Survey on Ulam Stability with Respect to n-Norms and (n, β)-Norms

Abstract

This article is a survey of the results published so far on Ulam stability of functional equations in n-normed spaces and (n, β)-normed spaces. We present and examine them, highlighting some traps they contain and outlining potential straightforward generalizations. We also draw attention to certain symmetries present in the results discussed. In this way, we complement two earlier surveys on Ulam stability in two-normed spaces.

1. Introduction and Preliminaries

One of the main issues in Ulam stability can be roughly formulated in the following way: How far an approximate solution to an equation deviates from the exact solutions to it. It is connected to the problems considered, e.g., in the theories of shadowing, approximation, optimization, perturbation, and fixed points. Since this topic is already quite well known, we will not provide more detailed historical data, but only refer interested readers to the monographs [1,2].

However, as one of the reviewers suggested, it is worth adding that over the past two decades, various aspects of Ulam stability have been extensively studied in the context of difference and differential Equations (also of fractional order), and related nonlocal and time-scale models, motivated by applications in physics, biology, engineering, and control theory (cf, e.g., [3,4,5,6,7,8,9,10]). This rapidly growing body of work has played a key role in revitalizing the theory and, in many cases, contributed to the extension of Ulam stability to the more abstract framework of functional analysis.

Such problems have already been considered for numerous equations and in many spaces in which distance has been measured in various ways (e.g., by fuzzy norms, random norms, seminorms, quasinorms, non-Archimedean metrics, gauges, modulars). In this paper, we present and discuss the results on Ulam stability considered for functions taking values in n-normed spaces and $(n,\beta )$-normed spaces. We point to the various pitfalls they may contain (e.g., several cited results omit essential assumptions, such as the appropriate type of completeness, or contain minor inaccuracies in variable domains) and show possible simple generalizations. In this way, we complement analogous earlier surveys (see [11]) concerning Ulam stability in 2-normed spaces.

Since the publication of the survey [11], several further works on stability in 2-normed spaces have been published, but we do not include them in this review. We focus on stability in n-normed and $(n,\beta )$-normed spaces with general n.

Let us recall that one of the main directions in Ulam stability has been initiated by the results of D.H. Hyers [12], T. Aoki [13] and Th.M. Rassias [14], who considered stability of the additive Cauchy equation

$$A(x+y)=A\left(x\right)+A\left(y\right)$$

(1)

in Banach spaces. All these results are included in the subsequent theorem (see, e.g., [1,2,15]), which we will yet refer to in the last section.

Theorem 1.

Let X and Y be normed spaces, $X_0:=X\setminus \left\{0\right\}$, and $\eta \ge 0$ and $p\ne 1$ be fixed real numbers. Let $g:X\to Y$ satisfy the inequality

$$\parallel g(x+y)-g\left(x\right)-g\left(y\right)\parallel \le \eta ({\parallel x\parallel }^p+{\parallel y\parallel }^p),x,y\in X_0.$$

(2)

Then the following two statements are valid.

(i) 

If $p\ge 0$ and Y is complete, then there exists a unique $A:X\to Y$ that is additive (i.e., fulfills (1) for all $x,y\in X$) and such that

$$\parallel g\left(x\right)-A\left(x\right)\parallel \le {\displaystyle \frac{\eta {\parallel x\parallel }^p}{|1-2^{p-1}|}},x\in X_0.$$

(ii) 

If $p<0$, then g is additive.

Statement (i) is often said to express the Hyers–Ulam (sometimes some other names are also added here) stability of Equation (1). The result depicted by (ii) is called a hyperstability property of (1).

It has been shown in [16] that, for $p=1$, an analogous outcome is not possible (see also [17]).

Through the article, we use $\mathbb{R}$, $\mathbb{Q}$, $\mathbb{Z}$ and $\mathbb{N}$ to denote the sets of reals, rational numbers, integers, and positive integers, respectively. Furthermore, ${\mathbb{R}}_{+}$ stands for the set of nonnegative reals, ${\mathbb{R}}_0:=\mathbb{R}\setminus \left\{0\right\}$, and ${\mathbb{N}}_0:=\mathbb{N}\cup \left\{0\right\}$. $A^B$ always denotes the family of all functions mapping a set $B\ne \varnothing$ into a set $A\ne \varnothing$.

In the next section, we provide auxiliary information on $(n,\beta )$-normed (and also on n-normed) spaces, and in Section 3 we prove some simple, but general results on the stability of some functional equations, which are very useful in further parts of the paper. The main survey of different stability results of functional equations in $(n,\beta )$-normed and n-normed spaces is included in Section 4. In Section 5, we give some final remarks. The last section presents some open problems and suggestions for future research.

2. Auxiliary Information

Let $n\in \mathbb{N}$, $\beta \in (0,1]$ and $\mathfrak{L}$ be a real linear space, which is at least n-dimensional. We say that a function ${\parallel ·,\dots ,·\parallel }_{\beta }$, mapping ${\mathfrak{L}}^n$ into ${\mathbb{R}}_{+}$, is an $(n,\beta )$-norm on $\mathfrak{L}$ if the following four conditions hold:

(N1) 

$\parallel l_1,\dots ,l_n{\parallel }_{\beta }=0$ if and only if the vectors $l_1,\dots ,l_n$ are linearly dependent;

(N2) 

$\parallel l_1,\dots ,l_n{\parallel }_{\beta }$ is invariant under permutation of $l_1,\dots ,l_n$;

(N3) 

$\parallel \alpha l_1,\dots ,l_n{\parallel }_{\beta }={\left|\alpha \right|}^{\beta }{\parallel l_1,\dots ,l_n\parallel }_{\beta }$;

(N4) 

$\parallel k_1+k_2,l_2,\dots ,l_n{\parallel }_{\beta } \le \parallel k_1,l_2,\dots ,l_n{\parallel }_{\beta }+{\parallel k_2,l_2,\dots ,l_n\parallel }_{\beta }$,

for every $\alpha \in \mathbb{R}$ and $k_1,k_2,l_1,\dots ,l_n\in \mathfrak{L}$. If ${\parallel ·,\dots ,·\parallel }_{\beta }$ is an $(n,\beta )$-norm on $\mathfrak{L}$, then the pair $(\mathfrak{L},\parallel ·,\dots ,·{\parallel }_{\beta })$ is said to be an $(n,\beta )$-normed space.

The notion of $(n,\beta )$-normed space was first introduced in [18]. It is an extension of the idea of n-normed space, which is an $(n,\beta )$-normed space with $\beta =1$.

The concept of n-normed spaces was initiated in [19] and next investigated in [20]; it generalizes an earlier concept of 2-normed spaces (i.e., n-normed spaces with $n=2$) introduced by S. Gähler [21,22] (see [23] for further information).

If $n\in \mathbb{N}$, $n>1$, and $(\mathfrak{L},<·,·>)$ is a real inner product space (and, as before, $\mathfrak{L}$ is at least n-dimensional), then an n-norm on $\mathfrak{L}$ can be defined by the formula

$$\parallel l_1,\dots ,l_n\parallel =\mathrm{abs}{\left(\left|\begin{array}{cccc}<l_1,l_1>& <l_1,l_2>& \dots & <l_1,l_n>\\ ⋮& ⋮& \ddots & ⋮\\ <l_n,l_1>& <l_n,l_2>& \dots & <l_n,l_n>\end{array}\right|\right)}^{1/2}$$

for $l_1,\dots ,l_n\in \mathfrak{L}$. Here, for every real number a, $\mathrm{abs}\left(a\right)$ denotes the module (absolute value) of a.

In the case $\mathfrak{L}={\mathbb{R}}^n$ (with the usual inner product), this n-norm is sometimes denoted by ${\parallel ·,\dots ,·\parallel }_E$ and also can be expressed by the formula

$$\parallel l_1,\dots ,l_n{\parallel }_E=|det\left(l_{ij}\right)|=\mathrm{abs}\left(\left|\begin{array}{cccc}{l}_{11}& l_{12}& \dots & l_{1n}\\ ⋮& ⋮& \ddots & ⋮\\ l_{n1}& l_{n2}& \dots & l_{nn}\end{array}\right|\right),$$

for $l_i=(l_{i1},\dots ,l_{in})\in {\mathbb{R}}^n,i\in \{1,\dots ,n\}.$ It is called the Euclidean n-norm on ${\mathbb{R}}^n$.

Each n-norm $\parallel ·,\dots ,·\parallel$ on ${\mathbb{R}}^n$ has the form: $\parallel u_1,\dots ,u_n\parallel =\gamma \parallel u_1,\dots ,u_n{\parallel }_E$ for $u_1,\dots ,u_n\in {\mathbb{R}}^n$, with some real constant $\gamma >0$ (see, e.g., ([19], Satz 3) or ([23], Remark 2)).

Remark 1.

According to ([23], Theorem 1), for every $n\in \mathbb{N}$ and each real linear space X of dimension at least n, there exists an n-norm on X (provided we assume the axiom of choice if the dimension of X is not finite). In the proof of this theorem it is shown how to construct such n-norms (see also [24]).

Remark 2.

