Approximate Cubic Lie Derivations

Abstract

We study the stability and hyperstability of cubic Lie derivations on normed algebras. At the end, we write some additional observations about our results.

1. Introduction

A classical question in the theory of functional equations is 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? We say that a functional equation ℰ is stable if any approximate solution of ℰ is near to a true solution of ℰ. The study of stability problem for functional equations is related to a question of Ulam [1] concerning the stability of group homomorphisms. This question was affirmatively answered for Banach spaces by Hyers [2]. Subsequently, the result of Hyers was generalized by Aoki [3] for additive mappings and by Rassias [4] for linear mappings by considering an unbounded Cauchy difference. Rassias’ paper has provided a lot of influence in the development of what we now call the Hyers-Ulam-Rassias stability of functional equations. For further information about the topic, we refer the reader to [5, 6].

Jun and Kim [7] introduced the following functional equation: 𝑓(𝑥),2𝑥+𝑦+𝑓2𝑥−𝑦=2𝑓𝑥+𝑦+2𝑓𝑥−𝑦+12𝑓 (1) and established a general solution for it. Note that, if we replace 𝑥=𝑦=0 in (1), we get 𝑓(0)=0. Therefore, setting 𝑥=0 in (1), we obtain 𝑓(−𝑦)=−𝑓(𝑦). Moreover, writing 𝑦=0 in (1), we get 𝑓(2𝑥)=8𝑓(𝑥). Now, it is easy to see that a function 𝑓(𝑥)=𝜆𝑥³ is a solution of (1), where 𝜆 is a fixed scalar. Thus, it is natural that (1) is called a cubic functional equation. Moreover, every solution of the cubic functional equation is said to be a cubic mapping. In the last few decades, a number of mathematicians worked on the stability of some type of cubic functional equations (see, e.g., [8–13]).

In [14], the authors investigated the stability of cubic derivations in a connection with the functional equation (1). Recently, Yang et al. [15] studied the stability of cubic ∗-derivations on Banach ∗-algebras and in [16] the authors proved the generalized Hyers-Ulam-Rassias stability of ternary cubic derivations on ternary Fréchet algebras. Motivated by these results, we investigate the stability of cubic Lie derivations. In particular, we show that such derivations can be generated by functions which satisfy some quite natural and simple conditions.

2. Stability of Cubic Lie Derivations

Before stating our first theorem, let us recall some basic definitions and known results which we will use in the sequel. Throughout the paper, 𝒜 will be a complex normed algebra and ℳ a Banach 𝒜-bimodule. We will use the same symbol ‖⋅‖ in order to represent the norms on a normed algebra 𝒜 and a normed 𝒜-bimodule ℳ. For all 𝑥,𝑦∈𝒜, the symbol [𝑥,𝑦] will denote the commutator 𝑥𝑦−𝑦𝑥. We say that a mapping 𝑓∶𝒜→ℳ is cubic homogeneous if 𝑓(𝜆𝑥)=𝜆³𝑓(𝑥) for all 𝑥∈𝒜 and all scalars 𝜆∈ℂ. In the following, Λ will stand for the set of all complex units; that is, Λ={𝜆∈ℂ∶|𝜆|=1}. (2)

A cubic homogeneous mapping 𝑑∶𝒜→ℳ is called a cubic derivation if 𝑑𝑥𝑦=𝑑(𝑥)𝑦³+𝑥³𝑑𝑦 (3) holds for all 𝑥,𝑦∈𝒜. Following this notion, we introduce a cubic Lie derivation as a cubic homogeneous mapping 𝑙∶𝒜→ℳ satisfying 𝑙𝑥,𝑦=𝑙(𝑥),𝑦³+𝑥³,𝑙𝑦 (4) for all 𝑥,𝑦∈𝒜.

