KYUNGPOOK Math. J. 2019; 59(4): 689-702
On the Stability of a Higher Functional Equation in Banach Algebras
Yong-Soo Jung
Department of Mathematics, Sun Moon University, Asan, Chungnam 336-708, Korea
e-mail : ysjung@sunmoon.ac.kr
Received: July 25, 2018; Accepted: May 10, 2019; Published online: December 23, 2019.

This is an Open Access article distributed under the terms of the Creative Commons Attribution Non-Commercial License (http://creativecommons.org/licenses/by-nc/3.0/) which permits unrestricted non-commercial use, distribution, and reproduction in any medium, provided the original work is properly cited.
Abstract

Let and ℬ be real (or complex) algebras. We investigate the stability of a sequence F = {f0, f1, · · ·, fn, · · ·} of mappings from into ℬ satisfying the higher functional equation: $fn(x+y+zw)=fn(x)+fn(y)+∑i+j=ni≤j[fi(z)fj(w)+cijfi(w)fj(z)]$

for each n = 0, 1, · · · and all , where $cij={1if i≠j,0if i=j.$

Keywords: higher functional equation, stability.
1. Introduction and Preliminaries

The study of stability problems originated from a famous talk given by S.M. Ulam [18] in 1940 : Under what condition does there exists a homomorphism near an approximate homomorphism ? In the next year 1941, D. H. Hyers [8] was answered affirmatively the question of Ulam for Banach spaces, which states that if δ > 0 andis a map witha normed space, a Banach space such that

$‖f(x+y)−f(x)−f(y)‖≤δ$

for all, then there exists a unique additive mappingsuch that

$‖f(x)−T(x)‖≤δ$

for all. A generalized version of the theorem of Hyers for approximately additive mappings was first given by T. Aoki [1] in 1950. In 1978, Th.M. Rassias [15] independently introduced the unbounded Cauchy difference and was the first to prove the stability of the linear mapping between Banach spaces: if there exist a θ ≥ 0 and 0 ≤ p < 1 such that

$‖f(x+y)−f(x)−f(y)‖≤θ(‖x‖p+‖y‖p)$

for all, then there exists a unique additive mapsuch that

$‖f(x)−T(x)‖≤2θ2−2p‖x‖p$

for all.

In 1991, Z. Gajda [6] answered the question for the case p > 1, which was raised by Rassias. Gajda [6] also gave an example that the Rassias’ stability result is not valid for p = 1.

In 1992, a generalization of the Rassias theorem was obtained by P. Gǎvruţǎ [7]:

Suppose ( , +) is an abelian group, is a Banach space and the so-called admissible control functionsatisfies

$ψ(x,y):=12∑k=0∞ϕ(2kx,2ky)2k<∞,$

for all. Ifis a mapping such that

$‖f(x+y)−f(x)−f(y)‖≤ϕ(x,y)$

for all, then there exists a unique additive mappingsuch that

$‖f(x)−T(x)‖≤ψ(x,x)$

for all.

Throughout this note, let ℕ be the set of natural numbers and we assume that and ℬ are algebras over the real or complex field . An additive mapping is said to be a ring homomorphism if the functional equation h(xy) = h(x)h(y) holds for all . An additive mapping is said to be a ring left derivation (resp. ring derivation) if the functional equation d(xy) = xd(y) + yd(x) (resp. d(xy) = xd(y) + d(x)y) holds for all . In addition, d is called a linear left derivation (resp. linear derivation) if the functional equation d(λx) = λd(x) is valid for all and for all .

M. Brešar and J. Vukman [5, Proposition 1.6] showed that every ring left derivation on a semiprime ring is a ring derivation which maps the ring into its center.

