Superstability of Generalized Higher Derivations

Abstract

We define the notion of an approximate generalized higher derivation and investigate the superstability of strong generalized higher derivations.

1. Introduction and Preliminaries

The problem of stability of functional equations was originally raised by Ulam [1, 2] in 1940 concerning the stability of group homomorphisms. Hyers [3] gave an affirmative answer to the question of Ulam. Superstability, the result of Hyers, was generalized by Aoki [4], Bourgin [5], and Rassias [6]. During the last decades, several stability problems for various functional equations have been investigated by several authors. We refer the reader to the monographs [7–10].

Let (𝐸,‖⋅‖) be a complex normed space, and let 𝑘∈ℕ. We denote by 𝐸^𝑘 the linear space 𝐸⊕⋅⋅⋅⊕𝐸 consisting of 𝑘-tuples (𝑥₁,…,𝑥_𝑘), where 𝑥₁,…,𝑥_𝑘∈𝐸. The linear operations on 𝐸^𝑘 are defined coordinatewise. The zero element of either 𝐸 or 𝐸^𝑘 is denoted by 0. We denote by ℕ_𝑘 the set {1,2,…,𝑘} and by ℭ_𝑘 the group of permutations on 𝑘 symbols.

Definition 1.1. A multi-norm on {𝐸^𝑘∶𝑘∈ℕ} is a sequence (‖⋅‖_𝑘)=(‖⋅‖_𝑘∶𝑘∈ℕ) such that ‖⋅‖_𝑘 is a norm on 𝐸^𝑘 for each 𝑘∈ℕ, ‖𝑥‖₁=‖𝑥‖ for each 𝑥∈𝐸, and the following axioms are satisfied for each 𝑘∈ℕ with 𝑘≥2:(𝑀1)‖(𝑥_𝜎(1),…,𝑥_𝜎(𝑘))‖_𝑘=‖(𝑥₁,…,𝑥_𝑘)‖_𝑘(𝜎∈ℭ_𝑘,𝑥₁,…,𝑥_𝑘∈𝐸);(𝑀2)‖(𝛼₁𝑥₁,…,𝛼_𝑘𝑥_𝑘)‖_𝑘≤(max_{𝑖∈ℕ𝑘}|𝛼_𝑖|)‖(𝑥₁,…,𝑥_𝑘)‖_𝑘(𝛼₁,…,𝛼_𝑘∈ℂ,𝑥₁,…,𝑥_𝑘∈𝐸);(𝑀3)‖(𝑥₁,…,𝑥_𝑘−1,0)‖_𝑘=‖(𝑥₁,…,𝑥_𝑘−1)‖_𝑘−1(𝑥₁,…,𝑥_𝑘∈𝐸);(𝑀4)‖(𝑥₁,…,𝑥_𝑘−1,𝑥_𝑘−1)‖_𝑘=‖(𝑥₁,…,𝑥_𝑘−1)‖_𝑘−1(𝑥₁,…,𝑥_𝑘∈𝐸).In this case, we say that ((𝐸^𝑘,‖⋅‖)𝑘∈ℕ) is a multi-normed space.

We recall that the notion of multi-normed space was introduced by Dales and Polyakov in [11]. Motivations for the study of multi-normed spaces and many examples are given in [11].

Suppose that ((𝐸^𝑘,‖⋅‖_𝑘)𝑘∈ℕ) is a multi-normed space, and 𝑘∈ℕ. The following properties are almost immediate consequences of the axioms:(i)‖(𝑥,…,𝑥)‖_𝑘=‖𝑥‖(𝑥∈𝐸);(ii)max_{𝑖∈ℕ𝑘}‖𝑥_𝑖‖≤‖(𝑥₁,…,𝑥_𝑘)‖_𝑘≤∑^𝑘_𝑖=1‖𝑥_𝑖‖≤𝑘max_{𝑖∈ℕ𝑘}‖𝑥_𝑖‖(𝑥₁,…,𝑥_𝑘∈𝐸).

