The stability of the exponential equation

Authors:
Roman Ger and Peter Semrl

Journal:
Proc. Amer. Math. Soc. **124** (1996), 779-787

MSC (1991):
Primary 39B72

DOI:
https://doi.org/10.1090/S0002-9939-96-03031-6

MathSciNet review:
1291769

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Abstract: We generalize the well-known Baker's superstability result for exponential mappings with values in the field of complex numbers to the case of an arbitrary commutative complex semisimple Banach algebra. It was shown by Ger that the superstability phenomenon disappears if we formulate the stability question for exponential complex-valued functions in a more natural way. We improve his result by showing that the maximal possible distance of an -approximately exponential function to the set of all exponential functions tends to zero as tends to zero. In order to get this result we have to prove a stability theorem for real-valued functions additive modulo the set of all integers .

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Additional Information

**Roman Ger**

Affiliation:
Institute of Mathematics, Silesian University, Bankowa 14, 40-007 Katowice, Poland

**Peter Semrl**

Affiliation:
TF, University of Maribor, Smetanova 17, P.O. BOX 224, 62000 Maribor, Slovenia

DOI:
https://doi.org/10.1090/S0002-9939-96-03031-6

Keywords:
Exponential functions,
congruentialy additive functions,
stability

Received by editor(s):
February 1, 1994

Received by editor(s) in revised form:
July 24, 1994

Communicated by:
J. Marshall Ash

Article copyright:
© Copyright 1996
American Mathematical Society