STABILITY OF A AFFINE TYPE AQ FUNCTIONAL EQUATION IN VARIOUS BANACH SPACES S. PINELAS1, M. ARUNKUMAR2, E. SATHYA3, V. ALEXPANDIYAN4, V. CHANDIRAN5, T. VELMURUGAN6 1Departamento de Ciencias Exatas e Engenharia, Academia Militar , Av. Conde Castro Guimaraes, 2720-113 Amadora, Portugal Center for Research and Development in Mathematics and Applications (CIDMA), Departamento de Matemtica, Universidade de Aveiro, 3810-193 Aveiro. e-mail:sandra.pinelas@gmail.com; 2,3,4,5Department of Mathematics, Kalaignar Karunanidhi Government Arts College, (Affiliated to Thiruvalluvar University), Tiruvannamalai - 606 603, TamilNadu, India. e-mail: drarun4maths@gmail.com; sathya24mathematics@gmail.com; e-mail: valexpandiyan98@gmail.com; chandhiranphd@gmail.com; 6Department of Mathematics, MRK College of Arts and Science, Pazhanchanallur, Kattumannarkoil - 608 301,Tamil Nadu, India. f-e-mail:smmuruganvel@gmail.com 1. INTRODUCTION S.M. Ulam’s question [31] in 1940 rewoke the journey of the research in the stability theory of func- tional equations. Many mathematicians have studied and published several novel results in the field of stability theory, such as, D.H. Hyers (1941) [14] , T. Aoki (1950) [2], Th.M. Rassias (1978) [24], J.M. Rassias (1982) [23], P. Gavruta (1994) [13], and K. Ravi, M. Arunkumar, J.M. Rassias (2008) [26]. Famous functional equations for additive and quadratic functions are F (w1 + w2) = F (w1) +F (w2), (1.1) and F (w1 + w2) +F (w1 − w2) = 2F (w1) + 2F (w2). (1.2) S.M. Jung [15], PL. Kannappan [16], and Th.M. Rassias [25] discussed the general solution and gen- eralized Ulam - Hyers stability of different forms of functional equations in various normed spaces. In fact, M. Arunkumar et. al., [3], M. Arunkumar, J.M. Rassis [4], M. Arunkumar et. al., [5, 6], A. Bodaghi [8], and references therein establish the general solution and generalized Hyers-Ulam stability of the several AQ functional equations. 2010 Mathematics Subject Classification. :39B52, 32B72, 32B82 . Key words and phrases. : Mixed functional equations, Generalized Ulam - Hyers stability, Direct Method, Fixed Method, Banach space, Intuitionistic Fuzzy Banach space. Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 32 No. 10s(2025) https://internationalpubls.com 2186 ABSTRACT. In this article, a new affine type AQ functional equations is proposed. The generalized Ulam-Hyers stability of this equations is analyzed using the product, sum, and mixed product-sum of powers of norms, as well as the general control function. The stability analysis is carried out in Banach space and Intu-itionistic Fuzzy Banach spaces using Hyers direct method. Also, we examine the stability of same functional equation by using Radus Fixed point method in both the spaces. Article History: Received: 14-01-2025 Revised: 16-02-2025 Accepted: 05-03-2025 L. Lucht, C. Methfessel [17] proposed affine functional equations and recurrent sequences in 1993. Additionally, in 2013, L. Cadariu, L. Gavruta, and P. Gavruta [11] demonstrated the generalized Hyers- Ulam stability and obtained the general solution for an affine functional equation of the form f (2x + y) + f (x + 2y) + f (x) + f (y) = 4 f (x + y + z) (1.3) by using the direct method as well as the fixed point method. Infact, in 2014, M. Mursaleen, KJ. Ansari [19] considered the following affine functional equation f (3x + y + z) + f (x + 3y + z) + f (x + y + 3z) + f (x) + f (y) + f (z) = 6 f (x + y + z) (1.4) and find its general solution and proved some stability results by using direct method as well as the fixed point method. Also in 2015, Md. Nasiruzzaman [21] provide the fuzzy version Hyers-Ulam- Rassias stability of (1.4) . Further, in 2016 M. Mursaleen, KJ. Ansari[20] prove the general solution of the following affine functional equation f (kx1 + x2 + · · ·+ xk) + f (x1 + kx2 + · · ·+ xk) + · · ·+ f (x1 + x2 + · · ·+ kxk) + f (x1) + f (x2) + · · ·+ f (xk) = 2k f (x1 + x2 + · · ·+ xk), k ≥ 2. (1.5) and established the Hyers-Ulam-Rassias stability of the above functional equation in the fuzzy normed spaces which as an generalized version of (1.4). Recently, C. Benzarouala et.al., [9, 10] proved the general Ulam stability result for the functional equation m ∑ i=1 Ai f ( n ∑ j=1 aijxj ) = D(x1, · · · , xn), (1.6) in the class of functions f mapping a module X, over a commutative ring K, into a Banach space Y, where m and n are fixed positive integers, aij ∈ K for every i ∈ {1, · · · , m} and j ∈ {1, · · · n}, A1, · · · , Am are scalars, and the function D : Xn → Y is fixed. Numerous important functional equations are particular cases of A′is the homogeneous version of (1.6) are Cauchy, Jensen, Jordanvon Neumann, Drygas, Frechet, Popoviciu, Wright and many others. Also, for particular cases of A′is in (1.6), we get functional equations , like equation in a single variable, cohomological equation, Schroder equation, Abel equation and many others. The stability of (1.6) in random normed spaces has been studied by C. Benzarouala et.al., [10]. Inspired by the aforementioned information and study findings, in this paper we present a novel affine type additive quadratic mixed functional equation of the form F (3w1 + w2 + w3) +F (w1 + 3w2 + w3) +F (w1 + w2 + 3w3) = 6F ( 3 ∑ ψ=1 wψ ) + 1 2 { F ( 3 ∑ ψ=1 wψ ) +F ( − 3 ∑ ψ=1 wψ )} − 3 ∑ ψ=1 { F (wψ)− 5 2 [ F (wψ) +F (−wψ) ]} . (1.7) We analyze the stability in the sense of Ulam, Hyers, Rassias’s, Gavruta and Radu of the above affine type AQ Functional Equation in Banach Space and Intuitionistic Fuzzy Banach Space using Direct and Fixed Methods. Remark 1.1. The homogeneous version of (1.6) for m = n = 3, A1 = A2 = A3 = 1, a11 = a22 = a33 = 3 and a12 = a13 = a21 = a23 = a31 = a32 = 1 is f (3x1 + x2 + x3) + f (x1 + 3x2 + x3) + f (x1 + x2 + 3x3) = 0. (1.8) So, we cant get our functional equation (1.7) from (1.6). Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 32 No. 10s(2025) https://internationalpubls.com 2187 Remark 1.2. In the functional equation (1.6), for n = 3, m = 15, D = 0, A1 = A2 = A3 = 1, a11 = a22 = a33 = 3, a12 = a13 = a21 = a23 = a31 = a32 = 1, A4 = −6, a41 = a42 = a43 = 1, A5 = A6 = −1 2 , a51 = a52 = a53 = 1, a61 = a62 = a63 = −1 A7 = A8 = A9 = 1, a71 = 1, a72 = a73 = 0, a81 = a83 = 0, a82 = 1, a91 = a92 = 0, a93 = 1, A10 = A11 = A12 = A13 = A14 = A15 = −5 2 , a10 1 = 1, a10 2 = a10 3 = 0, a11 1 = a11 3 = 0, a11 2 = 1, a12 1 = a12 2 = 0, a12 3 = 1, a13 1 = −1, a13 2 = a13 3 = 0, a14 1 = a14 3 = 0, a14 2 = −1, a15 1 = a15 2 = 0, a15 3 = −1. So, after giving particular values to A′is and a′is, we get our functional equation (1.7) from (1.6). Moreover, the results in the manuscript under review complement of the results in [9, 10]. Lemma 1.3. [21] Let A and B be real vector spaces. Suppose F : A → B be an odd mapping satisfying (1.7). Then F is additive. Lemma 1.4. [12] Let A and B be real vector spaces. Suppose F : A → B be an even mapping satisfying (1.7). Then F is quadratic. Now, we present the result due to Margolis, Diaz [18] and Radu [22] for fixed point theory. Theorem 1.5. [18, 22] Suppose that for a complete generalized metric space (Ω, δ) and a strictly contractive mapping T : Ω −→ Ω with Lipschitz constant L. Then, for each given x ∈ Ω , either d(Tnx, Tn+1x) = ∞ ∀ n ≥ 0, or there exists a natural number n0 such that (FPC1) d(Tnx, Tn+1x) < ∞ for all n ≥ n0 ; (FPC2) The sequence (Tnx) is convergent to a fixed point y∗ of T (FPC3) y∗ is the unique fixed point of T in the set ∆ = {y ∈ Ω : d(Tn0 x, y) < ∞}; (FPC4) d(y∗, y) ≤ 1 1−L d(y, Ty) for all y ∈ ∆. 2. STABILITY OF (1.7) IN BANACH SPACES In this section, we explore the generalized Ulam - Hyers stability of the functional equation (1.7) in Banach space. To prove the stability results, let us takeW1 be a normed space andW2 be a Banach space. Suppose that F :W1 →W2 and Ψ :W3 1 → [0, ∞) satisfying the following functional inequalities ∥∥∥F (3w1 + w2 + w3) +F (w1 + 3w2 + w3) +F (w1 + w2 + 3w3)− 6F ( 3 ∑ ψ=1 wψ ) − 1 2 { F ( 3 ∑ ψ=1 wψ ) +F ( − 3 ∑ ψ=1 wψ )} + 3 ∑ ψ=1 { F (wψ)− 5 2 [ F (wψ) +F (−wψ) ]} ∥∥∥ ≤ Ψ (w1, w2, w3) , (2.1) Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 32 No. 10s(2025) https://internationalpubls.com 2188 and ∥∥∥F (3w1 + w2 + w3) +F (w1 + 3w2 + w3) +F (w1 + w2 + 3w3)− 6F ( 3 ∑ ψ=1 wψ ) − 1 2 { F ( 3 ∑ ψ=1 wψ ) +F ( − 3 ∑ ψ=1 wψ )} + 3 ∑ ψ=1 { F (wψ)− 5 2 [ F (wψ) +F (−wψ) ]} ∥∥∥ ≤  δ, δ 3 ∑ ψ=1 ∣∣wψ ∣∣ϕ , δ 3 ∑ ψ=1 ∣∣wψ ∣∣ϕψ , δ 3 ∏ ψ=1 ∣∣wψ ∣∣ϕ , δ 3 ∏ ψ=1 ∣∣wψ ∣∣ϕψ , δ { 3 ∑ ψ=1 ∣∣wψ ∣∣3ϕ + 3 ∏ ψ=1 ∣∣wψ ∣∣ϕ} , (2.2) for all w1, w2, w3 ∈ W1, δ be a positive constant and ϕ be any real number. 