Generalized Hyers-Ulam Stability of Generalized (𝑁,𝐾)-Derivations

Abstract

Let 3≤𝑛, and 3≤𝑘≤𝑛 be positive integers. Let 𝐴 be an algebra and let 𝑋 be an 𝐴-bimodule. A ℂ-linear mapping 𝑑∶𝐴→𝑋 is called a generalized (𝑛,𝑘)-derivation if there exists a (𝑘−1)-derivation 𝛿∶𝐴→𝑋 such that 𝑑(𝑎₁𝑎₂⋯𝑎_𝑛)=𝛿(𝑎₁)𝑎₂⋯𝑎_𝑛+𝑎₁𝛿(𝑎₂)𝑎₃⋯𝑎_𝑛+⋯+𝑎₁𝑎₂⋯𝑎_𝑘−2𝛿(𝑎_𝑘−1)𝑎_𝑘⋯𝑎_𝑛+𝑎₁𝑎₂⋯𝑎_𝑘−1𝑑(𝑎_𝑘)𝑎_𝑘+1⋯𝑎_𝑛+𝑎₁𝑎₂⋯𝑎_𝑘𝑑(𝑎_𝑘+1)𝑎_𝑘+2⋯𝑎_𝑛+𝑎₁𝑎₂⋯𝑎_𝑘+1𝑑(𝑎_𝑘+2)𝑎_𝑘+3⋯𝑎_𝑛𝑎+⋯+₁⋯𝑎_𝑛−1𝑑(𝑎_𝑛) for all 𝑎₁,𝑎₂,…,𝑎_𝑛∈𝐴. The main purpose of this paper is to prove the generalized Hyers-Ulam stability of the generalized (𝑛,𝑘)-derivations.

1. Introduction

It seems that the stability problem of functional equations introduced by Ulam [1]. Let (𝐺₁,⋅)be a group and let (𝐺₂,∗)be a metric group with the metric 𝑑(⋅,⋅).Given 𝜖>0,does there exist a 𝛿>0,such that if a mapping ℎ∶𝐺₁→𝐺₂satisfies the inequality 𝑑(ℎ(𝑥⋅𝑦),ℎ(𝑥)∗ℎ(𝑦))<𝛿,for all 𝑥,𝑦∈𝐺₁,then there exists a homomorphism 𝐻∶𝐺₁→𝐺₂with 𝑑(ℎ(𝑥),𝐻(𝑥))<𝜖,for all 𝑥∈𝐺₁? In other words, under what condition does there exist a homomorphism near an approximate homomorphism? The concept of stability for functional equations arises when one replaces the functional equation by an inequality which acts as a perturbation of the equation. In 1941, Hyers [2] gave the first affirmative answer to the question of Ulam for Banach spaces 𝐸 and 𝐸^′. Let 𝑓∶𝐸→𝐸^′ be a mapping between Banach spaces such that ‖𝑓(𝑥+𝑦)−𝑓(𝑥)−𝑓(𝑦)‖≤𝛿(1.1) for all 𝑥,𝑦∈𝐸, and for some 𝛿>0. Then there exists a unique additive mapping 𝑇∶𝐸→𝐸^′ such that ‖𝑓(𝑥)−𝑇(𝑥)‖≤𝛿(1.2) for all 𝑥∈𝐸. By the seminal paper of Th. M. Rassias [3] and work of Gadja [4], if one assumes that 𝐸 and 𝐸^′ are real normed spaces with 𝐸^′ complete, 𝑓∶𝐸→𝐸^′ is a mapping such that for each fixed 𝑥∈𝐸 the mapping 𝑡↣𝑓(𝑡𝑥) is continuous in real 𝑡 for each fixed 𝑥 in 𝐸, and that there exists 𝛿≥0 and 𝑝≠1 such that ‖𝑓(𝑥+𝑦)−𝑓(𝑥)−𝑓(𝑦)‖≤𝛿(‖𝑥‖^𝑝+‖𝑦‖^𝑝)(1.3) for all 𝑥,𝑦∈𝐸. Then there exists a unique linear map 𝑇∶𝐸→𝐸^′ such that ‖𝑓(𝑥)−𝑇(𝑥)‖≤2𝛿‖𝑥‖^𝑝|2^𝑝−2|(1.4) for all 𝑥∈𝐸.

