Generalized Hyers-Ulam Stability of an Additive–Quadratic–Cubic–Quartic Functional Equation

Abstract

We will prove the generalized Hyers-Ulam stability of an additive–quadratic–cubic–quartic functional equation ${\sum }_{m=1}^5{(-1)}^{m-1}[{\sum }_{1\le i_1<\cdots <i_m\le 5}f(x_{i_1}+x_{i_2}+\cdots +x_{i_m})]=0$ in the spirit of Găvruţa.

1. Introduction

Throughout this paper, let V and W be real vector spaces, X be a real normed space, and Y be a real Banach space. A mapping ${\widehat{f}}_k:V\to W$ satisfying the equation

$$\sum _{s=0}^k\left({\displaystyle \genfrac{}{}{0pt}{}{k}{s}}\right){(-1)}^{k-s}{\widehat{f}}_k(x+sy)=k!{\widehat{f}}_k(y)$$

(1)

for all $x,y\in V$ is called additive, quadratic, cubic, and quartic for $k=1,2,3$, and 4, respectively.

A mapping f is called an additive–quadratic–cubic–quartic mapping if it can be expressed as the sum of additive, quadratic, cubic, and quartic mappings. Furthermore, a functional equation $Df=0$ is referred to as an additive–quadratic–cubic–quartic functional equation if its general solution set coincides with the set of all additive–quadratic–cubic–quartic mappings.

Consider the functional equation

$$\sum _{m=1}^5{(-1)}^{m-1}\left(\sum _{1\le i_1<\cdots <i_m\le 5}f(x_{i_1}+x_{i_2}+\cdots +x_{i_m})\right)=0$$

(2)

for all $x_1,x_2,\dots ,x_5\in V$. For example, the function $f:\mathbb{R}\to \mathbb{R}$ defined by $f(x)={\displaystyle \sum _{s=1}^4}{a}_s{x}^s$, where the $a_s$ are real constants, is a solution of (2).

The stability of the functional Equation (2) can be considered a generalization of prior research on the additive–quadratic functional equation

$$\begin{array}{c}\hfill f(x_1+x_2+x_3)-f(x_1+x_2)-f(x_1+x_3)-f(x_2+x_3)+f(x_1)+f(x_2)+f(x_3)=0\end{array}$$

for all $x_1,x_2,x_3\in V$ (See [1,2,3,4,5,6,7] for more details).

Hyers’ work on the stability of the additive functional equation

$$f(x_1+x_2)-f(x_1)-f(x_2)=0,$$

has served as the starting point for stability research on a wide range of functional equations (see [8,9]). Hyers’ results were generalized by Rassias [10] and Găvruţa [11], and since then, many mathematicians have applied these generalized methods to study the stability of various functional equations.

The purpose of this paper is to show that the functional Equation (2) is an additive–quadratic–cubic–quartic functional equation and to prove its stability in the spirit of Găvruţa. Our work extends the existing stability results for the additive–quadratic functional equation, as seen in [2,3,4,5,6], to the more general case of (2). Furthermore, we apply this result to derive a theorem for another distinct additive–quadratic–cubic–quartic functional equation. As corollaries of these theorems, we establish the stability of both (2) and the latter equation in the sense of Hyers and Rassias.

The stability of a functional equation means that an exact solution exists in the vicinity of every approximate solution. Moreover, the uniqueness of the exact solution is an element addressed within the scope of stability. Existing stability results for additive–quadratic–cubic–quartic functional equations, distinct from (2), have been limited to cases corresponding to $\mathcal{l}=0$ and $\mathcal{l}=4$ in the condition (17) of Theorem 4 [12,13]. Consequently, the corollaries of those prior studies were effectively restricted to the ranges $p<1$ and $4<p$ in Theorem 5. Furthermore, in most existing results, the uniqueness of the solution in the vicinity of an approximate solution has not been established. In contrast, this paper provides a rigorous proof for both the existence and uniqueness of the mapping under more generalized conditions. Moreover, while the proofs in prior studies are detailed, they are often exceedingly lengthy. For instance, demonstrating stability for more complex cases, such as additive–quadratic–cubic–quartic–quintic–sextic functional equations, would require prohibitive space, posing significant difficulties for subsequent research. In this paper, we utilize the identity (5) to introduce a more generalized and concise methodology that is readily applicable to future studies.

2. Stability of an Additive–Quadratic–Cubic–Quartic Functional Equation

2.1. Preliminaries

For a given mapping $f:V\to W$, we use the following abbreviations:

$$\begin{array}{cc}\hfill E_{n}f(x_1,x_2,\dots ,x_n)& :=\sum _{m=1}^n{(-1)}^{n-m}\left(\sum _{1\le i_1<\cdots <i_m\le n}f(x_{i_1}+x_{i_2}+\cdots +x_{i_m})\right),\hfill \\ \hfill {\mathrm{\Delta }}_{x_1}{\mathrm{\Delta }}_{x_2}\dots {\mathrm{\Delta }}_{x_n}f(x):=& \sum _{m=1}^n{(-1)}^{n-m}\left(\sum _{1\le i_1<\cdots <i_m\le n}f(x+x_{i_1}+\cdots +x_{i_m})\right)+{(-1)}^{n}f(x),\hfill \\ \hfill f_{1,4}(x):=& {\omega }_1\left(f(8x)-28f(4x)+224f(2x)-512f(x)\right),\hfill \\ \hfill f_{2,4}(x):=& {\omega }_2\left(f(8x)-26f(4x)+176f(2x)-256f(x)\right),\hfill \\ \hfill f_{3,4}(x):=& {\omega }_3\left(f(8x)-22f(4x)+104f(2x)-128f(x)\right),\hfill \\ \hfill f_{4,4}(x):=& {\omega }_4\left(f(8x)-14f(4x)+56f(2x)-64f(x)\right),\hfill \\ \hfill \mathrm{\Gamma }f(x):=& f(16x)-30f(8x)+280f(4x)-960f(2x)+1024f(x)\hfill \end{array}$$

for all $x,x_1,x_2,\dots ,x_n\in V$, where ${\omega }_1,{\omega }_2,{\omega }_3,{\omega }_4$ are given by

$${\omega }_1=-{\displaystyle \frac{1}{168}},{\omega }_2={\displaystyle \frac{1}{96}},{\omega }_3=-{\displaystyle \frac{1}{192}},{\omega }_4={\displaystyle \frac{1}{1344}}.$$

