Abstract
We show that a very classical result, proved by T. Aoki, Z. Gajda and Th. M. Rassias and concerning the Hyers–Ulam stability of the Cauchy equation f(x+y)=f(x)+f(y), can be significantly improved. We also provide some immediate applications of it (among others for the cocycle equation, which is useful in characterizations of information measures). In particular, we give a solution to a problem that was formulated more than 20 years ago and concerned optimality of some estimations.
The proof of that result is based on a fixed point theorem.
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Brzdȩk, J. Hyperstability of the Cauchy equation on restricted domains. Acta Math Hung 141, 58–67 (2013). https://doi.org/10.1007/s10474-013-0302-3
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DOI: https://doi.org/10.1007/s10474-013-0302-3