Let $\beta \in (0,1]$, $n\in \mathbb{N}\setminus \left\{1\right\}$ be fixed and $(\mathfrak{L},\parallel ·,\dots ,·\parallel )$ be an n-normed space. Then it is easily seen that the formula

$$\parallel l_1,\dots ,l_n{\parallel }_{\beta }:={\parallel l_1,\dots ,l_n\parallel }^{\beta },l_1,\dots ,l_n\in \mathfrak{L},$$

defines an $(n,\beta )$-norm in $\mathfrak{L}$. Consequently (see Remark 1), for each real linear space X of dimension at least n, there exists an $(n,\beta )$-norm on X (again, as in Remark 1, provided we assume the axiom of choice if the dimension of X is not finite).

Furthermore, note that if ${\parallel ·,\dots ,·\parallel }_{\beta 1}$ and ${\parallel ·,\dots ,·\parallel }_{\beta 2}$ are $(n,\beta )$-norms in $\mathfrak{L}$ and $a,b\in {\mathbb{R}}_{+}$, $a+b>0$, then the mappings ${\parallel ·,\dots ,·\parallel }_{\beta +},{\parallel ·,\dots ,·\parallel }_{\beta m}:{\mathfrak{L}}^n\to {\mathbb{R}}_{+}$, given by

$$\parallel l_1,\dots ,l_n{\parallel }_{\beta +}:=a\parallel l_1,\dots ,l_n{\parallel }_{\beta 1}+b{\parallel l_1,\dots ,l_n\parallel }_{\beta 2},$$

$$\parallel l_1,\dots ,l_n{\parallel }_{\beta m}:=max\left\{a\parallel l_1,\dots ,l_n{\parallel }_{\beta 1},b{\parallel l_1,\dots ,l_n\parallel }_{\beta 2}\right\}$$

for $l_1,\dots ,l_n\in {\mathfrak{L}}^n$, are $(n,\beta )$-norms in $\mathfrak{L}$, too.

If $n\in \mathbb{N}\setminus \left\{1\right\}$ is fixed and $(\mathfrak{L},\parallel ·,\dots ,·{\parallel }_{\beta })$ is an $(n,\beta )$-normed space, then (to simplify some formulas) we write in the rest of this paper:

$$\begin{array}{c}\hfill {\parallel x,z\parallel }_{\beta }:={\parallel x,z_1,\dots ,z_{n-1}\parallel }_{\beta },x\in \mathfrak{L},z=(z_1,\dots ,z_{n-1})\in {\mathfrak{L}}^{n-1}.\end{array}$$

(3)

This also applies to the case $\beta =1$, i.e., when the $(n,\beta )$-norm is simply an n-norm.

Remark 3.

Analogously as in [24] (for n-norms), it can be noticed that every $(n,\beta )$-norm generates an $(n-1,\beta )$-norm and finally also a β-norm (i.e., $(1,\beta )$-norm). For a somewhat analogous observation we refer to ([25], Remark 2).

Let $\beta \in (0,1]$, $n\in \mathbb{N}\setminus \left\{1\right\}$ and $(\mathfrak{L},\parallel ·,\dots ,·{\parallel }_{\beta })$ be an $(n,\beta )$-normed space. Let $(b_1,\dots ,b_n)\in {\mathfrak{L}}^n$ be a sequence of n linearly independent vectors in $\mathfrak{L}$ and define $\mathcal{B}\subset {\mathfrak{L}}^{n-1}$ by

$$\mathcal{B}:=\{(e_{i,1},\dots ,e_{i,n-1}):i=1,\dots ,n\},$$

with

$$e_{i,j}=\left\{\begin{array}{cc}{b}_j,\hfill & i>j;\hfill \\ b_{j+1},\hfill & i\le j,\hfill \end{array}\right.i\in \{1,\dots ,n\},j\in \{1,\dots ,n-1\}.$$

Then it is easy to check that the formula

$$\begin{array}{c}\hfill {\parallel x\parallel }_{\mathcal{B}}=\sum _{d\in \mathcal{B}}\parallel x,d\parallel ,x\in \mathfrak{L},\end{array}$$

(4)

defines a $(1,\beta )$-norm in $\mathfrak{L}$, which is just a norm when $\beta =1$ (cf. ([25], Remark 2) for the case $\beta =1$).

Moreover, given $\xi :\mathcal{B}\to (0,\infty )$, we also can define a $(1,\beta )$-norm ${\parallel ·\parallel }_{\mathcal{B},\xi }$ in $\mathfrak{L}$ by

$$\begin{array}{c}\hfill {\parallel x\parallel }_{\mathcal{B},\xi }=\sum _{d\in \mathcal{B}}\xi \left(d\right)\parallel x,d\parallel ,x\in \mathfrak{L}.\end{array}$$

(5)

We also have the following definitions.

Definition 1.

Let $\beta \in (0,1]$, $n\in \mathbb{N}\setminus \left\{1\right\}$ and $(\mathfrak{L},\parallel ·,\dots ,·{\parallel }_{\beta })$ be an $(n,\beta )$-normed space. A sequence ${\left(v_k\right)}_{k\in \mathbb{N}}$ in $\mathfrak{L}$ is called an $(n,\beta )$-Cauchy sequence if

$$\underset{m,k\to \infty }{lim}{\parallel v_m-v_k,z\parallel }_{\beta }=0,z\in {\mathfrak{L}}^{n-1}.$$

A sequence ${\left(v_k\right)}_{k\in \mathbb{N}}$ in $\mathfrak{L}$ is said to be $(n,\beta )$-convergent if there is $v\in \mathfrak{L}$ (called the $(n,\beta )$-limit of ${\left(v_k\right)}_{k\in \mathbb{N}}$) such that

$$\underset{k\to \infty }{lim}{\parallel v_k-v,z\parallel }_{\beta }=0,z\in {\mathfrak{L}}^{n-1}.$$

This $(n,\beta )$-limit is unique, denoted by ${lim}_{k\to \infty }{v}_k$, and we write $v={lim}_{k\to \infty }{v}_k$.

We say that an $(n,\beta )$-normed space is $(n,\beta )$-complete (or is an $(n,\beta )$-Banach space) if every $(n,\beta )$-Cauchy sequence in it is $(n,\beta )$-convergent; if $\beta =1$, then we may simply say that the space is n-complete (or n-Banach space), instead of $(n,1)$-complete (or $(n,1)$-Banach space).

The notion of Ulam stability in $(n,\beta )$-normed spaces can be understood as in the subsequent definition.

Definition 2.

Let $\beta \in (0,1]$, $n\in \mathbb{N}\setminus \left\{1\right\}$, $(\mathfrak{L},\parallel ·,\dots ,·{\parallel }_{\beta })$ be an $(n,\beta )$-normed space and $L\subset {\mathfrak{L}}^{n-1}$ be nonempty. Let D and T be nonempty sets, ${\mathcal{U}}_0\subset \mathcal{U}\subset {\mathfrak{L}}^T$ and $\mathcal{E}\subset {{\mathbb{R}}_{+}}^{D\times L}$ be nonempty, $\mathcal{S}:\mathcal{E}\to {{\mathbb{R}}_{+}}^{T\times L}$, ${\chi }_0\in {\mathfrak{L}}^D$ and $\mathcal{F}:\mathcal{U}\to {\mathfrak{L}}^D$. The equation

$$\mathcal{F}\left(\varphi \right)={\chi }_0$$

is $(\mathcal{S},L)$–stable in ${\mathcal{U}}_0$ provided, for any $\varphi \in {\mathcal{U}}_0$ and $\delta \in \mathcal{E}$ with

$$\parallel \left(\mathcal{F}\varphi \right)\left(s\right)-{\chi }_0\left(s\right),z{\parallel }_{\beta }\le \delta (s,z),s\in D,z\in L,$$

there is a solution $\psi \in \mathcal{U}$ of the equation with

$$\parallel \varphi \left(t\right)-\psi \left(t\right),z{\parallel }_{\beta }\le \left(\mathcal{S}\delta \right)(t,z),t\in T,z\in L.$$

For some further information on n-normed spaces we refer to, e.g., [24,26,27].

3. Preliminary Hyperstability Results

From now on we assume that $n\ge 2$ is an integer, $\beta \in (0,1]$ and $(\mathfrak{L},\parallel ·,\dots ,·{\parallel }_{\beta })$ is an $(n,\beta )$-normed space.

Some stability problems are nearly trivial in $(n,\beta )$-normed spaces. To show this, we provide two examples of very simple (but general) hyperstability results. They can be generalized further very easily.

Proposition 1.

Let D and E be nonempty sets, $b_1,\dots ,b_n\in \mathfrak{L}$ be linearly independent vectors, $\mathcal{B}$ be defined as in Remark 3, $\mathcal{U}\subset {\mathfrak{L}}^E$, $\mathcal{F}:\mathcal{U}\to {\mathfrak{L}}^D$, $\varphi \in \mathcal{U}$, ${\chi }_0\in {\mathfrak{L}}^D$ and $\phi :D\times {\mathfrak{L}}^{n-1}\to {\mathbb{R}}_{+}$. Assume that there is a sequence ${\left(l_k\right)}_{k\in \mathbb{N}}$ in $\mathbb{R}$ such that

$$\begin{array}{c}\hfill \underset{k\to \infty }{lim}{\displaystyle \frac{\phi (s,l_{k}b)}{l_k^{(n-1)\beta }}}=0,s\in D,b\in \mathcal{B},\end{array}$$

(6)

where $a(z_1,\dots ,z_{n-1}):=(a{z}_1,\dots ,a{z}_{n-1})$ for $a\in \mathbb{R}$, $(z_1,\dots ,z_{n-1})\in {\mathfrak{L}}^{n-1}$. If

$$\parallel \left(\mathcal{F}\varphi \right)\left(s\right)-{\chi }_0\left(s\right),l_{k}b{\parallel }_{\beta }\le \phi (s,l_{k}b),s\in D,b\in \mathcal{B},k\in \mathbb{N},$$

then

$$\begin{array}{c}\hfill \left(\mathcal{F}\varphi \right)\left(s\right)={\chi }_0\left(s\right),s\in D.\end{array}$$

(7)

Proof. 

