On Ulam Stability of the Davison Functional Equation in m-Banach Spaces

Abstract

We prove new Ulam stability results for the Davison functional equation, in the class of mappings h from a ring F into an m-Banach space. In this way, we complement several earlier outcomes, by extending them to the case of m-normed spaces. Our proofs are based on an earlier Ulam stability result obtained for some functional equation in a single variable.

1. Introduction

In recent years, Ulam’s idea of stability has gained popularity as a research topic (cf. [1,2,3,4,5,6,7,8,9]) and been extended in several directions. Let us mention that it is connected to the subjects studied in some other fields of mathematics, including theories of optimization, approximation, perturbation, and shadowing (see [10]). It mainly deals with different kinds of equations (difference, differential, integral, functional, etc.) and considers the following issue: How much does an equation’s approximate solution vary from its exact solution? Numerous works were inspired by this topic, and we refer to monographs [11,12,13,14] for further information on this subject.

The main early outcomes in this area were obtained by, e.g., D.H. Hyers [1], T. Aoki [15], and Th. M. Rassias [16]. They motivated numerous mathematicians worldwide to continue analogous studies. The outcomes obtained in these works can be summarized in the following theorem (see, e.g., [13]):

Theorem 1. 

Assume that V is a normed space, $V_0:=V\setminus \left\{0\right\}$, W is a Banach space, and $c\ge 0$ and $t\ne 1$ are real numbers. Let $\chi :V\to W$ satisfy the inequality:

$$\parallel \chi (s+v)-\chi \left(s\right)-\chi \left(v\right)\parallel \le {c(\parallel s\parallel }^t+{\parallel v\parallel }^t),s,v\in V_0.$$

(1)

Then there exists exactly one additive mapping $\mathsf{\Gamma }:V\to W$ such that

$$\parallel \chi \left(s\right)-\mathsf{\Gamma }\left(s\right)\parallel \le {\displaystyle \frac{{c\parallel s\parallel }^t}{|1-2^{t-1}|}},s\in V_0.$$

(2)

Let us recall that $\mathsf{\Gamma }:V\to W$ is additive if

$$\begin{array}{c}\hfill \mathsf{\Gamma }(s+v)=\mathsf{\Gamma }\left(s\right)+\mathsf{\Gamma }\left(v\right)\end{array}$$

(3)

for all $s,v\in V$ (here, as before, V and W denote linear spaces). For information on more recent related results, we refer to, e.g., [11,12,13,14].

The next quite abstract definition shows that the notion of Ulam stability can be made a bit more precise (${\mathbb{R}}_{+}$ means the set of all non-negative reals, and $C^D$ stands for the family of all maps from a nonempty set D into a nonempty set C).

Definition 1. 

Assume that $k>0$ is an integer, $(L,\rho )$ is a metric space, $H_0\subset H$ are nonempty sets, ${\mathcal{K}}_0\subset \mathcal{K}\subset L^H$ and $\mathcal{T}\subset {\mathbb{R}}_{+}^{H_0^k}$ are nonempty, $\mathcal{S}:\mathcal{T}\to {\mathbb{R}}_{+}^{H_0}$, and ${\mathcal{F}}_1,{\mathcal{F}}_2:\mathcal{K}\to L^{H^k}$. The equation

$$\begin{array}{c}\hfill \left({\mathcal{F}}_1\varphi \right)(s_1,\dots ,s_k)=\left({\mathcal{F}}_2\varphi \right)(s_1,\dots ,s_k)\end{array}$$

(4)

is said to be $\mathcal{S}$-stable in ${\mathcal{K}}_0$ if, for every $\varphi \in {\mathcal{K}}_0$ and $\xi \in \mathcal{T}$ fulfilling the inequality

$$\rho (\left({\mathcal{F}}_1\varphi \right)(s_1,\dots ,s_k),\left({\mathcal{F}}_2\varphi \right)(s_1,\dots ,s_k))\le \xi (s_1,\dots ,s_k),s_1,\dots ,s_k\in H_0,$$

there exists a solution $\psi \in \mathcal{K}$ of Equation (4) with

$$\rho (\psi \left(s\right),\varphi \left(s\right))\le \left(\mathcal{S}\xi \right)\left(s\right),s\in H_0.$$

If $\left(\mathcal{S}\xi \right)\left(s\right)=0$ for every $\xi \in \mathcal{T}$ and $s\in H$, then the equation is said to be $\mathcal{T}$–hyperstable in ${\mathcal{K}}_0$.

Clearly, (4) is Equation (3) when $k=2$, $H=V$, $H_0=V_0$, $\left({\mathcal{F}}_1\mathsf{\Gamma }\right)(s,t)=\mathsf{\Gamma }(s+t)$ and $\left({\mathcal{F}}_2\mathsf{\Gamma }\right)(s,t)=\mathsf{\Gamma }\left(s\right)+\mathsf{\Gamma }\left(t\right)$ for $\mathsf{\Gamma }\in \mathcal{K}$ and $s,t\in H=V.$

According to Definition 1, Theorem 1 is stating that, for any real number $t\ne 1$, Equation (3) is $\mathcal{S}$-stable in ${\mathcal{K}}_0=\mathcal{K}=W^V$, for $\mathcal{S}:\mathcal{T}\to {\mathbb{R}}_{+}^{V_0}$ given by

$$\left(\mathcal{S}{\xi }_{\mu }\right)\left(s\right)={\displaystyle \frac{{\mu \parallel s\parallel }^t}{|1-2^{t-1}|}},{\xi }_{\mu }\in \mathcal{T},s\in V_0,$$

where

$${\xi }_{\mu }(s,v)={\mu (\parallel s\parallel }^t+{\parallel v\parallel }^t),s,v\in V_0,\mu \in {\mathbb{R}}_{+},$$

and

$$\mathcal{T}=\{{\xi }_{\mu }\in {\mathbb{R}}_{+}^{V_0\times V_0}:\mu \in {\mathbb{R}}_{+}\}.$$

Quite recently, an outcome more precise than Theorem 1 (but only for maps that take the real values) has been proved in [17] (with the Banach limit technique).

Let us yet add that the hyperstability phenomenon, mentioned at the end of Definition 1, occurs for the instance in the situation considered in Theorem 1 for $t<0$, which means that then every $h:V\to W$ satisfying (1) must be additive (see, e.g., [13] (Theorem 36)).

It is clear, that the notions of approximate solutions and closeness of two maps can be perceived in various ways. This means that it makes sense to consider the Ulam stability with respect to different ways of measuring distance. One of the interesting and non-classical methods of measuring distances can be introduced through the concept of m-norms that was proposed by Gähler [18] (see also [19,20,21]).

In this paper, we prove some new results concerning Ulam stability of the Davison functional equation (see [22]) in m-normed spaces. The Davison equation has the form

$$\begin{array}{c}\hfill h\left(sv\right)+h(s+v)=h(sv+s)+h\left(v\right)\end{array}$$

(5)

and can be considered for maps, e.g., from a ring into a linear space.

The first description of solutions to (5) has been provided by W. Benz [23], who has proved that every continuous solution $f:\mathbb{R}\to \mathbb{R}$ of (5) is of the form $f\left(x\right)=ax+b$ for $x\in \mathbb{R}$. Later, Girgensohn and Lajko [24] have obtained the general solution of the Davison functional equation without any regularity assumptions on f and proved that $f:\mathbb{R}\to \mathbb{R}$ satisfies (5) if and only if there exists an additive $h:\mathbb{R}\to \mathbb{R}$ and a real constant b such that $f\left(x\right)=h\left(x\right)+b$ for $x\in \mathbb{R}$. Some further results concerning solutions to (5) have been obtained in [25].

