Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 32 No. 9s (2025) 475 https://internationalpubls.com Investigating a Novel Stability Results of Generalized Alternate Cubic Functional Equations: Classical Method for Banach Spaces and Direct -Fixed Point Approaches for Fuzzy Normed Spaces P. Agilan πŸβˆ—, V. Vijayan 𝟐, M. Sophia πŸ‘ V.Banu Priya πŸ’ 1Department of Mathematics, St.Joseph’s College of Engineering, OMR, Chennai - 600 119, TamilNadu, India. 2Department of Electronics and Instrumentation Engineering, St.Joseph’s College of Engineering, OMR, Chennai - 600 119, TamilNadu, India. E-mail: vinvpn@gmail.com. 3 Department of Mathematics, SIMATS Engineering, Saveetha Nagar, Thandalam, Kanchipuram-Chennai Rd, Chennai- 602105, TamilNadu, India. E-mail: sophia.raj2005@gmail.com. 4Department of Mathematics, R.M.K College of Engineering and Technology, Kavaraipettai - 601 206, TamilNadu, India. E-mail:spriya.maths@gmail.com. *Corresponding Author: agilram@gmail.com. Article History: Received: 10-11-2024 Revised: 16-12-2024 Accepted: 11-01-2025 Abstract: This paper explores novel stability results for generalized alternate cubic functional equation(Fun Eq) using two distinct analytical frameworks: the classical method for Banach spaces and the direct and fixed point approaches for fuzzy normed spaces. The study examines the stability behavior of the generalized alternate cubic functional equation, focusing on how small deviations from exact solutions influence the overall stability in different normed environments. In Banach spaces, the classical approach is applied to derive conditions for Hyers-Ulam stability, providing insight into the equation’s behavior under small perturbations. For fuzzy normed spaces, both direct and fixed point methods are employed to account for the inherent uncertainties and fuzziness in the normed structure, offering a more flexible stability analysis. The results obtained highlight the differences and advantages of each approach, contributing to the broader understanding of functional equations in both deterministic and fuzzy frameworks. These findings have potential applications in various mathematical and applied fields, where both precise and imprecise data structures are considered. Keywords: Banach Spaces, Fuzzy Normed Spaces, Cubic Functional Equations, Ulam - Hyers Stability, Fixed point. 1 Introduction The Ulam-Hyers-Rassias stability deals with the stability of Fun Eq, which is a branch of mathematical analysis. Specifically, it focuses on determining under what conditions an approximate solution of a Fun Eq remains close to the exact solution. The stability concept was initiated by Stanislaw Ulam in 1940 [1] when he asked whether approximate homomorphisms on groups could be approximated by true homomorphisms. Later, in the 1940s and 1950s, Donald H. Hyers [2] and Th.M. Rassias [3] extended Ulam’s work to Fun Eq in Banach spaces. The Ulam-Hyers-Rassias stability theorem Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 32 No. 9s (2025) 476 https://internationalpubls.com provides conditions under which a functional equation approximately satisfies the equation. For further developments and the subsequent contributions by T. Aoki, P. Gavruta, J.M. Rassias, Isac and others [4, 5, 6, 7, 8, 9, 10]. It’s of great significance in many areas of mathematics, including functional analysis, operator theory, and mathematical physics. In practical terms, this stability theorem has applications in various fields, such as numerical analysis, optimization, control theory, and signal processing, where it’s crucial to understand how small errors in input data or parameters affect the output of a mathematical model or system. The concept of Hyers-Ulam stability has had a significant impact across various mathematical domains. Initially introduced in the context of Fun Eq, Hyers-Ulam stability focuses on whether small deviations from a functional equation still allow for an approximate solution that is close to an exact solution. This principle has since been applied to numerous areas such as: Differential Equations[11, 12]: Hyers-Ulam stability helps assess the stability of differential equations, especially in determining whether solutions to perturbed equations remain close to the solutions of the original equation. Integral Equations[13, 14]: In integral equations, the stability concept provides a framework to ensure that approximate solutions remain consistent even under perturbations. Operator Theory[15, 16]: Hyers- Ulam stability has been extended to operator equations, aiding in the analysis of bounded linear operators and their robustness under small changes. Approximation Theory[17]: It plays a role in approximation theory by ensuring that near solutions of