Some Remarks Concerning the General Octic Functional Equation

Lee, Yang-Hi; Roh, Jaiok

doi:https://doi.org/10.1155/2023/2930056

Journal of Mathematics

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Research Article | Open Access

Volume 2023 | Article ID 2930056 | https://doi.org/10.1155/2023/2930056

Some Remarks Concerning the General Octic Functional Equation

Yang-Hi Lee¹and Jaiok Roh²

Academic Editor: Gaetano Luciano

Received09 Dec 2022

Revised27 Mar 2023

Accepted15 Apr 2023

Published16 May 2023

Abstract

In this article, we study the stability of various forms for the general octic functional equation We first find a special way of representing a given mapping as the sum of eight mappings. And by using the above representation, we will investigate the hyperstability of the general octic functional equation. Furthermore, we will discuss the Hyers–Ulam–Rassias stability of the general octic functional equation.

1. Introduction

Let be a real normed space and be a real Banach space throughout this article. Consider the functional equation

This functional equation is called as Jensen, general quadratic, general cubic, general quartic, general quintic, general sextic, general septic, and general octic functional equation, respectively, for . And a function given by is a particular solution of equation (1), when a monomial mapping is a particular solution of the -monomial functional equation.

The -monomial functional equation is called as Cauchy, quadratic, cubic, quartic, quintic, sextic, septic, and octic functional equation, respectively, for .

We letfor all . Since and for all , if satisfies the functional equation or , then satisfies the functional equation . Therefore, we say that equation (1) is a more general functional equation than the -monomial functional (2). For additional information one can refer to [1]. Actually, the general octic functional equation is a generalization of Cauchy, Jensen, quadratic, general quadratic, cubic, general cubic, quartic, general quartic, quintic, general quintic, sextic, general sextic, septic, general septic, and octic functional equation.

The stability of equations is a very important topic in the field of mathematical analysis. The stability of various forms for functional equation has been developed by Ulam [2], Hyers [3], Rassias [4], and Găvruta [5]. Specially, Gilányi [6] investigated the stability of the monomial functional equations in real normed spaces. And Cădariu and Radu [7], Kaiser [8], and Lee [9] generalized the results of Gilányi [6]. Moreover, one can refer to [9–11] for the hyperstability of the monomial functional equation. The given functional equation is called hyperstable if any mapping satisfying the given equation approximately (in some sense) must be actually a solution to it (refer to [12]).

Recent surveys of the hyperstability of various equations can be found, among others, in [13–17]. Moreover, recent results of the stability of the functional equation (1) can be found, among others, in [18–24]. Also, very interesting applications to the Hyers–Ulam stability of equations can be found in [25–30]. Studies on the stability and hyperstability of the general octic functional equation have not yet been performed.

In this article, we investigated the hyperstability and Hyers–Ulam–Rassias stability of the general octic functional equation.

In Section 2, we will show that any given mapping is expressed as the sum of eight mappings , , , , , , , and derived from . The expression of the mapping in this way is an essential preliminary step in the proof of the hyperstability and stability of the general octic functional equation, which will be shown in Sections 3 and 4.

In Section 3, we will show the hyperstability of the functional general octic equation as follows:

Let be a given negative real number. If satisfies the inequalityfor all , then for all .

Here, means that , , , , , , , and defined in Section 2 are additive, quadratic, cubic, quartic, quintic, sextic, septic, and octic mappings, respectively.

In Section 4, we will show the stability of the functional general octic equation as follows:

Let be a given non-negative real number. If satisfies inequality (4) for all , then there exists a unique mapping satisfying , , andfor all , where is some constant.

As an example, we will consider and see that can only be obtained as the limit of .

2. Representation of a Mapping

In this section, we find a way of representing a given mapping as the sum of eight mappings , , , , , , , and derived from .

For a given mapping , we denote the following equation:

And for a given mapping , let us consider the following system of nonhomogeneous linear equations:for all . Then, we obtain the following two lemmas by the uniqueness of solution stated in Cramer’s rule.

Lemma 1. Let be a given mapping and

Then, we have the following equation:for all .

Lemma 2. Let be a given mapping and

Then, for all , we have the following equation:

From Lemma 1 and Lemma 2, the mappings , , , , , , , and used to prove the main theorem in Sections 3 and 4 can be calculated and summarized as follows.

Lemma 3. If , , , , , , , and are the mappings given in Lemma 1 and Lemma 2, then , , , , , , , and become as follows:and the following equation:holds for all .

Next, we will define the mappings needed to prove the main theorems.

Definition 1. For a given mapping , we define the following equation:for all .
Next, by tedious computations, one can obtain the following lemma:

Lemma 4. For a given mapping , the following equalities:hold for all , where is the mapping in Lemma 3.

