In 2006, C. Park proved the stability of homomorphisms in C∗-ternary algebras and of derivations on C∗-ternary algebras for the following
generalized Cauchy-Jensen additive mapping: 2f((∑j=1pxj/2)+∑j=1dyj)=∑j=1pf(xj)+2∑j=1df(yj). In this note, we improve and generalize some results concerning this functional equation.

1. Introduction and Preliminaries

The stability problem of functional equations is 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. 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)
for all x,y∈E, where ε and p are constants with ε>0 and p<1. Then the limit
L(x)=limn→∞f(2nx)2n
exists for all x∈E, and L:E→E' is the unique additive mapping which satisfies
∥f(x)-L(x)∥≤2ε2-2p∥x∥p
for all x∈E. If p<0, then inequality (1.1) holds for x,y≠0 and (1.3) for x≠0. Also, if for each x∈E the mapping f(tx) is continuous in t∈ℝ, then L is linear.

It was shown by Gajda [5] as well as by Rassias and Šemrl [6] that one cannot prove a Rassias’s type theorem when p=1. The counter examples of Gajda [5] as well as of Rassias and Šemrl [6] have stimulated several mathematicians to invent new definitions of approximately additive or approximately linear mappings; compare Găvruţa [7] and Jung [8], who among others studied the stability of functional equations. Theorem 1.1 provided a lot of influence in the development of a generalization of the Hyers-Ulam stability concept. This new concept is known as Hyers-Ulam-Rassiasstability of functional equations (cf. the books of Czerwik [9], Hyers et al. [10]).Theorem 1.2 (Rassias [<xref ref-type="bibr" rid="B33">11</xref>–<xref ref-type="bibr" rid="B35">13</xref>]).

Let X be a real normed linear space and Y a real Banach space. Assume that f:X→Y is a mapping for which there exist constants θ≥0 and p,q∈ℝ such that r=p+q≠1 and f satisfies the functional inequality (Cauchy-Găvruţa-Rassias inequality)∥f(x+y)-f(x)-f(y)∥≤θ∥x∥p∥y∥q
for all x,y∈X. Then there exists a unique additive mapping L:X→Y satisfying
∥f(x)-L(x)∥≤θ|2r-2|∥x∥r
for all x∈X. If, in addition, f:X→Y is a mapping such that the transformation t→f(tx) is continuous in t∈ℝ for each fixed x∈X, then L is linear.

For the case r=1, a counter example has been given by Găvruţa [14]. The stability in Theorem 1.2 involving a product of different powers of norms is called Ulam-Găvruţa-Rassias stability (see [15–17]). In 1994, a generalization of Theorems 1.1 and 1.2 was obtained by Găvruţa [7], who replaced the bounds ε(∥x∥p+∥y∥p) and θ∥x∥p∥y∥q by a general control function φ(x,y). During past few years several mathematicians have published on various generalizations and applications of generalized Hyers-Ulam stability to a number of functional equations and mappings (see [16–44]).

Following the terminology of [45], 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 [x1,x2,x3]=[xσ(1),xσ(2),xσ(3)] for all x1,x2,x3∈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, that is, 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 [46]).

A C*-ternary algebra is a complex Banach space A, equipped with a ternary product (x,y,z)↦[x,y,z] of A3 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 [45, 47]). 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, that is, 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 [23, 45, 48]).

Let (A,∘) be a C*-algebra and [x,y,z]:=x∘y*∘z for all x,y,z∈A. The mapping H:A→A defined by H(x)=-ix is a C*-ternary algebra isomorphism. Let a∈A with a*=a. The mapping δa:A→A defined by δa(x)=i(ax-xa) is a C*-ternary derivation. There are some applications, although still hypothetical, in the fractional quantum Hall effect, the nonstandard statistics, supersymmetric theory, and Yang-Baxter equation (cf. [49–51]).

Throughout this paper, assume that p, d are nonnegative integers with p+d≥3, and that A and B are C*-ternary algebras.

2. Stability of Homomorphisms in <inline-formula><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" id="M126"><mml:mrow><mml:msup><mml:mstyle displaystyle="true"><mml:mi>C</mml:mi></mml:mstyle><mml:mo>*</mml:mo></mml:msup></mml:mrow></mml:math></inline-formula>-Ternary Algebras

The stability of homomorphisms in C*-ternary algebras has been investigated in [31] (see also [37]). In this note, we improve some results in [31]. For a given mapping f:A→B, we define
Cμf(x1,…,xp,y1,…,yd):=2f(∑j=1pμxj2+∑j=1dμyj)-∑j=1pμf(xj)-2∑j=1dμf(yj)
for all μ∈𝕋1:={λ∈ℂ:|λ|=1} and all x1,…,xp,y1,…,yd∈A.

