# 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  concerning the stability of group homomorphisms. Hyers  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  for additive mappings and by Rassias  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 $x,y∈E$, 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 $x∈E$ 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 $x∈E$. If $p<0$ then inequality (1.1) holds for $x,y≠0$ and (1.2) for $x≠0$. Also, if for each $x∈E$ the mapping $f(tx)$ is continuous in $t∈R$, then L is linear.

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

Following the terminology of , a nonempty set G with a ternary operation $[⋅,⋅,⋅]:G×G×G→G$ 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 $[x,y,z]=(x∘y)∘z$ for all $x,y,z∈G$, 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 $x,y,z,u,v∈G$ (see ).

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 $∥[x,y,z]∥≤∥x∥⋅∥y∥⋅∥z∥$ and $∥[x,x,x]∥= ∥ x ∥ 3$ (see [12, 14]). Every left Hilbert $C ∗$-module is a $C ∗$-ternary algebra via the ternary product $[x,y,z]:=〈x,y〉z$.

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.

A $C$-linear mapping $H:A→B$ is called a $C ∗$-ternary algebra 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 $C$-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 [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. ).

Throughout this paper, assume that p, d are nonnegative integers with $p+d≥3$, 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 $x,y∈E$;

3. (iii)

$d(x,z)≤d(x,y)+d(y,z)$ for all $x,y,z∈E$.

Theorem 1.2 Let $(E,d)$ be a complete generalized metric space and let $J:E→E$ be a strictly contractive mapping with constant $L<1$. Then for each given element $x∈E$, 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 $Y=y∈E:d( J n 0 ,y)<∞$;

4. (4)

$d(y, y ∗ )≤ 1 1 − L d(y,Jy)$ for all $y∈Y$.

## 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 via direct method. In this note, we improve some results in via the fixed-point method. For a given mapping $f:A→B$, 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 $μ∈ T 1 :={λ∈C:|λ|=1}$ and all $x 1 ,…, x p , y 1 ,…, y d ∈A$.

One can easily show that a mapping $f:A→B$ 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 $x,y∈A$.

We will use the following lemma in this paper.

Lemma 2.1 ()

Let $f:A→B$ be an additive mapping such that $f(μx)=μf(x)$ for all $x∈A$ 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 $x,y,z∈A$ 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 $f:A→B$ 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 $x∈A$, then there exists a unique $C ∗$-ternary algebras homomorphism $H:A→B$ satisfying

$∥ f ( x ) − H ( x ) ∥ ≤ 1 ( 1 − L ) 2 γ φ(x,…,x)$
(2.3)

for all $x∈A$.

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 $x∈A$. Let $E:={g:A→B}$. We introduce a generalized metric on E as follows:

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

Now, we consider the mapping $Λ:E→E$ defined by

Let $g,h∈E$ 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 $x∈A$. 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 $x∈A$. So $d(Λg,Λh)≤Ld(g,h)$ for any $g,h∈E$. 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.,

$H:A→B,H(x)= lim n → ∞ ( Λ n f ) (x)= lim n → ∞ 1 γ n f ( γ n x )$
(2.5)

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

Therefore, by Lemma 2.1, the mapping $H:A→B$ is $C$-linear.

It follows from (2.2) and (2.5) that for all $x,y,z∈A$. Thus

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

for all $x,y,z∈A$. Therefore, the mapping H is a $C ∗$-ternary algebras homomorphism.

Now, let $T:A→B$ 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 $x∈A$, 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 $f:A→B$ 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 $x∈A$, then there exists a unique $C ∗$-ternary algebras homomorphism $H:A→B$ satisfying

$∥ f ( x ) − H ( x ) ∥ ≤ 1 ( 1 − L ) 2 γ φ(x,…,x)$

for all $x∈A$.

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 $x∈A$. Let $E:={g:A→A}$. We introduce a generalized metric on E as follows:

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

Now, we consider the mapping $Λ:E→E$ defined by

Let $g,h∈E$ 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 $x∈A$. 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 $x∈A$, and so $d(Λg,Λh)≤Ld(g,h)$ for any $g,h∈E$. 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.,

$H:A→B,H(x)= lim n → ∞ ( Λ n f ) (x)= lim n → ∞ γ n f ( x γ n )$

for all $x∈A$.

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

Corollary 2.5 ()

Let r and θ be nonnegative real numbers such that $r∉[1,3]$, and let $f:A→B$ 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 $H:A→B$ 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 $x∈A$.

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 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 ()

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

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

for all $x,y,z∈A$. 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 $f:A→B$ 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 $f:A→B$ 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 $f:A→B$ is a $C ∗$-ternary algebra homomorphism.

Proof By using the proof of Corollary 2.5, there exists a unique $C ∗$-ternary algebra homomorphism $H:A→B$ 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 $x∈A$ 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 $x∈A$. So $[H(x),H(y),H(z)]=[H(x),H(y),f(z)]$ for all $x,y,z∈A$. Letting $x=y= x 0$ in the last equality, we get $f(z)=H(z)$ for all $z∈A$. Similarly, one can show that $H(x)=f(x)$ for all $x∈A$ 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 $f:A→B$ is a $C ∗$-ternary algebra homomorphism. □

Theorem 3.3 Let $r>1$ and θ be nonnegative real numbers, and let $f:A→B$ 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 $f:A→B$ 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  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 $f:A→A$, let

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

for all $x,y,z∈A$.

Theorem 4.1 ()

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

$∥ D f ( x , y , z ) ∥ ≤θ ( ∥ x ∥ r + ∥ y ∥ r + ∥ z ∥ r )$

for all $x,y,z∈A$. Then there exists a unique $C ∗$-ternary derivation $δ:A→A$ 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 $x∈A$.