Note that

$$\begin{array}{cc}\hfill E_{n}f(x_1,x_2,\dots ,x_n)=& {\mathrm{\Delta }}_{x_1}{\mathrm{\Delta }}_{x_2}\cdots {\mathrm{\Delta }}_{x_n}f(0)-{(-1)}^{n}f(0),\hfill \\ \hfill E_{5}f(x_1,x_2,\dots ,x_5)=& \sum _{m=1}^5{(-1)}^{5-m}\left(\sum _{1\le i_1<\cdots <i_m\le 5}f(x_{i_1}+x_{i_2}+\cdots +x_{i_m})\right)\hfill \\ \hfill =& f(x_1+x_2+x_3+x_4+x_5)-f(x_1+x_2+x_3+x_4)\hfill \\ \hfill & -f(x_1+x_2+x_3+x_5)-f(x_1+x_2+x_4+x_5)\hfill \\ \hfill & -f(x_1+x_3+x_4+x_5)-f(x_2+x_3+x_4+x_5)\hfill \\ \hfill & +f(x_1+x_2+x_3)+f(x_1+x_2+x_4)+f(x_1+x_3+x_4)\hfill \\ \hfill & +f(x_2+x_3+x_4)+f(x_1+x_2+x_5)+f(x_1+x_3+x_5)\hfill \\ \hfill & +f(x_2+x_3+x_5)+f(x_1+x_4+x_5)+f(x_2+x_4+x_5)\hfill \\ \hfill & +f(x_3+x_4+x_5)-f(x_1+x_2)-f(x_1+x_3)-f(x_1+x_4)\hfill \\ \hfill & -f(x_1+x_5)-f(x_2+x_3)-f(x_2+x_4)-f(x_2+x_5)-f(x_3+x_4)\hfill \\ \hfill & -f(x_3+x_5)-f(x_4+x_5)+f(x_1)+f(x_2)+f(x_3)+f(x_4)+f(x_5)\hfill \end{array}$$

for all $x_1,x_2,x_3,x_4,x_5\in V$.

We can also confirm the following theorem, introduced by Albert and Baker [14], by employing Corollary 1, Theorem 3, and Corollary 3 in the paper by Djoković [15].

Theorem 1 

([14] Theorem C). For a given mapping $f:V\to W$, the following are equivalent:

$(i)$

f satisfies the functional equation ${\mathrm{\Delta }}_{x_1}{\mathrm{\Delta }}_{x_2}\cdots {\mathrm{\Delta }}_{x_n}f(x)=0$ for all $x,x_1,x_2,\dots ,x_n\in V$.

$(ii)$

There exist mappings ${\widehat{f}}_1,{\widehat{f}}_2,\dots ,{\widehat{f}}_{n-1}:V\to W$ satisfying (1) for all $x,y\in V$ and each $k\in \{1,2,\dots ,n-1\}$, such that $f(x)={\displaystyle \sum _{k=1}^{n-1}}{\widehat{f}}_k(x)+f(0)$. In particular, ${\widehat{f}}_k(rx)=r^k{\widehat{f}}_k(x)$ for all $x\in V$ and all $r\in \mathbb{Q}$.

Theorem 2. 

For a given mapping $f:V\to W$, the following are equivalent:

$(i)$

$E_{n}f(x_1,x_2,\dots ,x_n)=0$ for all $x_1,x_2,\dots ,x_n\in V$.

$(ii)$

${\mathrm{\Delta }}_{x_1}{\mathrm{\Delta }}_{x_2}\dots {\mathrm{\Delta }}_{x_n}f(x)=0$ for all $x,x_1,x_2,\dots ,x_n\in V$ and $f(0)=0$.

Proof. 

(i) ⇒ (ii): Assume that $f:V\to W$ satisfies $E_{n}f(x_1,x_2,\dots ,x_n)=0$ for all $x_1,x_2,\dots ,x_n\in V$. Since ${\mathrm{\Delta }}_0^{n}f(0)=0$ and $E_{n}f(0,\dots ,0)=0$, we have

$$f(0)={(-1)}^n{\mathrm{\Delta }}_0^{n}f(0)-{(-1)}^n{E}_{n}f(0,\dots ,0)=0$$

and

$${\mathrm{\Delta }}_{x_1}{\mathrm{\Delta }}_{x_2}\dots {\mathrm{\Delta }}_{x_n}f(0)=E_{n}f(x_1,x_2,\dots ,x_n)+{(-1)}^{n}f(0)=0$$

(3)

for all $x_1,x_2,\dots ,x_n\in V$. By setting $x_n=x$, Equation (3) implies

$${\mathrm{\Delta }}_{x_1}\dots {\mathrm{\Delta }}_{x_{n-1}}f(x)-{\mathrm{\Delta }}_{x_1}\dots {\mathrm{\Delta }}_{x_{n-1}}f(0)={\mathrm{\Delta }}_{x_1}\dots {\mathrm{\Delta }}_{x_{n-1}}{\mathrm{\Delta }}_{x}f(0)=0$$

(4)

for all $x,x_1,\dots ,x_{n-1}\in V$. Combining (3) and (4), we obtain

$$\begin{array}{cc}\hfill {\mathrm{\Delta }}_{x_1}\dots {\mathrm{\Delta }}_{x_n}f(x)& ={\mathrm{\Delta }}_{x_1}\dots {\mathrm{\Delta }}_{x_n}f(x)-{\mathrm{\Delta }}_{x_1}\dots {\mathrm{\Delta }}_{x_n}f(0)\hfill \\ \hfill & ={\mathrm{\Delta }}_{x_n}\left({\mathrm{\Delta }}_{x_1}\dots {\mathrm{\Delta }}_{x_{n-1}}f(x)-{\mathrm{\Delta }}_{x_1}\dots {\mathrm{\Delta }}_{x_{n-1}}f(0)\right)\hfill \\ \hfill & ={\mathrm{\Delta }}_{x_n}(0)=0\hfill \end{array}$$

for all $x,x_1,\dots ,x_n\in V$.

(ii) ⇒ (i): If f satisfies ${\mathrm{\Delta }}_{x_1}\dots {\mathrm{\Delta }}_{x_n}f(x)=0$ and $f(0)=0$, then $E_{n}f(x_1,\dots ,x_n)={\mathrm{\Delta }}_{x_1}\dots {\mathrm{\Delta }}_{x_n}f(0)-{(-1)}^{n}f(0)=0-0=0$. □

Remark 1. 

