Let 3≤n, and 3≤k≤n be positive integers. Let A be an algebra and let X be an A-bimodule. A ℂ-linear
mapping d:A→X is called a generalized (n,k)-derivation if
there exists a (k−1)-derivation δ:A→X such that
d(a1a2⋯an)=δ(a1)a2⋯an+a1δ(a2)a3⋯an+⋯+a1a2⋯ak−2δ(ak−1)ak⋯an+a1a2⋯ak−1d(ak)ak+1⋯an+a1a2⋯akd(ak+1)ak+2⋯an+a1a2⋯ak+1d(ak+2)ak+3⋯an+⋯+a1⋯an−1d(an) for all a1,a2,…,an∈A. The main purpose of this paper is
to prove the generalized Hyers-Ulam stability of the generalized
(n,k)-derivations.

1. Introduction

It seems that the stability problem of functional equations introduced by Ulam [1]. Let (G1,·)be a group and let (G2,*)be a metric group with the metric d(·,·).Given ϵ>0,does there exist a δ>0,such that if a mapping h:G1→G2satisfies the inequality d(h(x·y),h(x)*h(y))<δ,for all x,y∈G1,then there exists a homomorphism H:G1→G2with d(h(x),H(x))<ϵ,for all x∈G1? In other words, under what condition does there exist a homomorphism near an approximate homomorphism? The concept of stability for functional equations arises when one replaces the functional equation by an inequality which acts as a perturbation of the equation. In 1941, Hyers [2] gave the first affirmative answer to the question of Ulam for Banach spaces E and E′. Let f:E→E′ be a mapping between Banach spaces such that
∥f(x+y)-f(x)-f(y)∥≤δ
for all x,y∈E, and for some δ>0. Then there exists a unique additive mapping T:E→E′ such that
∥f(x)-T(x)∥≤δ
for all x∈E. By the seminal paper of Th. M. Rassias [3] and work of Gadja [4], if one assumes that E and E′ are real normed spaces with E′ complete, f:E→E′ is a mapping such that for each fixed x∈E the mapping t↣f(tx) is continuous in real t for each fixed x in E, and that there exists δ≥0 and p≠1 such that
∥f(x+y)-f(x)-f(y)∥≤δ(∥x∥p+∥y∥p)
for all x,y∈E. Then there exists a unique linear map T:E→E′ such that
∥f(x)-T(x)∥≤2δ∥x∥p|2p-2|
for all x∈E.

On the other hand J. M. Rassias [5] generalized the Hyers stability result by presenting a weaker condition controlled by a product of different powers of norms. If it is assumed that there exist constants Θ≥0 and p1,p2∈ℝ such that p=p1+p2≠1, and f:E→E′ is a map from a norm space E into a Banach space E′ such that the inequality
∥f(x+y)-f(x)-f(y)∥≤Θ∥x∥p1∥y∥p2
for all x,y∈E, then there exists a unique additive mapping T:E→E′ such that
∥f(x)-T(x)∥≤Θ2-2p∥x∥p
for all x∈E. If in addition for every x∈E,f(tx) is continuous in real t for each fixed x, then T is linear.

Suppose (G,+) is an abelian group, E is a Banach space, and that the so-called admissible control function φ:G×G→ℝ satisfies
φ̃(x,y):=2-1∑n=0∞2-nφ(2nx,2ny)<∞
for all x,y∈G. If f:G→E is a mapping with
∥f(x+y)-f(x)-f(y)∥≤φ(x,y)
for all x,y∈G, then there exists a unique mapping T:G→E such that T(x+y)=T(x)+T(y) and ∥f(x)-T(x)∥≤φ̃(x,x), for all x,y∈G (see [6]).

Generalized derivations first appeared in the context of operator algebras [7]. Later, these were introduced in the framework of pure algebra [8, 9].

Definition 1.1.

Let A be an algebra and let X be an A-bimodule. A linear mapping d:A→X is called

(i) derivation if d(ab)=d(a)b+ad(b), for all a,b∈A;

(ii) generalized derivation if there exists a derivation (in the usual sense) δ:A→X such that d(ab)=ad(b)+δ(a)b, for all a,b∈A.

Every right multiplier (i.e., a linear map h on A satisfying h(ab)=ah(b), for all a,b∈A) is a generalized derivation.

Definition 1.2.

