# The Hyers-Ulam-Rassias Stability of(m,n)(σ,τ)-Derivations on Normed Algebras

Authors: Ajda Fošner
Journal: Abstract and Applied Analysis
 (2012)

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## Abstract

We study the Hyers-Ulam-Rassias stability of(m,n)(σ,τ)-derivations on normed algebras.


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## Body

## 1.  Introduction

A classical question in the theory of functional equations is as follows.Under what conditions is it true that a mapping which approximately satisfies a functional equationℰmust be somehow close to an exact solution ofℰ? This problem was formulated by Ulam in 1940 (see [1, 2]). He investigated the stability of group homomorphisms.  Let  (𝒢1,∘)  be a group, and let(𝒢2,*,δ)  be a metric group with a metricδ(·,·). Suppose thatf:𝒢1→𝒢2  is a map and   ϵ>0  a fixed scalar. Does there existsλ>0  such that iffsatisfies the inequality(1.1)δ(f(x∘y),f(x)*f(y))≤λfor allx,y∈𝒢1, then there exists a group homomorphismF:𝒢1→𝒢2  with the property(1.2)δ(f(x),F(x))≤ϵfor allx∈𝒢1?One year later, Ulam's problem was affirmatively solved by Hyers [3] for the Cauchy  functional  equation f(x+y)=f(x)+f(y).:  Let𝒳1be a normed space,𝒳2a Banach space, andϵ>0a fixed scalar. Suppose that f  :𝒳1→𝒳2is a map with the property(1.3)‖f(x+y)-f(x)-f(y)‖<ϵfor allx,y∈𝒳1. Then there exists a unique additive mappingF:𝒳1→𝒳2  such that(1.4)‖f(x)-F(x)‖<ϵfor allx∈𝒳1. This gave rise to the stability theory of functional equations.The famous Hyers stability result has been generalized in the stability of additive mappings involving a sum of powers of norms by Aoki [4] which allowed the Cauchy difference to be unbounded. In 1978, Rassias [5] proved the stability of linear mappings in the following way. Let𝒳1be a real normed space and𝒳2a real Banach space. If there exist scalarsϵ≥0and0≤p<1such that(1.5)‖f(x+y)-f(x)-f(y)‖≤ϵ(‖x‖p+‖y‖p)for allx,y∈𝒳1,  then there exists a unique additive mappingF:𝒳1→𝒳2with the property(1.6)‖f(x)-F(y)‖≤2ϵ2-2p‖x‖pfor allx∈𝒳1. Moreover, if the mapr↦f(rx)is continuous onℝfor eachx∈𝒳1, thenFis linear.This result has provided a lot of influence in the development of what we now call the Hyers-Ulam-Rassias stability of functional equations.Later, Găvruţa [6] generalized the Rassias' theorem as follows:Let(𝒢,+)be an Abelian group and𝒳a Banach space. Suppose that the so-called admissible control functionφ:𝒢×𝒢→[0,∞)satisfies(1.7)∑k=0∞φ(2kx,2ky)2k+1<∞for allx,y∈𝒢. If  f:𝒢→𝒳  is a mapping with the property(1.8)‖f(x+y)-f(x)-f(y)‖≤φ(x,y)for allx,y∈𝒢, then there exists a unique additive mappingF:𝒢→𝒳such that(1.9)‖f(x)-F(x)‖≤∑k=0∞φ(2kx,2ky)2k+1for allx∈𝒢.In the last few decades, various approaches to the problem have been introduced by several authors. Moreover, it is surprising that in some cases theapproximate mapping is actually a true mapping. In such cases we call the equation ℰ superstable. For the history and various aspects of this theory we refer the reader to monographs [7–9].As we are aware, the stability of derivations was first investigated by Jun and Park [10]. During the past few years, approximate derivations were studied by a number of mathematicians (see [11–18] and references therein).Moslehian [19] studied the stability of (σ,τ)-derivations and generalized some results obtained in [18]. He also established the generalized Hyers-Ulam-Rassias stability of (σ,τ)-derivations on normed algebras into Banach bimodules. This motivated us to investigate approximate (m,n)(σ,τ)-derivations on normed algebras. The aim of this paper is to study the stability of (m,n)(σ,τ)-derivations and to generalize some results given in [19].


