Stability of Quartic Functional Equation in Modular Spaces via Hyers and Fixed-Point Methods

In this work, we introduce a new type of generalised quartic functional equation and obtain the general solution. We then investigate the stability results by using the Hyers method in modular space for quartic functional equations without using the Fatou property, without using the ∆b-condition and without using both the ∆b-condition and the Fatou property. Moreover, we investigate the stability results for this functional equation with the help of a fixed-point technique involving the idea of the Fatou property in modular spaces. Furthermore, a suitable counter example is also demonstrated to prove the non-stability of a singular case.


Introduction
Functional equations play a crucial role in the study of stability problems in several frameworks. Ulam was the first who questioned the stability of group homomorphisms and this opened the way to work on stability problems (see [1]). Using Banach spaces, Hyers [2] solved this stability problem by considering Cauchy's functional equation. Hyers' work was expanded upon by Aoki [3] by assuming an unbounded Cauchy difference. Rassias [4] presented work on additive mapping and these kinds of results are further presented by Gȃvruţa [5].
The concept of generalised Hyers-Ulam stability derives from historical contexts and this problem is found for different kinds of functional equations (FE). The functional equation is connected to a biadditive symmetric function (see [11,12]). Each equation is naturally referred to as a quadratic FE.Any solution of Equation (1) is a quadratic function. A function φ : E → E (E : real vector space) is said to be quadratic if there is a unique symmetric biadditive function T satisfying φ(u) = T(u, u) for all u (see [11,12]). The following functional equation was first presented by Jun and H. M. Kim [13]: which differs from Equation (1) in various ways. It is clear that the function φ(v) = cv 3 is a solution to Equation (2). As a consequence, it is natural to say that Equation (2) is a cubic FE and so every solution of Equation (2) is a cubic function. In [14], Lee et al. presented the quartic FE as: and found its solution and demonstrated the H-U-R stability. It is simple to demonstrate that φ(v) = cv 4 satisfies Equation (3) so this equality is called quartic FE, and its solution is called quartic mapping (QM). Except for direct approaches, the fixed-point method is the most often used method for establishing the stability of FEs (see [15][16][17]). In [18], the authors proposed a generalised quartic FE and investigated Hyers-Ulam stability in modular spaces using a fixed-point method as well as the Fatou property. Many research papers on different generalisations and the generalised H-U stability's implications for various functional equations have been recently published (see [19][20][21][22][23][24][25]).
To obtain our results, we define the quartic FE by We investigate certain stability results of the above quartic FE which will be based on Hyers and fixed-point methods involving the idea of the Fatou property and ∆ b -condition in the framework of modular spaces. Here, we consider the difference cases to obtain our results (i) with only the Fatou property, (ii) with only the ∆ b -condition, and (iii) without the Fatou property and the ∆ b -condition.
We begin by considering some fundamentally important concepts. Consider E to be a linear space over K (C or R). We call a functional ρ : If the inequality in (c) is replaced by (c') ρ(βu + γv) ≤ βρ(u) + γρ(v), then ρ is thus said to be convex modular.
Note that ρ is the following vector space which defined by a modular ρ: and E ρ is also known as a modular space. Let E ρ be a modular space and {u n } ∈ E ρ . One has (1) If ρ(u n − u) → 0 as n → ∞, {u n } is ρ-convergent to u ∈ E ρ and represented by The modular ρ is said to have the Fatou property if and only if ρ(u) ≤ lim n→∞ inf ρ(u n ) when the sequence {u n } in modular space E ρ is ρ-convergent to u.
In this case, k b is a ∆ b -constant related to ∆ b -condition.
Definition 2 ([34]). Suppose the sequence {v n } in a modular space V ρ . Then, we say that for any l, m ∈ A. The J orbit around a point u is Then, the quantity is known as the orbital diameter of J at u. If Υ ρ (J) < ∞ holds, J is said to has a bounded orbit at u (see [34]). Proposition 1 ([35]). In modular spaces, (1) If u n ρ − → u and is a constant vector, then u n + ρ − → u + , and It should be noted that if α is chosen from the equivalent scalar field with |α| > 1 in modular spaces, the convergence of a sequence {u n } to u does not mean that {αu n } converges to αu. Many mathematicians established additional criteria on modular spaces in order for the multiples of the convergent sequence {u n } in the modular spaces to naturally converge.
The modular ρ has the Fatou property if

Main Results
It follows, by replacing v with 3v in Equation (5), that Now, we obtain, by replacing v with 3v in Equation (6), that In general, for any n ∈ Z + (the set of positive integers), we have Thus, the function φ is even and has a solution of quartic FE. Therefore, φ is quartic. Finally, by replacing (v 1 , v 2 , v 3 , v 4 ) by (u, u, v, 0) in Equation (4), we obtain the Equation (3).

