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Superstability and Stability of the Pexiderized Multiplicative Functional Equation
Journal of Inequalities and Applications volume 2010, Article number: 486325 (2010)
We obtain the superstability of the Pexiderized multiplicative functional equation and investigate the stability of this equation in the following form: .
The superstability of the functional equation was studied by Baker et al. . 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  as follows.
Theorem 1.1 (Baker ).
Let , be a semigroup and satisfying
for all . Put Then for all or else for all .
A different generalization of the result of Baker et al. was given by Székelyhidi . 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 ).
Let be a commutative group with identity and let be functions such that there exist functions with
for all . Then is bounded or is an exponential and .
In this paper, we prove the superstability of the Pexiderized multiplicative functional equation (PMFE)
That is, we prove that if are functional on a semigroup with identity satisfying and
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 .
2. Superstability of the PMFE
In this section, let be a semigroup with identity and a function with
for all and
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 .
Let and also ,
for all and for some . Let . Then satisfy the conditions (1.4), (2.1), (2.2) and
In particular, we know that , and
Let be a semigroup with identity . If are functions with for some satisfying and condition (1.4), that is,
for all and .
If we replace by and also by in (1.4), we get
Also we replace by in (1.4), then we have
for all . An induction argument implies that for all ,
Indeed, if inequality (2.9) holds, using inequality (1.4) and (2.8) we have
for all . By (2.9), we have
Thus we can easily show that from as and thus as . By (1.4),
and thus we have
for all . Then, by (2.2),
for all . Thus we have as . Since
as , we can define by
for all . Then
for all .
Let be a semigroup with identity and functions satisfying the inequality
for all . If , then is bounded or else is exponential and
Let be a semigroup with identity and functions satisfying condition (1.4), that is,
If satisfies that for some and for some then
for all and .
Let and for every and . Then
for all , and where . By Theorem 2.3, we complete the proof.
Let be a semigroup with identity . If are nonzero functions satisfying condition (1.4), that is,
then either is bounded, or else
for all and .
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 . 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  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 . Since then, the stability problems of various functional equations have been investigated by many authors (see [6, 8–18]).
Ger  suggested another type of stability for the exponential equation in the following type:
In this section, the stability problem for the Pexiderized multiplicative functional equation in the following form:
will be investigated.
Throughout this section, we denote by a commutative semigroup and by a function such that
for all . Also we let
for all Inequality (3.3) implies that
(d)for all for
The following functions satisfy condition (3.3) above.
(a) for every and
(b), for every .
Let and also ,
for all and for some . Let . Then satisfy condition (3.3) and
In particular, we know that if we let then
If are functions such that
for all , then there exists a function and there exists a constant such that for all and
for all . Moreover, if is bounded, then
for all and for some constant .
If we define functions by
for all , then equality (3.13) may be transformed into
for all . For the case of , the above inequality implies
for all . Putting instead of and instead of in (3.20), respectively, we get
Letting by and by in (3.20), we have
From (3.21), (3.22) and (3.23),
for all . Now replacing by and by , respectively, we have
for all . Replacing by and by in (3.21), (3.22), and (3.23), respectively, one obtains
for all . Also from (3.22) and (3.23), we have
for all . Thus we have
for all . For arbitrary positive integer , putting instead of in (3.24) and instead of in (3.28), respectively, we see that
for all . By (3.29) with ,
for all . By (3.30), for every positive integer with , we have
as . This proves that is a Cauchy sequence in . Thus we can define a function by
for all . Then, by (3.20) and (3.31), we have
for all . Thus
for all . Now replacing by and then by in (3.20), respectively, we obtain
for all . By (3.22), (3.23), and (3.36), we have
for all and for fixed . By (3.3) and (3.30),
for all . By (3.20), (3.38) and (3.39), for all with , there exits a constant such that
Now we define a function by
for all . Then
for all . By (3.40), we have
and thus for all
If is bounded, there exist constants such that
and by the same method above, we have
for all . Therefore, we have
for all .
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Lee, Y. Superstability and Stability of the Pexiderized Multiplicative Functional Equation. J Inequal Appl 2010, 486325 (2010) doi:10.1155/2010/486325
- Banach Space
- Positive Integer
- Vector Space
- Commutative Group
- Functional Equation