1. Introduction

The stability problem of functional equations is originated from a question of Ulam [1] concerning the stability of group homomorphisms. Hyers [2] gave a first affirmative partial answer to the question of Ulam for Banach spaces. Hyers' Theorem was generalized by Aoki [3] for additive mappings and by Rassias [4] for linear mappings by considering an unbounded Cauchy difference. The paper of Rassias [4] 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 Gvruta [8] by replacing the unbounded Cauchy difference by a general control function in the spirit of Rassias' approach (see [2, 5–13]).

Jun and Kim [14] introduced and investigate the following functional equation:

(1.1)

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 [15], Park and Bae considered the following quartic functional equation:

(1.2)

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,

(1.3)

It is easy to show that the function satisfies the functional equation (1.3). We establish the general solution and prove the generalized Hyers-Ulam stability for the functional equation (1.3).

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.

Lemma 2.1.

If an even mapping satisfies (1.3), then is quartic.

Proof.

Putting in (1.3), we get . Setting in (1.3), by the evenness of , we obtain

(2.1)

for all . Hence (1.3) can be written as

(2.2)

for all . Replacing by in (1.3), we obtain

(2.3)

for all . By (2.1) and (2.3), we obtain

(2.4)

for all . According to (2.4), (2.2) can be written as

(2.5)

for all . This shows that is quartic, which completes the proof of the lemma.

Lemma 2.2.

If an odd mapping satisfies (1.3), then f is cubic-additive.

Proof.

We show that the mappings and , respectively, defined by and , are additive and cubic, respectively.

Since is odd, . Letting in (1.3), we obtain

(2.6)

for all . Hence (1.3) can be written as

(2.7)

for all . Replacing by and in (2.7), respectively, we get

(2.8)

for all . Replacing by in (2.7), we obtain

(2.9)

for all . Replacing by in (2.9), we get

(2.10)

for all . Replacing by and by in (2.9), we get

(2.11)

for all . Replacing by in (2.11), we get

(2.12)

for all .

Subtracting (2.12) from (2.10), we obtain

(2.13)

for all . By (2.8) and (2.13), we obtain

(2.14)

for all .

Replacing by in (2.14), we get

(2.15)

for all .

By (2.14) and (2.15), we obtain

(2.16)

for all .

By (2.7) and (2.16), we have

(2.17)

for all . Replacing by in (2.17), we get

(2.18)

for all . Replacing by in (2.18), respectively, we get

(2.19)

for all .

By (2.18) and (2.19), we obtain

(2.20)

for all . Replacing by in (2.17), respectively, we get

(2.21)

for all . Thus it follows from (2.20) and (2.21) that

(2.22)

for all . Replacing by in (2.22), we obtain

(2.23)

for all . Replacing by in (2.23), respectively, we get

(2.24)

for all . By (2.23) and (2.24), we obtain

(2.25)

for all . Adding (2.22) to (2.25) and using (2.17), we get

(2.26)

for all . The last equality means that

(2.27)

for all Thus the mapping is additive.

Replacing by in (2.17), respectively, we get

(2.28)

for all . Since for all

(2.29)

for all Hence it follows from (2.17) and (2.28) that

(2.30)

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.

Remark 2.3.

The functional equation (1.3) is equivalent to the functional equation

(2.31)

The left hand side is even with respect to and the right hand side is odd by the assumption of Lemma 2.2. Thus

(2.32)

So we conclude that , as desired.

Theorem 2.4.

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.

Proof.

Let satisfy (1.3). We decompose into the even part and the odd part by setting

(2.33)

for all By (1.3), we have

(2.34)

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

(3.1)

for all We consider the following functional inequality:

(3.2)

for an upper bound

Theorem 3.1.

Let be fixed. Suppose that an even mapping satisfies and

(3.3)

for all If the upper bound is a function such that

(3.4)

and that for all then the limit

(3.5)

exists for all and is a unique quartic mapping satisfying (1.3) and

(3.6)

for all

Proof.

Putting in (3.3), we obtain

(3.7)

for all On the other hand, replacing by in (3.3), we get

(3.8)

for all By (3.7) and (3.8), we get

(3.9)

for all Replacing by in (3.9), we get

(3.10)

for all . It follows from (3.10) that

(3.11)

for all . It follows from (3.11) that

(3.12)

for all .

This shows that is a Cauchy sequence in . Since is complete, the sequence converges. We now define by

(3.13)

for all It is clear that (3.6) holds, and for all By (3.3), we have

(3.14)

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

(3.15)

for all . By taking in this inequality, we have for all which gives the conclusion for the case Let Then by (3.9), we have

(3.16)

for all Replacing by in (3.16) and dividing by 16, we get

(3.17)

for all By (3.16) and (3.17), we obtain

(3.18)

for all It follows from (3.18) that

(3.19)

for all Dividing both sides of (3.19) by and then replacing by , we get

(3.20)

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

Theorem 3.2.

