A Functional Inequality in Restricted Domains of Banach Modules
© M.B. Moghimi et al. 2009
Received: 28 April 2009
Accepted: 16 August 2009
Published: 8 October 2009
We investigate the stability problem for the following functional inequality on restricted domains of Banach modules over a -algebra. As an application we study the asymptotic behavior of a generalized additive mapping.
1. Introduction and Preliminaries
The following question concerning the stability of group homomorphisms was posed by Ulam : Under what conditions does there exist a group homomorphism near an approximate group homomorphism?
Hyers  considered the case of approximately additive mappings , where and are Banach spaces and satisfiesHyers inequality
for all .
In 1950, Aoki  provided a generalization of the Hyers' theorem for additive mappings and in 1978, Rassias  generalized the Hyers' theorem for linear mappings by allowing the Cauchy difference to be unbounded (see also ). The result of Rassias' theorem has been generalized by Forti [6, 7] and G vruta  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)
for all and all , then is -linear.
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
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.
for all with and all .
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
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.
for all then is -linear.
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
for all with and .
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  to give the following counterexample.
which contradicts (3.48).
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.
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