For a given map 𝑓∶𝒜→ℳ, we consider Δ^𝜆_𝑓𝑥,𝑦∶=𝑓2𝜆𝑥+𝜆𝑦+𝑓2𝜆𝑥−𝜆𝑦−2𝜆³𝑓𝑥+𝑦−2𝜆³𝑓𝑥−𝑦−12𝜆³𝑓(𝑥), (5) where 𝑥,𝑦∈𝒜, 𝜆∈ℂ, and Δ_𝑓𝑥,𝑦∶=𝑓𝑥,𝑦−𝑓(𝑥),𝑦³−𝑥³,𝑓𝑦,𝑥,𝑦∈𝒜. (6) Moreover, for a function 𝜑∶𝒜×𝒜→[0,∞) we use the following abbreviation: 𝜙𝑥,𝑦∶=^∞_𝑘=0𝜑2^𝑘𝑥,2^𝑘𝑦8^𝑘,𝑥,𝑦∈𝒜. (7)

Theorem 1. Suppose that 𝑙∶𝒜→ℳ is a mapping for which there exists a function 𝜑∶𝒜×𝒜→[0,∞) such that 𝜙Δ𝑥,𝑦<∞,𝑥,𝑦∈𝒜,_𝜆𝑙𝑥,𝑦≤𝜑𝑥,𝑦,Δ_𝑙𝑥,𝑦≤𝜑𝑥,𝑦 (8)(9) for all 𝑥,𝑦∈𝒜 and 𝜆∈Λ. If for each fixed 𝑥∈𝒜 the mapping 𝑟↦𝑙(𝑟𝑥) from ℝ to ℳ is continuous, then there exists a unique cubic Lie derivation 𝐿∶𝒜→ℳ such that ‖𝑙(𝑥)(𝑥≤𝜙()−𝐿)‖𝑥,016 (10) for all 𝑥∈𝒜.

Proof. Writing 𝑦=0 and 𝜆=1 in the first inequality of (9), we obtain 𝑙(2𝑥)8(𝑥≤𝜑()−𝑙)𝑥,016 (11) for all 𝑥∈𝒜. Using the induction, it is easy to see that 𝑙2^𝑝𝑥8^𝑝−𝑙2^𝑞𝑥8^𝑞≤116^𝑝−1_𝑘=𝑞𝜑2^𝑘𝑥,08^𝑘 (12) for all 𝑥∈𝒜 and all 𝑝>𝑞≥0. By (8), it follows that, for all 𝑥∈𝒜, the sequence {𝑙(2^𝑛𝑥)/8^𝑛}^∞_𝑛=0 is Cauchy and, since ℳ is complete, it is convergent. Thus, we can define a mapping 𝐿∶𝒜→ℳ as 𝐿(𝑥)=lim_𝑛→∞𝑙2^𝑛𝑥8^𝑛,𝑥∈𝒜. (13) Writing 𝑝=𝑛 and 𝑞=0 in (12), we have 𝑙2^𝑛𝑥8^𝑛(𝑥≤1−𝑙)16^𝑛−1_𝑘=0𝜑2^𝑘𝑥,08^𝑘 (14) for 𝑛>0. Taking the limit as 𝑛 tends to ∞ we get (10).
In the following, we will show that 𝐿 is a unique cubic Lie derivation such that (10) holds true for all 𝑥∈𝒜.
Claim 1 (𝐿 is a cubic mapping). Recall that Δ^𝜆_𝐿𝑥,𝑦=lim_𝑛→∞Δ^𝜆_𝑙2^𝑛𝑥,2^𝑛𝑦8^𝑛≤lim_𝑛→∞𝜑2^𝑛𝑥,2^𝑛𝑦8^𝑛=0 (15) for all 𝑥,𝑦∈𝒜 and 𝜆∈Λ. Hence, taking 𝜆=1 in (15), it follows that 𝐿 is a cubic mapping.
Claim 2 (𝐿 is cubic homogeneous). By (15), we have Δ^𝜆_𝐿(𝑥,0)=0. Thus, 𝐿(2𝜆𝑥)=8𝜆³𝐿(𝑥),𝑥∈𝒜,𝜆∈Λ. (16)
Let 𝑥₀∈𝒜 be any fixed element. Firstly, we will show that 𝐿(𝑟𝑥₀)=𝑟³𝐿(𝑥₀) for all 𝑟∈ℝ. We will omit the details since the arguments are the same as in [17] (see also [18]). So, let us choose any linear functional 𝜌∈ℳ^∗, where ℳ^∗ denotes the dual space of ℳ. Then, we can define a function 𝜓∶ℝ→ℝ by 𝜓(𝑟)=𝜌𝐿𝑟𝑥₀,𝑟∈ℝ. (17) It is easy to see that 𝜓 is cubic. Set 𝜓_𝑛(𝑟)𝑙2=𝜌^𝑛𝑟𝑥₀8^𝑛,𝑛≥0. (18) Note that 𝜓 is a pointwise limit of a sequence of continuous functions 𝜓_𝑛, 𝑛≥0. Thus, 𝜓 is a continuous cubic function and 𝜓(𝑟)=𝑟³𝜓(1),𝑟∈ℝ. (19) Consequently, 𝜌𝐿𝑟𝑥₀=𝜓(𝑟)=𝑟³𝜓(1)=𝑟³𝜌𝐿𝑥₀=𝜌𝑟³𝐿𝑥₀ (20) for all 𝑟∈ℝ. Since 𝜌∈ℳ^∗ was an arbitrary linear functional, we conclude that 𝐿𝑟𝑥₀=𝑟³𝐿𝑥₀,𝑟∈ℝ. (21)
Let 𝜆∈ℂ. Then 𝜆/|𝜆|∈Λ, |𝜆|/2∈ℝ, and we have 𝐿𝜆𝑥₀2𝜆=𝐿|𝜆||𝜆|2𝑥₀𝜆=8|𝜆|³𝐿|𝜆|2𝑥₀𝜆=8³|𝜆|³|𝜆|2³𝐿𝑥₀=𝜆³𝐿𝑥₀. (22) Since 𝑥₀ was any element from 𝒜, we conclude that 𝐿 is cubic homogeneous.
Claim 3 (𝐿 is a cubic Lie derivation). Using the second inequality in (9), we have Δ_𝐿𝑥,𝑦=lim_𝑛→∞Δ_𝑙2^𝑛𝑥,2^𝑛𝑦8^2𝑛≤lim_𝑛→∞𝜑2^𝑛𝑥,2^𝑛𝑦8^2𝑛=0 (23) for all 𝑥,𝑦∈𝒜. This, together with Claim 2, yields that 𝐿 is a cubic Lie derivation.
Claim 4 (𝐿 is unique). Suppose that there exists another cubic Lie derivation 𝐿∶𝒜→ℳ such that 𝑙(𝑥)−𝐿(𝑥≤𝜙())𝑥,016 (24) for all 𝑥∈𝒜. Then 𝐿(𝑥)−𝐿(𝑥)=lim_𝑛→∞18^𝑛𝑙2^𝑛𝑥−𝐿2^𝑛𝑥≤lim_𝑛→∞1𝜙216^𝑛𝑥,08^𝑛=lim_𝑛→∞116^∞_𝑘=0𝜑2^𝑘+𝑛𝑥,08^𝑘+𝑛=lim_𝑛→∞116^∞_𝑘=𝑛𝜑2^𝑘𝑥,08^𝑘=0. (25) Therefore, 𝐿(𝑥)=𝐿(𝑥) for all 𝑥∈𝒜, which proves the uniqueness of 𝐿. The proof is completed.

