AAA Abstract and Applied Analysis 1687-0409 1085-3375 Hindawi Publishing Corporation 347478 10.1155/2012/347478 347478 Research Article The Hyers-Ulam-Rassias Stability of (m,n)(σ,τ)-Derivations on Normed Algebras Fošner Ajda Brillouet-Belluot Nicole Faculty of Management University of Primorska Cankarjeva 5 6104 Koper Slovenia upr.si 2012 15 7 2012 2012 08 04 2012 31 05 2012 2012 Copyright © 2012 Ajda Fošner. This 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.

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

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 equation must be somehow close to an exact solution of ? 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 metric δ(·,·). Suppose that f:𝒢1𝒢2  is a map and   ϵ>0  a fixed scalar. Does there exists λ>0  such that if fsatisfies the inequality(1.1)δ(f(xy),f(x)*f(y))λfor all x,y𝒢1, then there exists a group homomorphism F:𝒢1𝒢2  with the property(1.2)δ(f(x),F(x))ϵfor all x𝒢1?

One year later, Ulam's problem was affirmatively solved by Hyers  for the Cauchy  functional  equation f(x+y)=f(x)+f(y).: Let 𝒳1 be a normed space, 𝒳2 a Banach space, and ϵ>0 a fixed scalar. Suppose that f  : 𝒳1𝒳2 is a map with the property (1.3)f(x+y)-f(x)-f(y)<ϵfor all x,y𝒳1. Then there exists a unique additive mapping F:𝒳1𝒳2  such that(1.4)f(x)-F(x)<ϵfor all x𝒳1. 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  which allowed the Cauchy difference to be unbounded. In 1978, Rassias  proved the stability of linear mappings in the following way. Let 𝒳1 be a real normed space and 𝒳2 a real Banach space. If there exist scalars ϵ0 and 0p<1 such that(1.5)f(x+y)-f(x)-f(y)ϵ(xp+yp)for all x,y𝒳1,  then there exists a unique additive mapping F:𝒳1𝒳2 with the property(1.6)f(x)-F(y)2ϵ2-2pxpfor all x𝒳1. Moreover, if the map rf(rx) is continuous on for each x𝒳1, then F is 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  generalized the Rassias' theorem as follows: Let (𝒢,+) be an Abelian group and 𝒳 a Banach space. Suppose that the so-called admissible control function φ:𝒢×𝒢[0,) satisfies(1.7)k=0φ(2kx,2ky)2k+1<for all x,y𝒢. If f:𝒢𝒳  is a mapping with the property(1.8)f(x+y)-f(x)-f(y)φ(x,y)for all x,y𝒢, then there exists a unique additive mapping F:𝒢𝒳 such that(1.9)f(x)-F(x)k=0φ(2kx,2ky)2k+1for all x𝒢.

In the last few decades, various approaches to the problem have been introduced by several authors. Moreover, it is surprising that in some cases the approximate mapping is actually a true mapping. In such cases we call the equation superstable. For the history and various aspects of this theory we refer the reader to monographs .

As we are aware, the stability of derivations was first investigated by Jun and Park . During the past few years, approximate derivations were studied by a number of mathematicians (see  and references therein).

Moslehian  studied the stability of (σ,τ)-derivations and generalized some results obtained in . 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 .

2. Preliminaries

Throughout, 𝒜 will be a normed algebra and a Banach 𝒜-bimodule. Let σ and τ be two linear operators on 𝒜. An additive mapping d:𝒜 is called an (σ,τ)-derivation if (2.1)d(xy)=d(x)σ(y)+τ(x)d(y) holds for all x,y𝒜. Ordinary derivations from 𝒜 to and maps defined by xaσ(x)-τ(x)a, where a𝒜 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 , where further information about (σ,τ)-derivations can be found.

In  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.

Let m and n be nonnegative integers with m+n0. An additive mapping d:𝒜 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,y𝒜. 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 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, ). 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 that m and n are nonnegative integers with m+n0. We will use the same symbol · in order to represent the norms on a normed algebra 𝒜 and a Banach 𝒜-bimodule . For a given (admissible control) function φ:𝒜×𝒜[0,) we will use the following abbreviation: (2.3)ϕ(x,y):=k=0φ(2kx,2ky)2k+1,x,yA. Let us start with one well-known lemma.