According to Theorems 1 and 2, a mapping $f:V\to W$ satisfies the functional Equation (2) for all $x_1,\dots ,x_5\in V$ if and only if there exist an additive mapping ${\widehat{f}}_1$, a quadratic mapping ${\widehat{f}}_2$, a cubic mapping ${\widehat{f}}_3$, and a quartic mapping ${\widehat{f}}_4$ such that $f(x)={\sum }_{k=1}^4{\widehat{f}}_k(x)$ for all $x\in V$. Hence, the functional equation $E_{5}f(x_1,\dots ,x_5)=0$ is an additive–quadratic–cubic–quartic functional equation.

The following lemma provides an identity that plays a crucial role in the proof of our main theorem.

Lemma 1. 

For any mapping $f:V\to W$, the equality

$$\begin{array}{cc}\hfill \mathrm{\Gamma }f(x)& =E_{5}f(4x,4x,4x,2x,2x)+2E_{5}f(4x,4x,2x,2x,2x)\hfill \\ \hfill & +8E_{5}f(4x,2x,2x,2x,2x)+24E_{5}f(2x,2x,2x,2x,2x)\hfill \\ \hfill & +40E_{5}f(2x,2x,2x,x,x)+80E_{5}f(2x,2x,x,x,x)\hfill \\ \hfill & +96E_{5}f(2x,x,x,x,x)+64E_{5}f(x,x,x,x,x)\hfill \end{array}$$

(5)

holds for all $x\in V$.

Proof. 

The identity (5) follows from the following direct evaluations:

$$\begin{array}{cc}\hfill E_{5}f(4x,4x,4x,2x,2x)=& f(16x)-2f(14x)-2f(12x)+6f(10x)-6f(6x)\hfill \\ \hfill & +2f(4x)+2f(2x),\hfill \\ \hfill 2E_{5}f(4x,4x,2x,2x,2x)=& 2f(14x)-6f(12x)+2f(10x)+10f(8x)-10f(6x)\hfill \\ \hfill & -2f(4x)+6f(2x),\hfill \\ \hfill 8E_{5}f(4x,2x,2x,2x,2x)=& 8f(12x)-32f(10x)+40f(8x)-40f(4x)+32f(2x),\hfill \\ \hfill 24E_{5}f(2x,2x,2x,2x,2x)=& 24f(10x)-120f(8x)+240f(6x)-240f(4x)+120f(2x),\hfill \\ \hfill 40E_{5}f(2x,2x,2x,x,x)=& 40f(8x)-80f(7x)-80f(6x)+240f(5x)-240f(3x)\hfill \\ \hfill & +80f(2x)+80f(x),\hfill \\ \hfill 80E_{5}f(2x,2x,x,x,x)=& 80f(7x)-240f(6x)+80f(5x)+400f(4x)-400f(3x)\hfill \\ \hfill & -80f(2x)+240f(x),\hfill \\ \hfill 96E_{5}f(2x,x,x,x,x)=& 96f(6x)-384f(5x)+480f(4x)-480f(2x)+384f(x),\hfill \\ \hfill 64E_{5}f(x,x,x,x,x)=& 64f(5x)-320f(4x)+640f(3x)-640f(2x)+320f(x)\hfill \end{array}$$

for all $x\in V$. Summing these expressions leads to the desired equality. □

From the definitions of $f_{k,4}$ ($k=1,2,3,4$) and $\mathrm{\Gamma }f$, we obtain the following lemma.

Lemma 2 

([16] Theorem 2.5). For any mapping $f:V\to W$, the following equalities hold for all $x\in V$ and each $k\in \{1,2,3,4\}$:

$$\begin{array}{cc}\hfill & f_{k,4}(x)-{\displaystyle \frac{f_{k,4}(2x)}{2^k}}=-{\displaystyle \frac{{\omega }_k}{2^k}}\mathrm{\Gamma }f(x),\hfill \end{array}$$

(6)

$$\begin{array}{cc}\hfill & f_{k,4}(x)-2^k{f}_{k,4}\left({\displaystyle \frac{x}{2}}\right)={\omega }_k\mathrm{\Gamma }f\left({\displaystyle \frac{x}{2}}\right),\hfill \end{array}$$

(7)

$$\begin{array}{cc}\hfill & f(x)=\sum _{k=1}^4{f}_{k,4}(x).\hfill \end{array}$$

(8)

Proof. 

By setting $n=4, a=2$, and $A=\{2,2^2,2^3,2^4\}$ in [16] Theorem 2.5, and replacing $f_{k,a,A}, {\omega }_k(A), a_k$, and ${\mathrm{\Gamma }}_{a,A}f$ with $f_{k,4}, {\omega }_k, 2^k$, and $\mathrm{\Gamma }f$, respectively, we can verify that equalities (6), (7), and (8) hold. One can also verify these equalities by direct computation. □

Lemma 3. 

If $f:V\to W$ satisfies the functional equation $E_{5}f(x_1,\dots ,x_5)=0$ for all $x_1,\dots ,x_5\in V$, then the mappings $f_{k,4}:V\to Y$ satisfy

$$f_{k,4}(2x)=2^k{f}_{k,4}(x)$$

(9)

for all $x\in V$ and each $k\in \{1,2,3,4\}$.

Proof. 

If a mapping $f:V\to W$ satisfies $E_{5}f(x_1,\dots ,x_5)=0$, it also satisfies $\mathrm{\Gamma }f(x)=0$ by Lemma 1. Therefore, equality (9) follows from (6) and the fact that

$$2^k{f}_{k,4}(x)-f_{k,4}(2x)=-2^k\left(f_{k,4}(x)-{\displaystyle \frac{f_{k,4}(2x)}{2^k}}\right)={\omega }_k\mathrm{\Gamma }f(x)=0$$

for all $x\in V$ and each $k\in \{1,2,3,4\}$. □

The following lemma is a special case of [17] Corollary 6 with $a=2$ and $A=\{2,2^2,2^3,2^4\}$.