approximation problems can still yield good approximations, thus enhancing the reliability of numerical methods. Control Theory[18, 19]: In systems governed by control equations, Hyers-Ulam stability contributes to the robustness analysis, determining how systems behave when subject to small external disturbances. Overall, Hyers-Ulam stability provides a foundational tool to understand the resilience of mathematical models in various applied and theoretical settings, ensuring that minor errors or perturbations do not drastically alter solutions. In fuzzy normed spaces, stability results are typically established using fixed-point methods or direct analytical approaches. The fuzzy nature of the space allows for handling vagueness or uncertainty in the norm, which is critical for real-world applications where data may not always be exact. The fixed- point method, for instance, is a powerful tool used to prove the existence of a stable cubic mapping, which satisfies the functional equation under these conditions [20, 21, 22, 23, 24]. Recent studies show that fuzzy normed spaces provide a more flexible framework for analyzing the stability of functional equations. In this setting, the stability of cubic functional equations is guaranteed even when deviations occur, provided the system adheres to specific constraints. This makes fuzzy stability particularly relevant in fields like applied mathematics, economics, and engineering, where imprecision often exists. By focusing on these modern methods, researchers have successfully derived new stability results for cubic equations, contributing to both theoretical mathematics and practical problem-solving in uncertain environments The study focuses on a generalized alternate cubic functional equation, which is a more intricate form compared to traditional cubic equations. Exploring the stability of such equations in Banach and fuzzy Banach spaces is essential because these spaces are widely used in various branches of functional analysis, optimization, and differential equations. Recently Agilan et.al exploring the stability results Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 32 No. 9s (2025) 477 https://internationalpubls.com in various additive functional equation through various normed spaces such as [25, 26, 27, 28, 29, 30, 31, 32]. In this paper, the authors investigate the generalized Ulam-Hyers stability of a Alternate cubic functional equation β„±(β„œπ‘Žπ‘£ + 𝑀) Β±β„œ 𝑏 β„±(𝑣 βˆ’β„œ π‘Žπ‘€) = ( β„œπ‘Ž(1Β±β„œπ‘Ž+𝑏) 2 ) [β„±(𝑣 + 𝑀) + β„±(𝑣 βˆ’ 𝑀)] + ( β„œπ‘Ž(β„œπ‘Žβˆ“β„œπ‘) 2 ) [β„±(𝑣 + 𝑀) βˆ’ β„±(𝑣 βˆ’ 𝑀)] +(β„œ2π‘Ž βˆ’ 1)[(β„œπ‘Ž βˆ“β„œ 𝑏)β„±(𝑣) βˆ“ (β„œπ‘Ž+𝑏 Β± 1)β„±(𝑀)] (1) where β„œ, π‘Ž, 𝑏 are integers with β„œ β‰  0, Β±1 and π‘Ž β‰  𝑏 β‰  0, Β±1 in Banach and fuzzy Banach spaces. Lemma 1.1 let us consider X and Y be real vector spaces. An odd function satisfies the functional equation β„±(π‘šπ‘£ + 𝑀) + β„±(π‘šπ‘£ βˆ’ 𝑀) = π‘šβ„±(𝑣 + 𝑀) +π‘šβ„±(𝑣 βˆ’ 𝑀) + 2(π‘š3 βˆ’ π‘š)β„±(𝑣) (2) for all 𝑣, 𝑀 ∈ 𝑋 if satisfies the Fun Eq(1) for all 𝑣,𝑀 ∈ 𝑋 . Proof. Assume 𝑓: 𝑋 β†’ π‘Œ satisfies the functional equation (2). Letting 𝑣 = 𝑀 = 0 in (2), we get β„±(0) = 0 . Setting in (2), we have β„±(βˆ’π‘€) = βˆ’β„±(𝑀) and 𝑀 = 0 we get β„±(π‘šπ‘£) = π‘š3β„±(𝑣) (3) for all 𝑣 ∈ 𝑋. In particular replace π‘š by β„œ π‘Ž in (2), we get β„±(β„œπ‘Žπ‘£ + 𝑀) + β„±(β„œπ‘Žπ‘£ βˆ’ 𝑀) = β„œ π‘Ž β„±(𝑣 + 𝑀) + β„œ π‘Ž β„±(𝑣 βˆ’ 𝑀) + 2(β„œ3π‘Ž βˆ’β„œ π‘Ž)β„±(π‘₯) (4) for all 𝑣, 𝑀 ∈ 𝑋. Replace 𝑀 by β„œ π‘Žπ‘€ in (4), we obtain β„±(β„œπ‘Ž(𝑣 + 𝑀)) + β„±(β„œπ‘Ž(𝑣 βˆ’ 𝑀)) = β„œ π‘Ž[β„±(𝑣 +β„œ π‘Žπ‘€) + β„±(𝑣 βˆ’β„œ π‘Žπ‘€)] + 2(β„œ3π‘Ž βˆ’β„œ π‘Ž)β„±(𝑣) (5) for all 𝑣, 𝑀 ∈ 𝑋. Using (3) in (5), we have β„œ 3π‘Ž[β„±(𝑣 + 𝑀) + β„±(𝑣 βˆ’ 𝑀)] = β„œ π‘Ž[β„±(𝑣 + β„œ π‘Žπ‘€) + β„±(𝑣 βˆ’β„œ π‘Žπ‘€)] + 2(β„œ3π‘Ž βˆ’ β„œ π‘Ž)β„±(𝑣) (6) for all 𝑣, 𝑀 ∈ 𝑋. Divide the above equation by β„œ π‘Ž , we get β„±(𝑣 +β„œ π‘Žπ‘€) + β„±(𝑣 βˆ’β„œ π‘Žπ‘€) = β„œ 2π‘Ž[β„±(𝑣 + 𝑀) + β„±(𝑣 βˆ’ 𝑀)] βˆ’ 2(β„œ2π‘Ž βˆ’ 1)β„±(𝑣) (7) for all 𝑣, 𝑀 ∈ 𝑋. Replace 𝑣 by 𝑀 and 𝑀 by 𝑣 in (7) and using oddness of 𝐢, we obtain β„±(β„œπ‘Žπ‘£ βˆ’ 𝑀) = β„±(β„œπ‘Žπ‘£ + 𝑀) βˆ’β„œ 2π‘Ž[β„±(𝑣 + 𝑀) βˆ’ β„±(𝑣 βˆ’ 𝑀)] + 2(β„œ2π‘Ž βˆ’ 1)β„±(𝑀) (8) for all 𝑣, 𝑀 ∈ 𝑋. Substitute (8) in (4), we get β„±(β„œπ‘Žπ‘£ + 𝑀) = β„œπ‘Ž 2 [β„±(𝑣 + 𝑀) + β„±(𝑣 βˆ’ 𝑀)] + β„œ2π‘Ž 2 [β„±(𝑣 + 𝑀) βˆ’ β„±(𝑣 βˆ’ 𝑀)] +(β„œ3π‘Ž βˆ’β„œ π‘Ž)β„±(𝑣) βˆ’ (β„œ2π‘Ž βˆ’ 1)β„±(𝑀) (9) for all 𝑣, 𝑀 ∈ 𝑋. Replace 𝑣 by βˆ’π‘€ and 𝑀 by 𝑣 in (9) and using oddness of 𝐢, we obtain Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 32 No. 9s (2025) 478 https://internationalpubls.com β„±(𝑣 βˆ’β„œ π‘Žπ‘€) = β„œπ‘Ž 2 [β„±(𝑣 βˆ’ 𝑀) βˆ’ β„±(𝑣 βˆ’ 𝑀)] + β„œ2π‘Ž 2 [β„±(𝑣 βˆ’ 𝑀) + β„±(𝑣 + 𝑀)] βˆ’(β„œ3π‘Ž βˆ’β„œ π‘Ž)β„±(𝑀) βˆ’ (β„œ2π‘Ž βˆ’ 1)β„±(𝑣) (10) for all 𝑣, 𝑀 ∈ 𝑋. Both side multiply by