Fixed points and the stability of the linear functional equations in a single variable

TL;DR

This paper demonstrates that a 2014 stability result for a linear functional equation is a special case of a fixed point theorem from 2012, providing new insights into function approximation within this context.

Contribution

It shows that a known stability result is a specific instance of a more general fixed point theorem and offers a characterization of approximable functions.

Findings

The 2014 stability result is a particular case of a 2012 fixed point theorem.

Provides a characterization of functions approximable by solutions to the linear functional equation.

Establishes a connection between stability results and fixed point theory.

Abstract

We prove that an interesting result concerning generalized Hyers-Ulam-Rassias stability of a linear functional equation obtained in 2014 by S.M. Jung, D. Popa and M.T. Rassias in Journal of Global Optimization is a particular case of a fixed point theorem given by us in 2012. Moreover, we give a characterization of functions that can be approximated with a given error, by the solution of the previously mention linear equation.