Stability of a General Functional Equation

Abstract

In this paper, we investigate a general multivariable functional equation. We prove, using the fixed-point method, the generalized Hyers–Ulam stability of this equation in Banach spaces. In this way, we obtain sufficient conditions for the stability of a wide class of functional equations and control functions. We also show, using examples, how some additional assumptions imposed on the function when examining the Hyers–Ulam stability of a functional equation affect the size of the approximating constant and limit the number of considered solutions for this equation. The functional equation studied in this paper has symmetric coefficients (with precision up to the sign), and it is a generalization of an equation characterizing n-quadratic functions, as well as many other functional equations with symmetric coefficients: for example, the multi-Cauchy equation and the multi-Jensen equation. Our results generalize many known outcomes.

1. Introduction

The theory of the stability of functional equations began with Hyers’ answer to his famous question posed in 1940: “When is it true that the solution of an equation differing slightly from a given one must of necessity be close to the solution of the given equation?”. Since then, many authors have investigated this topic (see, for example, books [1,2,3,4]), generalizing Ulam’s problem and Hyers’ theorem in different directions and dealing with various equations: for example, functional, differential, fractional differential, difference, and integral equations (see, for example, recent papers [5,6,7,8,9,10,11]).

When modeling physical processes described by functional equations, various errors and deviations often appear. The question then arises as to what extent a small perturbation of a state affects that state, which leads to the consideration of stability problems.

In [12], the author introduced the following equation:

$$\begin{array}{cc}\hfill \sum _{i_1,i_2,\dots ,i_n\in \{-1,1\}}f(a_{1,i_1,i_2,\dots ,i_n}(x_{11}+i_1{x}_{12}),\dots ,a_{n,i_1,i_2,\dots ,i_n}& (x_{n1}+i_n{x}_{n2}))\hfill \\ \hfill =\sum _{j_1,j_2,\dots ,j_n\in \{1,2\}}{A}_{j_1,\dots ,j_n}f(x_{1j_1},\dots ,x_{n{j}_n}),& x_{11},x_{12},\dots ,x_{n1},x_{n2}\in X\hfill \end{array}$$

(1)

for some $a_{1,i_1,i_2,\dots ,i_n},\dots ,a_{n,i_1,i_2,\dots ,i_n}\in \mathbb{F},$$A_{j_1,\dots ,j_n}\in \mathbb{K}$, where $f:X^n\to Y$, X is a linear space over the field $\mathbb{F}$, and Y is a linear space over the field $\mathbb{K}$.

In [13], Ciepliński proved the generalized Hyers–Ulam stability of functional Equation (1) in Banach spaces assuming additionally that $f:X^n\to Y$ is a function such that

$$\begin{array}{cc}\hfill f(x_1,\dots ,x_n)=0\mathrm{for} \mathrm{any}{x}_1,\dots ,x_n\in X& \mathrm{with} \mathrm{at} \mathrm{least} \mathrm{one} \mathrm{component}\hfill \\ \hfill \mathrm{which} \mathrm{is} \mathrm{equal} \mathrm{to} \mathrm{zero}.\end{array}$$

(A)

Equation (1) was introduced as a natural generalization of the following equation:

$$\begin{array}{cc}\hfill \sum _{i_1,\dots ,i_n\in \{-1,1\}}f\left(x_{11}+i_1{x}_{12},\dots ,x_{n1}+i_n{x}_{n2}\right)& \hfill \\ \hfill =\sum _{j_1,\dots ,j_n\in \{1,2\}}{2}^{n}f(x_{1j_1},\dots ,x_{n{j}_n}),& x_{11},x_{12},\dots ,x_{n1},x_{n2}\in X,\hfill \end{array}$$

(2)

which characterizes the so-called multi-quadratic functions (a quadratic in each variable).

In [14], the general solution of Equation (1) for $n\in \{1,2\}$ was described. It does not follow from this description that assumption (A) is natural. None of the six examples of solutions given there relative to some special forms of Equation (1) for $n=2$ have this property. However, assumption (A) is natural in some special examples of equations of form (1), e.g., for multi-Cauchy, multi-quadratic equations. The price of accepting this assumption is that we cannot apply the theorem to the solutions of (1), which does not satisfy assumption (A), and this severely limits the number of solutions that can be considered for (1). Assumption (A) is crucial to the proof given in [13], and it implies that the sum

$$\sum _{i_1,\dots ,i_n\in \{-1,1\}}f\left(a_{1,i_1,\dots ,i_n}(x_1+i_1{x}_1),\dots ,a_{n,i_1,\dots ,i_n}(x_n+i_n{x}_n)\right)$$

reduces to a single term $f\left(2a_{1,1,\dots ,1}{x}_1,\dots ,2a_{n,1,\dots ,1}{x}_n\right),$ which greatly simplifies the proof.

Let us also mention that Equation (1), which has symmetric coefficients (with precision up to the sign), generalizes many well-known functional equations (one of them is clearly (2), because substituting

$$g(x_1,x_2,\dots ,x_n):=f(x_1,x_2,\dots ,x_n)-f(0,\dots ,0),x_1,x_2,\dots ,x_n\in X,$$

(3)

from (1), we get, for example, multi-Cauchy (4), multi-Jensen (5), multi-Cauchy–Jensen (6), multi-Cauchy–quadratic (7), and multi-Jensen–quadratic (8) equations:

$$g(x_{11}+x_{12},\dots ,x_{n1}+x_{n2})=\sum _{j_1,\dots ,j_n\in \{1,2\}}g(x_{1j_1},\dots ,x_{n{j}_n}),$$

(4)

$$g\left({\displaystyle \frac{1}{2}}{x}_{11}+{\displaystyle \frac{1}{2}}{x}_{12},\dots ,{\displaystyle \frac{1}{2}}{x}_{n1}+{\displaystyle \frac{1}{2}}{x}_{n2}\right)=\sum _{j_1,\dots ,j_n\in \{1,2\}}{\displaystyle \frac{1}{2^n}}g(x_{1j_1},\dots ,x_{n{j}_n}),$$

(5)

$$\begin{array}{cc}\hfill g(x_{11}+x_{12},& \dots ,x_{k1}+x_{k2},{\displaystyle \frac{1}{2}}{x}_{k+1,1}+{\displaystyle \frac{1}{2}}{x}_{k+1,2},\dots ,{\displaystyle \frac{1}{2}}{x}_{n1}+{\displaystyle \frac{1}{2}}{x}_{n2})\hfill \\ \hfill & =\sum _{j_1,\dots ,j_n\in \{1,2\}}{\displaystyle \frac{1}{2^{n-k}}}g(x_{1j_1},\dots ,x_{n{j}_n}),\hfill \end{array}$$

(6)

$$\begin{array}{cc}\hfill \sum _{i_{k+1},\dots ,i_n\in \{-1,1\}}& f\left(x_{11}+x_{12},\dots ,x_{k1}+x_{k2},x_{k+1,1}+i_{k+1}{x}_{k+1,2},\dots ,x_{n1}+i_n{x}_{n2}\right)\hfill \\ \hfill & =\sum _{j_1,\dots ,j_n\in \{1,2\}}{2}^{n-k}f(x_{1j_1},\dots ,x_{n{j}_n}),\hfill \end{array}$$

(7)

$$\begin{array}{cc}\hfill \sum _{i_{k+1},\dots ,i_n\in \{-1,1\}}& f\left({\displaystyle \frac{1}{2}}{x}_{11}+{\displaystyle \frac{1}{2}}{x}_{12},\dots ,{\displaystyle \frac{1}{2}}{x}_{k1}+{\displaystyle \frac{1}{2}}{x}_{k2},x_{k+1,1}+i_{k+1}{x}_{k+1,2},\dots ,x_{n1}+i_n{x}_{n2}\right)\hfill \\ \hfill & =\sum _{j_1,\dots ,j_n\in \{1,2\}}{2}^{n-2k}f(x_{1j_1},\dots ,x_{n{j}_n}),\hfill \end{array}$$

(8)

where $k\in \{0,\dots ,n\}$ is fixed.

