The Hyers-Ulam-Rassias Stability of (𝑚,𝑛)_(𝜎,𝜏)-Derivations on Normed Algebras

Abstract

We study the Hyers-Ulam-Rassias stability of (𝑚,𝑛)_(𝜎,𝜏)-derivations on normed algebras.

1. Introduction

A classical question in the theory of functional equations is as follows. Under what conditions is it true that a mapping which approximately satisfies a functional equation ℰ must be somehow close to an exact solution of ℰ? This problem was formulated by Ulam in 1940 (see [1, 2]). He investigated the stability of group homomorphisms. Let  (𝒢₁,∘)  be a group, and let (𝒢₂,∗,𝛿)  be a metric group with a metric 𝛿(⋅,⋅). Suppose that 𝑓∶𝒢₁→𝒢₂  is a map and   𝜖>0  a fixed scalar. Does there exists 𝜆>0  such that if 𝑓satisfies the inequality𝛿𝑓𝑥∘𝑦,𝑓(𝑥)∗𝑓𝑦≤𝜆 (1.1)for all 𝑥,𝑦∈𝒢₁, then there exists a group homomorphism 𝐹∶𝒢₁→𝒢₂  with the property𝛿𝑓(𝑥),𝐹(𝑥)≤𝜖 (1.2)for all 𝑥∈𝒢₁?

One year later, Ulam's problem was affirmatively solved by Hyers [3] for the Cauchy  functional  equation   𝑓(𝑥+𝑦)=𝑓(𝑥)+𝑓(𝑦).: Let 𝒳₁ be a normed space, 𝒳₂ a Banach space, and 𝜖>0 a fixed scalar. Suppose that f  : 𝒳₁→ 𝒳₂ is a map with the property 𝑓𝑥+𝑦−𝑓(𝑥)−𝑓𝑦<𝜖 (1.3)for all 𝑥,𝑦∈𝒳₁. Then there exists a unique additive mapping 𝐹∶𝒳₁→𝒳₂  such that𝑓(𝑥)−𝐹(𝑥)<𝜖 (1.4)for all 𝑥∈𝒳₁. This gave rise to the stability theory of functional equations.

The famous Hyers stability result has been generalized in the stability of additive mappings involving a sum of powers of norms by Aoki [4] which allowed the Cauchy difference to be unbounded. In 1978, Rassias [5] proved the stability of linear mappings in the following way. Let 𝒳₁ be a real normed space and 𝒳₂ a real Banach space. If there exist scalars 𝜖≥0 and 0≤𝑝<1 such that𝑓𝑥+𝑦−𝑓(𝑥)−𝑓𝑦≤𝜖‖𝑥‖^𝑝+𝑦^𝑝 (1.5)for all 𝑥,𝑦∈𝒳₁,  then there exists a unique additive mapping 𝐹∶𝒳₁→𝒳₂ with the property𝑓(𝑥)−𝐹𝑦≤2𝜖2−2^𝑝‖𝑥‖^𝑝 (1.6)for all 𝑥∈𝒳₁. Moreover, if the map 𝑟↦𝑓(𝑟𝑥) is continuous on ℝ for each 𝑥∈𝒳₁, then 𝐹 is linear. This result has provided a lot of influence in the development of what we now call the Hyers-Ulam-Rassias stability of functional equations.

Later, Găvruţa [6] generalized the Rassias' theorem as follows: Let (𝒢,+) be an Abelian group and 𝒳 a Banach space. Suppose that the so-called admissible control function 𝜑∶𝒢×𝒢→[0,∞) satisfies^∞_𝑘=0𝜑2^𝑘𝑥,2^𝑘𝑦2^𝑘+1<∞ (1.7)for all 𝑥,𝑦∈𝒢. If 𝑓∶𝒢→𝒳  is a mapping with the property𝑓𝑥+𝑦−𝑓(𝑥)−𝑓𝑦≤𝜑𝑥,𝑦 (1.8)for all 𝑥,𝑦∈𝒢, then there exists a unique additive mapping 𝐹∶𝒢→𝒳 such that𝑓(𝑥)−𝐹(𝑥)≤^∞_𝑘=0𝜑2^𝑘𝑥,2^𝑘𝑦2^𝑘+1 (1.9)for all 𝑥∈𝒢.

