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Abstract and Applied Analysis
Volume 2008, Article ID 136592, 11 pages
http://dx.doi.org/10.1155/2008/136592
Research Article

Functional Inequalities Associated with Additive Mappings

1Department of Mathematics, Hallym University, Chuncheon 200-702, South Korea
2Department of Mathematics, Mokwon University, Daejeon 302-729, South Korea

Received 13 May 2008; Revised 25 June 2008; Accepted 1 August 2008

Academic Editor: John Rassias

Copyright © 2008 Jaiok Roh and Ick-Soon Chang. 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

The functional inequality is investigated, where is a group divisible by and are mappings, and is a Banach space. The main result of the paper states that the assumptions above together with (1) and (2) , or , imply that is additive. In addition, some stability theorems are proved.

1. Introduction and Preliminaries

The concept of stability for a functional equation arises when we replace the functional equation by an inequality which acts as a perturbation of the equation. The study of stability problems had been formulated by Ulam [1] during a talk in 1940: under what condition does there exist a homomorphism near an approximate homomorphism? In the following year 1941, Hyers [2] had answered affirmatively the question of Ulam for Banach spaces, which states that if and is a mapping with a normed space, a Banach space such that for all then there exists a unique additive mapping such that for all Then, Aoki [3] in 1950 and Rassias [4] in 1978 proved the following generalization of Hyers' theorem [2] by considering the case when the inequality (1.1) is unbounded.

Proposition 1.1. Let be a mapping from a normed space into a Banach space subject to the inequality for all where and are constants with and Then, there exists a unique additive mapping such that for all If then inequality (1.3) holds for and (1.4) for

Following the techniques of the proof of the corollary of Hyers [2], we observed that Hyers introduced (in 1941) Hyers continuity condition about the continuity of the mapping in for each fixed and then he proved homogenouity of degree one and, therefore, the famous linearity. This condition has been assumed further till now through the complete Hyers direct method in order to prove linearity for generalized Hyers-Ulam stability problem forms (cf., [5]).

In 1991, Gajda [6] provided an affirmative answer to Rassias' question whether his theorem can be extended for values of greater than one. However, it was shown by Gajda [6] as well as by Rassias and Šemrl [7] that one cannot prove a theorem similar to [4] when On the other hand, Rassias [810] generalized Hyers' stability result by presenting a weaker condition controlled by (or involving) a product of different powers of norms (from the right-hand side of assumed conditions) as follows.

Proposition 1.2. Suppose that there exist constants and such that and is a mapping with a normed space, is a Banach space such that the inequality holds for all Then, there exists a unique additive mapping such that for all

Since then, more generalizations and applications of the generalized Hyers-Ulam stability to a number of functional equations and mappings have been investigated in [1132].

Recently, Roh and Shin [33] proved that if is a mapping from a normed space into a Banach space satisfying the inequality for all and some then it is additive. In addition, they investigated the stability in Banach spaces.

In this paper, we will consider a mapping on a group instead of a normed space which satisfies the following inequality: for all where is a group, and are mappings, and is a Banach space.

Inequalities of the type above and some corresponding equations were examined by several authors [3436]. In [35], it has been proved that if is a (not necessarily 2-divisible) group, is an inner product space and the mapping satisfies for all then it is additive. By replacing in (1.8), we obtain for all which implies the inequality (1.9). Therefore, Theorem 2.1 in this paper is a special case of the result cited above for mappings which maps into Hilbert spaces. However, Theorem 2.1 is also valid for Banach spaces, while the result in [35] is not. (A counter example was constructed in [34]).

2. Main Results

Theorem 2.1. Let be a 2-divisible group and a Banach space. Assume that a mapping satisfies the assumptions
(1) (2) or , and that the mapping satisfies the inequality (1.8). Then, is additive.