The stability and generalized Hyers–Ulam stability of the special cases of functional Equation (1), including those mentioned above, have been studied by many mathematicians (see, e.g., [15,16,17,18,19,20,21,22,23,24]).

In this paper, we prove the generalized Hyers–Ulam stability of Equation (1) in Banach spaces without assuming condition (A) (see Theorem 2). To obtain the stability and generalized stability of Equations (2), (4), and (6)–(8), we prove Theorem 3, in which we additionally assume that $f(0,\dots ,0)=0.$

We also show, using examples, how additional assumptions imposed on function f, such as $f(0,\dots ,0)=0$ or condition (A), when studying the Hyers–Ulam stability of a functional equation affect the size of the approximating constant and limit the number of considered solutions for this equation.

In the proof of our stability results, we use the fixed-point method, which was first used to study the Hyers–Ulam stability of functional equations by Baker [25]. For more information on this method, we refer the reader to the reviews [26,27,28,29].

Our results can be applied to prove the stability and generalized stability of various functional equations with general form (1): for example, Equations (2), (4), and (6)–(8). In this way, our results generalize several known facts.

Throughout this paper, $\mathbb{N}$ denotes the set of all positive integers; $n,k\in \mathbb{N}$, ${\mathbb{N}}_0=\mathbb{N}\cup \left\{0\right\}$, ${\mathbb{R}}_{+}:=[0,+\infty ),$ and $C^D$ denote the set of functions $f:D\to C.$ Furthermore, we assume that X is a vector space over the field $\mathbb{F}\in \{\mathbb{R},\mathbb{C}\}$, Y is a Banach space over the field $\mathbb{K}\in \{\mathbb{R},\mathbb{C}\}$, and the coefficients in (1) are such that

$$\begin{array}{c}\hfill A:=\sum _{j_1,\dots ,j_n\in \{1,2\}}{A}_{j_1,\dots ,j_n}\ne 0\end{array}$$

(9)

and

$$\begin{array}{c}\hfill a_{l,1,\dots ,1}\ne {\displaystyle \frac{1}{2}}\mathrm{for} \mathrm{some}l\in \{1,\dots ,n\}.\end{array}$$

(10)

In the next section, we prove our main result (Theorem 2), which provides a criterion for the generalized Hyers–Ulam stability of the equations with form (1) in Banach spaces. In Section 3, we discuss some consequences of Theorem 2. Namely, first, we present Theorem 3 and apply it to obtain the well-known results of the stability of Equations (2), (4), and (6)–(8). Then, we show, using examples, how some additional assumptions imposed on function f when examining the stability of Equation (1) affect the size of the approximating constant and limit the number of considered solutions for this equation. Finally, we use our criterion to prove the generalized Hyers–Ulam stability of two functional equations with a popular control function.

2. Generalized Stability of Equation (1) in Banach Spaces

In this section, we prove the generalized (in the spirit of D.G. Bourgin and P. Găvruţa) Hyers–Ulam stability of Equation (1) in Banach spaces. The proof is based on the fixed-point result, which can be derived from Theorem 1 [30]. To present this, we introduce the following three hypotheses.

Hypothesis 1.

E is a nonempty set; Y is a Banach space; $f_1,\dots ,f_k:E\to E$ and $L_1,\dots ,L_k:E\to {\mathbb{R}}_{+},$$k\in \mathbb{N}$.

Hypothesis 2.

$\mathcal{T}:Y^E\to Y^E$ is an operator satisfying the following inequality:

$$\begin{array}{c}\hfill \parallel \mathcal{T}\xi \left(x\right)-\mathcal{T}\mu \left(x\right)\parallel \le \sum _{i=1}^k{L}_i\left(x\right)\parallel \xi \left(f_i\left(x\right)\right)-\mu \left(f_i\left(x\right)\right)\parallel ,\\ \hfill \xi ,\mu \in Y^E,x\in E.\end{array}$$

Hypothesis 3.

$\Lambda :{{\mathbb{R}}_{+}}^E\to {{\mathbb{R}}_{+}}^E$ is an operator defined by the following:

$$\Lambda \delta \left(x\right):=\sum _{i=1}^k{L}_i\left(x\right)\delta \left(f_i\left(x\right)\right),\delta \in {{\mathbb{R}}_{+}}^E,x\in E.$$

Now, we are in a position to present the mentioned fixed-point theorem.

Theorem 1.

Let Hypotheses 1–3 hold, and let functions $\epsilon :E\to {\mathbb{R}}_{+}$ and $\phi :E\to Y$ fulfill the following two conditions:

$$\parallel \mathcal{T}\phi \left(x\right)-\phi \left(x\right)\parallel \le \epsilon \left(x\right),x\in E$$

and

$${\epsilon }^{*}\left(x\right):=\sum _{m=0}^{\infty }{\Lambda }^m\epsilon \left(x\right)<\infty ,x\in E.$$

Then, there exists a unique fixed-point ψ of $\mathcal{T}$ such that

$$\parallel \phi \left(x\right)-\psi \left(x\right)\parallel \le {\epsilon }^{*}\left(x\right),x\in E.$$

Moreover,

$$\psi \left(x\right):=\underset{m\to \infty }{lim}{\mathcal{T}}^m\phi \left(x\right),x\in E.$$

We denote

$$\begin{array}{cc}\hfill S:& =\{(i_1,\dots ,i_n)\in {\{-1,1\}}^n\},\hfill \\ \hfill (\Phi f)& (x_{11},\dots ,x_{n1},x_{12},\dots ,x_{n2}):=\sum _{(i_1,\dots ,i_n)\in S}f\left(a_{1,i_1,\dots ,i_n}(x_{11}+i_1{x}_{12}),\dots ,a_{n,i_1,\dots ,i_n}(x_{n1}+i_n{x}_{n2})\right)\hfill \\ \hfill & -\sum _{j_1,\dots ,j_n\in \{1,2\}}{A}_{j_1,\dots ,j_n}f(x_{1j_1},\dots ,x_{n{j}_n}).\hfill \end{array}$$

With this notation, we have the following result.

Theorem 2.

Let $f:X^n\to Y$ and $\theta :X^{2n}\to {\mathbb{R}}_{+}$ be mappings satisfying the inequality

$$\begin{array}{cc}\hfill \parallel (\Phi f)& (x_{11},\dots ,x_{n1},x_{12},\dots ,x_{n2})\parallel \le \theta (x_{11},\dots ,x_{n1},x_{12},\dots ,x_{n2})\hfill \end{array}$$

(11)

for $x_{11},x_{12},\dots ,x_{n1},x_{n2}\in X.$

Assume also that there exists ${\omega }_{i_1,\dots ,i_n}\in {\mathbb{R}}_{+}$ such that 

$$\begin{array}{cc}\hfill \theta ((1+i_1)a_{1,i_1,\dots ,i_n}& x_{11},\dots ,(1+i_n)a_{n,i_1,\dots ,i_n}{x}_{n1},(1+i_1)a_{1,i_1,\dots ,i_n}{x}_{12},\dots ,(1+i_n)a_{n,i_1,\dots ,i_n}{x}_{n2})\hfill \\ \hfill & \le {\omega }_{i_1,\dots ,i_n}\theta (x_{11},\dots ,x_{n1},x_{12},\dots ,x_{n2})\hfill \end{array}$$

(12)

for $(i_1,\dots ,i_n)\in S$ and

$$\gamma :={\displaystyle \frac{1}{\left|A\right|}}\left(\sum _{(i_1,\dots ,i_n)\in S}{\omega }_{i_1,\dots ,i_n}\right)<1.$$

Then, there exists a unique solution $F:X^n\to Y$ of (1) such that

$$\begin{array}{c}\hfill \parallel f(x_1,\dots ,x_n)-F(x_1,\dots ,x_n)\parallel \le {\displaystyle \frac{\theta (x_1,\dots ,x_n,x_1,\dots ,x_n)}{\left|A\right|(1-\gamma )}},x_1,\dots ,x_n\in X.\end{array}$$

(13)

Moreover, F is a unique solution of (1) such that there exists a constant $B\in (0,+\infty )$ with

$$\begin{array}{c}\hfill \parallel f(x_1,\dots ,x_n)-F(x_1,\dots ,x_n)\parallel \le B\theta (x_1,\dots ,x_n,x_1,\dots ,x_n),x_1,\dots ,x_n\in X.\end{array}$$

(14)

Proof. 

