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The Hyers-Ulam-Rassias Stability of -Derivations on Normed Algebras
We study the Hyers-Ulam-Rassias stability of -derivations on normed algebras.
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 must be somehow close to an exact solution of This problem was formulated by Ulam in 1940 (see [1, 2]). He investigated the stability of group homomorphisms. Let be a group, and let be a metric group with a metric . Suppose that is a map and a fixed scalar. Does there exists such that if satisfies the inequalityfor all , then there exists a group homomorphism with the propertyfor all ?
One year later, Ulam's problem was affirmatively solved by Hyers  for the Cauchy functional equation .: Let be a normed space, a Banach space, and a fixed scalar. Suppose that f : → is a map with the property for all . Then there exists a unique additive mapping such thatfor all . 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  which allowed the Cauchy difference to be unbounded. In 1978, Rassias  proved the stability of linear mappings in the following way. Let be a real normed space and a real Banach space. If there exist scalars and such thatfor all , then there exists a unique additive mapping with the propertyfor all . Moreover, if the map is continuous on for each , then 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  generalized the Rassias' theorem as follows: Let be an Abelian group and a Banach space. Suppose that the so-called admissible control function satisfiesfor all . If is a mapping with the propertyfor all , then there exists a unique additive mapping such thatfor all .
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 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 . During the past few years, approximate derivations were studied by a number of mathematicians (see [11–18] and references therein).
Moslehian  studied the stability of -derivations and generalized some results obtained in . He also established the generalized Hyers-Ulam-Rassias stability of -derivations on normed algebras into Banach bimodules. This motivated us to investigate approximate -derivations on normed algebras. The aim of this paper is to study the stability of -derivations and to generalize some results given in .
Throughout, will be a normed algebra and a Banach -bimodule. Let and be two linear operators on . An additive mapping is called an -derivation if holds for all . Ordinary derivations from to and maps defined by , where is a fixed element and are endomorphisms on , are natural examples of -derivations on . Moreover, if is an endomorphism on , then is a -derivation on . We refer the reader to , where further information about -derivations can be found.
Let and be nonnegative integers with . An additive mapping is called a -derivation if holds for all . Clearly, -derivations are one of the natural generalizations of -derivations (the case ). If , where denotes the identity map on , and an additive mapping satisfies (2.2), then is called a -derivation. In the last few decades a lot of work has been done on the field of -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 -derivations.
In the following, we will assume that and are nonnegative integers with . We will use the same symbol in order to represent the norms on a normed algebra and a Banach -bimodule . For a given (admissible control) function we will use the following abbreviation: Let us start with one well-known lemma.
Lemma 2.1 (see ). Suppose that a function satisfies , . If is a mapping with for all , then there exists a unique additive mapping such that for all .
We say that an additive mapping is -linear if for all and all scalars . In the following, will denote the set of all complex units, that is, For a given additive mapping , Park  obtained the next result.
Lemma 2.2. If for all and all , then is -linear.
3. The Results
Our first result is a generalization of [19, Theorem 2.1] (the case ). We use the direct method to construct a unique -linear mapping from an approximate one and prove that this mapping is an appropriate -derivation on . This method was first devised by Hyers . The idea is taken from .
Theorem 3.1. Let and be mappings with . Suppose that there exists a function such that for all and for all and . Then there exist unique -linear mappings satisfying for all , and a unique -linear -derivation such that for all .
Proof. Taking in (3.1) and using Lemma 2.1, it follows that there exists a unique additive mapping such that holds for all . More precisely, using the induction, it is easy to see that
for all , all positive integers , and all . According to the assumptions on , it follows that the sequence is Cauchy. Thus, by the completeness of , this sequence is convergent and we can define a map as
Using (3.1), we get
This yields that
for all and . Using Lemma 2.2, it follows that the map is -linear. Moreover, according to inequality (3.7), we have
for all .
Next, we have to show the uniqueness of . So, suppose that there exists another -linear mapping such that for all . Then Therefore, for all , as desired.
Similarly we can show that there exist unique -linear mappings defined by Furthermore, for all .
It remains to prove that is an -derivation. Writing in the place of and in the place of in (3.4), we obtain This yields that for all . Thus, mappings and satisfy (2.2). The proof is completed.
Remark 3.2. If there exists such that and the map are continuous at point , then is continuous on . Namely, if was not continuous, then there would exist an integer and a sequence such that and , . Let . Then since is continuous at point . Thus, there exists an integer such that for every we have Therefore, for every . Letting and using the continuity of the map at point , we get a contradiction.
Let and . Applying Theorem 3.1 for the case
Corollary 3.3. Let and be mappings with . Suppose that (3.1), (3.2), (3.3), and (3.4) hold true for all and , where a function is defined as above. Then there exist unique -linear mappings satisfying for all and a unique -linear -derivation such that
Proof . Note that for all and
Remark 3.4. Recall that we can actually take any map in the form where . In this case we have
Before stating our next result, let us write one well-known lemma about the continuity of measurable functions (see, e.g., ).
Lemma 3.5. If a measurable function satisfies for all , then is continuous.
Now we are in the position to state a result for normed algebras which are spanned by a subset of . For example, can be a -algebra spanned by the unitary group of or the positive part of
Theorem 3.6. Let be a normed algebra which is spanned by a subset of and , mappings with . Suppose that there exists a function such that for all and (3.1), (3.2), (3.3) holds true for all and . Moreover, suppose that (3.4) holds true for all . If for all the functions , , and are continuous on , then there exist unique -linear mappings satisfying for all and a unique -linear -derivation such that for all .
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 defined by , . Moreover, for all .
Writing , in (3.1), we get Therefore, This yields that for all . In the next step we will show that is -linear, that is, for all and all .
Since is additive, we have for every and all rational numbers . Let us fix elements and , where denotes the dual space of . Then we can define a function by Firstly, we would like to prove that is continuous. Recall that for all . Furthermore, for all . Set Obviously, is a sequence of continuous functions and is its pointwise limit. This yields that is a Borel function and, by Lemma 3.5 it is continuous. Therefore, we have for all . This implies . Since was an arbitrary element from , we proved that is -linear.
Now, let . Then for some real numbers . Using (3.31), we have for all . This means that is -linear.
Similarly we can show that there exist unique -linear mappings satisfying for all . Moreover, (2.2) holds true for all . Since is linearly generated by , we conclude that is an -derivation on . The proof is completed.
Remark 3.7. As above, we can apply Theorem 3.6 for the case where and .
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