# A Functional Inequality in Restricted Domains of Banach Modules

- M. B. Moghimi
^{1}, - Abbas Najati
^{1}and - Choonkil Park
^{2}Email author

**2009**:973709

https://doi.org/10.1155/2009/973709

© M.B. Moghimi et al. 2009

**Received: **28 April 2009

**Accepted: **16 August 2009

**Published: **8 October 2009

## Abstract

## 1. Introduction and Preliminaries

The following question concerning the stability of group homomorphisms was posed by Ulam [1]: *Under what conditions does there exist a group homomorphism near an approximate group homomorphism?*

Hyers [2] considered the case of approximately additive mappings
, where
and
are Banach spaces and
satisfies*Hyers inequality*

In 1950, Aoki [3] provided a generalization of the Hyers' theorem for additive mappings and in 1978, Rassias [4] generalized the Hyers' theorem for linear mappings by allowing the Cauchy difference to be unbounded (see also [5]). The result of Rassias' theorem has been generalized by Forti [6, 7] and G vruta [8] who permitted the Cauchy difference to be bounded by a general control function. During the last three decades a number of papers have been published on the generalized Hyers-Ulam stability to a number of functional equations and mappings (see [9–23]). We also refer the readers to the books [24–28].

Throughout this paper, let
be a unital
-algebra with unitary group
, unit
and norm
. Assume that
is a left
-module and
is a left Banach
-module. An additive mapping
is called
*-linear* if
for all
and all
. In this paper, we investigate the stability problem for the following functional inequality:

on restricted domains of Banach modules over a -algebra, where are nonzero positive real numbers. As an application we study the asymptotic behavior of a generalized additive mapping.

## 2. Solutions of the Functional Inequality (1.2)

Theorem 2.1.

for all and all , then is -linear.

Proof.

for all and all . Hence and it follows from (2.2) and (2.3) that and for all and all Therefore for all Hence for all and all rational numbers .

for all Since is an arbitrary nonzero element in in the previous paragraph, one can replace instead of in (2.5). Thus we have for all and all So is -linear.

The following theorem is another version of Theorem 2.1 on a restricted domain when

Theorem 2.2.

Let and be left -modules and let be nonzero positive real numbers. Assume that a mapping satisfies and the functional inequality (2.1) for all with and all . Then is -linear.

Proof.

Hence satisfies (2.8) and we infer that satisfies (2.2) for all and all . By Theorem 2.1, is -linear.

## 3. Generalized Hyers-Ulam Stability of (1.2) on a Restricted Domain

In this section, we investigate the stability problem for -linear mappings associated to the functional inequality (1.2) on a restricted domain. For convenience, we use the following abbreviation for a given function and

Theorem 3.1.

Proof.

for all . It follows from the definition of and (3.2) that and for all with and all . Hence is -linear by Theorem 2.2.

We apply the result of Theorem 3.1 to study the asymptotic behavior of a generalized additive mapping. An asymptotic property of additive mappings has been proved by Skof [32] (see also [30, 33]).

Corollary 3.2.

Proof.

for all and all The uniqueness of implies for all Hence letting in (3.16), we obtain that is -linear.

The following theorem is another version of Theorem 3.1 for the case

Theorem 3.3.

Proof.

Hence is additive. Since for all , we have from (3.22) that for all and all Letting , we get . Therefore for all and all This proves that is -linear. Also, satisfies inequality (3.19) for all with , by the definition of .

For the case we use the Gajda's example [35] to give the following counterexample.

Example 3.4.

which contradicts (3.48).

## Declarations

### Acknowledgments

The third author was supported by Basic Science Research Program through the National Research Foundation of Korea funded by the Ministry of Education, Science and Technology (NRF-2009-0070788). The authors would like to thank the referees for a number of valuable suggestions regarding a previous version of this paper.

## Authors’ Affiliations

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