One can easily show that a mapping f:A→B satisfies
Cμf(x1,…,xp,y1,…,yd)=0
for all μ∈𝕋1 and all x1,…,xp,y1,…,yd∈A if and only if
f(μx+λy)=μf(x)+λf(y)
for all μ,λ∈𝕋1 and all x,y∈A.

We will use the following lemmas in this paper.

Lemma 2.1 (see [<xref ref-type="bibr" rid="B29">30</xref>]).

Let f:A→B be an additive mapping such that f(μx)=μf(x) for all x∈A and all μ∈𝕋1. Then the mapping f is ℂ-linear.

Lemma 2.2.

Let {xn}n,{yn}n and {zn}n be convergent sequences in A. Then the sequence {[xn,yn,zn]}n is convergent in A.

Proof.

Let x,y,z∈A such that
limn→∞xn=x,limn→∞yn=y,limn→∞zn=z.
Since
[xn,yn,zn]-[x,y,z]=[xn-x,yn-y,zn-z]+[xn-x,yn,z]+[x,yn-y,zn]+[xn,y,zn-z]
for all n, we get
∥[xn,yn,zn]-[x,y,z]∥≤∥xn-x∥∥yn-y∥∥zn-z∥+∥xn-x∥∥yn∥∥z∥+∥x∥∥yn-y∥∥zn∥+∥xn∥∥y∥∥zn-z∥
for all n. So
limn→∞[xn,yn,zn]=[x,y,z].
This completes the proof.

Theorem 2.3 (see [<xref ref-type="bibr" rid="B30">31</xref>]).

Let r and θ be nonnegative real numbers such that r∉[1,3], and let f:A→B be a mapping such that
∥Cμf(x1,…,xp,y1,…,yd)∥B≤θ(∑j=1p∥xj∥Ar+∑j=1d∥yj∥Ar),∥f([x,y,z])-[f(x),f(y),f(z)]∥B≤θ(∥x∥Ar+∥y∥Ar+∥z∥Ar)
for all μ∈𝕋1 and all x,y,z,x1,…,xp,y1,…,yd∈A. Then there exists a unique C*-ternary algebra homomorphism H:A→B such that
∥f(x)-H(x)∥B≤2r(p+d)θ|2(p+2d)r-(p+2d)2r|∥x∥Ar
for all x∈A.

In the following theorem we have an alternative result of Theorem 2.3.

Theorem 2.4.

Let r, s, and θ be nonnegative real numbers such that 0<r<1, 0<s<3 (resp., r>1, s>3), and let d≥2. Suppose that f:A→B is a mapping with f(0)=0, satisfying (2.8) and
∥f([x,y,z])-[f(x),f(y),f(z)]∥B≤θ(∥x∥As+∥y∥As+∥z∥As)
for all μ∈𝕋1 and all x,y,z∈A. Then there exists a unique C*-ternary algebra homomorphism H:A→B such that
∥f(x)-H(x)∥B≤dθ2|d-dr|∥x∥Ar
for all x∈A.

Proof.

We prove the theorem in two cases.Case 1.

0<r<1 and 0<s<3.

