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C -ternary 3-derivations on C -ternary algebras

  • 1,
  • 2,
  • 3 and
  • 4Email author
Journal of Inequalities and Applications20132013:124

https://doi.org/10.1186/1029-242X-2013-124

  • Received: 31 October 2012
  • Accepted: 5 March 2013
  • Published:

Abstract

In this paper, we prove the Hyers-Ulam stability of C -ternary 3-derivations and of C -ternary 3-homomorphisms for the functional equation

f ( x 1 + x 2 , y 1 + y 2 , z 1 + z 2 ) = 1 i , j , k 2 f ( x i , y j , z k )

in C -ternary algebras.

MSC:17A40, 39B52, 46Lxx, 46K70, 46L05, 46B99.

Keywords

  • Hyers-Ulam stability
  • 3-additive mapping
  • C -ternary algebra
  • C -ternary 3-derivation
  • C -ternary 3-homomorphism

1 Introduction and preliminaries

Ternary algebraic operations were considered in the nineteenth century by several mathematicians such as Cayley [1] who introduced the notion of a cubic matrix, which in turn was generalized by Kapranov, Gelfand and Zelevinskii [2]. The simplest example of such non-trivial ternary operation is given by the following composition rule:
{ a , b , c } i j k = 1 l , m , n N a n i l b l j m c m k n ( i , j , k = 1 , 2 , , N ) .
Ternary structures and their generalization, the so-called n-ary structures, raise certain hopes in view of their applications in physics. Some significant physical applications are as follows (see [3, 4]).
  1. (1)
    The algebra of nonions generated by two matrices
    ( 0 1 0 0 0 1 1 0 0 ) and ( 0 1 0 0 0 ω ω 2 0 0 ) ( ω = e 2 π i 3 )
     
was introduced by Sylvester as a ternary analog of Hamiltons quaternions (cf. [5]).
  1. (2)

    The quark model inspired a particular brand of ternary algebraic systems. The so-called Nambu mechanics is based on such structures (see [6]).

     

There are also some applications, although still hypothetical, in the fractional quantum Hall effect, the nonstandard statistics, supersymmetric theory, and Yang-Baxter equation (cf. [3, 5, 7]).

A C -ternary algebra is a complex Banach space A, equipped with a ternary product ( x , y , z ) [ x , y , z ] of A 3 into A, which is -linear in the outer variables, conjugate -linear in the middle variable, and associative in the sense that [ x , y , [ z , w , v ] ] = [ x , [ w , z , y ] , v ] = [ [ x , y , z ] , w , v ] , and satisfies [ x , y , z ] x y z and [ x , x , x ] = x 3 (see [8]). Every left Hilbert C -module is a C -ternary algebra via the ternary product [ x , y , z ] : = x , y z .

If a C -ternary algebra ( A , [ , , ] ) has an identity, i.e., an element e A such that x = [ x , e , e ] = [ e , e , x ] for all x A , then it is routine to verify that A, endowed with x y : = [ x , e , y ] and x : = [ e , x , e ] , is a unital C -algebra. Conversely, if ( A , ) is a unital C -algebra, then [ x , y , z ] : = x y z makes A into a C -ternary algebra.

Throughout this paper, assume that C -ternary algebras A and B are induced by unital C -algebras with units e and e , respectively.

A -linear mapping H : A B is called a C -ternary homomorphism if H ( [ x , y , z ] ) = [ H ( x ) , H ( y ) , H ( z ) ] for all x , y , z A . If, in addition, the mapping H is bijective, then the mapping H : A B is called a C -ternary algebra isomorphism. A -linear mapping δ : A A is called a C -ternary 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 [9]).

