Some Comments on the Superstability of a General Functional Equation

Abstract

In this paper, we prove a superstability theorem for a general functional equation ${\sum }_{j=1}^{\infty }{a}_{j}f\left({\gamma }_j(t,s)\right)=h\left(t\right)g\left(s\right)$, with the unknown functions $g:T\to X$, $h:S\to \mathbb{K}$ and $f:S\to X$, such that the series ${\sum }_{j=1}^{\infty }f\left({\gamma }_j(t,s)\right)$ is convergent for every $(s,t)\in S\times T$, where S and T are nonempty sets, and X is a Banach space over a field $\mathbb{K}$, which is either the set of real numbers $\mathbb{R}$ or the set of complex numbers $\mathbb{C}$. Namely, we show that if h is unbounded, and the difference ${\sum }_{j=1}^{\infty }{a}_{j}f\left({\gamma }_j(t,s)\right)-h\left(t\right)g\left(s\right)$ is bounded, then h and g satisfy the equation ${\sum }_{j=1}^{\infty }{a}_{j}g\left({\gamma }_j(t,s)\right)=h\left(t\right)g\left(s\right)$. Next, we show that the superstability of pexiderizations and radical versions of several well-known functional equations (e.g., of Cauchy, d’Alembert, Wilson, Reynolds, and homogeneity) is a consequence of this simple outcome. In this way, we generalize several classical superstability results and, in particular, the first superstability outcome of J. Baker, J. Lawrence, and F. Zorzitto, which states that every unbounded mapping f, from a real vector space X into the set of real numbers $\mathbb{R}$, satisfying the inequality $\left|f\right(x+y)-f(x\left)f\right(y\left)\right|\le \delta$ for every $x,y\in X$, with some real $\delta >0$, is an exponential function, i.e., satisfies the equality $f(x+y)=f\left(x\right)f\left(y\right)$ for all $x,y\in X$. In order to make this publication more accessible to a wider range of readers, we limit various related information, avoid very abstract generalizations, and provide some simple examples.

1. Introduction

The well-known problem of Ulam stability concerns the size of the difference between the solutions and approximate solutions to various equations (e.g., difference, differential, functional, integral, etc.). This question was asked by S. M. Ulam in 1940, for the first time, with regard to homomorphisms and approximate homomorphisms of some groups. The first answer was given by D. H. Hyers [1] in Banach spaces. Further results in this area were soon published in [2,3,4] on approximate isometries and then in papers [5,6,7]. A generalization of Hyers’ result [1] was published in 1950 in [8] and rediscovered much later in [9] in a somewhat modified form.

Afterward, this problem attracted the attention of numerous researchers, and for relevant information on this topic and other references, we refer to monographs [10,11,12]. More up-to-date results can be found in [13,14] (also for difference, differential, and integral equations).

Clearly, this idea of stability has much to do with the issues studied in some other fields of mathematics, including theories of optimization, approximation, perturbation, and shadowing (see [15,16]). The following theorem (see, e.g., refs. [10,12] or [14] (Theorem 36)) can be considered to be one of the main outcomes in Ulam stability theory:

Theorem 1.

Let V and U be normed spaces, $V_0:=V\setminus \left\{0\right\}$, and $\eta \ge 0$ and $\mu \ne 1$ be real numbers. Assume that $\xi :V\to U$ satisfies the following inequality:

$$\parallel \xi (a_1+a_2)-\xi \left(a_1\right)-\xi \left(a_2\right)\parallel \le \eta (\parallel a_1{\parallel }^{\mu }+\parallel a_2{\parallel }^{\mu }),\forall a_1,a_2\in V_0.$$

(1)

Then, the following two statements are valid.

(i) 

If $\mu <0$, then ξ is additive, i.e.,

$$\begin{array}{c}\hfill \xi (a_1+a_2)=\xi \left(a_1\right)+\xi \left(a_2\right),\forall a_1,a_2\in V.\end{array}$$

(ii) 

If U is complete, and $\mu \ge 0$, then there is a unique additive mapping $A:V\to U$, such that

$$\parallel \xi \left(a\right)-A\left(a\right)\parallel \le {\displaystyle \frac{{\eta \parallel a\parallel }^{\mu }}{|1-2^{\mu -1}|}},\forall a\in V_0.$$

The case $\mu =0$ of this theorem was proved in [1]. For this reason, results concerning inequalities modeled on (1), with a constant on the right-hand side of the inequality, are usually called the Ulam–Hyers (or Hyers–Ulam) stability. The case $\mu \ne 0$ was first considered by T. Aoki [8], who proved the result contained in Theorem 1 for $\mu \in [0,1)$. His result was later rediscovered by Th. M. Rassias [9], in a somewhat different form. The paper of Rassias [9] became so influential that results of this type, concerning inequalities modeled on (1), with certain functions on the right-hand side of the inequality, came to be quite often referred to as Ulam–Hyers–Rassias stability (see, e.g., [10,11,12]). Sometimes, the names of Aoki, Bourgain, Gavruţa, and Forti were also added there (see [10,11,12] for some results of those authors and the relevant references). A somewhat different terminology was used, e.g., in [14,17,18].

Recently, results much finer than those in Theorem 1 have been obtained in [19], but only for mappings taking real values.

During research in this area, a special phenomenon was discovered, which is now called superstability. The first such result was found by J. Baker, J. Lawrence, and F. Zorzitto [17] (see also [20,21,22]) while studying the Ulam stability of the exponential equation

$$f(x+y)=f\left(x\right)f\left(y\right).$$

(2)

Namely, they proved that if f is a function mapping a real vector space X into the set of real numbers $\mathbb{R}$, and

$$\left|f\right(x+y)-f(x\left)f\right(y\left)\right|\le \delta ,\forall x,y\in X,$$

with some real $\delta >0$, then f is bounded or must be an exponential function (i.e., $f(x+y)=f\left(x\right)f\left(y\right)$ for all $x,y\in X$). It is exactly this property that has usually been named superstability.

Afterward, numerous other similar results were obtained for various equations. Some of them concern the so-called pexiderizations of several well-known equations. In the case of the Cauchy exponential Equation (2) such pexiderization means the following equation:

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

For example, such results have already been obtained in [18,23,24] for the following well-known functional equations:

$$\psi (s+t)+\psi (s-t)=2\psi \left(t\right)\psi \left(s\right),$$

(3)

$$\psi (s+t)+\psi (s-t)=h\left(t\right)\psi \left(s\right),$$

(4)

$$\begin{array}{c}\hfill \psi {\left({\displaystyle \frac{s+t}{2}}\right)}^2-\psi {\left({\displaystyle \frac{s-t}{2}}\right)}^2=\psi \left(t\right)\psi \left(s\right),\end{array}$$

(5)

$$\begin{array}{c}\hfill \psi {\left({\displaystyle \frac{s+t}{2}}\right)}^2-\psi {\left({\displaystyle \frac{s-t}{2}}\right)}^2=h\left(t\right)\psi \left(s\right).\end{array}$$

(6)

More involved and general outcomes in this area have been proved in [22,25]. For more abstract versions of them, we refer to [20,26,27,28]. More recent related results have been published in [28,29,30]. Various similar outcomes for several other equations can be found in [31] (for the Dhombres equation), refs. [32,33] (for the Gołąb-Schinzel equation), ref. [34] (for some systems of functional equations for generalized hyperbolic functions and generalized cosine function), refs. [35,36,37] (for the homogeneity equation), ref. [38] (for a Cauchy type equation), ref. [39] (for the d’Alembert type functional equation), and [37,40] (for other equations). Let us add that, in many of those results, the constants on the right-hand side of the considered inequalities have been replaced by functions of one variable.

Equations (3) and (5) are generally called the d’Alembert (or cosine) and Wilson (or sine) equations, respectively. Equation (6) is sometimes called the first generalization of the Wilson equation (see, e.g., [41]); Equation (4) was introduced by Wilson [42]. For more information related to those equations, we refer, e.g., to [41,43,44].

