (x)-{\mathcal {T}}_m\mu (x),x_2,\ldots ,x_n\big \Vert _{*,\beta }\\ {}&= \left\| \xi \left( \root k \of {(1+m^k)x^k}\right) -\xi (mx)-\mu \left( \root k \of {(1+m^k)x^k}\right) +\mu (mx),x_2,\ldots ,x_n\right\| _{*,\beta } \\&\le \max \left\{ \big \Vert \xi (f_{1}(x))-\mu (f_{1}(x)),x_2,\ldots ,x_n\big \Vert _{*,\beta },\big \Vert \xi (f_{2}(x))-\mu (f_{2}(x)),x_2,\ldots ,x_n\big \Vert _{*,\beta }\right\} \\&= \max _{1\le i\le 2}K_i(x,x_2,\ldots ,x_n)\big \Vert (\xi -\mu )(f_{i}(x)),x_2,\ldots ,x_n\big \Vert _{*,\beta }, \end{aligned}$$

where $(\xi -\mu )(x)\equiv \xi (x)-\mu (x)$. Therefore, (H4) is valid for ${\mathcal {T}}_m$ with $m\in {\mathbb {N}}.$ Note that, in view of (5.2), we have:

$$\begin{aligned} \Lambda _m\varepsilon _{l}(x,x_2,\ldots ,x_n)\le a_m\varepsilon _{l}(x,x_2,\ldots ,x_n),\quad l,m\in {\mathbb {N}},\;\;x\in {\mathbb {R}}_0,\;\;x_2,\ldots ,x_n\in X. \end{aligned}$$

(5.8)

Using mathematical induction, we will show that for each $x \in {\mathbb {R}}$ and $x_2,\ldots ,x_n\in X$, we have:

$$\begin{aligned} \Lambda _m^\ell \varepsilon _{l}(x,x_2,\ldots ,x_n)\le a_m^{\ell }\varepsilon _{l}(x,x_2,\ldots ,x_n) \end{aligned}$$

(5.9)

for all $\ell \in {\mathbb {N}}$ and $m\in {\mathcal {M}}$. From (5.8), we obtain that the inequality (5.9) holds for $\ell = 1.$ Next, we will assume that (5.9) holds for $\ell = r$, where $r \in {\mathbb {N}}$. Then, we have:

$$\begin{aligned} \Lambda _m^{r+1}\varepsilon _{l}(x,x_2,\ldots ,x_n)=&\Lambda _m\left( \Lambda _m^{r}\varepsilon _{l}(x,x_2,\ldots ,x_n)\right) \\ =&\;\max \left\{ \Lambda _m^{r}\varepsilon _{l}\left( \root k \of {(m^k+1)x^k},x_2,\ldots ,x_n\right) ,\Lambda _m^{r}\varepsilon _{l}(mx,x_2,\ldots ,x_n)\right\} \\ \le&\;a_m^{r} \max \left\{ \varepsilon _{l}\left( \root k \of {(m^k+1)x^k},x_2,\ldots ,x_n\right) ,\varepsilon _{l}(mx,x_2,\ldots ,x_n)\right\} \\ \le&\;a_m^{r+1}\varepsilon _{l}(x,x_2,\ldots ,x_n). \end{aligned}$$

This shows that (5.9) holds for $\ell = r + 1$. Now, we can conclude that the inequality (5.9) holds for all $\ell \in {\mathbb {N}}$. Therefore, by (5.9), we obtain that:

$$\begin{aligned} \lim _{\ell \rightarrow \infty }\Lambda _m^{\ell }\varepsilon _m(x,x_2,\ldots ,x_n)=0 \end{aligned}$$

for all $x\in {\mathbb {R}}_0$ and $m\in {\mathcal {M}}$. Furthermore, for each $\ell \in {\mathbb {N}}_0$, $m \in {\mathcal {M}}$, $x\in {\mathbb {R}}_0$ and $x_2,\ldots ,x_n\in X$, we have:

$$\begin{aligned}&\sup _{\ell \in {\mathbb {N}}_0} \Lambda _m^{\ell }\varepsilon _m(x,x_2,\ldots ,x_n)=\varepsilon _m(x,x_2,\ldots ,x_n), \\&\sup _{\ell \in {\mathbb {N}}_0} \Lambda _m^{\ell +1}\varepsilon _m(x,x_2,\ldots ,x_n)=\Lambda _m\varepsilon _m(x,x_2,\ldots ,x_n). \end{aligned}$$

