A hyperstability result for the Cauchy equation

Authors

  • J. Brzdek Pedagogical University, Krakow, Poland

Keywords:

Hyperstability, Cauchy equation, additive function, restricted domain

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 E1 into a normed space E2, and for every real numbers r,s with r+s>0 one of the following two conditions must be valid: supx,y∈E1‖f(x+y)−f(x)−f(y)‖‖x‖r‖y‖s=∞,supx,y∈E1‖f(x+y)−f(x)−f(y)‖‖x‖r‖y‖s=0. In particular, we present a new method for proving stability for functional equations, based on a fixed point theorem. 10.1017/S0004972713000683

Author Biography

J. Brzdek, Pedagogical University, Krakow, Poland

Department of Mathematics

Published

2013-12-02

Issue

Vol. 89 No. 1 (2014)

Section

Articles
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