Since

$$\begin{array}{c}\hfill \parallel \left(\mathcal{F}\varphi \right)\left(s\right)-\varphi \left(s\right),b{\parallel }_{\beta }=l_k^{-(n-1)\beta }\parallel \left(\mathcal{F}\varphi \right)\left(s\right)-\varphi \left(s\right),l_{k}b{\parallel }_{\beta }\\ \hfill \le l_k^{-(n-1)\beta }\phi (s,l_{k}b),s\in D,b\in \mathcal{B},k\in \mathbb{N},\end{array}$$

letting $k\to \infty$, by (6), we get $\parallel \left(\mathcal{F}\varphi \right)\left(s\right)-\varphi \left(s\right),b{\parallel }_{\beta }=0$ for every $s\in D$ and $b\in \mathcal{B}$, whence (by (N1)) (7) holds. □

Note that a particular case of (7) is, for example, the inhomogeneous Cauchy functional equation

$$\phi (x+y)=\phi \left(x\right)+\phi \left(y\right)+\eta (x,y)$$

that can be considered for $\phi :E\to \mathfrak{L}$, with some $\eta :E^2\to \mathfrak{L}$, where $(E,+)$ is a groupoid. It is enough to take $D=E^2$, $\mathcal{U}={\mathfrak{L}}^E$, $\left(\mathcal{F}\varphi \right)(s_1,s_2)=\varphi (s_1+s_2)-\varphi \left(s_1\right)-\varphi \left(s_2\right)$ for $(s_1,s_2)\in D$, $\varphi \in {\mathfrak{L}}^E$, and ${\chi }_0=\eta$.

Analogously, (7) may take one of the forms of the Jensen functional equation

$$2f(x+y)=f\left(2x\right)+f\left(2y\right),$$

$$\begin{array}{c}\hfill f\left({\displaystyle \frac{x+y}{2}}\right)={\displaystyle \frac{1}{2}}\left(f\left(x\right)+f\left(y\right)\right),\end{array}$$

(8)

or the form of the quadratic functional equation

$$\begin{array}{c}\hfill f(x+y)+f(x-y)=2f\left(x\right)+2f\left(y\right).\end{array}$$

(9)

Observe that if $\phi$ (in (6)) does not depend on the second variable, i.e., $\phi (s,b_1)=\phi (s,b_2)$ for $s\in D$ and $b_1,b_2\in {\mathfrak{L}}^{n-1}$, then (6) holds with $l_k=k$ for $k\in \mathbb{N}$. Consequently, analogously as Proposition 1, we can easily obtain the following.

Proposition 2.

Let D be a nonempty set, $b_1,\dots ,b_n\in \mathfrak{L}$ be linearly independent vectors and $\mathcal{B}$ be defined as in Remark 3. If $F:D\to \mathfrak{L}$ is such that

$$\begin{array}{c}\hfill \parallel F\left(s\right),kb{\parallel }_{\beta }\le \psi \left(s\right),s\in D,b\in \mathcal{B},k\in \mathbb{N},\end{array}$$

(10)

with some $\psi :D\to {\mathbb{R}}_{+}$, then

$$\begin{array}{c}\hfill F\left(s\right)=0,s\in D.\end{array}$$

(11)

Remark 4.

It should be noted that in Proposition 2 it is very important that condition (10) holds for $k\in \mathbb{N}$ (in a similar role, condition (6) uses the real sequence ${\left(l_k\right)}_{k\in \mathbb{N}}$). Without this, the final statement would not be true in such a general form, as it is shown below.

Namely, let $f:D\to \mathfrak{L}$ be a constant function: $f\left(s\right)\equiv c$, with some $c\in \mathfrak{L}\setminus \left\{0\right\}$. Then, with any $\varphi :D\to D$ and $a\in \mathbb{R}\setminus \left\{1\right\}$, we have

$$\begin{array}{c}\hfill \parallel f\left(\varphi \left(s\right)\right)-af\left(s\right),b{\parallel }_{\beta }={|1-a|}^{\beta }{\parallel c,b\parallel }_{\beta }=:\psi \left(s\right),s\in D,b\in \mathcal{B};\end{array}$$

(12)

that is

$$\begin{array}{c}\hfill \parallel F\left(s\right),b{\parallel }_{\beta }\le \psi \left(s\right),s\in D,b\in \mathcal{B},\end{array}$$

(13)

with $F\left(s\right):\equiv f\left(\varphi \right(s\left)\right)-af\left(s\right)$, but $F\left(s\right)\ne 0$ for each $s\in D$.

An analogous situation occurs when T is a nonempty set, $g:T\to \mathfrak{L}$ is a constant function: $g\left(s\right)\equiv c$ with some $c\in \mathfrak{L}\setminus \left\{0\right\}$, $\eta :T^2\to T$, $D=T^2$ and $F(x,y):=g\left(\eta \right(x,y\left)\right)-g\left(x\right)-g\left(y\right)$ for $(x,y)\in D$. Then (with the same $\mathcal{B}$ as above)

$$\begin{array}{c}\hfill \parallel g\left(\eta (x,y)\right)-g\left(x\right)-g\left(y\right),b{\parallel }_{\beta }={\parallel c,b\parallel }_{\beta }=:\psi (x,y),(x,y)\in D,b\in \mathcal{B};\end{array}$$

that is, (13) holds, but again $F\left(s\right)=c\ne 0$ for each $s\in D$.

Proposition 2 yields at once the following corollary.

Corollary 1.

Let E be a linear space over a field $\mathbb{K}$, $k,m\in \mathbb{N}$, $a_{ij}\in \mathbb{K}$ for $i=1,\dots ,m$ and $j=1,\dots ,k$, $A_1,\dots ,A_m\in \mathbb{R}$, $\psi :E^k\to [0,\infty )$, and let $g_1,\dots ,g_m:E\to \mathfrak{L}$, $H:E^k\to \mathfrak{L}$ be mappings satisfying the inequality

$$\begin{array}{cc}\hfill \parallel \sum _{i=1}^m{A}_i{g}_i\left(\sum _{j=1}^k{a}_{ij}{x}_j\right)-H(x_1,\dots ,x_k),z{\parallel }_{\beta }\le & \psi (x_1,\dots ,x_k)\hfill \end{array}$$

for $x_1,\dots ,x_k\in E$ and $z\in {\mathfrak{L}}^{n-1}$. Then

$$\begin{array}{c}\hfill \sum _{i=1}^m{A}_i{g}_i\left(\sum _{j=1}^k{a}_{ij}{x}_j\right)=H(x_1,\dots ,x_k),x_1,\dots ,x_k\in E.\end{array}$$

(14)

Proof. 

It is enough to apply Proposition 2 with $D=E^k$ and

$$F(x_1,\dots ,x_k)=\sum _{i=1}^m{A}_i{g}_i\left(\sum _{j=1}^k{a}_{ij}{x}_j\right)-H(x_1,\dots ,x_k),x_1,\dots ,x_k\in E.$$

□

Note that Equation (14) is a generalization (the so-called pexiderization) of the functional equation

$$\begin{array}{c}\hfill \sum _{i=1}^m{A}_{i}g\left(\sum _{j=1}^k{a}_{ij}{x}_j\right)=H(x_1,\dots ,x_k),x_1,\dots ,x_k\in E,\end{array}$$

(15)

the stability of which (with some H) was studied, e.g., in [28,29].

Note that using Proposition 2, we can also easily obtain numerous other hyperstability results in $(n,\beta )$-normed spaces.

For more information on the functional equations mentioned above (and many other) we refer to [30,31,32].

4. Ulam Stability in n-Normed Spaces and $(n,\beta )$-Normed Spaces

In this section, we discuss some published results on Ulam stability of functional equations in n-normed spaces and $(n,\beta )$-normed spaces.

Let us start with the outcomes in [33], where the author has considered stability of some multi equations, i.e., some particular cases of systems of functional equations of the following general form

$$\begin{array}{c}{\mathsf{\Phi }}_i(f\left(E_i(x,y)\right),f\left(F_i(x,y)\right))\hfill \\ ={\mathsf{\Psi }}_i\left(f\left(G_i(x,y)\right),f\left(H_i(x,y)\right)\right),i=1,\dots ,k,\hfill \end{array}$$

(16)

that can be considered for $f:V^k\to \mathfrak{L}$, where $k\in \mathbb{N}$ is fixed, $V\ne \varnothing$ is a set, and ${\mathsf{\Phi }}_i,{\mathsf{\Psi }}_i:{\mathfrak{L}}^2\to \mathfrak{L}$, $E_i,F_i,G_i,H_i:V^k\times V^k\to V^k$ for $i=1,\dots ,k$ are given.