Consequently, by Theorem 4.1 (with $W={\mathbb {R}}_0$ and $\varphi = f$), for each $m \in {\mathcal {M}}$ the mapping $F'_{m}: {\mathbb {R}}_0\rightarrow X$, given by $F'_m(x)=\lim _{\ell \rightarrow \infty }{\mathcal {T}}_m^{\ell }f(x)$ for $x\in {\mathbb {R}}_0$, is a fixed point of ${\mathcal {T}}_m$, that is:

$$\begin{aligned} F'_m(x)=F'_m\left( \root k \of {( m^k+1)x^k}\right) - F'_m(mx),\quad \;x\in {\mathbb {R}}_0,\;\;m\in {\mathcal {M}}. \end{aligned}$$

Moreover:

$$\begin{aligned} \big \Vert f(x)-F'_{m}(x),x_2,\ldots ,x_n\big \Vert _{*,\beta }\le \sup _{\ell \in {\mathbb {N}}_0} \Lambda _m^{\ell }\varepsilon _m(x,x_2,\ldots ,x_n) \end{aligned}$$

for all $x\in {\mathbb {R}}_0,$ $x_2,\ldots ,x_n\in X$, and $m \in {\mathcal {M}}_0.$

Define $F_{m}: {\mathbb {R}}\rightarrow X$ by $F_m(0)=F'_m(0)$ and $F_{m}(x)=F'_{m}(x)$ for $x\in {\mathbb {R}}_0$ and $m\in {\mathcal {M}}.$ Then, it is easily seen that, by (5.7):

$$\begin{aligned} F_m(x)=\lim _{\ell \rightarrow \infty }{\mathcal {T}}_m^{\ell }f(x),\quad x\in {\mathbb {R}},\;\;m\in {\mathcal {M}}. \end{aligned}$$

Next, we show that:

$$\begin{aligned} \Big \Vert {\mathcal {T}}_m^{\ell }f\left( \root k \of {x^k+y^k}\right) -{\mathcal {T}}_m^{\ell }f(x)-{\mathcal {T}}_m^{\ell }f(y),x_2,\ldots ,x_n\Big \Vert _{*,\beta }\le a_m^{\ell } L\big (x^k,y^k,x_2,\ldots ,x_n\big ) \end{aligned}$$

(5.10)

for every $x, y\in {\mathbb {R}}_0$, $x_2,\ldots ,x_n\in X,$ $\ell \in {\mathbb {N}}_0$, and $m\in {\mathcal {M}}$.

Clearly, if $\ell =0$, then (5.10) is simply (5.3). Therefore, fix $\ell \in {\mathbb {N}}_0$ and suppose that (5.10) holds for n and every $x, y \in {\mathbb {R}}_0$ and $x_2,\ldots ,u_n\in X$. Then, for every $x, y\in {\mathbb {R}}_0$ and $x_2,\ldots ,u_n\in X:$

$$\begin{aligned}&\Big \Vert {\mathcal {T}}_m^{\ell +1}f\left( \root k \of {x^k+y^k}\right) -{\mathcal {T}}_m^{\ell +1}f(x)-{\mathcal {T}}_m^{\ell +1}f(y),x_2,\ldots ,x_n\Big \Vert _{*,\beta } \\&\quad =\;\Big \Vert {\mathcal {T}}_m^{\ell }f\Big (\root k \of {(m^k+1)(x^k+y^k)}\Big )-{\mathcal {T}}_m^{\ell }f\Big (m\root k \of {x^k+y^k}\Big ) -{\mathcal {T}}_m^{\ell }f\Big (\root k \of {(m^k+1)x^k}\Big )+{\mathcal {T}}_m^{\ell }f(mx)\\&\quad \quad -{\mathcal {T}}_m^{\ell }f\Big (\root k \of {(m^k+1)y^k}\Big )+{\mathcal {T}}_m^{n}f(my),x_2,\ldots ,x_n\Big \Vert _{*,\beta } \\&\quad \le \max \bigg \{ \Big \Vert {\mathcal {T}}_m^{\ell }f\Big (\root k \of {(m^k+1)(x^k+y^k)}\Big )-{\mathcal {T}}_m^{\ell }f\Big (\root k \of {(m^k+1)x^k}\Big )\\&\quad \quad -{\mathcal {T}}_m^{\ell }f\Big (\root k \of {(m^k+1)y^k}\Big ),x_2,\ldots ,x_n\Big \Vert _{*,\beta }, \Big \Vert {\mathcal {T}}_m^{\ell }f\Big (m\root k \of {x^k+y^k}\Big )-{\mathcal {T}}_m^{\ell }f(mx)\\&\qquad -{\mathcal {T}}_m^{\ell }f(my),x_2,\ldots ,x_n\Big \Vert _{*,\beta }\bigg \} \\&\quad \le \max \Big \{a_m^{\ell }L\big ((1+m^k)x^k,(1+m^k)y^k,x_2,\ldots ,x_n\big ),a_m^{\ell }L\big (m^kx^k,m^ky^k,x_2,\ldots ,x_n\big )\Big \} \\&\quad \le a_m^{\ell +1}L\big (x^k,y^k,x_2,\ldots ,x_n\big ). \end{aligned}$$