In 1940, Ulam [10] gave a talk before the Mathematics Club of the University of Wisconsin in which he discussed a number of unsolved problems. Among these was the following question concerning the stability of homomorphisms:

We are given a group G and a metric group G with metric ρ ( , ) . Given ϵ > 0 , does there exist a δ > 0 such that if f : G G satisfies ρ ( f ( x y ) , f ( x ) f ( y ) ) < δ for all x , y G , then a homomorphism h : G G exists with ρ ( f ( x ) , h ( x ) ) < ϵ for all x G ?

In 1941, Hyers [11] gave the first partial solution to Ulam’s question for the case of approximate additive mappings under the assumption that G and G are Banach spaces. Then, Aoki [12] and Bourgin [13] considered the stability problem with unbounded Cauchy differences. In 1978, Rassias [14] generalized the theorem of Hyers [11] by considering the stability problem with unbounded Cauchy differences. In 1991, Gajda [15], following the same approach as that by Rassias [14], gave an affirmative solution to this question for p > 1 . It was shown by Gajda [15] as well as by Rassias and Šemrl [16], that one cannot prove a Rassias-type theorem when p = 1 . Gǎvruta [17] obtained the generalized result of the Rassias theorem which allows the Cauchy difference to be controlled by a general unbounded function. During the last two decades, a number of articles and research monographs have been published on various generalizations and applications of the Hyers-Ulam stability to a number of functional equations and mappings, for example, Cauchy-Jensen mappings, k-additive mappings, invariant means, multiplicative mappings, bounded n th differences, convex functions, generalized orthogonality mappings, Euler-Lagrange functional equations, differential equations, and Navier-Stokes equations. The stability problems 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 [1831]).

Let X and Y be complex vector spaces. A mapping f : X × X × X Y is called a 3-additive mapping if f is additive for each variable, and a mapping f : X × X × X Y is called a 3--linear mapping if f is -linear for each variable.

A 3--linear mapping H : A × A × A B is called a C -ternary 3-homomorphism if it satisfies
H ( [ x 1 , y 1 , z 1 ] , [ x 2 , y 2 , z 2 ] , [ x 3 , y 3 , z 3 ] ) = [ H ( x 1 , x 2 , x 3 ) , H ( y 1 , y 2 , y 3 ) , H ( z 1 , z 2 , z 3 ) ]

for all x 1 , y 1 , z 1 , x 2 , y 2 , z 2 , x 3 , y 3 , z 3 A .

For a given mapping f : A 3 B , we define
D λ , μ , ν f ( x 1 , x 2 , y 1 , y 2 , z 1 , z 2 ) : = f ( λ x 1 + λ x 2 , μ y 1 + μ y 2 , ν z 1 + ν z 2 ) λ μ ν 1 i , j , k 2 f ( x i , y j , z k )

for all λ , μ , ν S 1 : = { λ C : | λ | = 1 } and all x 1 , x 2 , y 1 , y 2 , z 1 , z 2 A .

Bae and Park [32] proved the Hyers-Ulam stability of 3-homomorphisms in C -ternary algebras for the functional equation
D λ , μ , ν f ( x 1 , x 2 , y 1 , y 2 , z 1 , z 2 ) = 0 .

Lemma 1.1 [32]

Let X and Y be complex vector spaces, and let f : X × X × X Y be a 3-additive mapping such that f ( λ x , μ y , ν z ) = λ μ ν f ( x , y , z ) for all λ , μ , ν S 1 and all x , y , z X . Then f is 3--linear.

Theorem 1.2 [32]

Let p , q , r ( 0 , ) with p + q + r < 3 and θ ( 0 , ) , and let f : A 3 B be a mapping such that
D λ , μ , ν f ( x 1 , x 2 , y 1 , y 2 , z 1 , z 2 ) θ max { x 1 , x 2 } p max { y 1 , y 2 } q max { z 1 , z 2 } r
(1)
and
f ( [ x 1 , y 1 , z 1 ] , [ x 2 , y 2 , z 2 ] , [ x 3 , y 3 , z 3 ] ) [ f ( x 1 , x 2 , x 3 ) , f ( y 1 , y 2 , y 3 ) , f ( z 1 , z 2 , z 3 ) ] θ i = 1 3 x i p y i q z i r
(2)
for all λ , μ , ν S 1 and all x 1 , x 2 , x 3 , y 1 , y 2 , y 3 , z 1 , z 2 , z 3 A . Then there exists a unique C -ternary 3-homomorphism H : A 3 B such that
f ( x , y , z ) H ( x , y , z ) θ 2 3 2 p + q + r x p y q z r
(3)

for all x , y , z A .

