EUROPEAN JOURNAL OF PURE AND APPLIED MATHEMATICS 2025, Vol. 18, Issue 4, Article Number 6699 ISSN 1307-5543 – ejpam.com Published by New York Business Global Hyers-Ulam Stability of Generalized Quartic Mapping in Non-Archimedean (n, β)-Normed Spaces Senthil Gowri1, Siriluk Donganont2,∗, S. Karthick3, Radhakrishnan Balaanandhan4, Kandhasamy Tamilvanan5 1 Department of Mathematics, Saveetha School of Engineering, Saveetha Institute of Medical and Technical Sciences, Saveetha University, Tandalam, Chennai 602105, Tamil Nadu, India 2 School of Science, University of Phayao, Phayao 56000, Thailand 3 Department of Mathematics, College of Engineering and Technology, SRM Institute of Science and Technology, Kattankulathur 603203, Tamil Nadu, India 4 Department of Mathematics, Sri Sankara Arts and Science College (Autonomous), Enathur 631561, Kanchipuram, Tamil Nadu, India 5 Department of Mathematics, Faculty of Science and Humanities, R.M.K. Engineering College, Kavaraipettai, Tiruvallur 601206, Tamil Nadu, India Abstract. In this article, we introduce a novel structure termed as the non-Archimedean (n, β)- normed space, formulated over a non-Archimedean field. This generalization extends the concept of classical normed spaces by integrating a parameterized framework involving n-tuples and an exponent β. We delve into the fundamental characteristics of these spaces, demonstrating how they connect to standard non-Archimedean n-normed and n-quasi-normed structures. Moreover, we provide examples that support the theory and help show some fixed point results, making these spaces easier to use in real problems. 2020 Mathematics Subject Classifications: 39B52, 39B72, 46S40 Key Words and Phrases: Quartic functional equation, Hyers-Ulam stability, non-Archimedean (n, β)-normed spaces, generalized control function 1. Introduction Due to its extensive applicability in many other domains and its deep consequences in mathematical analysis, the study of functional equations and their stability qualities has attracted a lot of attention. Ulam [1] raised a basic query about the stability of group homomorphisms in 1940, which is when this topic first emerged. The theory of functional ∗Corresponding author. DOI: https://doi.org/10.29020/nybg.ejpam.v18i4.6699 Email addresses: gowrisenthil.sse@saveetha.com (S. Gowri), siriluk.pa@up.ac.th (S. Donganont), karthickmaths007@gmail.com (S. Karthick), balaanandhanmaths@gmail.com (R. Balaanandhan), tamiltamilk7@gmail.com (K. Tamilvanan) https://www.ejpam.com 1 Copyright: © 2025 The Author(s). (CC BY-NC 4.0) S. Gowri et al. / Eur. J. Pure Appl. Math, 18 (4) (2025), 6699 2 of 14 equation stability was developed as a result of this investigation. Hyers [2] in 1941 provided a crucial answer by establishing the first stability result for linear functional equations, which is now known