APPROXIMATE QUARTIC LIE *-DERIVATIONS

Abstract

We will show the general solution of the functional equation f(x + ay) + f(x − ay) + 2(a² − 1)f(x) = a²f(x + y) + a²f(x − y) + 2a²(a² − 1)f(y) and investigate the stability of quartic Lie *-derivations associated with the given functional equation.

1. INTRODUCTION

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) = x4 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 : ℝ4 → 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.

 

2. A QUARTIC 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 An : Xn → Y is called n-additive if it is additive in each of its variables. A function An is said to be symmetric if An(x1 , ⋯ , xn) = An(xσ(1) , ⋯ , xσ(n)) for every permutation {σ(1) , ⋯ , σ(n)} of {1 , 2 , ⋯ , n} . If An(x1 , x2 , ⋯ , xn) is an n-additive symmetric map, then An(x) will denote the diagonal An(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 x1 = x2 = ⋯ = xs = x and xs+1 = xs+2 = ⋯ = xn = y in An(x1 , x2 , ⋯ , xn) will be denoted by As,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) = A4(x) for all x ∈ X , where A4(x) is the diagonal of the 4-additive symmetric mapping A4 : X4 → 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 A0(x) = A0 is an arbitrary element of Y , and Ai(x) is the diagonal of the i-additive symmetric mapping Ai : Xi → Y for i = 1, 2, 3, 4 . By f(0) = 0 and f(−x) = f(x) for all x ∈ X ; we get A0(x) = A0 = 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) = A4(x) for all x ∈ X , where A4(x) is the diagonal of a 4-additive symmetric mapping A4 : X4 → Y . Note that

where 1 ≤ s, t ≤ 3 and c ∈ ℚ . Thus we may conclude that f satisfies the equation (1.3).

 

3. QUARTIC LIE *-DERIVATIONS

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) = μ4f(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) = a4 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, 𝕋1 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 : A5 → [0, ∞) such that

for all and all a, b, x, y, z ∈ A in which n0 ∈ 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 ΔμL(0, b) = 0 . Hence

for all b ∈ A and . Let μ ∈ 𝕋1 = {λ ∈ ℂ | |λ| = 1} . Then μ = eiθ , where 0 ≤ θ ≤ 2π . Let . Hence we have 이미지 . Then

for all μ ∈ 𝕋1 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 gk 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 skx, sky , respectively, and z = 0 in the inequality (3.4), we have

for all x, y ∈ A . Hence we have ΔL(x, y) = 0 for all x, y ∈ A. That is, L is a quartic Lie derivation. Letting x = y = 0 and replacing z by skz 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

(1) d(x, y) = 0 if and only if x = y ; (2) d(x, y) = d(y, x) for all x, y ∈ X ; (3) d(x, z) ≤ d(x, y) + d(y, z) for all x, y, z ∈ X .

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 n0 such that

(1) d(Tnx, Tn+1x) < ∞ for all n ≥ n0 ; (2) The sequence (Tnx) is convergent to a fixed point y* of T ; (3) y* is the unique fixed point of T in the set(4) for all y ∈ △ .

Theorem 3.8. Let f : A → M be a continuous even mapping with f(0) = 0 and let ϕ : A5 → [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 limn→∞ 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 Δμf(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 .