We need to take account of the superstability for generalized left
derivations (resp., generalized derivations) associated with a Jensen-type
functional equation, and we also deal with problems for the Jacobson radical
ranges of left derivations (resp., derivations).

1. Introduction

Let 𝒜 be an algebra over the real or complex field 𝔽. An additive mapping d:𝒜→𝒜 is said to be a left derivation (resp., derivation) if the functional equation d(xy)=xd(y)+yd(x) (resp., d(xy)=xd(y)+d(x)y) holds for all x,y∈𝒜. Furthermore, if the functional equation d(λx)=λd(x) is valid for all λ∈𝔽 and all x∈𝒜, then d is a linear left derivation (resp., linear derivation). An additive mapping G:𝒜→𝒜 is called a generalized left derivation (resp., generalized derivation) if there exists a
left derivation (resp., derivation) δ:𝒜→𝒜 such that the functional equation G(xy)=xG(y)+yδ(x) (resp., G(xy)=xG(y)+δ(x)y) is fulfilled for all x∈𝒜. In addition, if the functional equations G(λx)=λG(x) and δ(λx)=λδ(x) hold for all λ∈𝔽 and all x∈𝒜, then G is a linear generalized left derivation (resp., linear generalized derivation).

It is of interest to consider the concept of stability
for a functional equation arising when we replace the functional equation by an
inequality which acts as a perturbation of the equation. The study of stability
problems had been formulated by Ulam [1] during a talk in 1940: “Under what condition does there exists a
homomorphism near an approximate homomorphism?” In the following year
1941, Hyers [2]
was answered affirmatively the question of Ulam for Banach spaces, which states
that if ε>0 and f:𝒳→𝒴 is a map with 𝒳, a normed space, 𝒴, a Banach space, such that∥f(x+y)−f(x)−f(y)∥≤ε,for all x,y∈𝒳, then there exists a unique additive mapT:𝒳→𝒴such that∥f(x)−T(x)∥≤ε,for all x∈𝒳. Moreover, if f(tx) is continuous in t∈ℝ for each fixed x in 𝒳, where ℝ denotes the set of real numbers, then T is 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 [3] and for approximate linear mappings was presented by
Rassias [4] in
1978 by considering the case when the inequality (1.1) is unbounded. Due to the
fact that the additive functional equation f(x+y)=f(x)+f(y) is said to have the Hyers-Ulam-Rassias stability property.
The stability result concerning derivations between operator algebras was first
obtained by Šemrl [5]. Recently, Badora
[6] gave a
generalization of the Bourgin's result [7]. He also dealt with the Hyers-Ulam stability and the
Bourgin-type superstability of derivations in [8].

In 1955, Singer and Wermer [9] obtained a fundamental
result which started investigation into the ranges of linear derivations on
Banach algebras. The result, which is called the Singer-Wermer theorem, states
that any continuous linear derivation on a commutative Banach algebra maps into
the Jacobson radical. They also made a very insightful conjecture, namely, that
the assumption of continuity is unnecessary. This was known as the
Singer-Wermer conjecture and was proved in 1988 by Thomas [10]. The Singer-Wermer
conjecture implies that any linear derivation on a commutative semisimple
Banach algebra is identically zero which is the result of Johnson
[11]. After then, Hatori
and Wada [12] showed
that a zero operator is the only derivation on a commutative semisimple Banach
algebra with the maximal ideal space without isolated points. Note that this
differs from the above result of B.E. Johnson. Based on these facts and a
private communication with Watanabe [13], Miura
et al. proved the Hyers-Ulam-Rassias
stability and Bourgin-type superstability of derivations on Banach algebras in
[13]. Various
stability results are given by Moslehian and Park, see, for example,
[14–18].

The main purpose of the present paper is to consider
the superstability of generalized left derivations (resp., generalized
derivations) on Banach algebras associated to the following Jensen type
functional equation:f(x+yk)=f(x)k+f(y)k,where k>1 is an integer. This functional equation is
introduced in [19].
Moreover, we will investigate the problems for the Jacobson radical ranges of
left derivations (resp., derivations) on Banach algebras. We use the direct
method and some ideas of Amyari et al. [19].