Substituting $x_{j1}=x_{j2}=x_j$ for $j\in \{1,\dots ,n\}$ in (11), we get

$$\parallel \sum _{(i_1,\dots ,i_n)\in S}f\left((1+i_1)a_{1,i_1,\dots ,i_n}{x}_1,\dots ,(1+i_n)a_{n,i_1,\dots ,i_n}{x}_n\right)-Af(x_1,\dots ,x_n)\parallel$$

$$\le \theta (x_1,\dots ,x_n,x_1,\dots ,x_n)\mathrm{for}\mathrm{all}{x}_1,\dots ,x_n\in X,$$

where

$$\begin{array}{cc}\hfill \parallel & {\displaystyle \frac{1}{A}}\sum _{(i_1,\dots ,i_n)\in S}f\left((1+i_1)a_{1,i_1,\dots ,i_n}{x}_1,\dots ,(1+i_n)a_{n,i_1,\dots ,i_n}{x}_n\right)-f(x_1,\dots ,x_n)\parallel \hfill \\ \hfill & \le {\displaystyle \frac{1}{\left|A\right|}}\theta (x_1,\dots ,x_n,x_1,\dots ,x_n)\mathrm{for}\mathrm{all}{x}_1,\dots ,x_n\in X.\hfill \end{array}$$

(15)

Define

$$\left(\mathcal{T}\xi \right)(x_1,\dots ,x_n):={\displaystyle \frac{1}{A}}\sum _{(i_1,\dots ,i_n)\in S}\xi \left((1+i_1)a_{1,i_1,\dots ,i_n}{x}_1,\dots ,(1+i_n)a_{n,i_1,\dots ,i_n}{x}_n\right)$$

for all $\xi \in Y^{X^n},x_1,\dots ,x_n\in X$ and

$$\epsilon (x_1,\dots ,x_n):={\displaystyle \frac{1}{\left|A\right|}}\theta (x_1,\dots ,x_n,x_1,\dots ,x_n)$$

for all $x_1,\dots ,x_n\in X$. Then, for any $\xi ,\mu :X^n\to Y,$$x_1,\dots ,x_n\in X$, we have

$$\begin{array}{cc}\hfill \parallel \left(\mathcal{T}\xi \right)(x_1,\dots ,x_n)& -\left(\mathcal{T}\mu \right)(x_1,\dots ,x_n)\parallel \hfill \\ \hfill \le \sum _{(i_1,\dots ,i_n)\in S}{\displaystyle \frac{1}{\left|A\right|}}& \parallel \xi \left((1+i_1)a_{1,i_1,\dots ,i_n}{x}_1,\dots ,(1+i_n)a_{n,i_1,\dots ,i_n}{x}_n\right)\hfill \\ & -\mu \left((1+i_1)a_{1,i_1,\dots ,i_n}{x}_1,\dots ,(1+i_n)a_{n,i_1,\dots ,i_n}{x}_n\right)\parallel ,\hfill \end{array}$$

Thus, Hypothesis 2 is valid, and by (15),

$$\parallel \left(\mathcal{T}f\right)(x_1,\dots ,x_n)-f(x_1,\dots ,x_n)\parallel \le \epsilon (x_1,\dots ,x_n)$$

for all $x_1,\dots ,x_n\in X$.

Next, substitute

$$(\Lambda \eta )(x_1,\dots ,x_n):={\displaystyle \frac{1}{\left|A\right|}}\sum _{(i_1,\dots ,i_n)\in S}\eta \left((1+i_1)a_{1,i_1,\dots ,i_n}{x}_1,\dots ,(1+i_n)a_{n,i_1,\dots ,i_n}{x}_n\right)$$

for all $\eta \in {\mathbb{R}}_{+}^{X^n}$, $x_1,\dots ,x_n\in X$. It is easily seen that $\Lambda$ has the form described in Hypothesis 3.

Now, we prove

$$\begin{array}{c}\hfill \sum _{m=0}^{\infty }\left({\Lambda }^m\epsilon \right)(x_1,\dots ,x_n)\le {\displaystyle \frac{\epsilon (x_1,\dots ,x_n)}{1-\gamma }},x_1,\dots ,x_n\in X.\end{array}$$

(16)

First, we show that for every $x_1,\dots ,x_n\in X$ and $m\in {\mathbb{N}}_0$,

$$\begin{array}{c}\hfill \left({\Lambda }^m\epsilon \right)(x_1,\dots ,x_n)\le {\gamma }^m\epsilon (x_1,\dots ,x_n).\end{array}$$

(17)

The above inequality is fulfilled for $m=0$. Take $x_1,\dots ,x_n\in X$, $m\in {\mathbb{N}}_0$, and assume (17). Thus,

$$\begin{array}{ccc}\hfill \left({\Lambda }^{m+1}\epsilon \right)(x_1,\dots ,x_n)& =& \Lambda \left({\Lambda }^m\epsilon \right)(x_1,\dots ,x_n)\hfill \\ & =& {\displaystyle \frac{1}{\left|A\right|}}\sum _{(i_1,\dots ,i_n)\in S}\left({\Lambda }^m\epsilon \right)\left((1+i_1)a_{1,i_1,\dots ,i_n}{x}_1,\dots ,(1+i_n)a_{n,i_1,\dots ,i_n}{x}_n)\right)\hfill \\ & \le & {\displaystyle \frac{1}{\left|A\right|}}\sum _{(i_1,\dots ,i_n)\in S}{\gamma }^m\epsilon \left((1+i_1)a_{1,i_1,\dots ,i_n}{x}_1,\dots ,(1+i_n)a_{n,i_1,\dots ,i_n}{x}_n\right)\hfill \\ & \le & {\gamma }^m\left(\sum _{(i_1,\dots ,i_n)\in S}{\displaystyle \frac{1}{\left|A\right|}}{\omega }_{i_1,\dots ,i_n}\right)\epsilon (x_1,\dots ,x_n)\hfill \\ & =& {\gamma }^{m+1}\epsilon (x_1,\dots ,x_n)\hfill \end{array}$$

and by induction, the proof of condition (17) is completed. Condition (16) is a consequence of the convergence of the power series.

The operators $\mathcal{T}:Y^{X^n}\to Y^{X^n}$ and $\Lambda :{\mathbb{R}}_{+}^{X^n}\to {\mathbb{R}}_{+}^{X^n}$ satisfy the assumptions of Theorem 1. Therefore, there exists a unique fixed point $F:X^n\to Y$ of $\mathcal{T}$ such that (13) holds. Moreover,

$$F(x_1,\dots ,x_n)=\underset{m\to \infty }{lim}\left({\mathcal{T}}^{m}f\right)(x_1,\dots ,x_n),x_1,\dots ,x_n\in X.$$

(18)

Now, we prove that for any $x_{11},x_{12},\dots ,x_{n1},x_{n2}\in X$ and $m\in {\mathbb{N}}_0$, we have

$$\begin{array}{c}{\displaystyle \parallel \left(\Phi \left({\mathcal{T}}^{m}f\right)\right)(x_{11},\dots ,x_{n1},x_{12},\dots ,x_{n2})\parallel \le {\gamma }^m\theta \left(x_{11},\dots ,x_{n1},x_{12},\dots ,x_{n2}\right).}\hfill \end{array}$$

(19)

