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Homomorphisms and derivations in C -ternary algebras via fixed point method

Abstract

Park (J. Math. Phys. 47:103512, 2006) proved the Hyers-Ulam stability of homomorphisms in C -ternary algebras and of derivations on C -ternary algebras for the following generalized Cauchy-Jensen additive mapping:

2f ( j = 1 p x j 2 + j = 1 d y j ) = j = 1 p f( x j )+2 j = 1 d f( y j ).

In this paper, we improve and generalize some results concerning this functional equation via the fixed-point method.

MSC:39B52, 17A40, 46B03, 47Jxx.

1 Introduction and preliminaries

The stability problem of functional equations originated from a question of Ulam [1] concerning the stability of group homomorphisms. Hyers [2] gave a first affirmative partial answer to the question of Ulam for Banach spaces. Let X and Y be Banach spaces. Hyers’ theorem was generalized by Aoki [3] for additive mappings and by Rassias [4] for linear mappings by considering an unbounded Cauchy difference.

Theorem 1.1 (Th.M. Rassias)

Let f:E E be a mapping from a normed vector space E into a Banach space E subject to the inequality

f ( x + y ) f ( x ) f ( y ) ϵ ( x p + y p )
(1.1)

for all x,yE, where ϵ and p are constants with ϵ>0 and p<1. Then the limit

L(x)= lim n f ( 2 n x ) 2 n

exists for all xE and L:E E is the unique additive mapping which satisfies

f ( x ) L ( x ) 2 ϵ 2 2 p x p
(1.2)

for all xE. If p<0 then inequality (1.1) holds for x,y0 and (1.2) for x0. Also, if for each xE the mapping f(tx) is continuous in tR, then L is linear.

Rassias [5] during the 27th International Symposium on Functional Equations asked the question whether such a theorem can also be proved for p1. Gajda [6] following the same approach as in Rassias [4], gave an affirmative solution to this question for p>1. It was shown by Gajda [6], as well as by Rassias and Šemrl [7] that one cannot prove a Rassias’ type theorem when p=1. The counterexamples of Gajda [6], as well as of Rassias and Šemrl [7] have stimulated several mathematicians to invent new definitions of approximately additive or approximately linear mappings, cf. Găvruta [8], Jung [9], who among others studied the Hyers-Ulam stability of functional equations. The inequality (1.1) that was introduced for the first time by Rassias [4] provided a lot of influence in the development of a generalization of the Hyers-Ulam stability concept (cf. the books of Czerwik [10], Hyers, Isac, and Rassias [11]).

Following the terminology of [12], a nonempty set G with a ternary operation [,,]:G×G×GG is called a ternary groupoid and is denoted by (G,[,,]). The ternary groupoid (G,[,,]) is called commutative if [ x 1 , x 2 , x 3 ]=[ x σ ( 1 ) , x σ ( 2 ) , x σ ( 3 ) ] for all x 1 , x 2 , x 3 G and all permutations σ of {1,2,3}.

If a binary operation is defined on G such that [x,y,z]=(xy)z for all x,y,zG, then we say that [,,] is derived from . We say that (G,[,,]) is a ternary semigroup if the operation [,,] is associative, i.e., if [[x,y,z],u,v]=[x,[y,z,u],v]=[x,y,[z,u,v]] holds for all x,y,z,u,vG (see [13]).

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 are C-linear in the outer variables, conjugate C-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 [12, 14]). 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.

A C-linear mapping H:AB is called a C -ternary algebra 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 C-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 [12, 15]).

There are some applications, although still hypothetical, in the fractional quantum Hall effect, the nonstandard statistics, supersymmetric theory, and Yang-Baxter equation (cf. [1618]).

Throughout this paper, assume that p, d are nonnegative integers with p+d3, and that A and B are C -ternary algebras.

The aim of the present paper is to establish the stability problem of homomorphisms and derivations in C -ternary algebras by using the fixed-point method.

Let E be a set. A function d:E×E[0,1] is called a generalized metric on E if d satisfies

  1. (i)

    d(x,y)=0 if and only if x=y;

  2. (ii)

    d(x,y)=d(y,x) for all x,yE;

  3. (iii)

    d(x,z)d(x,y)+d(y,z) for all x,y,zE.