2.1. Oddness of F : Additive Case Stability Results : Direct Method. Theorem 2.1. Suppose that an odd function F : W1 → W2 satisfy the functional inequality (2.1) where Ψ :W3 1 → [0, ∞) with the condition lim `→∞ Ψ ( 5`mw1, 5`mw2, 5`mw3 ) 5`m = 0; µ = ±1, (2.3) for all w1, w2, w3 ∈ W1. Then there exists a unique additive mapping A(w1) : W1 → W2 which satisfies (1.7) and the functional inequality ‖F (w1)−A(w1)‖ ≤ 1 5 ∞ ∑ η= 1−µ 2 1 5ηµ ΨA (5ηµw1) (2.4) = 1 5 ∞ ∑ η= 1−µ 2 1 5ηµ { 1 3 { Ψ (5ηµw1, 5ηµw1, 5ηµw1) + 3Ψ (5ηµw1, 5ηµw1,−5ηµw1) }} , (2.5) and the mapping A(w1) is obtained by A(w1) = lim `→∞ 1 5`mF ( 5`mw1 ) , (2.6) for all w1 ∈ W1. Proof. Using oddness of F in (2.1), we get∥∥∥F (3w1 + w2 + w3) +F (w1 + 3w2 + w3) +F (w1 + w2 + 3w3)− 6F ( 3 ∑ ψ=1 wψ ) + 3 ∑ ψ=1 F (wψ) ∥∥∥ ≤ Ψ (w1, w2, w3) , ∀ w1, w2, w3 ∈ W1. (2.7) Interchanging (w1, w2, w3) by (w1, w1, w1) in (2.7), we obtain∥∥∥3F (5w1)− 6F (3w1) + 3F (w1) ∥∥∥ ≤ Ψ (w1, w1, w1) , ∀ w1 ∈ W1. (2.8) Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 32 No. 10s(2025) https://internationalpubls.com 2189 Again interchanging (w1, w2, w3) by (w1, w1,−w1) in (2.7), we have∥∥∥2F (3w1)− 6F (w1) ∥∥∥ ≤ Ψ (w1, w1,−w1) ⇒ ∥∥∥6F (3w1)− 18F (w1) ∥∥∥ ≤ 3Ψ (w1, w1,−w1) , ∀ w1 ∈ W1. (2.9) Combining (2.8) and (2.9), we arrive∥∥∥3F (5w1)− 15F (w1) ∥∥∥ ≤ ∥∥∥3F (5w1)− 6F (3w1) + 3F (w1) ∥∥∥+ ∥∥∥6F (3w1)− 18F (w1) ∥∥∥ ≤ Ψ (w1, w1, w1) + 3Ψ (w1, w1,−w1) , ∀ w1 ∈ W1. (2.10) One can see from (2.10) that∥∥∥F (5w1)− 5F (w1) ∥∥∥ ≤ 1 3 { Ψ (w1, w1, w1) + 3Ψ (w1, w1,−w1) } = ΨA (w1) , ∀ w1 ∈ W1. (2.11) It follows from (2.11) that ∥∥∥1 5 F (5w1)−F (w1) ∥∥∥ ≤ 1 5 ΨA (w1) , ∀ w1 ∈ W1. (2.12) Generalizing for a positive integer `, we get∥∥∥ 1 5` F (5`w1)−F (w1) ∥∥∥ ≤ 1 5 ` ∑ η=0 1 5η ΨA (5ηw1) , ∀ w1 ∈ W1. (2.13) Now, changing w1 by 5`1 w1 in (2.13), we obtain∥∥∥ 1 5`+`1 F (5`+`1 w1)− 1 5`1 F (5`1 w1) ∥∥∥ = 1 5`1 ∥∥∥ 1 5` F (5`+`1 w1)−F (5`1 w1) ∥∥∥ ≤ 1 5 ` ∑ η=0 1 5η+`1 ΨA ( 5η+`1 w1 ) → 0 as `1 → ∞, ∀ w1 ∈ W1. (2.14) Therefore, the sequence { 1 5` F (5`w1) } , is a Cauchy sequence and it converges to A(w1) inW2. So, we define A(w1) = lim `→∞ 1 5` F ( 5`w1 ) , ∀ w1 ∈ W1. (2.15) Taking limit `→ ∞ in (2.13), we have∥∥∥A(w1)−F (w1) ∥∥∥ ≤ 1 5 ∞ ∑ η=0 1 5η ΨA (5ηw1) , ∀ w1 ∈ W1. (2.16) Thus, (2.4) and (2.5) holds for µ = 1. Interchanging (w1, w2, w3) = ( 5`w1, 5`w2, 5`w3 ) , we arrive 1 5` ∥∥∥F (5`(3w1 + w2 + w3)) +F (5`(w1 + 3w2 + w3)) +F (5`(w1 + w2 + 3w3))− 6F ( 3 ∑ ψ=1 5`wψ ) − 1 2 { F ( 3 ∑ ψ=1 5`wψ ) +F ( − 3 ∑ ψ=1 5`wψ )} + 3 ∑ ψ=1 { F (5`wψ)− 5 2 [ F (5`wψ) +F (−5`wψ) ]} ∥∥∥ ≤ 1 5` Ψ ( 5`w1, 5`w2, 5`w3 ) , ∀w1, w2, w3 ∈ W1. (2.17) Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 32 No. 10s(2025) https://internationalpubls.com 2190 Taking limit `→ ∞ in (2.17), using (2.15) and (2.3), we get A(3w1 + w2 + w3) +A(w1 + 3w2 + w3) +A(w1 + w2 + 3w3) = 6A ( 3 ∑ ψ=1 wψ ) + 1 2 { A ( 3 ∑ ψ=1 wψ ) +A ( − 3 ∑ ψ=1 wψ )} − 3 ∑ ψ=1 { A(wψ)− 5 2 [ A(wψ) +A(−wψ) ]} , for all w1, w2, w3 ∈ W1. So, A(w1) satisfies (1.7). In order to confirm that A(w1) is unique, suppose B(w1) be another mapping (1.7), (2.15) and (2.16), we obtain∥∥∥A(w1)−B(w1) ∥∥∥ = ∥∥∥ 1 5` A ( 5`w1 ) − 1 5` B ( 5`w1 ) ∥∥∥ ≤ 1 5` ∥∥∥A (5`w1 ) −F ( 5`w1 ) ∥∥∥+ 1 5` ∥∥∥F (5`w1 ) −B ( 5`w1 ) ∥∥∥ ≤ 2 5 ∞ ∑ η=0 1 5η+` ΨA ( 5η+`w1 ) → 0 as `1 → ∞, for all w1 ∈ W1. Therefore A(w1) is unique. So, the Theorem holds for µ = 1. Changing w1 = w1 5 in (2.11), we have∥∥∥F (w1)− 5F (w1 5 ) ∥∥∥ ≤ ΨA (w1 5 ) , ∀ w1 ∈ W1. (2.18) Generalizing for a positive integer `, we get∥∥∥F (w1)− 5`F (w1 5η ) ∥∥∥ ≤ 1 5 ` ∑ η=1 5η ΨA (w1 5η ) , ∀ w1 ∈ W1. (2.19) The rest of the proof is similar to that of above case. So, the Theorem holds for µ = −1. Hence the proof is complete � Corollary 2.2. Suppose that an odd function F : W1 → W2 satisfy the functional inequality (2.2) for all w1, w2, w3 ∈ W1. Then there exists a unique additive mapping A(w1) :W1 →W2 which satisfies (1.7) and the functional inequality ‖F (w1)−A(w1)‖ ≤  δ |3| , 4δ|w1|ϕ |5−5ϕ | ; ϕ 6= 1, 4δ 3 3 ∑ ψ=1 |wψ | ϕψ |5−5 ϕψ | ; ϕ1, ϕ2, ϕ3 6= 1, 4δ|w1|3ϕ 3|5−53ϕ | ; 3ϕ 6= 1, 4δ|wψ | 3 ∑ ψ=1 ϕψ 3 ∣∣∣5−5 3 ∑ ψ=1 ϕψ ∣∣∣ ; 3 ∑ ψ=1 ϕψ 6= 1, 16δ|w1|3ϕ 3|5−53ϕ | ; 3ϕ 6= 1, (2.20) for all w1 ∈ W1. 2.2. Evenness of F : Quadratic Case Stability Results : Direct Method. Theorem 2.3. Suppose that an even function F : W1 → W2 satisfy the functional inequality (2.1) where Ψ :W3 1 → [0, ∞) with the condition lim `→∞ Ψ ( 5`mw1, 5`mw2, 5`mw3 ) 25`m = 0; µ = ±1, (2.21) Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 32 No. 10s(2025) https://internationalpubls.com 2191 for all w1, w2, w3 ∈ W1. Then there exists a unique quadratic mapping Q(w1) :W1 →W2 which satisfies (1.7) and the functional inequality ‖F (w1)−Q(w1)‖ ≤ 1 25 ∞ ∑ η= 1−µ 2 1 25ηµ ΨQ (5ηµw1) (2.22) = 1 25 ∞ ∑ η= 1−µ 2 1 25ηµ { 1 3 { Ψ (5ηµw1, 5ηµw1, 5ηµw1) + 7 2 Ψ (5ηµw1, 5ηµw1,−5ηµw1) }} , (2.23) and the mapping Q(w1) is obtained by Q(w1) = lim `→∞ 1 25`mF ( 5`mw1 ) , (2.24) for all w1 ∈ W1. Proof. Using evenness of F in (2.1), we get ∥∥∥F (3w1 + w2 + w3) +F (w1 + 3w2 + w3) +F (w1 + w2 + 3w3)− 7F ( 3 ∑ ψ=1 wψ ) − 4 3 ∑ ψ=1 F (wψ) ∥∥∥ ≤ Ψ (w1, w2, w3) , ∀ w1, w2, w3 ∈ W1. (2.25) Interchanging (w1, w2, w3) by (w1, w1, w1) in (2.25), we obtain∥∥∥3F (5w1)− 7F (3w1)− 12F (w1) ∥∥∥ ≤ Ψ (w1, w1, w1) , ∀ w1 ∈ W1. (2.26) Again interchanging (w1, w2, w3) by (w1, w1,−w1) in (2.25), we have∥∥∥2F (3w1)− 18F (w1) ∥∥∥ ≤ Ψ (w1, w1,−w1) ⇒ ∥∥∥7F (3w1)− 63F (w1) ∥∥∥ ≤ 7 2 Ψ (w1, w1,−w1) , ∀ w1 ∈ W1. (2.27) Combining (2.26) and (2.27), we arrive∥∥∥3F (5w1)− 75F (w1) ∥∥∥ ≤ ∥∥∥3F (5w1)− 7F (3w1)− 12F (w1) ∥∥∥+ ∥∥∥7F (3w1)− 63F (w1) ∥∥∥ ≤ Ψ (w1, w1, w1) + 7 2 Ψ (w1, w1,−w1) , ∀ w1 ∈ W1. (2.28) One can see from (2.28) that∥∥∥F (5w1)− 25F (w1) ∥∥∥ ≤ 1 3 { Ψ (w1, w1, w1) + 7 2 Ψ (w1, w1,−w1) } = ΨQ (w1) , ∀ w1 ∈ W1. (2.29) It follows from (2.29) that ∥∥∥ 1 25 F (5w1)−F (w1) ∥∥∥ ≤ 1 25 ΨQ (w1) , ∀ w1 ∈ W1. (2.30) The rest of the proof is similar to that of Theorem 2.1. Hence the proof is complete. � Corollary 2.4. Suppose that an even function F : W1 → W2 satisfy the functional inequality (2.2) for all w1, w2, w3 ∈ W1. Then there exists a unique quadratic mapping Q(w1) : W1 → W2 which satisfies (1.7) and Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 32 No. 10s(2025) https://internationalpubls.com 2192 the functional inequality ‖F (w1)−Q(w1)‖ ≤  3δ 2|24| , 27δ|w1|ϕ 6|25−5ϕ | ; ϕ 6= 2, 9δ 6 3 ∑ ψ=1 |wψ | ϕψ |25−5 ϕψ | ; ϕ1, ϕ2, ϕ3 6= 2, 9δ|w1|3ϕ 6|25−53ϕ | ; 3ϕ 6= 2, 9δ|wψ | 3 ∑ ψ=1 ϕψ 6 ∣∣∣25−5 3 ∑ ψ=1 ϕψ ∣∣∣ ; 3 ∑ ψ=1 ϕψ 6= 2, 36δ|w1|3ϕ 6|25− 53ϕ| ; 3ϕ 6= 2, (2.31) for all w1 ∈ W1. 2.3. Oddness and Evenness of F : Additive Quadratic Case Stability Results : Direct Method. Theorem 2.5. Suppose that a function F : W1 → W2 satisfy the functional inequality (2.1) where Ψ : W3 1 → [0, ∞) with the conditions (2.3) and (2.21) for all w1, w2, w3 ∈ W1. Then there exists a unique additive mapping A(w1) :W1 →W2 and a unique quadratic mappingQ(w1) :W1 →W2 which satisfies (1.7) and the functional inequality ‖F (w1)−A(w1)−Q(w1)‖ ≤ 1 2 1 5 ∞ ∑ η= 1−µ 2 1 5ηµ { ΨA (5ηµw1) + ΨA (−5ηµw1) } + 1 25 ∞ ∑ η= 1−µ 2 1 25ηµ { ΨQ (5ηµw1) + ΨQ (−5ηµw1) } ≤ 1 2 1 5 ∞ ∑ η= 1−µ 2 1 5ηµ { 1 3 { Ψ (5ηµw1, 5ηµw1, 5ηµw1) + 3Ψ (5ηµw1, 5ηµw1,−5ηµw1) } + 1 3 { Ψ (−5ηµw1,−5ηµw1,−5ηµw1) + 3Ψ (−5ηµw1,−5ηµw1, 5ηµw1) }} + 1 25 ∞ ∑ η= 1−µ 2 1 25ηµ { 1 3 { Ψ (5ηµw1, 5ηµw1, 5ηµw1) + 7 2 Ψ (5ηµw1, 5ηµw1,−5ηµw1) } + 1 3 { Ψ (−5ηµw1,−5ηµw1,−5ηµw1) + 7 2 Ψ (−5ηµw1,−5ηµw1, 5ηµw1) }}} , (2.32) and the mapping A(w1) and Q(w1) are given in (2.6) and (2.24) for all w1 ∈ W1. Proof. Consider a function Fodd(w1) by Fodd(w1) = 1 2 { F (w1)−F (−w1) } , ∀ w1 ∈ W1, (2.33) which gives Fodd(0) = 0; Fodd(−w1) = −Fodd(w1), ∀ w1 ∈ W1. (2.34) Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 32 No. 10s(2025) https://internationalpubls.com 2193 By Theorem 2.1, it follows from (2.33), (2.1), (2.5) and (2.6), we arrive ‖Fodd(w1)−A(w1)‖ ≤ 1 2 · 1 5 ∞ ∑ η= 1−µ 2 1 5ηµ { ΨA (5ηµw1) + ΨA (−5ηµw1) } (2.35) = 1 2 · 1 5 ∞ ∑ η= 1−µ 2 1 5ηµ { 1 3 { Ψ (5ηµw1, 5ηµw1, 5ηµw1) + 3Ψ (5ηµw1, 5ηµw1,−5ηµw1) } + 1 3 { Ψ (−5ηµw1,−5ηµw1,−5ηµw1) + 3Ψ (−5ηµw1,−5ηµw1, 5ηµw1) }} , (2.36) for all w1 ∈ W1. Consider a function Feven(w1) by Feven(w1) = 1 2 { F (w1) +F (−w1) } , ∀ w1 ∈ W1, (2.37) which gives Feven(0) = 0; Feven(−w1) = Feven(w1), ∀ w1 ∈ W1. (2.38) By Theorem 2.3, it follows from (2.37), (2.1), (2.22) and (2.23), we see ‖Feven(w1)−Q(w1)‖ ≤ 1 2 · 1 25 ∞ ∑ η= 1−µ 2 1 25ηµ { ΨQ (5ηµw1) + ΨQ (−5ηµw1) } (2.39) = 1 2 · 1 