Superstability and Stability of the Pexiderized Multiplicative Functional Equation

Abstract

We obtain the superstability of the Pexiderized multiplicative functional equation and investigate the stability of this equation in the following form: .

1. Introduction

The superstability of the functional equation was studied by Baker et al. [1]. They proved that if is a functional on a real vector space satisfying for some fixed and all , then is either bounded or else for all . This result was genealized with a simplified proof by Baker [2] as follows.

Theorem 1.1 (Baker [2]).

Let , be a semigroup and satisfying

(1.1)

for all . Put Then for all or else for all .

A different generalization of the result of Baker et al. was given by Székelyhidi [3]. It involves an interesting generalization of the class of bounded functions on a group or semigroup and may be stated as follows.

Theorem 1.2 (Székelyhidi [3]).

Let be a commutative group with identity and let be functions such that there exist functions with

(1.2)

for all . Then is bounded or is an exponential and .

In this paper, we prove the superstability of the Pexiderized multiplicative functional equation (PMFE)

(1.3)

That is, we prove that if are functional on a semigroup with identity satisfying and

(1.4)

for all and for a function with some coditions, then is bounded or else is an exponential and . This is a generalization of the result of Székelyhidi. Also we investigate the stability of the Pexiderized multipicative functional equation (1.3) in the sense of Ger [4].

2. Superstability of the PMFE

In this section, let be a semigroup with identity and a function with

(2.1)

for all and

(2.2)

for all

Example 2.1.

The following functions satisfy conditions (2.1) and (2.2) above.

(a) for every and

(b), for every and is a functional on .

(c), for every .

(d), for every .

Example 2.2.

Let and also ,

(2.3)

for all and for some . Let . Then satisfy the conditions (1.4), (2.1), (2.2) and

(2.4)

In particular, we know that , and

Theorem 2.3.

Let be a semigroup with identity . If are functions with for some satisfying and condition (1.4), that is,

(2.5)

then

(2.6)

for all and .

Proof.

If we replace by and also by in (1.4), we get

(2.7)

Also we replace by in (1.4), then we have

(2.8)

for all . An induction argument implies that for all ,

(2.9)

Indeed, if inequality (2.9) holds, using inequality (1.4) and (2.8) we have

(2.10)

for all . By (2.9), we have

(2.11)

Thus we can easily show that from as and thus as . By (1.4),

(2.12)

and thus we have

(2.13)

for all . Then, by (2.2),

(2.14)

and so

(2.15)

for all . Thus we have as . Since

(2.16)

as , we can define by

(2.17)

for all . Then

(2.18)

for all .

Corollary 2.4.

Let be a semigroup with identity and functions satisfying the inequality

(2.19)

for all . If , then is bounded or else is exponential and

Theorem 2.5.

Let be a semigroup with identity and functions satisfying condition (1.4), that is,

(2.20)

If satisfies that for some and for some then

(2.21)

for all and .

Proof.

Let and for every and . Then

(2.22)

for all , and where . By Theorem 2.3, we complete the proof.

Corollary 2.6.

Let be a semigroup with identity . If are nonzero functions satisfying condition (1.4), that is,

(2.23)

then either is bounded, or else

(2.24)

for all and .

Proof.

Let for some . If is unbounded, then there exists such that . By Theorem 2.5, we complete the proof.

3. Stability of the PMFE

In 1940, Ulam gave a wide-ranging talk in the Mathematical Club of the University of Wisconsin in which he discussed a number of important unsolved problems [5]. One of those was the question concerning the stability of homomorphisms.

Let be a group and let be a metric group with a metric . Given , does there exist a such that if a mapping satisfies the inequality for all , then there exists a homomorphism with for all ?

In the next year, Hyers [6] answered the Ulam's question for the case of the additive mapping on the Banach spaces . Thereafter, the result of Hyers has been generalized by Rassias [7]. Since then, the stability problems of various functional equations have been investigated by many authors (see [6, 8–18]).

Ger [4] suggested another type of stability for the exponential equation in the following type:

(3.1)

In this section, the stability problem for the Pexiderized multiplicative functional equation in the following form:

(3.2)

will be investigated.

Throughout this section, we denote by a commutative semigroup and by a function such that

(3.3)

for all . Also we let

(3.4)

for all Inequality (3.3) implies that

(a)for all

(3.5)

(b)for all

(3.6)

(c)for all

(3.7)

(d)for all for

(3.8)

because

(3.9)

Example 3.1.

The following functions satisfy condition (3.3) above.

(a) for every and

(b), for every .

Example 3.2.

Let and also ,

(3.10)

for all and for some . Let . Then satisfy condition (3.3) and

(3.11)

In particular, we know that if we let then

(3.12)

Theorem 3.3.

If are functions such that

(3.13)

for all , then there exists a function and there exists a constant such that for all and

(3.14)

for all . Moreover, if is bounded, then

(3.15)

for all and for some constant .

Proof.

If we define functions by

(3.16)

for all , then equality (3.13) may be transformed into

(3.17)

and thus

(3.18)

for all . For the case of , the above inequality implies

(3.19)

and so

(3.20)

for all . Putting instead of and instead of in (3.20), respectively, we get

(3.21)

Letting by and by in (3.20), we have

(3.22)

and also

(3.23)

From (3.21), (3.22) and (3.23),

(3.24)

for all . Now replacing by and by , respectively, we have

(3.25)

for all . Replacing by and by in (3.21), (3.22), and (3.23), respectively, one obtains

(3.26)

for all . Also from (3.22) and (3.23), we have

(3.27)

for all . Thus we have

(3.28)

for all . For arbitrary positive integer , putting instead of in (3.24) and instead of in (3.28), respectively, we see that

(3.29)

for all . By (3.29) with ,

(3.30)

for all . By (3.30), for every positive integer with , we have

(3.31)

as . This proves that is a Cauchy sequence in . Thus we can define a function by

(3.32)

for all . Then, by (3.20) and (3.31), we have

(3.33)

for all . Thus

(3.34)

for all . Now replacing by and then by in (3.20), respectively, we obtain

(3.35)

and so

(3.36)

for all . By (3.22), (3.23), and (3.36), we have

(3.37)

and thus

(3.38)

for all and for fixed . By (3.3) and (3.30),

(3.39)

for all . By (3.20), (3.38) and (3.39), for all with , there exits a constant such that

(3.40)

Now we define a function by

(3.41)

for all . Then

(3.42)

for all . By (3.40), we have

(3.43)

and thus for all

(3.44)

If is bounded, there exist constants such that

(3.45)

and so

(3.46)

and by the same method above, we have

(3.47)

for all . Therefore, we have

(3.48)

for all .