Generalized Stability of a General Septic Functional Equation

Abstract

In this paper, we generalize previous results on the generalized stability of the general septic functional equation ${\Delta }_y^{8}f\left(x\right)=0$ for all $x,y\in V$.

1. Introduction

In this paper, let V, X, and Y be a real vector space, a real normed space, and a real Banach space, respectively.

Throughout this paper, for a given mapping $f:V\to Y$ and a given positive integer n, let ${\Delta }^{n}f:V^2\to Y$ be the mapping defined by

$$\begin{array}{cc}\hfill \underset{y}{\overset{n}{\Delta }}f\left(x\right):=& \sum _{i=0}^n\left({\displaystyle \genfrac{}{}{0pt}{}{n}{i}}\right){(-1)}^{n-i}f(x+iy)\hfill \end{array}$$

for all $x,y\in V$. We consider the functional equation:

$$\underset{y}{\overset{n}{\Delta }}f\left(x\right)-n!f\left(y\right)=0$$

(1)

for all $x,y\in V$. The functional Equation (1) is called a monomial functional equation. For example, the function $f:\mathbb{R}\to \mathbb{R}$ given by $f\left(x\right)=a{x}^n$ is a particular solution of the functional Equation (1), where a is a real constant and $\mathbb{R}$ is the set of real numbers. In particular, the functional Equation (1) is called an additive (quadratic, cubic, quartic, quintic, sextic, and septic) functional equation for the case $n=1$ ($n=2$, $n=3$, $n=4$, $n=5$, $n=6$, and $n=7$, respectively).

A question of the stability of group isomorphism raised by S. M. Ulam [1] motivated the study on the stability of the additive functional equation by Hyers [2], and the study of Hyers became the beginning of the study of the stability of various functional equations. Th. M. Rassias [3] generalized Hyers’ results and P. Găvruta [4] generalized Rassias’ results. Subsequently, the stability of the quadratic functional equation was proven by F. Skof [5] and P. W. Cholewa [6]. Many mathematicians have obtained many results on the stability of the functional equation equivalent to the functional Equation (1) for $n<30$ (refer to [7,8,9]).

For the case of any n, the stability of the monomial functional Equation (1) was shown by Z. Kaiser [10], A. Gilányi [11,12,13], L. Cădariu and V. Radu [14], and Y.-H. Lee [15].

Consider the functional equation

$$\underset{y}{\overset{n}{\Delta }}f\left(x\right)=0$$

(2)

for all $x,y\in V$. For example, the function $f:\mathbb{R}\to \mathbb{R}$ given by $f\left(x\right)={\displaystyle \sum _{i=0}^{n-1}}{a}_i{x}^i$ is a particular solution of the functional Equation (2), where $a_i$ are real constants. In particular, the functional Equation (2) is called a Jensen (general quadratic, general cubic, general quartic, general quintic, general sextic, and general septic) functional equation for the case $n=2$ ($n=3$, $n=4$, $n=5$, $n=6$, $n=7$, and $n=8$, respectively).

For the case of any n, the Hyers–Ulam stability of the functional Equation (2) was established by Hyers [16] and by Albert and Baker [17] through the direct method used by Hyers in [2]. The Hyers–Ulam–Rassias stability of the functional Equation (2) for the cases of $n=2,3,4,5,6,7,8$ has been shown by a few mathematicians, but the general case of n has not been proven yet. The generalized stability of the functional Equation (2) for the cases of $n=2,3,4,5,6,7,8$ has also been shown by some few mathematicians (refer to [18,19,20,21]). Moreover, only partial results have been obtained for the generalized stability of the general septic functional equation, which cannot be considered complete results.

If $f:V\to Y$ is a solution mapping of the general septic functional equation,

$$\underset{y}{\overset{8}{\Delta }}f\left(x\right)=0$$

(3)

for all $x,y\in V$, then we call the mapping f a general septic mapping.

From the identities

$$\begin{array}{cc}\hfill \underset{y}{\overset{8}{\Delta }}f\left(x\right)=& \underset{y}{\overset{7}{\Delta }}f(x+y)-\underset{y}{\overset{7}{\Delta }}f\left(x\right),\hfill \\ \hfill \underset{y}{\overset{8}{\Delta }}f\left(x\right)=& \underset{y}{\overset{7}{\Delta }}f(x+y)-(n-1)!f\left(y\right)-\left(\underset{y}{\overset{7}{\Delta }}f\left(x\right)-(n-1)!f\left(y\right)\right),\hfill \end{array}$$

for all $x,y\in V$, we know that the stability of the general septic functional equation is a generalization for the stability of the septic functional equation and a generalization for the stability of the general sextic functional equation.

S.-S Jin and Y.-H. Lee [22] showed the Hyers–Ulam–Rassias stability of the general septic functional Equation (3) for all $x,y\in V$, where the Hyers–Ulam–Rassias stability is the proof method used by Rassias in [3]. I.-S. Chang et al. [23] showed partial results of the generalized stability of the functional Equation (3). The generalized stability introduced here by P. Găvruta [4] is a generalization of the Hyers–Ulam–Rassias stability.

In this paper, we show concise results that improve the previous results on the stability of the general septic functional equation in the spirit of P. Găvruta through another clearer proof. In particular, we obtain complete results instead of the partial results for the generalized stability of the general septic functional equation obtained by Chang et al. [23].

2. Stability of a General Septic Functional Equation

Throughout this paper, for a given mapping $f:V\to Y$ and a fixed positive integer n, we use the following abbreviations:

$$\begin{array}{cc}\hfill \tilde{f}\left(x\right):=& f\left(x\right)-f\left(0\right),\hfill \\ \hfill f_1\left(x\right):=& {\displaystyle \frac{1}{{\alpha }_1}}(\tilde{f}\left(64x\right)-252\tilde{f}\left(32x\right)+20832\tilde{f}\left(16x\right)-714240\tilde{f}\left(8x\right)\hfill \\ \hfill & +10665984\tilde{f}\left(4x\right)-66060288\tilde{f}\left(2x\right)+134217728\tilde{f}\left(x\right)),\hfill \\ \hfill f_2\left(x\right):=& {\displaystyle \frac{1}{{\alpha }_2}}(\tilde{f}\left(64x\right)-250\tilde{f}\left(32x\right)+20336\tilde{f}\left(16x\right)-674560\tilde{f}\left(8x\right)\hfill \\ \hfill & +9396224\tilde{f}\left(4x\right)-49807360\tilde{f}\left(2x\right)+67108864\tilde{f}\left(x\right)),\hfill \\ \hfill f_3\left(x\right):=& {\displaystyle \frac{1}{{\alpha }_3}}(\tilde{f}\left(64x\right)-246\tilde{f}\left(32x\right)+19368\tilde{f}\left(16x\right)-600960\tilde{f}\left(8x\right)\hfill \\ \hfill & +7286784\tilde{f}\left(4x\right)-29097984\tilde{f}\left(2x\right)+33554432\tilde{f}\left(x\right)),\hfill \\ \hfill f_4\left(x\right):=& {\displaystyle \frac{1}{{\alpha }_4}}(\tilde{f}\left(64x\right)-238\tilde{f}\left(32x\right)+17528\tilde{f}\left(16x\right)-475456\tilde{f}\left(8x\right)\hfill \\ \hfill & +4487168\tilde{f}\left(4x\right)-15597568\tilde{f}\left(2x\right)+16777216\tilde{f}\left(x\right)),\hfill \\ \hfill f_5\left(x\right):=& {\displaystyle \frac{1}{{\alpha }_5}}(\tilde{f}\left(64x\right)-222\tilde{f}\left(32x\right)+14232\tilde{f}\left(16x\right)-300480\tilde{f}\left(8x\right)\hfill \\ \hfill & +2479104\tilde{f}\left(4x\right)-8060928\tilde{f}\left(2x\right)+8388608\tilde{f}\left(x\right)),\hfill \\ \hfill f_6\left(x\right):=& {\displaystyle \frac{1}{{\alpha }_6}}(\tilde{f}\left(64x\right)-190\tilde{f}\left(32x\right)+9176\tilde{f}\left(16x\right)-168640\tilde{f}\left(8x\right)\hfill \\ \hfill & +1301504\tilde{f}\left(4x\right)-4096000\tilde{f}\left(2x\right)+4194304\tilde{f}\left(x\right)),\hfill \\ \hfill f_7\left(x\right):=& {\displaystyle \frac{1}{{\alpha }_7}}(\tilde{f}\left(64x\right)-126\tilde{f}\left(32x\right)+5208\tilde{f}\left(16x\right)-89280\tilde{f}\left(8x\right)\hfill \\ \hfill & +666624\tilde{f}\left(4x\right)-2064384\tilde{f}\left(2x\right)+2097152\tilde{f}\left(x\right)),\hfill \\ \hfill \Gamma f\left(x\right):=& f\left(128x\right)-254f\left(64x\right)+21336f\left(32x\right)-755904f\left(16x\right)\hfill \\ \hfill & +12094464f\left(8x\right)-87392256f\left(4x\right)+266338304f\left(2x\right)-268435456f\left(x\right),\hfill \end{array}$$