Lemma 2.1 (see [<xref ref-type="bibr" rid="B9">6</xref>]).

Suppose that a function φ:𝒜×𝒜[0,) satisfies ϕ(x,y)<, x,y𝒜. If f:𝒜 is a mapping with (2.4)f(x+y)-f(x)-f(y)φ(x,y) for all x,y𝒜, then there exists a unique additive mapping F:𝒜 such that (2.5)f(x)-F(x)ϕ(x,x) for all x𝒜.

We say that an additive mapping f:𝒜 is -linear if f(λx)=λf(x) for all x𝒜 and all scalars λ. In the following, Λ will denote the set of all complex units, that is, (2.6)Λ={λC:|λ|=1}. For a given additive mapping f:𝒜, Park  obtained the next result.

Lemma 2.2.

If f(λx)=λf(x) for all x𝒜 and all λΛ, then f is -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 -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 . The idea is taken from .

Theorem 3.1.

Let d:𝒜 and f,g:𝒜𝒜 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,y𝒜 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,y𝒜 and λΛ. Then there exist unique -linear mappings σ,τ:𝒜𝒜 satisfying (3.5)f(x)-σ(x)ϕ(x,x),g(x)-τ(x)ϕ(x,x) for all x𝒜, and a unique -linear (m,n)(σ,τ)-derivation D:𝒜 such that (3.6)d(x)-D(x)ϕ(x,x) for all x𝒜.

Proof.

Taking λ=1 in (3.1) and using Lemma 2.1, it follows that there exists a unique additive mapping D:𝒜 such that d(x)-D(x)ϕ(x,x) holds for all x𝒜. 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)2qk=qp-1φ(2kx,2kx)2k+1 for all x𝒜, all positive integers l, and all 0q<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 , this sequence is convergent and we can define a map D:𝒜 as (3.9)D(x):=limkd(2kx)2k,xA. Using (3.1), we get (3.10)D(λx+λy)-λD(x)-λD(y)=limk2-kd(λ2kx+λ2ky)-λd(2kx)-λd(2ky)limk2-kφ(2kx,2ky)=0. This yields that (3.11)D(λx+λy)=λD(x)+λD(y) for all x,y𝒜 and λΛ. Using Lemma 2.2, it follows that the map D is -linear. Moreover, according to inequality (3.7), we have (3.12)d(x)-D(x)=limkd(x)-d(2kx)2kϕ(x,x) for all x𝒜.

Next, we have to show the uniqueness of D. So, suppose that there exists another -linear mapping D~:𝒜 such that d(x)-D~(x)ϕ(x,x) for all x𝒜. Then (3.13)D(x)-D~(x)=limk2-kd(2kx)-D~(2kx)limk2-kϕ(2kx,2kx)=limk2-kj=0φ(2j+kx,2j+kx)2j+1=limkj=kφ(2jx,2jx)2j+1=0. Therefore, D(x)=D~(x) for all x𝒜, as desired.

Similarly we can show that there exist unique -linear mappings σ,τ:𝒜𝒜 defined by (3.14)σ(x):=limkf(2kx)2k,xA,τ(x):=limkg(2kx)2k,  xA. Furthermore, (3.15)f(x)-σ(x)ϕ(x,x),g(x)-τ(x)ϕ(x,x) for all x𝒜.

It remains to prove that D 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)=limk4-k(m+n)d(4kxy)-2md(2kx)f(2ky)-2ng(2kx)d(2ky)limk4-kφ(2kx,2ky)=0 for all x,y𝒜. Thus, mappings D and σ,τ satisfy (2.2). The proof is completed.

Remark 3.2.

If there exists x0𝒜 such that d and the map xϕ(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 limkxk=0 and D(xk)>1/C, k0. Let t>C(2ϕ(x0,x0)+1). Then (3.18)limkd(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 xϕ(x,x) at point x0, we get a contradiction.

Let ϵ0 and 0p<1. Applying Theorem 3.1 for the case (3.21)φ(x,y):=ϵ(xp+yp),x,yA.

Corollary 3.3.

