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Hyperstability of the Cauchy equation on restricted domains

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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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Authors and Affiliations

  1. Department of Mathematics, Pedagogical University, Podchora̧żych 2, 30–084, Kraków, Poland

    Janusz Brzdȩk

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  1. Janusz Brzdȩk
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Correspondence to Janusz Brzdȩk.

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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

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