Thus, by induction, we have shown that (5.10) holds for all $x,y\in {\mathbb {R}}_0$, $x_2,\ldots ,x_n\in X$ and for all $\ell \in {\mathbb {N}}_0$. Letting $\ell \rightarrow \infty$ in (5.10), we obtain that:

$$\begin{aligned} F_m\left( \root k \of {x^k+y^k}\right) =F_m(x)+F_m(y),\;\;x,y\in {\mathbb {R}}_0,\;\;m\in {\mathcal {M}}. \end{aligned}$$

(5.11)

Therefor, we have proved that for each $m\in {\mathcal {M}}$, there exists a function $F_m:{\mathbb {R}}\rightarrow X$ satisfying (1.2) for $x,y\in {\mathbb {R}}_0$, such that:

$$\begin{aligned} \big \Vert f(x)-F_{m}(x),x_2,\ldots ,x_n\big \Vert _{*,\beta }\le \sup _{\ell \in {\mathbb {N}}_0} \Lambda _m^{\ell }\varepsilon _m(x,x_2,\ldots ,x_n)= \varepsilon _m(x,x_2,\ldots ,x_n) \end{aligned}$$

(5.12)

for all $x\in {\mathbb {R}}_0$, $x_2,\ldots ,x_n\in X$, and $m\in {\mathcal {M}}$.

Now, we show that $F_{m}=F_{l}$ for all $m,l\in {\mathcal {M}}$. Therefore, fix $m,l\in {\mathcal {M}}$. Note that $F_{l}$ satisfies (5.11) with m replaced by l. Hence, taking $y=mx$ in (5.11), we get ${\mathcal {T}}_mF_{j}=F_{j}$ for $j=m,l$ and:

$$\begin{aligned} \big \Vert F_{m}(x)-F_{l}(x),x_2,\ldots ,x_n\big \Vert _{*,\beta }\le \max \{\varepsilon _m(x,x_2,\ldots ,x_n),\varepsilon _l(x,x_2,\ldots ,x_n)\} \end{aligned}$$

for all $x\in {\mathbb {R}}_0$ and $x_2,\ldots ,x_n\in X$. Whence, by (5.9):

$$\begin{aligned} \big \Vert F_{m}(x)-F_{l}(x),x_2,\ldots ,x_n\big \Vert _{*,\beta }&=\big \Vert {\mathcal {T}}_m^{\ell }F_{m}(x)-{\mathcal {T}}_m^{\ell }F_{l}(x),x_2,\ldots ,x_n\big \Vert _{*,\beta } \\&\le \max \big \{\Lambda _m^{\ell }\varepsilon _{m}(x,x_2,\ldots ,x_n),\Lambda _m^{\ell }\varepsilon _{l}(x,x_2,\ldots ,x_n)\big \} \\&\le a_m^{\ell }\max \big \{\varepsilon _{m}(x,x_2,\ldots ,x_n),\varepsilon _{l}(x,x_2,\ldots ,x_n)\big \} \end{aligned}$$

for all $x\in {\mathbb {R}}_0$, $x_2,\ldots ,x_n\in X$ and $\ell \in {\mathbb {N}}_0.$ Letting $\ell \rightarrow \infty$, we get $F_{m}=F_{l}=:F$. Thus, in view of (5.12), we have proved that:

$$\begin{aligned} \big \Vert f(x)-F(x),x_2,\ldots ,x_n\big \Vert _{*,\beta }\le \varepsilon _m(x,x_2,\ldots ,x_n),\quad x\in {\mathbb {R}}_0,\;\;x_2,\ldots ,x_n\in X,\;\;m\in {\mathcal {M}}. \end{aligned}$$

Since, in view of (5.11), it is easy to notice that F is a solution to (1.2) and, by Theorem 3.1, the function $F:{\mathbb {R}}\rightarrow X$ has the form $F(x)=T\big (x^k\big )$ with some additive function T. Therefore, we derive (5.4).