If $(V,+)$ is a semigroup and, for $i=1,\dots ,k$,

$$\begin{array}{c}\hfill {\mathsf{\Phi }}_i(y,z)=2y,{\mathsf{\Psi }}_i(y,z)=y+z,y,z\in \mathfrak{L},\end{array}$$

and

$$\begin{array}{cc}\hfill & E_i(x,y)=(x_1,\dots ,x_{i-1},x_i+y_i,x_{i+1},\dots ,x_k),\hfill \\ \hfill & G_i(x,y)=(x_1,\dots ,x_{i-1},2x_i,x_{i+1},\dots ,x_k),\hfill \\ \hfill & H_i(x,y)=(x_1,\dots ,x_{i-1},2y_i,x_{i+1},\dots ,x_k)\hfill \end{array}$$

for $x=(x_1,\dots ,x_k),y=(y_1,\dots ,y_k)\in V^k$, then solutions $f:V^k\to \mathfrak{L}$ of system (16) are called k-Jensen maps or multi-Jensen maps (see, e.g., [34,35]). If we assume (well-defined) divisibility by 2 in V, then this particular case of (16) can be rewritten in the following form (patterned on the Jensen Equation (8)):

$$\begin{array}{cc}\hfill f(x_1,\dots & ,x_{i-1},{\displaystyle \frac{x_i+x_{i+1}}{2}},x_{i+2},\dots ,x_{k+1})\hfill \\ \hfill =& {\displaystyle \frac{1}{2}}(f(x_1,\dots ,x_i,x_{i+2},\dots ,x_{k+1})\hfill \\ \hfill & +f(x_1,\dots ,x_{i-1},x_{i+1},\dots ,x_{k+1})),i\in \{1,\dots ,k\}.\hfill \end{array}$$

In [33], the author has also considered stability of two other multi-equations, called there the multi-Euler–Lagrange additive equation and the multi-Euler–Lagrange quadratic equation, which are both of form (16). For more details we refer to [33]. Here, we only provide a simple hyperstability theorem for (16) that generalizes all stability outcomes in [33].

Theorem 2.

Let V be a nonempty set, $k\in \mathbb{N}$ be fixed, and ${\mathsf{\Phi }}_i,{\mathsf{\Psi }}_i:{\mathfrak{L}}^2\to \mathfrak{L}$, $E_i,F_i,G_i,H_i:V^k\times V^k\to V^k$, for $i=1,\dots ,k$, be given.

Assume that $f:V^k\to \mathfrak{L}$ satisfies the inequalities

$$\begin{array}{c}\hfill \parallel {\mathsf{\Phi }}_i(f\left(E_i(x,y)\right),f\left(F_i(x,y)\right)-{\mathsf{\Psi }}_i\left(f\left(G_i(x,y)\right),f\left(H_i(x,y)\right)\right),z{\parallel }_{\beta }\\ \hfill \le {\delta }_i(x,y),x,y\in V^k,z\in {\mathfrak{L}}^{n-1},i=1,\dots ,k,\end{array}$$

(17)

with some ${\delta }_1,\dots ,{\delta }_k:V^k\times V^k\to {\mathbb{R}}_{+}$. Then (16) holds for all $x,y\in V^k$.

Proof. 

For each $i\in \{1,\dots ,k\}$, it is enough to use Proposition 2 with $D=V^k\times V^k$ and

$$F(x,y)={\mathsf{\Phi }}_i(f\left(E_i(x,y)\right),f\left(F_i(x,y)\right)-{\mathsf{\Psi }}_i\left(f\left(G_i(x,y)\right),f\left(H_i(x,y)\right)\right)$$

for $x,y\in V^k$. □

Remark 5.

It appears that, similarly to what was noticed in Remark 4, Theorem 2 will not be true when condition (17) is assumed only for, e.g., $z\in \mathcal{B}$, where $\mathcal{B}$ is defined as in Remark 3. The same applies to Theorems 11 and 12, which are presented a little further on.

In [26], the authors have studied stability of the Cauchy, Jensen, and quadratic functional equations in n-Banach spaces. The main results concerning stability of the Cauchy Equation (1) (Theorems 3.1 and 3.3 in [26]) can be stated as follows.

Theorem 3.

Assume that X is a linear space, $\beta =1$, $\mathfrak{L}$ is n-complete, $s\in \{1,-1\}$ and $\phi :X^{n+1}\to {\mathbb{R}}_{+}$ is such that

$$\sum _{i=0}^{\infty }{\displaystyle \frac{\phi (2^{\alpha \left(i\right)}x,2^{\alpha \left(i\right)}x,x_2,\dots ,x_n)}{2^{\alpha \left(i\right)+1}}}<\infty ,\underset{k\to \infty }{lim}{\displaystyle \frac{\phi (2^{sk}x,2^{sk}y,x_2,\dots ,x_n)}{2^{sk}}}=0,$$

for all $x,y,x_2,\dots ,x_n\in X$, where $\alpha \left(i\right):=is+{\displaystyle \frac{1}{2}}(s-1)$ for $i\in {\mathbb{N}}_0$.

If $f:X\to \mathfrak{L}$ is a surjection that satisfies the inequality

$$\parallel f(x+y)-f\left(x\right)-f\left(y\right),f\left(x_2\right),\dots ,f\left(x_n\right){\parallel }_1\le \phi (x,y,x_2,\dots ,x_n)$$

for all $x,y,x_2,\dots ,x_n\in X$, then there is a unique additive $A:X\to \mathfrak{L}$ with

$$\parallel f\left(x\right)-A\left(x\right),f\left(x_2\right),\dots ,f\left(x_n\right){\parallel }_1\le \sum _{i=0}^{\infty }{\displaystyle \frac{\phi (2^{\alpha \left(i\right)}x,2^{\alpha \left(i\right)}x,x_2,\dots ,x_n)}{2^{\alpha \left(i\right)+1}}}$$

for all $x,x_2,\dots ,x_n\in X$.

Actually, the case $s=-1$ has not been proved in [26]; the authors claim that it is analogous to $s=1$, but give a somewhat misleading suggestion.

Motivated by the remarks of one of the reviewers, let us mention that the assumption of surjectivity of f in Theorem 3 (and in some next results) seems to be necessary only to ensure that all the values of $(f\left(x_2\right),\dots ,f\left(x_n\right))$ give the whole set ${\mathfrak{L}}^{n-1}$, which is convenient in the proofs. It appears that it could be weakened in some ways.

As a consequence of Theorem 3, the authors have obtained in [26] the following corollary (it is stated in [26] without n-completeness of $\mathfrak{L}$, but it appears that the n-completeness is necessary there).

Corollary 2.

Assume that $(X,\parallel ·\parallel )$ is a real normed space, $\beta =1$, and $\mathfrak{L}$ is n-complete. Let $\theta ,p,q,r\in {\mathbb{R}}_{+}$, $p+q\ne 1$.

If $f:X\to \mathfrak{L}$ is a surjection satisfying the inequality

$$\begin{array}{cc}\hfill \parallel f(x+y)& -f\left(x\right)-f\left(y\right),f\left(x_2\right),\dots ,f\left(x_n\right){\parallel }_1\hfill \\ \hfill & \le {\theta \parallel x\parallel }^p{\parallel y\parallel }^q\parallel x_2{\parallel }^r·\dots ·{\parallel x_n\parallel }^r,x,y,x_2,\dots ,x_n\in X,\hfill \end{array}$$

(18)

then there is a unique additive $A:X\to \mathfrak{L}$ such that

$$\parallel f\left(x\right)-A\left(x\right),f\left(x_2\right),\dots ,f\left(x_n\right){\parallel }_1\le {\displaystyle \frac{\theta {\parallel x\parallel }^{p+q}\parallel x_2{\parallel }^r·\dots ·{\parallel x_n\parallel }^r}{|2-2^{p+q}|}}$$

(19)

for all $x,x_2,\dots ,x_n\in X$.

Similar results have also been obtained in [26] for the Jensen and quadratic Equations (8) and (9) (it appears that in the corollaries the n-completeness of the n-normed space is necessary and in ([26], Theorem 5.3) the assumption $f\left(0\right)=0$ is missing).

The subsequent hyperstability result for the Cauchy Equation (1) has been proven in [25].

Theorem 4.

Let $\beta =1$, $\mathfrak{L}$ be n-complete, $(V,\parallel ·\parallel )$ be a normed space, $S\subset V\setminus \left\{0\right\}$ be nonempty, ${\mathfrak{L}}_0:=\mathfrak{L}\setminus \left\{0\right\}$, $\rho :{\mathfrak{L}}_0^{n-1}\to {\mathbb{R}}_{+}$, and $p,q\in \mathbb{R}$ be such that $p+q<0$. Let

$$\begin{array}{c}\hfill lx\in S,x\in S,l\in \mathbb{N},l\ge l_0,\end{array}$$

(20)

with some $l_0\in \mathbb{N}$.

Assume that $g:S\to \mathfrak{L}$ fulfills the inequality

$$\begin{array}{c}\hfill {\parallel g(x+y)-g\left(x\right)-g\left(y\right),z\parallel }_1\le \rho \left(z\right){\parallel x\parallel }^q{\parallel y\parallel }^p\end{array}$$

(21)

for $z\in {\mathfrak{L}}_0^{n-1}$ and $x,y\in S$ such that $x+y\in S$. Then g is additive on S, i.e., $g(x+y)=g\left(x\right)+g\left(y\right)$ for every $x,y\in S$ such that $x+y\in S$.