Lemma 4 

([17] Corollary 6). For a given mapping $f:V\to Y$, assume that there exist a mapping $F:V\to Y$ and a function $\varphi :V\to [0,\infty )$ satisfying any of the following conditions:

$$\begin{array}{cc}\hfill & \parallel f(x)-F(x)\parallel \le \sum _{i=0}^{\infty }{\displaystyle \frac{1}{2^i}}\varphi (2^{i}x)<\infty ,\hfill \end{array}$$

(10)

$$\begin{array}{cc}\hfill & \parallel f(x)-F(x)\parallel \le \sum _{i=0}^{\infty }{\displaystyle \frac{1}{2^{(\mathcal{l}+1)i}}}\varphi (2^{i}x)+\sum _{i=0}^{\infty }{2}^{\mathcal{l}i}\varphi \left({\displaystyle \frac{1}{2^i}}x\right)<\infty \hfill \end{array}$$

(11)

for a fixed $\mathcal{l}\in \{1,2,3\}$, or

$$\begin{array}{cc}\hfill & \parallel f(x)-F(x)\parallel \le \sum _{i=0}^{\infty }{2}^{4i}\varphi \left({\displaystyle \frac{1}{2^i}}x\right)<\infty ,\hfill \end{array}$$

(12)

for all $x\in V$. If, in addition, there exist mappings $F_1,F_2,F_3,F_4:V\to Y$ such that $F(x)={\sum }_{k=1}^4{F}_k(x)$ and $F_k(2x)=2^k{F}_k(x)$ for all $x\in V$ and each $k\in \{1,2,3,4\}$, then the mapping F is uniquely determined.

Combining Lemmas 3 and 4, we obtain the following theorem regarding the uniqueness of the mapping F.

Theorem 3. 

For a given mapping $f:V\to Y$, if there exist an additive–quadratic–cubic–quartic mapping $F:V\to Y$ and a function $\varphi :V\to [0,\infty )$ satisfying any of the conditions (10)–(12) for all $x\in V$, then the mapping F is uniquely determined.

The inequalities and identity provided in the following lemma facilitate the simplification of inequality (18) in the main theorem.

Lemma 5. 

The following inequalities hold for all $i\in \{0,1,2,\dots \}$:

$$\begin{array}{cc}\hfill \left|\sum _{k=\mathcal{l}}^4{\displaystyle \frac{{\omega }_k}{2^{ki}}}\right|\le {\displaystyle \frac{|{\omega }_{\mathcal{l}}|}{2^{\mathcal{l}i}}}& \mathit{for}\mathcal{l}\in \{1,2,3\},\hfill \end{array}$$

(13)

$$\begin{array}{cc}\hfill \left|\sum _{k=1}^{\mathcal{l}}{2}^{ki}{\omega }_k\right|\le 2^{\mathcal{l}i}\left|{\omega }_{\mathcal{l}}\right|& \mathit{for}\mathcal{l}\in \{2,3,4\}.\hfill \end{array}$$

(14)

Moreover, the following identity holds for all $i\in \{0,1,2\}$:

$$\begin{array}{c}\hfill 2^i{\omega }_1+2^{2i}{\omega }_2+2^{3i}{\omega }_3+2^{4i}{\omega }_4=0.\end{array}$$

(15)

Proof. 

In the case $n=4$ with $a=2$ and $A=\{2,2^2,2^3,2^4\}$, the identity (15) is verified by replacing ${\omega }_k(A)$ and $a_k$ with ${\omega }_k$ and $2^k$, respectively, in accordance with [16] Lemma 2.3. Inequality (13) follows from the following evaluations for all $i\ge 0$:

$$\begin{array}{cc}\hfill \left|\sum _{k=3}^4{\displaystyle \frac{{\omega }_k}{2^{ki}}}\right|& ={\displaystyle \frac{1}{8^i·192}}-{\displaystyle \frac{1}{{16}^i·1344}}\le {\displaystyle \frac{1}{8^i·192}},\hfill \\ \hfill \left|\sum _{k=2}^4{\displaystyle \frac{{\omega }_k}{2^{ki}}}\right|& ={\displaystyle \frac{1}{4^i·96}}-{\displaystyle \frac{1}{8^i·192}}+{\displaystyle \frac{1}{{16}^i·1344}}\le {\displaystyle \frac{1}{4^i·96}},\hfill \\ \hfill \left|\sum _{k=1}^4{\displaystyle \frac{{\omega }_k}{2^{ki}}}\right|& ={\displaystyle \frac{1}{2^i·168}}-{\displaystyle \frac{1}{4^i·96}}+{\displaystyle \frac{1}{8^i·192}}-{\displaystyle \frac{1}{{16}^i·1344}}\le {\displaystyle \frac{1}{2^i·168}}.\hfill \end{array}$$

Similarly, inequality (14) is derived from

$$\begin{array}{cccc}\hfill \left|\sum _{k=1}^2{2}^{ki}{\omega }_k\right|& ={\displaystyle \frac{4^i}{96}}-{\displaystyle \frac{2^i}{168}}\le {\displaystyle \frac{4^i}{96}}\hfill & \hfill & \mathrm{for}i\ge 0,\hfill \\ \hfill \left|\sum _{k=1}^3{2}^{ki}{\omega }_k\right|& ={\displaystyle \frac{8^i}{192}}-{\displaystyle \frac{4^i}{96}}+{\displaystyle \frac{2^i}{168}}\le {\displaystyle \frac{8^i}{192}}\hfill & \hfill & \mathrm{for}i\ge 0,\hfill \\ \hfill \left|\sum _{k=1}^4{2}^{ki}{\omega }_k\right|& ={\displaystyle \frac{{16}^i}{1344}}-{\displaystyle \frac{8^i}{192}}+{\displaystyle \frac{4^i}{96}}-{\displaystyle \frac{2^i}{168}}\le {\displaystyle \frac{{16}^i}{1344}}\hfill & \hfill & \mathrm{for}i\ge 3,\hfill \\ \hfill \left|\sum _{k=1}^4{2}^{ki}{\omega }_k\right|& =0\le {\displaystyle \frac{{16}^i}{1344}}\hfill & \hfill & \mathrm{for}i\in \{0,1,2\}.\hfill \end{array}$$

 □

2.2. Main Result

As our main result, we prove the generalized Hyers–Ulam stability of the additive–quadratic–cubic–quartic functional Equation (2) in the spirit of Găvruţa. To ensure readability and a smooth logical flow, we present the proof with ℓ fixed; we recommend following the argument under this assumption.

Theorem 4. 

