SUPERSTABILITY FOR GENERALIZED MODULE LEFT DERIVATIONS AND GENERALIZED MODULE DERIVATIONS ON A BANACH MODULE (II)

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Generalized Module Derivations Huai-Xin Cao, Ji-Rong Lv

and J.M. Rassias vol. 10, iss. 3, art. 85, 2009

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SUPERSTABILITY FOR GENERALIZED MODULE LEFT DERIVATIONS AND GENERALIZED MODULE

DERIVATIONS ON A BANACH MODULE (II)

HUAI-XIN CAO, JI-RONG LV

College of Mathematics and Information Science Shaanxi Normal University, Xi’an 710062, P. R. China EMail:caohx@snnu.edu.cn r981@163.com

J.M. RASSIAS

Pedagogical Department, Section of Mathematics and Informatics National and Capodistrian University of Athens

Athens 15342, Greece

EMail:jrassias@primedu.uoa.gr

Received: 12 January, 2009

Accepted: 12 May, 2009

Communicated by: S.S. Dragomir

2000 AMS Sub. Class.: Primary 39B52; Secondary 39B82.

Key words: Superstability, Generalized module left derivation, Generalized module derivation, Module left derivation, Module derivation, Banach module.

Abstract: In this paper, we introduce and discuss the superstability of generalized module left derivations and generalized module derivations on a Banach module.

Acknowledgements: This subject is supported by the NNSFs of China (No. 10571113, 10871224).

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1. Introduction

The study of stability problems was formulated by Ulam in [28] during a talk in 1940: “Under what conditions does there exist a homomorphism near an approximate homomorphism?” In the following year 1941, Hyers in [12] answered the question of Ulam for Banach spaces, which states that ifε >0andf : X →Y is a map with a normed spaceXand a Banach spaceY such that

(1.1) kf(x+y)−f(x)−f(y)k ≤ε,

for allx, yinX, then there exists a unique additive mappingT :X →Y such that

(1.2) kf(x)−T(x)k ≤ε,

for allx in X. In addition, if the mapping t 7→ f(tx)is continuous in t ∈ R for each fixed x in X, then the mapping T is real linear. This stability phenomenon is called the Hyers-Ulam stability of the additive functional equation f(x+y) = f(x) + f(y). A generalized version of the theorem of Hyers for approximately additive mappings was given by Aoki in [1] and for approximate linear mappings was presented by Th. M. Rassias in [26] by considering the case when the left hand side of the inequality (1.1) is controlled by a sum of powers of norms [25]. The stability of approximate ring homomorphisms and additive mappings were discussed in [6,7,8,10,11,13,14,21].

The stability result concerning derivations between operator algebras was first obtained by P. Semrl in [27]. Badora [5] and Moslehian [17,18] discussed the Hyers- Ulam stability and the superstability of derivations. C. Baak and M. S. Moslehian [4]

discussed the stability ofJ^∗-homomorphisms. Miura et al. proved the Hyers-Ulam- Rassias stability and Bourgin-type superstability of derivations on Banach algebras in [16]. Various stability results on derivations and left derivations can be found in

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[3, 19, 20, 2, 9]. More results on stability and superstability of homomorphisms, special functionals and equations can be found in J. M. Rassias’ papers [22,23,24].

Recently, S.-Y. Kang and I.-S. Chang in [15] discussed the superstability of generalized left derivations and generalized derivations. In the present paper, we will discuss the superstability of generalized module left derivations and generalized module derivations on a Banach module.

To give our results, let us give some notations. LetA be an algebra over the real or complex fieldFandX be anA-bimodule.

Definition 1.1. A mappingd:A →A is said to be module-Xadditive if (1.3) xd(a+b) =xd(a) +xd(b) (a, b∈A, x∈X).

A module-Xadditive mappingd:A →A is said to be a module-X left derivation (resp., module-X derivation) if the functional equation

(1.4) xd(ab) =axd(b) +bxd(a) (a, b∈A, x∈X) (resp.,

(1.5) xd(ab) =axd(b) +d(a)xb (a, b∈A, x∈X)) holds.