β„œ 𝑏 in (10), we get β„œ 𝑏 β„±(𝑣 βˆ’β„œ π‘Žπ‘€) = β„œπ‘Ž+𝑏 2 [β„±(𝑣 βˆ’ 𝑀) βˆ’ β„±(𝑣 βˆ’ 𝑀)] + β„œ2π‘Ž+𝑏 2 [β„±(𝑣 βˆ’ 𝑀) + β„±(𝑣 + 𝑀)] βˆ’(β„œ2π‘Ž+𝑏 βˆ’ β„œ 𝑏)β„±(𝑣) βˆ’ (β„œ3π‘Ž+𝑏 βˆ’β„œ π‘Ž+𝑏)β„±(𝑀) (11) for all 𝑣, 𝑀 ∈ 𝑋. Adding (9) and (11), we arrive β„±(β„œπ‘Žπ‘£ + 𝑀) +β„œ 𝑏 β„±(𝑣 βˆ’β„œ π‘Žπ‘€) = ( β„œπ‘Ž(1+β„œπ‘Ž+𝑏) 2 ) [β„±(𝑣 + 𝑀) + β„±(𝑣 βˆ’ 𝑀)] + ( β„œπ‘Ž(β„œπ‘Žβˆ’β„œπ‘) 2 ) [β„±(𝑣 + 𝑀) βˆ’ β„±(𝑣 βˆ’ 𝑀)] +(β„œ2π‘Ž βˆ’ 1)[(β„œπ‘Ž βˆ’β„œ 𝑏)β„±(𝑣) βˆ’ (β„œπ‘Ž+𝑏 + 1)β„±(𝑀)] (12) for all 𝑣, 𝑀 ∈ 𝑋. Subtracting (9) and (11), we arrive β„±(β„œπ‘Žπ‘£ + 𝑀) βˆ’β„œ 𝑏 β„±(𝑣 βˆ’β„œ π‘Žπ‘€) = ( β„œπ‘Ž(1βˆ’β„œπ‘Ž+𝑏) 2 ) [β„±(𝑣 + 𝑀) + β„±(𝑣 βˆ’ 𝑀)] + ( β„œπ‘Ž(β„œπ‘Ž+β„œπ‘) 2 ) [β„±(𝑣 + 𝑀) βˆ’ β„±(𝑣 βˆ’ 𝑀)] +(β„œ2π‘Ž βˆ’ 1)[(β„œπ‘Ž +β„œ 𝑏)β„±(𝑣) + (β„œπ‘Ž+𝑏 βˆ’ 1)β„±(𝑀)] (13) for all 𝑣, 𝑀 ∈ 𝑋. Combining both (12) and (13) we arrive (1). 2 Banach space stability results direct method Theorem 2.1 Assume X be normed linear space and Y be Banach space. Suppose that the function β„±:X β†’ Y satisfice ‖𝐷ℱ(𝑣,𝑀)β€– ≀ 𝔔(𝑣, 𝑀) (1) βˆ€π‘£,𝑀 ∈ 𝑋 and Let 𝔔:𝑋 Γ— 𝑋 β†’ [0,∞) be a function such that lim π‘›β†’βˆž 𝔔(β„œπ‘Žπ‘›π‘£,β„œπ‘Žπ‘›π‘€) β„œ3π‘Žπ‘› = 0 (2) βˆ€π‘£,𝑀 ∈ 𝑋, then βˆƒ Cubic map β„±: 𝑋 β†’ π‘Œ with the the FE (??) and β€–β„±(𝑣) βˆ’ β„±(𝑣)β€– ≀ 1 β„œ3π‘Žβˆ‘ ∞ π‘ž=1 𝔔(β„œπ‘Ž(π‘žβˆ’1)𝑣,0) β„œ3π‘Žπ‘ž (3) βˆ€π‘£ ∈ 𝑋. Let β„±(𝑣) is defined as β„±(𝑣) = lim π‘›β†’βˆž β„±(β„œπ‘Žπ‘›π‘’) β„œ3π‘Žπ‘› (4) βˆ€π‘£ ∈ 𝑋. Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 32 No. 9s (2025) 479 https://internationalpubls.com Proof. Considering (𝑣,𝑀) by (0,0) in (1), then we have β„±(𝑣) = 0. Switching (𝑣,𝑀) by (𝑣, 0) in (1), we get β€–β„±(β„œπ‘Žπ‘£) βˆ’β„œ 3π‘Ž β„±(𝑣)β€– ≀ 𝔔(𝑣, 0) (5) βˆ€π‘£ ∈ 𝑋, We replace 𝑣 by β„œ π‘Ž(π‘žβˆ’1)𝑣 (for π‘ž ∈ β„• and π‘ž β‰₯ 1) in (5) , and we obtain β€–β„±(β„œπ‘Žπ‘žπ‘£) βˆ’β„œ 3π‘Ž β„±(β„œπ‘Ž(π‘žβˆ’1)𝑣)β€– ≀ 𝔔(β„œπ‘Ž(π‘žβˆ’1)𝑣, 0) βˆ€π‘£ ∈ 𝑋. By multiplying both sides of the aforementioned inequality by 1 β„œ3π‘Žπ‘ž, we get the consequence of adding 𝑛 inequalities. βˆ‘π‘›π‘ž=1 1 β„œ3π‘Žπ‘ž β€–β„±(β„œπ‘Žπ‘žπ‘£) βˆ’β„œ 3π‘Žπ‘ž β„±(β„œπ‘Ž(π‘žβˆ’1)𝑣)β€– ≀ βˆ‘π‘›π‘ž=1 𝔔(β„œπ‘Ž(π‘žβˆ’1)𝑣,0) β„œ3π‘Žπ‘ž Making use of the triangle inequality |𝐴 + 𝐡| ≀ |𝐴| + |𝐡| After simplifying, we get at the left side of the inequality. β€– 1 β„œ3π‘Žπ‘› β„±(β„œπ‘Žπ‘›π‘£) βˆ’ β„±(𝑣)β€– ≀ βˆ‘π‘›π‘ž=1 𝔔(β„œπ‘Ž(π‘žβˆ’1)𝑣,0) β„œ3π‘Žπ‘ž (6) Since βˆ‘π‘›π‘ž=1 𝔔(β„œπ‘Ž(π‘žβˆ’1)𝑣,0) β„œ3π‘Žπ‘ž ≀ βˆ‘βˆž π‘ž=1 𝔔(β„œπ‘Ž(π‘žβˆ’1)𝑣,0) β„œ3π‘Žπ‘ž the inequality (6) yields β€– 1 β„œ3π‘Žπ‘› β„±(β„œπ‘Žπ‘›π‘£) βˆ’ β„±(𝑣)β€– ≀ βˆ‘βˆž π‘ž=1 𝔔(β„œπ‘Ž(π‘žβˆ’1)𝑣,0) β„œ3π‘Žπ‘ž βˆ€π‘£ ∈ 𝑋. It will be proven by induction that (6) exists βˆ€ β„•. Here π‘š > 𝑛 > 0, then π‘š βˆ’ 𝑛 ∈ β„• and let 𝑛 by π‘š βˆ’ 𝑛 in (6), then β€– 1 β„œ3π‘Ž(π‘šβˆ’π‘›) β„±(β„œπ‘Ž(π‘šβˆ’π‘›)𝑣) βˆ’ β„±(𝑣)β€– ≀ βˆ‘βˆž π‘ž=1 𝔔(β„œπ‘Ž(π‘žβˆ’1)𝑣,0) β„œ3π‘Žπ‘ž (7) which is β€– 1 β„œ3π‘Žπ‘š β„±(β„œπ‘Ž(π‘šβˆ’π‘›)𝑣) βˆ’ 1 β„œ3π‘Žπ‘› β„±(𝑣)β€– ≀ 1 β„œ3π‘Žπ‘› βˆ‘ ∞ π‘ž=1 𝔔(β„œπ‘Ž(π‘žβˆ’1)𝑣,0) β„œ3π‘Žπ‘ž (8) βˆ€π‘’ ∈ 𝑋. Interchanging 𝑒 by β„œ π‘Žπ‘›π‘£ in (8), we obtain β€– 1 β„œ3π‘Žπ‘š β„±(β„œπ‘Žπ‘šπ‘£) βˆ’ 1 β„œ3π‘Žπ‘› β„±(β„œπ‘Žπ‘›π‘£)β€– ≀ 1 β„œ3π‘Žπ‘› βˆ‘ ∞ π‘ž=1 𝔔(β„œπ‘Ž(π‘ž+π‘›βˆ’1),0) β„œ3π‘Žπ‘ž (9) Since lim π‘›β†’βˆž 1 β„œ3π‘Žπ‘› = 0 and hence from (9), we obtain Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 32 No. 9s (2025) 480 https://internationalpubls.com lim π‘›β†’βˆž β€– 1 β„œ3π‘Žπ‘š β„±(β„œπ‘Žπ‘šπ‘£) βˆ’ 1 β„œ3π‘Žπ‘› β„±(β„œπ‘Žπ‘›π‘£)β€– = 0 Finally { β„±(β„œπ‘Žπ‘›π‘£) β„œ3π‘Žπ‘› } 𝑛=1 ∞ is Cauchy sequence. The sequence then has a limit in 𝑋. Define 𝐴(𝑣) = lim π‘›β†’βˆž β„±(β„œπ‘Žπ‘›π‘£) β„œ3π‘Žπ‘› βˆ€π‘’ ∈ 𝑋. we prove 𝐴: 𝑋 β†’ 𝑋 is a Linear mapping. β€–β„±(β„œπ‘Žπ‘£ + 𝑀) Β±β„œ 𝑏 β„±(𝑣 βˆ’β„œ π‘Žπ‘€) βˆ’ ( β„œπ‘Ž(1Β±β„œπ‘Ž+𝑏) 2 ) [β„±(𝑣 + 𝑀) + β„±(𝑣 βˆ’ 𝑀)] βˆ’ ( β„œπ‘Ž(β„œπ‘Žβˆ“β„œπ‘) 2 ) [β„±(𝑣 + 𝑀) βˆ’ β„±(𝑣 βˆ’ 𝑀)] βˆ’(β„œ2π‘Ž βˆ’ 1)[(β„œπ‘Ž βˆ“β„œ 𝑏)β„±(𝑣) βˆ“ (β„œπ‘Ž+𝑏 Β± 1)β„±(𝑀)]β€– = 1 β„œπ‘Žπ‘› β€–β„±(β„œπ‘Žπ‘£ + 𝑀) Β±β„œ 𝑏 β„±(𝑣 βˆ’ β„œ π‘Žπ‘€) βˆ’ ( β„œπ‘Ž(1Β±β„œπ‘Ž+𝑏) 2 ) [β„±(𝑣 + 𝑀) + β„±(𝑣 βˆ’ 𝑀)] βˆ’ ( β„œπ‘Ž(β„œπ‘Žβˆ“β„œπ‘) 2 ) [β„±(𝑣 + 𝑀) βˆ’ β„±(𝑣 βˆ’ 𝑀)] βˆ’(β„œ2π‘Ž βˆ’ 1)[(β„œπ‘Ž βˆ“β„œ 𝑏)β„±(𝑣) βˆ“ (β„œπ‘Ž+𝑏 Β± 1)β„±(𝑀)]β€– ≀ lim π‘›β†’βˆž 𝔔(β„œπ‘Žπ‘›π‘£,β„œπ‘Žπ‘›π‘€) β„œ3π‘Žπ‘› = 0 Hence β„±(β„œπ‘Žπ‘£ + 𝑀) Β±β„œ 𝑏 β„±(𝑣 βˆ’β„œ π‘Žπ‘€) = ( β„œπ‘Ž(1Β±β„œπ‘Ž+𝑏) 2 ) [β„±(𝑣 + 𝑀) + β„±(𝑣 βˆ’ 𝑀)] + ( β„œπ‘Ž(β„œπ‘Žβˆ“β„œπ‘) 2 ) [β„±(𝑣 + 𝑀) βˆ’ β„±(𝑣 βˆ’ 𝑀)] +(β„œ2π‘Ž βˆ’ 1)[(β„œπ‘Ž βˆ“β„œ 𝑏)β„±(𝑣) βˆ“ (β„œπ‘Ž+𝑏 Β± 1)β„±(𝑀)] βˆ€π‘’ ∈ 𝑋. Next, we consider ||𝐴(𝑣) βˆ’ β„±(𝑣)|| = || lim π‘›β†’βˆž β„±(β„œπ‘Žπ‘›π‘’) β„œπ‘Žπ‘› βˆ’ β„±(𝑣)|| = lim π‘›β†’βˆž || β„±(β„œπ‘Žπ‘›π‘’) β„œ3π‘Žπ‘› βˆ’ β„±(𝑣)|| Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 32 No. 9s (2025) 481 https://internationalpubls.com ≀ lim π‘›β†’βˆž 1 β„œ3π‘Žβˆ‘ ∞ π‘ž=1 