### 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/S0004972713000683

\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/S0004972713000683

#### Keywords

Hyperstability, Cauchy equation, additive function, restricted domain

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Bulletin of the Aust. Math. Soc., copyright Australian Mathematical Publishing Association Inc.