for all $x,y\in V$, where

$$\begin{array}{cc}\hfill & {\alpha }_1=39372480,{\alpha }_2=-19998720,{\alpha }_3=30965760,{\alpha }_4=-115605504,\hfill \\ \hfill & {\alpha }_5=990904320,{\alpha }_6=-20478689280,{\alpha }_7=1290157424640.\hfill \end{array}$$

By laborious but monotonous calculations, we can obtain some useful equations in the following lemma.

Lemma 1. 

For a given mapping $f:V\to Y$, the equalities

$$\begin{array}{ccc}\hfill & \underset{y}{\overset{8}{\Delta }}\tilde{f}\left(x\right)=\underset{y}{\overset{8}{\Delta }}f\left(x\right),\hfill & \hfill \\ \hfill & \Gamma \tilde{f}\left(x\right):=\underset{-16x}{\overset{8}{\Delta }}f\left(128x\right)+8\underset{8x}{\overset{8}{\Delta }}f\left(48x\right)+64\underset{8x}{\overset{8}{\Delta }}f\left(40x\right)+260\underset{8x}{\overset{8}{\Delta }}f\left(32x\right)\hfill & \hfill \\ \hfill & +736\underset{8x}{\overset{8}{\Delta }}f\left(24x\right)+1688\underset{8x}{\overset{8}{\Delta }}f\left(16x\right)+3424\underset{8x}{\overset{8}{\Delta }}f\left(8x\right)+6180\underset{-8x}{\overset{8}{\Delta }}f\left(64x\right)\hfill & \hfill \\ \hfill & +9408\underset{4x}{\overset{8}{\Delta }}f\left(24x\right)+75264\underset{4x}{\overset{8}{\Delta }}f\left(20x\right)+273728\underset{4x}{\overset{8}{\Delta }}f\left(16x\right)\hfill & \hfill \\ \hfill & +609280\underset{4x}{\overset{8}{\Delta }}f\left(12x\right)+948416\underset{4x}{\overset{8}{\Delta }}f\left(8x\right)+1114624\underset{4x}{\overset{8}{\Delta }}f\left(4x\right)\hfill & \hfill \\ \hfill & +1010240\underset{-4x}{\overset{8}{\Delta }}f\left(32x\right)+630784\underset{2x}{\overset{8}{\Delta }}f\left(12x\right)+5046272\underset{2x}{\overset{8}{\Delta }}f\left(10x\right)\hfill & \hfill \\ \hfill & +17762304\underset{2x}{\overset{8}{\Delta }}f\left(8x\right)+36126720\underset{2x}{\overset{8}{\Delta }}f\left(6x\right)+46821376\underset{2x}{\overset{8}{\Delta }}f\left(4x\right)\hfill & \hfill \\ \hfill & +39796736\underset{2x}{\overset{8}{\Delta }}f\left(2x\right)+20883456\underset{-2x}{\overset{8}{\Delta }}f\left(16x\right)+4325376\underset{x}{\overset{8}{\Delta }}f\left(6x\right)\hfill & \hfill \\ \hfill & +34603008\underset{x}{\overset{8}{\Delta }}f\left(5x\right)+119930880\underset{x}{\overset{8}{\Delta }}f\left(4x\right)+232783872\underset{x}{\overset{8}{\Delta }}f\left(3x\right)\hfill & \hfill \\ \hfill & +270401536\underset{x}{\overset{8}{\Delta }}f\left(2x\right)+181403648\underset{x}{\overset{8}{\Delta }}f\left(x\right)+56229888\underset{-x}{\overset{8}{\Delta }}f\left(8x\right),\hfill & \hfill \end{array}$$

(4)

$$\begin{array}{cc}\hfill & {\tilde{f}}_k\left(x\right)-{\displaystyle \frac{{\tilde{f}}_k\left(2x\right)}{2^k}}={\displaystyle \frac{-\Gamma \tilde{f}\left(x\right)}{2^k{\alpha }_k}},{\tilde{f}}_k\left(x\right)-2^k{\tilde{f}}_k\left({\displaystyle \frac{x}{2}}\right)={\displaystyle \frac{1}{{\alpha }_k}}\Gamma \tilde{f}\left({\displaystyle \frac{x}{2}}\right),\hfill \end{array}$$

(5)

$$\begin{array}{cc}\hfill & f\left(x\right)=f_1\left(x\right)+f_2\left(x\right)+f_3\left(x\right)+f_4\left(x\right)+f_5\left(x\right)+f_6\left(x\right)+f_7\left(x\right),\hfill \end{array}$$

(6)

hold for all $x\in V$ and $k=1,2,\dots ,7$.

Since Equation (6) holds for any mapping $f:V\to Y$, applying the lemma below, we know that if f is an additive-cubic-quint-septic mapping, then $f_1$ is an additive mapping, $f_2\equiv 0$, $f_3$ is a cubic mapping, $f_4\equiv 0$, $f_5$ is a quintic mapping, $f_6\equiv 0$, and $f_7$ is a septic mapping. Therefore, if $f:\mathbb{R}\to \mathbb{R}$ is a function defined by $f\left(x\right)=3x+4x^3-2x^5+x^7$, then $f_1\left(x\right)=3x$, $f_2\left(x\right)=0$, $f_3\left(x\right)=4x^3$, $f_4\left(x\right)=0$, $f_5\left(x\right)=-2x^5$, $f_6\left(x\right)=0$, and $f_7\left(x\right)=x^7$.

Lemma 2. 

If $f:V\to Y$ satisfies the functional equation $\underset{y}{\overset{8}{\Delta }}f\left(x\right)=0$ for all $x,y\in V$, then the mappings ${\tilde{f}}_1,\dots ,{\tilde{f}}_7:V\to Y$ satisfies

$$\begin{array}{c}\hfill {\tilde{f}}_k\left(2x\right)=2^k{\tilde{f}}_k\left(x\right)\end{array}$$

(7)

for all $x\in V$ and each $k\in \{1,2,3,4,5,6,7\}$.