Proof. By letting in (1.8), we get ; and by letting and in (1.8), we have for all Also, by letting and in (1.8), we obtain for all
Next, we are in the position to show that is additive. We will consider two different cases for second assumption of
Case 1. Assume for all We get by (2.1) and (2.2) for all positive integer and all Therefore, we can define for all Due to (1.8), (2.1), and (2.2), we obtain for all Thus,
Case 2. Assume for all We get by (2.1) and (2.2) for all positive integer and all Therefore, we can define for all Due to (1.8), (2.1), and (2.2), we obtain for all Thus,

Next, we will study the generalized Hyers-Ulam stability of functional inequality (1.8).

Theorem 2.2. Let be a 2-divisible abelian group and a Banach space. Assume that a mapping satisfies the assumptions
(1) (2) ,and the inequality (1.8). Then, there exists a unique additive mapping such that for all

Proof. Letting in (1.8), we get By assumption, we should have since Hence, So, by letting and in (1.8), we have for all Letting , and in (1.8), we obtain for all Therefore, we have for all nonnegative integers and with and all It means that the sequence is a Cauchy sequence. Since is complete, the sequence converges. So we can define a mapping by for all Moreover, by letting and passing we get (2.7).
Now, we claim that the mapping is additive. We note by (2.9) that So we have By (1.8), (2.8), and (2.9), we have for all
Now, to prove uniqueness of the mapping let us assume that is an additive mapping satisfying (2.7). Then, we obtain due to (2.7) that for all as

Theorem 2.3. Let be a 2-divisible abelian group and a Banach space. Assume that a mapping satisfies the assumptions
(1) (2) and the inequality (1.8). Then, there exists a unique additive mapping such that for all where

Proof. By letting in (1.8), we get We also have, by letting and in (1.8), that for all Next, by letting , and in (1.8), we obtain for all Therefore, we have for all nonnegative integers and with and all It follows that the sequence is a Cauchy and so it is convergent since is complete. So one can define a mapping by for all By letting and taking the limit we arrive at (2.14).
Now, we claim that the mapping is additive. By (2.18), one notes So we have By (1.8), (2.17), and (2.18), we obtain for all
Now, to show uniqueness of the mapping let us assume that is another additive mapping satisfying (2.14). Then, by (2.14), and assumptions of we have for all , as

With the help of Theorems 2.2 and 2.3, we obtain the following corollaries. Corollary 2.4. Suppose that is a mapping from a normed space into a Banach space subject to the inequality for all where and are constants with and Then, there exists a unique additive mapping such that for all Corollary 2.5. Suppose that is a mapping from a normed space into a Banach space subject to the inequality (2.23) for all where and are constants with and Then, there exists a unique additive mapping such that for all

Theorem 2.6. Let be a 2-divisible abelian group and a Banach space. Assume that a mapping satisfies the assumptions
(1) (2) , and the inequality (1.8). Then, there exists a unique additive mapping such that for all where

Proof. Letting in (1.8), we get By assumption, we should have since Hence, we obtain By letting and in (1.8), we have for all Let Then, we obtain for all Hence, for all nonnegative integers and with and all So the sequence is a Cauchy sequence. Due to the completeness of this sequence is convergent. Let be a mapping defined by for all Letting and sending we get (2.26).
Next, we claim that the mapping is additive. We first note that because So, by (1.8) and (2.28), we obtain for all So we have
The proof of uniqueness for is similar to the proof of Theorem 2.2.

Theorem 2.7. Let be a 2-divisible abelian group and a Banach space. Assume that a mapping satisfies the assumptions
(1) (2) , and the inequality (1.8). Then, there exists a unique additive mapping such that for all where

Proof. Due to (2.17), we have for all So, we obtain for all nonnegative integers and with and all This means that the sequence is a Cauchy sequence. Since is complete, the sequence converges. Thus, we may define a mapping by for all Letting and passing the limit we get (2.32).
Now, we claim that the mapping is additive. By (2.16), (1.8), and (2.34), we have for all
The proof of uniqueness for is similar to the proof of Theorem 2.3.

Acknowledgments

The authors would like to thank referees for their valuable comments regarding a previous version of this paper. This work was supported by the Korea Research Foundation Grant funded by the Korean Government (KRF-2008-531-C00008).

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