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
In this paper, we consider the additive-cubic-quartic functional equation and prove the generalized Hyers-Ulam stability of the additive-cubic-quartic functional equation in Banach spaces.
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,
2. An Additive-Cubic-Quartic Functional Equation
Throughout this section, and will be real vector spaces. Before proceeding the proof of Theorem 2.4 which is the main result in this section, we shall need the following two lemmas.
If an even mapping satisfies (1.3), then is quartic.
for all . This shows that is quartic, which completes the proof of the lemma.
If an odd mapping satisfies (1.3), then f is cubic-additive.
We show that the mappings and , respectively, defined by and , are additive and cubic, respectively.
Since is odd, . Letting in (1.3), we obtain
for all .
Subtracting (2.12) from (2.10), we obtain
for all .
Replacing by in (2.14), we get
for all .
By (2.14) and (2.15), we obtain
for all .
By (2.7) and (2.16), we have
for all .
By (2.18) and (2.19), we obtain
for all Thus the mapping is additive.
Replacing by in (2.17), respectively, we get
for all Thus the mapping is cubic.
On the other hand, we have for all This means that is cubic-additive. This completes the proof of the lemma.
The following is suggested by an anonymous referee.
So we conclude that , as desired.
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.
3. Stability of an Additive-Cubic-Quartic Functional Equation
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
for all We consider the following functional inequality:
for an upper bound
for all .
for all Hence by Lemma 2.1, is quartic.
It remains to show that is unique. Suppose that there exists a quartic mapping which satisfies (1.3) and (3.6). Since and for all we conclude that
for all By taking in (3.20), is a Cauchy sequence in . Then exists for all It is easy to see that (3.6) holds for
The rest of the proof is similar to the case
for all This means that satisfies (1.3). Then by Lemma 2.2, is additive. Thus (3.31) implies that is additive.
To prove the uniqueness of , suppose that is an additive mapping satisfying (3.24). Then for every we have and Hence it follows that
for all . This shows that for all
for all . Since is an odd mapping, satisfies (2.6). By (3.44), we conclude that for all Then is cubic.
We have to show that is unique. Suppose that there exists another cubic mapping which satisfies (1.3) and (3.39). Since and for all we have
for all By letting in the above inequality, we get for all which gives the conclusion.
for all So we get (3.50) by letting and for all
To prove the uniqueness of and let be other additive and cubic mappings satisfying (3.50). Let Then
for all Since , by (3.55), we obtain that for all Again by (3.55), we have for all
Now we prove the generalized Hyers-Ulam stability of the functional equation (1.3).
for all .
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.
for all .
It follows from Theorem 3.5 by taking for all .
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.
for all .
The proof is similar to the proof of Theorem 3.5.
for all .
for all .
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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