The proof of this result is based on the fixed-point theorem in [36], stated for operators acting on some classes of functions, with values in n-Banach spaces. Unfortunately, in ([25], Theorem 3), which contains the result depicted by Theorem 4, the assumption that $\mathfrak{L}$ is n-complete has been forgotten. This assumption is necessary in the proof, because of the application of the fixed-point theorem from [36]. However, in the next section, we show that such an assumption is superfluous in Theorem 4.

Ref. [25] also contains the following hyperstability result for (1).

Theorem 5.

Let $\beta =1$, the dimension of $\mathfrak{L}$ be at least $n+2$, $S\subset \mathfrak{L}$ be nonempty, and $c,p,q\in \mathbb{R}$ be such that $p+q\ne 1$ and $c\ge 0$. Assume that $g:S\to \mathfrak{L}$ fulfills the inequality

$${\parallel g(x+y)-g\left(x\right)-g\left(y\right),z\parallel }_1\le {c\parallel x,z\parallel }_1^q{\parallel y,z\parallel }_1^p$$

for $z\in {\mathfrak{L}}^{n-1}$ and $x,y\in S$ such that $x+y\in S$ and ${\parallel x,z\parallel }_1{\parallel y,z\parallel }_1\ne 0$. Then g is additive on S.

In [18], the authors have studied stability of the pexiderized Cauchy functional equation

$$f(x+y)=g\left(x\right)+h\left(y\right)$$

in $(n,\beta )$-Banach spaces and obtained the following main result.

Theorem 6.

Let X be a linear space, $\mathfrak{L}$ be $(n,\beta )$-complete, $\psi :{\mathfrak{L}}^{n-1}\to {\mathbb{R}}_{+}$ and $\phi :X^2\to {\mathbb{R}}_{+}$ be such that

$$\begin{array}{c}\hfill \mathsf{\Phi }\left(x\right)=\sum _{i=1}^{\infty }{2}^{-i\beta }\left(\phi (2^{i-1}x,0)+\phi (0,2^{i-1}x)+\phi (2^{i-1}x,2^{i-1}x)\right)<\infty \end{array}$$

$$\begin{array}{c}\hfill \underset{m\to \infty }{lim}{2}^{-m\beta }\phi (2^{m}x,2^{m}y)=0\end{array}$$

for all $x,y\in X$. If $f,g,h:X\to \mathfrak{L}$ satisfy the inequality

$$\parallel f(x+y)-g\left(x\right)-h\left(y\right),z_1,\dots ,z_{n-1}{\parallel }_{\beta }\le \phi (x,y)\psi (z_1,\dots ,z_{n-1})$$

(22)

for all $x,y\in X$ and $z_1,\dots ,z_{n-1}\in \mathfrak{L}$, then there exists a unique additive $A:X\to \mathfrak{L}$ such that

$$\begin{array}{cc}\hfill \parallel f\left(x\right)-A\left(x\right),& z_1,\dots ,z_{n-1}{\parallel }_{\beta }\hfill \\ \hfill & \le \mathsf{\Phi }\left(x\right)\psi (z_1,\dots ,z_{n-1})+{\parallel h\left(0\right),z_1,\dots ,z_{n-1}\parallel }_{\beta }\hfill \\ \hfill & +\parallel g\left(0\right),z_1,\dots ,z_{n-1}{\parallel }_{\beta },\hfill \end{array}$$

$$\begin{array}{cc}\hfill \parallel g\left(x\right)-A\left(x\right),& z_1,\dots ,z_{n-1}{\parallel }_{\beta }\hfill \\ \hfill & \le \mathsf{\Phi }\left(x\right)\psi (z_1,\dots ,z_{n-1})+{\parallel g\left(0\right),z_1,\dots ,z_{n-1}\parallel }_{\beta }\hfill \\ \hfill & +2\parallel h\left(0\right),z_1,\dots ,z_{n-1}{\parallel }_{\beta }+\phi (x,0)\psi (z_1,\dots ,z_{n-1}),\hfill \end{array}$$

$$\begin{array}{cc}\hfill \parallel h\left(x\right)-A\left(x\right),& z_1,\dots ,z_{n-1}{\parallel }_{\beta }\hfill \\ \hfill & \le \mathsf{\Phi }\left(x\right)\psi (z_1,\dots ,z_{n-1})+{\parallel h\left(0\right),z_1,\dots ,z_{n-1}\parallel }_{\beta }\hfill \\ \hfill & +2\parallel g\left(0\right),z_1,\dots ,z_{n-1}{\parallel }_{\beta }+\phi (0,x)\psi (z_1,\dots ,z_{n-1})\hfill \end{array}$$

for all $x\in X$ and $z_1,\dots ,z_{n-1}\in \mathfrak{L}$.

Let us add here that, in [18], the stability of the Cauchy functional Equation (1) and the Jensen functional Equation (8) was also studied in non-Archimedean $(n,\beta )$-normed spaces.

The fixed-point theorem in [36] was also used in [37] to prove the subsequent stability result for the Davison functional equation

$$\begin{array}{c}\hfill h\left(ab\right)+h(a+b)=h(ab+a)+h\left(b\right),\end{array}$$

(23)

that can be considered, e.g., for mappings h from a ring $\mathcal{R}$ into $\mathfrak{L}$.

Theorem 7.

Let $\beta =1$, $\mathfrak{L}$ be n-complete, $\mathcal{R}$ be a ring with the unit element 1 and $\gamma :{\mathcal{R}}^2\times {\mathfrak{L}}^{n-1}\to {\mathbb{R}}_{+}$ be such that, for every $a,b\in \mathcal{R}$, $u\in {\mathfrak{L}}^{n-1}$,

$$\xi (a,u):=\sum _{i=0}^{\infty }{\displaystyle \frac{\mu (2^{i}a,u)}{2^i}}<\infty ,\underset{n\to \infty }{lim}{\displaystyle \frac{{\mu }^{\prime }(2^{n}a,2^{n}b,u)}{2^n}}=0,$$

where

$$\begin{array}{cc}\hfill \mu (a,u)=& {\displaystyle \frac{1}{2}}[\gamma (4a,-4a,u)+\gamma (4a,-4a+1,u)+\gamma (8a,-2a,u)\hfill \\ \hfill & +\gamma (3a,0,u)+\gamma (3a,1,u)+\gamma (6a,0,u)+\gamma (7a,-4a,u)\hfill \\ \hfill & +\gamma (7a,-4a+1,u)+\gamma (14a,-2a,u)],\hfill \end{array}$$

$$\begin{array}{cc}\hfill {\mu }^{\prime }(a,b,u)=& \gamma (4a+3b,-4a,u)+\gamma (4a+3b,-4a+11,u)\hfill \\ \hfill & +\gamma (8a+6b,-2a,u)+\gamma (3a+4b,-4b,u)\hfill \\ \hfill & +\gamma (3a+4b,-4b+1,u)+\gamma (6a+8b,-2b,u)\hfill \\ \hfill & +2\gamma (7a,-4a,u)+2\gamma (7a,-4a+1,u)+2\gamma (14a,-2a,u).\hfill \end{array}$$

If $g:\mathcal{R}\to \mathfrak{L}$ fulfills the inequality

$${\parallel g\left(ab\right)+g(a+b)-g(ab+a)-g\left(b\right),u\parallel }_1\le \gamma (a,b,u)$$

for every $a,b\in \mathcal{R}$ and $u\in {\mathfrak{L}}^{n-1}$, then there is a unique additive $\mathsf{\Psi }:\mathcal{R}\to \mathfrak{L}$ with

$$\parallel g\left(6a\right)-\mathsf{\Psi }\left(a\right)-g\left(0\right),u{\parallel }_1\le \xi (a,u),a\in \mathcal{R},u\in {\mathfrak{L}}^{n-1}.$$

In [38], the authors have investigated stability of the following two functional equations:

$$\begin{array}{c}\hfill g(2x+y)+g(2x-y)=2g(x+y)+2g(x-y)+12g\left(x\right),\end{array}$$

(24)

$$\begin{array}{c}\hfill g(2x+y)+g(2x-y)=4g(x+y)+4g(x-y)+24g\left(x\right)+6g\left(y\right),\end{array}$$

(25)

for maps g from a real linear space into an n-Banach space.

Below we recall only the results stated for (24), because those formulated for (25) are very similar and we refer to [38] for more details.

Theorem 8.

Assume that X is a real linear space, $\beta =1$, and $\mathfrak{L}$ is n-complete. Let $s\in \{1,-1\}$ and $\phi :X^{n+1}\to {\mathbb{R}}_{+}$ be such that

$$\sum _{i=0}^{\infty }{\displaystyle \frac{\phi (2^{\alpha \left(i\right)}x,0,x_2,\dots ,x_n)}{8^{\alpha \left(i\right)+1}}}<\infty ,\underset{k\to \infty }{lim}{\displaystyle \frac{\phi (2^{sk}x,2^{sk}y,x_2,\dots ,x_n)}{8^{sk}}}=0$$

for all $x,y,x_2,\dots ,x_n\in X$, where $\alpha \left(i\right):=is+{\displaystyle \frac{1}{2}}(s-1)$ for $i\in {\mathbb{N}}_0$. Suppose that $f:X\to \mathfrak{L}$ is surjective and

$$\parallel D_f(x,y),f\left(x_2\right),\dots ,f\left(x_n\right){\parallel }_1\le \phi (x,y,x_2,\dots ,x_n),x,y,x_2,\dots ,x_n\in X,$$

where $D_f(x,y):=f(2x+y)+f(2x-y)-2f(x+y)-2f(x-y)-12f\left(x\right).$

Then there is a unique $g:X\to \mathfrak{L}$ fulfilling (24) for $x,y\in X$ such that

$$\parallel f\left(x\right)-g\left(x\right),f\left(x_2\right),\dots ,f\left(x_n\right){\parallel }_1\le {\displaystyle \frac{1}{2}}\sum _{i=0}^{\infty }{\displaystyle \frac{\phi (2^{\alpha \left(i\right)}x,0,x_2,\dots ,x_n)}{8^{\alpha \left(i\right)+1}}}$$

for all $x,x_2,\dots ,x_n\in X$.