It follows from (ii) that if (𝐸,‖⋅‖) is a Banach space, then ((𝐸^𝑘,‖⋅‖_𝑘) is a Banach space for each 𝑘∈ℕ. In this case, ((𝐸^𝑘,‖⋅‖_𝑘)𝑘∈ℕ) is a multi-Banach space.

By (ii), we get the following lemma.

Lemma 1.2. Suppose that 𝑘∈ℕ and (𝑥₁,…,𝑥_𝑘)∈𝐸^𝑘. For each 𝑗∈ℕ_𝑘, let {𝑥^𝑗_𝑛}_𝑛∈ℕ be a sequence in 𝐸 such that lim_𝑛→∞𝑥^𝑗_𝑛=𝑥_𝑗. Then for each (𝑦₁,…,𝑦_𝑘)∈𝐸^𝑘, one has lim_𝑛→∞𝑥¹_𝑛−𝑦₁,…,𝑥^𝑘_𝑛−𝑦_𝑘=𝑥₁−𝑦₁,…,𝑥_𝑘−𝑦_𝑘. (1.1)

Definition 1.3. Let ((𝐸^𝑘,‖⋅‖_𝑘)𝑘∈ℕ) be a multi-normed space. A sequence {𝑥_𝑛} in 𝐸 is a multinull sequence if, for each 𝜖>0, there exists 𝑛₀∈ℕ such that sup_𝑘∈ℕ𝑥_𝑛,…,𝑥_{𝑛+𝑘−1𝑘}<𝜖𝑛≥𝑛₀. (1.2) Let 𝑥∈𝐸. We say that lim_𝑛→∞𝑥_𝑛=𝑥 if {𝑥_𝑛−𝑥} is a multi-null sequence.

Definition 1.4. Let (𝒜,‖⋅‖) be a normed algebra such that ((𝒜^𝑘,‖⋅‖_𝑘)𝑘∈ℕ) is said to be a multi-normed space. Then ((𝒜^𝑘,‖⋅‖_𝑘)𝑘∈ℕ) is a multi-normed algebra if 𝑥₁𝑦₁,…,𝑥_𝑘𝑦_𝑘𝑘≤𝑥₁,…,𝑥_𝑘𝑘𝑦₁,…,𝑦_𝑘𝑘, (1.3) for 𝑘∈ℕ and 𝑥₁,…,𝑥_𝑘,𝑦₁,…,𝑦_𝑘∈𝒜. Furthermore, if ((𝒜^𝑘,‖⋅‖_𝑘)𝑘∈ℕ) is a multi-Banach space, then ((𝒜^𝑘,‖⋅‖_𝑘)𝑘∈ℕ) is a multi-Banach algebra.

Let 𝒜 be an algebra and 𝑘₀∈{0,1,…,}∪{∞}. A family {𝐷_𝑗}^𝑘0_𝑗=0 of linear mappings on 𝒜 is said to be a higher derivation of rank 𝑘₀ if the functional equation 𝐷_𝑗(𝑥𝑦)=∑^𝑗_𝑖=0𝐷_𝑖(𝑥)𝐷_𝑗−𝑖(𝑦) holds for all 𝑥,𝑦∈𝒜, 𝑗=0,1,2,…,𝑘₀. If 𝐷₀=𝑖𝑑_𝒜, where 𝑖𝑑_𝒜 is the identity map on 𝒜, then 𝐷₁ is a derivation and {𝐷_𝑗}^𝑘0_𝑗=0 is called a strong higher derivation. A standard example of a higher derivation of rank 𝑘₀ is {𝐷^𝑗/𝑗!}^𝑘0_𝑗=0, where 𝐷∶𝒜→𝒜 is a derivation. The reader may find more information about higher derivations in [12–18].