2 C -ternary 3-homomorphisms in C -ternary algebras

Theorem 2.1 Let p , q , r ( 0 , ) with p + q + r < 3 and θ ( 0 , ) , and let f : A 3 B be a mapping satisfying (1) and (2). If there exists an ( x 0 , y 0 , z 0 ) A 3 such that lim n 1 8 n f ( 2 n x 0 , 2 n y 0 , 2 n z 0 ) = e , then the mapping f is a C -ternary 3-homomorphism.

Proof By Theorem 1.2, there exists a unique C -ternary 3-homomorphism H : A 3 B satisfying (3). Note that
H ( x , y , z ) : = lim n 1 8 n f ( 2 n x , 2 n y , 2 n z )
for all x , y , z A . By the assumption, we get that
H ( x 0 , y 0 , z 0 ) = lim n 1 8 n f ( 2 n x 0 , 2 n y 0 , 2 n z 0 ) = e .
It follows from (2) that
[ H ( x 1 , x 2 , x 3 ) , H ( y 1 , y 2 , y 3 ) , H ( z 1 , z 2 , z 3 ) ] [ H ( x 1 , x 2 , x 3 ) , H ( y 1 , y 2 , y 3 ) , f ( z 1 , z 2 , z 3 ) ] = H ( [ x 1 , y 1 , z 1 ] , [ x 2 , y 2 , z 2 ] , [ x 3 , y 3 , z 3 ] ) [ H ( x 1 , x 2 , x 3 ) , H ( y 1 , y 2 , y 3 ) , f ( z 1 , z 2 , z 3 ) ] = lim n 1 8 2 n f ( [ 2 n x 1 , 2 n y 1 , z 1 ] , [ 2 n x 2 , 2 n y 2 , z 2 ] , [ 2 n x 3 , 2 n y 3 , z 3 ] ) [ f ( 2 n x 1 , 2 n x 2 , 2 n x 3 ) , f ( 2 n y 1 , 2 n y 2 , 2 n y 3 ) , f ( z 1 , z 2 , z 3 ) ] lim n θ 2 n ( p + q ) 8 2 n i = 1 3 x i p y i q z i r = 0
for all x 1 , x 2 , x 3 , y 1 , y 2 , y 3 , z 1 , z 2 , z 3 A . So,
[ H ( x 1 , x 2 , x 3 ) , H ( y 1 , y 2 , y 3 ) , H ( z 1 , z 2 , z 3 ) ] = [ H ( x 1 , x 2 , x 3 ) , H ( y 1 , y 2 , y 3 ) , f ( z 1 , z 2 , z 3 ) ]

for all x 1 , x 2 , x 3 , y 1 , y 2 , y 3 , z 1 , z 2 , z 3 A . Letting x 1 = y 1 = x 0 , x 2 = y 2 = y 0 and x 3 = y 3 = z 0 in the last equality, we get f ( z 1 , z 2 , z 3 ) = H ( z 1 , z 2 , z 3 ) for all z 1 , z 2 , z 3 A . Therefore, the mapping f is a C -ternary 3-homomorphism. □

