Abstract

In this work we study the equation $x'=s(t,x)=sum_0^m a_i(t)x^i$. We give some criteria to estimate the number of limit cycles of the equation. Different from the fixed sign hypothesis for some coefficients $a_i(t)$ in the classical results, our criteria are only concerned with $S(t; x)$ on some non-intersecting curves. Applying these criteria, we obtain some new results on the limit cycles of the trigonometrial generalized Abel equation and the planar polynomial system with homogeneous nonlinearities.