The Wilson (or sine) Equation (5) is sometimes better known in the following form:

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

(7)

which is (5) with $x={\displaystyle \frac{1}{2}}(s+t)$ and $y={\displaystyle \frac{1}{2}}(s-t)$. Equation (7) is patterned on the following trigonometric identity:

$$sin(x+y)sin(x-y)={sin}^{2}x-{sin}^{2}y.$$

Let us also mention that in [45] the authors initiated a study of the superstability of several particular cases of the following (so-called radical) functional equations:

$$\begin{array}{c}\hfill {\psi }_1\left(\sqrt[n]{s^n+t^n}\right)+{\psi }_1\left(\sqrt[n]{s^n-t^n}\right)={\psi }_2\left(t\right){\psi }_3\left(s\right),\end{array}$$

(8)

$$\begin{array}{c}\hfill {\psi }_1\left(\sqrt[n]{s^n+t^n}\right)-{\psi }_1\left(\sqrt[n]{s^n-t^n}\right)={\psi }_2\left(t\right){\psi }_3\left(s\right),\end{array}$$

(9)

$$\begin{array}{c}\hfill \varphi {\left(\sqrt[n]{{\displaystyle \frac{s^n+t^n}{2}}}\right)}^2-\varphi {\left(\sqrt[n]{{\displaystyle \frac{s^n-t^n}{2}}}\right)}^2={\psi }_2\left(t\right){\psi }_3\left(s\right),\end{array}$$

(10)

for real or complex functions of real variable. More recent related results can be found in [46,47]. Note that actually (10) is (9), with

$${\psi }_1\left(u\right)\equiv \varphi {\left({\displaystyle \frac{1}{\sqrt[n]{2}}}u\right)}^2.$$

Throughout this paper, $\mathbb{N}$ stands, as usual, for the set of positive integers, and ${\mathbb{N}}_0:=\mathbb{N}\cup \left\{0\right\}$. Next, $\mathbb{K}$ is either the field of real numbers $\mathbb{R}$ or the field of complex numbers $\mathbb{C}$, X is a Banach space over $\mathbb{K}$, S and T are nonempty sets, and ${\mathbb{R}}_{+}:=[0,\infty )$.

Let $a_j\in \mathbb{K}$ and ${\gamma }_j:S\times T\to S$ for $j\in \mathbb{N}$. Motivated by several quite recent results on the superstability of functional equations in, e.g., [45,46,47], we investigate the superstability of the following very general functional equation:

$$\sum _{j=1}^{\infty }{a}_{j}f\left({\gamma }_j(t,s)\right)=h\left(t\right)g\left(s\right),$$

(11)

with the unknown functions $g:T\to X$, $h:S\to \mathbb{K}$ and $f:S\to X$, under the assumption that the series ${\sum }_{j=1}^{\infty }f\left({\gamma }_j(t,s)\right)$ is convergent for every $(s,t)\in S\times T$. In this way, we generalize one of the main outcomes in [48] (Theorem 1), which has been proved for (11) with $f=g$.

Note that all the above-mentioned equations are special cases of (11). For further examples of particular forms of (11), we refer to the next section.

2. Auxiliary Information

In this paper (in the next section), we prove a generalization of the result given below, obtained in [48] (Theorem 1) for the functional equation

$$\sum _{j=1}^{\infty }{a}_{j}f\left({\gamma }_j(t,s)\right)=h\left(t\right)f\left(s\right),$$

(12)

considered for unknown functions $h:T\to \mathbb{K}$ and $f\in \mathcal{F}$, where

$$\mathcal{F}:=\{f:S\to X:\sum _{j=1}^{\infty }{a}_{j}f\left({\gamma }_j(t,s)\right) \mathrm{is} \mathrm{uniformly} \mathrm{convergent} \mathrm{in} T\times S\}.$$

Theorem 2.

Suppose that $h:T\to \mathbb{K}$, $f\in \mathcal{F}$, $N\in \mathbb{N}$, ${\epsilon }_l:S\to \mathbb{R}$ for $l\in {\mathbb{N}}_0$, $\delta >0$,

$$\begin{array}{c}\hfill \parallel \sum _{j,k=1}^{N+l}{a}_j{a}_k[f\left({\gamma }_j(t_2,{\gamma }_k(t_1,s))\right)-f\left({\gamma }_k(t_1,{\gamma }_j(t_2,s))\right)]\parallel \le {\epsilon }_l\left(s\right),\\ \hfill \forall t_1,t_2\in T,s\in S,l\in {\mathbb{N}}_0,\end{array}$$

(13)

and

$$\parallel \sum _{j=1}^{\infty }{a}_{j}f\left({\gamma }_j(t,s)\right)-h\left(t\right)f\left(s\right)\parallel \le \delta ,\forall t\in T,s\in S.$$

(14)

Then, one of the following two conditions holds.

(i) 

h is bounded.

(ii) 

h and f satisfy the equation

$$\sum _{j=1}^{\infty }{a}_{j}f\left({\gamma }_j(t,s)\right)=h\left(t\right)f\left(s\right),$$

and there is $M\in \mathbb{N}$, such that

$$a_{M+j}f\left({\gamma }_{M+j}(t,s)\right)=0,\forall s\in S,t\in T,j\in \mathbb{N}.$$

Below, we provide various examples of functions ${\gamma }_j$ satisfying a stronger version of (13) (cf. [48], Example 1), i.e., the condition

$$\begin{array}{c}\hfill {\gamma }_j(t_2,{\gamma }_k(t_1,s))={\gamma }_k(t_1,{\gamma }_j(t_2,s)),\forall t_1,t_2\in T, s\in S, j,k\in \mathbb{N},\end{array}$$

(15)

which obviously implies

$$\begin{array}{c}\hfill \sum _{j,k=1}^{N+l}{a}_j{a}_k[f\left({\gamma }_j(t_2,{\gamma }_k(t_1,s))\right)-f\left({\gamma }_k(t_1,{\gamma }_j(t_2,s))\right)]=0,\forall t_1,t_2\in T, s\in S, l\in {\mathbb{N}}_0,\end{array}$$

(16)

for every function $f:S\to X$ and $N=1$.

Example 1.

Assume that $(U,+)$ is an abelian semigroup, $\eta :S\to U$ is a surjection, and $\rho :U\to S$ is a selection of $\eta$ (i.e., $\eta \left(\rho \right(x\left)\right)=x$ for $x\in U$). Fix ${\xi }_j:T\to U$ for $j\in \mathbb{N}$, and let

$${\gamma }_j(t,s)=\rho ({\xi }_j\left(t\right)+\eta \left(s\right)),\forall j\in \mathbb{N}, t\in T, s\in S.$$

(17)

Then,

$$\begin{array}{cc}\hfill {\gamma }_j(t_2,{\gamma }_k(t_1,s))& =\rho ({\xi }_j\left(t_2\right)+\eta \left({\gamma }_k(t_1,s)\right))=\rho ({\xi }_j\left(t_2\right)+\eta \left(\rho ({\xi }_k\left(t_1\right)+\eta \left(s\right))\right)\hfill \\ \hfill & =\rho ({\xi }_j\left(t_2\right)+{\xi }_k\left(t_1\right)+\eta \left(s\right))=\rho ({\xi }_k\left(t_1\right)+{\xi }_j\left(t_2\right)+\eta \left(s\right))\hfill \\ \hfill & ={\gamma }_k(t_1,{\gamma }_j(t_2,s)),\forall t_1,t_2\in T,s\in S,k,j\in \mathbb{N}.\hfill \end{array}$$

Consequently, condition (16) is fulfilled for each $f:S\to X$ (with $N=1$), and Equation (11) takes the form

$$\sum _{j=1}^{\infty }{a}_{j}f\left(\rho \left({\xi }_j\left(t\right)+\eta \left(s\right)\right)\right)=h\left(t\right)g\left(s\right).$$

(18)