A family {𝑓_𝑗}^𝑘0_𝑗=0 of linear mappings on 𝒜 is called a generalized strong higher derivation if 𝑓₀=𝑖𝑑_𝒜, and there exists a higher derivation {𝐷_𝑗}^𝑘0_𝑗=0 such that𝑓_𝑗𝑥𝑦=𝑥𝑓_𝑗𝑦+^𝑗_𝑖=1𝐷_𝑖(𝑥)𝑓_𝑗−𝑖𝑦, (1.4) for all 𝑥,𝑦∈𝒜 and 𝑗=0,1,2,…,𝑘₀.

The stability of derivations was studied by Park [19, 20]. In this paper, using some ideas from [21, 22], we investigate the superstability of generalized strong higher derivations in multi-Banach algebras.

2. Stability of Generalized Higher Derivations

In this section, we define the notion of an approximate generalized higher derivation. Then we show that an approximate generalized strong higher derivation on a multi-Banach algebra is a strong generalized higher derivation.

Lemma 2.1. Let (𝐸,‖⋅‖) be a normed space, and let ((𝐹^𝑘,‖⋅‖_𝑘∶𝑘∈ℕ) be a multi-Banach space. Let 𝑘∈ℕ, 𝜖>0, and 𝑓∶𝐸→𝐹 a mapping satisfying 𝑓(0)=0 and sup_𝑘∈ℕ𝑓𝑥₁+𝑦₁𝑡−𝑓𝑥₁𝑡−𝑓𝑦₁𝑡𝑥,…,𝑓^𝑘+𝑦_𝑘𝑡−𝑓𝑥_𝑘𝑡−𝑓𝑦_𝑘𝑡_𝑘≤𝜖, (2.1) for all integer 𝑡>1 and all 𝑥₁,…,𝑥_𝑘,𝑦₁,…,𝑦_𝑘∈𝐸, then there exists a unique additive mapping 𝑇∶𝐸→𝐹 such that 𝑓𝑥₁−𝑇𝑥₁,…,𝑓𝑥_𝑘−𝑇𝑥_𝑘≤𝜖𝑥₁,…,𝑥_𝑘∈𝐸. (2.2)