Suppose that an odd mapping satisfies

(3.21)

for all If the upper bound is a function such that

(3.22)

and that for all then the limit

(3.23)

exists for all and is a unique additive mapping satisfying (1.3) and

(3.24)

for all

Proof.

Set in (3.21). Then by the oddness of , we have

(3.25)

for all Replacing by in (3.21), we obtain

(3.26)

for all . Combining (3.25) and (3.26) yields that

(3.27)

for all Putting and for all we get

(3.28)

for all . It follows from (3.28) that

(3.29)

for all Multiplying both sides of (3.29) by and then replacing by , we get

(3.30)

for all So is a Cauchy sequence in . Put for all Then we have

(3.31)

for all . On the other hand, it is easy to show that

(3.32)

for all Hence it follows that

(3.33)

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

(3.34)

for all . This shows that for all

Theorem 3.3.

Suppose that an odd mapping satisfies

(3.35)

for all If the upper bound is a function such that

(3.36)

and that

(3.37)

for all then the limit

(3.38)

exists for all and is a unique cubic mapping satisfying (1.3), and

(3.39)

for all

Proof.

It is easy to show that satisfies (3.27). Setting and then putting in (3.27), we obtain

(3.40)

for all It follows from (3.40) that

(3.41)

for all . Replacing by in (3.41) and then multiplying both sides of (3.41) by we get

(3.42)

for all . Since the right hand side of the inequality (3.42) tends to 0 as the sequence is Cauchy. Now we define

(3.43)

for all Then we have

(3.44)

for all Let

(3.45)

for all . Then we have

(3.46)

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

(3.47)

for all By letting in the above inequality, we get for all which gives the conclusion.

Theorem 3.4.

Suppose that an odd mapping satisfies

(3.48)

for all If the upper bound is a function such that

(3.49)

and that for all then there exist a unique cubic mapping , and a unique additive mapping such that

(3.50)

for all

Proof.

By Theorems 3.2 and 3.3, there exist an additive mapping and a cubic mapping such that

(3.51)

for all Combining two equations in (3.51) yields that

(3.52)

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

(3.53)

for all Since

(3.54)

for all Hence (3.53) implies that

(3.55)

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).

Theorem 3.5.

Suppose that a mapping satisfies and for all If the upper bound is a function such that

(3.56)

and that for all then there exist a unique additive mapping a unique cubic mapping , and a unique quartic mapping such that

(3.57)

for all .

Proof.

Let for all Then and

(3.58)

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.

Corollary 3.6.

Let and let be a positive real number. Suppose that a mapping satisfies and

(3.59)

for all Then there exist a unique additive mapping a unique cubic mapping , and a unique quartic mapping satisfying

(3.60)

for all .

Proof.

It follows from Theorem 3.5 by taking for all .

Theorem 3.7.

Suppose that an odd mapping satisfies

(3.61)

for all If the upper bound is a function such that

(3.62)

and that for all then the limit

(3.63)

exists for all and is a unique additive mapping satisfying (1.3) and

(3.64)

for all

Proof.

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.

Theorem 3.8.

Suppose that an odd mapping satisfies

(3.65)

for all If the upper bound is a function such that

(3.66)

and that for all then the limit

(3.67)

exists for all and is a unique cubic mapping satisfying (1.3), and

(3.68)

for all

Theorem 3.9.

Suppose that an odd mapping satisfies

(3.69)

for all If the upper bound is a function such that

(3.70)

and that for all then there exist a unique additive mapping , and a unique cubic mapping such that

(3.71)

for all

Proof.

The proof is similar to the proof of Theorem 3.4.

Theorem 3.10.

Suppose that satisfies and

(3.72)

for all If the upper bound is a function such that

(3.73)

and that for all then there exist a unique additive mapping a unique cubic mapping , and a unique quartic mapping such that

(3.74)

for all .

Proof.

The proof is similar to the proof of Theorem 3.5.

Corollary 3.11.

Let and let be a positive real number. Suppose that satisfies and

(3.75)

for all Then there exist a unique additive mapping a unique cubic mapping , and a unique quartic mapping satisfying

(3.76)

for all .

Corollary 3.12.

Let be a positive real number. Suppose that a mapping satisfies and for all Then there exist a unique additive mapping a unique cubic mapping , and a unique quartic mapping such that

(3.77)

for all .