In the last few decades, various approaches to the problem have been introduced by several authors. Moreover, it is surprising that in some cases the approximate mapping is actually a true mapping. In such cases we call the equation ℰ superstable. For the history and various aspects of this theory we refer the reader to monographs [7–9].

As we are aware, the stability of derivations was first investigated by Jun and Park [10]. During the past few years, approximate derivations were studied by a number of mathematicians (see [11–18] and references therein).

Moslehian [19] studied the stability of (𝜎,𝜏)-derivations and generalized some results obtained in [18]. He also established the generalized Hyers-Ulam-Rassias stability of (𝜎,𝜏)-derivations on normed algebras into Banach bimodules. This motivated us to investigate approximate (𝑚,𝑛)_(𝜎,𝜏)-derivations on normed algebras. The aim of this paper is to study the stability of (𝑚,𝑛)_(𝜎,𝜏)-derivations and to generalize some results given in [19].

2. Preliminaries

Throughout, 𝒜 will be a normed algebra and ℳ a Banach 𝒜-bimodule. Let 𝜎 and 𝜏 be two linear operators on 𝒜. An additive mapping 𝑑∶𝒜→ℳ is called an (𝜎,𝜏)-derivation if 𝑑𝑥𝑦=𝑑(𝑥)𝜎𝑦+𝜏(𝑥)𝑑𝑦 (2.1) holds for all 𝑥,𝑦∈𝒜. Ordinary derivations from 𝒜 to ℳ and maps defined by 𝑥↦𝑎𝜎(𝑥)−𝜏(𝑥)𝑎, where 𝑎∈𝒜 is a fixed element and 𝜎,𝜏 are endomorphisms on 𝒜, are natural examples of (𝜎,𝜏)-derivations on 𝒜. Moreover, if 𝜓 is an endomorphism on 𝒜, then 𝜓 is a ((1/2)𝜓,(1/2)𝜓)-derivation on 𝒜. We refer the reader to [20], where further information about (𝜎,𝜏)-derivations can be found.

In [19] Moslehian studied stability of (𝜎,𝜏)-derivations. The natural question here is, whether the analogue results hold true for (𝑚,𝑛)_(𝜎,𝜏)-derivations. Theorem 3.1 answers this question in the affirmative.

Let 𝑚 and 𝑛 be nonnegative integers with 𝑚+𝑛≠0. An additive mapping 𝑑∶𝒜→ℳ is called a (𝑚,𝑛)_(𝜎,𝜏)-derivation if (𝑚+𝑛)𝑑𝑥𝑦=2𝑚𝑑(𝑥)𝜎𝑦+2𝑛𝜏(𝑥)𝑑𝑦 (2.2) holds for all 𝑥,𝑦∈𝒜. Clearly, (𝑚,𝑛)_(𝜎,𝜏)-derivations are one of the natural generalizations of (𝜎,𝜏)-derivations (the case 𝑚=𝑛). If 𝜎,𝜏=𝑖𝑑, where 𝑖𝑑 denotes the identity map on 𝒜, and an additive mapping 𝑑∶𝒜→ℳ satisfies (2.2), then 𝑑 is called a (𝑚,𝑛)-derivation. In the last few decades a lot of work has been done on the field of (𝑚,𝑛)-derivations on rings and algebras (see, e.g, [21–25]). This motivated us to study the Hyers-Ulam-Rassias stability of functional inequalities associated with (𝑚,𝑛)_(𝜎,𝜏)-derivations.

In the following, we will assume that 𝑚 and 𝑛 are nonnegative integers with 𝑚+𝑛≠0. We will use the same symbol ‖⋅‖ in order to represent the norms on a normed algebra 𝒜 and a Banach 𝒜-bimodule ℳ. For a given (admissible control) function 𝜑∶𝒜×𝒜→[0,∞) we will use the following abbreviation: 𝜙𝑥,𝑦∶=^∞_𝑘=0𝜑2^𝑘𝑥,2^𝑘𝑦2^𝑘+1,𝑥,𝑦∈𝒜. (2.3) Let us start with one well-known lemma.