Let ℓ be a fixed integer such that $0\le \mathcal{l}\le 4$. If a function $\phi :V^5\to [0,\infty )$ satisfies the condition

$$\left\{\begin{array}{cc}{\displaystyle \sum _{i=0}^{\infty }\frac{\phi (2^i{x}_1,\dots ,2^i{x}_5)}{2^i}<\infty }\hfill & \mathit{if}\mathcal{l}=0,\hfill \\ {\displaystyle \sum _{i=0}^{\infty }\left(\frac{\phi (2^i{x}_1,\dots ,2^i{x}_5)}{2^{(\mathcal{l}+1)i}}+2^{\mathcal{l}i}\phi \left(\frac{x_1}{2^i},\dots ,\frac{x_5}{2^i}\right)\right)<\infty }\hfill & \mathit{if}0<\mathcal{l}<4,\hfill \\ {\displaystyle \sum _{i=0}^{\infty }{16}^i\phi \left(\frac{x_1}{2^i},\dots ,\frac{x_5}{2^i}\right)<\infty }\hfill & \mathit{if}\mathcal{l}=4,\hfill \end{array}\right.$$

(16)

and a mapping $f:V\to Y$ satisfies the inequality

$$\begin{array}{c}\hfill ∥E_{5}f(x_1,x_2,x_3,x_4,x_5)∥\le \phi (x_1,x_2,x_3,x_4,x_5)\end{array}$$

(17)

for all $x_1,\dots ,x_5\in V$, then there exists a unique additive–quadratic–cubic–quartic mapping $F:V\to Y$ such that

$$\parallel f(x)-F(x)\parallel \le \left\{\begin{array}{cc}{\displaystyle |{\omega }_1|\sum _{i=0}^{\infty }\frac{\mathrm{\Phi }(2^{i}x)}{2^{i+1}}}\hfill & \mathit{if}\mathcal{l}=0,\hfill \\ {\displaystyle |{\omega }_{\mathcal{l}}|\sum _{i=0}^{\infty }{2}^{\mathcal{l}i}\mathrm{\Phi }\left(\frac{x}{2^{i+1}}\right)+|{\omega }_{\mathcal{l}+1}|\sum _{i=0}^{\infty }\frac{\mathrm{\Phi }(2^{i}x)}{2^{(\mathcal{l}+1)(i+1)}}}\hfill & \mathit{if}0<\mathcal{l}<4,\hfill \\ {\displaystyle |{\omega }_4|\sum _{i=3}^{\infty }{2}^{4i}\mathrm{\Phi }\left(\frac{x}{2^{i+1}}\right)}\hfill & \mathit{if}\mathcal{l}=4\hfill \end{array}\right.$$

(18)

for all $x\in V$, where $\mathrm{\Phi }:V\to [0,\infty )$ is defined by

$$\begin{array}{cc}\hfill \mathrm{\Phi }(x):=& \phi (4x,4x,4x,2x,2x)+2\phi (4x,4x,2x,2x,2x)+8\phi (4x,2x,2x,2x,2x)\hfill \\ \hfill & +24\phi (2x,2x,2x,2x,2x)+40\phi (2x,2x,2x,x,x)\hfill \\ \hfill & +80\phi (2x,2x,x,x,x)+96\phi (2x,x,x,x,x)+64\phi (x,x,x,x,x).\hfill \end{array}$$

Proof. 

From (5), (17), and the definition of $\mathrm{\Phi }$, we obtain

$$\begin{array}{cc}\hfill \parallel \mathrm{\Gamma }f(x)\parallel & =\parallel E_{5}f(4x,4x,4x,2x,2x)+2E_{5}f(4x,4x,2x,2x,2x)\hfill \\ \hfill & +8E_{5}f(4x,2x,2x,2x,2x)+24E_{5}f(2x,2x,2x,2x,2x)\hfill \\ \hfill & +40E_{5}f(2x,2x,2x,x,x)+80E_{5}f(2x,2x,x,x,x)\hfill \\ \hfill & +96E_{5}f(2x,x,x,x,x)+64E_{5}f(x,x,x,x,x)\parallel \hfill \\ \hfill & \le \mathrm{\Phi }(x)\hfill \end{array}$$

(19)

for all $x\in V$.

Let ℓ be a fixed integer in $\{0,1,2,3,4\}$. In what follows, we carry out the main part of the proof by considering two separate cases: $\mathcal{l}+1\le k\le 4$ and $1\le k\le \mathcal{l}$.

(1) Let k be an integer such that $\mathcal{l}+1\le k\le 4$. If $\phi$ satisfies (16), then since $2^{ki}\ge 2^{(\mathcal{l}+1)i}$ for $i\ge 0$, we have

$$\sum _{i=0}^{\infty }{\displaystyle \frac{\phi (2^i{x}_1,\dots ,2^i{x}_5)}{2^{ki}}}\le \sum _{i=0}^{\infty }{\displaystyle \frac{\phi (2^i{x}_1,\dots ,2^i{x}_5)}{2^{(\mathcal{l}+1)i}}}<\infty .$$

By the definition of $\mathrm{\Phi }$, it follows that

$$\sum _{i=0}^{\infty }{\displaystyle \frac{\mathrm{\Phi }(2^{i}x)}{2^{k(i+1)}}}\le \sum _{i=0}^{\infty }{\displaystyle \frac{\mathrm{\Phi }(2^{i}x)}{2^{(\mathcal{l}+1)(i+1)}}}<\infty$$

(20)

for all $x\in V$. Applying (6) and (19), we get

$$\begin{array}{cc}\hfill ∥{\displaystyle \frac{f_{k,4}(2^{m}x)}{2^{km}}}-{\displaystyle \frac{f_{k,4}(2^{m+n}x)}{2^{k(m+n)}}}∥& \le \sum _{i=m}^{m+n-1}∥{\displaystyle \frac{f_{k,4}(2^{i}x)}{2^{ki}}}-{\displaystyle \frac{f_{k,4}(2^{i+1}x)}{2^{k(i+1)}}}∥\hfill \\ \hfill & =\sum _{i=m}^{m+n-1}∥{\displaystyle \frac{{\omega }_k}{2^{k(i+1)}}}\mathrm{\Gamma }f(2^{i}x)∥\hfill \\ \hfill & \le \left|{\omega }_k\right|\sum _{i=m}^{m+n-1}{\displaystyle \frac{\mathrm{\Phi }(2^{i}x)}{2^{k(i+1)}}}\hfill \end{array}$$

for all $x\in V,m,n\in {\mathbb{N}}_0$. By (20), the sequence $\left\{{\displaystyle \frac{f_{k,4}(2^{i}x)}{2^{ki}}}\right\}$ is a Cauchy sequence. Since Y is a Banach space, the sequence converges for each $x\in V$. Thus, for $k\ge \mathcal{l}+1$, we can define $F_k:V\to Y$ as

$$F_k(x):=\underset{i\to \infty }{lim}{\displaystyle \frac{f_{k,4}(2^{i}x)}{2^{ki}}}.$$