25 ∞ ∑ η= 1−µ 2 1 25ηµ { 1 3 { Ψ (5ηµw1, 5ηµw1, 5ηµw1) + 7 2 Ψ (5ηµw1, 5ηµw1,−5ηµw1) } + 1 3 { Ψ (−5ηµw1,−5ηµw1,−5ηµw1) + 7 2 Ψ (−5ηµw1,−5ηµw1, 5ηµw1) }} , (2.40) for all w1 ∈ W1. Assume a function F (w1) by F (w1) = Fodd(w1) +Feven(w1), ∀ w1 ∈ W1. (2.41) Now, it follows from (2.35), (2.36), (2.39), (2.40) and (2.41), we have ‖F (w1)−A(w1)−Q(w1)‖ ≤ ‖Fodd(w1)−A(w1)‖+ ‖Feven(w1)−Q(w1)‖ ≤ 1 2 1 5 ∞ ∑ η= 1−µ 2 1 5ηµ { ΨA (5ηµw1) + ΨA (−5ηµw1) } + 1 25 ∞ ∑ η= 1−µ 2 1 25ηµ { ΨQ (5ηµw1) + ΨQ (−5ηµw1) } ≤ 1 2 1 5 ∞ ∑ η= 1−µ 2 1 5ηµ { 1 3 { Ψ (5ηµw1, 5ηµw1, 5ηµw1) + 3Ψ (5ηµw1, 5ηµw1,−5ηµw1) } + 1 3 { Ψ (−5ηµw1,−5ηµw1,−5ηµw1) + 3Ψ (−5ηµw1,−5ηµw1, 5ηµw1) }} + 1 25 ∞ ∑ η= 1−µ 2 1 25ηµ { 1 3 { Ψ (5ηµw1, 5ηµw1, 5ηµw1) + 7 2 Ψ (5ηµw1, 5ηµw1,−5ηµw1) } + 1 3 { Ψ (−5ηµw1,−5ηµw1,−5ηµw1) + 7 2 Ψ (−5ηµw1,−5ηµw1, 5ηµw1) }}} , for all w1, w2, w3 ∈ W1. � Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 32 No. 10s(2025) https://internationalpubls.com 2194 Corollary 2.6. Suppose that a functionF :W1 →W2 satisfy the functional inequality (2.2) for all w1, w2, w3 ∈ W1. Then there exists a unique additive mapping A(w1) :W1 →W2 and a unique quadratic mapping Q(w1) : W1 →W2 which satisfies (1.7) and the functional inequality ‖F (w1)−A(w1)−Q(w1)‖ ≤  δ |3| + 3δ 2|24| , 4δ|w1|ϕ |5−5ϕ | + 27δ|w1|ϕ 6|25−5ϕ | ; ϕ 6= 1, 2, 4δ 3 3 ∑ ψ=1 |wψ | ϕψ |5−5 ϕψ | + 9δ 6 3 ∑ ψ=1 |wψ | ϕψ |25−5 ϕψ | ; ϕ1, ϕ2, ϕ3 6= 1, 2, 4δ|w1|3ϕ 3|5−53ϕ | + 9δ|w1|3ϕ 6|25−53ϕ | ; 3ϕ 6= 1, 2, 4δ|wψ | 3 ∑ ψ=1 ϕψ 3 ∣∣∣5−5 3 ∑ ψ=1 ϕψ ∣∣∣ + 9δ|wψ | 3 ∑ ψ=1 ϕψ 6 ∣∣∣25−5 3 ∑ ψ=1 ϕψ ∣∣∣ ; 3 ∑ ψ=1 ϕψ 6= 1, 2, 16δ|w1|3ϕ 3|5−53ϕ | + 36δ|w1|3ϕ 6|25−53ϕ | ; 3ϕ 6= 1, 2, (2.42) for all w1 ∈ W1. 2.4. Oddness of F : Additive Case Stability Results : Fixed Point Method. Theorem 2.7. Suppose that an odd function F : W1 → W2 satisfy the functional inequality (2.1) where Ψ :W3 1 → [0, ∞) with the condition lim `→∞ Ψ ( τ` v w1, τ` v w2, τ` v w3 ) τ` v = 0; τv = { 5; ν = 0 1 5 ; ν = 1 , ∀ w1, w2, w3 ∈ W1. (2.43) If there exists L = L(ν) be a function have the property ΨA(w1) = ΨA (w1 5 ) and 1 τv ΨA (τvw1) = L ΨA(w1), ∀ w1 ∈ W1. (2.44) Then there exists a unique additive mappingA(w1) :W1 →W2 which satisfies (1.7) and the functional inequal- ity ‖F (w1)−A(w1)‖ ≤ L1−ν 1− L ΨA (w1) (2.45) = L1−ν 1− L { 1 3 { Ψ (w1, w1, w1) + 3Ψ (w1, w1,−w1) }} , (2.46) and the mapping A(w1) is obtained by A(w1) = lim `→∞ 1 τ` v F ( τ` v w1 ) , (2.47) for all w1 ∈ W1. Proof. Assume a set G = {F/F :W1 →W2, F (0) = 0}, (2.48) and introduce the generalized metric on the above set G as d(F ,F1) = inf{K ∈ (0, ∞) : ‖F (w1)−F1(w1)‖ ≤ K Ψ(w1, w1, w1), w1 ∈ W1}. (2.49) It is easy to see that (G, d) is complete. Define a functionH : G → G by HF (w1 ) = 1 τv F (τv w1 ), f or all w1 ∈ W1. (2.50) Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 32 No. 10s(2025) https://internationalpubls.com 2195 Now F , F1 ∈ G and w1 ∈ W1, we see d(F ,F1) ≤ K ⇒ ‖ F (w1)−F1(w1) ‖≤ K Ψ(w1, w1, w1), ⇒ ∥∥∥∥ 1 τv F (τvw1)− 1 τv F1(τvw1) ∥∥∥∥ ≤ 1 τv K Ψ(τvw1, τvw1, τvw1), ⇒ ‖ HF (w1)−HF1( w1 ) ‖≤ L K Ψ(w1, w1, w1), ⇒d(HF ,HF1) ≤ L K, i.e.,H is a strictly contractive mapping on G with Lipschitz constant L (see [18]). For the case ν = 0, it follows from (2.12) and with the help of (2.44), (2.50), (2.49), we get∥∥∥1 5 F (5w1)−F (w1) ∥∥∥ ≤ 1 5 ΨA (w1) ,⇒ d(HF ,F ) ≤ L = L1−ν, ∀ w1 ∈ W1. (2.51) For the case ν = 1, it follows from (2.18) and with the help of (2.44), (2.50), (2.49), we obtain∥∥∥F (w1)− 5F (w1 5 ) ∥∥∥ ≤ ΨA (w1 5 ) ,⇒ d(F ,HF ) ≤ 1 = L1−ν, ∀ w1 ∈ W1. (2.52) Combining (2.51) and (2.52), we have d(F ,HF ) ≤ 1 = L1−ν. (2.53) Therefore (FPC1) of Theorem 1.5 holds. The rest of the proof follows by Theorem 1.5. Hence the proof is complete. � Corollary 2.8. Suppose that an odd function F : W1 → W2 satisfy the functional inequality (2.2) for all w1, w2, w3 ∈ W1. Then there exists a unique additive mapping A(w1) :W1 →W2 which satisfies (1.7) and the functional inequality (2.20) for all w1 ∈ W1. Proof. If we take Ψ (w1, w2, w3) =  δ, δ ∑3 ψ=1 ∣∣wψ ∣∣ϕ , δ ∑3 ψ=1 ∣∣wψ ∣∣ϕψ , δ ∏3 ψ=1 ∣∣wψ ∣∣ϕ , δ ∏3 ψ=1 ∣∣wψ ∣∣ϕψ , δ { ∑3 ψ=1 ∣∣wψ ∣∣3ϕ + ∏3 ψ=1 ∣∣wψ ∣∣ϕ} , (2.54) in Theorem 2.7 and changing (w1, w2, w3) by ( τ` v w1, τ` v w2, τ` v w3 ) and dividing by τ` v in (2.54), one can see 1 τ` v Ψ ( τ` v w1, τvw` 2, τ` v w3 ) =  δ τ`v → 0 as ` to ∞, δ τ`v ∑3 ψ=1 ∣∣∣τ` v wψ ∣∣∣ϕ , → 0 as ` to ∞, δ τ`v ∑3 ψ=1 ∣∣∣τ` v wψ ∣∣∣ϕψ , → 0 as ` to ∞, δ τ`v ∏3 ψ=1 ∣∣∣τ` v wψ ∣∣∣ϕ , → 0 as ` to ∞, δ τ`v ∏3 ψ=1 ∣∣∣τ` v wψ ∣∣∣ϕψ , → 0 as ` to ∞, δ τ`v { ∑3 ψ=1 ∣∣∣τ` v wψ ∣∣∣3ϕ + ∏3 ψ=1 ∣∣∣τ` v wψ ∣∣∣ϕ} , → 0 as ` to ∞. Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 32 No. 10s(2025) https://internationalpubls.com 2196 Therefore (2.43) holds for all w1, w2, w3 ∈ W1. Now, from (2.44), we have ΨA(w1) = ΨA (w1 5 ) = 1 3 { Ψ (w1 5 , w1 5 , w1 5 ) + 3Ψ (w1 5 , w1 5 ,−w1 5 ) } =  4δ 3 , 12| w1 5 | ϕ 3 , 4δ 3 ∑3 ψ=1 ∣∣w1 5 ∣∣ϕψ , 4δ| w1 5 | 3ϕ 3 , 4δ| w1 5 | 3 ∑ ψ=1 ϕψ 3 , 16δ| w1 5 | 3ϕ 3 , (2.55) 1 τv ΨA (τvw1) = 1 τv 1 3 { Ψ (τvw1, τvw1, τvw1) + 3Ψ (τvw1, τvw1,−τvw1) } =  4δ τv ·3 , 12δ|τvw1|ϕ τv ·3 , 4δ τv ·3 ∑3 ψ=1 ∣∣τvwψ ∣∣ϕψ , 4δ|τvw1|3ϕ τv ·3 , 4δ|τvw1| ∑3 ψ=1 ϕψ τv ·3 , 16δ|τvw1|3ϕ τv ·3 , =  τ−1 v ΨA(w1), τ ϕ−1 v ΨA(w1), ∑3 ψ=1 τ ϕψ−1 v ΨA(w1), τ 3ϕ−1 v ΨA(w1), τ ∑3 ψ=1 ϕψ−1 v ΨA(w1), τ 3ϕ−1 v ΨA(w1), =  L ΨA(w1), L ΨA(w1), L ΨA(w1), L ΨA(w1), L ΨA(w1), L ΨA(w1), (2.56) for all w1 ∈ W1. For the case ν = 0, we have L = τ−1 0 = 5−1 and from (2.46), we arrive ‖F (w1)−A(w1)‖ ≤ L1−ν 1− L ΨA (w1) = L1−ν 1− L { 1 3 { Ψ (w1, w1, w1) + 3Ψ (w1, w1,−w1) }} = (5−1)1−0 1− 5−1 { 4δ 3 } = δ 3 . For the case ν = 1, we have L = τ−1 1 = ( 1 5 ) −1 = 5 and from (2.46), we obtain ‖F (w1)−A(w1)‖ ≤ L1−ν 1− L ΨA (w1) = L1−ν 1− L { 1 3 { Ψ (w1, w1, w1) + 3Ψ (w1, w1,−w1) }} = (5)1−1 1− 5 { 4δ 3 } = δ −3 . For the case ν = 0, we have L = τ ϕ−1 0 = 5ϕ−1 and from (2.46), we arrive ‖F (w1)−A(w1)‖ ≤ L1−ν 1− L ΨA (w1) = L1−ν 1− L { 1 3 { Ψ (w1, w1, w1) + 3Ψ (w1, w1,−w1) }} = (5ϕ−1)1−0 1− 5ϕ−1 { 12δ|w1 5 |ϕ 3 } = 4δ 5− 5ϕ . For the case ν = 1, we have L = τ ϕ−1 1 = ( 1 5 ) ϕ−1 = 51−ϕ and from (2.46), we get ‖F (w1)−A(w1)‖ ≤ L1−ν 1− L ΨA (w1) = L1−ν 1− L { 1 3 { Ψ (w1, w1, w1) + 3Ψ (w1, w1,−w1) }} = (51−ϕ)1−1 1− 51−ϕ { 12δ|w1 5 |ϕ 3 } = 4δ 5ϕ − 5 . Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 32 No. 10s(2025) https://internationalpubls.com 2197 For the case ν = 0, we have L = τ 3ϕ−1 0 = 53ϕ−1 and from (2.46), we arrive ‖F (w1)−A(w1)‖ ≤ L1−ν 1− L ΨA (w1) = L1−ν 1− L { 1 3 { Ψ (w1, w1, w1) + 3Ψ (w1, w1,−w1) }} = (53ϕ−1)1−0 1− 53ϕ−1 { 4δ|w1 5 |3ϕ 3 } = 4δ|w1|3ϕ 3(5− 53ϕ) . For the case ν = 1, we have L = τ 3ϕ−1 1 = ( 1 5 ) 3ϕ−1 = 51−3ϕ and from (2.46), we obtain ‖F (w1)−A(w1)‖ ≤ L1−ν 1− L ΨA (w1) = L1−ν 1− L { 1 3 { Ψ (w1, w1, w1) + 3Ψ (w1, w1,−w1) }} = (51−3ϕ)1−1 1− 51−3ϕ { 4δ|w1 5 |3ϕ 3 } = 4δ|w1|3ϕ 3(53ϕ − 5) . Similarly, we can prove the rest cases. � 2.5. Evenness of F : Quadratic Case Stability Results : Fixed Point Method. Theorem 2.9. Suppose that an even function F : W1 → W2 satisfy the functional inequality (2.1) where Ψ :W3 1 → [0, ∞) with the condition lim `→∞ Ψ ( τ` v w1, τ` v w2, τ` v w3 ) τ2` v = 0; τv = { 5; ν = 0 1 5 ; ν = 1 , ∀ w1, w2, w3 ∈ W1. (2.57) If there exists L = L(ν) be a function have the property ΨQ(w1) = ΨQ (w1 5 ) and 1 τ2 v ΨQ (τvw1) = L ΨQ(w1), ∀ w1 ∈ W1. (2.58) for all w1, w2, w3 ∈ W1. Then there exists a unique quadratic mapping Q(w1) :W1 →W2 which satisfies (1.7) and the functional inequality ‖F (w1)−Q(w1)‖ ≤ L1−ν 1− L ΨQ (w1) (2.59) = L1−ν 1− L { 1 3 { Ψ (w1, w1, w1) + 7 2 Ψ (w1, w1,−w1) }} , (2.60) and the mapping Q(w1) is obtained by Q(w1) = lim `→∞ 1 τ2` v F ( τ` v w1 ) , (2.61) for all w1 ∈ W1. Proof. By Theorem 2.7, define a functionH : G → G by HF (w1 ) = 1 τ2 v F (τv w1 ), f or all w1 ∈ W1. (2.62) Now F ,F1 ∈ G and w1 ∈ W1, we see d(F ,F1) ≤ K ⇒ ‖ F (w1)−F1(w1) ‖≤ K Ψ(w1, w1, w1), ⇒ ∥∥∥∥ 1 τ2 v F (τvw1)− 1 τ2 v F1(τvw1) ∥∥∥∥ ≤ 1 τ2 v K Ψ(τvw1, τvw1, τvw1), ⇒ ‖ HF (w1)−HF1( w1 ) ‖≤ L K Ψ(w1, w1, w1), ⇒d(HF ,HF1) ≤ L K, i.e.,H is a strictly contractive mapping on G with Lipschitz constant L (see [18]). The rest of the proof is similar to that of Theorem 2.7. Hence the proof is complete. � Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 32 No. 10s(2025) https://internationalpubls.com 2198 Corollary 2.10. Suppose that an even function F : W1 → W2 satisfy the functional inequality (2.2) for all w1, w2, w3 ∈ W1. Then there exists a unique quadratic mapping Q(w1) : W1 → W2 which satisfies (1.7) and the functional inequality (2.31) for all w1 ∈ W1. 