Letting μ=1, x1=⋯=xp=0 and y1=⋯=yd=x in (2.8), we get
∥f(dx)-df(x)∥B≤dθ2∥x∥Ar
for all x∈A. If we replace x by dnx in (2.13) and divide both sides of (2.13) to dn+1, we get
∥1dn+1f(dn+1x)-1dnf(dnx)∥B≤θ2d(r-1)n∥x∥Ar
for all x∈A and all nonnegative integers n. Therefore,
∥1dn+1f(dn+1x)-1dmf(dmx)∥B≤θ2∑i=mnd(r-1)i∥x∥Ar
for all x∈A and all nonnegative integers n≥m. From this it follows that the sequence {(1/dn)f(dnx)} is Cauchy for all x∈A. Since B is complete, the sequence {(1/dn)f(dnx)} converges. Thus one can define the mapping H:A→B by
H(x):=limn→∞1dnf(dnx)
for all x∈A. Moreover, letting m=0 and passing the limit n→∞ in (2.15), we get (2.12). It follows from (2.8) that
∥2H(∑j=1pμxj2+∑j=1dμyj)-∑j=1pμH(xj)-2∑j=1dμH(yj)∥B=limn→∞1dn∥2f(dn∑j=1pμxj2+dn∑j=1dμyj)-∑j=1pμf(dnxj)-2∑j=1dμf(dnyj)∥B≤limn→∞dnrdnθ(∑j=1p∥xj∥Ar+∑j=1d∥yj∥Ar)=0
for all μ∈𝕋1 and all x1,…,xp,y1,…,yd∈A. Hence
2H(∑j=1pμxj2+∑j=1dμyj)=∑j=1pμH(xj)+2∑j=1dμH(yj)
for all μ∈𝕋1 and all x1,…,xp,y1,…yd∈A. So H(λx+μy)=λH(x)+μH(y) for all λ,μ∈𝕋1 and all x,y∈A. Therefore by Lemma 2.1 the mapping H:A→B is ℂ-linear.

It follows from Lemma 2.2 and (2.11) that
∥H([x,y,z])-[H(x),H(y),H(z)]∥B=limn→∞1d3n∥f([dnx,dny,dnz])-[f(dnx),f(dny),f(dnz)]∥B=θlimn→∞dnsd3n(∥x∥As+∥y∥As+∥z∥As)=0
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 algebra homomorphism.

Now let T:A→B be another C*-ternary algebra homomorphism satisfying (2.12). Then we have
∥H(x)-T(x)∥B=limn→∞1dn∥f(dnx)-T(dnx)∥B≤dθ2|d-dr|limn→∞dnrdn∥x∥Ar=0
for all x∈A. So we can conclude that H(x)=T(x) for all x∈A. This proves the uniqueness of H. Thus the mapping H:A→B is a unique C*-ternary algebra homomorphism satisfying (2.12), as desired.

Case 2.

r>1 and s>3.

Similar to the proof of Case 1, we conclude that the sequence {dnf(d-nx)} is a Cauchy sequence in B. So we can define the mapping H:A→B by
H(x):=limn→∞dnf(d-nx)
for all x∈A. The rest of the proof is similar to the proof of Case 1.

Theorem 2.5 (see [<xref ref-type="bibr" rid="B30">31</xref>]).

Let r and θ be nonnegative real numbers such that r∉[1/(p+d),1], and let f:A→B be a mapping such that
∥Cμf(x1,…,xp,y1,…,yd)∥B≤θ∏j=1p∥xj∥Ar·∏j=1d∥yj∥Ar,∥f([x,y,z])-[f(x),f(y),f(z)]∥B≤θ∥x∥Ar∥y∥Ar∥z∥Ar
for all μ∈𝕋1 and all x,y,z,x1,…,xp,y1,…,yd∈A. Then there exists a unique C*-ternary algebra homomorphism H:A→B such that
∥f(x)-H(x)∥B≤2(p+d)rθ|2(p+2d)(p+d)r-2(p+d)r(p+2d)|∥x∥A(p+d)r
for all x∈A.

The following theorem shows that the mapping f:A→B in Theorem 2.5 is a C*-ternary algebra homomorphism when r>0.Theorem 2.6.

Let r,s,q,r1,…,rp,s1,…,sd, and θ be nonnegative real numbers such that r+s+q≠3 and rk>0(sk>0) for some 1≤k≤p, p≥2(1≤k≤d,d≥2).

Let f:A→B be a mapping satisfying
∥Cμf(x1,…,xp,y1,…,yd)∥B≤θ∏j=1p∥xj∥Arj·∏j=1d∥yj∥Asj,∥f([x,y,z])-[f(x),f(y),f(z)]∥B≤θ∥x∥Ar∥y∥As∥z∥Aq
for all μ∈𝕋1 and all x,y,z,x1,…,xp,y1,…,yd∈A. Then the mapping f:A→B is a C*-ternary algebra homomorphism. (We put ∥·∥A0=1).

Proof.

