Solutions and Stability of Generalized Mixed Type QC Functional Equations in Random Normed Spaces
© Yeol Je Cho et al. 2010
Received: 29 July 2010
Accepted: 31 August 2010
Published: 7 September 2010
In 1940, the stability problem of functional equations originated from a question given by Ulam , that is, the stability of group homomorphisms.
In other words, under what condition, does there exists a homomorphism near an approximate homomorphism? The concept of stability for functional equation arises when we replace the functional equation by an inequality which acts as a perturbation of the equation.
In 1941, Hyers  gave a first affirmative answer to the question of Ulam for Banach spaces.
In 1978, Rassias  provided a generalization of Hyers' Theorem which allows the Cauchy difference to be unbounded. In 1991, Gajda  answered the question for the case , which was raised by Rassias. This new concept is known as the Hyers-Ulam-Rassias stability of functional equations (see [5–17]).
Hyers-Ulam-Rassias stability problem for the quadratic functional equation (1.3) was proved by Skof  for a mapping , where is a normed space and is a Banach space. Cholewa  noticed that the theorem of Skof is still true if the relevant domain is replaced by an abelian group. In , Czerwik proved the Hyers-Ulam-Rassias stability of (1.3) and Grabiec  generalized these results mentioned above.
where is a mapping from a real vector space into a real vector space 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 is thus called a cubic functional equation. Every solution of the cubic functional equation is said to be a cubic function. Also, Jun and Kim proved that a function between real vector spaces and is a solution of (1.5) if and only if there exits a unique function such that for all and is symmetric for each fixed one variable and is additive for fixed two variables.
Definition 1.1 (see ).
Definition 1.2 (see ).
Theorem 1.4 (see ).
where on random normed spaces. It is easy to see that the mapping is a solution of the functional equation (1.10). In Section 2, we investigate the general solution of functional equation (1.10) when is a mapping between vector spaces and, in Section 3, we establish the stability of the functional equation (1.10) in RN-spaces.
2. General Solutions
Before proceeding to the proof of Theorem 2.3 which is the main result in this section, we need the following two lemmas.
A function with satisfies (1.10) for all if and only if there exist functions and such that for all where the function is symmetric for each fixed one variable and is additive for fixed two variables and is symmetric biadditive.
It is clear that for all and it is easy to show that the functions and satisfy (1.10). Hence, by Lemmas 2.1 and 2.2, we know that the functions and are quadratic and cubic, respectively. Thus there exist a symmetric biadditive function such that for all and the function such that for all where the function is symmetric for each fixed one variable and is additive for fixed two variables. Therefore, we get for all
Conversely, let for all where the function is symmetric for each fixed one variable and is additive for fixed two variables and is biadditive. By a simple computation, one can show that the functions and satisfy the functional equation (1.10). Thus the function satisfies (1.10). This completes the proof.
3. Stability Problems
Now, we show that is a quadratic mapping. In fact, replacing with and , respectively, in (3.3) and then taking the limit as , we find that satisfies (1.10) for all Therefore, the mapping is quadratic.
Obviously, (3.38) follows from (3.40) and (3.42). This completes the proof.
This paper was supported by the Korea Research Foundation Grant funded by the Korean Government (KRF-2008-313-C00050).
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