#### Abstract

we establish the general solution for a mixed type functional equation of aquartic and a quadratic mapping in linear spaces. In addition, we investigate the generalized Hyers-Ulam stability in -Banach spaces.

#### 1. Introduction and Preliminaries

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 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 [2] gave a first affirmative answer to the question of Ulam for Banach spaces. Let be a mapping between Banach spaces such that for all , and for some . Then, there exists a unique additive mapping such that for all .

The result of Hyers was generalized by Aoki [3] for approximate additive function and by Rassias [4] for approximate linear function by allowing the difference Cauchy equation to be controlled by . Taking into consideration a lot of influence of Ulam, Hyers and Rassias on the development of stability problems of functional equations, the stability phenomenon that was proved by Rassias may be called the Hyers-Ulam-Rassias stability (see [5, 6]). In 1994, a generalization of Rassias theorem was obtained by Gvruta [7], who replaced by a general control function . The functional equation is related to a symmetric biadditive function [810]. It is natural that this equation is called a quadratic functional equation. In particular, every solution of the quadratic equation (1.3) is said to be a quadratic function. It is well known that a function between real vector spaces is quadratic if and only if there exists a unique symmetric biadditive function such that for all in the vector space. The biadditive function is given by A Hyers-Ulam stability problem for the quadratic functional equation (1.3) was proved by Skof for functions , where is normed space and is Banach space (see [11]). In the paper [12], Czerwik proved the Hyers-Ulam-Rassias stability of (1.3).

Lee et al. [13] considered the following functional equation: In fact, they proved that a function between two real vector spaces and is a solution of (1.5) if and only if there exists a unique symmetric biquadratic function such that for all . The biquadratic function is given by It is easy to show that the function satisfies the functional equation (1.5), which is called the quartic functional equation (see also [14]).

Jun and Kim [15] have obtained the generalized Hyers-Ulam stability for a mixed type of cubic and additive functional equation. In addition, the generalized Hyers-Ulam stability for a mixed type of cubic, quadratic, and additive functional equation has been investigated by Gordji and Khodaei [16] (see also [17, 18]). The stability problems for several mixed types of functional equations have been extensively investigated by a number of authors and there are many interesting results concerning this problem [1927].

In this paper, we deal with the following functional equation derived from quartic and quadratic functions: for fixed integers , . It is easy to see that the function is a solution of the functional equation (1.7). In the sequel, we investigate the general solution of functional equation (1.7) when is a function between vector spaces, and then we prove the generalized Hyers-Ulam stability of (1.7) in the spirit of Hyers, Ulam, and Rassias using the direct method.

We recall some basic facts concerning quasi-Banach spaces and some preliminary results.

Definition 1.1 (see [28, 29]). Let be a real linear space. A quasinorm is a real-valued function on satisfying the following:(1) for all and if and only if ,(2) for all and all ,(3)there is a constant such that for all .
The pair is called a quasinormed space if is a quasinorm on .

The smallest possible is called the modulus of concavity of . A quasi-Banach space is a complete quasinormed space. A quasinorm is called a -norm if for all . In this case, a quasi-Banach space is called a -Banach space.

Given a -norm, the formula gives us a translation invariant metric on . By the Aoki-Rolewicz Theorem [29], each quasinorm is equivalent to some -norm (see also [28]). Since it is much easier to work with -norms, henceforth we restrict our attention mainly to -norms.

Lemma 1.2 (see [17]). Let be nonnegative real numbers. Then, one has for a positive real number with .

#### 2. General Solution

We here present the general solution of (1.7).

Theorem 2.1. Let both and be real vector spaces. A function satisfies (1.7) for all if and only if there exists a unique symmetric biquadratic function and a unique symmetric biadditive function such that for all .

Proof. Let satisfy (1.7) and let be functions defined by for all . We claim that the functions and are quadratic and quartic, respectively.
Letting in (1.7), we have . By putting in (1.7), one leads to the evenness of . Replacing by in (1.7), we have for all . Replacing by in (2.3), we obtain for all . Adding (2.3) to (2.4), we get by evenness of , for all . From the substitution in (1.7), we have by evenness of , for all . Replacing by in (2.6), we get for all . Adding (2.6) to (2.7), we get by evenness of , for all . By using (1.7) and (2.5), it follows from (2.8) that for all . If we replace by in (1.7), then we get that for all . It follows from (2.9) and (2.10) that for all . On the other hand, putting in (1.7), we get for all . Putting in (1.7), we get for all . Putting in (1.7) and using the evenness of , we obtain for all . Letting in (2.10), we have for all . It follows from (2.14) and (2.15) that for all . Now, by using (2.12), (2.13) and (2.16), we lead to for all . Finally, comparing (2.11) with (2.17), then we conclude that for all . Replacing by in (2.18), we get for all . Interchanging with in (2.18), one gets for all . It follows from (2.19) and (2.20) that for all . This means that for all . So the function defined by is quadratic.
To prove that defined by is quartic, we need to show that for all . Replacing and by and in (2.18), respectively, we obtain for all . But, since for all , where is a quadratic function defined above, we see that for all . Hence, according to (2.24) and (2.25), we get for all . By multiplying 4 on both sides of (2.18), we get that for all . If we subtract the last equation from (2.26), then we arrive at for all . This means that satisfies (2.23) and, therefore, the function is quartic. Thus, there exists a unique symmetric biquadratic function and a unique symmetric biadditive function such that and for all (see [8, 13]). Therefore, we obtain from (2.2) that for all .
The proof of the converse is trivial.

