The Stability of a General Sextic Functional Equation by Fixed Point Theory

Abstract

In this paper, we will consider the generalized sextic functional equation And by applying the fixed point theorem in the sense of Cdariu and Radu, we will discuss the stability of the solutions for this functional equation.

1. Introduction

In 1940, Ulam [1] remarked the problem concerning the stability of group homomorphisms. In 1941, Hyers [2] gave an answer to this question for additive mappings between Banach spaces. Subsequently, many mathematicians came to deal with this question (cf. [3–10]). Let and be real vector spaces, be a real normed space, be a real Banach space, (the set of natural numbers), and be a given mapping. Consider the functional equation

for every , where . The functional equation (1) is called an -monomial functional equation, and every solution of the functional equation (1) is called to be a monomial mapping of degree . The function given by is a particular solution of the functional equation (1). In particular, the functional equation (1) is called a sextic functional equation for the case , and every solution of the functional equation (1) is called to be a sextic mapping for the case . Many mathematicians [11–17] have previously investigated the stability of the sextic functional equation, and many authors [18–26] have studied the stability of the -monomial functional equation in various spaces.

The solution of the functional equation

is called a generalized polynomial mapping of degree (See Baker [27]). The function given by is a particular solution of the functional equation (3). Some mathematicians [28–31] have previously investigated the stability of the functional equation (3) for the cases . In particular, the functional equation is called a general sextic functional equation, and every solution of the functional equation (4) is said to be the general sextic mapping.

In this paper, we will partially generalize the results in [31] for the stability of the general sextic functional equation. For the details, one can refer Corollary 4 and Corollary 7 which are special cases of main theorems. Specifically, in this paper, we will show that there is only one solution of the general sextic functional equation (4) near the function , which approximates the functional equation (4) by using fixed point theorem [32–35]. Moreover, the solution mapping of the functional equation (4) can be explicitly constructed by the formula

or

which approximates the mapping .

2. Main Results

We first recall the following Margolis and Diaz fixed point theorem, which is necessary to obtain the main results of this paper.

Proposition 1 (see [36]). Suppose is a complete generalized metric space, which means that the metric may assume infinite values, and is a strictly contractive mapping with the Lipschitz constant . Then, for each given element , either or there exists an integer such that: (i) for all (ii)The sequence converge to a fixed point of (iii) is a unique fixed point of in (iv) for every In this paper, we let and be real vector spaces, be a real normed space, and be a real Banach space. For a mapping , we use the following abbreviations for every .

Now, we will see useful lemma for the proof of main theorem.

Lemma 2. Let be a real constant such that and . Let be a function for which there exists a constant such that for all . Then, , and the equality holds for all .

Proof. When and , it is not difficult to see that and in the trigonometric function table. So .
We can also obtain the equality (7) by the following calculation: And, to obtain the equality (8), by (7), we obtain the following calculation:

In the following main theorem, we will prove the generalized Hyers-Ulam stability of the functional equation (4) by using the direct method.

Theorem 3. Let , , and be as in Lemma 2. If is a mapping satisfying and the inequality then there exists the unique solution mapping of (4) such that for all , where In particular, is represented by for all .

Proof. We let the set be the set of the functions with . And we define a generalized metric on by Then, it is not so difficult to show that is a complete generalized metric space (see ([34], Theorem 2.5) or the proof of ([37], Theorem 3.1)). Next, we see the mapping , which is defined by for all
And, by using the oddness and the evenness of and and , due to mathematical induction, we can get holds for all and .
Let and we choose as an arbitrary constant with . Due to the definition of and (7) in Lemma 2, we have for every , which implies that for all , where . That is, with the Lipschitz constant , is a strictly contractive self-mapping of , where .
Now, after long and tedious calculation, we have And, by (11) we obtain for every . It implies that from the definition of and due to Proposition 1, the sequence converges to only one fixed point of in the set which implies (13). Moreover, by Proposition 1, we have which implies (12).
Also, by the equality (8) in Lemma 2, since one has and for all , we obtain for every .
Therefore, is the unique solution of the functional equation (4) with (12). Finally, we see that if is a solution of the sextic functional equation (4) with , then we can derive that is a fixed point of from the equality

In next corollary, we will consider special function in Theorem 3 to compare with the results in [31].

Corollary 4. Let be a real normed space, be as in Lemma 2, and be a fixed real number such that . If satisfies the equality and the inequality for all , then there exists a unique solution mapping of (4) satisfying the inequality for all .

Proof. If we put and , then we have the equalities for all . So the condition (6) in Lemma 2 holds for all . According to Theorem 3, there exists a unique solution mapping of (4) satisfying the inequality (16) for all .

Next, we will try to prove the stability of the sextic functional equation (4) from another point of view. For that, we first will introduce useful facts in the following lemma.

Lemma 5. Let be a real constant such that and . Let be a function for which there exists a constant such that for all . Then, we have , and the equality holds for all .

Proof. When and , it is not difficult to see that and in the trigonometric function table. Also, we obtain the equality (18) from the following calculation: And by (17) and (18), we have for all . Therefore, by taking the limit, we complete the proof of (19).

In the following theorem, we will prove the stability of the solution for the sextic functional equation (4) with different types of functions compared to Theorem 3.

Theorem 6. Let , , and be as in Lemma 5. If is a mapping such that the inequality (11) in Theorem 3 holds for every , then there exists only one solution of (4) satisfying the following inequality for every , where In particular, is represented by for any .

Proof. Similar to Theorem 3, we consider the set which contains all functions with and we define a generalized metric on as We now consider the mapping defined by for every . Then, similar to Theorem 3, by mathematical induction, we obtain that holds for every and .
Let and we assume as an arbitrary constant with . By the definition of and (18) in Lemma 5, we have for every , which implies that for all , where . So, with the Lipschitz constant , is a strictly contractive self-mapping of , where .
Moreover, by the definition of , with long and tedious calculation, we have And, by (11) in assumption, we obtain for every .
It implies that by the definition of . Therefore, according to Proposition 1, the sequence converges to only one fixed point of in the set , which is represented by (23) for every .
We also due to Proposition 1 obtain that which implies (22).
Now, by (19) in Lemma 5, since we have for all , due to the equality (23), we obtain which conclude that is a solution of the sextic functional equation (4).
Finally, we see that if is a solution of the sextic functional equation (4), then the equality implies that is a fixed point of .

In next corollary, we will consider special function in Theorem 6 to compare with the results in [31].

Corollary 7. Let be a real normed space, be as in Lemma 5, and be a fixed real number such that . If satisfies the equality and the inequality (15) for all , then there exists a unique solution mapping of (4) such that for all .

Proof. If we put and , then we have the equalities for all . So the condition (17) in Lemma 5 holds for all . According to Theorem 6, there exists a unique solution mapping of (4) satisfying the inequality (24) for all .

Data Availability

No data are available for this study.

Conflicts of Interest

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

Acknowledgments

This work was supported by Hallym University Research Fund (HRF-201910-014).