Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 32 No. 7s (2025) 806 https://internationalpubls.com Exploring Advanced Stability of Higher-Order Functional Equations in Neutrosophic Normed Spaces via Hyers-Ulam Methodologies P. Agilan πŸβˆ—, V. Vijayan 𝟐, M. Sophia πŸ‘ G. Ganapathy 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 Mathematical Studies, SIMATS Engineering, Chennai - 600 077, TamilNadu, India. E-mail: sophia.raj2005@gmail.com. 4Department of Mathematics, R.M.D Engineering College, Kavaraipettai - 601 206, TamilNadu, India. E-mail: barathganagandhi@gmail.com. *Corresponding Author: agilram@gmail.com. Article History: Received: 28-10-2024 Revised:12-11-2024 Accepted:19-12-2024 Abstract: In this article, focuses on examining the stability of higher-order functional equations within the framework of neutrosophic normed spaces, which incorporate elements of uncertainty and indeterminacy. By utilizing the Hyers-Ulam method, the study investigates how small perturbations in the functional equations impact their solutions. The research extends classical stability theories, such as Hyers-Ulam stability, into the neutrosophic normed space context, providing a broader understanding of how functional equations behave under uncertainty. The findings offer significant contributions to the field of functional equations and neutrosophic mathematics, opening up new pathways for applications in areas that require the handling of imprecise data. Keywords: Duodecic, Tridecic Functional Equations, Generalized Hyers - Ulam Stability. 1. Introduction The Ulam-Hyers stability, introduced by Stanislaw Ulam and Donald Hyers in the 1940s and 1941s respectively [1, 2], is a fundamental concept in functional analysis and mathematical stability theory. It examines the behavior of solutions to functional equations when subjected to small perturbations. Functional equations, which establish specific relationships between the values of functions, are prevalent across various fields, including mathematics, physics, engineering, and economics. The stability of such equations focuses on how slight changes in inputs or parameters influence their solutions. The Ulam-Hyers stability theory establishes conditions under which solutions to functional equations remain close to the original solutions despite small perturbations. It addresses the existence and uniqueness of solutions while analyzing their dependence on initial conditions and parameters. This concept is significant due to its extensive applicability and relevance to practical problems. It enhances the robustness and reliability of mathematical models and numerical algorithms, offering valuable insights into the behavior and predictability of dynamic systems. Ongoing research continues to expand the theory, uncovering connections with other mathematical domains and exploring its potential applications in various disciplines ([3, 4, 5, 6, 7]). Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 32 No. 7s (2025) 807 https://internationalpubls.com Lotfi A. Zadeh, a renowned mathematician and computer scientist, introduced the concept of fuzzy sets in his seminal 1965 paper titled "Fuzzy Sets" [8]. Zadeh developed this concept to overcome the constraints of classical set theory, which depends on sharp, well-defined boundaries for membership. Building upon this foundation, Atanassov proposed the notion of intuitionistic fuzzy sets (IFS) in 1983 [9, 10]. IFS extended fuzzy sets to incorporate a more nuanced framework for addressing uncertainty, vagueness, and hesitation. Further advancing these ideas, neutrosophic sets were introduced as an extension of classical set theory, providing a