- Research Article
- Open Access

# 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

## Keywords

- Banach Space
- Linear Mapping
- Functional Equation
- Rational Number
- Nonnegative Integer

## 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

## References

- Ulam SM:
*A Collection of the Mathematical Problems*. Interscience, New York, NY, USA; 1960.Google Scholar - Hyers DH:
**On the stability of the linear functional equation.***Proceedings of the National Academy of Sciences*1941,**27:**222-224. 10.1073/pnas.27.4.222MathSciNetView ArticleGoogle Scholar - Aoki T:
**On the stability of the linear transformation in Banach spaces.***Journal of the Mathematical Society of Japan*1950,**2:**64-66. 10.2969/jmsj/00210064MathSciNetView ArticleMATHGoogle Scholar - Rassias ThM:
**On the stability of the linear mapping in Banach spaces.***Proceedings of the American Mathematical Society*1978,**72:**297-300. 10.1090/S0002-9939-1978-0507327-1MathSciNetView ArticleMATHGoogle Scholar - Bourgin DG:
**Classes of transformations and bordering transformations.***Bulletin of the American Mathematical Society*1951,**57:**223-237. 10.1090/S0002-9904-1951-09511-7MathSciNetView ArticleMATHGoogle Scholar - Forti GL:
**An existence and stability theorem for a class of functional equations.***Stochastica*1980,**4:**23-30. 10.1080/17442508008833155MathSciNetView ArticleMATHGoogle Scholar - Forti GL:
**Hyers-Ulam stability of functional equations in several variables.***Aequationes Mathematicae*1995,**50**(1-2):143-190. 10.1007/BF01831117MathSciNetView ArticleMATHGoogle Scholar - Gavruta P:
**A generalization of the Hyers-Ulam-Rassias stability of approximately additive mappings.***Journal of Mathematical Analysis and Applications*1994,**184**(3):431-436. 10.1006/jmaa.1994.1211MathSciNetView ArticleMATHGoogle Scholar - Cholewa PW:
**Remarks on the stability of functional equations.***Aequationes Mathematicae*1984,**27**(1):76-86. 10.1007/BF02192660MathSciNetView ArticleMATHGoogle Scholar - Czerwik S:
**On the stability of the quadratic mapping in normed spaces.***Abhandlungen aus dem Mathematischen Seminar der Universität Hamburg*1992,**62**(1):59-64. 10.1007/BF02941618MathSciNetView ArticleMATHGoogle Scholar - Faiziev VA, Rassias ThM, Sahoo PK:
**The space of****-additive mappings on semigroups.***Transactions of the American Mathematical Society*2002,**354**(11):4455-4472. 10.1090/S0002-9947-02-03036-2MathSciNetView ArticleMATHGoogle Scholar - Grabiec A:
**The generalized Hyers-Ulam stability of a class of functional equations.***Publicationes Mathematicae Debrecen*1996,**48**(3-4):217-235.MathSciNetMATHGoogle Scholar - Hyers DH, Rassias ThM:
**Approximate homomorphisms.***Aequationes Mathematicae*1992,**44**(2-3):125-153. 10.1007/BF01830975MathSciNetView ArticleMATHGoogle Scholar - Isac G, Rassias ThM:
**Stability of****-additive mappings: applications to nonlinear analysis.***International Journal of Mathematics and Mathematical Sciences*1996,**19:**219-228. 10.1155/S0161171296000324MathSciNetView ArticleMATHGoogle Scholar - Jun K-W, Lee Y-H:
**On the Hyers-Ulam-Rassias stability of a pexiderized quadratic inequality.***Mathematical Inequalities and Applications*2001,**4**(1):93-118.MathSciNetView ArticleMATHGoogle Scholar - Kannappan Pl:
**Quadratic functional equation and inner product spaces.***Results in Mathematics*1995,**27:**368-372.MathSciNetView ArticleMATHGoogle Scholar - Najati A:
**Hyers-Ulam stability of an n-apollonius type quadratic mapping.***Bulletin of the Belgian Mathematical Society—Simon Stevin*2007,**14**(4):755-774.MathSciNetMATHGoogle Scholar - Najati A, Park C:
**Hyers-Ulam-Rassias stability of homomorphisms in quasi-Banach algebras associated to the Pexiderized Cauchy functional equation.