𝔔(β„œπ‘Ž(π‘žβˆ’1)𝑣,0) β„œ3π‘Žπ‘ž Hence, we get ||𝐴(𝑣) βˆ’ β„±(𝑣)|| ≀ 1 β„œ3π‘Žβˆ‘ ∞ π‘ž=1 𝔔(β„œπ‘Ž(π‘žβˆ’1)𝑣,0) β„œ3π‘Žπ‘ž βˆ€π‘’ ∈ 𝑋. Here to obtain 𝐴 is unique. Then another mapping 𝐡: 𝑋 β†’ π‘Œ occurs and ||𝐡(𝑣) βˆ’ β„±(𝑣)|| ≀ 1 β„œ3π‘Žβˆ‘ ∞ π‘ž=1 𝔔(β„œπ‘Ž(π‘žβˆ’1)𝑣,0) β„œ3π‘Žπ‘ž Hence ||𝐡(𝑣) βˆ’ 𝐴(𝑣)|| ≀ ||𝐡(𝑣) βˆ’ β„±(𝑣)|| + ||𝐴(𝑣) βˆ’ β„±(𝑣)|| ≀ 1 β„œ3π‘Žβˆ‘ 𝑛 π‘ž=1 𝔔(β„œπ‘Ž(π‘žβˆ’1)𝑣,0) β„œ3π‘Žπ‘ž + 1 β„œ3π‘Žβˆ‘ ∞ π‘ž=1 𝔔(β„œπ‘Ž(π‘žβˆ’1)𝑣,0) β„œ3π‘Žπ‘ž = 2 β„œ3π‘Žβˆ‘ ∞ π‘ž=1 𝔔(β„œπ‘Ž(π‘žβˆ’1)𝑣,0) β„œ3π‘Žπ‘ž Because the additive mappings are 𝐴 and 𝐡, we can observe ||𝐴(𝑣) βˆ’ 𝐡(𝑣)|| = 2 β„œ3π‘Žπ‘› ||𝐴(β„œ π‘Žπ‘›) βˆ’ 𝐡(β„œπ‘Žπ‘›π‘£)|| ≀ 2 β„œ3π‘Žπ‘› βˆ‘ ∞ π‘ž=1 𝔔(β„œπ‘Ž(π‘ž+π‘›βˆ’1)𝑣,0) β„œ3π‘Žπ‘ž (10) As a result (10), using the limit 𝑛 β†’ ∞ and obtain lim π‘›β†’βˆž ||𝐴(𝑣) βˆ’ 𝐡(𝑣)|| ≀ lim π‘›β†’βˆž 2 β„œ3π‘Žπ‘› βˆ‘ ∞ π‘ž=1 𝔔(β„œπ‘Ž(π‘ž+π‘›βˆ’1)𝑣,0) β„œ3π‘Žπ‘ž Hence ||𝐴(𝑣) βˆ’ 𝐡(𝑣)|| ≀ 0 we conclude that 𝐴(𝑣) = 𝐡(𝑣) βˆ€ 𝑣 ∈ 𝑋. At the end 𝐴 is unique. Corollary 2.2 Consider the map β„±:X β†’ Yfulfills β€–Dβ„±(v,w)β€– ≀ { π”˜, π”˜{||v||p + ||w||p}, p β‰  3; π”˜ {||v||p||w||p + {||v||2p + ||w||2p}} , 2p β‰  3; (11) and the function 𝐴: 𝑋 β†’ 𝑋, we arrive the result Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 32 No. 9s (2025) 482 https://internationalpubls.com β€–β„±(𝑣) βˆ’ 𝐴(𝑣)β€– ≀ { π”˜ |β„œ3π‘Žβˆ’1| , π”˜||𝑣||𝑝 |β„œ3π‘Žβˆ’β„œπ‘Žπ‘| , π”˜||𝑣||2𝑝 |β„œ3π‘Žβˆ’β„œ2π‘Žπ‘| (12) βˆ€ 𝑣 ∈ 𝑋. 3 Definitions of Fuzzy Normed spaces Definition 3.1 Let X be a real linear space. A function N:X Γ— ℝ β†’ [0,1](the so-called fuzzy subset) is said to be a fuzzy norm on X if for all v,w ∈ X and all s, t ∈ ℝ, (𝐹1) 𝒩(𝑣, 𝑐) = 0 for 𝑐 ≀ 0; (𝐹2) 𝑣 = 0 if and only if 𝒩(𝑣, 𝑐) = 1 for all 𝑐 > 0; (𝐹3) 𝒩(𝑐𝑣, 𝑑) = 𝒩 (𝑣, 𝑑 |𝑐| ) if 𝑐 β‰  0; (𝐹4) 𝒩(𝑣 + 𝑀, 𝑠 + 𝑑) β‰₯ π‘šπ‘–π‘›{𝒩(𝑣, 𝑠),𝒩(𝑀, 𝑑)}; (𝐹5) 𝒩(𝑣,β‹…) is a non-decreasing function on ℝ and π‘™π‘–π‘šπ‘‘β†’βˆžπ’©(𝑣, 𝑑) = 1; (𝐹6) for 𝑣 β‰  0,𝒩(𝑣,β‹…) is (upper semi) continuous on ℝ. The pair (𝑋,𝑁) is called a fuzzy normed linear space. One may regard 𝒩(𝑋, 𝑑) as the truth-value of the statement the norm of 𝑣 is less than or equal to the real number 𝑑’. Example 3.2 Let (X, || β‹… ||) be a normed linear space. Then 𝒩(𝑣, 𝑑) = { 𝑑 𝑑+‖𝑣‖ , 𝑑 > 0, 𝑣 ∈ 𝑋, 0, 𝑑 ≀ 0, 𝑣 ∈ 𝑋 is a fuzzy norm on 𝑋. 4 Direct method of fuzzy stability result 𝐷 β„±(𝑣,𝑀) = β„±(β„œπ‘Žπ‘£ + 𝑀) Β±β„œ 𝑏 β„±(𝑣 βˆ’β„œ π‘Žπ‘€) = ( β„œπ‘Ž(1Β±β„œπ‘Ž+𝑏) 2 ) [β„±(𝑣 + 𝑀) + β„±(𝑣 βˆ’ 𝑀)] + (β„œ2π‘Ž βˆ’ 1)[(β„œπ‘Ž βˆ“β„œ 𝑏)β„±(𝑣) βˆ“ (β„œπ‘Ž+𝑏 Β± 1)β„±(𝑀)] Theorem 4.1 Assume that X linear space, (Z,Nβ€²) fuzzy normed space and (Y,Nβ€²)fuzzy Banach space. Let Ξ² ∈ {βˆ’1,1} be fixed and let 𝔔:X2 β†’ Z be a mapping such that for some d with 0 < ( d β„œ3a) Ξ² < 1 𝑁′(𝔔(β„œπ‘Žπ›½π‘£,β„œπ‘Žπ›½π‘€), π‘Ÿ) β‰₯ 𝑁′(π‘‘π‘Žπ›½π””(𝑣,𝑀), π‘Ÿ) (1) for all 𝑣 ∈ 𝑋 and all π‘Ÿ > 0, 𝑑 > 0, and Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 32 No. 9s (2025) 483 https://internationalpubls.com lim π‘˜β†’βˆž 𝑁′(𝔔(β„œπ‘Žπ›½π‘˜π‘£,β„œπ‘Žπ›½π‘˜π‘€),β„œπ‘Žπ›½3π‘˜π‘Ÿ) = 1 (2) for all 𝑣, 𝑀 ∈ 𝑋 and all π‘Ÿ > 0. Suppose that a function 𝑓: 𝑋 β†’ π‘Œ satisfies the inequality 𝒩(𝐷ℱ(𝑣, 𝑀), π‘Ÿ) β‰₯ 𝑁′(𝔔(𝑣,𝑀), π‘Ÿ) (3) for all π‘Ÿ > 0 and all 𝑣,𝑀 ∈ 𝑋. Then the limit π’ž(𝑣) = 𝑁 βˆ’ lim π‘˜β†’βˆž β„±(β„œπ‘Žπ›½π‘˜π‘£) β„œπ‘Žπ›½3π‘˜ (4) exists for all 𝑣 ∈ 𝑋 and the mapping 𝐢: 𝑋 β†’ π‘Œ is a unique cubic mapping such that 𝒩(π’ž(𝑣) βˆ’ β„±(𝑣), π‘Ÿ) β‰₯ 𝑁′(𝔔(𝑣, 0), |β„œ3π‘Ž βˆ’ π‘‘π‘Ž|π‘Ÿ) (5) for all 𝑣 ∈ 𝑋 and all π‘Ÿ > 0. Proof. First assume 𝛽 = 1. Replacing (𝑣, 𝑀) by (𝑣, 0) in (3), we get 𝒩(β„±(β„œπ‘Žπ‘£) βˆ’β„œ 3π‘Ž β„±(𝑣), π‘Ÿ) β‰₯ 𝑁′(𝔔(𝑣, 0), π‘Ÿ) (6) for all 𝑣 ∈ 𝑋 and all π‘Ÿ > 0. Replacing 𝑣 by β„œ π‘Žπ‘˜π‘£ in (6), we obtain 𝒩( β„±(β„œπ‘Ž(π‘˜+1)𝑣) β„œ3π‘Ž βˆ’ β„±(β„œπ‘Žπ‘˜π‘£), π‘Ÿ β„œ3π‘Ž) β‰₯ 𝑁′(𝔔(β„œπ‘Žπ‘˜π‘£, 0), π‘Ÿ) (7) for all 𝑣 ∈ 𝑋 and all π‘Ÿ > 0. Using (1), (𝐹3) in (7), we arrive 𝒩( β„±(β„œπ‘Ž(π‘˜+1)𝑣) β„œ3π‘Ž βˆ’ β„±(β„œπ‘Žπ‘˜π‘£), π‘Ÿ β„œ3π‘Ž) β‰₯ 𝑁′ (𝔔(𝑣, 0), π‘Ÿ π‘‘π‘Žπ‘˜ ) (8) for all 𝑣 ∈ 𝑋 and all π‘Ÿ > 0. It is easy to verify from (8), that 𝒩( β„±(β„œπ‘Ž(π‘˜+1)𝑣) β„œ3π‘Ž(π‘˜+1) βˆ’ β„±(β„œπ‘Žπ‘˜π‘£) β„œ3π‘Žπ‘˜ , π‘Ÿ β„œ3π‘Žβ‹…β„œ3π‘Žπ‘˜) β‰₯ 𝑁′ (𝔔(𝑣, 0), π‘Ÿ π‘‘π‘Žπ‘˜ ) (9) holds for all 𝑣 ∈ 𝑋 and all π‘Ÿ > 0. Replacing π‘Ÿ by β„œ π‘Žπ‘˜π‘Ÿ in (9), we get 𝒩( β„±(β„œπ‘Ž(π‘˜+1)𝑣) β„œ3π‘Ž(π‘˜+1) βˆ’ β„±(β„œπ‘Žπ‘˜π‘£) β„œ3π‘Žπ‘˜ , π‘‘π‘Žπ‘˜ π‘Ÿ β„œ3π‘Žβ‹…β„œ3π‘Žπ‘˜) β‰₯ 𝑁′(𝔔(𝑣, 0), π‘Ÿ) (10) for all 𝑣 ∈ 𝑋 and all π‘Ÿ > 0. It is easy to see that β„±(β„œπ‘Žπ‘˜π‘£) β„œ3π‘Žπ‘˜ βˆ’ β„±(𝑣) = βˆ‘π‘˜βˆ’1 𝑖=0 [ β„±(β„œπ‘Ž(𝑖+1)π‘₯) β„œ3π‘Ž(𝑖+1) βˆ’ β„±(β„œπ‘Žπ‘–π‘£) β„œ3π‘Žπ‘– ] (11) for all 𝑣 ∈ 𝑋. From equations (10) and (11), we have 𝒩( β„±(β„œπ‘Žπ‘˜π‘£) β„œ3π‘Žπ‘˜ βˆ’ β„±(𝑣), βˆ‘π‘˜βˆ’1 𝑖=0 𝑑𝑖 π‘Ÿ β„œ3π‘Žβ‹…β„œ3π‘Žπ‘–) β‰₯ π‘šπ‘–π‘›β‹ƒπ‘˜βˆ’1 𝑖=0 { β„±(β„œπ‘Ž(𝑖+1)𝑣) β„œ3π‘Ž(𝑖+1) βˆ’ β„±(β„œπ‘–π‘Žπ‘£) β„œ3π‘Žπ‘– , 𝑑𝑖 π‘Ÿ β„œ3π‘Žβ‹…β„œ3π‘Žπ‘–} β‰₯ π‘šπ‘–π‘›β‹ƒπ‘˜βˆ’1 𝑖=0 {𝑁′(𝔔(𝑣, 0), π‘Ÿ)} β‰₯ 𝑁′(𝔔(𝑣, 0), π‘Ÿ) (12) Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 32 No. 9s (2025) 484 https://internationalpubls.com for all 𝑣 ∈ 𝑋 and all π‘Ÿ > 0. Replacing 𝑣 by β„œ π‘šπ‘Žπ‘£ in (12) and using (1), (𝐹3), we obtain 𝒩( β„±(β„œπ‘Ž(π‘˜+π‘š)𝑣) β„œ3π‘Ž(π‘˜+π‘š) βˆ’ β„±(β„œπ‘šπ‘Žπ‘£) β„œ3π‘Žπ‘š , βˆ‘π‘˜βˆ’1 𝑖=0 𝑑𝑖 π‘Ÿ β„œ3π‘Žβ‹…β„œ3π‘Ž(𝑖+π‘š)) β‰₯ 𝑁′ (𝔔(𝑣, 0), π‘Ÿ π‘‘π‘Žπ‘š ) (13) for all 𝑣 ∈ 𝑋 and all π‘Ÿ > 0 and all π‘š, π‘˜ β‰₯ 0. Replacing π‘Ÿ by π‘‘π‘Žπ‘šπ‘Ÿ in (13), we get 𝒩( β„±(β„œπ‘Ž(π‘˜+π‘š)𝑣) β„œ3π‘Ž(π‘˜+π‘š) βˆ’ β„±(β„œπ‘Žπ‘šπ‘£) β„œ3π‘Žπ‘š , βˆ‘π‘š+π‘˜βˆ’1 𝑖=π‘š π‘‘π‘Žπ‘– π‘Ÿ β„œ3π‘Žβ‹…β„œ3π‘Žπ‘–) β‰₯ 𝑁′(𝔔(𝑣, 0), π‘Ÿ) (14) for all 𝑣 ∈ 𝑋 and all π‘Ÿ > 0 and all π‘š, π‘˜ β‰₯ 0. Using (𝐹3) in (14), we obtain 𝒩( β„±(β„œπ‘Ž(π‘˜+π‘š)𝑣) β„œ3π‘Ž(π‘˜+π‘š) βˆ’ β„±(β„œπ‘Žπ‘šπ‘£) β„œ3π‘Žπ‘š , π‘Ÿ) β‰₯ 𝑁′(𝔔(𝑣, 0), π‘Ÿ βˆ‘π‘š+π‘˜βˆ’1 𝑖=π‘š π‘‘π‘Žπ‘– β„œ 3π‘Žβ‹…β„œ3π‘Žπ‘– ) (15) for all 𝑣 ∈ 𝑋 and all π‘Ÿ > 0 and all π‘š, π‘˜ β‰₯ 0. Since 0 < 𝑑 < β„œ 3π‘Ž and βˆ‘π‘˜π‘–=0 ( 𝑑 β„œ3π‘Ž) 𝑖 < ∞, the cauchy criterion for convergence and (𝐹5) implies that { β„±(β„œπ‘Žπ‘˜π‘£) β„œ3π‘Žπ‘˜ } is a Cauchy sequence in (π‘Œ,𝑁). Since (π‘Œ, 𝑁) is a fuzzy Banach space, this sequence converges to some point β„±(𝑣) ∈ π‘Œ. So one can define the mapping 𝐢: 𝑋 β†’ π‘Œ by β„±(𝑣) = 𝑁 βˆ’ lim π‘˜β†’βˆž β„±(β„œπ‘Žπ‘˜π‘£) β„œ3π‘Žπ‘˜ for all 𝑣 ∈ 𝑋. Letting π‘š = 0 in (15), we get 𝒩( β„±(β„œπ‘Žπ‘˜π‘£) β„œ3π‘Žπ‘˜ βˆ’ β„±(𝑣), π‘Ÿ) β‰₯ 𝑁′(𝔔(𝑣, 0), π‘Ÿ βˆ‘π‘˜βˆ’1 𝑖=0 π‘‘π‘Žπ‘– β„œ 3π‘–β‹…β„œ3π‘Žπ‘– ) (16) for all 𝑣 ∈ 𝑋 and all π‘Ÿ > 0. Letting π‘˜ β†’ ∞ in (16) and using (𝐹6), we arrive 𝒩(π’ž(𝑣) βˆ’ β„±(𝑣), π‘Ÿ) β‰₯ 𝑁′(𝔔(𝑣, 0), π‘Ÿ(β„œ3π‘Ž βˆ’ 𝑑)) for all 𝑣 ∈ 𝑋 and all π‘Ÿ > 0. To prove 𝐢 satisfies the (1), replacing (𝑣, 𝑀) by (β„œπ‘Žπ‘˜π‘£,β„œπ‘Žπ‘˜π‘€) in (3), respectively , we obtain 𝒩( 1 β„œ3π‘Žπ‘˜π·β„±(β„œπ‘Žπ‘˜π‘£,β„œπ‘Žπ‘˜π‘€), π‘Ÿ) β‰₯ 𝑁′(𝔔(β„œπ‘Žπ‘˜π‘£,β„œπ‘Žπ‘˜π‘€),β„œ3π‘Žπ‘˜π‘Ÿ) (17) for all π‘Ÿ > 0 and all 𝑣,𝑀 ∈ 𝑋. Now, 𝒩(β„±(β„œπ‘Žπ‘£ + 𝑀) Β±β„œ 𝑏 β„±(𝑣 βˆ’β„œ π‘Žπ‘€) βˆ’ ( β„œπ‘Ž(1Β±β„œπ‘Ž+𝑏) 2 ) [β„±(𝑣 + 𝑀) + β„±(𝑣 βˆ’ 𝑀)] βˆ’ ( β„œπ‘Ž(β„œπ‘Žβˆ“β„œπ‘) 2 ) [β„±(𝑣 + 𝑀) βˆ’ β„±(𝑣 βˆ’ 𝑀)] βˆ’(β„œ2π‘Ž βˆ’ 1)[(β„œπ‘Ž βˆ“ β„œ 𝑏)β„±(𝑣) βˆ“ (β„œπ‘Ž+𝑏 Β± 1)β„±(𝑀)], π‘Ÿ 6 ) β‰₯ π‘šπ‘–π‘› {𝒩 (π’ž(β„œπ‘Žπ‘£ + 𝑀) βˆ’ 1 β„œ3π‘Žπ‘˜ β„±(β„œπ‘Žπ‘£ + 𝑀), π‘Ÿ 6 ), Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 32 No. 9s (2025) 485 https://internationalpubls.com 𝒩 (Β±β„œ π‘π’ž(𝑣 βˆ’ β„œ π‘Žπ‘€) Β± 1 β„œ3π‘Žπ‘˜ β„œ 𝑏 β„±(𝑣 βˆ’β„œ π‘Žπ‘€), π‘Ÿ 6 ), 𝒩 (βˆ’( β„œπ‘Ž(1Β±β„œπ‘Ž+𝑏) 2 ) [π’ž(𝑣 + 𝑀) + π’ž(𝑣 βˆ’ 𝑀)] βˆ’ 1 β„œ3π‘Žπ‘˜ ( β„œπ‘Ž(1Β±β„œπ‘Ž+𝑏) 2 ) [β„±(𝑣 + 𝑀) + β„±(𝑣 βˆ’ 𝑀)], π‘Ÿ 6 ), 𝒩 (βˆ’( β„œπ‘Ž(β„œπ‘Žβˆ“β„œπ‘) 2 ) [π’ž(𝑣 + 𝑀) βˆ’ π’ž(𝑣 βˆ’ 𝑀)] βˆ’ 1 β„œ3π‘Žπ‘˜ ( β„œπ‘Ž(β„œπ‘Žβˆ“β„œπ‘) 2 ) [β„±(𝑣 + 𝑀) βˆ’ β„±(𝑣 βˆ’ 𝑀)], π‘Ÿ 6 ), 𝒩(βˆ’(β„œ2π‘Ž βˆ’ 1)[(β„œπ‘Ž βˆ“ β„œ 𝑏)π’ž(𝑣) βˆ“ (β„œπ‘Ž+𝑏 Β± 1)π’ž(𝑀)] βˆ’ 1 β„œ3π‘Žπ‘˜ (β„œ 2π‘Ž βˆ’ 1)[(β„œπ‘Ž βˆ“β„œ 𝑏)β„±(𝑣) βˆ“ (β„œπ‘Ž+𝑏 Β± 1)β„±(𝑀)], π‘Ÿ 6 ), 𝒩 ( 1 β„œ3π‘Žπ‘˜ β„±(β„œπ‘Žπ‘£ + 𝑀) Β± 1 β„œ3π‘Žπ‘˜ β„œ 𝑏 β„±(𝑣 βˆ’β„œ π‘Žπ‘€) βˆ’ 1 β„œ3π‘Žπ‘˜ ( β„œπ‘Ž(1Β±β„œπ‘Ž+𝑏) 2 ) [β„±(𝑣 + 𝑀) + β„±(𝑣 βˆ’ 𝑀)] βˆ’ 1 β„œ3π‘Žπ‘˜ ( β„œπ‘Ž(β„œπ‘Žβˆ“β„œπ‘) 2 ) [β„±(𝑣 + 𝑀) βˆ’ β„±(𝑣 βˆ’ 𝑀)] βˆ’ 1 β„œ3π‘Žπ‘˜ (β„œ 2π‘Ž βˆ’ 1)[(β„œπ‘Ž βˆ“β„œ 𝑏)β„±(𝑣) βˆ“ (β„œπ‘Ž+𝑏 Β± 1)β„±(𝑀)], π‘Ÿ 6 )} for all 𝑣, 𝑀 ∈ 𝑋 and all π‘Ÿ > 0. 𝒩(β„±(β„œπ‘Žπ‘£ + 𝑀) Β±β„œ 𝑏 β„±(𝑣 βˆ’β„œ π‘Žπ‘€) βˆ’ ( β„œπ‘Ž(1Β±β„œπ‘Ž+𝑏) 2 ) [β„±(𝑣 + 𝑀) + β„±(𝑣 βˆ’ 𝑀)] βˆ’ ( β„œπ‘Ž(β„œπ‘Žβˆ“β„œπ‘) 2 ) [β„±(𝑣 + 𝑀) βˆ’ β„±(𝑣 βˆ’ 𝑀)] βˆ’(β„œ2π‘Ž βˆ’ 1)[(β„œπ‘Ž βˆ“ β„œ 𝑏)β„±(𝑣) βˆ“ (β„œπ‘Ž+𝑏 Β± 1)β„±(𝑀)], π‘Ÿ 6 ) β‰₯ π‘šπ‘–π‘›{1,1,1,1,1,1,1𝑁′(𝔔(β„œπ‘Žπ‘˜π‘£, 0),β„œ3π‘Žπ‘˜π‘Ÿ)} β‰₯ 𝑁′(𝔔(β„œπ‘Žπ‘˜π‘£, 0),β„œ3π‘Žπ‘˜π‘Ÿ) (18) for all 𝑣, 𝑀 ∈ 𝑋 and all π‘Ÿ > 0. Letting π‘˜ β†’ ∞ in (18) and using (2), we see that 𝒩(β„±(β„œπ‘Žπ‘£ + 𝑀) Β±β„œ 𝑏 β„±(𝑣 βˆ’β„œ π‘Žπ‘€) βˆ’ ( β„œπ‘Ž(1Β±β„œπ‘Ž+𝑏) 2 ) [β„±(𝑣 + 𝑀) + β„±(𝑣 βˆ’ 𝑀)] βˆ’ ( β„œπ‘Ž(β„œπ‘Žβˆ“β„œπ‘) 2 ) [β„±(𝑣 + 𝑀) βˆ’ β„±(𝑣 βˆ’ 𝑀)] Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 32 No. 9s (2025) 486 https://internationalpubls.com βˆ’(β„œ2π‘Ž βˆ’ 1)[(β„œπ‘Ž βˆ“ β„œ 𝑏)β„±(𝑣) βˆ“ (β„œπ‘Ž+𝑏 Β± 1)β„±(𝑀)], π‘Ÿ 6 ) = 1 (19) for all 𝑣, 𝑀 ∈ 𝑋 and all π‘Ÿ > 0. Using (𝐹2) in the above inequality gives β„±(β„œπ‘Žπ‘£ + 𝑀) Β±β„œ 𝑏 β„±(𝑣 βˆ’β„œ π‘Žπ‘€) = ( β„œπ‘Ž(1Β±β„œπ‘Ž+𝑏) 2 ) [β„±(𝑣 + 𝑀) + β„±(𝑣 βˆ’ 𝑀)] + ( β„œπ‘Ž(β„œπ‘Žβˆ“β„œπ‘) 2 ) [β„±(𝑣 + 𝑀) βˆ’ β„±(𝑣 βˆ’ 𝑀)] +(β„œ2π‘Ž βˆ’ 1)[(β„œπ‘Ž βˆ“ β„œ 𝑏)β„±(𝑣) βˆ“ (β„œπ‘Ž+𝑏 Β± 1)β„±(𝑀)] for all 𝑣, 𝑀 ∈ 𝑋. Hence 𝐢 satisfies the cubic functional equation (1). In order to prove β„±(𝑣) is unique, let β„±β€²(𝑣) be another cubic functional equation satisfying (1) and (5). Hence, 𝒩(π’ž(𝑣) βˆ’ β„±β€²(𝑣), π‘Ÿ) = 𝒩 ( β„±(β„œπ‘Žπ‘˜π‘£) β„œ3π‘Žπ‘˜ βˆ’ β„±β€²(β„œπ‘Žπ‘˜π‘£) β„œ3π‘Žπ‘˜ , π‘Ÿ) β‰₯ π‘šπ‘–π‘›{𝒩 ( β„±(β„œπ‘Žπ‘˜π‘£) β„œ3π‘Žπ‘˜ βˆ’ β„±(β„œπ‘Žπ‘˜π‘£) β„œ3π‘Žπ‘˜ , π‘Ÿ 2 ) ,𝒩 ( β„±(β„œπ‘Žπ‘˜π‘£) β„œ3π‘Žπ‘˜ βˆ’ β„±β€²(β„œπ‘Žπ‘˜π‘£) β„œ3π‘Žπ‘˜ , π‘Ÿ 2 )} β‰₯ 𝑁′ (𝔔(β„œπ‘Žπ‘˜π‘’, 0), (β„œ3π‘Žπ‘˜βˆ’π‘‘π‘Ž)π‘Ÿ 2 ) β‰₯ 𝑁′ (𝔔(𝑣, 0), (β„œ3π‘Žπ‘˜βˆ’π‘‘π‘Ž)π‘Ÿ 2π‘‘π‘˜ ) for all 𝑒 ∈ 𝑋 and all π‘Ÿ > 0. Since lim π‘˜β†’βˆž (β„œ3π‘Žπ‘˜βˆ’π‘‘π‘Ž)π‘Ÿ 2π‘‘π‘˜ = ∞, we obtain lim π‘˜β†’βˆž 𝑁′(𝔔(𝑣, 0), (β„œ3π‘Žπ‘˜βˆ’π‘‘π‘Ž)π‘Ÿ 2π‘‘π‘˜ ) = 1. Thus 𝒩(β„±(𝑣) βˆ’ β„±β€²(𝑣), π‘Ÿ) = 1 for all 𝑣 ∈ 𝑋 and all π‘Ÿ > 0, hence β„±(𝑣) = β„±β€²(𝑣). Therefore β„±(𝑣) is unique. Corollary 4.2 Suppose that a function f:X β†’ Y satisfies the inequality 𝑁(𝐷ℱ(𝑣, 𝑀), π‘Ÿ) β‰₯ { 𝑁′(πœ–, π‘Ÿ), 𝑁′(πœ–||𝑣||𝑠 + ||𝑀||𝑠, π‘Ÿ), 𝑠 β‰  3; 𝑁′(πœ–(||𝑣||𝑠||𝑀||𝑠 + {||𝑣||2𝑠 + ||𝑀||2𝑠}), π‘Ÿ), 𝑠 β‰  3 2 ; (20) for all 𝑣, 𝑀 ∈ 𝑋 and all π‘Ÿ > 0, where πœ–, 𝑠 are constants with πœ– > 0. Then there exists a unique cubic mapping 𝐢: 𝑋 β†’ π‘Œ such that Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 32 No. 9s (2025) 487 https://internationalpubls.com 𝑁(π’ž(𝑣) βˆ’ β„±(𝑣), π‘Ÿ) β‰₯ { 𝑁′(πœ–, |β„œ3π‘Ž βˆ’ 1|π‘Ÿ), 𝑁′(πœ–||𝑣||𝑠, |β„œ3π‘Ž βˆ’β„œ π‘Žπ‘ |π‘Ÿ), 𝑁′(πœ–||𝑣||2𝑠, |β„œ3π‘Ž βˆ’β„œ 3π‘Žπ‘ |π‘Ÿ) (21) for all 𝑣 ∈ 𝑋 and all π‘Ÿ > 0. 5 Fixed Point Method of Fuzzy Stability Results For to prove the stability result we define the following: 𝛿𝑖 is a constant such that 𝛿𝑖 = { β„œ π‘Ž 𝑖𝑓 𝑖 = 0, 1 β„œπ‘Ž 𝑖𝑓 𝑖 = 1 and Ξ© is the set such that Ξ© = {𝑔 | 𝑔: 𝑋 β†’ π‘Œ, 𝑔(0) = 0}. Theorem 5.1 Let f:X β†’ Y be a mapping for which there exist a function 𝔔:X2 β†’ Z with the condition lim π‘˜β†’βˆž 𝑁′(𝔔(𝛿𝑖 π‘˜π‘£, 𝛿𝑖 π‘˜π‘€), 𝛿𝑖 3π‘˜π‘Ÿ) = 1 βˆ€ 𝑣, 𝑀 ∈ 𝑋, π‘Ÿ > 0 (22) and satisfying the functional inequality 𝒩(𝐷 β„±(𝑣, 𝑀), π‘Ÿ) β‰₯ 𝑁′(𝔔(𝑣,𝑀), π‘Ÿ) βˆ€ 𝑣, 𝑀 ∈ 𝑋, π‘Ÿ > 0. (23) If there exists 𝐿 = 𝐿(𝑖) such that the function 𝑣 β†’ 𝛽(𝑣) = 𝔔( 𝑣 β„œπ‘Ž , 0), has the property 𝑁′ (𝐿 1 𝛿𝑖 3𝛽(𝛿𝑖𝑣), π‘Ÿ) = 𝑁′(𝛽(𝑣), π‘Ÿ), βˆ€ 𝑣 ∈ 𝑋, π‘Ÿ > 0. (24) Then there exists unique cubic function 𝐢: 𝑋 β†’ π‘Œ satisfying the functional equation (1) and 𝒩(β„±(𝑣) βˆ’ π’ž(𝑣), π‘Ÿ) β‰₯ 𝑁′ ( 𝐿1βˆ’π‘– 1βˆ’πΏ 𝛽(π‘₯), π‘Ÿ) , βˆ€ 𝑣 ∈ 𝑋, π‘Ÿ > 0. (25) Proof. Let 𝑑 be a general metric on Ξ©, such that 𝑑(𝑔, β„Ž) = 𝑖𝑛𝑓{𝐾𝒩(0,∞)|𝒩(𝑔(𝑣) βˆ’ β„Ž(𝑣), π‘Ÿ) β‰₯ 𝑁′(𝐾𝛽(𝑣), π‘Ÿ), 𝑣 ∈ 𝑋, π‘Ÿ > 0}. It is easy to see that (Ξ©, 𝑑) is complete. Define 𝑇:Ξ© β†’ Ξ© by 𝑇𝑔(π‘₯) = 1 𝛿𝑖 3𝑔(𝛿𝑖𝑣), for all 𝑣 ∈ 𝑋. For 𝑔, β„Ž ∈ Ξ©, we have 𝑑(𝑔, β„Ž) ≀ 𝐾 β‡’ 𝒩(𝑔(𝑣) βˆ’ β„Ž(𝑣), π‘Ÿ) β‰₯ 𝑁′(𝐾𝛽(𝑣), π‘Ÿ) β‡’ 𝒩 ( 𝑔(𝛿𝑖𝑣) 𝛿𝑖 3 βˆ’ β„Ž(𝛿𝑖𝑣) 𝛿𝑖 3 , π‘Ÿ) β‰₯ 𝑁′ ( 𝐾 𝛿𝑖 3𝛽(𝛿𝑖𝑣), π‘Ÿ) Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 32 No. 9s (2025) 488 https://internationalpubls.com β‡’ 𝒩(𝑇𝑔(𝑣) βˆ’ π‘‡β„Ž(𝑣), π‘Ÿ) β‰₯ 𝑁′(𝐾𝐿𝛽(𝑣), π‘Ÿ) β‡’ 𝑑(𝑇𝑔(𝑣), π‘‡β„Ž(𝑣)) ≀ 𝐾𝐿 β‡’ 𝑑(𝑇𝑔, π‘‡β„Ž) ≀ 𝐿𝑑(𝑔, β„Ž) (26) for all 𝑔, β„Ž ∈ Ξ©. There fore 𝑇 is strictly contractive mapping on Ξ© with Lipschitz constant 𝐿. Replacing (𝑣, 𝑀) by (𝑣, 0) in (23), we get 𝒩(β„±(β„œπ‘Žπ‘£) βˆ’β„œ 3π‘Ž β„±(𝑣), π‘Ÿ) β‰₯ 𝑁′(𝔔(𝑣, 0), π‘Ÿ). (27) for all 𝑣 ∈ 𝑋, π‘Ÿ > 0. Using (F3) in (27), we arrive 𝒩( β„±(β„œπ‘Žπ‘£) β„œ3π‘Ž βˆ’ β„±(𝑣), π‘Ÿ) β‰₯ 𝑁′ ( 1 β„œ3π‘Žπ””(𝑣, 0), π‘Ÿ) (28) for all 𝑣 ∈ 𝑋, π‘Ÿ > 0 with the help of (24) when 𝑖 = 0, it follows from (28), we get β‡’ 𝒩 ( β„±(β„œπ‘Žπ‘£) β„œ3π‘Ž βˆ’ β„±(𝑣), π‘Ÿ) β‰₯ 𝑁′(𝐿𝛽(𝑣), π‘Ÿ) β‡’ 𝑑(𝑇𝐢, 𝐢) ≀ 𝐿 = 𝐿1 = 𝐿1βˆ’π‘– (29) Replacing 𝑣 by 𝑣 β„œπ‘Ž in (27), we obtain 𝒩(β„±(𝑣) βˆ’β„œ 3π‘Ž β„± ( 𝑣 β„œπ‘Ž) , π‘Ÿ) β‰₯ 𝑁′ (𝔔( 𝑣 β„œπ‘Ž , 0) , π‘Ÿ) (30) for all 𝑣 ∈ 𝑋, π‘Ÿ > 0 with the help of (24) when 𝑖 = 1, it follows from (30) we get β‡’ 𝒩 (β„±(𝑣) βˆ’β„œ 3π‘Ž β„±( 𝑣 β„œπ‘Ž) , π‘Ÿ) β‰₯ 𝑁′(𝛽(𝑣), π‘Ÿ) β‡’ 𝑑(𝐢, 𝑇𝐢) ≀ 1 = 𝐿0 = 𝐿1βˆ’π‘– (31) Then from (29) and (31) we can conclude, 𝑑(𝐢, 𝑇𝐢) ≀ 𝐿1βˆ’π‘– < ∞ Now from the fixed point alternative in both cases, it follows that there exists a fixed point 𝐢 of 𝑇 in Ξ© such that π’ž(𝑣) = 𝑁 βˆ’ lim π‘˜β†’βˆž β„±(π‘›π‘˜π‘£) 𝑛3π‘˜ , βˆ€π‘£ ∈ 𝑋, π‘Ÿ > 0. (32) Replacing (𝑣,𝑀) by (𝛿𝑖𝑣, 𝛿𝑖𝑀) in (23), we arrive 𝒩( 1 𝛿𝑖 3π‘˜π·β„±(𝛿𝑖𝑣, 𝛿𝑖𝑀), π‘Ÿ) β‰₯ 𝑁′(𝔔(𝛿𝑖𝑣, 𝛿𝑖𝑀), 𝛿𝑖 3π‘˜π‘Ÿ) (33) for all π‘Ÿ > 0 and all 𝑣,𝑀 ∈ 𝑋, we can prove the function, 𝐢: 𝑋 β†’ π‘Œ satisfies the functional equation (1). By fixed point alternative, since 𝐢 is unique fixed point of 𝑇 in the set Ξ” = {𝑓 ∈ Ξ©|𝑑(𝑓, 𝐢) < ∞}, therefore 𝐢 is a uniqe function such that Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 32 No. 9s (2025) 489 https://internationalpubls.com 𝒩(β„±(𝑣) βˆ’ β„±(𝑣), π‘Ÿ) β‰₯ 𝑁′(𝐾𝛽(𝑣), π‘Ÿ) (34) for all 𝑣 ∈ 𝑋, π‘Ÿ > 0 and 𝐾 > 0. Again using the fixed point alternative, we obtain 𝑑(𝐢, π’ž) ≀ 1 1βˆ’πΏ 𝑑(𝐢, 𝑇𝐢) β‡’ 𝑑(𝐢, π’ž) ≀ 𝐿1βˆ’π‘– 1βˆ’πΏ β‡’ 𝒩(β„±(𝑣) βˆ’ π’ž(𝑣), π‘Ÿ) β‰₯ 𝑁′ ( 𝐿1βˆ’π‘– 1βˆ’πΏ 𝛽(𝑣), π‘Ÿ), (35) for all 𝑣 ∈ 𝑋 and π‘Ÿ > 0. Corollary 5.2 Suppose that a function f:X β†’ Y satisfies theinequality 𝑁(𝐷ℱ(𝑣, 𝑀), π‘Ÿ) β‰₯ { 𝑁′(πœ–, π‘Ÿ), 𝑁′(πœ–||𝑣||𝑠 + ||𝑀||𝑠, π‘Ÿ), 𝑠 β‰  3; 𝑁′(πœ–(||𝑣||𝑠||𝑀||𝑠 + {||𝑣||2𝑠 + ||𝑀||2𝑠}), π‘Ÿ), 𝑠 β‰  3 2 ; (36) for all 