We will show the general solution of the functional equation f ( x + ay ) + f ( x − ay ) + 2( a ² − 1) f ( x ) = a ² f ( x + y ) + a ² f ( x − y ) + 2 a ² ( a ² − 1) f ( y ) and investigate the stability of quartic Lie *-derivations associated with the given functional equation. The stability problem of functional equations originated from a question of Ulam [17] concerning the stability of group homomorphisms. Hyers [7] gave a first affirmative partial answer to the question of Ulam. Afterwards, the result of Hyers was generalized by Aoki [1] for additive mapping and by Rassias [14] for linear mappings by considering a unbounded Cauchy difference. Later, the result of Rassias has provided a lot of in°uence in the development of what we call Hyers-Ulam stability or Hyers-Ulam-Rassias stability of functional equations. For further information about the topic, we also refer the reader to [10] , [8] , [2] and [3] . Recall that a Banach *-algebra is a Banach algebra (complete normed algebra) which has an isometric involution. Jang and Park [9] investigated the stability of *-derivations and of quadratic *-derivations with Cauchy functional equation and the Jensen functional equation on Banach *-algebra. The stability of *-derivations on Banach *-algebra by using fixed point alternative was proved by Park and Bodaghi and also Yang et al.; see [12] and [19] , respectively. Also, the stability of cubic Lie derivations was introduced by Fošner and Fošner; see [6] . Rassias [13] investigated stability properties of the following quartic functional equation It is easy to see that f ( x ) = x ⁴ is a solution of (1.1) by virtue of the identity For this reason, (1.1) is called a quartic functional equation. Also Chung and Sahoo [4] determined the general solution of (1.1) without assuming any regularity conditions on the unknown function. In fact, they proved that the function f : ℝ → ℝ is a solution of (1.1) if and only if f ( x ) = A ( x , x , x , x ) , where the function A : ℝ ⁴ → ℝ is symmetric and additive in each variable. In this paper, we deal with the following functional equation: for all x , y ∈ X and an integer a ( a ≠ 0 , ±1) . We will show the general solution of the functional equation (1.3), define a quartic Lie *-derivation related to equation (1.3) and investigate the Hyers-Ulam stability of the quartic Lie *-derivations associated with the given functional equation. In this section let X and Y be real vector spaces and we investigate the general solution of the functional equation (1.3). Before we proceed, we would like to introduce some basic definitions concerning n -additive symmetric mappings and key concepts which are found in [16] and [18] . A function A : X → Y is said to be additive if A ( x + y ) = A ( x ) + A ( y ) for all x , y ∈ X : Let n be a positive integer. A function A_n : Xⁿ → Y is called n - additive if it is additive in each of its variables. A function A_n is said to be symmetric if A_n ( x ₁ , ⋯ , x_n ) = A_n ( x _σ(1) , ⋯ , x _σ(n) ) for every permutation { σ (1) , ⋯ , σ ( n )} of {1 , 2 , ⋯ , n } . If A_n ( x ₁ , x ₂ , ⋯ , x_n ) is an n -additive symmetric map, then Aⁿ ( x ) will denote the diagonal A_n ( x , x , ⋯ , x ) and Aⁿ ( rx ) = rⁿAⁿ ( x ) for all x ∈ X and all r ∈ ℚ . such a function Aⁿ ( x ) will be called a monomial function of degree n (assuming Aⁿ ≢ 0). Furthermore the resulting function after substitution x ₁ = x ₂ = ⋯ = x_s = x and x _s+1 = x _s+2 = ⋯ = x_n = y in A_n ( x ₁ , x ₂ , ⋯ , x_n ) will be denoted by A ^s,n−s ( x , y ) . Theorem 2.1. A function f : X → Y is a solution of the functional equation (1.3) if and only if f is of the form f ( x ) = A ⁴ ( x ) for all x ∈ X , where A ⁴ ( x ) is the diagonal of the 4- additive symmetric mapping A ₄ : X ⁴ → Y . Proof . Assume that f satisfies the functional equation (1.3). Letting x = y = 0 in the equation (1.3), we have that is, f (0) = 0 . Putting x = 0 in the equation (1.3), we get for all y ∈ X . Replacing y by − y in the equation (2.1),we obtain for all y ∈ X . Combining two equations (2.1) and (2.2), we have f ( y ) = f (− y ) , for all y ∈ X . That is, f is even. We can rewrite the functional equation (1.3) in the form for all x , y ∈ X and an integer a ( a ≠ 0 , ±1) . By Theorem 3.5 and 3.6 in [18] , f is a generalized polynomial function of degree at most 4, that is, f is of the form for all x ∈ X , where A ⁰ ( x ) = A ⁰ is an arbitrary element of Y , and Aⁱ ( x ) is the diagonal of the i -additive symmetric mapping A_i : Xⁱ → Y for i = 1, 2, 3, 4 . By f (0) = 0 and f (− x ) = f ( x ) for all x ∈ X ; we get A ⁰ ( x ) = A ⁰ = 0 : Substituting (2.3) into the equation (1.3) we have for all x , y ∈ X . Note that Since a ≠ 0, ±1 , we have for all y ∈ X . Thus for all x ∈ X . Conversely, assume that f ( x ) = A ⁴ ( x ) for all x ∈ X , where A ⁴ ( x ) is the diagonal of a 4-additive symmetric mapping A ₄ : X ⁴ → Y . Note that where 1 ≤ s , t ≤ 3 and c ∈ ℚ . Thus we may conclude that f satisfies the equation (1.3). Throughout this section, we assume that A is a complex normed *-algebra and M is a Banach A -bimodule. We will use the same symbol || · || as norms on a normed algebra A and a normed A -bimodule M . A mapping f : A → M is a quartic homogeneous mapping if f ( μa ) = μ ⁴ f ( a ) ; for all a ∈ A and μ ∈ ℂ . A quartic homogeneous mapping f : A → M is called a quartic derivation if holds for all x , y ∈ A . For all x , y ∈ A , the symbol [ x , y ] will denote the commutator xy − yx . We say that a quartic homogeneous mapping f : A → M is a quartic Lie derivation if for all x , y ∈ A . In addition, if f satisfies in condition f ( x *) = f ( x )* for all x ∈ A , then it is called the quartic Lie *- derivation . Example 3.1. Let A = ℂ be a complex field endowed with the map z ↦ z * = (where is the complex conjugate of z ). We define f : A → A by f ( a ) = a ⁴ for all a ∈ A . Then f is quartic and for all a ∈ A . Also, for all a ∈ A . Thus f is a quartic Lie *-derivation. In the following, 𝕋 ¹ will stand for the set of all complex units, that is, For the given mapping f : A → M , we consider for all x , y ∈ A , μ ∈ ℂ and s ∈ ℤ ( s ≠ 0 , ±1) . Theorem 3.2. Suppose that f : A → M is an even mapping with f (0) = 0 for which there exists a function ϕ : A ⁵ → [0, ∞) such that for all and all a , b , x , y , z ∈ A in which n ₀ ∈ ℕ . Also, if for each fixed b ∈ A the mapping r ↦ f ( rb ) from ℝ to M is continuous, then there exists a unique quartic Lie *- derivation L : A → M satisfying Proof . Let a = 0 and μ = 1 in the inequality (3.3), we have for all b ∈ A . Using the induction, it is easy to show that for t > k ≥ 0 and b ∈ A . The inequalities (3.2) and (3.7) imply that the sequence is a Cauchy sequence. Since M is complete, the sequence is convergent. Hence we can define a mapping L : A → M as for b ∈ A . By letting