It remains to prove the statement concerning the uniqueness of F. Therefore, let $G:{\mathbb {R}}\rightarrow X$ be also a solution of (1.2) and:

$$\begin{aligned} \big \Vert f(x)-G(x),x_2,\ldots ,x_n\big \Vert _{*,\beta }\le \phi _L(x,x_2,\ldots ,x_n),\quad x\in {\mathbb {R}}_0,\;\;x_2,\ldots ,x_n\in X. \end{aligned}$$

Then:

$$\begin{aligned} \big \Vert F(x)-G(x),x_2,\ldots ,x_n\big \Vert _{*,\beta }\le \phi _L(x,x_2,\ldots ,x_n),\quad x\in {\mathbb {R}}_0,\;\;x_2,\ldots ,x_n\in X. \end{aligned}$$

Further ${\mathcal {T}}_mG=G$ for each $m\in {\mathbb {N}}$. Consequently, with a fixed $m\in {\mathcal {M}}$:

$$\begin{aligned} \big \Vert F(x)-G(x),x_2,\ldots ,x_n\big \Vert _{*,\beta }&=\big \Vert {\mathcal {T}}_m^{\ell }F(x)-{\mathcal {T}}_m^{\ell }G(x),x_2,\ldots ,x_n\big \Vert _{*,\beta } \\&\le \Lambda _m^{\ell }\phi _L(x,x_2,\ldots ,x_n) \\&\le \Lambda _m^{\ell }\varepsilon _{m}(x,x_2,\ldots ,x_n) \\&\le a_m^{\ell }\varepsilon _{m}(x,x_2,\ldots ,x_n) \end{aligned}$$

for all $x\in {\mathbb {R}}_0$, $x_2,\ldots ,x_n\in X$ and $\ell \in {\mathbb {N}}_0.$ Letting $\ell \rightarrow \infty$, we get $F=G$. This also confirms the uniqueness of T. The proof of the theorem is complete. $\square$

The following hyperstability result can be deduced from Theorem 5.1. This result is a generalization of many works referenced in [16,17,18].

Corollary 5.2

Let X be a non-Archimedean $(n,\beta )$-Banach space. Let $f : {\mathbb {R}} \rightarrow X$, $c:{\mathbb {N}}\rightarrow {\mathbb {R}}_+$ and $L:{\mathbb {R}}_{0}\times {\mathbb {R}}_{0} \times X^{n-1}\rightarrow {\mathbb {R}}_+$ be functions and the conditions (5.1), (5.2), and (5.3) be valid. Assume that:

$$\begin{aligned} \inf _{m\in {\mathcal {M}}}L\big (x^k,m^kx^k,x_2,\ldots ,x_n\big )=0 \end{aligned}$$

(5.13)

for all $x\in {\mathbb {R}}_0$ and $x_2,\ldots ,x_n\in X$, where $k\in {\mathbb {N}}$ is fixed. Then, f satisfies (1.2) for all $x,y\in {\mathbb {R}}$.

Proof

In view of (5.13), $\phi _L(x, x_2, \cdots , x_n)=0$ for each $x\in {\mathbb {R}}_0,\;x_2,\ldots ,x_n\in X$, where $\phi _L$ is defined by (5.5). Hence, from Theorems 3.1 and 5.1, we easily derive that f is a solution of (1.2) for all $x,y\in {\mathbb {R}}$. $\square$

Corollary 5.3

Let X be a non-Archimedean $(n,\beta )$-Banach space. Let $c:{\mathbb {N}}\rightarrow {\mathbb {R}}_+$ and $L:{\mathbb {R}}_{0}\times {\mathbb {R}}_{0} \times X^{n-1}\rightarrow {\mathbb {R}}_+$ be functions and the conditions (5.1), (5.2) and (5.13) be valid. Let $f : {\mathbb {R}} \rightarrow X$ and $F: {\mathbb {R}}^2 \rightarrow X$ be two functions, such that:

$$\begin{aligned} \Big \Vert f\left( \root k \of {x^k+y^k}\right) -f(x)-f(y)-&F(x,y),x_2,\ldots ,x_n\Big \Vert _{*,\beta }\le L\big (x^k,y^k,x_2\ldots ,x_n\big ),\\ {}&\quad x,y\in {\mathbb {R}}_0,\;x_2,\ldots ,x_n\in X, \end{aligned}$$

where $k\in {\mathbb {N}}$ is fixed. Assume that the functional equation:

$$\begin{aligned} h\left( \root k \of {x^k+y^k}\right) =h(x)+h(y)+F(x,y),\;\;\,x,y\in {\mathbb {R}}_0 \end{aligned}$$