Proof. 

If $f:V\to Y$ satisfies the functional equation $\underset{y}{\overset{8}{\Delta }}f\left(x\right)=0$ for all $x,y\in V$, then $f:V\to Y$ satisfies the functional equation $\Gamma \tilde{f}\left(x\right)=0$ from (4). Therefore, Equality (7) follows from Equality (5). □

According to Corollary 6 in [24], we obtain the following lemma.

Lemma 3. 

For a given mapping $f:V\to Y$, if there exists a mapping $F:V\to Y$ and a function $\varphi :V\setminus \left\{0\right\}\to [0,\infty )$ that satisfy either

$$\begin{array}{cc}\hfill & \parallel f\left(x\right)-F\left(x\right)\parallel \le \sum _{i=0}^{\infty }{\displaystyle \frac{1}{2^i}}\varphi \left(2^{i}x\right)<\infty or\hfill \\ \hfill & \parallel f\left(x\right)-F\left(x\right)\parallel \le \sum _{i=0}^{\infty }{\displaystyle \frac{1}{2^{(\mathsf{\ell }+1)i}}}\varphi \left(2^{i}x\right)+\sum _{i=0}^{\infty }{2}^{\mathsf{\ell }i}\varphi \left({\displaystyle \frac{1}{2^i}}x\right)<\infty or\hfill \\ \hfill & \parallel f\left(x\right)-F\left(x\right)\parallel \le \sum _{i=0}^{\infty }{2}^{7i}\varphi \left({\displaystyle \frac{1}{2^i}}x\right)<\infty ,\hfill \end{array}$$

for all $x\in V\setminus \left\{0\right\}$ and for some $\mathsf{\ell }\in \{1,2,3,4,5,6\}$, where $F\left(x\right)={\displaystyle \sum _{k=1}^7}{F}_k\left(x\right)$ and every $F_k$ has the Property (7), then the mapping F is uniquely determined.

Proof. 

In the case of $n=1$, $a=2$, ${\alpha }_1=1,{\alpha }_2=2,\dots ,{\alpha }_7=7$ in Corollary 6 of [24], we can obtain this theorem by simply rewriting it. □

Lemma 4. 

If a mapping $f:V\to Y$ satisfies the functional equation $\underset{y}{\overset{8}{\Delta }}f\left(x\right)=0$ for all $x,y\in V\setminus \left\{0\right\}$, then it is a general septic mapping.

Proof. 

It is clear that $\underset{0}{\overset{8}{\Delta }}f\left(x\right)=0$ for all $x\in V$ and

$$\underset{y}{\overset{8}{\Delta }}f\left(0\right)=\underset{-y}{\overset{8}{\Delta }}f\left(8y\right)=0$$

for all $y\in V\setminus \left\{0\right\}$. So, $\underset{y}{\overset{8}{\Delta }}f\left(x\right)=0$ for all $x,y\in V$ as desired. □

Now, using the above lemmas, we obtain the following main theorem for the generalized stability of the general septic functional equation.

Theorem 1. 

Let $\phi :{(V\setminus \left\{0\right\})}^2\to [0,\infty )$ be a function satisfying one of the following conditions:

$$\begin{array}{cc}\hfill \sum _{i=0}^{\infty }& {\displaystyle \frac{1}{2^i}}\phi (2^{i}x,2^{i}y)<\infty ,\hfill \end{array}$$

(8)

$$\begin{array}{cc}\hfill \sum _{i=0}^{\infty }& {\displaystyle \frac{1}{2^{(\mathsf{\ell }+1)i}}}\phi (2^{i}x,2^{i}y)<\infty and\sum _{i=0}^{\infty }{2}^{\mathsf{\ell }i}\phi \left({\displaystyle \frac{x}{2^i}},{\displaystyle \frac{y}{2^i}}\right)<\infty ,\hfill \end{array}$$

(9)

$$\begin{array}{cc}\hfill \sum _{i=0}^{\infty }& {128}^i\phi \left({\displaystyle \frac{x}{2^i}},{\displaystyle \frac{y}{2^i}}\right)<\infty ,\hfill \end{array}$$

(10)

for all $x,y\in V\setminus \left\{0\right\}$ and for some $\mathsf{\ell }\in \{1,2,3,4,5,6\}$. Suppose that $f:V\to Y$ is a mapping such that

$$\begin{array}{c}\hfill \parallel \underset{y}{\overset{8}{\Delta }}f\left(x\right)\parallel \le \phi (x,y)\end{array}$$

(11)

for all $x,y\in V\setminus \left\{0\right\}$. Then, there exists a unique general septic mapping F such that $F\left(0\right)=0$ and

$$\begin{array}{cc}\hfill \parallel \tilde{f}\left(x\right)-F\left(x\right)\parallel \le & {\displaystyle \frac{1}{2{\alpha }_1}}\sum _{i=0}^{\infty }{\displaystyle \frac{\mathrm{\Phi }\left(2^{i}x\right)}{2^i}},\hfill \end{array}$$

(12)

$$\begin{array}{ccc}\hfill \parallel \tilde{f}\left(x\right)-F\left(x\right)\parallel \le & \sum _{i=0}^{\infty }{\displaystyle \frac{2^{\mathsf{\ell }i}}{|{\alpha }_{\mathsf{\ell }}|}}\mathrm{\Phi }\left({\displaystyle \frac{x}{2^{i+1}}}\right)+\sum _{i=0}^{\infty }{\displaystyle \frac{\mathrm{\Phi }\left(2^{i}x\right)}{2^{(\mathsf{\ell }+1)(i+1)}\left|{\alpha }_{\mathsf{\ell }+1}\right|}},\hfill & \hfill \end{array}$$

(13)

$$\begin{array}{cc}\hfill \parallel \tilde{f}\left(x\right)-F\left(x\right)\parallel \le & \sum _{i=6}^{\infty }{\displaystyle \frac{{128}^i}{{\alpha }_7}}\mathrm{\Phi }\left({\displaystyle \frac{x}{2^{i+1}}}\right),\hfill \end{array}$$

(14)

for all $x\in V\setminus \left\{0\right\}$ if φ satisfies $\left(8\right)$, $\left(9\right)$, or $\left(10\right)$ for some $\mathsf{\ell }\in \{1,2,3,4,5,6\}$, respectively, where $\mathrm{\Phi }:V\setminus \left\{0\right\}\to [0,\infty )$ is the function defined by

$$\begin{array}{cc}\hfill \mathrm{\Phi }\left(x\right)& :=\varphi (128x,-16x)+8\varphi (48x,8x)+64\varphi (40x,8x)+260\varphi (32x,8x)+736\varphi (24x,8x)\hfill \\ \hfill & +1688\varphi (16x,8x)+3424\varphi (8x,8x)+6180\varphi (64x,-8x)+9408\varphi (24x,4x)\hfill \\ \hfill & +75264\varphi (20x,4x)+273728\varphi (16x,4x)+609280\varphi (12x,4x)+948416\varphi (8x,4x)\hfill \\ \hfill & +1114624\varphi (4x,4x)+1010240\varphi (32x,-4x)+630784\varphi (12x,2x)+5046272\varphi (10x,2x)\hfill \\ \hfill & +17762304\varphi (8x,2x)+36126720\varphi (6x,2x)+46821376\varphi (4x,2x)+39796736\varphi (2x,2x)\hfill \\ \hfill & +20883456\varphi (16x,-2x)+4325376\varphi (6x,x)+34603008\varphi (5x,x)+119930880\varphi (4x,x)\hfill \\ \hfill & +232783872\varphi (3x,x)+270401536\varphi (2x,x)+181403648\varphi (x,x)+56229888\varphi (8x,-x)\hfill \end{array}$$

for all $x\in V\setminus \left\{0\right\}$.

