Orthogonal Stability and Nonstability of a Generalized Quartic Functional Equation in Quasi- β -Normed Spaces

In this work, we examine the generalized Hyers-Ulam orthogonal stability of the quartic functional equation in quasi- β -normed spaces. Moreover, we prove that this functional equation is not stable in a special condition by a counterexample.


Introduction
In this paper, ℝ and ℂ 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 ðG 1 , * Þ, ðG 2 , ⋆Þ be two groups and d : G 2 × G 2 → ½0,∞Þ be a metric. Given δ > 0, does there exist ε > 0 such that if a function g : G 1 → G 2 satisfies the inequality for all x, y ∈ G 1 , then there is a homomorphism h : 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.
In [16], Xu et al. obtained the general solution and investigated the Ulam stability problem for the quintic functional equation in quasi-β-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-β -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 f : ℝ → ℝ is a solution of (3) if and only if f ðxÞ = Aðx, x, x, xÞ, where the function A : ℝ 4 → ℝ 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 f ðxÞ = x 4 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 f : E → F be a mapping which satisfies the inequality for all x, y ∈ E with x⊥y, where ε and p are constants with ε, , and −1 ≠ jmj p−1 < 1. Then, the limit lim n→∞ m −2n f ðm n xÞ exists for all x ∈ E, and Q : E → F is the unique orthogonally Euler-Lagrange quadratic mapping such that for all x ∈ E.
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][26][27][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  In this paper, we examine the generalized Hyers-Ulam orthogonal stability of the quartic functional equation as where m is a positive integer with ℕ − f0, 1, 2, 3, 4g. It is easy to see that the function ϕðvÞ = av 4 is a solution of the functional equation (7).

Orthogonal Hyers-Ulam Stability
Lemma 2 (see [32]. Let E and F be real vector spaces. If the mapping ϕ : E → F satisfies the functional equation (7) for Remark 3. Let E be a linear space and ϕ : ℝ → E be a function satisfies (7). Then, the following two assertions hold: (1) ϕðr k/4 vÞ = r k ϕðvÞ for all v ∈ ℝ, r ∈ ℚ and k integers.
Here, let us consider E to be a linear space over K and F is a ðβ, pÞ-Banach space with p-norm ∥·∥ Y .
Let K be the modulus concavity of ∥·∥ F .
For our convenience, we use the abbreviation for a function ϕ : E → F: for with v i ⊥v j , i ≠ j = 1, 2, ⋯, m, and the contractively subadditive function ψ and a constant L fulfilling 2 ð1−4βÞ L < 1. Then, there exists a unique mapping Q 4 : E → F which is quartic such that for all v ∈ E.
Proof. Setting ðv 1 , v 2 ,⋯,v m Þ by ðv, 0,⋯,0Þ in (9), we have for all v ∈ E. Replacing v in (11) by 2 m v and dividing by 2 4ðm+1Þβ in (11) we attain We have for all v ∈ E and m ≥ i > 0. Clearly, F is complete, the Cauchy sequence fϕð2 m vÞ/2 4m g converges for every v ∈ E. Next, we define a mapping Q 4 : E → F by for all v ∈ E. Letting i = 0 and taking m → ∞ in (13), we obtain (10). Next, we want to prove that Q 4 is quartic. From (9) and (14) that Therefore, by Lemma 2, we conclude that Q 4 is quartic. Next, to show that the function Q 4 is unique. Let us consider another quartic function R 4 : E → F which fulfils the inequality (10) we get This shows that Q 4 = R 4 ; therefore, Q 4 is unique mapping. This ends the proof of the theorem.

Corollary 5.
If β = 1 and τ be a positive real number and a function ϕ : E → F for which for all v 1 , v 2 , ⋯, v m ∈ E with v i ⊥v j , i ≠ j = 1, 2, ⋯, m. Then, there exists Q 4 : E → Fwhich 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 ϕ : E → F in which exists a mapping ψ : E m → ½0,∞Þ such that for all v 1 , v 2 , ⋯, v m ∈ E with v i ⊥v j , i ≠ j = 1, 2, ⋯, m, and the expansively superadditive function ψ and a constant L fulfilling 2 ð4β−1Þ L < 1. Then, there exists a unique mapping Q 4 : E → F which is quartic which fulfils 3 Journal of Function Spaces for all v ∈ E.
With the upcoming theorems, we establish the stability of equation (7) by using an idea of Gavruta in [24]. Theorem 7. Let a mapping ψ : E m → ½0,∞Þ such that If ϕ : E → F is a mapping which fulfils with v i ⊥v j , i ≠ j = 1, 2, ⋯, m, then there exists a unique mapping Q 4 : E → F which is quartic which satisfies for all v ∈ E.
Proof. From equation (11) in Theorem 4, we get Replacing v through 2 m v in inequality (25) and dividing by 2 4βðm+1Þ , we obtain Already, we know that F is a ðβ, pÞ-Banach space; we obtain for all v ∈ E with m ≥ i > 0. From inequalities (22) and (27) that the sequence fϕð2 m vÞ/2 4m g is Cauchy in F for every v ∈ E. We know that if F is complete, the sequence fϕð2 m vÞ/ 2 4m g converges for every v ∈ E. Now, we can define a map- for all v ∈ E. Letting i = 0 and taking m → ∞ in (27), we obtain the result (24). The remaining proof is the same as the proof of Theorem 4.

Theorem 8.
Let ψ : E m → ½0,∞Þ be a mapping such that with v i ⊥v j , i ≠ j = 1, 2, ⋯, m, and with v i ⊥v j , i ≠ j = 1, 2, ⋯, m. Then, there exists a unique function Q 4 : E → F which is quartic which fulfils Proof. From equation (11), we get Setting v by v/2 m+1 in (33) and multiply by 2 4βm , we have we have Then, we conclude from (42) and (34) that the sequence f2 4m ϕðv/2 m Þg is Cauchy in F for every v ∈ E.
As F is complete, the sequence f2 4m ϕðv/2 m Þg converges for every v ∈ E. Next, we define a mapping Q 4 : E → F by for all v ∈ E. Letting i = 0 and taking m → ∞ in (34), we