## 2. Preliminaries

Throughout,𝒜 will be a normed algebra and ℳ a Banach 𝒜-bimodule. Let σ and τ be two linear operators on 𝒜. An additive mapping d:𝒜→ℳ is called an (σ,τ)-derivation if
(2.1)d(xy)=d(x)σ(y)+τ(x)d(y)
holds for all x,y∈𝒜. Ordinary derivations from 𝒜 to ℳ and maps defined by x↦aσ(x)-τ(x)a, where a∈𝒜 is a fixed element and σ,τ are endomorphisms on 𝒜, are natural examples of (σ,τ)-derivations on 𝒜. Moreover, if ψ is an endomorphism on 𝒜, then ψ is a ((1/2)ψ,(1/2)ψ)-derivation on 𝒜. We refer the reader to [20], where further information about (σ,τ)-derivations can be found.In [19] Moslehian studied stability of (σ,τ)-derivations. The natural question here is, whether the analogue results hold true for (m,n)(σ,τ)-derivations. Theorem 3.1 answers this question in the affirmative.Letm and n be nonnegative integers with m+n≠0. An additive mapping d:𝒜→ℳ is called a (m,n)(σ,τ)-derivation if
(2.2)(m+n)d(xy)=2md(x)σ(y)+2nτ(x)d(y)
holds for all x,y∈𝒜. Clearly, (m,n)(σ,τ)-derivations are one of the natural generalizations of (σ,τ)-derivations (the case m=n). If σ,τ=id, where id denotes the identity map on 𝒜, and an additive mapping d:𝒜→ℳ satisfies (2.2), then d is called a (m,n)-derivation. In the last few decades a lot of work has been done on the field of (m,n)-derivations on rings and algebras (see, e.g, [21–25]). This motivated us to study the Hyers-Ulam-Rassias stability of functional inequalities associated with (m,n)(σ,τ)-derivations.In the following, we will assume thatm and n are nonnegative integers with m+n≠0. We will use the same symbol ∥·∥ in order to represent the norms on a normed algebra 𝒜 and a Banach 𝒜-bimodule ℳ. For a given (admissible control) function φ:𝒜×𝒜→[0,∞) we will use the following abbreviation:
(2.3)ϕ(x,y):=∑k=0∞φ(2kx,2ky)2k+1,x,y∈A.
Let us start with one well-known lemma.Lemma 2.1 (see [6]).
Suppose that a functionφ:𝒜×𝒜→[0,∞) satisfies ϕ(x,y)<∞, x,y∈𝒜. If f:𝒜→ℳ is a mapping with
(2.4)‖f(x+y)-f(x)-f(y)‖≤φ(x,y)
for all x,y∈𝒜, then there exists a unique additive mapping F:𝒜→ℳ such that
(2.5)‖f(x)-F(x)‖≤ϕ(x,x)
for all x∈𝒜.We say that an additive mappingf:𝒜→ℳ is ℂ-linear if f(λx)=λf(x) for all x∈𝒜 and all scalars λ∈ℂ. In the following, Λ will denote the set of all complex units, that is,
(2.6)Λ={λ∈C:|λ|=1}.
For a given additive mapping f:𝒜→ℳ, Park [26] obtained the next result.Lemma 2.2.
Iff(λx)=λf(x) for all x∈𝒜 and all λ∈Λ, then f is ℂ-linear.


