On the existence of periodic solutions of Rayleigh equations

In this paper, we study the existence of periodic solutions of Rayleigh equation 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 and f(x) satisfies sgn(x)(f(x) − p(t)) ≥ c, for t ∈R, |x| ≥ d with c, d being positive constants.Received: July 1, 2002; revised: February 19, 2003Research supported by the National Natural Science Foundation of China, No.10001025 and No.10471099, Natural Science Foundation of Beijing, No. 1022003 and by a postdoctoral Grant of University of Torino, Italy.