- Research Article
- Open Access
On the Stability of Generalized Quartic Mappings in Quasi- -Normed Spaces
© Dongseung Kang. 2010
- Received: 28 August 2009
- Accepted: 25 January 2010
- Published: 2 February 2010
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 such that , where , , , for all .
- 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
It is easy to see that is a solution of (1.1) by virtue of the identity
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:
for fixed integer with
Let be a real number with and let be either or We will consider the definition and some preliminary results of a quasi- -norm on a linear space.
Let be a linear space over a field A quasi- -norm is a real-valued function on satisfying the followings.
(1) for all and if and only if
(2) for all and all
(3) There is a constant such that for all
The pair is called a quasi- -normed space if is a quasi- -norm on The smallest possible is called the modulus of concavity of A quasi-Banach space is a complete quasi- -normed space.
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.
where for all
for all Thus they are equivalent.
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
for all By letting we immediately have the uniqueness of .
If is replaced by in the inequality (3.5), then the proof follows from the proof of Theorem 3.1.
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 .
A function having a domain and a codomain that are both closed under addition is called
(1)a subadditive function if
(2)a contractively subadditive function if there exists a constant with such that
(3)an expansively superadditive function if there exists a constant with such that
for all By letting we immediately have the uniqueness of .
for all The remains follow from the proof of Theorem 3.3.
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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