### Definition 1.1

A sequence H = {h0, h1, · · ·, hn, · · · } of additive mappings from into ℬ is called a higher ring left derivation (resp. higher ring derivation) from into ℬ if the functional equation

$hn(xy)=∑i+j=ni≤j[hi(x)hj(y)+cijhi(y)hj(x)] (resp. hn(xy)=∑i+j=nhi(x)hj(y))$

holds for each n = 0, 1, · · · and for all , where

$cij={1if i≠j,0if i=j.$

Let . The sequence H = {h1, h2, · · ·, hn, · · ·} of additive mappings on satisfying the relation (1.1), particulary, is called a strong higher ring left derivation (resp. strong higher ring derivation) on if h0 acts as an identity mapping on in (1.1). If each hn in H satisfies the functional equation hn(λx) = λhn(x) for all and all , then we say that H is a higher linear left derivation (resp. higher linear derivation).

### Remark 1.2

K.-H. Park [14] proved that every strong higher ring left derivation H = {h1, h2, · · ·, hn, · · · } on a semiprime ring is a strong higher ring derivation such that each hn in H maps the ring into its center. In Definition 1.1, the higher ring left derivation H from into ℬ (resp. the strong higher ring left derivation on ) is a ring homomorphism if n = 0 (resp. a ring left derivation if n = 1).

Consider a sequence F = {f0, f1, · · ·, fn, · · · } of mappings from into ℬ such that the functional equation

$fn(x+y+zw)=fn(x)+fn(y)+∑i+j=ni≤j[fi(z)fj(w)+cijfi(w)fj(z)]$

holds for each n = 0, 1, · · · and all , where

$cij={1if i≠j,0if i=j.$

For convenience’ sake, we will say that the relation (1.2) is a higher functional equation. In particular, if fn(0) = 0 for each n = 0, 1, · · ·, then we see that F is a higher ring left derivation from into ℬ.

### Remark 1.3

Let F = {f1, f2, · · ·, fn, · · ·} be a sequence of mappings on and f0 an identity mapping on . Then F satisfies the functional equation (1.2) if and only if F is a strong higher ring left derivation on . In fact, F is a strong higher ring left derivation on since it follows from induction that for each fn in F, we get fn(0) = 0.

### Example 1.4

Given any ring left derivation d on an algebra with unit and an invertible element , let be the mapping defined by δ(x) = cd(x) for all . Then δ(xy) = ϑ(x)δ(y) + ϑ(y)δ(x) for all , where the relation ϑ(x) = cxc−1, , defines an inner automorphism of . Let f0 = ϑ, fn = δ if n = m for some m ∈ ℕ, and fn = 0 if n ≥ 1 and nm. Then we see that the sequence F = {f0, f1, · · ·, fn, · · · } of mappings on satisfies the equation (1.2).

In 1949, D.G. Bourgin [4] proved the following stability result, which is sometimes called the superstability of ring homomorphisms: suppose thatandare Banach algebras with unit. Ifis a surjective mapping such that

$‖f(x+y)−f(x)−f(y)‖≤ε,‖f(xy)−f(x)f(y)‖≤δ$

for some ɛ > 0, δ > 0 and all, then f is a ring homomorphism.

R. Badora [2] gave a generalization of the Bourgin’s result. Badora [3] also obtained the following results for the stability in the sense of Hyers and Ulam and for the superstability of ring derivations: letbe a subalgebra of a Banach algebra. Assume thatis a mapping such that

$‖f(x+y)−f(x)−f(y)‖≤ε,‖f(xy)−xf(y)−f(x)Y‖≤δ$

for some ɛ ≥ 0, δ ≥ 0 and all. Then there exists a unique ring derivationsuch that

$‖f(x)−d(x)‖≤ε$

for all. Moreover,

$x{f(y)−d(y)}=0$

for all. In addition, ifandhave the unit element, then f is a ring derivation.

On the other hand, T. Miura et al. [13] proved the stability in the sense of Hyers, Ulam and Rassias and the superstability of ring derivations on Banach algebras.

In [11] and [12], we dealt with the stability of higher ring derivations and higher ring left derivations, respectively.

Here it is natural to ask that there exists an approximate sequence of mappings which is not an exactly sequence of mappings satisfying the functional equation (1.2). We observe the following example.