## 3. The Results

Our first result is a generalization of [19, Theorem 2.1] (the case m=n). We use the direct method to construct a unique ℂ-linear mapping from an approximate one and prove that this mapping is an appropriate (m,n)(σ,τ)-derivation on 𝒜. This method was first devised by Hyers [3]. The idea is taken from [19].Theorem 3.1.
Letd:𝒜→ℳ and f,g:𝒜→𝒜 be mappings with d(0)=f(0)=g(0)=0. Suppose that there exists a function φ:𝒜×𝒜→[0,∞) such that ϕ(x,y)<∞ for all x,y∈𝒜 and
(3.1)‖d(λx+λy)-λd(x)-λd(y)‖≤φ(x,y),(3.2)‖f(λx+λy)-λf(x)-λf(y)‖≤φ(x,y),(3.3)‖g(λx+λy)-λg(x)-λg(y)‖≤φ(x,y),(3.4)‖(m+n)d(xy)-2md(x)f(y)-2ng(x)d(y)‖≤φ(x,y)
for all x,y∈𝒜 and λ∈Λ. Then there exist unique ℂ-linear mappings σ,τ:𝒜→𝒜 satisfying
(3.5)‖f(x)-σ(x)‖≤ϕ(x,x),‖g(x)-τ(x)‖≤ϕ(x,x)
for all x∈𝒜, and a unique ℂ-linear (m,n)(σ,τ)-derivation D:𝒜→ℳ such that
(3.6)‖d(x)-D(x)‖≤ϕ(x,x)
for all x∈𝒜.Proof.
Takingλ=1 in (3.1) and using Lemma 2.1, it follows that there exists a unique additive mapping D:𝒜→ℳ such that ∥d(x)-D(x)∥≤ϕ(x,x) holds for all x∈𝒜. More precisely, using the induction, it is easy to see that
(3.7)‖d(2lx)2l-d(x)‖≤∑k=0l-1φ(2kx,2kx)2k+1,(3.8)‖d(2px)2p-d(2qx)2q‖≤∑k=qp-1φ(2kx,2kx)2k+1
for all x∈𝒜, all positive integers l, and all 0≤q<p. According to the assumptions on ϕ(x,y), it follows that the sequence {d(2kx)/2k}k=0∞ is Cauchy. Thus, by the completeness of  ℳ, this sequence is convergent and we can define a map D:𝒜→ℳ as
(3.9)D(x):=lim⁡k→∞d(2kx)2k,x∈A.
Using (3.1), we get
(3.10)‖D(λx+λy)-λD(x)-λD(y)‖=lim⁡k→∞⁡2-k‖d(λ2kx+λ2ky)-λd(2kx)-λd(2ky)‖≤lim⁡k→∞⁡2-kφ(2kx,2ky)=0.
This yields that
(3.11)D(λx+λy)=λD(x)+λD(y)
for all x,y∈𝒜 and λ∈Λ. Using Lemma 2.2, it follows that the map D is ℂ-linear. Moreover, according to inequality (3.7), we have
(3.12)‖d(x)-D(x)‖=lim⁡k→∞‖d(x)-d(2kx)2k‖≤ϕ(x,x)
for all x∈𝒜.
Next, we have to show the uniqueness ofD. So, suppose that there exists another ℂ-linear mapping D~:𝒜→ℳ such that ∥d(x)-D~(x)∥≤ϕ(x,x) for all x∈𝒜. Then
(3.13)‖D(x)-D~(x)‖=lim⁡k→∞2-k‖d(2kx)-D~(2kx)‖≤lim⁡k→∞2-kϕ(2kx,2kx)=lim⁡k→∞2-k∑j=0∞φ(2j+kx,2j+kx)2j+1=lim⁡k→∞∑j=k∞φ(2jx,2jx)2j+1=0.
Therefore, D(x)=D~(x) for all x∈𝒜, as desired.
Similarly we can show that there exist uniqueℂ-linear mappings σ,τ:𝒜→𝒜 defined by
(3.14)σ(x):=lim⁡k→∞f(2kx)2k,x∈A,τ(x):=lim⁡k→∞g(2kx)2k,x∈A.
Furthermore,
(3.15)‖f(x)-σ(x)‖≤ϕ(x,x),‖g(x)-τ(x)‖≤ϕ(x,x)
for all x∈𝒜.
It remains to prove thatD is an (m,n)(σ,τ)-derivation. Writing 2kx in the place of x and 2ky in the place of y in (3.4), we obtain