Theorem 1.2 Let (E,d) be a complete generalized metric space and let J:EE be a strictly contractive mapping with constant L<1. Then for each given element xE, either

d ( J n x , J n + 1 x ) =

for all nonnegative integers n or there exists a nonnegative integer n 0 such that

  1. (1)

    d( J n x, J n + 1 x)< for all n n 0 ;

  2. (2)

    the sequence J n x converges to a fixed point y of J;

  3. (3)

    y is the unique fixed point of J in the set Y=yE:d( J n 0 ,y)<;

  4. (4)

    d(y, y ) 1 1 L d(y,Jy) for all yY.

2 Stability of homomorphisms in C -ternary algebras

Throughout this section, assume that A is a unital C -ternary algebra with norm and unit e, and that B is a unital C -ternary algebra with norm and unit e .

The stability of homomorphisms in C -ternary algebras has been investigated in [19]via direct method. In this note, we improve some results in [19]via the fixed-point method. For a given mapping f:AB, we define

C μ f( x 1 ,, x p , y 1 ,, y d ):=2f ( j = 1 p μ x j 2 + j = 1 d μ y j ) j = 1 p μf( x j )2 j = 1 d μf( y j )

for all μ T 1 :={λC:|λ|=1} and all x 1 ,, x p , y 1 ,, y d A.

One can easily show that a mapping f:AB satisfies

C μ f( x 1 ,, x p , y 1 ,, y d )=0

for all μ T 1 and all x 1 ,, x p , y 1 ,, y d A if and only if

f(μx+λy)=μf(x)+λf(y)

for all μ,λ T 1 and all x,yA.

We will use the following lemma in this paper.

Lemma 2.1 ([20])

Let f:AB be an additive mapping such that f(μx)=μf(x) for all xA and all μ T 1 . Then the mapping f is C-linear.

Lemma 2.2 Let { x n } n , { y n } n and { z n } n be convergent sequences in A. Then the sequence {[ x n , y n , z n ]} is convergent in A.

Proof Let x,y,zA such that

lim n x n =x, lim n v n =y, lim n z n =z.

Since

[ x n , y n , z n ] [ x , y , z ] = [ x n x , y n y , z n , z ] + [ x n , y n , z ] + [ x , y n y , z n ] + [ x n , y , z n z ]

for all n, we get

[ x n , y n , z n ] [ x , y , z ] = x n x y n y z n z + x n x y n z + x y n y z n + x n y z n z

for all n. So

lim n [ x n , y n , z n ]=[x,y,z].

This completes the proof. □

Theorem 2.3 Let f:AB be a mapping for which there exist functions φ: A p + d [0,) and ψ: A 3 [0,) such that

(2.1)
(2.2)

for all μ T 1 and all x,y,z, x 1 ,, x p , y 1 ,, y d A, where γ= p + 2 d 2 . If there exists constant L<1 such that

φ(γx,,γx)γLφ(x,,x)

for all xA, then there exists a unique C -ternary algebras homomorphism H:AB satisfying

f ( x ) H ( x ) 1 ( 1 L ) 2 γ φ(x,,x)
(2.3)

for all xA.

Proof Let us assume μ=1 and x 1 == x p = y 1 == y d =x in (2.1). Then we get

f ( γ x ) γ f ( x ) 1 2 φ(x,,x)
(2.4)

for all xA. Let E:={g:AB}. We introduce a generalized metric on E as follows:

d(g,h):=inf { C [ 0 , ] : g ( x ) h ( x ) C φ ( x , , x )  for all  x A } .

It is easy to show that (E,d) is a generalized complete metric space.

Now, we consider the mapping Λ:EE defined by

(Λg)(x)= 1 γ g(γx),for all gE and xA.

