No abstract

In this paper, the problem of positive periodic solutions is studied for the Liénard equation with a singularity of repulsive type,where f : (0, +∞) → R is continuous, α, h are continuous with T-periodic and α(t) ≥ 0 for all t ∈ R, μ ∈ (0, +∞) is a constant. By means of a Manásevich-Mawhin's continuation theorem, a sufficient and necessary condition is obtained for the existence of positive T-periodic solutions of the equation. The interesting point is that the weak singularity of restoring forcex μ at x = 0 is allowed and f may have a singularity at x = 0.

In this paper, we study the existence of periodic solutions for Rayleigh equation with a singularity of repulsive typewhere α ≥ 1 is a constant, and ϕ and p are T-periodic functions. The proof of the main result relies on a known continuation theorem of coincidence degree theory. The interesting point is that the sign of the function ϕ(t) is allowed to change for t ∈ [0, T].