- Research Article
- Open Access
Stability of an Additive-Cubic-Quartic Functional Equation
© M. Eshaghi-Gordji et al 2009
- Received: 8 September 2009
- Accepted: 8 December 2009
- Published: 2 February 2010
- Functional Equation
- Additive Mapping
- Positive Real Number
- Cauchy Sequence
- Real Vector
The stability problem of functional equations is originated from a question of Ulam  concerning the stability of group homomorphisms. Hyers  gave a first affirmative partial answer to the question of Ulam for Banach spaces. Hyers' Theorem was generalized by Aoki  for additive mappings and by Rassias  for linear mappings by considering an unbounded Cauchy difference. The paper of Rassias  has provided a lot of influence in the development of what we callgeneralized Hyers-Ulam stability or asHyers-Ulam-Rassias stability of functional equations. A generalization of the Rassias theorem was obtained by G vruta  by replacing the unbounded Cauchy difference by a general control function in the spirit of Rassias' approach (see [2, 5–13]).
Jun and Kim  introduced and investigate the following functional equation:
and prove the generalized Hyers-Ulam stability for the functional equation (1.1). Obviously, the function satisfies the functional equation (1.1), which is called a cubic functional equation. Every solution of the cubic functional equation is said to be a cubic mapping. Jun and Kim proved that a mapping between two real vector spaces and is a solution of (1.1) if and only if there exists a unique mapping such that for all ; moreover, is symmetric for each fixed one variable and is additive for fixed two variables.
In , Park and Bae considered the following quartic functional equation:
In fact, they proved that a mapping between two real vector spaces and is a solution of (1.2) if and only if there exists a unique symmetric multi-additive mapping such that for all (see [7, 11]). It is easy to show that the function satisfies the functional equation (1.2), which is called a quartic functional equation. Every solution of the quartic functional equation is said to be a quartic mapping.
In this paper, we aim to deal with the next functional equation derived from additive, cubic, and quadric mappings,
Subtracting (2.12) from (2.10), we obtain
By (2.14) and (2.15), we obtain
By (2.7) and (2.16), we have
By (2.18) and (2.19), we obtain
The following is suggested by an anonymous referee.
If a mapping satisfies (1.3) for all , then there exist a unique additive mapping a unique mapping , and a unique symmetric multi-additive mapping such that for all and that is symmetric for each fixed one variable and is additive for fixed two variables.
for all This means that satisfies (1.3). Similarly we can show that satisfies (1.3). By Lemmas 2.1 and 2.2, and are quartic and cubic-additive, respectively. Thus there exist a unique additive mapping a unique mapping , and a unique symmetric multi-additive mapping such that and that for all and is symmetric for each fixed one variable and is additive for fixed two variables. Thus for all as desired.
We now investigate the generalized Hyers-Ulam stability problem of the functional equation (1.3). From now on, let be a real vector space and let be a Banach space. Now before taking up the main subject, given , we define the difference operator by
Now we prove the generalized Hyers-Ulam stability of the functional equation (1.3).
for all Hence in view of Theorem 3.1, there exists a unique quartic mapping satisfying (3.6). Let for all Then , and for all From Theorem 3.4, it follows that there exist a unique cubic mapping and a unique additive mapping satisfying (3.44). Now it is obvious that (3.57) holds for all and the proof of the theorem is complete.
The proof is similar to the proof of Theorem 3.2.
Employing a similar way to the proof of Theorem 3.3, we get the following theorem.
The proof is similar to the proof of Theorem 3.4.
The proof is similar to the proof of Theorem 3.5.
The authors would like to thank the referees for a number of valuable suggestions regarding a previous version of this paper. The third and corresponding author was supported by Basic Science Research Program through the National Research Foundation of Korea funded by the Ministry of Education, Science and Technology (NRF-2009-0070788).
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