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Journal of Convex Analysis 26 (2019), No. 2, 543--562
Copyright Heldermann Verlag 2019

Asymptotic Hyers-Ulam Stability or Superstability for Generalized Linear Equations by Unilateral Perturbations

Cathérine Peppo
Itescia, 8 rue P. de Coubertin, 95300 Pontoise, France


In relation to the famous problem of Ulam ``Give conditions in order for a linear mapping near an approximated linear mapping to exist'', we consider the stability or superstability of generalized linear equation \begin{center} $f(x+y)-f(x)-f(y)=B[\phi(x)+\phi(y)]$ \end{center} by left or right perturbations with some hypotheses of convexity or concavity, and -- in a forthcoming paper -- apply our conclusions to the generalized exponential equation $$ \frac{f(x+y)} {f(x)f(y)}= [\phi(x)\phi(y)]^{B}. $$

Keywords: Hyers-Ulam stability, superstability, asymptotic stability, linear equation, exponential equation.

MSC: 39B62, 26A51

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