In a simplified situation, when $U=S$ and $\eta \left(s\right)=s$ for $s\in S$, Equation (11) becomes

$$\sum _{j=1}^{\infty }{a}_{j}f({\xi }_j\left(t\right)+s)=h\left(t\right)g\left(s\right).$$

With appropriate assumptions about S, T and X (which can be easily guessed), we have the following special cases of it: Equation (2) and (when $(U,+)$ is the semigroup $(\mathbb{K},·)$) the multiplicative version of it

$$f\left(xy\right)=f\left(x\right)f\left(y\right),$$

the next two pexiderizations (see, e.g., [41], Chapter 2.2) of them

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

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

the d’Alembert (cosine) equation

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

its generalization

$$\begin{array}{c}\hfill f(x+y)+f(x+\sigma (y\left)\right)=h\left(y\right)g\left(x\right)\end{array}$$

(19)

(introduced by Wilson [42] with $\sigma \left(y\right)\equiv -y$) studied recently, e.g., in [28,49,50], the Reynolds exponential equation (cf. [51])

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

(20)

(closely related to the Reynolds operator considered in, e.g., [52,53]) and the subsequent pexiderized version of it:

$$\begin{array}{c}\hfill f\left(\sigma \right(x)+y)=h\left(x\right)g\left(y\right).\end{array}$$

(21)

Let us mention that a superstability result for (20) has been proved in [54].

If $n\in \mathbb{N}$, $n>1$, $S=\mathbb{R}$, $U=P:=\{x^n:x\in \mathbb{R}\}$, $\eta \left(s\right)=s^n$ for $s\in \mathbb{R}$, and $\rho \left(u\right)=\sqrt[n]{u}$ for $u\in P$, then (17) becomes

$${\gamma }_j(t,s)=\sqrt[n]{{\xi }_j\left(t\right)+s^n},\forall j\in \mathbb{N},t,s\in \mathbb{R}.$$

So, for $T=\mathbb{R}$ (whenever $\sqrt[n]{s^n+\sigma \left(t^n\right)}$ makes sense for all $s,t\in \mathbb{R}$), we have, for instance, the following particular radical cases of Equation (18) (and also of (11)):

$$f\left(\sqrt[n]{\sigma \left(x^n\right)+y^n}\right)=h\left(x\right)g\left(y\right),$$

$$a_{1}f\left(\sqrt[n]{x^n+y^n}\right)+a_{2}f\left(\sqrt[n]{x^n+\sigma \left(y^n\right)}\right)=h\left(y\right)g\left(x\right)$$

(22)

that are related to Equations (19) and (21) (cf., e.g., [28,45,46,47]). Clearly, Equation (22) is a generalization of (8) and (9).

If n is odd, and $X=\mathbb{K}$, then Equation (10) can also be considered a particular case of (22) with $a_1=1$, $a_2=-1$, and

$$f\left(s\right)=\psi {\left({\displaystyle \frac{1}{\sqrt[n]{2}}}s\right)}^2,\forall s\in \mathbb{R}.$$

Example 2.

Let $\mathbb{F}$ be a commutative ring and Y be a module over $\mathbb{F}$, $\xi :S\to Y$ be surjective, $\rho :Y\to S$ be a selection of $\xi$, ${\tau }_j:T\to \mathbb{F}$ for $j\in \mathbb{N}$, and ${\gamma }_j$ be given by ${\gamma }_j(t,s)=\rho \left({\tau }_j\left(t\right)\xi \left(s\right)\right)$ for $t\in T$, $s\in S$, $j\in \mathbb{N}$. Then,

$$\begin{array}{cc}\hfill {\gamma }_j(t_2,{\gamma }_k(t_1,s))& =\rho \left({\tau }_j\left(t_2\right)\xi \left({\gamma }_k(t_1,s)\right)\right)=\rho \left({\tau }_j\left(t_2\right)\xi \left(\rho \left({\tau }_k\left(t_1\right)\xi \left(s\right)\right)\right)\right)\hfill \\ \hfill & =\rho \left({\tau }_j\left(t_2\right){\tau }_k\left(t_1\right)\xi \left(s\right)\right)=\rho \left({\tau }_k\left(t_1\right){\tau }_j\left(t_2\right)\xi \left(s\right)\right)\hfill \\ \hfill & ={\gamma }_k(t_1,{\gamma }_j(t_2,s)),\forall t_1,t_2\in T,s\in S,k,j\in \mathbb{N}.\hfill \end{array}$$

Hence, condition (16) is satisfied for every $f\in X^S$ (with $N=1$).

If $S=Y$ and $\xi \left(s\right)=s$ for $s\in S$, then (11) has the form

$$\sum _{j=1}^{\infty }{a}_{j}f\left({\tau }_j\left(t\right)s\right)=h\left(t\right)g\left(s\right).$$

The homogeneity functional equation

$$f\left(\gamma \right(t\left)s\right)=h\left(t\right)f\left(s\right)$$

(23)

is a particular case of it. The superstability of (23) has been investigated, e.g., in [35,36,37,40]. Note that (23) is actually a partial pexiderization of the functional equation

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

i.e., the multiplicative version of the Reynolds Equation (20).

Example 3.

Assume that $\xi :S\to (0,\infty )$ is surjective, $\rho :(0,\infty )\to S$ is a selection of $\xi$, and ${\eta }_j:T\to \mathbb{R}$ for $j\in \mathbb{N}$ is given. Define ${\gamma }_j:T\times S\to S$ by ${\gamma }_j(t,s)=\rho \left(\xi {\left(s\right)}^{{\eta }_j\left(t\right)}\right)$ for $s\in S$, $t\in T$. Then,

$$\begin{array}{cc}\hfill {\gamma }_j(t_2,{\gamma }_k(t_1,s))& =\rho \left(\xi {\left({\gamma }_k(t_1,s)\right)}^{{\eta }_j\left(t_2\right)}\right)=\rho \left(\xi {\left(\rho \left(\xi {\left(s\right)}^{{\eta }_k\left(t_1\right)}\right)\right)}^{{\eta }_j\left(t_2\right)}\right)\hfill \\ \hfill & =\rho \left({\left(\xi {\left(s\right)}^{{\eta }_k\left(t_1\right)}\right)}^{{\eta }_j\left(t_2\right)}\right)=\rho \left(\xi {\left(s\right)}^{{\eta }_k\left(t_1\right){\eta }_j\left(t_2\right)}\right)\hfill \\ \hfill & =\rho \left(\xi {\left(s\right)}^{{\eta }_j\left(t_2\right){\eta }_k\left(t_1\right)}\right)={\gamma }_k(t_1,{\gamma }_j(t_2,s)),\forall t_1,t_2\in T, s\in S, k,j\in \mathbb{N}.\hfill \end{array}$$

This implies that condition (16) is valid for every $f:S\to X$ (with $N=1$).

For $S=(0,\infty )$ and $\xi \left(s\right)=s$ for $s\in S$, Equation (11) has the form

$$\sum _{j=1}^{\infty }{a}_{j}f\left(s^{{\eta }_j\left(t\right)}\right)=h\left(t\right)f\left(s\right).$$

A particular case of it is the functional equation

$$\begin{array}{c}\hfill f\left(s^t\right)=tf\left(s\right)\end{array}$$

(24)

considered in [40], where its superstability has been investigated.

3. The Main Result

Let

$$\mathcal{L}:=\{f:S\to X:\sum _{j=1}^{\infty }{a}_{j}f\left({\gamma }_j(t,s)\right)\mathrm{be}\mathrm{convergent}\mathrm{for}\mathrm{every}(t,s)\in T\times S\}.$$

In this section, we prove a superstability theorem for Equation (11), with $f\in \mathcal{L}$ and $h:T\to \mathbb{K}$, $g:S\to X$. Clearly, (11) seems to be ‘only’ a small modification (pexiderization) of (12), where f on the right-hand side of Equation (12) has been replaced by another unknown function g. However, this new result allows one to obtain many new superstability outcomes for functional equations that are particular cases of (11), including the equations mentioned in the previous section. Some appropriate examples are provided in the next section.

