The stability of the equilibrium of the damped oscillator with damping changing sign

In this paper, we prove a sufficient and necessary condition for the stability of the equilibrium x=x′=0x=x′=0 of the damped oscillator with damping changing sign x″+h(t)x′+a(t)x2n+1+e(t,x)=0,n≥1 where a(t),h(t) are continuous and 1-periodic with h¯=∫01h(t)dt=0, e(t,x)e(t,x) is continuous, 1-periodic in tt and dominated by the power x2n+2x2n+2 in a neighborhood of x=0x=0.