Let 𝜈,𝜀≥0 and 0≤𝑝<3. Applying Theorem 1 for the case 𝜑𝑥,𝑦∶=𝜈+𝜀‖𝑥‖^𝑝+𝑦^𝑝,𝑥,𝑦∈𝒜, (26) we have the next corollary.

Corollary 2. Suppose that 𝑙∶𝒜→ℳ is a mapping such that (9) holds true for all 𝑥,𝑦∈𝒜 and 𝜆∈Λ, where 𝜑 is a function defined as above. If for each fixed 𝑥∈𝒜 the mapping 𝑟↦𝑙(𝑟𝑥) from ℝ to ℳ is continuous, then there exists a unique cubic Lie derivation 𝐿∶𝒜→ℳ such that ‖𝑙(𝑥)(𝑥≤𝜈−𝐿)‖+𝜀‖𝑥‖1416−2^𝑝𝑝+1 (27) for all 𝑥∈𝒜.

Proof. Note that we have 𝜙𝑥,𝑦=8𝜈7+8𝜀‖𝑥‖^𝑝+𝑦^𝑝8−2^𝑝,𝑥,𝑦∈𝒜. (28)

3. Hyperstability of Cubic Lie Derivations

The investigation of the multiplicative Cauchy functional equation highlighted a new phenomenon, which is nowadays called superstability (see, e.g., [19]). In this case the so-called stability inequality implies that the observed function is either bounded or it is a solution of the functional equation. But it can also happen that each function 𝑓, satisfying the functional equation ℰ approximately, must actually be a solution of the proposed equation ℰ. In this case, we say that the functional equation ℰ is hyperstable. According to our best knowledge, the first hyperstability result was published in [20] and concerned the ring homomorphisms. However, the term hyperstability has been used for the first time probably in [21] (see also [22, 23]). For more information about the new results on the hyperstability, we refer the reader to [24–26].