Proof. We omit the proof.

3. Hyperstability of the General Octic Functional Equation

In this section, we will prove the hyperstability of the general octic functional equation. For the proof, we will use the mappings , , , , , , , , , and given in Lemma 3 and Lemma 4.

Theorem 1. Let and be real numbers. Suppose that is a mapping such thatfor all . Then,for all .

Proof. Step 1. Since and = 0, we have and . By equations (22), (16), and (17), we have the following equation:for all , whereFrom equations (18), (19), (20), (21), (24), and (25) and the definitions of , , …, , we obtain the following inequalities:which hold for all . Now, for any , we have the following equation:for all . Therefore, for all , we obtain the following equations:Hence, we can get the following equation:for all and . It follows from and equations (29)–(36) that the sequences , …, are Cauchy sequences for all . Also, since is complete and for all , the sequences , …, and converge. Now, we define the mappings as follows:for all .

Step 2. By equation (22) and with the definitions of , , , and = , we have the following equation:for all . Also, by the definition of , we have the following equation:for all . And by equation (38), we obtain the following equation:for all . Hence, by equations (38)–(40), we have for all . Similarly, we can show that , …, for all . Next, since for all , for all .

Step 3. By letting and passing the limit in equation (36), we get the following inequality:for all .
Also, since and for all , we have the following equality for any :for all . Therefore, by and equations (22) and (41), we have the following equation:for all , where . Hence, we have = 0 for all . Also, since , we have for all .
So, we have = = 0 for all . And by the fact, = , we have = = = 0 to complete the proof.

4. Hyers–Ulam–Rassias Stability of the General Octic Functional Equation

In this section, we will investigate the Hyers–Ulam–Rassias stability of the general octic functional equation, as follows:

Lemma 5. If is a mapping such that for all with , then for each , in Lemma 3 satisfies the following equality:for all .

Proof. By the definitions of , , , and , we can get , , and , because , for all .
Also, due to , , and Lemma 4, we obtained .

Theorem 2. Let be a given non-negative real number. And suppose that is a mapping such that

Then, there exists a unique mapping satisfying , , andfor all , where

Proof. From the definition of , we have that for all and . By equation (46) and the definitions of and , we have the following equations:for all .

Next, we want to find a mapping . To find , we will first find , where and make = . Depending on the value of given , we will have a different approach to finding the mapping .

Case 1. If , then it follows from Lemma 4 and equation (49) thatfor all . So, we obtain thatfor all and . It follows from and equation (52) that the sequence is a Cauchy sequence for all . Since is complete, the sequence converges for all . Hence, we can define the mapping by the following equation:Moreover, letting and passing the limit in equation (52), we get the following inequality:for all . From the definition of , we easily get for all andfor all .

Case 2. If , then it follows from Lemma 4 and equation (49) thatfor all . So, we have the following equation:for all and . It follows from and equation (57) that the sequence is a Cauchy sequence for all . Since is complete, the sequence converges for all . Hence, we can define the mapping by the following equation:Moreover, letting and passing the limit in equation (57), we get the following inequality:for all . From the definition of , we easily get for all and for all .
Hence, depending on the value of , we can choose the definition of as the mapping in equation (53) or equation (58).
Similarly, we can define the mappings for .

Case 3. If , then it follows from Lemma 4 and equation (50) thatfor all and . It follows from and equation (60) that the sequence is a Cauchy sequence for all . Since is complete, the sequence converges for all . Hence, we can define the mapping byMoreover, letting and passing the limit in equation (60), we get the following inequality:for all . From the definition of , we easily get for all and for all .

Case 4. If , then it follows from Lemma 4 and equation (50) thatfor all and . It follows from and equation (63) that the sequence is a Cauchy sequence for all . Since is complete, the sequence converges for all . Hence, we can define the mapping by the following equation:Moreover, letting and passing the limit in equation (63), we get the following inequality:for all . From the definition of , we easily get for all and for all .
Hence, depending on the value of , we can choose the definition of as the mapping in equation (61) or equation (64).

Case 5. If , then it follows from Lemma 4 and equation (49) thatfor all and . It follows from and equation (66) that the sequence is a Cauchy sequence for all . Since is complete, the sequence converges for all . Hence, we can define the mapping by the following equation:Moreover, letting and passing the limit in equation (66), we get the following inequality:for all . From the definition of , we easily get for all and for all .

Case 6. If , then it follows from Lemma 4 and equation (49) thatfor all and . It follows from and equation (69) that the sequence is a Cauchy sequence for all . Since is complete, the sequence converges for all . Hence, we can define the mapping by the following equation:Moreover, letting and passing the limit in equation (69), we get the following inequality:for all . From the definition of , we easily get for all and for all .
Hence, depending on the value of , we can choose the definition of as the mapping in equation (67) or equation (70).