On the other hand J. M. Rassias [5] generalized the Hyers stability result by presenting a weaker condition controlled by a product of different powers of norms. If it is assumed that there exist constants Θ≥0 and 𝑝₁,𝑝₂∈ℝ such that 𝑝=𝑝₁+𝑝₂≠1, and 𝑓∶𝐸→𝐸^′ is a map from a norm space 𝐸 into a Banach space 𝐸^′ such that the inequality ‖𝑓(𝑥+𝑦)−𝑓(𝑥)−𝑓(𝑦)‖≤Θ‖𝑥‖^𝑝1‖𝑦‖^𝑝2(1.5) for all 𝑥,𝑦∈𝐸, then there exists a unique additive mapping 𝑇∶𝐸→𝐸^′ such that Θ‖𝑓(𝑥)−𝑇(𝑥)‖≤2−2^𝑝‖𝑥‖^𝑝(1.6) for all 𝑥∈𝐸. If in addition for every 𝑥∈𝐸,𝑓(𝑡𝑥) is continuous in real 𝑡 for each fixed 𝑥, then 𝑇 is linear.

Suppose (𝐺,+) is an abelian group, 𝐸 is a Banach space, and that the so-called admissible control function 𝜑∶𝐺×𝐺→ℝ satisfies 𝜑(𝑥,𝑦)∶=2^∞−1_𝑛=02^−𝑛𝜑(2^𝑛𝑥,2^𝑛𝑦)<∞(1.7) for all 𝑥,𝑦∈𝐺. If 𝑓∶𝐺→𝐸 is a mapping with ‖𝑓(𝑥+𝑦)−𝑓(𝑥)−𝑓(𝑦)‖≤𝜑(𝑥,𝑦)(1.8) for all 𝑥,𝑦∈𝐺, then there exists a unique mapping 𝑇∶𝐺→𝐸 such that 𝑇(𝑥+𝑦)=𝑇(𝑥)+𝑇(𝑦) and ‖𝑓(𝑥)−𝑇(𝑥)‖≤𝜑(𝑥,𝑥), for all 𝑥,𝑦∈𝐺 (see [6]).

Generalized derivations first appeared in the context of operator algebras [7]. Later, these were introduced in the framework of pure algebra [8, 9].

Definition 1.1. Let 𝐴 be an algebra and let 𝑋 be an 𝐴-bimodule. A linear mapping 𝑑∶𝐴→𝑋 is called
(i) derivation if 𝑑(𝑎𝑏)=𝑑(𝑎)𝑏+𝑎𝑑(𝑏), for all 𝑎,𝑏∈𝐴;
(ii) generalized derivation if there exists a derivation (in the usual sense) 𝛿∶𝐴→𝑋 such that 𝑑(𝑎𝑏)=𝑎𝑑(𝑏)+𝛿(𝑎)𝑏, for all 𝑎,𝑏∈𝐴.

Every right multiplier (i.e., a linear map ℎ on 𝐴 satisfying ℎ(𝑎𝑏)=𝑎ℎ(𝑏), for all 𝑎,𝑏∈𝐴) is a generalized derivation.

