#### Abstract

We give an Ulam type stability result for the following functional equation: under a suitable condition. We also give a concrete stability result for the case taking up as a control function.

#### 1. Introduction

In 1940, Ulam [1] proposed the following stability problem: “When is it true that a function which satisfies some functional equation *approximately* must be close to one satisfying the equation *exactly*?” Next year, Hyers [2] gave an answer to this problem for additive mappings between Banach spaces. Furthermore, Aoki [3] and Rassias [4] obtained independently generalized results of Hyers’ theorem which allow the Cauchy difference to be unbounded.

Let and be normed spaces over , which denotes either the real field or the complex field . Throughout the paper, we fix scalars and vectors and . We say that a mapping of into is -additive if for all . When , we say it to be -additive. Aczél [5] specified what this generalized Cauchy equation is. The Ulam type stability problem for such an has been investigated in [6–8]. However, these results have been obtained in cases where either or (see Theorems A and B). In this paper, we will investigate the problem for -additive mappings, that is, in the case . In Section 2, we state the details of -additive mappings (Theorem 3). In Section 3, we give our main results about the stability for them (see Theorems 7–10). In the final section, we apply the results to some concrete examples, where we take up as a control function (see Corollaries 11–14).

#### 2. -Additive Mappings

The following result asserts that any -additive mapping is transformed into some -additive mapping by a certain translation and that any -additive mapping is an additive mapping in usual sense with some extra condition.

Proposition 1. *Let and be two mappings of into such that for all . Then the following three statements are equivalent: *(i) *is *-*additive*, (ii) *is *-*additive*, (iii) *is additive and * *for all *.

*Proof. *(i)(ii) Since
for all , it follows that -additive -additive.

(ii)(iii) Suppose that
for all . When , we have . Using this, and also for all . Therefore,
for all .

(iii)(ii) Because for all (see also the following remark), it is trivial.

*Remark 2. *We denote by the field of all rational numbers. It is well known that if is additive, then for every and , that is, is -linear. Hence, if is additive and continuous, must be -linear. On the other hand, when , we have a lot of continuous additive nonlinear mappings by considering the composition of linear transformations on and the -linear isometry .

The constant is a trivial -additive mapping of into . The following theorem says that unless it is a unique -additive mapping, discontinuous one always exists.

Theorem 3. *
(I) If , then there exists a discontinuous -additive mapping of into .**
(II) If , then the followings hold: *(i)*If both and are transcendental numbers, then there exists a discontinuous -additive mapping of into . *(ii)*If one of and is transcendental and the other is algebraic, then the constant is a unique -additive mapping of into . *(iii)*If both and are algebraic with a common minimal polynomial, then there exists a discontinuous -additive mapping of into . *(iv)*If and are algebraic with distinct minimal polynomials, then the constant is a unique -additive mapping of into . ** Moreover, when , there is no nontrivial continuous -additive mapping of into . On the other hand, when , if is not real and (the complex conjugate of ), then the mapping must be conjugate linear for every continuous -additive mapping of into . *

In order to show Theorem 3, we need some lemmas for -valued -additive functions defined on . For any , we denote by the subfield of generated by over .

By the following proofs of Lemma 6 and Theorem 3, if there is a discontinuous -additive mapping, then there are sufficiently many such mappings in the sense that there exists such a mapping which separates any -linear independent points of .

Lemma 4. *Any -additive of into itself satisfies for all and .*

*Proof. *Note that is additive and for each by Proposition 1. Let with . Since is -linear,
for all .

Lemma 5. *Let and be subfields of and an isomorphism of onto . Then, has an additive bijective extension to the full space such that
**
for all and . Moreover, one has a discontinuous one whenever .*

*Proof. *Let and be an -linear base and an -linear base of , respectively. Because both of them have same cardinality, we take . Moreover, we may assume without loss of generality that for some . For any , there exist a finite and such that
and this decomposition is unique. Hence, we can define by
This is a desired extension.