Theorem 2.2 Let p i , q i , r i ( 0 , ) ( i = 1 , 2 , 3 ) such that p i 1 or q i 1 or r i 1 for some 1 i 3 and θ , η ( 0 , ) , and let f : A 3 B be a mapping such that
D λ , μ , ν f ( x 1 , x 2 , y 1 , y 2 , z 1 , z 2 ) θ ( x 1 p 1 x 2 p 2 y 1 q 1 y 2 q 2 + y 1 q 1 y 2 q 2 z 1 r 1 z 2 r 2 + x 1 p 1 x 2 p 2 z 1 r 1 z 2 r 2 )
(4)
and
f ( [ x 1 , y 1 , z 1 ] , [ x 2 , y 2 , z 2 ] , [ x 3 , y 3 , z 3 ] ) [ f ( x 1 , x 2 , x 3 ) , f ( y 1 , y 2 , y 3 ) , f ( z 1 , z 2 , z 3 ) ] η x 1 p 1 x 2 p 2 x 3 p 3 y 1 q 1 y 2 q 2 y 3 q 3 z 1 r 1 z 2 r 2 z 3 r 3
(5)

for all λ , μ , ν S 1 and all x 1 , x 2 , x 3 , y 1 , y 2 , y 3 , z 1 , z 2 , z 3 A . Then the mapping f : A 3 B is a C -ternary 3-homomorphism.

Proof Letting x i = y j = z k = 0 ( i , j , k = 1 , 2 ) in (4), we get f ( 0 , 0 , 0 ) = 0 . Putting λ = μ = ν = 1 , x 2 = 0 and y j = z k = 0 ( j , k = 1 , 2 ) in (4), we have f ( x 1 , 0 , 0 ) = 0 for all x 1 A . Similarly, we get f ( 0 , y 1 , 0 ) = f ( 0 , 0 , z 1 ) = 0 for all y 1 , z 1 A . Setting λ = μ = ν = 1 , x 2 = 0 , y 2 = 0 and z 1 = z 2 = 0 , we have f ( x 1 , y 1 , 0 ) = 0 for all x 1 , y 1 A . Similarly, we get f ( x 1 , 0 , z 1 ) = f ( 0 , y 1 , z 1 ) = 0 for all x 1 , y 1 , z 1 A . Now letting λ = μ = ν = 1 and y 2 = z 2 = 0 in (4), we have
f ( x 1 + x 2 , y 1 , z 1 ) = f ( x 1 , y 1 , z 1 ) + f ( x 2 , y 1 , z 1 )

for all x 1 , x 2 , y 1 , z 1 A .

Similarly, one can show that the other equations hold. So, f is 3-additive.

Letting x 2 = y 2 = z 2 = 0 in (4), we get f ( λ x 1 , μ y 1 , ν z 1 ) = λ μ ν f ( x 1 , y 1 , z 1 ) for all λ , μ , ν S 1 and all x 1 , y 1 , z 1 A . So, by Lemma 1.1, the mapping f is 3--linear.

Without any loss of generality, we may suppose that p 1 1 .

Let p 1 < 1 . It follows from (5) that
f ( [ x 1 , y 1 , z 1 ] , [ x 2 , y 2 , z 2 ] , [ x 3 , y 3 , z 3 ] ) [ f ( x 1 , x 2 , x 3 ) , f ( y 1 , y 2 , y 3 ) , f ( z 1 , z 2 , z 3 ) ] = lim n 1 3 n f ( [ 3 n x 1 , y 1 , z 1 ] , [ x 2 , y 2 , z 2 ] , [ x 3 , y 3 , z 3 ] ) [ f ( 3 n x 1 , x 2 , x 3 ) , f ( y 1 , y 2 , y 3 ) , f ( z 1 , z 2 , z 3 ) ] η lim n 3 n p 1 3 n × ( x 1 p 1 x 2 p 2 x 3 p 3 y 1 q 1 y 2 q 2 y 3 q 3 z 1 r 1 z 2 r 2 z 3 r 3 ) = 0

for all x 1 , x 2 , x 3 , y 1 , y 2 , y 3 , z 1 , z 2 , z 3 A .