Definition 1.2. A mappingf :X →X is said to be module-A additive if (1.6) af(x1+x2) = af(x1) +af(x2) (x1, x2 ∈X, a∈A).

A module-A additive mappingf : X → X is called a generalized module-A left derivation (resp., generalized module-A derivation) if there exists a module-X left derivation (resp., module-X derivation)δ:A →A such that

(1.7) af(bx) =abf(x) +axδ(b) (x∈X, a, b ∈A)

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(resp.,

(1.8) af(bx) =abf(x) +aδ(b)x (x∈X, a, b ∈A)).

In addition, if the mappings f and δ are all linear, then the mapping f is called a linear generalized module-A left derivation (resp., linear generalized module-A derivation).

Remark 1. LetA =XandA be one of the following cases:

(a) a unital algebra;

(b) a Banach algebra with an approximate unit.

Then module-A left derivations, module-A derivations, generalized module-A left derivations and generalized module-A derivations on A become left derivations, derivations, generalized left derivations and generalized derivations on A as discussed in [15].

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2. Main Results

Theorem 2.1. Let A be a Banach algebra, X a Banach A-bimodule, k andl be integers greater than1, andϕ : X×X×A ×X → [0,∞)satisfy the following conditions:

(a) lim

n→∞k⁻ⁿ[ϕ(kⁿx, kⁿy,0,0) +ϕ(0,0, kⁿz, w)] = 0 (x, y, w ∈X, z ∈A).

(b) lim

n→∞k⁻²ⁿϕ(0,0, kⁿz, kⁿw) = 0 (z ∈A, w ∈X).

(c) ϕ(x) :=˜ P∞

n=0k⁻ⁿ⁺¹ϕ(kⁿx,0,0,0)<∞(x∈X).

Suppose that f : X → X and g : A → A are mappings such thatf(0) = 0, δ(z) := lim

n→∞

1

kⁿg(kⁿz)exists for allz ∈A and

(2.1)

∆¹_f,g(x, y, z, w)

≤ϕ(x, y, z, w) for allx, y, w ∈X andz ∈A where

∆¹_f,g(x, y, z, w) = fx k +y

l +zw +fx

k − y

l +zw

−2f(x)

k −2zf(w)−2wg(z).

Thenf is a generalized module-A left derivation andg is a module-X left deriva- tion.

Proof. By takingw=z = 0, we see from (2.1) that (2.2)

fx k + y

l

+fx k − y

l

− 2f(x) k

≤ϕ(x, y,0,0)

for allx, y ∈X. Lettingy= 0and replacingxbykxin (2.2), we get (2.3)

f(x)− f(kx) k

≤ 1

2ϕ(kx,0,0,0)

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for allx∈X. Hence, for allx∈X, we have from (2.3) that

f(x)−f(k²x) k²

≤

f(x)− f(kx) k

+

f(kx)

k −f(k²x) k²

≤ 1

2ϕ(kx,0,0,0) + 1

2k⁻¹ϕ(k²x,0,0,0).

By induction, one can check that (2.4)

f(x)− f(kⁿx) kⁿ

≤ 1 2

n

X

j=1

k^−j+1ϕ(k^jx,0,0,0)

for allxinXandn = 1,2, . . . .Letx∈Xandn > m. Then by (2.4) and condition (c), we obtain that

f(kⁿx)

kⁿ −f(k^mx) k^m

= 1 k^m

f(k^n−m·k^mx)

k^n−m −f(k^mx)

≤ 1 k^m · 1

2

n−m

X

j=1

k^−j+1ϕ(k^j·k^mx,0,0,0)

≤ 1 2

∞

X

s=m

k^−s+1ϕ(k^sx,0,0,0)

→0 (m→ ∞).