Let g,hE and let C[0,] be an arbitrary constant with d(g,h)C. From the definition of d, we have

g ( x ) h ( x ) Cφ(x,,x)

for all xA. By the assumption and the last inequality, we have

( Λ g ) ( x ) ( Λ h ) ( x ) = 1 γ g ( γ x ) h ( γ x ) C γ φ(γx,,γx)CLφ(x,,x)

for all xA. So d(Λg,Λh)Ld(g,h) for any g,hE. It follows from (2.4) that d(Λf,f) 1 2 γ . Therefore according to Theorem 1.2, the sequence { Λ n f} converges to a fixed point H of Λ, i.e.,

H:AB,H(x)= lim n ( Λ n f ) (x)= lim n 1 γ n f ( γ n x )
(2.5)

and H(γx)=γH(x) for all xA. Also H is the unique fixed point of Λ in the set E={gE:d(f,g)<} and

d(H,f) 1 1 L d(Λf,f) 1 ( 1 L ) 2 γ

i.e., the inequality (2.3) holds true for all xA. It follows from the definition of H that

for all μ T 1 and all x 1 ,, x p , y 1 ,, y d A. Hence

2H ( j = 1 p μ x j 2 + j = 1 d μ y j ) = j = 1 p μH( x j )+2 j = 1 d μH( y j )

for all μ T 1 and all x 1 ,, x p , y 1 ,, y d A. So H(λx+μy)=λH(x)+μH(y) for all λ,μ T 1 and all x,yA.

Therefore, by Lemma 2.1, the mapping H:AB is C-linear.

It follows from (2.2) and (2.5) that

for all x,y,zA. Thus

H ( [ x , y , z ] ) = [ H ( x ) , H ( y ) , H ( z ) ]

for all x,y,zA. Therefore, the mapping H is a C -ternary algebras homomorphism.

Now, let T:AB be another C -ternary algebras homomorphism satisfying (2.3). Since d(f,T) 1 ( 1 L ) 2 γ and T is C-linear, we get T E and (ΛT)(x)= 1 γ (Tγx)=T(x) for all xA, i.e., T is a fixed point of Λ. Since H is the unique fixed point of Λ E , we get H=T. □

Theorem 2.4 Let f:AB be a mapping for which there exist functions φ: A p + d [0,) and ψ: A 3 [0,) satisfying (2.1), (2.2),

for all x,y,z, x 1 ,, x p , y 1 ,, y d A, where γ= p + 2 d 2 . If there exists constant L<1 such that

φ ( 1 γ x , , 1 γ x ) 1 γ Lφ(x,,x)

for all xA, then there exists a unique C -ternary algebras homomorphism H:AB satisfying

f ( x ) H ( x ) 1 ( 1 L ) 2 γ φ(x,,x)

for all xA.

Proof If we replace x in (2.4) by x γ , then we get

f ( x ) γ f ( x γ ) 1 2 φ ( x γ , , x γ ) L 2 γ φ(x,,x)
(2.6)

for all xA. Let E:={g:AA}. We introduce a generalized metric on E as follows:

d(g,h):=inf { C [ 0 , ] : g ( x ) h ( x ) C φ ( x , , x )  for all  x A } .

It is easy to show that (E,d) is a generalized complete metric space.

Now, we consider the mapping Λ:EE defined by

(Λg)(x)=γg ( x γ ) ,for all gE and xA.

Let g,hE and let C[0,] be an arbitrary constant with d(g,h)C. From the definition of d, we have

g ( x ) h ( x ) Cφ(x,,x)

for all xA. By the assumption and the last inequality, we have

( Λ g ) ( x ) ( Λ h ) ( x ) = γ g ( x γ ) γ h ( x γ ) γCφ ( x γ , , x γ ) CLφ(x,,x)

for all xA, and so d(Λg,Λh)Ld(g,h) for any g,hE. It follows from (2.6) that d(Λf,f) 1 2 γ . Thus, according to Theorem 1.2, the sequence { Λ n f} converges to a fixed point H of Λ, i.e.,

H:AB,H(x)= lim n ( Λ n f ) (x)= lim n γ n f ( x γ n )

for all xA.

