EUROPEAN JOURNAL OF PURE AND APPLIED MATHEMATICS 2025, Vol. 18, Issue 2, Article Number 5897 ISSN 1307-5543 – ejpam.com Published by New York Business Global Stability of Hyper 3-Homomorphisms and Hyper 3-Derivations in Ternary Algebras EunHwa Shim1, Siriluk Donganont2,∗, Choonkil Park3 1 Department of Mathematics, Hanyang University, Seoul 04763, Korea 2 School of Science, University of Phayao, Phayao 56000, Thailand 3 Department of Mathematics, Research Institute for Convergence of Basic Science, Hanyang University, Seoul 04763, Korea Abstract. In this paper, we introduce hyper 3-homomorphisms and hyper 3-derivations in com- plex ternary algebras and we prove the Hyers-Ulam stability of hyper 3-homomorphisms and hyper 3-derivations in complex ternary algebras for the following 3-additive functional equation f(x1 + x2, y1 + y2, z1 + z2) = 2∑ i,j,k=1 f(xi, yj , zk). (1) Further, we investigate isomorphisms between complex ternary algebras, associated with the 3- additive functional equation. 2020 Mathematics Subject Classifications: 11E20, 39B52, 39B82 Key Words and Phrases: Hyers-Ulam stability, 3-additive functional equation, ternary algebra, hyper 3-homomorphism, hyper 3-derivation 1. Introduction and Preliminaries The first stability proplem was raised by Ulam [1] during his talk at University of Wisconsin in 1940. In 1941, Hyers [2] gave a first affirmative answer to the question of Ulam for Banach spaces. Let f : E → E′ be a mapping between Banach spaces such that ∥f(x+ y)− f(x)− f(y)∥ ≤ δ for all x, y ∈ E and for some δ > 0. Then, there exists a unique additive mapping l : E → E′ such that ∥f(x)− l(x)∥ ≤ δ ∗Corresponding author. DOI: https://doi.org/10.29020/nybg.ejpam.v18i2.5897 Email addresses: stareun97@hanyang.ac.kr (E. Shim), siriluk.pa@up.ac.th (S. Donganont), baak@hanyang.ac.kr (C. Park) https://www.ejpam.com 1 Copyright: © 2025 The Author(s). (CC BY-NC 4.0) E. Shim, S. Donganont, C. Park / Eur. J. Pure Appl. Math, 18 (2) (2025), 5897 2 of 14 for all x ∈ E. This stability phenomenon is called the Hyers-Ulam stability of the additive functional equation g(x+ y) = g(x)+ g(y). In 1978, Rassias [3] generalized the theorem of Hyers by considering the stability problem with unbounded Cauchy differences. Moreover if f(µx) is continuous in µ ∈ R for each fixed x ∈ E, then l is R-linear. Găvruta [4] obtained a generalized result of the Rassias theorem which allows the Cauchy difference to be controlled by a general unbounded function. The stability problems of various functional equations and functional inequalities have been extensively investigated by a number of authors (see [5–14]). Ternary structures