Definition 1.2. Let 𝑛≥2,𝑘≥3 be positive integers. Let 𝐴 be an algebra and let 𝑋 be an 𝐴-bimodule. A ℂ-linear mapping 𝑑∶𝐴→𝑋 is called
(i) 𝑛-derivation if 𝑑(𝑎₁𝑎₂⋯𝑎_𝑛)=𝑑(𝑎₁)𝑎₂⋯𝑎_𝑛+𝑎₁𝑑(𝑎₂)𝑎₃⋯𝑎_𝑛+⋯+𝑎₁⋯𝑎_𝑛−1𝑑(𝑎_𝑛)(1.9) for all 𝑎₁,𝑎₂,…,𝑎_𝑛∈𝐴;
(ii) generalized (𝑛,𝑘)-derivation if there exists a (𝑘−1)-derivation 𝛿∶𝐴→𝑋 such that 𝑑(𝑎₁𝑎₂⋯𝑎_𝑛)=𝛿(𝑎₁)𝑎₂⋯𝑎_𝑛+𝑎₁𝛿(𝑎₂)𝑎₃⋯𝑎_𝑛+⋯+𝑎₁𝑎₂⋯𝑎_𝑘−2𝛿(𝑎_𝑘−1)𝑎_𝑘⋯𝑎_𝑛+𝑎₁𝑎₂⋯𝑎_𝑘−1𝑑(𝑎_𝑘)𝑎_𝑘+1⋯𝑎_𝑛+𝑎₁𝑎₂⋯𝑎_𝑘𝑑(𝑎_𝑘+1)𝑎_𝑘+2⋯𝑎_𝑛+𝑎₁𝑎₂⋯𝑎_𝑘+1𝑑(𝑎_𝑘+2)𝑎_𝑘+3⋯𝑎_𝑛+⋯+𝑎₁⋯𝑎_𝑛−1𝑑(𝑎_𝑛)(1.10) for all 𝑎₁,𝑎₂,…,𝑎_𝑛∈𝐴.

By Definition 1.2, we see that a generalized (2,3)-derivation is a generalized derivation.

For instance, let 𝒜 be a Banach algebra. Then we take 𝒯=,0𝒜𝒜𝒜00𝒜𝒜000𝒜0000(1.11)𝒯 is an algebra equipped with the usual matrix-like operations. It is easy to check that every linear map from 𝒜 into 𝒜 is a (5,3)-derivation, but there are linear maps on 𝒯 which are not generalized derivations.

The so-called approximate derivations were investigated by Jun and Park [10]. Recently, the stability of derivations have been investigated by some authors; see [10–13] and references therein. Moslehian [14] investigated the generalized Hyers-Ulam stability of generalized derivations from a unital normed algebra 𝐴 to a unit linked Banach 𝐴-bimodule (see also [15]).

In this paper, we investigate the generalized Hyers-Ulam stability of the generalized (𝑛,𝑘)-derivations.

2. Main Result

In this section, we investigate the generalized Hyers-Ulam stability of the generalized (𝑛,𝑘)-derivations from a unital Banach algebra 𝐴 into a unit linked Banach 𝐴-bimodule. Throughout this section, assume that 𝐴 is a unital Banach algebra, 𝑋 is unit linked Banach 𝐴-bimodule, and suppose that 3≤𝑛, and 3≤𝑘≤𝑛.

We need the following lemma in the main results of the present paper.

Lemma 2.1 (see [16]). Let 𝑈,𝑉 be linear spaces and let 𝑓∶𝑈→𝑉 be an additive mapping such that 𝑓(𝜆𝑥)=𝜆𝑓(𝑥), for all 𝑥∈𝑈 and all 𝜆∈𝕋¹∶={𝜆∈ℂ;|𝜆|=1}. Then the mapping 𝑓 is ℂ-linear.

Now we prove the generalized Hyers-Ulam stability of generalized (𝑛,𝑘)-derivations.