Let n≥2,k≥3 be positive integers. Let A be an algebra and let X be an A-bimodule. A ℂ-linear mapping d:A→X is called

(i) n-derivation if
d(a1a2⋯an)=d(a1)a2⋯an+a1d(a2)a3⋯an+⋯+a1⋯an-1d(an)
for all a1,a2,…,an∈A;

(ii) generalized (n,k)-derivation if there exists a (k-1)-derivation δ:A→X such that
d(a1a2⋯an)=δ(a1)a2⋯an+a1δ(a2)a3⋯an+⋯+a1a2⋯ak-2δ(ak-1)ak⋯an+a1a2⋯ak-1d(ak)ak+1⋯an+a1a2⋯akd(ak+1)ak+2⋯an+a1a2⋯ak+1d(ak+2)ak+3⋯an+⋯+a1⋯an-1d(an)
for all a1,a2,…,an∈A.

By Definition 1.2, we see that a generalized (2,3)-derivation is a generalized derivation.

For instance, let 𝒜 be a Banach algebra. Then we take
𝒯=[0𝒜𝒜𝒜00𝒜𝒜000𝒜0000],𝒯 is an algebra equipped with the usual matrix-like operations. It is easy to check that every linear map from 𝒜 into 𝒜 is a (5,3)-derivation, but there are linear maps on 𝒯 which are not generalized derivations.

The so-called approximate derivations were investigated by Jun and Park [10]. Recently, the stability of derivations have been investigated by some authors; see [10–13] and references therein. Moslehian [14] investigated the generalized Hyers-Ulam stability of generalized derivations from a unital normed algebra A to a unit linked Banach A-bimodule (see also [15]).

In this paper, we investigate the generalized Hyers-Ulam stability of the generalized (n,k)-derivations.

2. Main Result

In this section, we investigate the generalized Hyers-Ulam stability of the generalized (n,k)-derivations from a unital Banach algebra A into a unit linked Banach A-bimodule. Throughout this section, assume that A is a unital Banach algebra, X is unit linked Banach A-bimodule, and suppose that 3≤n, and 3≤k≤n.

We need the following lemma in the main results of the present paper.

Lemma 2.1 (see [<xref ref-type="bibr" rid="B14">16</xref>]).

Let U,V be linear spaces and let f:U→V be an additive mapping such that f(λx)=λf(x), for all x∈U and all λ∈𝕋1:={λ∈ℂ;|λ|=1}. Then the mapping f is ℂ-linear.

Now we prove the generalized Hyers-Ulam stability of generalized (n,k)-derivations.

Theorem 2.2.

Suppose f:A→X is a mapping with f(0)=0 for which there exists a map g:A→X with g(0)=0 and a function φ:An+2→ℝ+ such that
max{∥f(λa+λb+a1a2⋯an)-λf(a)-λf(b)-a1⋯ak-1f(ak)ak+1⋯an-a1⋯akf(ak+1)ak+2⋯an-⋯-a1⋯an-1f(an)-g(a1)a2⋯an-a1g(a2)a3⋯an-⋯-a1a2⋯ak-2g(ak-1)ak⋯an∥,∥g(λa+λb+a1a2⋯an)-λg(a)-λg(b)-g(a1)a2⋯an-a1g(a2)a3⋯an-⋯-a1⋯ak-2g(ak-1)ak⋯an∥}≤φ(a,b,a1,a2,…,an),φ̃(a,b,a1,a2,…,an):=2-1∑i=0∞2-iφ(2ia,2ib,2ia1,…,2ian)<∞
for all a,b,a1,a2,…,an∈A and all λ∈𝕋1. Then there exists a unique generalized (n,k)-derivation d:A→X such that
∥f(a)-d(a)∥≤φ̃(a,a,0,0,0,…,0)
for all a∈A.

Proof.