2. Main Results

Throughout this paper, the element e of an algebra will denote a unit. We now
establish the superstability of a generalized left derivation associated with
the Jensen type functional equation as follows.

Theorem 2.1.

Let 𝒜 be a Banach algebra with unit. Suppose that f:𝒜→𝒜 is a mapping with f(0)=0 for which there exists a mapping g:𝒜→𝒜 such that the functional
inequality:∥f(x+yk+zw)−f(x)k−f(y)k−zf(w)−wg(z)∥≤ε,for all x,y,z,w∈𝒜. Then, f is a generalized left derivation, and g is a left derivation.

Proof.

Substituting w=0 in (2.1), we get∥f(x+yk)−f(x)k−f(y)k∥≤ε,for all x,y∈𝒜. Let us take y=0 and replace x by kx in the above relation. Then, it
becomes∥f(x)−f(kx)k∥≤ε,for all x∈𝒜. An induction implies that∥f(knx)kn−f(x)∥≤kk−1(1−1kn)ε,for all x∈𝒜. By virtue of (2.4), one can easily check that
for n>m,∥f(knx)kn−f(kmx)km∥=1km∥f(kn−m⋅kmx)kn−m−f(kmx)∥≤1km−1(k−1)(1−1kn−m)ε,for all x∈𝒜. So, the sequence {f(knx)/kn} is Cauchy. Since 𝒜 is complete, {f(knx)/kn} converges. Let d:𝒜→𝒜 be the mapping defined by (x∈𝒜)d(x):=limn→∞f(knx)kn.By letting n→∞ in (2.4), we get∥f(x)−d(x)∥≤kk−1ε,for all x∈𝒜.

Now, we assert that d is additive. Replacing x and y by knx and kny in (2.2), respectively, we
have∥1knf(knx+knyk)−1kf(knx)kn−1kf(kny)kn∥≤1knε,for all x,y∈𝒜, taking the limit as n→∞, we obtaind(x+yk)=d(x)k+d(y)k,for all x,y∈𝒜. Letting y=0 in the previous identity yields d(x/k)=d(x)/k for all x∈𝒜. So, (2.9) becomes d(x+y)=d(x)+d(y), for all x,y∈𝒜, namely, d is additive.

To demonstrate the uniqueness of the additive mapping d subject to (2.7), we assume that there exists
another additive mapping D:𝒜→𝒜 satisfying the inequality (2.7), for all x∈𝒜. Since D(knx)=knD(x) and d(knx)=knd(x), we see that∥D(x)−d(x)∥=1kn∥D(knx)−d(knx)∥≤1kn[∥D(knx)−f(knx)∥+∥f(knx)−d(knx)∥]≤2kn−1(k−1)ε,for all x∈𝒜. By letting n→∞ in this inequality, we conclude that D=d, that is, d is unique.

Next, we are going to prove that f is a generalized left derivation. If we take x=y=0 in (2.1), we also have∥f(zw)−zf(w)−wg(z)∥≤ε,for all z,w∈𝒜. Moreover, if we replace z and w with knz and knw in (2.11), respectively, and then divide both
sides by k2n, we get∥f(k2nzw)k2n−zf(knw)kn−wg(knz)kn∥≤1k2nε,for all z,w∈𝒜. Letting n→∞, we obtainlimn→∞wg(knz)kn=d(zw)−zd(w),for all z,w∈𝒜. Suppose that w=e in the above equation. Then, it
followslimn→∞g(knz)kn=d(z)−zd(e),for all z∈𝒜. Thus, if δ(z):=d(z)−zd(e), then by the additivity of d, we getδ(x+y)=d(x)+d(y)−xd(e)−yd(e)=(d(x)−xd(e))+(d(y)−yd(e))=δ(x)+δ(y),for all x∈𝒜. Hence, δ is additive.

