Abstract
In this work, we examine the generalized Hyers-Ulam orthogonal stability of the quartic functional equation in quasi--normed spaces. Moreover, we prove that this functional equation is not stable in a special condition by a counterexample.
1. Introduction
In this paper, and denote sets of all real numbers and complex numbers, respectively.
In the fall of 1940, Ulam [1] suggested the stability problem of functional equations concerning the stability of group homomorphisms as follows:
Ulam’s question: let , be two groups and be a metric. Given , does there exist such that if a function satisfies the inequality for all , then there is a homomorphism with
In other words, under what condition does there exist a homomorphism near an approximate homomorphism? The concept of stability for functional equation arises when we replace the functional equation by an inequality which acts as a perturbation of the equation. In 1941, Hyers [2] gave the first affirmative answer to the question of Ulam for Banach spaces. This result was generalized by Aoki [3] for additive mappings.
During the past few years, several mathematicians have published on various generalizations and applications of generalized Hyers-Ulam stability to a number of functional equations and mappings (see, for instance, [4–15]).
In [16], Xu et al. obtained the general solution and investigated the Ulam stability problem for the quintic functional equation in quasi--normed spaces via fixed point method. This method is different from the direct method, initiated by Hyers in [2]. And also, Eskandani et al. [17, 18] obtained the general solution for the mixed additive and quadratic functional equation and a cubic functional equation and established its generalized Hyers-Ulam stability in quasi--normed spaces.
The Ulam-type stability result for the quartic functional equation was first developed by Rassias [19]. Subsequently, Sahoo and Chung [20] determined the general solution of (3) without assuming any regularity conditions on the unknown function. In fact, they proved that the function is a solution of (3) if and only if , where the function is symmetric and additive in each variable. Since the solution of (3) is even, we can rewrite (3) as
Lee et al. [21] obtained the general solution of (4) and proved the Hyers-Ulam-Rassias stability of this equation. It is easy to show that the function satisfies the functional equation (4), which is called a quartic functional equation, and every solution of the quartic functional equation is said to be a quartic function. In [22] Ravi et al. have investigated the generalized Hyers-Ulam product-sum stability of functional equations and have the following theorem.
Theorem 1. Let be a mapping which satisfies the inequality for all with , where and are constants with and either , or , with , , , and . Then, the limit exists for all , and is the unique orthogonally Euler-Lagrange quadratic mapping such that for all .
In 1982, Rassias [23] provided generalizations of the Hyers-Ulam stability theorem which allows the Cauchy difference controlled by a product of different powers of norm. And then, the result of the Rassias theorem has been generalized by Gavruta [24] by replacing the unbounded Cauchy difference by a generalized control function. Also, Rassias (see [23, 25–28]) solved the Ulam problem for different mappings. In addition, Ravi et al. considered the mixed product-sum of powers of norms control function [22]. Note that the mixed product-sum function was introduced by Ravi et al. in 2008-2009 ([22, 29–31]).
In this paper, we examine the generalized Hyers-Ulam orthogonal stability of the quartic functional equation as where is a positive integer with . It is easy to see that the function is a solution of the functional equation (7).
2. Orthogonal Hyers-Ulam Stability
Lemma 2 (see [32]. Let and be real vector spaces. If the mapping satisfies the functional equation (7) for all with , then is quartic.
Remark 3. Letbe a linear space andbe a function satisfies (7). Then, the following two assertions hold:(1) for all and integers.(2) for all if is continuous.Here, let us consider to be a linear space over and is a -Banach space with -norm .
Let be the modulus concavity of .
For our convenience, we use the abbreviation for a function : for all .
Theorem 4. Let a function which there exists such that with , and the contractively subadditive function and a constant fulfilling . Then, there exists a unique mapping which is quartic such that for all .
Proof. Setting by in (9), we have
for all . Replacing in (11) by and dividing by in (11) we attain
We have
for all and . Clearly, is complete, the Cauchy sequence converges for every . Next, we define a mapping by
for all . Letting and taking in (13), we obtain (10). Next, we want to prove that is quartic. From (9) and (14) that
for all with . Therefore, by Lemma 2, we conclude that is quartic. Next, to show that the function is unique.
Let us consider another quartic function which fulfils the inequality (10) we get
This shows that ; therefore, is unique mapping. This ends the proof of the theorem.
Corollary 5. If and be a positive real number and a function for which for all with . Then, there exists which is a unique quartic mapping that fulfils
The following theorem is obtained by replacing the expansive superadditive instead of the contractive subadditive in Theorem 4.
Theorem 6. Let a function in which exists a mapping such that for all with , and the expansively superadditive function and a constant fulfilling . Then, there exists a unique mapping which is quartic which fulfils for all .
With the upcoming theorems, we establish the stability of equation (7) by using an idea of Gavruta in [24].
Theorem 7. Let a mapping such that for all with , and
If is a mapping which fulfils with , then there exists a unique mapping which is quartic which satisfies for all .
Proof. From equation (11) in Theorem 4, we get Replacing through in inequality (25) and dividing by , we obtain Already, we know that is a -Banach space; we obtain for all with . From inequalities (22) and (27) that the sequence is Cauchy in for every . We know that if is complete, the sequence converges for every . Now, we can define a mapping by for all . Letting and taking in (27), we obtain the result (24). The remaining proof is the same as the proof of Theorem 4.
Theorem 8. Let be a mapping such that with , and for all . If fulfils with . Then, there exists a unique function which is quartic which fulfils
Proof. From equation (11), we get
Setting by in (33) and multiply by , we have
we have
Then, we conclude from (42) and (34) that the sequence is Cauchy in for every .
As is complete, the sequence converges for every . Next, we define a mapping by
for all . Letting and taking in (34), we obtain (32). The remaining proof is the same as the proof of Theorem 4.
Corollary 9. Let be the positive real numbers such that or . If a mapping satisfies the inequality for all with , then there exists a unique quartic mapping which satisfies for all .
Corollary 10. Let be the positive real numbers such that or . If a mapping satisfies the inequality for all with , then the mapping is quartic.
3. Counterexample
Here, we proved the nonstability of equation (7) in a special condition by a counterexample which is a modified idea of Gajda [9].
Example 11. Let a mappingdefined bywhere then the function fulfils for all , but there does not exist a quartic mapping such that whereandare constants.
Proof. Clearly, is bounded by on . If or 0, then the left side of (29) is less than , and thus, (29) is true.
Next, we assume that
then there exists an integer such that
So that and for every . For ,
Next,
It follows from (43) that
for all . Thus, satisfies (29) for all with .
Assume that there is a contrary mappingwhich is quartic which fulfils (42). We know that, for every , is bounded and continuous and is bounded on any open interval containing the origin which is continuous at the origin.
In the view of Remark 3, must be . Thus, we have
But we can select an integer with . If , then for any , and for , we obtain
which contradicts.
Data Availability
No data were used to support the findings of the study.
Conflicts of Interest
The authors declare that they have no competing interests.
Authors’ Contributions
All authors contributed equally to this work. And all the authors have read and approved the final version of the manuscript.
Acknowledgments
This research was funded by the Deanship of Scientific Research at Princess Nourah Bint Abdulrahman University through the Fast-Track Research Funding Program.