Theorem 3. Let 𝒜 be a normed algebra with an element 𝑒 which is not a zero divisor. Suppose that 𝑙∶𝒜→𝒜 is a mapping for which there exists a function 𝜑∶𝒜×𝒜→[0,∞) such that lim_𝑛→∞𝜑𝑛𝑥,𝑦𝑛³𝑙(𝑥)𝑦=0,₃−𝑥³𝑙𝑦𝑙(𝑧)≤𝜑𝑥,𝑦,𝑥,𝑦³−𝑧³𝑙(𝑥),𝑦³+𝑥³,𝑙𝑦≤𝜑𝑧,𝑥,𝑦 (29)(30)(31) for all 𝑥,𝑦,𝑧∈𝒜. Then, 𝑙 is a cubic Lie derivation on 𝒜.

Proof. We divide the proof into several steps.
Claim 1 (𝑙 is a cubic mapping). Let 𝑥,𝑦,𝑧∈𝒜. Then, 𝑛³𝑧³𝑙(𝑥≤𝑛2𝑥+𝑦+𝑙2𝑥−𝑦−2𝑙𝑥+𝑦−2𝑙𝑥−𝑦−12𝑙)₃𝑧³𝑙2𝑥+𝑦−𝑙(𝑛𝑧)2𝑥+𝑦³+𝑛³𝑧³𝑙2𝑥−𝑦−𝑙(𝑛𝑧)2𝑥−𝑦³+2𝑙(𝑛𝑧)𝑥+𝑦³−𝑛³𝑧³𝑙𝑙(𝑥+𝑦+2𝑛𝑧)𝑥−𝑦³−𝑛³𝑧³𝑙𝑙()𝑥𝑥−𝑦+12𝑛𝑧³−𝑛³𝑧³𝑙(𝑥) (32) for all positive integers 𝑛. By (30), it follows that 𝑧³𝑙(𝑥2𝑥+𝑦+𝑙2𝑥−𝑦−2𝑙𝑥+𝑦−2𝑙𝑥−𝑦−12𝑙)≤𝑛⁻³𝜑().𝑛𝑧,2𝑥+𝑦+𝜑𝑛𝑧,2𝑥−𝑦+2𝜑𝑛𝑧,𝑥+𝑦+2𝜑𝑛𝑧,𝑥−𝑦+12𝜑𝑛𝑧,𝑥 (33) If we let 𝑛→∞, we conclude that 𝑧³𝑙(𝑥)2𝑥+𝑦+𝑙2𝑥−𝑦−2𝑙𝑥+𝑦−2𝑙𝑥−𝑦−12𝑙=0 (34) for all 𝑥,𝑦,𝑧∈𝒜. Putting 𝑧=𝑒 in the last equality, we conclude that 𝑙 satisfies a cubic functional equation (1).
Claim 2 (𝑙 is cubic homogeneous). Let 𝜆∈ℂ and 𝑥,𝑦∈𝒜. Then, 𝑛³𝑦³𝑙(𝜆𝑥)−𝜆³𝑙(𝑥)≤𝑛³𝑦³𝑙(𝜆𝑥)−𝑙𝑛𝑦𝜆³𝑥³+|𝜆|³𝑙𝑛𝑦𝑥³−𝑛³𝑦³𝑙(𝑥+|𝜆|)≤𝜑𝑛𝑦,𝜆𝑥³𝜑𝑛𝑦,𝑥 (35) for all positive integers 𝑛. Thus, 𝑦³𝑙(𝜆𝑥)−𝜆³𝑙(𝑥)≤𝑛⁻³𝜑𝑛𝑦,𝜆𝑥+|𝜆|³𝜑𝑛𝑦,𝑥. (36) Taking the limit when 𝑛 tends to ∞, we conclude that 𝑦³𝑙(𝜆𝑥)−𝜆³𝑙(𝑥)=0,𝑥,𝑦∈𝒜,𝜆∈ℂ. (37) Writing 𝑦=𝑒, we get 𝑙(𝜆𝑥)=𝜆³𝑙(𝑥) for all 𝑥∈𝒜 and 𝜆∈ℂ. In other words, 𝑙 is cubic homogeneous.
Claim 3 (𝑙 is a cubic Lie derivation on 𝒜). Let 𝑥,𝑦,𝑧∈𝒜. Then 𝑛³𝑧³𝑙𝑥,𝑦−𝑙(𝑥),𝑦³−𝑥³𝑦≤𝑛,𝑙₃𝑧³𝑙𝑥,𝑦−𝑙(𝑛𝑧)𝑥,𝑦³+𝑙(𝑛𝑧)𝑥,𝑦³−𝑛³𝑧³𝑙(𝑥),𝑦³+𝑥³,𝑙𝑦≤2𝜑𝑛𝑧,𝑥,𝑦 (38) for all positive integers 𝑛. Thus, 𝑧³𝑙𝑥,𝑦−𝑙(𝑥),𝑦³−𝑥³,𝑙𝑦≤𝑛⁻³2𝜑𝑛𝑧,𝑥,𝑦 (39) for all 𝑥,𝑦,𝑧∈𝐴 and 𝑛∈ℕ. Similarly as in Claims 1 and 2, we can show that 𝑙([𝑥,𝑦])=[𝑙(𝑥),𝑦³]+[𝑥³,𝑙(𝑦)] for all 𝑥,𝑦∈𝒜. The proof is completed.