(3.16)‖(m+n)d(4kxy)-2md(2kx)f(2ky)-2ng(2kx)d(2ky)‖≤φ(2kx,2ky).
This yields that
(3.17)‖((m+n)D(xy)-2mD(x)σ(y)-2nτ(x)D(y)‖=lim⁡k→∞⁡4-k‖(m+n)d(4kxy)-2md(2kx)f(2ky)-2ng(2kx)d(2ky)‖≤lim⁡k→∞⁡4-kφ(2kx,2ky)=0
for all x,y∈𝒜. Thus, mappings D and σ,τ satisfy (2.2). The proof is completed.Remark 3.2.
If there existsx0∈𝒜 such that d and the map x↦ϕ(x,x) are continuous at point x0, then D is continuous on 𝒜. Namely, if D was not continuous, then there would exist an integer C and a sequence {xk}k=0∞ such that limk→∞xk=0 and ∥D(xk)∥>1/C, k≥0. Let t>C(2ϕ(x0,x0)+1). Then
(3.18)lim⁡k→∞d(txk+x0)=d(x0)
since d is continuous at point x0. Thus, there exists an integer k0 such that for every k>k0 we have
(3.19)‖d(txk+x0)-d(x0)‖<1.
Therefore,
(3.20)2ϕ(x0,x0)+1<tC<‖D(txk)‖=‖D(txk+x0)-D(x0)‖≤‖D(txk+x0)-d(txk+x0)‖+‖d(txk+x0)-d(x0)‖+‖d(x0)-D(x0)‖<ϕ(txk+x0,txk+x0)+1+ϕ(x0,x0)
for every k>k0. Letting k→∞ and using the continuity of the map x↦ϕ(x,x) at point x0, we get a contradiction.Letϵ≥0 and 0≤p<1. Applying Theorem 3.1 for the case
(3.21)φ(x,y):=ϵ(‖x‖p+‖y‖p),x,y∈A.Corollary 3.3.
Letd:𝒜→ℳ and f,g:𝒜→𝒜 be mappings with d(0)=f(0)=g(0)=0. Suppose that (3.1), (3.2), (3.3), and (3.4) hold true for all x,y∈𝒜 and λ∈Λ, where a function φ:𝒜×𝒜→[0,∞) is defined as above. Then there exist unique ℂ-linear mappings σ,τ:𝒜→𝒜 satisfying
(3.22)‖f(x)-σ(x)‖≤2ϵ2-2p‖x‖p,‖g(x)-τ(x)‖≤2ϵ2-2p‖x‖p
for all x∈𝒜 and a unique ℂ-linear (m,n)(σ,τ)-derivation D:𝒜→ℳ such that
(3.23)‖d(x)-D(x)‖≤2ϵ2-2p‖x‖pProof .
Note thatϕ(x,y)<∞ for all x,y∈𝒜 and
(3.24)ϕ(x,y)=ϵ2-2p(‖x‖p+‖y‖p),x,y∈A.Remark 3.4.
Recall that we can actually take any mapφ:𝒜×𝒜→[0,∞) in the form
(3.25)φ(x,y):=ν+ϵ(‖x‖p+‖y‖p),x,y∈A,
where ν≥0. In this case we have
(3.26)ϕ(x,y)=ν+ϵ(‖x‖p+‖y‖p)(2-2p),x,y∈A.Before stating our next result, let us write one well-known lemma about the continuity of measurable functions (see, e.g., [27]).Lemma 3.5.
If a measurable functionψ:ℝ→ℝ satisfies ψ(r1+r2)=ψ(r1)+ψ(r2) for all r1,r2∈ℝ, then ψ is continuous.Now we are in the position to state a result for normed algebras𝒜 which are spanned by a subset 𝒮 of 𝒜. For example, 𝒜 can be a C*-algebra spanned by the unitary group of 𝒜 or the positive part of 𝒜Theorem 3.6.
Let𝒜 be a normed algebra which is spanned by a subset 𝒮 of 𝒜 and d:𝒜→ℳ, f,g:𝒜→𝒜 mappings with d(0)=f(0)=g(0)=0. Suppose that there exists a function φ:𝒜×𝒜→[0,∞) such that ϕ(x,y)<∞ for all x,y∈𝒜 and (3.1), (3.2), (3.3) holds true for all x,y∈𝒜 and λ=1,i. Moreover, suppose that (3.4) holds true for all x,y∈𝒮. If for all x∈𝒜 the functions r↦d(rx), r↦f(rx), and r↦g(rx) are continuous on ℝ, then there exist unique ℂ-linear mappings σ,τ:𝒜→𝒜 satisfying
