Regular solutions of a functional equation derived from the invariance problem of Matkowski means

TL;DR

This paper characterizes regular solutions to a functional equation linked to the invariance of generalized weighted quasi-arithmetic means, focusing on differentiable functions and exploring solutions beyond regularity.

Contribution

It provides a comprehensive solution to a functional equation from the invariance problem of Matkowski means, including regular and some irregular solutions.

Findings

Solutions when F is affine and g1, g2 are strictly monotone

Solutions when g1, g2 are invertible affine functions with common additive part

Extension to non-regular solutions

Abstract

The main result of the present paper is about the solutions of the functional equation \Eq{*}{ F\Big(\frac{x+y}2\Big)+f_1(x)+f_2(y)=G(g_1(x)+g_2(y)),\qquad x,y\in I, } derived originally, in a natural way, from the invariance problem of generalized weighted quasi-arithmetic means, where $F,f_1,f_2,g_1,g_2:I\to\mathbb{R}$ and $G:g_1(I)+g_2(I)\to\mathbb{R}$ are the unknown functions assumed to be continuously differentiable with $0\notin g'_1(I)\cup g'_2(I)$, and the set $I$ stands for a nonempty open subinterval of $\mathbb{R}$. In addition to these, we will also touch upon solutions not necessarily regular. More precisely, we are going to solve the above equation assuming first that $F$ is affine on $I$ and $g_1$ and $g_2$ are continuous functions strictly monotone in the same sense, and secondly that $g_1$ and $g_2$ are invertible affine functions with a common additive part.