robust mechanism for managing indeterminacy, uncertainty, and incomplete information [11, 12]. Fuzzy normed spaces (FNS) are mathematical structures that generalize classical normed spaces by incorporating fuzzy numbers. Katsaras first introduced the concept of FNS, defining them as vector spaces equipped with a fuzzy norm, where the norm values are represented as fuzzy numbers rather than real numbers [13]. In 2006, the concept of intuitionistic fuzzy normed spaces (IFNS) was introduced [14]. IFNS merge elements of fuzzy mathematics, intuitionistic fuzzy sets, and normed spaces, offering a flexible framework for addressing uncertainty and imprecision in mathematical modeling and analysis. Neutrosophic normed linear spaces [15, 16] extend neutrosophic set theory to linear algebraic structures, enabling a more comprehensive representation of uncertainty within vector spaces. Neutrosophic concepts have been widely applied in various branches of mathematics, including groups and subgroups [17], vector spaces [18], ring homomorphisms [19, 20], linear transformations [21], number theory [22], graph theory [23], measure theory, integral theory, and probability theory [24]. Additionally, neutrosophic normed linear spaces find practical applications in diverse fields such as decision-making, control systems, optimization, image processing, pattern recognition, medical diagnosis, finance and risk management, information retrieval, and artificial intelligence [25, 26, 27, 28, 29, 30, 31]. Agilan et al. have introduced novel functional equations and established the Hyers- Ulam stability of these equations across various normed spaces ([32, 33, 34, 35, 36, 37, 38]). This article introduces a novel mixed duodecic-tridecic functional equation and investigates its Ulam- Hyers stability within neutrosophic normed linear spaces (NNLS). Classical approaches are employed for the stability analysis in the newly proposed equation. Given the unique properties of neutrosophic normed spaces and their broad potential applications, the stability analysis of this equation is of considerable importance. Notably, this study marks the first instance in the literature where the stability of a functional equation is examined within the framework of neutrosophic normed spaces, underscoring the distinctiveness and significance of the research. This study sets out to accomplish the following key objectives: (i) To expand and advance the existing research on neutrosophic normed linear spaces. (ii) To establish the uniqueness of solutions for the newly introduced Higher-order functional equation. (ii) To investigate the Hyers-Ulam stability of the proposed equation within neutrosophic normed linear spaces, employing the direct method. 𝔉(11ℨ) = 1,88,30,57,02,60,326𝔉(ℨ) βˆ’ 1,56,92,14,18,83,605𝔉(βˆ’β„¨) (1) The above equation having solution 𝔉(ℨ) = π’œ1 ℨ 13 +π’œ2 ℨ 12. Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 32 No. 7s (2025) 808 https://internationalpubls.com Remark 1.1 Let us take an odd mapping 𝔉:𝒳𝑀 β†’ 𝒴𝑀 which satisfies the FE (1) then it is tridecic 𝔉(11ℨ) = 96,88,90,10,407𝔉(ℨ) = 1113𝔉(ℨ) for all ℨ ∈ 𝒳𝑀 Remark 1.2 Let us take an even mapping 𝔉:𝒳𝑀 β†’ 𝒴𝑀 which satisfies the FE (1) then it is duodecic 𝔉(11ℨ) = 13,84,12,87,201𝔉(ℨ) = 1112𝔉(ℨ) for all ℨ ∈ 𝒳𝑀 2 Definition of Neutrosophic normed spaces Definition 2.1 The Seven-tuple (𝔸,π’œ1,π’œ2, π’œ3 βˆ—,β‹„,⊘) is said to be a neutrosophic normed space (for short, NNS) if 𝔸 is a vector space, βˆ— is a continuous πœ…-norm, β‹„ and ⊘ is a continuous πœ… βˆ’ conorm, and π’œ1, π’œ2,π’œ3 are fuzzy sets on 𝔸 Γ— (0,∞) satisfying the following conditions. For every 𝑝, π‘ž ∈ 𝔸 and 𝑠, πœ… > 