The rest of the proof is similar to the proof of Theorem 2.3, and we omit it. □

Corollary 2.5 ([19])

Let r and θ be nonnegative real numbers such that r[1,3], and let f:AB be a mapping such that

C μ f ( x 1 , , x p , y 1 , , y d ) θ ( j = 1 p x j r + j = 1 d y j r )
(2.7)

and

f ( [ x , y , z ] ) [ f ( x ) , f ( y ) , f ( z ) ] θ ( x r + y r + z r )
(2.8)

for all μ T 1 and all x,y,z, x 1 ,, x p , y 1 ,, y d A. Then there exists a unique C -ternary algebra homomorphism H:AB such that

f ( x ) H ( x ) 2 r ( p + d ) θ | 2 ( p + 2 d ) r ( p + 2 d ) 2 r | x r
(2.9)

for all xA.

Proof The proof follows from Theorems 2.3 and 2.4 by taking

for all μ T 1 and all x,y,z, x 1 ,, x p , y 1 ,, y d A. Then we can choose L= 2 1 r ( p + 2 d ) r 1 , when 0<r<1 and L=2 2 1 r ( p + 2 d ) r 1 , when r>3 and we get the desired results. □

3 Superstability of homomorphisms in C -ternary algebras

Throughout this section, assume that A is a unital C -ternary algebra with norm and unit e, and that B is a unital C -ternary algebra with norm and unit e .

We investigate homomorphisms in C -ternary algebras associated with the functional equation C μ f( x 1 ,, x p , y 1 ,, y d )=0.

Theorem 3.1 ([19])

Let r>1 (resp., r<1) and θ be nonnegative real numbers, and let f:AB be a bijective mapping satisfying (2.1) and

f ( [ x , y , z ] ) = [ f ( x ) , f ( y ) , f ( z ) ]

for all x,y,zA. If lim n ( p + 2 d ) n 2 n f( 2 n e ( p + 2 d ) n )= e (resp., lim n 2 n ( p + 2 d ) n f( ( p + 2 d ) n 2 n e)= e ), then the mapping f:AB is a C -ternary algebra isomorphism.

In the following theorems we have alternative results of Theorem 3.1.

Theorem 3.2 Let r<1 and θ be nonnegative real numbers, and let f:AB be a mapping satisfying (2.7) and (2.8). If there exist a real number λ>1 (resp., 0<λ<1) and an element x 0 A such that lim n 1 λ n f( λ n x 0 )= e (resp., lim n λ n f( x 0 λ n )= e ), then the mapping f:AB is a C -ternary algebra homomorphism.

Proof By using the proof of Corollary 2.5, there exists a unique C -ternary algebra homomorphism H:AB satisfying (2.9). It follows from (2.9) that

H(x)= lim n 1 λ n f ( λ n x ) , ( H ( x ) = lim n λ n f ( x λ n ) )

for all xA and all real numbers λ>1 (0<λ<1). Therefore, by the assumption, we get that H( x 0 )= e .

Let λ>1 and lim n 1 λ n f( λ n x 0 )= e . It follows from (2.8) that

for all xA. So [H(x),H(y),H(z)]=[H(x),H(y),f(z)] for all x,y,zA. Letting x=y= x 0 in the last equality, we get f(z)=H(z) for all zA. Similarly, one can show that H(x)=f(x) for all xA when 0<λ<1 and lim n λ n f( x 0 λ n )= e .

Similarly, one can show the theorem for the case λ>1.

Therefore, the mapping f:AB is a C -ternary algebra homomorphism. □

Theorem 3.3 Let r>1 and θ be nonnegative real numbers, and let f:AB be a mapping satisfying (2.7) and (2.8). If there exist a real number 0<λ<1 (resp., λ>1) and an element x 0 A such that lim n 1 λ n f( λ n x 0 )= e (resp., lim n λ n f( x 0 λ n )= e ), then the mapping f:AB is a C -ternary algebra homomorphism.

Proof The proof is similar to the proof of Theorem 3.2 and we omit it. □

4 Stability of derivations on C -ternary algebras

Throughout this section, assume that A is a C -ternary algebra with norm .