(3.27)‖f(x)-σ(x)‖≤ϕ(x,x),‖g(x)-τ(x)‖≤ϕ(x,x)
for all x∈𝒜 and a unique ℂ-linear (m,n)(σ,τ)-derivation D:𝒜→ℳ such that
(3.28)‖d(x)-D(x)‖≤ϕ(x,x)
for all x∈𝒜.We will give just a sketch of the proof since most of the steps are the same as in the proof of Theorem3.1.Proof.
As in the proof of Theorem3.1, we can show that there exists a unique additive mapping D:𝒜→ℳ defined by D(x):=limk→∞(d(2kx)/2k), x∈𝒜. Moreover, ∥d(x)-D(x)∥≤ϕ(x,x) for all x∈𝒜.
Writingy=0, λ=i in (3.1), we get
(3.29)‖d(ix)-id(x)‖≤φ(x,0).
Therefore,
(3.30)‖D(ix)-iD(x)‖=lim⁡k→∞2-k‖d(2kix)-id(2kx)‖≤lim⁡k→∞2-kφ(2kx,0)=0.
This yields that
(3.31)D(ix)=iD(x)
for all x∈𝒜. In the next step we will show that D is ℝ-linear, that is,
(3.32)D(rx)=rD(x)
for all x∈𝒜 and all r∈ℝ.
SinceD is additive, we have D(qx)=qx for every x∈𝒜 and all rational numbers q. Let us fix elements x0∈𝒜 and ρ∈ℳ*, where ℳ* denotes the dual space of ℳ. Then we can define a function ψ:ℝ→ℝ by
(3.33)ψ(r)=ρ(D(rx0)),r∈R.
Firstly, we would like to prove that ψ is continuous. Recall that
(3.34)ψ(r1+r2)=ρ(D((r1+r2)x0))=ρ(D(r1x0))+ρ(D(r2x0))=ψ(r1)+ψ(r2)
for all r1,r2∈ℝ. Furthermore,
(3.35)ψ(r)=lim⁡k→∞ρ(d(2krx0)2k)
for all r∈ℝ. Set
(3.36)ψk(r)=ρ(d(2krx0)2k),k≥0.
Obviously, {ψk}k=0∞ is a sequence of continuous functions and ψ is its pointwise limit. This yields that ψ is a Borel function and, by  Lemma 3.5 it is continuous. Therefore, we have ψ(r)=rψ(1) for all r∈ℝ. This implies D(rx0)=rD(x0). Since x0 was an arbitrary element from 𝒜, we proved that D is ℝ-linear.
Now, letλ∈ℂ. Then λ=r1+ir2 for some real numbers  r1,r2. Using (3.31), we have
(3.37)D(λx)=D((r1+ir2)x)=D(r1x)+D(ir2x)=r1D(x)+ir2D(x)=λD(x)
for all x∈𝒜. This means that D is ℂ-linear.
Similarly we can show that there exist uniqueℂ-linear mappings σ,τ:𝒜→𝒜 satisfying
(3.38)‖f(x)-σ(x)‖≤ϕ(x,x),‖g(x)-τ(x)‖≤ϕ(x,x)
for all x∈𝒜. Moreover, (2.2) holds true for all x,y∈𝒮. Since 𝒜 is linearly generated by 𝒮, we conclude that D is an (m,n)(σ,τ)-derivation on 𝒜. The proof is completed.Remark 3.7.
As above, we can apply Theorem3.6 for the case
(3.39)φ(x,y):=ν+ϵ(‖x‖p+‖y‖p),x,y∈A,
where ν,ϵ≥0 and 0≤p<1.Remark 3.8.
Ifϵ≥0 and 0≤p<1/2, then we can use in Theorem 3.1 as well as in Theorem 3.6 a function φ:𝒜×𝒜→[0,∞) given by
(3.40)φ(x,y):=ϵ‖x‖p‖y‖p,x,y∈A.
In this case
(3.41)ϕ(x,y)=ϵ2-4p‖x‖p‖y‖p,x,y∈A.



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