(5.14)

admits a solution $f_0:{\mathbb {R}}\rightarrow X$ for $x,y\in {\mathbb {R}}_0,$ with $F(0,0)=-f_0(0)$. Then, f is a solution of (5.14) for all $x,y\in {\mathbb {R}}.$

Proof

Let $g(x):=f(x)-f_0(x)$ for $x\in {\mathbb {R}}$. Then:

$$\begin{aligned} \Big \Vert g\left( \root k \of {x^k+y^k}\right)&-g(x)-g(y),x_2,\ldots ,x_n\Big \Vert _{*,\beta } \\ =&\;\Big \Vert f\left( \root k \of {x^k+y^k}\right) -f_0\left( \root k \of {x^k+y^k}\right) -f(x)+f_0(x)-f(y)\\ {}&-F(x,y)+f_0(y)+F(x,y),x_2,\ldots ,x_n\Big \Vert _{*,\beta } \\ \le&\; \max \bigg \{\left\| f\left( \root k \of {x^k+y^k}\right) -f(x)-f(y)-F(x,y),x_2,\ldots ,x_n\right\| _{*,\beta } , \\&\quad \quad \left\| f_0\left( \root k \of {x^k+y^k}\right) -f_0(x)-f_0(y)-F(x,y),x_2,\ldots ,x_n\right\| _{*,\beta } \bigg \} \\ =&\;\left\| f\left( \root k \of {x^k+y^k}\right) -f(x)-f(y)-F(x,y),x_2,\ldots ,x_n\right\| _{*,\beta } \\ \le&\; L\big (x^k,y^k,x_2,\ldots ,x_n\big ),\quad x_2,\ldots ,x_n\in X,\; x, y\in {\mathbb {R}}_0. \end{aligned}$$

It follows from Corollary 5.2 that g satisfies the functional equation (1.2) for all $x, y\in {\mathbb {R}}$. Therefore:

$$\begin{aligned}&f\left( \root k \of {x^k+y^k}\right) -f(x)-f(y)-F(x,y) =g\left( \root k \of {x^k+y^k}\right) -g(x)-g(y) \\&\quad +f_0\left( \root k \of {x^k+y^k}\right) -f_0(x)-f_0(y)-F(x,y)=0 \end{aligned}$$

for all $x, y\in {\mathbb {R}}$. $\square$

6 Some Consequences

According to Theorem 5.1 and Corollaries 5.2, 5.3, we derive three natural examples of functions L and c satisfying the conditions (5.1) and (5.2). Namely, for:

• (i) $L(x,y,x_2,\ldots ,x_n) :=\varepsilon |x|^{p}|y|^{q}\Vert z,x_2,\ldots ,x_n\Vert _{*,\beta };$

• (ii) $L(x,y,x_2,\ldots ,x_n) :=\varepsilon \big (|x|^{p}|y|^{q}+|x|^{p+q}+|y|^{p+q}\big )\Vert z,x_2,\ldots ,x_n\Vert _{*,\beta };$

• (iii) $L(x,y,x_2,\ldots ,x_n) :=\varepsilon \big (\alpha _1|x|^{s_1}+\alpha _2|y|^{s_2}\big )^{w}\Vert z,x_2,\ldots ,x_n\Vert _{*,\beta }$

for all $x,y \in {\mathbb {R}}_0,$ $x_2,\ldots ,x_n \in X$, and for some arbitrary element $z\in X$ and $\varepsilon ,p,q,s_i,w\in {\mathbb {R}}$, such that $\varepsilon \ge 0$, $p+q<0,$ $\alpha _i>0$ and $ws_i<0$ for $i=1,2.$

Corollary 6.1

Let X be a non-Archimedean $(n,\beta )$-Banach space and $\varepsilon ,p,q\in {\mathbb {R}},$ with $\varepsilon \ge 0$ and $p+q<0$. Suppose that $f : {\mathbb {R}} \rightarrow X$ satisfies the inequality:

$$\begin{aligned} \left\| f\left( \root k \of {x^k+y^k}\right) -f(x)-f(y),x_2,\ldots ,x_n \right\| _{*,\beta }\le \varepsilon |x|^{p}|y|^{q}\Vert z,x_2,\ldots ,x_n\Vert _{*,\beta } \end{aligned}$$

for all $x,y\in {\mathbb {R}}_0$ and $x_2,\ldots ,x_n\in X$ and for some arbitrary element $z\in X,$ with $k\in {\mathbb {N}}$ is fixed. Then, the following two statements are valid.