0, (πœ™1) π’œ1(𝑝, πœ…) + π’œ2(𝑝, πœ…) +π’œ3(𝑝, πœ…) ≀ 3, (πœ™2) 0 ≀ π’œ1(𝑝, πœ…) ≀ 1,0 ≀ π’œ2(𝑝, πœ…) ≀ 1,0 ≀ π’œ3(𝑝, πœ…) ≀ 1, (πœ™3) π’œ1(𝑝, πœ…) > 0, (πœ™4) π’œ1(𝑝, πœ…) = 1, if and only if 𝑝 = 0. (πœ™5) π’œ1(𝛼𝑝, πœ…) = π’œ1 (𝑝, πœ… |𝛼| ) for each 𝛼 β‰  0, (πœ™6) π’œ1(𝑝, πœ…) βˆ— π’œ1(π‘ž, 𝑠) ≀ π’œ1(𝑝 + π‘ž, πœ… + 𝑠), (πœ™7) π’œ1(𝑝,β‹…): (0,∞) β†’ [0,1] is continuous, (πœ™8) lim πœ…β†’βˆž π’œ1(𝑝, πœ…) = 1 and lim πœ…β†’0 π’œ1(𝑝, πœ…) = 0, (πœ™9) π’œ2(𝑝, πœ…) < 1, (πœ™10) π’œ2(𝑝, πœ…) = 0, if and only if 𝑝 = 0. (πœ™11) π’œ2(𝛼𝑝, πœ…) = π’œ2 (𝑝, πœ… |𝛼| ) for each 𝛼 β‰  0, (πœ™12) π’œ2(𝑝, πœ…) β‹„ π’œ2(π‘ž, 𝑠) β‰₯ π’œ2(𝑝 + π‘ž, πœ… + 𝑠), (πœ™13) π’œ2(𝑝,β‹…): (0,∞) β†’ [0,1] is continuous, (πœ™14) lim πœ…β†’βˆž π’œ2(𝑝, πœ…) = 0 and lim πœ…β†’0 π’œ2(𝑝, πœ…) = 1 (πœ™15) π’œ3(𝑝, πœ…) < 1, (πœ™16) π’œ3(𝑝, πœ…) = 0, if and only if 𝑝 = 0. (πœ™17) π’œ3(𝛼𝑝, πœ…) = π’œ3 (𝑝, πœ… |𝛼| ) for each 𝛼 β‰  0, (πœ™18) π’œ3(𝑝, πœ…) βŠ˜π’œ3(π‘ž, 𝑠) β‰₯ π’œ3(𝑝 + π‘ž, πœ… + 𝑠), Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 32 No. 7s (2025) 809 https://internationalpubls.com (πœ™19) π’œ3(𝑝,β‹…): (0,∞) β†’ [0,1] is continuous, (πœ™20) lim πœ…β†’βˆž π’œ3(𝑝, πœ…) = 0 and lim πœ…β†’0 π’œ3(𝑝, πœ…) = 1. 3 Stability Results in neutrosophic normed space: Hyers Classical Direct Method Theorem 3.1 Assume that 𝒳𝑀 is a LS, (π’΅π‘š ,π’œ1β€²,π’œ2β€², π’œ3β€²) is a NNS and (𝒴𝑀 ,π’œ1,π’œ2, π’œ3) an NBS . Let Γ°:𝒳𝑀 ⟢ π’΅π‘š be a function such that for some 0 < ( 𝔙 1113 ) 𝐹 < 1 with 𝐹 ∈ {1, βˆ’1}. π’œ1β€²(Γ°(11 𝑛𝐹ℨ),β„³) β‰₯ π’œ1β€²(𝔙 𝑛𝐹ð(ℨ),β„³) π’œ2β€²(Γ°(11 𝑛𝐹ℨ),β„³) ≀ π’œ2β€²(𝔙 𝑛𝐹ð(ℨ),β„³) π’œ3β€²(Γ°(11 𝑛𝐹ℨ),β„³) ≀ π’œ3β€²(𝔙 𝑛𝐹ð(ℨ),β„³)} (1) for all ℨ ∈ 𝒳𝑀 and all β„³ > 0 and lim π‘›β†’βˆž π’œ1β€²(Γ°(11 𝐹𝑛ℨ), 1113𝐹𝑛ℳ) = 1 lim π‘›β†’βˆž π’œ2β€²(Γ°(11 𝐹𝑛ℨ), 1113𝐹𝑛ℳ) = 0 lim π‘›β†’βˆž π’œ3β€²(Γ°(11 𝐹𝑛ℨ), 1113𝐹𝑛ℳ) = 0} (2) for all ℨ ∈ 𝒳𝑀 and all β„³ > 0 . Let an odd function 𝔉:𝒳𝑀 βŸΆπ’΄π‘€ satisfying π’œ1(𝔉(11ℨ) βˆ’ 1,88,30,57,02,60,326𝔉(ℨ) + 1,56,92,14,18,83,605𝔉(βˆ’β„¨),β„³) β‰₯ π’œ1β€²(Γ°(ℨ),β„³) π’œ2(𝔉(11ℨ) βˆ’ 1,88,30,57,02,60,326𝔉(ℨ) + 1,56,92,14,18,83,605𝔉(βˆ’β„¨),β„³) ≀ π’œ2β€²(Γ°(ℨ),β„³) π’œ3(𝔉(11ℨ) βˆ’ 1,88,30,57,02,60,326𝔉(ℨ) + 1,56,92,14,18,83,605𝔉(βˆ’β„¨),β„³) ≀ π’œ3β€²(Γ°(ℨ),β„³)} (3) for all ℨ ∈ 𝒳𝑀 and all β„³ > 0. Then there exists a unique tridecic mapping 𝒯:𝒳𝑀 βŸΆπ’΄π‘€ satisfying (1) and π’œ1(𝔉(ℨ) βˆ’ 𝒯(ℨ),β„³) β‰₯ π’œ1β€²(Γ°(ℨ),β„³|1113 βˆ’ 𝔙|) π’œ2(𝔉(ℨ) βˆ’ 𝒯(ℨ),β„³) ≀ π’œ2β€²(Γ°(ℨ),β„³|1113 βˆ’π”™|) π’œ3(𝔉(ℨ) βˆ’ 𝒯(ℨ),β„³) ≀ π’œ3β€²(Γ°(ℨ),β„³|1113 βˆ’π”™|)} (4) for all ℨ ∈ 𝒳𝑀 and all β„³ > 0. Proof. For the first case 𝐹 = 1. Using oddness of 𝔉 in in (3), we obtain Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 32 No. 7s (2025) 810 https://internationalpubls.com π’œ1(𝔉(11ℨ) βˆ’ 11 13𝔉(ℨ),β„³) β‰₯ π’œ1β€²(Γ°(ℨ),β„³) π’œ2(𝔉(11ℨ) βˆ’ 11 13𝔉(ℨ),β„³) ≀ π’œ2β€²(Γ°(ℨ),β„³) π’œ3(𝔉(11ℨ) βˆ’ 11 13𝔉(ℨ),β„³) ≀ π’œ3β€²(Γ°(ℨ),β„³)} (5) for all ℨ ∈ 𝒳𝑀 and all β„³ > 0. Using (NNS5), (NNS11) and (NNS17) in (5), we have π’œ1( 𝔉(11ℨ) 1113 βˆ’ 𝔉(ℨ), β„³ 1113 ) β‰₯ π’œ1β€²(Γ°(ℨ),β„³) π’œ2( 𝔉(11ℨ) 1113 βˆ’ 𝔉(ℨ), β„³ 1113 ) ≀ π’œ2β€²(Γ°(ℨ),β„³) π’œ3( 𝔉(11ℨ) 1113 βˆ’ 𝔉(ℨ), β„³ 1113 ) ≀ π’œ3β€²(Γ°(ℨ),β„³)} (6) for all ℨ ∈ 𝒳𝑀 and all β„³ > 0. Let us take ℨ by 11𝑛ℨ in (6), we arrive π’œ1( 𝔉(11(𝑛+1)ℨ) 1113 βˆ’ 𝔉(11𝑛ℨ), β„³ 1113 ) β‰₯ π’œ1β€²(Γ°(11 𝑛ℨ),β„³) π’œ2( 𝔉(11(𝑛+1)ℨ) 1113 βˆ’ 𝔉(11𝑛ℨ), β„³ 1113 ) ≀ π’œ2β€²(Γ°(11 𝑛ℨ),β„³) π’œ3( 𝔉(11(𝑛+1)ℨ) 1113 βˆ’ 𝔉(11𝑛ℨ), β„³ 1113 ) ≀ π’œ3β€²(Γ°(11 𝑛ℨ),β„³) } (7) for all ℨ ∈ 𝒳𝑀 and all β„³ > 0. It is simple to confirm that (7) and using (1), (NNS5), (NNS11) and (NNS17) that π’œ1( 𝔉(11(𝑛+1)ℨ) 1113(𝑛+1) βˆ’ 𝔉(11𝑛ℨ) 1113𝑛 , β„³ 1113β‹…1113𝑛 ) β‰₯ π’œ1β€² (Γ°(ℨ), β„³ 𝔙𝑛 ) π’œ2( 𝔉(11𝑛+1ℨ) 1113(𝑛+1) βˆ’ 𝔉(11𝑛ℨ) 1113𝑛 , β„³ 1113β‹…1113𝑛 ) ≀ π’œ2β€² (Γ°(ℨ), β„³ 𝔙𝑛 ) π’œ3( 𝔉(11(𝑛+1)ℨ) 1113(𝑛+1) βˆ’ 𝔉(11𝑛ℨ) 1113𝑛 , β„³ 1113β‹…1113𝑛 ) ≀ π’œ3β€² (Γ°(ℨ), β„³ 𝔙𝑛 ) } (8) for all ℨ ∈ 𝒳𝑀 and all β„³ > 0. Swapping β„³ into 𝔙𝑛ℳ in (8), we have π’œ1( 𝔉(11𝑛+1ℨ) 1113(𝑛+1) βˆ’ 𝔉(11𝑛ℨ) 1113𝑛 , ℳ⋅𝔙𝑛 1113β‹…1113𝑛 ) β‰₯ π’œ1β€²(Γ°(ℨ),β„³) π’œ2( 𝔉(11𝑛+1ℨ) 1113(𝑛+1) βˆ’ 𝔉(11𝑛ℨ) 1113𝑛 , ℳ⋅𝔙𝑛 1113β‹…1113𝑛 ) ≀ π’œ2β€²(Γ°(ℨ),β„³) π’œ3( 𝔉(11𝑛+1ℨ) 1113(𝑛+1) βˆ’ 𝔉(11𝑛ℨ) 1113𝑛 , ℳ⋅𝔙𝑛 1113β‹…1113𝑛 ) ≀ π’œ3β€²(Γ°(ℨ),β„³)} (9) Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 32 No. 7s (2025) 811 https://internationalpubls.com for all ℨ ∈ 𝒳𝑀 and all β„³ > 0. It is simple to observe that 𝔉(11𝑛ℨ) 1113𝑛 βˆ’ 𝔉(ℨ) = βˆ‘π‘›βˆ’1𝑖=0 𝔉(11𝑖+1ℨ) 1113(𝑖+1) βˆ’ 𝔉(11𝑖ℨ) 1113𝑖 (10) for all ℨ ∈ 𝒳𝑀. It follows from (9) and (10), we get π’œ1 ( 𝔉(11𝑛ℨ) 1113𝑛 βˆ’ 𝔉(ℨ),βˆ‘π‘›βˆ’1𝑖=0 𝔙𝑖 β„³ 1113β‹…1113𝑖 ) = π’œ1 (βˆ‘ π‘›βˆ’1 𝑖=0 𝔉(11𝑖+1ℨ) 1113(𝑖+1) βˆ’ 𝔉(11𝑖ℨ) 1113𝑖 , βˆ‘π‘›βˆ’1𝑖=0 𝔙𝑖 β„³ 1113β‹…1113𝑖 ) π’œ2 ( 𝔉(11𝑛ℨ) 1113𝑛 βˆ’ 𝔉(ℨ),βˆ‘π‘›βˆ’1𝑖=0 𝔙𝑖 β„³ 1113β‹…1113𝑖 ) = π’œ2 (βˆ‘ π‘›βˆ’1 𝑖=0 𝔉(11𝑖+1ℨ) 1113(𝑖+1) βˆ’ 𝔉(11𝑖ℨ) 11𝑖 , βˆ‘π‘›βˆ’1𝑖=0 𝔙𝑖 β„³ 1113β‹…1113𝑖 ) π’œ3 ( 𝔉(11𝑛ℨ) 1113𝑛 βˆ’ 𝔉(ℨ),βˆ‘π‘›βˆ’1𝑖=0 𝔙𝑖 β„³ 1113β‹…1113𝑖 ) = π’œ3 (βˆ‘ π‘›βˆ’1 𝑖=0 𝔉(11𝑖+1ℨ) 1113(𝑖+1) βˆ’ 𝔉(11𝑖ℨ) 1113𝑖 , βˆ‘π‘›βˆ’1𝑖=0 𝔙𝑖 β„³ 1113β‹…1113𝑖 ) } (11) for all ℨ ∈ 𝒳𝑀 and all β„³ > 0. Using (NNS5), (NNS11) and (NNS17) in (11), we have π’œ1 ( 𝔉(11𝑛ℨ) 1113𝑛 βˆ’ 𝔉(ℨ), βˆ‘π‘›βˆ’1𝑖=0 𝔙𝑖ℳ 1113β‹…1113𝑖 ) β‰₯ βˆπ‘›βˆ’1 𝑖=0 π’œ1 ( 𝔉(11𝑖+1ℨ) 1113(𝑖+1) βˆ’ 𝔉(11𝑖ℨ) 1113𝑖 , 𝔙𝑖 β„³ 1113β‹…1113𝑖 ) π’œ2 ( 𝔉(11𝑛ℨ) 1113𝑛 βˆ’ 𝔉(ℨ), βˆ‘π‘›βˆ’1𝑖=0 𝔙𝑖ℳ 1113β‹…1113𝑖 ) ≀ βˆπ‘›βˆ’1 𝑖=0 π’œ2 ( 𝔉(11𝑖+1ℨ) 1113(𝑖+1) βˆ’ 𝔉(11𝑖ℨ) 1113𝑖 , 𝔙𝑖 β„³ 1113β‹…1113𝑖 ) π’œ3 ( 𝔉(11𝑛ℨ) 1113𝑛 βˆ’ 𝔉(ℨ), βˆ‘π‘›βˆ’1𝑖=0 𝔙𝑖ℳ 1113β‹…1113𝑖 ) ≀ βˆπ‘›βˆ’1 𝑖=0 π’œ3 ( 𝔉(11𝑖+1ℨ) 1113(𝑖+1) βˆ’ 𝔉(11𝑖ℨ) 1113𝑖 , 𝔙𝑖 β„³ 1113β‹…1113𝑖 )} (12) where ∏ π‘›βˆ’1 π’₯=0 𝑄𝑗 = 𝑄1 βˆ— 𝑄2 βˆ— β‹― βˆ— 𝑄𝑛 π‘Žπ‘›π‘‘ ∐ π‘›βˆ’1 π’₯=0 𝑅𝑗 = 𝑅1 β‹„ 𝑅2 β‹„ β‹―β‹„ π‘…π‘›π‘Žπ‘›π‘‘ ∐ π‘›βˆ’1 π’₯=0 𝑆𝑗 = 𝑆1βŠ˜π‘†2βŠ˜β‹―βŠ˜ 𝑆𝑛 for all ℨ ∈ 𝒳𝑀 and all β„³ > 0. Hence, from (12) and (9), we arrive π’œ1 ( 𝔉(11𝑛ℨ) 1113𝑛 βˆ’ 𝔉(ℨ), βˆ‘π‘›βˆ’1𝑖=0 𝔙𝑖 β„³ 1113β‹…1113𝑖 ) β‰₯ βˆπ‘›βˆ’1 𝑖=0 π’œ1β€²(Γ°(ℨ),β„³) = π’œ1β€²(Γ°(ℨ),β„³) π’œ2 ( 𝔉(11𝑛ℨ) 1113𝑛 βˆ’ 𝔉(ℨ), βˆ‘π‘›βˆ’1𝑖=0 𝔙𝑖 β„³ 1113β‹…1113𝑖 ) ≀ βˆπ‘›βˆ’1 𝑖=0 π’œ2β€²(Γ°(ℨ),β„³) = π’œ2β€²(Γ°(ℨ),β„³) π’œ3 ( 𝔉(11𝑛ℨ) 1113𝑛 βˆ’ 𝔉(ℨ), βˆ‘π‘›βˆ’1𝑖=0 𝔙𝑖 β„³ 1113β‹…1113𝑖 ) ≀ βˆπ‘›βˆ’1 𝑖=0 π’œ1β€²(Γ°(ℨ),β„³) = π’œ3β€²(Γ°(ℨ),β„³) } (13) for all ℨ ∈ 𝒳𝑀 and all β„³ > 0. Replacing ℨ by 11π‘šβ„¨ in (13) and using (1), (NNS5), (NNS11) and (NNS17), we obtain Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 32 No. 7s (2025) 812 https://internationalpubls.com π’œ1 ( 𝔉(11𝑛+π‘šβ„¨) 1113(𝑛+π‘š) βˆ’ 𝔉(11π‘šβ„¨) 1113π‘š , βˆ‘π‘›βˆ’1𝑖=0 𝔙𝑖 β„³ 1113β‹…1113(𝑖+π‘š) ) β‰₯ π’œ1β€²(Γ°(11 π‘šβ„¨),β„³) = π’œ1β€² (Γ°(ℨ), β„³ π”™π‘š ) π’œ2 ( 𝔉(11𝑛+π‘šβ„¨) 1113(𝑛+π‘š) βˆ’ 𝔉(11π‘šβ„¨) 1113π‘š , βˆ‘π‘›βˆ’1𝑖=0 𝔙𝑖 β„³ 1113β‹…1113(𝑖+π‘š) ) ≀ π’œ2β€²(Γ°(11 π‘šβ„¨),β„³) = π’œ2β€² (Γ°(ℨ), β„³ π”™π‘š ) π’œ3 ( 𝔉(11𝑛+π‘šβ„¨) 1113(𝑛+π‘š) βˆ’ 𝔉(11π‘šβ„¨) 1113π‘š , βˆ‘π‘›βˆ’1𝑖=0 𝔙𝑖 β„³ 1113β‹…1113(𝑖+π‘š) ) ≀ π’œ3β€²(Γ°(11 π‘šβ„¨),β„³) = π’œ1β€² (Γ°(ℨ), β„³ π”™π‘š ) } (14) for all ℨ ∈ 𝒳𝑀 and all β„³ > 0 also π‘š,𝑛 are positive numbers. Changing β„³ by π”™π‘šβ„³ in (14), we get π’œ1 ( 𝔉(11𝑛+π‘šβ„¨) 1113(𝑛+π‘š) βˆ’ 𝔉(11π‘šβ„¨) 1113π‘š , βˆ‘π‘›βˆ’1𝑖=0 𝔙𝑖+π‘š β„³ 1113β‹…1113(𝑖+π‘š) ) β‰₯ π’œ1β€²(Γ°(ℨ),β„³) π’œ2 ( 𝔉(11𝑛+π‘šβ„¨) 1113(𝑛+π‘š) βˆ’ 𝔉(11π‘šβ„¨) 1113π‘š , βˆ‘π‘›βˆ’1𝑖=0 𝔙𝑖+π‘š β„³ 1113β‹…1113(𝑖+π‘š) ) ≀ π’œ2β€²(Γ°(ℨ),β„³) π’œ3 ( 𝔉(11𝑛+π‘šβ„¨) 1113(𝑛+π‘š) βˆ’ 𝔉(11π‘šβ„¨) 1113π‘š , βˆ‘π‘›βˆ’1𝑖=0 𝔙𝑖+π‘š β„³ 1113β‹…1113(𝑖+π‘š) ) ≀ π’œ3β€²(Γ°(ℨ),β„³)} (15) which implies π’œ1 ( 𝔉(11𝑛+π‘šβ„¨) 1113(𝑛+π‘š) βˆ’ 𝔉(11π‘šβ„¨) 1113π‘š ,β„³) β‰₯ π’œ1β€² (Γ°(ℨ), β„³ βˆ‘π‘›βˆ’1𝑖=π‘š 𝔙𝑖 1113β‹…1113𝑖 ) π’œ2 ( 𝔉(11𝑛+π‘šβ„¨) 1113(𝑛+π‘š) βˆ’ 𝔉(11π‘šβ„¨) 1113π‘š ,β„³) ≀ π’œ2β€² (Γ°(ℨ), β„³ βˆ‘π‘›βˆ’1𝑖=π‘š 𝔙𝑖 1113β‹…1113𝑖 ) π’œ3 ( 𝔉(11𝑛+π‘šβ„¨) 1113(𝑛+π‘š) βˆ’ 𝔉(11π‘šβ„¨) 1113π‘š ,β„³) ≀ π’œ3β€² (Γ°(ℨ), β„³ βˆ‘π‘›βˆ’1𝑖=π‘š 𝔙𝑖 1113β‹…1113𝑖 ) } (16) Here { 𝔉(11𝑛ℨ) 1113𝑛 } is a Cauchy sequence in (𝒴𝑀 ,π’œ1, π’œ2,π’œ3) also a complete NNS-space is (𝒴𝑀 ,π’œ1,π’œ2, π’œ3) then this sequence converges to a particular point 𝒯(ℨ) ∈ π‘Œ. lim π‘›β†’βˆž π’œ1 ( 𝔉(11𝑛ℨ) 1113𝑛 βˆ’ 𝒯(ℨ),β„³) = 1, lim π‘›β†’βˆž π’œ2 ( 𝔉(11𝑛ℨ) 1113𝑛 βˆ’ 𝒯(ℨ),β„³) = 0 lim π‘›β†’βˆž π’œ3 ( 𝔉(11𝑛ℨ) 1113𝑛 βˆ’ 𝒯(ℨ),β„³) = 0 and 𝔉(11𝑛ℨ) 1113𝑛 ⟢ 𝑁𝑁𝑆 𝒯(ℨ), π‘Žπ‘  𝑛 β†’ ∞. Taking π‘š = 0 in (15), we reach Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 32 No. 7s (2025) 813 https://internationalpubls.com π’œ1 ( 𝔉(11𝑛ℨ) 1113𝑛 βˆ’ 𝔉(ℨ),β„³) β‰₯ π’œ1β€² (Ξ¨(ℨ), β„³ βˆ‘π‘›βˆ’1𝑖=0 𝔙𝑖 1113β‹…1113𝑖 ) π’œ2 ( 𝔉(11𝑛ℨ) 1113𝑛 βˆ’ 𝔉(ℨ),β„³) ≀ π’œ2β€² (Ξ¨(ℨ), β„³ βˆ‘π‘›βˆ’1𝑖=0 𝔙𝑖 1113β‹…1113𝑖 ) π’œ3 ( 𝔉(11𝑛ℨ) 1113𝑛 βˆ’ 𝔉(ℨ),β„³) ≀ π’œ3β€² (Ξ¨(ℨ), β„³ βˆ‘π‘›βˆ’1𝑖=0 𝔙𝑖 1113β‹…1113𝑖 ) } (17) for all ℨ ∈ 𝒳𝑀 and all β„³ > 0. Considering 𝑛 β†’ ∞ in (17), we arrive π’œ1(𝒯(ℨ) βˆ’ 𝔉(ℨ),β„³) β‰₯ π’œ1β€²(Γ°(ℨ),β„³|1113 βˆ’ 𝔙|) π’œ2(𝒯(ℨ) βˆ’ 𝔉(ℨ),β„³) ≀ π’œ2β€²(Γ°(ℨ),β„³|1113 βˆ’π”™|) π’œ3(𝒯(ℨ) βˆ’ 𝔉(ℨ),β„³) ≀ π’œ3β€²(Γ°(ℨ),β„³|1113 βˆ’π”™|)} (18) for all ℨ ∈ 𝒳𝑀 and all β„³ > 0 . Next, we have to show 𝔉 satisfies (1), letting ℨ by 11𝑛ℨ in (3) respectively, we have π’œ1 ( 1 1113𝑛 [𝔉(3 β‹… 11𝑛ℨ) βˆ’ 1,88,30,57,02,60,326𝔉(11𝑛ℨ) + 1,56,92,14,18,83,605𝔉(βˆ’11𝑛ℨ)],β„³) β‰₯ π’œ1β€²(Γ°(11 𝑛ℨ),1113𝑛ℳ) π’œ2 ( 1 1113𝑛 [𝔉(3 β‹… 11𝑛ℨ) βˆ’ 1,88,30,57,02,60,326𝔉(11𝑛ℨ) + 1,56,92,14,18,83,605𝔉(βˆ’11𝑛ℨ)],β„³) ≀ π’œ2β€²(Γ°(11 𝑛ℨ),1113𝑛ℳ) π’œ3 ( 1 1113𝑛 [𝔉(3 β‹… 11𝑛ℨ) βˆ’ 1,88,30,57,02,60,326𝔉(11𝑛ℨ) + 1,56,92,14,18,83,605𝔉(βˆ’11𝑛ℨ)],β„³) ≀ π’œ3β€²(Γ°(11 𝑛ℨ),1113𝑛ℳ) } (19) for all ℨ ∈ 𝒳𝑀 and all β„³ > 0 . Now, π’œ1(𝒯(11ℨ) βˆ’ 1,88,30,57,02,60,326𝒯(ℨ) + 1,56,92,14,18,83,605𝒯(βˆ’β„¨),β„³) β‰₯ π’œ1(𝒯(11ℨ) βˆ’ 1 1113𝑛 𝔉(11ℨ), β„³ 4 ) βˆ— π’œ1(βˆ’1,88,30,57,02,60,326𝒯(ℨ) + 1,88,30,57,02,60,326 1 1113𝑛 𝔉(ℨ), β„³ 4 ) βˆ— π’œ1(1,56,92,14,18,83,605𝒯(βˆ’β„¨) + 1,56,92,14,18,83,605 1 1113𝑛 𝔉(βˆ’β„¨), β„³ 4 ) βˆ— π’œ1( 1 1113𝑛 𝔉(11ℨ) βˆ’ 1,88,30,57,02,60,326 1 1113𝑛 𝔉(ℨ) + 1,56,92,14,18,83,605 1 1113𝑛 𝔉(βˆ’β„¨), β„³ 4 ) (20) Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 32 No. 7s (2025) 814 https://internationalpubls.com π’œ2(𝒯(11ℨ) βˆ’ 1,88,30,57,02,60,326𝒯(ℨ) + 1,56,92,14,18,83,605𝒯(βˆ’β„¨),β„³) β‰₯ π’œ1(𝒯(11ℨ) βˆ’ 1 1113𝑛 𝔉(11ℨ), β„³ 4 ) β‹„ π’œ2(βˆ’1,88,30,57,02,60,326𝒯(ℨ) + 1,88,30,57,02,60,326 1 1113𝑛 𝔉(ℨ), β„³ 4 ) β‹„ π’œ2(βˆ’1,56,92,14,18,83,605𝒯(βˆ’β„¨) + 1,56,92,14,18,83,605 1 1113𝑛 𝔉(βˆ’β„¨), β„³ 4 ) β‹„ π’œ2( 1 1113𝑛 𝔉(11ℨ) βˆ’ 1,88,30,57,02,60,326 1 1113𝑛 𝔉(ℨ) + 1,56,92,14,18,83,605 1 1113𝑛 𝔉(βˆ’β„¨), β„³ 4 ) (21) and π’œ3(𝒯(11ℨ) βˆ’ 1,88,30,57,02,60,326𝒯(ℨ) + 1,56,92,14,18,83,605𝒯(βˆ’β„¨),β„³) β‰₯ π’œ3(𝒯(11ℨ) βˆ’ 1 1113𝑛 𝔉(11ℨ), β„³ 4 ) βŠ˜π’œ3(βˆ’1,88,30,57,02,60,326𝒯(ℨ) + 1,88,30,57,02,60,326 1 1113𝑛 𝔉(ℨ), β„³ 4 ) βŠ˜π’œ3(1,56,92,14,18,83,605𝒯(βˆ’β„¨) + 1,56,92,14,18,83,605 1 1113𝑛 𝔉(βˆ’β„¨), β„³ 4 ) βŠ˜π’œ1( 1 1113𝑛 𝔉(11ℨ) βˆ’ 1,88,30,57,02,60,326 1 1113𝑛 𝔉(ℨ) + 1,56,92,14,18,83,605 1 1113𝑛 𝔉(βˆ’β„¨), β„³ 4 ) (22) for all ℨ ∈ 𝒳𝑀 and all β„³ > 0 . Also, lim π‘›β†’βˆž π’œ1( 1 1113𝑛 [𝔉(3 β‹… 11𝑛ℨ) βˆ’ 1,88,30,57,02,60,326𝔉(11𝑛ℨ) + 1,56,92,14,18,83,605𝔉(βˆ’11𝑛ℨ)], β„³ 4 ) = 1 lim π‘›β†’βˆž π’œ2( 1 1113𝑛 [𝔉(3 β‹… 11𝑛ℨ) βˆ’ 1,88,30,57,02,60,326𝔉(11𝑛ℨ) + 1,56,92,14,18,83,605𝔉(βˆ’11𝑛ℨ)], β„³ 4 ) = 0 lim π‘›β†’βˆž π’œ3( 1 1113𝑛 [𝔉(3 β‹… 11𝑛ℨ) βˆ’ 1,88,30,57,02,60,326𝔉(11𝑛ℨ) + 1,56,92,14,18,83,605𝔉(βˆ’11𝑛ℨ)], β„³ 4 ) = 0} (23) for all ℨ ∈ 𝒳𝑀 and all β„³ > 0 . By