### Example 1.5

Let A be a compact Hausdorff space and let C(A) be the commutative Banach algebra of complex-valued continuous functions on A under pointwise operations and the supremum norm ||·||. Assume that ϑ : C(A) → C(A) is a nonzero algebra homomorphism. We define g : C(A) → C(A) by

$g(z)(a)={ϑ(z)(a)log∣ϑ(z)(a)∣if ϑ(z)(a)≠0,0 if ϑ(z)(a)=0$

for all zC(A) and aA. It is easy to see that g(zw) = ϑ(z)g(w) + g(z)ϑ(w) for all z, wC(A). Let f0 = ϑ, fn = g if n = m for some m ∈ ℕ, and fn = 0 if n ≥ 1 and nm. Then we see that the sequence F = {f0, f1, · · ·, fn · · · } satisfies the relation

$fn(zw)=∑i+j=ni≤j[fi(z)fj(w)+cijfi(w)fj(z)]$

for each n = 0, 1, · · ·, and all z, wC(A), where

$cij={1if i≠j,0if i=j.$

We claim that

$‖fn(x+y+zw)−fn(x)−fn(y)−fn(zw)‖∞≤2(‖x‖∞+‖y‖∞+‖zw‖∞)$

for all x, y, z, wC(A).

Observe that for all with x + y + zw ≠ 0,

$∣(x+y+zw) log ∣x+y+zw∣−x log ∣x∣−y log ∣y∣−zw log ∣zw‖≤2(∣x∣+∣y∣+∣zw∣)$

Indeed, fix , x + y + zw ≠ 0 arbitrarily. Since log(1 + x) ≤ x for all x ≥ 0, we get

$∣(x+y+zw) log ∣x+y+zw∣−x log ∣x∣−y log ∣y∣−zw log ∣zw∣∣≤∣x∣|log∣x+y+zw∣∣x∣|+∣y∣|log∣x+y+zw∣∣y∣|+∣zw∣|log∣x+y+zw∣∣zw∣|≤∣x∣log(1+∣y+zw∣∣x∣)+∣y∣log(1+∣x+zw∣∣y∣)+∣zw∣log(1+∣x+y∣∣zw∣)≤∣x∣∣y+zw∣∣x∣+∣y∣∣x+zw∣∣y∣+∣zw∣∣x+y∣∣zw∣=∣y+zw∣+∣x+zw∣+∣x+y∣≤2(∣x∣+∣y∣+∣zw∣).$

This yields, for each n = 0, 1, · · ·,

$‖fn(x+y+zw)−fn(x)−fn(y)−fn(zw)‖∞≤2(‖x‖∞+‖y‖∞+‖zw‖∞)$

for all x, y, z, wC(A).

Now, it follows that

$‖fn(x+y+zw)−fn(x)−fn(y)−∑i+j=ni≤j[fi(z)fj(w)+cijfi(w)fj(z)]‖∞=‖fn(x+y+zw)−fn(x)−fn(y)−fn(zw)‖∞≤2(‖x‖∞+‖y‖∞+‖zw‖∞)$

for each n = 0, 1, · · · and all x, y, z, wC(A). Thus we may regard F as an approximate sequence of mappings on C(A) with respect to the equation (1.2).

2. Main results

Our objective is to investigate the stability of the higher functional equation (1.2) in the sense of the generalized version of Hyers-Ulam-Rassias due to [7]. Furthermore, we will show the superstability of the equation (1.2).