t = n and k = 0 in the inequality (3.7), we have for n > 0 and b ∈ A . By taking n → ∞ in the inequality (3.9), the inequalities (3.2) implies that the inequality (3.5) holds. Now, we will show that the mapping L is a unique quartic Lie *-derivation such that the inequality (3.5) holds for all b ∈ A . We note that for all a , b ∈ A and . By taking μ = 1 in the inequality (3.10), it follows that the mapping L is a quartic mapping. Also, the inequality (3.10) implies that Δ _μ L (0, b ) = 0 . Hence for all b ∈ A and . Let μ ∈ 𝕋 ¹ = {λ ∈ ℂ | |λ| = 1} . Then μ = e^iθ , where 0 ≤ θ ≤ 2π . Let . Hence we have 이미지 . Then for all μ ∈ 𝕋 ¹ and a ∈ A . Suppose that ρ is any continuous linear functional on A and b is a fixed element in A . Then we can define a function g : ℝ → ℝ by for all r ∈ ℝ . It is easy to check that g is cubic. Let for all k ∈ ℕ and r ∈ ℝ . Note that g as the pointwise limit of the sequence of measurable functions g_k is measurable. Hence g as a measurable quartic function is continuous (see [5] ) and for all r ∈ ℝ . Thus for all r ∈ ℝ . Since ρ was an arbitrary continuous linear functional on A we may conclude that for all r ∈ ℝ . Let μ ∈ ℂ ( μ ≠ 0) . Then . Hence for all b ∈ A and μ ∈ ℂ ( μ ≠ 0) . Since b was an arbitrary element in A , we may conclude that L is quartic homogeneous. Next, replacing x , y by s^kx , s^ky , respectively, and z = 0 in the inequality (3.4), we have for all x , y ∈ A . Hence we have Δ L ( x , y ) = 0 for all x , y ∈ A . That is, L is a quartic Lie derivation. Letting x = y = 0 and replacing z by s^kz in the inequality (3.4), we get for all z ∈ A . As n → ∞ in the inequality (3.11), we have for all z ∈ A . This means that L is a quartic Lie *-derivation. Now, assume L' : A → A is another quartic *-derivation satisfying the inequality (3.5). Then which tends to zero as k → ∞ , for all b ∈ A . Thus L ( b ) = L' ( b ) for all b ∈ A . This proves the uniqueness of L . Corollary 3.3. Let θ , r be positive real numbers with r < 4 and let f : A → M be an even mapping with f (0) = 0 such that for all and a , b , x , y , z ∈ A . Then there exists a unique quartic Lie *-derivation L : A → M satisfying for all b ∈ A . Proof . The proof follows from Theorem 3.2 by taking ϕ ( a , b , x , y , z ) = θ (|| a || ^r + || b || ^r + || x || ^r + || y || ^r + || z || ^r ) for all a , b , x , y , z ∈ A . In the following corollaries, we show the hyperstability for the quartic Lie *-derivations. Corollary 3.4. Let r be positive real numbers with r < 4 and let f : A → M be an even mapping with f (0) = 0 such that for all and a , b , x , y , z ∈ A . Then f is a quartic Lie *-derivation on A . Proof . By taking ϕ ( a , b , x , y , z ) = (|| a || ^r + || x || ^r )(|| b || ^r + || y || ^r || z || ^r ) in Theorem 3.2 for all a , b , x , y , z ∈ A , we have . Hence the inequality (3.5) implies that f = L , that is, f is a quartic Lie *-derivation on A . Corollary 3.5. Let r be positive real numbers with r < 4 and let f : A → M be an even mapping with f (0) = 0 such that for all and a , b , x , y , z ∈ A . Then f is a quartic Lie *-derivation on A . Proof . By taking ϕ ( a , b , x , y , z ) = (|| a || ^r + || x || ^r )(|| b || ^r + || y || ^r + || z || ^r ) in Theorem 3.2 for all a , b , x , y , z ∈ A , we have . Hence the inequality (3.5) implies that f = L , that is, f is a quartic Lie *-derivation on A . Now, we will investigate the stability of