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
We obtain the general solution and the generalized Hyers-Ulam-Rassias stability of the mixed type additive, quadratic, and cubic functional equation .
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.
If is even function, then is quadratic.
If is odd function, then is cubic-additive.
for an upper bound
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).
for all By using (3.26), we may define a mapping as for all Similar to Theorem 3.1, we can show that is the unique cubic function satisfying (1.6) and (3.25).
for all On the other hand and are cubic, then Therefore by (3.35) we obtain that for all Again by (3.35) we have for all
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.
for all .
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.
It follows from Theorem 3.6 by taking for all .
for all .
By Theorem 3.8, we are going to investigate the following stability problem for functional equation (1.6).
for all .
By Corollary 3.9, we solve the following Hyers-Ulam stability problem for functional equation (1.6).
for all .
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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