Stability of a conditional Cauchy equation

Abstract

Let ${\mathbb R}$ be the set of real numbers, ${f : \mathbb {R} \to \mathbb {R}}$,  ${\epsilon \ge 0}$ and d > 0. We denote by {(x ₁, y ₁), (x ₂, y ₂), (x ₃, y ₃), . . .} a countable dense subset of ${\mathbb {R}^2}$ and let

$$U_d:=\bigcup\nolimits_{j=1}^{\infty} \{(x, y)\in \mathbb {R}^2:\,|x|+|y| > d,\, |x-x_j| < 1,\, |y-y_j| < 2^{-j}\}.$$

We consider the Hyers-Ulam stability of the conditional Cauchy functional inequality

$$|f(x+y)-f(x)-f(y)|\le \epsilon$$

for all ${(x, y) \in U_d}$ .