Theorem 2.2. Suppose 𝑓∶𝐴→𝑋 is a mapping with 𝑓(0)=0 for which there exists a map 𝑔∶𝐴→𝑋 with 𝑔(0)=0 and a function 𝜑∶𝐴^𝑛+2→ℝ⁺ such that max{‖𝑓(𝜆𝑎+𝜆𝑏+𝑎₁𝑎₂⋯𝑎_𝑛)−𝜆𝑓(𝑎)−𝜆𝑓(𝑏)−𝑎₁⋯𝑎_𝑘−1𝑓(𝑎_𝑘)𝑎_𝑘+1⋯𝑎_𝑛−𝑎₁⋯𝑎_𝑘𝑓(𝑎_𝑘+1)𝑎_𝑘+2⋯𝑎_𝑛−⋯−𝑎₁⋯𝑎_𝑛−1𝑓(𝑎_𝑛)−𝑔(𝑎₁)𝑎₂⋯𝑎_𝑛−𝑎₁𝑔(𝑎₂)𝑎₃⋯𝑎_𝑛−⋯−𝑎₁𝑎₂⋯𝑎_𝑘−2𝑔(𝑎_𝑘−1)𝑎_𝑘⋯𝑎_𝑛‖,‖𝑔(𝜆𝑎+𝜆𝑏+𝑎₁𝑎₂⋯𝑎_𝑛)−𝜆𝑔(𝑎)−𝜆𝑔(𝑏)−𝑔(𝑎₁)𝑎₂⋯𝑎_𝑛−𝑎₁𝑔(𝑎₂)𝑎₃⋯𝑎_𝑛−⋯−𝑎₁⋯𝑎_𝑘−2𝑔(𝑎_𝑘−1)𝑎_𝑘⋯𝑎_𝑛‖}≤𝜑(𝑎,𝑏,𝑎₁,𝑎₂,…,𝑎_𝑛),(2.1)𝜑(𝑎,𝑏,𝑎₁,𝑎₂,…,𝑎_𝑛)∶=2^∞−1_𝑖=02^−𝑖𝜑(2^𝑖𝑎,2^𝑖𝑏,2^𝑖𝑎₁,…,2^𝑖𝑎_𝑛)<∞(2.2) for all 𝑎,𝑏,𝑎₁,𝑎₂,…,𝑎_𝑛∈𝐴 and all 𝜆∈𝕋¹. Then there exists a unique generalized (𝑛,𝑘)-derivation 𝑑∶𝐴→𝑋 such that ‖𝑓(𝑎)−𝑑(𝑎)‖≤𝜑(𝑎,𝑎,0,0,0,…,0)(2.3) for all 𝑎∈𝐴.