Stability of Quartic FE: Hyers Method
Consider a modular ρ as semi-convex. The Hyers-Ulam stability of Equation (4) in modular spaces is an important theorem in the absence of the Fatou condition.
For notational handiness, we define a mapping φ : E → F ρ (E: linear space; F ρ : ρ-complete semi-convex modular space) by Theorem 2. Let b ≥ 3 be an integer. Suppose F ρ satisfies the ∆ b -condition. If a mapping ψ : E 4 → [0, ∞) exists for which a mapping φ : then there is an unique QM Q : E → F ρ , defined by for all v ∈ E.
for all v ∈ E. So, even without utilising the Fatou property, the ∆ b -condition shows that the inequality holds for an integer l > 1 and for all v ∈ E. Taking l → ∞, we have the inequality (8). (7), we see that From the semi-convexity of ρ, it follows that and all non-negative integers l > 1. Taking the limit as l → ∞, we can see that Q is quartic. We suppose a QM Q : E → F ρ to demonstrate the uniqueness of Q. The function Q satisfies the inequality Taking l → ∞, we finally find that Q is unique, which completes the proof.
then there is an unique QM Q : Corollary 2. Let b ≥ 3 be an integer. Suppose that a normed space E with · and F ρ satisfies then there is an unique QM Q : An alternative stability theorem for Equation (4) in modular spaces will be proved without the ∆ b -condition, given below. Theorem 3. Let b ≥ 3 be an integer. Let F ρ satisfy the Fatou property. If a mapping φ : E → F ρ satisfies the inequality (7) and a mapping ψ : then there is an unique QM Q : Proof. By replacing v 1 = v and v 2 = v 3 = v 4 = 0 in Equation (7), we obtain Without using ∆ b -condition, the above inequality becomes for all v ∈ E and for all integers l > 1. This yields i.e., lim Then, based on the Fatou property, it follows that the inequality Now, we assert that Q satisfies the quartic FE. It should be noted that: and all l ∈ N. As a result of the semi-convexity of ρ, we can see that holds for all v 1 , v 2 , v 3 , v 4 ∈ E, and then taking l → ∞, we obtain ρ 1 361 Q(v 1 , v 2 , v 3 , v 4 ) = 0. As a result, Q must be quartic.
To demonstrate that the function Q is unique, we consider that Q : E → F ρ is an another quartic function which satisfies the inequality (10). As Q and Q are quartic, as evidenced by the previous equality, for all v ∈ E. Taking l → ∞, we conclude that Q = Q . Hence, Q is the only quartic mapping near φ that satisfies the inequality (10). Corollary 3. Let b ≥ 3 be an integer. Suppose that a normed space E with · and F ρ satisfy the Fatou property. For any λ > 0 and α ∈ (−∞, 4) are real numbers, if a mapping φ : then there is an unique QM Q : Corollary 4. Let b ≥ 3 be an integer. Suppose that a normed space E with · and F ρ satisfy the Fatou property. For any λ > 0 and 4α ∈ (−∞, 4) are given real numbers, if a mapping φ : E → F ρ such that then there is an unique QM Q : The upcoming proposition is a revised version of modular stability results of Theorem 3 in [36], which does not need the ∆ b -condition of ρ, which is given below.

Proposition 2.
Let F ρ satisfy the Fatou property. If a mapping φ : E → F ρ satisfy the inequality (7) and a mapping ψ : then there is an unique QM Q : Now, in modular spaces, we present an alternative stability Theorem 2 that does not utilise both the Fatou property and the ∆ b -condition.
Theorem 4. If a mapping φ : E → F ρ satisfy the inequality (7) and a mapping ψ : then there is an unique QM Q : for all v ∈ E.

