Three remarks on a question of Acz\'el

TL;DR

This paper investigates the monotonic, continuous, and differentiable solutions of a family of functional equations posed by Aczél, revealing parameter-dependent solution behaviors and explicitly characterizing solutions.

Contribution

It provides a complete analysis of solutions for different parameter values, including explicit forms for continuous and differentiable solutions, addressing a question posed by Aczél.

Findings

Monotonic solutions match expectations for certain parameter values.

For other parameter values, monotonic solutions differ from expectations.

Explicit continuous and differentiable solutions are derived.

Abstract

In the 49th International Symposium on Functional Equations, J. Acz\'el asked for the monotonic solutions of a certain one-parameter family of functional equations. In this short note we find that for a certain value of the parameter the monotonic solutions are the expected ones, and that for other values this is not so. Further, we find all continuous and continuously differentiable solutions to the equations.