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C -ternary 3-derivations on C -ternary algebras

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.

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]xyz 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,yz.

If a C -ternary algebra (A,[,,]) has an identity, i.e., an element eA such that x=[x,e,e]=[e,e,x] for all xA, then it is routine to verify that A, endowed with xy:=[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:AB is called a C -ternary homomorphism if H([x,y,z])=[H(x),H(y),H(z)] for all x,y,zA. If, in addition, the mapping H is bijective, then the mapping H:AB is called a C -ternary algebra isomorphism. A -linear mapping δ:AA 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,zA (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(xy),f(x)f(y))<δ for all x,yG , then a homomorphism h:GG exists with ρ(f(x),h(x))<ϵ for all xG ?

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×XY is called a 3-additive mapping if f is additive for each variable, and a mapping f:X×X×XY is called a 3--linear mapping if f is -linear for each variable.

A 3--linear mapping H:A×A×AB 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 :={λC:|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×XY be a 3-additive mapping such that f(λx,μy,νz)=λμνf(x,y,z) for all λ,μ,ν S 1 and all x,y,zX. 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,zA.

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,zA. 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 1i3 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 )=λμνf( 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 1i6 and

1 3 n ψ ( x 1 , , 3 n x i , , x 9 ) =0or 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 1i3 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,zA.

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,zA.

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,zA. So, we can conclude that δ(x,y,z)=T(x,y,z) for all x,y,zA. 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,zA.

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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).

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Correspondence to Dong Yun Shin.

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The authors declare that they have no competing interests.

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All authors conceived of the study, participated in its design and coordination, drafted the manuscript, participated in the sequence alignment, and read and approved the final manuscript.

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Dehghanian, M., Mosadegh, S.M.S.M., Park, C. et al. C -ternary 3-derivations on C -ternary algebras. J Inequal Appl 2013, 124 (2013). https://doi.org/10.1186/1029-242X-2013-124

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Keywords

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