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APPROXIMATE QUARTIC LIE *-DERIVATIONS
APPROXIMATE QUARTIC LIE *-DERIVATIONS
Journal of the Korean Society of Mathematical Education Series B The Pure and Applied Mathematics. 2015. Nov, 22(4): 389-401
Copyright © 2015, Korean Society of Mathematical Education
  • Received : October 15, 2015
  • Accepted : October 23, 2015
  • Published : November 30, 2015
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HEEJEONG KOH

Abstract
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.
Keywords
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
PPT Slide
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It is easy to see that f ( x ) = x 4 is a solution of (1.1) by virtue of the identity
PPT Slide
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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:
PPT Slide
Lager Image
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 ( x 1 , ⋯ , xn ) = An ( x σ(1) , ⋯ , x σ(n) ) for every permutation { σ (1) , ⋯ , σ ( n )} of {1 , 2 , ⋯ , n } . If An ( x 1 , x 2 , ⋯ , 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 x 1 = x 2 = ⋯ = xs = x and x s+1 = x s+2 = ⋯ = xn = y in An ( x 1 , x 2 , ⋯ , xn ) will be denoted by A s,ns ( 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
PPT Slide
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that is, f (0) = 0 . Putting x = 0 in the equation (1.3), we get
PPT Slide
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for all y X . Replacing y by − y in the equation (2.1),we obtain
PPT Slide
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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
PPT Slide
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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
PPT Slide
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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 Ai : 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
PPT Slide
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for all x , y X . Note that
PPT Slide
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Since a ≠ 0, ±1 , we have
PPT Slide
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for all y X . Thus
PPT Slide
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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
PPT Slide
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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 ) = μ 4 f ( a ) ; for all a A and μ ∈ ℂ . A quartic homogeneous mapping f : A M is called a quartic derivation if
PPT Slide
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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
PPT Slide
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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 * =
PPT Slide
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(where
PPT Slide
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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
PPT Slide
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for all a A . Also,
PPT Slide
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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,
PPT Slide
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For the given mapping f : A M , we consider
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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
PPT Slide
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PPT Slide
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PPT Slide
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for all
PPT Slide
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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 to M is continuous, then there exists a unique quartic Lie *- derivation L : A M satisfying
PPT Slide
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Proof . Let a = 0 and μ = 1 in the inequality (3.3), we have
PPT Slide
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for all b A . Using the induction, it is easy to show that
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for t > k ≥ 0 and b A . The inequalities (3.2) and (3.7) imply that the sequence
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is a Cauchy sequence. Since M is complete, the sequence is convergent. Hence we can define a mapping L : A M as
PPT Slide
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for b A . By letting t = n and k = 0 in the inequality (3.7), we have
PPT Slide
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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
PPT Slide
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for all a , b A and
PPT Slide
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. 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
PPT Slide
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for all b A and
PPT Slide
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. Let μ ∈ 𝕋 1 = {λ ∈ ℂ | |λ| = 1} . Then μ = e , where 0 ≤ θ ≤ 2π . Let
PPT Slide
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. Hence we have 이미지 . Then
PPT Slide
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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
PPT Slide
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for all r ∈ ℝ . It is easy to check that g is cubic. Let
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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
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for all r ∈ ℝ . Thus
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for all r ∈ ℝ . Since ρ was an arbitrary continuous linear functional on A we may conclude that
PPT Slide
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for all r ∈ ℝ . Let μ ∈ ℂ ( μ ≠ 0) . Then
PPT Slide
Lager Image
. Hence
PPT Slide
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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
PPT Slide
Lager Image
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 skz in the inequality (3.4), we get
PPT Slide
Lager Image
for all z A . As n → ∞ in the inequality (3.11), we have
PPT Slide
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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
PPT Slide
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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
PPT Slide
Lager Image
for all
PPT Slide
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and a , b , x , y , z A . Then there exists a unique quartic Lie *-derivation L : A M satisfying
PPT Slide
Lager Image
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
PPT Slide
Lager Image
for all
PPT Slide
Lager Image
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
PPT Slide
Lager Image
. 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
PPT Slide
Lager Image
for all
PPT Slide
Lager Image
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
PPT Slide
Lager Image
. 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 ifx=y;
  • (2)d(x,y) =d(y,x) for allx,y∈X;
  • (3)d(x,z) ≤d(x,y) +d(y,z) for allx,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
PPT Slide
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or there exists a natural number n 0 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 : A 5 → [0, ∞) be a continuous mapping such that
PPT Slide
Lager Image
PPT Slide
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for all
PPT Slide
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and a , b , x , y , z A . If there exists a constant l ∈ (0, 1) such that
PPT Slide
Lager Image
for all a , b , x , y , z A , then there exists a quartic Lie *-derivation L : A M satisfying
PPT Slide
Lager Image
for all b A .
Proof . Consider the set
PPT Slide
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and introduce the generalized metric on Ω,
PPT Slide
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It is easy to show that (Ω, d ) is complete. Now we define a function T : ­Ω ­→ Ω by
PPT Slide
Lager Image
for all b A . Note that for all g , h ∈ ­Ω , let c ∈ (0, ∞) be an arbitrary constant with d ( g , h ) ≤ c . Then
PPT Slide
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for all b A . Letting b = sb in the inequality (3.17) and using (3.14) and (3.16), we have
PPT Slide
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that is,
PPT Slide
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Hence we have that
PPT Slide
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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
PPT Slide
Lager Image
for all b A . This means that
PPT Slide
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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
PPT Slide
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for all b A . Hence
PPT Slide
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This implies that the inequality (3.15) holds for all b A . Since l ∈ (0, 1) , the inequality (3.14) shows that
PPT Slide
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Replacing a , b by sna , snb , respectively, in the inequality (3.12), we have
PPT Slide
Lager Image
Taking the limit as k tend to infinity, we have Δ μ f ( a , b ) = 0 for all a , b A and all
PPT Slide
Lager Image
. 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
PPT Slide
Lager Image
for all
PPT Slide
Lager Image
and a , b , x , y , z A . Then there exists a unique quartic Lie *-derivation L : A M satisfying
PPT Slide
Lager Image
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
PPT Slide
Lager Image
for all
PPT Slide
Lager Image
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
PPT Slide
Lager Image
. 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
PPT Slide
Lager Image
for all
PPT Slide
Lager Image
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
PPT Slide
Lager Image
. Hence the inequality (3.15) implies that f = L , that is, f is a quartic Lie *-derivation on A .
Acknowledgements
The present research was conducted by the research fund of Dankook University in 2014.
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