- Research Article
- Open Access
Superstability and Stability of the Pexiderized Multiplicative Functional Equation
© YoungWhan Lee. 2010
- Received: 14 August 2009
- Accepted: 28 December 2009
- Published: 12 January 2010
We obtain the superstability of the Pexiderized multiplicative functional equation and investigate the stability of this equation in the following form: .
- Banach Space
- Positive Integer
- Vector Space
- Commutative Group
- Functional Equation
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 ).
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 ).
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 .
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 .
In particular, we know that , and
for all and .
for all .
for all . If , then is bounded or else is exponential and
for all and .
for all , and where . By Theorem 2.3, we complete the proof.
for all and .
Let for some . If is unbounded, then there exists such that . By Theorem 2.5, we complete the proof.
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
The following functions satisfy condition (3.3) above.
(a) for every and
(b) , for every .
for all and for some constant .
for all .
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