In the case where $\mathcal{l}=0$ and $k=1$, the definition of $F_1$ and $f_{1,4}$ along with (17) yields

$$\begin{array}{cc}\hfill \parallel E_5{F}_1(x_1,\dots ,x_5)\parallel & =\underset{i\to \infty }{lim}∥{\displaystyle \frac{E_5{f}_{1,4}(2^i{x}_1,\dots ,2^i{x}_5)}{2^i}}∥=0\hfill \end{array}$$

for all $x_1,\dots ,x_5\in V$ (since ${lim}_{i\to \infty }{2}^{-i}\phi (2^{i}x)=0$). Similarly, $E_5{F}_k=0$ for all $k\in \{\mathcal{l}+1,\dots ,4\}$. Finally, by (13) and (19), we obtain

$$\begin{array}{cc}\hfill ∥\sum _{k=\mathcal{l}+1}^4\left(f_{k,4}(x)-F_k(x)\right)∥& \le \sum _{i=0}^{\infty }∥\sum _{k=\mathcal{l}+1}^4\left({\displaystyle \frac{f_{k,4}(2^{i}x)}{2^{ki}}}-{\displaystyle \frac{f_{k,4}(2^{i+1}x)}{2^{k(i+1)}}}\right)∥\hfill \\ \hfill & =\sum _{i=0}^{\infty }∥\sum _{k=\mathcal{l}+1}^4{\displaystyle \frac{{\omega }_k}{2^{k(i+1)}}}\mathrm{\Gamma }f(2^{i}x)∥\hfill \\ \hfill & =\sum _{i=0}^{\infty }\left|\sum _{k=\mathcal{l}+1}^4{\displaystyle \frac{{\omega }_k}{2^{k(i+1)}}}\right|\parallel \mathrm{\Gamma }f(2^{i}x)\parallel \hfill \\ \hfill & \le |{\omega }_{\mathcal{l}+1}|\sum _{i=0}^{\infty }{\displaystyle \frac{\mathrm{\Phi }(2^{i}x)}{2^{(\mathcal{l}+1)(i+1)}}}\hfill \end{array}$$

(21)

for all $x\in V$.

(2) Let k be an integer such that $1\le k\le \mathcal{l}$. If $\phi$ satisfies (16), then since $2^{ki}\le 2^{\mathcal{l}i}$ for $i\ge 0$, we have

$$\sum _{i=0}^{\infty }{2}^{ki}\phi \left({\displaystyle \frac{x_1}{2^i}},\dots ,{\displaystyle \frac{x_5}{2^i}}\right)\le \sum _{i=0}^{\infty }{2}^{\mathcal{l}i}\phi \left({\displaystyle \frac{x_1}{2^i}},\dots ,{\displaystyle \frac{x_5}{2^i}}\right)<\infty$$

for all $x_1,\dots ,x_5\in V$. By the definition of $\mathrm{\Phi }$, it follows that

$$\sum _{i=0}^{\infty }{2}^{ki}\mathrm{\Phi }\left({\displaystyle \frac{x}{2^i}}\right)\le \sum _{i=0}^{\infty }{2}^{\mathcal{l}i}\mathrm{\Phi }\left({\displaystyle \frac{x}{2^i}}\right)<\infty$$

(22)

for all $x\in V$.

Applying (7) and (19), we obtain

$$\begin{array}{cc}\hfill ∥f_{k,4}(x)-2^{km}{f}_{k,4}\left({\displaystyle \frac{x}{2^m}}\right)∥& \le \sum _{i=0}^{m-1}∥{\omega }_k{2}^{ki}\mathrm{\Gamma }f\left({\displaystyle \frac{x}{2^{i+1}}}\right)∥\hfill \\ & \le \left|{\omega }_k\right|\sum _{i=0}^{m-1}{2}^{ki}\mathrm{\Phi }\left({\displaystyle \frac{x}{2^{i+1}}}\right)\hfill \end{array}$$

for all $x\in V,m\in \mathbb{N}$, and $k\le \mathcal{l}$. This yields the Cauchy property:

$$∥2^{km}{f}_{k,4}\left({\displaystyle \frac{x}{2^m}}\right)-2^{k(m+n)}{f}_{k,4}\left({\displaystyle \frac{x}{2^{m+n}}}\right)∥\le \left|{\omega }_k\right|\sum _{i=m}^{m+n-1}{2}^{ki}\mathrm{\Phi }\left({\displaystyle \frac{x}{2^{i+1}}}\right)$$

(23)

for all $n\in {\mathbb{N}}_0$. Since Y is a Banach space, the sequence $\left\{2^{ki}{f}_{k,4}(x/2^i)\right\}$ converges for all $x\in V$. For $k\le \mathcal{l}$, we define

$$F_k(x):=\underset{i\to \infty }{lim}{2}^{ki}{f}_{k,4}\left({\displaystyle \frac{x}{2^i}}\right).$$

As in Case (1), (17) and the definition of $F_k$ imply $E_5{F}_k=0$ for all $k\in \{1,\dots ,\mathcal{l}\}$. By (7), (13), and (19), we obtain

$$\begin{array}{cc}\hfill \parallel \sum _{k=1}^{\mathcal{l}}\left(f_{k,4}(x)-F_k(x)\right)\parallel & \le \sum _{i=0}^{\infty }∥\sum _{k=1}^{\mathcal{l}}{\omega }_k{2}^{ki}\mathrm{\Gamma }f\left({\displaystyle \frac{x}{2^{i+1}}}\right)∥\hfill \\ & \le \left|{\omega }_{\mathcal{l}}\right|\sum _{i=0}^{\infty }{2}^{\mathcal{l}i}\mathrm{\Phi }\left({\displaystyle \frac{x}{2^{i+1}}}\right)\hfill \end{array}$$

(24)

for all $x\in V$. In the particular case where $\mathcal{l}=4$, the identity (15) implies ${\sum }_{k=1}^4{\omega }_k{2}^{ki}=0$ for $i\in \{0,1,2\}$. Thus,

$$\parallel \sum _{k=1}^4\left(f_{k,4}(x)-F_k(x)\right)\parallel \le \left|{\omega }_4\right|\sum _{i=3}^{\infty }{2}^{4i}\mathrm{\Phi }\left({\displaystyle \frac{x}{2^{i+1}}}\right)$$

(25)

holds for all $x\in V$ by applying (14) and (15).

