AAAAbstract and Applied Analysis1687-04091085-3375Hindawi Publishing Corporation43793110.1155/2009/437931437931Research ArticleGeneralized Hyers-Ulam Stability of Generalized (N,K)-DerivationsEshaghi GordjiM.1RassiasJ. M.2GhobadipourN.1CalvertBruce1Department of MathematicsSemnan UniversityP. O. Box 35195-363SemnanIransemnan.ac.ir2Section of Mathematics and InformaticsPedagogical DepartmentNational and Capodistrian University of Athens4, Agamemnonos St.Aghia Paraskevi15342 AthensGreeceuoa.gr20090906200920090502200907042009120520092009Copyright © 2009This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Let 3n, and 3kn be positive integers. Let A be an algebra and let X be an A-bimodule. A -linear mapping d:AX is called a generalized (n,k)-derivation if there exists a (k1)-derivation δ:AX such that d(a1a2an)=δ(a1)a2an+a1δ(a2)a3an++a1a2ak2δ(ak1)akan+a1a2ak1d(ak)ak+1an+a1a2akd(ak+1)ak+2an+a1a2ak+1d(ak+2)ak+3an++a1an1d(an) for all a1,a2,,anA. 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 . 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:G1G2  satisfies the inequality   d(h(x·y),h(x)*h(y))<δ,  for all   x,yG1,  then there exists a homomorphism   H:G1G2  with   d(h(x),H(x))<ϵ,  for all   xG1? 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  gave the first affirmative answer to the question of Ulam for Banach spaces E and E. Let f:EE be a mapping between Banach spaces such that f(x+y)-f(x)-f(y)δ for all x,yE, and for some δ>0. Then there exists a unique additive mapping T:EE such that f(x)-T(x)δ for all xE. By the seminal paper of Th. M. Rassias  and work of Gadja , if one assumes that E and E are real normed spaces with E complete, f:EE is a mapping such that for each fixed xE the mapping tf(tx) is continuous in real t for each fixed x in E, and that there exists δ0 and p1 such that f(x+y)-f(x)-f(y)δ(xp+yp) for all x,yE. Then there exists a unique linear map T:EE such that f(x)-T(x)2δxp|2p-2| for all xE.

On the other hand J. M. Rassias  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 Θ0 and p1,p2 such that p=p1+p21, and f:EE is a map from a norm space E into a Banach space E such that the inequality f(x+y)-f(x)-f(y)Θxp1yp2 for all x,yE, then there exists a unique additive mapping T:EE such that f(x)-T(x)Θ2-2pxp for all xE. If in addition for every xE,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 satisfies φ̃(x,y):=2-1n=02-nφ(2nx,2ny)< for all x,yG. If f:GE is a mapping with f(x+y)-f(x)-f(y)φ(x,y) for all x,yG, then there exists a unique mapping T:GE such that T(x+y)=T(x)+T(y) and f(x)-T(x)φ̃(x,x), for all x,yG (see ).

Generalized derivations first appeared in the context of operator algebras . 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:AX is called

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

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

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

Definition 1.2.

Let n2,k3 be positive integers. Let A be an algebra and let X be an A-bimodule. A -linear mapping d:AX is called

(i) n-derivation if d(a1a2an)=d(a1)a2an+a1d(a2)a3an++a1an-1d(an) for all a1,a2,,anA;

(ii) generalized (n,k)-derivation if there exists a (k-1)-derivation δ:AX such that d(a1a2an)=δ(a1)a2an+a1δ(a2)a3an++a1a2ak-2δ(ak-1)akan+a1a2ak-1d(ak)ak+1an+a1a2akd(ak+1)ak+2an+a1a2ak+1d(ak+2)ak+3an++a1an-1d(an) for all a1,a2,,anA.

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 . Recently, the stability of derivations have been investigated by some authors; see  and references therein. Moslehian  investigated the generalized Hyers-Ulam stability of generalized derivations from a unital normed algebra A to a unit linked Banach A-bimodule (see also ).

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 3n, and 3kn.

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:UV be an additive mapping such that f(λx)=λf(x), for all xU and all λ𝕋1:={λ;|λ|=1}. Then the mapping f is -linear.

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

Theorem 2.2.

Suppose f:AX is a mapping with f(0)=0 for which there exists a map g:AX with g(0)=0 and a function φ:An+2+ such that max{f(λa+λb+a1a2an)-λf(a)-λf(b)-a1ak-1f(ak)ak+1an-a1akf(ak+1)ak+2an--a1an-1f(an)-g(a1)a2an-a1g(a2)a3an--a1a2ak-2g(ak-1)akan,g(λa+λb+a1a2an)-λg(a)-λg(b)-g(a1)a2an-a1g(a2)a3an--a1ak-2g(ak-1)akan}φ(a,b,a1,a2,,an),φ̃(a,b,a1,a2,,an):=2-1i=02-iφ(2ia,2ib,2ia1,,2ian)< for all a,b,a1,a2,,anA and all λ𝕋1. Then there exists a unique generalized (n,k)-derivation d:AX such that f(a)-d(a)φ̃(a,a,0,0,0,,0) for all aA.

Proof.