and their generalization, the so-called n-ary structures, raise certain hopes in view of their applications in physics (see [15–17]). A general ternary algebra is defined as internal ternary multiplication in a vector space. Let A be a linear space over a complex number field equipped with a mapping [·, ·, ·] : A3 = A × A × A → A with (x, y, z) 7→ [x, y, z], which is C-inear in each outer variable and conjugate C-linear in the middle variable, and satisfies the following associative identity condition [[x, y, z], u, v] = [x, [y, z, u], v] = [x, y, [z, u, v]] for all x, y, z, u, v ∈ A. Then the pair (A, [·, ·, ·]) is called a complex ternary algebra. Assume that A is a complex ternary algebra. Then we say that A has a unit if there exist an element e ∈ A such that [e, e, a] = [e, a, e] = [a, e, e] = a for all a ∈ A. Park [18] and Moslehian [19] contributed works on the stability problem of ternary homomorphisms and ternary derivations and Bavand Savadkouhi [20] investigated the stability problem of ternary Jordan homomorphisms and ternary Jordan derivations. The stability probelms of several functional equations have been extensively investigated by a number of authors and there are many interesting results, containing ternary homomorphisms and ternary derivations, concerning this problem (see [21–26]). Let A and A′ be complex ternary algebras. A C-linear mapping H : A → A′ is called a ternary algebra homomorphism if H([x, y, z]) = [H(x), H(y), H(z)] for all x, y, z ∈ A. If, in addition, the C-linear mapping H is bijective, then the C-linear mapping H : A → A′ is called a ternary algebra isomorphism. A C-linear mapping δ : A → A is called a ternary algebra derivation if δ([x, y, z]) = [δ(x), y, z] + [x, δ(y), z] + [x, y, δ(z)] for all x, y, z ∈ A (see [27–30]). Let X be a complex ternary algebra. A mapping f : X3 → X is 3-additive if f(x1 + x2, y1 + y2, z1 + z2) = 2∑ i,j,k=1 f(xi, yj , zk) for all x1, y1, z1, x2, y2, z2 ∈ X. A mapping f : X3 → X is called 3-linear if f is 3-additive and C-linear for each variable. Throughout the paper, assume that X is a complex ternary algebra, Y is a complex ternary Banach algebra and t is a fixed nonzero real number with |t| < 1. E. Shim, S. Donganont, C. Park / Eur. J. Pure Appl. Math, 18 (2) (2025), 5897 3 of 14 2. Stability of hyper 3-homomorphisms in ternary algebras In this section, we prove the Hyers-Ulam stability of hyper 3-homomorphisms in com- plex ternary algebras and we investigate ternary algebra isomorphisms between complex ternary algebras, associated with the 3-additive functional equation (1). Definition 1. Let X and Y be complex ternary algebras. A 