Since case $m=0$ is just (11), fix $m\in {\mathbb{N}}_0$ and assume that (19) holds for any $x_{11},x_{12},\dots ,x_{n1},x_{n2}\in X$. Then, for any $x_{11},x_{12},\dots ,x_{n1},x_{n2}\in X$, we get

$$\begin{array}{cc}\hfill \parallel & {\displaystyle \left(\Phi \left({\mathcal{T}}^{m+1}f\right)\right)(x_{11},\dots ,x_{n1},x_{12},\dots ,x_{n2})\parallel }\hfill \\ & = \parallel \sum _{(i_1,\dots ,i_n)\in S}\left(\mathcal{T}\left({\mathcal{T}}^{m}f\right)\right)\left(a_{1,i_1,\dots ,i_n}(x_{11}+i_1{x}_{12}),\dots ,a_{n,i_1,\dots ,i_n}(x_{n1}+i_n{x}_{n2})\right)\hfill \\ & -\sum _{j_1,\dots ,j_n\in \{1,2\}}{A}_{j_1,\dots ,j_n}\left(\mathcal{T}\left({\mathcal{T}}^{m}f\right)\right)(x_{1j_1},\dots ,x_{n{j}_n})\parallel \hfill \\ & = \parallel \sum _{(i_1,\dots ,i_n)\in S}{\displaystyle \frac{1}{A}}(\sum _{(i_1^{*},\dots ,i_n^{*})\in S}({\mathcal{T}}^{m}f)((1+i_1^{*})a_{1,i_1^{*},\dots ,i_n^{*}}{a}_{1,i_1,\dots ,i_n}(x_{11}+i_1{x}_{12}),\dots ,\hfill \\ & (1+i_n^{*})a_{n,i_1^{*},\dots ,i_n^{*}}{a}_{n,i_1,\dots ,i_n}(x_{n1}+i_n{x}_{n2})\left)\right)\hfill \\ & -\sum _{j_1,\dots ,j_n\in \{1,2\}}{A}_{j_1,\dots ,j_n}{\displaystyle \frac{1}{A}}(\sum _{(i_1^{*},\dots ,i_n^{*})\in S}({\mathcal{T}}^{m}f)\left((1+i_1^{*})a_{1,i_1^{*},\dots ,i_n^{*}}{x}_{1j_1},\dots ,(1+i_n^{*})a_{n,i_1^{*},\dots ,i_n^{*}}{x}_{n{j}_n}\right))\parallel \hfill \\ & {\displaystyle =\frac{1}{\left|A\right|}}\parallel \sum _{(i_1^{*},\dots ,i_n^{*})\in S}\Phi ({\mathcal{T}}^{m}f)((1+i_1^{*})a_{1,i_1^{*},\dots ,i_n^{*}}{x}_{11},\dots ,(1+i_n^{*})a_{n,i_1^{*},\dots ,i_n^{*}}{x}_{n1},\hfill \\ & (1+i_1^{*})a_{1,i_1^{*},\dots ,i_n^{*}}{x}_{12},\dots ,(1+i_n^{*})a_{n,i_1^{*},\dots ,i_n^{*}}{x}_{n2})\parallel \hfill \\ & {\displaystyle \le \frac{1}{\left|A\right|}}(\sum _{(i_1^{*},\dots ,i_n^{*})\in S}\parallel \Phi ({\mathcal{T}}^{m}f)((1+i_1^{*})a_{1,i_1^{*},\dots ,i_n^{*}}{x}_{11},\dots ,(1+i_n^{*})a_{n,i_1^{*},\dots ,i_n^{*}}{x}_{n1},\hfill \\ & (1+i_1^{*})a_{1,i_1^{*},\dots ,i_n^{*}}{x}_{12},\dots ,(1+i_n^{*})a_{n,i_1^{*},\dots ,i_n^{*}}{x}_{n2})\parallel )\hfill \\ & {\displaystyle \le \frac{1}{\left|A\right|}}(\sum _{(i_1^{*},\dots ,i_n^{*})\in S}{\gamma }^m\theta ((1+i_1^{*})a_{1,i_1^{*},\dots ,i_n^{*}}{x}_{11},\dots ,(1+i_n^{*})a_{n,i_1^{*},\dots ,i_n^{*}}{x}_{n1},\hfill \\ & (1+i_1^{*})a_{1,i_1^{*},\dots ,i_n^{*}}{x}_{12},\dots ,(1+i_n^{*})a_{n,i_1^{*},\dots ,i_n^{*}}{x}_{n2}\left)\right)\hfill \\ & \le {\gamma }^m{\displaystyle \frac{1}{\left|A\right|}}(\sum _{(i_1^{*},\dots ,i_n^{*})\in S}{\omega }_{i_1^{*},\dots ,i_n^{*}}\theta (x_{11},\dots ,x_{n1},x_{12},\dots ,x_{n2}))\hfill \\ & ={\gamma }^{m+1}\theta (x_{11},\dots ,x_{n1},x_{12},\dots ,x_{n2}),\hfill \end{array}$$

Thus, (19) holds for any $x_{11},x_{12},\dots ,x_{n1},x_{n2}\in X$ and $m\in {\mathbb{N}}_0$.

Letting $m\to \infty$ in (19) and using (18), we finally obtain

$$(\Phi F)(x_{11},\dots ,x_{n1},x_{12},\dots ,x_{n2})=0$$

for all $x_{11},x_{12},\dots ,x_{n1},x_{n2}\in X$, which means that F satisfies (1).

Finally, suppose that there exist $B>0$ and $H:X^n\to Y$, which is a solution of (1), such that (14) is satisfied. By the triangle inequality and (13),

$$\begin{array}{cc}\hfill \parallel F(x_1,\dots ,x_n)& -H(x_1,\dots ,x_n)\parallel \hfill \\ \hfill & \le \parallel f(x_1,\dots ,x_n)-F(x_1,\dots ,x_n)\parallel +\parallel f(x_1,\dots ,x_n)-H(x_1,\dots ,x_n)\parallel \hfill \\ \hfill & \le {\displaystyle \frac{\epsilon (x_1,\dots ,x_n)}{1-\gamma }}+B\left|A\right|\epsilon (x_1,\dots ,x_n)=\left(1+B\left|A\right|(1-\gamma )\right)\epsilon (x_1,\dots ,x_n){\displaystyle \frac{1}{1-\gamma }}\hfill \\ \hfill & =\left(1+B\left|A\right|(1-\gamma )\right)\epsilon (x_1,\dots ,x_n)\sum _{p=0}^{\infty }{\gamma }^p,x_1,\dots ,x_n\in X.\hfill \end{array}$$

(20)

Let $C:=1+B\left|A\right|(1-\gamma )$. We show that for all $l\in {\mathbb{N}}_0$ and $x_1,\dots ,x_n\in X$,

$$\begin{array}{c}\hfill \parallel F(x_1,\dots ,x_n)-H(x_1,\dots ,x_n)\parallel \le C\epsilon (x_1,\dots ,x_n)\sum _{p=l}^{\infty }{\gamma }^p.\end{array}$$

(21)

The case $l=0$ is exactly (20). Thus, fix $x_1,\dots ,x_n\in X$ and assume that (21) holds for $l\in {\mathbb{N}}_0$; we will prove this for $l+1$. Observe that for every $x_1,\dots ,x_n\in X$,

$$\left(\mathcal{T}H\right)(x_1,\dots ,x_n)={\displaystyle \frac{1}{A}}\sum _{(i_1,\dots ,i_n)\in S}H\left((1+i_1)a_{1,i_1,\dots ,i_n}{x}_1,\dots ,(1+i_n)a_{n,i_1,\dots ,i_n}{x}_n\right)=H(x_1,\dots ,x_n)$$