Proof. By (2.1) we have ‖𝑓(𝜆𝑎+𝜆𝑏+𝑎₁𝑎₂⋯𝑎_𝑛)−𝜆𝑓(𝑎)−𝜆𝑓(𝑏)−𝑎₁⋯𝑎_𝑘−1𝑓(𝑎_𝑘)𝑎_𝑘+1⋯𝑎_𝑛−𝑎₁⋯𝑎_𝑘𝑓(𝑎_𝑘+1)𝑎_𝑘+2⋯𝑎_𝑛−⋯−𝑎₁⋯𝑎_𝑛−1𝑓(𝑎_𝑛)−𝑔(𝑎₁)𝑎₂⋯𝑎_𝑛−𝑎₁𝑔(𝑎₂)𝑎₃⋯𝑎_𝑛−⋯−𝑎₁𝑎₂⋯𝑎_𝑘−2𝑔(𝑎_𝑘−1)𝑎_𝑘⋯𝑎_𝑛‖≤𝜑(𝑎,𝑏,𝑎₁,𝑎₂,…,𝑎_𝑛),(2.4)‖𝑔(𝜆𝑎+𝜆𝑏+𝑎₁𝑎₂⋯𝑎_𝑛)−𝜆𝑔(𝑎)−𝜆𝑔(𝑏)−𝑔(𝑎₁)𝑎₂⋯𝑎_𝑛−𝑎₁𝑔(𝑎₂)𝑎₃⋯𝑎_𝑛−⋯−𝑎₁⋯𝑎_𝑘−2𝑔(𝑎_𝑘−1)𝑎_𝑘⋯𝑎_𝑛‖≤𝜑(𝑎,𝑏,𝑎₁,𝑎₂,…,𝑎_𝑛)(2.5) for all 𝑎,𝑏,𝑎₁,𝑎₂,…,𝑎_𝑛∈𝐴 and all 𝜆∈𝕋¹. Setting 𝑎₁,𝑎₂,…,𝑎_𝑛=0 and 𝜆=1 in (2.4), we have ‖𝑓(𝑎+𝑏)−𝑓(𝑎)−𝑓(𝑏)‖≤𝜑(𝑎,𝑏,0,0,…,0)(2.6) for all 𝑎,𝑏∈𝐴. One can use induction on 𝑛 to show that ‖2^−𝑚𝑓(2^𝑚𝑎)−𝑓(𝑎)‖≤2^−1𝑚−1_𝑖=02^−𝑖𝜑(2^𝑖𝑎,2^𝑖𝑎,0,0,…,0)(2.7) for all 𝑛∈ℕ and all 𝑎∈𝐴, and that ‖2^−𝑚𝑓(2^𝑚𝑎)−2^−𝑙𝑓(2^𝑙𝑎)‖≤2^−1𝑚−1_𝑖=𝑙2^−𝑖𝜑(2^𝑖𝑎,2^𝑖𝑎,0,0…,0)(2.8) for all 𝑚>𝑙 and all 𝑎∈𝐴. It follows from the convergence (2.2) that the sequence 2^−𝑚𝑓(2^𝑚𝑎) is Cauchy. Due to the completeness of 𝑋, this sequence is convergent. Set 𝑑(𝑎)∶=lim_𝑚→∞2^−𝑚𝑓(2^𝑚𝑎).(2.9) Putting 𝑎₁,𝑎₂,…,𝑎_𝑛=0 and replacing 𝑎,𝑏 by 2^𝑚𝑎,2^𝑚𝑏, respectively, in (2.4), we get ‖2^−𝑚𝑓(2^𝑚(𝜆𝑎+𝜆𝑏))−2^−𝑚𝜆𝑓(2^𝑚𝑎)−2^−𝑚𝜆𝑓(2^𝑚𝑏)‖≤2^−𝑚𝜑(2^𝑚𝑎,2^𝑚𝑏,0,0,…,0)(2.10) for all 𝑎,𝑏∈𝐴 and all 𝜆∈𝕋¹. Taking the limit as 𝑚→∞ we obtain 𝑑(𝜆𝑎+𝜆𝑏)=𝜆𝑑(𝑎)+𝜆𝑑(𝑏)(2.11) for all 𝑎,𝑏∈𝐴 and all 𝜆∈𝕋¹. So by Lemma 2.1, the mapping 𝑑 is ℂ-linear.
Using (2.5), (2.2), and the above technique, we get 𝛿(𝑎)∶=lim_𝑚→∞2^−𝑚𝑔(2^𝑚𝑎),𝛿(𝜆𝑎+𝜆𝑏)=𝜆𝛿(𝑎)+𝛿(𝑏)(2.12) for all 𝑎,𝑏∈𝐴 and all 𝜆∈𝕋¹. Hence by Lemma 2.1, 𝛿 is ℂ-linear. Moreover, it follows from (2.7) and (2.9) that ‖𝑓(𝑎)−𝑑(𝑎)‖≤𝜑(𝑎,𝑎,0,0,…,0), for all 𝑎∈𝐴. It is known that the additive mapping 𝑑 satisfying (2.3) is unique [17]. Putting 𝜆=1,𝑎=𝑏=0, and replacing 𝑎₁,𝑎₂,…,𝑎_𝑛 by 2^𝑚𝑎₁,2^𝑚𝑎₂,…,2^𝑚𝑎_𝑛, respectively, in (2.4), we get ‖𝑓(2^𝑛𝑚𝑎₁𝑎₂⋯𝑎_𝑛)−2^{(𝑛−1)𝑚}𝑎₁⋯𝑎_𝑘−1𝑓(2^𝑚𝑎_𝑘)𝑎_𝑘+1⋯𝑎_𝑛−2^{(𝑛−1)𝑚}𝑎₁⋯𝑎_𝑘𝑓(2^𝑚𝑎_𝑘+1)𝑎_𝑘+2⋯𝑎_𝑛−⋯−2^{(𝑛−1)𝑚}𝑎₁⋯𝑎_𝑛−1𝑓(2^𝑚𝑎_𝑛)−2^{(𝑛−1)𝑚}𝑔(2^𝑚𝑎₁)𝑎₂⋯𝑎_𝑛−2^{(𝑛−1)𝑚}𝑎₁𝑔(2^𝑚𝑎₂)𝑎₃⋯𝑎_𝑛−⋯−2^{(𝑛−1)𝑚}𝑎₁𝑎₂⋯𝑎_𝑘−2𝑔(2^𝑚𝑎_𝑘−1)𝑎_𝑘⋯𝑎_𝑛‖≤𝜑(0,0,2^𝑚𝑎₁,2^𝑚𝑎₂,…,2^𝑚𝑎_𝑛),(2.13) whence ‖2^−𝑛𝑚𝑓(2^𝑛𝑚𝑎₁𝑎₂⋯𝑎_𝑛)−2^−𝑚𝑎₁⋯𝑎_𝑘−1𝑓(2^𝑚𝑎_𝑘)𝑎_𝑘+1⋯𝑎_𝑛−2^−𝑚𝑎₁⋯𝑎_𝑘𝑓(2^𝑚𝑎_𝑘+1)𝑎_𝑘+2⋯𝑎_𝑛−⋯−2^−𝑚𝑎₁⋯𝑎_𝑛−1𝑓(2^𝑚𝑎_𝑛)−2^−𝑚𝑔(2^𝑚𝑎₁)𝑎₂⋯𝑎_𝑛−2^−𝑚𝑎₁𝑔(2^𝑚𝑎₂)𝑎₃⋯𝑎_𝑛−⋯−2^−𝑚𝑎₁𝑎₂⋯𝑎_𝑘−2𝑔(2^𝑚𝑎_𝑘−1)𝑎_𝑘⋯𝑎_𝑛‖≤2^−𝑛𝑚𝜑(0,0,2^𝑚𝑎₁,2^𝑚𝑎₂,…,2^𝑚𝑎_𝑛)(2.14) for all 𝑎₁,𝑎₂,…,𝑎_𝑛∈𝐴. By (2.9), lim_𝑚→∞2^−𝑛𝑚𝑓(2^𝑛𝑚𝑎)=𝑑(𝑎) and by the convergence of series (2.2), lim_𝑚→∞2^−𝑛𝑚𝜑(0,0,2^𝑚𝑎₁,2^𝑚𝑎₂,…,2^𝑚𝑎_𝑛)=0. Let 𝑚 tend to ∞ in (2.14). Then 𝑑(𝑎₁𝑎₂⋯𝑎_𝑛)=𝑎₁⋯𝑎_𝑘−1𝑑(𝑎_𝑘)𝑎_𝑘+1⋯𝑎_𝑛+𝑎₁⋯𝑎_𝑘𝑑(𝑎_𝑘+1)𝑎_𝑘+2⋯𝑎_𝑛+⋯+𝑎₁⋯𝑎_𝑛−1𝑑(𝑎_𝑛)+𝛿(𝑎₁)𝑎₂⋯𝑎_𝑛+𝑎₁𝛿(𝑎₂)𝑎₃⋯𝑎_𝑛+⋯+𝑎₁𝑎₂⋯𝑎_𝑘−2𝛿(𝑎_𝑘−1)𝑎_𝑘⋯𝑎_𝑛(2.15) for all 𝑎₁,𝑎₂,…,𝑎_𝑛∈𝐴.
Next we claim that 𝛿 is a (𝑘−1)-derivation. Putting 𝜆=1,𝑎=𝑏=0, and replacing 𝑎₁,𝑎₂,…,𝑎_𝑛 by 2^𝑚𝑎₁,2^𝑚𝑎₂,…,2^𝑚𝑎_𝑛, respectively, in (2.5), we get ‖𝑔(2^𝑛𝑚𝑎₁𝑎₂⋯𝑎_𝑛)−2^{(𝑛−1)𝑚}𝑔(2^𝑚𝑎₁)𝑎₂⋯𝑎_𝑛−2^{(𝑛−1)𝑚}𝑎₁𝑔(2^𝑚𝑎₂)𝑎₃⋯𝑎_𝑛−2^{(𝑛−1)𝑚}𝑎₁𝑎₂𝑔(2^𝑚𝑎₃)𝑎₄⋯𝑎_𝑛−⋯−2^{(𝑛−1)𝑚}𝑎₁⋯𝑎_𝑘−2𝑔(2^𝑚𝑎_𝑘−1)𝑎_𝑘⋯𝑎_𝑛‖≤𝜑(0,0,2^𝑚𝑎₁,2^𝑚𝑎₂,…,2^𝑚𝑎_𝑛),(2.16) whence ‖2^−𝑛𝑚𝑔(2^𝑛𝑚𝑎₁𝑎₂⋯𝑎_𝑛)−2^−𝑚𝑔(2^𝑚𝑎₁)𝑎₂⋯𝑎_𝑛−2^−𝑚𝑎₁𝑔(2^𝑚𝑎₂)𝑎₃⋯𝑎_𝑛−2^−𝑚𝑎₁𝑎₂𝑔(2^𝑚𝑎₃)𝑎₄⋯𝑎_𝑛−⋯−2^−𝑚𝑎₁⋯𝑎_𝑘−2𝑔(2^𝑚𝑎_𝑘−1)𝑎_𝑘⋯𝑎_𝑛‖≤2^−𝑛𝑚𝜑(0,0,2^𝑚𝑎₁,2^𝑚𝑎₂,…,2^𝑚𝑎_𝑛)(2.17) for all 𝑎₁,𝑎₂,…,𝑎_𝑛∈𝐴. Let 𝑚 tends to ∞ in (2.17). Then 𝛿(𝑎₁𝑎₂⋯𝑎_𝑘−1𝑎_𝑘𝑎_𝑘+1⋯𝑎_𝑛)=𝛿(𝑎₁)𝑎₂⋯𝑎_𝑛+𝑎₁𝛿(𝑎₂)𝑎₃⋯𝑎_𝑛+𝑎₁𝑎₂𝛿(𝑎₃)𝑎₄⋯𝑎_𝑛+⋯+𝑎₁𝑎₂⋯𝑎_𝑘−2𝛿(𝑎_𝑘−1)𝑎_𝑘⋯𝑎_𝑛(2.18) for all 𝑎₁,𝑎₂,…,𝑎_𝑛∈𝐴.
Setting 𝑎_𝑘=𝑎_𝑘+1=⋯=𝑎_𝑛=1 in (2.18). Hence the mapping 𝛿 is (𝑘−1)-derivation.

