# Generalized Hyers-Ulam Stability of  Generalized(N,K)-Derivations

Authors: M. Eshaghi Gordji; J. M. Rassias; N. Ghobadipour
Journal: Abstract and Applied Analysis
 (2009)

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## Abstract

Let3≤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.


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## Body

## 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 metricd(·,·).Givenϵ>0,does there exist aδ>0,such that if a mappingh:G1→G2satisfies the inequalityd(h(x·y),h(x)*h(y))<δ,for allx,y∈G1,then there exists a homomorphismH:G1→G2withd(h(x),H(x))<ϵ,for allx∈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
(1.1)∥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
(1.2)∥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
(1.3)∥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
(1.4)∥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
(1.5)∥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
(1.6)∥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
(1.7)φ̃(x,y):=2-1∑n=0∞2-nφ(2nx,2ny)<∞
for all x,y∈G. If f:G→E is a mapping with
(1.8)∥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.
LetA be an algebra and let X be an A-bimodule. A linear mapping d:A→X is called
(i) derivation ifd(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 maph on A satisfying h(ab)=ah(b), for all a,b∈A) is a generalized derivation.Definition 1.2.
Letn≥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
(1.9)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
(1.10)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 Definition1.2, we see that a generalized (2,3)-derivation is a generalized derivation.For instance, let𝒜 be a Banach algebra. Then we take
(1.11)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 [16]).
LetU,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.
Supposef: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
(2.1)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),(2.2)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
(2.3)∥f(a)-d(a)∥≤φ̃(a,a,0,0,0,…,0)
for all a∈A.Proof.
By (2.1) we have
(2.4)∥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),(2.5)∥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
(2.6)∥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
(2.7)∥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
(2.8)∥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
(2.9)d(a):=limm→∞2-mf(2ma).
Putting a1,a2,…,an=0 and replacing a,b by 2ma,2mb, respectively, in (2.4), we get
(2.10)∥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
(2.11)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
(2.12)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
(2.13)∥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
(2.14)∥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
(2.15)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 thatδ 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
(2.16)∥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
(2.17)∥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
(2.18)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.
Settingak=ak+1=⋯=an=1 in (2.18). Hence the mapping δ is (k-1)-derivation.Corollary 2.3.
Supposef: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
(2.19)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
(2.20)∥f(a)-d(a)∥≤θ∥a∥p1-2p-1
for all a∈A.Proof.
Putφ(a,b,a1,a2,…,an)=θ(∥a∥p+∥b∥p+∑i=1n∥ai∥p) in Theorem 2.2.Corollary 2.4.
Supposef: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
(2.21)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
(2.22)∥f(a)-d(a)∥≤θ
for all a∈A.Proof.
Lettingp=0 in Corollary 2.3, we obtain the above result of Corollary 2.4.



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*Source: 437931-2009-07-06.xml*