### Theorem 2.1

Letbe an algebra anda Banach algebra. For each n = 0, 1, 2, · · ·, letbe a function such that

$ψn(x,y,z,w)=12∑k=0∞ϕn(2kx,2ky,2kz,w)2k<∞$

for all. Suppose that F = {f0, f1, · · ·, fn, · · · } is a sequence of mappings fromintosuch that each n = 0, 1, · · ·,

$‖fn(x+y+zw)−fn(x)−fn(y)−∑i+j=ni≤j[fi(z)fj(w)+cijfi(w)fj(z)]‖≤ϕn(x,y,z,w)$

holds for all. Then there exists a unique higher ring left derivation H = {h0, h1, · · ·, hn, · · · } fromintosuch that for each n = 0, 1, · · ·,

$‖fn(x)−hn(x)‖≤ψn(x,x,0,0)+cn$

holds for all, where

$cn=‖∑i+j=ni≤j[fi(0)fj(0)+cijfi(0)fj(0)]‖$

for each n = 0, 1, · · ·. Moreover,

$∑i+j=ni≤jhi(x)[fj(y)−hj(y)]+∑i+j=ni≤jcij[fi(y)−hi(y)]hj(x)=0$

for each n = 0, 1, · · · and all.

Proof

Putting z = w = 0 in (2.2), we have

$‖fn(x+y)−fn(x)−fn(y)−∑i+j=ni≤j[fi(0)fj(0)+cijfi(0)fj(0)]‖≤ϕn(x,y,0,0)$

which yields

$‖fn(x+y)−fn(x)−fn(y)‖≤‖fn(x+y)−fn(x)−fn(y)−∑i+j=ni≤j[fi(0)fj(0)+cijfi(0)fj(0)]‖+‖∑i+j=ni≤j[fi(0)fj(0)+cijfi(0)fj(0)]‖≤ϕn(x,y,0,0)+cn,$

i.e.,

$‖fn(x+y)−fn(x)−fn(y)‖≤ϕn(x,y,0,0)+cn$

for each n = 0, 1, · · · and all .

Using Hyers’ direct method on inequality (2.5), it follows from induction on l that

$‖12lfn(2lx)−fn(x)‖≤12∑k=0l−1ϕn(2kx,2kx,0,0)+cn2k$

for each n = 0, 1, · · · and all and that

$‖12lfn(2lx)−12mfn(2mx)‖≤12∑k=ml−1ϕn(2kx,2kx,0,0)+cn2k$

for each l > m and all . Hence the convergence of (2.1) tells us that the sequence {$12lfn(2lx)$} is Cauchy for each n = 0, 1, · · · and all . Let

$hn(x)=liml→∞12lfn(2lx)$

for each n = 0, 1, · · · and all . Taking l → ∞ in (2.6), we obtain (2.3). In view of the same process as Hyers’ method [8], we see that each mapping hn, n = 0, 1, · · ·, is additive and unique.

Next, we need to show that the sequence H = {h0, h1, · · ·, hn, · · · } satisfies the identity

$hn(xy)=∑i+j=ni≤j[hi(x)hj(y)]+cijhi(y)hj(x)$

for each n = 0, 1, · · · and all . Setting x = y = 0 in (2.2), we get

$‖fn(zw)−2fn(0)−∑i+j=ni≤j[fi(z)fj(w)+cijfi(w)fj(z)]‖≤ϕn(0,0,z,w)$

which implies that

$‖fn(zw)−∑i+j=ni≤j[fi(z)fj(w)+cijfi(w)fj(z)]‖≤ϕn(0,0,z,w)+2‖fn(0)‖$

for each n = 0, 1, · · · and all . Let a function be defined by

$Δn(z,w)=fn(zw)−∑i+j=ni≤j[fi(z)fj(w)+cijfi(w)fj(z)]$

for each n = 0, 1, · · · and all . Using (2.1) and (2.8), we have

$liml→∞12lΔn(2lz,w)=0$

for each n = 0, 1, · · · and all . Now, from (2.7), (2.9) and (2.10), we deduce that

$hn(zw)=liml→∞12lfn(2l(zw))=liml→∞12lfn((2lz)w))=liml→∞12l{∑i+j=ni≤j[fi(2lz)fj(w)+cijfi(w)fj(2lz)]+Δn(2lz,w)}=liml→∞∑i+j=ni≤j12l[fi(2lz)fj(w)+cijfi(w)fj(2lz)]+liml→∞12lΔn(2lz,w)=∑i+j=ni≤j{limx→∞12lfi(2lz)fj(y)+cijliml→∞12lfi(w)fj(2lz)}=∑i+j=ni≤j[hi(z)fj(w)+cijfi(w)hj(z)]$