This shows that the sequence

nf(kⁿx) kⁿ

o

is a Cauchy sequence in the Banach A- moduleX and therefore converges for all x∈ X. Putd(x) = lim

n→∞

f(kⁿx)

kⁿ for every x∈Xandf(0) =d(0) = 0. By (2.4), we get

(2.5) kf(x)−d(x)k ≤ 1

2ϕ(x)˜ (x∈X).

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Next, we show that the mapping d is additive. To do this, let us replace x, y by kⁿx, kⁿyin (2.2), respectively. Then

1 kⁿf

kⁿx

k +kⁿy l

+ 1

kⁿf kⁿx

k −kⁿy l

− 1

k · 2f(kⁿx) kⁿ

≤k⁻ⁿϕ(kⁿx, kⁿy,0,0) for allx, y ∈ X. If we let n → ∞ in the above inequality, then the condition (a) yields that

(2.6) dx

k + y l

+dx k − y

l

= 2 kd(x)

for all x, y ∈ X. Since d(0) = 0, taking y = 0 and y = _k^lx, respectively, we see thatd ^x_k

= ^d(x)_k and d(2x) = 2d(x) for allx ∈ X, and then we obtain that d(x+y)+d(x−y) = 2d(x)for allx, y ∈X. Now, for allu, v ∈X, putx= ^k₂(u+v), y= ₂^l(u−v). Then by (2.6), we get that

d(u) +d(v) = dx k +y

l

+dx k − y

l

= 2

kd(x) = 2 kd

k

2(u+v)

=d(u+v).

This shows thatdis additive.

Now, we are going to prove that f is a generalized module-A left derivation.

Lettingx=y= 0in (2.1), we get

kf(zw) +f(zw)−2zf(w)−2wg(z)k ≤ϕ(0,0, z, w), that is

(2.7) kf(zw)−zf(w)−wg(z)k ≤ 1

2ϕ(0,0, z, w)

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for allz ∈A andw∈X. By replacingz, w withkⁿz, kⁿwin (2.7) respectively, we deduce that

(2.8)

1

k²ⁿf k²ⁿzw

−z 1

kⁿf(kⁿw)−w 1

kⁿg(kⁿz)

≤ 1

2k⁻²ⁿϕ(0,0, kⁿz, kⁿw)

for allz ∈A andw∈X. Lettingn → ∞, condition (b) yields that

(2.9) d(zw) =zd(w) +wδ(z)

for all z ∈ A and w ∈ X. Since d is additive, δ is module-X additive. Put

∆(z, w) = f(zw) − zf(w)− wg(z). Then by (2.7) we see from condition (a) that

k⁻ⁿk∆(kⁿz, w)k ≤ 1

2k⁻ⁿϕ(0,0, kⁿz, w)→0 (n→ ∞) for allz ∈A andw∈X. Hence

d(zw) = lim

n→∞

f(kⁿz·w) kⁿ

= lim

n→∞

kⁿzf(w) +wg(kⁿz) + ∆(kⁿz, w) kⁿ

=zf(w) +wδ(z)

for allz ∈A andw ∈X. It follows from (2.9) thatzf(w) =zd(w)for allz ∈A andw ∈X, and thend(w) =f(w)for all w∈X. Sincedis additive,f is module- A additive. So, for alla, b∈A andx∈X by (2.9),

af(bx) =ad(bx) =abf(x) +axδ(b)

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and

xδ(ab) = d(abx)−abf(x)

=af(bx) +bxδ(a)−abf(x)

=a(d(bx)−bf(x)) +bxδ(a)

=axδ(b) +bxδ(a).

This shows that if δ is a module-X left derivation on A, then f is a generalized module-A left derivation onX.

Lastly, we prove that g is a module-X left derivation on A. To do this, we compute from (2.7) that

f(kⁿzw)

kⁿ −zf(kⁿw)

kⁿ −wg(z)

≤ 1

2k⁻ⁿϕ(0,0, z, kⁿw)

for allz ∈A and allw∈X. By lettingn→ ∞, we get from condition (a) that d(zw) = zd(w) +wg(z)

for allz ∈A and allw∈X. Now, (2.9) implies thatwg(z) =wδ(z)for allz ∈A and allw ∈ X. Hence, g is a module-X left derivation onA. This completes the proof.

