A conjecture of Gy. Petruska
type: Item
instance of: scholarly article
title: A conjecture of Gy. Petruska
zbMATH Open document ID: 0707.39004
DOI: 10.1007/BF01833154
published in: Aequationes Mathematicae
publication date: 1990-01-01T00:00:00Z
full work available at URL: https://eudml.org/doc/137355
review text: The paper deals with the solution of the integral equation \(f(x)=(1/\phi (x))\int^{\phi (x)}_{0}f(t)dt\quad (0<x<1).\) A conjecture by \textit{Gy. Petruska} [Aequationes Math. 38, 66-72 (1989; Zbl 0679.39007)] states that if \(0<\phi (x)<1\) is a continuous and increasing function in (0,1) with \(\lim_{x\to 0}\phi (x)/x=0\), then the equation admits a nonconstant bounded solution. In the paper the conjecture is proved not to be consistent; it is shown that every bounded solution of the given equation must be constant, provided \(1\leq f(x)\leq 2\) for all \(0<x<1\).
Mathematics Subject Classification ID: 39B05
Mathematics Subject Classification ID: 45A05
zbMATH DE Number: 4161335
zbMATH Keywords: integro-functional equation
zbMATH Keywords: nonconstant bounded solution
Wikidata QID: Q123230683
author: Vilmos Totik
reviewed by: Vinicio C. Boffi
MaRDI profile type: Publication
cites work: On a functional equation with retarded mean
description: scientific article; zbMATH DE number 4161335