That is, we obtain that

$hn(zw)=∑i+j=ni≤j[hi(z)fj(w)+cijfi(w)hj(z)]$

for each n = 0, 1, · · · and all . Let l ∈ ℕ be fixed. Applying the relation (2.11) and the additivity of each hn, n = 0, 1, · · ·, we get

$∑i+j=ni≤j[hi(z)fj(2lw)+cijfi(2lw)hj(z)]=hn(z(2lw))=hn((2lz)w)=∑i+j=ni≤j[hi(2lz)fj(w)+cijfi(w)hj(2lz)]=2l∑i+j=ni≤j[hi(z)fj(y)+cijfi(w)hj(z)]$

Hence we get

$∑i+j=ni≤j[hi(z)fj(w)+cijfi(w)hj(z)]=∑i+j=ni≤j[hi(z)12lfj(2lw)+cij12lfi(2lw)hj(z)]$

for each n = 0, 1, · · · and all . Taking l→∞ in (2.12), we see that

$∑i+j=ni≤j[hi(z)fj(w)+cijfi(w)hj(z)]=∑i+j=ni≤j[hi(z)hj(w)+cijhi(w)hj(z)]$

for each n = 0, 1, · · · and all which means (2.4). Combining (2.11) with (2.13), it follows that H = {h0, h1, · · ·, hn, · · · } satisfies the relation

$hn(xy)=∑i+j=ni≤j[hi(z)hj(w)+cijhi(w)hj(z)].$

for each n = 0, 1, · · · and all , i.e., H is a higher ring left derivation from into ℬ. This completes the proof of the theorem.

### Remark 2.2

Let be an algebra and a function such that

$ψn(x,y,z,w)=12∑k=0∞2kϕn(2−kx,2−ky,2−kz,w)<∞$

for all . If we replace (2.10) in Theorem 2.1 by

$liml→∞2lΔn(2−lz,w)=0,$

then (2.4) in Theorem 2.1 does not generally hold in case of ψn(x, y, z, w) < ∞. For, we see that

$liml→∞2lΔn(2−lz,w)=0$

is not valid since limn→∞2l+1||fn(0)|| ≠ 0.

### Corollary 2.3

Letbe a normed algebra anda Banach algebra. Let θn ∈ (0,∞) for each n = 0, 1, · · · and p, q real numbers such that p ≠ 1. Suppose that F = {f0, f1, · · ·, fn, · · · } is a sequence of mappings fromintosuch that for each n = 0, 1, · · ·,

$‖fn(x+y+zw)−fn(x)−fn(y)−∑i+j=ni≤j[fi(z)fj(w)+cijfi(w)fj(z)]‖≤θn(‖x‖p+‖y‖p+‖z‖p‖w‖q)$

holds for all. Then there exists a unique higher ring left derivation H = {h0, h1, · · ·, hn, · · · } fromintosuch that for each n = 0, 1, · · ·,

$‖fn(x)−hn(x)‖≤{2θn2−2p‖x‖p+cnif p<1,2pθn2p−2‖x‖p+cnif p>1$

holds all, where

$cn=‖∑i+j=ni≤j[fi(0)fj(0)+cijfi(0)fj(0)]‖$

for each n = 0, 1, · · ·.

Proof

Let ϕn(x, y, z,w) = θn(||x||p + ||y||p + ||z||p ||w||q) for each n = 0, 1, · · · and all . Suppose that p < 1. Since we have

$ψn(x,x,0,0)=12∑k=0∞ϕn(2kx,2kx,0,0)2k=θn2∑k=0∞‖2kx‖p+‖2kx‖p2k=θn‖x‖p∑k=0∞2(p−1)k=θn‖x‖p11−2p−1=2θn2−2p‖x‖p,$

it follows from (2.3) in Theorem 2.1 that

$‖fn(x)−hn‖≤ψn(x,x,0,0)+cn=2θn2−2p‖x‖p+cn.$

for each n = 0, 1, · · · and all .

