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

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 $$\frac{f(x+y)} {f(x)f(y)}= [\phi(x)\phi(y)]^{B}.$$