as the Hyers-Ulam stability. Later, Rassias developed Hyers-Ulam- Rassias stability theory by adding a perturbation term that was dependent on the norm, which was a substantial generalization. Notable among the many functional equation classes are quartic functional equations, which generalize the behavior of fourth-degree polynomials. Approximation theory, infor- mation theory, theoretical physics, and differential equations all naturally produce these types of equations. The quartic functional equation τ(t1 + 2t2) + τ(t1 − 2t2) = 4[τ(t1 + t2) + τ(t1 − t2)] + 6τ(t1)− 24τ(t2) serves as a classical example whose general solution often involves quartic mappings such as τ(t) = at4. A lot of research has been done on how stable these kinds of equations are in both standard and extended normed settings [3–9]. A notable development in this field pertains to the application of non-Archimedean normed spaces. These spaces satisfy the strong triangle inequality: ∥t1 + t2∥ ≤ max{∥t1∥, ∥t2∥}, which endows them with distinct topological and algebraic properties (see, [10, 11]). Such spaces frequently occur in p-adic analysis, number theory, and information theory. Build- ing on this framework, the notion of (n, β)-normed spaces was introduced to capture a broader class of normed structures (Ref. [12–14]).These spaces provide a more compre- hensive analytical investigation by generalizing n-normed spaces (β = 1) and β-normed spaces (n = 1) (Ref. [15–17]). Recent research has expanded classical stability results to non-Archimedean (n, β)- normed spaces, analyzing quadratic, cubic, and quartic equations through novel fixed point methods and contractive conditions. These advancements offer enhanced understanding of the structure of functional equations in the context of perturbations within ultrametric environments ([18–27]). This paper aims to investigate the Hyers-Ulam stability of a generalized quartic func- tional equation within non-Archimedean (n, β)-normed spaces. By employing direct ana- lytical methods alongside fixed point techniques, we derive new stability results and outline conditions for the existence and uniqueness of quartic solutions. The findings present a notable extension of current research and pave the way for future investigations in abstract analysis and p-adic functional theory. The purpose of this study is to explore the Hyers-Ulam stability of the generalized quartic functional equation ϕ ( r∑ i=1 ti ) = ∑ 1≤i 3, then ∥2n∥ = 1 for all integer n. S. Gowri et al. / Eur. J. Pure Appl. Math, 18 (4) (2025), 6699 4 of 14 Remark 1. [28] A non-Archimedean (n, β) containing a sequence {tm} if and only if the negative absolute value of tm+1 converges to zero, then normed space E is a Cauchy sequence. Lemma 1. [28] Consider {tp} is a convergent sequence in a linear (n, β)-normed space E, lim