# Superstability of Generalized Higher Derivations

Authors: T. L. Shateri
Journal: Abstract and Applied Analysis
 (2011)

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## Abstract

We define the notion of an approximate generalized higher derivation and investigate the superstability of strong generalized higher derivations.


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## Body

## 1. Introduction and Preliminaries

The problem of stability of functional equations was originally raised by Ulam [1, 2] in 1940 concerning the stability of group homomorphisms. Hyers [3] gave an affirmative answer to the question of Ulam. Superstability, the result of Hyers, was generalized by Aoki [4], Bourgin [5], and Rassias [6]. During the last decades, several stability problems for various functional equations have been investigated by several authors. We refer the reader to the monographs [7–10].Let(E,∥·∥) be a complex normed space, and let k∈ℕ. We denote by Ek the linear space E⊕···⊕E consisting of k-tuples (x1,…,xk), where x1,…,xk∈E. The linear operations on Ek are defined coordinatewise. The zero element of either E or Ek is denoted by 0. We denote by ℕk the set {1,2,…,k} and by ℭk the group of permutations on k symbols.Definition 1.1.
A multi-norm on{Ek:k∈ℕ} is a sequence (∥·∥k)=(∥·∥k:k∈ℕ) such that ∥·∥k is a norm on Ek for each k∈ℕ, ∥x∥1=∥x∥ for each x∈E, and the following axioms are satisfied for each k∈ℕ with k≥2:(M1)
∥(xσ(1),…,xσ(k))∥k=∥(x1,…,xk)∥k(σ∈ℭk,x1,…,xk∈E);(M2)
∥(α1x1,…,αkxk)∥k≤(maxi∈ℕk|αi|)∥(x1,…,xk)∥k(α1,…,αk∈ℂ,x1,…,xk∈E);(M3)
∥(x1,…,xk-1,0)∥k=∥(x1,…,xk-1)∥k-1(x1,…,xk∈E);(M4)
∥(x1,…,xk-1,xk-1)∥k=∥(x1,…,xk-1)∥k-1(x1,…,xk∈E).In this case, we say that ((Ek,∥·∥)k∈ℕ) is a multi-normed space.We recall that the notion of multi-normed space was introduced by Dales and Polyakov in [11]. Motivations for the study of multi-normed spaces and many examples are given in [11].Suppose that((Ek,∥·∥k)k∈ℕ) is a multi-normed space, and k∈ℕ. The following properties are almost immediate consequences of the axioms:(i)
∥(x,…,x)∥k=∥x∥(x∈E);(ii)
maxi∈ℕk∥xi∥≤∥(x1,…,xk)∥k≤∑i=1k∥xi∥≤kmaxi∈ℕk∥xi∥(x1,…,xk∈E).It follows from(ii) that if (E,∥·∥) is a Banach space, then ((Ek,∥·∥k) is a Banach space for each k∈ℕ. In this case, ((Ek,∥·∥k)k∈ℕ) is a multi-Banach space.By (ii), we get the following lemma.Lemma 1.2.
Suppose thatk∈ℕ and (x1,…,xk)∈Ek. For each j∈ℕk, let {xnj}n∈ℕ be a sequence in E such that limn→∞xnj=xj. Then for each (y1,…,yk)∈Ek, one has
(1.1)lim⁡n→∞(xn1-y1,…,xnk-yk)=(x1-y1,…,xk-yk).Definition 1.3.
Let((Ek,∥·∥k)k∈ℕ) be a multi-normed space. A sequence {xn} in E is a multinull sequence if, for each ϵ>0, there exists n0∈ℕ such that
(1.2)sup⁡k∈N‖(xn,…,xn+k-1)‖k<ϵ(n≥n0).
Let x∈E. We say that limn→∞xn=x if {xn-x} is a multi-null sequence.Definition 1.4.
Let(𝒜,∥·∥) be a normed algebra such that ((𝒜k,∥·∥k)k∈ℕ) is said to be a multi-normed space. Then ((𝒜k,∥·∥k)k∈ℕ) is a multi-normed algebra if
(1.3)‖(x1y1,…,xkyk)‖k≤‖(x1,…,xk)‖k‖(y1,…,yk)‖k,
for k∈ℕ and x1,…,xk,y1,…,yk∈𝒜. Furthermore, if ((𝒜k,∥·∥k)k∈ℕ) is a multi-Banach space, then ((𝒜k,∥·∥k)k∈ℕ) is a multi-Banach algebra.Let𝒜 be an algebra and k0∈{0,1,…,}∪{∞}. A family {Dj}j=0k0 of linear mappings on 𝒜 is said to be a higher derivation of rank k0 if the functional equation Dj(xy)=∑i=0jDi(x)Dj-i(y) holds for all x,y∈𝒜, j=0,1,2,…,k0. If D0=id𝒜, where id𝒜 is the identity map on 𝒜, then D1 is a derivation and {Dj}j=0k0 is called a strong higher derivation. A standard example of a higher derivation of rank k0 is {Dj/j!}j=0k0, where D:𝒜→𝒜 is a derivation. The reader may find more information about higher derivations in [12–18].A family{fj}j=0k0 of linear mappings on 𝒜 is called a generalized strong higher derivation if f0=id𝒜, and there exists a higher derivation {Dj}j=0k0 such that(1.4)fj(xy)=xfj(y)+∑i=1jDi(x)fj-i(y),
for all x,y∈𝒜 and j=0,1,2,…,k0.The stability of derivations was studied by Park [19, 20]. In this paper, using some ideas from [21, 22], we investigate the superstability of generalized strong higher derivations in multi-Banach algebras.