Let rk>0 for some 1≤k≤p (we have similar proof when sk>0 for some 1≤k≤d). We now assume, without loss of generality, that r1>0. Letting x1=⋯=xp=y1=⋯=yd=0 in (2.26), we get that f(0)=0. Letting x2=2x and x1=x3=⋯=xp=y1=⋯=yd=0 in (2.26), we get
μf(2x)=2f(μx)
for all μ∈𝕋1 and all x∈A. Setting μ=1 in (2.28), we get that f(2x)=2f(x) for all x∈A. Therefore,
f(μx)=μf(x),f(2μx)=2μf(x)
for all μ∈𝕋1 and all x∈A. If we put x2=2x and y1=y and x1=x3=⋯=xp=y2=⋯=yd=0 in (2.26), we get
2f(μx+μy)=μf(2x)+2μf(y)
for all μ∈𝕋1 and all x∈A. It follows from (2.29) and (2.30) that
f(μx+λy)=μf(x)+λf(y)
for all λ,μ∈𝕋1 and all x,y∈A. Therefore, by Lemma 2.1 the mapping f:A→B is ℂ-linear. Let r+s+q>3. Then it follows from (2.27) that
∥f([x,y,z])-[f(x),f(y),f(z)]∥B=limn→∞8n∥f([x2n,y2n,z2n])-[f(x2n),f(y2n),f(z2n)]∥B≤θ∥x∥Ar∥y∥As∥z∥Aqlimn→∞(82r+s+q)n=0
for all x,y,z∈A. Therefore,
f([x,y,z])=[f(x),f(y),f(z)]
for all x,y,z∈A. Similarly, for r+s+q<3, we get (2.33).

In the rest of this section, assume that A is a unital C*-ternary algebra with norm ∥·∥A and unit e, and that B is a unital C*-ternary algebra with norm ∥·∥B and unit e'.

We investigate homomorphisms in C*-ternary algebras associated with the functional equation Cμf(x1,…,xp,y1,…,yd)=0.

Theorem 2.7 (see [<xref ref-type="bibr" rid="B30">31</xref>]).

Let r>1(r<1) and θ be nonnegative real numbers, and let f:A→B be a bijective mapping satisfying (2.8) such that
f([x,y,z])=[f(x),f(y),f(z)]
for all x,y,z∈A. If limn→∞((p+2d)n/2n)f(2ne/(p+2d)n)=e'(limn→∞(2n/(p+2d)n)f((p+2d)n/2n)e=e'), then the mapping f:A→B is a C*-ternary algebra isomorphism.

In the following theorems we have alternative results of Theorem 2.7.

Theorem 2.8.

Let r<1,s<2 and θ be nonnegative real numbers, and let f:A→B be a mapping satisfying (2.8) and (2.11). If there exist a real number λ>1(0<λ<1) and an element x0∈A such that limn→∞(1/λn)f(λnx0)=e'(limn→∞λnf(x0/λn)=e'), then the mapping f:A→B is a C*-ternary algebra homomorphism.

Proof.

By using the proof of Theorem 2.4, there exists a unique C*-ternary algebra homomorphism H:A→B satisfying (2.12). It follows from (2.12) that
H(x)=limn→∞1λnf(λnx),(H(x)=limn→∞λnf(xλn))
for all x∈A and all real numbers λ>1(0<λ<1). Therefore, by the assumption we get that H(x0)=e'. Let λ>1 and limn→∞(1/λn)f(λnx0)=e'. It follows from (2.11) that
∥[H(x),H(y),H(z)]-[H(x),H(y),f(z)]∥B=∥H[x,y,z]-[H(x),H(y),f(z)]∥B=limn→∞1λ2n∥f([λnx,λny,z])-[f(λnx),f(λny),f(z)]∥B≤θlimn→∞1λ2n(λns∥x∥As+λns∥y∥As+∥z∥As)=0
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=x0 in the last equality, we get f(z)=H(z) for all z∈A. Similarly, one can shows that H(x)=f(x) for all x∈A when 0<λ<1 and limn→∞λnf(x0/λn)=e'. Therefore, the mapping f:A→B is a C*-ternary algebra homomorphism.

3. Derivations on <inline-formula><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" id="M357"><mml:mrow><mml:msup><mml:mstyle displaystyle="true"><mml:mi>C</mml:mi></mml:mstyle><mml:mo>*</mml:mo></mml:msup></mml:mrow></mml:math></inline-formula>-Ternary Algebras

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

Park [31] proved the Hyers-Ulam-Rassias stability and Ulam-Găvruţa-Rassias stability of derivations on C*-ternary algebras for the following functional equation:
Cμf(x1,…,xp,y1,…,yd)=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 3.1 (see [<xref ref-type="bibr" rid="B30">31</xref>]).