3-linear mapping h : X3 → Y is called a hyper 3-additivec mapping if h satisfies 8h(x1, y1, z1) = 2∑ i,j,k=1 h(xi + (−1)ix2, yj + (−1)jy2, z1 + (−1)kz2) (2) for all x1, y1, z1, x2, y2, z2 ∈ X. Definition 2. Let X and Y be complex ternary algebras. A 3-linear mapping h : X3 → Y is called a hyper 3-homomorphism if h satisfies h([x1, y1, z1], [x2, y2, z2], [x3, y3, z3]) = [h(x1, x2, x3), h(y1, y2, y3), h(z1, z2, z3)] for all x1, x2, x3, y1, y2, y3, z1, z2, z3 ∈ X. Lemma 1. Let X and Y be complex ternary algebras. Let h : X3 → Y be a hyper 3-additive mapping and satisfy h(2x, 2y, 2z) = 8h(x, y, z) for all x, y, z ∈ X3, then h is 3-additive. Proof. For x1, x2, y1, y2, z1, z2 ∈ X, we define p1 := x1 + x2 2 , p2 := x1 − x2 2 , q1 := y1 + y2 2 , q2 := y1 − y2 2 , r1 := z1 + z2 2 and r2 := z1 − z2 2 . It follows from (2) that h(x1 + x2, y1 + y2, z1 + z2) = h(2p1, 2q1, 2r1) = 8h(p1, q1, r1) = 2∑ i,j,k=1 h(p1 + (−1)ip2, q1 + (−1)jq2, r1 + (−1)kr2) = 2∑ i,j,k=1 h(xi, yj , zk). This completes the proof. Lemma 2. [31] Let X and Y be complex vector spaces and f : X3 → Y be a 3-additive mapping such that f(λx, µy, νz) = λµνf(x, y, z) for all λ, µ, ν ∈ T1 := {κ ∈ R | |κ| = 1} and x, y, z ∈ X. Then f is 3-linear. E. Shim, S. Donganont, C. Park / Eur. J. Pure Appl. Math, 18 (2) (2025), 5897 4 of 14 Theorem 1. Let X and Y be complex ternary algebras and t be a real number satisfying |t| < 1. Assume that a mapping h : X3 → Y satisfies h(0, a, b) = h(a, 0, b) = h(a, b, 0) = 0 and ∥∥∥∥∥∥8h(x1, y1, z1)− 2∑ i,j,k=1 h(x1 + (−1)ix2, y1 + (−1)jy2, z1 + (−1)kz2) ∥∥∥∥∥∥ (3) ≤ ∥∥∥∥∥∥t 8 2∑ i,j,k=1 h ( x1 + (−1)ix2 2 , y1 + (−1)jy2 2 , z1 + (−1)kz2 2 ) − 8h(x1, y1, z1) ∥∥∥∥∥∥ for all a, b ∈ X and all (x1, y1, z1), (x2, y2, z2) ∈ X3. Then h is hyper 3-additive. Proof. Letting x1 = x2 := x, y1 = y2 := y and z1 = z2 := z in (3), we get ∥h(2x, 2y, 2z)− 8h(x, y, z)∥ ≤ 0 for all x, y, z ∈ X. So h(2x, 2y, 2z) = 8h(x, y, z) for all x, y, z ∈ X. It follows from (3) that∥∥∥∥∥∥8h(x1, y1, z1)− 2∑ i,j,k=1 h(x1 + (−1)ix2, y1 + (−1)jy2, z1 + (−1)kz2) ∥∥∥∥∥∥ ≤ ∥∥∥∥∥∥t 8h(x1, y1, z1)− 2∑ i,j,k=1 h(x1 + (−1)ix2, y1 + (−1)jy2, z1 + (−1)kz2) ∥∥∥∥∥∥ for all (x1, y1, z1), (x2, y2, z2) ∈ X3. Thus 8h(x1, y1, z1) = 2∑ i,j,k=1 h(x1 + (−1)ix2, y1 + (−1)jy2, z1 + (−1)kz2) for all (x1, y1, z1), (x2, y2, z2) ∈ X3, since |t| < 1. Thus, the mapping h is hyper 3-additive. Theorem 2. Let X be a complex ternary algebra, Y be a complex ternary Banach algebra and t be a real number satisfying |t| < 1. Let φ : X6 → [0,∞) and ψ : X9 → [0,∞) be functions such that +∞∑ j=1 8jφ ( x 2j , y 2j , z 2j , x 2j , y 2j , z 2j ) <∞ and +∞∑ j=1 83jψ ( x 2j , x 2j , x 2j , y 2j , y 2j , y 2j , z 2j , z 2j , z 2j ) <∞ E. Shim, S. Donganont, C. Park / Eur. J. Pure Appl. Math, 18 (2) (2025), 5897 5 of 14 for all x, y, z ∈ X. Assume that a mapping h : X3 → Y satisfies h(0, a, b) = h(a, 0, b) = h(a, b, 0) = 0 and ∥∥∥∥∥∥8µh(x1, y1, z1)− 2∑ i,j,k=1 h(µ(x1 + (−1)ix2), µ(y1 + (−1)jy2), µ(z1 + (−1)kz2)) ∥∥∥∥∥∥ (4) ≤ ∥∥∥∥∥∥t 8µh(x1, y1, z1)− 2∑ i,j,k=1 h ( µ x1 + (−1)ix2 2 , µ y1 + (−1)jy2 2 , µ z1 + (−1)kz2 2 )∥∥∥∥∥∥ +φ(x1, y1, z1, x2, y2, z2) for all a, b ∈ X and all (x1, y1, z1), (x2, y2, z2) ∈ X3 and all µ ∈ T1. Let h : X3 → X satisfy ∥h([x1, y1, z1], [x2, y2, z2], [x3, y3, z3])− [h(x1, x2, x3), h(y1, y2, y3), h(z1, z2, x3)]∥ (5) ≤ ψ(x1, x2, x3, y1, y2, y3, z1, z2, z3) for all x1, x2, x3, y1, y2, y3, z1, z2, z3 ∈ X. Then there exists a unique hyper 3-homomorphism H : X3 → Y such that ∥h(x, y, z)−H(x, y, z)∥ ≤ +∞∑ j=0 8jφ ( x 2j+1 , y 2j+1 , z 2j+1 , x 2j+1 , y 2j+1 , z 2j+1 ) (6) for all x, y, z ∈ X. Proof. Letting µ = 1, x1 = x2 := x, y1 = y2 := y and z1 = z2 := z in (4), we get ∥h(2x, 2y, 2z)− 8h(x, y, z)∥ ≤ φ(x, y, z, x, y, z) and so ∥∥∥h(x, y, z)− 8h (x 2 , y 2 , z 2 )∥∥∥ ≤ φ (x 2 , y 2 , z 2 , x 2 , y 2 , z 2 ) for all x, y, z ∈ X. Hence∥∥∥8lh( x 2l , y 2l , z 2l ) − 8l+kh ( x 2l+k , y 2l+k , z 2l+k )∥∥∥ (7) ≤ k−1∑ j=0 ∥∥∥8l+jh ( x 2l+j , y 2l+j , z 2l+j ) − 8l+(j+1)h ( x 2l+j+1 , y 2l+j+1 , z 2l+j+1 )∥∥∥ = k−1∑ j=0 8l+j ∥∥∥h( x 2l+j , y 2l+j , z 2l+j ) − 8h ( x 2l+j+1 , y 2l+j+1 , z 2l+j+1 )∥∥∥ E. Shim, S. Donganont, C. Park / Eur. J. Pure Appl. Math, 18 (2) (2025), 5897 6 of 14 ≤ k−1∑ j=0 8l+jφ ( x 2l+j+1 , y 2l+j+1 , z 2l+j+1 , x 2l+j+1 , y 2l+j+1 , z 2l+j+1 ) for all nonnegative integers l, k and all x, y, z ∈ X. It follows that { 8jh ( x 2j , y 2j , z 2j )} is a Cauchy sequence for each (x, y, z) ∈ X3. Since Y is complete, { 8jh ( x 2j , y 2j , z 2j )} converges. Thus one can define the mapping H : X3 → Y by H(x, y, z) := lim n→+∞ 8nh ( x 2n , y 2n , z 2n ) for all (x, y, z) ∈ X3. Moreover, letting l = 0 and passing the limit k → ∞ in (7), we get (6). It follows from (4) that∥∥∥∥∥∥8µH(x1, y1, z1)− 2∑ i,j,k=1 H(µ(x1 + (−1)ix2), µ(y1 + (−1)jy2), µ(z1 + (−1)kz2)) ∥∥∥∥∥∥ = lim n→+∞ 8n ∥∥∥8µh(x1 2n , y1 2n , z1 2n ) − 2∑ i,j,k=1 h ( µ x1 + (−1)ix2 2n , µ y1 + (−1)jy2 2n , µ z1 + (−1)kz2 2n )∥∥∥∥∥∥ ≤ lim n→+∞ 8n ∥∥∥t(8µh(x1 2n , y1 2n , z1 2n ) − 2∑ i,j,k=1 h ( µ x1 + (−1)ix2 2n , µ y1 + (−1)jy2 2n , µ z1 + (−1)kz2 2n )∥∥∥∥∥∥ + lim n→+∞ 8nφ (x1 2n , y1 2n , z1 2n , x2 2n , y2 2n , z2 2n ) = ∥t (8µH(x1, y1, z1) − 2∑ i,j,k=1 H(µ(x1 + (−1)ix2), µ(y1 + (−1)jy2), µ(z1 + (−1)kz2)) ∥∥∥∥∥∥ for all (x1, y1, z1), (x2, y2, z2) ∈ X3 and µ ∈ T1. Thus∥∥∥∥∥∥8µH(x1, y1, z1)− 2∑ i,j,k=1 H(µ(x1 + (−1)ix2), µ(y1 + (−1)jy2), µ(z1 + (−1)kz2)) ∥∥∥∥∥∥ ≤ ∥t (8µH(x1, y1, z1) (8) − 2∑ i,j,k=1 H(µ(x1 + (−1)ix2), µ(y1 + (−1)jy2), µ(z1 + (−1)kz2)) ∥∥∥∥∥∥ E. Shim, S. Donganont, C. Park / Eur. J. Pure Appl. Math, 18 (2) (2025), 5897 7 of 14 for all (x1, y1, z1), (x2, y2, z2) ∈ X3 and µ ∈ T1. Let µ = 1 in (8). By Theorem 1, the mapping H : X3 → X is 3-additive. It follows from (8) and the 3-additivity of H that∥∥∥∥∥∥8µH(x1, y1, z1)− 2∑ i,j,k=1 H(µ(x1 + (−1)ix2), µ(y1 + (−1)jy2), µ(z1 + (−1)kz2)) ∥∥∥∥∥∥ ≤ ∥t (8µH(x1, y1, z1) − 2∑ i,j,k=1 H(µ(x1 + (−1)ix2), µ(y1 + (−1)jy2), µ(z1 + (−1)kz2)) ∥∥∥∥∥∥ for all (x1, y1, z1), (x2, y2, z2) ∈ X3 and µ ∈ T1. Since |t| < 1, 8µH(x1, y1, z1) = 2∑ i,j,k=1 H(µ(x1 + (−1)ix2), µ(y1 + (−1)jy2), µ(z1 + (−1)kz2)), and H(µ(x1, y1, z1)) = µH(x1, y1, z1) for all (x1, y1, z1) ∈ X3 and µ ∈ T1. By Lemma 2, the mapping H : X3 → X is 3-linear. It follows from (5) and the 3-additivity of H that ∥H([x1, y1, z1], [x2, y2, z2], [x3, y3, z3])− [H(x1, x2, x3), H(y1, y2, y3), H(z1, z2, z3)]∥ = lim n→+∞ 83n ∥∥∥h( [x1, y1, z1] 8n , [x2, y2, z2] 8n , [x3, y3, z3] 8n ) − [ h (x1 2n , x2 2n , x3 2n ) , h ( y1 2n , y2 2n , y3 2n ) , h ( z1 2n , z2 2n , z3 2n )] ∥∥∥ ≤ lim n→+∞ 83nψ (x1 2n , x2 2n , x3 2n , y1 2n , y2 2n , y3 2n , z1 2n , z2 2n , z3 2n ) = 0. So H([x1, y1, z1], [x2, y2, z2], [x3, y3, z3]) = [H(x1, x2, x3), H(y1, y2, y3), H(z1, z2, z3)] for all x1, x2, x3, y1, y2, y3, z1, z2, z3 ∈ X. Therefore, the mapping H is a unique hyper 3-homomor-phism satisfying (6). Theorem 3. Let X be a complex ternary algebra, Y be a complex ternary Banach algebra and t be a real number satisfying |t| < 1. Let h : X3 → Y be a bijective mapping satisfying (4) such that h([x1, y1, z1], [x2, y2, z2], [x3, y3, z3]) = [h(x1, x2, x3), h(y1, y2, y3), h(z1, z2, z3)] (9) for all x1, x2, x3, y1, y2, y3, z1, z2, z3 ∈ X. If h(αx0, βy0, γz0) is continuous in α, β, γ ∈ R for each fixed (x0, y0, z0) ∈ X3 and limn→+∞ 8nh ( e 2n , e 2n , e 2n ) = e′, then the mapping h : X3 → Y is a hyper 3-isomorphism. E. Shim, S. Donganont, C. Park / Eur. J. Pure Appl. Math, 18 (2) (2025), 5897 8 of 14 Proof. Since h satisfies (9), the mapping h : X3 → Y satisfies (4) by Theorem 1, there exists a hyper 3-homomorphism H : X3 → Y satisfying (6). The mapping H : X3 → Y is defined by H(x, y, z) := lim n→+∞ 8nh ( x 2n , y 2n , z 2n ) for all x, y, z ∈ X. It follows from (9) that ∥[H(x1, x2, x3), H(y1, y2, y3), H(z1, z2, z3)]− [H(x1, x2, x3), H(y1, y2, y3), h(z1, z2, z3)]∥ = ∥H([x1, y1, z1], [x2, y2, z2], [x3, y3, z3])− [H(x1, x2, x3), H(y1, y2, y3), h(z1, z2, z3)]∥ = lim n→+∞ 82n ∥∥∥h([x1 2n , y1 2n , z1 ] , [x2 2n , y2 2n , z2 ] , [x3 2n , y3 2n , z3 ]) − [ h (x1 2n , x2 2n , x3 2n ) , h ( y1 2n , y2 2n , y3 2n ) , h(z1, z2, z3) ] ∥∥∥ ≤ lim n→+∞ 82nψ (x1 2n , x2 2n , x3 2n , y1 2n , y2 2n , y3 2n , z1, z2, z3 ) = 0 for all x1, x2, x3, y1, y2, y3, z1, z2, z3 ∈ X. So [H(x1, x2, x3), H(y1, y2, y3), H(z1, z2, z3)] = [H(x1, x2, x3), H(y1, y2, y3), h(z1, z2, z3)] for all x1, x2, x3, y1, y2, y3, z1, z2, z3 ∈ X. Letting x1 = y1 = x2 = y2 = x3 = y3 = e in the last equality, we get h(z1, z2, z3) = H(z1, z2, z3) for all z1, z2, z3 ∈ X. Therefore, the bijective mapping h : X3 → Y is a hyper 3-isomorphism. 3. Stability of hyper 3-derivations in ternary algebras In this section, we prove the Hyers-Ulam stability of hyper 3-derivations in complex ternary algebras. Definition 3. Let X be a ternary algebra. A 3-linear mapping f : X3 → X is called a hyper 3-derivation if f satisfies f([x1, y1, z1], [x2, y2, z2], [x3, y3, z3]) = [f(x1, x2, x3), [y1, y2, y3], [z1, z2, z3]] + [[x1, x2, x3], f(y1, y2, y3), [z1, z2, z3]] + [[x1, x2, x3], [y1, y2, y3], f(z1, z2, z3)] for all x1, x2, x3, y1, y2, y3, z1, z2, z3 ∈ X. Theorem 4. Let X be a ternary algebra and t be a real number satisfying |t| < 1. If a mapping f : X3 → X satisfies∥∥∥∥∥∥f(x1 + x2, y1 + y2, z1 + z2)− 2∑ i,j,k=1 f(xi, yj , zk) ∥∥∥∥∥∥ (10) E. Shim, S. Donganont, C. Park / Eur. J. Pure Appl. Math, 18 (2) (2025), 5897 9 of 14 ≤ ∥∥∥∥∥∥t 8f ( x1 + x2 2 , y1 + y2 2 , z1 + z2 2 ) − 2∑ i,j,k=1 f(xi, yj , zk) ∥∥∥∥∥∥ for all (x1, y1, z1), (x2, y2, z2) ∈ X3. Then f is 3-additive. Proof. Letting x1 = x2 := x, y1 = y2 := y and z1 = z2 := z in (10), we get ∥f(2x, 2y, 2z)− 8f(x, y, z)∥ ≤ 0 for all x, y, z ∈ X. So f(2x, 2y, 2z) = 8f(x, y, z) for all x, y, z ∈ X. It follows from (10) that ∥∥∥∥∥∥f(x1 + x2, y1 + y2, z1 + z2)− 2∑ i,j,k=1 f(xi, yj , zk) ∥∥∥∥∥∥ ≤ ∥∥∥∥∥∥t f(x1 + x2, y1 + y2, z1 + z2)− 2∑ i,j,k=1 f(xi, yj , zk) ∥∥∥∥∥∥ for all (x1, y1, z1), (x2, y2, z2) ∈ X3. Thus f(x1 + x2, y1 + y2, z1 + z2) = 2∑ i,j,k=1 f(xi, yj , zk) for all (x1, y1, z1), (x2, y2, z2) ∈ X3, since |t| < 1. Thus the mapping f is 3-additive. Theorem 5. Let X be a ternary Banach algebra and t be a real number satisfying |t| < 1. Let φ : X6 → [0,∞) and ψ : X9 → [0,∞) be functions such that +∞∑ j=1 8jφ ( x 2j , y 2j , z 2j , x 2j , y 2j , z 2j ) <∞ and +∞∑ j=1 83jψ ( x 2j , x 2j , x 2j , y 2j , y 2j , y 2j , z 2j , z 2j , z 2j ) <∞ for all x, y, z ∈ X. Let f : X3 → X be a mapping satisfying∥∥∥∥∥∥f(µ(x1 + x2, y1 + y2, z1 + z2))− µ 2∑ i,j,k=1 f(xi, yj , zk) ∥∥∥∥∥∥ (11) ≤ ∥∥∥∥∥∥t 8f ( µ ( x1 + x2 2 , y1 + y2 2 , z1 + z2 2 )) − µ 2∑ i,j,k=1 f(xi, yj , zk) ∥∥∥∥∥∥ E. Shim, S. Donganont, C. Park / Eur. J. Pure Appl. Math, 18 (2) (2025), 5897 10 of 14 +φ(x1, y1, z1, x2, y2, z2) for