## 2. Stability of Generalized Higher Derivations

In this section, we define the notion of an approximate generalized higher derivation. Then we show that an approximate generalized strong higher derivation on a multi-Banach algebra is a strong generalized higher derivation.Lemma 2.1.
Let(E,∥·∥) be a normed space, and let ((Fk,∥·∥k:k∈ℕ) be a multi-Banach space. Let k∈ℕ, ϵ>0, and f:E→F a mapping satisfying f(0)=0 and
(2.1)sup⁡k∈N‖(f(x1+y1t)-f(x1)t-f(y1)t,…,f(xk+ykt)-f(xk)t-f(yk)t)‖k≤ϵ,
for all integer t>1 and all x1,…,xk,y1,…,yk∈E, then there exists a unique additive mapping T:E→F such that
(2.2)‖(f(x1)-T(x1),…,f(xk)-T(xk))‖≤ϵ(x1,…,xk∈E).Proof.
Substitutingyi=0 for i=1,…,k and replacing x1,…,xk by tx1,…,txk in (2.1), we get
(2.3)sup⁡k∈N‖(f(x1)-f(tx1)t,…,f(xk)-f(txk)t)‖k≤ϵ.
Replacing x1,…,xk by tnx1,…,tnxk and dividing by tn in (2.3), it follows that
(2.4)sup⁡k∈N‖(f(tnx1)tn-f(tn+1x1)tn+1,…,f(tnxk)tn-f(tn+1xk)tn+1)‖k≤ϵtn.
An induction argument implies that
(2.5)sup⁡k∈N‖(f(tnx1)tnf(tnx1)tn-f(tn+mx1)tn+m,…,f(tnxk)tn-f(tn+mxk)tn+m)‖k≤ϵ(1tn+1+⋯1tn+m),
for x∈E and n,m∈ℕ. Hence, the sequence {f(tnx)/tn} is cauchy and hence is convergent in the complete multi-normed space F. Let T:E→F be the mapping defined by
(2.6)T(x):=lim⁡n→∞f(tnx)tn.
Hence, for each r>0, there exists N∈ℕ such that
(2.7)sup⁡k∈N‖(f(tnx1)tn-T(x),…,f(tn+k-1xk)tn+k-1-T(x))‖k≤r(n≥N).
In particular, the property (ii) of multi-norm implies that
(2.8)lim⁡n→∞‖f(tnx)tn-T(x)‖=0(x∈E).
We show that T is additive. Putting  n=0 in (2.5), we get
(2.9)sup⁡k∈N‖(f(x1)-f(tmx1)tm,…,f(xk)-f(tmxk)tm)‖k≤ϵ.
Taking the limit as m→∞, we obtain
(2.10)sup⁡k∈N‖(f(x1)-T(x1),…,f(xk)-T(xk))‖k≤ϵ.
Let x,y∈E, put x1=⋯=xk=tnx, y1=⋯=yk=tny in (2.1), and divide by tn, Then we have
(2.11)‖t-nf(tnx+tnyt)-t-1f(tnx)tn-t-1f(tny)tn‖k≤ϵtn.
By letting n→∞, we get
(2.12)T(x+yt)=T(x)t+T(y)t.
Letting y=0 in (2.12) yields T(x/t)=T(x)/t for all x∈E. Hence, we get T(x+y)=T(x)+T(y), that is, T is additive. Now, if T′ is another required additive mapping, we see that
(2.13)‖T′(x)-T(x)‖≤1tn‖T′(tnx)-T(tnx)‖≤1tn‖T′(tnx)-f(tnx)‖+1tn‖f(tnx)-T(tnx)‖≤2tn-1(t-1)ϵ,
for all x∈E. By letting n→∞ in this inequality, we conclude that T=T′. This proves the uniqueness assertion.Definition 2.2.
Let((𝒜k,∥·∥k)k∈ℕ) be a multi-Banach algebra. Suppose that ϵ>0, t>1 is an integer and ψ:𝒜×𝒜→[0,∞) is a control function such that