As a consequence, the authors have stated the following corollary (without assuming the n-completeness of $\mathfrak{L}$, but it appears that this assumption is necessary).

Corollary 3.

Assume that $(X,\parallel ·\parallel )$ is a real normed space, $\beta =1$, and $\mathfrak{L}$ is n-complete. Let $\theta \in {\mathbb{R}}_{+}$ and $p,q,r\in (0,\infty )$ be such that $p,q\in (3,\infty )$ or $p,q\in (0,3)$. Let $f:X\to \mathfrak{L}$ be a surjection satisfying the inequality

$$\begin{array}{cc}\hfill \parallel D_f(x,y)& ,f\left(x_2\right),\dots ,f\left(x_n\right){\parallel }_1\hfill \\ \hfill & \le \theta ({\parallel x\parallel }^p+{\parallel y\parallel }^q)\parallel x_2{\parallel }^r·\dots ·{\parallel x_n\parallel }^r,x,y,x_2,\dots ,x_n\in X.\hfill \end{array}$$

Then there is a unique $g:X\to \mathfrak{L}$ fulfilling (24) for $x,y\in X$ such that

$$\begin{array}{c}\hfill \parallel f\left(x\right)-g\left(x\right),f\left(x_2\right),\dots ,f\left(x_n\right){\parallel }_1\le {\displaystyle \frac{\theta {\parallel x\parallel }^p\parallel x_2{\parallel }^r·\dots ·{\parallel x_n\parallel }^r}{|16-2^{p+1}|}}\end{array}$$

for all $x,x_2,\dots ,x_n\in X$.

The authors in [39] investigated stability of the functional equation

$$\begin{array}{c}\hfill f(kx+y)+f(kx-y)=kf(x+y)+kf(x-y)+2f\left(kx\right)-2kf\left(x\right),\end{array}$$

(26)

where $k\notin \{-1,0,1\}$ is a fixed integer. They have proved several results. The first one is the following theorem.

Theorem 9.

Let X be a real linear space, $k\notin \{-1,0,1\}$ be an integer, $\beta =1$, and $\mathfrak{L}$ be n-complete. Let $f:X\to \mathfrak{L}$ be such that $f\left(0\right)=0$. Assume that there is $\phi :X^2\times {\mathfrak{L}}^{n-1}\to {\mathbb{R}}_{+}$ with

$$\sum _{j=0}^{\infty }{\displaystyle \frac{1}{2^j}}\phi \left(2^{j}x,2^{j}y,u_2,\dots ,u_n\right)<\infty ,$$

$$\parallel Df(x,y),u_2,\dots ,u_n{\parallel }_1\le \phi (x,y,u_2,\dots ,u_n)$$

for all $x,y\in X$, $u_2,\dots ,u_n\in \mathfrak{L}$, where $Df(x,y):=f(kx+y)+f(kx-y)-kf(x+y)-kf(x-y)-2f\left(kx\right)+2kf\left(x\right)$.

Then, there is a unique additive mapping $A:X\to \mathfrak{L}$ such that

$$\parallel f\left(2x\right)-8f\left(x\right)-A\left(x\right),u_2,\dots ,u_n{\parallel }_1\le \sum _{j=0}^{\infty }{\displaystyle \frac{1}{2^{j+1}}}\tilde{\phi }(2^{j}x,u_2,\dots ,u_n)$$

for all $x\in X$, $u_2,\dots ,u_n\in \mathfrak{L}$, where $\tilde{\phi }$ is given by the (rather long and complicated) formula (2.5) in [39].

The other results obtained in [39] are somewhat similar to Theorem 9. We should mention here that in [39] it is written several times (also in the result presented above) that $u_2,\dots ,u_n\in X$, but this seems to be just a simple mistake.

In ([40], Theorem 7) the following result has been proven.

Theorem 10.

Assume that $\beta =1$, $\mathfrak{L}$ is n-complete, X is a linear space over a field $\mathbb{F}$, $m\in \mathbb{N}$, $a_{11},a_{12},\dots ,a_{m1},a_{m2}\in \mathbb{F}$, $A_{i_1,\dots ,i_m}\in \mathbb{R}$ for $i_1,\dots ,i_m\in \{1,2\}$ and

$$\gamma :=\left|\sum _{i_1,\dots ,i_m\in \{1,2\}}{A}_{i_1,\dots ,i_m}\right|>1.$$

If $\epsilon >0$ and $f:X^m\to \mathfrak{L}$ satisfies the inequality

$$\begin{array}{cc}\hfill \parallel f(a_{11}{x}_{11}+a_{12}{x}_{12}& ,\dots ,a_{m1}{x}_{m1}+a_{m2}{x}_{m2})-\hfill \\ \hfill & \sum _{i_1,\dots ,i_m\in \{1,2\}}{A}_{i_1,\dots ,i_m}f(x_{1i_1},\dots ,x_{m{i}_m}),z{\parallel }_1\le \epsilon \hfill \end{array}$$

(27)

for $x_{11},x_{12},\dots ,x_{m1},x_{m2}\in X$ and $z\in {\mathfrak{L}}^{n-1}$, then there exists a unique $g:X^m\to \mathfrak{L}$ such that

$$\begin{array}{cc}\hfill g(a_{11}{x}_{11}+a_{12}{x}_{12},& \dots ,a_{m1}{x}_{m1}+a_{m2}{x}_{m2})\hfill \\ \hfill & =\sum _{i_1,\dots ,i_m\in \{1,2\}}{A}_{i_1,\dots ,i_m}g(x_{1i_1},\dots ,x_{m{i}_m})\hfill \end{array}$$

(28)

for $x_{11},x_{12},\dots ,x_{m1},x_{m2}\in X$ and

$${\parallel f\left(x\right)-g\left(x\right),z\parallel }_1\le {\displaystyle \frac{\epsilon }{\gamma -1}},x\in X^m,z\in {\mathfrak{L}}^{n-1}.$$

In [41], the authors noticed that the following two hyperstability results are valid (analogously, as in Theorem 10, X is a linear space over a field $\mathbb{F}$, $m\in \mathbb{N}$, $a_{11},a_{12},\dots ,a_{m1},a_{m2}\in \mathbb{F}$, $A_{i_1,\dots ,i_m}\in \mathbb{R}$ for $i_1,\dots ,i_m\in \{1,2\}$).

Theorem 11.

Let $\phi :X^{2m}\to {\mathbb{R}}_{+}$ and $g:X^m\to \mathfrak{L}$ satisfy the inequality

$$\begin{array}{cc}\hfill \parallel g(a_{11}{x}_{11}+a_{12}{x}_{12},& \dots ,a_{m1}{x}_{m1}+a_{m2}{x}_{m2})\hfill \\ \hfill & -\sum _{i_1,\dots ,i_m\in \{1,2\}}{A}_{i_1,\dots ,i_m}g(x_{1i_1},\dots ,x_{m{i}_m}),z{\parallel }_{\beta }\hfill \\ \hfill \le & \phi (x_{11},x_{12},\dots ,x_{m1},x_{m2})\hfill \end{array}$$

(29)

for $x_{11},x_{12},\dots ,x_{m1},x_{m2}\in X$ and $z\in {\mathfrak{L}}^{n-1}$. Then g satisfies (28) for all $x_{11},x_{12},\dots ,x_{m1},x_{m2}\in X$.

Theorem 12.

Let $\mathbb{F}=\mathbb{R}$, $\epsilon \in {\mathbb{R}}_{+}$ and ${\left\{{\alpha }_i\right\}}_{i=1}^m$, ${\left\{{\beta }_i\right\}}_{i=1}^m$, ${\left\{r_i\right\}}_{i=1}^m$ be sequences in ${\mathbb{R}}_{+}$ such that ${\max }_{1\le i\le m}{r}_i<1/\beta$ or ${\min }_{1\le i\le m}{r}_i>1/\beta$. Let $g:{\mathfrak{L}}^m\to \mathfrak{L}$ fulfill the inequality

$$\begin{array}{cc}\hfill \parallel g(a_{11}{x}_{11}+a_{12}{x}_{12},& \dots ,a_{m1}{x}_{m1}+a_{m2}{x}_{m2})\hfill \\ \hfill & -\sum _{i_1,\dots ,i_m\in \{1,2\}}{A}_{i_1,\dots ,i_m}g(x_{1i_1},\dots ,x_{m{i}_m}),z{\parallel }_{\beta }\hfill \\ \hfill \le & \epsilon +\sum _{i=1}^m[{\alpha }_i\parallel x_{i1}{,z\parallel }_{\beta }^{r_i}+{\beta }_i\parallel x_{i2},z{\parallel }_{\beta }^{r_i}]\hfill \end{array}$$

(30)

for $x_{11},x_{12},\dots ,x_{m1},x_{m2}\in \mathfrak{L}$ and $z\in {\mathfrak{L}}^{n-1}$. Then g satisfies (28) for all $x_{11},x_{12},\dots ,x_{m1},x_{m2}\in \mathfrak{L}$.