#### 3. Generalized Hyers-Ulam Stability

From this point on, assume that is a quasinormed space with quasinorm and that is a -Banach space with -norm . Let be the modulus of concavity of .

Before taking up the main subject, given a mapping , we define the difference operator by for all . Let for notational convenience.

Theorem 3.1. Let be fixed and let be a function such that for all and for all . Suppose that an even function with satisfies the inequality for all . Then, there exists a unique quadratic function such that for all , where

Proof. Let . Setting in (3.4), we have for all . Putting in (3.4), we obtain for all . Replacing by in (3.7), we see that for all . Setting by in (3.4) and using the evenness of , we get for all . It follows from (3.9) and (3.10) that for all . Also, it follows from (3.7) and (3.8) that for all . Finally, using (3.11) and (3.12), we obtain that where for all . Let be a function defined by for all . From (3.13), we conclude that for all . If we replace in (3.15) by and multiply both sides of (3.15) by , then we get for all and all non-negative integers . Since is a -Banach space, the inequality (3.16) gives for all nonnegative integers and with and all . Since , by Lemma 1.2 and (3.14), we conclude that for all . Therefore, it follows from (3.3) and (3.18) that for all . It follows from (3.17) and (3.19) that the sequence is a Cauchy for all . Since is complete, the sequence converges for all . So one can define a function by for all . Letting and passing the limit in (3.17), we get for all . Thus (3.5) follows from (3.18) and (3.21). Now we show that is quadratic. It follows from (3.16), (3.19) and (3.20) that for all . So, for all . On the other hand, it follows from (3.2), (3.4) and (3.20) that for all . Hence the function satisfies (1.7). Thus, by Theorem 2.1, the function is quadratic. Therefore, (3.23) implies that the function is quadratic.
Now, to prove the uniqueness property of , let be another quadratic function satisfying (3.5). It follows from (3.3) that for all . Hence, for all . It follows from (3.5), (3.20) and (3.26) that for all . So .
For , we can prove the theorem by a similar argument.

Corollary 3.2. Let be nonnegative real numbers such that , or , . Suppose that an even function with satisfies the inequality for all . Then there exists a unique quadratic function satisfying for all , where

Proof. In Theorem 3.1, putting for all , we get the desired result.

Corollary 3.3. Let and be real numbers such that . Suppose that an even function with satisfies the inequality for all . Then there exists a unique quadratic function satisfying for all .

Proof. In Theorem 3.1, taking , for all , we arrive at the desired result.

Theorem 3.4. Let be fixed and let be a function such that for all and for all . Suppose that an even function with satisfies the inequality for all . Then there exists a unique quartic function such that for all , where

Proof. Being similar to the proof of Theorem 3.1, we omit its proof.

Corollary 3.5. Let be nonnegative real numbers such that or . Suppose that an even function with satisfies the inequality (3.28) for all . Then there exists a unique quartic function satisfying for all , where for all .

Corollary 3.6. Let and be real numbers such that . Suppose that an even function with satisfies the inequality (3.31) for all . Then, there exists a unique quartic function satisfying for all .

Now, we are ready to prove the main theorem concerning the stability problem for (1.7).

Theorem 3.7. Let be fixed and let be a function such that for all and for all . Suppose that an even function with satisfies the inequality for all . Then, there exists a unique quadratic function and a unique quartic function such that for all , where

Proof. By Theorems 3.1 and 3.4, there exists a quadratic function and a quartic function such that for all . Therefore, it follows from (3.46) that for all . Thus we obtain (3.44) by letting and for all .
To prove the uniqueness property of and , let be another quadratic and quartic functions satisfying (3.44). Let and . Hence, for all . Since , for all , we figure out that for all . Therefore, we get and then .

Corollary 3.8. Let be nonnegative real numbers such that or , or . Suppose that an even function with satisfies the inequality (3.28), for all . Then, there exists a unique quadratic function and a unique quartic function such that for all , where and are defined as in Corollaries 3.2 and 3.5.

Corollary 3.9. Let and be non-negative real numbers such that . Suppose that an even function with satisfies the inequality (3.31) for all . Then there exist a unique quadratic function and a unique quartic function such that for all .

Corollary 3.10. Suppose that an even function with satisfies the inequality for all where . Then there exist a unique quadratic function and a unique quartic function such that for all .

#### Acknowledgment

Hark-Mahn Kim was supported by Basic Research Program through the National Research Foundation of Korea funded by the Ministry of Education, Science and Technology (no. 2011-0002614).