Taking 𝑛 β†’ ∞ in (21), (22) and using (23), we proved that 𝒯 satisfies (1). Therefore, 𝒯 is a tridecic mapping. Next, we need to prove 𝒯(ℨ) is unique, let 𝒯′(ℨ) be another tridecic FE satisfying (1) and (4), Then π’œ1(𝒯(ℨ) βˆ’ 𝒯′(ℨ),β„³) β‰₯ π’œ1 (𝒯(11 𝑛ℨ) βˆ’ 𝔉(11𝑛ℨ), β„³.1113𝑛 2 ) βˆ— π’œ1 (𝔉(11 𝑛ℨ) βˆ’ 𝒯′(11𝑛ℨ), β„³.1113𝑛 2 ) β‰₯ π’œ1β€² (Γ°(11 𝑛ℨ), 1113𝑛 β„³ |1113βˆ’π”™| 2 ) β‰₯ π’œ1β€² (Γ°(ℨ), 1113𝑛 β„³ |1113βˆ’π”™| 2⋅𝔙𝑛 ) π’œ2(𝒯(ℨ) βˆ’ 𝒯′(ℨ),β„³) ≀ π’œ2 (𝒯(11 𝑛ℨ) βˆ’ 𝔉(11𝑛ℨ), β„³.1113𝑛 2 ) β‹„ π’œ2 (𝔉(11 𝑛ℨ) βˆ’ 𝒯′(11𝑛ℨ), β„³.1113𝑛 2 ) ≀ π’œ2β€² (Γ°(11 𝑛ℨ), 1113𝑛ℳ |1113βˆ’π”™| 2 ) ≀ π’œ2β€² (Γ°(ℨ), 1113𝑛 β„³ |1113βˆ’π”™| 2⋅𝔙𝑛 ) Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 32 No. 7s (2025) 815 https://internationalpubls.com π’œ3(𝒯(ℨ) βˆ’ 𝒯′(ℨ),β„³) ≀ π’œ3 (𝒯(11 𝑛ℨ) βˆ’ 𝔉(11𝑛ℨ), β„³.1113𝑛 2 ) βˆ— π’œ3 (𝔉(11 𝑛ℨ) βˆ’ 𝒯′(11𝑛ℨ), β„³.1113𝑛 2 ) ≀ π’œ3β€² (Γ°(11 𝑛ℨ), 1113𝑛 β„³ |1113βˆ’π”™| 2 ) ≀ π’œ3β€² (Γ°(ℨ), 1113𝑛 β„³ |1113βˆ’π”™| 2⋅𝔙𝑛 ) for all ℨ ∈ 𝒳𝑀 and all β„³ > 0 . Since lim π‘›β†’βˆž 1113𝑛 β„³ |1113βˆ’π”™| 2 𝔙𝑛 = ∞, we obtain lim π‘›β†’βˆž π’œ1β€² (Γ°(ℨ), 1113𝑛 β„³ |1113βˆ’π”™| 2⋅𝔙𝑛 ) = 1 lim π‘›β†’βˆž π’œ2β€² (Γ°(ℨ), 1113𝑛 β„³ |1113βˆ’π”™| 2⋅𝔙𝑛 ) = 0 lim π‘›β†’βˆž π’œ3β€² (Γ°(ℨ), 1113𝑛 β„³ |1113βˆ’π”™| 2⋅𝔙𝑛 ) = 0} for all ℨ ∈ 𝒳𝑀 and all β„³ > 0 . Thus π’œ1(𝒯(ℨ) βˆ’ 𝒯′(ℨ),β„³) = 1 π’œ2(𝒯(ℨ) βˆ’ 𝒯′(ℨ),β„³) = 0 π’œ3(𝒯(ℨ) βˆ’ 𝒯′(ℨ),β„³) = 0 } for all ℨ ∈ 𝒳𝑀 and all β„³ > 0 . Hence, 𝒯(ℨ) = 𝒯′(ℨ). Therefore, 𝒯(ℨ) is unique. For second case, we have to take 𝐹 = βˆ’1. Considering ℨ by ℨ 13 in (5), we have π’œ1 (𝔉(ℨ) βˆ’ 11 13𝔉( ℨ 13 ) ,β„³) β‰₯ π’œ1β€² (Γ° ( ℨ 13 ) ,β„³) π’œ2 (𝔉(ℨ) βˆ’ 11 13𝔉 ( ℨ 13 ) ,β„³) ≀ π’œ2β€² (Γ° ( ℨ 13 ) ,β„³) π’œ3 (𝔉(ℨ) βˆ’ 11 13𝔉 ( ℨ 13 ) ,β„³) ≀ π’œ3β€² (Γ° ( ℨ 13 ) ,β„³)} (24) for all ℨ ∈ 𝒳𝑀 and all β„³ > 0. Corollary 3.2 Let β„’, π‘Ÿ are constants, with β„’ > 0 and π‘Ÿ β‰  12 and let an odd function 𝔉:𝒳𝑀 βŸΆπ’΄π‘€ satisfies π’œ1(𝔉(11ℨ) βˆ’ 1,88,30,57,02,60,326𝔉(ℨ) + 1,56,92,14,18,83,605𝔉(βˆ’β„¨),β„³) β‰₯ { π’œ1β€²(β„’,β„³), π’œ1β€²(β„’(||ℨ|| π‘Ÿ),β„³), π’œ2(𝔉(11ℨ) βˆ’ 1,88,30,57,02,60,326𝔉(ℨ) + 1,56,92,14,18,83,605𝔉(βˆ’β„¨),β„³) ≀ { π’œ2β€²(β„’,β„³), π’œ2β€²(β„’(||ℨ|| π‘Ÿ),β„³), π’œ3(𝔉(11ℨ) βˆ’ 1,88,30,57,02,60,326𝔉(ℨ) + 1,56,92,14,18,83,605𝔉(βˆ’β„¨),β„³) ≀ { π’œ3β€²(β„’,β„³), π’œ3β€²(β„’(||ℨ|| π‘Ÿ),β„³), } (25) for all ℨ ∈ 𝒳𝑀 and all β„³ > 0, Then there exists a unique tridecic function 𝒯:𝒳𝑀 βŸΆπ’΄π‘€ such that Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 32 No. 7s (2025) 816 https://internationalpubls.com π’œ1(𝔉(ℨ) βˆ’ 𝒯(ℨ),β„³) β‰₯ { π’œ1β€²(β„’, |11 13 βˆ’ 1|β„³), π’œ1β€²(β„’||ℨ|| π‘Ÿ , |1113 βˆ’ 11π‘Ÿ|β„³), π’œ2(𝔉(ℨ) βˆ’ 𝒯(ℨ),β„³) ≀ { π’œ2β€²(β„’, |11 13 βˆ’ 1|β„³), π’œ2β€²(β„’||ℨ|| π‘Ÿ , |1113 βˆ’ 11π‘Ÿ|β„³), π’œ3(𝔉(ℨ) βˆ’ 𝒯(ℨ),β„³) ≀ { π’œ3β€²(β„’, |11 13 βˆ’ 1|β„³), π’œ3β€²(β„’||ℨ|| π‘Ÿ , |1113 βˆ’ 11π‘Ÿ|β„³), } (26) for all ℨ ∈ 𝒳𝑀 and all β„³ > 0 . Theorem 3.3 Assume that 𝒳𝑀 is a LS, (π’΅π‘š ,π’œ1β€²,π’œ2β€², π’œ3β€²) is a NNS and (𝒴𝑀 ,π’œ1,π’œ2, π’œ3) an NBS . Let Γ°:𝒳𝑀 ⟢ π’΅π‘š be a function such that for some 0 < ( 𝔙 1112 ) 𝐹 < 1 with 𝐹 ∈ {1, βˆ’1}. π’œ1β€²(Γ°(11 𝑛𝐹ℨ),β„³) β‰₯ π’œ1β€²(𝔙 𝑛𝐹ð(ℨ),β„³) π’œ2β€²(Γ°(11 𝑛𝐹ℨ),β„³) ≀ π’œ2β€²(𝔙 𝑛𝐹ð(ℨ),β„³) π’œ3β€²(Γ°(11 𝑛𝐹ℨ),β„³) ≀ π’œ3β€²(𝔙 𝑛𝐹ð(ℨ),β„³)} (27) for all ℨ ∈ 𝒳𝑀 and all β„³ > 0 and lim π‘›β†’βˆž π’œ1β€²(Γ°(11 𝐹𝑛ℨ), 1112𝐹𝑛ℳ) = 1 lim π‘›β†’βˆž π’œ2β€²(Γ°(11 𝐹𝑛ℨ), 1112𝐹𝑛ℳ) = 0 lim π‘›β†’βˆž π’œ3β€²(Γ°(11 𝐹𝑛ℨ), 1112𝐹𝑛ℳ) = 0} (28) for all ℨ ∈ 𝒳𝑀 and all β„³ > 0 . Let an even function 𝔉:𝒳𝑀 βŸΆπ’΄π‘€ satisfying π’œ1(𝔉(11ℨ) βˆ’ 1,88,30,57,02,60,326𝔉(ℨ) + 1,56,92,14,18,83,605𝔉(βˆ’β„¨),β„³) β‰₯ π’œ1β€²(Γ°(ℨ),β„³) π’œ2(𝔉(11ℨ) βˆ’ 1,88,30,57,02,60,326𝔉(ℨ) + 1,56,92,14,18,83,605𝔉(βˆ’β„¨),β„³) ≀ π’œ2β€²(Γ°(ℨ),β„³) π’œ3(𝔉(11ℨ) βˆ’ 