Park [19] proved the Hyers-Ulam stability of derivations on C -ternary algebras for the functional equation C μ f( x 1 ,, x p , y 1 ,, y d )=0.

For a given mapping f:AA, let

Df(x,y,z)=f ( [ x , y , z ] ) [ f ( x ) , y , z ] [ x , f ( y ) , z ] [ x , y , f ( z ) ]

for all x,y,zA.

Theorem 4.1 ([19])

Let r and θ be nonnegative real numbers such that r[1,3], and let f:AA be a mapping satisfying (2.7) and

D f ( x , y , z ) θ ( x r + y r + z r )

for all x,y,zA. Then there exists a unique C -ternary derivation δ:AA such that

f ( x ) δ ( x ) 2 r ( p + d ) | 2 ( p + 2 d ) r ( p + 2 d ) 2 r | θ x r

for all xA.

In the following theorem, we generalize and improve the result in Theorems 4.1.

Theorem 4.2 Let φ: A p + d [0,) and ψ: A 3 [0,) be functions such that

(4.1)
(4.2)

for all x,y,z, x 1 ,, x p , y 1 ,, y d A, where γ= p + 2 d 2 . Suppose that f:AA is a mapping satisfying

(4.3)
(4.4)

for all μ T 1 and all x,y,z, x 1 ,, x p , y 1 ,, y d A. If there exists a constant L<1 such that

φ(γx,,γx)γφ(x,,x),

then the mapping f:AA is a C -ternary derivation.

Proof Let us assume μ=1 and x 1 == x p = y 1 == y d =x in (4.3). Then we get

f ( γ x ) γ f ( x ) 1 2 φ(x,,x)
(4.5)

for all xA. Let E:={g:AA}. We introduce a generalized metric on E as follows:

d(g,h):=inf { C [ 0 , ] : g ( x ) h ( x ) C φ ( x , , x )  for all  x A } .

It is easy to show that (E,d) is a generalized complete metric space.

Now, we consider the mapping Λ:EE defined by

(Λg)(x)= 1 γ g(γx),for all gE and xA.

Let g,hE and let C[0,] be an arbitrary constant with d(g,h)C. From the definition of d, we have

g ( x ) h ( x ) Cφ(x,,x)

for all xA. By the assumption and the last inequality, we have

( Λ g ) ( x ) ( Λ h ) ( x ) = 1 γ g ( γ x ) h ( γ x ) C γ φ(γx,,γx)CLφ(x,,x)

for all xA. Then d(Λg,Λh)Ld(g,h) for any g,hE. It follows from (2.4) that d(Λf,f) 1 2 γ . Thus according to Theorem 1.2, the sequence { Λ n f} converges to a fixed point δ of Λ, i.e.,

δ:AA,δ(x)= lim n ( Λ n f ) (x)= lim n 1 γ n f ( γ n x )
(4.6)

and δ(γx)=γδ(x) for all xA. Also δ is the unique fixed point of Λ in the set E={gE:d(f,g)<} and

d(δ,f) 1 1 L d(Λf,f) 1 ( 1 L ) 2 γ

i.e., the inequality (2.3) holds true for all xA. It follows from the definition of δ, (4.1), (4.3), and (4.6) that

for all μ T 1 and all x,y,z, x 1 ,, x p , y 1 ,, y d A. Hence,

2δ ( j = 1 p μ x j 2 + j = 1 d μ y j ) = j = 1 p μδ( x j )+2 j = 1 d μδ( y j )

for all μ T 1 and all x 1 ,, x p , y 1 ,, y d A. So δ(λx+μy)=λδ(x)+μδ(y) for all λ,μ T 1 and all x,yA.

Therefore, by Lemma 2.1 the mapping δ:AA is C-linear.

It follows from (4.2) and (4.4) that

D δ ( x , y , z ) = lim n 1 γ 3 n D f ( γ n x , γ n y , γ n z ) lim n 1 γ 3 n ψ ( γ n x , γ n y , γ n z ) =0

for all x,y,zA. Hence

δ ( [ x , y , z ] ) = [ δ ( x ) , y , z ] + [ x , δ ( y ) , z ] + [ x , y , δ ( z ) ]
(4.7)

for all x,y,zA. So the mapping δ:AA is a C -ternary derivation.

