Periodic solutions of Rayleigh equations via time-maps

In this paper, we study the existence of periodic solutions of Rayleigh equations x″+f(t,x′)+g(x)=e(t),x″+f(t,x′)+g(x)=e(t), where f:R2→R is continuous and TT-periodic with respect to the first variable, g,e:R→R are continuous and ee is TT-periodic. We prove that the given equation possesses at least one TT-periodic solution provided that either lim supc→+∞τ(c)+lim infc→−∞τ(c)>T or lim infc→+∞τ(c)+lim supc→−∞τ(c)>T is satisfied, where ττ is the time-map defined in Section 1.