By (2.1) we have
∥f(λa+λb+a1a2⋯an)-λf(a)-λf(b)-a1⋯ak-1f(ak)ak+1⋯an-a1⋯akf(ak+1)ak+2⋯an-⋯-a1⋯an-1f(an)-g(a1)a2⋯an-a1g(a2)a3⋯an-⋯-a1a2⋯ak-2g(ak-1)ak⋯an∥≤φ(a,b,a1,a2,…,an),∥g(λa+λb+a1a2⋯an)-λg(a)-λg(b)-g(a1)a2⋯an-a1g(a2)a3⋯an-⋯-a1⋯ak-2g(ak-1)ak⋯an∥≤φ(a,b,a1,a2,…,an)
for all a,b,a1,a2,…,an∈A and all λ∈𝕋1. Setting a1,a2,…,an=0 and λ=1 in (2.4), we have
∥f(a+b)-f(a)-f(b)∥≤φ(a,b,0,0,…,0)
for all a,b∈A. One can use induction on n to show that
∥2-mf(2ma)-f(a)∥≤2-1∑i=0m-12-iφ(2ia,2ia,0,0,…,0)
for all n∈ℕ and all a∈A, and that
∥2-mf(2ma)-2-lf(2la)∥≤2-1∑i=lm-12-iφ(2ia,2ia,0,0…,0)
for all m>l and all a∈A. It follows from the convergence (2.2) that the sequence 2-mf(2ma) is Cauchy. Due to the completeness of X, this sequence is convergent. Set
d(a):=limm→∞2-mf(2ma).
Putting a1,a2,…,an=0 and replacing a,b by 2ma,2mb, respectively, in (2.4), we get
∥2-mf(2m(λa+λb))-2-mλf(2ma)-2-mλf(2mb)∥≤2-mφ(2ma,2mb,0,0,…,0)
for all a,b∈A and all λ∈𝕋1. Taking the limit as m→∞ we obtain
d(λa+λb)=λd(a)+λd(b)
for all a,b∈A and all λ∈𝕋1. So by Lemma 2.1, the mapping d is ℂ-linear.

Using (2.5), (2.2), and the above technique, we get
δ(a):=limm→∞2-mg(2ma),δ(λa+λb)=λδ(a)+δ(b)
for all a,b∈A and all λ∈𝕋1. Hence by Lemma 2.1, δ is ℂ-linear. Moreover, it follows from (2.7) and (2.9) that ∥f(a)-d(a)∥≤φ̃(a,a,0,0,…,0), for all a∈A. It is known that the additive mapping d satisfying (2.3) is unique [17]. Putting λ=1,a=b=0, and replacing a1,a2,…,an by 2ma1,2ma2,…,2man, respectively, in (2.4), we get
∥f(2nma1a2⋯an)-2(n-1)ma1⋯ak-1f(2mak)ak+1⋯an-2(n-1)ma1⋯akf(2mak+1)ak+2⋯an-⋯-2(n-1)ma1⋯an-1f(2man)-2(n-1)mg(2ma1)a2⋯an-2(n-1)ma1g(2ma2)a3⋯an-⋯-2(n-1)ma1a2⋯ak-2g(2mak-1)ak⋯an∥≤φ(0,0,2ma1,2ma2,…,2man),
whence
∥2-nmf(2nma1a2⋯an)-2-ma1⋯ak-1f(2mak)ak+1⋯an-2-ma1⋯akf(2mak+1)ak+2⋯an-⋯-2-ma1⋯an-1f(2man)-2-mg(2ma1)a2⋯an-2-ma1g(2ma2)a3⋯an-⋯-2-ma1a2⋯ak-2g(2mak-1)ak⋯an∥≤2-nmφ(0,0,2ma1,2ma2,…,2man)
for all a1,a2,…,an∈A. By (2.9), limm→∞2-nmf(2nma)=d(a) and by the convergence of series (2.2), limm→∞2-nmφ(0,0,2ma1,2ma2,…,2man)=0. Let m tend to ∞ in (2.14). Then
d(a1a2⋯an)=a1⋯ak-1d(ak)ak+1⋯an+a1⋯akd(ak+1)ak+2⋯an+⋯+a1⋯an-1d(an)+δ(a1)a2⋯an+a1δ(a2)a3⋯an+⋯+a1a2⋯ak-2δ(ak-1)ak⋯an
for all a1,a2,…,an∈A.