Lemma 2.1 (see [6]). Suppose that a function 𝜑∶𝒜×𝒜→[0,∞) satisfies 𝜙(𝑥,𝑦)<∞, 𝑥,𝑦∈𝒜. If 𝑓∶𝒜→ℳ is a mapping with 𝑓𝑥+𝑦−𝑓(𝑥)−𝑓𝑦≤𝜑𝑥,𝑦 (2.4) for all 𝑥,𝑦∈𝒜, then there exists a unique additive mapping 𝐹∶𝒜→ℳ such that 𝑓(𝑥)−𝐹(𝑥)≤𝜙(𝑥,𝑥) (2.5) for all 𝑥∈𝒜.

We say that an additive mapping 𝑓∶𝒜→ℳ is ℂ-linear if 𝑓(𝜆𝑥)=𝜆𝑓(𝑥) for all 𝑥∈𝒜 and all scalars 𝜆∈ℂ. In the following, Λ will denote the set of all complex units, that is, Λ={𝜆∈C∶|𝜆|=1}. (2.6) For a given additive mapping 𝑓∶𝒜→ℳ, Park [26] obtained the next result.

Lemma 2.2. If 𝑓(𝜆𝑥)=𝜆𝑓(𝑥) for all 𝑥∈𝒜 and all 𝜆∈Λ, then 𝑓 is ℂ-linear.

3. The Results

Our first result is a generalization of [19, Theorem 2.1] (the case 𝑚=𝑛). We use the direct method to construct a unique ℂ-linear mapping from an approximate one and prove that this mapping is an appropriate (𝑚,𝑛)_(𝜎,𝜏)-derivation on 𝒜. This method was first devised by Hyers [3]. The idea is taken from [19].

Theorem 3.1. Let 𝑑∶𝒜→ℳ and 𝑓,𝑔∶𝒜→𝒜 be mappings with 𝑑(0)=𝑓(0)=𝑔(0)=0. Suppose that there exists a function 𝜑∶𝒜×𝒜→[0,∞) such that 𝜙(𝑥,𝑦)<∞ for all 𝑥,𝑦∈𝒜 and 𝑑𝜆𝑥+𝜆𝑦−𝜆𝑑(𝑥)−𝜆𝑑𝑦≤𝜑𝑥,𝑦,𝑓𝜆𝑥+𝜆𝑦−𝜆𝑓(𝑥)−𝜆𝑓𝑦≤𝜑𝑥,𝑦,𝑔𝜆𝑥+𝜆𝑦−𝜆𝑔(𝑥)−𝜆𝑔𝑦≤𝜑𝑥,𝑦,(𝑚+𝑛)𝑑𝑥𝑦−2𝑚𝑑(𝑥)𝑓𝑦−2𝑛𝑔(𝑥)𝑑𝑦≤𝜑𝑥,𝑦 (3.1)(3.2)(3.3)(3.4) for all 𝑥,𝑦∈𝒜 and 𝜆∈Λ. Then there exist unique ℂ-linear mappings 𝜎,𝜏∶𝒜→𝒜 satisfying 𝑓(𝑥)−𝜎(𝑥)≤𝜙(𝑥,𝑥),𝑔(𝑥)−𝜏(𝑥)≤𝜙(𝑥,𝑥) (3.5) for all 𝑥∈𝒜, and a unique ℂ-linear (𝑚,𝑛)_(𝜎,𝜏)-derivation 𝐷∶𝒜→ℳ such that ‖𝑑(𝑥)−𝐷(𝑥)‖≤𝜙(𝑥,𝑥) (3.6) for all 𝑥∈𝒜.