2.6. Oddness and Evenness of F : Additive Quadratic Case Stability Results : Fixed Point Method. Theorem 2.11. Suppose that a function F :W1 →W2 satisfy the functional inequality (2.1) where Ψ :W3 1 → [0, ∞) with the conditions (2.43) and (2.57) for all w1, w2, w3 ∈ W1. If there exists L = L(ν) be function have the properties (2.44) and (2.58) Then there exists a unique additive mapping A(w1) : W1 → W2 and a unique quadratic mapping Q(w1) :W1 →W2 which satisfies (1.7) and the functional inequality ‖F (w1)−A(w1)−Q(w1)‖ ≤ 1 2 · L1−ν 1− L { ΨA (w1) + ΨA (−w1) + ΨQ (w1) + ΨQ (−w1) } (2.63) = 1 2 · L1−ν 1− L { 1 3 { Ψ (w1, w1, w1) + 3Ψ (w1, w1,−w1) + Ψ (−w1,−w1,−w1) + 3Ψ (−w1,−w1, w1) +Ψ (w1, w1, w1) + 7 2 Ψ (w1, w1,−w1) + Ψ (−w1,−w1,−w1) + 7 2 Ψ (−w1,−w1, w1) }} , (2.64) and the mapping A(w1) and Q(w1) are given in (2.47) and (2.61) for all w1 ∈ W1. Proof. The proof is similar ideas to that of Theorem 2.5. � Corollary 2.12. Suppose that a functionF :W1 →W2 satisfy the functional inequality (2.2) for all w1, w2, w3 ∈ W1. Then there exists a unique additive mapping A(w1) : W1 → W2 and a unique quadratic mapping Q(w1) :W1 →W2 which satisfies (1.7) and the functional inequality (2.42) for all w1 ∈ W1. 3. STABILITY IN INTUITIONISTIC FUZZY BANACH SPACE OF (1.7) In this section, we explore the generalized Ulam - Hyers stability of the functional equation (1.7) in Intuitionistic Fuzzy Banach Space. In order to prove stability results, assume (W1, µ, ν) and (W2, µ′, ν′) are Intuitionistic Fuzzy normed space and Intuitionistic Fuzzy Banach space respectively. Suppose that F : W1 → W2 and Ψ : W3 1 → [0, ∞) satisfy the following functional inequalities µ ( F (3w1 + w2 + w3) +F (w1 + 3w2 + w3) +F (w1 + w2 + 3w3)− 6F ( ∑3 ψ=1 wψ ) − 1 2 { F ( ∑3 ψ=1 wψ ) +F ( −∑3 ψ=1 wψ )} + ∑3 ψ=1 { F (wψ)− 5 2 [ F (wψ) +F (−wψ) ]} , Λ ) ≥ µ′ (Ψ (w1, w2, w3) , Λ) ν ( F (3w1 + w2 + w3) +F (w1 + 3w2 + w3) +F (w1 + w2 + 3w3)− 6F ( ∑3 ψ=1 wψ ) − 1 2 { F ( ∑3 ψ=1 wψ ) +F ( −∑3 ψ=1 wψ )} + ∑3 ψ=1 { F (wψ)− 5 2 [ F (wψ) +F (−wψ) ]} , Λ ) ≤ ν′ (Ψ (w1, w2, w3) , Λ)  (3.1) µ ( F (3w1 + w2 + w3) +F (w1 + 3w2 + w3) +F (w1 + w2 + 3w3)− 6F ( ∑3 ψ=1 wψ ) − 1 2 { F ( ∑3 ψ=1 wψ ) +F ( −∑3 ψ=1 wψ )} + ∑3 ψ=1 { F (wψ)− 5 2 [ F (wψ) +F (−wψ) ]} , Λ ) ≥ µ′ (δ, Λ) , ν ( F (3w1 + w2 + w3) +F (w1 + 3w2 + w3) +F (w1 + w2 + 3w3)− 6F ( ∑3 ψ=1 wψ ) − 1 2 { F ( ∑3 ψ=1 wψ ) +F ( −∑3 ψ=1 wψ )} + ∑3 ψ=1 { F (wψ)− 5 2 [ F (wψ) +F (−wψ) ]} , Λ ) ≤ ν′ (δ, Λ) ,  (3.2) Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 32 No. 10s(2025) https://internationalpubls.com 2199 µ ( F (3w1 + w2 + w3) +F (w1 + 3w2 + w3) +F (w1 + w2 + 3w3)− 6F ( ∑3 ψ=1 wψ ) − 1 2 { F ( ∑3 ψ=1 wψ ) +F ( −∑3 ψ=1 wψ )} + ∑3 ψ=1 { F (wψ)− 5 2 [ F (wψ) +F (−wψ) ]} , Λ ) ≥ µ′ ( δ ∑3 ψ=1 ∣∣wψ ∣∣ϕ , Λ ) , ν ( F (3w1 + w2 + w3) +F (w1 + 3w2 + w3) +F (w1 + w2 + 3w3)− 6F ( ∑3 ψ=1 wψ ) − 1 2 { F ( ∑3 ψ=1 wψ ) +F ( −∑3 ψ=1 wψ )} + ∑3 ψ=1 { F (wψ)− 5 2 [ F (wψ) +F (−wψ) ]} , Λ ) ≤ ν′ ( δ ∑3 ψ=1 ∣∣wψ ∣∣ϕ , Λ ) ,  (3.3) µ ( F (3w1 + w2 + w3) +F (w1 + 3w2 + w3) +F (w1 + w2 + 3w3)− 6F ( ∑3 ψ=1 wψ ) − 1 2 { F ( ∑3 ψ=1 wψ ) +F ( −∑3 ψ=1 wψ )} + ∑3 ψ=1 { F (wψ)− 5 2 [ F (wψ) +F (−wψ) ]} , Λ ) ≥ µ′ ( δ ∑3 ψ=1 ∣∣wψ ∣∣ϕψ , Λ ) , ν ( F (3w1 + w2 + w3) +F (w1 + 3w2 + w3) +F (w1 + w2 + 3w3)− 6F ( ∑3 ψ=1 wψ ) − 1 2 { F ( ∑3 ψ=1 wψ ) +F ( −∑3 ψ=1 wψ )} + ∑3 ψ=1 { F (wψ)− 5 2 [ F (wψ) +F (−wψ) ]} , Λ ) ≤ ν′ ( δ ∑3 ψ=1 ∣∣wψ ∣∣ϕψ , Λ ) ,  (3.4) µ ( F (3w1 + w2 + w3) +F (w1 + 3w2 + w3) +F (w1 + w2 + 3w3)− 6F ( ∑3 ψ=1 wψ ) − 1 2 { F ( ∑3 ψ=1 wψ ) +F ( −∑3 ψ=1 wψ )} + ∑3 ψ=1 { F (wψ)− 5 2 [ F (wψ) +F (−wψ) ]} , Λ ) ≥ µ′ ( δ ∏3 ψ=1 ∣∣wψ ∣∣ϕ , Λ ) , ν ( F (3w1 + w2 + w3) +F (w1 + 3w2 + w3) +F (w1 + w2 + 3w3)− 6F ( ∑3 ψ=1 wψ ) − 1 2 { F ( ∑3 ψ=1 wψ ) +F ( −∑3 ψ=1 wψ )} + ∑3 ψ=1 { F (wψ)− 5 2 [ F (wψ) +F (−wψ) ]} , Λ ) ≤ ν′ ( δ ∏3 ψ=1 ∣∣wψ ∣∣ϕ , Λ ) ,  (3.5) µ ( F (3w1 + w2 + w3) +F (w1 + 3w2 + w3) +F (w1 + w2 + 3w3)− 6F ( ∑3 ψ=1 wψ ) − 1 2 { F ( ∑3 ψ=1 wψ ) +F ( −∑3 ψ=1 wψ )} + ∑3 ψ=1 { F (wψ)− 5 2 [ F (wψ) +F (−wψ) ]} , Λ ) ≥ µ′ ( δ 3 ∏ ψ=1 ∣∣wψ ∣∣ϕψ , Λ ) , ν ( F (3w1 + w2 + w3) +F (w1 + 3w2 + w3) +F (w1 + w2 + 3w3)− 6F ( ∑3 ψ=1 wψ ) − 1 2 { F ( ∑3 ψ=1 wψ ) +F ( −∑3 ψ=1 wψ )} + ∑3 ψ=1 { F (wψ)− 5 2 [ F (wψ) +F (−wψ) ]} , Λ ) ≤ ν′ ( δ 3 ∏ ψ=1 ∣∣wψ ∣∣ϕψ , Λ ) ,  (3.6) µ ( F (3w1 + w2 + w3) +F (w1 + 3w2 + w3) +F (w1 + w2 + 3w3)− 6F ( ∑3 ψ=1 wψ ) − 1 2 { F ( ∑3 ψ=1 wψ ) +F ( −∑3 ψ=1 wψ )} + ∑3 ψ=1 { F (wψ)− 5 2 [ F (wψ) +F (−wψ) ]} , Λ ) ≥ µ′ ( δ { ∑3 ψ=1 ∣∣wψ ∣∣3ϕ + ∏3 ψ=1 ∣∣wψ ∣∣ϕ} , Λ ) , ν ( F (3w1 + w2 + w3) +F (w1 + 3w2 + w3) +F (w1 + w2 + 3w3)− 6F ( ∑3 ψ=1 wψ ) − 1 2 { F ( ∑3 ψ=1 wψ ) +F ( −∑3 ψ=1 wψ )} + ∑3 ψ=1 { F (wψ)− 5 2 [ F (wψ) +F (−wψ) ]} , Λ ) ≤ ν′ ( δ { ∑3 ψ=1 ∣∣wψ ∣∣3ϕ + ∏3 ψ=1 ∣∣wψ ∣∣ϕ} , Λ ) ,  (3.7) for all w1, w2, w3 ∈ W1 and all Λ > 0 with δ be a positive constant and ϕ be any real number. Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 32 No. 10s(2025) https://internationalpubls.com 2200 3.1. Definitions and Notations of Intuitionistic Fuzzy Banach Space. Now, we recall the basic defini- tions and notations in the setting of intuitionistic fuzzy normed space given in [27]. Definition 3.1. [27] A binary operation ∗ : [0, 1]× [0, 1] −→ [0, 1] is said to be continuous t-norm if ∗ satisfies the following conditions: (∗1) ∗ is commutative and associative; (∗2) ∗ is continuous; (∗3) a ∗ 1 = a for all a ∈ [0, 1]; (∗4) a ∗ b ≤ c ∗ d whenever a ≤ c and b ≤ d for all a, b, c, d ∈ [0, 1] . Definition 3.2. [27] A binary operation � : [0, 1] × [0, 1] −→ [0, 1] is said to be continuous t-conorm if � satisfies the following conditions: (�1) � is commutative and associative; (�2) � is continuous; (�3) a � 0 = a for all a ∈ [0, 1]; (�4) a � b ≤ c � d whenever a ≤ c and b ≤ d for all a, b, c, d ∈ [0, 1] . Definition 3.3. [27] The five-tuple (X, µ, ν, ∗, �) is said to be an intuitionistic fuzzy normed space (for short, IFNS) if X is a vector space, ∗ is a continuous t-norm, � is a continuous t− conorm, and µ, ν are fuzzy sets on X× (0, ∞) satisfy the following conditions. For every x, y ∈ X and s, t > 0 (IFN1) µ(x, t) + ν(x, t) ≤ 1; (IFN2) µ(x, t) > 0; (IFN3) µ(x, t) = 1, if and only if x = 0; (IFN4) µ(dx, t) = µ ( x, t d ) for each d 6= 0; (IFN5) µ(x, t) ∗ µ(y, s) ≤ µ(x + y, t + s); (IFN6) µ(x, ·) : (0, ∞)→ [0, 1] is continuous; (IFN7) lim t→∞ µ(x, t) = 1 and lim t→0 µ(x, t) = 0; (IFN8) ν(x, t) < 1; (IFN9) ν(x, t) = 0, if and only if x = 0; (IFN10) ν(dx, t) = ν ( x, t d ) for each d 6= 0; (IFN11) ν(x, t) � ν(y, s) ≥ ν(x + y, t + s); (IFN12) ν(x, ·) : (0, ∞)→ [0, 1] is continuous; (IFN13) lim t→∞ ν(x, t) = 0 and lim t→0 ν(x, t) = 1. In this case, (µ, ν) is called an intuitionistic fuzzy norm. Example 3.4. [27] Let (X, ‖·‖) be a normed space. Let a ∗ b = ab and a � d = min {a + b, 1} for all a, b ∈ [0, 1]. For all x ∈ X and every t > 0, consider µ(x, t) = { t t+‖x‖ i f t > 0; 0 i f t ≤ 0; and ν(x, t) = { ‖x‖ t+‖x‖ i f t > 0; 0 i f t ≤ 0. Then (X, µ, ν, ∗, �) is an IFN-space. Definition 3.5. [27] Let (X, µ, ν, ∗, �) be an IFNS. Then, a sequence x = {xk} is said to be intuitionistic fuzzy convergent to a point L ∈ X if lim µ(xk − L, t) = 1 and lim ν(xk − L, t) = 0, for all ρ > 0. In this case, we write xk IF−→ L as k→ ∞. Definition 3.6. [27] Let (X, µ, ν, ∗, �) be an IFN-space. Then, x = {xk} is said to be intuitionistic fuzzy Cauchy sequence if µ ( xk+p − xk, t ) = 1 and ν ( xk+p − xk, t ) = 0, for all ρ > 0, and p = 1, 2 · · · . Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 32 No. 10s(2025) https://internationalpubls.com 2201 Definition 3.7. [27] Let (X, µ, ν, ∗, �) be an IFN-space. Then (X, µ, ν, ∗, �) is said to be complete if every intuitionistic fuzzy Cauchy sequence in (X, µ, ν, ∗, �) is intuitionistic fuzzy convergent (X, µ, ν, ∗, �). 