and H is a fixed point of $\mathcal{T}$. Consequently,

$$\begin{array}{cc}\hfill \parallel F(x_1,\dots ,x_n)& - H(x_1,\dots ,x_n)\parallel = \parallel \left(\mathcal{T}F\right)(x_1,\dots ,x_n)-\left(\mathcal{T}H\right)(x_1,\dots ,x_n)\parallel \hfill \\ \hfill =& \parallel {\displaystyle \frac{1}{A}}\sum _{(i_1,\dots ,i_n)\in S}F\left((1+i_1)a_{1,i_1,\dots ,i_n}{x}_1,\dots ,(1+i_n)a_{n,i_1,\dots ,i_n}{x}_n\right)\hfill \\ \hfill & -{\displaystyle \frac{1}{A}}\sum _{(i_1,\dots ,i_n)\in S}H\left((1+i_1)a_{1,i_1,\dots ,i_n}{x}_1,\dots ,(1+i_n)a_{n,i_1,\dots ,i_n}{x}_n\right)\parallel \hfill \\ \hfill \le & {\displaystyle \frac{1}{\left|A\right|}}\sum _{(i_1,\dots ,i_n)\in S}\parallel F\left((1+i_1)a_{1,i_1,\dots ,i_n}{x}_1,\dots ,(1+i_n)a_{n,i_1,\dots ,i_n}{x}_n\right)\hfill \\ \hfill & -H\left((1+i_1)a_{1,i_1,\dots ,i_n}{x}_1,\dots ,(1+i_n)a_{n,i_1,\dots ,i_n}{x}_n\right)\parallel \hfill \\ \hfill \le & {\displaystyle \frac{1}{\left|A\right|}}\sum _{(i_1,\dots ,i_n)\in S}C\epsilon \left((1+i_1)a_{1,i_1,\dots ,i_n}{x}_1,\dots ,(1+i_n)a_{n,i_1,\dots ,i_n}{x}_n\right)\sum _{p=l}^{\infty }{\gamma }^p\hfill \\ \hfill \le & C{\displaystyle \frac{1}{\left|A\right|}}\sum _{(i_1,\dots ,i_n)\in S}{\omega }_{i_1,\dots ,i_n}\epsilon (x_1,\dots ,x_n)\sum _{p=l}^{\infty }{\gamma }^p\hfill \\ \hfill =& C\epsilon (x_1,\dots ,x_n)\sum _{p=l+1}^{\infty }{\gamma }^p.\hfill \end{array}$$

Letting $l\to \infty$ in (21), we get $F=H$, and the proof is complete. □

Remark 1.

Theorem 2 has a rather general form, and from it, we obtain the generalized Hyers–Ulam stability of Equation (1) for an arbitrary coefficient $a_{l,i_1,\dots ,i_n}$ (assuming only that at least one of the coefficient $a_{l,1,\dots ,1}$ is different from ${\displaystyle \frac{1}{2}}$) and $A_{j_1,\dots ,j_n}$ (assuming their sum is nonzero) such that $\gamma <1$. The price of this general approach is the size of the approximating function.

3. Some Consequences of Theorem 2

To apply Theorem 2 to obtain the stability and generalized stability of the equations mentioned in the Introduction Section, we must assume that $f(0,\dots ,0)=0.$

We denote

$$\begin{array}{cc}\hfill K:& =\{(i_1,\dots ,i_n)\in S:\left((1+i_1)a_{1,i_1,\dots ,i_n},\dots ,(1+i_n)a_{n,i_1,\dots ,i_n}\right)\ne (0,\dots ,0)\}\hfill \end{array}$$

and we obtain the following.

Theorem 3.

Let $f:X^n\to Y$ and $\theta :X^{2n}\to {\mathbb{R}}_{+}$ be mappings satisfying inequality (11) for $x_{11},x_{12},\dots ,x_{n1},$$x_{n2}\in X$, and $f(0,\dots ,0)=0$ or $\theta (0,\dots ,0)=0.$

Assume also that there exists ${\omega }_{i_1,\dots ,i_n}\in {\mathbb{R}}_{+}$ such that (12) holds for every $(i_1,\dots ,i_n)\in K$ and

$$\gamma :={\displaystyle \frac{1}{\left|A\right|}}\left(\sum _{(i_1,\dots ,i_n)\in K}{\omega }_{i_1,\dots ,i_n}\right)<1.$$

Then, there exists a unique solution $F:X^n\to Y$ of (1) with $F(0,\dots ,0)=0$ if $f(0,\dots ,0)=0$ such that (13) holds.

Moreover, F is the unique solution of (1) with $F(0,\dots ,0)=0$ if $f(0,\dots ,0)=0$ such that there exists a constant $B\in (0,+\infty )$ with (14).

Proof. 

Let us first assume that $f(0,\dots ,0)=0$ and repeat the proof of Theorem 2, replacing the set S everywhere with the set $K.$ In this way, we prove Theorem 3 in the case when $f(0,\dots ,0)=0.$

• If $f(0,\dots ,0)\ne 0$ and $\theta (0,\dots ,0)=0$, we define

$$g(x_1,\dots ,x_n):=f(x_1,\dots ,x_n)-f(0,\dots ,0),x_1,\dots ,x_n\in X.$$

Then, $g(0,\dots ,0)=0$, and by (11), substituting $x_{11}=x_{12}=\dots =x_{n1}=x_{n2}=0$, we have

$$(2^n-A)f(0,\dots ,0)=0,$$

and hence

$$\parallel (\Phi g)(x_{11},\dots ,x_{n1},x_{12},\dots ,x_{n2})\parallel \le \theta (x_{11},\dots ,x_{n1},x_{12},\dots ,x_{n2})$$

for $x_{11},x_{12},\dots ,x_{n1},x_{n2}\in X.$ Consequently, according to our previous considerations, there exists a unique function $G:X^n\to Y$ satisfying (1) such that

$$\parallel g(x_1,\dots ,x_n)-G(x_1,\dots ,x_n)\parallel \le {\displaystyle \frac{\theta (x_1,\dots ,x_n,x_1,\dots ,x_n)}{\left|A\right|(1-\gamma )}},x_1,\dots ,x_n\in X.$$

Obviously, $F(x_1,\dots ,x_n):=G(x_1,\dots ,x_n)+f(0,\dots ,0)$, $x_1,\dots ,x_n\in X$ is the desired function. □

Theorem 2 and Theorem 3 can be used to prove the stability of various functional equations of general form (1). Below, we indicate suitable examples of its applications.

Let us start with the following observation. Condition (12) for $\theta \equiv \epsilon$ holds with ${\omega }_{i_1,\dots ,i_n}=1$ for every $(i_1,\dots ,i_n)\in S$; then,

$$\sum _{(i_1,\dots ,i_n)\in S}{\omega }_{i_1,\dots ,i_n}=2^n\mathrm{and}\sum _{(i_1,\dots ,i_n)\in K}{\omega }_{i_1,\dots ,i_n}=\stackrel{=}{K},$$

where $\stackrel{=}{K}:=\mathrm{card}\left(K\right)$.

From Theorem 3, substituting $f(0,\dots ,0)=0$ into (3), we can derive several consequences. For example, we obtain the well-known result of the stability of multi-Cauchy–quadratic Equation (7) (see [31]: Corollary 3).

Corollary 1.

Let $k\in \{0,\dots ,n\}$, $\epsilon >0$ and $f:X^n\to Y.$ Assume also that $f(0,\dots ,0)=0$, and for any $x_{11},x_{12},\dots ,x_{n1},x_{n2}\in X$,

$$\begin{array}{cc}\hfill \parallel \sum _{i_{k+1},\dots ,i_n\in \{-1,1\}}& f\left(x_{11}+x_{12},\dots ,x_{k1}+x_{k2},x_{k+1,1}+i_{k+1}{x}_{k+1,2},\dots ,x_{n1}+i_n{x}_{n2}\right)\hfill \\ \hfill & -\sum _{j_1,\dots ,j_n\in \{1,2\}}{2}^{n-k}f(x_{1j_1},\dots ,x_{n{j}_n})\parallel \le \epsilon .\hfill \end{array}$$

Then, there exists a unique multi-additive-quadratic function $F:X^n⟶Y$ such that

$$\parallel f(x_1,\dots ,x_n)-F(x_1,\dots ,x_n)\parallel \le {\displaystyle \frac{\epsilon }{2^{2n-k}(1-{\gamma }_k)}},x_1,\dots ,x_n\in X,$$

where

$${\gamma }_k:=\left\{\begin{array}{cc}{\displaystyle \frac{1}{2^n}}\hfill & fork\in \{1,\dots ,n\}\hfill \\ & \\ {\displaystyle \frac{2^{n-k}-1}{2^{2n-k}}}\hfill & fork=0\hfill \end{array}\right..$$

Proof. 

