#### Abstract

Let be real vector spaces. It is shown that an odd mapping satisfies for all if and only if the odd mapping is Cauchy additive. Furthermore, we prove the generalized Hyers-Ulam stability of the above functional equation in real Banach spaces.

#### 1. Introduction

The stability problem of functional equations originated from a question of Ulam [1] concerning the stability of group homomorphisms. Hyers [2] gave the first affirmative partial answer to the question of Ulam for Banach spaces. Hyers' Theorem was generalized by Aoki [3] for additive mappings and by Th. M. Rassias [4] for linear mappings by considering an unbounded Cauchy difference. The paper of Th. M. Rassias [4] has provided a lot of influence in the development of what we call * generalized Hyers-Ulam stability * of functional equations. A generalization of Th. M. Rassias' theorem was obtained by Gvruta [5] by replacing the unbounded Cauchy difference by a general control function.

The functional equation,
is called a * quadratic functional equation *. In particular, every solution of the quadratic functional equation is said to be a * quadratic mapping *. 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. The generalized Hyers-Ulam stability of the quadratic functional equation has been proved by Czerwik [8], J. M. Rassias [9], Gvruta [10], and others [11]. In [12], Th. M. Rassias proved that the norm defined over a real vector space is induced by an inner product if and only if for a fixed integer
holds for all An operator extension of this norm equality is presented in [13]. For more information on the recent results on the stability of quadratic functional equation, see [14]. Inner product spaces, Cauchy equation, Euler-Lagrange-Rassias equations, and Ulam-Gvruta-Rassias stability have been studied by several authors (see [15–27]).

In [28], C. Park, Lee, and Shin proved that if an even mapping satisfies

then the even mapping is quadratic. Moreover, they proved the generalized Hyers-Ulam stability of the quadratic functional equation (1.3) in real Banach spaces.

Throughout this paper, assume that is a fixed positive integer, and are real normed vector spaces.

In this paper, we investigate the functional equation and prove the generalized Hyers-Ulam stability of the functional equation (1.4) in real Banach spaces.

#### 2. Functional Equations Related to Inner Product Spaces

We investigate the functional equation (1.4).

Lemma 2.1. *Let and be real vector spaces. An odd mapping satisfies
**
for all if and only if the odd mapping is Cauchy additive, that is,
**
for all . *

*Proof. *Assume that satisfies (2.1).

Letting , in (2.1), we get

for all . Since is odd,
for all and . So
for all . Letting in (2.5), we get for all . Thus
for all .

It is easy to prove the converse.

For a given mapping , we define for all .

We are going to prove the generalized Hyers-Ulam stability of the functional equation in real Banach spaces.Theorem 2.2. *Let be a mapping satisfying for which there exists a function such that
**
for all . Then there exists a unique Cauchy additive mapping satisfying (2.1) such that*

for all .

*Proof. *Letting and in (2.9), we get
for all . Replacing by in (2.11), we get
for all . Let for all . It follows from (2.11) and (2.12) that
for all . So
for all . Hence
for all nonnegative integers and with and all . It follows from (2.8) and (2.15) that the sequence is Cauchy for all . Since is complete, the sequence converges. So one can define the mapping by
for all .

By (2.8) and (2.9),

for all . So . By Lemma 2.1, the mapping is Cauchy additive. Moreover, letting and passing the limit in (2.15), we get (2.10). So there exists a Cauchy additive mapping satisfying (2.1) and (2.10).

Now, let be another Cauchy additive mapping satisfying (2.1) and (2.10). Then we have

which tends to zero as for all . So we can conclude that for all . This proves the uniqueness of .Corollary 2.3. *Let and be positive real numbers, and let be a mapping such that
**
for all . Then there exists a unique Cauchy additive mapping satisfying (2.1) such that
**
for all .**Proof. *Define , and apply Theorem 2.2 to get the desired result.Corollary 2.4. *Let be an odd mapping for which there exists a function satisfying (2.8) and (2.9). Then there exists a unique Cauchy additive mapping satisfying (2.1) such that
**
or (alternative approximation)
**
for all , where is defined in (2.8).*Theorem 2.5. *Let be a mapping satisfying for which there exists a function satisfying (2.9) such that
**
for all . Then there exists a unique Cauchy additive mapping satisfying (2.1) such that
**
for all .*

*Proof. *It follows from (2.13) that
for all . So
for all nonnegative integers and with and all . It follows from (2.23) and (2.26) that the sequence is Cauchy for all . Since is complete, the sequence converges. So one can define the mapping by
for all .

By (2.9) and (2.23),

for all . So . By Lemma 2.1, the mapping is Cauchy additive. Moreover, letting and passing the limit in (2.26), we get (2.24). So there exists a Cauchy additive mapping satisfying (2.1) and (2.24).

The rest of the proof is similar to the proof of Theorem 2.2.

Corollary 2.6. *Let and be positive real numbers, and let be a mapping satisfying (2.19). Then there exists a unique Cauchy additive mapping satisfying (2.1) such that
**
for all .**Proof. *Define , and apply Theorem 2.5 to get the desired result.Corollary 2.7. *Let be an odd mapping for which there exists a function satisfying (2.9) and (2.23). Then there exists a unique Cauchy additive mapping satisfying (2.1) such that
**
or (alternative approximation),
**
for all , where is defined in (2.23).*The following was proved in [28].*Remark 2.8 ([28]). * Let be a mapping satisfying for which there exists a function satisfying (2.9) such that
for all . Then there exists a unique quadratic mapping satisfying (2.1) such that
for all .

Note that
Combining Theorem 2.2 and Remark 2.8, we obtain the following result.Theorem 2.9. *Let be a mapping satisfying for which there exists a function satisfying (2.9) and (2.32). Then there exist a unique Cauchy additive mapping satisfying (2.1) and a unique quadratic mapping satisfying (2.1) such that
**
for all , where and are defined in (2.8) and (2.32), respectively.*Corollary 2.10. *Let and be positive real numbers, and let be a mapping satisfying (2.19). Then there exist a unique Cauchy additive mapping satisfying (2.1) and a unique quadratic mapping satisfying (2.1) such that
**
for all .**Proof. *Define , and apply Theorem 2.9 to get the desired result.The following was proved in [28].*Remark 2.11 (see [28]). * Let be a mapping satisfying for which there exists a function satisfying (2.9) such that
for all . Then there exists a unique quadratic mapping satisfying (2.1) such that
for all .

Note that

Combining Theorem 2.5 and Remark 2.11, we obtain the following result.Theorem 2.12. *Let be a mapping satisfying for which there exists a function satisfying (2.9) and (2.23). Then there exist a unique Cauchy additive mapping satisfying (2.1) and a unique quadratic mapping satisfying (2.1) such that
**
for all , where and are defined in (2.23) and (2.37), respectively.*Corollary 2.13. *Let and be positive real numbers, and let be a mapping satisfying (2.19). Then there exist a unique Cauchy additive mapping satisfying (2.1) and a unique quadratic mapping satisfying (2.1) such that
**
for all .**Proof. *Define , and apply Theorem 2.12 to get the desired result.

#### Acknowledgment

The first author was supported by National Research Foundation of Korea (NRF-2009-0070788).