In order to get a discontinuous extension, we consider the base and such that for some . Because , take a rational sequence converging to . Suppose that is continuous. Then,
as . This is contradiction. Thus is discontinuous.

Lemma 6. *Suppose that . *(i)*If both and are transcendental numbers, then there exists a discontinuous -additive function of into itself. *(ii)*If one of and is transcendental and the other is algebraic, then the constant is a unique -additive function of into itself. *(iii)*If both and are algebraic with a common minimal polynomial, then there exists a discontinuous -additive function of into itself. *(iv)*If and are algebraic with distinct minimal polynomials, then the constant is a unique -additive function of into itself. ** Moreover, when , every nontrivial -additive function of into itself is discontinuous. On the other hand, when , if is not real and , then any continuous -additive function of into itself must be of form for some .*

*Proof. *(i) Suppose that both and are transcendental. Then, (resp., ) is isomorphic to the rational function field in indeterminate (resp., ). So, the substitution induces an isomorphism of fields. By Lemma 5, because , has a discontinuous additive extension to such that for every and . Then, for all , and, hence, is -additive by Proposition 1.

(ii) Let be any -additive function. If is transcendental and is algebraic with nonzero polynomial such that , then from Lemma 4, we have
for all . If is transcendental and is algebraic with nonzero polynomial such that , then from Lemma 4, we have
for all .

(iii) If is algebraic with minimal polynomial , then consists of all polynomials in of degree up to . So, if is also algebraic with the same minimal polynomial , then the substitution induces an automorphism . As same as (i), has a discontinuous -additive extension.

(iv) Suppose that and are algebraic with distinct minimal polynomials and over the field , respectively. Let be any -additive function. To show , we assume, on the contrary, that there is an with . Then, from Lemma 5, we have , and hence . This contradicts the prerequisite for and . Hence, must be zero.

When , since every continuous additive function is -linear and , there is no continuous -additive function by Proposition 1. Now, we consider the case . Let be a nontrivial continuous -additive function. Note that is -linear. If is not real and , we can easily see that for all . Thus, is conjugate linear, and hence , where .

*Proof of Theorem 3. *(I) We assume without loss of generality that with the help of Proposition 1. Given an -additive function , take a with and a nonzero functional in , the dual space of . Put
Then, we can easily see that is an -additive mapping of into . Also, if is discontinuous, then so is . In fact, if is discontinuous, we can find a sequence in such that and (). Choose an in with and put for each . Then, as and , so is discontinuous, as required. Therefore Theorem 3(I), (II)-(i), and (II)-(iii) follow easily from Lemmas 4 and 6.

Given an -additive mapping of into , take and arbitrarily, and put for each . Then, is -additive. If for each and , then by the Hahn-Banach theorem. Therefore, Theorem 3(II)-(ii) and (II)-(iv) follow easily from Lemma 6. The final assertion in (II) also follows from Lemma 6 and its proof.

#### 3. A Stability of Generalized Additive Mappings

In this section, we consider a couple of cases which are left out in [8] about the Ulam type stability. We take a nonnegative function (say a *control function*) on and also a certain nonnegative function on which depends on . We say that a system of all -additive mappings is *strictly *-*stable* whenever the following statement is true:

“*If a mapping ** of ** into ** satisfies**for all **, then there exists a unique **-additive mapping ** such that**for all **.” *

Throughout the remainder of this paper, we assume that is a Banach space. This is because all of our results depend on the following theorems whose proofs need the Banach fixed point theorem.

Theorem A (see [8, Theorem 3.1]). * Let and with . One takes a control function which satisfies
**
for all and puts
**
for each .**Then, the strict -stability holds for the system of -additive mappings.*

Theorem B (see [8, Theorem 3.2]). *Let and with . One takes a control function which satisfies
**
for all and puts
**
for each .**Then, the strict -stability holds for the system of -additive mappings.*

Both of these theorems do not say about -additive mappings at all; however, we will get the following stability theorems for them. Theorem 7 is of the case , Theorem 8 is of the case , and Theorems 9 and 10 are for -additive mappings. These cover all of the systems of -additive mappings.

