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Journal of Applied Mathematics

Volume 2014, Article ID 184680, 9 pages

http://dx.doi.org/10.1155/2014/184680
Research Article

On the Generalized Hyers-Ulam Stability of an -Dimensional Quadratic and Additive Type Functional Equation

Department of Mathematics Education, Gongju National University of Education, Gongju 314-711, Republic of Korea

Received 13 February 2014; Accepted 28 April 2014; Published 26 May 2014

Academic Editor: Sabri Arik

Copyright © 2014 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.

Abstract

We investigate the generalized Hyers-Ulam stability of a functional equation .

1. Introduction

Throughout this paper, let be a normed space and a Banach space. For a given mapping , we define for all . A mapping is called an additive mapping (a quadratic mapping, resp.) if satisfies the functional equation ( , resp.). If a mapping is represented by sum of an additive mapping and a quadratic mapping, we call the mapping a quadratic-additive mapping. For a functional equation if all of the solutions of are quadratic-additive mappings and all of quadratic-additive mappings are the solutions of , then we call the functional equation a quadratic-additive type functional equation.

In 1940, Ulam [1] raised a question concerning the stability of homomorphisms. Hyers [2], Aoki [3], Rassias [4], and Găvruţa [5] made important role to study the stability of the functional equation. During the last decades, the stability problems of functional equations have been extensively investigated by a number of mathematicians (see also [69]).

In 2006, Jun and Kim [10] obtained the stability of the functional equation for all ( ) (see also [1115]). The functional equation (2) is a quadratic-additive type functional equation (see Theorem 2.6 in [16]). For the case , Jung [17] proved the stability of the functional equation (2) (see also [1820]) and, for the case , Chang et al. [21] proved the stability of the functional equation (2) (see also [2225]).

In this paper, we will generalize the previous results of the stability problem of the functional equation (2) on the punctured domain. In particular, we will show the superstability (if ) of the functional equation (2) in the sense of Rassias.

2. Stability of the Functional Equation (2) ( Is Even)

Let be a fixed element in and let be a function satisfying the conditions: for all , where is a fixed even integer greater than 2 in this section. For convenience, we use the following abbreviations in this section for a given mapping : for all . From these, we get the equality for all and all nonnegative integers , where are the integers defined by for .

Lemma 1. If is a mapping such that for all , then for all and all nonnegative integers .

Proof. We can easily get for all and all nonnegative integers .

Theorem 2. Suppose that is a mapping such that for all with . Then, there exists a unique mapping satisfying (8) for all and for all with , where are the mappings defined by for all .

Proof. It follows from (6) and (11) that for all and all nonnegative integers , with . From (3), (4), and (14), it follows that the sequence is Cauchy for all . Since is complete, the sequence converges. From this and , we can define the mapping by for all . Moreover, letting and taking the limit as in (14), we get the inequality (12). Notice that , , and for all . Hence, it follows from (11) and the definition of that for all .

Now, let be another mapping satisfying (8) for all and (12) with . Using Lemma 1, (12), and , we obtain for all and all positive integers . It follows from (12) and (17) that for all and all positive integers . We can easily show that the terms on the right-hand side of the inequality (18) tend to 0 as for the cases and . For the case , we have for all and all positive even integers . So, we also show that the terms on the right-hand side of the inequality (18) tend to 0 as for the cases . Using the equality , we can conclude that for all . This proves the uniqueness of .

Corollary 3. Let be a real number. Suppose that is a mapping such that for all (with if ). Then, there exists a unique mapping satisfying (8) for all and for all with .

Proof. Put for all . By Theorem 2, there exists a unique mapping satisfying (8) for all and for all with . From these, we get the inequalities for all and all positive real numbers . Taking the limit as or in the above inequality, we have if . Hence, if , then the inequality for all follows from (23). If , then we get the inequalities for all and all positive integers . Taking the limit as in the above inequality, we get for all . Since , the equality holds for all . The result follows from this, (23), and (25).

Lemma 4. If is a mapping satisfying (8) for all with and is continuous in for each fixed , then is represented by for all and all .