By (2.1) we have f(λa+λb+a1a2an)-λf(a)-λf(b)-a1ak-1f(ak)ak+1an-a1akf(ak+1)ak+2an--a1an-1f(an)-g(a1)a2an-a1g(a2)a3an--a1a2ak-2g(ak-1)akanφ(a,b,a1,a2,,an),g(λa+λb+a1a2an)-λg(a)-λg(b)-g(a1)a2an-a1g(a2)a3an--a1ak-2g(ak-1)akanφ(a,b,a1,a2,,an) for all a,b,a1,a2,,anA and all λ𝕋1. 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,bA. One can use induction on n to show that 2-mf(2ma)-f(a)2-1i=0m-12-iφ(2ia,2ia,0,0,,0) for all n and all aA, and that 2-mf(2ma)-2-lf(2la)2-1i=lm-12-iφ(2ia,2ia,0,0,0) for all m>l and all aA. 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):=limm2-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,bA and all λ𝕋1. Taking the limit as m we obtain d(λa+λb)=λd(a)+λd(b) for all a,bA and all λ𝕋1. So by Lemma 2.1, the mapping d is -linear.

Using (2.5), (2.2), and the above technique, we get δ(a):=limm2-mg(2ma),δ(λa+λb)=λδ(a)+δ(b) for all a,bA and all λ𝕋1. Hence by Lemma 2.1, δ is -linear. Moreover, it follows from (2.7) and (2.9) that f(a)-d(a)φ̃(a,a,0,0,,0), for all aA. It is known that the additive mapping d satisfying (2.3) is unique . Putting λ=1,a=b=0, and replacing a1,a2,,an by 2ma1,2ma2,,2man, respectively, in (2.4), we get f(2nma1a2an)-2(n-1)ma1ak-1f(2mak)ak+1an-2(n-1)ma1akf(2mak+1)ak+2an--2(n-1)ma1an-1f(2man)-2(n-1)mg(2ma1)a2an-2(n-1)ma1g(2ma2)a3an--2(n-1)ma1a2ak-2g(2mak-1)akanφ(0,0,2ma1,2ma2,,2man), whence 2-nmf(2nma1a2an)-2-ma1ak-1f(2mak)ak+1an-2-ma1akf(2mak+1)ak+2an--2-ma1an-1f(2man)-2-mg(2ma1)a2an-2-ma1g(2ma2)a3an--2-ma1a2ak-2g(2mak-1)akan2-nmφ(0,0,2ma1,2ma2,,2man) for all a1,a2,,anA. By (2.9), limm2-nmf(2nma)=d(a) and by the convergence of series (2.2), limm2-nmφ(0,0,2ma1,2ma2,,2man)=0. Let m tend to in (2.14). Then d(a1a2an)=a1ak-1d(ak)ak+1an+a1akd(ak+1)ak+2an++a1an-1d(an)+δ(a1)a2an+a1δ(a2)a3an++a1a2ak-2δ(ak-1)akan for all a1,a2,,anA.

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(2nma1a2an)-2(n-1)mg(2ma1)a2an-2(n-1)ma1g(2ma2)a3an-2(n-1)ma1a2g(2ma3)a4an--2(n-1)ma1ak-2g(2mak-1)akanφ(0,0,2ma1,2ma2,,2man), whence 2-nmg(2nma1a2an)-2-mg(2ma1)a2an-2-ma1g(2ma2)a3an-2-ma1a2g(2ma3)a4an--2-ma1ak-2g(2mak-1)akan2-nmφ(0,0,2ma1,2ma2,,2man) for all a1,a2,,anA. Let m tends to in (2.17). Then δ(a1a2ak-1akak+1an)=δ(a1)a2an+a1δ(a2)a3an+a1a2δ(a3)a4an++a1a2ak-2δ(ak-1)akan for all a1,a2,,anA.

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

Corollary 2.3.

Suppose f:AX is a mapping with f(0)=0 for which there exists constant θ0,p<1 and a map g:AX with g(0)=0 such that max{f(λa+λb+a1a2an)-λf(a)-λf(b)-a1ak-1f(ak)ak+1an-a1akf(ak+1)ak+2an--a1an-1f(an)-g(a1)a2an-a1g(a2)a3an--a1a2ak-2g(ak-1)akan,g(λa+λb+a1a2an)-λg(a)-λg(b)-g(a1)a2an-a1g(a2)a3an--a1ak-2g(ak-1)akan}θ(ap+bp+i=1naip) for all a1,a2,,anA and all λ𝕋. Then there exists a unique generalized (n,k)-derivation d:AX such that f(a)-d(a)θap1-2p-1 for all aA.

Proof.

Put φ(a,b,a1,a2,,an)=θ(ap+bp+i=1naip) in Theorem 2.2.

Corollary 2.4.

Suppose f:AX is a mapping with f(0)=0 for which there exists constant θ0 and a map g:AX with g(0)=0 such that max{f(λa+λb+a1a2an)-λf(a)-λf(b)-a1ak-1f(ak)ak+1an-a1akf(ak+1)ak+2an--a1an-1f(an)-g(a1)a2an-a1g(a2)a3an--a1a2ak-2g(ak-1)akan,g(λa+λb+a1a2an)-λg(a)-λg(b)-g(a1)a2an-a1g(a2)a3an--a1ak-2g(ak-1)akan}θ for all a1,a2,,anA. Then there exists a unique generalized (n,k)-derivation d:AX such that f(a)-d(a)θ for all aA.

Proof.

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

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