Let p 1 > 1 . It follows from (5) that
f ( [ x 1 , y 1 , z 1 ] , [ x 2 , y 2 , z 2 ] , [ x 3 , y 3 , z 3 ] ) [ f ( x 1 , x 2 , x 3 ) , f ( y 1 , y 2 , y 3 ) , f ( z 1 , z 2 , z 3 ) ] = lim n 3 n f ( [ 1 3 n x 1 , y 1 , z 1 ] , [ x 2 , y 2 , z 2 ] , [ x 3 , y 3 , z 3 ] ) [ f ( 1 3 n x 1 , x 2 , x 3 ) , f ( y 1 , y 2 , y 3 ) , f ( z 1 , z 2 , z 3 ) ] η lim n 3 n 3 n p 1 × ( x 1 p 1 x 2 p 2 x 3 p 3 y 1 q 1 y 2 q 2 y 3 q 3 z 1 r 1 z 2 r 2 z 3 r 3 ) = 0
for all x 1 , x 2 , x 3 , y 1 , y 2 , y 3 , z 1 , z 2 , z 3 A . Therefore,
f ( [ x 1 , y 1 , z 1 ] , [ x 2 , y 2 , z 2 ] , [ x 3 , y 3 , z 3 ] ) = [ f ( x 1 , x 2 , x 3 ) , f ( y 1 , y 2 , y 3 ) , f ( z 1 , z 2 , z 3 ) ]

for all x 1 , x 2 , x 3 , y 1 , y 2 , y 3 , z 1 , z 2 , z 3 A . So, the mapping f : A 3 B is a C -ternary 3-homomorphism. □

Theorem 2.3 Let φ : A 6 [ 0 , ) and ψ : A 9 [ 0 , ) be functions such that
φ ( x 1 , , x 6 ) = 0
if x i = 0 for some 1 i 6 and
1 3 n ψ ( x 1 , , 3 n x i , , x 9 ) = 0 or 3 n ψ ( x 1 , , 1 3 n x i , , x 9 ) = 0 .
Suppose that f : A 3 B is a mapping satisfying
D λ , μ , ν f ( x 1 , x 2 , y 1 , y 2 , z 1 , z 2 ) φ ( x 1 , x 2 , y 1 , y 2 , z 1 , z 2 )
and
f ( [ x 1 , y 1 , z 1 ] , [ x 2 , y 2 , z 2 ] , [ x 3 , y 3 , z 3 ] ) [ f ( x 1 , x 2 , x 3 ) , f ( y 1 , y 2 , y 3 ) , f ( z 1 , z 2 , z 3 ) ] ψ ( x 1 , x 2 , x 3 , y 1 , y 2 , y 3 , z 1 , z 2 , z 3 )

for all λ , μ , ν S 1 and all x 1 , x 2 , x 3 , y 1 , y 2 , y 3 , z 1 , z 2 , z 3 A . Then the mapping f is a C -ternary 3-homomorphism.

Proof The proof is similar to the proof of Theorem 2.2. □

Corollary 2.4 Let p i , q i , r i ( 0 , ) ( i = 1 , 2 , 3 ) such that p i 1 or q i 1 or r i 1 for some 1 i 3 and θ , η ( 0 , ) , and let f : A 3 B be a mapping such that
D λ , μ , ν f ( x 1 , x 2 , y 1 , y 2 , z 1 , z 2 ) θ x 1 p 1 x 2 p 2 y 1 q 1 y 2 q 2 z 1 r 1 z 2 r 2
and
f ( [ x 1 , y 1 , z 1 ] , [ x 2 , y 2 , z 2 ] , [ x 3 , y 3 , z 3 ] ) [ f ( x 1 , x 2 , x 3 ) , f ( y 1 , y 2 , y 3 ) , f ( z 1 , z 2 , z 3 ) ] η x 1 p 1 x 2 p 2 x 3 p 3 y 1 q 1 y 2 q 2 y 3 q 3 z 1 r 1 z 2 r 2 z 3 r 3

for all λ , μ , ν S 1 and all x 1 , x 2 , x 3 , y 1 , y 2 , y 3 , z 1 , z 2 , z 3 A . Then the mapping f : A 3 B is a C -ternary 3-homomorphism.