Proof. 

Notice that, from (4) and (11), we have

$$\begin{array}{cc}\hfill \parallel \Gamma \tilde{f}& \left(x\right)\parallel =\parallel \underset{-16x}{\overset{8}{\Delta }}f\left(128x\right)+8\underset{8x}{\overset{8}{\Delta }}f\left(48x\right)+64\underset{8x}{\overset{8}{\Delta }}f\left(40x\right)+260\underset{8x}{\overset{8}{\Delta }}f\left(32x\right)+736\underset{8x}{\overset{8}{\Delta }}f\left(24x\right)\hfill \\ \hfill & +1688\underset{8x}{\overset{8}{\Delta }}f\left(16x\right)+3424\underset{8x}{\overset{8}{\Delta }}f\left(8x\right)+6180\underset{-8x}{\overset{8}{\Delta }}f\left(64x\right)+9408\underset{4x}{\overset{8}{\Delta }}f\left(24x\right)\hfill \\ \hfill & +75264\underset{4x}{\overset{8}{\Delta }}f\left(20x\right)+273728\underset{4x}{\overset{8}{\Delta }}f\left(16x\right)+609280\underset{4x}{\overset{8}{\Delta }}f\left(12x\right)+948416\underset{4x}{\overset{8}{\Delta }}f\left(8x\right)\hfill \\ \hfill & +1114624\underset{4x}{\overset{8}{\Delta }}f\left(4x\right)+1010240\underset{-4x}{\overset{8}{\Delta }}f\left(32x\right)+630784\underset{2x}{\overset{8}{\Delta }}f\left(12x\right)+5046272\underset{2x}{\overset{8}{\Delta }}f\left(10x\right)\hfill & \hfill \\ \hfill & +17762304\underset{2x}{\overset{8}{\Delta }}f\left(8x\right)+36126720\underset{2x}{\overset{8}{\Delta }}f\left(6x\right)+46821376\underset{2x}{\overset{8}{\Delta }}f\left(4x\right)+39796736\underset{2x}{\overset{8}{\Delta }}f\left(2x\right)\hfill \\ \hfill & +20883456\underset{-2x}{\overset{8}{\Delta }}f\left(16x\right)+4325376\underset{x}{\overset{8}{\Delta }}f\left(6x\right)+34603008\underset{x}{\overset{8}{\Delta }}f\left(5x\right)+119930880\underset{x}{\overset{8}{\Delta }}f\left(4x\right)\hfill \\ \hfill & +232783872\underset{x}{\overset{8}{\Delta }}f\left(3x\right)+270401536\underset{x}{\overset{8}{\Delta }}f\left(2x\right)+181403648\underset{x}{\overset{8}{\Delta }}f\left(x\right)+56229888\underset{-x}{\overset{8}{\Delta }}f\left(8x\right)\parallel \hfill \\ \hfill \le & \mathrm{\Phi }\left(x\right)\hfill \end{array}$$

(15)

for all $x\in V\setminus \left\{0\right\}$. We prove the theorem in two steps:

Step 1.

Let $k\in \{1,2,3,4,5,6,7\}$ and $\delta \in \{-1,1\}$, and let $\phi$ satisfy

$$\begin{array}{c}\hfill \sum _{n=0}^{\infty }{\displaystyle \frac{\phi (2^{\delta n}x,2^{\delta n}y)}{2^{\delta kn}}}<\infty \end{array}$$

(16)

for all $x,y\in V\setminus \left\{0\right\}$. Together with

$${\displaystyle \frac{{\tilde{f}}_k\left(2^{\delta n}x\right)}{2^{\delta kn}}}-{\displaystyle \frac{{\tilde{f}}_k\left(2^{\delta (n+m)}x\right)}{2^{\delta k(n+m)}}}=\sum _{i=n}^{n+m-1}\left({\displaystyle \frac{{\tilde{f}}_k\left(2^{\delta i}x\right)}{2^{\delta ki}}}-{\displaystyle \frac{{\tilde{f}}_k\left(2^{\delta (i+1)}x\right)}{2^{\delta k(i+1)}}}\right),$$

(5), and (15), we have the inequalities

$$\begin{array}{c}\hfill ∥{\displaystyle \frac{{\tilde{f}}_k\left(2^{n}x\right)}{2^{kn}}}-{\displaystyle \frac{{\tilde{f}}_k\left(2^{n+m}x\right)}{2^{k(n+m)}}}∥\le \sum _{i=n}^{n+m-1}∥{\displaystyle \frac{\Gamma \tilde{f}\left(2^{i}x\right)}{2^k{\alpha }_k{2}^{ki}}}∥\le {\displaystyle \frac{1}{|{\alpha }_k|}}\sum _{i=n}^{n+m-1}{\displaystyle \frac{\mathrm{\Phi }\left(2^{i}x\right)}{2^{k(i+1)}}}\end{array}$$

(17)

and

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

(18)

for all $x\in V\setminus \left\{0\right\}$, $k\in \{1,2,3,4,5,6,7\}$, and $n,m\in N\cup \left\{0\right\}$. It leads us to prove that $\left\{{\displaystyle \frac{{\tilde{f}}_k\left(2^{\delta n}x\right)}{2^{\delta kn}}}\right\}$ is a Cauchy sequence for all $x\in V\setminus \left\{0\right\}$ if $\phi$ satisfies (16). Moreover, since Y is complete and ${\tilde{f}}_k\left(0\right)=0$, the sequence converges for all $x\in V$. It follows that we can define a mapping $F_{\delta k}:V\to Y$ by

$$\begin{array}{c}\hfill F_{\delta k}\left(x\right):=\underset{n\to \infty }{lim}{\displaystyle \frac{{\tilde{f}}_k\left(2^{\delta n}x\right)}{2^{\delta kn}}}\end{array}$$

(19)

for all $x\in V$ if $\phi$ satisfies (16). Now, we observe that the equality

$$\begin{array}{cc}\hfill \underset{y}{\overset{8}{\Delta }}{F}_{\delta k}\left(x\right)=& \sum _{i=0}^8\left({\displaystyle \genfrac{}{}{0pt}{}{8}{i}}\right){(-1)}^{8-i}{F}_{\delta k}(x+iy)\hfill \\ \hfill =& \underset{n\to \infty }{lim}\sum _{i=0}^8\left({\displaystyle \genfrac{}{}{0pt}{}{8}{i}}\right){(-1)}^{8-i}{\displaystyle \frac{{\tilde{f}}_k\left(2^{\delta n}(x+iy)\right)}{2^{\delta kn}}}\hfill \end{array}$$

holds for all $x,y\in V\setminus \left\{0\right\}$, $k\in \{1,2,3,4,5,6,7\}$, and $\delta \in \{+1,-1\}$. Together with the definition of ${\tilde{f}}_1$, if $\phi$ satisfies (16) for $k=1$, then we have