all (x1, y1, z1), (x2, y2, z2) ∈ X3 and µ ∈ T1, and ∥f([x1, y1, z1], [x2, y2, z2], [x3, y3, z3])− [f(x1, x2, x3), [y1, y2, y3], [z1, z2, z3]] (12) −[[x1, x2, x3], f(y1, y2, y3), [z1, z2, z3]]− [[x1, x2, x3], [y1, y2, y3], f(z1, z2, z3)]∥ ≤ ψ(x1, x2, x3, y1, y2, y3, z1, z2, z3) for all x1, x2, x3, y1, y2, y3, z1, z2, z3 ∈ X. Then there exists a unique hyper 3-derivation D : X3 → X such that ∥f(x, y, z)−D(x, y, z)∥ ≤ +∞∑ j=0 8jφ ( x 2j+1 , y 2j+1 , z 2j+1 , x 2j+1 , y 2j+1 , z 2j+1 ) (13) for all x, y, z ∈ X. Proof. Letting µ = 1, x1 = x2 := x, y1 = y2 := y and z1 = z2 := z in (11), we get ∥f(2x, 2y, 2z)− 8f(x, y, z)∥ ≤ φ(x, y, z, x, y, z) for all x, y, z ∈ X. By induction, we have∥∥∥f(x, y, z)− 8nf ( x 2n , y 2n , z 2n )∥∥∥ ≤ n−1∑ j=0 8jφ ( x 2j , y 2j , z 2j , x 2j , y 2j , z 2j ) for all x, y, z ∈ X. Hence∥∥∥8lf ( x 2l , y 2l , z 2l ) − 8kf ( x 2k , y 2k , z 2k )∥∥∥ (14) ≤ k−1∑ j=l ∥∥∥8jf ( x 2j , y 2j , z 2j ) − 8j+1f ( x 2j+1 , y 2j+1 , z 2j+1 )∥∥∥ ≤ k−1∑ j=l 8jφ ( x 2j+1 , y 2j+1 , z 2j+1 , x 2j+1 , y 2j+1 , z 2j+1 ) for all nonnegative integers l, k(k > l) and all x, y, z ∈ X. It follows that the sequence{ 8kf ( x 2k , y 2k , z 2k )} is a Cauchy sequence for each (x, y, z) ∈ X3. Since X is complete, the sequence { 8kf ( x 2k , y 2k , z 2k )} converges. Thus one can define the mapping D : X3 → X by D(x, y, z) := lim n→+∞ 8nf ( x 2n , y 2n , z 2n ) for all (x, y, z) ∈ X3. Moreover, letting l = 0 and passing the limit k → ∞ in (14), we get (13). It follows from (11) that ∥D(µ(x1 + x2, y1 + y2, z1 + z2))− µ 2∑ i,j,k=1 D(xi, yj , zk)∥ E. Shim, S. Donganont, C. Park / Eur. J. Pure Appl. Math, 18 (2) (2025), 5897 11 of 14 = lim n→+∞ 8n ∥∥∥∥∥∥f µ(x1 + x2 2n , y1 + y2 2n , z1 + z2 2n ) − µ 2∑ i,j,k=1 f ( xi 2n , yj 2n , zk 2n )∥∥∥∥∥∥ ≤ lim n→+∞ 8n ∥∥∥∥∥∥t 8f ( µ ( x1 + x2 2n+1 , y1 + y2 2n+1 , z1 + z2 2n+1 )) − µ 2∑ i,j,k=1 f ( xi 2n , yj 2n , zk 2n )∥∥∥∥∥∥ + lim n→+∞ 8nφ (x1 2n , y1 2n , z1 2n , x2 2n , y2 2n , z2 2n ) = ∥∥∥∥∥∥t 8D ( µ ( x1 + x2 2 , y1 + y2 2 , z1 + z2 2 )) − µ 2∑ i,j,k=1 D(xi, yj , zk) ∥∥∥∥∥∥ for all (x1, y1, z1), (x2, y2, z2) ∈ X3 and µ ∈ T1. Thus ∥D(µ(x1 + x2, y1 + y2, z1 + z2))− µ 2∑ i,j,k=1 D(xi, yj , zk)∥ (15) ≤ ∥∥∥∥∥∥t 8D ( µ ( x1 + x2 2 , y1 + y2 2 , z1 + z2 2 )) − µ 2∑ i,j,k=1 D(xi, yj , zk) ∥∥∥∥∥∥ for all (x1, y1, z1), (x2, y2, z2) ∈ X3 and µ ∈ T1. Let µ = 1 in (15). By Theorem 4, the mapping D : X3 → X is 3-additive. It follows from (15) and the 3-additivity of D that ∥D(µ(x1 + x2, y1 + y2, z1 + z2))− µ 2∑ i,j,k=1 D(xi, yj , zk)∥ ≤ ∥∥∥∥∥∥t D(µ(x1 + x2, y1 + y2, z1 + z2))− µ 2∑ i,j,k=1 D(xi, yj , zk) ∥∥∥∥∥∥ for all (x1, y1, z1), (x2, y2, z2) ∈ X3 and µ ∈ T1. Since |t| < 