A partial positive answer to a Lijun-Yun conjecture
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.
scientific article; zbMATH DE number 7189725
scientific article