3.2. Oddness of F : Additive Case Stability Results : Direct Method. Theorem 3.8. Suppose that an odd function F : W1 → W2 satisfy the functional inequality (3.1) where Ψ :W3 1 → [0, ∞) with the conditions µ′ ( Ψ ( 5`mw1, 5`mw2, 5`mw3 ) , Λ ) ≥ µ′ ( I`mΨ (w1, w2, w3) , Λ ) ν′ ( Ψ ( 5`mw1, 5`mw2, 5`mw3 ) , Λ ) ≤ ν′ ( I`mΨ (w1, w2, w3) , Λ )  (3.8) and lim `→∞ µ′ ( Ψ ( 5`mw1, 5`mw2, 5`mw3 ) , 5`mΛ ) = 1 lim `→∞ ν′ ( Ψ ( 5`mw1, 5`mw2, 5`mw3 ) , 5`mΛ ) = 0  (3.9) for all w1, w2, w3 ∈ W1 and all Λ > 0 with m = ±1 and 0 < ( I 5 )µ < 1 . Then there exists a unique additive mapping A(w1) :W1 →W2 which satisfies (1.7) and the functional inequality µ (A(w1)−F (w1), Λ) ≥ µ′ ( ΨA (w1) , 3Λ 4 |5− I| ) = µ′ ( Ψ (w1, w1, w1) , 3Λ 4 |5− I| ) ∗ µ′ ( Ψ (w1, w1,−w1) , 3Λ 4 |5− I| ) ν (A(w1)−F (w1), Λ) ≤ ν′ ( ΨA (w1) , 3Λ 4 (5− I) ) = ν′ ( Ψ (w1, w1, w1) , 3Λ 4 (5− I) ) � ν′ ( Ψ (w1, w1,−w1) , 3Λ 4 (5− I) )  (3.10) and the mapping A(w1) is obtained by lim `→∞ µ ( 1 5`mF ( 5`mw1 ) −A(w1), Λ ) = 1 lim `→∞ ν ( 1 5`mF ( 5`mw1 ) −A(w1), Λ ) = 0  (3.11) for all w1 ∈ W1 and all Λ > 0. Proof. Using oddness of F in (2.1), we get µ ( F (3w1 + w2 + w3) +F (w1 + 3w2 + w3) +F (w1 + w2 + 3w3)− 6F ( ∑3 ψ=1 wψ ) + ∑3 ψ=1 F (wψ), Λ ) ≥ µ′ (Ψ (w1, w2, w3) , Λ) ν ( F (3w1 + w2 + w3) +F (w1 + 3w2 + w3) +F (w1 + w2 + 3w3)− 6F ( ∑3 ψ=1 wψ ) + ∑3 ψ=1 F (wψ), Λ ) ≤ ν′ (Ψ (w1, w2, w3) , Λ)  (3.12) for all w1, w2, w3 ∈ W1 and all Λ > 0 . Interchanging (w1, w2, w3) by (w1, w1, w1) in (3.12), we obtain µ (3F (5w1)− 6F (3w1) + 3F (w1), Λ) ≥ µ′ (Ψ (w1, w1, w1) , Λ) ν (3F (5w1)− 6F (3w1) + 3F (w1), Λ) ≤ ν′ (Ψ (w1, w1, w1) , Λ) } (3.13) for all w1 ∈ W1 and all Λ > 0 . Again interchanging (w1, w2, w3) by (w1, w1,−w1) in (3.12) and using (IFN4), (IFN10), we have µ (2F (3w1)− 6F (w1) , Λ) ≥ µ′ (Ψ (w1, w1,−w1) , Λ) ν (2F (3w1)− 6F (w1) , Λ) ≤ ν′ (Ψ (w1, w1,−w1) , Λ) } ⇒ µ (6F (3w1)− 18F (w1) , 3Λ) ≥ µ′ (Ψ (w1, w1,−w1) , Λ) ν (6F (3w1)− 18F (w1) , 3Λ) ≤ ν′ (Ψ (w1, w1,−w1) , Λ) } (3.14) Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 32 No. 10s(2025) https://internationalpubls.com 2202 for all w1 ∈ W1 and all Λ > 0. Combining (3.13) and (3.14) using (IFN5), (IFN11), we arrive µ (3F (5w1)− 15F (w1), 4Λ) ≥ µ (3F (5w1)− 6F (3w1) + 3F (w1), Λ) ∗ µ (6F (3w1)− 18F (w1) , 3Λ) ≥ µ′ (Ψ (w1, w1, w1) , Λ) ∗ µ′ (Ψ (w1, w1,−w1) , Λ) = µ′ (ΨA (w1) , Λ) ν (3F (5w1)− 15F (w1), 4Λ) ≤ ν (3F (5w1)− 6F (3w1) + 3F (w1), Λ) � ν (6F (3w1)− 18F (w1) , 3Λ) ≤ ν′ (Ψ (w1, w1, w1) , Λ) � ν′ (Ψ (w1, w1,−w1) , Λ) = ν′ (ΨA (w1) , Λ)  (3.15) for all w1 ∈ W1 and all Λ > 0. Using (IFN4), (IFN10), one can see from (3.15) that µ ( 1 5 F (5w1)−F (w1), 4 3 · 1 5 Λ ) ≥ µ′ (ΨA (w1) , Λ) ν ( 1 5 F (5w1)−F (w1), 4 3 · 1 5 Λ ) ≤ ν′ (ΨA (w1) , Λ)  (3.16) for all w1 ∈ W1 and all Λ > 0. Changing w1 by 5`w1 in (3.16), and using (IFN4), (IFN10), (3.8), we get µ ( 1 5`+1F (5 `+1w1)− 1 5` F (5`w1), 4 3 · 5 · 1 5` Λ ) ≥ µ′ ( ΨA ( 5`w1 ) , Λ ) ≥ µ′ ( I`ΨA (w1) , Λ ) = µ′ ( ΨA (w1) , 1 I` Λ ) ν ( 1 5`+1F (5 `+1w1)− 1 5` F (5`w1), 4 3 · 5 · 1 5` Λ ) ≤ ν′ ( ΨA ( 5`w1 ) , Λ ) ≤ ν′ ( I`ΨA (w1) , Λ ) = ν′ ( ΨA (w1) , 1 I` Λ )  (3.17) for all w1 ∈ W1 and all Λ > 0 also ` > 0. Changing Λ by I`Λ in (3.17), we see µ ( 1 5`+1F (5 `+1w1)− 1 5` F (5`w1), 4 3 · 5 · ( I 5 )` Λ ) ≥ µ′ (ΨA (w1) , Λ) ν ( 1 5`+1F (5 `+1w1)− 1 5` F (5`w1), 4 3 · 5 · ( I 5 )` Λ ) ≤ ν′ (ΨA (w1) , Λ)  (3.18) for all w1 ∈ W1 and all Λ > 0. It is easy to check that 1 5` F (5`w1)−F (w1) = `−1 ∑ η=0 1 5η+1F (5 η+1w1)− 1 5ηF (5 ηw1) (3.19) for all w1 ∈ W1. Using (IFN5), (IFN11), it follows from (3.18) and (3.19), we obtain µ ( 1 5` F (5`w1)−F (w1), `−1 ∑ η=0 4 3 · 5 · ( I 5 )η Λ ) = µ ( `−1 ∑ η=0 1 5η+1F (5 η+1w1)− 1 5ηF (5 ηw1), `−1 ∑ η=0 4 3 · 5 · ( I 5 )η Λ ) ≥ `−1 ∏ η=0 µ ( 1 5η+1F (5 η+1w1)− 1 5ηF (5 ηw1), 4 3 · 5 · ( I 5 )η Λ ) ≥ ∏`−1 η=0 µ′ (ΨA (w1) , Λ) = µ′ (ΨA (w1) , Λ) ν ( 1 5` F (5`w1)−F (w1), `−1 ∑ η=0 4 3 · 5 · ( I 5 )η Λ ) = ν ( `−1 ∑ η=0 1 5η+1F (5 η+1w1)− 1 5ηF (5 ηw1), `−1 ∑ η=0 4 3 · 5 · ( I 5 )η Λ ) ≤ `−1 ä η=0 ν ( 1 5η+1F (5 η+1w1)− 1 5ηF (5 ηw1), 4 3 · 5 · ( I 5 )η Λ ) ≤ ä`−1 η=0 ν′ (ΨA (w1) , Λ) = ν′ (ΨA (w1) , Λ)  (3.20) Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 32 No. 10s(2025) https://internationalpubls.com 2203 where `−1 ∏ η=0 µ = µ ∗ µ ∗ µ ∗ ... and `−1 ä η=0 ν = ν � ν � ν � ... for all w1 ∈ W1 and all Λ > 0. Again changing w1 by 5`1 w1 in (3.20), and using (IFN4), (IFN10), (3.8) in that changing Λ by I`1 Λ, we have µ ( 1 5`+`1 F (5`+`1 w1)− 1 5`1 F (`1w1), `−1 ∑ η=0 4 3 · 5 · ( I 5 )η+`1 Λ ) ≥ µ′ (ΨA (w1) , Λ) ν ( 1 5`+`1 F (5`+`1 w1)− 1 5`1 F (`1w1), `−1 ∑ η=0 4 3 · 5 · ( I 5 )η+`1 Λ ) ≤ ν′ (ΨA (w1) , Λ)  (3.21) for all w1 ∈ W1 and all Λ > 0 also `, `1 > 0. It follows from (3.21) that µ ( 1 5`+`1 F (5`+`1 w1)− 1 5`1 F (`1w1), Λ ) ≥ µ′ ΨA (w1) , Λ ∑`−1 η=0 4 3·5 · ( I 5 )η+`1  ν ( 1 5`+`1 F (5`+`1 w1)− 1 5`1 F (`1w1), Λ ) ≤ ν′ ΨA (w1) , Λ ∑`−1 η=0 4 3·5 · ( I 5 )η+`1   (3.22) for all w1 ∈ W1 and all Λ > 0. By data, the Cauchy criterion for convergence in Intuitionistic Fuzzy normed space gives that the sequence { 1 5` F (5`w1) } , is Cauchy in (W2, µ′, ν′) and it is a complete In- tuitionistic Fuzzy normed space, this sequence converges to some point A(w1) in (W2, µ′, ν′) for all w1 ∈ W1. So, by notation, we write lim `→∞ µ ( 1 5` F ( 5`w1 ) −A(w1), Λ ) = 1; lim `→∞ ν ( 1 5` F ( 5`w1 ) −A(w1), Λ ) = 0;  (3.23) for all w1 ∈ W1 and all Λ > 0. Letting `1 = 0 and `→ ∞ in (3.22) and using (3.23), we arrive µ (A(w1)−F (w1), Λ) ≥ µ′ ( ΨA (w1) , 3Λ 4 (5− I) ) = µ′ ( Ψ (w1, w1, w1) , 3Λ 4 (5− I) ) ∗ µ′ ( Ψ (w1, w1,−w1) , 3Λ 4 (5− I) ) ν (A(w1)−F (w1), Λ) ≤ ν′ ( ΨA (w1) , 3Λ 4 (5− I) ) = ν′ ( Ψ (w1, w1, w1) , 3Λ 4 (5− I) ) � ν′ ( Ψ (w1, w1,−w1) , 3Λ 4 (5− I) )  (3.24) for all w1 ∈ W1 and all Λ > 0. Thus, (3.10) and (3.11) holds for µ = 1. Interchanging (w1, w2, w3) = ( 5`w1, 5`w2, 5`w3 ) , in (3.1) and using (IFN4), (IFN10), we have µ ( 1 5` { F (5`(3w1 + w2 + w3)) +F (5`(w1 + 3w2 + w3)) +F (5`(w1 + w2 + 3w3))− 6F ( ∑3 ψ=1 5`wψ ) − 1 2 { F ( ∑3 ψ=1 5`wψ ) +F ( −∑3 ψ=1 5`wψ )} + ∑3 ψ=1 { F (5`wψ)− 5 2 [ F (5`wψ) +F (−5`wψ) ]}} , Λ ) ≥ µ′ ( Ψ ( 5`w1, 5`w2, 5`w3 ) , 5` Λ ) ν ( 1 5` { F (5`(3w1 + w2 + w3)) +F (5`(w1 + 3w2 + w3)) +F (5`(w1 + w2 + 3w3))− 6F ( ∑3 ψ=1 5`wψ ) − 1 2 { F ( ∑3 ψ=1 5`wψ ) +F ( −∑3 ψ=1 5`wψ )} + ∑3 ψ=1 { F (5`wψ)− 5 2 [ F (5`wψ) +F (−5`wψ) ]}} , Λ ) ≤ ν′ ( Ψ ( 5`w1, 5`w2, 5`w3 ) , 5` Λ )  (3.25) Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 32 No. 10s(2025) https://internationalpubls.com 2204 for all w1, w2, w3 ∈ W1 and all Λ > 0. Now, µ ( A(3w1 + w2 + w3) +A(w1 + 3w2 + w3) +A(w1 + w2 + 3w3)− 6A ( ∑3 ψ=1 wψ ) − 1 2 { A ( ∑3 ψ=1 wψ ) −A ( −∑3 ψ=1 wψ )} + ∑3 ψ=1 { A(wψ)− 5 2 [ A(wψ) +A(−wψ) ]} , Λ ) ≥ µ ( A(3w1 + w2 + w3)− 1 5` F (5`(3w1 + w2 + w3)), Λ 7 ) ∗ µ ( A(w1 + 3w2 + w3)− 1 5` F (5`(w1 + 3w2 + w3)), Λ 7 ) ∗ µ ( A(w1 + w2 + 3w3)− 1 5` F (5`(w1 + w2 + 3w3)), Λ 7 ) ∗ µ ( −6A ( ∑3 ψ=1 wψ ) + 1 5` 6F ( ∑3 ψ=1 5`wψ, Λ 7 )) ∗ µ ( 1 2 { A ( ∑3 ψ=1 wψ ) +A ( −∑3 ψ=1 wψ )} − 1 5` 1 2 { F ( ∑3 ψ=1 5`wψ ) +F ( −∑3 ψ=1 5`wψ )} , Λ 7 ) ∗ µ ( ∑3 ψ=1 { A(wψ)− 5 2 [ A(wψ) +A(−wψ) ]} − 1 5` ∑3 ψ=1 { F (5`wψ)− 5 2 [ F (5`wψ) +F (−5`wψ) ]} , Λ 7 ) ∗ µ ( 1 5` { F (5`(3w1 + w2 + w3)) +F (5`(w1 + 3w2 + w3)) +F (5`(w1 + w2 + 3w3)) −6F ( ∑3 ψ=1 5`wψ ) − 1 2 { F ( ∑3 ψ=1 5`wψ ) +F ( −∑3 ψ=1 5`wψ )} +∑3 ψ=1 { F (5`wψ)− 5 2 [ F (5`wψ) +F (−5`wψ) ]}} , Λ 7 ) ν ( A(3w1 + w2 + w3) +A(w1 + 3w2 + w3) +A(w1 + w2 + 3w3)− 6A ( ∑3 ψ=1 wψ ) − 1 2 { A ( ∑3 ψ=1 wψ ) −A ( −∑3 ψ=1 wψ )} + ∑3 ψ=1 { A(wψ)− 5 2 [ A(wψ) +A(−wψ) ]} , Λ ) ≤ ν ( A(3w1 + w2 + w3)− 1 5` F (5`(3w1 + w2 + w3)), Λ 7 ) � ν ( A(w1 + 3w2 + w3)− 1 5` F (5`(w1 + 3w2 + w3)), Λ 7 ) � ν ( A(w1 + w2 + 3w3)− 1 5` F (5`(w1 + w2 + 3w3)), Λ 7 ) � ν ( −6A ( ∑3 ψ=1 wψ ) + 1 5` 6F ( ∑3 ψ=1 5`wψ, Λ 7 )) � ν ( 1 2 { A ( ∑3 ψ=1 wψ ) +A ( −∑3 ψ=1 wψ )} − 1 5` 1 2 { F ( ∑3 ψ=1 5`wψ ) +F ( −∑3 ψ=1 5`wψ )} , Λ 7 ) � ν ( ∑3 ψ=1 { A(wψ)− 5 2 [ A(wψ) +A(−wψ) ]} − 1 5` ∑3 ψ=1 { F (5`wψ)− 5 2 [ F (5`wψ) +F (−5`wψ) ]} , Λ 7 ) � ν ( 1 5` { F (5`(3w1 + w2 + w3)) +F (5`(w1 + 3w2 + w3)) +F (5`(w1 + w2 + 3w3)) −6F ( ∑3 ψ=1 5`wψ ) − 1 2 { F ( ∑3 ψ=1 5`wψ ) +F ( −∑3 ψ=1 5`wψ )} +∑3 ψ=1 { F (5`wψ)− 5 2 [ F (5`wψ) +F (−5`wψ) ]}} , Λ 7 )  (3.26) for all w1, w2, w3 ∈ W1 and all Λ > 0. Taking limit `→ ∞ in (3.26), using (3.23) and (3.25), we get µ ( A(3w1 + w2 + w3) +A(w1 + 3w2 + w3) +A(w1 + w2 + 3w3)− 6A ( ∑3 ψ=1 wψ ) − 1 2 { A ( ∑3 ψ=1 wψ ) +A ( −∑3 ψ=1 wψ )} + ∑3 ψ=1 { A(wψ)− 5 2 [ A(wψ) +A(−wψ) ]} , Λ ) = 1 ν ( A(3w1 + w2 + w3) +A(w1 + 3w2 + w3) +A(w1 + w2 + 3w3)− 6A ( ∑3 ψ=1 wψ ) − 1 2 { A ( ∑3 ψ=1 wψ ) +A ( −∑3 ψ=1 wψ )} + ∑3 ψ=1 { A(wψ)− 5 2 [ A(wψ) +A(−wψ) ]} , Λ ) = 0  (3.27) for all w1, w2, w3 ∈ W1 and all Λ > 0. Using (IFN3), (IFN9) in (3.27), we see, A(w1) satisfies (1.7). In order to confirm that A(w1) is unique, suppose B(w1) be another mapping (1.7), (3.23) and (3.24), we Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 32 No. 10s(2025) https://internationalpubls.com 2205 obtain µ (A(w1)−B(w1), 2Λ) = µ ( A ( 5`w1 ) −B ( 5`w1 ) , 5` 2Λ ) ≥ µ ( A ( 5`w1 ) −F ( 5`w1 ) , 5`Λ ) ∗ µ ( F ( 5`w1 ) −B ( 5`w1 ) , 5`Λ ) ≥ µ′ ( ΨA ( 5`w1 ) , 3Λ 4 5`(5− I) ) ∗ µ′ ( ΨA ( 5`w1 ) , 3Λ 4 5`(5− I) ) ≥ µ′ ( ΨA (w1) , 3Λ 4 5` I` (5− I) ) ν (A(w1)−B(w1), 2Λ) = ν ( A ( 5`w1 ) −B ( 5`w1 ) , 5` 2Λ ) ≤ ν ( A ( 5`w1 ) −F ( 5`w1 ) , 5`Λ ) � ν ( F ( 5`w1 ) −B ( 5`w1 ) , 5`Λ ) ≤ ν′ ( ΨA ( 5`w1 ) , 3Λ 4 5`(5− I) ) � ν′ ( ΨA ( 5`w1 ) , 3Λ 4 5`(5− I) ) ≤ ν′ ( ΨA (w1) , 3Λ 4 5` I` (5− I) )  (3.28) for all w1 ∈ W1 