Abstract
We will show the general solution of the functional equation = and investigate the stability of cubic -derivations associated with the given functional equation on Banach -algebras.
1. Introduction
Ulam [1] proposed the famous problem concerning the stability of group homomorphisms. We say that a functional equation is said to be stable if any approximate solution to the functional equation is near a true solution of that functional equation. The question of Ulam was affirmatively answered for Banach spaces by Hyers [2]. 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. This result of Rassias leads mathematicians working in stability of functional equations to establish what is known today as Hyers-Ulam-Rassias stability and to introduce new definitions of stability concepts. During the last three decades, several stability problems of a large variety of functional equations have been extensively studied and generalized by a number of authors; see [5]. By using fixed point methods the stability problems of several functional equations have been extensively investigated by a number of authors; see [6–9]. Also, Jang and Park [10] investigated the stability of -derivations and of quadratic -derivations with Cauchy functional equation and the Jensen functional equation on Banach -algebra. In particular, the stability of -derivations on Banach -algebra by using fixed point alternative was proved by Park and Bodaghi; see [11].
Jun and Kim [12] introduced the following functional equation: and they established a general solution. Najati [13] investigated the cubic functional equation for a fixed positive integer .
In this paper, we deal with the functional equation for all and fixed . We will provide the general solution of the functional equation (3) and investigate the stability of cubic -derivations associated with the given functional equation on Banach -algebras.
2. Cubic Functional Equations
In this section let and be real vector spaces and we investigate the general solution of the functional equation (3). Before we proceed, we would like to introduce some basic definitions concerning -additive symmetric mappings and key concepts which are found in [14, 15]. A function is said to be additive if for all . Let be a positive integer. A function is called -additive if it is additive in each of its variables. A function is said to be symmetric if for every permutation of . If is an -additive symmetric map, then will denote the diagonal and for all and all . Such a function will be called a monomial function of degree (assuming ). Furthermore the resulting function after substitution and in will be denoted by .
Theorem 1. A function is a solution of the functional equation (3) if and only if is of the form for all , where is the diagonal of the -additive symmetric map .
Proof. Assume that satisfies the functional equation (3). Let in (3). We have
that is, . It is easy to show that is and an odd mapping for all . We can rewrite the functional equation (3) in the form
for all . By Theorems 3.5 and 3.6 in [15], is a generalized polynomial function of degree at most 3; that is, is of the form
for all , where is an arbitrary element of and is the diagonal of the -additive symmetric map for . By and for all , we get . Substituting (6) into (3) we have
for all . Note that
Then we have
for all . Letting in (9), we get for all . Hence we have for all . Thus
for all .
Conversely, assume that for all , where is the diagonal of a -additive symmetric map . Note that
where , , and . Thus we may conclude that satisfies (3).
For this reason, we call a function a generalized cubic functional equation if it satisfies (3).
3. Generalized Cubic -Derivations
Throughout this section, we assume that is a Banach -algebra. A mapping is a cubic derivation if is a cubic homogeneous mapping; that is, is cubic, , and for all and . In addition, if satisfies for all , then it is called a cubic -derivation. For a given mapping , we consider for all , , and , and .
Theorem 2. Suppose that is a mapping with for which there exists a function such that for all and all in which . Also assume that for each fixed the mapping from to is continuous; then there exists a unique generalized cubic -derivation on satisfying
Proof. Letting and in inequality (14), we have
for all . Using induction, it can be shown that
for and . Inequalities (13) and (18) imply that the sequence is a Cauchy sequence. Since is a Banach algebra, the sequence is convergent. Hence we can define a mapping as
for . Also, letting in inequality (18), we have
for and . Inequalities (13) and (20) imply that inequality (16) holds.
In the following, we will show that the mapping is a unique generalized cubic -derivation such that inequality (16) holds for all . We note that
for all and . By taking in inequality (21), it follows that the mapping is a generalized cubic mapping. Also, inequality (21) implies that . Hence
for all and . Let . Then , where . Let . Hence we have . Then
for all and . Suppose that is any continuous linear functional on and is a fixed element in . Then we can define a function by
for all . It is easy to check that is cubic. Let
for all and .
