1. Introduction

The stability problem of functional equations originated from a question of Ulam [1] in 1940, concerning the stability of group homomorphisms. Let be a group, and let be a metric group with the metric Given , does there exist a , such that if a mapping satisfies the inequality for all , then there exists a homomorphism with for all ? In other words, under what condition does there exist a homomorphism near an approximate homomorphism?

In 1941, Hyers [2] gave a first affirmative answer to the question of Ulam for Banach spaces. Let be a mapping between Banach spaces such that

(11)

for all and for some Then there exists a unique additive mapping such that

(12)

for all Moreover if is continuous in for each fixed then is linear (see also [3]). In 1950, Aoki [4] generalized Hyers' theorem for approximately additive mappings. In 1978, Th. M. Rassias [5] provided a generalization of Hyers' theorem which allows the Cauchy difference to be unbounded. This new concept is known as Hyers-Ulam-Rassias stability of functional equations (see [2–24]).

The functional equation

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is related to symmetric biadditive function. In the real case it has among its solutions. Thus, it has been called quadratic functional equation, and each of its solutions is said to be a quadratic function. Hyers-Ulam-Rassias stability for the quadratic functional equation (1.3) was proved by Skof for functions , where is normed space and Banach space (see [25–28]).

The following cubic functional equation was introduced by the third author of this paper, J. M. Rassias [29, 30] (in 2000-2001):

(14)

Jun and Kim [13] introduced the following cubic functional equation:

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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 explains why it is called cubic functional equation.

Jun and Kim proved that a function between real vector spaces and is a solution of (1.5) if and only if there exists a unique function such that for all and is symmetric for each fixed one variable and is additive for fixed two variables (see also [31–33]).

We deal with the following functional equation deriving from additive, cubic and quadratic functions:

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It is easy to see that the function is a solution of the functional equation (1.6). In the present paper we investigate the general solution and the generalized Hyers-Ulam-Rassias stability of the functional equation (1.6).

2. General Solution

In this section we establish the general solution of functional equation (1.6).

Theorem 2.1.

Let , be vector spaces, and let be a function. Then satisfies (1.6) if and only if there exists a unique additive function , a unique symmetric and biadditive function and a unique symmetric and 3-additive function such that for all .

Proof.

Suppose that for all , where is additive, is symmetric and biadditive, and is symmetric and 3-additive. Then it is easy to see that satisfies (1.6). For the converse let satisfy (1.6). We decompose into the even part and odd part by setting

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for all By (1.6), we have

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for all This means that satisfies (1.6), that is,

(23)

Now putting in (2.3), we get . Setting in (2.3), by evenness of we obtain

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Replacing by in (2.3), we obtain

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Comparing (2.4) with (2.5), we get

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By utilizing (2.5) with (2.6), we obtain

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Hence, according to (2.6) and (2.7), (2.3) can be written as

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With the substitution in (2.8), we have

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Replacing by in above relation, we obtain

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Setting instead of in (2.8), we get

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Interchanging and in (2.11), we get

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If we subtract (2.12) from (2.11) and use (2.10), we obtain

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which, by putting and using (2.7), leads to

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Let us interchange and in (2.14). Then we see that

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and by adding (2.14) and (2.15), we arrive at

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Replacing by in (2.8), we obtain

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Let us Interchange and in (2.17). Then we see that

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Thus by adding (2.17) and (2.18), we have

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Replacing by in (2.11) and using (2.7) we have

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and interchanging and in (2.20) yields

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If we add (2.20) to (2.21), we have

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Interchanging and in (2.8), we get

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and by adding the last equation and (2.8) with (2.19), we get

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Now according to (2.22) and (2.24), it follows that

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From the substitution in (2.25) it follows that

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Replacing by in (2.25) we have

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and interchanging and yields

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By adding (2.27) and (2.28) and then using (2.25) and (2.26), we lead to

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If we compare (2.16) and (2.29), we conclude that

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This means that is quadratic. Thus there exists a unique quadratic function such that for all On the other hand we can show that satisfies (1.6), that is,

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Now we show that the mapping defined by is additive and the mapping defined by is cubic. Putting in (2.31), then by oddness of we have

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Hence (2.31) can be written as

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From the substitution in (2.33) it follows that

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Interchange and in (2.33), and it follows that

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With the substitutions and in (2.35), we have

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Replace by in (2.34). Then we have

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Replacing by in (2.37) gives

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Interchanging and in (2.38), we get

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If we add (2.38) to (2.39), we have

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Replacing by in (2.36) gives

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By comparing (2.40) with (2.41), we arrive at

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Replacing by in (2.42) gives

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With the substitution in (2.43), we have

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and replacing by gives

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Let us interchange and in (2.45). Then we see that

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If we add (2.45) to (2.46), we have

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Adding (2.42) to (2.47) and using (2.33) and (2.35), we obtain

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for all The last equality means that

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for all Therefore the mapping is additive. With the substitutions and in (2.35), we have

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Let be the additive mapping defined above. It is easy to show that is cubic-additive function. Then there exists a unique function and a unique additive function such that for all and is symmetric and 3-additive. Thus for all , we have

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This completes the proof of theorem.