Actually, these results have been proven in [41], only in the case $\beta =1$, but their proofs can also be easily extended to the case $\beta \ne 1$ (somewhat similarly as in the proof of Proposition 1). Clearly, both theorems generalize Theorem 10.

In [41], the following two stability results have also been obtained (again X is a linear space over a field $\mathbb{F}$, $m\in \mathbb{N}$, $a_{11},a_{12},\dots ,a_{m1},a_{m2}\in \mathbb{F}$, $A_{i_1,\dots ,i_m}\in \mathbb{R}$ for $i_1,\dots ,i_m\in \{1,2\}$).

Theorem 13.

Assume that $\mathbb{F}\in \{\mathbb{R},\mathbb{C}\}$, $\epsilon ,\theta \in {\mathbb{R}}_{+}$, $\beta =1$, $\mathfrak{L}$ is n-complete, $\parallel ·\parallel$ is a norm in X, $h:X\to {\mathfrak{L}}^{n-1}$ is surjective and

$$\gamma :=\left|\sum _{i_1,\dots ,i_m\in \{1,2\}}{A}_{i_1,\dots ,i_m}\right|>1.$$

If $f:X^m\to \mathfrak{L}$ is such that

$$\begin{array}{cc}\hfill \parallel f(a_{11}{x}_{11}+a_{12}{x}_{12},\dots ,a_{m1}{x}_{m1}+a_{m2}{x}_{m2})-& \hfill \\ \hfill \sum _{i_1,\dots ,i_m\in \{1,2\}}{A}_{i_1,\dots ,i_m}f(x_{1i_1},\dots ,x_{m{i}_m}),h\left(z\right){\parallel }_1& \le \epsilon +\theta \parallel z\parallel \hfill \end{array}$$

(31)

for $x_{11},x_{12},\dots ,x_{m1},x_{m2},z\in X$, then there is a unique $g:X^m\to \mathfrak{L}$ fulfilling (28) for $x_{11},x_{12},\dots ,x_{m1},x_{m2}\in X$ and such that

$$\parallel f(x_1,\dots ,x_m)-g(x_1,\dots ,x_m),h\left(z\right){\parallel }_1\le {\displaystyle \frac{\epsilon +\theta \parallel z\parallel }{\gamma -1}},x_1,\dots ,x_m,z\in X.$$

Theorem 14.

Let $\mathbb{F}=\mathbb{R}$, $\beta =1$, $\mathfrak{L}$ be n-complete, $\epsilon \in {\mathbb{R}}_{+}$, and ${\left\{{\alpha }_i\right\}}_{i=1}^m$, ${\left\{{\beta }_i\right\}}_{i=1}^m$ be sequences in ${\mathbb{R}}_{+}$ such that

$$\gamma :=\left|\sum _{i_1,\dots ,i_m\in \{1,2\}}{A}_{i_1,\dots ,i_m}\right|>{\max }_{1\le j\le m}|a_{j1}+a_{j2}|.$$

Let $f:{\mathfrak{L}}^m\to \mathfrak{L}$ satisfy the inequality

$$\begin{array}{cc}\hfill \parallel f(a_{11}{x}_{11}+a_{12}{x}_{12},& \dots ,a_{m1}{x}_{m1}+a_{m2}{x}_{m2})\hfill \\ \hfill & -\sum _{i_1,\dots ,i_m\in \{1,2\}}{A}_{i_1,\dots ,i_m}f(x_{1i_1},\dots ,x_{m{i}_m}),z{\parallel }_1\hfill \\ \hfill \le & \epsilon +\sum _{i=1}^m\left[{\alpha }_i\parallel x_{i1},z\parallel +{\beta }_i\parallel x_{i2},z\parallel \right]\hfill \end{array}$$

(32)

for $x_{11},x_{12},\dots ,x_{m1},x_{m2}\in \mathfrak{L}$ and $z\in {\mathfrak{L}}^{n-1}$.

Then there is a unique $g:{\mathfrak{L}}^m\to \mathfrak{L}$ fulfilling (28) for $x_{11},x_{12},\dots ,x_{m1},x_{m2}\in \mathfrak{L}$ and such that

$$\parallel f(x_1,\dots ,x_m)-g(x_1,\dots ,x_m){,z\parallel }_1\le \sum _{i=1}^m{\displaystyle \frac{{\alpha }_i+{\beta }_i}{\gamma -|a_{i1}+a_{i2}|}}\parallel x_i,z\parallel$$

for all $x_1,\dots ,x_m\in \mathfrak{L}$ and $z\in {\mathfrak{L}}^{n-1}$.

We have corrected some small mistakes that occur in [41] in the original versions of Theorems 13 and 14.

In [42], the author has investigated (among other) stability in n-normed spaces of the following functional equation:

$$\begin{array}{cc}\hfill \sum _{i_1,i_2,\dots ,i_m\in \{-1,1\}}g(& a_{1,i_1,i_2,\dots ,i_m}(x_{11}+i_1{x}_{12}),\dots ,a_{m,i_1,i_2,\dots ,i_m}(x_{m1}+i_m{x}_{m2}))\hfill \\ \hfill & =\sum _{j_1,j_2,\dots ,j_m\in \{1,2\}}{A}_{j_1,\dots ,j_m}g(x_{1j_1},\dots ,x_{m{j}_m}),\hfill \end{array}$$

(33)

for g mapping a linear space X over the field $\mathbb{F}$ into an n-Banach space, where $a_{1,i_1,i_2,\dots ,i_m},\dots ,a_{m,i_1,i_2,\dots ,i_m}\in \mathbb{F}$ and $A_{j_1,\dots ,j_m}\in \mathbb{R}$ (cf. [43]). The main result (for the n-Banach space) can be formulated as follows:

Theorem 15.

Let $\beta =1$, $\mathfrak{L}$ be n-complete, $\epsilon \in {\mathbb{R}}_{+}$, and

$$\gamma :=\left|\sum _{j_1,j_2,\dots ,j_m\in \{1,2\}}{A}_{j_1,\dots ,j_m}\right|>1.$$

If $f:X^m\to \mathfrak{L}$ is such that $f(x_{11},\dots ,x_{m1})=0$ for any $(x_{11},\dots ,x_{m1})\in X^m$ that has at least one component equal to zero and

$$\begin{array}{cc}\hfill \parallel \sum _{i_1,i_2,\dots ,i_m\in \{-1,1\}}& f\left(a_{1,i_1,i_2,\dots ,i_m}(x_{11}+i_1{x}_{12}),\dots ,a_{m,i_1,i_2,\dots ,i_m}(x_{m1}+i_m{x}_{m2})\right)\hfill \\ \hfill & -\sum _{j_1,j_2,\dots ,j_m\in \{1,2\}}{A}_{j_1,\dots ,j_m}f(x_{1j_1},\dots ,x_{m{j}_m}),z{\parallel }_1\le \epsilon \hfill \end{array}$$

(34)

for $x_{11},x_{12},\dots ,x_{m1},x_{m2}\in X$ and $z\in {\mathfrak{L}}^{n-1}$, then there is $g:X^m\to \mathfrak{L}$ satisfying (33) for $x_{11},x_{12},\dots ,x_{m1},x_{m2}\in X$ such that

$$\parallel f(x_1,\dots ,x_m)-g(x_1,\dots ,x_m){,z\parallel }_1\le {\displaystyle \frac{\epsilon }{\gamma -1}},x_1,\dots ,x_m\in X,z\in {\mathfrak{L}}^{n-1}.$$

Clearly, as before, we can easily deduce from Proposition 2 that every $f:X^m\to \mathfrak{L}$, satisfying (34) for $x_{11},x_{12},\dots ,x_{m1},x_{m2}\in X$ and $z\in {\mathfrak{L}}^{n-1}$, must in fact satisfy (33) (with $g=f$) for all $x_{11},x_{12},\dots ,x_{m1},x_{m2}\in X$ (see also ([43], Theorem 2)).

Finally, we should mention the results in [44], where the authors investigated the stability of the K-cubic functional equation in the intuitionistic fuzzy n-normed spaces.

5. Final Remarks and General Methods

In this section, $\mathcal{B}\subset {\mathfrak{L}}^{n-1}$ is defined as in Remark 3, where $(b_1,\dots ,b_n)\in {\mathfrak{L}}^n$ is a fixed sequence of n linearly independent vectors in $\mathfrak{L}$. First, we show that the assumption of n-completeness of $\mathfrak{L}$ in Theorem 4 is superfluous; i.e., we prove the following:

Theorem 16.

Let $\beta =1$, $(V,\parallel ·\parallel )$ be a normed space, $S\subset V\setminus \left\{0\right\}$ be nonempty, (20) be valid with some $l_0\in \mathbb{N}$, ${\mathfrak{L}}_0:=\mathfrak{L}\setminus \left\{0\right\}$, $\rho :\mathcal{B}\to {\mathbb{R}}_{+}$, and $p,q\in \mathbb{R}$ be such that $p+q<0$.

If $g:S\to \mathfrak{L}$ fulfills inequality (21) for $z\in \mathcal{B}$ and $x,y\in S$ such that $x+y\in S$, then g is additive on S.