Corollary 2.2. Let A be a Banach algebra, X a Banach A-bimodule, ε ≥ 0, p, q, s, t ∈ [0,1)andk andl be integers greater than 1. Suppose thatf : X → X andg :A →A are mappings such thatf(0) = 0,δ(z) := lim

n→∞

1

kⁿg(kⁿz)exists for allz ∈A and

(2.10)

∆¹_f,g(x, y, z, w)

≤ε(kxk^p +kyk^q+kzk^skwk^t)

for allx, y, w ∈X and allz ∈A (0⁰ := 1). Thenf is a generalized module-A left derivation andg is a module-Xleft derivation.

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Proof. It is easy to check that the function

ϕ(x, y, z, w) =ε(kxk^p+kyk^q+kzk^skwk^t) satisfies conditions (a), (b) and (c) of Theorem2.1.

Corollary 2.3. Let A be a Banach algebra with unit e, ε ≥ 0, and k and l be integers greater than1. Suppose that f, g : A → A are mappings withf(0) = 0 such that

∆¹_f,g(x, y, z, w) ≤ε

for allx, y, w, z ∈A. Thenf is a generalized left derivation andg is a left deriva- tion.

Proof. By takingw= ein (2.8), we see that the limitδ(z) := lim

n→∞

1

kⁿg(kⁿz)exists for allz ∈ A. It follows from Corollary2.2 and Remark1that f is a generalized left derivation andgis a left derivation. This completes the proof.

Lemma 2.4. LetX, Y be complex vector spaces. Then a mappingf : X → Y is linear if and only if

f(αx+βy) = αf(x) +βf(y) for allx, y ∈Xand allα, β ∈T:={z ∈C:|z|= 1}.

Proof. It suffices to prove the sufficiency. Suppose thatf(αx+βy) =αf(x)+βf(y) for all x, y ∈ X and all α, β ∈ T := {z ∈ C : |z| = 1}. Then f is additive and f(αx) = αf(x) for all x ∈ X and all α ∈ T. Let α be any nonzero complex number. Take a positive integern such that|α/n| < 2. Take a real numberθ such that0 ≤a :=e^−iθα/n <2. Putβ = arccos^a₂. Thenα =n(e^i(β+θ)+e^{−i(β−θ)})and therefore

f(αx) = nf(e^i(β+θ)x) +nf(e^{−i(β−θ)}x)

=ne^i(β+θ)f(x) +ne^{−i(β−θ)}f(x) =αf(x)

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for allx∈X. This shows thatf is linear. The proof is completed.

Theorem 2.5. Let A be a Banach algebra, X a Banach A-bimodule, k andl be integers greater than1, andϕ : X×X×A ×X → [0,∞)satisfy the following conditions:

(a) lim

n→∞k⁻ⁿ[ϕ(kⁿx, kⁿy,0,0) +ϕ(0,0, kⁿz, w)] = 0 (x, y, w ∈X, z ∈A).

(b) lim

n→∞k⁻²ⁿϕ(0,0, kⁿz, kⁿw) = 0 (z∈A, w ∈X).

(c) ϕ(x) :=˜ P∞

n=0k⁻ⁿ⁺¹ϕ(kⁿx,0,0,0)<∞ (x∈X).

Suppose that f : X → X and g : A → A are mappings such thatf(0) = 0, δ(z) := lim

n→∞

1

kⁿg(kⁿz)exists for allz ∈A and

(2.11)

∆³_f,g(x, y, z, w, α, β)

≤ϕ(x, y, z, w)

for all x, y, w ∈ X, z ∈ A and all α, β ∈ T := {z ∈ C : |z| = 1}, where

∆³_f,g(x, y, z, w, α, β)stands for

f αx

k +βy l +zw

+f

αx k − βy

l +zw

− 2αf(x)

k −2zf(w)−2wg(z).

