A hyperstability result for the Cauchy equation

Abstract

We prove a hyperstability result for the Cauchy functional equation $f(x+y)=f(x)+f(y)$, which complements some earlier stability outcomes of J.M. Rassias. As a consequence, we obtain the slightly surprising corollary that for every function $f$, mapping a normed space $E_1$ into a normed space $E_2$, and for every real numbers $r,s$ with $r+s>0$ one of the following two conditions must be valid: \begin{align*} \sup_{x,y\in E_1} \|f(x+y)-f(x)-f(y)\| \|x\|^r \|y\|^s=\infty,\\ \sup_{x,y\in E_1} \|f(x+y)-f(x)-f(y)\|\, \|x\|^r \,\|y\|^s=0. \end{align*} In particular, we present a new method for proving stability for functional equations, based on a fixed point theorem. 10.1017/S000497271300068