We need the following hypothesis for the functions $f\in \mathcal{L}$.

$\left(\mathcal{B}\right)$

There exist $L\in {\mathbb{R}}_{+}$ and $M\left(L\right)\in \mathbb{N}$ such that

$$\parallel \sum _{j=M\left(L\right)+l}^{\infty }{a}_{j}f\left({\gamma }_j(t,s)\right)\parallel \le L,\forall t,s\in S,l\in \mathbb{N}.$$

(25)

We write

$$\mathcal{G}:=\{f\in \mathcal{L}:\mathrm{hypothesis} (\mathcal{B}) \mathrm{holds}\}.$$

It is easily seen that if the series ${\sum }_{j=1}^{\infty }{a}_{j}f\left({\gamma }_j(t,s)\right)$ is uniformly convergent in $T\times S$, then $\left(\mathcal{B}\right)$ holds. This means that $\mathcal{F}\subset \mathcal{G}$.

Now, we are in a position to prove the following generalization of Theorem 2 (i.e., of Theorem 1 in [48]).

Theorem 3.

Let $h:T\to \mathbb{K}$ be unbounded and $\delta >0$ be a real number. Let $g:S\to X$ and $f\in \mathcal{G}$ be such that

$$\underset{(t,s)\in T\times S}{sup}\parallel \sum _{j=1}^{\infty }{a}_{j}f\left({\gamma }_j(t,s)\right)-h\left(t\right)g\left(s\right)\parallel \le \delta .$$

(26)

Moreover, assume that there exist $N\in \mathbb{N}$ and ${\epsilon }_l:S\to \mathbb{R}$ for $l\in {\mathbb{N}}_0$, such that

$$\begin{array}{c}\hfill \parallel \sum _{j,k=1}^{N+l}{a}_j{a}_k[f\left({\gamma }_j(t_2,{\gamma }_k(t_1,s))\right)-f\left({\gamma }_k(t_1,{\gamma }_j(t_2,s))\right)]\parallel \le {\epsilon }_l\left(s\right),\\ \hfill \forall t_1,t_2\in T, s\in S, l\in {\mathbb{N}}_0.\end{array}$$

(27)

Then, h and g satisfy the equation

$$\sum _{j=1}^{\infty }{a}_{j}g\left({\gamma }_j(t,s)\right)=h\left(t\right)g_0\left(s\right),$$

(28)

with some $g_0:S\to X$ (given by (36)), and

$$a_{\mu +l}g\left({\gamma }_{\mu +l}(t,s)\right)=0,\forall s\in S, t\in T, l\in \mathbb{N},$$

(29)

where $\mu :=max\{N,M(L\left)\right\}$.

Moreover, if $f=g$, then $g_0=g=f$.

Proof. 

Since $f\in \mathcal{G}$, hypothesis $\left(\mathcal{B}\right)$ holds, which means that there exist $L:S\to (0,\infty )$ and $M\left(L\right)\in \mathbb{N}$, such that (25) is valid. Consequently, for each $t\in T$, $s\in S$, $m\in {\mathbb{N}}_0$,

$$\begin{array}{cc}\hfill \parallel \sum _{j=1}^{M\left(L\right)+m}& a_{j}f\left({\gamma }_j(t,s)\right)-h\left(t\right)g\left(s\right)\parallel \hfill \\ \hfill & =\parallel \sum _{j=1}^{\infty }{a}_{j}f\left({\gamma }_j(t,s)\right)-h\left(t\right)g\left(s\right)-\sum _{j=M\left(L\right)+m+1}^{\infty }{a}_{j}f\left({\gamma }_j(t,s)\right)\parallel \hfill \\ \hfill & \le \parallel \sum _{j=1}^{\infty }{a}_{j}f\left({\gamma }_j(t,s)\right)-h\left(t\right)g\left(s\right)\parallel +\parallel \sum _{j=M\left(L\right)+m+1}^{\infty }{a}_{j}f\left({\gamma }_j(t,s)\right)\parallel \hfill \\ \hfill & \le \delta +L.\hfill \end{array}$$

(30)

h is assumed to be unbounded; therefore, there exists a sequence $(t_n:n\in {\mathbb{N}}_0)$ in T such that $h\left(t_n\right)\ne 0$ for $n\in {\mathbb{N}}_0$ and ${lim}_{n\to \infty }\left|h\left(t_n\right)\right|=\infty$. It is easily seen that (30), with t replaced by $t_n$, yields

$$\begin{array}{c}\hfill \parallel \sum _{i=1}^{M\left(L\right)+m}{\displaystyle \frac{a_i}{h\left(t_n\right)}}f\left({\gamma }_i(t_n,s)\right)-g\left(s\right)\parallel \le {\displaystyle \frac{\delta +L}{\left|h\right(t_n\left)\right|}},\forall m,n\in {\mathbb{N}}_0, s\in S,\end{array}$$

whence

$$\begin{array}{cc}\hfill \parallel \underset{n\to \infty }{lim}\sum _{i=1}^{M\left(L\right)+m}{\displaystyle \frac{a_i}{h\left(t_n\right)}}f\left({\gamma }_i(t_n,s)\right)-g\left(s\right)\parallel & =\underset{n\to \infty }{lim}\parallel \sum _{i=1}^{M\left(L\right)+m}{\displaystyle \frac{a_i}{h\left(t_n\right)}}f\left({\gamma }_i(t_n,s)\right)-g\left(s\right)\parallel \hfill \\ \hfill & \le \underset{n\to \infty }{lim}{\displaystyle \frac{\delta +L}{\left|h\right(t_n\left)\right|}}=0,\forall s\in S, m\in {\mathbb{N}}_0.\hfill \end{array}$$

This means that

$$g\left(s\right)=\underset{n\to \infty }{lim}\sum _{k=1}^{M\left(L\right)+m}{\displaystyle \frac{a_k}{h\left(t_n\right)}}f\left({\gamma }_k(t_n,s)\right),\forall s\in S, m\in {\mathbb{N}}_0.$$

(31)

Hence, for every $t\in T$, $s\in S$, $m\in {\mathbb{N}}_0$, we obtain

$$\begin{array}{cc}\hfill \sum _{j=1}^{M\left(L\right)+m}{a}_{j}g\left({\gamma }_j(t,s)\right)& =\sum _{j=1}^{M\left(L\right)+m}{a}_j\underset{n\to \infty }{lim}{\displaystyle \frac{1}{h\left(t_n\right)}}\sum _{k=1}^{M\left(L\right)+m}{a}_{k}f\left({\gamma }_k(t_n,{\gamma }_j(t,s))\right)\hfill \\ \hfill & =\underset{n\to \infty }{lim}{\displaystyle \frac{1}{h\left(t_n\right)}}\sum _{k,j=1}^{M\left(L\right)+m}{a}_k{a}_{j}f\left({\gamma }_k(t_n,{\gamma }_j(t,s))\right).\hfill \end{array}$$

(32)