𝑣, 𝑀 ∈ 𝑋 and π‘Ÿ > 0, where πœ–, 𝑠 are constants with πœ– > 0. Then there exists a unique cubic mapping 𝐢: 𝑋 β†’ π‘Œ such that 𝒩(π’ž(𝑣) βˆ’ β„±(𝑣), π‘Ÿ) β‰₯ { 𝑁′(πœ–, |β„œ3π‘Ž βˆ’ 1|π‘Ÿ), 𝑁′(πœ–||𝑣||𝑠, |β„œ3π‘Ž βˆ’β„œ π‘Žπ‘ |π‘Ÿ), 𝑁′(πœ–||𝑣||2𝑠, |β„œ3π‘Ž βˆ’β„œ 3π‘Žπ‘ |π‘Ÿ), (37) for all 𝑣 ∈ 𝑋 and all π‘Ÿ > 0. Proof. Setting 𝔔(𝑣,𝑀) = { 𝑁′(πœ–, π‘Ÿ), 𝑁′(πœ–||𝑣||𝑠 + ||𝑀||𝑠, π‘Ÿ), 𝑁′(πœ–(||𝑣||𝑠||𝑀||𝑠 + {||𝑣||2𝑠 + ||𝑀||2𝑠}), π‘Ÿ) for all 𝑣, 𝑀 ∈ 𝑋. Then, 𝑁′(𝔔(𝛿𝑖 π‘˜π‘£, 0), 𝛿𝑖 3π‘˜π‘Ÿ) = { 𝑁′(πœ–, 𝛿𝑖 3π‘˜π‘Ÿ) 𝑁′(πœ–||𝑣||𝑠, 𝛿𝑖 (3βˆ’π‘ )π‘˜ π‘Ÿ) 𝑁′(πœ–||𝑣||2𝑠, 𝛿𝑖 (3βˆ’2𝑠)π‘˜ π‘Ÿ) = { β†’ 1 π‘Žπ‘  π‘˜ β†’ ∞, β†’ 1 π‘Žπ‘  π‘˜ β†’ ∞, β†’ 1 π‘Žπ‘  π‘˜ β†’ ∞. Thus, (22) is holds. But we have 𝛽(𝑣) = 𝔔( 𝑣 β„œπ‘Ž , 0) has the property 𝑁′ (𝐿 1 𝛿𝑖 3𝛽(𝛿𝑖𝑣), π‘Ÿ) β‰₯ 𝑁′(𝛽(𝑣), π‘Ÿ) βˆ€ 𝑣 ∈ 𝑋, π‘Ÿ > 0. Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 32 No. 9s (2025) 490 https://internationalpubls.com Hence 𝑁′(𝛽(𝑣), π‘Ÿ) = 𝑁′ (𝔔( π‘₯ β„œπ‘Ž , 0) , π‘Ÿ) = { 𝑁′(πœ–, π‘Ÿ), 𝑁′ ( πœ– β„œπ‘Žπ‘  ||𝑣|| 𝑠 , π‘Ÿ) , 𝑁′ ( πœ– β„œ2π‘Žπ‘  ||𝑣|| 2𝑠, π‘Ÿ) . Now, 𝑁′ ( 1 𝛿𝑖 3𝛽(𝛿𝑖𝑣), π‘Ÿ) = { 𝑁′ ( πœ– 𝛿𝑖 3 , π‘Ÿ) , 𝑁′ ( πœ– 𝛿𝑖 3 ( 1 β„œπ‘Žπ‘ ) ||𝛿𝑖𝑣|| 𝑠, π‘Ÿ) , 𝑁′ ( πœ– 𝛿𝑖 3 ( 1 β„œ2π‘Žπ‘ ) ||𝛿𝑖𝑣|| 2𝑠, π‘Ÿ) = { 𝑁′(𝛿𝑖 βˆ’3𝛽(𝑣), π‘Ÿ), 𝑁′(𝛿𝑖 π‘ βˆ’3𝛽(𝑣), π‘Ÿ), 𝑁′(𝛿𝑖 2π‘ βˆ’3𝛽(𝑣), π‘Ÿ). Now from (25), we prove the following cases for conditions (𝑖) and (𝑖𝑖). Case:1 𝐿 = β„œ βˆ’3π‘Ž for 𝑠 = 0 if 𝑖 = 0 𝒩(π’ž(𝑣) βˆ’ β„±(𝑣), π‘Ÿ) β‰₯ 𝑁′ ( β„œβˆ’3π‘Ž 1βˆ’β„œβˆ’3π‘Ž 𝛽(𝑣), π‘Ÿ) = 𝑁′ ( πœ– (β„œ3π‘Žβˆ’1) , π‘Ÿ) = 𝑁′(πœ–, (β„œ3π‘Ž βˆ’ 1)π‘Ÿ). Case:2 𝐿 = β„œ 3π‘Ž for 𝑠 = 0 if 𝑖 = 1 𝒩(π’ž(𝑣) βˆ’ β„±(𝑣), π‘Ÿ) β‰₯ 𝑁′ ( 1 1βˆ’β„œ3π‘Žπ›½(𝑣), π‘Ÿ) = 𝑁′ ( πœ– (1βˆ’β„œ3π‘Ž) , π‘Ÿ) = 𝑁′(πœ–, (1βˆ’β„œ 3π‘Ž)π‘Ÿ). Case:3 𝐿 = β„œ π‘Ž(π‘ βˆ’3) for 𝑠 > 3 if 𝑖 = 0 𝒩(π’ž(𝑣) βˆ’ β„±(𝑣), π‘Ÿ) β‰₯ 𝑁′ ( β„œπ‘Ž(π‘ βˆ’3) 1βˆ’β„œπ‘Ž(π‘ βˆ’3)𝛽(𝑣), π‘Ÿ) = 𝑁′ ( πœ– (β„œ3π‘Žβˆ’β„œπ‘Žπ‘ ) ||𝑣||𝑠, π‘Ÿ) = 𝑁′(πœ–||𝑣||𝑠, (β„œ3π‘Ž βˆ’β„œ π‘Žπ‘ )π‘Ÿ). Case:4 𝐿 = β„œ π‘Ž(3βˆ’π‘ ) for 𝑠 < 3 if 𝑖 = 1 𝒩(π’ž(𝑣) βˆ’ β„±(𝑣), π‘Ÿ) β‰₯ 𝑁′ ( 1 1βˆ’β„œπ‘Ž(3βˆ’π‘ ) 𝛽(𝑣), π‘Ÿ) = 𝑁′ ( πœ– (β„œπ‘Žπ‘ βˆ’β„œ3π‘Ž) ||𝑣||𝑠, π‘Ÿ) = 𝑁′(πœ–||𝑣||𝑠, (β„œπ‘Žπ‘  βˆ’β„œ 3π‘Ž)π‘Ÿ). Case:5 𝐿 = β„œ π‘Ž(2π‘ βˆ’3) for 𝑠 > 3 2 if 𝑖 = 0 𝒩(π’ž(𝑣) βˆ’ β„±(𝑣), π‘Ÿ) β‰₯ 𝑁′ ( β„œπ‘Ž(2π‘ βˆ’3) 1βˆ’β„œπ‘Ž(2π‘ βˆ’3)𝛽(𝑣), π‘Ÿ) = 𝑁′ ( πœ– (β„œ3π‘Žβˆ’β„œ2π‘Žπ‘ ) ||𝑣||𝑠, π‘Ÿ) = 𝑁′(πœ–||𝑣||𝑠 , (β„œ3π‘Ž βˆ’β„œ 2π‘Žπ‘ )π‘Ÿ). Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 32 No. 9s (2025) 491 https://internationalpubls.com Case:6 𝐿 = β„œ π‘Ž(3βˆ’2𝑠) for 𝑠 < 3 2 if 𝑖 = 1 𝒩(π’ž(𝑣) βˆ’ β„±(𝑣), π‘Ÿ) β‰₯ 𝑁′ ( 1 1βˆ’β„œπ‘Ž(3βˆ’2𝑠)𝛽(𝑣), π‘Ÿ) = 𝑁′ ( πœ– (β„œ2π‘Žπ‘ βˆ’β„œ3π‘Ž) ||𝑣||𝑠, π‘Ÿ) = 𝑁′(πœ–||𝑣||𝑠 , (β„œ2π‘Žπ‘  βˆ’β„œ 3π‘Ž)π‘Ÿ). 6 Conclusion In this paper, we have established novel stability results for generalized alternate cubic functional equations using the classical method for Banach spaces and both direct and fixed point approaches for fuzzy normed spaces. The classical method provided a clear pathway to demonstrate Hyers-Ulam stability in Banach spaces, revealing how small deviations affect the functional equation’s solutions. In contrast, the fuzzy normed space framework, enriched by direct and fixed point methods, allowed for a more nuanced stability analysis, accommodating uncertainties and imprecisions intrinsic to fuzzy systems. Our results highlight the effectiveness of combining classical techniques with fixed point theory in analyzing functional equations under different normed environments. The comparative analysis between deterministic Banach spaces and the more flexible fuzzy normed spaces underscores the adaptability of the generalized alternate cubic functional equation across various mathematical contexts. These findings offer significant contributions to the stability theory of functional equations and lay the groundwork for future applications in both pure and applied mathematical fields, particularly in scenarios involving uncertain or fuzzy data. Conflict of interest. The authors declare that they have no competing interests. References [1] S.M. Ulam, Problems in Modern Mathematics, Science Editions, Wiley, NewYork, 1964. [2] D.H. Hyers, On the stability of the linear functional equation, Proc.Nat. Acad.Sci.,U.S.A.,27, 1941, 222-224. [3] Th.M. Rassias, On the stability of the linear mapping in Banach spaces, Proc.Amer.Math.Soc., 72, 1978, 297-300. [4] T. Aoki, On the stability of the linear transformation in Banach spaces, J. Math. Soc. Japan, 2 ,1950, 64-66. [5] P. Gavruta, A generalization of the Hyers-Ulam-Rassias stability of approximately additive mappings , J. Math. Anal. Appl., 184 1994, 431-436. [6] J.M. Rassias, On approximately of approximately linear mappings by linear mappings, J. Funct. Anal. USA, 46, 1982, 126-130. [7] D.H. Hyers, G. Isac, Th.M. Rassias, Stability of Fun Eq in several