(2.14)ψ(tnx,tmy)≤αn+mψ(x,y),
for some 0<α<t, all nonnegative numbers m,n and all x,y∈𝒜. An (ϵ,ψ)-approximate generalized strong higher derivation of rank k0 is a family {fj}j=0k0 of mappings from 𝒜 into 𝒜 with fj(0)=0, f0=id𝒜, and there exists a family {gj}j=0k0 of mappings from 𝒜 into 𝒜 such that g0=id𝒜 and
(2.15)sup⁡k∈N‖(fj(x1+y1t+z1w1)-fj(x1)t-fj(y1)t-z1fj(w1)-gj(z1)w1,…,fj(xk+k1t+zkwk)-fj(xk)t-fj(yk)t-zkfj(wk)-gj(zk)wk)‖k≤ϵ,
for all 0≤j≤k0, t>1 and all x1,…,xk,y1,…,yk,z1,…,zk,w1,…,wk∈𝒜, and
(2.16)‖fj(xy)-xfj(y)-∑i=1jgi(x)fj-i(y)‖≤ψ(x,y),
for all 0≤j≤k0 and x,y∈𝒜.Theorem 2.3.
Let𝒜 be a Banach algebra with unit e, and let {fj}j=0k0 be a (ϵ,ψ)-approximate generalized strong higher derivation on a multi-Banach algebra ((𝒜k,∥·∥k)k∈ℕ), then {fj}j=0k0 is a strong higher derivation.Proof.
Lettingzi=wi=0 for i=1,…,k in (2.15), Lemma 2.1 implies that for each 0≤j≤k0, there is an additive mapping dj defined by dj(x)=limn→∞(fj(tnx)/tn) such that ∥dj(x)-fj(x)∥≤ϵ for all x∈𝒜. If j=1, [21, Theorem 2.2] implies that f1 and g1 are a generalized derivation and a derivation, respectively. Also by the proof of [21, Theorem 2.2], we have
(2.17)f1(xy)=xf1(y)+g1(x)y,lim⁡n→∞g1(tnx)tn=d1(x)-xd1(e)=g1(x).
By induction for 1≤i≤j-1, assume that
(2.18)fi=xfi(y)+∑l=1igl(x)fi-l(y),gi=∑l=0igl(x)gi-l(y),
for all x,y∈𝒜 such that
(2.19)lim⁡n→∞gi(tnx)tn=di(x)-xdi(e)-∑l=1igl(x)di-l(e)=gi(x).
It follows from (2.14) and (2.16) that
(2.20)‖fj(t2nxy)t2n-xfj(tny)-∑i=1jgi(tnx)tnfj-i(tny)tn‖≤ψ(tnx,tny)t2n≤(αt)2n.
Passing the limit as n→∞, we obtain
(2.21)lim⁡n→∞ygj(tnx)tn=dj(xy)-xdj(y)-∑i=1j-1gi(x)dj-i(y),
for all x,y∈𝒜. Put y=e in the above equation, then
(2.22)lim⁡n→∞gj(tnx)tn=dj(x)-xdj(e)-∑i=1j-1gi(x)dj-i(e).
If Dj(x)=dj(x)-xdj(e)-∑i=1j-1gi(x)dj-i(e), then by additivity of di and gi for 0≤i≤j-1, we get
(2.23)Dj(a+b)=dj(a+b)-(a+b)dj(e)-∑i=1jgi(a+b)dj-i(e)=dj(a)+dj(b)-adj(e)-bdj(e)-∑i=1jgi(a)dj-i(e)-∑i=1jgi(b)dj-i(e)=Dj(a)+Dj(b).
Therefore, Dj is additive. Now, let F(x,y)=fj(xy)-xfj(y)-∑i=1jgi(x)fj-i(y), if we take xi=yi=0 and zi=x, wi=y for i=1,…,k in (2.15), then limn→∞(F(tnx,y)/tn)=0. Hence,
(2.24)dj(xy)=lim⁡n→∞fj(tnxy)tn=lim⁡n→∞fj(tnx⋅y)tn=lim⁡n→∞tnxfj(y)+∑i=1jgi(tnx)fj-i(y)+F(tnx,y)tn=xfj(y)+∑i=1j-1gi(x)fj-i(y)+Dj(x)y,