Note that as the pointwise limit of the sequence of measurable functions is measurable. Hence as a measurable cubic function is continuous (see [16, 17]) and
for all . Thus
for all . Since was an arbitrary continuous linear functional on we may conclude that
for all . Let . Then . Hence
for all and . Since was an arbitrary element in , we may conclude that is cubic homogeneous.
Next, replacing by , respectively, and in inequality (15), we have
for all . Taking the limit as tends to infinity, we have for all . Letting and replacing by in inequality (15), we get
for all . As in inequality (31), we have
for all . This means that is a cubic -derivation. We now show that is unique. Assume is another generalized cubic -derivation satisfying inequality (16). Then
which tends to zero as , for all . Thus for all .
Corollary 3. Let be positive real numbers with and let be a mapping with such that for all and . Then there exists a unique generalized cubic -derivation on satisfying for all .
Proof. The proof follows from Theorem 2 by taking + + for all .
In the following corollaries, we show the hyperstability for the generalized cubic -derivations.
Corollary 4. Let be positive real numbers with and let be a mapping with such that for all and . Then is a generalized cubic -derivation on .
Proof. By taking in Theorem 2 for all , we have . Equation (16) implies that ; that is, is the generalized cubic -derivation on .
Corollary 5. Let be positive real numbers with and let be a mapping with such that for all and . Then is a generalized cubic -derivation on .
Proof. By taking in Theorem 2 for all , we have . Equation (16) implies that ; that is, is the generalized cubic -derivation on .
Now, we will investigate the stability of the given functional equation (12) using the alternative fixed point method. Before proceeding with the proof, we will state the theorem, the alternative of fixed point; see [18, 19].
Definition 6. Let be a set. A function is called a generalized metric on if satisfies the following:(1) if and only if ;(2) for all ;(3) for all .
Theorem 7 (the alternative of fixed point [18, 19]). Suppose that one is given a complete generalized metric space and a strictly contractive mapping with Lipschitz constant . Then, for each given , either or there exists a natural number such that (1) for all ;(2)the sequence is convergent to a fixed point of ;(3) is the unique fixed point of in the set (4) for all .
Theorem 8. Let be a continuous mapping with and let be a continuous mapping such that for all and . If there exists a constant such that for all , then there exists a unique generalized cubic -derivation on satisfying for all .
Proof. Consider the set and introduce the generalized metric on , It is easy to show that is complete. Now we define a function by for all . Note that, for all , one lets be an arbitrary constant with . Then for all . Letting in inequality (47) and using (42) and (46), we have that is, Hence we have that for all ; that is, is a strict self-mapping of with the Lipschitz constant . Letting , in inequality (40), we get for all . This means that We can apply the alternative of fixed point and, since , there exists a fixed point of in such that for all . Hence This implies that inequality (43) holds for all . Since , inequality (42) shows that Replacing by , , respectively, in inequality (40), we have Taking the limit as tends to infinity, we have for all and all . The remains are similar to the proof of Theorem 2.
Corollary 9. Let be positive real numbers with and let be a mapping with such that for all and . Then there exists a unique generalized cubic -derivation on satisfying for all .
Proof. The proof follows from Theorem 8 by taking = + + for all .
In the following corollary, we show the hyperstability for the generalized cubic -derivations.
Corollary 10. Let be positive real numbers with and let be a mapping with such that for all and . Then is a generalized cubic -derivation on .
Proof. By taking in Theorem 8 for all , we have . Equation (43) implies that ; that is, is the generalized cubic -derivation on .
Conflict of Interests
The authors declare that there is no conflict of interests regarding the publication of this paper.
Acknowledgments
The authors sincerely thank the reviewers for their careful reading and fruitful suggestions to improve the quality of the first draft of this paper. The present research was conducted by the research fund of Dankook University in 2014.