$$\begin{array}{cc}\hfill \parallel & \underset{y}{\overset{8}{\Delta }}{F}_{\delta 1}\left(x\right)\parallel =\underset{n\to \infty }{lim}{\displaystyle \parallel \frac{1}{2^{\delta n}{\alpha }_1}}\sum _{i=0}^8\left({\displaystyle \genfrac{}{}{0pt}{}{8}{i}}\right){(-1)}^{8-i}(\tilde{f}\left(2^{\delta n+6}(x+iy)\right)-252\tilde{f}\left(2^{\delta n+5}(x+iy)\right)\hfill \\ \hfill & +20832\tilde{f}\left(2^{\delta n+4}(x+iy)\right)-714240\tilde{f}\left(2^{\delta n+3}(x+iy)\right)+10665984\tilde{f}\left(2^{\delta n+2}(x+iy)\right)\hfill \\ \hfill & -66060288\tilde{f}\left(2^{\delta n+1}(x+iy)\right)+134217728\tilde{f}(2^{\delta n}(x+iy))\parallel \hfill \\ \hfill & =\underset{n\to \infty }{lim}{\displaystyle \parallel \frac{134217728{\Delta }_{2^{\delta n}y}^{8}f\left(2^{\delta n}x\right)}{{\alpha }_1·2^{\delta n}}}-{\displaystyle \frac{66060288\underset{2^{\delta n+1}y}{\overset{8}{\Delta }}f\left(2^{\delta n+1}x\right)}{{\alpha }_1·2^{\delta n}}}\hfill \\ \hfill & +{\displaystyle \frac{10665984{\Delta }_{2^{\delta n+2}y}^{8}f\left(2^{\delta n+2}x\right)}{{\alpha }_1·2^{\delta n}}}-{\displaystyle \frac{714240\underset{2^{\delta n+3}y}{\overset{8}{\Delta }}f\left(2^{\delta n+3}x\right)}{{\alpha }_1·2^{\delta n}}}\hfill \\ \hfill & +{\displaystyle \frac{20832\underset{2^{\delta n+4}y}{\overset{8}{\Delta }}f\left(2^{\delta n+4}x\right)}{{\alpha }_1·2^{\delta n}}}-{\displaystyle \frac{252\underset{2^{\delta n+5}y}{\overset{8}{\Delta }}f\left(2^{\delta n+5}x\right)}{{\alpha }_1·2^{\delta n}}}+{\displaystyle \frac{\underset{2^{\delta n+6}y}{\overset{8}{\Delta }}f\left(2^{\delta n+6}x\right)}{{\alpha }_1·2^{\delta n}}}\parallel \hfill \\ \hfill & \le \underset{n\to \infty }{lim}({\displaystyle \frac{134217728\phi (2^{\delta n}x,2^{\delta n}y)}{|{\alpha }_1|·2^{\delta n}}}+{\displaystyle \frac{66060288\phi (2^{\delta n+1}x,2^{\delta n+1}y)}{|{\alpha }_1|·2^{\delta n}}}\hfill \\ \hfill & +{\displaystyle \frac{10665984\phi (2^{\delta n+2}x,2^{\delta n+2}y)}{|{\alpha }_1|·2^{\delta n}}}+{\displaystyle \frac{714240\phi (2^{\delta n+3}x,2^{\delta n+3}y)}{|{\alpha }_1|·2^{\delta n}}}\hfill \\ \hfill & +{\displaystyle \frac{20832\phi (2^{\delta n+4}x,2^{\delta n+4}y)}{|{\alpha }_1|·2^{\delta n}}}+{\displaystyle \frac{252\phi (2^{\delta n+5}x,2^{\delta n+5}y)}{|{\alpha }_1|·2^{\delta n}}}+{\displaystyle \frac{\phi (2^{\delta n+6}x,2^{\delta n+6}y)}{|{\alpha }_1|·2^{\delta n}}})\hfill \\ \hfill & =0\hfill \end{array}$$

for all $x,y\in V\setminus \left\{0\right\}$. In a similar way, by the definition of ${\tilde{f}}_k$, if $\phi$ satisfies (16) for $k\in \{2,3,4,5,6,7\}$, then we obtain

$$\begin{array}{cc}\hfill & ∥\underset{y}{\overset{8}{\Delta }}{F}_{\delta 2}\left(x\right)∥\le \underset{n\to \infty }{lim}({\displaystyle \frac{67108864\phi (2^{\delta n}x,2^{\delta n}y)}{|{\alpha }_2|·2^{\delta 2n}}}+{\displaystyle \frac{49807360\phi (2^{\delta n+1}x,2^{\delta n+1}y)}{|{\alpha }_2|·2^{\delta 2n}}}\hfill \\ \hfill & +{\displaystyle \frac{9396224\phi (2^{\delta n+2}x,2^{\delta n+2}y)}{|{\alpha }_2|·2^{\delta 2n}}}+{\displaystyle \frac{674560\phi (2^{\delta n+3}x,2^{\delta n+3}y)}{|{\alpha }_2|·2^{\delta 2n}}}\hfill \\ \hfill & +{\displaystyle \frac{20336\phi (2^{\delta n+4}x,2^{\delta n+4}y)}{|{\alpha }_2|·2^{\delta 2n}}}+{\displaystyle \frac{250\phi (2^{\delta n+5}x,2^{\delta n+5}y)}{|{\alpha }_2|·2^{\delta 2n}}}+{\displaystyle \frac{\phi (2^{\delta n+6}x,2^{\delta n+6}y)}{|{\alpha }_2|·2^{\delta 2n}}})\hfill \\ \hfill & =0,\hfill \end{array}$$

$$\begin{array}{cc}\hfill & ∥\underset{y}{\overset{8}{\Delta }}{F}_{\delta 3}\left(x\right)∥\le \underset{n\to \infty }{lim}({\displaystyle \frac{33554432\phi (2^{\delta n}x,2^{\delta n}y)}{|{\alpha }_3|·2^{\delta 3n}}}+{\displaystyle \frac{29097984\phi (2^{\delta n+1}x,2^{\delta n+1}y)}{|{\alpha }_3|·2^{\delta 3n}}}\hfill \\ \hfill & +{\displaystyle \frac{7286784\phi (2^{\delta n+2}x,2^{\delta n+2}y)}{|{\alpha }_3|·2^{\delta 3n}}}+{\displaystyle \frac{600960\phi (2^{\delta n+3}x,2^{\delta n+3}y)}{|{\alpha }_3|·2^{\delta 3n}}}\hfill \\ \hfill & +{\displaystyle \frac{19368\phi (2^{\delta n+4}x,2^{\delta n+4}y)}{|{\alpha }_3|·2^{\delta 3n}}}+{\displaystyle \frac{246\phi (2^{\delta n+5}x,2^{\delta n+5}y)}{|{\alpha }_3|·2^{\delta 3n}}}+{\displaystyle \frac{\phi (2^{\delta n+6}x,2^{\delta n+6}y)}{|{\alpha }_3|·2^{\delta 3n}}})\hfill \\ \hfill & =0,\hfill \end{array}$$

$$\begin{array}{cc}\hfill & ∥\underset{y}{\overset{8}{\Delta }}{F}_{\delta 4}\left(x\right)∥\le \underset{n\to \infty }{lim}({\displaystyle \frac{16777216\phi (2^{\delta n}x,2^{\delta n}y)}{|{\alpha }_4|·2^{\delta 4n}}}+{\displaystyle \frac{15597568\phi (2^{\delta n+1}x,2^{\delta n+1}y)}{|{\alpha }_4|·2^{\delta 4n}}}\hfill \\ \hfill & +{\displaystyle \frac{4487168\phi (2^{\delta n+2}x,2^{\delta n+2}y)}{|{\alpha }_4|·2^{\delta 4n}}}+{\displaystyle \frac{475456\phi (2^{\delta n+3}x,2^{\delta n+3}y)}{|{\alpha }_4|·2^{\delta 4n}}}\hfill \\ \hfill & +{\displaystyle \frac{17528\phi (2^{\delta n+4}x,2^{\delta n+4}y)}{|{\alpha }_4|·2^{\delta 4n}}}+{\displaystyle \frac{238\phi (2^{\delta n+5}x,2^{\delta n+5}y)}{|{\alpha }_4|·2^{\delta 4n}}}+{\displaystyle \frac{\phi (2^{\delta n+6}x,2^{\delta n+6}y)}{|{\alpha }_4|·2^{\delta 4n}}})\hfill \\ \hfill & =0,\hfill \end{array}$$