Let Δ(z,w)=f(zw)−zf(w)−wg(z), for all z,w∈𝒜. Since, f and g satisfy the inequality given in (2.11),
thenlimn→∞Δ(knz,w)kn=0,for all z,w∈𝒜. We note thatd(zw)=limn→∞f(knzw)kn=limn→∞f(knz⋅w)kn=limn→∞knzf(w)+wg(knz)+Δ(knz,w)kn=limn→∞{zf(w)+wg(knz)kn+Δ(knz,w)kn}=zf(w)+wδ(z),for all z,w∈𝒜. Since δ is additive, we can rewrite (2.17)
asknzf(w)+knwδ(z)=d(knz⋅w)=d(z⋅knw)=zf(knw)+knwδ(z),for all z,w∈𝒜. Based on the above relation, one has zf(w)=z(f(knw)/kn),
for all z,w∈𝒜. Moreover, we can obtain zf(w)=zd(w), for all z,w∈𝒜 as n→∞. If z=e, we also have that f=d. Therefore, we getf(zw)=zf(w)+wδ(z),for all z,w∈𝒜.

We now want to verify that δ is a left derivation using the equations
developed in the previous part. Indeed, using the
facts that f satisfies (2.19), we haveδ(xy)=f(xy)−xyf(e)=xf(y)+yδ(x)−xyf(e)=x(f(y)−yf(e))+yδ(x)=xδ(y)+yδ(x),for all x,y∈𝒜, which means that f is a generalized left derivation.

We finally need to show that g is a left derivation. Let us replace w by knw in (2.11). Then,∥f(knzw)kn−zf(knw)kn−wg(z)∥≤1knε,for all z,w∈𝒜. Passing the limit as n→∞, we getd(zw)−zd(w)−wg(z)=0,for all z,w∈𝒜. This implies that d(zw)=zd(w)+wg(z), for all z,w∈𝒜, and thus if w=e, we deduce that g(z)+zd(e)=d(z), for all z∈𝒜. Hence, we get g(z)=d(z)−zd(e)=δ(z), for all z∈𝒜. Since, δ is a left derivation, we can conclude that g is a left derivation as well. This completes
the proof of the theorem.

Employing the similar way as in the proof of Theorem
2.1, we get the following result for a generalized derivation.

Theorem 2.2.

Let 𝒜 be a Banach algebra with unit. Suppose that f:𝒜→𝒜 is a mapping with f(0)=0 for which there exists a mapping g:𝒜→𝒜 such that∥f(x+yk+zw)−f(x)k−f(y)k−zf(w)−g(z)w∥≤ε,for all x,y,z,w∈𝒜. Then, f is a generalized derivation, and g is a derivation.

In view of the Thomas' result [10], derivations on Banach
algebras now belong to the noncommutative setting. Among various noncommutative
version of the Singer-Wermer theorem, Brešar and Vukman
[20] proved the
following. Any continuous linear
left derivation on a Banach algebra maps into its Jacobson radical and also any
left derivation on a semiprime ring is a derivation which maps into its center.

The following is the functional inequality with the
problem as in the above Brešar and Vukman's
result.

Theorem 2.3.

Let 𝒜 be a semiprime Banach algebra with unit.
Suppose that f:𝒜→𝒜 is a mapping with f(0)=0 for which there exists a mapping g:𝒜→𝒜 such that the functional
inequality:∥f(αx+βyk+zw)−αf(x)k−βf(y)k−zf(w)−wg(z)∥≤ε,for all x,y,z,w∈𝒜 and all α,β∈𝕌={z∈C:|z|=1}. Then, f is a linear generalized left derivation. In
this case, g is a linear derivation which maps 𝒜 into the intersection of its center Z(𝒜) and its Jacobson radical rad(𝒜).

Proof.