The stability of (5) (and some related issues) has been investigated in [26,27,28,29,30,31]. The most general stability result for (5) is Theorem 1 in [28], which can be stated as follows.

Theorem 2. 

Let F be a ring with unit element 1 and W be a Banach space. Let ${\eta }_0:F^2\to {\mathbb{R}}_{+}$ be such that

$$\sum _{i=1}^{\infty }{\displaystyle \frac{{\eta }_0(2^{i-1}v,2^{i-1}s+u)}{2^i}}<\infty ,v,s,u\in F.$$

Let $h:F\to W$ satisfy the inequality

$$\parallel h\left(vs\right)+h(v+s)-h(vs+v)-h\left(s\right)\parallel \le {\eta }_0(v,s),v,s\in F.$$

(6)

Then there exists a unique additive mapping $\psi :F\to W$ such that

$$\parallel h\left(6v\right)-\psi \left(v\right)-h\left(0\right)\parallel \le \sum _{i=0}^{\infty }{\displaystyle \frac{\mu \left(2^{i}v\right)}{2^i}},v\in F,$$

(7)

where

$$\begin{array}{cc}\hfill \mu \left(v\right)=& {\displaystyle \frac{1}{2}}[\eta (4v,-4v)+\eta (4v,-4v+1)+\eta (8v,-2v)\hfill \\ \hfill & +\eta (3v,0)+\eta (3v,1)+\eta (6v,0)+\eta (7v,-4v)\hfill \\ \hfill & +\eta (7v,-4v+1)+\eta (14v,-2v)],v\in F.\hfill \end{array}$$

(8)

Actually, the term $2\eta (7v,-4v)$ in (8) has in [28] the form $2\eta (7v,-v)$, but it is a mistake caused by a misprint made in the proof on p. 504 in [28] (while making a substitution in Formula (2.4)).

If ${\eta }_0$ is a constant function, i.e., (6) has the form

$$\parallel h\left(vs\right)+h(v+s)-h(vs+v)-h\left(s\right)\parallel \le c,v,s\in F,$$

with some fixed real $c>0$, then (7) becomes the following inequality:

$$\parallel h\left(6v\right)-\psi \left(v\right)-h\left(0\right)\parallel \le 9c,v\in F,$$

(9)

which is the best known result in this case even when F is the field of reals (see [27] and Corollary 1 in [28]).

Somewhat different versions of Theorem 2 have been provided in [26,29] (Theorem 2.1). The stability of (5) on restricted domains has been investigated in [31]. In [28,30,31], the authors considered stability of pexiderized versions of (5), i.e., some cases of the equation

$$\begin{array}{c}\hfill f_1\left(sv\right)+f_2(s+v)=f_3(sv+s)+f_4\left(v\right).\end{array}$$

(10)

In this paper, we prove a version of Theorem 2 for m-normed spaces, which can be seen as an extension of Theorem 1 in [28], because every norm can be considered to be a special case of the m-norm, with $m=1$ (see, e.g., [32]).

For several examples of related stability results in m-normed spaces (but for some other equations), we refer to [32,33,34,35,36,37,38,39,40] (and the references therein).

For more information on solutions to the functional equations that are considered in this paper, we refer to monographs [41,42,43,44].

Finally, let us explain that $\mathbb{R}$ always denotes the set of real numbers, ${\mathbb{R}}_0:=\mathbb{R}\setminus \left\{0\right\}$, ${\mathbb{R}}_{+}=[0,\infty )$, $\mathbb{Q}$ means the set of rational numbers, $\mathbb{Z}$ stands for the set of integers, $\mathbb{N}$ is the set of positive integers, and ${\mathbb{N}}_0:=\mathbb{N}\cup \left\{0\right\}$.

2. Basic Information on m-Normed Spaces

The idea of m-normed spaces (with $m\in \mathbb{N}$) was presented by S. Gähler [18] and next studied by A. Misiak [21]. It generalizes an earlier notion of 2-normed spaces (i.e., m-normed spaces with $m=2$) introduced by S. Gähler [19,20] (see also [45]).

Let us remind the reader of some basic information on m-normed spaces (see, e.g., [21,32,36,37,38,46] for further details).

Let $m\in \mathbb{N}$ and $\mathcal{P}$ be a real linear space. Let the dimension of $\mathcal{P}$ be at least m, and $\parallel ·,\dots ,·\parallel$ be a map from ${\mathcal{P}}^m$ to ${\mathbb{R}}_{+}$ satisfying, for all $\nu \in \mathbb{R}$ and $b_1,b_2,a_1,\dots ,a_n\in \mathcal{P}$, the subsequent four conditions:

(A)

$\parallel a_1,\dots ,a_m\parallel =0$ if and only if the vectors $a_1,\dots ,a_m$ are linearly dependent;

(B)

the value of $\parallel a_1,\dots ,a_m\parallel$ does not depend on a permutation of $a_1,\dots ,a_m$;

(C)

$\parallel \nu a_1,\dots ,a_m\parallel =\left|\nu \right|\parallel a_1,\dots ,a_m\parallel$;

(D)

$\parallel b_1+b_2,a_2,\dots ,a_m\parallel \le \parallel b_1,a_2,\dots ,a_m\parallel +\parallel b_2,a_2,\dots ,a_m\parallel$.

Then, $\parallel ·,\dots ,·\parallel$ is said to be an m-norm on $\mathcal{P}$, and the pair $(\mathcal{P},\parallel ·,\dots ,·\parallel )$ is called an m-normed space. If $m=1$, then it is easily seen that conditions (A)–(D) depict a classical normed space (cf., e.g., [32] (Remark 2)).

If $<·,·>$ is a real inner product in a real linear space $\mathcal{P}$ that is at least m-dimensional, then we can define an m-norm on $\mathcal{P}$ by:

$${\parallel a_1,\dots ,a_m\parallel }_S=\mathrm{abs}\left({\left|\begin{array}{cccc}<a_1,a_1>& <a_1,a_2>& \cdots & <a_1,a_m>\\ ⋮& ⋮& \ddots & ⋮\\ <a_m,a_1>& <a_m,a_2>& \cdots & <a_m,a_m>\end{array}\right|}^{1/2}\right)$$

for $a_1,\dots ,a_m\in \mathcal{P}$, where $\mathrm{abs}\left(d\right)$ stands for the module (absolute value) of a real number d (we refer to [32] (Remark 3) for further related information).

If $\mathcal{P}={\mathbb{R}}^m$ is endowed with the usual inner product, this m-norm takes the form:

$${\parallel a_1,\dots ,a_m\parallel }_E=|det\left(a_{ij}\right)|,a_i=(a_{i1},\dots ,a_{in})\in {\mathbb{R}}^m,i\in 1,\dots ,m,$$

where

$$det\left(a_{ij}\right)=\left|\begin{array}{cccc}{a}_{11}& a_{12}& \cdots & a_{1m}\\ ⋮& ⋮& \ddots & ⋮\\ a_{n1}& a_{m2}& \cdots & a_{mm}\end{array}\right|.$$

It is called the Euclidean m-norm on ${\mathbb{R}}^m$. Moreover, if $\parallel ·,\dots ,·\parallel$ is an m-norm in ${\mathbb{R}}^m$, then $\parallel a_1,\dots ,a_m\parallel =c\parallel a_1,\dots ,a_m{\parallel }_E$ for $a_1,\dots ,a_m\in {\mathbb{R}}^m$, with some positive real constant c (see, e.g., [18] (Satz 3) or [32] (Remark 2)).