***Journal of Mathematical Analysis and Applications*2007,**335**(2):763-778. 10.1016/j.jmaa.2007.02.009MathSciNetView ArticleMATHGoogle Scholar - Najati A, Park C:
**The Pexiderized Apollonius-Jensen type additive mapping and isomorphisms between****-algebras.***Journal of Difference Equations and Applications*2008,**14**(5):459-479. 10.1080/10236190701466546MathSciNetView ArticleMATHGoogle Scholar - Park C-G:
**On the stability of the linear mapping in Banach modules.***Journal of Mathematical Analysis and Applications*2002,**275**(2):711-720. 10.1016/S0022-247X(02)00386-4MathSciNetView ArticleMATHGoogle Scholar - Rassias ThM:
**On a modified Hyers-Ulam sequence.***Journal of Mathematical Analysis and Applications*1991,**158**(1):106-113. 10.1016/0022-247X(91)90270-AMathSciNetView ArticleMATHGoogle Scholar - Rassias ThM:
**On the stability of functional equations and a problem of Ulam.***Acta Applicandae Mathematicae*2000,**62:**23-130. 10.1023/A:1006499223572MathSciNetView ArticleMATHGoogle Scholar - Rassias ThM:
**On the stability of functional equations in Banach spaces.***Journal of Mathematical Analysis and Applications*2000,**251**(1):264-284. 10.1006/jmaa.2000.7046MathSciNetView ArticleMATHGoogle Scholar - Aczél J, Dhombres J:
*Functional Equations in Several Variables*. Cambridge University Press, Cambridge, UK; 1989.View ArticleMATHGoogle Scholar - Czerwik S:
*Functional Equations and Inequalities in Several Variables*. World Scientific, River Edge, NJ, USA; 2002.View ArticleMATHGoogle Scholar - Hyers DH, Isac G, Rassias ThM:
*Stability of Functional Equations in Several Variables*. Birkhäuser, Basel, Switzerland; 1998.View ArticleMATHGoogle Scholar - Jung S:
*Hyers-Ulam-Rassias Stability of Functional Equations in Mathematical Analysis*. Hadronic Press, Palm Harbor, Fla, USA; 2001.MATHGoogle Scholar - Rassias ThM:
*Functional Equations, Inequalities and Applications*. Kluwer Academic Publishers, Dordrecht, The Netherlands; 2003.View ArticleMATHGoogle Scholar - Kadison RV, Pedersen G:
**Means and convex combinations of unitary operators.***Mathematica Scandinavica*1985,**57:**249-266.MathSciNetMATHGoogle Scholar - Jung S-MO:
**Hyers-ulam-rassias stability of jensen's equation and its application.***Proceedings of the American Mathematical Society*1998,**126**(11):3137-3143. 10.1090/S0002-9939-98-04680-2MathSciNetView ArticleMATHGoogle Scholar - Jung S, Moslehian MS, Sahoo PK:
**Stability of a generalized Jensen equation on restricted domains.**http://arxiv.org/abs/math/0511320v1 - Skof F:
**Sull' approssimazione delle applicazioni localmente****-additive.***Atti della Accademia delle Scienze di Torino*1983,**117:**377-389.MathSciNetMATHGoogle Scholar - Hyers DH, Isac G, Rassias ThM:
**On the asymptoticity aspect of Hyers-Ulam stability of mappings.***Proceedings of the American Mathematical Society*1998,**126**(2):425-430. 10.1090/S0002-9939-98-04060-XMathSciNetView ArticleMATHGoogle Scholar - Skof F:
**On the stability of functional equations on a restricted domain and a related topic.**In*Stabiliy of Mappings of Hyers-Ulam Type*. Edited by: Rassias ThM, Tabor J. Hadronic Press, Palm Harbor, Fla, USA; 1994:41-151.Google Scholar - Gajda Z:
**On stability of additive mappings.***International Journal of Mathematics and Mathematical Sciences*1991,**14:**431-434. 10.1155/S016117129100056XMathSciNetView ArticleMATHGoogle Scholar

## Copyright

This article is published under license to BioMed Central Ltd. 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.