1,88,30,57,02,60,326𝔉(ℨ) + 1,56,92,14,18,83,605𝔉(βˆ’β„¨),β„³) ≀ π’œ3β€²(Γ°(ℨ),β„³)} (29) for all ℨ ∈ 𝒳𝑀 and all β„³ > 0. Then there exists a unique duodecic mapping π’Ÿ:𝒳𝑀 βŸΆπ’΄π‘€ satisfying (1) and Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 32 No. 7s (2025) 817 https://internationalpubls.com π’œ1(𝔉(ℨ) βˆ’ π’Ÿ(ℨ),β„³) β‰₯ π’œ1β€²(Γ°(ℨ), |11 12 βˆ’ 𝔙|β„³) π’œ2(𝔉(ℨ) βˆ’ 𝒯(ℨ),β„³) ≀ π’œ2β€²(Γ°(ℨ), |11 12 βˆ’ 𝔙|β„³) π’œ3(𝔉(ℨ) βˆ’ π’Ÿ(ℨ),β„³) ≀ π’œ3β€²(Γ°(ℨ), |11 12 βˆ’π”™|β„³)} (30) for all ℨ ∈ 𝒳𝑀 and all β„³ > 0. Proof. For the first case 𝐹 = 1. By applying the evenness condition of 𝔉 in in (29), we arrive π’œ1(𝔉(11ℨ) βˆ’ 1,56,92,14,18,83,605𝔉(ℨ),β„³) β‰₯ π’œ1β€²(Γ°(ℨ),β„³) π’œ2(𝔉(11ℨ) βˆ’ 1,56,92,14,18,83,605𝔉(ℨ),β„³) ≀ π’œ2β€²(Γ°(ℨ),β„³) π’œ3(𝔉(11ℨ) βˆ’ 1,56,92,14,18,83,605𝔉(ℨ),β„³) ≀ π’œ3β€²(Γ°(ℨ),β„³) } (31) for all ℨ ∈ 𝒳𝑀 and all β„³ > 0. From (31) we have π’œ1(𝔉(11ℨ) βˆ’ 11 12𝔉(ℨ),β„³) β‰₯ π’œ1β€²(Γ°(ℨ),β„³) π’œ2(𝔉(11ℨ) βˆ’ 11 12𝔉(ℨ),β„³) ≀ π’œ2β€²(Γ°(ℨ),β„³) π’œ3(𝔉(11ℨ) βˆ’ 11 12𝔉(ℨ),β„³) ≀ π’œ3β€²(Γ°(ℨ),β„³) } (32) for all ℨ ∈ 𝒳𝑀 and all β„³ > 0. Corollary 3.4 Let β„’, π‘Ÿ are constants, with β„’ > 0 and π‘Ÿ β‰  12 and let an even mapping 𝔉:𝒳𝑀 βŸΆπ’΄π‘€ satisfies π’œ1(𝔉(11ℨ) βˆ’ 1,88,30,57,02,60,326𝔉(ℨ) + 1,56,92,14,18,83,605𝔉(βˆ’β„¨),β„³) β‰₯ { π’œ1β€²(β„’,β„³), π’œ1β€²(β„’(||ℨ|| π‘Ÿ),β„³), π’œ2(𝔉(11ℨ) βˆ’ 1,88,30,57,02,60,326𝔉(ℨ) + 1,56,92,14,18,83,605𝔉(βˆ’β„¨),β„³) ≀ { π’œ2β€²(β„’,β„³), π’œ2β€²(β„’(||ℨ|| π‘Ÿ),β„³), π’œ3(𝔉(11ℨ) βˆ’ 1,88,30,57,02,60,326𝔉(ℨ) + 1,56,92,14,18,83,605𝔉(βˆ’β„¨),β„³) ≀ { π’œ3β€²(β„’,β„³), π’œ3β€²(β„’(||ℨ|| π‘Ÿ),β„³), } (33) for all ℨ ∈ 𝒳𝑀 and all β„³ > 0, Then there exists a unique duodecic function π’Ÿ:𝒳𝑀 βŸΆπ’΄π‘€ such that Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 32 No. 7s (2025) 818 https://internationalpubls.com π’œ1(𝔉(ℨ) βˆ’ π’Ÿ(ℨ),β„³) β‰₯ { π’œ1β€²(β„’, |11 12 βˆ’ 1|β„³), π’œ1β€²(β„’||ℨ|| π‘Ÿ , |1112 βˆ’ 11π‘Ÿ|β„³), π’œ2(𝔉(ℨ) βˆ’ π’Ÿ(ℨ),β„³) ≀ { π’œ2β€²(β„’, |11 12 βˆ’ 1|β„³), π’œ2β€²(β„’||ℨ|| π‘Ÿ , |1112 βˆ’ 11π‘Ÿ|β„³), π’œ3(𝔉(ℨ) βˆ’ π’Ÿ(ℨ),β„³) ≀ { π’œ3β€²(β„’, |11 12 βˆ’ 1|β„³), π’œ3β€²(β„’||ℨ|| π‘Ÿ , |1112 βˆ’ 11π‘Ÿ|β„³), } (34) for all ℨ ∈ 𝒳𝑀 and all β„³ > 0 . Theorem 3.5 Assume that 𝒳𝑀 is a LS, (π’΅π‘š ,π’œ1β€²,π’œ2β€², π’œ3β€²) is a NNS and (𝒴𝑀 ,π’œ1,π’œ2, π’œ3) an NBS . Let Γ°:𝒳𝑀 ⟢ π’΅π‘š be a function such that for some 0 < ( 𝔙 1112 ) 𝐹 < 1,0 < ( 𝔙 1113 ) 𝐹 < 1 with 𝐹 ∈ {1,βˆ’1}. Let 𝔉:𝒳𝑀 βŸΆπ’΄π‘€ be a function satisfying the inequality with conditions (1), (27), (2) and (28). Then . π’œ1(𝔉(11ℨ) βˆ’ 1,88,30,57,02,60,326𝔉(ℨ) + 1,56,92,14,18,83,605𝔉(βˆ’β„¨),β„³) β‰₯ π’œ1β€²(Γ°(ℨ),β„³) π’œ2(𝔉(11ℨ) βˆ’ 1,88,30,57,02,60,326𝔉(ℨ) + 1,56,92,14,18,83,605𝔉(βˆ’β„¨),β„³) ≀ π’œ2β€²(Γ°(ℨ),β„³) π’œ3(𝔉(11ℨ) βˆ’ 1,88,30,57,02,60,326𝔉(ℨ) + 1,56,92,14,18,83,605𝔉(βˆ’β„¨),β„³) ≀ π’œ3β€²(Γ°(ℨ),β„³)} (35) for all ℨ ∈ 𝒳𝑀 and all β„³ > 0. Then there exists a unique tridecic function 𝒯:𝒳𝑀 βŸΆπ’΄π‘€ and a unique duodecic function π’Ÿ:𝒳𝑀 βŸΆπ’΄π‘€ satisfying (1) and π’œ1(𝑔(ℨ) βˆ’ 𝒯(ℨ) βˆ’ π’Ÿ(ℨ),β„³) β‰₯ π’œ1β€²(Γ°(ℨ), |11 13 βˆ’π”™|β„³) βˆ— π’œ1β€²(Γ°(βˆ’β„¨), |11 13 βˆ’ 𝔙|β„³) βˆ— π’œ1β€²(Γ°(ℨ), |11 12 βˆ’π”™|β„³) βˆ— π’œ1β€²(Γ°(βˆ’β„¨), |11 12 βˆ’ 𝔙|β„³) π’œ2(𝑔(ℨ) βˆ’ 𝒯(ℨ) βˆ’ π’Ÿ(ℨ),β„³) ≀ π’œ2β€²(Γ°(ℨ), |11 13 βˆ’π”™|β„³) β‹„ π’œ2β€²(Γ°(βˆ’β„¨), |11 13 βˆ’π”™|β„³) β‹„ π’œ2β€²(Γ°(ℨ), |11 12 βˆ’π”™|β„³) β‹„ π’œ2β€²(Γ°(βˆ’β„¨), |11 12 βˆ’ 𝔙|β„³) π’œ3(𝑔(ℨ) βˆ’ 𝒯(ℨ) βˆ’ π’Ÿ(ℨ),β„³) ≀ π’œ3β€²(Γ°(ℨ), |11 13 βˆ’π”™|β„³)βŠ˜π’œ3β€²(Γ°(βˆ’β„¨), |11 13 βˆ’ 𝔙|β„³) βŠ˜π’œ3β€²(Γ°(ℨ), |11 12 βˆ’ 𝔙|β„³)βŠ˜π’œ3β€²(Γ°(βˆ’β„¨), |11 12 βˆ’π”™|β„³)} (36) for all ℨ ∈ 𝒳𝑀 and all β„³ > 0. Proof. Let π”‰π‘œ(ℨ) = 𝔉(ℨ)βˆ’π”‰(βˆ’β„¨) 2 for all ℨ ∈ 𝒳𝑀. Then π”‰π‘œ(0) = 0 and π”‰π‘œ(βˆ’β„¨) = βˆ’π”‰π‘œ(ℨ) for