1, D(µ(x1 + x2, y1 + y2, z1 + z2)) = µ 2∑ i,j,k=1 D(xi, yj , zk) and D(µ(x1, y1, z1)) = µD(x1, y1, z1) for all (x1, y1, z1) ∈ X3 and µ ∈ T1. By Lemma 2, the mapping D : X3 → X is 3-linear. It follows from (12) and the 3-additivity of D that ∥D([x1, y1, z1], [x2, y2, z2], [x3, y3, z3])− [D(x1, x2, x3), [y1, y2, y3], [z1, z2, z3]] −[[x1, x2, x3], D(y1, y2, y3), [z1, z2, z3]]− [[x1, x2, x3], [y1, y2, y3], D(z1, z2, z3)]∥ = lim n→+∞ 83n ∥∥∥f ( [x1, y1, z1] 8n , [x2, y2, z2] 8n , [x3, y3, z3] 8n ) − [ f (x1 2n , x2 2n , x3 2n ) , [y1, y2, y3] 8n , [z1, z2, z3] 8n ] E. Shim, S. Donganont, C. Park / Eur. J. Pure Appl. Math, 18 (2) (2025), 5897 12 of 14 − [ [x1, x2, x3] 8n , f ( y1 2n , y2 2n , y3 2n ) , [z1, z2, z3] 8n ] − [ [x1, x2, x3] 8n , [y1, y2, y3] 8n , f ( z1 2n , z2 2n , z3 2n )]∥∥∥ ≤ lim n→+∞ 83nψ (x1 2n , x2 2n , x3 2n , y1 2n , y2 2n , y3 2n , z1 2n , z2 2n , z3 2n ) = 0 for all x1, x2, x3, y1, y2, y3, z1, z2, z3 ∈ X. So D([x1, y1, z1], [x2, y2, z2], [x3, y3, z3]) = [D(x1, x2, x3), [y1, y2, y3], [z1, z2, z3]] −[[x1, x2, x3], D(y1, y2, y3), [z1, z2, z3]]− [[x1, x2, x3], [y1, y2, y3], D(z1, z2, z3)] for all x1, x2, x3, y1, y2, y3, z1, z2, z3 ∈ X. Therefore, the mapping H is a unique hyper 3-derivation satisfying (13). 4. Conclusion and future work In this paper, we introduced hyper 3-homomorphisms and hyper 3-derivations in ternary algebras and we proved the Hyers-Ulam stability of hyper 3-homomorphisms and hyper 3-derivations in ternary Banach algebras, associated with the 3-additive functional equation (1). We will provide suitable examples and useful applications in next work. Acknowledgements The authors are thankful to the editors and the anonymous reviewers for many valuable suggestions to improve this paper. Declarations Availablity of data and materials Not applicable. Human and animal rights We would like to mention that this article does not contain any studies with animals and does not involve any studies over human being. Conflict of interest The authors declare that they have no competing interests. Fundings S. Donganont was supported by the University of Phayao and Thailand Science Research and Innovation Fund (Fundamental Fund 2025, Grant No. 5020/2567). E. Shim, S. Donganont, C. Park / Eur. J. Pure Appl. Math, 18 (2) (2025), 5897 13 of 14 References [1] S M Ulam. Problems in Modern Mathematics. John Wiley & Sons, Inc., New York, 1964. [2] D H Hyers. On the stability of the linear functional equation. Proc. Natl. Acad. Sci. U.S.A., 27:222–224, 1941. [3] T M Rassias. On the stability of the linear mapping in Banach spaces. Proc. Amer. Math. 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