Theorem 1 in [32] states that, for each $m\in \mathbb{N}$ and every real linear space X that is at least m-dimensional, there exists an m-norm on X. Moreover, in the proof of this theorem, it is shown how to define such an m-norm.

Remark 1. 

H. Gunawan and M. Mashadi [46] showed that, when $m>1$, every m-norm yields an $(m-1)$-norm and finally also a norm.

Let us also mention that in [39] (Remark 2), the following formula has been given to define a norm in an m-normed space $(\mathcal{P},\parallel ·,\dots ,·\parallel )$:

$$\begin{array}{c}\hfill {\parallel x\parallel }_d=\sum _{i=1}^m\parallel c_{1i},\dots ,c_{mi}\parallel ,x\in \mathcal{P},\end{array}$$

(11)

where $d=(d_1,\dots ,d_m)\in {\mathcal{P}}^m$ is a sequence of m linearly independent vectors in $\mathcal{P}$ and

$$c_{ij}=\left\{\begin{array}{cc}x,\hfill & i=j\hfill \\ d_i,\hfill & i\ne j,\hfill \end{array}\right.i,j=1,\dots ,m.$$

Therefore, every m-norm in $\mathcal{P}$ generates a large family of norms in $\mathcal{P}$.

In what follows, $(\mathcal{P},\parallel ·,\dots ,·\parallel )$ always denotes an m-normed space and we assume that $m>1$. To simplify some formulas, we write

$$\parallel x,z\parallel :=\parallel x,z_1,\dots ,z_{m-1}\parallel ,x\in \mathcal{P},z=(z_1,\dots ,z_{m-1})\in {\mathcal{P}}^{m-1}.$$

We need the following definitions and properties.

Definition 2. 

A sequence ${\left(a_k\right)}_{k\in \mathbb{N}}$ in $\mathcal{P}$ is said to be a Cauchy sequence if

$$\underset{n,k\to \infty }{lim}\parallel a_n-a_k,w\parallel =0,w\in {\mathcal{P}}^{m-1}.$$

A sequence ${\left(a_k\right)}_{k\in \mathbb{N}}$ in $\mathcal{P}$ is convergent if there exists such $a\in \mathcal{P}$ that

$$\underset{k\to \infty }{lim}\parallel a_k-a,w\parallel =0,w\in {\mathcal{P}}^{m-1}.$$

This limit is unique, it is called the limit of the sequence ${\left(a_k\right)}_{k\in \mathbb{N}}$, and we denote it by ${lim}_{k\to \infty }{a}_k$ (we write $a={lim}_{k\to \infty }{a}_k$).

An m-normed space is an m-Banach space if each Cauchy sequence in it is convergent. The following properties have been stated in [38] (see also [36]).

Lemma 1. 

The subsequent four statements are valid.

(i) 

If ${\left(a_k\right)}_{k\in \mathbb{N}}$ is a convergent sequence in $\mathcal{P}$, then

$$\underset{k\to \infty }{lim}\parallel a_k,w\parallel =\parallel \underset{k\to \infty }{lim}{a}_k,w\parallel ,w\in {\mathcal{P}}^{m-1}.$$

(ii) 

If $s,v\in \mathcal{P}$ and $w\in {\mathcal{P}}^{m-1}$, then

$$|\parallel s,w\parallel -\parallel v,w\parallel |\le \parallel s-v,w\parallel .$$

(iii) 

If $s\in \mathcal{P}$ and

$$\parallel s,w\parallel =0,w\in {\mathcal{P}}^{m-1},$$

then $s=0.$

In m-normed spaces, the Ulam stability of equations can be understood, e.g., in the following way.

Definition 3. 

Let V and D be sets that are nonempty. Let ${\mathcal{U}}_0\subset \mathcal{U}\subset {\mathcal{P}}^V$ and $\mathcal{B}\subset {{\mathbb{R}}_{+}}^{D\times {\mathcal{P}}^{m-1}}$ be nonempty, $\mathcal{S}:\mathcal{B}\to {{\mathbb{R}}_{+}}^{V\times {\mathcal{P}}^{m-1}}$ and ${\mathcal{F}}_1,{\mathcal{F}}_2:\mathcal{U}\to {\mathcal{P}}^D$. The equation

$$\begin{array}{c}\hfill {\mathcal{F}}_{1}g={\mathcal{F}}_{2}g\end{array}$$

(12)

is $\mathcal{S}$-stable in ${\mathcal{U}}_0$ provided, for any $g\in {\mathcal{U}}_0$ and $\xi \in \mathcal{B}$ with

$$\begin{array}{c}\hfill \parallel \left({\mathcal{F}}_{1}g\right)\left(s\right)-\left({\mathcal{F}}_{2}g\right)\left(s\right),z\parallel \le \xi (s,z),s\in D,z\in {\mathcal{P}}^{m-1},\end{array}$$

(13)

there exists a solution $\psi \in \mathcal{U}$ of functional Equation (12) such that

$$\parallel g\left(v\right)-\psi \left(v\right),z\parallel \le \left(\mathcal{S}\xi \right)(v,z),v\in V,z\in {\mathcal{P}}^{m-1}.$$

For further information on Ulam stability in m-normed spaces, we refer to, e.g., [35,36,37,38]. The next proposition shows that, in some cases, the situation is nearly trivial (cf. [35]).

Proposition 1. 

Let V, D, ${\mathcal{U}}_0\subset \mathcal{U}$, and ${\mathcal{F}}_1,{\mathcal{F}}_2$ be as in Definition 3. Let $\xi :D\times {\mathcal{P}}^{m-1}\to {\mathbb{R}}_{+}$ and $g\in {\mathcal{U}}_0$ satisfy inequality (13). Assume that the following hypothesis holds:$(\mathcal{H}$) for every $s\in D$ and $z\in {\mathcal{P}}^{m-1}$, there is a sequence ${\left(a_n\right)}_{n\in \mathbb{N}}$ in $\mathbb{R}\setminus \left\{0\right\}$ with

$$\begin{array}{c}\hfill \underset{n\to \infty }{lim}{\displaystyle \frac{\xi (s,a_{n}z)}{|a_n{|}^{m-1}}}=0,\end{array}$$

(14)

where $az=a(z_1,\dots ,z_{m-1})=(a{z}_1,\dots ,a{z}_{m-1})$ for $a\in \mathbb{R}$, $z=(z_1,\dots ,z_{m-1})\in {\mathcal{P}}^{m-1}$.

Then (12) is valid.

Proof. 

Fix $s\in D$ and $z\in {\mathcal{P}}^{m-1}$. Let ${\left(a_n\right)}_{n\in \mathbb{N}}$ be a sequence in $\mathbb{R}$ that satisfies condition (14). It follows from (13) that, for each $n\in \mathbb{N}$,

$$\begin{array}{c}\hfill |a_n{|}^{m-1}\parallel \left({\mathcal{F}}_{1}g\right)\left(s\right)-\left({\mathcal{F}}_{2}g\right)\left(s\right),z\parallel =\parallel \left({\mathcal{F}}_{1}g\right)\left(s\right)-\left({\mathcal{F}}_{2}g\right)\left(s\right),a_{n}z\parallel \le \xi (s,a_{n}z),\end{array}$$

whence

$$\begin{array}{c}\hfill \parallel \left({\mathcal{F}}_{1}g\right)\left(s\right)-\left({\mathcal{F}}_{2}g\right)\left(s\right),z\parallel \le {\displaystyle \frac{\xi (s,a_{n}z)}{|a_n{|}^{m-1}}}.\end{array}$$

Now, letting $n\to \infty$, we get $\parallel \left({\mathcal{F}}_{1}g\right)\left(s\right)-\left({\mathcal{F}}_{2}g\right)\left(s\right),z\parallel =0$.

