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Hyers-Ulam-Rassias stability of generalized module left -derivations
Journal of Inequalities and Applications volume 2013, Article number: 208 (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.
Before stating our main theorem, let us write some basic definitions and known results which we will need in the sequel. First, an additive mapping is called a module left derivation if holds for all . Let with be some fixed integers. Then an additive mapping is called a module left -derivation if
for all . Clearly, module left -derivations are one of the natural generalizations of module left derivations (the case ). Furthermore, an additive mapping is called a generalized module left derivation if there exists a module left derivation such that is fulfilled for all . Motivated by this notion, we define a generalized module left -derivation as an additive mapping for which there exists a module left -derivation such that
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
Throughout the paper, we assume that m and n are nonnegative integers with . We say that an additive mapping is ℂ-linear (or just linear) if for all and all scalars . In the following, Λ denotes the set of all complex units, i.e.,
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 .
Theorem 2.2 Let be a normed algebra, ℳ be a Banach left -module, and 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
for all and . Then there exists a unique linear generalized module left -derivation such that
for all .
Proof Taking in (1), we obtain
for all x, y. Thus, by Corollary 3.2 in , we conclude that there exists a unique additive mapping such that (5) holds for all . Next, replacing y by 0 in (1), we get
and, consequently, for all and . Note also that for every and all , we have since G is additive. Therefore,
Letting , we conclude that for all and all . According to Lemma 2.1, this yields that G is linear.
Similarly, we can show that there exists a unique linear mapping such that
for all . Moreover, by Theorem 1 in , D is a module left -derivation.
It remains to prove that G is a generalized module left -derivation with an associated module left -derivation D, i.e.,
for all . So, let and . By (3), we have
Since ℳ is a Banach left -module, there exists a positive constant C such that
This yields that
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.
Remark 3.1 Let g and F be as in Theorem 2.2 and let . If the maps g and are continuous at point , then G is continuous on . Namely, if G was not continuous, then there would exist an integer C and a sequence such that and for . Write
Let . Then
since g is continuous at . Thus, there exists an integer such that for every , we have
for every . Letting and using the continuity of the map at point , we get a contradiction.
Remark 3.2 Let and . Then a function defined by
satisfies all the assumptions of Theorem 2.2. Namely, and for all . In this case, we can write (5) as
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 .
Theorem 4.2 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 (3) and (4) hold true for all and
for all . Then g is a generalized module left -derivation.
Proof We divide the proof into several steps.
Step 1. Firstly, we show that d is a module left -derivation on . By the proof of Theorem 2.2, there exists a unique module left -derivation D on satisfying (6). Moreover, according to Lemma 4.1, we have for all , . Thus,
for all and . Letting , we conclude that . In other words, d is a module left -derivation on .
Step 2. Let and . We claim that . Suppose that is a module left -derivation from the proof of Theorem 2.2 and . Recall that G is additive and therefore . Moreover, G satisfies (5) and, by (3), we have
This yields that
Letting , we get for all . In particular,
for all and all .
Step 3. We prove that . Namely, using (5), we obtain
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.
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The author would like to thank the referees for their useful comments.
The author declares that she has no competing interests.
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Fošner, A. Hyers-Ulam-Rassias stability of generalized module left -derivations. J Inequal Appl 2013, 208 (2013). https://doi.org/10.1186/1029-242X-2013-208
- Hyers-Ulam-Rassias stability
- normed algebra
- Banach left -module
- module left -derivation
- generalized module left -derivation