A partial positive answer to a Lijun-Yun conjecture
type: Item
instance of: scholarly article
title: A partial positive answer to a Lijun-Yun conjecture
zbMATH Open document ID: 1478.34032
author: Abílio Lemos
author: Alexandre Miranda Alves
author: Anderson De Araújo
published in: Bollettino dell'Unione Matematica Italiana
publication date: 2020-04-15T00:00:00Z
review text: In the paper of \textit{L. Yang} and \textit{Y. Tang} [J. Math. Anal. Appl. 261, No. 1, 100--112 (2001; Zbl 0995.34031)] the authors conjectured that the Abel differential equation \[ x'=f(t) x^3+g(t)x^2,\quad t\in[-1,1], \] where \(f\) is a polynomial and \(g(t)=2t,\) has a center at \(x=0\) if and only if \(f\) is odd. Remember that it is said that the above equation has a center at \(x=0\) if all the solutions in a neighbourhood of \(x=0\) satisfy \(x(-1)=x(1).\) The authors of the reviewed paper prove that for general analytic functions \(f\) and \(g\) a necessary condition for the origin to be a center is that the following three moments vanish \[ \int_{-1}^1 f(t)(G(t))^k\,dt=0,\quad k=0,1,2, \] where \(G(t)=\int_{-1}^t g(s)\,ds.\) They use this result to prove similar results to the one stated in the conjecture for some particular functions \(f\) and \(g.\) As far as the reviewer knows the conjecture remains open.
reviewed by: Armengol Gasull
Mathematics Subject Classification ID: 34C05
Mathematics Subject Classification ID: 34A34
Mathematics Subject Classification ID: 44A60
Mathematics Subject Classification ID: 37C60
zbMATH DE Number: 7189725
zbMATH Keywords: Abel differential equation
zbMATH Keywords: moment conditions
zbMATH Keywords: center
MaRDI profile type: Publication
Wikidata QID: Q123021938
full work available at URL: s40574-019-00203-x
OpenAlex ID: W2945404991
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DOI: 10.1007/S40574-019-00203-X
description: scientific article; zbMATH DE number 7189725