Proof. Substituting 𝑦_𝑖=0 for 𝑖=1,…,𝑘 and replacing 𝑥₁,…,𝑥_𝑘 by 𝑡𝑥₁,…,𝑡𝑥_𝑘 in (2.1), we get sup_𝑘∈ℕ𝑓𝑥₁−𝑓𝑡𝑥₁𝑡,…,𝑓𝑥_𝑘−𝑓𝑡𝑥_𝑘𝑡_𝑘≤𝜖. (2.3) Replacing 𝑥₁,…,𝑥_𝑘 by 𝑡^𝑛𝑥₁,…,𝑡^𝑛𝑥_𝑘 and dividing by 𝑡^𝑛 in (2.3), it follows that sup_𝑘∈ℕ𝑓𝑡^𝑛𝑥₁𝑡^𝑛−𝑓𝑡^𝑛+1𝑥₁𝑡^𝑛+1𝑓𝑡,…,^𝑛𝑥_𝑘𝑡^𝑛−𝑓𝑡^𝑛+1𝑥_𝑘𝑡^𝑛+1_𝑘≤𝜖𝑡^𝑛. (2.4) An induction argument implies that sup_𝑘∈ℕ𝑓𝑡^𝑛𝑥₁𝑡^𝑛−𝑓𝑡^𝑛+𝑚𝑥₁𝑡^𝑛+𝑚𝑓𝑡,…,^𝑛𝑥_𝑘𝑡^𝑛−𝑓𝑡^𝑛+𝑚𝑥_𝑘𝑡^𝑛+𝑚_𝑘1≤𝜖𝑡^𝑛+11+⋯𝑡^𝑛+𝑚, (2.5) for 𝑥∈𝐸 and 𝑛,𝑚∈ℕ. Hence, the sequence {𝑓(𝑡^𝑛𝑥)/𝑡^𝑛} is cauchy and hence is convergent in the complete multi-normed space 𝐹. Let 𝑇∶𝐸→𝐹 be the mapping defined by 𝑇(𝑥)∶=lim_𝑛→∞𝑓𝑡^𝑛𝑥𝑡^𝑛. (2.6) Hence, for each 𝑟>0, there exists 𝑁∈ℕ such that sup_𝑘∈ℕ𝑓𝑡^𝑛𝑥₁𝑡^𝑛(𝑥)𝑓𝑡−𝑇,…,^{𝑛+𝑘−1}𝑥_𝑘𝑡^{𝑛+𝑘−1}−𝑇(𝑥)_𝑘≤𝑟(𝑛≥𝑁). (2.7) In particular, the property (ii) of multi-norm implies that lim_𝑛→∞𝑓𝑡^𝑛𝑥𝑡^𝑛−𝑇(𝑥)=0(𝑥∈𝐸). (2.8) We show that 𝑇 is additive. Putting 𝑛=0 in (2.5), we get sup_𝑘∈ℕ𝑓𝑥₁−𝑓𝑡^𝑚𝑥₁𝑡^𝑚,…,𝑓𝑥_𝑘−𝑓𝑡^𝑚𝑥_𝑘𝑡^𝑚_𝑘≤𝜖. (2.9) Taking the limit as 𝑚→∞, we obtain sup_𝑘∈ℕ𝑓𝑥₁−𝑇𝑥₁,…,𝑓𝑥_𝑘−𝑇𝑥_𝑘𝑘≤𝜖. (2.10) Let 𝑥,𝑦∈𝐸, put 𝑥₁=⋯=𝑥_𝑘=𝑡^𝑛𝑥, 𝑦₁=⋯=𝑦_𝑘=𝑡^𝑛𝑦 in (2.1), and divide by 𝑡^𝑛, Then we have 𝑡^−𝑛𝑓𝑡^𝑛𝑥+𝑡^𝑛𝑦𝑡−𝑡⁻¹𝑓𝑡^𝑛𝑥𝑡^𝑛−𝑡⁻¹𝑓𝑡^𝑛𝑦𝑡^𝑛_𝑘≤𝜖𝑡^𝑛. (2.11) By letting 𝑛→∞, we get 𝑇𝑥+𝑦𝑡=𝑇(𝑥)𝑡+𝑇𝑦𝑡. (2.12) Letting 𝑦=0 in (2.12) yields 𝑇(𝑥/𝑡)=𝑇(𝑥)/𝑡 for all 𝑥∈𝐸. Hence, we get 𝑇(𝑥+𝑦)=𝑇(𝑥)+𝑇(𝑦), that is, 𝑇 is additive. Now, if 𝑇′ is another required additive mapping, we see that 𝑇^′(𝑥)(𝑥≤1−𝑇)𝑡^𝑛𝑇^′𝑡^𝑛𝑥−𝑇𝑡^𝑛𝑥≤1𝑡^𝑛𝑇^′𝑡^𝑛𝑥−𝑓𝑡^𝑛𝑥+1𝑡^𝑛𝑓𝑡^𝑛𝑥−𝑇𝑡^𝑛𝑥≤2𝑡^𝑛−1(𝑡−1)𝜖, (2.13) for all 𝑥∈𝐸. By letting 𝑛→∞ in this inequality, we conclude that 𝑇=𝑇′. This proves the uniqueness assertion.