$$\begin{array}{cc}\hfill & ∥\underset{y}{\overset{8}{\Delta }}{F}_{\delta 5}\left(x\right)∥\le \underset{n\to \infty }{lim}({\displaystyle \frac{8388608\phi (2^{\delta n}x,2^{\delta n}y)}{|{\alpha }_5|·2^{\delta 5n}}}+{\displaystyle \frac{8060928\phi (2^{\delta n+1}x,2^{\delta n+1}y)}{|{\alpha }_5|·2^{\delta 5n}}}\hfill \\ \hfill & +{\displaystyle \frac{2479104\phi (2^{\delta n+2}x,2^{\delta n+2}y)}{|{\alpha }_5|·2^{\delta 5n}}}+{\displaystyle \frac{300480\phi (2^{\delta n+3}x,2^{\delta n+3}y)}{|{\alpha }_5|·2^{\delta 5n}}}\hfill \\ \hfill & +{\displaystyle \frac{14232\phi (2^{\delta n+4}x,2^{\delta n+4}y)}{|{\alpha }_5|·2^{\delta 5n}}}+{\displaystyle \frac{222\phi (2^{\delta n+5}x,2^{\delta n+5}y)}{|{\alpha }_5|·2^{\delta 5n}}}+{\displaystyle \frac{\phi (2^{\delta n+6}x,2^{\delta n+6}y)}{|{\alpha }_5|·2^{\delta 5n}}})\hfill \\ \hfill & =0,\hfill \end{array}$$

$$\begin{array}{cc}\hfill & ∥\underset{y}{\overset{8}{\Delta }}{F}_{\delta 6}\left(x\right)∥\le \underset{n\to \infty }{lim}({\displaystyle \frac{4194304\phi (2^{\delta n}x,2^{\delta n}y)}{|{\alpha }_6|·2^{\delta 6n}}}+{\displaystyle \frac{4096000\phi (2^{\delta n+1}x,2^{\delta n+1}y)}{|{\alpha }_6|·2^{\delta 6n}}}\hfill \\ \hfill & +{\displaystyle \frac{1301504\phi (2^{\delta n+2}x,2^{\delta n+2}y)}{|{\alpha }_6|·2^{\delta 6n}}}+{\displaystyle \frac{168640\phi (2^{\delta n+3}x,2^{\delta n+3}y)}{|{\alpha }_6|·2^{\delta 6n}}}\hfill \\ \hfill & +{\displaystyle \frac{9176\phi (2^{\delta n+4}x,2^{\delta n+4}y)}{|{\alpha }_6|·2^{\delta 6n}}}+{\displaystyle \frac{190\phi (2^{\delta n+5}x,2^{\delta n+5}y)}{|{\alpha }_6|·2^{\delta 6n}}}+{\displaystyle \frac{\phi (2^{\delta n+6}x,2^{\delta n+6}y)}{|{\alpha }_6|·2^{\delta 6n}}})\hfill \\ \hfill & =0,\hfill \end{array}$$

and

$$\begin{array}{cc}\hfill & ∥\underset{y}{\overset{8}{\Delta }}{F}_{\delta 7}\left(x\right)∥\le \underset{n\to \infty }{lim}({\displaystyle \frac{2097152\phi (2^{\delta n}x,2^{\delta n}y)}{|{\alpha }_7|·2^{\delta 7n}}}+{\displaystyle \frac{2064384\phi (2^{\delta n+1}x,2^{\delta n+1}y)}{|{\alpha }_7|·2^{\delta 7n}}}\hfill \\ \hfill & +{\displaystyle \frac{666624\phi (2^{\delta n+2}x,2^{\delta n+2}y)}{|{\alpha }_7|·2^{\delta 7n}}}+{\displaystyle \frac{89280\phi (2^{\delta n+3}x,2^{\delta n+3}y)}{|{\alpha }_7|·2^{\delta 7n}}}\hfill \\ \hfill & +{\displaystyle \frac{5208\phi (2^{\delta n+4}x,2^{\delta n+4}y)}{|{\alpha }_7|·2^{\delta 7n}}}+{\displaystyle \frac{126\phi (2^{\delta n+5}x,2^{\delta n+5}y)}{|{\alpha }_7|·2^{\delta 7n}}}+{\displaystyle \frac{\phi (2^{\delta n+6}x,2^{\delta n+6}y)}{|{\alpha }_7|·2^{\delta 7n}}})\hfill \\ \hfill & =0\hfill \end{array}$$

for all $x,y\in V\setminus \left\{0\right\}$. And then, since $\underset{y}{\overset{8}{\Delta }}{F}_{\delta k}\left(x\right)=0$ for all $x,y\in V\setminus \left\{0\right\}$, the mapping $F_{\delta k}$ is a general septic mapping for all $k\in \{1,2,3,4,5,6,7\}$ and $\delta \in \{+1,-1\}$ by Lemma 2.4.

Step 2.

Now, we define the desired general septic mapping F for all cases.

(1)

Let $\phi$ satisfy the Condition (8) and let $F_1,F_2$, $F_3$, $F_4$, $F_5$, $F_6$, and $F_7$ be mappings defined by (19). We put a general septic mapping $F:V\to Y$ by

$$F\left(x\right):=F_1\left(x\right)+F_2\left(x\right)+F_3\left(x\right)+F_4\left(x\right)+F_5\left(x\right)+F_6\left(x\right)+F_7\left(x\right)$$

for all $x\in V$. Observe that, by (17) we have

$$\begin{array}{cc}\hfill & \parallel \sum _{k=1}^7{\tilde{f}}_k\left(x\right)-\sum _{k=1}^7{\displaystyle \frac{{\tilde{f}}_k\left(2^{n}x\right)}{2^{kn}}\parallel }\le \sum _{i=0}^{n-1}\left|\sum _{k=1}^7{\displaystyle \frac{1}{2^k{\alpha }_k·2^{ki}}}\right|\parallel \Gamma \tilde{f}\left(2^{i}x\right)\parallel \le {\displaystyle \frac{1}{2|{\alpha }_1|}}\sum _{i=0}^{n-1}{\displaystyle \frac{\mathrm{\Phi }\left(2^{i}x\right)}{2^i}}\hfill \end{array}$$

for all $x\in V\setminus \left\{0\right\}$, since $\left|{\displaystyle \sum _{k=1}^7}{\displaystyle \frac{1}{2^k{\alpha }_k·2^{ki}}}\right|\le {\displaystyle \frac{1}{2|{\alpha }_1|2^i}}$ for all $i\in \mathbb{N}\cup \left\{0\right\}$. If $n\to \infty$, we obtain the Inequality (12).

