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
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
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:
2. Solutions of the Functional Inequality (1.2)
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
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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