Thus, we have shown that

$$\begin{array}{c}\hfill \parallel \left({\mathcal{F}}_{1}g\right)\left(s\right)-\left({\mathcal{F}}_{2}g\right)\left(s\right),z\parallel =0,s\in D,z\in {\mathcal{P}}^{m-1},\end{array}$$

which means that ${\mathcal{F}}_{1}g={\mathcal{F}}_{2}g$. □

Note that in the case where $\xi$ in (13) does not depend on z (i.e., $\xi (s,z_1)=\xi (s,z_2)$ for every $s\in D$ and $z_1,z_2\in {\mathcal{P}}^{m-1}$) condition (14) is fulfilled with, e.g., $a_n=n$ for $n\in \mathbb{N}$.

Let F be a ring. Clearly, under the assumptions as in Definition 3, Equation (12) becomes (5), i.e., the equation

$$\begin{array}{c}\hfill h\left(sv\right)+h(s+v)=h(sv+s)+h\left(v\right),\end{array}$$

(15)

if $V=F$, $D=F^2$, $\left({\mathcal{F}}_{1}f\right)(s,v)=f\left(sv\right)+f(s+v)$ and $\left({\mathcal{F}}_{2}f\right)(s,v)=f(sv+s)+f\left(v\right)$ for $(s,v)\in F^2$, $f\in {\mathcal{P}}^F$. Consequently, Proposition 1 yields the following corollary.

Corollary 1. 

Let F be a ring and $h:F\to \mathcal{P}$ satisfy the inequality

$$\begin{array}{c}\hfill \parallel h\left(uv\right)+h(u+v)-h(uv+u)-h\left(v\right),z\parallel \le \xi (u,v,z),u,v\in F,z\in {\mathcal{P}}^{m-1},\end{array}$$

(16)

with a function $\xi :F^2\times {\mathcal{P}}^{m-1}$ fulfilling hypothesis $\left(\mathcal{H}\right)$ with $D=F^2$. Then h is a solution to Equation (15).

Remark 2. 

A very simple example of a function $\xi :F^2\times {\mathcal{P}}^{m-1}\to {\mathbb{R}}_{+}$ fulfilling hypothesis $\left(\mathcal{H}\right)$ is following:

$$\xi (s,v,z)={\xi }_0(s,v){\parallel z\parallel }_1^t,s,v\in F,z\in {\mathcal{P}}^{m-1},$$

where ${\xi }_0:F^2\times {\mathcal{P}}^{m-1}\to {\mathbb{R}}_{+}$ can be any function, $t\in {\mathbb{R}}_{+}$, $t\ne 1$, and ${\parallel ·\parallel }_1$ is an $(m-1)$-norm in $\mathcal{P}$ (see Remark 1).

In the rest of the paper we will consider more involved situations.

3. Auxiliary Results

If C is a nonempty set and $h:C\to C$, then $h^k:C\to C$ (for $k\in {\mathbb{N}}_0$) is defined by:

$$h^0\left(a\right)=a,h^{k+1}\left(a\right)=h\left(h^k\left(a\right)\right),a\in C,k\in {\mathbb{N}}_0.$$

We need the following stability result from [40] (Corollary 10), concerning a functional equation in one variable.

Corollary 2. 

Let D be a nonempty set, $l\in \mathbb{N}$, $L_j:D\to {\mathbb{R}}_{+}$ for $j=1,\dots ,l$, $\mathsf{\Psi }:D\times {\mathcal{P}}^l\to \mathcal{P}$, and

$$\parallel \mathsf{\Psi }(s,v_1,...,v_l)-\mathsf{\Psi }(s,u_1,...,u_l),w\parallel \le \sum _{i=1}^l{L}_i\left(s\right)\parallel v_i-u_i,w\parallel$$

(17)

for any $s\in D$, $w\in {\mathcal{P}}^{m-1}$ and $(v_1,...,v_l),(u_1,...,u_l)\in {\mathcal{P}}^l$. Assume that $\epsilon :D\times {\mathcal{P}}^{m-1}\to {\mathbb{R}}_{+}$ is such that

$${\epsilon }^{*}(s,w):=\sum _{j=0}^{\infty }{\mathsf{\Lambda }}^j\epsilon (s,w)<\infty ,s\in D,w\in {\mathcal{P}}^{m-1},$$

(18)

where $\mathsf{\Lambda }:{\mathbb{R}}_{+}^{D\times {\mathcal{P}}^{m-1}}\to {\mathbb{R}}_{+}^{D\times {\mathcal{P}}^{m-1}}$ is given by

$$\mathsf{\Lambda }\delta (s,w)=\sum _{k=1}^l{L}_k\left(s\right)\delta (f_k\left(s\right),w),\delta \in {\mathbb{R}}_{+}^{D\times {\mathcal{P}}^{m-1}},s\in D,w\in {\mathcal{P}}^{m-1},$$

(19)

with some $f_j:D\to D$ for $j=1,\dots ,l$. Let $\phi :D\to \mathcal{P}$ be such that

$$\parallel \phi \left(s\right)-\mathsf{\Psi }(s,\phi \left(f_1\left(s\right)\right),...,\phi \left(f_l\left(s\right)\right)),w\parallel \le \epsilon (s,w),s\in D,w\in {\mathcal{P}}^{m-1}.$$

(20)

Then, the limit

$$\psi \left(s\right):=\underset{n\to \infty }{lim}{\mathcal{L}}^n\phi \left(s\right)$$

(21)

exists for each $s\in D$ with

$$\mathcal{L}\phi \left(s\right):=\mathsf{\Psi }(s,\phi \left(f_1\left(s\right)\right),...,\phi \left(f_l\left(s\right)\right)),\phi \in {\mathcal{P}}^D,s\in D,$$

(22)

and the function $\psi :D\to \mathcal{P}$ defined by (21) is a unique solution of the functional equation

$$\mathsf{\Psi }(s,\psi \left(f_1\left(s\right)\right),...,\psi \left(f_l\left(s\right)\right))=\psi \left(s\right),s\in D,$$

(23)

such that

$$\parallel \phi \left(s\right)-\psi \left(s\right),w\parallel \le {\epsilon }^{*}(s,w),s\in D,w\in {\mathcal{P}}^{m-1}.$$

(24)

We need this corollary in the case where Equation (23) has the following very simple form:

$$a\psi \left(B\right(s\left)\right)=\psi \left(s\right),s\in D,$$

(25)

with some fixed $a\in \mathbb{R}$, $a\ne 0$, and $B:D\to D$. Then, the corollary can be stated as follows.

Corollary 3. 

Assume that $a\in \mathbb{R}$, $a\ne 0$, $B:D\to D$ and $\epsilon :D\times {\mathcal{P}}^{m-1}\to {\mathbb{R}}_{+}$ are such that

$${\epsilon }^{*}(s,w):=\sum _{j=0}^{\infty }{\left|a\right|}^j\epsilon (B^j\left(s\right),w)<\infty ,s\in D,w\in {\mathcal{P}}^{m-1}.$$

(26)

Let $\phi :D\to \mathcal{P}$ satisfy the inequality

$$\parallel \phi \left(s\right)-a\phi \left(B\left(s\right)\right),w\parallel \le \epsilon (s,w),s\in D,w\in {\mathcal{P}}^{m-1}.$$

(27)

Then, the limit

$$\psi \left(s\right):=\underset{n\to \infty }{lim}{a}^n\phi \left(B^n\left(s\right)\right)$$

(28)

exists for each $s\in D$ and the function $\psi :D\to \mathcal{P}$ defined by (28) is a unique solution of the functional equation

$$\psi \left(s\right)=a\psi \left(B\right(s\left)\right),s\in D,$$

(29)

such that (24) holds.