Let r and θ be nonnegative real numbers such that r∉[1,3], and let f:A→A a mapping satisfying (2.8) and
∥Df(x,y,z)∥A≤θ(∥x∥Ar+∥y∥Ar+∥z∥Ar)
for all x,y,z∈A. Then there exists a unique C*-ternary derivation δ:A→A such that
∥f(x)-δ(x)∥A≤2r(p+d)|2(p+2d)r-(p+2d)2r|θ∥x∥Ar
for all x∈A.

Theorem 3.2 (see [<xref ref-type="bibr" rid="B30">31</xref>]).

Let r and θ be nonnegative real numbers such that r∉[1/(p+d),1], and let f:A→A be a mapping satisfying (2.23) and
∥Df(x,y,z)∥A≤θ∥x∥Ar∥y∥Ar∥z∥Ar
for all x,y,z∈A. Then there exists a unique C*-ternary derivation δ:A→A such that
∥f(x)-δ(x)∥A≤2(p+d)r|2(p+2d)(p+d)r-(p+2d)2(p+d)r|θ∥x∥A(p+d)r
for all x∈A.

In the following theorems we generalize and improve the results in Theorems 3.1 and 3.2.

Theorem 3.3.

Let φ:Ap+d→[0,∞) and ψ:A3→[0,∞) be functions such that
φ̃(x):=∑n=0∞γ-nφ(γnx,…,γnx)<∞,limn→∞γ-nφ(γnx1,…,γnxp,γny1,…,γnyd)=0,limn→∞γ-3nψ(γnx,γny,γnz)=0,limn→∞γ-2nψ(γnx,γny,z)=0
for all x,y,z,x1,…,xp,y1,…,yd∈A where γ=(p+2d)/2. Suppose that f:A→A is a mapping satisfying
∥Cμf(x1,…,xp,y1,…,yd)∥A≤φ(x1,…,xp,y1,…,yd),∥Df(x,y,z)∥A≤ψ(x,y,z)
for all μ∈𝕋1 and all x,y,z,x1,…,xp,y1,…,yd∈A. Then the mapping f:A→A is a C*-ternary derivation.

Proof.

Let us assume μ=1 and x1=⋯=xp=y1=⋯=yd=x in (3.10). Then we get
∥2f(p+2d2x)-(p+2d)f(x)∥A≤φ(x,…,x)
for all x∈A. If we replace x in (3.12) by γnx and divide both sides of (3.12) to γn+1, then we get
∥1γn+1f(γn+1x)-1γnf(γnx)∥A≤12γn+1φ(γnx,…,γnx)
for all x∈A and all integers n≥0. Hence
∥1γn+1f(γn+1x)-1γmf(γmx)∥A≤12γ∑i=mn1γiφ(γix,…,γix)
for all x∈A and all integers n≥m≥0. From this it follows that the sequence {(1/γn)f(γnx)} is Cauchy for all x∈A. Since A is complete, the sequence {(1/γn)f(γnx)} converges. Thus we can define the mapping δ:A→A by
δ(x):=limn→∞1γnf(γnx)
for all x∈A. Moreover, letting m=0 and passing the limit n→∞ in (3.14), we get
∥δ(x)-f(x)∥A≤12γφ̃(x)
for all x∈A. It follows from (3.8) and (3.10) that
∥Cμδ(x1,…,xp,y1,…,yd)∥A=limn→∞1γn∥Cμf(γnx1,…,γnxp,γny1,…,γnyd)∥A≤limn→∞1γnφ(γnx1,…,γnxp,γny1,…,γnyd)=0
for all μ∈𝕋1 and all x,y,z,x1,…,xp,y1,…,yd∈A. Hence
2δ(∑j=1pμxj2+∑j=1dμyj)=∑j=1pμδ(xj)+2∑j=1dμδ(yj)
for all μ∈𝕋1 and all x1,…,xp,y1,…,yd∈A. So δ(λx+μy)=λδ(x)+μδ(y) for all λ,μ∈𝕋1 and all x,y∈A. Therefore, by Lemma 2.1 the mapping δ:A→A is ℂ-linear.

