The various types of stability for functional equations are very interesting material in the field of mathematical analysis. The stability problems of functional equations have been developed by some authors ; see [8,10,21,24]. In paricular, Gilányi [9] has investigated the stability of the monomial functional equation in real normed spaces. Subsequent studies have improved the results of Gilányi (for example, [6,11,13]). Moreover, the hyperstability of the monomial functional equation can be found in [1,13,14]. The hyperstability of functional equation means that arbitrary mappings satisfying equation approximately (in some sense) must be really solution to it (refer to [19]).

In this paper, we discuss the general functional equation :

This functional equation is called as the general nonic functional equation, specially, for $n=10$. The function $f\left(x\right):={\sum }_{i=0}^{n-1}{a}_i{x}^i$ is a particular solution of (1.1), while the function $f\left(x\right):=a{x}^n$ is a particular solution of the n-monomial functional equation

One can find more details in [3]. The functional equation (1.2) is said to be a nonic functional equation, specially, for $n=10.$ That is the reason that we call the equation (1) as the general nonic functional equation, specially, for $n=10.$

On the other hand, the rest of this paper is organized as follows. In section $2,$ we study a way of representing a given mapping as the sum of odd and even mappings. In section $3,$ we investigate the hyperstability of the general nonic functional equation, that is, (1.1) for $n=10.$ And then in section $4,$ we discuss the stability problem of the general nonic functional equation.

Not much study has been conducted on the general nonic functional equations. The big advantage of this paper is the uniqueness of the solution in the stability of the general nonic functional equation. The uniqueness of the solution in the stability of the monomial functional equation have been discussed in many researches. But, the uniqueness of the solution in the stability of the general nonic functional equations is more complicated problem. Considering the special representation of a given mapping, we then solved this problem.

In the papers [2,4,5,12,23], one can see the hyperstability result of the functional equations. The recent results of the stability of the general functional equation (1.1) can be found in [7,15,16,17,18,20,22].

Lastly, before going into the content of the paper, readers should recall that X is a normed space and Y is a Banach space throughout this paper.

In this section, we will introduce a way of representing a given mapping as the sum of odd and even mappings.

For a given mapping $f:X\to Y,$ we denote

Let us consider the following system of nonhomogeneous linear equations and for all $x\in X.$ Then, we obtain the following lemmas by the uniqueness of solution stated in Cramer’s rule.

From now on, we define the mappings needed to prove main theorems.

As results of tedious calculation, we obtain the following lemmas :

Proof. Since $Df(x,y)=0$ for all $x,y\in X,$ by the definition of $f_i,\mathsf{\Gamma }f$ and $\mathsf{\Delta }f,$ we have

Applying Lemma 4, we arrive at $f_i\left(2x\right)=2^i{f}_i\left(x\right).$□

In this section, we will prove the hyperstability of the general nonic functional equation. To prove main theorem, we will use the functions introduced in previous section.

Proof.•  STEP $1^{\circ }$: Using the definition of $\tilde{f}$ together with we have that

From the expression (3.1) and the definitions of $\mathsf{\Gamma }f$ and $\mathsf{\Delta }f,$ we get for all $x\in X\setminus \left\{0\right\},$ where

Then, Lemma 3 and the definitions of $f_k$ ($k\in \{1,2,\dots ,9\}$) ensure the inequalities for all $x\in X\setminus \left\{0\right\}$. Note that

This implies that for all $x\in X\setminus \left\{0\right\}$ and $n,m\in \mathbb{N}\cup \left\{0\right\}.$ Due to $p<0,$ the sequences $\left\{\frac{{\tilde{f}}_1\left(2^{n}x\right)}{2^n}\right\},\cdots ,\left\{\frac{{\tilde{f}}_9\left(2^{n}x\right)}{2^{9n}}\right\}$ and $\left\{{\sum }_{k=1}^9\frac{{\tilde{f}}_k\left(2^{n}x\right)}{2^{kn}}\right\}$ are Cauchy for all $x\in X\setminus \left\{0\right\}.$ Since Y is complete and $\tilde{f}\left(0\right)=0,$ the sequences $\left\{\frac{{\tilde{f}}_1\left(2^{n}x\right)}{2^n}\right\},\cdots ,\left\{\frac{{\tilde{f}}_9\left(2^{n}x\right)}{2^{9n}}\right\}$ and $\left\{{\sum }_{k=1}^9\frac{{\tilde{f}}_k\left(2^{n}x\right)}{2^{kn}}\right\}$ converge for all $x\in X.$ Hence, for each $k\in \{1,2,\dots ,9\},$ we can define mappings $F_k,F:X\to Y$ by