(2)

Let $\phi$ satisfy the Condition (9) for some $\mathsf{\ell }\in \{1,2,\dots ,6\}$ and let $F_{-1},\dots ,F_{-\mathsf{\ell }}$, $F_{\mathsf{\ell }+1},\dots ,F_7$ be mappings defined by (19). Putting a general septic mapping $F:V\to Y$ by

$$F\left(x\right):=F_{-1}\left(x\right)+\cdots +F_{-\mathsf{\ell }}\left(x\right)+F_{\mathsf{\ell }+1}\left(x\right)+\cdots +F_7\left(x\right)$$

for all $x\in V$. Since we can check the inequalities $\left|{\displaystyle \sum _{k=\mathsf{\ell }+1}^7}{\displaystyle \frac{1}{2^k{\alpha }_k·2^{ki}}}\right|\le {\displaystyle \frac{1}{2^{\mathsf{\ell }+1}\left|{\alpha }_{\mathsf{\ell }+1}\right|2^{(\mathsf{\ell }+1)i}}}$ and $\left|{\displaystyle \sum _{k=1}^{\mathsf{\ell }}}{\displaystyle \frac{2^{ki}}{{\alpha }_k}}\right|\le {\displaystyle \frac{2^{\mathsf{\ell }i}}{|{\alpha }_{\mathsf{\ell }}|}}$ for all $i\in \mathbb{N}\cup \left\{0\right\}$ through some cumbersome calculations, we can obtain the Inequality (13) from the Inequalities (17), (18), and

$$\begin{array}{cc}\hfill \parallel \sum _{k=1}^7& {\tilde{f}}_k\left(x\right)-\sum _{k=1}^{\mathsf{\ell }}{2}^{kn}{\tilde{f}}_k\left({\displaystyle \frac{x}{2^n}}\right)-\sum _{k=\mathsf{\ell }+1}^7{\displaystyle \frac{{\tilde{f}}_k\left(2^{n}x\right)}{2^{kn}}}\parallel \hfill \\ \hfill \le & \sum _{i=0}^{n-1}\left[\parallel \sum _{k=1}^{\mathsf{\ell }}\left(2^{ki}{\tilde{f}}_k\left({\displaystyle \frac{x}{2^i}}\right)-2^{k(i+1)}{\tilde{f}}_k\left({\displaystyle \frac{x}{2^{i+1}}}\right)\right)\parallel +\parallel \sum _{k=\mathsf{\ell }+1}^7\left({\displaystyle \frac{{\tilde{f}}_k\left(2^{i}x\right)}{2^{ki}}}-{\displaystyle \frac{{\tilde{f}}_k\left(2^{i+1}x\right)}{2^{k(i+1)}}}\right)\parallel \right]\hfill \\ \hfill \le & \sum _{i=0}^{n-1}\left|\sum _{k=1}^{\mathsf{\ell }}{\displaystyle \frac{2^{ki}}{{\alpha }_k}}\right|∥\Gamma \tilde{f}\left({\displaystyle \frac{x}{2^{i+1}}}\right)∥+\sum _{i=0}^{n-1}\left|\sum _{k=\mathsf{\ell }+1}^7{\displaystyle \frac{1}{2^k{\alpha }_k·2^{ki}}}\right|\parallel \Gamma \tilde{f}\left(2^{i}x\right)\parallel \hfill \\ \hfill \le & \sum _{i=0}^{n-1}{\displaystyle \frac{2^{\mathsf{\ell }i}}{|{\alpha }_{\mathsf{\ell }}|}}∥\Gamma \tilde{f}\left({\displaystyle \frac{x}{2^{i+1}}}\right)∥+\sum _{i=0}^{n-1}{\displaystyle \frac{1}{2^{\mathsf{\ell }+1}\left|{\alpha }_{\mathsf{\ell }+1}\right|2^{(\mathsf{\ell }+1)i}}}\parallel \Gamma \tilde{f}\left(2^{i}x\right)\parallel \hfill \\ \hfill \le & {\displaystyle \frac{1}{|{\alpha }_{\mathsf{\ell }}|}}\sum _{i=0}^{n-1}{2}^{\mathsf{\ell }i}\mathrm{\Phi }\left({\displaystyle \frac{x}{2^{i+1}}}\right)+{\displaystyle \frac{1}{2^{\mathsf{\ell }+1}\left|{\alpha }_{\mathsf{\ell }+1}\right|}}\sum _{i=0}^{n-1}{\displaystyle \frac{\mathrm{\Phi }\left(2^{i}x\right)}{2^{(\mathsf{\ell }+1)i}}}\hfill \end{array}$$

for all $x\in V\setminus \left\{0\right\}$.

(3)

Let $\phi$ satisfy the Condition (10) and let $F_{-1},F_{-2}$, $F_{-3}$, $F_{-4}$, $F_{-5}$, $F_{-6}$, and $F_{-7}$ be mappings defined by (19). Putting a general septic mapping

$$F\left(x\right):=F_{-1}\left(x\right)+F_{-2}\left(x\right)+F_{-3}\left(x\right)+F_{-4}\left(x\right)+F_{-5}\left(x\right)+F_{-6}\left(x\right)+F_{-7}\left(x\right)$$

for all $x\in V$. Since we can check the inequalities ${\displaystyle \sum _{k=1}^7}{\displaystyle \frac{2^{ki}}{{\alpha }_k}}=0$ for $i\in \{0,1,2,3,4,5\}$ and $\left|{\displaystyle \sum _{k=1}^7}{\displaystyle \frac{2^{ki}}{{\alpha }_k}}\right|\le {\displaystyle \frac{2^{7i}}{|{\alpha }_7|}}$ for all $i>5$, the Inequality (14) follows from the Inequalities (18) and

$$\begin{array}{cc}\hfill \parallel \sum _{k=1}^7{\tilde{f}}_k\left(x\right)-& \sum _{k=1}^7{2}^{kn}{\tilde{f}}_k\left({\displaystyle \frac{x}{2^n}}\right)\parallel \le \sum _{i=0}^{n-1}\parallel \sum _{k=1}^7\left(2^{ki}{\tilde{f}}_k\left({\displaystyle \frac{x}{2^i}}\right)-2^{k(i+1)}{\tilde{f}}_k\left({\displaystyle \frac{x}{2^{i+1}}}\right)\right)\parallel \hfill \\ \hfill & \le \sum _{i=0}^{n-1}\left|\sum _{k=1}^7{\displaystyle \frac{2^{ki}}{{\alpha }_k}}\right|∥\Gamma \tilde{f}\left({\displaystyle \frac{x}{2^{i+1}}}\right)∥\le \sum _{i=6}^{n-1}{\displaystyle \frac{2^{7i}}{|{\alpha }_7|}}∥\Gamma \tilde{f}\left({\displaystyle \frac{x}{2^{i+1}}}\right)∥\hfill \\ \hfill & \le {\displaystyle \frac{1}{|{\alpha }_7|}}\sum _{i=6}^{n-1}{2}^{7i}\mathrm{\Phi }\left({\displaystyle \frac{x}{2^{i+1}}}\right)\hfill \end{array}$$

for all $x\in V\setminus \left\{0\right\}$.

Moreover, by the definition, we easily obtain

$$F_{\delta k}\left(2x\right)=2^k{F}_{\delta k}\left(x\right)$$

and $\underset{y}{\overset{8}{\Delta }}{F}_{\delta k}\left(x\right)=0$ for all $x,y\in V$, $k\in \{1,2,3,4,5,6,7\}$ and $\delta \in \{+1,-1\}$. According to Lemma 2.4, F is the unique general septic mapping. □

The stability results for the functional Equation (2) proven by Chang et al. [23] only cover the Conditions (8) and (10) of Theorem 2.5.