p→∞ ∥tp, κ1, κ2, · · · , κn−1∥β = ∥∥∥∥ lim p→∞ tp, κ1, κ2, · · · , κn−1 ∥∥∥∥ β for all κ1, κ2, · · · , κn−1 ∈ E. Lemma 2. [28] Let (E, ∥·, · · · , ·∥β) be a linear (n, β)-normed space, 0 < β ≤ 1 and n ≥ 2. If t1 ∈ E and ∥t1, κ1, · · · , κn−1∥β = 0 for all κ1, · · · , κn−1 ∈ E, then t1 = 0. Theorem 1. [29] If a mapping ϕ : E → F satisfies the functional equation (1) for all t1, t2, · · · , tr ∈ E, then the function ϕ : E → F is quartic. 2.1. Structural Examples and Fundamental Properties of Non-Archimedean (n, β)-Normed Spaces In this subsection, we provide illustrative examples to demonstrate the structure and behavior of non-Archimedean (n, β)-normed spaces. These examples underline how such spaces extend the classical notions of normed vector spaces under ultrametric constraints. Example 2. [28, 30, 31] Let X = Kn be the n-dimensional vector space over a non- Archimedean field K. Define the mapping ∥·, . . . , ·∥β : Xn → R+ as ∥x1, x2, . . . , xn∥β := |det(x1, x2, . . . , xn)|β , where x1, x2, . . . , xn are vectors in X and the determinant is computed by treating them as rows of an n× n matrix. It can be verified that this function satisfies all the conditions of a non-Archimedean (n, β)-norm. Example 3. [28, 30, 31] Consider the vector space X = c0(K), the space of sequences converging to zero over a non-Archimedean field K. Define the (n, β)-norm by ∥x1, . . . , xn∥β := sup m∈N ∣∣∣det(x(m) 1 , x (m) 2 , . . . , x (m) n )∣∣∣β , where x (m) i denotes the m-th component of the sequence xi. This function defines a valid (n, β)-norm due to the ultrametric inequality and properties of determinants over K. We now list some fundamental properties that hold in any non-Archimedean (n, β)- normed space (X, ∥·, . . . , ·∥β). Proposition 1. [28] Let x1, . . . , xn, y ∈ X. Suppose that X is non-Archimedean (n, β)- normed space. Then: S. Gowri et al. / Eur. J. Pure Appl. Math, 18 (4) (2025), 6699 5 of 14 (i) If x1 is linearly dependent on {x2, . . . , xn}, then ∥x1, . . . , xn∥β = 0. (ii) If all vectors x1, x2, . . . , xn are linearly independent, then ∥x1, x2, . . . , xn∥β > 0. (iii) The (n, β)-norm is symmetric in all arguments. (iv) For any scalar ζ ∈ K, ∥ζx1, x2, x3, . . . , xn−1, xn∥β = |ζ|β · ∥x1, x2, x3, . . . , xn−1, xn∥β. (v) The strong triangle inequality holds: ∥x1 + y, x2, x3, . . . , xn−1, xn∥β ≤ max {∥x1, x2, x3, . . . , xn−1, xn∥β, ∥y, x2, x3, . . . , xn−1, xn∥β} . 3. Stability of the Generalized Quartic Functional Equation Consider E as a vector space and (F, ∥·, . . . , ·∥β) as an element of it. Rest assured that the space (n, β) is non-Archimedean, with n ≥ 2 and 0 < β, β1 ≤ 1. We consider the generalized quartic functional equation (1) defined via the following difference operator: ∆ϕ(t1, . . . , tr) = −ϕ ( r∑ i=1 ti ) + ∑ 1≤i β, and let ϖ : Fn−1 → [0,∞) be a control function. Assume