Further, according to (27), for every $r_1,r_2\in T,s\in S,l\in {\mathbb{N}}_0$, we have

$$\begin{array}{c}\hfill \parallel \sum _{j,k=1}^{N+l}{a}_k{a}_j[f\left({\gamma }_j(r_2,{\gamma }_k(r_1,s))\right)-f\left({\gamma }_k(r_1,{\gamma }_j(r_2,s))\right)]\parallel \le {\epsilon }_l\left(s\right),\end{array}$$

whence

$$\begin{array}{cc}\hfill \parallel \underset{n\to \infty }{lim}& {\displaystyle \frac{1}{h\left(t_n\right)}}\sum _{j,k=1}^{N+l}{a}_k{a}_j[f\left({\gamma }_j(r_2,{\gamma }_k(r_1,s))\right)-\underset{n\to \infty }{lim}{\displaystyle \frac{1}{h\left(t_n\right)}}f\left({\gamma }_k(r_1,{\gamma }_j(r_2,s))\right)]\parallel \hfill \\ \hfill & =\underset{n\to \infty }{lim}{\displaystyle \frac{1}{\left|h\right(t_n\left)\right|}}\parallel \sum _{j,k=1}^{N+l}{a}_k{a}_j[f\left({\gamma }_j(r_2,{\gamma }_k(r_1,s))\right)-f\left({\gamma }_k(r_1,{\gamma }_j(r_2,s))\right)]\parallel \hfill \\ \hfill & \le \underset{n\to \infty }{lim}{\displaystyle \frac{{\epsilon }_l\left(s\right)}{\left|h\right(t_n\left)\right|}}=0,\hfill \end{array}$$

and consequently,

$$\begin{array}{c}\hfill \underset{n\to \infty }{lim}{\displaystyle \frac{1}{h\left(t_n\right)}}\sum _{k,j=1}^{N+l}{a}_k{a}_{j}f\left({\gamma }_j(r_2,{\gamma }_k(r_1,s))\right)=\underset{n\to \infty }{lim}{\displaystyle \frac{1}{h\left(t_n\right)}}\sum _{k,j=1}^{N+l}{a}_k{a}_{j}f\left({\gamma }_k(r_1,{\gamma }_j(r_2,s))\right).\end{array}$$

Therefore, by (32),

$$\begin{array}{c}\hfill \sum _{j=1}^{\mu +l}{a}_{j}g\left({\gamma }_j(t,s)\right)=\underset{n\to \infty }{lim}{\displaystyle \frac{1}{h\left(t_n\right)}}\sum _{k,j=1}^{\mu +l}{a}_k{a}_{j}f\left({\gamma }_j(t,{\gamma }_k(t_n,s))\right),\forall t\in T, s\in S ,l\in {\mathbb{N}}_0,\end{array}$$

with $\mu :=max\{N,M(L\left)\right\}$. This implies that

$$\begin{array}{cc}\hfill \sum _{j=1}^{\mu +l}{a}_{j}g\left({\gamma }_j(t,s)\right)& =\sum _{k=1}^{\mu +l}{a}_k\underset{n\to \infty }{lim}{\displaystyle \frac{1}{h\left(t_n\right)}}\sum _{j=1}^{\mu +l}{a}_{j}f\left({\gamma }_j(t,{\gamma }_k(t_n,s))\right)\hfill \\ \hfill & =\underset{n\to \infty }{lim}{\displaystyle \frac{1}{h\left(t_n\right)}}\sum _{k=1}^{\mu +l}{a}_k\left[\left(\sum _{j=1}^{\mu +l}{a}_{j}f\left({\gamma }_j(t,{\gamma }_k(t_n,s))\right)\right)-h\left(t\right)g\left({\gamma }_k(t_n,s)\right)\right]\hfill \\ \hfill & +\underset{n\to \infty }{lim}h\left(t\right)\sum _{k=1}^{\mu +l}{\displaystyle \frac{a_k}{h\left(t_n\right)}}g\left({\gamma }_k(t_n,s)\right),\forall t\in T, s\in S ,l\in {\mathbb{N}}_0.\hfill \end{array}$$

(33)

Replacing s by ${\gamma }_k(t_n,s)$ in (30) we obtain

$$\begin{array}{c}\hfill \parallel \sum _{j=1}^{M\left(L\right)+l}{a}_{j}f\left({\gamma }_j(t,{\gamma }_k(t_n,s))\right)-h\left(t\right)g\left({\gamma }_k(t_n,s)\right)\parallel \le \delta +L,\\ \hfill \forall t\in T, s\in S, l,n\in {\mathbb{N}}_0, k\in \mathbb{N}.\end{array}$$

So, for every $t\in T, s\in S, l\in {\mathbb{N}}_0, k\in \mathbb{N}$,

$$\underset{n\to \infty }{lim}{\displaystyle \frac{1}{\left|h\right(t_n\left)\right|}}\parallel \sum _{j=1}^{M\left(L\right)+l}{a}_{j}f\left({\gamma }_j(t,{\gamma }_k(t_n,s))\right)-h\left(t\right)g\left({\gamma }_k(t_n,s)\right)\parallel \le \underset{n\to \infty }{lim}{\displaystyle \frac{\delta +L}{\left|h\right(t_n\left)\right|}}=0,$$

whence

$$\underset{n\to \infty }{lim}{\displaystyle \frac{1}{h\left(t_n\right)}}\left[\sum _{j=1}^{M\left(L\right)+l}{a}_{j}f\left({\gamma }_j(t,{\gamma }_k(t_n,s))\right)-h\left(t\right)g\left({\gamma }_k(t_n,s)\right)\right]=0.$$

(34)

Consequently, using (33) and (34), we have

$$\begin{array}{c}\hfill \sum _{j=1}^{\mu +l}{a}_{j}g\left({\gamma }_j(t,s)\right)=\underset{n\to \infty }{lim}h\left(t\right)\sum _{k=1}^{\mu +l}{\displaystyle \frac{a_k}{h\left(t_n\right)}}g\left({\gamma }_k(t_n,s)\right),\forall t\in T, s\in S, l\in {\mathbb{N}}_0,\end{array}$$

whence

$$\begin{array}{c}\hfill \sum _{j=1}^{\mu +l}{a}_{j}g\left({\gamma }_j(t,s)\right)=h\left(t\right)g_0\left(s\right),\forall t\in T, s\in S, l\in {\mathbb{N}}_0,\end{array}$$

(35)

where

$$\begin{array}{c}\hfill g_0\left(s\right):=\underset{n\to \infty }{lim}\sum _{k=1}^{\mu +l}{\displaystyle \frac{a_k}{h\left(t_n\right)}}g\left({\gamma }_k(t_n,s)\right),\forall s\in S, l\in {\mathbb{N}}_0.\end{array}$$

(36)

Hence, letting $l\to \infty$ in (35), we obtain that $g_0$ and h satisfy Equation (28). Clearly, if $f=g$, then (31) and (36) imply that $g_0=g$.

Finally, it is easily seen that (35) yields

$$\begin{array}{cc}\hfill a_{\mu +l}g\left({\gamma }_{\mu +l}(t,s)\right)& =\sum _{j=1}^{\mu +l}{a}_{j}g\left({\gamma }_j(t,s)\right)-\sum _{j=1}^{\mu +l-1}{a}_{j}g\left({\gamma }_j(t,s)\right)\hfill \\ \hfill & =h\left(t\right)g_0\left(s\right)-h\left(t\right)g_0\left(s\right)=0,\forall t\in T, s\in S, l\in \mathbb{N},\hfill \end{array}$$

which means that (29) holds. □

Theorem 3 yields the subsequent three corollaries.

Corollary 1.

Assume that $X=\mathbb{K}$, $c\in \mathbb{K}$, $c\ne 0$, $S=T$, $f\in \mathcal{G}$, there exist $N\in \mathbb{N}$ and ${\epsilon }_l:S\to \mathbb{R}$, for $l\in {\mathbb{N}}_0$, such that (27) holds, and

$$\rho :=\underset{s,t\in T}{sup}|\sum _{j=1}^{\infty }{a}_{j}f\left({\gamma }_j(t,s)\right)-cf\left(t\right)f\left(s\right)|<\infty .$$

(37)

Then, one of the following two conditions is valid.

(i) 

f is an unbounded solution to the equation

$$\sum _{j=1}^{\infty }{a}_{j}f\left({\gamma }_j(t,s)\right)=cf\left(t\right)f\left(s\right),$$

(38)

and

$$a_{\mu +l}f\left({\gamma }_{\mu +l}(t,s)\right)=0,\forall s, t\in T, l\in \mathbb{N},$$

with $\mu :=max\{N,M(L\left)\right\}$.

