#### 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  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  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  for approximate additive function and by Rassias  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 , who replaced by a general control function . The functional equation is related to a symmetric biadditive function . 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 ). In the paper , Czerwik proved the Hyers-Ulam-Rassias stability of (1.3).

Lee et al.  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 ).

Jun and Kim  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  (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 .

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 , each quasinorm is equivalent to some -norm (see also ). Since it is much easier to work with -norms, henceforth we restrict our attention mainly to -norms.

Lemma 1.2 (see ). 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).