that ϕ : E → F is a function such that ∥∆ϕ (t1, t2, . . . , tr) , ν1, . . . , νn−1∥β ≤ µ r∑ j=1 ∥tj∥sβ1 ϖ(ν1, . . . , νn−1) (2) for all t1, . . . , tr ∈ E and ν1, . . . , νn−1 ∈ F . Then there exists a unique quartic mapping Q4 : E → F satisfying ∥ϕ(t)−Q4(t), ν1, . . . , νn−1∥β ≤ µ|2−4β| ∥t∥sβ1 ϖ(ν1, . . . , νn−1) (3) for all t ∈ E and all ν1, . . . , νn−1 ∈ F . S. Gowri et al. / Eur. J. Pure Appl. Math, 18 (4) (2025), 6699 6 of 14 Proof. Replacing (t1, t2, . . . , tr) by (t, 0, . . . , 0) in (2), we obtain∥∥ϕ(2t)− 24ϕ(t), ν1, . . . , νn−1 ∥∥ β ≤ µ ∥t∥sβ1 ϖ(ν1, . . . , νn−1). (4) Dividing both sides of (4) by |24β| gives∥∥∥∥ϕ(2t)24 − ϕ(t), ν1, . . . , νn−1 ∥∥∥∥ β ≤ |2−4β|µ ∥t∥sβ1 ϖ(ν1, . . . , νn−1). (5) Replacing t by 2pt in (5) yields∥∥∥∥ϕ(2p+1t) 24(p+1) − ϕ(2pt) 24p , ν1, . . . , νn−1 ∥∥∥∥ β ≤ |2−4(p+1)β|µ∥2pt∥sβ1 ϖ(ν1, . . . , νn−1) ≤ |2−4β| ∣∣∣2sβ1−4β ∣∣∣p µ∥t∥sβ1 ϖ(ν1, . . . , νn−1). (6) Since sβ1 > β and |2| ≠ 1, the R.H.S. of (6) tends to zero as p→ ∞. Hence, the sequence{ ϕ(2pt) 24p } is Cauchy in F , which is complete. Therefore, we define Q4(t) := lim p→∞ ϕ(2pt) 24p for all t ∈ E. (7) To prove that Q4 is quartic, apply (2) and Lemma 1: ∥∆Q4(t1, . . . , tr), ν1, . . . , νn−1∥β = lim p→∞ ∣∣∣2−4pβ ∣∣∣ ∥∆ϕ(2pt1, . . . , 2ptr), ν1, . . . , νn−1∥β ≤ lim p→∞ µ ∣∣∣2sβ1−4β ∣∣∣p r∑ j=1 ∥tj∥sβ1 ϖ(ν1, . . . , νn−1) = 0. Thus, by Lemma 2, Q4 satisfies ∆Q4 = 0, and so Q4 is quartic. To estimate the difference between ϕ and Q4, we observe from (5) and similar recursive steps that ∥∥∥∥ϕ(t)− ϕ(2pt) 24p , ν1, . . . , νn−1 ∥∥∥∥ β ≤ |2−4β|µ∥t∥sβ1 ϖ(ν1, . . . , νn−1). (8) Letting p→ ∞ in (8) and applying the definition of Q4, we obtain the inequality (3). Finally, we show uniqueness. Consider another quartic mapping Q′ 4 satisfying (3). Then∥∥Q4(t)−Q′ 4(t), ν1, . . . , νn−1 ∥∥ β = ∣∣∣2−4pβ ∣∣∣ ∥∥Q4(2 pt)−Q′ 4(2 pt), ν1, . . . , νn−1 ∥∥ β ≤ ∣∣∣2−4pβ ∣∣∣ ·max { ∥Q4(2 pt)− ϕ(2pt), ν1, . . . , νn−1∥β ,∥∥ϕ(2pt)−Q′ 4(2 pt), ν1, . . . , νn−1 ∥∥ β } ≤ µ ∣∣∣2−4β ∣∣∣ ∣∣∣2sβ1−4β ∣∣∣p ∥t∥sβ1 ϖ(ν1, . . . , νn−1). Taking the limit as p → ∞, we conclude Q4(t) = Q′ 4(t) for all t ∈ E. Hence, Q4 is the only one quartic function satisfying (3). S. Gowri et al. / Eur. J. Pure Appl. Math, 18 (4) (2025), 6699 7 of 14 Theorem 3. Let µ ∈ [0,∞) and s ∈ (0,∞) with sβ1 < β. Let ϖ : Fn−1 → [0,∞) be a control function. Assume that the mapping ϕ : E → F satisfies ∥∆ϕ (t1, . . . , tr) , ν1, . . . , νn−1∥β ≤ µ r∑ j=1 ∥tj∥sβ1 ϖ(ν1, . . . , νn−1) (9) for all t1, . . . , tr ∈ E and all ν1, . . . , νn−1 ∈ F . Then there exists a unique quartic function Q4 : E → F satisfying ∥ϕ(t)−Q4(t), ν1, . . . , νn−1∥β ≤ µ|2−sβ1 | ∥t∥sβ1 ϖ(ν1, . . . , νn−1) (10) for all t ∈ E and all ν1, . . . , νn−1 ∈ F . Proof. Replacing (t1, . . . , tr) by (t, 0, . . . , 0) in (9), we have∥∥ϕ(2t)− 24ϕ(t), ν1, . . . , νn−1 ∥∥ β ≤ µ ∥t∥sβ1 ϖ(ν1, . . . , νn−1). (11) Replacing