Asymptotic stability of the Cauchy and Jensen functional equations

TL;DR

This paper studies the asymptotic stability of the Cauchy and Jensen functional equations, showing that approximate solutions for large arguments imply near solutions everywhere, with implications for hyperstability.

Contribution

It establishes new results on the asymptotic stability and hyperstability of the Cauchy and Jensen functional equations under approximate conditions.

Findings

Approximate solutions at large arguments extend to near solutions everywhere.

The error term in solutions can be bounded by a constant multiple of the original error.

Results demonstrate hyperstability properties of these equations.

Abstract

The aim of this note is to investigate the asymptotic stability behaviour of the Cauchy and Jensen functional equations. Our main results show that if these equations hold for large arguments with small error, then they are also valid everywhere with a new error term which is a constant multiple of the original error term. As consequences, we also obtain results of hyperstability character for these two functional equations.