Abstract

Consider the differential equation $dx/dt=a\_$_n(t)xⁿ+a_n-1(t)x^n-1+...+a₁(t) x+a₀(t), where $x\in \; R$, $t\in [0,1]$ and $a$₀,a₁,...,a_n are real smooth 1-periodic functions. This paper studies some particular cases of the above equation and gives criteria to give an upper bound of the number or periodic solutions (i.e. solutions satisfying $x(0)=x(1)$) that they can have. Our results extend known results for Abel equations $(n=3)$ and can be applied to control the number of limit cycles of some planar ordinary differential equations.

This is a joint paper with A. Guillamon.