•  STEP $2^{\circ }$: By (3.1) and the definitions of $Df,F_1$ and $f_1,$ since $D{f}_1(x,y)=D{\tilde{f}}_1(x,y),$ we then have for all $x,y\in X\setminus \left\{0\right\}.$ On the other hand, from the definition of $D{F}_1,$ we have for all $x\in X.$ And, in view of (3.3), we obtain

Therefore, the relations (3.3), (3.4) and (3.5) yield that $D{F}_1(x,y)=0$ for all $x,y\in X.$ Similarly, we can show that $D{F}_k(x,y)=0$ for each $k\in \{2,3,\cdots ,9\}$ and all $x,y\in X.$ Since $DF(x,y)={\sum }_{k=1}^{9}D{F}_k(x,y)$ for all $x,y\in X,$ we get $DF(x,y)=0$ for all $x,y\in X.$

•  STEP $3^{\circ }$: Observe that for all $x\in X,$

Then, we see that for any $n\in N$ and $x\in X\setminus \left\{0\right\}.$ Moreover, by letting $n=0$ and taking the limit $m\to \infty$ in (3.2), we get the inequalities for all $x\in X.$

Since $p<0,$ by (3.1), (3.6) and (3.7), we are forced to conclude that for all $x\in X\setminus \left\{0\right\},$ where

So, we have $\parallel \tilde{f}\left(x\right)-F\left(x\right)\parallel =0$ for all $x\in X\setminus \left\{0\right\}.$ Since $\tilde{f}\left(0\right)=0=F\left(0\right),$ we have $\tilde{f}\left(x\right)=F\left(x\right)$ for all $x\in X.$ Therefore, $D\tilde{f}(x,y)=DF(x,y)=0$ for all $x,y\in X.$ From the fact that $D\tilde{f}(x,y)=Df(x,y),$ we finally have which completes the proof. □

In this section, we will consider the stability of the general nonic functional equation

Proof. From the definition of $\tilde{f},$ we get $D\tilde{f}(x,y)=Df(x,y)$ and $\tilde{f}\left(0\right)=0.$ By (4.1) and the definition of $\mathsf{\Gamma }f$ and $\mathsf{\Delta }f,$ we have that for all $x\in X.$

Next, for $i\in \{1,2,\dots ,9\},$ we will find $F_k$ to make $F\left(x\right)$ = ${\sum }_{k=1}^9{F}_k\left(x\right).$ For each given $p\ne 1,2,3,4,5,6,7,8,9,$ we will use different approach to find the functions $F_k.$

•Setting $F_1:$ Let $0\le p<1$. It follows from Lemma 4 and (4.3) that for all $x\in X.$ Notice that for all $x\in X,$ which implies that for all $x\in X$ and $n,m\in \mathbb{N}\cup \left\{0\right\}$. By (4.5), the sequence $\left\{\frac{{\tilde{f}}_1\left(2^{n}x\right)}{2^n}\right\}$ is Cauchy for all $x\in X,$ because of the fact that $0\le p<1.$ Since Y is complete, the sequence $\left\{\frac{{\tilde{f}}_1\left(2^{n}x\right)}{2^n}\right\}$ converges. Hence, we may define a mapping $F_1:X\to Y$ by

Moreover, letting $n=0$ and taking the limit $m\to \infty$ in (4.5), we get the inequality

By the definition of $F_1,$ we easily get $F_1\left(2x\right)=2F_1\left(x\right)$ for all $x\in X$ and for all $x,y\in X.$