To this end, we need to recall the subsequent result from [45].

Theorem 17.

Let $(L,\parallel ·\parallel )$ and $(V,\parallel ·\parallel )$ be normed spaces, $S\subset V\setminus \left\{0\right\}$ be nonempty, (20) hold with some $l_0\in \mathbb{N}$, and $c,p,q\in \mathbb{R}$ be such that $p+q<0$ and $c\ge 0$.

If $g:S\to L$ fulfills the inequality

$$\begin{array}{c}\hfill \parallel g(x+y)-g\left(x\right)-g\left(y\right)\parallel \le {c\parallel x\parallel }^q{\parallel y\parallel }^p,x,y\in S,x+y\in S,\end{array}$$

(35)

then g is additive on S.

Proof of Theorem 16 .

Define a norm ${\parallel ·\parallel }_{\mathcal{B}}$ in $\mathfrak{L}$ by (4) (see Remark 3). Then

$$\begin{array}{c}\hfill {\parallel g(x+y)-g\left(x\right)-g\left(y\right)\parallel }_{\mathcal{B}}\le {\parallel x\parallel }^q{\parallel y\parallel }^p\sum _{b\in \mathcal{B}}\rho \left(b\right),x,y\in S,x+y\in S.\end{array}$$

This means that condition (35) is satisfied with $\parallel ·\parallel :={\parallel ·\parallel }_{\mathcal{B}}$ and $c:={\sum }_{b\in \mathcal{B}}\rho \left(b\right)$. Consequently, according to Theorem 17, g is additive on S. □

In a similar way, we can obtain the following analog of Theorem 1 (ii) for n-normed spaces.

Theorem 18.

Let X be a normed space, $X_0:=X\setminus \left\{0\right\}$, $\eta :X\to {\mathbb{R}}_{+}$, $\eta :\mathcal{B}\to {\mathbb{R}}_{+}$, and $p<0$ be a fixed real number. Let $g:X\to \mathfrak{L}$ satisfy the inequality

$$\begin{array}{c}\hfill {\parallel g(x+y)-g\left(x\right)-g\left(y\right),z\parallel }_1\le {\eta \left(z\right)(\parallel x\parallel }^p+{\parallel y\parallel }^p),\\ \hfill x,y\in X_0,z\in \mathcal{B}.\end{array}$$

Then g is additive.

Proof. 

It is enough to use Theorem 1 (ii), with a reasoning analogous to that in the proof of Theorem 16. □

Since one of the reviewers asked about another example of the application of the reasoning used in the proof of Theorem 16, we provide it using the subsequent result, which can be easily deduced from Theorem 2 in [46].

Theorem 19.

Let X be a normed space, Y be a Banach space, $X_0:=X\setminus \left\{0\right\}$, and $c>0$ and $p<0$ be fixed real numbers. If $f:X_0\to Y$ satisfies the inequality

$$\begin{array}{c}\hfill \parallel f(x+y)+f(x-y)-2f\left(x\right)-f\left(y\right)-f(-y)\parallel \le c(\parallel x{\parallel }^p+\parallel y{\parallel }^p),\\ \hfill x,y\in X_0,x+y,x-y\in X_0,\end{array}$$

(36)

then

$$\begin{array}{c}\hfill f(x+y)+f(x-y)=2f\left(x\right)+f\left(y\right)+f(-y),x,y\in X_0,x+y,x-y\in X_0.\end{array}$$

(37)

Namely, we have the following result.

Theorem 20.

Let X be a normed space, $X_0:=X\setminus \left\{0\right\}$, $\eta :\mathcal{B}\to {\mathbb{R}}_{+}$, and $p<0$ be a fixed real number. Let $f:X_0\to \mathfrak{L}$ satisfy the inequality

$$\begin{array}{c}\hfill {\parallel f(x+y)+f(x-y)-2f\left(x\right)-f\left(y\right)-f(-y),z\parallel }_1\le {\eta \left(z\right)(\parallel x\parallel }^p+{\parallel y\parallel }^p),\\ \hfill x,y\in X_0,x+y,x-y\in X_0,z\in \mathcal{B}.\end{array}$$

Then (37) holds.

Proof. 

We define a norm ${\parallel ·\parallel }_{\mathcal{B}}$ in $\mathfrak{L}$ by (4). Then

$$\begin{array}{c}\hfill {\parallel f(x+y)+f(x-y)-2f\left(x\right)-f\left(y\right)-f(-y)\parallel }_{\mathcal{B}}\le \sum _{b\in \mathcal{B}}\eta \left(b\right)\left({\parallel x\parallel }^p+{\parallel y\parallel }^p\right),\\ \hfill x,y\in X_0,x+y,x-y\in X_0,\end{array}$$

which means that inequality (36) is fulfilled with $c:={\sum }_{b\in \mathcal{B}}\eta \left(b\right)$.

Furthermore, it is well known that every normed space can be completed to a unique (up to isometric isomorphism) Banach space. That is, $\mathfrak{L}$, with the norm ${\parallel ·\parallel }_{\mathcal{B}}$, can be considered a subspace of a Banach space Y, and we can treat f as a map from X to Y. Hence, by Theorem 19, (37) holds. □

Clearly, we can apply this method to other hyperstability results (e.g., results from [28,29]), obtained for the normed spaces, to extend them to the n-normed spaces.

Below we show that a similar approach can also be used with regard to some superstability results (see, e.g., [1,2,3] for more information on superstability and some examples of such outcomes).

So, let us consider here the following superstability result that was proven by J. Baker [47].

Proposition 3.

Let $(S,★)$ be a semigroup and $h:S\to \mathbb{C}$ be such that

$$\begin{array}{c}\hfill \rho :=\underset{x,y\in S}{sup}|h(x★y)-h\left(x\right)h\left(y\right)|<\infty .\end{array}$$

(38)

Then

$$\begin{array}{c}\hfill h(x★y)=h\left(x\right)h\left(y\right),x,y\in S,\end{array}$$

(39)

or

$$\begin{array}{c}\hfill \underset{x\in S}{sup}\left|h\left(x\right)\right|\le {\displaystyle \frac{1+\sqrt{1+4\rho }}{2}}.\end{array}$$

(40)

The next proposition shows how to easily extend it to the case when $\mathbb{C}$ is endowed with a 2-norm.

Proposition 4.

Let $(S,★)$ be a semigroup, $\parallel ·,·\parallel$ be a 2-norm in $\mathbb{C}$, and $B\subset \mathbb{C}\setminus \left\{0\right\}$ be nonempty and such that $u{v}^{-1}$ is not a real number for some $u,v\in B$. Let $h:S\to \mathbb{C}$ be such that

$$\begin{array}{c}\hfill \underset{x,y\in S}{sup}\parallel h(x★y)-h\left(x\right)h\left(y\right),z\parallel <\infty ,z\in B.\end{array}$$

(41)

Then one of the following two statements is valid:

(i) 

h is a solution of Equation (39);

(ii) 

Conditions (38) and (40) hold and

$$\begin{array}{c}\hfill \underset{x\in S}{sup}\parallel h\left(x\right),z\parallel <\infty ,z\in B.\end{array}$$

(42)

Proof. 

Suppose that (i) is not valid, i.e., h is not a solution to Equation (39). Take $u,z_0\in B$ such that $z_0{u}^{-1}\notin \mathbb{R}$ and define a norm ${\parallel ·\parallel }_0$ in $\mathbb{C}$ by

$$\begin{array}{c}\hfill {\parallel w\parallel }_0=\parallel w,z_0\parallel +\parallel w,u\parallel ,w\in \mathbb{C}.\end{array}$$

(43)

Then, by (41),

$$\begin{array}{cc}\hfill \underset{x,y\in S}{sup}\parallel h(x★y)-h\left(x\right)h\left(y\right){\parallel }_0=\underset{x,y\in S}{sup}& (\parallel h(x★y)-h\left(x\right)h\left(y\right),z_0\parallel \hfill \\ \hfill & +\parallel h(x★y)-h\left(x\right)h\left(y\right),u\parallel )<\infty ,\hfill \end{array}$$

which implies (38) (because all norms in $\mathbb{C}$ are equivalent). Consequently, on account of Proposition 3, we get (40).

Observe yet that (40) yields ${sup}_{x\in S}{\parallel h\left(x\right)\parallel }_0<\infty$, which means that (42) holds (because $u,z_0\in B$ were taken arbitrarily). □

6. Open Problems and Future Directions

One of the reviewers insisted that this paper should indicate future directions for research on Ulam stability and suggested the following open problem: Can the fixed-point method in [36] be extended to $(n,\beta )$-normed spaces with $\beta <1$?

Another open problem that can be mentioned here is the question of optimality of estimates in the stability results obtained for the n-normed and $(n,\beta )$-normed spaces (see, e.g., (19) or (21)). This has already been considered for various equations in some spaces; and we refer to [2,3,4,10,48,49,50,51,52,53,54] for more information on this topic.

Next, it would be useful to find a method similar to that used in the proof of Theorem 16 that allows one to easily extend the hyperstability results from the normed spaces to the $(n,\beta )$-normed spaces with any positive $\beta <1$.

Let us also mention that it would be interesting to extend the reasoning given in the proof of Proposition 4 to other superstability results, also with respect to the $(n,\beta )$-norms. We refer to [55,56,57,58,59,60] for some examples of such superstability outcomes.