Proof. 

It is enough to use Corollary 2 with $l=1$, $L_1\left(s\right)=a$, $\mathsf{\Psi }(s,v_1)=a{v}_1$, $f_1\left(s\right)=B\left(s\right)$, $\mathcal{L}\phi \left(s\right)=a\phi \left(B\right(s\left)\right)$, $\mathsf{\Lambda }\delta (s,w)=a\delta \left(B\right(s),w)$. □

4. Stability Results for the Davison Functional Equation

In this section, F stands for a ring with the unit element 1. We prove Ulam stability results for Equation (5), for functions $h:F\to \mathcal{P}$. In this way, we extend Theorem 2 (i.e., Theorem 1 in [28]) to the case of m-normed spaces, because each norm can be considered to be an m-norm for $m=1$ (see, e.g., [32]).

The main tool in the proofs is a stability result for an equation in a single variable, i.e., Corollary 3. Such an approach is already known (see, e.g., the proof of Corollary 2.8 in [47]), but—as far as we know—it has not been applied so far either to the Davison equation or to m-normed spaces.

We have the following.

Theorem 3. 

Let $\eta :F^2\times \mathcal{P}\to {\mathbb{R}}_{+}$ be such that

$$\mu (v,u):=\sum _{i=0}^{\infty }{\displaystyle \frac{M(2^{i}v,u)}{2^i}}<\infty ,v\in F,u\in {\mathcal{P}}^{m-1},$$

(30)

$$\underset{n\to \infty }{lim}{\displaystyle \frac{M^{\prime }(2^{n}v,2^{n}s,u)}{2^n}}=0,v,s\in F,u\in {\mathcal{P}}^{m-1},$$

(31)

where

$$\begin{array}{cc}\hfill M(v,u)=& {\displaystyle \frac{1}{2}}[\eta (4v,-4v,u)+\eta (4v,-4v+1,u)+\eta (8v,-2v,u)\hfill \\ \hfill & +\eta (3v,0,u)+\eta (3v,1,u)+\eta (6v,0,u)+\eta (7v,-4v,u)\hfill \\ \hfill & +\eta (7v,-4v+1,u)+\eta (14v,-2v,u)],v\in F,u\in {\mathcal{P}}^{m-1},\hfill \end{array}$$

$$\begin{array}{cc}\hfill M^{\prime }(v,s,u)=& \eta (4v+3s,-4v,u)+\eta (4v+3s,-4v+11,u)\hfill \\ \hfill & +\eta (8v+6s,-2v,u)+\eta (3v+4s,-4s,u)\hfill \\ \hfill & +\eta (3v+4s,-4s+1,u)+\eta (6v+8s,-2s,u)\hfill \\ \hfill & +2\eta (7v,-4v,u)+2\eta (7v,-4v+1,u)\hfill \\ \hfill & +2\eta (14v,-2v,u),v,s\in F,u\in {\mathcal{P}}^{m-1}.\hfill \end{array}$$

Let $h:F\to \mathcal{P}$ satisfy the inequality

$$\parallel h\left(vs\right)+h(v+s)-h(vs+v)-h\left(s\right),u\parallel \le \eta (v,s,u),v,s\in F,u\in {\mathcal{P}}^{m-1}.$$

(32)

Then, there exists a unique additive mapping $\psi :F\to \mathcal{P}$ such that

$$\parallel h\left(6v\right)-\psi \left(v\right)-h\left(0\right),u\parallel \le \mu (v,u),v\in F,u\in {\mathcal{P}}^{m-1}.$$

(33)

Proof. 

Replacing s by $s+1$ in (32), we obtain

$$\begin{array}{c}\parallel h(vs+v)+h(v+s+1)-h(vs+2v)-h(s+1),u\parallel \\ \le \eta (v,s+1,u),v\in F,u\in {\mathcal{P}}^{m-1}.\end{array}$$

(34)

Hence

$$\begin{array}{c}\parallel h\left(vs\right)+h(v+s)+h(v+s+1)-h\left(s\right)-h(vs+2v)-h(s+1),u\parallel \\ \le \parallel h\left(vs\right)+h(v+s)-h(vs+v)-h\left(s\right),u\parallel \\ +\parallel h(vs+v)+h(v+s+1)-h(vs+2v)-h(s+1),u\parallel \\ \le \eta (v,s,u)+\eta (v,s+1,u),v,s\in F,u\in {\mathcal{P}}^{m-1}.\end{array}$$

Next, replacing s by $4s$ in (34), by (32), we obtain

$$\begin{array}{c}\parallel h\left(4vs\right)+h(v+4s)+h(v+4s+1)-h\left(4s\right)-h(4vs+2v)-h(4s+1),u\parallel \\ \le \eta (v,4s,u)+\eta (v,4s+1,u),v,s\in F,u\in {\mathcal{P}}^{m-1},\end{array}$$

whence, in view of (32),

$$\begin{array}{cc}\hfill \parallel h(v+4s)+& h(v+4s+1)-h(2v+2s)-h\left(4s\right)-h(4s+1)+h\left(2s\right),u\parallel \hfill \\ \hfill \le & \parallel h\left(4vs\right)+h(v+4s)+h(v+4s+1)-h\left(4s\right)-h(4vs+2v)-h(4s+1),u\parallel \hfill \\ \hfill & +\parallel h\left(4vs\right)+h(2v+2s)-h(4vs+2v)-h\left(2s\right),u\parallel \hfill \\ \hfill \le & \eta (v,4s,u)+\eta (v,4s+1,u)+\eta (2v,2s,u),v,s\in F,u\in {\mathcal{P}}^{m-1}.\hfill \end{array}$$

Now, replacing v by $v-s$ in the last inequality, we have

$$\begin{array}{cc}\hfill \parallel h(v+3s)& +h(v+3s+1)-h\left(2v\right)-h\left(4s\right)-h(4s+1)+h\left(2s\right),u\parallel \hfill \\ \hfill & \le \eta (v-s,4s,u)+\eta (v-s,4s+1,u)+\eta (2v-2s,2s,u),v,s\in F,u\in {\mathcal{P}}^{m-1},\hfill \end{array}$$

which, with $3v$ instead of v, yields

$$\begin{array}{cc}\hfill \parallel h(3v+3s)& +h(3v+3s+1)-h\left(6v\right)-h\left(4s\right)-h(4s+1)+h\left(2s\right),u\parallel \hfill \\ \hfill \le & \eta (3v-s,4s,u)+\eta (3v-s,4s+1,u)\hfill \\ \hfill & +\eta (6v-2s,2s,u),v,s\in F,u\in {\mathcal{P}}^{m-1}.\hfill \end{array}$$

(35)

Further, note that (35) with $s=-v$ becomes the inequality

$$\begin{array}{cc}\hfill \parallel h\left(0\right)+h\left(1\right)& -h\left(6v\right)-h(-4v)-h(-4v+1)+h(-2v),u\parallel \hfill \\ \hfill \le & \eta (4v,-4v,u)+\eta (4v,-4v+1,u)\hfill \\ & +\eta (8v,-2v,u),v\in F,u\in {\mathcal{P}}^{m-1},\hfill \end{array}$$