Thenf is a linear generalized module-A left derivation andg is a linear module-X left derivation.

Proof. Clearly, the inequality (2.1) is satisfied. Hence, Theorem 2.1 and its proof show thatf is a generalized left derivation andg is a left derivation onA with

(2.12) f(x) = lim

n→∞

f(kⁿx)

kⁿ , g(x) =f(x)−xf(e)

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for everyx∈X. Takingz=w= 0in (2.11) yields that (2.13)

f

αx k +βy

l

+f αx

k −βy l

− 2αf(x) k

≤ϕ(x, y,0,0)

for allx, y ∈X and allα, β ∈T. If we replacexandywithkⁿxandkⁿyin (2.13) respectively, then we see that

1 kⁿf

αkⁿx

k + βkⁿy l

+ 1

kⁿf

αkⁿx

k − βkⁿy l

− 1 kⁿ

2αf(kⁿx) k

≤k⁻ⁿϕ(kⁿx, kⁿy,0,0)

→0

asn → ∞for allx, y ∈Xand allα, β ∈T. Hence,

(2.14) f

αx k +βy

l

+f αx

k −βy l

= 2αf(x) k

for allx, y ∈Xand allα, β ∈T. Sincef is additive, takingy = 0in (2.14) implies that

(2.15) f(αx) =αf(x)

for allx ∈ X and all α ∈ T. Lemma2.4 yields thatf is linear and so is g. Next, similar to the proof of Theorem 2.3 in [15], one can show that g(A) ⊂ Z(A)∩ rad(A). This completes the proof.

Corollary 2.6. LetA be a complex semi-prime Banach algebra with unite,ε ≥0, p, q, s, t∈[0,1)andkandlbe integers greater than1. Suppose thatf, g :A →A are mappings withf(0) = 0and satisfy following inequality:

(2.16)

∆³_f,g(x, y, z, w, α, β)

≤ε(kxk^p+kyk^q+kzk^skwk^t)

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for allx, y, z, w ∈A and allα, β ∈T(0⁰ := 1). Thenf is a linear generalized left derivation andgis a linear left derivation which mapsA into the intersection of the center Z(A)and the Jacobson radical rad(A)ofA.

Proof. Since A has a unit e, letting w = e in (2.8) shows that the limit δ(z) :=

n→∞lim

1

kⁿg(kⁿz) exists for allz ∈ A. Thus, using Theorem 2.5 for ϕ(x, y, z, w) = ε(kxk^p+kyk^q +kzk^skwk^t)yields thatf is a linear generalized left derivation and g is a linear left derivation sinceA has a unit. Similar to the proof of Theorem 2.3 in [15], one can check that the mappinggmapsA into the intersection of the center Z(A)and the Jacobson radical rad(A)ofA. This completes the proof.

Corollary 2.7. LetA be a complex semiprime Banach algebra with unite, ε ≥ 0, kandl be integers greater than1. Suppose thatf, g :A →A are mappings with f(0) = 0and satisfy the following inequality:

∆³_f,g(x, y, z, w, α, β) ≤ε

for allx, y, z, w ∈A and allα, β ∈T. Thenf is a linear generalized left derivation and g is a linear left derivation which maps A into the intersection of the center Z(A)and the Jacobson radical rad(A)ofA.

Remark 2. Inequalities (2.10) and (2.16) are controlled by their right-hand sides by the “mixed sum-product of powers of norms", introduced by J. M. Rassias (in 2007) and applied afterwards by K. Ravi et al. (2007-2008). Moreover, it is easy to check that the function

ϕ(x, y, z, w) =Pkxk^p+Qkyk^q+Skzk^s+Tkwk^t

satisfies conditions (a), (b) and (c) of Theorem2.1and Theorem2.5, whereP, Q, T, S∈ [0,∞)andp, q, s, t∈[0,1)are all constants.

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