Proof.
Letting v 1 = v and v 2 = v 3 = v 4 = 0 in inequality (7), one has and then the semi-convexity of ρ and ∑ l−1 for all v ∈ E and all l > 0. By the similar argument of the proof of Theorem 3, we have a ρ-Cauchy sequence { φ(b l v) b 4l } and the limit of function Q : E → F ρ defined as i.e., lim for all v ∈ E without employing the Fatou property and the ∆ b -condition. Furthermore, as in the proof of Theorem 2, one may show that Q satisfies Equation (4). Now, without invoking the Fatou property and the ∆ b -condition, we verify the inequality (11) of φ by Q. By utilizing the semi-convexity of ρ and ∑ l−1 for all integer l > 1 and for all v ∈ E. We arrive to the conclusion by using l → ∞.
Corollary 5. Let b ≥ 3 be an integer. Suppose that a normed space E with · . Any λ > 0 and α ∈ (−∞, 4) are real numbers if a mapping φ : then there is an unique QM Q : where v = 0 if α < 0. Corollary 6. Let b ≥ 3 be an integer. Suppose that a normed space E with · . Any λ > 0 and 4α ∈ (−∞, 4) are real numbers, if a mapping φ : E → F ρ such that then there is an unique QM Q : for all v, v 1 , v 2 , v 3 , v 4 ∈ E and for some L ∈ (0, 4). If a mapping φ : E → F ρ satisfies Equation (7), then there is an unique QM Q :

Stability of Quartic FE: Fixed-Point Method
Theorem 5. Let b ≥ 3 be an integer and a mapping ψ : for all v i ∈ E; i = 1, 2, 3, 4, with 0 < L < 1. If an even mapping φ : for all v i ∈ E; i = 1, 2, 3, 4, then there is an unique QM Q 4 : for all v ∈ E.
Proof. We define the set and ρ is a function on Υ as Now, we need to demonstrate that the function ρ is a semi-convex modular on Υ. Clearly, ρ holds conditions (a) and (b). So, it is enough to verify that ρ is semi-convex modular. Given ε > 0, ∃ λ 1 > 0 such that Since ε > 0 was arbitrary, from above, we find that ρ is semi-convex modular on Υ. Next, we want to verify that Υ ρ is ρ-complete.
for all n, m ≥ n 0 . Thus, we have for all v ∈ E, and n, m ≥ n 0 . Therefore, a ρ- Now, let us define a mapping p : E → F ρ by We arrive by taking into account Equation (15) that since ρ holds the Fatou property. Thus, {p n } ρ-converges and so Υ ρ is ρ-complete.
We now want to prove that ρ holds Fatou property.
For all ε > 0, consider a constant λ n (n ∈ N) which is real such that for all v ∈ E. We know that ρ holds the Fatou property, so we obtain Thus, we obtain since ε > 0 was arbitrary. Hence, ρ also holds the Fatou property. Let us define a mapping χ : Υ ρ → Υ ρ by Suppose p, q ∈ Υ ρ and λ ∈ [0, 1] with ρ(p − q) < λ (λ is an arbitrary constant). Employing the definition of ρ, we write Using Equations (12) and (16), we have which means that χ is a ρ-contraction. Now, we will show that χ has a φ bounded orbit. In Replacing v with bv in inequality (17), we obtain (bv, 0, 0, 0), ∀v ∈ E.
By using Equations (17) and (18), we obtain Clearly, by induction, It follows from Equation (19) that for n, m ∈ N and all v ∈ E. We conclude that by defining ρ, This means that an orbit of χ at φ is bounded. The sequence of {χ n φ} ρ-converges into Q 4 ∈ Υ ρ , according to Theorem 1.5 in [34]. Now, we have the ρ-contractivity of χ, where Taking the limit n → ∞ and apply ρ Fatou property, we get Thus, we have Letting l → ∞, we obtain Theorem 1, Q 4 is quartic. So, the inequality (19) gives (14). Let Q 4 : E → F ρ be an another QM that meets inequality (14) to prove the uniqueness of Q 4 . Thus, Q 4 is a fixed point of χ, so This yields ρ Q 4 − Q 4 = 0. Consequently, Q 4 = Q 4 . which proves the uniqueness of function Q 4 .

