#### Abstract

We investigate the stability of the functional equation by using the fixed point theory in the sense of CΔdariu and Radu.

#### 1. Introduction

In 1940, Ulam [1] raised a question concerning the stability of homomorphisms as follow. Given a group , a metric group with the metric , and a positive number , does there exist a such that if a mapping satisfies the inequality
for all then there exists a homomorphism with
for all ? When this problem has a solution, we say that the homomorphisms from to are * stable. * In the next year, Hyers [2] gave a partial solution of Ulam's problem for the case of approximate additive mappings under the assumption that and are Banach spaces. Hyers' result was generalized by Aoki [3] for additive mappings and by Rassias [4] for linear mappings by considering the stability problem with unbounded Cauchyβs differences. The paper of Rassias had much influence in the development of stability problems. The terminology Hyers-Ulam-Rassias stability originated from this historical background. During the last decades, the stability problems of functional equations have been extensively investigated by a number of mathematicians, see [5β12].

Almost all subsequent proofs, in this very active area, have used Hyers' method of [2]. Namely, the mapping , which is the solution of a functional equation, is explicitly constructed, starting from the given mapping , by the formulae or . We call it * a direct method. * In 2003, CΔdariu and Radu [13] observed that the existence of the solution for a functional equation and the estimation of the difference with the given mapping can be obtained from the fixed point theory alternative. This method is called * a fixed point method. * In 2004, they applied this method [14] to prove stability theorems of * the Cauchy functional equation: *
In 2003, they [15] obtained the stability of * the quadratic functional equation*:
by using the fixed point method. Notice that if we consider the functions defined by and , where is a real constant, then satisfies (1.3), and holds (1.4), respectively. We call a solution of (1.3) * an additive map, * and a mapping satisfying (1.4) is called * a quadratic map. * Now we consider the functional equation:
which is called * the Cauchy additive and quadratic-type functional equation. * The function defined by satisfies this functional equation, where are real constants. We call a solution of (1.5) * a quadratic-additive mapping. *

In this paper, we will prove the stability of the functional equation (1.5) by using the fixed point theory. Precisely, we introduce a strictly contractive mapping with the Lipschitz constant . Using the fixed point theory in the sense of CΔdariu and Radu, together with suitable conditions, we can show that the contractive mapping has the fixed point. Actually the fixed point becomes the precise solution of (1.5). In Section 2, we prove several stability results of the functional equation (1.5) using the fixed point theory, see Theorems 2.3, 2.4, and 2.5. In Section 3, we use the results in the previous sections to get a stability of the Cauchy functional equation (1.3) and that of the quadratic functional equation (1.4), respectively.

#### 2. Main Results

We recall the following result of the fixed point theory by Margolis and Diaz.

Theorem 2.1 (see [16] or [17]). * Suppose that a complete generalized metric space , which means that the metric may assume infinite values, and a strictly contractive mapping with the Lipschitz constant are given. Then, for each given element , either
**
or there exists a nonnegative integer such that *(1)* for all , *(2)*the sequence is convergent to a fixed point of , *(3)* is the unique fixed point of in , *(4)* for all .*

Throughout this paper, let be a (real or complex) linear space, and let be a Banach space. For a given mapping , we use the following abbreviation:
for all . If is a solution of the functional equation , see (1.5), we call it a * quadratic-additive mapping*. We first prove the following lemma.

Lemma 2.2. *If is a mapping such that for all , then is a quadratic-additive mapping.*

*Proof. *By choosing , we get

Since , we easily obtain for all .

Now we can prove some stability results of the functional equation (1.5).

Theorem 2.3. *Let be a given function. Suppose that the mapping satisfies
**
for all . If there exist constants such that has the property
**
for all , then there exists a unique quadratic-additive mapping such that
**
for all . In particular, is represented by
**
for all .*

*Proof. *It follows from (2.5) that
for all , and
for all . From this, we know that . Let be the set of all mappings ββ with . We introduce a generalized metric on by
It is easy to show that is a generalized complete metric space. Now we consider the mapping , which is defined by
for all . Notice
for all and . Let , and let be an arbitrary constant with . From the definition of , we have
for all , which implies that
for any . That is, is a strictly contractive self-mapping of with the Lipschitz constant . Moreover, by (2.4), we see that
for all . It means that by the definition of . Therefore, according to Theorem 2.1, the sequence converges to the unique fixed point of in the set , which is represented by (2.7) for all . By the definition of , together with (2.4) and (2.7), it follows that
for all . By Lemma 2.2, we have proved that
for all .

Theorem 2.4. *Let be a given function. Suppose that the mapping satisfies (2.5) for all . If there exists a constant such that has the property
**
for all , then there exists a unique quadratic-additive mapping satisfying (2.6) for all . Moreover, if is continuous, then itself is a quadratic-additive mapping.*

*Proof. *It follows from (2.18) that
for all , and
for all . By the same method used in Theorem 2.3, we know that there exists a unique quadratic-additive mapping satisfying (2.6) for all . Since is continuous, we get
for all and for any fixed integers with . Therefore, we obtain
for all , where is defined by . Since , we have shown that . This completes the proof of this theorem.