(ii) 

f is bounded and, in the case $\alpha :={\sum }_{j=1}^{\infty }\left|a_j\right|<\infty$,

$$\left|f\left(t\right)\right|\le {\displaystyle \frac{\alpha +\sqrt{{\alpha }^2+4\left|c\right|\rho }}{2\left|c\right|}},\forall t\in T.$$

(39)

Proof. 

In view of Theorem 3 with $g=f$ and $h\left(t\right)=cf\left(t\right)$ for $t\in T=S$, it is enough to consider the case of bounded f. Write

$$\nu :=sup\left\{\right|f\left(t\right)|:t\in T\}.$$

Then, there is a sequence $(t_n:n\in \mathbb{N})$ in T such that ${lim}_{n\to \infty }\left|f\left(t_n\right)\right|=\nu$, and (37) implies that, for each $n\in \mathbb{N}$,

$$\left|c\right||f\left(t_n\right){|}^2\le \rho +\sum _{j=1}^{\infty }|a_j\left|\right|f\left({\gamma }_j(t_n,t_n)\right)|\le \rho +\alpha \nu .$$

Letting $n\to \infty$, we have

$$\left|c\right|{\nu }^2-\alpha \nu -\rho \le 0,$$

whence $\nu$ must be smaller than the positive root of the equation $\left|c\right|x^2-\alpha x-\rho =0$, which means that

$$\nu \le {\displaystyle \frac{\alpha +\sqrt{{\alpha }^2+4\left|c\right|\rho }}{2\left|c\right|}}.$$

□

Remark 1.

If ${\sum }_{i=1}^{\infty }{a}_i={\sum }_{i=1}^{\infty }\left|a_i\right|<\infty$, then estimation (39) is optimal. In fact, it is easy to check that, for f defined by

$$f\left(t\right)={\displaystyle \frac{\alpha +\sqrt{{\alpha }^2+4\left|c\right|\rho }}{2c}},\forall t\in T,$$

we have

$$\rho =|\sum _{j=1}^{\infty }{a}_{j}f\left({\gamma }_j(t,s)\right)-cf\left(t\right)f\left(s\right)|,\forall s,t\in T.$$

Remark 2.

Estimation (39) with $c=1$ is exactly the same as those obtained in [17,18] for the exponential functional equation (i.e., the special case of Equation (38) with $c=1$, $a_1=1$, and $a_j=0$ for $j>1$) and the d’Alembert Equation (3) (i.e., the special case of Equation (38) with $c=1$, $a_1=a_2={\displaystyle \frac{1}{2}}$, and $a_j=0$ for $j>2$). Therefore, in particular, Corollary 1 (ii) extends these estimations to the case of $c\ne 1$.

Corollary 2.

Let δ be a positive real number, $M\in \mathbb{N}$, $h:T\to \mathbb{K}$ be unbounded, $g:S\to X$, and $f:T\to X$. Assume that there exists $\epsilon :S\to \mathbb{R}$, such that

$$\begin{array}{c}\hfill \parallel \sum _{j,k=1}^M{a}_j{a}_k[f\left({\gamma }_j(t_2,{\gamma }_k(t_1,s))\right)-f\left({\gamma }_k(t_1,{\gamma }_j(t_2,s))\right)]\parallel \le \epsilon \left(s\right),\forall t_1, t_2\in T, s\in S.\end{array}$$

(40)

If

$$\underset{(t,s)\in T\times S}{sup}\parallel \sum _{j=1}^M{a}_{j}f\left({\gamma }_j(t,s)\right)-h\left(t\right)g\left(s\right)\parallel \le \delta ,$$

(41)

then h and g satisfy the equation

$$\sum _{j=1}^M{a}_{j}g\left({\gamma }_j(t,s)\right)=h\left(t\right)g_0\left(s\right),$$

with some $g_0:S\to X$. Moreover, if $f=g$, then $g_0=g=f$.

Proof. 

Clearly, (41) is (26) with $a_j=0$ for $j>M$. Analogously, (25) holds with any $L\in {\mathbb{R}}_{+}$ and $M\left(L\right)=M$, and (40) implies (27) with $N=M$. So, it is enough to use Theorem 3. □

4. Some Applications

Now, we show applications of the results from the previous section to some particular cases of functional Equation (11).

First, let us apply Corollary 2 to the equation

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

(42)

for functions $f,g:S\to X$ and $h:T\to \mathbb{K}$, where $F:T\times S\to S$ is given and fulfills the condition

$$\begin{array}{c}\hfill F(t_2,F(t_1,s))=F(t_1,F(t_2,s)),\forall t_1, t_2\in T, s\in S,\end{array}$$

(43)

i.e., condition (15) with ${\gamma }_j={\gamma }_k=F$. If $S=T$, then for functions F that are symmetric (i.e., $F(s,t)=F(t,s)$ for $s,t\in T$), equality (43) becomes the associativity condition

$$\begin{array}{c}\hfill F(t,F(s,u\left)\right)=F\left(F\right(t,s),u),\forall t, s, u\in T,\end{array}$$

and we refer to [55,56,57] for some information concerning functions fulfilling it.

According to Examples 1–3, Equation (42) can be a pexiderized version of the Reynolds functional Equation (20) (i.e., Equation (21)), the homogeneity Equation (23), Equation (24) and the exponential Equation (2). We have the following corollary.

Corollary 3.

Let $h:T\to \mathbb{K}$ be unbounded and δ be a positive real number. Assume that there is $x_0\in T$, such that $h\left(x_0\right)=1$ and the mapping $S\ni y\to F(x_0,y)\in S$ is surjective.

Then, $f, g:S\to X$ satisfy the inequality

$$\underset{(x,y)\in T\times S}{sup}\parallel f\left(F(x,y)\right)-h\left(x\right)g\left(y\right)\parallel \le \delta ,$$

(44)

if and only if h and g fulfill the equation

$$g\left(F(x,y)\right)=h\left(x\right)g\left(F(x_0,y)\right),$$

(45)

and

$$\underset{(x,y)\in T\times S}{sup}\parallel f\left(F(x,F(x_0,y))\right)-g\left(F(x,y)\right)\parallel \le \delta .$$

(46)

Proof. 

Clearly, (44) is (41) with $M=1$ and ${\gamma }_1=F$. Next, (43) is condition (15) (with ${\gamma }_j={\gamma }_k=F$). Hence (see Examples 1–3), we can use Corollary 2 to obtain that h and g fulfill the equation $g\left(F(x,y)\right)=h\left(x\right)g_0\left(y\right)$, with some function $g_0:S\to X$. It is easily seen that from the equation (with $x=x_0$) we obtain $g_0\left(y\right)=g\left(F(x_0,y)\right)$ for $y\in S$, which implies Equation (45) and, consequently, (46).

Conversely, if h and g satisfy Equation (45), and (46) is valid, then

$$\begin{array}{c}\hfill \parallel f\left(F(x,F(x_0,y))\right)-h\left(x\right)g\left(F(x_0,y)\right)\parallel =\parallel f\left(F(x,F(x_0,y))\right)-g\left(F(x,y)\right)\parallel \le \delta ,\end{array}$$

for every $x\in T,y\in S$. Since the mapping $S\ni y\to F(x_0,y)\in S$ is surjective, this yields (44). □

Taking $T=S$, $X=\mathbb{K}$, and $g=h$ in Corollary 3, we obtain the following result.

Corollary 4.

Let $(S,★)$ be an abelian group with neutral element e, $\sigma :S\to S$, $\sigma \left(e\right)=e$, δ be a positive real number, $f, h:S\to \mathbb{K}$, $h\left(e\right)=1$, and

$$\underset{x,y\in S}{sup}\parallel f(\sigma \left(x\right)★y)-h\left(x\right)h\left(y\right)\parallel \le \delta .$$

Then, one of the following two conditions is valid.

(i) 

f and h are unbounded, h is a solution to the equation

$$h\left(\sigma \right(x)★y)=h\left(x\right)h\left(y\right),$$

and

$$\underset{y\in S}{sup}\parallel f\left(y\right)-h\left(y\right)\parallel \le \delta .$$

(ii) 

f and h are bounded.

