On the existence of periodic solutions of Rayleigh equations

Abstract.

In this paper, we study the existence of periodic solutions of Rayleigh equation

$$ x'' + f(x') + g(x) = p(t) $$

where f, g are continuous functions and p is a continuous and 2π-periodic function. We prove that the given equation has at least one 2π-periodic solution provided that f(x) is sublinear and the time map of equation x′′ + g(x) = 0 satisfies some nonresonant conditions. We also prove that this equation has at least one 2π-periodic solution provided that g(x) satisfies $\lim_{|x|\to+\infty}sgn(x)g(x) = +\infty$ and f(x) satisfies sgn(x)(f(x) − p(t)) ≥ c, for t ∈R, |x| ≥ d with c, d being positive constants.