variables, Birkhauser, Basel, 1998. [8] J. Aczel and J. Dhombres, Fun Eq in Several Variables, Cambridge Univ,Press, 1989. [9] S. Czerwik, Fun Eq and Inequalities in Several Variables, World Scientific, River Edge, NJ, 2002. [10] S.M. Jung, Hyers-Ulam-Rassias Stability of Fun Eq in Mathematical Analysis, Hadronic Press, Palm Harbor,2001. [11] TunΓ§, Osman. New Results on the Ulam–Hyers–Mittag–Leffler Stability of Caputo Fractional-Order Delay Differential Equations." Mathematics 12, no. 9 2024: 1342. [12] Selvam, A., Sabarinathan, S., Sooppy Nisar, K., Ravichandran, C. and Senthil Kumar, B.V. Results on Ulam‐type stability of linear differential equation with integral transform. Mathematical Methods in the Applied Sciences, 47(4), 2024.pp.2311-2323. [13] TunΓ§, Osman, Cemil TunΓ§, and Jen-Chih Yao. New results on Ulam stabilities of nlinear integral equations." Mathematics 12, no. 5 2024: 682. Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 32 No. 9s (2025) 492 https://internationalpubls.com [14] TunΓ§, Osman, and Cemil TunΓ§. On Ulam stabilities of delay Hammerstein integral equation Symmetry 15, no. 9 2023: 1736. [15] Novac, Adela, Diana Otrocol, and Dorian Popa. On ulam stability of a partial differential operator in Banach spaces." Mathematics 11, no. 11 2023: 2488. [16] GarcΓ­a, Gonzalo, and Gaspar Mora. The degree of nondensifiability of linear bounded operators and its applications." Applied General Topology 25, no. 1 2024: 213-228. [17] Zada, Akbar, Peiguang Wang, Dhaou Lassoued, and Tongxing Li. Connections between Hyers-Ulam stability and uniform exponential stability of 2-periodic linear nonautonomoussystems Advances in Difference Equations 2017 (2017): 1-7. [18] Kumar, Bhim, and Muslim Malik. Existence, controllability and Hyers–Ulam stability of a hybrid neutral switched system with impulsive effects." International Journal of Systems Science 55, no. 3 2024: 517-534. [19] Almarri, Barakah, Xingtao Wang, and Ahmed M. Elshenhab. Controllability and Hyers–Ulam stability of fractional systems with pure delay." Fractal and Fractional 6, no. 10 ,2022: 611. [20] Agarwal, R. P., Cho, Y. J., Saadati, R., Wang, S, Nonlinear Fuzzy stability of cubic functional equations. Journal of Inequalities and Applications, 2012, 1-19. [21] Saadati, R., Cho, Y. J., Rassias, J. M, Nonlinear L-Fuzzy Stability of K-Cubic Functional Equation. Filomat, 29(5), (2015) 1137-1148. [22] Mohiuddine, S.A. and Alotaibi, A., Fuzzy stability of a cubic functional equation via fixed point technique. Advances in Difference Equations, 2012, pp.1-8. [23] Lee, Yang-Hi, and Soon-Mo Jung. Fuzzy stability of the cubic and quadratic functional equation. Appl. Math. Sci.(Ruse) 10 (2016): 2671-2686. [24] Javadi, S. and Rassias, J.M., Stability of general cubic mapping in fuzzy normed spaces. Analele ştiinΕ£ifice ale UniversitΔƒΕ£ii" Ovidius" ConstanΕ£a. Seria MatematicΔƒ, 2012,20(1), pp.129-150. [25] Pasupathi, A.; Konsalraj, J.; Fatima, N.; Velusamy, V.; Mlaiki, N.; Souayah, N. Direct and fixed-point stability– instability of additive functional equation in Banach and quasi-beta normed spaces. Symmetry 2022, 14, 1700. [26] Agilan, P.; Julietraja, K.; Mlaiki, N.; Mukheimer, A. Intuitionistic fuzzy stability of an Euler–Lagrange symmetry additive functional equation via direct and fixed point technique (FPT). Symmetry 2022, 14, 2454. [27] Agilan, P.; Almazah, M.A.A.; Julietraja, K.; Alsinai, A. Classical and fixed point approach to the stability analysis of a bilateral symmetric additive functional equation in fuzzy and random normed spaces. Mathematics 2023, 11, 681. [28] Agilan, P.; Julietraja.; K Almazah, M.A.A.; Alsinai, A. Stability analysis of a new class of series type additive functional equation in Banach spaces: direct and fixed point techniques Mathematics 2023, 11, 887. doi.org/10.3390/math11040887. [29] Aloqaily, Ahmad, P. Agilan, K. Julietraja, S. Annadurai, and Nabil Mlaiki. A novel stability analysis of functional equation in neutrosophic normed spaces. Boundary Value Problems, 2024, no. 1 (2024), 47. [30] Agilan, P., Julietraja, K., Kanimozhi, B. and Alsinai, A., Hyers Stability of AQC Functional Equation. Dynamics of Continuous, Discrete and Impulsive Systems Series B: Applications and Algorithms, 2024, 31, 63-75. [31] Agilan, P., Julietraja, K, Sarah Aljohani, Nabil Mlaiki, Generalised Ulam-Hyers Stability Analysis for System of Additive Functional Equation in Fuzzy and Random Normed Spaces:Direct and Fixed Point Approach. Int. J. Anal. Appl., 22 2024, 201. [32] Agilan.P, Vijayan.V, Sophia.M, Ganapathy.G., Exploring Advanced Stability of Higher-Order Functional Equations in Neutrosophic Normed Spaces via Hyers-Ulam Methodologies, Communications on Applied Nonlinear Analysis, 2025, Vol 32 No. 7s (2025), 806-822.