Case 7. If , then we can define the mapping by the following equation:Moreover, we get the following inequality:for all . From the definition of , we easily get for all and for all .

Case 8. If , then we can define the mapping by the following equation:Moreover, we get the following inequality:for all . From the definition of , we easily get for all and for all .
Hence, depending on the value of , we can choose the definition of as the mapping in equation (72) or equation (74).

Case 9. If , then we can define the mapping by the following equation:Moreover, we get the following inequality:for all . From the definition of , we easily get for all and for all .

Case 10. If , then we can define the mapping by the following equation:Moreover, we get the following inequality:for all . From the definition of , we easily get for all and for all .
Hence, depending on the value of , we can choose the definition of as the mapping in equation (76) or equation (78).

Case 11. If , then we can define the mapping by the following equation:Moreover, we get the following inequality:for all . From the definition of , we easily get for all and for all .

Case 12. If , then we can define the mapping by the following equation:Moreover, we get the following inequality:for all . From the definition of , we easily get for all and for all .
Hence, depending on the value of , we can choose the definition of as the mapping in equation (80) or equation (82).

Case 13. If , then we can define the mapping by the following equation:Moreover, we get the following inequality:for all . From the definition of , we easily get for all and for all .

Case 14. If , then we can define the mapping by the following equation:Moreover, we get the following inequality:for all . From the definition of , we easily get for all and for all .
Hence, depending on the value of , we can choose the definition of as the mapping in equation (84) or equation (86).

Case 15. If , then we can define the mapping by the following equation:Moreover, we get the following inequality:for all . From the definition of , we easily get for all and for all .

Case 16. If , then we can define the mapping by the following equation:Moreover, we get the following inequality:for all . From the definition of , we easily get for all and for all .
Hence, depending on the value of , we can choose the definition of as the mapping in equation (88) or equation (90).
Finally, we define the mapping as follows:Since , for all , we have the following equation:Next, we need to prove that the mapping satisfies inequality (49). Since and for all , we have inequality (49).
At last, we need to prove the uniqueness of . Suppose that be another mapping satisfying for all and inequality (49) for all with . By Lemma 5, the mappings satisfy the following equation:To prove the uniqueness of , we will assume for a given . For other cases, one can prove by a similar method. If , then we have the following equation:for all and all positive integer , whereTaking the limit in the above inequalities as , we obtain the equality for all , which means that for all .
Also, we obtain the following equation:for all and all positive integer . Taking the limit in the above inequalities as , we obtain the equality for all , which means that for all .
And we easily show that , , , , , and by the similar method. Since , we complete the proof of the uniqueness of for the case . For the other cases, we can easily prove the uniqueness of in much the same way as the proof for the case .

5. Conclusion

In this article, we discussed the hyperstability and Hyers–Ulam–Rassias stability of the general octic functional equation. To prove main theorems, we proved that any given mapping can be expressed as the sum of eight mappings , , , , , , , and derived from . The expression of the mapping in this way is an essential preliminary step in the proof of the stabilities of the general octic functional equation, which are shown in Sections 3 and 4. In particular, we proved that holds for if the mapping satisfies the general octic functional equation. This means that , , , , , , , and are additive, quadratic, cubic, quartic, quintic, sextic, septic, and octic mapping, respectively, if the mapping satisfies the general octic functional equation. Not much research has been conducted in the general octic functional equation. The big advantage of this article is that we proved the uniqueness of the solution in the Hyers–Ulam–Rassias stability of the general octic functional equation. The uniqueness of the solution in the Hyers–Ulam–Rassias stability of the monomial functional equation has been discussed in many researches. But the uniqueness of the solution in the Hyers–Ulam–Rassias stability of the general functional equations is a more complicated problem. As an example, for , the general octic mapping satisfying inequality (53) can be expressed only as the limit of the following function:as , while one obtain in the case of stability of the octic functional equation. As a follow-up research, we studied the stability of the general nonic functional equation. Finally, we will try to get generalized results of the stability for the general -th functional equation.

Data Availability

No data were used to support this study.

Conflicts of Interest

The authors declare that they have no conflicts of interest.

Authors’ Contributions

The authors conceived the study, participated in its design and coordination, drafted the manuscript, participated in the sequence alignment, and read and approved the final manuscript.

Acknowledgments

This work was supported by the National Research Foundation of Korea (NRF) Grant Funded by the Korea Government (MSIT) (no. 2021R1A2C109489611).

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Copyright © 2023 Yang-Hi Lee and Jaiok Roh. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

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