Proof. Taking 𝜆=1 in (3.1) and using Lemma 2.1, it follows that there exists a unique additive mapping 𝐷∶𝒜→ℳ such that ‖𝑑(𝑥)−𝐷(𝑥)‖≤𝜙(𝑥,𝑥) holds for all 𝑥∈𝒜. More precisely, using the induction, it is easy to see that 𝑑2^𝑙𝑥2^𝑙−𝑑(𝑥)≤^𝑙−1_𝑘=0𝜑2^𝑘𝑥,2^𝑘𝑥2^𝑘+1,𝑑2^𝑝𝑥2^𝑝−𝑑2^𝑞𝑥2^𝑞≤^𝑝−1_𝑘=𝑞𝜑2^𝑘𝑥,2^𝑘𝑥2^𝑘+1 (3.7)(3.8) for all 𝑥∈𝒜, all positive integers 𝑙, and all 0≤𝑞<𝑝. According to the assumptions on 𝜙(𝑥,𝑦), it follows that the sequence {𝑑(2^𝑘𝑥)/2^𝑘}^∞_𝑘=0 is Cauchy. Thus, by the completeness of ℳ, this sequence is convergent and we can define a map 𝐷∶𝒜→ℳ as 𝐷(𝑥)∶=lim_𝑘→∞𝑑2^𝑘𝑥2^𝑘,𝑥∈𝒜. (3.9) Using (3.1), we get 𝐷𝜆𝑥+𝜆𝑦−𝜆𝐷(𝑥)−𝜆𝐷𝑦=lim_𝑘→∞2^−𝑘𝑑𝜆2^𝑘𝑥+𝜆2^𝑘𝑦−𝜆𝑑2^𝑘𝑥−𝜆𝑑2^𝑘𝑦≤lim_𝑘→∞2^−𝑘𝜑2^𝑘𝑥,2^𝑘𝑦=0. (3.10) This yields that 𝐷𝜆𝑥+𝜆𝑦=𝜆𝐷(𝑥)+𝜆𝐷𝑦 (3.11) for all 𝑥,𝑦∈𝒜 and 𝜆∈Λ. Using Lemma 2.2, it follows that the map 𝐷 is ℂ-linear. Moreover, according to inequality (3.7), we have ‖𝑑(𝑥)−𝐷(𝑥)‖=lim_𝑘→∞𝑑(𝑥)−𝑑2^𝑘𝑥2^𝑘≤𝜙(𝑥,𝑥) (3.12) for all 𝑥∈𝒜.
Next, we have to show the uniqueness of 𝐷. So, suppose that there exists another ℂ-linear mapping 𝐷∶𝒜→ℳ such that ‖𝑑(𝑥)−𝐷(𝑥)‖≤𝜙(𝑥,𝑥) for all 𝑥∈𝒜. Then 𝐷(𝑥)−𝐷(𝑥)=lim_𝑘→∞2^−𝑘𝑑2^𝑘𝑥−𝐷2^𝑘𝑥≤lim_𝑘→∞2^−𝑘𝜙2^𝑘𝑥,2^𝑘𝑥=lim_𝑘→∞2^∞−𝑘_𝑗=0𝜑2^𝑗+𝑘𝑥,2^𝑗+𝑘𝑥2^𝑗+1=lim^∞_{𝑘→∞𝑗=𝑘}𝜑2^𝑗𝑥,2^𝑗𝑥2^𝑗+1=0. (3.13) Therefore, 𝐷(𝑥)=𝐷(𝑥) for all 𝑥∈𝒜, as desired.
Similarly we can show that there exist unique ℂ-linear mappings 𝜎,𝜏∶𝒜→𝒜 defined by 𝜎(𝑥)∶=lim_𝑘→∞𝑓2^𝑘𝑥2^𝑘𝜏(𝑥),𝑥∈𝒜,∶=lim_𝑘→∞𝑔2^𝑘𝑥2^𝑘,𝑥∈𝒜. (3.14) Furthermore, 𝑓(𝑥)−𝜎(𝑥)≤𝜙(𝑥,𝑥),𝑔(𝑥)−𝜏(𝑥)≤𝜙(𝑥,𝑥) (3.15) for all 𝑥∈𝒜.
It remains to prove that 𝐷 is an (𝑚,𝑛)_(𝜎,𝜏)-derivation. Writing 2^𝑘𝑥 in the place of 𝑥 and 2^𝑘𝑦 in the place of 𝑦 in (3.4), we obtain (𝑚+𝑛)𝑑4^𝑘𝑥𝑦−2𝑚𝑑2^𝑘𝑥𝑓2^𝑘𝑦−2𝑛𝑔2^𝑘𝑥𝑑2^𝑘𝑦≤𝜑2^𝑘𝑥,2^𝑘𝑦. (3.16) This yields that ((𝑚+𝑛)𝐷𝑥𝑦−2𝑚𝐷(𝑥)𝜎𝑦−2𝑛𝜏(𝑥)𝐷𝑦=lim_𝑘→∞4^−𝑘(𝑚+𝑛)𝑑4^𝑘𝑥𝑦−2𝑚𝑑2^𝑘𝑥𝑓2^𝑘𝑦−2𝑛𝑔2^𝑘𝑥𝑑2^𝑘𝑦≤lim_𝑘→∞4^−𝑘𝜑2^𝑘𝑥,2^𝑘𝑦=0 (3.17) for all 𝑥,𝑦∈𝒜. Thus, mappings 𝐷 and 𝜎,𝜏 satisfy (2.2). The proof is completed.