Let 𝜀≥0 and 0≤𝑝<3. Applying Theorem 3 for the case 𝜑𝑥,𝑦∶=𝜀‖𝑥‖^𝑝𝑦^𝑝,𝑥,𝑦∈𝒜, (40) we have the next corollary.

Corollary 4. Let 𝒜 be a normed algebra with an element 𝑒 which is not a zero divisor. Suppose that 𝑙∶𝒜→𝒜 is a mapping such that 𝑙(𝑥)𝑦³−𝑥³𝑙𝑦≤𝜀‖𝑥‖^𝑝𝑦^𝑝,𝑙(𝑧)𝑥,𝑦³−𝑧³𝑙(𝑥),𝑦³+𝑥³𝑦‖𝑧‖,𝑙≤𝜀^𝑝[𝑥,𝑦]^𝑝 (41) for all 𝑥,𝑦,𝑧∈𝒜. Then 𝑙 is a cubic Lie derivation on 𝒜.

4. Some Additional Remarks

In this section, we will write some additional observations about our results. We start with the theorem which is analogue to Theorem 1. Since the proof is similar, it is omitted.

Theorem 5. Suppose that 𝑙∶𝒜→ℳ is a mapping for which there exists a function 𝜑∶𝒜×𝒜→[0,∞) such that (9) holds true for all 𝑥,𝑦∈𝒜, 𝜆∈Λ, and ^∞_𝑘=08^𝑘𝜑𝑥2^𝑘,𝑦2^𝑘<∞,𝑥,𝑦∈𝒜. (42) If for each fixed 𝑥∈𝒜 the mapping 𝑟↦𝑙(𝑟𝑥) from ℝ to ℳ is continuous, then there exists a unique cubic Lie derivation 𝐿∶𝒜→ℳ such that (10) is valid for all 𝑥∈𝒜.

Similarly, the next result is analogue to Theorem 3.

Theorem 6. Let 𝒜 be a normed algebra with an element 𝑒 which is not a zero divisor. Suppose that 𝑙∶𝒜→𝒜 is a mapping for which there exists a function 𝜑∶𝒜×𝒜→[0,∞) such that (30) and (31) hold true for all 𝑥,𝑦,𝑧∈𝒜 and lim_𝑛→∞𝑛³𝜑𝑥𝑛,𝑦=0,𝑥,𝑦∈𝒜. (43) Then, 𝑙 is a cubic Lie derivation on 𝒜.

Let us end this paper with a remark on 𝑘-cubic functional equations. In [27], Najati introduced the following functional equations: 𝑓𝑘𝑘𝑥+𝑦+𝑓𝑘𝑥−𝑦=𝑘𝑓𝑥+𝑦+𝑘𝑓𝑥−𝑦+2³−𝑘𝑓(𝑥), (44) where 𝑘≥2 is a positive integer. For 𝑘=2, we obtain (1). Najati proved that a function 𝑓 satisfies the functional equation (1) if and only if 𝑓 satisfies the functional equation (44). Therefore, every solution of the functional equation (44) is also a cubic function. Following these results, it is easy to see that we can consider (44) in our theorems instead of (1).

Acknowledgment

The authors would like to thank the referees for their useful comments.