Definition 2.2. Let ((𝒜^𝑘,‖⋅‖_𝑘)𝑘∈ℕ) be a multi-Banach algebra. Suppose that 𝜖>0, 𝑡>1 is an integer and 𝜓∶𝒜×𝒜→[0,∞) is a control function such that 𝜓𝑡^𝑛𝑥,𝑡^𝑚𝑦≤𝛼^𝑛+𝑚𝜓𝑥,𝑦, (2.14) for some 0<𝛼<𝑡, all nonnegative numbers 𝑚,𝑛 and all 𝑥,𝑦∈𝒜. An (𝜖,𝜓)-approximate generalized strong higher derivation of rank 𝑘₀ is a family {𝑓_𝑗}^𝑘0_𝑗=0 of mappings from 𝒜 into 𝒜 with 𝑓_𝑗(0)=0, 𝑓₀=𝑖𝑑_𝒜, and there exists a family {𝑔_𝑗}^𝑘0_𝑗=0 of mappings from 𝒜 into 𝒜 such that 𝑔₀=𝑖𝑑_𝒜 and sup_𝑘∈ℕ𝑓_𝑗𝑥₁+𝑦₁𝑡+𝑧₁𝑤₁−𝑓_𝑗𝑥₁𝑡−𝑓_𝑗𝑦₁𝑡−𝑧₁𝑓_𝑗𝑤₁−𝑔_𝑗𝑧₁𝑤₁,…,𝑓_𝑗𝑥_𝑘+𝑘₁𝑡+𝑧_𝑘𝑤_𝑘−𝑓_𝑗𝑥_𝑘𝑡−𝑓_𝑗𝑦_𝑘𝑡−𝑧_𝑘𝑓_𝑗𝑤_𝑘−𝑔_𝑗𝑧_𝑘𝑤_𝑘𝑘≤𝜖, (2.15) for all 0≤𝑗≤𝑘₀, 𝑡>1 and all 𝑥₁,…,𝑥_𝑘,𝑦₁,…,𝑦_𝑘,𝑧₁,…,𝑧_𝑘,𝑤₁,…,𝑤_𝑘∈𝒜, and 𝑓_𝑗𝑥𝑦−𝑥𝑓_𝑗𝑦−^𝑗_𝑖=1𝑔_𝑖(𝑥)𝑓_𝑗−𝑖𝑦≤𝜓𝑥,𝑦, (2.16) for all 0≤𝑗≤𝑘₀ and 𝑥,𝑦∈𝒜.

Theorem 2.3. Let 𝒜 be a Banach algebra with unit 𝑒, and let {𝑓_𝑗}^𝑘0_𝑗=0 be a (𝜖,𝜓)-approximate generalized strong higher derivation on a multi-Banach algebra ((𝒜^𝑘,‖⋅‖_𝑘)𝑘∈ℕ), then {𝑓_𝑗}^𝑘0_𝑗=0 is a strong higher derivation.

