- Research Article
- Open Access
- InGoo Cho^{1},
- Dongseung Kang^{2}Email author and
- Heejeong Koh^{2}
https://doi.org/10.1155/2010/368981
© In Goo Cho et al. 2010
- Received: 12 March 2010
- Accepted: 23 April 2010
- Published: 26 May 2010
Abstract
Keywords
- Banach Space
- Real Number
- Functional Equation
- Linear Space
- Normed Space
1. Introduction
In 1940 Ulam [1] 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 [2] 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 [16] 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.
Definition 1.1.
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.
Definition 1.2.
Let be a linear space over a field A - -norm is a real-valued function on satisfying the followings.
(1) for all and if and only if .
(3)There is a constant such that for all .
The pair is called aquasi- -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.
A quasi- -norm is called a -norm ( if for all In this case, a quasi- -Banach space is called a -Banach space; see [17–19].
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 quintic function satisfying (1.5).
2. Quintic Functional Equations
3. Stabilities
Theorem 3.1.
Proof.
for all By letting we immediately have the uniqueness of
Theorem 3.2.
Proof.
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 [19].
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 ,
Theorem 3.3.
Proof.
for all By letting , we immediately have the uniqueness of .
Theorem 3.4.
Proof.
Authors’ Affiliations
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