The following corollary is an alternative result of Theorem 2.1.

Corollary 2.2.

Let , be vector spaces, and let be a function satisfying (1.6). Then the following assertions hold.

  1. (a)

    If is even function, then is quadratic.

  2. (b)

    If is odd function, then is cubic-additive.

3. Stability

We now investigate the generalized Hyers-Ulam-Rassias stability problem for functional equation (1.6). 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

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for all We consider the following functional inequality:

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for an upper bound

Theorem 3.1.

Let be fixed. Suppose that an even mapping satisfies and

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for all If the upper bound is a mapping such that

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and that

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for all then the limit

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exists for all and is a unique quadratic function satisfying (1.6), and

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for all

Proof.

Let Putting in (3.3), we get

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for all On the other hand by replacing by in (3.3), it follows that

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for all Combining (3.8) and (3.9), we lead to

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for all With the substitution in (3.10) and then dividing both sides of inequality by 2, we get

(311)

Now, using methods similar as in [8, 34, 35], we can easily show that the function defined by for all is unique quadratic function satisfying (1.6) and (3.7). Let Then by (3.10) we have

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for all And analogously, as in the case , we can show that the function defined by is unique quadratic function satisfying (1.6) and (3.7).

Theorem 3.2.

Let be fixed. Let is a mapping such that

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and that

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for all

Suppose that an odd mapping satisfies

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for all

Then the limit

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exists, for all and is a unique additive function satisfying (1.6), and

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for all

Proof.

Let set in (3.15). Then by oddness of we have

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for all Replacing by in (3.15) we get

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Combining (3.18) and (3.19), we lead to

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for all Putting and for all Then we get

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for all Now, in a similar way as in [8, 34, 35], we can show that the limit exists, for all and is the unique function satisfying (1.6) and (3.17). If , then the proof is analogous.

Theorem 3.3.

Let be fixed. Suppose that an odd mapping satisfies

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for all If the upper bound is a mapping such that

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and that for all then the limit

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exists, for all and is a unique cubic function satisfying (1.6) and

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for all

Proof.

We prove the theorem for When we have a similar proof. It is easy to see that satisfies (3.20). Set then by putting in (3.20), it follows that

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for all By using (3.26), we may define a mapping as for all Similar to Theorem 3.1, we can show that is the unique cubic function satisfying (1.6) and (3.25).

Theorem 3.4.

Suppose that an odd mapping satisfies

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for all If the upper bound is a mapping such that

(328)

and that for all then there exists a unique cubic function and a unique additive function such that

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for all

Proof.

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

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for all Combine the two equations of (3.30) to obtain

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for all So we get (3.29) by letting and for all To prove the uniqueness of and let be another additive and cubic maps satisfying (3.29). Let , and let So

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for all Since

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then

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for all Hence (3.32) implies that

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for all On the other hand and are cubic, then Therefore by (3.35) we obtain that for all Again by (3.35) we have for all

Theorem 3.5.

Suppose that an odd mapping satisfies

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for all If the upper bound is a mapping such that

(337)

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

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for all

Proof.

The proof is similar to the proof of Theorem 3.4.

Now we establish the generalized Hyers-Ulam-Rassias stability of functional equation (1.6) as follows.

Theorem 3.6.

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

(339)

and that for all then there exist a unique additive function a unique quadratic function and a unique cubic function such that

(340)

for all .

Proof.

Let for all Then and for all Hence in view of Theorem 3.1 there exists a unique quadratic function satisfying (3.7). Let for all Then and for all From Theorem 3.4, it follows that there exist a unique cubic function and a unique additive function satisfying (3.29). Now it is obvious that (3.40) holds true for all and the proof of theorem is complete.

Corollary 3.7.

Let Suppose that a mapping satisfies and

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for all Then there exist a unique additive function a unique quadratic function and a unique cubic function satisfying

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for all

Proof.

It follows from Theorem 3.6 by taking for all .

Theorem 3.8.

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

(343)

and that for all then there exists a unique additive function a unique quadratic function and a unique cubic function such that

(344)

for all .

By Theorem 3.8, we are going to investigate the following stability problem for functional equation (1.6).

Corollary 3.9.

Let Suppose that satisfies and

(345)

for all then there exist a unique additive function a unique quadratic function and a unique cubic function satisfying

(346)

for all .

By Corollary 3.9, we solve the following Hyers-Ulam stability problem for functional equation (1.6).

Corollary 3.10.

Let be a positive real number. Suppose that a mapping satisfies and for all then there exist a unique additive function a unique quadratic function and a unique cubic function such that

(347)

for all .