In addition, for Equation (19) we can obtain the following result that is analogous to Corollary 3.

Corollary 5.

Let $(S,+)$ be an abelian groupoid with the neutral element 0, $\sigma :S\to S$, $T=S$, $h:S\to \mathbb{K}$ be unbounded, $h\left(0\right)=2$, δ be a positive real number, and $\sigma \left(0\right)=0$. Then, $f,g:S\to X$ satisfy the inequality

$$\underset{(x,y)\in S\times S}{sup}\parallel f(x+y)+f(x+\sigma \left(y\right))-h\left(y\right)g\left(x\right)\parallel \le \delta ,$$

(47)

if and only if h and g satisfy the equation

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

(48)

and

$$\underset{y\in S}{sup}\parallel f\left(y\right)-g\left(y\right)\parallel \le {\displaystyle \frac{\delta }{2}}.$$

(49)

Proof. 

First, assume that (47) holds. Then, by Corollary 2, h and g satisfy the functional equation $g(x+y)+g(x+\sigma \left(y\right))=h\left(y\right)g_0\left(x\right)$ with some $g_0:S\to X$. Clearly, the equation with $y=0$ yields $2g\left(x\right)=h\left(0\right)g_0\left(x\right)$ for $x\in S$, which implies that Equation (48) is fulfilled. Consequently,

$$\underset{(x,y)\in S\times S}{sup}\parallel f(x+y)+f(x+\sigma \left(y\right))-g(x+y)-g(x+\sigma \left(y\right))\parallel \le \delta ,$$

which (with $y=0$) yields (49).

Now, assume that h and g satisfy Equation (48), and condition (49) is valid. Then,

$$\begin{array}{cc}\hfill \parallel f(x+y)& +f(x+\sigma (y\left)\right)-h\left(y\right)g\left(x\right)\parallel \hfill \\ \hfill & =\parallel f(x+y)+f(x+\sigma (y\left)\right)-g(x+y)-g(x+\sigma (y\left)\right)\parallel \hfill \\ \hfill & \le \parallel f(x+y)-g(x+y)\parallel +\parallel f(x+\sigma (y\left)\right)-g(x+\sigma (y\left)\right)\parallel \le \delta \hfill \end{array}$$

for every $x, y\in S$, which means that (47) holds. □

Analogously, we can obtain the subsequent similar result for the radical equation

$$a_{1}f\left(\sqrt[n]{x^n+y^n}\right)+a_{2}f\left(\sqrt[n]{x^n+\sigma \left(y^n\right)}\right)=h\left(y\right)g\left(x\right)$$

(50)

with $n\in \mathbb{N}$, $n>1$, $a_1, a_2\in \mathbb{K}$, $h:\mathbb{R}\to \mathbb{K}$, $f,g:\mathbb{R}\to X$, and $\sigma :\mathbb{R}\to \mathbb{R}$. We must assume here that n is odd or $\sigma \left(y\right)\ge 0$ for every $y\in {\mathbb{R}}_{+}$. Note that (50) is a generalization of radical Equations (8)–(10).

Corollary 6.

Let $n>1$ be an odd integer, $\sigma :\mathbb{R}\to \mathbb{R}$, $\sigma \left(0\right)=0$, $h:\mathbb{R}\to \mathbb{K}$ be unbounded, $h\left(0\right)\ne 0$, and δ be a positive real number. If $f,g:\mathbb{R}\to X$ satisfy the inequality

$$\underset{x,y\in \mathbb{R}}{sup}\parallel a_{1}f\left(\sqrt[n]{x^n+y^n}\right)+a_{2}f\left(\sqrt[n]{x^n+\sigma \left(y^n\right)}\right)-h\left(y\right)g\left(x\right)\parallel \le \delta ,$$

(51)

then h and g satisfy the equation

$$a_{1}g\left(\sqrt[n]{x^n+y^n}\right)+a_{2}g\left(\sqrt[n]{x^n+\sigma \left(y^n\right)}\right)=h\left(y\right)g_0\left(x\right)$$

(52)

with some $g_0:\mathbb{R}\to X$ and,

(i) 

in the case $h\left(0\right)=a_1+a_2\ne 0$, we have $g_0=g$ and

$$\underset{x\in \mathbb{R}}{sup}\parallel f\left(x\right)-g\left(x\right)\parallel \le {\displaystyle \frac{\delta }{|a_1+a_2|}};$$

(53)

(ii) 

in the case $a_1+a_2=0$, $g_0\left(x\right)\equiv 0$.

Moreover, if $a_1{a}_2>0$, then (52), with $g_0=g$, and (53) imply (51).

Proof. 

First, assume that (51) holds. Then, by Corollary 2 (see Example 1), h and g satisfy Equation (52) with some $g_0:\mathbb{R}\to X$.

If $h\left(0\right)=a_1+a_2\ne 0$, then (52) (with $y=0$) yields $g_0=g$, whence, by (51) and (52) (with $y=0$), we obtain (53). If $a_1+a_2=0$, then analogously we get $g_0\left(x\right)\equiv 0$.

Now, assume that $a_1{a}_2>0$, h and $g=g_0$ satisfy Equation (52), and (53) is valid. Then, $|a_1|+|a_2|=|a_1+a_2|$, and consequently,

$$\begin{array}{cc}\hfill \parallel a_{1}f\left(\sqrt[n]{x^n+y^n}\right)& +a_{2}f\left(\sqrt[n]{x^n+\sigma \left(y^n\right)}\right)-h\left(y\right)g\left(x\right)\parallel \hfill \\ \hfill =& \parallel a_{1}f\left(\sqrt[n]{x^n+y^n}\right)+a_{2}f\left(\sqrt[n]{x^n+\sigma \left(y^n\right)}\right)\hfill \\ \hfill & -a_{1}g\left(\sqrt[n]{x^n+y^n}\right)-a_{2}g\left(\sqrt[n]{x^n+\sigma \left(y^n\right)}\right)\parallel \hfill \\ \hfill \le & |a_1|\parallel f\left(\sqrt[n]{x^n+y^n}\right)-g\left(\sqrt[n]{x^n+y^n}\right)\parallel \hfill \\ \hfill & +|a_2|\parallel f\left(\sqrt[n]{x^n+\sigma \left(y^n\right)}\right)-g\left(\sqrt[n]{x^n+\sigma \left(y^n\right)}\right)\parallel \hfill \\ \hfill \le & \left(\right|a_1|+|a_2\left|\right){\displaystyle \frac{\delta }{|a_1+a_2|}}=|a_1+a_1|{\displaystyle \frac{\delta }{|a_1+a_2|}}=\delta ,\forall x, y\in \mathbb{R},\hfill \end{array}$$

which means that (51) holds. □

In the cases of Equations (9) and (10) (when $\sigma \left(t\right)=-t$ for $t\in \mathbb{R}$), we obtain better results than in Corollary 6 (ii). That is, we have the following two corollaries.

Corollary 7.

Let $n\in \mathbb{N}$ be odd, $n>1$, $h:\mathbb{R}\to \mathbb{K}$ be unbounded, and δ be a positive real number. Then the following two statements are valid.

(a) 

If $h\left(0\right)\ne 0$ and $f,g:\mathbb{R}\to X$ satisfy the inequality

$$\underset{x,y\in \mathbb{R}}{sup}\parallel f\left(\sqrt[n]{x^n+y^n}\right)-f\left(\sqrt[n]{x^n-y^n}\right)-h\left(y\right)g\left(x\right)\parallel \le \delta ,$$

(54)

then g is a constant function and

$$\underset{y\in \mathbb{R}}{sup}\parallel f\left(\sqrt[n]{2}y\right)-f\left(0\right)-h\left(y\right)g\left(0\right)\parallel \le \delta .$$

(55)

(b) 

If $f:\mathbb{R}\to X$ satisfies the inequality

$$\underset{x,y\in \mathbb{R}}{sup}\parallel f\left(\sqrt[n]{x^n+y^n}\right)-f\left(\sqrt[n]{x^n-y^n}\right)-h\left(y\right)f\left(x\right)\parallel \le \delta ,$$

(56)

then it is a solution to the equation

$$f\left(\sqrt[n]{x^n+y^n}\right)-f\left(\sqrt[n]{x^n-y^n}\right)=h\left(y\right)f\left(x\right).$$

(57)

Proof. 