To complete the proof, we now determine the additive–quadratic–cubic–quartic mapping F that satisfies (18). By defining a mapping

$$F(x):=\sum _{k=1}^4{F}_k(x)$$

for all $x\in V$, we see that F satisfies

$$E_{5}F(x_1,\dots ,x_5)=\sum _{k=1}^{\mathcal{l}}{E}_5{F}_k(x_1,\dots ,x_5)+\sum _{k=\mathcal{l}+1}^4{E}_5{F}_k(x_1,\dots ,x_5)=0,$$

which implies that F is an additive–quadratic–cubic–quartic mapping. Note that the inequality

$$\parallel f(x)-F(x)\parallel \le ∥\sum _{k=1}^{\mathcal{l}}\left(f_{k,4}(x)-F_k(x)\right)∥+∥\sum _{k=\mathcal{l}+1}^4\left(f_{k,4}(x)-F_k(x)\right)∥$$

(26)

holds for all $x\in V$ by the triangle inequality and (8).

If $\mathcal{l}=0$, inequality (18) follows from (21) and (26). If $\mathcal{l}\in \{1,2,3\}$, inequality (18) follows from (21), (24), and (26). If $\mathcal{l}=4$, inequality (18) follows from (25) and (26).

According to Theorem 3, the mapping $F:V\to Y$ is the unique solution of $E_{5}F(x_1,\dots ,x_5)=0$ satisfying inequality (18) for all $x\in V$. This completes the proof. □

2.3. Applications

From Theorem 4, we obtain the generalized stability of the Fréchet functional equation ${\mathrm{\Delta }}_{x_1}\dots {\mathrm{\Delta }}_{x_5}f(x)=0$ for all $x,x_1,\dots ,x_5\in V$ (see [1,18]).

Corollary 1. 

Let $\mathcal{l},\phi$, and Φ be as in Theorem 4. For a fixed $\mathcal{l}\in \{0,\dots ,4\}$, if a function $\phi :V^5\to [0,\infty )$ satisfies the condition in (16) and $f:V\to Y$ satisfies

$$\begin{array}{c}\hfill \parallel {\mathrm{\Delta }}_{x_1}{\mathrm{\Delta }}_{x_2}{\mathrm{\Delta }}_{x_3}{\mathrm{\Delta }}_{x_4}{\mathrm{\Delta }}_{x_5}f(x)\parallel \le \phi (x_1,\dots ,x_5)\end{array}$$

(27)

for all $x,x_1,\dots ,x_5\in V$, then there exists a unique additive–quadratic–cubic–quartic mapping $F:V\to Y$ satisfying (18) with f replaced by $\tilde{f}$, where $\tilde{f}(x):=f(x)-f(0)$.

Proof. 

Define $\tilde{f}:V\to Y$ by $\tilde{f}(x)=f(x)-f(0)$. Then $\tilde{f}(0)=0$. Since the Fréchet operator ${\mathrm{\Delta }}_{x_1}\dots {\mathrm{\Delta }}_{x_5}$ annihilates constant mappings, we have ${\mathrm{\Delta }}_{x_1}\dots {\mathrm{\Delta }}_{x_5}\tilde{f}(x)={\mathrm{\Delta }}_{x_1}\dots {\mathrm{\Delta }}_{x_5}f(x)$. By (27) and Theorem 2, it follows that $\parallel E_5\tilde{f}(x_1,\dots ,x_5)\parallel \le \phi (x_1,\dots ,x_5)$. The existence of a unique mapping F follows directly from Theorem 4. □

From Theorem 4, we obtain the Hyers–Ulam–Rassias stability of the additive–quadratic–cubic–quartic functional Equation (2) in the sense of Rassias.

Theorem 5. 

Let θ and p be positive real constants such that $p\notin \{1,2,3,4\}$. If $f:X\to Y$ satisfies the inequality

$$\parallel E_{5}f(x_1,x_2,x_3,x_4,x_5)\parallel \le \theta \sum _{j=1}^5{\parallel x_j\parallel }^p$$

for all $x_1,\dots ,x_5\in X$, then there exists a unique additive–quadratic–cubic–quartic mapping $F:X\to Y$ such that

$$\parallel f(x)-F(x)\parallel \le \left\{\begin{array}{cc}{\displaystyle \frac{M\theta }{168(2-2^p)}{\parallel x\parallel }^p}\hfill & \mathit{if}0<p<1,\hfill \\ {\displaystyle \left(\frac{1}{168(2^p-2)}+\frac{1}{96(4-2^p)}\right)M\theta {\parallel x\parallel }^p}\hfill & \mathit{if}1<p<2,\hfill \\ {\displaystyle \left(\frac{1}{96(2^p-4)}+\frac{1}{192(8-2^p)}\right)M\theta {\parallel x\parallel }^p}\hfill & \mathit{if}2<p<3,\hfill \\ {\displaystyle \left(\frac{1}{192(2^p-8)}+\frac{1}{1344(16-2^p)}\right)M\theta {\parallel x\parallel }^p}\hfill & \mathit{if}3<p<4,\hfill \\ {\displaystyle \frac{64}{21·8^p(2^p-16)}M\theta {\parallel x\parallel }^p}\hfill & ifp>4\hfill \end{array}\right.$$

(28)

for all $x\in X$, where $M:=15·4^p+536·2^p+1024$.

Proof. 

By setting $\phi (x_1,\dots ,x_5)=\theta {\sum }_{j=1}^5{\parallel x_j\parallel }^p$ in Theorem 4, we obtain $\mathrm{\Phi }(x)=(15·4^p+536·2^p+1024)\theta {\parallel x\parallel }^p$. We observe that $\phi$ satisfies condition (16) with $\mathcal{l}=0$ for $0<p<1$, $\mathcal{l}=1$ for $1<p<2$, $\mathcal{l}=2$ for $2<p<3$, $\mathcal{l}=3$ for $3<p<4$, and $\mathcal{l}=4$ for $p>4$. Applying inequality (18) for each corresponding ℓ yields the desired result (28). □

From Theorem 4, we obtain the Hyers–Ulam stability of the additive–quadratic–cubic–quartic functional Equation (2).

Theorem 6. 