It is enough to apply Theorem 3 with

$$\begin{array}{cc}\hfill a_{l,i_1,\dots ,i_n}& =\left\{\begin{array}{cc}1\hfill & \mathrm{for}{i}_1=\dots =i_k=1,i_{k+1},\dots ,i_n\in \{-1,1\},l\in \{1,\dots ,n\}\hfill \\ 0\hfill & \mathrm{for} \mathrm{others}\hfill \end{array}\right.,\hfill \\ \hfill A_{j_1,\dots ,j_n}& =2^{n-k}\mathrm{for}{j}_1,\dots ,j_n\in \{1,2\},\hfill \end{array}$$

because, in this case, $A=2^{2n-k}$ and $\stackrel{=}{K}=\left\{\begin{array}{cc}{2}^{n-k}\hfill & \mathrm{for}k\in \{1,\dots ,n\}\hfill \\ 2^{n-k}-1\hfill & \mathrm{for}k=0\hfill \end{array}\right..$ □

Remark 2.

Corollary 1 immediately provides the stability of multi-Cauchy Equation (4) with $n=k$ (see [15,32]) and the stability of multi-quadratic Equation (2) with $k=0$ (compare [24,31]), which we obtain with the additional assumption that $f(0,\dots ,0)=0$.

Thanks to this assumption, we have obtained a better control constant for the multi-quadratic equation than in [31], where this assumption is not made, and worse than in [24], where the function f additionally satisfies condition (A). Note that to obtain multi-quadratic Equation (7) from Equation (1), we do not need to assume that $f(0,\dots ,0)=0$, and we can obtain the same result as that in [31] using Theorem 2.

Remark 3.

Condition (10) imposed on the coefficient $a_{l,1,\dots ,1}$, where $l\in \{1,\dots ,n\}$ in Theorems 2 and 3, excludes its application to multi-Jensen Equation (5). However, this situation changes completely if at least one of the coefficient $a_{l,1,\dots ,1}$ is different from ${\displaystyle \frac{1}{2}}$: that is, for example, when we consider multi-Cauchy–Jensen (6) and multi-Jensen–quadratic (8) equations. Namely, we have the following.

Corollary 2.

Let $k\in \{1,\dots ,n\}$, $\epsilon >0$, and $f:X^n\to Y.$ Assume also that $f(0,\dots ,0)=0$ and for any $x_{11},x_{12},\dots ,x_{n1},x_{n2}\in X$:

$$\begin{array}{cc}\hfill \parallel f(x_{11}+x_{12},\dots & ,x_{k1}+x_{k2},{\displaystyle \frac{1}{2}}{x}_{k+1,1}+{\displaystyle \frac{1}{2}}{x}_{k+1,2},\dots ,{\displaystyle \frac{1}{2}}{x}_{n1}+{\displaystyle \frac{1}{2}}{x}_{n2})\hfill \\ \hfill & -\sum _{j_1,\dots ,j_n\in \{1,2\}}{\displaystyle \frac{1}{2^{n-k}}}f(x_{1j_1},\dots ,x_{n{j}_n})\parallel \le \epsilon .\hfill \end{array}$$

(22)

Then, there exists a unique multi-additive–Jensen function $F:X^n⟶Y$ such that

$$\parallel f(x_1,\dots ,x_n)-F(x_1,\dots ,x_n)\parallel \le {\displaystyle \frac{\epsilon }{2^k-1}},x_1,\dots ,x_n\in X.$$

Proof. 

It is enough to apply Theorem 3 with the following:

$$\begin{array}{cc}\hfill a_{l,i_1,\dots ,i_n}& =\left\{\begin{array}{cc}1\hfill & \mathrm{for}{i}_1=\dots =i_n=1,l\in \{1,\dots ,k\}\hfill \\ {\displaystyle \frac{1}{2}}\hfill & \mathrm{for}{i}_1=\dots =i_n=1,l\in \{k+1,\dots ,n\}\hfill \\ 0\hfill & \mathrm{for} \mathrm{others}\hfill \end{array}\right.,\hfill \\ \hfill A_{j_1,\dots ,j_n}& ={\displaystyle \frac{1}{2^{n-k}}}\mathrm{for}{j}_1,\dots ,j_n\in \{1,2\}.\hfill \end{array}$$

(MAJ)

Because, in this case, $\stackrel{=}{K}=1$ and $A=2^k$, substituting $\gamma ={\displaystyle \frac{1}{2^k}}$, we obtain our thesis. □

Remark 4.

The above corollary corresponds to Corollary 3.3 from [17], but in this paper, we started from the following form of condition (22):

$$\begin{array}{cc}\hfill \parallel 2^{n-k}f(x_{11}+x_{12},\dots & ,x_{k1}+x_{k2},{\displaystyle \frac{1}{2}}{x}_{k+1,1}+{\displaystyle \frac{1}{2}}{x}_{k+1,2},\dots ,{\displaystyle \frac{1}{2}}{x}_{n1}+{\displaystyle \frac{1}{2}}{x}_{n2})\hfill \\ \hfill & -\sum _{j_1,\dots ,j_n\in \{1,2\}}f(x_{1j_1},\dots ,x_{n{j}_n})\parallel \le {\epsilon }^{*},\hfill \end{array}$$

from which we obtained (22) by dividing both sides of this inequality by $2^{n-k}$ and substituting $\epsilon ={\displaystyle \frac{{\epsilon }^{*}}{2^{n-k}}}.$

Corollary 3.

Let $k\in \{0,\dots ,n-1\}$, $\epsilon >0$ and $f:X^n\to Y.$ Assume also that $f(0,\dots ,0)=0$, and for any $x_{11},x_{12},\dots ,x_{n1},x_{n2}\in X$,

$$\begin{array}{cc}\hfill \parallel \sum _{i_{k+1},\dots ,i_n\in \{-1,1\}}& f\left({\displaystyle \frac{1}{2}}{x}_{11}+{\displaystyle \frac{1}{2}}{x}_{12},\dots ,{\displaystyle \frac{1}{2}}{x}_{k1}+{\displaystyle \frac{1}{2}}{x}_{k2},x_{k+1,1}+i_{k+1}{x}_{k+1,2},\dots ,x_{n1}+i_n{x}_{n2}\right)\hfill \\ \hfill & -2^{n-2k}\sum _{j_1,\dots ,j_n\in \{1,2\}}f(x_{1j_1},\dots ,x_{n{j}_n})\parallel \le \epsilon .\hfill \end{array}$$

Then, there exists a unique multi-Jensen–quadratic function $F:X^n⟶Y$ such that

$$\parallel f(x_1,\dots ,x_n)-F(x_1,\dots ,x_n)\parallel \le {\displaystyle \frac{\epsilon }{2^{2n-2k}(1-{\gamma }_k)}},x_1,\dots ,x_n\in X,$$

where

$${\gamma }_k:=\left\{\begin{array}{cc}{\displaystyle \frac{1}{2^{n-k}}}\hfill & fork\in \{1,\dots ,n-1\}\hfill \\ & \\ {\displaystyle \frac{2^{n-k}-1}{2^{2n-2k}}}\hfill & fork=0\hfill \end{array}\right..$$

Proof. 

It is enough to apply Theorem 3 with

$$\begin{array}{cc}\hfill a_{l,i_1,\dots ,i_n}& =\left\{\begin{array}{cc}{\displaystyle \frac{1}{2}}\hfill & \mathrm{for}{i}_1=\dots =i_k=1,i_{k+1},\dots ,i_n\in \{-1,1\},l\in \{1,\dots ,k\}\hfill \\ 1\hfill & \mathrm{for}{i}_1=\dots =i_k=1,i_{k+1},\dots ,i_n\in \{-1,1\},l\in \{k+1,\dots ,n\}\hfill \\ 0\hfill & \mathrm{for} \mathrm{others}\hfill \end{array}\right.,\hfill \\ \hfill A_{j_1,\dots ,j_n}& =2^{n-2k}\mathrm{for}{j}_1,\dots ,j_n\in \{1,2\},\hfill \end{array}$$

because, in this case, $A=2^{2n-2k}$ and $\stackrel{=}{K}=\left\{\begin{array}{cc}{2}^{n-k}\hfill & \mathrm{for}k\in \{1,\dots ,n-1\}\hfill \\ 2^{n-k}-1\hfill & \mathrm{for}k=0\hfill \end{array}\right..$ □

Remark 5.

The above corollary is the well-known result (see Corollary 4.1 [16]).

