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Abstract and Applied Analysis
Volume 2013 (2013), Article ID 425784, 5 pages
Approximate Cubic Lie Derivations
1Faculty of Management, University of Primorska, Cankarjeva 5, 6104 Koper, Slovenia
2Faculty of Logistics, University of Maribor, Mariborska Cesta 7, 3000 Celje, Slovenia
Received 11 April 2013; Accepted 27 June 2013
Academic Editor: Janusz Brzdek
Copyright © 2013 Ajda Fošner and Maja Fošner. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
We study the stability and hyperstability of cubic Lie derivations on normed algebras. At the end, we write some additional observations about our results.
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 must be somehow close to an exact solution of ? We say that a functional equation is stable if any approximate solution of is near to a true solution of . The study of stability problem for functional equations is related to a question of Ulam  concerning the stability of group homomorphisms. This question was affirmatively answered for Banach spaces by Hyers . Subsequently, the result of Hyers was generalized by Aoki  for additive mappings and by Rassias  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  introduced the following functional equation: and established a general solution for it. Note that, if we replace in (1), we get . Therefore, setting in (1), we obtain . Moreover, writing in (1), we get . Now, it is easy to see that a function is a solution of (1), where 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 , the authors investigated the stability of cubic derivations in a connection with the functional equation (1). Recently, Yang et al.  studied the stability of cubic -derivations on Banach -algebras and in  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, will be a complex normed algebra and a Banach -bimodule. We will use the same symbol in order to represent the norms on a normed algebra and a normed -bimodule . For all , the symbol will denote the commutator . We say that a mapping is cubic homogeneous if for all and all scalars . In the following, will stand for the set of all complex units; that is,
A cubic homogeneous mapping is called a cubic derivation if holds for all . Following this notion, we introduce a cubic Lie derivation as a cubic homogeneous mapping satisfying for all .
For a given map , we consider where , , and Moreover, for a function we use the following abbreviation:
Theorem 1. Suppose that is a mapping for which there exists a function such that for all and . If for each fixed the mapping from to is continuous, then there exists a unique cubic Lie derivation such that for all .
Proof. Writing and in the first inequality of (9), we obtain
for all . Using the induction, it is easy to see that
for all and all . By (8), it follows that, for all , the sequence is Cauchy and, since is complete, it is convergent. Thus, we can define a mapping as
Writing and in (12), we have
for . Taking the limit as tends to we get (10).
In the following, we will show that is a unique cubic Lie derivation such that (10) holds true for all .
Claim 1 ( is a cubic mapping). Recall that for all and . Hence, taking in (15), it follows that is a cubic mapping.
Claim 2 ( is cubic homogeneous). By (15), we have . Thus,
Let be any fixed element. Firstly, we will show that for all . We will omit the details since the arguments are the same as in  (see also ). So, let us choose any linear functional , where denotes the dual space of . Then, we can define a function by It is easy to see that is cubic. Set Note that is a pointwise limit of a sequence of continuous functions , . Thus, is a continuous cubic function and Consequently, for all . Since was an arbitrary linear functional, we conclude that
Let . Then , , and we have Since was any element from , we conclude that is cubic homogeneous.
Claim 3 ( is a cubic Lie derivation). Using the second inequality in (9), we have for all . This, together with Claim 2, yields that is a cubic Lie derivation.
Claim 4 ( is unique). Suppose that there exists another cubic Lie derivation such that for all . Then Therefore, for all , which proves the uniqueness of . The proof is completed.
Let and . Applying Theorem 1 for the case we have the next corollary.
Corollary 2. Suppose that is a mapping such that (9) holds true for all and , where is a function defined as above. If for each fixed the mapping from to is continuous, then there exists a unique cubic Lie derivation such that for all .
Proof. Note that we have
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., ). 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 , satisfying the functional equation approximately, must actually be a solution of the proposed equation . In this case, we say that the functional equation is hyperstable. According to our best knowledge, the first hyperstability result was published in  and concerned the ring homomorphisms. However, the term hyperstability has been used for the first time probably in  (see also [22, 23]). For more information about the new results on the hyperstability, we refer the reader to [24–26].
Theorem 3. Let be a normed algebra with an element which is not a zero divisor. Suppose that is a mapping for which there exists a function such that for all . Then, is a cubic Lie derivation on .
Proof. We divide the proof into several steps.
Claim 1 ( is a cubic mapping). Let . Then, for all positive integers . By (30), it follows that If we let , we conclude that for all . Putting in the last equality, we conclude that satisfies a cubic functional equation (1).
Claim 2 ( is cubic homogeneous). Let and . Then, for all positive integers . Thus, Taking the limit when tends to , we conclude that Writing , we get for all and . In other words, is cubic homogeneous.
Claim 3 ( is a cubic Lie derivation on ). Let . Then for all positive integers . Thus, for all and . Similarly as in Claims 1 and 2, we can show that for all . The proof is completed.
Let and . Applying Theorem 3 for the case we have the next corollary.
Corollary 4. Let be a normed algebra with an element which is not a zero divisor. Suppose that is a mapping such that for all . Then is a cubic Lie derivation on .
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 is a mapping for which there exists a function such that (9) holds true for all , , and If for each fixed the mapping from to is continuous, then there exists a unique cubic Lie derivation such that (10) is valid for all .
Similarly, the next result is analogue to Theorem 3.
Theorem 6. Let be a normed algebra with an element which is not a zero divisor. Suppose that is a mapping for which there exists a function such that (30) and (31) hold true for all and Then, is a cubic Lie derivation on .
Let us end this paper with a remark on -cubic functional equations. In , Najati introduced the following functional equations: where is a positive integer. For , we obtain (1). Najati proved that a function satisfies the functional equation (1) if and only if 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).
The authors would like to thank the referees for their useful comments.
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