Corollary 2.3. Suppose 𝑓∶𝐴→𝑋 is a mapping with 𝑓(0)=0 for which there exists constant 𝜃≥0,𝑝<1 and a map 𝑔∶𝐴→𝑋 with 𝑔(0)=0 such that max{‖𝑓(𝜆𝑎+𝜆𝑏+𝑎₁𝑎₂⋯𝑎_𝑛)−𝜆𝑓(𝑎)−𝜆𝑓(𝑏)−𝑎₁⋯𝑎_𝑘−1𝑓(𝑎_𝑘)𝑎_𝑘+1⋯𝑎_𝑛−𝑎₁⋯𝑎_𝑘𝑓(𝑎_𝑘+1)𝑎_𝑘+2⋯𝑎_𝑛−⋯−𝑎₁⋯𝑎_𝑛−1𝑓(𝑎_𝑛)−𝑔(𝑎₁)𝑎₂⋯𝑎_𝑛−𝑎₁𝑔(𝑎₂)𝑎₃⋯𝑎_𝑛−⋯−𝑎₁𝑎₂⋯𝑎_𝑘−2𝑔(𝑎_𝑘−1)𝑎_𝑘⋯𝑎_𝑛‖,‖𝑔(𝜆𝑎+𝜆𝑏+𝑎₁𝑎₂⋯𝑎_𝑛)−𝜆𝑔(𝑎)−𝜆𝑔(𝑏)−𝑔(𝑎₁)𝑎₂⋯𝑎_𝑛−𝑎₁𝑔(𝑎₂)𝑎₃⋯𝑎_𝑛−⋯−𝑎₁⋯𝑎_𝑘−2𝑔(𝑎_𝑘−1)𝑎_𝑘⋯𝑎_𝑛‖}≤𝜃(‖𝑎‖^𝑝+‖𝑏‖^𝑝+^𝑛_𝑖=1‖𝑎_𝑖‖^𝑝)(2.19) for all 𝑎₁,𝑎₂,…,𝑎_𝑛∈𝐴 and all 𝜆∈𝕋. Then there exists a unique generalized (𝑛,𝑘)-derivation 𝑑∶𝐴→𝑋 such that ‖𝑓(𝑎)−𝑑(𝑎)‖≤𝜃‖𝑎‖^𝑝1−2^𝑝−1(2.20) for all 𝑎∈𝐴.

