A Functional Inequality in Restricted Domains of Banach Modules

Abstract

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 [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 satisfiesHyers inequality

(11)

for all .

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 Gvruta [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:

(12)

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.

Let and be left -modules and let be nonzero real numbers. If a mapping with satisfies the functional inequality

(21)

for all and all , then is -linear.

Proof.

Letting in (2.1) we get

(22)

for all and all . Letting (resp., ) in (2.2), we get

(23)

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 [29], there exist elements such that . Since is additive and for all all rational numbers and all , we have

(24)

for all . Replacing instead of in the above equation, we have

(25)

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.

Letting with in (2.1), we get

(26)

for all . Let and let Then

(27)

Therefore replacing and by and in (2.6), respectively, we get

(28)

for all with and all .

Similar to the proof of Theorem  3 of [30] (see also [31]), we prove that satisfies (2.8) for all and all . Suppose If , let with otherwise

(29)

Since it is easy to verify that

(210)

Therefore

(211)

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

(31)

for all

Theorem 3.1.

Let and be given. Assume that a mapping satisfies the functional inequality

(32)

for all with and all . Then there exist a unique -linear mapping and a constant such that

(33)

for all

Proof.

Let with . Then (3.2) implies that

(34)

Thus

(35)

for all with and all . Let and let Then Therefore it follows from (3.5) that

(36)

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

(37)

for all with and all . Hence

(38)

for all and all , where

(39)

Letting and in (3.8), respectively, we get

(310)

for all and all . It follows from (3.8) and (3.10) that

(311)

for all . By the results of Hyers [2] and Rassias [4], there exists a unique additive mapping given by such that

(312)

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.

Let be nonzero positive real numbers. Assume that a mapping with satisfies

(313)

for all then is -linear.

Proof.

It follows from (3.13) that there exists a sequence monotonically decreasing to zero, such that

(314)

for all with and all . Therefore

(315)

for all with and all . Applying (3.15) and Theorem 3.1, we obtain a sequence of unique -linear mappings satisfying

(316)

for all . Since the sequence is monotonically decreasing, we conclude

(317)

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.

Let be given and let be nonzero real numbers. Assume that a mapping with satisfies the functional inequality

(318)

for all with and all . Then there exists a unique -linear mapping such that

(319)

for all with and .

Proof.

Letting in (3.18), we get

(320)

for all with and all . Hence

(321)

for all with and all . It follows from (3.21) that

(322)

for all with and all . Adding (3.21) to (3.22), we get

(323)

for all with and all . Therefore

(324)

for all with . Let with . We may put in (3.24) to obtain

(325)

We can replace by in (3.25) for all nonnegative integers Then using a similar argument given in [4], we have

(326)

Hence we have the following inequality:

(327)

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

(328)

for all with . Hence

(329)

for all with . We extend the additivity of to the whole space by using an extension method of Skof [34]. Let and be given with Let be the smallest integer such that We define the mapping by

(330)

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

(331)

From the definition of and (3.29), we get that holds true for all Let and let be an integer such that Then

(332)

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

(333)

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.

Let be defined by

(334)

Consider the function by the formula

(335)

It is clear that is continuous, bounded by on and

(336)

for all (see [35]). It follows from (3.36) that the following inequality:

(337)

holds for all First we show that

(338)

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

(339)

Now suppose that Then there exists an integer such that

(340)

Therefore

(341)

Hence

(342)

for all From the definition of and (3.40), we have

(343)

Therefore satisfies (3.38). Now we prove that

(344)

for all and all where

(345)

It follows from (3.37) and (3.38) that

(346)

for all and all Thus satisfies inequality (3.18) for Let be a linear functional such that

(347)

for all where is a positive constant. Then there exists a constant such that for all rational numbers . So we have

(348)

for all rational numbers . Let with If , then for all So

(349)

which contradicts (3.48).