Remark 3.2. If there exists 𝑥₀∈𝒜 such that 𝑑 and the map 𝑥↦𝜙(𝑥,𝑥) are continuous at point 𝑥₀, then 𝐷 is continuous on 𝒜. Namely, if 𝐷 was not continuous, then there would exist an integer 𝐶 and a sequence {𝑥_𝑘}^∞_𝑘=0 such that lim_𝑘→∞𝑥_𝑘=0 and ‖𝐷(𝑥_𝑘)‖>1/𝐶, 𝑘≥0. Let 𝑡>𝐶(2𝜙(𝑥₀,𝑥₀)+1). Then lim_𝑘→∞𝑑𝑡𝑥_𝑘+𝑥₀=𝑑𝑥₀ (3.18) since 𝑑 is continuous at point 𝑥₀. Thus, there exists an integer 𝑘₀ such that for every 𝑘>𝑘₀ we have 𝑑𝑡𝑥_𝑘+𝑥₀−𝑑𝑥₀<1. (3.19) Therefore, 2𝜙𝑥₀,𝑥₀𝑡+1<𝐶<𝐷𝑡𝑥_𝑘=𝐷𝑡𝑥_𝑘+𝑥₀−𝐷𝑥₀≤𝐷𝑡𝑥_𝑘+𝑥₀−𝑑𝑡𝑥_𝑘+𝑥₀+𝑑𝑡𝑥_𝑘+𝑥₀−𝑑𝑥₀+𝑑𝑥₀−𝐷𝑥₀<𝜙𝑡𝑥_𝑘+𝑥₀,𝑡𝑥_𝑘+𝑥₀+1+𝜙𝑥₀,𝑥₀ (3.20) for every 𝑘>𝑘₀. Letting 𝑘→∞ and using the continuity of the map 𝑥↦𝜙(𝑥,𝑥) at point 𝑥₀, we get a contradiction.

Let 𝜖≥0 and 0≤𝑝<1. Applying Theorem 3.1 for the case 𝜑𝑥,𝑦∶=𝜖‖𝑥‖^𝑝+𝑦^𝑝,𝑥,𝑦∈𝒜. (3.21)

Corollary 3.3. Let 𝑑∶𝒜→ℳ and 𝑓,𝑔∶𝒜→𝒜 be mappings with 𝑑(0)=𝑓(0)=𝑔(0)=0. Suppose that (3.1), (3.2), (3.3), and (3.4) hold true for all 𝑥,𝑦∈𝒜 and 𝜆∈Λ, where a function 𝜑∶𝒜×𝒜→[0,∞) is defined as above. Then there exist unique ℂ-linear mappings 𝜎,𝜏∶𝒜→𝒜 satisfying 𝑓(𝑥)−𝜎(𝑥)≤2𝜖2−2^𝑝‖𝑥‖^𝑝,𝑔(𝑥)−𝜏(𝑥)≤2𝜖2−2^𝑝‖𝑥‖^𝑝 (3.22) for all 𝑥∈𝒜 and a unique ℂ-linear (𝑚,𝑛)_(𝜎,𝜏)-derivation 𝐷∶𝒜→ℳ such that ‖𝑑(𝑥)−𝐷(𝑥)‖≤2𝜖2−2^𝑝‖𝑥‖^𝑝 (3.23)

