- Research Article
- Open Access
© In Goo Cho et al. 2010
- Received: 12 March 2010
- Accepted: 23 April 2010
- Published: 26 May 2010
- Banach Space
- Real Number
- Functional Equation
- Linear Space
- Normed Space
In 1940 Ulam  proposed the problem concerning the stability of group homomorphisms as follows.Let be a group and let be a metric group with the metric . Given that , 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, when is it true that a mapping satisfying a functional equation approximately must be close to the solution of the given functional equation ? In 1941, Hyers  considered the case of approximately additive mappings under the assumption that and are Banach spaces.
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.
Similar to the quartic functional equation, we may define quintic functional equation as follows.
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 .
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