1. Introduction

The stability problem of functional equations originated from a question of Ulam [1] in 1940, concerning the stability of group homomorphisms. Let be a group, and let be a metric group with the metric Given , does there exist a , such that if a mapping satisfies the inequality for all , then there exists a homomorphism with for all ? In other words, under what condition does there exist a homomorphism near an approximate homomorphism?

In 1941, Hyers [2] gave a first affirmative answer to the question of Ulam for Banach spaces. Let be a mapping between Banach spaces such that for all and for some Then there exists a unique additive mapping such that for all Moreover if is continuous in for each fixed then is linear (see also [3]). In 1950, Aoki [4] generalized Hyers' theorem for approximately additive mappings. In 1978, Th. M. Rassias [5] provided a generalization of Hyers' theorem which allows the Cauchy difference to be unbounded. This new concept is known as Hyers-Ulam-Rassias stability of functional equations (see [2–24]).

The functional equation is related to symmetric biadditive function. In the real case it has among its solutions. Thus, it has been called quadratic functional equation, and each of its solutions is said to be a quadratic function. Hyers-Ulam-Rassias stability for the quadratic functional equation (1.3) was proved by Skof for functions , where is normed space and Banach space (see [25–28]).

The following cubic functional equation was introduced by the third author of this paper, J. M. Rassias [29, 30] (in 2000-2001):

Jun and Kim [13] introduced the following cubic functional equation: and they established the general solution and the generalized Hyers-Ulam-Rassias stability for the functional equation (1.5).

The function satisfies the functional equation (1.5), which explains why it is called cubic functional equation.

Jun and Kim proved that a function between real vector spaces and is a solution of (1.5) if and only if there exists a unique function such that for all and is symmetric for each fixed one variable and is additive for fixed two variables (see also [31–33]).

We deal with the following functional equation deriving from additive, cubic and quadratic functions:

It is easy to see that the function is a solution of the functional equation (1.6). In the present paper we investigate the general solution and the generalized Hyers-Ulam-Rassias stability of the functional equation (1.6).

2. General Solution

In this section we establish the general solution of functional equation (1.6).

Theorem 2.1. Let , be vector spaces, and let be a function. Then satisfies (1.6) if and only if there exists a unique additive function , a unique symmetric and biadditive function and a unique symmetric and 3-additive function such that for all .

Proof. Suppose that for all , where is additive, is symmetric and biadditive, and is symmetric and 3-additive. Then it is easy to see that satisfies (1.6). For the converse let satisfy (1.6). We decompose into the even part and odd part by setting for all By (1.6), we have for all This means that satisfies (1.6), that is, Now putting in (2.3), we get . Setting in (2.3), by evenness of we obtain Replacing by in (2.3), we obtain Comparing (2.4) with (2.5), we get By utilizing (2.5) with (2.6), we obtain Hence, according to (2.6) and (2.7), (2.3) can be written as With the substitution in (2.8), we have Replacing by in above relation, we obtain Setting instead of in (2.8), we get Interchanging and in (2.11), we get If we subtract (2.12) from (2.11) and use (2.10), we obtain which, by putting and using (2.7), leads to Let us interchange and in (2.14). Then we see that and by adding (2.14) and (2.15), we arrive at Replacing by in (2.8), we obtain Let us Interchange and in (2.17). Then we see that Thus by adding (2.17) and (2.18), we have Replacing by in (2.11) and using (2.7) we have and interchanging and in (2.20) yields If we add (2.20) to (2.21), we have Interchanging and in (2.8), we get and by adding the last equation and (2.8) with (2.19), we get Now according to (2.22) and (2.24), it follows that From the substitution in (2.25) it follows that Replacing by in (2.25) we have and interchanging and yields By adding (2.27) and (2.28) and then using (2.25) and (2.26), we lead to If we compare (2.16) and (2.29), we conclude that This means that is quadratic. Thus there exists a unique quadratic function such that for all On the other hand we can show that satisfies (1.6), that is, Now we show that the mapping defined by is additive and the mapping defined by is cubic. Putting in (2.31), then by oddness of we have Hence (2.31) can be written as From the substitution in (2.33) it follows that Interchange and in (2.33), and it follows that With the substitutions and in (2.35), we have Replace by in (2.34). Then we have Replacing by in (2.37) gives Interchanging and in (2.38), we get If we add (2.38) to (2.39), we have Replacing by in (2.36) gives By comparing (2.40) with (2.41), we arrive at Replacing by in (2.42) gives With the substitution in (2.43), we have and replacing by gives Let us interchange and in (2.45). Then we see that If we add (2.45) to (2.46), we have Adding (2.42) to (2.47) and using (2.33) and (2.35), we obtain for all The last equality means that for all Therefore the mapping is additive. With the substitutions and in (2.35), we have Let be the additive mapping defined above. It is easy to show that is cubic-additive function. Then there exists a unique function and a unique additive function such that for all and is symmetric and 3-additive. Thus for all , we have This completes the proof of theorem.

The following corollary is an alternative result of Theorem 2.1.

Corollary 2.2. Let , be vector spaces, and let be a function satisfying (1.6). Then the following assertions hold. (a)If is even function, then is quadratic.(b)If is odd function, then is cubic-additive.

3. Stability

We now investigate the generalized Hyers-Ulam-Rassias stability problem for functional equation (1.6). From now on, let be a real vector space, and let be a Banach space. Now before taking up the main subject, given , we define the difference operator by for all We consider the following functional inequality: for an upper bound

Theorem 3.1. Let be fixed. Suppose that an even mapping satisfies and for all If the upper bound is a mapping such that and that for all then the limit exists for all and is a unique quadratic function satisfying (1.6), and for all

Proof. Let Putting in (3.3), we get for all On the other hand by replacing by in (3.3), it follows that for all Combining (3.8) and (3.9), we lead to for all With the substitution in (3.10) and then dividing both sides of inequality by 2, we get
Now, using methods similar as in [8, 34, 35], we can easily show that the function defined by for all is unique quadratic function satisfying (1.6) and (3.7). Let Then by (3.10) we have for all And analogously, as in the case , we can show that the function defined by is unique quadratic function satisfying (1.6) and (3.7).

Theorem 3.2. Let be fixed. Let is a mapping such that and that for all
Suppose that an odd mapping satisfies for all
Then the limit exists, for all and is a unique additive function satisfying (1.6), and for all

Proof. Let set in (3.15). Then by oddness of we have for all Replacing by in (3.15) we get Combining (3.18) and (3.19), we lead to for all Putting