Proof . Note that 𝜙(𝑥,𝑦)<∞ for all 𝑥,𝑦∈𝒜 and 𝜙=𝜖𝑥,𝑦2−2^𝑝‖𝑥‖^𝑝+𝑦^𝑝,𝑥,𝑦∈𝒜. (3.24)

Remark 3.4. Recall that we can actually take any map 𝜑∶𝒜×𝒜→[0,∞) in the form 𝜑𝑥,𝑦∶=𝜈+𝜖‖𝑥‖^𝑝+𝑦^𝑝,𝑥,𝑦∈𝒜, (3.25) where 𝜈≥0. In this case we have 𝜙𝜖‖𝑥‖𝑥,𝑦=𝜈+^𝑝+𝑦^𝑝(2−2^𝑝),𝑥,𝑦∈𝒜. (3.26)

Before stating our next result, let us write one well-known lemma about the continuity of measurable functions (see, e.g., [27]).

Lemma 3.5. If a measurable function 𝜓∶ℝ→ℝ satisfies 𝜓(𝑟₁+𝑟₂)=𝜓(𝑟₁)+𝜓(𝑟₂) for all 𝑟₁,𝑟₂∈ℝ, then 𝜓 is continuous.

Now we are in the position to state a result for normed algebras 𝒜 which are spanned by a subset 𝒮 of 𝒜. For example, 𝒜 can be a 𝐶^∗-algebra spanned by the unitary group of 𝒜 or the positive part of 𝒜

Theorem 3.6. Let 𝒜 be a normed algebra which is spanned by a subset 𝒮 of 𝒜 and 𝑑∶𝒜→ℳ, 𝑓,𝑔∶𝒜→𝒜 mappings with 𝑑(0)=𝑓(0)=𝑔(0)=0. Suppose that there exists a function 𝜑∶𝒜×𝒜→[0,∞) such that 𝜙(𝑥,𝑦)<∞ for all 𝑥,𝑦∈𝒜 and (3.1), (3.2), (3.3) holds true for all 𝑥,𝑦∈𝒜 and 𝜆=1,𝑖. Moreover, suppose that (3.4) holds true for all 𝑥,𝑦∈𝒮. If for all 𝑥∈𝒜 the functions 𝑟↦𝑑(𝑟𝑥), 𝑟↦𝑓(𝑟𝑥), and 𝑟↦𝑔(𝑟𝑥) are continuous on ℝ, then there exist unique ℂ-linear mappings 𝜎,𝜏∶𝒜→𝒜 satisfying 𝑓(𝑥)−𝜎(𝑥)≤𝜙(𝑥,𝑥),𝑔(𝑥)−𝜏(𝑥)≤𝜙(𝑥,𝑥) (3.27) for all 𝑥∈𝒜 and a unique ℂ-linear (𝑚,𝑛)_(𝜎,𝜏)-derivation 𝐷∶𝒜→ℳ such that ‖𝑑(𝑥)−𝐷(𝑥)‖≤𝜙(𝑥,𝑥) (3.28) for all 𝑥∈𝒜.

We will give just a sketch of the proof since most of the steps are the same as in the proof of Theorem 3.1.