(36)

and (35), with $s=0$, is

$$\begin{array}{cc}\hfill \parallel h\left(3v\right)& +h(3v+1)-h\left(6v\right)-h\left(1\right),u\parallel \hfill \\ \hfill & \le \eta (3v,0,u)+\eta (3v,1,u)+\eta (6v,0,u),v\in F,u\in {\mathcal{P}}^{m-1}.\hfill \end{array}$$

(37)

Also, if we replace v by $2v$ in (35) and take $s=-v$, we obtain

$$\begin{array}{cc}\hfill \parallel h\left(3v\right)+h(3v+1)& -h\left(12v\right)-h(-4v)-h(-4v+1)+h(-2v),u\parallel \hfill \\ \hfill \le & \eta (7v,-4v,u)+\eta (7v,-4v+1,u)\hfill \\ \hfill & +\eta (14v,-2v,u),v\in F,u\in {\mathcal{P}}^{m-1}.\hfill \end{array}$$

(38)

Define $\xi :F\to \mathcal{P}$ by $\xi \left(v\right)=h\left(6v\right)-h\left(0\right)$ for $v\in F$. Then, from (36)–(38), it follows that

$$\begin{array}{cc}\hfill \parallel \xi \left(2v\right)-2& \xi \left(v\right),u\parallel =\parallel h\left(12v\right)-h\left(0\right)-2\left(h\right(6v)-h(0\left)\right),u\parallel \hfill \\ \hfill =& \parallel 2h\left(6v\right)-h\left(12v\right)-h\left(0\right),u\parallel \hfill \\ \hfill \le & \parallel h\left(0\right)+h\left(1\right)-h\left(6v\right)-h(-4v)-h(-4v+1)+h(-2v),u\parallel \hfill \\ \hfill & +\parallel h\left(3v\right)+h(3v+1)-h\left(6v\right)-h\left(1\right),u\parallel \hfill \\ \hfill & +\parallel h\left(3v\right)+h(3v+1)-h\left(12v\right)-h(-4v)-h(-4v+1)+h(-2v),u\parallel \hfill \\ \hfill \le & \eta (4v,-4v,u)+\eta (4v,-4v+1,u)+\eta (8v,-2v,u)\hfill \\ \hfill & +\eta (3v,0,u)+\eta (3v,1,u)+\eta (6v,0,u)\hfill \\ \hfill & +\eta (7v,-4v,u)+\eta (7v,-4v+1,u)\hfill \\ \hfill & +\eta (14v,-2v,u),v\in F,u\in {\mathcal{P}}^{m-1},\hfill \end{array}$$

(39)

which implies that

$$\begin{array}{c}\hfill \parallel {\displaystyle \frac{1}{2}}\xi \left(2v\right)-\xi \left(v\right),u\parallel \le M(v,u),v\in F,u\in {\mathcal{P}}^{m-1}.\end{array}$$

(40)

Note that, by (30), condition (26) is fulfilled with $\epsilon =M$, $a={\displaystyle \frac{1}{2}}$ and $B\left(v\right)=2v$. So, by Corollary 3, the limit

$$\psi \left(v\right):=\underset{n\to \infty }{lim}{\displaystyle \frac{1}{2^n}}\xi \left(2^{n}v\right)$$

(41)

exists for each $v\in F$ and the function $\psi :F\to \mathcal{P}$, defined by (41), is a unique solution of the functional equation

$$\psi \left(v\right)={\displaystyle \frac{1}{2}}\psi \left(2v\right),v\in F,$$

(42)

such that

$$\parallel \psi \left(v\right)-\xi \left(v\right),u\parallel \le \mu (v,u),v\in F,u\in {\mathcal{P}}^{m-1}.$$

(43)

This and the definition of $\xi$ imply (33).

Now, we show that $\psi$ is additive. To this end, replace v and s in (35), respectively, by $v+s$ and $-v$, $v+s$ and $-s$, $2s$ and $-s$, $2v$ and $-v$. Then, we obtain the following four inequalities:

$$\begin{array}{cc}\hfill \parallel h\left(3s\right)& +h(3s+1)-h(6v+6s)-h(-4v)-h(-4v+1)+h(-2v),u\parallel \hfill \\ \hfill & \le \eta (4v+3s,-4v,u)+\eta (4v+3s,-4v+1,u)+\eta (8v+6s,-2v,u),\hfill \end{array}$$

(44)

$$\begin{array}{cc}\hfill \parallel h\left(3v\right)& +h(3v+1)-h(6v+6s)-h(-4s)-h(-4s+1)+h(-2s),u\parallel \hfill \\ \hfill & \le \eta (3v+4s,-4s,u)+\eta (3v+4s,-4s+1,u)+\eta (6v+8s,-2s,u),\hfill \end{array}$$

(45)

$$\begin{array}{cc}\hfill \parallel h\left(3s\right)& +h(3s+1)-h\left(12s\right)-h(-4s)-h(-4s+1)+h(-2s),u\parallel \hfill \\ \hfill & \le \eta (7s,-4s,u)+\eta (7s,-4s+1,u)+\eta (14s,-2s,u),\hfill \end{array}$$

(46)

$$\begin{array}{cc}\hfill \parallel h\left(3v\right)& +h(3v+1)-h\left(12v\right)-h(-4v)-h(-4v+1)+h(-2v),u\parallel \hfill \\ \hfill & \le \eta (7v,-4v,u)+\eta (7v,-4v+1,u)+\eta (14v,-2v,u)\hfill \end{array}$$

(47)

for every $v,s\in F$, $u\in {\mathcal{P}}^{m-1}$.

Note that (44)–(47) yield

$$\begin{array}{cc}\hfill \parallel 2h(6v& +6s)-h(12v)-h(12s),u\parallel \hfill \\ \hfill \le & \parallel h(6v+6s)-h\left(3s\right)-h(3s+1)+h(-4v)+h(-4v+1)-h(-2v),u\parallel \hfill \\ \hfill & +\parallel h\left(3s\right)+h(3s+1)-h\left(12s\right)-h(-4s)-h(-4s+1)+h(-2s),u\parallel \hfill \\ \hfill & +\parallel h(6v+6s)-h\left(3v\right)-h(3v+1)+h(-4s)+h(-4s+1)-h(-2s),u\parallel \hfill \\ \hfill & +\parallel h\left(3v\right)+h(3v+1)-h\left(12v\right)-h(-4v)-h(-4v+1)+h(-2v),u\parallel \hfill \\ \hfill =& \parallel h\left(3s\right)+h(3s+1)-h(6v+6s)-h(-4v)-h(-4v+1)+h(-2v),u\parallel \hfill \\ \hfill & +\parallel h\left(3v\right)+h(3v+1)-h(6v+6s)-h(-4s)-h(-4s+1)+h(-2s),u\parallel \hfill \\ \hfill & +\parallel h\left(3s\right)+h(3s+1)-h\left(12s\right)-h(-4s)-h(-4s+1)+h(-2s),u\parallel \hfill \\ \hfill & +\parallel h\left(3v\right)+h(3v+1)-h\left(12v\right)-h(-4v)-h(-4v+1)+h(-2v),u\parallel \hfill \\ \hfill \le & \eta (4v+3s,-4v,u)+\eta (4v+3s,-4v+1,u)+\eta (8v+6s,-2v,u)\hfill \\ \hfill & +\eta (3v+4s,-4s,u)+\eta (3v+4s,-4s+1,u)+\eta (6v+8s,-2s,u)\hfill \\ \hfill & +\eta (7s,-4s,u)+\eta (7s,-4s+1,u)+\eta (14s,-2v,u)\hfill \\ \hfill & +\eta (7v,-4v,u)+\eta (7v,-4v+1,u)+\eta (14v,-2v,u)=M^{\prime }(v,s,u)\hfill \end{array}$$