Proof. We will prove the equality for all integers . First, we will use the induction on to prove the equality (28) for all nonnegative integers . Note that . We can easily prove it for the cases . For the case , we can show that for all . Assume that (28) holds for all and all nonnegative integers . Then, we obtain which completes (28) for all nonnegative integers . Using the similar method, we also can prove the equality (28) for all negative integers . By (28), we get the equalities for all and all integers . Hence, for all and all integers . If , then there exists a rational sequence satisfying . Since is continuous in for each fixed , we have for all .

3. Stability of the Functional Equation (2) ( Is Odd)

Let , , , and be as in Section 2. In this section, let be an odd integer greater than 2. For convenience, we use the following abbreviations in this section for a given mapping : for . From these, we get for all .

Using (35) and a similar method in the proof of Lemma 1, we get the following lemma.

Lemma 5. If is a mapping satisfying (8) for all , then for all .

From (35), Lemma 5, and similar methods used in Theorem 2, we get the following theorem.

Theorem 6. If is a unique mapping satisfying (11) for all with , then there exists a unique mapping satisfying (8) for all and for all with , where are the mappings defined by for all .

From Theorem 6 and similar methods used in Corollary 3, we get the following corollary.

Corollary 7. Let be a real number. Suppose that is a mapping satisfying (20) for all (with if ). Then, there exists a unique mapping satisfying (8) for all and for all with .

Proof. Put for all . By Theorem 6, there exists a unique mapping satisfying (8) for all and for all with . From these, we get the inequalities for all and all positive real numbers . Taking the limit as or in the above inequality, we have if . Hence, if , then the inequality for all follows from (40). If , then we get the inequalities for all and all positive integers . Taking the limit as in the above inequality, we get for all . Since , the equality holds for all . The result follows from this, (40), and (42).

From similar methods used in Lemma 4, we get the following lemma.

Lemma 8. If is a mapping satisfying (8) for all with and is continuous in for each fixed , then is represented by for all and all .

Proof. We will use the induction on to prove (44) for all nonnegative integers . Note that . We can easily prove it for the cases . For the case , we can show that for all . Assume that (44) holds for all and all nonnegative integers . Then, we obtain which completes the proof of (44). The remainder of the proof is the same in the proof of Lemma 4.

Corollary 9. If is a mapping satisfying (8) for all , then .

Proof. Put . Then, we have for all . By Corollaries 3 and 7, for all with . So, we get the desired result.

Corollary 10. Let be a real number and an integer. Suppose that is a mapping satisfying for all and is continuous. Then, is represented by for all and for all .

Proof. If is even, then the equality (49) follows from Corollary 3 and Lemma 4. If is odd, then the equality (49) follows from Corollary 7 and Lemma 8. And we can easily show that the function defined by (49) satisfies the functional equation for all .

4. Another Proof for the Stability of the Functional Equation (2)

Let , , be as in Section 2. In this section, Let be a fixed integer greater than 2 and let be a function satisfying the conditions (3) and (4) for all . For convenience, we use the following abbreviations in this section for a given mapping : for all . From these, we get for all . Using (51) and a similar method in the proof of Lemma 1, we get the following lemma.

Lemma 11. If is a quadratic-additive mapping, then for all .

Theorem 12 (compare with Theorem 3.1 in [15]). Suppose that is a mapping such that for all . Then, there exists a unique quadratic-additive mapping such that for all , where are the mappings defined by for all .

Proof. Note that for all positive integers . It follows from (3) that . From this, (51), Lemma 11, and similar methods used in Theorem 2, we obtain this theorem.

Corollary 13 (compare with Corollary 3.3 in [15]). Let be a positive real number. Suppose that is a mapping such that for all . Then, there exists a unique quadratic-additive mapping such that for all .

Proof. Since , we get . From Theorem 12 and similar methods used in Corollary 3, we obtain this corollary.

From Theorem 12 and similar methods used in Corollary 3, we get the following corollary.

Corollary 14 (compare with Corollary 3.2 in [15]). Suppose that is a mapping such that for all . Then, there exists a unique quadratic-additive mapping such that for all .

Conflict of Interests

The author declares that there is no conflict of interests regarding the publication of this paper.

Acknowledgments

This research was supported by Basic Science Research Program through the National Research Foundation of Korea (NRF) funded by the Ministry of Education, Science and Technology (no. 2012R1A1A4A01002971).

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