Let δ be a positive real constant. If $f:V\to Y$ satisfies the inequality

$$\parallel E_{5}f(x_1,x_2,x_3,x_4,x_5)\parallel \le \delta$$

for all $x_1,\dots ,x_5\in V$, then there exists a unique additive–quadratic–cubic–quartic mapping $F:V\to Y$ such that

$$\parallel f(x)-F(x)\parallel \le {\displaystyle \frac{15}{8}}\delta$$

for all $x\in V$.

Proof. 

By setting $\phi (x_1,\dots ,x_5)=\delta$ in Theorem 4, we observe that $\phi$ satisfies condition (16) for the case $\mathcal{l}=0$ and $\mathrm{\Phi }(x)=315\delta$. Given $\left|{\omega }_1\right|={\displaystyle \frac{1}{168}}$, it follows from inequality (18) for the case $\mathcal{l}=0$ that

$$\parallel f(x)-F(x)\parallel \le {\displaystyle \frac{1}{168}}\sum _{i=0}^{\infty }{\displaystyle \frac{315\delta }{2^{i+1}}}={\displaystyle \frac{315}{168}}\delta ={\displaystyle \frac{15}{8}}\delta$$

for all $x\in V$. □

In a similar manner to the derivation of Theorem 5, the following Hyers–Ulam–Rassias stability of the Fréchet functional equation ${\mathrm{\Delta }}_{x_1}\dots {\mathrm{\Delta }}_{x_5}f(x)=0$ follows from Corollary 1.

Corollary 2. 

Let θ and p be positive real numbers such that $p\notin \{1,2,3,4\}$. If $f:X\to Y$ satisfies

$$\parallel {\mathrm{\Delta }}_{x_1}{\mathrm{\Delta }}_{x_2}{\mathrm{\Delta }}_{x_3}{\mathrm{\Delta }}_{x_4}{\mathrm{\Delta }}_{x_5}f(x)\parallel \le \theta \sum _{j=1}^5{\parallel x_j\parallel }^p$$

for all $x,x_1,\dots ,x_5\in X$, then there exists a unique additive–quadratic–cubic–quartic mapping $F:X\to Y$ such that

$$\parallel f(x)-f(0)-F(x)\parallel \le \left\{\begin{array}{cc}{\displaystyle \frac{M\theta }{168(2-2^p)}}{\parallel x\parallel }^p\hfill & \mathit{if}0<p<1,\hfill \\ {\displaystyle \frac{(20-3·2^p)M\theta }{672(2^p-2)(4-2^p)}}{\parallel x\parallel }^p\hfill & \mathit{if}1<p<2,\hfill \\ {\displaystyle \frac{(12-2^p)M\theta }{192(2^p-4)(8-2^p)}}{\parallel x\parallel }^p\hfill & \mathit{if}2<p<3,\hfill \\ {\displaystyle \frac{(104-6·2^p)M\theta }{1344(2^p-8)(16-2^p)}}{\parallel x\parallel }^p\hfill & \mathit{if}3<p<4,\hfill \\ {\displaystyle \frac{64M\theta }{21·8^p(2^p-16)}}{\parallel x\parallel }^p\hfill & \mathit{if}p>4,\hfill \end{array}\right.$$

where $M:=15·4^p+536·2^p+1024$.

Similarly, the Hyers–Ulam stability of the Fréchet functional equation follows from Corollary 1 and Theorem 6.

Corollary 3. 

Let δ be a positive real constant. If $f:V\to Y$ satisfies

$$\parallel {\mathrm{\Delta }}_{x_1}{\mathrm{\Delta }}_{x_2}{\mathrm{\Delta }}_{x_3}{\mathrm{\Delta }}_{x_4}{\mathrm{\Delta }}_{x_5}f(x)\parallel \le \delta$$

for all $x,x_1,\dots ,x_5\in V$, then there exists a unique additive–quadratic–cubic–quartic mapping $F:V\to Y$ such that

$$\parallel f(x)-f(0)-F(x)\parallel \le {\displaystyle \frac{15}{8}}\delta$$

for all $x\in V$.

3. Conclusions

Additive, quadratic, cubic, quartic, and other hybrid mappings, such as additive–quadratic or additive-cubic-quartic mappings, are all categorized as additive–quadratic–cubic–quartic mappings. Therefore, proving the stability of an additive–quadratic-cubic-quartic functional equation implies establishing the stability of each of its constituent mappings. By extension, it becomes necessary to sequentially investigate the stability of higher-order equations, such as additive–quadratic–cubic–quartic–quintic and additive–quadratic–cubic–quartic–sextic functional equations.

Previous studies on the general stability of additive–quadratic–cubic–quartic functional equations have several limitations, which are summarized as follows:

Major Limitations of Previous Research:

• Restricted Stability Range: Existing studies only addressed the cases $\mathcal{l}=0$ and $\mathcal{l}=4$ within the conditions of (16). Consequently, the stability results were confined to the specific ranges of $0<p<1$ and $p>4$ in Theorem 5. In fact, (12) in Theorem 5 of [12] corresponds to the case where $p<{log}_2\left(\sqrt{41}-5\right)\approx 0.4886$, while (25) in Theorem 6 of [12] corresponds to the case where $p>{log}_2\left({\displaystyle \frac{32}{\sqrt{41}-5}}\right)\approx 4.5114$.
• Lack of Uniqueness Proof: Rather than establishing the uniqueness of the mapping F, prior research only demonstrated the existence of individual additive, quadratic, cubic, and quartic components. Consequently, they failed to guarantee the uniqueness of the integrated additive–quadratic–cubic–quartic mapping F satisfying inequality (18).
• Complexity as a Barrier: The complexity of the right-hand side of inequality (18) hindered further research on more generalized functional equations. This imposed significant constraints on the study of higher-order equations, such as the additive–quadratic–cubic–quartic–quintic–sextic type.

Contributions and Significance of This Study: By applying the proof techniques developed in Theorem 4, it becomes possible to overcome the primary limitations of previous research and investigate the general stability of more complex and higher-order functional equations.

Methodological Limitations: However, a limitation of the current approach remains: while the methodology in Theorem 4 enables the investigation of the stability of $E_{n}f(x_1,\dots ,x_n)=0$ in the sense of Găvruţa for small n, the computational complexity of deriving equality (5) becomes prohibitively high as n increases (e.g., $n>20$). Therefore, this method remains insufficient for proving the stability of $E_{n}f=0$ for an arbitrary natural number n.