(48)

for every $v,s\in F$, $u\in {\mathcal{P}}^{m-1}$. Replacing $v,s$ by $2^{n}v,2^{n}s$ in (48) and then dividing the resulting inequality by $2^n$, we get:

$$\begin{array}{cc}\hfill \parallel {\displaystyle \frac{h\left(2^n(6v+6s)\right)}{2^{n-1}}}& -{\displaystyle \frac{h(2^n·12v)}{2^n}}-{\displaystyle \frac{h(2^n·12s)}{2^n}},u\parallel \hfill \\ \hfill & \le {\displaystyle \frac{M^{\prime }(2^{n}v,2^{n}s,u)}{2^n}},v\in F,u\in {\mathcal{P}}^{m-1},n\in \mathbb{N},\hfill \end{array}$$

whence, by (31) and the definition of $\psi$, it follows that

$$2\psi (v+s)-\psi \left(2v\right)-\psi \left(2s\right)=0,v,s\in F.$$

Consequently, in view of (42), $\psi (v+s)=\psi \left(v\right)+\psi \left(s\right)$ for every $v,s\in F$.

It remains to show that $\psi$ is unique. Therefore, let $\beta :F\to \mathcal{P}$ also be an additive mapping satisfying the inequality

$$\parallel h\left(6v\right)-\beta \left(v\right)-h\left(0\right),u\parallel \le \mu (v,u),v\in F,u\in {\mathcal{P}}^{m-1}.$$

Then, by the additivity of $\psi$ and $\beta$,

$$\begin{array}{cc}\hfill \parallel \psi \left(v\right)-\beta \left(v\right),u\parallel & ={\displaystyle \frac{1}{2^n}}\parallel \psi \left(2^{n}v\right)-\beta \left(2^{n}v\right),u\parallel \hfill \\ \hfill & \le 2{\displaystyle \frac{\mu (2^{n}v,u)}{2^n}}=2\sum _{i=n}^{\infty }{\displaystyle \frac{M(2^{i}v,u)}{2^i}},v\in F,u\in \mathcal{P},n\in \mathbb{N},\hfill \end{array}$$

whence, letting $n\to \infty$, we obtain that $\psi =\beta$. □

We say that F is uniquely divisible by 2 if, for every $v\in F$, there exists a unique $s_v\in F$ such that $v=2s_v$. In what follows, we write $2^{0}v=v$, $2^{-1}v:=s_v$, and $2^{-n-1}v:=2^{-1}\left(2^{-n}v\right)$ for $n\in \mathbb{N}$.

The next theorem complements Theorem 3.

Theorem 4. 

Let F be uniquely divisible by 2 and $\eta :F^2\times \mathcal{P}\to {\mathbb{R}}_{+}$ be such that

$$\mu (v,u):=\sum _{i=1}^{\infty }{2}^{i}M(2^{-i}v,u)<\infty ,v\in F,u\in {\mathcal{P}}^{m-1},$$

(49)

$$\underset{n\to \infty }{lim}{2}^n{M}^{\prime }(2^{-n}v,2^{-n}s,u)=0,v,s\in F,u\in {\mathcal{P}}^{m-1},$$

(50)

where M and $M^{\prime }$ are defined as in Theorem 3. Let $h:F\to \mathcal{P}$ satisfy inequality (32).

Then there exists a unique additive mapping $\psi :F\to \mathcal{P}$ such that (33) holds.

Proof. 

In the same way as in the proof of Theorem 3, we define $\xi :F\to \mathcal{P}$ and show that (40) is valid, which yields

$$\begin{array}{c}\hfill \parallel \xi \left(v\right)-2\xi \left(2^{-1}v\right),u\parallel \le 2M(2^{-1}v,u),v\in F,u\in {\mathcal{P}}^{m-1}.\end{array}$$

(51)

This means that (27) is valid and, by (49), condition (26) is fulfilled with $D=F$, $\epsilon (v,u)=2M(2^{-1}v,u)$, $a=2$ and $B\left(v\right)=2^{-1}v$. Hence, by Corollary 3, the limit

$$\psi \left(v\right):=\underset{n\to \infty }{lim}{2}^n\xi \left(2^{-n}v\right)$$

(52)

exists for each $v\in F$, and the function $\psi :F\to \mathcal{P}$, defined by (52), is a unique solution of the functional equation

$$\psi \left(v\right)=2\psi \left(2^{-1}v\right),v\in F,$$

(53)

such that

$$\parallel \psi \left(v\right)-\xi \left(v\right),u\parallel \le \mu (v,u),v\in F,u\in {\mathcal{P}}^{m-1},$$

(54)

which implies (33).

Again, in the same way as in the proof of Theorem 3, we show that (48) holds, i.e.,

$$\begin{array}{c}\hfill \parallel 2h(6v+6s)-h\left(12v\right)-h\left(12s\right),u\parallel \le M^{\prime }(v,s,u),v,s\in F,u\in {\mathcal{P}}^{m-1}.\end{array}$$

(55)

Replacing $v,s$ by $2^{-n}v,2^{-n}s$ in (48) and then multiplying the resulting inequality by $2^n$, we obtain

$$\begin{array}{cc}\hfill \parallel 2^{n}h\left(2^{-n}(6v+6s)\right)& -2^{n}h\left(2^{-n}\left(12v\right)\right)-2^{n}h\left(2^{-n}\left(12s\right)\right),u\parallel \hfill \\ \hfill & \le 2^n{M}^{\prime }(2^{-n}v,2^{-n}s,u),v\in F,u\in {\mathcal{P}}^{m-1},n\in \mathbb{N}.\hfill \end{array}$$

Consequently, on account of (50) and the definition of $\psi$,

$$2\psi (v+s)-\psi \left(2v\right)-\psi \left(2s\right)=0,v,s\in F,$$

whence, by (53), we obtain $\psi (v+s)=\psi \left(v\right)+\psi \left(s\right)$ for every $v,s\in F$.

To complete the proof suppose that $\beta :F\to \mathcal{P}$ also is an additive mapping with

$$\parallel h\left(6v\right)-\beta \left(v\right)-h\left(0\right),u\parallel \le \mu (v,u),v\in F,u\in {\mathcal{P}}^{m-1}.$$

Then, the additivity of $\psi$ and $\beta$ imply that,

$$\begin{array}{cc}\hfill \parallel \psi \left(v\right)-\beta \left(v\right),u\parallel & =2^n\parallel \psi \left(2^{-n}v\right)-\beta \left(2^{-n}v\right),u\parallel \hfill \\ \hfill & \le 22^n\mu (2^{-n}v,u)=2\sum _{i=n+1}^{\infty }{2}^{i}M(2^{-i}v,u),v\in F,u\in \mathcal{P},n\in \mathbb{N},\hfill \end{array}$$

whence, with $n\to \infty$, we obtain $\psi =\beta$. □

5. Some Applications and Examples

The situation is very simple when the function $\eta$ in (32) is constant with respect to the first two variables (cf. Corollary 1). Namely, we have the following corollary, which extends Corollary 1 in [28] to the case of m-normed spaces.

Corollary 4. 