Next we claim that δ is a (k-1)-derivation. Putting λ=1,a=b=0, and replacing a1,a2,…,an by 2ma1,2ma2,…,2man, respectively, in (2.5), we get
∥g(2nma1a2⋯an)-2(n-1)mg(2ma1)a2⋯an-2(n-1)ma1g(2ma2)a3⋯an-2(n-1)ma1a2g(2ma3)a4⋯an-⋯-2(n-1)ma1⋯ak-2g(2mak-1)ak⋯an∥≤φ(0,0,2ma1,2ma2,…,2man),
whence
∥2-nmg(2nma1a2⋯an)-2-mg(2ma1)a2⋯an-2-ma1g(2ma2)a3⋯an-2-ma1a2g(2ma3)a4⋯an-⋯-2-ma1⋯ak-2g(2mak-1)ak⋯an∥≤2-nmφ(0,0,2ma1,2ma2,…,2man)
for all a1,a2,…,an∈A. Let m tends to ∞ in (2.17). Then
δ(a1a2⋯ak-1akak+1⋯an)=δ(a1)a2⋯an+a1δ(a2)a3⋯an+a1a2δ(a3)a4⋯an+⋯+a1a2⋯ak-2δ(ak-1)ak⋯an
for all a1,a2,…,an∈A.

Setting ak=ak+1=⋯=an=1 in (2.18). Hence the mapping δ is (k-1)-derivation.

Corollary 2.3.

Suppose f:A→X is a mapping with f(0)=0 for which there exists constant θ≥0,p<1 and a map g:A→X with g(0)=0 such that
max{∥f(λa+λb+a1a2⋯an)-λf(a)-λf(b)-a1⋯ak-1f(ak)ak+1⋯an-a1⋯akf(ak+1)ak+2⋯an-⋯-a1⋯an-1f(an)-g(a1)a2⋯an-a1g(a2)a3⋯an-⋯-a1a2⋯ak-2g(ak-1)ak⋯an∥,∥g(λa+λb+a1a2⋯an)-λg(a)-λg(b)-g(a1)a2⋯an-a1g(a2)a3⋯an-⋯-a1⋯ak-2g(ak-1)ak⋯an∥}≤θ(∥a∥p+∥b∥p+∑i=1n∥ai∥p)
for all a1,a2,…,an∈A and all λ∈𝕋. Then there exists a unique generalized (n,k)-derivation d:A→X such that
∥f(a)-d(a)∥≤θ∥a∥p1-2p-1
for all a∈A.

Proof.

Put φ(a,b,a1,a2,…,an)=θ(∥a∥p+∥b∥p+∑i=1n∥ai∥p) in Theorem 2.2.

Corollary 2.4.

Suppose f:A→X is a mapping with f(0)=0 for which there exists constant θ≥0 and a map g:A→X with g(0)=0 such that
max{∥f(λa+λb+a1a2⋯an)-λf(a)-λf(b)-a1⋯ak-1f(ak)ak+1⋯an-a1⋯akf(ak+1)ak+2⋯an-⋯-a1⋯an-1f(an)-g(a1)a2⋯an-a1g(a2)a3⋯an-⋯-a1a2⋯ak-2g(ak-1)ak⋯an∥,∥g(λa+λb+a1a2⋯an)-λg(a)-λg(b)-g(a1)a2⋯an-a1g(a2)a3⋯an-⋯-a1⋯ak-2g(ak-1)ak⋯an∥}≤θ
for all a1,a2,…,an∈A. Then there exists a unique generalized (n,k)-derivation d:A→X such that
∥f(a)-d(a)∥≤θ
for all a∈A.

Proof.

Letting p=0 in Corollary 2.3, we obtain the above result of Corollary 2.4.

UlamS. M.HyersD. H.On the stability of the linear functional equationRassiasTh. M.On the stability of the linear mapping in Banach spacesGajdaZ.On stability of additive mappingsRassiasJ. M.On approximation of approximately linear mappings by linear mappingsGăvruţaP.A generalization of the Hyers-Ulam-Rassias stability of approximately additive mappingsMathieuM.WeiF.XiaoZ.Generalized Jordan derivations on semiprime ringsHvalaB.Generalized derivations in ringsJunK.-W.ParkD.-W.Almost derivations on the Banach algebra Cn[0,1]AmyariM.BaakC.MoslehianM. S.Nearly ternary derivationsBadoraR.On approximate derivationsParkC.-G.Linear derivations on Banach algebrasMoslehianM. S.Hyers-Ulam-Rassias stability of generalized derivationsGordjiM. E.GhobadipourN.Nearly generalized Jordan derivationsto appear in Mathematica SlovacaParkC.-G.Homomorphisms between Poisson JC∗-algebrasBaakC.MoslehianM. S.On the stability of J∗-homomorphisms