3 C -ternary 3-derivations on C -ternary algebras

Definition 3.1 A 3--linear mapping D : A 3 A is called a C -ternary 3-derivation if it satisfies
D ( [ x 1 , y 1 , z 1 ] , [ x 2 , y 2 , z 2 ] , [ x 3 , y 3 , z 3 ] ) = [ D ( x 1 , x 2 , x 3 ) , [ y 1 , y 2 , y 3 ] , [ z 1 , z 2 , z 3 ] ] + [ [ x 1 , x 2 , x 3 ] , D ( y 1 , y 2 , y 3 ) , [ z 1 , z 2 , z 3 ] ] + [ [ x 1 , x 2 , x 3 ] , [ y 1 , y 2 , y 3 ] , D ( z 1 , z 2 , z 3 ) ]

for all x 1 , x 2 , x 3 , y 1 , y 2 , y 3 , z 1 , z 2 , z 3 A .

Theorem 3.2 Let p , q , r ( 0 , ) with p + q + r < 3 and θ ( 0 , ) , and let f : A 3 A be a mapping such that
D λ , μ , ν f ( x 1 , x 2 , y 1 , y 2 , z 1 , z 2 ) θ max { x 1 , x 2 } p max { y 1 , y 2 } q max { z 1 , z 2 } r
(6)
and
f ( [ x 1 , y 1 , z 1 ] , [ x 2 , y 2 , z 2 ] , [ x 3 , y 3 , z 3 ] ) [ f ( x 1 , x 2 , x 3 ) , [ y 1 , y 2 , y 3 ] , [ z 1 , z 2 , z 3 ] ] [ [ x 1 , x 2 , x 3 ] , f ( y 1 , y 2 , y 3 ) , [ z 1 , z 2 , z 3 ] ] [ [ x 1 , x 2 , x 3 ] , [ y 1 , y 2 , y 3 ] , f ( z 1 , z 2 , z 3 ) ] θ i = 1 3 x i p y i q z i r
(7)
for all λ , μ , ν S 1 and all x 1 , x 2 , x 3 , y 1 , y 2 , y 3 , z 1 , z 2 , z 3 A . Then there exists a unique C -ternary 3-derivation δ : A 3 A such that
f ( x , y , z ) δ ( x , y , z ) θ 2 3 2 p + q + r x p y q z r
(8)

for all x , y , z A .

Proof By the same method as in the proof of [[32], Theorem 1.2], we obtain a 3--linear mapping δ : A 3 A satisfying (8). The mapping δ ( x , y , z ) : = lim j 1 8 j f ( 2 j x , 2 j y , 2 j z ) for all x , y , z A .

It follows from (7) that
δ ( [ x 1 , y 1 , z 1 ] , [ x 2 , y 2 , z 2 ] , [ x 3 , y 3 , z 3 ] ) [ δ ( x 1 , x 2 , x 3 ) , [ y 1 , y 2 , y 3 ] , [ z 1 , z 2 , z 3 ] ] [ [ x 1 , x 2 , x 3 ] , δ ( y 1 , y 2 , y 3 ) , [ z 1 , z 2 , z 3 ] ] [ [ x 1 , x 2 , x 3 ] , [ y 1 , y 2 , y 3 ] , δ ( z 1 , z 2 , z 3 ) ] = lim n 1 8 3 n f ( 2 3 n [ x 1 , y 1 , z 1 ] , 2 3 n [ x 2 , y 2 , z 2 ] , 2 3 n [ x 3 , y 3 , z 3 ] ) [ f ( 2 n x 1 , 2 n x 2 , 2 n x 3 ) , [ 2 n y 1 , 2 n y 2 , 2 n y 3 ] , [ 2 n z 1 , 2 n z 2 , 2 n z 3 ] ] [ [ 2 n x 1 , 2 n x 2 , 2 n x 3 ] , f ( 2 n y 1 , 2 n y 2 , 2 n y 3 ) , [ 2 n z 1 , 2 n z 2 , 2 n z 3 ] ] [ [ 2 n x 1 , 2 n x 2 , 2 n x 3 ] , [ 2 n y 1 , 2 n y 2 , 2 n y 3 ] , f ( 2 n z 1 , 2 n z 2 , 2 n z 3 ) ] lim n θ 2 n ( p + q + r ) 8 3 n i = 1 3 x i p y i q z i r = 0