First, assume that $h\left(0\right)\ne 0$ and (54) holds. Then, by Corollary 2, h and g satisfy the equation

$$g\left(\sqrt[n]{x^n+y^n}\right)-g\left(\sqrt[n]{x^n-y^n}\right)=h\left(y\right)g_0\left(x\right)$$

with some $g_0:\mathbb{R}\to X$, whence (with $y=0$) we obtain $g_0\left(x\right)\equiv 0$. Consequently, from the equation we deduce that g is constant. Now, (54) (with $x=y$) yields (55).

If (56) holds, then Corollary 2 (with $f=g$) implies that f satisfies Equation (57). □

Corollary 8.

Let $n\in \mathbb{N}$ be odd, $n>1$, $h:\mathbb{R}\to \mathbb{K}$ be unbounded, and δ be a positive real number. Then the following two statements are valid.

(a) 

If $h\left(0\right)\ne 0$ and $\psi ,g:\mathbb{R}\to X$ satisfy the inequality

$$\underset{x,y\in \mathbb{R}}{sup}\parallel \psi {\left(\sqrt[n]{{\displaystyle \frac{y^n+x^n}{2}}}\right)}^2-\psi {\left(\sqrt[n]{{\displaystyle \frac{x^n-y^n}{2}}}\right)}^2-h\left(y\right)g\left(x\right)\parallel \le \delta ,$$

(58)

then g is a constant function and

$$\underset{y\in \mathbb{R}}{sup}\parallel \psi {\left(y\right)}^2-\psi {\left(0\right)}^2-h\left(y\right)g\left(0\right)\parallel \le \delta .$$

(b) 

If $\psi :\mathbb{R}\to X$ satisfies the inequality

$$\underset{x,y\in \mathbb{R}}{sup}\parallel \psi {\left(\sqrt[n]{{\displaystyle \frac{y^n+x^n}{2}}}\right)}^2-\psi {\left(\sqrt[n]{{\displaystyle \frac{x^n-y^n}{2}}}\right)}^2-h\left(y\right)\psi \left(x\right)\parallel \le \delta ,$$

(59)

then it is a solution to the equation

$$\psi {\left(\sqrt[n]{{\displaystyle \frac{y^n+x^n}{2}}}\right)}^2-\psi {\left(\sqrt[n]{{\displaystyle \frac{x^n-y^n}{2}}}\right)}^2=h\left(y\right)\psi \left(x\right).$$

Proof. 

Define $f:\mathbb{R}\to X$ by $f\left(t\right)=\psi {\left(2^{-1/n}t\right)}^2$ for $x\in \mathbb{R}$. Then, it is easily seen that (58) and (59) are just (54) and (56), respectively. Consequently, it is enough to use Corollary 7. □

We end this paper with an example of a somewhat surprising consequence of Corollary 5.

Corollary 9.

Let $(S,+)$ be an abelian group, $h:S\to \mathbb{K}$ be unbounded, $h\left(0\right)=2$, and $h{\left(x_0\right)}^2\ne h\left(2x_0\right)+2$ for some $x_0\in S$. Then,

$$\underset{x,y\in S}{sup}\parallel f(x+y)+f(x-y)-h\left(y\right)h\left(x\right)\parallel =\infty$$

for every function $f:S\to \mathbb{K}$.

Proof. 

Suppose that

$$\underset{x,y\in S}{sup}\parallel f(x+y)+f(x-y)-h\left(y\right)h\left(x\right)\parallel <\infty .$$

Then, by Corollary 5 (with $S=T$, $X=\mathbb{K}$, $\sigma \left(y\right)\equiv -y$ and $g=h$), h is a solution to the equation

$$h(x+y)+h(x-y)=h\left(y\right)h\left(x\right).$$

(60)

Taking $x=y=x_0$ in (60), we obtain $h{\left(x_0\right)}^2=h\left(2x_0\right)+2$, which is a contradiction. □

5. Conclusions

The well-known problem of Ulam stability concerns the size of the difference between the solutions and approximate solutions to various equations. During the study of it, a particular phenomenon was discovered that is now called superstability, which means that every mapping, for which the difference between the left- and right-hand sides of an equation is bounded, must be either bounded or be a solution to the equation.

In this paper, we prove a superstability theorem for the pexiderized general functional Equation (11) with the unknown functions $g:T\to X$, $h:S\to \mathbb{K}$ and $f:S\to X$, under the assumption that the series ${\sum }_{j=1}^{\infty }f\left({\gamma }_j(t,s)\right)$ is convergent for every $(s,t)\in S\times T$. We present some examples showing that this outcome implies various new superstability results for pexiderized versions of several well-known equations, as well as for their radical versions (see, e.g., Corollaries 5 and 6, with $a_1=a_2=1$).

One of our main assumptions is inequality (14), and there arises a natural question as to whether this condition can be weakened for instance in the following way:

$$\parallel \sum _{j=1}^{\infty }{a}_{j}f\left({\gamma }_j(t,s)\right)-h\left(t\right)f\left(s\right)\parallel \le \eta \left(s\right),\forall t\in T, s\in S,$$

(61)

with fixed $\eta :S\to {\mathbb{R}}_{+}$ (cf., e.g., [28,46,47]). Unfortunately, this does not seem to be possible, because we assume only a fairly weak condition on functions ${\gamma }_k$, i.e., condition (13). However, this weak condition allows us to include a wider range of equations in our considerations.

Further research in this area will focus on the possibility of replacing inequality (14) (in the main result of this paper) with this weaker assumption (61). However, this will probably require a more restrictive condition than (14), thus narrowing the class of equations considered.

In some cases (see, e.g., Corollaries 3 and 5), our main result allows one to obtain fairly complete descriptions of functions satisfying the inequalities considered.

Moreover, our outcomes show that there is a significant symmetry between the results on the superstability of numerous functional equations with one unknown function and those concerning their pexiderizations or radical versions. For example, from Corollary 6 (with $a_1=a_2=1$), we obtain the following result:

Corollary 10.

Let $n>1$ be an odd integer, $\sigma :\mathbb{R}\to \mathbb{R}$, $\sigma \left(0\right)=0$, $h:\mathbb{R}\to \mathbb{K}$ be unbounded, $h\left(0\right)=2$, and δ be a positive real number. Then, $f,g:\mathbb{R}\to X$ satisfy the inequality

$$\underset{x,y\in \mathbb{R}}{sup}\parallel f\left(\sqrt[n]{x^n+y^n}\right)+f\left(\sqrt[n]{x^n+\sigma \left(y^n\right)}\right)-h\left(y\right)g\left(x\right)\parallel \le \delta ,$$

if and only if h and g satisfy the equation

$$g\left(\sqrt[n]{x^n+y^n}\right)+g\left(\sqrt[n]{x^n+\sigma \left(y^n\right)}\right)=h\left(y\right)g\left(x\right),$$

and

$$\underset{x\in \mathbb{R}}{sup}\parallel f\left(x\right)-g\left(x\right)\parallel \le {\displaystyle \frac{\delta }{2}}.$$

It is easily seen that Corollaries 10 and 5 have very symmetric forms. We also note that there is some symmetry between the results of J. Baker, J. Lawrence, and F. Zorzitto [17] and Corollaries 3 and 4.

Finally, let us mention that from Theorem 3 one can derive (in an analogous way) various similar results for some other equations (which are special cases of Equation (11)), in which analogous symmetries will appear.