- Research Article
- Open Access
A Functional Inequality in Restricted Domains of Banach Modules
Advances in Difference Equations volume 2009, Article number: 973709 (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 Gvruta  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)
Let and be left -modules and let be nonzero real numbers. If a mapping with satisfies the functional inequality
for all and all , then is -linear.
Letting in (2.1) we get
for all and all . Letting (resp., ) in (2.2), we get
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 .
Now let and let be an integer number with . Then by Theorem 1 of , there exist elements such that . Since is additive and for all all rational numbers and all , we have
for all . Replacing instead of in the above equation, we have
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.
Letting with in (2.1), we get
for all . Let and let Then
Therefore replacing and by and in (2.6), respectively, we get
for all with and all .
Since it is easy to verify that
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
Let and be given. Assume that a mapping satisfies the functional inequality
for all with and all . Then there exist a unique -linear mapping and a constant such that
Let with . Then (3.2) implies that
for all with and all . Let and let Then Therefore it follows from (3.5) that
for all with and all . For the case let be an element of which is defined in the proof of Theorem 2.2. It is clear that Using (2.11) and (3.6), we get
for all with and all . Hence
for all and all , where
Letting and in (3.8), respectively, we get
for all and all . It follows from (3.8) and (3.10) that
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.
Let be nonzero positive real numbers. Assume that a mapping with satisfies
for all then is -linear.
It follows from (3.13) that there exists a sequence monotonically decreasing to zero, such that
for all with and all . Therefore
for all with and all . Applying (3.15) and Theorem 3.1, we obtain a sequence of unique -linear mappings satisfying
for all . Since the sequence is monotonically decreasing, we conclude
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
Let be given and let be nonzero real numbers. Assume that a mapping with satisfies the functional inequality
for all with and all . Then there exists a unique -linear mapping such that
for all with and .
Letting in (3.18), we get
for all with and all . Hence
for all with and all . It follows from (3.21) that
for all with and all . Adding (3.21) to (3.22), we get
for all with and all . Therefore
for all with . Let with . We may put in (3.24) to obtain
We can replace by in (3.25) for all nonnegative integers Then using a similar argument given in , we have
Hence we have the following inequality:
for all with and all integers Since is complete, (3.27) shows that the limit exists for all with . Letting and in (3.27), we obtain that satisfies inequality (3.19) for all with . It follows from the definition of and (3.24) that
for all with . Hence
for all with . We extend the additivity of to the whole space by using an extension method of Skof . Let and be given with Let be the smallest integer such that We define the mapping by
Let be given with and let be the smallest integer such that Then is the smallest integer satisfying If , we have and . Therefore For the case , it follows from the definition of that
From the definition of and (3.29), we get that holds true for all Let and let be an integer such that Then
It remains to prove that is -linear. Let and let be a positive integer such that Since for all and satisfies (3.28), we have
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.
Let be defined by
Consider the function by the formula
It is clear that is continuous, bounded by on and
for all (see ). It follows from (3.36) that the following inequality:
holds for all First we show that
for all If satisfies (3.38) for all then satisfies (3.38) for all To see this, let (the result is obvious when ). Then for all Replacing by , we get that for all Hence we may assume that If or then
Now suppose that Then there exists an integer such that
for all From the definition of and (3.40), we have
Therefore satisfies (3.38). Now we prove that
for all and all where
It follows from (3.37) and (3.38) that
for all and all Thus satisfies inequality (3.18) for Let be a linear functional such that
for all where is a positive constant. Then there exists a constant such that for all rational numbers . So we have
for all rational numbers . Let with If , then for all So
which contradicts (3.48).
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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.