In [14], as an example of Equation (1), the following equation was considered:

$$\begin{array}{c}\hfill f\left(2(x_1+x_2),{\displaystyle \frac{\sqrt{2}}{2}}(y_1+y_2)\right)+f\left(2(x_1+x_2),{\displaystyle \frac{\sqrt{2}}{2}}(y_1-y_2)\right)+f\left(0,\left(4-{\displaystyle \frac{\sqrt{2}}{2}}\right)(y_1+y_2)\right)\\ \hfill +f\left(0,-{\displaystyle \frac{\sqrt{2}}{2}}(y_1-y_2)\right)=2f(x_1,y_1)+2f(x_1,y_2)+2f(x_2,y_1)+2f(x_2,y_2),\end{array}$$

(23)

and its solutions are in the form

$$f(x,y)=h_{aq}(x,y)+{\phi }_a\left(x\right)+{\psi }_a\left(y\right),$$

where $h_{aq}$ is additive with respect to the first variable and quadratic with respect to the second variable, and ${\phi }_a$ and ${\psi }_a$ are additive functions. We notice that all functions f satisfying Equation (23) have the property $f(0,0)=0$. Imposing the additional condition $f(x,0)=f(0,y)=0,x,y\in X$ on a function f satisfying Equation (23) significantly reduces the number of solutions to only additive–quadratic functions.

In this example, we will show how additional assumptions imposed on function f, limiting the number of functions considered, affect the size of the approximating constant. We compare the following corollaries.

Corollary 4.

Assume that $\epsilon >0$ and $f:X^2\to Y$ is a function satisfying the inequality

$$\begin{array}{c}\hfill \parallel f\left(2(x_1+x_2),{\displaystyle \frac{\sqrt{2}}{2}}(y_1+y_2)\right)+f\left(2(x_1+x_2),{\displaystyle \frac{\sqrt{2}}{2}}(y_1-y_2)\right)+f\left(0,\left(4-{\displaystyle \frac{\sqrt{2}}{2}}\right)(y_1+y_2)\right)\\ \hfill +f\left(0,-{\displaystyle \frac{\sqrt{2}}{2}}(y_1-y_2)\right)-2f(x_1,y_1)-2f(x_1,y_2)-2f(x_2,y_1)-2f(x_2,y_2)\parallel \le \epsilon \end{array}$$

(24)

for $x_1,x_2,y_1,y_2\in X.$

Then, there exists a unique solution $F:X^2\to Y$ of (23) such that

$$\parallel f(x,y)-F(x,y)\parallel \le {\displaystyle \frac{\epsilon }{4}},x,y\in X.$$

(25)

Proof. 

It is enough to apply Theorem 2 with $\theta \equiv \epsilon$, $x_{11}=x_1,x_{12}=x_2,x_{21}=y_1,x_{22}=y_2,$ and

$$\begin{array}{cc}\hfill a_{l,i_1,i_2}& =\left\{\begin{array}{cc}2\hfill & \mathrm{for}(i_1,i_2)\in \{(1,1),(1,-1)\}\mathrm{and}l=1\hfill \\ {\displaystyle \frac{\sqrt{2}}{2}}\hfill & \mathrm{for}(i_1,i_2)\in \{(1,1),(1,-1)\}\mathrm{and}l=2\hfill \\ 0\hfill & \mathrm{for}(i_1,i_2)\in \{(-1,1),(-1,-1)\}\mathrm{and}l=1\hfill \\ 4-{\displaystyle \frac{\sqrt{2}}{2}}\hfill & \mathrm{for}(i_1,i_2)=(-1,1)\mathrm{and}l=2\hfill \\ -{\displaystyle \frac{\sqrt{2}}{2}}\hfill & \mathrm{for}(i_1,i_2)=(-1,-1)\mathrm{and}l=2\hfill \end{array}\right.,\hfill \\ \hfill A_{j_1,j_2}& =2\mathrm{for}{j}_1,j_2\in \{1,2\}.\hfill \end{array}$$

(E)

Because, in this case, $\stackrel{=}{S}=4$ and $A=8$, substituting $\gamma ={\displaystyle \frac{1}{2}}$, we obtain our thesis. □

Corollary 5.

Let $\epsilon >0$ and $f:X^2\to Y.$ Assume also that $f(0,0)=0$ and f satisfies inequality (24) for $x_1,x_2,y_1,y_2\in X$.

Then, there exists a unique solution $F:X^2\to Y$ of (23) such that

$$\parallel f(x,y)-F(x,y)\parallel \le {\displaystyle \frac{\epsilon }{5}},x,y\in X.$$

(26)

Proof. 

It suffices to apply Theorem 3 with $x_{11}=x_1,x_{12}=x_2,x_{21}=y_1,x_{22}=y_2,$ and (E). Because, in this case, $\stackrel{=}{K}=3$ and $A=8$, substituting $\gamma ={\displaystyle \frac{3}{8}}$, we obtain our thesis. □

Corollary 6.

Let $\epsilon >0$ and $f:X^2\to Y.$ Assume also that $f(0,0)=f(x,0)=f(0,y)=0$ for $x,y\in X$ and f satisfies inequality (24) for $x_1,x_2,y_1,y_2\in X$.

Then, there exists a unique solution $F:X^2\to Y$ of (23) such that

$$F(x,0)=F(0,y)=0and\parallel f(x,y)-F(x,y)\parallel \le {\displaystyle \frac{\epsilon }{7}},x,y\in X.$$

(27)

Proof. 

It suffices to apply Corollary 10 from [13] with (E) and $A=8$ to obtain our thesis. □

Remark 6.

We notice that in Corollary 6, we conduct our study only for certain solutions of Equation (23), namely, only for those that are additive–quadratic functions, i.e., for a limited group of solutions of Equation (23). The above corollaries show that, in some cases, at the cost of limiting the number of considered solutions of the equation (e.g., to those that satisfy (A)), we can obtain a better approximation constant.

In the case where the assumptions of Corollary 6 are satisfied, the assumptions of Corollaries 4 and 5 are also satisfied. Applying Corollary 6 to the function f satisfying (24) and (A), we obtain the best approximation constant. Regardless of which corollary we use, we obtain the same solution F of Equation (23) (it also satisfies condition (A)), which is close to f (i.e., satisfying condition (27) and also conditions (25) and (26)) because there is exactly one such function.

Remark 7.

We observe that Corollary 10 from [13] can be obtained in a way analogous to Theorem 3, assuming additionally condition (A) and replacing the set K with the set

$$P:=\{(i_1,\dots ,i_n)\in S:(1+i_1)a_{1,i_1,\dots ,i_n}·\dots ·(1+i_n)a_{n,i_1,\dots ,i_n}\ne 0\}.$$

Theorems 2 and 3 can be used to prove the generalized Hyers–Ulam stability of various functional equations of the general form (1) with a wide class of control functions. Among the frequently occurring control functions in (11) is the following:

$$\theta (x_{11},\dots ,x_{n1},x_{12},\dots ,x_{n2})=D\prod _{l\in \{1,\dots ,n\}}\parallel x_{l1}{\parallel }^{p_l}·{\parallel x_{l2}\parallel }^{q_l},$$

with some $D,p_l,q_l\in (0,+\infty )$ for $l\in \{1,\dots ,n\}$. We note that, in this case, $\theta (0,\dots ,0)=0$, and condition (12) is fulfilled with

$${\omega }_{i_1,\dots ,i_n}=\prod _{l\in \{1,\dots ,n\}}{\left((1+i_l)\left|a_{l,i_1,\dots ,i_n}\right|\right)}^{p_l+q_l}for(i_1,\dots ,i_n)\in S.$$

From Theorem 3, using the above control function $\theta$, we obtain the following results.

Corollary 7.