Assume that p > 1. Since we have

$ψn(x,x,0,0)=12∑k=0∞2kϕn(2−kx,2−kx,0,0)=θn2∑k=0∞2k(‖2−kx‖p+‖2−kx‖p)=θn‖x‖p∑k=0∞2(1−p)k=θn‖x‖p11−21−p=2pθn2p−2‖x‖p,$

it follows from (2.3) in Theorem 2.1 that

$‖fn(x)−hn‖≤ψn(x,x,0,0)+cn=2pθn2p−2‖x‖p+cn.$

for each n = 0, 1, · · · and all .

By setting ϕn(x, y, z,w) = ɛn for each n = 0, 1, · · · and all , Theorem 2.1 also gives us the following corollary.

### Corollary 2.4

Letbe an algebra anda Banach algebra. Suppose that F = {f0, f1, · · ·, fn, · · · } is a sequence of mappings fromintosuch that for each n = 0, 1, · · ·, there exists ɛn > 0 such that

$‖fn(x+y+zw)−fn(x)−fn(y)−∑i+j=ni≤j[fi(z)fj(w)+cijfi(w)fj(z)]‖≤εn$

holds for all. Then there exists a unique higher ring left derivation H = {h0, h1, · · ·, hn, · · · } fromintosuch that for each n = 0, 1, · · · and all,

$‖fn(x)−hn(x)‖≤εn+cn,$

where

$cn=‖∑i+j=ni≤j[fi(0)fj(0)+cijfi(0)fj(0)]‖$

for each n = 0, 1, · · ·.

As a consequence of Corollary 2.4, we get the following Bourgin-type superstability [4] of the higher functional equation (1.2).

### Theorem 2.5

Letandbe Banach algebras with unit. Suppose that F = {f0, f1, · · ·, fn, · · · } is a sequence of mappings fromintosatisfying the inequality (2.14), where f0is onto. Then F is a higher ring left derivation frominto ℬ.

Proof

As in (2.5) and (2.8), the relation (2.14) yields that

$‖fn(x+y)−fn(x)−fn(y)‖≤εn+cn$

for each n = 0, 1, · · · and all , where

$cn=‖∑i+j=ni≤j[fi(0)fj(0)+cijfi(0)fj(0)]‖$

for each n = 0, 1, · · ·, and that

$‖fn(zw)−∑i+j=ni≤j[fi(z)fj(w)+cijfi(w)fj(z)]‖≤εn+2‖fn(0)‖$

for each n = 0, 1, · · · and all . By induction, we lead the conclusion. From the Bourgin’s theorem [4], we see that f0 is a ring homomorphism from onto ℬ and so (2.7) gives that $h0(z)=liml→∞12lf0(2lz)=f0(z)$ for all , i.e., f0 = h0. If n = 1, then it follows from the relation (2.4) that f1(z) = h1(z) holds for all since h0 is onto. Let us assume that fm(z) = hm(z) is valid for all and all m < n. Then (2.4) implies that h0(z){fn(w) − hn(w)} = 0 for all z, . Since h0 is onto, we have fn(w) = hn(w) for all . Hence we conclude that fn(z) = hn(z) holds for all n = 0, 1, · · · and all . Now, Corollary 2.4 tells us that F is a higher ring left derivation from into ℬ. The proof of the theorem is complete.

We continue the next result.

### Theorem 2.6

Letbe a semiprime Banach algebra and f0an identity mapping on. Suppose that F = {f1, f2, · · ·, fn, · · · } is a sequence of mappings onsatisfying the inequality (2.14). Then F is a strong higher ring left derivation on. Furthermore, F is a strong higher ring derivation onsuch that each fn in F mapsinto its center.

Proof

For all , we have, by (2.7),

$h0(z)=liml→∞12lf0(2lz)=z$

and so h0 is an identity mapping on .