t by t/2 in (11), we obtain∥∥∥ϕ(t)− 24ϕ ( t 2 ) , ν1, . . . , νn−1 ∥∥∥ β ≤ µ|2−sβ1 | ∥t∥sβ1 ϖ(ν1, . . . , νn−1). (12) Switching t by t/2p in (12), we have∥∥∥24pϕ( t 2p ) − 24(p+1)ϕ ( t 2p+1 ) , ν1, . . . , νn−1 ∥∥∥ β ≤ µ|2−sβ1 | ∣∣∣24β−sβ1 ∣∣∣p ∥t∥sβ1 ϖ(ν1, . . . , νn−1). (13) Since sβ1 < β and |2| ≠ 1, the R.H.S. of (13) tends to zero as p→ ∞. Thus, the sequence {24pϕ(t/2p)} is Cauchy in F , which is complete. Hence, define Q4(t) := lim p→∞ 24pϕ ( t 2p ) (14) for all t ∈ E. We now show that Q4 is quartic. From (9) and Lemma 1, we get ∥∆Q4(t1, . . . , tr), ν1, . . . , νn−1∥β = lim p→∞ ∣∣∣24pβ∣∣∣ ∥∥∥∆ϕ( t1 2p , . . . , tr 2p ) , ν1, . . . , νn−1 ∥∥∥ β ≤ lim p→∞ µ ∣∣∣24β−sβ1 ∣∣∣p r∑ j=1 ∥tj∥sβ1 ϖ(ν1, . . . , νn−1) = 0. Hence, by Lemma 2, the function Q4 is quartic. To prove inequality (10), note from (12) and similar reasoning (induction or recursion) that ∥∥∥ϕ(t)− 24pϕ ( t 2p ) , ν1, . . . , νn−1 ∥∥∥ β ≤ µ|2−sβ1 | ∥t∥sβ1 ϖ(ν1, . . . , νn−1). (15) S. Gowri et al. / Eur. J. Pure Appl. Math, 18 (4) (2025), 6699 8 of 14 Taking the limit as p→ ∞ in (15) and using the definition of Q4 in (14), we obtain (10). To prove uniqueness, assume that another quartic function Q′ 4 : E → F satisfying (10). Then∥∥Q4(t)−Q′ 4(t), ν1, . . . , νn−1 ∥∥ β = ∣∣∣24pβ∣∣∣ ∥∥∥Q4 ( t 2p ) −Q′ 4 ( t 2p ) , ν1, . . . , νn−1 ∥∥∥ β ≤ ∣∣∣24pβ∣∣∣max {∥∥∥Q4 ( t 2p ) − ϕ ( t 2p )∥∥∥ β ,∥∥∥ϕ( t 2p ) −Q′ 4 ( t 2p )∥∥∥ β } ≤ µ|2−sβ1 | ∣∣∣24β−sβ1 ∣∣∣p ∥t∥sβ1 ϖ(ν1, . . . , νn−1). As p→ ∞, the R.H.S. tends to zero. Thus,∥∥Q4(t)−Q′ 4(t), ν1, . . . , νn−1 ∥∥ β = 0, which implies, by Lemma 2, that Q4 = Q′ 4. Hence, Q4 is unique. Theorem 4. Let a function ψ : Er → [0,∞) such that lim p→∞ ∣∣∣∣ 1 24pβ ∣∣∣∣ψ (2pt1, . . . , 2 ptr) = 0 (16) for all t1, . . . , tr ∈ E, and let ϖ : Fn−1 → [0,∞) be a control function. Assume that the mapping ϕ : E → F satisfies ∥∆ϕ(t1, . . . , tr), ν1, . . . , νn−1∥β ≤ ψ(t1, . . . , tr)ϖ(ν1, . . . , νn−1) (17) for all t1, . . . , tr ∈ E and ν1, . . . , νn−1 ∈ F . Then there exists a unique quartic mapping Q4 : E → F satisfying ∥ϕ(t)−Q4(t), ν1, . . . , νn−1∥β ≤ ψ̃(t)ϖ(ν1, . . . , νn−1) (18) where ψ̃(t) := lim p→∞ max {∣∣∣2−4iβ ∣∣∣ψ (2i−1t, 0, 0, . . . , 0 ) : 1 ≤ i ≤ p } (19) for all t ∈ E. Moreover, if lim t→∞ lim p→∞ max {∣∣∣2−4iβ ∣∣∣ψ (2i−1t, 0, 0, . . . , 0 ) : 1 + t ≤ i ≤ p+ t } = 0 (20) for every t ∈ E, then the function Q4 is unique. Proof. Replacing (t1, . . . , tr) by (t, 0, . . . , 0) in (17) and dividing both sides by |24β| gives ∥∥∥∥ϕ(2t)24 − ϕ(t), ν1, . . . , νn−1 ∥∥∥∥ β ≤ |2−4β|ψ(t, 0, . . . , 0)ϖ(ν1, . . . , νn−1). (21) S. Gowri et al. / Eur. J. Pure Appl. Math, 18 (4) (2025), 6699 9 of 14 Replacing t with 2it and dividing by |24iβ|, we get∥∥∥∥ϕ(2i+1t) 24(i+1) − ϕ(2it) 24i , ν1, . . . , νn−1 ∥∥∥∥ β ≤ |2−4(i+1)β|ψ ( 2it, 0, . . . , 0 ) ϖ(ν1, . . . , νn−1). (22) By (16), the right-hand side tends