Existence and multiplicity of periodic solutions of semilinear resonant Duffing equations with singularities

In this paper, we deal with the existence of positive periodic solutions of singular resonant Duffing equations x′′ + (1)/4n2x+g(x) = p(t), where g has a singularity at x = 0 and n is a positive integer. We give an explicit condition to ensure the existence of positive 2π-periodic solutions when the limit limx→+∞g(x) = g(+∞) exists and is finite. On the basis of this conclusion, we give an answer to the problem raised by Del Pino and Manásevich. We also study the multiplicity of positive periodic solutions of singular Duffing equations x′′+g(x) = p(t). When g satisfies the semilinear condition at infinity and the time map satisfies an oscillation condition, we prove that the given equation possesses infinitely many positive 2π-periodic solutions by using the Poincaré–Birkhoff theorem