From the similar way as in the proof of Theorem 2.5 using the induction and the relation (2.4), we get

$z{fn(w)−hn(w)}=0$

for all n ∈ ℕ and all . This implies that

${fn(w)−hn(w)}z{fn(w)−hn(w)}=0$

for all n ∈ ℕ and all . Since is semiprime, it follows that fn(w) = hn(w) for all n ∈ ℕ and all . Therefore, F is a strong higher ring left derivation on . It follows from [14] that F is a strong higher ring derivation on such that each fn in F maps into its center. This completes the proof.

The Singer-Wermer theorem [16], which is a fundamental result in a Banach algebra theory, states that every continuous linear derivation (or linear left derivation) on a commutative Banach algebra maps into the Jacobson radical. They also conjectured that the assumption of continuity is unnecessary. M.P. Thomas [17] proved the conjecture. According to the Thomas’ result, it is easy to see that every linear derivation (or linear left derivation) on a commutative semisimple Banach algebra is identically zero which is the result of B.E. Johnson [9].

The following is similar to B.E. Johnson’ result [9] in the sense of Hyers-Ulam [8].

### Theorem 2.7

Letbe a semisimple Banach algebra and f0an identity mapping on. Suppose that F = {f1, f2 · · ·, fn · · · } is a sequence of mappings onsatisfying the following:

For each n ∈ ℕ, there exists ɛn > 0 such that

$‖fn(αx+βy+zw)−αfn(x)−βfn(y)−∑i+j=ni≤j[fi(z)fj(w)+cijfi(w)fj(z)]‖≤εn$

holds for alland all. Then we have F = {0}n∈ℕon.

Proof

Put in (2.15). Then it follows from Theorem 2.6 that F is a strong higher ring left derivation on . So, each fn, n ∈ ℕ, is additive on . Setting y = x, z = w = 0 in (2.15) and then following the same process as in (2.5), we obtain that

$‖fn((α+β)x)−(α+β)fn(x)‖≤εn+cn$

for each n ∈ ℕ and all , where

$cn=‖∑i+j=ni≤j[fi(0)fj(0)+cijfi(0)fj(0)]‖$

for each n ∈ ℕ. Thus we see that

$12l‖fn(2l(α+β)x)−(α+β)fn(2lx)‖→0$

as l→∞ which implies that for each n ∈ ℕ,

$fn((α+β)x)=liml→∞12lfn(2l(α+β)x)=(α+β)liml→∞12lfn(2lx)=(α+β)fn(x)$

for all and all .

Clearly, fn(0x) = 0 = 0fn(x) for each n ∈ ℕ and all . Now, let λ ∈ ℂ (λ ≠ 0), and let N ∈ ℕ greater than |λ|. By appying a geometric argument, we see that there exist such that $2λN=λ1+λ2$. By the additivity of each fn, n ∈ ℕ, we get $fn(12x)=12fn(x)$ for each n ∈ ℕ and all .

Therefore, we have

$fn(λx)=fn(N2·2·λNx)=Nfn(12·2·λNx)=N2fn((λ1+λ2)x)=N2(λ1+λ2)fn(x)=N2·2·λNfn(x)=λhf(x)$

for each n ∈ ℕ and all , so that fn is ℂ-linear for each n ∈ ℕ. This means that that F is a strong higher linear left derivation on .

Since f1 is a linear left derivation on , we have f1 = 0 on by [10, Corollary 3.7]. Assume that n ≥ 2 and fm = 0 for all m < n. Since we have

$fn(xy)=xfn(y)+yfn(x)+∑i+j=ni≤j,i≠0,n[fi(x)fj(y)+cijfi(y)fj(x)],$

it follows from the hypothesis that

$fn(xy)=xfn(y)+yfn(x)$

for all . This implies that fn is a linear left derivation on . Therefore, [10, Corollary 3.7] again gives fn = 0 on . By the induction, we have F = {0}n∈ℕ on which completes the proof.

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