Superstability and Stability of the Pexiderized Multiplicative Functional Equation
© YoungWhan Lee. 2010
Received: 14 August 2009
Accepted: 28 December 2009
Published: 12 January 2010
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 ).
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 ).
In this paper, we prove the superstability of the Pexiderized multiplicative functional equation (PMFE)
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
The following functions satisfy conditions (2.1) and (2.2) above.
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.
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.
The following functions satisfy condition (3.3) above.
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