for all x,y∈𝒜. Since g1,…,gj-1,f1,…,fj-1 and Dj are additive, we can write
(2.25)tnxfj(y)+∑i=1j-1tngi(x)fj-i(y)+tnDj(x)y=dj(tnx⋅y)=dj(x⋅tny)=xfj(tny)+∑i=1j-1tngi(x)fj-i(y)+tnDj(x)y,
for all x,y∈𝒜. We conclude that xfj(y)=x(fj(tny)/tn), so we can obtain xfj(y)=xdj(y), for all x,y∈𝒜 as n→∞. If x=e, we have fj=dj. Therefore,
(2.26)fj(xy)=xfj(y)+∑i=1j-1gi(x)fj-i(y)+Dj(x)y,
for all x,y∈𝒜. Now, we replace y by tny in (2.16), then
(2.27)‖fj(tnxy)tn-xfj(tny)tn-∑i=1jgi(x)fj-i(y)‖≤ψ(x,tny)tn≤(αt)n,
for all x,y∈𝒜. Letting n→∞, we get
(2.30)dj(xy)=xdj(y)+∑i=1jgi(x)fj-i(y).
Thus if y=e, we conclude that
(2.31)dj(x)=xdj(e)+∑i=1jgi(x)fj-i(e),
for all x∈𝒜. Hence,
(2.32)gj(x)=dj(x)-xdj(e)-∑i=1j-1gi(x)fj-i(e)=Dj(x).
But for all x,y∈𝒜, we have
(2.33)Dj(xy)=fj(xy)-xyfj(e)-∑i=1j-1gi(xy)fj-i(e)=xfj(y)+∑i=1j-1gi(x)fj-i(y)+Dj(x)y-xyfj(e)-∑i=1j-1(∑l=1igl(x)gi-l(y))fj-i(e)=g1(x)(fj-1(y)-yfj-1(e)-∑l=1j-1gl(y)fj-l(e))+⋯+gj-1(f1(y)-yf1(e))=xDj(y)+Dj(x)y+∑i=1j-1gi(x)gj-i(y),
and by (2.32), it follows that gj(xy)=Dj(xy)=∑i=0jgi(x)gj-i(y); therefore {gj} is a strong higher derivation. By (2.26), we can conclude that {fj} is a generalized strong higher derivation.Remark 2.4.
Recall that a control function is an operation that controls the recording or processing or transmission of interpretation of data. A typical example of the control functionψ is ψ(x,y)=αϵ(∥x∥p+∥y∥q)+δ∥x∥p∥y∥q, such that ϵ,δ≥0 and 0≤p, q<1.Corollary 2.5.
Every(ϵ,ψ)-approximate generalized derivation (regarded as an approximate generalized strong higher derivation of rank 1) on a multi-Banach algebra ((𝒜k,∥·∥k)k∈ℕ) is a derivation.The following theorem generalizes Theorem2.3. The arguments are similar to those in the proof of [21, Theorem 2.3].Theorem 2.6.
Let𝒜 be a Banach algebra with unit e, and let {fj}j=0k0 be a family {fj}j=0k0 of mappings from 𝒜 into 𝒜 with fj(0)=0 and f0=id𝒜 for which there exists a family {gj}j=0k0 of mappings in which g0=id𝒜 on 𝒜 such that
(2.34)sup⁡k∈N‖(fj(βx1+γy1t+z1w1)-βfj(x1)t-γfj(y1)t-z1fj(w1)-gj(z1)w1,…,fj(βxk+γykt+zkwk)-βfj(xk)t-γfj(yk)t-zkfj(wk)-gj(zk)wk)‖k≤ϵ,
for all 0≤j≤k0, t>1 and all β,γ∈𝕋={z∈ℂ:|z|=1} and
(2.35)‖fj(xy)-xfj(y)-∑i=1jgi(x)fj-i(y)‖≤ψ(x,y),
for all 0≤j≤k0 and x,y∈𝒜, then {fj}j=0k0 is a strong higher derivation.



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*Source: 239849-2011-09-28.xml*