It follows from (3.9) and (3.11) that
∥Dδ(x,y,z)∥A=limn→∞1γ3n∥Df(γnx,γny,γnz)∥A≤limn→∞1γ3nψ(γnx,γny,γnz)=0
for all x,y,z∈A. Hence
δ([x,y,z])=[δ(x),y,z]+[x,δ(y),z]+[x,y,δ(z)]
for all x,y,z∈A. So the mapping δ:A→A is a C*-ternary derivation.

It follows from (3.9) and (3.11)
∥δ[x,y,z]-[δ(x),y,z]-[x,δ(y),z]-[x,y,f(z)]∥A=limn→∞1γ2n∥f[γnx,γny,z]-[f(γnx),γny,z]-[γnx,f(γny),z]-[γnx,γny,f(z)]∥A≤limn→∞1γ2nψ(γnx,γny,z)=0
for all x,y,z∈A. Thus
δ[x,y,z]=[δ(x),y,z]+[x,δ(y),z]+[x,y,f(z)]
for all x,y,z∈A. Hence we get from (3.20) and (3.22) that
[x,y,δ(z)]=[x,y,f(z)]
for all x,y,z∈A. Letting x=y=f(z)-δ(z) in (3.23), we get
∥f(z)-δ(z)∥A3=∥[f(z)-δ(z),f(z)-δ(z),f(z)-δ(z)]∥A=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 3.4.

Let r<1,s<2, and θ be nonnegative real numbers, and let f:A→A be a mapping satisfying (2.8) and
∥Df(x,y,z)∥A≤θ(∥x∥As+∥y∥As+∥z∥As)
for all x,y,z∈A. Then the mapping f:A→A is a C*-ternary derivation.

Proof.

Define
φ(x1,…,xp,y1,…,yd)=θ(∑j=1p∥xj∥Ar+∑j=1d∥yj∥Ar),ψ(x,y,z)=θ(∥x∥As+∥y∥As+∥z∥As)
for all x,y,z,x1,…,xp,y1,…,yd∈A, and apply Theorem 3.3.

Corollary 3.5.

Let r,s, and θ be nonnegative real numbers such that s,r(p+d)<1, and let f:A→A be a mapping satisfying (2.23) and
∥Df(x,y,z)∥A≤θ∥x∥As∥y∥As∥z∥As
for all x,y,z∈A. Then the mapping f:A→A is a C*-ternary derivation.

Proof.

Define
φ(x1,…,xp,y1,…,yd)=θ∏j=1p∥xj∥Ar∏j=1d∥yj∥Ar,ψ(x,y,z)=θ∥x∥As∥y∥As∥z∥As
for all x,y,z,x1,…,xp,y1,…,yd∈A, and apply Theorem 3.3.

Theorem 3.6.

Let φ:Ap+d→[0,∞) and ψ:A3→[0,∞) be functions such that
φ̃(x):=∑n=1∞γnφ(xγn,…,xγn)<∞,limn→∞γnφ(x1γn,…,xpγn,y1γn,…,ydγn)=0,limn→∞γ3nψ(xγn,yγn,zγn)=0,limn→∞γ2nψ(xγn,yγn,z)=0
for all x,y,z,x1,…,xp,y1,…,yd∈A where γ=(p+2d)/2. Suppose that f:A→A is a mapping satisfying (3.10) and (3.11). Then the mapping f:A→A is a C*-ternary derivation.

Proof.

If we replace x in (3.12) by x/γn+1 and multiply both sides of (3.12) by γn, then we get
∥γn+1f(xγn+1)-γnf(xγn)∥A≤γn2φ(xγn+1,…,xγn+1)
for all x∈A and all integers n≥0. Hence
∥γn+1f(xγn+1)-γmf(xγm)∥A≤12γ∑i=m+1n+1γiφ(xγi,…,xγi)
for all x∈A and all integers n≥m≥0. From this it follows that the sequence {γnf(x/γn)} is Cauchy for all x∈A. Since A is complete, the sequence {γnf(x/γn)} converges. Thus we can define the mapping δ:A→A by
δ(x):=limn→∞γnf(xγn)
for all x∈A. The rest of the proof is similar to the proof of Theorem 3.3, and we omit it.

Corollary.

Let r,s, and θ be nonnegative real numbers such that s,r(p+d)>1, and let f:A→A be a mapping satisfying (2.23) and (3.27). Then the mapping f:A→A is a C*-ternary derivation.

Acknowledgment

The authors would like to thank the referees for their useful comments and suggestions. The corresponding author was supported by Daejin University Research Grant in 2009.