Let $k\in \{1,\dots ,n\}$, $D,p_l,q_l\in (0,+\infty )$ for $l\in \{1,\dots ,n\}$, ${\sum }_{l\in \{1,\dots ,k\}}(p_l+q_l)<k$ and $f:X^n\to Y$. Assume also that $f(0,\dots ,0)=0$, and for any $x_{11},x_{12},\dots ,x_{n1},x_{n2}\in X$,

$$\begin{array}{cc}\hfill \parallel f(x_{11}+x_{12},\dots & ,x_{k1}+x_{k2},{\displaystyle \frac{1}{2}}{x}_{k+1,1}+{\displaystyle \frac{1}{2}}{x}_{k+1,2},\dots ,{\displaystyle \frac{1}{2}}{x}_{n1}+{\displaystyle \frac{1}{2}}{x}_{n2})\hfill \\ \hfill & -\sum _{j_1,\dots ,j_n\in \{1,2\}}{\displaystyle \frac{1}{2^{n-k}}}f(x_{1j_1},\dots ,x_{n{j}_n})\parallel \le D\prod _{l\in \{1,\dots ,n\}}\parallel x_{l1}{\parallel }^{p_l}·{\parallel x_{l2}\parallel }^{q_l}.\hfill \end{array}$$

(28)

Then, there exists a unique multi-additive–Jensen mapping $F:X^n⟶Y$ such that

$$\parallel f(x_1,\dots ,x_n)-F(x_1,\dots ,x_n)\parallel \le {\displaystyle \frac{D\prod _{l\in \{1,\dots ,n\}}{\parallel x_l\parallel }^{p_l+q_l}}{2^k-2^{{\sum }_{l\in \{1,\dots ,k\}}(p_l+q_l)}}},x_1,\dots ,x_n\in X.$$

Proof. 

Using Theorem 3 with (MAJ) and substituting $\gamma ={\displaystyle \frac{2^{{\sum }_{l\in \{1,\dots ,k\}}(p_l+q_l)}}{2^k}}$ because, in this case, $A=2^k$ and

$$\sum _{(i_1,\dots ,i_n)\in K}{\omega }_{i_1,\dots ,i_n}={\omega }_{1,\dots ,1}=\prod _{l\in \{1,\dots ,k\}}{2}^{p_l+q_l}=2^{{\sum }_{l\in \{1,\dots ,k\}}(p_l+q_l)},$$

we obtain our thesis. □

Remark 8.

The above Corollary is a particular case of Corollary 7 from [18], but in this paper, the authors started with the following form of condition (28):

$$\begin{array}{cc}\hfill \parallel 2^{n-k}f(x_{11}+x_{12},\dots & ,x_{k1}+x_{k2},{\displaystyle \frac{1}{2}}{x}_{k+1,1}+{\displaystyle \frac{1}{2}}{x}_{k+1,2},\dots ,{\displaystyle \frac{1}{2}}{x}_{n1}+{\displaystyle \frac{1}{2}}{x}_{n2})\hfill \\ \hfill & -\sum _{j_1,\dots ,j_n\in \{1,2\}}f(x_{1j_1},\dots ,x_{n{j}_n})\parallel \le C\prod _{l\in \{1,\dots ,n\}}\parallel x_{l1}{\parallel }^{p_l}·{\parallel x_{l2}\parallel }^{q_l},\hfill \end{array}$$

where $C\in (0,+\infty )$, from which we obtain (28) by dividing both sides of this inequality by $2^{n-k}$ and substituting $D={\displaystyle \frac{C}{2^{n-k}}}.$

Remark 9.

Corollary 7, with $n=k$, immediately provides the generalized Hyers–Ulam stability of multi-Cauchy Equation (4), and in this case, this corollary is a special case of Corollary 3.5 from [15].

Remark 10.

In Corollary 7, the assumption $f(0,\dots ,0)=0$ follows from the fact that, in order to obtain Equation (6) from Equation (1), we have to assume that $f(0,\dots ,0)=0$. In general, when using Theorem 3 with the control function

$$\theta (x_{11},\dots ,x_{n1},x_{12},\dots ,x_{n2})=D\prod _{l\in \{1,\dots ,n\}}\parallel x_{l1}{\parallel }^{p_l}·{\parallel x_{l2}\parallel }^{q_l},$$

we do not need the assumption of zeroing function f at $(0,\dots ,0)$ because $\theta (0,\dots ,0)=0$.

Corollary 8.

Let $D,p_1,p_2,q_1,q_2\in (0,+\infty )$, $2(p_1+q_1)+{\displaystyle \frac{1}{2}}(p_2+q_2)<3$ and $f:X^2\to Y$. Assume also that for any $x_1,x_2,y_1,y_2\in X$,

$$\begin{array}{cc}\hfill \parallel f\left(2(x_1+x_2),{\displaystyle \frac{\sqrt{2}}{2}}(y_1+y_2)\right)& +f\left(2(x_1+x_2),{\displaystyle \frac{\sqrt{2}}{2}}(y_1-y_2)\right)+f\left(0,\left(4-{\displaystyle \frac{\sqrt{2}}{2}}\right)(y_1+y_2)\right)\hfill \\ \hfill +f\left(0,-{\displaystyle \frac{\sqrt{2}}{2}}(y_1-y_2)\right)& -2f(x_1,y_1)-2f(x_1,y_2)-2f(x_2,y_1)-2f(x_2,y_2)\parallel \hfill \\ \hfill & \le D\parallel x_1{\parallel }^{p_1}\parallel x_2{\parallel }^{q_1}\parallel y_1{\parallel }^{p_2}{\parallel y_2\parallel }^{q_2}.\hfill \end{array}$$

Then, there exists a unique solution $F:X^2\to Y$ of (23) such that

$$\parallel f(x,y)-F(x,y)\parallel \le {\displaystyle \frac{{D\parallel x\parallel }^{p_1+q_1}{\parallel y\parallel }^{p_2+q_2}}{8-2^{2(p_1+q_1)+\frac{1}{2}(p_2+q_2)}}},x,y\in X.$$

(29)

Proof. 

Using Theorem 3 with $x_{11}=x_1,x_{12}=x_2,x_{21}=y_1,x_{22}=y_2$, and (E) and substituting $\gamma ={\displaystyle \frac{2^{2(p_1+q_1)+\frac{1}{2}(p_2+q_2)}}{8}}$ because, in this case, $A=8$ and

$$\sum _{(i_1,i_2)\in K}{\omega }_{i_1,i_2}={\omega }_{1,1}+{\omega }_{1,-1}+{\omega }_{-1,1}={\omega }_{1,1}=4^{p_1+q_1}·{\sqrt{2}}^{p_2+q_2}=2^{2(p_1+q_1)+{\displaystyle \frac{1}{2}}(p_2+q_2)},$$

we obtain our thesis. □

Remark 11.

In the proof of the above corollary, regardless of whether we use Theorem 2 or Theorem 3, we will obtain the same estimate in (29) because, in this case,

$$\sum _{(i_1,i_2)\in S}{\omega }_{i_1,i_2}={\omega }_{1,1}=\sum _{(i_1,i_2)\in K}{\omega }_{i_1,i_2}.$$

Remark 12.

Analyzing the proof of Theorem 2, we deduce the following results:

• For multi-Cauchy Equation (4), it is enough to assume that X is a commutative semigroup with the identity element 0.
• For multi-Cauchy–Jensen Equation (5), it is enough to assume that X is a commutative semigroup that is uniquely divisible by 2, with the identity element 0.
• For multi-quadratic (2) and multi-Cauchy–quadratic Equation (7), it is enough to assume that X is a commutative group.
• For multi-Jensen–quadratic Equation (8), it is enough to assume that X is a commutative group that is uniquely divisible by $2.$

4. Conclusions

In this paper, we proved, by applying the fixed-point approach, the generalized Hyers–Ulam stability of Equation (1) in Banach spaces without assumption (A). We obtain sufficient conditions for the stability of a wide class of functional equations and control functions. Our results may be used for proving the stability and generalized Hyers–Ulam stability of different functional equations of general form (1): for example, for Equations (2), (4), and (6)–(8). In this way, our outcomes generalize several known facts.

We used examples to show how additional assumptions imposed on function f, such as $f(0,\dots ,0)=0$ or condition (A), when examining the stability of Equation (1) affect the size of the approximating constant and limit the number of considered solutions for this equation.

The problem of the generalized Hyers–Ulam stability of multivariable functional equations is currently the subject of many studies, and it seems interesting to prove a theorem that is analogous to Theorem 2, which would provide a criterion for the generalized stability of a broader class of functional equations.