Substituting $\phi (x,y)$ as $\theta (\parallel x{\parallel }^p+\parallel y{\parallel }^p)$ in Theorem 1, we obtain the following corollary for the stability of the Hyers–Ulam–Rassias stability of the general septic functional equation.

Corollary 1. 

Let θ be a positive real constant and p a real number such that $p\ne 1,2,3,4,5,6,7$. If $f:X\to Y$ satisfies the inequality

$$\begin{array}{c}\hfill \parallel \underset{y}{\overset{8}{\Delta }}f\left(x\right)\parallel \le {\theta (\parallel x\parallel }^p+{\parallel y\parallel }^p)\end{array}$$

for all $x,y\in X\setminus \left\{0\right\}$, then there exists a unique mapping general septic mapping F such that $F\left(0\right)=0$ and

$$\begin{array}{ccc}\hfill \parallel \tilde{f}\left(x\right)-F\left(x\right)\parallel \le & {\displaystyle \frac{{M\theta \parallel x\parallel }^p}{|{\alpha }_1|(2-2^p)}}\hfill & \hfill (ifp<1),\\ \hfill \parallel \tilde{f}\left(x\right)-F\left(x\right)\parallel \le & {\displaystyle \frac{{M\theta \parallel x\parallel }^p}{|{\alpha }_{\mathsf{\ell }}|(2^p-2^{\mathsf{\ell }})}}+{\displaystyle \frac{{M\theta \parallel x\parallel }^p}{|{\alpha }_{\mathsf{\ell }+1}|(2^{\mathsf{\ell }+1}-2^p)}}\hfill & \hfill (if\mathsf{\ell }<p<\mathsf{\ell }+1),\\ \hfill \parallel \tilde{f}\left(x\right)-F\left(x\right)\parallel \le & {\displaystyle \frac{{M\theta \parallel x\parallel }^p}{|{\alpha }_7|2^{6(p-7)}(2^p-128)}}\hfill & \hfill (if7<p),\end{array}$$

(20)

for all $x\in X\setminus \left\{0\right\}$ and for some $\mathsf{\ell }\in \{1,2,3,4,5,6\}$, where

$$\begin{array}{cc}\hfill M:=& {128}^p+6180·{64}^p+8·{48}^p+64·{40}^p+1010500·{32}^p+10144·{24}^p\hfill \\ \hfill & +75264·{20}^p+21158873·{16}^p+1240064·{12}^p+5046272·{10}^p\hfill \\ \hfill & +74956392·8^p+40452096·6^p+34603008·5^p+171907840·4^p\hfill \\ \hfill & +232783872·3^p+477265920·2^p+1081081856.\hfill \end{array}$$

Remark 1. 

Specifically, if we compare Corollary 1 and Theorem 2.2 in [22] for the case $4<p<5$, the Inequality (20) in Corollary 1 is

$$\begin{array}{cc}\hfill \parallel \tilde{f}\left(x\right)-F\left(x\right)\parallel \le & {\displaystyle \frac{{M\theta \parallel x\parallel }^p}{115605504(2^p-16)}}+{\displaystyle \frac{{M\theta \parallel x\parallel }^p}{990904320(32-2^p)}}\hfill \end{array}$$

for all $x\in X$ and the inequality inequality (2.13) in Theorem 2.2 in [22] is

$$\begin{array}{cc}\hfill \parallel \tilde{f}\left(x\right)-F\left(x\right)\parallel \le & {\displaystyle \frac{K^{\prime }\theta {\parallel x\parallel }^p}{22680|2-2^p|}}+{\displaystyle \frac{{K\theta \parallel x\parallel }^p}{720|4-2^p|}}+{\displaystyle \frac{K^{\prime }\theta {\parallel x\parallel }^p}{17280|8-2^p|}}+{\displaystyle \frac{{K\theta \parallel x\parallel }^p}{576|16-2^p|}}\hfill \\ \hfill & +{\displaystyle \frac{K^{\prime }\theta {\parallel x\parallel }^p}{69120|32-2^p|}}+{\displaystyle \frac{{K\theta \parallel x\parallel }^p}{2880|64-2^p|}}+{\displaystyle \frac{K^{\prime }\theta {\parallel x\parallel }^p}{1451520|2^p-128|}}\hfill \end{array}$$

for all $x\in X$, where K and $K^{\prime }$ are constants given by

$$\begin{array}{cc}\hfill K:=& \left(8^p+8·6^p+196·4^p+1280·3^p+4317·2^p+16224\right),\hfill \\ \hfill K^{\prime }:=& \left(8^p+8·6^p+196·4^p+1280·3^p+4317·2^p+16224\right).\hfill \end{array}$$

From this, we can see that the Inequality (20) in Corollary 1 is simpler and clearer than the inequality (2.13) in Theorem 2.2 of [22] for the Hyers–Ulam–Rassias stability of the general septic functional equation. Also, since p is a real number in Corollary 1 and a positive real number in Theorem 2.2 in [22], Corollary 1 is a generalization of Theorem 2.2 of [22]. However, since ${\parallel 0\parallel }^p$ is not defined when $p<0$ in Corollary 1, Inequality (20) holds for $x\in X\setminus \left\{0\right\}$.

Note that if we obtain the Hyers–Ulam–Rassias stability of the functional Equation (2) by the corollary of Theorem 3 and Theorem 4 proven by Chang et al., this is obtained only for the cases $p<1$ and $p>7$.

As a corollary of 1, the Hyers–Ulam stability of the general septic functional equation is obtained as follows.

Corollary 2. 

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

$$\begin{array}{c}\hfill \parallel \underset{y}{\overset{8}{\Delta }}f\left(x\right)\parallel \le \delta \end{array}$$

for all $x,y\in X$, then there exists a unique general septic mapping F such that

$$\begin{array}{cc}\hfill \parallel \tilde{f}\left(x\right)-F\left(x\right)\parallel \le & {\displaystyle \frac{1070799177}{39372480}}\delta \hfill \end{array}$$

(21)

for all $x\in X$.

Proof. 

According to Corollary 1 when $\theta ={\displaystyle \frac{\delta }{2}}$ and $p=0$, there exists a unique general septic mapping F satisfying $F\left(0\right)=0$ and the Inequality (21) for all $x\in X\setminus \left\{0\right\}$. Since $\tilde{f}\left(0\right)=0$, F satisfyies the Inequality (21) for all $x\in X$. □

3. Conclusions

In this paper, we investigated the generalized stability of the general septic functional equation. Precisely, if $f:V\to Y$ is a mapping such that $\parallel \underset{y}{\overset{8}{\Delta }}f\left(x\right)\parallel \le \phi (x,y)$ for all $x,y\in V\setminus \left\{0\right\}$, where $\phi :{(V\setminus \left\{0\right\})}^2\to [0,\infty )$ holds the Conditions (8), (9) or (10), then there exists a unique general septic mapping F such that the difference $\parallel \tilde{f}\left(x\right)-F\left(x\right)\parallel$ satisfies the Conditions (12), (13) or (14), respectively.

The generalized stability of the functional Equation (2) is a generalization for the stability of various functional equations that have additive, quadratic, additive–quadratic, additive, quadratic–cubic, cubic, quartic, quintic, sextic, and cubic–septic, septic, Jensen, general quadratic, general cubic, general quartic, general quintic, and general octic mappings as solutions. Therefore, proving the generalized stability of the functional Equation (2) occupies a very important position in the study of the stability of functional equations. However, since there are many obstacles to overcome in proving the generalized stability of the functional Equation (2) for general n, it is necessary to continue research on the generalized stability of the functional Equation (2) for the cases $n=7,9,10,\dots$ for now.