for all x 1 , x 2 , x 3 , y 1 , y 2 , y 3 , z 1 , z 2 , z 3 A .

Now, let T : A 3 A be another 3-derivation satisfying (8). Then we have
δ ( x , y , z ) T ( x , y , z ) = 1 8 n δ ( 2 n x , 2 n y , 2 n z ) T ( 2 n x , 2 n y , 2 n z ) 1 8 n δ ( 2 n x , 2 n y , 2 n z ) f ( 2 n x , 2 n y , 2 n z ) + 1 8 n f ( 2 n x , 2 n y , 2 n z ) T ( 2 n x , 2 n y , 2 n z ) θ 2 ( p + q + r 3 ) n + 1 2 3 2 p + q + r x p y q z r ,

which tends to zero as n for all x , y , z A . So, we can conclude that δ ( x , y , z ) = T ( x , y , z ) for all x , y , z A . This proves the uniqueness of δ.

Therefore, the mapping δ : A 3 A is a unique C -ternary 3-derivation satisfying (8). □

Corollary 3.3 Let ϵ ( 0 , ) , and let f : A 3 A be a mapping satisfying
D λ , μ , ν f ( x 1 , x 2 , y 1 , y 2 , z 1 , z 2 ) ϵ
and
f ( [ x 1 , y 1 , z 1 ] , [ x 2 , y 2 , z 2 ] , [ x 3 , y 3 , z 3 ] ) [ f ( x 1 , x 2 , x 3 ) , [ y 1 , y 2 , y 3 ] , [ z 1 , z 2 , z 3 ] ] [ [ x 1 , x 2 , x 3 ] , f ( y 1 , y 2 , y 3 ) , [ z 1 , z 2 , z 3 ] ] [ [ x 1 , x 2 , x 3 ] , [ y 1 , y 2 , y 3 ] , f ( z 1 , z 2 , z 3 ) ] 3 ϵ
for all λ , μ , ν S 1 and all x 1 , x 2 , x 3 , y 1 , y 2 , y 3 , z 1 , z 2 , z 3 A . Then there exists a unique C -ternary 3-derivation δ : A 3 A such that
f ( x , y , z ) δ ( x , y , z ) ϵ 7

for all x , y , z A .

Declarations

Acknowledgements

CP was supported by Basic Science Research Program through the National Research Foundation of Korea funded by the Ministry of Education, Science and Technology (NRF-2012R1A1A2004299) and DYS was supported by Basic Science Research Program through the National Research Foundation of Korea funded by the Ministry of Education, Science and Technology (NRF-2010-0021792).

Authors’ Affiliations

(1)
Department of Mathematics, Sirjan University of Technology, Sirjan, Iran
(2)
Department of Mathematics, University of Yazd, P.O. Box 89195-741, Yazd, Iran
(3)
Department of Mathematics, Research Institute for Natural Sciences, Hanyang University, Seoul, 133-791, Korea
(4)
Department of Mathematics, University of Seoul, Seoul, 130-743, Korea

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© Dehghanian et al.; licensee Springer 2013

This article is published under license to BioMed Central Ltd. This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

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