and all Λ > 0. Taking limit `→ ∞ in (3.28), and using (IFN7), (IFN13), we arrive µ (A(w1)−B(w1), 2Λ) = 1 ν (A(w1)−B(w1), 2Λ) = 0 } (3.29) for all w1 ∈ W1 and all Λ > 0. By (IFN4) and (IFN10), we get A(w1) is unique. So, the Theorem holds for m = 1. Changing w1 = w1 5 in (3.15) and using (IFN4), (IFN10), (3.8), in that changing Λ by Λ I , we have µ ( F (w1)− 5F (w1 5 ) , 4 3 · I Λ ) ≥ µ′ (ΨA (w1) , Λ) ν ( F (w1)− 5F (w1 5 ) , 4 3 · I Λ ) ≤ ν′ (ΨA (w1) , Λ)  (3.30) for all w1 ∈ W1 and all Λ > 0. Changing w1 by w1 5` in (3.30), and using (IFN4), (IFN10), (3.8) in that changing Λ by Λ I` , we get µ ( 5`F (w1 5` ) − 5`+1F ( w1 5`+1 ) , 4 3 · I (5 I )` Λ ) ≤ µ′ (ΨA (w1) , Λ) ν ( 5`F (w1 5` ) − 5`+1F ( w1 5`+1 ) , 4 3 · I (5 I )` Λ ) ≤ ν′ (ΨA (w1) , Λ)  (3.31) for all w1 ∈ W1 and all Λ > 0 also ` > 0. It is easy to check that F (w1)− 5`F (w1 5` ) = ` ∑ η=1 5η−1F ( w1 5η−1 ) − 5ηF (w1 5η ) (3.32) for all w1 ∈ W1. The rest of the proof is similar to that of above case. So, the Theorem holds for m = −1. Hence the proof is complete. � Corollary 3.9. Suppose that an odd function F : W1 → W2 satisfy the functional inequality (3.2) for all w1, w2, w3 ∈ W1 and all Λ > 0 with δ be a positive constant and ϕ be any real number. Then there exists a unique additive mapping A(w1) :W1 →W2 which satisfies (1.7) and the functional inequality µ (A(w1)−F (w1), Λ) ≥ µ′ (δ, |3| Λ) , ν (A(w1)−F (w1), Λ) ≤ ν′ (δ, |3| Λ) , } (3.33) for all w1 ∈ W1. Corollary 3.10. Suppose that an odd function F : W1 → W2 satisfy the functional inequality (3.3) for all w1, w2, w3 ∈ W1 and all Λ > 0 with δ be a positive constant and ϕ be any real number. Then there exists a Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 32 No. 10s(2025) https://internationalpubls.com 2206 unique additive mapping A(w1) :W1 →W2 which satisfies (1.7) and the functional inequality µ (A(w1)−F (w1), Λ) ≥ µ′ ( δ|w1|ϕ, Λ 4 |5− 5ϕ| ) , ϕ 6= 1, ν (A(w1)−F (w1), Λ) ≤ ν′ ( δ|w1|ϕ, Λ 4 |5− 5ϕ| ) , ϕ 6= 1,  (3.34) for all w1 ∈ W1. Corollary 3.11. Suppose that an odd function F : W1 → W2 satisfy the functional inequality (3.4) for all w1, w2, w3 ∈ W1 and all Λ > 0 with δ be a positive constant and ϕ be any real number. Then there exists a unique additive mapping A(w1) :W1 →W2 which satisfies (1.7) and the functional inequality µ (A(w1)−F (w1), Λ) ≥ µ′ ( δ ∑3 ψ=1 |w1| ϕψ , 3Λ 4 ∑3 ψ=1 |5− 5ϕψ | ) , ϕ1, ϕ2, ϕ3 6= 1, ν (A(w1)−F (w1), Λ) ≤ ν′ ( δ ∑3 ψ=1 |w1| ϕψ , 3Λ 4 ∑3 ψ=1 |5− 5ϕψ | ) , ϕ1, ϕ2, ϕ3 6= 1,  (3.35) for all w1 ∈ W1. Corollary 3.12. Suppose that an odd function F : W1 → W2 satisfy the functional inequality (3.5) for all w1, w2, w3 ∈ W1 and all Λ > 0 with δ be a positive constant and ϕ be any real number. Then there exists a unique additive mapping A(w1) :W1 →W2 which satisfies (1.7) and the functional inequality µ (A(w1)−F (w1), Λ) ≥ µ′ ( δ|w1|3ϕ, 3Λ 4 |5− 53ϕ| ) , 3ϕ 6= 1, ν (A(w1)−F (w1), Λ) ≤ ν′ ( δ|w1|3ϕ, 3Λ 4 |5− 53ϕ| ) , 3ϕ 6= 1,  (3.36) for all w1 ∈ W1. Corollary 3.13. Suppose that an odd function F : W1 → W2 satisfy the functional inequality (3.6) for all w1, w2, w3 ∈ W1 and all Λ > 0 with δ be a positive constant and ϕ be any real number. Then there exists a unique additive mapping A(w1) :W1 →W2 which satisfies (1.7) and the functional inequality µ (A(w1)−F (w1), Λ) ≥ µ′ ( δ|wψ|∑ 3 ψ=1 ϕψ , 3Λ 4 ∣∣∣5− 5∑3 ψ=1 ϕψ ∣∣∣) , ∑3 ψ=1 ϕψ 6= 1, ν (A(w1)−F (w1), Λ) ≤ ν′ ( δ|wψ|∑ ψ=13 ϕψ , 3Λ 4 ∣∣∣5− 5∑3 ψ=1 ϕψ ∣∣∣) , ∑3 ψ=1 ϕψ 6= 1,  (3.37) for all w1 ∈ W1. Corollary 3.14. Suppose that an odd function F : W1 → W2 satisfy the functional inequality (3.7) for all w1, w2, w3 ∈ W1 and all Λ > 0 with δ be a positive constant and ϕ be any real number. Then there exists a unique additive mapping A(w1) :W1 →W2 which satisfies (1.7) and the functional inequality µ (A(w1)−F (w1), Λ) ≥ µ′ ( 2δ|w1|3ϕ, 3Λ 4 |5− 53ϕ| ) , 3ϕ 6= 1, ν (A(w1)−F (w1), Λ) ≤ ν′ ( 2δ|w1|3ϕ, 3Λ 4 |5− 53ϕ| ) , 3ϕ 6= 1,  (3.38) for all w1 ∈ W1. 3.3. Evenness of F : Quadratic Case Stability Results : Direct Method. Theorem 3.15. Suppose that an even function F : W1 → W2 satisfy the functional inequality (3.1) where Ψ :W3 1 → [0, ∞) with the conditions (3.8) and lim `→∞ µ′ ( Ψ ( 5`mw1, 5`mw2, 5`mw3 ) , 25`mΛ ) = 1 lim `→∞ ν′ ( Ψ ( 5`mw1, 5`mw2, 5`mw3 ) , 25`mΛ ) = 0  (3.39) Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 32 No. 10s(2025) https://internationalpubls.com 2207 for all w1, w2, w3 ∈ W1 and all Λ > 0 with m = ±1 and 0 < ( I 25 )µ < 1 . Then there exists a unique quadratic mapping Q(w1) :W1 →W2 which satisfies (1.7) and the functional inequality µ (Q(w1)−F (w1), Λ) ≥ µ′ ( ΨQ (w1) , 7Λ 3 |25− I| ) = µ′ ( Ψ (w1, w1, w1) , 7Λ 3 |25− I| ) ∗ µ′ ( Ψ (w1, w1,−w1) , 7Λ 3 |25− I| ) ν (Q(w1)−F (w1), Λ) ≤ ν′ ( ΨQ (w1) , 7Λ 3 (25− I) ) = ν′ ( Ψ (w1, w1, w1) , 7Λ 3 (25− I) ) � ν′ ( Ψ (w1, w1,−w1) , 7Λ 3 (25− I) )  (3.40) and the mapping Q(w1) is obtained by lim `→∞ µ ( 1 25`mF ( 5`mw1 ) −Q(w1), Λ ) = 1 lim `→∞ ν ( 1 25`mF ( 5`mw1 ) −Q(w1), Λ ) = 0  (3.41) for all w1 ∈ W1 and all Λ > 0. Proof. Using evenness of F in (2.1), we get µ ( F (3w1 + w2 + w3) +F (w1 + 3w2 + w3) +F (w1 + w2 + 3w3)− 7F ( 3 ∑ ψ=1 wψ ) − 4 3 ∑ ψ=1 F (wψ), Λ ) ≥ µ′ (Ψ (w1, w2, w3) , Λ) ν ( F (3w1 + w2 + w3) +F (w1 + 3w2 + w3) +F (w1 + w2 + 3w3)− 7F ( 3 ∑ ψ=1 wψ ) − 4 3 ∑ ψ=1 F (wψ), Λ ) ≤ ν′ (Ψ (w1, w2, w3) , Λ)  (3.42) for all w1, w2, w3 ∈ W1 and all Λ > 0 . Interchanging (w1, w2, w3) by (w1, w1, w1) in (3.42), we obtain µ (3F (5w1)− 7F (3w1)− 12F (w1), Λ) ≥ µ′ (Ψ (w1, w1, w1) , Λ) ν (3F (5w1)− 7F (3w1)− 12F (w1), Λ) ≤ ν′ (Ψ (w1, w1, w1) , Λ) } (3.43) for all w1 ∈ W1 and all Λ > 0 . Again interchanging (w1, w2, w3) by (w1, w1,−w1) in (3.42) and using (IFN4), (IFN10), we have µ (2F (3w1)− 18F (w1) , Λ) ≥ µ′ (Ψ (w1, w1,−w1) , Λ) ν (2F (3w1)− 18F (w1) , Λ) ≤ ν′ (Ψ (w1, w1,−w1) , Λ) } ⇒ µ ( 7F (3w1)− 63F (w1) , 2 7 Λ ) ≥ µ′ (Ψ (w1, w1,−w1) , Λ) ν ( 7F (3w1)− 63F (w1) , 2 7 Λ ) ≤ ν′ (Ψ (w1, w1,−w1) , Λ) } (3.44) for all w1 ∈ W1 and all Λ > 0. Combining (3.43) and (3.44) using (IFN5), (IFN11), we arrive µ ( 3F (5w1)− 75F (w1), 9 7 Λ ) ≥ µ (3F (5w1)− 7F (3w1)− 12F (w1), Λ) ∗ µ ( 7F (3w1)− 63F (w1) , 2 7 Λ ) ≥ µ′ (Ψ (w1, w1, w1) , Λ) ∗ µ′ (Ψ (w1, w1,−w1) , Λ) = µ′ ( ΨQ (w1) , Λ ) ν ( 3F (5w1)− 15F (w1), 9 7 Λ ) ≤ ν (3F (5w1)− 7F (3w1)− 12F (w1), Λ) � ν ( 7F (3w1)− 63F (w1) , 2 7 Λ ) ≤ ν′ (Ψ (w1, w1, w1) , Λ) � ν′ (Ψ (w1, w1,−w1) , Λ) = ν′ ( ΨQ (w1) , Λ )  (3.45) for all w1 ∈ W1 and all Λ > 0. Using (IFN4), (IFN10), one can see from (3.45) that µ ( 1 25 F (5w1)−F (w1), 9 7 · 3 · 1 25 Λ ) ≥ µ′ ( ΨQ (w1) , Λ ) ν ( 1 25 F (5w1)−F (w1), 9 7 · 3 · 1 25 Λ ) ≤ ν′ ( ΨQ (w1) , Λ )  (3.46) Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 32 No. 10s(2025) https://internationalpubls.com 2208 for all w1 ∈ W1 and all Λ > 0. The rest of the proof is similar to that of Theorem 3.8. Hence the proof is complete. � Corollary 3.16. Suppose that an even function F : W1 → W2 satisfy the functional inequality (3.2) for all w1, w2, w3 ∈ W1 and all Λ > 0 with δ be a positive constant and ϕ be any real number. Then there exists a unique quadratic mapping Q(w1) :W1 →W2 which satisfies (1.7) and the functional inequality µ (Q(w1)−F (w1), Λ) ≥ µ′ (δ, |8| 7Λ) , ν (Q(w1)−F (w1), Λ) ≤ ν′ (δ, |8| 7Λ) , } (3.47) for all w1 ∈ W1 and all Λ > 0. Corollary 3.17. Suppose that an even function F : W1 → W2 satisfy the functional inequality (3.3) for all w1, w2, w3 ∈ W1 and all Λ > 0 with δ be a positive constant and ϕ be any real number. Then there exists a unique quadratic mapping Q(w1) :W1 →W2 which satisfies (1.7) and the functional inequality µ (Q(w1)−F (w1), Λ) ≥ µ′ ( δ|w1|ϕ, 7Λ 9 |25− 5ϕ| ) , ϕ 6= 2, ν (Q(w1)−F (w1), Λ) ≤ ν′ ( δ|w1|ϕ, 7Λ 9 |25− 5ϕ| ) , ϕ 6= 2,  (3.48) for all w1 ∈ W1 and all Λ > 0. Corollary 3.18. Suppose that an even function F : W1 → W2 satisfy the functional inequality (3.4) for all w1, w2, w3 ∈ W1 and all Λ > 0 with δ be a positive constant and ϕ be any real number. Then there exists a unique quadratic mapping Q(w1) :W1 →W2 which satisfies (1.7) and the functional inequality µ (Q(w1)−F (w1), Λ) ≥ µ′ ( δ ∑3 ψ=1 |w1| ϕψ , 7Λ 3 ∑3 ψ=1 |25− 5ϕψ | ) , ϕ1, ϕ2, ϕ3 6= 2, ν (Q(w1)−F (w1), Λ) ≤ ν′ ( δ ∑3 ψ=1 |w1| ϕψ , 7Λ 3 ∑3 ψ=1 |25− 5ϕψ | ) , ϕ1, ϕ2, ϕ3 6= 2,  (3.49) for all w1 ∈ W1 and all Λ > 0. Corollary 3.19. Suppose that an even function F : W1 → W2 satisfy the functional inequality (3.5) for all w1, w2, w3 ∈ W1 and all Λ > 0 with δ be a positive constant and ϕ be any real number. Then there exists a unique quadratic mapping Q(w1) :W1 →W2 which satisfies (1.7) and the functional inequality µ (Q(w1)−F (w1), Λ) ≥ µ′ ( δ|w1|3ϕ, 7Λ 3 |25− 53ϕ| ) , 3ϕ 6= 2, ν (Q(w1)−F (w1), Λ) ≤ ν′ ( δ|w1|3ϕ, 7Λ 3 |25− 53ϕ| ) , 3ϕ 6= 2,  (3.50) for all w1 ∈ W1 and all Λ > 0. Corollary 3.20. Suppose that an even function F : W1 → W2 satisfy the functional inequality (3.6) for all w1, w2, w3 ∈ W1 and all Λ > 0 with δ be a positive constant and ϕ be any real number. Then there exists a unique quadratic mapping Q(w1) :W1 →W2 which satisfies (1.7) and the functional inequality µ (Q(w1)−F (w1), Λ) ≥ µ′ ( δ|wψ|∑ 3 ψ=1 ϕψ , 7Λ 3 ∣∣∣25− 5∑3 ψ=1 ϕψ ∣∣∣) , ∑3 ψ=1 ϕψ 6= 2, ν (Q(w1)−F (w1), Λ) ≤ ν′ ( δ|wψ|∑ 3 ψ=1 ϕψ , 7Λ 3 ∣∣∣25− 5∑3 ψ=1 ϕψ ∣∣∣) , ∑3 ψ=1 ϕψ 6= 2,  (3.51) for all w1 ∈ W1 and all Λ > 0. Corollary 3.21. Suppose that an even function F : W1 → W2 satisfy the functional inequality (3.7) for all w1, w2, w3 ∈ W1 and all Λ > 0 with δ be a positive constant and ϕ be any real number. Then there exists a unique quadratic mapping Q(w1) :W1 →W2 which satisfies (1.7) and the functional inequality µ (Q(w1)−F (w1), Λ) ≥ µ′ ( 2δ|w1|3ϕ, 7Λ 3 |25− 53ϕ| ) , 3ϕ 6= 2, ν (Q(w1)−F (w1), Λ) ≤ ν′ ( 2δ|w1|3ϕ, 7Λ 3 |25− 53ϕ| ) , 3ϕ 6= 2,  (3.52) for all w1 ∈ W1 and all Λ > 0. Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 32 No. 10s(2025) https://internationalpubls.com 2209 3.4. Oddness and Evenness of F : Additive Quadratic Case Stability Results : Direct Method. Theorem 3.22. Suppose that a function F :W1 →W2 satisfy the functional inequality (3.1) where Ψ :W3 1 → [0, ∞) with the conditions (3.8), (3.9), and (3.39) for all w1, w2, w3 ∈ W1 and all Λ > 0 with m = ±1 and 0 < ( I 5 )µ < 1, 0 < ( I 25 )µ < 1 . Then there exists a unique additive mappingA(w1) :W1 →W2 and a unique quadratic mapping Q(w1) :W1 →W2 which satisfies (1.7) and the functional inequality µ (F (w1)−A(w1)−Q(w1), 4Λ) ≥ µ′ ( ΨA (w1) , 3Λ 4 |5− I| ) ∗ µ′ ( ΨA (−w1) , 3Λ 4 |5− I| ) ∗ µ′ ( ΨQ (w1) , 7Λ 3 |25− I| ) ∗ µ′ ( ΨQ (−w1) , 7Λ 3 |25− I| ) = µ′ ( Ψ (w1, w1, w1) , 3Λ 4 |5− I| ) ∗ µ′ ( Ψ (w1, w1,−w1) , 3Λ 4 |5− I| ) ∗ µ′ ( Ψ (−w1,−w1,−w1) , 3Λ 4 |5− I| ) ∗ µ′ ( Ψ (−w1,−w1, w1) , 3Λ 4 |5− I| ) ∗ µ′ ( Ψ (w1, w1, w1) , 7Λ 3 |5− I| ) ∗ µ′ ( Ψ (w1, w1,−w1) , 7Λ 3 |5− I| ) ∗ µ′ ( Ψ (−w1,−w1,−w1) , 7Λ 3 |5− I| ) ∗ µ′ ( Ψ (−w1,−w1, w1) , 7Λ 3 |5− I| ) ν (F (w1)−A(w1)−Q(w1), 4Λ) ≤ ν′ ( ΨA (w1) , 3Λ 4 |5− I| ) � ν′ ( ΨA (−w1) , 3Λ 4 |5− I| ) � ν′ ( ΨQ (w1) , 7Λ 3 |25− I| ) � ν′ ( ΨQ (−w1) , 7Λ 3 |25− I| ) = ν′ ( Ψ (w1, w1, w1) , 3Λ 4 |5− I| ) � ν′ ( Ψ (w1, w1,−w1) , 3Λ 4 |5− I| ) � ν′ ( Ψ (−w1,−w1,−w1) , 3Λ 4 |5− I| ) � ν′ ( Ψ (−w1,−w1, w1) , 3Λ 4 |5− I| ) � ν′ ( Ψ (w1, w1, w1) , 7Λ 3 |5− I| ) � ν′ ( Ψ (w1, w1,−w1) , 7Λ 3 |5− I| ) � ν′ ( Ψ (−w1,−w1,−w1) , 7Λ 3 |5− I| ) � ν′ ( Ψ (−w1,−w1, w1) , 7Λ 3 |5− I| )  (3.53) and the mapping A(w1) and Q(w1) are given in (3.11) and (3.41) for all w1 ∈ W1. Proof. By Theorem 3.8, it follows from (2.33), (3.1) and (3.10), we arrive µ (A(w1)−Fodd(w1), 2Λ) ≥ µ′ ( ΨA (w1) , 3Λ 4 |5− I| ) ∗ µ′ ( ΨA (−w1) , 3Λ 4 |5− I| ) ν (A(w1)−Fodd(w1), 2Λ) ≤ ν′ ( ΨA (w1) , 3Λ 4 |5− I| ) � ν′ ( ΨA (−w1) , 3Λ 4 |5− I| )  (3.54) for all w1 ∈ W1 and all Λ > 0. By Theorem 3.15, it follows from (2.37), (3.1), and (3.40), we see µ (Q(w1)−Feven(w1), 2Λ) ≥ µ′ ( ΨQ (w1) , 7Λ 3 |25− I| ) ∗ µ′ ( ΨQ (−w1) , 7Λ 3 |25− I| ) ν (Q(w1)−Feven(w1), 2Λ) ≤ ν′ ( ΨQ (w1) , 7Λ 3 |25− I| ) � ν′ ( ΨQ (−w1) , 7Λ 3 |25− I| )  (3.55) for all w1 ∈ W1 and all Λ > 0. Now, it follows from (3.54), (3.55) and (2.40), we have µ (F (w1)−A(w1)−Q(w1), 4Λ) ≥ µ (A(w1)−Fodd(w1), 2Λ) ∗ µ (Q(w1)−Feven(w1), 2Λ) ≥ µ′ ( ΨA (w1) , 3Λ 4 |5− I| ) ∗ µ′ ( ΨA (−w1) , 3Λ 4 |5− I| ) ∗ µ′ ( ΨQ (w1) , 7Λ 3 |25− I| ) ∗ µ′ ( ΨQ (−w1) , 7Λ 3 |25− I| ) ν (F (w1)−A(w1)−Q(w1), 4Λ) ≤ ν (A(w1)−Fodd(w1), 2Λ) � ν (Q(w1)−Feven(w1), 2Λ) ≤ ν′ ( ΨA (w1) , 3Λ 4 |5− I| ) � ν′ ( ΨA (−w1) , 3Λ 4 |5− I| ) � ν′ ( ΨQ (w1) , 7Λ 3 |25− I| ) � ν′ ( ΨQ (−w1) , 7Λ 3 |25− I| )  (3.56) Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 32 No. 10s(2025) https://internationalpubls.com 2210 for all w1 ∈ W1 and all Λ > 0. � Corollary 3.23. Suppose that a functionF :W1 →W2 satisfy the functional inequality (3.2) for all w1, w2, w3 ∈ W1 and all Λ > 0 with δ be a positive constant and ϕ be any real number. Then there exists a unique additive mapping A(w1) :W1 →W2 and a unique quadratic mapping Q(w1) :W1 →W2 which satisfies (1.7) and the functional inequality µ (F (w1)−A(w1)−Q(w1), 4Λ) ≥ µ′ (2δ, (|3|+ 7 |8|)Λ) ν (F (w1)−A(w1)−Q(w1), 4Λ) ≤ ν′ (2δ, (|3|+ 7 |8|)Λ) } (3.57) for all w1 ∈ W1. Corollary 3.24. Suppose that a functionF :W1 →W2 satisfy the functional inequality (3.3) for all w1, w2, w3 ∈ W1 and all Λ > 0 with δ be a positive constant and ϕ be any real number. Then there exists a unique additive mapping A(w1) :W1 →W2 and a unique quadratic mapping Q(w1) :W1 →W2 which satisfies (1.7) and the functional inequality µ (F (w1)−A(w1)−Q(w1), 4Λ) ≥ µ′ ( 2δ|w1|ϕ, Λ { 1 4 |5− 5ϕ|+ 7 9 |25− 5ϕ| }) , ϕ 6= 1, 2, ν (F (w1)−A(w1)−Q(w1), 4Λ) ≤ ν′ ( 2δ|w1|ϕ, Λ { 1 4 |5− 5ϕ|+ 7 9 |25− 5ϕ| }) , ϕ 6= 1, 2,  (3.58) for all w1 ∈ W1. Corollary 3.25. Suppose that a functionF :W1 →W2 satisfy the functional inequality (3.4) for all w1, w2, w3 ∈ W1 and all Λ > 0 with δ be a positive constant and ϕ be any real number. Then there exists a unique additive mapping A(w1) :W1 →W2 and a unique quadratic mapping Q(w1) :W1 →W2 which satisfies (1.7) and the functional inequality µ (F (w1)−A(w1)−Q(w1), 4Λ) ≥ µ′ ( 2δ ∑3 ψ=1 |wψ| ϕψ , { 3 4 ∑3 ψ=1 |5− 5ϕψ |+ 7 3 ∑3 ψ=1 |25− 5ϕψ | }) , ϕ1, ϕ2, ϕ3 6= 1, 2, ν (F (w1)−A(w1)−Q(w1), 4Λ) ≤ ν′ ( 2δ ∑3 ψ=1 |wψ| ϕψ , { 3 4 ∑3 ψ=1 |5− 5ϕψ |+ 7 3 ∑3 ψ=1 |25− 5ϕψ | }) , ϕ1, ϕ2, ϕ3 6= 1, 2,  (3.59) for all w1 ∈ W1. Corollary 3.26. Suppose that a functionF :W1 →W2 satisfy the functional inequality (3.5) for all w1, w2, w3 ∈ W1 and all Λ > 0 with δ be a positive constant and ϕ be any real number. Then there exists a unique additive mapping A(w1) :W1 →W2 and a unique quadratic mapping Q(w1) :W1 →W2 which satisfies (1.7) and the functional inequality µ (F (w1)−A(w1)−Q(w1), 4Λ) ≥ µ′ ( 2δ|w1|3ϕ, Λ { 3 4 |5− 53ϕ|+ 7 3 |25− 53ϕ| }) , 3ϕ 6= 1, 2, ν (F (w1)−A(w1)−Q(w1), 4Λ) ≤ ν′ ( 2δ|w1|3ϕ, Λ { 3 4 |5− 53ϕ|+ 7 3 |25− 53ϕ| }) , 3ϕ 6= 1, 2, } (3.60) for all w1 ∈ W1. Corollary 3.27. Suppose that a functionF :W1 →W2 satisfy the functional inequality (3.6) for all w1, w2, w3 ∈ W1 and all Λ > 0 with δ be a positive constant and ϕ be any real number. Then there exists a unique additive mapping A(w1) :W1 →W2 and a unique quadratic mapping Q(w1) :W1 →W2 which satisfies (1.7) and the functional inequality µ (F (w1)−A(w1)−Q(w1), 4Λ) ≥ µ′ ( 4δ|wψ|∑ 3 ψ=1 ϕψ , 2Λ { 3 4 ∣∣∣5− 5∑3 ψ=1 ϕψ ∣∣∣+ 7 3 ∣∣∣25− 5∑3 ψ=1 ϕψ ∣∣∣}) , ∑3 ψ=1 ϕψ 6= 1, 2, ν (F (w1)−A(w1)−Q(w1), 4Λ) ≤ ν′ ( 4δ|wψ|∑ 3 ψ=1 ϕψ , 2Λ { 3 4 ∣∣∣5− 5∑3 ψ=1 ϕψ ∣∣∣+ 7 3 ∣∣∣25− 5∑3 ψ=1 ϕψ ∣∣∣}) , ∑3 ψ=1 ϕψ 6= 1, 2,  (3.61) for all w1 ∈ W1. Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 32 No. 10s(2025) https://internationalpubls.com 2211 Corollary 3.28. Suppose that a functionF :W1 →W2 satisfy the functional inequality (3.7) for all w1, w2, w3 ∈ W1 and all Λ > 0 with δ be a positive constant and ϕ be any real number. Then there exists a unique additive mapping A(w1) :W1 →W2 and a unique quadratic mapping Q(w1) :W1 →W2 which satisfies (1.7) and the functional inequality µ (F (w1)−A(w1)−Q(w1), 4Λ) ≥ µ′ ( 4δ|w1|3ϕ, Λ { 3 4 |5− 53ϕ|+ 7 3 |25− 53ϕ| }) , 3ϕ 6= 1, 2, ν (F (w1)−A(w1)−Q(w1), 4Λ) ≤ ν′ ( 4δ|w1|3ϕ, Λ { 3 4 |5− 53ϕ|+ 7 3 |25− 53ϕ| }) , 3ϕ 6= 1, 2, } (3.62) for all w1 ∈ W1. 