the given functional equation (3.1) using the alternative fixed point method. Before proceeding the proof, we will state the theorem, the alternative of fixed point; see [11] and [15] . Definition 3.6. Let X be a set. A function d : X × X → [0, ∞] is called a generalized metric on X if d satisfies Theorem 3.7 (The alternative of fixed point [11] , [15] ). Suppose that we are given a complete generalized metric space (Ω, d ) and a strictly contractive mapping T : ­Ω → Ω­ with Lipschitz constant l . Then for each given x ∈ Ω ­, either or there exists a natural number n ₀ such that Theorem 3.8. Let f : A → M be a continuous even mapping with f (0) = 0 and let ϕ : A ⁵ → [0, ∞) be a continuous mapping such that for all and a , b , x , y , z ∈ A . If there exists a constant l ∈ (0, 1) such that for all a , b , x , y , z ∈ A , then there exists a quartic Lie *-derivation L : A → M satisfying for all b ∈ A . Proof . Consider the set and introduce the generalized metric on Ω, It is easy to show that (Ω, d ) is complete. Now we define a function T : ­Ω ­→ Ω by for all b ∈ A . Note that for all g , h ∈ ­Ω , let c ∈ (0, ∞) be an arbitrary constant with d ( g , h ) ≤ c . Then for all b ∈ A . Letting b = sb in the inequality (3.17) and using (3.14) and (3.16), we have that is, Hence we have that for all g , h ∈ Ω ­, that is, T is a strictly self-mapping of Ω­ with the Lipschitz constant l . Letting μ = 1 , a = 0 in the inequality (3.12), we get for all b ∈ A . This means that We can apply the alternative of fixed point and since lim _n→∞ d ( Tⁿ f , L ) = 0 , there exists a fixed point L of T in ­Ω such that for all b ∈ A . Hence This implies that the inequality (3.15) holds for all b ∈ A . Since l ∈ (0, 1) , the inequality (3.14) shows that Replacing a , b by sⁿa , sⁿb , respectively, in the inequality (3.12), we have Taking the limit as k tend to infinity, we have Δ _μ f ( a , b ) = 0 for all a , b ∈ A and all . The remains are similar to the proof of Theorem 3.2. Corollary 3.9. Let θ , r be positive real numbers with r < 4 and let f : A → M be a mapping with f (0) = 0 such that for all and a , b , x , y , z ∈ A . Then there exists a unique quartic Lie *-derivation L : A → M satisfying for all b ∈ A . Proof . The proof follows from Theorem 3.8 by taking ϕ ( a , b , x , y , z ) = θ (|| a || ^r + || b || ^r + || x || ^r + || y || ^r + || z || ^r ) for all a , b , x , y , z ∈ A . In the following corollaries, we show the hyperstability for the quartic Lie *-derivations. Corollary 3.10. Let r be positive real numbers with r < 4 and let f : A → M be an even mapping with f (0) = 0 such that for all and a , b , x , y , z ∈ A . Then f is a quartic Lie *-derivation on A Proof . By taking ϕ ( a , b , x , y , z ) = (|| a || ^r + || x || ^r )(|| b || ^r + || y || ^r || z || ^r ) in Theorem 3.8 for all a , b , x , y , z ∈ A , we have . Hence the inequality (3.15) implies that f = L , that is, f is a quartic Lie *-derivation on A . Corollary 3.11. Let r be positive real numbers with r < 4 and let f : A → M be an even mapping with f (0) = 0 such that for all and a , b , x , y , z ∈ A . Then f is a quartic Lie *-derivation on A Proof . By taking ϕ ( a , b , x , y , z ) = (|| a || ^r + || x || ^r )(|| b || ^r + || y || ^r + || z || ^r ) in Theorem 3.8 for all a , b , x , y , z ∈ A , we have . Hence the inequality (3.15) implies that f = L , that is, f is a quartic Lie *-derivation on A .