On periodic solutions of nonlinear differential equations with singularities

Abstract

Necessary and sufficient conditions for existence of periodic solutions of differential equations containing singularities are given. Our theorems apply to u + 1 / u α = h ( t ) ≡ h ( t + T ) u + 1/{u^\alpha } = h\left ( t \right ) \equiv h\left ( {t + T} \right ) for all α > 0 \alpha > 0 and to u − 1 / u α = h ( t ) u - 1/{u^\alpha } = h\left ( t \right ) if α ≥ 1 \alpha \geq 1 , and for this case α ≥ 1 \alpha \geq 1 is an essential condition.