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A Counterexample to Matkowski's Conjecture for Quasi Graph-Additive Functions

Tibor Kiss

Abstract

In this paper we investigate a conjecture of Janusz Matkowski concerning the continuous solutions of the functional equation \[ f\big(f(-x)+x\big)=f\big(-f(x)\big)+f(x),\qquad x\in\mathbb{R}. \] Matkowski conjectured that all continuous solutions must necessarily be linear on both the negative and the positive half-line. We show, however, that the family of continuous solutions to the equation in question is far richer than anticipated: there exist continuous solutions that admit an arbitrary part. In addition, we provide a sufficient condition which, in the continuous setting, enforces the conclusion predicted by Matkowski's Conjecture.

A Counterexample to Matkowski's Conjecture for Quasi Graph-Additive Functions

Abstract

In this paper we investigate a conjecture of Janusz Matkowski concerning the continuous solutions of the functional equation Matkowski conjectured that all continuous solutions must necessarily be linear on both the negative and the positive half-line. We show, however, that the family of continuous solutions to the equation in question is far richer than anticipated: there exist continuous solutions that admit an arbitrary part. In addition, we provide a sufficient condition which, in the continuous setting, enforces the conclusion predicted by Matkowski's Conjecture.
Paper Structure (2 sections, 7 theorems, 27 equations)

This paper contains 2 sections, 7 theorems, 27 equations.

Table of Contents

  1. Introduction
  2. Solutions of (\ref{['Ekadd']}) with non-linear components

Key Result

Theorem 1

Let $F:\mathbb{R}\times\mathbb{R}\to\mathbb{R}$ be translative and $f:\mathbb{R}\to\mathbb{R}$ be defined as $f(x):=F(x,0)$. The binary operation $F$ is weakly associative if and only if is satisfied for all $x\in\mathbb{R}$.

Theorems & Definitions (14)

  • Theorem : J. Matkowski, Mat25
  • Theorem : W. Jarczyk, Jar88
  • Conjecture : J. Matkowski, Remark 8. and Conjecture 2. of Mat25
  • Proposition 1
  • proof
  • Corollary 1
  • proof
  • Proposition 2
  • proof
  • Remark
  • ...and 4 more