We continue our investigation with the next result.

Theorem 2.5. *Let . Suppose that satisfies the inequality for all . If there exists such that the mapping has the property
**
for all , then there exists a unique quadratic-additive mapping such that
**
for all . In particular, is represented by
**
for all .*

*Proof. *By the similar method used to prove in the proof of Theorem 2.3, we can easily show that . Let the set be as in the proof of Theorem 2.3. Now we consider the mapping defined by
for all and . Notice that
and , for all . Let , and let be an arbitrary constant with . From the definition of , we have
for all . So
for any . That is, is a strictly contractive self-mapping of with the Lipschitz constant . Also we see that
for all , which implies that . Therefore, according to Theorem 2.1, the sequence converges to the unique fixed point of in the set , which is represented by (2.25). Since
the inequality (2.24) holds. From the definition of , (2.4), and (2.23), we have
for all . By Lemma 2.2, is quadratic additive.

*Remark 2.6. *If satisfies the equality for all in Theorems 2.3, 2.4, and 2.5, then the inequalities (2.6) and (2.24) can be replaced by
for all , respectively.

#### 3. Applications

For a given mapping , we use the following abbreviations: for all . Using Theorems 2.3, 2.4, and 2.5 we will show the stability results of the additive functional equation and the quadratic functional equation in the following corollaries.

Corollary 3.1. *Let , be mappings for which there exist functions , such that
**
for all . If there exists such that
**
for all , then there exist unique additive mappings , such that
**
for all . In particular, the mappings , are represented by
**
for all . Moreover, if is continuous, then is itself an additive mapping.*

*Proof. *Notice that
for all and . Put
for all and , then satisfies (2.5), satisfies (2.18), and satisfies (2.23). Therefore, , for all and . According to Theorem 2.3, there exists a unique mapping satisfying (3.6), which is represented by (2.7). Observe that, by (3.2) and (3.3),
as well as
for all . From this and (2.7), we get (3.9). Moreover, we have
for all . Taking the limit as in the above inequality and using , we get
for all . According to Theorem 2.4, there exists a unique mapping satisfying (3.7), which is represented by (2.7). By using the similar method to prove (3.9), we can show that is represented by (3.10). In particular, if is continuous, then is continuous on , and we can say that is an additive map by Theorem 2.4. On the other hand, according to Theorem 2.5, there exists a unique mapping satisfying (3.8) which is represented by (2.25). Observe that, by (3.2) and (3.5),
as well as
for all . From these and (2.25), we get (3.11). Moreover, we have
for all . Taking the limit as in the above inequality and using , we get
for all .

Corollary 3.2. * Let , be mappings for which there exist functions , such that
**
for all . If there exists such that the mapping satisfies (3.3), satisfies (3.4), and satisfies (3.5) for all , then there exist unique quadratic mappings , such that
**
for all . In particular, the mappings , are represented by
**
for all . Moreover, if is continuous, then itself is a quadratic mapping.*

*Proof. *Notice that
for all and . Put , for all and , then satisfies (2.5), satisfies (2.18), and satisfies (2.23). Moreover,
for all and . According to Theorem 2.3, there exists a unique mapping satisfying (3.23) which is represented by (2.7). Observe that
as well as
for all . From this and (2.7), we get (3.26) for all . Moreover, we have
for all . Taking the limit as in the above inequality, we get
for all . Using , we have
for all . Therefore, for all .

Next, by Theorem 2.4, there exists a unique mapping satisfying (3.24), which is represented by (2.7). By using the similar method to prove (3.26), we can show that is represented by (3.27). In particular, is continuous, then is continuous on , and we can say that is a quadratic map by Theorem 2.4. On the other hand, according to Theorem 2.5, there exists a unique mapping satisfying (3.25) which is represented by (2.25). Observe that
for all . It leads us to get
for all . From these and (2.25), we obtain (3.28). Moreover, we have
for all . Taking the limit as in the above inequality and using , we get
for all .

Now, we obtain Hyers-Ulam-Rassias stability results in the framework of normed spaces using Theorems 2.3 and 2.4.

Corollary 3.3. *Let be a normed space, and let be a Banach space. Suppose that the mapping satisfies the inequality
**
for all , where and . Then there exists a unique quadratic-additive mapping such that
**
for all . Moreover if , then is itself a quadratic-additive mapping.*

*Proof. *This corollary follows from Theorems 2.3, 2.4, and 2.5, and Remark 2.6, by putting
for all with if , if , and if .

Corollary 3.4. *Let be a normal space let and be a Banach space. Suppose that the mapping satisfies the inequality
**
for all , where and . Then there exists a unique quadratic-additive mapping such that
**
for all . Moreover, if , then is itself a quadratic-additive mapping.*

*Proof. *This corollary follows from Theorems 2.3, 2.4, 2.5, and Remark 2.6, by putting
for all with if , if , and if .