Hyers-Ulam-Rassias stability of generalized module left -derivations
© Fošner; licensee Springer 2013
Received: 24 September 2012
Accepted: 14 April 2013
Published: 26 April 2013
The generalized Hyers-Ulam-Rassias stability of generalized module left -derivations on a normed algebra into a Banach left -module is established.
We say that a functional equation ℰ is stable if any function f which approximately satisfies the equation ℰ is near to an exact solution of ℰ. The problem of stability of functional equations was formulated by Ulam in 1940 for group homomorphisms (see [1, 2]). One year later, Ulam’s problem was affirmatively solved by Hyers  for the Cauchy functional equation . This gave rise to the stability theory of functional equations. Later, Aoki  and Rassias  considered mappings from a normed space into a Banach space such that the norm of the Cauchy difference is bounded by the expression for all x, y, some , and . The terminology Hyers-Ulam-Rassias stability indeed originates from Rassias’s paper.
In the last few decades, the stability problem of several functional equations has been extensively studied by many authors. For the history and various aspects of the theory, we refer to monographs [6–8]. We also refer the reader to the paper , where a precise description of the Hyers-Ulam-Rassias stability is given.
As we are aware, the stability of derivations was first investigated by Jun and Park . During the past few years, approximate derivations were studied by a number of mathematicians (see [11–16] and references therein). The stability result concerning derivations between operator algebras was first obtained by Šemrl . Later, Moslehian  studied approximate generalized derivations on unital Banach algebras into Banach bimodules, and in  Jung examined the stability of module left derivations. Recently, the author studied the generalized Hyers-Ulam-Rassias stability of a functional inequality associated with module left -derivations . The natural question here is whether we can generalize these results in the setting of generalized module left -derivations. Theorem 2.2 answers this question in the affirmative.
In the following, and ℳ will represent a complex normed algebra and a Banach left -module, respectively. Recall that if has a unit element e such that for all , then a left -module ℳ is called unitary. Here, ⋅ denotes the module multiplication on ℳ. We will use the same symbol to represent the norms on a normed algebra and a normed left -module ℳ.
for all . Obviously, if , then every generalized module left -derivation is a generalized module left derivation.
In the last few decades, a lot of work has been done in the field of left derivations and generalized left derivations (see, for example, [19, 20] and references therein). Recently also -derivations and generalized -derivations have been defined and investigated [21–24]. This motivated us to study the generalized Hyers-Ulam-Rassias stability of functional inequalities associated with generalized module left -derivations.
2 The main result
For a given additive mapping , Park  obtained the next result.
Lemma 2.1 If for all and all , then f is ℂ-linear.
Our first result is a generalization of Theorem 1 in .
for all .
Letting , we conclude that for all and all . According to Lemma 2.1, this yields that G is linear.
for all . Moreover, by Theorem 1 in , D is a module left -derivation.
Letting , we conclude that . The proof is completed. □
3 Some additional remarks
In this section we write some additional results and observations about our main theorem.
for every . Letting and using the continuity of the map at point , we get a contradiction.
where and .
Remark 3.3 Let denote the family of all sets such that each additive function bounded on Γ is continuous. The question which subsets belong to has been a subject of many papers. It is known that every non-empty open subset Γ of ℂ is a member of . Moreover, if and , then . The same is true when has a positive inner Lebesgue measure or contains a subset of the second category with the Baire property. For more information and further references concerning the subject, we refer the reader to [27–29].
Let be a bounded set. Using standard techniques, it is easy to see that every additive function with the property for all and all must be ℂ-linear (see, for example, [, Lemma 1]). Therefore, we can show that Theorem 2.2 is valid even if we replace Λ with any set which contains a bounded subset . Namely, as in the proof of Theorem 2.2, we can show that there exists a unique generalized module left -derivation satisfying (5). Moreover, for all and all . Using the above mentioned arguments, it follows that G is ℂ-linear.
4 Superstability of generalized module left -derivations
We end this paper with some observations on superstability. We say that a functional equation ℰ is superstable if each function f, satisfying the equation ℰ approximately, must actually be a solution of it. The notion of superstability has appeared in connection with the investigation of stability of the exponential equation . The first result for the superstability of this equation was proved by Bourgin . Later, this problem was renewed and investigated by Baker, Lawrence, and Zorzitto  (for more information, see  and references therein). Our last result shows that this kind of properties are valid also for conditions involving generalized module left -derivations.
In the following, and ℳ will be a normed algebra with a unit e and a unitary Banach left -module, respectively. Assume that is a function satisfying and for some nonnegative scalars ξ, η with . Then we have the next lemma.
Lemma 4.1 [, Lemma 1]
If is a mapping satisfying (4) and , , then for all and all .
Our last result is a generalization of Theorem 2 in .
for all . Then g is a generalized module left -derivation.
Proof We divide the proof into several steps.
for all and . Letting , we conclude that . In other words, d is a module left -derivation on .
for all and all .
for all and . Taking the limit when , we conclude that , as desired. Therefore, g is a generalized module left -derivation on . This completes the proof. □
Corollary 4.3 Let be a normed algebra with a unit e, let ℳ be a unitary Banach left -module, and let be a function such that and for some nonnegative scalars ξ, η with . Suppose that is a mapping for which there exists a mapping such that (1)-(4) hold true for all and . Then g is a linear generalized module left -derivation.
The author would like to thank the referees for their useful comments.
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