It follows from (4.2) and (4.4)

for all x,y,zA. Thus

δ[x,y,z]= [ δ ( x ) , y , z ] + [ x , δ ( y ) , z ] + [ x , y , f ( z ) ]
(4.8)

for all x,y,zA. Hence, we get from (4.7) and (4.8) that

[ x , y , δ ( z ) ] = [ x , y , f ( z ) ]
(4.9)

for all x,y,zA. Letting x=y=f(z)δ(z) in (4.9), we get

f ( z ) δ ( z ) 3 = [ f ( z ) δ ( z ) , f ( z ) δ ( z ) , f ( z ) δ ( z ) ] =0

for all zA. Hence, f(z)=δ(z) for all zA. So the mapping f:AA is a C -ternary derivation, as desired. □

Corollary 4.3 Let r<1, s<2 and θ be nonnegative real numbers, and let f:AA be a mapping satisfying (2.7) and

D f ( x , y , z ) A θ ( x A s + y A s + z A s )

for all x,y,zA. Then the mapping f:AA is a C -ternary derivation.

Proof Defining

φ( x 1 ,, x p , y 1 ,, y d )=θ ( j = 1 p x j A r + j = 1 d y j A r )

and

ψ(x,y,z)=θ ( x A s + y A s + z A s )

for all x,y,z, x 1 ,, x p , y 1 ,, y d A, and applying Theorem 4.2, we get the desired result. □

Theorem 4.4 Let φ: A p + d [0,) and ψ: A 3 [0,) be functions such that

for all x,y,z, x 1 ,, x p , y 1 ,, y d A where γ= p + 2 d 2 . Suppose that f:AA is a mapping satisfying (4.3) and (4.4). If there exists a constant L<1 such that

φ ( x γ , , x γ ) L γ φ(x,,x),

then the mapping f:AA is a C -ternary derivation.

Proof If we replace x in (4.5) by x γ , then we get

f ( x ) γ f ( x γ ) A 1 2 φ ( x γ , , x γ )

for all xA. Let E:={g:AA}. We introduce a generalized metric on E as follows:

d(g,h):=inf { C [ 0 , ] : g ( x ) h ( x ) C φ ( x , , x )  for all  x A }

It is easy to show that (E,d) is a generalized complete metric space.

Now, we consider the mapping Λ:EE defined by

(Λg)(x)=γg ( x γ ) ,for all gE and xA.

Let g,hE and let C[0,] be an arbitrary constant with d(g,h)C. From the definition of d, we have

g ( x ) h ( x ) Cφ(x,,x)

for all xA. By the assumption and last inequality, we have

( Λ g ) ( x ) ( Λ h ) ( x ) = γ g ( x γ ) γ h ( x γ ) γCφ ( x γ , , x γ ) CLφ(x,,x)

for all xA. Then d(Λg,Λh)Ld(g,h) for any g,hE. It follows from (4.5) that d(Λf,f) 1 2 γ . Therefore according to Theorem 1.2, the sequence { Λ n f} converges to a fixed point δ of Λ, i.e.,

δ:AA,δ(x)= lim n ( Λ n f ) (x)= lim n γ n f ( x γ n )

and δ(γx)=γδ(x) for all xA.

The rest of the proof is similar to the proof of Theorem 4.2, and we omit it. □

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Acknowledgements

The authors are grateful to the reviewers for their valuable comments and suggestions.

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Correspondence to Choonkil Park.

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All authors carried out the proof. All authors conceived of the study, and participated in its design and coordination. All authors read and approved the final manuscript.

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Kenari, H., Saadati, R. & Park, C. Homomorphisms and derivations in C -ternary algebras via fixed point method. Adv Differ Equ 2012, 137 (2012). https://doi.org/10.1186/1687-1847-2012-137

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Keywords

  • C -ternary algebra isomorphism
  • generalized Cauchy-Jensen functional equation
  • Hyers-Ulam stability
  • C -ternary derivation
  • fixed-point method