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 $f:A→A$ 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 $f:A→A$ 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 $x∈A$. Let $E:={g:A→A}$. We introduce a generalized metric on E as follows:

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

Now, we consider the mapping $Λ:E→E$ defined by

Let $g,h∈E$ 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 $x∈A$. 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 $x∈A$. Then $d(Λg,Λh)≤Ld(g,h)$ for any $g,h∈E$. 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.,

$δ:A→A,δ(x)= lim n → ∞ ( Λ n f ) (x)= lim n → ∞ 1 γ n f ( γ n x )$
(4.6)

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

Therefore, by Lemma 2.1 the mapping $δ:A→A$ 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 $x,y,z∈A$. Hence

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

for all $x,y,z∈A$. So the mapping $δ:A→A$ is a $C ∗$-ternary derivation.

It follows from (4.2) and (4.4) for all $x,y,z∈A$. Thus

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

for all $x,y,z∈A$. Hence, we get from (4.7) and (4.8) that

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

for all $x,y,z∈A$. 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 $z∈A$. Hence, $f(z)=δ(z)$ for all $z∈A$. So the mapping $f:A→A$ is a $C ∗$-ternary derivation, as desired. □

Corollary 4.3 Let $r<1$, $s<2$ and θ be nonnegative real numbers, and let $f:A→A$ 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 $x,y,z∈A$. Then the mapping $f:A→A$ 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 $f:A→A$ 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 $f:A→A$ 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 $x∈A$. Let $E:={g:A→A}$. We introduce a generalized metric on E as follows:

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

Now, we consider the mapping $Λ:E→E$ defined by

Let $g,h∈E$ 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 $x∈A$. 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 $x∈A$. Then $d(Λg,Λh)≤Ld(g,h)$ for any $g,h∈E$. 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.,

$δ:A→A,δ(x)= lim n → ∞ ( Λ n f ) (x)= lim n → ∞ γ n f ( x γ n )$

and $δ(γx)=γδ(x)$ for all $x∈A$.

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

## References

1. 1.

Ulam SM: A Collection of the Mathematical Problems. Interscience, New York; 1960.

2. 2.

Hyers DH: On the stability of the linear functional equation. Proc. Natl. Acad. Sci. USA 1941, 27: 222–224. 10.1073/pnas.27.4.222

3. 3.

Aoki T: On the stability of the linear transformation in Banach spaces. J. Math. Soc. Jpn. 1950, 2: 64–66. 10.2969/jmsj/00210064

4. 4.

Rassias TM: On the stability of the linear mapping in Banach spaces. Proc. Am. Math. Soc. 1978, 72: 297–300. 10.1090/S0002-9939-1978-0507327-1

5. 5.

Rassias TM: Problem 16; 2. Report of the 27th international symp. on functional equations. Aequ. Math. 1990, 39: 292–293.

6. 6.

Gajda Z: On stability of additive mappings. Int. J. Math. Math. Sci. 1991, 14: 431–434. 10.1155/S016117129100056X

7. 7.

Rassias TM, Šemrl P: On the behaviour of mappings which do not satisfy Hyers-Ulam stability. Proc. Am. Math. Soc. 1992, 114: 989–993. 10.1090/S0002-9939-1992-1059634-1

8. 8.

Gǎvruta P: A generalization of the Hyers-Ulam-Rassias stability of approximately additive mappings. J. Math. Anal. Appl. 1994, 184: 431–436. 10.1006/jmaa.1994.1211

9. 9.

Jung S: On the Hyers-Ulam-Rassias stability of approximately additive mappings. J. Math. Anal. Appl. 1996, 204: 221–226. 10.1006/jmaa.1996.0433

10. 10.

Czerwik P: Functional Equations and Inequalities in Several Variables. World Scientific, Singapore; 2002.

11. 11.

Hyers DH, Isac G, Rassias TM: Stability of Functional Equations in Several Variables. Birkhäuser, Basel; 1998.

12. 12.

Amyari M, Moslehian MS: Approximately ternary semigroup homomorphisms. Lett. Math. Phys. 2006, 77: 1–9. 10.1007/s11005-005-0042-6

13. 13.

Bazunova N, Borowiec A, Kerner R: Universal differential calculus on ternary algebras. Lett. Math. Phys. 2004, 67: 195–206.

14. 14.

Zettl H: A characterization of ternary rings of operators. Adv. Math. 1983, 48: 117–143. 10.1016/0001-8708(83)90083-X

15. 15.

Moslehian MS:Almost derivations on $C ∗$-ternary rings. Bull. Belg. Math. Soc. Simon Stevin 2007, 14: 135–142.

16. 16.

Kerner, R: Ternary algebraic structures and their applications in physics. Preprint

17. 17.

Vainerman L, Kerner R: On special classes of n -algebras. J. Math. Phys. 1996, 37: 2553–2565. 10.1063/1.531526

18. 18.

Abramov V, Kerner R, Le Roy B:Hypersymmetry: a $Z 3$-graded generalization of supersymmetry. J. Math. Phys. 1997, 38: 1650–1669. 10.1063/1.531821

19. 19.

Park C:Isomorphisms between $C ∗$-ternary algebras. J. Math. Phys. 2006., 47(10): Article ID 103512

20. 20.

Park C:Homomorphisms between Poisson $J C ∗$-algebras. Bull. Braz. Math. Soc. 2005, 36: 79–97. 10.1007/s00574-005-0029-z

## Acknowledgements

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

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Correspondence to Choonkil Park.

### Competing interests

The authors declare that they have no competing interests.

### Authors’ contributions

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

• $C ∗$-ternary algebra isomorphism
• $C ∗$-ternary derivation 