to zero as i→ ∞, so the sequence { ϕ(2mt) 24m } is Cauchy. As F is complete, define Q4(t) := lim m→∞ ϕ(2mt) 24m . We next show Q4 is quartic. From (17), Lemma 1, and the definition of Q4, we have: ∥∆Q4(t1, . . . , tr), ν1, . . . , νn−1∥β = lim p→∞ ∥∥∥∥ 1 24p ∆ϕ (2pt1, . . . , 2 ptr) , ν1, . . . , νn−1 ∥∥∥∥ β ≤ lim p→∞ |2−4pβ|ψ (2pt1, . . . , 2 ptr) ϖ(ν1, . . . , νn−1) = 0. Hence, by Lemma 2, Q4 is quartic. From (21), we obtain:∥∥∥∥ϕ(t)− ϕ(24t) 28 , ν1, . . . , νn−1 ∥∥∥∥ β ≤ max { |2−4β|ψ(t, 0, . . . , 0), |2−8β|ψ(2t, 0, . . . , 0) } ϖ(ν1, . . . , νn−1). Inductively, for all p ∈ N, we get:∥∥∥∥ϕ(t)− ϕ(2pt) 24p , ν1, . . . , νn−1 ∥∥∥∥ β ≤ max {∣∣∣2−4tβ ∣∣∣ ψ (2t−1t, 0, . . . , 0 ) : 1 ≤ t ≤ p } ϖ(ν1, . . . , νn−1). (23) Letting p→ ∞ in (23), we obtain (18) by the definition of ψ̃(t) in (19). To prove uniqueness, suppose another quartic mapping Q′ 4 also satisfies (18). Then∥∥Q4(t)−Q′ 4(t), ν1, . . . , νn−1 ∥∥ β = ∣∣∣2−4tβ ∣∣∣ ∥∥Q4(2 tt)−Q′ 4(2 tt), ν1, . . . , νn−1 ∥∥ β ≤ ∣∣∣2−4tβ ∣∣∣max {∥∥Q4(2 tt)− ϕ(2tt) ∥∥ β , ∥∥ϕ(2tt)−Q′ 4(2 tt) ∥∥ β } ≤ ∣∣∣2−4tβ ∣∣∣ ψ̃(2tt)ϖ(ν1, . . . , νn−1). By assumption (20), the last term tends to zero as t→ ∞. Therefore,∥∥Q4(t)−Q′ 4(t), ν1, . . . , νn−1 ∥∥ β = 0, and by Lemma 2, we conclude Q4 = Q′ 4. Thus, Q4 is unique. S. Gowri et al. / Eur. J. Pure Appl. Math, 18 (4) (2025), 6699 10 of 14 Theorem 5. Let ψ : Er → [0,∞) be a control function satisfying lim p→∞ ∣∣∣24pβ∣∣∣ψ ( t1 2p , t2 2p , . . . , tr 2p ) = 0 (24) for all t1, . . . , tr ∈ E, and let ϖ : Fn−1 → [0,∞) be a control mapping. Assume that a function ϕ : E → F satisfies ∥∆ϕ(t1, . . . , tr), ν1, . . . , νn−1∥β ≤ ψ(t1, . . . , tr)ϖ(ν1, . . . , νn−1) (25) for all t1, . . . , tr ∈ E and ν1, . . . , νn−1 ∈ F . Then there exists a unique quartic mapping Q4 : E → F satisfying ∥ϕ(t)−Q4(t), ν1, . . . , νn−1∥β ≤ ψ̃(t)ϖ(ν1, . . . , νn−1), (26) where ψ̃(t) := lim p→∞ max {∣∣∣24(i−1)β ∣∣∣ψ (2−it, 0, 0, . . . , 0 ) : 1 ≤ i ≤ p } . (27) Moreover, if lim t→∞ lim p→∞ max {∣∣∣24(i−1)β ∣∣∣ψ (2−it, 0, 0, . . . , 0 ) : 1 + t ≤ i ≤ p+ t } = 0 (28) for all t ∈ E, then the quartic mapping Q4 is unique. Proof. Setting (t1, t2, . . . , tr) by (t, 0, . . . , 0) in (25), we obtain∥∥ϕ(2t)− 24ϕ(t), ν1, ν2, . . . , νn−1 ∥∥ β ≤ ψ(t, 0, . . . , 0)ϖ(ν1, ν2, . . . , νn−1). (29) Replacing t by t 2 and multiplying by |24β| repeatedly, we define a sequence:{ 24pϕ ( t 2p )} . (30) Using (24) and similar arguments as in Theorem 4, we conclude this sequence is Cauchy in F and hence convergent, due to completeness. Define Q4(t) := lim p→∞ 24pϕ ( t 2p ) , for all t ∈ E. Following the structure of (30), we can show ∥Q4(t)− ϕ(t), ν1, ν2, . . . , νn−1∥β ≤ ψ̃(t)ϖ(ν1, ν2, . . . , νn−1), establishing (26). To prove that Q4 is quartic, observe that: ∥∆Q4(t1, . . . , tr), ν1, . . . , νn−1∥β = lim p→∞ ∣∣∣24pβ∣∣∣ ∥∥∥∆ϕ( t1 2p , . . . , tr 2p ) , ν1, . . . , νn−1 ∥∥∥ β , S. Gowri et al. / Eur. J. Pure Appl. Math, 18 (4) (2025), 6699 11 of 14 which tends to zero by (24), hence ∆Q4 = 0 and Q4 is quartic. For uniqueness, suppose another quartic mapping Q′ 4 satisfies (26). Then∥∥Q4(t)−Q′ 4(t), ν1, . . . , νn−1 ∥∥ β = ∣∣∣24tβ∣∣∣ ∥∥∥Q4 ( t 2t ) −Q′ 4 ( t 2t ) , ν1, . . . , νn−1 ∥∥∥ β ≤ ∣∣∣24tβ∣∣∣ ψ̃ ( t 2t ) ϖ(ν1, . . . , νn−1) → 0 as t → ∞ by assumption (28). Hence, ∥Q4(t)−Q′ 4(t), ν1, . . . , νn−1∥β = 0 for all t ∈ E, implying Q4 = Q′ 4 by Lemma 2. 4. Consequences and Illustrative Example in Qp Corollaries and Examples Corollary 4.1 (Classical Non-Archimedean Case) Let E and F be non-Archimedean normed spaces, i.e., (n, β)-normed spaces with n = 1, β = 1. Assume that a mapping ϕ : E → F fulfills ∥∆ϕ(t1, t2, . . . , tr)∥ ≤ µ r∑ j=1 ∥tj∥s for some constants µ ≥ 0, s > 1, and all t1, t2, . . . , tr ∈ E. Then there is a unique quartic mapping Q4 : E → F fulfilling ∥ϕ(t)−Q4(t)∥ ≤ µ|2−4|∥t∥s, for every t ∈ E. Corollary 4.2 (Stability in β-Normed Ultrametric Spaces) Let E be a β-normed space and F a complete non-Archimedean (n, β)-normed space with 0 < β < 1. If a function ϕ : E → F satisfies ∥∆ϕ(t1, t2, . . . , tr), ν1, . . . , νn−1∥β ≤ µ r∑ j=1 ∥tj∥sβ for all tj ∈ E, νk ∈ F , then there is a unique quartic function Q4 : E → F fulfilling ∥ϕ(t)−Q4(t), ν1, . . . , νn−1∥β ≤ µ|2−4β|∥t∥sβ. Example 4.3 (Mapping on a Qp Space) Let p > 3 be a prime number and E = Qp, the field of p-adic numbers. Define the function ϕ : Qp → Qp by ϕ(t) = t4 + ϵ(t), where |ϵ(t)|p ≤ δ|t|sp for some δ > 0 and s > 4. Then ϕ satisfies the condition of Theorem 2 for suitable µ, and there is only one quartic function Q4(t) = t4 fulfilling ∥ϕ(t)−Q4(t)∥β ≤ µ∥t∥s. S. Gowri et al. / Eur. J. Pure Appl. Math, 18 (4) (2025), 6699 12 of 14 5. Conclusion This work examines the Hyers-Ulam stability of a generalized quartic functional equa- tion within non-Archimedean (n, β)-normed spaces. These spaces, which generalize tradi- tional normed and ultrametric structures, offer a comprehensive framework for examining the behaviour of functional equations under perturbations. Theorem 2 and Theorem 3 examined stability in the context of a non-Archimedean β1-normed space as the domain and a full non-Archimedean (n, β)-normed space as the codomain. Theorem 4 and Theorem 5 broadened these findings to encompass more com- plex control functions, hence permitting enhanced flexibility in the assumptions regarding perturbations. Our results establish the existence and uniqueness of quartic mappings that resem- ble the original functional equation, therefore validating its Ulam-type stability in this extended non-Archimedean context. 6. Conflict of interest The authors declare that they have no competing interests. Availability of data and materials Not applicable. 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