3.5. Oddness of F : Additive Case Stability Results : Fixed Point Method. Theorem 3.29. Suppose that an odd function F : W1 → W2 satisfy the functional inequality (3.1) where Ψ :W3 1 → [0, ∞) with the condition lim `→∞ µ′ ( Ψ ( τ` v w1, τ` v w2, τ` v w3 ) , τ` v Λ ) = 1 lim `→∞ ν′ ( Ψ ( τ` v w1, τ` v w2, τ` v w3 ) , τ` v Λ ) = 0  ; τv = { 5; v = 0; 1 5 ; v = 1; (3.63) for all w1, w2, w3 ∈ W1 and all Λ > 0 If there exists L = L(ν) be a function have the property µ (ΨA(w1), Λ) = µ ( ΨA (w1 5 ) , Λ ) ν (ΨA(w1), Λ) = ν ( ΨA (w1 5 ) , Λ ) } and µ ( 1 τv ΨA (τvw1) , Λ ) = µ (L ΨA(w1), Λ) ν ( 1 τv ΨA (τvw1) , Λ ) = ν (L ΨA(w1), Λ)  , (3.64) for all w1 ∈ W1 and all Λ > 0. Then there exists a unique additive mapping A(w1) :W1 →W2 which satisfies (1.7) and the functional inequality µ (A(w1)−F (w1), Λ) ≥ µ′ ( L1−v 1− L ΨA (w1) , 3Λ 4 ) = µ′ ( L1−v 1− L Ψ (w1, w1, w1) , 3Λ 4 ) ∗ µ′ ( L1−v 1− L Ψ (w1, w1,−w1) , 3Λ 4 ) ν (A(w1)−F (w1), Λ) ≤ ν′ ( L1−v 1− L ΨA (w1) , 3Λ 4 ) = ν′ ( L1−v 1− L Ψ (w1, w1, w1) , 3Λ 4 ) � ν′ ( L1−v 1− L Ψ (w1, w1,−w1) , 3Λ 4 )  (3.65) and the mapping A(w1) is obtained by lim `→∞ µ ( 1 τ` v F ( τ` v w1 ) −A(w1), Λ ) = 1 lim `→∞ ν ( 1 τ` v F ( τ` v w1 ) −A(w1), Λ ) = 0  (3.66) for all w1 ∈ W1 and all Λ > 0. Proof. Assume a set G as in Theorem 2.7 of (2.48) and introduce the generalized metric on the above set G as d(F ,F1) = inf { K ∈ (0, ∞) : { µ (F (w1)−F1(w1), Λ) ≥ µ (K Ψ(w1, w1, w1), Λ) ν (F (w1)−F1(w1), Λ) ≤ ν (K Ψ(w1, w1, w1), Λ) }} . (3.67) Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 32 No. 10s(2025) https://internationalpubls.com 2212 for all w1 ∈ W1 and all Λ > 0. It is easy to see that (G, d) is complete. Define a function H : G → G as by Theorem 2.7 of (2.50) and for F ,F1 ∈ G and w1 ∈ W1 and all Λ > 0, we see d(F ,F1) ≤ K ⇒ { µ (F (w1)−F1(w1), Λ) ≥ µ (K Ψ(w1, w1, w1), Λ) ν (F (w1)−F1(w1), Λ) ≤ ν (K Ψ(w1, w1, w1), Λ) } ⇒  µ (∥∥∥ 1 τv F (τvw1)− 1 τv F1(τvw1) ∥∥∥ , Λ ) ≥ µ ( τvK Ψ( 1 τv w1, τvw1, τvw1), Λ ) ν (∥∥∥ 1 τv F (τvw1)− 1 τv F1(τvw1) ∥∥∥ , Λ ) ≤ ν ( τvK Ψ( 1 τv w1, τvw1, τvw1), Λ )  ⇒ { µ (HF (w1)−HF1(w1), Λ) ≥ µ (L K Ψ(w1, w1, w1), Λ) ν (HF (w1)−HF1(w1), Λ) ≤ ν (L K Ψ(w1, w1, w1), Λ) } ⇒d(HF ,HF1) ≤ L K, i.e.,H is a strictly contractive mapping on G with Lipschitz constant L (see [18]). For the case ν = 0, it follows from (3.16) and with the help of (3.64), (2.50), (3.67), we get µ ( 1 5 F (5w1)−F (w1), 4 3 Λ ) ≥ µ′ ( 1 5 ΨA (w1) , Λ ) ν ( 1 5 F (5w1)−F (w1), 4 3 Λ ) ≤ ν′ ( 1 5 ΨA (w1) , Λ ) ⇒ d(HF ,F ) ≤ L = L1−v, (3.68) for all w1 ∈ W1 and all Λ > 0. For the case ν = 1, it follows from (3.22) and with the help of (3.64), (2.50), (3.67), we obtain µ ( F (w1)− 5F (w1 5 ) , 4 3 · I Λ ) ≥ µ′ ( ΨA (w1 5 ) , Λ ) ν ( F (w1)− 5F (w1 5 ) , 4 3 · I Λ ) ≤ ν′ ( ΨA (w1 5 ) , Λ ) ⇒ d(F ,HF ) ≤ 1 = L1−v, (3.69) for all w1 ∈ W1 and all Λ > 0. Combining (3.68) and (3.69), we have d(F ,HF ) ≤ 1 = L1−v. (3.70) Therefore (FPC1) of Theorem 1.5 holds. The rest of the proof follows by Theorem 1.5. Hence the proof is complete. � Corollary 3.30. Suppose that an odd function F : W1 → W2 satisfy the functional inequalities (3.2), (3.3), (3.4), (3.5), (3.6), (3.7) for all w1, w2, w3 ∈ W1 with δ be a positive constant and ϕ be any real number. Then there exists a a unique additive mapping A(w1) :W1 →W2 which satisfies (1.7) and the functional inequalities (3.33), (3.34), (3.35), (3.36), (3.37), (3.38), respectively for all w1 ∈ W1. 3.6. Evenness of F : Quadratic Case Stability Results : Fixed Point Method. Theorem 3.31. Suppose that an even function F : W1 → W2 satisfy the functional inequality (3.1) where Ψ :W3 1 → [0, ∞) with the condition lim `→∞ µ′ ( Ψ ( τ` v w1, τ` v w2, τ` v w3 ) , τ2` v Λ ) = 1 lim `→∞ ν′ ( Ψ ( τ` v w1, τ` v w2, τ` v w3 ) , τ2` v Λ ) = 0  ; τv = { 5; v = 0; 1 5 ; v = 1; (3.71) for all w1, w2, w3 ∈ W1 and all Λ > 0. If there exists L = L(ν) be function have the property µ ( ΨQ(w1), Λ ) = µ ( ΨQ (w1 5 ) , Λ ) ν ( ΨQ(w1), Λ ) = ν ( ΨQ (w1 5 ) , Λ ) } and µ ( 1 τ2 v ΨQ (τvw1) , Λ ) = µ ( L ΨQ(w1), Λ ) ν ( 1 τ2 v ΨQ (τvw1) , Λ ) = ν ( L ΨQ(w1), Λ )  , (3.72) Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 32 No. 10s(2025) https://internationalpubls.com 2213 for all w1 ∈ W1 and all Λ > 0. Then there exists a unique quadratic mapping Q(w1) : W1 → W2 which satisfies (1.7) and the functional inequality µ (Q(w1)−F (w1), Λ) ≥ µ′ ( L1−v 1− L ΨQ (w1) , 7Λ 3 ) = µ′ ( L1−v 1− L Ψ (w1, w1, w1) , 7Λ 3 ) ∗ µ′ ( L1−v 1− L Ψ (w1, w1,−w1) , 7Λ 3 ) ν (Q(w1)−F (w1), Λ) ≤ ν′ ( L1−v 1− L ΨQ (w1) , 7Λ 3 ) = ν′ ( L1−v 1− L Ψ (w1, w1, w1) , 7Λ 3 ) � ν′ ( L1−v 1− L Ψ (w1, w1,−w1) , 7Λ 3 )  (3.73) and the mapping Q(w1) is obtained by lim `→∞ µ ( 1 τ2` v F ( τ` v w1 ) −Q(w1), Λ ) = 1 lim `→∞ ν ( 1 τ2` v F ( τ` v w1 ) −Q(w1), Λ ) = 0  (3.74) for all w1 ∈ W1 and all Λ > 0. Proof. Define a function H : G → G as by Theorem 2.9 of (2.62) and for F ,F1 ∈ G and w1 ∈ W1 and all Λ > 0, we see d(F ,F1) ≤ K ⇒ { µ (F (w1)−F1(w1), Λ) ≥ µ (K Ψ(w1, w1, w1), Λ) ν (F (w1)−F1(w1), Λ) ≤ ν (K Ψ(w1, w1, w1), Λ) } ⇒  µ (∥∥∥ 1 τ2 v F (τvw1)− 1 τ2 v F1(τvw1) ∥∥∥ , Λ ) ≥ µ (τvK Ψ(τvw1, τvw1, τvw1), Λ) ν (∥∥∥ 1 τ2 v F (τvw1)− 1 τ2 v F1(τvw1) ∥∥∥ , Λ ) ≤ ν ( τ2 v K Ψ(τvw1, τvw1, τvw1), Λ )  ⇒ { µ (HF (w1)−HF1(w1), Λ) ≥ µ (L K Ψ(w1, w1, w1), Λ) ν (HF (w1)−HF1(w1), Λ) ≤ ν (L K Ψ(w1, w1, w1), Λ) } ⇒d(HF ,HF1) ≤ L K, i.e.,H is a strictly contractive mapping on G with Lipschitz constant L (see [18]). The rest of the proof is similar to that of Theorem 3.29. Hence the proof is complete. � Corollary 3.32. Suppose that an even function F : W1 → W2 satisfy the functional iinequalities (3.2), (3.3), (3.4), (3.5), (3.6), (3.7) for all w1, w2, w3 ∈ W1 with δ be a positive constant and ϕ be any real number. Then there exists a unique quadratic mapping Q(w1) : W1 → W2 which satisfies (1.7) and the functional inequalities (3.47), (3.48), (3.49), (3.50), (3.51), (3.52), for all w1 ∈ W1. 3.7. Oddness and Evenness of F : Additive Quadratic Case Stability Results : Fixed Point Method. Theorem 3.33. Suppose that a function F :W1 →W2 satisfy the functional inequality (3.1) where Ψ :W3 1 → [0, ∞) with the conditions (3.63) and (3.71) for all w1, w2, w3 ∈ W1 and all Λ > 0. If there exists L = L(ν) be function have the properties (3.64) and (3.72) for all w1 ∈ W1 and all Λ > 0. Then there exists a unique additive mapping A(w1) :W1 →W2 and a unique quadratic mapping Q(w1) :W1 →W2 which satisfies (1.7) and the Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 32 No. 10s(2025) https://internationalpubls.com 2214 functional inequality µ (F (w1)−A(w1)−Q(w1), 4Λ) ≥ µ′ ( L1−v 1−L ΨA (w1) , 3Λ 4 ) ∗ µ′ ( L1−v 1−L ΨA (−w1) , 3Λ 4 ) ∗ µ′ ( L1−v 1−L ΨQ (w1) , 7Λ 3 ) ∗ µ′ ( L1−v 1−L ΨQ (−w1) , 7Λ 3 ) = µ′ ( L1−v 1−L Ψ (w1, w1, w1) , 3Λ 4 ) ∗ µ′ ( L1−v 1−L Ψ (w1, w1,−w1) , 3Λ 4 ) ∗ µ′ ( L1−v 1−L Ψ (−w1,−w1,−w1) , 3Λ 4 ) ∗ µ′ ( L1−v 1−L Ψ (−w1,−w1, w1) , 3Λ 4 ) ∗ µ′ ( L1−v 1−L Ψ (w1, w1, w1) , 7Λ 3 ) ∗ µ′ ( L1−v 1−L Ψ (w1, w1,−w1) , 7Λ 3 ) ∗ µ′ ( L1−v 1−L Ψ (−w1,−w1,−w1) , 7Λ 3 ) ∗ µ′ ( L1−v 1−L Ψ (−w1,−w1, w1) , 7Λ 3 ) ν (F (w1)−A(w1)−Q(w1), 4Λ) ≤ ν′ ( L1−v 1−L ΨA (w1) , 3Λ 4 ) � ν′ ( L1−v 1−L ΨA (−w1) , 3Λ 4 ) � ν′ ( L1−v 1−L ΨQ (w1) , 7Λ 3 ) � ν′ ( L1−v 1−L ΨQ (−w1) , 7Λ 3 ) = ν′ ( L1−v 1−L Ψ (w1, w1, w1) , 3Λ 4 ) � ν′ ( L1−v 1−L Ψ (w1, w1,−w1) , 3Λ 4 ) � ν′ ( L1−v 1−L Ψ (−w1,−w1,−w1) , 3Λ 4 ) � ν′ ( L1−v 1−L Ψ (−w1,−w1, w1) , 3Λ 4 ) � ν′ ( L1−v 1−L Ψ (w1, w1, w1) , 7Λ 3 ) � ν′ ( L1−v 1−L Ψ (w1, w1,−w1) , 7Λ 3 ) � ν′ ( L1−v 1−L Ψ (−w1,−w1,−w1) , 7Λ 3 ) � ν′ ( L1−v 1−L Ψ (−w1,−w1, w1) , 7Λ 3 )  (3.75) and the mapping A(w1) and Q(w1) are given in (3.65) and (3.74) for all w1 ∈ W1 and all Λ > 0. Proof. The proof is similar ideas to that of Theorem 3.22. � Corollary 3.34. Suppose that a function F : W1 → W2 satisfy the functional inequalities (3.2), (3.3), (3.4), (3.5), (3.6), (3.7) for all w1, w2, w3 ∈ W1 and all Λ > 0 with δ be a positive constant and ϕ be any real number. Then there exists a unique additive mapping A(w1) : W1 → W2 and a unique quadratic mapping Q(w1) : W1 → W2 which satisfies (1.7) and the functional inequalities (3.57), (3.58), (3.59), (3.60), (3.61), (3.62) for all w1 ∈ W1. CONCLUSION In this paper, we analyze the generalized Ulam-Hyers stability of a affine type AQ Functional Equa- tion in Banach Space and Intuitionistic Fuzzy Banach Space with the help of classical Hyers direct and Radus fixed methods. The results are new, since we are getting better possible upper bound than previ- ous stability analysis (see [6]). 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Oddness of F: Additive Case Stability Results : Direct Method 2.2. Evenness of F: Quadratic Case Stability Results : Direct Method 2.3. Oddness and Evenness of F: Additive Quadratic Case Stability Results : Direct Method 2.4. Oddness of F: Additive Case Stability Results : Fixed Point Method 2.5. Evenness of F: Quadratic Case Stability Results : Fixed Point Method 2.6. Oddness and Evenness of F: Additive Quadratic Case Stability Results : Fixed Point Method 3. Stability In Intuitionistic Fuzzy Banach Space of (1.7) 3.1. Definitions and Notations of Intuitionistic Fuzzy Banach Space 3.2. Oddness of F: Additive Case Stability Results : Direct Method 3.3. Evenness of F: Quadratic Case Stability Results : Direct Method 3.4. Oddness and Evenness of F: Additive Quadratic Case Stability Results : Direct Method 3.5. Oddness of F: Additive Case Stability Results : Fixed Point Method 3.6. Evenness of F: Quadratic Case Stability Results : Fixed Point Method 3.7. Oddness and Evenness of F: Additive Quadratic Case Stability Results : Fixed Point Method Conclusion Acknowledgment Conflict of Interest References