• (a) If $q\ge 0$, then there exists a unique additive function $T : {\mathbb {R}} \rightarrow X$ for all $x,y\in {\mathbb {R}},$ such that:

  $$\begin{aligned} \left\| f(x)-T(x^k),x_2,\ldots ,x_n\right\| _{*,\beta }&\le \varepsilon |x|^{p+q}\Vert z,x_2,\ldots ,x_n\Vert _{*,\beta }, \\&\quad x\in {\mathbb {R}}_0,\;\; x_2,\ldots ,x_n\in X. \end{aligned}$$

• (b) If $q< 0$, then f satisfies (1.2) for all $x,y\in {\mathbb {R}}$.

Proof

Let $L\big (x^k,y^k,x_2,\ldots ,x_n\big ) :=\varepsilon |x|^{p}|y|^{q}\Vert z,x_2,\ldots ,x_n\Vert _{*,\beta }$ and $c(t)=t^{(p+q)/k}$ in Theorem 5.1 for all $x,y\in {\mathbb {R}}_0$ and $x_2,\ldots ,x_n\in X$ and for some arbitrary element $z\in X$, where $t\in {\mathbb {N}}$ and $p,q,\varepsilon \in {\mathbb {R}}$, such that $\varepsilon \ge 0$ and $p+q<0$, then we get that the condition (5.2) is valid. Obviously, (5.13) holds if $q<0$, but if $q\ge 0,$ then $\inf _{m\in {\mathcal {M}}}L\big (x^k,m^kx^k,x_2,\ldots ,x_n\big )=L\big (x^k,x^k,x_2,\ldots ,x_n\big )$. On the other hand, there exists $m_0\in {\mathbb {N}}$, such that:

$$\begin{aligned} \max \{c(m^k),c(m^k+1)\}=m^{p+q}<1,\;\;m\ge m_0. \end{aligned}$$

Therefore, we obtain (5.1), as well. Then, by Theorem 5.1 and Corollary 5.2, we get the desired results. $\square$

Corollary 6.2

Let X be a non-Archimedean $(n,\beta )$-Banach space and $\varepsilon ,p,q\in {\mathbb {R}},$ such that $\varepsilon \ge 0,$ $p+q<0$ and $q<0$. Let $f : {\mathbb {R}} \rightarrow X$ and $F: {\mathbb {R}}^2 \rightarrow X$ be two functions, such that:

$$\begin{aligned} \left\| f\left( \root k \of {x^k+y^k}\right) -f(x)-f(y)-F(x,y),x_2,\ldots ,x_n \right\| _{*,\beta }\le \varepsilon |x|^{p}|y|^{q}\Vert z,x_2,\ldots ,x_n\Vert _{*,\beta } \end{aligned}$$

for all $x,y\in {\mathbb {R}}_0$ and $x_2,\ldots ,x_n\in X$ and for some arbitrary element $z\in X,$ with $k\in {\mathbb {N}}$ is fixed. Assume that the functional equation:

$$\begin{aligned} g\left( \root k \of {x^k+y^k}\right) =g(x)+g(y)+F(x,y),\;\;\,x,y\in {\mathbb {R}}_0 \end{aligned}$$

(6.1)

admits a solution $g_0:{\mathbb {R}}\rightarrow X$ for $x,y\in {\mathbb {R}}_0$ with $F(0,0)=-g_0(0)$. Then, f is a solution of (6.1) for all $x, y\in {\mathbb {R}}$.