all ℨ ∈ 𝒳𝑀. Hence by Theorem 3.1, we have Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 32 No. 7s (2025) 819 https://internationalpubls.com π’œ1(π”‰π‘œ(ℨ) βˆ’ 𝒯(ℨ),β„³) β‰₯ π’œ1β€²(Γ°(ℨ), |11 13 βˆ’ 𝔙|β„³) βˆ— π’œ1β€²(Γ°(βˆ’β„¨), |11 13 βˆ’ 𝔙|β„³) π’œ2(π”‰π‘œ(ℨ) βˆ’ 𝒯(ℨ),β„³) ≀ π’œ2β€²(Γ°(ℨ), |11 13 βˆ’π”™|β„³) β‹„ π’œ2β€²(Γ°(βˆ’β„¨), |11 13 βˆ’π”™|β„³) π’œ3(π”‰π‘œ(ℨ) βˆ’ 𝒯(ℨ),β„³) ≀ π’œ3β€²(Γ°(ℨ), |11 13 βˆ’π”™|β„³)βŠ˜π’œ3β€²(Γ°(βˆ’β„¨), |11 13 βˆ’ 𝔙|β„³) } (37) for all ℨ ∈ 𝒳𝑀 and all β„³ > 0. Also, let 𝔉𝑒(ℨ) = 𝔉(ℨ)+𝔉(βˆ’β„¨) 2 for all ℨ ∈ 𝒳𝑀. Then 𝔉𝑒(0) = 0 and 𝔉𝑒(βˆ’β„¨) = 𝔉𝑒(ℨ) for all ℨ ∈ 𝒳𝑀. Hence by Theorem 3.3, we have π’œ1(𝔉𝑒(ℨ) βˆ’ π’Ÿ(ℨ),β„³) β‰₯ π’œ1β€²(Γ°(ℨ), |11 12 βˆ’ 𝔙|β„³) βˆ— π’œ1β€²(Γ°(βˆ’β„¨), |11 12 βˆ’ 𝔙|β„³) π’œ2(𝔉𝑒(ℨ) βˆ’ π’Ÿ(ℨ),β„³) ≀ π’œ2β€²(Γ°(ℨ), |11 12 βˆ’π”™|β„³) β‹„ π’œ2β€²(Γ°(βˆ’β„¨), |11 12 βˆ’π”™|β„³) π’œ3(𝔉𝑒(ℨ) βˆ’ π’Ÿ(ℨ),β„³) ≀ π’œ3β€²(Γ°(ℨ), |11 12 βˆ’π”™|β„³)βŠ˜π’œ1β€²(Γ°(βˆ’β„¨), |11 12 βˆ’π”™|β„³) } (38) for all ℨ ∈ 𝒳𝑀 and all β„³ > 0. Define 𝑔(ℨ) = π”‰π‘œ(ℨ) + 𝔉𝑒(ℨ) (39) for all ℨ ∈ 𝒳𝑀. From (37),(38) and (39), we arrive π’œ1(𝔉(ℨ) βˆ’ 𝒯(ℨ) βˆ’ π’Ÿ(ℨ),2β„³) = π’œ1(π”‰π‘œ(ℨ) + 𝔉𝑒(ℨ) βˆ’ 𝒯(ℨ) βˆ’ π’Ÿ(ℨ),2β„³) β‰₯ π’œ1(π”‰π‘œ(ℨ) βˆ’ 𝒯(ℨ),β„³) βˆ— π’œ1(𝔉𝑒(ℨ) βˆ’ π’Ÿ(ℨ),β„³) β‰₯ π’œ1β€²(Γ°(ℨ), |11 13 βˆ’ 𝔙|β„³) βˆ— π’œ1β€²(Γ°(βˆ’β„¨), |11 13 βˆ’π”™|β„³) βˆ— π’œ1β€²(Γ°(ℨ), |11 12 βˆ’π”™|β„³) βˆ— π’œ1β€²(Γ°(βˆ’β„¨), |11 12 βˆ’π”™|β„³) and π’œ2(𝔉(ℨ) βˆ’ 𝒯(ℨ) βˆ’ π’Ÿ(ℨ),2β„³) = π’œ2(π”‰π‘œ(ℨ) + 𝔉𝑒(ℨ) βˆ’ 𝒯(ℨ) βˆ’ π’Ÿ(ℨ),2β„³) ≀ π’œ2(π”‰π‘œ(ℨ) βˆ’ 𝒯(ℨ),β„³) β‹„ π’œ2(𝔉𝑒(ℨ) βˆ’ π’Ÿ(ℨ),β„³) ≀ π’œ2β€²(Γ°(ℨ), |11 13 βˆ’ 𝔙|β„³) β‹„π’œ2β€²(Γ°(βˆ’β„¨), |11 13 βˆ’ 𝔙|β„³) β‹„ π’œ2β€²(Γ°(ℨ), |11 12 βˆ’ 𝔙|β„³) β‹„π’œ2β€²(Γ°(βˆ’β„¨), |11 12 βˆ’π”™|β„³) and π’œ3(𝔉(ℨ) βˆ’ 𝒯(ℨ) βˆ’ π’Ÿ(ℨ),2β„³) = π’œ3(π”‰π‘œ(ℨ) + 𝔉𝑒(ℨ) βˆ’ 𝒯(ℨ) βˆ’ π’Ÿ(ℨ),2β„³) ≀ π’œ2(π”‰π‘œ(ℨ) βˆ’ 𝒯(ℨ),β„³)βŠ˜π’œ3(𝔉𝑒(ℨ) βˆ’ π’Ÿ(ℨ),β„³) ≀ π’œ3β€²(Γ°(ℨ), |11 13 βˆ’π”™|β„³)βŠ˜π’œ3β€²(Γ°(βˆ’β„¨), |11 13 βˆ’ 𝔙|β„³) ⊘ π’œ3β€²(Γ°(ℨ), |11 12 βˆ’ 𝔙|β„³)βŠ˜π’œ3β€²(Γ°(βˆ’β„¨), |11 12 βˆ’π”™|β„³) for all ℨ ∈ 𝒳𝑀 and all β„³ > 0. Corollary 3.6 Let β„’, π‘Ÿ are constants with β„’ > 0 and π‘Ÿ β‰  13,12 and let the function 𝔉:𝒳𝑀 βŸΆπ’΄π‘€ satisfies Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 32 No. 7s (2025) 820 https://internationalpubls.com π’œ1(𝔉(11ℨ) βˆ’ 1,88,30,57,02,60,326𝔉(ℨ) + 1,56,92,14,18,83,605𝔉(βˆ’β„¨),β„³) β‰₯ { π’œ1β€²(β„’,β„³), π’œ1β€²(β„’(||ℨ|| π‘Ÿ),β„³), π’œ2(𝔉(11ℨ) βˆ’ 1,88,30,57,02,60,326𝔉(ℨ) + 1,56,92,14,18,83,605𝔉(βˆ’β„¨),β„³) ≀ { π’œ2β€²(β„’,β„³), π’œ2β€²(β„’(||ℨ|| π‘Ÿ),β„³), π’œ3(𝔉(11ℨ) βˆ’ 1,88,30,57,02,60,326𝔉(ℨ) + 1,56,92,14,18,83,605𝔉(βˆ’β„¨),β„³) ≀ { π’œ3β€²(β„’,β„³), π’œ3β€²(β„’(||ℨ|| π‘Ÿ),β„³), } (40) for all ℨ ∈ 𝒳𝑀 and all β„³ > 0. Then there exists a unique tridecic function 𝒯:𝒳𝑀 βŸΆπ’΄π‘€ and a unique duodecic function π’Ÿ:𝒳𝑀 βŸΆπ’΄π‘€ such that π’œ1(𝔉(ℨ) βˆ’ 𝒯(ℨ) βˆ’ π’Ÿ(ℨ),β„³) β‰₯ { π’œ1β€²(β„’, |11 13 βˆ’ 1|β„³) βˆ— π’œ1β€²(β„’, |11 12 βˆ’ 1|β„³), π’œ1β€²(β„’||ℨ|| π‘Ÿ , |1113 βˆ’ 11π‘Ÿ|β„³) βˆ— π’œ1β€²(β„’||ℨ|| π‘Ÿ , |1112 βˆ’ 11π‘Ÿ|β„³), π’œ2(𝔉(ℨ) βˆ’ 𝒯(ℨ) βˆ’ π’Ÿ(ℨ),β„³) ≀ { π’œ2β€²(β„’, |11 13 βˆ’ 1|β„³) β‹„ π’œ2β€²(β„’, |11 12 βˆ’ 1|β„³), π’œ2β€²(β„’||ℨ|| π‘Ÿ , |1113 βˆ’ 11π‘Ÿ|β„³) β‹„ π’œ2β€²(β„’||ℨ|| π‘Ÿ , |1112 βˆ’ 11π‘Ÿ|β„³), π’œ3(𝔉(ℨ) βˆ’ 𝒯(ℨ) βˆ’ π’Ÿ(ℨ),β„³) ≀ { π’œ3β€²(β„’, |11 13 βˆ’ 1|β„³)βŠ˜π’œ3β€²(β„’, |11 12 βˆ’ 1|β„³), π’œ3β€²(β„’||ℨ|| π‘Ÿ , |1113 βˆ’ 11π‘Ÿ|β„³)βŠ˜π’œ3β€²(β„’||ℨ|| π‘Ÿ , |1112 βˆ’ 11π‘Ÿ|β„³), } (41) for all ℨ ∈ 𝒳𝑀 and all β„³ > 0 . 4 Conclusion The numerical stability analysis conducted in this study underscores the viability and robustness of mixed-type duodecic and tridecic functional equations in neutrosophic normed spaces. 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