Corollary 7. Let b ≥ 3 be an integer and a mapping
with 0 < L < 1. If φ : E → F is an even mapping with φ(0) = 0 such that for all v i ∈ E; i = 1, 2, 3, 4, so there is an unique QM Q 4 : E → F having i=1 v i p and taking L = b p−4 in the last corollary, then we arrive at the stability result for the sum of norms as where p (p < 4) and α are constants.
Theorem 6. Let b ≥ 3 be an integer. Suppose a mapping ψ : with 0 < L < 1. If a mapping φ : E → F ρ is even with φ(0) = 0 such that the inequality (13) holds, then there is an unique QM Q 4 : E → F ρ having Proof. Consider the set Let ρ be a function on Υ, defined by We have the same evidence as Theorem 5: (a) The function ρ is a convex modular on Υ.
Let us define a mapping χ : Υ ρ→Υ ρ for all v ∈ E and for p ∈ Υ ρ by Let p, q ∈ Υ ρ and λ ∈ [0, 1] with ρ(p − q) < λ (λ is an arbitrary constant). Consequently, for all v ∈ E. We obtain by assumption and the above inequality that which proves that χ is a ρ-contraction.
We will now show that χ has a bounded orbit at φ. Setting (v 1 , v 2 , v 3 , v 4 ) by (v, 0, 0, 0) in Equation (13), we obtain It follows by replacing v with v b in Equation (21) that Again, replacing v by v b in Equation (22), we obtain Considering Equations (21)- (23), for all v ∈ E, we obtain We can easily determine by induction that (24) gives for all v ∈ E, and all n, m ∈ N. We can conclude that by defining ρ, This means that the χ orbit is limited to φ. The sequence {χ n φ} ρ-converges to Q 4 ∈ Υ ρ from Theorem 1.5 in [31].
We have from the ρ-contractivity of χ that Letting n → ∞ together with Fatou property, we have Therefore, the function Q 4 is a fixed point of χ. (13), we obtain Passing to the limit l → ∞, we obtain Therefore, Q 4 is quartic from Theorem 7. Using the inequality (24), we obtain the inequality (20).
It is only left to show the uniqueness of Q 4 . For this, consider another QM Q 4 : E → F ρ which satisfies the inequality (14). Then, Q 4 is a fixed point of χ. So, we write Corollary 8. Let b ≥ 3 be an integer and also let ψ : E 4 → [0, +∞) be a mapping such that , for all v i ∈ E; i = 1, 2, 3, 4, with 0 < L < 1. If a mapping φ : E → F is even with φ(0) = 0 satisfies the inequality (7), then there is an unique QM Q 4 : E → F satisfying i=1 v i p and taking L = b 4−p in Corollary 8, we fairy have the stability results for the sum of norms as follows: where p (p > 4) and α are constants.

Illustrative Examples
Here, in this section, we investigate a suitable example to verify that the stability of quartic FE (4) fails for a singular case. Following by the example of Gajda (see [37]), we examine the following counter-example which proves the instability in a particular conditions b = 3 and α = 4 in Corollaries 3 and 5 of Equation (4).

Remark 4.
If a mapping φ : R → E satisfies the functional Equation (4), then (C1) φ(m c/4 v) = m c φ(v), for all v ∈ R, m ∈ Q and c ∈ Z, where Suppose that the function φ defined in Equation (25) which satisfies for all v 1 , v 2 , v 3 , v 4 ∈ R. We here obtain that there does not exist a QM Q : R → R satisfying for all v ∈ R,, where λ and δ are constants.
Suppose that the function φ defined in Equation (29)

Conclusions and Discussion
Many mathematicians obtain the stability results of various kinds of additive, quadratic, and cubic functional equations in various spaces. In our investigations, we first defined a new kind of quartic FN in the first section of this paper and obtained the general solution of our newly defined quartic FN. Additionally, we explored the stability results of this quartic FN in the setting of modular space using Hyers' technique by taking into our account three cases, that are: without utilising the Fatou property, without using the ∆ b -condition, and without using the ∆ b -condition and Fatou property. Moreover, by taking into our account the Fatou property and fixed-point approach, we established some stability results of our quartic FN in the framework of modular spaces. In addition, an appropriate counter-example is provided to demonstrate the non-stability of the singular case.
It is worth mentioning that one can further determine the stability results of this quartic FN in various frameworks, namely, quasi-β-normed spaces, fuzzy normed space, non-Archimedean spaces, random normed spaces, probabilistic normed spaces, intuitionistic fuzzy normed space and so on. The findings and techniques used in this study might be valuable to other researchers who want to conduct further work in this area.