Corollary 6.3

Let X be a non-Archimedean $(n,\beta )$-Banach space and $\varepsilon ,p,q\in {\mathbb {R}},$ such that $\varepsilon \ge 0$ and $p+q<0$. Suppose that $f : {\mathbb {R}} \rightarrow X$ satisfies the inequality:

$$\begin{aligned} \left\| f\left( \root k \of {x^k+y^k}\right) -f(x)-f(y),x_2,\ldots ,x_n \right\| _{*,\beta }\le \varepsilon \big (|x|^{p}|y|^{q}+|x|^{p+q}+|y|^{p+q}\big )\Vert z,x_2,\ldots ,x_n\Vert _{*,\beta } \end{aligned}$$

for all $x,y\in {\mathbb {R}}_0$ and $x_2,\ldots ,x_n\in X$ and for some arbitrary element $z\in X,$ with $k\in {\mathbb {N}}$ is fixed. Then, the following two statements are valid.

• (a) If $q> 0$, then there exists a unique additive function $T : {\mathbb {R}} \rightarrow X$ for all $x,y\in {\mathbb {R}},$ such that:

  $$\begin{aligned} \left\| f(x)-T(x^k),x_2,\ldots ,x_n\right\| _{*,\beta }\le 3\varepsilon |x|^{p+q}\Vert z,x_2,\ldots ,x_n\Vert _{*,\beta }, \quad x\in {\mathbb {R}}_0,\;\; x_2,\ldots ,x_n\in X. \end{aligned}$$

• (b) If $q= 0$, then there exists a unique additive function $T : {\mathbb {R}} \rightarrow X$ for all $x,y\in {\mathbb {R}},$ such that:

  $$\begin{aligned} \left\| f(x)-T(x^k),x_2,\ldots ,x_n\right\| _{*,\beta }\le 2\varepsilon |x|^{p+q}\Vert z,x_2,\ldots ,x_n\Vert _{*,\beta }, \quad x\in {\mathbb {R}}_0,\;\; x_2,\ldots ,x_n\in X. \end{aligned}$$

• (c) If $q< 0$, then there exists a unique additive function $T : {\mathbb {R}} \rightarrow X$ for all $x,y\in {\mathbb {R}},$ such that:

  $$\begin{aligned} \left\| f(x)-T(x^k),x_2,\ldots ,x_n\right\| _{*,\beta }\le \varepsilon |x|^{p+q}\Vert z,x_2,\ldots ,x_n\Vert _{*,\beta }, \quad x\in {\mathbb {R}}_0,\;\; x_2,\ldots ,x_n\in X. \end{aligned}$$

Proof

Let $L\big (x^k,y^k,x_2,\ldots ,x_n\big ) :=\varepsilon \big (|x|^{p}|y|^{q}+|x|^{p+q}+|y|^{p+q}\big )\Vert z,x_2,\ldots ,x_n\Vert _{*,\beta }$ and $c(t)=t^{(p+q)/k}$ in Theorem 5.1 for all $x,y\in {\mathbb {R}}_0$ and $x_2,\ldots ,x_n\in X$ and for some arbitrary element $z\in X,$ where $t\in {\mathbb {N}}$ and $\varepsilon ,p,q\in {\mathbb {R}}$, such that $\varepsilon \ge 0$ and $p+q<0$, then we get that the condition (5.2) is valid. Obviously:

$$\begin{aligned}&\inf _{m\in {\mathcal {M}}}L\big (x^k,m^kx^k,x_2,\ldots ,x_n\big )=\varepsilon |x|^{p+q}\Vert z,x_2,\ldots ,x_n\Vert _{*,\beta } \;\;\text {if}\;\; q<0, \\&\inf _{m\in {\mathcal {M}}}L\big (x^k,m^kx^k,x_2,\ldots ,x_n\big )=2\varepsilon |x|^{p+q}\Vert z,x_2,\ldots ,x_n\Vert _{*,\beta } \;\;\text {if}\;\; q=0 \end{aligned}$$

and

$$\begin{aligned} \inf _{m\in {\mathcal {M}}}L\big (x^k,m^kx^k,x_2,\ldots ,x_n\big )=3\varepsilon |x|^{p+q}\Vert z,x_2,\ldots ,x_n\Vert _{*,\beta } \;\;\text {if}\;\; q>0. \end{aligned}$$

On the other hand, there exists $m_0\in {\mathbb {N}}$, such that:

$$\begin{aligned} \max \{c(m^k),c(m^k+1)\}=m^{p+q}<1,\;\;m\ge m_0. \end{aligned}$$

Therefore, we obtain (5.1), as well. Then, by Theorem 5.1, we get the desired results. $\square$

