Solution and Stability of a Mixed Type Additive, Quadratic, and Cubic Functional Equation
© M. Eshaghi Gordji et al. 2009
Received: 24 January 2009
Accepted: 26 June 2009
Published: 17 August 2009
The stability problem of functional equations originated from a question of Ulam  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?
for all Moreover if is continuous in for each fixed then is linear (see also ). In 1950, Aoki  generalized Hyers' theorem for approximately additive mappings. In 1978, Th. M. Rassias  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]).
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]).
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]).
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).
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 .
This completes the proof of theorem.
The following corollary is an alternative result of Theorem 2.1.
The proof is similar to the proof of Theorem 3.4.
Now we establish the generalized Hyers-Ulam-Rassias stability of functional equation (1.6) as follows.
Let for all Then and for all Hence in view of Theorem 3.1 there exists a unique quadratic function satisfying (3.7). Let for all Then and for all From Theorem 3.4, it follows that there exist a unique cubic function and a unique additive function satisfying (3.29). Now it is obvious that (3.40) holds true for all and the proof of theorem is complete.
By Theorem 3.8, we are going to investigate the following stability problem for functional equation (1.6).
By Corollary 3.9, we solve the following Hyers-Ulam stability problem for functional equation (1.6).
The authors would like to express their sincere thanks to referees for their invaluable comments. The first author would like to thank the Semnan University for its financial support. Also, the fourth author would like to thank the office of gifted students at Semnan University for its financial support.
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