#### Abstract

We investigate the following typical form of a certain class of quadratic functional equations: . Furthermore, we provide a systematic program to prove the generalized Hyers-Ulam stability for the class of functional equations via the stability for the typical form.

#### 1. Introduction

In 1964, Ulam [1] proposed the following stability problem:

“let be a group and a metric group with the metric . Given a constant , does there exist a constant such that if a mapping satisfies for all , then there exists a unique homomorphism with for all ?”

In 1941, Hyers [2] answered this problem under the assumption that the groups are Banach spaces. Aoki [3] and Rassias [4] generalized the result of Hyers. Rassias [4] solved the generalized Hyers-Ulam stability of the functional inequality
for some and with and for all , where is a function between Banach spaces. The paper of Rassias [4] has provided a lot of influence in the development of what we call *the generalized Hyers-Ulam stability or the Hyers-Ulam-Rassias stability* of functional equations. A generalization of the Rassias theorem was obtained by Găvruţa [5] by replacing the unbounded Cauchy difference by a general control function in the spirit of Rassias approach.

The functional equation
is called *a quadratic functional equation* and a solution of a quadratic functional equation is called *quadratic*. A generalized Hyers-Ulam stability problem for the quadratic functional equation was proved by Skof [6] for mappings , where is a normed space and is a Banach space. Cholewa [7] noticed that the theorem of Skof is still true if the relevant domain is replaced by an Abelian group. Czerwik [8] proved the generalized Hyers-Ulam stability for the quadratic functional equation, and Park [9] proved the generalized Hyers-Ulam stability of the quadratic functional equation in Banach modules over a -algebra. Also, the stability problems of functional equations related to quadratic functions can be found in many papers (e.g., [10, 11], etc.).

Rassias [12] investigated the following Euler-Lagrange functional equation: and Gordji and Khodaei [13] investigated other Euler-Lagrange functional equations for fixed integers with , and for fixed integers with and .

In this paper, we consider the sum of two functional equations (2) and (5), that is, for fixed nonzero real numbers with and , and prove the generalized Hyers-Ulam stability for it.

As applications of theorems in Sections 2 and 3, we have a systematic program to prove the generalized Hyers-Ulam stability for functional inequalities which can be deformed into the following functional inequality:

Throughout this paper, assume that is a normed space and is a Banach space.

#### 2. Solutions of (6)

In this section, we investigate solutions of (6). In Corollary 5, it can be concluded that any solution of (6) is quadratic if is a rational number. We start with the following lemma.

Lemma 1. *Let be a mapping with . Suppose that satisfies (6); then the following equation holds:
**
for all .*

*Proof. *Letting in (6), we have
for all . Setting and in (6), we have
for all . Letting in (10) and adding the two equations, we have
for all .

Replacing by in (6), we have
for all , and letting in (12), we have
for all .

Replacing and by and in (6), respectively, we have
for all , and letting in (14), we have
for all . By (12), (13), (14), and (15), we have
for all . By (6) and (16), we have
for all . Now, just simplifying this equation, we can get the result.

Next three theorems deal with (6) for the different cases.

Theorem 2. *Let be a mapping with . Suppose that satisfies (6). If and , then is quadratic.*

*Proof. *By (10) and (11) in the proof of Lemma 1, we have
for all . Since , we have
for all . By (10) and (19), we have
for all . Replacing and by and in (6), respectively, by (20), we have
for all . Replacing and by and in (6), respectively, by (19), we have
for all . By (21) and (22), we have
for all . Since , then
for all . Hence, is quadratic.

Theorem 3. *Let be a mapping with . Suppose that satisfies (6) and is a rational number. If , then is quadratic.*

*Proof. *Since , by the first few lines in the proof of Theorem 2, is even. Hence, in this case, we can easily check that (8) can be reduced to
for all . Since , we have
for all . By [14], a function satisfying (26) is quartic-quadratic. But in our case, also satisfies (26) and since , is quadratic.

Theorem 4. *Let be a mapping with . Suppose that satisfies (6) and is a rational number. If , then is quadratic.*

*Proof. *Suppose that . By (8), we have
for all . By [15], is quadratic-cubic and since , is quadratic.

Combining Theorems 2, 3, and 4 we can get the following corollary as the conclusion of this section.

Corollary 5. *Let be a mapping with . Suppose that satisfies (6) and is a rational number. Then is quadratic.*

#### 3. The Generalized Hyers-Ulam Stability for (6)

In this section, we will prove the generalized Hyers-Ulam stability for (6).

