Hyperstability of the general linear equation on restricted domains (Q1677510)

The authors investigate the hyperstability of the generalized Cauchy functional equation \[ f(ax+by)= Af(x)+Bf(y)+C, \] \smallskip for functions \(f:X\to Y\) where \(X\) and \(Y\) are real or complex linear space, \(C\in Y\), \(a,b,A,B\) are complex numbers and \(ab\neq 0\). They prove, under some assumptions, that the function satisfying the generalized Cauchy equation approximately on a restricted domain, must be a solution.