Corollary 6.4

Let X be a non-Archimedean $(n,\beta )$-Banach space and $\varepsilon ,s_i,w,\alpha _i\in {\mathbb {R}},$ such that $\varepsilon \ge 0$, $\alpha _i>0$ and $ws_i<0$ for $i=1,2$. Suppose that $f : {\mathbb {R}} \rightarrow X$ satisfies the inequality:

$$\begin{aligned} \left\| f\left( \root k \of {x^k+y^k}\right) -f(x)-f(y),x_2,\ldots ,x_n \right\| _{*,\beta }\le \varepsilon \big (\alpha _1|x|^{s_1}+\alpha _2|y|^{s_2}\big )^{w}\Vert z,x_2,\ldots ,x_n\Vert _{*,\beta } \end{aligned}$$

for all $x,y\in {\mathbb {R}}_0$ and $x_2,\ldots ,x_n\in X$ and for some arbitrary element $z\in X,$ with $k\in {\mathbb {N}}$ is fixed. Then, the following two statements are valid.

• (a) If $w>0,$ then there exists a unique additive function $T : {\mathbb {R}} \rightarrow X$ for all $x,y\in {\mathbb {R}},$ such that:

  $$\begin{aligned} \left\| f(x)-T(x^k),x_2,\ldots ,x_n\right\| _{*,\beta }\le \varepsilon \alpha _{1}^{w}|x|^{s_1w}\Vert z,x_2,\ldots ,x_n\Vert _{*,\beta }. \end{aligned}$$

• (b) If $w<0,$ then there exists a unique additive function $T : {\mathbb {R}} \rightarrow X$ for all $x,y\in {\mathbb {R}},$ such that:

  $$\begin{aligned} \left\| f(x)-T(x^k),x_2,\ldots ,x_n\right\| _{*,\beta }\le \varepsilon (\alpha _1+\alpha _2)^w|x|^{s_0w}\Vert z,x_2,\ldots ,x_n\Vert _{*,\beta } \end{aligned}$$

for all $x\in {\mathbb {R}}_0,$ and all $x_2,\ldots ,x_n\in X$ and for some arbitrary element $z\in X,$ where:

$$\begin{aligned} s_0:= {\left\{ \begin{array}{ll} \max \{s_1,s_2\}\quad &{}\text{ if } w>0;\\ \min \{s_1,s_2\}\quad &{}\text{ if } w<0. \end{array}\right. } \end{aligned}$$

Proof

Let $L\big (x^k,y^k,x_2,\ldots ,x_n\big ) :=\varepsilon \big (\alpha _1|x|^{s_1}+\alpha _2|y|^{s_2}\big )^{w}\Vert z,x_2,\ldots ,x_n\Vert _{*,\beta }$ and $c(t)=t^{s_0w/k}$ in Theorem 5.1 for all $x,y\in {\mathbb {R}}_0$ and $x_2,\ldots ,x_n\in X$ and for some arbitrary element $z\in X,$ where $t\in {\mathbb {N}}$ and $\varepsilon ,\alpha _i,s_i,w\in {\mathbb {R}}$, such that $ws_i<0$, $\alpha _i>$ and $\varepsilon \ge 0$ for $i\in \{0,1,2\},$ with:

$$\begin{aligned} s_0:= {\left\{ \begin{array}{ll} \max \{s_1,s_2\}\quad &{}\text{ if } w>0;\\ \min \{s_1,s_2\}\quad &{}\text{ if } w<0. \end{array}\right. } \end{aligned}$$

Then, we get that the condition (5.2) is valid. Obviously:

$$\begin{aligned} {\left\{ \begin{array}{ll} \inf _{m\in {\mathcal {M}}}L\big (x^k,m^kx^k,x_2,\ldots ,x_n\big )=\varepsilon \alpha _{1}^{w}|x|^{s_1w}\Vert z,x_2,\ldots ,x_n\Vert _{*,\beta }\quad &{}\text{ if } w>0;\\ \inf _{m\in {\mathcal {M}}}L\big (x^k,m^kx^k,x_2,\ldots ,x_n\big )=\varepsilon (\alpha _1+\alpha _2)^w|x|^{s_0w}\Vert z,x_2,\ldots ,x_n\Vert _{*,\beta }\quad &{}\text{ if } w<0.\\ \end{array}\right. } \end{aligned}$$

On the other hand, there exists $m_0\in {\mathbb {N}}$, such that:

$$\begin{aligned} \max \{c(m^k),c(m^k+1)\}=m^{ws_0}<1,\;\;m\ge m_0. \end{aligned}$$

Therefore, we obtain (5.1), as well. Then, by Theorem 5.1, we get the desired results. $\square$