Theorem 7. *Let and with . One takes a control function which satisfies
**
for all , and puts
**
for each .**Then, the strict -stability holds for the system of -additive mappings.*

*Proof. *Put and for each . Then, changes into
where
for all . By denoting , , , , and in (19), changes up to the following estimate of by the control function :
for all .

Under these transformations, , and are equipped with
for all . The latter follows from the inequality in which must satisfy because by using (20), we get
for all . Since (21) and (22) hold, it follows from Theorem that there exists a unique -additive mapping such that
for all . However, we can easily see the following two assertions: (i) is -additive if and only if is -additive, (ii)(24) is equivalent to .

This completes the proof.

Theorem 8. *Let and with . One takes a control function which satisfies
**
for all and puts
**
for each .**Then the strict -stability holds for the system of -additive mappings.*

*Proof. *We consider the same transformations and the same estimate (21) of by in the proof of Theorem 7. Under these transformations, we have
Moreover, for every we have
because
Since (21), (27) and (28) hold, it follows from Theorem that there exists a unique -additive mapping such that
for all . This means and is -additive.

Theorem 9. *Let . One takes a control function which satisfies
**
for all and puts
**
for each .**Then the strict -stability holds for the system of -additive mappings.*

*Proof. *Put for each . Then, changes into the following estimate of by the control function :
where
for all . Put .

Under these transformations, , and are equipped with
for all . The latter follows from the following inequality:
for all . Since (34), (36) hold, it follows from Theorem that there exists a unique -additive mapping such that
for all . However we can easily see the following two assertions: (i) is -additive if and only if is -additive; (ii)(38) is equivalent to .

This completes the proof.

Theorem 10. *Let . One takes a control function which satisfies
**
for all and puts
**
for each .**Then the strict -stability holds for the system of -additive mappings.*

*Proof. *We consider the same transformation and the same estimate (34) of by in the proof of Theorem 9. Under these transformations, and are equipped with
for all . So it follows that
for all . Since (34), (41) and (43) hold, it follows from Theorem that there exists a unique -additive mapping such that
for all . This means and is -additive.

#### 4. Concrete Examples

Throughout this section, let . We fix nonnegative constants and and take the control function defined by for every .

Corollary 11. *When and , one puts
**
for each .**Then, the system of -additive mappings is strictly -stable.*

*Proof. *Put . Then and
for all . By Theorem 7, for a mapping of into satisfying () for and , there exists a unique -additive mapping such that
for all . Because of , we have the corollary.

Corollary 12. *When and , one puts
**
for each .**Then the system of -additive mappings is strictly -stable.*

*Proof. *Put . Then and
for all . By Theorem 8, for a mapping of into satisfying (†) for and , there exists a unique -additive mapping such that
for all . Because of , we have the corollary.

Corollary 13. *When , one puts
**
for each .**Then, the system of -additive mappings is strictly -stable.*

*Proof. *Put . Then,
for all . Since , we also have . By Theorem 9, for a mapping of into satisfying () for and , there exists a unique -additive mapping such that
for all . Because of , we have the corollary.

Corollary 14. *When , one puts
**
for each .**Then, the system of -additive mappings is strictly -stable. *

*Proof. *Put . Then,
for all . Since , we also have . By Theorem 10, for a mapping of into satisfying () for and , there exists a unique -additive mapping such that
for all . Because of , we have the corollary.

*Remark 15. *In Corollary 14, taking , we can easily observe that the corollary is just the stability result due to Hyers [2, Theorem 1].

#### Acknowledgment

The third and fifth authors are partially supported by Grant-in-Aid for Scientific Research, Japan Society for the Promotion of Science.