#### Abstract

If a mapping can be expressed by sum of a septic mapping, a sextic mapping, a quintic mapping, a quartic mapping, a cubic mapping, a quadratic mapping, an additive mapping, and a constant mapping, we say that it is a general septic mapping. A functional equation is said to be a general septic functional equation provided that each solution of that equation is a general septic mapping. In fact, there are a lot of ways to show the stability of functional equations, but by using the method of Gvruta, we examine the stability of general septic functional equation which considered. The method of Gvruta as just mentioned was given in the reference Gavruta (1994).

#### 1. Introduction

The concept of stability for a functional equation arising when replacing the functional equation by an inequality which acts as a perturbation of the equation. Ulam  posed the question concerning the stability of group homomorphisms. Hyers  gave the first partial affirmative answer to the question of Ulam, which states that if and is a mapping with a normed space, a Banach space such that then there exists a unique additive mapping such that

Thereafter, many authors have generalized the Hyers’ result (refer to ). On the other hand, a generalization of the Hyers result was also obtained by Gvruta . That is, let be an abelian group and let be a Banach space. Suppose that a function satisfies the assumption for all and that is a mapping such that the inequality holds for all Then, there exists a unique additive mapping with

We, meanwhile, call the functional equation the -monomial functional equation. The -monomial, the -monomial, the -monomial, the -monomial, the -monomial, the -monomial, and the -monomial functional equation are called additive, quadratic, cubic, quartic, quintic, sextic, and septic functional equation, respectively. In this case, we say that each solution of the previous equation is additive, quadratic, cubic, quartic, quintic, sextic, and septic mapping, respectively. The function defined by is a particular solution of the -monomial functional equation. Quite recently, Lee  showed the stability of the -monomial functional equation in the sense of Gvruta.

The following functional equation is called as Jensen, general quadratic, general cubic, general quartic, general quintic, general sextic, and general septic functional equation, respectively, for The solution of the general septic functional equation is said to be a general septic mapping. The function given by is a particular solution of the general septic mapping. More detailed term for the concept of a general septic mapping can be found in Baker’s paper  by the term generalized polynomial mapping of degree at most 7.

For a number of years now, many interesting results of the stability problems to several functional equations (or involving the range from additive functional equation to sextic functional equation) have been investigated; see, e.g., .

Our principal purpose is to consider the following general septic functional equation and then we are going to obtain the stability theorems of the functional equation (8) in the spirit of Gvruta approach.

#### 2. Stability of the General Septic Functional Equation (8)

In this section, we let and be a real Banach space and a real vector space, respectively. For a given mapping and all , we use the following abbreviations

On the other hand, if is a mapping defined by , we know that

In addition, through tedious computation, we then get the following expressions for all

Lemma 1. Let be a mapping with Suppose that and are mappings given by for all and all integers and for all and all integers . Then, holds for all and all integers and is fulfilled for all and all integers

Proof. By using (11) and the definitions of and , we can obtain the result after tedious calculations. Therefore, the proof will be omitted here.

Lemma 2. Assume that is a mapping with subject to the equation for all Then, we have for all and all positive integers

Proof. We have by the definitions of and that So, we figure out which implies for all and all positive integers Similarly, we get the equality for all and all positive integers .

We are now in a position to prove the following theorem.

Theorem 3. Suppose that a function satisfies the condition for all Assume that is a mapping subject to the inequality for all Then, there exists a unique general septic mapping with such that for all , where and are functions defined by

Proof. Considering a mapping defined by , we see that satisfies the properties Then, by (11) and the definitions of and , we obtain that hold for all It follows from (15) and (26) that for all . This gives that for all and all nonnegative integers and By the definition of and together with (21) and (29), the sequence is Cauchy in And since is complete, the sequence converges. Therefore, we can define a mapping by Note that and follow from Furthermore, by letting and passing the limit as in (29), we arrive at the inequality (23). On the other hand, from (21) and the definition of , we yield the following inequality for all .
In order to prove the uniqueness of , we suppose that is another general septic mapping satisfying (23) and However, it is also possible to show uniqueness by replacing condition (23) with weaker condition. So, we want to prove that there is a unique mapping satisfying the weaker condition for all . According to Lemma 2, we get for all positive integers Thus, by using the condition (32) and the definition of , after tedious calculations, we have for all and all positive integer Taking the limit as in the last inequality, we then have This implies that

Theorem 4. Suppose that a function satisfies the condition for all Assume that is a mapping subject to the inequality for all Then, there exists a unique general septic mapping with such that for all , where and are the functions in Theorem 3.

Proof. We first define a mapping by Then, we find that With help of the definitions of and , the relations (16), (26), and (36) guarantee that for all , which leads to the inequality for all and all nonnegative integers and In view of (35) and (40), the sequence is Cauchy in . By completeness of , the sequence converges. Thereby, we can define a mapping by Since , we have Note that for all Letting and sending the limit in (40) as , we obtain the desired result (37).
But, we intend to claim that The expression (35) and the definition of provide that To show the aforementioned uniqueness, let us assume that is another general septic mapping with (37) and Then, this proof is analogue to that of Theorem 3. That is, instead of the condition (37), it suffices to prove that there is a unique mapping satisfying the weaker condition for all . From Lemma 2, the equality holds for all positive integer So, we have by (43) that for all and all positive integer Taking the limit as in the previous inequality, we get the relation which means that

#### 3. Discussion

In this paper, we investigated the stability of general septic functional equation by using the method of Gvruta. In Theorem 3, we proved that if the function satisfies the inequality where a function satisfies the condition for all , then there exists a unique general septic mapping near the function .

And in Theorem 4, we proved similar result when the function satisfies the condition

Lee  proved the stability of the general sextic functional equation for the mapping such that

For the future work, we want to see if we can get similar results for the general septic functional equation. And by using our results, we want to know what we can say for the stability of the general octic functional equation.

Also, by fixed point theorem, Roh et al.  showed the stability of the general sextic functional equation for the mapping such that where is a function for which there exists a constant such that for all , and is a real constant such that and . In fact, for the stability of the septic functional equation, we first tried this method, but we failed to get the good result. The calculation became complicated because it resulted in a quadratic equation problem. So, we leave this problem as an open problem.

#### Data Availability

For the manuscript, there is no data we need.

#### Conflicts of Interest

The authors declare that there are no conflicts of interest regarding the publication of this article.

#### Authors’ Contributions

All authors contributed equally to the writing of this paper. All authors read and approved the final manuscript.

#### Acknowledgments

The authors would like to thank the referee for his/her time and efforts. This work was supported by the Hallym University Research Fund (HRF-202011-012), and Jaiok Roh was partially supported by the Data Science Convergence Research Center of Hallym University.