We consider α=β=1∈𝕌 in (2.24) and then f satisfies the inequality (2.1). It follows
from Theorem 2.1 that f is a generalized left derivation, and g is a left derivation, wheref(x):=limn→∞f(knx)kn,g(x):=f(x)−xf(e),for all x∈𝒜. Letting w=0 in (2.24), we have∥f(αx+βyk)−αf(x)k−βf(y)k∥≤ε,for all x,y∈𝒜 and all α,β∈𝕌. If we also replace x and y with knx and kny in (2.26), respectively, and then divide both
sides by kn, we see that∥1knf(αknx+βknyk)−α1kf(knx)kn−β1kf(kny)kn∥≤1knε⟶0,for all x,y∈𝒜 and all α,β∈𝕌, as n→∞. So, we getf(αx+βyk)=αf(x)k+βf(y)k,for all x,y∈𝒜 and all α,β∈𝕌. From the additivity of f, we find thatf(αx+βy)=αf(x)+βf(y),for all x,y∈𝒜 and all α,β∈𝕌. Let us now assume that λ is a nonzero complex number and that L a positive integer greater than |λ|. Then by applying a geometric argument, there
exist λ1,λ2∈𝕌 such that 2(λ/L)=λ1+λ2.
In particular, due to the additivity of f, we obtain f((1/2)x)=(1/2)f(x) for all x∈𝒜. Thus, we havef(λx)=f(L2⋅2⋅λLx)=Lf(12⋅2⋅λLx)=L2f((λ1+λ2)x)=L2(λ1+λ2)f(x)=L2⋅2⋅λLf(x)=λf(x),for all x∈𝒜. Also, it is obvious that f(0x)=0=0f(x), for all x∈𝒜, that is, f is ℂ-linear. Therefore, f is a linear generalized left derivation, and
so g is also a linear left derivation. According to
the Brešar and Vukman's
result which tells us that g is a linear derivation which maps 𝒜 into its center Z(𝒜). Since Z(𝒜) is a commutative Banach algebra, the
Singer-Wermer conjecture tells us that g|Z(𝒜) maps Z(𝒜) into rad(Z(𝒜))=Z(𝒜)∩rad(𝒜) and thus g2(𝒜)⊆rad(𝒜). Using the semiprimeness of rad(𝒜) as well as the identity, we
have2g(x)yg(x)=g2(xyx)−xg2(yx)−g2(xy)x+xg2(y)x,for all x,y∈𝒜, we have g(𝒜)⊆rad(𝒜), that is, g is a linear derivation which maps 𝒜 into the intersection of its center Z(𝒜) and its Jacobson radical rad(𝒜). The proof of the theorem is ended.

The next corollary is the Brešar and Vukman's
result.

Corollary 2.4.

Let 𝒜 be a Banach algebra with unit. Suppose that f:𝒜→𝒜 is a continuous mapping with f(0)=0 for which there exists a mapping g:𝒜→𝒜 such that the functional inequality (2.26).
Then, f:𝒜→𝒜 is a linear generalized left derivation. In
this case, g maps 𝒜 into its Jacobson radical rad(𝒜).

Proof.

On account of Theorem 2.3, g is a linear left derivation on 𝒜. Hence, g maps 𝒜 into its Jacobson radical rad(𝒜) by the Brešar and Vukman's
result, which completes the proof.

With the help of Theorem 2.2, the following property
can be derived along the same argument in the proof of Theorem 2.3.

Theorem 2.5.

Let 𝒜 be a commutative Banach algebra with unit.
Suppose that f:𝒜→𝒜 is a mapping with f(0)=0 for which there exists a mapping g:𝒜→𝒜 such that the functional inequality:∥f(αx+βyk+zw)−αf(x)k−βf(y)k−zf(w)−g(z)w∥≤ε,for all x,y,z,w∈𝒜 and all α,β∈𝕌={z∈C:|z|=1}. Then, f is a linear generalized derivation. In this
case, g maps 𝒜 into its Jacobson radical rad(𝒜).

Remark 2.6.

We can generalize our results by substituting another
functions or another forms satisfying suitable conditions (see, e.g.,
[19, 21]) for the bound ε of the functional inequalities connected to
the Jensen type functional equation.

Acknowledgments

The authors would like to thank referees for their
valuable comments regarding a previous version of this paper. The corresponding
author dedicates this paper to his late father.

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