Journal of Applied Mathematics

Journal of Applied Mathematics / 2011 / Article

Research Article | Open Access

Volume 2011 |Article ID 817079 | 16 pages | https://doi.org/10.1155/2011/817079

A Fixed Point Approach to the Stability of the Cauchy Additive and Quadratic Type Functional Equation

Academic Editor: Yuri Sotskov
Received02 Jun 2011
Accepted18 Jul 2011
Published15 Sep 2011

Abstract

We investigate the stability of the functional equation by using the fixed point theory in the sense of Cădariu 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 Cauchy’s 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 [5–12].

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, Cădariu 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 Cădariu 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    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 .

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Copyright © 2011 Sun Sook Jin and Yang-Hi Lee. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.


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