Proof. Put 𝜑(𝑎,𝑏,𝑎₁,𝑎₂,…,𝑎_𝑛)=𝜃(‖𝑎‖^𝑝+‖𝑏‖^𝑝+∑^𝑛_𝑖=1‖𝑎_𝑖‖^𝑝) in Theorem 2.2.

Corollary 2.4. Suppose 𝑓∶𝐴→𝑋 is a mapping with 𝑓(0)=0 for which there exists constant 𝜃≥0 and a map 𝑔∶𝐴→𝑋 with 𝑔(0)=0 such that max{‖𝑓(𝜆𝑎+𝜆𝑏+𝑎₁𝑎₂⋯𝑎_𝑛)−𝜆𝑓(𝑎)−𝜆𝑓(𝑏)−𝑎₁⋯𝑎_𝑘−1𝑓(𝑎_𝑘)𝑎_𝑘+1⋯𝑎_𝑛−𝑎₁⋯𝑎_𝑘𝑓(𝑎_𝑘+1)𝑎_𝑘+2⋯𝑎_𝑛−⋯−𝑎₁⋯𝑎_𝑛−1𝑓(𝑎_𝑛)−𝑔(𝑎₁)𝑎₂⋯𝑎_𝑛−𝑎₁𝑔(𝑎₂)𝑎₃⋯𝑎_𝑛−⋯−𝑎₁𝑎₂⋯𝑎_𝑘−2𝑔(𝑎_𝑘−1)𝑎_𝑘⋯𝑎_𝑛‖,‖𝑔(𝜆𝑎+𝜆𝑏+𝑎₁𝑎₂⋯𝑎_𝑛)−𝜆𝑔(𝑎)−𝜆𝑔(𝑏)−𝑔(𝑎₁)𝑎₂⋯𝑎_𝑛−𝑎₁𝑔(𝑎₂)𝑎₃⋯𝑎_𝑛−⋯−𝑎₁⋯𝑎_𝑘−2𝑔(𝑎_𝑘−1)𝑎_𝑘⋯𝑎_𝑛‖}≤𝜃(2.21) for all 𝑎₁,𝑎₂,…,𝑎_𝑛∈𝐴. Then there exists a unique generalized (𝑛,𝑘)-derivation 𝑑∶𝐴→𝑋 such that ‖𝑓(𝑎)−𝑑(𝑎)‖≤𝜃(2.22) for all 𝑎∈𝐴.

Proof. Letting 𝑝=0 in Corollary 2.3, we obtain the above result of Corollary 2.4.