Proof. As in the proof of Theorem 3.1, we can show that there exists a unique additive mapping 𝐷∶𝒜→ℳ defined by 𝐷(𝑥)∶=lim_𝑘→∞(𝑑(2^𝑘𝑥)/2^𝑘), 𝑥∈𝒜. Moreover, ‖𝑑(𝑥)−𝐷(𝑥)‖≤𝜙(𝑥,𝑥) for all 𝑥∈𝒜.
Writing 𝑦=0, 𝜆=𝑖 in (3.1), we get ‖𝑑(𝑖𝑥)−𝑖𝑑(𝑥)‖≤𝜑(𝑥,0). (3.29) Therefore, ‖𝐷(𝑖𝑥)−𝑖𝐷(𝑥)‖=lim_𝑘→∞2^−𝑘𝑑2^𝑘𝑖𝑥−𝑖𝑑2^𝑘𝑥≤lim_𝑘→∞2^−𝑘𝜑2^𝑘𝑥,0=0. (3.30) This yields that 𝐷(𝑖𝑥)=𝑖𝐷(𝑥) (3.31) for all 𝑥∈𝒜. In the next step we will show that 𝐷 is ℝ-linear, that is, 𝐷(𝑟𝑥)=𝑟𝐷(𝑥) (3.32) for all 𝑥∈𝒜 and all 𝑟∈ℝ.
Since 𝐷 is additive, we have 𝐷(𝑞𝑥)=𝑞𝑥 for every 𝑥∈𝒜 and all rational numbers 𝑞. Let us fix elements 𝑥₀∈𝒜 and 𝜌∈ℳ^∗, where ℳ^∗ denotes the dual space of ℳ. Then we can define a function 𝜓∶ℝ→ℝ by 𝜓(𝑟)=𝜌𝐷𝑟𝑥₀,𝑟∈ℝ. (3.33) Firstly, we would like to prove that 𝜓 is continuous. Recall that 𝜓𝑟₁+𝑟₂=𝜌𝐷𝑟₁+𝑟₂𝑥₀=𝜌𝐷𝑟₁𝑥₀+𝜌𝐷𝑟₂𝑥₀=𝜓𝑟₁+𝜓𝑟₂ (3.34) for all 𝑟₁,𝑟₂∈ℝ. Furthermore, 𝜓(𝑟)=lim_𝑘→∞𝜌𝑑2^𝑘𝑟𝑥₀2^𝑘 (3.35) for all 𝑟∈ℝ. Set 𝜓_𝑘(𝑟)𝑑2=𝜌^𝑘𝑟𝑥₀2^𝑘,𝑘≥0. (3.36) Obviously, {𝜓_𝑘}^∞_𝑘=0 is a sequence of continuous functions and 𝜓 is its pointwise limit. This yields that 𝜓 is a Borel function and, by Lemma 3.5 it is continuous. Therefore, we have 𝜓(𝑟)=𝑟𝜓(1) for all 𝑟∈ℝ. This implies 𝐷(𝑟𝑥₀)=𝑟𝐷(𝑥₀). Since 𝑥₀ was an arbitrary element from 𝒜, we proved that 𝐷 is ℝ-linear.
Now, let 𝜆∈ℂ. Then 𝜆=𝑟₁+𝑖𝑟₂ for some real numbers 𝑟₁,𝑟₂. Using (3.31), we have 𝐷(𝜆𝑥)=𝐷𝑟₁+𝑖𝑟₂𝑥=𝐷𝑟₁𝑥+𝐷𝑖𝑟₂𝑥=𝑟₁𝐷(𝑥)+𝑖𝑟₂𝐷(𝑥)=𝜆𝐷(𝑥) (3.37) for all 𝑥∈𝒜. This means that 𝐷 is ℂ-linear.
Similarly we can show that there exist unique ℂ-linear mappings 𝜎,𝜏∶𝒜→𝒜 satisfying 𝑓(𝑥)−𝜎(𝑥)≤𝜙(𝑥,𝑥),𝑔(𝑥)−𝜏(𝑥)≤𝜙(𝑥,𝑥) (3.38) for all 𝑥∈𝒜. Moreover, (2.2) holds true for all 𝑥,𝑦∈𝒮. Since 𝒜 is linearly generated by 𝒮, we conclude that 𝐷 is an (𝑚,𝑛)_(𝜎,𝜏)-derivation on 𝒜. The proof is completed.

Remark 3.7. As above, we can apply Theorem 3.6 for the case 𝜑𝑥,𝑦∶=𝜈+𝜖‖𝑥‖^𝑝+𝑦^𝑝,𝑥,𝑦∈𝒜, (3.39) where 𝜈,𝜖≥0 and 0≤𝑝<1.

Remark 3.8. If 𝜖≥0 and 0≤𝑝<1/2, then we can use in Theorem 3.1 as well as in Theorem 3.6 a function 𝜑∶𝒜×𝒜→[0,∞) given by 𝜑𝑥,𝑦∶=𝜖‖𝑥‖^𝑝𝑦^𝑝,𝑥,𝑦∈𝒜. (3.40) In this case 𝜙=𝜖𝑥,𝑦2−4^𝑝‖𝑥‖^𝑝𝑦^𝑝,𝑥,𝑦∈𝒜. (3.41)