Let d:𝒜 and f,g:𝒜𝒜 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,y𝒜 and λΛ, where a function φ:𝒜×𝒜[0,) is defined as above. Then there exist unique -linear mappings σ,τ:𝒜𝒜 satisfying (3.22)f(x)-σ(x)2ϵ2-2pxp,g(x)-τ(x)2ϵ2-2pxp for all x𝒜 and a unique -linear (m,n)(σ,τ)-derivation D:𝒜 such that (3.23)d(x)-D(x)2ϵ2-2pxp

Proof .

Note that ϕ(x,y)< for all x,y𝒜 and (3.24)ϕ(x,y)=ϵ2-2p(xp+yp),x,yA.

Remark 3.4.

Recall that we can actually take any map φ:𝒜×𝒜[0,) in the form (3.25)φ(x,y):=ν+ϵ(xp+yp),  x,yA, where ν0. In this case we have (3.26)ϕ(x,y)=ν+ϵ(xp+yp)(2-2p),x,yA.

Before stating our next result, let us write one well-known lemma about the continuity of measurable functions (see, e.g., ).

Lemma 3.5.

If a measurable function ψ: satisfies ψ(r1+r2)=ψ(r1)+ψ(r2) for all r1,r2, then ψ is continuous.

Now we are in the position to state a result for normed algebras 𝒜 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 𝒜

Theorem 3.6.

Let 𝒜 be a normed algebra which is spanned by a subset 𝒮 of 𝒜 and d:𝒜, f,g:𝒜𝒜 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,y𝒜 and (3.1), (3.2), (3.3) holds true for all x,y𝒜 and λ=1,i. Moreover, suppose that (3.4) holds true for all x,y𝒮. If for all x𝒜 the functions rd(rx), rf(rx), and rg(rx) are continuous on , then there exist unique -linear mappings σ,τ:𝒜𝒜 satisfying (3.27)f(x)-σ(x)ϕ(x,x),g(x)-τ(x)ϕ(x,x) for all x𝒜 and a unique -linear (m,n)(σ,τ)-derivation D:𝒜 such that (3.28)d(x)-D(x)ϕ(x,x) for all x𝒜.

We will give just a sketch of the proof since most of the steps are the same as in the proof of Theorem 3.1.

Proof.

As in the proof of Theorem 3.1, we can show that there exists a unique additive mapping D:𝒜 defined by D(x):=limk(d(2kx)/2k), x𝒜. Moreover, d(x)-D(x)ϕ(x,x) for all x𝒜.

Writing y=0, λ=i in (3.1), we get (3.29)d(ix)-id(x)φ(x,0). Therefore, (3.30)D(ix)-iD(x)=limk2-kd(2kix)-id(2kx)limk2-kφ(2kx,0)=0. This yields that (3.31)D(ix)=iD(x) for all x𝒜. In the next step we will show that D is -linear, that is, (3.32)D(rx)=rD(x) for all x𝒜 and all r.

Since D is additive, we have D(qx)=qx for every x𝒜 and all rational numbers q. Let us fix elements x0𝒜 and ρ*, where * denotes the dual space of . Then we can define a function ψ: by (3.33)ψ(r)=ρ(D(rx0)),rR. 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. Furthermore, (3.35)ψ(r)=limkρ(d(2krx0)2k) for all r. Set (3.36)ψk(r)=ρ(d(2krx0)2k),k0. 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. This implies D(rx0)=rD(x0). Since x0 was an arbitrary element from 𝒜, we proved that D is -linear.

Now, let λ. 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 x𝒜. This means that D is -linear.

Similarly we can show that there exist unique -linear mappings σ,τ:𝒜𝒜 satisfying (3.38)f(x)-σ(x)ϕ(x,x),g(x)-τ(x)ϕ(x,x) for all x𝒜. Moreover, (2.2) holds true for all x,y𝒮. 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 Theorem 3.6 for the case (3.39)φ(x,y):=ν+ϵ(xp+yp),x,yA, where ν,ϵ0 and 0p<1.

Remark 3.8.

If ϵ0 and 0p<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):=ϵxpyp,x,yA. In this case (3.41)ϕ(x,y)=ϵ2-4pxpyp,x,yA.

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