UlamS. M.HyersD. H.On the stability of the linear functional equationAokiT.On the stability of the linear transformation in Banach spacesRassiasTh. M.On the stability of the linear mapping in Banach spacesGajdaZ.On stability of additive mappingsRassiasTh. M.ŠemrlP.On the behavior of mappings which do not satisfy Hyers-Ulam stabilityGăvruţaP.A generalization of the Hyers-Ulam-Rassias stability of approximately additive mappingsJungS.-M.On the Hyers-Ulam-Rassias stability of approximately additive mappingsCzerwikS.HyersD. H.IsacG.RassiasTh. M.RassiasJ. M.On approximation of approximately linear mappings by linear mappingsRassiasJ. M.On approximation of approximately linear mappings by linear mappingsRassiasJ. M.Solution of a problem of UlamGăvruţaP.An answer to a question of John M. Rassias concerning the stability of Cauchy equationBouikhaleneB.ElqorachiE.Ulam-Găvruta-Rassias stability of the Pexider functional equationNakmahachalasintP.On the generalized Ulam-Găvruta-Rassias stability of mixed-type linear and Euler-Lagrange-Rassias functional equationsNakmahachalasintP.Hyers-Ulam-Rassias and Ulam-Găvruta-Rassias stabilities of an additive functional equation in several variablesBaakC.MoslehianM. S.On the stability of J∗-homomorphismsJunK.-W.KimH.-M.RassiasJ. M.Extended Hyers-Ulam stability for Cauchy-Jensen mappingsKimH.-M.JunK.-W.RassiasJ. M.Extended stability problem for alternative Cauchy-Jensen mappingsNajatiA.Hyers-Ulam stability of an n-Apollonius type quadratic mappingNajatiA.Stability of homomorphisms on JB∗-triples associated to a Cauchy-Jensen type functional equationNajatiA.RanjbariA.On homomorphisms between C∗-ternary algebrasNajatiA.On the stability of a quartic functional equationNajatiA.MoghimiM. B.Stability of a functional equation deriving from quadratic and additive functions in quasi-Banach spacesNajatiA.ParkC.Hyers-Ulam-Rassias stability of homomorphisms in quasi-Banach algebras associated to the Pexiderized Cauchy functional equationNajatiA.ParkC.The Pexiderized Apollonius-Jensen type additive mapping and isomorphisms between C∗-algebrasParkC.-G.Lie ∗-homomorphisms between Lie C∗-algebras and Lie ∗-derivations on Lie C∗-algebrasParkC.-G.Homomorphisms between Lie JC∗-algebras and Cauchy-Rassias stability of Lie JC∗-algebra derivationsParkC.-G.Homomorphisms between Poisson JC∗-algebrasParkC.Isomorphisms between C∗-ternary algebrasParkC.-G.Hyers-Ulam-Rassias stability of a generalized Euler-Lagrange type additive mapping and isomorphisms between C∗-algebrasParkC.NajatiA.Homomorphisms and derivations in C∗-algebrasRassiasJ. M.On a new approximation of approximately linear mappings by linear mappingsRassiasJ. M.On the stability of the Euler-Lagrange functional equationRassiasJ. M.Solution of a Cauchy-Jensen stability Ulam type problemRassiasJ. M.KimH.-M.Approximate homomorphisms and derivations between C∗-ternary algebrasRassiasJ. M.Refined Hyers-Ulam approximation of approximately Jensen type mappingsKimH.-M.RassiasJ. M.Generalization of Ulam stability problem for Euler-Lagrange quadratic mappingsRassiasTh. M.The problem of S. M. Ulam for approximately multiplicative mappingsRassiasTh. M.On the stability of functional equations in Banach spacesRassiasTh. M.On the stability of functional equations and a problem of UlamRassiasTh. M.SkofF.Proprietà locali e approssimazione di operatoriAmyariM.MoslehianM. S.Approximate homomorphisms of ternary semigroupsBazunovaN.BorowiecA.KernerR.Universal differential calculus on ternary algebrasZettlH.A characterization of ternary rings of operatorsMoslehianM. S.Almost derivations on C∗-ternary ringsAbramovV.KernerR.Le RoyB.Hypersymmetry: a ℤ3-graded generalization of supersymmetryKernerR.Ternary algebraic structures and their applications in physicspreprintVainermanL.KernerR.On special classes of n-algebras