Proof. Letting 𝑧_𝑖=𝑤_𝑖=0 for 𝑖=1,…,𝑘 in (2.15), Lemma 2.1 implies that for each 0≤𝑗≤𝑘₀, there is an additive mapping 𝑑_𝑗 defined by 𝑑_𝑗(𝑥)=lim_𝑛→∞(𝑓_𝑗(𝑡^𝑛𝑥)/𝑡^𝑛) such that ‖𝑑_𝑗(𝑥)−𝑓_𝑗(𝑥)‖≤𝜖 for all 𝑥∈𝒜. If 𝑗=1, [21, Theorem 2.2] implies that 𝑓₁ and 𝑔₁ are a generalized derivation and a derivation, respectively. Also by the proof of [21, Theorem 2.2], we have 𝑓₁𝑥𝑦=𝑥𝑓₁𝑦+𝑔₁(𝑥)𝑦,lim_𝑛→∞𝑔₁𝑡^𝑛𝑥𝑡^𝑛=𝑑₁(𝑥)−𝑥𝑑₁(𝑒)=𝑔₁(𝑥). (2.17) By induction for 1≤𝑖≤𝑗−1, assume that 𝑓_𝑖=𝑥𝑓_𝑖𝑦+^𝑖_𝑙=1𝑔_𝑙(𝑥)𝑓_𝑖−𝑙𝑦,𝑔_𝑖=^𝑖_𝑙=0𝑔_𝑙(𝑥)𝑔_𝑖−𝑙𝑦, (2.18) for all 𝑥,𝑦∈𝒜 such that lim_𝑛→∞𝑔_𝑖𝑡^𝑛𝑥𝑡^𝑛=𝑑_𝑖(𝑥)−𝑥𝑑_𝑖(𝑒)−^𝑖_𝑙=1𝑔_𝑙(𝑥)𝑑_𝑖−𝑙(𝑒)=𝑔_𝑖(𝑥). (2.19) It follows from (2.14) and (2.16) that 𝑓_𝑗𝑡^2𝑛𝑥𝑦𝑡^2𝑛−𝑥𝑓_𝑗𝑡^𝑛𝑦−^𝑗_𝑖=1𝑔_𝑖𝑡^𝑛𝑥𝑡^𝑛𝑓_𝑗−𝑖𝑡^𝑛𝑦𝑡^𝑛≤𝜓𝑡^𝑛𝑥,𝑡^𝑛𝑦𝑡^2𝑛≤𝛼𝑡^2𝑛. (2.20) Passing the limit as 𝑛→∞, we obtain lim_𝑛→∞𝑦𝑔_𝑗𝑡^𝑛𝑥𝑡^𝑛=𝑑_𝑗𝑥𝑦−𝑥𝑑_𝑗𝑦−^𝑗−1_𝑖=1𝑔_𝑖(𝑥)𝑑_𝑗−𝑖𝑦, (2.21) for all 𝑥,𝑦∈𝒜. Put 𝑦=𝑒 in the above equation, then lim_𝑛→∞𝑔_𝑗𝑡^𝑛𝑥𝑡^𝑛=𝑑_𝑗(𝑥)−𝑥𝑑_𝑗(𝑒)−^𝑗−1_𝑖=1𝑔_𝑖(𝑥)𝑑_𝑗−𝑖(𝑒). (2.22) If 𝐷_𝑗(𝑥)=𝑑_𝑗(𝑥)−𝑥𝑑_𝑗(𝑒)−∑^𝑗−1_𝑖=1𝑔_𝑖(𝑥)𝑑_𝑗−𝑖(𝑒), then by additivity of 𝑑_𝑖 and 𝑔_𝑖 for 0≤𝑖≤𝑗−1, we get 𝐷_𝑗(𝑎+𝑏)=𝑑_𝑗(𝑎+𝑏)−(𝑎+𝑏)𝑑_𝑗(𝑒)−^𝑗_𝑖=1𝑔_𝑖(𝑎+𝑏)𝑑_𝑗−𝑖(𝑒)=𝑑_𝑗(𝑎)+𝑑_𝑗(𝑏)−𝑎𝑑_𝑗(𝑒)−𝑏𝑑_𝑗(𝑒)−^𝑗_𝑖=1𝑔_𝑖(𝑎)𝑑_𝑗−𝑖(𝑒)−^𝑗_𝑖=1𝑔_𝑖(𝑏)𝑑_𝑗−𝑖(𝑒)=𝐷_𝑗(𝑎)+𝐷_𝑗(𝑏). (2.23) Therefore, 𝐷_𝑗 is additive. Now, let 𝐹(𝑥,𝑦)=𝑓_𝑗(𝑥𝑦)−𝑥𝑓_𝑗(𝑦)−∑^𝑗_𝑖=1𝑔_𝑖(𝑥)𝑓_𝑗−𝑖(𝑦), if we take 𝑥_𝑖=𝑦_𝑖=0 and 𝑧_𝑖=𝑥, 𝑤_𝑖=𝑦 for 𝑖=1,…,𝑘 in (2.15), then