Periodic solutions of Duffing equations with semi-quadratic potential and singularity

In this paper, we deal with the existence of periodic solutions of the second order differential equations x +g(x) = p(t) with singularity. We prove that the given equation has at least one periodic solution when g(x) has singularity at origin, G(x)(= x 1 g(s) ds) satisfies n2 4 < lim inf x→+∞ 2G(x) x2 lim sup x→+∞ 2G(x) x2 < (n+ 1)2 4 for large positive x, where n 0 is an integer and G(x) tends to positive infinity as x→0+. © 2005 Elsevier Inc. All rights reserved.