Theorem 6. *Let be a function such that
**
for all . Let be a mapping such that
**
for a fixed rational number and fixed nonzero real numbers , with and . Then there exists a unique quadratic mapping satisfying (6) and
**
for all .*

*Proof. *Let . Then and
for all . Setting in (31), we have
for all , where . Replacing by in (32) and dividing (32) by , we have
for all and all nonnegative integers . For with ,
for all . Since (28) holds for , goes to as . So is a Cauchy sequence in , and since is a Banach space, there exists a mapping such that
for all and
for all . Replacing and by and in (31), respectively, and dividing (31) by , we have
for all and letting in the above inequality, we can show that satisfies (6). By Corollary 5, is quadratic.

Now, we show the uniqueness of the quadratic mapping . Suppose that is a quadratic mapping satisfying (6) and (30). Then we have
for all and for all positive integers . Hence, letting in the above inequality by (28) the tail part goes to . So we have
for all .

We remark that if in Theorem 6, inequality (30) can be replaced by

Related with Theorem 6, we can also have the following theorem. And the proof is similar to that of Theorem 6.

Theorem 7. *Let be a function such that
**
for all . Let be a mapping such that
**
for a fixed rational number and fixed nonzero real numbers , with and . Then there exists a unique quadratic mapping such that
**
for all .*

For the stability problem of quadratic functional equations, we can show that many quadratic functional equations turn out to be types of (6) or to be deformed into the type of (6). For example, Gordji and Khodaei [13] investigated the following functional equation: Indeed, the functional equation (44) can be written as where . Hence the functional equations (44) and (45) are special cases of the functional equation (6).

As another example, Jun et al. [16] investigated the following functional equation: where is an integer with . Suppose that satisfies (47). Then clearly, is even, and hence the functional equation (47) can be deformed into for all . That is, we can transform (47) into the type of (6).

As an example of in Theorems 6 and 7, we can take which appeared in [17]. Then we can formulate the following corollary.

Corollary 8. *Let be a real number with . Let be a mapping such that
**
for a fixed rational number and fixed nonzero real numbers , with and . Then there exists a unique quadratic mapping such that
**
for all .*

We remark that the functional equation (6) is not stable for in Corollary 8. The following example, which is a special case of the example in [18], shows that (6) is not stable for especially in the case of , , and . We give a proof for the reader’s convenience.

*Example 9. *Let be a mapping defined by
and define a mapping by

We will show that satisfies the functional inequality
for all , but there do not exist a quadratic mapping and a positive constant such that
for all .

Note that for all . For any mapping , let for all .

First, suppose that . Then for all .

Now suppose that . Then there is a positive integer such that and so Hence, we have Hence for any , and so Thus satisfies (53).

Suppose that there exist a quadratic mapping and a positive constant with (54). Since , for all , and since is quadratic, for all and all natural numbers . Hence, we have for all , and so, by (54), we have for all .

Take a positive integer such that , and pick with . Then which contradicts (64).

#### 4. Deforming Inequalities into the Type of (29)

It turns out that lots of functional inequalities can be deformed into inequality (29). So we can regard inequality (29) as a typical form of a certain class of functional inequalities. In this point of view, we have a following systematic program to prove the generalized Hyers-Ulam stability of certain functional inequalities.

*Step 1. *Deform a given inequality into the type of (29) and get a modified bound function.

*Step 2. *Apply Theorem 6 for the modified bound function.

It should be remarked that if a functional inequality can be deformed into the type of (29), then a solution of the original functional equation is quadratic. And, it can be easily checked that the resulting unique quadratic mapping in Step 2 also satisfies the original functional equation. So we don’t need to worry anything about the given functional equation in our program. In this section, we illustrate just two of them.

First, we consider the following functional equation: for some rational number with .

Theorem 10. *Let be a function with (28). Let be a mapping satisfying and
**
for some rational number with . Then there exists a unique quadratic mapping such that satisfies (66) and
**
for all .*

*Proof. *Setting in (67), we have
and by (67) and (69), we have
for all . Letting in (67), we have
for all . Hence, by (70) and (71), we have
for all , where . So by Theorem 6, we get the result.

*Remark 11. *It would be interesting to see how Theorem 10 works well for a simple case of . Take . Then the original inequality in Theorem 10 is
After the deforming process, inequality (73) turns into the following new inequality which is standard in our sense:
With , apply Theorem 6 or Theorem 7 ( in the theorems) to inequality (74); we get the following conclusion.

There exists a unique quadratic mapping such that satisfies (66) and

Now, we consider the following functional equation:

Theorem 12. *Let be a function with (28) for . Let be a mapping satisfying and
**
Then there exists a unique quadratic mapping such that satisfies (76) and
**
for all .*

*Proof. *Setting in (77), we have
for all . Letting in (79), we have
for all . By (79) and (80), we have
for all . Hence by (81), we have
for all .

Letting in (77), we get
and by (77) and (83), we have
for all , and so by (82), we have
for all , where . So by Theorem 6, we get the result.

#### Conflict of Interests

The authors declare that there is no conflict of interests regarding the publication of this paper.

#### Acknowledgment

The first author was supported by the research fund of Dankook University in 2013.