Let $ϵ:{\mathcal{P}}^{m-1}\to {\mathbb{R}}_{+}$. If a function $h:F\to \mathcal{P}$ satisfies the inequality

$$\parallel h\left(vs\right)+h(v+s)-h(vs+v)-h\left(s\right),u\parallel \le ϵ\left(u\right),v,s\in F,u\in {\mathcal{P}}^{m-1},$$

then there exists a unique additive function $\psi :F\to \mathcal{P}$ such that

$$\parallel h\left(6v\right)-\psi \left(v\right)-h\left(0\right),u\parallel \le 9ϵ\left(u\right),v\in F,u\in {\mathcal{P}}^{m-1}.$$

Proof. 

It is enough to use Theorem 3 with $\eta (v,s,u)\equiv ϵ\left(u\right)$ and notice that $M(v,u)\equiv {\displaystyle \frac{9}{2}}ϵ\left(u\right)$, which implies

$$\mu (v,u)=\sum _{n=0}^{\infty }{2}^{-n}M(2^{n-1}v,2^{n-1}s+z,u)={\displaystyle \frac{9}{2}}\sum _{n=0}^{\infty }{2}^{-n}ϵ\left(u\right)=9ϵ\left(u\right)$$

for $v\in F$ and $u\in {\mathcal{P}}^{m-1}$. □

A somewhat involved example of function $\eta :F^2\times {\mathcal{P}}^{m-1}\to {\mathbb{R}}_{+}$ satisfying (30) and (31) is:

$$\begin{array}{c}\hfill \eta (v,s,u)=\chi (v,s)\rho \left(u\right),v,s\in F,u\in {\mathcal{P}}^{m-1},\end{array}$$

(56)

where $\rho :{\mathcal{P}}^{m-1}\to {\mathbb{R}}_{+}$ is an arbitrary mapping (cf. Remark 2) and $\chi :F^2\to {\mathbb{R}}_{+}$ satisfies the following conditions:

$$\begin{array}{c}\hfill \chi (v,s+z)\le \chi (v,s)+\chi (v,z),v,s,z\in F,\end{array}$$

(57)

$$\begin{array}{c}\hfill \chi (kv,s)={\left|k\right|}^p\chi (v,s),\chi (v,ks)={\left|k\right|}^q\chi (v,s),v,s\in F,k\in \mathbb{Z},k\ne 0,\end{array}$$

(58)

with some $p,q\in \mathbb{R}$, $p+q<1$ and $p<1$.

In fact, it is easily seen that (58) implies that $\chi (v,0)=0$ for each $v\in F$, whence using (57) we find

$$\begin{array}{cc}\hfill M(v,u)=& {\displaystyle \frac{1}{2}}[\eta (4v,-4v,u)+\eta (4v,-4v+1,u)+\eta (8v,-2v,u)\hfill \\ \hfill & +\eta (3v,1,u)+\eta (7v,-4v,u)+\eta (7v,-4v+1,u)+\eta (14v,-2v,u)]\hfill \\ \hfill =& {\displaystyle \frac{1}{2}}[4^p{4}^q\chi (v,v)+4^p\chi (v,-4v+1)+8^p{2}^q\chi (v,v)\hfill \\ \hfill & +3^p\chi (v,1)+7^p{4}^q\chi (v,v)+7^p\chi (v,-4v+1)+{14}^p{2}^q\chi (v,v)]\rho \left(u\right)\hfill \\ \hfill =& {\displaystyle \frac{\rho \left(u\right)}{2}}[(4^p{4}^q+8^p{2}^q+7^p{4}^q+{14}^p{2}^q)\chi (v,v)+(4^p+7^p)\chi (v,-4v+1)+3^p\chi (v,1)]\hfill \end{array}$$

and consequently

$$\begin{array}{cc}\hfill \mu (v,u)=& \sum _{n=0}^{\infty }{\displaystyle \frac{M(2^{n}v,u)}{2^n}}\hfill \\ \hfill =& {\displaystyle \frac{\rho \left(u\right)}{2}}(4^p{4}^q+8^p{2}^q+7^p{4}^q+{14}^p{2}^q)\sum _{n=0}^{\infty }{2}^{-n}{2}^{(p+q)n}\chi (v,v)\hfill \\ \hfill & +{\displaystyle \frac{\rho \left(u\right)(4^p+7^p)}{2}}\sum _{n=0}^{\infty }{2}^{-n}{2}^{pn}\chi (v,-2^{n+2}v+1)+{\displaystyle \frac{3^p\rho \left(u\right)}{2}}\sum _{n=0}^{\infty }{2}^{-n}{2}^{pn}\chi (v,1)\hfill \\ \hfill \le & {\displaystyle \frac{\chi (v,v)\rho \left(u\right)(4^p{4}^q+8^p{2}^q+7^p{4}^q+{14}^p{2}^q)}{2-2^{p+q}}}+{\displaystyle \frac{3^p\chi (v,1)\rho \left(u\right)}{2-2^p}}\hfill \\ \hfill & +{\displaystyle \frac{\rho \left(u\right)(4^p+7^p)}{2}}\left(\sum _{n=0}^{\infty }{2}^{2q}{2}^{(p+q-1)n}\chi (v,v)+\sum _{n=0}^{\infty }{2}^{(p-1)n}\chi (v,1)\right)\hfill \\ \hfill =& {\displaystyle \frac{\chi (v,v)\rho \left(u\right)(2^{2(p+q)}+2^{3p+q}+7^p{2}^{2q}+7^p{2}^{p+q})}{2-2^{p+q}}}+{\displaystyle \frac{3^p\chi (v,1)\rho \left(u\right)}{2-2^p}}\hfill \\ \hfill & +{\displaystyle \frac{\chi (v,v)\rho \left(u\right)(2^{2(p+q)}+7^p{2}^{2q})}{2-2^{p+q}}}+{\displaystyle \frac{\chi (v,1)\rho \left(u\right)(2^{2p}+7^p)}{2-2^p}}\hfill \\ \hfill =& {\displaystyle \frac{\eta (v,v,u)(2^{2(p+q)+1}+2^{3p+q}+7^p{2}^{2q+1}+7^p{2}^{p+q})}{2-2^{p+q}}}\hfill \\ \hfill & +{\displaystyle \frac{\eta (v,1,u)(2^{2p}+3^p+7^p)}{2-2^p}}\hfill \end{array}$$

for $v\in F$ and $u\in {\mathcal{P}}^{m-1}$.

Analogously, it can be shown that (31) is true.

Note that, if for instance $(F,\parallel ·,·{\parallel }_2)$ is also a 2-normed space and $\chi (v,s)={\parallel v,s\parallel }_2^p$ for $v,s\in F$, then (57) and (58) are valid for $q=p\in (0,1]$.

If F is a real linear space, ${\parallel ·\parallel }_1$ and ${\parallel ·\parallel }_2$ are norms in F (see, e.g., Remark 1), and $\alpha >0$ is a fixed real number, then we can take $\chi (v,s)={\alpha \parallel v\parallel }_1^p{\parallel s\parallel }_2^q$ for $v,s\in F$, with fixed $p,q\in (0,1]$, and then (57) and (58) are fulfilled.

Clearly, one can easily find some other more involved appropriate examples of $\chi$.

6. Conclusions

In this paper, we have studied the stability of the Davison functional equation in m-Banach spaces using some useful recent Ulam stability results for a functional equation in one variable. We have also presented some examples showing particular cases of our results. In this way, we have complemented several earlier outcomes. Potential future work could be to investigate the stability of the Davison functional equation in some other spaces like, e.g., $dq$-metric spaces or b-metric spaces.