lim_𝑛→∞(𝐹(𝑡^𝑛𝑥,𝑦)/𝑡^𝑛)=0. Hence, 𝑑_𝑗𝑥𝑦=lim_𝑛→∞𝑓_𝑗𝑡^𝑛𝑥𝑦𝑡^𝑛=lim_𝑛→∞𝑓_𝑗𝑡^𝑛𝑥⋅𝑦𝑡^𝑛=lim_𝑛→∞𝑡^𝑛𝑥𝑓_𝑗𝑦+∑^𝑗_𝑖=1𝑔_𝑖𝑡^𝑛𝑥𝑓_𝑗−𝑖𝑦+𝐹𝑡^𝑛𝑥,𝑦𝑡^𝑛=𝑥𝑓_𝑗𝑦+^𝑗−1_𝑖=1𝑔_𝑖(𝑥)𝑓_𝑗−𝑖𝑦+𝐷_𝑗(𝑥)𝑦, (2.24) for all 𝑥,𝑦∈𝒜. Since 𝑔₁,…,𝑔_𝑗−1,𝑓₁,…,𝑓_𝑗−1 and 𝐷_𝑗 are additive, we can write 𝑡^𝑛𝑥𝑓_𝑗𝑦+^𝑗−1_𝑖=1𝑡^𝑛𝑔_𝑖(𝑥)𝑓_𝑗−𝑖𝑦+𝑡^𝑛𝐷_𝑗(𝑥)𝑦=𝑑_𝑗𝑡^𝑛𝑥⋅𝑦=𝑑_𝑗𝑥⋅𝑡^𝑛𝑦=𝑥𝑓_𝑗𝑡^𝑛𝑦+^𝑗−1_𝑖=1𝑡^𝑛𝑔_𝑖(𝑥)𝑓_𝑗−𝑖𝑦+𝑡^𝑛𝐷_𝑗(𝑥)𝑦, (2.25) for all 𝑥,𝑦∈𝒜. We conclude that 𝑥𝑓_𝑗(𝑦)=𝑥(𝑓_𝑗(𝑡^𝑛𝑦)/𝑡^𝑛), so we can obtain 𝑥𝑓_𝑗(𝑦)=𝑥𝑑_𝑗(𝑦), for all 𝑥,𝑦∈𝒜 as 𝑛→∞. If 𝑥=𝑒, we have 𝑓_𝑗=𝑑_𝑗. Therefore, 𝑓_𝑗𝑥𝑦=𝑥𝑓_𝑗𝑦+^𝑗−1_𝑖=1𝑔_𝑖(𝑥)𝑓_𝑗−𝑖𝑦+𝐷_𝑗(𝑥)𝑦, (2.26) for all 𝑥,𝑦∈𝒜. Now, we replace 𝑦 by 𝑡^𝑛𝑦 in (2.16), then 𝑓_𝑗𝑡^𝑛𝑥𝑦𝑡^𝑛−𝑥𝑓_𝑗𝑡^𝑛𝑦𝑡^𝑛−^𝑗_𝑖=1𝑔_𝑖(𝑥)𝑓_𝑗−𝑖𝑦≤𝜓𝑥,𝑡^𝑛𝑦𝑡^𝑛≤𝛼𝑡^𝑛, (2.27) for all 𝑥,𝑦∈𝒜. Letting 𝑛→∞, we get 𝑑_𝑗𝑥𝑦=𝑥𝑑_𝑗𝑦+^𝑗_𝑖=1𝑔_𝑖(𝑥)𝑓_𝑗−𝑖𝑦. (2.30) Thus if 𝑦=𝑒, we conclude that 𝑑_𝑗(𝑥)=𝑥𝑑_𝑗(𝑒)+^𝑗_𝑖=1𝑔_𝑖(𝑥)𝑓_𝑗−𝑖(𝑒), (2.31) for all 𝑥∈𝒜. Hence, 𝑔_𝑗(𝑥)=𝑑_𝑗(𝑥)−𝑥𝑑_𝑗(𝑒)−^𝑗−1_𝑖=1𝑔_𝑖(𝑥)𝑓_𝑗−𝑖(𝑒)=𝐷_𝑗(𝑥). (2.32) But for all 𝑥,𝑦∈𝒜, we have 𝐷_𝑗𝑥𝑦=𝑓_𝑗𝑥𝑦−𝑥𝑦𝑓_𝑗(𝑒)−^𝑗−1_𝑖=1𝑔_𝑖𝑥𝑦𝑓_𝑗−𝑖(𝑒)=𝑥𝑓_𝑗𝑦+^𝑗−1_𝑖=1𝑔_𝑖(𝑥)𝑓_𝑗−𝑖𝑦+𝐷_𝑗(𝑥)𝑦−𝑥𝑦𝑓_𝑗(𝑒)−^𝑗−1_𝑖=1^𝑖_𝑙=1𝑔_𝑙(𝑥)𝑔_𝑖−𝑙𝑦𝑓_𝑗−𝑖(𝑒)=𝑔₁(𝑥)𝑓_𝑗−1𝑦−𝑦𝑓_𝑗−1(𝑒)−^𝑗−1_𝑙=1𝑔_𝑙𝑦𝑓_𝑗−𝑙(𝑒)+⋯+𝑔_𝑗−1𝑓₁𝑦−𝑦𝑓₁(𝑒)=𝑥𝐷_𝑗𝑦+𝐷_𝑗(𝑥)𝑦+^𝑗−1_𝑖=1𝑔_𝑖(𝑥)𝑔_𝑗−𝑖𝑦, (2.33) and by (2.32), it follows that 𝑔_𝑗(𝑥𝑦)=𝐷_𝑗(𝑥𝑦)=∑^𝑗_𝑖=0𝑔_𝑖(𝑥)𝑔_𝑗−𝑖(𝑦); therefore {𝑔_𝑗} is a strong higher derivation. By (2.26), we can conclude that {𝑓_𝑗} is a generalized strong higher derivation.

