- Research Article
- Open Access
© Dongseung Kang. 2010
- Received: 28 August 2009
- Accepted: 25 January 2010
- Published: 2 February 2010
- Banach Space
- Functional Equation
- Linear Space
- Normed Space
- Regularity Condition
One of the interesting questions concerning the stability problems of functional equations is as follows: when is it true that a mapping satisfying a functional equation approximately must be close to the solution of the given functional equation? Such an idea was suggested in 1940 by Ulam  as follows. Let be a group and let be a metric group with the metric . Given , does there exist a such that if a function satisfies the inequality for all , then there is a homomorphism with for all ? In other words, we are looking for situations when the homomorphisms are stable; that is, if a mapping is almost a homomorphism, then there exists a true homomorphism near it. In 1941, Hyers  considered the case of approximately additive mappings in Banach spaces and satisfying the well-known weak Hyers inequality controlled by a positive constant. The famous Hyers stability result that appeared in  was generalized in the stability involving a sum of powers of norms by Aoki . In 1978, Rassias  provided a generalization of Hyers Theorem which allows the Cauchy difference to be unbounded. During the last decades, stability problems of various functional equations have been extensively studied and generalized by a number of authors [5–10]. In particular, Rassias  introduced the quartic functional equation
For this reason, (1.1) is called a quartic functional equation. Also Chung and Sahoo  determined the general solution of (1.1) without assuming any regularity conditions on the unknown function. In fact, they proved that the function is a solution of (1.1) if and only if where the function is symmetric and additive in each variable. Lee and Chung  introduced a quartic functional equation as follows:
In this paper, we consider the following the generalized quartic functional equation:
for fixed integers and such that for all We investigate the generalized Hyers-Ulam-Rassias stability problem in quasi- -normed spaces and then the stability by using a subadditive function for the generalized quartic function satisfying (1.4).
For the same reason as (1.1) and (1.2), we call (1.4) generalized quartic functional equation.
Throughout this section, let be a quasi- -normed space and let be a quasi- -Banach space with a quasi- -norm . Let be the modulus of concavity of . We will investigate the generalized Hyers-Ulam-Rassias stability problem for the functional equation (1.4). After then we will study the stability by using a subadditive function. For a given mapping and all fixed integers and with let
Now we will recall a subadditive function and then investigate the stability under the condition that the space is a -Banach space. The basic definitions of subadditive functions follow from .
The author would like to thank referees for their valuable suggestions and comments. The present research was conducted by the research fund of Dankook University in 2009.
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