Remark 2.4. Recall that a control function is an operation that controls the recording or processing or transmission of interpretation of data. A typical example of the control function 𝜓 is 𝜓(𝑥,𝑦)=𝛼𝜖(‖𝑥‖^𝑝+‖𝑦‖^𝑞)+𝛿‖𝑥‖^𝑝‖𝑦‖^𝑞, such that 𝜖,𝛿≥0 and 0≤𝑝, 𝑞<1.

Corollary 2.5. Every (𝜖,𝜓)-approximate generalized derivation (regarded as an approximate generalized strong higher derivation of rank 1) on a multi-Banach algebra ((𝒜^𝑘,‖⋅‖_𝑘)𝑘∈ℕ) is a derivation.

The following theorem generalizes Theorem 2.3. The arguments are similar to those in the proof of [21, Theorem 2.3].

Theorem 2.6. Let 𝒜 be a Banach algebra with unit 𝑒, and let {𝑓_𝑗}^𝑘0_𝑗=0 be a family {𝑓_𝑗}^𝑘0_𝑗=0 of mappings from 𝒜 into 𝒜 with 𝑓_𝑗(0)=0 and 𝑓₀=𝑖𝑑_𝒜 for which there exists a family {𝑔_𝑗}^𝑘0_𝑗=0 of mappings in which 𝑔₀=𝑖𝑑_𝒜 on 𝒜 such that sup_𝑘∈ℕ𝑓_𝑗𝛽𝑥₁+𝛾𝑦₁𝑡+𝑧₁𝑤₁𝑓−𝛽^𝑗𝑥₁𝑡𝑓−𝛾^𝑗𝑦₁𝑡−𝑧₁𝑓_𝑗𝑤₁−𝑔_𝑗𝑧₁𝑤₁,…,𝑓_𝑗𝛽𝑥_𝑘+𝛾𝑦_𝑘𝑡+𝑧_𝑘𝑤_𝑘𝑓−𝛽^𝑗𝑥_𝑘𝑡𝑓−𝛾^𝑗𝑦_𝑘𝑡−𝑧_𝑘𝑓_𝑗𝑤_𝑘−𝑔_𝑗𝑧_𝑘𝑤_𝑘𝑘≤𝜖, (2.34) for all 0≤𝑗≤𝑘₀, 𝑡>1 and all 𝛽,𝛾∈𝕋={𝑧∈ℂ∶|𝑧|=1} and 𝑓_𝑗𝑥𝑦−𝑥𝑓_𝑗𝑦−^𝑗_𝑖=1𝑔_𝑖(𝑥)𝑓_𝑗−𝑖𝑦≤𝜓𝑥,𝑦, (2.35) for all 0≤𝑗≤𝑘₀ and 𝑥,𝑦∈𝒜, then {𝑓_𝑗}^𝑘0_𝑗=0 is a strong higher derivation.